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1.1359790.pdf | High frequency spin dynamics in magnetic heterostructures (invited)
R. L. Stamps
Citation: Journal of Applied Physics 89, 7101 (2001); doi: 10.1063/1.1359790
View online: http://dx.doi.org/10.1063/1.1359790
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155.33.16.124 On: Sat, 29 Nov 2014 02:18:41High frequency spin dynamics in magnetic heterostructures invited
R. L. Stampsa)
Department of Physics, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia
Fast reversal processes in magnetic particles and arrays involve fundamental magnetic dynamic and
relaxation processes. Exchange and dipolar interactions determine equilibrium ground states andstronglyinfluencelinearandnonlineardynamics.Calculationsareusedtoshowhowhighfrequencyresonancesinarraysofdenselypackedmagneticparticlescanaffectreversaltimes,possiblyleadingto dramatic decreases in switching rates. High frequency excitations and dynamic processes ininterface exchange coupled magnets are also discussed, with emphasis on exchange biasedmaterials. The exchange bias effect is closely related to interface magnetic structure andmagnetization processes in systems of ferromagnets exchange coupled to antiferromagnets. It isshown how magnetization processes in the antiferromagnet can be studied through observation ofdynamic effects in the ferromagnetic component. © 2001 American Institute of Physics.
@DOI: 10.1063/1.1359790 #
I. INTRODUCTION
Time scales for interesting dynamic behavior in mag-
netic materials cover several decades. Resonance phenomenaassociated with linear response in ferromagnets are typicallystudied at GHz frequencies, and some ferrimagnets and an-tiferromagnets show resonance at frequencies into the infra-red. Processes associated with coherent reversal are highlynonlinear types of dynamics with characteristic times on theorder of nanoseconds. Domain walls in ferromagnets can dis-play dynamic oscillations at MHz frequencies. Thermallydriven magnetization processes, involving coherent reversalor domain nucleation and growth, have features that are stud-ied over time intervals from picoseconds to minutes.
Three examples of phenomena will be described here
through calculations of linear and nonlinear spin dynamicsfor small particles and exchange coupled films. For experi-mental studies, linear response studies are useful for obtain-ing values of local effective exchange and anisotropy fields.These work by probing the restoring forces, or torques, act-ing on spins slightly disturbed from equilibrium. It is pos-sible to associate particular features with surface and inter-face effects, making these techniques especially importantfor studies of buried interfaces and strongly coupled systems.Nonlinear response covers a range of phenomena includingdynamic soliton formation and reversal processes. Examplesare presented here for reversal dynamics of single domainparticles and exchange coupled systems. These dynamics de-scribe switching of spins from one equilibrium configurationto another, and relevant time scales are determined by relax-ation mechanisms.
The three examples described in this article are ~1!
switching dynamics of single domain magnetic dots; ~2!long
wavelength spin wave and domain wall resonance probes ofexchange coupling at interfaces; and ~3!spin dynamics as
probes of exchange bias mechanisms in ferromagnet/antiferromagnet layered films.II. MAGNETIC DOTS
A number of techniques have been developed for growth
or construction of magnetic particles with nanometer dimen-sions. The size and uniformity of these structures can becontrolled to a remarkable degree, and geometries that sup-port stable single magnetic domain configurations have beenreported.
1
A general description of spin dynamics for single do-
main particles is often made using a version of the Landau–Lifshitz torque equations. This model is a semiclassical treat-ment of the magnetization produced by local moments withdamping processes represented by one of a number of pos-sible terms. A useful representation is the Gilbert form of theequations of motion:
d
dtm5gm3heff2am3d
dtm. ~1!
A local moment is represented by m,gis the gyromag-
netic ratio, and ais a parameter representing dissipation of
energy out of the spin system. The dissipation is assumed tobe small. The effective field acting on mish
effwhich con-
tains contributions from applied magnetic fields, local mag-netocrystalline anisotropies, shape anisotropies, and interac-tions with other magnetic moments. The form assumed herefor the effective field containing these contributions is
h
eff5zˆS2H~t!12K
MmzD1xˆb~t!1hd. ~2!
This effective field contains time dependent applied
fields along the xandzdirections, an anisotropy Kfor a
uniaxis along the zdirection, and a field hdrepresenting in-
teractions with other particles.
Approximate solutions can be found for the case where
there are no interactions. b!Hand both applied fields are
applied at t50 and kept constant thereafter. The initial ori-
entation of the moment is in the 1zdirection. The applied
fields create torques that cause the moment to precess, anda!Electronic mail: stamps@pd.uwa.edu.auJOURNAL OF APPLIED PHYSICS VOLUME 89, NUMBER 11 1 JUNE 2001
7101 0021-8979/2001/89(11)/7101/6/$18.00 © 2001 American Institute of Physics
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155.33.16.124 On: Sat, 29 Nov 2014 02:18:41dissipation allows the moment to eventually reverse into a
low energy configuration aligned along the new effectivefield direction.
There are two time scales involved for small dissipation:
the precession time and the reversal time. This motivates theapproximation which consists of separating the long timefrom short time behavior in Eq. ~1!, and averaging over the
high frequency components. The result is an equation for thereversal time that can be solved exactly in terms of the com-ponent of the moment along the ydirection.
2Defining the
angle ubetweenmand theyaxis, the result for K50i s
cosu52tanh~agH!. ~3!
The characteristic time associated with the reversal in
this limit is tswitch51/(agH). In the case where bis not
neglible, tswitch5(12b/H)/(agH). The role of bis to ac-
celerate the switching of the moment by providing a torquealong a direction perpendicular to a plane containing Hand
m. The minimum switching time occurs for pure precession,
and the ‘‘bias field,’’ b, exerts a torque on min such a way
thatmis able to precess through much of the reversal by
torques from Hand the anisotropy K.
Precession during reversal is shown in Fig. 1 for switch-
ing of a single magnetic moment. The results are calculatedby numerical integration of Eq. ~1!using the effective field
of Eq. ~2!. Here the uniaxial anisotropy is K55
pM2/2. The
high frequency precession is centered around a reversalcurve described by a function of the form given in Eq. ~3!.
Herebis large enough to create a precession dominated re-
versal with precession effects limited to the final half of theswitching.
Interesting effects occur when interactions between mag-
netic dots are included through the h
dterm. An array of
densely packed dots can have local dipolar fields with mag-nitudes on the order of several hundred oersteds for dots with100 nm diameters spaced 10 nm apart.
3The dipole field act-
ing on a particular magnetic moment can be calculated bysumming over the instantaneous dipole fields produced by allother dots in an array. The time evolution of the entire arraycan be followed by numerically integrating the set of coupledequations of motion.
4
A variety of interesting dynamics occur, including routes
to chaos, in response to large amplitude rf driving fields. Aresult of particular relevance to switching dynamics was
found when the dot array density was varied. The array den-sity controls the magnitude of h
dand strongly affects the
reversal dynamics of an array of dots. An example is shownin Fig. 2 where the reversal time of an array of magnetic dotsis shown as a function of dipolar coupling h
d.4
The reversal time is shortest for small and large values
of the interdot coupling, but has a maximum in a range of hd
strengths. The reason for this maximum is that the array
supports excitations in the dipolar field analogous to magne-tostatic spin waves in thin films. The energies of these exci-tations are controlled by the magnitude of h
d, and the
switching time peaks when reversal involves excitations ofthese modes. This interaction is optimal for a restricted rangeof interaction strengths.
III. EXCHANGE COUPLING I: SPIN WAVES AND
DOMAIN WALLS
The above example illustrates how weak interactions be-
tween spins affect linear response and impact nonlinear dy-namics. A different class of structures involve strong ex-change interactions across interfaces between dissimilarmaterials. The nature of the coupling is not well-understoodin all cases, and it is possible to identify unique features inlinear response frequencies associated with the coupling.Some features are strongly dependent on particular func-tional forms used to represent the coupling and can be usedto distinguish between different models.
5In this section ex-
amples for two types of linear excitations, spin waves anddomain wall resonances, are discussed as probes of interlayerexchange coupling between two exchange coupled ferromag-nets. The case of interlayer exchange between a ferromagnetand antiferromagnet is discussed in the next section.
Small fluctuations of spins about their ground state equi-
librium results in restoring torques. The magnitude of thetorques determines frequencies of experimentally observableresonances, and torques which involve exchange energy pro-vide direct measures of the strength of the effective exchangefield.
Magnetic trilayers are useful for identifying exchange
FIG. 1. Reversal of a single domain particle accelerated by a perpendicu-
larly oriented bias field b. The components of mare plotted as a function of
time. Note the high frequency precession terms modulating the reversal.
FIG. 2. Switching time as a function of interparticle interaction hdfor an
array of single domain particles. Resonance with a band of magnetostaticexcitations in the array lead to an increase in the time for reversal driven by
application of a constant applied field at time t50.7102 J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 R. L. Stamps
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155.33.16.124 On: Sat, 29 Nov 2014 02:18:41contributions to the magnetic resonance frequencies because
this structure facilitates direct observation of frequency shiftsdue to exchange coupling.
6The idea is that each ferromag-
netic film has an associated resonance mode. When the twofilms are exchange coupled, the resonance modes are notdegenerate in frequency but instead appear as an acousticand optic mode with frequencies split by the exchange.
7The
acoustic mode corresponds to in phase precession of thespins in each film and does not involve interlayer exchange.The optic mode corresponds to out of phase precession, andthe difference between the optic and acoustic mode frequen-cies provides a measure of the interlayer exchange.
In metallic trilayers with ferromagnetic films separated
by nonmagnetic transition metal spacers, interlayer magneticcoupling is mediated by conduction electrons through anRuderman–Kittel–Kasuya–Yosida type interaction. Theform of this coupling can be described by an energy of theform
8
Eex5J1cos~Du!1J2cos2~Du!. ~4!
The relative orientation of the magnetization of the two
films is specified by the angle Du. TheJ1term represents a
simple exchange term between two moments and the J2term
describes contributions from competing ferromagnetic andantiferromagnetic coupling. The model energy of Eq. ~4!has
been proposed for trilayers wherein the sign of the exchangecoupling alternates in sign along the interface. Another en-ergy has been proposed for trilayer structures in which thespacer is a two sublattice antiferromagnet, and the spacerthickness varies along the interface. This energy has theform
9,10
Es5C1Du21C2~Du2p!2. ~5!
Spin wave and resonance frequencies corresponding to
these mechanisms for trilayers can be calculated once theequilibrium configuration of the magnetic components is de-termined. The equilibrium configuration is found by search-ing for minima of the total energy of the structure includingcontributions from the external applied field, anisotropies,demagnetizing fields, and the exchange energy. The eigen-frequencies of small oscillations about the equilibrium canthen be calculated.
An example is given in Fig. 3, where frequencies of theacoustic and optic modes are shown as functions of applied
magnetic field Hfor an exchange coupled trilayer. The
closed circles correspond to an exchange energy of the formgiven by Eq. ~5!and the open circles correspond to an ex-
change energy of the form in Eq. ~4!withJ
250. The most
distinctive difference between the two modes is the modesoftening of the acoustic mode due to the J
1term in Eq. ~4!.
This feature corresponds to alignment of the magnetizationsin the two ferromagnet films, and does not occur for theantiferromagnet spacer modeled by Eq. ~5!. This point is
discussed in detail in Ref. 5.
Other types of magnetic resonance associated with do-
main walls exist in unsaturated trilayers. An interesting con-sequence is that exchange coupling can serve as a self-pinning mechanism for domain walls in separate magneticfilms. An example is sketched in Fig. 4 for domain walls intwo antiferromagnetically coupled ferromagnet films. Atequilibrium the walls center above one another in such a wayas to minimize the interlayer exchange energy.
Deviations from equilibrium involve restoring forces due
to the interlayer exchange. This leads to the possibility ofresonances in the motion of the domain walls about equilib-rium. The frequency of the resonances depends upon the signand magnitude of the interlayer coupling. Acoustic and optictype modes are possible, although acoustic oscillations re-quire some sort of additional pinning in order to have non-zero frequency, otherwise this mode corresponds to transla-tion of the walls. The frequency of the optic mode provides adirect measure of the interlayer exchange coupling:
11
v5A4pHexM. ~6!
The exchange field Hexis proportional to J1for an in-
terlayer coupling similar in form to the first term in Eq. ~4!.
An important feature is that this frequency corresponds tomotion of walls that depends on the overlap of domain walls.These resonances therefore represent local probes of ex-change coupling on length scales determined by the widthsof domain walls. For materials such as Fe or Co, the lengthscales are roughly 10 nm, and the resonance frequencies areon the order of 1 Ghz.
FIG. 3. Acoustic and optic modes for resonances in an exchange coupled
trilayer. The open circles are calculated using bilinear and biquadratic termsfor exchange as in Eq. ~4!. The closed circles are calculated using an energy
of the form Eq. ~5!suggested by Slonczewski ~Ref. 9 !.
FIG. 4. Illustration of domain wall pair pinning due to antiferromagnetic
exchange coupling between ferromagnetic films. In ~a!domain walls in the
two separate films orient such as to minimize exchange energy coupling thefilms. In ~b!the domain walls experience restoring forces due to the ex-
change coupling when displaced from equilibrium.7103 J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 R. L. Stamps
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155.33.16.124 On: Sat, 29 Nov 2014 02:18:41IV. EXCHANGE COUPLING II: MECHANISMS FOR
EXCHANGE BIAS
The previous examples illustrate features of linear and
nonlinear dynamics that can be used as probes of local ex-change fields, domain wall dynamics, and effects of interac-tions on reversal processes. These features are also rele-vant to studies of the interface between a ferromagnetand an antiferromagnet. Exchange coupled ferromagnet/antiferromagnet bilayers can exhibit the phenomena of ex-change bias, and have become of great interest recently forpotential application in spin electronic devices.
The mechanism of exchange bias is not completely un-
derstood despite several decades of study. Key questions areconcerned with the role of domain wall formation and pin-ning at the interface, interface anisotropies in the antiferro-magnet, and the importance of a magnetic and geometricalstructure near the interface.
It is possible to show that an energy of the form in Eq.
~4!can describe the interlayer exchange energy for a ferro-
magnet in contact with both sublattices of a two sublatticeantiferromagnet.
12The exchange constants have the interpre-
tation that J15Ja2JbandJ25(Ja1Jb)2/s, whereJaand
Jbare the average values of the exchange coupling to the a
andbsublattices, respectively. A complete energy describing
a uniformly magnetized ferromagnet exchange coupled toboth sublattices of the antiferromagnet is
E
bias52HMtfcos~u2r!1J1cos~u2a!
2J2cos2~u2a!1s~12cosa!. ~7!
The first term is the Zeeman energy of a ferromagnetic
film of thickness tfin an applied magnetic field Haligned
along the rdirection. The angle uspecifies the orientation of
the ferromagnet magnetization M. The angle ais the orien-
tation of a vector lassociated with the antiferromagnet order.
lis defined as the vector difference of the antiferromagnet
sublattice magnetizations, and ais the angle between land
the easy axis of the antiferromagnet anisotropy uniaxis. Thesecond term is the exchange coupling and the third term isthe energy of a partial wall pinned at the interface andformed in the antiferromagnet. The existence of the magneticstructure in the antiferromagnet is important for understand-ing the magnitude of the exchange bias, and has been pro-posed by several authors based on general energyconsiderations.
13–15
An example of how the exchange energies J1andJ2
control the equilibrium configuration of the magnetization in
an applied field His shown in Fig. 5. Here, hysteresis is
calculated for different ratios of r5Ja/Jb, corresponding to
different fractions of antiferromagnet sublattices present atthe ferromagnet surface. The hysteresis is determined byminimizing Eq. ~7!with respect to
uandafor given values
ofHandr. The applied field for this figure is oriented at r
5p/6 from the anisotropy uniaxis. For this orientation hys-
teresis appears because the domain wall formed near the in-terface in the antiferromagnet depins from the interface.
As in the discussion for trilayers, the frequencies of spin
waves and resonances depend upon the equilibrium configu-ration of the ferromagnet film’s magnetization. The fre-quency of the ferromagnet resonance mode can be found
from an energy Fbased on Eq. ~7!but generalized to allow
out of plane fluctuations of the ferromagnet. Under the as-sumption that the antiferromagnet contributes to the effectivefields acting on the ferromagnet ~the dynamics of the antifer-
romagnet spins are negligible !the ferromagnet resonance
can be calculated according to
16,17
v
g51
MA]2F
]f2]2F
]u22S]2F
]f]uD. ~8!
The resonance frequency then includes contributions
from exchange coupling to the antiferromagnet and equil-brium angles that are found by minimizing the energy. Theresonance frequencies for the examples in Fig. 5 are shownin Fig. 6.
6
The resonances vary continuously with applied field ex-
cept when the magnetic configuration changes due to depin-ning of a wall. Discontinuous jumps in the frequencies ap-pear at these points. Because the hysteresis loops havedifferent coercive fields in the forward and reverse magneti-zation directions, the effective fields governing the frequen-cies of resonance also differ for the forward and reverse fielddirections. The frequencies are likewise hysteretic.
FIG. 5. Magnetic hysteresis observed through the ferromagnet component
of an exchange coupled ferromagnet/antiferromagnet bilayer. Partial wallformation, pinning, and depinning in the antiferromagnet is responsible forthe hysteresis loop shift and coercive fields. The different curves correspondto different ratios of antiferromagnet sublattices present at the interface.
FIG. 6. Resonance mode frequencies as a function of field for the magne-tization loops shown in Fig. 5. Discontinuities appear when a wall depins inthe antiferromagnet and the ferromagnet magnetization changes orientation.7104 J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 R. L. Stamps
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155.33.16.124 On: Sat, 29 Nov 2014 02:18:41Because the internal fields are very sensitive to the rela-
tive magnitudes of JaandJb, the resonance frequencies are
also sensitive to the ratio Ja/Jb. This ratio is likely to vary
across an interface and therefore lead to a number of differ-ent corresponding resonance frequencies. This feature mayprovide an explanation for large linewidths observed in lightscattering and resonance measurements of exchange biasedbilayers.
18,19
Most importantly, the measurement of frequency shifts
of resonance and spin wave modes can provide detailed in-formation regarding internal fields acting in theantiferromagnet.
20Exchange coupling of the ferromagnet to
the antiferromagnet means that antiferromagnet spins aredriven off-resonance, but still experience torques due to ef-fective anisotropy fields near the interface. Measurements ofspin wave frequency can be used to identify these contribu-tions and thereby provide information on in plane and out ofplane anisotropies in the antiferromagnet. The magnitudes ofthese anisotropies are very important for understanding pin-ning and depinning events in mechanisms for exchange bias.Recent experimental work along these lines has been re-ported by Ercole et al.
21
The role of anisotropies in the pinning of walls at the
interface is particularly important for thermal stability of theexchange bias effect.
22,23There are dynamics associated with
magnetization reversals due to depinning events and theseoccur on time scales controlled by energy barriers whosemagnitudes are determined by anisotropies in theantiferromagnet.
15A model for the barrier energy can be
constructed by assuming that the depinning occurs via out ofplane rotation of the antiferromagnet spins involved in par-tial wall formation. An estimate of the barrier energy is
15
Eb5a@KoD11
2s~11cosa!#. ~9!
The barrier depends on Ko, the out of plane anisotropy
andD, the domain wall length in the antiferromagnet. The
probability that a reversal will occur in time Dtis
P5expF2NDt
texpS2Eb
kBTDG. ~10!
Dynamics of this process can be studied in ac suscepti-
bility experiments and applied field rate experiments.24,25An
example of hysteresis curves calculated for field rate experi-ments is shown in Fig. 7. An ensemble of exchange biasedgrains is considered using a Monte Carlo approach, and theaverage magnetization is plotted as a function of appliedfield as the field is cycled at different rates R. The system
begins in the same initial equilibrium configuration for eachloop. The coercivity fields in the reverse cycle occur at dif-ferent values depending on the field rate. The reason is thatthe antiferromagnet spins are only weakly influenced by theapplied field, and are mainly sensitive to the orientation ofthe ferromagnet. Thermal processes begin to occur as theferromagnet angle
uchanges, causing ato change as the
antiferromagnet moves to a new equilibrium. The barrieralso changes according to Eq. ~9!, leading to a new distribu-
tion of ferromagnet grain orientations. Furthermore, a slowfield rate means that the reversal occurs over long times,thereby allowing a large number of thermal events. Thus the
biggest change in the reverse cycle coercive field occurs forslow rates.
V. SUMMARY
Calculations of spin dynamics associated with magneti-
zation reversal and linear response have been discussed withan emphasis on how experimental measurements can be usedas probes of local effective fields. Examples have been pre-sented for switching dynamics of arrays of weakly coupledmagnetic dots, exchange coupled trilayers, and exchangebias bilayer structures. Resonance frequencies, measurableusing ferromagnetic resonance and Brillouin light scatteringtechniques, have been calculated with an emphasis on fea-tures that can serve as probes of exchange coupling andanisotropies. Dynamics associated with domain wall reso-nances and thermally driven reversal processes were shownto provide unique information regarding local exchange andanisotropy pinning fields.
ACKNOWLEDGMENT
This work was supported by the Australian Research
Council.
1K. Ounadjela and R. L. Stamps, in Handbook of Nanostructured Materials
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FIG. 7. Magnetic hysteresis loops for collection of exchange biased grains
calculated for different applied field rates R. The system of grains is initially
at equilibrium in the positive field direction, and one field cycle is plotted.The value of the coercive field on the reverse path depends on the field rate.A slower field rate means that there is more time for thermal fluctuations tooccur that can depin partial walls from the interface.7105 J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 R. L. Stamps
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155.33.16.124 On: Sat, 29 Nov 2014 02:18:4115R. L. Stamps, Phys. Rev. B 61, 12174 ~2000!.
16J. Smit and H. G. Beljers, Philips Res. Rep. 10,1 1 3 ~1955!.
17H. Suhl, Phys. Rev. 97,5 5 5 ~1955!.
18C. Mathieu, M. Bauer, B. Hillebrands, J. Fassbender, G. Gu ¨ntherodt, R.
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Bland, Phys. Rev. B 62, 6429 ~2000!.
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2994 ~1999!.
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1.372487.pdf | Micromagnetic simulation of thermal effect in longitudinal thin film disk media
Qingzhi Peng
Citation: Journal of Applied Physics 87, 5678 (2000); doi: 10.1063/1.372487
View online: http://dx.doi.org/10.1063/1.372487
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Published by the AIP Publishing
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Micromagnetic simulation is utilized to study thermal effects in longitudinal thin film media. A
Monte Carlo method is adopted to compute the thermally assisted switching process in a large timescale ~from seconds to years !. Throughout the study, the ratio of film thickness to grain size is fixed
to be 1.67 as grain size varies between 8 and 15 nm. The effective media bulk coercivity issimulatedbyassuminganappliedfielddurationof1s.Thethermaldecayofboththebulkremanentmagnetization and recorded transitions is studied. The thermal effects are compared for media withdifferent intergranular exchange couplings. Media with larger exchange coupling is less susceptibleto thermal effect. © 2000 American Institute of Physics. @S0021-8979 ~00!83408-8 #
I. INTRODUCTION
As medium grain size and film magnetic moment are
reduced in order to improve media signal-to-noise ratio~SNR!performance, thermal effects play more important
roles in the magnetization switching process.
1,2The thermal
effects have two aspects: the time dependence of media ap-parent effective coercivity due to the thermal assistance andthe decay of media magnetization.
3–5The first aspect is re-
lated to the fundamental recording issues such as writabilitythat will affect media recording performance. The secondone is associated with long term stability of recorded infor-mation. Therefore the fundamental understanding of thoseissues is essential to the further advancement of magneticrecording areal density. In this article, micromagnetic mod-eling is used to study the thermal effect in longitudinal thinfilm media.
II. MICROMAGNETIC MODELING
A rectangular array of grains is used to simulate a sec-
tion of longitudinal media. Each grain is assumed to besingle domain and to possess a two-dimensional ~2D!ran-
dom uniaxial caxis in the film plane. Throughout the study,
the ratio of grain height hto its size Dis fixed to be h/D
51.67 for each value of Dranging from 8 to 15 nm. With a
rectangular shape, the grain volume is given by hD
2
51.67D3. The grain size is assumed to be uniform. The
anisotropy field is Hk52K/Ms58500Oe, where Msis me-
dium saturation magnetization equal to 300 emu/cc. Themagnetization reversal dynamics is simulated by solving theLandau–Lifshitz–Gilbert equation. The gyromagnetic ratiois 1.76 310
7s21Oe21and the Gilbert damping constant ais
chosen to be 0.1.
A Monte Carlo method is adopted to compute the ther-
mally assisted magnetization reversals in a large time dura-tion from seconds to years.
4The probability of magnetization
switching due to thermal perturbation is described byArrhenius–Neel formalism:P
i5f0exp~2DEi/kBT!,
wheref0is the attempt frequency chosen to be 109s21and
kBis the Boltzmann constant. A constant temperature of T
5300K is utilized. DEiis the energy barrier that the mag-
a!Electronic mail: qpeng@us.ibm.com
FIG. 1. Simulated M–Hloops for media with different grain size. ~a!he
50.031, ~b!he50.062, ~c!he50.078.JOURNAL OF APPLIED PHYSICS VOLUME 87, NUMBER 9 1 MAY 2000
5678 0021-8979/2000/87(9)/5678/3/$17.00 © 2000 American Institute of Physics
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155.33.120.167 On: Fri, 05 Dec 2014 05:51:05netization has to overcome along a given reversal path. In
this work, it is assumed that the magnetization will attemptto overcome the energy barrier via all paths with equal prob-ability. The magnetization switching probability Pis ob-
tained by averaging all the individual probabilities P
i.
Medium bulk magnetic properties ~M–Hloops !and the
thermal decay of remanent magnetization are simulated byusing a 64 364 array. Medium effective bulk coercivity is
obtained from the simulated M–Hloops by assuming an
applied field duration o f1sa teach data point. A larger array
size of 128 3128 is utilized to simulate the square wave re-
cording and the thermal decay aftereffects of the recordedbits. In the recording simulation, all recording parameters arescaled relative to grain size. Media is initially dc magnetized.Then, a square wave pattern consisting of four transitions isat first dynamically written using a flying height FHT/D
;1.5 and track width W/D;67. The transitions are written
at two densities with the bit spacing B/D55 and 10. For
example, when D58 nm,B/D55 and 10 correspond to a
recording density of 508 and 254 kfci, respectively. For boththe remanent magnetization and recorded transitions, thethermal decay is computed over a time span of up to 5–6years. The thermal effect is compared for media with differ-ent exchange coupling constant h
e.
III. RESULTS AND DISCUSSION
The simulated M–Hloops with and without thermal
effects for media with different grain size exchange couplingare plotted in Fig. 1. Table I summarizes simulated mediabulk magnetic properties such as M
rt, the intrinsic and ef-
fective coercivities. Media effective coercivity typically islower than the intrinsic coercivity due to thermal assistance.ForD515, 12, 10, and 8 nm, the corresponding media sta-
bility factors KV/k
BTare 174, 89, 51, and 26, respectively.
Figure 1 ~a!shows the M–Hloops with an exchange cou-
plingheof 0.031. Compared with the intrinsic coercivity
Hc054200Oe, the effective media bulk coercivity Hcis re-
duced to 3728 Oe with D515nm. As Ddecreases to 8 nm,
Hcis reduced to 805 Oe. When the intergranular exchange
coupling increases, the Hcreduction due to thermal effect
becomes smaller as shown in Figs. 1 ~b!and 1 ~c!. Withhe
50.062 @Fig. 1 ~b!#, media intrinsic coercivity Hc0becomes
3900 Oe. In this case, compared with Hc0, the effective
coercivity Hcis reduced to 3581 and 1391 Oe as Ddecreases
from 15 to 8 nm. As heis increased further to 0.078 @Fig.
1~c!#, it becomes more evident that media effective coerciv-
ityHcis less affected by thermal effect.
In Fig. 2, the decay of media remanent magnetization is
FIG. 2. Thermal decay of media bulk remanent magnetization vs time.
FIG. 3. Thermal decay of recorded magnetization transition. ~a!he
50.031, ~b!he50.078.TABLE I. Summary of calculated media bulk magnetic properties.
he50.031 he50.062 he50.078
D515D512D510D58D515D512D510D58D515D512D510D58
Mrt0.57 0.46 0.37 0.26 0.61 0.49 0.41 0.33 0.63 0.50 0.42 0.33
Hc04200 4200 4200 4200 3900 3900 3900 3900 3700 3700 3700 3700
Hc 3728 3342 2589 805 3581 3137 2426 1391 3581 2943 2447 1779
Note: ~1!Mrtin memu/cm2,~2!Hc0andHcin Oe, ~3!Din nm.5679 J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 Qingzhi Peng
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
155.33.120.167 On: Fri, 05 Dec 2014 05:51:05plotted. The solid lines ~curves A–C !in Fig. 2 correspond to
a smaller exchange coupling of he50.031. In this case, after
6 years, the remanent magnetization value decreases by 3.1%and 12.1% for D512 and 10 nm, respectively. However,
whenDis reduced to 8 nm, severe thermal decay was ob-
served and media becomes thermally demagnetized after 0.5day. When the exchange coupling h
eis increased to 0.062
~dotted curves D and E in Fig. 2 !, the remanent magnetiza-
tion value decreases by 0.14% for D510nm and 25.6% for
D58 nm after 6 years. Media thermal stability improves
with increased exchange coupling. Since the thermal fluctua-tion is completely random for each grain, the exchange cou-pling effectively introduces additional thermal energy bar-rier.
Figure 3 shows the thermal decay of recorded magneti-
zation transitions. The magnetization transition decay is cal-culated by averaging the magnetization amplitude betweentransitions over the time. In Fig. 3 ~a!, the thermal decay is
plotted for transitions with bit spacing B/D510 and 5 and
media with exchange coupling h
e50.031. With a grain size
D512nm, after 5.5 years, the magnetization transition de-
cays by 1.6% and 3.6% at density B/D510 and 5, respec-
tively. When Dis reduced to 10 nm, the decay of media
magnetization transition increases to 31.1% and 83.7% fordensityB/D510 and 5, respectively. In Fig. 3 ~b!, a lower
recording density B/D510 is used to compared the thermal
decay for media with different exchange coupling. With h
e
50.031, as grain size Dis reduced to 8 nm, the magnetiza-
tion transition is thermally demagnetized after 12 min. Whenthe exchange coupling h
eis increased to 0.078, as shown in
Fig. 3 ~b!, media becomes more thermally stable. With he
50.078 and D58 nm, media magnetization transition de-
cays by 66.1% after 5.5 years. For media with D510nm,the decay is 0.57%, which is significantly smaller than
31.1% in the prior case with he50.031 shown in Fig. 3 ~a!.I n
addition, compared to the case of bulk remanent magnetiza-tion, the decay of magnetization transition is larger and in-creases with recording density due to the existence of de-magnetization field between transitions.
As discussed above, large exchange coupling and grain
size reduces media thermal effects. However, this improve-ment is achieved at the cost of media SNR performance,especially when increasing exchange coupling.
6The thermal
effect is a major challenge for the continuing pace of arealdensity growth ~.60% per year !using longitudinal thin film
media.
IV. CONCLUSION
The thermal effect in magnetization switching process in
longitudinal thin film media is studied by using numericalmicromagnetics. Media effective coercivity is calculated andcompared with intrinsic coercivity for various grain size andintergranular exchange coupling. The decay of media rema-nent magnetization and recorded magnetization transition iscalculated over a time span of up to 5–6 years. Media withincreased intergranular exchange coupling is less subject tothermal fluctuation.
1S. H. Charap, P.-L. Lu, and Y. He, IEEE Trans. Magn. MAG-33 ,9 7 8
~1997!.
2H. N. Bertram, H. Zhou, and R. Gustafson, IEEE Trans. Magn. MAG-34 ,
1845 ~1998!.
3M. P. Sharrock, J. Appl. Phys. 76, 6413 ~1994!.
4P.-L. Lu and S. H. Charap, J. Appl. Phys. 75, 5768 ~1994!.
5R. W. Chantrell and A. Lyberatos, J. Appl. Phys. 76, 6407 ~1994!.
6J. G. Zhu and H. N. Bertram, J. Appl. Phys. 63,3 2 4 8 ~1988!.5680 J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 Qingzhi Peng
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155.33.120.167 On: Fri, 05 Dec 2014 05:51:05 |
1.4807460.pdf | Electric detection of the thickness dependent damping in Co90Zr10 thin
films
Hang Chen, Xiaolong Fan, Wenxi Wang, Hengan Zhou, Y. S. Gui et al.
Citation: Appl. Phys. Lett. 102, 202410 (2013); doi: 10.1063/1.4807460
View online: http://dx.doi.org/10.1063/1.4807460
View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v102/i20
Published by the American Institute of Physics.
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Downloaded 24 Jun 2013 to 128.226.37.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsElectric detection of the thickness dependent damping in Co 90Zr10thin films
Hang Chen,1Xiaolong Fan,1,a)Wenxi Wang,1Hengan Zhou,1Y. S. Gui,2C.-M. Hu,2
and Desheng Xue1
1The Key Lab for Magnetism and Magnetic Materials of Ministry of Education, Lanzhou University,
Lanzhou 730000, People’s Republic of China
2Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
(Received 13 April 2013; accepted 5 May 2013; published online 24 May 2013)
In this letter, we propose a dc electrical detection method for investigating the spin dynamics of
ferromagnetic thin films. Based on anomalous Hall effect (AHE), the out-of-plane component of the
dynamic magnetization can directly rectify the rf current into a time-independent Hall voltage at theferromagnetic resonance. This method is applied for studying the damping mechanism in Co
90Zr10
films. The thickness dependent zero-frequency linewidth and the effective Gilbert damping arerelated to the surface roughness and microstructure evolution. Compared with standard cavityferromagnetic resonance, the AHE rectification is more suitable for studying the dynamic properties
of local magnetic moment.
VC2013 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4807460 ]
Increasing the switching speed of the magnetic cells in
magnetic random access memory (MRAM) devices has been
a challenge. The physics of magnetization reversal is
described by the Landau-Lifshitz-Girbert (LLG) equation, inwhich the damping parameter is a critical factor that deter-
mines the switching speed and relaxation process.
1,2So far,
the damping mechanism in ferromagnetic thin films is notwell understood. Usually, the damping can be experimentally
determined from the linewidth of ferromagnetic resonance
(FMR) via the relation DH¼DH
0þax/c. The first term is
caused by nonintrinsic magnetic damping induced by inhomo-
geneities, which is frequency independent. And the second
term comes from the linearization of LLG equation, which isrelated to the intrinsic magnetic dissipation. In order to sepa-
rate the intrinsic and extrinsic magnetic damping, one needs
to measure FMR at different frequencies. While it is difficultfor traditional resonant cavity FMR measurements due to the
fixed frequency,
3,4the separation can be achieved by broad-
band techniques such as stripline, vector network analyzer,pulse inductive microwave magnetometer techniques,
5as well
as recently developed electric detection of FMR.6,7
Electric detection of FMR based on the spin rectification
has demonstrated good reliability and validity to study mag-
netic damping.8,9The anisotropy magnetoresistance (AMR)
which couples spin and charge in ferromagnets would resultin a dc electric signal at the FMR.
6,7Due to its high sensitiv-
ity and measurement flexibility, the spin rectification effect
has been successfully applied in the studies of spin dynamicsincluding FMR and spin wave resonances.
8–11However, this
approach is limited to those materials with appropriate AMR
effect. In order to extend spin rectification to broad magneticmaterials, we propose here a rectification mechanism based
on Anomalous Hall Effect (AHE). It is found that a transverse
dc Hall voltage appears when a radio frequency current (rf)flowing along the longitudinal direction of a ferromagnetic
Hall device. Therefore, resonance peaks associated with spin
dynamics can be measured by a dc electrical way. Assistedby the surface and microstructure analysis, the thicknessdependent magnetic damping mechanism in Co
90Zr10films
has been investigated by the AHE spin rectification.
Co90Zr10films are soft magnetic material with excellent
high frequency responding and large permeability in GHzranges. The fabrication process of the films has been intro-
duced in Ref. 12. By using laser exposure and lift-off tech-
nique, Hall crosses with 100 lm in width and 4.5 mm in
length were made from Co
90Zr10films with the thickness
ranging from 5 to 100 nm. The Hall geometry and the coordi-
nate system we used are shown in the inset of Fig. 1(d). The
longitudinal resistivity qxxand Hall resistivity qxywere
measured by using lock-in amplifiers (SR830, Stanford) with
a modulation frequency at 1.31 kHz and a current of 100 lA.
A microwave generator (E8257D, Agilent) was used to inject
modulated rf current (2–18 GHz) into the Hall device, and
the rectified voltage was measured by the lock-in amplifier.Transmission Electron Microscope (TEM, F30, FEI) and
Atomic Force Microscopy (AFM, MFP-3D, Asylum
Research) were employed to investigate the microstructureand the surface topologies of the films.
Figure 1shows the static transport properties of the
Co
90Zr10films. The films present fairly weak AMR effect as
shown in Fig. 1(a). The AMR resistivity Dq, which is defined
as the difference between the longitudinal resistivities when
the magnetization Mis parallel and perpendicular to the cur-
rent, are in the order of nXcm as shown in Fig. 1(c).
Although Dqincreases slightly with the film thickness, the
largest amplitude of the AMR ratio is only 0.0032% which isthree orders smaller than that of usual magnets such as
1%–3% for FeNi films. On the other hand, the AHE is quite
evident in these films. Figure 1(b) shows the Hall resistivity
as a function of perpendicular applied magnetic field, from
which the saturated AHE resistivity q
H¼2.30lXcm can be
obtained. The saturation magnetization M0estimated from
the curve is 10.5 kOe, which coincides with the result from
vibrating magnetometer measurements (not shown here). As
shown in Fig. 1(d), the qHvalues of the films are in the
lXcm range which is three orders larger than Dq. This result
means the AHE is the dominating effect that couples spin
and the charge in the Co 90Zr10films.a)E-mail: fanxiaolong@lzu.edu.cn
0003-6951/2013/102(20)/202410/4/$30.00 VC2013 AIP Publishing LLC 102, 202410-1APPLIED PHYSICS LETTERS 102, 202410 (2013)
Downloaded 24 Jun 2013 to 128.226.37.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsFormer studies on AMR spin rectification have indicated
the fact that the rectified dc electric signal due to FMR isproportional to the AMR ratio of ferromagnetic films.
6,7
Therefore, the AMR spin rectification is not suitable for the
magnetic materials with weak AMR effect, such as theCo
90Zr10films here. Nevertheless, based on the AHE which
couples the spin and charge by spin-orbit coupling, we pro-
pose here a spin rectification enabled by the AHE. We beginour theory from the generalized Ohm’s law, E¼q
?
jþDqm(j/C1m)2qH(j/C2m),13where jis current density
vector and m¼M/M0is the unit vector of the magnetization.
Based on the coordinate system shown in Fig. 1(d), the
ycomponent of the electric field is
Ey¼DqmymxjxþqHmzjx: (1)
By considering the fact that Dq/C28qHin the Co 90Zr10films,
the first term which is the so called “planar Hall effect” can beignored. If we send a rf current ~j¼j
xe/C0ixtalong the longitudi-
nal direction into the cross, it will simultaneously induce a rf
magnetic field he/C0iðxt/C0UÞ.H e r e x¼2pfis the frequency, and
Uis the relative phase of the rf field with respect to ~j.D u et o
the torque of the rf field, the magnetization will precess around
its equilibrium direction, i.e., m¼m0þmte/C0iðxt/C0uÞ,w h e r e
mtis the amplitude of the dynamic magnetization unit vector
anduis the phase lag between ~jandm. Consequently, a
Hall voltage appears as VyðtÞ¼Ðw
0Eydt¼qHwjx½m0zcosxt
þmtzcosxtcosðxtþuÞ/C138,w h e r e w¼100lm is the distance
between Hall contact leads. After a time averaging of Vy(t), a
time-independent Hall voltage is generated
Vy¼1
TðT
0VyðtÞdt¼qHwjx
2mtzcosu; (2)
where T¼2p/xis the period of the rf current. It is clear that
the rectified dc Hall voltage is proportional to the amplitude ofthe out-of-plane dynamic magnetization. Therefore, if V
yis
measured as a function of frequency or magnetic field, peaks
associated with magnetization resonance should appear.Figure 2(a)shows a typical Vy(H) curve on which a reso-
nance peak can be observed. Because of the asymmetric line
shape, the resonance position and the linewidth cannot be
directly determined. Therefore, we have to solve the H
dependent expression of Vybased on Eq. (2). The further
analysis of the resonant line shape of Vy(H) depends on a
detailed calculation of mtzandu. Both parameters can be
obtained by solving the LLG equation. The detailed calcula-
tion can be found in Refs. 7and14, so we only provide the
final expression
Vy¼VDDHðH/C0H0Þ
ðH/C0H0Þ2þDH2þVLDH2
ðH/C0H0Þ2þDH2:(3)
Based on this equation, the resonant signal due to the AHE
spin rectification shows a linear combination of a dispersive
line shape (D) which is proportional to DHðH/C0H0Þ=
½ðH/C0H0Þ2þDH2/C138and a Lorentz one (L) which is propor-
tional to DH2=½ðH/C0H0Þ2þDH2/C138. The VDand VLare line
shape amplitudes, which depend on the values of qHand the
properties of the rf signal, such as the amplitude of ~jandh,
the direction of h, and the relative phase U. However, H0
andDHare the position and linewidth of the FMR, which
depend on the frequency and dynamic properties of the mag-net itself.
The resonance line shapes in Fig. 2(a)was fitted by using
Eq.(3)where V
D¼0.25lV,VL¼/C00.14lV,DH¼43.0 Oe,
andH0¼/C01.36 kOe. The shadow areas represent the contribu-
tion of D and L. The in-plane angular dependent VDandVL
could be used to determine the direction and phase of the
induced rf field, which has been discussed in Refs. 14and15.
Here we only concentrate on the resonance properties. The
Vy(H) curves measured at different frequency (4–16 GHz) have
b e e nfi t t e db yu s i n gE q . (3)in a similar way. The frequency de-
pendent H0andDHare shown in Figs. 2(b)and2(c). The reso-
nant field H0can be fitted by using the Kittel equation of
FIG. 1. (a) Open circles are experimental data of longitudinal resistivity qxx
of the film (100 nm) as a function of in-plane angle hwhich is the angle
between Mandxaxis. The solid line is a fit according to the AMR principle
q¼qk/C0Dqsin2h. (b) AHE resistivity qxyas a function of perpendicularly
applied field. (c) and (d) are thickness dependent jDqjandqH. The inset
shows the hall geometry and the coordination system.
FIG. 2. (a) Open circles are raw data of the AHE rectified Hall voltage as afunction of magnetic field applied along the xdirection, which is fitted by
using Eq. (3). The D and L denote the dispersive and Lorentz line shape. (b)
The relation between the frequency and the resonance position, which indi-
cates the FMR nature of the peaks in the V
y(H) curves. (c) The frequency de-
pendent linewidth. All the data come from a 40 nm sample.202410-2 Chen et al. Appl. Phys. Lett. 102, 202410 (2013)
Downloaded 24 Jun 2013 to 128.226.37.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsin-plane FMR x¼cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðH0þHKþM0ÞðH0þHKÞp
,w h e r e
Hk¼17.4 Oe is the in-plane effective uniaxial anisotropy
field. Besides, M0¼10.4 kOe and gyromagnetic ratio
c¼19.1 GHz/kOe also can be determined from the fitting. As
s h o w ni nF i g . 2(b), the symbols coincide with the fitted curve
quite well, which is a proof that the resonant peaks in Vy(H)
curves originate from FMR. Moreover, the FMR linewidth DH
presents a linear dependence on frequency, which has been fit-
ted by using DH¼DH0þax/cas the solid line in Fig. 3(c).
The interception of the line gives the value of DH0¼6.64 Oe
which is the extrinsic contribution to FMR linewidth;16–18
furthermore, by using the cvalue fitted from the resonance dis-
persion, the Gilbert damping parameter a¼0.0103 can be
obtained from the slope of the solid line plot in Fig. 2(c).
It is clear therefore that the AHE spin rectification is
capable for studying the spin dynamics and damping param-
eters by dc electric measurement, especially for those mag-
netic materials exhibiting weak AMR effect. It is alsonoteworthy that for an arbitrarily chosen magnetic material,
both AMR and AHE would contribute to the rectified Hall
voltage, as elucidated by Eq. (1). Besides, although the line
shape amplitudes V
LandVDwould depend on Dq,qH, and
the properties of the rf signal in a more complicated way,
Eq.(3)is a general expression applicable for fitting the FMR
electrically detected via both AMR and AHE.
The frequency dependent linewidth of all five samples
are shown in Fig. 3(a). The zero-frequency linewidth DH0
and the effective Gilbert damping awere obtained by the lin-
ear fits. The DH0is found to decrease with the thickness, as
shown in Fig. 3(b), which is consistent with previous
reports.19TheDH0is empirically related to the “magnetic
roughness” which is caused by the surface quality in the ultra
thin films.20Figure 4(a) shows the values of the root meansquare roughness rof the films characterized by AFM.
Generally, larger ralways corresponds to rougher surface,
as shown in the insets of Fig. 4(a). The 5 nm film presents a
coarser surface feature in which many islands with those
heights less than 3 nm can be observed. It is clear that both r
andDH0decrease with the film thickness in a similar trend,
which indicates that the zero-frequency linewidth DH0
observed here depends on the surface topography.
For the FMR of in-plane magnetized films, the extrinsic
linewidth can be understood by the two magnon scattering
(TMS) mechanism.21In such a picture, the extrinsic line-
width has a strong dependence on applied magnetic field(or frequency as well), which is related to the perturbations
of Zeeman term, dipolar energy, and surface anisotropy
caused by surface roughness. Strictly speaking, the extrinsiclinewidth based on the TMS vanishes at zero field or zero
frequency, so the “zero-frequency linewidth” usually used is
quite inappropriate in theory. However, this empirical termis successful to semi-quantitatively describe the surface qual-
ity. This is due to the fact that the extrinsic contribution to-
gether with the Gilbert damping is usually treated by a linearfrequency dependent fit on the total linewidth in the relative
high frequency region (usually >5 GHz). Therefore, the ex-
trinsic contribution to the linewidth would be roughly treatedas a linear frequency dependent case, wherein the intercep-
tion gives birth to the “zero-frequency linewidth” and the
slope would result in an additional effective Gilbert dampingterm. Based on this picture, it is reasonable to predict that
the effective aof the Co
90Zr10film should decrease with the
film thickness as the surface getting smooth. However, theFIG. 3. (a) Frequency dependent FMR linewidth DHfor the films with
different thickness. Symbols are experimental data, which follow linear
relations indicated by solid lines. (b) and (c) show the thickness dependent
zero-frequency linewidth DH0and Gilbert damping a. (d) Comparison on
the values of thickness dependent DHbetween cavity FMR (8.984 GHz) and
AHE rectification (9.0 GHz). The inset in (d) shows the cavity FMR spec-
trum of 5 nm sample.
FIG. 4. (a) Surface roughness r(solid squares) decreases with the thickness
of the Co 90Zr10films. Two insets are 3D surface morphologies for 5 nm and
100 nm samples; the scan area for each is 4 lm2. (b)-(e) TEM images of the
Co90Zr10films. The 5 nm film presents an amorphous state. When the thick-
ness increases to 20 nm, a few Co nano-grains can be observed as the inset
in (c) shows. As the film becomes thicker, the Co grains become larger and
better crystallized.202410-3 Chen et al. Appl. Phys. Lett. 102, 202410 (2013)
Downloaded 24 Jun 2013 to 128.226.37.5. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsdata in Fig. 3(c) show an inverse trend, which means there
are other mechanisms affecting awhile the film is getting
thick rather than the TMS.
Figures 4(b)–4(e) show the microstructure of Co 90Zr10
films with different thickness. It is evident that the micro-
structure changes from a complete amorphous state (5 nm) to
a heterogeneous state (60 nm) wherein a few Co nano-grainsembed into an amorphous matrix when the film thickness is
increasing. It is known that the Gilbert damping in ferromag-
netic materials generally originate from spin-orbit interaction
in combination with impurity scattering that transfers
magnetic energy to itinerant quasiparticles.
22Therefore,
magnetic disorder characterized by the distribution of Co
nano-grains will play an important role in determining the
value of intrinsic Gilbert damping constant in the Co 90Zr10
film. As shown in Fig. 3(c), the avalue increases from
0.0040 (5 nm) to 0.0103 (40 nm), very likely resulting from
the increasing magnetic and crystalline disorders, becausethe Co nano-grains start crystallizing in the uniform amor-
phous matrix while the films become thick.
Finally we compare the values of DHobtained by AHE
rectification with that by cavity FMR which is the conven-
tional way for measuring the damping parameter. The thick-
ness dependent DHof the films are measured by using a
cavity FMR system (ESR JEOL, JES-FA300, 8.984 GHz)
and are plotted in Fig. 3(d). Comparatively, the results
obtained by the AHE rectification at 9.0 GHz are also plot-ted. Two methods reveal a similar thickness dependent trend,
which indicates the validity of our method in studying the
magnetic damping. On the other hand, the DHvalues of cav-
ity FMR are 15–20 Oe larger than that of AHE rectification.
This discrepancy may result from the fact that the absorption
signal in the cavity FMR measurements comes from theentire thin film samples with the area of a few millimeter
square; however, the AHE rectified electric signal comes
from a much smaller area (0.01 mm
2in this work) defined by
the Hall bar structure. The larger the sample area that con-
tributes to the signal, the more the inhomogeneities would be
involved in extrinsic contribution to the linewidth.Moreover, because the absorption signal of the cavity FMR
is directly proportional to the sample volumes, the FMR am-
plitude of the 5 nm sample (with 100 lm in width and 3 mm
in length) almost met the sensitivity limit of the equipment,
as shown in the inset of Fig. 3(d). In contrast, based on
Eq.(2), the AHE rectified voltage is independent of the sam-
ple volume. The only geometry related parameter in that
equation is w, which originates from the integration of the
Hall electrical field along the width of the stripe. With thefixed current amplitude, the reduction in width would
enhance current density, which results in a geometry-
independent voltage. Therefore, the AHE rectification ismore suitable for studying the dynamic properties of local
magnet moment.
In conclusion, the AHE spin rectification effect has been
used for studying the thickness dependent damping inCo
90Zr10films. Based on the AHE, the out-of-plane compo-
nent of the dynamic magnetization can directly rectify the rf
current into a dc Hall voltage at the FMR. By using lineshape fitting, the frequency and film thickness dependences
of the FMR linewidth of the Co
90Zr10film have been
obtained. Together with the surface and microstructure anal-
ysis, the zero-frequency linewidth has been explained by the
TMS due to surface roughness. The effective Gilbert damp-ing parameter, which is found to increase with the thickness,
has been attributed to the microstructure evolution while the
film is getting thick.
This work was supported by the National Basic
Research Program of China (Grant No. 2012CB933101),NSFC (Grant Nos. 11034004, 50925103, 61102001, and
11128408), and FRFCU (No. lzujbky-2013-ct01).
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1.355609.pdf | Magnetic viscosity in highdensity recording
PuLing Lu and Stanley H. Charap
Citation: Journal of Applied Physics 75, 5768 (1994); doi: 10.1063/1.355609
View online: http://dx.doi.org/10.1063/1.355609
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/75/10?ver=pdfcov
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141.210.2.78 On: Wed, 26 Nov 2014 03:19:48Magnetic viscosity in high-density recording
Pu-Ling Lu and Stanley H. Charap
Data Storage Systems Center, Department of Electrical and Computer Engineering, Carnegie Mellon
Giversi&, Pittsburgh, Pennsylvania I5213
For future ultrahigh-density magnetic recording, the magnetic viscosity in thin-film media will
become an issue due to the drastic reduction in grain size. An algorithm combining a Monte Carlo
method and molecular dynamics was employed to study the thermal effects in thin-film media. The
component of the field perpendicular to the plane defined by the axes of shape anisotropy and
uniaxial crystalline anisotropy makes it necessary to use the three-dimensional energy surface to
find the minimum energy barrier. This barrier is used to sample the reversal rate and the elapsed
time. Hysteresis loops for various K,V/kT ratios and sweep times are simulated. Isolated and di-bit
transitions are written, taking into account thermally assisted switching. After the head field is
turned off, the subsequent thrrmal decay is computed for time spans as long as 6 months. Significant
aftereffect is found for grain volumes about twice that for ordinary superparamagnetism.
1. INTRODUCTION
With the assistance of thermal energy, the magnetization
of a particle can surmount an energy barrier and switch from
one stable direction to another. This process will take a cer-
tain time compared with the quick approach of the particle
magnetization to a local minimum when subjected a large
external field. This phenomenon is an inherent behavior of
ferromagnets and is well known as the magnetic aftereffect
or viscosity.’ The ratio of the energy barrier to the thermal
energy kT (k is the Boltzmann’s constant, T is the absolute
temperature) determines the magnitude of the aftereffect. A
comprehensive treatment of thermal fluctuations was given
by Brown.” In magnetic recording, the media must be ad-
equately resistant to thermal fluctuations. To maintain a cer-
tain amount of written signal so as to have adequate signal-
to-noise ratio (SNR) after thermal decay, generally requires a
K,V/kT much higher than the commonly known superpara-
magnetic limit of about 25.” Studies of the magnetic afteref-
fect have been widely reported for particulate recording
media.“-” In their work on granular Fe-(Si02), Kanai and
Charap6 first implemented an algorithm combining a molecu-
lar dynamics method and a Monte Carlo method to study the
aftereffect and transition broadening in the media, This algo-
rithm was introduced to treat an ensemble of spherically
shaped uniaxial particles with a distribution of easy-axis di-
rections and with particle magnetizations free to orient in
three dimensions. Good agreement with vibrating sample
magnetometer (VSM) experimental results was found. While
no significant thermal effect is expected for current thin-film
media, which typically have k’,V/kT values of more than
1000, it will be an important issue for future ultrahigh-
density magnetic recording media due to the dramatic reduc-
tion in grain size in order to maintain reasonable. jitter per-
formance. In this investigation, an algorithm was introduced
to study the magnetic viscosity in thin-film media under con-
ditions of ultrahigh-density recording.
II. MODEL
A computer simulation model has been developed on
DEC 3100 and 5000 workstations, based on a combined
molecular-dynamics model and the Monte Carlo simulation
of aftereffect. The molecular-dynamics part of the mode1 is similar to
the micromagnetic model of Zhu and Bertramc7 In the model,
the film is considered to consist of a planar array of hexago-
nally shaped grains. The grains are hexagonal close packed
and every grain is assumed to be a single-domain particle
with a nonmagnetic boundary; within each grain only coher-
ent rotation is assumed. Crystalline uniaxial anisotropy, mag-
netostatic interactions, and the se.lf-demagnetizing field of
each grain are inc1ude.d in the calculation. intergranular ex-
change interaction across the boundaries is not included in
this study. All grams have the same anisotropy energy con-
stant K, and saturation magnetization M, . We have chosen
arrays of grains with random distribution of easy-axis orien-
tations, confined to the plane.
The Landau-Lifshitz equation with Gilbert damping is
employed to describe the time development of the magneti-
zation of each grain,
dk - =- d7 1 iar (6fxHj - I Ta2 [+x(&xHj]. (1)
Here 6I is the unit vector in the direction of the magnetiza-
tion of the grain, y is the gyromagnetic ratio, r is the time
normalized to the period ( yHk) -l, and o is the damping
constant, which is chosen to be 1 for numerical convenience
in all calculations. Previous studies7 have found that the
macroproperties such as coercivity and remanence are insen-
sitive to the damping constant. Although the precession time
does depe.nd on its value, this time is usually of the order of
nanoseconds which is very short compared with the elapsed
time in the Monte Carlo simulations. H is the effective field
acting on the grain, normalized to H, . H, is the crystalline
anisotropy field, H,=2K,IM,, and M, is the saturation
magnetization. It is through H that the equations for the in-
dividual grains are coupled. The integration of the Landau-
Lifshitz equations is conducted by a fourth-order Runge-
Kutta method.
In this Monte Carlo method,s the energy barriers AE for
all grains are calculated first. The probability per unit time
for reversal of each grain can be obtained by utilizing Niel’s
formalism: ri= l/r=fa exp( - AEi/kT). Here r is the time
constant, AEi is the energy barrier for the ith grain, and fn is
5768 J. Appl. Phys. 75 (lo), 15 May 1994 0021-8979/94/75(10)/5768/3/$+.X00 6 1994 American institute of Physics
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
141.210.2.78 On: Wed, 26 Nov 2014 03:19:48a frequency constant chosen to be lo” s- ’ as an
approximation.’ The reversal rate R of the assembly can be
found by summing all the reversal probabilities,
N
R=x ri.
jzI
According to the probability ri/R, one of the magnetizations
is selected to reverse. The time LYE needed for this reversal to
happen can be sampled from the exponential probability den-
sity function: R espi - tR), ix., At= -lni&l?, where .$ is a
random number uniformly distributed from 0 to 1.
This algorithm uses, as the time increment, the average
tinltl between successful reversals instead of the constant
amount of simulation time increment in the conventional
Monte C’arlo method. For the latter method, the number of
time increments yielding a successful reversal may be but a
small fraction of the increments tested. For this method, the
time increment will tend to increase as the total activity de-
creases, reficcting the increased time between successful
magnetization reversals. This method eliminates the unsuc-
cessful switching attempts a priori and makes the simulation
practical for any time length of interest although it starts
slower than the conventional hJonte Garlo method.
Generslly~ a grain has two forms of anisotropy. One is
the crystalline uniaxial anisotropy and the other is the shape
anisotropy which is also uniaxial if an ellipsoid approxima-
tion of the grain shape is ass.umed. The latter one can vary
from zero to comparable~ or even larger than the former, and
can be in the film plane or out of the plane, both depending
on the ratio of grain height to its in-plane dimension and M, .
The con1bine.d effects of these two anisotropies defines a
plane; fcor magnetizations confined to that plane the aniso-
tropy is effectively uniaxial; but, thin-film media usually
have a distribution of the crystalline easy-axis orientations.
The magnetizations, and so the interaction fields, are free to
orient in the tilm plane or space. The effective field on each
grain, including the applied field and the interaction field. is
usually not in the plane defined by the crystalline easy axis
and the easy axis from shape anisotropy. The grains are no
longer uniaxial. In general, a three-dimensional energy sur-
ease must be used in order to find the minimum energy bar-
rier which is the difference between the energy at a saddle
point 011 the surface and the energy at the local minimum. A
two-dimensional secant method is employed to search for
this barrier.
The molecular dynamics and Monte Carlo method are
used alternately. The former is employed first to find the
local equilibrium configuration and then, by the Monte Carlo
method, one grain magnetization is chosen to reverse and the
elapsed time for this step is sanlpled. The total e1apse.d time
equals the accumulated 4.t for all steps. This process is re-
peated till the desired simulation time is reached.
Ill. RESULTS AND DlSCUSSlONS
The system WC used for calculation was a 60X60X1
array of hcp gmins. Easy axes were randomly distributed in
the film plane. The boundary conditions in the track direction
for d&bit transitions and across a track were set to be peri- “F’co r- - -.~ I l-1 -. &“-“-----.- :\ \\ 0.8 - *:s ---‘~.-----& --____._____ d
a ‘\,). o.fJ - * 0.6 a a”“,%T-axo
I 1 -.._( * h’,,“/tT =4? -. 0.2 -i_ ‘---.__ --. r- --_ d 0. I 18lX)
rime-xalelogit),in senmls Sweep times
FIG. 1. The simulated coercivity H, isolid). normalized by the c<ltxcivify
without thermal effect H rU, against logarithmic time scale for K,?r/kT=42,
K,V/kT=XJ, K,V/kT=1000. The dashed curves are calculated from Eq.
(2). The dotted curves are a linttar depcndcnce on log(,t).
odic, while an antiperiodic boundary condition was em-
ployed in track direction for isolated transitions. The head
field used to write transitions was a Karlquist field produced
by a head with a gap length g= 120 nm and a head media
separation d=38.5 nm. The thermal effect during the writing
process was included. The 3D energy surface was usc.d to
find the minimum energy barriers. The subsequent thermal
decay was observed for as long as 6 months. The tempera-
ture was always 300 K.
Figure 1. shows coercivities normalized to the value HCo
in the limit of zero sweep time for K,V!kT= 1000, 53, 42,
corresponding to grains with both diameters D and film
thicknesses Sof 23, 10, and 8 nm and sweep times of 10 7 s,
0.1 s, and 30 min, corresponding to recording, MH loop, and
VSM measurements. Here L) is the diameter of the circle
inscribed within the hexagonal grain. For all cases, k’, was
4X 10” erg/cm?‘, M, cSIN$ =O.l (Lzi,lli, =O. 1 j, and there
was no exchange interaction between grains. The step size of
the applied field was chosen as that in a VSM measurement:
a small one of about 90 Oe in the vicinity of coercivity and
a big one of about 900 Oe for the remaining range. At zero
sweep time, or without thermal effect, the coercivities for
three cases are all equal to a same value: H,,,=1).43SHk.
For K,V/kT= 1000, no significant thermal effect on coerciv-
ity is observed. However, when K,,V/kT=83, which is very
near the value of the likely ratios for the future single layer
media, the coercivity has pronounced time dependence. The
H, obtained from the simulated VSM measurement is about
65% of that without thermal effect. For K,,VikT-42, the
value is further reduced to less than half ofHCtr. The rema-
nence squareness and the coercivity squareness, however,
show little change for all cases. In his article,” Sharrock de-
rived the following time dependence formula of coercivity
for particulate media without interactions among particles:
y +[g ln(&]]i12, (2)
This formula instead of the linear dependence on log(time)
was found to fiit experimental data well; hut, our simulation
results for thin-film media fit with the logitime) curve for
7> l/f,. Interactions among grains and lack of orientation
may contribute to the difference.
Figure 2 shows the thermal effect on (a) an isolated tran-
sition and (bj a di-bit transition for a single-layer film with
J. Appf. Phys., Vol. 75, No. 10, 15 May 1994 P.-L. Lu and S. H. Charap 5769
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
141.210.2.78 On: Wed, 26 Nov 2014 03:19:48M/M
I head Odd
FIG. 2. <a) An isolated transition and ibi a di-bit transition in a single-layer
film for cases without thermal effect, with thermally assisted switchcs dur-
ing writing, and after 1 s, 30 mm, and 6 month decay. Here K,V/kT=60 and
magnetostatic interaction parameter bf,S/u;P =O.l (M,IH~=O.l).
D = S=9 nm, and hf,SJHfl =O.l. These dimensions lead to
K,V~kT==ljO. This value is more than twice that for ordinary
superparamagnetism. In the figure, the horizontal axis is the
position represented by the number of the row along the
track direction and vertical axis is the magnetization compo-
nent along the recording track direction averaged across the
track and normalized by hf, . A writing head moves from
left- to right-hand side. Nead fields are reversed at row 40 for
the isolated transition and at rows 40 and 48 for the di-bit
transition, respectively. There are usually 60 rows in our
model, and the magnetizations from rows 1 to 9 and from 51
ta 60 are not shown here. The bit length, i.e., the distance
behveen two transitions in the di-bit transition, is about 62
nm (S rows). The positions of transition centers shift about
one row, corresponding to about 8 nm, due to the thermdlIy
assisted switches during writing. The magnitude of the mag-
netization away from a transition decreases from around
0.74M,$ to about 0.61M, after 6 months of decay. The de-
cay in the region between two transitions of a di-bit becomes worse when they become closer and interact with each other.
The magnetization at the bit center is reduced from about
0.6M, without thermal aftereffect to 0.54n;l, when includ-
ing the thermally assisted switches during writing. It further
decays to about 0.38M, after 6 months. The magnetostatic
interaction from the magne.tization beyond the two transi-
tions tends to reverse the recorded magnetization between
them with the help of thermal energy. Some of the first tran-
sition is erased by the head field while writing the second
one.
IV. CONCLUSIONS
A combined molecular-dynamics and Pvlonte Carlo com-
puter simulation method has been developed to study the
thermal effect in magnetic thin-film recording media. The
crystalline anisotropy, shape anisotropy, applied field, and
magnetostatic interaction field are included in the mode.]. A
thre.e-dimensional energy surface is used to find the mini-
mum energy barrier in general. The writing process was
simulated with the.rmally assisted switches taken into consid-
eration and subsequent thermal decay was observed for long
time spans. Significant aftereffect was found both in hyster-
esis loop simulations and transition decays for grain volumes
more than twice that for ordinary superparamagnetism.
ACKNOWLEDGMENTS
The authors would like to express their gratitude to Dr.
Yasuhisa Kanai for discussions. This material is based (in
part) upon work supported by the National Science Founda-
tion under Grant No. ECD-X907068. The government has
certain rights in this material.
r S. Chikazumi and S. Charap, P&&s of~f~grrctism (Krieger, Malabar, FL.,
1964), Chap. 15.
“W. F. Brown, IEEE Trans. Magn. iW%G-15, 1196 i’lY7Y’L
3M. P. Sharrock, IEEE Trans. Magn. MAG-26, 193 (1990).
‘S. B. Oseroff, D. Clark, S. Schultz, and S. Shtrikman, IEEE Trans. Magn.
IL&W&21, 1495 (1985).
“S. H. Champ, J. Appl. Phys. 63, 2054 t 1988).
6Y. Kanai and S. IL Champ, IEEE Trans. Magn. MAG-27, 4972 il991).
‘J. Zhu and N. Bertram, J. Appi. Phys. 63, 3248 <19X8).
“K. Binder, ~Vfonre Carlo Method in Statistical Physics (Springer, Berlin,
19X6), Chap. 1, p. 32.
5770 J. Appl. Phys., Vol. 75, No. 10, 15 May 1994 P.-L. Lu and S. H. Charap
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
141.210.2.78 On: Wed, 26 Nov 2014 03:19:48 |
5.0017320.pdf | AIP Conference Proceedings 2265 , 030337 (2020); https://doi.org/10.1063/5.0017320 2265 , 030337
© 2020 Author(s).Structural and magnetization dynamic
properties of single crystalline Bi-doped YIG
thin film grown on GGG substrate having
different planes
Cite as: AIP Conference Proceedings 2265 , 030337 (2020); https://doi.org/10.1063/5.0017320
Published Online: 05 November 2020
G. Gurjar , Vinay Sharma , S. Patnaik , and Bijoy K. Kuanr
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Substrate having different Planes.
G. Gurjar1, Vinay Sharma2
, S. Patnaik1,a, Bijoy K. Kuanr2
1School of physical sciences, Jawaharlal Nehru University, New Delhi-110067, India
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Abstract. Structural and magnetization dynamic properties of Bi 0.1Y2.9Fe5O12(BYIG) thin film grown over single
crystalline Gadolinium gallium garnet (GGG, [111]& [100]) subst rate by excimer pulsed laser deposition (PLD) are
reported. X-ray diffraction and Ferromagnetic resonance(FMR) ha ve been performed on in-situ annealed film.BYIG over
GGG with [100] is grown structura lly better than film grown ove r GGG having [111] plane . It has a very narrow FMR
line width (23.50 Gauss)compared with film grown over GGG ([111 ]) (51.60 Gauss) and BYIG thin film grown over
GGG with [100] orientationhave very low Gilbert damping constan t (1.42×10-5) as compared with film grown over GGG
with [111] orientation(2.5×10-4).Bi-doped YIG thin film grown on GGG [100] substrate shows ver y low damping and
hence can be used for microwave devices such as micro wave filt er etc.
INTRODUCTION
Yttrium iron garnet (YIG) is a ferrimagnetic insulator at room temperature (having Tc around 560K). it hascubic
structure (space group Ia 3തd) and its chemical formula is Y 3Fe5O12, where Y ions occupy the 24c sites ( in the
Wyckoff notation), Fe octahedral 16a and tetrahedral 24d sites, and oxygen the 96h sites1. YIG has wide
applications inspintronic devices, microwave devices like reson ators, circulators and microwave filters etc.2 Y I G
shows room temperatureferrimagnetism very narrow resonance line width etc. so it is used primarily in microwave
devices.The coercivity of YIG is increases when Bismuth is dope d(Bi 0.1Y2.9Fe5O12(BYIG))7.So, to use the structural
inheritance of BYIG we have deposited a good quality single cry stalline BYIG thin film on GGG ([100] and [111])
substrate using PLD. From Ferromagnetic resonance (FMR) techniq ue is used to measure magnetization dynamic
properties of magnetic materials, here we have did the FMR spec troscopic experiment to measure the FMR
linewidth and damping parameters of Bi-doped YIG film over GGG with [100] and [111] planes5. Here we have
reported the structural and magnetic properties of Bi-doped YIG film over GGG substrate with [100] and [111]
planes.
EXPERIMENTAL DETAILS
U s i n g t h e B i s m u t h d o p e d Y I G t a r g e t , t h e B Y I G t h i n f i l m s w e r e g r o wn by the Pulsed laser deposition (PLD)
technique on single crystalline gadolinium gallium garnet (GGG, [ 1 0 0 ] & [ 1 1 1 ] ) s u b s t r a t e a t 8 0 0oC substrate
temperature and the oxygen pressure kept at0.15mbar.The deposit ion chamber was cleaned and evacuated to a base
vacuum of 1.5×10-6 mbar. We have used excimer laser (248nm) to ablate the BYIG tar get with frequency 5 Hz and
5-6 ns pulse width. The BYIG film was grown at a rate of 1.5nm/ min. The Target to substrate heater distance was
fixed to 4 cm. The as-grown thin film was in-situ annealed for 2 hours at 800 oC with 0.15mbar oxygen pressure.
DAE Solid State Physics Symposium 2019
AIP Conf. Proc. 2265, 030337-1–030337-4; https://doi.org/10.1063/5.0017320
Published by AIP Publishing. 978-0-7354-2025-0/$30.00030337-1The str u
Ferroma g
duroid m
plane.
The X-r a
BYIG fi l
XRD co n
impurity
YIG/GG G
YIG/GG G
The FM R
parallel t
experim e
BYIG/G G
best can ductural prope r
gnetic resona n
made microstri
ay diffraction (
lm grown ove r
FIGU R
nfirm the sin g
in figure 1(b )
TAB
L
G [111] 1
G [100] 1
R spectrosco p
to the film pl a
ental S11 dat a
GG[100] as c
didate for ma g
(a) rties of BYI
nce measure m
ipline in a fli p
(XRD) patter n
r GGG substr a
RE 1. Shows th e
gle crystalline
) is denoted b y
Par
BLE 1. Shows
Lattice param e
12.378
12.376
py is observe d
ane (inplane) a
a, we have us e
compared to t h
gnonics and m
G thin film
ments were o
p chip mode w
RESUL T
(a)
n were perfor m
ate with [111 ]
e BYIG film gr o
nature of B Y
y *
rameters obtai
the derived pa r
eter (Å) V o
18
18
(b) Magn e
d f o r B Y I G / G
and perpendi c
ed the lorentz i
he BYIG/GG
microwave de v
were studie d
observed by t
with dc magn e
TS AND D I
Structura l
med at room t
] and [100] pl a
own o n GGG s u
YIG thin film g
ned from XR D
rameters of YI G
olume of unit c
896.49383
895.57469
etization d y
GGG ([100] &
cular to the fi l
ian fit as sho w
G[111]. This
vice applicati o
d by X-ray
theKeysight V
etic field appl i
ISCUSSI O
l propertie s
temperature a n
anes.
ubstrate with ( a
grown on GGG
D are listed i n
G target and YI G
cell (Å3)
ynamic pr o
&[111]) thin
lm plane (ou t
wn in fig. 2. W
low Gilbert d
ons. All the d e
diffraction u
Vector Netw o
ied perpendic u
ON
s
nd figure 1 s h
a) [111] and (b)
G substrate wit h
n table 1.
G thin film fro m
operties
film when d c
t of plane(OO
We observed v
damping ma k
erived parame t
(b) using Cu K α
ork Analyzer
ular and para l
hows the XR D
[100] planes.
h [111] and [1 0
m XRD
c magnetic fi
P)) and from
very low FM R
kes BYIG/GG G
ters are listed
α1 (1.5406Å)
(VNA) usin g
llel to the fil m
D pattern of
00] planes . The
eld is applie d
the calibrate d
R linewidth i n
G[100] film a
in Table 2.
.
g
m
e
d
d
n
a
030337-2FIG
The FM R
magneti c
experim e
and α fo r
equation observe
d
F
GURE 2 . FMR
R linewidth (
c field linewi d
ental data. A ft
r both bulk an d
is given by
d the microwa v
IGURE 3 . sho w
spectroscopy ( l
BYIG
(ΔH) and Gil b
dth (∆H), res o
fter calibratin g
d YIG thin fi l
equation (1); w
ve absorption
ws fitted LLG e
lorentzian fit) o
G/GGG [111] o u
bert dampin g
onance field
g all the FM R
lm usingLand a
where, α is t h
parameters a r
equation for (a )
fH(
of (a) BYIG/G G
ut-of-plane (d)
g c o n s t a n t o f
(Hr) were ca l
R data at diff e
au–Lifshitz– G
he damping p
re better for Y
) BYIG/GGG [ 1
H f)0
GG [100] out-o f
BYIG/GGG [1
BYIG/GGG
lculated from
erent resonan c
Gilbert equati o
parameter an d
YIG thin film o
100] out-of-pla n
f
34
f-plane(b) BYI G
11] inplane
is shown in t
the lorentzi a
ce frequencie s
on(LLG). The
d γ i s t h e g y r
on GGG subs t
ne(b) BYIG/G G
G/GGG [100] i n
ta b l e 2 . 0 r e s p
an fits to the
s we have ca l
e correspondi n
romagnetic r a
trate with [10 0
GG [111] out- o
nplane (c)
pectively. Th e
calibrated S 11
lculated ∆H( f)
ng
atio. We hav e
0] plane.
of-plane
e
1
f)
e
030337-3Landau–Lifshitz–Gilbert equationfitting is shown in figure 3 fo r BYIG/GGG ([100] &[111]) thin
film when dc magnetic field is applied perpendicular to the fil m plane (out of plane(OOP)) and
obtained parameters as in figure inset.
Table 2. FMR data of (a)YIG target (b) YIG film grown on GGG substrate
FMR linewidth (ΔH ) (gauss) Gilbert damping constant (α)
YIG/GGG [100] OOP 23.50 1.4×10-5
YIG/GGG [111] OOP 51.60 2.5×10-4
CONCLUSION
In summary,Bi-doped YIG film grown on GGG substratewith [100] shows better magnetization and FMR linewidth
as compared with film grown on GGG substrate with [111]. From F MR spectroscopy there is a lower Gilbert
damping parameter in BYIG thin film grown on GGG with [100] pla ne. This lower value of Gilbert damping and
FMR linewidth allows the use of single crystalline BYIG film as the best candidate for microwave applications.
ACKNOWLEDGEMENT
G. Gurjar, Vinay Sharma thanks CSIR-UGC for fellowship. We ackn owledge AIRF, JNU for access of PPMS
facility.
REFERENCES
1.) V. Sharma and B.K. Kuanr, J. Alloys Compd. 748, (2018).
2.) G. Gurjar, V. Sharma, S. Patnaik, and B.K. Kuanr, in AIP Conf. Proc. (2018).
3.) B. Bhoi, N. Venkataramani, R.P.R.C. Aiyar, and S. Prasad, I EEE Trans. Magn. 49, 990 (2013).
4.) N. Kumar, S. Prasad, D.S. Mi sra, N. Venkataramani, M. Bohra, and R. Krishnan, J. M agn. Magn. Mater. 320,
2233 (2008). 5.) V. Sharma, J. Saha, S. Patnai k, and B.K. Kuanr, J. Magn. Magn. Mater. 439, (2017).
6.) K. Momma and F. Izumi , J. Appl. Crystallogr. 44, 1272 (2011).
7.) Gene siegel et al. Scientific reports 4 (2014): 4429.
030337-4 |
1.1455602.pdf | Spin-polarized current induced magnetization switch: Is the modulus of the magnetic
layer conserved? (invited)
J.-E. Wegrowe, X. Hoffer, Ph. Guittienne, A. Fábián, L. Gravier, T. Wade, and J.-Ph. Ansermet
Citation: Journal of Applied Physics 91, 6806 (2002); doi: 10.1063/1.1455602
View online: http://dx.doi.org/10.1063/1.1455602
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/91/10?ver=pdfcov
Published by the AIP Publishing
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155.97.178.73 On: Tue, 02 Dec 2014 15:07:42Spin-polarized current induced magnetization switch: Is the modulus
of the magnetic layer conserved? invited
J.-E. Wegrowe,a)X. Hoffer, Ph. Guittienne, A. Fa ´bia´n, L. Gravier, T. Wade,
and J.-Ph. Ansermet
Institut de Physique Expe ´rimentale, Ecole Polytechnique Fe ´de´rale de Lausanne,
CH-1015 Lausanne, Switzerland
The direct effect of spin-polarized current on magnetization states is studied on various
electrodeposited single contacted nanowires ~diameter about 60 nm !. Three kinds of samples have
been studied: ~1!Homogeneous Ni nanowires, ~2!nanowires composed of both a homogeneous Ni
part and a multilayered Co ~10 nm !/Cu~10 nm !part, ~3!pseudospin-valve pillars Co ~30 nm !/Cu~10
nm!/Co~10!electrodeposited in Cu wires. The magnetization reversal due to the current injection is
observed in the three cases. The effect is observed with using different experimental protocols,including current activated after-effect measurements. The results obtained suggest that twodifferent mechanisms are able to account for the magnetization reversal: exchange torque and spintransfer. We propose a definition of the two mechanisms based on the conservation ornonconservation of the magnetic moment of the ferromagnetic nanostructure. © 2002 American
Institute of Physics. @DOI: 10.1063/1.1455602 #
I. INTRODUCTION
The problem discussed in this article concerns the inter-
play between spin-dependent transport effects in metals and
the magnetization reversal in nanostructures. Recent experi-mental results show that it is possible to control the magne-tization reversal with injection of a spin-polarizedcurrent.
1–12After the pioneering work of Berger13and
Slonczewski,14various theoretical models were
proposed.15–19Two different approaches can be distin-
guished. First, the ‘‘exchange torque’’ concerns two ferro-magnetic layers separated by a nonmagnetic metallic layer~thick with respect to the exchange interaction length, e.g.,
above 5 nm !. The effect of the current is to provoke an ef-
fective coupling or an effective field between the two layers.Such a model was applied to the results obtained on pseudospin-valve structures of the form Co ~30 nm !/Cu~10 nm !/
Co~1.5 nm !.
5,8In the second approach, the system is com-
posed of a thick ferromagnetic layer ~thick with respect to
the spin-diffusion length, e.g., 10 nm !in which a spin-
polarized current is injected. Due to spin-flip scattering, thespin-polarization of the current is not conserved between thetwo interfaces of the ferromagnet. Instead, spin magnetiza-tion is transferred to the layer, in a relaxation process that isnot equivalent to a torque. This process was invoked in spin-polarized electron transmission experiments
7,20and applied
to current-induced magnetization reversal in magneticnanowires.
12,17Recent results of current induced magnetiza-
tion switching ~CIMS !suggest that both mechanisms coex-
ists. The aim of this article is to review the results obtainedwith electrodeposited nanowires, and to discuss a unified pic-ture for interpreting the data.II. SAMPLES
The samples were obtained by the method of elec-
trodeposition in nanoporous track-etched membrane tem-
plates. Polycarbonate membranes of 6000 nm thickness, andpore diameters of about 60–80 nm were used. Gold is sput-tered on the membrane surfaces, before electroplating. Wiresof controlled morphology were obtained by depositing layersof various materials ~Cu, Co, Ni !and sizes ~down to a few
nanometers !.
21The next step was to contact a single nano-
wire in the membrane. This was performed in a Cu electro-lytic bath by controlling the voltage between the top and thebottom of the membrane. The method is described in Ref.22. Three kinds of samples have been studied in the frame-work of this study ~see Fig. 1 !:~1!homogeneous Ni
nanowires,
2~2!nanowires composed of both a homogeneous
a!Electronic mail: jean-eric.wegrowe@epfl.ch
FIG. 1. ~1!Ni homogeneous nanowires. ~2!Hybrid structures composed of
homogeneous Ni part as a probe, and a @Co/Cu/Co #multilayered part as a
spin polarizer. ~3!Pseudospin-valve structure.JOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 10 15 MAY 2002
6806 0021-8979/2002/91(10)/6806/6/$19.00 © 2002 American Institute of Physics
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155.97.178.73 On: Tue, 02 Dec 2014 15:07:42Ni part and a multilayered Co/Cu part,10,12and~3!spin-valve
pillars Co ~30 nm !/Cu~10 nm !/Co~10!electrodeposited on Cu
wires.23
A comparative study of CIMS in samples ~1!and~3!is
shown in Fig. 2. In order to evidence an effect of the spin-polarized current on the magnetization, it is necessary toquantify small variations of the magnetization ~of the order
of 10
213emu or 10216Am2!.Asample of type ~1!has been
chosen because the magnetization reversal mechanisms arewell characterized.
24–27However, in this type of sample
~composed by a single magnetic domain !the origin of the
spin-polarization of the current is not controlled. It could bespin-polarized by spin accumulation effects at the normal/ferromagnetic interface, by nanoinhomogeneities, or may notbe spin-polarized at all. In sample ~2!, the current is spin-
polarized by a Co/Cu multilayered part above the Ni. Themagnetic behavior of the Ni part ~equilibrium states and ir-
reversible switching !is shown to be very similar to that of
sample ~1!~not shown here !.
10,12Sample ~3!is also com-
posed of a spin polarizer and a ferromagnetic layer to probethe magnetization reversal. This pseudospin-valve is com-posed of a ‘‘pinned’’ magnetic layer of high coercivity ~po-
larizer !, and a ‘‘free’’ magnetic layer of low coercivity
~analyser !~see Fig. 1 !. It is important to note that, in contrast
to the previous studies reported,
5,8the thickness of the free
layer here is larger than the spin diffusion length. The mag-netic characterization is based on the magnetoresistive ~MR!
properties @Figs. 2 ~a!and 2 ~c!#. The MR hysteresis loops
were measured with a usual lock-in detection ~ac current of
0.5
mA, about 104A/cm2density !. In the case of samples of
type~1!, the magnetization states are deduced from the an-
isotropic magnetoresistance ~AMR !hysteresis loop.The hys-
teresis loop is composed of a reversible part, which can befitted by a uniform rotation of the magnetization, and anirreversible jump, described by the switching field H
sw@seeFig. 2 ~a!#.25,26In sample ~3!, the giant MR ~GMR !accounts
for the relative orientation of the two Co layers @see Fig.
2~c!#. For each kind of sample, a strong effect of the current
injection on the magnetization reversal is observed.
III. MAGNETIZATION REVERSAL DUE TO CURRENT
INJECTION
Protocol ~A!. In a first approach, the effect of high cur-
rent injection is observed by measuring the deviation of theswitching field with and without current injection.At a fixedfield, the samples define a two state system. The state of themagnetization is first measured at a fixed field just beforecurrent injection and about 0.5 s after injection. The durationof the current pulse is 0.5
ms. Reversible processes are,
hence, not accessible with this protocol. The relevant param-eter in this context is the maximal distance DH
max5uH0
2Hswubetween the switching field Hswwithout current in-
jection and the field H0at which the current still provokes
the magnetization reversal. In the results presented in Fig. 2,the current density is about 2 310
7A/cm2. The maximal ef-
fect observed is 40% of the switching field ~orDHmax
550 mT !for samples of type ~1!@Fig. 2 ~b!#25% for type
~2!,10,12and 80% of the hysteresis width for samples of type
~3!23@Fig. 2 ~d!#. Note that the maximum field induced by the
current ~Oersted field !is about 5 mT. In the case of uniform
magnetic configurations the parameter DHgives the rotation
of the magnetization Dwdue to current injection @Fig. 2 ~b!#.
In the analysis developed in Ref. 12, the rotation of the mag-netization is reversible as long as the critical angle
wcis not
reached.17If the critical angle is reached, the magnetization
jumps irreversibly, and the jump can be measured at any timeafter current injection. The limit between the two regimes~reversible versus irreversible !is, hence, given by the param-
eterDH
max. This parameter has been measured as a function
of the current amplitude, the current direction, and as a func-tion of the angle of applied field,
u.10,12The results are sum-
marized in Sec. V.
In the case of the pseudospin-valve structures @samples
of type ~3!#, the magnetization states are much more difficult
to describe due mainly to the fact that, in contrast to thenanowires ~cylindrical geometry !, the position of the magne-
tization of each layer must be defined by two angles ~in and
out of the plane of the Co layers !. However, these structures
allow us to gain crucial information about CIMS, becausethe spin polarization of the current is given by the directionof the magnetization of the layer of higher coercivity. Theeffect of the current is then studied for both transitions of thesame layer ~the layer of lower coercivity, or free layer; the
pinned layer being fixed !: transition from antiparallel to par-
allel states ~AP–P !and from parallel to antiparallel ~P–AP !
states @see Figs. 2 ~c!and 2 ~d!#. The necessity of working
with a fixed layer implies to work on the minor loop @inset of
Fig. 2 ~c!#.
Protocol ~B!.Another protocol was defined,
4,5,8in which
a ramp of current is applied at fixed field H~i.e., at distance
DHfrom the switching field without current !.Ahysteresis as
a function of the current amplitude can then be obtained atfixed magnetic field. The relevant parameter in this case isthe critical current I
c, at which the magnetization jumps.
FIG. 2. ~a!AMR hysteresis loop of sample of type ~1!measures at different
angles of the applied field. ~b!Zoom on the irreversible part of the hysteresis
loop with the effect of current injections ~arrows !. The magnetization states
are sketched. ~c!CPP–GMR hysteresis loop of a pseudospin-valve, with
minor loop in the inset. ~d!Minor loop with current injection ~arrows !for
the two transitions.6807 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Wegrowe et al.
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155.97.178.73 On: Tue, 02 Dec 2014 15:07:42The relation between the function Ic(DH) and the function
DHmax(I) is not trivial and would necessitate a better under-
standing of the phenomenon. However, the qualitative analy-sis can be performed with both functions by observing thesymmetry of the transitions ~AP–Pand P–AP !with respect
to current directions.
IV. CURRENT ACTIVATED ESCAPE OF A
METASTABLE STATE
Protocol ~C!:The protocols described earlier are not able
to take into account the effect of the temperature ~which
depends on the amplitude of the injected current via theJoule effect !, and the effect of the pulse duration. The meta-
stable characteristic of the process must be taken into ac-count by a thermal activation approach. In terms of thermalactivation, the switching field H
swis the field at which the
potential barrier vanishes within the thermal fluctuations, sothat the parameter DH
maxgives a measure of the potential
barrier height overcome by the magnetization due to currentinjection ~see Fig. 3 !. In the simple cases,
24,28,29the potential
barrier can be written in the form DE5E0@1
2(Heff)/(Hsw)#awhereHeffis the magnetic field possibly
corrected by a current dependent term HI~discussed in Sec.
VI!,ais about 3/2, and E0is the barrier height ~mainly due
to anisotropy !for a symmetric double-well potential. The
potential is also defined by its degree of asymmetry, the pa-rameter
dwE~which is also a measure of the irreversibility of
the process !. The dynamics is described by an exponential
decay where the relaxation time tis given by the Kramers–
Brown law.30
t5t0Le~DE/kT!1t0Re~DER/kT!, ~1!
whereTis the temperature, kis the Boltzmann constant, t0is
the waiting time without barrier around the left ~L!or the
right ~R!~Fig. 3 !of the potential minimum, and DER
5E0@11(Heff)/(Hsw)#ais the barrier for the activation of the
inverted process. In usual situations, and in the following,the second term of the right side of Eq. ~1!is neglected with
respect to the first term. However, in the context of CIMS,this term may allow us to understand the possible switchingof the magnetization at a fixed external field from a meta-stable state to a more stable state, and switching back to theinitial state with current injection.
In contrast to the previous approaches, the time resolved
magnetization reversal is now explicitelly defined by twoprocesses; the conservative effective field H
effwhich derivesfrom the potential, and a stochastic process which is due to
the action of the other degree of freedom whose dynamicsare characterized by much faster relaxation times. The cur-rent dependence could be accounted for by a term in theeffective magnetic field ~which acts on the parameters DE
and
dwE!and leads then to a hysteretic behavior for the
magnetization curve M(I) as a function of the current similar
to usual hysteresis loops M(H). On the other hand, the cur-
rent dependence could also be accounted for by the effect ofcurrent dependent dissipative processes. The term ( E
0/kT)
3(J) would then be current dependent, e.g., with T5Tjoule
1Tsf, where the temperature of the sample Tjouleis due to
thermostat and Joule effect, and Tsfis due to current depen-
dent spin-flip scattering ~Fig. 3 !. The two processes sketched
here~conservative and dissipative processes !are detailed in
Sec. VI.
Current dependent magnetic after-effect measurements
were performed as a function of the current density onsamples of type ~1!in the submicrosecond range.The experi-
ments are detailed in Ref. 9. A square excitation of currentdensityJis injected at a distance DHfrom the switching
field without current. The resistance variation is measuredduring the 8
ms of the excitation @see Fig. 4 ~a!#.Astationary
thermal regime is reached some few hundreds of nanosec-onds after the beginning of the pulse. The temperature of thesample, during current injection, is deduced from the mea-sured linear temperature dependence of the resistance. Thejump of the magnetization, measured by its AMR signal,occurs at a time
tafter the beginning of the excitation @see
Fig. 4 ~b!#. An histogram is performed. The mean value is
identified to the waiting time needed to overcome the energybarrier for a given current excitation @the error bars in Fig.
4~c!accounts for the width of the histogram #. The waiting
time
tis strongly dependent to the magnetic field as de-
scribed in Eq. ~1!, so that it can be kept in our observation
window of some few microseconds for current variationsrange between 0.5 and 2.5 310
7A/cm2@see Fig. 4 ~c!#.
FIG. 3. Current induced escape out of metastable states.
FIG. 4. Ater-effect measurements with homogeneous Ni nanowires. ~a!Cur-
rent injection ~right scale !, and response of the resistance ~left scale !.~b!
Zoom of the AMR response. ~c!Mean switching time as a function of the
applied field for various current injections. ~d!Variation of the parameter E0
as a function of the current applitude.6808 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Wegrowe et al.
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155.97.178.73 On: Tue, 02 Dec 2014 15:07:42Note that only large waiting times ~tabove 1 or 2 ms!are
taken into account, in order to neglect the few hundreds ofnanoseconds where the temperature is not constant in thesample. It is not possible to fit the data with using the acti-vation law Eq. ~1!with the current dependence contained in
the parameter H
Ionly. The fit @Fig. 4 ~c!#is, hence, per-
formed with current dependence contained in the parameterE
0. The energy E0varies from 6 3104Kt o0 . 8 3104K be-
tween 1 and 4 3107A/cm2@Fig. 4 ~d!#.
V. RESULTS
The study of current activated escape of a metastable
state shows that there is a direct effect of the current on themagnetization reversal, independent of the thermal activationdue to Joule heating. The comparative study of CIMS be-tween samples ~1!and~2!as a function of the angle of the
applied field shows that the observed effect is related to thespin polarization of the current.
12This means that all CIMS
effects observed in our samples ~and probably the other mea-
sured structures !,3,11have to be described in the framework
of a unified picture. However, if CIMS is clearly evidencedin the Ni nanowires ~and also Co not shown here !, the diffi-
culty of describing spin-polarization and spin accumulationeffects at both interfaces makes any microscopic descriptionunrealistic at this stage of the study.
The study of CIMS in samples of type ~3!as a function
of the configuration of the layers ~parallel and antiparallel !,
and as a function of the two current directions, shows that astrong asymmetry exists, which is qualitatively differentfrom that observed in other Co/Cu/Co pseudospin-valve sys-tems with thinner free layers.
5,8These results are presented
in terms of the parameter DHmax(I) for both transitions:
AP–P and P–AP in Fig. 5. Except for a rather weak contri-bution, which will be discussed later, only the transitionAP–Pis allowed, and only for the current direction I
1which
corresponds to the electrons flowing from the pinned layer~the polarizer !to the free layer ~the analyzer !. However, the
‘‘weak contribution’’is confirmed with the time-resolved ~or
‘‘reversible’’ !measurements:
23the two transitionsAP–P and
P–AP can be observed at some weak fixed field, with thesymmetry already observed by others and interpreted interms of exchange torque
5,8~i.e., AP–P transition with I1
and P–AP with I2!.
These observations are also confirmed by the ramp in
currentR(I)@protocol ~B!#in a sample of type ~3!~Fig. 6 !.
The measurements are performed at 50 K, with nanovoltme-ter detection. The sample is saturated at H524 T before the
field is set for each current ramp. The R(I) curves show that
for the large majority of applied fields, the only allowedtransition is for antiparallel to parallel states ~AP–P !, what-
ever the current direction, the initial magnetic state, and thedirection of the ramp @see Fig. 6 ~a!#. This is true except for a
few particular cases, corresponding to weak external fieldsand intermediate transitions ~‘‘pseudotransition’’ P–AP !.I n
the case shown in Fig. 6 ~b!, the total loop with the pseudo
transition P–AP and AP–P is performed at 240 mT. The
pseudotransitions correspond to about 0.5 VGMR while the
complete transition corresponds to 1 VGMR ~see Fig. 6 !.These results suggest that the CIMS pseudotransition P–AP
would be possible only if
dwE@or (dwE)/(E0)#is small
enough in the absence of current injection.
VI. PHENOMENOLOGICAL MODEL
The aim of the following discussion is to clarify the
possible existence of two CIMS mechanisms with radicallydifferent symmetry properties. In this first approach, fluctua-tion is not taken into account. The dynamics of the magne-tization is then described by the Landau–Lifshitz–Gilbertequation
dM
dt5g8M~MˆHeff!1h8~MˆHeff!3M, ~2!
where the first and second terms on the right-hand side are,
respectively, the precession term ~or transverse relaxation !
and the longitudinal relaxation term. The phenomenologicalparameters h
8andg8are linked to the gyromagnetic ratio g
and the Gilbert damping coefficient aby the relation h8
5(ga)/@(11a2)Ms1#andg85g/@(11a2)Ms1#. The ef-
fective field Heff5Hd2!11Ha1Hincludes dipole field Hd2!1
FIG. 5. Results obtained in pseudospin-valves samples with protocol A.
DHmaxis plotted as a function of the current for both transitions AP–P
~upper graph !and P–AP ~lower graph !, and both current directions. I1
corresponds to the electrons flowing from the pinned layer ~or polarizer !to
the free layer ~or analyzer !.6809 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Wegrowe et al.
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
155.97.178.73 On: Tue, 02 Dec 2014 15:07:42due to the pinned layer, the anisotropy field Ha~including
magnetostatic field !, and an applied field H. the magnetiza-
tion of the free ferromagnetic layer is described by Mof
modulus Ms. It is clear from Eq. ~2!that the time variation
of the magnetization is always perpendicular to the magne-tization, or in other words, that the modulus of the magneti-zation is always constant.
Without any loss of generality ~as far as the magnetiza-
tion is an extensive variable !, it is possible to show
17that the
effect of the current can be taken into account by adding athird term to the equation
dM˜
dt5g8Ms1~MˆHeff!1h8~MˆHeff!3M1f~I,M!e.
~3!
Now,M˜is the magnetization of the free layer plus the mag-
netization due to the spin transfer, if any.The specification ofthe scalar function f(I,M), and the definition of the unit
vectoreneed further hypothesis about the microscopicmechanisms involved. However, a first hypothesis related to
the structure of the equation can be formulated by two op-tions. ~1!We want to keep the modulus of the magnetization
constant, and Eq. ~3!reduces to Eq. ~2!provided that the
effective field is modified by an additive term H
Ipropor-
tional to f(I,M).~2!In the other case, M˜ÞMand Eq. ~3!
has the structure of a conservation equation. There is a trans-fer of magnetization from the current to the ferromagneticlayer, and this transfer is irreversible.
In the exchange torque picture we are in case ~1!, the
two layers are coupled by the torque:
14,16e5(M/Ms)3@eI
3(M/Ms)#, whereeIis the direction of the spin polarization
of the current. If the pinned layer is fixed, the two transitions~P–AP !and~AP–P !play a quasisymmetric role with chang-
ing the current direction, as previously measured.
5,8The
R(I) curves can then be though of as R(Heff) magnetoresis-
tance hysteresis loops and only a small asymmetry is ex-pected due to the nonlinearity of the function f(I,M).
14,8
In the case of spin transfer due to spin relaxation,17e
5eIis the direction of the spin polarization of the current,
i.e., the orientation of the magnetization of the pinned layer if
the conduction electrons are flowing from the pinned layer tothe free layer, i.e., with I
1. In this case, the free layer is
stabilized if the pinned layer is parallel to the magnetizationof the free layer ~vanishing DH
max!, whereas the transition is
favored ~large DHmax!if the pinned layer is antiparallel. In
the case of I2, the current is flowing from the free layer to
the pinned layer and no effect ~vanishing DHmax!is expected
on the free layer, no matter what transitionAP–Por P–APisstudied,becausetheinjectedcurrententeringinthefreelayeris not spin polarized. This typical asymmetry was observed~Fig. 5 !. Furthermore, the observations with protocol ~B!of
the hysteresis R(I)@Fig. 6 ~a!#can hardly be interpreted in
terms of hysteresis R
˜(HI), because of the strong asymmetry.
Therefore, it may be ascribed to a contribution of a spintransfer
17due to spin relaxation from the spin-polarized cur-
rent to the magnetization of the ferromagnetic layer.
Note that in contrast to the results shown in Fig. 5, both
current directions provoke the magnetization reversal in Fig.6. This is due to the fact that the minor loop was not identi-fied in these measurements ~because the two layers are too
strongly coupled, with respect to the anisotropy difference.In other words, both functions ‘‘polarizer’’ and ‘‘analyzer’’are permuted by changing the current direction.
VII. CONCLUSION
A comparative study has been performed on various
single contacted magnetic nanostructures in order to under-stand the effect of CIMS.Arelation of the CIMS effect withthe spin polarization of the current has been shown by com-paring CIMS of Ni nanowires with and without spin polar-izer. On the other hand, the thermal activation includingJoule heating and the direct effect of the current on the po-tential barrier has been quantified separately with measure-ments of current activated escape out of a metastable state.The process cannot be described in terms of a current depen-dent effective field only. Measurements performed on pseudospin-valve structures with thick free layers show that the
FIG. 6. Pseudospin-valve sample measured at T550 K. The R(I) curve is
plotted as a function of the current amplitude at fixed applied field. ~a!High
fields, only the transition AP–P is allowed. ~b!Magnetic field H
5220 mT, both pseudotransitions P–AP and AP–P are observed. Two
ramps are performed, first starting for increasing current ~1–4!, and then for
decreasing current (1 8–58). Inset: hysteresis loop as a function of the ex-
ternal field.6810 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Wegrowe et al.
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155.97.178.73 On: Tue, 02 Dec 2014 15:07:42transition from P–AP state is not equivalent to the transition
from AP–P state. This qualitative asymmetry is also con-firmed by the hysteresis obtained in ramping the current atfixed field. These results suggest that another mechanism,that is not equivalent to a torque, is responsible for CIMS inour structures. A phenomenological description was pro-posed in order to define this spin transfer mechanism interms of nonconservation of modulus of the magnetic mo-ment.
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1.3582149.pdf | Magnetotransport in nanostructures: The role of inhomogeneous currents
Tiago S. Machado, M. Argollo de Menezes, Tatiana G. Rappoport, and Luiz C. Sampaio
Citation: Journal of Applied Physics 109, 093904 (2011); doi: 10.1063/1.3582149
View online: http://dx.doi.org/10.1063/1.3582149
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158.42.28.33 On: Thu, 18 Dec 2014 09:45:35Magnetotransport in nanostructures: The role of inhomogeneous currents
Tiago S. Machado,1M. Argollo de Menezes,2Tatiana G. Rappoport,3
and Luiz C. Sampaio1,a)
1Centro Brasileiro de Pesquisas Fı ´sicas, Xavier Sigaud, 150, Rio de Janeiro, RJ, 22.290-180, Brazil
2Instituto de Fı ´sica, Universidade Federal Fluminense, Rio de Janeiro, RJ, 24.210-346, Brazil
3Instituto de Fı ´sica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, 68.528-970, Brazil
(Received 4 January 2011; accepted 27 March 2011; published online 3 May 2011)
In the study of electronic transport in nanostructures, electric current is commonly considered
homogeneous along the sample. We use a method to calculate the magnetoresistance of magnetic
nanostructures where the current density may vary in space. The current distribution is numericallycalculated by combining micromagnetic simulations with an associated resistor network and by
solving the latter with a relaxation method. As an example, we consider a Permalloy disk
exhibiting a vortex-like magnetization profile. We find that the current density is inhomogeneousalong the disk, and that during the core magnetization reversal it is concentrated toward the center
of the vortex and is repelled by the antivortex. We then consider the effects of the inhomogeneous
current density on spin-torque transfer. The numerical value of the critical current densitynecessary to produce a vortex core reversal is smaller than the one that does not take the
inhomogeneity into account.
VC2011 American Institute of Physics . [doi: 10.1063/1.3582149 ]
I. INTRODUCTION
Electric transport in magnetic nanostructures is a useful
tool both for probing and for manipulating the magnetiza-
tion. In the low current density regime, magnetoresistance
curves are useful for probing the sample’s magnetizationstate while, in the high current density regime, magnetization
patterns can be modified by a spin-transfer torque.
1–3Magne-
toresistance measurements have the advantage of being rela-tively simple and fast, serving as an efficient magnetic
reading mechanism.
4,5
Depending upon their thickness and diameter, small fer-
romagnetic disks exhibit stable topological defects known as
magnetic vortices.6,7These vortices can be manipulated by
picosecond pulses of a few (tens of) oersted in-plane mag-netic fields that switch their polarity,
8–13making them good
candidates for elementary data storage units.9
For their use as storage units, the most viable form of
manipulation of the magnetization is through spin-torque
transfer, with the injection of high density electrical cur-
rents.1The effect of these currents in the magnetization dy-
namics is described theoretically by the incorporation of
adiabatic and nonadiabatic spin-torque terms in the Landau-
Lifshitz-Gilbert (LLG) equation.14,15These two terms are
proportional to the injected current density and it is normally
considered an homogeneous current distribution inside the
disk. Although theoretical predictions using this approachqualitatively agree with experimental results, there is a lack
of quantitative agreement between theoretical and experi-
mental results regarding the current densities necessary tomodify the magnetic structures.
17–19
In this paper we investigate the effect of nonuniform cur-
rent distributions on electronic transport and spin-torquetransfer in ferromagnetic systems exhibiting vortices. We
numerically calculate the magnetoresistance (MR) and local
current distribution of a ferromagnetic disk by separating the
time scales for magnetic ordering and electronic transport. Weconsider an effective anisotropic magnetoresistance (AMR)
that depends on the local magnetization. We discretize the disk
in cells and solve the Landau-Lifshitz-Gilbert (LLG) equa-tion
20numerically with the fourth-order Runge-Kutta,21
thereby obtaining the magnetization profile of the disk. Thispattern is used to calculate the magnetoresistance of each cellas a fixed current, I, is applied at two symmetrically distributed
electrical contacts, resulting in a voltage drop and an inhomo-
geneous current distribution along the disk.
This method couples the electric and magnetic proper-
ties of the metallic nanomagnets and can be used to analyze
the effect of inhomogeneous current distributions in differentcontexts. First, we discuss the limit of low current density
where transport measurements can be used to probe the mag-
netic structure. We compare the magnetic structure with themagnetoresistance curves and show how the magnetoresist-
ance measurements could be interpreted to obtain informa-
tion on the magnetization profile and its dynamics during thevortex core magnetization reversal. Moreover, we discuss
the consequences of a nonhomogeneous current distribution
on the spin-torque transfer and find that the critical currentdensity that produces the vortex core reversal is reduced by
one order of magnitude whenever such noninhomogeneity is
taken into account. This result can be seen as a new route tounderstanding why the experimental values of the critical
current densities are usually lower than the ones obtained in
the LLG calculations.
17–19
This article is organized as follows: In Sec. IIwe discuss
the model and method for the calculation of the magnetore-
sistance and current distribution. In Sec. III, we exemplify
the calculations by considering the magnetoresistance and
current distributions of a Permalloy disk exhibiting aa)Author to whom correspondence should be addressed. Electronic mail:
sampaio@cbpf.br.
0021-8979/2011/109(9)/093904/6/$30.00 VC2011 American Institute of Physics 109, 093904-1JOURNAL OF APPLIED PHYSICS 109, 093904 (2011)
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158.42.28.33 On: Thu, 18 Dec 2014 09:45:35magnetic vortex. In Sec. IV, we study the consequences of a
nonhomogeneous current distribution on the spin-torque
transfer. In Sec. Vwe summarize the main results.
II. MAGNETORESISTANCE AND CURRENT
DISTRIBUTION CALCULATIONS
Let us consider a 36 nm-thick Permalloy disk with a diam-
eter of 300 nm discretized into a grid of 4 /C24/C24n m3cells.
The dynamics of the magnetization vector associated with eachcell is given by the Landau-Lifsh itz-Gilbert equation, which we
numerically integrate with the fourth-order Runge-Kutta
and discretization step, h¼10
/C04.21The parameters associated
with the LLG equation are the saturation magnetization Ms
¼8:6/C2105A/m, the exchange coupling, A¼1:3/C210/C011J/
m, and the Gilbert damping constant a¼0:05.13
By varying the external in-plane magnetic field, H, from
negative to positive saturation we obtain a hysteresis curve,
as depicted in Fig. 1, which is consistent with experimental
observations.7As shown in Fig. 1(a), in static equilibrium
and in the absence of magnetic fields, a vortex structure with
a core magnetized perpendicular to the disk plane is formedin the center of the disk. If a small in-plane magnetic field,
H, is applied, the core is displaced from the center [Fig.
1(b)]. At a critical field, H
c1, the vortex is expelled from the
disk, resulting in a discontinuity in the hysteresis loop. As
the external field, H, is lowered back, the vortex structure
reappears, but at a lower field, Hc2<Hc1.
In order to investigate the electronic transport on the
nanomagnetic disk we consider the magnetization profile,
f~Mig, obtained as the stationary solution of the LLG equa-
tion, as a starting point to calculate the magnetoresistance,
Ri, in each cell, i, of the disk. It is well established that in rel-
atively clean magnetic metals the main source of magnetore-sistance is the anisotropic magnetoresistance (AMR),
22
which can be expressed as q¼q?þðqk/C0q?Þcos2u,
where uis the angle between the local magnetization and
the electric current and q?andqkare the resistivities when
the magnetization is perpendicular and parallel to the cur-
rent, respectively. We decompose the current into orthogonalcomponents, xandy, such that if the normalized projection
of the magnetization, ~Mi, on the current direction ^u(u¼
x;y)i smu
i¼cosu, and the cell geometrical factor is taken
into account, the magnetoresistance, Ri, is split into orthogo-
nal components as Ru
i¼R?
iþðRk
i/C0R?
iÞðmu
iÞ2in every cell,
i, of the disk (Fig. 2). Thus, we obtain a resistor network
where the resistances depend on the local magnetization andare assumed to be approximately constant at the time scale
of electronic scattering processes.
Guided by recent experiments
23we allow a constant
current, I, to flow along the disk by attaching symmetrically
FIG. 1. (Color online) Magnetic hysteresis obtained with a micromagnetic
simulation of a Permalloy disk with a diameter of 300 nm and a thickness of
36 nm, subject to a static in-plane magnetic field, H. Two configurations for
the vortex core, corresponding to different external fields (0 and 75 mT), are
also depicted.
FIG. 2. Original cells used in the LLG simulation with the associated resist-
ance network.
FIG. 3. (Color online) (a) Magnetoresistance for the magnetic configura-
tions obtained in Fig. 1for uniform (circle) and nonuniform (square) current
distribution. Bottom: electric current map for (b) zero field and (c) H¼75
mT. The red (blue) color corresponds a current density which is about
1/C02%larger (smaller) than the uniform current at the saturation field. The
red color is in the center of the vortex.093904-2 Machado et al. J. Appl. Phys. 109, 093904 (2011)
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158.42.28.33 On: Thu, 18 Dec 2014 09:45:35placed electrodes on it (see Fig. 3). The voltage drop along
the resistors and the associated current map of the disk are
obtained by solving Kirchhoff’s equation iteratively at eachnode of the grid with a relaxation method
24,25
Vðnþ1Þ
i ¼X
jðÞ1=Rij0
@1
A/C01X
jðÞVðnÞ
j
Rijþbi0
@1
A; (1)
where RijisRx
iðRy
iÞifiandjare horizontal (vertical) neigh-
bors and biis the boundary current, assumed to be Ið/C0IÞat
the leftmost (rightmost) cells, and zero otherwise (see
Fig.2). Here, Vn
iis the voltage at site iafter niterations and
the sums run over the nearest-neighbors jðÞof node i. Start-
ing with a random initial condition, fVð0Þ
ig, at each site we it-
erate Eq. (1)until each VðnÞ
ibecomes stationary (within 9
decimal digits precision). After convergence, we calculate
the equivalent resistance, the ratio, Req¼DV=I, between the
voltage drop, DV, between the electrodes, given by
DV¼X
ikbi¼IVi/C0X
jkbj¼/C0IVj; (2)
and the current, I, entering the disk.
III. MAGNETO-STRUCTURE AND
MAGNETORESISTANCE
A. Hysteresis and magnetoresistance
In order to obtain the magnetoresistance curves, the cal-
culation discussed in the previous section is performed at dif-
ferent fields. The magnetoresistance and current distribution
for the same points of the hysteresis loop in Fig. 1are
depicted in Fig. 3. Figure 3(a) displays the magnetoresist-
ance curves for both homogeneous (without using the resist-
ance network26) and nonhomogeneous current distributions.
The vortex expulsion and its formation at a different critical
field are clearly identified and, with qk¼155Xnm and
q?¼150Xnm, we obtain a MR of 1 :2%for the nonhomo-
geneous distribution, which is a typical value found in previ-
ous experiments.23,27
One also observes that the magnetoresistance curves for
uniform and nonuniform current distributions differ signifi-
cantly, the latter being more comparable to experimental
results with the same contact geometry.23As expected, a ho-
mogeneous current overestimates the magnetoresistance,
since the current will flow through regions of high resistance,
whereas with the current found by solving Laplace’s equa-tion on the associated resistor network, one finds a preferen-
tial path (higher current density) on regions of low
resistance. This difference is more pronounced in the pres-ence of a vortex, since the magnetization of the disk is highly
nonhomogeneous on such configurations.
In light of the discussion above, one sees in Figs. 3(b)
and3(c) that the current is not homogeneously distributed
inside the disk, being stronger toward the center of the vortex
core. In the center of the disk the magnetization either pointsin the ^zdirection, perpendicular to the direction of current
flow, or loops about the vortex core. In both cases, the currenthas a path where its direction is always perpendicular to the
magnetization, reducing the local magnetoresistance. Above
the saturation field, the magnetization is uniform and at thedisk center the same applies to the current. The red (blue)
region has a current density 1 /C02%larger (smaller) than the
current, I, at saturation. The red region is in the center of the
vortex. This effect might be enhanced if other sources of mag-
netoresistance are considered, such as giant magnetoresist-
ance, for example. Similar approaches using different sourcesof magnetoresistance and geometries have been used to calcu-
late the magnetoresistance in nanomagnets.
27–32
B. Dynamics
Next, we study the dynamics of the vortex core magnet-
ization reversal by the application of short in-plane magneticfields. Under a pulsed in-plane magnetic field or spin polar-
ized current excitation, the vortex with a given polarity (V
þ)
dislocates from the center of the disk with the nucleation of avortex (V
/C0)-antivortex (AV/C0) pair with opposite polarity af-
ter the vortex attains a critical velocity of rotation about the
disk center.3,33The original Vþthen annihilates with the
AV/C0, and a vortex with reversed core magnetization (V/C0)
(Ref. 9) remains. If a low-density electronic current is made
to flow through the sample (without disturbing the magnet-
ization dynamics), we observe changes in the magnetoresist-
ance, as the vortices nucleate and annihilate. In Fig. 4(a) we
depict the dynamics of the magnetoresistance as a pulsed in-
plane magnetic field is applied in the ^xdirection at t¼20 ps
for different pulse intensities. The pulses have their shapesketched in gray in Fig. 4(a) with a full width at half maxi-
mum of t¼250 ps. Depending on the pulse intensity, the
vortex core magnetization does not reverse at all ( l
0H<43
mT), reverses once (54 mT <l0H<64 mT), or multiple
times ( l0H>64 mT).12,13
During the application of a field pulse in the ^xdirection,
i.e., parallel to the current flow, the vortex core is pushed to
the^ydirection, breaking the rotation symmetry of the disk’s
magnetization, increasing both the total mxcomponent and
the disk’s equivalent resistance [see Fig. 4(a)]. At t¼340 ps,
the field is practically zero and, from the decay of the mag-
netoresistance to its equilibrium (initial) value, one can inferwhether there was reversal of the vortex core polarization or
not: for pulses that induce reversal, the value of the magneto-
resistance just after the pulse is always larger than its initialvalue. If there is no reversal the magnetoresistance attains a
minimum value that is lower than its initial value, i.e., before
the application of the pulse, and oscillates about it.
In Figs. 4(b)–4(e) we depict snapshots of current (color
map) and magnetization (arrows) distributions at time steps
marked with black dots in Fig. 4(a), in situations with or
without vortex core magnetization reversal. Whenever the
pulse decreases its intensity, the total m
xcomponent and the
equivalent resistance of the disk follow the same pattern(although with some time delay), because the vortex core
tends to return to the disk center, where m
x¼0. Figure 4(b)
shows the current distribution and magnetization at amoment corresponding to the minimum of the resistance
curve, for a field intensity, l
0H¼27 mT, for which there is093904-3 Machado et al. J. Appl. Phys. 109, 093904 (2011)
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158.42.28.33 On: Thu, 18 Dec 2014 09:45:35no vortex core reversal. There is a large region with mymag-
netization (and small mx) in the center of the disk. This
region, together with the vortex core, creates a low resistance
path for the electronic current, decreasing the equivalent re-sistance toward a value below the equilibrium resistance.
Figures 4(c)–4(e) show the magnetization and current distri-
butions at different moments of the vortex core magnetiza-tion reversal for a situation where there is a single reversal
(l
0H¼64 mT). In Fig. 4(c) we depict the current distribu-
tion at the exact moment of nucleation of the V/C0-AV/C0pair,
the initial stage of vortex core magnetization reversal. Figure
4(d) shows the spin waves emitted just after the Vþ-AV/C0
annihilation, which is a process that occurs with energy dissi-
pation. Such energy loss drives the vortex core to the disk
center along with some small oscillations, mainly due to the
reflections of spin waves at the edges of the disk. It turns outthat the resistance follows an equivalent behavior: itdecreases toward the initial resistance value and remains
always above it. Figure 4(e) shows the current distribution
after the field pulse has vanished. As can be seen, the time
dependent resistance curves can give us an indication of thevortex reversal process.
Let us discuss in further detail the interplay between the
magnetization pattern and the current distribution. In Fig. 5( a )
we show a snapshot of the current distribution during the vor-
tex core magnetization reversal process, with the V
þand the
V/C0-AV/C0pair with negative polarity. As shown in Figs. 3,4,
and 5(b) the current is pushed to the vortex core, where
mx¼0 and, consequently, the local resistance is minimum.
With the nucleation of the AV/C0vortex [Fig. 5(a)],mxbecomes
larger than zero around it, with my!0. As the current flows
in the ^xdirection, it is repelled from the antivortex core.
In the latter analysis we considered a particular orienta-
tion of the AV. However, as can be seen in Figs. 5(c) and
5(d), depending on their orientation, antivortices can either
attract (in the first case) or repel currents (in the latter case).Vortices are rotation invariant, and always attract current
toward their cores. It is important to point out that this differ-
ence in current distributions might have important conse-quences in the high-density current spin-torque transfer
acting on either a vortex or an antivortex. For instance,
although the inversion process through spin-torque for anAV is equivalent to the one for a V, we should expect differ-
ent current densities in each one, since currents can only pen-
etrate the AV core at a particular orientation.
IV. SPIN-TORQUE TRANSFER
In this section, we discuss the consequences of inhomo-
geneous currents in the spin-torque transfer. In order to
determine how the current distribution is incorporated in the
spin-torque terms of the modified LLG equation, we need toreview a few steps of their derivations. It is important to note
that in our approach, the only source of nonhomogeneous
FIG. 4. (Color online) (a) Evolution of magnetoresistance after the applica-
tion of pulsed in-plane magnetic fields (the shape is shown in gray) with dif-
ferent intensities. Snapshots of magnetization (arrows) and current distribution
(color map) for pulse fields (b) without and (c)–(e) with (a vortex core mag-
netization reversal. The white cross shows the position of the disk center.
Both the current and the magnetic field are applied in the ^xdirection.
FIG. 5. (Color online) (a)–(d) Snapshots of magnetization (arrows) and cur-
rent distribution (color map) of (a) the vortex core magnetization reversalprocess at t¼214 ps, (b) a vortex and its associated current distribution,
and (c) and (d) antivortices rotated by 45
/C14with respect to each other and
their associated current distribution. Depending on the relative orientation of
the antivortex it can either focus (c) or repel (d) currents away from the cen-
ter of the core.093904-4 Machado et al. J. Appl. Phys. 109, 093904 (2011)
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158.42.28.33 On: Thu, 18 Dec 2014 09:45:35current distribution is the anisotropic magnetoresistance, as
discussed in Sec. II. All other effects are neglected.
The itinerant electron spin operator satisfies the continu-
ity equation
d
dthsiþr/C1h bJi¼/C0i
/C22hðh½s;H/C138iÞ (3)
where Jis the spin current operator. The Hamiltonian, H,i s
the s-d Hamiltonian ( Hsd¼/C0 Jexs/C1S), where sand
S=S¼/C0M=Msare the spins of itinerant and localized elec-
trons, and Jexis the exchange coupling strength between
them. We define the spin current density, J¼h^Ji¼
/C0ðglBP=eM sÞjeðrÞ/C10M,w h e r e jeðrÞis the current density,
and the electron spin density is given by m¼hsi.14We use
the same approximations previously used to calculate the spin-torque,
14,15with the new ingredient of nonhomogeneous cur-
rent density. We obtain
d
dtm¼lbP
eM sMr/C1jeðrÞ ½/C138 þ jeðrÞ/C1r½/C138 fg
/C0JexS
Msm/C2M; (4)
where Mis the matrix magnetization, gis the Lande ´ factor
splitting, lBis the Bohr magneton, Pis the spin current
polarization of the ferromagnet, and eis the electron charge.
From the continuity equation for charges, the term contain-
ingr/C1jeðrÞis always zero, even if the current density is not
constant. As we discussed previously, the same divergent is
used to determine the current distribution in Sec. II. This
expression is exactly the same expression obtained previ-ously, but with j
eðrÞin the second term of the right hand side
of the equation varying with r. This current distribution is
introduced at the modified LLG that considers spin-torquetransfer. Therefore, we obtain a spin-torque transfer where
the current distribution is not uniform.
To consider the spin-torque transfer effects we include
adiabatic and nonadiabatic spin torque terms in the LLG
equation,
d
dtm¼/C0c0m/C2Heffþam/C2d
dtm/C0ðu/C1r Þm
þbm/C2½ ðu/C1r Þm/C138; (5)
where, m¼M/Msis the normalized local magnetization, ais
a phenomenological damping constant, c0is the gyroscopic
ratio, and Heffis the effective field, which is composed of
the applied external field, the demagnetization field, the ani-
sotropy field, and the exchange field. The first term describes
the precession of the normalized local magnetization aboutthe effective field. The second term describes the relaxation
of the normalized local magnetization and bis a dimension-
less parameter that describes the strength of the nonadiabaticterm, which we consider to be 0.5.
15,16The velocity
u(r)¼ðgPlB=2eM sÞje(r) is a vector pointing parallel to the
direction of the electron flow and je(r) is calculated using the
procedure discussed in Sec. II.
To explain the importance of our assumption about the
current distribution let us analyze the critical current density,jc
e, which is the minimal current density needed to produce a
vortex core reversal. For this purpose, we simulated the mag-
netization dynamics of a system subjected to a DC current
with the modified LLG equation [Eq. (5)]. In Fig. 6(a) one
sees the vortex core polarity as a function of current density,
je. The different curves represent situations of homogeneous
current (squares) and three different values of AMR wherethe magnetoresistance ranges from 2 to 10 %. Such AMR, as
discussed in the previous sections, determines the degree of
current inhomogeneity throughout the disk. One can see thatthe critical current density, j
c
ein our model is 3 /C010%
smaller than the one obtained for uniform currents. These
results suggest a new route, together with the nonadiabaticterm, to explain the discrepancy between the experimental
results and theoretical calculations of the critical current den-
sity,j
c
e.
Even though the value of jc
eis reduced, the vortex-core
reversal process for nonhomogenous current distributions is
similar to the homogenous case. Figures 6(b) and6(c) show
the magnetic configurations for the moment just before the
Vþ-AV/C0annihilation, at the critical current density, for the
model with nonhomogeneous and homogeneous current dis-tributions, respectively. In the case of inhomegeneous cur-
rents, the fact that a V(AV) attracts (repels) the current
affects the velocity and separation distance of the V-AV pairduring the vortex core reversal. As the vortices attract cur-
rent, the current densities at their core are higher than the av-
erage and can reach values higher than j
c
eof the homogenous
case. As a result, the vortex gains the necessary velocity to
produce the vortex core switching for a lower jc
e.
FIG. 6. (Color online) (a): Core polarity as a function of current density for
homogeneous (squares) and nonhomogeneous current distributions (with dif-
ferent MR). (b) and (c): Magnetization profiles during the inversion process
at the critical current for both (b) nonhomogeneous and (c) homogeneous (c)
current distributions. The color map represent the out-of-plane magnetiza-
tion, mz, and the arrows represent the in-plane component.093904-5 Machado et al. J. Appl. Phys. 109, 093904 (2011)
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158.42.28.33 On: Thu, 18 Dec 2014 09:45:35Alternatively, as the antivortices repel currents, the current
density at their core is smaller, making them slower thanthe
vortices. As a result, after the nucleation of the V/C0-AV/C0
pair, their separation occurs faster than in the case where the
current density in the center of a V or an AV is the same, as
is usually considered in micromagnetic simulations. Conse-quently, not only is j
c
ereduced, but the inversion time is also
reduced.
Our analysis might also have important technological
implications, since we observe a (almost linear) correlation
between the current density necessary to produce a core
inversion and the anisotropic magnetoresistance of the mate-rial. Thus, by increasing the AMR of the sample, one can
decrease the critical current, j
c
e, which is strongly desirable in
memory devices for the sake of low energy comsumptionand minimal heat waste.
V. CONCLUSIONS
We performed a realistic calculation of the magnetore-
sistance effects in magnetic nanostructures that takes into
account inhomogeneous current densities. For that purpose,we adapted a numerical relaxation scheme for the Laplace
equation to the solution of the LLG equation for the magnet-
ization profile along a Permalloy disk. Our results suggestthat resistance measurements might be useful to probe the
dynamics of the vortex core magnetization reversal, induced
by short in-plane magnetic pulses. Moreover, we note thatthe difference between current distributions close to the vor-
tices and anti-vortices have significant consequences for the
spin-torque transfer effect. The inhomogeneous current dis-tribution inside the magnet substantially reduces the critical
current density necessary to produce a vortex core reversal.
We conclude that materials with large anisotropic magneto-resistance need lower current densities to modify their mag-
netic structure, a much desired feature for most modern
memory devices.
ACKNOWLEDGMENTS
This work was supported by CNPq and FAPERJ. L.C.S.
and T.G.R. acknowledge the ‘‘INCT de Foto ˆnica’’ and
‘‘INCT de Informaca ˜o Qua ˆntica’’, respectively, for financial
support.
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158.42.28.33 On: Thu, 18 Dec 2014 09:45:35 |
1.3537906.pdf | Level Density Calculation using Collective Enhanced Parameter on Several Deformed
Light Nuclei
Y. S. Perkasa, A. Waris, and R. Kurniadi
Citation: AIP Conference Proceedings 1325, 236 (2010); doi: 10.1063/1.3537906
View online: http://dx.doi.org/10.1063/1.3537906
View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1325?ver=pdfcov
Published by the AIP Publishing
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128.184.220.23 On: Tue, 11 Aug 2015 00:59:15Level Density Calculation using Collective Enhanced
Parameter on Several Deformed Light Nuclei
Y. S. Perkasa, A. Waris, R. Kurniadi
Nuclear Physics Research Group,
Faculty of Mathematics and Natural Sciences,
Institut Teknologi Bandung
Email: rijalk@fi.itb.ac.id
Abstract. Level density parameter (LDP) has been provided t o calculate level density on several deformed light nuclei.
These LDP are calculated using collective enhanceme nt including rotational and vibrational ground stat e modes at
neutron binding energy and obtained by equidistant method of collective coupled state bands each nucle us [1]. Level
density from these calculation has been compared wi th other result from different model and found many agreement in
their values..
Keywords: Level density, Level density parameter (LDP), colle ctive enhancement.
PACS: 21.10.Ma
INTRODUCTION
Level density has been evaluated for many several
years ago since bethe formulated the first expressi on for
level density based on Fermi gas theory that assume s
neutrons and protons could fill up higher levels at any
excitation energy. Since then, several model of lev el density
calculation has been formulated based on Fermi gas model
such as Constant Temperature model [2] that divide energy
range into low energy part and high energy part whe re they
are separated by matching energy E M . Other model such as
Back-shifted Fermi Gas model (BFM) [7] where Fermi gas
formulation is used in all energy range also taken into
account.
These model above considered to be the simple model
due to lack of nuclear interaction such as shell ef fect, pairing
effect, deformation and finite size effect, etc. In this work,
level density would be calculated using collective
enhancement from vibrational and rotational effect where
fermi gas model couldn’t describe first excited lev els that
result from coherent excitation of fermion. This
enhancement would be treated through level density
parameter.
FERMI GAS MODEL
Fermi gas model had two basic assumption,
first, excited levels from single particle states a re
equally spaced (equidistant), second, collective le vels
are negligible. Formulation that could be the first point in deriving level density is Fermi gas state densit y for
two fermion system
( )[ ]
4 / 54 / 12exp
12UaaUExtot
Fπω = (1)
Where U defined as
∆− =xEU (2)
Level density parameter from above equation could b e
obtained through
( )ν ππgga + =62 (3)
Fermi level density derived under assumption that t he
projection of total angular momentum are randomly
coupled [6]
( )( ) [ ]
+−+=Π4 / 54 / 1 22
32exp
1222 / 1exp
2212
21,,
UaaU J JJExFπ
σ σπρ(4)
Total Fermi gas level density defined as sum of all
spins and parities
( )[ ]
4 / 54/ 12exp
1221
UaaUExtot
Fπ
σπρ = (5)
and related to the total fermi gas state density th rough
( )()
σπωρ
2xtot
F
xtot
FEE= (6)
Level density parameter in this model could obtaine d
from experimentally parameterized formulation
( )∑+=
−Π =2 / 1
2/ 1 0,,1IJ
IJnFJSDρ (7)
236
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128.184.220.23 On: Tue, 11 Aug 2015 00:59:15Where 0D is the average s-wave level spacing at the
neutron separation energy nS that obtained from s-
wave resonances experiment and I is the spin of
target nucleus.
CONSTANT TEMPERATURE MODEL
Constant Temperature Model (CTM) divide
energy range into two parts, low energy part from 0
MeV to matching energy ME and high energy part
above ME.
() ()
( )Mxxtot
FMxxtot
Txtot
EEEEEEE
≥ = ≤ =
,,
ρρ ρ (8)
Constant Temperature law is applied for energy belo w
ME while Fermi gas model applied for energy above
it.
( ) ( ) ( )
( )Mx xFMx xtot
TxF x
EE JEEEEJERJE
≥ Π = ≤ =Π
,,,, ,21,,
ρρ ρ (9)
Where
( )( )
+−+=22
222 / 1exp212,σ σJ JJERxF (10)
First discrete levels exponential form of constant
temperature law lead to the formulation of total le vel
densities of constant temperature part
( )()
−= =TEE
TdEEdNEx
xx
xtot
T0exp1ρ (11)
Level density for fermi gas part could be obtained
using continuinity principles at the matching energ y
point. This effort lead to the two equation with th ree
unknown variables
() [ ]Mtot
M ETTEE
Fρln0 − = (12)
( )M
xtot
FEdEd
Tρln1= (13)
Solution to this set of equation require another
constraint, that is tot
Tρ should obey with
( ) ∫+ =U
LE
Extot
x LU EdENN ρ (14)
and finally yields
( ) 0 exp exp exp = − +
−
−
ULL U M
Mtot
F NNTE
TE
TEETρ (15)
BACK SHIFT FERMI GAS MODEL
BFM used to implement Fermi gas expression for
all energy range and pairing energy treated as
adjustable parameter
( )[ ]
4 / 54 / 12exp
1221
UaaUExtot
Fπ
σπρ = (16) ( )( ) [ ]
4/ 54 / 1 22
32exp
12 22 / 1exp
2212
21,,
UaaU J JJExFπ
σ σπρ
+−+=Π(17)
Total level density for BFM is defined as
( )( ) ( )1
01 1−
+ =tEE
xtot
Fxtot
BFMρ ρρ (18)
while level density for each state reads
( )( )( )xtot
BFM xBFM EJ JJE ρσ σρ
+−+=Π22
222 / 1exp212
21,, (19)
COLLECTIVE ENHANCEMENT IN
LEVEL DENSITY
Effect of collective enhancement can be seen at
deformed Fermi gas level density through intrinsic
level density
( ) ()() ( )Π =Π ,, ,,int, , JEEKEKJExFxvibxrot xdefF ρ ρ (20)
Where `rotKand vibK are rotational and vibrational
factor respectively.
Vibrational factor is approximated by [5]
( ) [ ]tUS Kvib / exp δ δ− = (21)
Where Sδ and Uδ are changes in entropy and
excitation energy. Another expression [4] could be
used for liquid drop model.
Rotational factor has strong influence than vibrati onal
one and depends on nuclear shape or deformation
factor. However, this factor is vanished at higher
excitation energy, hence the final formulation of l evel
density should contains damping factor.
( ) ()() ( )() ( ) [ ]
( ) ( ) ( ) Π = Π +Π −=Π
,,,, ,, 1,,
int,, int,
JEEKEKJEEfJEEKEf JE
xFxvibxrotxdefFx xFxvibx x
ρρ ρ ρ
(21)
Where
() [ ]() ( )1 . 1 1 max2+ − =⊥ x xrot Ef EK σ (22)
( )
−+=
. ...
exp11
sg
colsg
colxx
dEEEf (23)
This collective enhancement could be easily applied to
all level density model such as described above.
RESULTS AND DISCUSSIONS
Level density parameter that required to calculate
level density on some deformed nuclei are obtained
from [3] where it had been refitted to get the line ar
form. This linearized LDP originated from Neutron
Resonance Data (NRD). Level density model that used
in all calculation is Back-Shifted Fermi Gas (BFM).
TABEL 1. Level density parameter for some
237
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Nucleus NRD BSFG Collective
Ca44
20 6.34 5.93 6.93
Ti48
22 6.93 5.67 7.02
Cr54
24 6.96 5.73 8.01
Zn68
30 9.75 8.42 8.74
Se78
34 11.88 10.25 12.24
Ca-44 Level Density
05001000150020002500300035004000
0 2 4 6 8 10 12
Excitation (MeV)LDNRD
Coll
BFSG
FIGURE 1. Total level density of Ca-44
Ti-48 Total Level Density
0100200300400500600700800900
0 2 4 6 8
Excitation (MeV)Level DensityNRD
Coll
BSFG
FIGURE 2. Total level density of Ti-48 Cr-54 Total Level density
0100002000030000400005000060000
0 2 4 6 8 10 12 14
Excitation (MeV)Level DensityNRD
Coll
BSFG
FIGURE 3. Total level density of Cr-54
Zn-68 Total Level Density
05001000150020002500300035004000
0 1 2 3 4 5 6 7 8
Excitation (MeV)Level DensityNRD
Coll
BSFG
FIGURE 4. Total level density of Zn-68
Se-78 Total Level Density
01000200030004000500060007000
0 1 2 3 4 5 6 7
Excitation (MeV)Level DensityNRD
Coll
BSFG
FIGURE 5. Total level density of Se-78
It is clear from above figures that level density
value had very strong dependence on LDP. In most
cases, level density that calculated using collecti ve
enhanced factor is higher than the other while leve l
density from BSFG LDP model is much lower
compared to NRD data. This lower factor of BSFG
238
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128.184.220.23 On: Tue, 11 Aug 2015 00:59:15LDP model could be occurred due to usability of fer mi
gas formulation in all energy range.
Collective enhancement level density could be
recalculated to achieve better approximation especi ally
at higher energies by considering adjustable
deformation parameter and approximated value of
damping function parameters. This damping function
parameters are not reproduced well and had to be
refitted due to its importance on rotational factor at
higher energy.
The discrepancies at higher energy range could also
occurred because collective LDP parameter that used
in this calculation is obtained from equidistant mo del
on lower energy range only [1].
CONCLUSIONS
Collective enhanced level density calculation
at higher energy range require more advanced
treatment to achive better approximation with data
from NRD especially at deformation and collective
damping factor due to its importance on rotational
effect.
ACKNOWLEDGMENTS
This research is supported by Riset Kelompok
Keahlian ITB No kontrak : 239/K01.7/PL/2010.
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5.0004649.pdf | Appl. Phys. Lett. 116, 192407 (2020); https://doi.org/10.1063/5.0004649 116, 192407
© 2020 Author(s).Linear dependence of skyrmion velocity
on response resonance frequency of local
magnetization
Cite as: Appl. Phys. Lett. 116, 192407 (2020); https://doi.org/10.1063/5.0004649
Submitted: 13 February 2020 . Accepted: 04 May 2020 . Published Online: 13 May 2020
Lingwen Kong , Lan Bo , Rongzhi Zhao , Chenglong Hu , Lianze Ji , Yanhui Zhang , and Xuefeng Zhang
Linear dependence of skyrmion velocity on
response resonance frequency of local
magnetization
Cite as: Appl. Phys. Lett. 116, 192407 (2020); doi: 10.1063/5.0004649
Submitted: 13 February 2020 .Accepted: 4 May 2020 .
Published Online: 13 May 2020
Lingwen Kong,1LanBo,1Rongzhi Zhao,1,2Chenglong Hu,1,2Lianze Ji,1,2Yanhui Zhang,1,3and Xuefeng Zhang1,2,a)
AFFILIATIONS
1Key Laboratory for Anisotropy and Texture of Materials (MOE), School of Materials Science and Engineering, Northeastern
University, Shenyang 110819, China
2Institute of Advanced Magnetic Materials, College of Materials and Environmental Engineering, Hangzhou Dianzi University,Hangzhou 310012, China
3State Key Laboratory of Rolling Technology and Automation, Northeastern University, Shenyang 110819, China
a)Author to whom correspondence should be addressed: zhangxf@atm.neu.edu.cn
ABSTRACT
Spin waves (SWs) have been proven effective in driving the magnetic skyrmion motion, while the physical correlation between skyrmion
velocity and the resonance frequency of local magnetization remains unknown. Here, we theoretically investigate the skyrmion motion in amagnetic Co/Pt nanotrack with the perpendicular magnetic anisotropy, which is driven by SWs. The results show that magnetic skyrmions
move along the propagation direction of SWs in a specific frequency range (50–175 GHz). It is evidenced that there is a linear relationship
between the response resonance frequency ( f
r) of local magnetization and the skyrmion velocity (v), and the motion of skyrmions could also
be manipulated by controlling the amplitude and location of the exciting source. The present study provides a fundamental insight intounderstanding the intrinsic physics of SW-driven skyrmion-based devices.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0004649
Magnetic skyrmions were proposed theoretically by Tony
Skyrme in 1962 and discovered experimentally by neutron scatter-
ing in 2009.
1,2They are spin magnetic nanostructures with topolog-
ical protection and generally exist in chiral magnetic materials with
the Dzyaloshinskii–Moriya interaction (DMI).3–5Due to their
topological protection, small size, and easy manipulation, sky-
rmions are promising in applications of high-density, low-energy
consumption, non-volatile computing, and memory devices.6–11In
this context, various approaches have been devoted to developing
skyrmion-based devices by applying external triggers.12–16For
example, spin-polarized current has been proven effective for themanipulation of skyrmions because it is relatively accessible to inte-
grate with the existing semiconductor devices.
7,14–20Although
having remarkable achievements,8,21,22there is still an unexpected
generation of Joule heat, which could increase the energy loss and
destroy the stored information or even the device. In order to over-
come such an issue, spin waves (SWs) have been recently proposed
to drive skyrmion motion, ascribed to the momentum exchange
between magnons and skyrmions.23–31Under the excitation of SWs, the velocity of skyrmions is found
to increase in the initial stage and then decrease.23,28Such a phenome-
non could be explained by the Thiele equation, which treats skyrmions
as rigorous particles and ignores the inner freedom of skyrmions.
However, the inner freedom plays an important role in the magnetiza-
tion dynamics. For example, low energy resonance modes are observed
on the annular domain in the process of center-of-mass motion of
skyrmions.32In this work, we investigated the relationship between
the local resonance modes of magnetization and the velocity change
by analyzing the SW-driven skyrmion motion in a perpendicular mag-
netic anisotropy of the Co/Pt film.33,34It is found that there is a linear
relationship between the response resonance frequency ( fr)o fl o c a l
magnetization and the skyrmion velocity. Furthermore, the motion of
skyrmions could also be manipulated by changing the amplitude and
location of the exciting source, mediated by the control of the response
resonance frequency ( fr).
Our micromagnetic simulation is performed using the Object
Oriented Micromagnetic Framework (OOMMF)35software contain-
ing the extension module of the interfacial DMI.36The magnetization
Appl. Phys. Lett. 116, 192407 (2020); doi: 10.1063/5.0004649 116, 192407-1
Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldynamics can be obtained by solving the Landau–Lifshitz–Gilbert
equation,37
dM
dt¼/C0 cM/C2Heff/C0aM/C2dM
dt/C18/C19
; (1)
where cis the gyromagnetic ratio, ais the Gilbert damping constant,
Mis the magnetization vector, and Heffi st h ee f f e c t i v efi e l d ,w h i c h
includes exchange, demagnetization, magnetic anisotropy, DMI, andexternal applied magnetic fields. The DMI energy in a continuous
magnetization model is expressed as
e
DM¼Dm z@mx
@x/C0mx@mz
@x/C18/C19
þmz@my
@y/C0my@mz
@y/C18/C19"#
;(2)
where mx,my,a n d mzare components of normalized magnetization
and D is the continuous effective DMI constant. As shown in Fig. 1 ,a
skyrmion with a diameter of 10 nm is located in a racetrack with alength of 1000 nm, a width of 40 nm, and a thickness of 0.4 nm, which
is stable under zero field due to the strong perpendicular magnetic
anisotropy and the competition between exchange energy and DMIenergy. Notice that the canted spins induced by DMI at the boundaries
could exert a repulsive force on the skyrmion, which helps to stabilize
the skyrmion in the nanotrack. The excitation source is located atd
s¼150 nm away from the skyrmion and 20 nm-width (see the yellow
region in Fig. 1 ), where SWs are excited by an ac magnetic field along
the y-direction, Hy¼Hmsin(2 pft), with Hmbeing the amplitude and f
t h ef r e q u e n c yo ft h ea cm a g n e t i cfi e l d .T h ef r e q u e n c yo fe x c i t e dS W si s
consistent with the frequency of the ac magnetic field because SWs are
synchronously excited by the microwave magnetic field.38,39The
material parameters we used correspond to the Co/Pt film:24,33,34the
saturation magnetization Ms¼5.8/C2105A/m, the exchange stiffness
constant A¼1.5/C210/C011J/m, the interfacial DMI constant
D¼3.0 mJ/m2, the perpendicular magnetic anisotropy constant
Ku¼8.0/C2105J/m3, and the Gilbert damping constant a¼0.02.
The dipolar coupling becomes local in the zero-thickness limit40for
ultrathin films, which could be seen as the shape anisotropy40,41by
K¼Ku/C01
2l0Ms2in analytic methods. The mesh size is 1 /C21/C20.4 nm3,
w h i c hi sl e s st h a nt h ee x c h a n g el e n g t h , lex¼ffiffiffiffi ffi
A
Kuq
¼4.3 nm.23,42In orderto avoid the reflection of SWs at ends of the nanotrack, we set up
two 10 nm-width buffer zones (see light-green and deep-blue regions in
Fig. 1 ), and damping coefficients are set to 0.5 and 1 from inside to
outside, respectively.
We first study the propagation characteristics of SWs in the race-
track without skyrmions by the sinc excitation field Hy¼Hmsin(2 pft)/
(2pft), with Hm¼200 mT and f¼85 GHz. The FFT of this excitation
field is a rectangular function, which is used to excite a range offrequencies and calculate the propagation properties of SWs in thenanotrack. The SW spectrum of the nanotrack is shown in Fig. 2(a) .
In order to avoid the influence of the edge on the spectrum, we average
the magnetization in the y-direction. Because the magnetization of the
nanotrack is out of plane and the exciting magnetic field is in plane,i.e., the angle between the magnetization and the field is p/2, the asym-
metry of the dispersion vanishes.
43,44It can be seen from the disper-
sion in Fig. 2(a) that only when the frequency f>56 GHz, SWs can
propagate in the nanotrack.27Based on this, we study the motion of
skyrmions driven by SWs with frequencies of 55 GHz, 85 GHz, and115 GHz. We fix the amplitude H
m¼200 mT, which is sufficient to
excite SWs without destruction of the skyrmion texture,45as shown in
Fig. 2(b) . Due to the boundary force caused by the narrow nanotrack,
the skyrmion does not move along the y-direction but only in the x-
direction.24,46It can be seen that the motion of skyrmions shows a
trend of gradual increase and subsequent decrease with the increase intime, associated with the maximum velocity of 17 m/s at 3.8 ns at85 GHz. The position of skyrmions is calculated by tracing the center
of the circular domain wall (m
z¼0). The instantaneous velocity is
computed by v ¼Ds/Dt, where Dt is the time interval ( Dt¼0.005 ns)
andDs is the distance of motion in this interval. Furthermore, the
motion of the skyrmion in the nanotrack is investigated in the fre-
quency range of 0 – 200 GHz, as shown in Fig. 2(c) . It is indicated that
only when the frequency of the exciting field is in the range from50 GHz to 175 GHz, SWs can drive the skyrmion motion
24and the
a v e r a g ev e l o c i t yr e a c h e st h ep e a k1 7m / sa t8 5 G H z .T h ea v e r a g evelocity (
v) is also calculated by v¼S/T, where S and T are the total
distance and total time (20 ns) of the skyrmion motion, respectively.
FIG. 1. (a) Schematic of the micromagnetic model: a skyrmion is driven to move
along the racetrack under the excitation of SWs. (b) The enlarged view of sky-rmions in the racetrack. Color scale: out-of-plane component (m
z) of magnetization.
(c) The ac magnetic field imposed to excite SWs.
FIG. 2. (a) The SW spectrum of the racetrack without skyrmions. (b) The instanta-
neous velocity as a function of time at different SW frequencies (55 GHz, 85 GHz,and 115 GHz). (c) The average velocity of skyrmions as a function of SWfrequency.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 116, 192407 (2020); doi: 10.1063/5.0004649 116, 192407-2
Published under license by AIP PublishingWe show the position of skyrmions at selected times, as shown in
Fig. 3(a) . The skyrmion moves to the end of the racetrack after 20 ns at
85 GHz due to the large average velocity (10 m/s). The response reso-
nance frequency ( fr) is calculated by performing the FFT of m zin each
mesh of the nanotrack, where the frequency with the maximum
amplitude is selected in the centerline (y ¼20 nm). The instantaneous
velocity of skyrmions as a function of the position is presented in
Fig. 3(b) at different frequencies (55 GHz, 70 GHz, 85 GHz, 100 GHz,
and 115 GHz). It is found that the response resonance frequency ( fr)o f
local magnetization increases gradually with the motion and reaches
the peak value when the velocity is maximum. Thus, it is inferred that
the change in velocity is dominated by the response resonance fre-quency (f
r)of local magnetization in the racetrack. The stronger the
resonance frequency, the greater the velocity of the skyrmion. At the
end of the movement [marked by the circle in Fig. 3(b) ], the deviation
between velocity and the resonance frequency could originate from
the inertia of skyrmions.28InFig. 3(c) , a damping oscillation of the
skyrmion diameter is observed from 0 ns to 0.5 ns to reach an equilib-
rium state.47However, the skyrmion diameter remains constant in the
process of motion, which has no influence on the skyrmion motion.
In order to further clarify the relationship between the velocity
(v) and the response resonance frequency ( fr), we analyze the motion
process at 85 GHz. The snapshots of skyrmion motion are shown in
Fig. 4(a) , which is divided into two parts: the increase in velocity from
the initial position to x ¼46 nm and the decrease in velocity from
x¼46 nm to the end. Figures 4(b) and4(c)show the velocity (v) and
the response resonance frequency ( fr) of skyrmions as a function of
the position, respectively. It can be seen that in the initial stage, the
velocity gradually increases and reaches the peak value v ¼17.3 m/s at
x¼46 nm. The response frequency ( fr) increases gradually with the
increase in the moving distance, which reaches the peak value of
2.1 GHz at x ¼46 nm and then decreases gradually until the end of the
skyrmion motion. There is a similar trend between the velocity (v) andresonance frequency ( f
r)a ss h o w ni n Figs. 4(b) and4(c), and the veloc-
ity (v) as a function of the resonance frequency ( fr)i ss h o w ni n
Fig. 4(d) . The relationship could be obtained by a linear fitting, and each
frcorresponds to a specific velocity according to v ¼/C02.95þ8.63/C2fr(fr>0.342 GHz). In other words, the motion velocity of skyrmions
could be manipulated by controlling the resonance frequency ( fr).
The velocity of skyrmions could also be manipulated by adjusting
the position of the exciting source ( ds). We fix the amplitude of the
exciting field Hmto 200 mT, and the distance of motion increases
gradually with the decrease in dsinFig. 5(a) . The maximum of velocity
for each dsalso increases from 7 m/s to 28 m/s with the decrease in ds
f r o m 2 0 0n m t o 5 0n m , a s s h o w n i n Fig. 5(b) . The skyrmions could
move at a constant velocity of 7 m/s when ds¼200 nm. Because the
intrinsic damping of the nanotrack induces the decay of SWs with
the increase in the propagation distance (as shown in Fig. S1 in the
supplementary material ), the velocity of skyrmions decreases with the
increase in damping coefficients. When the damping is fixed, the sky-rmion velocity (v) and the resonance frequency ( f
r) still satisfy a linear
relationship, as shown in Fig. S2 in the supplementary material .T h e
response resonance frequency ( fr) of local magnetization along the
path of movement is shown in Fig. 5(c) , and the trend of resonance
frequency is consistent with the change in velocity as demonstrated in
Fig. 3 . The average velocity ( v) decreases from 16 m/s to 7 m/s with
the increase in dsfrom 50 nm to 200 nm as plotted in Fig. 5(d) .
Similarly, we fit the average velocity curve of skyrmions, presenting
the linear relationship between the average velocity ( v) and ds,
FIG. 3. (a) The position of skyrmions at selected times at different frequencies of SWs (55 GHz, 70 GHz, 85 GHz, 100 GHz, and 115 GHz). (b) The response resonance fre-
quency ( fr) of local magnetization and velocity as a function of the position. (c) The skyrmion diameter as a function of time.
FIG. 4. Analysis of the skyrmion motion process. (a) The snapshots of the skyrmion
motion within 20 ns at 85 GHz. (b) The velocity v and (c) response resonance fre-quency f
rof skyrmions as a function of the position. (d) The velocity of skyrmions
as a function of the response resonance frequency.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 116, 192407 (2020); doi: 10.1063/5.0004649 116, 192407-3
Published under license by AIP Publishingexpressed by v¼18.58–0.057 /C2ds(ds>0). These results demonstrate
that the manipulation of velocity by the position of the exciting sourcecould also be realized, which is mediated by controlling the resonance
frequency of ( f
r) local magnetization.
In addition, the amplitude of the exciting field could have a dra-
matic impact on the motion of skyrmions. As shown in Fig. 6(a) ,t h e
distance of movement increases gradually with the decrease in Hm
when the frequency of SWs is fixed at 85 GHz. The maximum velocity
of the skyrmion increases from 12 m/s to 22 m/s with the increase in
Hmfrom 150 mT to 250 mT, as shown in Fig. 6(b) . The response reso-
nance frequency ( fr) of local magnetization along the path of move-
ment is shown in Fig. 6(c) . The increase in excitation amplitude
enables skyrmions to obtain the enhanced driving force through
momentum exchange, which increases the velocity of skyrmions at the
macro-level. The average velocity ( v) also increases from 5 m/s to
15.5 m/s with the increase in Hmfrom 150 mT to 250 mT, as plotted
inFig. 6(d) . We fit the average velocity curve of skyrmions, presenting
the linear relationship between the average velocity ( v) and Hm,
expressed by v¼/C01.19þ0.048/C2Hm(Hm>25 mT). When v¼0,
we get the critical amplitude Hm¼25 mT, which is consistent with the
reported work.24
Although our work is focused on N /C19eel skyrmions, similar results
could be expected in Bloch skyrmions and anti-skyrmions because a
similar momentum transfer is also observed in Bloch skyrmions underthe action of SWs.
25For anti-skyrmions, only the topological number
Qantiis opposite to skyrmions Q sky(Qanti¼/C0Qsky), which induces an
opposite chirality motion48and has no influence on the momentum
transfer.In summary, we theoretically investigated the skyrmion motion
in the magnetic Co/Pt nanotrack with the perpendicular magnetic
anisotropy (PMA), which is driven by SWs. We found that skyrmions
only move in a specific frequency range of SWs (50 – 175 GHz), and
the velocity (v) is linearly correlated with the response resonance fre-
quency ( fr) of local magnetization in the nanotrack. In addition, the
average velocity ( v) can be manipulated by the amplitude and location
of the exciting fields. Our results evidence the physical origin for thetunability of skyrmion velocity under the excitation of SWs, which is
significant for designing the SW-driven skyrmion-based devices.
See the supplementary material for the propagation of SWs in
the nanotrack and the influence of damping constant on the motion.
The authors gratefully acknowledge the National Natural
Science Foundation of China (No. U1704253), the Natural Science
Foundation of Zhejiang Province (No. LR18E010001), and the
LiaoNing Revitalization Talents Program (No. XLYC1807177).
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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Published under license by AIP Publishing |
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Citation: The Journal of Chemical Physics 148, 222834 (2018); doi: 10.1063/1.5023916
View online: https://doi.org/10.1063/1.5023916
View Table of Contents: http://aip.scitation.org/toc/jcp/148/22
Published by the American Institute of PhysicsTHE JOURNAL OF CHEMICAL PHYSICS 148, 222834 (2018)
Quantum chemistry in arbitrary dielectric environments:
Theory and implementation of nonequilibrium Poisson
boundary conditions and application to compute vertical
ionization energies at the air/water interface
Marc P . Coonsa)and John M. Herbertb)
Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, USA
(Received 29 January 2018; accepted 6 April 2018; published online 25 April 2018)
Widely used continuum solvation models for electronic structure calculations, including popular
polarizable continuum models (PCMs), usually assume that the continuum environment is isotropic
and characterized by a scalar dielectric constant, ". This assumption is invalid at a liquid/vapor inter-
face or any other anisotropic solvation environment. To address such scenarios, we introduce a more
general formalism based on solution of Poisson’s equation for a spatially varying dielectric function,
"(r). Inspired by nonequilibrium versions of PCMs, we develop a similar formalism within the con-
text of Poisson’s equation that includes the out-of-equilibrium dielectric response that accompanies
a sudden change in the electron density of the solute, such as that which occurs in a vertical ion-
ization process. A multigrid solver for Poisson’s equation is developed to accommodate the large
spatial grids necessary to discretize the three-dimensional electron density. We apply this methodol-
ogy to compute vertical ionization energies (VIEs) of various solutes at the air/water interface and
compare them to VIEs computed in bulk water, finding only very small differences between the two
environments. VIEs computed using approximately two solvation shells of explicit water molecules
are in excellent agreement with experiment for F (aq), Cl (aq), neat liquid water, and the hydrated
electron, although errors for Li+(aq) and Na+(aq) are somewhat larger. Nonequilibrium corrections
modify VIEs by up to 1.2 eV , relative to models based only on the static dielectric constant, and are
therefore essential to obtain agreement with experiment. Given that the experiments (liquid microjet
photoelectron spectroscopy) may be more sensitive to solutes situated at the air/water interface as
compared to those in bulk water, our calculations provide some confidence that these experiments
can indeed be interpreted as measurements of VIEs in bulk water. Published by AIP Publishing.
https://doi.org/10.1063/1.5023916
I. INTRODUCTION
Fundamental aspects of ion solvation at the air/water
interface have attracted significant attention in recent years,1–7
including investigations of how ion coordination motifs, con-
centrations, and reactivity may differ at the interface versus
bulk water. At the same time, the development of liquid micro-
jet photoelectron spectroscopy has opened the way to experi-
mental measurements of vertical ionization energies (VIEs) of
molecules in solution,8–11,17as opposed to the gas-phase VIEs
afforded by traditional photoelectron spectroscopy. However,
interpretation of solution-phase photoelectron spectra is com-
plicated by the possibility that the ejected electron may be
scattered and/or recaptured by the liquid and thus detected
with reduced kinetic energy or possibly not detected at all.
As such, the microjet experiments are likely more sensitive to
species solvated at the liquid/vapor interface than they are to
the same species in a bulk liquid environment. The wavelength-
a)Present address: The Dow Chemical Company, 1776 Building, Midland, MI
48674, USA.
b)herbert@chemistry.ohio-state.edudependent nature of the electron attenuation length12,13(a mea-
sure of the likelihood that the emitted electron is recaptured)
leads to solution-phase photoelectron spectra that depend on
the wavelength of the photodetachment laser.14,15For these
reasons and others,16theoretical prediction of VIEs in solu-
tion is desirable in order to facilitate the interpretation of
the experiments. From a quantum chemistry point of view,
one may expect a significant polarization response from the
medium upon ionization of the solute, so the question arises
how this effect can be incorporated in a tractable way. A contin-
uum representation of the solvent represents one cost-effective
strategy.
The most common continuum solvation models in quan-
tum chemistry are based upon the framework of the polarizable
continuum model (PCM),19–23in which the continuum sol-
vent’s electrostatic interaction with the atomistic solute is
parameterized in terms of a single, scalar dielectric constant, ".
For accurate solvation energies, nonelectrostatic interactions
must be included as well, but the electrostatic contribution
can still be obtained from a PCM.23–27These models are sim-
ple, efficient, and—assuming nonelectrostatic corrections are
included—reasonably accurate,20,23–27and are therefore wide-
ly used in quantum chemistry. As conventionally formulated,
0021-9606/2018/148(22)/222834/21/ $30.00 148, 222834-1 Published by AIP Publishing.
222834-2 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
however, these models assume that the solvation environment
is isotropic, as appropriate for solvation in bulk liquid but not
at an interface.
There have been some attempts to modify the PCM for-
malism for use in anisotropic environments, including a formu-
lation that uses a dielectric tensor in place of a scalar dielectric
constant.28–31This is useful, e.g., in the case of liquid crystals
where the dielectric “constant” is strongly direction-dependent
and therefore a diagonal tensor with different values "xx,
"yy, and"zzmight afford a reasonable description. Mennucci
et al.32–34and others35,36have developed PCMs designed
for liquid/vapor interfaces by modifying certain matrix ele-
ments of the PCM equations within the interfacial region.
Mennucci et al. used a smooth switching function to inter-
polate between liquid and vapor values of ",32–34as is also
done in the approach presented here. For complete general-
ity, however, an anisotropic continuum environment should be
described theoretically using Poisson’s equation,22not with a
scalar (or tensor) dielectric constant but rather with a spatially
varying dielectric function ,"(r).
A simplified version of such a model, in which "(r) is
replaced by a set of distinct dielectric constants "1,"2,:::in
different spatial regions, was introduced long ago by Saku-
raiet al.37,38and used in semi-empirical electronic structure
calculations.38At its core, this model amounts to solution of
Poisson’s equation in each spatial region, subject to appropri-
ate boundary conditions. In the present work, we introduce an
even more general formalism and computational algorithm in
which the function "(r) is allowed to be completely arbitrary.
It is ultimately defined by the value of "at each point on a
discretization grid.
Other solvers for Poisson’s equation have been reported
recently,39–43including several for use with quantum chem-
istry.40–43What is novel in the present work is the introduction
of nonequilibrium corrections. These account for the response
of the continuum solvent to a sudden change in the electron
density of the solute, such as that which occurs upon (vertical)
ionization.44–47We have previously formulated this nonequi-
librium theory for use with PCMs,48–50and here we make
the appropriate modifications for use with Poisson’s equation.
A preliminary version of this methodology was reported in
Ref. 42, but whereas that formulation was perturbative (follow-
ing along the lines of our group’s previous work on PCMs48),
the present version includes the full solvent response. We have
also made significant improvements to our grid-based Poisson
solver, as compared to the one described in Ref. 42.
This work aims to evaluate the limitations of nonequi-
librium anisotropic Poisson boundary conditions in quantum
chemistry calculations, by comparing to aqueous-phase VIEs
measured using liquid microjet photoelectron spectroscopy.11
Perhaps unsurprisingly, VIEs for atomic ions computed using
nothing but a PCM representation of the solvent afford
extremely poor agreement with experiment;18hence, we will
include explicit water molecules in the atomistic, quantum-
mechanical (QM) region. In PCM calculations, where the
solute cavity that defines the solute/continuum interface is usu-
ally constructed from atom-centered spheres,19–23inclusion
of a large number of explicit solvent molecules sometimes
leads to erratic convergence with respect to the size of theQM region.51This occurs because the dielectric medium arti-
ficially intrudes into the interstices between explicit solvent
molecules, which should properly be characterized by "= 1
since these are part of the QM region. We avoid such artifacts
by using a single, spherical solute cavity around the entire QM
region or alternatively using a novel cavity construction that
is described herein. Finally, we consider whether VIEs com-
puted in bulk water differ appreciably from those obtained
at the air/water interface. This is easily addressed computa-
tionally but less trivial to interrogate experimentally, although
experimental insight might be gained from angle-resolved
photoelectron spectroscopy.52–55
II. NONEQUILIBRIUM POISSON FORMALISM
A. Self-consistent equilibrium solvation
Solution of the gas-phase Poisson equation,
ˆr2'sol(r)= 4sol(r), (2.1)
affords the electrostatic potential 'sol(r) arising from the elec-
tronic and nuclear components of the solute’s charge density,
sol(r). The quantity sol(r) is to be computed from a quantum
chemistry calculation, but we make no assumptions about the
level of electronic structure theory. The quantities 'solandsol
can be partitioned into electronic and nuclear components,
'sol(r)='nuc(r) +'elec(r), (2.2a)
sol(r)=nuc(r) +elec(r). (2.2b)
The solute’s internal energy in a vacuum is
Eint=1
2
dr'sol(r)sol(r) (2.3)
and includes the electron–electron, electron–nuclear, and
nuclear–nuclear interactions. Upon immersion of the solute in
continuum solvent, characterized by a spatially varying dielec-
tric function "(r), the solute–solvent interaction is governed
by the most general form of Poisson’s equation,
ˆrf
"(r)ˆr'tot(r)g
= 4sol(r), (2.4)
where
'tot(r)='sol(r) +'pol(r) (2.5)
includes an induced polarization potential, 'pol(r).
For electronic structure methods using atom-centered
Gaussian basis functions g(r), the electronic contribution to
the electrostatic potential is
'elec(r)= X
P'(r), (2.6)
where Pis the one-electron density matrix and '(r) is the
electrostatic potential generated by the shell pair g(r)g(r),
'(r)=
dr0g(r)g(r0)
jr r0j. (2.7)
Nuclear charges are smeared out using Gaussian functions to
avoid problems with discretizing them onto a grid. The elec-
trostatic potential generated by these Gaussian nuclear charges
is
'nuc(r)=atomsX
Z
jr R jerf jr R jp
2!
, (2.8)222834-3 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
where Z andR are the charge and position of nucleus and
is the standard deviation of the Gaussian, which is an input
parameter to the method.
The solute’s charge density is obtained from 'sol(r)
according to
sol(r)= 1
4ˆr2'sol(r). (2.9)
In this work, ˆr2'sol(r) is computed using an eighth-order cen-
tral finite-difference scheme, as described in Sec. III A. Unlike
our original implementation of Poisson boundary conditions,42
which required the direct evaluation of the electron density
on the real-space grid, the present implementation evaluates
'elec(r) on the grid, via Eq. (2.6), and then computes sol(r)
from Eq. (2.9). We find that the present approach is more robust
with respect to changes in the grid size and spacing.
To solve the Poisson problem in Eq. (2.4), we adapt a
procedure outlined in Refs. 40 and 41 for obtaining the solvent
polarization response, which is characterized by the quantities
'pol(r) andpol(r). Equation (2.4) is first recast as a vacuum-
like Poisson equation,
ˆr2'tot(r)= 4tot(r), (2.10)
where the total charge density is
tot(r)=sol(r) +pol(r). (2.11)
Note carefully the difference between Eq. (2.10) and Eq. (2.1).
The effects of the inhomogeneous dielectric function "(r) are
contained in the polarization charge density pol(r), the form
of which is40,41
pol(r)="1 "(r)
"(r)#
sol(r) +iter(r). (2.12)
The first term on the right side of Eq. (2.12) is the solute charge
density scaled by a dielectric-dependent factor that is only
non-zero outside of the atomistic region (solute cavity), where
">1. The second term iter(r) is a charge density induced by
the inhomogeneous dielectric in regions where it transitions
from"= 1 near the solute molecule to a value appropriate
for bulk solvent outside of the solute cavity. As the notation
implies, this correction is obtained iteratively, and its value at
thekth iteration can be expressed as40,41
(k)
iter(r)=1
4ˆrln"(r)ˆr'(k)
tot(r). (2.13)
Algorithm 1 outlines a procedure for the iterative solu-
tion of Eq. (2.10) to obtain 'tot(r),'pol(r), andpol(r). We
call this the “equilibrium” Poisson-equation solver (PEqS)
method, which we now describe. The quantities 'sol(r) and
sol(r) are initialized using Eqs. (2.2a) and (2.6)–(2.9), then
(k)
iter(r) is computed using Eq. (2.13), and Eq. (2.12) is then
used to generate (k)
pol(r). With the total density now in hand, the
total electrostatic potential is obtained via the numerical solu-
tion of Eq. (2.10) using a multigrid conjugate gradient (CG)
procedure that is described in Sec. III B. This affords '(k+1)
tot(r),
and the iterative part of the charge density is then updated using
Eq. (2.13). However, rather than using this directly to defineAlgorithm 1. Equilibrium PEqS method.
1: Initialize h=0.
2:forj= 1, 2,:::do until SCF error< SCF
3: Diagonalize F(j)=F(j)
0+h(j)to obtain P(j)
4: Compute sol(r) and'sol(r)
5: ifj= 1then
6:tot(r) =sol(r)
7:'tot(r) ='sol(r)
8: else
9:tot(r) =sol(r) +pol(r)
10:'tot(r) ='sol(r) +'pol(r)
11: end if
12: Initialize (0)
iter(r) using'(0)
tot(r)
13: fork= 0, 1,:::dountiliter< PEqS
14: Compute '(k+1)
tot(r)
15: Update (k+1)
iter(r),pol(r), andtot(r)
16: end for
17: Update h(j)
18: Compute E=Eint+Gelst
19:end for
(k+1)
iter(r), we instead use a damping procedure to stabilize the
update between iterations kandk+ 1,
(k+1)
iter(r)=
4ˆrln"(r)ˆr'(k+1)
tot(r)
+(1 )(k)
iter(r). (2.14)
We take= 0.6 as in Refs. 40 and 41. Convergence of the
solvent polarization response is achieved when the residual
iter=
(k+1)
iter(r) (k)
iter(r)
(2.15)
falls below a threshold, PEqS.
Operationally, the solvent polarization response is
included in the QM calculations by augmenting the gas-phase
Fock matrix F0with a correction hto its one-electron part.
This correction has matrix elements
h=
dr'(r)pol(r). (2.16)
Finally, the converged solution of Eq. (2.10) affords a total
energy
E=Eint+1
2
dr'pol(r)sol(r), (2.17)
which consists of the solute’s internal energy Eintfrom the
electronic structure calculation, plus the electrostatic contri-
bution
Gelst=1
2
dr'sol(r)pol(r) (2.18)
to the solvation free energy.
B. State-specific nonequilibrium solvation
To incorporate solvent polarization effects following ver-
tical ionization of the solute, we have adapted the nonequi-
librium solvation approach developed for PCMs,45,48–50,56–58
for use with three-dimensional charge densities rather than
the apparent surface charges used by PCMs. In previous
work, we developed a perturbative approach to correcting
the solute/continuum interaction for a sudden change in the
electronic state of the solute, either electronic excitation or222834-4 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
ionization.48–50The perturbative approach is tantamount to
“freezing” the inertial components of a reference-state solvent
reaction field such that, upon vertical ionization, this frozen
reaction field governs the solvent response to the ionized state’s
charge distribution, which is prevented from relaxing. In the
case of electronic excitation, the perturbative nature of the
correction avoids a state-switching problem that arises in the
case of near-degeneracies when using a state-specific Hamilto-
nian,59and in Ref. 42 we introduced a Poisson equation solver
based on the perturbative approach.
For VIEs, the state-switching problem is not an issue and
in this work we develop a Poisson equation solver based on a
state-specific, nonequilibrium treatment of solvent polariza-
tion, rather than a perturbative approach. Within the state-
specific approach, the final (ionized) state’s charge distribution
relaxes in the presence of the slow inertial component of the
reference-state reaction field as well as the fast noninertial
component of its own reaction field.
For the state-specific method, the solute wave function for
state | iiis obtained by solving the Schr ¨odinger equation
ˆHSS
ij i =ESS
ij i (2.19)
with a state-specific Hamiltonian of the form
ˆHSS
i=ˆHvac
i+ˆVslow
0+ˆVfast
i. (2.20)
Subscripts indicate whether a particular quantity originates
from the equilibrium reference state ( i= 0) or else the nonequi-
librium final state. (For ionization to the electronic ground
state, there is only one possible final state that we will indicate
byi= 1 in what follows.) Superscripts “slow” versus “fast”
in Eq. (2.20) indicate which part of the solvent response is
considered: either the slow inertial part, representing nuclear
degrees of freedom (orientational and vibrational fluctuations
of the solvent), or else the fast electronic part. The quantity
ˆHvac
iis the molecular Hamiltonian that affords the solute’s vac-
uum internal energy Eint,ifor state i, and the operators ˆVslow
i
and ˆVfast
igenerate the indicated components of the solvent
polarization response, i.e., 'slow
pol,iand'fast
pol,i, where
'slow=fast
pol,i(r)=
dr0slow=fast
pol,i(r0)
jr r0j. (2.21)
As in previous work,48–50we use the Marcus partition of
the fast and slow components of the polarization response. (See
Ref. 49 for a comparison to the common alternative, Pekar par-
titioning, with the conclusion that this choice makes essentially
no difference for solvation energies.) Within this approach, the
slow component of the reference-state polarization charge den-
sity,slow
pol,0(r), which affords 'slow
pol,0(r) according to Eq. (2.21),
is computed according to48,49
slow
pol,0(r)= slow
!
pol,0(r), (2.22)
where = slow+ fastis the static susceptibility, partitioned
into slow and fast components,
slow="solv "1
4, (2.23a)
fast="1 1
4. (2.23b)Here,"solvis the static dielectric constant of the solvent and
"1=n2is its optical dielectric constant, where ndenotes the
solvent’s index of refraction. The quantity "1encodes the fast
electronic contribution to the solvent polarization response.
To obtain the fast components of the ionized state’s polar-
ization response, 'fast
pol,1(r) andfast
pol,1(r), within the Marcus
partition,49the Poisson equation is modified such that the total
source-charge density is the ionized solute’s charge density,
sol,1(r) +slow
pol,0(r), and the dielectric function is the optical
one. This modified form of Eq. (2.4) is
ˆrf
"1(r)ˆr'fast
tot,1(r)g
= 4sol,1(r) +slow
pol,0(r). (2.24)
Here,'fast
tot,1(r) is the total fast component of the ionized
state’s electrostatic potential. To apply the equilibrium PEqS
procedure introduced in Sec. II A, Eq. (2.24) is rewritten as
ˆr2'fast
tot,1(r)= 4fast
tot,1(r), (2.25)
wherefast
tot,1(r) and'fast
tot,1(r) are to be computed self-
consistently. The total nonequilibrium source charge density
is
fast
tot,1(r)=sol,1(r) +slow
pol,0(r) +fast
pol,1(r). (2.26)
For the Marcus partitioning scheme,49fast
pol,1(r) takes the form
fast
pol,1(r)= 1 "1(r)
"1(r)!sol,1(r) +slow
pol,0(r)+iter,1(r),
(2.27)
whereiter,1(r) is computed iteratively according to
(k)
iter,1(r)=1
4ˆrln"1(r)ˆr'fast,( k)
tot,1(r). (2.28)
Between iterations, we apply the damping procedure of Eq.
(2.14).
Finally, the nonequilibrium free energy is48,49
Gelst,1=W0,1+
dr'sol,1(r)slow
pol,0(r)
+1
2
dr'sol,1(r)fast
pol,1(r)
1
2
dr'sol,0(r)slow
pol,0(r). (2.29)
The term
W0,1=1
2
dr'slow
pol,0(r)fast
pol,1(r) fast
pol,0(r)(2.30)
arises within the Marcus partitioning scheme49due to
Coulomb interactions between fast and slow components of
the solvent polarization response,19,46,48,49in which the slow
components of the reference-state response affect the fast com-
ponents of the final-state response.45,60The quantity Gelst,1
defined in Eq. (2.29) is added to the gas-phase internal energy
Eint,1to generate the electrostatic interaction energy of the final
(ionized) state, Eelst,1. The state-specific nonequilibrium VIE
is then evaluated as the difference between the ionized- and
reference-state electrostatic energies, Gelst,1 Gelst,0.48,49The222834-5 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
result is
VIE=E1,0+W0,1+1
2
dr'sol,1(r)fast
pol,1(r)
+
dr'sol,1(r) 'sol,0(r)slow
pol,0(r)
1
2
dr'sol,0(r)fast
pol,0(r), (2.31)
where
E1,0=Eint,1 Eint,0 (2.32)
is the difference between the ionized- and reference-state
internal energies.
Operationally, the gas-phase Fock matrix for the ionized
state must be corrected for the solvent response using a matrix
h1whose elements are
h,1=
dr'(r)slow
pol,0(r) +fast
pol,1(r). (2.33)
The state-specific nonequilibrium PEqS method is summa-
rized in Algorithm 2.
C. Dielectric function
This section describes the construction of the dielectric
function"(r) that appears in Eqs. (2.4) and (2.24), for both
bulk solvation and the liquid/vapor interface.
1. Bulk environment
In conventional PCM calculations, the solute cavity is a
two-dimensional surface constructed from a union of atom-
centered spheres, possibly with additional surface elements
added to smooth the seams where those spheres intersect. In
any case, it is assumed that the dielectric constant changes
abruptly at the cavity surface, switching from its vacuum
value ("= 1) inside the cavity to a value appropriate for
Algorithm 2. Nonequilibrium PEqS method.
1: Proceed with Algorithm 1 and save data to disk.
2: Reference data: E0,Gelst,0,'slow
pol,0(r), andslow
pol,0(r)
3: Initialize h1=0
4:forj= 1, 2,:::do until SCF error< SCF
5: Diagonalize F(j)=F(j)
0+h(j)
1to obtain P(j)
6: Compute sol,1(r) and'sol,1(r)
7: ifj= 1then
8:fast
tot,1(r)=sol,1(r) +slow
pol,0(r)
9:'fast
tot,1(r)='sol,1(r) +'slow
pol,0(r)
10: else
11:fast
tot,1(r)=sol,1(r) +slow
pol,0(r) +fast
pol,1(r)
12:'fast
tot,1(r)='sol,1(r) +'slow
pol,0(r) +'fast
pol,1(r)
13: end if
14: Initialize (0)
iter,1(r) using'fast,(0)
tot,1(r)
15: fork= 0, 1,:::dountiliter,1< PEqS
16: Compute 'fast,( k+1)
tot,1(r)
17: Update (k+1)
iter,1(r),fast
pol,1(r), andfast
tot,1(r)
18: end for
19: Update h(j)
1
20: Compute E1=Eint,1+Gelst,1
21:end for
22: Compute VIE = E1 E0bulk solvent ( "solv) outside. This abrupt change in "poses
no problems within the PCM formalism but is problematic in
the present context, where it is necessary to discretize three-
dimensional space. As such, the sharp transition in "(r) must be
smoothed.61
Several groups have proposed dielectric functions that are
functionals of the electron density and thus conform automat-
ically to molecular shape,40,62–64analogous to using an iso-
density contour to define the cavity in a PCM calculation.65–67
Such cavities (or dielectric functions) must be self-consistently
updated at each self-consistent field (SCF) cycle. Instead, we
choose the rigid cavity model of Ref. 41 that uses a prod-
uct of spherically symmetric atomic switching functions s to
smooth the discontinuous function "(r) that is used (implicitly)
in PCM calculations and is based on atom-centered spheres.
The resulting dielectric function is
"(r)="vac+ ("solv "vac)atomsY
s d ,;jr R j. (2.34)
For generality, we have written this in terms of an arbitrary
“vacuum” dielectric constant "vacinside the cavity. For any
choice"vac,1, however, the electronic structure program
ought properly to be modified to use a Coulomb operator
("vacr) 1rather than r 1. All numerical calculations presented
here use"vac= 1.
The switching functions in Eq. (2.34) are defined as
s d ,;jr R j=1
2"
1 + erf jr R j d
!#
, (2.35)
where d is the radius of the atomic sphere centered at R .
With s chosen in this way, the dielectric function transitions
smoothly from "vacto"solvover a region whose length is 4
and is centered at a distance d from nucleus . Following
Ref. 41, we set = 0.265 Å, and following standard PCM
convention we take d = 1.2 rvdW, ,19,22,56where the atomic
van der Waals (vdW) radii rvdW, are taken from Bondi’s set,68
except that for hydrogen we use the updated value of 1.1
Å.69As such, Eq. (2.34) mimics the dielectric function that
is used implicitly in PCM calculations, except that the former
is continuous everywhere.
However, this dielectric function poses a problem when
explicit solvent molecules are included as part of the solute.
An egregious example is the case of the hydrated electron,
e (aq),70represented in the following example as a (H 2O)
28
cluster model extracted from a condensed-phase simulation.71
As shown in Fig. 1, this cluster model consists of approx-
imately two solvation shells of water molecules coordinated
around an unpaired electron. Cluster models of this type, com-
bined with a PCM to capture long-range solvation effects, have
previously been used to estimate the VIE of e (aq),72but this
is potentially problematic because the vdW cavity that is con-
ventionally used in PCM calculations places high-dielectric
regions in between water molecules.
This can be seen explicitly by plotting the dielectric func-
tion"(r) in Eq. (2.34), for a vdW cavity corresponding to the
water configuration shown in Fig. 1. (We emphasize that up
to a switching function to smooth the transition between "= 1
and"= 78, this is the dielectric function that is used, implicitly,
in PCM calculations.) A two-dimensional slice through "(r) is222834-6 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
FIG. 1. Singly occupied molecular orbital of a (H 2O)
28cluster model of
e (aq). The opaque and translucent isosurfaces encapsulate 50% and 95% of
the probability density, respectively. Positive values of the orbital are shown
in blue, and the very small negative regions are shown in green. The latter are
only visible in the 95% isoprobability contour.
plotted in Fig. 2(a). Although the cavity correctly conforms to
the molecular shape of the water cluster, the dielectric function
is problematic in the region near the cluster’s center of mass
(c.o.m.). There, blue and green regions indicate a solvent-like
value of"that penetrates into the region of space occupied by
the unpaired electron that, as part of the solute, ought instead to
experience"= 1. While e (aq) might seem like an unusual case
due to the esoteric nature of the solute, the problem is a general
one, as illustrated by the dielectric function for a F (H2O)31
cluster that is plotted in Fig. 3. High-dielectric regions can
once again be found inside of the atomistic QM region.
To address this problem, we pursue an approach used
also in the context of PCMs, in which a fictitious spherical
“solvent probe” is rolled along the surface of the vdW cavity
(constructed from unscaled Bondi radii); the locus of points
traced out by the center of this probe sphere defines the solvent-
accessible surface (SAS).73Equivalently, the SAS is simply a
vdW surface constructed using radii d =rvdW, +rprobe that
are equal to vdW (Bondi) radii augmented by the probe radius.
For aqueous solvation, the standard choice is rprobe= 1.4 Å,19,73
representing half the distance to the first peak in the oxygen–
oxygen radial distribution function of liquid water.74Using
d =rvdW, +rprobe in Eq. (2.34) successfully removes values
" > 1 in the interstices between water molecules, however,
the resulting VIEs are quite poor and in some cases the PEqS
procedure is difficult to converge. A “modified” SAS (mSAS)
construction, using the reduced value rprobe = 0.7 Å, allevi-
ates the convergence problems and affords more reasonable
VIEs but does not entirely eliminate artifactual high-dielectric
regions between water molecules, as shown in Fig. 2(b).
To rectify this, we introduce a “hybrid” cavity that retains
the conformity to molecular shape exhibited by the vdW
and SAS cavities but eliminates problematic high-dielectric
regions in this e (aq) test case. The hybrid cavity is built upon
the mSAS cavity ( rprobe = 0.7 Å), adding a geometric con-
straint that exploits the roughly spherical nature of the solute
configurations to ensure that the dielectric function assumes
the value"= 1 inside the solute region. To this end, we adapt a
procedure from Ref. 75 that was used to characterize binding
FIG. 2. Two-dimensional slices through the function "(r), for the (H 2O)
28
cluster that is shown in Fig. 1, which was extracted from a simulation of e (aq).
The dielectric function is constructed using either (a) the vdW solute cavity,
Eq. (2.34) with parameters d set to scaled Bondi radii; (b) a “modified” SAS
construction, created by setting d =rvdW, + 0.7 Å, which differs from the
usual SAS choice, rprobe = 1.4 Å; or (c) a “hybrid” cavity, which is described in
the context of Eq. (2.36). Each panel plots "(r) in the xzplane that contains the
cluster center of mass (c.o.m.). The dielectric function transitions smoothly
from"= 1 inside the cavity to "= 78.39 outside.
motifs of excess electrons in (H 2O)
nclusters. The shape of
each solute configuration (water cluster) is approximated as
an ellipsoid centered at the cluster c.o.m. ( x0,y0,z0), whose
surface is defined by the equation S(x,y,z)1, where
S(x,y,z)=(x x0)2
a2+(y y0)2
b2+(z z0)2
c2. (2.36)
The volume enclosed by S(x,y,z) is treated as the solvent-
excluded region, and the hybrid cavity, whose dielectric func-
tion is plotted in Fig. 2(c), is constructed from a mSAS cavity
by enforcing the condition that "(r) ="vacifS(x,y,z)<1.
The parameter ais set equal to the maximum atomic-to-c.o.m.
distance along the xaxis, plus a distance d 2that centers
the switching function in Eq. (2.35) at a distance d from the
nucleus. This furthermore ensures that "(r)"vacat a distance222834-7 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
FIG. 3. Two-dimensional slices through the function "(r) for a cluster
F (H2O)31extracted from a simulation of F (aq). The vdW cavity is used
to construct the dielectric function, so this plot is analogous to that shown in
Fig. 2(a) for the hydrated electron.
2from any atomic center. The parameters bandcare defined
similarly, for the yandzdirections.
Figure 2(c) illustrates the hybrid cavity dielectric function
for (H 2O)
28that is obtained using this procedure. This model
affords a satisfactory description of the dielectric environ-
ment, in the sense that high-dielectric regions between water
molecules are eliminated. As such, we use this definition of
the cavity and corresponding dielectric function for all PEqS
calculations reported in Sec. V.
2. Interfacial environment
The dielectric function for the liquid/vapor interface is
defined as in our previous work.42Specifically, we interpo-
late"(r) from"solv= 78.4 to"vac= 1.0 across the Gibbs
dividing surface (GDS). The periodic slabs used for molecular
dynamics (MD) simulations at the interface extend infinitely
in the xandydirections, and the location zGDSof the GDS
is determined over the course of the simulation by computing
ensemble-averaged solvent density profiles. These are com-
puted individually, for each solute, using 0.5 Å bins along the
zdirection, and the resulting density profiles are then fit to the
following functional form:32,76,77
(z)=1
2solv(
1tanh (z zGDS))
. (2.37)
Here,solvis the bulk liquid density (treated here as a fitting
parameter) and is a parameter such that the thickness of the
liquid/vapor interface is 4/ . The hyperbolic tangent term in
Eq. (2.37) is positive if z>zGDSand negative if z<zGDS.
Best-fit parameters solv, , and zGDSare listed in Table I
for each solute considered in this work. Fitted values of solv
TABLE I. Parameters for Eq. (2.37), obtained by fitting ensemble-averaged
solvent density profiles from MD simulations.
Solute solv(g/cm3) (Å 1) zGDS(Å)
Li+0.976 0.652 9.205
Na+0.979 0.604 9.154
H2O 0.986 0.724 8.963
e 1.016 0.668 1.508
F 0.976 0.635 9.105
Cl 0.969 0.626 9.328
FIG. 4. Two-dimensional slice through "(r) for a (H 2O)
24cluster represent-
inge (aq) at the liquid/vapor interface. A hybrid cavity is first constructed,
as described in the discussion surrounding Eq. (2.36), and then Eq. (2.38) is
used to interpolate the dielectric from "solv!"vacacross the GDS, which is
indicated by the dashed black line ( zGDS= 1.508 Å). Out of 24 explicit H 2O
molecules inside the cavity, only 2–3 lie above the GDS.
are in reasonable agreement with the actual density of liquid
water at 298 K. Fitted values of demonstrate that the inter-
facial region for the ionic solutes is discernibly thicker than
the liquid/vapor interface for neat water, an observation that is
also reported in other studies of anions at interfaces.77,78Tak-
ing parameters from Table I, we describe the z-dependence of
the interfacial dielectric function as in previous work,32,42,76
"(z)=1
2"solv(
1tanh (z zGDS))
. (2.38)
Figure 4 illustrates the dielectric function "(r) for a (H 2O)
24
cluster extracted from an interfacial configuration of e (aq),79
with the c.o.m. placed at the origin.
III. NUMERICAL SOLUTION OF POISSON’S EQUATION
Solution of the linear equations that define PCMs is often
(though not always22,80–82) accomplished via matrix inver-
sion. Matrix diagonalization incurs a cost that is O(N3
grid) in
floating-point operations and O(N2
grid) in memory, for Ngrid
discretization points, and for PCMs this is typically trivial in
comparison to the cost of the QM calculation for the solute. An
exception is QM/MM/PCM calculations, where the QM/MM
solute is potentially quite large, and conjugate gradient (CG)
procedures have been developed to handle such cases.22,80For
the PEqS method, however, direct inversion is prohibitively
expensive from the start, as s106Cartesian grid points might
be required to discretize three-dimensional space, with a mem-
ory cost alone that would exceed 7 Tb to store the discretized
Laplacian. It is therefore essential to employ relaxation tech-
niques such as iterative CG procedures. Here, we discuss
finite-difference discretization schemes for solving Poisson’s
equation on large Cartesian grids and also discuss improving
the efficiency of PEqS using a multigrid method. Much of this
work, including the multigrid method, is new since the pilot
implementation of PEqS that was reported in Ref. 42.222834-8 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
A. Finite-difference scheme
For the discussion that follows, let us rewrite Eqs. (2.10)
and (2.25) in a generic form
ˆr2'(r)=(r), (3.1)
where the factor of 4that ordinarily appears in Poisson’s
equation is instead included in (r). Writing out the Laplacian
operator explicitly, this is
@2'
@x2+@2'
@y2+@2'
@z2=(x,y,z), (3.2)
subject to the Dirichlet boundary condition
'(x,y,z)=0,8(x,y,z)2
(3.3)
at the surface boundary
(see below).
For a uniform rectangular grid centered at the origin, with
side lengths fLx,Ly,Lzgcontaining fNx,Ny,Nzggrid points
(so that Ngrid=NxNyNz), the domain
is defined as
=( Lx=2<x<Lx=2), ( Ly=2<y<Ly=2),
( Lz=2<z<Lz=2). (3.4)
The surface boundary is defined by the collection of rectangu-
lar planes
=(
[Lx,y,z], [x,Ly,z], [x,y,Lz])
. (3.5)
For convenience, we assume in what follows that the grid is
cubic, with equal spacing hin each direction. Cartesian coor-
dinates are then mapped onto grid coordinates as xi= Lx/2 +
ih, where i= 0,:::, (Nx 1).
The value of the electrostatic potential '(x,y,z) at the grid
point ( xi,yj,zk) is denoted as
'i,j,kB'(xi,yj,zk), (3.6)
with a similar notation for other discretized quantities. Expres-
sions for the discretized first and second derivatives of 'i,j,kare
obtained using a central finite-difference scheme. A general
expression for an nth-order approximation to the mth-order
derivative, whose finite-difference approximation exhibits
error of O(h2n), is
@m'i,j,k
@xm=nX
p= ncm,p'i+p,j,k
hm
, (3.7)
for certain coefficients cm,p. We use an eighth-order ( n= 4)
finite-difference approximation for the first ( m= 1) and second
(m= 2) derivatives. Coefficients cm,nfor this approximation
are given in Table II.
TABLE II. Central finite-difference coefficients cm,pfor the discretized first
(m= 1) and second ( m= 2) derivatives in Eq. (3.7). These coefficients afford
an eighth-order approximation scheme whose accuracy is O(h8).
cm,0 cm,1 cm,2 cm,3 cm,4
m= 1 0 4/51/54/1051/280
m= 2 205/72 8/5 1/5 8/315 1/560B. Multigrid approach
We employ a multigrid method to improve the efficiency
of the iterative CG procedure. To facilitate the following dis-
cussion, let us rewrite Poisson’s equation [Eq. (3.1)] in a
discretized form involving a matrix–vector product,
Lh'h=h. (3.8)
TheNgridNgridmatrixLhcontains the discretized Laplacian
operator, and the vectors 'handhcontain the discretized val-
ues'h
i,j,k,and 4h
i,j,k,, respectively. The quantities 'handh
are subject to appropriate boundary conditions, and the super-
script hsignifies the level (fineness) of the discretization. The
CG algorithm avoids the prohibitive memory cost associated
with forming Lhexplicitly; only its action on the vector 'his
required.
Via Fourier analysis of the discretization errors
vh
i,j,k='exact
i,j,k 'h
i,j,k, (3.9)
it has been shown that the spectrum of errors contains wave-
lengthsthat are comparable to, or larger than, the grid resolu-
tion, h.83The CG routine efficiently eliminates discretization
errors with'hbut struggles with error components for which
>h.83Iterative techniques such as CG therefore effectively
smooth out short-wavelength discretization errors, but they do
not perform well (or at least, efficiently) for obtaining a fully
converged solution due to long-wavelength error components.
To illustrate this, the CG routine was employed to compute 'h
in Eq. (3.8) for a single H 2O molecule placed at the center of
a (15 Å)3cubic Cartesian grid with a resolution h= 0.074 Å.
Figure 5 shows the Euclidean norm of the residual error in the
electrostatic potential,
rh=h Lh'h, (3.10)
FIG. 5. Comparison of the performance of several iterative schemes for solv-
ing Eq. (3.8) to an accuracy PEqS = 10 5a.u. (indicated by the horizontal
black line), for a single water molecule whose c.o.m. resides at the center of a
(15 Å)3Cartesian grid with h= 0.074 Å. The slow decay of the residual error
for the standard CG routine is indicative of >h, and over 250 iterations are
required for convergence. The multigrid methods achieve convergence more
rapidly, with the W-cycle implementation performing best. Both multigrid
algorithms exhibit an exponential decrease of rhwith respect to the iteration
number, in stark contrast to results from the standard CG method.222834-9 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
as a function of iteration number. There is a rapid decrease in
krhkduring the first few iterations, but in the end more than
250 iterations are required to achieve convergence, defined as
krhk<10 5a.u. The inability of the CG routine to eliminate
the long-wavelength error components furthermore manifests
as a broad and slowly decaying shoulder, evident in iterations
10–110 in Fig. 5.
Since the charge density (r) computed by the electronic
structure calculation is known essentially to arbitrary accuracy,
at least compared to the s10 5a.u. residual convergence error
in the CG approach, we can safely assume that (r) is free
of discretization error upon formation of h. Supposing that
Eq. (3.8) can be solved exactly for the electrostatic potential,
thenh=Lh'exact. Thus Eq. (3.10) can be written in terms of
vh[defined in Eq. (3.9)], even when the exact solution is not
known,83
rh=Lh'exact Lh'h
=Lh('exact 'h)
=Lhvh. (3.11)
Despite never actually acquiring 'exact, Eq. (3.11) provides an
avenue for computing the quantity vh, which is an integral part
of the multigrid method.The multigrid method seeks to obviate undesired compu-
tational effort spent eliminating long-wavelength error com-
ponents that results in the slow convergence exhibited by the
CG routine. The ultimate goal is a solution to Eq. (3.8) on a fine
rectangular grid, and we refer to this as the “target” grid reso-
lution, denoted by h. The schematic for a two-level “V-cycle”
multigrid approach83is illustrated in Fig. 6. (The nomenclature
is explained below, where we introduce an alternative “W-
cycle” approach as well.) The multigrid method is designed to
relax the iterative solution on the target grid, where the compu-
tational cost is highest, as few times as possible. This is step 1
in the algorithm outlined in Fig. 6, and it serves to reduce short-
wavelength errors. The resulting residual rhobtained using the
target grid is then restricted to a grid with only half as many
grid points in each Cartesian direction. The resolution of this
grid is denoted as H= 2h, and the iterative solution on the
coarser grid serves to reduce longer-wavelength components
of the error. The restriction rh!rHis illustrated in step 2 of
Fig. 6 and is accomplished using a restriction matrix IH
h,
rH=IH
hrh. (3.12)
In practice, IH
his not formed and its action on rhto generate
rHis expressed as
rH
I,J,K=1
8rh
i,j,k+1
16 rh
i+1,j,k+rh
i 1,j,k+rh
i,j+1,k+rh
i,j 1,k+rh
i,j,k+1+rh
i,j,k 1+1
32 rh
i+1,j+1,k+rh
i+1,j 1,k+rh
i 1,j+1,k+rh
i 1,j 1,k
+1
32 rh
i+1,j,k+1+rh
i+1,j,k 1+rh
i 1,j,k+1+rh
i 1,j,k 1+1
32 rh
i,j+1,k+1+rh
i,j+1,k 1+rh
i,j 1,k+1+rh
i,j 1,k 1
+1
64 rh
i+1,j+1,k+1+rh
i+1,j+1,k 1+rh
i+1,j 1,k+1+rh
i+1,j 1,k 1+1
64 rh
i 1,j+1,k+1+rh
i 1,j+1,k 1+rh
i 1,j 1,k+1+rh
i 1,j 1,k 1. (3.13)
The notation ( I,J,K) inrH
I,J,Kis introduced to denote that a
different mapping scheme for the coarse-grid coordinates is
required, namely, xI= Lx/2 +IHxforI= 0,:::, (Nx 1)/2
andHx= 2hx. Equation (3.13) is valid for three dimensions
and shows that a coarse grid point takes its value from all
neighboring points on the target grid, with a weight determined
by its proximity to rH
I,J,K.
After forming rH, a Poisson-like equation is solved to
obtain the coarse grid discretization error, vH,
LHvH=rH. (3.14)
This is illustrated in step 3 of Fig. 6. Relaxing vHon the
coarse grid reduces the problematic long-wavelength error
components with which the primitive CG routine struggles;
consequently, vHprovides a better correction to 'h. How-
ever, vHcannot be used directly to correct 'hbecause the
former is defined only on the coarser grid and must be inter-
polated back to the target grid. This process of inverse restric-
tion is illustrated in step 4 of Fig. 6. Interpolation of vH
to form vhis accomplished via the inverse iteration matrix
Ih
H,
vh=Ih
HvH. (3.15)The action of Ih
His expressed by the following set of
equations:
vh
i,j,k=vH
I,J,K, (3.16a)
vh
i+1,j,k=1
2 vH
I,J,K+vH
I+1,J,K, (3.16b)
vh
i,j+1,k=1
2 vH
I,J,K+vH
I,J+1,K, (3.16c)
vh
i,j,k+1=1
2 vH
I,J,K+vH
I,J,K+1, (3.16d)
vh
i+1,j+1,k=1
4 vH
I,J,K+vH
I+1,J,K+vH
I,J+1,K+vH
I+1,J+1,K,
(3.16e)
vh
i+1,j,k+1=1
4 vH
I,J,K+vH
I+1,J,K+vH
I,J,K+1+vH
I+1,J,K+1,
(3.16f)
vh
i,j+1,k+1=1
4 vH
I,J,K+vH
I,J+1,K+vH
I,J,K+1+vH
I,J+1,K+1,
(3.16g)
vh
i+1,j+1,k+1=1
8 vH
I,J,K+vH
I+1,J,K+vH
I,J+1,K+vH
I,J,K+1
+1
8 vH
I+1,J+1,K+vH
I+1,J,K+1+vH
I,J+1,K+1
+vH
I+1,J+1,K+1. (3.16h)
The interpolated discretization error vhcorrects'haccording
to
'h!'h+vh. (3.17)222834-10 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
FIG. 6. Illustration of a two-level V-cycle multigrid algo-
rithm applied to solve Eq. (3.8). The input is a source
charge density hand the output is 'h, both of which are
discretized on a target grid whose resolution is h. Steps 1
and 2 show the formation of the residual error rhon the
target grid and subsequently its restriction to a coarser
grid whose resolution is H= 2h. On the coarser grid, rH
is used in a CG routine to compute a relaxed residual
vector rHin step 3. The relaxed residual is then interpo-
lated back to the target grid to form vhin step 4. In step
5,vhis used to correct the solution on the target grid,
and this process is repeated (starting from step 2) until
convergence.
FIG. 7. Flow diagrams illustrating four-level multigrid
algorithms of either the (a) V-cycle or (b) W-cycle variety.
Either approach uses a target grid of resolution hand three
coarser grids of resolutions H= 2h, 4h, and 8 h. Downward
arrows represent the restriction of the residual error vector
from a finer to a coarser grid. After relaxing
0and
1
times, the discretization error is then interpolated from
coarser to finer grids, indicated by an upward arrow. The
interpolated discretization error is further relaxed on the
finer grids
2or
3times. Convergence of the solution on
the target grid is then tested, and the process is repeated
until the desired accuracy is achieved.
This is step 5 of Fig. 6. Convergence is achieved when krh
new
rh
oldkfalls below a desired threshold, at which point 'his
the fully relaxed solution to Eq. (3.8). Otherwise the process
shown in Fig. 6 repeats with step 2.
In contrast to the two-level algorithm outlined in Fig. 6,
the PEqS method implemented here actually uses a four-level
method that employs two additional coarse grids whose res-
olutions are H= 4handH= 8h. These four-level schemes
improve upon the two-level scheme by eliminating error com-
ponents on multiple length scales, affording more effective
corrections at each grid level and resulting in a fully relaxed,
target-grid solution in fewer iterations.83Schematics for “V-
cycle” and “W-cycle” variants of the four-level method are
provided in Fig. 7, which introduces parameters
0,
1,
2, and
3that control the maximum number of CG iterations spent
at various grid levels. The strategy is to always fully relax the
error vector vH=8hat grid level 3 (the coarsest grid), so we set
the parameter
0equal to the maximum number of allowed
iterations,
0= 500 here. The other parameters are set to
1
= 2,
2= 3, and
3=
1+
2, as in Ref. 83. Passing through
either cycle outlined in Fig. 7, one performs CG iterations of
the equation Lnhvnh=rnhin order to compute vnh, for n= 1,
2, 4, or 8 as appropriate. Iterations continue until the residual
rnhis reduced below threshold or until the maximum number
of iterations is reached. At grid levels finer than H= 8h, this
means that vnhneed not be fully relaxed at each step. (Conver-
gence failure on the coarsest grid, however, implies the failure
of the whole algorithm, though we have not found this to be aproblem with the parameters described herein.) Proceeding in
this way, the solution 'hon the target grid, which is the most
expensive to compute, is relaxed a total of
1+
2times, in
either the V- or W-cycle approach.
In more detail, the four-level algorithms proceed as fol-
lows. At the target grid level, the residual error rhcorrespond-
ing to'hafter
1CG iterations is restricted to grid level
1, signified by a downward arrow in Fig. 7. Using rH=2hin
Eq. (3.14), vH=2his relaxed
1times and is stored in memory.
This process of restriction is repeated for each downward arrow
until reaching the coarsest level of discretization (grid level
3), where the solution is then fully relaxed. Upward arrows in
Fig. 7 signify interpolation of the discretization error from a
coarser grid to a finer one using Eq. (3.15), and the resulting
error vector is then used to update the residual error at grid
levels 2 and 1, or the electrostatic potential on the target grid.
Each of these is relaxed
2times. Returning to the example in
Fig. 5, we note that the four-level methods require about 75
iterations (V-cycle) and 35 iterations (W-cycle) on the target
grid, a remarkable improvement over the standard CG routine.
We select the four-level W-cycle method for PEqS calculations
due to its superior performance in this example.
IV. COMPUTATIONAL METHODS
In this work, we compute VIEs for neat liquid water and
for five aqueous ions: F , Cl , Li+, Na+, and the hydrated
electron. Experimental VIEs are available for each of these222834-11 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
species,11,15and while we examine e (aq) due mainly to
our group’s long-standing interest in this species,42,70,75,84–87
and because preliminary calculations on e (aq) were previ-
ously reported using a perturbative version of PEqS,42the
four atomic ions are selected because their simple structure
should eliminate any concerns about adequate MD sampling.
This is especially true given that an accurate polarizable force
field ( amoeba ) is available for these species.88,89This allows
us to focus on the role of the continuum model in VIE
calculations.
A. Molecular dynamics simulations
Simulations of neat liquid water were performed with 222
water molecules in a periodic cell 18.8 Å on a side, correspond-
ing to a density of 0.9995 g/cm3atT= 300 K, using the amoeba
force field88,89as implemented in the tinker software pack-
age,90v. 7.1.2. Electrostatic interactions were computed using
Ewald summation with a real-space cutoff of 9.4 Å. The neat
liquid water simulations were equilibrated for 1 ns with the
final 500 ps extracted for further use. For simulations of the
neat liquid/vapor interface, the final snapshot from the bulk
water simulation is used as a starting point and the simula-
tion box was extended to 90.0 Å in the zdirection so that the
dimensions of the simulation cell measured 18.8 Å 18.8 Å
90.0 Å. The resulting water “slab” was equilibrated for an
additional 1 ns at T= 300 K.
MD simulations for the aqueous halides and alkali cations
were initialized starting from the equilibrated neat liquid water
simulation, replacing either the H 2O molecule nearest to the
center of the cell (in the bulk simulations) or that nearest to the
interface (in slab simulations) with an ion. The bulk simula-
tions were then equilibrated for an additional 250 ps at T= 300
K followed by a 500 ps production run with snapshots stored
every 5 ps. In contrast, simulations at the liquid/vapor inter-
face were not equilibrated, and a 500 ps production run began
immediately after insertion of the ion, again with a stride of 5
ps between saved snapshots. The snapshots for e (aq) in liquid
water and at the air/water interface were taken from QM/MM
simulations reported in Refs. 71 and 79. They are the same
snapshots used in some of our previous studies of e (aq).42,87
Solute configurations for subsequent PEqS and PCM cal-
culations were generated from the stored snapshots by select-
ing a QM region that includes all water molecules within a
sphere of radius 5.5 Å centered at the ion, or in the case of
e (aq), centered at the centroid of the spin density. Extensive
convergence tests in previous work suggest that larger QM
regions are unnecessary for VIE calculations.42For neutral
water, the H 2O molecule nearest to the center of the simulation
cell is chosen as the center for the bulk liquid configurations,
whereas the water molecule nearest to the GDS is chosen for
the liquid/vapor configurations. VIEs reported here are aver-
ages over 101 snapshots, each separated in time by 5 ps, except
in the case of e (aq) where we use 87 snapshots, each separated
in time by 100 fs.
B. Continuum solvation models
The dielectric function, charge densities, and electrostatic
potentials required for PEqS calculations were discretized on a
(25 Å)3Cartesian grid with spacing x=y=z= 0.24 Å. Thehybrid cavity model described in Sec. II C 1 is used to construct
"(r), and for the interfacial configurations the dielectric func-
tion is modified according to Eq. (2.38), with solute-specific
parameters taken from Table I. Concurrent acquisition of the
polarization response charge density and electrostatic poten-
tial through Eqs. (2.10) and (2.25) is accomplished using the
four-level W-cycle multigrid technique (Sec. III B), with relax-
ation parameters
0= 500,
1= 2,
2= 3, and
3=
1+
2.
The iterative charge density is updated until the residual iter
falls below a threshold PEqS = 10 5a.u.
To complement the PEqS calculations, we also compute
VIEs of the bulk water configurations using a nonequilibrium
version48–50of IEF-PCM,91the “integral equation formalism”
(IEF) version of PCM.22,91–93The solute cavity in these calcu-
lations is defined in one of the two ways. One approach uses
a single sphere to encapsulate the QM region; this region was
carved out of the solution using a radius of 5.5 Å and we set
the spherical cavity radius to 7.5 Å. This sphere is then dis-
cretized using a Lebedev integration grid with 5294 points.
Alternatively, we use the SAS cavity constructed from a union
of atom-centered spheres with radii rvdW+rprobe, where rvdW
is an unscaled Bondi radius68,69andrprobe = 1.4 Å. (This is
the standard SAS definition,19,73not the modified one used in
Sec. II C 1 to define the hybrid cavity.) In this case, each atomic
sphere is discretized with a 302-point Lebedev grid using the
switching/Gaussian algorithm.94
For isotropic solvation in bulk water, we expect that the
PEqS and PCM methods should afford similar results, up to
discretization errors that can be made arbitrarily small,22,91
and neglecting charge penetration effects (i.e., volume polar-
ization95,96) arising from the part of the solute’s charge density
that extends beyond the cavity. The latter effects are mit-
igated within the IEF-PCM framework,92,97,98and through
the inclusion of explicit water molecules around the anions.
Unfortunately, the PEqS method in its current implementa-
tion is sensitive to the Gaussian width parameter that is
used to blur the nuclear charges [Eq. (2.8)], so we exploit
the expected numerical equivalence with IEF-PCM to set this
parameter. Setting = 0.300 Å (F ), 0.525 Å (H 2O), or
0.570 Å (Li+) affords PEqS and IEF-PCM solvation ener-
gies that are in good agreement, for a few test configura-
tions. The F value ofwas used for all three anions and
the Li+value was used for both cations. Values of deter-
mined in bulk water were used also for the interfacial PEqS
calculations.
C. Electronic structure calculations
The state-specific PEqS method has been implemented in
a locally modified version of the q-chem program99and will be
released with v. 5.1. Electronic structure calculations were per-
formed at the level of second-order Møller-Plesset perturbation
theory (MP2) within the resolution-of-identity (RI) approxi-
mation.100All electrons were correlated for the Li+(aq) and
Na+(aq) calculations, whereas other calculations use a frozen
core. The PEqS part of the calculation resides in the Hartree-
Fock iterations and uses the Hartree-Fock electron density.
(Nonequilibrium PCM results for excitation energies suggest
that a fully self-consistent use of the correlated electron density
has negligible effect on the results.50) The SCF convergence222834-12 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
threshold is set to SCF= 10 5a.u. in all calculations, with an
integral and shell-pair screening threshold of 10 8a.u.
We use the 6-311+Gbasis set for all H 2O molecules,
except in the case of e (aq) where we use 6-311++Ginstead,
to ensure that the interstices between the water molecules are
supported by basis functions. (We have previously concluded
that one set of diffuse functions on all atoms is sufficient to
support a hydrated electron that occupies an excluded volume
in the structure of liquid water.84,86) For Li and Na, we use
the cc-pCVTZ basis set, which includes functions to describe
core/valence correlation, whereas for F and Cl we use aug-cc-
pVTZ. In all cases, we employ the auxiliary basis set designed
for either cc-pVTZ or aug-cc-pVTZ,101as appropriate.
The valence photoelectron spectrum of liquid water con-
sists of a broad absorption feature centered at 11.3 eV ,102
attributed to ionization of a 1 b1MO localized on a single
H2O molecule.11,102–105Experiments to determine the VIE of
F (aq) are complicated by the fact that the fluoride signal is
embedded in the 1 b1band of water.11,16When explicit water
molecules are included in the QM region, the corresponding
VIE calculation is problematic as well, not only for F (aq)
but also for Na+(aq), Li+(aq), or any solute whose VIE lies
near or above that of water. In such cases, a simple calcula-
tion of the lowest-energy state of the ionized system results in
ionization of water rather than the desired solute,18as shown
for Li+(aq) in Fig. 8(a) and for F (aq) in Fig. 8(b). In these
examples, the VIE is computed for a system consisting of the
atomic ion surrounded by about 30 explicit water molecules
and spherical PCM boundary conditions. In both cases, how-
ever, it is a water molecule that is ionized rather than the atomic
ion. Figure 8(c) shows that even neat liquid water is problem-
atic, as in this case the lowest-energy ionized system places
the cation hole on an “edge” water molecule that lies near the
QM/continuum boundary and does not fully participate in the
hydrogen-bonding network. Computed VIEs are in very poor
agreement with experiment, e.g., 8.3 eV for neat liquid water.
To ionize F , Cl , Li+, or Na+embedded in a water cluster,
one must remove an electron from an orbital other than the
HOMO of the full QM system. At the same time, one still wants
to include orbital relaxation effects upon ionization yet prevent
variational collapse of the wave function to the lowest-energy
solution, which is the one depicted in Figs. 8(a)–8(c). The
maximum overlap method106(MOM) was designed precisely
for this purpose and has been used, for example, to compute
core-excited states (K-edge spectra) by moving an electron
from a core orbital into an extra-valence (virtual) orbital and
then relaxing the orbitals while searching for the maximum
overlap solution.107
Here, we start from an initial guess generated using the so-
called fragment MO (FragMO) procedure108and then use the
MOM method to preserve the character of the initial orbitals
during subsequent SCF iterations. The FragMO procedure first
computes MOs on isolated fragments (here, either H 2O or an
atomic ion), which affords an easy means to remove an elec-
tron from an MO associated with a particular fragment, e.g.,
the 1 b1orbital of a particular water molecule. A superposi-
tion of fragment density matrices is then used as the initial
guess for the supersystem SCF calculation. The converged
SCF solution obtained from this FragMO/MOM procedure
FIG. 8. Spin densities spin= following ionization of cluster con-
figurations representing (a) Li+(aq), (b) F (aq), and (c) neat liquid water.
(Essentially 100% of spinis encapsulated within each surface.) In each case,
a standard SCF calculation of the ionized ground state results in a hole that is
localized on a water molecule near the continuum boundary and disconnected
from the hydrogen-bonding network. Computed VIEs are in poor agreement
with experiment. Panels (d)–(f) show the spin densities obtained from the
FragMO/MOM SCF approach, which ionizes the atomic ion in (d) and (e),
and a central water molecule in (f). In these cases, computed VIEs are in rea-
sonable agreement with measured values. All VIEs were computed using the
nonequilibrium IEF-PCM with a spherical cavity.
contains a “hole,” which manifests as a single virtual orbital
with an energy below that of the HOMO. For Li+(aq), the
core hole occasionally leads to accidental quasi-degeneracies
amongst the Hartree-Fock eigenvalues such that the MP2
energy denominator becomes very small. For this species only,
we therefore omit this “hole” orbital from the MP2 calculation
of the correlation energy.
VIEs recomputed using nonequilibrium IEF-PCM in con-
junction with this FragMO/MOM SCF procedure are shown in
Figs. 8(d)–8(f). It is first of all clear that this approach success-
fully ionizes the desired species, which is a centrally located
H2O molecule in the case of neat liquid water. The computed
VIEs are also in far better agreement with experiment, e.g., for
the snapshots shown in Fig. 8 they are 11.8 eV (computed) for
H2O(aq) versus 11.3 eV (experiment);10211.3 eV versus 11.6
eV for F (aq);11and 62.0 eV versus 60.4 eV for Li+(aq).18We
use the FragMO/MOM SCF procedure for all systems except
e (aq), where the VIE lies well below that of liquid water and
a standard SCF procedure is adequate.
It is worth emphasizing that the use of the FragMO/MOM
procedure does not force the converged, singly occupied MO222834-13 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
to be localized on any single monomer; only the initial-guess
MOs are localized and in subsequent SCF iterations the orbitals
are free to delocalize, as seen, for example, in Fig. 8(e). This is
likely to be important, e.g., to simulate the full valence photo-
electron spectrum of liquid water. Although the lowest-energy
feature is assigned to ionization of the 1 b1orbital localized
on a single water molecule, the higher-energy 3 a1feature is
broadened into an (unresolved) doublet,11,102–105assigned to
ionization from bonding and anti-bonding combinations of the
3a1orbitals on two hydrogen-bonded H 2O molecules.105,109
In this work, we consider only the lowest VIE of each solute,
but in principle it should be possible to simulate the splitting
of the 3 a1feature by constructing the appropriate initial-guess
orbitals on a water dimer.
V. RESULTS
A. VIEs in bulk water
Configurationally averaged VIEs in bulk liquid water,
computed using the nonequilibrium PEqS and PCM meth-
ods, are shown in Table III along with experimental values
obtained from liquid microjet experiments.11,15,18,102Also
listed is the average number hNiof explicit water molecules
in the QM region, which is about 30 in each case, amounting
to roughly two solvation shells. In previous work on e (aq),42
we observed that increasing hNiup to 90, corresponding to an
increase in the radius of the QM region from 5.5 Å to 8.0 Å,
changed VIEs by0.1 eV , both in bulk liquid water and at the
liquid/vapor interface. Amongst the solutes considered here,
we anticipate that e is most acutely affected by the size of the
QM region and thus we regard the present calculations to be
adequately converged in this respect.
Agreement with experimental VIEs is excellent for the
anionic solutes and for H 2O, especially for the PEqS treat-
ment of solvation. Although we do not expect exact agreement
between the PEqS and PCM calculations, primarily because
the solute cavities differ but also due to the approximate man-
ner in which IEF-PCM accounts for volume polarization,95–97
results from all three solvation models agree to within about0.4 eV . (In fairness, the Gaussian widths for the PEqS nuclear
charges were determined in order to match IEF-PCM solvation
energies for a few snapshots.) For e (aq), both PCM methods
underestimate the VIE by 0.5 eV , whereas the PEqS calcu-
lations are spot on, at 3.7 eV; arguably, this is the system for
which PCM boundary conditions are most questionable110due
to the delocalized nature of the solute. For reasons that are not
clear, computed VIEs for Li+(aq) and Na+(aq) exhibit much
larger errors of 0.7–1.0 eV (PEqS) or 0.9–1.4 eV (PCM) with
respect to experiment.
Uncertainties on the calculated VIEs in Table III repre-
sent one standard deviation across MD snapshots and provide
an estimate of the inhomogeneous broadening arising from
thermal sampling of solvent configurations. We character-
ize the width of the computed spectrum in terms of the full
width at half maximum (FWHM = 2.355 standard devi-
ation), assuming a Gaussian distribution of the computed
values, and comparison to experiment should provide some
insight regarding the quality of the underlying MD simula-
tions. Computed FWHMs for Li+and Na+are 0.9–1.1 eV , in
good agreement with experimental widths of 1.11–1.24 eV .18
Peak width measurements are not available for F , but for Cl
the measured width is 0.60 eV .18Our computational uncer-
tainties for F (aq) and Cl (aq) correspond to a FWHM of
0.8–0.9 eV , slightly larger than what is observed experimen-
tally for Cl (aq) but in keeping with the trend that the halide
anions have narrower photoelectron spectra as compared to
the alkali cations. The halides also have narrower photoelec-
tron spectra as compared to neat liquid water, with the lat-
ter at 1.45–1.47 eV (experiment102,104,105) versus 1.1–1.2 eV
(theory).
One note of caution is in order with regard to spectral
widths. Although the favorable comparison between the com-
puted VIEs and the experimental values demonstrates the suc-
cess of the FragMO/MOM approach, the effect of the lifting
ofp-orbital degeneracy cannot easily be elucidated using this
approach. However, the magnitude of the p-orbital splitting
resulting from an asymmetric distribution of water molecules
has been estimated to be rather small,18viz., 0.03 eV for
Na+(aq), 0.11 eV for F (aq), and 0.12 eV for Cl (aq).
TABLE III. Average VIEs computed with nonequilibrium PEqS and PCM methods at the (RI)MP2 level using
triple-basis sets as described in Sec. IV C. Computed VIEs are averages over MD snapshots, and uncertainties
represent one standard deviation. Experimental error bars, which come from the references indicated, represent
uncertainty in the peak position and are not peak widths.
Computed VIE (eV)
ExperimentalPEqS PCM
SolutehNi VIE (eV) Hybrid Spherical SAS
Li+30 60.40 0.07a61.410.45 61.85 0.45 61.59 0.41
Na+29 35.40 0.04a36.090.43 36.54 0.44 36.34 0.40
H2O 30 11.31 0.04b11.530.51 11.64 0.51 11.55 0.45
e 30 3.7 0.1c3.750.55 3.16 0.32 3.18 0.28
F 30 11.58d11.660.36 11.37 0.39 11.52 0.37
Cl 32 9.60 0.07a9.650.37 9.36 0.38 9.41 0.35
aReference 18.
bReference 102.
cReference 15.
dReference 11.222834-14 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
TABLE IV . Nonequilibrium solvation correction to the VIE [Eq. (5.3)],
averaged over MD snapshots.
hVIE noneqi(eV)
PEqS PCM
Solute Hybrid Spherical SAS
Li+0.65 0.53 0.59
Na+0.65 0.53 0.59
H2O 0.11 0.55 0.62
e 1.17 0.54 0.61
F 0.88 0.53 0.58
Cl 0.88 0.54 0.59
The magnitude of the nonequilibrium correction to the
VIE warrants consideration. It might be assumed that this cor-
rection, which comes from the boundary conditions, would
be rather small given the number of explicit water molecules
in our calculations, but in fact this is not the case. The
magnitude of the nonequilibrium correction is not available
directly from the solvation models (i.e., it is not separable
in the total nonequilibrium energy expression) but can be
deduced by considering two strategies by which a VIE might be
computed.
(1) Employ a nonequilibrium method that uses the optical
dielectric constant "1for the ionized state and the static
dielectric constant "solvfor the initial state.
(2) Perform two equilibrium solvation free energy calcula-
tions, both of which use "solv, and compute the VIE as
the difference between the free energies of the initial
and the ionized states.
The free energy of the ionized state computed using the
equilibrium strategy (2) includes within it the polariza-
tion effects resulting from both the slow (nuclear) and fast(electronic) contributions from the solvent. We might express
this VIE as
VIE eq=Efinal("solv) Einitial("solv), (5.1)
where the notation indicates that "solvis the dielectric constant
of merit in both calculations. This differs from nonequilibrium
strategy (1), which accounts only for the fast component,
VIE noneq=Efinal("1) Einitial("solv). (5.2)
The difference between these two calculations provides a
measure of the nonequilibrium correction to the VIE,
VIE noneq=VIE noneq VIE eq
=Efinal("1) Efinal("solv).(5.3)
The average value of VIE noneq from each set of calcu-
lations is listed in Table IV. This correction ranges from 0.5
to 1.2 eV , and this value characterizes the error that would be
made if only equilibrium solvation models were available. As
such, computational strategies for vertical ionization energies
that are based on equilibrium PCMs should not be trusted,
although they are sometimes encountered in the literature.
B. VIEs at the liquid/vapor interface
Figure 9 shows the time-dependent VIE, computed using
the nonequilibrium PEqS method, for a single MD trajectory
of Li+and of Na+, initialized at the liquid/vapor interface.
Also plotted is the distance dGDS between the ion and the
instantaneous GDS.
Within 50 ps, Li+moves away from the interface in favor
of a more bulk-like environment, and for the remainder of the
simulation its position fluctuates in the range 4.0 Å <dGDS
<8.5 Å. Na+departs the interface on a similar time scale
anddGDSthen fluctuates from 6.0 to 9.0 Å for most of the
simulation, except when the ion briefly drifts back to the inter-
face around 250–300 ps, before quickly descending again into
FIG. 9. VIEs computed using the nonequilibrium PEqS method [Eq. (2.31)] along a single MD trajectory for (a) Li+(aq) and (b) Na+(aq), along with the distance
dGDSfrom the Gibbs dividing surface that defines the interface, again for (c) Li+(aq) and (d) Na+(aq). Also shown on the VIE plots is a best-fit line to the data
(in blue) and the average bulk VIE (in red).222834-15 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
FIG. 10. VIEs computed using the nonequilibrium PEqS method [Eq. (2.31)] along a single MD trajectory for (a) F (aq) and (b) Cl (aq), along with the distance
dGDS from the Gibbs dividing surface that defines the interface, again for (c) F (aq) and (d) Cl (aq). Also shown on the VIE plots is a best-fit line to the data
(in blue) and the average bulk VIE (in red).
a bulk-like solvation environment by 350 ps. Linear regres-
sions of the VIE( t) data (shown in blue in Fig. 9) are nearly
flat and almost indistinguishable from the average bulk VIE
(shown in red), demonstrating the extent to which the VIE is
insensitive to the position of the ion relative to the liquid/vapor
interface.
Figure 10 presents the same data for representative tra-
jectories of F (aq) and Cl (aq) initialized at the interface. The
fluoride ion remains within 2–4 Å of the interface for the first
50 ps but then moves away, with dGDSfluctuating from 6 to
8 Å. This behavior is similar to the cation data in Fig. 9. In
contrast, the migration of Cl away from the interface is notice-
ably slower and occurs over 350 ps, with the ion then returning
rapidly to within 2 Å of the GDS near the end of the simulation.
Linear fits to the instantaneous VIE data are not quite as flat
as in the case of the cations, indicating that the VIE exhibits a
minor dependence on the location of the anion relative to the
interface.
These interfacial simulations were initialized by replacing
a water molecule with an ion in a previously equilibrated sim-
ulation of neat liquid water, so the early-time dynamics reflect
the rearrangement of the solvent to accommodate the ion. We
therefore attribute the slope in the linear VIE fits for F and
Cl , which is not observed for Li+or Na+(where the slopes are
essentially zero) as evidence of greater disruption of the water
network when the larger and more polarizable halide ions are
inserted. Following an equilibration period of roughly 100 ps,
however, the interfacial VIEs fluctuate around mean values of
11.63 eV (for F ) and 9.62 eV (for Cl ), which are nearly iden-
tical to the average bulk values of 11.66 eV and 9.65 eV . As
such, the slight difference in the early-time dynamics of the
anions relative to that of the cations seems insignificant and
mostly an artifact of the simulation procedure, i.e., the fact that
the ion is not equilibrated at the interface at t= 0.
Table V compares the average VIEs and average value
ofdGDSfor nonequilibrium PEqS calculations of neat liquidwater and e (aq). In contrast to the calculations for the inter-
facial halide anions and alkali cations (e.g., Figs. 9 and 10),
where the ion starts at the interface at t= 0 but quickly diffuses
deeper into the liquid, the simulations leading to Table V are
more truly interfacial. The average VIE for liquid water that is
reported in Table V is obtained by ionizing the H 2O molecule
that is closest to the instantaneous GDS at each time step. For
e (aq), an electron initialized at the interface remains there
long enough to generate a meaningful interfacial trajectory,79
and for the snapshots used to compute the average e (aq) VIE
in Table V the centroid of the spin density is no farther than
2.5–3.0 Å from the liquid surface. In contrast, halide anions
and alkali cations initialized at the interface sample values of
dGDS in the range 6–8 Å even in the early-time dynamics,
as can be seen from the representative trajectories in Figs. 9
and 10.
Average VIEs for water and for e (aq) reported in Table V
can thus be cleanly identified as interfacial VIEs for these
species, and the interfacial VIE for liquid water (11.61 0.52
eV) is indistinguishable from the bulk value (11.53 0.51 eV).
Fore (aq), the interfacial VIE of 3.35 0.46 eV is discernibly
lower than the bulk value of 3.75 0.55, albeit not by much.
The latter simulations, which are based on the same trajectory
data as our previous ones in Ref. 42 but with a slightly better
treatment of the electronic structure (including a state-specific
PEqS approach rather than a perturbative one) generally
TABLE V . Average distance hdGDSibetween the c.o.m. of the solute and
the GDS, along with the corresponding average VIE, for configurations
extracted from a liquid/vapor MD simulation. Uncertainties reflect one
standard deviation.
Solute hdGDSi(Å) hVIEi(eV)
H2O 0.28 0.31 11.61 0.52
e 1.820.35 3.35 0.46222834-16 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
support our previous conclusion that the hydrated electron at
the air/water interface would be difficult to distinguish from its
counterpart in bulk water using liquid microjet photoelectron
spectroscopy.
VI. DISCUSSION
Winter et al.18have used liquid microjet photoelectron
spectroscopy, in conjunction with a variety of computational
strategies, to investigate the VIEs of aqueous halide anions and
alkali cations. In the following discussion, we consider the
QM/MM equilibrium PCM calculations reported in Ref. 18
alongside results from the present work. Errors relative to
experiment, for the calculations reported in Ref. 18 and for
the present work, are listed in Table VI.
Due to complications arising from water ionization in
the presence of explicit solvent molecules, as discussed in
Sec. IV C, the equilibrium PCM calculations in Ref. 18 include
only the bare ion in the QM region. The equilibrium nature
of the PCM used in that work affords an adiabatic ioniza-
tion energy (AIE) because all solvent degrees of freedom
are (implicitly) relaxed following ionization. (For a molec-
ular solute, or if explicit solvent molecules are included in the
QM region, the use of an equilibrium PCM without geometry
optimization in the ionized state affords something in between
a VIE and an AIE because the implicit solvent degrees of
freedom are relaxed but the explicit nuclear degrees of free-
dom are not.) QM/MM calculations reported in Ref. 18 also
include only the solute in the QM region, surrounded by clas-
sical point charges representing water molecules. Unlike the
equilibrium PCM calculations, ionization energies computed
using this approach are indeed VIEs, albeit ones that lack any
electronic polarization contribution from the solvent because
the point charges cannot be polarized upon ionization of the
solute.
For the alkali cations, VIEs computed at the QM/MM
level err by 6.4 eV (Li+) and 4.2 eV (Na+) and are clearly unac-
ceptable. AIEs computed with the equilibrium PCM approach
are larger than experimental VIEs, by 1.8 eV for Li+(aq) but
only by 0.05 eV for Na+(aq). This large discrepancy in accu-
racy is puzzling and is likely fortuitous, but in any case the
TABLE VI. Errors in computed VIEs, relative to experimental values from
Table III. Positive values indicate that the theoretical result is larger than the
experimental VIE.
Signed error vs. experiment (eV)
PEqS (noneq.)aPCM (noneq.)aPCM (eq.)
Solute Hybrid Spherical SAS IsodensitybQM/MM
Li+1.01 1.45 1.19 1.83c6.43c
Na+0.69 1.14 0.94 0.05c4.15c
H2O 0.22 0.33 0.24 ::: :::
e 0.0 0.5 0.5::: :::
F 0.08 0.21 0.06 3.78d 0.39d
Cl 0.05 0.24 0.19 2.75d 0.93d
aMP2 results from this work.
bSolute cavity determined as an isocontour of the electron density.
cCCSD(T)/cc-pV5Z results from Ref. 18.
dCCSD(T)/aug-cc-pVTZ results from Ref. 18.juxtaposition of calculated AIEs with experimental VIEs is
not reasonable, especially for the halides where the hydration
structure of X differs considerably from that of neutral X.
Overall, the enormity of the errors for this approach and for
the QM/MM calculations suggests that a proper description
of “specific” solvent effects, by means of explicit QM water
molecules, is essential, even when using a PCM. (This fact is
well known, e.g., in p Kacalculations.111) With respect to the
nonequilibrium PCM results with explicit solvent, for which a
direct comparison to experiment is appropriate, errors for both
Li+(aq) and Na+(aq) are 0.94–1.45 eV , which we consider to be
surprisingly large given the rather simple electronic structure
of these solutes.
MP2/PEqS calculations including 30 explicit QM water
molecules represent our best attempt for these systems, yet
these calculations still overestimate the cation VIEs by 1.0 eV
(Li+) and 0.7 eV (Na+). (That the error for Na+is more compa-
rable to that for Li+, as compared to the calculations reported
in Ref. 18, suggests that the accuracy of the equilibrium PCM
result for Na+is indeed fortuitous.) The reasons behind this
remaining error remain a topic for further study; a more thor-
ough examination of cavity construction is probably warranted
at the very least.
As compared to the alkali cations, where all methods con-
sidered here and in Ref. 18 overestimate the experimental VIE,
both positive and negative errors are observed for the halide
ions, perhaps simply because the errors are closer to zero.
(Exceptions are the AIEs computed with an equilibrium PCM,
which are much too small.) The QM/MM results are much
more accurate than they were for the cations, but this seems
fortuitous given that an anionic QM solute likely suffers more
from overpolarization by the point charges than does a cation
solute due to the more diffuse nature of the wave function.
Notably, the equilibrium PCM results for the anions are far less
accurate than those for the cations, but the relative accuracy
for cations represents a form of error cancellation as suggested
in Ref. 18. Namely, for the cations, only minor reorientation of
the solvent molecules is required to accommodate the resulting
divalent ion, due to the pre-existing favorable alignment of the
solvent dipoles in the monovalent state, but ionization of the
anions results in a charge-neutral solute and thus significant
reorientation of the solvent. As a result, one expects the AIE to
be much smaller than the VIE for the anions, but more similar
to the VIE for the cations.
Considering e (aq), we note that the experimental VIE
of 3.70.1 eV15that is provided in Table III represents an
upward revision of many previously reported VIEs in the range
3.3–3.4 eV .14,112–116The newer value has been called the “gen-
uine” binding energy of e (aq),15as it includes corrections
for scattering of the ejected electron that lead to wavelength
dependence in the photoelectron spectrum.14,15MP2/PEqS
calculations also afford a VIE of 3.7 eV , which is rather remark-
able given the complexity of this species though not out of
line with the accuracy that we obtain for the halide anions.
(Our calculation also represents an upward revision of the
MP2/6-31++Gvalue that we reported previously, based on
a perturbative version of the nonequilibrium PEqS method.42)
Assuming that the new experimental VIE withstands further
scrutiny, then the present MP2/PEqS calculation would seem222834-17 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
to affirm the simulation procedures71,79used to generate the
MD snapshots for this species. It also adds to the ongoing
debate regarding the detailed structure of the aqueous elec-
tron,70,71,117–124and in particular the question of whether this
species is cavity-forming or not; the MD snapshots used here
correspond to a cavity-forming electron, as can be seen in
Fig. 1. Finally, we note that amongst the solutes examined
here e (aq) affords the largest discrepancy between PEqS and
PCM values of the VIE. This may represent a failure of IEF-
PCM to adequately describe volume polarization by this highly
delocalized non-classical solute.110
Finally we consider the 1 b1state of liquid water. Errors
in the VIE, obtained with nonequilibrium methods, are
0.2–0.3 eV . Crucial to the success of this method is the
FragMO/MOM SCF technique, in order to ionize a central
H2O molecule not too near the QM/continuum boundary.
This approach will require some care if the full photoelec-
tron spectrum is desired, to select the appropriate orbitals for
ionization, and may be hopeless in cases where the MO picture
of photoionization completely breaks down.125Unfortunately,
alternative electronic structure techniques that afford ioniza-
tion energies directly,126–130and can deal with “shake-up”
ionization events,125,126,130are considerably more expensive.
Alternatively, a simple technique to correct the Kohn-Sham
eigenvalue spectrum has recently been shown to afford valence
photoelectron spectra that compare well with experiment131
and is no more expensive than standard density-functional
calculations. In our estimation, however, this “potential adjus-
tors” technique131is unlikely to work across the entire
(core + valence) photoelectron spectrum.
VII. CONCLUSIONS
We have presented a detailed description of the theory
and implementation of the state-specific nonequilibrium PEqS
method and its application to compute aqueous-phase VIEs.
In contrast to PCMs, which are the de facto implicit solva-
tion models in electronic structure calculations, PEqS calcu-
lations require discretization of three-dimensional space and
not simply a two-dimensional cavity surface. This makes PEqS
considerably more expensive than PCM calculations despite
the efficient multigrid approach described here. Computational
expense notwithstanding, the PEqS approach has the advan-
tage that it treats volume polarization (charge leakage outside
of the QM region) exactly , up to discretization errors. That said,
among the solutes considered here this seems to matter only
fore (aq). For more “classical” solutes, nonequilibrium PCM
calculations with the solvent accessible surface construction73
are within 0.2 eV of PEqS results, thus validating the more
affordable PCM approach in bulk solution.
A more important advantage of the PEqS approach is that
it is naturally applicable to arbitrary (and therefore anisotropic)
dielectric environments, defined by a dielectric function "(r).
This function could be defined based on the electron den-
sity,40,62–67but here we adopt an approach analogous to PCM
calculations and define a surface to delineate the boundary
between the atomistic QM region and its continuum envi-
ronment. The usual PCM prescription using atom-centered
vdW spheres, however, proves to be problematic when explicitwater molecules are included in the QM region, leading to
unphysical high-dielectric regions between these explicit sol-
vent molecules. Oddly, this problem is not often discussed in
the quantum chemistry literature although a similar problem
in biomolecular simulation has been widely discussed,132–136
where in the context of Poisson-Boltzmann electrostatics
calculations the vdW cavity construction may leave high-
dielectric regions in the hydrophobic interior of a protein. In
this work, we introduced a “hybrid” cavity model that avoids
this problem.
Here, we used the PEqS approach, in conjunction with
QM regions containing 30 explicit water molecules, to com-
pute VIEs for neat liquid water as well as F (aq), Cl (aq),
Na+(aq), Li+(aq), and e (aq), both in bulk liquid water and at
the air/water interface. Ionization energies for most of these
systems lie below (or are similar to) that of liquid water itself,
and a na ¨ıve calculation of the lowest-energy ionized state
thus results in ionization of H 2O rather than the solute of
interest. We circumvent this problem by means of a fragment-
based initial guess combined with the maximum overlap
method.106
We find that nonequilibrium corrections to VIEs, which
are missing from continuum models based only on the static
dielectric constant, amount to 0.5–1.2 eV for each system
investigated in this work. VIEs computed at the MP2/PEqS
(noneq.) level for liquid water, F (aq), Cl (aq), and e (aq)
agree with experimental results to within 0.2 eV , with slightly
larger errors when nonequilibrium PCMs are used as a sub-
stitute for PEqS. For reasons that remain unclear, however,
errors for alkali cations are larger, e.g., 1.0 eV for Li+(aq) at
the MP2/PEqS (noneq.) level. Consistent with our previous
work on e (aq),42there is very little difference between VIEs
computed at the air/water interface versus those in bulk water,
for any of the solutes considered here. For liquid water, the
same conclusion has recently been reported based on G0W0
calculations.137
All of the QM calculations in this work were performed
at the MP2 level, but the PEqS method works equally well
in the context of density functional theory, and also for
other correlated wave functions, if the Hartree-Fock density
is used in Poisson’s equation. A nonequilibrium treatment
of vertical excitation energies within the PEqS framework
could be accomplished by adapting PCM algorithms described
previously in the context of time-dependent density func-
tional theory48,49and the algebraic diagrammatic construc-
tion.50Finally, it should be possible to adapt this methodol-
ogy to develop nonequilibrium versions of density-dependent
dielectric solvation models in which "(r) is a functional of
(r).40,62–64,138This would eliminate some of the arbitrariness
in construction of the QM/continuum boundary. We hope to
address this, along with the sensitivity of PEqS results to the
Gaussian smearing of the nuclear charges, in future work.
ACKNOWLEDGMENTS
This work was supported by National Science Founda-
tion Grant Nos. CHE-1300603 and CHE-1665322. Calcula-
tions were performed at the Ohio Supercomputer Center under
Project No. PAA-0003.139J.M.H. is a fellow of the Alexander222834-18 M. P . Coons and J. M. Herbert J. Chem. Phys. 148, 222834 (2018)
von Humboldt Foundation and serves on the Board of Directors
of Q-Chem, Inc.
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1.4932356.pdf | Scattering of high-energy magnons off a magnetic skyrmion
Sarah Schroeter and Markus Garst
Citation: Low Temp. Phys. 41, 817 (2015); doi: 10.1063/1.4932356
View online: http://dx.doi.org/10.1063/1.4932356
View Table of Contents: http://aip.scitation.org/toc/ltp/41/10
Published by the American Institute of Physics
Scattering of high-energy magnons off a magnetic skyrmion
Sarah Schroeter and Markus Garsta)
Institut f €ur Theoretische Physik, Universit €at zu K €oln, Z €ulpicher Str. 77a, K €oln 50937, Germany
(Submitted March 30, 2015)
Fiz. Nizk. Temp. 41, 1043–1053 (October 2015)
We discuss the scattering of high-energy magnons off a single magnetic skyrmion within the
field-polarized ground state of a two-dimensional chiral magnet. For wavevectors larger than the
inverse skyrmion radius, krs/C291 the magnon scattering is dominated by an emerging magnetic
field whose flux density is essentially determined by the topological charge density of the skyrmiontexture. This leads to skew and rainbow scattering characterized by an asymmetric and oscillating
differential cross section. We demonstrate that the transversal momentum transfer to the skyrmion
is universal due to the quantization of the total emerging flux while the longitudinal momentumtransfer is negligible in the high-energy limit. This results in a magnon-driven skyrmion motion
approximately antiparallel to the incoming magnon current and a universal relation between current
and skyrmion-velocity.
VC2015 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4932356 ]
1. Introduction
The experimental discovery of skyrmions in chiral
magnets1–7and in magnetic monolayers8–10has triggered an
increasing interest in the interaction of spin currents withtopological magnetic textures.
11–30It has been demon-
strated13,16that skyrmions can be manipulated by ultralow
electronic current densities of 106A/m2, which is five orders
of magnitudes smaller than in conventional spintronic appli-cations using domain walls. The adiabatic spin-alignment ofelectrons moving across a skyrmion texture results in anemergent electrodynamics implying a topological
11,12,30as
well as a skyrmion-flow Hall effect.17In insulators, the inter-
play of thermal magnon currents and skyrmions is marked
by a topological magnon Hall effect and a magnon-drivenskyrmion motion.
23–25The topological nature of the mag-
netic skyrmions is responsible for a peculiar dynamics31–35
that is also at the origin of these novel spintronic and calori-
tronic phenomena, which are at the focus of the fledgling
field of skyrmionics.22
In two spatial dimensions, skyrmions are identified by
the topological charge density
qtop¼1
4p^n@x^n/C2@y^n/C0/C1; (1)
where ^nis the orientation of the magnetization vector. For a
magnetization homogeneously polarized at the boundary, the
spatial integralÐd2rqtop¼Wis quantized, W2Z, and thus
allows to count skyrmions within the sample. In turn, a finitewinding number Wtranslates to a gyrocoupling vector Gin
the Thiele equation of motion of the skyrmion,
36and the
resulting gyrotropic spin-Magnus force governs its dynam-ics.
37As a consequence, in the presence of an applied
electronic spin current, the skyrmions will acquire a veloc-
ity14,15,17that remains finite in the limit of adiabatic spin-
transfer torques and small Gilbert damping a, giving rise to auniversal current-velocity relation.
18
In order to address the interaction of magnon currents
with magnetic textures, a corresponding adiabatic approxi-
mation has been recently invoked on the level of theLandau–Lifshitz–Gilbert equation by Kovalev and
Tserkovnyak.38This approximation has been used in Refs.
23and24to derive an effective Thiele equation of motion
for the skyrmion coordinate Rin the presence of a magnon
current density J
G/C2_R¼/C0G/C2veffþbveffþ/C1/C1/C1 ; (2)
with b¼0 in the adiabatic limit. The effective velocity veff
¼glBJ=ð/C22hm0Þis related to the current density via the
g-factor g, the Bohr magneton lB>0 and the local magnet-
ization m0. The gyrocoupling vector is given by G
¼/C04p^z/C22hm0=ðglBÞwith units of spin density corresponding
to a flux of /C02p/C22hp e ra r e ao fas p i n /C01
2in a two-
dimensional system with the unit normal vector ^z. The dots
in Eq. (2)represent further terms omitted for the purpose of
the following discussion, that is, in particular, a damping
force proportional to the Gilbert constant a.N e g l e c t i n g
these additional terms, Eq. (2)predicts for b¼0, similar to
the skyrmion-driven motion by electronic currents, a uni-
versal current-velocity relation _R¼/C0veff¼/C0glBJ=ð/C22hm0Þ
with a skyrmion velocity that is antiparallel to J.
Consequently, a magnon current generated by a thermal
gradient will induce a skyrmion motion towards the hotregion of the sample, which was indeed observed numeri-
cally.
23,24,27Mochizuki et al.25also used Eq. (2)with b¼0
to account for the experimental observation of a thermallyinduced rotation of a skyrmion crystal.
However, the question arises as to when the adiabatic
limit of Eq. (2)is actually applicable and under what condi-
tions. The validity regime of the adiabatic approximation for
magnon-driven motion of magnetic textures has not beenexplicitly discussed in Ref. 38. In fact, in order to account
quantitatively for their numerical experiment Lin et al .
24
introduced the bparameter in Eq. (2)on phenomenological
grounds calling it a measure for non-adiabaticity.
Subsequently, Kovalev28argued that a finite bparameter
arises due to dissipative processes.
In contrast, we have recently shown by considering the
magnon–skyrmion scattering problem29that a monochromatic
1063-777X/2015/41(10)/9/$32.00 VC2015 AIP Publishing LLC 817LOW TEMPERATURE PHYSICS VOLUME 41, NUMBER 10 OCTOBER 2015
magnon current with energy ewill give rise to a reactive
momentum-transfer force in the Thiele equation which readsin linear response
G/C2_R¼kr
?ðeÞð^z/C2JeÞþkrkðeÞJeþ/C1/C1/C1 ; (3)
where the magnon dispersion is e¼egapþð/C22hkÞ2=ð2MmagÞ
with the magnon gap egapand the magnon mass Mmag. This
force on the right-hand side of Eq. (3)is determined by the
two-dimensional transport scattering cross sections
rkeðÞ
r?eðÞ/C18/C19
¼ðp
/C0pdv1/C0cosv
/C0sinv/C18/C19dr
dv; (4)
where dr=dvis the energy-dependent differential scattering
cross section of the skyrmion. In the limit of low-energies krs
/C281, where rsis the skyrmion radius, s-wave scattering is found
to dominate so that r?ðeÞ! 0 and, as shown in Ref. 23,t h e
force becomes longitudinal to Je.T h i s ,i nt u r n ,i m p l i e sa
skyrmion motion approximatel y perpendicular to the magnon
current, _R!krkðeÞ
jGj^z/C2Je, thus maximally violating the predic-
tions of the adiabatic limit of Eq. (2). This implies that Eq. (2)is
not valid for low-energy magnons whose wavevector is compa-
rable or smaller than the inverse size of the texture.
It is one of the aims of this work to demonstrate explic-
itly that in the high-energy limit, krs/C291, on the other hand,
the momentum-transfer force of Eq. (3)due to a monochro-
matic magnon wave indeed reduces to the form of Eq. (2).
The effective velocity in this case, however, is to be identi-
fied with veff¼jAj2ð/C22hk=MmagÞwhere Ais the amplitude of
the incoming magnon wave. In the high-energy limit themagnon-skyrmion interaction is dominated by a scatteringvector potential, i.e., an emerging orbital magnetic fieldwhose flux is quantized and related to the skyrmion topol-ogy. As a result, the transversal momentum transfer assumesa universal value in the high-energy limit kr
?ðeÞ! 4pas
anticipated in Ref. 25. Moreover, the longitudinal momen-
tum transfer yields a reactive contribution, bs, to the b
parameter that, in this limit, is determined by the square ofthe classical deflection function H(b) integrated over the
impact parameter b, see Fig. 1(b)
b
e¼jGj
8pkð1
/C01dbHbðÞðÞ2: (5)
As the scattering is in forward direction at high energies,
HðbÞ/C241=k, the parameter vanishes as be/1=kso that it is
indeed small for large jrs/C291.
The outline of the paper is as follows. In Sec. 2we
shortly review the definition of the magnon–skyrmion scat-tering problem and some of the main results of Ref. 29.I n
Sec. 3we examine the scattering properties of high-energy
magnons including the skew and rainbow effects, the totaland transport scattering cross sections, and the magnon pres-
sure on the skyrmion leading to Eq. (2). We finish with a
short discussion in Sec. 4.
2. Skyrmionic soliton and its spin-wave excitations
This section closely follows Ref. 29and reviews the
magnon-skyrmion scattering problem in a two-dimensionalchiral magnet. We start with the standard model for a cubic
chiral magnet restricted to a two-dimensional plane that isdescribed by the energy functional
39,40
E¼qs
2@a^nj/C0/C12þ2Qeiaj^ni@a^nj/C02j2^n^Bhi
; (6)
with spatial index a2f1;2g¼f x;ygand i;j2f1;2;3g,
2iajis the totally antisymmetric tensor with 2123¼1, and qs
is the stiffness. The two length scales are given by the wave-
vectors Qandj. The former determines the strength of the
spin-orbit Dzyaloshinskii–Moriya interaction, that we choseto be positive, Q>0. The latter, j>0, measures the
strength of the applied magnetic field, that is applied perpen-
dicular to the two-dimensional plane, ^B¼^z. We neglect
cubic anisotropies, dipolar interactions as well as magneticanisotropies for simplicity. The latter can be easily includedresulting in an additional length scale.
2.1. Skyrmionic saddle-point solution
The theory (6)possesses a topological soliton solution,
i.e., a skyrmion, as first pointed out by Bogdanov andHubert.
41,42With the standard parameterization of the unit
vector ^nT
s¼ðsinhcosu;sinhsinu;coshÞ, the skyrmion
obeys
h¼hqðÞ;u¼vþp
2; (7)
where qandvare polar coordinates of the two-dimensional
spatial vector r¼qðcosv;sinvÞ. The polar angle hobeys
the differential equation
h00þh0
q/C0sinhcosh
q2þ2Qsin2h
q/C0j2sinh¼0;(8)
with the boundary conditions hð0Þ¼pand lim q!1hðqÞ
¼0. At large distances qj/C291, the polar angle obeys the
asymptotics hðqÞ/C24e/C0jq=ffiffiffiqp. The resulting skyrmion tex-
ture is illustrated in Fig. 1(a). The associated topological
charge density
FIG. 1. (a) A chiral magnetic skyrmion texture of linear size rs. (b)
Illustration of a classical magnon trajectory within the xyplane scattering
off a skyrmion positioned at Rwith impact parameter band classical deflec-
tion angle H.818 Low Temp. Phys. 41(10), October 2015 S. Schroeter and M. Garst
qs
top¼1
4p^ns@x^ns/C2@y^ns/C0/C1¼1
4ph0sinh
q; (9)
integrates toÐd2rqtop¼/C01 identifying the solution as a
skyrmion. The skyrmion radius rscan be defined with the
help of the area
ð
d2rð1/C0^nzÞ=2¼pr2
s;
and it is found to approximately obey rs/C241=j2.
The skyrmion is a large-amplitude excitation of the fully
polarized ground state as long as its energy is positive, whichis the case for j>j
crwhere j2
cr/C250:8Q2, which is the re-
gime we focus on. For smaller values of j, skyrmions prolif-
erate resulting in the formation of a skyrmion crystal groundstate.
2.2. Magnon-skyrmion scattering problem
Magnon wavefunction . The magnons correspond to
spin-wave excitations around the skyrmion solution ^nsthat
can be analyzed in the spirit of previous work by Ivanov andcollaborators.
43–46We introduce the local orthogonal frame
^ei^ej¼dijwith ^e1/C2^e2¼^e3, where ^e3ðrÞ¼ ^nsðrÞtracks the
skyrmion profile. For the two orthogonal vectors we use
^eT
1¼ð /C0 sinu;cosu;0Þand
^eT
2¼/C0 ð /C0 coshcosu;sinhsinu;coshÞ:
The excitations are parameterized in the standard fashion
^n¼^e3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1/C02jwj2q
þ^eþwþ^e/C0w/C3; (10)
where wis the magnon wavefunction and ^e6¼1ffiffi
2p^e16i^e2Þ ð .
For large distances, q/C29rs, this parameterization assumes
the form
^n/C25^zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1/C02jwj2q
þ1ffiffiffi
2p ^xþi^yðÞ /C0e/C0ivw/C0/C1
/C0e/C0ivw/C0/C1
þc:c:/C18/C19
:
(11)
It is important to note that the local frame ^eicorresponds to
a rotating frame even at large distances reflected in the phasefactor /C0e
/C0ivin the second term. For the discussion of mag-
non scattering, it will be convenient to introduce a wave-function w
labwith respect to a frame that reduces to the
laboratory frame at large distances, that is simply obtainedby the gauge transformation
w
labðr;tÞ¼/C0 e/C0ivwðr;tÞ: (12)
Magnon Hamiltonian . In order to derive an effective
Hamiltonian for w, we consider the Landau–Lifshitz equation
@t^n¼/C0c^n/C2Beff; (13)
with c¼glB=/C22h, where the effective magnetic field Beffðr;tÞ
¼/C01
m0dE
d^nðr;tÞis determined by the functional derivative of the
integrated energy density E¼ÐdtdrE. Expanding (13) in
lowest order in w, one finds that the spinor WT¼ðw;w/C3Þisgoverned by a bosonic Bogoliubov–deGennes (BdG)
equation
i/C22hsz@tW¼HW; (14)
with the Hamiltonian
H¼/C22h2/C0i1r/C0 sza ðÞ2
2Mmagþ1V0þsxVx; (15)
where rT¼ð@x;@yÞ, and sxandszare Pauli matrices. The
potentials are given by
V0qðÞ¼egap
j2/C0sin2h
2q2/C0Qsin 2 hðÞ
2q/C0Q2sin2h
þj2cosh/C0Qh0/C0h02
2!
; (16)
V0qðÞ¼egap
j2/C0sin2h
2q2/C0Qsin 2 hðÞ
2q/C0Qh0/C0h02
2 !
:(17)
The magnon energy gap is defined by
egap¼glBqsj2
m0¼/C22h2j2
2Mmag; (18)
which also identifies the magnon mass Mmag. The vector
potential reads a¼avðqÞ^vwith ^vT¼ð /C0 sinv;cosvÞand
av¼cosh
q/C0Qsinh: (19)
It obeys the Coulomb gauge ra¼0. The polar angle in all
potentials is the soliton solution, h¼h(q), and depends on
the distance q.
Effective magnetic flux . Far away from the skyrmion the
Hamiltonian simplifies H!H 0forq!1 with
H0¼/C22h2/C0i1r/C0 sz1
q^v/C16/C172
2Mmagþ1egap: (20)
The remaining vector potential is attributed to the choice of
the rotating orthogonal frame in the definition of the magnon
wavefunction, see Eq. (11). It can be easily eliminated by
the gauge transformation (12)
W!Wlab¼e/C0iszðvþpÞW; (21)
av!av
lab¼av/C01
q¼cosh/C01
q/C0Qsinh: (22)
With respect to this laboratory orthogonal frame, the vector
scattering potential alab¼av
lab^vvanishes exponentially for
large distances, q/C29rs.
The associated flux B¼r/C2ð /C22halabÞ¼B ^zwill play an
important role in the following discussion, where BðrÞ
¼/C22h
q@qðqav
labðqÞÞ. According to Stokes’ theorem the total fluxÐd2rBðrÞ¼0 vanishes as alabis exponentially confined to
the skyrmion radius. However, there is an interesting spatial
flux distributionLow Temp. Phys. 41(10), October 2015 S. Schroeter and M. Garst 819
BðrÞ¼/C0 4p/C22hdðrÞþB regðjrjÞ; (23)
BregqðÞ¼/C04p/C22h/C0qs
top/C0Q
4pq@qqsinhðÞ/C18/C19
: (24)
Since for small distances av
labðqÞ!/C0 2=q, there is a singular
flux contribution at the skyrmion origin with quantizedstrength /C04p/C22h. As it is quantized, this singular flux will not
contribute to the magnon scattering. The regular part of theeffective magnetic flux, B
reg, only depends on the radius q
and is spatially confined to the skyrmion area. Its spatial
distribution can be related with the help of Eq. (9)to the top-
ological charge density qs
topof the skyrmion in addition to a
term proportional to Q. While /C0qs
topis always positive, the
latter term can also be negative so that Bregas a function
of distance qeven changes sign for lower values of j2, see
Fig. 2. The spatial integral over the second term of Eq. (24)
however vanishes so that the total regular flux
ð
d2rBregðqÞ¼/C0 4p/C22hð
d2rqs
top¼4p/C22h;
is quantized and determined by the topological charge of the
skyrmion.25,47
2.3. Magnon spectrum
In order to solve Eq. (14) for the magnon eigenvalues
and eigenfunctions, one uses the angular momentum basisWðr;tÞ¼expð/C0iet=/C22hþimvÞg
mðqÞwith positive energy
e/C210. The angular momentum /C22hmturns out to be a good
quantum number and the wave equation (14) reduces to a ra-
dial eigenvalue problem for gmðqÞthat can be solved with
the help of the shooting method.29In order to obtain positive
expectation values of the Hamiltonian, one has to look foreigenfunctions with a positive norm
ð
1
0dqqg†
mðqÞszgmðqÞ>0: (25)
The resulting spectrum is shown in Fig. 3as a function
of the parameter j2=Q2that measures the strength of the
magnetic field. The magnon continuum with the scatteringstates are confined to energies larger than the magnon gap
egap/j2which increases linearly with the field (black solid
line). In the field range shown, there are three subgap statesthat correspond to bound magnon–skyrmion modes. Whilethe breathing mode with angular momentum m¼0 exists
over the full field range, a quadrupolar mode with m¼/C0 2
emerges for lower fields just before the field-polarized statebecomes globally unstable at j
2
cr/C250:8Q2(dashed-dotted
line). The eigenenergy of the latter finally vanishes atj
2
bimeron /C250:56Q2, indicating a local instability of the theory
with respect to quadrupolar deformations of the skyrmion,i.e., the formation of a bimeron.
48Furthermore, a sextupolar
mode with m¼/C03 only exists within the metastable regime.
The corresponding eigenfunctions of these modes do notpossess any nodes. We have not yet found bound modeswith a single or more nodes, which might however emergeform¼/C01 at larger fields.
Apart from the modes shown in Fig. 3, the spectrum
ofHalso contains a zero mode with angular momentum
m¼/C01 given by
g
zm
/C01¼1ffiffiffi
8psinh
q/C0h0
sinh
qþh00
BBB@1
CCCA: (26)
This zero mode is related to the translational invariance
of the theory (6)that is explicitly broken by the skyrmion
solution. The real and imaginary part of the amplitude ofthe eigenfunction (26) correspond to translations of the
skyrmion within the two-dimensional plane.
3. High-energy scattering of magnons
The properties of the magnon scattering states for arbi-
trary energies, e/C21egap, have been discussed in Ref. 29.I n
the present work, we elaborate on the scattering of magnons
FIG. 2. Regular part of the effective magnetic flux density (24) for various
values of j2=Q2. For lower values of j2=Q2density close to the skyrmion
center is suppressed and even becomes negative for j2=Q2/H113511:3. As a result,
the effective local Lorentz force evaluated along a classical magnon trajec-
tory with b¼0 changes sign resulting in a suppression of the deflection
angle.
FIG. 3. Magnon spectrum in the presence of a single skyrmion excitation as
a function of j2=Q2measuring the strength of the magnetic field.29The
magnon gap egap¼eDMj2=Q2increases linearly with the field (black solid
line). The field-polarized state becomes unstable at j2
cr/C250:8Q2(dashed-
dotted line) while the theory (14) becomes locally unstable at
j2
bimeron /C250:56Q2. Apart from the zero mode (not shown), there exist three
subgap modes with angular momentum m¼0,/C02,/C03.820 Low Temp. Phys. 41(10), October 2015 S. Schroeter and M. Garst
in the high-energy limit, e/C29egap, which corresponds to
magnon wavevectors much larger than the inverse skyrmion
radius, krs/C291. In this limit, the treatment of the scattering
simplifies considerably allowing for a transparent discussionof characteristic features.
In the high-energy limit the magnon-skyrmion interac-
tion is governed by the scattering vector potential
aðrÞ¼a
vðqÞ^vof Eq. (19) so that the scattering has a purely
magnetic character. In particular, in this limit one canneglect the anomalous potential V
x, and the BdG equation
(14) reduces to a Schrodinger equation for the magnon
wavefunction
i/C22h@tw¼/C22h2/C0i$2a ðÞ2
2Mmagþegap !
w: (27)
Setting wðr;tÞ¼expð/C0iekt=/C22hÞexpðimvÞgmðqÞwith the dis-
persion ek¼egapþ/C22h2k2
2Mmagand wavevector k>0, one obtains
the radial wave equation for gmðqÞ
/C0@2
qþ@q
q/C18/C19
þm/C0qavqðÞ/C0/C12
q2/C0k2"#
gm¼0: (28)
For large distances qavðqÞ! 1, which identifies the angular
momentum of the incoming wave to be Lz¼/C22hðm/C01Þ.
3.1. Eikonal approximation
As we are interested in the high-energy limit, we can
treat this wave equation in the eikonal approximation.However, in order to make contact with Ref. 29, we first
give the resulting phase shift within the WKB approximation
that is obtained by following Langer
49,50
dWKB
m¼ð1
q0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2/C0m/C0qavqðÞ/C0/C12
q2s
/C0k0
@1
Adq
þp
2jm/C01j/C0kq0; (29)
where q0is the classical turning point. The eikonal approxi-
mation for the phase shift is then obtained by taking the limit
k!1 while keeping the impact parameter b¼Lz=ð/C22hkÞ
fixed, dWKB
m!d1ðbÞ, yielding
d1bðÞ¼bð1
jbjav
labqðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q2/C0b2p dq¼bð1
1av
labsjbjðÞffiffiffiffiffiffiffiffiffiffiffiffiffi
s2/C01p ds; (30)
where we used qav
labðqÞ¼qavðqÞ/C01, see Eq. (22), and in
the last equation we substituted s¼q=jbj. This phase shift is
odd with respect to b, i.e., d1¼/C0d1ð/C0bÞ. Note that the
scattering is non-perturbative even in the high-energy limitin the sense that the phase shift d
1ðbÞcovers the entire inter-
valð/C0p;pÞas a function of b, see Fig. 4. In particular, in the
limit of small impact parameter b!0:
d1bðÞ!bð1
1/C02=sjbjðÞffiffiffiffiffiffiffiffiffiffiffiffiffi
s2/C01p ds¼/C0psgnbðÞ: (31)
For impact parameters larger than the skyrmion radius,
b/C29rs, the phase shift vanishes exponentially.The deflection angle in the eikonal approximation is
given by the derivative of d1ðbÞ
H1bðÞ¼2/C22h@d1bðÞ
@Lz¼2
kd0
1bðÞ¼Hreg
1bðÞ/C04p
kdbðÞ:(32)
The step of d1(b) for head-on collisions, see Eq. (31),
leads to the delta function d(b). The classical deflection func-
tion is given by the regular part, which reads
Hreg
1bðÞ/C02
/C22hkð1
1sjbjBregsjbjðÞffiffiffiffiffiffiffiffiffiffiffiffiffi
s2/C01p ds; (33)
¼1
/C22hkð1
/C0kiBregffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2þx2p/C16/C17
dx; (34)
where in the last equation we substituted x¼jbjffiffiffiffiffiffiffiffiffiffiffiffiffi
s2/C01p
and
used that the integrand is an even function of x. It is deter-
mined by the regular part of the flux density, Breg, given in
Eq.(24), integrated along a straight trajectory shifted from
thex-axis by the impact parameter b. Its behavior as a func-
tion of bis shown in Fig. 5for various values of j2=Q2. The
FIG. 4. Scattering phase shift for high-energy magnons (30) as a function of
impact parameter bfor different values of j2=Q2. The scattering is nonper-
turbative as the phase shift assumes values within the entire interval
ð/C0p;pÞ.
FIG. 5. Classical deflection angle for scattering of high-energy magnons
(33) as a function of impact parameter for different values of j2=Q2. In the
high-energy limit, the scattering is in the forward direction with a deflection
angle decreasing with increasing wavevector kasHreg
1ðbÞ/C241=k. The inset
focuses on the change of curvature at b¼0 for j2/C251:6Q2with the same
units on the vertical axis.Low Temp. Phys. 41(10), October 2015 S. Schroeter and M. Garst 821
deflection angle is always positive implying that, classically,
the Lorentz force attributed to Bregalways skew scatters the
magnons to the right-hand side from the perspective of theincoming wave even for negative impact parameters, seeFig. 1(b). Note that the deflection angle possesses a local
minimum at b¼0 for j
2/H113511:6Q2that however gets filled
and transitions into a maximum for larger values of j. This
change of curvature at b¼0 is related to the change of cur-
vature of the flux density B00
regðqÞat the origin p ¼0, see
Fig.2, that happens for a similar value of j. As the total flux
ofBregis quantized, the deflection angle integrated over the
impact parameter is just given by the universal value
ð1
/C01dbHreg
1ðbÞ¼4p=k:
3.2. Differential cross section
In the following, we consider a magnon scattering setup
where an on-shell magnon plane wave with wavevector k¼
k^xalong the x-direction and amplitude Adefined within the
laboratory orthogonal frame, see Eq. (12), is impinging on
the skyrmion, see also Fig. 1(b). At large distances this
wavefunction assumes the asymptotic behavior
wlabr;tðÞ¼Ae/C0iek=/C22heikrþfvðÞeikq
ffiffiffiqp !
; (35)
where the scattering amplitude is given by
fvðÞ¼e/C0ip=4
ffiffiffiffiffiffiffiffi
2pkpX1
m¼/C01eim/C01ðÞ vei2dm/C01 ðÞ : (36)
Note that the additional phase factor e/C0ivarises from the
gauge transformation (12). The differential cross section is
then obtained by@r
@v¼jfðvÞj2.
High-energy limit of the scattering amplitude . In the
high-energy limit, we can replace the sum over angular mo-mentum numbers by an integral over the impact parameter,b¼ðm/C01Þ=k, so that the scattering amplitude reads
approximately
f
1vðÞ¼e/C0ip=4
ffiffiffiffiffiffiffiffi
2pkp kð1
/C01dbeibkvei2d1bðÞ/C01 ðÞ ; (37)
with d1ðbÞdefined in Eq. (30). The differential cross section
in this limit
@r1
@v¼jf1vðÞj2¼k
Q2Skv=QðÞ ; (38)
is then determined by the dimensionless function S, which is
shown in Fig. 6.
The support of the differential cross section is approxi-
mately limited by the extremal values of the classical deflec-tion angle of Eq. (33) and Fig. 5. Note that the angle vis
defined in a mathematically positive sense so that a positiveHtranslates to a negative value of v. It is strongly asymmet-
ric with respect to forward scattering reflecting the skewscattering arising from the Lorentz force of the emerging
magnetic field B
reg.
Rainbow scattering and Airy approximation . Moreover,
the differential cross section exhibits oscillations. These canbe attributed to an effect known as rainbow scattering. Asthe function H
reg
1(b) is even in b, there exist for a given clas-
sically allowed deflection angle Halways at least one pair
6bclof impact parameters that solve Hreg
1ð6bclÞ¼H. For a
given angle Hthe magnons might, therefore, either pass the
skyrmion on its right- or left-hand side; these classical trajec-tories interfere leading to the oscillations in dr=dv.
First, consider values j
2/H114071:6Q2for which Hreg
1(b) pos-
sesses only a single maximum at b¼0. The maximum value
Hreg
1(0) is known as rainbow angle and for values of vclose
to/C0Hreg
1(0), the interference effect of classical trajectories
can be illustrated with the help of the Airy approximation for
the scattering amplitude. For such values of v, the/C01 in the
integrand of Eq. (37) can be neglected as it only contributes
to forward scattering. Expanding the exponent of the remain-ing integrand up to third order in bone then obtains
f
1vðÞjAiry
¼e/C0ip=4
ffiffiffiffiffiffiffiffi
2pkp kð1
/C01dbexp ibkvþHreg
10ðÞ/C0/C1
þik
6H00reg
10ðÞb3/C20/C21
¼ffiffiffiffiffiffiffiffi
2pkp
e/C0ip=4
kjH00reg
10ðÞj=2/C2/C31=2Ai/C0kvþHreg
10ðÞ/C0/C1
kjH00reg
10ðÞj=2/C2/C31=3 !
; (39)
where in the last equation we identified the integral represen-
tation of the Airy function Ai using that H00reg
1<ð0Þ. In the
inset of Fig. 6, we compare the differential cross section
atj2¼2Q2with the Airy approximation resulting from
Eq.(39). The latter reproduces the exponential decrease for
large angles v</C0Hreg
1ð0Þcorresponding to the dark side
and also the oscillations on the bright side, v>/C0Hreg
1ð0Þ,o f
the rainbow angle. It of course fails close to forward scatter-
ing and for positive angles v>0 where the classical deflec-
tion angle has lost its support.
Close to j2/C251:6Q2even the derivative H00reg
1ð0Þvan-
ishes, see inset of Fig. 5, giving rise to a cubic rainbow
effect.51Finally, for smaller values of j2there also exist two
FIG. 6. Differential cross section of high-energy magnons (38) for various
values of j2=Q2. It is asymmetric with respect to v¼0 due to skew scatter-
ing, and the oscillations are attributed to rainbow scattering. The inset com-
pares the curve for j2=Q2¼2 with the Airy approximation (39) (green solid
line) with the same units on the vertical axis; the arrow indicates the position
of the corresponding rainbow angle /C0kHreg
1ð0Þ=Q.822 Low Temp. Phys. 41(10), October 2015 S. Schroeter and M. Garst
pairs of classical trajectories that interfere in the differential
cross section.
3.3. Total and transport scattering cross section
We continue with a discussion of the total, rtot
¼Ðp
/C0pdvdr=dv, and the transport scattering cross section
defined in Eq. (4). In order to determine their high-energy
limit, one first expresses dr=dv¼jfðvÞj2in terms of the
exact representation (36) for the scattering amplitude fðvÞ
and evaluates the integral over v. Afterwards one takes the
high-energy limit k!1 with keeping the impact parameter
b¼(m/C01)/kfixed.
The total scattering cross section of the skyrmion then
reduces to
r1
tot¼4ð1
/C01dbðsind1ðbÞÞ2: (40)
It saturates to a finite value in the high-energy limit, and its
dependence on jis shown in Fig. 7. It decreases with
increasing jand thus decreasing skyrmion radius rsas
expected. One might expect that r1
tot/C24rswhich however
only holds approximately.
Using that d1(b) is an odd function of b, we obtain for
the transport scattering cross section r?ðeÞin the high-
energy limit
r1
?eðÞ¼8
kð1
0dbd0
1bðÞsind1bðÞ ðÞ2; (41)
¼8
kd1
2/C0sin 2 d1ðÞ
4/C18/C190
/C0p¼4p
k: (42)
In the last line, we further used the boundary values of the
function d1ðbÞ. It vanishes r1
?ðeÞ/C241=k, but with a univer-
sal prefactor that is independent of j.
Finally, for the longitudinal transport scattering cross
section we obtain for krs/C291
r1
keðÞ¼4
k2ð1
0db2d0
1ðÞ2sind0
1ðÞ2/C0d00
1sind1cosd1/C16/C17
:
(43)After integrating by parts this simplifies to
r1
keðÞ¼4
k2ð1
0dbd0
1bðÞðÞ2¼ð1
/C01db1
2Hreg
1bðÞ/C0/C12:(44)
It is given by the square of the classical deflection angle (33)
integrated over the impact parameter b. It vanishes as r1
k
/C241=k2in the high-energy limit with a prefactor whose j
dependence is shown in Fig. 8. On dimensional grounds
one might expect k2r1
k/C241=rswhich again only holds
approximately.
3.4. Magnon pressure in the high-energy limit
We have shown in Ref. 29by considering the energy-
momentum tensor of the field theory that the monochromaticplane wave of (35) with wavevector k¼k^xleads to a
momentum-transfer force in the Thiele equation of motionof the form given in Eq. (3)with the magnon current
J
e¼^xjAj2m0/C22h
glB/C22hk
Mmag¼jGj
4pveff: (45)
In the second equation, we have introduced the effective ve-
locity veff¼^xjAj2/C22hk
MmagandjGj¼4pm0/C22h=ðglBÞwith the pur-
pose of comparing with Eq. (2).
This momentum transfer is illustrated in Fig. 9. In the
high-energy limit, the transversal and longitudinal forces aregiven by
F
?¼kr1
?ðeÞð^z/C2JeÞ¼4pð^z/C2JeÞ¼/C0 G/C2veff;(46)
Fk¼kr1
keðÞJe¼jGj
8pkð1
/C01dbHreg
1bðÞ/C0/C12veff; (47)
where we used Eqs. (41) and (44) as well as G¼/C0 j Gj^z.
They are indeed of the form given in Eq. (2). The transversal
momentum-transfer force, F?, is universal, and Fkis deter-
mined by the bparameter of Eq. (5)after identifying H(b)
with the classical deflection angle Hreg
1(b).
Is there an intuitive classical interpretation of these
momentum-transfer forces? From the classical limit of theSchrodinger equation (27) follows the equation of motion
for the coordinate r(t) of a classical magnon particle
25
FIG. 7. Total scattering cross section of the skyrmion in the high-energy
limit, Eq. (40), as a function of j2=Q2. It decreases for increasing external
magnetic field strength, j.FIG. 8. The longitudinal transport scattering cross section, Eq. (44),
vanishes as r1
k/C241=k2in the high-energy limit. The panel shows the
K-dependence of the prefactor.Low Temp. Phys. 41(10), October 2015 S. Schroeter and M. Garst 823
Mmag€r¼_r/C2ð^zBregðjrjÞÞ; (48)
with the regular part of the effective magnetic flux distribu-
tionBregof Eq. (24). Note that we have chosen in Eq. (27)
the charge to be þ1. Consider the change of momentum, dp,
of this magnon particle after scattering off the static sky-rmion by integrating the left-hand side of Eq. (48)
dpðbÞ¼ð
1
/C01dtM mag€rðtÞ¼Mmagð_rð1Þ /C0 _rð/C01ÞÞ
¼pcosHðbÞ/C01
/C0sinHðbÞ !
: (49)
In the last equation, we have exploited that at large distances
the magnitude of momentum Mmagj_rð61Þj ¼ premains
unchanged due to energy conservation, while the orientationof velocity is determined by the scattering angle HðbÞ, see
Fig. 1(b), that depends on the impact parameter bof the
trajectory.
This momentum dp(b) is transferred to the skyrmion.
The momentum-transfer force on the skyrmion due to a cur-rent of classical magnon particles along ^xwith density
m
0=ðglBÞand velocity veff¼jveffjis then given by
F¼Fk
F?/C18/C19
¼/C0veffm0
glBð1
/C01dbdpbðÞ; (50)
with Fk=?¼jFk=?j. In the high-energy limit, the scattering
is in forward direction so that we can expand Eq. (49) in the
deflection angle H(b) and the force becomes with p¼/C22hk
F¼veffm0
glB/C22hkð1
/C01db1
2HbðÞðÞ2
HbðÞ0
@1
A: (51)
Finally using that the integralÐ1
/C01dbHðbÞ¼4p=kis quan-
tized in the high-energy limit, that we already know from thediscussion in the context of Eq. (33), we recover Eqs. (46)
and(47).For the understanding of the universality of F
?, it is also
instructive to consider alternatively the right-hand side ofthe classical equations of motion (48). By integrating the
right-hand side, one obtains for the transversal momentumchange
dp
y¼ð1
/C01dtð/C0_xÞBregðjrjÞ /C25 /C0ð1
/C01dxBregðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2þx2p
Þ:(52)
In the last equation we employed the high-energy approxi-
mation by straightening the magnon trajectory. It follows
then for the transversal force
F?¼veffm0
glBð1
/C01dbð1
/C01dxBregffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2þx2p/C16/C17
; (53)
¼veffm0
glB4p/C22h; (54)
where its universality is now directly related to the quantized
total flux of Breg.
4. Summary
The scattering of high-energy magnons with wavevec-
tors krs/C291 off a magnetic skyrmion of linear size rsis
governed by a vector scattering potential. The associatedeffective magnetic field is related to the topological chargedensity of the skyrmion and is exponentially confined to theskyrmion area. The total flux is determined by the topologi-cal skyrmion number and is quantized.
When a magnon traverses the skyrmion, classically
speaking, it experiences the resulting Lorentz force and isdeflected to a preferred direction determined by the sign ofthe emergent magnetic flux. This results in skew scatteringwith a differential cross section that is asymmetric withrespect to forward scattering, see Fig. 6. As the flux distribu-
tion is rotationally symmetric, the classical deflection angleH(b) as a function of the impact parameter bis even in the
high-energy limit, H(b)¼H(/C0b). As a consequence, for a
given deflection angle Hthere exist corresponding classical
trajectories with positive as well as negative b, i.e., that pass
the skyrmion on the left-hand as well as on the right-handside. These trajectories interfere which leads to oscillationsin the differential cross section, an effect known as rainbowscattering.
Magnons hitting the skyrmion also transfer momentum
giving rise to a force in the Thiele equation of motion, seeEq.(3). In the high-energy limit, this force can be interpreted
classically and assumes the form of Eq. (2). While the trans-
versal momentum-transfer force, F
?is universal and deter-
mined by the total emergent magnetic flux, the longitudinalmomentum-transfer force, F
kis obtained by integrating
(H(b))2over the impact parameter bleading to the parameter
beof Eq. (5). Since for large energies the classical deflection
angle is small, H(b)/C241/k, the momentum transfer is mainly
transversal, Fk=F?/C241=k. This leads to a skyrmion motion
@tRapproximately antiparallel to the magnon current Jewith
a small skyrmion Hall angle U¼be=jGjdefined in Fig. 9
FIG. 9. An incoming monochromatic magnon current Jeleads to a
momentum-transfer force Fthat is determined by the transport scattering
cross sections, see Eq. (3). The image shows the magnon wavefunction in
the WKB approximation with the skyrmion being represented by the circlewith radius r
s.29For high-energy magnons with wavevector krs/C291, the
transversal force dominates, FkF?/C241=k, resulting in a skyrmion motion
@tRapproximately antiparallel to Jewith a small skyrmion Hall angle
U/C241=k.824 Low Temp. Phys. 41(10), October 2015 S. Schroeter and M. Garst
U¼1
2ð1
/C01HbðÞðÞ2db
ð1
/C01HbðÞðÞ db¼k
8pð1
/C01HbðÞðÞ2db/1
k; (55)
where the integralÐ1
/C01HðbÞdb¼4p=kis universal in the
high-energy limit. Interestingly, the Hall angle Uat high
energies increases with decreasing skyrmion radius rs, which
is shown in Fig. 8identifying U¼kr1
kðeÞ=4p.
While the skyrmion Hall angle Uis small at high ener-
gies krs/C291, we note that it increases with decreasing
energy and assumes the maximum value29U¼p=2 in the
low-energy limit krs/C281 where s-wave scattering prevails
and Eq. (2)ceases to be valid.
We acknowledge helpful discussions with M. Mostovoy
and A. Rosch.
a)Email: mgarst@uni-koeln.de
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|
1.2837800.pdf | Thermal fluctuation effects on spin torque induced switching: Mean and variations
Xiaobin Wang, Yuankai Zheng, Haiwen Xi, and Dimitar Dimitrov
Citation: Journal of Applied Physics 103, 034507 (2008); doi: 10.1063/1.2837800
View online: http://dx.doi.org/10.1063/1.2837800
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/103/3?ver=pdfcov
Published by the AIP Publishing
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131.181.251.130 On: Sun, 23 Nov 2014 13:01:14Thermal fluctuation effects on spin torque induced switching:
Mean and variations
Xiaobin Wang,a/H20850Yuankai Zheng, Haiwen Xi, and Dimitar Dimitrov
Seagate Technology, 7801 Computer Avenue, Bloomington, Minnesota 55435, USA
/H20849Received 8 October 2007; accepted 29 November 2007; published online 12 February 2008 /H20850
Thermal fluctuation effects on mean and variation of spin torque induced magnetic element
switching are analyzed. Asymptotic forms of the switching time distribution from the stochasticLandau–Lifshitz–Gilbert equation, and numerical solutions of the first and second moments ofswitching time from the corresponding Fokker–Planck equation, are used to characterize switchingtime and switching current density for the whole time range, from the second thermal reversalregion to the nanosecond dynamic reversal region. It is shown that as time scales become shorter,switching time distributions become narrower, whereas switching current distributions may becomebroader. This paper provides a physical understanding of these different scaling behaviors. © 2008
American Institute of Physics ./H20851DOI: 10.1063/1.2837800 /H20852
I. INTRODUCTION
Fast magnetization switching under spin torque current
excitation and sufficient thermal stability at room tempera-ture are two main design criterions for spin torque magneticrandom access memory /H20849MRAM /H20850. When characterizing spin
torque induced switching, both the averaged switching be-havior and the switching variation are critical. In this articlewe will analyze both mean and variation of switching behav-ior for a spin torque MRAM element for the whole timerange, from the second thermal reversal region to the nano-second dynamic reversal region. The analysis is based uponsolving the Fokker–Planck equation of the stochasticLandau–Lifshitz–Gilbert /H20849LLG /H20850equation with a spin torque
term. With the help of asymptotic switching time probabilitydistributions at long and short time scales, numerical solu-tions of mean switching time, and second moment of switch-ing time are used to characterize averaged switching behav-ior and switching variation. It is found that as time scalesbecome shorter, switching time distributions become nar-rower, whereas switching current distributions may becomebroader. This article discusses the physical reasons behindthis. Whether switching time variation or current densityvariation should be used to characterize spin torque inducedmagnetization switching variation depends on the particularapplication of the spin torque switching phenomenon.
II. STOCHASTIC MODEL FOR SPIN TORQUE
MAGNETIZATION SWITCHING VARIATION
The magnetization dynamics in the free layer of a spin
torque MRAM is described by stochastic LLG equation atfinite temperature.
dm
dt=/H9251m/H11003/H20849m/H11003/H20849heff+hfluc/H20850/H20850−m
/H11003/H20849/H20849heff+hfluc/H20850+/H9252m/H11003p/H20850, /H208491/H20850
where mis the normalized magnetization, heff=Heff/Ms=/H11509/H9255//H11509mis the normalized magnetic field with normalized
energy density /H9255, and hflucis the thermal fluctuation field. /H9251
is the damping parameter, pis a unit vector pointing to the
spin polarization direction and /H9252=/H9257hJ /2eMs2dis the normal-
ized spin torque polarization magnitude, where /H9257is the po-
larization, dis film thickness, and Jis current density. Notice
the spin torque term in Eq. /H208491/H20850only includes an adiabatic
term. Another term proportional to m/H11003pcan be added to
Eq. /H208491/H20850for nonadiabatic spin torque effects. For the sake of
simplicity, in this article we only consider the adiabatic spintorque term. Our methods can be easily extended to include anonadiabatic term.
If the spin polarization points in the direction of easy
axis of a rectangular free layer /H20849Fig. 1/H20850, the energy of the
magnetic system is /H9255=E/M
s2V=/H208491/2/H20850Nxmx2+/H208491/2/H20850Ny2my2
+/H208491/2/H20850Nzmz2and the spin polarization direction is p=ez,
where Nx,Ny, and Nzare demagnetization factors. The
magnitude of the stochastic term in Eq. /H208491/H20850is determined by
fluctuation-dissipation condition at room temperature as inRef. 1. Here the stochastic fluctuation magnitude is repre-
sented by
/H9254=kBT/2KV. For spin torque MRAM magnetiza-
tion switching, we are interested in the switching time for agiven magnetic element shape and a given polarization cur-rent density for the whole time range, from the short time
nanosecond region /H20849writing data /H20850to the long time second
a/H20850Electronic mail: xiaobin.wang@seagate.com.
FIG. 1. /H20849Color online /H20850Shape and coordinates of a MRAM free layer
element.JOURNAL OF APPLIED PHYSICS 103, 034507 /H208492008 /H20850
0021-8979/2008/103 /H208493/H20850/034507/4/$23.00 © 2008 American Institute of Physics 103 , 034507-1
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131.181.251.130 On: Sun, 23 Nov 2014 13:01:14thermal switching region /H20849data reliability and testing /H20850. Due to
thermal fluctuations, this magnetization reversal time is a
stochastic quantity. The first moment of this reversal timerepresents the averaged magnetization switching speed.However, in order to characterize switching variations,higher order moments, or even the probability distributionfunction of switching times, are required. Although it is dif-ficult to obtain the switching time probability distributionfunction covering whole time range /H20849from nanosecond dy-
namic switching to second thermal switching /H20850directly from
Fokker–Planck equation corresponding to Eq. /H208491/H20850, the
switching time probability distribution function asymptoti-cally approaches a certain functional form that can be ap-proximately described by a few parameters /H20849or moments /H20850in
the long time thermal reversal region and in the short timedynamic reversal region. Based upon this understanding, nu-merical solutions of the moments of the switching time in the
whole time range enables a detailed analysis of both themean and variation of spin torque switching.
The detailed technique of solving for the exit time /H20849or
switching time /H20850of the stochastic LLG equation can be found
in Refs. 2and3. In Ref. 2, a stochastic average technique is
used to simplify the numerical solution of reversal time ofthe stochastic magnetic dynamic equation /H20849different from
LLG /H20850for small damping case. Following the same math-
ematical procedure, we integrate the stochastic LLG withspin torque term, Eq. /H208491/H20850around a constant energy level to
obtain the following stochastic differential equation:
d/H9255=
A/H20849/H9255/H20850dt+/H20881B/H20849/H9255/H20850dW /H20849t/H20850, /H208492/H20850
where A/H20849/H9255/H20850and B/H20849/H9255/H20850are convection and diffusion coeffi-
cients. dW /H20849t/H20850is the increment of a Brownian process. A/H20849/H9255/H20850
can be explicitly represented as
A/H20849/H9255/H20850=/H20886d/H9272sin/H9258
/H11509/H9255//H11509/H9258/H20877−/H9251/H20875/H20873/H11509/H9255
/H11509/H9258/H208742
+1
sin2/H9258/H20873/H11509/H9255
/H11509/H9272/H208742/H20876+/H9257hJ
2eMs2dsin/H9258/H11509/H9255
/H11509/H9258+/H9254
2/H20873/H115092/H9255
/H11509/H92582+1
sin2/H9258/H115092/H9255
/H11509/H92722/H20874/H20878
/H20886d/H9272sin/H9258
/H11509/H9255//H11509/H9258, /H208493/H20850
where /H9258,/H9272are magnetization angles in spherical coordinates.
/H20859is the integration of gyromagnetic motion around constant
energy level /H9255/H20849/H9258,/H9272/H20850=/H9255.B/H20849/H9255/H20850can be explicitly represented as
B/H20849/H9255/H20850=/H20886d/H9272/H9254sin/H9258
/H11509/H9255//H11509/H9258/H20875/H20873/H11509/H9255
/H11509/H9258/H208742
+1
sin2/H9258/H20873/H11509/H9255
/H11509/H9272/H208742/H20876
/H20886d/H9272sin/H9258
/H11509/H9255//H11509/H9258. /H208494/H20850
Explicit formulas for calculating moments of exit time for
stochastic differential Eq. /H208492/H20850can be found in Sec. 5 of Ref.
3and formula /H2084941/H20850in Ref. 2.
III. SWITCHING DISTRIBUTION AND VARIATION
FOR LONG TIME THERMAL REVERSAL REGION
For long time thermal switching, the solution of the
Fokker–Planck equation corresponding to Eq. /H208491/H20850or Eq. /H208492/H20850
gives a switching time probability distribution approachingthe Poisson distribution with characteristic time
/H9270/H20849Refs. 1
and2/H20850:
p/H20849t/H20850=1
/H9270e−t//H9270. /H208495/H20850
The switching probability can be found by integrating
Eq. /H208495/H20850:P/H20849t/H20850=1− e−t//H9270. For the Poisson distribution, the sec-
ond moment of the switching time distribution is twice that
of the mean switching time. Thus, the standard deviation ofthe switching time is the same as mean switching time. Thecoefficient of variation, which is defined as the ratio of stan-dard deviation to the mean, is a measure of relative variabil-
ity of switching time. For the Poisson distribution, the coef-ficient of variation is one. Figure 2shows current density and
coefficient of variation of switching time versus meanswitching time for a rectangular magnetic element /H20849Fig. 1/H20850
for the whole time range. The element dimension is 90 nmby 60 nm by 2 nm. The saturation magnetization is1200 emu /cm
3. The damping parameter is 0.0025 and the
spin polarization efficiency is 0.3. The result is based upon
FIG. 2. Switching current density vs mean switching time and switching
time coefficient of variation vs mean switching time.034507-2 Wang et al. J. Appl. Phys. 103 , 034507 /H208492008 /H20850
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131.181.251.130 On: Sun, 23 Nov 2014 13:01:14the numerical solution of the first and second moments of the
exit time of the Fokker–Planck equation corresponding toEq. /H208491/H20850, as discussed in the previous section. Figure 2indeed
shows that the coefficient of variation approaches 1 for longtime thermal decay.
More information can be obtained by considering the
switching time probability dependence on spin torque currentdensity. The mean switching time due to thermal reversal isproportional to the exponential of the energy barrier to theleading order:
/H9270=f0−1e/H9004E/kBT, where f0is the attempt fre-
quency for thermal reversal. For spin torque switching, thedependence of the energy barrier on spin current density canbe approximated by
/H9004E=1
2MsHcV/H208731−J
Jc/H20874 /H208496/H20850
for spin polarization in the direction of easy axis of free
layer.4In the previous formula, Hcis the magnetic elementcoercivity and Jcis the intrinsic zero temperature switching
current density. The probability density function of theswitching time as a function of spin polarized current densitycan be obtained by differentiating the switching probability
P/H20849t/H20850with respect to current density. From Eq. /H208496/H20850, the fol-
lowing formula is obtained:
p/H20849J/H20850=e
−t//H9270/H20849J/H20850t
/H9270/H20849J/H20850MsHcV
2kBT1
Jc, /H208497/H20850
where
/H9270=MsHcV
2kBT/H208731−J
Jc/H20874.
Figure 3shows the switching time probability density depen-
dence upon spin torque current density at different timescales. Interestingly, we see that the spin torque current den-sity distribution is not narrowed as time scales down, al-though Fig. 3shows the coefficient of variation decreases as
time scales down.
IV. SWITCHING DISTRIBUTION AND VARIATION
FOR SHORT TIME DYNAMIC REVERSAL REGION
For short time dynamic switching, the switching time
probability density approaches the shape of a skewed Gauss-ian, instead of the exponential shape of a Poisson distribu-tion. We could not obtain an analytical formula for theswitching time distribution for spin torque switching in theshort time dynamic region. However, the formula for thehitting time distribution of an Ornstein-Uhlenbeck process/H20849Brownian motion in a parabolic potential /H20850from a given ex-
cited energy level to equilibrium position is well known.
5
This distribution function has an asymmetric Gaussianshape. Similarly, the switching time probability density forspin torque switching in the short time dynamic region hasan asymmetric Gaussian shape. For this asymmetricGaussian-like distribution function, the second moment ofthe distribution characterizes the switching time variation.Figure 4shows schematically the probability distribution
function of switching time for the short time dynamic region
FIG. 3. Switching probability distribution as a function of spin torque cur-
rent density for different time scales.
FIG. 4. Schematic picture of the switching time prob-ability distribution function for long time thermalswitching and short time dynamic switching.034507-3 Wang et al. J. Appl. Phys. 103 , 034507 /H208492008 /H20850
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131.181.251.130 On: Sun, 23 Nov 2014 13:01:14and the long time thermal region. They differ significantly.
Thus the approximating method of obtaining current densityvariation based upon the Poisson distribution and the energybarrier for the long time thermal region in previous sectioncannot be extended to the short time dynamic region. Itshould be pointed out that both the mean and variation ofswitching time in the short time dynamic region in Fig. 2are
well justified, because they are solutions of exit time mo-ments of the stochastic LLG equation with spin torque forthe whole time region.
In order to estimate switching current density variations
for the whole time range, we consider the mapping of meanswitching time versus switching current, T=T/H20849J/H20850, at different
time scales. This mapping can be used to link current varia-
tion to switching time variation in an approximate way:
/H9254J
J=/H20841dJ/dT/H20841
J/T/H9254T
T. /H208498/H20850
Figure 5shows the switching current density coefficient
of variation and standard deviation using Eq. /H208498/H20850. The almost
constant current density standard deviation for the long timethermal reversal region is consistent with the results in Fig.3. Because switching time decreases rapidly as spin torque
increases, a smaller coefficient of variation in switching timestill results in a bigger coefficient of variation in switchingcurrent as time scales down to the short time dynamic region.This indicates that the coefficient of variation of switchingtime decreases as time scales down from the thermal regionto the dynamic region, whereas at the same time the switch-ing current density distributions are broadened. Whetherswitching time variation or current density variation shouldbe used to characterize spin torque induced switching varia-tions depends on the particular application. For a memoryelement switching with a constant current pulse, it is impor-tant to control the write current distribution to prevent non-switching events. For a current pulse with well-controlledmagnitude, the switching time variation for constant currentamplitude ultimately determines the memory writing perfor-mance at short dynamic time.
V. CONCLUSIONS
Thermal fluctuation effects on spin torque induced
STRAM switching are studied. Both switching time varia-tions and switching current variations are obtained basedupon the switching time probability distribution function andthe solution of moments of switching time of the stochasticLLG equation for the whole time range. The switching timedistributions are narrowed as time scales down from the longtime thermal reversal region to the short time dynamic rever-sal region. However, due to the rapid decrease of switchingtime as spin torque or external magnetic field increases, theswitching current density is broadened as time scales downfrom the long time thermal reversal region to the short timedynamic region.
ACKNOWLEDGMENT
The authors give special thanks to Dr. Gene Sandler for
previewing this paper.
1W. F. Brown, Phys. Rev. 130, 1677 /H208491963 /H20850.
2X. Wang, N. H. Bertram, and V. L. Vladimir, J. Appl. Phys. 92, 2064
/H208492002 /H20850.
3C. G. Gardiner, Handbook of Stochastic Methods: for Physics, Chemistry
and Natural Sciences /H20849Springer, New York, 1985 /H20850.
4Z. Li and S. Zhang, Phys. Rev. B 69, 134416 /H208492004 /H20850.
5L. Alili, P. Patie, and J. L. Pederson, Stoch. Models 21, 967 /H208492005 /H20850.
FIG. 5. Switching current density coefficient of variation vs mean switching
time and switching current density standard deviation vs mean switchingtime.034507-4 Wang et al. J. Appl. Phys. 103 , 034507 /H208492008 /H20850
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131.181.251.130 On: Sun, 23 Nov 2014 13:01:14 |
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The Journal of the Acoustical Society of America 147, EL326 (2020); https://doi.org/10.1121/10.0001037Multiple two-step oscillation regimes produced by the alto
saxophone
Tom Colinot,1,a)Philippe Guillemain,1Christophe Vergez,1Jean-Baptiste Doc,2and Patrick Sanchez1
1Aix Marseille University, French National Centre for Scientific Research, Centrale Marseille, Laboratory of Mechanics and Acoustics,
4, Impasse Nikola Tesla, 13013 Marseille, France
2Laboratoire de M /C19ecanique des Structures et des Syste `mes coupl /C19es, Conservatoire National des Arts et M /C19etiers, 292 rue Saint-Martin,
F-75141 Paris Cedex 03, France
ABSTRACT:
A saxophone mouthpiece fitted with sensors is used to observe the oscillation of a saxophone reed, as well as the
internal acoustic pressure, allowing to identify qualitatively different oscillating regimes. In addition to the standard
two-step regime, where the reed channel successively opens and closes once during an oscillation cycle, the experi-mental results show regimes featuring two closures of the reed channel per cycle, as well as inverted regimes, where
the reed closure episode is longer than the open episode. These regimes are well-known on bowed string instruments
and some were already described on the Uilleann pipes. A simple saxophone model using measured input impedanceis studied with the harmonic balance method, and is shown to reproduce the same two-step regimes. The experiment
shows qualitative agreement with the simulation: in both cases, the various regimes appear in the same order as the
blowing pressure is increased. Similar results are obtained with other values of the reed opening control parameter,as well as another fingering.
VC2020 Acoustical Society of America .https://doi.org/10.1121/10.0001109
(Received 8 January 2020; revised 29 March 2020; accepted 31 March 2020; published online 20 April 2020)
[Editor: Thomas R. Moore] Pages: 2406–2413
I. INTRODUCTION
Various oscillating regimes, defined as the pattern of
oscillations both mechanical and acoustical that correspondto the production of a periodic sound, have been observed
and classified on bowed string instruments ( Schelleng,
1973 ). The strong non-linear friction law between bow and
string leads to an oscillation pattern known as stick-slipmotion, where the string sticks to the bow for a part ofthe period and then slips for another part of the period. The
stick-slip phases may occur twice per period, leading to the
so-called “double stick-slip” motion.
Reed conical instruments have often been compared to
bowed strings, by virtue of the cylindrical saxophone approxi-
mation, which replaces the conical resonator with two parallelcylinders ( Ollivier et al. ,2 0 0 4 ) because their impedance is
similar in low frequency. In reed instruments, the analogousmotion to stick-slip is called two-step motion ( Ollivier et al. ,
2005 ). It consists of a beating reed regime, where the reed
channel is closed for part of the period and open for the rest
of the period. The most common case, where the reed closureepisode is shorter than half the period, is called a standardtwo-step motion. Otherwise, the regime is called inverted.Standard and inverted two-step motions have been observedexperimentally on a saxophone and predicted analytically on
a cylindrical equivalent ( Dalmont et al. , 2000 ). Oscillating
regimes showing more than one closure of the reed per periodwere never studied on the saxophone to our knowledge. Theyhave been observed on a double reed instrument, the IrishUilleann pipes ( Dalmont and Le Vey, 2014 ). To observe the
signals produced by a wind instrument in a playing situationwith a musician, an instrumented mouthpiece fitted with areed displacement and pressure sensors can be used.Instrumented mouthpieces can help explain features of theproduced sound, for instance, spectral content on a saxo-
phone ( Guillemain et al. , 2010 ) or transient descriptors on a
clarinet ( P/C18amies-Vil /C18aet al. , 2018 ). They also provide a
means to estimate some of the parameters of a physicalmodel based on the dynamical behavior of the system(Mu~noz Aranc /C19onet al. , 2016 ).
This paper reports experiments in playing conditions
exhibiting classic standard and inverted regimes, as well asdouble two-step motions, where the reed channel closestwice per period. To complete the study, we show that asimple saxophone model based on the input impedance of
the saxophone used for the experiment is able to reproduce
these double two-step regimes. The Harmonic BalanceMethod (HBM) associated with continuation (AsymptoticNumerical Method) is used to obtain periodic signals corre-sponding to several control parameter combinations. Thenumerical simulations, in addition to experimental data, pro-
vide insights about the possible ways of transition between
single and double two-step regimes, as well as the secondregister of the instrument. We also show that similar behav-ior occurs for neighboring fingerings and control parametervalues. Describing and categorizing the oscillation regimes
of the saxophone, as well as the musician’s actions needed
to obtain them, is among the first steps toward objectivecharacterization of the ease of playing of an instrument.
a)Electronic mail: colinot@lma.cnrs-mrs.fr
2406 J. Acoust. Soc. Am. 147(4), April 2020 VC2020 Acoustical Society of America 0001-4966/2020/147(4)/2406/8/$30.00
ARTICLE ...................................II. EXPERIMENTAL OBSERVATION OF DOUBLE TWO-
STEP MOTIONS ON A SAXOPHONE
A. Experimental apparatus
An instrumented mouthpiece is used to monitor the blow-
ing pressure, the pressure inside the mouthpiece, and the posi-
tion of the reed. It is shown in Fig. 1. It consists of a modified
saxophone mouthpiece (Buffet-Crampon, Mantes-la-Ville,
France) incorporating two pressure probes: one going into the
mouth of the musician and one into the mouthpiece, as well asan optical sensor (Everlight ITR8307, New Taipei City,Taiwan) measuring the displacement of the reed. The
pressure probe tubes are connected to a Honeywell
TSCDRRN005PDUCV (Charlotte, North Carolina) pressuresensor. The tubes have a radius of 0.55 mm and a length of
20 mm (mouth pressure) and 62 mm (pressure in the mouth-
piece). According to Guillemain et al. (2010) , the transfer
function of these capillary tubes is well represented by a
model with non-isothermal boundary conditions ( Keefe,
1984 ). An inverse filtering was performed on the pressure sig-
nals to compensate the effect of the probe tubes. Signals are
t h e na c q u i r e du s i n ga nN IU S B - 9 2 3 4c a r db yN a t i o n a l
Instruments, Austin, Texas at a 51.2 kHz sampling rate.Experimental signals displayed hereafter are not scaled or con-
verted as this work focuses on a qualitative study of the
regime types. The instrumented mouthpiece is equipped witha saxophone reed (Rico Royal strength 2) and mounted on a
commercial alto saxophone (Buffet-Crampon Senzo).
Throughout the remainder of the paper, a low Bfinger-
ing (written pitch) is studied. In concert pitch, the funda-
mental note expected with this fingering is a D3 at the
frequency 146.83 Hz. The input impedance of the saxophonefor this fingering has been measured using the CTTM(Centre de Transfert de Technologie du Mans, Le Mans,
France) impedance sensor ( Dalmont and Le Roux, 2008 ). Its
modulus is displayed in Fig. 2. The B fingering, which pro-
duces the second lowest note on the instrument, is chosen
because the double two-step regimes studied in this work
tend to appear more easily on the lowest notes of the saxo-phone. Note that for this fingering, the note most commonly
expected by musicians is the first register, whose frequency
is around the first impedance peak. On this fingering, thefirst register is often hard to produce, especially for beginner
musicians. This can be understood when looking at the
impedance modulus curve in Fig. 2, where the first peak is
lower than the next three peaks: the upper resonances of thebore play a large part in the sound production, leading to a
complicated sound production behavior. This profile ofamplitude of the first few impedance peaks is also found insoprano and tenor saxophone ( Chen et al. , 2009 ). The lowest
fingering ( B[) was not chosen, although it was tested,
because it is more subject to producing undesired multi-phonics and quasi-periodic regimes.
B. Observation of single and double two-step osc illating
regimes
The main oscillating regimes of a saxophone are beat-
ing, which means that the reed channel closes completelyduring part of the cycle. They can be thought of as two-stepmotions ( Ollivier et al. , 2004 ) and classified as standard or
inverted, depending on the relative duration of the open andclosed episode. Different regimes can be obtained for thesame fingering just by varying the control parameters suchas the blowing pressure. Figure 3shows measured examples
of these two-step regimes. The reed displacement signal waspost-processed by subtracting its moving average over aperiod, to be centered around 0. The standard regime ischaracterized by an open episode and a short closed episode.As can be seen in Fig. 3(a), the reed is opened—and displays
FIG. 1. (Color online) Instrumented alto saxophone mouthpiece including
pressure probes for the pressure in the mouth of the musician and in themouthpiece, and an optical sensor measuring the displacement of the reed.
The reed is pulled back so that the optical sensor is uncovered.FIG. 2. Input impedance modulus measured for the studied fingering of thealto saxophone: low Bin written pitch. The modulus of the impedance is
normalized by the characteristic impedance at the input of the instrument.
FIG. 3. (Color online) Measured reed position for simple two-step motions:standard (a) and inverted (b). The reed channel is closed when xis low.
These waveforms correspond to different blowing pressures (see circle
markers in Fig. 5).
J. Acoust. Soc. Am. 147(4), April 2020 Colinot et al. 2407https://doi.org/10.1121/10.0001109small amplitude oscillations around the highest values of
x—for about 6 ms. Its closure corresponds to the main dip in
the waveform and it lasts for about 1 ms per period. For the
inverted motion in Fig. 3(b), the duration ratio is reversed:
the reed channel is almost at its narrowest about 6 ms and
opens wide briefly for about 1 ms. Note that the standard
regime is obtained for lower values of the blowing pressurethan the inverted regime.
The analogy with bowed string instruments suggests the
apparition of other types of regimes. For example, under a
given excitation condition, bowed strings are subject to the
double stick-slip phenomenon ( Woodhouse, 2014 ), an oscil-
lation regime where the string slips under the bow twice per
period (instead of once for the standard Helmholtz motion).When transposed to conical reed instruments, this phenome-
non corresponds to two closures of the reed channel per
period. These regimes are observed experimentally on the
low fingerings of the saxophone and they can be standard or
inverted, as shown in Fig. 4. This oscillating regime can be
called “double two-step.” Note that the double two-step
regime is distinct from second register regimes: it is a first
register regime, as it produces the same note as the standard
two-step regime. For the standard version of the double two-step regime, the closure episodes are about 1 ms, almost the
same duration as in the single standard two-step motion
[Fig. 3(a)]. For the inverted double two-step regime, the
short openings of the reed channel also last for about 1 ms.
For illustration purposes, the audible sound outside the
instrument was recorded and short clips are provided as
Mms. 3, 4, 1, and 2. Note that the audible sound correspond-
ing to these double two-step regimes ( Mm. 3 andMm. 4 )i s
clearly different from single regimes ( Mm. 1 andMm. 2 ).
The difference in audible sound is less clear between a stan-
dard regime and its inverted counterpart.
Mm. 1. Sound recorded outside the resonator for the
standard two-step motion, corresponding to the mea-sured displacement shown in Fig. 3(a)(220.9 ko).
Mm. 2. Sound recorded outside the resonator for the
inverted two-step motion, corresponding to the mea-
sured displacement shown in Fig. 3(b)(294.3 ko).Mm. 3. Sound recorded outside the resonator for the
double two-step motion, corresponding to the measured
displacement shown in Fig. 4(a)(509.2 ko).
Mm. 4. Sound recorded outside the resonator for the
inverted double two-step motion, corresponding to themeasured displacement shown in Fig. 4(b)(570.0 ko).
In order to estimate the relative regions of production of
each type of regime in the control parameter space, a blow-ing pressure ramp is performed by a musician and recorded
using an instrumented mouthpiece for the B fingering of
the test saxophone. The musician sees the evolution of theblowing pressure parameter in real-time on a screen. Theplayer makes as little embouchure adjustments as possibleand focuses on increasing the blowing pressure progres-sively. Results are shown in Fig. 5. This ramp was obtained
in a single breath after several tries. For clarity, the blowingpressure signal is smoothed by a moving average with arectangular window, adjusted to reject the fundamental fre-quency of the oscillations and keep only the slowly varying
value of the signal. Regimes are classified automatically
based on the ratio of duration of the open and closed reedepisodes. The reed displacement signal is high-pass filteredin order to remove the direct current (DC) component. Thereed is then considered “open” when the displacement signalis above 0 and “closed” when it is below 0. The ratiobetween closed duration and oscillation period is then com-puted and averaged over four periods. Thresholds aredefined arbitrarily to separate between the different types of
regimes, at 0.1, 0.25, 0.5, 0.6, and 0.8 (see dotted lines in
Fig.5). Looking at the pressure ramp in its entirety shows a
FIG. 4. (Color online) Measured reed position for double two-step motions:
standard (a) and inverted (b). These waveforms correspond to different
blowing pressures (see circle markers in Fig. 5).
FIG. 5. (Color online) Result of a blowing pressure increase (low B fingering,
alto saxophone) recorded with the instrumented mouthpiece. Left yaxis (red
online): measured smoothed blowing pressure in Pa. Right yaxis: ratio
between closure episode duration and oscillation period (solid line), and
regime separation thresholds (dotted lines). Grayed areas emphasize the dura-tion of each type of regime. Circles correspond to reed displacement signals
in Figs. 3and4and pressure signals in Fig. 6.
2408 J. Acoust. Soc. Am. 147(4), April 2020 Colinot et al.https://doi.org/10.1121/10.0001109possible order of the regimes when increasing the blowing
pressure: standard and double two-step motions, second reg-ister, and inverted double then inverted two-step motions.Note that in this ramp, the episode between 1 and 2 s with aclosure ratio of little above 0.25 is actually a quasi-periodic
oscillation, with the actual double two-step oscillation start-
ing at around 2.3 s.
III. NUMERICAL STUDY OF THE REGIMES USING A
PHYSICAL MODEL
A. Saxophone model
A simplified saxophone model consists of three main
elements: the resonator, the reed channel, and reed dynam-ics. Here all variables are dimensionless and obtained fromtheir physical counterparts (denoted with a hat) as
p¼
^p
pM;u¼Zc^u
pM;x¼^x
H; (1)
where pMis the static pressure necessary to close the reed
completely, Zcis the characteristic impedance at the input of
the resonator, and His the distance separating the reed from
the mouthpiece lay at rest. Note that x¼0 denotes the reed
at equilibrium, and x¼/C01 corresponds to a closed reed
channel.
The resonator is represented by its dimensionless input
impedance, decomposed as a sum of modes
ZðxÞ¼PðxÞ
UðxÞ¼XNm
n¼0Cn
ix/C0snþ/C22Cn
ix/C0/C22sn; (2)
where Cnare the complex residues and snthe complex poles.
These modal parameters are estimated from measured saxo-phone input impedance ( Taillard et al. , 2018 ). Equation (2)
can be transformed into the temporal evolution of the modalcomponents p
n, since jxtranslates into a time-domain deriv-
ative by inverse Fourier transform
_pnðtÞ¼snpnðtÞþCnuðtÞ: (3)
The acoustic pressure pat the input of the tube is expressed
as a sum including the modal components
pðtÞ¼2XNm
n¼1ReðpnðtÞÞ: (4)
The number of modes Nmis chosen as Nm¼12, sufficiently
large to represent the main resonances of the resonator.Results obtained using N
m¼6 lead to similar conclusions.
The flow uat the input of the resonator is governed by the
nonlinear characteristic ( Wilson and Beavers, 1974 )
u¼fxþ1½/C138þsignðc/C0pÞffiffiffiffiffiffiffiffiffiffiffiffiffi
jc/C0pjp
; (5)
where ½xþ1/C138þ¼maxðxþ1;0Þ. This nonlinear characteris-
tic uses the dimensionless control parameters of reedopening at rest fand blowing pressure c. The expression of
these parameters are
f¼wHZ cffiffiffiffiffiffiffiffiffi
2
qpMs
;c¼^c
pM; (6)
where wis the effective width of the reed channel, qthe
density of air, and ^cis the physical value of the blowing
pressure. For this study the parameter fis fixed at f¼0:6,
unless otherwise specified. Following the values of reedchannel height at rest H¼17/C210
/C05m and reed stiffness K
¼6:4/C2106Pa m provided in Mu~noz Aranc /C19onet al. (2016) ,
with an approximate effective width of w¼1/C210/C02m and
characteristic impedance Zc¼3/C2106Pa s/m3, one finds
f¼Zcwffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2H=qKp
¼0:58 which justifies studying f’0:6
in this work. To use HBM and Asymptotic Numerical
Method, described in Sec. III B, it is convenient to regular-
ize the characteristic of Eq. (5)using j/C1j’ffiffiffiffiffiffiffiffiffiffiffiffi
/C12þgp
, where
the parameter gis fixed at 10/C03(Kergomard et al. , 2016 ).
The reed is modeled as a single degree of freedom
oscillator driven by the pressure difference between the
input of the resonator and the mouth of the resonator
€x
x2
rþqr_x
xrþx¼/C0 ð c/C0pÞ; (7)
where xrand qrare the angular frequency and damping
coefficient of the reed, chosen at xr¼4224 rad/s based on
Mu~noz Aranc /C19onet al. (2016) andqr¼1. In this model, the
impact of the reed on the mouthpiece lay is ignored(Dalmont et al. , 2000 ;Doc et al. , 2014 ). For further details
on the effect of ignoring reed impact in a saxophone model,
see (Colinot et al. , 2019 ).
B. Numerical resolution with HBM
Periodic solutions to the system of Eqs. (2),(5), and (7)
are found using the HBM, under the formalism proposed inCochelin and Vergez (2009) . The HBM was pioneered by
Krylov and Bogoliubov (1949) and Nakhla and Vlach
(1976) , and was applied to musical instrument models first
inGilbert et al. (1989) . Each variable X(where Xcan stand
forp
n,u,x…) is assumed to be periodic and thus decom-
posed into its Fourier series truncated at order H
XðtÞ¼X1
k¼/C01Xkexpðikx0tÞ’XH
k¼/C0HXkexpðikx0tÞ;(8)
where x0is the angular frequency. This yield an algebraic
system where the unknowns are the Fourier coefficients andthe angular frequency. Hereafter, H¼20 is chosen because
it appears sufficient for a good representation of the studied
regimes. The emergence of these different regimes dependson the value of the blowing pressure parameter c. To com-
pare the value of cleading to each regime to the experimen-
tal results of Fig. 5, a Taylor-series based continuation
method (Asymptotic Numerical Method) is applied to the
J. Acoust. Soc. Am. 147(4), April 2020 Colinot et al. 2409https://doi.org/10.1121/10.0001109algebraic system obtained by harmonic balance ( Guillot
et al. , 2019 ). The source code for this method may be found
online at http://manlab.lma.cnrs-mrs.fr/ . The continuation
yields possible periodic solutions, as well as their stability
(Bentvelsen and Lazarus, 2018 ;Lazarus and Thomas,
2010 ). This may be displayed as a bifurcation diagram
representing the evolution of one descriptor of the periodic
solutions as a function of the blowing pressure. The bifurcation
diagrams displayed here do not change when adding more har-monics, but their computation is more time consuming.
C. Results
Depending on the value of the blowing pressure param-
eterc, all types of two-step regimes observed experimentally
are found to be stable periodic solutions of the model.
Figure 6compares the regime types found in measurement
and simulation from their pressure waveforms. No a posteriori
adjustment of the model is performed, and therefore no pre-
cise agreement of the waveforms is expected. Many differ-
ences between synthesized and measured signals could beexplained by the reed opening parameter fbeing constant
and not adjusted in the model, and the response of the pres-
sure probe tube affecting the measured pressure signal.Some high frequency components of synthesized signal can
also be misrepresented due to the modal truncation of the
impedance. However, several main features of the measuredsignals can be identified on the synthesized signals, such as
the duration of the short low-pressure episodes on the stan-
dard and double two-step regimes, and the short high-pressure episodes on the inverted double and inverted
two-step regimes. It can also be noted that both synthesized
and measured signals exhibit secondary fast oscillations ofsmall amplitude during the long episodes (open or closed).
A similar “minor oscillations” phenomenon is known to
appear on bowed strings ( Kohut and Mathews, 1971 ). The
opening duration of the synthesized inverted two-step
regime presented in Fig. 6(g)is longer than the closure dura-
tion of the synthesized standard two-step of Fig. 6(a), which
is contrary to the usual Helmholtz motion formulation in
which both durations are determined only by the geometry
of the resonator. This is always the case with the model ofthis paper, with both time-domain synthesis and the har-
monic balance: the synthesized and standard and inverted
two-step display a whole range of opening or closure dura-tions depending on the value of the blowing pressure. This
phenomenon is further detailed below, in Mm. 5 , Fig. 7, and
the corresponding commentary.
The bifurcation diagram summarizing the evolution of
the different oscillating regimes depending on the blowing
pressure parameter cis presented in Fig. 7. A parameter of
the oscillating regimes, the amplitude of the first cosine, i.e.,
the real part of the first Fourier coefficient of Eq. (8), of the
first modal pressure p
1is displayed. This parameter was cho-
sen because it allows for clear separation of the branches
corresponding to each regime. Note that the sign of this
coefficient can be either positive or negative dependingsolely on a choice of phase of the oscillation. On the dia-
grams displayed hereafter, the sign of p1was chosen so that
the different solution branches are as easy to distinguish aspossible. The most important part of the branches is stable
regimes (thick lines in the figure). Each branch is labeled
with the type of regime it corresponds to. The regime type isdetermined manually by observing the waveform, which can
be done exhaustively using animations such as Mm. 5. Note
that the animation shows the standard two-step regimemorphing gradually into the inverted two-step regime, on
the same branch. The closure duration of the reed increasesFIG. 6. Synthesized and measured pressure signals in the mouthpiece for
two-step regimes. Arbitrary units.
2410 J. Acoust. Soc. Am. 147(4), April 2020 Colinot et al.https://doi.org/10.1121/10.0001109progressively with the blowing pressure parameter c,i n
clear contradiction with the Helmholtz motion approxima-
tion. The topic of continuous transition between standardand inverted regimes for a conical woodwind remains to be
fully understood, although experimental explorations point
to similar results ( Dalmont, 2007 ). All the other branches
correspond to only one type of regime each.
Mm. 5. Animation: Evolution of the acoustic pressure
waveform and spectrum following the stable branches
of the bifurcation diagram in Fig. 7(1.0 Mo).
Figure 7is qualitatively coherent with the experimental
findings in Fig. 5in terms of order of emergence of the sta-
ble regimes when varying the blowing pressure. Starting
with a low blowing pressure, the first stable regime is the
standard two-step. When the blowing pressure increases, thestable branch is followed until its end, and then the system
jumps on another stable branch. At the end of the standard
two-step branch, around c¼0:69, there are two coexisting
branches: the inverted two-step and the double two-step.
Note that for the parameter values where two stable regimescoexist, different initial conditions may lead to one or the
other. Describing the conditions leading to one or the other
regime (called their “attraction basin”) exhaustively is
almost impossible. Consequently, when using the bifurca-
tion diagram to predict which regimes can be producedwhen increasing the blowing pressure, several scenarios can
be devised, and it is extremely difficult to decide which one
is the most probable without checking it experimentally. For
instance, according to this bifurcation diagram, it would be
possible for the system to start from the standard two-step,jump to an inverted two-step regime, and follow this branch
until extinction at high blowing pressure ( c’1:5), with no
production of double two-step regimes. However, we couldnot obtain this scenario experimentally. Another possible
order suggested by the bifurcation diagram, after thestandard two-step, is jumping to double two-step, second
register, inverted double two-step, and then inverted two-step, when it is the only stable branch (for c>1:5). The
experiment shows that it is possible to obtain all theseregimes in this order of emergence when increasing theblowing pressure.
Figure 7shows that the double two-step branches are
linked to the second register branch: a continuum of solu-tions exist between second register and double two-stepmotion—even though some of the solutions on the path areunstable. The junction between these branches can be seenas a period-doubling of the second register. Inverted regimesappear at high blowing pressure, which is coherent with thestatic behavior as the reed tends to close more and more
when the blowing pressure is higher. During the oscillation,
the reed closes for a longer and longer portion of the period,thus transitioning from standard to inverted motion. A highblowing pressure leads to extinction of the oscillation: thereed channel stays closed. Figure 7(b) shows the same met-
ric as Fig. 5, the duration ratio between closure episode and
period. It can be noted that the thresholds between the dif-ferent regimes are not the same as those fixed empirically.Additionally, the model predicts that the inverted two-stepcan appear at relatively low closure ratios, but these werenever found experimentally. This may be due to the inverted
double two-step being very stable in this blowing pressure
regions, thus making it hard to find other solutions.
It is worth noting that the same oscillating regimes
appear in the same order for other values of the reed openingparameter f, around the one used in Fig. 7(f¼0:6).
Figure 8shows two bifurcation diagrams, obtained for
f¼0:5 and f¼0:75, respectively. The stability region of
the regimes are affected by the value of f. In particular, a
lower f
enlarges the zone of stability of the second register
while a greater freduces it. It can also be noted that in this
particular case, a higher fvalue leads to an uninterrupted
single two-step branch, where standard and invertedFIG. 7. (Color online) Bifurcation diagram: (a) amplitude of the first cosine of the first modal pressure p1and (b) ratio between closure episode duration and
oscillation period; with respect to the blowing pressure parameter c, for the low Bfingering of an alto saxophone. In (a), the line aspect denotes stability of
the regimes: thick black line is stable, dotted gray line is unstable. Circle markers correspond to the plots in Fig. 6.f¼0:6.
J. Acoust. Soc. Am. 147(4), April 2020 Colinot et al. 2411https://doi.org/10.1121/10.0001109two-step are connected by stable regimes. Another comment
can be made on the bifurcation diagram obtained forf¼0:5 [Fig. 8(a)], on the inverted double two-step branch.
In this case, the inverted double-two-step branch that is con-nected to the second register branch only contains unstableregimes—in Fig. 8(a) it is the small branch of negative p
1,
between c¼0:86 and c¼1:04. This branch corresponds to
the branch in Fig. 7where the inverted double two-step
becomes stable. However, in Fig. 8(a), another inverted dou-
ble two-step branch shows stable regimes, that are indicatedby the inverted double two-step arrow. This other branch isnot connected to the second register, but to the inverted sin-gle two-step branch, by a long unstable portion of branch.Therefore it appears that double two-step regimes can beconsidered as degenerate from the single two-step or the
second register, depending on the value of the control
parameters.
A similar behavior is also observed for neighboring fin-
gerings. Figure 9shows the bifurcation diagram for thefingering just above the one used for Figs. 7and8: the low
Cfingering. The bifurcation diagram in Fig. 9has the same
structure as the others, although the inverted double two-step regime is unstable. In particular, the transition between
standard two-step and inverted two-step regimes is an unsta-
ble portion of branch featuring two-fold bifurcations (twopoints where two solutions collide and disappear, which can
be seen as turning-up points on the bifurcation diagram),
similar to that of Fig. 8, up, and Fig. 7. It is also worth not-
ing that on this fingering, the double two-step branch and
second register branch are connected by stable regimes
only: the thick lines connect at c¼0:8. This indicates that
for this fingering, it is possible to have continuous transition
between double two-step and second register using only sta-
ble regimes. A synthesized example of this transition is
shown in Mm. 6 .
Mm. 6. Animation: Evolution of the acoustic pressure
waveform and spectrum during a continuous transition
between double two-step regime and second register for
the low Cfingering of an alto saxophone, following
branches of the bifurcation diagram in Fig. 9(313.9 ko).
The double two-step regime becomes unstable on fin-
gerings D and higher for the main value of f¼0:6 studied
here. This may be a sign that its production is linked to the
high amplitude of the second and third resonances of the res-onator, which is a characteristic of the low fingerings of the
saxophone.
IV. CONCLUSION
Alto saxophones are able to produce double two-steps
motions that seem analogous to double stick-slip motions in
bowed strings ( Woodhouse, 2014 ). The production region of
these regimes appears linked to the second register of the
resonator. The appearance of the many oscillating regimes
on the studied fingerings may be due to the strong role ofFIG. 8. Bifurcation diagram: amplitude of the first cosine of the first modal pressure p1with respect to the blowing pressure parameter c, for the low Bfin-
gering of an alto saxophone. (a) f¼0:5 and (b) f¼0:75. The line aspect denotes stability of the regimes: thick black line is stable, dotted gray line is
unstable.
FIG. 9. Bifurcation diagram: amplitude of the first cosine of the first modalpressure p
1with respect to the blowing pressure parameter c, for the low C
fingering of an alto saxophone. f¼0:6, same as in Fig. 7.
2412 J. Acoust. Soc. Am. 147(4), April 2020 Colinot et al.https://doi.org/10.1121/10.0001109the second and third mode of the resonator. The simple sax-
ophone model used in this paper is capable of reproducing
these regimes, even though it ignores the impact betweenthe reed and the mouthpiece lay. The model also corrobo-
rates the order of appearance of these regimes when
increasing the blowing pressure on a real saxophone.Complementary numerical studies show that the double
two-step phenomenon is not restricted to a particular set of
parameters, but appears for several combinations of controlparameters and several fingerings. The description of the
playability of a saxophone in the low fingerings may take
these regimes into account, whether they are undesirable, asis the case for the double fly-back motion in violins, or a
useful tool of expressivity for the musician. Acoustical or
geometrical characteristics of the resonator remain to belinked to the ease of production of double two-step regimes.
ACKNOWLEDGMENTS
The authors would like to thank Louis Guillot for
advice and guidance in the construction of the bifurcationdiagram. This work has been carried out in the framework
of the Labex MEC (Contract No. ANR-10-LABX-0092) and
of the A*MIDEX project (Contract No. ANR-11-IDEX-0001-02), funded by the Investissements d’Avenir French
Government program managed by the French National
Research Agency (ANR). This study has been supported bythe French ANR 659 LabCom LIAMFI (Contract No. ANR-
16-LCV2-007-01).
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motion analogy,” Acta Acust. Acust. 86(4), 671–684.Dalmont, J.-P., and Le Roux, J. C. ( 2008 ). “A new impedance sensor for
wind instruments,” J. Acoust. Soc. Am. 123(5), 3014–3014.
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lore, hell and hard D,” in International Symposium on Musical Acoustics ,
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steady-state oscillations of a clarinet using the harmonic balancetechnique,” J. Acoust. Soc. Am. 86(1), 35–41.
Guillemain, P., Vergez, C., Ferrand, D., and Farcy, A. ( 2010 ). “An instru-
mented saxophone mouthpiece and its use to understand how an experi-enced musician plays,” Acta Acust. Acust. 96(4), 622–634.
Guillot, L., Cochelin, B., and Vergez, C. ( 2019 ). “A Taylor series-based
continuation method for solutions of dynamical systems,” Nonlinear
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string,” J. Acoust. Soc. Am. 49(2B), 532–537.
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Mechanics (Princeton University Press, Princeton, NJ).
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puting the stability of periodic solutions of dynamical systems,” Comptes
Rendus M /C19ecanique 338(9), 510–517.
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Ollivier, S., Kergomard, J., and Dalmont, J.-P. ( 2005 ). “Idealized models of
reed woodwinds. Part II: On the stability of ‘two-step’ oscillations,” ActaAcust. Acust. 91(1), 166–179.
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2413https://doi.org/10.1121/10.0001109 |
1.1988971.pdf | Magnetic switching of ferromagnetic thin films under thermal perturbation
Di Liu and Carlos Garcia-Cervera
Citation: Journal of Applied Physics 98, 023903 (2005); doi: 10.1063/1.1988971
View online: http://dx.doi.org/10.1063/1.1988971
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/98/2?ver=pdfcov
Published by the AIP Publishing
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131.230.73.202 On: Sat, 20 Dec 2014 17:52:53Magnetic switching of ferromagnetic thin films under thermal perturbation
Di Liua/H20850
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street,
New York, New York 10012
Carlos Garcia-Cerverab/H20850
Mathematics Department, University of California, Santa Barbara, California 93106
/H20849Received 12 January 2005; accepted 3 June 2005; published online 21 July 2005 /H20850
In this paper, we study the magnetic switching of submicron-sized ferromagnetic thin films under
thermal noise and external field. It is shown that the presence of the noise makes the switching easierwith weaker external fields and induces more intermediate metastable states in the switchingpathways. Different switching pathways are preferred at different temperatures. A quantitativerelation between the temperature and the switching field for different metastable states is giventhrough an adjusted Arrhenius formula near the critical field. Based on this, preferred switchingpathways at different temperatures are obtained by comparing the energy barriers along differentpathways. © 2005 American Institute of Physics ./H20851DOI: 10.1063/1.1988971 /H20852
I. INTRODUCTION
The process of magnetic switching of nanoscale ferro-
magnetic materials is a very important subject in the study ofthe magnetic recording process. Applications include tech-nologies of manufacturing computer disks and memory cells,such as magnetoresistive random access memory/H20849MRAM /H20850.
1,2As the size of the magnetic devices decreases,
thermal fluctuations become significant and must be includedin realistic analysis and simulations. An attempt to incorpo-rate thermal effects in micromagnetics was done by Brown,Jr. in /H20849Ref. 3 /H20850for the special case of single-domain particles
which have uniform magnetization. In Refs. 4 and 5, thehysteresis loops of single-domain particles at finite tempera-ture were studied with large deviation theory. For more gen-eral cases, experiments and numerical simulations have beenextensively carried on to understand the mechanism of theprocess /H20849see Refs. 6–8 and the references therein /H20850and vari-
ous observations have been obtained. On the other hand, themagnetic switching under external field is known to be verynonlinear and hysteretic. So what is obviously of interest isthe switching of magnetic field under the influences of boththermal perturbation and external field. Problems under in-vestigation are the switching fields and the switching path-ways at finite temperature in the hysteresis loops.
The paper is organized as follows. In Sec. II, we will
show the thermal effects on the hysteresis loops of ferromag-netic thin films by solving the full stochastic Landau-Lifshitz/H20849SLL /H20850equation. It is observed that under thermal noise, the
magnetization switches in the hysteresis loops with weakerexternal fields than without the noise. The noise may inducemore intermediate stages in the switching pathways by driv-ing the process into adjacent metastable states. And at differ-ent temperatures, the switching follows different pathways.Next, in Sec. III, we will analyze the overdamped SLL equa-tion for single-domain particles and give a quantitative rela-tion between the strength of the noise and the switching field
for different metastable states using an adjusted Arrheniusformula near the critical field. We will also show that thesystem can be approximated by a reduced discrete Markovchain switching between metastable states. Finally, in Sec.IV , we will apply the same method developed in Sec. III toferromagnetic thin films to predict the switching field be-tween different metastable states in the hysteresis loops atfinite temperature and give the preferred pathways at differ-ent temperatures. This is done by using the zero-temperaturestring method for micromagnetics to find the energy barriersbetween different metastable states.
II. MICROMAGNETICS UNDER THERMAL NOISE
A. Dynamics and numerical schemes
Based on the Landau-Lifshitz theory, the dynamics of
the magnetization distribution in a ferromagnetic material isdescribed by the following Landau-Lifshitz equation:
M˙=−
/H9253M/H11003H−/H9253/H9251
MsM/H11003/H20849M/H11003H/H20850, /H208491/H20850
where /H20841M/H20841=Ms/H20849const /H20850is the saturation magnetization far
from the Curie temperature. /H9251is a dimensionless damping
coefficient. /H9253=ge//H208492me/H20850is the gyromagnetic ratio where e
and meare the positive charge and mass of the electron. His
the local field computed as the following unconstrained firstvariation:
H=−/H9254F
/H9254M, /H208492/H20850
where Fis the Landau-Lifshitz energy functional,a/H20850Electronic mail: dliu@alumni.princeton.edu
b/H20850Electronic mail: cgarcia@math.ucsb.eduJOURNAL OF APPLIED PHYSICS 98, 023903 /H208492005 /H20850
0021-8979/2005/98 /H208492/H20850/023903/10/$22.50 © 2005 American Institute of Physics 98, 023903-1
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
131.230.73.202 On: Sat, 20 Dec 2014 17:52:53F/H20851M/H20852=1
2/H20885
/H9024/H20877Cex
Ms2/H20841/H11612M/H208412+/H9021/H20873M
Ms/H20874
−2/H92620HeM+/H92620/H20841/H11612U/H208412/H20878dx. /H208493/H20850
/H9024is the volume occupied by the material. Cexis the ex-
change constant, and /H9021/H20849M/Ms/H20850=Ku/H20849M22+M32/H20850/Ms2is the an-
isotropy. −2 /H92620HeMis the energy due to the external field He
where /H92620is the permeability of vacuum. The last term in the
energy is due to the field induced by the magnetization insidethe material such that
/H9004U=/H20877/H11612M,x/H33528/H9024
0, x/H33528/H9024c,/H20878 /H208494/H20850
together with the jump condition at the boundary /H20851U/H20852=0,
/H20851/H11612U·/H9263/H20852=−M·/H9263.
To introduce the thermal effects, we replace Hin /H208491/H20850
with H+/H20881/H9260w˚where wis a space-time white noise and ˚
denotes the Stratonovich integral in time. By the fluctuation-dissipation theorem, the strength of the noise should satisfy
/H9260=2/H9251KBT
/H208491+/H92512/H20850/H9253Ms, /H208495/H20850
where KBis the Boltzmann constant and Tis the tempera-
ture. Then Eq. /H208491/H20850becomes the SLL equation,
M˙=−/H9253/H11003M/H20849H+/H20881/H9260w˚/H20850−/H9253/H9251
MsM/H11003M/H11003/H20849H+/H20881/H9260w˚/H20850./H208496/H20850
It is shown in Appendix A that the strength of the noise we
add as in /H208495/H20850is consistent with the case when the thin-film
sample is reduced to single-domain particles,3for which both
the magnetization and stray field are uniform: therefore, theexchange energy vanishes. This happens when the size of thematerial sample is very small.
9We apply a quasistatic exter-
nal field such that
He=/H20849−1+/H9253⌊t//H9004⌋/H9004/H20850Hmax,t/H33528/H208750,2
/H9253/H20876, /H208497/H20850
where /H9004is the time interval during which each external field
is applied. We use ⌊x⌋to denote the biggest integer no larger
than xso the external field applied as above is changed qua-
sistatically with a constant value on each subinterval/H20851k/H9004,/H20849k+1/H20850/H9004/H20850.
/H9253/H110220 measures the speed with which the ex-
ternal field is changed.
We solve the Landau-Lifshitz equation /H20851Eq. /H208491/H20850/H20852with an
Euler-projection scheme. The Euler method is adopted forthe time discretization and the magnetic field is renormalizedto the sphere of constant magnetization at each time step:
M
*=Mti+/H9004t/H20873−/H9253Mti/H11003Hti−/H9253/H9251
MsMti/H11003Mti/H11003Hti/H20874,
Mti+1=MsM*
/H20841M*/H20841. /H208498/H20850
For the space discretization, we divide the computational do-
main into cells. In each cell we approximate the magnetiza-tion by a vector with constant magnitude, but free to rotate inany direction. We approximate the stray field by its average
value in each cell. This stray field is computed using fastFourier transform /H20849FFT /H20850. The details of this computation can
be found in /H20849Ref. 10 /H20850. We also solve the stochastic Landau-
Lifshitz equation /H20851Eq. /H208496/H20850/H20852with the Euler-projection scheme:
M
*=Mti+/H9004t/H20873−/H9253Mti/H11003H˜ti−/H9253/H9251
MsMti/H11003Mti/H11003H˜ti/H20874,
Mti+1=MsM*
/H20841M*/H20841, /H208499/H20850
where
H˜ti=Hti+/H20881/H9260//H9263/H9004wti, /H2084910/H20850
with/H9263being the volume of the computational cell. The time
discretization for the thermal noise is performed in the fol-lowing Stratonovich sense:
/H9004w
ti=wti+1−wti−1
/H208812, /H2084911/H20850
where /H20853wti/H20854’s are independent and identically distributed
standard random walks on the real line.
We choose the sample to be a 200 /H11003200/H1100310 nm3square
permalloy film and the computational grid to be 64 /H1100364. The
time step is chosen to be 10−13s for both the Landau-Lifshitz
equation /H20851Eq. /H208491/H20850/H20852and the stochastic Landau-Lifshitz equa-
tion /H20851Eq. /H208496/H20850/H20852. We use the same physical parameters as in
Ref. 7 such that
/H9251=1 ,
/H92620=4/H9266/H1100310−7N/A2,/H9253= 1.76 /H110031011T−1s−1,
Ms= 9.0/H11003105A/m, Ku= 1.0/H11003102J/m3,
Cex= 1.3/H1100310−11J/m, KB= 1.380 65 /H1100310−23J/K.
/H2084912/H20850
The external field is changed along the direction /H20849cos/H92580,
sin/H92580,0 /H20850,/H208490/H11021/H92580/H112701/H20850from − Hmax=−300 Oe to Hmax
=300 Oe. For the loop without noise, we initialize the mag-
netization uniformly with /H20849−1, 0, 0 /H20850in the loop. Each exter-
FIG. 1. Hysteresis loops of thin film at T=0 K and T=300 K.023903-2 Liu, Garcia-Cervera, and E J. Appl. Phys. 98, 023903 /H208492005 /H20850
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131.230.73.202 On: Sat, 20 Dec 2014 17:52:53nal field is run until the process reaches a stable state. This is
guaranteed by small thresholds for the magnitudes of thegradient field
/H9254F//H9254Mand the relative changes of Fand M.
For the loop with noise, we initialize the magnetization withthe stable state for the initial external field obtained in theloop without the noise. For each external field, the stochasticLandau-Lifshitz dynamics is run for 5 ns.
B. Hysteresis loops at zero and finite temperature
For ferromagnetic thin films, there are two stable con-
figurations at zero temperature, commonly known as Sstate
and Cstate. For Sstates, the magnetization in two end do-
mains is parallel to each other and the magnetization formsan S-shape configuration. For Cstates, the magnetization in
two end domains is opposite to each other and the magneti-zation forms a C-shape configuration.
11In Fig. 1, we show
the hysteresis loops without and with the thermal noise attemperature T=300 K. The switching in the zero-
temperature loop has two phases. The first one is from S
stage S
1toSstage S2and the second one is from Sstage S2
to another Sstage S3. At finite temperature, there are two
thermal effects on the loop. First, the switching occurs at alower field in the S
1→S2switching. Second, with the noise,
the magnetization switches along the S2→C1→S3pathway,
with one more intermediate Cstate stage C1. We apply
decreasing/increasing external fields on stage C1without
noise and get the stable states at zero temperature, which isshown in Fig. 2.
In Fig. 2, we also show the loops for different realiza-
tions of the thermal noise at the same temperature T
=300 K. It can be seen that the thermal effect that the mag-netization switches with a weaker field does not change with
respect to realizations. But instead of switching to C
1, for
some realizations with the same probability, the magnetiza-tion switches to another different Cstate stage C
2. For each
external field, states in C1and C2stages have the same mean
magnetization. Figure 3 gives the Sstate in stage S2and the
Cstates in stages C1and C2. It can be seen that there is a
symmetry between C1and C2with respect to S2. The mag-
netization in the top domains of S2and C1are opposite and
the magnetization in the bottom domains of S2and C1are
parallel. Meanwhile, the magnetization in the bottom do-mains of S
2and C2are opposite and the magnetization in the
top domains of S2and C2are parallel. In other words, from
S2, the magnetization switches to C1if the top domain of S2
changes the direction and switches to C2if the bottom do-
main of S2changes the direction. For very few realizations,
the intermediate Cstates are not observed. The same obser-
vation holds for temperatures up to T=900 K. We give in
Fig. 4 the loops from 0 to 900 K. It can be seen that thehigher the temperature, the weaker the external field neededfor the S
1→S2switching is.
Now we raise the temperature up to T=1200 K to see
the thermal effect on the hysteresis loops at higher tempera-tures. In Fig. 5, we show some realizations of the loops atT=1200 K in which the switching follows the S
2→C1/C2
→S3pathways. Consistent with the above observations, the
S1→S2switching happens at much lower fields. The only
difference from the lower temperatures is that after the mag-netization settles at the S
3stage, it may switch to another C
state which has a mean magnetization close to that of the S3
states. Figure 6 shows a different switching pattern for other
realizations at T=1200 K. The magnetization first switches
FIG. 2. Hysteresis loops of thin film for different realizations of the noise at
T=300 K.
FIG. 3. Sstate in stage S2and Cstates in stage C1and
C2.
FIG. 4. Hysteresis loops of thin film at different temperatures.023903-3 Liu, Garcia-Cervera, and E J. Appl. Phys. 98, 023903 /H208492005 /H20850
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131.230.73.202 On: Sat, 20 Dec 2014 17:52:53at a negative external field to a Cstate. There are two differ-
ent configurations for this Cstate, which we denote by CA
and CB. Then at a positive external field, the magnetic field
switches to a vortex state VAorVB, whose configuration
depends on the previous Cstate. After the vortex state, the
magnetization will switch to another Cstate stage CA/H11032orCB/H11032
depending on the path it has followed. Again, after the
CA/H11032/CB/H11032stages, the magnetization may switch to another Cor
Sstate with a close mean magnetization. In short, for these
realizations, the switching follows the S1→CA→VA→CA/H11032or
S1→CB→VB→CB/H11032pathway. Figures 7 and 8 give the states
in these pathways. The observation that the switching be-tween Cstates is through vortex nucleation and propagation
is consistent with Ref. 8.
In conclusion, the above results show that the thermal
noise makes the magnetic switching easier with weaker ex-ternal fields and induces different switching pathways. Thehigher the temperature, the more different pathways theswitching may follow. A quantitative study will be given insubsequent sections.
III. MAGNETIC SWITCHING OF SINGLE-DOMAIN
PARTICLES
In this section, we want to give quantitatively the rela-
tion between the strength of thermal noise and the switchingfield for different metastable states in the hysteresis loops ofsingle-domain particles in which the magnetization is uni-form. This relation is given through Eq. /H2084922/H20850.I f
/H9251/H112711, the
first term /H20849the gyromagnetic term /H20850on the right-hand side of
the Landau-Lifshitz equation /H20851Eq. /H208491/H20850/H20852is dominated by the
second term /H20849the damping term /H20850and hence can be dropped.Transforming into angular variables and introducing the ther-
mal noise, we get the following random perturbed gradientsystem describing the overdamped Landau-Lifshitz dynam-ics for single-domain particles under thermal noise andchanging external field:
/H9278˙=−1
/H9254/H11612/H9278V/H20849/H9278,t/H20850+/H208812/H9280
/H9254w˙, /H2084913/H20850
where
V/H20849/H9278,t/H20850=−1
4cos 2 /H20849/H9278−/H92580/H20850−h0/H20849−1+ ⌊t//H9004⌋/H9004/H20850cos/H9278. /H2084914/H20850
Here/H9278represents the angle between the magnetization and
the external field. /H9254is a constant obtained from variable
transformation and depends on the physical parameters ofthe system.
/H9280is the strength of the noise determined by
fluctuation-dissipation theorem and wis a standard Brownian
motion. The energy function V/H20849/H9278,t/H20850is chosen to be the
Stoner-Wohlfarth potential for single-domain ferromagnetic
materials. It is a special case of the Landau-Lifshitz potentialwhen the magnetization is uniform across the sample. We areassuming no crystalline anisotropy here for simplicity andthe fact that it is usually smaller than the shape anisotropy inpermalloy. The first term of V/H20849
/H9278,t/H20850is the anisotropy and the
second is the energy due to the following time-dependent
external field:
h=h0/H20849−1+ ⌊t//H9004⌋/H9004/H20850,t/H33528/H208510,2 /H20852, /H2084915/H20850
which is changed from − h0toh0./H9004represents the observa-
tion time for each external field. ⌊x⌋is defined as before to be
the biggest integer no larger than x. Notice that we can also
write V=Vh/H20849/H9278/H20850./H92580gives the preferred direction of magneti-
FIG. 6. Hysteresis loops at T=1200 K following the vortex pathways.
FIG. 5. Hysteresis loops at T=1200 K following the S2→C1/C2→S3
pathways.
FIG. 7. Cand vortex states in CA→VA→CA/H11032pathway
with increasing He.023903-4 Liu, Garcia-Cervera, and E J. Appl. Phys. 98, 023903 /H208492005 /H20850
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131.230.73.202 On: Sat, 20 Dec 2014 17:52:53zation by the sample. For simplicity and numerical tractabil-
ity, we choose the parameters such that
/H20849/H9254,h0,/H9004,/H92580/H20850=/H208492.0/H1100310−5, ± 1,2.0 /H1100310−3,/H9266/10 /H20850. /H2084916/H20850
There are several time scales in system /H2084913/H20850. The first
one is the relaxation time scale for the process to reach thesteady states of V
h/H20849/H9278/H20850. The second, represented by /H9004,i st h e
observation time scale for each external field. The third one
is the exit time scale which is the waiting time for the pro-cess to switch from one steady state of V
h/H20849/H9278/H20850to another
driven by the noise. This time scale depends on /H9280, the
strength of the noise. The fourth time scale is the switchingtime scale for the execution of the switching. By choosing
/H9254/H11270/H9004,/H9280/H112701, /H2084917/H20850
we have the relaxation and switching time scales much
smaller than the other two. Hence we discuss the behavior ofthe system when the observation time scale and exit timescale change with /H9004and
/H9280to study the effect of the thermal
perturbation on the hysteresis loops. For /H9280=0 and /H9280
=0.0025 /H20851Fig. 9 /H20852, plotting cos /H9278/H20849h/H20850against hwhen h0=±1
gives the hysteresis loops generated by the separation of the
local minima of Vh/H20849/H9278/H20850. The loops under noise are the nu-
merical solutions of Ecos /H20851/H9278/H20849t/H20850/H20852obtained by first solving the
forward Fokker-Planck equation for the probability distribu-
tion of /H9278/H20849t/H20850with periodic boundary conditions, then taking
the numerical integration for /H9278/H20849t/H20850with respect to the solution
of the Fokker-Planck equation. It can be seen that the noise
makes the switching easier with a weaker external field.
Now we want to analyze the relation between the
switching field and /H20849/H9004,/H9280/H20850.Vh/H20849/H9278/H20850can be numerically com-puted very easily. First, we have the observation that func-
tion Vh/H20849/H9278/H20850exhibits different properties for different h. Let-
ting
hc= 0.590, /H2084918/H20850
which is the switching field in the hysteresis loop without
noise, we have the following:
/H208491/H20850forh/H33355−hcand h/H33356hc,Vh/H20849/H9278/H20850has one minimum and one
maximum, and
/H208492/H20850for − hc/H11021h/H11021hc,Vh/H20849/H9278/H20850has two local minima and two
local maxima.
The energy landscapes of Vfor different values of hare
shown in Fig. 10. Due to symmetry, we only analyze theswitching near the critical field h
cwhen the external field is
applied from −1 to 1. For − hc/H11021h/H11021hc, we denote by /H92741/H20849h/H20850
the local minimum of Vh/H20849/H9278/H20850shown in Fig. 9 as the bottom
loop without noise when his changed from −1 to 1 and
denote by /H92742/H20849h/H20850the other local minimum. We also denote by
/H92581/H20849h/H20850and/H92582/H20849h/H20850the local maxima. Let /H9258/H20849h/H20850/H33528/H20853/H92581/H20849h/H20850,/H92582/H20849h/H20850/H20854
be the critical point with a lower-energy barrier such that
Vh/H20851/H9258/H20849h/H20850/H20852=min /H20853Vh/H20851/H92581/H20849h/H20850/H20852,Vh/H20851/H92582/H20849h/H20850/H20852/H20854. We define the energy
barrier /H9004V/H20849h/H20850from/H92741to/H92742to be
/H9004V/H20849h/H20850=Vh/H20851/H9258/H20849h/H20850/H20852−Vh/H20851/H92741/H20849h/H20850/H20852. /H2084919/H20850
Numerical results given in Fig. 11 show that /H9004V/H20849h/H20850is mono-
tonically decreasing with respect to hon the interval
/H20851−hc,hc/H20852.
Notice that on each time subinterval /H20851i/H9004,/H20849i+1/H20850/H9004/H20850, dy-
namics /H2084913/H20850is homogeneous, i.e., the coefficients are time
FIG. 8. Cand vortex states in CB→VB→CB/H11032pathway
with increasing He.
FIG. 9. Hysteresis loops of single-domain particle with and without the
noise.
FIG. 10. Energy landscapes of the single-domain particles for different ex-ternal fields.023903-5 Liu, Garcia-Cervera, and E J. Appl. Phys. 98, 023903 /H208492005 /H20850
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131.230.73.202 On: Sat, 20 Dec 2014 17:52:53independent. It is shown in Appendix B that for each external
field near hc, the mean exit time /H9270for the process to over-
come the energy barrier /H9004Vto switch from /H92741to/H92742is given
by the following adjusted Arrhenius formula:
/H9270/H20849h/H20850=/H92630/H9254
/H92801/3exp/H20875/H9004V/H20849h/H20850
/H9280/H20876, /H2084920/H20850
where /H92630=/H208480/H11009e−x3/6dx2//H20841V/H208493/H20850/H20851lim h→hc−/H92741/H20849h/H20850/H20852/H208412/3. Direct nu-
merical computation shows that /H92630=2.127. Due to the expo-
nential dependence of the exit time scale /H9270on/H9004Vand the
monotonicity of /H9004V, the exit time scale decreases when h
approaches hc. The switching should be most probable when
the exit time scale is comparable to the observation timescale such that
/H9270/H20849h*/H20850=/H9004. /H2084921/H20850
The monotonicity of /H9004V/H20849h/H20850implies an inverse function
/H20849/H9004V/H20850−1for/H9004Vand a unique solution for Eq. /H2084921/H20850. Then we
have the following equation for the switching field in the
hysteresis loops under random perturbation:
h*=/H20849/H9004V/H20850−1/H20875/H9280ln/H20873/H92801/3/H9004
/H92630/H9254/H20874/H20876. /H2084922/H20850
We solve the Fokker-Planck equation to get the loops for
different strengths of the noise and pick up the switchingfield in the loop to be the first field where the magnetizationis fully switched. Table I gives this result and the switchingfield predicted by /H2084922/H20850. The relative error is /H110210.03. The hys-
teresis loop and the predicted switching field when
/H9280=0.01
are shown in Fig. 12.
By the fact that the dynamics is homogeneous on each
subinterval, we know12that for each external field, the
nth/H20849n/H33528N/H20850moment /H9270n/H20849/H9278/H20850of the exit time satisfies the equa-
tion−1
/H9254/H11509V
/H11509/H9278/H11509/H9270n
/H11509/H9278+/H92802
/H9254/H115092/H9270n
/H11509/H92782=−n/H9270n−1. /H2084923/H20850
Boundary layer analysis as in Ref. 13 can show that except
on a boundary layer of thickness of n/H92802,
/H9270n/H20849/H9278/H20850=/H20849nK/H20850/H9270n−1/H20849/H9278/H20850= const. /H2084924/H20850
This means that away from the boundary, the exit time can
be approximated by an exponentially distributed randomvariable with parameter
/H9270/H20849/H9278/H20850=Kin the sense that the mo-
ments are asymptotically close. Similar results are also given
in Ref. 14 by a more subtle analysis. The above analysisimplies that we can approximate dynamics /H2084913/H20850by a discrete
Markov chain in which the process switches between
/H92741/H20849h/H20850
and/H92742/H20849h/H20850with exponentially distributed transition probabili-
ties, which provides a method to dramatically reduce the
computation by simulating the Markov chain instead of solv-ing the Fokker-Planck equation. Figure 13 gives the expec-tations of the magnetization with respect to this Markovchain using the mean exit time
/H9270given by /H2084920/H20850when /H9280
=0.01. It can be seen that it is almost identical with the result
by solving the Fokker-Planck equation.
IV. SWITCHING OF FERROMAGNETIC THIN FILMS
From Sec. II, we see that the thermal noise has two
effects on the magnetic switching. The first is to make theswitching easier with weaker external fields. The second is toinduce different switching pathways at different tempera-tures. The questions that need to be answered are what theswitching field under noise is and which pathways are pre-ferred at different temperatures. In this section, we will applythe adjusted Arrhenius formula /H20851Eq. /H2084920/H20850/H20852to these problems.
A. Energy landscapes under different external
fields
First we want to study the energy landscapes of thin
films under different external fields and the implications inmagnetic switching. For a given energy function or func-tional V/H20849x/H20850, the minimum energy path /H20849MEP /H20850
/H9272between dif-
ferent metastable states Aand Bis defined to be the curve
connecting Aand Band satisfying the following equation:15
FIG. 11. The energy barrier /H9004V/H20849h/H20850for − hc/H11021h/H11021hc.
TABLE I. Simulated and predicted switching fields of single-domain par-
ticles.
/H9280 0.00125 0.0025 0.005 0.01
Switching field 0.580 0.566 0.542 0.500
Predicted field 0.574 0.563 0.544 0.513
FIG. 12. Predicted switching field for single-domain particle when /H9280=0.01.023903-6 Liu, Garcia-Cervera, and E J. Appl. Phys. 98, 023903 /H208492005 /H20850
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131.230.73.202 On: Sat, 20 Dec 2014 17:52:53/H11612/H11036V/H20849/H9272/H20850=0 , /H2084925/H20850
where/H11036denotes the projection onto the hyperplane perpen-
dicular to /H9272. It it known16that, for Landau-Lifshitz potential
and dynamics, the MEP shares the same critical points andhence the same energy barriers with the transition pathwaysbetween different metastable states at finite temperature. Thezero-temperature string method for micromagnetics
16solves
the MEP by evolving the following gradient flow in the pathspace:
/H9278t=−/H11612/H11036V/H20849/H9278/H20850+rtˆ, /H2084926/H20850
where /H9278=/H9278/H208510,1 /H20852is a curve connecting Aand B.tˆ=/H9278/H9251//H20841/H9278/H9251/H20841is
the unit tangent along /H9278.ris a Lagrangian multiplier which
keeps a certain parametrization of the evolving curve. In thefollowing, we are going to choose rsuch that
/H20841
/H9278/H20841/H9251//H20841/H9278/H208411= sin /H20849/H9251/H9257/H9266/2/H20850,/H9251/H33528/H208510,1 /H20852, /H2084927/H20850
where /H20841/H9278/H20841/H9251is the arclength of the evolving curve. /H9257/H110221i sa
fixed number. This parametrization allows to focus on thenucleation of the switchings. The string method can be easilyparallelized by dividing the evolving curve into a certainnumber of subcurves, all evolving with /H2084926/H20850. This can be
implemented by an message passing interface /H20849MPI /H20850
structure.
In Fig. 14, we give the energy barriers between S
1and S2
stages for different external fields obtained by using the
string method. It can be seen that consistent with the case ofsingle-domain particles, the energy barrier decreases whenthe external field is increased. This means that the higher thetemperature, the lower the field needed for the switching,which is also observed in direct simulations.B. Switching field at finite temperature
Now we want to study the switching field in the hyster-
esis loops of thin films with the adjusted Arrhenius formula.We do the following substitution for /H2084920/H20850in the context of
the stochastic Landau-Lifshitz dynamics:
/H9254→1
/H9251/H9253Ms,/H9280→kBT. /H2084928/H20850
Then we have the adjusted Arrhenius formula for ferromag-
netic thin film which gives the mean exit time from theneighborhood of one metastable state near the critical field:
/H9270=/H92630
/H9251/H9253Ms/H20849kBT/H208501/3exp /H20851/H9004F//H20849kBT/H20850/H20852, /H2084929/H20850
where /H9004Fis the energy barrier between different metastable
states. /H92630is a prefactor depending on the third-order curva-
ture of the energy landscape at the critical field. In the hys-teresis loops with thermal noise, the switching should happenwhen the exit time scale is equal to the observation timescale, i.e.,
/H9270=/H9004. /H2084930/H20850
Due to the difficulty arising from the infinite dimensional
nature of the Landau-Lifshitz dynamics, we still do not haveefficient tools to estimate
/H92630. Since the dependence of /H9270on
the energy barrier /H9004Fis exponential and much more signifi-
cant than the dependence of /H9270on/H92630, we assume /H92630=1. By
the monotonicity of /H9004F, we have the following equation for
the switching field at temperature T:
He*=/H20849/H9004F/H20850−1/H20853kBTln/H20851/H9004/H9251/H9253Ms/H20849kBT/H208501/3/H20852/H20854. /H2084931/H20850
TABLE II. Simulated and predicted switching fields of thin films.
Temperature /H20849K/H20850 100 200 300 400 500 600
Mean switching field /H20849Oe/H20850 132.7 125.2 124.4 120.7 103.4 96.7
Predicted field /H20849Oe/H20850 127.1 121.2 116.3 111.7 107.5 103.5
FIG. 13. Simulation of the hysteresis loop with the reduced Markov chain
when /H9280=0.01.
FIG. 14. The energy barriers between S1and S2stages for different external
fields.023903-7 Liu, Garcia-Cervera, and E J. Appl. Phys. 98, 023903 /H208492005 /H20850
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131.230.73.202 On: Sat, 20 Dec 2014 17:52:53Now we focus on the S1→S2switching. We use formula
/H2084931/H20850to predict the switching field for temperatures from 100
to 600 K. At the same time, we compute eight realizations ofthe loops for each temperature and take the mean switchingfield. Table II and Fig. 15 give the mean switching fields andthe predicted fields. The relative error is /H110210.08.
C. Preferred pathways at different temperatures
Now we want to study the preference of different path-
ways at different temperatures. We first focus on the lowertemperatures. From the results in Sec. II, we know that atzero temperature, the switching follows the S
2→S3pathway
while at finite temperature, the switching follows the S2
→C1/C2→S3pathways. Figure 16 gives the energy barriers
of the S2→S3and S2→C1switchings for different external
fields. Due to symmetry, the energy barriers for the S2→C2
switching will be the same as those for the S2→C1switch-
ing. It can be seen that the energy barriers for the S2
→C1/C2switchings are much smaller than the barriers of
theS2→S3switching. This means that the S2→C1/C2→S3
pathways are more preferred than the S2→S3pathway. InFig. 17, we give the energy along the minimum-energy paths
ofS2→S3and S2→C1→S3switchings when the external
field is 90 Oe.
Now we move to the higher temperature of 1200 K.
Figure 18 gives the energy barriers for the S1→CApathways
for different external fields. Again for the reason of the sym-metry between C
Aand CBwith respect to S1, the energy
barriers for the S1→CBis the same. It can be seen that the
energy barrier increases when external field is increased fromnegative to positive. Since this switching is away from thecritical field, we can use the original Arrhenius formula with-out the adjustment in the prefactor:
/H9270=/H92630
/H9251/H9253Msexp /H20851/H9004F//H20849kBT/H20850/H20852. /H2084932/H20850
Assuming /H92630=1 again in switching condition /H2084930/H20850gives
He*=/H20849/H9004F/H20850−1/H20851kBTln/H20849/H9004/H9251/H9253Ms/H20850/H20852. /H2084933/H20850
The predicted switching field for S1→CA/CBatT=1200 K
using formula /H2084933/H20850isHe,1200*=−129.1197 Oe. At T=600 K,
the predicted field is He,600*=−348.9502 Oe. Notice that
He,600*/H11021−Hmax/H11021He,1200*where Hmaxis the maximum exter-
FIG. 16. Energy barriers of S2→S3and S2→C1switchings for different
external fields.
FIG. 15. Simulated and predicted switching field between S1and S2stages.
FIG. 17. Energy along the MEPs of S2→S3and S2→C1→S3switchings
when He=90 Oe.
FIG. 18. Energy barriers for the S1→CAswitching under different external
fields.023903-8 Liu, Garcia-Cervera, and E J. Appl. Phys. 98, 023903 /H208492005 /H20850
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131.230.73.202 On: Sat, 20 Dec 2014 17:52:53nal field applied on the sample. This means that, under the
external field we are applying, the S1→CA/CBswitching is
impossible at T=600 K and is more preferred at T=1200 K.
In other words, when the external field is negatively largeand the temperature is high, the energy barriers separating S
1
and CA/CBare so small that they are ignored by the fluctua-
tion. This also by symmetry explains the observation that theprocess may switch to other Sand Cstates after C
A/H11032/CB/H11032when
the external field gets positively large and the fluctuationbecomes important again. In Fig. 19, we give the energyalong the minimum energy path along the C
A→VA→CA/H11032
switching when the external field is 180 Oe. The increasing
energy barrier also implies that the S1→CA/CBswitching
becomes difficult when the external field is increased fromnegative to positive. This is why for some of the realizations,when the S
1→CA/CBswitching does not happen at large
negative external fields, the switching will follow the S1
→S2pathway later when the external field gets positive.
V. CONCLUSIONS
So far, we have studied the effects of incorporating ther-
mal noise into the full Landau-Lifshitz dynamics for ferro-magnetic thin films and the relation between the thermalnoise and switching field in the hysteresis loops. The sto-chastic Landau-Lifshitz equation with Stratonovich noise issolved with an Euler-projection method. The energy barriersbetween different metastable states under different externalfields are obtained with the string method. Hence the switch-ing field can be predicted for different temperatures by theadjusted Arrhenius formula and preferred pathways at differ-ent temperatures can be given. Future work involves the as-ymptotics for the prefactor of the Arrhenius formula in infi-nite dimensions.
ACKNOWLEDGMENTS
We are grateful to Weinan E, Robert Kohn and Eric
Vanden-Eijnden for stimulating discussions and insightfulcomments. One of the authors /H20849D.L. /H20850partially acknowledges
partial support by NSF via Grant No. DMS97-29992 and byONR via Grant No. N00014-01-1-0674. We want to thankPrinceton Institute for Computational Science and Engineer-
ing for providing the computing resources.
APPENDIX A: THE STRENGTH OF THE THERMAL
NOISE
Here we want to give an argument for the strength of the
noise we added in the Landau-Lifshitz equation as given by/H208495/H20850. In Ref. 3, the following Gilbert equation was used to
describe the dynamics of the magnetization for single-domain particles:
M
t=/H92530M/H11003/H20873−1
/H9263/H11509F//H11509M−/H9257Mt/H20874=/H92530M/H20849H−/H9257Mt/H20850,
/H20849A1 /H20850
where Fis the Landau-Lifshitz potential and /H9263is the volume
of the particle. Comparing the coefficients with /H208491/H20850gives
/H92530=−/H9253/H20849/H92512+1 /H20850,/H9257=/H9251
/H9253Ms/H20849/H92512+1 /H20850. /H20849A2 /H20850
To incorporate the thermal noise, in Ref. 3, the field Hin
/H20849A1/H20850was replaced by H+h˚twhere h˚tis a Stratonovich noise
white in time. It is shown3that for the above equation to
have an equilibrium distribution e−F/H20849M/H20850//H20849kBT/H20850, the strength of
the noise htshould satisfy the following:
/H20855hti/H20856=0 , /H20855htiht+/H9270j/H20856=2kBT/H9257
/H9263/H9254/H20849/H9270/H20850=2/H9251kBT
/H9253Ms/H20849/H92512+1 /H20850/H9263/H9254/H20849/H9270/H20850
=D/H9254/H20849/H9270/H20850. /H20849A3 /H20850
If we add the noise in /H208496/H20850such that /H9260=2/H9251kBT//H9253Ms/H20849/H92512+1/H20850.
Then after space discretization, the noise on each computa-
tional cell ksatisfies
/H20855wti/H20856=0 , /H20855wti,wt+/H9270j/H20856=D/H9254ij/H9254/H20849/H9270/H20850, /H20849A4 /H20850
which is consistent with /H20849A3/H20850and fluctuation-dissipation
theorem.
APPENDIX B: ADJUSTED ARRHENIUS FORMULA
NEAR THE CRITICAL FIELD
In this Appendix, we want to give the adjusted Arrhenius
formula for the single-domain particles near the critical fieldh
c. Notice that by the definition Vh/H20851/H9258/H20849h/H20850/H20852=min /H20853Vh/H20851/H92581/H20849h/H20850/H20852,
Vh/H20851/H92582/H20849h/H20850/H20852/H20854. With a much higher probability, the process
switches by overcoming /H9258/H20849h/H20850and/H9004V/H20849h/H20850instead of overcom-
ing the other critical point /H20853/H92581,/H92582/H20854//H9258, which has a higher
energy barrier. Hence we can impose a reflecting boundarycondition at /H20853
/H92581,/H92582/H20854//H9258. Solving directly /H20851Eq. /H2084923/H20850/H20852forn=1,12
we have
/H9270/H11015/H9254
/H9280/H20885
/H92741/H92742
d/H9251exp /H20851V/H20849/H9251/H20850//H9280/H20852/H20885
/H20853/H92581,/H92582/H20854//H9258/H9258
d/H9252exp /H20851−V/H20849/H9252/H20850//H9280/H20852.
/H20849B1/H20850
It can be seen that when h→hc−,/H9258/H20849h/H20850→/H9258*and/H92741/H20849h/H20850→/H9258*for
some/H9258*. Notice that by the stability condition,
FIG. 19. Energy along the MEP of CA→VA→CA/H11032switching when He
=180 Oe.023903-9 Liu, Garcia-Cervera, and E J. Appl. Phys. 98, 023903 /H208492005 /H20850
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131.230.73.202 On: Sat, 20 Dec 2014 17:52:53Vh/H11032/H20851/H9258/H20849h/H20850/H20852=Vh/H11032/H20851/H92741/H20849h/H20850/H20852=0 , Vh/H11033/H20851/H9258/H20849h/H20850/H20852/H333550/H33355Vh/H11033/H20851/H92741/H20849h/H20850/H20852.
/H20849B2/H20850
By continuity, we have for i=1, 2,
Vh/H20849i/H20850/H20849/H9258*/H20850= lim
h→hc−Vh/H20849i/H20850/H20851/H9258/H20849h/H20850/H20852= lim
h→hc−Vh/H20849i/H20850/H20851/H92741/H20849h/H20850/H20852=0 , /H20849B3/H20850
while direct computation shows that V/H208493/H20850/H20849/H9258*/H20850/H110210. We make
the following third-order Taylor expansion for /H92741/H11021/H9278/H11021/H92742:
Vh/H20849/H9278/H20850/H11015Vh/H20849/H9258/H20850+1
2Vh/H208492/H20850/H20849/H9258/H20850/H20849/H9278−/H9258/H208502+1
6Vh/H208493/H20850/H20849/H9258/H20850/H20849/H9278−/H9258/H208503
/H11015Vh/H20849/H9258/H20850+1
6Vh/H208493/H20850/H20849/H9258*/H20850/H20849/H9278−/H9258*/H208503, /H20849B4/H20850
and for /H20853/H92581,/H92582/H20854//H9278/H11021/H9278/H11021/H9258, we have
Vh/H20849/H9278/H20850/H11015Vh/H20849/H92741/H20850+1
6Vh/H208493/H20850/H20849/H9258*/H20850/H20849/H9278−/H9258*/H208503. /H20849B5/H20850
From Fig. 20, we can see that we only need to integrate on
half of the real line. Hence we have/H9270/H11015/H9254
/H9280exp/H20875Vh/H20849/H9258/H20850−Vh/H20849/H92741/H20850
/H9280/H20876
/H11003/H20885
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6Vh/H208493/H20850/H20849/H9258*/H20850/H20849/H9278−/H9258*/H208503//H9280/H20876d/H9278
/H11003/H20885
−/H11009/H9258*
exp/H20875−1
6Vh/H208493/H20850/H20849/H9258*/H20850/H20849/H9272−/H9258*/H208503//H9280/H20876d/H9272
=/H9254/H20873/H20885
0/H11009
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/H92801/3/H20841V/H208493/H20850/H20849/H9258*/H20850/H208412/3exp/H20873/H9004V
/H9280/H20874. /H20849B6/H20850
1J. Daughton, Thin Solid Films 216, 162 /H208491992 /H20850.
2G. Pinz, Science 282, 1660 /H208491998 /H20850.
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/H20849submitted /H20850.
6R. H. Koch, G. Grinstein, G. A. Keefe, Yu. Lu, P. L. Trouilloud, W. J.
Gallagher, and S. S. P. Parkin, Phys. Rev. Lett. 84, 5419 /H208492000 /H20850.
7J. Li and J. Shi, Appl. Phys. Lett. 93, 3821 /H208492001 /H20850.
8C. J. García-Cervera and W. E, IEEE Trans. Magn. 39, 1766 /H208492003 /H20850.
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240,5 9 9 /H208491948 /H20850.
10C. J. García-Cervera, Z. Gimbutas, and W. E, J. Comput. Phys. 184,3 7
/H208492003 /H20850.
11J. Miltat, G. Albuquerque, and A. Thiaville, in Topics in Applied Physics ,
Spin Dynamics in Confined Magnetic Structures I, edited by B. Hill-ebrands and K. Ounadjela /H20849Springer, Berlin, 2002 /H20850,p .8 3 .
12C. W. Gardiner, Handbooks of Stochastic Methods for Physics, Chemistry
and Natural Sciences , Springer Series in Synergetics 13 /H20849Springer, Berlin,
1983 /H20850.
13B. J. Matkowsky and Z. Schuss, SIAM J. Appl. Math. 22,3 6 5 /H208491977 /H20850.
14F. Martinelli, E. Olivieri, and E. Scoppola, J. Stat. Phys. 55,4 7 7 /H208491989 /H20850.
15H. Jónsson, G. Mills, and K. W. Jacobsen, in Classical and Quantum
Dynamics in Condensed Phase Simulations , edited by B. J. Berne, G.
Ciccotti, and D. F. Cokder /H20849World Scientific, Singapore, 1998 /H20850,p .3 8 5 .
16W. E, W. Ren, and E. Vanden-Eijnden, J. Appl. Phys. 93,2 2 7 5 /H208492003 /H20850.
FIG. 20. Energy landscape for single-domain particle near the critical field.023903-10 Liu, Garcia-Cervera, and E J. Appl. Phys. 98, 023903 /H208492005 /H20850
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1.4801037.pdf | Objective evaluation of musical instrument quality: A grand challenge in musical
acoustics.
Murray Campbell
Citation: Proc. Mtgs. Acoust. 19, 032003 (2013); doi: 10.1121/1.4801037
View online: https://doi.org/10.1121/1.4801037
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Volume 19, 2013
http://acousticalsociety.org/
ICA 2013 Montreal
Montreal, Canada
2 - 7 June 2013
Interdisciplinary
Session 3aID: Plenary Lecture: Objective Evaluation of Musical Instrument Quality: A
Grand Challenge in Musical Acoustics
3aID1.
Objective evaluation of musical instrument quality: A grand challenge in
musical acoustics.
Murray Campbell*
*Corresponding author's address: University of Edinburgh, Edinburgh, EH9 3JZ, Scotland, United Kingdom,
d.m.campbell@ed.ac.uk
Over the last few decades, increasingly sophisticated experimental and computational studies have clarified the processes involved in sound
production in musical instruments. One of the principal goals of this research effort has, however, remained tantalisingly elusive: the
establishment of clear and unambiguous relationships between objectively measured properties of an instrument and judgements of its musical
qualities by an experienced player. This is partly because player evaluation is a subtle and highly subjective process in which many different
aspects of the instrument's performance may be tested. Early studies concentrated on the steady state spectra of sound recorded in the far field of
the instrument. More recently it has been recognised that transient aspects of an instrument's performance are important in judgements of quality
made by performers. These aspects include the ease with which a stable regime of oscillation can be initiated,and the flexibility with which
pitch, amplitude and timbre can be modified during performance. Attempts to define "playability" of an instrument in scientific terms, and to
relate these scientific metrics to the vocabulary used by performers in judgements of playability, have been partially successful, but many
questions remain unanswered.
Published by the Acoustical Society of America through the American Institute of PhysicsM. Campbell
© 2013 Acoustical Society of America [DOI: 10.1121/1.4801037]
Received 10 Feb 2013; published 2 Jun 2013
Proceedings of Meetings on Acoustics, Vol. 19, 032003 (2013) Page 1INTRODUCTION
Why Study Musical Acoustics?
Music, in its widest sense, is perhaps the most universal human art form. The first exchanges between a
mother and her new-born child are already musical in nature [1], and our most profound emotional andspiritual experiences throughout life are often enriched and transformed by music. It is therefore hardly
surprising that from the time of the Ancient Greeks to the present day philosophers and scientists have beenattracted to the study of musical sound and musical sound sources. This is the field of research which is nowgenerally described as musical acoustics.
Most of the current work in musical acoustics falls into one of three broad categories. The first is
concerned with the physics of musical instruments and other sound sources; the second deals with thetransmission of sound from source to listener , including the important topics of concert hall acoustics andsound recording and reproduction; the third considers the psychoacoustics of musical sound perception.
There are many motivations for research work in these different areas, but two of the strongest are common
to almost all studies in musical acoustics. One is simple scientific curiosity: the desire to understand how therelatively simple laws of physics lead to such rich and complex phenomena as are evident in a musicalperformance. Musically important aspects of sound production, transmission or perception often involve verysubtle features of the underlying mechanisms, and therefore provide particularly stringent tests of
simplifications and approximations. The other is the desire to be able to offer well-founded guidance to those
engaged in the practical business of music making: players, teachers, musical instrument makers, soundrecording engineers and designers of spaces for musical performance. Such professionals have developed awealth of experience and skill, often the fruit of generations of trial and error . The hope of the musicalacoustician is to find ways of supplementing this practical knowledge with scientific principles and toolswhich will lead to more cost-effective and reliable methods of achieving the goal of musical excellence.
Can Musical Instrument Quality Be Measured Scientifically?
Factors Involved in Judgements of Quality
This talk reviews attempts to isolate and describe scientifically the factors which make an instrument
musically excellent. It is important at the outset to recognise that what may seem to be an obvious
improvement in an instrument from the scientific point of view may not be accepted as such by musicians,who must be the ultimate arbiters of musical quality . An instructive example is provided by the work of thenineteenth century instrument maker and acoustician Theobald Boehm, who in 1832 revolutionised thedesign of the flute by introducing a cylindrical bore incorporating much larger tone holes than earlier
instruments. The increase in tone hole area resulted in a significant increase in radiated sound power , which
was welcomed by musicians and widely adopted. In 1844 Louis August Buffet, assisted by Boehm, introduceda new design of oboe with similarly enlarged tone holes; the redesigned instrument was certainly louder , butits sound was generally considered too bright and strident. It did not replace the traditional design of oboe inthe orchestra, although it had some use in outdoor performances by military bands [2].
The example of the Boehm oboe illustrates that the timbre of an instrument is at least as important as
its sound power in musical quality evaluation. Initial scientific studies of musical instrument timbre were
largely concerned with spectral analysis of steady state sounds. Early attempts in musical sound synthesis
revealed the importance of transients in the perception of instrumental sounds and the identification ofspecific instruments. In recent years it has been increasingly recognised that factors relating to the player’sinteraction with the instrument are of particular importance in judgement of instrument quality by a player .Thes factors include the ease with which a note may be initiated, the uniformity of response over the playingrange, and the flexibility which is available for modulating the pitch, loudness and timbre of the sound.
These factors are frequently grouped together under the general term ‘playability’.M. Campbell
Proceedings of Meetings on Acoustics, Vol. 19, 032003 (2013) Page 2Requirements for the Investigation of Instrument Quality
The major problem in evaluating musical instrument quality is that of relating subjective judgements by
players to objective scientific measurements. This task requires that a number of conditions are satisfied:
1. Experimental arrangements for scientific measurements must be devised which are not too divorced
from musical practice.
2. A sufficiently high level of accuracy must be achieved to explore subtle but musically significant effects.3. Numerical models must be developed which are not too simplified to be musically relevant.4. The language(s) used by players to describe judgements of musical instrument quality must be
interpreted and related to scientific terminology .
5. Musicians of sufficiently high calibre must be willing to collaborate if musically meaningful results are
to be obtained from player tests.
6. Psychoacoustic evaluation studies must be devised which take fully into account possible player bias
and inconsistency .
To illustrate the ways in which these problems have been tackled, the results that have been obtained
and the areas still requiring study , we focus on three areas in which a significant amount of work has beendone, and which are still fields of active research: pianos, bowed string instruments and brass instruments.
STUDIES OF PIANO QUALITY
A Simple Model of Piano Sound Production
At first sight, the science of sound production in a piano seems straightforward. A force applied to one of
the keys is transmitted to a pivoted hammer by a lever system known as the action. The construction of the
action is such that the hammer is accelerated to a velocity which depends on the applied key force, and thenreleased; after release it swings freely , and its felt-covered head hits and rebounds from either one string or asmall group of strings tuned in unison. A check mechanism holds the rebounding hammer , preventing adouble hit. The impulsive excitation of a string by a blow from the hammer head imparts energy to thenormal modes of the string, which decay in free vibration. The string is coupled through a bridge to the piano
soundboard, which in turn vibrates and radiates sound.
The pianist has no contact with the hammer or string at the point of impact, and the nature of the string
vibration is determined by a single variable, the hammer head velocity . It thus appears possible to modelpiano sound production in terms of a linear system taking account of the position of striking on the string andthe resonance characteristics of the soundboard. To make such a model musically realistic, however , anumber of complicating factors have to be taken into account [3].
Refinements of the Simple Piano Model
The hammer-String Interaction
Modern piano hammer heads are covered by several layers of felt, which has nonlinear compressive
behaviour [4, 5]. The way in which the loudness and timbre of the piano sound changes with key force
therefore depends strongly on the way in which the hammer head has been treated, and on its history of use.The spectral content of the string vibration is affected by the time of contact of the hammer head with thestring, which also depends on the state of the felt covering. The flexibility of the hammer shank has also beenshown to affect the transfer of energy from hammer head to string [6].M. Campbell
Proceedings of Meetings on Acoustics, Vol. 19, 032003 (2013) Page 3String Inharmonicity
The finite stiffness of steel piano strings leads to inharmonicity of the string normal mode frequencies.
[7]. The deviation from perfectly harmonic frequency components increases with mode number , and has a
significant effect on the timbre of piano sound. String inharmonicity also affects the way that pianos are
tuned: careful measurements of expertly tuned concert grand pianos have shown that the tuning is‘stretched’, with an octave ratio slightly greater than 2. Interestingly , these deviations from perfectharmonicity contribute desirable features to the piano sound quality; synthesised sounds with harmonicfrequency components are judged by musicians to lack liveness and warmth [8].
The sound quality of upright pianos is generally considered to be inferior to that of full sized grand
pianos, especially in the bass register . This has frequently been attributed to the fact that the bass strings on
a upright piano are much shorter than those on a grand piano, and therefore have a much higher degree of
inharmonicity . Recent research has suggested that differences in spectral envelope related to the absence oflow frequency resonances in the soundboard of the upright may be more influential than inharmonicity insuch comparisons of piano quality [9].
Effect of String Coupling on Amplitude Envelopes
One of the most desirable features of piano sound is the ‘singing’ quality associated with a slow decay
rate. Piano strings typically display a double decay rat e - a fast intial decay followed by a slowly decaying
aftersound. This is partly due to differences in the way that vertically and horizontally polarized string
vibrations couple to the bridge and soundboard, but Gabriel Weinreich showed that coupling between strings
nominally in unison could lead to an extended decay time through mode locking if the strings were slightlymistuned [10]. In fact small deviations (of the order of 1.5 cents) from strict unison tuning of the three stringsin piano trichords had already been observed in the practice of expert tuners [11]. The perceived smoothnessof the decay can also be enhanced by subtle mistuning of trichords [12, 13].
Piano Actions and Piano Touch
The term ‘touch’ is used by pianists to describe the nature of the gesture used to depress a key on the
piano. Many players believe that by altering the gesture (for example, by stroking rather than hitting thekey) it is possible to change the timbre of a single piano tone without modifying its loudness. It is difficult tofind objective justification for this, since the only variable which affects the string vibration is the speed of thehammer head at impact.
One factor to bear in mind is that the player is very close to the instrument, and will be able to hear
subtle effects, such as the noise made by the key hitting its bed, which will not be significantly radiated intothe far field. More importantly , it must be recognised that the player depends totally on the action and
damping mechanisms of the piano to create and control the musical performance. The preceived playabilityof the instrument is intimately bound up with its mechanical performance, and cross-modal interactionbetween kinesthetic and auditory feedback can lead players to misinterpret the sources of quality cues.
A striking example of the effect of cross-modality was provided by experiments reported by Alexander
Galembo [14]. In the late 1970s the Leningrad piano factory conducted a number of tests in which twelveprofessional pianists played three different grand pianos (Steinway , Bechstein and Leningrad) in theLeningrad Conservatory concert hall. The pianists were asked to play freely , and to rate the quality of thepianos in terms of tone quality , dynamic range and playing comfort. The players agreed that the Steinwaywas much superior in tone to the Leningrad piano, but made no clear distinction between the playing comfortof the instruments. However , when the players were asked to listen to single tones, scales and chords playedbehind an acoustically transparent curtain they were unable to identify which instrument was being played.
In a further test, the three pianos were put into a triangular arrangement with a rotating piano stool at the
centre, and each blindfolded pianist was presented with the pianos in a random order . In this test theperformers could correctly identify the pianos even when deafened by white noise in headphones.
The conclusion drawn by Galembo was that the quality judgements, which the players had attributed to
timbral differences in the free playing test, were in fact based on differences in mechanical response. ThisM. Campbell
Proceedings of Meetings on Acoustics, Vol. 19, 032003 (2013) Page 4illustrates one of the most significant problems in scientific evaluation of musical instrument quality:
judgements by musicians are clearly of paramount importance, but are often hard to interpret.
STUDIES OF QUALITY OF BOWED STRING INSTRUMENTS
Descriptions of Violin Timbre
The question of musical instrument quality has special significance when the instrument under
consideration is the violin. The financial value of a Stradivari or Guarneri violin is typically two orders ofmagnitude greater than the cost of an instrument from a modern maker of the highest reputation. Manynon-musical factors are involved in this price differential, but a lively debate continues as to whether oldItalian violins are superior in musical quality to the best violins of the present day . Acousticians havecontributed to this debate for several decades, but objective criteria for assessing violin quality have provedelusive.
A number of studies have attempted to relate musical quality judgements to the strengths of sound
radiation in specific frequency bands. Dünnwald [15] reported measurements of frequency spectra of the
sound radiation from several hundred violins of varying age and quality . The bridge of each instrument was
excited sinusoidally by an electromagnetic driver . He identified four frequency bands, and proposed that eachwas associated with a specific tonal characteristic (See Table 1).
TABLE 1:Dünnwald Frequency Bands
Band number Frequency range Timbral characteristic
1 190–650 Hz fullness of sound
2 650–1300 Hz nasality
3 1300–4200 Hz brilliance, clarity4 4200–6400 Hz harshness, lack of clarity
Dünnwald’s approach has had a considerable influence on later work on violin timbre, but his association
of timbral character with frequency bands has not proved robust. For example, the violin maker andacoustician Martin Schleske has proposed a somewhat different scheme [16], illustrated in Table 2. Schleskenotes that these judgements are likely to vary significantly from one listener to another .
TABLE 2:Schleske Frequency Bands
Band number Frequency range Too strong Too weak
1 270 Hz dull, hollow thin, chirpy
2 450–550 Hz hollow , wolf tendency flat, weak3 700–1000 Hz not specified not specified4 1000–1800 Hz vulgar , nasal powerless, covered
5 2000–3500 Hz harsh, vulgar dull, covered
Input Admittance of Violins
In attempting to relate the perceived quality of the sound of bowed string instruments to physical
characteristics of the instruments, measurement of the bridge admittance (also known as mobility) hasproved to be one of the most useful techniques [17, 18, 19]. In one version of this technique, a calibrated
impulse is applied to the treble corner of the violin bridge and the resulting bridge velocity is recorded by a
laser vibrometer . The admittance is defined as the frequency domain ratio of bridge velocity to applied force.A typical violin bridge admittance curve is shown in Figure 1.
Specific features of the admittance curve can be related to structural vibration properties of the violin
[20], but while an admittance curve without significant resonance peaks would be an indication of a very poorM. Campbell
Proceedings of Meetings on Acoustics, Vol. 19, 032003 (2013) Page 5FIGURE 1:Bridge admittance |Y|for a good quality violin (courtesy of J. Woodhouse).
instrument, it has not proved possible to find features which discriminate unambiguously between violins
judged excellent and instruments which are of only average quality . After an exhaustive study of 17 violins ofwidely varying quality , including mobility and radiativity measurements, George Bissinger [21] concluded
that the only features which appeared to characterise the very best instruments were a relatively uniformspread of resonances and a strong response in the lowest frequency band.
A recent set of studies by Fritz et al. [22] revisited the relationship between spectral response and
quality judgements using the “virtual violin” technique [23]. In this approach the acoustical response of agood quality modern violin was modelled as a sum of 54 vibration modes. The amplitudes, frequencies and
quality factors of the modes were deduced from a bridge admittance measurement. The model was then usedto compute a finite impulse response digital filter simulating the transient response of the violin. A force
signal recorded at the bridge of a violin when played normally was fed through the filter to a pair ofheadphones. The amplitudes of the model modes in specified frequency bands were then increased ordiminished, and a panel of 14 musically expert listeners assessed the resulting timbral changes. Threedescriptors were chosen on the basis of a previous semantic study: these were bright ,harsh , and nasal .F o r
comparison with the results of Dünnwald, the descriptor clear was also included.
TABLE 3:Fritz et al. Frequency Bands
Band number Frequency range Increased Amplitude
1 190–380 Hz more nasal (Group 1n), less nasal (Group 2n)
2 380–760 Hz less bright, more nasal (Group 1n), less nasal (Group 2n)3 760–1520 Hz less bright, more nasal (Group 1n), less nasal (Group 2n)
4 1520–3040 Hz brighter , harsher , less nasal (Group 1n), more nasal (Group 2n)
5 3040–6080 Hz brighter , harsher , less nasal (Group 1n), more nasal (Group 2n)
The results of Fritz et al. are summarised in Table 3. Evaluations of clarity and brightness appeared to
be judgements on the same timbral dimension in these tests, so the dependence of clarity on frequency bandhas not been included in Table 3. Judgements of brightness and harshness were broadly consistent across thegroup of test subjects. Judgements of nasality were clearly made on a different basis by two approximately
eaqual subgroups: Group 1n found nasality to increase with increased amplitude in Bands 1-3 and decreasedamplitude in Bands 4-5, while Group 2n found the reverse. Neither subgroup associated nasality specifically
with Band 3, as would have been expected from the results of Dünnwald. Thes inconsistencies are furtherexamples of the problems involved in relating timbral properties to acoustical response.M. Campbell
Proceedings of Meetings on Acoustics, Vol. 19, 032003 (2013) Page 6Playability of Bowed String Instruments
In the last two decades, considerable research effort has been devoted to studying the criteria which
players use when judging violins and other bowed string instruments. Woodhouse [24, 25] noted that a violin
player evaluating a new instrument considers not only its sound quality but also the various aspects of the
physical interaction between player and instrument which contribute to judgements of playability . Oneimportant issue is the ease and smoothness with which the periodic string vibration known as Helmholtzmotion can be initiated and sustained. Schelleng’s work [26] on theoretical maximum and minimum limits onthe force exerted by the bow hair on the string has been extended by Guettler [27] and Woodhouse [28] toconsider the interdependent effects of bow force and bow acceleration on the length of starting transients in
stringed instruments. Identifying other aspects of playability which are salient in player judgements of
quality , and relating these to structural properties of the instrument, remain outstanding challenges in violinacoustics.
A recent study of player evaluations of high quality violins [29] has highlighted the problem of
interpreting correctly the results of such tests. 21 experienced violinists were asked to compare six violins,three of which were Stradivari or Guarneri instruments and three of which were by leading contemporarymakers. The players compared the instruments in free playing in a dry room, with low illumination and
wearing goggles which prevented visual identification of the instruments. Players were asked to rate the
instruments using various criteria, including playability , response and tone colour . The new violins wererated more highly than the old Italian instruments for playability and response, but there was no significantdistinction for tone colour . It has frequently been stated that an experienced player can immediatelydistinguish an antique violin from a new one [30], but the players in this test were generally unable to tell
whether the instrument they were playing was old or new . The shielding from visual cues was obviouslycrucial in the experiment, not only because the players might identify the instruments visually but also
because of the possibility of cross-modal effects altering the players’ perceptions of timbre and playability .
STUDIES OF QUALITY OF BRASS INSTRUMENTS
Descriptions of Brass Instrument Timbre
There is one aspect of brass instrument timbre which has been recognised as strongly characteristic of
the family in both musical and acoustical studies [31, 32, 33, 34]. This is the increase in brightness of the
sound which occurs during a crescendo on a brass instrument. Brightness is associated with a high value of
the spectral centroid SC , defined for a sound with discrete spectral component frequencies fiand amplitudes
Aias
SC=/summationtext
iAifi/summationtext
iAi. (1)
The spectrum of a brass instrument played quietly is typically dominated by a few low amplitude harmonics.
As the dynamic level increases, higher frequency harmonics become increasingly important; for a trumpetblown fortissimo more than 40 components of significant amplitude can be observed [35]. For other types ofbrass instrument, such as the saxhorn, the brightness increases much more gradually with increasingloudness.
It is significant that this most striking characteristic of a brass instrument is not a fixed degree on a
perceptual timbre scale (brightness), but rather a relationship between two dimensions (brightness and
loudness) which colours the perception of transient features related to musical expression. Experimentscarried out in the 1970s using computer synthesised versions of recorded musical instrument sounds [36, 37]revealed that transient features were particularly important in the recognition of a musical instrument fromits timbre. Low frequency , low amplitude inharmonic components in the attack transient were associatedwith the brass family , as was tapering of the onsets of higher harmonics and associated spectral fluctuations.M. Campbell
Proceedings of Meetings on Acoustics, Vol. 19, 032003 (2013) Page 7Input impedance
The input impedance of a wind instrument is defined as
Z(f)=pi(f)
ui(f), (2)
where piis the acoustic pressure at the input and uiis the acoustic volume velocity into the instrument for a
sine wave input at frequency f. The input impedance curve shown in Figure 2 illustrates the linear acoustic
response of a tenor trombone measured at the mouthpiece entrance plane. Measurements of this type havebeen used for several decades in studies of brass instruments, and are currently used for quality control inthe brass instrument manufacturing industry [38, 39].
FIGURE 2:Input impedance curve for a Conn 8H orchestral tenor trombone
FIGURE 3:Input impedance curves for an alphorn in A /flatM. Campbell
Proceedings of Meetings on Acoustics, Vol. 19, 032003 (2013) Page 8Acoustic Resonances and Playing Frequencies
The input impedance curve for a brass instrument resembles the bridge admittance curve for a stringed
instrument in that each describes the linear acoustic response of the instrument to a sinusoidal input signal.
There is however a fundamental difference in the relationship between the resonances shown in the two type
of curve and the playing characteristics of the instruments concerned. In the case of a stringed instrumentthe coupling between the body and the sound source (the vibrating string) is relatively weak, and only in thepathological case of a wolf note does feedback from body vibrations significantly perturb the stick-slipinteraction between bow hair and string [28]. In contrast, there is a strong nonlinear coupling between theair column resonances in a brass instrument and the vibrations of the player’s lips. Since the lip vibration
generates the sound through modulation of the air flow into the instrument, the acoustic resonances of the
brass instrument air column play a major role in determining both the frequency and the timbre of a playednote.
The nature of the input impedance curve is determined by the internal bore profile of the instrument.
Inspection of the trombone input impedance curve in Figure 2 shows that there are around 15 recognisablepeaks. From the second to the fifteenth peak, the frequencies satisfy the approximate harmonic relationship
f
n/similarequal57.7 n, n≥2. (3)
The lowest peak, at 39 Hz, is not a member of this series. This feature is characteristic of instruments with a
large proportion of cylindrical tubing. For instruments in which the tubing is mostly conical the lowest peak
also satisfies the approximate harmonic relationship; Figure 3 illustrates the example of the alphorn.
To sound a note, a brass player normally adjusts the muscles controlling the mechanical resonance
frequencies of the lips in such a way that the flow of air between them induces a bifurcation to an oscillatingregime with frequency close to one of the acoustic resonances. The lips and air column then lock into a stableperiodic vibration. It is important to note, however , that the playing frequency is not simply the frequency ofthe nearest acoustic resonance. The nonlinear nature of the coupling means that higher frequency
resonances can also exert an influence on the intonation. It is even possible to play a note for which there is
no acoustic resonance close to the playing frequency . An important example is the trombone pedal note. Onthe instrument whose input impedance is shown in Figure 2, a strong note can be played at a frequency of57.7 Hz, the oscillation regime being supported by the acoustic resonances at integer multiples of the playingfrequency . The timbre is characteristically bright, with little spectral energy at the fundamental frequency .
Measurements of acoustic resonance frequencies thus provide a valuable guide to intonation quality of
brass instruments, and have been used to generate targets for optimization programs [40, 41, 42]. Questions
remain, however , about the detailed relationship between measured acoustic resonance frequencies and
musical judgements of intonation accuracy [43, 44].
Wall material
The walls of wind instruments vibrate when the instruments are played. Wall vibrations of a clarinet or
trombone played loudly can be felt by the fingers of the musician, and many players and instrument makersbelieve that these vibrations also contribute significantly to the sound quality of the instrument. Recent
theoretical and experimental work on a simplified clarinet model [45] has suggested that although wall
vibrations can couple to acoustic resonances to produce audible changes in input impedance and radiatedsound, these effects are unlikely to have a noticeable influence on the sound quality of normal woodwindinstruments. In an important set of experiments on brass instruments, Kausel et al. [46] found measurablechanges in sound spectrum caused by damping the walls while the instruments were artificially blown. It issuggested that this is not due to direct sound radiation from the walls, but rather to to coupling of acoustic
resonances with relatively broad axial resonances of the bell [47].
Further work is required to establish the extent to which the views of instrument makers and players
about the effects of changes in wall thickness, metal composition and finish on timbral quality can beunderstood scientifically . A salutory warning of the potential difficulty of such a task is provided by a studyundertaken over three decades ago by Richard Smith [48]. A set of six trombone bells were made on the samemandrel, but with varying wall thickness (from 0.3 mm to 0.5 mm). Tests with an artificial sound sourceM. Campbell
Proceedings of Meetings on Acoustics, Vol. 19, 032003 (2013) Page 9showed that spectral differences of the order of several decibels at a position corresponding to a player’s left
ear were related to differences in bell wall thickness. He then carried out a double blind playing testinvolving ten leading trombonists. The players were blindfolded, and precautions taken to equalise theweight and balance of the instruments. Under these conditions the players were unable to distinguishbetween instruments with different bell thicknesses. When a pure copper bell was included in the test set, itwas not recognised as significantly different in timbral quality , although Smith reports that “whensubsequently played in non-blind tests it gained magical properties!” This is yet another example of theimportance of cross-modal influences on judgements of musical instrument quality .
Nonlinear sound propagation
The amplitude of the pressure waves generated in a brass instrument played loudly can exceed 10 kPa.
At this level, the timbre of the instrument is significantly affected by nonlinear sound propagation. In
instruments with long sections of cylindrical tubing (trumpets and trombones) wave steepening can even leadto shock wave formation, with concomitant transfer of energy to very high frequency air column modes [49].Nonlinear propagation is then the dominant factor in the sound timbre, which is described as ‘brassy’ (French‘cuivré’). Since the distortion arising from nonlinear propagation is cumulative, its effects are less importantin instruments with shorter tube length. Instruments with conical bores are also less affected by nonlinear
propagation, since the sound pressure level diminishes as the cross-sectional area increases.
While the most striking consequence of nonlinear propagation is the brilliant blare of a fortissimo
trumpet, it is also an important aspect of brass instrument quality even at moderate sound levels. Myers etal. [50] have proposed a taxonomy of brass instruments based on the rate at which nonlinear distortionincreases during a crescendo. Each instrument is characterised by a brassiness potential parameter B , which
takes into account the variation of bore diameter Dwith axial distance xfrom the entrance plane:
B
=1
Lecl/integraldisplayL
0D0
D(x)dx, (4)
where D0is the minimum bore diameter and Leclis the equivalent cone length of the instrument (equal to
c/2fwhere cis the speed of sound and fis the nominal fundamental frequency of the instrument). For all
conventional brass instrument bores, Bis a number between 0 and 1. Figure 4 illustrates the spread of
values for a number of instruments playing in approximately the same pitch range as the trombone.
The development of brightness in a crescendo also depends directly on the absolute radial scale of the
bore, for two reasons. To achieve a given radiated sound level, the player of a narrow bore instrument must
generate a larger input pressure amplitude than a player of a wider bore instrument with similar relative
bore profile; this leads to greater nonlinear distortion in the narrower instrument. However viscothermallosses are also greater for narrower diameter tubes, and since these losses increase with frequency theypreferentially damp the higher harmonics. For the normal range of bores used in brass instruments theformer effect dominates, so that the French horns shown in Figure 4 have a more rapid development of
brightness that their low Bvalues would suggest because of their small input diameter .
Many questions remain to clarify concerning the relationships between these aspects of brass instrument
timbre and players’ judgements of quality . In the twentieth century most orchestral trombone players
adopted large bore diameter trombones in the quest for higher sound output without excessive brassiness;some players, however , consider that these instruments lack character at low dynamic levels compared withthe narrow bored instruments commonly used in earlier periods. The musical context must be carefullyconsidered in discussions of instrumental quality .
Playability of Brass Instruments
The relationship between player and instrument is particularly intimate in the case of brass instruments,
since the player’s lips are the essential components in the sound source [51]. When a player starts a noteseveral periods of the lip vibration may occur before the sound wave reflected from the bell returns to themouthpiece and a standing wave is established [52]. An important aspect of playability is the ease with whichM. Campbell
Proceedings of Meetings on Acoustics, Vol. 19, 032003 (2013) Page 10FIGURE 4:Scatter plot of the brassiness potential parameter Bcomputed from physical measurements, plotted against the minimum
diameter D0, for 26 instruments of conventional design in Edinburgh University Collection of Historic Musical Instruments [50].
a note may be started. The strength of a regime of oscillation is enhanced when the acoustic resonances are
harmonically aligned; players then talk of the notes being ‘well centred’. It seems plausible that harmonicalignment of the modes is linked to ease of starting a note, but this hypothesis remains to be fully tested.
Strong centring of notes is usually considered a desirable quality in a brass instrument, but the
relationship between centring and flexibility requires further investigation. The latter quality is related tothe ease with which the player can ‘bend’ the pitch of a note, or make a smooth transition from one regime of
oscillation to another [53]. It has been suggested that harmonically aligned resonances with high Q values
may result in an instrument being judged ‘stiff ’ rather than flexible. This could be a particular problem ininstruments such as the baroque trumpet, on which the eleventh acoustic mode has to support two pitches asemitone apart [54].
On an instrument in which the acoustic resonances are significantly inharmonic, the nonlinear nature of
the coupling between lip and air column can lead to instability and pitch drift as the dynamic level isincreased. An example of such an instrument is the serpent. Although this instrument is usually made from
wood and is equipped with finger holes, it is considered acoustically to be part of the brass family since it is
excited by lip vibration in a cup mouthpiece. Figure 5 shows the input impedance for a serpent with thelowest three finger holes open. The strongly inharmonic nature of the resonances is a consequence of thesmall diameter and irregular spacing of the side holes, and is typical of measurements carried out on manyoriginal and reproduction instruments. A virtuoso performer can play the serpent with good intonation andan attractive mellow timbre; exactly how this unlikely feat is achieved is a subject currently under study .
One playing technique which could partly explain the ability of an expert player to sound notes without
apparent support from the acoustic resonances of the instrument involves modification of the vocal tract. The
player’s mouth cavity and throat have resonances which are upstream of the players lips, but which couple tothe lips in the same way as the downstream resonances of the instrument. It has been shown that expertsaxophone players tune vocal tract resonances in high register performance [55], and recent measurementshave confirmed that vocal tract resonances appear to be significant in trombone playing over most of theregister [56].
The final word on brass instrument quality evaluation must be on mouthpiece design, which mostM. Campbell
Proceedings of Meetings on Acoustics, Vol. 19, 032003 (2013) Page 11FIGURE 5:Input impedance curves for a serpent with three finger holes open
players believe to have a strong effect on the overall quality of the instrument. A brass instrument
mouthpiece serves a number of functions; it acts as a support to the lips, and its rim defines the limit ofpossible lip vibration. The mouthpiece also boosts the heights of impedance peaks in a range slightly belowits own acoustic resonance frequency: this effect is strongly evident in Figure 3. The aeroacoustics of the
mouthpiece are less well understood. The jet of air which emerges from the lips of the player is usually
considered to dissipate its energy without pressure recovery some distance before it reaches the throat whichconnects the mouthpiece to the main tubing of the instrument, but this hypothesis requires verification.Many players and manufacturers believe that the wall thickness and overall mass of the mouthpiece have amajor effect on the playability and timbre of the instrument, but possible causes for such a dependence arehard to identify . The relationship between mouthpiece design parameters and musical quality judgements is
a topic ripe for serious scientific scrutiny .
CONCLUSION
The basic physics of the musical instruments discussed here is well understood, but many musically
important aspects require finer measurements and greater understanding of the language and requirementsof musicians. Time domain modelling is poised to play an important role in exploring the perceptualsignificance of specified small changes in the design of an instrument, but models of the sound generating
mechanisms require further refinement before the sound output from a complete instrument model is
suficiently realistic to be musically useful. Optimisation methods for musical instruments are improving, butneed more musically relevant targets. The ultimate aim of this work is to explain scientifically why aninstrument is judged to be musically excellent, and to offer guidance to makers wishing to achieve andmaintain excellence in musical instrument manufacture.
ACKNOWLEDGMENTS
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1.5098453.pdf | Appl. Phys. Lett. 114, 232407 (2019); https://doi.org/10.1063/1.5098453 114, 232407
© 2019 Author(s).Ultrafast magnetization switching in
nanoscale magnetic dots
Cite as: Appl. Phys. Lett. 114, 232407 (2019); https://doi.org/10.1063/1.5098453
Submitted: 02 April 2019 . Accepted: 01 June 2019 . Published Online: 14 June 2019
Amal El-Ghazaly
, Brandon Tran
, Alejandro Ceballos , Charles-Henri Lambert , Akshay Pattabi , Sayeef
Salahuddin
, Frances Hellman
, and Jeffrey Bokor
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magnetic dots
Cite as: Appl. Phys. Lett. 114, 232407 (2019); doi: 10.1063/1.5098453
Submitted: 2 April 2019 .Accepted: 1 June 2019 .
Published Online: 14 June 2019
Amal El-Ghazaly,1,a)
Brandon Tran,2
Alejandro Ceballos,3Charles-Henri Lambert,1Akshay Pattabi,1
Sayeef Salahuddin,1
Frances Hellman,2,3
and Jeffrey Bokor1
AFFILIATIONS
1Department of Electrical Engineering and Computer Science, University of California, Berkeley, California 94720, USA
2Department of Physics, University of California, Berkeley, California 94720, USA
3Department of Materials Science and Engineering, University of California, Berkeley, California 94720, USA
a)Electronic mail: aelghazaly@berkeley.edu
ABSTRACT
Ultrafast magnetization switching at picosecond and sub-picosecond time scales has tremendous technological potential but still poses numerous
questions regarding the underlying quantum mechanical phenomena, including the roles of and interactions between the electrons, spins, and
phonons (lattice). At the nanometer-scale dimensions relevant for modern applications, these phenomena become increasingly more pronounced.
Until now, helicity-independent all-optical switching (HI-AOS) has been largely limited to amorphous Gd-Fe-Co alloys, for which scaling waschallenging due to their relatively low anisotropies. In this work, we demonstrate HI-AOS in amorphous GdCo and scale it to nanometer dimen-sions while still maintaining uniform out-of-plane magnetization. Single shot HI-AOS is demonstrated in these patterned samples down to a mini-mum optically detectable magnetic dot size of 200 nm. The ultrafast switching behavior was also confirmed using time-resolved magneto-optic
Kerr effect measurements and found to settle to its opposite magnetization state at faster rates for smaller dot diameters, passing a threshold of
75% magnetization reversal within approximately 2 ps for a 200 nm dot compared to approximately 40 ps for a 15 lm pattern. The size depen-
dence of the ultrafast switching is explained in terms of the electron-phonon and spin-lattice interactions.
Published under license by AIP Publishing. https://doi.org/10.1063/1.5098453
Magnetic nanodots serve as the basis for most modern magnetic
device applications. As magnetic phenomena are discovered inresearch, scaling down of these phenomena to nanometer sizes mustfollow such that they can be exploited to meet the competitivedemands of today’s electronics. For many of these magnetic phenom-ena, nanoscale behavior is strongly influenced by dimensions, makingthe fundamental understanding of the interplay between sizes and theunderlying physical effects increasingly important for such small devi-ces. Applications for magnetic materials in consumer electronics growdaily, with the largest interest in the areas of sensors,
1logic,2,3and
memory.4,5For example, magnetic random access memory (MRAM)
provides unique advantages to memory systems by combining highareal density with device nonvolativity. In addition, recent discoveriesin ultrafast magnetism potentially offer orders of magnitude fasterswitching speeds than current MRAM devices by seemingly avoidingthe conventional precessional switching behavior.
6–8Given the highly
advantageous nature of ultrafast picosecond switching speeds, we
address the question of scaling and not only demonstrate this phe-nomenon at nanoscale dimensions but also analyze the variousquantum mechanical effects that influence the switching speed at
smaller dimensions.
The field of ultrafast magnetism began with the discovery of sub-
picosecond demagnetization of ferromagnetic Ni in 1996 (Ref. 9)a n d
has since diversified to include complete ultrafast reversal of magnetiza-
tion via either helicity-dependent
10–12or helicity-independent all-optical
switching (HI-AOS).6,8Thus far, amorphous Gd-Fe-Co (a-Gd-Fe-Co)
has been the most widely studied ferrimagnetic alloy for ultrafast switch-ing due to its ability to reliably reverse large areas of its magnetization
with single shots of a laser pulse irrespective of laser polarization. While
t h em e c h a n i s mf o rs i n g l es h o tH I - A O Si na - G d - F e - C oi sn o ty e tcompletely understood, it is attributed to the femtosecond heating of
conduction electrons, resulting in the rapid demagnetization of the
material. Through a combination of electron-phonon coupling, spin-lattice coupling, and the large negative exchange interaction between
the two sublattices, reversal of the magnetization takes place.
13–15The
ability to switch its magnetization in this manner at picosecond timescales, much faster than conventional precessional switching, gives
a-Gd-Fe-Co a competitive edge for future electronics.
Appl. Phys. Lett. 114, 232407 (2019); doi: 10.1063/1.5098453 114, 232407-1
Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplTaking advantage of ultrafast magnetization switching for device
applications requires that the phenomenon be understood at the nano-
scale, where the competition between various quantum mechanical
interactions could lead to observable differences in behavior compared
to that at the microscale. Confinement of magnetization switching to
nanoscale regions of large continuous a-Tb-Fe-Co alloy films has been
shown possible using tightly focused laser beams16and plasmonic
nanoantennas;17however, precise control of the magnetization switch-
ing location and behavior proved to be difficult as a result of inhomo-geneity in the local anisotropy. On the other hand, nanometer-scale
patterning of individual a-Gd-Fe-Co dots could lead to a more precise
definition of the switching location and was attempted by LeGuyader
et al. ,
18but large regions at the edges of the patterns were found to
have lost perpendicular magnetic anisotropy (PMA), thereby compli-
cating the overall net magnetization of the structures. In this work, we
present scaling of ultrafast magnetization switching in nanoscale pat-
terned magnetic dots and do so by the way of uncovering a family of
materials, that of a-Gd-Co alloys, also capable of single shot HI-AOS.
Since cobalt thin films can be grown with large PMA when inter-
faced with Pt,19we replaced Fe in the original a-Gd-Fe-Co entirely with
Co and sandwiched the film between Pt layers in order to contribute to
interfacial PMA from the Pt/Co interfaces to develop a resilient PMA
capable of withstanding sample patterning. Surprisingly, the Ta (3 nm)/Pt
(3 nm)/a-Gd 30Co70(10 nm)/Pt (3 nm) material stack not only maintained
PMA up to approximately 10 nm in thickness but also reversed its mag-
netization ultrafast on picosecond time scales in response to a femtosec-
ond laser pulse excitation. Previous investigations of a-Gd 17Co83films,20
in contrast to a-Gd-Fe-Co, had observed only ultrafast demagnetizationb e h a v i o r .W ea t t r i b u t et h ed i f f e r e n c ei nu l t r a f a s tb e h a v i o ra n dt h ed i s c o v -ery of switching in our alloy to the higher concentration of Gd, leading to
the composition having a compensation temperature near but below
room temperature.
21Although a-Gd-Fe-Co is known to switch for com-
pensation temperatures on either side of room temperature, it was
believed that a compensation temperature above room temperature was a
prerequisite for AOS in new materials.15,22,23However, the compensation
temperature of our a-Gd-Co film was found to be T m/C24230 K, therefore
behaving similar to a-Gd-Fe-Co, contradicting previous assumptions, and
confirming the theory put forth by Moreno et al. that crossing the com-
pensation temperature is not necessary for HI-AOS.24
Arrays of magnetic dots ranging in size from 15 lmd o w nt o
50 nm, as shown in Fig. 1 , were fabricated to characterize the effect of
scaling on ultrafast switching dynamics. Each of the samples was pre-
pared using a lift-off technique where the dot arrays were prepatterned
using electron-beam lithography, and then, the material stack was
sputter deposited and finally lifted off to yield the magnetic patterns.
Two samples were used: one of the previously mentioned a-Gd-Co
s t a c ka n daT a( 5 n m ) / a - G d 27Fe66Co7(10 nm)/Pt (5 nm) stack as the
reference. Measurements of the sample magnetization were conducted
using laser Magneto-Optic Kerr Effect (MOKE) microscopy. Although
all films exhibited both PMA and single shot HI-AOS in their continu-
ous unpatterned film form, only a-Gd-Co maintained its out-of-plane(OOP) magnetization in the smallest nanoscale dots measured
(200 nm); OOP magnetization could be measured down to just
900 nm diameters for a-Gd-Fe-Co. Here, we therefore utilize the
a-Gd-Co nanodots to investigate nanoscale ultrafast behavior.
For each dot diameter, d, an area of size 25 lm/C225lmw a s
filled with equally spaced dots of spacing 2d. The array pitch waschosen for maximum areal density but minimum magnetostatic cou-
pling between the dots, as verified by COMSOL simulations (see thesupplementary material ). By eliminating the coupling between dots, the
array could be measured altogether and treated as a summation of inde-
pendent dots of the same size. A Ti-Sapphire laser with a center wave-length of 810 nm and a pulse width of about 70 fs was split to provideboth the high power pump excitation for ultrafast switching and the
lower power probe for laser MOKE detection. The polarization of the
probe was modulated at 50 kHz by a photoelastic modulator to ensure ahigher signal to noise ratio. The pump beam full width at half maximum(FWHM) diameter was /C2495lm which, for the chosen incident fluence
of 6.89 mJ/cm
2, ensured a switching area larger than the 25 lmm a g -
netic array area; the probe beam FWHM diameter was 15 lm, making
it sufficiently small to measure a signal from only a single array of dots.
Hysteresis loops with OOP MOKE magnetization sensitivity,
shown in Fig. 2 , were measured for the dot arrays of different diame-
ters for a-Gd-Fe-Co and a-Gd-Co and used to determine if PMA was
sustained. a-Gd-Fe-Co dots are seen to lose their OOP magnetization
FIG. 1. Nanoscale patterning of (a) 25 lm/C225lm areas filled with uniformly pat-
terned dots of diameter d and spacing 2d. The red circle represents the FWHM
size of the probe beam centered on the dot array. (b) SEM image showing a regionof the sample with 15 lm, 5lm, 1lm, and 900 nm dot arrays and (c) a close up of
a 200 nm dot array.
FIG. 2. OOP hysteresis loops of the a-Gd-Fe-Co (left) and a-Gd-Co (right) samples
taken by laser MOKE.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 114, 232407 (2019); doi: 10.1063/1.5098453 114, 232407-2
Published under license by AIP Publishingsignal beginning from 1 lm and disappearing below 900 nm diame-
ters, but for a-Gd-Co, PMA was sustained down to the smallest nano-
scale dimensions. In these experiments, magnetic behavior could be
confirmed down to 200 nm.
In subsequent experiments, only the a-Gd-Co patterned dots
were utilized to test ultrafast HI-AOS. Figure 3(a) confirms single shot
switching in large 4 lm diameter dots, visually represented as a con-
trast change in the image MOKE with each shot of the pump laserbeam. Smaller dots, whose magnetization could not be resolved in acamera image, were measured with laser MOKE to verify their abilityto characteristically toggle their magnetization up and down in
response to each laser pulse. The laser MOKE measurement of 500 nmdots in Fig. 3(b) clearly indicates single shot switching for ten consecu-
tive laser shots. In comparison, the 200 nm measurement in Fig. 3(c) ,
while still demonstrating single shot switching, shows evidence of the
higher noise levels observed nearer to the laser detection limit. (See the
supplementary material for discussion of reduced signal contrast for
smaller dot sizes.) Variation of its magnetization signal during the sin-
gle shot measurement and of the nanodot coercive field compared to
the 5 lm dot in the hysteresis loop ( Fig. 2 right) can be attributed to
drift in the laser power, position, and spatial interference patterns withthe nanodot array, irregular magnetization effects at the pattern edges,
and slight dot-to-dot disparities in magnetic behavior. Nevertheless,
we observed single shot HI-AOS in nanoscale dots as small as 200 nm
in diameter.
Since the technological impact of HI-AOS is derived from its
ultrafast switching speed, it is important to address the effect of scaling
also on the switching dynamics. For this, we carried out time-resolvedMOKE (TR-MOKE) measurements for each dot diameter, once again,
down to the laser’s 200 nm limit of detection. The time-resolved ultra-
fast switching behavior of both the continuous unpatterned film and
the 15 lm square pattern is shown in Fig. 4(a) as well as the micro and
nanoscale patterns in Fig. 4(b) . As seen from the two graphs, the ultra-
fast behavior is uninhibited by scaling. Even the smallest, 200 nm
diameter dots switch in picosecond time scales. Although the larger
features in Fig. 4(a) could be measured at the pump incident fluence of
5.17 mJ/cm
2, which was found to be sufficient for switching both the
continuous and the 15 lm pattern, the patterns in Fig. 4(b) (including
the 15 lm square) were all measured at the previously quoted fluence
of 6.89 mJ/cm2, which was found to be necessary for guaranteeing
switching of the smallest dots. The higher incident fluence required bysmaller dots is likely due to the nonuniform light absorption profile in
the case of obliquely incident light onto a patterned feature
25and inter-
ference patterns generated by the light hitting the array of nanodots
(see the supplementary material ). Such effects make the calculation of
absorbed fluence in the nanodots nontrivial. Greater noise in the
T R - M O K Ed a t af o rd o t ss m a l l e rt h a n1 5 lm is most likely due to the
individual dot size being smaller than the probe beam diameter (approx.
15lm), therefore requiring that the measurement be a summation of
many dots in an array ( Fig. 1 ) instead of a single continuous film.
Additional details regarding the nanodot TR-MOKE measurement and
analysis methods can be found in the supplementary material .
A further trend can be seen from the TR-MOKE curve fit data,
indicating that rather than the overall switching speed being impeded
by the reduced dimensions, it is in fact enhanced. The magnetization
can be seen to settle to the reverse direction at a faster rate moving
from the continuous film to 15 lm[Fig. 4(a) ]a n df r o m1 5 lmt o
much smaller dimensions [ Fig. 4(b) ] such that 200 nm dots reach 75%
of their saturation magnetization within approximately just 2 ps as
compared to 15 lm patterns which only require 40 ps. The fact that
smaller dots achieve a greater amount of magnetization reversal in the
first few picoseconds suggests that the source of faster switching for
smaller dot patterns is related to their underlying subpicoseconddynamics and energy transfer rates.
This clear experimental trend, summarized in Fig. 5 , can be
understood by considering the various contributions to the better heat
diffusion and energy dissipation of smaller features. A typical ultrafast
switching transient can be divided into several separate behaviors in
the time domain. First, the femtosecond heating stimulus causes a
FIG. 3. Single shot switching results. (a) (left) SEM image of 4 lm dots and (mid-
dle, right) MOKE images of single shot HI-AOS after subsequent pulses, wheremagnetization switching is represented by the contrast change between up (light)and down (dark) magnetization states. (b) 500 nm and (c) 200 nm single shot
switching experiments taken by laser MOKE, indicating toggle switching between
up and down magnetization states with each subsequent pulse. Greater amounts ofnoise in the 200 nm measurement indicate the measurement approaching the limitof detection.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 114, 232407 (2019); doi: 10.1063/1.5098453 114, 232407-3
Published under license by AIP Publishingrapid demagnetization on the time scale of a few hundred femtosec-
onds. Then, the magnetization reversal on each of the transition metal
and rare earth sublattices begins to occur at extremely fast rates as theelectron-phonon and spin-lattice coupling effects cause the electronsand spins to equilibrate with the rest of the system. This process takesplace on the order of just a few hundred femtoseconds and 1–2 pico-
seconds for the transition metal and rare-earth sublattices, respec-
tively.
13–15,20The remaining slower transient is rate-limited first by the
spin-lattice coupling which describes the continuous transfer of heatfrom the spins to the lattice and then by the subsequent ambient diffu-sion of that heat from the lattice, allowing the magnetization to settle
to its thermally equilibrated value after hundreds of picoseconds.COMSOL heat transfer simulations as a function of pattern size
(see the supplementary material , Fig. 2.4) suggest a heat diffusion from
the magnetic stack to its neighboring environment (the Si substrate
and air) at long time scales, which matches the trend seen to begin inthe experimental size-dependent TR-MOKE data at shorter time scales(Fig. 4 ). The temperature of the patterned feature settles more quickly
for smaller sizes, as the surface area to volume ratio increases. This is
due to the fact that for a given unit of volume, a larger surface area
yields a faster thermal diffusion. While the COMSOL simulationresults may explain the long-time behavior, the short time scale, ultra-fast behavior cannot be understood purely by classical heat diffusion.
Although not entirely understood, dissipation of energy at the
ultrafast time scale is believed to be mostly determined by theelectron-phonon coupling and spin-lattice coupling. In continuoussamples with effectively infinite lateral dimensions, electrons and pho-
nons equilibrate within /C241p s .
26However, as the lateral dimensions
decrease, the electron-phonon coupling in metals is known toincrease.
27–29This directly results from the increase in surface scatter-
ing and the resulting reduction in the mean-free path as the lateraldimensions are reduced (see the supplementary material ). With higher
electron-phonon coupling, the efficiency of the energy transfer from
electrons to phonons is increased and the system equilibrates faster.Moreover, in the rapid initial demagnetization step, the spin tempera-ture tends to track the electron temperature.
6Spin-lattice coupling
increases significantly at higher temperatures, particularly for Gd
where the lattice is indirectly coupled to the 4f spins through the 5dconduction electrons;
6,26,30at high temperatures, the 4f-5d coupling
of spin-flip processes is enhanced. Thus, we suggest that in the initial
1–2 ps of the ultrafast process, both the electron-phonon and
FIG. 4. Time resolved magnetization switching behavior of (a) an unpatterned, continu-
ous film and 15 lmx1 5 lm square pattern of a-Gd-Co, both taken at an incident flu-
ence of 5.17 mJ/cm2. (b) TR-MOKE measurements of micro and nanoscale dot
patterns of a-Gd-Co, all taken at an incident fluence of 6.89 mJ/cm2. The normalized
experimental data are represented by the shaded regions with the width of one stan-dard deviation. The curve fits [based on Eq. (1) of the supplementary material ] summa-
rize the behavior for each size. A clear trend can be seen where smaller patterns settle
to the reversed magnetization state at faster rates than larger dots.FIG. 5. The time required for the magnetization to cross a 75% reversal threshold
based on fitting of the TR-MOKE data for different dot diameters (see the supple-
mentary material ). The 75% reversal threshold can be used to classify switching in
digital electronics. The red curve is a guide to the eye for the data as a function ofsize. The inset shows an expanded view of the smaller nanodot data. Asymmetricerror bars arise from the uncertainty in the 75% reversal time, particularly given the
long tail in the remagnetization TR-MOKE data as compared to the much more
rapid demagnetization.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 114, 232407 (2019); doi: 10.1063/1.5098453 114, 232407-4
Published under license by AIP Publishingspin-lattice coupling increase to allow faster energy dissipation,
thereby causing the spins and magnetization to settle faster to the
reversed magnetization direction. Alternatively, we may also consider a
more macroscopic picture where despite the effective OOP uniaxialanisotropy slightly decreasing for smaller lateral dimensions (see thesupplementary material ), it is accompanied by a dramatic increase in
the Gilbert damping coefficient, as found by Song et al. ,
31which also
suggests that the magnetization will settle faster for smaller dot sizes.These results present great promise for future applications since smallernanoscale magnetic memory devices would reach their equilibriummagnetization faster and allow shorter write times, as shown in Fig. 5 .
In conclusion, we have demonstrated single shot ultrafast mag-
netization switching in nanoscale dots by introducing the materiala-Gd-Co to the helicity-independent all-optical switching family.Using nanoscale a-Gd-Co patterns with OOP magnetization, singleshot HI-AOS was demonstrated down to the minimum optically
detectable size of 200 nm diameter. Time-resolved MOKE experi-
ments confirmed the ultrafast characteristic of the nanodot magneti-zation switching. The results further demonstrated that smallernanoscale patterns settle to their final magnetization states fasterthan larger patterns due to their faster heat dissipation. 200 nm dots
were found to reverse to 75% of their saturation magnetization
within just 2 ps. We attribute the faster switching speed in smallerdots to the increase in electron-phonon coupling with greater surfacescattering and the increase in spin-lattice coupling with higher spintemperatures in smaller patterns.
See the supplementary material for detailed methodology
describing the measurement and analysis procedure for laser MOKEnanodot hysteresis loops, single shot switching, and TR-MOKE.Analytical density of states calculation and simulation parameters andgeometries for both the COMSOL and JOOMMF simulations are also
included.
The authors thank Daisy O’Mahoney, P. Nigel Brown, Claudia
Robinson, and Travis Butler for their assistance with experiments. The
authors also thank Bert Koopmans for his helpful discussions. Thiswork was primarily funded by the U.S. Department of Energy, Officeof Science, Office of Basic Energy Sciences, Materials Sciences andEngineering Division under Contract No. DE-AC02-05-CH11231 and
the National Science Foundation Award No. 0939514 within the
Nonequilibrium Magnetic Materials Program (MSMAG). A.E. isgrateful for support from the University of California President’sPostdoctoral Fellowship Program. Support for nanoscale fabricationand laser experiments was provided by the Center for Energy Efficient
Electronics Science. Fabrication was performed at the Marvell
Nanofabrication Laboratory at the University of California Berkeleyand the Stanford Nanofabrication Facility and Stanford Nano SharedFacilities at Stanford University.
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Appl. Phys. Lett. 114, 232407 (2019); doi: 10.1063/1.5098453 114, 232407-5
Published under license by AIP Publishing |
1.19254.pdf | Coulomb crystals of oil droplets
Scott Robertson and Richard Younger
Citation: American Journal of Physics 67, 310 (1999); doi: 10.1119/1.19254
View online: http://dx.doi.org/10.1119/1.19254
View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/67/4?ver=pdfcov
Published by the American Association of Physics Teachers
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128.138.73.68 On: Tue, 23 Dec 2014 01:02:16Coulomb crystals of oil droplets
Scott Robertson and Richard Younger
Department of Physics, University of Colorado, Boulder, Colorado 80309-0390
~Received 26 May 1998; accepted 17 September 1998 !
Coulomb crystals of charged liquid droplets have been created in a Paul trap at atmospheric
pressure. The technique improves upon similar experiments with solid dust particles by having acontrolled and reproducible charge-to-mass ratio. The charge-to-mass ratio of the droplets, thespatial configurations of small crystals, and the frequency of the center-of-mass mode of oscillationhave been determined. ©
1999 American Association of Physics Teachers.
I. INTRODUCTION
In 1959, Wuerker et al.1described the trapping of charged
micron-sized aluminum particles in the Paul trap.2The par-
ticles were initially trapped in clouds with large random mo-tion and gas dynamic drag reduced this motion until thecloud froze into a lattice. These experiments demonstratedthat macroscopic particles could form a crystalline array as aresult of the interparticle Coulomb repulsion and that thearray could be ‘‘melted’’ by changing the amplitude of thealternating trap voltage. Simpler versions of this experimentfor classroom use have been described which do not requirea vacuum and which use alternating voltage at the linefrequency.
3,4Quantitative studies in these experiments have
been hampered by a lack of control over the charge-to-massratio of the particles. We describe a new technique for thecreation of Coulomb crystals in which the particles are oildroplets with nearly identical charge and mass. A video sys-tem allows experiments to be recorded and shown to largeaudiences. The relative simplicity of the apparatus makes itvaluable in demonstrating the physical concepts of charge-to-mass ratio, melting and crystallization, ponderomotiveforce, and normal modes of oscillation. It may also serve asa simple alternative to the ion trap for research where thetopic rests upon classical mechanics rather than quantum me-chanics.
A motivation for the development of charged particle traps
is ion spectroscopy and its application to improved atomicclocks.
5Coulomb crystals are of interest as a means to re-
duce uncertainties in frequency arising from the Doppler ef-fect of thermal motion. There may also be applications inparticle accelerators where the intersection of aligned crys-talline beams would give a high collision rate.
6Crystalliza-
tion occurs when the ratio of the interparticle potential en-ergy to the thermal energy is sufficiently large. The coupling
parameter is G5Q
2/4pe0dTwhereQis the particle charge,
dis the interparticle spacing, and Tis the temperature in
energy units. An infinitely large system will crystallize into abody-centered-cubic lattice at G5178.
7The structure of
small crystals is modified by surface effects and particlestend to lie on concentric shells which are locally hexagonalin two dimensions. The number of layers in the smallestdimension needed to obtain ‘‘infinite’’ behavior at the centeris estimated to be approximately 60.
8
The lowest energy configuration of Nparticles in a spheri-
cal potential has been studied computationally for Nas large
as 5000.9,10Particularly simple configurations are N54i n
which the particles are at the vertices of a regular tetrahedron
andN58 in which the particles lie at the vertices of a cube
having one face rotated 45°. The configurations for N.12are concentric spherical shells. The second shell begins at
N513 with one particle in the center of an icosahedral shell
and the third shell begins at N561 with shell occupation
numbers of 1, 12, and 48. ‘‘Magic numbers’’ corresponding
to closed shells appear frequently in atomic and nuclearphysics and molecular clusters with closed shells are formedin greater abundance in condensing flows.
11
Recent experimental studies in the Paul trap include the
suspension of individual liquid droplets for studies offluorescence
12and the suspension of two macroscopic
particles13and of small numbers of ions14,15for studies of
regular and chaotic motion. In the Penning trap, which has
similar equations of motion, 2.7 3105ions have been stored,
electrostatic modes of oscillation have been predicted and
observed,16and the Bragg scattering of laser light is consis-
tent with a body-centered-cubic lattice surrounded by con-centric shells.
17,18Crystals of macroscopic particles have
also been observed in the radio-frequency plasma dischargesdeveloped for semiconductor processing.
19In these experi-
ments the Coulomb force is modified by the Debye shieldingof the plasma, therefore the interparticle force has the formof the Yukawa potential and the shielding must be includedin calculating the coupling parameter.
20Experiments have
been performed with monodisperse spherical particles21,22
and with particles of silicon grown in situfrom silane gas.23
The particles are suspended in layers in the electrostatic
sheath above a planar electrode. The number of particles in a
plane may be very large ( .103) but the number of planes is
limited to about 20 by the thickness of the sheath.
Melting24,25and charge density waves26–28have been ob-
served. Particles heated in flames and charged by thermionicemission may also form crystals.
29
II. MATHEMATICAL BACKGROUND
Motion of particles in the Paul trap has been analyzed
extensively in the literature.30–33An alternating potential V
at frequency Vand an optional constant potential Uare ap-
plied between the ring electrode and end cap electrodes ~Fig.
1!. The alternating potential pulls particles toward the elec-
trodes and pushes them away resulting in an oscillation ormicromotion superimposed upon the thermal motion. Thepull half of the cycle brings the particle nearer to an elec-trode where the inhomogeneous electric field is stronger,thus the subsequent push half of the cycle occurs in a stron-ger field. The time averaged force is away from each of theelectrodes and toward the center of the trap. This is an ex-ample of the ponderomotive force,
34which may be described
310 310 Am. J. Phys. 67~4!, April 1999 © 1999 American Association of Physics Teachers
This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.138.73.68 On: Tue, 23 Dec 2014 01:02:16by an effective potential. For electrodes which are hyperbo-
loidal, the sum of the ponderomotive and static potentials isquadrupolar and can be written
31
C~r,z!5FQV2
MV2~r0212z02!21U
r0212z02Gr2
1F4QV2
MV2~r0212z02!222U
r0212z02Gz2
51
2Q@krr21kzz2#, ~1!
whereMis the particle mass, r0is the midplane radius of the
ring electrode and 2 z0is the separation of the end electrodes.
The force is F52QCandkrandkzare the ‘‘spring con-
stants’’ associated with this force. The force arising from the
static potential Uis inward in the radial direction and out-
ward in the zdirection or vice versa. The ponderomotive
force is greater for lower frequencies, however, the fre-quency must be sufficiently high for stability. The motionalong each axis is described by the Mathieu equation and thenature of the solutions depends upon dimensionless param-eters. For motion along the zaxis, for example, these param-
eters are given by
q
z58VQ
MV2~r0212z02!, ~2!
and
az5216UQ
MV2~r0212z02!. ~3!
The motion is stable over a limited range of qzandazand
beyond this range particles are lost.
In the case U50, potential surfaces are oblate spheroids
and small numbers of particles tend to form planar rather
than spherical arrays. Two particles, for example, will find
equilibrium positions in the z50 plane. In this case, the
motion is stable for qz,0.9. The potential well may be made
spherical by adjusting Uto the value
Ut5QV2
MV2~r0212z02!. ~4!
ForU.Ut, the equipotentials are prolate and two particles
will become aligned along the zaxis rather than in the z
50 plane.
A very large number of particles constitute a non-neutral
plasma. An equilibrium requires a number density, n,i n
which the self-potential of the charges cancels the effectivetrapping potential:
n512
e0V2
MV2~r0212z02!2, ~5!
which is independent of Q. A measurement of the mean in-
terparticle separation, n21/3, thus provides a means for de-
termining the particle mass. In the continuum limit, the par-
ticle cloud supports oscillations at the plasma frequency:vp52)QV
MV~r0212z02!. ~6!
III. EXPERIMENTAL APPARATUS
A. Paul trap
The Paul trap, Fig. 1, is of similar construction to that in
Ref. 4. The hyperboloidal electrodes in the standard Paul trapare replaced by a ring and two spheres.
35The ring is fabri-
cated from a 5-cm brass plate. A hole 3.80 cm in diameter is
cut in the center of the plate, thus r051.9cm. The plate is
blackened and soldered to a 3.1-mm-diam brass rod. The
upper and lower electrodes are 12.7-mm-diam metal spheressupported by brass rods pressed into drilled holes. The rodsare attached to insulating supporting posts such that the sepa-
ration 2z
052.6cm. The ring is connected t oa5k Va c high
voltage transformer operated from a variable autotransformer
for which V5377s21. The spheres may be ~1!grounded, ~2!
held at a dc potential 2U,o r~3!connected to a low fre-
quency ac source to excite oscillations of the droplet cloud.
A 2-M Vresistor capable of dissipating 10 w is placed in
series with each power supply to limit the current in the caseof an accidental short circuit or contact with an operator. Aclear plastic storage container or cake cover is placed overthe trap to prevent air currents from disturbing the arrays andto prevent accidental shocks. A soldering iron with a cuttingtip is used to melt a 1-cm-diam hole in the cover for injectingthe oil droplets.
B. Oil droplet generator
Charged droplet generators
36–40are often called electrohy-
drodynamic sprays or electrosprays. A flow of liquid from aneedle in an electrostatic field ~Fig. 2 !has a surface charge
that appears on subsequent droplets. At sufficiently low val-ues of charge density, the flow is not modified by the elec-trostatic force and a steady stream of droplets is produced@Fig. 2 ~b!#whose motion is determined primarily by gravity
and atmospheric drag. In this low-potential case, the dropletdiameter is approximately 1.9 times the diameter of the flowat the point at which the stream breaks into a droplet. Theflow diameter decreases with distance from the needle. Jonesand Thong
41have found empirically that the charge on the
droplets is approximately
Q59&pe0rd2E, ~7!
E5&f
rnln~4h/rn!, ~8!
whereEis the magnitude of the electric field at the needle
tip,rnis the needle radius, rdis the droplet radius, fis the
Fig. 1. Diagram of the Paul trap.
311 311 Am. J. Phys., Vol. 67, No. 4, April 1999 S. Robertson and R. Younger
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128.138.73.68 On: Tue, 23 Dec 2014 01:02:16needle potential, and his the separation between the needle
and the ground plane. For h57 mm,rn50.20mm, rd
50.12mm, and f53.8kV, one finds that E55.3
3106V/m and Q52.7310211C(1.6 3108electrons !.
At higher values of potential, the electrostatic force will
cause the flow to break into a fine spray @Fig. 2 ~c!#called the
cone-jet mode.42Detailed measurements have shown that the
spray consists of a core of relatively monodisperse dropletssurrounded by smaller satellite droplets.
43The diameter of
droplets is usually in the range 1–200 mm and depends upon
the flow rate, needle diameter, and electric potential. Thestandard deviation in the diameters of core droplets is re-ported to be 6% except at the smallest sizes where it in-creases to 12%. There is no theory for the charge on thesedroplets, however, there is a well-known upper bound, theRayleigh limit, for the charge at which the droplets willfission
44,45given by
Q2564p2e0grd3, ~9!
where gis the surface tension. Half this value is often used
as a crude estimate of the charge on droplets from the cone-
jet mode. For rd50.12mm, the Rayleigh limit for glycerin
droplets is 2.5 310211C.
The droplet generator @Fig. 2 ~a!#is a hypodermic needle
biased to high potential and placed above a grounded alumi-num plate having a 7-mm-diam hole to pass the droplets.A second grounded aluminum plate is placed below the firstin order to select the more uniform core of droplets
in the cone-jet mode of operation. Glycerin (density
51.26310
3kg/m3,g50.063N/m) is used because it is com-
monly available, nontoxic, and nonflammable. Droplet for-
mation is more reproducible if the conductivity of the glyc-erin is increased by adding approximately 15% vinegar ~to
reduce viscosity and resistivity !and a few drops of concen-trated acetic acid ~to further reduce resistivity !. A crude mea-
surement of resistivity made with an ohmmeter and parallel
copper electrodes indicates 10
4ohmmeters.
The needle is 20 gauge and the end is made blunt by
grinding. The inner diameter is 0.58 mm and the outer diam-eter is 0.91 mm. The needle is the locking variety and isattached to a 3-cc plastic syringe. The syringe is held in aninsulating block by a thumbscrew. The block is mountedabove the first ground plane on 63-mm spacers and the sec-ond ground plane is attached to the first with 25.4-mm alu-minum spacers. The distance from the needle tip to theground plane is manually adjusted to ;5 mm. A wire with a
small alligator clip is used to attach the needle to the highvoltage supply and a ground wire is permanently connectedto the ground plane. The needle is operated at positive po-larity to reduce the threshold for a corona discharge. In thelow voltage mode @Fig. 2 ~b!#a steady stream of droplets is
produced and crystals of ;10 particles are easily trapped. In
the high voltage mode @Fig. 2 ~c!#a spray is produced and
;50 particles may be trapped.
C. Video system
The video camera is a camcorder manufactured for home
use. Larger images are obtained by attaching a No. 4close-up lens ~focal length 25 cm !. Trapped particles oscil-
late about their central position due to the alternating electricfield and this causes them to appear as streaks. These streaksbecome dots when the camera is used in the fast-exposure or‘‘sports’’ mode. Video photographs of an oscilloscope tracemade in the fast-exposure mode show a segment of the traceindicating an exposure duration of 0.1 ms. A standard35-mm slide projector provides sufficient light to make drop-lets visible. A blank metal slide with a circular opening isplaced in the projector to reduce stray light striking the elec-trodes so that the automatic gain control in the camera usesthe maximum gain setting. An auxiliary lens ~focal length 20
cm!in front of the projector is used to bring this aperture in
focus within the trapping volume. The projector and cameraare aimed obliquely so that the electrodes do not block theview of the particles. Black paper is used as a backdrop toincrease the contrast of the images. Video images are digi-tized by a plug-in card for a PC, converted to gray-scale bitmaps, then converted to negatives for increased clarity.
IV. EXPERIMENTS
A. Observations of small crystalline arrays
Small crystalline arrays, Fig. 4, are generated by operating
the needle in the low voltage mode ~;4k V!and adjusting
the ac trapping voltage to ;5 kV. The droplet stream is
directed into the trap from a height of 0.8 m and at an angleto avoid the droplets striking the upper end cap. Dropletsgradually accumulate inside the trap. After the desired num-ber of trapped particles is obtained, the needle potential isremoved, which stops the flow. The number of trapped drop-lets is limited to about 10 because new droplets often pushother droplets out of the trap. The first few particles form aplanar array and additional particles fill a prolate spheroidalvolume. This may be made spherical by adding the dc bias,
U
t. The arrays are most easily observed on a video monitor.
The three-dimensional configuration may be deduced from
Fig. 2. ~a!Diagram of the oil droplet generator. ~b!Video photograph of the
generator in the low field mode taken with the fast-exposure mode of thecamera. ~c!Video photograph of the high voltage mode of the needle in the
normal exposure mode of the camera.
312 312 Am. J. Phys., Vol. 67, No. 4, April 1999 S. Robertson and R. Younger
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128.138.73.68 On: Tue, 23 Dec 2014 01:02:16the two-dimensional video images by causing the crystals to
rotate slowly. This is done by blowing into a plastic hose anddirecting the flow tangentially.
The interparticle separation is determined by measure-
ments made on the video monitor. The magnification is de-termined by viewing a ruler placed in the trap. The couplingparameter Gis calculated from the interparticle separation,
the charge, and the temperature. The latter is assumed to beroom temperature. Gis thus determined to be on the order of
10
7.
B. Measurement of charge-to-mass ratio
Three methods are used to determine the charge-to-mass
ratio of the droplets: ~1!measurement of the displacement of
droplets in response to a potential difference on the end caps,~2!direct measurement of charge and mass, and ~3!measure-
ment of the ac trapping potential at which the particle motionresults in escape. The equation for the displacement, s,o fa
particle from the center of the trap due to a potential drop
DVbetween the end caps is
46
s5aDVMV2~r0212z02!2
16z0QV2, ~10!
where ais a geometry factor relating the electric field to DV
andz0witha>0.8 for hyperboloidal electrodes. To measure
charge-to-mass ratio, the trap potential is lowered until only
a single particle remains in the trap. A known potential DVis
applied to the end caps and the displacement is observed
with the video system. Typical experimental values are DV
5390V,s50.5cm,V54 kV, and Q/M51.331023C/kg.
This displacement is superimposed upon a constant displace-
ment from gravity of 2g/vz2which is about 1 mm.
The charge on droplets may be determined by directing
the flow into a calibrated Faraday cup connected to a sensi-tive amplifier.
47The resulting train of pulses is observed on
an oscilloscope and the data are digitized and stored ~Fig. 3 !.
The charge on the droplets is determined from the relation
Q5CVp, whereCis the capacitance charged by the droplets
andVpis the observed peak voltage. The rate of consump-
tion of liquid overestimates the flow rate because many drop-
lets fall outside the area intercepted by the Faraday cup. Theflow rate is therefore determined by capturing the droplets ona small weighing paper. The frequency of the droplets isdetermined from the oscillogram. A typical set of data for
glycerin with 17% vinegar indicates a drop every 25 63m s
and a pulse-height analysis indicates a charge of 1.0060.03310
212C. The flow rate of the liquid is 24 mg/min,
which indicates a mean droplet radius of 0.12 mm, a mean
mass of 9.3 31029kg, and a mean charge-to-mass ratio of
1.131023C/kg. Pulse frequency and mass are dependent
upon the concentrations of glycerin, water, and acetic acid in
the droplet mixture. The standard deviation of the chargevaries from 4% to 16%, with less deviation at higher waterconcentration.
The third method to determine the charge-to-mass ratio is
to measure the ac trapping voltage at which the particle mo-tion become unstable. The ac voltage is gradually increased
until the particles escape and the value for Q/Mis deter-
mined from
Q
M5qV2~r0212z02!
8V, ~11!
where the value of qzis that at the instability threshold. In
vacuum this value is 0.908, however, this threshold may be
increased for oil droplets in air. Stokes’ law determines the
drag force Fd526prdmnwhere nis droplet velocity and
m51.8131024gm/cms is the viscosity of air. The damping
time is td5M/6prdmwhich is of the order of 0.1 s for
typical droplets. Winter and Ortjohann4have presented a
table giving the threshold value for qzas a function of a
damping parameter b59m/rrd2Vwhere ris the fluid den-
sity. For typical conditions, b50.02, the qzat the instability
threshold is not significantly increased above the vacuum
value, the threshold ac potential is V57.5kV, and thus
Q/M51.531023C/kg.
C. Excitation of center-of-mass oscillations
Individual particles may oscillate in the ponderomotive
potential well at the axial frequency vz5(kz/M)1/2as well
as at a radial frequency determined by kr. ForU50, the
axial oscillation frequency is
vz52&QV
MV~r0212z02!. ~12!
From Eqs. ~2!and~12!we find that vz/V5qz/2A2, thus vz
is below the ac line frequency. The resonant frequency is
found experimentally by placing an additional ac voltage onone of the two end caps and by varying the frequency toobtain the maximum amplitude of oscillation. The frequen-cies involved are at the low end of the audio range and aregenerated by a sine wave oscillator which is dc-coupled to anoperational amplifier capable of delivering 615 V. Measure-
ments are made on single particles by first trapping a groupof particles and then momentarily lowering the trapping po-
tential until one particle remains. The largest values of
vzare
obtained by increasing the ac voltage to a point just below
the stability limit. This has the advantage of making the os-cillation period much shorter than the damping time. For a
typical particle,
vzincreases linearly from 16 to 26 Hz as V
is increased from 4.4 to 6.2 kV with the measured vzbeing
within 20% of the calculated value. The effect of drag is to
reduce the resonant frequency to a value only slightly below
Fig. 3. Oscillogram of the pulse-height data used to determine charge-to-
mass ratio.
313 313 Am. J. Phys., Vol. 67, No. 4, April 1999 S. Robertson and R. Younger
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128.138.73.68 On: Tue, 23 Dec 2014 01:02:16that given by Eq. ~12!. Drag also allows higher experimental
values of vzby allowing stable operation for qzhigher than
0.9.
D. High voltage mode of the needle
In the high voltage mode a fine mist is obtained which
slows to a terminal velocity within a short distance of theneedle. This mist is allowed to float into the trap and then thetrapping voltage is applied. Typically about 40 particles aretrapped @Fig. 4 ~c!#. The droplet charge is too small and ar-
rives too irregularly to allow a measurement of charge andrate by means of the Faraday cup. Measurement of displace-
ment gives a charge-to-mass ratio of 4.4 310
23C/kg. The
mean interparticle separation gives, from Eq. ~5!, a mass of
8310211kg with an uncertainty of about a factor of 2. This
implies a charge of 6.4 310213C and a droplet radius of
0.025 mm.
Clouds obtained with the high voltage mode of the needle
show chaotic motion ~melting !whenqzis increased to near
the instability threshold and the crystalline state is recovered
~freezing !whenqzis lowered. Computer simulations of ions
in the Paul trap have shown continuous random motion at
qz50.8 and freezing at qz50.3.48At the higher qzvalues,
the heating by the micromotion maintains the cloud in the
chaotic state. In the case of oil droplets in air, quantitativeagreement with simulations for vacuum is not expected be-cause of the effects of gas dynamic drag.
V. SUMMARY AND CONCLUSION
The Paul trap has been used to study Coulomb crystals of
charged oil droplets in air. The droplets are generated by anelectrospray which can be operated to produce a steady drop-let stream or a fine mist. The charge of droplets has beendetermined from Faraday cup measurements and the stan-dard deviation is a few percent. The mass has been deter-mined from weighing accumulated droplets and the charge-
to-mass ratio determined from the displacement caused by anadditional voltage on the trap end caps. The maximum trap-ping potential for which motion is stable has been used foran independent measurement of the charge-to-mass ratio.Axial oscillations of single particles have been excited by anadditional ac potential on the end caps. Transitions of cloudsfrom a chaotic ~melted !state to a crystalline state have been
induced by varying the trapping potential.
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Fig. 4. Video photographs of crystalline arrays. ~a!Photograph in the stan-
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freezes the particle motion. ~c!Photograph of ;40 particles obtained from
the high voltage ~cone-jet !mode of the electrospray.
314 314 Am. J. Phys., Vol. 67, No. 4, April 1999 S. Robertson and R. Younger
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reference, however, the feedback resistance is reduced from 200 to 5 me-gohms to increase the frequency response.
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FLAT FEET
Oppenheimer @was#then professor of theoretical physics at Berkeley, later famous for his part
in building the atomic bomb, for his political activity, and for his unjust victimization. At the time,he was considered a demigod by himself and others at Berkeley, and as such he spake in learnedand obscure fashions. Besides, he knew quantum mechanics well, and in this he was unique atBerkeley. He taught it in none too easy a fashion, which showed off his prowess and attracted anumber of gifted students. His course later formed the basis of Leonard Schiff’s well-knowntreatise on quantum mechanics. Oppenheimer’s loyal disciples hung on his words and put oncorresponding airs. Just as we in Rome had acquired Fermi’s intonation, in Berkeley Oppenhe-imer’s students walked as if they had flat feet, an infirmity of their master’s.
Emilio Segre `,A Mind Always in Motion—The Autobiography of Emilio Segre `~University of California Press, Berkeley,
1993!, p. 138.
315 315 Am. J. Phys., Vol. 67, No. 4, April 1999 S. Robertson and R. Younger
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128.138.73.68 On: Tue, 23 Dec 2014 01:02:16 |
1.3562519.pdf | Micromagnetic method of s-parameter characterization of magnonic
devices
M. Dvornik, A. N. Kuchko, and V. V. Kruglyak
Citation: J. Appl. Phys. 109, 07D350 (2011); doi: 10.1063/1.3562519
View online: http://dx.doi.org/10.1063/1.3562519
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v109/i7
Published by the American Institute of Physics.
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Downloaded 12 Jul 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsMicromagnetic method of s-parameter characterization of magnonic devices
M. Dvornik,1A. N. Kuchko,2and V. V . Kruglyak1,a)
1School of Physics, University of Exeter, Exeter, United Kingdom
2Physics Department, Donetsk National University, Donetsk, Ukraine
(Presented 15 November 2010; received 22 September 2010; accepted 8 December 2010; published
online 13 April 2011)
Designers of nano-scale magnonic devices would benefit from methods of their evaluation that do
not require one to access the microscopic level of description or to construct device prototypes.
Here, we propose a numerical micromagnetics version of such a method, in which magnonic devices
are considered as two-port linear networks and can therefore be described in terms of theirs-parameters (i.e., reflection and transmission characteristics). In the micromagnetic model, the
sample is composed from a magnonic device-under-test situated between input and output magnonic
waveguides. First, dispersion relations and amplitudes of spin waves in the input and outputwaveguides are calculated from the simulations. The results are then compared to derive the
s-parameters of the device-under-test. We use a simple rectangular magnetic nonuniformity, for
which analytical results are readily obtained, to evaluate the efficiency and limitations of thetechnique in the sub-terahertz band.
VC2011 American Institute of Physics . [doi: 10.1063/1.3562519 ]
Nano-scale magnonic devices open an intriguing path to-
ward analog signal processing in the sub-terahertz frequency
band.1–3In particular, the use of propagating spin waves4,5
offers direct processing, in which no frequency conversion is
required prior to the signal manipulation, thereby reducing
processing time and facilitating real-time devices. This is in
addition to the long known opportunity to combine the signalprocessing with the re-configurability and data storage func-
tionality within the same chip. As the number of concepts and
modifications of magnonic devices continues to rapidlygrow,
1,2,6–11there also grows the demand for methods of per-
formance evaluation without building prototypes or develop-
ing theoretical models at the microscopic level of description.
Here, we propose a numerical micromagnetics version
of such a method, in which magnonic devices are considered
as two-port linear networks and can therefore be described interms of their s-parameters (i.e., reflection and transmission
characteristics). The magnonic “device-under-test” is situ-
ated between input and output magnonic waveguides. Thedispersion relations and amplitudes of spin waves in the
input and output are calculated from micromagnetic simula-
tions using the methodology described, e.g., in Refs. 12and
13. The results are then used to derive the reflection and trans-
mission characteristics (coefficients) of the evaluated device
as a function of the spin wave frequency. The calculations inthis paper have been performed using the Object Oriented
Micromagnetic Framework (OOMMF).
14However, any of
the existing micromagnetic packages (e.g., see Refs. 15–18)
could also be used for this purpose, at least in principle, pro-
vided that the required data analysis software is developed.
The geometry of the micromagnetic problem is shown
in Fig. 1. The sample has a total length of 10.5 lm, a width
of 100 nm, and a thickness of 10 nm. Its inner part
consists of the 2.5 lm long “input” and “output” waveguides
(marked as I and III, respectively) and the 100 nm long“device-under-test” (marked as II). The input and output
waveguides are made of Permalloy19and the device-under-
test is represented by a uniform cobalt layer.20The Gilbert
damping constant ais set to 0.001 in the three layers. The
outer layers (marked as “D”) have the same magnetic parame-ters as the input and output waveguides, except the damping
constants. The latter are now set to 0.1 in order to absorb spin
waves reaching the layers and thereby to suppress backreflec-tion from the layers and hence also from the ends of the sam-
ple. No anisotropy other than that naturally resulting from the
magneto-dipole energy is included in the calculation.
First, the ground state is obtained by relaxation from a
perfect saturated state along the length of the sample to the
state at the bias field H
bof 1 kOe applied in the same direc-
tion. Then, the sample is excited by applying a highly local-
ized transient magnetic field with temporal profile
h¼h0sin 2pflðt/C0t0Þ ½/C138
2pflðt/C0t0Þ; (1)
at the boundary between the left damped layer and the input
waveguide (I). The transient field is perpendicular to theplane of the sample and has amplitude of h
0¼50 Oe. Ideally,
the field defined by temporal profile (1) should lead to excita-
tion of propagating spin waves of nearly equal amplitude atfrequencies up to the cut-off value of f
l¼4 THz, which is
not the case in practice due to the limited duration of the
simulation. In order to partly suppress the corresponding dis-tortion of the excitation spectrum, the center of the transient
field is delayed relative to the start of the simulation by time
t
0equal to 10 periods of the “sinc” function. Each simulation
is run for 8 ns and the data are recorded every dt¼120 fs.
The corresponding frequency bandwidth flimof the simula-
tions is equal to flim¼0.5/dt¼4.17 THz. So, condition
fl<flimnecessary to prevent aliasing is satisfied.
The cell size of the rectangular mesh is equal
sx/C2sy/C2sz¼1/C2100/C210 nm3, and so, the width and thick-
ness of the mesh cell coincide with the correspondinga)Electronic mail: v.v.kruglyak@exeter.ac.uk.
0021-8979/2011/109(7)/07D350/3/$30.00 VC2011 American Institute of Physics 109, 07D350-1JOURNAL OF APPLIED PHYSICS 109, 07D350 (2011)
Downloaded 12 Jul 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsdimensions of the sample. So, wave vectors ki,krandktof the
incident, reflected, and transm itted spin waves respectively are
parallel to the length of the samp le and therefore to the bias
magnetic field and the static ma gnetization. The wave vector
bandwidth klimof the simulations is klim/2p¼0.5/sx¼0.5/C2109
m-1. Prior to the Fourier analysis, the static magnetization pro-
file is subtracted from the dynam ical data, in order to extract
the pure dynamic component of the magnetization m(r,t). The
method described below is then applied to the z-component of
the dynamic magnetization mz(r,t).
The implemented here method of the s-parameter extrac-
tion is based on the method of the magnonic dispersion calcu-
lation described in Refs. 12and13. The main idea is to make
use of information about spin wave amplitudes m(k,f)t h a ti s
obtained as a result of the calculation of spin waves dispersion
f(k). By applying this method separately to the input and out-
put waveguides, one can calculate the complex Fourier ampli-tudes of the input and output signals and then complex
transmission and reflection coefficients as their ratios.
The results of the time domain simulations are obtained
as a 2D array of data m
z(xi,tj), where iandjare integer indi-
ces of the mesh cells and time steps respectively. By per-
forming a 2D Fourier transform of the data, spin waveamplitudes m(k
i,fj) are calculated as a function of discrete
valued wave vector kiand frequency fj. First, we find the dis-
persion in the form of a continuously valued frequencydefined on the discrete mesh of the wave number, f
i¼f(ki).
We assume that dispersion f(k) of spin waves is equivalent
for the forward ( k>0) and backward ( k<0) propagating
spin waves, i.e., f(k)¼f(/C0k).21For each | i|, we use cubic
interpolation to find frequencies fi¼f(ki) as points at which
functions m(ki,f)þm(k-i,f) of continuously valued frequency
freach their local maxima.22Then, we extract the amplitudes
of the backward and forward propagating spin wavesseparately from the k<0a n d k>0 branches of the dispersion
respectively using bilinear interpolation of m(ki,fj)t omi(ki,fi),
with the latter now being a discrete 1D set of data. The inter-polation algorithm is adjusted so that the discretization of the
frequency rather than wave vector remains equidistant. This
allows us to use the same frequency mesh to compare ampli-tudes of spin waves extracted from different simulations. This
is preferred since we are interested in the frequency (rather
than wave number) dependence of the s-parameters.
Due to effects connected with the finite damping and
group velocity of spin waves in the input and output wave-
guides, two different simulations have to be performed. First,we perform a reference simulation for a sample like the
tested one but in which the device-under-test is replaced by a
uniform layer with the properties, width and thickness of theinput and output waveguides and the length of the device-
under-test. Reference amplitudes m
f
R,I(f) and mf
R,III(f) of the
forward spin waves in the input and output waveguides,respectively, are then calculated as functions of the fre-
quency. Then, simulations for the sample with the device-
under-test are performed, and amplitudes m
b
I(f) and mf
III(f)
of the reflected from and transmitted through the device-
under-test spin waves propagating in the input and output
waveguides respectively are obtained. Finally, the values ofs-parameters S
11andS21are calculated as
S11¼mb
I
mf
R;I¼RðfÞ;
S21¼mb
III
mf
R;III¼TðfÞ;(2)
where T(f) and R(f) are the transmission and reflection coeffi-
cients respectively. The damped regions must be excludedfrom the analysis. For each particular problem (i.e., mag-
nonic device tested), the length of the input and output wave-
guides has to be optimized so as to obtain the requiredspectral resolution of the subsequent Fourier analyses and to
allow spin waves with the smallest group velocities to reach
the device-under-test well within the simulation time.
We apply the method to calculation of the reflection and
transmission characteristics of a device-under-test repre-
sented by a uniform inclusion of Co. Figure 1shows the dis-
persion of spin waves calculated for the reference sample
(excluding damped regions) and then “digitized” using the
described method in dipole-exchange, dipole (magnetostatic)and exchange approximations. The exchange and dipole-
exchange approximations produce very similar results at
frequencies above about 100 GHz, while the dipole anddipole-exchange curves agree only in close vicinity of the
uniform ferromagnetic resonance frequency, i.e., at wave
numbers up to about 6 p/C210
4m/C01.
In the exchange approximation, the dispersion can also
be easily calculated analytically, facilitating verification of the
method. The comparison appears to show an excellent agree-ment between the theory and simulations at frequencies up to
about 0.5 THz. At higher frequencies, the simulated dispersion
curve has a downward curvature as a result of the discrete na-ture of the numerically solved problem. Indeed, the highest
FIG. 1. (Color online) The dispersion curves of spin waves calculated from
simulations for the reference sample (excluding damped regions) and then
digitized is shown for dipole-exchange (DE), dipole (D) and exchange (E)
approximations. The curves are marked by “s” in the legend. The analytical
curve (t) calculated in the exchange approximation is also shown. The upperinset shows the same dispersion on a greater scale together with the geome-
try of the considered problem. The bottom inset shows, as a function of the
spin wave frequency, the absolute value of the difference between the ana-
lytical and simulated curves calculated in the exchange approximation in the
units of the Co layer thickness.07D350-2 Dvornik, Kuchko, and Kruglyak J. Appl. Phys. 109, 07D350 (2011)
Downloaded 12 Jul 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsfrequency and wave number accessible in the simulations cor-
respond to the edge of the Brillouin zone of the spectrum of
the 1D chain of spins with a period equal to the cell size.Hence, the frequency range of agreement between the theory
and simulations can be expanded by reducing the cell size.
Figure 2shows the squared amplitudes of the reflection
and transmission coefficients simulated and analytically cal-
culated in the exchange approximation. The curves are char-
acterized by a quasiperiodic alteration of regions of high andlow transmission and reflection. The alteration originates
from the Fabry-Perot resonance of spin waves in the cavity
represented by the Co layer, with the frequency of alterationdetermined by its thickness. The predictions of the si-
mulations and analytical theory for positions of the minima
and maxima of reflection and transmission agree atfrequencies <200 GHz, while slowly “dephasing” at higher
frequencies and then coming into phase again at frequencies
about 1 THz. The dephasing originates from the differencebetween simulated and analytical dispersion curves illus-
trated in the bottom inset of Fig. 1. The amplitude of varia-
tion of the simulated transmission coefficient agrees wellwith the analytical theory, while the simulations tend to
underestimate the variation in the reflection coefficient.
The latter discrepancy could be attributed to the effect
of (small but finite) damping in the simulations and then to
accumulation of spin wave energy in and in the vicinity of
the Co layer, which is expected to lead to appearance oflocalized “defect” modes.
23The localized spin waves gain
their energy from the incident spin wave and hence might
lead to the observed discrepancy.
The frequency dependences of the reflection and trans-
mission coefficients calculated in the dipole-exchange and
exchange approximations (not shown) agree well at frequen-cies above about 200 GHz, as expected from the similarity of
the corresponding dispersion curves in the frequency range,
as shown in Fig. 1. This demonstrates the applicability of the
exchange approximation at the high frequencies, and allows
one to exploit the analogy existing between the exchangespin waves and the motion of an electron in a nonuniform
potential, pointed out and exploited, e.g., in Refs. 23–25.
In summary, we have proposed a micromagnetic method
by which to evaluate performance of magnonic (spin wave)
devices. We have applied the method to a simple rectangular
magnetic nonuniformity and have successfully calculated itsreflection and transmission coefficients. The technique is very
efficient in the sub-terahertz band, albeit faces some difficul-
ties in the low gigahertz band associated with the low groupvelocity of spin waves. We have shown that the exchange
approximation is well suited for description of propagating
spin waves at THz frequencies. However, the accuracy of theapproximation in a particular problem depends upon the rela-
tive value of the spin wave wavelength and the characteristic
scale of the nonuniformities in the problem. At THz frequen-cies, the dispersion obtained from the simulations deviates
from that calculated using the continuous medium approxi-
mation. This deviation not only emphasizes the discrete na-ture of the micromagnetic simulations, but also suggests that
micromagnetic solvers based on truly atomistic models are
required and might well be feasible computationally in future.
The research leading to these results has received funding
from the EC’s 7th Framework Programme (FP7/2007-2013)
under GA 233552 (DYNAMAG) and from the Engineeringand Physical Research Council (EPSRC) of the UK.
1V. V. Kruglyak et al.,J. Phys. D: Appl. Phys. 43, 264001 (2010).
2A. Khitun et al.,J. Phys. D: Appl. Phys. 43, 264005 (2010).
3The International Technology Roadmap for Semiconductors, available at
http://www.itrs.net/Links/2009ITRS/Home2009.htm .
4A. I. Akhiezer et al.,Spin Waves (North-Holland, Amsterdam, 1968).
5A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves
(Chemical Rubber Corp., New York, 1996).
6S. V. Vasiliev et al.,J. Appl. Phys. 101, 113919 (2007).
7T. Schneider et al.,J. Nanoelectron. Optoelectron. 3, 69 (2008).
8A. A. Serga et al.,Appl. Phys. Lett. 94, 112501 (2009).
9S. K. Kim et al.,Appl. Phys. Lett. 95, 082507 (2009).
10V. E. Demidov et al.,Appl. Phys. Lett. 95, 262509 (2009).
11H. Al-Wahsh, Eur. Phys. J. B 73, 527 (2010).
12V. V. Kruglyak and R. J. Hicken, J. Magn. Magn. Mater. 306, 191 (2006).
13S.-K. Kim, J. Phys. D: Appl. Phys. 43, 264004 (2010).
14M. J. Donahue, IEEE Trans. Magn. 45, 3923 (2009); M. Donahue and D.
G. Porter, OOMMF User’s guide, version 1.0, Interagency Report NIS-
TIR6376 (NIST, Gaithersburg, MD, 1999), available at http://math.nist.gov/
oommf/.
15D. V. Berkov and N. L. Gorn, J. Phys. D Appl. Phys. 41, 164013 (2008);
See also http://www.micromagus.de/.
16B. C. Choi et al.,IEEE Trans. Magn. 43, 2 (2007); See also http://llgmicro.-
home.mindspring.com/.
17S. Bance et al.,J. Appl. Phys. 103, 07E735 (2008); See also http://mag-
net.atp.tuwien.ac.at/scholz/magpar/.
18H. Fangohr et al .,J. Appl. Phys. 105, 07D529 (2009); See also http://
www.soton.ac.uk/ /C24fangohr/nsim/nmag/.
19The magnetization of saturation Ms¼8/C2105A/m, the gyromagnetic ratio
g¼2.1, and the exchange constant A¼1.3/C210/C011J/m.
20Ms¼14/C2105A/m, g¼2.1, and A¼3/C210/C011J/m.
21We note that, in general, the corresponding amplitudes are not equal since
they correspond to spin waves propagating in opposite directions.
22The summation of the two amplitudes is required to avoid errors connectedwith the decreases of amplitude of either reflected or transmitted wave due
to physical reasons (interference, etc.).
23V. V. Kruglyak et al.,J. Appl. Phys. 99, 08C906 (2006).
24E. Schlo ¨mann, J. Appl. Phys. 35, 159 (1963); Ernst Schlo ¨mann and R. I.
Joseph, ibid.35, 167 (1964).
25H. Al-Wahsh et al.,Phys. Rev. B 78, 075401 (2008).
FIG. 2. (Color online) The squared amplitudes of the simulated (s) and ana-
lytical (t) reflection (R) and transmission (T) coefficients are shown for the
exchange approximation.07D350-3 Dvornik, Kuchko, and Kruglyak J. Appl. Phys. 109, 07D350 (2011)
Downloaded 12 Jul 2012 to 130.63.180.147. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions |
1.4737126.pdf | Dynamics of magnetic nanoparticle in a viscous liquid: Application to
magnetic nanoparticle hyperthermia
N. A. Usov and B. Ya. Liubimov
Citation: J. Appl. Phys. 112, 023901 (2012); doi: 10.1063/1.4737126
View online: http://dx.doi.org/10.1063/1.4737126
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v112/i2
Published by the American Institute of Physics.
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Downloaded 13 Mar 2013 to 129.174.21.5. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsDynamics of magnetic nanoparticle in a viscous liquid: Application
to magnetic nanoparticle hyperthermia
N. A. Usov1,2and B. Y a. Liubimov1
1Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, Russian Academy of Sciences,
(IZMIRAN), 142190 Troitsk, Moscow, Russia
2Magnetic and Cryoelectronic Systems Ltd., 142190 Troitsk, Moscow, Russia
(Received 22 March 2012; accepted 10 June 2012; published online 18 July 2012)
It is shown that the magnetic dynamics of an assembly of nanoparticles dispersed in a viscous
liquid differs significantly from the behavior of the same assembly of nanoparticles immobilized in
a solid matrix. For an assembly of magnetic nanoparticles in a liquid two characteristic mode for
stationary magnetization oscillations are found that can be called the viscous and magnetic modes,respectively. In the viscous mode, which occurs for small amplitude of the alternating magnetic
field H
0as compared to the particle anisotropy field Hk, the particle rotates in the liquid as a whole.
In a stationary motion the unit magnetization vector and the director, describing the spatialorientation of the particle, move in unison, but the phase of oscillations of these vectors is shifted
relative to that of the alternating magnetic field. Therefore, for the viscous mode the energy
absorption is mainly due to viscous losses associated with the particle rotation in the liquid. In theopposite regime, H
0/C21Hk, the director oscillates only slightly near the external magnetic field
direction, whereas the unit magnetization vector sharply jumps between magnetic potential wells.
Thus, a complete orientation of the assembly of nanoparticles in the liquid occurs in the alternatingmagnetic field of sufficient amplitude. As a result, large specific absorption rates, of the order of
1 kW/g, can be obtained for an assembly of magnetic nanoparticles in viscous liquid in the
transient, H
0/C240.5Hk, and magnetic modes at moderate frequency and alternating magnetic field
amplitude. VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4737126 ]
I. INTRODUCTION
Superparamagnetic nanoparticles have actually found
important applications in biomedicine, particularly for tar-geted drug delivery and magnetic nanoparticle hyperther-
mia.
1,2To select an assembly of magnetic nanoparticles
suitable in hyperthermia, it is important2–4to clarify the con-
ditions that provide sufficiently high specific absorption rate
(SAR) of the assembly in alternating external magnetic field
of moderate amplitude H0and frequency f. From a theoreti-
cal point of view, the behavior of an assembly of superpara-
magnetic nanoparticles in an alternating external magnetic
field has been studied in detail5–9for the case when the uni-
axial nanoparticles are immobilized in a surrounding solid
matrix. In this case, only the particle magnetic moments can
respond to the alternating external magnetic field, whereas therotation of the particles as a whole is impossible. The descrip-
tion of the power absorption process
5–9takes into account the
thermal fluctuations of the particle magnetic moments at a fi-nite temperature and uses the methods developed to study the
Neel– Brown magnetization relaxation.
10–12One of the most
important results obtained for an assembly of immobilizednanoparticles is a significant SAR dependence on the mag-
netic parameters and average sizes of the nanoparticles, since
these parameters determine the characteristic relaxation times
Nof the particle magnetic moment.
However, the most of the SAR measurements13–26were
carried out for assemblies of magnetic nanoparticles dis-persed in aqueous solutions or liquid mixtures of various vis-cosities. In magnetic hyperthermia, the assembly of
nanoparticles is likely to operate in a liquid medium,although in some cases the particles probably will be immo-
bilized at the vessel walls or inside the biological cells.
16It
is therefore important to generalize the theoretical results5–9
to the case of an assembly of nanoparticles in a viscous liq-
uid. In a viscous medium, the particles can rotate as a whole
both under the influence of the regular torque associatedwith the magnetic interactions and due to thermal fluctua-
tions in the surrounding liquid. The process of the second
type is usually described in the theory of Brownian relaxa-tion
27,28by introducing the corresponding relaxation time sB.
The behavior of an assembly of magnetic nanoparticles
in a viscous liquid is studied in details, in particular, in thetheory of magnetic fluids.
27–30In early papers by Newman
and Yarbrough31,32the relaxation of the magnetic moment
of an assembly of particles in a constant external magneticfield was considered neglecting the thermal fluctuations.
Later, more general approach was developed to study
magnetization relaxation
33,34and complex magnetic
susceptibility35–38of an assembly of magnetic nanoparticles
in an alternating external magnetic field. It should be noted,
however, that the low-frequency hysteresis loops of the as-sembly of nanoparticles in an alternating magnetic field of fi-
nite amplitude are still not investigated in detail. As a rule,
the theoretical interpretation
3,4,16,17,19,23,24of the behavior of
an assembly of nanoparticles in a liquid in the alternating
external magnetic field is based on the linear approxima-
tion.39Besides, it uses an assumption27,39that the magnetic
0021-8979/2012/112(2)/023901/11/$30.00 VC2012 American Institute of Physics 112, 023901-1JOURNAL OF APPLIED PHYSICS 112, 023901 (2012)
Downloaded 13 Mar 2013 to 129.174.21.5. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsresponse of the assembly in a liquid can be characterized by
the so-called effective relaxation time, sef¼sBsN/(sBþsN).
Note that the linear approximation breaks down already inthe moderate fields, H
0¼200–300 Oe, which are often used
in the experiment. Besides, the concept of the effective
relaxation time is not strictly justified.
Surely, both relaxation times, sBandsN, are essential to
describe various aspects of the magnetic nanoparticle behav-
ior in a liquid. However, the introduction of the effectiverelaxation time s
efoften oversimplifies the physical situation,
since it does not take into account the complex dynamics of
a magnetic nanoparticle in liquid. An attempt to overcomethis difficulty was made in Ref. 40, where the necessary set
of equations has been written down. However, the low fre-
quency hysteresis loops of the assembly have not been con-structed. The dynamics of magnetic nanoparticles in viscous
liquid in alternating magnetic field was considered also in
recent Ref. 41. Unfortunately, the authors
41used the wrong
expression for the regularly torque, Nm(see Eq. (2)below)
in their equation for the rotational motion of the particle in
liquid (Eq. (11) in Ref. 41). This makes their results
doubtful.
In this paper, the problem is studied using the stochastic
equations of motion33,34for the unit magnetization vector a
and the unit vector n, which determines the space orientation
of a magnetic nanoparticle with uniaxial anisotropy. By
solving these equations, one can describe in detail both therelaxation of the magnetic moment of a dilute assembly of
superparamagnetic particles in a constant magnetic field and
the behavior of the assembly in the low-frequency alternat-ing magnetic field of finite amplitude. In this paper, we show
that there are basically two regimes of the stationary magnet-
ization oscillations, depending on the amplitude of the alter-nating magnetic field. They can be characterized as viscous
and magnetic modes, respectively. The viscous regime
occurs for low magnetic field amplitudes, H
0/C28Hk, where
Hk¼2K1/Msis the particle anisotropy field, K1is the
magnetic anisotropy constant and Msis the saturation
magnetization. In the viscous mode the unit vectors abn
move in unison and out of phase with respect to the phase of
the alternating magnetic field. In the opposite limit H0/C21Hk,
the vector noscillates only slightly, while the unit magnet-
ization vector jumps between the states a¼6h0, where h0is
the unit vector along the direction of the external magnetic
field.
Interestingly, in both cases for stationary magnetization
oscillations a partial or complete orientation of the assembly
in viscous liquid occurs. The transition between the oscilla-tion regimes occurs within the range 0.5 H
k/C20H0<Hk,
depending on the magnetic field frequency and the liquid vis-
cosity. In this paper, we describe in detail the behavior of thelow-frequency hysteresis loops of the assembly as a function
of the alternating magnetic field amplitude, frequency, and
the liquid viscosity. The SAR of the assembly is calculatedas a function of the frequency, viscosity, and other relevant
parameters. Based on these calculations, we discuss the opti-
mal conditions for an assembly of superparamagnetic nano-particles in a liquid to absorb the energy of the alternating
external magnetic field.II. BASIC EQUATIONS
Let us consider a behavior of a uniformly magnetized
spherical nanoparticle with uniaxial magnetic anisotropy in a
viscous liquid under the influence of a constant or alternatingexternal magnetic field. Let nbe the unit vector firmly
attached to the particle. It shows the direction of the particle
easy anisotropy axis. The kinematic equation of motion forthis vector is given by
d~n
dt¼½~x;~n/C138; (1)
where xis the angular velocity of the particle rotation as a
whole. The rotational motion of the particle is described by a
stochastic equation of motion33
Id~x
dtþn~x¼~Nmþ~Nth; (2)
where Iis the moment of inertia of a spherical particle,
n¼6gVis the friction coefficient, gis the dynamic viscosity
of the liquid, and Vis the particle volume. The friction coef-
ficient is determined by solving the problem42of rotation of
a particle in a viscous liquid in the Stokes approximation for
a small Reynolds number. In Eq. (2),Nmis the regularly tor-
que associated with the particle magnetic moment and theN
this the fluctuating torque that leads to a free Brownian
rotation of the particle in a liquid in the absence of external
magnetic field.
Dynamics of the unit magnetization vector aof a single-
domain nanoparticle is described by the stochastic Landau-
Lifshitz equation10–12
@~a
@t¼/C0c1½~a;~Hefþ~Hth/C138/C0jc1½~a;½~a;~Hefþ~Hth/C138/C138;(3)
where c1¼jc0j/(1þj2),jis the damping constant and c0is
the gyromagnetic ratio. In Eq. (3),Hefis a vector of the
effective magnetic field and Hthis a random magnetic field
associated with the presence of thermal fluctuations in thesystem.
The total magnetic energy of the particle in an external
uniform magnetic field H
0is given by
W¼/C0K1Vð~a~nÞ2/C0MsVð~a~H0Þ: (4)
In the alternating magnetic field of a frequency fthe vector
H0is replaced by H0cos(xt), where x¼2pfis the angular
frequency. The effective magnetic field in Eq. (3)is the de-
rivative of the total energy
~Hef¼/C0@W
VM s@~a¼~H0þHkð~a~nÞ~n: (5)
Similarly, the regularly torque in Eq. (2)can be calculated as
~Nm¼@W
@~n;~n/C20/C21
¼/C02K1Vð~a~nÞ½~a;~n/C138: (6)
Note that Eq. (6)is a direct consequence of the general
Lagrange principle.42It differs from the wrong expression,023901-2 N. A. Usov and B. Y a. Liubimov J. Appl. Phys. 112, 023901 (2012)
Downloaded 13 Mar 2013 to 129.174.21.5. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions~Nm¼½~l;~H0/C138¼MsV½~a;~H0/C138, used in Eq. (11) of Ref. 41to
describe the mechanical rotation of a magnetic nanoparticle
in a viscous liquid.
In accordance with the fluctuation-dissipation theorem11
the components of the fluctuating torque Nthhave the follow-
ing statistical properties33,43(i,j¼x,y,z):
hNth;iðtÞi ¼ 0;hNth;iðtÞNth;jðt1Þi ¼ 2kBTndijdðt/C0t1Þ;(7)
where kBis the Boltzmann constant and Tis the absolute
temperature. For the components of the fluctuating magneticfieldH
thsimilar relations10,11are supposed to be valid
hHth;iðtÞi ¼ 0;hHth;iðtÞHth;jðt1Þi ¼2kBTj
jc0jMsVdijdðt/C0t1Þ:(8)
The set of Eqs. (1)–(8)can be simplified, if one takes into
account31that because of a very small size of the magnetic
nanoparticle it is possible to neglect in Eq. (2)the effective
moment of inertia, assuming I/C250. Then, one obtains from
Eqs. (1),(2), and (6)an equation of motion for the vector n
as follows:
@~n
@t¼Gð~a~nÞ/C16
~a/C0ð~a~nÞ~n/C17
/C01
n½~n;~Nth/C138; (9)
where G¼2K1V/n¼K1/3g. It is interesting to note that the
coefficient Gdoes not depend on the particle radius. Equa-
tions (3)and(9), together with Eqs. (7)and(8)constitute a
complete set of equations describing the behavior of a mag-
netic nanoparticle in a viscous liquid under the influence ofthe external ac or dc magnetic field. They have to be solved
together using the corresponding numerical procedure (see
Appendix).
In this paper, the illustrative calculations are performed
for an assembly of uniaxial nanoparticles with magnetic
parameters typical of the particles of iron oxides, K
1¼105
erg/cm3,Ms¼400 emu/cm3. Therefore, the anisotropy field
of the particle is given by Hk¼500 Oe. The magnetic damp-
ing constant is assumed to be j¼0.5. The viscosity of the
liquid varies from the value typical for water, g¼0.01
g/(cm s), when the coefficient G¼3.3/C2106s/C01, to a suffi-
ciently large value g¼1.0 g/(cm s). In the latter case, one has
G¼3.3/C2104s/C01.
Let us assume the value C¼2/C110/C06erg/cm for the
exchange constant of the magnetic material. Then, the char-acteristic single-domain radius of the nanoparticle can be
estimated using Brown’s
44,45lower and upper estimates. As
shown by Brown,44for the lower bound to the single-domain
radius one can take alow
c¼x1R0=ffiffiffiffi
Np
, where R0¼ffiffiffiffi
Cp
=Msis
the exchange length, N¼4p/3 is the demagnetizing factor of
a sphere and x1¼2.0816 is the minimal root of the spherical
Bessel function derivative. As an upper bound for a soft
magnetic nanoparticle ( K1<M2
s) one can use the critical ra-
dius of stability45of a uniform magnetization,
aup
c¼x1R0=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N/C02K=M2
sp
. Then, for a particle with given
magnetic parameters one finds R0¼35.4 nm, aclow¼36 nm,
acup¼43 nm. Therefore, the single-domain diameter of the
nanoparticles studied lies in the range 72 nm <Dc<86 nm.
In this paper, the calculations are carried out for particleswith diameters D/C2060 nm. Therefore, it can be safely
assumed that the magnetic behavior of a particle is mainly
determined by a uniform rotation mode.46
III. MAGNETO-DYNAMICS (MD) APPROXIMATION
It is instructive to consider first the dynamics of mag-
netic nanoparticles in a liquid neglecting the thermal fluctua-
tions, i.e., dropping in Eqs. (3)and(9)the fluctuating terms
NthandHth. This approximation31,32may be called the MD
one. It is similar to the Stoner–Wohlfarth46approximation in
the theory of a single-domain particle, when the influence of
the thermal fluctuations on the particle behavior is com-pletely neglected. We shall see below, in Sec. IV, that the
MD approximation is very useful for a better understanding
of the results of numerical simulation of the complete set ofEqs. (3)and(7)–(9), which take into account the effect of
thermal fluctuations. In the MD approximation, the behavior
of the particles in a liquid depends significantly on the ampli-tude of the external alternating magnetic field. Let us con-
sider two characteristic cases.
A. Small magnetic field amplitude, H0<<Hk
This limit may be called a viscous mode of magnetiza-
tion oscillations, because in this case the power absorption is
mainly due to viscous losses associated with the particle
rotation in a liquid. Let us fix a relatively small amplitude ofthe alternating magnetic field, H
0¼30 Oe, as compared to
the particle anisotropy field, Hk¼500 Oe, and will gradually
increase the frequency, assuming the liquid viscosity to below, g¼0.01 g/(cm s). In a stationary motion, which occurs
after several periods of oscillation of the alternating mag-
netic field elapsed, the information about the initial positionsof the vectors aandnis completely lost. As Figure 1(a)
shows, at low frequency, f¼10 kHz, in stationary motion the
vectors abnoscillate between the directions 6h
0and
remain almost parallel to each other. Consequently, the mag-
netic moment of the assembly is changed due to the rotation
of magnetic nanoparticles in a liquid as a whole. As seen inFig.1(a), the phase of the stationary oscillations of the vec-
torsaandnis shifted by p/2 with respect to that of the ac
magnetic field.
As Fig. 1(b) shows, with increase in the frequency the
amplitude of the oscillation of the vectors aandnalong the
magnetic field direction is greatly reduced. This means thatthese unit vectors rotate almost perpendicular to the vector
h
0. In this position, they oscillate in unison with relatively
small amplitude. The phase of the oscillations is still shiftedwith respect to the phase of the alternating magnetic field. It
is found that the amplitude of the oscillations increases as a
function of H
0. On the contrary, the increase in viscosity (see
Fig. 1(c)) leads to a decrease of the oscillation amplitude
again. Also, a relative phase shift appears for aandn
oscillations.
Nevertheless, for all cases considered the motion of the
vectors aandnin the viscous mode is qualitatively the
same. In fact, at a sufficiently high frequency, f/C21100 kHz,
for every nanoparticle in stationary movement the vectors a
andnoscillate being nearly perpendicular to the vector h0.023901-3 N. A. Usov and B. Y a. Liubimov J. Appl. Phys. 112, 023901 (2012)
Downloaded 13 Mar 2013 to 129.174.21.5. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsThus, there is a partial orientation of a dilute assembly of
nanoparticles in a liquid, as in the spherical coordinates(h,u) the distribution of the directors of different nanopar-
ticles is concentrated near the spherical angle h¼p/2.As Fig. 2shows, in the stationary magnetization oscilla-
tions the area of MD hysteresis loop is finite. As we have
seen above, in the viscous mode the vectors aandnmove in
unison, so that the deviation of the magnetic vector from the
easy anisotropy axis is small, as a rule. Nevertheless, there is
a phase shift between the magnetic moment oscillations andthe oscillations of the alternating magnetic field. Thus, there
is a nonzero absorption of the energy of alternating external
magnetic field. It is clear that in the viscous mode the powerabsorption is related mainly with the viscous friction during
the particle rotation in a liquid.
It is well known
5,9,39that the specific absorption rate is
proportional to the area of the assembly hysteresis loop. As
Fig.2(a)shows, in the viscous mode the MD hysteresis loop
area decreases rapidly with increasing frequency. Indeed, themagnetic field cannot rotate the particles to large angles due
to their inertia associated with the viscous friction. Similarly,
as shown in Fig. 2(b), in the viscous mode the hysteresis
loop area decreases sharply with increasing of viscosity, as
the amplitude of the unit magnetization vector oscillations
along the magnetic field direction decreases.
FIG. 1. The stationary MD oscillations of the components of the unit mag-
netization vector (2) and the particle director (3) along the magnetic field
direction for various cases: (a) H0¼30 Oe, f¼10 kHz; (b) H0¼30 Oe,
f¼100 kHz; (c) H0¼100 Oe, f¼100 kHz; g¼0.1 g/(cm s). Curve (1) shows
the oscillations of the reduced alternating magnetic field.
FIG. 2. (a) MD hysteresis loops of a particle in liquid in the viscous mode,H
0<Hk, for various frequencies: (1) f¼10 kHz, (2) f¼50 kHz, (3)
f¼100 kHz, (4) f¼200 kHz; (b) MD loops as a function of the liquid viscos-
ity: (1) g¼0.01 g/(cm s), (2) g¼0.03 g/(cm s), (3) g¼0.1 g/(cm s).023901-4 N. A. Usov and B. Y a. Liubimov J. Appl. Phys. 112, 023901 (2012)
Downloaded 13 Mar 2013 to 129.174.21.5. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsB. Large magnetic field amplitude, H0/C21Hk
In this case, the mode of the stationary oscillations is
simpler. As calculations show, for any initial direction of the
vector n, after a few oscillation periods this vector sets
almost parallel to the field direction. This means that n/C25h0,
orn/C25/C0h0, depending on the initial conditions. As Fig. 3(a)
shows, the vector noscillates in this position with a very
small amplitude, while the magnetic vector jumps betweenthe states a¼6h
0, moving from one magnetic potential well
to another. The regime of magnetic oscillations observed in
the limit H0/C21Hkmay be called a magnetic mode, since in
this case the behavior of the magnetic vectors is similar to
that for an assembly of oriented nanoparticles immobilized
in the solid matrix.
It is important to note that in the magnetic mode there is
almost complete orientation of an assembly of magnetic
nanoparticles in a liquid, because the directors of variousnanoparticles point along the magnetic field direction. As a
result, the shape of the MD hysteresis loop in the magnetic
mode is close to rectangular. Clearly, in this case, the energyabsorption is certainly determined by the magnetizationrelaxation process, since the contribution to the energy
absorption of the viscous friction is small.
In contrast to the viscous mode, the increase of the field
frequency or the liquid viscosity does not change much the
shape of the MD hysteresis loop in the magnetic mode. It is
found that at H
0¼Hkthe MD hysteresis loop shape is
weakly dependent on frequency in the frequency range
f¼100–800 kHz, and on the liquid viscosity in the range
g¼0.01–0.1 g/(cm s). The effect of viscosity becomes appre-
ciable only at g¼1.0 g/(cm s).
At low frequencies, f/C2410 kHz, and viscosity g¼0.01
the viscous mode exists for H0<0.5Hk, and the magnetic
mode is already realized at H0/C210.5Hk.H o w e v e r ,i nt h eM D
approximation the critical field for transition to the magnetic
mode increases gradually as a function of frequency. AsFig.3(b) shows, at H
0¼0.5Hkthe MD hysteresis loop is still
nearly rectangular at f¼100 kHz, but its area decreases gradu-
ally as the function of frequency. Actually, it is found that atf¼500 kHz the magnetic mode is realized at H
0/C210.7Hkonly.
For the most interesting viscous mode of the magnetiza-
tion oscillations one can get an approximate analytical solu-tion, which confirms the numerical results presented in Figs.
1and2. As shown below, for the viscous mode the intrinsic
magnetic damping of the particle is negligible, so that thebehavior of the vectors aandnis approximately described
by a pair of equations that follow from Eqs. (3)and(9)
@~a
@t¼/C0c½~a;~H0cosðxtÞþHkð~a~nÞ~n/C138; (10a)
@~n
@t¼Gð~a~nÞ/C16
~a/C0ð~a~nÞ~n/C17
: (10b)
As Fig. 1shows, for the viscous mode, H0<Hk, the vectors
aandnare always close, so that it is reasonable to put
~n¼~aþ½~a;~e/C138;j~ej/C281: (11)
Eq.(10) can be rewritten as follows:
@~a
@t¼/C0c½~a;~H0cosðxtÞþHk½~a;~e/C138/C138; (12a)
@
@tð~aþ½~a;~e/C138Þ ¼ /C0 G½~a;~e/C138: (12b)
In the left-hand side of Eq. (12b) , the correction term propor-
tional to a small vector ecan be omitted. Then, substituting
Eq.(12b) in Eq. (12a) , one obtains an equation for the
vector a
@~a
@t¼/C0c½~a;~H0cosðxtÞ/C138 þcHk
G~a;@~a
@t/C20/C21
: (13a)
Having in hand the solution of Eq. (13a) , one can obtain vec-
tornby means of the relation
~n¼~a/C0G@~a
@t: (13b)
Note that equation (13a) is the Landau-Lifshitz-Gilbert
equation,10–12which describes a precession of the unit
FIG. 3. (a) The stationary MD oscillations of the unit magnetization vector
(2) and the particle director (3) at f¼10 kHz in the magnetic mode,
H0¼0.5Hk, in comparison with the oscillations of the reduced magnetic
field, curve 1); (b) MD hysteresis loops at H0¼0.5Hkfor different frequen-
cies: (1) f¼100 kHz, (2) f¼300 kHz, (3) f¼500 kHz.023901-5 N. A. Usov and B. Y a. Liubimov J. Appl. Phys. 112, 023901 (2012)
Downloaded 13 Mar 2013 to 129.174.21.5. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsmagnetization vector ain the magnetic field H0in the pres-
ence of a large effective damping l¼cHk/G¼6cg/Ms/C291.
Indeed, setting c¼1.7/C2107Oe/C01s/C01,g¼0.01 g/(cm s), and
Ms¼400 emu/cm3one obtains l¼2.55/C2103. Evidently,
the effective damping only increases as a function of the
liquid viscosity.
Equation (13a) has an exact solution. In the spherical
coordinates ( h,/) with the polar axis along the vector h0, the
components of the unit magnetization vector are given bya
x¼sinhcosu,ay¼sinhsinu, and az¼cosh. For the spheri-
cal angles, one obtains from Eq. (13a) the set of equations
dh
dt¼/C0lsinhd/
dt;d/
dt¼cH0cosðxtÞ
1þl2: (14)
The integration of Eq. (14)gives
/ðtÞ¼/0þBsinðxtÞ;tanhðtÞ
2¼Cexpð/C0lBsinðxtÞÞ;
B¼xH
xð1þl2Þ: (15)
Here, xH¼cH0;/0andCare the constants of integration. It
is convenient to set C¼1. Then, at t¼pn/x,n¼0, 1,…,
when the external magnetic field is given by H0(t)/
H0¼6h0, the angle h(t)¼p/2, i. e., the vector ais perpen-
dicular to h0. According to Eq. (15), the angle hvaries
between the limits hmin¼2arctan ðexpð/C0lBÞÞ, and
hmax¼2arctan ðexpðlBÞÞ. The amplitude of these oscilla-
tions is determined by the dimensionless parameter lB/C25
xH/xl. It is clear that with increasing xthe parameter lB
decreases and the oscillations of the angle hoccur near the
value h¼p/2. Note that for a typical case H0¼100 Oe,
f¼100 kHz, and g¼0.01 g/(cm s), one obtains lB/C251.
Thus, in this case the constant B/C251/l/C281, so that the vec-
torsaandnoscillate nearly in the same plane, /(t)/C25/0.
IV. BEHAVIOR OF SUPERPARAMAGNETIC
NANOPARTICLES
The MD approximation is useful for understanding the
basic features of the dynamics of magnetic nanoparticles in
a viscous liquid. However, the thermal fluctuations have asignificant impact on the nanoparticle behavior. Neverthe-
less, in this section, we show that the MD hysteresis loops
are often reproduced in the limit of large particle diameterswhen their magnetic moments are relatively stable with
respect to thermal agitation. To analyze the behavior of
superparamagnetic nanopartic les in an alternating external
magnetic field, the complete set of the stochastic equations
(3),(7)–(9)should be considered. The solution of the set is
performed by using the well known algorithms [ 47–49]( s e e
also Appendix).
To ensure the accuracy of the simulations performed,
we use Milshtein scheme
47,48and keep the physical time
step lower than 1/50 of the characteristic particle precession
time. For every particle diameter, a time-dependent particle
magnetization M(t)¼Msa(t) is calculated in a sufficiently
large series of the numerical experiments, Nexp¼500–1000,
for the same frequency and magnetic field amplitude.Because various runs of the calculations are statistically in-
dependent, the component of the dilute assembly magnetiza-
tion along the magnetic field direction is obtained12as an
average value, hMe(t)i. Note, that the hysteresis loops shown
below correspond to a stationary regime that is achieved af-
ter sufficient number of periods of the alternating magneticfield has been elapsed. All calculations are carried out at a
room temperature, T¼300 K, for particles with the same
magnetic parameters as in Sec. III.
A. Viscous mode
Fig. 4(a) shows the evolution of the thermal hysteresis
loop as a function of the particle diameter in a typical vis-cous mode, f¼100 kHz, H
0¼100 Oe. It is seen that with
increasing particle diameter the thermal hysteresis loops
approach ultimate MD loop, because the effect of the ther-mal fluctuations on the dynamics of the unit magnetization
vector decreases. However, the MD loop is not reached even
for a rather large single-domain nanoparticle with diameter
FIG. 4. (a) The thermal hysteresis loops of an assembly in liquid in the vis-
cous mode as a function of the particle diameter: (1) D¼16 nm, (2)
D¼20 nm, (3) D¼28 nm, (4) D¼60 nm, (5) ultimate MD loop; (b) The
hysteresis loops of oriented assembly of immobilized nanoparticles with the
same magnetic parameters as a function of the diameter: (1) D¼18 nm, (2)
D¼20 nm, (3) D¼22 nm, (4) D¼24 nm.023901-6 N. A. Usov and B. Y a. Liubimov J. Appl. Phys. 112, 023901 (2012)
Downloaded 13 Mar 2013 to 129.174.21.5. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsD¼60 nm, because the thermal fluctuations of the director
remain appreciable at room temperature, T¼300 K. Indeed,
the characteristic Brownian relaxation time27,28sB¼3gV/
kBThas only power dependence on the particle diameter. It
changes slowly in comparison to the Neel–Brown relaxation
time that has an exponential dependence on this parame-ter,
10,11sN¼s0exp(K1V/kBT), where s0is a pre-exponential
constant.
One can see in Fig. 4(a) that in the viscous mode the
hysteresis loop area increases monotonically with increase in
the nanoparticle diameter. This behavior is completely dif-
ferent from that of an assembly of superparamagnetic nano-particles immobilized in a solid matrix.
5Indeed, as Fig. 4(b)
shows, for an assembly of immobilized nanoparticles with
the same magnetic parameters, the hysteresis loop area firstincreases and then decreases rapidly, because in the limit
H
0/C28Hkthe external magnetic field is unable to remagnetize
the nanoparticles of sufficiently large diameters. Thus, for anassembly of nanoparticles in a solid matrix there is a rather
narrow range of diameters (in the given case D¼20–22 nm)
where the area of the assembly hysteresis loop has a maxi-mum. It can be shown that, similarly to the MD approxima-
tion (see Fig. 2) in the viscous mode the area of the thermal
hysteresis loop decreases as a function of the frequency orliquid viscosity.
B. Magnetic mode
Fig. 5(a) shows the thermal hysteresis loops of the as-
sembly for a magnetic mode, H0¼Hk, for different particle
diameters. With increase in diameter the influence of thermal
fluctuations of the magnetic vector decreases rapidly. As a
result, the thermal hysteresis loops monotonically approachto the corresponding MD loop. However, the thermal hyster-
esis loops do not reach the ultimate MD one again, since the
thermal fluctuations of the director are still significant at aroom temperature. Due to thermal fluctuations of the director
the earlier switching of the unit magnetization vector occurs.
Thus, the coercive force of the assembly is reduced in com-parison with the ultimate MD loop. Note that the thermal
hysteresis loop shape is approximately rectangular for all
particle diameters, because in the magnetic mode the nano-particle assembly in the liquid is oriented along the magnetic
field direction.
In Fig. 5(b), we compare the thermal hysteresis loops of
an assembly in a liquid with that of oriented assembly of
nanoparticles immobilized in a solid matrix. Note that the
hysteresis loops for an assembly of nanoparticles in liquidhave a lower coercive force. The difference of the loops of
the two assemblies in Fig. 5(b) is associated with a slight
deviation of the director nof the particle in liquid at the
moments when the external magnetic field is close to the par-
ticle coercive force. Due to this deviation, which is impossi-
ble for an immobilized nanoparticle, the effective energybarrier is reduced and the unit magnetization vector jumps to
another well in the lower magnetic field, as compared to the
case of immobilized particle. Although in the magneticmode the difference of the hysteresis loops for the two
assemblies is not as striking as for the viscous mode, yet thecoercive force of the assembly in a liquid is smaller than that
for the corresponding assembly of immobilized nanopar-
ticles. As we have seen in Sec. III, the MD hysteresis loop in
the magnetic regime, H
0/C21Hk, are relatively weakly depend-
ent on the field frequency and the liquid viscosity. In the
magnetic mode, the thermal hysteresis loops show the samebehavior too.
C. The intermediate case, H0/C250.5Hk
It is interesting to consider this case separately, as in the
MD approximation the transition between the magnetization
oscillation modes occurs in the field interval
0.5Hk/C20H0/C20Hk, depending on the field frequency and the
liquid viscosity. As Fig. 6(a) shows, at a low frequency,
f¼100 kHz and H0¼300 Oe the thermal hysteresis loops
approach to the MD one with increase of the particle diame-ter. All these loops correspond to the magnetic mode. How-
ever, it is found that when frequency increases up to
f¼500 kHz, for the thermal hysteresis loops magnetic mode
is realized at lower amplitudes as compared to the MD
approximation. Indeed, as Fig. 6(b) shows, in this case the
FIG. 5. (a) The thermal hysteresis loops of an assembly in liquid in the mag-
netic mode, H0¼Hk,f¼500 kHz, as a function of the particle diameter: (1)
D¼16 nm, (2) D¼20 nm, (3) D¼24 nm, (4) D¼40 nm, (5) ultimate MD
loop; (b) The thermal hysteresis loop (1) of an assembly in liquid at
g¼0.01 g/(cm s) in comparison with that of oriented assembly (2) of the
same nanoparticles immobilized in a solid matrix.023901-7 N. A. Usov and B. Y a. Liubimov J. Appl. Phys. 112, 023901 (2012)
Downloaded 13 Mar 2013 to 129.174.21.5. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsthermal hysteresis loops are roughly rectangular and differ
considerably from the corresponding MD loop. Therefore, at
high enough frequency the presence of the thermal fluctua-tions lowers the characteristic field for the transition to the
magnetic mode.
For practical applications in hyperthermia, it is impor-
tant to provide a maximum squareness of the assembly hys-
teresis loop. Indeed, the SAR of the assembly is given by
5,9
SAR¼Af/q, where Ais the area of the assembly hysteresis
loop calculated in the variables ( M,H), and qis the density
of the magnetic material. Therefore, the hysteresis loop hav-
ing maximal squareness gives the maximum possible assem-bly SAR at a given frequency.
5,9In Ref. 9a dimensionless
ratio A/Amaxis introduced where Amax¼4MsH0is the ulti-
mate value of the hysteresis loop area. Evidently, the assem-bly hysteresis loops with sufficiently large ratios A/A
max/C251
are optimal for hyperthermia. However, the loop area is also
frequency dependent.
In the viscous regime at a low frequency for the nano-
particles of sufficiently large diameter the hysteresis loopshape is close to rectangular. However, at a low frequency,
f¼10 kHz, the SAR of the assembly is relatively small. As
we have seen in the Sec. IVA, in the viscous mode the
increase in frequency leads to a significant decrease in the
area of the thermal hysteresis loop. This implies that the vis-
cous mode is hardly optimal for an assembly of magneticnanoparticles in viscous liquid. Much higher SAR values for
this assembly can be obtained in the magnetic mode, which
also provides the high loop squareness due to the orientationof the assembly in the alternating magnetic field. However,
the magnetic field of large amplitude, H
0/C24Hk, is undesirable
from both technical and medical points of view.1–4Actually,
it is important to have appreciable SAR values at as small
magnetic fields as possible, because the magnetic field
strength decreases rapidly as a function of the distance fromthe magnetic field source. This fact can be essential for
tumors located deeply in the living body. Therefore, the in-
termediate regime, H
0/C240.5Hk, seems promising for mag-
netic hyperthermia, because in this case the hysteresis loop
can be rectangular at sufficiently high frequencies.
It is important to note that for an assembly of nanopar-
ticles in a liquid, contrary to an assembly of immobilized
nanoparticles, the increase of the nanoparticle diameters
only increases the hysteresis loop area. Accordingly, thelarge SAR values are expected for magnetic nanoparticle
assemblies with sufficiently large particle diameters. They
have to be close to the particle absolute single-domain size,when the magnetization reversal processes occurs by means
of the uniform rotation mode.
46
In Fig. 7, we show the calculated SAR values for a
dilute assembly of nanoparticles in water, g¼0.01 g/(cm s).
The particle diameter is fixed at D¼40 nm, the anisotropy
field is equal to Hk¼500 Oe, the density of particles is
assumed to be q¼5 g/cm3. In accordance with the above
arguments, Fig. 7(a) shows that in the viscous mode,
H0<0.5Hk, the SAR of the assembly is relatively small and
practically does not depend on the frequency. However, after
the transition to the magnetic mode, even at H0¼0.5Hk, the
SAR almost linearly increases with the frequency, since thehysteresis loop of the assembly in the magnetic mode is close
to rectangular. Fig. 7(b) shows the dependence of the SAR
on the magnetic field amplitude at a fixed frequency. Againone can see that at the frequency f¼500 kHz a significant
increase of the SAR occurs at H
0¼0.5Hk, after the transition
to the magnetic mode. One can see that for the given assem-bly the SAR increases more than twice in a relatively small
field interval, 200 Oe <H
0<250 Oe.
V. DISCUSSION AND CONCLUSIONS
In this paper, the dynamics of a dilute assembly of
superparamagnetic nanoparticles in a viscous liquid under
the influence of external alternating magnetic field is studied
theoretically. The results are obtained using numerical simu-lation of the stochastic equations of motion for the unit mag-
netization vector aand the director n. The latter describes
the space orientation of the particle as a whole. It is shownthat the dynamics of the particle in a liquid depends on the
amplitude of the alternating magnetic field. In the viscous
FIG. 6. (a) The thermal hysteresis loops of an assembly in liquid as a func-
tion of particle diameter at H0¼300 Oe, f¼100 kHz: (1) D¼20 nm, (2)
D¼28 nm, (3) D¼48 nm, (4) ultimate MD loop; (b) the same as in (a) but
forH0¼250 Oe, f¼500 kHz: (1) D¼20 nm, (2) D¼30 nm, (3) D¼40 nm,
(4) MD loop.023901-8 N. A. Usov and B. Y a. Liubimov J. Appl. Phys. 112, 023901 (2012)
Downloaded 13 Mar 2013 to 129.174.21.5. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsmode, H0/C28Hk, the particle generally rotates in a liquid as a
whole, the vectors aandnmove in unison, and their oscilla-
tions are shifted in phase relative to that of the magnetic fieldoscillation. Therefore, the power absorption of the assembly
is due mainly to the viscous losses in the liquid. The viscous
regime is characterized by a sharp decrease in the hysteresisloop area with increasing both the frequency and viscosity of
the liquid, because in both cases, the amplitude of the oscil-
lations of the components of vectors aandnparallel to the
field direction, is dramatically reduced. In a stationary
motion these vectors fluctuate being nearly perpendicular to
the magnetic field direction. As a result, there is a partial ori-entation of an assembly of nanoparticles in the liquid at a
sufficiently high frequency of the alternating magnetic field.
In the opposite regime, H
0/C21Hk, in a stationary motion
the director noscillates slightly near the external magnetic
field direction, whereas the unit magnetization vector a
sharply jumps between the states 6h0. Thus, a complete ori-
entation of the assembly of nanoparticles in a liquid occurs
in the alternating magnetic field of sufficient amplitude. The
hysteresis loop shape of the assembly is nearly rectangular.It has relatively weak dependence on the magnetic field fre-
quency and the liquid viscosity.Thus, the magnetic dynamics of an assembly of mag-
netic nanoparticles in a liquid differs significantly from the
behavior of the same assembly of nanoparticles immobilizedin a solid matrix. In particular, for an assembly of nanopar-
ticles in the liquid the hysteresis loop area increases monot-
onically with increasing particle diameter, since themagnetic moments of superparamagnetic nanoparticles with
larger diameters are less susceptible to the thermal fluctua-
tions. In contrast, for an assembly of nanoparticles immobi-lized in a solid matrix
5there is a narrow range of diameters,
where the hysteresis loop area has a maximum. Indeed, for
H0<Hk, the magnetization reversal for the particles of suffi-
ciently large diameter is not possible due to high value of the
effective energy barriers.
For practical applications in magnetic nanoparticle
hyperthermia, the assemblies with sufficiently large SAR are
promising.2–4,9The SAR of the assembly in a liquid can be
significantly increased by selecting a suitable mode of mag-netization oscillations. It is shown in the present paper that
for an assembly of nanoparticles in a liquid the intermediate
excitation regime, H
0/C250.5Hk, is preferable. Theoretical esti-
mate gives for this case quite large SAR values, of the order
of 1 kW/g, for an assembly with magnetic parameters typical
for iron oxides, and for moderate values of H0¼200–300 Oe
andf¼300–500 kHz.
The present study shows that the usual analysis of the ex-
perimental data3,16,17,19,23,24on the power absorption in a liquid
made in a linear approximation,39and using the assumption of
the effective relaxation time27,39is hardly adequate. Indeed,
the introduction of the effec tive relaxation time is a formal
receipt that does not take into account the complex dynamics
of magnetic nanoparticles in a viscous liquid in an alternating
external magnetic fie ld of finite amplitude.
In fact, the orientation of an assembly of magnetic nano-
particles in a liquid in a strong alternating magnetic field has
been experimentally observed in Refs. 22,25, and 26. In par-
ticular, it has been found22that in sufficiently strong mag-
netic field there is even a spatial redistribution of the
nanoparticles, so that they are self- organized into levitatingneedles elongated along the magnetic field direction. The
authors of Refs. 25and26claim that the approximate analyt-
ical expressions for the assembly hysteresis loop area and forthe SAR (Ref. 9) are in qualitative agreement with their ex-
perimental measurements. Meanwhile, the analytical esti-
mates
9are derived under the implicit assumption that there
is no rotation of the magnetic nanoparticles as a whole. On
the other hand, the experimental results22,25,26were obtained
for the assembly of nanoparticles dispersed in the liquid.Evidently, a significant difference in the behavior of the
assemblies of nanoparticles dispersed in a liquid and immo-
bilized in a solid matrix stated in the present paper has to betaken into account for a convincing interpretation of the ex-
perimental data. For a dense assembly of magnetic nanopar-
ticles, it is necessary to take into account also the effect ofstrong magnetic dipole interactions between the magnetic
nanoparticles. It has been shown experimentally
50that the
average demagnetizing field, which is determined by thedemagnetizing factor of the whole nanoparticle assembly,
has a very significant effect on the measured SAR value.
FIG. 7. SAR of a dilute assembly of uniaxial magnetic nanoparticles in a
viscous liquid: (a) as a function of frequency at different H0values and (b)
as a function of field amplitude at various frequencies.023901-9 N. A. Usov and B. Y a. Liubimov J. Appl. Phys. 112, 023901 (2012)
Downloaded 13 Mar 2013 to 129.174.21.5. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsIn conclusion, we would like to note that all calculations
in this work are carried out under the assumption that the
magnetic particle diameter is equal to its full diameter in theliquid. The rotational particle diameter in liquid is greater if
there is a non-magnetic layer at the particle surface. How-
ever, the existence of a thin non-magnetic layer only leads toa small change in the regular and random torques in Eqs. (6)
and(7). This effect can hardly significantly alter the results
obtained in this paper.
ACKNOWLEDGMENTS
Partial financial support from the Russian Foundation
for Basic Research (Grant @10-02-01394-a) is gratefully
acknowledged.
APPENDIX: SOLUTION OF THE STOCHASTIC
EQUATIONS
The procedure for solving the set of the stochastic
equations (3),(7)–(9)is as follows. First, we introduce a
dimensionless time t/C3¼tc1Htin Eqs. (3)and(9)using a
characteristic magnetic field
Ht¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H2
0þH2
kq
: (A1)
The reason for this is a high procession frequency of the unit
magnetization vector. This vector is always moving much
faster relative to the motion of the director n. Next, we
introduce the dimensionless fields in Eq. (3) setting
hef;i¼Hef;i=Ht, and hth;i¼Hth;i=Ht,(i¼x,y,z)
@~a
@t/C3¼/C0 ½ ~a;~hefþ~hth/C138/C0j½~a;½~a;~hefþ~hth/C138/C138: (A2)
For the average components of the reduced random magnetic
field, one obtains from Eq. (8)the relations
hhth;iðtÞi ¼ 0;hhth;iðt/C3Þhth;jðt/C3
1Þi ¼kc1
Htdijdðt/C3/C0t/C3
1Þ;
k¼2kBTj
jc0jMsV: (A3)
In the integration of the stochastic Eq. (A2) by the known
algorithm,47,48it is necessary to use Gaussian random
numbers
DWm;i¼ðt/C3þdt/C3
t/C3dt0hth;iðt0Þ:
The statistical properties of these numbers follow from
Eq.(A3)
hDWm;ii¼0;hDWm;iDWm;ji¼kc1
Htdt/C3dij¼r2
mdij;
where the corresponding dispersion is given byrm¼ffiffiffiffiffiffiffiffiffiffiffiffiffi
kc1
Htdt/C3s
¼2j
1þj2kBT
MsHtVdt/C3/C18/C191=2
:
Similarly, the dimensionless equation for the director is
given by
@~n
@t/C3¼2K1V
nc1Htð~a~nÞ/C16
~a/C0ð~a~nÞ~n/C17
/C01
e0½~n;~Nth/C138; (A4)
where e0¼nc1Htis the characteristic energy. It is convenient
to introduce dimensionless components of the random tor-
que, setting ~Nth;i¼Nth;i=e0. They have the following statisti-
cal properties
h~Nth;ii¼0;h~Nth;iðt/C3Þ~Nth;jðt/C3
1Þi ¼2kBT
e0dijdðt/C3/C0t/C3
1Þ:(A5)
In the integration of the stochastic Eq. (A4), another Gaus-
sian random numbers have to be used
DWn;i¼ðt/C3þdt/C3
t/C3dt0~Nth;iðt0Þ:
Their statistical properties follow from Eq. (A5)
hDWn;ii¼0;hDWn;iDWn;ji¼2kBT
e0dt/C3dij¼r2
ndij:
Here the corresponding dispersion is given by
rn¼2kBT
e0dt/C3/C18/C191=2
:
To integrate the stochastic Eqs. (A2) and(A4), a small incre-
ment of the dimensionless time, dt*¼10/C02–1 0/C03, has been
used12in order to keep the physical time step sufficiently
small in comparison with the characteristic particle preces-
sion time.
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1.4906599.pdf | Reducing the switching current with a Gilbert damping constant
in nanomagnets with perpendicular anisotropy
Keisuke Y amada,a)Kiyoaki Oomaru, Satoshi Nakamura, Tomonori Sato,
and Y oshinobu Nakatani
Graduate School of Informatics and Engineering, University of Electro-Communications, Chofu,
Tokyo 182-8585, Japan
(Received 27 October 2014; accepted 14 January 2015; published online 28 January 2015)
We report on current-induced magnetization switching in a nanomagnet with perpendicular anisot-
ropy, and investigate the effects of the damping constant ( a) on the switching current ( Isw) by vary-
ing the nanosecond-scale pulse current duration ( tp), the saturation magnetization ( Ms), and the
magnetocrystalline anisotropy ( Ku). The results show that reduction of abelow a certain threshold
(ac) is ineffective in reducing Iswfor short tp. When tpis short, it is necessary to reduce both aand
Mssimultaneously until acis reached to reduce Isw. The results presented here offer a promising
route for the design of ultrafast information storage and logic devices using current-induced mag-netization switching.
VC2015 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4906599 ]
Current-induced magnetization switching in nanomag-
nets via spin-transfer torque has attracted attention as a novel
technique for memory development.1,2This technique can be
used to develop non-volatile memory with both high-speed
magnetization switching and low power consumption, e.g.,spin-torque magnetoresistive random access memory (spin-
RAM). There have been many theoretical reports on magnet-
ization switching
3,4and experimental studies of in-plane5–7
and out-of-plane8–15magnetization magnets. To realize
high-density spin-RAM, a low-magnetization switching cur-
rent is required for the writing process, and a high thermal
stability factor ( D) is required to retain the information re-
cord. Magnetic materials with perpendicular magnetic ani-
sotropy can meet the above requirements.8–15These
materials can reduce the switching current ( Isw) and sustain
the magnetization direction at room temperature, even for
device sizes of less than a few tens of nanometers.8–10In a
practical spin-RAM device with size of several tens of nano-
meters, currents of <100lA are required, and the main
memory and cache memory must be accessed within time-
frames of <10 ns and <1 ns, respectively.16,17
Theoretically, the switching current for spin-transfer tor-
que reversal of the out-of-plane magnetization using a cur-
rent of infinite duration (DC current) in a thin-film geometry
is given by5,7,8,10
Isw¼2eMsV
lBgP/C1acHeff
k; (1)
where e,Ms,V,lB,g,P,a,c, and Hkeffare the electron
charge, saturation magnetization, magnet volume, Bohr mag-
neton, g-factor, spin polarization, Gilbert damping constant,
gyromagnetic ratio, and effective uniaxial anisotropy field,
respectively. The thermal stability factor Dis given by
D¼DE
kBT¼Keff
uV
kBT¼MsHeff
kV
2kBT; (2)where DE,kB,T, and Kueffare the energy barrier for magnet-
ization switching, the Boltzmann constant, temperature, and
the effective magnetocrystalline anisotropy, respectively.From Eqs. (1)and(2),I
swis proportional to a/C1Ms/C1Hkeff,8,10
while the thermal stability factor is also proportional to
Ms/C1Hkeff.8It may thus be possible to use a low Gilbert damp-
ing constant to reduce Iswwithout changing D.18In initial
studies, Iswwas reported to decrease in proportion to a;19,20
however, this effect of aonIswwas reported for DC currents.
There have also been studies using currents with finite dura-tion,
3,7,11,13–15which reported that Iswis inversely propor-
tional to the pulse duration ( tp) and that the magnetization
switching is affected by temperature during the current pulse.However, these studies did not discuss the I
swdependence
onain detail. Thus, the effect of aonIswwith varying tp,
Ms, and Hkeffis not well understood.
In this work, we simulated spin-current-induced mag-
netization switching in a perpendicularly magnetized nano-magnet using the macrospin model, which is the simplestmodel available that is comparable with the analyticalmodel. We investigated the effect of aonI
swwith varying
tp,Msand magnetocrystalline anisotropy Ku. The values of
KueffandKuare different in the thin-film geometry because
the demagnetization field in Kueffinduced by the shape
anisotropy must be considered. Because the effect of shapeanisotropy is lacking from the macrospin model, thedemagnetization field is not generated. Therefore, weassume that K
u¼Kueffand Hk¼Hkeffin the macrospin
model. We varied the value of Kubecause Kueff(¼Ku)i s
proportional to Hkeff(¼Hk), as shown in Eq. (2).T h e
results show that Iswhas a threshold switching current
(Iswc) that is dependent on tp,a n d Iswcappeared when awas
below a specific threshold ( ac). We also showed that both a
and Msshould be reduced simultaneously until acis
reached to reduce Isw.Iswcalso showed a small dependence
onKu; however, it was not simply proportional to the value
ofKu. To understand these results, we derived an empirical
formula based on the analytical equation for the DC switch-ing current (Eq. (1)).
a)E-mail: yamada@gilbert.cs.uec.ac.jp
0003-6951/2015/106(4)/042402/5/$30.00 VC2015 AIP Publishing LLC 106, 042402-1APPLIED PHYSICS LETTERS 106, 042402 (2015)
A macrospin model was used in the simulation.3,4,21
Magnetization motion was ca lculated using the Landau–
Lifshitz–Gilbert equation with a spin-transfer torque term:1,3,4
d~m
dt¼/C0 j cj~m/C2~Heff/C0/C1
þa~m/C2d~m
dt/C0lBgPI
2eMsV~m/C2~m/C2~nS ðÞ :
(3)
Here, ~m,~Heff,I, and ~nSare the magnetic moment, effective
magnetic field ( ~Heff¼Hk), current, and unit vector of the
spin-transfer torque, respectively. The material parametersused for the simulation were similar to those of
CoFeB.
12,13,22,23Specifically, we used Ms¼600 emu/cm3,
Ku¼1.76/C2106erg/cm3, and V¼11.223nm3.aranged from
1 to 0.0001, which matches the range of the experimentally
measured values of a.22,23The value of Kuwas estimated
from the thermal stability factor D¼60 (T¼300 K) and an
easy axis of magnetization existed along the z-axis. The
value of Kuwas similar that for CoFeB.23The volume Vhad
the same volume as a free layer with diameter of 30 nm and
thickness of 2 nm. These size parameters enable realization
of a device with a capacity of 10 Gbit/cm2.16,17The current
with spin polarization P¼1.0 flowed in the z-axis direction.
A pulse current with a square waveform (rise and fall time of
0 ns) was used.
For the calculations, the initial magnetization angle was
tilted by hinit¼7.40/C14(0.129 rad) with respect to the z-axis, as
shown in the inset of Fig. 1. The angle hinitwas estimated
fromDas follows:24
hinit¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kBT
MsHeff
ks
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kBT
MsH?þHk/C0/C1=2s
¼ffiffiffiffiffiffiffiffiffiffiffi
2kBT
MsHkr
¼ffiffiffiffi
1
Dr
;
(4)
where H?andHkare the out-of-plane and in-plane effective
anisotropy fields, respectively. Specifically, H?¼Hkand
Hk¼0. The initial angle was the average tilt angle owing to
thermal fluctuation of the magnetization at room temperature
(T¼300 K), and was consistent with hinitas obtained by
Taniguchi.25The critical angle of magnetization switching
angle was tilted by hcrit¼172.6/C14(¼180/C14/C0hinit) with respectto the z-axis. This angle was also determined from the aver-
age tilt angle owing to thermal fluctuation. Magnetizationswitching began when the magnetization angle was greaterthan h
crituntil the current pulse was cut off. The Oersted field
generated by the current was ignored. No thermally excited
magnetization process was introduced.
Figure 1(a)shows simulation results of the effect of aon
Iswfor various pulse durations ( tp¼1, 10, and 100 ns). Iswis
proportional to afrom a¼1 until a/C240.003, when
tp¼100 ns. In addition, Iswis also proportional to afrom
a¼1 until a/C240.1 when tp¼1 ns. However, Iswdoes not
decrease below a/C240.01 for tp¼1 ns. These results indicate
thatIswhas a saturation value regardless of a, which is less
than the value for a/C240.01. The saturation current value is
defined as the threshold switching current ( Iswc).
Because Iswcchanges with tpanda, the lower limit of a,
which is effective for Iswreduction, is examined. The avalue
where Iswis close to Iswc(i.e., where Isw¼1.025 /C2Iswc)i s
defined as the threshold, ac. Figure 1(b) shows that both Iswc
andacare inversely proportional to tp(Iswcandac/1/tp).
Here, Iswcandacare 80.18 lA and 0.011, respectively, for
tp¼1 ns, indicating that Iswdoes not decrease when ais less
than/C240.01. These results show that reducing abelow a spe-
cific threshold is ineffective in reducing Iswwhen tp¼1 ns.
At present, magnetization switching is required with
tp<1 ns for high-speed switching, and materials with
a/C240.01 are commonly used.14,15From this perspective, our
results show that it is unnecessary to study avalues below ac
forIswreduction.
Because it is difficult to derive an analytical equation
forIswthat solves Eq. (3)directly, we compared the simula-
tion results and equations by solving Eq. (3)using two
assumptions, which led to an empirical equation.
We first assumed cylindrical symmetry, writing Eq. (3)
in spherical coordinates as follows:14
1
sinhdh
dt¼acHkcoshþlBgPI
2eMsV: (5)
Equation (5)can be rewritten using separation of variables.
Integrating Eq. (5)with respect to time (angle) in the interval
from 0 to tp(or from hinittohcrit), we find:
FIG. 1. (a) Plot of Iswas function of awith tpof 1 ns (squares), 10 ns (dots), and 100 ns (triangles). The solid black and green lines are the results for Eqs. (1)
and(7)fortp¼1 ns, respectively. The dotted lines show the results for Eq. (8)for each tp. Inset: Spherical coordinate system for the macroscopic spin ~m. The
initial and critical angles of the magnetization are tilted by hinitandhcritwith respect to the z-axis, respectively. The spin current pulse flows in the z-axis direc-
tion. (b) Variation of Iswcandacwith tp, showing that Iswcandacare inversely proportional to tp(Iswc,ac/1/tp).042402-2 Y amada et al. Appl. Phys. Lett. 106, 042402 (2015)1
tp¼2A2/C0B2 ðÞ
A/C1C/C0B/C1D/C02/C1A/C1ln BþA/C1EÞ ð: (6)
A¼acHk;
B¼lBgPI=ð2eMsVÞ;
C¼lnð1/C0coshcritÞ/C1lnð1þcoshcritÞ
/C0lnð1/C0coshinitÞ/C1lnð1þcoshinitÞ;
D¼lnð1/C0coshcritÞ=lnð1þcoshcritÞ
/C0lnð1/C0coshinitÞ=lnð1þcoshinitÞ;
E¼coshcrit/C0coshinit:
We then solve Eq. (6)based on the assumptions that the
current pulse has infinite duration, and ahas a limit of 0.
For infinite tp, the left side of Eq. (6)is 0, and thus, the
right side must also be 0. From these results, B ¼6Ai s
obtained, and B ¼A is equivalent to Eq. (1), where Kueffis
replaced by Ku. Note that we do not consider B ¼/C0A,
because this expression means that the current is applied in
the opposite direction, which is inconsistent with our simula-
tion conditions. This result is shown in Fig. 1(a) as a black
line. These results are also consistent with the simulation
results for tp¼1 ns when aranges from 1 to /C240.1.
When ais sufficiently close to 0, A ¼0 in Eq. (6)and
B¼D/(2tp). Then, Iswcan be expressed by
Isw¼2eMsV
lBgPD=2
tp !
¼2eMsV
lBgPC1
tp/C18/C19
: (7)
Under our simulation conditions, C1is 5.48. From
Eq.(7),Iswis inversely proportional to tp, i.e., Iswincreases
astpdecreases. Iswis shown for tp¼1 ns in Fig. 1(a) as the
green line. Also, when tp¼1 ns, Iswis 80.11 lA, which is
almost identical to Iswc¼80.18 lA, which was obtained
from the simulation results for tp¼1 ns.
After comparing the simulation results with the assump-
tions above, we derive the following empirical equation by
adding Eqs. (1)and(7):
Isw¼2eMsV
lBgPacHkþC1
tp/C18/C19
: (8)
Equation (8)is indicated by dotted lines for each tpin Fig.
1(a). The results for Eq. (8)largely agree with the simulation
results; however, Iswfrom the simulation results and fromEq.(8)fortp¼1 ns do not agree because Iswis centered
around a/C240.053, i.e., at the intersection of Eqs. (1)and(7).
This difference occurs because Eq. (7)is assumed to exist in
a region where ais sufficiently low. Consequently, the Isw
value obtained from Eq. (8)is almost consistent with the
simulation results. In contrast, the assumption of Eq. (7)no
longer holds for increasing a, and the results derived from
Eq. (8)and from the simulations deviate accordingly.
Because the first term of Eq. (8)increases with increasing a,
the deviation becomes small enough to be ignored, and the
results of Eq. (8)agree with those of the simulation for larger
values of a.
As described earlier, Iswis proportional to a/C1Ms/C1Hkwhen
an infinite current pulse is applied;8,10therefore, Iswalways
decreases with decreasing MsorHk(/Ku) with an infinite
current pulse. MsandHkcan vary because the order parame-
ters and the alloy composition ratios can be varied.26These
material parameters can also be varied by varying the thick-
ness of the nonmagnetic layers adjacent to the magnetic
layer.23,26Also, attempts have been made to reduce Iswby
reducing Ms27andD(/Ku/Hk)28,29in experiments. To
discuss these phenomena when using a finite pulse current,
we investigated the effects of MsandKuonIswfortp¼1 ns.
Figure 2(a) shows Iswas a function of aand Msat
tp¼1 ns and Ku¼1.76/C2106erg/cm3, while Fig. 2(b) shows
Iswcandacas functions of Ms; these results were obtained in
the same manner as Fig. 1(b). From Fig. 2(a),Iswis almost
constant regardless of the value of Mswhen aranges from
a¼1 until a/C240.3; in contrast, Iswvaries with the value of
Mswhen ais below /C240.3. Also, Iswis equal to Iswcwhen ais
below ac, and Iswcincreases with increasing Ms. These
results show that Msandashould be reduced simultaneously
when aranges from a/C240.3 to acto reduce Isw.
Figure 2(b) shows the proportional dependence of Mson
Iswcandac(i.e., Iswc,ac/Ms) obtained from Fig. 2(a). These
results can be explained using Eq. (8). The first term of Eq.
(8)is not dependent on Msbecause Hkcan be replaced in Eq.
(1)byHk¼2Ku/Msin the macrospin model. However, Iswc
is proportional to Mswhen ais sufficiently low. The first
term can therefore be ignored because the second term
includes Ms. This also explains why acis proportional to Ms.
Iswcand acfor high-speed switching are large when
compared with the corresponding values for slow-speed
switching. For high-speed switching, Iswccan be reduced by
reducing Ms, and this reduction in Mscan lead to a reduction
FIG. 2. (a) Plot of Iswas a function of a
with Msof 300 emu/cm3(red), 600 emu/
cm3(blue), and 1200 emu/cm3(black)
attp¼1n sa n d Ku¼1.76/C2106erg/cm3.
The dotted lines show the results for Eq.
(8)for each Ms.( b )V a r i a t i o no f Iswc
andacwith Msattp¼1 ns, showing that
Iswcandacare proportional to Ms(Iswc,
ac/Ms).042402-3 Y amada et al. Appl. Phys. Lett. 106, 042402 (2015)ofac, indicating that the lower limit of ais also reduced.
Therefore, reduction of both Msand aare important for
high-speed switching.
Figure 3(a) shows Iswas a function of aand Kufor
tp¼1 ns and Ms¼600 emu/cm3, and the Kudependence on
Iswcandacis shown in Fig. 3(b). Here, the thermal stability
factor ( D/Ku)f o r Ku¼0.88, 1.76, and 3.52 Merg/cm3is
D¼30, 60, and 120, respectively. In this calculation, Dis
varied; therefore, we change the initial angle obtained fromEq.(4)for each D. Figure 3(a) shows that I
swis proportional
to both aandKuwhen aranges from 1 to /C240.1. In contrast,
Iswdoes not decrease with decreasing awhen ais below
/C240.01. Note that Iswcdepends only slightly on Ku:Iswcis
increased by /C2410% when Kuis doubled. Because the tilt
angle due to thermal fluctuation changes when Kuis varied,
the angle required for magnetization switching also changes
(see Eqs. (2)and(4)). From these results, Iswcis dependent
on the value of Ku; specifically, Iswcis proportional to the
value of Ku, but it changes very little with respect to the vari-
ation of Ku. These results show that Iswcan be reduced by
reducing both aandKuwhen aranges from 1 to /C240.1. Also,
Iswis nearly constant and is independent of Dwhen a<ac.
We found that acis inversely proportional to Ku(ac/
1/Ku), as shown in Fig. 3(b). This result can be explained
using Eq. (8). Because Kuis included in the first term, Iswis
proportional to Kuwhen aranges from 1 to /C240.1, and Iswis
primarily governed by the first term of Eq. (8).
In this letter, we have investigated nanosecond current-
induced magnetization switching in a nanomagnet withperpendicular magnetic anisotropy via macrospin model sim-ulations. We demonstrated the effect of aonI
swby varying
tp,Ms, and Ku. First, we studied the effect of aandtponIsw.
Fortp¼100 ns, Iswdecreased with decreasing a, whereas Isw
saturated with decreasing afortp¼1 ns. These results
showed that Iswhad a threshold switching current ( Iswc) for
tp¼1 ns when a/H113510.01, i.e., reducing abelow a certain
threshold was ineffective in reducing Iswfor high-speed
switching ( tp¼1 ns). Using these results, we showed the va-
lidity of our simple experimental equation for Isw, which was
obtained by adding a term that was inversely proportional tot
p, and which was determined by the magnetization switch-
ing angle. Then, we investigated the effects of aandMson
Isw. When aranged from 1 to /C240.3,Iswwas independent of
the value of Ms; in contrast, Iswwas not proportional to the
value of Mswhen a/H113510.3, but Iswcwas proportional to thevalue of Ms. This result showed that simultaneous reduction
of both aandMswas required when aranged from /C240.3 to
acto reduce Isw. Finally, we showed the effects of aandKu
onIswto confirm the thermal stability factor ( D/Ku). When
aranged from 1 to /C240.1,Iswwas dependent on the value of
Ku. In contrast, Iswcwas only slightly dependent on the value
ofKuwhen a/H113510.01, i.e., Iswwas nearly constant and inde-
pendent of Dwhen a<ac.
This effective method for reduction of Iswfor high-
speed switching ( tp¼1 ns) is achieved by using small values
ofMs, and by reducing auntil acis reached. Iswccan be
reduced by 50% when the value of Msis similarly reduced.
For example, because the actual aof CoFeB is close to
0.01,23Iswis 43.4 lA for high-speed switching when Msis
300 emu/cm3. This Isw(<100lA) is quite small for practical
implementation in spin-RAM.16,17
The practical spin-RAM will have a relatively small di-
ameter and will be used to increase the memory capacity. Asmall device is necessary to achieve a high K
uto maintain
thermal stability. However, it can be adapted to the macro-
spin model and can also yield Iswvia Eq. (8), because the
magnetization switching mechanism does not change.
Nevertheless, the following effects on current-induced mag-
netization switching must be considered for any practicalspin-RAM size: (a) the effects of shape anisotropy and the
non-uniformity of the magnetic structure; (b) the effects of
the rise and fall times of the current pulse; and (c) thermalfluctuations of magnetization during application of the cur-rent pulse. The effects of (a) are relatively small when a
small device is used. The effect of the current pulse shape of
(b) may change I
swbecause the switching time required for
magnetization switching is varied. Further theoretical study
is required for the case of the effects of thermal fluctuations
in (c). Because the magnetization switching is either assistedor inhibited by the thermal fluctuations, it is necessary to
consider the switching probability. Therefore, it is necessary
to perform similar simulations when taking thermal fluctua-tions into account, and thereafter derive an equation for I
sw
that takes the switching probability into account.
This study was supported by the New Energy and
Industrial Technology Development Organization (NEDO).
The authors would like to thank Dr. T. Taniguchi for fruitfuldiscussions. K.Y. is supported by JSPS Postdoctoral
Fellowships for Research.FIG. 3. (a) Plot of Iswas a function of
afor Kuof 0.88 Merg/cm3(red),
1.76 Merg/cm3(blue), and 3.52 Merg/
cm3(black) at tp¼1 ns and
Ms¼600 emu/cm3. The dotted lines
show the results for Eq. (8)for each
Ku. (b) Variation of IswcandacforKu,
showing that Iswcis proportional to Ms
(Iswc/Ku) and that acis inversely pro-
portional to Ku(ac/1/Ku).042402-4 Y amada et al. Appl. Phys. Lett. 106, 042402 (2015)1L. Berger, Phys. Rev. B 54, 9353 (1996).
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1.4768446.pdf | Magnetic vortex echoes
F. Garcia, J. P. Sinnecker, E. R. P. Novais, and A. P. Guimarães
Citation: Journal of Applied Physics 112, 113911 (2012); doi: 10.1063/1.4768446
View online: http://dx.doi.org/10.1063/1.4768446
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/112/11?ver=pdfcov
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130.160.4.77 On: Fri, 19 Dec 2014 17:10:40Magnetic vortex echoes
F . Garcia,1J. P . Sinnecker,2E. R. P . Novais,2and A. P . Guimar ~aes2,a)
1Laborat /C19orio Nacional de Luz S /C19ıncrotron, 13083-970 Campinas, SP, Brazil
2Centro Brasileiro de Pesquisas F /C19ısicas, 22290-180 Rio de Janeiro, RJ, Brazil
(Received 21 August 2012; accepted 6 November 2012; published online 7 December 2012)
The dynamic properties of magnetic vortices have many potential applications in fast magnetic
devices. Here we present a micromagnetic study of the motion of magnetic vortices in arrays of 100nanodisks that have a normal distribution of diameters, as expected in real array systems, e.g.,
produced by nanolithography. The micromagnetic simulated experiments follow a protocol with an
initial preparation and magnetic pulses that enable the control of the magnetic vortices initialpositions and circular motion direction. The results show a new effect—the magnetic vortex echo
(MVE) that arises from the refocusing of the overall array magnetization. We show, by using arrays
with different interdisk separations, that MVE affords a means of characterizing them as regards thehomogeneity and intensity of the interaction between its elements, properties that are relevant for
device applications. We also show that a simple analytical model, analogous to the one that describes
the spin echo in magnetic resonance, can be used to explain most features of the simulated magneticvortex echo.
VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4768446 ]
I. INTRODUCTION
Many magnetic nanoobjects have as ground state a mag-
netic vortex configuration, i.e., a pattern of magnetization
tangential to concentric circles with a singularity at the cen-ter, where the magnetization points out of plane, the vortex
core.
1–3When a core is excited, i.e., displaced from its equi-
librium, it performs a spiral-like motion back to the originalposition, with a well defined eigenfrequency of several hun-
dred MHz.
4Because of these dynamic properties, the vorti-
ces have many potential applications.1,4–8Usually, in these
applications, one desires high speed dynamics and high den-
sities; therefore, the vortices should be organized in as com-
pact as possible arrays, and the optimization of theperformance of the device requires an adequate physical
characterization of their dynamic properties. The magnetic
vortices have two main features: one is the sense of magnet-ization curling, i.e., its circulation, which can be counter-
clockwise (c ¼1) or clockwise (c ¼/C01); and, the second
one, the core polarity (p), being p ¼þ1(/C01) for upward
(downward) magnetization direction of the vortex core. The
vortex core translation eigenfrequency (usually called gyro-
tropic frequency) is closely related to the geometry ofthe nanoobject and, e.g., for a thin nanodisk, is given by
x
G/C25ð20=9ÞcMsb(Msis the saturation magnetization, cis
the Gilbert gyromagnetic ratio, and b¼h=Ris the aspect ra-
tio).2The sense of the gyrotropic core motion (or the sign
of the gyrotropic frequency) is determined by the core polarity
and, for an upward (downwar d) core magnetization, p ¼þ1
(/C01), the core will precess in the c ounter-clockwise (clockwise)
direction. Therefore, by controlling the vortex polarity, it is pos-
sible to control the sense of gyrotropic vortex core motion.
Until recently most studies neglected any dipolar cou-
pling between nanoobjetcs with vortices, since they present a
magnetic flux closure in the relaxed form. However, anout-of-equilibrium core generates sufficient magnetostatic
energy to dynamically couple neighbor vortices, as demon-
strated in some very recent studies.8–16Particularly interest-
ing is the fact that it is possible to transfer energy, with
negligible loss, between two neighbor vortices by stimulated
gyrotropic motion.9This dynamic coupling is strongly de-
pendent on the distance dbetween the centers of the vortices.
This has been shown by Vogel and co-workers,9using ferro-
magnetic resonance (FMR), who obtained for a 4 /C2300
array a dependence of the form d/C0n, with n¼6. The same
was found by Sugimoto et al.8using a pair of disks excited
with rf current. On the other hand, Jung et al. ,11studying a
pair of nanodisks with time-resolved X-ray spectroscopy,
found n¼3:9160:07. Likewise Sukhostavets et al.,12also
for a pair of disks, in this case studied by micromagneticsimulation, obtained n¼3.2 and 3.6 for the xandyinterac-
tion terms, respectively.
Most works deal with idealized systems containing one,
two or no more than few array elements, and effects such as
magnetic vortex coupling, inhomogeneities, magnetic stabil-
ity in large arrays of nanostructures, among others, havebeen neglected so far. The question of how to characterize
the dynamic properties of large area arrays of magnetic vorti-
ces has thus very important implications.
In the present work we are proposing a new phenomenon,
the magnetic vortex echo (MVE), and have developed an ana-
lytical model that describes its main features. This analyticaldescription is analogous to that used for the spin echo
observed in nuclear magnetic resonance (NMR), essential in
applications such as magnetic resonance imaging (MRI).
17
We present a micromagnetic study of the motion of
magnetic vortices in arrays of 100 nanodisks that have a nor-
mal distribution of diameters, as expected in real array sys-tems, e.g., produced by nanolithography. The results show
the magnetic vortex echo effect arising from the collective
magnetic vortex cores motion which leads to refocusing ofthe overall array magnetization, as shown in Fig. 1. Using
a)Author to whom correspondence should be addressed:apguima@cbpf.br.
0021-8979/2012/112(11)/113911/5/$30.00 VC2012 American Institute of Physics 112, 113911-1JOURNAL OF APPLIED PHYSICS 112, 113911 (2012)
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
130.160.4.77 On: Fri, 19 Dec 2014 17:10:40large arrays with different interdisk separations, MVE
affords a means of characterizing large arrays as regards the
homogeneity and intensity of the interaction between the
array elements, properties that are relevant for deviceapplications.
II. THE MODEL
The formulation of the model begins considering an infi-
nite array of magnetic nanoelements with distance dbetween
their centers. Let us now consider that their vortex cores per-
form gyrotropic motion, after being excited by the action ofsome external perturbation, e.g., an in-plane magnetic field
applied along the ydirection, which has displaced all cores
along the xaxis, increasing the overall M
ymagnetization. As
in a real vortex array, we assume that the disks do not have
exactly the same gyrotropic frequencies, arising, for
instance, from their size distribution.
To derive the time dependence of the array magnetiza-
tion we will assume first that the gyrotropic frequencies vary
continuously and have a Gaussian distribution PðxÞwith
standard deviation Dx. Second, we will also assume that the
circulation and polarity are initially the same for every vor-
tex:c¼þ1 and p¼þ1. This is not an issue, as will be clear
in Sec. III; however, this configuration can be easily
achieved by proper procedures (Antos et al.18and the refer-
ences therein).
After the vortices are displaced at t¼0, they will relax
toward the equilibrium position in a spiral-like gyrotropic
motion, with different frequencies x, generating an oscilla-
tory behavior of both in-plane magnetization components
(MxðtÞandMyðtÞ). After a given elapsed time, since we are
considering a distribution of gyrotropic frequencies, thecores will be completely out of phase, and as consequence,
the overall array magnetization will be reduced and eventu-
ally will be damped to zero. Using the approach employed inthe description of magnetic resonance (e.g., see (Refs. 19
and 20)), one can derive the array ycomponent of the
magnetization,
M
yðtÞ¼Myð0Þð1
/C01e/C01
2ðx/C0x0Þ2
Dx2
Dxffiffiffiffiffi ffi
2pp cosðxtÞdx; (1)an integral that is the Fourier transform of the gyrotropic fre-
quency distribution PðxÞ.21One is now able to express MyðtÞ
as a function of an important relaxation time T/C3
2¼1=Dx,
MyðtÞ¼Myð0Þe/C01
2t2
T/C32
2cosðx0tÞ: (2)
The same reasoning can be applied to MxðtÞ. This result
shows that the total magnetization tends to zero, as the dif-
ferent contributions of both MyðtÞand MxðtÞget gradually
out of phase. This damping, with a characteristic time T/C3
2,i s
analogous to the free induction decay (FID) in NMR.
After an elapsed time t, the angle rotated by each vortex
core will be xt;i fa t t¼swe invert the polarities of the vor-
tices in the array, e.g., using an appropriate magnetic pulse,
the motion of the cores will change direction (i.e.,
x!/C0 x). Therefore, for t>sone obtains
Myðt/C0sÞ¼Myð0Þð1
/C01e/C01
2ðx/C0x0Þ2
Dx2
Dxffiffiffiffiffi ffi
2pp cos½xðs/C0tÞ/C138dx:(3)
Theycomponent of the magnetization becomes
MyðtÞ¼Myð0Þe/C01
2ðt/C02sÞ2
T/C32
2cosðx0tÞ: (4)
Equation (4)means that MyðtÞ(and MxðtÞ) increases for
s<t<2s, reaching a maximum at a time t¼2s: this maxi-
mum is the magnetic vortex echo, analogous to the spin echoobserved in magnetic resonance (Fig. 1). In the case of the
NMR spin echo, the maximum is due to the refocusing of the
in-plane components of the nuclear magnetization.
Up to now we have only considered a frequency distri-
bution arising from geometric inhomogeneities.
22,25In a real
vortex array other irreversible processes should also be con-sidered, and the decrease of M
yðtÞand MxðtÞcomponents
will also be affected by these additional processes that we
define to be characterized by a relaxation time T2. Consider-
ing this, MyðtÞwill be
MyðtÞ¼Myð0Þe/C01
2ðt/C02sÞ2
T/C32
2e/C0t/C0s
T2cosðx0tÞ: (5)
T2is a relaxation time analogous to the spin transverse
relaxation time (or spin-spin relaxation time) T2in magnetic
resonance: now 1 =T/C3
2¼Dxþ1=T2.T2can be measured by
determining the decay of the echo amplitude for different val-
ues of the interval s. The processes contributing to T2are: (a)
the interaction between the nanoelements, which, in the first
approximation, amounts to random magnetic fields that will
increase or decrease xof a given element, producing a fre-
quency spread of width Dx0¼1=T0
2and (b) the loss in mag-
netization (of rate 1 =TaÞarising from the energy dissipation
related to the Gilbert damping constant athat appears in the
Landau-Lifshitz-Gilbert1equation. Identifying Tato the NMR
longitudinal relaxation time T1, one has191=T2¼1=T0
2
þ1=2Ta. Therefore the relaxation rate 1 =T/C3
2is given by
1
T/C3
2¼Dxþ1
T2¼Dxþ1
T0
2þ1
2Ta: (6)
FIG. 1. Formation of magnetic vortex echoes: superposition of the individ-
ual simulated disks of the 10 /C210 matrix at different instants. (a) Top view
of the disk at t¼0, (b) top view at t¼s/C0/C15(before the inverting magnetic
pulse), (c) bottom view at t¼sþ/C15(after the inverting magnetic pulse), and
(d) bottom view at t¼2s(at the moment of the vortices refocalization). All
disks initially with same circulation c¼þ1 and polarity p¼þ1. (enhanced
online) [URL: http://dx.doi.org/10.1063/1.4768446.1 ].113911-2 Garcia et al. J. Appl. Phys. 112, 113911 (2012)
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130.160.4.77 On: Fri, 19 Dec 2014 17:10:40The vortex echo maximum at t¼2s, from Eq. (5),i s
Myð2sÞ/expð/C0s=T2Þ; one should note that the maximum
magnetization recovered at a time 2 sdecreases exponentially
with T2, i.e., this maximum is only affected by the homoge-
neous part of the total decay rate given by Eq. (6). In other
words, the vortex echo cancels the loss in MyðtÞdue to the
inhomogeneity Dx, but it does not cancel the decrease in
MyðtÞdue to the interaction between the nanoelements (the
homogeneous relaxation term 1 =T0
2), or due to the energy
dissipation (term 1 =2Ta).
Note also that if one attempted to estimate the inhomo-
geneity of an array of nanoelements using another method,e.g., measuring the linewidth of a FMR spectrum, one would
have the contribution of this inhomogeneity together with
the other terms that appear in Eq. (6), arising from interac-
tion between the elements and from the damping. On the
other hand, measuring the vortex echo it would be possible
to separate the intrinsic inhomogeneity from these contribu-tions, since T
2can be measured separately, independently of
the term Dx.
III. MICROMAGNETIC SIMULATION
In order to confirm the validity of the MVE model, we
have performed micromagnetic simulations of an assemblyof 100 magnetic nanodisks employing the
OOMMF code.23
The simulated system was a square array of 10 /C210 Permal-
loy disks, thickness 20 nm, with distance dfrom center to
center. This distance was varied from d¼350 nm up to 1,
in which case the simulations were made on disks one at a
time, adding the individual magnetic moments liðtÞ.T o
account for the inhomogeneities expected in a real vortex
array, we have introduced a Gaussian distribution of diame-
ters, centered on 250 nm with standard deviation r¼10 nm
and r¼20 nm ;r¼10 nm corresponds to Dx/C251:5
/C2108s/C01. The disks were placed at random on a square lat-
tice. We have also used different values of a.
The simulation initial state was prepared by applying a
perpendicular magnetic field pulse of Bz¼þ300 mT to set
all disks to the polarity p¼þ1, followed by an in-plane field
of 25 mT along the ydirection in order to displace all the
vortex cores from the equilibrium positions. The system was
then allowed to precess freely until t¼s, when the vortex
polarities were inverted by the action of a Gaussian magnetic
pulse of amplitude Bz¼/C0300 mT, with width 100 ps.
Simulations were performed either with random circula-
tion or with c¼þ1; the result is that the value of cis irrele-
vant, as we can verify by comparing Figs. 2(a) and2(b). For
disks having different circulations ( c¼61), the cores will
be displaced in opposite directions, but the MVE will be the
same, since all the magnetizations will point along the samedirection. On the other hand, in a configuration where the po-
larity of the disks is initially random (i.e., p¼61) the
p¼/C01 disks would not invert their polarities under the influ-
ence of the negative B
zfield pulse at t¼s, therefore they
would not contribute to the echoes, and the echo amplitude
would be reduced. However, since the preparation of the sys-tem involves an initial positive B
zpulse, all disks will have
initially the same polarity ( p¼þ1), as assumed in Sec. II.We have chosen to present the simulations performed
preparing all disks with same circulation ( c¼þ1) and polar-
ity (p¼þ1), without loss of generality.
As expected from the model, the array simulated overall
in-plane magnetization is markedly damped as a result of thedefocusing from the initial state, showing a clear FID with a
characteristic time T
/C3
2. Moreover, the micromagnetic simula-
tions have also confirmed the occurrence of the echoes at theexpected times ( t¼2s). For different values of r, the T
/C3
2
time, and consequently the duration of the FID and the width
of the echo are modified (Figs. 2(b) and2(d)); increasing a
results in a faster decay of the echo intensity (Figs. 2(b) and
2(e)). We have also obtained multiple echoes, by exciting
the system with two pulses (Fig. 2(c)).24
Figure 3shows the dependence of 1 =T2onafor
r¼10 nm; essentially the same result is obtained for
r¼20 nm, since T2does not depend on Dx(Eq.(6)). Taking a
linear approximation, 1 =T2/C25Aa, and since for d¼1 there is
no interaction between the disks, 1 =T2¼1=2Ta, and therefore
1
Ta¼2Aa: (7)
From the least squares fit (Fig. 3),A¼1:6/C21010s/C01. This
relation can be used to determine experimentally a, meas-
uring T2with vortex echoes, for an array of well-separated
disks. Note that we can only obtain directly the TaandDx
contributions to T/C3
2using the echoes.FIG. 2. Magnetic vortex echoes: simulations (black line) for 100 nanodisks,
with d¼1 (a)r¼10 nm ;s¼30 ns ;a¼0,p¼þ1, and random c;
(b)r¼10 nm ;s¼30 ns ;a¼0 (in red, fit using Eq. (2)plus Eq. (5)); (c)
r¼20 nm ;s¼10 ns, and s¼40 ns (two pulses), and a¼0:001;
(d)r¼20 nm ;s¼20 ns ;a¼0; (e) r¼10 nm ;s¼20 ns ;a¼0:005. The
inversion pulses ( Bz¼/C0300 mT) are also shown (in blue). Disks in (b) to
(e) initially with same circulation c¼þ1 and polarity p¼þ1.113911-3 Garcia et al. J. Appl. Phys. 112, 113911 (2012)
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
130.160.4.77 On: Fri, 19 Dec 2014 17:10:40Regarding the problem of determination of the interac-
tion between nanoelements, from our micromagnetic simula-
tions we could describe the dependence of the contributionto 1=T
/C3
2as a function of the distance dbetween the nanodisks
as
T/C3
2¼BþCd/C0n: (8)
Using Eq. (8)we found, from the best fit (Fig. 4),
n¼4:160:4, in good agreement with Jung et al.11and rea-
sonable agreement with Sukhostavets et al.12
In Fig. 5we show the results of the simulations with
r¼10 nm and a¼0:001. Assuming n¼4, a reasonable lin-
ear fit can be obtained with B¼ð6:560:1Þ/C210/C09s and
C¼/C0 ð 4:260:3Þ/C210/C026m4s. From T/C3
2ðd¼1 Þ and using
Eq. (7), we get Dx/C25ð1:5360:1Þ/C2108s/C01, as expected
from our initial choice of the Gaussian distribution of diame-
ters (Dx/C251:5/C2108s/C01).
Combining Eqs. (6),(7), and (8), one can obtain the
interaction term 1 =T0
2. The computation of 1 =T0
2required thedetermination of the other individual contributions to 1 =T/C3
2,
which was done through the simulation of vortex echoes. To
derive simply the dependence of the interaction on dit is suf-
ficient to measure 1 =T/C3
2as a function of d, since the interac-
tion term 1 =T0
2is the only contribution that is dependent on
d; this does not need the use of the echoes, only requiring the
determination of the relaxation rate 1 =T/C3
2.
IV. CONCLUSIONS
Micromagnetic simulated experiments in large nanodisk
arrays reveal a new effect—the magnetic vortex echo—that
arises from the refocusing of the overall array magnetization.
We have shown the MVE potential as a characterizationtechnique, since it is a direct way of obtaining important pa-
rameters such as T
2, related to the interaction between the
nanoelements with vortex ground states, and the Gilbertdamping constant a; it therefore can be used to determine a
in these systems. Applications of the MVE include the mea-
surement of the inhomogeneity, such as the distribution ofdimensions, aspect ratios, perpendicular magnetic fields, and
so on, in a planar array of nanoelements with vortices; it may
be used to study arrays of nanowires or nanopillars contain-ing thin layers of magnetic material. These properties cannot
be obtained directly, for example, from the linewidth of
FMR absorption. In an actual MVE experiment the sequenceof external magnetic field pulses has to be repeated many
times (as in NMR), and the echo signals added to improve
the signal to noise ratio.
We also show that a simple analytical model, analogous
to the one that describes the spin echo in magnetic reso-
nance, can be used to explain most features of the MVE.This model has validated the micromagnetic simulations of
the new phenomenon and confirmed the applicability of the
MVE as a useful tool for the characterization of large arraysof magnetic nanoobjects with ground state magnetic vortex
configuration.FIG. 3. Variation of 1 =T2obtained by fitting the curves of echo intensity
versus stoM0expð/C0s=T2Þ, as a function of a, for D¼250 nm, r¼10 nm ;
d¼1 ; the continuous line is a linear fit.
FIG. 4. Variation of T/C3
2versus d/C01for an array of 10 /C210 nanodisks with a
distribution of diameters centered on D¼250 nm ðr¼10 nm Þ;a¼0:001
and separation d; the continuous line is the best fit to Eq. (8).FIG. 5. Variation of T/C3
2versus d/C04for an array of 10 /C210 nanodisks with a
distribution of diameters centered on D¼250 nm ðr¼10 nm Þ;a¼0:001
and separation d; the continuous line is a linear fit. Inset (a) shows an echo
simulation for d¼550 nm, s¼30 ns ;a¼0:001.113911-4 Garcia et al. J. Appl. Phys. 112, 113911 (2012)
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
130.160.4.77 On: Fri, 19 Dec 2014 17:10:40The authors would like to thank G.M.B. Fior for the col-
laboration; we are also indebted to the Brazilian agenciesCNPq, CAPES, FAPERJ, and FAPESP.
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22The sources of inhomogeneity are the spread in radii, in thickness, or thepresence of defects. An external perpendicular field Hadds a contribution
tox,x¼x
GþxH, with xH¼x0p(H/Hs), where pis the polarity and Hs
the field that saturates the nanodisk magnetization.25A distribution DHis
another source of the spread Dx.
23Seehttp://math.nist.gov/oommf/ for information on the OOMMF micro-
magnetic simulation program.
24These echoes, however, are not equivalent to the stimulated echoesobserved in NMR with two 90
/C14pulses.17
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177602 (2009).113911-5 Garcia et al. J. Appl. Phys. 112, 113911 (2012)
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
130.160.4.77 On: Fri, 19 Dec 2014 17:10:40 |
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International Journal of Control
ISSN: 0020-7179 (Print) 1366-5820 (Online) Journal homepage: https://www.tandfonline.com/loi/tcon20
Semi-globally practical finite-time stability for
uncertain nonlinear systems based on dynamic
surface control
Yang Liu, Xiaoping Liu, Yuanwei Jing & Ziye Zhang
To cite this article: Yang Liu, Xiaoping Liu, Yuanwei Jing & Ziye Zhang (2019): Semi-globally
practical finite-time stability for uncertain nonlinear systems based on dynamic surface control,
International Journal of Control, DOI: 10.1080/00207179.2019.1598579
To link to this article: https://doi.org/10.1080/00207179.2019.1598579
Accepted author version posted online: 26
Mar 2019.
Published online: 02 Apr 2019.
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INTERNATIONAL JOURNAL OF CONTROL
https://doi.org/10.1080/00207179.2019.1598579
Semi-globally practical finite-time stability for uncertain nonlinear systems based on
dynamic surface control
Yang Liua,c, Xiaoping Liub,c, Yuanwei Jingaand Ziye Zhangd
aCollege of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, People’s Republic of China;bSchool of Information and
Electrical Engineering, Shandong Jianzhu University, Shandong, People’s Republic of China;cDepartment of Electrical Engineering, Lakehead
University, Thunder Bay, Canada;dCollege of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, People’s
Republic of China
ABSTRACT
In this paper, a semi-globally practical finite-time stability (SGPFS) problem is investigated for a class
of uncertain nonlinear systems. Two well-known control techniques, dynamic surface control (DSC) andadding a power integrator (AAPI), are combined to obtain the semi-globally practical finite-time controller.With the aid of DSC, a less-complex finite-time control algorithm is presented, which makes the closed-loopsystem SGPF-stable. Two examples are provided to explain the feasibility and effectiveness of the proposeddesign technique.ARTICLE HISTORY
Received 29 January 2018
Accepted 17 March 2019
KEYWORDS
Semi-globally practicalfinite-time stability; dynamicsurface control; adding a
power integrator;
backstepping
1. Introduction
Duringthepast20years,thefinite-timecontrol(FTC)problem
hasbeenstudiedwell.Moreover,amajorityoffinite-timedesignresultsarebasedon ˙V(x)+cV
α(x)≤0withc>0and0 <α<
1p r o p o s e di nB h a ta n dB e r n s t e i n( 2000), which together with
the adding a power integrator (AAPI), a novel FTC methodis presented in Huang, Lin, and Yang ( 2005). Afterwards, a
largequantityoffinite-timeachievementsdependonthedesignidea of Huang et al. ( 2005)( s e eD u ,C h e n g ,H e ,&J i a ,2014 ;
Du, Qian, Frye, & Li, 2012;G a o&W u ,2016 ,2017;H u a n g ,
Wen, Wang, & Song, 2016;H u a n g&X i a n g ,2016 and refer-
ences therein). However, due to the utilisation of AAPI strat-egy and estimations of derivatives of virtual controllers, which
m a k e st h ed e s i g np r o c e s sv e ryc o m p l i c a t e d .H e n c e ,s o m el o w -
complexity methods are considered to simplify the design of
the FTC controller (Seo, Shim, & Seo, 2008;S o n g ,W a n g ,H o l -
loway, & Krstic, 2017) .A u t h o r si nS o n ge ta l .( 2017)p r o p o s e
an entirely novel method to solve a FTC problem by employ-ing an unbounded strictly-increasing function, by which the
setting time dose not depend on the initial condition. A sim-ple design method is introduced by utilising dynamic expo-
nentscalingandanovelconceptcalleddegreeindicatorinSeo
et al. (2008 ) where a global smooth feedback controller is con-
structed. Apart from the abovementioned achievements, there
still exist many other FTC algorithms (Ding, Qian, & Li, 2010;
Hong,2002;H o ng&J iang,2006 ;H ong,W ang,&Cheng,2006 ;
Shen&Huang, 2009;W u,Chen,&Li,2016 ).Withtheassump-
tion that nonlinear systems are homogeneous, a homogeneous
finite-time local controller is proposed for a family of nonlin-ear systems in Hong ( 2002), Hong and Jiang ( 2006), and Hong
et al. (2006 ). Authors of Shen and Huang ( 2009)d e v e l o pa
CONTACT Xiaoping Liu xliu2@lakeheadu.ca Department of Electrical Engineering, Lakehead University, P7B 5E1 Thunder Bay, Canada; School of Information
and Electrical Engineering, Shandong Jianzhu University, Shandong, People’s Republic of Chinanew finite-time stability condition, that is, ˙V(x)+cVα(x)+
kV(x)≤0w i t hk>0, which makes the system converge faster
than Huang et al. ( 2005). Wu et al. (2016 ) address a global
finite-time stability problem for nonlinear systems with multi-ple unknown control directions. Ding et al. ( 2010)i st h efi r s t
todealwithaFTCissueforakindofupper-triangularsystems.TheFTCproblemwithinput-to-statestabilityisinvestigatedinHong,Jiang,andFeng( 2008).
Itiswellknownthatthedynamicsurfacecontrol(DSC)can
beusedtoestimatethederivativesofvirtualcontrollaws.InNi,Liu, Liu, Hu, and Shen ( 2016), Liu, Wang, and Zhang ( 2015),
Han, Ha, and Lee ( 2016), and Wang and Song ( 2017), FTC
problems are studied by virtue of DSC. Specifically, authors inLiu et al. ( 2015) discuss the finite-time DSC stability problem
for a class of high-order uncertain nonlinear systems with theaid of sign functions and a set of surface gains. A fixed-time
dynamic surface control issue is addressed for power systems
based on high-order sliding mode control in Ni et al. ( 2016).
Similar to Liu et al. ( 2015), both DSC and the fuzzy control
a r ec o m b i n e dt os o l v efi n i t e - t i m ec o n t r o li s s u ef o rn o n l i n e a rlarge-scale systems in Han et al. ( 2016). A so-called fraction-
DSC (F-DSC) approach and neural networks are employed
to deal with a finite-time containment control problem for
multi-agent systems with pure feedback structure in Wangand Song ( 2017). Although finite-time DSC control problems
have been considered in Ni et al. ( 2016), Liu et al. ( 2015), Han
et al. (2016 ), Wang and Song ( 2017) ,t h er e s u l t sa r es t i l ll i m -
ited, so it is necessary to explore this field. Besides, to the best
of our knowledge, there has no report on semi-globally prac-
tical finite-time stability (SGPFS) problem based on AAPI andDSC.
© 2019 Informa UK Limited, trading as Taylor & Francis Group2 Y. LIU ET AL.
Contributions of this work. Inspired by the aforementioned
analysis, this paper will attempt to provide a solution to cope
with the SGPFS problem via AAPI and DSC for the following
non-strictfeedbackuncertainnonlinearsystems.
˙xi=xi+1+fi(x),i=1,...,n−1
˙xn=u+fn(x) (1)
wherex=[x1,...,xn]T∈Rnstands for the state vector, u∈R
denotes the control input. fi(·)(i=1,...,n)a r et h eu n k n o w n
C1nonlinearfunctions.
By comparingwith the relevant results, the contributionsof
thisworkarelistedasfollows:
(1) AccordingtoWang,Song,Holloway,andKrstic( 2017),the
AAPI proposed in Huang et al. ( 2005)m a k e st h efi n i t e -
time control design complicated. Therefore, in order to
simplify the design process of the controller, DSC is usedto eliminate repeated differentiation of virtual controllers.
NotethatthispaperisfirsttosolvetheSGPFSproblemby
combiningwithAAPIandDSC.
(2) Ontheotherhand,theassumptionthattheunknownnon-
linearfunctionsareboundedbyapositivestrictlyincreas-ing function is made in Huang et al. ( 2005)a n dW u
etal.(2016 ).However,theproposedassumptioncondition
inthispaperislessrestrictivethanHuangetal.(2005 )and
Wu et al. ( 2016) ,w h i c hr e s u l t si nal e s sc o n s e r v a t i v ec o n -
troller.ThespecificexplanationcanbeseeninRemark2.1ofthiswork.
(3) Compared with the existing finite-time control with DSC
(see Han et al., 2016;S u n ,R e n ,&L i ,2013 ;W a n g
&S o n g ,2017), (i) the first difference is on the systems in
consideration. Wang and Song ( 2017)c o n s i d e rt h em u l t i -
agent systems with a nonaffine pure-feedback form; Hanet al. (2016 ) investigate nonlinear large-scale intercon-
nected systems; Sun et al. ( 2013) focuses on a class of
nonlinearpure-feedbacksystems.(ii)theseconddifferenceis on the design method adopted. Except for the dynamicsurfacecontrol(DSC)method,WangandSong( 2017)and
Han et al. ( 2016) use the neuroadaptive and fuzzy con-
t r o lm e t h o d st oa d d r e s st h efi n i t e - t i m ec o n t r o lp r o b l e m ,respectively;Sunetal.( 2013)employL
∞andtheextended
stateobservertohandlethefinite-timecontrolissue.How-ever,theaddingapowerintegratorisutilisedtoachievethe
finite-time control by combining with DSC for a kind of
strict-feedbacknonlinearsystemsinthiswork.
2. Assumptions, lemmas and definitions
To achieve the control objectives, several basic assumptions,lemmasanddefinitionsaremadeinthissection.
Assumption2.1: TherearesomeC
1knownfunctions μij(¯xi)≥
0suchthat
/vextendsingle/vextendsinglefi(x)/vextendsingle/vextendsingle≤|x
1|μi1(¯xi)+···+ |xi|μii(¯xi)(2)
withi =1,...,nand ¯xi=[x1,...,xi]T.Remark2.1: Assumption2.1meansthattheunknownnonlin-
ear function fi(x)is bounded by a positive function with the
formof/summationtexti
j=1|xj|μij(¯xj).ItisworthnotingthatAssumption2.1
is less restrictive than the assumption in Huang et al. ( 2005),
which has also been made in Wu et al. ( 2016), because it is
required that μk1=···= μkiwithi=1,...,nin the existing
papers.
Definition2.1(Wang,Chen,Lin,Sun,&Wang, 2017):Con-
sideranonlinearsystemdefinedby
˙κ=f(κ) (3)
where κis the state vector. It is assumed that f(κ):/Omega1→Rn
is continuous on an open neighbourhood /Omega1of the origin with
f(0)=0.Ifthereare ς>0and0 <T(κ0)<∞foreachinitial
condition κ(t0)=κ0suchthatthefollowingholds
/bardblκ(t)/bardbl≤ς,t≥t0+T
withT(κ0)being a settling time, then the origin of system (3)
issemi-globallypracticalfinite-timestability(SGPFS).Lemma 2.1 (Yu, Shi, & Zhao, 2018):The trajectory of system
(3)isSGPFS,ifthereexistaC
1functionV (κ)>0withV (0)=0
andthreepositivenumbersc >0,k>0,0<α<1and0</rho1<
∞suchthat
˙V(κ)+cVα(κ)+kV(κ)≤/rho1 (4)
whereV (κ)isdefinedonaneighbourhoodU ⊂Rnoftheorigin.
If U=Rn,thenthetrajectoryofsystem(3)isPFS.
Remark 2.2: It follows from Remark 5 of Yu et al. (2018 )t h a t
(1)˙V≤−cVα−kV+/rho1is better than ˙Vn≤−cVα+/rho1,s i n c e
ithasfasterconvergenceratetotheequilibriumwhenthestateis
farawayfromtheequilibrium;(2)ifsetting c=0,thentheused
sufficient condition ˙V≤−cVα−kV+/rho1is reduced to ˙V≤
−kV+/rho1,whichistheordinarycontrolschemeandimpliesthat
the sufficient condition in this paper includes the condition in
ordinarymethodasaspecialcase.Basedontheaboveanalysis,itisclearthattheconvergencerateinthisworkisfasterthanW ang
etal.(2017 ),Wang,Chen,Liu,andLin( 2018),Sun,Chen,Lin,
Wang,andZhou( 2016),andtheproposedfinite-timecontroller
ismoregeneralthanWangetal.( 2017),Wangetal.(2018 ),Sun
etal.(2016 ),andtheordinarycontrolapproach.
Lemma 2.2 (Huang et al., 2005):For any x
i∈Rw i t h
i=1,...,nand0 </epsilon1≤1,(5)holds
(|x1|+|x2|+···|xn|)/epsilon1≤|x1|/epsilon1+|x2|/epsilon1+···|xn|/epsilon1(5)
Lemma 2.3 (Huang et al., 2005):For any x ∈R, y∈Ra n d
0</pi1=/pi11//pi12≤1with/pi11and/pi12being odd integers, the
followinginequalityholds
/vextendsingle/vextendsinglex/pi1−y/pi1/vextendsingle/vextendsingle≤21−/pi1/vextendsingle/vextendsinglex−y/vextendsingle/vextendsingle
/pi1(6)INTERNATIONAL JOURNAL OF CONTROL 3
Lemma 2.4 (Huang et al., 2016;W ue ta l . , 2016):Forν>
0,λ>0,ι>0, θ≥0,δ≥0,a n dπ ≥0,t h ef o l l o w i n gt w o
inequalitieshold
θνδλπ≤ιθν+λ+λ
ν+λ/bracketleftbiggν
ι(ν+λ)/bracketrightbiggν
λ
πν+λ
λδν+λ(7)
and
νλ≤ν1+ι+λ1+1
ι (8)
3. Finite-time DSC algorithm and stability analysis
In this section, a novel finite-time control scheme with low-
complexity is shown by utilising DSC. To solve the SGPFSp r o b l e mf o rt h es y s t e m( 1 ) ,as e to fp a r a m e t e r s ,s u r f a c ee r r o r s
and boundary layer errors are defined by (9), (10) and (11),
respectively.
σ
i=(2n+3−2i)/(2n+1) (9)
and
ξi=(xi)1/σi−/parenleftbig
x/trianglesolid
i/parenrightbig1/σi(10)
and
yi=ωi−/parenleftBig
x/triangle
i/parenrightBig1/σi(11)
wherei=1,...,n,x/triangle
iisthevirtualcontroller, ωiistheoutputof
thelow-passfilter(12)with (x/triangle
i)1/σibeingtheinputand τibeing
atimeconstant, x/trianglesolid
i=ωσi
i,andy1=ω1=x/triangle
1=x/trianglesolid
1=0.
τi˙ωi+ωi=/parenleftBig
x/triangle
i/parenrightBig1/σi,ωi(0)=/parenleftBig
x/triangle
i(0)/parenrightBig1/σi(12)
withi=2,...,n.Moreover,thevirtualcontrollers x/triangle
iisdefined
in(13).
x/triangle
i=−ξσi
i−1βi−1(¯xi−1) (13)
where βi−1(¯xi−1)is theC1function which will be specified in
the sequel. It follows from x/trianglesolid
i=ωσi
ithat(x/trianglesolid
i)1/σi=ωi.A sa
result,(11)canberewrittenas
yi=/parenleftbig
x/trianglesolid
i/parenrightbig1/σi−/parenleftBig
x/triangle
i/parenrightBig1/σi(14)
In addition, the derivative of (x/trianglesolid
i)1/σiwith respect to time can
becalculatedby
d/bracketleftBig/parenleftbig
x/trianglesolid
i/parenrightbig1/σi/bracketrightBig
dt=˙ωi=/parenleftBig
x/triangle
i/parenrightBig1/σi−ωi
τi
=/parenleftBig
x/triangle
i/parenrightBig1/σi−/parenleftbig
x/trianglesolid
i/parenrightbig1/σi
τi=−yi
τi(15)
Itfollowsfrom(9),(13),(14)and(15)that(16)and(17)hold.
σ1=1>σ2>···>σn+1=1/(2n+1)(16)
˙yi=−yi
τi+d/bracketleftBig
ξi−1β1/σi
i−1(¯xi−1)/bracketrightBig
dt(17)Step1.ChooseaL yapunovfunction
V1=1
2x2
1 (18)
Differentiatingitgives
˙V1=x1x2+x1f1(x)
≤x1x2+x2
1μ11(x1)
≤x1/parenleftBig
x2−x/triangle
2/parenrightBig
+x1x/triangle
2+xd
1¯μ11(x1)(19)
where ¯μ11(x1)=(1+x2
1)μ11(x1)is aC1function and
d=4n/(2n +1).Thevirtualcontroller x/triangle
2canbeselectedas
x/triangle
2=−xd−1
1/bracketleftBig
¯c+n−1+¯μ11(x1)+m/parenleftbig
1+x2
1/parenrightbig2−d/bracketrightBig
=−ξσ2
1β1(x1) (20)
withβ1(x1)=¯c+n−1+¯μ11(x1)+m(1+x2
1)2−dbeingaC1
functionand ¯c>0.Therefore, ˙V1canberewrittenas
˙V1≤−(¯c+n−1)xd
1+x1/parenleftBig
x2−x/triangle
2/parenrightBig
−mxd
1/parenleftbig
1+x2
1/parenrightbig2−d
(21)
It follows from Liu, Liu, Jing, and Zhang ( 2018)t h a t
(1+x2
1)2−d≥(2x1)2−d=22−dx2−d
1,whichindicates
−mxd
1/parenleftbig
1+x2
1/parenrightbig2−d≤−¯mx2
1
with¯m=22−dm.Therefore,(21)canberewrittenas
˙V1≤−(¯c+n−1)xd
1−¯mx2
1+x1/parenleftBig
x2−x/triangle
2/parenrightBig
(22)
Step2.TheL yapunovfunction V2isdefinedby
V2=V1+Ϝ2+1
dyd
2 (23)
with Ϝ2=/integraltextx2
x/trianglesolid
2(υ1/σ2−(x/trianglesolid
2)1/σ2)2−σ2dυ. Differentiating the
Lyapunov function V2,itproduces
˙V2≤−(¯c+n−1)xd
1−¯mx2
1−yd
2
τ2
+x1/parenleftBig
x2−x/triangle
2/parenrightBig
+ξ2−σ2
2/parenleftBig
x3−x/triangle
3/parenrightBig
+ξ2−σ2
2x/triangle
3+ξ2−σ2
2f2+∂Ϝ2
∂x1˙x1+η2/parenleftbig
x1,y2/parenrightbig
(24)
where η2(x1,y2)=d[ξ1β1/σ2
1(x1)]
dtyd−1
2is a continuous function
withrespectto x1andy2.
In what follows, the items x1(x2−x/triangle
2),ξ2−σ2
2f2and∂Ϝ2
∂x1˙x1
in (24) can be estimated and the corresponding results are
shownin(25),(26)and(27).
x1/parenleftBig
x2−x/triangle
2/parenrightBig
≤1
2ξd
1+φ21ξd
2+φ22yd
2 (25)
ξ2−σ2
2f2≤1
2ξd
1+ϕ2(¯x2)ξd
2+1
2yd
2(26)4 Y. LIU ET AL.
∂Ϝ2
∂x1˙x1≤ψ21/parenleftbig
y2/parenrightbig
ξd
2+ψ22yd
2+ψ23(27)
Thespecificcomputingprocessof(25)–(27)aswellas φ21,φ22,
ϕ2(¯x2),ψ21(y2),ψ22,andψ23canbefoundin Appendix1 .Based
on(25)–(27), ˙V2canbeestimatedby
˙V2≤−(¯c+n−1)ξd
1−/parenleftbigg1
τ2−φ22−1
2/parenrightbigg
yd
2
+ξ2−σ2
2/parenleftBig
x3−x/triangle
3/parenrightBig
+B2/parenleftbig¯x2,y2/parenrightbig
+ξ2−σ2
2/parenleftBig
x/triangle
3+ξd−2+σ2
2[φ21
+ϕ2(¯x2)+ψ21/parenleftbig
y2/parenrightbig/bracketrightbig/parenrightbig
(28)
whereB2(¯x2,y2)=ψ23+η2(x1,y2)+ψ22yd
2i sac o n t i n u o u s
function.Thevirtualcontroller x/triangle
3isdefinedby
x/triangle
3=−ξd−2+σ2
2/bracketleftBig
(¯c+n−2)+φ21+ϕ2(¯x2)
+ψ21/parenleftbig
y2/parenrightbig
+m/parenleftbig
1+ξ2
2/parenrightbig2−d/bracketrightBig
=−ξσ3
2β2(¯x2) (29)
with β2(¯x2)=(¯c+n−2)+φ21+ϕ2(¯x2)+ψ21(y2)+m
(1+ξ2
2)2−dbeing aC1function. Similar to Step 1, −mξd
2(1+
ξ2
2)2−d≤−¯mξ2
2.Substituting(29)into(28),ityields
˙V2≤−(¯c+n−2)2/summationdisplay
j=1ξd
j−¯m2/summationdisplay
j=1ξ2
j−/parenleftbigg1
τ2−φ22−1
2/parenrightbigg
yd
2
+ξ2−σ2
2/parenleftBig
x3−x/triangle
3/parenrightBig
+B2/parenleftbig¯x2,y2/parenrightbig
(30)
Inductive step .S u p p o s ea tS t e p (i−1),t h e r ee x i s tas e to ffi l -
ters (12) and virtual controllers (13) such that Vi−1=Vi−2+
Ϝi−1+1
dyd
i−1satisfiesthefollowinginequality
˙Vi−1≤−(¯c+n−(i−1))i−1/summationdisplay
j=1ξd
j−¯m2/summationdisplay
j=1ξ2
j
−i−1/summationdisplay
j=2/parenleftbigg1
τj−φj2−1
2/parenrightbigg
yd
j
+i−1/summationdisplay
j=2Bj/parenleftbig
xj−1,yj/parenrightbig
+ξ2−σi−1
i−1/parenleftBig
xi−x/triangle
i/parenrightBig
+ηi−1/parenleftbig¯xi−2,yi−1/parenrightbig
(31)
whereBj(xj−1,yj)=ηj(xj−1,yi)+ψj3+ψj2yd
jwithηi−1(¯xi−2,
yi−1)=d[ξi−2β1/σi−1
i−2(¯xi−2)]
dtyd−1
i−1i sac o n t i n u o u sf u n c t i o na n d
Ϝi−1=/integraltextxi−1
x/trianglesolid
i−1(υ1/σi−1−(x/trianglesolid
i−1)1/σi−1)2−σi−1dυ.
In the sequel, it will be claimed that (31) also holds at
Stepi.I no r d e rt ov e r i f yt h ec l a i m ,t a k i n gt h et i m e - d e r i v a t i v e sof aC1Lyapunov function Vi=Vi−1+Ϝi+1
dyd
iwithϜi=/integraltextxi
x/trianglesolid
i(υ1/σi−(x/trianglesolid
i)1/σi)2−σidυresultsin
˙Vi≤−(¯c+n−(i−1))i−1/summationdisplay
j=1ξd
j
−¯mi−1/summationdisplay
j=1ξ2
j−yd
i
τi+ξ2−σi−1
i−1/parenleftBig
xi−x/triangle
i/parenrightBig
+ξ2−σi
i/parenleftBig
xi+1−x/triangle
i+1/parenrightBig
+ξ2−σi
ix/triangle
i+1+ξ2−σi
ifi
−i−1/summationdisplay
j=2/parenleftbigg1
τj−φj2−1
2/parenrightbigg
yd
j
+i−1/summationdisplay
j=2Bj/parenleftbig
xj−1,yj/parenrightbig
+i−1/summationdisplay
j=1∂Ϝi
∂xj˙xj+ηi/parenleftbig¯xi−1,yi/parenrightbig
(32)
Similar to the previous steps, it follows from Appendix 2
t h a ts o m ei t e m si n( 3 2 )c a nb ee s t i m a t e db yt h ef o l l o w i n g
inequalities.
ξ2−σi−1
i−1/parenleftBig
xi−x/triangle
i/parenrightBig
≤1
2ξd
i−1+φi1ξd
i+φi2yd
i (33)
ξ2−σi
ifi≤1
2i/summationdisplay
j=1ξd
j+1
2yd
i+/Delta1i(¯xi)ξd
i(34)
i−1/summationdisplay
j=1∂Ϝi
∂xj˙xj≤ψi1/parenleftbig
yi/parenrightbig
ξd
i+ψi2yd
i+ψi3(35)
where φi1,φi2,/Delta1i(¯xi),ψi1(yi),ψi2,a n dψ i3can be found in
Appendix 2 .
Remark 3.1: Owing to applying the low-pass filters, the pre-
sented method is simpler than Huang et al. ( 2005)f o rt h e
estimationof/summationtexti−1
j=1∂Ϝi
∂xj˙xj.Itfollowsfrom(A9)that
i−1/summationdisplay
j=1∂Ϝi
∂xj˙xj=(2−σi)yi
τi/integraldisplayxi
x/trianglesolid
i/parenleftBig
υ1/σi−/parenleftbig
x/trianglesolid
i/parenrightbig1/σi/parenrightBig2−σidυ
which makes the estimation much easier than in Huang
etal.(2005 ).
Accordingto(33),(34)and(35), ˙Vicanbeexpressedas
˙Vi≤−(¯c+n−i)i−1/summationdisplay
j=1ξd
j+ξ2−σi
i/parenleftBig
xi+1−x/triangle
i+1/parenrightBig
−i/summationdisplay
j=2/parenleftbigg1
τj−φj2−1
2/parenrightbigg
yd
j
+ξ2−σi
i/parenleftBig
x/triangle
i+1+ξd−2+σ2
i (φi1+/Delta1i(¯xi)
+ψi1/parenleftbig
yi/parenrightbig
+1
2/parenrightbigg/parenrightbigg
+i/summationdisplay
j=2Bj/parenleftbig
xj−1,yj/parenrightbig
(36)INTERNATIONAL JOURNAL OF CONTROL 5
Thevirtualcontroller x/triangle
i+1is
x/triangle
i+1=−ξd−2+σi
i/parenleftbigg
(¯c+n−i)+/Delta1i(¯xi)+φi1
+ψi1+1
2+m/parenleftbig
1+ξ2
i/parenrightbig2−d/parenrightbigg
=−ξσi+1
iβi(¯xi) (37)
with βi(¯xi)=(¯c+n−i)+/Delta1i(¯xi)+φi1+ψi1+1
2+m(1+
ξ2
i)2−dbeing aC1function as well. Substituting (37) into (36),
itgives
˙Vi≤−(¯c+n−i)i/summationdisplay
j=1ξd
j−¯mi/summationdisplay
j=1ξ2
j
−i/summationdisplay
j=2/parenleftbigg1
τj−φj2−1
2/parenrightbigg
yd
j
+i/summationdisplay
j=2Bj/parenleftbig
xj−1,yj/parenrightbig
+ξ2−σi
i/parenleftBig
xi+1−x/triangle
i+1/parenrightBig
(38)
Sofar,theproofoftheinductivestepiscompleted.Accordingto
theinductiveargumentabove,thecontroller uisdefinedasthe
formof(39),whichcanbederivedfrom(37)bysetting iton.
u=x/triangle
n+1=−ξ1
2n+1nβn(¯xn) (39)
Therefore,
˙Vn≤−¯cn/summationdisplay
j=1ξd
j−¯mn/summationdisplay
j=1ξ2
j−n/summationdisplay
j=2/parenleftbigg1
τj−φj2−1
2/parenrightbigg
yd
j
+n/summationdisplay
j=2Bj/parenleftbig
xj−1,yj/parenrightbig
(40)
It follows from Huang et al. ( 2005)t h a tVn≤2/summationtextn
j=1ξ2
j+
2/summationtextn
j=2yd
j.DuetoLemma2.2,thefollowinginequalityholds.
Vα
n≤2n/summationdisplay
j=1ξd
j+2n/summationdisplay
j=2yd
j+2(n−1)(1−α)αα/(1−α)
withα=2n
2n+1.W iththisinmind,itisclearthat
κ1Vα
n+κ2Vn≤2κ1n/summationdisplay
j=1ξd
j+2κ2n/summationdisplay
j=1ξ2
j
+2(κ1+κ2)n/summationdisplay
j=2yd
j+¯/Theta1(41)
where ¯/Theta1=2κ1(n−1)(1−α)αα/(1−α),κ1>0a n dκ 2>0.
Hence, it can be easily calculated from (40) and (41) that (42)holds.
˙Vn≤−¯cn/summationdisplay
j=1ξd
j−¯mn/summationdisplay
j=1ξ2
j−n/summationdisplay
j=2/parenleftbigg1
τj−φj2−1
2/parenrightbigg
yd
j
+n/summationdisplay
j=2Bj/parenleftbig
xj−1,yj/parenrightbig
≤−κ⎛
⎝2κ1n/summationdisplay
j=1ξd
j+2κ2n/summationdisplay
j=1ξ2
j+2(κ1+κ2)n/summationdisplay
j=2yd
j⎞⎠
+n/summationdisplay
j=2Bj/parenleftbig
xj−1,yj/parenrightbig
≤−κ/parenleftbig
κ1Vα
n+κ2Vn−¯/Theta1/parenrightbig
+n/summationdisplay
j=2Bj/parenleftbig
xj−1,yj/parenrightbig
=−¯κ1Vα
n−¯κ2Vn+/Theta1 (42)
where ¯κ1=κκ1,¯κ2=κκ2,/Theta1=/summationtextn
j=2Bj(xj−1,yj)+κ¯/Theta1and
κ=min⎧
⎨
⎩¯c
2κ1,¯m
2κ2,1
2(κ1+κ2)n/summationdisplay
j=2/parenleftbigg1
τj−φj2−1
2/parenrightbigg⎫
⎬
⎭(43)
0<τj<2
2+φj2(44)
Moreover, for any given positive constant p,t h es e t /Omega1n=
{/summationtextn
j=1ξ2
j+/summationtextnj=2yd
j≤2p}is compact in Rn×(n−1). Hence,
thereexists M>0suchthat |/Theta1|≤Mon/Omega1n.Therefore,(42)can
bechangedto
˙Vn+¯κ1Vα
n+¯κ2Vn≤M (45)
Remark 3.2: Due to Liu et al. ( 2015), let¯κ1>λ1+M
2pαand
¯κ2>λ2+M
2pwithλ1>0a n dλ 2>0, then ˙Vn≤−λ1Vα
n−
λ2VnonVn=p. Hence, Vn≤pis an invariant set, i.e. if
Vn(0)≤p,thenVn(t)≤pforallt≥0.Furthermore,thestates
convergetoanarbitrarilysmallzonebyincreasingthevaluesof
λ1andλ2.
Remark 3.3: It is well known that the time constant τjshould
beverysmall.Therefore,therangeof τjisreasonabledueto0 <
τj≤2
2+φj2,j=2,...,n.
Sofar,itfollowsfromLemma2.1and(45)thatthenon-strict
feedback uncertain nonlinear system (1) is SGPFS. Moreover,themainresultissummarisedasTheorem3.1.
Theorem 3.1: Under the conditions that Assumption 2.1holds
andV
n(0)≤pforanyinitialconditionswithpbeinganarbitrary
positive constant, if there exist the design parameters κin(43),
τi(i=2,...,n) in(44),κ1>0,κ2>0,m>0a n d ¯c>0,t h e n
the non-strict feedback nonlinear system (1)is SGPF-stable with
thecontrollawuin (39),thevirtualcontrollersin (13)aswellas
thefirst-orderlow-passfilters (12).6 Y. LIU ET AL.
Remark 3.4: It follows from Step 1 to Step nthat the pre-
sented design procedure of semi-globally practical finite-time
controller is simpler than Huang et al. (2005 ,2016), Huang
and Xiang ( 2016), Gao and Wu ( 2016), Du et al. (2012 ,2014),
and Gao and Wu ( 2017) due to avoiding the repeated differen-
tiationsforthevirtualcontrollaws.
4. Simulation results
4.1 Example 1
ItfollowsfromShunsuke,Nami,Hisakazu,andHirokazu( 2011)
thatthemodelofrobotmanipulatorsisasfollows.
˙x1(t)=x2(t)
˙x2(t)=1
J/parenleftbig
rgsinx1(t)−Dx2(t)−F/parenrightbig
+1
Ju(t)(46)
The physical meaning of the parameters is given in Shunsuke
etal.(2011 ).AccordingtoTheorem3.1, u=−ξσ3
2β2(¯x2),and
β1(x1)=¯c+n−1+¯μ11(x1)+m/parenleftbig
1+x2
1/parenrightbig2−d,
β2(¯x2)=¯c+n−2+φ21+ϕ2(¯x2)
+ψ21/parenleftbig
y2/parenrightbig
+m/parenleftbig
1+ξ2
2/parenrightbig2−d,
¯μ11(x1)=/parenleftbig
1+x2
1/parenrightbig
μ11,
¯γ2(¯x2)=/parenleftbig
1+x2
1/parenrightbig
μ21(¯x2).
withn=2,¯c=1,m=0.01, the parameters φ21,ϕ2(¯x2)and
ψ21(y2)can be found in Appendix 1 . Furthermore, it fol-
lows from Assumption 2.1 that μ11(¯x1)=0,μ21(¯x2)=rg
J,
μ22(¯x2)=D
J.
Simulationiscarriedoutwith J=3.2870(kg·m2),r=2.3126
(kg·m),g=9.8(m/s2),D=18.6916(N·m· sec),F=24.2500,
τ2=0.01,x(0)=[0.8,0]T.
The simulation results are demonstrated by Figures 1–4.
Figures1and2show the states of (46) and control input u,
respectively. It can be seen from Figures 3and4that the sur-
faceerrorsandboundarylayererrorapproachtozero.Besides,Figures3and4also verify that all the states and errors of (46)
areSGFTB.
4.2 Example 2
According to Liu et al. (2015 ), a third-order system is given as
follows⎧
⎪⎨
⎪⎩˙x
1=x2+x2
1sin(x1)
˙x2=x3
˙x3=u(47)
It follows from (47) that μ11(¯x1)=x2
1,μ21(¯x2)=μ22(¯x2)=
0,μ31(¯x3)=μ32(¯x3)=μ33(¯x3)=0,τ2=0.01,τ3=0.01and
x(0)=[−3,0.5,0]T.Besides,theotherparametersaregivenas
follows:
β1=¯c+n−1+m/parenleftbig
1+x2
1/parenrightbig2−d,
β2=¯c+n−2+φ21+ψ21+ϕ2+m/parenleftbig
1+ξ2
2/parenrightbig2−d,Figure 1. The states x1and x2of the system.
Figure 2. The control input u.
Figure 3. The surface error ξ1andξ2.INTERNATIONAL JOURNAL OF CONTROL 7
Figure 4. The boundary layer error y2.
Figure 5. The state x1of the system.
β3=¯c+n−3+/Delta13(¯x3)+φ31+ψ31+m/parenleftbig
1+ξ2
3/parenrightbig2−d,
u=x/triangle
4=−ξ1
2n+1
3β3(¯x3)
wheren=3,¯c=0.5,m=0.01,φ21,φ31,ψ21,ψ31,ϕ2and
/Delta13(¯x3)canbeseeninA ppendices1and2.
Moreover, the proposed control method is compared with
the finite-time DSC control in Liu et al. ( 2015)w h e r et h ec o r -
responding parameters are set to K1=20,K2=30,K3=30,
γ1=γ1=5,α=0.7,τ2=τ3=0.01.
The simulation results are demonstrated by Figures 5–10.
Figures5and7showthecomparedresultsofthestates,respec-
tively.Thetrajectoriesoftheboundarylayererrorsareexhibited
inFigures 8and9.Figure10demonstratesthecurveofthecon-
trol input u. It can be seen from Figures 5to10that (1) the
proposed approach makes the system states and layer errorsconverge faster and the overshoot be smaller; (2) the control
input is smaller than Liu et al. (2015 ); and (3) all the signals of
theclosed-loopsystemareSGPF-stable.Figure 6. The state x2of the system.
Figure 7. The state x3of the system.
Figure 8. The first boundary layer error.8 Y. LIU ET AL.
Figure 9. The second boundary layer error.
Figure 10. The control input u.
5. Conclusion
The SGPFS problem has been solved for a class of non-strict
feedback uncertain nonlinear systems in this paper. Further-
more, the continuous finite-time feedback controller has been
constructed. Owing to the use of DSC technique, the pre-
s e n t e dd e s i g np r o c e s si ss i m p l e rt h a nt h ee x i s t i n gr e s u l t s( e . g .
see Du et al., 2012,2014;G a o&W u , 2016,2017;H u a n g
et al.,2005,2016;H u a n g&X i a n g , 2016). The assumption
for the unknown nonlinearities is less restrictive than Huangetal.(2005 )andWuetal.( 2016).Besides,thedevelopedmethod
canalsobeusedtosolvethecorrespondingcontrolproblemfor
stochastic nonlinear systems, time-delay nonlinear systems or
large-scalenonlinearsystems,justtonameafew.
Acknowledgements
T h ea u t h o r sw o u l dl i k et ot h a n kE d i t o r - i n - c h i e f ,A s s o c i a t eE d i t o ra n dt h e
reviewersfortheirvaluablecommentsandhelpfulsuggestions.Disclosure statement
Nopotentialconflictofinterestwasreportedbytheauthors.
Funding
Meanwhile, this work was supported in part by the China ScholarshipCouncil [grant no. 201606080044], the National Natural Science Founda-tionofChina[grantnos.61773108and61503222]andtheNaturalSciencesand Engineering Research Council of Canada [grant no. RGPIN-2017-05367].
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Appendices
Appendix 1. The estimations of x1(x2−x/triangle
2),ξ2−σ 2
2f2
and∂Ϝ2
∂x1˙x1
(1) The item x1(x2−x/triangle
2)is estimated by the following inequality based
on(14),Lemmas2.2–2.4.
x1/parenleftBig
x2−x/triangle
2/parenrightBig
≤|ξ1|21−σ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle(x
2)1/σ2−/parenleftBig
x/triangle
2/parenrightBig1/σ2/vextendsingle/vextendsingle/vextendsingle/vextendsingleσ2
=21−σ2|ξ1|/vextendsingle/vextendsingle/vextendsingle/bracketleftBig
(x
2)1/σ2−/parenleftbig
x/trianglesolid
2/parenrightbig1/σ2/bracketrightBig
+y2/vextendsingle/vextendsingle/vextendsingleσ2
≤2|ξ1||ξ2|σ2+2|ξ1|/vextendsingle/vextendsingley2/vextendsingle/vextendsingleσ2
≤1
2ξd
1+φ21ξd
2+φ22yd
2 (A1)
withφ21=[2n−1
4n(2n+1
n)2n+1
2n−124n
2n−1]andφ 22=φ21.
(2)Theestimationof ξ2−σ2
2f2.Accordingto(10),(14),Lemmas2.2–2.4,
|x2|≤|ξ2|σ2+/vextendsingle/vextendsinglex/trianglesolid
2/vextendsingle/vextendsingle≤|ξ
2|σ2+/vextendsingle/vextendsingle/vextendsingle/vextendsingley2+/parenleftBig
x/triangle
2/parenrightBig1/σ2/vextendsingle/vextendsingle/vextendsingle/vextendsingleσ2
≤|ξ2|σ2+/vextendsingle/vextendsingle/vextendsinglex/triangle
2/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingley
2/vextendsingle/vextendsingleσ2
≤|ξ2|σ2+|ξ1|σ2β1(x1)+/vextendsingle/vextendsingley2/vextendsingle/vextendsingleσ2.( A2)
It can be deduced from Assumption 2.1, (A2) the following inequality
holds.
/vextendsingle/vextendsinglef2/vextendsingle/vextendsingle≤|x
1|μ21(¯x2)+|x2|μ22(¯x2)
≤|ξ1|σ2γ2(¯x2)+|ξ2|σ2μ22(¯x2)+/vextendsingle/vextendsingley2/vextendsingle/vextendsingleσ2μ22(¯x2)
≤/parenleftbig|ξ1|σ2+|ξ2|σ2+/vextendsingle/vextendsingley2/vextendsingle/vextendsingleσ2/parenrightbig
¯γ2(¯x2) (A3)where γ2(¯x2)=(1+ξ2
1)μ21(¯x2)+β1(x1)μ22(¯x2)and ¯γ2(¯x2)=max
{γ2(¯x2),μ22(¯x2)}areC1functions.Then,
ξ2−σ2
2f2≤ξ2−σ2
2/parenleftbig|ξ1|σ2+|ξ2|σ2+/vextendsingle/vextendsingley2/vextendsingle/vextendsingleσ2/parenrightbig
¯γ2(¯x2)
≤|ξ2|2−σ2|ξ1|σ2−2
2n+1/parenleftbig/parenleftbig
1+ξ2
1/parenrightbig
¯γ2(¯x2)/parenrightbig
+|ξ2|2−σ2|ξ2|σ2−2
2n+1/parenleftbig/parenleftbig
1+ξ2
2/parenrightbig
¯γ2(¯x2)/parenrightbig
+|ξ2|2−σ2/vextendsingle/vextendsingley
2/vextendsingle/vextendsingleσ2−2
2n+1/parenleftbig/parenleftbig
1+y2
2/parenrightbig
¯γ2(¯x2)/parenrightbig
≤1
2ξd
1+1
2yd
2+ϕ2(¯x2)ξd
2 (A4)
where ϕ2(¯x2)=ϕ21(¯x2)+ϕ22(¯x2)+ϕ23(¯x2)and
˜ϕ2=2n+3
4n/parenleftbiggn−1
n/parenrightbigg2n−2
2n+3
ϕ21(¯x2)=˜ϕ2/bracketleftbig/parenleftbig
1+ξ2
1/parenrightbig
¯γ2(¯x2)/bracketrightbig4n
2n+3,
ϕ22(¯x2)=/parenleftbig
1+ξ2
2/parenrightbig
¯γ2(¯x2),
ϕ23(¯x2)=˜ϕ2/bracketleftbig/parenleftbig
1+y2
2/parenrightbig
¯γ2(¯x2)/bracketrightbig4n
2n+3.
(3)Itfollowsfrom(15)andLemma2.4that∂Ϝ2
∂x1˙x1canbeestimated.
∂Ϝ2
∂x1˙x1=⎡
⎣−(2−σ2)∂/bracketleftBig/parenleftbig
x/trianglesolid
2/parenrightbig1/σ2/bracketrightBig
∂x1
×/integraldisplayx2
x/trianglesolid
2/parenleftBig
υ1/σ2−x/trianglesolid1/σ2
2/parenrightBig1−σ2dυ/bracketrightBigg
˙x1
≤2(2−σ2)/vextendsingle/vextendsingley2/vextendsingle/vextendsingle|ξ
2|
τ2
≤/parenleftBig
1+ξd
2/parenrightBig/bracketleftbigg2(2−σ2)
τ2/vextendsingle/vextendsingley
2/vextendsingle/vextendsingle/bracketrightbigg
≤ψ
21/parenleftbig
y2/parenrightbig
ξd
2+ψ22yd
2+ψ23 (A5)
with˜ψ2=4n+6
τ2(2n+1),ψ21(y2)=˜ψ2|y2|andψ22=ψ23=˜ψ2.
Appendix 2. The estimation results of ξ2−σ i−1
i−1(xi−x/triangle
i),
ξ2−σ i
ifiand/summationtexti−1
j=1∂Ϝi
∂xj˙xj
SimilartotheprocessofA ppendices1and2.
ξ2−σi−1
i−1 (xi−x/triangle
i),ξ2−σi
ifiand/summationtexti−1
j=1∂Ϝi
∂xj˙xjare estimated based on (10),
(14), (15), Lemmas 2.2–2.4. The specific process are shown in (A6),
(A8),(A9).
(1)ξ2−σi−1
i−1 (xi−x/triangle
i)isestimatedby
ξ2−σi−1
i−1/parenleftBig
xi−x/triangle
i/parenrightBig
≤|ξ2−σi−1
i−1 |21−σi/vextendsingle/vextendsingle/vextendsingle/vextendsingle(x
i)1/σi−/parenleftBig
x/triangle
i/parenrightBig1/σi/vextendsingle/vextendsingle/vextendsingle/vextendsingleσi
=|ξ2−σi−1
i−1 |21−σi/vextendsingle/vextendsingle/vextendsingle/bracketleftBig
(x
i)1/σi−/parenleftbig
x/trianglesolid
i/parenrightbig1/σi/bracketrightBig
+yi/vextendsingle/vextendsingle/vextendsingleσi
≤2|ξ2−σi−1
i−1 ||ξi|σi+2|ξ2−σi−1
i−1 |/vextendsingle/vextendsingley
i/vextendsingle/vextendsingleσi
≤1
2ξd
i−1+φi1ξd
i+φi2yd
i (A6)
withφi1=2n+3−2i
4n(2n+2i−3
n)2n−3+2i
2n+3−2i24n
2n+3−2i,φi2=φi1.
(2)Theitem ξ2−σi
ificanbecalculatedby(A7)and(A8).
/vextendsingle/vextendsinglef
i/vextendsingle/vextendsingle≤|x
1|μi1(¯xi)+|x2|μi2(¯xi)+···+ |xi|μii(¯xi)
≤i−1/summationdisplay
j=1/bracketleftbig/vextendsingle/vextendsingleξ
j/vextendsingle/vextendsingleσjμij(¯xi)+/vextendsingle/vextendsingleξ
j/vextendsingle/vextendsingleσj+1βj/parenleftbig¯xj−1/parenrightbig
μi,j+1(¯xi)/bracketrightbig10 Y .L I UE TA L .
+i/summationdisplay
j=2/vextendsingle/vextendsingleyi/vextendsingle/vextendsingleσi/bracketleftBig/vextendsingle/vextendsingley
i/vextendsingle/vextendsingle1−σi/vextendsingle/vextendsingley
j/vextendsingle/vextendsingleσjμij(¯xi)/bracketrightBig
+|ξi|σiμii(¯xi)
≤i−1/summationdisplay
j=1/vextendsingle/vextendsingleξj/vextendsingle/vextendsingleσiγij(¯xi)+|ξi|σiμii(¯xi)
+/vextendsingle/vextendsingley
i/vextendsingle/vextendsingleσii/summationdisplay
j=2/bracketleftBig/parenleftbig
1+y2
i/parenrightbig/parenleftBig
1+y2
j/parenrightBig
μij(¯xi)/bracketrightBig
≤i/summationdisplay
j=1/vextendsingle/vextendsingleξj/vextendsingle/vextendsingleσi¯γij(¯xi)+/vextendsingle/vextendsingley
i/vextendsingle/vextendsingleσi˜γij(¯xi) (A7)
where ¯γij(¯xi)=max{γij(¯xi),μii(¯xi)}and˜γij(¯xi)=/summationtexti
j=2[(1+y2
i)(1+y2
j)
μij(¯xi)]areC1function,and γij(¯xi)=[(1+ξ2
j)μij(¯xi)+(1+ξ2
j)βj(¯xj−1)
μi,j+1(¯xi)].
Then,
ξ2−σi
ifi≤|ξ2−σi
i|i/summationdisplay
j=1/vextendsingle/vextendsingleξj/vextendsingle/vextendsingleσi¯γij(¯xi)+|ξ2−σi
i|/vextendsingle/vextendsingley
i/vextendsingle/vextendsingleσi˜γij(¯xi)
≤i/summationdisplay
j=1|ξi|2−σi/vextendsingle/vextendsingleξ
j/vextendsingle/vextendsingleσi−2
2n+1/parenleftBig
1+ξ2
j/parenrightBig
¯γij(¯xi)
+|ξi|2−σi/vextendsingle/vextendsingley
i/vextendsingle/vextendsingleσi−2
2n+1/parenleftbig
1+y2
i/parenrightbig
˜γij(¯xi)
≤1
2i/summationdisplay
j=1ξd
j+1
2yd
i+/Delta1i(¯xi)ξd
i (A8)
with˜ϕi=2n−1+2i
4n(2n+1−2i
2n)2n+1−2i
2n−1+2i,and
ϕij1(¯xi)=˜ϕi/bracketleftBig/parenleftBig
1+ξ2
j/parenrightBig
¯γij(¯xi)/bracketrightBig4n
2n−1+2i,ϕij2(¯xi)=˜ϕi/bracketleftbig/parenleftbig
1+y2
i/parenrightbig
˜γij(¯xi)/bracketrightbig4n
2n−1+2i,
/Delta1i(¯xi)=i/summationdisplay
j=1/parenleftbig
ϕij1(¯xi)+ϕij2(¯xi)/parenrightbig
.
(3)Theestimationof/summationtexti−1
j=1∂Ϝi
∂xj˙xjcanbeshownin(A9).
i−1/summationdisplay
j=1∂Ϝi
∂xj˙xj=−(2−σi)/integraldisplayxi
x/trianglesolid
i/parenleftBig
υ1/σi−/parenleftbig
x/trianglesolid
i/parenrightbig1/σi/parenrightBig2−σidυ
×⎡
⎣i−1/summationdisplay
j=1∂/bracketleftBig/parenleftbig
x/trianglesolid
i/parenrightbig1/σi/bracketrightBig
∂xj˙xj⎤⎦
=−(2−σ
i)/integraldisplayxi
x/trianglesolid
i/parenleftBig
υ1/σi−/parenleftbig
x/trianglesolid
i/parenrightbig1/σi/parenrightBig2−σidυ
×/bracketleftBig/parenleftbig
x/trianglesolid
i/parenrightbig1/σi/bracketrightBig/prime
≤|ξi|2(2−σi)
τi/vextendsingle/vextendsingleyi/vextendsingle/vextendsingle
≤/parenleftBig
1+ξd
i/parenrightBig/bracketleftbigg2(2−σi)
τi/vextendsingle/vextendsingleyi/vextendsingle/vextendsingle/bracketrightbigg
≤2(2−σ
i)
τi/vextendsingle/vextendsingleyi/vextendsingle/vextendsingleξd
i
+2(2−σi)
τi/parenleftBig
1+yd
i/parenrightBig
=ψi1/parenleftbig
yi/parenrightbig
ξd
i+ψi2yd
i+ψi3 (A9)
where ˜ψi=2(2n−1+2i)
τi(2n+1),ψi1(yi)=˜ψi|yi|andψi2=ψi3=˜ψi. |
1.3702631.pdf | Relativistic explicit correlation: Coalescence conditions and practical suggestions
Zhendong Li, Sihong Shao, and Wenjian Liu
Citation: The Journal of Chemical Physics 136, 144117 (2012); doi: 10.1063/1.3702631
View online: http://dx.doi.org/10.1063/1.3702631
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This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.114.34.22 On: Tue, 02 Dec 2014 01:47:47THE JOURNAL OF CHEMICAL PHYSICS 136, 144117 (2012)
Relativistic explicit correlation: Coalescence conditions
and practical suggestions
Zhendong Li,1,a)Sihong Shao,2,a)and Wenjian Liu1,b)
1Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry,
State Key Laboratory of Rare Earth Materials Chemistry and Applications, College of Chemistry andMolecular Engineering, and Center for Computational Science and Engineering, Peking University, Beijing100871, People’s Republic of China
2LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
(Received 25 January 2012; accepted 26 March 2012; published online 13 April 2012)
To set up the general framework for relativistic explicitly correlated wave function methods,
the electron-electron coalescence conditions are derived for the wave functions of the Dirac-
Coulomb (DC), Dirac-Coulomb-Gaunt (DCG), Dirac-Coulomb-Breit (DCB), modified Dirac-Coulomb (MDC), and zeroth-order regularly approximated (ZORA) Hamiltonians. The manipula-
tions make full use of the internal symmetries of the reduced two-electron Hamiltonians such that
the asymptotic behaviors of the wave functions emerge naturally. The results show that, at the coa-lescence point of two electrons, the wave functions of the DCG Hamiltonian are regular, while those
of the DC and DCB Hamiltonians have weak singularities of the type r
ν
12withνbeing negative and
ofO(α2). The behaviors of the MDC wave functions are related to the original ones in a simple
manner, while the spin-free counterparts are somewhat different due to the complicated electron-
electron interaction. The behaviors of the ZORA wave functions depend on the chosen potential in
the kinetic energy operator. In the case of the nuclear attraction, the behaviors of the ZORA wave
functions are very similar to those of the nonrelativistic ones, just with an additional correction of
O(α2) to the nonrelativistic cusp condition. However, if the Coulomb interaction is also included,
the ZORA wave functions become close to the large-large components of the DC wave functions.
Note that such asymptotic expansions of the relativistic wave functions are only valid within an ex-
tremely small convergence radius RcofO(α2). Beyond this radius, the behaviors of the relativistic
wave functions are still dominated by the nonrelativistic limit, as can be seen in terms of direct per-
turbation theory (DPT) of relativity. However, as the two limits α→0 and r12→0 do not commute,
DPT is doomed to fail due to incorrect descriptions of the small-small component /Psi1SSof the DC
wave function for r12<Rc. Another deduction from the possible divergence of /Psi1SSatr12=Rcis
that the DC Hamiltonian has no bound electronic states, although the last word cannot be said. These
findings enrich our understandings of relativistic wave functions. On the practical side, it is shownthat, under the no-pair approximation, relativistic explicitly correlated wave function methods can be
made completely parallel to the nonrelativistic counterparts, as demonstrated explicitly for MP2-F12.
Yet, this can only be achieved by using an extended no-pair projector. © 2012 American Institute of
Physics .[http://dx.doi.org/10.1063/1.3702631 ]
I. INTRODUCTION
It has been well recognized, already from the early
days of quantum mechanics, that orbital products (Slater de-
terminants) fail to model the exact wave function of thenonrelativistic Schrödinger-Coulomb equation at short inter-
electronic distances. The direct consequence is that electron
correlation energies calculated by orbital-based methods con-
verge extremely slowly with respect to the basis set size. The
situation can only be improved by using explicitly correlatedtrial wave functions that depend explicitly on the interelec-
tronic distances r
ij. This was demonstrated by Hylleraas,1al-
ready in 1929, for the ground state of helium. However, hisansatz was motivated by the observation that the helium
1S
a)Z. Li and S. Shao contributed equally to this work.
b)Author to whom correspondence should be addressed. Electronic mail:
liuwjbdf@gmail.com.state depends only on the shape of the electron-nucleus tri-
angle rather than by a consideration of the Coulomb singu-
larities. Much later, it was proved rigorously by Kato2that
the Schrödinger-Coulomb Hamiltonian is self-adjoint on thesecond Sobolev space and the corresponding wave function is
bounded, continuous, and must satisfy the following “corre-
lation cusp condition”:
lim
r12→0/parenleftbigg∂/Psi1
∂r12/parenrightbigg
av=1
2/Psi1(r12=0). (1)
Here, the subscript “av” represents the average over the an-
gular part of the relative coordinate /vectorr12. This condition arises
directly from the requirement that the divergent Coulomb
interaction at the electron-electron coalescence point
(r12=0) should precisely be compensated by the local
kinetic energy, so as to result in a finite local energy. The
most important implication of this condition lies in that the
0021-9606/2012/136(14)/144117/23/$30.00 © 2012 American Institute of Physics 136, 144117-1
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
128.114.34.22 On: Tue, 02 Dec 2014 01:47:47144117-2 Li, Shao, and Liu J. Chem. Phys. 136, 144117 (2012)
exact wave function is linear in r12and hence has discon-
tinuous first order derivatives around the coalescence point.
More general cusp conditions were then derived by Packand Brown
3for molecular systems. Based on such analytic
structures of the exact wave functions, a bunch of explicitly
correlated wave function methods have been developed in thelast decades.
4–8In particular, by augmenting the conventional
excitations into products of unoccupied one-electron orbitals
by just a small set of explicitly correlated configurations andcarefully factorizing the difficult many-electron integrals into
products of simple one- and two-electron integrals through
the resolution of the identity (RI) (Ref. 9) with a complemen-
tary auxiliary basis set (CABS),
10the R12/F12 methods11,12
have now evolved into practical tools for general molecules.
The request for relativistic explicitly correlated wave
function methods for systems containing heavy atoms is even
more imperative, as the relativistic corrections converge moreslowly with respect to the basis set size than the nonrelativistic
correlation energies.
13–15Moreover, even the medium-quality
basis sets for heavy atoms are already too large, such that fur-ther increasing the basis set size would result in formidable
computational costs. However, at variance with the significant
advances in nonrelativistic explicitly correlated methods, thedevelopment of the relativistic counterparts lags far behind.
The increased complexities and the reduced symmetries cer-
tainly result in substantial technical difficulties but they arenot really an issue. Rather, it is the lack of knowledge on the
analytical structures of the relativistic wave functions that is
the major obstacle. To put relativistic explicit correlated wave
function methods on a firm ground, at least the following
problems must be addressed seriously:
P(1) Spectral properties of a given relativistic many-
electron Hamiltonian (e.g., Dirac-Coulomb (DC),
Dirac-Coulomb-Gaunt (DCG), Dirac-Coulomb-Breit(DCB), or their approximate variants).
P(2) Asymptotic behaviors of the exact relativistic wave
functions at the electron-nucleus and electron-electron
coalescence points as well as the three-particle coales-
cence points.
P(3) Essential impacts of the asymptotic behaviors on the
rates of convergence of relativistic corrections.
P(4) Practical strategies to incorporate the asymptotic be-
haviors into explicitly correlated trial wave functions.
The properties and solutions of the one-electron Dirac equa-
tion have been well understood.
16In particular, it has been
proven17,18that the Dirac resolvent is holomorphic in the fine
structure αaround the nonrelativistic limit (nrl). This provides
a firm basis for the α-expansion of one-electron wave func-
tions and energies. In contrast, the situation for the many-
body problem is very different. The α-holomorphy of the
Dirac resolvent has only been shown for two particles sub-
ject to an attractive and bounded interaction.19The very basic
spectral properties of the first-quantized, configuration-spaceDC, DCG, and DCB Hamiltonians, e.g., the self-adjointness
and the existence of a point spectrum, still remain to be
very hard mathematical problems and have recently pro-voked much attention of hardcore mathematicians.
20Fortu-
nately, such problems are of no physical relevance as oneshould not try to solve the first-quantized, configuration-space
DC/DCG/DCB equation per se. Rather, it is the “potential-
independent no-pair approximation +perturbative QED”
approach21that should be adopted. This is because only QED
(an intrinsically time-dependent approach), but neither the
configuration-space (CS; associated with the empty Dirac pic-ture and particle conserving)
22nor the Fock-space (FS; asso-
ciated with the filled Dirac picture and charge conserving)23
approach of relativistic quantum chemistry, provides the cor-
rect prescription on the correlation aspects of the Dirac nega-
tive energy states (NES): The NES are correlating in QED but
anticorrelating (i.e., energy increasing when included in the
correlation treatment) in both CS and FS! CS and FS even
fail to describe correctly the one-body terms involving theNES, but which are precisely the terms that are responsible
for removing the intrinsic errors of order ( Zα)
3of the no-pair
DC/DCG/DCB equation. Therefore, the term “exact relativis-tic wave function” is physically meaningless. However, the
analytical structures of the projected relativistic wave func-
tions are more difficult, if not impossible, to be investigatedthan those of the non-projected ones, for a unique and ex-
act projector does not exist. Moreover, the analytical struc-
tures of the wave functions with different projectors are notnecessarily the same. Again fortunately, the direct knowledge
on the analytical structures of the projected relativistic wave
functions is not really needed. Rather, that of the “exact rel-ativistic wave functions” can directly be transplanted to the
no-pair approximation ( vide post ). Note also that the math-
ematical form of the asymptotic behavior of a wave func-
tion at the coalescence point of two electrons is indepen-
dent of the physical nature of the state, bound or scattering.The first analysis of the “exact relativistic wave functions”
was made by Kutzelnigg,
24who found that the wave func-
tion of the DC equation has a weak singularity of the type
rν
12, with ν=/radicalBig
1−α2
4−1 being only slightly less than zero,
while those of the DCG ( ν=√
1+α2−1>0) and DCB
(ν=0) equations are both regular at the electron-electron co-
alescence. Yet, no detailed derivations were provided therein.Therefore, more general and detailed analyses of the rela-
tivistic many-electron wave functions are still highly desired.
In particular, it will be shown that the two results obtainedby Kutzelnigg
24for the DCG and DCB equations should be
revised.
At variance with the little information on the electron-
electron coalescence conditions, the electron-nucleus coales-
cence conditions can readily be established. As an illustration,
we consider the radial part of the large component ψLof the
Dirac bispinor ( ψL,ψS)T. It can be expanded in power of the
electron-nucleus distance r
ψL(r)=Nrν(f(0)+f(1)r+f(2)r2+··· ), (2)
where Nandf(i)are the respective normalization constant
and expansion coefficients, all of which are dependent onthe principle quantum number nand the relativistic angu-
lar momentum quantum number κ=(2j+1)(l−j). The
singularity in the nuclear attraction potential dictates that
ν=/radicalbig
κ2−(Zα)2−1, with Zbeing the nuclear charge. A
straightforward manipulation of the exact solution25of the
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one-electron Dirac equation leads to the following results:
ν=l+O(α2),f(0)=2(l+1)+O(α2),
f(1)
f(0)=−Z
l+1+O(α2), (3)
forκ=−(l+1) (i.e., s1/2,p3/2,d5/2,f7/2, etc.), and
ν=l−1+O(α2),f(0)=O(α2),
f(1)=l−n
2l+1+O(α2),f(2)
f(1)=−Z
l+1+O(α2),
(4)
forκ=l(i.e., p1/2,d3/2,f5/2, etc.). Thus, in the limit of
α→0, both cases reduce to the well-known nonrelativistic
nuclear cusp condition3
f(1)
nrl/slashbig
f(0)
nrl=−Z
l+1. (5)
Note that, for the case of κ=l, the leading term f(0)vanishes
atα→0. Therefore, it is the ratio f(2)/f(1)instead of f(1)/f(0)
that reduces to the nonrelativistic nuclear cusp condition.
As for the rates of convergence of relativistic corrections,
it was shown recently by Kutzelnigg26that the leading partial
wave increment (PWI) of the first order relativistic correctionE
2to the ground state of helium-like ions goes as ( l+1
2)−2,
which is much slower than the leading PWI (Ref. 27)o ft h e
nonrelativistic correlation energy that goes as ( l+1
2)−4.A s
emphasized by Kutzelnigg, both types of PWI are entirely de-
termined by the nonrelativistic correlation cusp equation (1).
More insights can be gained by means of double perturbationtheory that treats both relativity and electron-electron inter-
action as perturbations, with the nonrelativistic bare-nuclear
Hamiltonian as zeroth order, viz.,
/Psi1=/Psi1
(0,0)+λ/Psi1(0,1)+α2/Psi1(2,0)+λ2/Psi1(0,2)+α2λ/Psi1(2,1)
+α4/Psi1(4,0)+···, (6)
E=E(0,0)+Ecorr+Erel+Erel/corr, (7)
Ecorr=∞/summationdisplay
n=1λnE(0,n), (8)
Erel=∞/summationdisplay
m=1α2mE(2m,0), (9)
Erel/corr=∞/summationdisplay
m,n=1α2mλnE(2m,n). (10)
Here, the first and second superscripts denote the respec-
tive orders in relativity and electron-electron interaction. Inparticular, /Psi1
(0, 0)is simply the antisymmetrized product of
nonrelativistic hydrogenic orbitals. For the ground state of
helium-like ions, the ( l+1
2)−4type of leading PWI of
the nonrelativistic correlation energy Ecorr,E q . (8),i s
known28,29to arise from the second order term E(0,2)=/angbracketleft/Psi1(0,0)|1
r12|/Psi1(0,1)/angbracketright, with /Psi1(0, 1)ofO(r1
12). As for the rel-
ativistic corrections, only the leading order ( m=1) terms
can be considered here as some of the higher order terms
are singular ( vide post ). The first order uncorrelated relativis-
tic correction E(2, 0)inErel,E q . (9), can directly be eval-
uated and hence does not contribute to the PWI. It is also
straightforward to show that only the l=0 terms contribute
toE(2, 1),E q . (10). The leading PWI of the cross term E(2, 2),
Eq.(10) goes as ( l+1
2)−2due to the mass-velocity term
/angbracketleft/Psi1(0, 1)|T1T2|/Psi1(0, 1)/angbracketright. It is therefore clear that the observed
leading PWI (Ref. 26)o fE2=/summationtext∞
n=1E(2,n), i.e., first order
in relativity but infinite order in correlation, is due to the low-
est order interplay E(2, 2)between relativity and correlation.
Note in passing that the two-electron Darwin term, going also
as (l+1
2)−2, only appears in the Breit-Pauli Hamiltonian15,26
but not in direct perturbation theory (DPT).30Although only
sketched, the above results can readily be understood in terms
of the rule of thumb: If the integrand is of O(r−k
12), the lead-
i n gP W Iw o u l dg oa s( l+1
2)−4+k. It is certainly worthwhile
to carry out similar analysis on other states than1S. In partic-
ular, the leading PWI of the energy that is finite order in cor-
relation but nonexpanded in relativity is highly wanted. Suchresults will be very helpful for designing relativistic extrapo-
lation methods.
Given the above as yet unresolved theoretical issues,
there have been attempts towards relativistic explicitly corre-
lated wave function methods, based on either the DC Hamil-
tonian or its approximate variants. The former includes the
relativistic extensions of the Hylleraas-type CI (Refs. 31–33)
and the free iterative complement interaction.
34As indi-
cated above, the energies by such methods are always a
bit too high by order ( Zα)3due to the incorrect treatment
of NES. In addition, such methods cannot readily be ap-plied to general molecules due to the involvement of compli-
cated integrals. The latter includes the quantum Monte Carlo
(QMC) method
35,36combined with the spin-free zeroth-
order regular approximation (ZORA) (Refs. 37and38)a s
well as the relativistic R12/F12 approach combined with
the first order DPT,14relativistic effective core potentials,39
and spin-free one-electron second order Douglas-Kroll-Hess
Hamiltonian.40Note that only the s-wave electron-nucleus
coalescence condition ( rνwithν=/radicalbig
l(l+1)+1−(Zα)2
−1) has been considered in the ZORA-QMC implementa-
tion. Such methods based on the approximate Hamiltonians
work well only for not too heavy atoms. For heavy and super-heavy atoms, four-component relativistic explicitly correlated
methods based on the no-pair DC/DCG/DCB Hamiltonian
should be developed. As pointed out before,
21anextended
no-pair projection has to be introduced here to avoid the con-
taminations of NES.
Among the four problems, P(2) is the basis of P(3) and
P(4) and may also be helpful for resolving P(1). Therefore, it
is going to be the focus of the present account. The consid-ered Hamiltonians include DC, DCG, DCB, spin-free part of
modified DC,
41,42as well as ZORA.37,38The manipulations
will make full use of the underlying symmetries of the equa-tions, such that the asymptotic behaviors of the wave func-
tions emerge naturally. Based on the theoretical findings, a
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four-component explicitly correlated second order Møller-
Plesset perturbation theory (MP2-F12) is proposed in
conjunction with the no-pair DC/DCG/DCB Hamiltonian.Throughout the paper, use of the atomic units ( e=m=¯
=1) and the Einstein summation convention over repeated
indices will be made.
II. THE DC, DCB, AND DCG HAMILTONIANS
TheN-electron Dirac equation reads
ˆH/Psi1(1,2,...,N )=E/Psi1(1,2,...,N ), (11)
where the Hamiltonian is defined in the framework of
clamped nuclei as
ˆH=/summationdisplay
kˆhD
k+/summationdisplay
k>lˆgkl, (12)
ˆhD
k=c/vectorαk·/vectorpk+βkc2+φk, (13)
φk=−/summationdisplay
AZA
rkA,r kA=| /vectorrk−/vectorrA|. (14)
Here, ˆhD
kis the one-electron Dirac operator for electron ksub-
ject to the nuclear attraction φk. The constant c=1/αis the
speed of light and /vectorp=−i/vector∇is the linear momentum opera-
tor./vectorαandβare the usual 4 ×4 Dirac matrices. The electron-
electron interaction operator ˆgklcan be put into a generic form
ˆgkl=dC1
rkl+dG/vectorαk·/vectorαl
rkl+dR(/vectorαk·ˆrkl)(/vectorαl·ˆrkl)
rkl,(15)
with ˆrkl=/vectorrkl/rkl. The Coulomb interaction is represented by
ˆgklwith (1, 0, 0) for the coefficients ( dC,dG,dR). Likewise,
the Coulomb-Gaunt and Coulomb-Breit interactions are re-
covered by the coefficients (1, −1, 0) and (1, −1/2,−1/2),
respectively. In addition, the attractive Coulomb interactionbetween electron-positron pairs can also be covered by using
the coefficients ( −1, 0, 0).
TheN-electron relativistic wave function /Psi1in Eq. (11)
has 4
Ncomponents, each of which depends on 3 Nspatial co-
ordinates of electrons,
/Psi1X1X2···XN(/vectorr1,/vectorr2,...,/vectorrN),X k∈{Lα,L β,Sα,S β }.(16)
These components are not completely independent, since the
antisymmetry principle dictates that they must satisfy the fol-
lowing relation:
/Psi1X1X2···Xk···Xl···XN(/vectorr1,/vectorr2,...,/vectorrk,...,/vectorrl,...,/vectorrN)
=−/Psi1X1X2···Xl···Xk···XN(/vectorr1,/vectorr2,...,/vectorrl,...,/vectorrk,...,/vectorrN).
(17)
Note in passing that this antisymmetry relation holds only for
the individual components but not for the blocks (cf. Eq. (74)).
This feature is not really obvious. Therefore, more detaileddiscussions on the antisymmetrization of orbital products are
given in supplementary material.
43A. The reduced two-electron problem
To investigate the electron-electron coalescence condi-
tions, suffice it to concentrate only on the relative motion of
two electrons at small interelectronic distances. For this pur-
pose, the coordinates /vectorr1and/vectorr2are transformed to the center
of mass /vectorR12and relative /vectorr12coordinates of two coalescing
electrons, viz.,
/vectorR12=1
2(/vectorr1+/vectorr2),/vectorr12=/vectorr1−/vectorr2, (18)
from which the corresponding momenta can be derived:
/vectorP12=/vectorp1+/vectorp2,/vectorp12=1
2(/vectorp1−/vectorp2). (19)
In terms of such transformations, Eq. (11) can be rewritten as
ˆh12/Psi1=ˆW/Psi1, (20)
where
ˆh12=ˆt12+ˆg12,ˆt12=c(/vectorα1−/vectorα2)·/vectorp12, (21)
and ˆWcontains all the remaining terms
ˆW=E−/parenleftbigg/summationdisplay
k≥3ˆhD
k+/summationdisplay
k>l≥3ˆgkl/parenrightbigg
−/summationdisplay
k=1,2/parenleftbigg1
2c/vectorαk·/vectorP12+βkc2/parenrightbigg
−/summationdisplay
k=1,2/parenleftbigg
φk+/summationdisplay
l≥3ˆgkl/parenrightbigg
. (22)
The operators ˆt12and ˆg12describe, respectively, the relative
kinetic energy and interaction energy of electrons 1 and 2,while ˆWprovides damping on the interaction between the two
electrons due to the rest of the system, including the electro-
static interaction of the two electrons with the rest of the sys-tem, the kinetic and potential terms of the other electrons, as
well as the kinetic energy of the center of mass motion.
Consider the region of configuration space where elec-
trons 1 and 2 are close together and all the other electrons
and nuclei are well separated from these two electrons and
from each other, viz., 0 ≤r
12≤/epsilon1andrRA=|/vectorR12−/vectorrA|,
rRk=|/vectorR12−/vectorrk|,rkl=| /vectorrk−/vectorrl|/greatermuch/epsilon1fork,l≥3. Here, /epsilon1is an
arbitrary small positive number. Within this region, the wave
function /Psi1can be expanded in power of r12as
/Psi1=/Psi1(ν)+/Psi1(ν+1)+···, (23)
where νis the lowest power of nonvanishing /Psi1inr12.T h e
operators ˆh12,E q . (21), and ˆW,E q . (22), can also be expanded
in the same way. In particular, both ˆt12and ˆg12inˆh12lower
the power of r12by one and can hence be labeled as ˆt(−1)
12and
ˆg(−1)
12, leading to ˆh(−1)
12. For the potential terms in Eq. (22),t h e
following partial wave expansions can be invoked:
1
r1A=1/vextendsingle/vextendsingle/vectorrRA+1
2/vectorr12/vextendsingle/vextendsingle=1
rRA+∞/summationdisplay
l=0(−1)l/parenleftbiggr12
2rRA/parenrightbiggl
Pl(cosθA),
1
r2A=1/vextendsingle/vextendsingle/vectorrRA−1
2/vectorr12/vextendsingle/vextendsingle=1
rRA+∞/summationdisplay
l=0/parenleftbiggr12
2rRA/parenrightbiggl
Pl(cosθA),
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128.114.34.22 On: Tue, 02 Dec 2014 01:47:47144117-5 Li, Shao, and Liu J. Chem. Phys. 136, 144117 (2012)
1
r1k=1/vextendsingle/vextendsingle/vectorrRk+1
2/vectorr12/vextendsingle/vextendsingle=1
rRk+∞/summationdisplay
l=0(−1)l/parenleftbiggr12
2rRk/parenrightbiggl
Pl(cosθk),
1
r2k=1/vextendsingle/vextendsingle/vectorrRk−1
2/vectorr12/vextendsingle/vextendsingle=1
rRk+∞/summationdisplay
l=0/parenleftbiggr12
2rRk/parenrightbiggl
Pl(cosθk), (24)
where θAis the angle between /vectorrRAand/vectorr12,θkis the angle
between /vectorrRkand/vectorr12, and Plare the Legendre polynomials.
The operator ˆWthen becomes
ˆW=ˆW(0)+O(/epsilon12)+···, (25)
ˆW(0)=E−/parenleftbigg/summationdisplay
k≥3ˆhD
k+/summationdisplay
k>l≥3ˆgkl/parenrightbigg
−/summationdisplay
k=1,2/parenleftbigg1
2c/vectorαk·/vectorP12+βkc2/parenrightbigg
−/summationdisplay
k=1,2/parenleftbigg
φ(0)
k+/summationdisplay
l≥3ˆg(0)
kl/parenrightbigg
, (26)
where φ(0)
kandˆg(0)
klfork=1, 2 arise from the s-wave ( l=0)
terms in Eq. (24) and can be summed up as
φ(0)
1+φ(0)
2=−2/summationdisplay
AZA
rRA:=2φ(0), (27)
ˆg(0)
1l+ˆg(0)
2l=dC2
rRl+dG(/vectorα1+/vectorα2)·/vectorαl
rRl
+dR(/vectorα1·ˆrRl)(/vectorαl·ˆrRl)+(/vectorα2·ˆrRl)(/vectorαl·ˆrRl)
rRl.
(28)
It is important to realize that all the odd order terms ˆW(2n+1)
(n∈N)i nE q . (25) vanish identically due to the cancelation
of the odd lterms for electrons 1 and 2.
By collecting the terms of the same order, Eq. (20) gives
rise to a set of equations, with the lowest three orders being
O(/epsilon1ν−1):/parenleftbigˆt(−1)
12+ˆg(−1)
12/parenrightbig
/Psi1(ν)=0, (29)
O(/epsilon1ν):/parenleftbigˆt(−1)
12+ˆg(−1)
12/parenrightbig
/Psi1(ν+1)=ˆW(0)/Psi1(ν), (30)
O(/epsilon1ν+1):/parenleftbigˆt(−1)
12+ˆg(−1)
12/parenrightbig
/Psi1(ν+2)=ˆW(0)/Psi1(ν+1).(31)
The coalescence condition (29) is essential for ensuring that
the local energy
EL=ˆH/Psi1
/Psi1(32)
remains finite at the coalescence point. This can readily be
understood as follows:
lim
r12→0EL=lim
r12→0(ˆh12−ˆW+E)/Psi1
/Psi1(33)
=lim
r12→0ˆh(−1)
12/Psi1(ν)+/bracketleftbigˆh(−1)
12/Psi1(ν+1)+(−ˆW(0)+E)/Psi1(ν)/bracketrightbig
+···
/Psi1(ν)+/Psi1(ν+1)+···
(34)=lim
r12→0ˆh(−1)
12/Psi1(ν)
/Psi1(ν)+lim
r12→0/parenleftBigg
E+ˆh(−1)
12/Psi1(ν+1)−ˆW(0)/Psi1(ν)
/Psi1(ν)/parenrightBigg
+···,
(35)
where Eq. (33) arises from Eq. (32) and the relation
ˆH=ˆh12−ˆW+E. The first term of Eq. (35) is ofO(r−1
12)
and hence will diverge if the wave function does not satisfy
condition (29) properly.
Before going into details, it is at this stage instructive to
compare Eqs. (29)–(31) with the nonrelativistic counterparts:
O(/epsilon1ν−2):ˆtS(−2)
12/Psi1(ν)
S=0, (36)
O(/epsilon1ν−1):ˆtS(−2)
12/Psi1(ν+1)
S+ˆgS(−1)
12/Psi1(ν)
S=0, (37)
O(/epsilon1ν):ˆtS(−2)
12/Psi1(ν+2)
S+ˆgS(−1)
12/Psi1(ν+1)
S=ˆWS(0)/Psi1(ν)
S, (38)
O(/epsilon1ν+1):ˆtS(−2)
12/Psi1(ν+3)
S+ˆgS(−1)
12/Psi1(ν+2)
S=ˆWS(0)/Psi1(ν+1)
S,
(39)
where
ˆtS(−2)
12=/vectorp2
12,ˆgS(−1)
12=1
r12, (40)
ˆWS(0)=E−/parenleftbigg/summationdisplay
k≥3ˆhS
k+/summationdisplay
k>l≥3ˆgS
kl/parenrightbigg
−1
4/vectorP2
12
−/summationdisplay
k=1,2/parenleftbigg
φ(0)
k+/summationdisplay
l≥3ˆgS(0)
kl/parenrightbigg
. (41)
The most important difference in between is that the non-
relativistic kinetic energy operator ˆtS(−2)
12 is a second order
differential operator, while the relativistic one ˆt(−1)
12 is only
first order. Consequently, the lowest order equation (36) for
the Schrödinger equation is of O(/epsilon1ν−2), which determines the
asymptotic behavior of the wave function /Psi1(ν)
Sasrl
12Yml
l, with
ν=landYml
lbeing the spherical harmonics. The next or-
der equation (37) involves ˆgS(−1)
12, whose singularity results in
discontinuous ( l+1)th order derivatives characterized by the
cusp condition3
/parenleftBigg
∂l+1/Psi1S
∂rl+1
12/parenrightBigg
r12=0=1
2/parenleftbigg∂l/Psi1S
∂rl
12/parenrightbigg
r12=0, (42)
which reduces to the Kato cusp condition (1)forl=0. Both
conditions (36)and(37) have to be satisfied for a finite local
energy (32) of the Schödinger equation. The next two order
equations (38)and(39) can be employed to derive the second
and third order coalescence conditions44that are no longer
universal but system and state dependent and vary throughoutconfiguration space due to the involvement of the ˆW
S(0)oper-
ator. In contrast, in the relativistic case, the lowest order equa-
tion(29) is only of O(/epsilon1ν−1), to which the singular term ˆg(−1)
12
has already entered. At variance with the universality of the
first order nonrelativistic cusp condition (42), the relativistic
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counterpart, i.e., the relation between /Psi1(ν+1)and/Psi1(ν)deter-
mined by Eq. (30), cannot be universal due to the appearance
ofˆW(0),E q . (26), that depends on the system and state. Note
also that the ˆW(0)operator (26) is more complicated than the
nonrelativistic counterpart ˆWS(0),E q . (41), since the former
does not commute with all the symmetry operations of ˆh12
(vide post ). This will result in great difficulties in manipulat-
ing Eqs. (30)and(31). In short, there does not exist a simple
relativistic analog of the nonrelativistic cusp condition (42).
Another significant difference between the relativistic
and nonrelativistic cases lies in that the relativistic wave func-
tion/Psi1in Eq. (20) has 16 components depending on the rela-
tive coordinate /vectorr12and the spin degrees of freedom, while the
nonrelativistic one /Psi1Sis simply a scalar function. As such,
the relativistic local energy (32) is also a multi-component
function, with each component being the ratio between the
corresponding components of ˆH/Psi1 and/Psi1. Therefore, making
full use of the internal symmetries of the reduced Hamiltonian
ˆh12is essential to simplify the manipulations.
B. Symmetries of ˆh12
In this section we mainly focus on the homogeneous
equation (29). The superscript ( ν)o f/Psi1can hence be dropped
for simplicity. Equation (29) can be rewritten in block form
⎛
⎜⎜⎜⎜⎝V
C −c/vectorσ2·/vectorp12c/vectorσ1·/vectorp12 VB
−c/vectorσ2·/vectorp12 VC VB c/vectorσ1·/vectorp12
c/vectorσ1·/vectorp12 VB VC −c/vectorσ2·/vectorp12
VB c/vectorσ1·/vectorp12−c/vectorσ2·/vectorp12 VC⎞
⎟⎟⎟⎟⎠
×⎛
⎜⎜⎜⎜⎝/Psi1
LL
/Psi1LS
/Psi1SL
/Psi1SS⎞
⎟⎟⎟⎟⎠=0, (43)
where each block /Psi1XY(X,Y∈{L,S}) is a four-component
function. The operators VCandVBread
VC=dC1
r12,V B=dG/vectorσ1·/vectorσ2
r12+dR(/vectorσ1·ˆr12)(/vectorσ2·ˆr12)
r12.
(44)
To reveal the symmetry properties of ˆh12, we rewrite it as43
ˆh12=c/vectorσ1·/vectorp12C1−c/vectorσ2·/vectorp12C2+VCE+VBC12(45)
in terms of the following “block operators” that merely inter-
change the blocks /Psi1XYof the wave function:
E=I4◦I4=/parenleftbiggI20
0I2/parenrightbigg
◦/parenleftbiggI20
0I2/parenrightbigg
=⎛
⎜⎜⎝I4000
0I400
00 I40
000 I4⎞
⎟⎟⎠,
(46)C1=γ5◦I4=⎛
⎝0I2
I20⎞
⎠◦⎛
⎝I20
0I2⎞
⎠=⎛
⎜⎜⎜⎜⎜⎝00 I
40
000 I4
I4000
0I400⎞
⎟⎟⎟⎟⎟⎠,
(47)
C
2=I4◦γ5=⎛
⎝I20
0I2⎞
⎠◦⎛
⎝0I2
I20⎞
⎠=⎛
⎜⎜⎜⎜⎜⎝0I
400
I4000
000 I4
00 I40⎞
⎟⎟⎟⎟⎟⎠,
(48)
C
12=γ5◦γ5=C1C2=C2C1=⎛
⎜⎜⎜⎜⎜⎝000 I
4
00 I40
0I400
I4000⎞
⎟⎟⎟⎟⎟⎠,
(49)
where the symbol ◦represents the Tracy-Singh product, 45,46
which is a generalization of the standard Kronecker product
(⊗) for partitioned matrices. The multiplications between the
“component operators” (e.g., c/vectorσ1·/vectorp12) and the “block op-
erators” (e.g., C1)i nE q . (45) are similar to those between
numbers and matrices (for more details see Ref. 43). Such
a formulation is particularly advantageous in that the block
structure of the wave function in Eq. (43) can always be re-
tained and the symmetry properties of ˆh12can readily be de-
duced. To do so, we first introduce an Abelian group G
G={E,C1,C2,C12}, (50)
which arises as a direct product of groups {E,C1}and
{E,C2}. To the best of our knowledge, such a group has never
been considered before. Being Abelian, each element of G
commutes with ˆh12. It can further be shown43that the follow-
ing operators:
{ˆh12,C12,ˆP12,ˆI,ˆj2
12,ˆj12,z}, (51)
are mutually commutative and hence share the same eigen-
functions. Here, ˆP12is the permutation operator for electrons
1a n d2( vide post ),ˆIis the space inversion for the relative
coordinate /vectorr12, and/vectorj12is the angular momentum
/vectorj12=/vectorl12+/vectors,/vectors=/vectors1+/vectors2, (52)
where /vectorl12=/vectorr12×/vectorp12is the orbital angular momentum of the
relative motion and /vectorskis the spin of electron k. The eigenval-
ues of operators (51)can be employed to classify the solutions
of Eq. (43).
To start off, the eigenfunctions of {ˆj2
12,ˆj12,z}, denoted
as|(ls),jmj/angbracketright, are first constructed via the LScoupling
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equation (52),
|(ls),jm j/angbracketright=+s/summationdisplay
ms=−s|lml/angbracketright|sms/angbracketright/angbracketleftlmlsms|jmj/angbracketright, (53)
where /angbracketleftlmlsms|jmj/angbracketrightare the Clebsch-Gordan coefficients. Note
that the quantum number sof the total spin-angular momen-
tum/vectorscan only be 0 (singlet) or 1 (triplet) for two electrons.
Given j, the quantum number lof/vectorl12can only be jfors=0
and can be j+1,j,o rj−1f o r s=1. For simplicity, the four
possible eigenfunctions are to be denoted as /Omega1i
/Omega11=|(l=j,s=0),jm j/angbracketright, (54)
/Omega12=|(l=j,s=1),jm j/angbracketright, (55)
/Omega13=|(l=j−1,s=1),jm j/angbracketright, (56)
/Omega14=|(l=j+1,s=1),jm j/angbracketright, (57)
which form an orthonormal basis set for the subspace of given
jandmj. As the parity of |(ls),jmj/angbracketrightis (−1)l, the four functions
/Omega1ican be classified into two groups, one with parity ( −1)j
(i.e., l=j) including /Omega11and/Omega12, and the other with parity
(−1)j+1(i.e., l=j±1) including /Omega13and/Omega14. Thus, for given
jandmj, the components /Psi1XYof/Psi1can be expressed as
/Psi1XY
+=fXY
1/Omega11+fXY
2/Omega12,X , Y ∈{L,S}, (58)
with parity +(−1)jor as
/Psi1XY
−=fXY
3/Omega13+fXY
4/Omega14,X , Y ∈{L,S}, (59)
with parity −(−1)j. While the amplitudes fXY
iare dependent
not only on the radial part of /vectorr12but also on the center of
mass coordinate /vectorR12of the two electrons as well as all the
coordinates of the rest of the system, the /Omega1idepend only on
the spin-angular part of the relative coordinate /vectorr12. Therefore,
the notation2s+1lj(s=0, 1;l=s,p,d,...; j=0 ,1 ,...)f o rt h e
states to be used below should not be confused with the true
spectroscopic terms that involve the total angular momenta of
all the electrons. By noting that /Psi1LLand/Psi1SShave the same
parity, and /Psi1LSand/Psi1SLalso have the same parity but different
from that of /Psi1LLand/Psi1SS(see Ref. 43), the wave function /Psi1
for given jandmjcan be constructed as
/Psi1+=⎛
⎜⎜⎜⎜⎜⎝/Psi1
LL
+
/Psi1LS
−
/Psi1SL
−
/Psi1SS
+⎞
⎟⎟⎟⎟⎟⎠,/Psi1
−=⎛
⎜⎜⎜⎜⎜⎝/Psi1
LL
−
/Psi1LS
+
/Psi1SL
+
/Psi1SS
−⎞
⎟⎟⎟⎟⎟⎠, (60)
with parities +(−1)
jand−(−1)j, respectively.
The function /Psi1+or/Psi1−still has eight unknowns but
which can further be reduced by using C12and ˆP12. Since
the eigenvalues of C12and ˆP12can only be +1o r−1, the
spaces for /Psi1+and/Psi1−can, respectively, be decomposed as
direct sums ( ⊕) of the eigensubspaces
V+=VA
(+,+)⊕VS
(+,+)⊕VA
(+,−)⊕VS
(+,−), (61)
V−=VA
(−,+)⊕VS
(−,+)⊕VA
(−,−)⊕VS
(−,−), (62)where the second subscript +(−) indicates the corresponding
eigenvalue +1(−1) of C12, while the superscript A(S) indi-
cates antisymmetric (symmetric) under the permutation ˆP12.
In addition, the following identities:43
C12C1=C1C12, (63)
ˆIC1=−C1ˆI, (64)
ˆP12C1=C2ˆP12=C1C12ˆP12, (65)
imply that an arbitrary function /Psi1with eigenvalues
{η(C12),η(ˆI),η(ˆP12)}will be transformed to a function C1/Psi1
with eigenvalues {η(C12),−η(ˆI),η(C12)η(ˆP12)}. Therefore,
the following relations can be established for functions inspaces V
+andV−:
VA
(−,+)=C1VA
(+,+),VS
(−,+)=C1VS
(+,+), (66)
VA
(−,−)=C1VS
(+,−),VS
(−,−)=C1VA
(+,−). (67)
It can then immediately be deduced that the asymptotic be-
havior of the wave function /Psi1−constructed as
/Psi1−=C1/Psi1+=C1⎛
⎜⎜⎜⎜⎜⎝/Psi1LL
+
/Psi1LS
−
/Psi1SL
−
/Psi1SS
+⎞
⎟⎟⎟⎟⎟⎠=⎛
⎜⎜⎜⎜⎜⎝/Psi1
SL
−
/Psi1SS
+
/Psi1LL
+
/Psi1LS
−⎞
⎟⎟⎟⎟⎟⎠(68)
is exactly the same as that of /Psi1
+. For instance, the1s0(=/Psi1+)
and3p0(=C1/Psi1+) states will have the same asymptotic be-
haviors. Note that the presentations so far hold for both twoidentical fermions (electrons or positrons) and an electron-
positron pair. For an electronic system, only the antisymmet-
ric parts V
A
(+,+)andVA
(+,−)ofV+,E q . (61), andVA
(−,+)and
VA
(−,−)ofV−,E q . (62), are relevant. Furthermore, because of
the first equality of Eq. (66), we need to only consider the
wave function /Psi1+belonging to VA
(+,+),VA
(+,−), andVS
(+,−).T h e
asymptotic behaviors of /Psi1−belonging to VA
(−,+)andVA
(−,−)
are the same as those of /Psi1+inVA
(+,+)andVS
(+,−), respectively.
To construct explicitly the electronic wave functions /Psi1+,
we first note that the eigenfunctions of C12are simply
⎛
⎜⎜⎜⎜⎝ϕ
1
ϕ2
ϕ2
ϕ1⎞
⎟⎟⎟⎟⎠,⎛
⎜⎜⎜⎜⎝ϕ
1
ϕ2
−ϕ2
−ϕ1⎞
⎟⎟⎟⎟⎠, (69)
with eigenvalues of +1 and −1, respectively. Additional re-
strictions on the amplitudes f
XY
iare further imposed by the
antisymmetry principle. To see this, we write the permutationoperator ˆP
12as
ˆP12=ˆπ12ˆ/Pi112/Pi112, (70)
where ˆ π12interchanges the spatial coordinates, viz.,
ˆπ12f(/vectorr1,/vectorr2)=f(/vectorr2,/vectorr1),ˆπ12f(/vectorr12,/vectorR12)=f(−/vectorr12,/vectorR12),
(71)
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while the “component operator” ˆ/Pi112acts on the blocks
/Psi1XY(X,Y∈{L,S}) of/Psi1,
ˆ/Pi112=1
2(I4+/vectorσ1·/vectorσ2)=⎛
⎜⎜⎝1000
0010
01000001⎞
⎟⎟⎠. (72)
The “block operator” /Pi1
12in Eq. (70) is given as
/Pi112=⎛
⎜⎜⎝I4000
00 I40
0I400
000 I4⎞
⎟⎟⎠. (73)
Then the antisymmetry principle ˆP12/Psi1(1,2)=−/Psi1(1,2) dic-
tates that
ˆπ12ˆ/Pi112/Psi1YX(1,2)=−/Psi1XY(1,2),X , Y ∈{L,S}.(74)
Straightforward manipulations further reveal that the respec-
tive actions of ˆ π12and ˆ/Pi112on/Omega1iare
ˆπ12/Omega1i=(−1)l/Omega1i, (75)
ˆ/Pi112/Omega1i=[s(s+1)−1]/Omega1i=(−1)s+1/Omega1i,s∈{0,1}.
(76)
We therefore have
ˆπ12ˆ/Pi112/Omega1i=(−1)l+s+1/Omega1i. (77)
That is, /Omega1iis the eigenfunction of ˆ π12ˆ/Pi112with eigenvalue
(−1)l+s+1.F o r/Psi1XY
+,E q . (58), the action of ˆ π12ˆ/Pi112leads to
ˆπ12ˆ/Pi112/Psi1XY
+=(−1)j+1fXY
1/Omega11+(−1)jfXY
2/Omega12.(78)
In view of Eq. (74), the coefficients must satisfy
fYX
1=(−1)jfXY
1,fYX
2=(−1)j+1fXY
2. (79)
Therefore, fXX
1is nonzero only for even j, while fXX
2is
nonzero only for odd j. Similarly, for /Psi1XY
−,E q . (59), the ac-
tion of ˆ π12ˆ/Pi112leads to
ˆπ12ˆ/Pi112/Psi1XY
−=(−1)j+1fXY
3/Omega13+(−1)j+1fXY
4/Omega14,(80)
such that the coefficients are subject to
fYX
3=(−1)jfXY
3,fYX
4=(−1)jfXY
4. (81)
That is, both fXX
3andfXX
4are nonzero only for even j. These
results together with Eq. (69) lead immediately to the follow-
ing forms for functions /Psi1+inVA
(+,+)andVA
(+,−):
/Psi1A,e
(+,+)=⎛
⎜⎜⎜⎜⎝f
LL
1/Omega11
fLS
3/Omega13+fLS
4/Omega14
fLS
3/Omega13+fLS
4/Omega14
fLL
1/Omega11⎞
⎟⎟⎟⎟⎠, (82)
/Psi1
A,o
(+,+)=⎛
⎜⎜⎜⎜⎝fLL
2/Omega12
0
0
fLL
2/Omega12⎞
⎟⎟⎟⎟⎠, (83)/Psi1A,e
(+,−)=⎛
⎜⎜⎜⎜⎝fLL
1/Omega11
0
0
−fLL
1/Omega11⎞
⎟⎟⎟⎟⎠, (84)
/Psi1A,o
(+,−)=⎛
⎜⎜⎜⎜⎝fLL
2/Omega12
fLS
3/Omega13+fLS
4/Omega14
−/parenleftbig
fLS
3/Omega13+fLS
4/Omega14/parenrightbig
−fLL
2/Omega12⎞
⎟⎟⎟⎟⎠(85)
for even j(denoted by a superscript e) and odd j(denoted by a
superscript o), respectively. The forms for functions in VS
(+,−)
are the same as those in VA
(+,−)if the parity of jis reversed,
viz.,
/Psi1S,e
(+,−)∼/Psi1A,o
(+,−),/Psi1S,o
(+,−)∼/Psi1A,e
(+,−). (86)
Therefore, the asymptotic behaviors of the relativistic wave
functions can simply be deduced from Eqs. (82)–(85).N o -
ticeably, the number of unknowns in /Psi1has been reduced from
16 to 1 for Eqs. (83)and(84) and to 3 for Eqs. (82)and(85).
These results facilitate greatly the subsequent analysis of the
asymptotic behaviors. For completeness, all the eight typesof functions in Eqs. (61)and(62), i.e., the common eigen-
functions of the operators (51), are explicitly documented in
Appendix.
C. Asymptotic behaviors determined by ˆh(−1)
12/Psi1(ν)=0
Having determined the structures of /Psi1(ν),E q s . (82)–(85),
we can now insert /Psi1(ν)into Eq. (43)and integrate out the spin-
angular part /Omega1ito obtain equations for the radial part fXY
i.T o
do so, the actions of /vectorσk·/vectorp12,/vectorσ1·/vectorσ2, and/vectorσk·ˆr12on functions
fXY
i/Omega1ihave to first be evaluated. The identity
/vectorσk·/vectorp12=−i(/vectorσk·ˆr12)/parenleftbigg∂
∂r12−/vectorσk·/vectorl12
r12/parenrightbigg
(87)
shows that only the formulas for /vectorσk·ˆr12and/vectorσk·/vectorl12acting
on/Omega1iare needed to evaluate /vectorσk·/vectorp12(fXY
i/Omega1i). Being scalar
operators, the actions of /vectorσk·ˆr12,/vectorσk·/vectorl12, and/vectorσ1·/vectorσ2on/Omega1ican
be expressed through the RI
ˆQ/Omega1i=4/summationdisplay
i/prime=1/Omega1i/prime/angbracketleft/Omega1i/prime|ˆQ|/Omega1i/angbracketright,ˆQ∈{ /vectorσk·ˆr12,/vectorσk·/vectorl12,/vectorσ1·/vectorσ2},
(88)
where the matrix elements /angbracketleft/Omega1i/prime|ˆQ|/Omega1i/angbracketrightcan systematically be
evaluated using the Wigner-Eckart theorem for compositeoperators.
47The resulting matrices can be summarized as
follows:
[/vectorσk·ˆr12]=⎛
⎜⎜⎝00 ±b∓a
00 −a−b
±b−a 00
∓a−b 00⎞
⎟⎟⎠, (89)
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a=/radicalBigg
j+1
2j+1,b=/radicalBigg
j
2j+1, (90)
[/vectorσk·/vectorl12]=⎛
⎜⎜⎝0 ±√j(j+1) 0
±√j(j+1) −10
00 j−1
00 0 −(j+2)⎞
⎟⎟⎠,
(91)
and
[/vectorσ1·/vectorσ2]=⎛
⎜⎜⎝−3000
01 0 0
00 1 000 0 1⎞
⎟⎟⎠, (92)
where the upper and lower signs in the matrix elements of
[/vectorσ
k·ˆr12] and [ /vectorσk·/vectorl12] correspond to k=1 and 2, respectively.
To expedite subsequent manipulations, we combine the
functions (82)–(85) with different eigenvalues of C12to form
eigenfunctions of {ˆP12,ˆI,ˆj2
12,ˆj12,z}, i.e.,
/Psi1A,e
+=⎛
⎜⎜⎜⎜⎝f
LL
1/Omega11
fLS
3/Omega13+fLS
4/Omega14
fLS
3/Omega13+fLS
4/Omega14
fSS
1/Omega11⎞
⎟⎟⎟⎟⎠, (93)
/Psi1
A,o
+=⎛
⎜⎜⎜⎜⎝fLL
2/Omega12
fLS
3/Omega13+fLS
4/Omega14
−/parenleftbig
fLS
3/Omega13+fLS
4/Omega14/parenrightbig
fSS
2/Omega12⎞
⎟⎟⎟⎟⎠, (94)
each of which has 4 unknowns. These two expressions cover
Eqs. (82)–(85). Substituting /Psi1A,e
+,E q . (93), into Eq. (43) and
integrating the spin-angular part /Omega1igive rise to four equations
forfLL
1,fLS
3,fLS
4, andfSS
1
2i/parenleftbig
−bFLS
3+aFLS
4/parenrightbig
+dCα
r12fLL
1+(−3dG−dR)α
r12fSS
1=0,
(95)
−iFLLSS
1−+(dC+dG+qdR)α
r12fLS
3+(pdR)α
r12fLS
4=0,
(96)
−iFLLSS
1++(pdR)α
r12fLS
3+(dC+dG−qdR)α
r12fLS
4=0,
(97)
2i/parenleftbig
−bFLS
3+aFLS
4/parenrightbig
+dCα
r12fSS
1+(−3dG−dR)α
r12fLL
1=0,
(98)while substituting /Psi1A,o
+,E q . (94), into Eq. (43) leads to an-
other four equations for fLL
2,fLS
3,fLS
4, andfSS
2:
−2i/parenleftbig
aFLS
3+bFLS
4/parenrightbig
+dCα
r12fLL
2+(dG+dR)α
r12fSS
2=0,
(99)
iFLLSS
2−+(dC−dG−qdR)α
r12fLS
3−(pdR)α
r12fLS
4=0,
(100)
iFLLSS
2+−(pdR)α
r12fLS
3+(dC−dG+qdR)α
r12fLS
4=0,
(101)
2i/parenleftbig
aFLS
3+bFLS
4/parenrightbig
+dCα
r12fSS
2+(dG+dR)α
r12fLL
2=0.
(102)
The intermediate quantities in the above equations are defined
as
FLS
3=/parenleftbiggd
dr−j−1
r/parenrightbigg
fLS
3, (103)
FLS
4=/parenleftbiggd
dr+j+2
r/parenrightbigg
fLS
4, (104)
FLLSS
1−=b/parenleftbiggd
dr+j+1
r/parenrightbigg/parenleftbig
fLL
1+fSS
1/parenrightbig
, (105)
FLLSS
1+=a/parenleftbigg
−d
dr+j
r/parenrightbigg/parenleftbig
fLL
1+fSS
1/parenrightbig
, (106)
FLLSS
2−=−a/parenleftbiggd
dr+j+1
r/parenrightbigg/parenleftbig
fLL
2−fSS
2/parenrightbig
,(107)
FLLSS
2+=b/parenleftbigg
−d
dr+j
r)/parenleftbig
fLL
2−fSS
2/parenrightbig
, (108)
p=2√j(j+1)
2j+1=2ab, (109)
q=1
2j+1=a2−b2(110)
with aandbgiven in Eq. (90). The asymptotic behaviors of
/Psi1A,e
+and/Psi1A,o
+can be obtained by inserting the expansions
fXY
i=rν
12fXY(0)
i+O(/epsilon1ν+1),X , Y ∈{L,S},(111)
into the corresponding equations. The value for νand the
mutual relations between fXY(0)
i are determined by the
requirement that the algebraic equations for fXY(0)
i have
nontrivial solutions. If the determinant of the coefficient
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matrix of the algebraic equations is zero for arbitrary ν, the desired νhas to be determined from the next order equation (30).
This particular situation will be discussed in Sec. II D.
1. Algebraic equations for /Psi1A,e
(+,+)
The algebraic equations for fXY(0)
i ,E q . (82), can be obtained from Eqs. (95)–(98) as
Me⎛
⎜⎜⎝fLL(0)
1
fLS(0)
3
fLS(0)
4⎞
⎟⎟⎠=0, (112)
with
Me=⎛
⎜⎜⎝α(dC−3dG−dR)−2ib(ν−j+1) 2 ia(ν+j+2)
−2ib(ν+j+1)α(dC+dG+qdR) αpdR
−2ia(−ν+j) αpdR α(dC+dG−qdR)⎞
⎟⎟⎠. (113)
The determinant of the coefficient matrix Meis
det(Me)=4α{(dC+dG+dR)[(ν+1)2−1]−d},
(114)
with
d=(dC+dG−dR)/braceleftbigg
j(j+1)−[(dC−dG)2
−(2dG+dR)2]α2
4/bracerightbigg
. (115)
Note that nontrivial solutions can only be obtained if
det(Me)=0, from which the value of νcan be determined
ifdC+dG+dR/negationslash=0. The situations for the DC, DCG, and
DCB Hamiltonians are summarized below.
a. The DC Hamiltonian. Setting dC=1 and dG=dR
=0i nE q s . (114) and(115) leads to
det(Me)=4α/braceleftbigg
ν2+2ν−j(j+1)+α2
4/bracerightbigg
=0,(116)
ν=/radicalbigg
j(j+1)+1−α2
4−1. (117)
The value of νin Eq. (117) w i t han e g a t i v es i g ni nf r o n to f
the square root must be discarded, because otherwise the cor-
responding wave functions would not be normalizable. Therelations among fXY(0)
i read
fLS(0)
3=2ib
α(ν+j+1)fLL(0)
1, (118)
fLS(0)
4=2ia
α(−ν+j)fLL(0)
1. (119)
Forj=0, Eq. (117) reduces to
ν=/radicalbigg
1−α2
4−1=−α2
8+O(α4), (120)
indicating that the wave function for the1s0state has a weak
singularity at r12=0, as already found by Kutzelnigg.24Asa
=1 and b=0f o r j=0( s e eE q . (90)), Eqs. (118) and(119)
reduce to
fLS(0)
3=0, (121)
fLS(0)
4=iα
4fLL(0)
1+O(α3). (122)
Further in view of action (68) ofC1on/Psi1A,e
(+,+),E q . (82),f o r
j=0, one sees that the3p0state has the same singularity as
Eq.(120) .
b. The DCG Hamiltonian. Setting dC=1,dG=−1, and
dR=0i nE q . (113) leads to
Me=⎛
⎜⎜⎜⎜⎜⎝4α −2ib(ν−j+1) 2 ia(ν+j+2)
−2ib(ν+j+1) 0 0
−2ia(−ν+j)0 0⎞
⎟⎟⎟⎟⎟⎠. (123)
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As det( Me) is identically zero, there always exist non-
trivial solutions for fXY(0)
i . Actually, there can be two cases,
j=0o rj/negationslash=0.
IffLL(0)
1/negationslash=0, it can be deduced from the second and third
rows of Eqs. (112) together with (123) that
j=ν=b=0, (124)
with the relations among amplitudes being
fLS(0)
3=0, (125)
fLS(0)
4=−iαfLL(0)
1. (126)
Equation (125) follows directly from the fact that the function
/Omega13,E q . (56), does not exist for j=0. Note that the present
zero value of νis different from that (√
1+α2−1) obtained
by Kutzelnigg.24
On the other hand, if fLL(0)
1=0, only the relation
b(ν−j+1)fLS(0)
3=a(ν+j+2)fLS(0)
4 (127)
can be obtained. Note that j/negationslash=0 in this case, because other-
wisefLS(0)
3 and hence fLS(0)
4 would also vanish, contradict-
ing the requirement for nontrivial solutions. To determine the
power ν, the algebraic equations of O(/epsilon1ν) must be consid-
ered, as the Coulomb and Gaunt singularities happen to cancel
out at O(/epsilon1ν−1), see Eqs. (96)and(97) fordC=1,dG=−1,
anddR=0. We postpone the discussion of this situation to
Sec. II D.
c. The DCB Hamiltonian. Setting dC=1 and dG
=dR=−1
2in Eqs. (114) and(115) leads to
det(Me)=−4αj(j+1). (128)
Obviously, a nontrivial solution can only be obtained for j
=0, for which Me,E q . (113) , becomes
Me=⎛
⎝3α02i(ν+2)
00 0
2iν0 α⎞
⎠. (129)
SincefLS(0)
3=0 again because of the nonexistence of /Omega13,E q .
(56),f o r j=0, the requirement of nontrivial solution is only
fulfilled if the minor Me
22
Me
22=det/parenleftbigg3α2i(ν+2)
2iνα/parenrightbigg
=3α2+4ν(ν+2)
(130)vanishes, leading to
ν=/radicalbigg
1−3α2
4−1=−3α2
8+O(α4). (131)
The relation between the amplitudes is then
fLS(0)
4=−2iν
αfLL(0)
1=3iα
4fLL(0)
1+O(α3).(132)
It is seen from Eq. (131) that the wave function of the DCB
Hamiltonian is also singular at r12=0, somewhat worse than
that of the DC Hamiltonian.
Curiously, if one had chosen dC=1 and dG=− dR
=− 1 / 2i nE q s . (114) and(115) , corresponding to an arti-
ficial interaction consisting of the Coulomb potential mi-nus the gauge part of the Breit term, one would obtain ν
=0 independently of jas well as the simple relation f
LS(0)
4
=iα
2fLL(0)
1 for the1s0state. This is the result actually
obtained by Kutzelnigg,24originally claimed for the DCB
Hamiltonian. Of course, this unfortunate mistake was already
noticed by himself,26two decades after the work though.
2. Algebraic equations for /Psi1A,e
(+,−)
In the case of /Psi1A,e
(+,−),E q . (84), the only nontrivial alge-
braic equation that can be obtained from Eqs. (95)–(98) is
α(dC+3dG+dR)fLL(0)
1=0. (133)
Since the prefactor is different from zero for the DC, DCG,
and DCB Hamiltonians, we have fLL(0)
1=0. That is, there
exist no nontrivial solutions for all the three Hamiltonians.
This occurs also to /Psi1S,o
(+,−)and/Psi1A,o
(−,−)for/Psi1S,o
(+,−)has the same
form as /Psi1A,e
(+,−)(see Eq. (86)) and /Psi1A,o
(−,−)=C1/Psi1S,o
(+,−)(see
Eq.(67)).
3. Algebraic equations for /Psi1A,o
(+,−)
The algebraic equations for fXY(0)
i ,E q . (85), are obtained
from Eqs. (99)–(102) as
Mo⎛
⎜⎜⎜⎝fLL(0)
2
fLS(0)
3
fLS(0)
4⎞
⎟⎟⎟⎠=0, (134)
where
Mo=⎛
⎜⎜⎜⎜⎜⎜⎜⎝α(d
C−dG−dR)−2ia(ν−j+1)−2ib(ν+j+2)
−2ia(ν+j+1)α(dC−dG−qdR) −αpdR
2ib(−ν+j) −αpdR α(dC−dG+qdR)⎞
⎟⎟⎟⎟⎟⎟⎟⎠. (135)
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The determinant of the coefficient matrix Mois
det(Mo)=4α[(dC−dG+dR)(ν+1)2−d],(136)
with
d=(dC−dG−dR)/braceleftbigg
j(j+1)−/bracketleftbig
(dC−dG)2−d2
R/bracketrightbigα2
4/bracerightbigg
.
(137)
The situations for the DC, DCG, and DCB Hamiltonians are
summarized as follows.
a. The DC Hamiltonian. Setting dC=1 and dG=dR=0
in Eq. (136) leads to
det(Mo)=4α/braceleftbigg
(ν+1)2−j(j+1)+α2
4/bracerightbigg
=0,(138)
ν=/radicalbigg
j(j+1)−α2
4−1. (139)
The relations between the amplitudes read
fLS(0)
3=2ia
α(ν+j+1)fLL(0)
2, (140)
fLS(0)
4=2ib
α(ν−j)fLL(0)
2. (141)
b. The DCG Hamiltonian. Setting dC=1,dG=−1, and
dR=0i nE q . (136) leads to
det(Mo)=8α{(ν+1)2−j(j+1)+α2}=0,(142)
ν=/radicalbig
j(j+1)−α2−1. (143)
The relations between the amplitudes read
fLS(0)
3=ia
α(ν+j+1)fLL(0)
2, (144)
fLS(0)
4=ib
α(ν−j)fLL(0)
2. (145)
c. The DCB Hamiltonian. Setting dC=1 and
dG=dR=−1
2in Eq. (136) leads to
det(Mo)=4α[(ν+1)2−2j(j+1)+α2]=0,(146)
ν=/radicalbig
2j(j+1)−α2−1. (147)
The relations between the amplitudes read
fLS(0)
3=ia
α(ν+2j+1)fLL(0)
2, (148)
fLS(0)
4=ib
α(ν−2j−1)fLL(0)
2. (149)In sum, as jis odd in Eqs. (139) ,(143) , and (147) ,t h e
wave functions /Psi1A,o
(+,−)are all regular at the coalescence point
for all the three Hamiltonians. These results hold also for/Psi1
S,e
(+,−)and/Psi1A,e
(−,−)for/Psi1S,e
(+,−)has the same form as /Psi1A,o
(+,−)(see
Eq.(86)) and/Psi1A,e
(−,−)=C1/Psi1S,e
(+,−)(see Eq. (67)). It is just that,
for/Psi1A,e
(−,−)with even j, the restriction j≥2 should be imposed,
because det( Mo)>0f o r j=0, which implies no nontrivial
solutions.
4. Algebraic equations for /Psi1A,o
(+,+)
In the case of /Psi1A,o
(+,+),E q . (83), the only nontrivial alge-
braic equation that can be obtained from Eqs. (99)–(102) is
α(dC+dG+dR)fLL(0)
2=0. (150)
The prefactor dC+dG+dRequals one for the DC Hamilto-
nian and hence fLL(0)
2=0. Therefore, no nontrivial solutions
exist for /Psi1A,o
(+,+)of the DC Hamiltonian. In contrast, the pref-
actor is zero for the DCG and DCB Hamiltonians, such thatf
LL(0)
2 cannot be determined. This situation will further be
discussed below.
D. Asymptotic behaviors determined by
ˆh(−1)
12/Psi1(ν+1)=ˆW(0)/Psi1(ν)
As discussed before, there are two situations where the
solutions cannot be determined by the lowest order equation
(29).O n ei s /Psi1A,e
(+,+),E q . (82), with fLL(0)
1=0f o rt h eD C G
Hamiltonian, and the other is /Psi1A,o
(+,+),E q . (83),f o rt h eD C G
and DCB Hamiltonians. The desired solutions can only be
found by resorting to Eq. (30) ofO(/epsilon1ν). To do so, we first
rewrite the operator ˆW(0),E q . (26), in block form
ˆW(0)=ˆw(0)
0E−c2B12+ˆw(0)
1C1+ˆw(0)
2C2, (151)
where
B12=β◦I4+I4◦β=⎛
⎜⎜⎜⎜⎝2I
400 0
00 0 000 0 000 0 −2I
4⎞
⎟⎟⎟⎟⎠,(152)
ˆw
(0)
0=E−/parenleftBigg/summationdisplay
k≥3ˆhD
k+/summationdisplay
k>l≥3ˆgkl/parenrightBigg
−2φ(0)−/summationdisplay
l≥3dC2
rRl,
(153)
ˆw(0)
k=−/summationdisplay
l≥3/bracketleftbigg
dG/vectorσk·/vectorαl
rRl+dR(/vectorσk·ˆrRl)(/vectorαl·ˆrRl)
rRl/bracketrightbigg
−1
2c/vectorσk·/vectorP12,
k=1,2. (154)
Since the spin-angular functions /Omega1iof given jandmjare not
eigenfunctions of /vectorσk(k=1, 2) in ˆ w(0)
k,E q . (154) , the ampli-
tudesfXY(0)
i with different jandmjwill get coupled. In addi-
tion, both the spins and orbital angular momenta of electrons
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1 and 2 will be entangled individually with those of the other
electrons due to the presence of /vectorαlandˆrRlin ˆw(0)
k. Therefore,
the reduced two-electron problem becomes truly a many-bodyproblem and no simple solutions can be found for Eq. (30).
The situation is only simplified by neglecting the couplings
between the f
XY(0)
i with different jandmj. That is, /Psi1(ν)are
assumed to be eigenfunctions of {ˆj2
12,ˆj12,z}.
1./Psi1A,e
(+,+)with fLL(0)
1=0for the DCG Hamiltonian
In order to determine the asymptotic behavior of /Psi1A,e
(+,+),
Eq.(82), with fLL(0)
1=0 for the DCG Hamiltonian, it is as-
sumed that /Psi1(ν)is an eigenfunction of {ˆj2
12,ˆj12,z}, viz.,
/Psi1(ν)=/Psi1A,e
(+,+)=rν
12⎛
⎜⎜⎜⎜⎜⎝0
fLS(0)
3/Omega13+fLS(0)
4/Omega14
fLS(0)
3/Omega13+fLS(0)
4/Omega14
0⎞
⎟⎟⎟⎟⎟⎠,(155)
with the relation between fLS(0)
3 andfLS(0)
4 given by
Eq.(127) . The action of ˆW(0)on/Psi1(ν)is
ˆW(0)/Psi1(ν)=rν
12⎛
⎜⎜⎜⎜⎜⎝0
ˆw
(0)
0/parenleftbig
fLS(0)
3/Omega13+fLS(0)
4/Omega14/parenrightbig
ˆw(0)
0/parenleftbig
fLS(0)
3/Omega13+fLS(0)
4/Omega14/parenrightbig
0⎞
⎟⎟⎟⎟⎟⎠
+r
ν
12⎛
⎜⎜⎜⎜⎜⎝/parenleftbig
ˆw
(0)
1+ˆw(0)
2/parenrightbig/parenleftbig
fLS(0)
3/Omega13+fLS(0)
4/Omega14/parenrightbig
0
0
/parenleftbig
ˆw(0)
1+ˆw(0)
2/parenrightbig/parenleftbig
fLS(0)
3/Omega13+fLS(0)
4/Omega14/parenrightbig⎞
⎟⎟⎟⎟⎟⎠.
(156)
The first and second parts of ˆW
(0)/Psi1(ν)correspond to differ-
ent angular momenta for ˆ w(0)
1and ˆw(0)
2do not commute with
{ˆj2
12,ˆj12,z}. Therefore, to determine the power ν, suffice it
to only consider the first part of ˆW(0)/Psi1(ν),E q . (156) , which
shares the same symmetry as /Psi1A,e
(+,+). In this case, /Psi1(ν+1)can
still be chosen as the form of /Psi1A,e
(+,+),E q . (82). Substituting
the expression for /Psi1(ν+1)into Eq. (30) and integrating out the
angular parts /Omega13and/Omega14yield two algebraic equations
−2ib(ν+j+2)fLL(1)
1=αˆw(0)
0fLS(0)
3, (157)
−2ia(−ν+j−1)fLL(1)
1=αˆw(0)
0fLS(0)
4, (158)
where fLL(1)
1 represents the first order unknown in /Psi1(ν+1).
Equations 157,158, and 127together give rise to
ν=j−1, (159)
fLS(0)
4=0. (160)
Note that the value of jhere cannot be 0, because otherwise
fLS(0)
3 would also vanish, contradicting the requirement that
νbe the lowest power of /Psi1with at least one nonvanishing
amplitude fXY(0)
i .Finally, it is interesting to see from Eqs. (124) and(159)
that the wave functions /Psi1A,e
(+,+)of the DCG Hamiltonian are
of integral powers of r12and hence free of singularities. In the
case of fLL(0)
1=0,/Psi1LLand/Psi1SSare of order rν+1
12, one order
higher than that of /Psi1LSand/Psi1SL. This is quite different from
all the other cases where all the components share the samepower.
2./Psi1A,o
(+,+)for the DCG and DCB Hamiltonians
The yet undetermined /Psi1A,o
(+,+),E q . (83),f o rt h eD C G
and DCB Hamiltonians can be analyzed in the same way as
before. Again assuming that /Psi1A,o
(+,+)is an eigenfunction of
{ˆj2
12,ˆj12,z}, viz.,
/Psi1(ν)=/Psi1A,o
(+,+)=rν
12⎛
⎜⎜⎜⎜⎜⎝f
LL(0)
2/Omega12
0
0
fLL(0)
2/Omega12⎞
⎟⎟⎟⎟⎟⎠. (161)
The action of ˆW
(0)on/Psi1(ν)leads to
ˆW(0)/Psi1(ν)=rν
12⎛
⎜⎜⎜⎜⎜⎝/parenleftbig
ˆw(0)
0−2c2/parenrightbig
fLL(0)
2/Omega12
0
0
/parenleftbig
ˆw(0)
0+2c2/parenrightbig
fLL(0)
2/Omega12⎞
⎟⎟⎟⎟⎟⎠
+rν
12⎛
⎜⎜⎜⎜⎜⎝0
/parenleftbig
ˆw
(0)
1+ˆw(0)
2/parenrightbig
fLL(0)
2/Omega12/parenleftbig
ˆw(0)
1+ˆw(0)
2/parenrightbig
fLL(0)
2/Omega12
0⎞
⎟⎟⎟⎟⎟⎠.(162)
To determine the power ν, only the first part of ˆW
(0)/Psi1(ν),
Eq.(162) , is to be considered, which conserves antisymmetry
and has the same parity as /Psi1A,o
(+,+), but breaks the C12sym-
metry. Therefore, /Psi1(ν+1)has to take form (94). Substituting
/Psi1(ν+1)into Eq. (30) and integrating out /Omega12gives rise to
αfLL(1)
2−2ia(ν−j+2)fLS(1)
3−2ib(ν+j+3)fLS(1)
4
−αfSS(1)
2=α/parenleftbig
ˆw(0)
0−2c2/parenrightbig
fLL(0)
2, (163)
−αfLL(1)
2+2ia(ν−j+2)fLS(1)
3+2ib(ν+j+3)fLS(1)
4
+αfSS(1)
2=α/parenleftbig
ˆw(0)
0+2c2/parenrightbig
fLL(0)
2. (164)
Note that the left hand side of Eq. (163) is just the neg-
ative of that of Eq. (164) , which is also evident from
Eqs. (99)and(102) . The sum of Eqs. (163) and(164) then
leads to
2αˆw(0)
0fLL(0)
2=0. (165)
Since ˆw(0)
0cannot always be zero, fLL(0)
2 must be zero. There-
fore, there exist no nontrivial solutions for /Psi1A,o
(+,+)of the DCG
and DCB Hamiltonians.
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III. APPROXIMATE RELATIVISTIC HAMILTONIANS
Having discussed in depth the wave functions of the
DC, DCG, and DCB Hamiltonians, we now turn to the ap-
proximate variants, including the spin-free part of the modi-
fied DC Hamiltonian as well as the ZORA Hamiltonian. Thefindings are to be compared closely with the nonrelativistic
counterparts.
A. The spin-free modified Dirac equation
The so-called modified Dirac equation41,42can formally
be written as
ˆHZ/Psi1Z=EˆSZ/Psi1Z, (166)
with
ˆHZ=ˆZ†ˆHˆZ, ˆSZ=ˆZ†ˆZ, /Psi1 Z=ˆZ−1/Psi1. (167)
The one-electron transformation operator ˆZis simply
ˆZk=/parenleftBigg
I2 0
0/vectorσk·/vectorpk
2c/parenrightBigg
, (168)
in terms of which the many-electron ˆZoperator can system-
atically be constructed through the Tracy-Singh product.43In
the case of two electrons, ˆZreads
ˆZ=ˆZ1◦ˆZ2=⎛
⎜⎜⎜⎜⎜⎝I
400 0
0/vectorσ2·/vectorp2
2c00
00/vectorσ1·/vectorp1
2c0
00 0(/vectorσ1·/vectorp1)(/vectorσ2·/vectorp2)
4c2⎞
⎟⎟⎟⎟⎟⎠.(169)
The modified Hamiltonian ˆH
Zand the metric ˆSZcan then be
written explicitly as
ˆHZ=⎛
⎜⎜⎜⎜⎜⎝ˆV
12ˆT2ˆT1 0
ˆT2ˆV2
12 0ˆT1ˆT2
2c2
ˆT1 0 ˆV1
12ˆT1ˆT2
2c2
0ˆT1ˆT2
2c2ˆT1ˆT2
2c2ˆV12
12⎞
⎟⎟⎟⎟⎟⎠, (170)
ˆS
Z=⎛
⎜⎜⎜⎜⎜⎝I
400 0
0ˆT2
2c200
00ˆT1
2c2 0
00 0ˆT1ˆT2
4c4⎞
⎟⎟⎟⎟⎟⎠, (171)
where
ˆV
12=(φ1+φ2)+ˆg12, (172)
ˆV1
12=/parenleftbigg
φ1
1+ˆT1φ2
2c2/parenrightbigg
+ˆg1
12−ˆT1, (173)
ˆV2
12=/parenleftbiggφ1ˆT2
2c2+φ2
2/parenrightbigg
+ˆg2
12−ˆT2, (174)ˆV12
12=/parenleftbiggφ1
1ˆT2+ˆT1φ2
2
2c2/parenrightbigg
+ˆg12
12−ˆT1ˆT2
c2, (175)
ˆTk=1
2(/vectorσk·/vectorpk)(/vectorσk·/vectorpk)=1
2/vectorp2
k, (176)
φk
k=1
4c2(/vectorσk·/vectorpk)φk(/vectorσk·/vectorpk), (177)
ˆg1
12=1
4c2(/vectorσ1·/vectorp1)ˆg12(/vectorσ1·/vectorp1), (178)
ˆg2
12=1
4c2(/vectorσ2·/vectorp2)ˆg12(/vectorσ2·/vectorp2), (179)
ˆg12
12=1
16c4(/vectorσ1·/vectorp1)(/vectorσ2·/vectorp2)ˆg12(/vectorσ2·/vectorp2)(/vectorσ1·/vectorp1).
(180)
Note in passing that the above transformation renders the
nrl particularly transparent. Taking the limit c→∞ in
Eqs. (170) and(171) leads to
ˆHnrl
Z=⎛
⎜⎜⎜⎜⎜⎝ˆV
12ˆT2ˆT10
ˆT2−ˆT2 00
ˆT1 0−ˆT10
00 0 0⎞
⎟⎟⎟⎟⎟⎠, (181)
ˆS
nrl
Z=⎛
⎜⎜⎜⎜⎝I
4000
00 0 000 0 000 0 0⎞
⎟⎟⎟⎟⎠, (182)
and hence
/Psi1
LL
Z,nrl=/Psi1LS
Z,nrl=/Psi1SL
Z,nrl=/Psi1LL
nrl. (183)
Equation (166) then reduces to the two-electron Schrödinger
equation
(ˆT1+ˆT2+φ1+φ2+ˆg12)/Psi1LL
Z,nrl=E/Psi1LL
Z,nrl.(184)
Another advantage of the transformation lies in that it al-
lows an exact separation of the spin-free and spin-dependent
t e r m so fE q . (166) through the Dirac identity
(/vectorσ·/vectorA)(/vectorσ·/vectorB)=/vectorA·/vectorB+i/vectorσ·(/vectorA×/vectorB). (185)
In view of relation (167) , the asymptotic behaviors of /Psi1Zof
the modified DC equation can directly be deduced from those
of the original /Psi1. Since the components /Psi1XYof/Psi1have the
same power νinr12, the components of /Psi1Zbehave asymptot-
ically as
/Psi1LL
Z∼rν
12,/Psi1LS
Z,/Psi1SL
Z∼rν+1
12,/Psi1SS
Z∼rν+2
12.(186)
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The asymptotic behaviors of the spin-free modified wave
function /Psi1Zare to be determined by the homogeneous equa-
tion
⎛
⎜⎜⎜⎜⎝1
r1
2/vectorp2 1
2/vectorp20
1
2/vectorp2 1
4c2pi1
rpi 01
8c2/vectorp4
1
2/vectorp201
4c2pi1
rpi1
8c2/vectorp4
01
8c2/vectorp4 1
8c2/vectorp4 1
16c4pipj1
rpipj⎞
⎟⎟⎟⎟⎠
×⎛
⎜⎜⎜⎜⎝/Psi1LL
Z
/Psi1LS
Z
/Psi1SL
Z
/Psi1SS
Z⎞
⎟⎟⎟⎟⎠=0,i , j ∈{x,y,z }, (187)
where the subscript 12 in /vectorr
12and/vectorp12has been omitted for
simplicity. Noting that {ˆl2
12,ˆl12,z}commute with the Hamilto-
nian in Eq. (187) , the asymptotic behaviors of the spin-free
/Psi1Zcan, similar to Eq. (186) ,b ea s s u m e da s
/Psi1LL
Z=rν
12Yml
lfLL(0)+O(/epsilon1ν+1), (188)
/Psi1LS
Z=rν+1
12Yml
lfLS(0)+O(/epsilon1ν+2), (189)
/Psi1SL
Z=rν+1
12Yml
lfSL(0)+O(/epsilon1ν+2), (190)
/Psi1SS
Z=rν+2
12Yml
lfSS(0)+O(/epsilon1ν+3). (191)
Thel=0 and l=1 cases are of most interest here, since
they are related with the respective singlet and triplet states,
whose spatial wave functions are, respectively, symmetric andantisymmetric with respect to the permutation of electrons.
Forl=0, the value of νis found to be
ν=/radicalBigg
2−α2
8−/radicalbigg
1+α4
8−1=−α2
16+O(α4),(192)
showing that the1sstate is somewhat less singular than the
1s0state of the original DC Hamiltonian, see Eq. (120) .T h e
corresponding amplitudes fXY(0)are related by
fLS(0)=fSL(0)=1
2fLL(0)+O(α2), (193)
fSS(0)=−2c2fLL(0)+O(α0), (194)
which are also different from the unmodified wave function,
see Eqs. (121) and(122) .
In the case of l=1, the value of νis to be determined by
(t−1)(t−4)(t−8)+α2
4(t−2)(t−7)=0,t=(ν+1)2.
(195)
This equation has closed solutions but which are too cumber-
some to be presented here. Therefore, only the leading order
terms of νare given explicitly
ν=1−α2
32+O(α4). (196)The corresponding amplitudes fXY(0)are related by
fLS(0)=fSL(0)=1
4fLL(0)+O(α2), (197)
fSS(0)=−2
3c2fLL(0)+O(α0), (198)
which are not fundamentally different from the previous case,
see Eq. (193) and(194) . Note that both Eqs. (192) and(196)
coincide with the nonrelativistic results upon taking the nrlα=0. However, the ratios between the f
XY(0)
i are different
from the nrl equation (183) . This simply means that the two
limits r12→0 and c→∞ do not commute. The relations
between fSS(0)andfLL(0)in Eqs. (194) and(198) reveal that,
near the coalescence point, /Psi1SS
Zis larger than /Psi1LL
Zby a factor
ofc2. This is not surprising, since /Psi1SS
Zis related to c2/Psi1SS
(see Eqs. (167) and(169) ) while /Psi1SS,/Psi1LL, and/Psi1LL
Zare of
the same order of magnitude near the coalescence point.
B. The zeroth-order regular approximation
The two-electron ZORA equation reads
/bracketleftbigg
ˆTZORA
1+ˆTZORA
2+1
r12/bracketrightbigg
/Psi1LL=ˆWLL/Psi1LL, (199)
where
ˆWLL=E−φ1−φ2, (200)
ˆTZORA
k=(/vectorσk·/vectorpk)c2
2c2−Vext
k(/vectorσk·/vectorpk) (201)
=c2
2c2−Vext
k/vectorp2
k+c2
/parenleftbig
2c2−Vext
k/parenrightbig2/parenleftbig
/vectorpkVext
k/parenrightbig
·/vectorpk
+c2
/parenleftbig
2c2−Vext
k/parenrightbig2i/vectorσk·/bracketleftbig/parenleftbig
/vectorpkVext
k/parenrightbig
×/vectorpk/bracketrightbig
. (202)
Equation (202) amounts to separating the spin-free and spin-
dependent terms through identity (185) . Two choices are pos-
sible for the external field Vext
k, the simple nuclear attraction
(Vext
k=φk) or both the nuclear attraction and the Coulomb
interaction ( Vext
k=φk+1
r12). They result in different asymp-
totic behaviors of the wave functions at small r12and are thus
discussed separately.
1. ZORA with Vext
k=φk
In terms of the partial wave expansion (24) forφk,t h e
ZORA kinetic energy operator can be written as
ˆTZORA=+∞/summationdisplay
k=−2ˆt(k)
12, (203)
ˆt(−2)
12=2c2
2c2−φ(0)/vectorp2
12, (204)
ˆt(−1)
12=c2
(2c2−φ(0))2(/vectorσ1−/vectorσ2)·(/vectorφ(0)×/vectorp12),(205)
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φ(0)=−/summationdisplay
AZA
rRA,/vectorφ(0)=/summationdisplay
AZA
r3
RA/vectorrRA. (206)
Note that the second term in Eq. (202) does not contribute
toˆt(−1)
12due to the cancelation between terms for electrons 1
and 2 at this order. The equations determining the asymptotic
behaviors of the wave functions read
O(/epsilon1ν−2):ˆt(−2)
12/Psi1(ν)=0, (207)
O(/epsilon1ν−1):ˆt(−2)
12/Psi1(ν+1)+/parenleftBig
ˆt(−1)
12+ˆg(−1)
12/parenrightBig
/Psi1(ν)=0,(208)
which are formally similar to Eqs. (36)and(37) for the
Schrödinger equation, except for the spin-orbit term ˆt(−1)
12,
Eq.(205) ,i nE q . (208) .A s ˆt(−2)
12,E q . (204) , is a scalar op-
erator, Eq. (207) dictates that the asymptotic behavior of the
ZORA wave function should be /Psi1(ν)=rν
12Yml
lfLL(0)withν
=l, i.e., the same as the nonrelativistic case. Neglecting ˆt(−1)
12
in Eq. (208) gives rise to the following cusp condition for
spin-free ZORA:
fLL(1)=1
2(l+1)/parenleftbigg
1−φ(0)
2c2/parenrightbigg
fLL(0). (209)
In view of the expansion /Psi1=rl
12(fLL(0)+fLL(1)r12+
···)Yml
l,E q . (209) can be rewritten as
/parenleftBigg
∂l+1/Psi1
∂rl+1
12/parenrightBigg
r12=0=1
2/parenleftbigg
1−φ(0)
2c2/parenrightbigg/parenleftbigg∂l/Psi1
∂rl
12/parenrightbigg
r12=0,(210)
which reduces to Eq. (42) in the nrl. That is, spin-free ZORA
introduces a correctionφ(0)
2c2ofO(α2) to the nonrelativistic
cusp condition. The correction is not universal but depends
on the positions of the electrons and atoms and is always neg-
ative (see Eq. (206) ), such that the ZORA correlation cusp
equation (210) is always acuter than the nonrelativistic one,
especially for the situation where two electrons are close to
the nuclei. The spin-orbit term ˆt(−1)
12,E q . (205) , can couple
the amplitudes fLL(0)with different jandmj, leading to a cou-
pled set of linear equations. However, as ˆt(−1)
12 is ofO(α2),
theO(α0) term in Eq. (209) will not be affected, such that
the nonrelativistic cusp still dominates even in the presenceof spin-orbit couplings. As such, it is reasonable to combine
the ZORA Hamiltonian directly with nonrelativistic R12/F12
methods. In particular, the fixed amplitude ansatz
12derived
from the nonrelativistic correlation cusp equation (42) should
work well. Finally, it deserves to be mentioned that further in-
cluding the electron-electron cusp equation (210) into the Jas-
trow factor in the spin-free ZORA-QMC method35,36should
yield more accurate results.
2. ZORA with Vext
k=φk+1
r12
By first making the following expansion with respect
tor12
c2
2c2−φk−1
r12=−c2r12−c2(2c2−φ(0))r2
12+O(/epsilon13),
(211)the ZORA kinetic energy operator (201) can be expanded as
ˆTZORA=+∞/summationdisplay
k=−1ˆt(k)
12, (212)
where the lowest order term is
ˆt(−1)
12
=−c2(/vectorσ1·/vectorp12)r12(/vectorσ1·/vectorp12)−c2(/vectorσ2·/vectorp12)r12(/vectorσ2·/vectorp12)
(213)
=−2c2r12p2
12+ic2[(/vectorσ1·ˆr12)(/vectorσ1·/vectorp12)+(/vectorσ2·ˆr12)(/vectorσ2·/vectorp12)]
(214)
=−2c2r12p2
12+2ic2ˆr12·/vectorp12−c2(/vectorσ1+/vectorσ2)·(ˆr12×/vectorp12).
(215)
The equations determining the asymptotic behaviors of the
ZORA wave functions read
O(/epsilon1ν−1):/parenleftbigˆt(−1)
12+ˆg(−1)
12/parenrightbig
/Psi1(ν)=0, (216)
O(/epsilon1ν):/parenleftbigˆt(−1)
12+ˆg(−1)
12/parenrightbig
/Psi1(ν+1)+ˆt(0)
12/Psi1(ν)=ˆWLL(0)/Psi1(ν),
(217)
which are formally similar to Eqs. (29)and(30) for the Dirac
equation, except for the presence of ˆt(0)
12in Eq. (217) .A s
Eq.(217) is not to be invoked, the involved expression of ˆt(0)
12
is not documented here.
First consider the spin-free case, which amounts to ne-
glecting the third term in Eq. (215) ,
ˆt(−1)
12=−2c2/vectorp12·r12/vectorp12=−2c2/bracketleftbigg
r12p2
12−∂
∂r12/bracketrightbigg
.(218)
The power νof/Psi1(ν)=rν
12Yml
lfLL(0)can readily be derived
from Eq. (216) as
ν=/radicalbigg
l(l+1)+1−α2
2−1, (219)
which does not have the correct nrl ( ν=l), except for l=0.
This peculiar feature stems from expansion (211) , the conver-
gence of which requires that r12<Rcwith the convergence
radius Rc=1
|2c2−φ(0)
k|≤α2
2≈2.7×10−5. However, if the ex-
pansion had first been taken with respect to α, the conver-
gence would require that r12>Rc. Therefore, the two limits
r12→0 and c→∞ do not commute.
The inclusion of spin-orbit couplings is more straightfor-
ward by starting with Eq. (214) , since the actions of /vectorσk·ˆr12
and/vectorσk·/vectorp12have already been known when examining the
Dirac equation. By noting that ˆt(−1)
12,E q . (213) , commutes
with the permutation of electrons 1 and 2 as well as the space
inversion of /vectorr12, the ZORA wave functions can be written as
/Psi1LL,e
+=fLL
1/Omega11,/Psi1LL,o
+=fLL
2/Omega12,
/Psi1LL,e
−=fLL
3/Omega13+fLL
4/Omega14, (220)
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which satisfy manifestly both the antisymmetry principle and
the parity as can be verified from the /Psi1LLcomponents of /Psi1
shown in Appendix. The above fLL
ican be expanded in the
same way as in Eq. (111) . Substituting /Psi1LL,e
+ and/Psi1LL,o
+ in
Eq.(216) leads, respectively, to
ν=/radicalbigg
j(j+1)+1−α2
2−1, (221)
and
ν=/radicalbigg
j(j+1)−α2
2−1. (222)
Substituting /Psi1LL,e
− into Eq. (216) gives rise to two solutions:
ν=/radicalbigg
j2−α2
2−1,fLL
4=0, (223)
and
ν=/radicalbigg
(j+1)2−α2
2−1,fLL
3=0. (224)
According to the definition of /Omega1iEqs. (54)–(57),E q . (221)
holds for l=jands=0, Eq. (222) forl=jands=1,
Eq.(223) forl=j−1 and s=1, while Eq. (224) forl=j+1
ands=1. Therefore, Eq. (221) is identical with Eq. (219) ,b u t
Eqs. (222) –(224) are different from Eq. (219) . That is, spin-
orbit couplings lead to changes in the asymptotic behaviors of
triplet wave functions.IV. DISCUSSION
For clarity, the lowest order coalescence conditions for
the various relativistic wave functions are summarized in
Table I. It deserves to be mentioned that the values of ν
for the wave functions of the DC and DCG Hamiltoniansare the same as those obtained by Malenfant
48for particle-
antiparticle pairs. The most salient feature is that, except for
ZORA with Vext
k=φkand the other cases with j=0o rl=
0, the powers νdo not have the correct nrl ( ν=l). That is, the
two limits r12→0 and c→∞ generally do not commute. To
further scrutinize this peculiarity, we consider the exact wavefunction of the DC Hamiltonian for a two-electron system
⎛
⎜⎜⎝V
C c/vectorσ2·/vectorp2c/vectorσ1·/vectorp1 0
c/vectorσ2·/vectorp2VC 0 c/vectorσ1·/vectorp1
c/vectorσ1·/vectorp1 0 VC c/vectorσ2·/vectorp2
0 c/vectorσ1·/vectorp1c/vectorσ2·/vectorp2VC⎞
⎟⎟⎠⎛
⎜⎜⎝/Psi1LL
/Psi1LS
/Psi1SL
/Psi1SS⎞
⎟⎟⎠
=⎛
⎜⎜⎝W/Psi1LL
(W+2c2)/Psi1LS
(W+2c2)/Psi1SL
(W+4c2)/Psi1SS⎞
⎟⎟⎠, (225)
where
W=E−2c2−φ1−φ2. (226)
TABLE I. Coalescence conditions rν
12for the wave functions of the Dirac-Coulomb (DC), Dirac-Coulomb-Gaunt (DCG), Dirac-Coulomb-Breit (DCB), spin-
free part of the modified DC (sf-MDC), and zeroth-order regularly approximated (ZORA) Hamiltonians. For other explanations see the text.
Hamiltonian Wave function ν jorl
DC /Psi1A,e
(+,+),/Psi1A,e
(−,+)/radicalBig
j(j+1)+1−α2
4−1 0 ,2 ,4 ,. . .
/Psi1A,o
(+,−),/Psi1A,e
(−,−)/radicalBig
j(j+1)−α2
4−1 1 ,2 ,3 ,. . .
DCG /Psi1A,e
(+,+),/Psi1A,e
(−,+)00
/Psi1A,e
(+,+),/Psi1A,e
(−,+)j−1 2 ,4 ,6 ,...
/Psi1A,o
(+,−),/Psi1A,e
(−,−)/radicalbig
j(j+1)−α2−1 1 ,2 ,3 ,. . .
DCB /Psi1A,e
(+,+),/Psi1A,e
(−,+)/radicalBig
1−3α2
4−10
/Psi1A,o
(+,−),/Psi1A,e
(−,−)/radicalbig
2j(j+1)−α2−1 1 ,2 ,3 ,...
sf-MDC /Psi1LL
Z−α2
16+O(α4)0
/Psi1LL
Z1−α2
32+O(α4)1
ZORAa/Psi1LLl 0, 1, 2, . . .
sf-ZORAa/Psi1LLl 0, 1, 2, . . .
ZORAb/Psi1LL,e
+ (l=j,s=0)/radicalBig
j(j+1)+1−α2
2−1 0 ,2 ,4 ,...
/Psi1LL,o
+ (l=j,s=1)/radicalBig
j(j+1)−α2
2−1 1 ,3 ,5 ,...
/Psi1LL,e
− (l=j−1,s=1)/radicalBig
j2−α2
2−1 2 ,4 ,6 ,...
/Psi1LL,e
− (l=j+1,s=1)/radicalBig
(j+1)2−α2
2−1 0 ,2 ,4 ,...
sf-ZORAb/Psi1LL/radicalBig
l(l+1)+1−α2
2−1 0 ,1 ,2 ,...
aVext
k=φk, the nuclear attraction for electron k.
bVext
k=φk+1
r12.
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Equation (225) gives rise to the following relations between
the components /Psi1XY:
/Psi1SL
/Psi1LS=/vectorσ1·/vectorp1/Psi1LL+/vectorσ2·/vectorp2/Psi1SS
/vectorσ2·/vectorp2/Psi1LL+/vectorσ1·/vectorp1/Psi1SS, (227)
/Psi1SS
/Psi1LL=/vectorσ1·/vectorp1/Psi1LS+/vectorσ2·/vectorp2/Psi1SL
/vectorσ2·/vectorp2/Psi1LS+/vectorσ1·/vectorp1/Psi1SLR(c,r 12),(228)
R(c,r 12)=W−VC
W−VC+4c2. (229)
Here, the ratios (227) and(228) should be understood in the
same sense as that in Eq. (32), i.e., between the corresponding
components of the numerator and denominator. In particular,Eq.(227) implies that /Psi1
SLand/Psi1LSare of the same orders in c
andr12. Therefore, the ratio/Psi1SS
/Psi1LL,E q . (228) , at the two limits is
determined mainly by the function R(c,r12), Eq. (229) , which
behaves (see Fig. 1(a))a s
lim
c→∞R(c,r 12)=lim
c→∞/braceleftbigg
c−2/parenleftbigg
−1
4r12+W
4/parenrightbigg
+c−4/parenleftbigg
−1
16r2
12+W
8r12−W2
16/parenrightbigg
+O(c−6)/bracerightbigg
(230)
=0, (231)
lim
r12→0R(c,r 12)=lim
r12→0/braceleftbig
1+4c2r12+O(r2
12)/bracerightbig
(232)
=1. (233)
That is,
lim
r12→0lim
c→∞R(c,r 12)=0, (234)
lim
c→∞lim
r12→0R(c,r 12)=1. (235)
The following limits can then readily be deduced:
lim
c→∞/Psi1SS
/Psi1LL=0, (236)
lim
r12→0/Psi1SS
/Psi1LL=lim
r12→0/vectorσ1·/vectorp1/Psi1LS+/vectorσ2·/vectorp2/Psi1SL
/vectorσ2·/vectorp2/Psi1LS+/vectorσ1·/vectorp1/Psi1SL·lim
r12→0R(c,r 12)
(237)
=lim
r12→0/vectorσ1·/vectorp12/Psi1LS(ν)−/vectorσ2·/vectorp12/Psi1SL(ν)
/vectorσ1·/vectorp12/Psi1SL(ν)−/vectorσ2·/vectorp12/Psi1LS(ν)(238)
=lim
r12→0−VC/Psi1SS(ν)
−VC/Psi1LL(ν)(239)
=1. (240)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
−10
(a)
(b)−8−6−4−20246810R(c, r12)r12/α2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
−10−8−6−4−20246810R(c, r12)r12/α2
FIG. 1. Plot of the function R(c,r12), Eq. (229) , with W=1. (a)VC=1
r12
and (b) VC=−1
r12. Red solid line: full R(c,r12); blue solid line: R(c,r12)
expanded at small r12, i.e., 1 +4c2
VC; black solid line: R(c,r12) expanded at
large c, i.e.,1
4c2(W−VC); and black dashed line: r12=Rc=1
|W+4c2|.
Equation (238) arises from the coordinate transformation (18)
followed by expansions (23) of/Psi1XYinr12, while Eq. (239)
follows directly from the first and fourth rows of the homo-geneous equation (43) determining the asymptotic behaviors
of/Psi1. Finally, Eq. (240) stems from the relation lim
r12→0/Psi1SS(ν)
/Psi1LL(ν)
=limr12→0rν
12fSS(0)
j=0
rν
12fLL(0)
j=0=1, where the first equality holds because
thej=0 state is singular while other states vanish at r12=0,
and the second equality is implied directly by structure (82)
arising from the C12operation. It is therefore clear that the
two limits do not commute, viz.,
0=lim
r12→0lim
c→∞/Psi1SS
/Psi1LL/negationslash=lim
c→∞lim
r12→0/Psi1SS
/Psi1LL=1.(241)
This is quite different from the situation for the electron-
nucleus coalescence. Consider, e.g., the 1 s1
2state of a hy-
drogenic ion, for which the radial wave function is known
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exactly
ψL=/radicalBigg
(ν+2)Z
/Gamma1(2ν+3)(2Z)ν+1rνe−Zr,ν=/radicalbig
1−(Zα)2−1,
(242)
ψS=−i/radicalBigg
−νZ
/Gamma1(2ν+3)(2Z)ν+1rνe−Zr. (243)
Although the two limits c→∞ andr→0 also do not com-
mute for the individual components, e.g.,
lim
r→0lim
c→∞ψL=lim
r→02Z3
2e−Zr=2Z3
2,lim
c→∞lim
r→0ψL=∞,
(244)
they do commute for the ratio
ψS
ψL=−i/radicalbigg
−ν
ν+2=−i1−/radicalbig
1−(Zα)2
Zα=−i
2Zα+O(α3),
(245)
which is independent of r. That is,
lim
r→0lim
c→∞ψS
ψL=lim
c→∞lim
r→0ψS
ψL=0. (246)
It is to be noted that expansion (232) ofR(c,r12) around
r12=0 has a finite convergence radius Rc=1
|4c2+W|, which is
aboutα2
4for the usual states of interest ( W∼O(c0)). The pre-
viously obtained asymptotic behaviors of the relativistic wave
functions are only valid for r12<Rc, except for the particular
case of ZORA with Vext
k=φk, for which the asymptotic be-
haviors are valid for all r12. As for the behaviors of the wave
functions at r12>Rc, the following expansion of R(c,r12)
around r12=+ ∞ should be adopted:
R(c,r 12)=W
4c2+W−4c2
(4c2+W)2r12
−4c2
(4c2+W)3r2
12+···. (247)
For the situation |W|<4c2, the right hand side can further be
expanded around c=∞ , leading formally to Eq. (230) .H o w -
ever, it should be noted that r12>Rcand|W|<4c2together
span only a subdomain of expansion (230) , viz., |W−VC
4c2|<1
or equivalently W−4c2<1
r12<W+4c2. That is, expan-
sions (230) and(247) agree with each other only if both |W|
<4c2andr12>Rchold.
Expansion (230) ofR(c,r12) around c=∞ is inti-
mately linked to the explicitly correlated DPT.30Because of
the change of the metric, DPT assumes the expansion
¯/Psi1SS=c2/Psi1SS=¯/Psi1SS
0+c−2¯/Psi1SS
2+O(c−4), (248)
which amounts to using Eq. (230) for the scaled quantity
¯R(c,r 12)=c2R(c,r 12)=¯R0(r12)+c−2¯R2(r12)+O(c−4).
As already mentioned, for |W|<4c2, the usual energy
range of interest ¯R(c,r 12) only converges for r12>Rc.
Therefore, the extension of ¯R(c,r 12) to the domain r12<Rcis doomed to fail. Such a failure manifests already
to the second order relativistic correction c−4E4, which
involves the term /angbracketleft¯/Psi1SS
0|1
r12|¯/Psi1SS
0/angbracketright. In view of the relations
¯/Psi1SS
0∼/Psi1LL
0¯R0(r12),/Psi1LL
0∼r0
12, and ¯R0(r12)=−1
4r12+W
4,i t
can immediately be seen that ¯/Psi1SS
0goes as r−1
12, in agreement
with the previous result30deduced from the nonrelativistic
correlation cusp condition (1)as well as the kinetic balance
conditions. Therefore, the integrand of /angbracketleft¯/Psi1SS
0|1
r12|¯/Psi1SS
0/angbracketrightgoes
asr−3
12at small r12, resulting in divergence. To higher orders
even more severe divergences would arise, as can be seen
from the more singular term ¯R2(r12) going as r−2
12(see
Eq.(230) ). In contrast, the exact /Psi1SSis much better behaved
than ¯/Psi1SS
0atr12<Rc,s e eE q . (232) . More specifically, the
only singular term of /Psi1SSgoes as fSS(0)rν
12withνgiven in
Eq.(120) forj=0. Besides, as dictated by the C12symmetry,
fSS(0)has the same magnitude as fLL(0), implying that the
leading term of ¯/Psi1SS=c2/Psi1SSshould be of O(c2), larger by
two orders of magnitude than the presumed O(c0) starting
point of DPT in Eq. (248) . This is quite different from
the one-electron case, where the leading term of the ratio
ψS/ψLis of O(c−1) (see Eq. (245) ), such that the leading
term of the scaled small component ¯ψS=cψSis indeed of
O(c0). It is therefore not surprising that the α-morphology
of one-electron wave functions is guaranteed17,18but that
of many-electron DC wave functions holds only for somebounded interaction.
19As a specific form for such a bounded
interaction was not discussed explicitly therein, here we
consider a regularized Coulomb potential Vμ
C=erf(μr12)
r12, with
μbeing a fixed, positively valued parameter and erf( x) being
the error function. This amounts to using a Gaussian-type
“finite electron model,” i.e., ρe(r)=(μ2
π)3
2e−μ2r2. Because of
the finite limit
lim
r12→0erf(μr12)
r12=2√πμ, (249)
the Coulomb singularity at r12=0 is removed. The domain
for expansion (230) around c=∞ now becomes W−4c2
<Vμ
C<W+4c2. It can immediately be deduced that, for
|W|<4c2, expansion (230) converges for all r12∈[0,+∞)
if the length-scale parameter μis smaller than μ0=√π
2(W
+4c2)∈(0,4√πc2). This implies that DPT can be applied
for such a regularized electron-electron interaction Vμ
C. Fur-
ther in view of the energy-time uncertainty relation, viz.,
/Delta1E/Delta1t =mc2/Delta1t=mc
μ0≥¯
2, the upper limit of μ0should be
sharpened to μ0≤2cto avoid creations of electron-positron
pairs.
Another point to be mentioned here is that, regardless
of the above regularization, the denominator of R(c,r12),
Eq.(229) , becomes zero at r12=Rc, such that the ratio
/Psi1SS
/Psi1LLis divergent if the factor/vectorσ1·/vectorp1/Psi1LS+/vectorσ2·/vectorp2/Psi1SL
/vectorσ2·/vectorp2/Psi1LS+/vectorσ1·/vectorp1/Psi1SLcannot off-
set this singularity. As there is no obvious reason for this
factor as well as /Psi1LLto always be zero on the (3 N−1)-
dimensional hypersurface determined by the constraint r12
=Rc,/Psi1SSmight be divergent at r12=Rc. It can hence be
conjectured that the DC wave functions cannot be normalized .
In other words, the DC Hamiltonian has no bound electronic
states . This formal argument for the exact DC wave functions
is in line with the (in)famous Brown-Ravenhall disease or
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continuum dissolution deduced from a second order pertur-
bation analysis22and numerical calculations32,33with the DC
Hamiltonian itself, as well as a quasi-solvable model with asimplified Hamiltonian.
49However, this is hardly an issue21
as stressed in the Introduction. Note in passing that the nor-malizability problem does not occur to the positronium, wherethe denominator in R(c,r
12)i sW+4c2+VC. In this case, the
singularity appears on the negative radial axis (see Fig. 1(b))
and hence does not affect the wave function. This is preciselythe reason behind the success of the finite-element method for
low-lying bound states of the positronium.
50
If one really wants to solve the many-electron Dirac
equation in spite of the strong cons,21one should consider
a rational correlation factor for the components /Psi1XY, e.g.,
f12=rν
12/parenleftbigga11+a12r12
a21+a22r12/parenrightbigg
, (250)
where the parameters aijare to be optimized. Both the expan-
sion regions of r12<Rcandr12>Rccan then be covered so
as to mimic the effect of R(c,r12), Eq. (229) . In contrast, the
traditional correlation factors, either linear11or nonlinear12in
r12, could not do this precisely.
V. TECHNICALITIES FOR RELATIVISTIC EXPLICITLY
CORRELATED METHODS
The situation with relativistic explicitly correlated meth-
ods is at first glance rather puzzling. On one hand, irre-
spective of the self-adjointness or boundedness, it makes no
sense to directly solve the first-quantized, configuration-space
many-body Dirac equation.21Instead, it is the no-pair pro-
jected equation that should be solved. On the other hand,the standard no-pair Hamiltonian
51,52/Lambda1++ˆH/Lambda1++has a fi-
nite spectrum, such that it is incompatible with any explic-
itly correlated methods, just like the algebraic exact two-component counterparts
53–55or any second-quantized Hamil-
tonian. Moveover, the analysis of the no-pair wave functions
is a formidable task due to the non-uniqueness of the pro-jector. Nonetheless, these problems can be avoided to a large
extent. First, the incompatibility problem can be resolved by
introducing an extended no-pair Hamiltonian
21
ˆH+=/parenleftBiggˆPˆHˆP ˆPˆHˆQ
ˆQˆHˆP ˆQˆHˆQ/parenrightBigg
, (251)
where the first-quantized, configuration-space Hamiltonian ˆH
has been defined in Eq. (12), while the projectors ˆPand ˆQ
are to act, respectively, on the conventional ( ˆT|0/angbracketright) and explicit
(ˆC|0/angbracketright) correlation subspaces. To construct such projectors, we
first decompose the identity operator as
1=/Lambda1++/Lambda1−,/Lambda1+=O+
S+V+
S+˜V+
S,/Lambda1−=V−
S+˜V−
S,
(252)
where O+
S,V+
S, andV−
Sare the respective projectors for the
occupied positive energy states (PES), unoccupied PES, and
NES defined by a given basis (denoted as BasS), whereas ˜V+
S
and ˜V−
Sare the corresponding complements. We then have
ˆP12=(O+
S(1)+V+
S(1))(O+
S(2)+V+
S(2)), (253)ˆQ12=/Lambda1++
12Q++12, (254)
/Lambda1++
12=/Lambda1+(1)/Lambda1+(2), (255)
Q++
12=(1−O+
S(1))(1 −O+
S(2))(1 −V+
S(1)V+
S(2)),
(256)
where Q++
12is to ensure strong orthogonality to the reference
and orthogonality to the conventional products of virtual PES.There are two possible choices for /Lambda1
++
12in the spirit of “dual
basis projectors.”21One is to approximate the positive energy
complement ˜V+
SasV+
L−V+
S, with V+
Lconstructed with an
enlarged basis (BasL) consisting of the BasS as an orthogonal
subset, such that
/Lambda1++
12,a≈(O+
S(1)+V+
L(1))(O+
S(2)+V+
L(2)),(257)
ˆQ12,a≈/Lambda1++
12,aQ++
12=V+
L(1)V+
L(2)−V+
S(1)V+
S(2).
(258)
The other choice is to approximate the negative energy com-
plement ˜V−
SasV−
L−V−
Sagain with V−
Lconstructed with the
BasL, such that
/Lambda1++
12,b=(1−/Lambda1−(1))(1−/Lambda1−(2))≈(1−V−
L(1))(1−V−
L(2)),
(259)
ˆQ12,b≈/Lambda1++
12,bQ++
12=(1−O+
S(1)−V−
L(1))(1 −O+
S(2)
−V−
L(2))(1 −V+
S(1)V+
S(2)). (260)
The subtle difference between ˆQ12,aand ˆQ12,blies in that
the former is free of contaminations of NES but at the price
of a reduced explicit correlation space ( V+
L−V+
S⊂˜V+
S),
whereas the latter has the full explicit correlation space ˜V+
S
but at the price of some contaminations of NES, hardly of any
significance though. Interestingly, ˆQ12,bis formally in line
with the filled Dirac picture: Both O+
SandV−
Lcan be viewed
as occupied. As a whole, ˆQ12,bensures strong orthogonality
to the “reference” O+
S+V−
Las well as orthogonality to the
conventional correlation subspace. This kind of projector is
known as “ ansatz 3” in the nonrelativistic F12 methods.10If a
single basis set is to be used (i.e., BasL =BasS and hence
V+
L=V+
SandV−
L=V−
S), only the second choice ˆQ12,b,
Eq.(260) , shall work. From now on we will not distinguish
between /Lambda1++
12,aand/Lambda1++
12,b(denoted simply as /Lambda1++
12) and be-
tween ˆQ12,aand ˆQ12,b(denoted simply as ˆQ12). Note in pass-
ing that the present extended no-pair projector is conceptually
different from the one56constructed by the all-positive en-
ergy part of the complex-coordinate-rotated (CCR) spectrum
of the core Hamiltonian. The former works with the originalHamiltonian whereas the latter, albeit of the same dimension
as the present ˆQ
12,a,E q . (258) , invokes the CCR Hamilto-
nian. Moreover, we disagree with their viewpoint56that the
(positive-valued) difference between the non-projected and
projected CI energies represents one of the QED corrections.
The non-projected CI treatment of NES is simply incorrect.21
The Hamiltonian H+,E q . (251) , alongside with the pro-
jectors ˆP12,E q . (253) , and ˆQ12,E q s . (258) or(260) , can
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128.114.34.22 On: Tue, 02 Dec 2014 01:47:47144117-21 Li, Shao, and Liu J. Chem. Phys. 136, 144117 (2012)
now act on the unprojected many-electron basis functions
(ˆT|0/angbracketright,ˆC|0/angbracketright)T,independently of the wave function ansatz .
Taking MP2 as an example, we have the following Hylleraasfunctional for a given occupied electron pair |ij/angbracketright:
H
(ij)=/angbracketleftuij|ˆF12−εi−εj|uij/angbracketright+/angbracketleftuij|ˆg12|ij/angbracketright+/angbracketleftij|ˆg12|uij/angbracketright,
(261)
ˆF12=ˆF1+ˆF2, (262)
|uij/angbracketright=|ab/angbracketrighttij
ab+ˆQ12f12|kl/angbracketrightcij
kl, (263)
where ˆFpis the Fock operator and εpis the corresponding
eigenvalue. The following convention for labeling the molec-
ular orbitals has been employed here: { i,j,k,l, ...} f o r t h e
occupied PES, { a,b,c,d,...}f o rt h ev i r t u a lP E S ,a n d{ p,q,
r,s,...}f o ru n s pecified orbitals. Functional (261) can be de-
coupled into a conventional (MP2) part and an F12 correction
term by omitting their couplings. The amplitudes cij
klare then
obtained by solving the linear system of equations
Bkl(ij)
mnCij
kl=−Vij
mn, (264)
Bkl(ij)
mn=/angbracketleftmn|f12ˆQ12(ˆF12−εi−εj)ˆQ12f12|kl/angbracketright,(265)
Vij
mn=/angbracketleftmn|f12ˆQ12g12|ij/angbracketright. (266)
Clearly, the four-component relativistic MP2-F12 formulated
in this way is completely parallel to the nonrelativistic coun-terpart.
Alternatively, one can start from the first order equation
(ˆF
12−εi−εj)|ωij/angbracketright+ˆg12|ij/angbracketright=0, (267)
for the unprojected pair function |ωij/angbracketrightthat is strongly orthog-
onal to |ij/angbracketrightbut contaminated by NES. Thanks to the relation
[ˆF12,/Lambda1++
12]=0, multiplying Eq. (267) from the left by /Lambda1++
12
leads to
/Lambda1++
12(ˆF12−εi−εj)|uij/angbracketright+/Lambda1++
12ˆg12/Lambda1++
12|ij/angbracketright=0,
|uij/angbracketright=/Lambda1++
12|ωij/angbracketright, (268)
which is just the stationarity condition for the projected func-
tional (261) , further approximated to Eq. (264) though. This
simple manipulation shows that one can first construct the un-
projected pair function |ωij/angbracketrightand then from it the projected
one|uij/angbracketright. This trivial result has an important implication: The
knowledge on the analytic structures of the projected wave
functions is not really needed. Rather, that of the “exact rel-
ativistic wave functions” can directly be transplanted to theno-pair approximation. This holds at least to first order in
the electron-electron interaction. According to the previous
results (see Table I), the singularities of the “exact relativis-
tic wave functions” are rather weak, with the lowest power
νofr
12being roughly O(α2). More importantly, they only
affect an extremely small neighborhood of the coalescencepoint of two electrons, such that they are only of minor im-
portance for the calculation of the electronic energy, thanks to
the suppression by the volume element 4 πr
2
12for very small
r12. Instead, it is the extended region away from the coales-
cence point and the overall shape of the correlation hole thatare really important.57This region is still governed by the be-
haviors in the nrl. Therefore, it should be sufficient to directly
use, e.g., the nonrelativistic Slater-type correlation factor12
f12=−1
γexp(−γr12)=−1
γ+r12+O/parenleftbig
r2
12/parenrightbig
.(269)
Apart from the usual two-electron integrals, the following
kinds of integrals
f12,f12
r12,f2
12,[ˆT1,f12],[[ˆT1+ˆT2,f12],f12] (270)
are also required. They can all be evaluated analytically for
Gaussian-type spinors. Plugging the Dirac kinetic operator in[ˆT
1,f12] yields
[/vectorα1·/vectorp1,f12]=[/vectorα1·/vectorp12,f12]=−if/prime
12(/vectorα1·ˆr12), (271)
such that
[[ˆT1+ˆT2,f12],f12]=0. (272)
This strongly suggests the use of “approximation C”58when
evaluating the integral /angbracketleftij|f12ˆF12f12|kl/angbracketrightinvolving the ex-
change operator K12, viz.,
/angbracketleftij|f12ˆF12f12|kl/angbracketright=/angbracketleftij|f12(ˆF12+ˆK12)f12|kl/angbracketright
−/angbracketleftij|f12ˆK12f12|kl/angbracketright
=1
2/angbracketleftij|[f12,[ˆF12+ˆK12,f12]]|kl/angbracketright
+1
2/angbracketleftij|[ˆF12+ˆK12,f2
12]+|kl/angbracketright
−/angbracketleftij|f12ˆK12f12|kl/angbracketright
=1
2/angbracketleftij|(ˆF12+ˆK12)f2
12|kl/angbracketright+1
2/angbracketleftij|f2
12(ˆF12
+ˆK12)|kl/angbracketright−/angbracketleftij|f12ˆK12f12|kl/angbracketright.(273)
All the three terms can then be approximated by the RI with
a kinetically balanced CABS.
Finally, it deserves to be emphasized again that the no-
pair energy is dependent on the mean-field potential gener-ating the orbitals, leading to an intrinsic uncertainty of order
(Zα)
3. However, this uncertainty can readily be removed by
including the counter one-body terms derived from QED, seeEqs. (100) and (105) in Ref. 21. We then have an “potential-
independent no-pair approximation.” All beyond this can be
ascribed to QED effects.
VI. CONCLUSION AND OUTLOOK
The coalescence behaviors of relativistic wave functions
have been analyzed in depth for the whole spectrum of rel-ativistic many-electron Hamiltonians. The results are indis-
pensable for establishing relativistic many-body theories on
a firm basis. In particular, some formal evidence is foundto show that the configuration-space Dirac-Coulomb Hamil-
tonian has no bound electronic states. This is of course
hardly of any physical consequence as the no-pair approxi-
mation is mandatory in solving the many-body Dirac equa-
tion. It is then shown that, by introducing an extended
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128.114.34.22 On: Tue, 02 Dec 2014 01:47:47144117-22 Li, Shao, and Liu J. Chem. Phys. 136, 144117 (2012)
no-pair projector, four-component relativistic explicitly cor-
related wave function methods can be made fully parallel to
the nonrelativistic counterparts. The potential dependence ofthe so-calculated energy can further be removed so as to meet
QED seamlessly. That is, the “(time-independent) potential-
independent no-pair approximation + (time-dependent)perturbative QED” should be taken as the ultimate approach
for structural calculations of heavy atoms and molecules.
These findings (i.e., coalescence conditions, extended-no-pairprojected Hamiltonian, explicitly correlated treatment of PES,
and QED description of NES) open up new and exciting per-
spectives in relativistic molecular quantum mechanics, the
union of relativistic quantum chemistry and QED. Further
combined with the recently developed exact two-componentHamiltonians that fit any orbital-based correlation methods,
the “big picture” of relativistic molecular quantum mechanics
can be regarded as established.
ACKNOWLEDGMENTS
The research of this work was supported by grants from
the National Natural Science Foundation of China (ProjectNos. 21033001 and 11101011) and the Research Fund for
the Doctoral Program of Higher Education (Project No.
20110001120112).
APPENDIX: COMMON EIGENFUNCTIONS
OF {ˆh12,C12,ˆP12,ˆI,ˆj2
12,ˆj12,z}
Since the operators {ˆh12,C12,ˆP12,ˆI,ˆj2
12,ˆj12,z},E q . (51),
are mutually commutative,43their common eigenfunctions
can be classified as follows. Superscript AorSindicates anti-
symmetric or symmetric under the permutation ˆP12, with the
former for two electrons (or two positrons) and the latter for
an electron-positron pair (i.e., positronium). Superscript eor
orefers to even or odd j. The first subscript +or−refers to
the parity +(−1)jor−(−1)j, whereas the second subscript +
or−indicates the eigenvalue +1o r−1o fC12(cf. Eq. (69)).
The spin-angular functions /Omega1i(i∈{1, 2, 3, 4}) have been
defined in Eqs. (54)–(57).
1.VA
(+,+):
/Psi1A,e
(+,+)=⎛
⎜⎜⎜⎜⎝f
LL
1/Omega11
fLS
3/Omega13+fLS
4/Omega14
fLS
3/Omega13+fLS
4/Omega14
fLL
1/Omega11⎞
⎟⎟⎟⎟⎠,/Psi1
A,o
(+,+)=⎛
⎜⎜⎜⎜⎝f
LL
2/Omega12
0
0
fLL
2/Omega12⎞
⎟⎟⎟⎟⎠.
(A1)
2.V
S
(+,+):
/Psi1S,e
(+,+)=⎛
⎜⎜⎜⎜⎝fLL
2/Omega12
0
0
fLL
2/Omega12⎞
⎟⎟⎟⎟⎠,/Psi1S,o
(+,+)=⎛
⎜⎜⎜⎜⎝fLL
1/Omega11
fLS
3/Omega13+fLS
4/Omega14
fLS
3/Omega13+fLS
4/Omega14
fLL
1/Omega11⎞
⎟⎟⎟⎟⎠.
(A2)3.VA
(+,−):
/Psi1A,e
(+,−)=⎛
⎜⎜⎜⎜⎝f
LL
1/Omega11
0
0
−fLL
1/Omega11⎞
⎟⎟⎟⎟⎠,/Psi1
A,o
(+,−)=⎛
⎜⎜⎜⎜⎝f
LL
2/Omega12
fLS
3/Omega13+fLS
4/Omega14
−fLS
3/Omega13−fLS
4/Omega14
−fLL
2/Omega12⎞
⎟⎟⎟⎟⎠.
(A3)
4.V
S
(+,−):
/Psi1S,e
(+,−)=⎛
⎜⎜⎜⎜⎝fLL
2/Omega12
fLS
3/Omega13+fLS
4/Omega14
−fLS
3/Omega13−fLS
4/Omega14
−fLL
2/Omega12⎞
⎟⎟⎟⎟⎠,/Psi1S,o
(+,−)=⎛
⎜⎜⎜⎜⎝fLL
1/Omega11
0
0
−fLL
1/Omega11⎞
⎟⎟⎟⎟⎠.
(A4)
5.VA
(−,+):
/Psi1A,e
(−,+)=⎛
⎜⎜⎜⎜⎝f
LL
3/Omega13+fLL
4/Omega14
fLS
1/Omega11
fLS
1/Omega11
fLL
3/Omega13+fLL
4/Omega14⎞
⎟⎟⎟⎟⎠,/Psi1
A,o
(−,+)=⎛
⎜⎜⎜⎜⎝0
f
LS
2/Omega12
fLS
2/Omega12
0⎞
⎟⎟⎟⎟⎠.
(A5)
6.V
S
(−,+):
/Psi1S,e
(−,+)=⎛
⎜⎜⎜⎜⎝0
fLS
2/Omega12
fLS
2/Omega12
0⎞
⎟⎟⎟⎟⎠,/Psi1 S,o
(−,+)=⎛
⎜⎜⎜⎜⎝fLL
3/Omega13+fLL
4/Omega14
fLS
1/Omega11
fLS
1/Omega11
fLL
3/Omega13+fLL
4/Omega14⎞
⎟⎟⎟⎟⎠.
(A6)
7.VA
(−,−):
/Psi1A,e
(−,−)=⎛
⎜⎜⎜⎜⎝fLL
3/Omega13+fLL
4/Omega14
fLS
2/Omega12
−fLS
2/Omega12
−fLL
3/Omega13−fLL
4/Omega14⎞
⎟⎟⎟⎟⎠,/Psi1 A,o
(−,−)=⎛
⎜⎜⎜⎜⎝0
fLS
1/Omega11
−fLS
1/Omega11
0⎞
⎟⎟⎟⎟⎠.
(A7)
8.VS
(−,−):
/Psi1S,e
(−,−)=⎛
⎜⎜⎜⎜⎝0
f
LS
1/Omega11
−fLS
1/Omega11
0⎞
⎟⎟⎟⎟⎠,/Psi1
S,o
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128.114.34.22 On: Tue, 02 Dec 2014 01:47:47144117-23 Li, Shao, and Liu J. Chem. Phys. 136, 144117 (2012)
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128.114.34.22 On: Tue, 02 Dec 2014 01:47:47 |
1.4816271.pdf | Manipulation of domain propagation dynamics with the magnetostatic
interaction in a pair of Fe-rich amorphous microwires
P. Gawroński, V. Zhukova, A. Zhukov, and J. Gonzalez
Citation: J. Appl. Phys. 114, 043903 (2013); doi: 10.1063/1.4816271
View online: http://dx.doi.org/10.1063/1.4816271
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Downloaded 02 Aug 2013 to 132.174.255.116. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsManipulation of domain propagation dynamics with the magnetostatic
interaction in a pair of Fe-rich amorphous microwires
P . Gawro /C19nski,1,2,a)V. Zhukova,2A. Zhukov,2,3and J. Gonzalez2
1Faculty of Physics and Computer Science, AGH University of Science and Technology, al. Mickiewicza 30,
30-059 Cracow, Poland
2Department of Material Physics, Chemistry Faculty, Universidad del Pa /C19ıs Vasco/Euskal Herriko
Unibertsitatea (UPV/EHU), P.O. Box 1072, 220080 San Sebasti /C19an, Spain
3IKERBASQUE, Basque Foundation for Science, Bilbao 48011, Spain
(Received 26 April 2013; accepted 5 July 2013; published online 23 July 2013)
We studied the domain wall dynamics in a system of two magnetostatically interacting Fe-rich
glass coated amorphous microwires paying attention on the influence of the interaction and the
external tensile stress on the velocity of the domain wall propagation. We measured and analyzednumerically the dependence of the shape of the hysteresis loops on the frequency of the applied
field considering its origin related with the finite domain wall velocity. The critical condition for
the disappearance of the plateau on the hysteresis loops separating two remagnetization events in asystem of two microwires was investigated.
VC2013 AIP Publishing LLC .
[http://dx.doi.org/10.1063/1.4816271 ]
I. INTRODUCTION
Success of existing and potential applications of the
amorphous microwires, in highly sensitive magnetic sensors1
and magnetic logic and memory devices2stimulates the
research and development of new methods of experimental
and theoretical analysis of the magnetic properties of suchmicrowires.
3,4Studies of glass-coated microwires with ferro-
magnetic nucleus attracted considerable attention owing to
their reduced dimensions and unusual magnetic propertiessuch as spontaneous magnetic bistability as well as Giant
magnetoimpedance effect.
3,4Spontaneous magnetic bistabil-
ity observed in Fe-rich microwires is particularly interestingfor studies of the domain walls (DW) propagation within the
inner core of microwire.
3,4
As reported elsewhere, DWs propagation can be driven
by the magnetic field or by the electric current flowing through
the sample. For the case of magnetic field driven DW dynam-
ics the domain wall velocity is determined by the magneticfield value and by the wires dimensions.
4–8Extremely fast
DW propagation has been reported for amorphous microwires
with circular cross section (usually exceeding 1000 m =s).9
Additionally, DW velocity, v, depends on the magnetoelastic
anisotropy depending on the magnetostriction constant value
as well as on the internal or applied stress.10
Abovementioned magnetic bistability of amorphous
glass-coated microwires with positive magnetostriction con-
stant is related with fast magnetization switching of a largesingle axially magnetized domain.
3,4Onset of such peculiar
domain structure consisting of a single large axially magne-
tized domain surrounded by outer radially magnetized shell isdetermined by the stresses arising during solidification of
composite wire consisting of metallic nucleus surrounded by
the glass coating.
4,11–13The strength of these stresses is deter-
mined by the relative volumes of solidifying metallic nucleusand glass coating.11–13Consequently, the magnetic properties
of glass-coated microwires are determined by the magnetoe-lastic anisotropy originated from the composite.
In a number of applications, arrays of closely spaced
magnetic wires are of interest. In this case, their magneticresponse will be affected by the stray fields from surrounding
microwires.
14–16Magneto dipole interactions are quite signif-
icant in assembly of bistable wires (typically of Fe-basedcomposition) resulting in change of the shape of the hystere-
sis loops.
14–16This change of the shape of the hysteresis loop
of the arrays has been attributed to the effect of superpositionof the external magnetic field and stray field produced by
neighboring microwires. After switching the microwire with
lower H
s, it produces a stray field in the opposite to the exter-
nal field sense, so as the total field becomes insufficient to
switch the magnetization of the second wire with larger Hs,
resulting in splitting of the hysteresis loop.14–16
The numerical simulation of the remagnetization process
in magnetic materials is usually performed by solving
Landau-Lifshitz Gilbert equation either by finite differencemethod or finite element method.
17Unfortunately for the
microwires with dimensions usually larger than 5 lm such
simulation is difficult to realize. On the other hand, for theglass coated microwires such simulations must consider the
distributions of the internal stresses. However, we developed
recently the phenomenological approach for numerical studyof the remagnetization reversal of the microwire systems.
18,19
In this paper we are applying our scheme of calculation
to investigate the dependence of the shape of the hysteresisloop on the frequency of the applied field in a single micro-
wire, as well as in the system of two microwires. The impor-
tant elements of our numerical approach are the experimentalparameters such as the values of the switching field and the
mobility of the domain wall.
We are also studying the influence of the magnetostatic
interaction between the microwires on the velocity of the
propagating domain wall. Achieving fast domain wall
a)gawron@newton.ftj.agh.edu.pl
0021-8979/2013/114(4)/043903/9/$30.00 VC2013 AIP Publishing LLC 114, 043903-1JOURNAL OF APPLIED PHYSICS 114, 043903 (2013)
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tial technological applications such as memory or logic devi-
ces.20,21We want to demonstrate that the interaction
between the microwires could serve as a velocity control pa-
rameter, decreasing or increasing the domain wall propaga-
tion in a carefully designed many microwires systems.
II. EXPERIMENTAL METHOD AND SAMPLES
We investigated the magnetization reversal process and
the domain wall dynamics of the cylindrical glass coated
amorphous microwires of the nominal compositionFe
75B15Si10fabricated by Taylor-Ulitovsky technique as
described elsewhere.3,4The length of all measured samples
was equal to L¼10 cm, but they differed in the diameter of
the metallic nucleus ( d) as well as the total diameter ( D).
The sample Ahasd¼18lm and D¼24:8lm, the sample
Bhasd¼6:8lm and D¼21:3lm, and the sample Chas
d¼6:1lm and D¼25:2lm. Producing samples with dif-
ferent ratio, q, of metallic nucleus diameter, d, and total di-
ameter, D;q¼d=D, allowed us to control residual stresses,
since the strength of internal stresses is determined by the ra-
tioq.11–13Additionally, the geometrical differences result in
distinct magnetic properties as the remanence and theswitching field. We measured the axial hysteresis loops by
the induction method and the domain wall velocity by the
Sixtus-Tonks like method in a single microwire and the sys-tem of two microwires, as previously described in Refs. 14,
15, and 18.
III. EXPERIMENTAL RESULTS AND DISCUSSION
The axial hysteresis loops of a single microwire, meas-
ured at different frequencies of the applied field, are pre-
sented in Fig. 1. The shape of the hysteresis loops depends
on the relation between the velocity of the domain wallwhich propagates along the microwire and the rate of the
changes of the applied field. The velocity of the domain wall
in this range of the frequencies of the applied field is almostconstant. As the frequency of the applied field grows the
applied field changes faster than the domain wall travelsalong the microwire, it manifests itself as the gradual loss of
the squareness of the hysteresis loops, accompanied by the
increase of the switching field value.
The switching field ( H
s) of the amorphous microwire
can be expressed as22
Hs/C24cw
Msd; (1)
where Msis the saturation magnetization and cwis the do-
main wall surface energy,22,23
cw¼ðAKÞ0:5; (2)
where Ais the exchange constant and Kis the magnetoelastic
energy22,23
K¼3=2kðriþrappÞ: (3)
The magnetoelastic energy Kis a product of the magne-
tostriction constant k, positive and non-vanishing in case of
the Fe-rich amorphous microwire, and the sum of the internal
residual stresses ( ri) coming from the quenching and the
drawing during the fabrication process and the applied stress
(rapp). The switching field ( Hs) is inversely proportional to
the diameter d,s o Hsincreases when the diameter dof the
metallic nucleus decreases. Another contribution to Hscomes
from distribution of the stresses. The strength of the internal
stresses increases when the ratio q¼d=Ddecreases.24So in
our case for sample A of d¼18lm and ratio q¼0:72 the
switching field at f¼50 Hz is equal to 24 :8A=m while for
sample C of d¼6:1lm and ratio q¼0:24 the switching
field at f¼50 Hz is 186 :72 A =m.
The dependence of the switching field on the frequency
of the applied field for all the measured samples is gatheredin Fig. 2.
Similar to previous studies realized in different kinds of
amorphous magnetic materials presenting rectangular hyster-esis loop we observed increase of the switching field with
frequency.
25
We have measured the hysteresis loops of two Fe-rich
amorphous microwires using the experimental setup sche-
matically presented in Fig. 3. The microwires were fixed
FIG. 1. Dependence of the shape of the hysteresis loop of a single microwire
on the frequency of the applied field, for the amplitude of the applied field
Hm¼360 A =m.FIG. 2. Dependence of the switching field ( Hs) on the frequency of the
applied field for the samples A,B, and C.043903-2 Gawro /C19nski et al. J. Appl. Phys. 114, 043903 (2013)
Downloaded 02 Aug 2013 to 132.174.255.116. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionstogether, so the distance between two magnetic nucleus (d) is
double glass coating thickness, in order to assure their paral-
lel placement and to maximize the dipolar interactionbetween the neighboring microwires. In such systems two
magnetization jumps related with fast magnetization switch-
ing are visible in the remagnetization process (see Fig. 4),
each one corresponds to the reversal of the magnetization of
the individual microwire. Due to the aforementioned magne-
tostatic interaction, the first magnetization jump occurs at thelower value of the applied field than the switching field ( Hs)
for a single microwire, and the second magnetization jump at
the higher applied field.
In case of pair of microwires with the same diameters of
the metallic nucleus and the glass coating, the magnetostaticinteraction ( d) between them is symmetrical, so the first
magnetization switching takes place at H
CC1¼HsC/C0dand
the second switching at HCC2¼HsCþd. The separation of
the magnetization jumps, the length of the plateau, equals in
this case to HCC2/C0HCC1¼2d, where dis the experimental
measure of the strength of the magnetostatic interactionbetween the neighboring microwires. The magnetostatic
interaction, that is a direct consequence of the stray field sur-
rounding the microwires, depends mainly on two parameters:the distance between their metallic nuclei and the magnetiza-
tion of the neighboring microwire. The magnetostatic inter-
action decreases quite abruptly with the increase of the interwire distance ( d), usually when d>2:5 mm the hysteresis
loop loses its two magnetization jumps shape and d¼0. On
the other hand the magnetostatic interaction increases withthe growth of the magnetization of the neighboring micro-
wire due to the increase of the stray field of such micro-
wire.
14It is worth noting that in the case of the hysteresis
loop of the two identical microwires the magnetization of the
plateau, that is separating these two magnetization jumps, is
equal to zero because these two microwires are magnetizedin the opposite directions.
In Fig. 4, we present the example of hysteresis loops for
a system of two microwires with the different qratio of the
metallic nucleus diameter, d, and the total diameter, D. In the
case of pair of microwires consisting of the samples AandC,
we observed the remagnetization of the microwire Aat lower
magnetic field since it has the smaller value of the switching
field ( Hs
A¼26 A =m). However owing to the magneticinteraction of the microwire Cwith the microwire Athe
applied field sufficient to flip the magnetization of the sampleAis lower: it decreases to H
AC1¼HsA/C0d1¼16 A =m.
Moreover the second magnetization jump of the microwire C
takes place at applied field ( HAC21¼HsCþd2¼276 A =m).
This value of applied field is higher than the switching field of
the single microwire C(HsC¼184 A =m) due to the magneto-
static interaction with the microwire A. It is worth mentioning
that the separation of the magnetization jumps, i.e., the width
of the plateau is greater than the sum of the magnetostaticinteractions between the microwires H
AC2/C0HAC1>d1þd2.
The influence of the microwire Aon the actual value of the
switching field of the wire C(HAC2) is bigger than in case of
the microwire Cacting on the microwire A. This asymmetry
of the interactions is mainly due to the difference in the mag-
netization of the microwires, that is a direct result of the dif-ference in the diameters of the metallic nuclei of the
microwires Aand C. Usually for Fe-rich microwires (e.g.,
Fe
75B15Si10) as the diameter of the metallic nucleus decreases
the switching field increases and the magnetization decreases.
This increase of the switching field with decreasing of the me-
tallic nucleus diameter is usually attributed to the growth ofthe internal stress with the reduction of the qratio.
11–13Since
the microwires have different magnetizations, the magnetiza-
tion of the plateau is non-zero even though the microwires arestill magnetized in the opposite directions.
We have studied the dependence of the shape of the hys-
teresis loops of pair of microwires on the frequency of theapplied field. We considered two microwires systems of the
same metallic nucleus diameters (see Fig. 5(a)) and of the dif-
ferent diameters (Fig. 5(b)). Regardless of the diameters of
the microwires, the slope of the hysteresis during the magnet-
ization jumps decreases with the frequency of the applied
field. In case of pair of microwires of the same diameter, theslope corresponding to the first magnetization jump is differ-
ent from the slope of the second magnetization jump due to
the dependence of the velocity of the domain wall on theapplied field, e.g., for the frequency of 1000 Hz in Fig. 5(a).
The width of the plateau separating the magnetization jumps
FIG. 3. The scheme of the experimental setup for measuring the hysteresis
loops of the systems of the microwires. S—15 cm long solenoid, PC—the
pick-up coil, d—the distance between the neighboring microwires, z ¼L—
the length of the microwires.
FIG. 4. Influence of the magnetostatic interaction between the microwires
on the shape of the hysteresis loops: A$C—two different microwires,
C$C—two the same microwires, A—single microwire, C—single micro-
wire, Hm¼360 A =m.043903-3 Gawro /C19nski et al. J. Appl. Phys. 114, 043903 (2013)
Downloaded 02 Aug 2013 to 132.174.255.116. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsdecreases monotonically as the frequency of the applied field
increases. Eventually, for each value of the amplitude of the
magnetic field ( Hm) exists a critical frequency ( xcr)o ft h e
disappearance of the plateau. This critical frequency decreases
with the amplitude of the applied field (see Fig. 6). Thisdependence must be attributed to the effect of the growth of
the velocity of the domain wall with the applied field.
Using Sixtus-Tonks like experimental setup, described
in details elsewhere,7,9,26–29we measured the dependence of
the DW velocity on the applied field in a single microwire
and pair of microwires (see Fig. 7). Our experimental setup
is schematically presented in Fig. 8. The 14 cm long solenoid
(S) serves as an exciting coil providing the magnetic field
necessary to nucleate and propagate the reverse domainalong the microwire. The detection system consists of three
pick-up coils ( p
1;p2;p3) connected to separate channels of
the oscilloscope. The typical electromotive force (emf) sig-nals induced in the subsequent pick-up coils by the advanc-
ing domain wall are presented in Fig. 9. The presented
temporal order of the three emf picks positively validates theassumption that during the remagnetization process the sin-
gle domain wall is propagating along the microwire. As long
as the assumption is hold the domain wall velocity is calcu-lated as v¼Ds=Dt, where Dsis the distance between two
subsequent pick-up coils, and Dtis the time difference
between the maximum of the induced emf signals.
The domain wall is propagating with the velocity pro-
portional to the driving magnetic field, this linear relation
was expressed by Sixtus and Tonks as
29
v¼SðHs/C0H0Þ; (4)
where Hs>H0(Ref. 22) is the switching field and H0is the
critical magnetic field for the domain wall displacement, and
Sis the domain wall mobility.
The dependence of the DW velocity on the applied field
for a single microwire in the case of samples BandCis non-
linear. In both cases the dependence can be divided into three
linear regimes. The viscous regime starts at 250 A =m for the
sample Band 350 A =m for the sample C, and finishes about
390 A =ma n d4 9 5 A =m, respectably. In this regime theFIG. 5. Dependence of the shape of the hysteresis loops for a system of (a)
two the same microwires ( B) (b) two different microwires ( BandC) on the
frequency of the applied field, Hm¼360 A =m.
FIG. 6. Dependence of the critical frequency on the amplitude of the applied
field.FIG. 7. Influence of the magnetostatic interaction between the microwires on
the velocity of the domain wall. C!B—velocity of the DW in microwire B
under the influence of microwire C;A!B—velocity of the DW in micro-
wire Bunder the influence of microwire A;C!C—velocity of the DW in
microwire Cunder the influence of microwire C;A!C—velocity of the
DW in microwire Cunder the influence of microwire A,B,andC—single
microwires.043903-4 Gawro /C19nski et al. J. Appl. Phys. 114, 043903 (2013)
Downloaded 02 Aug 2013 to 132.174.255.116. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsestimated values of the DW mobilities and the critical propa-
gation fields are SB¼1:74 m2=As and SC¼
1:56 m2=As;HB0¼22 A =ma n d HC0¼106 A =m. In the sec-
ond regime the velocity of the domain wall abruptly increases
from 600 m =sa t3 6 0 A =m to over 1200 m =s at 420 A =mf o r
the sample Band similarly for the sample C. Such growth of
the velocity could be explained by the domain wall interaction
with the distributed defects and Walker breakdown.26,27In the
third regime for the applied fields higher than 420 A =mf o r
the sample Band 550 A =m for the sample Cthe DW mobili-
ties are similar to the ones in the vicious regime, SB¼
2:58 m2=As and SC¼1:96 m2=As.
Two microwire systems either consisting of the same or
different microwire samples are suitable for an investigationof the influence of the magnetostatic interaction between the
microwires on the DW velocity. In our case, no modification
of the existing experimental setup was necessary for twomicrowire systems measurement, because the hysteresis
loops for two Fe-rich microwire systems presented in Fig. 4
show that two magnetization jumps are well separated. Thisseparation means that the electromotive force (emf) peaks
induced in the pick-up coils when the domain wall is propa-
gating, needed for evaluation of the DW velocity, for the firstand the second magnetization reversals are separated and
can be easily identified. When the applied field is below theswitching threshold for the second magnetization jump, the
pick-up coils receive signal only from the microwire which
is being remagnetized first. When the applied field is largerthan the switching field needed for the second magnetization
jump the pick-up coils gather the signal from both micro-
wires, but the separate emf peaks from the first and the sec-ond magnetization reversals take place at the different points
in the time scale. The amplitude and the shape of the peaks
are different for the microwires with a different diameter ofthe metallic nucleus. The curves denoted as C!CandC!B
in Fig. 7represent the dependence of the DW velocity on the
applied field for the microwire, remagnetizing at lowerapplied field in the two-microwires system. The dependences
C!CandC!Bcorrespond to the magnetostatic interac-
tion of the microwire Con the microwires CandB, respec-
tively. In both cases the nonlinear shape of the curve, for the
microwire under the influence of the neighboring microwire
is the same as in the case of the single microwire. However,there are two main differences, the DW starts to move at
lower applied field and the domain wall velocity is higher
then in the case of the single microwire. The neighboringmicrowire serves as a source of the additional magnetic field
and in the case of the first magnetization jump it promotes
remagnetization of the affected microwire, by effectivelydecreasing the switching field value and increasing the do-
main wall velocity. The wide separation between the first
and the second magnetization jump presented in Fig. 4, for A
andC, and similarity for AandBis a combination of a huge
difference of the switching field values of the microwires
and the magnetostatic interaction between the microwires.Such separation allows measuring the domain wall velocity
in the microwires CandBat the second magnetization jump
under the influence of the interaction of the microwire A.
The curves denoted as A!CandA!Bin Fig. 7present
the dependence of the velocity of the microwire that was
remagnetized during the second magnetization jump. Thenonlinear shape of the dependence is the same as for a single
microwire measurement, but we managed to observe only
the part for the higher applied field. After the completedremagnetization of the first microwire during the first mag-
netization jump, this microwire produces the additional mag-
netic field of the opposite direction to the applied field, andthus effectively increases the switching field and decreases
the DW velocity of the neighboring microwire.
The experimental setup presented in Fig. 8, allows to
study the influence of the external tensile stress on the do-
main wall velocity in a single microwire as well as two
microwire systems, by fixing one end of the microwire to asample holder (SH), while attaching a mechanical load (ML)
to the other end of the microwire. In Fig. 10we present the
magnetic field dependence of the DW velocity for the single
microwire Bin the presence of an additional tensile stress.
Damping is the most relevant parameter describing the
domain wall dynamics and it is usually expressed as23,30
S¼2l0Ms=b; (5)
where Msis a magnetization saturation and bis the domain
wall damping. In case of the amorphous microwires usually
FIG. 8. The scheme of the experimental setup for measuring the domain
wall velocity of the systems of the microwires. S—15 cm long solenoid,
p1;p2;p3—2 mm pick-up coils, ML—mechanical load, SH—sample holder,
d—the distance between the neighboring microwires, z ¼L - the length of
the microwires.
FIG. 9. The electromotive force (emf) signals induced in the subsequent pick-up coils during the propagation of a single domain wall along the microwire.043903-5 Gawro /C19nski et al. J. Appl. Phys. 114, 043903 (2013)
Downloaded 02 Aug 2013 to 132.174.255.116. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsthe spin-relaxation damping bris considered, that is inversely
proportional to the domain wall width dw(Refs. 22and23)
br/C241=dw/C24ðK=AÞ1=2: (6)
The damping brincreases when the applied mechanical
stress increases as well as the internal stress controlled by ra-
tioq¼d=D. The domain wall mobility ( S) and as a result
domain wall velocity, which are inversely proportional to
damping, decrease when the applied stress increases. In
order to achieve higher domain wall velocity it is necessaryto relax the stress and decrease the damping. On the other
hand, since the domain wall velocity is proportional to H
s
(Eq. (4)) and Hsis inversely proportional to diameter d(Eq.
(1)), the microwires with smaller diameter shows higher Hs
and higher domain wall velocity.
The breaking points of the nonlinear dependence of the
domain wall velocity on the applied field move to the higher
fields, as the applied stress increases. The viscous linear re-
gime that ends for the zero applied stress at 250 A =m, for the
case of the applied stress of 23 :5 MPa prolongs to 650 A =m,
and for the case of the applied stress of 58 :8 MPa persists up
to 900 A =m. The observed domain wall mobility in the vis-
cous linear regime decreases from SB¼1:74 m2=As for zero
applied stress to SB¼0:38 m2=As for 58 :8 MPa, and the crit-
ical propagation field changes the value and the sing toH
B0¼/C0732 A =m. As the linear regime extends with the
applied stress, the domain wall velocity drops drastically, e.
g., for zero applied stress at 500 A =m the domain wall veloc-
ity is about 1400 m =s and for the stress of 59 MPa is only
about 450 m =s. The high domain wall velocity at low applied
fields is due to the existence of the nonlinear behavior that iseliminated by the application of the external tensile stress.
The same behavior is observed for sample C.
In Fig. 11(a) we present the dependence of the DW ve-
locity for the microwire Bmeasured under the influence of
an external tensile stress and in the presence of the micro-
wire C. In this configuration the additional magnetic field
produced by the neighboring microwire Cdiminishes the
influence of the applied stress. The DW velocity, measuredunder 2 :2 MPa stress is still higher than in the case of the sin-
gle microwire C. We can obtain the same dependence of the
DW velocity on the applied magnetic field as in the case of
the single microwire when we apply the external stress about
6:5 MPa in the presence of the magnetostatic interaction of
the microwire C. Further increase of the external stress leads
to gradual decrease of the DW velocity, because the external
stress has the stronger influence on the domain wall dynam-ics. In Fig. 11(b) we present the dependence of the DW ve-
locity for the microwire Bmeasured under the influence of
the external tensile stress and in the presence of the micro-wire A. In this configuration the external tensile stress as
well as the magnetostatic interaction result in decreasing the
DW velocity.
IV. NUMERICAL CALCULATIONS
Our previously developed calculation scheme18,19can
be applied to the numerical reconstruction of the dependence
of the shape of the hysteresis loops on the frequency of the
applied field and the theoretical analysis of the condition ofthe disappearance of the plateau in the hysteresis loops for
two-microwires systems. Our phenomenologically basedFIG. 10. Influence of the applied stress on the velocity of the domain wall of
the single microwire B.
FIG. 11. Influence of the applied stress on the velocity of the domain wall in
microwire B (a) under influence of microwire C and the applied tensilestress, (b) under influence of microwire A and the applied tensile stress.043903-6 Gawro /C19nski et al. J. Appl. Phys. 114, 043903 (2013)
Downloaded 02 Aug 2013 to 132.174.255.116. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsnumerical approach for a single microwire can be briefly
summarized as follows. In order to activate the remagnetiza-
tion reversal, the applied field ( Happ) acting on a single
microwire must exceed the value of the switching field ( Hs).
So the starting time ( t0) of the reversal process is obtained
from the following equation:
Happ¼Hmsinðxt0Þ/C0HsðxÞ; (7)
where HsðxÞis the experimentally obtained switching field
for a given microwire presented in Fig. 2. During the remag-
netization process the domain wall unpins from one end of
the microwire ( z0ðt¼t0Þ) and propagates along the micro-
wire. This movement of the domain wall is usually described
by a linear dependence of the velocity of the domain wall on
the applied field, in the following form:
vDW¼dz
dt¼SðHmsinðxt0Þ/C0H0Þ; (8)
where H0is a critical propagation field and Sis a domain
wall mobility.
The integration of Eq. (2)gives us the time ( t1) when the
remagnetization is completed, e.g., when the domain wallreaches the other end of the microwire ( z
1ðt¼t1Þ¼L). Both
parameters SandH0are obtained from the experimental de-
pendence of the DW velocity on the applied field and arespecific for a given microwire, see Fig. 7. During the propa-
gation of the DW the single domain inner core is no longer
uniformly magnetized up ( Ms
up) or down ( Msdown). As the
DW propagates through the microwire, the area magnetized
parallel to the applied field increases at the expense of the
opposite magnetized one. The actual value of the magnetiza-tion of the microwires, when the DW is at a given point
(z
DW) in the microwire can be calculated from our proposed
scheme of the local magnetization profile, presented in Fig.12. The local magnetization profile informs us how the mag-
netization is distributed along the microwire. We previously
measured and successfully applied for the calculation thelocal magnetization profile for Fe-rich wire with diameter of
125lm.
19We assumed that the local magnetization profile
for the microwire is more square and more steep at the endsthan the one for the thicker wire, since the penetration length
of closure domains for the microwires is assumed to be
much shorter than for thicker wires.4The local magnetiza-
tion profile for a microwire when the domain wall is at agiven point ( z
DW) is a composition of two local magnetiza-
tion profiles; one of the microwire magnetized up ( Msup) and
FIG. 12. Local magnetization profile of the microwire when the domain
wall is propagating along the wire.FIG. 13. The calculated dependence of the shape of the hysteresis loops of a
single microwire on the frequency of the applied field, Hm¼360 A =m.
FIG. 14. Calculated dependence of the shape of the hysteresis loops for a
system of (a) two the same microwires ( B) (b) two different microwires ( B
andC) on the frequency of the applied field, Hm¼360 A =m.043903-7 Gawro /C19nski et al. J. Appl. Phys. 114, 043903 (2013)
Downloaded 02 Aug 2013 to 132.174.255.116. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsthe other of microwire magnetized down ( Msdown). The posi-
tion of the DW calculated from Eq. (2)controls the length of
each of the components of the total profile. The integrationof the proposed local magnetization profile gives us the de-
pendence of the magnetization of the microwire, with the
propagating DW, on the applied field
mðHðtÞÞ ¼ðzDW
0Mupðz0Þdz0þðL
zDWMdownðz0Þdz0
ðL
0Mðz0Þdz0:(9)
The combination of the solutions of Eqs. (1)–(3)allows to
reproduce numerically the dependence of the shape of the hys-
teresis loop on the frequency of the applied field for a singlemicrowire. The characteristic loss of the squareness of the hys-
teresis loops with the growth of the frequency of the applied
field measured in Fig. 1is numerically reproduced in Fig. 13.
The same calculation scheme can be applied to reconstruction
of the hysteresis loops for two-microwire systems. Let us
assume that both microwires (e.g., AandC) are magnetized
down. The reversal of the magnetization begins when the DW
in the microwire A(a smaller switching field, see Fig. 4), starts
to propagate from one end of the microwire z
0ðt¼t0Þ.T h e
applied field ( Hmsinðxt0Þ) must exceed the switching field of
microwire A, dependent on the frequency of the applied field
(HsAðxÞ), diminished in this case by the magnetic field cre-
ated by the neighboring microwire C(HintCA). The starting
time t0can be derived from the following equation:
Hmsinðxt0Þ¼HsAðxÞ/C0HintCA: (10)
Once the domain wall started in the microwire A, it propa-
gates according to the following equation of motion
dz
dt¼SA½HmsinðxtÞ/C0HA0þHintCA/C138; (11)
where HA0is a propagation field of the microwire A. From
Eq.(5)we can calculate the time t1when the remagnetiza-
tion of the microwire Ais completed, when the DW reaches
the other end of the microwire z1ðt¼t1Þ. Applying the same
scheme for the microwire C, the starting time ( t2) can be cal-
culated fromHmsinðxt2Þ¼HsCðxÞþHintAC; (12)
in this case the applied field must exceed the switching field
of the microwire C(HsCðxÞ) augmented by the magnetic
field created by the neighboring microwire C(HintAC).
Similarly the time t3when the remagnetization reversal in
the microwire Cis completed can be calculated from
dz
dt¼SC½HmsinðxtÞ/C0HC0/C0HintAC/C138: (13)
The numerical solution of the above equations allow suc-
cessfully reconstruct in Fig. 14the characteristic behavior of
the measured dependence of the hysteresis loops on the fre-
quency of the applied field in two microwires systems pre-sented in Fig. 5.
When the plateau disappears, the width of the plateau
depends on the time difference between two events: stoppingof the DW at the end of the first microwire ( t
1) and starting
the propagation of DW in the second microwire ( t2). So
when the remagnetization processes in the first microwire(A) finish at the very same moment when the remagnetiza-
tion in the second microwire ( C) begins, that is t
1¼t2, then
the plateau disappears. The analytical condition for the dis-appearance of the plateau can be obtained in the following
manner. By transforming Eqs. (4)and(6)we get the expres-
sions for the starting times
t
0¼1
xarcsinHsAðxÞ/C0HintCA
Hm; (14)
t1¼1
xarcsinHsCðxÞþHintAC
Hm: (15)
The integration of the equation of motion (5)gives us
z1¼z0þSAHm
xðcosðxt0Þ/C0cosðxt1ÞÞ/C20
/C0ðHA0/C0HintCAÞðt1/C0t0Þ/C21
: (16)
The analytical condition for the critical frequency xcr¼fðHmÞ
can be obtain from combining Eqs. (7)–(10) and substituting
t1¼t2
xcr¼SA
LHmcos arcsinHsAðxcrÞ/C0HintCA
Hm/C18/C19
/C0cos arcsinHsCðxcrÞþHintAC
Hm/C18/C19 /C20/C21/C20
/C0ðHA0/C0HintCAÞ/C1arcsinHsCðxcrÞþHintAC
Hm/C0arcsinHsAðxcrÞ/C0HintCA
Hm/C18/C19 /C21
; (17)
where L¼z1/C0z0is the length of the microwires. From the
solution of Eq. (11) we obtained a good agreement between
the calculated and the measured dependence of the critical
frequency on the applied field presented in Fig. 6.V. CONCLUSIONS
We presented the experimental study of the influence of
the frequency of the applied field on the shape of the043903-8 Gawro /C19nski et al. J. Appl. Phys. 114, 043903 (2013)
Downloaded 02 Aug 2013 to 132.174.255.116. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionshysteresis loops for a single microwire and two-microwire
system. We analyzed the experimental results within the the-
oretical framework and derived the condition of the disap-pearance of the plateau between two subsequent
magnetization jumps, calculated and compared with the ex-
perimental data. We showed, that this critical condition is aninterplay between the rate of the changes of the applied field
and the DW velocity. Results obtained in this work might be
important for the experiments performed on microwiresarrays when the frequency or the magnetic field amplitude is
relatively high.
We analyzed the DW dynamics in a single microwire
and two-microwire system with different diameters and stud-
ied the effect of the magnetostatic interaction on the DW ve-
locity. The interaction between the neighboring microwiresdepends on the geometrical features of the microwires.
Depending on the microwires’ dimensions the temporal mag-
netic configuration of the microwires’ results in decreasingor increasing the DW velocity.
The manipulation of the DW velocity can be achieved
by controlling of the magnetostatic interaction between themicrowires by means of the precise control of the distance
between the microwires, since we know that the interaction
is proportional to the distance between the neighboringmicrowires and the magnetization of the microwire that pro-
duces the additional magnetic field. The increase of the DW
velocity can be also obtained by carefully designing the mag-netostatically interacting system of many microwires. We
also demonstrated that the application of the external tension
is an important control parameter for the DW dynamics for asingle microwire as well as for two-microwire systems.
ACKNOWLEDGMENTS
Partially supported by the Polish Ministry of Science
and Higher Education and its grants for scientific research.This work was supported by EU ERA-NET programme
under project “SoMaMicSens” (MANUNET-2010-Basque-
3), by EU under FP7 “EM-safety” project, by SpanishMinistry of Science and Innovation, MICINN under Project
MAT2010-18914, by the Basque Government under Saiotek
10 MIMAGURA project (S-PE11UN087), and by federaltarget program “Scientific and scientific-pedagogical person-
nel of innovative Russia,” state contracts no 14.A18.21.0762.
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1.4914068.pdf | Micromagnetic study of interaction between achiral and homochiral domain
walls in ultrathin ferromagnetic strips
/C19Oscar Alejos1,a)and Eduardo Mart /C19ınez2
1Dpto. Electricidad y Electr /C19onica, Universidad de Valladolid, Valladolid 47011, Spain
2Dpto. F /C19ısica Aplicada, Universidad de Salamanca, Salamanca 37008, Spain
(Presented 7 November 2014; received 22 September 2014; accepted 30 October 2014; published
online 9 March 2015)
Magnetic domain walls have been repetitively proposed for its use in memory and logic devices.
Most promising devices are based on ferromagnetic/heavy-metal bilayers, with perpendicularmagnetic anisotropy. The characteristics of the walls in these devices are influenced by the strength
of the Dzyaloshinskii-Moriya interaction. When this interaction is strong, it results in the formation
of homochiral N /C19eel walls, while its practical absence allows the formation of Bloch walls, either in
parallel or antiparallel configurations. For isolated domain walls, a one-dimensional model can be
successfully derived from the dynamic equations, which are of great help in order to understand
their dynamics under different stimuli. However, a thorough study of the interactions betweendomain walls is required if such models are to be extended to two or more close walls. The present
work studies the coexistence of two close nucleated domain walls by means of micromagnetic
simulations, either in the case of Bloch walls, both parallel and antiparallel, or in the case ofhomochiral N /C19eel walls, when a strong Dzyaloshinskii-Moriya interaction is present. Two
interaction mechanisms between such walls have been revealed. The first one seems to be relevant
for relatively distant walls as being inversely proportional to the square of distance, in ratheragreement with the mechanism proposed by other authors. The second one, which can be straightly
characterized in the case of N /C19eel walls, has been estimated as inversely proportional to the fourth
power of distance, then dominating for relatively close walls. Such dipolar-like interactionhas been associated with the equivalent magnetic moments of domain walls. Finally, numerical
simulations of the interaction in time of domain walls are shown, which can be appropriately
explained by means of the mechanisms here described.
VC2015 AIP Publishing LLC .
[http://dx.doi.org/10.1063/1.4914068 ]
I. INTRODUCTION
Magnetic domain walls (DWs) are of importance
because of their potential application for memory and logic
devices.1–4Most promising ones are based on ultrathin ferro-
magnetic films with perpendicular magnetic anisotropy
(PMA), where two DWs are simultaneously nucleated by
means of the Oersted field created by a current line. Thecharacteristics of these walls are governed by the presence or
absence of the Dzyaloshinskii-Moriya interaction (DMI).
5
This interaction, present in some ferromagnetic/heavy-metal
bilayers, results in the formation of homochiral N /C19eel walls
(NWs), while its absence allows the formation of Bloch
walls (BWs), either in parallel (P) or antiparallel (AP)configurations.
Several efforts have been made in order to understand
the dynamics of such DWs under different stimuli. For asingle DW, a one-dimensional model can be successfully
derived from the dynamic equation. However, its extension
to two or more DWs requires a thorough study of the interac-tions between them.
In the present work, the interaction between pairs of
DWs in ultra-thin ferromagnetic strips is studied by meansof micromagnetic ( lMag) simulations, both in the presence
and in the absence of DMI. This interaction is characterizedwith the help of external magnetic uniform fields, and ana-lyzed in terms of two different mechanisms. The first one,similar in a certain range of distances to the one proposed inthe literature,
6seems to be relevant for relatively distant
walls, as being inversely proportional to the square of the
distance between DWs. The second one, proposed here, isunderstood as a dipolar interaction between DW equivalentmagnetic moments, then being inversely proportional to thefourth power of distance, and dominating for relatively closewalls. This term, which had not been previously consideredin, accounts for the differences in the interactions betweenBWs and homochiral NWs, and is relevant from both funda-mental and technological points of view because it points outthe limit of the data-storage density achieved with theseDW-based devices.
II. MICROMAGNETIC MODEL: DIMENSIONS AND
MATERIAL PARAMETERS
In the framework of the lMag model, the magnetization
Mis a continuous vectorial function in time and space,
whose dynamics is governed by the Landau-Lifshitz (LLG)equation
a)Electronic mail: oscaral@ee.uva.es
0021-8979/2015/117(17)/17D509/4/$30.00 VC2015 AIP Publishing LLC 117, 17D509-1JOURNAL OF APPLIED PHYSICS 117, 17D509 (2015)
dm
dt¼/C0c0m/C2Hef fþam/C2dm
dt/C18/C19
; (1)
where c0,a, and mbeing, respectively, the gyromagnetic ra-
tio, the Gilbert damping parameter, and the normalized local
magnetization, the latter defined as mðr;tÞ¼Mðr;tÞ
Ms, where
Msis the saturation magnetization. Hef fis the effective field,
derived from system energy density /C15in the following way:
Hef f¼/C01
l0Msd/C15
dm, which along with the standard exchange,
magnetostatic, uniaxial anisotropy, and Zeeman contribu-
tions may also include the anisotropy exchange DMI.7–9In
the thin-film approach ( Lz/C28Ly;Lx, where Lz,Ly, and Lxare,
respectively, the dimensions of the strip in the Z, Y, and X
directions) the interfacial DMI energy density /C15DMis given
by9,10
/C15DM¼QD½mzr/C1m/C0ðm/C1r Þmz/C138; (2)
where Dis the absolute DMI parameter accounting for its in-
tensity and Q¼61 defines the chirality of this interaction.
In long strips with PMA, the presence of this interaction
determines the orientation of the magnetization within the
DWs. For example, the absence of this interaction allows the
existence of achiral BW, where the magnetization rotates
within planes perpendicular to the strip longitudinal axis.
Multiple BWs may appear along the strip, each pair of adja-
cent walls being either P or AP from the point of view of the
orientation of the magnetization, due to the achiral nature ofthe considered interactions. On the other hand, there is a
critical value D
c,10so that the magnetization rotates along
the wall within the plane that contains both the easy and the
longitudinal strip axis (N /C19eel walls or NWs). In the case of
NWs, the chiral nature of the DMI forces adjacent walls to
be always AP, their chirality depending on the sign of Qin
Eq.(2). The critical value Dcis given in absolute terms by
Dc¼2l0M2
sDNx/C0Ny ðÞ
p, where NxandNy, defined so that Nx>Ny,
are the wall in-plane demagnetizing factors, and Dis the
wall width, defined as D¼ffiffi ffi
A
kq
,Aandkbeing, respectively,
the exchange constant, and the effective anisotropy resulting
from both the magnetocrystalline and the magnetostatic
terms.
III. ANALYTICAL DESCRIPTION OF CHIRAL AND
ACHIRAL WALLS
The magnetization profile of steady DWs can be analyti-
cally written for isolated walls in long strips. For a PMA strip
with dimensions Lz/C28Ly/C28Lx,t h eo r i e n t a t i o n hof the mag-
netization respective to the out-of-plane axis changes within
the DW according to the following expression:11
cosh¼/C0tanhx/C0q
D; (3)
qbeing the wall position along the strip. The parameter x
defines the point where the orientation of the magnetization
is calculated, so that the magnetization points up within the
domain located at x<Xand points down within the domain
at the other side of the wall. The opposite transition can bestraightforwardly obtained by a change of sign of the previ-
ous expression. The use of such expression along with a
second angle U, which defines the orientation of the in-plane
magnetization, allows to develop the well-known one-
dimensional model (1DM) of the DW dynamics.10,12When
several DWs are present along the strip, the 1DM model asitself can be independently applied to each DW if the
distance between them is sufficiently high.
13Nevertheless,
an interaction appears between two DWs as one DWapproaches each other. This interaction has been character-
ized for BWs, both P and AP, and one analytical expression
6
has been proposed accounting for a mutual repulsion
between such walls. However, the expression dramatically
fails for the shortest distances prior to DWs collapsing.
Since the torque due to the application of a uniform
magnetic field Bzperpendicular to the plane of the strip does
not alter the inner structure of isolated DWs, the virtual work
principle can be of help in order to characterize the interac-tion. As an example, Figure 1shows the profile of the mag-
netization obtained from lMag calculations by using the
GPMagnet code
14for a strip where two AP NWs are present.
For these and any latter lMag calculations, a CoFe strip with
high PMA and a cross section Ly/C2Lz¼160 nm /C20:6n m
has been considered. Common and typical high PMA param-
eters were taken:9saturation magnetization Ms¼7/C2105A
m,
exchange constant A¼10/C011J
m, and uniaxial anisotropy
constant Ku¼4:8/C2105J
m3. These values result in a DW
width parameter D/C257.32 nm. Additionally, the nucleation
FIG. 1. lMag calculation (dots) and analytical description (continuous lines)
of the magnetization profile for one pair of AP NWs nucleated in a long strip
with PMA under the application of four different out-of-plane external mag-
netic fields ranging from 4 mT to 112 mT. The analytical description (contin-
uous plots) is given by Eqs. (4)and(5).17D509-2 /C19O Alejos and E. Mart /C19ınez J. Appl. Phys. 117, 17D509 (2015)of NWs is possible since a DMI constant D¼1:2mJ
m2,a si n
the case of Pt/CoFe bi-layers, has been chosen, a value which
is several times bigger than the corresponding critical valueD
cfor this geometry. Dots in this figure represent the steady
components mzandmxat every computational cell along the
strip in a limited range of 100 nm ( Lxwas chosen several
times larger than this length) and the four graphs from down
to up are obtained for different applied fields ranging fromB
z¼4m T t o Bz¼112 mT. From the depicted mzspatial
dependence, it can be inferred that there exist two lateralup-domains and one central down-domain separated by the
aforementioned AP NWs. At this point, it must be noted that
DWs seem to keep its structural parameters even for distan-ces between them as short as approximately 3 D, as shown in
the graph on top. This follows from the fact that the profilescan be satisfactorily described by the analytical expressions
m
z¼cosðh1þh2Þ¼cosh1cosh2/C0sinh1sinh2;(4)
mx¼sinðh1þh2Þ¼cosh1sinh2þsinh1cosh2;(5)
where angles h1andh2are defined as in Eq. (3), by consider-
ing the Dtheoretically calculated, with q1andq2accounting
for the DWs positions. The analytical profile of the ensembleof two DWs, represented with continuous lines, is then
defined from a simple sum of the orientation angles charac-
terizing the individual DWs. Additionally, this analyticalprofile allows to straightforwardly calculate the distancebetween DWs as d¼q
2–q1.
IV. NUMERICAL RESULTS
A. Interaction between chiral and achiral walls
As stated in Sec. III, interaction between DWs can be
characterized from the application of an uniform out-of-
plane magnetic field. Figure 2shows the results obtainedfrom lMag simulations. The graph at the up left corner
presents the equilibrium distance dbetween DWs as a func-
tion of the applied field Bzfor P BWs, AP BWs, and AP
NWs. In the range of distances where any pair of these types
of DWs may exist, the interaction seems to be independent
of the wall type. An important difference between BWs and
NWs must be noted: due to the DMI,5NWs are much more
stable than BWs, since a much more intense field is required
in order to annihilate the pair of NWs. The other three graphs
in this figure show the particular results obtained for every
type of DWs. The graphs additionally present the numerical
fitting of these results. Since results are represented in
log-log scale, it can be inferred that the interaction can be
characterized by curves of the type d¼aB/C0n
z, rather ndeter-
mining the slope of the fitting plots. While only one fitting
plot with slope n/C250.5 seems to adequately represent the
interaction between BWs, then defining an interaction inver-
sely proportional to the square of the distance rather similar
to the one proposed elsewhere6in a certain range of distan-
ces, two different plots are required in order to appropriately
fit the results obtained for AP NWs. As aforementioned, the
interaction is similar to the one obtained for BWs in the
range of large distances. However, as the AP NWs become
closer, the slope seems to tend to a value close to n¼0.25,
which establishes an inverse dependence of the interaction
with the fourth power of the distance. This type of depend-
ence, for example, appears between magnetic dipoles. It
must be taken into account that each DW can be character-
ized by an equivalent magnetic dipolar momentum, i.e., two
co-aligned, oppositely oriented, and then repelling dipoles in
this case. As the DWs become closer, these dipolar-like
interactions mask other possible interactions, but rapidly
vanish as the DWs get farther away each other.
B. Dynamics of the repulsion between AP N /C19eel walls
The aforementioned 1DM predicts that, for bi-layers
with strong DMI as these considered in this study, the iner-
tial term associated with the DW acceleration plays a negli-
gible role in the DW dynamics. According to this statement,
a differential equation of the type _q¼s
qrcan be proposed in
order to define this dynamics, qrepresenting the absolute
position of any of the two interacting DWs in a pair with
respect to the intermediate position between both DWs, and
sbeing a certain coefficient of proportionality. This equation
can be easily worked out, in particular, for r¼2 (long
distance interaction)
qðtÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q3
0þst3q
; (6)
and for r¼4 (short distance interaction)
qðtÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q5
0þst5q
; (7)
q0being in both cases the starting DW position. The validity
of this simple model can be confirmed by a couple of exam-
ples. In this way, two certain external magnetic fields are
applied in order to define in each simulation the initial posi-
tion of a pair of NWs. The magnetic fields are then removedFIG. 2. Characterization of interactions between P and AP BWs, and AP
NWs. The graph on the up-left corner shows distance between DWs as a
function of applied uniform magnetic field Bz, obtained from lMag simula-
tions. The other three graphs present individually the same lMag results
along with their respective numerical fittings, then revealing the dependence
on distance of the interaction between DWs.17D509-3 /C19O Alejos and E. Mart /C19ınez J. Appl. Phys. 117, 17D509 (2015)to promote the evolution in time of each pair under the only
influence of the interaction of the DWs in the pair. This is
shown in Figure 3. In the first example, red circles represent
the position qðtÞof one of the DWs forming part of a pair as
a function of time obtained from lMag simulations, when
the initial distance between DWs is of 80 nm ( q0¼40 nm).
Besides, the continuous red plot represents the fitting of thisevolution by means of Eq. (6). For such large distances, the
inverse quadratic interaction dominates the whole process,
and the mentioned equation appropriately accounts for thisphenomenon.
Finally, blue triangles are obtained from lMag simula-
tions when the initial distance between DWs is as short as20 nm ( /C253D), and then q
0¼10 nm. In the first instants, the
inverse bi-quadratic interaction dominates, and then Eq. (7)
has been used for the numerical fitting (see the inset). Asmall discrepancy can be seen between simulation and the
analytical description, but this can be associated with the
fact that the numerical fitting has been forced to pass throughthe first and the last point in the inset graph. It must be taken
into account that at the instants immediately previous to
t¼2 ns, the two determined interaction mechanisms have
similar weights, and then neither Eq. (6)nor Eq. (7)repre-
sent the exact solution of the dynamics. In any case, from
this instant on, the dynamics can be exactly described as inthe previous example by means of Eq. (6), both examples
with almost identical scoefficients, as it can be expected.V. CONCLUSIONS
The work here presented studies by means of lMag sim-
ulations the coexistence of two close nucleated DWs, either
in the case of BWs, both in P and AP configurations, or in thecase of homochiral NWs. From a general point of view, the
interaction between two DWs would consist of magnetostatic
and exchange interactions, the former including the demag-netizing interaction between the lateral domains on the cen-
tral one, which depends on the distance dbetween DWs as
1
d2,
and explains the interaction between two achiral BWs for anypossible configuration. Additionally, chiral AP NWs alsodepict a short distance regime where the interaction goes as
1
d4, which can be attributed to the dipolar force that one wall
exerts on each other.13Finally, the present analysis leads to
the conclusion that the exchange interaction might be being
masked by the others of magnetostatic origin in the particular
case of chiral AP NWs, as it has been revealed by the numeri-cal simulations of the interaction in time of domain walls,which have been appropriately explained by means of the
two interaction regimes here described.
ACKNOWLEDGMENTS
This work has been supported by Project No.
MAT2011-28532-C03-01 from Spanish Government and
Project No. SA282U14 from Junta de Castilla y Le /C19on. We
also acknowledge support by WALL project, FP7-PEOPLE-2013-ITN, Grant Agreement No. 608031.
1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008).
2K.-J. Kim, J.-C. Lee, S.-J. Yun, G.-H. Gim, K.-S. Lee, S.-B. Choe, and K.-
H. Shin, Appl. Phys. Express 3, 083001 (2010).
3A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. 8, 152 (2013).
4J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nat.
Nanotechnol. 8, 839 (2013).
5S. Emori, U. Bauer, S.-M. Ahn, E. Mart /C19ınez, and G. S. D. Beach, Nat.
Mater. 12, 611 (2013).
6K.-J. Kim, K.-W. Moon, K.-S. Lee, and S.-B. Choe, Nanotechnology 22,
025702 (2011).
7T. Moriya, Phys. Rev. Lett. 4, 228 (1960).
8M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G.
Bihlmayer, A. Kubetzka, O. Pietzsch, S. Blugel, and R. Wiesendanger,
Nature 447, 190 (2007).
9M. Heide, G. Bihlmayer, and S. Blugel, Phys. Rev. B 78, 140403 (2008).
10A. Thiaville, S. Rohart, E. Jue, V. Cros, and A. Fert, Europhys. Lett. 100,
57002 (2012).
11Equation (3)is, in fact, the analytical Bloch profile, which can be used to
accurately describe the magnetization of chiral N /C19eel walls after neglecting
the small deviations at the tails.10
12E. Mart /C19ınez, J. Phys.: Condens. Matter. 24, 024206 (2012).
13E. Mart /C19ınez and O. Alejos, J. Appl. Phys. 116, 023909 (2014).
14L. L /C19opez-D /C19ıaz, D. Aurelio, L. Torres, E. Mart /C19ınez, M. A. Hern /C19andez-
L/C19opez, J. Gomez, O. Alejos, M. Carpentieri, G. Finocchio, and G.
Consolo, J. Phys. Appl. Phys. 45, 323001 (2012).FIG. 3. Two examples of the evolution in time of one DW forming part of a
pair, under the only influence of its mutual repulsion. Dotted plots represent
lMag results, while continuous plots represent the respective numerical fit-
tings according to Eqs. (6)and(7). The inset zooms in on the first instants of
the evolution for the pair of closest DWs.17D509-4 /C19O Alejos and E. Mart /C19ınez J. Appl. Phys. 117, 17D509 (2015)Journal
of
Applied
Physics
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|
1.2947322.pdf | AIP Conference Proceedings 18, 222 (1974); https://doi.org/10.1063/1.2947322 18, 222
© 1974 American Institute of Physics.A Model For Dynamic
Conversion in Bubble Domains
Cite as: AIP Conference Proceedings 18, 222 (1974); https://
doi.org/10.1063/1.2947322
Published Online: 23 July 2008
F. B. Hagedorn
222
A MODEL FOR DYNAMIC CONVERSION IN BUBBLE DOMAINS
F. B. Hagedorn
Bell Laboratories, Murray Hill, N.J. 07974
ABSTRACT
Erratic propagation behavior of magnetic bubble domains,
as reported previously by Vella-Coleiro et al., has been
attributed to dynamic conversion of a normal bubble domain
wall into a relatively immobile but temporary state. A
model for dynamic conversion is presented. This model is
based on the nucleation and propagation of Bloch lines
within the moving bubble domain wall. Material imperfec-
tions are postulated to play an important role in nucle-
ating these Bloch lines, thereby accounting for the erratic
behavior of the propagating bubbles.
INTRODUCTION
Dynamic conversion of a normal bubble domain wall into a tempo-
rary state that is relatively immobile has been recently reported.l, 2
This phenomenon was originally identified during bubble transport
measurements, where a single bubble domain was repetitively trans-
lated back and forth over a distance of several ~mby the application
of a carefully controlled magnetic field gradient.3 Erratic bubble
propagation behavior was observed in this repetitive experiment, even
though the same bubblewas being subjected to a series of identical
pulse sequences. Skew propagation (as seen in hard bubble dynamics 4)
was observedl, 2 to occur during some of the pulses, and ~ue distance
over which the bubble was translated during a fixed pulse length and
amplitude was observed to vary by as much as a factor of i0 from one
pulse sequence to the next.
QUALITATIVE MODEL FOR DYNAMIC CONVERSION
This model is based on nucleation and propagation of horizontal
Bloch lines (HBL) within the bubble domain wall. For planar walls,
Slonczewski5 has shown that HBL can originate near one surface of the
magnetic film and move through the domain wall to the other surface~
While moving, such HBL can dissipate a large amount of energy.
One essential point of the present model is the hypothesis that
the nucleation of such HBL is the result of interactions between the
moving bubble wall and material inhomogeneities or imperfections, the
size-scale of which is comparable with the domain wall thickness.
Inspection of the equations of motion has not revealed an intrinsic
instability for HBL nucleation during steady-state motion of the wall,
so this imperfection-interaction mechanism has been postulated to
account for a portion of the erratic behavior which is observed.
A ~cond essential feature of the present model arises from the
geometric differences between HBL motion in planar as o~posed to
cylindrical domain walls. Thiele's g-force formalism 6,~ provides a
223
(a) - (b) (e) (d)
Fig. 1-Schematic representation of the ma~etization ~ the ce~er of
a bubble domain wall during Bloch line growth.
convenient method of e~loring this ~fference. Figure ia~ follo~ng
Ref. 7~ shows a moving bubble domain in which a HBL is propag~ing
downward. ~e g-force~ which is ~iving the ~L, v~ishes where the
wa~ tangents are para~el to the velocity direction~ Consequent,
the HBL is curved in the bubble domain wall. In Fig~ ~, the rear
half of the curved domain wa~ is represe~ed schem~ical~ in a
pl~ar sketch~ again showing the cu~ed HBL. Figure ic shows the
result of further ~L motion~ a~er which a 2~ ~L lies along the
bottom s~face of the wa~ in combination with a pair of vertical
Blo~ lines (VBL) throu~ the thic~ess of the fi~. Figure id shows
the spin configuration which results a~er a second HBL nucle~es
from Fig. ic ~d then pr~agates to the top of the film. In Fig. id~
there are two pairs of VBL~ in addition to the 2~ HBL d~amical~
trapped ~ the top ~d bottom film surfaces by the dema~etizing field
gradient.
Similar nucle~ion ~d propagation of Bloch lines c~ occur inde-
pendently in the front half og the bubble domain wall shown in Fig.
la. Analysis of the g-forces b,7 shows that all of the VBL are forced
to the sides of the moving bubble w~l. Each generated pair of VBL~
as in Fig. lc, contains one ri~t-h~ded ~d one le~-handed line.
All ri~t-h~ded lines end up on one side~ while ~i le~-handed lines
~e forced to the other side of the moving bubble wall. D~ic sta-
bilization of a static~ ~st~le spin config~ation thus results.
The motion of this st~ilized spin confi~ration is slowed down both
by subsequently created HBL moving across the thic~ess of the film,
as indicated schematically in Fig. i, and by the existence of pre-
vious~ generated VBL. The l~ter cre~e an added dissipation, which
c~ be easi~ seen from Thiele's dissipation dyadic.6,7 Conse~ent~,
the reduced velocity observed in dyn~ic conversion is acco~ted for
by the combin~ion of these two factors.
The third essential feat~e of this model is to postul~e yet
mother resu~ from the interaction between the moving bubble domain
wall ~d imperfections near the magnetic film surfaces. Shown in
Fig. 2a is a somewh~ more detailed represent~ion of Fig. id. Each
line in Fig. 2 a represents a conto~ of const~t direction for the
ma~etization. In Fig. 2b it is assumed that the lower 2~ HBL shown
in Fig. la has interacted with ~ imperfection ~d reached the film
sur~ce~ that the VBL on the le~ are now attached to the s~face im
exact~ the way th~ occurs in hard bubbles, and that the VBL on the
224
(a) (b) (c) (d)
Fig. 2-Contours of the magnetization in a section of the bubble
domain wall before and after severing the Bloch lines.
right terminate in a vortex which is somewhere away from the film
surface. Surface tension will cause the vortex to be pulled toward
the upper surface, as shown in Fig. 2c, after which there will be a
net transverse force due to the unequal numbers of right-handed and
left-handed VBL. This force has been calc~l~ted in detail to account
for the dynamic properties of hard bubbles6,8,9 and can explain the
occasional skew propagation effects which are observedl, 2 in dynamic
conversion. According to the model being presented herein, skew
propagation is erratic because of the statistical nature of the
interactions which lead to the severing of the Bloch line structure
shown in Fig. 2. It is conjectured for this part of the model
that the configuration of Fig. 2c may be dynamically stabilized
but that the vortex is pulled around the bubble wall by surface
tension as soon as motion terminates, thereby reverting to the
initial normal configuration as shown in Fig. 2d.
QUANTITATIVE ASPECTS OF DYNAMIC CONVERSION
While a more complete quantitative description is given else-
where, I0 it will be instructive to summarize some of these results
here. It is shown in Ref. i0, for the case of a planar wall, that
the wall velocity as a function of the driving magnetic field (Hd)
would be expected to be as shown in Fig. 3. Below a critical vel-
ocity (Vp) defined in Ref. 5 to be i
= 24171j , (1)
Fig. 3 is linear. In Eq. (i),171 is the gyro-
magnetic ratio of the electron, A is the mag-
v netic exchange constant, h is the film thick-
ness, and K u is the uniaxial anisotropy
constant of the material. Equation (i) defines v p the minimum velocity for propagating a HBL away
from the film surface, once it has been
V nucleated, and this velocity will be attained
in a given material when H d = ~, with
i
i = 24 A /h (2)
H H d P
P where ~ is the Gilbert damping parameter for
Fig. 3-Sketch of the the magnetic film.
wall velocity as a For Hd>HD, the wall motion becomes non-
function of the uniform in time~ depending on the location of driving field.
225
the moving HBL. The average wall velocity (V) has been calculated lO
using the assumption that there is always one HBL moving through the
film thickness, and V is shown in Fig. 3 as the lower bound of the
cross-hatched region. In Ref. lO, it is shown from the equations of
motion that V = 0.55 Vp. ll If there are no HBL moving through the
wall, then the dashed line (i.e., an extrapolation of the initial
linear region) pertains. If a HBL is moving only part of the time
during the motion of the wall, a time-average must be done; the result
will fall somewhere in the cross-hatched region. Figure 3, therefore,
pertains to a moving planar wall in which HBLmay be nucleated spo-
radically. The observed 2 variations in the measured wall velocity
can be accounted for in terms of Fig. 3, and this fact provides the
motivation to consider the nucleation-from-imperfections hypothesis.
In addition, Fig. 3 suggests the spread in measured velocity should
increase with increasing drive field; the experimental results 2 are
also consistent with this feature of the model.
For bubble domains, the curved wall complicates a corresponding
analysis. However, Vp would appear to be a lower limit for the
velocity required to propagate a HBL, once it has been nucleated. It
is possible that a somewhat larger velocity could be required in
order to achieve the "stretching" indicated schematically in Fig. 1.
The effects of VBL have not been included in Fig. 3, though, and can
be shown by using Thiele's7 dissipation dyadic results and the ob-
served 12 maximum Bloch line densities to be equivalent to an added
dissipation of not more than a few times that due to the mo~ion of a
simple planar wall. The net result is that the average velocity of
a bubble domain can appear below V in Fig. 3, as well as anywhere in
the cross-hatched region.
DISCUSSION
Quantitative comparisons probably require one to take into
account the excitation of the domain wall modes, as discussed by
Thiele.13 These modes represent additional damping and also are
excited by interactions between the moving wall and imperfections in
the magnetic film. Dynamic conversion and wall-mode excitation can
and probably do coexist in a given bubble domain when it is moving,
and a clear experimental separation of the two has not been demon-
strated. However, erratic skew propagation would appear to be a
unique sign of dynamic conversion, and the observed threshold value
of about I000 cm/sec as reported in Fig. i is in reasonable agreement
with the value of ii00 cm/sec as calculated from Eq. (i) and the
material parameters which pertain to Ref. I.
Another aspect of the model which can be experimentally checked
is related to the in-plane anisotropy dependence of Bloch line
nucleation. Slonczewski's5 model for the twisted wall structure
makes it clear that a large anisotropy in the plane of the film,
either from an externally applied ma~Detic field or from a material
anisotropy~ should make HBL nucleation much more difficult. Complete
experimental verification of this point is not yet available, but the
fact that dynamic conversion was never observed in orthoferrite
supports this aspect of the model.
226
A final comment is that the model proposed herein leads one to
suggest a new kind of film structure for bubble domains. With two
similar magnetic films separated by a very thin nonmagnetic film~ it
is expected that the exchange coupling necessary to propagate the
Bloch line will be interrupted but that a pair of bubbles in the two
magnetic films will be strongly magnetostatically coupled and will
appear approximately as one bubble. The wall structure will be
dynamically stabilized as shown in Fig. 4, however. Only a single
vertical Bloch line will appear on each side of the bubble~resulting
in relatively little added damping. Experiments using such a multi-
layered structure are presently in progress.
Fig.4-Dynamically stabilized configuration in a 3-
layer structure which suppresses dynamic conversion.
ACKNOWLEDGMENTS
The author has benefited from discussions with
A. A. Thiele, G. P. Vella-Coleiro, J. E. Geusic and
A. H. Bobeck.
REFERENCES
i. G. P. Vella-Coleiro, F. B. Hagedorn, Y. S. Chen and S. L. Blank 3
Appl. Phys. Lett. 22, 324 (1973).
2. G. P. Vella-Coleiro, paper presented at 1973 M3 Conference.
3. G. P. Vella-Coleiro and W. J. Tabor, Appl. Phys. Lett. 21, 7
(1972).
4. W. J. Tabor~ A. H. Bobeck~ G. P. Vella-Coleiro and A. Rosencwaig~
Bell Syst. Teeh. J. 51, 1427 (1972).
5. J. C. Slonczewski, J. Appl. Phys. 44, 1759 (1973).
6. A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).
7. A. A. Thiele, J. Appl. Phys.~ scheduled for December 1973.
8. J. C. Slonczewski, Phys. Rev. Lett. 29, 1679 (1972).
9. A. A. Thiele, F. B. Hagedorn and G. P. Vella-Coleiro~ Phys. Rev.
B 8, 241 (1973).
i0. F. B. Hagedorn~ J. Appl. Phys.~ to be published.
Ii. The velocity saturation effect shown in Fig. 3 was previously
obtained in Ref. 5. In Ref. 5, however~ the saturation velocity
was calculated to be 0.3 yD. Although the origin of this differ-
ence is discussed in Ref. I0~ it is not important for the present
discussion, which is predominantly qualitative.
12. D. H. Smith and A. A. Thiele, paper presented at 1973 M3 Conf-
erence.
13. A. A. Thiele, Phys. Rev. B ~, 391 (1973).
|
1.89114.pdf | Walkertype velocity oscillations of magnetic domain walls
G. P. VellaColeiro
Citation: Applied Physics Letters 29, 445 (1976); doi: 10.1063/1.89114
View online: http://dx.doi.org/10.1063/1.89114
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/29/7?ver=pdfcov
Published by the AIP Publishing
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Appl. Phys. Lett. 99, 122504 (2011); 10.1063/1.3641884
Kinetic depinning of a magnetic domain wall above the Walker field
Appl. Phys. Lett. 98, 042502 (2011); 10.1063/1.3543844
Manipulating magnetic moment in a magnetic domain wall under transverse magnetic fields near Walker
threshold
J. Appl. Phys. 108, 063904 (2010); 10.1063/1.3488011
Magnetic domain wall collision around the Walker breakdown in ferromagnetic nanowires
J. Appl. Phys. 106, 103926 (2009); 10.1063/1.3264642
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131.230.68.4 On: Fri, 05 Dec 2014 11:36:30Walker-type velocity oscillations of magnetic domain walls
G. P. Vella-Coleiro
Bell Laboratories, Murray Hill, New Jersey 07974
(Received 4 June 1976)
We report stroboscopic observations of the radial motion of a magnetic bubble domain wall in an epitaxial
LuGdAI iron garnet film. At high drive fields, initial velocities up to 9500 em/sec were measured, and the
domain wall was observed to move backwards during the field pulse, in agreement with calculations based
on the Walker model.
PACS numbers: 75.60.Fk
A number of authorsl-4 have recently discussed theo
retically the motion of magnetic domain walls in uni
axial media in fields exceeding the Walker critical
field3 "w = 21TM a (M is the saturation moment and a is
the Gilbert damping parameter). It was shown that the
wall velocity should oscillate rapidly between large
positive and negative values, the wall moving alternate
ly forwards and backwards but with a net positive dis
placement provided Q"* O. To date, no observations of
these oscillations have been reported, although some
experiments in epitaxial garnet films have been de
scribed5-' where very large wall velocities were ob
served, and some of these results6 were interpreted in
terms of a Walker breakdown of the wall motion. In
this paper we report direct stroboscopic observations
of wall motion which confirm the occurrence of very
high velocities and also show oscillations of the Walker
type.
The experiments were performed in an epitaxial
LuGdaAlo.6Fe4.40t2 film on a (111) GGG substrate, with
the following parameter values: thickness h = 9. 4 Mm,
41TM = 189 G, wall width parameter t:. = (A/K)l /2 = 7.4
X 10-6 cm, gyromagnetic ratio y = 1. 4 X 10' sec-1 Oe-1,
dynamic coercivity He = O. 4 Oe, and ferromagnetic reso
nance damping parameter a = O. 023. An isolated mag
netic bubble domain was viewed in a microscope with
the pulsed output of a mode-locked argon ion laser
(514.5 nm) as illumination. The optical pulses were
approximately 1 nsec in duration and they could be
triggered to occur before, during, or after a magnetic
field pulse was applied to the bubble. The laser and
magnetic field were pulsed at a repetition rate of
approximately 3 kHz and the instantaneous bubble diam
eter was measured at an optical magnification of 500
with a filar eyepiece at various times during and after
the application of the field pulse. The field pulse had
a rise and fall time of approximately 2 nsec and its
polarity was such as to cause a reduction in bubble
diameter. The stationary bubble diameter was 8.1 Mm.
Three sets of data are shown in Fig. 1, where the
measured change in bubble radius, t:.R, is plotted ver
sus the time delay between the start of the field pulse
and the peak of the optical pulse. The shape and mag
nitude of the field pulse is shown in the lower part of
each section of Fig. 1. In each case, the wall initially
moves with a very high velocity, the highest instanta
neous velocity reached being -9500 cm/sec, as indi
cated in Fig. 1. At high drive fields [Figs. l(b) and
1(c) 1 backwards motion of the wall occurs during the
445 Applied Physics Letters, Vol. 29, No.7, 1 October 1976 field pulse. After the end of the pulse, the wall re
turns to its original position at a relatively low velo
city (-1600 cm/sec). This part of the motion starts as
soon as the pulse terminates at low drive field [Fig.
0 0
-02 -2 -;
E
.:!--04
~ ~
Q.
-4 L> x
-0.6
-6
-08
~ 10
x
(a) 0
0 0
-0.2 -2
-04
E
~ -0.6
~ -4
~
-6 ~ x
-8
-1.0
-10
0 0
-0.1 -1 E ~ ~
0: -2 ~
<I x
-0.3 .. -3
-0.4 . • . , ......... . . . . . -4
~
~
x (c)O
zo 40 60 80
TIns'
FIG. 1. Change in bubble radius, t:.R, vs time after the appli
cation of a field pulse in the same direction as the bias field.
In (a) the pulse field amplitude is -13 Oe, in (b) it is -24 Oe,
and in (c) it is -44 Oe. The shape of the field pulse is shown
in the lower part of each section. The right-hand scale shows
the variation of the bubble potential field Hb) with t:.R (see text).
Copyright © 1976 American Institute of Physics 445
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131.230.68.4 On: Fri, 05 Dec 2014 11:36:301(a)], but at high drive fields the wall remains prac
tically stationary for several tens of nanoseconds.
We compare the data shown in Fig. 1 with the pre
dictions of the Walker model. 3 This model is not strict
ly applicable to domain walls in thin films since it
treats a purely Bloch wall in an unbounded medium,
whereas in a thin film the spin structure of the wall
is twisted8,9 due to the magnetic field associated with
the surface charges. Nonetheless, our results, in as
far as they show very high initial velocities and velo
city oscillations, strongly suggest that the type of spin
precession involved is similar to that treated in the
Walker model. It is instructive, therefore, to compare
the data with the predictions of the Walker model, while
bearing in mind that perfect agreement is not to be ex
pected due to the different spin structures involved. We
also ignore the fact that the bubble wall is curved
whereas the theory treats planar walls, since this dif
ference is not expected to have any substantial effect
on the results. The time dependence of the wall dis
placement was obtained from a numerical integration
of Eqs. (12a) and (13b) of Ref. 3. The magnetic field
pulse was approximated by a trapezoidal shape with a
linearly rising portion, a portion with a constant mag
nitude, and a linearly falling portion. As the bubble
diameter changes, the bubble stability provides a re
storing force tending to return the diameter to its
equilibrium value. This effect, expressed as a bubble
potential field Hbp, was included in the calculation, and
its value corresponding to the value of AR is shown on
the right-hand side of Fig. 1. It was obtained from
measurements of the equilibrium bubble diameter at
various bias fields. With each value of AR one can then
associate a bias field value Hd which would be required
to stabilize the bubble at that diameter, and HbP is just
the difference between the bias field used and Hd•
The results of the calculations are represented by the
continuous lines in Fig. 1. The material parameter
values used were those given above, except for the val
ue of the damping parameter o!. If the FMR value of O!
were used, the Walker critical field, 3 above which velo
city oscillations are expected to occur, would have the
value Hw= 21TM O! = 2.2 De, and for the pulse field of
Fig. 1(a) the Walker model would predict velocity os
cillations occurring after the end of the field pulse,
since the bubble potential field is greater than 2.2 De
(the pulse duration of Fig. 1(a) is too short for oscilla
tions to occur during the pulse at that amplitude].
Also, the calculated wall displacement would be 1. 22
/.lm. The data in Fig. 1(a) show no sign of oscillation
and a wall displacement of 0.55 /.lm. Clearly, then, the
F MR value of O! is not appropriate for wall motion at
high velocity. We have, therefore, derived an appro
priate value of O! by fitting the data in Fig. l(a) for
t < 10 nsec to the calculation, the value obtained being
O! = 0.11. This value is in good agreement with the
value of 0.1 derived from high drive bubble translation
al velocity measurements in Ref. 6. The continuous
lines in Figs. l(b) and l(c) were calculated with O! = 0.11
and in each case good agreement with the data is ob
served during the first few nanoseconds of the motion.
Furthermore, the data in Figs. l(b) and 1(c) show an
oscillation, i. e., backward motion of the wall during
446 Appl. Phys. Lett., Vol. 29, No.7, 1 October 1976 the field pulse, in qualitative agreement with the cal
culation. Large discrepancies between the data and the
calculation develop in all cases for t? 10 nsec. This
effect might be due to a dynamic conversion10,l1 of the
spin structure of the bubble wall. Bloch lines are pre
sumed to be nucleated, resulting in a large reduction
of the wall velocity. The stationary region in Figs. 1(b)
and 1(c) could be the result of a nonuniform spin pre
cession which causes different parts of the wall to
move out of phase with one another, thus resulting in
little or no net motion. The data in Fig. 1 suggest that
the disturbance occurs 10-15 nsec after the wall motion
starts.
The close agreement between the data and the calcu
lations in Fig. 1 for t;:;: 10 nsec is quite remarkable in
view of the fact that the theory of wall motion in thin
films, as developed by Slonczewski8 and Hubert, 12 re
quires the presence of a horizontal Bloch line when the
drive field exceeds the critical value H~ "'V~O!/YA
"'1.4 De (0!=0.11), where V~ is the critical velocity
defined by Slonczewski. 8 Also, at drive fields in excess
of Hp the wall velocity is supposed to have the satura
tion value Vs = aV~, where a is a factor whose numerical
value lies in the range 0.3_0.5.8,11 For our film
Vs"'400-700 cm/sec. Our results strongly suggest that,
at least during the first 10 nsec or so of the motion,
no Bloch lines are present (this possibility is discussed
in Ref. 11) and the spins in the wall precess in a manner
very similar to that considered in the Walker model.
Thus the initial velocity can be much greater than Vs
and a wall oscillation of the Walker type can develop.
Since the measurements reported here involve the
direct stroboscopic observation of domain walls in
motion, they are not subject to some experimental un
certainties such as overshoot, distortion of the domain
shape, etc. However, they provide only the average
response of the wall, and no definite statement can be
made regarding velocity fluctuations from pulse to
pulse other than to note that little blurring of the dynam
ic wall position was observed, indicating that any fluc
tuations which might be present are of a type which have
a well-defined average value. It should also be noted
that due to the very high Faraday contrast obtained with
the argon laser illumination, the reproducibility of the
measure,ments was better than 0.1 /.lm. The data show
unambiguously that very high velocities are present
during the initial phase of dynamic bubble collapse,
in agreement with previous deductions from indirect
measurements. 6 Since a wall displacement approaching
1 /.lm can occur during this initial phase, we feel that
the transient part of the motion should be taken into
consideration in any wall motion experiment where
displacements of only a few micrometers or less are
measured, as is very often the case.
It is a pleasure to thank F. B. Hagedorn for helpful
discussions.
IJ.C. Slonczewski, Int. J. Magn. 2, 85 (1972).
2J. A. Cape, W. F. Hall, and G. W. Lehman, J. Appl. Phys.
45,3572 (1974).
3N.L. Schryer and L.R. Walker, J. Appl. Phys. 45, 5406
(1974) •
G.P. Vella-Coleiro 446
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131.230.68.4 On: Fri, 05 Dec 2014 11:36:30(H.C. Bourne, Jr. and D.S. Bartran, IEEE Trans. Magn.
MAG-I 0 , 1081 (1974).
sF.H. de Leeuw, J. Appl. Phys. 45, 3106 (1974).
sG. p. Vella-Colelro, AlP Conf. Proc. 24, 595 (I975).
7G. J. Zimmer, L. Gal, and F. B. Humphrey, AlP Conf.
Proc. 29, 85 (1975).
8J.C. Slonczewski, J. Appl. Phys. 44, 1759 (1973). 9E. Schlomann, J. Appl. Phys. 44, 1837 (1973).
10G.p. Vella-Coleiro, F.B. Hagedorn, Y.S. Chen, and
S. L. Blank, Appl. Phys. Lett. 22, 324 (1973): G. p. Vella
Coleiro, AlP Conf. Proc. 18, 217 (1974).
UF. B. Hagedorn, J. Appl. Phys. 45, 3129 (1974).
12A. Hubert, J. Appl. Phys. 46, 2276 (1975).
Observation of the optoacoustic effect in the microwave
region
Gerald Diebold* and David L. McFadden
Department of Chemistry, Boston College. Chestnut Hill, Massachusetts 02167
(Received 6 July 1976)
The microwave analog of the optoacoustic effect has been observed. Collisional relaxation of absorbed
microwave energy between Zeeman magnetic sublevels of gaseous molecular oxygen results in the
production of an acoustical signal which is detected by a sensitive microphone.
PACS numbers: 32.20.Es, 33.90.+h, 43.35.+d
When molecules in a gas absorb electromagnetic
energy, part of this energy is ultimately transformed
by colliSions into kinetic energy. As a result the tem
perature of the gas increases. Periodic temperature
variations can be produced when the incident light is
amplitude modulated or alternatively when the energy
levels of the gas are modulated by external electric or
magnetic fields. If the gas is contained in a closed
vessel, temperature OSCillations are equivalent to
periodic pressure disturbances which can be detected
as sound if the modulation frequency is in the audio
range. This effect, known as the optoacoustic effect,
has been observed over a broad range of excitation
wavelengths extending from the ultraviolet where atomic
or molecular electronic transitions are excited to the
infrared where excitation corresponds to transitions
between vibration-rotation states. Applications of the
optoacoustic effect are numerous, one of the first being
the photophone of Alexander Graham Bell. 1 More recent
applications include the detection of gaseous impurities
which are present at concentrations as low as 0.1
ppb,2-5 the measurement of vibrational relaxation
times, 6,7 optical absorption spectroscopy of solids, 8-10
and the study of mechanisms in molecular photochemis
try. 10-14 In this letter we report the detection and re
acoustic Signal resulting from the absorption and re
laxation of microwave energy between Zeeman magnetic
sublevels of molecular oxygen using a conventional
electron paramagnetic resonance spectrometer.
The magnitude of the optoacoustic effect varies con
siderably with the frequency of the incident photons.
The intensity of the acoustical signal and thus of the
microphone response should be proportional to the
optical power absorbed by the sample. Although the
factors which determine the rate of absorption of energy
from the radiation field strongly favor optical transi-
447 Applied Physics Letters, Vol. 29, No.7. 1 October 1976 tions over microwave tranSitions, a sufficiently strong
microwave absorption is made possible by the high
powers available from conventional klystron Sources
which are capable of producing several hundred milli
watts of continuous power in a narrow bandwidth (typi
cally 1 part in 105). Power absorption on a single EPR
line can be estimated to be as large as 1 mW for our
experimental conditions, which should be more than
adequate for acoustic detection. 15b,16 The negligible rate
of spontaneous emission at microwave frequencies
ensures that Virtually all of the absorbed power is con
verted to acoustic energy. It has also been shown that
RECORDER
FIG. 1. Schematic diagram of experimental apparatus.
Acoustical cavity shown in the center of the figure is 2.5 cm
in diameter and 33 cm long. Microphone is General Radio
model 1961 electret. Preamp is Analog Devices model 45J
operational amplifier wired for a gain of 9 in the noninverting
configuration. Lock-in amplifier is PAR HR-8. A Varian E-9
spectrometer is employed, and the E-235 cylindrical cavity
operates in the TEOln mode. N denotes the location of one of
the pressure nodes in the 1000-Hz acoustical standing wave.
The dashed circle represents the electromagnet pole face.
Copyright © 1976 American Institute of Physics 447
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131.230.68.4 On: Fri, 05 Dec 2014 11:36:30 |
1.5054123.pdf | Perspective: (Beyond) spin transport in insulators
Yaroslav Tserkovnyak
Citation: Journal of Applied Physics 124, 190901 (2018); doi: 10.1063/1.5054123
View online: https://doi.org/10.1063/1.5054123
View Table of Contents: http://aip.scitation.org/toc/jap/124/19
Published by the American Institute of PhysicsPerspective: (Beyond) spin transport in insulators
Yaroslav Tserkovnyak
Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
(Received 29 August 2018; accepted 3 November 2018; published online 21 November 2018)
Insulating materials with dynamical spin degrees of freedom have recently emerged as viable
conduits for spin flows. Transport phenomena harbored therein are, however, turning out to be much
richer than initially envisioned. In particular, the topological properties of the collective order-
parameter textures can give rise to conservation laws that are not based on any speci fic symmetries.
The emergent continuity relations are thus robust against structural imperfections and anisotropies,
which would be detrimental to the conventional spin currents (that rely on approximate spin-rotational symmetries). The underlying fluxes thus supersede the notion of spin flow in insulators,
setting the stage for nonequilibrium phenomena termed topological hydrodynamics. Here, we
outline our current understanding of the essential ingredients, based on the energetics of the electri-
cally-controlled injection of topological flows through interfaces, along with a reciprocal signal
generation by the out flow of the conserved quantity. We will focus on two examples for the latter:
winding dynamics in one-dimensional systems, which supplants spin super fluidity of axially-
symmetric easy-plane magnets, and skyrmion dynamics in two-dimensional Heisenberg-type
magnets. These examples will illustrate the essential common aspects of topological flows and hint
on generic strategies for their generation and detection in spintronic systems. Generalizations toother dimensions and types of order-parameter spaces will also be brie fly discussed. Published by
AIP Publishing. https://doi.org/10.1063/1.5054123
I. INTRODUCTION
Understanding electricity, which concerns phenomena
deriving from the motion of electric charge, has been a cor-
nerstone of solid-state physics. Studying and quantifying
such motion, e.g., through the measurements of electricalconductivity, provided fundamental probes of materials that
lead to some of the central discoveries of the 20th-century
physics, such as superconductivity and quantum Hall effect.Being primarily carried by electrons, electric charge flows
can be used to differentiate between some of the basic
electronic states of crystals, such as metals, insulators, andsemiconductors. Generally, whenever electronic charge corre-
lations bear some key signatures of the underlying phase or
state of a material, we can expect the electrical conductivity
to offer a valuable probe thereof. Conversely, a material
known to have some striking electrical response can be tai-lored for electronic applications.
A broad range of complex materials, however, have their
key dynamic properties rooted in different physics. In partic-ular, magnetic materials may exhibit essentially no electrical
dynamics, up to very high frequencies (determined by the
gap for charge excitations), while having their prevalent low-frequency fluctuations governed by the spin degrees of
freedom. This concerns, more generally, systems with strong
spin correlations and/or frustration, where the low-energyproperties are either dominated or, at least, strongly affected
by the correlated spin dynamics.
Spintronics has recently emerged as a field that exploits
these spin degrees of freedom to either study the underlying
materials and heterostructures or employ the associated func-
tionality in novel devices and computing architectures.
1–4One
feature that distinguishes spintronics from other spin-baseddisciplines, such as various spin-resonance and scattering
spectroscopies, is a focus on transport regimes, where thenet spin angular momentum in the system is conserved. In
this case, supported by the reasoning that is similar to that
underlying Kirchhoff ’s circuit laws for charge flows in elec-
trical circuits, one can construct spin- flow-based principles
for spin dynamics.
5Interfaces or junctions in a spin-active
heterostructure would then serve as nodes that transmit spin
flows.6The spin flow over a certain region (e.g., an interface
between two materials or a section of a single material), whichserves as a basic building block for the circuit perspective, canbe driven by an effective spin bias. Thermodynamically, thelatter can be understood as a drop in the spin (chemical)
potential, which is locally conjugate to the spin density. The
spin conservation would dictate a homogeneity of the spinpotential in equilibrium.
While a finite spin flow across a heterointerface may
have to be transmuted between physically disparate entities,
such as electron-hole pairs on one side and magnons on the
other,
7,8it can still be conserved. Such conservation, along a
specific axis, relies in general on the corresponding spin-
rotational symmetry, which must be satis fied in both materi-
als as well as at the interface itself. In practice, this is ofcourse always an approximation, which might explain why
the basic notion of spin transport
9was not widely accepted
for a long time. One important issue is that the spin signalscarried by decaying quasiparticles are exponentially sup-pressed beyond the associated spin-diffusion length.
2
In this perspective, I will start by recapping some recent
developments in our understanding of spin flows through
magnetic insulators. We will, for concreteness, su ppose that
the spin bias is produced by a nonequilibrium electron spinJOURNAL OF APPLIED PHYSICS 124, 190901 (2018)
0021-8979/2018/124(19)/190901/9/$30.00 124, 190901-1 Published by AIP Publishing.
accumulation, which can be controlled electrically.10–13It turns
out, however, that an ordinary spin flow is not the only trans-
port process that can be triggered by such spin biases. Thinkingmore broadly about the coherent (magnetic) order-parameter
dynamics, which can be controlled and detected electrically,
will bring us to the notion of the conserved topological flows.
An idealized concept of spin super fluidity
14–16is perhaps the
simplest example thereof, which will be relied heavily on for
pedagogical purposes. We will discuss how the interplay ofcurrent-induced work, topology, and coherent spin dynamics
can conspire to yield robust long-distance and low-dissipation
information flows through magnetic insulators.
II. BACKGROUND
A. Spin- flow nodes and circuitry
In a simple illustration of spin flows in solid-state hetero-
structures, consider a junction between a nonmagnetic metaland a magnetic insulator, as depicted in Fig. 1. This junction
can be viewed as a node in a larger circuit, which could be
ultimately driven by a combination of electrical and thermalmeans (through, e.g., the so-called spin Hall
9,10and spin
Seebeck17,19effects, respectively). In a nonequilibrium
steady state, we can have a situation, in which the itinerantelectrons in the metal obey the Fermi-Dirac statistics with the
spin-dependent distribution function
n
FD(ϵ)"=#
¼1
eβL(ϵ+μs=2)þ1, (1)
while the magnons follow the Bose-Einstein distribution
nBE(ϵ)¼1
eβR(ϵ/C0μm)/C01: (2)
β;1=kBTstands for the inverse temperature, on each side,
μsis the spin potential (also known in the literature as the
spin accumulation5,6,13) in the metal, while μmis the spin
potential (which corresponds simply to the bosonic chemicalpotential8) in the magnet. Orienting the spin quantization
axis here along a symmetry axis in spin space (which, in the
case of a collinear spin order, must be along the order param-eter), the spin flow is continuous across the interface. In
linear response, it should generally obey the following phe-
nomenology:
J
s¼G(μs/C0μm)þS(TL/C0TR), (3)
in close analogy with thermoelectricity.20Ghere is the interfa-
cial spin conductance and Sis the spin Seebeck coef ficient. In
thermodynamic equilibrium, μs¼μmand TL¼TR,s ot h a t
Js¼0. Microscopically, the values of GandSdepend on the
strength of the (Heisenberg) spin exchange at the interface,
between the itinerant electron spins on the left and localizedmagnetic moments on the right.
8,18,21–23These parameters, fur-
thermore, depend on the ambient temperature, typically increas-
ing with temperature, due to the bosonic statistics of magnons.
B. Energetics of the coherent spin transfer
Let us now look into the process of spin injection at an
interface between a normal metal and a dynamic magnet. At
sufficiently low temperatures, we can neglect thermal spin
excitations, like those underlying Eq. (3), and instead focus
on the coherent spin dynamics as well as the spin transport
driven by a (vectorial) spin bias μsin the normal metal.5Its
absolute value is jμsj¼μsand the direction is determined
by the spin-quantization axis for which the electron occupa-
tion follows Eq. (1).
As a starting point, consider a simple collinear ordering
in the magnet, whose dynamic state is described by a direc-
tional order parameter l(t), s.t. jl(t)j;1. Writing the (vecto-
rial) spin current Jsacross the interface in terms of μsand a
slowly-varying l(t) then gives5
Js¼g
2πl/C2μs/C2l/C0/C22h_l/C0/C1
: (4)
lhere can physically stand for the magnetic order in a ferro-
magnet or the Néel order in an antiferromagnet.24,25The
interfacial coef ficient gis known as the spin-mixing conduc-
tance.5,6,13The expression (4)is isotropic in spin space,
obeys Onsager reciprocity26,27(when viewed as relating the
spin flow into the normal metal with the order-parameter
dynamics in the magnet24), and vanishes when the frequency
of rotation matches the spin bias (which is easily understood
in the rotating frame of reference5). This expression, further-
more, breaks the (macroscopic) time invariance, as Js!Js,
l!/C0 l, and μs!/C0 μs, under time reversal. This underlines
its dissipative character, which we can exploit in order topump energy into the magnetic dynamics.
Spin transfer (4)across the interface signi fies a torque,
J
s!τ, when viewed from the point of view of the magnetic
dynamics, which translates into work
_W;τ/C1l/C2_l¼g
2πμs/C2l/C0/C22h_l/C0/C1
/C1_l (5)
on the magnetic order, per unit time. The second term,
//C0(_l)2, on the right-hand side contributes to the generic
Gilbert damping28of the magnetic dynamics, while the first
term, which is sometimes referred to as the antidamping
FIG. 1. A schematic of an elementary spin-transport node between a non-
magnetic metal (left) and a magnetic insulator (right). Electrons can flip their
spin at the interface, while transmitting (or absorbing) a magnon. The spincurrent J
sis driven by the thermodynamic bias of ( μs/C0μm), in spin sector,
and ( TL/C0TR), in heat sector. Such a bias across the interfacial node can be
established in response to thermoelectric controls of a larger circuit, in a self-
consistent steady state.17,18190901-2 Yaroslav Tserkovnyak J. Appl. Phys. 124, 190901 (2018)torque,29may effectively reverse the sign of the natural
damping, leading to a dynamic instability. We can under-
stand Eq. (5)from the Hamilton equations of motion for the
order-parameter ldynamics. To this end, we modify the rate
of change of the conjugate momentum πlas
_πl¼/C0@H
@lþτ/C2l, (6)
in the presence of an interfacial torque τ,w h e r e H(l,πl)i s
the Hamilton function. The reason for this is that πl¼ρs/C2l,
with the spin density ρsbeing the generator of rotations.30Its
dynamics are modi fied by the spin torque as _ρl!_ρlþτ.
The work production (5)by the torque is then finally
obtained as _H¼_πl/C1@H=@πlþ_l/C1@H=@l¼τ/C1l/C2_l, invok-
ing also the other Hamilton equation: _l¼@H=@πl.
More generally, for a noncollinear spin order that can be
parametrized by an SO(3) rotation matrix ^R(t), the appropri-
ate torques in the equation of motion can be derived from the
following Rayleigh dissipation function:31
R¼1
2h(μs/C0/C22hω)^g(μs/C0/C22hω), (7)
which corresponds to the (half of the) net dissipation in the
combined nonequilibrium system (i.e., the magnet plus theadjacent metal). Here, h;2π/C22hand ω;iTr[^R
T^L@t^R]=2
is the (vectorial) angular velocity of the spin dynamics,
defined in terms of the vector ^Lof SO(3) generators:
{^Lα}βγ;/C0iϵαβγ, the Levi-Civita symbol. ^gis a symmetric
real-valued 3 /C23 matrix, whose diagonalization de fines three
principal axes along with the associated (nonnegative) spinconductances, which generalize the scalar (spin-mixing) con-
ductance gdiscussed above. This treatment may be applied,
e.g., to noncollinear antiferromagnets and spin glasses withan (effective) SU(2) symmetry.
31,32In the simplest case of an
isotropic spin glass, ^g/^1. Figure 2shows a schematic ofthe nonequilibrium system at hand. The Rayleigh dissipation
function (7)encodes the information about the dissipation of
the magnetic dynamics into the normal-metal reservoir aswell as the reciprocal work done by a nonequilibrium spin
accumulation μ
sapplied to it.31
In closing this section, we would like to recall that a
straightforward way to establish an effective spin accumula-
tionμsat a boundary of a generic conductor is by using the
spin Hall effect.9,10Namely, on general symmetry grounds,
we may write
μs¼ϑsHz/C2j, (8)
where zis the normal to the interface and jis the (tangential)
electric current density. ϑsHis a material-dependent parame-
ter that depends on the strength of spin-orbit interactions near
the interface, vanishing in the absence thereof. Some heavy
metals and, particularly, the so-called topological-insulatormaterials are known to engender a sizable ϑ
sH.33
In the presence of a proximal magnetic material, which
modifies the spin-related boundary condition according to,
e.g., Eq. (4), the spin accumulation μsgenerally needs to be
calculated self-consistently, together with solving the mag-
netic equations of motion.5In certain special cases, however,
particularly in the limit of very fast spin relaxation in the
metal, the latter may be treated as a good spin reservoir that
is not signi ficantly affected by the spin flow in and out of the
adjacent magnet.
III. TOWARD TOPOLOGICAL FIELD FLOWS
A. Spin flow through an arbitrary insulator
Following the preceding discussion, we are now
equipped to subject an arbitrary insulating material to a spin
bias, by one or more voltage-controlled spin reservoirs. Thisis sketched in Fig. 3, where metallic spin reservoirs are
attached to supply arbitrarily oriented spin accumulations μ
(i)
s
via, e.g., the spin Hall effect. These spin biases can trigger
FIG. 3. A general spin circuit, in which the input terminals labeled by i¼
1, 2, establish local spin biases μ(i)
s, which drive spin dynamics in the mate-
rial or system of interest. The readout terminal (bottom) performs a measure-
ment of the resultant dynamic steady state via reciprocal means. For
example, if the input is accomplished by the spin Hall effect, by applying
electrical currents that induce spin accumulations (8), the output can be
achieved by measuring the inverse spin Hall voltage.10The instantaneous
state of the driven system can be described, e.g., by the position-dependent
rotation matrix ^R(r), supposing a rigid local order.
FIG. 2. A schematic of a noncollinear spin system (right) in contact with a
metallic spin reservoir (left). The nonequilibrium spin state of the metal is
parametrized by the (vectorial) spin accumulation μs. The magnet, whose
spin arrangement is determined by some isotropic exchange Hamiltonian, is
described, near the interface, by uniform (and essentially rigid) rotations of
all spins. Its instantaneous nonequilibrium state is thus characterized by the(vectorial) frequency of SO(3) rotation ω. The 3 /C23 matrix ^ggeneralizes the
concept of the spin-mixing conductance gpertinent to the collinear case.
Adapted from Y. Tserkovnyak and H. Ochoa, Phys. Rev. B 96, 100402(R)
(2017). Copyright 2017 American Physical Society.190901-3 Yaroslav Tserkovnyak J. Appl. Phys. 124, 190901 (2018)magnetic dynamics in the material, whose propagation can
be detected by one or more output contacts, which operate
reciprocally to the input ones.24,34Speci fically, we rely here
on the Onsager reciprocity,26according to which, loosely
speaking, if a metallic contact can trigger spin dynamics in
response to, e.g., an applied current, the same contact shouldbe able to pick up a voltage in response to similar spin
dynamics.
35–39
This philosophy can similarly be employed to study spin
currents carried by thermal magnons in magnetic insulators,
as has been demonstrated in Refs. 40and41. Here, different
platinum contacts were used for injecting and detecting spinflows transmitted by a ferrimagnetic insulator (yttrium iron
garnet). According to the bosonic statistics of magnons, this
spin-transport regime can be considered to be thermally acti-vated and incoherent. Furthermore, due to a finite lifetime of
the spin-carrying excitations, one can generally expect an
exponential suppression of the detected signal with distance.In the diffusive transport regime, the latter corresponds to the
spin-diffusion length of magnons, λ
s¼ffiffiffiffiffiffiffi ffiDτsp, where Dis
the diffusion coef ficient of thermal magnons and τsis their
characteristic lifetime.
B. Spin super fluidity
More interesting and potentially useful regimes of spin
transport concern spin flows that can be carried by coherent
order-parameter dynamics, in analogy to charge flows in
superconductors, mass flows in super fluid4He, and mass and
spinflows in3He.42This can be illustrated by considering an
easy-plane magnet, whose local con figuration can be parame-
trized by a canonical pair of variables ( w,ρs), where wis the
polar angle parametrizing the U(1) order-parameter within
the easy plane and ρsis the (nonequilibrium component of
the) spin density out of this plane. The canonical conjugacy
is evident as ρsis the generator of rotations within the easy
plane.14The simplest Hamiltonian describing a smooth
order-parameter field is
H¼ρ2
s
2χþA(∇w)2
2, (9)
where we truncated the expansion at the leading, quadratic
order in the deviations from the equilibrium. Ahere is the
order-parameter stiffness against long-wavelength distortions
andχis the local spin susceptibility. (Supposing the spin-
rotational symmetry within the easy plane, the Hamiltonian
should not depend on the absolute value of w.) The corre-
sponding Hamilton equations of motion are given by
@tw;δρsH¼ρs
χand@tρs;/C0δwH¼A∇2w: (10)
Thefirst equation can be interpreted as the Josephson relation
for the phase w, while the second equation can be understood
as the continuity equation:
@tρsþ∇/C1js¼0, where js;/C0A∇w: (11)
The underlying conservation law is dictated by the symmetry
under uniform rotations within the easy plane. The boundary
conditions at an interface with a spin-biased metal can beobtained from Eq. (4), in the case of a collinear local order
[or, more generally, from Eq. (7)]. Projecting this on the
easy-plane dynamics and supposing μsis parallel to the hard
axis, we get24
js¼gμs/C0/C22h@tw ðÞ , (12)
where the spin conductance gis normalized per unit area of
the interface. This is closely analogous to Andreev re flection
at a metal/superconductor interface, which is /(2eV/C0/C22h@tw),
in terms of the voltage Vapplied to the normal metal and
phase wdynamics of the condensate.
Combining Eq. (10) results in the wave equation for
angular dynamics:
@2
t/C0u2∇2/C0/C1
w¼0, (13)
with the sound velocity u;ffiffiffiffiffiffiffiffi
A=χp
. The linearly-dispersing
elementary excitations are akin to the first sound in a neutral
super fluid.
C. Role of anisotropies and dissipation
With the above idealized discussion setting the stage for
a super fluid-like treatment of easy-plane spin dynamics, there
are at least two ways in which it will differ from the genuinesuper fluidity, in practice. The crux of the matter is that the
latter is rooted in the fundamental gauge symmetry of the
underlying condensate, while the former is constructed interms of an approximate (structural) U(1) symmetry.
43
Breaking this symmetry microscopically, while preservingit on average, introduces a Rayleigh-Gilbert damping
28
R¼αs(@tw)2=2(αbeing a dimensionless parameter and sa
normalization prefactor in units of spin density), which modi-fies the Hamilton equation for spin density as @
tρs;
/C0δwH/C0δ@twR. This spoils the continuity equation:
@tρsþ∇/C1js¼/C0ρs
τα, (14)
where τα;χ=αsis understood as the spin relaxation time.
Breaking, furthermore, the spin-rotational symmetry
macroscopically adds anisotropies to the energy (9), which
can now depend on the absolute value of w. For example,
introducing an easy-axis anisotropy within the easy plane
results in H(w)!H(w)/C0Kcos2w. This, together with the
above damping, turns the wave equation (13) into the
damped sine-Gordon equation:15,16
@2
tþτ/C01
α@t/C0u2∇2/C0/C1
wþK
χsin 2w¼0: (15)
Injecting a spin current, as before, at an end of such a system
will now trigger dynamics that are qualitatively distinct from
an ordinary super flow. Rather than simply generating a
uniform spiraling flow (in a steady state), there is now a finite
threshold for inducing the dynamics (if we neglect, for
the moment, thermal activation44), upon which a train of
(domain-wall) solitons of size λ/differenceffiffiffiffiffiffiffiffiffi
A=Kp
propagates away
from the injector. Their density ngrows upon increasing the
input bias, coalescing into a state that mimics the originalsuper flow, when n/differenceλ
/C01.15,45As the pressure needed to
push the train (against the viscous Gilbert damping)190901-4 Yaroslav Tserkovnyak J. Appl. Phys. 124, 190901 (2018)decreases away from the source, the steady-state soliton
density will also decrease. Different stages of the collective
spin-flow evolution from the perfect super fluid (a), as we
turn on the magnetic anisotropies microscopically (b) and
macroscopically (c), are illustrated in Fig. 4.
We remark, in the passing, that for the internal consis-
tency of the above discussion, the easy-plane anisotropy
needs to be stronger than the parasitic anisotropy K. In this
case, the aforementioned threshold bias is lower than theupper critical bias dictated by the (Landau) stability of the
steady state against small perturbations.
15,45
D. Topological-charge hydrodynamics in 1D
Adding Gilbert damping and macroscopic anisotropies
to an idealized spin super flow(11) introduces additional
terms that spoil the continuity equation for spin dynamics
[cf. Eq. (15)]. In one spatial dimension (1D), this results in a
viscous solitonic transport, which, at a finite temperature and
dilute limit, may be expected to generically exhibit Brownian
motion.44It turns out, however, that even in this regime, a
hydrodynamic description in terms of a robust conservationlaw is possible. To this end, we are switching from the
hydrodynamics of spin density, ρ
s¼χ@tw, which is no
longer conserved, to the (dual) hydrodynamics of thewinding density, ρ
w;/C0@xw=2π, which is conserved, as
long as the large-angle out-of-plane excursions of the order
parameter are penalized by a strong easy-plane anisotropyand can be neglected. Irrespective of the details of the
damping and weak in-plane anisotropies, the continuity
equation,
@
tρwþ@xjw¼0 , (16)
with jw;@tw=2π, is automatically satis fied for a quasi-1D
spin texture, so long as the azimuthal angle w(x,t) is well
defined. This is guaranteed if the order parameter never
crosses the north or south pole in spin space. The preciseconditions for these are dictated by the energetics, the
strength of the driving, and thermal fluctuations.The continuity equation (16) sets the departure point for
constructing topological hydrodynamics, namely, a transport
theory for the conserved topological density, ρ
w(x,t). A
natural way to understand the injection mechanism for the
associated flowjwis offered by the energetic considerations.
(The detection then follows generally from the Onsager reci-procity.) Namely, projecting the spin-transfer power (5)onto
the easy-plane dynamics, we get
_W¼g
2πμs@tw/C0/C22h(@tw)2/C2/C3
¼gμsjw/C0hj2
w/C0/C1
: (17)
Thefirst term, /μsjw, stems from the torque by the spin bias
μsapplied to the adjacent reservoir. It is formally analogous
to the input power P¼VIof an electronic circuit subjected
to voltage V, when it carries charge current I. The second
term//C0j2
wdescribes dissipation due to spin pumping,5
which is analogous to Joule heating in the electronic counter-
part. We thus see that applying a spin bias μsnormal to an
easy-plane magnet translates into an energetic bias for the
injection of the topological flowjw. This would generate
dynamic magnetic textures as those depicted in Fig. 4, with
the details governed by magnetic anisotropies and damping.
We emphasize that this hydrodynamic construction is dic-
tated entirely by the topology associated with the windingdynamics, not making any simplifying assumptions about the
material and structural symmetries of the system.
By the Onsager reciprocity, if the spin bias μ
sinjects
flowjw(e.g., at the left contact depicted in Fig. 4), the topo-
logical out flowjwat the right contact will eject spin current
/jw,44which would in turn generate a measurable voltage V
by the inverse spin Hall effect.11The value of jw, in the
steady state, is determined by the microscopic details of the
magnetic conduit of the topological density ρw. In a number
of generic cases,24,44however, it can be written in linear
response as
jw/μs
rlþrrþrb, (18)
where rl,rparametrize the injection impedance at the contacts
andrb/Lthe bulk impedance for the propagation of the
FIG. 4. Three regimes of collective spin and winding flows, jsand jw, from the injector terminal (left) to the detector (right), connected by a
quasi-one-dimensional easy-plane magnetic strip of length L: (a) A perfect spin super flow, where the winding gradient @xwis uniform in a dynamic steady
state. (b) Turning on Gilbert damping αintroduces a negative gradient in js, accounting for the leakage of the angular momentum to the substrate. (c)
Additionally, adding a small easy-axis anisotropy Kalong the xaxis disrupts a smooth spin flow, by breaking the spin texture down into topological solitons of
size λ/differenceffiffiffiffiffiffiffiffiffi
A=Kp
. A steady-state motion of such solitons requires a diminishing pressure as they move along the xaxis, corresponding to their decreasing
density. The shaded regions highlight magnetic textures with the net (winding) charge of Qw¼þ1=2. In all three cases, there is a net topological charge flow
jwto the right.190901-5 Yaroslav Tserkovnyak J. Appl. Phys. 124, 190901 (2018)winding density along the magnetic channel of length L
(cf. Fig. 4). For the idealized spin-super flow regime
[Fig. 4(a)],24r¼g/C01,a te a c hi n t e r f a c e ,w h i l e rb¼0. This
mimics an electronic normal-metal/superconductor/normal-
metal heterostructure,46with greplacing the contact Andreev
conductance. Adding a Gilbert damping αto this [Fig. 4(b)]
gives24rb/αL,r eflecting the leakage of the angular momen-
tum into the substrate at a rate that scales with the system size
(since, in the steady state of coherent dynamics, @twmust be
uniform throughout the system). Finally, adding in-plane
anisotropies [Fig. 4(c)] results in44rb/L=D,w h e r e Dis the
diffusion coef ficient of the domain-wall solitons. Within the
Landau-Lifshitz-Gilbert phenomenology of magnetic dynam-
ics,28,47D/α/C01, which can be further modi fied by pinning
effects and the associated creep transport in disorderedwires.
48In this solitonic case, at elevated temperatures
(so quantum-tunneling effects play no role), the proportional-
ity coef ficient in Eq. (18) involves a Boltzmann factor e/C0βEdw,
where Edwis the free-energy cost to add a single domain wall
into a uniform system. The topological flow thus gets expo-
nentially suppressed at low temperatures, as the solitons,which carry both the winding density ρ
wand its flowjw,g e t
depleted from the magnetic wire. As already mentioned,15,45a
threshold bias then needs to be applied in order to overcomethe energy barrier E
dwfor injecting domain walls. Above this
critical bias, the solitons fill the system and establish a collec-
tive drift toward the detector [cf. Fig. 4(c)].
One salient feature of the collective response underlying
Eq.(18) concerns the algebraic, jw/L/C01, scaling of the non-
local response, in the limit of L!1. This is in stark con-
trast to the exponential suppression of the signals mediated
by a diffusive spin transport carried by magnons40or other
decaying quasiparticles. Here, in essence, in invoking topo-logical arguments for easy-plane dynamics, we have sup-
posed that magnetic solitons (or some arbitrary winding)
have an in finite lifetime. In reality, however, this lifetime is
effectively finite, albeit exponentially long, /e
βEw, where Ew
is the energy barrier for thermally-activated phase slips.49
These correspond microscopically to strong local deviations
of the magnetic order away from the easy plane, reaching the
north/south poles (in spin space) and thus undoing the
winding density ρw.50In the limits depicted in Figs. 4(a)and
4(b), such phase slips can locally unwind the smooth
winding density, while in Fig. 4(c), they can flip the chirality
(and thus the sign of the topological charge, +1=2) associ-
ated with each domain wall or spontaneously produce or
annihilate pairs of domain walls with the same chirality.
To summarize, the topological protection relies on a
large energy barrier Ew, which sets an exponentially long
lengthscale eβEwfor the validity of the continuity equation
(16) and the associated topological hydrodynamics. We do
not expect the nonlocal algebraic signals (18) to persist
beyond this lengthscale. It is useful to remark that in the case
when the solitonic transport of Fig. 4(c) is itself thermally
activated,44solitonic diffusion that preserves topological
charge can be established at intermediate temperatures,
Edw,kBT,Ew. The bene ficial disparity Edw/C28Ewis gen-
erally guaranteed, so long as the dominant magnetic anisot-
ropy in the system is of the easy-plane type (which isnaturally assumed throughout). This follows from the depen-
dence E/ffiffiffiffi
Kp
, for either of these two energies, on the rele-
vant anisotropy K.44,50At very low temperatures, quantum
phase slips ultimately take over in relaxing phase winding.51
In magnetic systems, this can be sensitive to microscopicdetails and, in particular, on whether the constituent spins areinteger or half-odd-integer.
52Apart from this, the quantum
regime of topological hydrodynamics remains largely unex-
plored. It should be clear, e.g., from the coherent-spinpath-integral perspective,
53that at least some of the robust
features underlying the continuity equation (16) and the
ensuing long-range transport should survive in the extremequantum regimes.
E. Higher-dimensional generalizations
One immediate generalization of the (topological)
winding hydrodynamics follows the structure of the homo-
topy group
πn(Sn)¼Z: (19)
Forn¼1, the integer corresponds to the number of the
winding twists discussed in the above one-dimensional case.
Forn¼2, this generalizes to the number of skyrmions that
characterize topological classes of two-dimensional magnetictextures.
54Forn¼3, the underlying topological textures (in
three spatial dimensions) are realized by placing the order
parameter on a hypersphere.55Alternatively, and more rele-
vant for spin systems, the order-parameter space here may be
given by SO(3), i.e., the group of rigid rotations in Euclidean
space. This is because π3(S3)¼π3[SO(3)], with SO(3) being
equivalent (according to the quaternion representation) to the
(real) projective space RP3, so essentially a 3-sphere (with
diametrically opposite points identi fied). One potential
physical realization of this is provided by the coherent
spin glasses32(or analogous noncollinear frustrated spin
systems56), in which three independent rotations of random
but locally frozen magnetic textures yield three phononic
(Goldstone-mode like) branches.57
We will illustrate a generalization of the winding hydro-
dynamics ( n¼1) to higher dimensions, as guided by the
homotopy (19), by considering the next simplest case of
n¼2. Physically, this concerns nonlinear σmodels (such as
Heisenberg ferro- or antiferromagnet) in two spatial dimen-
sions. The skyrmionic 3-current jskunderlying the topologi-
cal hydrodynamics is given by Ref. 58
jμ
sk¼1
8πϵμνρϵabcla@νlb@ρlc: (20)
Here, jl(x,y,t)j;1 describes a directional order-parameter
field. The fully-antisymmetric Levi-Civita symbols ϵare
accompanied with summations over repeated indices, withthe Greek letters ν,μ,ρlabeling three space-time coordinates
and the Roman letters a,b,cdesignating three spin-space
components. One easily checks that the current (20) obeys
the continuity equation:
@
μjμ
sk¼0: (21)190901-6 Yaroslav Tserkovnyak J. Appl. Phys. 124, 190901 (2018)The conserved (topological) charge,
Qsk;ð
dxdy j0
sk¼1
4πð
dxdy l/C1@xl/C2@yl, (22)
can be recognized as the skyrmion number, which is quan-
tized in integer values (if the order parameter is fixed at the
boundary or at in finity to point in the same direction).54This
integer is the degree of the R2!S2mapping, corresponding
to the number of times the sphere is covered by the magnetic
texture. Qskcan be thought as the two-dimensional generali-
zation of the winding number, which is the degree of the
R1!S1mapping.
In the special case of a ferromagnetic order parameter l,
we can easily establish an energetic bias for the skyrmionic
spin injection from a metallic contact using the adiabatic
spin-transfer torque.59Namely, applying an electric current ~j
tangential to the interface, the torque (per unit length of the
contact)
τ¼/C22hP
2e~j/C1~∇l (23)
would generally arise in the proximity to a smooth magnetic
texture. This torque follows from the (proximal) exchangeinteraction between electrons in the normal-metal contact and
the (insulating) ferromagnet. Pis a dimensionless parameter
parametrizing the strength of this exchange (with jPj!1i n
the extreme case of a very strong interaction that would
polarize and lock electron spins to the magnetic texture
59).
The work done by the torque (23) can be evaluated to
yield the power
_W¼ð
drτ/C1l/C2_l¼hPj
e^z/C1ð
d~r/C2~jsk, (24)
where the integration is performed along the length of the
current- jcarrying contact. We see from this that the electric
current tangential to a magnetic interface produces an ener-getic bias for the transverse skyrmion-density injection. We
can thus expect that a nonequilibrium skyrmion charge (22)
would generally develop over time, in the presence of such abias. The details of the ef ficiency of this skyrmion injection
depend of course on the physical regime of the system. In
particular, such skyrmionic injection and subsequent flow
were studied in Ref. 34in the regime of a thermally-activatedBrownian motion of a dilute gas of rigid (solitonic) sky-
rmions. In Ref. 60, the ensuing skyrmion flow was suggested
as a probe for different textured phases of chiral magnets(such as collinear, helical, and skyrmion-crystal phases),
which would yield different skyrmionic responses. In particu-
lar, in the crystalline phase, the work (24) would translate
into a boundary pressure that could trigger a gyrotropic
sliding motion of the skyrmionic crystal as a whole. One
could easily envision other physical scenarios, where suchtopological hydrodynamic probes may give useful informa-
tion about a nontrivial magnetic ordering, which would
otherwise not be directly accessible via other transportmeasurements.
In Fig. 5(a), we schematically depict this spin-torque-
induced skyrmion injection into a magnetic insulator. Thelatter could be either an ordinary Heisenberg ferromagnet or
a chiral magnet with propensity to form skyrmion textures
due to the Dzyaloshinski-Moriya interaction.
61Panel (b)
of the figure illustrates geometrical analogy between the
current-induced skyrmion flow and the Magnus force (which
is produced by the turbulent wake aft of a rotating bodysubjected to a hydrodynamic flow). Panel (c) (cf. Ref. 34
for more details) shows a nonlocal electrical measurement,
which probes a nonequilibrium skyrmion flux between two
metal contacts. Similarly to Fig. 4, the left metal contact
injects the topological hydrodynamics (now of the sky-
rmionic flavor). The right contact detects an electromotive
force Eproduced by the skyrmionic out flow through the
right contact, as dictated by the Onsager reciprocity:
35
E¼/C22hP
2eð
d~rl/C1~∇l/C2_l¼/C0hP
e^z/C1ð
d~r/C2~jsk: (25)
In the diffusive regime of solitonic propagation of skyrmion
density, as sketched in Fig. 5(c), the resultant transconduc-
tance scales algebraically as L/C01with the length Lof the
topological transport channel, similarly to the previous
winding example, Eq. (18). This stems from the conserved
character of the topological flow and the generic (Ohmic)
scaling/Lof its impedance. The latter is determined by the
solitonic diffusion coef ficient, which depends on Gilbert
damping, impurity potential, etc.
FIG. 5. (a) Electric-current induced injection of skyrmion flux into the magnetic region ( x.0), according to the work (24). Geometrically, this is analogous to
the Magnus force (b). Panel (c) shows a four-terminal electrical (drag) transconductance measurement, which could detect the injected (nonequ ilib-
rium) skyrmion flow from the left to the right metal. Panel (c) from H. Ochoa, S. K. Kim, and Y. Tserkovnyak, Phys. Rev. B 94, 024431 (2016). Copyright
2016 American Physical Society.190901-7 Yaroslav Tserkovnyak J. Appl. Phys. 124, 190901 (2018)IV. SUMMARY AND DISCUSSION
Spin transport in magnetic insulators may be carried
either by spin-carrying quasiparticles, such as magnons inordered spin systems, or coherent order-parameter dynamics,
such as an easy-plane super flow. In either case, the notion is
strictly-speaking meaningful when there is a spin-rotationsymmetry axis, along which the spin angular momentum is
conserved, at least approximately. It is remarkable that, while
the continuity equation for spin flow breaks down in the
opposite regime, broad classes of magnetic materials may
still exhibit robust collective transport behavior. The latter
can emerge, for example, when the real-space order-parameter textures can be classi fied into classes distinguished
by an extensive topological invariant. Here, we illustrated
this by focusing on two simple examples: winding dynamicsin one spatial dimension and skyrmion dynamics in two
dimensions. Noncollinear magnetic textures parametrized by
three Euler angles can allow one to also extend these ideas tothree-dimensional materials, such as spin glasses.
31,32
One could also envision other types of topological
hydrodynamics, which could be guided by the homotopyconsiderations for the coherent order-parameter fields. With
the key relevant mathematical concepts already established in
other areas of research, including both high and low ener-gies,
42the tools of spintronics are opening opportunities to
explore broad classes of magnetic materials from the perspec-
tive of topological transport. The first steps in this direction
are already being made.62–64The topological hydrodynamics
appears appealing both as a tool to probe complex phases of
quantum materials64and, eventually, as a utilitarian resource
within spintronics.65
ACKNOWLEDGMENTS
I am grateful to Benedetta Flebus, Se Kwon Kim,
Hector Ochoa, So Takei, Pramey Upadhyaya, and Ricardo
Zarzuela for insightful discussions and collaborations. Thework was supported in part by the National Science
Foundation (NSF) under Grant No. DMR-1742928 and
the Army Research Of fice (ARO) under Contract No.
W911NF-14-1-0016.
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1.3077204.pdf | Landau–Lifshitz or Gilbert damping? That is the question
W. M. Saslow
Citation: Journal of Applied Physics 105, 07D315 (2009); doi: 10.1063/1.3077204
View online: http://dx.doi.org/10.1063/1.3077204
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84.196.132.222 On: Thu, 15 May 2014 10:08:41Landau–Lifshitz or Gilbert damping? That is the question
W. M. Saslowa/H20850
Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA
/H20849Presented 14 November 2008; received 16 September 2008; accepted 4 January 2009;
published online 15 April 2009 /H20850
In their seminal 1935 paper on magnetics, Landau and Lifshitz /H20849LL/H20850proposed a form for
magnetization damping. In 1955 Gilbert proposed another form, introducing a dimensionlessparameter
/H9251. We derive LL damping using the theory of irreversible thermodynamics, summarize an
unbiased Langevin theory of fluctuations that yields LL damping, and argue that inhomogeneousbroadening might explain the nonresonance data that led Gilbert to formulate his theory. LL versusGilbert damping takes on special relevance in the context of bulk spin transfer torque and bulk spinpumping, where the form of damping affects the values of the “adiabatic” and “nonadiabatic” terms.We argue that the adiabatic and nonadiabatic terms are dissipative and reactive, respectively.©2009 American Institute of Physics ./H20851DOI: 10.1063/1.3077204 /H20852
I. INTRODUCTION
Landau and Lifshitz’s seminal 1935 paper on magnetics1
proposes, for the dynamics of the magnetization M/H6023of a uni-
form ferromagnet with gyromagnetic ratio /H9253in a net field H/H6023,
/H11509tM/H6023=−/H9253M/H6023/H11003H/H6023−/H9261Mˆ/H11003/H20849M/H6023/H11003H/H6023/H20850/H20849LL/H20850. /H208491/H20850
The first term is the expected precessional dynamics. With /H9261
a quantity with the same units as /H9253, the second term provides
a phenomenological form for the damping.
In the 1955 MMM conference proceedings, Gilbert ar-
gued that LL damping fails for large enough damping.2In-
stead, he proposed the form
/H11509tM/H6023=−/H9253GM/H6023/H11003H/H6023+/H9251Mˆ/H11003/H11509tM/H6023/H20849Gilbert /H20850. /H208492/H20850
With/H9251=/H9261//H9253and/H9253G=/H9253/H208491+/H92512/H20850, the LL and Gilbert forms are
mathematically equivalent. Good samples in ferromagnetic
resonance satisfy /H9251/H112701.3,4
Additional possible equations of motion were considered
in the 1950s,5of which we note only one by Callen.6In
practice, for small damping, LL and Gilbert are very nearlythe same, but at issue is a question of principle.
This work first shows that irreversible thermodynamics
predicts that magnetization damping takes the LL form. Itthen discusses Ref. 2and the experimental basis and theoret-
ical analysis on which Gilbert’s large damping argument isbased. Finally it discusses the implications of irreversiblethermodynamics for the additional physics associated withspin transfer torque and with spin pumping in nonuniformferromagnets. An unbiased Langevin theory of fluctuationsleads to LL damping, with a specific expression for /H9261.
7
II. IRREVERSIBLE THERMODYNAMICS AND LL
DAMPING
Irreversible thermodynamics has been applied to numer-
ous other condensed matter systems. A number of indepen-dent workers have already applied it for ferromagnets.
8–10
All obtain LL damping for low frequency, long wavelengthdynamics. A recent work on damping in nonuniform ferro-
magnetic insulators, including a magnetism-directed intro-duction to irreversible thermodynamics, finds that nonunifor-mity introduces as many as four new damping terms, butreduces to the LL form in the uniform case.
11Here we
present a derivation restricted to the uniform case.
Irreversible thermodynamics imposes the condition that
if local thermodynamics holds at the initial time, then the
equations of motion /H20849here, for /H9255,s, and M/H6023/H20850maintain local
thermodynamics at all future times.11
The differential of the internal energy density /H9255includes
an internal field H/H6023intvia the term H/H6023int·dM/H6023; the total energy
density also includes the interaction term − H/H6023·dM/H6023, where H/H6023
includes the external field H/H60230, lattice anisotropy from the
spin-orbit interaction, and anisotropy from the dipolar inter-
action. For a uniform system H/H6023intis along M/H6023, due to a uni-
form exchange field, so that M/H6023/H11003H/H6023int=0/H6023. Using a vector gen-
eralization of Johnson and Silsbee,12we define H/H6023/H11569=H/H6023−H/H6023int.
Then we take the basic thermodynamic relation to be
d/H9255=Tds−H/H6023/H11569·dM/H6023. /H208493/H20850
Here the temperature Tand the entropy density sboth are
even under time reversal. Both M/H6023andH/H6023/H11569are odd under time
reversal. In equilibrium H/H6023/H11569=0/H6023, so that H/H6023=H/H6023int.
The energy density, a conserved quantity, satisfies
/H11509t/H9255+/H11509iji/H9255=0 . /H208494/H20850
Here ji/H9255is the as-yet-unknown energy flux density. There is
no energy source because energy is conserved. The intrinsicsignature under time-reversal Tofj
i/H9255is odd. Dissipation oc-
curs for terms in ji/H9255that are even under T.
The entropy density s, a nonconserved quantity, satisfies
/H11509ts+/H11509ijis=R
T/H113500. /H208495/H20850
Here jisis as-yet-unknown entropy flux density and R/Tis
the as-yet-unknown entropy source density, where Ris the
volume rate of heating. The intrinsic signatures under timereversal of j
isandRare odd. Dissipation occurs for terms ina/H20850Electronic mail: wsaslow@tamu.edu.JOURNAL OF APPLIED PHYSICS 105, 07D315 /H208492009 /H20850
0021-8979/2009/105 /H208497/H20850/07D315/3/$25.00 © 2009 American Institute of Physics 105 , 07D315-1
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84.196.132.222 On: Thu, 15 May 2014 10:08:41jisandRthat are even under T. Heating is irreversible so in
practice Rcontains only terms that are even under T.
The /H20849nonconserved /H20850magnetization M/H6023satisfies
/H11509tM/H6023=−/H9253M/H6023/H11003H/H6023+N/H6023. /H208496/H20850
The first term is the /H20849known /H20850Larmor torque, with gyromag-
netic ratio /H9253/H110220 taken to include the effect of spin-orbit in-
teractions. It is even under T. The as-yet-unknown magneti-
zation source /H20849or torque density /H20850N/H6023, has even intrinsic time-
reversal signature. Dissipation occurs for terms in N/H6023that are
odd under T. We take /H20841M/H6023/H20841to be constant so N/H6023is normal to M/H6023.
Hence N/H6023·H/H6023/H11569=N/H6023·H/H6023.
In irreversible thermodynamics each part of the un-
known fluxes or sources must be proportional to the drivingterms in the thermodynamic variables, called forces or affini-
ties. Here the driving terms are
/H11509iT,Mˆ·H/H6023/H11569, and M/H6023/H11003H/H6023.I n
H/H6023/H11569=H/H6023−H/H6023intthe first term is along H/H6023and the second is along
M/H6023,s oH/H6023/H11569/H20849−/H9253M/H6023/H11003H/H6023/H20850=0, to be used shortly.
Employing Eqs. /H208493/H20850,/H208494/H20850, and /H208496/H20850,Rin Eq. /H208495/H20850becomes
0/H11349R=T/H11509ts+T/H11509ijis=/H11509t/H9255+T/H11509ijis+H/H6023/H11569·/H11509tM/H6023=− /H11509iji/H9255
+T/H11509ijis+H/H6023/H11569·/H20849−/H9253M/H6023/H11003H/H6023+N/H6023/H20850=− /H11509i/H20849ji/H9255−Tjis/H20850
−jis/H11509iT+N/H6023·H/H6023. /H208497/H20850
The divergence, if nonzero, could have either sign. To satisfy
R/H113500, we eliminate the divergence by setting
ji/H9255=Tjis. /H208498/H20850
Consistent with jisbeing a vector in real space, the only
allowed form proportional to /H11509iT,Mˆ·H/H6023/H11569, and M/H6023/H11003H/H6023is
jis=−/H9260
T/H11509iT, /H208499/H20850
where /H9260is a constant called the thermal conductivity. /H11509iTis
even under time reversal, and thus /H20849being opposite jis’s intrin-
sic time-reversal signature /H20850is dissipative.
Consistent with N/H6023being /H20849a/H20850a vector in spin space, /H20849b/H20850
normal to Mˆ, and /H20849c/H20850not changing the gyromagnetic ratio,
the only allowed form is
N/H6023=−/H9261Mˆ/H11003/H20849M/H6023/H11003H/H6023/H20850, /H2084910/H20850
where /H9261is a constant. This is, of course, LL damping. It is
odd under time reversal, and thus /H20849being opposite N/H6023’s intrin-
sic time-reversal signature /H20850is dissipative.
We now can determine R. From Eqs. /H208497/H20850–/H2084910/H20850we have
R=/H9260
T/H20849/H11509iT/H208502+/H9261
/H20841M/H6023/H20841/H20841M/H6023/H11003H/H6023/H208412. /H2084911/H20850
Thermodynamic stability /H20849R/H113500/H20850implies /H9260/H113500 and /H9261/H113500.
A recent study of magnetization damping using Lange-
vin theory,7where the dominant fluctuations are character-
ized by thermodynamic parameters taking on nonequilibriumvalues, found LL damping and an expression for the LLdamping constant in terms of near-equilibrium fluctuations.In contrast, the theory of Brown
13inputs Gilbert damping to
bias the fluctuations, and thus is not a Langevin theory.
III. GILBERT THEORY, KELLY’S ROTATIONAL
TORQUE DATA, GILBERT’S ANALYSIS
The original literature on Gilbert theory is difficult to
trace. The abstract for Gilbert’s talk at a American PhysicalSociety meeting in 1955 does not appear on the APSwebsite,
14although it does appear in bound copies of The
Physical Review. This abstract has been of such continuinginterest that it had been copied by a website available at thetime of the present submission.
15Unfortunately, the abstract
is not terribly revealing.
Another early reference to Gilbert theory is an unpub-
lished report.16Recently, Gilbert presented a retrospective,
which was part of his Ph.D. dissertation.17He argues, by
analogy to damping of a particle using the Rayleigh dissipa-tion function in a modified Hamiltonian formulation of me-
chanics, that the damping form should go as Mˆ/H11003
/H11509tM/H6023.17The
most revealing article we have found is in the MMM Con-ference Proceedings of 1955, which presents Kelley’smethod and data, and Gilbert’s analysis.
2
Kelly employed a nonresonant rotating field /H20849from
crossed coils /H20850in the plane of a Permalloy disk of thickness
h=3.3/H9262m and diameter /H20849perhaps radius /H20850d=1.3 cm and
measured the torque acting on the disk. Gilbert first em-ployed LL damping for fixed
/H9253and frequency-dependent /H9261,
but the theory could not fit the data. He then introducedanother form /H20849Gilbert damping /H20850. Using fixed
/H9253Gand
frequency-dependent /H9251,18he found that data for the four fre-
quencies /H20849in MHz /H20850of 2.0, 1.0, 0.032, and 0.015 could be fit
with values of /H9251given by 0.3, 0.3, 3, and 9. He notes that
/H20849/H9261//H9253/H20850//H208511+/H20849/H9261//H9253/H208502/H20852should not exceed 0.5, and then states
that using the LL form this value was exceeded for the lower
two frequencies.
If/H9253Gis constant, then /H9253=/H9253G//H208491+/H92512/H20850varies. Assuming
constant /H9253G, the values /H9251=3,9 at the lower two frequencies
imply that /H9253takes on values of about 0.1 and 0.01 of its high
frequency value. We find it difficult to believe that dissipa-tive processes can cause
/H9253to decrease by such enormous
factors. We think it more likely that an LL-based analysisfailed because 1955 sample-preparation techniques led tolarge inhomogeneous broadening, which dominated at thelower frequencies. Inhomogenous broadening can be incor-porated with Gilbert damping in a simple manner by taking/H9261→/H9261+A/f, where Acharacterizes the inhomogeneous
broadening and fis the frequency. Even the “high” fre-
quency value of
/H9251=0.3 indicates a poor sample relative to
modern ones. Such a poor sample should not be the basis forabandoning LL theory. Nevertheless, Gilbert’s use of the di-mensionless quantity
/H9251/H20849proportional to the inverse of the
quality factor Q/H20850is a valuable addition to the literature; in
LL damping the form /H9261=/H9253/H9251should be employed.
Note that Mˆ/H11003/H11509tM/H6023/H208491/H20850is not proportional to a thermo-
dynamic force, unlike what occurs in irreversible thermody-namics; /H208492/H20850does not have a unique time-reversal signature,07D315-2 W. M. Saslow J. Appl. Phys. 105 , 07D315 /H208492009 /H20850
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84.196.132.222 On: Thu, 15 May 2014 10:08:41and thus introduces a reactive response in addition to a dis-
sipative response; and /H208493/H20850gives the equation of motion a
curious self-referential character.
IV. SPIN TRANSFER TORQUE AND SPIN PUMPING—
ADIABATIC AND NONADIABATIC
The distinction between LL and Gilbert damping is usu-
ally insignificant for small /H9251. However, for spin transfer
torque, this distinction matters even for small /H9251.7In spin
transfer torque, when the magnetization is nonuniform /H20849e.g.,
at a surface or in a magnetic domain or vortex /H20850, conduction
of spin-polarized conduction electrons transfersmagnetization.
19,20There are two forms of spin transfer
torque, one called adiabatic and the other nonadiabatic,
where adiabatic refers to slow spatial variations.
Including the spin transfer torque term for current den-
sityjalong x, the LL equation reads7
/H11509tM/H6023=−/H9253M/H6023/H11003H/H6023−/H9261Mˆ/H11003/H20849M/H6023/H11003H/H6023/H20850/H20849LL/H20850−v/H20851/H11509xM/H6023
−/H9252Mˆ/H11003/H11509xM/H6023/H20852. /H2084912/H20850
Here vis proportional to jand the fractional magnetization P
and/H9252is a constant. For a two-band model, the form is even
more complex than Eq. /H2084912/H20850.21Here vhas units of velocity
and is proportional to current, which for ordinary conductingmagnets should be thought of as proportional to a gradient inthe electrochemical potential, which is even under T. There-
fore for ordinary conducting magnets the term in
v/H11509xM/H6023,
called adiabatic, has the same time-reversal properties as the
LL damping term, and leads to damping. The term in v/H9252Mˆ
/H11003/H11509xM/H6023, called nonadiabatic, has the same time-reversal prop-
erties as the Larmor term and is reversible. In the irreversiblethermodynamics, a term equivalent to
vappears in the dissi-
pation rate, but not a term equivalent to v/H9252.21
Using vector identities and /H9251/H11013/H9261//H9253, Eq. /H2084912/H20850can be re-
written as the corresponding Gilbert equation,
/H11509tM/H6023=−/H9253/H208491+/H92512/H20850M/H6023/H11003H/H6023+/H9251Mˆ/H11003/H11509tM/H6023/H20849Gilbert /H20850
−v/H208491−/H9252/H9251/H20850/H11509xM/H6023+v/H20849/H9252+/H9251/H20850Mˆ/H11003/H11509xM/H6023. /H2084913/H20850
The choice of LL or Gilbert damping clearly leads to signifi-
cant differences in the assessing the two types of spin trans-fer torque. Microscopic theory and data analysis should in-dicate which of LL or Gilbert is employed.
A number of recent works consider spin pumping /H20849of the
current /H20850for a system with nonuniform magnetization, three
of them
22–24using spin-Berry phase arguments25and one us-
ing the methods of irreversible thermodynamics.21Spin-
pumping is closely related to spin transfer torque.21In the
spin-Berry phase-based works22–24current and spin current
are driven by phase gradients, and thus are odd under timereversal, as for a magnetic superconductor, where they arenondissipative. It follows that for a magnetic superconductor,the adiabatic spin pumping and adiabatic spin transfer torqueterms are nondissipative, whereas the nonadiabatic spinpumping and nonadiabatic spin transfer torque terms are dis-sipative. In this case, just opposite to what one has for an
ordinary conducting magnet, a term proportional to
v/H9252
would appear in the dissipation rate, but not a term propor-tional to
valone.
V. CONCLUSIONS
The present work argues for LL rather than Gilbert
damping, as follows from many independent studies usingthe methods of irreversible thermodynamics, a near-equilibrium Langevin theory for magnetization damping, andan examination of the original arguments of Ref. 2. In par-
ticular, the data from Ref. 2was obtained by a nonresonant
method that to our knowledge has not been employed sincethen, and the neglect of inhomogeneous damping may nothave been valid. Recent work by Smith
26favors Gilbert
damping; we have not determined its relationship to that ofirreversible thermodynamics.
ACKNOWLEDGMENTS
I am grateful for valuable discussions with Carl Patton,
Sam Bhagat, Mark Stiles, and Tony Arrott. I am especiallyindebted to Carl Patton for emphasizing the need to providedetails of how LL damping follows from irreversible thermo-dynamics. Communications with Neil Smith are gratefullyacknowledged. This work was supported by the Departmentof Energy through DOE Grant No. DE-FG02-06ER46278.
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[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
84.196.132.222 On: Thu, 15 May 2014 10:08:41 |
1.4765668.pdf | Effects of spin-polarized current on pulse field-induced precessional magnetization
reversal
Guang-fu Zhang , Guang-hua Guo, , Xi-guang Wang , Yao-zhuang Nie , and Zhi-xiong Li
Citation: AIP Advances 2, 042127 (2012); doi: 10.1063/1.4765668
View online: http://dx.doi.org/10.1063/1.4765668
View Table of Contents: http://aip.scitation.org/toc/adv/2/4
Published by the American Institute of PhysicsAIP ADV ANCES 2, 042127 (2012)
Effects of spin-polarized current on pulse field-induced
precessional magnetization reversal
Guang-fu Zhang,1,2Guang-hua Guo,1,aXi-guang Wang,1Y ao-zhuang Nie,1
and Zhi-xiong Li1
1School of Physics and Electronics, Central South University, Changsha 410083, China
2School of Communication and Electronic Engineering, Hunan City University, Yiyang
413000, China
(Received 29 August 2012; accepted 19 October 2012; published online 26 October 2012)
We investigate effects of a small DC spin-polarized current on the pulse field-induced
precessional magnetization reversal in a thin elliptic magnetic element by micromag-
netic simulations. We find that the spin-polarized current not only broadens thetime window of the pulse duration, in which a successful precessional reversal is
achievable, but also significantly suppresses the magnetization ringing after the re-
versal. The pulse time window as well as the decay rate of the ringing increasewith increasing the current density. When a spin-polarized current with 5 MA/cm
2
is applied, the time window increases from 80 ps to 112 ps, and the relaxation timeof the ringing decreases from 1.1 ns to 0.32 ns. Our results provide useful informa-tion to achieve magnetic nanodevices based on precessional switching. Copyright
2012 Author(s). This article is distributed under a Creative Commons Attribution 3.0
Unported License .[http://dx.doi.org/10.1063/1.4765668 ]
I. INTRODUCTION
Ultrafast magnetization reversal is a key issue for the development of several forthcoming
nanoscale magnetic devices, such as magnetic data storage,1–3nonvolatile memories,4and logic
devices.5The conventional magnetization reversal has been realized by applying a magnetic field
anti-parallel to the initial magnetization state, and the reversal is achieved after several precessional
oscillations due to dissipative effects. Typical reversal time for this kind of reversal is of the order
of nanoseconds. Alternatively, the precessional magnetization reversal, in which the driving field isapplied orthogonally to the initial magnetization so that the created torque leads to a rapid large-angle
magnetization precession, is a promising approach toward the ultrafast magnetization reversal. The
fastest reversal is obtained when the magnetization precession is stopped after a half period of afull precession, and the switching time is only a few hundred picoseconds.
6–14T. Gerrits et al .7
demonstrated that a reliable precessional reversal in micrometre-sized elliptical permalloy element
is possible at switching time of about 200 ps. H. Schumacher et al.9also realized a quasi-ballistic
reversal with switching time was as short as 165 ps in a spin valve. Moreover, the reliable ultrafast
precessional reveral has been achieved by a picosecond spin-polarized current pulse in the spin
valves and magnetic tunnel junctions.15–21
It has been demonstrated theoretically and experimentally that the accurate control of the pulse
duration is necessary to realize a reliable precessional reversal either by magnetic field or by spin-
polarized current. There is a time window for the pulse duration. When the pulse duration is beyondthis time window, the precessional reversal can not be achieved because either the magnetization
precessional angle is too small or it is too large and the magnetization shots back again. In practice,
a large time window is required for magnetic nanodevices as the precessional period, and hencethe pulse time window is sensitive to the size and shape of nanomagnet.
22,23In addition, after the
aAuthor to whom correspondence should be addressed; Electronic mail: guogh@mail.csu.edu.cn
2158-3226/2012/2(4)/042127/7 C/circlecopyrtAuthor(s) 2012
2, 042127-1042127-2 Zhang et al. AIP Advances 2, 042127 (2012)
FIG. 1. Illustration of the thin magnetic element with the geometry and dimensions (a). The time evolution of the normalized
magnetization components mxfor pulse duration τp=100 ps (black circular dot), 58 ps (blue triangle dot) and 146 ps (red
square dot) with spin-polarized current density J=0( b )a n d J=5 MA/cm2(c). The inset of (b) is the time evolution of the
normalized magnetization components mxandmzunder the action of a long time pulse field without spin current. The inset
of (c) shows the time evolution of the normalized magnetization components myforτp=100 ps with J=0 (black line) and
J=5 MA/cm2(red line).
precessional reversal, a residual magnetization oscillation or ringing around the new equilibrium
state persists for several nanoseconds.12–14,24This increases the time needed to execute consecutive
switching events. Effective suppression of the magnetization ringing is necessary and still remains
to be a problem for the application of precessional reversal in ultrafast magnetic devices.
In the paper, we study the precessional magnetization reversal of a magnetic element by using
micromagnetic simulations. A strategy is presented for broadening the pulse time window and
suppressing the magnetization ringing.
II. SIMULATION DETAILS
The magnetic element under study is a thin elliptic element with lateral dimensions 150 nm
×70 nm and thickness t=3 nm as shown in Fig 1(a). The dynamics of magnetization follows the
Landau–Lifshitz–Gilbert–Slonczewski equation:25
d/vectorm
dt=−γ/vectorm×/vectorHeff+α/vectorm×d/vectorm
dt−aJ/vectorm×(/vectorm×/vectorp)( 1 )
Where /vectorm=/vectorM/Ms,Msis saturation magnetization. Heffis the total effective field, which is the sum of
the exchange field, the anisotropic field, the demagnetizing field and the external pulse field. γis the
gyromagnetic ratio. αdenotes the damping parameter. The last term of equation (1)is Slonczewski
spin torque with the magnitude aJ=(γ¯hJη)/eμ0tMs.Jis the current density. ηand/vectorprepresent
the spin polarization and the spin polarization direction of the current, respectively. Micromagnetic042127-3 Zhang et al. AIP Advances 2, 042127 (2012)
FIG. 2. The switching probability versus the field pulse duration without spin-polarized current (black solid line) and with
current J=5 MA/cm2(red dot line).
simulations presented here are performed with the micromagnetic code of OOMMF.26The simulation
cell size is chosen to be 2 ×2×3n m3. The magnetic parameters corresponding to the permalloy are
used: saturation magnetization Ms=8.6×105A/m, exchange stiffness constant A=1.05×10-12J/m,
magnetocrystalline constant k1=0J / m3, and damping coefficient α=0.01. The initial magnetization
of the element is along the x-direction. To realize the precessional reversal, a rectangular magnetic
field pulse Hpwith amplitude 50 mT is applied in the ydirection. The pulse duration is varied.
The spin-polarized current with palong the x-axis flows perpendicularly to the element along the
negative z-direction as shown in Fig. 1(a).
III. RESULTS AND DISCUSSION
We first study the pulse field-induced precessional reversal without the spin-polarized current.
Figure 1(b) shows the temporal evolutions of the x-component of the magnetization mxfor the
pulse durations τp=58 ps, 100 ps, and 146 ps. The temporal evolution under the action of a long
pulse field is also given as an inset of Fig. 1(b), which gives a magnetization precessional period of
230 ps. It is evident that the precessional reversal is sensitive to the pulse duration. A successful
reversal is achieved for τp=100 ps. While for τp=58 ps and 146 ps, the magnetization rotates back to
the initial state. We investigate the reversal behavior under different pulse duration from 2 ps to 230 pswith an interval of 2 ps. The switching probability P
swversus the pulse duration τpis shown in Fig. 2.
There are two pulse time windows for successful reversal. One time window (denoted as TW1) is
from 74 ps to142 ps in the neighborhood of the half precessional period time and another one (TW2)
appears at 30 ps - 42 ps. The mechanism of precessional reversal has been studied extensively.6–9For
a very thin magnetic element, when a field pulse Hpis applied to it, the magnetization immediately
starts to precess around the applied field direction. This precession quickly causes the magnetization
to tilt out of the plane, creating a demagnetizing field /vectorHdin the direction perpendicular to the plane.
Then the magnetization begins to rotate mainly in the plane under the action of the torque /vectorLd
=-γ/vectorm×/vectorHd. Therefore, the amplitude and the direction of /vectorHd(and hence the average z-component
of the magnetization mz) play a crucial role in the precessional reversal. When the magnetization
precesses just half of a period (that is 115 ps), the magnetization rotates close to the reversal state andthe demagnetizing field /vectorH
d(ormz) is very small as seen from the inset of Fig. 1(b). If the external
field is cut off at this moment, a successful reversal is achieved. This just corresponds to the pulse
time window TW1. If the pulse duration is larger than the upper limit of TW1, when the field is cutoff, the demagnetizing field is large enough to drive the magnetization rotating back clockwise to
the initial state as the m
zis positive (seen the inset of Fig. 1(b)) even if the magnetization is in the
third quadrant at the moment of field cutoff. Similarly, for the pulse duration smaller than the lowerlimit of TW1 (but larger than the upper limit of the time window TW2), when the field is cut off, the
magnetization is in the second quadrant, but under the action of the demagnetizing field it rotates
back counterclockwise to the original state as the m
zis negative. It can be seen from Fig. 2that the042127-4 Zhang et al. AIP Advances 2, 042127 (2012)
FIG. 3. Illustration of the direction of the damping and spin-transfer torques.
pulse time window TW2 is much smaller than half of a precessional period (115 ps). After cutoff
of the pulse field the magnetization is still in the first quadrant, but the demagnetizing field is largerenough to rotate it to the reversal state. It should be noted that TW2 is much narrower comparing
with TW1 and it disappears when the strength of pulse field is small (but the strength is still large
enough to reverse the magnetization when it is cut off at the moment of a half period, meaning the
TW1 still exists).
Following, we will see that the pulse time window can be effectively broadened and the magne-
tization ringing can be significantly suppressed if a small DC spin-polarized current is applied to the
element during the precessional reversal. Figure 1(c) shows the temporal evolutions of the magneti-
zation m
xfor the same field pulses as in Fig. 1(b) but a spin-polarized current with J=5M A / c m2
is added. J=5M A / c m2is much smaller than the critical density Jc=35 MA/cm2for the spin-
transfer torque-induced reversal. It can be seen that a successful precessional reversal is realized for
the field pulse τp=58 ps. The switching probability Pswversus the pulse duration τpwith assistance
of the spin-polarized current J=5M A / c m2is shown in Fig. 2. It can be seen that the precessional
reversal is possible for τpfrom 30 ps to 142 ps. Comparing the temporal evolutions of the magneti-
zation with and without the spin-polarized current, we can see that the small spin-polarized currentplays a neglectable role in magnetization rotation process before cutting off the pulse field. But
after switching off the field, the average z-component of the magnetization rapidly decreases, the
effect of the spin-transfer torque resulting from the spin-polarized current becomes remarkable. Asmentioned above, for the pulse field with τ
pin the range between TW1 and TW2, the magnetization
is in the second quadrant when the field is switched off. In the case without spin-polarized current,
the demagnetization field forces it rotate back counterclockwise to the initial state. In contrast, when
the current is applied, the resulted spin-transfer torque acting on the magnetization is directed toward
the negative x-axis (reversal equilibrium state) as indicated in Fig. 1(a), and hinders the magneti-
zation rotates back to the original state. As a result, the magnetization reversal is possible in wider
pulse time window. The expansion of the time window is dependent on the current density. With
the increase of the current density, TW1 is gradually expanded downward, while TW2 is expandedupward. When J≥5M A / c m
2, two time windows connect together.
The spin-polarized current can also significantly suppress the magnetization ringing after the
precessional reversal. This is clearly indicated by the inset of Fig. 1(c), in which the temporal
evolutions of the average magnetization myfor the field pulse τp=100 ps with and without the
spin-polarized current are depicted. The magnetization ringing is mainly caused by the following
two factors: first, the magnetization is not just in the reversal equilibrium state when the fieldis cut off, and the demagnetizing field and/or anisotropic field drive the magnetization oscillate
around the reversal equilibrium state. Second, the magnetization rotation is usually not uniform,
accompanied by the generation of spin waves. These spin waves persist for a long time after thereversal. The decay rate of magnetic ringing depends on the dissipation of the energy, and hence
on the damping parameter. Studies have confirmed that the spin-polarized current can increase or
decrease the magnetic damping.
27–29The mechanism is schematically illustrated in Fig. 3. When042127-5 Zhang et al. AIP Advances 2, 042127 (2012)
FIG. 4. The relaxation times as a function of spin-polarized current density. The black square dot is the total relaxation time
obtained by fitting mywith the exponential function. The red circular and blue triangle dots denote the relaxation time for
edge spin-wave modes 0-EM and 1-EM.
FIG. 5. The frequency spectra of the magnetization ringing after reversal. The inset is the spatial distribution of FFT powerfor three peaks, corresponding to the edge modes 0-EM, 1-EM and fundamental mode F.
the spin-polarized electrons penetrate into a magnetic element, the produced spin-transfer torque
predicated by the last term of Eq. (1)is either parallel to the damping torque or antiparallel to it,
depending on the direction of the current. When the spin-transfer torque is the same direction as
the damping, the spin-polarized current increases the value of the effective damping, leading toa faster dissipation of the magnetization oscillation energy.
27This is just in our case, where the
injected spin-polarized current flows perpendicularly to the element along the negative z-direction.
The spin-transfer torque is directed toward the reversal equilibrium (negative x-axis) as indicated in
Fig. 1(a), making the magnetization spiral more rapidly to the reversal direction. By fitting the my∼t
curves with exponential form e−t/τ, we get relaxation time τ=1.1 ns and 0.32 ns for J=0 and
5M A / c m2, respectively. Therefore, the spin-polarized current effectively increases the decay rate of
the magnetization ringing. The current density dependence of the relaxation time is shown in Fig. 4.
Magnetic ringing relaxation is accompanied by spin wave attenuation. To shed more light
on the suppression of the magnetization ringing, we carry out the frequency spectrum analysisof the magnetic ringing. By the Fast Fourier Transformation of the temporal evolutions of the
magnetization, the spectrums of the ringing are obtained as shown in Fig. 5. Three peaks are found
in 0-15 GHz range, which correspond to three different spin-wave eigenmodes. From the spatialdistributions of the FFT powers (shown as inset of Fig. 5), the eigenmodes are identified as edge
mode 0-EM, 1-EM and fundamental mode F.
30For the edge modes 0-EM and 1-EM, the magnetic
oscillation is mainly localized in the edge region of the element, while the oscillation of fundamental042127-6 Zhang et al. AIP Advances 2, 042127 (2012)
mode F is mainly concentrated in the central area. It should be noted that except the edge mode and
fundamental mode, high order spin-wave modes may exist depending on the field pulse duration, but
the edge mode is always the strongest, other high-frequency modes decay rapidly. It can be seen from
Fig. 5that the spin-polarized current reinforces the attenuation of the spin waves. The strength of the
peak decreases and the linewidth increases with the increase of the current density. The linewidth
/Delta1fmode can be obtained by fitting the frequency spectrum with Lorentzian functions. The relaxation
time for each spin-wave mode is evaluated by the formula τmode=γμ 0Ms/4π2fmode/Delta1fmode.31Figure
4shows the relaxation time τmode and its change with the current density. The edge mode 0-EM
has the largest relaxation time, meaning it is the main contribution to the magnetization ringing.
Furthermore, the relaxation times for all spin-wave modes decrease with the increasing current
density, indicating the spin-transfer torque effectively enhances the damping and speeding up the
relaxing of magnetization ringing.
IV. CONCLUSION
We have studied the pulse field-induced precessional reversal in a thin elliptic magnetic element
by micromagnetic simulations. The reversal is sensitive to the pulse duration. There are time windows
for the pulse duration. When the pulse duration is in the range of the time windows, a successfulprecessional reversal is realized. Otherwise, the magnetization either does not rotate enough to the
opposite equilibrium state or rotates over and shots back again to the initial state. The time window
can be effectively broadened by applying a small DC spin-polarized current. Furthermore, the spin-polarized current can significantly suppress the magnetization ringing after reversal. Frequency
spectrum analysis shows that the ringing is composed of several spin-wave modes, but the edge
mode has the largest contribution to the ringing. The spin-transfer torque can shorten the relaxationtimes of spin-wave modes and reinforce the magnetic damping. The results obtained in this work
may find their use in designing ultrafast magnetic nano-devices.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (No 60571043),
Doctoral Fund of Ministry of Education of China and the Scientific Plane Project of Hunan Province
of China (No 2011FJ3193). G. F. Z. acknowledges the support of the Scientific Research Fund of
Hunan Provincial Education Department (No 11C0254).
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1.2830964.pdf | Micromagnetic analysis of current driven domain wall motion in nanostrips with
perpendicular magnetic anisotropy
S. Fukami, T. Suzuki, N. Ohshima, K. Nagahara, and N. Ishiwata
Citation: Journal of Applied Physics 103, 07E718 (2008); doi: 10.1063/1.2830964
View online: http://dx.doi.org/10.1063/1.2830964
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/103/7?ver=pdfcov
Published by the AIP Publishing
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[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
132.174.255.116 On: Thu, 27 Nov 2014 17:56:17Micromagnetic analysis of current driven domain wall motion in nanostrips
with perpendicular magnetic anisotropy
S. Fukami,a/H20850T . Suzuki, N. Ohshima, K. Nagahara, and N. Ishiwata
Device Platforms Research Laboratories, NEC Corporation, 1120 Shimokuzawa,
Sagamihara 229-1198, Japan
/H20849Presented on 8 November 2007; received 7 September 2007; accepted 17 October 2007;
published online 31 January 2008 /H20850
Current driven domain wall motion in nanostrips with perpendicular magnetic anisotropy was
analyzed by using micromagnetic simulation. The threshold current density of perpendicularanisotropy strips in adiabatic approximation was much smaller than that of in-plane anisotropystrips, and it reduced with thickness reduction. The differences originate from the differences indomain wall width and hard-axis anisotropy. Also, the threshold current density of perpendicularanisotropy strips required to depin from a pinning site was quite small although the threshold fieldof the strips was sufficiently large relative to those of in-plane anisotropy strips. © 2008 American
Institute of Physics ./H20851DOI: 10.1063/1.2830964 /H20852
I. INTRODUCTION
Current driven domain wall motion /H20849DWM /H20850, first pre-
dicted by Berger,1has been intensively studied both
experimentally2–11and theoretically12–14in recent years. Ap-
plications of current driven DWM to storage,15logic,16and
memory17devices have been also presented. These applica-
tions require that two primary problems be solved. One ofthem is the difficulty in achieving stable control of the do-main wall. Some reports have pointed out structural changesin the domain wall, bidirectional displacement, and the sto-chastic nature of DWM, some of which originate from thethermal effect of Joule heating or the local pinning effect.
5,7–9
The other problem is large drive current. According to Ref.
18, the writing current must be reduced to under 0.5 mA for
a magnetic random access memory /H20849MRAM /H20850. Such a low
current cannot be achieved with widely used NiFe nanos-trips, in which critical current density has been reported to bearound 1 /H1100310
12A/m2. These problems can be largely solved
by reducing the critical current density required to depin thedomain wall from a pinning site. Relatively small criticalcurrent density has been reported for nanostrips with perpen-dicular magnetic anisotropy /H20849PMA /H20850.
10,11However, the differ-
ences between in-plane magnetic anisotropy /H20849IMA /H20850and
PMA have not been discussed quantitatively yet, and there-fore the reason for the small current density of PMA has notbeen clarified. In this paper, we report on micromagneticcalculation results of current driven DWM in nanostrips withPMA and IMA. We also discuss quantitative differences be-tween them.
II. CALCULATION METHOD
In order to calculate current driven DWM, we used the
generalized Landau-Lifshitz-Gilbert equation with adiabaticand non-adiabatic spin-transfer torque terms, i.e.,m˙=− /H20841
/H9253/H20841m/H11003H+/H9251m/H11003m˙−/H20849u·/H11612/H20850m+/H9252m/H11003/H20849u·/H11612/H20850m,
/H208491/H20850
where mis the local magnetization, /H9253is the gyromagnetic
ratio, His the micromagnetic effective field, /H9251is the Gilbert
damping constant, and /H9252is a coefficient of the nonadiabatic
effect.14The vector uis regarded as spin-polarized current
density, defined as u=/H20849gP/H9262B/2eM s/H20850j, where Pis the polar-
ization, Msis the magnetization, and jis the current density.
Based on Eq. /H208491/H20850, micromagnetic simulation was performed
with an OOMMF simulator19to which we made slight modi-
fications. Material parameters were Ms=8/H11003105A/m and
Ku=0 for IMA strips, Ms=6/H11003105A/m and Ku=4
/H11003105J/m3for PMA strips, and A=1.0/H1100310−11J/m and /H9251
=0.02 for both types of strips. The width, length, and thick-
ness of the strips were mainly 120 nm, 12 /H9262m, and 5 nm,
respectively. We chose a grid size of 4 /H110034n m2. This grid
size was checked in advance to generate few errors of criticalcurrent density. Stable domain wall structure was first calcu-lated without any external field or current, after that currentwas applied. Only a steady current with zero rise time wasconsidered for use.
III. RESULTS AND DISCUSSION
Figure 1shows the relationship between DWM velocity
and current density for both IMA and PMA strips. In thefigure, one can easily see the qualitative similarity and quan-titative dissimilarity between them, i.e., shapes of corre-sponding curves appear similar, while critical current densi-ties are different orders of magnitude. For example, criticalcurrent density uis 600 m /s for IMA strip and 42 m /s for
PMA strip in the case
/H9252=0. We then focused on this critical
current density of /H9252=0, which corresponds to the threshold
current density for DWM in the adiabatic approximation.
Figure 2shows the dependence of the threshold current
density on the thickness. This dependence are apparently op-posite for PMA and IMA strips, i.e., threshold current densitydecreases with reduced strip thickness for the former, whileit increases for the latter.
a/H20850Electronic mail: s-fukami@bu.jp.nec.comJOURNAL OF APPLIED PHYSICS 103, 07E718 /H208492008 /H20850
0021-8979/2008/103 /H208497/H20850/07E718/3/$23.00 © 2008 American Institute of Physics 103 , 07E718-1
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132.174.255.116 On: Thu, 27 Nov 2014 17:56:17Here, we analyze what these differences originate from.
In Ref. 12, the threshold current density of /H9252=0,jth,i se x -
pressed as jth/H11008Kh.a/H9004, where Kh.a.is hard-axis anisotropy and
/H9004is domain wall width. Therefore, we estimated the domain
wall width and hard-axis anisotropy for both IMA and PMAstrips.
Domain wall widths were obtained by fitting the mag-
netic profiles of calculated stable domain walls to
/H9258
=2 tan−1exp/H20849x//H9004/H20850.20The derived values are shown in Fig. 3.
The domain wall widths of PMA strips are several times
narrower than those of IMA strips. This difference is causedby the difference of anisotropy constant K
u, since /H9004
=/H9266/H20849A/Ku/H208501/2.
Next, we estimated the hard-axis anisotropy by comput-
ing the Walker breakdown field, HW=/H9251HK/2, using field-
driven simulation, where HKis the hard-axis anisotropy field
with HK=2Kh.a. /Ms.21Figure 4shows the thickness depen-
dence of the Walker breakdown field HWand the correspond-
ing hard-axis anisotropy field HK. In the figure, HKof PMA
strips are smaller than those of IMA strips in the thin region.Additionally, the opposite dependence on thicknesses forPMA and IMA strips, which was seen in Fig. 2, is clearly
shown here as well. These differences can be explained byconsidering the influences of magnetic charges induced bythe rotation of domain wall magnetization.
22
From the obtained calculation results, let us consider the
quantitative differences in DWM between PMA and IMAstrips. In the adiabatic case, spin-polarized current inducesthe rotation of magnetization in the domain wall to the hard-axis direction, which results in the pinning force of DWM.Above the threshold, spin-transfer torque overcomes thispinning effect and steady DWM occurs. These sequences arecommon for both IMA and PMA strips, therefore, they be-came qualitatively similar. It should be noted here that thespin-transfer torque, represented by the domain wall width,became larger for narrower domain walls and the pinningeffect, represented by the hard-axis anisotropy, becameweaker for smaller hard-axis anisotropy. Thus, we can under-stand the quantitative discrepancy between PMA and IMAstrips. By substituting the obtained domain wall widths /H20849Fig.
3/H20850and hard-axis anisotropy fields /H20849Fig. 4/H20850into j
th/H11008Kh.a./H9004,
deriving the ratios of threshold current density between PMAand IMA, u
IMA /uPMA, and comparing them with simulated
results in Fig. 2, theoretical ratios of uIMA /uPMAwere 30%–
60% larger than simulated ones. This disagreement probablyoriginates from the one-dimensional approximation in thetheory where two-dimensional degree of freedom was notconsidered. We concluded here that PMA strips show lowerdrive current density than IMA ones in the adiabatic case dueto the differences in domain wall width and hard-axis aniso-tropy. Furthermore, the thickness dependence of threshold
FIG. 1. /H20849Color online /H20850Relations between DWM velocity vand current
density ufor various /H9252values. /H20849a/H20850IMA and /H20849b/H20850PMA strips.
FIG. 2. /H20849Color online /H20850Thickness tdependence of threshold current density
uthin the adiabatic case.
FIG. 3. /H20849Color online /H20850Domain wall width /H9004derived from the profiles of
stable state, as a function of strip thickness t.
FIG. 4. /H20849Color online /H20850Walker breakdown field HWand the corresponding
hard-axis anisotropy field HKas a function of strip thickness t.HWwas
obtained from field driven simulation.07E718-2 Fukami et al. J. Appl. Phys. 103 , 07E718 /H208492008 /H20850
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132.174.255.116 On: Thu, 27 Nov 2014 17:56:17current density shown in Fig. 2can be attributed to the thick-
ness dependence of the hard-axis anisotropy /H20849Fig.4/H20850.
We now discuss the DWM from an artificial pinning site,
taking the nonadiabatic effect into consideration. For actualapplications, it is desirable to have small threshold currentdensity and large threshold field required to depin from thepinning site. It is known that threshold current density be-comes zero when the nonadiabatic term /H20849
/H9252term /H20850is taken
into account /H20849see Fig. 1/H20850, and it becomes finite again in a
system with threshold field.14Figure 5shows the threshold
field and current density of the domain wall pinned by anotch. In the figure, threshold current density of the PMAstrips is much smaller than that of IMA strips, although thethreshold field of the PMA strips was much larger than thatof the IMA ones. Specifically, very thin PMA strips showsignificantly small threshold current density of u/H1101110 m /s,
which corresponds to j/H110112/H1100310
11A/m2andI/H110110.05 mA, if
P=0.5 is assumed. We did calculation at various notch
depths, resulting in different threshold fields, and found thatthe threshold current density was independent on the thresh-old field in PMA strips, while it increased when the thresholdfield increased in IMA strips. We also found that the
/H9252value
does not affect the threshold current density of the PMAstrips, in contrast to the IMA strips case.
14We revealed the
origin of these differences by investigating magnetic struc-ture change of domain wall during DWM as follows. InPMA strips, the domain wall was displaced with breakdownby adiabatic term due to the small critical current density asmentioned above. On the other hand, in IMA strips, it wasdisplaced without breakdown by nonadiabatic effect, in thesame way as in field-driven DWM. This difference in drivingmechanism surely results in the difference in the relationshipbetween threshold current density and threshold field. Wethus concluded that nonadiabatic term drive is not suitablefor obtaining both low critical current density and high fieldrobustness, which could be possible with adiabatic termdrive, and, therefore, PMA is preferable to IMA for applica-tion to devices such as MRAMs.
Finally, we describe other factors which were not con-
sidered in this study but might be relevant to some extent.First, the approximation of the spin polarization of the cur-rent being along the local magnetization might be no longercorrect for PMA strips, in which domain wall width isaround 10 nm. Furthermore, Tatara and Kohno pointed out
that in narrow domain walls the current exerts a force on thewalls due to the reflection of conduction electrons.
12It might
be necessary to incorporate these effects into fundamentalequations.
IV. SUMMARY
We have investigated the current driven DWM in PMA
strips by using micromagnetic simulation and compared itwith DWM in IMA strips. It was found that the thresholdcurrent density of PMA strips in the adiabatic approximationwas much smaller than that of IMA strips, and that it reducedwith thickness reduction. These differences originate fromdifferences in the domain wall width and the hard-axis an-isotropy. Calculations on systems with a pinning site re-vealed that the threshold current density of PMA strips re-quired to depin the domain wall was quite small although thethreshold field was sufficiently large relative to those of IMAstrips.
The authors would like to thank Professor Y . Nakatani,
Professor T. Ono, and Professor G. Tatara for thoughtful dis-cussion. A portion of this work was supported by NEDO.
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FIG. 5. /H20849Color online /H20850Threshold field Hthand threshold current density uth
as a function of strip thickness t. The inset is the strip patterns used for the
calculation. Strip dimensions were 120 /H11003720 nm2and notch depth was
12 nm. /H9252was set to 0.04.07E718-3 Fukami et al. J. Appl. Phys. 103 , 07E718 /H208492008 /H20850
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132.174.255.116 On: Thu, 27 Nov 2014 17:56:17 |
1.3687726.pdf | In situ multifrequency ferromagnetic resonance and x-ray magnetic circular
dichroism investigations on Fe/GaAs(110): Enhanced g-factor
F. M. Römer, M. Möller, K. Wagner, L. Gathmann, R. Narkowicz et al.
Citation: Appl. Phys. Lett. 100, 092402 (2012); doi: 10.1063/1.3687726
View online: http://dx.doi.org/10.1063/1.3687726
View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v100/i9
Published by the American Institute of Physics.
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Downloaded 05 Jun 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsIn situ multifrequency ferromagnetic resonance and x-ray magnetic circular
dichroism investigations on Fe/GaAs(110): Enhanced g-factor
F. M . R o ¨mer,1,a)M. Mo ¨ller,1K. Wagner,1L. Gathmann,1R. Narkowicz,2H. Za¨hres,1
B. R. Salles,3P . Torelli,3R. Meckenstock,1J. Lindner,1and M. Farle1
1Faculty of Physics and Center for Nanointegration (CeNIDE), University of Duisburg-Essen, Lotharstr. 1,
47057 Duisburg, Germany
2Department of Physics, Technical University of Dortmund, Otto-Hahn-Str. 4, 44227 Dortmund, Germany
3Laboratorio TASC, IOM-CNR, S.S. 14 km 163.5, Basovizza, I-34149 Trieste, Italy
(Received 22 December 2011; accepted 3 February 2012; published online 1 March 2012)
We determined the magnetic anisotropy energy and g-factor of an uncapped 10 nm thick
Fe/GaAs(110) film using a setup that allows frequency (1.5–26.5 GHz) as well as angular
dependent ferromagnetic resonance measurements under ultrahigh vacuum conditions. The g-factorg¼2:6160:1 is unusually high at room temperature and can be interpreted as the result of an
increased orbital moment due to strain. This inter pretation is supported by more surface sensitive
x-ray magnetic circular dichroism measurements which yield g ¼2:2160:02 measured at
remanence. The difference in gmay be the result of magnetic field dependent magnetostriction which
influences the orbital moment.
VC2012 American Institute of Physics . [doi: 10.1063/1.3687726 ]
Beside Brillouin light scattering (BLS), ferromagnetic
resonance (FMR) has been established as one of the most
powerful techniques to investigate static and dynamic mag-
netic properties and is mostly treated in the framework of theLandau-Lifshitz formalism.
1–3Modern topics of spintronics,
e.g., spin torque driven processes,4are as well addressed5as
magnetic anisotropies. The latter has become a standard fieldof application for FMR so that nowadays almost every mag-
netic thin film system has been studied and successfully
characterized
6by FMR. It has been demonstrated that FMR
measurements can be performed under ultra high vacuum
(UHV) conditions. However, only cavities at fixed micro-
wave (mw) frequencies (e.g., 4 GHz, 10 GHz, and 35 GHz)have been used so far.
2Especially for quantitative measure-
ments of the g-factor and the unambiguous identification of
damping mechanisms, a quasi continuous frequency ( f) de-
pendent measurement is required to identify different relaxa-
tion channels of the precessing magnetization1,7and to
identify the effects of exchange coupling effects or an inho-mogeneous magnetization.
6The g-factor is a measure of the
ratio of orbital ( ll) to spin ( ls) magnetic moment and can be
compared to results from x-ray magnetic circular dichroism(XMCD) which allows for a direct determination of these
moments.
In air, broad continuous frequency bands are often cov-
ered using a microwave (mw) stripline structure and a vector
network analyser.
8However, the sensitivity is inferior to the
one achieved by cavity measurements and has not been usedin UHV. Another method for fdependent FMR measure-
ments is to use tunable cavities,
9which are limited to a small
bandwidth of about 1 to 3 GHz at 10 GHz, for example.
In the following, we present a setup which employs an
alternative approach to detect FMR and demonstrate its per-
formance by discussing results on the resonance position of f
dependent FMR measured on uncapped epitaxial 10 nm Fe/
GaAs(110) under UHV conditions.The experimental FMR setup is described in Fig. 1.F e
films are deposited at room temperature as described in Ref.
10. Using “intensity vs. electron energy” low energy electron
diffraction (IV-LEED), an out of plane lattice parameter of0.2889 nm with an expansion of 0 :860:3%compared to the
bulk value of 0.2866 nm
11was determined. The sample is
placed at /C250:3 mm distance to the FMR probe at the center
of both the glass part and the Helmholtz coil pair for external
field modulation. A conventional electromagnet provides
fields of up to 1.3 T. The FMR probe is a semi rigid mwcable (SRMC), whose one end is short-circuited; the micro-
wave current in the short induces a high fmagnetic field in
the film plane perpendicular to B
ext.
Compared to typical coplanar waveguide measurements,
the length of the short is always smaller than the wavelength
FIG. 1. (Color online) Schematic setup of the in situ multifrequency
approach. Thick (green) line: digital communication, dashed (red) line: ana-
logue signal, thin (orange) line: mw signal. A glass-to-metal adapter (G) is
mounted to the main UHV chamber, while an electromagnet (B) can be posi-
tioned around it. A pair of coils (M) is mounted to modulate the external
field. A mw synthesizer is connected via a circulator (C) to a mw feed-through. The reflected mw power can be measured at the Schottky diode
(SD) connected to a lock-in amplifier. The optimum measurement position
is marked by the black square near (P). The sample is fixed on the sample
holder (S). More details in Fig. 2.
a)Electronic mail: florian.roemer@uni-due.de.
0003-6951/2012/100(9)/092402/4/$30.00 VC2012 American Institute of Physics 100, 092402-1APPLIED PHYSICS LETTERS 100, 092402 (2012)
Downloaded 05 Jun 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsof the applied mw of up to 26.5 GHz (see Fig. 2). The sample
is located in a near field region of the short, in the area of
most homogeneous distribution of the microwave magneticfield. This is essential for a homogeneous excitation of the
magnetisation, as it may otherwise change the linewidth.
The field distribution was determined using the micro-
wave simulator software HFSS V.11 (
ANSOFT ). A snapshot of
the standing wave inside the microwave cable at f¼12 GHz
with 1 W of microwave power is shown in Fig. 2. Details are
provided in the figure caption. Power from a Rohde &
Schwarz SMR 40 microwave generator is fed into circulators
covering the frequency range of 1.5 to 26.5 GHz. The powerreflected from the short is detected using a broadband
Schottky diode. The UHV feedthrough is rated from DC to
18 GHz only and causes distortions to the FMR signal above18 GHz.
XMCD measurements were performed at the Advanced
Photoelectric-effect Experiments (APE) beamline at theELETTRA synchrotron facility in Trieste in remanence on
in-situ grown samples similarly prepared as those used for
FMR. The XMCD spectra were recorded in total electronyield mode.
12
The results of the in situ angular and fdependent meas-
urements are shown in Fig. 3(for a description see the figure
caption) and Fig. 4, respectively. The simultaneously fitteddependencies yield a crystalline anisotropy K4¼40:5k J=m3,
an uniaxial out of plane anisotropy K2?¼972 kJ =m3,a n da n
uniaxial in plane anisotropy K2k¼17:1k J=m3with the easy
direction parallel to [001].
The frequency dependence (Fig. 4) shows measurements
along the easy and intermediate direction of the Fe film. The
signal of the upper branch starts off at low external field at10 GHz and rises with a small curvature up to 26 GHz at
0.32 T and the magnetization is aligned parallel to the exter-
nal magnetic field. These resonances are the aligned or col-
linear modes. The same applies for the signals which are
shifted parallel towards higher external fields and correspondto measurements along the intermediate direction. The inset
shows the spectrum of the intermediate direction at
8.692 GHz. Ex situ hysteresis measurements confirmed that
forB
ext>0:02T, Mis in the fully saturated state.
An effect which is special for (110)-oriented thin films
is that the lower branch for the intermediate direction sud-denly ends at an external magnetic field l
0H¼Bext>0.13
Atf/C255 GHz and B /C250:02 T, a second resonance appears
as the external field is reduced. Here, the resonance conditionis fulfilled even for M not being parallel to the external field
that is the nonaligned or noncollinear mode. At the point
where the aligned mode vanishes and the nonaligned oneappears, the magnetisation suddenly switches to another
direction, as confirmed by hysteresis measurements. Without
this sudden change of the direction of M, the mode wouldshow the dispersion of the branch VI (red: g¼2.61, cyan:
g¼2.09). The blue and green lines (at position V), which
look like being mirrored at the abscissa, are solutions forwhich Mwould be oriented antiparallel to B. In our experi-
ment, the magnetisation does change its direction at about
25 mT, so that the modes V and VI at lower fields cannot beobserved.
To determine the different anisotropy constants like
crystalline cubic anisotropy K
4, uniaxial out of plane anisot-
ropy K2?and uniaxial in plane anisotropy K2k, one has to
evaluate the free energy which for a cubic system in [110]-
orientation reads6
F¼þK4=4/C1cos4hþsin4h/C1cos4/þsin22//C0/C1 /C2
þsin22h/C1sin2//C01=2/C1cos2//C0/C1 /C3
þK2ksin2h/C1cos2//C0d ðÞ þ K2?sin2h
/C0l0M2=2/C1cos2h/C0~M~Bext; (1)
Mis the magnetisation, /is the in plane angle of Mmeas-
ured with respect to the [11
/C220] direction (in Ref. 6[100],
respectively), /Bis the in plane angle of the external field,
anddis the angle between cubic and uniaxial direction. For
d=9 0/C14andK2k<0, the easy direction of K2kis parallel to
[11
/C220]. Equation (1)has to be inserted into the resonance con-
dition as described in Ref. 6, where Fxy¼@2F=@x@ywith
ðx;y¼h;uÞ
x
c¼1
MSjsinh0jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðFhhFuu/C0F2
huÞq
;c¼glB
/C22h: (2)
h/C14denotes the polar equilibrium angle of M. The numerical
solution using the bulk magnetisation ( M¼1:7M A =m) and
FIG. 2. (Color online) Simulation of the mw field distribution of a short-
circuited semi rigid cable for 12 GHz at 1 W of mw power. Color/gray scale
of magnetic field amplitudes is indicated on the right hand side. The short is
positioned between inner conductor and outer cladding of the cable. For a
front view, see Fig. 1.
FIG. 3. (Color online) Angular dependent measurement at 12.878 GHz of
10 nm Fe/GaAs(110). The resonance field is plotted versus the in plane angle
of the external magnetic field. The grayscale reflects the normalized signal
amplitude of the measurement; the red (solid) line shows the best fit to the
data (green dots) yielding the values in Table I. On the upper x-scale, the
crystallographic directions of the substrate are indicated. The graph reflects
the expected behaviour for cubic and uniaxial anisotropy constants of com-parable magnitude. [001] is the “easy,” [1
/C2210] “intermediate” and [1
/C2211] the
“hard” direction.092402-2 Ro ¨mer et al. Appl. Phys. Lett. 100, 092402 (2012)
Downloaded 05 Jun 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionslB¼Bohr magneton yields the theoretical resonance field.
The anisotropy values and gare varied until the best fit to the
experimental data is obtained. This is done simultaneously
for all the angular and fdependent data. The result of this
procedure is listed in Table Iand presented as coloured lines
in Figs. 3and4.
We find a large value of g¼2.61, which is uncommon
and extremely high for ferromagnetic Fe. To verify the valid-ity, green (II) and cyan (IV) indicate the fdependent simula-
tion data which result from the best angular dependent fit
using g ¼2.09, i.e., the bulk value.
14A reasonable fit in
angular and fdomain was only feasible with the increased
g-factor. The high value comes as a surprise and cannot be
attributed to technical difficulties of the setup, since othersamples measured in with our new apparatus showed the
same results as measured in conventional setups. For Fe
3Si/
MgO(001), for example, a value of g¼2.1 was obtained
which is in good agreement with literature.15
A g-factor of 2.09 of capped Fe/GaAs(001) is well
known.16,17Interestingly, a larger g¼2.26 was reported18
for uncapped 1.15 nm Fe/GaAs(001) indicating magnetic dif-
ferences between capped and uncapped Fe on GaAs(001). In
Fe/GaAs(110), we observe a large uniaxial in plane anisot-ropy K2kwhich is not present in bulk and indicates uniaxial
distortions. This deviation from cubic symmetry may explaina reduced quenching of the orbital moment l
l. Using
ll=ls¼ðg/C02Þ=2, the value of ll=lsis about 0.3.
For a better understanding and confirmation of the un-
usual g-factor, we performed XMCD measurements at rema-
nence.19The resulting g¼2/C1ð1þll=lsÞ¼2:2160:02
supports the increased value determined by FMR. Using thesum rules and assuming 90% polarization of the x-rays,
one calculates for a 10 nm film l
l¼0:2260:05 and
ls¼2:0860:05, where lsis equal to the one of bulk Fe.20
The difference between the FMR and XMCD results
might be due to the higher magnetic field in FMR, where
magnetostrictive effects21causing field dependent anisotropy
parameters22,23may complicate the FMR analysis or even
change the g-factor. Also, the surface layers dominantly
probed by the surface sensitive XMCD in total electron yieldmode are most likely less strained yielding the reduced
g¼2.21 (smaller l
l). From scanning tunneling microscopy
(STM), we know that Fe on GaAs(110) forms islands ofabout 7 nm diameter for a 6 nm thick film, with a roughness
of/C2510%. A capping layer will affect the magnetostrictive
properties of these surface islands by reducing their motionaldegrees of freedom or modifying the electronic structure by
the formation of alloys. Consequently, capped and uncapped
(110) Fe films might show different g-values.
In situ BLS measurements on 2 nm Fe/GaAs(110), using
comparable preparation conditions
24found the [11
/C220] direc-
tion as the easy axis of magnetisation due to the well-knownTABLE I. Best fit parameters of the simulation.
K4½kJ=m3/C138 K2jj½kJ=m3/C138 K2?½kJ=m3/C138 g
40:5621 7 :161 972 6100 2 :6160:1
FIG. 4. (Color online) Frequency dependent FMR of 10 nm Fe/GaAs(110) in the range of 1.5–26.5 GHz as a function of external magnetic field along the
[001] and [1
/C2210] direction. The grayscale indicates the signal amplitude. Both measurements were performed separately. Only the significant region around the
signals of [001] is shown here, and the background is the full data of [1
/C2210]. Dashed lines indicate different simulation parameters marked by (I) to (IV), where
those corresponding to blue (I) and red (III) are listed in Table I. At frequencies >18GHz, reflections at the feedthrough change the mw signals’ phase. Each
spectrum is measured at fixed f with decreasing field and then normalized to 61. For f <4 GHz, where there is no FMR-signal, just the noise is seen.092402-3 Ro ¨mer et al. Appl. Phys. Lett. 100, 092402 (2012)
Downloaded 05 Jun 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsthickness dependent in-plane reorientation.25The BLS fre-
quency dependence24yields g¼2.1. For our data, a fit with
g¼2.1 results in a systematic underestimation of the data
points as shown in Figure 4. A fit with g>2:1 leads to a
much better agreement. We finally note that previous FMR
measurements on capped Fe/GaAs(110) thin films found
bulk like g-factors ( g¼2.09). Re-evaluating the data of an
Al capped Fe film13,26deposited at p ¼10/C08mbar with our
fit routine, we also found g¼2.09, indicating that these
layers were in a structurally relaxed cubic state.
In summary, we have shown that our setup is suitable
for investigating magnetic properties in UHV and to investi-gate the influence of capping layers on the magnetic proper-
ties of ferromagnetic monolayers in situ . As an example to
emphasize the importance of combined angular and fre-quency dependent FMR measurements, we have identified a
surprisingly high value of the g-factor for uncapped Fe/
GaAs(110), which may result from the lattice deformationmeasured by IV-LEED. Surface sensitive XMCD measure-
ments support an increased g-factor.
We would like to thank the Deutsche Forschungsge-
meinschaft (DFG) SFB491 for financial support and C. Backfor establishing the contact between coauthors.
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Downloaded 05 Jun 2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions |
1.4922858.pdf | Comparisons of characteristic timescales and approximate models for Brownian
magnetic nanoparticle rotations
Daniel B. Reeves and John B. Weaver
Citation: Journal of Applied Physics 117, 233905 (2015); doi: 10.1063/1.4922858
View online: http://dx.doi.org/10.1063/1.4922858
View Table of Contents: http://aip.scitation.org/toc/jap/117/23
Published by the American Institute of PhysicsComparisons of characteristic timescales and approximate models
for Brownian magnetic nanoparticle rotations
Daniel B. Reevesa)and John B. Weaverb)
Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA
(Received 8 April 2015; accepted 11 June 2015; published online 19 June 2015)
Magnetic nanoparticles are promising tools for a host of therapeutic and diagnostic medical
applications. The dynamics of rotating magnetic nanoparticles in applied magnetic fields dependstrongly on the type and strength of the field applied. There are two possible rotation mechanisms
and the decision for the dominant mechanism is often made by comparing the equilibrium
relaxation times. This is a problem when particles are driven with high-amplitude fields becausethey are not necessarily at equilibrium at all. Instead, it is more appropriate to consider the
“characteristic timescales” that arise in various applied fields. Approximate forms for the character-
istic time of Brownian particle rotations do exist and we show agreement between several analyti-cal and phenomenological-fit models to simulated data from a stochastic Langevin equation
approach. We also compare several approximate models with solutions of the Fokker-Planck equa-
tion to determine their range of validity for general fields and relaxation times. The effective fieldmodel is an excellent approximation, while the linear response solution is only useful for very low
fields and frequencies for realistic Brownian particle rotations.
VC2015 AIP Publishing LLC .
[http://dx.doi.org/10.1063/1.4922858 ]
I. DESCRIBING DRIVEN NANOPARTICLE ROTATIONS
In many magnetic nanoparticle (MNP) applications like
biosensing,1–7hyperthermia,8,9and magnetic particle imag-
ing,10–13nanoparticles are driven to rotate by oscillating
magnetic fields.14Understanding the resulting magnetic par-
ticle dynamics is important to advance these applications. Atypical way to discuss the dynamics is through the timescalesof the nanoparticle rotations.
15–17In particular, we often con-
sider the relaxation time: the timescale for a sample of par-
ticles to return to equilibrium after some perturbation (e.g.,
alignment with a field). Conventional magnetic particles areunderstood to have two rotational mechanisms. The entireparticle can rotate as a rigid body by Brownian rotations,
18
and the particle’s magnetic moment can also rotate internallydue to restructuring of electronic states in N /C19eel rotation.
19,20
The equilibrium relaxation time is different for each mecha-
nism and depends on many parameters.21,22However,
because most applications involve magnetically excited par-
ticles, it is more important to examine non-equilibrium time-
scales determining the speed of movements in varyingdriving fields—these timescales can be very different fromthe relaxation time. One only needs to imagine that in astronger field, the particles will align faster to see why this is
true. We will hence refer to those non-equilibrium timescales
as the “characteristic times” of the rotations.
In reality, the possibility for N /C19eel rotations complicates
the matter and it is important to understand which mecha-nism is dominant for chosen nanoparticles.
23,24This is an
open problem because these processes will in general not be
decoupled. If the processes did truly happen independently(in parallel) the more prevalent relaxation mechanism would
be that with the shorter relaxation time; but, because theequilibrium relaxation time is not a precise metric, simplycomparing these times will not immediately determine thedominant rotation method.
25–27The notion then of purely
N/C19eel or purely Brownian particles is unrealistic, particularly
in nanoparticles with a wider size distribution.
It is possible to create a fully general model for the time
dynamics of magnetically driven magnetic particles includ-ing varying rotation methods as well as the specific condi-tions the particles experience in various applications.
23Two
main formalisms exist: The Langevin equation formalismdescribes a single particle’s dynamics with a stochastic dif-ferential equation that can be solved repeatedly to describethe average properties of an ensemble of particles. TheFokker-Planck formulation instead describes the time evolu-tion of the probability distribution of a sample of magnetiza-tions, so that ensemble averages can be found at any timefrom the distribution function. In this work, concentrating onBrownian particle rotations that are used in biosensing appli-cations, we solve both types of equations, using them toassess various models for nanoparticle relaxation times andcharacteristic times as well as approximate models for timedynamics.
II. THE FOKKER-PLANCK EQUATION FOR BROWNIAN
NANOPARTICLE ROTATIONS AND ASSOCIATED
APPROXIMATE MODELS
The Fokker-Planck equation (FPE) governs the distribu-
tion function Wðh;/;tÞof an ensemble of particle magnet-
izations. It can be derived heuristically from a continuityequation with an additional diffusion term.
19Each nanopar-
ticle’s magnetization is imagined to be a vector moving ona)Electronic mail: dbr@Dartmouth.edu
b)Present address: Radiology Department, Geisel School of Medicine,
Hanover, New Hampshire 03755, USA.
0021-8979/2015/117(23)/233905/7/$30.00 VC2015 AIP Publishing LLC 117, 233905-1JOURNAL OF APPLIED PHYSICS 117, 233905 (2015)
the unit sphere, and the diffusion is parameterized by D. The
general FPE is then written
@W
@t¼r/C1 Dr/C0dm
dt/C20/C21
W; (1)
where the magnetization time dynamics are given by different
differential equations for Brownian and N /C19eel rotation. We
focus on the Brownian case because fewer assumptions must
be made (e.g., constraining the anisotropy axis). The distribu-
tion is used to determine magnetization statistics usingthe definition of the probability momentsÐm
jWðh;/;tÞdX
¼hmjðtÞi. The magnetization dynamics of Brownian particle
rotations are dominated by torques caused by an applied field
and the viscous drag from the fluid. Several papers go throughderivations for the equations of motion.
15,16,22,28We choose a
compact form for the equation
dm
dt¼m/C2n ðÞ /C2 m
2sB; (2)
in terms of the equilibrium Brownian relaxation time sB
sB¼3gVh
kBT; (3)
determined by the suspension viscosity g, the hydrodynamic
volume of the particle Vh, and the local temperature Twith
Boltzmann’s constant kB. The unitless magnetic field nis a
vector quantity
n¼lH
kBT; (4)
incorporating the nanoparticle’s magnetic moment land
an applied field H. The magnetization mis normalized
and therefore unitless. Replacing the velocity of the magnet-
ization in Eq. (1)with that from Eq. (2)and assuming a
Maxwell-Boltzmann distribution at equilibrium (when
@W
@t¼0) we find D¼1/2sBand write
@W
@t¼1
2sBr/C1 r/C0 nþmm /C1nðÞ ½/C138 W; (5)
for which a general solution is not currently analytically pos-
sible. Since many applications use a single oscillating field(see, for example, magnetic particle imaging
11or magnetic
nanoparticle spectroscopy4–6), we choose n!nðtÞ^zand
simplify the FPE to only depend on the polar angle and timeW(h,t). Writing out letting x¼coshthe 1-D FPE is written
@W
@t¼1
2sB@
@x1/C0x2 ðÞ@W
@x/C0ntðÞW/C18/C19/C20/C21
: (6)
A solution to this equation is possible by expanding with
Legendre polynomials.17,23,29
A. Linear response
Following Debye,30it is possible to obtain an analytical
solution to the FPE assuming a small amplitude oscillatingfield n¼n
0eixt. In the small amplitude case, it is fair toassume the distribution function is linear in x, with the gen-
eral form Wlin¼AþBx. Inputting Wlininto Eq. (6)leads to
the average normalized magnetization in the direction of theoscillating field
hmi¼n0
3eixt
1þixsB: (7)
The susceptibility or slope of this equation is not realis-
tic for larger fields when n0>3 because the magnetization is
normalized to be on the unit sphere. The results are slightlybetter (see Fig. 5) if we use
hmi¼L n
0ðÞeixt
1þixsB; (8)
where for small fields, the Taylor expansion of the Langevin
function provides the equivalent susceptibility including thefactor of 1/3 and for large fields, the magnetization does notgrow above unity.
B. Moment equations from the FPE
Moment equations are found from Eq. (6)by multiplica-
tion with powers of xand subsequent integration over x. The
normalization condition defined by the probability distribu-tion W(x,t) and the definition of the statistical moments are
used. For example, after multiplying by xand two steps of
integration by parts we find
2s
B@hxi
@t¼n/C02hxi/C0nhx2i: (9)
The dynamics of the mean thus depend on the second
moment and a similar procedure gives the second moment31
sB@hx2i
@t¼1þnhxi/C03hx2i/C0nhx3i; (10)
and so on. An infinite series of coupled differential equations
emerge that can be truncated by a clever closure techniquetermed the “effective field” method.
32
C. Truncating the moment equation
The moment equation is truncated by assuming a distri-
bution function that is similar to the equilibrium distributionexcept having an “effective field” n
e. The field is free to be
slightly different than the applied field.31,32The advantages
of the effective field model are the simpler more intuitiveform and the ease of implementation relative to the stochas-tic or FPE methods. It is clear that the model describes an ex-ponential decay when the applied field is zero, and when theeffective field is equal to the actual field, the mean magnet-ization does not change. This is equivalent to assuming equi-librium. We note that the calculational simplicity only holdsfor 1D modeling. Each moment can be computed from theeffective normalized distribution function
W
e¼ne
2 sinh neexne: (11)233905-2 D. B. Reeves and J. B. Weaver J. Appl. Phys. 117, 233905 (2015)The mean and the second moments are functions of the
Langevin function with respect to the effective field
hxie¼L ð neÞ¼cothne/C01=ne (12)
and
hx2ie¼1/C02
neLneðÞ: (13)
Using the first moment equation (Eq. (9)), we now have a
differential equation for the mean magnetization purely in
terms of the Langevin function and the effective field
d
dthxie¼/C0hxie
sB1/C0n
nehxie/C0/C1 !
; (14)
where the effective field neis a function of the mean value.
To solve this implicit differential equation, the effective field
at each time can be found by inverting the Langevin functionfor a given magnetization using a Pad /C19e approximant.
33
D. Analytical characteristic timescales
Martensyuk, Raikher, and Shliomis (MRS) develop an
approximation to the characteristic timescale for a generaleffective field that is a small perturbation on the equilibriumfield.
32Here, we show the characteristic time to begin
aligned in the perpendicular direction and then evolve toalign with the field, so that we may compare with simulateddata. The perpendicular characteristic time is
sMRS
sB¼2LnðÞ
n/C0L nðÞ: (15)
And from this, low- and high-amplitude field approximations
to the perpendicular characteristic time can be easily written
slow
sB¼1/C01
10n2;shigh
sB¼2
n: (16)
In practice, we find that for fields of n>5, the large field
approximation suffices. We also summarize the characteris-tic timescales in Table I.
E. Fully general N /C19eel relaxation time
While this paper is designed to focus on Brownian nano-
particle rotations, it is important to consider the fully generalexpressions for N /C19eel relaxation times, which are not always
used in their complete forms. In particular, the N /C19eel event
time s0is sometimes determined solely from experiments,
but, in principle, can be broken down into several other pa-rameters for more specific measurements as
s
0¼l
2kBTc1þa2
a; (17)
with the Gilbert damping parameter a, the gyromagnetic ra-
tioc, and the magnetic moment l.21
Depending on the unitless anisotropy constant
r¼KVc/kBT, where Kis the anisotropy constant and Vcis
the magnetic core volume, two approximations exist forthe equilibrium N /C19eel relaxation time
s
N¼s0r1/C02
5rþ48
875r2/C18/C19/C01
ifr<1
s0
2ffiffiffiffiffip
r3r
exprðÞ ifr/C211:8
>>><
>>>:(18)
N/C19eel rotations are more likely in smaller single domain
nanoparticles where the energy scale to reverse the magnet-
ization is comparable to the thermal energy (i.e., r/C251a si n
superparamagnetic nanoparticles), or in large magnetic fieldswhere n>rN/C19eel rotations are also certainly possible.
Especially, in poly-disperse samples, it is less likely thatboth nanoparticle rotation mechanisms are not simultane-
ously occurring.
34
III. SIMULATION RESULTS
A. Comparison of characteristic timescales
A stochastic Langevin equation can be developed from
the magnetization equation (2)in different fashions, though
it is important to note that there is no completely generalway to add in thermal fluctuations. Typically, a Gaussian
fluctuating term is appended to the differential equation
defined by
hktðÞi¼0;hk
itðÞkjt0ðÞi¼dijdt/C0t0ðÞ
sB; (19)
with i,j2x,y,z. Simulations of the Brownian Langevin
equation (Eq. (2)with additional stochastic torques) can be
completed15,16,28to examine the characteristic time and com-
pare with the analytical expressions.
Our first result is intuitively obvious. If particles are ini-
tiated in a state completely aligned with some axis, and afield is turned on perpendicularly, the magnetizations align
with the field. Several example magnetizations simulated
with averages over 10
4particles using an Euler-Marayuma
integration scheme for the stochastic differential equationare shown in Fig. 1. They illustrate the decreasing character-
istic time for increasing field strength as n¼0!30. The
large changes in the dynamics indicate that it is incorrect touse the Brownian relaxation time to describe rotations.
The alignment to the field perpendicular to the original
state results in magnetization curves that can be fitted withTABLE I. Summary of characteristic timescales with descriptions.
Abbreviation Expression Description
sB3gVh
kBTEquilibrium relaxation time18
sMRS2LðnÞ
n/C0L ð nÞsB Timescale to align to a perpendicularly
applied static field of amplitude
n(Ref. 32)
sYE1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ0:21n2p sBPhenomenological fit to FPE simulations
of perpendicularly applied static field
of amplitude n(Ref. 29)233905-3 D. B. Reeves and J. B. Weaver J. Appl. Phys. 117, 233905 (2015)an exponential of the form hmzi/1/C0e/C0t=sc. Thus, we can
extract the approximate characteristic time for each appliedfield strength though for strong enough fields the exponential
form even breaks down. Yoshida and Enpuku also found the
characteristic time using FPE simulations.
29From their
simulated data, they developed a phenomenological fit to the
relationship between characteristic time and field strength as
sYE¼sBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ0:21n2p : (20)
The YE form is a good approximation and in the low- and
high-amplitude field limits approaches the analytical forms
from the effective field characteristic times Eq. (16). The
values of the characteristic time (with respect to the appliedfield) from our simulations agree with both the form of s
YE
and the approximate characteristic time sMRS. This is shown
in Fig. 2.
B. Validity regimes between models in oscillating
fields
When the applied field is oscillatory n¼n0cosxt,a si n
many applications, we can compare the various model
approximations at different field strengths n0, frequencies
f¼x/2p, and relaxation times sB. It has been shown that a
complete description of the dynamics using the FPE can be
parameterized by the combination variable fsB.17
To connect with biosensing experiments, we use typical
values of the relaxation time (500 ls for 100 nm particles at
room temperature and water viscosity), fields of 1 kHz then
lead to fsB/C250.5 and moments of 70 emu/g, and fields of
10 mT/ l0lead to n0/C2510. As mentioned previously, when
the unitless field is greater than the unitless anisotropy
(n0>r), however, N /C19eel rotations are expected and thedynamics are more complicated. Typical magnetite nanopar-
ticles may have anisotropies of the order of 1 kJ/m3and
10 nm core radii leading to r/C2510 as well.13,35
In principle, the Langevin equation and the FPE
approaches should be identical, and averaged solutions of
the Langevin equation15compared to the FPE truncated after
fifty iterations of the series solution moments17lead to van-
ishingly small error. We also note that the advantage of the
stochastic model is that it is amenable to different field geo-
metries and additional physics, while even the FPE is onlysolvable for very specific cases.
23However, if computation
time is a problem, it can certainly be useful to employ the ap-
proximate models.
We calculate the error as the squared error Ebetween
functions atandbt
E¼ð
Tðat/C0btÞ2dt/C20/C211=2
; (21)
where Tis a period of the oscillating field. The error between
the Langevin and FPE approaches is constant over fields andcan be made as small as desired by more averaging, or
shorter time steps, or both. An example is shown in Fig. 3.
We see that neither the field nor the frequency relaxationFIG. 1. Stochastic Langevin equation simulations of normalized mean mag-
netizations that begin aligned in the x-direction (perpendicular) and then
evolve with static fields applied in the z-direction (parallel) at various ampli-
tudes. As the amplitude increases, the particles align faster to the field in the
parallel direction and go to zero faster in the perpendicular direction, so that
magnetic saturation occurs in a fraction of the original relaxation time.FIG. 2. Comparison of the analytical expression (MRS, Eq. (15)) for the
characteristic time with the data fit model (YE, Eq. (20)) and stochastic
Langevin simulations.
FIG. 3. The error of the Langevin equation simulation with respect to the
FPE solution at different magnetic fields and for two fsBcombinations. In
principle, this error can be made arbitrarily small by increased averaging.233905-4 D. B. Reeves and J. B. Weaver J. Appl. Phys. 117, 233905 (2015)time combination effect the error percentage. This is unlike
the other models that have inherent approximations that con-
strain their range of validity. These simulations use 105aver-
ages of the Langevin equation and thirty components of theseries expansion solution to the FPE.
Fig.4qualitatively demonstrates the problems with the
models. For small fields, the Debye model is accurate but forlarger fields the amplitude is too large. The amplitude errorcan be slightly corrected by using the Langevin function to
choose the susceptibility. The purely equilibrium Langevin
function model matches the correct amplitude for largerfields but does not account for phase lags and thus only
begins to approach the correct solution at the large fields.
For practical purposes, we conclude that this model isuseless.
Fig.5quantitatively shows the agreement of the various
approximate models against the FPE solution. The data
include comparisons against the linear response modelEq.(8)with (Debye2) and without (Debye1) the Langevin
function susceptibility (see Sec. II A), the equilibrium
Langevin function model hmi¼L ½ nðtÞ/C138and the effective
field model Eq. (14) over a large range of applied fields and
for several field and relaxation time combinations. The error
is calculated using Eq. (21).
The results show that the effective field of MRS
32works
quite well over a very large range of the variables. The equi-librium Langevin function model begins to be reasonable
only at very high fields and never reaches the accuracy of the
effective field. The challenge of the Debye model is in deter-mining the susceptibility. If this number is chosen as the
value of the Langevin function, the model works very well
for low fields. In fact, it even surpasses the accuracy of theeffective field model when the amplitude is low, and the fre-quency relaxation time combination is high. The altered
Debye model is thus an accurate predictor of dynamics for a
small range of AC susceptibility biosensing.
IV. CONCLUSIONS
We have shown that the useful concept of equilibrium
relaxation times for magnetic nanoparticle rotations can be
extended to include the amplitude of a field driving the par-ticles. These “characteristic times” are a more general wayto describe the timescales of non-equilibrium rotations andare summarized in Table I. Our Langevin equation simula-
tions can be used to calculate the characteristic times, andfor dynamics where the distribution function is close to equi-librium (e.g., the driving field is almost adiabatically rotatingthe nanoparticles) our simulations agree with a numericalapproximation ( s
YE) proposed by Yoshida and Enpuku.29
We also demonstrate that our simulations, as well as those of
Yoshida and Enpuku can be characterized using the analyti-
cal approximation sMRS originally derived by Martsenyuk,FIG. 4. Example magnetizations for the various models (summarized in
Table IIat different fields and fsB¼1 compared to the FPE solution. The
amplitude error of the Debye models and the phase error of the Langevin
model are clear.FIG. 5. Errors calculated with Eq. (21) for the approximate models with
respect to the FPE. Many magnetic field amplitudes are used and the value
offsBis varied from 0.1 to 10.233905-5 D. B. Reeves and J. B. Weaver J. Appl. Phys. 117, 233905 (2015)Raikher, and Shliomis.32This is no mistake as the high- and
low-field approximations to sMRSappear numerically similar
to the equivalent expressions for sYE. These results highlight
the importance of considering the characteristic time instead
of simply the equilibrium relaxation time in describing par-ticles that are driven to rotate as opposed to those freelyrelaxing to equilibrium.
Multiple commonly used approximate forms for oscil-
lating dynamics of particles have also been shown, as well as
the appropriate ranges of validity for each model. The mod-els are collected in Table II. Each model was compared
against the FPE solution simulation. We found that thecombination of the Langevin function for the susceptibilityand the Debye model to account for relaxation (Debye2 in
Table II) is a reasonable approximation (below 1% error)
especially at low fields when fs
Bis large and nis small. The
effective field model (EF in Table II) is consistently accurate
to within 1% for lower field amplitudes and is simpler to cal-culate in a 1D geometry. However, in realistic biosensingapplications that require knowledge of Brownian nanopar-
ticle dynamics, and especially if full 3D simulations are
required, we conclude that it is likely necessary to use thestochastic Langevin equation model because it is easily ame-nable to variable field geometries or additional physics.
ACKNOWLEDGMENTS
This work was supported by NIH-NCI Grant No.
1U54CA151662-01 and the William H. Neukom Institute for
Computational Science.
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Abbreviation hm½nðtÞ/C138i Description and validity range
Langevin L½nðtÞ/C138 ¼ cothnðtÞ/C01=nðtÞ Equilibrium, frequency is low or field is extremely high
Debye1n0
3cosxt
1þðxsBÞ2þxsBsinxt
1þðxsBÞ2 !
Linear response, frequency is high or field is extremely low
Debye2 Lðn0Þcosxt
1þðxsBÞ2þxsBsinxt
1þðxsBÞ2 !
Linear response with Langevin function susceptibility, frequency
is high or field is low
EF /C0ð
dthmðtÞi
sB1þnðtÞ
LiðhmðtÞiÞ/C18/C19
Effective field model, quasi equilibrium conditions, requires
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1.123191.pdf | Microwave characterization of Nd 0.67 Sr 0.33 MnO 3−x thin films for magnetically
tunable filters
J. Wosik, L.-M. Xie, M. Strikovski, J. H. Miller Jr., and P. Przyslupski
Citation: Applied Physics Letters 74, 750 (1999); doi: 10.1063/1.123191
View online: http://dx.doi.org/10.1063/1.123191
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134.129.164.186 On: Sat, 20 Dec 2014 19:43:27Microwave characterization of Nd 0.67Sr0.33MnO32xthin films
for magnetically tunable filters
J. Wosik,a)L.-M. Xie, M. Strikovski, and J. H. Miller, Jr.
Texas Center for Superconductivity at University of Houston, Houston, Texas 77204
P. Przyslupski
Institute of Physics of Polish Academy of Sciences, 00-429 Warszawa, Poland
~Received 24 July 1998; accepted for publication 30 November 1998 !
We report on the microwave properties of Nd 0.67Sr0.33MnO32x~NSMO !and NSMO/YBa 2Cu3O72x
thin film heterostructures. The quality factor ~Q!and center frequency of a 13 GHz shielded
dielectric cavity resonator, with the film comprising one surface, were measured as functions oftemperature and direct current magnetic field. The Qversus field data was theoretically simulated
using the Landau–Lifschitz–Gilbert dynamic permeability equation thus demonstrating that themicrowave losses are determined by the ferromagnetic properties of the films. Our results indicatethat the field-dependent permeability
m(B) of NSMO films holds the potential to create
magnetically tunable microwave devices. © 1999 American Institute of Physics.
@S0003-6951 ~99!03605-0 #
Microwave filters have emerged recently as one of the
most promising high- Tcsuperconductor ~HTS!applications.1
Most filters developed thus far have fixed resonant frequen-
cies. If actively tunable HTS filters were designed and fab-ricated, additional novel applications with greater utilitywould arise. Reports on tunable HTS filters included the uti-lization of either the electric field-dependent permittivity offerroelectric materials
2,3or their magnetic field-dependent
permeability m(B).4,5Ferroelectrically tuned filters, al-
though satisfactory in terms of tunability range and speed,
have a limited quality factor Qdue to high dielectric losses
of the ferroelectric materials. Ferrites or ferrite garnets6used
in passive superconducting microwave devices also providesome degree of tunability, but they are not crystallographi-cally compatible with HTS materials. Moreover, garnet sub-strates require relatively high magnetic fields to tune the de-vices, thus reducing tunability and switching speed.
5
Therefore, there is a need for the development of alternativematerials.
In recent years, Mn-based perovskites exhibiting colos-
sal magnetoresistance phenomena and a paramagnetic-to-ferromagnetic phase transition, have attracted considerableinterest due to their potential for use in device applications.Despite numerous studies of these materials, there has beenonly limited investigation of high frequency properties. Theunderstanding of microwave losses in these perovskites is farfrom complete. It was reported that,
7,8the high-frequency
absorption in direct current ~dc!magnetic field B~below Cu-
rie temperature Tf) is dominated by the monBdependence.
The frequency shift due to the change of mwas also ob-
served at low frequencies in bulk and single crystalmaterials.
9,10For thin films, important for microwave device
applications, microwave absorption data are available but thefrequency dependence on Bis lacking.
Here, we describe investigations of field- and
temperature-dependent microwave properties ofNd
0.67Sr0.33MnO32x~NSMO !thin films, exploring their po-
tentials for tunable HTS microwave filters and related de-vices.
We grew 200-nm-thick NSMO single layers and
NSMO/YBa
2Cu3O72x~YBCO !bilayers on ~100!oriented
LaAlO 3substrates using a high-pressure on-axis dc sputter-
ing deposition system.11Both the single-layer and bilayer
films were deposited at 760°C and 3 mbar oxygen, and wereoxygenated in situat 460°C for 20 min. The YBCO layer in
the NSMO/YBCO bilayers had zero resistance below a criti-cal temperature T
cof 89 K. The individual NSMO layers
exhibited a peak in the temperature-dependent resistivity at atemperature T
pof 210 K. The Curie temperature Tfof the
NSMO films was about 200 K.12
The microwave measurements were carried out using a
13 GHz shielded dielectric cavity which is described in detailelsewhere.
13The measurement system consisted of a HP
8250C vector analyzer, a Janis cold head cryostat, and aVarian electromagnet with a four-quadrant power supply. Inorder to characterize the NSMO film losses, this film wasattached to the top of the dielectric disk, which was sand-wiched between two copper plates. The unloaded quality fac-tor~Q!was measured in one-port or two-port configurations
using software employing the Ginzton–Kajfez method
14with
a refined loss calibration procedure. A dc magnetic field wasapplied parallel to the film surface.
Measurements of the Qfactor and center frequency of
the resonator, as functions of temperature, were first donewith zero magnetic field. Then, the magnetic field depen-dence of the Qwas measured at several selected tempera-
tures ~Fig. 1 !. The insets show the dependence of Qon mag-
netic field at four selected temperatures: 14, 105, 165, and210 K. For reference, the measured magnetization M(T)
curve is also shown in Fig. 1. As the temperature is reduced,the magnetization begins to increase below T
f, and reaches
its saturation value below ;100 K. The magnetic field de-
pendencies of the Qfactor and resonant frequency fof the
cavity at different temperatures are shown in Fig. 2. Thecurves are displaced for convenience, and only the changesa!Also with Electrical and Computer Engineering Department. Electronic
mail: jarek@uh.eduAPPLIED PHYSICS LETTERS VOLUME 74, NUMBER 5 1 FEBRUARY 1999
750 0003-6951/99/74(5)/750/3/$15.00 © 1999 American Institute of Physics
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
134.129.164.186 On: Sat, 20 Dec 2014 19:43:27inQandfare presented. Above the Curie temperature, no
magnetic field dependence of the Qor resonant frequency is
observed. The shapes of these curves, both Q(H) andf(H),
depend strongly on temperature.
We theoretically simulated the field-dependent quality
factor by assuming that the NSMO film is a single domainferromagnet of resistivity
r, with a dc magnetic field Bap-
plied along the film’s surface and the radio frequency ~rf!
field perpendicular to the dc field, Brf'B. The dynamic per-
meability mcould then be calculated using the Landau–
Lifschitz–Gilbert equation15m~B,M!511Nm0MSB1Nm0M1jva
gD
SB1jva
gDSB1Nm0M1jva
gD2Sv
gD2,
~1!
whereMis the saturation magnetization, N54pis the de-
magnetisation factor, va/gis the width parameter, ais the
Gilbert damping parameter, g5gmB,gthe gyromagnetic
ratio, and mBis the Bohr magneton. We used the transmis-
sion line theory16to analyze the entire layered structure of
the cavity, consisting of a copper plate, sapphire disk,NSMO film, substrate, and another copper plate. The reso-nant frequency can be found, for such a structure, using theimpedance transformation method. Substituting the value of
mobtained from Eq. ~1!into the impedance matching equa-
tion and solving for the resonant frequency we can determinetheQfactor of such a system from the following equation:
Q
~m,r!5Re@f~m,r!#
Im@f~m,r!#. ~2!
The calculated field-dependent Qfactor, for various lev-
els of saturation magnetization, is presented in Fig. 3. Thesimulation shows very good qualitative agreement with theexperimental data from Fig. 2 ~a!. It indicates that the micro-
wave losses are influenced by the magnetic field through achange in the imaginary part of the magnetic permeability
m(B), which also affects the resonant frequency.
From Fig. 2 we can see that, in the range of temperatures
between 45 and 100 K, a relatively large change of resonantfrequency with magnetic field was observed without signifi-cant change in Q. Figure 4 shows the change of the resonant
frequency and Qvs dc magnetic field at 45 K. The measure-
ments were done with two films on both sides of the cavitydisk. Such a configuration, different from the one used in theprevious measurements ~Fig. 2 !, was used to increase sensi-
tivity. For the fields ranging from 2150 to 150 mT, the Q
values are relatively constant while the center frequency f
0
shift is large. This change Df0of 0.3 MHz is equivalent to
an effective mvalue changing from 1 to 1.75.
When the applied dc field was set to 300 mT at 45 K, the
change Dfof the resonant frequency f0of the dielectric
cavity was measured as 0.6 MHz ~see Fig. 4 !. Using a trans-
FIG. 1. Quality factor vs temperature for the dielectric cavity with a NSMO
thin film, measured at zero applied field. The insets show dc magnetic fielddependence of the quality factor at four selected temperatures ~14, 105, 165,
and 210 K !. The magnetization measured as a function of temperature is
also shown.
FIG. 2. ~a!Change of the quality factor DQas a function of applied dc
magnetic field for different temperatures in the range 15–240 K. ~b!Change
of resonant frequency Dfof the cavity at different temperatures. The curves
are displaced for clarity and the lines are eye guides. Tick markers spacingon the vertical axes correspond to 400 and 0.5 MHz for changes of Qand
frequency, respectively.
FIG. 3. Theoretical simulations showing the field dependence of the cavity
quality factor Q(B) with different saturation magnetizations Mof the
NSMO film as a parameter.751 Appl. Phys. Lett., Vol. 74, No. 5, 1 February 1999 Wosik et al.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
134.129.164.186 On: Sat, 20 Dec 2014 19:43:27mission line model, we estimated the relative change of ef-
fective permeability per unit field as Dm/DB;0.005/mT.
Such a small frequency change can be expected for a dielec-tric cavity because of the small film-to-cavity volume ratio.Using the Ansoft Maxwell program, we have modeled theinfluence of the same effective
mchange on center frequency
of a microstrip line, with the NSMO layer on top. The figureof merit
3K52QDf/f0for tuning was calculated to be about
500. This Kvalue is sufficient to tune superconducting filters
at relatively small magnetic fields.
Figure 5 shows Qplotted as a function of temperature
for the bilayered structure NSMO/YBCO. Above 90 K it isclear that Q(T) has essentially the same temperature depen-
dence as in Fig. 1. A large increase of Qbelow 90 K indi-
cates that the YBCO film becomes superconducting. We ob-served different Q(H) andf
0(H) behaviors than in the case
of the NSMO/Cu configuration. This is probably due to thepresence of weak links in the superconducting YBCO layerand to an increase of its surface resistance at high magneticfields.
The properties of perovskite thin films for tunable HTS
resonators have to be optimized for T
pandTfto be at or near
the critical temperature of YBCO. Presumably, a good can-didate for a tuning material would be an underdoped perov-skite film that shows transition from the paramagnetic insu-lator to ferromagnetic insulator state.
17Nevertheless, it is
clear that, in order to use these materials instead of ferritesfor magnetic tuning of resonators, better understanding of themicrowave loss mechanisms in perovskite materials is
needed.
In summary, we have measured the resonant frequency
shift and Q-factor change as functions of applied dc mag-
netic field and temperature for NSMO films. Using a singleferromagnetic domain approximation, our theoretical simula-tions of the field dependence of the microwave losses inmanganite NSMO thin films agreed qualitatively with theexperimental data. Our results indicate that field-dependentpermeability
m(B) of NSMO films opens prospects for fab-
rication of magnetically tunable microwave devices. If opti-mized for operating parameters such as T
fandTp, thin film
CMR perovskites appear to be excellent candidates for cre-ating monolithic tunable HTS filters because of their crystal-line compatibility with YBCO and related HTS films. In ad-dition, the ferromagnetic properties of these materials can besignificantly modified by changing their composition, oxy-gen content, and/or magnetic domain structure.
18
The authors are thankful to Andrei Strikovski and Liping
Ji for their technical assistance. This work was supported, inpart, by the Texas Higher Education Coordinating Board Ad-vanced Research Program and Advanced Technology Pro-gram, by the Texas Center for Superconductivity at the Uni-versity of Houston, and by the Robert A. Welch Foundation~E-1221 !.
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FIG. 4. The quality factor Qand center frequency f0as a function of
applied dc field at 45 K.
FIG. 5. Measured quality factor vs temperature of the dielectric cavity witha YBCO/NSMO bilayer thin film.752 Appl. Phys. Lett., Vol. 74, No. 5, 1 February 1999 Wosik
et al.
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1.2177049.pdf | Micromagnetic simulations of nanosecond magnetization reversal processes in
magnetic nanopillar
G. Finocchio, M. Carpentieri, B. Azzerboni, L. Torres, E. Martinez, and L. Lopez-Diaz
Citation: Journal of Applied Physics 99, 08G522 (2006); doi: 10.1063/1.2177049
View online: http://dx.doi.org/10.1063/1.2177049
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/99/8?ver=pdfcov
Published by the AIP Publishing
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129.21.35.191 On: Tue, 23 Dec 2014 00:31:37Micromagnetic simulations of nanosecond magnetization reversal
processes in magnetic nanopillar
G. Finocchio,a/H20850M. Carpentieri, and B. Azzerboni
Dipartimento di Fisica della Materia e Tecnologie Fisiche Avanzate, University of Messina,
Salita Sperone 31, 98166 Messina, Italy
L. T orres, E. Martinez, and L. Lopez-Diaz
Departamento de Fisica Aplicada, Universidad de Salamanca, Plaza de la Merced s/n,37008 Salamanca, Spain
/H20849Presented on 2 November 2005; published online 28 April 2006 /H20850
In this paper we study by means of the spin torque model the fast switching behavior of the
Co/H2084920 nm /H20850/Cu /H208495n m /H20850/Co /H208492.5 nm /H20850magnetic multilayers of two different cross sections: ellipse
/H20849130/H1100370 nm2/H20850and ellipse /H20849130/H1100340 nm2/H20850. Simulations have been performed at zero and room
/H20849300 K /H20850temperatures, these point out that the magnetization inversion occurs by nucleation
processes in three main steps, for both structures. In particular, for zero temperature the third step
of the switching depends on the value of the spin-polarized current. Furthermore, for all of thesimulated currents the switching processes are thermally activated and smoother with respect to zerotemperature. © 2006 American Institute of Physics ./H20851DOI: 10.1063/1.2177049 /H20852
Magnetization reversal by spin-polarized current intro-
duces a mechanism for writing magnetoresistive random ac-cess memory /H20849MRAM /H20850.
1–3Recent experiments focus their
attention on fast /H20849nanosecond /H20850switching processes with no
applied field;4,5a pulse of current is applied and depending
on its duration and amplitude, it drives or not the switchingprocesses.
4Although single domain models are useful to un-
derstand the general trends of the behavior of thesedevices,
6,7in some experimental works, multiple domain
configurations and domain wall motion are invoked as theunderlying cause of the observed magnetization dynamics.
8,9
In this paper, we will focus our attention on nanopillars
with a ferromagnet /H20851free layer /H20849FL/H20850/H20852normal metal/
ferromagnet /H20851pinned layer /H20849PL/H20850/H20852 /H20849FNF /H20850geometry. When PL
and FL are parallel /H20851parallel state /H20849PS/H20850/H20852, the structure presents
low electrical resistance, while for PL and FL antiparallel/H20851antiparallel state /H20849APS /H20850/H20852a high resistance state is
observed.
1–4,8We study how the spin-polarized current
/H20849SPC /H20850drives a fast magnetization reversal in the
Co/H2084920 nm /H20850/Cu /H208495n m /H20850/Co /H208492.5 nm /H20850multilayer of two differ-
ent cross sections: S1/H20849ellipse 130 /H1100370 nm2/H20850andS2/H20849ellipse
130/H1100340 nm2/H20850. The spin torque model /H20849STM /H20850based on
three-dimensional /H208493D/H20850dynamical micromagnetic code
which includes the SPC term has been used for thesimulations.
10,11Recent time-domain measurements of nano-
magnet dynamics driven by SPC confirm that the STM pre-dicts the magnetization dynamics correctly.
12In this paper,
these have been computed by solving the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation that includes the Slonczewski term:dM
dt=−/H9253/H11032M/H11003Heff−2/H9262BJ
/H208491+/H92512/H20850dMs3eg/H20849M,P/H20850M/H11003/H20849M/H11003P/H20850
+2/H9262B/H9251J
/H208491+/H92512/H20850dMs2eg/H20849M,P/H20850M/H11003P
−/H9251/H9253/H11032
MsM/H11003/H20849M/H11003Heff/H20850, /H208491/H20850
where Mis the magnetization of the FL, Heffis the effective
field, /H9253/H11032=/H9253//H208491+/H92512/H20850,/H9253is the electron gyromagnetic ratio,
and/H9251is the dimensionless damping parameter. Regarding
the SPC term, /H9262Bis the Bohr magneton, Jis the current
density /H20849positive when electrons flow from the FL to the PL /H20850,
dis the thickness of the free layer, eis the electron charge
/H20849positive scalar /H20850, andPis the magnetization of the PL. The
scalar function g/H20849M,P,/H9257/H20850was deduced by Slonczewski,1for
cobalt /H9257is 0.35.1The effective field includes the following
contributions:
Heff=Hexch+Hani+Hext+HM+HAmp+HAF, /H208492/H20850
where Hexch,Hani,Hext, and HMare the standard micromag-
netic contributions from exchange, anisotropy, external, anddemagnetizing fields. The anisotropy constant of thin Co /H20849k
u/H20850
is 1.74 /H11003105J/m3obtained by fitting the frequency of mi-
crowave oscillations in similar nanopillars.8,11HAmpandHAF
are the ampere field and the magnetostatic coupling between
PL and FL.10,11Previous works show that both HAmpand
HAFplay a crucial role in the dynamics of magnetic
nanostructures.10,11A damping parameter /H9251=0.005, a time
step of 60 fs, and a cubic cell size of 2.5 /H110032.5/H110032.5 nm3are
employed.
We focus our attention on the fast switching behavior of
the magnetization for zero applied field, performing thesimulations in order to reproduce the main experimental
a/H20850Electronic mail: gfinocchio@ingegneria.unime.itJOURNAL OF APPLIED PHYSICS 99, 08G522 /H208492006 /H20850
0021-8979/2006/99 /H208498/H20850/08G522/3/$23.00 © 2006 American Institute of Physics 99, 08G522-1
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129.21.35.191 On: Tue, 23 Dec 2014 00:31:37features.4,5,13The zero temperature simulations show that the
switching process occurs by means of a nucleation processfor both structures; since the behavior is qualitatively thesame, we point out the ones for S1. Figure 1 shows the
temporal evolution of /H20855m
X/H20856from the PS to APS, for different
values of SPC, J=−1/H11003108A/cm2/H20849solid line /H20850,J=−1.65
/H11003108A/cm2/H20849dashed line /H20850, and J=−2/H11003108A/cm2/H20849dotted
line/H20850. As can be seen for the intermediate current the magne-
tization oscillates in a metastable state before it reachesthe APS /H20849similar state is present for other currents, i.e.,
J=−1.5 /H1100310
8A/cm2/H20850.
For all of the simulations performed /H20849−2.1/H11003108
A/cm2/H11021J/H11021−0.8/H11003108A/cm2/H20850, the nucleation can be de-
scribed in three main steps: firstly, the applied current in-
duces an oscillation of the spins at the boundary of the struc-ture, giving rise to the formation of three domains, if thepulse duration and amplitude are large enough /H20851Fig. 2 /H20849a/H20850/H20852.
Secondly, the two little domains at the boundary expandquickly, while the central domain size decreases /H20851Fig. 2 /H20849b/H20850/H20852.
Finally, the last step of the switching occurs either by theexpulsion of the central domain or it is confined inside thestructure before and is finally destroyed /H20851Fig. 2 /H20849c/H20850/H20852. The
simulations show different expulsion mechanisms. Startingby a SPC J=−0.8 /H1100310
8A/cm2the expulsion occurs as
shown in Fig. 3, /H20849top left /H20850, increasing the current before the
central domain is expulsed in the left side of the structure/H20849Fig. 3, top right, i.e., J=−1.5 /H1100310
8A/cm2/H20850and then in the
right side /H20849Fig. 3, bottom left, i.e., J=−1.65 /H11003108A/cm2/H20850,
these last two kinds of expulsion mechanism give rise to ametastable state. Increasing again the current, the expulsionoccurs as shown in Fig. 3 /H20849bottom right, i.e., J=−1.75
/H1100310
8A/cm2/H20850. For higher values of current /H20849i.e., J/H33356−2
/H11003108A/cm2/H20850the third step of switching changes in the one
of Fig. 2 /H20849c/H20850. The expulsion mechanism depends on the value
of the current, it is a complex trade-off between the amperefield on one side and the torque of the SPC on the other. Inorder to confirm our conjecture, we have performed simula-tions with no ampere field; the inset of Fig.1 shows the tem-poral evolution of /H20855m
X/H20856from the PS to APS for the same
values of current of Fig. 1. The main qualitative difference is
the third step of the nucleation process; it occurs by means ofexpulsion of the central domain as shown in Fig. 3 /H20849top left /H20850
until a value of current J/H11015−1.6/H1100310
8A/cm2forS1, and
then it occurs in the same way as shown in Fig. 2 /H20849c/H20850. In fact,
no metastable states are observed.
The effect of the temperature on the switching processes
of these structures has been studied at room temperature/H20849300 K /H20850. Considering that experimental data are well de-
scribed by current dependent activation barriers that agree
with the prediction of the LLG-based models,
13–15we in-
clude in our micromagnetic simulations a thermal field as anadditive random field to the deterministic effective field for
FIG. 1. Temporal evolution of /H20855mX/H20856from the PS to the APS due to three
different values of SPC, J=−1/H11003108A/cm2/H20849solid line /H20850,J=−1.65
/H11003108A/cm2/H20849dashed line /H20850, and J=−2/H11003108A/cm2/H20849dotted line /H20850. Inset:
Same simulations performed with no ampere field.
FIG. 2. Nucleation process: /H20849a/H20850Initial configuration, /H20849b/H20850second step, and /H20849c/H20850
final step with confined domain.
FIG. 3. Expulsion mechanisms in the nucleation process: Top left /H20849J=−1
/H11003108A/cm2/H20850, top right /H20849J=−1.5 /H11003108A/cm2/H20850, bottom left /H20849J=−1.5
/H11003108A/cm2/H20850, and bottom right /H20849J=−1.75 /H11003108A/cm2/H20850.08G522-2 Finocchio et al. J. Appl. Phys. 99, 08G522 /H208492006 /H20850
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129.21.35.191 On: Tue, 23 Dec 2014 00:31:37each cell; this leads to a definition of the stochastic
Langevin-Landau-Lifshitz-Gilbert /H20849LLLG /H20850equation.14In or-
der to take into account the SPC terms in this formulation,the main hypotheses are that the spin torque does not containa fluctuating field, the fluctuating field is independent of thespin torque,
13,15and the magnetization configuration of the
PL is not affected by the temperature.
The thermal field Hthis a random fluctuating three-
dimensional vector quantity given by
Hth=/H9264/H208812/H9251
1+/H92512kBT
/H92620/H9253/H9004VM s/H9004t, /H208493/H20850
where kBis the Boltzmann constant, /H9004Vis the volume of the
computational cubic cell, /H9004tis the simulation time step, Tis
the temperature of the sample,13,15and/H9264is a Gaussian sto-
chastic process. The thermal field Hthsatisfies the following
statistical properties:
/H20855Hth,k/H20849t/H20850/H20856=0 ,
/H20855Hth,k/H20849t/H20850,Hth,l/H20849t/H11032/H20850/H20856=D/H9254kl/H9254/H20849t−t/H11032/H20850, /H208494/H20850
where kandlrepresent the Cartesian coordinates x,y, and z.
According to that, each component of Hth=/H20849Hth,x,
Hth,y,Hth,z/H20850are space and time independent random Gaussian
distributed numbers /H20849Wiener process /H20850with zero mean value.
The constant Dmeasures the strength of thermal fluctuations,
and its value is obtained from the Fokker-Planck equation.For our simulations, we have used a second order Heunscheme to solve numerically the LLLG equation; thisscheme converges directly to the Stratonovich solution of theLLLG equation.
14Figure 4 shows temporal evolution of
/H20855mx/H20856at room temperature from the PS to the APS
/H20851J=−1/H11003108A/cm2/H20849solid line /H20850,J=−1.65 /H11003108A/cm2
/H20849dashed line /H20850, and J=−2/H11003108A/cm2/H20849dotted line /H20850/H20852, aver-
aged on 60 iterations. For all the simulated currents, thesimulations confirm that the switching processes are ther-mally activated, in agreement with Refs. 13 and 16, but indisagreement with Ref. 4. Some intermediate oscillatorymagnetization states are suppressed by the thermal activa-tion, giving rise to switching processes smoother with re-spect to the zero temperature itself. Furthermore, the thermalactivation deletes the metastable state present for some val-ues of current at zero temperature.
In summary, we have simulated fast switching processes
using the STM based on a 3D micromagnetic model, whichinclude the effect of the SPC. For the two structures studied
and for the simulated currents at zero temperature, we foundthat the inversion of magnetization occurs by means of a
nucleation process which depend on the value of the current.Furthermore, the switching is thermally activated and themagnetization inversion is smoother at room temperature.
1J. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850;195, L261
/H208491999 /H20850;247, 324 /H208492002 /H20850.
2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850.
3J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph,
Phys. Rev. Lett. 84, 3149 /H208492000 /H20850.
4A. A. Tulapurkar et al. , Appl. Phys. Lett. 85,5 3 5 8 /H208492004 /H20850.
5T. Devolder et al. Appl. Phys. Lett. 86, 062505 /H208492005 /H20850.
6J. Z. Sun, Phys. Rev. B 62, 570 /H208492000 /H20850.
7Bertotti et al. , Phys. Rev. Lett. 94, 127206 /H208492005 /H20850.
8S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoe-
lkopf, R. A. Buhrman, and D. C. Ralph, Nature /H20849London /H20850425,3 8 0
/H208492003 /H20850.
9J. Miltat, G. Albuquerque, A. Thiaville, and C. V ouille, J. Appl. Phys. 89,
6982 /H208492001 /H20850.
10L. Torres, L. Lopez-Diaz, E. Martinez, M. Carpentieri, and G. Finocchio,
J. Magn. Magn. Mater. 286, 381 /H208492005 /H20850.
11M. Carpentieri et al. , J. Appl. Phys. 97, 10C713 /H208492005 /H20850.
12I. Krivorotov et al. , Science 307, 228 /H208492005 /H20850.
13I. N. Krivorotov et al. , Phys. Rev. Lett. 93, 166603 /H208492004 /H20850.
14J. L. Garcia-Palacios and F. J. Lazaro, Phys. Rev. B 58, 14937 /H208491996 /H20850.
15Z. Li and S. Zhang, Phys. Rev. B 69, 134416 /H208492004 /H20850.
16E. B. Myers et al. , Phys. Rev. Lett. 89, 196801 /H208492002 /H20850.
FIG. 4. Temporal evolution of /H20855mX/H20856from the PS to the APS at room tem-
perature /H20849300 K /H20850due to three different values of SPC: J=−1/H11003108A/cm2
/H20849solid line /H20850,J=−1.65 /H11003108A/cm2/H20849dashed line /H20850, and J=−2/H11003108A/cm2
/H20849dotted line /H20850.08G522-3 Finocchio et al. J. Appl. Phys. 99, 08G522 /H208492006 /H20850
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1.3369213.pdf | Micromagnetic simulations of linewidths and nonlinear frequency shift
coefficient in spin torque nano-oscillators
Mario Carpentieri and Luis Torres
Citation: J. Appl. Phys. 107, 073907 (2010); doi: 10.1063/1.3369213
View online: http://dx.doi.org/10.1063/1.3369213
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v107/i7
Published by the American Institute of Physics.
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Downloaded 05 Jun 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsMicromagnetic simulations of linewidths and nonlinear frequency shift
coefficient in spin torque nano-oscillators
Mario Carpentieri1,a/H20850and Luis T orres2
1Department of Elettronica, Informatica e Sistemistica, University of Calabria, I-87036, Arcavacata
di Rende, Cosenza, Italy
2Department of Fisica Aplicada, University of Salamanca, E-37008 Salamanca, Spain
/H20849Received 17 December 2009; accepted 17 February 2010; published online 8 April 2010 /H20850
The dependence of the linewidth on the temperature and the applied magnetic field angle is studied
in spin torque nano-oscillators /H20849STNOs /H20850by means of full micromagnetic simulations. The analyzed
spin valve is the experimental one by Sankey et al. /H20851Phys. Rev. Lett. 96, 227601 /H208492006 /H20850/H20852and the
magnetic parameters are given by magnetoresistance fitting. Linewidth behavior increases with thetemperature, in agreement with the analytical predictions by Tiberkevich et al. /H20851Phys. Rev. B 78,
092401 /H208492008 /H20850/H20852, and its slope depends on the applied field angle. Also, the nonlinear frequency shift
coefficient, which gives a measure of the nonlinearity degree of STNO and indicates the strength ofthe transformation of amplitude into phase fluctuations, is found. The understanding of the nonlinearfrequency shift allows one to tune the generation frequency of the STNO, but, at the same time,creates an additional source of the phase noise, which leads to a significant broadening of thelinewidth generation. Narrow linewidths /H20849around 10 MHz a t 0 K and 100 MHz at 300 K /H20850are found
in our shape-anisotropy nanopillars by applying close to in-plane magnetic field at an angle of 45°between in-plane easy and hard axes. © 2010 American Institute of Physics .
/H20851doi:10.1063/1.3369213 /H20852
I. INTRODUCTION
Thermal fluctuations generate phase noise, which im-
plies the generation of a linewidth, which is a fundamentalparameter to characterize the spectrum of a nonlinear oscil-lator. Spin-transfer torque from a dc spin-polarized currentcan provide magnetic switching or excite periodic oscillationof the magnetization in spin-valve nanostructures.
1–3Recent
experiments and several theories have been carried out toinvestigate the oscillation modes of magnetic multilayerednanostructures.
4
Spin torque nano-oscillators /H20849STNOs /H20850are magnetic
nanopillars where the “free” magnetic layer has finite lateralsizes and reflecting boundaries in the plane of the layer andrepresents a thin magnetic resonator. In agreement with theexperimental and theoretical works,
4,5the linewidth depends
strongly on the temperature /H20849T/H20850. Furthermore, a recent work
by Slavin et al.6analytically demonstrated that the compen-
sation of nonlinear phase noise provided minimum linewidthof a STNO and this was achieved for in-plane hard-axis mag-netization bias field. On the other hand, a complete linewidthstudy of STNO varying the applied field angle from out ofplane to in plane has been not reported to date.
In this paper, the temperature dependence of the line-
width for a nonlinear auto-oscillator has been fully investi-gated from a micromagnetic point of view in the experimen-tal device by Sankey et al.
7This is an individual ellipsoidal
PyCu nanomagnets of as small as 30 /H1100390/H110035.5 nm3and it
consists on a 20 nm thick pinned layer of Permalloy /H20849Py/H20850,a
12 nm Cu spacer, and a free layer /H20849FL/H20850ofd=5.5 nm thick
Py65Cu35alloy. Our simulations have been performed by amicromagnetic three-dimensional /H208493D /H20850dynamical code de-
veloped by our group.8The magnetic parameters used for the
FL simulations have been found by magnetoresistancefitting
7and they are saturation magnetization MS=2.785
/H11003105A/m and exchange constant A=1.0/H1100310−11J/m. The
free layer has been discretized in computational cells of2.5/H110032.5/H11003d/H20849thickness of the free layer /H20850nm
3. The ther-
mal fluctuation has been taken into account as an additive
stochastic contribution to the deterministic effective field foreach computational cell.
9External field will be applied in
different directions from perpendicular to in-plane direction,whereas the dynamics simulation of the FL is computed intwo dimensions. The magnetization dynamics is described bythe phenomenological Landau–Lifshitz–Gilbert equationaugmented by the Slonczewski’s spin-transfer torque.
1
The thermal field,9which is a random fluctuating 3D
vector quantity, is given by
Hth=/H9264/H20849t/H20850/H208812/H9251
1+/H92512KBT
/H92620/H9253/H11032/H9004VM s/H9004t, /H208491/H20850
where KBis the Boltzmann constant, /H9004Vis the volume of the
computational cubic cell, /H9004tis the simulation time step, Tis
the temperature of the sample,10,11and/H9264/H20849t/H20850is a Gaussian
stochastic process. The thermal field Hthsatisfies the follow-
ing statistical properties:
/H20877/H20855Hth,k/H20849r/H6023,t/H20850/H20856=0
/H20855Hth,k/H20849r/H6023,t/H20850Hth,l/H20849r/H6023/H11032,t/H11032/H20850/H20856=D/H9254kl/H9254/H20849t−t/H11032/H20850/H9254/H20849r/H6023−r/H6023/H11032/H20850,/H20878 /H208492/H20850
where kandlrepresent the Cartesian coordinates x,y,z. Ac-
cording to that, each component of Hth=/H20849Hth,x,Hth,y,Hth,z/H20850is
space and time independent random Gaussian distributed
number /H20849Wiener process /H20850with zero mean value. The constanta/H20850Electronic mail: mcarpentieri@deis.unical.it.JOURNAL OF APPLIED PHYSICS 107, 073907 /H208492010 /H20850
0021-8979/2010/107 /H208497/H20850/073907/4/$30.00 © 2010 American Institute of Physics 107 , 073907-1
Downloaded 05 Jun 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsDmeasures the strength of thermal fluctuations, and its value
is obtained from the fluctuation dissipation theorem.12
In the precessional regime, the thermal activation is
manifested by the “inhomogeneous” broadening of the line-width of the magnetization spectrum that provides a decreas-ing of the coherence degree of the phase noise.
In this work, a study of the influence of the temperature
and the applied field angle from out of plane to in plane andalong 45° with respect to the in-plane easy and hard axes onthe STNO linewidth will be presented. Furthermore, an esti-mate of the nonlinearity degree by a computation of the non-linear frequency shift coefficient will also be given.
II. LINEWIDTH AND NONLINEAR FREQUENCY SHIFT
COMPUTATION
The power spectrum of a nonlinear auto-oscillator in the
presence of noise has a finite width, which is configurablewith the linewidth generation. The measurement of the line-width is related to the full width at half maximum /H20849FWHM /H20850
of the power spectrum. From a practical point of view, thegeneration of the FWHM is one of the most important pa-rameters of nano-oscillators. In general, the linewidth ex-pression for a linear oscillator is given by
5
FWHM = /H9003+/H20849p0/H20850KBT
/H9255/H20849p0/H20850, /H208493/H20850
where /H9003+/H20849p0/H20850is the damping rate, KBTis the thermal energy,
and the /H9255/H20849p0/H20850is the energy that increases with the oscillation
power and it increases with the bias current. Since STNOs
are strongly nonlinear oscillators, the expression /H208493/H20850cannot
describe the linewidth behavior quantitatively. In fact, ac-cording to the analytical theory of Kim et al. ,
5amplitude
fluctuations are transformed into phase fluctuations, the non-linear frequency shift generates a source of phase noise thatimplies a broadening of the linewidth, which is not taken intoaccount in the previous equation. To evaluate the FWHM innonlinear oscillators, the power dependence of the frequencyhas to be considered and the additional “nonlinear” noiseterm − N
/H9254p/H20849t/H20850has to be added to expression /H208493/H20850. In this case,
introducing the nonlinearities, the linewidth will be greater
than the linear oscillator by a factor /H208491+/H92712/H20850and the linewidth
will be given by
FWHM =1
2/H208491+/H92712/H20850/H9003+/H20849p0/H20850KBT
/H9255/H20849p0/H20850. /H208494/H20850
The coefficient /H9271is the normalized nonlinear frequency shift
given by
/H9271=N
G+−G−, /H208495/H20850
where Nis the nonlinear frequency shift coefficient /H20849its sign
and magnitude depend on the direction and amplitude of theapplied magnetic field /H20850andGis the nonlinear damping.
Before introducing the thermal field, a study of the pre-
cessional regime characteristics varying current amplitudeand external field out-of-plane angle /H20849
/H9258,/H9258=0 means perpen-
dicular to plane /H20850will be given. The dependence of the fre-
quency on the dc bias /H20849Idc/H20850for different applied field anglesis shown in the left column of Fig. 1/H20849no thermal field is
considered /H20850. Both blue and red frequency shifts /H20849respectively,
increasing and decreasing frequency with the applied cur-rent /H20850depend on the demagnetizing and in-plane anisotropy
effects. Under suitable conditions, the nonlinear frequencyshift is suppressed by compensation between the red- andblueshift and a nonlinearity reduction is obtained.
We observe blue frequency shift with increasing I
dcfor
all the out-of-plane applied fields /H20849in the considered current
range /H20850, being the slope of the curve strongly dependent on
the field angle. This fact is clearly indicative of the linearitydegree, increasing slope points out an increasing nonlinearityof the sample that, as it will be shown below, is related to astrong broadening of the linewidth. In this case, for
/H9258=5°,
the blue frequency shift slope is high, this means that theprefactor related to the nonlinear component in Eq. /H208494/H20850is
quite large and the behavior is different to the linear oscilla-tor devices. When increasing the
/H9258value /H20849this means going
to the in-plane direction /H20850, the blueshift is still evident but its
slope decreases, and for greater current values, the oscillationcharacteristic frequency decreases /H20849not shown /H20850. For example,
considering an applied current of 0.75 mA, the precessionalfrequency runs from 12.3 GHz for out-of-plane field angle/H20849
/H9258=5° /H20850to 5.7 GHz for in-plane field angle /H20849/H9258=85° /H20850. The
inverse power with respect to the dc bias in the near-
threshold range of currents at T=150 K and for different
field angles is shown in the right column of Fig. 1. The
intersection of the dashed line with the x-axis gives the value
of the threshold current that discerns the below thresholdbehavior from above threshold behavior.
13,14We indicate the
threshold current value in the left panels of Fig. 1by an
arrow. A change in the blue frequency shift slope is observ-able after that value.
As described in our previous paper
15for out-of-plane
applied field in different structures and demonstratedanalytically,
16the linewidth depends strongly on the tem-
FIG. 1. Left column: precession frequency vs applied current amplitude for
different applied field angles /H20849/H92620Happ=420 mT, T=0 K /H20850. Right column:
inverse power behavior vs applied dc bias for different applied field angles/H20849
/H92620Happ=420 mT, T=150 K /H20850. The dashed line in the right column indicates
the intercept with the x-axis which gives the threshold current value /H20849indi-
cated by an arrow in left column /H20850.073907-2 M. Carpentieri and L. T orres J. Appl. Phys. 107 , 073907 /H208492010 /H20850
Downloaded 05 Jun 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsperature. Figure 2shows the linewidth behavior when in-
creasing the temperature for different applied field angles. Inorder to obtain 10 MHz frequency resolution, for each pointa long simulation time of 110 ns is performed /H20849the first 10 ns
are not considered in the Fourier analysis /H20850. Generally, ther-
mal activation is manifested by the inhomogeneous broaden-ing of the linewidth of the magnetization spectrum that pro-vides a decreasing of the coherence degree of the phasenoise. The strong temperature dependence indicates that ther-mal effects determine the coherence time for the phase fluc-
tuations of spin-transfer driven precession, and this time is“correlated” with the line shape. Indeed, the line shape fittingis Lorentzian at low temperature, on the other hand, at hightemperature regime, the line shape is better described by aGaussian function. The broken point from Lorentzian toGaussian behavior is strongly dependent of the field angle. Infact, changing
/H9258to the in-plane direction, nonlinearities de-
crease and the broken point from Lorentzian to Gaussianmoves toward high temperatures /H20851see for comparison Figs.
2/H20849a/H20850–2/H20849d/H20850/H20852.
Considering out-of-plane angles /H20849
/H9258=5° /H20850, the change
from Lorentzian to Gaussian occurs at very low temperatures
/H20851T=20 K, see Fig. 2/H20849a/H20850/H20852and this fact indicates that STNO
behavior is strongly nonlinear and the effect is a broadeningof the line shape. Moving toward the in-plane direction, thedevice behavior is more linear /H20849for
/H9258=45° the broken point
occurs at T=75 K /H20850and the FWHM decreases becoming
about half of the value with respect to /H9258=5°. Considering
field angles close to the in-plane direction /H20849/H9258=75° and 85° /H20850,
the STNO behavior is quite linear, the line shape fitting isLorentzian up to room temperature, and a very narrowFWHM is obtained. It is possible to think that the maximumFWHM value with respect to the temperature depends on thenonlinearity degree and this value moves downward at lowtemperature when the behavior is strongly nonlinear. A verynarrow FWHM by applying a magnetic field with
/H9258=85° and
an in-plane angle at 45° between easy and hard axes isfound. Its minimum value is around 10 MHz at T=5 K,while its maximum value is about 100 MHz at room tem-
perature, indicating that nonlinear frequency shift decreases.To give a complete picture of the nonlinear behavior of spintorque oscillators, the normalized nonlinear frequency shiftin Eq. /H208494/H20850has been computed.
In the above threshold regime, the linewidth is given by
Eq. /H208494/H20850, on the other hand, in below threshold the linewidth
is given by Eq. /H208493/H20850. Equations /H208493/H20850and /H208494/H20850indicate that the
linewidth is proportional to the inverse normalized power inthe asymptotic regions. According to this, from the ratio ofEq. /H208494/H20850to Eq. /H208493/H20850, the normalized nonlinear frequency shift
/H9271
can be extracted by
s/H11022/s/H11021=/H208491+/H92712/H20850/2, /H208496/H20850
where the coefficient s /H11022and s /H11021represent the slopes of the
asymptotes above and below thresholds, respectively. Fol-lowing the experimental method by Kudo et al. ,
17the simu-
lations shown in Fig. 3are used to compute these coeffi-
cients. Here the linewidth behavior with respect to theinverse power for different applied field angles, for a fixedtemperature T=150 K, and varying the applied current, is
shown. Two different regimes are found: for low values ofthe inverse power, the linewidth behavior is related to abovethreshold bias and the slope of the points are indicated by s
/H11022.
Vice versa, the high values of the inverse power show belowthreshold regime and the asymptote slope is quantified by s
/H11021.
The current values used below and above threshold regimesare the ones given in Figs. 1/H20849b/H20850,1/H20849d/H20850,1/H20849f/H20850, and 1/H20849h/H20850. The
threshold current is also the one obtained by the dashed linein the same figures.
It is known that nonlinear frequency shift coefficient
strongly depends on the applied field angle and it can be bothpositive and negative.
6For out-of-plane applied field angle
/H20851see Fig. 3/H20849a/H20850/H20852, the FWHM increases with the applied current
and its slope is negative. Moving in the direction of the in-plane field angle
/H9258/H20851Figs. 3/H20849b/H20850–3/H20849d/H20850/H20852, FWHM decreases with
increasing applied current and its slope changes sign. Re-garding the normalized coefficient
/H9271, by using Eq. /H208496/H20850, the
value is about 14.5 considering out-of-plane field angle /H20849/H9258
=5° /H20850, whereas this value strongly decreases and it assumes
FIG. 2. Temperature dependence of the FWHM under the action of a mag-
netic field /H92620Happ=420 mT and for different applied field angles from
nearly perpendicular to plane /H20849/H9258=5° /H20850to nearly in-plane /H20849/H9258=85° /H20850.
FIG. 3. Linewidths vs inverse power for different applied field angles in
below and above threshold regimes at T=150 K.073907-3 M. Carpentieri and L. T orres J. Appl. Phys. 107 , 073907 /H208492010 /H20850
Downloaded 05 Jun 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsthe value 3.4 for /H9258=45°. Moving toward in-plane angles, the
nonlinear coefficient further decreases to 2.7 /H20849/H9258=75° /H20850and it
assumes a value of 1.9 for more in-plane angles /H20849/H9258=85° /H20850.I n
this case, Eq. /H208494/H20850is close to the form /H208493/H20850related to linear
oscillators. This fact confirms that the nonlinearity degreedecreases moving downward for in-plane applied field anglesto give rise to very narrow linewidths.
Nonlinear frequency shift computations shown in Fig. 3
are in agreement with the device oscillations behavior at T
=150 K shown in Fig. 4. Here, the frequency dependence on
the applied current for different applied field angles is re-ported. Regarding out-of-plane field angles /H20849
/H9258=5° /H20850, it is pos-
sible to see that below threshold the frequency is quite con-
stant, whereas above threshold blue frequency shift isevident. Furthermore, for this angle, linewidth increases withthe applied current and more strongly at above thresholdregime /H20851see Fig. 4/H20849a/H20850/H20852. For the rest of the applied field angles
/H20849
/H9258=45°, 75°, 85° /H20850, the frequency behavior with the cur-
rent is quite linear and the FWHM decreases with the applied
current /H20851Figs. 4/H20849b/H20850–4/H20849d/H20850/H20852. The different behaviors of the
FWHM dependence on the applied current at /H9258=5° in Fig. 4
is in agreement with the different slopes /H20849s/H11022and s /H11021negative /H20850
found in the computation of the nonlinear coefficients in Fig.3.
III. CONCLUSIONS
In summary, a study of the linewidth behavior as a func-
tion of the temperature in STNO has been performed bymicromagnetic simulations. In agreement with the analytictheory, we show two different behaviors of the linewidth: atlow thermal noise, the line shape is Lorentzian and it isGaussian at high temperatures. The temperature where achange between Lorentzian and Gaussian behavior occursstrongly depends on the nonlinearity degree of the nano-oscillator. The behavior of the linewidth varying the applied
field angle from out of plane to in plane is also found. Mov-ing toward in-plane angles between easy and hard axis thenonlinearities decrease and a very narrow linewidth is found.Since STNOs are characterized by nonlinear behavior andthe nonlinear frequency shift coefficient gives a measure ofthese nonlinearities, the computation of this parameter fordifferent applied field angles at T=150 K is done. To this
aim, the behavior of the linewidth with respect to the inversepower is computed. The normalized nonlinear frequencyshift is high for out-of-plane angle /H20849about 15 /H20850and this factor
is close to one for in-plane angles. This means that the non-linear prefactor is very low and the STNO behavior is likethe one of linear oscillators. Nonlinear frequency shift com-putations confirm that this is the underlying physical cause ofthe linewidth broadening for out-of-plane field angles andgive an explanation of the narrow linewidth for in-plane ap-plied fields.
ACKNOWLEDGMENTS
This work was partially supported by Spanish Project
under Contract Nos. MAT2008-04706/NAN and SA025A08.The authors would like to thank S. Greco for his supportwith this research.
1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850.
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Finocchio, B. Azzerboni, L. Torres, E. Martinez, and L. Lopez-Diaz, Mat.Sci. Eng. B 126, 190 /H208492006 /H20850.
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Lopez-Diaz, J. Appl. Phys. 99, 08G522 /H208492006 /H20850.
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FIG. 4. Frequency dependence on the applied current for different applied
field angles with amplitude of /H92620Happ=420 mT at T=150 K. Inset: FWHM
vs applied current.073907-4 M. Carpentieri and L. T orres J. Appl. Phys. 107 , 073907 /H208492010 /H20850
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1.4870919.pdf | Emergent spin electromagnetism induced by magnetization textures in the presence of
spin-orbit interaction (invited)
Gen Tatara and Noriyuki Nakabayashi
Citation: Journal of Applied Physics 115, 172609 (2014); doi: 10.1063/1.4870919
View online: http://dx.doi.org/10.1063/1.4870919
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/17?ver=pdfcov
Published by the AIP Publishing
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203.92.101.73 On: Sat, 20 Dec 2014 13:34:39Emergent spin electromagnetism induced by magnetization textures
in the presence of spin-orbit interaction (invited)
Gen Tatara1,a)and Noriyuki Nakabayashi1,2
1RIKEN Center for Emergent Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama 351-0198 Japan
2Graduate School of Science and Engineering, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397
Japan
(Presented 8 November 2013; received 23 September 2013; accepted 28 October 2013; published
online 9 April 2014)
Emergent electromagnetic field which couples to electron’s spin in ferromagnetic metals is
theoretically studied. Rashba spin-orbit interaction induces spin electromagnetic field which is in
the linear order in gradient of magnetization texture. The Rashba-induced effective electric andmagnetic fields satisfy in the absence of spin relaxation the Maxwell’s equations as in the charge-
based electromagnetism. When spin relaxation is taken into account besides spin dynamics, a
monopole current emerges generating spin motive force via the Faraday’s induction law. Themonopole is expected to play an important role in spin-charge conversion and in the integration of
spintronics into electronics.
VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4870919 ]
I. INTRODUCTION
Our technology is based on various electromagnetic
phenomena. For designing electronics devices, thus the
Maxwell’s equation is of essential importance. The mathe-matical structure of the electromagnetic field is governed by
a U(1) gauge symmetry, i.e., an invariance of physical laws
under phase transformations. The gauge symmetry is equiva-lent to the conservation of the electric charge and was estab-
lished when a symmetry breaking of unified force occurred
immediately after the big bang. The beautiful mathematicalstructure of charge electromagnetism was, therefore, deter-
mined when our universe started, and there is no way to
modify its laws.
Fortunately, charge electromagnetism is not the only
electromagnetism allowed in the nature. In fact, electromag-
netism arises whenever there is a U(1) gauge symmetry asso-ciated with conservation of some effective charge. In solids,
there are several systems which have the U(1) gauge symme-
try as a good approximation. Solids could thus display severaltypes of effective electromagnetic fields. A typical example is
a ferromagnetic metal. In ferromagnetic metals, conduction
electron spin (mostly selectron) is coupled to the magnetiza-
tion (or localized spins of delectrons) by an interaction called
thesdinteraction, which tends to align the electron spin paral-
lel (or anti-parallel) to the localized spin. This interaction isstrong in most 3 dferromagnetic metals, and as a result, con-
duction electron’s spin originally consisting of three compo-
nents reduces to a single component along the localized spindirection. The remaining component is invariant under a phase
transformation, i.e., has a U(1) gauge symmetry just like the
electric charge does. A spin electromagnetic field thusemerges and couples to conduction electron’s spin. The sub-
ject of the present paper is this spin electromagnetic field. Theworld of spin electromagnetic field is richer than that of
electric charge, since the electron’s spin in solids is under
influence of various interactions such as spin-orbit interaction.We will in fact show that magnetic monopole emerges from
spin relaxation processes. Spin electromagnetic field
drives electron’s spin and thus plays an essential role inspintronics.
Let us discuss spin transport in ferromagnetic metals.
The conduction electron’s spin aligns with local spin direc-tion, n, almost perfectly because the sdinteraction is strong.
This limit is called the adiabatic limit. If the magnetization is
uniform, nothing particular happens since there is no scatter-ing of electron spin. Non-trivial effects are expected in the
presence of local spin structures. Because of the sdinterac-
tion, spin of electron traveling through a magnetizationstructure follows the local spin and rotates with it (Fig. 1).
There are two significant effects from this spin rotation. The
first is that the spin acquires a geometric phase.
1,2In the adi-
abatic limit, the non-commutative phase electron spin has
accumulated and is projected on a single diagonal compo-
nent proportional to n, resulting in the well-known spin
Berry’s phase. In the language of spin electromagnetism, the
spin Berry’s phase is the magnetic component of the spin
electromagnetic fields. (The electronic component is thespin motive force.) The other effect is a rotation of localized
spin induced when an electric current is applied, the spin-
transfer torque effect.
3,4These two effects are reciprocal to
each other.5–8
In this paper, we focus on the Berry’s phase effect in
spin electromagnetic field. The effect was theoretically dis-cussed in 1986 by Berger, who discussed a voltage generated
by a canting of wall plane of a driven domain wall.
3
Emergence of effective electromagnetism coupling to elec-
tron’s spin was pointed out by use of gauge field argument
by Volovik in 1987 (Ref. 9). Stern discussed the motive
force in the context of the spin Berry’s phase and discussedsimilarity to the Faraday’s law.
2Spin motive force wasa)Author to whom correspondence should be addressed. Electronic mail:
gen.tatara@riken.jp.
0021-8979/2014/115(17)/172609/6/$30.00 VC2014 AIP Publishing LLC 115, 172609-1JOURNAL OF APPLIED PHYSICS 115, 172609 (2014)
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
203.92.101.73 On: Sat, 20 Dec 2014 13:34:39rederived in Ref. 10in the case of domain wall motion. It
was argued in the context of topological pumping in Ref. 11.
Duine discussed spin electric field including the effect of
spin relaxation by use of non-adiabaticity parameter ( b).5,12
The Hall current induced by a spin electric field in the pres-
ence of spin-orbit interaction was theoretically studied by
Shibata and Kohno.6,13The effect of Rashba interaction on
spin electric field was discussed in Refs. 14and15. These
works10,12–15have focused solely on the spin electric field.
This is due to a technical difficulty; spin electric field Esis
accessible by studying longitudinal transport or force as alinear response with respect to E
s, while its magnetic coun-
terpart requires to take both EsandEinto account (or the
second order in Es). A calculation of Bswas carried out
recently.16
II. SPIN BERRY’S PHASE AS AN EFFECTIVE
ELECTROMAGNETIC FIELD
Let us consider conduction electron transport in the
presence of inhomogeneous magnetization texture. As a
result of a strong sdinteraction, the electron wave function
acquires a geometric quantum phase (Fig. 2). The phase is
written as an integral of an effective gauge field, As, along
its path Casu¼e
/C22hð
Cdr/C1As; (1)
where eis electron charge and /C22his the Planck’s constant di-
vided by 2 p. Existence of the phase means that there is an
effective magnetic field, Bs, as seen by rewriting the integral
over a closed path using the Stokes theorem
u¼e
/C22hð
SdS/C1Bs; (2)
where Bs/C17r/C2 Asrepresents curvature. This phase u
attached to electron spin is called the spin Berry’s phase.
Time-derivative of phase is equivalent to a voltage, and thus,
we have effective electric field defined by
_u¼/C0e
/C22hð
Cdr/C1Es; (3)
where Es/C17/C0 _As.EsandBsare called spin electric and mag-
netic field, respectively. They satisfy the Faraday’s law,
r/C2 Esþ_Bs¼0; (4)
as a trivial result of their definitions. (There is, however,
a possibility of topological monopole as discussed in
Sec. II B.) Further, if we define a charge by r/C1Es/C17qand a
current by1
lsr/C2 Bs/C0_Es/C17j(choosing permittivity to be
1), we see that a conservation law _qþr/C1 j¼0 is satis-
fied.17The spin charge qand current jappearing in these
equations are induced by the spin motive force, but they are
proportional to the electric charge and current, respectively,
since spin and charge are proportional to each other in ferro-magnetic metals; for example, j
s¼Pj(Pis spin polarization).
The fields thus have a structure of electromagnetism, and we
now have a spin electromagnetic field coupled to electron’sspin in ferromagnetic metals. One should note that those
fields are real or observable ones coupling to real electric
charge and current and not just “fictitious fields.”
A. Adiabatic spin gauge field
Here, we derive the explicit form of the spin gauge field
in the adiabatic limit, AðadÞ
s. We denote spin electric and
magnetic fields in this limit as EðadÞ
sandBðadÞ
s, respectively.
We consider a conduction electron hopping from a site rto a
neighboring site at r0/C17rþa(ais a vector connecting
neighboring sites) (Fig. 1). The localized spin directions at
those sites are nðrÞandnðrþaÞ, respectively, and the elec-
tron’s wave function at the two sites are18jni¼cosh
2j"i
þsinh
2ei/j#iandjn0i¼cosh0
2j"i þ sinh0
2ei/0j#i, where j"i
andj#irepresent the state with spin direction in the zand/C0z
directions, respectively, and h,/andh0;/0are the polar
angles of nðrÞandnðr0Þ, respectively (i.e., cos h¼nzðr) and
cosh0¼nzðr0Þ). The wave functions are concisely written
using matrices, UðrÞandUðr0Þ, which rotate the spin state j"i
tojni(Fig. 3), asjni¼UðrÞj"iandjn0i¼Uðr0Þj"i. The rota-
tion matrix is1(neglecting irrelevant phase factors)
FIG. 1. In the adiabatic limit, spin of conduction electron aligns with local
spin direction nðrÞat each site r. The effect of a difference of nðrÞandnðr0Þ
when a conduction electron hops from site rtor0is expressed by a unitary
matrix Uðr0Þ/C01UðrÞwhich acts on the spin wave function.
FIG. 2. (a) A closed path Cin the coordinate space in the presence of a
background magnetization texture (thick arrows). (b) The spin of a conduc-
tion electron is rotated by a strong sdinteraction with magnetization as it
moves along the path C, resulting in a Berry’s phase factor eiu.172609-2 G. Tatara and N. Nakabayashi J. Appl. Phys. 115, 172609 (2014)
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203.92.101.73 On: Sat, 20 Dec 2014 13:34:39UðrÞ¼ei
2ð//C0pÞrzei
2hrye/C0i
2ð//C0pÞrz
¼cosh
2sinh
2ei/
/C0sinh
2e/C0i/cosh
20
BB@1
CCA: (5)
The overlap of the electron wave functions at the two sites is
thushn
0jni ¼ h"j Uðr0Þ/C01UðrÞj" i. When localized spin tex-
ture is slowly varying, we can expand Uðr0Þ/C01UðrÞ’1
/C0UðrÞ/C01ða/C1r ÞUðrÞ(neglecting contributions of the order
ofa2) to obtain
hn0jni’1/C0h " j UðrÞ/C01ða/C1r ÞUðrÞj"i ’ eiu; (6)
where u/C17ia/C1h " jUðrÞ/C01rUðrÞj"i /C17 a/C1AðadÞ
s. A vector
AðadÞ
shere plays a role of a gauge field, similarly to that of
the electromagnetism. By use of Eq. (5), this gauge field
reads (the factor of1
2represents the magnitude of electron
spin)
AðadÞ
s¼/C22h
2eð1/C0coshÞr/: (7)
The matrix Uappearing in Eq. (5)is unitary, i.e., satis-
fiesU†U¼1(†is complex conjugate transpose) and has de-
terminant of 1. A 2 /C22 matrix having these characters is
called an SU(2) matrix (SU stands for special unitary), and
therefore, /C0iUðrÞ/C01rUðrÞis an SU(2) gauge field having
three components. Its adiabatic component, AðadÞ
s, arising
from a projection onto the "component, is called a U(1)
gauge field. (A matrix corresponding to a phase transforma-
tion, eiu,i sa1 /C21 unitary matrix, and so it is a U(1) matrix.)
The U(1) gauge field found here indicates that the whole
structure of charge electromagnetism applies also for the
spin electromagnetism. The argument here provides a mathe-matical ground for the phenomenological argument at the be-
ginning of the section. In contrast to the phase factor or
adiabatic gauge field, the spin-transfer torques arise whennon-adiabatic components such as h#jUðrÞ
/C01rUðrÞj"i are
induced by the applied electric field.19
B. Topological monopole
Using explicit expression for the spin gauge field in the
adiabatic limit, Eq. (7), adiabatic spin electromagnetic fields
becomeEðadÞ
s;i¼/C0/C22h
2en/C1ð_n/C2r inÞ;
BðadÞ
s;i¼/C22h
4eX
jk/C15ijkn/C1ð r jn/C2r knÞ: (8)
In terms of polar coordinates, the magnetic field here reads
BðadÞ
s;i¼/C22h
2eXi, where Xi/C17P
jk/C15ijksinhðrjhÞðr k/Þis a solid
angle of vector n. Let us see if these fields satisfy the
Maxwell’s equations. We can see that
r/C1BðadÞ
s¼/C22h
4eX
ijk/C15ijkrin/C1ð r jn/C2r knÞ/C17qðadÞ
m;(9)
which may tempt one to think that there is finite monopole
density. We should, however, be careful since the right-hand
side vanishes as a local quantity if nðrÞis a smooth function,
since nhas only two independent variables (two polar angles
ofhand/). Nevertheless, there remains an intriguing possi-
bility of a topological monopole.9In fact, the total monopole
charge Qm, defined by a volume integral of qm, is written by
use of Gauss’s law as (Ð
dSrepresents a surface integral)
Qm¼h
4peð
dS/C1X; (10)
and it follows that Qmis an integer multiple of h/esince
1
4pÐdS/C1Xis a winding number of a mapping from a sphere
in the coordinate space to a sphere in the spin space. This
monopole charge is finite when the magnetization structure,
nðrÞ, has a hedgehog-like singular structure (Fig. 4).
The Faraday’s induction law similarly reads ðr /C2 EðadÞ
sÞi
þ_BsðadÞ
;i¼/C22h
4eP
ijk/C15ijk_n/C1ð r jn/C2r knÞ/C17jðadÞ
m,w h i c hv a n i s h e s
locally but is finite when integrated, indicating that topologi-cal monopole currents exist.
C. Detection of spin electromagnetic fields
The spin electromagnetic fields are detectable in trans-
port measurements. The spin magnetic field causes an anom-
alous Hall effect of spin, i.e., spin Hall effect. This spin Halleffect due to topological spin Berry’s phase is also called
FIG. 3. A unitary transformation Uðh;/Þrelates the two spin configurations
j"iandjniasjni¼Uj"i.
FIG. 4. A magnetization structure, nðrÞ, of a hedgehog monopole having a
monopole charge of Qm¼1. At the center, nðrÞhas a singularity (the direc-
tion of nð0Þcannot be defined if nðrÞis a smooth function), and this gives
rise to a finite monopole charge.172609-3 G. Tatara and N. Nakabayashi J. Appl. Phys. 115, 172609 (2014)
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203.92.101.73 On: Sat, 20 Dec 2014 13:34:39topological Hall effect. When the magnetization structure
flows, the spin electric field arises due to the Lorentz force
according to Es¼v/C2Bs,w h e r e vis the velocity of the struc-
ture. Since spin current ( js) in ferromagnetic metals is always
accompanied with electric current ( j¼js/P), those effects are
observable in standard electric transport measurements. In thecase of motion of domain walls and vortices, the voltage sig-
nals of the order of lV have been observed.
20,21The topologi-
cal Hall effect due to skyrmion lattice turned out to induce aHall resistivity of 4 n Xcm.
22,23Although those signals are not
large, they confirm the existence of spin electromagnetic fields
experimentally. It was recently shown theoretically that spinBerry’s phase couples to helicity of circularly polarized light,
h,g e n e r a t i n ga nm a g n e t i cfi e l do f B/ðB
s/C1hÞn(nis local-
ized spin direction), suggesting optical detection of spinBerry’s phase in the inverse Faraday effect.
24
III. SPIN-ORBIT EFFECTS ON SPIN
ELECTROMAGNETIC FIELD
The adiabatic spin electromagnetic field arises in the pres-
ence of a strong sdinteraction and in the absence of spin relaxa-
tion. An issue we discuss in this section is how the spin-orbitinteraction affects the spin electromagnetic fields. Since the spin-
orbit interaction mixes the orbi tal motion of conduction electrons
and the spin, the system is no longer in the adiabatic limit, and anovel spin electromagnetic field is expected to emerge.
While in the adiabatic case, the gauge field argument in
Sec. II Awas useful, the approach fails when the spin-orbit
interaction is included. Several other ways to calculate spin
electric field has been proposed, such as estimating the force
acting on the electron spin using F¼
im
e/C22hnh½H;^j/C138i(nis elec-
tron density, mis electron mass, and square bracket and hi
denote commutator and quantum average, respectively).15
Here, we identify the spin electromagnetic fields by studying
transport properties following Refs. 16and25. We use the
following two basic equations. The first is
j
P¼1
lsr/C2 BsþrsEsþDsrqs; (11)
which is a result of the Maxwell’s equations and the Ohm’s law.
Here, rsis spin conductivity, lsis spin magnetic permeability,
Dsis spin diffusion constant, and qsis spin density. The second
is the Hall effect when an external electric field, E, is applied
jH¼rHðE/C2BsÞ; (12)
where rHis the Hall conductivity.
We consider a system with Rashba interaction and
strong sdinteraction. We first neglect spin relaxation. The
Hamiltonian we consider is then
H¼/C0/C22h2
2mr2þDsdn/C1rðÞ /C0iaR/C1ð r/C2 rÞ/C20/C21
; (13)
where aRis a vector representing the Rashba field, Dsdis the
strength of the sdinteraction, and ris a vector of Pauli mat-
rices. The left-hand side of the two equations, Eqs. (11) and
(12), is calculated using the Keldysh Green’s functionmethod summing over several Feynman diagrams. The result
at the linear order in the Rashba interaction is16
j¼n1f$/C2½$/C2ðaR/C2nÞ/C138g þ n2ðaR/C2_nÞ; (14)
jH¼X
6ð7Þs6
/C22hrB6ðE/C2ð$/C2ða/C2nÞÞÞ; (15)
where n1¼e/C22h
12mP
6ð6Þ/C236ð1þ1
5D2
sdð/C152
F;6/C05/C15F;6/C15F;7ÞÞ;n2
¼/C02e
3/C22hP
6ð6Þ/C15F6/C236s6, and rB6/C172e2
3m/C15F6/C236s6is the spin-
resolved Boltzmann conductivity ( kF6;/C236, and s6are spin-
dependent Fermi wave vector, density of states, and elastic
lifetime, respectively). From these results, we find that (wedenote Rashba-induced fields by E
RandBR)
ER¼/C0m
e/C22haR/C2_n; (16)
BR¼m
e/C22h$/C2ðaR/C2nÞ; (17)
andrH¼P
6ð7Þes6
mrB6. A significant feature of ERis that
it emerges even from the dynamics of uniform magnetization,
in contrast to the case of conventional Berry’s phase contribu-
tion, EðadÞ
s. As easily checked, these fields ERandBRsatisfy
the Maxwell’s equations, and they can be written using the
spin gauge field as ER¼/C0 _ARandBR¼$/C2AR,w h e r e
AR/C17m
e/C22hðaR/C2nÞ: (18)
Thus, we can define an effective U(1) gauge field even in the
presence of the Rashba interaction as far as its linear contri-
bution concerns. This fact is understood by noting that the
Rashba interaction in Eq. (13), linear in the momentum, can
be regarded as an effective spin gauge field if higher-order
contributions are neglected. The expression for ERwas dis-
cussed in Ref. 14using a chiral derivative argument.
A. Discussions
We have discussed the Rashba-induced spin electromag-
netic field, which is an extension of the spin Berry’s phasegeneralized to include the Rashba interaction. These fields
are linear in the gradient in space or in time, and thus, they
become dominant in slowly varying magnetization struc-tures. (Conventional spin Berry’s phase, Eq. (8), is propor-
tional to second order of gradients.) Let us estimate the
magnitude. Rashba interaction is strong on the surface ofheavy metals in particular when doped with Bi.
26Choosing
aR¼3e VA ˚(Ref. 14), we expect spin electric and magnetic
fields of the order of
ER¼maR
e/C22hx¼2:5k V =m;BR¼maR
e/C22hk¼2:5k T ;(19)
for a structure having a typical length scale kof 1 nm and dy-
namics with angular frequency xof 1 GHz. The Rashba
interaction is thus useful in creating extremely high effective
spin electromagnetic fields.
Let us consider the case of a domain wall in a wire (in the
xyplane) with Rashba field in the zdirection. For an in-plane172609-4 G. Tatara and N. Nakabayashi J. Appl. Phys. 115, 172609 (2014)
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203.92.101.73 On: Sat, 20 Dec 2014 13:34:39domain wall profile changing in the xdirection h¼p
2and
cos/¼tan hx
k(kis thickness of the wall), we have BR
¼maR
e/C22hk1
cos h2x
k^z(^zis unit vector in the zdirection). The spin mag-
netic field is, therefore, localized around the wall and corre-
sponds to a high field of 250 T if k¼10 nm. This localized
strong field would be detected as a local spin Hall voltage intheydirection when current is applied in the xdirection.
When a magnetic structure flow s, a spin electric field is
induced. In the case of a flow without deformation, the magnet-ization vector depends on the time as nðr;tÞ¼nðr/C0vtÞ,
where vis velocity of the flow. The spin electric field then reads
E
R¼m
e/C22hðaR/C2ðv/C1r ÞnÞ: (20)
When the in-plane domain wall flows in the xdirection
with a speed of vx, a spin voltage in the ydirection is
induced; Vy/C17Ð
dxE R;y¼2maR
e/C22hvx. For aR¼3e VA ˚and
vx¼100 m/s, the voltage is 0.5 mV. This value is 1000 times
larger than the conventional Berry’s phase contributionobserved in a permalloy, 0.4 lV at 130 m/s (Ref. 20). Even
for a system having a moderate Rashba field of a
R¼3m e V
A˚, the Rashba-induced signal is, therefore, comparable to the
conventional signals.
So far, we have calculated the spin magnetic field
induced by magnetization structures. Equation (17) indicates
that it arises also when the Rashba field, aR, has a gradient.
This case is indeed what is expected at the surfaces ofthin films.
27–29We consider a thin film in the xyplane with
aR¼ð0;0;aRðzÞÞalong the zaxis. When nis within the xy
plane, we obtain BR¼/C0m
e/C22hðrzaRÞn. If the Rashba interac-
tion decays at the length d, we might approximate
rzaR/C242aR=d, resulting in a magnetic field of 2.5 kT if
d¼1 nm. This leads to an intriguing possibility of spin
manipulation in very thin films.
Since the fields satisfy the Maxwell’s equations, the spin
electromagnetic fields propagate through the medium; wavesolutions propagating with a velocity
c
s/C171ffiffiffiffiffiffiffiffi/C15slsp ; (21)
where1
ls¼e/C22h
mn1, and electric permittivity in the diffusive
case is /C15s¼P
6ð6Þr6s6. It should be noted that the signs
of/C15sandlsmay be negative depending on the material. If
the product of the /C15sandlsis positive, the spin electromag-
netic wave propagates. However, it does not propagate if theproduct is negative.
As an example, let us consider the limit of a strong sd
coupling, /C23
/C0¼0( i . e . , Dsd¼/C15F). Approximating1
ls
’3e2/C22h2
20m2/C23and /C15s’e2/C22h2
3m2k2
F/C23s2(/C23,kF,a n d shere are spin-
averaged quantities), we obtain cs¼3
2ffiffi
5p1
kFs.F o r k/C01
F¼1:5
A˚ands¼10/C015s, the spin electromagnetic wave propa-
gates with a speed of cs¼1/C2105m/s. The propagating
wave here is a collective mode of conduction electron and
magnetization. The speed estimated here happens to be
accidentally of the same order as that for spin wavesobserved in a ferrimagnetic insulator,
30but the propagationmode here provides an informa tion transmission path dif-
ferent from spin waves.
IV. EFFECT OF SPIN RELAXATION: SPIN DAMPING
MONOPOLE
According to our result, Eq. (17), spin electromagnetic
fields satisfy a conventional Faraday’s induction law,
$/C2ERþ@BR
@t¼0. There is, therefore, no monopole terms in
ERandBR. Let us examine the effect of spin relaxation on the
Rashba-induced field. The spin relaxation is modeled using
the Hamiltonian for random potential due to impurities, vso,a s
Hsr¼/C0iX
ijk/C15ijkðrivsoÞrjrk: (22)
As shown in Ref. 15, the spin relaxation leads to a new con-
tribution to the spin electric field
E0
R¼/C0m
e/C22hbRðaR/C2ðn/C2_nÞÞ; (23)
where bRis a parameter representing spin relaxation
strength. We see that spin relaxation replaces _nin Eq. (17)
byn/C2_n, i.e., induces a perpendicular component to _n. The
role of spin relaxation here is thus just the same as in thecase of equation of motion for spin (the Landau-Lifshits-
Gilbert equation), where spin relaxation induces the Gilbert
damping term proportional to n/C2_n. A significant feature of
Eq.(23) is that a DC component of spin motive force arises
from precession of uniform magnetization, in sharp contrast
to the case without spin relaxation, Eq. (17). In fact, for a
small precession around the zaxis, ðn
x;ny;nzÞ
¼ðdncosxt;dnsinxt;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1/C0ðdnÞ2q
Þ, the time-average of
hn/C2_ni’2xðdnÞ2^zis finite, while hERi/h _ni¼0.
The spin relaxation contribution, E0
R, has a significant
feature that it is not possible to write it as a time derivative of
a local vector. A direct consequence of this fact is that wenow have a monopole term since r/C2 E
0
Rþ_B0
R/C17jmis non-
vanishing for any local vector B0
R. Thus, the Faraday’s induc-
tion law does not hold but a monopole current, jm, emerges,
as was first pointed out in Ref. 25. The monopole is called
spin damping monopole. It arises only when magnetization is
dynamic, and it is thus different from monopoles in particlephysics
31,32and the hedgehog monopole in ferromagnets9dis-
cussed in Sec. II B. Unlike those monopoles, spin damping
monopole is not a topological object, and hence its charge isnot quantized. Although spin damping monopole is not a topo-
logical object, it appears in the Faraday’s induction law, and it
has a physical meaning; it converts spin dynamics to chargedynamics by inducing E
0
Rand the other way round. The
monopole thus plays an essential role in integrating spin-
tronics into conventional electronics.
V. SUMMARY AND DISCUSSIONS
We have presented a brief review of the spin Berry’s
phase from the viewpoint of effective electromagnetic
fields that couple to electron’s spin and discussed the172609-5 G. Tatara and N. Nakabayashi J. Appl. Phys. 115, 172609 (2014)
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203.92.101.73 On: Sat, 20 Dec 2014 13:34:39effects of Rashba spin-orbit interaction and spin relaxation in
the regime of strong sdexchange interaction. A summary of
the conventional (topological) spin Berry’s phase contribution
and Rashba-induced contribution is shown in Table I.
The total electromagnetic field s acting on conduction elec-
trons in ferromagnetic metals is the sum of the electromagnetic
fields coupled to charge ( EandB) and the spin electromag-
netic fields, i.e., Eeff¼EþEsand Beff¼BþBs.T h e
spin electromagnetic fields read Es¼EðadÞ
sþERþE0
Rand
Bs¼BðadÞ
sþBRþB0
Rif we take account of the adiabatic
and Rashba-induced contributions we have discussed. TheAmp /C19ere’s law then reads r/C2 B
eff/C0/C15l_Eeff¼lj,w h e r e /C15
andlare the electric permittivity and magnetic permeabil-
ity for charge, respectively. This equation is written also as
r/C2 B/C0/C15l_E¼lðjþjðSEMF ÞÞ; (24)
where
jðSEMF Þ/C17/C01
lr/C2 Bsþ/C15sls
l_Es (25)
is the current generated by the spin electromagnetic fields. In
the low frequency limit, xs/C281, where xis the angular fre-
quency, the second term on the right-hand side reduces to
rsEs. Let us focus on the contribution from ERandE0
Rin
this limit. jðSEMF Þthen reads
jðSEMF Þ’/C0m
e/C22hrsaR/C2_nþbRðn/C2_nÞ ðÞ ½/C138 : (26)
This expression is interesting from the viewpoint of the
inverse spin Hall effect.33In fact, we see that jðSEMF Þarises
from _nþbRðn/C2_nÞ ðÞ , and this quantity was argued to be the
source for a spin current in diffusive regime in Ref. 34.I nE q .
(26), therefore, the Rashba interaction is playing a role of con-
verting the spin current source to a charge current. It would be
interesting to extend the present work on a uniform system tothe case of a junction and discuss the spin pumping and the
i n v e r s es p i nH a l le f f e c ti nm o r ed e t a i l .
ACKNOWLEDGMENTS
The authors thank H. Kohno, Shibata, and H. Saarikoski for
valuable discussions. N.N. is fina ncially supported by the JapanSociety for the Promotion of Science for Young Scientists. This
work was supported by a Grant-in-Aid for Scientific Research(C) (Grant No. 25400344) from Japan Society for the Promotion
of Science and UK-Japanese Collaboration on Current-Driven
Domain Wall Dynamics from JST.
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34A. Takeuchi, K. Hosono, and G. Tatara, Phys. Rev. B 81, 144405 (2010).TABLE I. Summary of conventional (topological) spin electromagnetic field and Rashba-induced spin electromagnetic field. The conventional one aris es from
non-coplanar structures, while the Rashba one arises from general inhomogeneous structures. All the contributions satisfy the Maxwell’s equation s (with or
without monopole terms). The conventional field and the Rashba-induced field with spin relaxation may contain monopoles, whose origin is topological in the
former case and non-topological in the latter case.
Topological Rashba With relaxation
Es/C0/C22h
2en/C1ð_n/C2r inÞ /C0m
e/C22hðaR/C2_nÞ/C0bRðaR/C2ðn/C2_nÞÞ
Bs /C22h
4e/C15ijkn/C1ð r jn/C2r knÞm
e/C22h½r /C2 ð aR/C2nÞ/C138To be done
Magnetization structures Non-coplanar General
Maxwell’s equation /H17034/H17034 /H17034
Monopole Topological /C2 Non-topological172609-6 G. Tatara and N. Nakabayashi J. Appl. Phys. 115, 172609 (2014)
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203.92.101.73 On: Sat, 20 Dec 2014 13:34:39 |
1.1607306.pdf | Reaction pathway and free energy barrier for defect-induced water dissociation on the
(101) surface of TiO 2 -anatase
Antonio Tilocca and Annabella Selloni
Citation: The Journal of Chemical Physics 119, 7445 (2003); doi: 10.1063/1.1607306
View online: http://dx.doi.org/10.1063/1.1607306
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/119/14?ver=pdfcov
Published by the AIP Publishing
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On: Tue, 28 Oct 2014 17:11:39Reaction pathway and free energy barrier for defect-induced water
dissociation on the 101surface of TiO 2-anatase
Antonio Tiloccaa)and Annabella Selloni
Department of Chemistry, Princeton University, Princeton, New Jersey 08544
~Received 21 April 2003; accepted 17 July 2003 !
The adsorption of a water molecule on a partially reduced TiO 2anatase ~101!surface has been
studied by first-principles molecular-dynamics simulations. At variance with the stoichiometricsurface, dissociation of water close to the oxygen vacancy is energetically favored compared tomolecular adsorption. However, no spontaneous dissociation was observed in a simulation ofseveral picoseconds, indicating the presence of an energy barrier between the molecular anddissociated states. The free energy profile along a possible dissociation path has been determinedthrough constrained molecular dynamics runs, from which a free energy barrier for dissociation of;0.1eVisestimated.Onthebasisoftheseresults,amechanismforthedissociationofwateratlow
coverage is proposed. © 2003 American Institute of Physics. @DOI: 10.1063/1.1607306 #
I. INTRODUCTION
The adsorption and reactivity of water on TiO 2surfaces
is an essential aspect of many applications of this material. Inparticular, water–surface interactions play a key role in pho-tocatalytic processes like hydrogen production
1and decon-
tamination of polluted water.1–3A central issue in under-
standing these processes is the nature of water adsorption,i.e., whether water is adsorbed in molecular or dissociatedform.
4The surface hydroxyl groups produced by dissociation
change the chemical properties of the surface. For instance, arecent study has shown that bridging OH groups are directlyinvolved in scavenging photoexcited electrons by reactingwith molecular oxygen, and molecular water hydrogen-bonded to surface OH groups may negatively affect this pro-cess by preventing access of O
2to the hydroxyl groups.5
Many experimental,6,7and theoretical8–11studies have ad-
dressed the nature of water adsorption on rutile TiO 2(110).
However the picture remained controversial until recentscanning tunneling microscopy ~STM!experiments,
12,13sup-
ported by density-functional theory ~DFT!calculations,13
clearly showed that, at low coverage, water dissociation on
the rutile ~110!surface occurs exclusively on oxygen vacan-
cies.
Although the anatase polymorph of TiO 2is known to be
more active than rutile for several photocatalyticapplications,
14so far only a few studies of water–anatase
surface interactions have been published. For the most stableanatase ~101!surface,
15DFT calculations found that, irre-
spective of coverage, molecular adsorption is always favoredin the absence of defects.
16This result is consistent with
recent temperature-programmed desorption ~TPD!and x-ray
photoelectron spectroscopy ~XPS!experiment,17which
found that the adsorbed water is predominantly bound to thesurface in a molecular state on anatase ~101!.
Similarly to rutile ~110!, the anatase ~101!surface showsboth fully coordinated (6 c) and under-coordinated (5 c)T i
atoms, as well as threefold (3 c) and twofold (2 c) coordi-
nated oxygens. Compared to rutile, the surface of anataseshows less tendency to form oxygen vacancies,
18presumably
because the removal of a bridging oxygen leads to the for-mation of a fourfold coordinated titanium atom (Ti
4c),
which is less stable than the Ti 5csites formed at oxygen
vacancies on rutile ~110!.17However, undercoordinated Ti 4c
sites are actually present at the step edges of anatase ~101!.
Such sites have been experimentally identified by STM18and
found to play an important role in the chemistry of thissurface.
17,18
To obtain insight into the role of low-coordinated defect
sites in the surface chemistry of anatase, and, more generally,of titania surfaces, in this paper we study the adsorption of aH
2O molecule on partially reduced anatase ~101!using first-
principles Car–Parrinello molecular dynamics ~CPMD !
simulations.19A few investigations of water on defect-free
and defective oxide surfaces have been already carried outusing this approach.
9,10,20–24For instance, spontaneous disso-
ciation of water on defect-free rutile TiO 2(110) was reported
to occur in CPMD trajectories at 500 K.9In a subsequent
study,21however, molecular water was found to be stable at
350 K on the ~110!surface, whereas spontaneous dissocia-
tion was observed at oxygen vacancies of the defectiveTiO
2(100) surface. Other CPMD investigations showed that
isolated water molecules spontaneously dissociate at defec-tive MgO ~100!surfaces, but not on the defect-free surface.
20
In more recent work,10,24mixed molecular–dissociated water
layers have been reported to occur on both TiO 2(110) and
MgO ~100!at high coverages. Altogether, these studies show
that CPMD simulations undoubtedly represent a powerfultool to explore sorbate dynamics on surfaces. However, sincethe accessible time scales are still very limited, the dynami-cal simulations should be complemented by structural opti-mizations as well as calculation of ~free!energy barriers.
22,23 a!Electronic mail: atilocca@princeton.eduJOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 14 8 OCTOBER 2003
7445 0021-9606/2003/119(14)/7445/6/$20.00 © 2003 American Institute of Physics
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On: Tue, 28 Oct 2014 17:11:39II. COMPUTATIONAL APPROACH
Calculations have been performed using the Perdew–
Burke–Ernzerhof ~PBE!25functional for the exchange-
correlation term of the electron–electron interaction. The useof a gradient corrected functional is essential to obtain areliable description of hydrogen bonds and water–surfaceintermolecular interactions; in addition, the PBE functionalhas been shown to perform well for bulk and surface TiO
2
anatase.15Vanderbilt ultrasoft26pseudopotentials have been
used to describe electron–core interactions, and valenceelectrons included the O2 s,2pand Ti3s,3p,3d,4s
shells. The smooth part of the wave functions was expandedin plane waves up to a kinetic-energy cutoff of 25 Ry, whilethe augmented density cutoff was 200 Ry. The large size ofthe periodic supercell ~see below !allowed us to restrict the
k-sampling to the Gpoint. All these approximations have
been extensively tested, and found adequate in previous stud-ies of titania surfaces.
15,16,27,28We modeled the anatase sur-
face using a supercell approach; each supercell exposes asurface area of 10.24 311.36 Å
2~corresponding to three sur-
face cells of the undefected surface !, and includes a periodi-
cally repeated slab of four Ti 6O12layers, corresponding to a
thickness of ;6 Å. We checked that geometries and adsorp-
tion energies do not change significantly if they are calcu-lated using a thicker slab.
27,28The slabs are separated by a
vacuum region of ;10 Å along the perpendicular direction.
A point defect was created by removing a bridging oxygenfrom the top layer, so that the net composition of the super-cell~without water !is Ti
24O47. A rather large surface cell is
needed here to effectively isolate the defect, by minimizingthe interaction with its periodic replicas. The optimized slabgeometry is shown in Fig. 1.Awater molecule was adsorbedon the top layer only, while the atoms of the bottom layerwere fixed in their bulk positions. Geometry optimizationswere carried out through damped dynamics until every com-ponent of the ionic forces was less than 0.05 eV/Å. In eachcase, the initial configuration was heated to ;400 K and then
left free to evolve in a short MD run; a low potential energyconfiguration during such run was used as the starting pointfor the damped dynamics. At the end of the minimization,additional MD runs were carried out to test the stability ofthe optimized structure.
In most simulations, a fictitious electronic mass
m5500
a.u. and a time step dt55 a.u.(0.121fs) were used, together
with the ‘‘true’’hydrogen mass of 1 amu. With these param-eters, the total energy was well conserved in all trajectorieswhere water remained undissociated. Instead, instabilitiesoccurred in the case of dissociation, probably connected tothe fast electronic rearrangements brought about by the pro-cess, as well as to the low energy gap between occupied andempty electronic states typical of partially reduced TiO
2
surfaces.29–31Therefore, more conservative parameters ( dt
54 a.u., hydrogen mass 52 amu, and m5700 a.u. !were cho-
sen for studying the dissociation dynamics. MD trajectorieswere run at an average temperature around 300 K ~RT!. Al-
thoughTPD experiments
17show that the desorption tempera-
ture of water coordinated to Ti 5cis;250 K, in our simula-
tions the water molecule is coordinated to Ti 4c, with a much
larger binding energy ~see below !. Indeed no desorption hasbeen observed throughout the calculated trajectories of this
work.
III. RESULTS
A. Molecularly versus dissociatively adsorbed water:
Structure and energetics
The stoichiometric anatase ~101!surface has a corru-
gated profile, showing alternate rows of Ti 6c,O2c,T i5c, and
O3crunning along @010#~Fig. 1 !. On this surface, water ad-
sorbs in molecular form,16with the oxygen atom above a
Ti5csite, and the hydrogens forming H-bonds with two sur-
face O 2catoms. When the surface is partially reduced by
removal of an O 2c, the Ti 6cand Ti 5catoms to which this
bridging oxygen was originally coordinated turn into fivefold(Ti
5c) and fourfold coordinated (Ti 4c) sites, respectively. We
started our study of molecular water adsorption close to thevacancy, by successively placing the molecule above theseundercoordinated Ti sites @Figs. 1 ~a!and 1 ~b!#, as well as
above an additional Ti
5catom @Fig. 1 ~c!#. The final, opti-
mized structures—that we shall call M1, M2, M3—areshown in Fig. 2, together with their corresponding adsorptionenergies. A further starting geometry with the water oxygenroughly replacing the missing surface O
2cconverged to the
same final state as M2. It appears that the chemisorption onTi
4cis much stronger than on Ti 5c. The most stable structure
is M2, with the water oxygen bonded to Ti 4cand the two
FIG. 1. ~top!The defective anatase ~101!slab used in the calculations ~op-
timized geometry !. The three undercoordinated titanium ions considered as
possible adsorption sites are shown as large gray spheres.Atoms a and c arefivefold coordinated, while ~b!is fourfold coordinated. ~bottom !Side view
of the same structure. Note the upward relaxation of the oxygen atom belowthe vacancy.7446 J. Chem. Phys., Vol. 119, No. 14, 8 October 2003 A. Tilocca and A. Selloni
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On: Tue, 28 Oct 2014 17:11:39hydrogens forming strong H-bonds (R H–O2c 51.81Å) with
two surface bridging oxygens. Structure M1, in which wateris coordinated to both Ti
4cand Ti 5cand forms a weaker
H-bond with an O 2c(RH–O2c 52 Å) has a somewhat smaller
adsorption energy. The water oxygen in M1 roughly replacesthe missing O
2c; the stabilization resulting from restoring
the coordination shell of two surface titania is counterbal-anced by a less effective arrangement of the hydrogen atomscompared to M2. In the latter, a small shift of the moleculeaway from the vacancy allows the formation of two stronghydrogen bonds with the surface. Moreover, in M2 the bondbetween the surface Ti
4catom and the water oxygen is much
stronger than the two Ti–O bonds in M1, as shown by thecorresponding bondlengths, 2.11 Å for the Ti
4c–O
bondlength in M2 versus 2.32 and 2.55 Å for the Ti 4c–O and
Ti5c–O bondlengths in M1. Finally, the adsorption energy of
a water molecule bonded only to Ti 5c~M3!is much lower,
and close to the 0.74 eV value for Ti 5c-coordinated water on
the stoichiometric surface.16
Structures with dissociated water were generated starting
from M1 and M2 and moving a water proton to theH-bonded O
2c. The corresponding energy-minimized struc-
tures are shown in Fig. 3. The D1 geometry originated fromM1 shows two bridging hydroxyl groups (OH
b) and no
H-bonds; this structure is more stable than the one ~labeled
D2!obtained by dissociation of M2, featuring one OH band
one terminal hydroxyl (OH t). This is not surprising, as sin-
gly coordinated hydroxyl groups are expected to be lessstable than bridging ones.
12,21Indeed, when a RT molecular
dynamics simulation was started from D2, the OH tbonded to
Ti4cgradually migrated towards the vacancy in a bridging
position between Ti 4cand Ti 5c~structure D3 !, where it re-
mained stable during a 1.5 ps trajectory. Optimization of thisgeometry led to an adsorption energy of 1.85 eV, essentiallythe same as that of D1. Other possible dissociated configu-
rations were also considered, but found to be substantiallyless stable than those in Fig. 3.
In summary, similarly to what was found for rutile
TiO
2(110),12,13at an oxygen vacancy of the anatase ~101!
surface dissociative adsorption of water is thermodynami-cally favored with respect to molecular adsorption. The sta-bility of the dissociated configurations D1 and D3 was fur-ther tested by running RT simulations; no recombination tomolecular water was observed in 1.5 ps.
B. Dissociation pathway and barrier
On the basis of the calculated molecular and dissociated
configurations for adsorbed water, a direct dissociation pathlinking M1 with D1 seems plausible. However, we neverobserved a dissociation of this type in our molecular dynam-ics simulations, and found rather that M1 transforms to theother, more stable, molecular structure M2. ~This occurs in a
time of ;1.5 ps at RT: In the first picosecond the H-bond is
broken and the water molecule moves along @010#forming a
new H-bond with the O
2con the opposite side of the va-
cancy. Then this H-bond is broken as well, and the molecule
migrates along @1¯01#, ending with its two hydrogen pointing
towards the two O 2c’ s ,a si nM 2 . !Thus, M2 is the most
appropriate starting point for studying the dissociation. How-ever, starting from M2, the molecule only showed local os-cillation about the adsorption site, with no dissociation in asimulation of 6 ps. This suggests the presence of an energybarrier to the dissociation, larger than the thermal energy at300 K.
In order to identify a possible reaction path for the acti-
vated dissociation, we used the Blue Moon ensemble
method,
32,33and took the distance between the transferred
FIG. 2. Optimized geometries of molecular water close
to the vacancy. The reported adsorption energies are
calculated as DHads52@Etot2Ebare2Ewat#, where
Etot(Ebare) are the energy of the slab with ~without !ad-
sorbate and Ewatis the energy of an isolated water mol-
ecule calculated in the same supercell.
FIG. 3. Optimized geometries of dissociated waterclose to the vacancy. D1 and D2 are obtained aftermoving one proton of M1 and M2, respectively, to theH-bonded surface bridging oxygen. D3 is obtained afterreoptimizing the final configuration of an MD trajectorystarted from D2.7447 J. Chem. Phys., Vol. 119, No. 14, 8 October 2003 Defect-induced water dissociation on anatase
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On: Tue, 28 Oct 2014 17:11:39proton and the target surface O 2cas the reaction coordinate
~qin the following !. Ten constrained simulations were car-
ried out. In each trajectory, qwas constrained to a fixed value
and after a short equilibration the running average of theconstraint force was calculated until it converged to a con-stant value. This required between 0.5 and 1.5 ps: Longertrajectories were needed to achieve convergence of the con-straint force close to the transition state ~TS!. In this way, the
reaction coordinate was gradually decreased from 1.86 Å~M2 state !to 0.987 Å ~D2 or D3 states !, i.e., the system was
smoothly driven from the reactant to product.
Within this approach, the free energy difference between
statesq
aandqbis obtained by integrating the mean con-
straint force along q:33
DG52E
qaqbdq8^f&q8. ~1!
This method, in conjunction with CP molecular dynam-
ics, has been successfully applied to study many reactiveprocesses in complex systems.
22,23,34–39It is generally more
efficient than performing a constrained ~0K!structure opti-
mization at each point along the reaction path, because theconvergence of the mean force is usually fast, i.e., relativelyshort trajectories are required. Moreover, the finite tempera-ture allows to include entropic and anharmonic effects ex-plicitly, and a reliable estimate of the free energy is obtained,provided a meaningful reaction coordinate is selected.
22
The mean constraint force and its integral are shown in
Fig. 4.The qvalue corresponding to theTS can be located as
the point in which the mean constraint force changes sign.35
Looking at Fig. 4 this occurs at qTS;1.25Å. We checked
that this is a reliable TS by starting two unconstrained MDtrajectories from the qpoints closest to it, on the reactant and
product sides. In both cases the free dynamics led to thecorresponding molecular or dissociated species in a veryshort time. The activation free energy for the dissociation(DG
‡) is 0.1 eV. As a further consistency check, a con-
strained geometry optimization with q5qTSwas carried out.The adsorption energy of the resulting structure ~labeled TS
in Fig. 6 !is 1.35 eV, corresponding to a potential energy
barrier of 0.12 eV, close to the free energy barrier. Besidesconfirming the accuracy of the free energy calculations, thisallow estimating a low entropy difference: DS
‡;6.7
31025eV/K.
Forq,qTS, the mean force is large and negative, indi-
cating that the proton is strongly attracted by the surfaceoxygen.
36Whenq51.03Å, the mean force is still large due
to the O 2c–H bond being constrained to a value larger to the
equilibrium bond distance. In the last constrained trajectory,qhas been chosen equal to the O
2c–H bond distance in the
D3 structure ~0.987 Å !. The mean force decays to zero,
showing that an equilibrium configuration was reached. Inthis last run, the expected migration of O
tH to the vacancy
was completed in 1.4 ps.
The dissociation free energy ( DGdiss) that we obtain
from this calculation ~20.107 eV !is smaller, in absolute
value, than the potential energy difference between the mo-lecular M2 and the dissociated D3 state. In fact, during theconstrained dynamics towards the products, before reachingthe final equilibrium state, the system probes many configu-rations corresponding to the metastable dissociated D2 state,whose potential energy is very close to the one of the mo-lecular state M2. As the free energy basin of the productsincludes configurations close to either D2 and D3 minima, itwill receive a ~potential energy !contribution from both. In-
deed the chosen reaction coordinate, while suitable to followthe proton transfer up to the TS, cannot discern between D2and D3 states. A different reaction coordinate, like the dis-tance between the water oxygen and O
2c, would be more
suitable in this context, but presumably not so effective indescribing the proton transfer. The point is that a single con-straint is in some cases not enough to control the full reactionpath from reactants to products.
35,36,40The reaction coordi-
nate that we choose is most effective for evaluating the freeenergy barrier, which is our main interest in this work. Wealso note that some positive entropic contribution to DG
diss
cannot be ruled out, but it is unlikely that they are the sole
responsible of the observed difference between free and po-tential energy of dissociation.
FIG. 4. Free energy profile ~top!and mean constaint force ~bottom !along
the reaction coordinate q. Labels as in Figs. 2 and 3.
FIG. 5. Time dependence of the OwH~thick line !and O2cH~thin line !bond
distances, following heating of the system to 700 K.7448 J. Chem. Phys., Vol. 119, No. 14, 8 October 2003 A. Tilocca and A. Selloni
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On: Tue, 28 Oct 2014 17:11:39IV. DISCUSSION AND CONCLUSIONS
The kinetic rate constant can be estimated by the
transition-state theory ~TST!expression:
k5kBT
he2DG‡/kBT, ~2!
wherekBandhare Boltzmann’s and Planck’s constants, re-
spectively. Equation ~2!yieldsk50.11ps21at 300 K. Thus,
it isa posteriori not surprising that no dissociation of the
molecular M2 configuration was observed in the above-mentioned 6 ps unconstrained trajectory at 300 K. As a con-sistency check, we performed a further MD simulation inwhich, after raising the temperature to ;700 K, we let the
system evolve without constraints. We found that after only0.2 ps a water proton moved to an O
2c~Fig. 5 !; the transfer
was followed by two quick recombinations, with the protonmoving between the two oxygens, before ending in the dis-sociated state.Then the migration of the terminal hydroxyl tothe vacancy takes place, as shown by the increase of theO
w–H distance in Fig. 5.
A possible mechanism for the dissociation of water ad-
sorbed at low coverage on defective anatase ~101!can be
sketched on the basis of these findings ~see Fig. 6 !. Molecu-
lar adsorption on Ti 4cis favored initially. It can in principle
occur in two different modes:With the water oxygen roughlyreplacing the missing bridging oxygen ~M1!or in the more
stable mode M2.Adirect dissociation of the molecule insidethe vacancy
21is unlikely: Molecular migration from M1 to
the other stable site is observed instead. Dissociation of wa-ter in this site, leading to a terminal and a bridging hydroxyl,is practically thermoneutral. The driving force of the disso-ciative process is the subsequent migration of the OH
tto the
vacancy site, leading to a more stable dissociated state withtwo bridging hydroxyls ~D3!.While the last diffusive process
is spontaneous at T5300K, the initial proton transfer to a
surface O
2crequires the crossing of a somewhat larger bar-
rier. Whereas this barrier is high enough to hinder the disso-ciation on the time scales available by ab initio moleculardynamics, on macroscopic time scales it is likely that all
water molecules near vacancy sites are dissociated at lowcoverage.
In conclusion, our first principles molecular-dynamics
simulations have elucidated the dissociation pathway of awater molecule adsorbed close to a low-coordinated defectsite on the TiO
2anatase ~101!surface. We have found that
the dissociation does not follow a direct pathway. Eventhough the overall barrier is small, the process is complex,involving a few different intermediate states.
ACKNOWLEDGMENTS
The calculations of this work have been performed on
the Lemieux Terascale Computing System at the PittsburghSupercomputer Center and on the IBM-SP3 computer at theKeck Computational Materials Science Laboratory in Princ-eton. We acknowledge support by the National ScienceFoundation under Grant No. CHE-0121432.
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Citation: Appl. Phys. Lett. 108, 112403 (2016); doi: 10.1063/1.4944419
View online: http://dx.doi.org/10.1063/1.4944419
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Published by the American Institute of Physics
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J.Sampaio,1,a)A. V. Khvalkovskiy,2,3M.Kuteifan,4M.Cubukcu,1D.Apalkov,2V.Lomakin,4
V.Cros,1and N. Reyren1,b)
1Unit/C19e Mixte de Physique, CNRS, Thales, Univ. Paris-Sud, Universit /C19e Paris-Saclay, 91767, Palaiseau, France
2Samsung Electronics, Semiconductor R&D Center (Grandis), San Jose, California 95134, USA
3Moscow Institute of Physics and Technology, State University, Moscow 141700, Russia
4Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla,
California 92093-0407, USA
(Received 22 October 2015; accepted 5 March 2016; published online 16 March 2016)
In order to increase the thermal stability of a magnetic random access memory cell, materials with
high spin-orbit interaction are often introduced in the storage layer. As a side effect, a strong
Dzyaloshinskii-Moriya interaction (DMI) may arise in such systems. Here, we investigate the
impact of DMI on the magnetic cell performance, using micromagnetic simulations. We find thatDMI strongly promotes non-uniform magnetization states and non-uniform switching modes of the
magnetic layer. It appears to be detrimental for both the thermal stability of the cell and its switch-
ing current, leading to considerable deterioration of the cell performance even for a moderate DMIamplitude.
VC2016 Author(s). All article content, except where otherwise noted, is licensed under a
Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/ ).
[http://dx.doi.org/10.1063/1.4944419 ]
Recently, the development of magnetic random access
memories (MRAM) for dense memory products such as
dynamic or static random access memories became focused
on magnetic cells with a high perpendicular magnetic anisot-
ropy (PMA). These designs are believed to offer an
improved thermal stability at very advanced technological
nodes of 20 nm and below.1,2The PMA storage (a.k.a.
“free”) magnetic layer is based on magnetically soft CoFeB,
which has a good lattice matching with the MgO barrier. The
interface between MgO and CoFeB provides sufficientlystrong PMA to hold perpendicular a CoFeB layer about 1 nm
thick.
3In order to further enhance the thermal stability of the
cell, elements with a strong spin-orbit coupling (SOC), such
as W, Pt, Ta, or Ir are often introduced into the free layer
(FL).4–7However, recent studies demonstrated that a very
large Dzyaloshinskii-Moriya exchange interaction (up to a
large fraction of the Heisenberg exchange) may arise at the
FM/SOC film interface.8,9Dzyaloshinskii-Moriya interaction
(DMI) can dramatically change the magnetic state of thefilm. It was shown to induce a significant spin tilt at the bor-
ders,
10,11for large DMI amplitude, it can stabilize cycloidal
states and skyrmion lattices.12One might expect that the
switching current could be reduced by the tilt produced even
by the smallest DMI because the initial spin-transfer torque
(STT) would be more efficient. Detailed simulations prove
the situation to be more complicated. DMI also drasticallychanges the domain wall (DW) energy and, thus, the mag-
netic switching process,
13both under field and under STT.
Consequently, it can then be anticipated that DMI may affect
the landscape of stable states and the reversal mechanisms,
which are critical to the operation of MRAM cells. In thisletter, we aim to analyze the influence of DMI on MRAM
cells with perpendicular magnetization, in the range of DMImagnitude that may exist in typical material stacking usedfor MRAM elements.
DMI describes the chiral exchange interaction that
favors rotations between neighboring spins.
14,15The energy
of an interfacial DMI between two neighboring spins S1and
S2can be written as
EDM¼~d12/C1ð~S1/C2~S2Þ; (1)
where ~d12is the DMI vector for these spins. For an interface
between perfectly isotropic films, ~d12is given by d^ez/C2~r12,
where dis the atomic DMI magnitude, ^ezthe unit vector nor-
mal to the interface, and ~r12the unit vector pointing from S1
toS2. In the micromagnetic approximation of continuous
magnetization, the interfacial DMI can be written as a vol-ume energy density
10
EDM¼Dðmz@xmx/C0mx@xmzþmz@ymy/C0my@ymzÞ;(2)
where D¼Cd/(at) is the micromagnetic DMI magnitude, C,
a, and tare a geometric factor dependent on the film stack-
ing, the lattice constant, and the thickness of the ferromag-netic film, respectively.
The DMI magnitude in thin magnetic films similar to
those used in MRAM structures may reach up to a fewmJ/m
2.16,17For example, recent measurements showed that
D¼0.053 mJ/m2for Ta/CoFe 0.6 nm/MgO,181.2 mJ/m2for
Pt/CoFe 0.6 nm/MgO,18a n d7m J / m2for Ir/Fe monolayer.8As
we show below, even for Din a range 0.3–1 mJ/m2,w es e ea
considerable impact on the MRAM cell performance.Performance of an STT-MRAM cell is characterized by two
key parameters: the thermal stability factor D, and the critical
switching current density j
c0.2Dequals to the energy barrier
height between the two magnetic states Ebnormalized for thea)Present address: Laboratoire de Physique des Solides, Univ. Paris-Sud,
Universit /C19e Paris-Saclay, CNRS, UMR 8502, 91405 Orsay Cedex, France.
b)Electronic mail: nicolas.reyren@thalesgroup.com
0003-6951/2016/108(11)/112403/4 VCAuthor(s) 2016.
108, 112403-1APPLIED PHYSICS LETTERS 108, 112403 (2016)
operating temperature D¼Eb/(kBT), where kBis the
Boltzmann constant; it defines the information retention timeast
0expðDÞ,w h e r e t0is typically of the order of 1 ns. jc0is
the zero-temperature instability threshold current density,
which defines the scale of the currents required for read and
write operations. In our study, we investigate how Dandjc0
change in presence of strong DMI effect using micromagnetic
simulations. We exploit three numeric techniques: static anddynamic micromagnetic simulations using Mumax3
20and
OOMMF19(for preliminary studies at T¼0) open source
codes, and nudged-elastic band (NEB) simulation of switchingpaths, using the FastMag code.
21W eu s ea sam o d e ls y s t e ma
perpendicularly magnetized disk of 32 nm diameter and 1 nmthickness, with the following material parameters: saturation
magnetization ( M
S) of 1.03 MA/m, exchange stiffness ( A)o f
10 pJ/m, perpendicular magnetocrystalline anisotropy ( Ku)o f
0.770 MJ/m3, and a Gilbert damping factor ( a)o f0 . 0 1 .T h e s e
parameters are typical of a perpendicularly magnetized CoFeBactive layer in a magnetic tunnel junction (MTJ) used in an
MRAM cell. With these values, we get an effective anisotropy
for the disk K
eff¼Ku/C01
2Nz/C0NxÞl0M2
s¼187 kJ =m3/C0
(where Niare the demagnetization factors of the disk22), corre-
sponding to l0HKeff¼364 mT, a threshold DMI Dc¼1.7 mJ/
m2,a n da n D¼KeffV/(kBT)¼36, calculated in a uniform rota-
tion approximation.
We first analyze how DMI affects the equilibrium quasi-
uniform states. In these simulations made using the MuMax3
code (version 3.6.1), the magnetization was initially set up
and let to completely relax. Once Dincreases, we see that
DMI induces a radial tilt of the magnetization on the bordersof the disk. As a result, the total micromagnetic energy (thesum of exchange, dipolar, anisotropy, and DMI energies)
reduces with D(Fig. 1(a)). This observation is in agreement
with other theoretical results reported for similarsystems.
10,11
Next, we study the evolution with Dof the system
energy once the magnetic disk has a straight DW in the mid-dle,E
DW,see Fig. 1(a). In this simulation, the magnetization
distribution was generated manually. (For metastable states,the system relaxes in the illustrated states, and the values for
the unstable states were obtained using an ideal straight
wall.) Even though this is not a true relaxed state, since it issymmetric it represents an energy extremum state on apossible switching path. We observe that the DMI lowers
E
DWand stabilizes a N /C19eel domain wall even if we started
from a Bloch wall (for D/C210.05 mJ/m2). The rate of varia-
tion of EDWwith Dfollows closely the theoretical value of
/C0pS(¼/C010/C016J/(Jm/C02)), where Sis the DW surface.11For
lowD, the DW state has a higher energy than that of a uni-
form state and is unstable (open circles in Fig. 1(a)). But for
D>1.8 mJ/m2, the DW state becomes meta-stable, which
means that a DW may by trapped in the disk center if it gets
there. For even larger D(D>2.6 mJ/m2), DW energy
becomes lower than the energy of the uniform state; thus, itbecomes the system ground state. This is an important resultas this meta-stable DW state may force the use of higher
writing currents, and impairs completely the required binary
operation of a typical MRAM cell (as the system no longerhas only two stable states).
From Fig. 1(a), we see that the energy difference
between the uniform and DW state diminishes with D.T o
accurately estimate the dependence of the energy barrier onD, we exploit the NEB simulations
23,24implemented in the
FastMag code. NEB is a method to calculate a minimum
energy path (MEP), i.e., the path in a configurational space
connecting two ground states (up and down states for ourdisk) with a trajectory having minimum energy span. Usingthis method, it was shown recently that for PMA MRAM
cells of sizes of even 20 nm or less, the domain-wall switch-
ing rather than the uniform rotation may be the primary ther-mal switching mechanism.
2
In Fig. 1(b), we show the MEP calculated using the
NEB method for Dbetween 0 and 3.5 mJ/m2, showing the
intermediate magnetic states as insets. These simulationsshow that MEP is the DW-mediated reversal for all consid-ered values of D(0–3.5 mJ/m
2), confirming the qualitative
conclusion from Fig. 1(a). It also confirms existence of the
metastable states for the DW for D/H114072 mJ/m2(correspond-
ing to the appearance of an intermediate energy minimum inthe curve of Fig. 1(b)). For larger D(D>2.5 mJ/m
2), DW at
the center of the disk becomes the ground state, and the high-
est energy point on MEP becomes an intermediate DW state
close to an edge (see the magnetization distribution in theinsets to Fig. 1(b)).
The energy barrier E
BandDas a function of Dcalcu-
lated by NEB simulations is plotted in Fig. 1(c). For D¼0
FIG. 1. Energy of static states in a nanodisk with DMI. (a) Energy of the quasi-uniform state (red line, circles) and of the DW state (black line, diamond s) ver-
sus DMI magnitude D, determined by micromagnetic simulations. The inset images show some of the stable and metastable configurations (where the colors
red/white/blue correspond to the z magnetization component). The open/filled circles denote (meta) stable/unstable DW states. (b) Minimum energy p aths of
magnetic reversal for D¼0–3 mJ/m2calculated with NEB, showing the DW-mediated reversal. (c) Barrier height calculated from (b). The right axis shows
the corresponding value of the room temperature thermal stability factor D.112403-2 Sampaio et al. Appl. Phys. Lett. 108, 112403 (2016)we get D¼33, which is close to the analytical result D¼36.
This shows that without DMI, the energy difference betweenthe uniform rotation and DW-mediated reversal is small, andDMI strongly promotes the DW-mediated reversal. Once D
increases, Ddramatically drops even for moderate values of
D: by 20% to 27 with D¼0.5 mJ/m
2, and by 40% for D¼1
mJ/m2, with a corresponding six orders of magnitude reduc-
tion of the retention time. For even larger D, we see that the
barrier vanishes completely.
We now investigate the effects of DMI on the STT-
induced switching performance. First, we simulate STTswitching of our FL at zero temperature, assuming a perpen-dicular current uniformly distributed in the disk, with a spinpolarization P¼40%, producing an in-plane torque ~m/C2~p
/C2~m;~pbeing the orientation of the injected spins (perpendic-
ular to the plane), and no out-of-plane torque.
25,26We con-
sidered square current pulses (instantaneous rise-time). Thesimulations show that the switching process starts by anexcitation of oscillations that increase in amplitude, untilmagnetization breaks into the two-domain state with a subse-
quent reversal of the disk by a DW propagation. For D¼0,
the amplitude of the oscillations increases gradually and uni-formly in the disk, while, for finite D, the oscillations are
uneven in amplitude, and strongly localized at the border ofthe disk. This may make the reversal process quite sensitiveon the border properties, such as its shape and roughness, but
also on the spatial discretization of the simulation (see sup-
plementary material
27). To avoid the artefacts related to the
boundary discretization, we used the FastMag code in thesesimulations; its finite element micromagnetic solver allowsdefining the simulated disk with a smooth border.
For each value of the current density j, we extract the
switching time t
swof our FL, defined as the time when the
FL magnetization crosses the equatorial plane (plane z¼0).
In Fig. 2(a), we show the simulation result for 1/ tswas a
function of j, for Dranging between 0 and 2 mJ/m2. We find
that the switching time at a given current density is alwayslarger for larger D. For an MRAM cell, the FL switching
time t
swvaries inversely with jas follows:28
t/C01
sw/j=jc0/C01: (3)
We use Eq. (3)to fit the switching data and extract jc0.I t
appears that even for large Dthe data is reasonably linear in
j, which allows us to fit this data using Eq. (3). The fit result,
jc0, is shown in as a function of Din the inset to Fig. 2.W e
observe that jc0increases with D: moderate at first with 15%
atD¼0.5 mJ/m2, but at a striking pace for larger D, reach-
ing 70% for D¼1 mJ/m2and 110% for D¼1.5 mJ/m2. For
even higher values of D(>2 mJ/m2), the system reaches of-
ten metastable states (with a DW), which impedes the deter-mination of switching times.
As we mentioned above, DMI promotes switching via
very non-uniform modes. Consequently, the cell switchingperformance and it dependence on Dmay become sensitive
to the shape of the sample. In order to verify this suggestion,we perform additional simulations of the STT switching of
the cells with different shapes. We find that while for D¼0,
j
c0does not depend much on the cell, for finite Dthis de-
pendence is considerable. For instance, jc0for 1 mJ/m2ranged from 3.3 up to 8.3 MA/cm2. These findings support
the importance of border resonant modes in the reversal pro-cess in the presence of DMI.
29See supplementary material
for more information about the study of the role of edges andthe dynamics of the switching.
27
We see that DMI leads to an increase in critical switching
current ( jc0) with simultaneous decrease in the thermal stabil-
ity factor ( D). These opposing effects suggests that switching
with STT at finite temperature might be very different fromtheT¼0 K case that we calculate in Fig. 2. To take the ther-
mal effects and DMI into account in determining j
c0, we per-
formed stochastic dynamical simulations, where weintroduced a random magnetic field with a Gaussian amplitudedistribution to simulate the effects of temperature.
20We simu-
lated repeatedly (at least twenty times) a current pulse withthe same STT parameters as before for each set of parameters(D,j,a n d T), and calculated the mean switching time s
sw.I n
the inset of Fig. 3,w es h o w jversus 1/ sswatD¼1m J / m2for
various values of temperature. We extrapolated jc0as before.
In Fig. 3, we show the variation of jc0with Dfor various
values of the temperature. We observe that jc0always
increases with D, with this increase being larger for higher T.
The rise of jc0is exacerbated by temperature: while at 0 K
thejc0atD¼2 mJ/m2is twice that of D¼0, at 300 K the dif-
ference is fivefold. For D¼0, we see that jc0decreases for
higher temperature. This result is in agreement with the sto-chastic macrospin simulations, which also show that even ina uniform switching mode and with a great statistical qualityj
c0is expected to decrease with the temperature (see supple-
mentary material for details27). However, for large D, we see
that this dependence is reversed, and jc0becomes larger for
larger T.
Finally, the influence of DMI on both the MRAM
switching current and thermal stability, quantified by jc0and
D, can also be seen in Figs. 3and1(c). We see readily thatFIG. 2. Switching under current (STT) at zero temperature. Simulated
applied current versus reciprocal switching time for different Dvalues. The
lines are linear fits to Eq. (3). The inset plot shows the extracted jc0as a
function of D.112403-3 Sampaio et al. Appl. Phys. Lett. 108, 112403 (2016)even a moderate DMI of D/C240.5 mJ/m2leads to an increase
injc0and a large decrease in the thermal stability by tens of
percent. This result emphasizes the importance of quantifica-tion and minimization of the DMI magnitude in materials
used for the free layers in MRAM cells, possibly using mate-
rials that induce DMI of opposing sign.
30
During the preparation of this article, an article by Jang
et al. appeared,31which discusses some of the points also
included here.
This work was supported by the Samsung Global
MRAM Innovation Program, and by the NSF Grant Nos.
DMR-1312750 and CCF-1117911.
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details about the effects related to edge roughness, the intermediate states
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Lett. 107, 202401 (2015).FIG. 3. Effects of DMI on the thermal stability and current induced switch-
ing of MRAMs. jc0versus DforT¼0, 50, 100, and 300 K. The inset plot is
the current versus the reciprocal mean switching time ( ssw) for D¼1 mJ/m2,
for temperatures of 50, 100, 200, and 300 K, extracted from multiple (60 to
80) stochastic simulations; the data for T¼0 are also shown.112403-4 Sampaio et al. Appl. Phys. Lett. 108, 112403 (2016) |
1.2431574.pdf | Phase-field model for epitaxial ferroelectric and magnetic nanocomposite thin films
J. X. Zhang, Y. L. Li, D. G. Schlom, L. Q. Chen, F. Zavaliche, R. Ramesh, and Q. X. Jia
Citation: Applied Physics Letters 90, 052909 (2007); doi: 10.1063/1.2431574
View online: http://dx.doi.org/10.1063/1.2431574
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/90/5?ver=pdfcov
Published by the AIP Publishing
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128.59.222.12 On: Thu, 27 Nov 2014 22:56:28Phase-field model for epitaxial ferroelectric and magnetic
nanocomposite thin films
J. X. Zhang,a/H20850Y . L. Li, D. G. Schlom, and L. Q. Chen
Department of Materials Science and Engineering, Pennsylvania State University, University Park,
Pennsylvania 16802
F . Zavaliche and R. Ramesh
Department of Materials Science and Engineering and Department of Physics, University of California,Berkeley, California 94720
Q. X. Jia
MP A-STC, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
/H20849Received 21 September 2006; accepted 9 December 2006; published online 31 January 2007 /H20850
A phase-field model was developed for studying the magnetoelectric coupling effect in epitaxial
ferroelectric and magnetic nanocomposite thin films. The model can simultaneously take intoaccount the ferroelectric and magnetic domain structures, the electrostrictive and magnetostrictiveeffects, substrate constraint, as well as the long-range interactions such as magnetic, electric,and elastic interactions. As an example, the magnetic-field-induced electric polarization inBaTiO
3–CoFe 2O4nanocomposite film was analyzed. The effects of the film thickness, morphology
of the nanocomposite, and substrate constraint on the degree of magnetoelectric coupling werediscussed. © 2007 American Institute of Physics ./H20851DOI: 10.1063/1.2431574 /H20852
Magnetoelectric materials, which are simultaneously
magnetic and ferroelectric, have drawn increasing interestdue to their multifunctionality.
1,2However, natural magneto-
electric single-phase crystals are rare and exhibit weak mag-netoelectric coupling.
3As a result, there have been many
efforts to prepare synthetic magnetoelectrics, i.e., compositesor solid solutions of ferroelectric and magnetic materials.
4–7
In addition to possessing the ferroelectricity and magnetism
in each individual phase, composites are shown to exhibit anextrinsic magnetoelectric coupling. Recently, epitaxial
BaTiO
3–CoFe 2O4/H20849Ref. 8/H20850and BiFeO 3–CoFe 2O4/H20849Ref. 9/H20850
nanocomposite films have been deposited by using pulsedlaser deposition, and magnetoelectric coupling phenomenahave been observed directly. Calculations by Nan et al.
10
and Liu et al.11,12have shown that large magnetic-field-
induced electric polarization /H20849MIEP /H20850could be produced in
nanocomposite films due to the enhanced elastic couplinginteraction.
The main purpose of this letter is to develop a phase-
field model for predicting the magnetoelectric coupling ef-fect for ferroelectric and magnetic nanocomposite thin films.The model simultaneously takes into account the ferroelec-tric and magnetic domain structures, the electrostrictive andmagnetostrictive effects, substrate constraint, as well as thelong-range interactions such as magnetic, electric, and elasticinteractions. As an example, we will study the magnetoelec-tric response in the BaTiO
3–CoFe 2O4nanocomposite films,
i.e., the magnetic-field-induced electric polarization. The ef-fects of film thickness, morphology of nanocomposite, andsubstrate constraint on the magnetoelectric coupling will beinvestigated.
In the model, a given microstructure state is described by
three fields: a local magnetization field M=M
sm=Ms
/H20849m1,m2,m3/H20850, a local polarization field P=/H20849P1,P2,P3/H20850, and an
order parameter field /H9257, which describes the spatial distribu-
tions of the two phases in the composite with /H9257=1 for themagnetic phase and /H9257=0 for the ferroelectric phase. Msis
the saturation magnetization. The total free energy of aferroelectric/magnetic composite is, then, expressed by
F=F
anis /H20849M/H20850+Fexch /H20849M/H20850+Fms/H20849M/H20850+Fexternal /H20849M,He/H20850
+Fbulk /H20849P/H20850+Fwall /H20849P/H20850+Felec /H20849P/H20850+Felas /H20849P,M/H20850, /H208491/H20850
where Fanis,Fexch,Fms,Fexternal ,Fbulk,Fwall,Felec, and Felasare
the magnetocrystalline anisotropy energy, magnetic ex-change energy, magnetostatic energy, external magnetic fieldenergy, ferroelectric bulk free energy, ferroelectric domainwall energy, electrostatic energy, and elastic energy, respec-tively. H
eis the externally applied magnetic field.
The elastic energy can be calculated with
Felas=1
2/H20885cijkleijekldV=1
2/H20885cijkl/H20849/H9255ij−/H9255ij0/H20850/H20849/H9255kl−/H9255kl0/H20850dV, /H208492/H20850
where eijis the elastic strain, /H9255ijis the total strain, and cijklis
the elastic stiffness tensor. /H9255ij0is the stress-free strain due to
the electostrictive effect or magnetostrictive effect, and isgiven by
/H9255
ij0=/H20902/H9257/H208753
2/H9261100/H20873mimj−1
3/H20874/H20876+/H208491−/H9257/H20850/H20849QijklPkPl/H20850/H20849i=j/H20850,
/H9257/H208733
2/H9261111mimj/H20874+/H208491−/H9257/H20850QijklPkPl /H20849i/HS11005j/H20850,
/H208493/H20850
where Qijklare the electrostrictive coefficients, and /H9261100and
/H9261111are the magnetostrictive constants. The summation con-
vention for the repeated indices is employed and i, j, k, l /H110051,
2, 3. The calculation of elastic energy for a film-substratesystem13is obtained using a combination of Khachaturyan’s
mesoscopic elasticity theory14and Stroh’s formalism of an-
isotropic elasticity.15
The mathematical expressions for the magnetocrystalline
anisotropy energy, magnetic exchange energy, magnetostaticenergy, external magnetic field energy, ferroelectric bulk free
a/H20850Electronic mail: jzz108@psu.eduAPPLIED PHYSICS LETTERS 90, 052909 /H208492007 /H20850
0003-6951/2007/90 /H208495/H20850/052909/3/$23.00 © 2007 American Institute of Physics 90, 052909-1
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128.59.222.12 On: Thu, 27 Nov 2014 22:56:28energy, ferroelectric domain wall energy, and electrostatic
energy are exactly the same as those given in Refs. 16and
17.
The temporal evolution of the magnetization configura-
tion is described by the Landau-Lifshitz-Gilbert equation,
/H208491+/H92512/H20850/H11509M
/H11509t=−/H92530M/H11003Heff−/H92530/H9251
MsM/H11003/H20849M/H11003Heff/H20850, /H208494/H20850
where /H92530is the gyromagnetic ratio, /H9251is the damping con-
stant, and Heffis the effective magnetic field, which is given
byHeff=− /H208491//H92620Ms/H20850/H20849/H11509F//H11509m/H20850.
The temporal evolution of the polarization field is de-
scribed by the time-dependent Ginzburg-Landau equation,
/H11509Pi
/H11509t=−L/H9254F
/H9254Pi, /H208495/H20850
where Lis a kinetic coefficient related to the domain evolu-
tion.
We used a BaTiO 3–CoFe 2O4nanocomposite film as an
example for our numerical simulations. The coefficients em-ployed in the simulations are listed in Ref. 18.19–23The sys-
tem was modeled by discretizing it into a three-dimensionalarray of cubic cells of 64 /H9004x/H1100364/H9004x/H11003128/H9004x, and periodic
boundary conditions were applied along the x
1andx2axes.
The cell size in real space was chosen to be /H9004x=l0, where
l0=/H20881G110//H92510and/H92510=/H20841/H92511/H20841T=25 °C . We chose the gradient en-
ergy coefficient as G11/G110=0.6. If l0=1 nm, G110=3.71
/H1100310−11C−2m4N, and the domain wall energy density is
about 5 /H1100310−3Jm−2for 180° domain wall, which is in line
with existing experimental measurement and theoreticalcalculation.24In this work, we ignored the misfit strain along
the ferroelectric-magnetic interface due to the lattice constantdifference between the two phases for simplicity.
One measure of magnetoelectric response is the appear-
ance of electric polarization upon applying an external mag-netic field. The initial polarization of BaTiO
3phase was
chosen to be along the x3axis /H20849P1=P2=0, P3/H110220/H20850, which
corresponds to the epitaxially grown single tetragonalc-phased BaTiO
3under in-plane compressive substrate
strain.25An external magnetic field Heis applied, which is
large enough to saturate the magnetic phase. By rotating themagnetic field from x
1axis to x3axis, we simulated the evo-
lution of the polarization in the ferroelectric phase, fromwhich the MIEP, i.e., /H9004
P3=P3−P3/H20849He/H20648x1/H20850, was calculated,
where P3is the effective /H20849average /H20850polarization of the entire
composite film.
We started with 1-3 type BaTiO 3–CoFe 2O4nanocom-
posite film, with the CoFe 2O4pillars embedded in the
BaTiO 3matrix as shown in Fig. 1/H20849a/H20850. The volume fraction of
CoFe 2O4is chosen to be f=0.35 /H20849similar to those studied in
the experiments in Ref. 8/H20850, the thickness of the film is h
=16 nm, and only one magnetic pillar was included in ourmodel; therefore the distance between neighboring magneticphases is d=64 nm and the radius of the pillar is r
=21.4 nm. The constraint strains from the substrate were
/H9255
11s=/H925522s=−0.005. The calculated effective /H20849average /H20850polar-
ization of the composite was P3/H20849He/H20648x1/H20850=0.180 C m−2when
the applied magnetic field was along the x1axis, which is
larger than that of a bulk single crystal sample /H208490.65
/H110030.260 C m−2=0.169 C m−2/H20850due to the compressive sub-
strate strains. As shown in Fig. 1/H20849b/H20850, with the rotation of the
applied magnetic field, the effective /H20849average /H20850polarization ofthe composite increases gradually. To clarify the origin of
MIEP, the stress distributions in the nanocomposite film werecalculated. Since the film consists of single ferroelectric/magnetic domains, stress components
/H926811and/H926822are almost
constant along the film thickness direction. However, as canbe seen in Fig. 2/H20849a/H20850, component
/H926833varies significantly with
the film thickness, as it has to be zero at the film surface tosatisfy the stress-free boundary condition. The change of thestress along the cross section at one-half of the film thicknesswith the applied magnetic field rotating from x
1axis to x3
axis is plotted in Figs. 2/H20849b/H20850–2/H20849d/H20850. It is seen that the rotation
of the applied magnetic field changes the stress distributionin the ferroelectric phase. As a result of the magnetostrictiveeffect, the magnetic phase deforms its shape with a change inmagnetization. As /H9261
100is negative for CoFe 2O4, the length of
the magnetic phase increases along the x1axis and decreases
along the x3axis after the rotation of the applied magnetic
field, and consequently, the stress distribution in the neigh-boring ferroelectric phase changes through the elastic inter-action between the two phases. Because of the piezoelectriceffect, the change in stress distribution leads to a change inthe polarization of the ferroelectric phase. For the electros-trictive constants we used, the decrease of
/H926811 /H20849/H926822/H20850
/H20849/H9004/H926811,/H9004/H926822/H110210/H20850in the ferroelectric phase increases the po-
larization along the x3axis /H20849P3/H20850, while the decrease of /H926833
FIG. 1. /H20849a/H20850Schematic illustration of 1-3 type BaTiO3–CoFe2O4nanocom-
posite film with CoFe2O4pillars /H20849shaded /H20850embedded in BaTiO3matrix
/H20849white /H20850. The applied magnetic field Heis in the x1-x3plane, and /H9251is the
angle between Heandx1axis. /H20849b/H20850Dependence of the magnetic-field-induced
electric polarization /H9004P3=P3−P3/H20849He/H20648x1/H20850on the direction of the applied
magnetic field /H20849f=0.35, h=16 nm, d=64 nm, and /H925511s=/H925522s=−0.005 /H20850.
FIG. 2. /H20849Color online /H20850/H20849a/H20850Stress distribution /H20849/H926833/H20850when He/H20648x1and /H20851/H20849b/H20850–/H20849d/H20850/H20852
the change of stress distributions /H20849/H9004/H926811,/H9004/H926822,a n d/H9004/H926833/H20850when the applied
magnetic field rotates from x1axis to x3axis /H20851/H9004/H9268ii=/H9268ii/H20849He/H20648x3/H20850−/H9268ii/H20849He/H20648x1/H20850,
f=0.35, h=16 nm, d=64 nm, and /H925511s=/H925522s=−0.005 /H20852.052909-2 Zhang et al. Appl. Phys. Lett. 90, 052909 /H208492007 /H20850
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128.59.222.12 On: Thu, 27 Nov 2014 22:56:28/H20849/H9004/H926833/H110210/H20850reduces it. Therefore, /H9004P3is determined by the
competition of /H9004/H926811/H20849/H9004/H926822/H20850and/H9004/H926833. In this example, /H9004/H926811
is dominant in enhancing the polarization of the nanocom-
posite film.
The dependence of MIEP on the film thickness was stud-
ied, and the results of /H9004P3*=P3/H20849He/H20648x3/H20850−P3/H20849He/H20648x1/H20850were
presented in Fig. 3/H20849a/H20850. With the increase of the film thick-
ness, the effect of /H9004/H926833becomes more important as the in-
fluence of the film surface is less significant. It was seen thatthe/H9004
P3*decreases with the increase of the film thickness,
and even becomes negative above a certain critical filmthickness, since the decrease of
/H926833/H20849/H9004/H926833/H110210/H20850reduces P3as
we discussed above. Recent studies26–28have shown that dif-
ferent morphologies of epitaxial nanocomposite films couldbe obtained by controlling the volume fractions of the phasesor the substrate’s thickness and orientation. Therefore, westudied as well the MIEP for two stripelike nanocompositesas shown in the inset of Fig. 3/H20849a/H20850./H20849The volume fraction of
CoFe
2O4was fixed to be f=0.35. /H20850From Fig. 2/H20849b/H20850we can see
that the change of /H926811is mostly along the sides of the mag-
netic phase in the x1direction. The stripelike morphologies
could enhance or decrease the effect of /H9004/H926811depending on
the orientation of its periodic distribution. As shown in Fig.3/H20849a/H20850, compared to the 1-3 type nanocomposite with magnetic
pillars in a ferroelectric matrix, /H9004
P3*becomes larger for the
stripelike morphology that distributes periodically along thex
1axis, while /H9004P3*is smaller for the stripelike morphology
that distributes periodically along the x2axis. The difference
is more significant for thin films for which the effect of /H9004/H926811
dominates.
It is expected that the constraint of the substrate will also
play an important role in the MIEP since it can affect thestress distribution in the film dramatically. Figure 3/H20849b/H20850shows
/H9004
P3*obtained under various compressive substrate strains
for 1-3 type nanocomposite films with two different thick-nesses. With the increase of the magnitude of compressivesubstrate strains, the magnitude of /H9004
P3*decreases for both
films. This indicates that under a large substrate compressivestrain, it becomes difficult to change the polarization of theferroelectric phase through elastic coupling.
It should be emphasized that the phase-field approach
presented here is three-dimensional and considers the micro-structure of the nanocomposite that is proved to be critical tothe magnetoelectric coupling in the nanocomposite. The elas-tic energy in the constrained thin film was incorporated, in-cluding the effect of free film surface and the constraint fromthe substrate. All prior studies essentially considered two-dimensional structures and the effect of thin film boundary
condition was included only approximately.
In summary, we have developed a phase-field model to
predict the magnetoelectric coupling in a nanocompositethin film made up of ferroelectric and magnetic materials.The magnetic-field-induced electric polarization /H20849MIEP /H20850in
BaTiO
3–CoFe 2O4nanocomposite films was analyzed. The
simulation showed that the MIEP is highly dependent on thefilm thickness, morphology of the nanocomposite, and sub-strate constraint, which provide a number of degrees of free-dom in controlling coupling in nanocomposite films.
The authors are grateful for the financial support of the
National Science Foundation under Grant Nos. DMR-0507146 and DMR 01-22638, Penn State MRI seed grant,and Laboratory-Directed Research and Development at LosAlamos National Laboratory. One of the authors /H20849L.Q.C. /H20850
would also like to acknowledge the support from theGuggenheim Foundation through a fellowship.
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/H20849Wiley, New York, 1983 /H20850, p. 198.
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18For BaTiO3,/H92511=4.124 /H20849T−115 /H20850/H11003105,/H925111=−2.097 /H11003108,/H925112=7.974
/H11003108,/H9251111=1.294 /H11003109,/H9251112=−1.950 /H11003109,/H9251123=−2.500 /H11003109,/H92511111
=3.863 /H110031010,/H92511112=2.529 /H110031010,/H92511122=1.637 /H110031010,/H92511123=1.367
/H110031010,Q11=0.10, Q12=−0.034, and Q44=0.029. For CoFe2O4,Ms=4
/H11003105,/H9261100=−590 /H1100310−6,/H9261111=120/H1100310−6,K1=3/H11003105,K2=0, and A=7
/H1100310−12.T=25 °C. For simplicity, we assumed elastic homogeneity in this
work, and the elastic constants of BaTiO3are used, i.e., c11=1.78 /H110031011,
c12=0.96 /H110031011, and c44=1.22 /H110031011/H20849in SI units /H20850.
19Y. L. Li, L. E. Cross, and L. Q. Chen, J. Appl. Phys. 98, 064101 /H208492005 /H20850.
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FIG. 3. /H20849a/H20850Dependence of magnetic-field-induced electric polarization
/H9004P3*=P3/H20849He/H20648x3/H20850−P3/H20849He/H20648x1/H20850on the film thickness h/H20849f=0.35 and /H925511s=/H925522s
=−0.005 /H20850./H20849b/H20850Dependence of the magnetic-field-induced electric polariza-
tion/H9004P3*on the substrate strains /H20849f=0.35 and d=64 nm /H20850.052909-3 Zhang et al. Appl. Phys. Lett. 90, 052909 /H208492007 /H20850
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1.3386468.pdf | Motion of transverse domain walls in thin magnetic nanostripes under transverse
magnetic fields
J. Lu and X. R. Wang
Citation: Journal of Applied Physics 107, 083915 (2010); doi: 10.1063/1.3386468
View online: http://dx.doi.org/10.1063/1.3386468
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/107/8?ver=pdfcov
Published by the AIP Publishing
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144.173.6.37 On: Tue, 11 Aug 2015 18:06:39Motion of transverse domain walls in thin magnetic nanostripes under
transverse magnetic fields
J. Lua/H20850and X. R. Wang
Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong
Kong Special Administrative Region, China
/H20849Received 17 December 2009; accepted 12 March 2010; published online 27 April 2010 /H20850
The motion of transverse magnetic domain walls /H20849TDW /H20850in thin magnetic nanostripes under
transverse magnetic fields /H20849TMF /H20850is investigated. In the absence of axial fields, an approximate static
TDW profile is obtained under a TMF with an arbitrary orientation. This profile becomes exact if theTMF is parallel or perpendicular to the stripe plane. Under nonzero axial fields, the TDW becomesasymmetric and twisted, and it moves along the wire axis with two different propagation modes,rigid-body mode and precession mode, depending on the strength of the axial field. The criticalstrength separating these two modes is called modified Walker limit H
W/H11032. The TMF dependence of
HW/H11032, the TDW velocity and maximum twisting angle at HW/H11032were investigated both numerically and
analytically. Moreover, it is shown that an early proposed velocity-field relationship fits well to theaverage velocities of a TDW above H
W/H11032. These results should be important for future developments
of magnetic nanodevices based on DW propagation. © 2010 American Institute of Physics .
/H20851doi:10.1063/1.3386468 /H20852
I. INTRODUCTION
Field-induced domain wall /H20849DW /H20850propagation in mag-
netic nanowires has attracted much attention in recentyears
1–15because of its fundamental importance in nanomag-
netism and potential applications in high density storage andspintronic devices. For a head-to-head /H20849HH /H20850or tail-to-tail
DW in a magnetic nanowire with its easy-axis along the wireaxis, it is known that the DW must propagate along the wireunder an axial magnetic field. The roadmap of DW propaga-tion is obtained recently.
14,15A static DW cannot exist in a
static field, thus it must move with time. A moving DW mustdissipate energy due to the Gilbert damping. As a result, theDW will propagate in the field direction along the wire, re-leasing the wire Zeeman energy to compensate the dissipatedenergy. Therefore, the time-averaged DW propagation veloc-ity must be proportional to the energy dissipation rate thatdepends on the axial field strength.
Since Schryer and Walker
1published their one-
dimensional DW motion work of an infinite uniaxially aniso-tropic medium under an external magnetic field, extensivestudies of the field-induced DW propagation in magneticnanowires have been conducted both experimentally
3–6and
numerically.7–10Suppose the wire axis is along z-direction,
the axial field-dependent average wall velocity curve, v¯−Hz,
has been measured or calculated for various nanowires.Many general features of DW propagation were discovered.Among them, the existence of a so-called Walker breakdownfield /H20849H
W/H20850separating the regions with high and low mobili-
ties was well established. It is known that the DW type var-
ies, depending on the width and thickness of thenanowires.
16–18Generally speaking, transverse /H20849vortex /H20850DWs
are more stable in narrow /H20849wide /H20850nanowires. When Hz
/H11021HW, after some slight modifications of its profile, a DW /H20849ofwhatever type /H20850propagates eventually along the wire like a
rigid body, with a mobility proportional to the DW width andinversely proportional to the damping constant. However,when H
zexceeds HW, the average wall velocity is reduced
dramatically. The dynamics depends strongly on the geom-etry of the nanowires. For nanowires with large cross-sections, a DW propagates along the wire axis accompaniedby the transformation between transverse DW /H20849TDW /H20850and
vortex/antivortex DWs. This behavior is confirmed by anumber of simulations in recent years.
10,18–22For nanowires
with small cross-sections, an initially stable TDW wouldmaintain its transverse profile during its propagation becausethe energy barrier between TDW and vortex/antivortex DWis too high. According to Walker’s analysis, the TDW planeshall rotate around the wire axis. The transverse magneticanisotropy /H20849TMA /H20850of the wire then modulates the width and
energy of the TDW periodically, resulting in a periodic os-cillation in DW velocity and a decrease in velocity mobility.
Recent advances in nanofabrication technology enable
us to study magnetic nanowires with length of micrometers.The most commonly used ones are the so-called nanostripesprepared by thermal evaporation
3or direct current /H20849dc/H20850mag-
netron sputtering4and followed by focused-ion-beam
milling.23Typically, the nanostripes are several nanometers
thick and hundreds of nanometers wide. For nanodevicesmade of such technique, the driving field should not be toosmall and usually higher than the Walker breakdown fieldH
Win order to overcome the pinning due to the imperfect-
ness of sample geometry or magnetic impurities. Therefore,the dominant dynamics for DW propagation would be theoscillatory ones. For nanostripes of the above mentionedcross-section dimension, the high field /H20849/H11022H
W/H20850dynamics of
DWs is the internal structure transformation one.18,19How-
ever, the key problem is the operating speed because the DWvelocity in this regime is usually much smaller than that
a/H20850Electronic mail: glnlj@yahoo.com.JOURNAL OF APPLIED PHYSICS 107, 083915 /H208492010 /H20850
0021-8979/2010/107 /H208498/H20850/083915/9/$30.00 © 2010 American Institute of Physics 107 , 083915-1
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144.173.6.37 On: Tue, 11 Aug 2015 18:06:39below HW. Many attempts have been made to increase the
DW velocity. In 2003, Nakatani et al.7proposed a way to
suppress the velocity breakdown using the edge roughness ofpatterned nanostripes. Four years later, Lee et al.
20proposed
an alternative way to increase the breakdown field HWby
simply using a magnetic underlayer of strong perpendicularmagnetic crystalline anisotropy. In the past two years, atransverse magnetic field /H20849TMF /H20850, either lying in
10or perpen-
dicular to21,22the nanostripe plane, has been proposed to sup-
press the DW velocity reduction.
The integration level of nanodevices is another impor-
tant issue. For nanodevices based on DW propagation inmagnetic nanostripes, the increase in integration level meansthe decrease in wire width. A propagating DW in a nanos-tripe with a small enough cross-section should always be aTDW. This is due to the fact that the exchange energy offorming a vortex/antivortex core is too high in these stripes./H20849The critical wire width for forming a vortex wall should be
order of DW width /H20850. Thus, it shall be interesting to study
TDW propagation along narrow wires. The strategies foundfor nanowires with large cross-sections may provide hints tothe solutions of this problem. Edge roughness, additional un-derlayer of strong crystalline TMA or simply a TMF can beused to suppress or delay the precession of the TDW plane,which leads to the DW velocity reduction. From the techni-cal point of view, a TMF should be the most convenient way,and this is the main focus of this paper.
This paper is organized as follows. In Sec. II, the model
of a magnetic nanostripe is introduced. Sec. III presents themain results. The static TDW profile under an arbitrary TMFin the absence of axial fields is first discussed. Then theeffects of TMFs on propagating TDW profile, the asymmetryand twisting are considered. Then detailed investigations ofTDW dynamics are performed numerically with the help ofthe micromagnetic simulation package
OOMMF .24Finally,
some approximate analysis are presented to understand thenumerical results /H20849detailed calculations are provided in Ap-
pendices A and B /H20850. The summary is given in Sec. IV.
II. MODEL
As shown in Fig. 1, a HH TDW is nucleated in a long
magnetic nanostripe of thickness tand width w. The z-axis is
assumed to be along the stripe. x-axis is chosen to be per-
pendicular to the stripe plane. The xyz-coordinate system iswhat is shown in Fig. 1./H9258/H20849r/H6023/H20850and/H9278/H20849r/H6023/H20850are the polar and
azimuthal angles of the magnetization vector M/H6023/H20849r/H6023/H20850, respec-
tively. /H9004denotes the DW width.
This nanostripe is modeled by the following energy den-
sity function:
E=−/H92620M/H6023·H/H6023/H20648+J
Ms2/H20849/H11612M/H6023/H208502−/H92620
2k1Mz2+E/H11036/H20849Mx,My/H20850,
/H208491/H20850
where Jis the exchange constant, H/H6023/H20648=Hzeˆzis the axial ex-
ternal field, k1is the crystalline anisotropy constant along
wire axis that measures the potential barrier, and E/H11036denotes
the TMA energy that is a function of MxandMyin general.
TMA could come from crystalline magnetic anisotropy ormagnetostatic energy due to the shape anisotropy or externalTMFs, etc. For a wire with inversion symmetry, the crystal-line anisotropy is an even function of M
xand My. In this
study, we assume it to be /H92620k2Mx2/2 with a characteristic
constant k2. In our previous works,15it has been shown that
in thin and narrow enough nanostripes, the magnetostaticenergy can be described by quadratic terms via its three av-
erage demagnetization factors D¯x,y,z. Thus after slightly
modifying the two anisotropy constants
k1/H11032=k1+/H20849D¯y−D¯z/H20850,k2/H11032=k2+/H20849D¯x−D¯y/H20850, /H208492/H20850
the magnetostatic energy can be absorbed into the crystalline
one.
A uniform TMF can be described by its magnitude Ht
and its azimuthal angle /H9023Hso that H/H6023/H11036is
H/H6023/H11036=/H20849Hx,Hy,0/H20850=Ht/H20849cos/H9023H,sin/H9023H,0/H20850. /H208493/H20850
The TMA due to a TMF is linear in MxandMy. Thus, for a
thin magnetic nanostripe, magnetic density function includ-ing the TMF takes the following form:
E=/H92620Ms2
2/H20851P/H20849/H9258/H110322+ sin2/H9258/H9278/H110322/H20850+/H20849k1/H11032+k2/H11032cos2/H9278/H20850sin2/H9258
−2htsin/H9258cos/H20849/H9278−/H9023H/H20850−2hzcos/H9258/H20852, /H208494/H20850
with
/H9258/H11032/H20849/H9278/H11032/H20850=/H11509/H9258/H20849/H9278/H20850
/H11509z,P=2J
/H92620Ms2,
hi=Hi
Ms,i=t,z. /H208495/H20850
The dynamics of the magnetization vector M/H6023/H20849x/H6023/H20850
=Msm/H6023/H20849x/H6023/H20850is known to be governed by the Landau–Lifshitz–
Gilbert /H20849LLG /H20850equation25
/H11509m/H6023
/H11509t=−/H9253m/H6023/H11003H/H6023eff+/H9251m/H6023/H11003/H11509m/H6023
/H11509t, /H208496/H20850
where Msis the saturation magnetization, /H9253is the gyromag-
netic ratio, and H/H6023effis the effective field which is the func-
tional derivative of energy density function with respect to
M/H6023,H/H6023eff=−/H208491//H92620Ms/H20850/H9254E//H9254m/H6023. Equation /H208496/H20850is a highly nonlin-
ear equation. For a single-domain magnetic particle, it is aw
tx
∆zM(r)
(r)φ(r)θ
y
FIG. 1. Sketch of a HH TDW in a nanostripe with thickness tand width w.
The coordinate system is as follows: z-axis is along the stripe, x-axis is
perpendicular to the stripe plane while y-axis is along zˆ/H11003xˆ. The DW has a
wall width /H9004./H9258/H20849r/H6023/H20850and/H9278/H20849r/H6023/H20850are the polar and azimuthal angles of the
magnetization M/H6023/H20849r/H6023/H20850, respectively.083915-2 J. Lu and X. R. Wang J. Appl. Phys. 107 , 083915 /H208492010 /H20850
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144.173.6.37 On: Tue, 11 Aug 2015 18:06:39nonlinear ordinary differential equation. Early studies26show
that many concepts from nonlinear dynamics are useful inunderstanding magnetization reversal. For an extended mag-netic system, the LLG equation becomes a nonlinear partialdifferential equation. Except for a few special cases,
1,27it is
difficult to obtain the analytical solution of the equation. Onthe other hand, experiments are restricted by sample prepa-ration, field generation, and instrumental limitation, etc.Therefore, numerical simulations are often used in modernmicromagnetic studies when analytical and experimental ap-proaches are limited.
III. RESULTS AND DISCUSSIONS
A. Static profile of a TDW under TMFs
A TMF does not create an energy difference between the
two domains separated by a HH TDW. Thus, the TDWwould eventually evolve into a stationary profile that can be
obtained by setting M
/H6023/H20648H/H6023effeverywhere. We find first the
magnetization orientations /H20849specified by /H9258Dand/H9278D/H20850inside
the two domains where the exchange energy does not con-
tribute to H/H6023eff.M/H6023/H20648H/H6023effis equivalent to
/H9254g
/H9254/H9258=/H9254g
/H9254/H9278=0 ,
g=/H20849k1/H11032+k2/H11032cos2/H9278/H20850sin2/H9258−2htsin/H9258cos/H20849/H9278−/H9023H/H20850. /H208497/H20850
It has the following solutions:
Forht/H11021hc=/H20851sin2/H9023H/k1/H110322+cos2/H9023H//H20849k1/H11032+k2/H11032/H208502/H20852−1/2,
sin/H9258D=ht
hc,
sin/H9278D=hcsin/H9023H
k1/H11032. /H208498/H20850
For
ht/H11022hc,
cos/H9258D=0 ,
k2/H11032
ht=cos/H9023H
cos/H9278D−sin/H9023H
sin/H9278D. /H208499/H20850
The above solutions imply the existence of a critical
TMF strength, hc. The magnetization orientations in the two
domains are symmetric with respect to the TDW center /H20849/H9258
=/H9266/2/H20850in the case of ht/H11021hc. The polar angle of the left
domain is /H9258D=sin−1/H20849ht/hc/H20850while that of the right domain is
/H9266−/H9258D. The azimuthal angles are the same, /H9278D
=sin−1/H20849hcsin/H9023H/k1/H11032/H20850. Besides the natural result that /H9258Din-
creases as htincreases, it is interesting to notice that the
azimuthal angles do not vary with ht. In fact, one can check
that in eˆ/H9278direction, the torque increment due to the TMF is
just balanced by that from the redistribution of magnetizationinside this fixed
/H9278D-plane. If htexceeds hc, the magnetization
of the two domains all point normally to the stripe axis andlie inside a uniform
/H9278D-plane according to Eq. /H208499/H20850/H20849/H9278D
→/H9023Hasht→/H11009/H20850. Thus, the TDW between them ceases toexist and the stripe becomes single-domained, a case of not
current interest. Therefore, we will consider the case of ht
/H11021hcbelow only.
In the DW region, the exchange energy contributes to
H/H6023effand static M/H6023satisfies
0= /H20851k2/H11032sin/H9258sin/H9278cos/H9278−htsin/H20849/H9278−/H9023H/H20850/H20852
+P/H208732 cos/H9258/H11509/H9278
/H11509z+ sin/H9258/H115092/H9278
/H11509z2/H20874,
0=− /H20851/H20849k1/H11032+k2/H11032cos2/H9278/H20850sin/H9258cos/H9258−htcos/H9258cos/H20849/H9278
−/H9023H/H20850/H20852+P/H20875/H115092/H9258
/H11509z2− sin/H9258cos/H9258/H20873/H11509/H9278
/H11509z/H208742/H20876. /H2084910/H20850
Equation /H2084910/H20850is usually hard to solve. However, when k2/H11032
/H11270htandP,“k2/H11032sin/H9258sin/H9278cos/H9278” in the first equation can be
replaced by “ k2/H11032sin/H9258Dsin/H9278cos/H9278,” and then Eq. /H2084910/H20850has
the following solution:
sin/H9258= sin/H9258D+cos2/H9258D
cosh/H9264+ sin/H9258D,/H9264=zcos/H9258D
/H9004/H20849/H9023H/H20850,
/H9004/H20849/H9023H/H20850=/H90040/H20881sin2/H9023H
k1/H110322+cos2/H9023H
/H20849k1/H11032+k2/H11032/H208502
sin2/H9023H
k1/H110322+cos2/H9023H
k1/H11032/H20849k1/H11032+k2/H11032/H20850,
/H9278=/H9278D. /H2084911/H20850
where /H90040=/H208812J//H20849/H92620k1/H11032Ms2/H20850. In general, the azimuthal angle at
the center of the TDW takes different value as that inside two
domains. Equation /H2084911/H20850is an approximate solution of the
TDW profile for an arbitrary /H9023Hsince it neglects the twist-
ing of the TDW. However, if k2/H11032=0 or /H9023H=n/H9266/2/H20849nis an
integer, meaning that the TMF is either inside or normal tothe stripe plane /H20850, Eq. /H2084911/H20850becomes exact because the TDW
plane does not twist in this case. In fact, profile, Eq. /H2084911/H20850, had
been obtained before.
28,29
B. Effects of TMFs on propagating TDWs: Emergence
of asymmetry and twisting
Suppose there is a static TDW in a narrow magnetic
nanostripe modeled by Eq. /H208494/H20850in the absence of an axial
magnetic fields. After applying an axial field H/H6023/H20648=Hzeˆz,a n
energy density difference is created between the two do-mains separated by the TDW. According to the roadmapfound earlier,
14,15the TDW must propagate along the stripe
axis toward the domain that has a higher energy density.During the propagation of the TDW, a TMF has strong ef-fects on DW motion. In this section, we show analyticallythat asymmetry and twisting of the TDW profile inevitablyappear due to the TMF.
To show these effects, we examine the magnetization
orientations in the two domains. By setting the variationalderivatives of the energy density function Eq. /H208494/H20850with re-
spect to
/H9258and/H9278to zeros, one obtains083915-3 J. Lu and X. R. Wang J. Appl. Phys. 107 , 083915 /H208492010 /H20850
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144.173.6.37 On: Tue, 11 Aug 2015 18:06:39sin/H9278=htsin/H9023H
hztan/H9258+k1/H11032sin/H9258,
cos/H9278=htcos/H9023H
hztan/H9258+/H20849k1/H11032+k2/H11032/H20850sin/H9258. /H2084912/H20850
/H9258has two solutions. One is /H9258Lfor the left domain in the
range of /H208490,/H9266/2/H20850. The other is /H9258Rfor the right domain in the
range of /H20849/H9266/2,/H9266/H20850. Equation /H2084912/H20850is a set of transcendental
equations with trigonometric functions and is hard to solve
analytically. However, it is easy to prove that /H9258L+/H9258R/H11021/H9266for
arbitrary /H9023H. This means that the TDW must be asymmetric
with respect to its center /H9258=/H9266/2. Furthermore, if /H9023H
/HS11005n/H9266/2, combined with /H9258L+/H9258R/H11021/H9266, Eq. /H2084912/H20850gives that /H9278L
/HS11005/H9278R, which results in the twisting of the TDW /H9278-plane. For
/H9023H=n/H9266/2, although /H9278L=/H9278R, numerical simulations show
that there is still twisting, the details will be presented in theSec. III C.
C. Numerical investigation of field-driven TDW
dynamics under TMFs
Due to the highly nonlinear nature of the LLG equation,
exact solution is hard to obtain. Since the TMF furthercauses asymmetry and twisting to the propagating TDW pro-file, the problem becomes even more complicate. Thus, weperform
OOMMF /H20849Ref. 24/H20850simulations for the TDW dynam-
ics under TMFs. In our simulations, the nanostripes are10
/H9262m long, 4 nm thick, and 20 nm wide. In these geom-
etries, both the initial stable and the propagating DWs aretransverse. The following magnetic parameters are used:
M
s=500 kA /m,J=20/H1100310−12J/m, and K1=200 kJ /m3.
The Gilbert damping coefficient is chosen as /H9251=0.1 to speed
up the simulations. We use K2=/H92620k2Ms2/2=80 kJ /m3to
model the crystalline TMA. During the simulation, the struc-ture is spatially discretized into 4 /H110034/H110034n m
3cubic ele-
ments /H20849called cell /H20850with no mesh outside the structure. The
demagnetization fields in each cell are calculated under theassumption that the magnetization is constant in each cell.Here we neglect the thermal effect and assume the physicalsystem is a t 0 K temperature.
For a specific choice of TMF H
/H6023/H11036, at the beginning of the
simulation, an ideal Neel TDW is nucleated in the stripeplane at 1
/H9262m from its left end and then relaxed to its equi-
librium profile without the axial driving field. This relaxationprocess is very fast and the TDW center hardly moves. An
axial field H
/H6023/H20648=Hzeˆzis then applied and the TDW is driven to
move. The evolution of the TDW profile is simulated and theinstantaneous velocities are obtained from Eq. /H208495/H20850of Ref. 15.
The average velocity of the TDW is calculated in two ways.In the first way, the average velocity is the time average ofthe instantaneous velocity within 4–5 full periods, which isreferred as calculated velocity. In the second way, the aver-age velocity is obtained by the best linear fit to the temporalevolution of the cell with
/H9258=/H9266/2 in the OOMMF simulation
over a long time /H20849at least more than five periods of velocity
oscillation /H20850, referred as simulated velocity.
Numerical simulations show that there are two propaga-
tion modes of a TDW under TMFs. Under a certain H/H6023/H11036,when the driving field Hzis below a critical value /H20849called
modified Walker limit HW/H11032which depends on H/H6023/H11036/H20850, the TDW
propagates like a rigid body with its velocity saturating to afixed value. However, if H
zexceeds HW/H11032, the TDW center
precesses around the stripe axis and the entire TDW propa-gates along the wire axis in a backward-and-forward fashion,reducing the time-average velocity dramatically. To see thetwo modes, TDW propagations under various H
zforHt
=1000 Oe and /H9023H=/H9266/2 are simulated. Numerically, HW/H11032
/H11015460 Oe with an error bar of /H110065 Oe is found. Below it,
after a transient process of several nanoseconds, the TDWpropagates like a rigid body. Figures 2/H20849a/H20850and2/H20849b/H20850show the
stationary TDW profile for H
z=300 Oe. Under this TMF, the
magnetization orientations in the two domains are /H9258L
=6.014° , /H9278L=90° and /H9258R=173.58° , /H9278R=90°, which are
the same as that calculated from Eq. /H2084912/H20850. It confirms that
magnetostatic interaction can be described by a local qua-dratic anisotropic energy in narrow nanowires. As shown inFig. 2/H20849a/H20850,
/H9258changes smoothly from /H9258Lto/H9258R. However, al-
though /H9278L=/H9278Rin this case, the TDW plane is still twisted
and its center has the largest azimuthal angle /H20851shown in Fig.
2/H20849b/H20850/H20852. The azimuthal angle difference between the magnetic
moment at the TDW center and those in the two domains isdenoted as /H20849/H9004
/H9278/H20850tw. It describes the maximum twisting of the
TDW plane. Its dependence on Hzis shown by the open
circles in Fig. 2/H20849c/H20850./H20849/H9004/H9278/H20850twincreases as Hzincreases from 0
toHW/H11032. Owing to the relatively small TMF /H20849Ht/HK/H110150.1/H20850, the
/H9258-asymmetry of the TDW is not apparent, but it does exist.
It is well-known that without TMFs, the TDW plane has
no twisting in rigid-body mode.1The increment in the azi-
muthal angle of the TDW plane, /H20849/H9004/H9278/H20850cant=/H9278plane−/H9278initial,i s
defined as the canting angle of the TDW plane. Its depen-
dence on Hzis known as1
/H20849/H9004/H9278/H20850cant=1
2sin−1Hz
HW,HW=/H9251k2/H11032Ms
2. /H2084913/H20850
The gray curve in Fig. 2/H20849c/H20850is Eq. /H2084913/H20850under the present
magnetic parameters. It is clear that as Hzincreases, /H20849/H9004/H9278/H20850tw
gradually departs from Eq. /H2084913/H20850and can even exceed 45° asFIG. 2. Rigid TDW profile under Hz=300 Oe /H20849HW/H11032/H11015460 Oe /H20850forHt
=1000 Oe and /H9023H=/H9266/2./H20849a/H20850/H9258-profile and /H20849b/H20850/H9278-profile. /H20849/H9004/H9278/H20850twis the larg-
est twisting azimuthal angle with respect to the two domains. /H20849c/H20850/H20849/H9004/H9278/H20850twvs
Hzbelow HW/H11032. The gray curve is the canting angle dependence on Hzin
rigid-body mode without TMFs for the same nanostripe.083915-4 J. Lu and X. R. Wang J. Appl. Phys. 107 , 083915 /H208492010 /H20850
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144.173.6.37 On: Tue, 11 Aug 2015 18:06:39Hzapproaches HW/H11032. This is due to the fact that the presence of
the TMF provides an extra torque /H20849damping torque /H20850in −eˆ/H9278
direction and hence holds a larger twisting angle at the TDW
center.
Above HW/H11032, the rigid-body mode does not exist and TDW
plane must precess around wire axis while the TDW propa-gates along the wire. Numerical simulations show that theTDW, after some transient process, moves backward-and-forward periodically. As an example, Fig. 3/H20849a/H20850is the time
evolution of the TDW center /H20849
/H9258=/H9266/2/H20850/H20849solid curve /H20850and the
TDW instantaneous velocity /H20849dash dot curve /H20850for Hz
=2000 Oe /H20849/H11022HW/H11032/H20850. It is clear that the twofold symmetry
around the wire axis is broken due to the presence of TMFs.
During one period, seven snapshots of the DW profile att
i,i=1,...7 indicated in Fig. 3/H20849a/H20850are plotted in Figs.
3/H20849b/H20850–3/H20849h/H20850.t1/H20851Fig.3/H20849b/H20850/H20852is the moment when the magnetiza-
tion in TDW is parallel to the TMF. /H9258/H20849z/H20850is the polar angle
distribution and /H9004/H9278/H20849z/H20850/H33528/H20849−180° ,180° /H20850is the azimuthalangle distribution with respect to the plane /H9278=90°. As time
goes by, the magnetic moments in the TDW precess aroundthe wire axis, leaving those inside the two domains un-changed, as shown in Figs. 3/H20849c/H20850and3/H20849d/H20850. During this pro-
cess, the twisting of the TDW plane is strengthened and/H9004
/H9278/H20849z/H20850turns to a bell shape. When the central magnetic mo-
ment of TDW goes back to the stripe plane but antiparallel to
the TMF, the twisting of the TDW plane becomes the high-est. Two kinks, which correspond to
/H9258=0° and /H9258=180°,
emerge, as indicated by the gray circles in Fig. 3/H20849e/H20850. At this
moment, /H9004/H9278/H20849z/H20850turns to a piecewise function, with its two
discontinuities corresponding to the two kinks. Figure 4
graphically illustrates the magnetization distribution at thismoment. As the central moment keeps on rotating around thewire axis, the two kinks are annihilated and the twisting ofTDW plane is weakened. /H9004
/H9278/H20849z/H20850becomes an inverted bell, as
shown by Figs. 3/H20849f/H20850and3/H20849g/H20850. When the central magnetic
moment is back to the stripe plane and parallel to the TMFagain, the twisting nearly disappears /H20851Fig.3/H20849h/H20850/H20852. Comparing
/H9258/H20849z/H20850in Figs. 3/H20849b/H20850and3/H20849h/H20850, the TDW has a positive displace-
ment along the wire within one period, which leads to a
positive average velocity. The above process repeats until theTDW reaches one stripe end.
It would be interesting to compare this DW oscillation
with those in wider and thicker nanostripes above the Walkerfields.
10,18–22In wider and thicker stripes, the DW oscilla-
tions come from the gyrotropic motion of nonlinear excita-tions, that is, magnetic topological solitons /H20849vortices and an-
tivortices /H20850. The DW propagations are accompanied by
periodic emission, motion, and absorption of several mag-netic solitons with integer and fractional topologicalcharges.
19In thin enough magnetic nanostripes, which are
the main concern in this work, the situation is different. Thehigh exchange energy barrier makes the vortices/antivorticeshard to exist. TDW becomes the only type of propagatingDWs. The oscillatory behavior of a TDW comes from thetransformation between external Zeeman energy and internalmagnetic energies of the nanostripe during the procession ofthe TDW center. The existence of the TMF lifts the twofoldsymmetry around the stripe axis, which makes the precessionand oscillation periods equal to each other.
The velocity simulation results over a large range of H
z
are obtained and shown in Fig. 5. The calculated /H20849simulated /H20850
velocities are denoted by crosses /H20849open circles /H20850with their
error bars smaller than the symbols. Their good overlap con-firms again the feasibility of absorbing the shape anisotropyFIG. 3. /H20849Color online /H20850TDW profile evolution in one period during its pre-
cession mode under Hz=2000 Oe. /H20849a/H20850Temporal evolution of TDW position
/H20849solid curve /H20850and its instantaneous velocity /H20849dash dot curve /H20850./H20851/H20849b/H20850–/H20849h/H20850/H20852Seven
snapshots of TDW profiles in one period. The gray circles in /H20849e/H20850indicate the
two kinks when TDW magnetization is antiparallel to the TMF. The nanos-tripe and the TMF are the same with those used in Fig. 2.0π∆φ(z)
z
FIG. 4. /H20849Color online /H20850Graphical view of the two kinks when the magneti-
zation of the TDW center lies in the stripe plane and is antiparallel to theTMF, the case of Fig. 3/H20849e/H20850.083915-5 J. Lu and X. R. Wang J. Appl. Phys. 107 , 083915 /H208492010 /H20850
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144.173.6.37 On: Tue, 11 Aug 2015 18:06:39into the crystalline one in thin and narrow enough nanos-
tripes. Meanwhile, although the TDW dynamics under TMFsis quite complex, the time-average velocities above H
W/H11032can
still be fitted by v¯=a/H20849Hz−H0/H208502/Hz+b/Hz/H20851Eq. /H208499/H20850of Ref. 15/H20852
very well, as shown by the gray curve in Fig. 5.
Now, we would like to investigate the TMF dependence
of the modified Walker limit HW/H11032of the same nanostripe used
early. In the simplest case, we set /H9023H=/H9266/2 and allow Htto
vary from 0 to HK=k1/H11032Ms/H110159.24 kOe. HW/H11032/H20849Ht/H20850data are ob-
tained and plotted in Fig. 6/H20849a/H20850by open squares. First, for
Ht/H110225 kOe, HW/H11032does not exist since before the rigid-body
mode breaks, the solution /H9258Rof Eq. /H2084912/H20850is no longer an
energy minimum but a maximum. The two domains wouldconverge to the same state characterized by /H20849
/H9258L,/H9278L/H20850and the
TDW between them will disappear. Then the corresponding
field-driven TDW propagation problem becomes meaning-
less. Figure 6/H20849a/H20850shows that HW/H11032increases as Htincreases.
When Ht/H11270HK,HW/H11032increases linearly with Ht. Meanwhile the
saturated velocity at HW/H11032,v/H20849HW/H11032/H20850, also grows up linearly, as
illustrated by open circles in Fig. 6/H20849b/H20850.A s Htincreases fur-
ther, HW/H11032departs from the linear trend and concaves upward,
meaning that the same increment of Htcan support a larger
Hzin the rigid-body mode. This is quite different from the
results of Sobolev et al.29While for v/H20849HW/H11032/H20850, our simulation
data show that it concaves downward, which is qualitatively
consistent with theirs. Finally, the Htdependence of the
maximum twisting angle /H20849/H9004/H9278/H20850maxatHW/H11032is shown in Fig. 6/H20849c/H20850
by open triangles. /H20849/H9004/H9278/H20850maxalways exceeds 45° and increases
linearly with Htover the entire Htrange where HW/H11032exists.
Due to the highly nonlinear nature of the LLG equation,
exact results of HW/H11032,v/H20849HW/H11032/H20850, and the maximum twisting angle
are difficult to obtain. Several kinds of approximate analysis
can be performed. In Appendix A, the well-known Sloncze-wski approach /H20849SA/H20850is introduced. In Appendix B, a slight
modified one /H20849modified SA, MSA /H20850is presented. The final
results are shown in Fig. 6by solid and dash dot curves,
respectively. Both approaches provide good results underlow TMFs but get worse as the TMF increases. Detailed
deductions and discussions are presented in the AppendicesA and B.
In the past a few years, several experimental investiga-
tions of DW propagation under TMFs in Permalloy /H20849Py/H20850
nanostripes have been performed.
30–32Numerical
simulations16–18have obtained that the critical cross-section
of Py nanowires, under /H20849above /H20850which the stable DW is
transverse /H20849vortex /H20850type in the absence of external fields, is
around 1500 nm2. The corresponding critical cross-section
for propagation DWs should be even smaller. However, thelowest cross-section of Py nanostripes prepared in these ex-periments is around 20 /H11003160 nm
2, which is too large. Then
the observed propagating DWs should be the magnetic topo-logical solitons /H20849vortices and antivortices /H20850, not the pure
TDWs. The experimental verification of the results presentedin this work needs further preparations and measurements ofthinner and narrower magnetic nanostripes.
IV. SUMMARY
In summary, we systematically investigated the field-
driven motion of TDWs in narrow magnetic nanostripes un-der TMFs. An approximate static TDW profile in an arbitraryTMF is obtained if the twisting of the TDW plane in
/H9278-direction is neglected. This approximate profile becomes
exact when the TMF is inside or normal to the stripe plane.As an axial field is applied, an energy density difference isFIG. 5. v¯vsHzdata over a large range of Hz. The nanostripe and the TMF
are the same with those used in Fig. 2. The crosses /H20849open circles /H20850are the
calculated /H20849simulated /H20850time-average velocities. The gray curve is the fitting
by Eq. /H208499/H20850of Ref. 15.
FIG. 6. /H20849Color online /H20850TheHtdependence of /H20849a/H20850HW/H11032,/H20849b/H20850v/H20849HW/H11032/H20850,a n d /H20849c/H20850the
maximum twisting angle of TDW plane at HW/H11032. The nanostripe is the same as
that used in Fig. 2. The open symbols denote the data from OOMMF simula-
tions. The solid /H20849dash dot /H20850curves are those from SA /H20849MSA /H20850, which can be
referred to Appendices A and B. The dashed line in /H20849c/H20850is the linear fit to
OOMMF data.083915-6 J. Lu and X. R. Wang J. Appl. Phys. 107 , 083915 /H208492010 /H20850
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144.173.6.37 On: Tue, 11 Aug 2015 18:06:39created between the two domains separated by the TDW. The
TDW then must propagate along the stripe axis because astatic TDW cannot exist.
14,15We showed that a propagating
TDW must be asymmetric with respect to its center and theTDW plane must be twisted, a direct consequence of theTMF.
The micromagnetic simulations /H20849by
OOMMF /H20850revealed
the details of TDW propagation. Under a given TMF, thereare two propagation modes, depending on the strength of theaxial field. For fields below a critical value called the modi-fied Walker limit H
W/H11032, the TDW propagates like a rigid-body
at a constant velocity. Different from that without TMFs, theTDW plane is twisted with a maximal twisting angle at DWcenter even in this rigid-body mode. Above H
W/H11032, TMA cannot
support the rigid-body mode anymore and the TDW centerbegin to precess around the stripe axis, leaving the magneti-zation orientations in the two domains unchanged. When themagnetization at TDW center is antiparallel to the transversecomponent of the magnetization inside domains, two kinkswith large exchange energies appear. The periodic change ofTMA due to the precession leads to the velocity oscillateswhich greatly reduces the average TDW velocity. This aver-age velocity in the precession mode can be described well byour previous high field velocity formula derived in the ab-sence of a TMF. Finally two approximate analysis, the tradi-tional SA and its modification, were carried out. Their advan-tages and disadvantages are discussed. The TMFdependences of H
W/H11032,v/H20849HW/H11032/H20850and the maximum twisting angle
atHW/H11032are obtained and compared with data from OOMMF
simulations. These results should be important for devices
based on DW propagation with high integration levels.
ACKNOWLEDGMENTS
This work is supported by Hong Kong UGC/CERG
/H20849Grant Nos. 603007, 603508, 604109, and HKU10/CRF/08-
HKUST17/CRF/08 /H20850.
APPENDIX A: SA
To understand the numerical results obtained in Sec.
III C, analytical investigations are performed. The dynamicalprofile of the TDW under TMFs is difficult, if not impos-sible, to solve rigorously. To proceed, approximations for thedynamical TDW profile are often used. In this section, theSA is introduced.
Following several previous works,
29,33a reasonable trial
dynamical profile of the polar angle /H9258is:
sin/H9258= sin/H92580+cos2/H92580
cosh u+ sin/H92580,u=z−q/H20849t/H20850
/H9004/H20849t/H20850,
/H9278=/H92780+/H9274/H20849t/H20850/H11003U/H20849u/H20850, /H20849A1/H20850
which is based on the static profile, Eq. /H2084911/H20850. In this profile,
/H92580and/H92780are the solutions of Eq. /H2084912/H20850in the left domain.
q/H20849t/H20850denotes the center position of TDW and /H9004/H20849t/H20850is the TDW
width parameter, which is obtained from the minima condi-
tion of the TDW energy. U/H20849u/H20850is a piecewise function, satis-
fying U/H20849u/H20850=1 when /H20841u/H20841/H11021/H9266/2 and U/H20849u/H20850=0 when /H20841u/H20841/H11022/H9266/2.
The present choice of the TDW profile have two main ap-proximations. First, the asymmetry of /H9258-profile is neglected.
Second, the continuous twisting in /H9278-profile has been sim-
plified into a piecewise function. Only the time evolution of
/H9278inside the interval − /H9266/H9004/2/H11021z−q/H20849t/H20850/H11021/H9266/H9004/2 is considered.
At the same time, for self-consistency, we assume that
/H9258/H20873u=−/H9266
2,t/H20874/H11013/H92580,/H9258/H20873u=/H9266
2,t/H20874/H11013/H9266−/H92580. /H20849A2/H20850
This choice of dynamical TDW profile is equivalent to divide
the entire wire sharply into three regions as follows: the leftdomain, the TDW region, and the right domain.
According to Slonczewski, the LLG Eq. /H208496/H20850is expressed
in the integral form as
/H20885/H20873/H11509/H9258
/H11509tsin/H9258+/H9251/H11509/H9278
/H11509tsin2/H9258/H20874dz=−/H9253
/H92620Ms/H20885/H9254E
/H9254/H9278dz,
/H20885/H20873/H11509/H9278
/H11509tsin/H9258−/H9251/H11509/H9258
/H11509t/H20874dz=/H9253
/H92620Ms/H20885/H9254E
/H9254/H9258dz. /H20849A3/H20850
Substituting Eqs. /H208494/H20850,/H20849A1/H20850, and /H20849A2/H20850into Eq. /H20849A3/H20850and inte-
grating zover q/H20849t/H20850−/H9266/H9004/H20849t/H20850/2−/H9280/H11021z/H11021q/H20849t/H20850+/H9266/H9004/H20849t/H20850/2+/H9280,
where /H9280is an infinitesimal, one gets the following equations:
/H9274˙=/H9275M
1+/H92512r1/H20849/H92580/H20850f1/H20849/H92580/H20850/H20851hz−/H9251r1/H20849/H92580/H20850p1/H20849/H92580/H20850/H20852,
q˙=/H9275M
1+/H92512r1/H20849/H92580/H20850f1/H20849/H92580/H20850/H20851/H9251hzf1/H20849/H92580/H20850+p1/H20849/H92580/H20850/H20852, /H20849A4/H20850
with
p1/H20849/H92580/H20850=htsin/H20849/H9274+/H92780−/H9023H/H20850/H11003g1/H20849/H92580/H20850−k2/H11032sin/H20849/H9274
+/H92780/H20850cos/H20849/H9274+/H92780/H20850/H11003f1/H20849/H92580/H20850,
f1/H20849/H92580/H20850=/H9266sin2/H92580+ 2 cos2/H92580+/H20849/H9266−2/H92580/H20850sin/H92580cos/H92580
2 cos/H92580,
g1/H20849/H92580/H20850=/H9266sin/H92580+/H20849/H9266−2/H92580/H20850cos/H92580
2 cos/H92580,
r1/H20849/H92580/H20850=/H9266−2/H92580
/H9266sin/H92580+/H20849/H9266−2/H92580/H20850cos/H92580,
/H9275M=/H9253Ms. /H20849A5/H20850
On the other hand, after integrating the energy density Eq.
/H208494/H20850over q/H20849t/H20850−/H9266/H9004/H20849t/H20850/2/H11021z/H11021q/H20849t/H20850+/H9266/H9004/H20849t/H20850/2 and minimizing
it, the time-dependent TDW width parameter has the form
/H9004=/H90040/H20851s1/H20849/H92580/H20850/n1/H20849/H92580/H20850/H208521/2,
s1/H20849/H92580/H20850=2 cos/H92580−/H20849/H9266−2/H92580/H20850sin/H92580
2 cos2/H92580,
n1/H20849/H92580/H20850=/H208751+k2/H11032
k1/H11032cos2/H20849/H9274+/H92780/H20850/H20876f1/H20849/H92580/H20850−2ht
k1/H11032cos/H20849/H9274+/H92780
−/H9023H/H20850g1/H20849/H92580/H20850. /H20849A6/H20850083915-7 J. Lu and X. R. Wang J. Appl. Phys. 107 , 083915 /H208492010 /H20850
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144.173.6.37 On: Tue, 11 Aug 2015 18:06:39The rigid-body motion of the TDW demands that /H9274˙=0
andq˙=const, which eventually leads to
/H9251p1/H20849/H92580/H20850r1/H20849/H92580/H20850=hz,q˙=/H9004/H9275Mhz
/H9251r1/H20849/H92580/H20850. /H20849A7/H20850
From Eq. /H20849A7/H20850, for a TMF specified by Htand/H9023H, the modi-
fied Walker limit HW/H11032, the corresponding maximum twisting
angle and TDW velocity at HW/H11032can be calculated through the
following procedure: starting from a small hz, put it into Eq.
/H2084912/H20850and one can obtain the corresponding /H92580and/H92780. Then
put/H92580and/H92780into the first equation of Eq. /H20849A7/H20850.I f
max /H20853/H9251p1/H20849/H92580/H20850r1/H20849/H92580/H20850,∀/H9274/H33528/H208510,2/H9266/H20850/H20854/H11022hz, then hz/H11021HW/H11032/Ms.
One can gradually increase hzand repeat the above process,
until max /H20853/H9251p1/H20849/H92580/H20850r1/H20849/H92580/H20850,∀/H9274/H33528/H208510,2/H9266/H20850/H20854=hz. By multiplying
Msto the present hz, one obtains HW/H11032. The corresponding /H9274
value, at which /H9251p1/H20849/H92580/H20850r1/H20849/H92580/H20850gets maximum and equals to
HW/H11032/Ms, is the maximum twisting angle at HW/H11032. At last, v/H20849HW/H11032/H20850
is given by the second equation of Eq. /H20849A7/H20850.
Following the above procedure, the Htdependence of
HW/H11032,v/H20849HW/H11032/H20850, and maximum twisting angle for /H9023H=/H9266/2 are
calculated and shown by solid curves in Figs. 6/H20849a/H20850–6/H20849c/H20850. For
HW/H11032, the SA results consist well with OOMMF data under low
Ht. Under high Ht, SA results become poor and the concavity
even becomes contrary to the OOMMF data. For v/H20849HW/H11032/H20850,S A
results seem not bad. While for the maximum twisting angle,
except for those under very low Ht, SA results are poor. This
is understandable. In SA, a piecewise function is used tomimic the continuous twisting of the
/H9278-plane. Thus, it is
harder for a given TMF to maintain a long platform of twist-ing than a continuous one with its peak at the center. So themaximum twisting angle obtained from SA must underesti-mate the genuine one. As H
tincreases, the difference be-
comes more and more apparent, leading to the concavedownward behavior of the solid curve in Fig. 6/H20849c/H20850.
APPENDIX B: MODIFIED SA
In SA, the continuous twisted /H9278-profile is roughly simu-
lated by a piecewise function. However, one can loosen thisassumption of
/H9278by inserting a weight function W/H20849z/H20850into the
integral kernels of Eq. /H20849A3/H20850, which results in the “MSA.”
Generally W/H20849z/H20850should have its peak inside the TDW region
and converge to zero rapidly enough in the two domains.
Based on the trial /H9258-profile shown in Eq. /H20849A1/H20850, one can found
the following function satisfies these conditions well:
W0/H20849z/H20850=/H9004/H11509/H9258
/H11509z=sin/H9258− sin/H92580
cos/H92580. /H20849B1/H20850
At the same time, we demand the /H9278-profile to be symmetric
with respect to its center and smooth enough, that is
/H9278=/H92780+/H9274/H20849z,t/H20850,/H9274/H20849q+z,t/H20850=/H9274/H20849q−z,t/H20850,
/H9274/H20849z→/H11006/H11009,t/H20850=0 , /H20849/H11509/H9274//H11509z/H20850/H20849z→/H11006/H11009,t/H20850=0 . /H20849B2/H20850
Based on the above preparation works, the integrals of
the LLG equation multiplied by W0/H20849z/H20850are performed over
the entire wire /H20849−/H11009/H11021z/H11021+/H11009/H20850. Therefore, we obtain the fol-
lowing equations:/H9274¯˙=/H9275M
f2/H20849/H92580/H20850+/H92512r2/H20849/H92580/H20850/H20851f2/H20849/H92580/H20850hz−/H9251r2/H20849/H92580/H20850p2/H20849/H92580/H20850/H20852,
q˙=/H9275M
f2/H20849/H92580/H20850+/H92512r2/H20849/H92580/H20850/H20851/H9251hz+p2/H20849/H92580/H20850/H20852, /H20849B3/H20850
with
p2/H20849/H92580/H20850=htsin/H20849/H9274¯+/H92780−/H9023H/H20850g2/H20849/H92580/H20850−k2/H11032sin/H20849/H9274¯
+/H92780/H20850cos/H20849/H9274¯+/H92780/H20850,
f2/H20849/H92580/H20850=/H20849/H9266−2/H92580/H20850− sin 2 /H92580
cos/H92580/H20851/H20849/H9266−2/H92580/H20850+ sin 2 /H92580/H20852,
g2/H20849/H92580/H20850=4 cos/H92580
/H20849/H9266−2/H92580/H20850+ sin 2 /H92580,
r2/H20849/H92580/H20850=2 cos/H92580−/H20849/H9266−2/H92580/H20850sin/H92580
2 cos2/H92580, /H20849B4/H20850
and/H9274¯is the first order approximation of the maximum twist-
ing angle. Meanwhile, after integrating the energy densityEq. /H208494/H20850multiplied by W
0/H20849z/H20850over the entire wire and mini-
mizing it, the time-dependent TDW width parameter has the
following form:
/H9004=/H90040/H20875s2/H20849/H92580/H20850
n2/H20849/H92580/H20850/H208761/2
,
s2/H20849/H92580/H20850=1
cos2/H92580/H20875/H20873sin2/H92580+1
2/H20874/H20849/H9266−2/H92580/H20850−3
2sin 2/H92580/H20876,
n2/H20849/H92580/H20850=/H208751+k2/H11032
k1/H11032cos2/H20849/H9274¯+/H92780/H20850/H20876/H20849/H9266−2/H92580/H20850+ sin 2 /H92580
2
−4ht
k1/H11032cos/H20849/H9274¯+/H92780−/H9023H/H20850cos/H92580. /H20849B5/H20850
The rigid-body motion of the TDW demands /H9274¯˙=0 and
q˙=const, which eventually leads to
/H9251p2/H20849/H92580/H20850r2/H20849/H92580/H20850=f2/H20849/H92580/H20850hz,q˙=/H9004/H9275Mhz
/H9251r2/H20849/H92580/H20850. /H20849B6/H20850
Following the similar procedure introduced in Appendix A,
theHtdependence of HW/H11032,v/H20849HW/H11032/H20850, and/H9274¯/H20849HW/H11032/H20850for/H9023H=/H9266/2
via MSA are calculated and shown by dash dot curves in
Figs. 6/H20849a/H20850–6/H20849c/H20850. For HW/H11032, although the absolute values are
worse than those from SA, the concavity is reproduced. For
v/H20849HW/H11032/H20850, data from MSA are better than those from SA under
low Ht, however, they become poor as Htbecomes quite
high. For the maximum twisting angle, data from MSA arebetter than those of SA but still lower than the simulated
values. This is understandable since
/H9274¯is indeed an average
twisting angle.
Both SA and MSA give deteriorating results as Htin-
creases. This is natural because in the trial /H9258-profile, Eq.
/H20849A1/H20850, we totally neglect the asymmetry effect due to the
TMF. As Htincreases, the asymmetry in /H9258-profile cannot be083915-8 J. Lu and X. R. Wang J. Appl. Phys. 107 , 083915 /H208492010 /H20850
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144.173.6.37 On: Tue, 11 Aug 2015 18:06:39ignored any more. Further theoretical investigations are still
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1.3680537.pdf | Noise-induced synchronization in spin torque nano oscillators
K. Nakada, S. Yakata, and T. Kimura
Citation: J. Appl. Phys. 111, 07C920 (2012); doi: 10.1063/1.3680537
View online: http://dx.doi.org/10.1063/1.3680537
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v111/i7
Published by the American Institute of Physics.
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Downloaded 04 Jan 2013 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsNoise-induced synchronization in spin torque nano oscillators
K. Nakada,1,a)S. Y akata,1,2and T. Kimura1,2
1Advanced Electronics Research Division, INAMORI Frontier Research Center, Kyushu University,
744 Motooka, Fukuoka, 819-0395, Japan
2CREST, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan
(Presented 3 November 2011; received 23 September 2011; accepted 4 January 2012; published
online 14 March 2012)
We have numerically studied the stochastic magnetization dynamics of a pair of spin torque nano
oscillators (STNOs) under noisy current injection by using the Landau-Lifshitz-Gilbert-Slonczewski(LLGS) equation with a macro-spin approximation. Common noisy current injection into both
STNOs is found to induce the phase synchronizations, where two STNOs show in-phase or
anti-phase locked precession depending on the sequences of Gaussian white noise. The noise-inducedsynchronization could be a possible application for controlling the output power in the array of the
STNOs.
VC2012 American Institute of Physics . [doi: 10.1063/1.3680537 ]
I. INTRODUCTION
In view of advanced information processing systems, a
spin-torque nano oscillator (STNO) is greatly expected as
a promising device for submicron microwave generator
because of its high tunable oscillation properties.1–3One of
the most important issues for practical realization of such
devices is the low output oscillation power. Mutual phase-
locking phenomena in serial and parallel array architectureshave been applied to synchronization schemes for increasing
the output power.
3–5In these architectures, each STNO inter-
acts with others via spin wave,3spin vortex,4and electric
currents.5To enhance such direct interactions, it is necessary
to optimize the spatial and geometric parameters of the array
architectures. However, even after such optimization, phaselocking phenomena occur only in the limited conditions, giv-
ing rise to a narrow frequency band and broad linewidth.
Moreover, phase drift among oscillators becomes a seriousobstacle as the number of the oscillators increases.
5–8
In the present study, we focus on the constructive effects
of noise as indirect interactions on synchronization amongnonlinear oscillators.
9–11We numerically investigate the sto-
chastic magnetization dynamics in a pair of STNOs under a
noisy current injection, and discuss the possibility of thenoise-induced synchronization in STNO arrays.
II. NOISE-INDUCED SYNCHRONIZATION SCHEME
Let us consider a synchronization scheme based on com-
mon noise-induced phase synchronization.9–11In general,
stochastic synchronization occurs in uncoupled nonlinear
oscillators subjected to common noise. The stochastic dy-namics of the oscillators are formulated as:
dXi
dt¼FðXiðtÞÞ þ gðtÞ; (1)
whereXirepresents the state vector of the ith oscillator, and
gthe vector of the noise. The function F(X) represents theintrinsic dynamics of the oscillator. In this case, the state Xi
synchronizes with each other due to the effects of the noise.
In view of practical applications, this phenomenon has
attracted great interest in the engineering field. In practice,an array of uncoupled nonlinear oscillator circuits has been
implemented on a silicon platform, and its validity is demon-
strated in real world environments.
11
From this point of view, common noise-induced phase
synchronization can be considered as a synchronization
scheme for an array of STNOs. Since random applied mag-netic fields and injected currents are multiplicative noise on
the dynamics of the STNOs, we should consider the stochas-
tic dynamics described by the following general form:
dXi
dt¼FðXiðtÞÞ þ rGðXiðtÞÞgðtÞ; (2)
where rrepresents the intensity of the noise g(t). In this
case, multiplicative noise on the noise-dependent function
G(X) causes noise-induced clustering, in which the phases of
the oscillators are distributed stochastically at cluster
states.10In the present study, we apply this formalism to
describing the dynamics of a pair of STNOs.
III. MODEL AND METHOD
Let us consider the magnetization dynamics of a pair of
uncoupled STNOs driven by common noise. First, we explain
a standard STNO consisting of a fixed and a free ferromagnetic
layer, and a nonmagnetic spac er layer, as shown in Fig. 1(a).
The motion of the magnetization in the free layer is well
described by the LLGS equation, which is given by:3,8
dm
dt¼/C0 j cjm/C2Hþam/C2dm
dtþjcjbJm/C2ðm/C2MÞ;(3)
where mandMrepresent the magnetization vector in the
free and fixed layer, respectively [Fig. 1(b)],athe Gilbert
damping parameter, cthe gyromagnetic ratio, ba material
parameter including fundamental constants, and the current J
flows from the fixed to the free layer in a positive direction.a)Electronic mail: nakada@ifrc.kyushu-u.ac.jp.
0021-8979/2012/111(7)/07C920/3/$30.00 VC2012 American Institute of Physics 111, 07C920-1JOURNAL OF APPLIED PHYSICS 111, 07C920 (2012)
Downloaded 04 Jan 2013 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsThe effective magnetic field is defined by H¼Haeˆxþ
(Hkmxeˆx/C0Hdzmzeˆz)/jmj, and Harepresents an external
applied magnetic field, Hka uniaxial anisotropy field, and
Hdzan demagnetization field normalized by the magnetic
permeability.5,6
The dynamics of each STNO are described by the LLGS
equation in spherical coordinates.5–7
dh
ds¼Ucoshcos/þaðHdzþHkcos2/Þsinhcosh
/C0Vsin//C0Hksin/cos/sinh; (4)
sinhd/
ds¼/C0Usin//C0ðHdzþHkcos2/Þsinhcosh
/C0Vcos/cosh/C0aHksin/cos/sinh; (5)
where hand/are the polar angles, the normalized time
s¼½c=ð1þa2Þ/C138t,and the transformation of the variables are
given by U¼aHa/C0bJandV¼HaþabJ.5,6
We consider a pair of STNOs subjected to Gaussian
white noise as a common injected current. For each STNO,the polar angles are represented by h
iand/i(i¼1, 2). Wesimulated the solution of the stochastic LLGS equation with
MATLAB using the Euler-Maruyama scheme. The Mers-
enne Twister function built in MATLAB was used as a nor-mal random number generator. The random numbers as
noise sequences were generated from different random seeds
for each trial.
To ensure the stability and accuracy in the simulation of
the LLGS equation with Gaussian white noise, we used Ito
calculus, which is widely utilized for numerical integrationof stochastic differential equations (SDEs). We converted
the LLGS equation as an SDE in the Stratonovich formula
into the Ito formula in accordance with the Ito-Stratonovichdrift conversion because random current injection essentially
acts as multiplicative noise on the magnetization dynamics.
IV. RESULTS AND DISCUSSION
Let us consider the stochastic magnetization dynamics
of two uncoupled STNOs driven by common noise, as shownin Fig. 1. In the numerical simulations, we set the parameters
as:a¼0.01, b¼1.0, H
a¼0.2 T, Hk¼0.01 T, Hdz¼1.6 T,
andJ¼0.01 A. We assumed that c¼1.0 and the time step
FIG. 1. (Color online) (a) Schematic illustration of a
pair of uncoupled STNOs where each STNO has a
standard structure that consists of a fixed ferromagnetic
layer, a free ferromagne tic layer, and a nonmagnetic
spacer layer. A common curre nt noise is injected into a
pair of STNOs. (b) Magnetization vectors in fixed andfree layers in spherical coordinates.
FIG. 2. (Color online) Stochastic synchronization of two uncoupled STNOs
in in-phase (a) and in anti-phase (b).
FIG. 3. (Color online) Time course of difference between polar angles h1
andh2in in-phase (a) and anti-phase (b) synchronization states.07C920-2 Nakada, Y akata, and Kimura J. Appl. Phys. 111, 07C920 (2012)
Downloaded 04 Jan 2013 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionswas set to 0.05 for simplicity in computation. Under these
conditions, each STNO shows in-plane large-angle preces-
sion at a steady state.5–8
Figure 2shows the time evolutions of a pair of STNOs,
where the intensity of noise dJwas set to 0.1 J.The wave-
forms of both STNOs were synchronized in in-phase at asteady state for a certain noise sequence [Fig. 2(a)]. In con-
trast, the waveforms synchronized in antiphase at a steady
state for a different noise sequence even under the same ini-tial conditions [Fig. 2(b)].
Figure 3shows the time evolutions of the relative phase
of two STNOs for the in-phase and anti-phase synchroniza-tion. Here, the relative phase is defined by the difference
between the polar angles h
1andh2. The noise starts to be
injected when the normalized time tnis zero. Since we
assume the same initial condition in the cases of Figs. 3(a)
and3(b), the relative phases in the two cases show almost
the same behavior in the transient regime ( tnis up to
0.5/C2104). After the transient states, the relative phase
approached the steady state where the STNOs synchronize
in-phase or anti-phase.
To clarify the importance of the common noise
injection, we compare the portraits of the polar angles in
phase-plane for several noisy situations. Figure 4shows the
phase-plane portraits of the polar angles at the steady states.
In the absence of the noise injection, the relative phasebetween two STNOs is fixed by the initial value [Fig. 4(a)].
When the uncommon noise is injected into each STNO inde-
pendently, the STNOs could not show the synchronization[Fig. 4(b)]. In contrast, the stochastic synchronization in in-
phase and anti-phase were induced by the common noise
injections, as shown in Figs. 4(c)and4(d), respectively.
We confirmed that the pair of the STNOs under the vari-
ous common noise sequences takes one of two synchroniza-
tion modes, either in-phase or anti-phase. The probability foreach synchronization mode in the present simulations is
almost same. However, the probability of the stochastic syn-
chronization can be modified by changing the parameters: a,
b, J, and H
a,Hk, and Hdz, that determine the precession
modes. These facts imply that the synchronization among
the array of the STNOs can be controlled by utilizing thecommon current noise injection.
V. CONCLUSION
We have numerically investigated the stochastic mag-
netization dynamics of two uncoupled STNOs under noisycurrent injection. In spite of no direct coupling, the STNOs
synchronized in anti-phase as well as in-phase at a steady
state. Such phenomena can be regarded as noise-inducedsynchronization and clustering as predicted for stochastic
systems with multiplicative noise. From the results, we will
investigate the optimization of noise-enhanced synchroniza-tion scheme for STNO arrays.
ACKNOWLEDGMENTS
This work was partially supported by CREST, Japan
Science and Technology Corporation (JST).
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FIG. 4. (Color online) Phase plane portraits of STNOs under (a) no noiseinjection, (b) independent noise injection, and (c) and (d) common noise
injections. In the case of common noise injection, in-phase and anti-phasesynchronization occur with equal probability.07C920-3 Nakada, Y akata, and Kimura J. Appl. Phys. 111, 07C920 (2012)
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1.2364384.pdf | Magnetostatic waves in layered materials and devices
Pedram Khalili Amiri and Behzad Rejaei
Citation: Journal of Applied Physics 100, 103909 (2006); doi: 10.1063/1.2364384
View online: http://dx.doi.org/10.1063/1.2364384
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/100/10?ver=pdfcov
Published by the AIP Publishing
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134.129.164.186 On: Sun, 21 Dec 2014 07:12:54Magnetostatic waves in layered materials and devices
Pedram Khalili Amiria/H20850and Behzad Rejaeib/H20850
Laboratory of High Frequency Technology and Components (HiTeC), Delft Institute of Microelectronics
and Submicron Technology (DIMES), Delft University of Technology, LB 2.380, Mekelweg 4,NL-2628 CD Delft, The Netherlands
/H20849Received 21 April 2006; accepted 16 August 2006; published online 29 November 2006 /H20850
Magnetostatic wave propagation in multilayers of ferro-/ferrimagnetic and nonmagnetic, dielectric
slabs is investigated using an effective medium theory and the transfer matrix method. Thepropagation in multilayers with antiparallel directions of magnetization is analyzed, in particular.Antiparallel multilayers support /H20849overall /H20850bulk waves at frequencies much higher than single layers
or parallel-magnetization structures. As possible applications of these multilayers, waveguides andresonators are proposed and discussed. © 2006 American Institute of Physics .
/H20851DOI: 10.1063/1.2364384 /H20852
I. INTRODUCTION
High frequency characteristics of radio frequency /H20849rf/H20850
and microwave devices employing magnetostatic waves inmagnetic materials are largely determined by the frequencydependence of the permeability of such materials. By usingmultilayers of different magnetic materials instead of a singlemagnetic core, it is therefore possible to engineer the effec-tive permeability of the medium in a way to enhance its highfrequency properties.
Since the work by Damon and Eshbach,
1a substantial
amount of work has been devoted to the analysis of magne-tostatic wave propagation in layered structures composed ofmagnetic materials and nonmagnetic dielectrics.
2–8Grünberg
and Mika2,3analyzed multilayers of both parallel and anti-
parallel magnetizations. Emtage and Daniel4discussed the
effect of gap modes on the propagation in parallel multilay-ers. Zhang and Zinn
8experimentally demonstrated the spin
wave modes of a magnetic double layer in both parallel andantiparallel configurations, verifying the predictions of Refs.2and3. They also noted that the antiparallel double layer
supports waves of a bulklike character with unusually highfrequencies.
High frequency magnetostatic waves in the antiparallel
configuration are the main interest of this work. To simplifythe analysis, an effective /H20849averaged /H20850permeability tensor will
be obtained and used to compute the magnetostatic modes ofthe antiparallel multilayers. Essentially this implies treatingthe medium as a metamaterial with one effective, homog-enous permeability tensor.
9The results of this effective me-
dium theory /H20849EMT /H20850will be compared to the transfer matrix
method /H20849TMM /H20850and the conditions for the averaging to be
valid will be examined.
While parallel-magnetization multilayers and single
magnetic layers support propagation of both surface- andbulklike magnetostatic waves,
1–8it will be shown that in the
antiparallel case overall surface magnetostatic waves do not
exist in the multilayer. Instead, bulklike solutions exist atfrequencies higher than those of any type of magnetostatic
wave in parallel or single layers, in agreement with the re-sults of Ref. 8. The presence of bulklike magnetostatic waves
at high frequencies can potentially lead to devices employingantiparallel magnetic multilayer cores. Two sample applica-tions, i.e., waveguides and resonators, are suggested and ana-lyzed. An advantage of such devices is the possibility of veryhigh frequency operation even without the application of anexternal dc magnetic field.
The paper is organized as follows: Section II introduces
an effective permeability tensor for the multilayers. Thecharacter of bulk and surface wave propagations is investi-gated using the effective permeability in Sec. III. Results ofthe effective medium analysis are verified with TMM and theconditions for its applicability are examined. Section IV dis-cusses waveguides and resonators based on the antiparallelmultilayers, and Sec. V summarizes and concludes the paper.The TMM analysis of the multilayers is expounded in anappendix.
II. EFFECTIVE MEDIUM ANALYSIS OF THE
MULTILAYER CONFIGURATION
Consider a superlattice composed of alternating mag-
netic and dielectric layers as shown in Fig. 1. The easy axis
of each magnetic layer is assumed to lie in the plane of thefilm and along the zaxis, but we include the possibility of
parallel as well as antiparallel orientation of magnetizationsin adjacent magnetic layers.
10Thus, the period of the super-
a/H20850FAX: /H1100131-15-2623271; electronic mail: p.khalili@dimes.tudelft.nl
b/H20850Electronic mail: b.rejaei@ewi.tudelft.nl
FIG. 1. Structure of the magnetic multilayer. Magnetization is along the z
axis.JOURNAL OF APPLIED PHYSICS 100, 103909 /H208492006 /H20850
0021-8979/2006/100 /H2084910/H20850/103909/9/$23.00 © 2006 American Institute of Physics 100 , 103909-1
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
134.129.164.186 On: Sun, 21 Dec 2014 07:12:54lattice is /H5129=2tM+2tD, where tMandtDdenote the thickness
of magnetic and dielectric layers, respectively.
Inside each magnetic layer, the permeability tensor /H9262↔is
given by
/H9262↔=/H20900/H9262 i/H9262a0
−i/H9262a/H92620
00 1 /H20901, /H208491/H20850
in which
/H9262=/H9275H/H20849/H9275H+/H9275M/H20850−/H92752
/H9275H2−/H92752, /H208492/H20850
/H9262a=/H9275/H9275 M
/H9275H2−/H92752, /H208493/H20850
/H9275M=/H9253M,/H9275H=/H9253Ha, /H208494/H20850
where /H9253is the gyromagnetic ratio, Mis the saturation mag-
netization, and Hais the magnetic anisotropy field. For the
case of an antiparallel multilayer, the orientation of the mag-netization in each layer determines the signs of MandH
a.
As shown in the Appendix, it is possible to directly treat
such a superlattice, e.g., in the magnetostatic approximation,using a TMM. This technique yields exact results but doesnot provide sufficient physical insight into the problem.Moreover, it does not easily lend itself to the analysis ofmore complicated structures based on these multilayers.Therefore, in this paper, we adopt a different approach basedon the EMT. This method is equivalent to treating the super-lattice as a metamaterial,
9thus considering it as a homog-
enous medium the /H20849magnetic /H20850properties of which are ob-
tained from those of the individual layers. It is important tonote that EMT is justified only for wavelengths much biggerthan the periodic variations of permeability in a multilayeredstructure, since, in that case, the wave does not seethe fast
variations of the individual layers but only feeltheir overall
effect. In this section we derive the effective permeabilitytensor of the superlattice within the EMT approximation anduse it in subsequent sections to analyze the propagation ofbulk and surface magnetostatic waves. A similar effectivemedium theory is presented in Refs. 11and12and is further
discussed in Ref. 13.
In the structure of Fig. 1, consider the relation between
the rf components of BandHinside each layer. Using /H208491/H20850,
one can write
B
x=/H92620/H20849/H9262Hx+i/H9262aHy/H20850, /H208495/H20850
By=/H92620/H20849−i/H9262aHx+/H9262Hy/H20850, /H208496/H20850
Bz=/H92620Hz, /H208497/H20850
where we take /H9262=/H92620,/H9262a=0 inside a dielectric layer. As an
estimation /H20849and considering that ultimately only wavelengths
much bigger than the period of the superlattice are of con-cern /H20850, one may use the condition of continuity of B
y,Hx, and
Hzon the interfaces to propose that they are nearly constant
over one period of the superlattice /H20849two magnetic and two
dielectric layers /H20850and thus equal to their average values. Bytransforming /H208495/H20850and /H208496/H20850such as to write BxandHyin terms
ofByandHx, and averaging the resulting relations over y,
one obtains
/H20855Bx/H20856=/H92620/H20883/H20873/H9262−/H9262a2
/H9262/H20874Hx/H20884+/H20883i/H9262a
/H9262By/H20884
=/H92620/H20883/H9262−/H9262a2
/H9262/H20884Hx+i/H20883/H9262a
/H9262/H20884By, /H208498/H20850
/H20855Hy/H20856=/H20883i/H9262a
/H9262Hx/H20884+1
/H92620/H208831
/H9262By/H20884
=i/H20883/H9262a
/H9262/H20884Hx+1
/H92620/H208831
/H9262/H20884By, /H208499/H20850
/H20855Bz/H20856=/H92620/H20855Hz/H20856. /H2084910/H20850
Here the symbol /H20855/H20856denotes averaging over one period of the
superlattice, i.e., for any quantity Q
/H20855Q/H20856=/H20873/H20858
k=14
Qktk/H20874/H20882/H20873/H20858
k=14
tk/H20874, /H2084911/H20850
where tkis the thickness of the kth layer and Qkdenotes the
value of Qinside that layer. By writing the components of B
in terms of those of Hagain, we can obtain the averaged
/H20849relative /H20850permeability tensor,
/H9262↔EMT=/H20855/H9262↔/H20856=/H20900m1ima0
−imam20
00 1 /H20901, /H2084912/H20850
where we have
m1=/H20883/H9262−/H9262a2
/H9262/H20884+/H20883/H9262a
/H9262/H208842/H20882/H208831
/H9262/H20884, /H2084913/H20850
m2=1/H20882/H208831
/H9262/H20884, /H2084914/H20850
ma=/H20883/H9262a
/H9262/H20884/H20882/H208831
/H9262/H20884. /H2084915/H20850
Consider now the case where the magnetization vectors in
neighboring magnetic layers are parallel to each other. Usingthe above relations, one can show that
m
1=1
1+/H9267/H20873/H9267+/H9262−/H9267/H9262a2
1+/H9267/H9262/H20874=1+/H208731
1+/H9267/H20874/H9275M/H9275e
/H9275H/H9275e−/H92752,
/H2084916/H20850
m2=/H208491+/H9267/H20850/H9262
1+/H9267/H9262=/H9275/H110362−/H92752
/H9275H/H9275e−/H92752, /H2084917/H20850
ma=/H9262a
1+/H9267/H9262=/H208731
1+/H9267/H20874/H9275M/H9275
/H9275H/H9275e−/H92752, /H2084918/H20850
/H9275e=/H9275H+/H9267
1+/H9267/H9275M, /H2084919/H20850103909-2 P . Khalili Amiri and B. Rejaei J. Appl. Phys. 100 , 103909 /H208492006 /H20850
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134.129.164.186 On: Sun, 21 Dec 2014 07:12:54/H9275/H11036=/H20881/H9275H/H20849/H9275H+/H9275M/H20850, /H2084920/H20850
where /H9267=tD/tMis the ratio of the thickness of the dielectric
and magnetic layers. Obviously, by letting /H9267→0/H20849dielectrics
much thinner than ferromagnetic layers /H20850we end up with the
original permeability tensor /H208491/H20850. In general, however, we
have a diluted permeability tensor due to the presence of the
dielectric layers.
The situation for antiparallel magnetizations in adjacent
magnetic layers, however, is very different. Here we have
m1=1
1+/H9267/H20873/H9267+/H9262−/H9262a2
/H9262/H20874=1+/H208731
1+/H9267/H20874/H9275M/H20849/H9275H+/H9275M/H20850
/H9275/H110362−/H92752,
/H2084921/H20850
m2=/H208491+/H9267/H20850/H9262
1+/H9267/H9262=/H9275/H110362−/H92752
/H9275H/H9275e−/H92752, /H2084922/H20850
ma=0 . /H2084923/H20850
Thus, the nondiagonal components of the permeability tensor
vanish. In the limit where the dielectrics are much thinnerthan the ferromagnetic layers, i.e.,
/H9267→0, we have
m1=/H9262−/H9262a2
/H9262, /H2084924/H20850
m2=/H9262. /H2084925/H20850
Note that m1and m2are in general dissimilar, unlike the
diagonal terms in /H208491/H20850. This, as we will see, affects the fre-
quency range of bulk magnetostatic waves propagating insuch materials.
III. MAGNETOSTATIC WAVES IN THE MAGNETIC/
DIELECTRIC SUPERLATTICE
To study the propagation of magnetostatic waves in the
framework of EMT, we define a magnetostatic potential /H9274in
the form:1,14,15
H=/H11612/H9274, /H2084926/H20850
where His the /H20849rf/H20850magnetic field vector. This leads to Walk-
er’s equation
/H11612·/H20849/H9262IEMTH/H20850=m1/H115092/H9274
/H11509x2+m2/H115092/H9274
/H11509y2+/H115092/H9274
/H11509z2=0 /H2084927/H20850
inside the superlattice, now viewed as a uniform medium
described by the effective permeability tensor /H9262JEMT. Assum-
ing magnetostatic waves propagating in the x-zplane with
the wave vector /H20849kx,kz/H20850, we can write the solution for /H9274
inside the superlattice in the form
/H9274=/H20851Aexp /H20849/H9264y/H20850+Bexp /H20849−/H9264y/H20850/H20852exp /H20849−ikxx−ikzz/H20850, /H2084928/H20850
where
/H9264=k/H20881m1
m2sin2/H20849/H9258/H20850+1
m2cos2/H20849/H9258/H20850,
k=/H20881kx2+kz2, /H2084929/H20850/H9258= tan−1/H20873kx
kz/H20874,
with kthe wave number and /H9258the angle of propagation with
respect to the easy axis /H20849zdirection /H20850. Outside the magnetic
layer we always have
/H9274=/H20851Cexp /H20849ky/H20850/H20852exp /H20849−ikxx−ikzz/H20850/H20849y/H333550/H20850, /H2084930/H20850
/H9274=/H20851Dexp /H20849−ky/H20850/H20852exp /H20849−ikxx−ikzz/H20850/H20849y/H33356d/H20850, /H2084931/H20850
where dis the overall thickness of the superlattice.
Imposing boundary conditions /H20849continuity of Hx,Hz,By/H20850
on the above solution at y=0 and y=dleads to the following
condition for magnetostatic wave propagation:
tanh /H20849/H9264d/H20850=2 sgn /H20849m2/H20850/H20881m2/H20851m1sin2/H20849/H9258/H20850+ cos2/H20849/H9258/H20850/H20852
/H20849ma2−m1m2/H20850sin2/H20849/H9258/H20850−m2cos2/H20849/H9258/H20850−1. /H2084932/H20850
In what follows we use the above equation to investigate the
conditions for propagation of volume and surface waves inthe superlattice.
A. Volume waves
V olume magnetostatic waves are characterized by sinu-
soidal dependence of the magnetic field on position insidethe film, in the direction normal to the plane of the film /H20849y
direction /H20850. Such solutions require
/H9264being a purely imaginary
number, i.e., /H9264=iqand are only possible if
m2/H20851m1sin2/H20849/H9258/H20850+ cos2/H20849/H9258/H20850/H20852/H110210, /H2084933/H20850
The resulting equation for qthen reads
tan/H20849qd/H20850=2 sgn /H20849m2/H20850/H20881−m2/H20851m1sin2/H20849/H9258/H20850+ cos2/H20849/H9258/H20850/H20852
/H20849ma2−m1m2/H20850sin2/H20849/H9258/H20850−m2cos2/H20849/H9258/H20850−1. /H2084934/H20850
The above equation has an infinite number of solutions for q,
corresponding to different “modes” of propagation for agiven frequency /H20849implicitly included in m
1,m2,ma/H20850.
In the case of parallel magnetizations, the range of fre-
quencies where the condition /H2084933/H20850holds is given by
/H20881/H9275e/H20875/H9275H+sin2/H20849/H9258/H20850
1+/H9267/H9275M/H20876/H11021/H9275/H11021/H9275/H11036,
tan2/H20849/H9258/H20850/H11021/H208491+/H9267/H20850/H9275H
/H9267/H9275M, /H2084935/H20850
/H9275/H11036/H11021/H9275/H11021/H20881/H9275e/H20875/H9275H+sin2/H20849/H9258/H20850
1+/H9267/H9275M/H20876,
tan2/H20849/H9258/H20850/H11022/H208491+/H9267/H20850/H9275H
/H9267/H9275M. /H2084936/H20850
Taking into account all possible propagation angles, one ob-
tains the overall frequency range for volume waves in thesuperlattice with parallel magnetizations,103909-3 P . Khalili Amiri and B. Rejaei J. Appl. Phys. 100 , 103909 /H208492006 /H20850
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134.129.164.186 On: Sun, 21 Dec 2014 07:12:54/H20881/H9275H/H9275e/H11021/H9275/H11021/H20881/H9275e/H20873/H9275H+1
1+/H9267/H9275M/H20874. /H2084937/H20850
In the limit /H9267→0, i.e., a single ferrite slab, one obtains the
well-known result,
/H9275H/H11021/H9275/H11021/H9275/H11036. /H2084938/H20850
Note that /H20849effective /H20850volume waves in the magnetic/dielectric
superlattice can propagate at frequencies higher than that of a
single magnetic slab /H20851/H9275/H11036=/H20881/H9275H/H20849/H9275H+/H9275M/H20850/H20852, up to a frequency
of/H9275=/H9275H+/H9275M/2, which happens when /H9267=1.
In the case of antiparallel alignment of the magnetiza-
tions the condition /H2084933/H20850translates into the following range of
frequencies for bulk magnetostatic waves in the effectivepermeability medium:
/H20881/H9275H/H9275e/H33355/H9275/H33355/H20881/H20849/H9275H+/H9275M/H20850/H20875/H9275H+sin2/H20849/H9258/H20850
/H9267+1/H9275M/H20876, /H2084939/H20850
where we have used Eqs. /H2084921/H20850–/H2084923/H20850. The overall propagation
frequency range, which coincides with the frequency rangeof volume waves propagating perpendicular to the magneti-zation /H20849
/H9258=/H9266/2/H20850, is given by
/H20881/H9275H/H9275e/H33355/H9275/H33355/H20881/H20849/H9275H+/H9275M/H20850/H20873/H9275H+1
/H9267+1/H9275M/H20874. /H2084940/H20850
The /H20849effective /H20850volume waves in this case can propagate at
frequencies up to /H9275=/H9275H+/H9275M, which is the case when /H9267
→0. Thus, even in the absence of dielectric layers, the anti-
parallel orientation of magnetizations in adjacent layers leadsto the existence of high frequency volume waves.
The higher upper bound on the frequency of propagation
of volume waves in the magnetic/dielectric superlattice whencompared to a single magnetic slab is due to the fact thatgenerally m
1/HS11005m2in/H2084912/H20850, whereas in /H208491/H20850the diagonal terms
are equal. In the direction perpendicular to the direction ofmagnetization, from /H2084927/H20850it is seen that volume wave propa-
gation is only possible when m
1andm2have different signs.
For a single layer, however, volume wave propagation /H20849along
the easy axis /H20850translates to the condition /H9262/H110210. Accounting
for the frequency dependence of /H9262and/H9262a, these translate
into different ranges of propagation for bulk waves, which isthe reason that multilayers can go beyond a single layer interms of bulk wave propagation frequency. Physically, this isbecause the effective volume waves propagating in the
multilayer at frequencies higher than
/H9275/H11036=/H20881/H9275H/H20849/H9275H+/H9275M/H20850
/H20849upper limit for volume waves in a single magnetic layer /H20850are
actually formed by a combination of surface waves in theindividual layers. Note that no volume waves propagate inthezdirection /H20849
/H9258=0/H20850in the multilayer at frequencies higher
than/H9275/H11036/H20851see eqs. /H2084935/H20850and /H2084939/H20850/H20852, essentially because a single
magnetic layer does not support surface waves along the di-rection of magnetization.
In order to verify the above results, which were based on
the EMT approximation, we have also carried out simula-tions using the exact TMM. Figure 2shows the propagation
band diagram of a magnetic/dielectric superlattice in the an-tiparallel configuration, for different thickness ratios
/H9267.I ti s
easy to verify that the propagation frequency range predictedby TMM for small values of kexactly coincides with that
obtained from /H2084940/H20850. This is consistent with the observation
that EMT should be in agreement with the /H20849exact /H20850TMM
solution if the wavelength is big enough /H20849i.e.,kis small /H20850.
The TMM calculation of the magnetostatic potential
wave form for a volume wave propagating perpendicular tothe magnetization in a multilayer is shown in Fig. 3. The
effectively sinusoidal profile of the potential is composed ofexponentially decaying surface waves on individual mag-netic layers, as argued above.
Finally, to further compare EMT and TMM we look at
the dispersion curves predicted by both methods. Figure 4
shows such a comparison, in which it is seen that with in-creasing kthe EMT solution starts to deviate from the /H20849exact /H20850
TMM solution. For small k, however, EMT results can be
trusted.
B. Surface waves
Solutions of Eq. /H2084932/H20850with a real, positive /H9264represent
waves decreasing exponentially inside the effective magnetic
FIG. 2. Propagation band diagram for a superlattice of 200 magnetic layers
in the antiparallel configuration /H20849/H9258=/H9266/2/H20850. The saturation magnetization and
anisotropy field of the magnetic layers are M=1 T and Ha=100 Oe, respec-
tively. The band diagram is plotted for three different ratios of dielectric tomagnetic layer thickness: /H20849a/H20850
/H9267=10, /H20849b/H20850/H9267=1, and /H20849c/H20850/H9267=0.1. The propaga-
tion constant is written in the normalized form k/H5129. We have /H9275H
/H11229280 MHz, /H9275/H11036/H112292.8 GHz, and /H9275M/H1122928 GHz. Unlike a multilayer with
parallel magnetizations, where reversing the ratio of the dielectric and fer-romagnetic layer thickness
/H9267has no effect on the band diagram except for a
single mode at /H9275H+/H9275M/2/H20849Ref. 4/H20850, no similarity between the bands for /H9267
=10 and /H9267=0.1 exists here. This is because of the absence of the so-called
gap modes in the antiparallel case.103909-4 P . Khalili Amiri and B. Rejaei J. Appl. Phys. 100 , 103909 /H208492006 /H20850
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134.129.164.186 On: Sun, 21 Dec 2014 07:12:54medium. Such solutions, characterized by the concentration
of the magnetic field near the surface, define the surfacemagnetostatic waves. A real value for
/H9264requires
m2/H20851m1sin2/H20849/H9258/H20850+ cos2/H20849/H9258/H20850/H20852/H333560. /H2084941/H20850
Furthermore, due to the constraint 0 /H33355tanh /H20849/H9264d/H20850/H110211i ti sr e -
quired that
0/H333552 sgn /H20849m2/H20850/H20881m2/H20851m1sin2/H20849/H9258/H20850+ cos2/H20849/H9258/H20850/H20852
/H20849ma2−m1m2/H20850sin2/H20849/H9258/H20850−m2cos2/H20849/H9258/H20850−1/H110211. /H2084942/H20850
From the above conditions, it can be shown that in the su-
perlattice with parallel alignment of magnetizations, the sur-face waves exist if
tan/H20849
/H9258/H20850/H11022/H9275H
/H9275Mand/H9267/H110211, /H2084943/H20850
with the range of frequencies given by
/H9275/H11036/H11021/H9275/H11021/H9275M
2sin/H20849/H9258/H20850+/H9275H/H208511 + sin2/H20849/H9258/H20850/H20852
2 sin /H20849/H9258/H20850,
tan2/H20849/H9258/H20850/H11021/H208491+/H9267/H20850/H9275H
/H9267/H9275M,/H20881/H9275e/H20875/H9275H+sin2/H20849/H9258/H20850
1+/H9267/H9275M/H20876
/H11021/H9275/H11021/H9275M
2sin/H20849/H9258/H20850+/H9275H/H208511 + sin2/H20849/H9258/H20850/H20852
2 sin /H20849/H9258/H20850,
tan2/H20849/H9258/H20850/H11022/H208491+/H9267/H20850/H9275H
/H9267/H9275M. /H2084944/H20850
Turning to the superlattice with antiparallel magnetiza-
tions, one should have
m2/H110210 and /H20851m1sin2/H20849/H9258/H20850+ cos2/H20849/H9258/H20850/H20852/H110210 /H2084945/H20850
for surface wave solutions to exist. The above inequalities
follow from Eqs. /H2084941/H20850and /H2084942/H20850together with the relation
ma=0. From the relations /H2084921/H20850and /H2084922/H20850form1andm2, re-
spectively, it is easy to show that these inequalities cannot besatisfied simultaneously. Therefore, no surface waves exist inthe superlattice. We conclude that in a superlattice consistingof equally thick but oppositely magnetized ferromagneticlayers, only volume magnetostatic waves can propagate.
IV. APPLICATIONS
As bulk magnetostatic waves in an ordinary ferrite are
only supported below /H20881/H9275H/H20849/H9275H+/H9275M/H20850, they are limited to
relatively low-frequency applications, or need the application
of a large external field to increase /H9275H.14The results of the
previous section, however, suggest that one can have /H20849over-
all/H20850bulk waves supported by parallel multilayers at frequen-
cies up to /H9275H+/H9275M/2/H20849for/H9267=1/H20850or antiparallel multilayers at
frequencies up to /H9275H+/H9275M/H20849in the limit of thin dielectrics,
i.e.,/H9267→0/H20850. This would mean that for a ferrite with M
/H110110.4T,16operation at frequencies up to /H1101110 GHz is achiev-
able in an antiparallel magnetic/dielectric superlattice, possi-bly even without the application of an external field, as dis-cussed below.
While the realization of parallel multilayers is straight-
forward /H20849e.g., by applying an external dc magnetic field /H20850,
antiparallel order of magnetization in a multilayer can beachieved in a number of ways. One approach is to use twodifferent materials with different coercivities, such as NiFe/H20849soft /H20850and Co /H20849hard /H20850, or two otherwise similar layers of NiFe,
the coercivities of which have been made different by vary-ing the deposition conditions. One could saturate them in onedirection and then reverse the direction of magnetization inhalf of the layers by a relatively small magnetic field. Thiswould also enable one to switch the superlattice from parallelto antiparallel and vice versa. This is, in fact, the methodused in Ref. 8to make an antiparallel double layer. Another
interesting option is that for appropriate geometrical condi-tions /H20849e.g., dielectric and ferromagnetic layer thicknesses /H20850
and /H20849shape and/or internal /H20850anisotropies, the multilayers
might arrange in an antiparallel manner all by themselves,creating single-domain films with antiparallel magnetizationsas a result of the flux closure at the edges of the magneticfilms.
17–20One could thus consider multilayers with antipar-
allel configuration with a very small /H20849and even without /H20850a
external dc magnetic field.
FIG. 3. Magnetostatic potential /H9274for 20 magnetic layers in antiparallel
configuration separated by dielectrics. M=1 T, Ha=100 Oe, /H9267=1, and /H9258
=/H9266/2. Magnetostatic potential is normalized to the value B0in Eq. /H20849A4/H20850,
distance is in units of /H5129/4. Sinusoidal fit is also shown. The frequency of
propagation corresponding to this mode is 3.5 GHz.
FIG. 4. Comparison of EMT /H20849/H9004/H20850and TMM /H20849/H11001/H20850solutions for the two
highest-frequency modes in a superlattice of 20 magnetic layers with anti-parallel orientations of magnetization with M=1 T, H
a=100 Oe, and /H9267=1.
The deviation of EMT from the /H20849exact /H20850TMM results increases with k.103909-5 P . Khalili Amiri and B. Rejaei J. Appl. Phys. 100 , 103909 /H208492006 /H20850
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134.129.164.186 On: Sun, 21 Dec 2014 07:12:54The high propagation frequency of volume waves /H20849up to
/H9275H+/H9275M/H20850, possibility of realization of single-domain films,
and the absence of any need for an external dc field makeantiparallel magnetic/dielectric superlattices quite attractiveas core material of rf magnetostatic devices. In what follows,we consider two examples: waveguides and resonators basedon effective magnetostatic volume waves in antiparallel mul-tilayers.
A. Waveguides
Using the effective permeability approach, we can treat
the problem of a rectangular magnetostatic waveguide filledwith a multilayer composed of identical magnetic layers inthe antiparallel configuration. Figure 5shows such a wave-
guide with the magnetizations parallel /H20849top/H20850or perpendicular
/H20849bottom /H20850to the waveguide axis. Consider first the case where
the magnetizations are perpendicular to the direction of thewaveguide. Employing boundary conditions of zero normalmagnetic fields on the metal plates, it is straightforward toshow that
/H9274has the following form:
/H9274=Acos/H20873n/H9266
Wz/H20874cos/H20873m/H9266
Hy/H20874exp /H20849−ikx/H20850, /H2084946/H20850
where His the height and Wis the width of the waveguide
defined along the yandzaxes, respectively. Substituting /H2084946/H20850
in/H2084927/H20850, one obtains
m1k2+m2/H20873m/H9266
H/H208742
+/H20873n/H9266
W/H208742
=0 . /H2084947/H20850
Thus we have
k2=−1
m1/H20875m2/H20873m/H9266
H/H208742
+/H20873n/H9266
W/H208742/H20876. /H2084948/H20850
Using Eqs. /H2084921/H20850and /H2084922/H20850one can show that the propagation
frequency spans the entire range given by Eq. /H2084940/H20850.In the absence of an external field, it is often reasonable
to assume /H9275H/H11270/H9275M. It then becomes possible to obtain an
approximate dispersion relation at frequencies much higherthan
/H9275/H11036using the following approximations:14
m1=1+/H208731
1+/H9267/H20874/H9275M/H20849/H9275H+/H9275M/H20850
/H9275/H110362−/H92752/H112291−1
1+/H9267/H20873/H9275M
/H9275/H208742
,
/H2084949/H20850
m2=/H9275/H110362−/H92752
/H9275H/H9275e−/H92752/H112291. /H2084950/H20850
This gives the following result for /H2084948/H20850at high frequencies:
/H9275/H11229/H9275M
/H208811+/H9267k
/H20881k2+/H20849m/H9266/H/H208502+/H20849n/H9266/W/H208502. /H2084951/H20850
We consider next a waveguide with height Halong the y
axis and width Walong the xaxis, thus assuming the direc-
tion of propagation to be along the magnetization axis z.
Similar to the previous case we now have
/H9274=Acos/H20873n/H9266
Wx/H20874cos/H20873m/H9266
Hy/H20874exp /H20849−ikz/H20850. /H2084952/H20850
By substitution of this equation in /H2084927/H20850we get
m1/H20873n/H9266
W/H208742
+m2/H20873m/H9266
H/H208742
+k2=0 . /H2084953/H20850
As in the previous case the waveguide modes span the fre-
quency range /H2084940/H20850. It is interesting to note, however, that
laterally uniform /H20849slablike /H20850modes /H20849n=0/H20850can only exist at
frequencies below /H9275/H11036where m2/H110210. At frequencies much
higher than /H9275/H11036one can simplify the dispersion equation us-
ing /H2084949/H20850and /H2084950/H20850as in the previous case,
/H9275=/H9275M
/H208811+/H9267n/H9266/W
/H20881k2+/H20849m/H9266/H/H208502+/H20849n/H9266/W/H208502/H2084954/H20850
The dispersion diagrams of the waveguide modes for m=1
andn=0, ... ,2 are shown in Figs. 6and7, respectively, for
propagation parallel and normal to the magnetization. Theydemonstrate the applicability of the approximate relations/H2084951/H20850and /H2084954/H20850at high frequencies by comparing them to /H2084948/H20850
and /H2084953/H20850. Note that while /H20849at high frequencies /H20850
/H9275is a de-
creasing function of k/H20849negative dispersion /H20850for propagation
parallel to the magnetizations /H20849/H9258=0/H20850, it is an increasing func-
FIG. 5. Waveguides with multilayer magnetic cores in the antiparallel con-
figuration with magnetizations parallel /H20849top/H20850or perpendicular /H20849bottom /H20850to the
direction of propagation. The intermediate dielectric layers are not shown.
FIG. 6. EMT solutions for rectangular waveguide with exact /H20849solid /H20850and
approximate /H20849/H11001/H20850dispersion relations. M=1 T, Ha=100 Oe, /H9267→0, and /H9258
=0.103909-6 P . Khalili Amiri and B. Rejaei J. Appl. Phys. 100 , 103909 /H208492006 /H20850
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134.129.164.186 On: Sun, 21 Dec 2014 07:12:54tion of k/H20849positive dispersion /H20850for propagation normal to the
magnetizations /H20849/H9258=/H9266/2/H20850.21
B. Resonators
By a similar approach, we consider resonators employ-
ing multilayer antiparallel magnetic cores. Considering thedimensions in the x,y, and zdirections to be L,H, and W,
respectively, the condition of zero normal magnetic fields onthe metal plates translates into
/H9274=Acos/H20873n/H9266
Wz/H20874cos/H20873m/H9266
Hy/H20874cos/H20873l/H9266
Lx/H20874. /H2084955/H20850
Substituting /H2084955/H20850in/H2084927/H20850, we have
m1/H20873l/H9266
L/H208742
+m2/H20873m/H9266
H/H208742
+/H20873n/H9266
W/H208742
=0 . /H2084956/H20850
An approximation for resonance frequencies much higher
than/H9275/H11036can be written as
/H9275=/H9275M
/H208811+/H9267l/H9266/L
/H20881/H20849l/H9266/L/H208502+/H20849n/H9266/W/H208502+/H20849m/H9266/H/H208502. /H2084957/H20850
As expected, all approximate relations predict the same up-
per frequency bound /H9275M//H208811+/H9267for the devices.
It should be emphasized that all results of the EMT ap-
proach are only correct if kis small. If the wavelength of
oscillations in the waveguides or resonators becomes smallenough to be comparable with the layer thicknesses in themultilayer, the results will not be valid, and more accuratemethods /H20849e.g., TMM /H20850must be used. EMT is thus only valid
for modes lower than a certain set of numbers l,m, and n.
The exact values of these are determined by the relation of L,
H, and Wwith t
MandtDin Fig. 1. While small values of k
improve the accuracy of the EMT approximation, one musttake into account that kshould still be large enough in order
for the magnetostatic approach to be reliable.
14As an esti-
mate, superlattice periods smaller than a few microns fallwithin the range of validity of the presented calculations.These are typical values of interest, especially for integratedapplications, and further reducing the superlattice period/H20849and thus individual layer thicknesses /H20850only improves the
accuracy of the EMT results. The above discussion ofwaveguides and resonators is thus applicable to a fairly largeset of practical situations.V. CONCLUSION
The propagation of magnetostatic waves in layered me-
dia and, in particular, the case of alternating directions ofmagnetization in a magnetic/dielectric superlattice were in-vestigated. An effective permeability approach was used toanalyze wave propagation in the multilayers and verified bycomparison with a transfer matrix solution. Possible applica-tions were suggested in the form of waveguides and resona-tors based on antiparallel magnetic multilayers. It was shownthat in such multilayers, one can obtain operation at frequen-cies which are unattainable, or hard to achieve /H20849e.g., by a
large external dc magnetic field /H20850using conventional mag-
netic materials. This is because of the possibility of /H20849overall /H20850
bulk magnetostatic wave propagation at high frequencies inlayered media with antiparallel magnetizations.
The above analysis neglects magnetic relaxation losses
/H20849which can be included, however, through the transformation
/H9275H→/H9275H+i/H9251/H9275, where /H9251is the Gilbert damping factor14/H20850,a s
well as conductivity of the magnetic films. While the latter isreasonable for ferrites, it is not true for materials currentlybeing investigated for on-chip integration with silicon, whichare mostly metallic /H20849e.g., Ref. 22–27/H20850. However, as we are
considering multilayers, rather than single layers of magneticfilms, the flow of eddy currents is impeded due to the pres-ence of the dielectrics. The effect of conductivity becomessmaller with further reduction of the individual magneticlayer thicknesses, simultaneously also improving the accu-racy of EMT, as discussed before. The thickness of the di-electrics should be big enough to eliminate the possibility ofexchange interaction among the magnetic layers, otherwise itwould reduce the accuracy of our analysis, which excludesexchange interaction. Although it has been shown that theantiparallel configuration may, in fact, be the natural state ofa multilayer /H20849due to flux closure at the edges /H20850,
17–20realization
of the antiparallel configuration for a particular material andgeometrical design and the conductivity of many magneticmaterials which could be of interest remain the practicalproblems to be solved before the permeability engineeringcan be used in practice.
ACKNOWLEDGMENTS
The authors would like to thank M. Vroubel and Y .
Zhuang from the Delft Institute of Microelectronics and Sub-micron Technology and Professor J. N. Burghartz from theInstitute for Microelectronics in Stuttgart for helpful discus-sions and suggestions. This work is part of a project sup-ported by the “Stichting voor de Technische Wetenschappen/H20849STW /H20850” of the Netherlands.
APPENDIX
The transfer matrix method /H20849TMM /H20850for a magnetic/
dielectric superlattice can be easily formulated within thequasistatic approximation. Inside each layer /H20849designated by a
subscript m/H20850, the magnetostatic potential defined by H=/H11612
/H9274
satisfies the Walker equation,
FIG. 7. EMT solutions for rectangular waveguide with exact /H20849solid /H20850and
approximate /H20849/H11001/H20850dispersion relations. M=1 T, Ha=100 Oe, /H9267→0, and /H9258
=/H9266/2.103909-7 P . Khalili Amiri and B. Rejaei J. Appl. Phys. 100 , 103909 /H208492006 /H20850
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134.129.164.186 On: Sun, 21 Dec 2014 07:12:54/H9262/H20873/H115092/H9274
/H11509x2+/H115092/H9274
/H11509y2/H20874+/H115092/H9274
/H11509z2=0 , /H20849A1/H20850
where /H9262=1 for nonmagnetic layers.
The boundary conditions /H20849continuity of tangential Hand
normal Bcomponents /H20850between layers mandm+1 can be
expressed in terms of /H9274as14
/H11509/H9274m
/H11509x=/H11509/H9274m+1
/H11509x, /H20849A2/H20850
−i/H9262a,m/H11509/H9274m
/H11509x+/H9262m/H11509/H9274m
/H11509y=−i/H9262a,m+1/H11509/H9274m+1
/H11509x+/H9262m+1/H11509/H9274m+1
/H11509y.
/H20849A3/H20850
/H20849For nonmagnetic layers we take /H9262a=0. /H20850Assuming the di-
rection of propagation to lie inside the x-zplane, the solution
in a magnetic or dielectric layer with index mis given by
/H9274m=/H20851Amexp /H20849−/H9264my/H20850+Bmexp /H20849/H9264my/H20850/H20852exp /H20849−ikxx−ikzz/H20850,/H9264m2=kx2+kz2
/H9262m. /H20849A4/H20850
From /H20849A2/H20850and /H20849A3/H20850, one can relate the wave amplitudes in
adjacent layers according to
/H20873Am+1
Bm+1/H20874=QJm+1,m/H20873Am
Bm/H20874, /H20849A5/H20850
QJm+1,m=1
2/H9262m+1/H9264m+1/H20875/H20849/H9257m+1−+/H9257m+/H20850exp /H20849/H9264m+1ym−/H9264mym/H20850/H20849/H9257m+1−−/H9257m−/H20850exp /H20849/H9264mym+/H9264m+1ym/H20850
/H20849/H9257m+1+−/H9257m+/H20850exp /H20849−/H9264mym−/H9264m+1ym/H20850/H20849/H9257m+1++/H9257m−/H20850exp /H20849/H9264mym−/H9264m+1ym/H20850/H20876, /H20849A6/H20850
where /H9257m±=/H9262m/H9264m±/H9262a,mkx, and ymdenotes the vertical posi-
tion of the interface between the layers.
For a total number of ferromagnetic layers N, there are
2Ninterfaces in the multilayer, and the overall transfer ma-
trix of the system is given by
QJ=/H20863
m=02N−1
QJm+1,m. /H20849A7/H20850
It relates the wave amplitudes in the two infinite layers be-
low and above the multilayer, denoted by the subscripts 0and 2 N, respectively. Since A
0=B2N=0, we have
/H20875A2N
0/H20876=/H20875Q11Q12
Q21Q22/H20876/H208750
B0/H20876, /H20849A8/H20850
leading to the relation
Q22=0 . /H20849A9/H20850
Due to the frequency dependence of permeability of the
magnetic layers, solution of the above equation results in thedispersion relation of the multilayer.
One can further make use of this formulation to derive
dispersion relations for an infinite stack of layers. Designat-
ing the transfer matrix of a unit cell /H20849i.e.,d
mtodm+4/H20850with QJu,
the Bloch theorem28can be used. This means that the eigen-
values of QJumust be of the form exp /H20849−ikB/H5129/H20850, where kBis theBloch wave propagation constant and /H5129is the period of the
superlattice. This is equivalent to the following dispersionrelation:
cos/H20849k
B/H5129/H20850=1
2/H20849Q11u+Q22u/H20850, /H20849A10 /H20850
where Q11uandQ22uare functions of both kand/H9275. Using /H20849A5/H20850
one can also plot the wave form of the magnetostatic poten-tial
/H9274for given kand/H9275.
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10The method used throughout this work can also be applied to situations
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tropy field /H20850and thickness of neighboring magnetic layers are different.
Nevertheless, we restrict ourselves to identical magnetic layers for thesake of simplicity.
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15740 /H208491993 /H20850.
FIG. 8. EMT solutions for rectangular waveguide with M=1 T, Ha
=100 Oe, /H9267→0,/H9258=/H925811=sin−1/H208491//H208813/H20850/H1122935°. Note the mode at /H927511=/H9275M//H208813
/H1122916.5 GHz.103909-8 P . Khalili Amiri and B. Rejaei J. Appl. Phys. 100 , 103909 /H208492006 /H20850
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
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/H20849CRC, New York, 1996 /H20850.
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York, 1993 /H20850.
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printed /H20849Lucent Technologies, Murray Hill, 1978 /H20850.
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21For a mode /H20849m,n/H20850it is possible to find an angle /H9258mnand a corresponding
frequency /H9275mnwhere the propagation is dispersionless. Using the modified
Walker’s equation /H2084927/H20850, and the approximate relations /H2084949/H20850and /H2084950/H20850, one
finds /H9258mn=sin−1/H20853/H208512+/H20849m/H9266/H/H208502//H20849n/H9266/W/H208502/H20852−1/2/H20854 and /H9275mn
=/H20849/H9275M//H208811+/H9267/H20850sin/H9258mn. It is easy to see from these relations that we alwayshave 0 /H11021/H9258mn/H11021/H9266/4 and /H9275mn/H11021/H9275M//H208812/H208491+/H9267/H20850. The case /H9258=/H925811
=sin−1/H208491//H208813/H20850/H1122935° for /H9267→0 is shown in Fig. 8.
22M. Vroubel, Y . Zhuang, B. Rejaei, and J. N. Burghartz, Trans. Magn. Soc.
Jpn. 2, 371 /H208492002 /H20850.
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Electron Devices Meet. 2002 , 18.06/1.
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J. Appl. Phys. 97,1 /H208492005 /H20850.
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28C. Kittel, Introduction to Solid State Physics , 5th ed. /H20849Wiley, New York,
1976 /H20850.103909-9 P . Khalili Amiri and B. Rejaei J. Appl. Phys. 100 , 103909 /H208492006 /H20850
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
134.129.164.186 On: Sun, 21 Dec 2014 07:12:54 |
1.5042567.pdf | Staggered field driven domain walls motion in antiferromagnetic heterojunctions
Y. L. Zhang , Z. Y. Chen , Z. R. Yan , D. Y. Chen , Z. Fan , and M. H. Qin
Citation: Appl. Phys. Lett. 113, 112403 (2018); doi: 10.1063/1.5042567
View online: https://doi.org/10.1063/1.5042567
View Table of Contents: http://aip.scitation.org/toc/apl/113/11
Published by the American Institute of PhysicsStaggered field driven domain walls motion in antiferromagnetic
heterojunctions
Y. L . Zhang, Z. Y . Chen, Z. R. Y an, D. Y. Chen, Z.Fan, and M. H. Qina)
Institute for Advanced Materials, South China Academy of Advanced Optoelectronics and Guangdong
Provincial Key Laboratory of Quantum Engineering and Quantum Materials, South China Normal University,
Guangzhou 510006, China
(Received 2 June 2018; accepted 25 August 2018; published online 10 September 2018)
In this work, we study the antiferromagnetic (AFM) spin dynamics in heterostructures which
consist of two coupled AFM layers, i.e., AFM1 layers (describing CuMnAs or Mn 2Au) with field-
like N /C19eel spin-orbit torque (NSOT) and AFM2 layers with easy-axis anisotropy orthogonal to that
in AFM1 layers. Our micromagnetic simulations demonstrate that through the interface coupling,
the AFM2 domain wall (DW) can be effectively driven by the AFM1 DW which is driven by theelectrical current induced NSOT [Gomonay et al. , Phys. Rev. Lett. 117, 017202 (2016)].
Furthermore, the two DWs detach from each other when the torque increases above a critical value.
The critical field and the highest possible velocity of the AFM2 DW depend on several factors,which are investigated and discussed in detail. Based on the calculated results, we propose a
method of efficiently modulating the multi DWs in antiferromagnets, which definitely provides use-
ful information for future AFM spintronics device design. Published by AIP Publishing.
https://doi.org/10.1063/1.5042567
Antiferromagnets are attracting widespread attention due
to their potential applications in the field of antiferromagnetic(AFM) spintronics. On the one hand, replacing ferromagnetsby antiferromagnets in spintronics devices offers insensitivityto external magnetic field perturbations and produces no per-turbing stray fields due to zero net magnetic moment in theAFM element.
1–3Thus, information stored in AFM domains
or domain walls (DWs) is robust, and the AFM elements can
be arranged with high density.4On the other hand, antiferro-
magnets generally exhibit ultrafast spin dynamics with charac-teristic frequencies in the THz range.
5More importantly, the
velocity of an AFM domain wall (DW) is only limited by theg r o u pv e l o c i t yo fs p i nw a v e s ,
6–8which is almost two orders of
magnitude larger than the velocity of a typical ferromagnetic
(FM) DW which is limited by the Walker breakdown.9–12As
a result, AFM spintronics is very promising in future storagedevices, and several efficient methods of modulating AFMdomains and driving DW motion have been revealed experi-mentally and/or theoretically.
13–22
Recent experiments reported that the applied electrical
current induces the local staggered effective field (or field-likeN/C19eel spin-orbit torque, NSOT)
23,24in CuMnAs25–27and
Mn2Au28due to the spin-orbit effect and in turn modulates the
orientation of the AFM moments. More interestingly, a high
velocity /C2410–100 km/s of the AFM DW motion driven by the
NSOT has been predicted in theory.13Specifically, for an
AFM DW under the staggered field B 1N(B2N) along the up
(down) direction coupling to the spin m1(m2)o nt h em a g n e t i c
sublattice 1 (2), the N /C19eel spin-orbit torque C1N(C2N)i s
induced and cants m1(m2) forward, as depicted in Fig. 1(a).
Subsequently, a rather large precession torque Cp
exfrom the
strong exchange interaction is generated due to the spin devia-tion, and drives the DW motion. Furthermore, AFM DWmotion has also been theoretically predicted by several exter-
nal stimuli including spin wave excitations,
7,21spin-orbit
torques,6,19temperature gradient,15–17and asymmetric field
pulses.14,20For example, it has been proven that the competi-
tion between the entropic torque and the Brownian force
determines the AFM DW motion towards the hotter or the
colder region under an applied temperature gradient.15–17
These proposed methods do provide important informa-
tion for future applications, while several shortcomingsdeserve to be overcome, especially considering that one ofthe major challenges in future AFM spintronics applications
is fast transport of multi AFM DWs for information storage.
For example, NSOT only arises in these AFM materials withparticular crystal structures. More importantly, NSOT drivesthe neighboring AFM DWs to approach to each other and
annihilates them,
13,29which strongly hinders its application
in future device where efficient motions of multi DWs are
FIG. 1. (a) Illustration of torques exerted on the center of AFM DW and (b)
schematic illustration of AFM1 and AFM2 DWs in the heterojunction.a)Electronic mail: qinmh@scnu.edu.cn
0003-6951/2018/113(11)/112403/5/$30.00 Published by AIP Publishing. 113, 112403-1APPLIED PHYSICS LETTERS 113, 112403 (2018)
indispensable. For other proposed methods, the drift veloci-
ties of AFM DWs are not so large, and strict restrictions on
these external stimuli are suggested. For example, our earlier
work has demonstrated that considerable velocity of the
AFM DW can be obtained only near the resonance frequency
of the oscillation magnetic field in cooperation with a static
field.14In addition, it has been proven in earlier work that
the interplay between the AFM and FM DWs in the FM-
AFM double layers can shift the AFM DW, when the FM
DW is driven by a spin current.18However, the velocity is
also limited by the Walker breakdown. Thus, there is still anurgent need in searching for efficient methods of driving
multi AFM DWs motion at a high speed.
Based on these previous studies, we investigate the AFM
dynamics in exchange coupled AFM1-AFM2 heterojunction
in which only the DW in AFM1 layers is driven by the
NSOT, as simplistically depicted in Fig. 1(b).W efi g u r eo u t
that the AFM2 DW can be driven efficiently by the interface
coupling. Furthermore, the critical field above which the two
DWs are detached and the highest possible velocity of the
AFM2 DW relevant to several factors have been clarified and
explained in detail. Based on this property, we put forward a
proposal for controlling multi-DWs in antiferromagnets.
In this work, the heterojunction system is considered to
be a three-dimensional cuboidal lattice with the free bound-
ary conditions applied in the x-,y-, and z-axis directions. The
model Hamiltonian is given by
H¼H
AF1þHAF2þHinter; (1)
where
HAF1¼JAF1X
hi;jimi/C1mj/C0dxX
imx
i/C0/C12
/C0dzX
imz
iðÞ2þX
iBN/C1mx
i; (2)
HAF2¼JAF2X
hn;mimn/C1mm/C0DxX
nmx
n/C0/C12/C0DzX
nmz
nðÞ2;
(3)
Hinter¼JinterX
hi;nimi/C1mn; (4)
where JAF1/JAF2is the AFM exchange interaction between
the nearest neighbors in AFM1/AFM2 layers, and Jinteris the
interface coupling between the nearest neighbors, mk¼lk/ls
(k¼i,j,m,n) is the normalized magnetic moment at site k
with the saturated moment ls,mr
k(r¼x,y,z) is the rcompo-
nent of mk,dx/J>dz/JandDz/J>Dx/Jare the anisotropy
constants, B N¼B1N(B2N) is the staggered field along the x
(–x)-axis direction on the magnetic sublattice 1 (2) in AFM1
layers. Here, the anisotropy constants could be modulated
through tuning the thickness of the heterojunction. The simu-
lation is performed on a Lx/C2Ly/C2Lzlattice (20 /C230/C2350,
a multilayer of 10 AFM1 and 10 AFM2 with a size of 30/
350 along the y/zdirection per layer) by solving the Landau-
Lifshitz-Gilbert equation30,31
@mi
@t¼/C0c
1þa2 ðÞmi/C2Hiþami/C2Hi ðÞ ½/C138 ; (5)where cis the gyromagnetic ratio, a¼a1(a2) is the Gilbert
damping constant in AFM1 (AFM2) layers, and the internal
field is Hi¼/C0@H/@mi. Unless stated elsewhere, JAF1¼JAF2
¼J(J¼1 is the energy unit), Jinter¼0.5J,dx¼Dz¼0.1J,
and a1¼a2¼0.01 are chosen. Moreover, the fourth-order
Runge-Kutta method is used to solve the equation with atime step Dt¼5.0/C210
/C04jls/cJj. After sufficient relaxation
of the N /C19eel-type DWs, we apply the staggered field and
study the DWs motion. The local staggered magnetization2n¼m
1/C0m2is calculated to describe the spin dynamics.
First, we study the case of the uniaxial anisotropy
(dz¼Dx¼0). The initial spin configuration is presented in
Fig. 2(a) which clearly shows two N /C19eel-type AFM DWs.
Specifically, AFM1 DW at the position z¼30aand AFM2
DW at the position z¼60a(ais the lattice constant) are
observed, as clearly shown in Fig. 2(b) which gives the three
components of the local magnetization n1andn2.F u r t h e r m o r e ,
theycomponent of the magnetization in the whole system
equals to zero ny
1¼ny
2¼0, and a small nz
1(nx
2) of the local
magnetization in AFM1 (AFM2) domains is observed due to
the interface coupling.
The electrical current induced B Ncan drive the motion
of the AFM1 DW. The velocity of the DW increases withthe increase in B
N, as clearly shown in Fig. 2(c)which gives
the displacements of the AFM1 DW (solid lines) and AFM2
DW (dashed lines) as functions of time for various B N.
Similar to the earlier report, the AFM1 DW can also shift theAFM2 DW at the expense of its velocity through the inter-play between the two DWs when they are close enough.
Subsequently, the two DWs connect and shift at a same
speed, as clearly shown in the supplementary material Movie
S1 which gives the motion of the DWs for B
N¼0.03. To
some extent, this phenomenon can be understood qualita-
tively by the analogue between the DWs interplay and an
inelastic collision of two quasiparticles, as detailedlyexplained in the earlier work.
18Taking CuMnAs as AFM1
FIG. 2. (a) The initial spin configuration and (b) the transverse component
of AFM1 and AFM2 DWs. (c) The displacement of the DWs dependent oftime under various staggered fields. The position of AFM1/AFM2 DW is
marked by the solid/dashed line.112403-2 Zhang et al. Appl. Phys. Lett. 113, 112403 (2018)layers, B N¼0.01 corresponds to the field of 30 mT induced
by a current density of 1.5 /C2108A/cm2and may result in the
DW motion at a speed of 200 m/s. Interestingly, the AFM2
DW detaches from the AFM1 DW when B Nfurther increases
above the critical value /C240.048, as clearly shown in the sup-
plementary material Movie S2 and Movie S3, which present
the local magnetization and spin configuration for B N¼0.05,
respectively. It is clearly shown that the AFM1/AFM2 DW
rotates by 180/C14around the x-/z-axis after the detachment.
The equilibrium velocities of the DWs for various B N
are summarized in Fig. 3(a). On the one hand, the velocity
increases nonlinearly with the increase in B N(below the
critical value) due to the spin-wave excitation as shown in
Fig. 3(b) which gives the local magnetization for
BN¼0.04. It is noted that the spin-wave excitation acting
as an additional energy dissipation is enhanced as B N
increases, resulting in the decrease of the acceleration withB
N, as confirmed in our simulations. As a matter of fact, the
AFM2 DW is driven by the AFM1 DW through the damp-
ing torque /C24/C0S/C2(S/C2Hinter), which results from the cou-
pling between the two DWs, as depicted in Fig. 3(c).A sB N
increases, the two DWs get close to each other, resulting in
the increase of the damping torque. The maximum value of
the torque is obtained at the critical B Nunder which the
positions of the two DWs coincide with each other [middle
layer in Fig. 3(c)], and the AFM2 DW reaches its highest
possible velocity vc. In this case, the coupling energy
between the two DWs is rather high, reducing the stability
of the DWs. As B Nfurther increases, the wall linking is not
stable and the detachment occurs. Furthermore, the spins in
the AFM1/AFM2 DW are driven out of the easy plane androtate by 180/C14around the x-/z-axis to save the interface
exchange coupling energy.
Subsequently, we investigate the effects of the interac-
tion couplings on the spin dynamics and the critical velocityv
c. Figure 4(a) gives the velocities of the DWs (solid and
empty circles) under B N¼0.04 and vc(blue asterisks under
labeled critical B N) for various JAF2. It is well noted that the
energy of the DW determines its stabilization, i.e., higher
energy results in more stable DW. Thus, the AFM2 DW is
further stabilized as JAF2increases, leading to the increase in
the critical B Nandvc, as clearly shown in our simulations.
Furthermore, for JAF2>1, the velocities of the DWs under
BN¼0.04 slightly decrease with the increasing JAF2due to
the reduction of their mobility. Furthermore, the effects ofJ
interare also studied, and the corresponding results are pre-
sented in Fig. 4(b). Both vcand the critical B Nsharply
increase as Jinterincreases and then slowly increase for Jinter
>0.5. Thus, it is suggested that vcis mainly determined by
JAF2for a rather large Jinter.
When the anisotropies of the individual layers are the
same (along the xorzdirection), the two DWs approach to
each other to save the interaction coupling energy even atB
N¼0, and the detachment of the two DWs is hardly real-
ized due to the strong interface couplings even when B N
increases above 0.1. These phenomena have been confirmed
in our simulations, although the corresponding results are
not shown here for brevity. Furthermore, biaxial anisotropydoes exist in some real materials and is suggested to play anessential role in the current-induced orientation of the AFMorder in an antiferromagnet.
32Thus, we also studied the
effects of the intermediate anisotropies dzand Dxon the
AFM dynamics for integrity. Figure 4(c)gives the simulated
velocity as a function of dzfor various Dxunder B N¼0.04.
FIG. 3. (a) The velocities of the DWs as functions of the staggered field B N,
and (b) the components of the local magnetization near the attached DWs
under B N¼0.04, and the calculated nz
1and nx
2demonstrate an spin-wave
emission, and (c) the depictions of the spin configurations of the DWs under
various staggered fields.FIG. 4. The velocities of the DWs (solid and dashed circles) under
BN¼0.04 as functions of (a) JAF2/Jand (b) Jinter/J. The critical B Nand
velocity are also presented by the blue number and asterisks, respectively.
The velocities of the DWs as functions of (c) dzfor various Dxfora1
¼a2¼0.01, and (d) a2for various a1fordz¼Dx¼0.02. The critical B Nand
velocity (asterisks) for several parameters are also given as an example.112403-3 Zhang et al. Appl. Phys. Lett. 113, 112403 (2018)It is clearly shown that the velocity is increased with the
increasing dzand/or Dx, demonstrating the role of the inter-
mediate anisotropy in the motion of the AFM DWs. Onemay note that the energy gap between the AFM DW anddomain prominently determines the mobility of the DW.When the intermediate anisotropy is considered, the energygap is reduced which enhances the reversal of the local spinsin the domains and results in a higher mobility of the DW.Thus, both d
zandDxcan speed up the AFM DWs, as con-
firmed in our simulations. Furthermore, both vc(asterisks)
and the critical B Nincrease due to the increases in the DW
energy when dzand/or Dxis increased. For example, the crit-
ical velocity for dz¼0.01 and Dx¼0.05 (red asterisk)
increases by /C2425% compared to the uniaxial case.
At last, the effects of the damping constants on the
velocities are also investigated and the simulated results arepresented in Fig. 4(d)which gives the velocities as functions
ofa
2for various a1under B N¼0.04. It is clearly shown that
the velocity decreases as the damping constant increases.One may note that the damping torque C
a
ex[Fig. 1(a)]i s
enhanced with the increase in the damping constant, whichreduces the effect of precession torque C
p
ex. As a result, the
effects of B Nare significantly suppressed, which speeds
down the AFM DWs. Furthermore, the critical B Nis
increased, while vcsignificantly decreases with the increase
in the damping constant, as shown in our simulations.
So far, it has been proven that the AFM2 DW can be
efficiently driven by the AFM1 DW through the interfacecoupling, and the two DWs are detached from each otherwhen B
Nincreases above a critical value. As a matter of
fact, these phenomena could be used in future AFM spin-tronics device design. For example, the bidirectional controlof AFM2 multi DWs can be realized through elaboratelymodulating B
Non the AFM1 layers. The proposed structure
is shown in Fig. 5(a) in which the different domains (blue
and red arrays in the top layer) are used to store information
bits (0 or 1) and their lengths determine the bit numbers. The
AFM1 layer (bottom layer) with single DW under B Nis used
to be a driving bar. The AFM2 DW gradually approaches toits neighbor, and both of the two DWs are annihilated finallywhen all the applied pulses are in the small B
Nrange.
Interestingly, the multi DWs can be effectively driven byalternately applying small and large B N, as depicted in the
inset of Fig. 5. Specifically, the first AFM2 DW can be
driven by the driving DW under a small B N, and detaches
from the driving DW (middle layer in the inset of Fig. 5)
when applying another B Nlarger than the critical value.
Subsequently, a small B Nis used to drive the motion of the
second AFM2 DW. As a result, the multi DWs can be indi-rectly driven by the driving AFM1 DW, as shown in Fig.
5(b). Then, the driving DW can shift back to the initial posi-
tion by applying a large opposite B
Nas depicted in Fig. 5(c).
Moreover, the reverse motion of the multi DWs can bedriven by the opposite B
Nthrough the inverse processes, and
the picture is not shown here for brevity. It is noted that themagnetization switching is hardly realized in the proposedstructure, and one may refer to other methods. For example,the reversal of magnetic domains has been experimentallyrealized in antiferromagnet NiO by anti-damping torqueinduced by applied electrical current.
32
Up till now, there is still an urgent need in modulating
AFM multi DWs to provide useful information for futurepractical applications, especially in theory, considering thelimitation of current experiments. In this work, the AFMDW has been proven to be driven efficiently by the drivingDW under the staggered field through the interface coupling
in heterostructures. The critical field above which the two
DWs detach from each other and the highest possible veloc-ity of the DW relevant to several factors have been clarifiedand explained in detail. Moreover, the control of multi DWsis proposed based on the detachment of the two DWs under astaggered field larger than the critical value, which definitelyprovides useful information for future applications.
Seesupplementary material for Movie S1: The motion
of the AFM DWs for B
N¼0.03. The simulation parameters
used are JAF1¼JAF2¼J,Jinter¼0.5J,dx¼Dz¼0.1J, and
a1¼a2¼0.01. Movie S2: The motion of the AFM DWs for
BN¼0.05. The simulation parameters used are JAF1¼JAF2
¼J,Jinter¼0.5J,dx¼Dz¼0.1J, and a1¼a2¼0.01. Movie
S3: The spin configuration in the motion of AFM DWs for
BN¼0.05. The simulation parameters used are JAF1¼JAF2
¼J,Jinter¼0.5J,dx¼Dz¼0.1J, and a1¼a2¼0.01.
The work was supported by the National Key Projects
for Basic Research of China (Grant No. 2015CB921202),and the Natural Science Foundation of China (Grant Nos.51332007 and 11204091), the Science and TechnologyPlanning Project of Guangdong Province (Grant No.2015B090927006), the Natural Science Foundation ofGuangdong Province (Grant No. 2016A030308019), andGuangdong Provincial Engineering Technology ResearchCenter for Transparent Conductive Materials.
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1.3701585.pdf | Enhancement of perpendicular magnetic anisotropy through reduction of Co-Pt
interdiffusion in (Co/Pt) multilayers
S. Bandiera, R. C. Sousa, B. Rodmacq, and B. Dieny
Citation: Applied Physics Letters 100, 142410 (2012); doi: 10.1063/1.3701585
View online: http://dx.doi.org/10.1063/1.3701585
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/100/14?ver=pdfcov
Published by the AIP Publishing
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This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 157.211.3.38
On: Fri, 28 Nov 2014 06:16:10Enhancement of perpendicular magnetic anisotropy through reduction
of Co-Pt interdiffusion in (Co/Pt) multilayers
S. Bandiera,a)R. C. Sousa, B. Rodmacq, and B. Dieny
SPINTEC, UMR 8191 CEA/CNRS/UJF-Grenoble 1/Grenoble-INP, INAC, F-38054 Grenoble, France
(Received 22 February 2012; accepted 21 March 2012; published online 6 April 2012)
We demonstrate that the effective magnetic anisotropy of sputtered (Co/Pt) multilayers can be
doubled by limiting the interdiffusion occurring at Co/Pt interfaces. We present a way to decrease
the interdiffusion by inserting an ultra-thin Cu layer at or near the Co/Pt interfaces. When such amaterial is sputtered on Co prior to the Pt deposition, the perpendicular magnetic anisotropy, as
well as the thermal stability, is enhanced for Co layer thicknesses smaller than 1 nm. This is of
great interest for out-of-plane magnetized spintronic devices which require high perpendicularmagnetic anisotropy for down-size scalability reasons together with a free layer as thin as possible
to reduce the writing energy when switched by spin transfer torque.
VC2012 American Institute of
Physics .[http://dx.doi.org/10.1063/1.3701585 ]
The control of perpendicular magnetic anisotropy
(PMA) is crucial in many of magnetic or spin-electronics
devices. Depending on the device, the anisotropy must beadjusted in order to match the technological requirements. In
the case of data storage media such as hard disk drives or
magnetoresistive random access memories (MRAMs), thisanisotropy has to be as large as possible in order to be able to
increase the storage density while keeping a sufficiently high
thermal stability (i.e., retention) of the bit magnetization.Typically, the condition K
effV>50–70 k BT must be fulfilled
for 10 yr retention, where Keffdenotes the effective anisot-
ropy of the magnetic material and V the magnetic volume ofa bit of information. Materials with PMA attract much inter-
est since their larger magnetic anisotropy compared to in-
plane magnetized materials allows scaling down to smallerbit dimensions. Moreover, in spin transfer torque (STT)
MRAMs, using perpendicular magnetized magnetic tunnel
junctions (p-MTJs) allows a priori STT switching at lower
current density. This, however, is only true if the ratio of Gil-
bert damping to current polarization remains of the same
order of magnitude in p-MTJs compared to in-plane MTJs.
1
STT switching offers better down-size scalability in MRAMs
since the switching current scales with the cell area.
A wide variety of materials present PMA. It can have a
bulk origin as in L1 0ordered alloys or rare earth/transition
metal alloys, or an interfacial origin as in NM/CoFeB/oxide
stacks2or (Co/NM) multilayers,3,4where NM denotes a non
magnetic metal. In these latter cases, the interfacial anisot-
ropy may arise from the hybridization of Co and NM orbitals
or from magnetoelastic effects.5Usually, non magnetic
materials with strong spin-orbit coupling such as Au, Pd, or
Pt allow obtaining large PMA. Recently, it was shown that
the PMA of a Pt/Co/Pt stack is mainly induced at the bottominterface, i.e., when the Co layer is deposited on a Pt buffer.
6
In addition, interdiffusion occurring at the top Co/Pt inter-face can deteriorate the PMA (Refs. 6and7) of the stack
when the Co layer is thinner than 1 nm.
In this letter, we show that the PMA of (Co/Pt) multi-
layers can be doubled by inserting an ultrathin Cu layer at
the Co/Pt interfaces. Such an insert prevents the diffusion of
Pt atoms into the Co layer, leading to an improved PMA.This enhancement is of great interest for MTJ electrodes for
MRAMs since it leads to an increased thermal stability while
minimizing the magnetic volume of the free layer.
Metallic layers were deposited on thermally oxidized sili-
con substrates by DC magnetron s puttering with an Ar pressure
of 2/C210
/C03mbar at room temperature. Samples were subse-
quently annealed at different temperatures T Aunder 10/C06mbar
vacuum for 30 min. All measurements were carried out at room
temperature on full sheet samples. Out-of-plane hysteresisloops were measured by extraord inary Hall effect (EHE), while
the magnetization (M
S) of the layers was measured by vibrat-
ing sample magnetometry (VSM). The effective anisotropy(K
eff) was calculated from the determination of both the anisot-
ropy field H Kand the measured M Sas Keff¼HKMS/2, H K
being measured by EHE with an in-plane applied field. In order
to study the interdiffusion occurring between the different met-
als in the stack, samples comprising a single Co layer have
been grown. The thickness of this Co layer has been chosen insuch a way that the interdiffusion effect is emphasized.
In order to probe the Co/Pt interdiffusion, and consider-
ing that Co and Cu are almost immiscible,
8a first series of
samples of composition Ta(3)/Pt(5)/Co(0.3)/Pt(t Pt)/Cu(2)/
Pt(3) (thickness in nm) were prepared and characterized in
the as-deposited state. Fig. 1shows typical hysteresis loops
obtained with a varying thickness of the Pt layer inserted
between the Co and Cu ones, measured by EHE in the out-
of-plane direction. While the samples with t Pt<1 nm present
perfectly square hysteresis loops, both remanence and coer-
civity disappear when t Ptis further increased. For 2 nm of Pt,
the saturation field increases up to 2 kOe. These latter sam-ples are still out-of-plane magnetized since hysteresis loops
measured by VSM show that the out-of-plane susceptibility
is larger than the in-plane one. The absence of remanence isa)Present address: Crocus Technology, 4 Place Robert Schuman, 38025
Grenoble, France. Electronic mail: sebastien.bandiera@cea.fr.
0003-6951/2012/100(14)/142410/4/$30.00 VC2012 American Institute of Physics 100, 142410-1APPLIED PHYSICS LETTERS 100, 142410 (2012)
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 157.211.3.38
On: Fri, 28 Nov 2014 06:16:10thus due to the formation of perpendicular magnetic
domains. Fig. 2presents the evolution of both Keffand M Sof
these samples as a function of t Pt. The increase of t Ptinduces
a reduction of the effective anisotropy Keff, together with a
decrease of the magnetization M S. Such a behavior can be
ascribed to an increase of Co-Pt interdiffusion when t Pt
increases, this interdiffusion being absent when the Cu layer
is in direct contact with Co, since Cu and Co are almost im-
miscible.8From Fig. 2, the deduced Co-Pt interdiffusion
length in the as-deposited state is of the order of 1.0 to
1.5 nm, in agreement with previous measurements obtained
by tomographic atomic probe.9It is noticeable that the maxi-
mum of PMA is obtained with t Pt¼0 nm, indicating that the
top Co/Pt interface does not induce a significant PMA com-
pared to the bottom one, as shown in previous studies,6and
that a Cu layer deposited on top of the Co layer allows
increasing the PMA of the stack compared to a Pt layer.
Ideally, a Cu monolayer should be inserted between the
Co layer and the Pt protection layer. However, since Cu and
Pt are miscible,10the benefits of such an insertion can be
cancelled if the Pt atoms diffuse to the Co layer. The evolu-tion of K
effand M Sas a function of the Cu insertion thick-
ness is studied in Ta(3)/Pt(5)/Co(0.4)/Cu(t Cu)/Pt(3) samples.
The evolution of both Keffand M Sis plotted in Fig. 3.Keff
and M Sincrease with t Cuand reach a saturation at about
tCu¼0.8 nm. Once again, Keffand M Sseem strongly corre-
lated indicating that the evolution of the effective anisotropyis governed by the interdiffusion occurring at the top Co/Pt
interface. It can be thus stated that the typical Cu-Pt interdif-
fusion length is of about 0.8 nm.
On the other hand, inserting a Cu layer between the bot-
tom Pt buffer and the Co layers is not suitable since the bot-tom Pt/Co interface is the main contributor to the PMA. Fig. 4
presents the evolution of K
effand M Sas a function of the Cu
thickness in Ta(3)/Pt(5)/Cu(t Cu)/Co(0.5)/Pt(3) stacks. These
measurements indicate that in that case, Keffand M Sare still
correlated and both decrease upon bottom Cu insertion. A sig-
nificant contribution to PMA is observed when t Cuis thinner
than 0.5 nm. This may be due to Cu-Pt interdiffusion leadingto the formation of a CuPt alloy at the bottom interface, induc-
ing an interfacial PMA lower than pure Pt but higher than
pure Cu. One should expect that the magnetization of the Colayer would increase by inserting a Cu layer immiscible with
Co at the bottom interface. However, interdiffusions in Pt/Co/
Pt stacks are negligible at the bottom Pt/Co interface,
6so that
MScannot be significantly enhanced by inserting a Cu layer
prior to the Co deposition. The observed reduction of M Swith
increasing t Cuis believed to be due to an increase of rough-
ness. Such a roughness increase may enhance the Co-Pt inter-
mixing arising at the top interface, but it may also deteriorate
the continuity of the Co layer (island growth rather than 2Dgrowth), reducing its Curie temperature and thus its magnet-
ization at room temperature.
These experiments on single Co layers show that (1)
the main contribution to interfacial PMA is due to the bottom
Pt/Co interface, (2) Co-Pt intermixing at the top interface
deteriorates the PMA of thin Co layers, and (3) such an inter-mixing can be avoided by inserting a thin Cu layer between
the Co layer and the top Pt capping layer, leading to an
increase of K
eff.
In order to obtain a sufficient thermal stability KeffV, V
can be increased by stacking n repeats of Co/Pt bilayers, Keff
FIG. 1. Out-of-plane hysteresis loops of Ta(3)/Pt(5)/Co(0.3)/Pt(t Pt)/Cu(2)/
Pt(3) stacks in the as-deposited state.
FIG. 2. Effective anisotropy Keff(a) and saturation magnetization M S(b) of
Ta(3)/Pt(5)/Co(0.3)/Pt(t Pt)/Cu(2)/Pt(3) as a function of the top Pt layer thick-
ness t Ptin the as-deposited state.
FIG. 3. Effective anisotropy Keff(a) and saturation magnetization M S(b) of
Ta(3)/Pt(5)/Co(0.4)/Cu(t Cu)/Pt(3) stacks as a function of the Cu layer thick-
ness t Cuin the as-deposited state.
FIG. 4. Effective anisotropy Keff(a) and saturation magnetization M S(b) of
Ta(3)/Pt(5)/Cu(t Cu)/Co(0.5)/Pt(3) stacks as a function of the Cu layer thick-
ness t Cuin the as-deposited state.142410-2 Bandiera et al. Appl. Phys. Lett. 100, 142410 (2012)
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 157.211.3.38
On: Fri, 28 Nov 2014 06:16:10remaining constant as a first approximation. On the other
hand, Cu insertions are of great interest in those (Co/Pt) n
multilayers, since it also increases the effective anisotropyK
effof each Co/Pt bilayer leading thus to an additional
increase of the overall thermal stability. The proposed mag-
netic multilayers consist of stacking (Co/Cu/Pt) trilayers
which present enhanced PMA compared to standard (Co/Pt)bilayers. However, magnetic coupling between the different
Co layers must be ensured. A good compromise between
PMA enhancement and strong enough magnetic coupling is
obtained with t
Cu¼0.4 nm. With such a Cu thickness, the
PMA of the stack is significantly increased while the mag-netic coupling is strong enough to totally pull out-of-plane a
2 nm thick magnetic layer such as CoFeB deposited on the
top of the (Co/Cu/Pt)
nmultilayer. Using thicker Cu inser-
tions results in CoFeB layers presenting a tilted magnetiza-
tion with respect to the normal of the thin film plane.
Moreover, for MRAM applications, annealing has to be
carried out in order to increase the tunnel magnetoresistance
(TMR) of the MTJ. However, such an annealing is detrimen-
tal to the PMA of metallic multilayers since it favors inter-diffusion. It is thus of great importance to check that these
multilayers can stand annealing temperatures of at least
300
/C14C. In order to study the improvement of PMA induced
by the Cu insertions, two types of multilayers were grown:
The first one is a standard Ta(3)/Pt(5)/[Co(t Co)/Pt(0.4)] /C25/
Cu(2)/Pt(2) multilayer and the second one consists of Ta(3)/Pt(5)/[Co(t
Co)/Cu(0.4)/Pt(0.4)] /C25/Cu(2)/Pt(2) stacking. Hys-
teresis loops indicate that both multilayers keep their out-of-
plane magnetization for annealing temperatures up to at least350
/C14C (not shown). Fig. 5shows the evolution of Kefffor
both types of multilayers as a function of Co thickness t Co
and annealing temperature T A. The optimal t Cowhich maxi-
mizes the PMA is 0.60( 60.05) nm for the standard (Co/Pt)
multilayers and 0.40( 60.05) nm for the (Co/Cu/Pt) ones. As
we showed above, Co layers with good magnetic propertiescan be grown thinner with a Cu insertion thanks to the reduc-
tion of Co-Pt interdiffusion. A moderate annealing below
250
/C14C slightly enhances the PMA of these multilayers, prob-
ably due to interface smoothing. The optimal annealing tem-
perature is about 200/C14C for the (Co/Pt) multilayers and
increases to 250/C14C for the (Co/Cu/Pt) ones. Such an
improvement is probably due to decreased interdiffusion at
the (Co/Pt) interfaces at moderate annealing temperatures,
since Co and Cu are immiscible. Replacing Cu by a materialwhich is immiscible with both Co and Pt should lead to an
even better stability against annealing. More interestingly,
the maximal K
effobtained for the (Co/Cu/Pt) multilayers is
17(62)/C2106erg. cm/C03while the standard (Co/Pt) ones
present a Keffof at most 8.6( 60.8)/C2106erg. cm/C03. The
PMA obtained for the optimized (Co/Cu/Pt) multilayer ishigher than that of the optimized (Co/Pt) stack for all anneal-
ing temperatures. Even for T
A¼350/C14C, Keffis still above
107erg.cm/C03when Cu insertions are used.
In order to reach TMR ratios around 100%, (Co/Pt) mul-
tilayers are not suitable candidates since their (111) fcc struc-
ture does not match the (001) bcc structure required forcrystalline MgO barriers. In order to overcome this issue, a
CoFeB layer can be inserted between the (Co/Pt) multilayer
and the MgO barrier.
11,12The crystallization upon annealingof initially amorphous CoFeB is indeed an efficient way to
obtain MTJ with high TMR ratio. Although interfacial PMAarises from the magnetic metal/oxide interface,
2,13such a
CoFeB insertion usually decreases the PMA of the stack due
to the increase of the demagnetizing energy. This can resultin an anisotropy reorientation in the thin film plane when the
PMA arising from the (Co/Pt) multilayer no longer over-
comes the demagnetizing energy. Perpendicular magnetiza-tion can be restored by either increasing the number of
repeats in the (Co/Pt) multilayer or enhancing its effective
anisotropy. The second solution is preferable since therequired critical current for STT switching is proportional to
the magnetic thickness.
1Designing a multilayer with a maxi-
mal PMA is thus the key factor to obtain highly scalableMTJs with low power consumption.
In conclusion, we demonstrated that the insertion of ultra-
thin Cu layers acting as a diffusion barrier at the Co/Pt interfa-ces in (Co/Pt) multilayers can double the anisotropy energy by
limiting Co-Pt interdiffusion at these interfaces. This effect
could be further enhanced by replacing the Cu layer by a ma-terial which is immiscible with both Pt and Co. Such stacks
are of great interest for MTJ electrodes which require with-
standing annealing temperatures above 250
/C14Cs t i l lk e e p i n g
maximal perpendicular anisotropy for downsize scalability.
This work was partly supported by the European Com-
mission through the Adv ERC HYMAGINE grant and by the
ANR-10-NANO PATHOS project.
FIG. 5. Effective anisotropy of (Co/Pt) (a) and (Co/Cu/Pt) (b) multilayers as
a function the Co thickness t Coand annealing temperature T A.142410-3 Bandiera et al. Appl. Phys. Lett. 100, 142410 (2012)
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On: Fri, 28 Nov 2014 06:16:10 |
1.4950971.pdf | Magnetic anisotropy, damping, and interfacial spin transport in Pt/LSMO bilayers
H. K. Lee, , I. Barsukov , A. G. Swartz , B. Kim , L. Yang , H. Y. Hwang , and I. N. Krivorotov
Citation: AIP Advances 6, 055212 (2016); doi: 10.1063/1.4950971
View online: http://dx.doi.org/10.1063/1.4950971
View Table of Contents: http://aip.scitation.org/toc/adv/6/5
Published by the American Institute of Physics
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Magnetic anisotropy, damping, and interfacial spin
transport in Pt/LSMO bilayers
H. K. Lee,1,aI. Barsukov,1A. G. Swartz,2B. Kim,2L. Yang,1H. Y. Hwang,2,3
and I. N. Krivorotov1
1Physics and Astronomy, University of California, Irvine, California 92697, USA
2Geballe Laboratory for Advanced Materials, Department of Applied Physics,
Stanford University, Stanford, California 94305, USA
3Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator
Laboratory, Menlo Park, California 94025, USA
(Received 11 April 2016; accepted 9 May 2016; published online 16 May 2016)
We report ferromagnetic resonance measurements of magnetic anisotropy and damp-
ing in epitaxial La 0.7Sr0.3MnO 3(LSMO) and Pt capped LSMO thin films on SrTiO 3
(001) substrates. The measurements reveal large negative perpendicular magnetic
anisotropy and a weaker uniaxial in-plane anisotropy that are una ffected by the
Pt cap. The Gilbert damping of the bare LSMO films is found to be low α
=1.9(1)×10−3, and two-magnon scattering is determined to be significant and
strongly anisotropic. The Pt cap increases the damping by 50% due to spin pumping,
which is also directly detected via inverse spin Hall e ffect in Pt. Our work demon-
strates e fficient spin transport across the Pt /LSMO interface. C2016 Author(s). All
article content, except where otherwise noted, is licensed under a Creative
Commons Attribution (CC BY) license (http: //creativecommons.org /licenses /by /4.0 /).
[http: //dx.doi.org /10.1063 /1.4950971]
Spin transport across an interface between nonmagnetic metal (NM) and ferromagnet (FM) by
spin Hall e ffect (SHE)1and spin pumping2–4is central to manipulation of magnetization dynamics
driven by pure spin currents. To date, significant focus has been set on NM /FM heterostructures
comprising 3d materials5–8with a recent extension to yttrium iron garnet (YIG).9–11Further identi-
fication of new material platforms for the e fficient generation, transmission, and conversion of spin
currents is of great importance for enriching this emerging field.
In this context, perovskite manganite La 0.7Sr0.3MnO 3(LSMO) is an attractive FM for dynami-
cally excited spin currents. LSMO is half-metallic with Curie temperature ( TC) above room temper-
ature, low saturation magnetization, and colossal magnetoresistance.12,13Its half-metallic nature
is expected to result in low magnetic damping.14,15Furthermore, this oxide system can be grown
epitaxially with atomically sharp interfaces16,17such that it can potentially o ffer tunable platform for
interfacial engineering.18In this Letter, we present detailed measurements of magnetic anisotropy
and damping in epitaxial LSMO films grown on SrTiO 3(001) (STO) substrates and investigate
interfacial spin transport in Pt /LSMO bilayers. We observe low magnetic damping and e fficient
interfacial spin transport in this system, which makes Pt /LSMO a promising candidate for spintronic
devices based on pure spin currents.
LSMO thin films were grown on TiO 2-terminated STO(001) substrates by pulsed laser depo-
sition (PLD) as described in Ref. 12. Films grown under these conditions exhibit enhanced metal-
licity in the thin limit ( ≥7 unit cells) with high Curie temperature TC≈360 K.12During growth
the LSMO film thickness was monitored by in situ reflection high-energy electron di ffraction
(RHEED). Fig. 1(a) shows X-ray di ffraction (XRD) structural characterization of the LSMO(25 nm)
thin film, which were grown under the same conditions as other films with di fferent thickness
reported in this Letter. The θ-2θscan around the STO (002) peak shows finite thickness fringe
aAuthor to whom correspondence should be addressed. Electronic mail: hankl@uci.edu
2158-3226/2016/6(5)/055212/7 6, 055212-1 ©Author(s) 2016.
055212-2 Lee et al. AIP Advances 6, 055212 (2016)
FIG. 1. X-ray di ffraction (XRD) of epitaxial LSMO(25 nm) on STO(001) substrate. (a) θ-2θscan near the (002) peak.
(b) Reciprocal space map (RSM) near the (103) peak.
patterns in accordance with a uniform, highly crystalline, epitaxial LSMO film. The reciprocal
space map (RSM) of the (103) peak in Fig. 1(b) confirms that the LSMO thin films have been grown
along the (001) orientation epitaxially and fully strained to the substrate. For Pt /LSMO bilayer
films, Pt layer was deposited ex situ using an e-beam evaporator.
We employ coplanar waveguide (CPW) broadband ferromagnetic resonance (FMR)19with
magnetic field modulation20to measure magnetic properties of LSMO films and Pt /LSMO bilayers
at room temperature. A typical FMR spectrum shown in Fig. 2(a) is well fit by a single absorption
profile described by the field-derivative of a sum of symmetric and antisymmetric Lorentzians.20
Previous studies have shown that LSMO thin films typically exhibit a strong satellite absorption
peak.21This mode has negligible amplitude in our samples and we focus our discussion on the
dominant FMR mode.
First, we study the magnetic anisotropy of uncapped LSMO(30 nm) thin films. Fig. 2(b) shows
the resonance field as a function of in-plane magnetic field angle φHwith respect to the [100] axis.
The data reveal a pronounced uniaxial magnetic anisotropy (UMA) with its easy axis parallel to the
[010] crystallographic axis. Frequency-dependent FMR measurements shown in Fig. 2(c) confirm
the uniaxial character of the in-plane magnetic anisotropy. Based on these observations, we model
the free energy density22of magnetization by:
FIG. 2. LSMO(30 nm) (a) A typical field-modulated FMR spectrum. (b) FMR resonance field as a function of in-plane angle
φHmeasured at 4 GHz. (c) Frequency-dependent FMR for easy axis (squares) and hard axis (circles). (d) AFM topography of
the LSMO surface shows terraces with step-edge orientation of 125◦with respect to [100]. Data are taken at room temperature
and all error bars are smaller than the symbol size.055212-3 Lee et al. AIP Advances 6, 055212 (2016)
F=−⃗M·⃗H+1
2MH 1cos2θ
−1
16MH mc(7+cos 4φ)sin4θ
−1
2MH unicos2(φ−φuni)sin2θ, (1)
whereθandφare the polar and azimuthal angles of the magnetization ⃗Mmeasured from [001] and
[100], respectively, and ⃗His the external magnetic field. The first term in Eq. (1) is the Zeeman en-
ergy. The second term is the e ffective out-of-plane magnetic anisotropy with H1
=4πM−2K/M−Hmc, where Kis the perpendicular magnetic anisotropy (PMA). The third term
describes the four-fold magnetocrystalline anisotropy (MCA) with e ffective field Hmc=2Kmc/M.
The last term is the in-plane UMA with anisotropy field Huniand its easy axis at φuni.
We use the Smit and Beljers formalism22,23to fit the FMR data:
(2πf
γ)2
=1
M2sin2θ∂2F
∂θ2∂2F
∂φ2−(∂2F
∂θ∂φ)2, (2)
where fis the resonance frequency, γ=gµB/~is the spectroscopic splitting factor and µBis the
Bohr magneton. Eq. (2) is evaluated at the equilibrium angles θeqandφeqobtained from minimiza-
tion of the free energy density in Eq. (1).
We employ Eq. (2) to simultaneously fit the frequency- and angle-dependent FMR data in
Fig. 2(b) 2(c) with g,H1,Hmc,Huniandφunias fitting parameters. The small di fferences be-
tween the experimental data and the fit in Fig. 2(b) cannot be reduced even by introducing the
second-order MCA term (not shown here). The best fit returns g=1.975 and H1=6380 Oe,
which are similar to the values reported in Refs. 24–26, respectively. MCA field is found to be
negligibly small ( |Hmc|<1 Oe) at room temperature despite the epitaxial nature of our LSMO
films. The in-plane magnetic anisotropy is dominated by the UMA term with Huni=42 Oe and
φuni=90◦given by the best fit. With room-temperature value of M≈265 emu/cm3,12our epitaxial
LSMO films on STO(001) exhibit negative PMA ( K≈−4.0×105erg/cm3) comparable to previous
reports.25,26
The UMA was previously observed in LSMO films grown on STO(001) and its easy axis was
found to be parallel to the atomic terrace edges of the miscut substrate.27,28In Fig. 2(d), we show
atomic force microscope (AFM) topography of our film studied by FMR. It shows step-and-terrace
features with 0.39 nm step height consistent with single LSMO unit cell,12and approximately
250 nm terrace width stemming from a slight miscut of the STO substrate. The step-edges of the
terraces are oriented at 125◦with respect to [100]. This step-edge orientation is not correlated with
either the symmetry axes of the crystal or the measured uniaxial magnetic anisotropy. While we
cannot unambiguously determine the origin of the observed UMA in our films, we find that it is not
related to the substrate miscut.
We analyze the measured FMR linewidth to quantify magnetic damping of our LSMO films.
The linewidth is found to be strongly anisotropic in the film plane with a four-fold and a two-fold
components as shown in Fig. 3(a). Such anisotropic linewidth has been observed in other epitaxial
film systems and explained in terms of two-magnon scattering that follows the in-plane symmetry of
defects in the film.29–32In particular, the four-fold contribution in cubic (001)-films stems from crys-
talline defects and typically presents maxima along <100>axes.31The two-fold contribution arises
from defects with uniaxial, stripe-like symmetry and presents maxima at directions perpendicular to
the uniaxial symmetry axis.32Based on Refs. 31 and 32, we can formulate the following ansatz for
the FMR linewidth ∆H(half width at half maximum):
∆H=∆HLF+∆Hinh+2πfα
γΨ+
jΓi j
2m
γΨ(3)
The first term describes the low-frequency contribution that stems from inhomogeneous micro-
wave field of the CPW. It has the form ∆HLF∝f−ρwithρ∈R+.33,34The second term represents
the line broadening due to inhomogeneity of the sample and has two components: i) a constant055212-4 Lee et al. AIP Advances 6, 055212 (2016)
FIG. 3. (a) FMR linewidth ( ∆H) as a function of in-plane magnetic field angle φHfor LSMO (squares) and Pt /LSMO
(circles) films at 4 GHz. (b) Frequency-dependent FMR linewidth for the LSMO film at three values of φH. The lines show
the best fit.
term and ii) a mosaicity term of the form ∝∂Hr/∂φ H, where Hris the resonance field.31,35The
third term describes Gilbert-type damping which is proportional to the Gilbert constant α. It in-
cludes a correction factor Ψ=cos(φ−φH)accounting for the field dragging e ffect.35The last term
reflects the two-magnon scattering with Γi j
2m=Γj
iξj
i(φ)ζ(f), where iandjare indices labeling the
symmetry of the two-magnon scattering channel and the axis of the maximum scattering rate for
this channel, respectively. The corresponding scattering rates are Γj
i. As described in Refs. 29–32,
ξj
2(φ)=cos4(φ−φmax
2,j)for the two-fold symmetry channel and ξj
4(φ)=cos2(2(φ−φmax
4,j))for the
four-fold symmetry channel, where φmax
i,jis the angle of the maximum scattering rate. The frequency
dependence ζ(f)of the two-magnon scattering is described in Ref. 29. Due to the distinctive
angular- and frequency-dependence of each term in Eq. (3), we can unambiguously fit the data in
Fig. 3 and extract all damping parameters.
From the fit shown in Fig. 3(a), the rates of two-magnon scattering with four-fold and two-fold
contributions are Γ⟨100⟩
4=2.4(3)×108Hz,Γ⟨110⟩
4=0.9(3)×108Hz, and Γ[010]
2=2.5(4)×108Hz,
respectively. The four-fold two-magnon scattering shows maxima along ⟨100⟩, as expected for the
(001) film.31The two-fold term Γ[010]
2presents maximum at [010]. This direction does not corre-
spond to either the hard axis of the UMA in contrast to the expected behavior32or to the terrace
orientation observed in AFM topography. In fact, the stripe-like terraces of the LSMO film generate
a weak additional two-fold two-magnon scattering channel with Γ⊥step
2=0.4(4)×108Hz.
The best fit gives the Gilbert constant αLSMO =1.9(1)×10−3. This value is low among reported
values of LSMO films on STO(001)21,36and is comparable to the lowest values reported for metallic
ferromagnetic films: α=2.1×10−3in epitaxial Fe-V alloy,372.3×10−3in epitaxial Fe 1−xSix,38055212-5 Lee et al. AIP Advances 6, 055212 (2016)
and 1.0×10−3in Co 2FeAl.39The inhomogeneous line broadening is found to be small for our
LSMO films ∆Hinh=1.3 Oe, with a negligible mosaicity contribution ≤0.7 Oe.
Next, we study the e ffect of adding a Pt capping layer to LSMO films. The best fit to the reso-
nance field data of Pt(5 nm) /LSMO(30 nm) film returns g=1.975, H1=6410 Oe, and |Hmc|<3
Oe. These values are very similar to those of the bare LSMO film. The UMA field Huni=36 Oe
decreases by 14% while retaining its easy axis along [010]. The FMR linewidth analysis reveals
that the two-fold and four-fold two-magnon scattering rates significantly increase compared to the
bare LSMO film and retain their symmetry: Γ[010]
2=7.3(2)×108Hz and Γ⟨100⟩
4=5.7(1)×108Hz.40
These findings suggest a modification of the LSMO surface due to Pt deposition, which impacts the
two-magnon scattering.
The FMR linewidth of the Pt /LSMO film versus frequency exhibits multiple peaks, as illus-
trated in Fig. 4(a). We note that these peaks are absent for the bare LSMO film in Fig. 3(b). Near
the frequency values marked as A and B in Fig. 4(a), the FMR absorption profile is significantly
distorted as shown in Fig. 4(b), 4(c). Previously, a similar e ffect was reported for permalloy (Py)
films with a periodic array of stripe-like defects. For Py, the peaks in the linewidth were found to
disappear when the film was magnetized parallel to the stripe-like defects.41,42The absence of the
peaks in our linewidth data for magnetization along the [100] axis in Fig. 4(a) suggests that the
Pt/LSMO bilayer films develop stripe-like magnetic defects oriented along this axis.
Another important e ffect of the Pt layer is the increase of the Gilbert damping constant due to
spin pumping – a process in which spin momentum is dynamically injected from the LSMO into
the adjacent Pt layer.3,4We fit the frequency- and angle-dependent linewidth data for the Pt /LSMO
bilayer to quantify the Gilbert constant. In this fitting procedure, we omit the linewidth data in the
frequency intervals that exhibit peaks (such as frequencies marked as A and B in Fig. 4. We estimate
the Gilbert constant to be αLSMO/Pt≈2.9(5)×10−3, which is∼50% higher than the value of the
bare LSMO film but still low compared to a typical Py film system.
The e ffective interfacial spin mixing conductance g↑↓
effcan be determined2,3from:
g↑↓
eff=4πM t LSMO
gµB(αPt/LSMO−αLSMO ) (4)
FIG. 4. Pt(5 nm) /LSMO(30 nm) bilayer (a) Frequency-dependent FMR linewidth for three values of φH. Multiple peaks
seen in the FMR linewidth as a function of frequency are due to distortions of the FMR absorption profile evident in (b) and
(c): color plots of the measured FMR signal versus frequency and magnetic field near frequencies marked A and B in (a).055212-6 Lee et al. AIP Advances 6, 055212 (2016)
FIG. 5. Field-modulated ISHE signal (red) and the corresponding FMR signal (black) of Pt(9 nm) /LSMO(20 nm) film
measured at 12 GHz.
For the 30 nm thick film, tLSMO =30×10−7cm with M≈265 emu/cm3, we estimate g↑↓
eff
≈0.55×1015cm−2. This number is comparable to the mixing conductance of Pt /Py films (2 .1
×1015cm−2)43,44that reflects significant spin transport across the Pt /LSMO interface despite of the
ex situ deposition of Pt.
For direct confirmation of the spin pumping process, we measure direct voltage induced in the
Pt film via the inverse spin Hall e ffect (ISHE).45We measure the ISHE voltage in Pt /LSMO bilayer
in the direction perpendicular to the bias magnetic field at the drive frequency of 12 GHz. As shown
in Fig. 5, the ISHE voltage lineshape closely tracks that of the absorptive FMR signal and changes
sign upon reversal of the magnetic field polarity, as expected for an ISHE signal.
In conclusion, we measured room-temperature in-plane magnetic anisotropy and damping in
epitaxial LSMO films and Pt /LSMO bilayers grown on STO(001) substrates. We find significant
uniaxial magnetic anisotropy, weak magnetocrystalline anisotropy, and strong negative perpendic-
ular magnetic anisotropy that remain una ffected by the Pt cap. Both LSMO and Pt /LSMO systems
presented significant anisotropic magnetic damping with four-fold and two-fold symmetry compo-
nents which we attribute to two-magnon scattering. The relatively high spin-mixing conductance
combined with very low Gilbert damping (comparable with the best reported values of other
metallic ferromagnets) make Pt /LSMO an attractive system for spintronic applications such as
spin Hall memories7and oscillators.10,46–49Lastly, these results indicate that LSMO is a promising
perovskite building block for all-oxide multifuncitonal high-frequency spintronics devices.
Research primarily supported as part of the SHINES, an Energy Frontier Research Center
funded by the U.S. Department of Energy (DOE), O ffice of Science, Basic Energy Sciences (BES),
under Award # SC0012670 (ferromagnetic resonance studies), by FAME, one of six centers of
STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA
(multilayer growth and structural characterization), and by the Nanoelectronics Research Corpo-
ration (NERC), a wholly owned subsidiary of the Semiconductor Research Corporation (SRC),
through the Center for Nanoferroic Devices (CNFD), an SRC-NRI Nanoelectronics Research Initia-
tive Center under Task ID 2398.003 (electrical transport studies).
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1.2190450.pdf | Micromagnetic investigation of the dynamics of magnetization switching induced by a
spin polarized current
Kyung-Jin Lee and Bernard Dieny
Citation: Applied Physics Letters 88, 132506 (2006); doi: 10.1063/1.2190450
View online: http://dx.doi.org/10.1063/1.2190450
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/88/13?ver=pdfcov
Published by the AIP Publishing
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130.88.90.110 On: Mon, 22 Dec 2014 18:55:09Micromagnetic investigation of the dynamics of magnetization switching
induced by a spin polarized current
Kyung-Jin Leea/H20850
Department of Materials Science and Engineering, Korea University, Seoul 136-713, Korea
Bernard Dieny
SPINTEC-URA CEA/CNRS, 38054 Grenoble, France
/H20849Received 26 November 2005; accepted 9 February 2006; published online 27 March 2006 /H20850
Using micromagnetic modeling, we tested a prediction of single-domain spin-torque theory which
switching current density depends only weakly on magnetic cell size. The switching time andcurrent density are strongly affected by the cell size for low spin polarization. Larger samples witha small length-to-width ratio and small spin polarization can exhibit a nonmonotonous dependenceof switching time on current. Excitation of incoherent spin waves caused by the circular Oerstedfield due to the current is responsible for this nonmonotonous dependence. However, the magneticdynamics recovers a single-domain-like behavior when the spin polarization is high and/or the cellsize is small. © 2006 American Institute of Physics ./H20851DOI: 10.1063/1.2190450 /H20852
Current induced magnetization switching /H20849CIMS /H20850has
been observed in various spin-valve structures.
1,2It is as-
cribed to the transfer of spin-angular momentum, i.e., spin-transfer torque, from spin polarized incoming electrons to alocal magnetization.
3CIMS provides a scalable write scheme
in magnetic random access memory /H20849MRAM /H20850below 150 nm
since the switching current is determined by a critical currentdensity. CIMS also solves the issue of write selectivity of thefield induced magnetization switching using two orthogonalcurrent lines. Indeed, in the present write scheme of MRAM,undesired switching of a half-selected cell may happen,whereas in CIMS, the current only flows through the ad-dressed cell so that the risk of write error is strongly reduced.
The single-domain spin-torque theory
3,4predicts another
important merit of CIMS which is a very weak dependenceof the critical current density /H20849J
c/H20850for the onset of magnetic
excitations on a possible wafer-level distribution of cell size.
It is because Jcis proportional to /H208492/H9266Ms+Hc/H20850, where Msis
the saturation magnetization and Hcis the coercivity of the
free layer. Hcis sensitive to the cell size but is much smaller
than 2/H9266Ms. A very weak cell-size dependence of Jcis crucial
for a mass production of such devices. However, micromag-netic simulations have revealed that the magnetic dynamicsinduced by the spin-transfer torque in a nanopillar could behighly nonlinear.
5–7The nonlinearity is caused by nonuni-
form magnetic fields, resulting in spatially distributed pre-cession frequency.
6Therefore, the prediction from the single-
domain spin-torque theory should be rigorously tested in theframework of micromagnetics.
In this work, we study the influence of magnetic cell size
on switching current density and switching time using micro-magnetic simulations. As a good approximation, the effect ofthe spin-transfer torque can be captured by an additionalterm in the conventional Landau-Lifshitz-Gilbert /H20849LLG /H20850
equation,dM
dt=−/H9253M/H11003Heff+/H9251
MSM/H11003dM
dt+/H9253/H6036
2eP
MS2tJM
/H11003/H20849M/H11003p/H20850, /H208491/H20850
where /H9253is the gyromagnetic ratio, Mis the magnetization
vector of the free layer, pis the unit vector along the direc-
tion of spin polarization /H20849"x/H20850,Msis the saturation magneti-
zation /H20849=800 emu/cm3/H20850,/H9251is the damping constant at zero
current /H20851=0.025 /H20849Ref. 8 /H20850/H20852,Pis the spin polarization factor, J
is the current density, and Heffis the effective magnetic field
including the anisotropy /H20849Hk=10 Oe /H20850, the exchange, the
magnetostatic, the thermal fluctuations, and the current in-
duced magnetic fields. The free layer has an elliptical shape,its thickness is 3 nm, the exchange stiffness constant is0.827/H1100310
−6erg/cm at room temperature /H20849RT /H20850/H208491.0
/H1100310−6erg/cm at the zero temperature; we adopted a renor-
malized exchange stiffness to take into account the effect ofnonzero temperature9/H20850, and the unit cell size is 4 nm. We
assumed no external field, no dipolar field, and a pulse widthof 2 ns with a current rise/fall time of 0.1 ns for the pulsedcurrent injection. All switching events have been calculatedat RT. Averaged switching time and standard deviation werestatistically analyzed from an ensemble consisting of 100switching events.
The average /H20849t
sw/H20850and the standard deviation of switch-
ing time as a function of current /H20849I/H20850were calculated for vari-
ous spin polarization factors, P/H20849Fig. 1, sample size=120
/H1100356 nm2/H20850. Without considering HOe, a monotonous decay of
tswwith current is observed. It shows an inverse proportion-
ality to /H20849I-Ic/H20850as predicted by the macrospin model4where Ic
is the theoretical critical current for the onset of magnetic
excitations. However, if HOeis taken into account, the depen-
dence of tswon current changes dramatically. It can even
become nonmonotonous /H20851for instance, P=0.2 in Fig. 1 /H20849a/H20850/H20852
and exhibits a kink as experimentally observed by Emley et
al.2The standard deviation also increases for currents larger
than the critical value for the kink /H20851Fig. 1 /H20849b/H20850/H20852. In this tested
sample, the switching is delayed over the full investigatedrange of current when taking into account H
Oe. Figure 2 /H20849a/H20850
shows the time variations of the normalized magnetizationa/H20850Electronic mail: kj /H6018lee@korea.ac.krAPPLIED PHYSICS LETTERS 88, 132506 /H208492006 /H20850
0003-6951/2006/88 /H2084913/H20850/132506/3/$23.00 © 2006 American Institute of Physics 88, 132506-1
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
130.88.90.110 On: Mon, 22 Dec 2014 18:55:09/H20849/H20855Mx/H20856/Ms/H20850for the two different switching events at RT
/H20849P=0.2, I=3/H11003Ic, and sample size=120 /H1100356 nm2/H20850. The sec-
ond switching event /H20849switching with vortex /H20850shows longer
switching time than the first one. The increase in tswis
caused by a vortex formation6in the time window from
1.0 to 1.6 ns of the second event. Figures 2 /H20849b/H20850and 2 /H20849c/H20850show
the spatial distributions of time averaged magnetic charge/H20849=/H20855/H20841/H11612M/H20841M
s/H20856/H20850for each switching. The higher value of
/H20855/H20841/H11612M/H20841/Ms/H20856indicates larger spatial inhomogeneities of mag-
netization. Because of the thermal fluctuations and the inco-
herent spin waves, the spatial distributions of /H20855/H20841/H11612M/H20841/Ms/H20856are
complex in both cases. An apparent difference in /H20855/H20841/H11612M/H20841/Ms/H20856
is that only the second event presents a darker region at
bottom left edge indicated by a white arrow /H20851Fig. 2 /H20849c/H20850/H20852. The
darker region corresponds to the vortex formation. A vortexin a magnetic cell with a lateral size of about 100 nm and athickness of 3 nm is rather difficult to form in field inducedswitching.
10However, it can be easily formed in CIMS be-
cause of HOein interplay with the spin torque when the cur-
rent is sufficiently high. When a vortex is formed, the spintorque stabilizes a part of the magnetization, but excites an-other part. As a result, the vortex core does not stay at a placeand wanders around the cell, which delays the switching.Note that the kink disappears when Pis sufficiently large/H20851P=0.7, Fig. 1 /H20849a/H20850/H20852. It indicates that the kink is caused by the
circular Oersted field which results in the excitation of inco-herent spin waves including the dynamic vortex. However,when the spin-transfer torque is much larger than that due toH
Oe, the incoherence is suppressed and the magnetic dynam-
ics recovers a single-domain-like behavior.
The maximum amplitude of the circular magnetic field
within the magnetic cell is proportional to the cell size /H20849L,
along in-plane long axis and W, along in-plane short axis /H20850.
Therefore, the cell size could affect the switching statistics.We calculated t
swfor various cell areas /H20851=/H9266/H20849LW /H20850/4/H20852and as-
pect ratios /H20849AR= L/W/H20850. For large samples /H20849L/H11022100 nm /H20850with
low spin polarization /H20849P=0.2 /H20850, we observed an anomalous
dependence of tswon current /H20851Fig. 3 /H20849a/H20850/H20852. In the high current
regime, there is a significant increase in tswas AR decreases.
This occurs because a lower AR yields an easier vortex for-mation. In the extreme cases where I/H110223I
cfor AR=1.33, we
could not determine tswsince no switching was observed
even with 100 ns current injection in some switching events.Note that there is a deep indicated by an arrow forAR=1.33. As already reported,
7the incoherent spin waves
sometimes help a faster switching. However, the incoherencegenerally delays the switching for samples having a practicalAR which is not close to unity. For a high spin polarization/H20849P=0.7 /H20850, we observed almost identical variations of t
sw
whatever the value of the aspect ratio /H20851Fig. 3 /H20849b/H20850/H20852.
When Lis smaller than 100 nm /H20849AR=2.0 /H20850, the anoma-
lous switching statistics is much less pronounced even for
low spin polarization /H20851Fig. 3 /H20849c/H20850/H20852. The proportionality of HOe
toLis responsible for the reduced incoherence. Furthermore,
a higher energy for spin waves excitation is required in asmaller cell because the energy is proportional to 1/ L
2in the
assumption of one-dimensional /H208491D /H20850infinite potential well.
The switching for a given current becomes faster as the cellarea decreases because the thermal fluctuations become cru-
cial in smaller cells /H20851Figs. 3 /H20849c/H20850and 3 /H20849d/H20850/H20852.
Furthermore, we studied the probability of switching
/H20849P
sw/H20850as a function of pulsed current for various aspect ratios
and cell areas. For P=0.2, we observed a difference in the
distributions of Pswwith varying AR. More importantly, the
switching probability never reached 100% at high currentsdue to vortex formation /H20851Fig. 4 /H20849a/H20850/H20852. For P=0.7, however, the
distributions of P
sware almost identical and the magnetiza-
FIG. 1. Average and standard deviation of switching time as a function of
current for various spin polarization factors, P;/H20849a/H20850average and /H20849b/H20850standard
deviation /H20849cell size=120 /H1100356 nm2/H20850.Icis the theoretical critical current for
onset of magnetic excitations.
FIG. 2. Delay in switching due to vortex formation. /H20849a/H20850Time-dependent
variations of normalized magnetization /H20849/H20855Mx/H20856/Ms/H20850for two different
switching events at room temperature /H20849P=0.2, I=3/H11003Ic, and sample
size=120 /H1100356 nm2/H20850,/H20849b/H20850spatial distributions of time averaged magnetic
charges /H20849=/H20855/H20841/H11612M/H20841/Ms/H20856/H20850for a switching event without vortex /H20851solid square of
Fig. 2 /H20849a/H20850/H20852, and /H20849c/H20850spatial distributions of /H20855/H20841/H11612M/H20841/Ms/H20856for a switching event
with vortex formation /H20851open circle of Fig. 2 /H20849a/H20850/H20852.
FIG. 3. Average of switching time as a function of current. Constant cell
area: /H20849a/H20850P=0.2 and /H20849b/H20850P=0.7. Constant aspect ratio: /H20849c/H20850P=0.2 and /H20849b/H20850
P=0.7.132506-2 K.-J. Lee and B. Dieny Appl. Phys. Lett. 88, 132506 /H208492006 /H20850
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
130.88.90.110 On: Mon, 22 Dec 2014 18:55:09tion completely switches for all switching events at high
enough currents /H20851Fig. 4 /H20849b/H20850/H20852. When Lis smaller than 100 nm
/H20849AR=2.0 /H20850, almost identical distributions of Pswwere ob-
tained for both low /H20849P=0.2 /H20850and high /H20849P=0.7 /H20850spin polariza-
tions /H20851Figs. 4 /H20849c/H20850and 4 /H20849d/H20850/H20852. An exceptional case is for
P=0.2 and /H20849L/H11003W/H20850=/H2084996/H1100348 nm2/H20850where a few incomplete
switchings due to vortex formation were still observed.
In conclusion, the cell size can significantly affect the
switching statistics for a large cell with low spin polarization.Therefore, an important figure of merit to determine the de-gree of magnetic incoherence in the magnetization dynamicsis/H20849lateral cell size /H20850//H20849spin polarization /H20850/H20849=L/P/H20850. The largerL/P, the more incoherent dynamics. Since in spin valves,
L/Pis larger in parallel magnetic configuration than in the
antiparallel case, a more distributed switching statistics isexpected for switching from parallel to antiparallel configu-ration than in the reciprocal case. The increase in the spinpolarization and/or the reduction in the cell size are essentialnot only for reducing the switching current density but alsofor controlling the switching current and pulse width withinacceptable margins.
This work was supported by the Korea University grant
and the RTB project of CEA/LETI.
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FIG. 4. Probability of switching as a function of current. Constant cell area:
/H20849a/H20850P=0.2 and /H20849b/H20850P=0.7. Constant aspect ratio: /H20849c/H20850P=0.2 and
/H20849b/H20850P=0.7.132506-3 K.-J. Lee and B. Dieny Appl. Phys. Lett. 88, 132506 /H208492006 /H20850
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130.88.90.110 On: Mon, 22 Dec 2014 18:55:09 |
1.5119787.pdf | Appl. Phys. Lett. 116, 042403 (2020); https://doi.org/10.1063/1.5119787 116, 042403
© 2020 Author(s).Spin current propagation through ultra-
thin insulating layers in multilayered
ferromagnetic systems
Cite as: Appl. Phys. Lett. 116, 042403 (2020); https://doi.org/10.1063/1.5119787
Submitted: 12 July 2019 . Accepted: 17 January 2020 . Published Online: 28 January 2020
C. Swindells
, A. T. Hindmarch , A. J. Gallant
, and D. Atkinson
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Absence of evidence of spin transport through amorphous Y 3Fe5O12
Applied Physics Letters 116, 032401 (2020); https://doi.org/10.1063/1.5119911Spin current propagation through ultra-thin
insulating layers in multilayered ferromagnetic
systems
Cite as: Appl. Phys. Lett. 116, 042403 (2020); doi: 10.1063/1.5119787
Submitted: 12 July 2019 .Accepted: 17 January 2020 .
Published Online: 28 January 2020
C.Swindells,1
A. T. Hindmarch,1A. J. Gallant,2
and D. Atkinson1,a)
AFFILIATIONS
1Department of Physics, Durham University, Durham DH1 3LE, United Kingdom
2Department of Engineering, Durham University, Durham DH1 3LE, United Kingdom
a)Electronic mail: del.atkinson@durham.ac.uk
ABSTRACT
Spin current pumping from a ferromagnet through an insulating layer into a heavy metal was studied in a CoFeB/SiO 2/Pt system in relation
to the thickness and interfacial structure of the insulating layer. The propagation of spin current from the ferromagnet into the heavy metal
falls rapidly with sub-nanometer thicknesses of SiO 2and is suppressed beyond a nominal thickness of 2 nm. Structural analysis shows that
SiO 2only forms a complete barrier layer beyond around 2 nm, indicating that the presence of a discontinuous insulating barrier, and not
tunneling or diffusion, explains the main observations of spin-pumping with thin insulating layers.
Published under license by AIP Publishing. https://doi.org/10.1063/1.5119787
The manipulation of spin currents across ferromagnetic (FM)
and non-magnetic (NM) interfaces is key to spintronic applications
and remains an active area of research.1–3Precessing magnetization in
a ferromagnetic layer can transfer spin angular momentum, in theform of a spin current, into an adjacent NM layer,
4a process referred
to as spin pumping. One of the main manifestations of this spin
pumping mechanism is an increase in the precessional damping of a
system,5–7and while details remain to be understood, the basis of this
process is well described for ferromagnetic/metallic systems.5,8
However, the propagation of spin current through an insulating bar-
rier has led to conflicting results in the literature. Initial theoretical pre-
dictions of spin pumping required a transparent interface between theFM and NM layers for a large increase in damping;
9however, early
experimental results by Moriyama et al.10suggested an enhancement
in the damping with an insulating barrier being present. This contrasts
with later works by Kim et al.11and Mosendz et al. ,12who observed
the suppression of spin pumping with the insertion of nano-oxide andMgO layers, respectively. Studies of both Si and oxide semiconduc-tors
13have also shown some suppression of spin pumping and suggest
that the carriers may continue to allow spin diffusion through the bar-
rier. Baker et al.14also observed the suppression of spin pumping but
with dynamic exchange between two FM layers across the insulatingbarrier in CoFe/MgO/Ni trilayers. Most recently, Mihalceanu et al.
15
reported a rapid decrease in the damping due to reduced spinpumping with the addition of an ultra-thin MgO barrier layer between
Fe and Pt, from which it was concluded that spin current can tunnel
through a few monolayers of an insulating oxide barrier. The workwas supported by transmission electron microscopy (TEM) imaging,which is limited to sampling very small areas and provides a projectionof a thin 3D sample volume that may not show pinhole defects, and
any defects present may be difficult to directly image.
16
The discrepancies between the previous studies may be associ-
ated with the details of the multilayered structure. In particular, the
nature of the interface structure in such systems is known to beimportant for spin-pumping,
17,18and in the ultra-thin film regime,
the presence of a continuous intermediate layer needs to be estab-lished when studying such interlayer effects.
19Both spin pumping
and d/C0dhybridization across a FM/NM interface lead to addi-
tional magnetic energy loss and increased precessional damping.
An increase in damping linked to spin pumping across a continu-ous insulating layer implies some form of spin current tunneling;however, even small discontinuities, such as pinholes, within theinsulating layer can allow for d/C0dhybridization between the fer-
romagnetic and heavy metal (HM) layers, leading to an increase in
the damping,
20,21and limited channels for spin current propaga-
tion. A detailed understanding of the role of the structure at theinterface is therefore needed to fully characterize the dynamic mag-netic behavior with an insulating barrier.
Appl. Phys. Lett. 116, 042403 (2020); doi: 10.1063/1.5119787 116, 042403-1
Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplThis study investigates the evolution of spin-pumping from a
thin-film ferromagnet into a heavy metal layer as a function of the
thickness of an oxide spacer layer. The spin transport was determined
by broadband ferromagnetic resonance (FMR) and the sample struc-ture was analyzed using x-ray reflectivity (XRR) in order to understand
the extent of the interfacial regions between the oxide and the FM and
NM layers. The study shows here that spin pumping can be fully sup-
pressed when a complete layer of the insulating material is formed.
The enhancement of damping by spin pumping depends upon
the interface and the NM material. Spin pumping leads to spin accu-mulation within the NM layer that decays over a characteristic length-
scale, the spin diffusion length. The transparency of the interface,
which governs the efficiency of spin pumping, is characterized by the
effective spin-mixing conductance.
22–24The enhancement in damping
also depends upon the thickness of both the FM and NM layers. TheFM thickness dependence of the damping, a
tot, is commonly given by
atot¼a0þc/C22h
4pMstFMg"#
eff; (1)
with a0being the bulk intrinsic Gilbert damping parameter, g"#
effthe
effective spin-mixing conductance, which is valid for a given NM thick-ness and other parameters, and cis the gyromagnetic ratio that can be
expressed in terms of the spectroscopic g-factor using c¼gl
B=/C22h.T h e
largest enhancement in the damping is obtained with a combination of
a small FM thickness and a large NM thickness, i.e., above the spin dif-
fusion length. However, in multilayered systems, it may be beneficialfor controlling the damping of the FM layers by manipulating the flow
of spin current across interfaces. One method to achieve this may be to
use insulating barriers; however, this requires the nature of spin trans-
port associated with an insulating barrier to be understood.
Magnetron sputtering was used to grow a series of samples varying
the SiO
2thickness in a CoFeB ð10 nm Þ=SiO 2ð0/C05n mÞ=Ptð10 nm Þ
structure, along with a reference sample with no Pt. Dynamic and direct
structural measurements on the reference samples can be found in the
supplementary material .
XRR was used to extract interfacial structure information. This
method measures over a large area, of the order of square centimeters,
unlike transmission electron microscopy, providing an averaged view
of both the layers and interfaces within a sample. Figure 1 shows the
examples of both the measured reflectivity data and the best-fitting
simulations obtained using the GenX code.25The scattering length den-
sity (SLD) profiles were extracted from the best-fitting model for a sam-
ple of CoFeB ð10 nm Þ=SiO 2ð2n mÞ=Ptð10 nm Þand CoFeB ð10 nm Þ=
SiO 2ð5n mÞ=Ptð10 nm Þ. The interface width in such multilayered
structures results from a combination of the topographical roughness
of the interface between the layers and some chemical intermixing
between these different layers, and here, the interface width between
the insulating and FM layers largely reflects chemical intermixing
across the interface. A value of the interface width can be estimated
from the slope of the scattering length density (SLD) where it changesfrom 90% to 10% of its value from one layer to the next. For the CoFeB
and SiO
2interface, this analysis gives an interface width of 2.4 nm, and
below this thickness, the SiO 2layer is discontinuous.
The damping was obtained from the measurements of magnetic
field-swept FMR as a function of SiO 2thickness. In this setup, the
sample was placed face down onto an impedance-matched microstri-
pline, driven at fixed excitation frequency, f, by an RF signal generator,with an external biasing magnetic field applied parallel to the transmis-
sion line and hence orthogonal to the RF excitation field. Helmholtz
coils were used to modulate the bias field and the time-varying output
voltage of a diode power detector across the line, proportional to thefield derivative of the transmitted RF power, and hence, microwave
absorption, v
00, by the sample, was measured using a lock-in amplifier.
The inset in Fig. 2(a) shows typical spectra around resonance as a
function of magnetic field for various excitation frequencies, f.
The relationship between the field swept linewidth, DH, and res-
onant frequency allows for the separation of intrinsic and extrinsic
contributions to the damping using
DH¼DH0þ4pa
cf; (2)
where 4 pa=cis the intrinsic linewidth and DH0is the extrinsic line-
w i d t h ,w h i c hi sr e l a t e dt od e f e c t sa n dl e a d st ot w o - m a g n o ns c a t t e r i n g .
An example fit to the linewidth data used to separate these contribu-
t i o n st ot h ed a m p i n gi ss h o w ni n Fig. 2(a) .
The effect of increasing the thickness of a SiO 2spacer layer on
both the intrinsic and extrinsic contributions to the precessionaldamping in CoFeB ð10 nm Þ=SiO
2ðxnmÞ=Ptð10 nm Þmultilayers is
shown in Figs. 2(b) and2(c). As the nominal thickness of the oxide
layer between the ferromagnet and the heavy metal spin-sinkincreases, the intrinsic linewidth decreases. This decrease is at a similar
rate to that observed for an MgO spacer layer.
15The intrinsic damping
decreases toward the value in the case where no spin-sink is present,as indicated in the figure by the orange square data point. No change
FIG. 1. (a) XRR data and best fit for CoFeB ð10 nm Þ=SiO 2ð2n mÞ=Ptð10 nm Þ(top)
and CoFeB ð10 nm Þ=SiO 2ð5n mÞ=Ptð10 nm Þ(bottom). (b) Real part of the scatter-
ing length density (SLD) profile from the best fit to the data for the 2 nm oxide bar-
rier. (c) Same as (b) for the 5 nm barrier.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 116, 042403 (2020); doi: 10.1063/1.5119787 116, 042403-2
Published under license by AIP Publishingin intrinsic damping is observed by varying the SiO 2thickness without
a Pt layer. The continued enhancement of damping with thin insulat-ing barrier thicknesses was previously attributed to the tunneling ofspin current through the insulating spacer layer.
However, an understanding of the interfacial structure is impor-
tant. As shown in Fig. 3 , by superimposing the normalized structural
SLD profile of the CoFeB/SiO
2interface on the same nominal SiO 2
thickness-axis as for the damping, the relationship between the structureof the insulating layer and the measured damping response can be com-pared. At low SiO
2thicknesses (below 2.4 nm), the SiO 2l a y e ri sd i s c o n -
tinuous, enabling some localized direct contact and d/C0dhybridization
between the ferromagnet and the heavy metal (HM), where the spacer
layer is incomplete, and creates direct pathways for the propagation ofspin current from the ferromagnet into the spin-sink. These two mecha-nisms enhance the damping above that of the pure ferromagnet
20but
decrease rapidly as the area of HM in direct contact with the FM is
reduced. However, when the insulating spacer layer continuously covers
the ferromagnet, above 2.4 nm, there is no measured enhancement ofthe intrinsic damping from the heavy metal layer.
The effects of the discontinuous interface are also observed in the
SiO
2thickness dependence of the extrinsic contribution to the damping,
seeFig. 3(b) . An increase in the extrinsic contribution to the linewidthindicates an increase in defects that mediate two-magnon scattering pro-
cesses. As a function of SiO 2thickness, the extrinsic contribution
increases in a single large step with the thinnest oxide layer and then
d e c r e a s e sa st h et h i c k n e s si n c r e a s e sf u r t h e r ,a n dt h i sd e c r e a s ei sc o m p a -
rable with the form of the scattering length density. The extrinsic contri-bution provides evidence further supporting the interpretation of the
nominal thickness dependence as a consequence of the presence of a dis-
continuous insulating layer, as it has been previously shown that the dis-
continuous coverage of a ferromagnet with a heavy metal layer leads to
enhanced extrinsic damping.
21A slight enhancement in extrinsic damp-
ing was also found without a Pt layer, which may be attributed to the
partial oxidation of the FM surface due to a discontinuous interface. The
common dependence of intrinsic and extrinsic damping upon the dis-
continuous SiO 2is further evidenced by the linear correlation between
the extrinsic and intrinsic contributions for samples lacking a full surfacecoverage of the SiO
2layer (i.e., below 2.4 nm), as shown in Fig. 3(c) .
Here, as discussed, regions with a low surface coverage allow for a large
increase in both the extrinsic and intrinsic contributions, both of whichare suppressed with the same functional form as the layer becomes com-
plete. Direct surface measurements are unable to distinguish between
defects such as pinholes, which would lead to this effect and topographi-
cal roughness, due to the lack of element specificity.FIG. 2. (a) The inset shows absorption derivative profiles at four frequencies with fits,
obtained from lock-in amplifier field-swept FMR, for CoFeB ð24 nm Þ=Ptð10 nm Þ=
SiO 2ð5n mÞ. The rest of (a) shows the measured linewidth from field-swept FMR, fitted
to Eq. (2)to extract both intrinsic and extrinsic damping contributions. (b) Decrease in
intrinsic contributions to the FMR linewidth as a function of SiO 2thickness for
CoFeB ð10 nm Þ=SiO 2ðxnmÞ=Ptð10 nm Þ(blue circles) with a reference sample without
platinum (orange square). (c) Decrease in extrinsic contributions as a function of SiO 2
thickness, where the orange square at 0 nm denotes a reference sample with no SiO 2,
and at 5 nm, it denotes a reference sample without Pt.FIG. 3. (a) Same as Fig. 2(b) but with extracted SLD for the 5 nm SiO 2barrier from
Fig. 1(c) superimposed shown in red dashed lines. The horizontal gray bar indi-
cates the damping equivalent to that of the ferromagnetic layer only. (b) Same asFig. 2(c) with SLD for a 5 nm SiO
2barrier superimposed on top given by the red
dashed line. (c) Correlation between the intrinsic and extrinsic contributions for
samples without a full surface coverage of the insulating layer.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 116, 042403 (2020); doi: 10.1063/1.5119787 116, 042403-3
Published under license by AIP PublishingIn conclusion, the link between the structure of the interface and
the spin transport with a SiO 2spacer layer was examined. It was found
that spin-pumping was observed for nominal SiO 2thicknesses up to
around 2 nm, but this correlates with the length-scale correspondingto the interface width of the barrier, such that structurally the insulat-ing layer was discontinuous when spin-pumping was observed and no
enhancement of the damping was measured when the SiO
2layer was
complete ( >2.4 nm). Thus, the experimentally observed spin-
pumping signals with ultra-thin insulators are due to the discontinu-ous insulating layer rather than requiring models involving tunnelingof pure spin-current. The incomplete SiO
2layer also leads to enhanced
extrinsic damping resulting from direct coupling between the FM and
HM layers when the insulating layer is discontinuous. It is also shownthat when the SiO
2layer is continuous, it represents a significant bar-
rier to spin transport, which allows for the suppression of spin currentin multilayered structures.
See the supplementary material for dynamic and direct structural
measurements on CoFeB/SiO
2bilayers.
Funding from EPSRC for the studentship for CR Swindells
1771248, Ref. EP/P510476/1, is acknowledged. Data presented
within this article can be found at https://doi:10.15128/r2cf95jb46b .
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Published under license by AIP Publishing |
1.50109.pdf | Formation and evolution of the cathode sheath on the streamer
arrival
Mirko Černák
Citation: AIP Conference Proceedings 363, 136 (1996); doi: 10.1063/1.50109
View online: https://doi.org/10.1063/1.50109
View Table of Contents: http://aip.scitation.org/toc/apc/363/1
Published by the American Institute of Physics
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3D PIC-MCC simulations of positive streamers in air gaps
Physics of Plasmas 24, 102112 (2017); 10.1063/1.5003666Formation and Evolution
of the Cathode Sheath on the Streamer Arrival
Mirko Cem~ik
lnstitite of Physics Faculty of Mathematics and Physics, Comenius University,
842 15 Bratislawa, Slovakia
Abstract. Dynamics of the cathode region formed by the streamer arrival is clarified on the basis
of a computer simulation model and experimental investigations. Prebreakdown streamers in
positive corona discharges, negative corona pulses, and "breakups" of cathode sheaths during
glow-to-arc transition breakdown processes are discussed in this context.
INTRODUCTION
At near-atmospheric pressures, the sequence of events leading to breakdown
consists of the bridging the gap by primary streamers, and the subsequent
heating of the channel created by the streamers. The arrival of the primary
streamer on the cathode, forming an active cathode region that produces the
electrons and ions by direct impact ionisation within the cathode fall, marks an
important turning point in the streamer-initiated breakdown process. There
seems to be consensus that upon the streamer arrival the main electron
generation must take place very near (~0.1 mm) to the cathode surface, and
that substantial charge transfer and neutralisation occur on a time scale of 10 -9
- 10 -8 s. Nevertheless, the details of the evolution of the primary streamer head
via a streamer-cathode interface to the cathode region remain somewhat
obscure.
The incomplete theoretical understanding of the streamer-cathode interaction
seems to result from the fact that, because of instabilities introduced by
numerical differentiation, the recent computer simulations of streamer-initiated
breakdown processes have only been continued to the point when the streamer
approaches the cathode. Probably the only theoretical models to date which
provide a detailed description of the cathode region development at near-
© 1996 American Institute of Physics
136
atmospheric pressures are the one-dimensional fluid model by Belasri et al. (1)
and the two-dimensional fluid model by Simon and BOtticher (2). The one-
dimensional approximation, however, is not adequate for the streamer-to-arc
transition, where the discharge has a small cross section, and in the model by
Simon and BOtticher an external circuit and discharge current computations are
not included because of the computing time restrictions.
Development of the theoretical models is hampered also by experimental
constraints, particularly from technical difficulties of viewing the small
streamer-cathode interface with nanosecond time resolution . In the most
commonly used plane-plane geometry of electrodes, a difficulty of the electrical
diagnostic is that the streamer arrival generates only a small current hump on the
discharge current growth waveform (4,5).
It is the main purpose of the present work to show how, combining computer
simulations with experimental investigations, some insight can be obtained into
the processes taking place at the streamer arrival at the cathode. In addition, it
will be illustrated that the understanding of the streamer-cathode interaction is
fundamental to the understanding of effects of the cathode surface properties on
the glow-to-arc transition and negative corona (Trichel) current pulses.
STREAMER - CATHODE INTERACTION
The streamer-cathode interaction will be discussed for the discharge in a short
positive point-plane gap, where the well-pronounced current signal induced by
the genuine primary streamer-cathode contact can be measured using a small
central cathode probe. The signal is relatively insensitive to gas composition
and is characterised by a fast rising (the rise time of-0.5 ns at atmospheric
pressure) current peak of amplitude in 0.0 - 0.1A scale (3-5). This is followed in
some 10 - 50 ns by a current hump and, subsequently, by a current portion
associated with the formation of a glow-discharge-type cathode spot (6,7).
1.5-D Fluid Model for a Short Positive Point-to-Plane Gap
The one and half dimensional equilibrium simulation model is based on the
numerical solution of Poisson's equation in conjunction with the continuity
equations for electrons, positive ions, and negative ions. The effects of
ionisation, attachment, electron diffusion, and photoemission and ion secondary
electron emission from the cathode are included.
The cathode probe current Ip was computed as:
137
Ip = J (q (Ji -Je ) + eo.0E/0 t) dS (1)
S
where Je Ji , and E are the electron and ions flux and the field intensity,
respectively, taken at the cathode probe surface S. Anode current I a was
computed according Rama-Shockley theorem. Figs. 1-4, show results computed
for the streamer-cathode interaction in a short positive point-plane gap in N 2 at
a pressure of 26.7 kPa, gap spacing of 10 ram, gap voltage of 4 kV, values of
secondary emission coefficients ~'i = 2-10-3 and ~'ph = 10-2, and the streamer
channel radius expanding from 0.5 mm near the anode to 1.2 mm at the cathode.
The good agreement obtained between the discharge behaviour observed
experimentally and those in our simulation model indicates that 1.5-D models
provides an adequate physical picture of streamer arrival at the cathode and,
taken together with experimental investigations, can serve as a basis for
developing a 2-D model.
The reader is referred to (7) for more details. Here, for brevity, we shall
restrict ourselves to the results in Figs. 1-4 and to discussion of principal
conclusions.
Conclusions and Comparison with Experimental Observations
Since, for pressures above say 10 kPa, the streamer discharge behaviour prior
to the glow-to-arc transition is very similar to that in air (8), the model is
believed to qualitatively describe the streamer-cathode contact also in air at
near-atmospheric pressures, where the vast majority of the experimental studies
have been made. In addition, based on the results by Kennedy (Ref 10, see
Fig.5.6b there) and by Martin et al. (11) which show striking similarities
between the current signal induced in the cathode probe hit by the streamer in a
short point-plane gap and that in a parallel-plane gap, we suppose that the model
provides an insight into the streamer-cathode interaction also for this gap
geometry
The computed time evolution of electron and ion densities near the cathode
surface at the streamer arrival shown in Figs. 1-2 illustrate the transformation of
the streamer front structure to an abnormal cathode fall of roughly 1 kV, which
is consistent with the experimental results by Cavenor and Mayer (12) indicating
that cathode spot created by the streamer arrival operates in the abnormal glow
regime. Also, the time evolution of the cathode sheath thickness indicated by
Figs. 1-3 is in fair agreement with that observed by Bertault et al. (6).
The results reveal that the dominant component of a sharp current spike
induced by the streamer arrival in a cathode probe (see Fig.4) is the
138
9.6
60.0
50.0
40.0
30.0 9.7 9.8 ' ' ' ' ' ' ' ' ' ' ' 'I' '
Electron densi[y [lO19m -3] 9.9
J i0.0
60.0
50.0
%-~
40.0
g
30.0
i ~ i ~ 9J7 i ~ ~ i~ i h ~ ~ 9/9 ~ ~ ~ ~ 20,0 20"09,6 9.8 10.0
Distance [mm]
FIGURE1. Spatio-temporal development of electron density during streamer-cathode contact
(the cathode is situated at the distance 10 mm).
9.6 9.7 9.8 10.0
60.0 ~ ~ F ~ ~ r ¢ ~ ~ , p 60.0
50.0
40.0
E-~
30.0
20.0
9.6 9.9 i i ~1 i
50.0
40.0 qJ
[-~
Ion densiky [1019m a] 30.0
I I I I I I I I i ; I I I I I I I I I 20.0
9.7 9. 9 9 10.0
Distance tram]
FIGURE 2. Spatio-temporal development of ion density during streamer-cathode contact
139
9.6
600
5O 0
&
40.0
30.0 9.7 9.8 9.9
~9
o
/
Ionization rate [lOaVm-as I] /~. 10.0
60.0
50.0
~g
40 0 ¢~
[c
30.0
' ' ' ' ' ' [ J J 18 ' ' ' ' 9.'9 ' i , i 20.0 20"09.6 9.7 9. 10.0
Distance [ram]
FIGURE 3. Spatio-temporal development of the ionisation rate during streamer-cathode
interaction.
Ip, I,,, G 60
40
20 Ip ~~,I~ 1111
0 20 40
t[.,] 0
60 30
E
[MY~m]
20
10
FIGURE 4. Ip, I a, I x, are the total probe, anode and conductive probe currents, respectively; E -
intensity of electric field at the axis of the probe.
140
displacement current and that this current signal is not very sensitive to
cathode emission.
Also, the sudden increase in ionisation activity near the cathode surface seen in
Fig.3 (and observed experimentally as the bright flash of light as the streamer hit
the cathode (6)), is not sensitive to secondary electron emission. The conductive
current due electron emission processes and to positive ion collection by the
cathode becomes the dominant part of the cathode probe current some 10 -20 ns
after the streamer arrival marked by the cathode-probe current spike. This is in
contrast to the commonly held belief (5,13) that the streamer arrival at the
cathode is associated with a sudden burst of electrons leading to the
neutralisation of the positive charge in the streamer head.
It is interesting to refer here to several implications of the model, which were
tested experimentally and can be used as a tool for further studies on the cathode
sheath evolution:
i) An implication of the model is that an increase in secondary photoemission
coefficient has little effect on the current spike induced at the streamer arrival,
but results in a reduction in the time lapse between this spike and the following
current hump due to incoming positive ions (see Fig.6 in Ref.7). This is in
conformity with the results in Ref. 14, Fig. 17, where values of the photoemission
coefficient were increased using CuI-coated cathode surfaces.
ii) The results in Fig.4 indicate that the field at the cathode surface reaches its
maximum several ns before the current hump due to incoming positive ions.
This is in striking conformity with the fact that "streamer-like-instabilities of the
cathode sheath" discussed in the following Section (see Figs. 5 and 6) take place
just at this moment (14), Figs. 15 and 22.
iii) Our computer simulations, and also the results by Morrow (15), Fig.11,
indicate that under certain conditions the discharge current can exhibit damped
oscillations with period of the order of 10 ns reflecting the back and forth
motion of the electric field in the cathode sheath. This phenomenon was
experimentally observed in (5), Fig5a; (14), Figs. 5 and 9, and (16), Fig.3.
Finally, it may be noted that a transition from capacitive to resistive behaviour
of the cathode sheath seen in Fig.4 is closely analogous to that in Ref. 1.
STREAMER FORMATION IN THE CATHODE SHEATH
It is generally accepted that ionisation instabilities in the cathode region
("breakups" of the cathode sheath (17)) play a crucial role in limiting the
duration of the high-pressure space glow discharge used, for example, in gas
lasers excited by transverse discharges (17-19). The ionisation instabilities result
141
in the "hot spots" formation near the cathode, which trigger filaments of strongly
enhanced current densities, leading to the arcing. Up to now no adequate model
of the ionisation instabilities in the cathode sheath of high-pressure glow
discharges has been developed. The most applicable model to the ionisation
instabilities seems to be that of Bityurin et al. (20), which predicts the possibility
of a cathode-directed ionising wave driven cathode-sheath instability with a
repetition period ~ 25 ns in N 2 at atmospheric pressure. In (1) streamers in the
cathode fall have been simulated, which propagate from the cathode surface
towards the positive column. This is, however, in contrast to the experimental
observations (19) indicating that the 'streamers' bridging the cathode sheath
begin from the positive column side.
It is believed that the formation of ionisation instabilities in the cathode sheath
can be at least partially understood in terms of the "positive-streamer-like
instabilities of the cathode sheath" discussed below.
In Fig. 5 the Ip waveform measured in CO 2 at a pressure of 13.33 kPa, gap
spacing of 10 mm, and gap voltage of 3kV, using the conditioned cathode
surface (waveform 1) is compared with that measured using unconditioned (i.e.,
freshly polished) cathode surface under the same experimental conditions.
The interpretation of the waveform 1 in terms of our computer model is clear:
The initial sharp current spike corresponds to the displacement current
generated at the streamer arrival at the cathode, the subsequent current hump is
due to incoming positive ions, and the following current portion corresponds to
a filamentary (abnormal) glow discharge. By comparing the spikes denoted by
X, which are seen on the waveform measured using unconditioned cathode
surface, with the 1, it can be seen that they are remarkably similar in shape to
the Ip waveform generated by the streamer arrival at the cathode (They are,
practically identical to Ip waveforms taken at a reduced gap voltage, see (21)).
For this reason they can be called 'positive-streamer-like'. The same
phenomenon can be observed using point cathode (14, 21,22), where the
ionisation is confined to a thin cathode region. This is why we use the term
'positive-streamer-like instabilities of the cathode sheath' (PSLI) to refer to the
phenomenon. We refer to the papers (14, 21,22) for a more complete analysis
and a possible explanation for the phenomenon.
Despite of its incomplete understanding the PSLI could well have practical
importance. Figs. 5 and 6 show that the appearance of the PSLI due to the use of
unconditioned cathodes (Fig.5) and due to 'ageing' of the cathode surface (Fig6)
resulted in transition of the discharge to spark. This phenomenon can limit
performance characteristics of pulsed corona devices (4) and wire chambers
used as detectors for ionising particles. This is in contrast to most switching
applications, where the glow discharge phase is undesirable because of its low
142
._>
E V"
...... x .... ~ ..... 2i .... xi ....... ~, ....... ~: ....... TX .... i ........
Time ( 50 ns/div)
FIGURE 5. Current signal induced in the cathode probe hit by the streamer measured
using the conditioned (1) and freshly polished (2) cathode surface.
E
"¢ ............................... i ............................... i ...............
C)
-E
k._
0
Time (200 ns/div)
"a ein " FIGURE 6. Spark development in a coaxial wire chamber due to g g. Conditions:
Ar +10% CH4, 100 kPa; anode and cathode diam.: 20 #m and 8 ram; gap voltage of
2500 V.
143
conductivity. Thus, for example, based on Ref. 11, one may speculate that an
artificial ignition of PSLI, accelerating the arc formation, could improve
operation characteristics of switching spark gaps. Also, we hypothesise (22,23)
that, streamers resulting in the "hot spots" formation in high pressure glow
discharges are due to the same phenomena.
POSITIVE-STREAMER MECHANISM FOR NEGATIVE
CORONA CURRENT PULSES
Evidence has been accumulating over the past twenty years that the steep
negative corona current rise is associated with the development of a positive
streamer (14, 16, 22,24-28) and its arrival at the cathode. It is notable, however,
that this mechanism does not seem to be generally accepted by the workers in
the field of corona discharges. Perhaps, the most common objection raised
against the positive-streamer mechanism for negative corona pulses, is that the
time for positive ions to move to the cathode is much longer than the recorded
pulse rise time (29). This objection is based on the unrealistic assumption that
the streamer arrival at the cathode surface is associated with the immediate
neutralisation of the positive ions at the cathode (see Fig.4).
A detailed discussion of the mechanisms for the negative corona pulse rise is
beyond the scope of this paper. It is of interest, however, to note here broad
similarities observed between the negative corona current pulses, current pulses
generated at the arrival of positive streamers on the cathode, and a current signal
corresponding to the positive-streamer-like instabilities of the cathode sheath,
which is indicative of the same physical mechanisms (14, 21).
ACKNOWLEDGEMENTS
I would like to express my particular gratitude to Prof. T. Hosokawa and
Dr. E. Marode for their substantial contributions to this work. The computer
simulation model presented was the PhD thesis ofI. Odrobina.
REFERENCES
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8. Odrobina I. and (~emhk M,"Formation of cathode region of filamentary
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International Symposium on High Pressure Low Temperature Plasma
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9. (~ernak M., Marode E, and Odrobina I.,"Comparison of current waveforms induced
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10. Kennedy J.T.," Study of the avalanche to streamer transition in insulating gases"
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18. B6tticher W., "Modelling of discharge pumped XeCI lasers. Open questions" in
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19. Makarov M., J. Phys. D: Appl. Phys. 28, 1083-93 (1985).
20. Bityurin V.A, Kulikovski A., and Lyubimov GA. Zh. Tekh. Fiz. 59, 50-
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21. (~ern~k M. and Hosokawa T., "Positive-streamer-like instabilities of the cathode
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I1 Cioeco 1991, pp.917-18.
22. (~errmk M., Hosokawa T., and Inoshima M., Appl. Phys. Lett. 57, 339-40 (1990)
23. Hosokawa T. and (~ernb& M, "Acceleration of filamentary glow-to-arc transition in
CO due to cathode sheath instabilities" in Proceedings of the lOth Int. Conf. on
Gas Discharges and their Appl., Swansea 1992, pp.452-55.
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26. (~ernhk M. and Hosokawa T., Phys. Rev. A 43, 1107-9(1991)
27. Cern~k M., Hosokawa T., and Odrobina I., J. Phys. D: AppL Phys. 26, 607-18(1993)
28. Liu J. and Govinda Raju R.R., IEEE Trnas.on Dielectrics andE1. Insul. 1, 520-29(1994)
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145
|
1.1697641.pdf | Spin wave contributions to the high-frequency magnetic response of thin films
obtained with inductive methods
G. Counil, Joo-Von Kim, T. Devolder, C. Chappert, K. Shigeto, and Y. Otani
Citation: Journal of Applied Physics 95, 5646 (2004); doi: 10.1063/1.1697641
View online: http://dx.doi.org/10.1063/1.1697641
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/95/10?ver=pdfcov
Published by the AIP Publishing
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128.248.155.225 On: Tue, 25 Nov 2014 06:16:27Spin wave contributions to the high-frequency magnetic response of thin
films obtained with inductive methods
G. Counil,a)Joo-Von Kim, T. Devolder, and C. Chappert
Institut d’Electronique Fondamentale, UMR CNRS 8622, Universite ´Paris-Sud, 91405 Orsay cedex, France
K. Shigeto and Y. Otani
FRS, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
~Received 1 December 2003; accepted 15 February 2004 !
The high-frequency magnetic response of Permalloy thin films have been measured using
network-analyzer ferromagnetic resonance.We demonstrate that the excitation of spin waves by thecoplanar wave-guide modify the magnetic response appreciably, in particular, by causing afrequency shift and broadening of the resonance peak.An analytic theory is presented to account forthe experimental observations and provides a quantitative tool to accurately determine the Gilbertdamping constant. © 2004 American Institute of Physics. @DOI: 10.1063/1.1697641 #
I. INTRODUCTION
As data transfer rates increase in magnetic recording
~hard disks, magnetic memories !, the high frequency perfor-
mance of related magnetic devices becomes increasingly im-portant. Controlling magnetization dynamics requires the un-derstanding and tuning of relaxation mechanisms, andreliable techniques are needed to observe such processes athigh frequencies.
One of the fundamental issues requiring closer attention
is the nature of magnetic relaxation at subnanosecond times-cales. For physical processes that conserve the norm of themagnetization M, the Landau–Lifshitz–Gilbert equation is
appropriate for describing its time-evolution
1
]M
]t52gM3Heff1a
MSM3]M
]t, ~1!
where the damping term, proportional to a, represents a phe-
nomenological dissipation term in the magnetization motion.Here,H
effis the total effective field seen by the magnetiza-
tionM,g[gmB/\is the gyromagnetic ratio, and MSis the
saturation magnetization. While the damping constant arep-
resents a crude average over the ensemble of microscopicprocesses responsible for energy dissipation, it remains auseful parameter for characterizing reversal times anddomain-wall velocities.
A measure of the damping constant can be obtained in
experiment by measuring the lifetime of linear magnetic ex-citations. In ferromagnetic resonance ~FMR !or Brillouin
light scattering ~BLS!experiments, for example, the lifetime
of the excited mode is extracted from the linewidth of themeasured magnetic susceptibility.
2In conventional FMR, a
magnetic sample is placed in a microwave cavity and is sub-jected to a time-varying electromagnetic field that is uniformover the sample dimensions. In such experiments, the line-widths observed give a measure of the lifetime of the uni-form precession mode that is excited. In BLS, photons in thevisible range are scattered from the surface magnetic film
and the corresponding energy shifts are recored. Typically,these processes involve the scattering of finite wave-vectorspin waves, so the linewidths measured give an indication ofthe lifetime of the surface magnetostatic waves excited.
While these methods have been extremely fruitful for the
study of linear dynamics, they are limited to large samplesizes or large arrays by the respective signal–to–noise char-acteristics. Recently, Silva et al.demonstrated that the mag-
netic response in the time domain of single patterned mag-netic elements
3can be studied using pulsed inductive
microwave magnetometry ~PIMM !.4In addition to vastly im-
proved sensitivity for measuring magnetic fluctuations insmall structures, the technique allows the magnetic responseto be probed over a continuous range of frequencies when
attached to a network analyzer ~NA-FMR !, which provides a
substantial advantage over conventional FMR. Furthermore,the microscopic scale of the experimental setup makes pos-sible the excitation and detection of nonuniform laterallyquantized spin-wave modes in micron-sized structures.
5,6
However, this last point must be taken into careful con-
sideration when studying continuous films with inductivetechniques such as PIMM or NA-FMR. Fields generated bythe coplanar waveguide structure will necessarily be spatiallynonuniform when viewed by a magnetic sample whose di-mensions exceed those of the stripline. The excitation ofnonuniform modes that follow in such systems must be takeninto account in the analysis of the magnetic response. To ourknowledge, little attention has been directed toward this as-pect in such inductive experiments. In this article, we studythe effects of such nonuniform modes on the total magneticresponse in frequency domain ~NA-FMR !. Moreover, such
effects are shown to be important to certain configurations,where a subsequent determination of the damping constantwithout the corrections we suggest may lead to significanterrors.
This article is organized as follows. In Sec. II, we
present the geometry of the setup and the measurement pro-cedure. In Sec. III, we discuss the advantages of the fre-
a!Electronic mails: counil@ief.u-psud.frJOURNAL OF APPLIED PHYSICS VOLUME 95, NUMBER 10 15 MAY 2004
5646 0021-8979/2004/95(10)/5646/7/$22.00 © 2004 American Institute of Physics
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128.248.155.225 On: Tue, 25 Nov 2014 06:16:27quency domain characterization to measure precisely the
evolution of the harmonic response. This is followed by adiscussion of the excitation of nonuniform spin waves inSec. IV. We observe a significant shift of the resonance fre-quency as well as an enhancement of the damping factorwhile varying the angle between the pumping field and themagnetization, which cannot be accounted for by the smallin-plane anisotropy field H
a. Moreover, we observe a mono-
tonic increase of the damping factor versus frequency at lowfrequency, which cannot be attributed to inhomogeneousbroadening due to the dispersion in H
ain our rather high
quality Permalloy thin films. We present a model that takesinto account the excitation of nonuniform spin-wave modesdue to the finite width of the ‘‘antenna’’ that emits and col-lects the magnetization precession modes. We analyze quan-titatively the contribution of this experimental source of line-width broadening, and show how to extract the value of the‘‘intrinsic’’ damping factor. A discussion and some conclud-ing remarks is given in Sec. V.
II. EXPERIMENTAL SETUP AND MEASUREMENT
PROCEDURE
We have studied a series of thin Permalloy films
~Sapphire/Ni 80Fe20dPy/Au10nm, dPy53, 6, 10, 50 nm !
with lateral dimensions 0.1 cm 31 cm. Since surface
anisotropies may exist for such thin films, the demagnetizingfield induced by out-of-plane motion of the magnetization isdecreased by the usual surface anisotropy term
m0Meff5m0MS22KS
MSdPy, ~2!
where m0is the permeability of free space, Ksthe surface
anisotropy, and dPythe film thickness. Here we have ne-
glected the contribution of any volume terms for the perpen-dicular anisotropy. The effective demagnetizing field
m0Meff, determined as the field applied perpendicular to the
sample surface necessary to saturate the sample, was mea-sured by the polar magneto-optic Kerr effect. The saturationmagnetization was found to be
m0MS51.04T, and the sur-
face anisotropy field 2 KS/(MSdPy) was found to scale with
the inverse thickness from 0.03 T for dPy550nm to 0.43 T
fordPy53 nm. A small in-plane uniaxial anisotropy field
m0Ha5131024T was obtained from longitudinal ~MOKE !
measurements.
The inductive measurement is made with an aluminum
coplanar waveguide patterned on a high-resistivity siliconsubstrate ~see Fig. 1 !. The length of the waveguide is L
55mm and the width of the center conductor is w
545
mm. The coplanar waveguide is contacted at both ends
with two microwave probes, connected to high frequencyK-cables. For time domain measurements ~PIMM !, one
probe is connected to a pulse generator that provides short~<200 ps !voltage pulses up to 10 V with 50 ps, and the
other probe is connected to a 50 GHz oscilloscope. For fre-quency domain measurements ~NA-FMR !, the probes are
connected to the two ports of a HP8753D network analyzer.The pulse or harmonic voltage sent through the waveguidecreates a pumping field hperpendicular to the waveguide,which excites the precessional motion of the magnetization.
The motion of the magnetic moment creates a mean fluxvariation,
4,7–12and the corresponding additional induced
voltage is proportional to the time derivative of the compo-nent of the magnetization perpendicular to the waveguide.The induced voltage is measured with the oscilloscope fortime domain measurements, and with the network analyzerfor frequency domain measurements. An electromagnetplaced under the sample support creates an in-plane rotatingstatic field
m0H0up to 0.04 T.
For both time- and frequency-domain measurements, the
transmitted voltage is first recorded with a 0.04 T static fieldparallel to the pumping field. For a pumping field hwith
small amplitude, no spin waves are generated in this configu-ration in principle. We then record the additional magneticresponse with the static field applied in different directions,and deduce the magnetic part of the response by subtractingthe two recorded signals. Such stringent experimentalconditions—strictly parallel pumping and dc fields for thecalibration, and swapping from the frequency domain to thetime domain without moving the sample and without discon-necting the coplanar waveguide—were found crucial tomake very precise comparisons. The differences found in thesamples response can thus unambiguously be ascribed to thevariation of the magnitude and direction of the applied staticfield.
III. HARMONIC RESPONSE MEASUREMENTS
In this section, we present some measurements of the
magnetic response of our Permalloy films in the frequencydomain. We probe the response to harmonic driving fieldgenerated by the stripline. To describe the magnetization dy-namics for arbitrary orientations of the Mrelative to the
stripline, we introduce a rotated coordinate system xyzas
defined in Fig. 1. In this rotated frame, the static magnetiza-tion vector is always parallel to the y-axis and fluctuations
about this direction ~spin waves !are confined to the
xz-plane:M(r,t).M
Syˆ1m(r,t), where m(r,t)5mx(r,t)xˆ
1mz(r,t)zˆ. From the geometry of our experiment and the
film thicknesses under consideration, it is sufficient to con-sider only the transverse magnetization components aver-aged over the film thickness d
Py
FIG. 1. Geometry of the setup. The stripline runs parallel along the Y-axis,
so the induced rf pumping field is always generated along the X-axis.w
denotes the width of the stripline. The magnetization Mlies in the XY-plane
~film plane !at equilibrium. udenotes the angle between Mand theX-axis in
the film plane. The rotated frame xyzfollows the equilibrium magnetization
orientation, where Mdefines the positive y-axis. All static applied fields H0
are restricted to the film plane.5647 J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 Counilet al.
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128.248.155.225 On: Tue, 25 Nov 2014 06:16:27mx,z~x,y;t!5E
0dPymx,z~x,y,z;t!dz
dPy. ~3!
In the weak damping limit, a!1, the linearized ~LLG!
equation for uniform precession motion reads, in the pres-ence of an in-plane driving field h(t)
F]
]t2vB2a]
]t
vH1a]
]t]
]tGFmx~t!
mz~t!G5vMF0
hx~t!G,~4!
as no linear excitation is generated from the Y-component of
the driving field. A straightforward Fourier transform of thelinearized LLG equation in Eq. ~4!gives
F2iv 2vB1iav
vH2iav 2ivGFmx~v!
mz~v!G5vMF0
hx~v!G,~5!
where we have defined vM[gm0Meff,vH[gm0H0, and
vB[vM1vH. From this equation we can derive the
frequency-dependent magnetic susceptibility tensor xJ(v),
defined by
Fmx~v!
mz~v!G5Fxxx~v!xxz~v!
xzx~v!xzz~v!GFhx~v!
0G. ~6!
For a given magnetization direction u, the only efficient
pumping field is hx5hZsinu. We measure the Xcomponent
of the magnetization, that is mX5mxsinu. Therefore, we
measure xxx(v)sin2uwith
xxx~v![x8~v!1ix9~v!, ~7!
where
x8~v!5vBvM~v022v2!
~v022v2!21a2v2~vH1vB!2,
~8!
x9~v!5avvM~vB212v022v2!
~v022v2!21a2v2~vH1vB!2.
v02[vHvBis the ferromagnetic resonance frequency. The
imaginary part x9~v!is a Lorentzian function with a maxi-
mum at v5v0, and a linewidth Dvdirectly related to the
damping factor
Dv52avM. ~9!
To probe the frequency response in the experiment a rf
currentirfis applied by the network analyzer to the stripline,
generating a field m0hrf.m0irf/2w5131024T. This field is
sufficiently small such that we remain in the linear regime ofexcitation. The magnetic susceptibility is determined fromthe inductance of the coplanar waveguide through
x(v)
5DZ2w/(ivm0dPyL), where DZis the change of the line
impedance Zatv, due to the presence of the magnetic
material.7The scattering parameters S11andS12, i.e., the
reflection and the transmission coefficients of the line, aredirectly measured as a function of frequency by the networkanalyzer. Simple transmission line theory leads, for example,toS
115Z/(Z12Zc), where Zc550Vis the characteristicimpedance of the line. We found no difference between the
sensitivity of the setup when making either reflection ( S11)
or transmission ( S12) measurements.
The frequency dependence of the magnetic susceptibility
is presented in Fig. 2 for the 50 nm thick sample. The fre-quency resolution is 7 MHz, as the number of points is 801in a range of 6 GHz. As a check, we also verified that theresonance frequency obtained from our susceptibility mea-surements as a function of applied external field is consistentwith Kittel’s formula: The resonance frequency squared is
proportional to H
0asv02.g2m02H0Meff, in the limit where
H0!Meff. The results are shown in the inset of Fig. 2. Here
we notice that the lock-in detection of the signal by the net-work analyzer allows very small amplitude measurements.For example, the technique is sufficiently sensitive to detectmagnetic oscillations in our 3 nm thick Permalloy sample,which is not possible using time domain measurements withour setup. A comparison of the measured susceptibilitiesfrom a range of film thicknesses is given in Fig. 3. While theresults for the d
Py53 nm sample were taken with a low
signal–to–noise ratio, the susceptibility curves are stillclearly distinguishable. We estimate that the uncertainty inthe damping factor obtained for the d
Py53 nm sample
would exceed 10%, in comparison to samples between dPy
56 nm and dPy550nm were the uncertainty in the line-
width measurement is better than 5%.
Determining afrom linewidth measurements has several
advantages. The amplitude of the pumping field, and conse-quently the amplitude of the magnetization motion, is con-
FIG. 2. Real and imaginary parts of the measured susceptibility xxxfor
dPy550nm. Inset: the square of the measured resonance frequency as a
function of applied field.The ~linear !fit is based on Kittel’s formula at small
fields, v02.g2m02H0Meff.
FIG. 3. Real and imaginary parts of the measured susceptibility xxx(v) for
dPy53, 6, and 50 nm, illustrating the sensitivity of the NA-FMR for thin
films.5648 J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 Counilet al.
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128.248.155.225 On: Tue, 25 Nov 2014 06:16:27stant and can be chosen such that we remain in the linear
regime. This is made possible by the lock-in detection of thenetwork analyzer. This provides an unambiguous character-ization of the damping factor, and nonlinear effects may bestudied separately by increasing the amplitude of the pump-ing field. Here we notice that no limit in frequency has beenobserved so far up to 6 GHz ~Fig. 2 !. Second, the data points
can be conveniently positioned over the frequency range toenhance the frequency resolution in a window of particularinterest. However, as we will discuss shortly, an unambigu-ous measure of the Gilbert damping requires nonuniformspin wave modes to be taken into account. Such spin wavesare generated because the excitation field is not uniform overthe surface of the sample.
IV. EXCITATION OF k‚¯0 SPIN WAVES
An important aspect of the PIMM geometry that war-
rants further discussion is the width of the stripline used toexcite and detect magnetization oscillations. In our case, thewidth of the stripline ~45
mm!is much smaller than the lat-
eral sample dimensions ~0.131.0 cm !. As such, the excita-
tion fields are by no means uniform over the entire sample;the fields are restricted to a region close to the stripline itself.Consequently, it is possible to excite spin waves with a finitewave-vector k
iin the film plane. From the translational in-
variance in the film plane, we include plane wave solutionsfor the lateral components of the magnetization,
m
x,z~x,y;t!51
AAdPy(
kimx,z~ki;t!exp~jkir!, ~10!
whereAis the surface of the magnetic film.
As the width of the stripline is much greater than the
exchange length ( lex.3 nm), only long wavelength spin
waves ~magnetostatic waves !are excited. Furthermore, the
stripline is much longer than the sample, so the excitationfields generated along the X-axis can be taken to be uniform
across the sample along Ydirection. As such, we expect the
generated spin waves will have a wave vector only along the
X-axis,k
i5kiXˆ. Since the magnetocrystalline anisotropy in
our Permalloy films is very small, the magnetization orien-tation can be considered to be always parallel to the staticapplied field direction. Thus, the relation between the spinwave propagation direction and the magnetization orienta-tion can be studied simply by varying the applied field ori-entation with respect to the stripline. For a given angle
u
betweenkiandH0, the spin wave frequency ~without ex-
change terms !is given by13
vs2~ki,u!5v0221
2gm0vM~H02@H01Meff#
3sin2u!kidPy. ~11!
Note that this expression is valid only in the thin-film limit
kidPy!1, where the surface and bulk magnetostatic waves
are degenerate. For our stripline, the maximum kipossible is
estimated from the width of the stripline to be p/w;1
3105m21, so our thin films dPy<50nm should satisfy this
limit.
In Fig. 4, we present the dispersion curves calculatedfrom Eq. ~11!foru510°, 45°, and 90° in the frequency/
wave-vector range appropriate for our experiment. Note thatfor
u50°, the pumping field is parallel to the magnetization,
and no linear excitations are generated in such a configura-tion. For
u510°, the frequencies of the excited modes are
almost constant over the range of wave vectors probed by thestriplinek
max’p/w, and so no change in the response of the
magnetic film is expected. This assumes, of course, that eachfinitek
imode excited in the range kmaxconsidered has a
similar lifetime to the uniform mode.13Foru>10°, the fre-
quency of the excited modes varies over a continuous rangefromf
0up tofu(kmax) that can attain a few hundred maga-
hertz, as seen in Fig. 4. In the linear response regime, thetotal measured response can be taken to be as a sum over allexcitedk
i~i.e., harmonic oscillator !states, so a shift in the
measured resonance peak is expected in addition to a line-width broadening.
As the angle
ubetween the static and rf fields is varied
~Fig. 1 !, we observe indeed a significant frequency shift
df05f(90°) 2f(0°) of up to 400 MHz accompanied by
linewidth broadening, in addition to the expected sin2u
variations of the pumping efficiency. We present the normal-ized amplitude of the imaginary part of the susceptibility inFig. 5, and an example of the signal for different angles
uin
the insert.
The magnetocrystalline anisotropy of the permalloy
samples cannot alone account for these shifts, since
dv0(Ha)52pdf0.2pgm0HavMeff/v0is less than 30 MHz
at 4 GHz for an anisotropy field m0Ha5131024T, while
the measured shift at 4 GHz is 160 MHz. To illustrate this
FIG. 4. Illustration of the dispersion relation of magnetostatic spin wave
modes for various angle u510°,u545°, and u590° at m0H05531023T.
All the modes from f0up tof(kmax,u) are excited.
FIG. 5. Normalized amplitude of the imaginary part of the measured sus-
ceptibility x9~v!as a function of the pumping field orientation u. The fit is a
sin2ucurve. Inset: frequency dependence of x9~v!foru510°, 45°, and 90°.
The frequency shift df0is 230 MHz.5649 J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 Counilet al.
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128.248.155.225 On: Tue, 25 Nov 2014 06:16:27point, we chose the orientation of the easy axis to be at
uHa545°. Therefore, the configurations with the static field
atu.0° and u.90° are symmetric about the easy-axis, so
any shifts in the resonance frequency due to the anisotropyshould mirror this symmetry. While micromagnetic simula-tions were called upon to show evidence of the excitation offinitekspin waves elsewhere,
5,9we argue that our model for
nonuniform modes can account for the experimental obser-vations. We assume that only spin-waves with wave-vector
ukiu<kmaxare excited, where the upper limit kmaxis deter-
mined by the width of the stripline. The total absorption inthe linear response regime can be obtained by summing overall contributions from the excited spin wave modes
xtot9~v,u!}E
0kmax r~k!
11$2@v2vs~k,u!#/Dvint%2dk,~12!
where rrepresents the constant density of modes in
wavevector space, and Dvintis the ‘‘intrinsic’’linewidth cor-
responding to the ‘‘intrinsic’’ damping factor a0given by
Dvint5a0gm0Meff. As discussed before, only spin waves
with wave vector in the xdirection may be excited. Hence,
the relevant density of modes is one-dimensional in wave-vector space. The corresponding density of modes in fre-quency space
r~v!
r~v!5rS]v
]kD21
, ~13!
is thus a constant, as the dispersion relation is linear for
small wavevector @Fig. 4 ~a!#.We can sum up the contribution
of each mode in the frequency space
xtot9~v,u!}E
v0vs~kmax,u! dv
11$2@v2vs~k,u!#/Dvint%2,
~14!
where v0[v(k50) is the measured resonance frequency
for small angles. From this, an analytical expression for thetotal absorption can be found,
xtot9~v,u!}arctanSvs~kmax,u!2v
Dvint/2D2arctanSv02v
Dvint/2D.
~15!
Straightforward algebra leads to a shift of the resonance fre-
quency dv0
dv0~kmax,u!51
2@vs~kmax,u!2v0#. ~16!
Note that this expression is consistent with the simple physi-
cal picture presented in Fig. 4. From this analysis we canalso derive an expression for the total linewidth D
v(kmax)
Dv~kmax!5DvintA11Svs~kmax,u!2v0
DvintD2
. ~17!
Then the good parameter to be considered here is the
ratio of the frequency shift to the intrinsic linewidth, whichcan be also understood from the simple picture presentedhere. We now give the approximate expressions of
dv0(kmax) and Dv(kmax) for u5p/2. Indeed, the measure-ments are performed in such a configuration to enhance the
sensitivity of the setup as seen in Sec. III. In the limit ofsmall dispersion in wavevector we have
dv0~kmax!.vMvMeff
8SkmaxdPy
v0D; ~18!
Dv~kmax!.Dvint1CSkmaxdPy
v0D2
, ~19!
whereCis a constant depending on vMandDvint.
From this, it is then possible to separate the artificial
contributions from the finite kispin wave modes aext(kmax)
and the ‘‘intrinsic’’ damping factor a0
atot~kmax!.Dvint
vMeff1C
vMeffSkmaxdPy
v0D2
,
~20!
[a01aext~kmax!.
The resonance frequency shift dv0scales linearly with the
inverse resonance frequency as shown in Fig. 6 for a thick-nessd
Py550nm, in agreement with the analysis presented
above @Eq.~18!#. We chose the thicker film to get the largest
resonance frequency shift possible and to enhance the preci-sion on the measurement, as
dv0scales linearly with the
thickness @Eq.~18!#. A linear fit, where kmaxis the only pa-
rameter, leads to the measured value of the dispersion in thewave vector. Note that we have
p/kmax5(3062)mm;w,i n
good agreement with the assumption that the dispersion inthe wave vector results only from the finite transverse dimen-sion of the waveguide. We also observe a linewidth enhance-ment while changing the angle. However, the pumping effi-ciency at small angles is too low to obtain a precisemeasurement of the linewidth.
We now apply this analysis to obtain a more accurate
determination of the damping constant from our measure-ments. In Fig. 7, the totaldamping constant
atot, as obtained
from a raw measurement of the resonance linewidth, isshown as a function of frequency for a series of film thick-nesses. We observe an enhancement in the damping constantat low frequencies of approximately 100% for d
Py550nm,
but lower than the measurement precision for dPy56 nm.
Such a monotonic increase in damping factor below 2 GHzhas already been observed by other groups,
4,12where it has
been attributed to inhomogeneous linewidth broadening
FIG. 6. Linear fit of the resonance frequency shift as a function of the
inverse resonance frequency, i.e., 1/ f0.5650 J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 Counilet al.
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128.248.155.225 On: Tue, 25 Nov 2014 06:16:27caused by the dispersion in Ha. Such a large dispersion is
unlikely here, however, due to the rather high quality of thePermalloy films used.
We believe that this increase of the damping factor at
low frequencies results from an artifact of the measurementprocedure, rather from the intrinsic properties of the sample.To illustrate this, we demonstrate here that the observed in-crease of
atotat low frequencies scales well with our model.
In Fig. 8, we present the measured total damping factor ofthed
Py550nm thick film. The exact expression of the line-
widthenhancement @Eq.~17!#isusedtofit atot(kmax)downto
lower frequencies.The free parameters here are a0andkmax,
so that a self-consistent measurement of kmaxcan be ob-
tained. For dPy550nm, we find a05(6.260.3)31023and
p/kmax5(4564)mm. The value of the dispersion in wave
vector scales well with the expected value p/w. Moreover,
the value of the damping factor is consistent with the valuesobtained from standard FMR measurements.
8
The same procedure is applied for other film thicknesses.
FordPy56 nm, the frequency dependence of atotis negli-
gible, consistent with the negligible linewidth enhancementpredicted by our model. In this case, the value of
a0is the
mean value of aover the frequency range. A summary is
given in Table I. Note that the value of the dispersion in thewave vector obtained from the frequency shift measurementswas found to be greater. However, one must keep in mindthat the excitation profile was assumed to be a gate functionin order to provide an analytical expression of the magneticresponse, as well as a simple physical picture of the observedextrinsic source of linewidth enhancement. While the totaldamping factor
atotincreases with the film thickness ~Fig. 7 !,
the opposite trend is found for the intrinsic damping constanta0, as shown in the insert of Fig. 8. Such a trend is consis-
tent with damping processes that originate from surfaces orinterfaces.
13–15
V. DISCUSSION AND CONCLUDING REMARKS
The inductive measurement of the susceptibility of thin
magnetic films proves to be a precise and reliable tool todetermine the value of the damping factor, provided the gen-eration of spin waves is taken into account. However, thedata obtained from such measurements must be treated withsome care. We have shown that the observed linewidthbroadening and frequency shifts are due to the excitation ofnonuniform spin wave modes, which result from applying aninhomogeneous driving field to the magnetic sample. Wewould like to emphasize, however, that one must be carefulin translating these results into the time domain. For in-stance, the broadening of the resonance peak here does notsimply translate into a simple enhancement of the decay con-stant in the time domain, but rather represents a more com-plicated distortion of the damped oscillations of the magne-tization.
In fact, an estimate of the equivalent time-resolved re-
sponse can be made as follows. The solution to the homoge-neous problem @Eq.~4!#is given by a damped sinusoid
m
x~t!5(
kimkieikire2~ivs11/t!t, ~21!
where, as usual, the coefficients mkiare determined by the
initial conditions and vsis the corresponding spin wave fre-
quency given by Eq. ~11!. The decay constant tis related to
the damping constant in the following way:
t52
a~vH1vB!, ~22!
so the time decay of a single mode gives a direct measure of
a. Here, it is implicitly assumed that each excited spin wave
mode decays with the same damping constant. However, it isclear from the sum over all excited k
istates in Eq. ~21!that
the time dependence of the magnetization will be compli-cated, since the spin wave frequencies are wave-vector de-pendent. Neglecting the spatial dependence for the moment,the total magnetic response can be obtained by integratingover all excited modes
m
x~t!}E
v0vs~kmax!
e2~ivs11/t!tdvs, ~23!
}sin~dv0t!
te2~iv¯11/t!t, ~24!
FIG. 7. Frequency dependence of the total damping factor atotfordPy56,
10, and 50 nm, as obtained from measurements of the resonance linewidth.
FIG. 8. Fit of the measured total damping factor for dPy550nm with a0
andkmaxas free parameters. Inset: evolution of a0with the inverse film
thickness.TABLE I. Corrected values of the intrinsic damping factor a0and wave-
vector spread kmax.
dPy(nm) a0(31023) p/kmax(mm)
6 8.2 60.4 fl
10 7.5 60.3 48 64
50 6.2 60.3 45 645651 J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 Counilet al.
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128.248.155.225 On: Tue, 25 Nov 2014 06:16:27where v¯[v01dv0is the mean frequency with dv0
[@vs(kmax)2v0#/2. This is consistent with our analysis pre-
sented earlier, where we see a frequency shift proportional tothe spread in spin wave frequencies and a prefactor thatmodifies the exponential decay. Thus, the actual time decayobserved is necessarily shorter than the true value governedby
abecause of destructive interference effects within a time
scale p/dv0. Similar results are obtained if phase shifts due
to the finite duration of the field pulse are taken into account.Some aspects of this behavior have been reported else-where.
17,18
The model presented here provides a quantitative analy-
sis of an apparent source of damping enhancement that arisesfrom the measurement geometry of the NA-FMR setup. Theaim of the analysis is to allow a much better estimate of the‘‘intrinsic’’or true damping factor to be obtained.We wish toreiterate that ‘‘intrinsic’’ here refers to the damping constantobtained if one were to perform a traditional FMR experi-ment, i.e., the damping corresponding to the lifetime of theuniform precession mode of the ferromagnet. This constantmay contain contributions from two-magnon processes
13and
spin transfer torques,16where such contributions are labeled
as ‘‘extrinsic’’ by other authors.
In summary, we have presented a study of high-
frequency magnetic response of thin Permalloy films mea-sured by NA-FMR. For inhomogeneous excitation fields, itis shown that propagating spin waves can lead to importantchanges in the measured magnetic susceptibility. We havedeveloped a model to account for such effects, in particular,for facilitating an accurate measure of the Gilbert damping inthin films.
ACKNOWLEDGMENTS
The authors would like to thank L. Lagae and Z. Celin-
ski for helpful discussions. J.V.K. would like to acknowledgefinancial support from the French Ministry of Research and
the CNRS. The work was supported by the NEDO contract‘‘Nanopatterned magnet,’’and by the European CommunitiesHuman Potential Program under Contract No HRPN-CT-2002-00318 ULTRASWITCH.
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~2003!.
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B63, 214416 ~2001!.
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Hurdequint, and C. Chappert, J. Magn. Magn. Mater. ~to be published !.
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Belgium, 2003.5652 J. Appl. Phys., Vol. 95, No. 10, 15 May 2004 Counilet al.
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1.3624900.pdf | Hybrid spintronics and straintronics: A magnetic technology for ultra low
energy computing and signal processing
Kuntal Roy, Supriyo Bandyopadhyay, and Jayasimha Atulasimha
Citation: Appl. Phys. Lett. 99, 063108 (2011); doi: 10.1063/1.3624900
View online: http://dx.doi.org/10.1063/1.3624900
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Downloaded 09 Jul 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsHybrid spintronics and straintronics: A magnetic technology for ultra low
energy computing and signal processing
Kuntal Roy,1,a)Supriyo Bandyopadhyay,1and Jayasimha Atulasimha2
1Department of Electrical and Computer Engineering, Virginia Commonwealth University, Richmond,
Virginia 23284, USA
2Department of Mechanical and Nuclear Engineering, Virginia Commonwealth University, Richmond,
Virginia 23284, USA
(Received 11 January 2011; accepted 25 July 2011; published online 9 August 2011)
The authors show that the magnetization of a 2-phase magnetostrictive/piezoelectric multiferroic
single-domain shape-anisotropic nanomagnet can be switched with very small voltages thatgenerate strain in the magnetostrictive layer. This can be the basis of ultralow power computing
and signal processing. With appropriate material choice, the energy dissipated per switching event
can be reduced to /C2445 kT at room temperature for a switching delay of /C24100 ns and /C2470 kT for a
switching delay of /C2410 ns, if the energy barrier separating the two stable magnetization directions
is/C2432 kT. Such devices can be powered by harvesting energy exclusively from the environment
without the need for a battery.
VC2011 American Institute of Physics . [doi: 10.1063/1.3624900 ]
The primary obstacle to continued downscaling of digi-
tal electronic devices in accordance with Moore’s law is the
excessive energy dissipation that takes place in the device
during switching of bits. Every charge-based device [e.g.,metal-oxide-semiconductor field-effect-transistor (MOS-
FET)] has a fundamental shortcoming in this regard. They
are switched by injecting or extracting an amount of chargeDQfrom the device’s active region with a potential gradient
DV, leading to an inevitable energy dissipation of DQ/C2DV.
Spin based devices, on the other hand, are switched by flip-ping spins without moving any charge in space ( DQ¼0) and
causing a current flow. Although some energy is still dissi-
pated in flipping spins, it can be considerably less than theenergy DQ/C2DVassociated with current flow. This gives
“spin” an advantage over “charge” as a state variable.
Recently, it has been shown that the minimum energy
dissipated to switch a charge-based device like a transistor at
a temperature Tis/C24NkTln (1/p), where Nis the number of in-
formation carriers (electrons or holes) in the device and pis
the bit error probability.
1On the other hand, the minimum
energy dissipated to switch a single-domain nanomagnet
(which is a collection of Mspins) can be only /C24kTln(1/p),
since the exchange interaction between spins makes Mspins
rotate together in unison like a giant classical spin.1,2This
gives the magnet an advantage over the transistor.
Unfortunately, the magnet’s advantage is lost if the
method adopted to switch, it is so inefficient that the energy
dissipated in the switching circuit far exceeds the energy dis-sipated in the magnet. Regrettably, this is often the case. A
magnet is usually flipped with either a magnetic field gener-
ated by a current
3or a spin polarized current exerting either
a spin transfer torque4or causing domain-wall motion.5The
energy dissipated to switch a magnet with current-generated
magnetic field was reported in Ref. 3as 1011–1012kT for a
switching delay of /C241ls, which clearly makes it impracti-
cal. In fact, it will make the magnet inferior to the transistorwhich can be switched in sub-ns while dissipating 107–108
kT of energy in a circuit.6Domain-wall motion induced by a
spin-polarized current can switch a nanomagnet in 2 ns while
dissipating 104–105kT of energy,7but there is still a need to
identify more energy-efficient mechanisms for switching a
magnet.
Recently, we have shown that the magnetization of a
shape-anisotropic piezoelectric/magnetostrictive multiferroic
nanomagnet can be switched with a small voltage applied to
the piezoelectric layer.8Such multiferroic systems have now
become commonplace9–11and there are proposals for using
them in magnetic logic and memory.8,12In this method, the
electrostatic potential generates uniaxial strain in the piezo-electric layer, and that is elastically transferred to the magne-
tostrictive layer if the latter is considerably thinner. The
nanomagnet is clamped along the hard axis. This makes themagnetization of the magnetostrictive layer rotate. Such
rotations have been demonstrated experimentally.
10
Consider an ellipsoidal multiferroic magnet with uniax-
ial shape anisotropy as shown in Fig. 1. The piezoelectric
layer is 40 nm thick, and the magnetostrictive layer is 10 nm
thick, which is thin enough that strain does not relax. Weassume that the piezoelectric layer is lead-zirconate-titanate
(PZT) and the magnetostrictive layer is polycrystalline
nickel or cobalt or Terfenol-D. For Terfenol-D, the majoraxis is assumed to be /C24102 nm and the minor axis is
/C2498 nm. Because of shape anisotropy, the two magnetiza-
tion orientations parallel to the easy axis (major axis of theellipse or the z-axis) are stable and can store the binary bits
0 and 1. We keep the potential energy barrier between these
two orientations (i.e., the shape anisotropy barrier) 0.8 eV or/C2432 kT at room temperature by choosing the appropriate
parameters, which makes the static bit error probability e
/C032.
Let us assume that the magnetization is initially oriented
along the /C0z-axis. Our task is to switch the nanomagnet so
that the final orientation is along the þz-axis. We do this by
applying a voltage Vacross the thickness of the piezoelectric
layer that generates uniaxial stress along the easy axisa)Electronic mail: royk@vcu.edu.
0003-6951/2011/99(6)/063108/3/$30.00 VC2011 American Institute of Physics 99, 063108-1APPLIED PHYSICS LETTERS 99, 063108 (2011)
Downloaded 09 Jul 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions(z-axis) via d31coupling. The energy dissipated in the
switching circuit during turn-on is (1/2) CV2while that dissi-
pated during turn-off is (1/2) CV2, where Cis the capacitance
of the piezoelectric layer plus any line capacitance. Since the
piezoelectric layer has a very large relative dielectric con-
stant (1000), its capacitance will dominate over the line ca-
pacitance which can be neglected.
There is an additional dissipation Edin the nanomagnet
due to Gilbert damping.13The total energy dissipated in the
switching process is, therefore, Etotal¼CV2þEd. Thus, in
order to calculate Etotalas a function of switching delay, we
have to calculate four quantities: (1) the stress needed to
switch the magnetization within a given delay, (2) the volt-
ageVneeded to generate this stress, (3) the capacitance C,
and (4) Edwhich is calculated by following the prescription
of Ref. 13.
In order to find the stress rrequired to switch a magne-
tostrictive nanomagnet in a given time delay s, we solve the
Landau-Lifshitz-Gilbert (LLG) equation for a single-domain
magnetostrictive nanomagnet subjected to stress r.We then
relate rto the strain ein the nanomagnet from Hooke’s law
(e¼r/Y, where Yis the Young’s modulus of the nanomag-
net) and find the voltage Vthat generates that strain in the
piezoelectric layer based on its d31coefficient and thickness.
Finally, we calculate the capacitance of the multiferroic sys-
tem by treating it as a parallel-plate capacitor. This allows usto find the energy dissipated in the switching circuit ( CV
2)a s
a function of the switching delay s.
In the supplementary material accompanying this let-
ter,14we show that both stress and shape anisotropy act like
a torque on the magnetization of the nanomagnet. This tor-
que per unit volume of the nanomagnet is
TEðtÞ¼/C0 nmðtÞ/C2r E½hðtÞ;/ðtÞ/C138; (1)
where E[h(t),/(t)] is the total potential energy of the nanomag-
net at an instant of time t. It is the sum of shape anisotropy
energy and stress anisotropy energy, both of which depend on
the magnetization orientation at t he given instant determined by
the polar angle h(t) and azimuthal angle /(t)o ft h em a g n e t i z a -
tion vector which is assumed to be in the radial direction.
We can write the torque as
TEðtÞ¼/C0 f 2Bð/ðtÞÞsinhðtÞcoshðtÞge^
/
/C0fB0eð/ðtÞÞsinhðtÞge^
h;(2)
where e^
hande^
/are unit vectors in the h- and /-directions,
and
B0ð/ðtÞÞ ¼l0
2M2
sX½Nxxcos2/ðtÞþNyysin2/ðtÞ/C0Nzz/C138;(3a)Bstress¼ð3=2ÞksrX; (3b)
Bð/ðtÞÞ ¼ B0ð/ðtÞÞ þ Bstress ; (3c)
B0eð/ðtÞÞ ¼l0
2M2
sXðNxx/C0NyyÞsinð2/ðtÞÞ: (3d)
Here Msis the saturation magnetization of the nanomagnet,
Xis its volume, l0is the permeability of free space, ksis the
magnetostrictive coefficient of the magnetostrictive layer,and N
bbis the demagnetization factor in the bdirection,
which can be calculated from the shape and size of the nano-
magnet (see the supplementary material14).
The magnetization dynamics of the single-domain nano-
magnet (neglecting thermal fluctuations) is described by the
LLG equation
dnmðtÞ
dtþanmðtÞ/C2dnmðtÞ
dt/C18/C19
¼c
MVTEðtÞ; (4)
where nm(t) is the normalized magnetization, ais the dimension-
less phenomenological Gilbert damping constant, c¼2lBl0//C142is
the gyromagnetic ratio for electrons, and MV¼l0MsX.
From this equation, we can derive two coupled equa-
tions that describe the h- and /-dynamics,
ð1þa2Þh0ðtÞ¼/C0c
MV½B0eð/ðtÞÞsinhðtÞ
þ2aBð/ðtÞÞsinhðtÞcoshðtÞ/C138;(5)
ð1þa2Þ/0ðtÞ¼c
MV½aB0eð/ðtÞÞ /C0 2Bð/ðtÞÞcoshðtÞ/C138
ðsinhðtÞ6¼0Þ:(6)
Clearly, the h- and /-motions are coupled and hence these
equations have to be solved numerically. We assume that the
initial orientation of the nanomagnet is close to the /C0z-axis
(h¼179/C14). It cannot be exactly along the /C0z-axis ( h¼180/C14)
since then the torque acting on it will be zero [see Eq. (2)]
and the magnetization will never rotate under any stress.Similarly, we cannot make the final state align exactly along
theþz-axis ( h¼0
/C14) in a reasonable time since there too the
torque vanishes. Hence, we assume that the final state ish¼1
/C14. Thus, both initial and final states are 1/C14off from the
easy axis. Thermal fluctuations can easily deflect the mag-
netization by 1/C14(see Ref. 15).
We apply the voltage generating stress abruptly at time
t¼0. This rotates the magnetization away from near the easy
axis ( h¼179/C14) to the new energy minimum at h¼90/C14.W e
maintain the stress until hreaches 90/C14which places the mag-
netization approximately along the in-plane hard axis(y-axis). Then, we reduce the voltage to zero abruptly. Sub-
sequently, shape anisotropy takes over and the magnetization
vector rotates towards the easy axis since that now becomesthe minimum energy state. The question is which direction
along the easy axis will the magnetization vector relax to. Is
it the /C0z-axis at h¼179
/C14(wrong state) or the þz-axis at
h¼1/C14(correct state)? That is determined by the sign of
B0e(/(t)) when hreaches 90/C14.I f/at that instant is less than
FIG. 1. (Color online) An elliptical multiferroic nanomagnet stressed with
an applied voltage.063108-2 Roy, Bandyopadhyay, and Atulasimha Appl. Phys. Lett. 99, 063108 (2011)
Downloaded 09 Jul 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions90/C14, then B0e(/(t)) is positive which makes the time deriva-
tive of hnegative (see Eq. (5)), so that hcontinues to
decrease and the magnetization reaches the correct state
close to the þz-axis. The coupled h- and /-dynamics ensures
that this is the case as long as the stress exceeds a minimum
value. Thus, successful switching requires a minimum stress.
Once we have found the switching delay sfor a given
stress rby solving Eqs. (5)and(6), we can invert the rela-
tionship to find rversus sand hence the energy dissipated
versus s. This is shown in Fig. 2where we plot the energy
dissipated in the switching circuit ( CV2), as well as the total
energy dissipated ( Etotal) versus delay for three different
magnetostrictive materials. For Terfenol-D, the stress
required to switch in 100 ns is 1.92 MPa and that required to
switch in 10 ns is 2.7 MPa.
Note that for a stress of 1.92 MPa, the stress anisotropy
energy Bstress is 32.7 kT while for 2.7 MPa, it is 46.2 kT. As
expected, they are larger than the shape anisotropy barrier of/C2432 kT which had to be overcome by stress to switch. A
larger excess energy is needed to switch faster. The energy
dissipated and lost as heat in the switching circuit ( CV
2)i s
only 12 kT for a delay of 100 ns and 23.7 kT for a delay of
10 ns. The total energy dissipated is 45 kT for a delay of 100
ns and 70 kT for a delay of 10 ns. Note that in order toincrease the switching speed by a factor of 10, the dissipation
needs to increase by a factor of 1.6. Therefore, dissipation
increases sub-linearly with speed, which bodes well for
energy efficiency.
With a nanomagnet density of 10
10cm/C02in a memory
or logic chip, the dissipated power density would have beenonly 2 mW/cm
2to switch in 100 ns and 30 mW/cm2toswitch in 10 ns, if 10% of the magnets switch at any given
time (10% activity level). Note that unlike transistors, mag-
nets have no leakage and no standby power dissipation,
which is an important additional benefit.
Such extremely low power and yet high density magnetic
logic and memory systems, composed of multiferroic nano-
magnets, can be powered by existing energy harvesting sys-tems
16–19that harvest energy from the environment without
the need for an external battery. These processors are
uniquely suitable for implantable medical devices, e.g., those
implanted in a patient’s brain that monitor brain signals to
warn of impending epileptic seizures. They can run onenergy harvested from the patient’s body motion. For such
applications, 10-100 ns switching delay is adequate. These
hybrid spintronic/straintronic processors can be also incorpo-rated in “wrist-watch” computers powered by arm move-
ment, buoy-mounted computers for tsunami monitoring (or
naval applications) that harvest energy from sea waves, orstructural health monitoring systems for bridges and build-
ings that are powered solely by mechanical vibrations due to
wind or passing traffic.
1S. Salahuddin and S. Datta, Appl. Phys. Lett. 90, 093503 (2007).
2R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E. Welland, and D. M.
Tricker, Phys. Rev. Lett. 83, 1042 (1999).
3M. T. Alam, M. J. Siddiq, G. H. Bernstein, M. T. Niemier, W. Porod, and
X. S. Hu, IEEE Trans. Nanotechnol. 9, 348 (2010).
4D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008).
5M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno, Nature (London)
428, 539 (2004).
6See CORE9GPLL_HCMOS9_TEC_4.0 Databook for information about
propagation delay and energy dissipation, UNICAD2.4, STMicroelec-
tronics (2003).
7S. Fukami, T. Suzuki, K. Nagahara, N. Ohshima, Y. Ozaki, S. Saito, R.Nebashi, N. Sakimura, H. Honjo, K. Mori, C. Igarashi, S. Miura, N. Ishiwata,
a n dT .S u g i b a y a s h i ,D i g .T e c h .P a p .-S y m p .V L S IT e c h n o l . 2009 , 230.
8J. Atulasimha and S. Bandyopadhyay, Appl. Phys. Lett. 97, 173105
(2010).
9F. Zavaliche, T. Zhao, H. Zheng, F. Straub, M. P. Cruz, P. L. Yang, D.Hao, and R. Ramesh, Nano Lett. 7, 1586 (2007).
10T. Brintlinger, S. H. Lim, K. H. Baloch, P. Alexander, Y. Qi, J. Barry, J.
Melngailis, L. Salamanca-Riba, I. Takeuchi, and J. Cumings, Nano Lett.
10, 1219 (2010).
11W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature (London) 442, 759
(2006).
12S. A. Wolf, J. Lu, M. R. Stan, E. Chen, and D. M. Treger, Proc. IEEE 98,
2155 (2010).
13B. Behin-Aein, S. Salahuddin, and S. Datta, IEEE Trans. Nanotechnol. 8,
505 (2009).
14See supplementary material at http://dx.doi.org/10.1063/1.3624900 for
detailed derivations and additional simulation results.
15D. E. Nikonov, G. I. Bourianoff, G. Rowlands, and I. N. Krivorotov, J.
Appl. Phys. 107, 113910 (2010).
16S. Roundy, Ph.D. thesis, Mech. Eng., University of California, Berkeley,
California, 2003.
17S. R. Anton and H. A. Sodano, Smart Mater. Struct. 16, R1 (2007).
18F. Lu, H. P. Lee, and S. P. Lim, Smart Mater. Struct. 13, 57 (2004).
19Y. B. Jeon, R. Sood, J. Jeong, and S. G. Kim, Sens. Actuators, A 122,1 6
(2005).
FIG. 2. (Color online) Energy dissipated in the switching circuit ( CV2) and
the total energy dissipated ( Etotal) as functions of delay for three different
materials used as the magnetostrictive layer in the multiferroic nanomagnet.063108-3 Roy, Bandyopadhyay, and Atulasimha Appl. Phys. Lett. 99, 063108 (2011)
Downloaded 09 Jul 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions |
1.1708839.pdf | Experiments on Surface Wave Propagation along Annular Plasma Columns
S. F. Paik, R. J. Briggs, and J. M. Osepchuk
Citation: Journal of Applied Physics 37, 2475 (1966); doi: 10.1063/1.1708839
View online: http://dx.doi.org/10.1063/1.1708839
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/37/6?ver=pdfcov
Published by the AIP Publishing
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to ] IP: 69.166.47.134 On: Wed, 03 Dec 2014 05:52:42JOURNAL OF APPLIED PHYSICS VOLUME 37, NUMBER 6 MAY 1966
Experiments on Surface Wave Propagation along Annular Plasma Columns*
S. F. PAIKt
Department of Electrical Engineering, Northwestern University, Evanston, Illinois
AND
R. J. BRIGGst
Lawrence RaJiation Laboratory, Univers~ty of California, Livermore, California
AND
J. M. OSEPCHUK
Raytheon Research Division, Waltham, Massachusetts
(Received 29 August 1965; in final form 17 December 1965)
The propagation of surface waves along a plasma column of annular cross section was investigated ex
perimentally. The laboratory plasma used for the experiment was a mercury-vapor dc discharge. The
properties of the experimental discharge tube were examined to show some basic differences between the
laboratory plasma and the idealized model of the plasma used for the analysis. In particular, a radially non
uniform density distribution and the variation of the distribution with the applied magnetic field was noted.
In spite of the radial-density inhomogeneity, the experimentally determined phase constants of the back
ward-surface wave are in good agreement with values predicted from the uniform density theory. The
attenuation in the absence of the magnetic field is consistent with collisional losses, predominantly with the
walls of the container. The effect of an axial magnetic field on the surface-wave characteristics is examined.
Experimental results show that in the presence of a magnetic field the attenuation of the backward wave is
markedly increased; this enhancement of the damping is not consistent with simple collision theory within
the framework of the uniform-density model.
I. INTRODUCTION
GUIDED waves in bounded plasma columns have
been studied by many authors in recent years,
and characteristics of various modes of propagation in
plasma waveguides have been well documented.l The
configurations which have been studied most extensively
are cylindrical plasma columns of circular cross sections
and rectangular slabs. More recently, it has been shown
that, in plasma columns of annular cross section, there
are two sets of circularly symmetric surface-wave modes
in the absence of a dc magnetic field.2,3 One propagates
along the outer surface of the annulus and is a forward
wave similar to the surface wave in ciruclar plasma
columns. The other surface wave is a backward wave
which propagates along the inner surface. In Fig. 1,
dispersion diagrams (w-(3) for these circularly symmetric
surface waves are presented. These backward-wave
modes on annular plasma columns are not of the variety
predicted by Trivelpiece and others for anisotropic
plasmas,4,5 nor are they related to the azimuthally
varying modes investigated by Carlile and others.6,7 Instead, these backward waves are surface waves
similar to the ones predicted by Oliner and Tamir8 in
isotropic plasma slabs. The existence of the backward
waves in annular plasmas has been demonstrated experi
mentally by Napoli and Swartz3 in a cesium discharge.
* This work was done at Raytheon Research Division, Waltham,
Mass., under the U. S. Army (Signal Corps) Contract No.
DA 36-039-AMC-02362 (E). t Formerly with Raytheon Research Division, Waltham, Mass. t Formerly with Research Laboratory of Electronics, MIT,
Cambridge, Mass., and Raytheon Research Division, Waltham, The purpose of this paper is to present some new
experimental results on the propagation of surface
waves in annular plasma columns. The laboratory
plasma used in our experiments was the positive column
of a mercury-vapor discharge contained between two
coaxial glass envelopes. In addition to the measurement
FIG. 1. w-f3 curves for
annular plasma column. 0.7
0.6
alf 0.5 ___ ~~~'2.0
%'15
(fffj.' ...
.• :......... Plasma
: .. :' Air
:." .0
b'
4
yo
Mass. on Electronics Waveguides (Polytechnic Press Brooklyn N Y
1 W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Aniso- 1958), p. 227. " . .,
tropic Plasmas (Technology Press, Cambridge, Mass., 1963). 613-. N. Carlile, J. Appl. Phys. 35, 1384 (1964); and (a) R. N.
2 S. F. Paik, J. Electron. Control 13, 515 (1962). Carhle, Tech. Rept. No. TM-30 Electronics Research Lab
3 L. S. Napoli and G. A. Schwartz, Phys. Fluids 6, 918 (1963). University of California, Berkeley, '1963. .,
4 A. W. Trivelpiece and R. W. Gould, J. App!. Phys. 30, 1784 7 V. L. Granatstein and S. P. Schlesinger J. Appl. Phys 35 (1959). 2846 (1964). ,. ,
6 L. D. Smullin and P. Chorney, Proceedings of the Symposium 8 T. Tamir and A. A. Oliner, Proc. IEEE 51, 315 (1963).
2475
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to ] IP: 69.166.47.134 On: Wed, 03 Dec 2014 05:52:422476 P;\1K, BRIGGS, AND OSEPCHUK
of the phase and attenuation constants of the waves, we
consider, in the present study, the effect of an external
magnetic field and the radical density inhomogeneity in
the experimental plasma tube.
Propagation characteristics of guided waves in
plasmas have been analyzed, in most cases, using an
idealized model of the plasma. In the ideal model, the
plasma is assumed to be a uniform stable medium in
which the effect of the collisions between particles and
the thermal motions may be neglected. The dispersion
diagram presented in Fig. 1 was obtained using such a
model. Positive columns of dc discharges are often used
for experiments on guided waves, but it is well known
that there are a number of differences between this
type of laboratory plasma and the ideal model. In
particular, the assumption of the uniform density dis
tribution has been questioned recently,9---12 and it has
been shown by several authors that the density inhomo
geneity does, in general, cause a significant change in the
propagation characteristics of guided waves.
Before presenting the experiments on the propagation
of waves at radio frequencies, therefore, we will dis
cuss the results of a detailed experimental investigation
of the properties of the discharge tube. Of particular
interest is the plasma density and its distribution in the
radial direction. In Sec. II, the results of ex-perimental
measurements of the average-number density and the
radial-density distribution with an applied magnetic
field was investigated in an effort to determine a pos
sible relationship between an anomalous behavior of the
surface waves in the presence of a magnetic field and the
radial-density distribution.
The phase and attenuation constants of the surface
backward waves were measured and compared with the
values predicted from the simple uniform density theory
in Sec. III. Our experimental results are in agreement
with earlier measurements of Napoli and Swartz.3 In
the absence of an external magnetic field, the experi
mentally determined dispersion diagram is closely
approximated by the dispersion characteristic obtained
from the simple uniform-density theory using the
average density. The attenuation of the surface back
ward wave in the absence of the magnetic field can be
quantitatively accounted for in terms of the collisional
damping, with the predominant collision being the
"wall" scattering.
In order to reduce the effect of the "wall collision,"
an axial magnetic field of small magnitude was applied.
The effect of this weak-axial magnetic field on the sur
face backward wave mode of propagation is described
in Sec. IV. Contrary to the expected reduction of the
attenuation, the attenuation of the wave increased
9 R. J. Briggs and S. F. Paik, Bull. Am. Phys. Soc. 10, 210
(1965) .
10 S. Jha and G. S. Kino, J. Electron. Control 14, 167 (1963).
II H. L. Stover, Tech. Rept. No. M-l140, Microwave Labora
tory, Stanford University, Stanford, Calif., 1964.
12 P. de Santis, Nuovo Cimento 34, 823 (1965). rapidly with the applied field. The anomalous damping
mechanism cannot be explained on the basis of simple
collision theory and the uniform-density model. In the
discussion of Sec. V, we consider some possible explan
ations for the enhanced damping observed.
II. CHARACTERISTICS OF THE ANNULAR
de DISCHARGE
A. Experimental Tube
The experimental tube with a hollow annular cross
section was constructed with two coaxially placed
precision-bore glass tubings. The circular symmetry of
the cross section is maintained by centering the two
glass envelopes with a Kovar anode at one end, and the
cathode assembly at the other end as shown in Fig. 2.
The glass tubing is made of thin-walled (0.025 in.) low
loss glass (7070). A thoria-coated tungsten wire was used
as the cathode. In addition to the Kovar anode there
is another auxilliary electrode near the cathode which
was used to initiate the discharge. A cylindrical probe
parallel to the axis of the tube is mounted on a bellow
assembly so that it can be moved laterally across the
tube. Table I gives pertinent dimensions of the experi
mental tube and the probe.
B. Measurement of Plasma Density
The backward-wave mode in an annular plasma
column is predicted to exist over a narrow band of fre
quencies between fp and fi(1+e)t. Since the wave
attenuates rapidly near the edges of the pass band due
to slow group velocity, the effect passband would not
be expected to be much greater than approximately 20%
of the plasma frequency. In order to find the passband
experimentally, therefore, it is essential to have an
accurate measure of the plasma density.
The dipole resonance technique was used primarily
for the measurement of the average electron density.Is
This diagnostic technique was particularly well suited
for the present purpose, since one of the dipole-resonance
TABLE I. Description of the hlbe and the probe.
A. Tube
Inner envelope:
Outer envelope:
Cathode:
Gas:
Over-all length :
B. Probe
Material:
Active length:
Distance of travel: 0.2S0-in. i.d. precision-bore 7070 tubing
(0.020 in. waH)
O.790-in. i.d. precision-bore 7070 tubing
(0.025 in. wall)
O.OlS-in. tungsten wire with 500F
(Raytheon No.) thoria coating
Mercury vapor at room temperature
(pressure 2 J1. Hg)
From probe to collector 12 in.
O.015-in. tungsten wire
0.250 in.
0.2 in.
13 F. W. Cra'Wiord, G. S. Kino, S. A. Self, and J. Spalter, J.
Appl. Phys. 34, 2186 (1963).
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded
to ] IP: 69.166.47.134 On: Wed, 03 Dec 2014 05:52:42SI'RF;\CE \\,\\'ES ;\I.():\(; .\:--.l:\I~I .. \R I'I..\S\\.\ COLI'\l:\S 24i7
FJ(;. 2. Schematic dia
gram of the tube ",ith ~-~~---~- 0,040 Tungsten wire (coated wilh gloss)
Screws for adjusting prob~ position
5" 8 Diamefer bellow
Kovar support
-0,015" Tungslen probe
0,250 lang Heliarc
weld
mo\'ul,\c probe. -r;==~I==-='-E============i;:===;~~====::i3::';~;;
Cathode
frequencies (which are the same as the cut otT frequencies
of the wave with thl: angular dl:pendence 11= 1), occurs
at the middle of the passband of the circularly 5\'111-
metric surface-wavl: modl:. Therefore, without knowing
the exact plasma frequency, the passband of the
circularh' symmet ric mode can he located readilv from
this n1e'~SU;Tment. The Langmuir probe described in
Table I is use(] as an Hmilliary tool for checking the
result of the rf measurement and also to determine the
radial distribution of the declron density.
The dipole-resonance frequencies for the annular
plasma column are expected to be much dilTerent from
t he resonant frequencies of solid plasma cylinders. The
resonant frequencies were calculated for the annular
column hy following the method of Messiaen and
Vandenplasl4 who cOllsidu-ed a cross-sectional geometry
almost like t he one \\T are considering here. Assuming
quasislatics and a uniform density distribution, we can
du-ive the resonant frequencies by writing the potential
functions in each region and matching boundary con
ditions. For thl: dimensions of thl: eXjlerimental ap
paratus given in Table I, the resonant frequencies were
calculated numerically. As in the case considered by
:'IIessiaen and \'andenplas, \\'l: obtained two resonant
irequmcies, at /l=O.i67 I" and /2=0.+85 fp'
The dipolc-resonance measurements were made using
a strip line similar to the one described in detail by
Crawford Cllll."l in the frequency mnge of 12 Ccsec.
:\ t I'[lical resonance curve is shown in Fig. 3. (Only one
res;mance is shown, since the rangl: of discharge current.
l! .\, \1. \lessiacn and 1'. E, Vandcnplus, J. Nuel. Energy, Pt. C
-j,2('7 (1962). \ " \ '---Inner envelope 0.0.-0 290 ~ 0.025 Ihick 7070910ss
Mercury pool Ouler envelope 10.-0.790"
0.025 Ihick 7070 glass Kavor
colleclor
was not broad enough to observe the two resonances
predicted.) The value of the discharge current at reso
nance is appr()'\imatdy 300 mA and the width of the
resonance curve shown here is roughly 50 111;\. In Fig.
3 the measured resonant frequencies are plotted as a
function of the discharge current. The plasma frequency
is calculated \lsing one of the numerical results given
above. To decide which one of thl: dipole resonances are
being observed, it is necessary to vary the discharge
Plasma frequency
calculated from
the lower curve
oFirs1 run
ASecond run
Idischarge (rnA) 500
FIG. 3. ;\ typical resonance cur\'c. Plasma frequency \'5 discharge
curren1. measured hy thr strip-line resonance Illcthod.
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current over a wider range. In our experiments, the
value of the plasma frequency was double-checked
against the Langmuir probe data. The uncertainty of
about five percent indicated in Fig. 3 represents the
variation of the experimental results over a long period
of time. Note that in the range of discharge currents
shown here, the plasma frequency, not the density, is
linearly related to the discharge current.
C. Radial-Density Distribution
The dipole resonance technique described above is an
effective method of measuring the average plasma
density. It is well known, however, that in dc discharges
such as the experimental tube discussed above, the
density distribution in the radial direction is not uni
form. In the mercury-vapor discharge tube described
above, the mean free paths of electrons and ions are at
least one order of magnitude greater than the transverse
dimensions of the container. The plasma in such a dis
charge should be considered a "collisionless" plasma.
The radial-density distribution in a "collisionless"
plasma can be calculated exactly in the absence of an
external magnetic field with the aid of the recently
developed theory,15 but the analytical method of
calculating the density distribution in collisionless
plasmas in a magnetic field has not yet been developed.
Since the electron mobility in the radial direction is
severely affected by a magnetic field of the order 100 G,
the density distribution is expected to vary appreciably
with a variation of the axial magnetic field of this
magnitude. We will present, in the following discussion,
the results of an experimental study of the density dis
tribution and the variation of the density profile in the
annular plasma column with a weak-axial magnetic
field.
The density profile was measured using the movable
probe described above with the tube placed in a solenoid.
The density measurement was made in the ion-satu
ration region, since the ion current is less affected by
the magnetic field. The Langmuir probe data in the ion
saturation region can be interpreted in the usual way if
the ion-Larmor radius is much larger than the probe
dimensions and the electron temperature is assumed to
remain constant. The ion-saturation current was meas
ured as a function of the probe voltage at various radial
positions, and the electron density at each position was
calculated from the measured data in the conventional
manner. The variation of the relative magnitude of the
plasma density as a function of the magnetic field is
presented in Fig. 4. The data were calculated from the
average data of several measurements with the dis
charge current held constant at 300 rnA. The ratio of
the "wall" density nw to the measured maximum density
no,16 is shown as a function of the axial magnetic field
IS J. V. Parker, Phys. Fluids 6, 1957 (1963).
16 no here refers to the density measured near the center of the
discharge. It is not necessarily the actual maximum density. 40
f 3.0
~
"'E
~ 2.0
:i
c:" 1.0 40
iii
IQ 3.0
8=230 {2.0
c: .\.0 ~~~:~g
8=150
Gauss
8=30
8=0
°0
Inner
wall ~~~- ~~
1
005 010 0.15 020 025 °0 Q05 0.10 ub Q20 025
Distance in inches outer Inner Distance in inches ruler
wall wall wall
FIG. 4. Probe measurement of density vs position.
in Fig. 5. As is illustrated in Figs. 5 and 6, the radial
density inhomogeneity becomes more pronounced as the
magnetic field is increased up to approximately 100 G.
The ratio of the "wall" density to the maximum density
decreases rapidly in the interval 0-100 G and remains
virtually constant as the magnetic field is increased
beyond this value. Qualitatively, this result is not at
variance with the earlier experimental results on helium
discharges reported by Bickerton and von EngelP
Note that the plasma frequencies corresponding to the
densities measured at zero magnetic field lies within the
range of 1.15 to 1.4 Gc/sec. This result is in excellent
agreement with the average plasma frequency measured
by the dipole-resonance method (Fig. 3).
III. SURFACE WAVE DISPERSION
CHARACTERISTICS
In this section we describe some measurements of the
wave propagation characteristics of the annular column.
As was pointed out in the introduction, the theoretical
model used to derive the dispersion characteristic shown
in Fig. 1 assumes a uniform plasma density. In view of
the substantial density inhomogeneity which was meas-
1.50r-------,----.-----,
1.25
1.00
0.5 W~min
/~
/
0.25
0. 100 200 300
B Gauss
FIG. 5. The ratio of "wall" density to the density at the "center"
(nw/no) vs magnetic field.
17 R. J. Bickerton and A. von Engel, Proc. Phys. Soc. (London)
B69,468 (1956).
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ured, it is of interest to determine whether the pre
dicted backward-wave mode can be observed experi
mentally, and also how well the measured dispersion
characteristic correlates with the simple theory when
the average density is used.
The experimental arrangement for the measurement
of the wavelength and attenuation of the wave in the
annular column is shown in Fig. 6. The experimental
tube is virtually identical to the one described in
Sec. II; the only difference being the design of the col
lector. The collector design was modified so that it
could be used as a transition from the coaxial line to the
plasma column. The new collector design is shown in
Fig. 7. The signal was coupled onto the plasma column
from a coaxial line with the aid of this transition.
Throughout the experiment, a conducting shield was
placed on the outer surface of the discharge tube. The
detecting probe was made of a section of a miniature
coaxial line.
The output of the detector in the arrangement shown
35t"b74~~~~~ tuner r=;
150.0. coaxia!
hoe
FIG. 6. Experimental:arrangement for measurement of wavelength.
in Fig. 6 is proportional to
A2e-Zaz+ B2+2ABe-az cos(/3z+ifJ), (1)
where A is the amplitude of the signal at z=O, and B is
the amplitude of the reference signal. The variation of
the amplitude of the detected signal is plotted as a
function of the axial distance, z, directly on an X-V
recorder, and the phase constant is determined from the
plot of the traveling-wave pattern.
Two sets of experimental results are plotted in a
normalized form (w/wp vs {3a) in Fig. 8. In calculating
the ratio w/wp, the "average" value of Wp as measured
by the dipole resonance method (Fi~. 3) was used. A
theoretical dispersion curve derived for the uniform
plasma is superimposed on the diagram to comp;tre the
experimental results with the simplified theory. Note
that one set of experimental points does lie quite dose
to the theoretical curve. The other set of experimental
points seems to be displaced vertically from the first
set, and this could be caused in part by a slight dif
ference in the setting on the x axis of the recorder or a
change in the discharge conditions. The agreement is Glass to Kovar seal
50nline
Outer glass lubOlg
FIG. 7. Design of the collector and the rf coupling scheme.
quite good, considering the approximation inherent in
using a uniform-density model and an "average"
density.
In another series of experiments, the forward surface
waves predicted by the simple theory (see Fig. 1) were
observed. Since the forward waves are predicted to
propagate in the frequency range below the passband
of the backward wave, the plasma density required for
the forward mode was much higher than for the back
ward wave at a given frequency.
The degree of coupling to either the forward or the
backward wave depended strongly on the type of
coupling probe used. This is reasonable, since the field
patterns of these two modes are distinctly different.
The radial (rf) potential distribution for the forward
and the backward surface waves are of the form sketched
in Fig. 9(a), and the corresponding electric field patterns
are sketched in Fig. 9(b). This sketch illustrates the
possible "mode-selective" properties of the radial and
longitudinal probe used as rf couplers. The experimental
results show that the forward wave is most easily ex
cited with the radial probe, and the backward wave with
the type of coupling scheme illustrated in Fig. 7.
In Eq. (1) it is assumed implicitly that there is a
finite attenuation. The experimental results showed that
the attenuation of the wave amplitude is indeed sig-
l
/.0
09
f 08 Q.
II
" II
07
0.6
050.
FIG. 8. w-{3 curve for the annular discharge column.
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Plasma
r--"'--
Forward- wave Backward-wave
Forward wave
(a)
Backward wave
(b)
FIG. 9. (a) Radial distribution of potentials; (b) ap
proximate electric field pattern.
nificantly high. Typically, attenuation constant was the
order of eight to nine dB/cm. This high attenuation is
attributed to the scattering of particles by the "walls"
and the relatively low-group velocity of the wave.18
The attenuation constant for waves in plasma col
umns is, in the first-order approximation6
(2)
where Vc is the collision frequency and Vg the group
velocity. Since the mean free paths for the electron
neutrai, electron-ion, and electron-electron collisions
are far greater than the distance between walls of the
discharge tube in the mercury discharge at room temper
ature, it is expected that the loss is predominantly due
to the electron scattering by the walls. An order of
magnitude calculation can be carried out to predict
the amount at attenuation expected from the wall
collision. First, the electron-wall collision frequency is
vcvT/(b-a), (3)
where Vr is the thermal velocity of electrons. If we
consider electron temperature to be 30000oK, then
for our experimental discharge tube where (b-a) is
approximately 0.6 cm, we obtain the collision frequency
18 The scattering, of course, actually occurs in the sheath region.
Stover-Kino have considered this problem in some detail, and
have pointed out that this "wall collision" mechanism can explain
the high attenuation which has also been found for the surface
waves on solid cylindrical plasma columns. CR. Stover and G. S.
Kino, Bull. Am. Phys. Soc. 9, 336 (1964)]. of 108 secl. The attenuation constant is, therefore,
of the order of 1 Np/cm or 9 dB/cm, if a group velocity
of 108 cm/sec is used in Eq. (2). This is the right order
of magnitude to explain the measured value of the at
tenuation constant.
IV. EFFECT OF A de MAGNETIC FIELD ON
THE BACKWARD SURFACE WAVE
If the loss of the signal in the plasma is indeed due to
the scattering of particles by the walls as postulated, a
significant improvement in the attenuation character
istics of the plasma column would be expected when a
dc magnetic field is applied in the axial direction. In the
presence of a magnetic field the plasma is no longer an
isotropic medium, and its dielectric constant must be
given in a tensor form. The magnetic field, therefore,
will not only affect the attenuation characteristic but
also the over-all dispersion characteristics of the plasma
waveguide. The dispersion characteristics of the surface
backward wave in uniform annular plasma columns in
the presence of an axial magnetic field are described in
the Appendix.
In the presence of an axial magnetic field, the back
ward surface wave has a reduced passband since
the frequency for resonance is raised from wp/Vl to
[(wl+w c2)/2J!. If the magnetic field is weak so that
the cyclotron frequency is much lower than the plasma
frequency, then this mode of propagation will be
virtuallv unaffected. In addition to the shift of the
resona;ce of the surface waves, two sets of volume waves
are introduced when the magnetic field is present. For
wc>wp, there is no surface wave.
To study the effect of the magnetic field on the back
ward-wave mode of the plasma column, the discharge
tube was placed inside a solenoid as before. An experi
mental arrangement identical to the one shown in Fig.
6 was used. The traveling-wave pattern for the back
ward-wave mode in the absence of the magnetic field
was obtained, and then the magnetic field was gradually
increased. Since the average density of the plasma
changed with the magnetic field, as discussed in the
previous section, it was necessary to vary the discharge
current to adjust the plasma density with each step of
change in the magnetic field to continue to observe the
wave pattern. The most significant result of this experi
ment was that the attenuation of the surface wave
increased with the increasing magnetic field, contrary
to the expected results. However, the existence of the
backward wave for small magnetic fields was confirmed.
A set of traveling-wave patterns obtained in the
presence of a magnetic field is shown in Fig. 10. The
maximum field at which the traveling-wave pattern is
recognizable is approximately 90 G. For each magnetic
field value, the discharge current was adjusted for
minimum attenuation of the wave. The percent vari
ation of the discharge current required was approxi
mately 50% when the magnetic field was raised to 90 G.
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FIG. 10. Traveling-wave pattern of the backward-surface
wave mode in a finite axial magnetic field.
This reflects the change in the discharge current neces
sary to keep the plasma frequency relatively constant.
When an axial magnetic field of 90 G is applied, the
Larmor radius of electrons is much smaller than the
distance between the walls of the discharge tube. For
this case, the attenuation due to the wall scattering
should be negligible. The observed damping is, therefore,
apparently not consistent with the collisional damping
expected on the basis of the usual uniform-density
models of the plasma column.
V. DISCUSSIONS
Surface waves in annular plasma columns were pre
dicted originallv from the analysis of an idealized
plasma model. Some of the prope;ties of the laboratory
plasma used for our experiment, a dc discharge plasm~
were investigated in Sec. II to point out some basi~
differences between the laboratory plasma and the ideal
model. In particular, it was shown that the radial
density distribution is nonuniform and that the form
of the density profile was dependent on the axial
magnetic field.
In spite of the radial inhomogeneity, forward and
backward modes of propagation in the annular column
were observed in the absence of the external magnetic
field. The measured dispersion characteristics have the
same form as is predicted by the uniform-density model.
In addition, the attenuation of the waves in th; absence
of the magnetic field can be quantitatively accounted for
on the basis of collisional damping, with the predomi
nant collision mechanism being the wall scattering.
When a small magnetic field in the axial direction is
introduced, however, the behavior of the backward
wave mode on the annular column is changed. The
wave becomes strongly damped, and the traveling-wave
pattern cannot be recognized as the magnetic field is
increased to about 90 G. The enhanced attenuation of
surface waves due to the presence of a magnetic field
has been observed by others.6.,19 In the experiment of Napoli et al.,19 the loss of the signal was attributed to
the dispersal of energy into side bands generated by
low-frequency fluctuations of the plasma. In the pres
ence of large fluctuations, such as striations in Napoli
et al.'s experiment, the low-frequency fluctuations can
be expected to cause such a damping. In the experiment
discussed here, however, the excessive damping observed
was due purely to the presence of the magnetic field.
The magnitude of the low-frequency oscillations in
mercury-vapor discharges is far less than the fluctu
ations accompanying striations.
One possible explanation for this anomalous damping
is suggested by the density profile measurements dis
cussed in Sec. II. The measured variation of the density
profile with magnetic field occurs mostly in the same
range of magnetic field where the rapid increase in the
attenuation of the surface wave was noted. If one writes
down the differential equations for a cold, collision less
plasma column with a finite density gradient, one finds
that in certain frequency bands the eigenmodes are
singular at critical radii where the plasma is locally in
"resonance." In most previous studies of guided waves
in inhomogeneous plasma columns,j()-12 the effect of this
singularity has not been considered. When such a
"resonance" is present, it is a clear warning of the
breakdown of the simple cold, collisionless theory. Stix20
has considered a similar problem for free waves in
homogeneous plasmas and showed that absorption can
take place at the critical layer. In the limit of a cold
plasma approximation with collisions, it is possible to
show9 that guided waves in plasma columns are also
subject to a similar mechanism of "resonance absorp
tion," if the radial dielectric constant becomes zero at
the signal frequency. Therefore, this absorption mecha
nism is predicted to occur within the frequency range
[Wpmax2+wl]!>w> [Wmin2+wl]!. (4)
The "resonance absorption" frequency band is de
termined by the radial-density profile and the magnetic
field. There are really two possibilities, however, for an
apparent "threshold" magnetic field for the onset of
"resonan.ce absorption." The simple (zero-temperature)
formulatlOn of the "resonance absorption"9 is only
expected to occur when the "nonlocal" effects due to
a finite temperature are negligible. This assumption is
probably not valid in the absence of a magnetic field in
~he type of plasma used for our experiments. However,
m the presence of a sufficiently strong magnetic field
the "local" relation between transverse-plasma curren~
and the electric field should be recovered. Thus a . . ' certam maXlmUm Larmor radius of electrons and
hence, a minimum magnetic field, may be requi;ed fo;
this type of mechanism to occur. There is also of course , ,
1. L. S. Napoli, G. 1\. Swartz, and H, T. Wexler Phys Fluids
8, 1142 (1965). ' .
2() T. H. Stix, Theory of Plasma Waves (McGraw-Hill Book Co.
Inc., New York, 1962), Chap. 10. '
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1.50r-------,------,---------.
1.25
0.50
0.25
o 100
B (gauss) Backward -wave
passband
Lower edge of
resonance absorrion
band jw~min+w /wpo
200 300
FIG. 11. Lower limit of the resonance absorption band and the
backward-wave passband vs magnetic field.
the possible threshold magnetic field due to the change
in the absorption bandwidth according to Eq. (4).
This threshold magnetic field can be calculated from the
experimentally measured density profile. The lower edge
of the resonance absorption band given by Eq. (4) is
plotted as a function of the axial magnetic field in Fig.
11. If we superimpose on this diagram a plot of the lower
edge of the backward wave passband, we note that
the entire passband of the backward wave is within
the region of resonance absorption above the magnetic
field strength of about SO G. It has been noted experi
mentally that the strong attenuation of the surface
waves begins to occur in the same range of magnetic
field.
Under ideal conditions, the backward-wave charac
teristics of the annular plasma column can be utilized
for a plasma backward-wave oscillator. The coaxial
geometry is particularly well suited for this application,
since an electron beam can be injected conveniently
into the hollow region inside the annulus. It has been
proposed that such an oscillator would provide the usual
advantage of being "structureless" as well as being
a widely tunable device, since the tuning can be
accomplished by changing the beam voltage or the
plasma density. In another possible application of the
annular plasma column, one takes advantage of its
geometrical shape as a device to couple rf energy from
a coaxial cable to other types of plasma waveguides.21
In considering both of these applications, the attenu
ation of the wave in the annular column was assumed to
be negligible. In view of the experimental results pre
sented in this paper, it may be concluded that the use
fulness of the annular plasma is limited due to its high
21 G. A. Schwartz and L. S. Napoli, Tech. Rept. No. AL-TDR
64-155, RCA Laboratories, Princeton, N. I., 1964. rate of attenuation. The method of eliminating the cause
for the high attenuation must be developed before the
plasma column can be useful in practical applications.
ACKNOWLEDGMENTS
The authors wish to acknowledge the technical help
received during the course of this work from Mrs. G.
Carota and C. Harney. Others who participated in
various phases of this experimental program are J. K.
Silk, J. Lotus, K. D. Gilbert, and J. Gallagher.
APPENDIX
(,)-~ Diagram for Annular Plasma Column in a
Finite Magnetic Field
In this Appendix we determine the nature of the
dispersion curves for the uniform annular plasma column
in a finite magnetic field. Since the backward-surface
wave along the inner surface of the plasma is of prime
interest, a configuration of the plasma column with the
outer surface shorted by a conductor is considered.2
We adopt the quasistatic approximation, and write
the electric field as a gradient of a scalar potential.
(A1)
where the potential and cf> satisfies the two-dimensional
Helmholz equation.
'\!2cf>+p2cf>=0 for a<r<b, (AZ)
V'2cf>+.B2cf>=0 for r<a,
where a and b are inner and outer radii of the plasma
(0) We < wp Shaded area for p2>O.
(b) we >wp "Volume" waves propagate
in this region
FIG. 12. K 1 and KII as functions of frequency.
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column, respectively, and
(A3)
Kl and K/I are the elements of the tensor dielectric
constant of a plasma in a finite-axial magnetic field,
K/I= 1-(wllw2)
K1= 1-[wil(w2-wl)]. (A4)
For the circularly symmetric mode, the dispersion
relations for the annular plasma column are
j3[I1 ((Ja) II o(j3a)]
=K1P[H10(pa,pb)/Hoo(pa,pb)] p2>0 (AS)
and
j3[I 1 ({3a)/ I o (j3a) ]
=K1"rfGlO(l'a,l'b)IGoo(ya,l'b)] f<O, (A6)
where
and
H mn(X,y)= ] m(x)N n(Y)-(-l)m-nN m(X)] n(Y)
Gmn(X,y) =I m(x)Kn(Y)- (-l)m-nK m(x)I n(Y)·
Equation (AS) is the dispersion relation for the so
called "volume" waves whose fields have a "harmonic"
radial variation within the plasma, and Eq. (A6) is the
dispersion relation for the surface-wave whose fields
decay away from the plasma-dielectric intersurface.
From Eqs. (A3), (AS), and (A6), we can obtain a
qualitative picture of the w-{3 diagrams for various
conditions.
The surface wave cannot exist if wp<we.1 This can
be seen by examining Eqs. (A3) and (A6). For f to
be negative and (32 to be positive finite, Kl and KII
must have different signs. But, since the Goo function is
always negative for real I'a and I'b, K/I must be nega
tive to satisfy Eq. (A6). This does not happen if
wp<w c• The "volume" wave, however, propagates in the
frequency ranges indicated in Fig. 12 regardless of the
relative magnitudes of Wp and We'
The qualitative picture of the w-f3 diagram can be
obtained when the resonances (i.e., fJ ---+ 00) and cutoffs
(i.e., fJ ---+ 0) of the waves are known. There are only
three ways in which j3 can tend to infinity.
1. K/I---+O,
2. Kl ---+ 00,
3. ipi---+ 00. (p finite)
(p finite) "Volume" waves
Wp ,
jw~+Wi Surface wave
2 "VolllTle" woves wc--------
~-----------------~
FIG. 13. Approximate w-fJ curves for annular plasma
columns in a finite magnetic field.
The resonant frequency corresponding to 1 and 2 are
Wp and We, respectively. In case 3, we can show from
dispersion equation, Eq. (A6), that such a resonance is
possible only when wc<wp. This is the surface-wave
resonance which occurs atl
(A7)
For cutoff, fJ ---+ 0, we must have one of the following
conditions:
(1) Kl---+O
(2) K/I ---+ 00
(3) Ipl---+O. (p finite)
(p finite)
The cutoff frequencies for the conditions 1 and 2 are
w= (wi+w c2)i and w=O, respectively. For ipi---+ 0, we
must have, from Eq. (A6)
That is, we can have i pi ---+ ° as j3 ---+ 0, but this must
also occur at the frequency for which Kl=O.
The information obtained so far on cutoffs and reso
nance presents a fairly complete picture of the dispersion
diagram. For we<wp, the dispersion diagram for the
annular plasma column has the form shown in Fig. 13.
Comparing this figure with the dispersion diagram
given in Fig. 1, we note that the backward surface
wave now has a reduced bandwidth since the resonance
frequency is raised from wp/v2 for no magnetic field to
[(wp2+w})j2]t. If the magnetic field is weak so that
the cyclotron frequency is much lower than the plasma
frequency, then this mode of propagation will be
virtually unaffected. In addition to the shift of the
resonance of the surface waves, two sets of volume
waves are introduced when the magnetic field is
present. For wc>w p, there is no surface wave.
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1.3532041.pdf | Stress-based control of magnetic nanowire domain walls in artificial multiferroic
systems
J. Dean , , M. T. Bryan , , T. Schrefl , and , and D. A. Allwood
Citation: Journal of Applied Physics 109, 023915 (2011); doi: 10.1063/1.3532041
View online: http://dx.doi.org/10.1063/1.3532041
View Table of Contents: http://aip.scitation.org/toc/jap/109/2
Published by the American Institute of Physics
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Applied Physics Letters 107, 142405 (2015); 10.1063/1.4932057Stress-based control of magnetic nanowire domain walls in artificial
multiferroic systems
J. Dean,1,a/H20850M. T. Bryan,1T. Schrefl,2and D. A. Allwood1
1Department of Materials Science and Engineering, University of Sheffield, Sheffield S1 3JD,
United Kingdom
2St. Poelten University of Applied Sciences, A-3100 St. Poelten, Austria
/H20849Received 18 October 2010; accepted 29 November 2010; published online 24 January 2011 /H20850
Artificial multiferroic systems, which combine piezoelectric and piezomagnetic materials, offer
novel methods of controlling material properties. Here, we use combined structural and magneticfinite element models to show how localized strains in a piezoelectric film coupled to apiezomagnetic nanowire can attract and pin magnetic domain walls. Synchronous switching ofaddressable contacts enables the controlled movement of pinning sites, and hence domain walls, inthe nanowire without applied magnetic field or spin-polarized current, irrespective of domain wallstructure. Conversely, domain wall-induced strain in the piezomagnetic material induces a localpotential difference in the piezoelectric, providing a mechanism for sensing domain walls. Thisapproach overcomes the problems in magnetic nanowire memories of domain wallstructure-dependent behavior and high power consumption. Nonvolatile random access or shiftregister memories based on these effects can achieve storage densities /H110221 Gbit /In
2, sub-10 ns
switching times, and power consumption /H11021100 keV per operation. © 2011 American Institute of
Physics ./H20851doi:10.1063/1.3532041 /H20852
I. INTRODUCTION
Long-term data storage applications, such as hard drives,
traditionally use magnetic order to represent information. Incontrast, random access memory /H20849RAM /H20850uses mainly
semiconductor-based systems with fast local access times.Recent proposals that use magnetic nanowire /H20849NW /H20850devices
offer the prospect of nonvolatile shift register and randomaccess memories.
1–4However, this technology is still ham-
pered by a number of issues, including the power required togenerate magnetic fields or electric currents for device op-eration. Multiferroic systems that exhibit both ferroelectricand magnetic order may circumvent this problem by using anelectric potential to control magnetization.
5,6Already this has
been demonstrated in patterned magnetic elements.7
In the proposed magnetic NW devices, regions of con-
stant magnetization /H20849“domains” /H20850in a magnetically-soft NW
are oriented preferentially along the wire length with theboundaries of these regions forming “domain walls” /H20849DWs /H20850.
DWs can be injected into NWs and moved using appliedmagnetic fields
3,8,9or spin-polarized currents.10Most NW
memory or logic systems use a series of DWs to represent adata stream
1–4or a single DW in a bistable element to form
a RAM cell.11In these systems, the data are represented
either by the orientation and position of either magnetic do-mains or DWs.
In all NW memories, the separation of DWs is the key to
defining the data density. This separation must be preservedto ensure data is not lost by annihilation of DW pairs. Thishas been achieved by using turns in the NWs with DWsdriven by a rotating magnetic field vector
2or using notches
in wire edges,3,12–14wire junctions,15,16or adjacent magneticelements17,18to pin DWs at regular intervals. These local
features present a DW with either a potential barrier or wellof height/depth that depends on the DW magnetizationconfiguration.
13This can cause difficulties in systems where
the DW structure is not perfectly controlled. Furthermore,depinning a trapped DW is usually a temperature-dependent,stochastic process, which causes further uncertainties. An at-tempt to circumvent these difficulties has been to use NWsof exquisite quality with very few defects and maintain theseparation of DWs using well-defined current pulses.
10Al-
though this was demonstrated for two DWs, this systemwould be highly vulnerable to defects, as a single DW pin-ning site could cause data loss in a shift register. Here weshow how an electrical potential applied locally to an “arti-ficial” multiferroic material can be used to precisely controlDW motion for memory applications. The same local contactconfiguration allows the DW to be stored, moved, and readwithout using a magnetic field or current. This addressablepinning offers highly reproducible operations by removingany DW structural dependence and stochastic depinning.
II. MODELING
Permalloy /H20849Ni81Fe19/H20850is used in most NW technologies,
as it has near zero magnetocrystalline anisotropy and mag-
netostriction in bulk form, allowing DW behavior to be de-termined solely by NW shape. If the NWs were made from apiezeomagnetic material coupled to a piezoelectric to forman artificial multiferroic, then an induced anisotropy withinthe magnetic layer could in principle be formed from anelectric field applied to the piezoelectric layer, via the gener-ated strain. Here, the
COMSOL finite element package19and a
finite element/boundary element micromagnetic code20were
used to mode la5n m thick, 100 nm wide ferromagnetic NWa/H20850Electronic mail: j.dean@shef.ac.uk.JOURNAL OF APPLIED PHYSICS 109, 023915 /H208492011 /H20850
0021-8979/2011/109 /H208492/H20850/023915/5/$30.00 © 2011 American Institute of Physics 109, 023915-1with Young’s modulus, YFeGa=100 GPa, Poisson’s ratio
/H9263FeGa=0.3,21,22magnetostriction constants of /H9261100=/H9261111
=200 ppm, magnetocrystalline anisotropy K=0 J m−3, ex-
change constant A=10 pJ m−1, saturation magnetization
Ms=800 kA m−1, and Gilbert damping constant /H9251=0.02.
This means that the NW has magnetic properties similar toPermalloy but the magnetoelastic properties of FeGa. Asshown in Fig. 1/H20849a/H20850, the NW sits on top of a silicon substrate
/H20849Young’s modulus, Y
S=100 GPa and Poisson’s ratio /H9263S
=0.3 /H20850and covered with a 200 nm thick layer of the piezo-
electric lead zirconate titanate /H20849PZT; variant PZT-5G,
Young’s modulus, YPZT=100 GPa, and Poisson’s ratio /H9263PZT
=0.3 and coupling constants taken from Ref. 23/H20850that is
poled along the wire length /H20849x-axis /H20850. On top of the PZT are a
number of 100 nm wide electronic contacts, separated by200 nm, that are used to apply electric potentials to the pi-ezoelectric. The resultant Cauchy stress tensor at every pointthroughout the NW was calculated in
COMSOL , forming a
“map” of the nonuniform stress. The Cauchy stress tensormap was then incorporated into the micromagnetic code
24
solving the Landau–Lifshitz–Gilbert equation to find themagnetic response of the NW.
A DW trap is created when contacts 1 and 3 are con-
nected to ground /H20849voltages V
1=V 3=0 V /H20850and a potential
V2=0.5 V is applied to contact 2 /H20851Fig.1/H20849b/H20850/H20852. This creates alarge electric field of 2.5 MV/m which is nonetheless lower
than the 6 MV/m dielectric breakdown strength of PZT-5H.25
The complete Cauchy stress map generated from these po-tential differences is shown in Fig. 1/H20849c/H20850, highlighting regions
of compressive and tensile stress. As the poled direction ofthe PZT-5H is along the x-axis, the positive potential differ-
ence generates a compressive x-axis stress between contact
pairs /H20849CP/H208501–2 /H20851white region in Fig. 1/H20849c/H20850-XX component /H20852
while the negative potential leads to a tensile x-axis stress CP
2–3 /H20851dark region in Fig. 1/H20849c/H20850-XX component /H20852. Other compo-
nents of the stress tensor are less significant, except for theXZ torsional stress component.
III. SWITCHABLE DW PINNING AND MOVEMENT
Figure 1/H20849d/H20850shows the total energy of a DW centered at
difference points along the NW. When the DW is in an un-stressed region /H20849away from the contacts /H20850, the total NW en-
ergy is 10.4 eV, due to the contributions of the exchange,demagnetization, and magnetocrystalline anisotropy ener-gies. The various piezoelectrically induced strain terms /H20851Fig.
1/H20849c/H20850/H20852induce additional magnetic anisotropies in the NW.
These create a potential well and barrier at different DWpositions /H20851Fig.1/H20849d/H20850/H20852, with the potential well forming a DW
trap. The DW energy decreases to a minimum of 8.2 eVbetween CP 1–2 /H20851Fig.1/H20849d/H20850/H20852, establishing a potential well of
depth E
w=2.2 eV. As the magnetization of the DW core is
orthogonal to the length of the wire, it is aligned with thelocal stress-induced anisotropy axis from the compressivestress at the energy minimum. If the DW is moved furtheralong the wire, the energy increases to a maximum of 12.8eV between CP 2–3 /H20851Fig. 1/H20849d/H20850/H20852, where the local stress-
induced anisotropy axis from the tensile stress is perpendicu-lar to the core magnetization. Once at the energy minimum,the DW can depin in two directions: away from the trap byovercoming the energy well, or through the trap by overcom-ing the combined effect of the potential well and barrier/H20849E
wb=4.4 eV /H20850. Assuming that the rate of thermally assisted
switching fis described by the Arrhenius–Néel relation f
=f0exp/H20849−Ew/kBT300/H20850, where f0is an attempt frequency, the
DW memory would remain stable almost indefinitely for any
reasonable value of f0/H20849108–1012s−1/H20850, provided the contact
potentials are maintained.
Using the same contact configuration, a DW is initialized
100 nm from contact 1 /H20851Fig.2/H20849a/H20850/H20852. If the trap is activated at
0 ns, the DW accelerates toward the energy minimum /H20851Fig.
2/H20849b/H20850/H20852, achieving a velocity of 600 m s−1, and reaching the
energy minimum position after 1.5 ns. The DW then under-goes damped oscillation about this position before stoppingaltogether after about 3 ns. In contrast, when the trap is leftinactive, the DW remains stationary /H20851Fig.2/H20849b/H20850/H20852.
The trap acts as a switchable pinning site, able to trap
DWs even in the presence of a magnetic field. In this con-figuration the depinning field needed to force the DWthrough the barrier /H20849positive x-direction /H20850is found to be lin-
early dependent on the contact potential, increasing at a rateof 80 Oe V
−1up to 1 V. In contrast, only 40 Oe V−1/H20849i.e., 20
Oe for V 2=0.5 V /H20850is required to depin the DW in the oppo-
site direction, as the DW does not need to overcome the
FIG. 1. /H20849Color online /H20850/H20849a/H20850Schematic of the complete DW trap system. /H20849b/H20850
Top view of the NW highlighting the location of the contacts on the PZT-5Hlayer. /H20849c/H20850The Cauchy stress matrix tensor maps generated from
COMSOL .
The stress indicates the maximum and minimum stresses for the tensile/H20849dark /H20850and compressive /H20849light color /H20850components. /H20849d/H20850The total energy den-
sity of a DW in the system when the trap is activated; a minima /H20849well /H20850and
a maximum /H20849barrier /H20850forms the complete “trap” system.023915-2 Dean et al. J. Appl. Phys. 109, 023915 /H208492011 /H20850potential barrier. The ability to trap and release DWs by ap-
plication and removal of an electric field, thus circumventsthe problem of depinning fields and current densities beingdependent upon DW structure encountered with geometricand magnetostatic DW traps.
13,26–31
IV. STRESS-BASED MEMORY CELLS
The principle of a fully switchable mechanism to control
DW motion can be extended further to demonstrate amemory element that operates without using an applied mag-netic field or a spin-polarized current. The element is basedupon the DW trap discussed above but contains five contacts/H20851Fig.3/H20849a/H20850/H20852. The contacts are arranged such that switching the
applied potentials in particular configurations /H20851Fig.3/H20849a/H20850/H20852lo-
cates the energy well in one of two positions /H20851Fig. 3/H20849b/H20850/H20852.
With the trap activated as by the “state 0” potentials in Fig.3/H20849a/H20850, we initialize the system with a DW located outside the
contacted region adjacent to contact 1 /H20851Fig.3/H20849a/H20850/H20852. The con-
tact potential configuration forms a potential well betweenCP 2–3 attracting the DW /H20851Fig. 3/H20849c/H20850/H20852. Energy barriers sur-
round this potential well between CP 1–2 and CP 4–5 toprevent the DW escaping. When the contact potential con-figuration is changed to state 1 /H20851Fig.3/H20849a/H20850/H20852, the energy mini-
mum switches position to between CP 3–4. The change inenergy gradient causes the DW to move to the position of thenew energy well. We alternated the contact potential configu-ration between states 0 and 1 every 6 ns to allow the DW tocome to rest in each configuration. The DW switches be-tween the two positions in a controllable manner /H20851Fig.3/H20849d/H20850/H20852.
This means that the two trapping positions can definememory states for data storage applications.
The memory state could be read using standard magne-
toresistance measurements, although this would require ad-ditional contacts embedded with the NW. Alternatively, datacan be read using CP 2–3 or C-P 3–4 /H20851Fig.3/H20849c/H20850/H20852by detecting
the strain in the NW due to the DW. As the magnetization inthe DW is orthogonal to the domain magnetization, a localchange in the magnetostrictive strain is generated in the vi-cinity of the DW. Coupling between the NW and the piezo-electric layer allows the strain in the wire to be convertedinto a potential difference in the piezoelectric. Themagnetostriction-induced potential difference was simulatedacross the contacts for a DW in the “0” memory state. Forsimplicity, the magnetization across the 100 nm wide DWwas assumed to be uniform, yielding a potential differencebetween CP 2–3 of 80
/H9262V. On the other hand, when the wall
is moved to the “1” memory state position, the potential inthis region falls to zero while the potential between CP 3–4increases to 80
/H9262V. These potential differences are of the
order that can be detected experimentally, indicating that themagnetostriction-induced strain can be used to detect thepresence or absence of a DW. Charge relaxation due to leak-
FIG. 2. /H20849Color online /H20850/H20849a/H20850Schematic of the DW trap system, with the DW is
initialized 100 nm from contact 1. /H20849b/H20850The velocity of the DW as it move
under the influences of the local stress toward the energy minima /H20849top/H20850and
the movement of a DW under the influence of an activated and inactive trap/H20849bottom /H20850.
FIG. 3. /H20849Color online /H20850/H20849a/H20850Schematic of the memory element highlighting
two contact potentials defined as state 0 and state 1, where G=0 V and V=0.5 V. /H20849b/H20850The total energy of a DW at different positions along the wire
when the system is in state 0 and 1. /H20849c/H20850A micromagnetic simulation of the
DW under the different contact potential configurations. Overlaid is a sche-matic of a possible read-back contact. /H20849d/H20850The DW position and simulated
read-back signal of the memory state, when switching between the twocontact potential configurations every 6 ns.023915-3 Dean et al. J. Appl. Phys. 109, 023915 /H208492011 /H20850age currents within the PZT will limit the availability of this
signal to a few milliseconds at most. Using this as a readoutmechanism in memory elements will require the DW to bemoved in order to refresh the signal. Models of larger cross-section NWs /H20849not shown here /H20850indicate that vortex DWs cre-
ate a stress-induced signal that is similar to transverse DWs,showing that wall structure should not significantly affect theread capability of the device.
V. MULTIBIT MEMORY
The principles of the memory cell operation can be ex-
panded to form a multibit storage memory. Figure 4/H20849a/H20850shows
a device with 13 contacts separated equally by 200 nm. Thecontacts are grouped in sets of three, such that from contact3, every third contact is held at 0.5 V /H20849activated /H20850while the
other contacts are at ground /H20849inactive /H20850, initially forming DW
traps just before contacts 3, 6, 9, and 12. To demonstrate thatthis configuration can lead to DW propagation along thelength of a NW, a DW is initialized between CP 2–3. Theactivated contact potential is then cascaded systematicallyalong the wire by synchronously activating only every thirdcontact. The DW responds by moving progressively alongthe NW. This is shown in Fig. 4/H20849b/H20850, in which the activated
contact potential is cascaded every 5 ns. As the contact po-tential configuration repeats every third contact, every thirdcycle activates identical contacts. Thus stress-induced DWmotion could be sustained along the whole length of the NW/H20851Fig.4/H20849a/H20850I–III /H20852, assuming enough contacts are present. Wallmotion is independent of the wall type /H20849vortex or transverse /H20850,
configuration /H20849head-to-head or tail-to-tail /H20850, and chirality, as
movement is driven by a localized anisotropy along the wireaxis. In many ways, the system is analogous to surfing, as theDW is continuously falling down a potential energy gradientthat moves like a “wave” in the same direction. If multipleDWs are present, they propagate synchronously along thewire in the same direction /H20851Fig.4/H20849a/H20850IV–XI /H20852, creating shift
register operation. Inverting the applied voltage or reversingthe cascade sequence reverses the propagation direction /H20851Fig.
4/H20849a/H20850XI–XIV /H20852. Alternatively, if the contacts are addressed
individually, it is possible to control a single DW indepen-dently /H20851Fig.4/H20849a/H20850XIV–XVI /H20852, enabling the device to operate
as RAM.
The three-cycle period of the applied potentials requires
storage of a bit occupying a total length of 900 nm of NW.Assuming 500 nm separation between neighboring wires toavoid unwanted DW interactions
32gives a storage density of
1.15 Gbits /In2. For the NW dimensions used here, contact
separations less than 200 nm lead to strong magnetostaticinteractions between DWs, requiring a progressively largerstress to maintain operation. Below 100 nm separations, thestress required to maintain wall separation /H20849/H110220.5 GPa /H20850be-
comes unfeasible due to the likelihood of structural fracture
occurring.
VI. DEVICE OPERATIONAL SPEED AND ENERGY
EFFICIENCY
The time to change the contact potential configuration
depends on the total device capacitance, Ctot=CN, where C
is the capacitance of a single trap and Nis the number of
traps in the device. Since the electrical contact area is abovethe piezoelectric layer, the capacitance of the system islargely determined by the dielectric coating used. For a SiO
2
coating /H20849dielectric constant /H9260=4.6 /H20850, 50 nm thick contacts,
200 nm contact separation, and 500 nm NW separation, C
=6.0/H1100310−18F/trap. Therefore ,a1M B shift register
charged through a load of resistance R=10/H9024, will have a
response time of RC tot/H110150.5 ns for device activation. The
device switching time is governed by the 0.5 ns for a DW totravel between traps and the time required for the DW todissipate energy /H20849ringing time /H20850. The latter can be approxi-
mated as the period of one ring cycle, 1.0 ns, after which96% of the energy has dissipated. Combining the times forcapacitance charging, moving the DW, ringing and capaci-tance discharge gives the total switching time of a single trapas 2.5 ns. A shift register with three cells per data bit wouldthen require 7.5 ns per full bit operation.
The work done in straining the piezoelectric layer to
create a pinning site was estimated to be 74 keV /H2084912 fJ /H20850per
operation by considering bulk measurements of PZT-5H/H20849Ref.33/H20850and integrating the electric field in the piezoelectric
layer. This is much greater than the energy per trap operation
due to capacitance of
1
2CVc2/H110154.7 eV but is competitive
with memory devices in published technology roadmaps.34
We have made no attempt to optimize the geometry of the
FIG. 4. /H20849Color online /H20850/H20849a/H20850Micromagnetic simulations of a 13 contact device
and “bit” generation. DWs being introduced and cascaded synchronouslydown the length of the wire /H20849I–XI /H20850. Reversing the voltage causes a reversal
in the DW propagation direction /H20849XI–XIV /H20850. Asynchronous activation of con-
tacts enables DWs to be moved independently /H20849XIV–XVI /H20850. The bit configu-
ration for XVI is shown. /H20849b/H20850The position of the DW along the wire when
the activated contact potential is cycled every 5 ns.023915-4 Dean et al. J. Appl. Phys. 109, 023915 /H208492011 /H20850system but expect further improvements to the data density,
switching speed, and energy with changes to the NW, piezo-electric layer, and contact dimensions.
VII. CONCLUSIONS
We have proposed a design for combining piezoelectric
and piezomagnetic materials to create stress-based memorydevices. Using strain generated by a localized electric poten-tial in a piezoelectric material, a change in magnetic responseis created in magnetostrictive NWs to trap DWs. This allowsa fast switchable mechanism for the creation of moving DWpinning sites. Both mechanisms are insensitive to DW struc-ture. A reciprocal effect is that the change inmagnetostriction-induced stress in the NW due to the pres-ence of a DW produces a change in the potential differenceacross contacts on the piezoelectric layer. This forms a novelmethod of detecting DWs. This allows the device to be op-erated without an applied magnetic field or spin-polarizedcurrent, reducing the power consumption. The ability to se-lectively move, pin, and sense DWs in one device with lowpower consumption indicates that this system has greatpromise as a future memory platform.
ACKNOWLEDGMENTS
We would like to thank EPSRC for their financial sup-
port under Grant Nos. EP/F069359/1 and EP/G032300/1.
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1.5035258.pdf | Effect of the dipolar coupling on the precessional magnetization switching in two-
dimensional arrays of single-domain nano-ellipses
J. C. S. Gomes , D. Toscano , E. L. M. Paixão , C. I. L. de Araujo , F. Sato , R. A. Dias , P. Z. Coura , and S. A.
Leonel
Citation: AIP Advances 8, 095017 (2018); doi: 10.1063/1.5035258
View online: https://doi.org/10.1063/1.5035258
View Table of Contents: http://aip.scitation.org/toc/adv/8/9
Published by the American Institute of Physics
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Effect of the dipolar coupling on the precessional
magnetization switching in two-dimensional arrays
of single-domain nano-ellipses
J. C. S. Gomes,1,aD. Toscano,1,bE. L. M. Paix ˜ao,1,cC. I. L. de Araujo,2,d
F. Sato,1,eR. A. Dias,1,fP . Z. Coura,1,gand S. A. Leonel1,h
1Departamento de F ´ısica, Laborat ´orio de Simulac ¸ ˜ao Computacional, Universidade Federal
de Juiz de Fora, Juiz de Fora, Minas Gerais 36036-330, Brazil
2Departamento de F ´ısica, Laborat ´orio de Spintr ˆonica e Nanomagnetismo,
Universidade Federal de Vic ¸osa, Vic ¸osa 36570-900, Minas Gerais, Brazil
(Received 13 April 2018; accepted 28 August 2018; published online 18 September 2018)
Various spintronic devices use single-domain magnetic nanoparticles as unit cells.
Herein, we investigated interparticle dipole-dipole interactions in arrays of Permalloy
single-domain nano-ellipses through micromagnetic simulations. In this study, the
variation is introduced not only to the aspect ratio and the spacing between ellipses
but to the magnetization distribution and the 2D lattice type as well. When integrating
the Landau-Lifshitz-Gilbert equation with zero external magnetic field, equilibrium
magnetic configurations were obtained for each array. For small values of the spac-
ing between ellipses, they interact strongly, such that the shape anisotropy is locally
modified by the distribution of the magnetization. Moreover, the effect of the dipolar
coupling on the precessional magnetization reversal is also studied. The minimum field
strength required to switch the magnetization depends on the magnetization distribu-
tion in a strongly interacting magnetic system. Consequently, we have assessed the
minimum spacing between particles in which single-domain nano-ellipses becomes a
non-interacting magnetic system. © 2018 Author(s). All article content, except where
otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5035258
I. INTRODUCTION
Studies involving nanotechnology have allowed not only the manufacture of magnetic samples
at the nanoscale, but also the evolution of experimental techniques to measure their properties.1–3
These properties are extended to several future technological applications, such as high-density
magnetic recording media,4–7magnetic random access memory (MRAM)8–11and data processing
devices.12–14
Ferromagnetic nanomaterials present interesting properties for memory technology in order to
substitute the current silicon-based static random-access memory (SRAM) and embedded Flash.15
Such samples can be manufactured in the cylinder, disc, prism, strip or ellipse shape with magnetic
materials such as cobalt and Permalloy. Permalloy is a highly targeted material for technological
applicability due to its high magnetic permeability, low coercivity, and negligible magnetostriction
aAuthor to whom correspondence should be addressed. Electronic mail: jcsgomes@fisica.ufjf.br
bElectronic mail: danilotoscano@fisica.ufjf.br
cElectronic mail: elmpaixao@fisica.ufjf.br
dElectronic mail: dearaujo@ufv.br
eElectronic mail: sjfsato@fisica.ufjf.br
fElectronic mail: radias@fisica.ufjf.br
gElectronic mail: pablo@fisica.ufjf.br
hElectronic mail: sidiney@fisica.ufjf.br
2158-3226/2018/8(9)/095017/7 8, 095017-1 ©Author(s) 2018
095017-2 Gomes et al. AIP Advances 8, 095017 (2018)
and magnetocrystalline anisotropy, while cobalt alloy is often used in magnetic tunnel junctions due
to its crystalline match with MgO or in perpendicular magnetic anisotropy based devices.
Due to the shape anisotropy, the remanent state of an elliptical nanomagnet made of Permalloy
can be a single-domain.16,17In the precessional magnetization reversal, the magnetization is switched
by coherent rotation by applying a magnetic field parallel to the ellipse short axis (magnetization hard
axis). This mechanism is very well discussed in the literature,18,23,24being the ideal method to obtain
the fastest magnetization reversal.
The basis of many spintronic devices consists of an array of non-interacting single-domain
magnetic nanoparticles and the concept of bit patterned media (BPM) is very known in the scientific
community.19–22Although the magnetization reversal in individual single-domain nanoparticles has
been intensively investigated,23–31interparticle interactions have been received less attention.32–34
Understanding and controlling the magnetization reversal of a single nanoparticle is not enough for
technological applications, since interparticle dipole-dipole interactions can affect the performance
of the devices.
In a previous paper,31the authors studied the magnetization reversal in an isolated Permalloy
single-domain nano-ellipse. It was verified that by adjusting the geometric factors of a single ellipse
and the parameters of the magnetic field pulse simultaneously, the highest degree of coherence
occurs when applying a magnetic field perpendicular to the magnetization easy axis. In that paper,
the authors point out the need to study the behavior of magnetization reversal in an array of identical
nano-ellipses, due to the influence of interparticle interactions. Thus, in the present work, we study
the magnetization behavior in arrays of Permalloy nano-ellipses, taking into account interparticle
dipole-dipole interactions.
Due to the long-range of the dipolar interaction, the interparticle magnetostatic coupling cannot
be underestimated. Obviously there is a minimum spacing between particles such that single-domain
nano-ellipses become a non-interacting magnetic system, whose equilibrium magnetization states
correspond to the Ising-like states. The information about the minimum spacing between ellipses
is crucial to increase the bit density in magnetic nanodevices. In order to save time and decrease
fabrication process expenses, optimal device designs can be theoretically investigated by simulation
in a broad set of ellipse dimensions and interparticle spacing. The results can be used to adjust
the interparticle dipolar interactions such that it is possible to engineer ultra-fast and high-density
spintronic devices with practically zero interactions between cells.
II. MODEL AND METHODOLOGY
In this work we considered ellipses arrays made of Permalloy with dimensions such that the
magnetic state of an isolated nano-ellipse is a single-domain (quasi-uniform state). Once the mag-
netocrystalline anisotropy is negligible in materials like Permalloy, the shape anisotropy imposes a
magnetization easy axis in nanomagnets. In the absence of an external magnetic field, the magnetic
moments of an isolated ellipse are confined to the plane of the ellipse and they are largely aligned
along the longest axis. Our magnetic system consists of 9 nano-ellipses arranged in such a way that
they are coplanar. The Fig. 1 shows how the nano-ellipses were arranged in the array. The variables
bxandbyare defined as the edge-to-edge separations between ellipses and the variable cxandcyare
defined as the center-to center separations between them. The Fig. 1 is only a schematic view of the
array and does not represent an equilibrium configuration.
For the geometric parameters of the ellipses, we have considered two different aspect ratios q:
70505 nm3(q= 1.40) and 110605 nm3(q= 1.83). In order to systematically explore the
dipolar coupling between single-domains, we have considered the nano-ellipses arranged into two
types of 2D-lattice: square and rectangular. The periodicity of arrays was varied by the edge-to-edge
separation, bxranging from 10 to 300 nm in steps of 10 nm. For the case in which the nano-ellipses
were arranged into a rectangular lattice, we have assumed the same separation in both axes, that is,
by=bx. When considering the case of a square lattice, cy=cx, we have the following constraint
by= (Lx Ly) +bx.
The following Hamiltonian containing the exchange, Zeeman and dipole-dipole interactions was
used to describe the array of Permalloy single-domain nano-ellipses:095017-3 Gomes et al. AIP Advances 8, 095017 (2018)
FIG. 1. Schematic view of the array of single-domain nano-ellipses. The aspect ratio of the ellipsis is given by q=Lx/Ly,
where LxandLyare are the dimensions with respect to the major and minor axes, respectively. Geometrical centers of the
ellipses are described by position vectors ~Rk=m cxˆi+n cyˆj, where mandnare integers, cxandcyare center-to-center
separations of the ellipses. bxandbyare the edge-to-edge separations of the ellipses. Naturally, we have the following relations
cx=Lx+bxandcy=Ly+by.
H=J(
1
2NX
<i,j>ˆmiˆmj Z
JNX
iˆmi~bext
i+D
2JNX
i,jˆmiˆmj 3( ˆmiˆrij)( ˆmjˆrij)
(rij=a)3)
(1)
where ˆ miand ˆmjare unit vectors representing the magnetic moments located at the iandjsites,
rijis the distance between them and arepresents lattice parameter. The summation in the first term
includes only the nearest magnetic moments of the same ellipse, whereas the summation in the
last term covers all possibles dipole-dipole interactions. In order to switch the magnetization of
the system, a magnetic field pulse was applied only in the central ellipse. In the micromagnetic
approach, the interaction constants not only depend on the material parameters but also the manner in
which the system is partitioned into cells. The size of the micromagnetic cell is chosen based on the
exchange length =q
2A
0M2sand each cell has an effective magnetic moment ~mi=(MsVcel) ˆmi. For
the case in which the system is discretized into cubic cells Vcel=a3, such as used in this work, the
interaction constants are given by J= 2Aa,D
J=1
4a
2andZ
J=a
2. It was used the parameters for
Permalloy-79 (Ni 79Fe21): saturation magnetization Ms= 8.6105A/m, exchange stiffness constant
A= 1.310 11J/m and damping parameter = 0.01. The cell size used in the simulations was
Vcell= 555 nm3.
Micromagnetic simulation results were obtained using our own computational code, which solves
the dimensionless version of the Landau-Lifshitz-Gilbert equation (LLG):
dˆmi
d = 1
1 + 2f
ˆmi~bi+ ˆmi( ˆmi~bi)g
(2)
where ~bi= @H
@ˆmiis the dimensionless effective field located at the cell i. The dimensionless time
interval is given by =!0t, where!0=
a20
eMsandtis the real-time interval.
In order to obtain the remanent states for arrays of dipolar coupled nano-ellipses, we have chosen
as initial conditions the distributions of magnetization which correspond to arrays of non-interacting
single-domain nano-ellipses. Four possible initial configurations are schematically shown in Figure 2.
In case 1, the magnetization of each ellipse was chosen randomly. In case 2, the magnetization
of the central ellipse is aligned in one direction, whereas magnetizations of the other ellipses are
aligned in the opposite direction. In case 3, the magnetization direction of the ellipses are aligned in
alternated directions. Finally, in case 4, the magnetization of all the ellipses are aligned in the same
direction.
The magnetic field required to switch the magnetization of the central ellipse can be used to
determine the minimum spacing in which the ellipses are uncoupled. To excite the precessional
switching of the central ellipse magnetization, we apply a pulse of magnetic field perpendicular to
the magnetization easy axis, given by
~B(t)=ˆj Bexte (t t0)2
22 (3)095017-4 Gomes et al. AIP Advances 8, 095017 (2018)
FIG. 2. Schematic view of a few possible magnetic states for arrays of non-interacting single-domain nano-ellipses (Ising-like
magnetization states). Due to the shape anisotropy, which originates in dipole-dipole interactions, the magnetization of each
ellipse can point in any direction of the easy axis; red arrows represent magnetic moments which point to the right, whereas
blue arrows represent magnetic moments which point to the left. Figures (a) to (d) represent initial configurations which were
used to obtain the magnetic state of the dipolarly coupled system by integrating the LLG equation with zero magnetic field.
The reason for using a field pulse with a Gaussian profile is due to the experimental impossibility
of the rise and fall times of the pulse being zero.28In the simulations, we used the pulse duration of
60 ps, i.e., the full width at half maximum WB=FWHM =(2p
2 ln 2 )0.05887 ns. The strength
of the magnetic field pulse, Bext, were varied during the investigations.
III. RESULTS AND DISCUSSION
Initially, we studied interparticle dipole-dipole interactions in arrays of single-domain nano-
ellipses. We have varied not only the aspect ratio and the spacing between ellipses but also the
configuration of magnetization and the array grid (rectangular or square). We have obtained the
equilibrium magnetic configurations for each array when integrating the LLG equation with zero
external magnetic field. For some magnetization configurations (case 1, for example), after the system
reaches the equilibrium magnetic state, we can observe that the interparticle dipolar coupling is
responsible by the shift of magnetization vector from the easy axis in the central ellipse as shown in
Fig. 3. It was noted 0 for magnetization configurations which present some kind of symmetry, for
example, cases 3 and 4. In this cases, the magnetization vectors got stuck in their equilibrium position
(easy axis). Since does not appear in all magnetization configurations and has an appreciable
value only for bxsufficiently small it is not a good parameter to quantify the dipolar coupling
strength.
The present study also investigated the precessional magnetization reversal in arrays of dipolarly
coupled nano-ellipses. Before all we have considered an isolated single-domain nano-ellipse and
determined the minimum field strength to switch its magnetization, these values are shown in the
Table I.
There is another consequence of the interparticle dipole-dipole interactions, that is, the distribu-
tion of magnetization can assist or hinder the magnetization reversal. In other words, the magnetic
field required to switch the magnetization of the central ellipse depends on the distribution of mag-
netization of the neighboring ellipses for samples that they are sufficiently close. It is evident that in
a real situation it would not be practical to check the magnetization of the ellipses near the ellipse in
which it is desired to cause the reversion and then we choose an ideal field for this. This choice of
field, for this absurd case, must be done with great care because if the applied field is too small the
reversal does not occur, and if the field is too large, two or more reversals may occur. For a better
technological acceptance, we must ensure that the reversal happens, be unique and the field applied095017-5 Gomes et al. AIP Advances 8, 095017 (2018)
FIG. 3. The snapshot shows the equilibrium magnetic state of a strongly interacting magnetic system. This equilibrium
configuration was obtained starting from the initial configuration of the case 1, using ellipse of dimensions 70 505 nm3
arranged in a rectangular array with bx= 10 nm. Interparticle dipole-dipole interactions are strong enough to reduce locally
the shape anisotropy. Evidently the magnetization vector of the central ellipse makes an angle of with the magnetization
easy axis.
be the as small as possible. After all, since this ellipses array has as an application the data storage
and the direction of the magnetic moments is used as the information bit, the 4 cases studied for each
ellipses configuration will appear as the magnetization reversals happen. In practical applications it is
desirable to know the minimum spacing in which the ellipses are uncoupled. Thus, the stability of the
magnetization state is not compromise and the same magnetic field strength can be used to switch the
magnetization of any magnetic state. In order to switch the magnetization of the central ellipse and
also to determine the minimum spacing in which the ellipses are uncoupled, all arrays were submitted
to a single magnetic field pulse, so that the minimum field to reverse the magnetization of the central
ellipse in the array coincides with the minimum field to reverse the magnetization of a single isolated
ellipse shown in Table I. Figure 4 shows the magnetization controllability diagram of the central
ellipse in arrays of dipolary coupled single-domains. From these diagrams it is possible to know
the minimum spacing such that the ellipses are uncoupled. For example, considering ellipses of size
70505 nm3arranged into a rectangular grid, the minimum spacing in which all the magnetization
distributions are uncoupled is bmin
x=210 nm. If the same single-domain nano-ellipses were arranged
into a square grid, the minimum spacing is bmin
x=230 nm. On the other hand, considering ellipses
of size 110605 nm3arranged into a rectangular grid, the minimum spacing in which all the
magnetization distributions are uncoupled is bmin
x=350 nm. If the same single-domain nano-ellipses
were arranged into a square grid, the minimum spacing is bmin
x=220 nm. We can realize that ellipses
which present the largest aspect ratio arranged into a rectangular lattice remain strongly coupled for
larger values of bx. This is due to the fact that ellipses are more compacted in a rectangular grid than
in a square grid, considering the same value of bx. The discussions in the Figure 4 were based on the
horizontal separation bx. In the case of square distribution it is enough, but in rectangular distribution
the vertical separation is relevant and can affect qualitatively the conclusions for bmin
x. Thus, if in a
rectangular array in which the ellipses magnetization configuration neighboring the central ellipse
TABLE I. Table containing the minimum field to switch the magnetization of isolated ellipses.
Dimensions (nm3) Bext
min.(mT)
70505 31
705010 48
705015 61
705020 72
110605 43
1106010 67
1106015 85
1106020 100095017-6 Gomes et al. AIP Advances 8, 095017 (2018)
FIG. 4. Magnetization controllability diagram of the central ellipse in arrays of dipolary coupled single-domains. Black circles
represent situations in which the arrays are uncoupled, thus a single switch occurs. Red squares represent situations in which
the arrays are strongly coupled and the distribution of magnetization hinder the magnetization reversal, thus the magnetization
dynamics is accomplished either without switching. Blue triangles represent situations in which the arrays are strongly coupled
and the distribution of magnetization assist the magnetization reversal. Array containing 9 ellipses of dimensions: a) 70 50
5 nm3arranged into a rectangular lattice, b) 70 505 nm3arranged into a square lattice, c) 110 605 nm3arranged
into a rectangular lattice, d) 110 605 nm3arranged into a square lattice.
prevents the switching, this effect will be more evident because they are more compact than in a
square matrix. Being more compact, the dipole interaction is stronger. As mentioned in the previous
section, the vertical separation in a rectangular distribution was varied assuming the same separation
in both axes ( by=bx); thus, cy=cx (Lx Ly).
IV. CONCLUSION
In this paper, we have performed micromagnetic simulations to investigate the dipolar coupling
between Permalloy single-domain nano-ellipses arranged on rectangular and square lattices. Besides,
considering ellipses with different aspect ratios, we have explored not only the interparticle separation
but also the magnetization distribution on the lattice points. Starting from Ising-like magnetization
states, we obtained the equilibrium magnetic configurations for arrays of interacting particles. The
equilibrium configurations obtained in this way were saved and used as initial configurations in other
simulations, where a single magnetic field pulse was applied to study the precessional magnetization
switching of the central ellipse. The main goal in this paper is to estimate the minimum spacing
between particles in which single-domain ellipses becomes a non-interacting magnetic system. The
minimum separations observed are the order of 3 Lxapproximately, and they strongly depend on the
ellipse dimensions, ellipse aspect ratio, and the array arrangement. We observed that our results agree
qualitatively with the behavior of the experimental results of the references 33 and 34. We would
like to emphasize that ellipses with larger aspect ratios are easily decoupled when they are arranged
in a square grid rather than in a rectangular grid. From the technological point of view, interparticle
dipole-dipole interactions in an array of identical single-domain nano-ellipses impose a restriction
on how far the miniaturization of spintronic devices can reach. Although, we have studied a finite095017-7 Gomes et al. AIP Advances 8, 095017 (2018)
array of a few elliptical elements, the chosen arrangement is the basis of many potential applications,
where the ellipses are considered as non-interacting.
ACKNOWLEDGMENTS
This work was partially supported by CAPES, CNPq, FAPEMIG and FINEP (Brazilian Agen-
cies). Numerical works were done at the Laborat ´orio de Simulac ¸ ˜ao Computacional do Departamento
de F´ısica da UFJF. We greatfully thank to our friend Saif Ullah for making the English corrections
in this paper.
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1.3182351.pdf | Origin of the spatial resolution in atom probe microscopy
Baptiste Gault, Michael P. Moody, Frederic de Geuser, Daniel Haley, Leigh T. Stephenson et al.
Citation: Appl. Phys. Lett. 95, 034103 (2009); doi: 10.1063/1.3182351
View online: http://dx.doi.org/10.1063/1.3182351
View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v95/i3
Published by the American Institute of Physics.
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Downloaded 12 Mar 2013 to 131.211.208.19. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsOrigin of the spatial resolution in atom probe microscopy
Baptiste Gault,1,2, a/H20850Michael P . Moody,1Frederic de Geuser,3Daniel Haley,1
Leigh T . Stephenson,1and Simon P . Ringer1
1Australian Key Centre for Microscopy and Microanalysis, The University of Sydney,
New South Wales 2006, Australia
2Department of Materials, University of Oxford, Parks Road, Oxford OX13PH, United Kingdom
3SIMaP , Grenoble INP , CNRS, UJF, 1130 rue de la Piscine - BP 75 - F-38402, Saint Martin d’Heres
Cedex, France
/H20849Received 5 May 2009; accepted 25 June 2009; published online 21 July 2009 /H20850
Atom-probe microscopy offers unprecedented insights on the subnanometer structure and chemistry
of materials in three dimensions. The actual spatial resolution achievable is however still anuncertain parameter, as no comprehensive study has been undertaken to unveil the physicsunderpinning how key parameters impact the performance. Here, we present a comprehensiveinvestigation of the in-depth and lateral resolution of the technique. We discuss methods to estimatethe resolution and show a resolution better than 20 pm in-depth. Models to support our results weredeveloped and are discussed in the present letter. © 2009 American Institute of Physics .
/H20851DOI: 10.1063/1.3182351 /H20852
The extreme complexity of the design of modern mate-
rials makes their characterization challenging owing to theneed to know the structure and chemistry of the material inthree dimensions at the atomic scale.
1Few techniques are
presently capable of making atomic-level information rou-tinely accessible.
1Transmission electron microscopy may
eventually meet the required criteria in the future, throughcontinued improvements in spatial resolution and analyticalcapabilities,
2–5and the capacity for tomographic analyses.6
Atom probe microscopy /H20849APM /H20850is a strong alternative can-
didate. APM maps the distribution of chemically identifiedatoms from bulk material in three dimensions and with near-atomic resolution.
7Owing to recent breakthroughs in instru-
ment design,8,9APM has become increasingly prominent in
the atomic-scale analysis of a range of structural and func-tional materials.
10–12Vurpillot et al.13defined the spatial res-
olution as the width of the damping function that limits theintensity of the peaks in the reciprocal space, whereas Kellyet al.
14proposed a way to access the limit of information15
based on the Fourier image analysis of a thin slice of a spa-tial distribution map /H20849SDM /H20850. The SDM is a data treatment
technique that reveals the average local neighborhood of theatoms, in a manner similar to a three-dimensional autocorre-lation function.
16,17Despite these previous efforts, a lack of
detailed understanding concerning the limits of APM’s in-trinsic spatial resolution is a conspicuous shortcoming of thisburgeoning technique. Here we show a limit of resolution inAPM images better than 20 pm and in doing so, we presenta treatment for the origins of resolution in atom probe byinvestigating the effect of key experimental parameters onspatial resolution.
APM spatial resolution is known to be anisotropic. This
is not surprising for a truly three-dimensional technique.Nonetheless this is an area which needs greater understand-ing. Resolution in the direction of the analysis /H20849z-dimension /H20850
is very high and individual atomic planes can be resolveddirectly.
18The origins of resolution in the z-dimension areinextricably linked to the field-evaporation process via the
reconstruction procedure. As the in-depth coordinates aremodified via sequential increments,
19a change in the evapo-
ration sequence must affect resolution in this dimension. Dueto a combination of trajectory aberrations during the earlystages of the ionic flight
20–22and momentum transfer arising
from thermal vibration at the surface, the lateral /H20849x-y/H20850reso-
lution is limited and structures in the plane perpendicular to
the direction of analysis appear blurred.18Let us consider the
separate cases of resolution in the z- and x-ydimensions.
Using a definition equivalent to that proposed by Vurpillot et
al.,13we propose that the width of a peak in the SDM reflects
the local spatial resolution in the respective real space direc-tions.
In the z-dimension, we first isolate the central peak of
thez-SDM and fit a Gaussian function /H20851Fig. 1/H20849a/H20850, top left
inset /H20852. The resolution is then defined as
/H9254=2/H9268, where /H9268is the
standard deviation of the fitted Gaussian function. By pre-cisely adjusting the direction of the z-SDM analysis relative
to the crystallographic planes /H20851Fig. 1/H20849a/H20850, lower right inset /H20852,
this procedure was repeated and applied to characterize sev-eral atomic plane families identified within a pure aluminum
a/H20850Author to whom correspondence should be addressed. Electronic mail:
baptiste.gault@materials.ox.ac.uk. Tel.: /H1100144 1865 273711.
FIG. 1. /H20849Color online /H20850Plot of the resolution as a function of the interspacing
dhklalong a given crystallographic direction hklfor several atomic plane
families /H20849/H20853002 /H20854,/H20853202 /H20854,/H20853113 /H20854,/H20853204 /H20854,/H20853115 /H20854, and /H20853206 /H20854/H20850. Gaussian
fitting procedure used to estimate the resolution /H20849top inset /H20850and procedure to
get the resolution for several atomic plane family /H20849bottom inset /H20850./H20849b/H20850Plot of
/H9004for the 002 planes as a function of the temperature.APPLIED PHYSICS LETTERS 95, 034103 /H208492009 /H20850
0003-6951/2009/95 /H208493/H20850/034103/3/$25.00 © 2009 American Institute of Physics 95, 034103-1
Downloaded 12 Mar 2013 to 131.211.208.19. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsspecimen prepared by electrochemical polishing with per-
chloric acid, and analyzed using an Imago LEAP 3000X Si at40 K, 0.025 ions per pulse in average and 25% pulse fractionusing voltage pulsing. Values for in-depth resolution as lowas 12 pm along the /H20853201 0 /H20854planes are reached. Interestingly,
a linear relation is revealed when
/H9254is plotted against the
relative lattice spacing, as shown in Fig. 1/H20849a/H20850.
How can this previously undocumented trend be ex-
plained? The specimen surface is formed of successive ter-races due to the intersection between the specimen tip shape/H20849a quasihemispherical cap /H20850and the crystalline lattice. The
atoms located at the edges of these terraces are less stronglybonded to the surface due to a lower number of neighbors.
23
They are thus the most likely to be field evaporated. Forwider terraces, nonkink-site atoms are, however, increasinglyprone to field evaporation.
23,24Early work by Drechsler
et al.25also demonstrated that the size of a terrace is propor-
tional to the lattice spacing of the particular crystallographicdirection and inversely proportional to the radius of curva-ture. As stated, the z-coordinate is assigned by successive
increments, combined with a position-dependant zoffset ac-
counting for the specimen’s curvature.
19Atomic planes ex-
hibiting larger terraces should statistically present a largeruncertainty in the order in which atoms are field evaporated,leading to corruption of the atomic planes during reconstruc-tion. Furthermore, increased temperature makes the statisti-cal process of field evaporation less dependant upon the ab-solute value of the binding energy of surface atoms.Temperature should thus worsen the resolution via increaseduncertainty in the sequence of evaporation. An expression forthe resolution that accounts for both effects can be derived:
/H9254=/H92540+dhklexp /H20849−/H9004E/kBT/H20850, where /H92540is defined as the mini-
mum resolution and accounts for systematic errors, kBis the
Boltzman constant and /H9004Eis the average binding energy at
which atoms are evaporated. Let us also define /H9004as a dimen-
sionless resolution: /H9004=/H9254/dhkl=/H92540/dhkl+exp /H20849−/H9004E/kBT/H20850.
The resolution of the /H20853002 /H20854planes was estimated across
a series of experiments on the same specimen at varioustemperatures, and plotted against temperature in Fig. 1/H20849b/H20850.
Unexpectedly, the resolution remains almost constant at tem-peratures below 80 K, indicating that systematic errors pre-dominate over detrimental effects induced by increasing tem-perature. Temperature-assisted field evaporation of nonkink-site atoms will result in progressive corruption of the atomic
planes, as the order in which atoms reach the detector be-comes increasingly less certain. To better understand the ef-fect of specimen temperature on the evaporation sequence, itis necessary to estimate the number of atoms likely to befield evaporated at a given temperature and how this numberchanges with temperature.
A face-centered cubic lattice was modeled and subse-
quently reduced into a tip shape, providing a theoreticalthree-dimensional volume that emulates actual experiments.The nearest neighbors directly surrounding each surfaceatom, up to the third, were identified and the energy binding
each atom to the surface was estimated using Lennard–Jonespotentials for pure aluminum, as described in Fig. 2/H20849a/H20850. The
discretised distribution of this binding energy is charted inFig. 2/H20849b/H20850. Atoms on different crystallographic facets, as
shown in the inset, have different average binding energies,which is consistent with previous experiments.
26,27Based on
existing models of field evaporation,23,26the number of sur-
face atoms, Nat, with a high enough probability to be field
evaporated, was estimated as a function of temperature,shown in the inset in Fig. 2/H20849c/H20850.
Concomitantly, the in-depth resolution was estimated in
a series of reconstructions in which the experimental se-quence of detected atoms was randomized within successivewindows of N
randatoms. As depicted in Fig. 2/H20849c/H20850, as the size
ofNrand, increases, so does the overall randomness of the
detection sequence, yet the resolution remains nearly con-stant below randomized intervals of 10 000 ions. However,the resolution deteriorates quickly for randomized intervalsof 25 000 ions and above. This value approximately corre-sponds to the total number of atoms on a single surface layerfor the reconstruction. Thus in practice, N
atis smaller than
the number of atoms on a complete layer even at tempera-tures customarily considered too high for atom-probe experi-ments. This is the origin of APM’s remarkably high reso-lution in the z-dimension: atoms are removed nearly atomic
layer by atomic layer even at relatively high temperature,inducing a high degree of order within the z-coordinate of
the reconstructed atoms.
Finally, advanced xy-SDMs have been used to reveal the
average two-dimensional atomic neighborhood within aplane perpendicular to a particular direction. A radial-
FIG. 2. /H20849Color online /H20850/H20849a/H20850Generation of an ideal tip apex /H208491/H20850. First a volume is defined and filled up by atoms arranged in a face-centered cubic lattice /H208492/H20850.
Then a hemispherical cap is cut out with a radius of curvature equivalent to the one of the experiment presented in Fig. 1. Finally, for each atom on the surface,
the distance to its first, second, and third nearest neighbors are calculated and the binding energy is computed /H208493/H20850. The color of the atoms relate to their binding
energy as revealed by the histogram in /H20849b/H20850./H20849c/H20850Resolution as a function of Nrand, with the different temperatures studied marked. The dotted line corresponds
to the Rayleigh criterion for the /H20853002 /H20854planes. /H20849Inset /H20850Number of atoms contributing to the detection rate against temperature.034103-2 Gault et al. Appl. Phys. Lett. 95, 034103 /H208492009 /H20850
Downloaded 12 Mar 2013 to 131.211.208.19. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsdistribution function /H20849RDF /H20850was calculated from the
xy-SDM, showing peaks at distances characteristic of the
local lateral atomic distribution /H20851Fig. 3/H20849a/H20850/H20852. The first peak of
the RDF corresponds to the interaction between an atom andits first shell of nearest neighbors. The width of this peak isthe direct representation of the lateral x-yresolution, /H9261,
where /H9261=2
/H9268of the fitted Gaussian function. We have esti-
mated /H9261across several crystallographic directions and it is
plotted against dhklin Fig. 3/H20849b/H20850. A trend of increasing reso-
lution /H20849/H9261decreasing /H20850with increasing lattice plane spacing
emerges. Areas of less densely packed atoms on the speci-men surface seem to exhibit inferior lateral resolution. Theimpact of temperature on the lateral x-yresolution of the 002
planes is revealed in Fig. 3/H20849c/H20850. As expected, the resolution
improves with decreasing temperature. The lateral resolutionseems to plateau at low temperature as contributions fromtrajectory aberrations are likely to predominate.
16,18The
change in the thermal agitation from 20 to 80 K is not suf-ficient to strongly affect the lateral resolution. Above 140 K,however, the structure of the xy-SDM was lost, a likely in-
dication of surface migration prior to evaporation.
16
Further enhancing the resolution will require improve-
ment of the general understanding of the nature of the trajec-tory aberrations, which appear to be the limiting factor ofboth the lateral and in-depth resolution. We have shown herethat crystallography and temperature strongly influences thepositioning performance of APM. Continued research willsee these methods and models progressively evolve, to cor-rect for the detrimental effects of these aberrations, and togive insights on the evolution of the spatial resolution in thecase of multicomponent materials and nonmetallic materials.Understanding the origins of resolution is the ultimate path-way to improve the performance of APM, which representone of the most promising atomic-scale microscopy and mi-croanalysis techniques available today.
The authors would like to thank Dr. Ross Marceau, Dr.
Tim Petersen, and Dr. Kyle Ratinac, as well as Mr. Alex LaFontaine, for fruitful discussions. The authors also thank Dr.F. Vurpillot and G. da Costa for provision of the Fouriertransform calculation software. The authors are grateful forfunding support from the Australian Research Council,which partly sponsored this work. The authors are gratefulfor scientific and technical input and support from the Aus-tralian Microscopy and Microanalysis Research Facility
/H20849AMMRF /H20850at The University of Sydney.
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P. Tiemeijer, M. van der Stam, S. von Harrach, M. Stekelenburg, M.Haider, S. Uhlemann, H. Müller, P. Hartel, B. Kabius, D. Miller, I. Petrov,E. A. Olson, T. Donchev, E. A. Kenik, A. R. Lupini, J. Bentley, S. J.Pennycook, I. M. Anderson, A. M. Minor, A. K. Schmid, T. Duden, V .Radmilovic, Q. M. Ramasse, M. Watanabe, R. Erni, E. A. Stach, P. Denes,and U. Dahmen, Microsc. Microanal. 14, 469 /H208492008 /H20850.
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Nature /H20849London /H20850363, 432 /H208491993 /H20850.
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FIG. 3. /H20849Color online /H20850/H20849a/H20850RDF calculated from the xy-SDM and the Gaussian function fitted to the first peak for four different crystallographic directions. /H20849b/H20850
Corresponding lateral resolution measured from the fit for several atomic plane families, /H20849c/H20850and several temperatures.034103-3 Gault et al. Appl. Phys. Lett. 95, 034103 /H208492009 /H20850
Downloaded 12 Mar 2013 to 131.211.208.19. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions |
1.1667796.pdf | Dynamic anisotropy in amorphous CoZrTa films
Andreas Neudert, Jeffrey McCord, Rudolf Schäfer, and Ludwig Schultz
Citation: Journal of Applied Physics 95, 6595 (2004); doi: 10.1063/1.1667796
View online: http://dx.doi.org/10.1063/1.1667796
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/95/11?ver=pdfcov
Published by the AIP Publishing
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130.102.42.98 On: Sun, 12 Oct 2014 12:52:11Dynamic anisotropy in amorphous CoZrTa films
Andreas Neudert,a)Jeffrey McCord,b)Rudolf Scha ¨fer, and Ludwig Schultz
Leibniz Institute for Solid State and Materials Research IFW Dresden, Helmholtzstrasse 20,
D-01069 Dresden, Germany
~Presented on 6 January 2004 !
The high-frequency response of amorphous CoZrTa thin films was measured by using a pulsed
inductive microwave magnetometer. The anisotropy of the magnetic films was varied by magneticfield annealing. Static anisotropy field values ranging from H
k5100 to 1920 A/m were obtained.
The dynamically determined anisotropy field is shifted to higher values compared to the staticanisotropy by an additional isotropic internal field H
add. This internal field is independent of the
strength of the static anisotropy field. We determined a value of about Hadd5510 A/m. © 2004
American Institute of Physics. @DOI: 10.1063/1.1667796 #
I. INTRODUCTION
The magnetization dynamics of soft magnetic films is
increasingly interesting for applications in magnetic devicesas the operating speeds approach the giga hertz regime. Twomain parameters determine the high-frequency ~hf!response
of magnetic films: the ferromagnetic resonance frequency
f
resand the magnetic damping parameter a. The ferromag-
netic resonance frequency1
fres5gm0
2pAMs~Hbias1Hk! ~1!
(g51.7631011T21s21) can be shifted to higher frequen-
cies by increasing the saturation magnetization Msor the
anisotropy field Hk. The major role falls to the anisotropy, as
it can be adjusted by orders of magnitude depending on ma-terial and process conditions. The resonance frequency alsocan be increased by an external magnetic field H
biasapplied
parallel to the easy axis.
Recent investigations on sputtered permalloy films
showed a significant difference between the statically anddynamically measured anisotropy fields.
2This can phenom-
enologically be described by introducing an additional fieldH
addin the Kittel equation ~1!, which is independent of the
direction of Hbiaswith respect to the easy axis:
fres,i5gm0
2pAMs~Hbias1Hk1Hadd!, ~2!
fres,’5gm0
2pAMs~Hbias2Hk1Hadd!. ~3!
The index at fdescribes the orientation of the external field
with respect to the easy axis. Equation ~3!is only valid for
Hbias.Hk. For amorphous CoZrTa thin films, similar results
have been reported.3In this article, we investigate the influ-
ence of the anisotropy strength on the additional field Hadd
for amorphous CoZrTa films.II. EXPERIMENTAL PROCEDURE
Extended amorphous CoZrTa films were deposited by rf
sputtering in argon atmosphere (1022mbar) onto circular
glass substrates with a diameter of 18 mm. The sputter targetconsists of 91.7 at.% Co, 2.2 at.% Zr, and 6.1 at.% Ta.During sputtering, an external magnetic field was applied toinduce a uniaxial magnetic anisotropy. The sputtered CoZrTafilms with thicknesses of t580, 150, and 300 nm possess a
saturation polarization of J
s51.35 T. Due to the amorphous
state, the electrical resistivity of these samples is relativelylarge ~about 110
mVcm!. This value corresponds to a skin
depth of about 500 nm for a magnetic ac field of 2 GHz, wellbeyond the maximum film thickness for this investigation.
Hysteresis loops were measured quasistatically using an
induction-field magnetometer at an operating frequency of10 Hz. To determine the static anisotropy field H
k,stat, the
anisotropy energy between easy and hard axis
Hk,stat52E
0Hup
@mea~H!2mha~H!#dH ~4!
was calculated from the measured easy axis ( mea) and hard
axis (mha) loops of the reduced magnetization ( m
5M/Ms). The upper integration boundary Hupwas chosen
higher than the anisotropy field Hk,stat.
The dynamic properties were investigated using a pulsed
inductive microwave magnetometer as described in Ref. 4.Avoltage pulse of 35 V is guided into a coplanar waveguidewith a width of the center conductor of 0.5 mm. The strengthof the pulsed magnetic field acting on the film is around 320A/m. The rise-time is about 75 ps and the first 5 ns of the 20ns pulse are detected by a 20 GHz sampling oscilloscope.The precessional motion of the magnetization induces a volt-age in the coplanar waveguide, which is extracted from themeasured signal by subtracting the pulse background, mea-sured with the saturated sample. To saturate the sample anexternal field of 4000A/m was applied in the direction of thepulse field. The Fourier transform of the induced voltage isproportional to the complex susceptibility
x~v!.5The ferro-
magnetic resonance frequency was derived from the zero-crossing of the real-part of the complex susceptibility
6as aa!Electronic mail: a.neudert@ifw-dresden.de
b!Electronic mail: j.mccord@ifw-dresden.deJOURNAL OF APPLIED PHYSICS VOLUME 95, NUMBER 11 1 JUNE 2004
6595 0021-8979/2004/95(11)/6595/3/$22.00 © 2004 American Institute of Physics
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130.102.42.98 On: Sun, 12 Oct 2014 12:52:11function of an external dc-field Hbias. By rotation of the
sample, the external field was applied parallel or perpendicu-lar to the easy axis of the magnetic film.
To vary the anisotropy of the films, the samples were
annealed in a magnetic field of 800 kA/m at temperaturesranging from 150 to 300°C. The direction of the externalfield was applied with different orientations to the alreadypresent easy axis of magnetization, thus resulting eventuallyin a switch of orientation and strength of anisotropy in thecase of a nonparallel alignment between external field andeasy axis.
7After each annealing step, the static and dynamic
properties of the samples were measured.
In Fig. 1, hysteresis loops of a CoZrTa film in the as-
prepared state are shown. In the as-deposited state values forH
k,statbetween 1520 and 1920 A/m for all thicknesses were
observed. After annealing, the anisotropy field was in therange of 100–1920 A/m. The effect of the decreased anisot-ropy on the dynamic response can be seen in Fig. 2, whichdisplays the induced voltages for the same film but afterdifferent annealing treatments. An increased amplitude withsmaller resonance frequency for the low anisotropy film be-comes visible. The effective damping parameter
awas de-
termined by fitting of the induced voltages with a numericalLandau–Lifshitz–Gilbert simulation. The value of the damp-ing increases slightly with increasing thickness ~from 0.012to 0.016 !. However, annealing did not alter the value of the
damping parameter. No dependency between
aand anisot-
ropy was found.
In Fig. 3, the square of the measured frequencies fresfor
Hbiasparallel and perpendicular to the easy axis of an 80-nm-
thick film are shown. Without an additional isotropic field
Hadd@see Eqs. ~2!and~3!#, the linear extrapolations of fres,i2
andfres,’2should cross the Hbiasaxis at the same values. This
is obviously not the case. The zero-crossing of the linearextrapolations are at different values for the measurementwithH
biasparallel and perpendicular to the easy axis ~similar
results for Ni 81Fe19are reported in Ref. 2 !. The anisotropy
field value Hk,Dfwas derived from the difference between
fres,’2andfres,i2~see Fig. 3 for illustration !. Note that this
differentially obtained value by definition leads to the anisot-
ropy field Hk. The value of Hk,iderived from fres,i2@Eq.~2!#
does not yield the anisotropy. Instead,
Hk,i5Hadd1Hk,Df ~5!
is obtained, including the additional isotropic field value
Hadd. Complementary measurements with a calibrated
hf-permeameter8confirm the results ~see Fig. 4 !. The real
FIG. 1. Easy ~ea!and hard axis ~ha!loop of a 300-nm-thick CoZrTa film in
the as-deposited state. The anisotropy field Hk,statand coercivity field values
Hcare indicated.
FIG. 2. Dynamic response of a 300-nm-thick film in the as-deposited state~dashed line !and after annealing at 225°C fo r2hi nafi e l do f8 0 0k A / m
applied perpendicular to the easy axis ~solid line !, resulting in different
values of H
k,statandfresas indicated. The indicated values for fresis the
measured value as described. The displayed response was measured without
external field Hbias.
FIG. 3. Square of the resonance frequency for Hbiasparallel ~i!and perpen-
dicular ~’!to the easy axis. The value of Hbiasat the zero-crossings of fres,i2
is named Hk,i, and the value of the anisotropy field extracted from the
horizontal displacement is called Hk,Df. The sample was an 80-nm-thick
film after two annealing treatments ~225 and 250°C) with the external field
parallel to the existing easy axis during annealing. The statically measured
anisotropy field Hk,statwas 1690 A/m, as compared to Hk,i51980 A/m and
Hk,Df51710 A/m.
FIG. 4. Frequency-dependent permeability measurement of a 300 nm
CoZrTa film. Real part m8and imaginary part m9of the frequency spectra
are shown. The low-frequency part of m8corresponds to Hk52280 A/m
compared to Hk,i52130 A/m and Hk,Df51480 A/m.6596 J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004 Neudert et al.
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130.102.42.98 On: Sun, 12 Oct 2014 12:52:11part of the low-frequency permeability m8and the resonance
frequency fresare in agreement with an increased Hk,i, sig-
nificantly larger than Hk,stat.
In Fig. 5, Hk,i,Hk,Df, andHk,statfor one 300 nm film
after different annealing steps are plotted.The statically mea-suredH
k,statis comparable to Hk,Df.Hk,iis significantly
larger and relatively independent of the anisotropy values.The additional offset field H
addis plotted in the bottom fig-
ure. The variation of Hk,iwithHk,Dfis examined in Fig. 6.
Hk,ias function of Hk,Dffor various films after different
annealing conditions is shown. The straight line representsthe linear equation ~5!. By extrapolating to H
k,Df50, ananisotropy-independent value of Hadd5510 A/m for the 300
nm film is found. For the films with 80 and 150 nm thick-ness, the value of H
addis smaller.
III. DISCUSSION
As shown in Sec. II, the dynamically determined anisot-
ropy field Hk,Dfis in good agreement with the statically de-
termined Hk,stat. The value obtained by extrapolating the
fres,i2plot toHbias50 A/m includes an additional isotropic
fieldHadd, which is independent of the anisotropy field Hk.
This additional field is also included in the measurement ofthe frequency-dependent susceptibility. In Ref. 9, a termsimilar to H
addwas introduced into the Kittel equation and
was explained by ripple effects.10Similar results on amor-
phous Co-based films with the same conclusions were ob-tained in Ref. 11. There, the anisotropy was determined fromthe dependency of the transverse initial susceptibility on theexternal biasing field. The inverse of the susceptibility de-pends on the anisotropy field in a similar way as the squareof the resonance frequency plotted in Fig. 3. High-frequencymeasurements comparing both measurement principles arediscussed in Ref. 3. According to Ref. 12, the effect of theasymmetrical crossing is caused mainly by a term that cor-responds to Hoffmann’s ripple theory,
10including an addi-
tional effective field.The existence of magnetization ripple isdue to the isotropic distribution of the local randomanisotropies,
11not related to the induced anisotropy. This ad-
ditional effective field exists in both static and dynamic mea-surements. Thus, the additional field might not be caused bythe dynamic measurement principle.
ACKNOWLEDGMENTS
The authors thank J. Paul for sample preparation, I. Ki-
witz for the annealing, H. Vinzelberg for the resistivity mea-surements, M. Frommberger and M. Thewes for the perme-ability measurement, and the DFG Schwerpunktprogramm1133 ‘‘Ultrafast Magnetization Processes’’ for financial sup-port.
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FIG. 5. Measured values of Hkfor different annealing conditions for a
300-nm-thick sample. The orientation of the external field during annealingand the temperature are shown in the upper part of the plot.
FIG. 6.Hk,ias a function of Hk,Df. The field Hk,iis the sum of Hk,Dfand
Hadd@Eq.~5!#. The extrapolation to Hk,Df50 yieldsHadd5510 A/m for the
samples with the 300 nm film.6597 J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004 Neudert et al.
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130.102.42.98 On: Sun, 12 Oct 2014 12:52:11 |
1.4921270.pdf | Axial vibrations of brass wind instrument bells and their
acoustical influence: Theory and simulations
Wilfried Kausel and Vasileios Chatziioannou
Institute of Music Acoustics (Wiener Klangstil), University of Music and Performing Arts, Vienna, Austria
Thomas R. Moore,a)Britta R. Gorman, and Michelle Rokni
Department of Physics, Rollins College, Winter Park, Florida 32789, USA
(Received 6 October 2014; revised 26 April 2015; accepted 28 April 2015)
Previous work has demonstrated that structural vibrations of brass wind instruments can audibly
affect the radiated sound. Furthermore, these broadband effects are not explainable by assumingperfect coincidence of the frequency of elliptical structural modes with air column resonances. In
this work a mechanism is proposed that has the potential to explain the broadband influences of
structural vibrations on acoustical characteristics such as input impedance, transfer function, andradiated sound. The proposed mechanism involves the coupling of axial bell vibrations to the inter-
nal air column. The acoustical effects of such axial bell vibrations have been studied by extending
an existing transmission line model to include the effects of a parasitic flow into vibrating walls, aswell as distributed sound pressure sources due to periodic volume fluctuations in a duct with oscil-
lating boundaries. The magnitude of these influences in typical trumpet bells, as well as in a com-
plete instrument with an unbraced loop, has been studied theoretically. The model results inpredictions of input impedance and acoustical transfer function differences that are approximately
1 dB for straight instruments and significantly higher when coiled tubes are involved or when very
thin brass is used.
VC2015 Acoustical Society of America .[http://dx.doi.org/10.1121/1.4921270 ]
[JW] Pages: 3149–3162
I. INTRODUCTION
Most makers and players of brass wind instruments are
convinced that wall material, wall thickness, and the posi-tions of the bends and braces can affect both the sensation
the player experiences and the sound the instrument pro-
duces when played. Because it is not obvious how theseaspects of the instrument could affect the sound, there hasbeen an ongoing debate concerning the validity of theclaims. An extensive review of the history of this debate haspreviously been presented by Kausel et al.
1
The results of experiments performed over the past dec-
ade have provided strong evidence that structural vibrationsdo indeed influence the radiated sound of certain brass windinstruments. Specifically, experiments on trumpets haveyielded results that clearly indicate effects attributable to
vibrations of the bell.
1,2Although it is now generally
accepted that structural vibrations can affect the sound pro-duced by a brass wind instrument, to our knowledge notheory has yet been presented that can qualitatively explainand quantitatively predict the effect. However, there doesseem to be a common understanding concerning which
mechanisms do not contribute to the observed effects.
The vibrational modes of brass wind instrument bells
that have shapes with radial nodes, referred to here as ellipti-
cal modes , as well as similar modes that are present in the
cylindrical tubes of woodwinds and organ pipes, have been
studied by Nief et al.
3–5Elliptical modes are easily stimu-
lated mechanically and during performance they can bestimulated by the vibration of the lips or by the vibrations of
the air column. In either case the displacement of the metalat the antinodes of these modes can be significant. However,
it can be assumed that elliptical modes are not the source of
the observed timbre differences that become apparent in thesound produced by the instrument when wall vibrations aredamped.
4
The resonances associated with elliptical modes have
quality factors typically exceeding 102and therefore their
effect is limited to a narrow band of frequencies, which isnot consistent with the broad-band effects observed in sev-eral recent experiments.
1Also, elliptical modes do not radi-
ate efficiently due to acoustic short-circuiting and therefore
the effects attributable to direct radiation are at least two
orders of magnitude below those of the air column in straighttubes.
6Similar results have been shown for these mode
shapes occurring in the flaring section of trombone bells.7
Finally, the area of an elliptical cross-section with consider-able amplitude is very close to that of a perfect circle, mak-
ing periodic variations of the characteristic impedance a
second-order effect at best.
1
Bending modes can also be observed in musical instru-
ments and have been investigated by Whitehouse.8
However, these modes can be ruled out as an explanation fortimbre differences for the same reasons. The exception isthat in coiled instruments they can lead to significant longitu-dinal bell displacements, as will be discussed later.
Mouthpiece vibrations and their interaction with the
player’s oscillating lips have been proposed as an explana-
tion for timbre differences caused by structural vibrationsobserved during experiments with both artificial lips and
a)Electronic mail: tmoore@rollins.edu
J. Acoust. Soc. Am. 137(6), June 2015 VC2015 Acoustical Society of America 3149 0001-4966/2015/137(6)/3149/14/$30.00
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.173.72.87 On: Wed, 24 Jun 2015 20:38:32with real players.2,9Mechanical feedback of this nature may
indeed have an effect on the sound, and while it is not dis-cussed in this work it can be studied using the theoreticalframework presented here. However, even without the pres-ence of oscillating lips, consistent differences between theinput impedance and acoustic transfer function (ATF) meas-ured with and without damping of the bell of a trumpet havebeen observed using excitation by a loudspeaker.
1Therefore,
while it is likely that mechanical feedback to the lips of theplayer cannot be ignored, there are significant acousticeffects that are not associated with this mechanism.
One possible explanation for the origin of the effects
due to vibrations of the walls of wind instruments was sug-gested by the authors in Ref. 1, where it was proposed that
variations in the diameter of the pipe at the frequency of theoscillating air column couple to the internal pressure wave.In the work reported here we expand on this theory and pres-ent results demonstrating that the presence of structuralresonances associated with circular modes without nodaldiameters explains the broad-band characteristics of theacoustic effects attributable to wall vibrations. We refer tothese structural resonances as axial modes. These axial
modes are related to a one-dimensional (1D) wall displace-ment profile in the axial direction. The displacement is dueto longitudinal strain oscillations or whole body motion.
Such resonances can be shown to be broad-band in the flar-ing bell of a modern brass wind instrument.
In this work we present models of the King Silver Flair
trumpet used in Ref. 1and of a simplified brass wind instru-
ment consisting of only the straight bell section of a trumpetwith an attached mouthpiece. The models predict that axialresonances exist and that the axial wall motion that occurs ina relatively wide range around the first resonance frequencyhas the potential to affect the enclosed air column stronglyenough to make an audible difference. It is possible thatother axial resonances affect the air column as well.
Initially, we present a comparison between the calcu-
lated acoustical transfer function of a straight bell when it isfree to vibrate and compare it to the calculated transfer func-tion when the bell is fixed and unable to vibrate. This com-parison shows that the wall vibrations can increase ordecrease the amplitude of the radiated sound in a frequencyrange containing several air column resonances. Whether anincrease or decrease occurs depends on whether the fre-quency of oscillation is above or below the structural reso-nance frequency. We also present a comparison between thecalculated acoustical input impedance in the damped andundamped case. All of these results predict effects attribut-able to the proposed vibro-acoustic coupling.
II. STRUCTURALVIBRATIONS
Using estimates for the local mass and stiffness of a typ-
ical trumpet bell, a 1D model was introduced in Ref. 1that
demonstrated the plausibility of the hypothesis that axisym-metric vibrations can affect the radiated sound. Below wedescribe a more rigorous structural model, which is global,2D, and axisymmetric. The bore shape, wall thickness pro-file, Young’s modulus, and Poisson’s ratio are also included.This model has been implemented using an implicit finite-
difference scheme of distributed point masses, with forces in
both the axial and radial directions acting upon them.
External masses, springs, and dampers can be added at
any point on the bore profile to represent axisymmetricapproximations of braces and fittings of an experimental
arrangement. These same parameters can also be used to
estimate the effect of the hands, lips, and head of the player.Initial explorations of this vast parameter space haverevealed a sensitivity of some acoustic parameters to theseboundary conditions, which agrees with the long-held opin-ions of players and instrument makers.
In what follows we present the results of structural simu-
lations of a straightened trumpet with a physical length of137 cm and constant wall thickness of 0.4 mm. The bore listwas that of a Silver Flair Trumpet in B [. The wall thickness is
changed only in the region encompassing the rim, where themass of a typical rim wire has been added. Predictions of thatmodel are initially compared to 2D and 3D finite element sim-ulations performed in
COMSOL , a commercial finite element
analysis program that is widely used and often validated.
The model is then extended to include the interaction
with the internal sound field, producing a vibro-acoustic sim-ulation. This simulation includes the effects of wall vibra-tions on the acoustical characteristics such as inputimpedance and sound pressure transfer function.
A. Proposed vibration mechanism
Structural vibrations that have the potential to influence
the radiated sound of brass wind instruments must exhibitsignificant vibration amplitudes over a frequency range aswide as several hundreds of Hz. Narrow band mechanicalresonances, which are known to exist in brass instruments,can only affect a single note or partial and only then if
the mechanical resonance frequency coincides with one of
the air column resonances. Although these narrow-bandresonances have been proposed as the causal mechanism forvibro-acoustic interactions in brass instruments, experimentshave shown that acoustic effects, such as the differences intimbre that can be attributed to wall vibrations, occur overbandwidths much larger than those of these high-Qresonances.
10
A second requirement is that these structural vibrations
must be able to effectively modulate the cross-sectional areaof the air column. Unlike pure bending modes or ellipticalmodes, which only very weakly translate into bore area fluc-tuations, mechanical vibrations responsible for the experi-mentally observed coupling between wall vibrations and theenclosed air must have no radial nodes.
1
It will be shown that mechanical resonances with axi-
symmetric mode shapes, but with no radial nodes, meet bothof these requirements. Figure 1illustrates how such axial
vibrations can translate into bore area fluctuations. The mag-nitude of such fluctuations is largest inside steeply flaring
regions such as the bell of a trumpet. The rim is an open end
of the distributed vibratory system and can be expected to bean antinode of the strongest mechanical resonances. Theregion near the rim is also the bore region with the steepest
3150 J. Acoust. Soc. Am., Vol. 137, No. 6, June 2015 Kausel et al. : Bell vibrations: Theory
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.173.72.87 On: Wed, 24 Jun 2015 20:38:32flare, and therefore it is reasonable that interaction with the
air column will be most pronounced in this region. This
agrees with the experimental observation that the vibrations
of purely cylindrical tubing do not effectively couple withthe internal air column.
To understand why axial bell vibrations translate into
effective bore diameter oscillations it is convenient to use a
coordinate system that is connected to the ambient air space.
In this coordinate system both fluid particles of the air col-
umn as well as wall segments can have a velocity relative to
the static inertial system. The diameter of an air column sliceinside a flaring bore segment can change dynamically in this
coordinate system, even when the wall velocity is purely
axial.
The model presented here includes viscous losses, but
not the viscous losses due to the additional velocity gradient
inside the boundary layer that axial wall motion may create.
Comparing axial wall velocities induced by the sound pres-sure to the corresponding air column velocities demonstrates
that this loss component can be ignored in situations typi-
cally found in brass wind instruments.
One problem with using a coordinate system that is in-
dependent of the bell is that the oscillations of the bell cause
the boundary of the last air-column slice to vanish periodi-
cally. This situation results in an undefined cross-sectionalarea of the last air column slice. This slice has a thickness
corresponding to the peak-to-peak amplitude of the axial rim
displacement. Fortunately there is a significant difference
between the magnitude of the air velocity of the standing
wave at the open mouth of the bell and the axial velocity ofthe rim itself. This difference is approximately an order of
magnitude, therefore, this undefined but very thin final air
column slice can be safely ignored.
As noted above, an appropriate coupling mechanism
must explain acoustical effects which occur over a frequency
range spanning several air column resonances. Axial strain
oscillations can satisfy this requirement. The mechanismwill be discussed qualitatively first, then the effects will be
demonstrated by quantitative simulations described at the
end of this section.
For any axial mode of vibration, the applied forces and
inertial forces of all oscillating mass elements must be in
equilibrium at all times. Therefore, according to Newton’s
second law, the total momentum of all oscillating mass ele-
ments must compensate the external momentum that excites
the system. By accumulating all partial moments left and rightof a single structural node, it can be shown that the equilib-
rium position of that node must shift as the frequency of oscil-lation changes to maintain the equilibrium of moments. Thismovement is due to the gradients of the axial velocity and
mass distribution, both of which increase significantly in the
flaring region near the rim. Therefore, there are many axialmodes within a range of frequencies that contribute to a
broad-band resonance. Mathematically such broad-band
effects can be described as infinitely many axial modes, whichare infinitesimally spaced in the frequency domain and whichexhibit modal shapes with nodes that are infinitesimally
shifted in their axial position. The width of this frequency
range depends on the mass and stiffness distribution along theaxis, which is primarily determined by the bore profile.
This mechanism results in an apparent broadband reso-
nance that can have a considerable amplitude in a frequencyrange that can span multiple adjacent air column resonances.
Usually there is more than one such axial broadband reso-
nance for any given bore profile, but typically only the low-est frequency resonance is below the cutoff-frequency of atrumpet bell. We will refer to these resonances as axial
resonances.
As will be discussed later, the vibrations described
above can affect acoustical air column properties in the
range of several dB even when there is only an acoustical
stimulus, i.e., the sound pressure in the mouthpiece. If oneassumes additional structural excitation by the vibrating lips,
the effect can be increased or decreased depending on the
force amplitude and the phase relationship between the lipmotion and the wall vibrations.
It can be expected that axial bell vibrations of an instru-
ment with a bend, similar to that shown in Fig. 1, will exhibit
a much larger influence on the acoustical characteristics thanwill occur in a straight instrument without bends. This differ-
ence is attributable to the reduced axial stiffness associated
with the bends.
When the mouthpiece is fixed, the strongly flaring end
section of a straight bell can only move when the whole
instrument is stretched or compressed against the axial stiff-ness of the structure, which is determined by the Young’s
modulus, the wall thickness and the bore profile. Treating
the steeply flaring end of the bell, including the rim wire, asa mass and the remaining nearly cylindrical part as a spring,one can estimate this spring constant. Assuming a length of
40 cm, a bore diameter of 12 mm, wall thickness of 0.5 mm,
and Young’s modulus of 100 GPa, the spring constant can beestimated to be
c¼E
A0
L0¼100 GPa /C2p/C212 mm
/C20:5m m =40 cm /C254:7k N =mm:
The equivalent tangential spring constant of a single
coil of the same tube with a coil diameter of 13 cm was
determined experimentally by loading the exit cross-sectiontangentially with a mass and measuring the static displace-ment. The value was determined to be c/C253.4 N/mm. If such
a coil were not stabilized by the manufacturer using a brace,
shown in Fig. 1as a stiff external spring, this low stiffnessFIG. 1. (Color online) Brass wind instrument with bends and braces.
J. Acoust. Soc. Am., Vol. 137, No. 6, June 2015 Kausel et al. : Bell vibrations: Theory 3151
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.173.72.87 On: Wed, 24 Jun 2015 20:38:32would result in significant bell displacement amplitudes and
a very low resonance frequency ( <30 Hz in this case). The
predicted influence of a single unbraced coil on the acousti-cal input impedance is discussed in Sec. IV.
B. Finite difference model
As with the model presented in Ref. 1, the model
described here includes a vibrating structure with distributed
mass and stiffness pairs. But here the elastic forces in boththe radial and axial direction are included. They interactwith the radial and axial displacement components accordingto Poisson’s ratio and they are part of the local linear differ-ential equations of motion. This implicit system of differen-
tial equations is discretized and then solved numerically.
Although both the axial and elliptical modes are sym-
metric about the axis of the bell, for ease of discussion wewill use the term axisymmetric to refer only to mode shapes
that are independent of radial angle. These mode shapes
affect the circular cross section of the bell equally and with aconstant phase.
As previously noted, although the narrow-band elliptical
resonances are easily excited, only axial resonances canaffect the internal air column efficiently enough and in awide enough frequency range to account for the observedeffects. In modeling the mechanical motion of the bell wetherefore only consider motion that is independent of the ra-dial angle. Similarly, all external and internal forces are con-
sidered to be perfectly axisymmetric.
The masses of thin cylindrical slices, or so-called hoop
segments, are represented by point masses. They are con-
nected to adjacent masses by springs representing the resist-
ance of the wall against in-plane stress perpendicular to thecircumference of the hoop segment. Circumferential elasticforces resist expansion or constriction of the bore due to aninternal or external overpressure. This kind of stiffness isrepresented by a spring that keeps the point mass at the dis-tance from the center of the hoop segment required by the
bore radius.
Both equivalent spring constants can be calculated for
the quasi-static case using Hooke’s law. To obtain the effec-
tive radial spring constant, knowledge of the radial wall dis-
placement of a single hoop segment due to a static inner airpressure is required. This relationship has been derived inRef. 1.
The resulting equivalent spring constants derived below
only depend on the Young’s modulus of the wall materialand some local geometric parameters. The quasi-staticassumption can therefore be dropped since the air pressure isnot relevant. However, discretization of the bore profile mustbe fine enough to allow for a sufficient number of masspoints over the wave length of both the structural waves and
sound waves. An axial bore resolution of 1 mm has been
used for the sake of bore accuracy. This resolution also satis-fies the acoustic sampling restriction.
An external mass attached to the instrument can be
added to any mass point. If the corresponding radial springstays unmodified this extra mass changes the local inertia butdoes not change the local stiffness. However, a modificationof the local wall thickness will change both the inertia and
the stiffness. The mechanics associated with the rim wire atthe bell can therefore be included by adding extra mass andradial stiffness.
External springs, forces, and dampers acting on any
mass point in the axial direction can also be added. As longas these external masses, springs, forces or dampers do not
break the axial symmetry they can be modeled realistically.
In this way braces, hands supporting the instrument, or theplayer’s head, all of which are coupled to the instrument, canbe taken into account.
We note that shear stress and bending moments have
not been included in the model as yet. Usually this simplifi-cation is justified because of the small displacements leadingto still smaller bending angles. But there is one case, where
this assumption obviously fails. This case will be discussed
in Sec. II D.
The discretization of the continuous distribution of mass
and stiffness in the bell using a finite number of masses andsprings for the purpose of numerical treatment is shown inFig.2. The equations of motion in the radial and axial direc-
tions containing all the forces acting on each mass point lead
to two systems of partial differential equations that can be
solved using a finite-difference frequency-domain approach.
The radial and axial displacements are related through
the Poisson effect, therefore both systems of differentialequations cannot be solved independently. However, radialdisplacements due to an expansion or constriction of thebore caused by the internal sound pressure are much smaller
than the axial displacements in brass wind instruments.
Therefore, we solve both systems of equations independentlyand take the Poisson effect of the axial displacement on theradial displacement into account in a post processing step.
Due to the axial symmetry, each lumped mass corre-
sponds to the mass of the equivalent circular segment ofbrass and is given by
m
i¼pqhð2ribþb2Þ=coshi; (1)
FIG. 2. Mass-spring model of the vibrating trumpet wall (Ref. 11). The sym-
bols are defined in the text.
3152 J. Acoust. Soc. Am., Vol. 137, No. 6, June 2015 Kausel et al. : Bell vibrations: Theory
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.173.72.87 On: Wed, 24 Jun 2015 20:38:32where miis the mass of segment i, riits internal radius, bits
thickness, and hthe axial distance between the masses. The
distance his taken to be the axial grid spacing, which is cho-
sen to be 2 mm. The length of each segment at rest is givenbyL
i¼h/coshi, where his the flare angle. The force due to
the internal pressure papplied on the walls of each segment
is given by
FpðiÞ¼2pripih=coshi; (2)
and is always perpendicular to the wall. Therefore, the radial
and axial components of the force due to internal pressurecan be calculated as
F
rðiÞ¼2pripih; (3)
FxðiÞ¼/C0 2pripihtanhi: (4)
The spring constants of the springs between the masses, cal-
culated using Hooke’s law, are given by
ci¼2pribE
h=coshi¼2pribEcoshi=h: (5)
The radial spring constants can be calculated using the defi-
nition of a spring constant as ki¼Fr(i)/si, where si¼pir2
i=
ðEbcos3hiÞis the amplitude of the radial wall displacement
and Fr(i) the radial pressure force.1Therefore, the radial
spring constants are given by
ki¼2pbEhcos3hi=ri: (6)
At the rim of the bell the brass is folded around a wire,
referred to as the rim wire. This constitutes an extra massthat can affect the structural resonances. Including this in themodel requires increasing the mass of the final segment andmodifying the stiffness of the last radial spring.
The model also includes the Poisson effect, which
describes the stretch in one dimension caused by a strain inanother dimension. As noted above, the displacements in theaxial direction are much greater than those in the radialdirection, therefore, only processes in which the radial dis-
placement is affected by the axial displacement are consid-
ered. This simplification allows one to solve for the axialdisplacement first, neglecting any effects due to radial dis-placement. The accompanying radial displacement can thenbe calculated using Poisson’s ratio.
One equation of motion for each direction is necessary.
For the axial displacement
m
i€x¼FRxðiÞþFLxðiÞþFxðiÞ
¼FRðiÞcoshiþ1þFLðiÞcoshiþFxðiÞ
¼ciþ1DLiþ1coshiþ1/C0ciDLicoshiþFxðiÞ; (7)
where xis the axial displacement, FRx(i) and FLx(i) are the
axial components of the spring forces to the right and left ofmass m
i, andDLiis the deformation of the spring with stiff-
ness ci. Substituting a single-frequency solution of the form
xi¼Xiejxtand simplifying yieldsciXi/C01þðmix2/C0ci/C0ciþ1ÞXiþciþ1Xiþ1þFxðiÞ¼0;
(8)
where Xicorresponds to the complex amplitude of the axial
displacement of mass miandxis the angular frequency.
Similarly, for the radial displacement, the correspondingequation of motion is
c
iYi/C01þðmix2/C0ci/C0ciþ1/C0kiÞYiþciþ1Yiþ1þFrðiÞ¼0:
(9)
The total displacement in the radial direction can be calcu-
lated by adding the contribution from the axial displacement,
Ytot¼YiþYXi¼Yi/C0ri/C23Xiþ1/C0Xi/C01
2h; (10)
where /C23is Poisson’s ratio. Solving Eqs. (8),(9), and (10)for
each frequency makes it possible to determine the displace-ment at any point on the wall. Results of this model arecompared to corresponding finite-element simulations inSec. II D.
C. Finite-element analysis
The model introduced in Sec. II Bcan be used to predict
many of the experimental effects reported previously,1,2,12as
well as predicting new phenomena that can be tested experi-mentally. However, it is useful to compare these predictions
with those of a fully 3D finite-element (FE) model. In so
doing it is possible to understand some of the limitations ofthe mass-spring model as well as determine how necessary afully 3D calculation is to predict the observed effects.
The implementation of a finite element model of the
straight trumpet bell with the attached mouthpiece describedin Sec. II Awas performed using
COMSOL . The thickness and
bore profile were provided by the manufacturer.
The simplification of using a straight bell eliminates
several degrees of freedom due to the simple symmetry. It
also assists the manufacturer in maintaining precise dimen-
sions and material properties. Both of these should improvethe agreement between modeling and experimental resultscompared to the previous attempt reported in Ref. 13, where
modeling results were compared with measurements madeon a complete trumpet.
Due to the axial symmetry, finite element modeling in
2D should be sufficient to capture all of the behavior thatcan be compared to the mass-spring model. However, a full3D frequency domain analysis has also been performed toshow the elliptical modes of vibration. These ellipticalmodes can serve as a cross-check of the structural parame-ters because it is easy to verify those frequencies experimen-tally. Additionally, while it is known that these vibrationalmodes do not significantly affect the radiated far-field sound
of brasses unless they are tuned to an air column reso-
nance,
3,5these modes can be used to validate the material
constants used in the simulations.
One of the most difficult parts of the bell to model is the
rim. This is because the rim is made by folding the metal
J. Acoust. Soc. Am., Vol. 137, No. 6, June 2015 Kausel et al. : Bell vibrations: Theory 3153
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.173.72.87 On: Wed, 24 Jun 2015 20:38:32back over the rim wire, which is then soldered in place.
Rather than attempting to determine the appropriate physical
parameters, the frequencies of the elliptical resonances ofthe bell were measured using decorrelated electronic specklepattern interferometry
14and the radius and mass of the rim
enclosing the rim wire was chosen so that the frequency ofthe (2,1) elliptical mode, corresponding to two nodal diame-ters and one nodal circle, matched the measured frequency.
Using a constant bell thickness and standard values for
brass density (8400 kg/m
3), Young’s modulus (110 GPa),
and Poisson’s ratio ( /C23¼0.35), the frequencies of many ellip-
tical modes were predicted to be close to the frequencies
measured using electronic speckle pattern interferometry.Any discrepancies can be explained by the imperfect circularcross-section of the bell.
Variations in the thickness of the walls is especially im-
portant because they are assumed to be constant in themodel. The thickness of the straight bells, while reported asbeing constant by the manufacturer, exhibited variations ofup to 17% along the circumference of the bell and up to 12%along the axis when measured using a Magna Mika 8500
VR
thickness gauge.
These thickness variations should be expected given the
manner in which the bells of brass instruments are manufac-tured, and they will undoubtedly shift resonance frequencies
and change operating deflection shapes. Along with the sol-
der seam, which adds a line with different material proper-ties to the contour of the bell, these variations can break theaxial symmetry. Therefore, mode splitting of axial vibrationmodes is expected to occur in bells manufactured in the tra-ditional manner.
D. Results and comparison of different methods
Although the 3D FE model provides a more precise
model of the bell under investigation than the mass-springmodel introduced in Sec. II B, the two models should com-
pare well in situations where the essential physics is captured
by the more simple model. Therefore, it is useful to compare
the results of the two models.
The 3D finite element model of the trumpet bell was
simulated as being stimulated at the mouthpiece plane by a
sinusoidal force acting in the axial direction with an ampli-tude of 1N. This amplitude is on the order of what isexpected to be present due to lip motion during actual per-formance. A small perturbation with 1 mN orthogonal to themain stimulus force was included to break the symmetry ofthe model. This perturbation ensured that elliptical modes
could also be excited.
The predicted displacement of the rim of the bell is plot-
ted in Fig. 3, along with the prediction of a corresponding
axisymmetric (2D) finite element simulation. Clearly the
assumption of symmetry along the axis only results in the
failure to predict some narrow-band resonances at a limitednumber of frequencies, each of which corresponds toresonances having elliptic or bending mode shapes.
There is one resonance corresponding to an axisymmetric
mode, which we will term the rim mode ,t h a ti sp r e d i c t e db y
both of the finite element simulations, but not predicted whenusing the finite difference scheme described in Sec. II B.T h i s
deflection shape is characterized by an antinode at the rim
with a nodal circle just a few centimeters away from the rim.It significantly deforms the structure near the end of the bell
and its frequency is determined primarily by the bending stiff-
ness rather than the strain resistance of the brass sheet. Thisdeflection shape can be described as a rotational motion of all
rim segments around the circular nodal line. Since this motion
involves rotational forces and rotational moments of inertia,which are not included in the finite difference model, the
model cannot predict such resonances.
Although the finite difference model cannot predict this
kind of motion, which is shown in Fig. 4,t h i sm o t i o na tt h e
extremity of the bell will not radiate efficiently due to the
dipole nature with dimensions small compared to the wave-length of excitation. Therefore, we do not expect a significant
acoustic effect. It is, however, possible that this resonance can
coincide with and de-tune the second longitudinal resonance.FIG. 3. (Color online) Rim displacement amplitude as a function of fre-
quency stimulated by a force applied at the mouthpiece calculated using a2D axisymmetric (dashed) and a 3D (solid) finite element model.
FIG. 4. (Color online) Second axial resonance deformation, i.e., the rim-
mode, scaled by a factor of 3000 to show the existence of a node very close
to the rim, calculated using a 2D finite element model.
3154 J. Acoust. Soc. Am., Vol. 137, No. 6, June 2015 Kausel et al. : Bell vibrations: Theory
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.173.72.87 On: Wed, 24 Jun 2015 20:38:32In this case it will be difficult to predict the frequencies of this
resonance accurately with the simple finite-difference model.
Table Ishows predictions of all three models for the first
two longitudinal axial resonances and for the rim mode
described above. Note that the frequency of the second axialresonance calculated by the FE models varies significantlyfrom that calculated by the mass-spring model. Presumablythis is because the second resonance has a node very close tothe rim and is therefore affected by rotational motion.Removing the rim-wire from the simulation detunes this res-
onance significantly, with the result being that the predic-
tions of all three models agree. These results are shown inthe second column of Table I. Animations of the first axial
and (2,1) elliptic mode shapes are shown in Figs. 5and6.
As a final test of the simple finite difference model, the
vibrational response to the distributed force stimulus of a re-alistic acoustic sound pressure profile has been calculatedand compared to a corresponding
COMSOL result. The sound
pressure profile of the enclosed air column, calculated using
BIAS,15was applied as a boundary load to the interior of the
bell walls in the mass-spring model and in a correspondingaxisymmetric 2D finite element model.
A comparison of the axial and radial displacement
amplitudes along the bore profile of the King Silver Flairtrumpet modeled without bends predicted by the two modelsis depicted in Fig. 7, where the solid lines represents the
results of the mass-spring model and the dashed lines repre-sent the results of the 2D FE model. The sinusoidal pressurein the mouthpiece was 250 Pa. The excitation frequencies of486 and 1069 Hz correspond to the fourth and ninth peak of
the input impedance.
Clearly the mass-spring model can capture much of the
essential physics of the situation. The fact that the rotationaldegree of freedom of the rim is not captured by the mass-spring model explains the difference between the two meth-ods in predicting the displacement in close proximity of the
rim. Since the instrument is excited at the frequency of an
air-column resonance, the frequency difference between theexcitation and the second structural resonance, which is arim mode in one of the cases, depends on the model used.
This results in different predictions for the displacement
amplitudes, but as noted previously, it is unlikely that thisresonance affects the radiated sound of the instrument.
Figure 8shows how operating deflection shapes smoothly
vary with frequency, a property of the proposed axial vibra-
tion mechanism which was discussed qualitatively above.These curves were obtained using the mass-spring model of astandard trumpet bell without bends, connected to a mouth-piece with a total physical length of 73 cm and discretized
into bore slices of 1 mm. The graph shows the magnitudes of
the axial vibration amplitudes plotted as a function of theaxial distance from the mouthpiece plane, when stimulated atthe mouthpiece end by an axial, sinusoidally oscillating me-chanical force of 1 N and an in-phase mouthpiece pressure of
250 Pa. The amplitude and phase relationship have been cho-
sen arbitrarily. The mechanical stimulus represents a conserv-ative estimate for a possible contact force applied by aplayer’s lips.
The model parameters have been chosen to match an
existing standard trumpet bell made from brass by an instru-ment maker. The bell was straight, without the usual bend. Thesimulation parameters were: Young’s modulus E¼100 GPa,
density q¼8440 kg/m
3, Poisson’s ratio /C23¼0.35, and damping
factor tan d¼0.05. These results clearly indicate that the pro-
posed vibration mechanism can result in significant motion inthe bell region over a rela tively wide frequency band.
The wall thickness of 0.55 mm is slightly larger than
that found in most trumpets, but still typical for some instru-ments. Since it was expected that wall vibration effects willTABLE I. First and second axial resonance frequencies for a trumpet with
and without a rim, as calculated using a 2D or 3D finite element model
(FEM) and the presented mass-spring model (MS).
Rim-wire No rim-wire
FEM 3D FEM 2D MS FEM 3D FEM 2D MS
f1 994 991 1018 1133 1134 1125
frim 1799 1754
f2 2648 2658 2413 2519 2546 2543
FIG. 5. (Color online) Animation of the motion of the first axial resonance.
The predicted frequency is 994 Hz (see Ref. 27).
FIG. 6. (Color online) Animation of the (2,1) elliptical mode shape. The pre-
dicted frequency is 472 Hz (see Ref. 27).
J. Acoust. Soc. Am., Vol. 137, No. 6, June 2015 Kausel et al. : Bell vibrations: Theory 3155
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wall material helps to estimate a lower bound for the magni-
tude of the effect. We expect the effects to exceed these pre-dictions in real instruments.
The unusually high inner damping factor of 0.05 was
determined experimentally by connecting the bell to a shakerand sweeping through the expected first axial resonance. The
amplitude of the axial rim displacement relative to that of
the driving point was measured at several different points,averaged in order to eliminate the effects of elliptic modes,
and plotted against the theoretical curve. The inner damping
factor was then used as a fitting parameter and the value waschosen to produce the best agreement between theory and
experiment.
This inner damping factor is usually written as tan( d)
and it refers to the tangent of the argument of a complex
Young’s modulus. Its value is normally assumed to be on the
order of 0.001 for brass, which is fifty times smaller than thevalue determined experimentally as described above. The
reason for this significant deviation has yet to be determined,
but two possible reasons are discussed below.
First, inner damping of metals is not well addressed in
the literature and to our knowledge there is no report that
posits the dependence of tan( d) on common metal treatments
such as molding, bending and annealing. It is worth noting
that the people who manufacture brass musical instruments
appear to be universally convinced of the importance ofthese processes in determining the final sound.
It is also possible that the uneven wall thickness profiles
around the perimeter and along the axis, combined with thegeneral deviation from a perfect circular symmetry that is in-
evitable in the manufacturing process, may have an overall
effect which can be predicted by adding a factor into theimaginary part of the complex Young’s modulus. Given the
importance of mechanical resonances and their bandwidth to
the final sound of brass instruments indicated by the workFIG. 7. (Color online) Axial (top) and radial (bottom) wall displacement amplitude as a function of position, caused by a 250 Pa sinusoidal mouthpiece pres-
sure at (a) 486 Hz and (b) 1069 Hz, calculated using the mass-spring model (solid) and a 2D finite element model (dashed).
FIG. 8. (Color online) Axial vibration amplitude profiles at various frequencies of a straight trumpet bell connected to a mouthpiece when stimulated acousti-
cally and mechanically at the mouthpiece end.
3156 J. Acoust. Soc. Am., Vol. 137, No. 6, June 2015 Kausel et al. : Bell vibrations: Theory
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ancy be determined in the near future.
Finally, it should be noted that other critical parameters,
such as Young’s modulus and the density may vary with com-position and treatment of the wall material. Reconstructingthe actual values by a parameter matching optimization rou-tine using measured structural characteristics, as has beendone in this work, may be the only practical way to accuratelydetermine all of the important parameters.
III. VIBRO-ACOUSTIC INTERACTION
Wave propagation inside a wind instrument has been
extensively studied and successfully modeled in the past.16–18
In these cases, the walls of the instrument are usually consid-
ered to be perfectly rigid, however, this is not the case whenbrass wind instruments are actually used in performance.Apart from the possibility of mechanical feedback to theplayer’s lips, there is a coupling between the vibrating wallsand the air column inside the instrument that can affect itsinput impedance.
3,4,19–22
The structural model described above can be used to
predict wall vibrations induced by the sound pressure inside
the instrument as well as by oscillating forces applied to anypart of the instrument. This allows one to study the effect ofwall material, mechanical damping, mass and stiffness distri-bution, and oscillating forces exerted by the vibrating lips onthe mouthpiece rim.
In this section we address the question of how to incor-
porate the structural model described above into an algo-rithm that can calculate the input impedance and ATF ofwind instruments, taking into account the effects of wallvibrations stimulated by the interior acoustic field as well asby external oscillating forces.
23A vibro-acoustic interaction
model was proposed by the authors in Ref. 1, but this model
was derived for the isothermal case. Here we address the adi-abatic case, which is more applicable to the conditions foundin wind instruments.
As noted above and illustrated in Fig. 1,a x i a lv i b r a t i o n s
translate into radial air column boundary oscillations insideflaring sections of the bore. Additionally, there is a smallercontribution due to the Poisson effect. These radial boundaryvibrations affect the enclosed air column through two separatemechanisms. First, they create a parasitic acoustic volumeflowcDuinto the vibrating wall as discussed in Sec. III B.
Second, they modulate the volume of all bore segments,which periodically changes the local air density and thereforethe local air pressure by an amount of cDp, which is addressed
in Sec. III C. All quantities marked by a caret (such as bA)a r e
complex, frequency-dependent amplitudes; they are the coef-ficients of the usually omitted term e
jxtand represent harmon-
ically oscillating values with a constant magnitude and phase.
Both contributions can be treated as distributed sound
flow and pressure sources with wavelets that propagate andinterfere with each other along the bore to produce an accu-mulated effect at the mouthpiece plane. Each local pointsourcecDpðxÞtransmits a wavelet towards the input plane
being modified by its distance-dependent transfer functionbAðxÞto generate an accumulated extra sound pressure cDp
0at
the mouthpiece plane,
cDp0¼ðL
0cDpðxÞbAðxÞdx: (11)
The distributed volume flow is accumulated in a similar
way, back-propagated to the entrance plane by the flowrelated transfer function bBðxÞ. In a discretized bore profile
consisting of purely cylindrical or conical segments thetransfer functions bAðxÞandbBðxÞfor sound pressure and flow
can be obtained from the product of the transfer matrices of
all segments,
15,16which back-propagate pandufrom the
plane at axial position xto the entry plane at x¼0.
As we are integrating both contributions over the axial
length of the instrument, care must be taken so as to not inte-grate the volume flow utwice. When calculating the flow u,a s
shown below, it is the contribution lost into the walls of a shortcylindrical segment of length h.If we wish to integrate this
contribution we first must divide it by the length hto obtain a
flow contribution per unit length. This can then be integrated
dDu
0¼ðL
0cDuxðÞ
hbBxðÞdx: (12)
A. Modified transmission line model
The starting point of the vibro-acoustic interaction
model is the 1D plane-wave transmission-line model asimplemented in Ref. 23and reviewed in Ref. 15. It allows
the calculation of input impedance and pressure transfer
function of acoustic ducts such as brass or woodwind instru-ments when their bore profile and radiation conditions areknown.
The model provides complex, frequency-dependent
transmission matrices
bT¼bT
abTb
bTcbTd !
(13)
for each cylindrical or conical slice of the bore profile. The
variable pair pressure bpand volume flow buare then transmit-
ted from the right side of the element to the left side accord-ing to
bp
1¼bTabp2þbTbbu2;
bu1¼bTcbp2þbTdbu2: (14)
Assuming unity sound pressure at the open mouth of the
bell, a volume flow of buBell¼1=bZradis enforced by the radia-
tion impedance bZrad. Back propagating bpandbuusing the trans-
mission matrices of all bore elements, an input pressure bp0and
volume flowbu0at the mouthpiece end of the bore can be
obtained. From this result the input impedance bZin¼bp0=bu0
and pressure transfer function bTp¼bprad=bp0can be derived.
Taking the vibrating walls in to account can be achieved
by modifying all inner bpi,buipairs according to bp/C3
i¼bpiþcDpi,
J. Acoust. Soc. Am., Vol. 137, No. 6, June 2015 Kausel et al. : Bell vibrations: Theory 3157
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i¼buiþcDui,w i t hcDpiandcDuibeing the complex
sound pressure and volume flow amplitudes lost due to theoscillating wall. This addition results in
bp
1¼bTaðbp2þcDp2ÞþbTbðbu2þcDu2Þ;
bu1¼bTcðbp2þcDp2ÞþbTdðbu2þcDu2Þ: (15)
Alternatively the wall vibration effect can be taken into
account by correcting the conventional transfer matrixelementsbT
aandbTcby multiplying them by the factor
ðbp2þcDp2Þ=bp2. Likewise, the elements bTbandbTdcan be
adjusted using the factor ðbu2þcDu2Þ=bu2.
B. Flow into the vibrating wall
An effective velocity bvðxÞ¼bdðxÞxcan be calculated
with the effective wall displacement amplitude
bdðxÞ¼bdradialðxÞ/C0bdaxialðxÞtanð/ðxÞÞ; (16)
where /is the flare angle and bdis the radial and axial com-
ponents of the local wall displacement amplitudes. Note that
a positive axial displacement in conjunction with a positive
flare angle actually reduces the effective boundary diameterfor a given air column slice, which explains the negativesign in Eq. (16).
The volume flow cDuinto the vibrating wall of a short
hoop segment with length his given by
cDuðxÞ¼bvðxÞ2rðxÞph; (17)
where r(x) is the local bore radius. This contribution has a
positive instantaneous value when the momentary wall ve-locity vis also positive, which is the case when the hoop seg-
ment expands. A positive instantaneous in-flow into the leftboundary of an element can therefore be either compensatedby a positive instantaneous out-flow out of the right bound-ary of that element or by some positive parasitic flow intothe walls. If the total flow is not balanced the pressure willchange ðdbp=dt¼bu
Left/C0buRight/C0cDuÞ.
The simplifying assumption made here is that only mass
continuity (in-flow equals total out-flow) is taken intoaccount while the balance of momentum and energy isneglected. This is justified because the mean thermal veloc-ity of air, which is given by v
rms¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3kBT=mp
in m/s, where
kBis Boltzmann’s constant, Tis temperature in K, and mis
the mass of an air molecule in kg, dominates all velocitiesrelated to flow that may result in other pressure related
forces.
C. Thermodynamic pressure modulation
During an adiabatic process the ideal gas equation is
given by
pðtÞVcðtÞ¼nðtÞRT; (18)
where the pressure p(t), volume V(t), and number of moles
n(t) all vary with time. Tis the equilibrium temperature, c
the heat capacity ratio, and Rthe universal gas constant. Inthe presence of the vibrating walls the volume oscillations
with an amplitude bVmust be included in the pressure
variation.
Following Ref. 1, but considering adiabatic conditions,
the effective time varying pressure deviation ( pþ) can be
obtained from the equilibrium pressure peqby means of a
Taylor series expansion. Neglecting second order terms thisexpansion becomes
bpþe
jxt¼bpejxt/C0cpeq
VeqbVejxt; (19)
where Veqis the equilibrium volume of the air column and bp
the complex amplitude of the oscillating air column pressure
without the presence of wall vibrations. The phase difference
between this internal pressure and the wall oscillations isreflected in the phase difference between the complex ampli-tudesbpandbV. Hence the extra pressure amplitude due to
wall oscillations is given by
/C0cDp¼c
peq
VeqbV¼cpeq
pr2h2prhbsðÞ ¼2cpeq
rbs; (20)
wherebsthe complex amplitude of the effective radial dis-
placement of the air column boundary and we have sup-
pressed the position dependence for notational clarity. Thenegative sign indicates that an in-phase radial wall displace-ment actually reduces the local sound pressure amplitude.
IV. RESULTS AND DISCUSSION
In all of the vibro-acoustic simulations reported here we
have used parameters and bore shapes derived from two sim-ilar straight trumpet bells with wall thicknesses of 0.5 mmand 0.55 mm manufactured by Musik Spiri. Measurements of
the thicker bell were used to produce the curves shown inFig.8, and the material properties stated above were applied
to both bells, unless otherwise stated.
A typical simulation result for the 0.55 mm bell without
a mouthpiece is shown in Fig. 9. The mouthpiece end was
rigidly fixed by adding an extremely heavy mass at thatpoint. The top plot depicts the two ATF curves representingthe ratio of the sound pressure amplitude in the bell plane tothat in the entrance plane. One calculation takes wall vibra-tions into account while the other one represents the case ofa completely rigid wall. The differences between the ATFcorresponding to the rigid case and that corresponding to thecase where the bell is allowed to vibrate freely are small butnoticeable. The ratio of the two ATFs is shown in the bottomplot using a dB-scale.
The results shown in Fig. 9indicate that the differences
in the ATF below approximately 800 Hz are always positivewhile those above that cross-over frequency are always neg-ative. The magnitudes of these differences reach their maxi-mum in close proximity to the frequency that corresponds tothe phase transition at the first axial resonance. In this exam-ple this structural resonance does not coincide with any ofthe air resonances, but occurs between the second and thirdacoustic resonance. Differences due to wall vibrations are
3158 J. Acoust. Soc. Am., Vol. 137, No. 6, June 2015 Kausel et al. : Bell vibrations: Theory
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ing to approximately one octave.
Another kind of influence is shown in Fig. 10, where the
calculated difference in the input impedance that results
from bell vibrations is plotted. In this case the 0.5 mm bell
has been chosen with a standard mouthpiece attached to it.
The mouthpiece was loaded with an extra mass of 2 kg to
represent the weight of a horn driver, which may be requiredwhen performing an experiment. To help orient the reader,
the frequencies of the maxima of the underlying input im-
pedance curve have been shown in the plot as vertical lines.This plot is the result of two vibro-acoustic finite element
simulations run in
COMSOL , again one with a rigid bell and
one with a bell free to vibrate.
In the finite-element model used to produce the curves
in Fig. 10a thin boundary layer next to the wall, where the
flow is retarded due to frictional losses,13was discretized
using a boundary layer mesh with each element’s thickness
being approximately 1 lm. The air domain was discretized
using a frequency-dependent mesh-size, ensuring that atleast ten elements per wavelength are present. A detail oft h em e s hi ss h o w ni nF i g . 11, where the sound pressure
level is also plotted for the first air resonance of the straight
bell. Finally, a perfectly matched layer was simulated assurrounding the semi-spherical radiation space to enforce
anechoic conditions. A baffle was also simulated to avoid
feedback to the input pressure, as is often the case whenperforming experiments.
1,2,12,24
The relative size of the differences at the third and fourth
air resonance in Fig. 10, where the input impedance is
approximately 150 M X, is again on the order of one decibel.
However, in this case the effect of the cross-over frequency
between the third and the fourth air resonance is different.
We can observe an alternating influence below the structuralresonance (minus-plus-minus) and an alternating influenceabove it (minus-plus-minus). Around the structural resonance
this alternation is toggled, which leads to two adjacent peaks
with the same negative difference surrounding the cross-overfrequency corresponding to the first axial resonance.
These observations lead to the question of why there are
two, and possibly more, fundamentally different kinds ofinfluences on acoustical characteristics which are obviously
due to the same structural resonance. To answer this question
we must consider all possible excitation mechanisms andtheir effects on the acoustic field inside the instrument.
FIG. 10. (Color online) Input impedance of the free and damped bell (top)
and the difference between the two curves (bottom) calculated numerically.
The dashed lines indicate the locations of the impedance peaks and the dot-
ted line indicates the frequency of the axial structural resonance of the bell.FIG. 9. (Color online) Simulated ATF of a straight bell without a mouth-
piece from the entrance plane to bell plane as a function of frequency, for
the case of a free vibrating bell (solid) and completely damped bell (dashed).
The bottom plot shows the difference in dB.
FIG. 11. (Color online) Sound pressure level for a free vibrating bell, show-
ing details of the underlying finite-element mesh.
J. Acoust. Soc. Am., Vol. 137, No. 6, June 2015 Kausel et al. : Bell vibrations: Theory 3159
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.173.72.87 On: Wed, 24 Jun 2015 20:38:32A brass wind instrument bell is an acoustic duct with
one end closed and the other end open. This means that the
sound pressure magnitude profile must have a maximum atthe mouthpiece and a minimum near the open end of thebell. At the lowest resonance frequency the mouthpiece pres-sure and the pressure inside the bell are in phase. At the nextresonance frequency a second pressure minimum finds itsplace inside the instrument. This inverts the phase relation-ship and causes the pressure at the mouthpiece and bell to be
out of phase. At still higher resonances each new pressure
minimum inside the instrument again alternates the phase ofthe sound pressure in the steeply flaring part of the bell rela-tive to the sound pressure present in the mouthpiece.
A longitudinal structural stimulus at the mouthpiece will
cause an in-phase displacement of the bell if the frequency islow. Near DC there will be whole-body motion and all partsof the bell will move synchronously and at the same veloc-ity. At higher frequencies periodic stress will cause axialstrain which adds length oscillations. Below a structural res-onance the bell displacement will still be almost in phasewith the mouthpiece stimulus. At a structural resonance,
however, the phase changes and above the resonance fre-
quency the axial bell motion will be out of phase with themotion at the mouthpiece.
The internal sound field is mainly affected in the flaring
bell region, but there are two possibilities to stimulate struc-tural vibrations by the interior sound pressure. In the mouth-piece the sound pressures may be up to a factor of 10
3higher
than in the bell region and the area that the pressure can acton is approximately 60 times smaller than the comparablearea in the bell region. Therefore, the dominating structuralstimulus can be situated either in the mouthpiece or in the
bell region, or it can be a combination of the two depending
on the frequency-dependent parameters related to the boreprofile and boundary conditions.
If the dominating structural stimulus mechanism is in
the bell region there will be an acoustical effect exhibiting
the same phase for all air resonances below structural stimu-lus, with the opposite phase for air resonances above it. Thestructural resonance frequency will be the cross-over fre-quency for any effect of wall vibrations on any acousticcharacteristic. In the simulation shown in Fig. 9this behavior
has been enforced by applying a large mass to the entrance
plane, thus fixing it in place.
I nt h ec a s ew h e r et h em o u t h p i e c ei sa t t a c h e dt ot h eb e l l
but is free to vibrate it is possible that the structural stimu-
lus of the sound pressure inside the mouthpiece cup domi-
nates the effects attributable to the bell motion. This mayoccur if the mouthpiece diameter is large and the axial me-chanical admittance of the mouthpiece is much higher thanthat of the bell. In this case, the behavior shown in Fig. 10
is expected because the alternating phase of the effect isrelated to the alternating phase of the sound pressure in thebell compared to the sound pressure in the mouthpiece,
which is synchronously exciting both the acoustical and
mechanical systems.
For the time being, only straight axisymmetric bells
can be simulated using the mass-spring model presented
here. However, an application to a complete instrumentwith a single loop similar to that shown in Fig. 1is shown
in Fig. 12. Using a coiled brass tube with a coil diameter of
14 cm, a bore diameter of 10.8 mm, and a wall thickness of0.4 mm, the effective stiffness of a spring linking the leadpipe and the straight part of the bell in axial direction wasdetermined experimentally to be 3400 N/m. A spring with
this spring constant was added to the model by adjusting
the stiffness of the cylindrical tube section between 19 and90 cm. This has been accomplished by reducing the wallthickness in this region to 1 lm and compensating for the
removed mass of the wall. A crosscheck of the equivalentspring constant yields c¼100 GPa /C2p/C210.8 mm /C21lm/
71 cm /C254.6 kN/mm which is close to what was measured.
The first axial resonance of this arrangement was deter-
mined to be approximately 30 Hz.
Figure 12shows that the acoustic influence of the
vibrating bell potentially can make a more significant dif-ference in this configuration. In this case the vibrating bell
acts to damp all resonances above the structural resonance
and it is likely that an instrument maker will try to shuntthat spring using a brace with much higher stiffness. Thebehavior is consistent with Fig. 9because a vibrating bell
which transmits better above the structural resonance does
FIG. 12. (Color online) Input imped-
ance of a trumpet with one coil for thecase of a bell free to vibrate (solid) and
when the vibrations are completely
damped (dashed).
3160 J. Acoust. Soc. Am., Vol. 137, No. 6, June 2015 Kausel et al. : Bell vibrations: Theory
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the input impedance.
V. CONCLUSIONS
The model discussed here predicts that axial resonances
have a measurable effect on the sound of brass wind instru-ments. It is likely that these effects explain the sensitivity toconstruction details that are commonly claimed by musiciansand makers of musical instruments. As presented, the modelcan predict most aspects of the essential structural behaviorof axisymmetric brass wind instrument bells and their effecton the acoustical characteristics. This model can predict theacoustical and mechanical transfer functions as well as the
acoustical input impedance in the presence of vibrations
induced by external sources or by the pressure fluctuationsof the internal air column.
The model can include external structural excitations,
such as those of the player’s vibrating lips, and allows oneto include external masses, springs, and damping at any partof the bore profile to reflect how the instrument is held,clamped, or stimulated. Subtle details like a non-constantwall thickness profile along the axis, a mouthpiece mass pro-
file, or a unique rim wire construction can also be specified.
Exploring this vast parameter space is beyond the scope ofthis work, but as a preliminary result it can be stated thatmost of these external influences, such as mechanical stimu-lation by player’s lips, braces that stiffen loops and bends,different rim wire constructions, and extra masses attachedto the mouthpiece can have an even stronger influence onacoustical parameters than what is shown in the simulationsreported here.
We expect that this model will aid in understanding the
effects of the different kinds of influences that wall vibra-tions have on acoustic characteristics of brass wind instru-ment bells. To facilitate these investigations, the model hasbeen implemented in the Brasswind Instrument AnalysisSystem (
BIAS)25and can be downloaded from Ref. 26.
Within this implementation of the simulation it is possibleto specify arbitrary bore and wall thickness profiles as wellas user-specified boundary conditions and materialconstants.
Using such hybrid models, which combine measured
mechanical transfer functions and physical structural modelsof some axisymmetric parts, the effect of wall vibrations onthe sound of real instruments can be predicted. This caneven include structural excitation by the player’s lips. Oncethe vibration state of the mouthpiece can be predicted it willbe possible to study its effect on the lip oscillator and there-fore on the oscillation threshold and response of aninstrument.
Predictions of this model have yet to be completely
validated by experiments. Of particular importance is theprediction that the acoustical effects of wall vibrationsshould be inverted at the frequencies of axial resonancesdue to the change in phase between the oscillating air col-umn and the oscillating wall that occurs at these frequen-cies. Preliminary experimental results indicate that this doesindeed occur.
12ACKNOWLEDGMENTS
The authors thank Werner Spiri and the Musik Spiri
company for manufacturing the straight trumpet bells. The
portion of this work performed at Rollins College was
supported by Grant No. PHY-1303251 from the NationalScience Foundation.
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Acusticum 2011 , European Acoustics Association, Aalborg, Denmark
(2011), pp. 527–532.
12B. Gorman, M. Rokni, T. Moore, W. Kausel, and V. Chatziioannou, “Bellvibrations and how they affect the sound of the modern trumpet,” in
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Institut Technologique Europen des Mtiers de la Musique, Le Mans,France (2014), pp. 215–218.
13V. Chatziioannou and W. Kausel, “Modelling the wall vibrations of brasswind instruments,” in Proceedings of the COMSOL Conference 2011,
Stuttgart, Germany (2011).
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thermal boundary conditions,” J. Acoust. Soc. Am. 75, 58–62 (1984).
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ed. (Addison-Wesley, New York, 1990), pp. 190–232.
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J. Acoust. Soc. Am., Vol. 137, No. 6, June 2015 Kausel et al. : Bell vibrations: Theory 3161
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.173.72.87 On: Wed, 24 Jun 2015 20:38:3221V. Chatziioannou, W. Kausel, and T. Moore, “The effect of wall vibra-
tions on the air column inside trumpet bells,” in Proceedings of the
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22W. Kausel, V. Chatziioannou, B. Gorman, M. Rokni, and T. Moore,
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23A. Braden, D. Chadefaux, V. Chatziioannou, S. Siddiq, C. Geyer, and W.
Kausel, “Acoustic Research Tool (ART),” http://sourceforge.net/projects/
artool (Last viewed 05/18/2015).24F. Gautier and N. Tahani, “Vibroacoustic behaviour of a simplified musi-cal wind instrument,” J. Sound Vib. 213, 107–125 (1998).
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27See supplemental material at http://dx.doi.org/10.1121/1.4921270 foranimation of the motion of the first axial resonance (predicted frequency is
994 Hz) and of the (2,1) elliptical mode shape (predicted frequency is
472 Hz).
3162 J. Acoust. Soc. Am., Vol. 137, No. 6, June 2015 Kausel et al. : Bell vibrations: Theory
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1.3686321.pdf | Gigabar material properties experiments on nif and omega
Damian Swift, James Hawreliak, David Braun, Andrea Kritcher, Siegfried Glenzer, G. W. Collins, Stephen
Rothman, David Chapman, and Steven Rose
Citation: AIP Conference Proceedings 1426, 477 (2012); doi: 10.1063/1.3686321
View online: http://dx.doi.org/10.1063/1.3686321
View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1426?ver=pdfcov
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134.124.28.17 On: Tue, 11 Aug 2015 09:42:07GIGABAR MATERIAL PROPERTIES EXPERIMENTS
ON NIF AND OMEGA
Damian C. Swift∗, James A. Hawreliak†, David Braun†, Andrea Kritcher†, Siegfried
Glenzer†, Gilbert Collins†, Stephen D. Rothman∗∗, David Chapman∗∗and Steven Rose‡
∗PLS-CMMD, Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94551, U.S.A.
†Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94551, U.S.A.
∗∗Atomic Weapons Establishment, Aldermaston, Reading, RG7 4PR, U.K.
‡Department of Physics, Imperial College, London, SW7 2AZ, U.K.
Abstract. The unprecedented laser capabilities of the National Ignition Facility (NIF) make it possible for
the first time to countenance laboratory-scale experiments in which gigabar pressures can be applied to a
reasonable volume of material, and sustained long enough for percent level equation of state measurements
to be made. We describe the design for planned experiments at the NIF, using a hohlraum drive to induce
a spherically-converging shock in samples of different materials. Convergence effects increase the shockpressure to several gigabars over a radius of over 100 microns. The shock speed and compression will be
measured radiographically over a range of pressures using an x-ray streak camera. In some cases, we will
use doped layers to allow a radiographic measurement of particle velocity.
Keywords: shock, equation of state, laser
PACS: 07.35.+k, 71.00.00, 91.45.Bg, 91.60.Gf
INTRODUCTION
Pressures in the gigabar (100 TPa) regime are pre-
dicted to occur in the cores of massive exoplanets
[1, 2]. Besides helping to interpret the structure of ex-
oplanets, equations of state (EOS) in this regime areimportant in the study of brown dwarf formation, and
thus to guide estimates of non-luminous mass given
the observable stars, which is necessary to determinewhether new physics is needed to explain galactic ro-
tation curves and thus support the existence of exotic
dark matter [3]. Technologically, these pressures andhigher occur in the implosion of thermonuclear fuel
capsules, and EOS are therefore relevant in the de-
velopment of inertial confinement fusion.
Pressures over a few megabars (100 GPa) are too
high to be induced using static laboratory techniques
such as diamond anvil cells, and must be generatedand studied in a transient way by shock or ramp load-ing. In the past, nuclear explosions have been used to
induce shocks of pressures up to several gigabars by
ablation [4, 5], but few such measurements have been
made. Laser-induced ablation on ∼10 kJ class lasers
such as Omega is now fairly routine for the genera-
tion of pressures up to several megabars, with directlaser irradiation of an ablator or laser heating of an x-
ray hohlraum [6]. The equivalent experiments at the
megajoule class NIF have demonstrated pressures of
∼5 TPa at less than full energy [7].
On any facility, the maximum pressure that can
be induced depends on the power available and the
volume into which it can be delivered. For a use-
ful EOS measurement, a sufficient volume of matter
must be prepared in the high-pressure state and pre-served long enough for a measurement to be made,
thus in practice the maximum pressure for a useful
experiment depends also on the total energy avail-able. Many laser platforms are capable of preparing
Shock Compression of Condensed Matter - 2011
AIP Conf. Proc. 1426, 477-480 (2012); doi: 10.1063/1.3686321
2012 American Institute of Physics 978-0-7354-1006-0/$0.00
477
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134.124.28.17 On: Tue, 11 Aug 2015 09:42:07matter at high pressures and temperatures, but the
volume and duration required for measurements ofreasonable accuracy are extremely challenging. Even
with the megajoule energies available at NIF, pres-
sures induced by ablation are unlikely to exceed acouple of hundred megabars for EOS experiments.
Converging shocks have been used previously to
increase the pressure available from chemical ex-plosives from a few tens of gigapascals to the ter-
apascal regime [8, 9], in 1D (cylindrical or spheri-
cal) or 2D (Mach reflection) configurations. Here weemploy converging compression waves to show that
EOS measurements can reasonably be performed at
pressures into the gigabar regime.
EXPERIMENTAL CONFIGURATION
At NIF, a large effort has been devoted to devel-
oping hohlraum platforms with exquisite spherical
drive symmetry, in order to induce symmetric implo-
sion of the thermonuclear fuel capsule. We proposeto take advantage of this work by replacing the fuel
capsule with a solid sample assembly of the same
diameter – 2 mm – and driving a shock into it. Com-pared with capsule implosions, convergent compres-
sion waves are generally more stable, as perturba-
tions from asymmetry tend to damp out.
It is desirable to use a plastic ablator (CH), as it
couples relatively well to the hohlraum radiation, and
can be doped to absorb hard x-rays that would pre-heat the sample. Most sample materials of interest
have a higher shock impedance than CH, thus in-
creasing the pressure in the sample.
Ignition hohlraum temperatures can reach 250-
300 eV , which should induce ablation pressures of
20-30 TPa in CH. Using existing EOS, we have pre-dicted the effect of convergence on the pressure of
shock and ramp waves. For shocks, the pressure
passes through 100 TPa at a radius of 100-200
μm,
which should allow multi-gigabar pressures to be ex-
plored with readily achievable radiographic resolu-
tions of around 20 μm (Fig. 1).
The principal diagnostic for the mechanical EOS
is streaked x-ray radiography, using a laser-heatedplasma backlighter. A complementary diagnostic
will be x-ray Thomson scattering,which uses spec-
troscopy of scattered x-rays to deduce the compres-sion, temperature and electron density [10, 11, 12].FIGURE 1. Pressure profiles for a diamond sample,
driven using a 500 μm thick CH ablator and a hohlraum
temperature of 250 eV . Black lines: intervals of 1 ns; grey
lines: 0.1 ns. 1 gigabar is 100 TPa.
He-like radiation from a Zn backlighter, at 8.95 keV ,
seems to be optimal for both diagnostics simulta-
neously, based on current backlighter development
for NIF. Further development is needed to be able
to interpret the Thomson scattering signal from theradially-varying states behind the converging shock.
Collimation of the x-ray source and the line-of-sight
to the spectrometer would collect scattered x-raysfrom a spatially-localized region. though collimation
of the x-ray source would impede the use of radiog-
raphy on the same shot. Windows are needed in the
hohlraum wall to allow the x-rays to pass. It is most
efficient to use windows no larger than necessary for
the diagnostics, but the windows then act as a 3D
source of cooling, which makes full hohlraum simu-
lations less tractable: we may use a cylindrical win-dow to improve azimuthal and simulation symmetry.
(Fig. 2.)
It would be valuable to obtain gigabar data on a
variety of materials. For the initial experiments, it is
desirable to choose a material that allows the most
accurate measurement to be made. One limiting fac-tor is the x-ray signal level, controlled by the sample
opacity and thickness. Low density samples such as
plastics are predicted to reach a high enough shocktemperature that the opacity is likely to fall signifi-
cantly from its initial value. Quantitative radiography
would be possible only by introducing marker lay-ers doped with a higher Z: this greatly complicates
sample fabrication, and the EOS of the doped ma-
478
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134.124.28.17 On: Tue, 11 Aug 2015 09:42:07FIGURE 2. Experimental configuration.
FIGURE 3. Simulated x-ray transmission through 1 mm
diamond sample within CH ablator.
terial may be significantly different. Solid matter of
Z=13 (Al) and higher generally remains cooler, but
the x-ray transmission would be low. The transmis-
sion through the sample can be increased by decreas-
ing its radius, using a much thicker ablator layer, at
the cost of a smaller region reaching gigabar pres-sures. The best compromise found so far is to use a
C (diamond) sample of 1 mm diameter, inside a CH
sphere of 2 mm diameter. The smallest x-ray trans-missions are predicted to be several percent through
the region exceeding 100 TPa (Fig. 3).
If possible, additional radiographic images will betaken along the hohlraum axis, to measure azimuthal
symmetry.
OMEGA EXPERIMENTS
A complementary series of experiments will be
performed at the Omega laser facility, to developand refine experimental and analysis methods. The
hohlraum and sample will be smaller, the sample
0.6 mm in diameter. The first experiments will usea solid CH sample, and are expected to provide EOS
data up to several tens of terapascals. The opacity of
CH is predicted to remain close to its initial value
over most of the range.
RADIOGRAPHIC ANALYSIS
EOS measurements from the converging shock
proposed here require radiographic measurement of
the shock speed and compression as a function of ra-dius, which in turn require the location of the shock
to be determined as a function of time, and the den-
sity jump at that location. Unlike a supported shock
in plane geometry, the density varies with radius
behind the shock, so an adequate spatial resolution
is needed to infer the value at the shock. It is notpractical to capture the whole radial profile of the
solid sample, ablated material from the sample and
hohlraum wall, and the residual wall on the x-raystreak record, so it is not possible to perform an Abel
inverse of the onion-skin type. Further, any radio-
graphic unfold process which works radially inwardwill accumulate errors. Fortunately, the unshocked
region at the center of the sample provides a strong
constraint on the unfold in the region of the shock,and the shape of the transmission profile in the cen-
ter contains information about the attenuation further
out. This information can be extracted by looking for
long spatial modes in the transmission profile [13].
An alternative approach is to use a parameterized
Bayesian profile-matching process, where the den-
sity in the central region is fixed. The position and
amplitude of the shock, and the density profile fur-ther out, are described with adjustable parameters,
which are optimized iteratively to give the best match
to the attenuation profile.
479
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134.124.28.17 On: Tue, 11 Aug 2015 09:42:07FIGURE 4. Unfolds of simulated radiographic data, for
the initial Omega experiments on CH, using different al-gorithms. The profile match was not allowed enough de-grees of freedom to reproduce the low density region at
∼0.17 mm radius, but the reconstruction of the shock was
still accurate.
We investigated the use of different algorithms,
including Abel inversion and profile matching, for
simulated radiographic data to which random noise
was added. Abel inversion magnified the noise by
over an order of magnitude around the radius of the
shock. Profile matching was capable of reproducing
the position and amplitude of the shock to within
a few percent, and converged reliably even with an
initial guess which was very different from the ac-tual profile. The reconstructed profile in the region of
the shock was not sensitive to inaccuracies in recon-
structing the profile in the outer, low density region(Fig. 4).
Generally, the shock speed is expected to vary
smoothly with radius, and the density profile behindthe shock should also vary smoothly with time, so
smoothing in the temporal direction can be used to
reduce the noise in the unfolded density profiles. Weestimate this smoothing to reduce the uncertainty in
EOS measurements by a factor of several.
CONCLUSIONS
We have designed an experimental platform for
NIF which should access gigabar pressures using
hohlraum-driven spherical samples, and streak radio-
graphy. Because of pulse-length constraints and theneed to base the design on thermonuclear ignition
configurations in order to take advantage of the largeeffort on drive symmetry, ramp loading is not feasi-
ble initially. However, with a converging shock, ra-
diographic measurements explore a range of statesalong the principal shock Hugoniot in each experi-
ment. Radiographic measurement of the shock com-
pression requires an accurate knowledge of the opac-ity of the sample, which means that the sample tem-
perature must remain well below the energy of the
electronic Kedge. For higher temperatures, radio-
graphic marker layers could be used. We have de-
veloped designs for the first series of experiments at
NIF and OMEGA. Analysis so far suggests that ac-curacies o(1%)should be achievable in shock speed
and compression.
ACKNOWLEDGMENTS
We would like to thank Dr Damien Hicks
(Lawrence Livermore National Laboratory) foradvice on radiography of hohlraum-driven samples.
This work was performed under the auspices of
the U.S. Department of Energy under contract #DE-AC52-07NA27344.
REFERENCES
1. Seager, S., Kuchner, M., Hier-Majumder, C.-A., and
Militzer, B., Astrophys. J. 669, 1279-1297 (2007).
2. Swift, D. C., et al., Astrophys. J. (in press) and
preprint arXiv:1001.4851 .
3. Bertone, G., Hooper, D., and Silk, J., Phys. Rep., 405,
279 (2005).
4. Ragan III, C. E., Phys. Rev. A 25, 3360-3375 (1982).
5. Avrorin, E. N., V odolaga, B. K., Simonenko, V . A.,
and Fortov, V . E., Phys.-Uspekhi 36, 5 (1993).
6. Collins, G. W., et al., Science 281, 1178-1181 (1998).
7. Eggert, J., et al., these AIP Conf. Proc.
8. Kanzleiter, R. J., et al., IEEE Trans. Plasma Sci., 30,
1755-1763 (2002).
9. Swift, D. C. and Ruiz, C. R., AIP Conf. Proc. 845,
1297-1300 (2006).
10. Kritcher, A. L., et al., Science 322, 69-71 (2008).
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1625 (2009).
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480
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1.1853517.pdf | Precharging strategy to accelerate spin-transfer switching below the nanosecond
T. Devolder, C. Chappert, P. Crozat, A. Tulapurkar, Y. Suzuki, J. Miltat, and K. Yagami
Citation: Applied Physics Letters 86, 062505 (2005); doi: 10.1063/1.1853517
View online: http://dx.doi.org/10.1063/1.1853517
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/86/6?ver=pdfcov
Published by the AIP Publishing
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130.113.69.48 On: Fri, 28 Nov 2014 21:08:00Precharging strategy to accelerate spin-transfer switching below
the nanosecond
T. Devolder,a!C. Chappert, and P. Crozat
Institut d’Electronique Fondamentale, UMR 8622 CNRS, Université Paris Sud, Bâtiment 220,
91405 Orsay, France
A. Tulapurkar and Y. Suzuki
NanoElectronics Research Institute, National Institute of Advanced Industrial Science and Technology,Tsukuba 305-8568, Japan
J. Miltat
Laboratoire de Physique des Solides, UMR 8502 CNRS, Université Paris Sud, Bâtiment 510,91405 Orsay, France
K. Yagami
MSNC, Semiconductor Technology Development Group, SONY Corporation, Atsugi, Kanagawa, Japan
sReceived 9 September 2004; accepted 23 November 2004; published online 2 February 2005 d
We compared different ways of inducing magnetization switching by spin momentum transfer in
pillar shaped CoFe/Cu/CoFe trilayers using sub-ns-current pulses. In comparison with switchinginduced by a single sub-ns pulse, precharging the device with a bias current prior to the applicationof the pulse proved to lower the required peak current. Precharging is efficient for pulses rangingfrom 2 ns down to at least 200 ps. Simulations indicate that the bias current prepares themagnetization in a precession state that provides an enhanced susceptibility to the spin torque of thepulsed current. The precession settling time is typically 2 ns, hence the precharging strategy losesits efficiency for longer pulses, in agreement with experiments. © 2005 American Institute of
Physics.fDOI: 10.1063/1.1853517 g
The spin-transfer magnetization switching
1,2has been
proposed to restore the scalability in high density magneticrandom access memories sMRAM dbeyond several Gbit/
chip. The proof of concept has been done
3–5but many issues
for applications are waiting for answers. Among the majorconcerns are the operating frequency limitations and the cor-related magnitude of the needed write current I
C. In a prac-
tical memory circuit, the peak value of the write current IC
determines the size of the write transistor, which sets a limit
on the memory areal density.
In year 2000, Sun predicted6that the switching speed
1/tin spin-transfer induced magnetization switching should
scale with uI–ICulnu0, i.e., both with the overdrive current
I–ICand with the initial misalignment u0between the trans-
ported spin polarization and the macrospin to be reversed.Full micromagnetic calculations
7have recently confirmed
this law. However in experiments so far u0was the misalign-
mentofthemagnetizationofthefreelayerfromitseasyaxis,mostly arising from finite temperature fluctuations. Increas-ing the switching speed can thus be done by increasing eitherthe current pulse I, which is not desirable or by preparing a
more favorable initial condition with
u0Þ0.7A straightfor-
ward strategy is to change u0by a field pulse transverse to
the easy axis, as classically done in magnetic fieldswitching.
8However in practical memory architectures, this
strategy would require additional addressing lines and largetransistors to provide enough current, which would signifi-cantly increase the technological complexity. Another pos-sible strategy is to exchange bias the pinned layer at an angleu0with the free layer’s easy axis. However in this case a
substantial part of the magnetoresistance signal is wasted atthe expense of the SNR at the reading step.
In this letter, we present a strategy to gain in switching
speed while keeping the full magnetoresistance ratio and notrequiring applying any magnetic field. We precharge the de-
vice with a dc bias current to excite a steady state precessionso that the magnetization is very unlikely to be near
u0=0
when the write current pulse is then sent. The so-preparedprecession increases the efficiency of the pulsed current andsignificantly accelerates the reversal for given current ampli-tude. Equivalently, it reduces the total current needed to re-verse in a certain duration.
We use pseudospin valves Co
75Fe25s2.5 nm d/
Cus6n m d/Co75Fe25s40 nm d, patterned as described
elsewhere.9The top sfree and thin dlayer is patterned into an
ellipse of size 2 a32b=173 380 nm2, while the bottom
layer is left unpatterned in the vicinity of the top ellipse. Thedevice is contacted in the current perpendicular to planesCPPdgeometry to sense its giant magnetoresistance sGMR d.
The bottom contact is short-circuited to the ground. The re-sistance of the device under test sDUT dis circaR=11 V
while the GMR between the parallel sPdand antiparallel
sAPdconfigurations is R
AP−RP=130 m V.The overall device
has a bandwidth of 13 GHz.
In spin transfer experiments, no external magnetic field
is applied. The setup aims at measuring the slow skHzdtime
dependence Rstdof the GMR while simultaneously submit-
ting the device to short current pulses. The instruments allow
simultaneous application of three electrical currents Ibias,Iac,
andIpulsein separate frequency domains. The bias current
sIbias,,mAdi sad c sf,5H z dtriangularly ramped currentadAuthor to whom correspondence should be addressed; electronic mail:
thibaut.devolder@ief.u-psud.frAPPLIED PHYSICS LETTERS 86, 062505 s2005 d
0003-6951/2005/86 ~6!/062505/3/$22.50 © 2005 American Institute of Physics 86, 062505-1
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130.113.69.48 On: Fri, 28 Nov 2014 21:08:00aimed at switching the magnetization of the top thin layer
through the spin-transfer effect sFig. 1, top loop d. The
switching are indicated by resistance jumps at a critical cur-
rent equal to ICAP!P=2.9 mA si.e., 2.6 3107A/cm2dfor the
antiparallel sAPdto parallel sPdtransition. The P to AP re-
quires more current, typically ICP!AP=−4.7 mA si.e., −3.8
3107A/cm2d. A small modulation sIac=36 mArms dis su-
perimposed for a measurement of the GMR with lock-in
technique. These low frequency currents Ibias+Iacare routed
in the device though the inductive port of a bias tee.
The capacitive port of the bias tee is fed by a pulse
generator. Current pulses are superimposed to the slow cur-rent sweep with a repetition frequency of 100 kHz and aduration
tpulse. Subsequent measurement of the hysteresis
loop shows either the regular jump at Ic, or a full jump at
another current Ibiasif a pulse has succeeded in switching the
magnetization of the free layer before Icis reached sFig. 1,
bottom loops d. The position of the full jump in the hysteresis
loop is used to identify all trio hIbias,Ipulse,tpulsejleading to
switching events. The studied intervals are respectively
ICAP!P,Ibias,ICP!AP;0,Ipulse,5ICP!AP; and 0.2 ,tpulse
,10 ns.
The required pulse duration for spin-transfer switching is
displayed in Fig. 2. When the DUT is not precharged si.e.,
Ibias=0d, the needed pulse duration decreases when the over-
drive is increased. In the interval ranging from200 ps to 2 ns, the needed pulse duration sthe “reversalspeed” dare well described by the rule of thumb 1/
t
<uI–ICun, withn<1.2±0.1. Note that this exponent is higher
than predicted by Refs. 6 and 7; the reason for this differenceis not understood.
The effect of precharging is described in Fig. 2 for I
bias
Þ0. The total current cost Ibias+Ipulseof a reversal event is
plotted against the way the system is prepared by Ibiasprior
to the application of the current pulse Ipulse.
For quite long pulse, e.g., t.5 ns, the reversal is not
significantly affected by the precharging strategy: the totalcurrent cost is almost constant, whatever the precharging cur-rentI
bias. It varies from 4.2 to 3.6 mA when Ibiasis increased
from −4 to 2 mA sFig. 2 d.
The effect of precharging becomes more significant for
pulses shorter than 2 ns: there appears an unequivocal nega-tive slope in the dependence of I
bias+IpulsevsIbias. This slope
shows that the reversal is eased when the DUT is positively
sIbias.0dprecharged, whereas the reversal is rendered more
difficult when the DUT is negatively precharged. Hence,
there is a net acceleration/deceleration obtained by positive/negative precharging.
In the regime of switching with strongly subnanosecond
pulses, the benefit of precharging increases substantially. Forinstance, a switching within 300 ps requires I
pulse=7mA
without precharge, and only Ipulse=3.9 mA with precharge
2 mA; 1.1 mA has been saved. Equivalently, with the totalapplied current of 7 mA, a precharge of 2 mA speeds up thereversal from 300 to 200 ps. The same precharge appliednegatively to the sample slows down the reversal far above300 ps sFig. 2 d.
In addition, there is a gradual slope change around I
bias
=2.1 mA ssee Fig. 2 d.The effect of precharging is even more
pronounced between that threshold and IC.
Before discussing these results, it is worth recalling that
the magnetization of a DUTsubmitted to a dc current smallerthan the critical current I
Chas already been studied theoreti-
cally by Sun et al. sRef. 6, Fig. 6 din the macrospin approxi-
mation at zero temperature. Three regimes were identified.At low current, the magnetization state lays at rest. Aspointed out later by Miltat et al.
10the lifetime of any fluc-
tuation diverges when the current increase to an instabilitycurrentI
instability. AboveIinstabilitythe easy axis magnetization
is driven unstable and turns into a steady-state precessionabout the easy axis,6with an in-plane fanning angle uthat
grows with the current above Iinstability.9A typically 20%–
30% larger current is then required for uto reach 90 deg. and
for the magnetization to switch. We carried on the same cal-culation in our case where
m0Hk=48 mT, and MS=1.5
3106A/m. The spin polarization and the damping param-
eter were chosen equal to 30% and 0.006 so that the criticalcurrent fits with our data, i.e., 2.7 310
7A/cm2sIC
=2.9 mA d. At 0 K, the instability current is found to be 2.2
3107A/cm2sIinstability=2.4 mA d.
Taking into account these predictions sFig. 6 in Ref. 6 d,
the effect of precharging with a current above Iinstabilityis
clear: it excites a steady state precession with a finite coneangle, so that the magnetization is very unlikely to be andstay near
u0=0 when the write current pulse is then sent.The
so-prepared precession increases the torque efficiency of thepulsed current and significantly accelerates the reversalevent. This assessment clearly correlates with the experi-ment: the precharging has a much more dramatic influenceon the reversal speed as we approach the switching current
FIG. 1. Hysteresis loops of the giant magnetoresistance vs dc bias current.
The top curve is recorded without any current pulse. The other curves arerecorded while current pulses are continuously sent through the device dur-ing the loop. Inset: sketch of the experimental procedure with the appliedcurrent vs time.
FIG. 2. Map of the switching duration tas a function of the total current
magnitude Ipulse+Ibiasand the precharging current Ibias. The vertical dotted
line is at a current Ibias=2.1mA sj=1.9 3107A/cm2dabove which the ef-
ficiency of precharging is dramatically enhanced.062505-2 Devolder et al. Appl. Phys. Lett. 86, 062505 ~2005 !
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130.113.69.48 On: Fri, 28 Nov 2014 21:08:00ssee the slope change near the vertical dotted line in Fig. 2 d.
Despite this nice correlation, the zero temperature model suf-fers from quantitative deficiencies. It overestimates the cur-rent above which the reversal speed is largely boosted. It alsodoes not explain the non vanishing effect of precharging withI
bias,Iinstability.
In order to better understand the accelerator/decelerator
effect of the bias current, we added the temperature in theLandau-Lifshitz-Gilbert equation modified5with the spin-
torque term. Since the sample is small, we still make themacrospin approximation for the thin layer and we assume afixed thick layer magnetization.Astochastic field is added ateach time step to account for the temperature, so that, insteadof having stable magnetization or perfectly periodic preces-sion trajectories, the magnetization now explores part of thephase space around the zero temperature positions. In thecalculations, the initial magnetization state is taken slightlyoff the easy axis. The starting magnetization position s
u0
=2.6 deg. dis chosen consistent from fluctuation-dissipation
theorem leading to u0rms<˛kT/m0MSHkVat 300 K.11
Figure 3 saddisplays a representative calculated magneti-
zation trajectory for an applied current above IinstabilityatT
=300 K.The precession excited by the spin flow is no longerstrictly periodic: the off-easy axis magnetization excursionfluctuates with time.
In Fig. 3 sbd, we have gathered the standard deviation of
the in-plane fanning angle
u0rmsof the magnetization for vari-
ous values of Ibiasat 300 K. Due to the temperature, the
precession angle is no longer zero at small currents. It en-larges dramatically when the current density exceeds a
threshold 1.9 3107A/cm2sIbias.2.1 mA d. The three re-
gimes formerly identified at 0 K6still qualitatively exist at
300 K, but the random nature of thermal activation blurs thetransitions between regimes and slightly diminishes I
instability.
Note that this finite temperature model gives a much morequantitative agreement with the experimental efficiency ofthe precharge strategy. The strategy yields indeed some ben-
efit below I
bias=2.1 mA, i.e., when u0rmsis predicted to be
only slightly above its thermal level, while prechargingyields a dramatically higher benefit above I
bias.2.1 mA
sFig. 2 dwhen u0rmsis predicted to quickly increase.
The room temperature model sFig. 3 dalso helps to un-
derstand why precharging is not that efficient for pulseslonger than 2–5 ns. Indeed Fig. 3 sadindicates that it takes
typically 2 ns for the precession to warm up from the initialmagnetization state. In addition, the excursion fluctuates insuch a way that one maxima is very likely to be reached inany 5 ns interval.As a result, even if no precharge is done, apulse longer than this characteristic warm-up time ensuresthat the precession will have enough delay to settle up during
the pulse. Hence, precharging should not give any significantbenefit unless being in the sub-2-ns regime, in agreementwith the experiments sFig. 2 d.
Pulses shorter than 200 ps could not be studied experi-
mentally. However, since the acceleration relies on the abil-ity of the pulse to sample an initial magnetization state offthe easy axis, the precharging strategy should remain effi-cient as long as the pulse duration exceeds the half preces-
sion period
p/g0˛MSHkwhere g0=221 kHzA−1m, i.e.,
typically down to 60 ps.
In summary, we have presented a strategy to decrease
the current pulse duration needed for a spin-transfer switch-ing event. It requires to precharge the device with a biascurrent prior to the switching pulse. When the prechargingcurrent has the same polarity than the pulse current, it easesthe reversal, while it inhibits the reversal in the oppositecase. In the acceleration case, the precharge current excitesthe magnetization to a precession trajectory out of its easyaxis, which dramatically increases the susceptibility of themagnetization to the subsequent pulsed current.
This strategy was proven efficient for pulse duration be-
tween 200 ps and 2 ns, with potential usefulness down to60 ps. Potential applications of this reversal scheme are an-ticipated for future MRAM based on spin-transfer switching,especially since precharging decreases the write current andconsequently allows us to use smaller write transistors.
1J. Slonczewski, J. Magn. Magn. Mater. 159,1s1996 d.
2M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407 s2002 d.
3E. B. Myers, F. J. Albert, J. C. Sankey, E. Bonet, R. A. Buhrman, and D.
C. Ralph, Phys. Rev. Lett. 89, 196801 s2002 d.
4F. J. Albert, N. C. Emley, E. B. Myers, D. C. Ralph, and R. A. Buhrman,
Phys. Rev. Lett. 89, 226802 s2002 d.
5J. Grollier, V. Cros, H. Jaffrès, A. Hamzic, J. M. George, G. Faini, J. Ben
Youssef, H. Le Gall, and A. Fert, Phys. Rev. B 67, 174402 s2003 d.
6J. Z. Sun, Phys. Rev. B 62, 570 s2000 d.
7Z. Li and S. Zhang, Phys. Rev. B 68, 024404 s2003 d.
8B. C. Choi, M. Belov, W. K. Hiebert, G. E. Ballentine, and M. R. Free-
man, Phys. Rev. Lett. 86, 728 s2001 d.
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Fukushima, and Y. Suzuki, Appl. Phys. Lett. 85, 5358 s2004 d.
10J. Miltat and A. Thiaville, Invited Talk at the International Conference on
Magnetism, Rome, July 2003.
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s2003 d.
FIG. 3. sadCalculated macrospin trajectory at T=300 K for an initial mag-
netization slightly off the easy axis su=2.6 deg dand for an applied bias
current of 2.1 3107A/cm2;sbdrms value of the in-plane excursion u0for
various bias currents, macrospin calculation at 300 K. The vertical dotted
line is drawn at j=1.9 3107A/cm2sIbias=2.1 mA d, where u0rmsrms under-
goes a steep increase.062505-3 Devolder et al. Appl. Phys. Lett. 86, 062505 ~2005 !
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1.3355903.pdf | Optimization of magnetic anisotropy and applied fields for hyperthermia applications
Hweerin Sohn and R. H. Victora
Citation: Journal of Applied Physics 107, 09B312 (2010); doi: 10.1063/1.3355903
View online: http://dx.doi.org/10.1063/1.3355903
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/107/9?ver=pdfcov
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128.114.34.22 On: Tue, 25 Nov 2014 21:40:16Optimization of magnetic anisotropy and applied fields for hyperthermia
applications
Hweerin Sohn and R. H. Victoraa/H20850
Department of Electrical and Computer Engineering, The Center for Micromagnetics and Information
Technologies (MINT), University of Minnesota, Minneapolis, Minnesota, USA
/H20849Presented 20 January 2010; received 30 October 2009; accepted 5 December 2009;
published online 3 May 2010 /H20850
Magnetic anisotropy and applied fields for hyperthermia applications have been optimized for iron
cobalt nanocrystalline particles using numerical micromagnetics. The optimized anisotropy energyis 7.6 k
BTat 500 KHz and the hysteresis loss at this optimized energy is approximately
120/H11003106ergs //H20849s/H11569g/H20850for a very small oscillating field of magnitude 10 Oe. We have also
investigated the effects of varying the applied field and find that the addition of a 20 Oe static field
applied perpendicular to the oscillating field approximately doubles the energy loss withoutsubjecting the patient to additional radiation. This is an important benefit for magnetichyperthermia. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3355903 /H20852
I. INTRODUCTION
Cancer is a leading contributor to mortality rates in most
countries. Biomedical application using fine magnetic par-ticles in alternating magnetic fields for thermotherapy hasattracted interest for more than half a century.
1Among many
applications, hyperthermia using magnetic nanoparticles is avery attractive and feasible therapy for tumor or cancer celltreatment. Studies of hyperthermia show that cancer growthcan be delayed or terminated at temperatures in the range of42–48 °C, while normal cells can tolerate highertemperatures.
2Magnetic nanocrystalline particles in alternat-
ing magnetic fields may have great heating efficienciescaused by hysteresis losses from the magnetization reversalprocesses. Recently nonoxide, high magnetic moment mate-rials were produced experimentally for a hyperthermia appli-cation. The Fe
70Co30nanoparticles had a cubic shape with
side length 12 m, which include da2n m nonmagnetic shell,
and a core magnetization of 1845 emu /cm3.3The specific
loss power /H20849SLP /H20850of magnetic nanoparticles depends on
magnetic moment, anisotropy energy density, particle size,and size distribution.
4Among those components, SLP is
strongly dependent on magnetic moment and anisotropy con-stant. In this paper, the interesting region where the thermalenergies and the energy barriers are comparable has beenexplored through numerical micromagnetics and explainedanalytically.
There are two different source of energy loss. Néel re-
laxation occurs due to the reorientation of the magnetic mo-ment inside of the particles. The characteristic relaxationtime is given by
/H9270N=/H92700e/H20849KV /kBT/H20850,
where /H92700is thought to be 10−9s,5Kis an anisotropy con-
stant, and Vis the volume of particle. The equation shows
the relaxation time depends on the ratio of the anisotropyenergy to the thermal energy. The hysteresis loss power maybe easily estimated from the measured hysteresis loop. The
comparison between estimated values of hysteresis losspower and results of calorimetric determination of the heatpower has been made and shown to be in good agreement.
6
The other source of energy loss is caused by the reorientationof the whole particle itself in a fluid solution of magneticnanoparticles /H20849Brown relaxation /H20850. In general, the two differ-
ent relaxation mechanisms occur at the same time and aneffective characteristic time is given by
/H9270eff=/H9270N/H9270B
/H9270N+/H9270B,
where /H9270Bis the Brown relaxation. In an alternating magnetic
field the frequency response of the magnetic nanoparticlescan be investigated by measuring susceptibility spectra. Sus-ceptibility is composed of real and imaginary parts. Magneticloss is related to the imaginary part of susceptibility
/H9273/H11033
which is described by,
/H9273/H11033=/H92730/H9275/H9270
1+/H20849/H9275/H9270/H208502,/H92730=MS2V
akBT,
where MSis saturation magnetization, /H92730is the static suscep-
tibility, and ais a factor depending on the ratio of the aniso-
tropy energy to thermal energy in the range of 1–3.7The loss
power density in alternating magnetic field is proportional tomagnetic field amplitude, field frequency, and
/H9273/H11033.
P=H2/H9275/H9273/H11033/2.
As the loss power density increases, the heating generation
increases as well. Increasing magnetic field amplitude andfrequency is the easiest way to raise the loss power density.However, magnetic hyperthermia has physical limits for bio-compatibility such as maximum rf-magnetic field amplitudesand frequency. Brezovich has discussed the upper limit of theproduct between magnetic field amplitude and frequency.
8
Magnetic losses of different magnetic nanoparticles show avariety of nonlinear dependences on field amplitude, fieldfrequency, volume, etc. Therefore high magnetic moment
a/H20850Electronic mail: victora@umn.edu.JOURNAL OF APPLIED PHYSICS 107, 09B312 /H208492010 /H20850
0021-8979/2010/107 /H208499/H20850/09B312/3/$30.00 © 2010 American Institute of Physics 107 , 09B312-1
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128.114.34.22 On: Tue, 25 Nov 2014 21:40:16nanoparticles such as iron cobalt with gold coating are very
promising for hyperthermia application. Their performancewill be optimized using micromagnetics to determine Néelrelaxation by integrating minor hysteresis loops.
II. RESULTS AND DISCUSSION
Micromagnetics allows prediction for Néel relaxation
loss while avoiding the oversimplification inherent to ana-lytic theories that can produce erroneous answers or missimportant discoveries. The simulation was based on the nu-merical integration of the stochastic Landau–Lifshitz–Gilbert/H20849LLG /H20850using the thermal fluctuation formalism developed by
Brown.
9The stochastic LLG equation is given by
dm
dt=/H9253
1+/H92512m/H11003/H20849Heff+Hth/H20850−/H9253/H9251
1+/H92512m/H11003/H20851m/H11003/H20849Heff
+Hth/H20850/H20852,
where /H9253is the gyromagnetic ratio, mis the unit vector
M/Msand/H9251is the damping constant. H effrepresents an
effective field including the uniaxial anisotropy field and theapplied field. The thermal fluctuation field, H
thfollows a
Gaussian distribution with mean equal to zero and is givenby
9
/H20855hi/H20849t/H20850hj/H20849t+/H9270/H20850/H20856=/H92682/H9254ij/H9254/H20849/H9270/H20850, where /H92682=2kBT/H9251
/H9253MsV/H9004t.
The iron cobalt nanoparticles are assumed uniformly magne-
tized. A uniaxal shape anisotropy energy is dominant andgiven by
E
uniaxial =Kuniaxial /H11569SIN2/H20849/H9258/H20850/H11569V,
where /H9258is the angle to the easy axis and the volume Vis
520 nm3. The calculation included magnetostatic interac-
tions between ten particles with random easy axes and wasrepeated 1000 times to generate adequate statistics.
Figure 1shows a variety of minor hysteresis loops
evaluated at comstant sweep rate for different anisotropy en-ergy densities, K
u. In the case of hysteresis loops with smallanisotropy constants, superparamagnetic effects are shown in
Figs. 1/H20849a/H20850–1/H20849c/H20850. With increasing anisotropy energy densities,
the area of a hysteresis loop increases as shown in Figs.1/H20849e/H20850–1/H20849g/H20850. Eventually, the anisotropy becomes too large for
thermal switching, and the loop reverts to a straight line asshown in Figs. 1/H20849h/H20850and1/H20849i/H20850. Interestingly, we find the at-
tempt frequency A to be about 2 /H1100310
7Hz. This agrees well
with Ref. 10but contradicts the usual assumption of 1 //H92700
=109Hz.5
Figure 2/H20849a/H20850shows the total energy loss for different an-
isotropy constants for the case of a small oscillating fieldwith magnitude 10 Oe. The energy loss per unit volume canbe obtained from /H20859H/H20849t/H20850dM.
11The highest peak shows ap-
proximately 120 /H11003106ergs //H20849s/H11569g/H20850/H20849=12 W /g/H20850hysteresis
loss energy at the anisotropy of 6 /H11003105ergs /cm3. We have
explored the effects of varying the applied field and find thatthe combination of a relatively small static field applied per-pendicular to the oscillating field approximately doubles theenergy loss for a given applied power. This is an importantbenefit for magnetic hyperthermia. Figure 2/H20849b/H20850shows the
total energy loss with a 20 Oe static applied field and alter-nating magnetic field ranging from /H1100210 to 10 Oe for differ-
ent anisotropy constants. The highest peak shows approxi-mately 220 /H1100310
6ergs //H20849s/H11569g/H20850per particle at the same
anisotropy energy density. Similar results are shown in Figs.
3/H20849a/H20850and3/H20849b/H20850under two different alternating magnetic field
ranges. Figure 3/H20849a/H20850shows the total energy per unit volume
with a 15 Oe static applied field and alternating magneticfield ranging from /H110025 to 5 Oe and Fig. 3/H20849b/H20850shows the total
energy per unit volume with a 25 Oe static applied field andalternating magnetic field ranging from /H1100220 to 20 Oe. The
highest peaks show approximately 94 /H1100310
6ergs //H20849s/H11569g/H20850and
710/H11003106ergs //H20849s/H11569g/H20850per particle at the same anisotropy en-
FIG. 1. Minor hysteresis loops of randomly oriented iron cobalt nanopar-
ticles for different anisotropy energy densities, Ku/H20849ergs /cm3/H20850which are
/H20849a/H208501/H11003105,/H20849b/H208502/H11003105,/H20849c/H208503/H11003105,/H20849d/H208504/H11003105,/H20849e/H208505/H11003105,/H20849f/H208506/H11003105,/H20849g/H20850
7/H11003105,/H20849h/H208508/H11003105,a n d /H20849i/H208509/H11003105.
FIG. 2. Hysteresis losses of randomly oriented iron cobalt nanoparticles for
different anisotropy energy densities, Kufrom 1 /H11003105to 1/H11003106ergs /cm3,
and alternating magnetic field range from /H1100210 to 10 Oe /H20849a/H20850without the
static field and /H20849b/H20850with the static magnetic field.
FIG. 3. Hysteresis losses of randomly oriented iron cobalt nanoparticles for
different anisotropy energy densities, Kufrom 1 /H11003105to 1/H11003106ergs /cm3,
and alternating magnetic field range /H20849a/H20850from/H110025t o5O ea n d /H20849b/H20850from/H1100220
to 20 Oe with the static magnetic field, respectively.09B312-2 H. Sohn and R. H. Victora J. Appl. Phys. 107 , 09B312 /H208492010 /H20850
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128.114.34.22 On: Tue, 25 Nov 2014 21:40:16ergy density, respectively. However, for large oscillating
fields, the contribution of the static applied field becomemuch less. For example, the contribution of the static fieldwhen the oscillating field is greater than 50 Oe is less than a10% increase in loss power for this size particle.
The total energy density of a nanoparticle is given by
E
tot=Ksin2/H20849/H9258/H20850−HM scos/H20849/H9258−/H9274/H20850,
where Kis anisotropy energy density, /H9274represents the angle
between the easy axis and the magnetic field and /H9258represents
the angle between the magnetization and the anisotropy axis.For randomly oriented easy axes of the particles, an energybarrier /H9004Ecan be analytically obtained for the case of H
small compared to 2 K/M
s
dE
dsin/H20849/H9258/H20850=0 ,→sin/H20849/H9258/H20850/H11015HM ssin/H20849/H9274/H20850
2K,
/H11030Emin=/H11006HM scos/H20849/H9274/H20850,
dE
dcos/H20849/H9258/H20850=0 ,→cos/H20849/H9258/H20850/H11015−HM scos/H20849/H9274/H20850
2K,
/H11030Emax=K−HM s/H20841sin/H20849/H9274/H20850/H20841,
/H9004E=K−HM s/H20841sin/H20849/H9274/H20850/H20841/H11006HM scos/H20849/H9274/H20850.
The magnetization direction of each particle can be seen to
align with the easy axis for very small field. The magnetiza-tion reversal process takes place by transition over this en-ergy barrier. The switching probability is determined by therelation between measuring time and relaxation time. Forsufficiently large barrier, the loss power from the energy bar-rier distribution is given by
P=4AHM
scos/H20849/H9274/H20850sinh/H20875VHM scos/H20849/H9274/H20850
kBT/H20876
/H11003e−/H20851K−HMs/H20841sin/H20849/H9274/H20850/H20841/H20852V/kBT.
This expression substantially differs from the usual linear
theory as described in the introduction. However, ifVHM
s//H20849kBT/H20850is not too large, then the loss power can be
approximated by
P=4AV/H20849HM s/H208502
kBTcos2/H20849/H9274/H20850e−KV /kBT,
which resembles the usual linear expression at optimal /H20849and
high /H20850frequencies.
The linear expression is tested in Fig. 4, which shows the
hysteresis losses versus the square of the oscillating mag-netic field at a zero static field. It can be seen that for lowfields below about 50 Oe, the loss depends linearly on H
osc2.This corresponds to HVM s/H11011/H11349kBTwhich makes sense in
view of the sinh function and provides an upper limit to thelinear theory.
III. CONCLUSIONS
We have micromagnetically simulated the Néel relax-
ation of superparamagnetic nanoparticles subject to an oscil-lating field. We calculate the optimized anisotropy energy ofthe simulated nanocrystalline iron cobalt particles to be3.142/H1100310
−13ergs which corresponds to an energy barrier of
7.6kBTat room temperature. We find that application of a
small field enhances the loss without the necessity for en-hanced radiation. The frequency of magnetization attempts tosurmount the energy barrier is shown to be about two ordersof magnitude smaller than previously estimated. Finally, weestablish upper limits for the applied field magnitude, beyondwhich the normal linear theory will fail.
ACKNOWLEDGMENTS
We wish to thank the University of Minnesota Super-
computing Institute for computer time. This work was par-tially supported by the Medical Device Center of Institute ofEngineering in Medicine at University of Minnesota and Na-tional Science Foundation under Grant No. BME 0730825.
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nanoparticles vs Hosc2at a zero static field.09B312-3 H. Sohn and R. H. Victora J. Appl. Phys. 107 , 09B312 /H208492010 /H20850
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1.2051789.pdf | Exchange spring structures and coercivity reduction in Fe Pt ∕ Fe Rh bilayers: A
comparison of multiscale and micromagnetic calculations
F. Garcia-Sanchez, O. Chubykalo-Fesenko, O. Mryasov, R. W. Chantrell, and K. Yu. Guslienko
Citation: Applied Physics Letters 87, 122501 (2005); doi: 10.1063/1.2051789
View online: http://dx.doi.org/10.1063/1.2051789
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129.24.51.181 On: Sat, 29 Nov 2014 05:53:14Exchange spring structures and coercivity reduction in FePt/FeRh
bilayers: A comparison of multiscale and micromagnetic calculations
F . Garcia-Sancheza/H20850and O. Chubykalo-Fesenko
Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, E-28049 Madrid, Spain
O. Mryasov and R. W. Chantrell
Seagate Research, 1251 Waterfront Place, Pittsburgh, Pennsylvania 15222
K. Yu. Guslienko
Argonne National Laboratory, Argonne, Illinois 60439
/H20849Received 20 June 2005; accepted 26 July 2005; published online 12 September 2005 /H20850
Calculations of magnetization reversal mechanism and coercivity reduction in exchange coupled
FePt/FeRh bilayers are presented. It is shown by comparison with atomistic model calculations thatthe use of a standard micromagnetic model leads to an underestimation of the exchange energy atthe interface, leading to a reduced coercivity decrease for small interfacial exchange energyconstant. This is due to the failure of the domain wall /H20849DW /H20850to penetrate the hard FePt phase in the
micromagnetic calculations. A multiscale model is proposed based an atomic level simulation in theinterface region coupled with a micromagnetic approach elsewhere. This leads to improvedcalculations of DW structures at the interface, allowing a detailed study of the magnetizationreversal mechanism. The new approach predicts a saturation in the coercivity reduction as a functionof interface exchange energy at 4% of the bulk value, which is associated with complete continuityof the DW across the interface. © 2005 American Institute of Physics ./H20851DOI: 10.1063/1.2051789 /H20852
The traditional approach to increase areal density in
magnetic recording is based on a scaling approach in whichthe grain size of the medium is decreased in order to achievethe required signal to noise ratio. However, a reduction in thegrain size leads to a reduction in the value KV/k
BT, where K
is the anisotropy constant, Vthe grain volume, and kBis
Boltzmann’s constant, which determines the thermal stabilityof the written information. Essentially, values of KV/k
BT
/H1102260 are required to ensure the long-term stability of written
information. Clearly, a reduction in the grain size can becompensated for by an increase in K, as in FePt materials.
1
However, associated with an increase in Kis a consequent
increase in the anisotropy field of the medium, given byH
k=2K/Ms, with Msthe saturation magnetization. This leads
to increasing medium coercivity and the requirement oflarger write fields. Although the trend to perpendicular re-cording with its larger write field will alleviate this problemto some extent, it is clear that write field limitations providea limit on the areal density in conventional recording. Oneapproach proposed to circumvent this problem is thermallyor heat-assisted magnetic recording.
2This is based on the
fact that the anisotropy constant decreases with increasingtemperature at a faster rate than the magnetization, leading toa reduction in the anisotropy field and coercivity. However,as pointed out by Thiele et al. ,
3the exponents of the power
law variations of KandMwith temperature are such that the
anisotropy field varies more slowly than K, requiring tem-
peratures close to or above the Curie temperature to writeinformation. This leads to significant practical problems as-
sociated with the head-disk interface, and especially the lossof lubricant.
4As a solution, Thiele et al.3proposed the idea
of a composite medium of FePt and FeRh. It has been estab-lished that the ordered bcc alloy FeRh undergoes a metamag-netic transition from antiferromagnetic to ferromagnetic
state.
5Our earlier calculations6using a one-dimensional /H208491D/H20850
model showed that the coercivity reduction results from anexchange spring mechanism.
7However, the exact reversal
mechanism and the degree of interfacial energy required formaximum coercivity reduction in a soft/hard magnetic mate-rial are questions requiring a three-dimensional /H208493D/H20850
calculation.
8Such a calculation is the subject of this letter.
We show that the straightforward application of micromag-netic approach seriously underestimates the coercivity reduc-tion for small interfacial exchange energy strengths. A mul-tiscale approach is proposed leading to an improveddescription of the domain wall /H20849DW /H20850structure across the
FePt/FeRh boundary. It is shown that the maximum reduc-tion in coercivity occurs when the DW structure across theinterface becomes continuous.
We use two computational approaches. The first is a
standard micromagnetic calculation, within which the systemis discretized into cubes of length 1.5 nm /H20849smaller than the
DW width in FePt of 4 nm /H20850. The system size is 80 /H1100380
/H1100330 cells to give a total size of 120 nm /H11003120 nm /H1100345 nm.
The thickness /H20849in the zdirection /H20850is divided into ten cells of
FePt /H2084915 nm /H20850and 20 cells of FeRh /H2084930 nm /H20850. The demagne-
tization field is calculated using the dynamic alternating di-
rection implicit
9/H20849DADI /H20850method, which solves the Poisson
equation using a fictitious time step. Periodic boundary con-ditions are used in x,ydirections, but not in the zdirection in
which several cells are added to simulate free space /H20849zero-
padding /H20850. The expression used for error is the maximum of
the normalized local torque given by max /H20648M
i
/H11003Hieff/H20648//H20851/H20648Mi/H20648/H20648Hieff/H20648/H20852/H11021/H9280, and the calculation finishes accord-
ing to the criterion /H9280=10−5. The minimization method used is
the dissipative dynamics resulting from Landau-Lifshitz-Gilbert with large damping constant. The following value ofthe material constants were used: A=10
−6erg/cm, KFePt=2a/H20850Electronic mail: fgarcias@icmm.csic.esAPPLIED PHYSICS LETTERS 87, 122501 /H208492005 /H20850
0003-6951/2005/87 /H2084912/H20850/122501/3/$22.50 © 2005 American Institute of Physics 87, 122501-1
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129.24.51.181 On: Sat, 29 Nov 2014 05:53:14/H11003107erg/cm3,KFeRh=0, MsFePt=1100 emu/cm3,MsFeRh=
1270 emu/cm3.
The variation of the coercivity with interfacial exchange
energy Jsfor the micromagnetic model is shown in Fig. 1.
First we discuss the comparison with atomistic quasi-one-dimensional model used in our earlier publication.
6This
model assumes uniform rotation in plane but allows atomicresolution in perpendicular direction. This model will be re-ferred as the 1D model. Clearly, the micromagnetic modelpredictions differ significantly from the 1D atomic scalemodel. It can be seen in Fig. 1 that there appears to be acritical value of exchange energy below which the micro-magnetic model shows small reduction in H
c, whereas the
1D model shows a continuous decrease of Hcwith Js.I n
addition, the 1D model shows saturation in the coercivityreduction at around J
s/J=0.2 /H20849with Jthe bulk value /H20850whereas
the micromagnetic model predicts a continuous /H20849albeit slow /H20850
increase up to the bulk value of exchange. Examination ofthe DW structures across the FePt/FeRh interface indicatedthat the micromagnetic model allowed large changes in mag-netization across the interface, which was likely to arise be-cause the micromagnetic approximation underestimates theexchange energy associated with large changes of magneti-zation. To test this hypothesis we have developed a multi-scale model, including an atomistic calculation in the inter-
face region.
The multiscale model is based on the partitioning of the
computational cell into atomistic and micromagnetic regions,as shown schematically in Fig. 2. The atomistic scale dis-cretization is used in the interface regions, where spatiallyrapid changes in magnetization might be expected. In thisregion, the exchange is treated exactly within the Heisenbergmodel, allowing the system access to the entire spectrum ofmagnetic excitations. The lattice structure is taken into ac-count explicitly, including best lattice-matching orientationof the FePt and FeRh. For the calculation of the magneto-static field, the atomistic region is partitioned into macrocellsof the same size as the micromagnetic region. Inside eachmacrocell the volume charges are neglected. For computa-tional simplicity the thickness of the atomistic region is com-mensurate with the size of the micromagnetic cells. In thiscase we use twice the macromagnetic cell size, giving athickness of 3 nm treated atomistically on both the FePt andFeRh sides of the interface. The average magnetization ofeach macrocell is calculated and used in the micromagnetic
evaluation of the magnetostatic field. In the micromagneticregion we use a discretization into cells in the usual way witha size of 1.5 /H110031.5/H110031.5 nm. The magnetostatic field is cal-
culated using DADI, including the averaged magnetizationof the micromagnetic cells within the atomistic region in thetotal calculation. Finally, we have to consider the interfacebetween the atomistic and micromagnetic region. On thisboundary we use Heisenberg exchange between the actualatoms in the atomistic region and virtual atoms in the micro-magnetic region having the direction of the average magne-tization in the micromagnetic cell projected onto the physicallattice. We would like to point out here that the necessity ofthe multiscale approach for small intergranular exchange val-ues has been suggested in Ref. 10.
The results of calculations of DW structures using the
multiscale model are shown in Fig. 3. The most importantfeatures of the calculations are twofold. Firstly, it can be seenthat there is a transition from the discontinuous DW structureat low J
s/Jto a continuous wall at a critical value of Jc/J
=0.04. For Js/H11022Jcthere is little evolution of the DW struc-
ture. The predictions of the multiscale model are in markedcontrast to those of the micromagnetic approach. This isdemonstrated in the inset of Fig. 3, which compares the DW
FIG. 1. The variation of coercivity /H20849given in units of anisotropy field Hk/H20850
with interfacial exchange energy for the micromagnetic and multiscale mod-els in comparison with a 1D atomic scale model.
FIG. 2. Schematic diagram showing the basis of the multiscale model interms of the partitioning of the system into micromagnetic and atomisticregions.
FIG. 3. DW structure calculated as a function of interfacial exchange /H20849Js/H20850
given in units of bulk exchange /H20849J/H20850. Results of multiscale calculations are
shown in comparison with micromagnetic /H20849inset /H20850.122501-2 Garcia-Sanchez et al. Appl. Phys. Lett. 87, 122501 /H208492005 /H20850
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129.24.51.181 On: Sat, 29 Nov 2014 05:53:14structures calculated using the multiscale and micromagnetic
models for Js/J=0.1. Although this value is greater than the
critical value Jcpredicted by the multiscale model, the mi-
cromagnetic calculation still predicts a discontinuous DWstructure. This is presumably because the micromagnetic de-termination of the exchange energy relies on a long-wavelength approximation, which underestimates the ex-change energy associated with rapid spatial variation themagnetization, thereby allowing discontinuous DW struc-tures under conditions where these are not supported by themultiscale calculations.
We have also studied the magnetization reversal process
itself using the multiscale model. The magnetization reversalprocess is found to essentially involve DW propagation, butis somewhat complex, and takes place in two distinct stages.In stage 1 there is a gradual propagation of the DW into thehard FePt phase. During this stage the magnetization of theFeRh layer changes relatively slowly. Complete reversal ofthe FeRh layer is not necessary to induce propagation of theDW into the FePt phase. During the second stage of themagnetization reversal process the magnetization in the FePtbecomes more negative than that in the FeRh. This estab-lishes a reverse DW that propagates back into the FeRhlayer, resulting in complete reversal of the whole system /H20849see
Fig. 4 /H20850.
Finally, we return to the calculation of the coercivity
reduction using the multiscale model. The results are givenin Fig. 1 in comparison with the 1D calculations and theresults of the 3D micromagnetic model. The predicted reduc-tion in H
cfrom the multiscale calculations as a function of
Js/Jis more rapid than that of the micromagnetic model, for
reasons which can now be understood in terms of the limi-tations of the micromagnetic approach. These limitations arealso responsible for the failure of the micromagnetic modelto predict saturation of the coercivity reduction until ex-
tremely large values of interfacial energy. In the case of themultiscale calculations it is interesting to note that there is acorrespondence between the onset of the continuous interfa-cial DW structure and the saturation of the coercivity reduc-tion shown in Fig. 1. Clearly, a discontinuous DW structure,which has an interfacial energy larger than that required toachieve a continuous DW, requires a larger field to initiatethe propagation of the DW leading to magnetization reversal.The prediction of saturation at relatively low interfacial ex-change energy is of practical significance. Clearly, the pro-duction of bilayer systems with this level of exchange energyare necessary in order to maximize the reduction in coerciv-ity. A further consideration arises from the fact that the re-duction in H
cis very rapid for small Js/J. This means that in
this region any local fluctuations in the exchange energystrength will give a contribution to the switching field distri-bution /H20849SFD /H20850over and above those arising from the disper-
sion of the intrinsic properties /H20849principally the anisotropy and
the grain volume /H20850. Given that a narrow SFD is required for
good recording properties, it would appear to be desirable todevelop bilayers with exchange energy in the saturation re-gion. The degree of exchange energy in bilayer systems suchas studied here is not well known and difficult to quantify.However, the interfacial exchange is generally rather small,and it may be that even such relatively small values asJ
s/J=0.04 may be beyond techniques such as sputtering, and
thus it may be possible that successful composite media mayrequire molecular-beam epitaxy or some other advanced ep-itaxial technique. The interfacial exchange energy strength isthus an open and important question, which requires seriousinvestigation if such composite media are to be achievedpractically.
1H. Zeng, S. Sun, T. S. Vedantam, J.-P. Liu, Z.-R. Dai, and Z.-L. Wang,
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5J. S. Kouvel, J. Appl. Phys. 37, 1257 /H208491966 /H20850.
6K. Yu. Guslienko, O. Chubykalo-Fesenko, O. Mryasov, R. Chantrell, and
D. Weller, Phys. Rev. B 70, 104405 /H208492004 /H20850.
7E. F. Kneller and R. Hawig, IEEE Trans. Magn. 27, 3588 /H208491991 /H20850;E .E .
Fullerton, J. S. Jiang, M. Grimsditch, C. H. Sowers, and S. D. Bader, Phys.Rev. B 58, 12193 /H208491998 /H20850.
8R. H. Victora and X. Shen, IEEE Trans. Magn. 41,5 3 7 /H208492005 /H20850; D. Suess,
T. Schrefl, R. Dittrich, M. Kirschner, F. Dorfbauer, G. Hrkac, and J. Fidler,J. Magn. Magn. Mater. 290–291 , 551 /H208492005 /H20850.
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FIG. 4. Propagation of the DW structure near the FePt/FeRh interface. The
calculations are carried out using the multiscale model with Js/J=0.8.122501-3 Garcia-Sanchez et al. Appl. Phys. Lett. 87, 122501 /H208492005 /H20850
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129.24.51.181 On: Sat, 29 Nov 2014 05:53:14 |
1.4799143.pdf | Magnetic and transport properties of tetragonal- or cubic-Heusler-type
Co-substituted Mn-Ga epitaxial thin films
T. Kubota,1,a)S. Ouardi,2S. Mizukami,1G. H. Fecher,2C. Felser,2Y . Ando,3
and T. Miyazaki1
1WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
2Department of Inorganic Chemistry, Max Planck Institute for Chemical Physics of Solids,
Dresden 01187, Germany
3Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan
(Presented 17 January 2013; received 1 November 2012; accepted 7 January 2013; published online
3 April 2013)
The composition dependence of the structural, magnetic, and transport properties of epitaxially
grown Mn-Co-Ga films were investigated. The crystal structure was observed to change from
tetragonal to cubic as the Co content was increased. In terms of the dependence of saturation
magnetization on the Co content, relatively small value was obtained for the Mn 2.3Co0.4Ga1.3film
at a large Kuvalue of 9.2 Merg/cm3. Electrical resistivity of Mn-Co-Ga films was larger than that
of pure Mn-Ga film. The maximum value of the resistivity was 490 lXcm for Mn 2.2Co0.6Ga1.2
film. The high resistivity of Mn-Co-Ga might be due to the presence of localized electron states in
the films due to chemical disordering caused by the Co substitution. VC2013 American Institute of
Physics .[http://dx.doi.org/10.1063/1.4799143 ]
I. INTRODUCTION
Mn-Ga ordered alloys with tetragonal distortion, which
are primarily used in spintronics applications, are known to
possess high magnetic anisotropy; ( Ku)1–5this property of
Mn-Ga alloys is essential to the retention of stored data inspintronics-based memory devices with nanometer-scale ele-
ments. In addition, materials with high spin polarization
2,6,7
and small Gilbert damping constant ( a)8are also particularly
attractive for realizing spin-transfer-torque (STT)-type mag-
netoresistive random access memory (MRAM).9The pri-
mary issue to be addressed in MRAM applications is toreduce the critical current ( I
c) required to cause STT-induced
magnetization switching. The current Icis proportional to the
damping constant aand the saturation magnetization ( Ms)o f
the free layer present in the magnetic tunnel junctions of the
MRAM,10,11and thus magnetic material with small Msvalue
is important as well. D022-type Mn-Ga alloys have been
known to exhibit relatively small Msvalue of about
250 emu/cm3, and moreover Alijani et al. discovered that Co
substitution of Mn in D022-type Mn 3Ga can reduce the Ms
value further.12In a recent study, we obtained a high Ku
value of the order of 107erg/cm3for an epitaxially grown
D022Mn-Co-Ga film.13Our preliminary work in the study
demonstrated attractive possibilities of using such films for
future memory applications; however, systematic investiga-
tions of epitaxially grown Mn-Co-Ga films over a wide com-position range are still required. Therefore, we investigated
structural, magnetic, and transport properties of Mn-Co-Ga
films in this study.II. EXPERIMENTAL
All the films were prepared by using an ultra-high-vac-
uum magnetron sputtering system. The stacking structure of
the samples was as follows: MgO(100) substrate/Mn-Co-Ga
(100 nm)/MgO(2 nm)/Al(2 nm). The Mn-Co-Ga layer was de-posited via a co-sputtering technique from a Mn-Ga alloy tar-
get and an elemental Co target. The substrate was heated to
500
/C14C during deposition. In this study, primarily focused on
the Co content dependences of structural, magnetic, and elec-
tric properties, and, thus, the film compositions under investi-
gations were Mn 2.6Ga1.4,M n 2.3Co0.4Ga1.3,M n 2.3Co0.5Ga1.2,
Mn2.2Co0.6Ga1.2,M n 2.1Co0.8Ga1.1,a n dM n 1.8Co1.2Ga1.0;t h e
Co content in the films was varied from 0 to 1.2. The film
compositions were determined by using inductively coupledplasma (ICP) mass spectroscopy for Mn
2.3Co0.4Ga1.3,
Mn2.3Co0.5Ga1.2,a n dM n 1.8Co1.2Ga1.0. The compositions of
the other films were estimated via the ratio of the sputteringpower used for each of the targets. The structural and mag-
netic properties of the films were investigated by using an
x-ray diffractometer (XRD) and a vibrating sample magne-tometer (VSM), respectively. Electrical resistivity was meas-
ured using the van der Pauw technique.
14All the experiments
were performed at room temperature.
III. RESULTS AND DISCUSSION
Fig.1shows the out-of-plane XRD patterns of Mn-Co-
Ga films. Only the peaks that originate from Mn-Co-Ga
(002), (004), and MgO (002) planes appear in the XRD spec-tra. Peaks marked with ?and/H11623represent diffractions from
the tetragonal ( D0
22)2and cubic ( Xa, the so-called inverse
Heusler)15,16structures, respectively. The epitaxial growth
of the films and the superlattice diffractions of (011) (for
D022) and (111) (for Xa) were confirmed via a /-scans for alla)Author to whom correspondence should be addressed. Electronic mail:
takahide@wpi-aimr.tohoku.ac.jp
0021-8979/2013/113(17)/17C723/3/$30.00 VC2013 American Institute of Physics 113, 17C723-1JOURNAL OF APPLIED PHYSICS 113, 17C723 (2013)
the films (not shown here). The lattice constants ( /H11623: a-axis;
/H17005: c-axis) and the c/aratio are summarized as a function of
Co content of the Mn-Co-Ga films in Figs. 2(a) and2(b),
respectively. The corresponding bulk values for Mn 3/C0x
GaCo x, as reported in Ref. 12, are also indicated by the open
symbols. The Mn 2.6Ga1.4and Mn 2.3Co0.4Ga1.3films exhib-
ited a tetragonal structure while the Mn 2.3Co0.5Ga1.2film
exhibited both tetragonal and cubic phases; and the structure
changed to cubic for films with larger Co content. The struc-tural transition point of the film samples depending on the
Co content is consistent with that for bulk ones, even though
the present Mn-Ga composition used in our study(Mn
2.6Ga1.4) was different from the bulk one (Mn 3.0Ga1.0),
which implies that band dispersion is not very sensitive to
theoff-stoichiometry of the Mn-Co-Ga alloys; further tetrag-
onal distortion occurs due to electronic instabilities corre-
sponding to a band-type Jahn-Teller effect.17
Fig.3shows the hysteresis loops of the Mn-Co-Ga films.
A magnetic field was applied perpendicular to the film plane
direction for the curves indicated by ?, and it was applied
along the in-plane direction for those indicated by k. The
hysteresis loops of the tetragonal samples show hard mag-
netic behavior with perpendicular anisotropy, while those of
the cubic ones show soft magnetic behavior with in-plane an-isotropy. The hysteresis loops of the Mn
2.3Co0.5Ga1.2which
contains both tetragonal and cubic structures exhibited unde-
fined loop shapes of small values of magnetization.The dependences of the magnetic moment ( l), magnetic
anisotropy energy ( Ku), and effective anisotropy field ( Heff
k)
as a function of the number of valence electron ( Zt)o fM n -
Co-Ga films are shown in Fig. 4. The lvalue of correspond-
ing to bulk Mn 3/C0xGaCo x(Ref. 12) and bulk Mn 3/C0xGa
(x¼0:6;1) (Ref. 3) are also plotted in the figure.18In addi-
tion, expected Slater-Pauling behavior19of Half-metallic
Heusler compounds is also indicated for the cubic composi-tions. The values of K
uwere determined by using the relation
Ku¼MsHeff
k=2þ2pM2
sin the same manner as described in
our previous work.5The dependence of lfor the film sam-
ples is similar to that of the reported bulk dependence. The l
exhibited minimum value around the boundary between the
tetragonal and cubic structures. On the other hand, the valuesofK
uandHeff
kdid not widely differ for the samples with tet-
ragonal structures. It is noteworthy that the magnetic moment
(l) of the Mn 2.3Co0.4Ga1.3film was as small as 0 :55lB
(/C24190 emu/cm3) at a large Kuvalue of 9.2 Merg/cm3. In this
case, lwas reduced to less than half of that for the filmFIG. 1. Out-of-plane x-ray diffraction spectra of Mn-Co-Ga films with vari-
ous composition ratios. Peaks marked with ?and/H11623represent the diffrac-
tions from the tetragonal phase and cubic phase, respectively. The large
peak appearing at 2 h=x/C2442/C14is originated from the (002) plane of the
MgO substrate.
FIG. 2. (a) lattice constants ( a- and c-axis) and (b) c/aratio as a function of
Co content, x. Lines are just guide to the eyes for data of the film samples.FIG. 3. Hysteresis loops of Mn-Co-Ga films for various composition ratios.
The magnetic field was applied perpendicular to the film plane direction for
the curves indicated by ?, and it was applied along the in-plane direction for
those indicated by k.
FIG. 4. (a) Magnetic moment ( l) and (b) uniaxial magnetic anisotropy
energy ( Ku) and effective anisotropic field ( Heff
k) as a function of the number
of valence electron ( Zt) in Mn-Co-Ga films. The lines serve as a visual guide
to indicate the data curve for tetragonal samples. The lines for cubic samples
indicate the Slater-Pauling rule.1917C723-2 Kubota et al. J. Appl. Phys. 113, 17C723 (2013)without Co content, while a large Kuclose to 10 Merg/cm3
was still maintained.
Subsequently, the electrical resistivity ( q) of the Mn-
Co-Ga films was investigated; the Co-content dependence of
resistivity is shown in Fig. 5. The qvalue of Mn 2.6Ga1.4was
of the same order as that of Mn-Ga films with different com-position ratio.
20With increasing Co content, qcorrespond-
ingly increased, and a maximum value of 490 lXcm was
obtained for the Mn 2.2Co0.6Ga1.2film. As the Co content
increased beyond 0.6, the qslightly decreased; however, it
was still larger than that of pure Mn 2.6Ga1.4. There are two
possible explanations for the Co content dependence of qfor
the Mn-Co-Ga films. The dependence could be extrinsic:
films for Co content of around 0.5, the smoothness and conti-
nuity of the film become poor, e.g., the roughness and peak-to-valley values of the Mn
2.2Co0.6Ga1.2film were about
10 nm and 100 nm, respectively. These values were about 10
times larger than the corresponding value of the Mn 2.6Ga1.4
film. Thus, increased electron scattering due to the presence
of discontinuities in the film might be one reason for the
observed qbehavior. The other possibility is that the q
behavior is intrinsic to the chemical nature of the film.
Chadov et al. have recently hypothesized that the chemical
disordering of Mn 3Ga alloy by Mn-Co substitution can cause
localization of the minority-spin channel around the Fermi
level.17They speculated that the electrical conductivity
reduces because of the reduced mobility of electrons.According to their calculation, the localization is not suffi-
ciently strong for the Mn
3Ga and Mn 2CoGa cases, while the
electrons are strongly localized in both tetragonal and cubicMn
2.5Co0.5Ga compounds. In our study, the films with higher
resistivity exhibit chemically disordered regions, and, thus,
such a localization can also be considered as the cause forthe observed Co content dependence of q.
IV. SUMMARY
Epitaxially grown Mn-Co-Ga tetragonal or cubic
Heusler compound thin films were successfully fabricated,
and their structural, magnetic, and electrical transport prop-
erties were investigated. The dependences of lattice parame-ters and saturation magnetization on Co content in the films
were similar to those reported for bulk Mn
3/C0xCoxGacompounds. A minimum saturation magnetization of 0 :55lB
(/C24190 emu/cm3) was obtained for the Mn 2.3Co0.4Ga1.3film
at a large Kuvalue of 9.2 Merg/cm3; this magnetization value
is acceptable in the light of future STT-MRAM applications.
The resistivity of the Mn-Co-Ga films was larger than that of
pure Mn-Ga. The observed higher resistivity subsequent toCo substitution might originate due to reduced electron mo-
bility because of the presence of localized electron states
around the Fermi level (intrinsic factor) as well as due toincrease in surface discontinuities of the film (extrinsic
factor).
ACKNOWLEDGMENTS
This work was partly supported by the ASPIMATT pro-
gram (JST), Grant for Industrial Technology Research from
NEDO, Grant-in-Aid for Scientific Research (JSPS), WorldPremier International Research Center Initiative (MEXT),
and the Casio foundation.
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Express 4, 043002 (2011).
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X. Zhang, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, Phys.
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9T. Kishi et al., Tech. Dig. - Int. Electron Devices Meet. 2008 , 309.
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17S. Chadov, J. Kiss, and C. Felser, Adv. Funct. Mater. 23, 832–838 (2013).
18Note that, in Fig. 4, all the values of l,Ku, and Heff
kwere not deduced for
the Mn 2.3Co0.5Ga1.2film because the loops did not indicate a state of satu-
ration, and no clear anisotropy.
19J. K€ubler, Physica B 127, 257 (1984).
20F. Wu, E. P. Sajitha, S. Mizukami, D. Watanabe, T. Miyazaki, H.
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line serves as a visual guide.17C723-3 Kubota et al. J. Appl. Phys. 113, 17C723 (2013)Journal of Applied Physics is copyrighted by the American Institute of Physics (AIP). Redistribution of journal
material is subject to the AIP online journal license and/or AIP copyright. For more information, see
http://ojps.aip.org/japo/japcr/jsp |
5.0050062.pdf | Appl. Phys. Lett. 118, 212408 (2021); https://doi.org/10.1063/5.0050062 118, 212408
© 2021 Author(s).Density functional theory study of chemical
pressure in multicaloric MTX compounds
Cite as: Appl. Phys. Lett. 118, 212408 (2021); https://doi.org/10.1063/5.0050062
Submitted: 11 March 2021 . Accepted: 25 April 2021 . Published Online: 27 May 2021
Timothy Q. Hartnett , Vaibhav Sharma , Radhika Barua , and
Prasanna V. Balachandran
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Applied Physics Letters 118, 213903 (2021); https://doi.org/10.1063/5.0048588Density functional theory study of chemical
pressure in multicaloric MTX compounds
Cite as: Appl. Phys. Lett. 118, 212408 (2021); doi: 10.1063/5.0050062
Submitted: 11 March 2021 .Accepted: 25 April 2021 .
Published Online: 27 May 2021
Timothy Q. Hartnett,1Vaibhav Sharma,2Radhika Barua,2and Prasanna V. Balachandran1,3,a)
AFFILIATIONS
1Department of Materials Science and Engineering, University of Virginia, Charlottesville, Virginia 22904, USA
2Department of Mechanical and Nuclear Engineering, Virginia Commonwealth University, Richmond, Virginia 23284, USA
3Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, Virginia 22904, USA
a)Author to whom correspondence should be addressed: pvb5e@virginia.edu
ABSTRACT
The MTX -based compounds are promising rare-earth-free candidates for multicaloric applications due to the proximity of their structural
and magnetic phase transitions. In this paper, we use first principles calculations to study how chemical pressure affects the energetics,saturation magnetization, and volume change. Our calculations reveal the presence of a complex interplay between the M-,T-, and X-site
elements in tuning the properties. The choice of elements for rational alloy design should be informed by the site-specific response. Our
work motivates future synthesis and characterization efforts to focus on uncovering site-specific data to tailor strategies for maximizing thecaloric response and bridge the knowledge-gap.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0050062
Solid-state cooling devices based on the “caloric” class of func-
tional materials are considered a promising alternative to conventional
vapor compression technologies. These materials completely eliminate
high-global warming potential (GWP) refrigerants and have high
energy efficiency.
1By definition, multicaloric materials exhibit revers-
ible thermal changes that can be driven either concurrently or in
sequence by more than one type of external energy source (magnetic
field, electric field, or strain/pressure). The use of multiple driving
forces can bring about larger thermal changes with smaller field mag-
nitudes over broader operating temperature ranges.2
Many multicaloric materials under current investigation contain
strategically limited and toxic elements that pose long-term sustain-
ability challenges [e.g., Gd 5(GeSi) 4, FeRh, and MnAs], or require
complex synthesis and processing to realize an acceptable, or even
marginal functional response.3–6Against this backdrop, the MTX fam-
ily of compounds [ M¼transition metal elements (Mn, Fe, or Co); T
¼transition metal elements (Ni, Fe, or Co); X¼main group p-block
element (Si, Ge, or Sn)] are poised to overcome these limitations since
they are made of earth-abundant and nontoxic elements and are scal-
able for powder production using low-cost, conventional solid-stateprocessing techniques.
7The two crystal structures in the MTX family
that are most critical for the multicaloric application are shown in
Fig. 1 . Select MTX alloys exhibit a first-order magnetostructural phase
transition between low temperature, ferromagnetic (FM) TiNiSi-typeorthorhombic ( Pnma ) phase and high temperature, paramagnetic
Ni2In-type hexagonal ( P63=mmc ) phase. The discontinuous nature of
this phase transition provides a large isothermal magnetic entropy
change and adiabatic temperature change leading to giant caloric
effect.8One of the fundamental questions of interest is how do we syn-
ergistically tune the structural and magnetic phases to maximize the
multicaloric response. The application of “chemical pressure” (i.e., the
incorporation of larger or smaller atoms with similar or different
valence electrons to expand or contract the crystal structure9,10)i sa
viable and practical approach to design MTX materials with targeted
properties.
In the MTX materials family, MnNiGe and MnNiSi are the two
most commonly investigated systems in the literature. In the pure
MnNiGe and MnNiSi compounds, a diffusionless structural phase
transformation from Pnma toP63=mmc occurs on heating ( Tt)a t4 9 3
and 1206 K, respectively.11Moreover, these compounds are also mag-
netic at room temperature. We can use the MnNiSi 1/C0xGexsolid solu-
tion as an example to demonstrate the role of chemical pressure,
where a larger-sized Ge-atom substitutes for the smaller Si-atom in the
X-site. Experimental measurements12have shown that we can tune
the FM Curie temperature ( Tc) of MnNiSi 1/C0xGexas a function of x
until x¼0.8. In pure MnNiSi, the Tcis 616 K, and it reduces to 410 K
in MnNiSi 0.2Ge0.8solid solution.12Pure MnNiGe undergoes a heli-
magnetic-to-paramagnetic phase transition at 346 K.13
Appl. Phys. Lett. 118, 212408 (2021); doi: 10.1063/5.0050062 118, 212408-1
Published under an exclusive license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplInterest in exploring the coupled nature of the structural and
magnetic phase transitions in these systems began more than four dec-
ades ago.13,14Since then, many efforts have been made to design the
material via composition engineering and hydrostatic pressure suchthat the T
tand Tccoincide at or near room temperature.15–26
However, there are fewer studies that have explored the electronic
structure of these materials using density functional theory (DFT) andshow how that may relate to caloric properties. One DFT study
explored the impact of Mn-substitution for Fe in Mn
xFe1/C0xNiSi com-
pound (where x¼0;0:35;0:75;1) and found a nonmonotonic
change in the total magnetic moment.27A more comprehensive DFT
study was conducted by Biswas et al.,24where they explored the role of
Mn-site substitution with Ti, V, Cr, Fe, and Co in MnNiSi. In thesame work, Biswas et al. then proceeded to study the impact of X-site
substitution of the Mn
0.5Fe0.5NiXsystem with X¼A l ,S i ,G a ,G e ,I n ,
Sn, and Sb.24The authors finally predicted a series of Mn 0.5Fe0.5
NiSi 1/C0xAlxcompositions, which were also experimentally verified.
More recently, Garcia et al. developed a novel DFT-based approach
and extended the concept of magnetic deformation proxy descriptor28
to study the magnetocaloric properties of more complex, disordered
solid solutions of Mn(Co 1/C0xFex)Ge and (Mn 1/C0xNix)CoGe.29
Despite some of the outstanding contributions from experimen-
tal and computational research on the MTX materials family, several
key questions still remain unaddressed where DFT calculations can
shed more light. In both the low-temperature Pnma and high-
temperature P63=mmc crystal structures, the M-a n d T-sites are crys-
tallographically inequivalent. These sites host the magnetic transition
metal atoms ( Fig. 1 ). When solid solutions of MTX compounds are
explored, it is unclear how the transition metal atoms at the M-a n d
T-sites contribute to the overall magnetostructural response. For
example, G €uc¸l€uet al. explored the magnetic properties of the
FeMn 1/C0xNixGe solid solution.30The crystal structure data reported in
the International Crystal Structure Database (collection codes 187298–
187301) showed partial site-occupancies between the Fe-, Mn-, and
Ni-atoms in both the M-a n d T-sites.31This begs the question: do the
transition metal atoms exhibit similar properties irrespective of their
M-o r T-site occupation? Our work is motivated to shed light on the
local site-substitutions for the rational design of multicaloric MTX
compounds.
Here, we perform DFT calculations for a range of end member
compounds to study the impact of transition metal site substitutionson the M-a n d T-sites. In addition, we also study the role of X-site sub-
stitutions in the MnNi Xsystems, where X¼Si, Ge, Sn, Al, and Ga.We specifically focus on exploring how these site substitutions affect
three properties that can be reliably calculated from the semi-local
functionals within DFT: (i) total energy difference between the FM
Pnma and P6
3=mmc phases ( DE)i nm e V / a t o m .B i s w a s et al. sug-
gested that DEcan be correlated with the martensite–austenite transi-
tion temperature.24(ii) Saturation magnetization ( Ms)o ft h e Pnma
structure in erg/cc. (iii) Volume difference between the Pnma and
P63=mmc structures per formula unit ( DV,i nA ˚3/f.u.). The Msand
DVare recognized as the two key macroscopic properties that will
impact the caloric response. In magnetostructural materials thatundergo first-order phase transition, the total entropy change ( DS
tot)
can be expressed as DStot¼DSmagþDSst,w h e r e DSmagandDSstare
the magnetic and structural entropy changes, respectively.32The Msis
a measure of magnetism per unit volume and is related to the DSmag
term; a large Msis a desired property for magnetocalorics.33TheDVis
related to DSst, as well as pressure, and its role can be discussed in
terms of the Clausius–Clayperon equation, dS¼(dTt/dP)dV .18
Spin-polarized DFT calculations were carried out using the plane
wave pseudopotential code, Quantum ESPRESSO.34–36For the 3 dele-
ments, all spins were treated as collinear in FM order since most
experimentally explored substitutions have resulted in FM alloys.24
Core and valence electrons were treated using the ultrasoft pseudopo-
tentials.37The exchange–correlation functionals were described using
the Perdew–Burke–Ernzerhof parameterization of the generalized
gradient approximation modified for solids.38The plane wave cutoff
energy was set to 60 Ry and a C-centered 14 /C214/C210 Monkhorst-
Pack k-point mesh39was used to sample the Bouillon Zone of the
P63=mmc phase and 10 /C214/C210 mesh size was used for the Pnma
phase. The atomic positions and cell volume were relaxed until forceswere less than 2 meV/A ˚and the total energy converged to 10
/C08eV. A
key objective of this work is to determine trends showing how the
chemical substitution in the M-,T-, and X-sites will affect the DE,Ms,
andDV. Since our focus is on making modifications to the parent
MnNiSi and MnNiGe compounds using solid solution strategies, wedo not calculate the thermodynamic stability for the line compounds
explored in this work.
In Table S1 (in the supplementary material ), we compare the
experimental and DFT calculated structure and magnetic properties
for pure MnNiSi and MnNiGe in the Pnma structure. The DFT results
show good agreement with experimental data, which gives confidence
in the methodology. The total density of states (DOS) and atom-
projected DOS for pure MnNiSi and MnNiGe materials in the low
energy Pnma structures with FM ordering are shown in Fig. 2 .I n
MnNiSi ( Fig. 2 , top panel), the orbital overlap between Si-3 p,M n - 3 d,
and Ni-3 dstates occurs at /C244 eV below the Fermi level ( E
F). The
minority spin states of Mn- and Ni-3 dorbitals dominate the EF.
Similarly, in MnNiGe ( Fig. 2 , bottom panel) the bonding between Ge-
4p,M n - 3 d,a n dN i - 3 dstates occurs at /C244e Vb e l o wt h e EF. Similar to
MnNiSi, the minority Mn-3 dand Ni-3 dstates dominate the EFin
pure MnNiGe. However, the total number of states at the EFis far
greater in MnNiSi compared to MnNiGe. In MnNiSi, there are 6.71minority states and 1.30 majority states at E
F. In MnNiGe, there are
2.11 minority states and 1.34 majority states at EF. This difference can
potentially have an influence on the intrinsic Gilbert damping in these
materials.40,41However, there is no data in the literature that correlate
the spin dynamics relaxation at the magnetic phase transition captured
by damping to the magnetocaloric response.
FIG. 1. The orthorhombic ( Pnma ) and hexagonal ( P63=mmc ) crystal structures of
theMTX compounds.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 118, 212408 (2021); doi: 10.1063/5.0050062 118, 212408-2
Published under an exclusive license by AIP PublishingThe impact of Mn- and Ni-site substitutions on DFT calculated
DE;DV,a n d Msproperties are outlined in Figs. 3(a)–3(c) .T h er a w
data are given in Table I . In both MnNiSi and MnNiGe, we can see
that Msis increased for Fe- and Co-substitution at the Ni-site; Msis
maximized with Fe-substitution [ Fig. 3(a) ]. The Msis decreased when
Fe and Co substitute the Mn-site. When both Mn- and Ni-sites weresubstituted simultaneously by Fe or Co, the behavior was mixed; M
s
FIG. 3. DFT calculated (a) and (d) saturation magnetization ( Ms) in the Pnma
phase, (b) and (e) DEbetween Pnma and P63=mmc phases, (c) and (f) DV
between Pnma andP63=mmc phases for various MTX compounds. In (a)–(c), we
fix the X-site as Si and Ge, but vary the Mn- and Ni-sites. In (d)–(f), we have MnNi-
Xcompounds, where X¼Si, Ge, Sn, Al, and Ga. Although Pnma is not the lowest
energy structure in CoCoGe, we show it here for consistency. Zero Mscorresponds
to a nonmagnetic phase. Negative sign in DEindicates that the P63=mmc phase
is lower in energy than the Pnma phase at 0 K. Negative sign in DVindicates that
theP63=mmc phase is larger at 0 K than the Pnma .FIG. 2. The total density of states (DOS) and atom-projected DOS spectra for pure
MnNiSi (top) and MnNiGe (bottom) in the FM Pnma structure.
TABLE I. Summary of space group, magnetism, and bond length data for various MTX compounds explored in this work. The lattice constants (a, b, and c) are in unit A ˚; volume
is in A ˚3; magnetic moments ( MmagandTmag) are in Bohr magnetons; bond lengths (M–M, T–T, X–X, M–T, M–X, and T–X) are in A ˚.
Compound Space group a b c Volume Mmag Tmag M–M T–T X–X M–T M–X T–X
MnNiGe Pnma 5.87 3.63 7.01 149.26 2.81 0.13 3.12 2.60 3.52 2.78 2.53 2.32
P63=mmc 4.12 4.12 4.91 72.26 2.29 0.12 3.42 2.46 3.42 2.68 2.38 2.68
MnFeGe Pnma 6.40 3.17 7.5 152.07 2.89 2.02 3.10 2.48 3.17 2.73 2.52 2.38
P63=mmc 4.12 4.12 4.96 69.88 2.30 2.18 3.44 2.48 3.44 2.68 2.38 2.68
MnCoGe Pnma 5.69 3.73 7.00 148.46 2.94 0.68 3.11 2.60 3.51 2.77 2.53 2.32
P63=mmc 4.10 4.10 4.94 71.92 2.32 1.09 3.42 2.47 3.42 2.67 2.37 2.67
FeNiGe Pnma 5.11 3.79 7.16 138.72 1.71 0.12 2.60 2.70 3.26 2.67 2.50 2.35
P63=mmc 4.03 4.03 4.91 68.99 1.27 0.14 3.38 2.45 3.38 2.63 2.33 2.63
CoNiGe Pnma 5.06 3.74 7.18 135.82 0.28 0.06 2.59 2.69 3.22 2.65 2.48 2.34
P63=mmc 3.97 3.97 4.96 67.86 0.00 0.00 3.38 2.48 3.38 2.61 2.29 2.61
FeFeGe Pnma 6.62 2.76 7.83 142.80 2.24 1.71 2.69 2.73 2.76 2.65 2.54 2.37
P63=mmc 4.06 4.06 4.91 69.97 1.29 2.24 3.39 2.46 3.39 2.64 2.34 2.64
CoCoGe Pnma 6.65 2.66 7.76 137.40 1.08 1.08 2.70 2.60 2.66 2.60 2.48 2.38
P63=mmc 3.95 3.95 4.94 66.81 0.00 0.00 3.36 2.47 3.36 2.59 2.28 2.59
MnNiSi Pnma 5.74 3.53 6.82 137.87 2.76 0.30 3.01 2.54 3.42 2.71 2.49 2.25
P63=mmc 4.00 4.00 4.77 66.07 2.77 0.38 3.32 2.38 3.32 2.60 2.31 2.60
MnFeSi Pnma 5.59 3.54 7.04 139.48 2.52 1.68 3.03 2.49 3.19 2.73 2.46 2.29Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 118, 212408 (2021); doi: 10.1063/5.0050062 118, 212408-3
Published under an exclusive license by AIP Publishingincreased in FeFeGe, but decreased in the CoCoGe compound. The
CoCoSi compound is found to be nonmagnetic in both Pnma and
P63=mmc structures ( Ms¼0). In MnNiSi, the Co-substitution at the
Ni-site maximized DV, while Fe-substitution has minimal impact
[Fig. 3(c) ]. The large increase in DVfor MnCoSi is due to a significant
decrease in the volume of the P63=mmc phase. In MnNiSi, substituting
Co in the Ni-site (MnCoSi) resulted in relatively smaller DVcompared
to the Co-substitution in the Mn-site (CoNiSi). This appears to indi-
cate that the Ni-site substitution exerts less chemical pressure on
the lattice in the MnNiSi system. When the system is MnNiGe,
Fe-substitution at Ni-site increased DV, while Co-substitution
decreased DV.
In MnNiGe, simultaneous substitution of Co-atoms in both Mn-
and Ni-sites had the largest impact in minimizing the DE[Fig. 3(b) ].
We find that CoCoGe is the only compound where the P63=mmc
phase was lower in energy than the Pnma phase. The total and atom-
projected DOS for the nonmagnetic and FM CoCoGe in thelow energy P6
3=mmm and high energy Pnma structures are shown in
Fig. 4 , respectively. In the high energy Pnma structure, CoCoGe has
3.05 majority and 16.95 minority states at the EF. This drastic increase
in the minority states at EFis due to increased contribution from the
Co-dstates that occupy the T-site in the Pnma structure. In the
MnNiSi and MnNiGe ( Fig. 2 ), the density of minority states at EFwas
6.74 and 2.11, respectively. In the nonmagnetic CoCoGe, the
P63=mmc structure has a total of 5.34 states at EF. Despite being non-
magnetic in the P63=mmc structure, it appears that the fewer DOS at
theEFcauses the P63=mmc structure to have lower total energy than
thePnma structure. This behavior is not mimicked by CoCoSi indicat-
ing that the X-site is important in Co-based MTX systems.
Inspired by the impact of X-site substitution, most obvious in
CoCo-X, we explored the role of various X-site elements on DE,Ms,
andDVby substituting MnNi- Xwith X¼Sn, Al, and Ga (all of which
have been experimentally explored in the literature). The results areshown in Figs. 3(d)–3(f) . All MnNi- Xcompounds have the Pnma as
the lowest energy structure [ DE>0i n Fig. 3(e) ]. We find trends in
the properties because of X-site substitutions. Moving down group 14
on the periodic table from Si to Sn, both MsandDVdecreased. This is
shown in Fig. 3(f) .T h ed e c r e a s ei n Ms, however, is not due to
decreased magnetic moment but rather increasing unit cell volume in
the low energy Pnma phase. The increase in volume seems to be
mostly driven by the increasing Ni- Xbond length ( Table I ).TABLE I. (Continued. )
Compound Space group a b c Volume Mmag Tmag M–M T–T X–X M–T M–X T–X
P63=mmc 4.01 4.01 4.80 66.86 1.77 1.90 3.33 2.40 3.33 2.61 2.32 2.61
MnCoSi Pnma 5.61 3.61 6.79 137.39 2.62 0.57 3.02 2.64 3.28 2.68 2.48 2.25
P63=mmc 3.96 3.96 4.77 64.91 1.45 0.42 3.31 2.40 3.31 2.58 2.29 2.58
FeNiSi Pnma 5.26 3.62 6.84 130.24 1.51 0.15 2.75 2.55 3.31 2.63 2.40 2.26
P63=mmc 3.91 3.91 4.81 63.64 0.82 0.11 3.30 2.40 3.30 2.56 2.26 2.56
CoNiSi Pnma 4.92 3.65 6.99 125.72 0.00 0.00 2.52 2.63 3.13 2.59 2.42 2.29
P63=mmc 3.87 3.87 4.87 63.03 0.00 0.00 3.30 2.43 3.30 2.54 2.23 2.54
FeFeSi Pnma 6.43 2.65 7.67 130.85 2.08 1.25 2.61 2.63 2.65 2.58 2.48 2.30
P63=mmc 3.95 3.95 4.79 64.64 1.01 1.95 3.31 2.39 3.31 2.57 2.28 2.57
CoCoSi Pnma 4.81 3.67 6.99 123.40 0.00 0.00 2.44 2.64 3.09 2.57 2.42 2.34
P63=mmc 3.85 3.85 4.84 62.07 0.00 0.00 3.29 2.42 3.29 2.53 2.22 2.53
MnNiSn Pnma 5.36 3.98 7.49 173.52 3.16 0.19 2.68 3.66 3.66 2.81 2.83 2.56
P63=mmc 4.40 4.40 5.24 87.95 2.93 0.17 3.65 2.62 3.65 2.86 2.54 2.86
MnNiAl Pnma 5.04 3.98 7.21 144.76 2.73 0.36 2.52 3.01 3.10 2.68 2.68 2.44
P63=mmc 4.13 4.13 4.96 73.41 2.77 0.38 3.44 2.48 3.44 2.69 2.39 2.69
MnNiGa Pnma 5.09 4.06 7.05 145.88 2.86 0.30 2.55 3.46 3.46 2.67 2.67 2.43
P63=mmc 4.14 4.14 4.98 73.92 2.88 0.32 3.45 2.49 3.45 2.69 2.39 2.69
FIG. 4. The total density of states (DOS) and atom-projected DOS for CoCoGe.
Top: nonmagnetic P63=mmc structure, bottom: FM Pnma structure.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 118, 212408 (2021); doi: 10.1063/5.0050062 118, 212408-4
Published under an exclusive license by AIP PublishingSubstitutions involving group 13 elements increase Ms, but results in
negative DV[Fig. 3(f) ]. The MnNiAl showed the largest value of Msat
793.24 erg/cc [ Fig. 3(d) ]. Among the five MnNi- Xcompounds,
MnNiGa had the lowest DE(only 181.78 meV/atom).
In summary, our results show that site-specific chemical substitu-
tions have nontrivial effects on the properties of MTX compounds.
Although Fe-substitution was found to consistently decrease the DE,
tailoring its location in the M-o r T-site is found to be important for
tuning the Ms. The behavior of Co-containing compounds was found
to be more sensitive to the site-specific substitution than Fe. TheCoCoGe compound exemplifies this character. Tuning properties via
X-site substitution were also found to be a viable option to compensate
for losses due to the trade-off between DE,M
s,a n dDVas a result of
M-a n d T-site substitutions. Our work emphasizes the need for using
characterization methods that provide insights into the site-specificstructure and magnetic properties, so that the processing protocols
can be tailored to optimize the multicaloric response. Although this
work revealed hitherto unknown site-specific trends, future workshould focus on its relationship to T
c,DS, and thermal hysteresis.42
See the supplementary material for a table comparing the experi-
mental vs DFT calculated structure and magnetic properties for pure
MnNiSi and MnNiGe in the low energy Pnma structure.
T.Q.H. and P.V.B. acknowledge support from the Defense
Advanced Research Project Agency (DARPA) program on
Topological Excitations in Electronics (TEE) under Grant No.
D18AP00009. Work at VCU was partially funded by the NationalScience Foundation under Award No. 1726617. All DFTcalculations were performed in the Rivanna high-performancecomputing cluster maintained by the Advanced Research
Computing Service at the University of Virginia.
DATA AVAILABILITY
The data that support the findings of this study are available
within the article and its supplementary material .
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Appl. Phys. Lett. 118, 212408 (2021); doi: 10.1063/5.0050062 118, 212408-5
Published under an exclusive license by AIP Publishing |
1.5086476.pdf | Appl. Phys. Lett. 114, 172403 (2019); https://doi.org/10.1063/1.5086476 114, 172403
© 2019 Author(s).Inducing out-of-plane precession of
magnetization for microwave-assisted
magnetic recording with an oscillating
polarizer in a spin-torque oscillator
Cite as: Appl. Phys. Lett. 114, 172403 (2019); https://doi.org/10.1063/1.5086476
Submitted: 21 December 2018 . Accepted: 10 April 2019 . Published Online: 29 April 2019
W. Zhou
, H. Sepehri-Amin
, T. Taniguchi
, S. Tamaru
, Y. Sakuraba , S. Kasai , H. Kubota
, and K.
Hono
Inducing out-of-plane precession of magnetization
for microwave-assisted magnetic recording with
an oscillating polarizer in a spin-torque oscillator
Cite as: Appl. Phys. Lett. 114, 172403 (2019); doi: 10.1063/1.5086476
Submitted: 21 December 2018 .Accepted: 10 April 2019 .
Published Online: 29 April 2019
W.Zhou,1,a)
H.Sepehri-Amin,1
T.Taniguchi,2
S.Tamaru,2
Y.Sakuraba,1,b)S.Kasai,1H.Kubota,2
and K. Hono1
AFFILIATIONS
1Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan
2National Institute of Advanced Industrial Science and Technology (AIST), Spintronics Research Center, Tsukuba 305-8568, Japan
a)Electronic mail: ZHOU.Weinan@nims.go.jp
b)Electronic mail: SAKURABA.Yuya@nims.go.jp
ABSTRACT
The dynamics of a simple design of a spin-torque oscillator (STO) compatible with microwave-assisted magnetic recording were investi-
gated. The STO with Ni 80Fe20(NiFe) used as a polarizer and Fe 67Co33(FeCo) used as a field generating layer was fabricated and measured.
As the bias voltage increased, the magnetization reversal of NiFe occurred, then, multiple signals appeared in the power spectra. The signalsreflected out-of-plane precession (OPP) mode oscillation of both the FeCo and NiFe layers, as well as the magnetoresistance effect of the
STO, which had a frequency equal to the difference between the oscillation frequencies of the NiFe and FeCo layers. Such dynamics were
reproduced by micromagnetic simulation. The results of the experiment and simulation demonstrate the merit of realizing OPP modeoscillation with a simple and thin structure suitable for a narrow gap recording head. In particular, the experimental results obtained withthis STO design revealed that the cone angle for OPP mode oscillation of the FeCo layer (estimated by using the macrospin model) was
large, namely, /C2470
/C14.
Published under license by AIP Publishing. https://doi.org/10.1063/1.5086476
Energy-assisted recording technologies, e.g., microwave-assisted
magnetic recording (MAMR), have become indispensable in regard tomaintaining the continuous improvement in recording density of harddisk drives.
1,2To fulfill the requirements of the signal-to-noise ratio
and thermal stability simultaneously, materials with increasing magne-tocrystalline anisotropy ( K
u) are being exploited as recording media.
MAMR grants writability to high Kumedia by applying an additional
ac magnetic field ( hac) to induce precessional motion of magnetic
moments, which results in magnetization switching under a muchsmaller magnetic field ( H) than the coercivity.
3One technical chal-
lenge concerning MAMR is to generate high-frequency, large-ampli-tude h
acwithin a nanosized area. It has been shown that for recording
media with an effective anisotropy field of 2 T, hacwith a frequency of
18 GHz and an amplitude of 0.1 T is necessary for sufficiently reducingthe switching field.
3However, since achieving higher recording density
by using MAMR is being targeted, a higher frequency and a largeramplitude will be required. It is expected that h
accan be generated
using a spin-torque oscillator (STO), which is a nanometer-scaledradio-frequency oscillator with the potential for a wide range of appli-
cations.4–10Utilized for MAMR, the STO is placed in the narrow gap
between the main pole and the trailing shield of the recording head.During recording, the magnetization of the main pole and the trailingshield results in an Hvalue of /C241 T to the STO along the perpendicu-
lar direction. As the current passes through the STO, spin-polarizedelectrons exert a spin-transfer torque (STT)
11,12to a soft magnetic
layer to cancel the damping torque, which makes the magnetizationundergo out-of-plane precession (OPP) mode oscillation
4,13forhac
generation. This layer is called the field generating layer (FGL).
Previously, it was proposed to use a perpendicularly magnetized layeras the polarizer for stable oscillation.
14–16STOs using a 3-nm-thick
Co2Fe(Ga 0.5Ge0.5) layer exchange-coupled with a 10-nm-thick L10-
FePt layer as the polarizer were experimentally studied, and the combi-nation of the materials with high spin polarization and high K
u
showed oscillation performance close to that required for practical
MAMR application.17,18However, the thick structure of the polarizer
requires a wide gap between the main pole and the trailing shield,
Appl. Phys. Lett. 114, 172403 (2019); doi: 10.1063/1.5086476 114, 172403-1
Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplwhich broadens the field distribution from the main pole and results
in recording transition noise.19
Recently, a simple design of an STO compatible with MAMR
application, where only a soft magnetic thin layer is exploited as thepolarizer, was proposed by Zhu et al.
20Under perpendicular H,t h e
magnetization of both the FGL and the polarizer will align along
the perpendicular H, as schematically illustrated in Fig. 1(a) .A st h e
electrons flow from top to bottom, the antiparallel configuration ofmagnetization is favored, and the STT will reverse the magnetizationof the polarizer [ Fig. 1(b) ]. After the reversal of the polarizer magneti-
zation, the electrons are spin polarized in the direction of the polarizer
(opposite to H) and then exert a torque to the FGL as they pass
through. This torque tries to rotate the magnetization of the FGLtoward the /C0z direction, and it is balanced by the damping torque of
the FGL, leading to OPP mode oscillation of the FGL. Since this opera-tion mechanism no longer requires a perpendicularly magnetized
polarizer or layer to pin the polarizer, it makes it possible to generate
h
acwith a thin and simple STO.21The dynamics of STOs consisting of
only soft magnetic layers have been studied under large perpendicularHand high current density.
4,22–24However, OPP mode oscillation has
not been clearly demonstrated and discussed from the viewpoint of
MAMR application.
In this study, OPP mode oscillation by the aforementioned
mechanism was experimentally demonstrated. The results of theexperiment show that both the polarizer and the FGL were in OPPmode oscillation at different frequencies [ Fig. 1(c) ]. The change in
resistance due to the magnetoresistance (MR) effect had a frequency
equal to the difference between those of the polarizer and FGL [ Fig.
1(d)]. Such dynamics were reproduced by micromagnetic simulation.In addition, the cone angle ( h) of OPP mode oscillation was estimated
using the macrospin model.
Fe
67Co33(FeCo) was used as the FGL, due to its large saturation
magnetization ( Ms), and Ni 80Fe20(NiFe) was used as the polarizer.
The STOs were microfabricated from a blanket thin film with the
following stacking structure: MgO (100) subs.//Cr (10)/Ag (100)/FeCo
(7)/Ag (5)/NiFe (7)/Ag (5)/Ru (8) (thickness in nanometers). A sche-
matic illustration of the circular pillar of the STO is shown in Fig. 1(e) .
The fabrication process for the STO is described in detail in the sup-
plementary material . Because the small pillars were covered by thick
electrodes and could not be clearly observed using a scanning electron
microscope (SEM), pillar diameters were estimated by using the diam-
eter of large size pillars ( D/C24140 and 350 nm) measured by SEM on
the same substrate and the change in resistance ( DR) obtained from
the MR curves, based on the linear relationship between DRand the
reciprocal of the area of the pillar ( DR/1=A). The experimental
results reported here were measured from a device with the diameter
of/C2428 nm. During the measurement, the substrate was mounted on a
sample fixture having a two-axis rotary stage and equipped with a cus-
tom high-frequency probe, which was inserted into an electromagnet.
This setup allowed us to apply an external Hin arbitrary directions.25
To measure the power spectral density (PSD) of the STO, a bias DC
voltage ( U) was applied to the STO through a bias-tee. The generated
signal was amplified using a low-noise amplifier and captured using a
commercial spectrum analyzer. The amplifier gain was not subtracted
from the results of the PSD. In addition to the DC-voltage source, a
lock-in amplifier was connected to the DC port during the measure-
ment of Rand d V/dI. The positive voltage and current density were
defined as electrons flowing from the top NiFe layer to the bottomFeCo layer. All measurements were carried out at room temperature.
The MR curve of the STO under perpendicular Hand a low U
value of /C01m V i s s h o w n i n Fig. 1(f) .A tz e r o H, the NiFe and FeCo
layers have their magnetization in-plane with an antiparallel configu-
ration due to the dipolar interaction, resulting in a high Rstate. As H
increases in the perpendicular direction, the magnetization of both
layers aligns toward HandRgradually decreases. The MR ratio of this
STO is /C246.2%. In addition, Rreaches a minimum at l
0H<1T ,
w h i c hi sm u c hs m a l l e rt h a n l0Msof FeCo, indicating a large reduction
of the demagnetization factor due to the small lateral size of the pillar.The alignment of the magnetization of both layers along the perpen-
dicular direction under l
0H¼1 T was also confirmed by micromag-
netic simulation. Figures 2(a) ,2(c),a n d 2(e) show d V/dIand R as a
function of Uand mappings of the PSD of the STO under perpendicu-
larl0H¼0.81 T (the angle between Hand the z-axis, i.e., H¼0/C14).
For the measurement, Hwas kept constant, while Uwas increased
from 0 to 150 mV. Here, the magnetization of the NiFe layer isreversed at U/C2422 mV, as indicated by the peaks in the d V/dIcurve
and the increase in R.
22–24,26After the reversal, the d V/dIcurve dips
slightly at U¼30 mV (as marked by the gray dashed line), which is
the threshold Ufor the appearance of the strong microwave signal
labeled I0.T h ed i pi nt h ed V/dIcurve corresponds to the decrease in R,
which was also observed in previous studies,23,24and is attributable to
the dynamics excitation of the FeCo layer. As Uincreases, the fre-
quency of I0decreases (i.e., a red-shift occurs). The same measurement
was also carried out with the same value of Hslightly tilted as H¼2/C14,
and the measurement results are shown in Figs. 2(b) ,2(d),a n d 2(f).A s
for the d V/dIcurve, the peaks and dip shift toward higher values,
FIG. 1. (a) Schematic illustration of magnetization of both NiFe and FeCo aligned
along H. NiFe was used as the polarizer and FeCo as the FGL. (b) NiFe is
reversed by STT. (c) Both NiFe and FeCo are in OPP mode oscillation. (d) If thexy-plane rotates with FeCo at the same frequency around the z-axis, in this coordi-nate system (x
0,y0, and z), FeCo stays still, while NiFe oscillates with frequency
equal to fNiFe/C0fFeCo, which is also the frequency of the change in resistance due
to the MR effect ( fMR). (e) Schematic illustration of the circular pillar of the STO. (f)
MR curve of the STO under perpendicular Hand a Uvalue of /C01 mV. The arrows
indicate the direction in which His swept of the corresponding curves.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 114, 172403 (2019); doi: 10.1063/1.5086476 114, 172403-2
Published under license by AIP Publishingnamely, U/C2424 and 34 mV, respectively. As for the mapping of the
PSD, a similar strong signal (labeled I) appears after the dip in thedV/dIcurve, together with other weak signals. Here, we emphasize
the appearance of the signals labeled II and III.
To better understand the dynamics, micromagnetic simula-
tion was carried out using the software magnum.fe ,
27which can
calculate the coupled dynamics of magnetization and the spinaccumulation simultaneously by solving the Landau-Lifshitz-Gilbert (LLG) equation and the time-dependent 3D-spin-diffusion
equation. A 28-nm-diameter circular pillar, consisting of a 7-nm-thick
NiFe layer and a 7-nm-thick FeCo layer separated by a 5-nm-thicknonmagnetic layer, was used as the model for micromagnetic simula-tion. l
0Ms, exchange stiffness ( A), and spin polarization ( b)o fN i F e
were set as 1.0 T, 13 pJ/m,28and 0.4, respectively, while l0Ms¼2.3 T,
A¼30 pJ/m,29andb¼0.5 were used for FeCo. A damping constant
(a) of 0.01 was used for both NiFe and FeCo. The dipolar interaction
between the NiFe layer and the FeCo layer was considered in thesimulation. The simulation was done at 0 K, and the thermal-noisefield was not considered. The spin-diffusion model
30was used for
the implantation of STT, and the detailed description can be foundin Refs. 21and31. The time evolution of the averaged magnetization
vector ( m) of the FeCo and NiFe layers in a stable oscillation state
under a perpendicular l
0Hof 0.81 T and a current density ( J)o f
3.2/C2108A/cm2is shown in Figs. 3(a) and3(c), respectively. (This J
approximately corresponds to U¼80 mV used in the experiment.)
Electrical potential ( E) between the top and bottom of the circular
pillar is shown in Fig. 3(e) . The time evolution of the x and y com-
ponents of mFeCoandmNiFeindicates that both the FeCo and NiFe
layers oscillate in OPP mode with the same rotation sense. The zcomponent is positive for m
FeCo, while negative for mNiFe, indicating
that the magnetization of the NiFe layer is reversed. The discreteFourier transform (DFT) was used to calculate the correspondingspectra in the frequency domain as shown in Figs. 3(b) ,3(d), and
3(f). As for the FeCo layer, the peak with the largest magnitudeappears at 5.17 GHz, which is the frequency of OPP mode oscilla-
tion of FeCo ( f
FeCo); as for the NiFe layer, it appears at fNiFe
¼30.33 GHz. On the other hand, as for the spectrum of E, which
corresponds to the experimentally measured PSD, the peak with thelargest magnitude appears at f
MR¼25.16 GHz, which differs from
either fFeCoorfNiFebut equals the difference between fNiFeandfFeCo,
i.e.,fMR¼fNiFe/C0fFeCo . It is worth mentioning that weak coupling
exists between the polarizer and the FGL, attributable to the dipolar
interaction and the STT, which results in the small peak at
30.33 GHz for jmFeCo ;xðfÞjthat equals fNiFeinFig. 3(b) and the small
peak at 5.17 GHz for jmNiFe ;xðfÞjthat equals fFeCoinFig. 3(d) .
The relationship fMR¼fNiFe/C0fFeCo was also observed experi-
mentally. The frequencies of the signals labeled I0,I ,I I ,a n dI I Ii nm a p -
ping of the PSD were extracted from Figs. 2(e) and2(f)and are plotted
as a function of Uas shown in Fig. 4(a) . As for the strong red-shift
signals, I of H¼2/C14(purple hollowed circles) overlaps I0ofH¼0/C14
(black circles) with little deviation, indicating that the oscillation
dynamics are not fundamentally changed by the tilting of H.
Furthermore, the difference in frequency between the signals labeled II
(red hollowed circles) and III (blue hollowed circles) equals to that ofI, as shown in Fig. 4(c) . The corresponding results from simulation
under l
0H¼0.81 T are shown in Figs. 4(b) and4(d). Here, the fre-
quencies of the peaks with the largest magnitude in the spectra in Figs.
3(b),3(d),a n d 3(f)are plotted as a function of J.T h ec o m p a r i s o n
between experiment and simulation indicates that the strong red-shiftsignals labeled I
0and I observed in the experiment are due to the MR
effect of the STO, as schematically illustrated in Fig. 1(d) .T h es i g n a l s
labeled II and III reflect OPP mode oscillation of the NiFe and FeCo
layers, respectively. As the magnetization dynamics of the FeCo layer
is excited, the STT to the NiFe layer changes accordingly. As the direc-
tions of the magnetizations of both the NiFe and FeCo layers vary, a
stable oscillation state is eventually reached, where the STT balancesthe damping torque, leading to OPP mode oscillation of both the NiFe
FIG. 2. (a) d V/dIof the STO as a function of Uunder perpendicular l0H¼0.81 T
in theH¼0/C14and (b) H¼2/C14directions. (c) Ras a function of Uunder perpen-
dicular l0H¼0.81 T in the H¼0/C14and (d) H¼2/C14directions. (e) Mappings of
the PSD under perpendicular l0H¼0.81 T in the H¼0/C14and (f) H¼2/C14
directions.
FIG. 3. (a) Time evolution of mFeCo in a stable oscillation state under perpendicular
l0H¼0.81 T and J¼3.2/C2108A/cm2obtained from micromagnetic simulation.
(b) DFT magnitude of the x component of mFeCo. (c) Time evolution of mNiFe. (d)
DFT magnitude of the x component of mNiFe. (e) Time evolution of the correspond-
ingE. (f) DFT magnitude of E.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 114, 172403 (2019); doi: 10.1063/1.5086476 114, 172403-3
Published under license by AIP Publishingand FeCo layers. The appearance of the signals for fNiFeand fFeCo
might be caused by distortion of the trajectory of OPP mode oscilla-
tion due to the tilting of H, which leads to the change in Rin every
period of oscillation. The red-shift of fMRandfNiFeand the blue-shift
offFeCo were qualitatively reproduced in the micromagnetic simula-
tion. It is worth mentioning that the mapping of the PSD shown inFig. 2(f) exhibits two weak microwave signals from additional modes,
which are not identified from the micromagnetic simulation. Themicrowave signals have the frequencies approximately equal to
2/C2f
FeCoandfNiFe/C02/C2fFeCo. The appearance of these signals may
be caused by the pillar of the STO having a finite deviation from theperfect circular shape. On the other hand, in micromagnetic simula-tion, a perfect circular pillar was used and the additional modes maynot appear.Some of the peaks and dips in the d V/dIcurves and Rof the STO
are mapped on the U/C0Hplane in Figs. 5(a) and5(b), respectively. At
zero H, the STO shows high R.A sHincreases, Rdecreases to a mini-
mum value under low U; however, it suddenly increases as Uincreases
to/C2420 mV. This behavior corresponds to the peaks in the d V/dI
curves [black circles in Fig. 5(a) ], which are caused by the reversal of
the NiFe layer. After the reversal of the NiFe layer, under high l
0H
/C241.5 T, the STO shows a high value of Rclose to the one under zero
H, indicating that NiFe and FeCo are antiparallel but are oriented
along the z-axis.26In the region between the high and low H,t h eS T O
shows intermediate R. Within that region, the area colored red in Fig.
5(a) represents the condition under which the signal due to OPP
mode oscillation of both layers was observed from the mapping of
PSD. The threshold Uon the left side of the red area coincides with
the dips in the d V/dIcurves, as marked by the gray hollowed circles in
Fig. 5(a) .
For MAMR application, a large hfor OPP mode oscillation is
important since it determines the amplitude of generated hac.hfrom
the positive z-axis of both NiFe and FeCo ( hNiFeandhFeCo) was esti-
mated based on fNiFeand fFeCo obtained from the experiment. The
macrospin model was used under the assumption that the frequencyof OPP mode oscillation is proportional to the effective field of the
layer, which is the sum of the external H, demagnetizing field, and
dipole field generated from the other layer.
32Figure 6(a) shows the
estimated husing the results from Fig. 4(a) (seesupplementary mate-
rialfor the detailed description of the estimation). A large hFeCo/C2460/C14
appears at U/C2440 mV and gradually increases to /C2470/C14asUincreases.
This trend is attributable to the increase in STT with increasing U.O n
the other hand, hNiFe/C24120/C14and slightly decreases as Uincreases.
The field-dependence of hwas also investigated. fNiFe,fFeCo,a n d fMR
are plotted in Fig. 6(b) as a function of Hobtained from mapping of
FIG. 4. (a) Frequencies of the microwave signals labeled I0, I, II, and III in mapping
of the PSD from Figs. 2(e) and2(f)as a function of U. (b) Frequencies of the peaks
with the largest magnitude from Figs. 3(b) ,3(d), and 3(f)as a function of J. (c)
(fNiFe/C0fFeCo)/C0fMRfrom the experiment as a function of Uand (d) from the simu-
lation as a function of J.U¼80 mV used in the experiment corresponds to
J¼3.3/C2108A/cm2.
FIG. 5. (a) Peaks (black circles) and dips (gray hollowed circles) in the d V/dIcurves
mapped on the U/C0Hplane. The red area marks the condition under which the
signals due to OPP mode oscillation of both layers were observed from mapping ofthe PSD. (b) Rof the STO mapped on the U/C0Hplane. Hin (a) and (b) was
applied in the perpendicular direction ( H¼0/C14).
FIG. 6. (a)hof NiFe and FeCo estimated from the results shown in Fig. 4(a) . (b)
fNiFe,fFeCo, and fMRas a function of Hunder U¼80 mV. f00NiFe was calculated
using the relationship of fMR¼fNiFe/C0fFeCo. (c)hof NiFe and FeCo estimated from
the results shown in (b).Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 114, 172403 (2019); doi: 10.1063/1.5086476 114, 172403-4
Published under license by AIP Publishingthe PSD under U¼80 mV, which corresponds to J¼3.3/C2108A/cm2.
At high H/C241 T, the microwave signal of fNiFewas so weak that it can-
not be distinguished in the power spectra, and the values calculatedfrom the relationship f
MR¼fNiFe/C0fFeCowere used, as indicated by the
red hollowed circles. fNiFe,fMR,a n d fFeCoincrease as Hincreases, and
fFeCoshows a maximum value of /C2416 GHz. hestimated by using these
results is shown in Fig. 6(c) .H e r e , hFeCoexhibits values of /C2470/C14with
small changes due to H,w h i l e hNiFeincreases as Hincreases. It is
worthwhile mentioning that for the mechanism of STO studied here,because the magnetization of the polarizer is reversed opposite to H
[Fig. 1(c) ], its demagnetizing field has a positive z component, while a
negative z component for that of the FGL. This leads to usually ahigher effective field for the polarizer and thus higher frequency ofOPP mode oscillation than that of the FGL.
In conclusion, the dynamics of another design of STO for
MAMR, in which only a soft magnetic thin layer is exploited as thepolarizer, were investigated. Using a NiFe layer as the polarizer and aFeCo layer as the FGL, the experimental and simulation results showthat both layers oscillate in OPP mode at different frequencies, namely,f
NiFeand fFeCo, respectively. Such dynamics also generated a micro-
wave signal due to the MR effect as fMR¼fNiFe/C0fFeCo. The macrospin
model was used to estimate hof OPP mode oscillation, and the estima-
tion results suggest that the FeCo layer has a large hof/C2470/C14at high
fFeCo/C2416 GHz.
Seesupplementary material for a detailed description of the fabri-
cation process for the STOs and the estimation of h.
This work was supported by the Advanced Storage Research
Consortium (ASRC), Japan, and JSPS KAKENHI Grant Nos. JP17H06152,
JP17K14802, and JP19K05257. The authors thank H. Suto, S. Tsunegi,
T. M. Nakatani, and R. Iguchi for the valuable discussions and N. Kojima
for the technical support.
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Published under license by AIP Publishing |
1.4972231.pdf | Constricted variational density functional theory for spatially clearly separated
charge-transfer excitations
Florian Senn and Young Choon Park
Citation: J. Chem. Phys. 145, 244108 (2016); doi: 10.1063/1.4972231
View online: http://dx.doi.org/10.1063/1.4972231
View Table of Contents: http://aip.scitation.org/toc/jcp/145/24
Published by the American Institute of Physics
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THE JOURNAL OF CHEMICAL PHYSICS 145, 244108 (2016)
Constricted variational density functional theory for spatially clearly
separated charge-transfer excitations
Florian Senn1,a)and Young Choon Park2
1Department of Chemistry, University of Calgary, 2500 University Drive NorthWest,
Calgary, Alberta T2N 1N4, Canada
2Department of Chemistry, Korea Advanced Institute of Science and Technology, Daejeon 305-701, South Korea
(Received 28 September 2016; accepted 2 December 2016; published online 28 December 2016)
Constricted Variational Density Functional Theory (CV-DFT) is known to be one of the success-
ful methods in predicting charge-transfer excitation energies. In this paper, we apply the CV-DFT
method to the well-known model systems ethylene-tetrafluoroethylene (C 2H4C2F4) and the
zincbacteriochlorin-bacteriochlorin complex (ZnBC BC). The analysis of the CV-DFT energies
enables us to understand the 1=Rcharge-transfer behaviour in CV-DFT for large separation distances
R. With this we discuss the importance of orbital relaxations using the relaxed version of CV( 1)-DFT,
the R-CV(1)-DFT method. Possible effects of the optimization of the transition matrix for the relaxed
self-consistent field version of CV( 1)-DFT, RSCF-CV( 1)-DFT in the case of large fragment separa-
tions are shown and we introduce two possible gradient restrictions to avoid the unwanted admixing
of other transitions. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4972231]
I. INTRODUCTION
Time-Dependent Density Functional Theory (TD-DFT)
with its relatively high accuracy has become a “‘workhorse’
of numerical quantum chemistry for computations of exci-
tation spectra and molecular response properties from first
principles,”1and due to its relatively low computational costs,
compared to other correlated quantum chemical approaches,
TD-DFT is “especially well suited to study large sys-
tems.”2Despite its popularity, TD-DFT applied with standard
exchange and correlation functionals fxchas difficulties to
produce charge-transfer (CT) excitation energies correctly.1
Indeed, it has even been demonstrated that for excitations
involving substantial CT-character the calculated energies
deviate from the experimental value “by up to several elec-
tron volts.”3So it is now well-known that TD-DFT based
on standard gradient-corrected functionals affords a quantita-
tively as well as a qualitatively incorrect picture of transitions
with CT-character between two spatially separated regions.4
We mention in passing that for the occurrence of the “charge-
transfer problem,” as understood in this work, no net charge
transfer between the different entities is necessary.5,6The rea-
son for these difficulties has been seen by many authors to
lie in the exchange and correlation functional.1,3,5,7,8Indeed,
functionals including a certain Hartree-Fock exchange part
improve the situation, and a functional like CAM-B3LYP9
clearly improved the accuracy of TD-DFT in excitations with
CT character;7,8,10Rudolph et al.11state that it “is now estab-
lished that hybrid functionals with range-separated exchange
can effectively handle the CT problem.” While the mentioned
CAM-B3LYP functional does not reproduce the correct 1=R
long-range limit,7there are successful methods improving the
a)Electronic mail: florian.senn@ucalgary.caasymptote of the exchange-correlation potential as, e.g., the
long-range corrected hybrid scheme (e.g., Refs. 12–15) and the
asymptotically corrected model potential scheme,16,17which
should exhibit the correct decay by construction,18but popu-
lar representatives as LB9419and LB 20cannot capture the
long-range Coulomb interaction.15
There are also DFT-based approaches which do not mod-
ify the functional but tackle this problem differently, where
all have their advantages and assets. Some of these, having
been applied for the study of CT excitations, are, e.g., Con-
strained Orthogonality Method (COM),21,22Maximum Over-
lap Method (MOM),23Constricted Variational Density Func-
tional Theory (CV-DFT)24and its extensions,25,26constrained
density functional theory,27Self-Consistent Field DFT
(SCF-DFT),28Orthogonality Constrained DFT (OCDFT),29
Ensemble-DFT,30,31subsystem DFT (FDE-ET).32
One of these mentioned alternative DFT-based methods is
CV-DFT, a variational approach for the description of excited
states, reviewed in Ref. 25. In Ref. 33 it has been shown how
the theoretical framework of this method is able to cope with
excitations including a CT character, and in Refs. 4 and 34
it has been shown how “the well-known failure can be traced
back to the use of linear response theory.”34Thus one can
also obtain reasonably accurate excitation energies for transi-
tions with CT-character using functionals based on the local
density approximation when going beyond linear response.
But it has not been shown how CV-DFT performs for excita-
tions where the charge donor and acceptor entities are clearly
spatially separated, while Ref. 8 shows how the orbital over-
lap has an influence on the performance of a functional for
excitation energies, at least when using TD-DFT, and that
CT excitations span a wide range of orbital overlap values.
We would like to demonstrate how the 1=Rbehaviour can
also be obtained even by a local density approximation func-
tional. We will do this by means of two examples which are
0021-9606/2016/145(24)/244108/10/ $30.00 145, 244108-1 Published by AIP Publishing.
244108-2 F. Senn and Y. C. Park J. Chem. Phys. 145, 244108 (2016)
well-studied in the literature concerning excitations with CT-
character; first, the model system ethylene-tetrafluoroethylene
C2H4C2F4to study the excitation energy dependence with
respect to the separation distance of the donor and accep-
tor, a system which has already been used several times for
this purpose.3,10,12,32,35,36As a second example we choose
the zincbacteriochlorin-bacteriochlorin complex ZnBC BC,
which is important as a “paradigmatic photosynthetic com-
plex.”37Additionally ZnBC BC is also “one of the earliest
charge-transfer systems for which TD-DFT difficulties have
been unraveled and discussed”37and so it is not surprising
that it has been studied with different approaches, e.g., TD-
DFT with different types of functionals (local,7,38,39hybrid,7,40
Coulomb attenuated hybrid,7and range separated hybrid41),
charge constrained DFT calculations,42pragmaticSCF-
DFT correction to CIS,39CIS(D),43and the Bethe-Salpeter
formalism.37,44
II. THEORETICAL PART
We will review the CV-DFT method in a nutshell under
the focus of explaining our findings and implementations; for
a detailed description of the CV-DFT method, we refer to
Refs. 25 and 26 (for the inclusion of the orbital relaxation
of triplet excited states).
We shall start from a closed-shell ground state density
described by a single Slater determinant 0=j 1 2::: i:::
nocc (with nocc: number of occupied orbitals, nvir: number of
virtual orbitals, and f gground state orbitals). The basic con-
cept is to construct a set of “occupied” excited state orbitals
f 0gby mixing the occupied and virtual ground state orbitals
through a unitary transformation Y. The unitary transformation
Yis expressed through a skew symmetric transition matrix U
as the expansion shown in the following:45
Y occ
vir!
=*
,mX
k=0Uk
k!+
- occ
vir!
= 0
occ
0
vir!
. (1)
Thereby m=1leads to Y=exp(U), which is the most general
form for this mapping (we deal with real orbitals)26and for
a summation with m=nwe talk about CV( n)-DFT25(lead-
ing to CV(1)-DFT for m=1). With this new set of “occu-
pied” excited state orbitals we obtain an excited state density.
The substitution of the density difference between the excited
state and the ground state density into the Kohn-Sham energy
expression yields the CV-DFT excitation energy, E. This
excitation energy Eis optimized variationally with respect to
Uin the SCF-CV(1)-DFT scheme.46To ensure that the energy
does not collapse to a lower state and has exactly one fully
transferred electron from the occupied to the virtual space, a
constraint is applied in CV-DFT,45
virX
aPaa= occX
iPii=1, (2)
wherePaa:=Pocc
iYaiYaiandPii:=Pocc
jYijYij.
Excited state orbitals which are not directly participat-
ing in the transition are frozen within the SCF-CV( 1)-DFT
scheme. In order to allow these orbitals to react to the excita-
tion, thus allowing all orbitals to relax, we apply a transforma-
tion (described by the matrix R) on the ground state orbitalsto obtain a set of relaxed orbitals f g,
i(1)= i(1)+virX
cRci c(1) 1
2virX
coccX
kRciRck k(1),
a(1)= a(1) occX
kRak k(1) 1
2virX
coccX
kRakRck c(1).(3)
The unitary transformation Yis now applied on f gto obtain
the relaxed excited state orbitals (see Refs. 25 and 26), leading
to the relaxed version of the method SCF-CV( 1)-DFT, RSCF-
CV(1)-DFT, or when R-CV( 1)-DFT when the transition
matrix is not optimized.
For infinite order CV-DFT, CV( 1)-DFT, the concept of
natural transition orbitals47(NTOs) has been introduced, as
the NTOs allow one to describe the excitations in a more
compact form than with canonical orbitals. A singular value
decomposition of the transition matrix Ugives
U =V(W )T. (4)
whereii=
iand= or . Therewith NTOs are obtained
as
o
i=occX
j(W )ji
j, (5)
v
i=virX
a(V)ai
a, (6)
where jgoes over the occupied ground state orbitals and agoes
over the virtual ground state orbitals. The “occupied” excited
state orbitals, resulting out of the unitary transformation Eq. (1)
taken to infinite order ( n=1), can be written with the NTOs
as45
0
j=cos[
j] o
j+sin[
j] v
j; j2foccg, (7)
whereis defined by the normalization condition assuring
that exactly one electron is transferred from the occupied to
the virtual space,45
occX
isin2(
i)=1 . (8)
We note that Eq. (8) corresponds to Eq. (2) when going to infi-
nite order.45In the case of only one non-zero singular value,
9!i:
i,0, we have a single NTO transition, meaning a tran-
sition from a single occupied NTO to a single virtual NTO, see
also Refs. 25 and 45. In general more than one singular value
is non-zero, so we refer to an excitation as being dominated
by single orbital replacement if
max>1.0, while in the case
max<1.0, the transition is best described by several NTO
replacements.25
The CV-DFT energy is determined once the matrices U
and, if applied, Rare determined. But, as we show in Sec. IV
we found that for an optimized transition matrix U, the correct
1=Rbehaviour is not guaranteed anymore for some spatially
separated charge-transfer excitations and this originates from
an unwanted mixing in the transition matrix U. To restrict the
optimization of the transition matrix Uminimizing the exci-
tation energy in an unwanted way (see Sec. IV), two different
approaches have been implemented for triplet excitations.
244108-3 F. Senn and Y. C. Park J. Chem. Phys. 145, 244108 (2016)
A. Version (a): Based on the Slater determinant overlap
The optimization of the transition matrix Uin order to
minimize the excitation energy Eis done in a self-consistent
way,46where we note the starting transition matrix U0and
its related Slater determinant 00, corresponding to the tran-
sition matrix of CV(2)-DFT.25This initial transition matrix
shall now determine which molecular orbitals are involved in
the final Slater determinant of the optimized excited state 0
as follows. When we optimize the transition matrix, we have
in iteration ma transition matrix update Um, which can be
decomposed into different NTO excitations (corresponding to
a singular value
i):Um=Pnocc
iUm
i, resulting in nocc
different contributions Um
i. From such a Um
ifol-
lows a Slater determinant 0k;
i,. These Slater determinants
have an overlap
jS 0:,j=h 00,j 0m;
i,i. (9)
The Slater determinant has contributions from the occupied
and virtual ground state orbitals, making it possible to decom-
pose the overlap of 00and 0m;
i,into an overlap of the
occupied and virtual ground state orbitals involved in this tran-
sition: S 0:,
o,o and S 0:,
v,v , respectively. In the case of a clear
single charge-transfer excitation, all occupied ground-state
orbitals are located on fragment A and all virtual ground-state
orbitals are located on fragment B. Therefore the contribu-
tionUm
itoUmis kept only when S 0:,
o,o >and
S 0:,
v,v >, otherwise it is removed from Um.
B. Version (b): Based on the density overlap
In Ref. 48 the density overlap is used as a measure of the
differential overlap to decide when the correction for the long-
range part is switched on. We will adopt this density overlap
with the following idea: A single entry of the transition matrix
U, the matrix element Uai, corresponds to a single canonical
orbital replacement transition (SOR-R-CV( 1)-DFT,26where
Usor:ai
bj=abij) and therewith a density f
Usor:aig
. We
decompose the transition matrix update U
U=occ,virX
i,a(Uai)USOR: ai. (10)
A transition matrix Udescribes an excitation with a result-
ing density change. We will use the overlap of such density
changes as the following criteria. If we have for a spin
an overlap between the density change of the initial transi-
tion matrix U0,
0, and a given SOR-CV (1)-DFT-density
change being larger than a threshold, thus S:,[sor:ai]
=sf
r;U0g
f
r;Usor:aig
dr> , for both, the part
where the electron is coming from (occupied ground state
orbitals: o,o) and where it is going to (virtual ground state
orbitals:3,3), the transition-matrix element Uaiwill lead to
acceptable transitions.
This gives us for each U-entry aithe overlaps
S:,
o,o[sor:ai]=
of
r;U0g
of
r;Usor:aig
dr(11)
S:,
v,v[sor:ai]=
vf
r;U0g
vf
r;Usor:aig
dr(12)with which we define
O(ai)=(1 if S:,
o,o[sor:ai]> ^S:,
v,v[sor:ai]>
0 else .
(13)
During the optimization, we multiply each matrix ele-
mentUaiwith O(ai), thus we will accept possible elements
Uaifrom the gradient only if the overlap-criteria is fulfilled,
otherwise we set Uai=0.
There are several ways to indicate a possible charge-
transfer character of a given excitation; Moore et al. in Ref. 49
provided simple schemes which they tested for several organic
molecules. We implemented the charge transfer parameter DCT
for the triplet excitations according to the idea of Ref. 50
with the following differences: Over the transition matrix we
know the density change and thus integrate directly over the
density change and not the total density. But when we allow
orbital relaxation, we admix occupied and virtual ground state
orbitals, see Eq. (3). Doing so, the domain (thus where an
electron is taken from) density is now built with contributions
of virtual ground state orbitals; thus orbitals with no electron
density in the ground state now obtain a certain electron den-
sity. Thus, even while being part of the domain, we have
points rwhere(r)>0, due to these contributions. A similar
argument holds for the domain where an electron is excited
to. Therefore we calculate the charge transfer parameter DCT
using the character of a triplet excitation by
DCT= q q , (14)
where we use
q =
r (r)dr
(r)dr, (15)
q =
r (r)dr
(r)dr. (16)
It is clear that in case of no orbital relaxation for a
triplet excitation, our modified DCTmatches the original
implementation of Ref. 50.
III. COMPUTATIONAL DETAILS
All calculations are done using an implementation in a
developer’s version of the ADF 2016 program.51–53We used
for our calculations an all-electron51standard triple- Slater
type orbital basis with one set of polarisation functions.54
The functionals adapted in this work are the local density
approximation in the V osko, Wilk, and Nusair parametrisa-
tion (LDA)55and the generalized gradient approximation of
Becke’s exchange56combined with Lee, Yang, and Parr (LYP)
correlation57–59(BLYP). Excitation energies are obtained
within the Tamm-Dancoff approximation (TDA).60In order
to compare with previous investigations, the calculations of
the ZnBC BC complex and its isolated fragments were car-
ried out using the ground state optimized geometries reported
in Ref. 39.
244108-4 F. Senn and Y. C. Park J. Chem. Phys. 145, 244108 (2016)
For convergence we used a CV-DFT tolerance of 10 3and
for damping, Xi=Xi 1+ X, we applied variable damping
factors = 0+~Xi~Xi 1, where XisUnorRnof the
actual iteration n=ior previous iteration n=i 1 (and = ~0
for iteration 0) with 02[0.15, 0.25 ]. Nevertheless we could
not obtain converged transitions for the calculation of C 2H4
C2F4using a gradient restriction based on the density overlap
criteria with =10 2at a distance of R= 5.0 Å (for the
meaning of we refer to Sec. II).
IV. RESULTS AND DISCUSSION
A. C 2H4C2F4and the 1/Rproblem
It has been demonstrated and explained why TD-DFT with
local exchange functionals fails to predict the excitation ener-
gies correctly for transitions with CT-character and also fails
to give the correct donor-acceptor distance long-range 1=R
dependence,35even for excitation processes between two non-
overlapping subsystems in which no net charge transfer takes
place6as in the example of C 2H4C2H4out of Ref. 5.
To examine the charge-transfer excitation energy depen-
dence with respect to the separation distance of the donor and
acceptor, the model system C 2H4C2F4has been used several
times.10,12,35,36With these different studies as a background,
we will use this prominent example to examine the behaviour
of CV-DFT and its versions. There are two transitions of partic-
ular interest, the transitions HOMO !LUMO and HOMO 1
!LUMO+1, for which the excited states are of b1 symme-
try. As visible in Fig. 1 these excitations have a clear charge
transfer between the two molecules C 2H4and C 2F4. We note
here that in the following when we classify a transition to be of
one of the mentioned types HOMO !LUMO or HOMO 1
!LUMO+1; not necessarily only these mentioned ground-
state orbitals contribute, but from all the contributing ground-
state orbitals they contribute most. Also we would like to
mention that a possible delocalization of a given orbital over
both fragments is highly functional dependent (compare, e.g.,
the results of Ref. 48).
Our results as well as selected reference values for com-
parison are shown in Fig. 2.
We will now have a closer look at the different versions
of CV-DFT and we will start with singlet excitations. Let us
begin with CV(1)-DFT, where the transition matrix Ucorre-
sponds to the one in TD-DFT with the Tamm-Dancoff approx-
imation (TDA),60but now applied to infinite order.25First we
notice that we obtain a 1=R-like behaviour; assuming a E(R)
= c1=R+cofunction, we obtain fitting coefficients c1for the
results presented in Fig. 2 of 1.1 and 0.9 E ha0. Further we can
see that CV(1)-DFT gives for both transitions singlet exci-
tation energies with values similar to the values reported for
the revised Hessian in Ref. 36 (we note here that for distances
R<5.0 Å the values deviate more).
Allowing the relaxation of all orbitals not participat-
ing directly in the transition, thus using the R-CV( 1)-DFT
method,2the excitation energies decrease significantly, but we
still obtain a 1=Rbehaviour (fitting coefficients c1for the
results presented in Fig. 2: 1.1 and 0.9 E ha0). For the HOMO
!LUMO transition we can see from Fig. 2 that our val-
ues agree with the values from LC-BLYP obtained in Ref. 12
FIG. 1. C 2H4C2F4: Representation of ground-state KS-orbitals (LDA),
R= 5.0 Å.
(MAD = 0.2 eV , RMSD = 0.2 eV). It is therefore not surpris-
ing that with the assumed E(R)= c1=R+c0behaviour, our
value for an extrapolated infinite separation, ER!1=12.7 eV ,
is quite close to the ER!1=12.5 eV reported in Ref. 12.
Let us now have a look at the triplet excitations of
R-CV(1)-DFT. For larger distances no spin interaction is
expected and so it is of no surprise that the triplet R-CV( 1)-
DFT transition energies match with the energies obtained for
the singlet excitations. While the HOMO 1!LUMO+1 tran-
sition energies are quasi-identical to the ones of the singlet
excitations (MAD¡0.1 eV , RMSD¡0.1 eV), we notice that for
the HOMO!LUMO transition, this is only the case for
R6.5 Å (MAD¡0.1 eV , RMSD¡0.1 eV , but MAD = 0.7 eV ,
RMSD = 1.4 eV if all data points are considered). Similar
findings result for CV( 1)-DFT.
We can say that for all these transitions, except the HOMO
!LUMO triplet excitations for R<6 Å, we obtain a nice
1=Rbehaviour. All these calculations have in common that
the transition matrix Uhas not been optimized and therefore
the character of the transition itself has also not changed.
CV-DFT is a variational method and we can optimize the
transition matrix Uin the sense of minimizing the excitation
energy, thus applying the method RSCF-CV( 1)-DFT.2,45,46
244108-5 F. Senn and Y. C. Park J. Chem. Phys. 145, 244108 (2016)
FIG. 2. C 2H4C2F4vertical excitation energies for singlets (circles) and triplets (triangles) using CV( 1)-DFT (orange), R-CV( 1)-DFT (red), and RSCF-
CV(1)-DFT (dark red). The values for the revised Hessian out of Ref. 36 (purple filled circles), LC-BLYP out of Ref. 12 (black filled circles), and SAC-CI
out of Ref. 12 (grey filled circles) are given as reference. The lines serve as a guide for the eyes and when the excitation is not dominated by one of the charge
transfer excitations, we set its value to zero (and are therewith not visible in the figure).
From Fig. 2 we see that the energy is clearly minimized, but the
excitation energies are nearly distance independent and clearly
the expected 1=Rlong-range dependence is now lost. This
(unwanted) energy gain comes from the optimization of the
transition matrix U. Thus, for simplicity of the argumentation,
we will forget the orbital relaxation. We can also see from Fig. 2
that this energy gain occurs for both, singlet and triplet excita-
tions, thus we can restrict the analysis to the triplet excitations
where we do not need to go over the mixed state excitations.
A closer look at the composition of the excited states
shows that the two transitions, HOMO !LUMO and
HOMO 1!LUMO+1, mix with contributions close to
=4:
12[0.82, 0.95 ](average value = 0.85) and
22
[0.63, 0.75 ](average value = 0.72), respectively (and no
significant further contribution of other NTO transitions).
Thus the two charge transfer excitations, which were clearlyseparated before, mix now in a way that we obtain the
transitions
1(HOMO )+
2(HOMO 1)!
1(LUMO )
+
2(LUMO +1)and
1(HOMO 1)+
2(HOMO )
!
1(LUMO+1 )+
2(LUMO ).
We can confirm this also by looking at the charge
transfer parameter DCTin Fig. 3: for CV( 1)-DFT and R-
CV(1)-DFT we obtain a DCTvalue corresponding to the sep-
aration distance Rof the two fragments of C 2H4C2F4, thus
indicating a charge transfer from one fragment to the other.
For RSCF-CV(1)-DFT we obtain a DCTvalue being signifi-
cantly lower, thus confirming that we have nearly no net charge
transfer.
To understand why this mixing of different excitations
happens, we rewrite the formula for the excitation energy for
the triplet state without orbital relaxation using NTOs, Eq. (29)
in Ref. 25,
ET=occ=2X
i=1sin2
i iv io +1
2occ=2X
i=1occ=2X
j=1sin2
isin2f
jg
Kio io jo jo +Kiv iv jv jv 2Kio io jv jv
+occ=2X
i=1occ=2X
j=1sin
icos
isinf
jg
cosf
jg
Kio iv jo jv , (17)
where we adopt the notation of Ref. 25 and therefore omarks
an occupied orbital and 3an unoccupied orbital, while i,jgo
over NTOs, Kare two electron integrals including a Coulomb
FIG. 3. C 2H4C2F4DCTvalues for vertical triplet excitations using CV-
DFT (orange triangles), R-CV( 1)-DFT (red circles), and RSCF-CV( 1)-DFT
(dark red squares). The lines serve as a guide for the eyes.as well as an exchange correlation part (for the precise defini-
tion we refer to Ref. 25). Ziegler and Krykunov already identi-
fied the term KKS
ii,aaas the one responsible for the asymptotic
1=Rbehaviour in CV(4)-DFT.4In CV(1)-DFT the corre-
sponding term is a part of the term Kio io jv jv of Eq. (17),
when io andjv are located on different fragments as we will
explain in the following part.
We have seen before that for the not optimized transition
matrix U, the excitations we are looking at here can be seen as
a single NTO transition (and thus 9!i:
i,0). In such a case
Eq. (17) simplifies to
E1 NTO
T=iv io +1
2(Kio io io io
+Kiv iv iv iv 2Kio io iv iv . (18)
The terms Kio io io io andKiv iv iv iv are two electron inte-
grals where all functions are located on the same fragment,
so the terms contributing with a positive sign have a large
contribution. In contrast the term Kio io iv iv , where for the
244108-6 F. Senn and Y. C. Park J. Chem. Phys. 145, 244108 (2016)
TABLE I. K-integrals of the energy composition for C 2H4C2F4atR= 8.0 Å.
Excitation HOMO !LUMOc/eV Excitation HOMO 1!LUMO+1c/eV
K-integral terma,bR-CV(1)-DFT RSCF-CV( 1)-DFT R-CV( 1)-DFT RSCF-CV( 1)-DFT
1=2Kio io jo jo 4.93 3.06 5.44 3.10
1=2Kiv iv jv jv 4.52 2.90 5.24 2.97
Kio io jv jv 1.79 6.29 1.80 6.36
Total 7.67 0.34 8.88 0.29
aThe values given are total contributions from all K-integrals of the same type (and thus include the cases i=j).
bIn the given implementation, the energy contributions due to orbital relaxation are separated, therefore the K-integral values given
here stem from the density change due to the final transition matrix U.
cDue to the mixing, for RSCF-CV( 1)-DFT these excitations are slightly dominated by the corresponding transition, but include
a nearly as important fraction of the second transition (in both excitations
i=0.82 and
j=0.75).
charge transfer excitations studied here the functions iv and
io are located on different fragments, and so the integral
contributing with a negative sign has a smaller absolute value
(for an example see the values for R-CV( 1)-DFT in Table I),
but it is the distance dependent term relevant for the 1=R
behaviour.In the case of the optimized transition matrix Uwe have
mainly two NTO transitions contributing and for simplicity
we approximate their contribution as
1=
2==4. For
the given calculations only the terms of the first two lines
in Eq. (17) contribute significantly and therewith we can
write
E2 NTOs
T1=2 iv io +1=2
jv jo
+1
2 1=4 Kio io io io +Kiv iv iv iv 2Kio io iv iv
+1=4
Kjo jo jo jo +Kjv jv jv jv 2Kjo jo jv jv
+1=4
Kio io jo jo +Kiv iv jv jv 2Kio io jv jv
+1=4
Kjo jo io io +Kjv jv iv iv 2Kjo jo iv iv
. (19)
Let us have a look at the different K-integrals for our case,
where two different charge transfer transitions mix: HOMO
!LUMO and HOMO 1!LUMO+1 (see Fig. 1). With a
positive sign we have contributions of terms involving either
only occupied or only virtual ground state NTOs. The NTOs
io andjv are located on one fragment, and iv andjo on
the other one. Therefore the terms in the third line of Eq. (19),
involving the NTOs from transition iandj, have only a small
contribution (going to zero for R!1 ) while the terms in the
second line involve only one transition and correspond to the
terms in Eq. (18), but due to the prefactor (sin2(
1)sin2(
2)
=1=4) their contribution is reduced. When we consider the
K-integrals contributing with negative sign, we see that the
terms from the second line in Eq. (19) include the occu-
pied ground state NTO io and the virtual ground state NTO
iv , which are, due to the charge transfer character, on dif-
ferent fragments and so contribute less as they behave as
1=R. These terms correspond to the term in Eq. (18). But
the remaining two K-integrals with a negative sign in the
third line of Eq. (19) include the occupied ground state NTO
and the virtual ground state NTO from the other transition,
located on the same fragment and therefore contribute with
a larger absolute amount and will not vanish for R!1 .
In total this mixing of the two different charge transfer
excitations results in a smaller destabilization ( K-integrals
with positive sign contribute less) and a larger stabiliza-
tion ( K-integral with negative sign result in larger values),a clear reduction of the total excitation energy E. For
clarification the values of the different K-integrals for the
example of C 2H4C2F4at a distance of R= 8.0 Å are shown
in Table I.
We have seen that the energy lowering due to the optimiza-
tion of the transition matrix Ucomes from a mix of different
charge-transfer excitations. Such a mix brings in additional
energy stabilizing K-terms of orbitals located at the same frag-
ment are while the additional destabilizing K-terms are located
on different fragments and contribute less. As a result we have
a partial charge cA2(0, 1)on fragment A and a partial charge
(1 cA)on fragment B, also when the two fragments are fur-
ther apart. We will therefore have to restrict the gradient in
such a way that for further separated fragments the two dif-
ferent charge-transfer excitations do not mix any more. From
the above analysis of the different K-integrals it is clear that
for a criterion, a simple distance dependence will not help. In
the following we will apply two approaches to restrict the gra-
dient from this unwanted mixing of different NTO transitions:
(a) based on the Slater determinant overlap (b) based on the
density overlap (see Sec. II).
The resulting excitation energies when applying these
restrictions are shown in Fig. 4.
As visible from Fig. 4, both versions of a gradient
restriction depend highly on the applied threshold parame-
ter. For a too small threshold value the transition we
wanted to block stays allowed and can be mixed in, thus
244108-7 F. Senn and Y. C. Park J. Chem. Phys. 145, 244108 (2016)
FIG. 4. C 2H4C2F4vertical triplet excitation energies with gradient restrictions; using the Slater determinant overlap (green) with threshold value = 10 1
(down-pointing triangles), 10 2(squares), 10 3(diamonds) and the density overlap (blue) with a threshold value = 10 1(crosses), 10 2(circles), 10 3(plus).
The values for R-CV( 1)-DFT (red filled triangles) triplet excitations and LC-BLYP singlet excitations out of Ref. 12 (black filled circles) are given as reference.
The lines serve as a guide for the eyes; when the excitation is not dominated by one of the charge transfer excitations, we set its value to zero (and are therewith
not visible in the figure, e.g., all HOMO 1!LUMO+1 excitation energies with the density overlap restriction using a threshold of =10 3).
FIG. 5. C 2H4C2F4DCTvalues for vertical triplet excitations with gradient restrictions; using the Slater determinant overlap (green) with threshold value
= 10 1(down-pointing triangles), 10 2(squares), 10 3(diamonds) and the density overlap (blue) with a threshold value = 10 1(crosses), 10 2(circles), 10 3
(plus). The values for R-CV( 1)-DFT (red filled triangles) are given as reference. The lines serve as a guide for the eyes. When the excitation is not dominated
by one of the charge transfer excitations, we set its value to zero.
we can obtain transitions corresponding to the ones obtained
with RSCF-CV(1)-DFT (happening, e.g., for the Slater
determinant overlap criteria with = 0.001 and R<8.0 Å:
MAD¡0.1 eV). These mixed in transitions can even become
dominant in a way that the excitation does not fulfill our clas-
sification any more (happening, e.g., for the density overlap
criteria with = 0.001). On the other hand a too high threshold
value blocks any change (visible, e.g., for the density over-
lap criteria with =0.1). While this is intuitively understand-
able, the HOMO!LUMO excitations at R= 5.5–6.5 Å with
the density overlap restriction and =0.01 deserve a further
look: We see in Fig. 4 that although we optimize the transition
matrix U, the excitation energy is higher than the excitation
energies obtained with R-CV( 1)-DFT. The excitations we
look at here are not the lowest excitations of a given sym-
metry and the transition matrix optimization therefore affects
the lower lying excitations (in our case there are two initially
TABLE II.!?Q-band excitation energies (in eV) for BC and ZnBC.
TD-DFT R-CV( 1)-DFT RSCF-CV( 1)-DFT References
LDA BLYP LDA BLYP LDA BLYP CAM-B3LYP7GW-BSE37Expt.61,62
BC 2.26 2.27 2.23 2.26 1.47 1.48 1.92 1.63 1.60
BC 2.51 2.51 2.51 2.53 2.08 2.10 2.53 2.24 2.30
ZnBC 2.30 2.31 2.23 2.27 1.36 1.37 1.87 1.59 1.65
ZnBC 2.49 2.52 2.71 2.57 2.18 2.19 2.59 2.27 2.20lower lying excitations). Through the condition of orthogo-
nality between different transitions, the transitions of interest
are affected, in this case resulting in a clear increase of the
excitation energy.
To underline our finding of a mix between different
charge-transfer excitations, we plot the charge transfer param-
eterDCTin Fig. 5 and compare it with DCT-value resulting
from the initial, unoptimized transition matrix. We can clearly
see how only some restrictions result in a clear charge transfer
between the two fragments and thus prohibit a mixing of the
excitations.
B. Application to ZnBC BC
The phenylene linked zincbacteriochlorin (ZnBC) bac-
teriochlorin (BC) complex (ZnBC BC) serves as a model
system for the study of energy transfer in photosynthesis.7,39
244108-8 F. Senn and Y. C. Park J. Chem. Phys. 145, 244108 (2016)TABLE III. Phenylene-linked ZnBC BC complex: Excitation energies and oscillator strengths f for the 10 lowest TD-DFT singlet excitations and corresponding R-CV( 1)-DFT excitation energies.
TD-DFT/LDA R-CV( 1)-DFT/LDA TD-DFT/BLYP R-CV( 1)-DFT/BLYP
Excitation E (eV) f Type E (eV) Type E (eV) f Type E (eV) Type
1 1.27 0.00 ZnBC !BC CT 2.19 BC !BC Q 1.28 0.00 ZnBC !BC CT 2.23 BC !BC Q
2 1.39 0.00 BC !ZnBC CT 2.19 ZnBC !ZnBC Q 1.41 0.00 BC !ZnBC CT 2.24 ZnBC !ZnBC Q
3 1.84 0.00 BC !ZnBC CT 2.47 BC !BC Q 1.85 0.00 BC !ZnBC CT 2.49 BC !BC Q
4 1.92 0.00 ZnBC !BC CT 2.50 ZnBC !ZnBC Q 1.92 0.00 ZnBC !BC CT 2.52 ZnBC !ZnBC Q
5 2.23 0.48 BC !BC Q 3.24 ZnBC !BC CT 2.25 0.44 BC !BC Q 3.22 ZnBC !BC CT
6 2.28 0.21 ZnBC !ZnBC Q 3.35 BC !ZnBC CT 2.30 0.24 ZnBC !ZnBC Q 3.35 BC !ZnBC CT
7 2.39 0.00 ZnBC !BC CT 3.83 BC !ZnBC CT 2.38 0.00 ZnBC !BC CT 3.80 BC !ZnBC CT
8 2.49 0.07 BC !BC Q 3.90 ZnBC !BC CT 2.49 0.07 BC !BC Q 3.87 ZnBC !BC CT
9 2.50 0.01 ZnBC !ZnBC Q 4.29 ZnBC !BC CT 2.50 0.01 ZnBC !ZnBC Q 4.24 ZnBC !BC CT
10 2.62 0.00 ZnBC !BC CT 5.08 ZnBC !BC CT 2.65 0.00 BC !ZnBC CT 4.55 BC !ZnBC CTThis donor-acceptor complex is also one of the first systems
revealing the difficulties of TD-DFT with charge-transfer exci-
tations.37,39,44For the isolated fragments, the ZnBC and the BC
molecule, experimental spectra are known, see, e.g., Refs. 61
and 62. The absorption spectra consist of two weakly allowed
bands in the red spectral region, the Q bands, and two strong
absorptions in the blue spectral region, the B (or Soret) bands.63
These excitations are explained as !?transitions.63From
Table II we see that our excitation energies obtained with R-
CV(1)-DFT are close to the corresponding TD-DFT results
but deviate from the findings of Ref. 7 (using LDA: MAD = 0.19
eV , using BLYP: MAD = 0.20 eV) and even more when com-
pared to the values obtained by Ref. 37 (but we note here that
Duchemin et al. used different coordinates), where especially
the lower lying Q-band excitation of each fragment deviates.
As expected the optimization of the transition matrix U, thus
using the RSCF-CV( 1)-DFT method, results in lower excita-
tion energies and these energies deviate more from the findings
by Ref. 7 (using LDA: MAD = 0.45 eV , using BLYP: MAD =
0.44 eV) but are clearly closer to the values from Ref. 37.
Dreuw and Head-Gordon mentioned nicely in Ref. 39 that
thesystems of the phenylene linkage are perpendicular to
thesystems of the ZnBC and BC fragment and therefore the
phenylene ring is expected to have only a minor influence on
the energetically low-lying electronic !?excitations of
the complex which are located on the ZnBC or BC fragment. In
the excitation spectra of ZnBC BC we can therefore find the Q
bands of the constituting fragments, intramolecular excitations
where the promoted electron remains on the same fragment
of the molecule (ZnBC or BC), with “almost exactly identical
energies.”39Additionally there are charge transfer excitations,
where the photoexcited electron goes from the donor to the
acceptor fragment of the molecule.44
We will now recalculate the singlet excitation energies of
ZnBC BC with CV-DFT. From Sec. IV A we have seen how
orbital relaxation is important. While for short distance intra-
molecular CT excitations the optimization of the transition
matrix Uis beneficial,34we have also seen in Sec. IV A that
for medium and long-range distances, this optimization can
lead to an unwanted mix of transitions. We apply therefore
the method R-CV( 1)-DFT, and the values for ZnBC BC are
given in Table III.
As expected, the excitation energies for the Q bands are
very similar to those obtained for the isolated fragments (where
only the second ZnBC excitation using LDA deviates more
than 0.05 eV). When we compare the values obtained with
R-CV(1)-DFT, given in Table III, with the corresponding
ones obtained with TD-DFT, we see a significant increase
of all CT excitation energies, while the Q state energies
change by less than 0.1 eV . As a result the four lowest exci-
tations are now all Q bands, in agreement with the findings in
Refs. 7, 37, 39, and 44 with the blemish of obtaining the BC
!BC Q band as lowest excitation and not the ZnBC
!ZnBC Q band (although our obtained energy differ-
ence between these two transitions is negligible). From
Table III we also see that our seventh transition for TD-
DFT using BLYP is a CT transition, in contrast to the
results obtained by Kobayashi et al. , a difference stem-
ming from using the same functional but a different
244108-9 F. Senn and Y. C. Park J. Chem. Phys. 145, 244108 (2016)
basis-set; additionally we use the Tamm-Dancoff approxima-
tion (we noted that not using the Tamm-Dancoff approxi-
mation can result in a change of order for some excitations
within TD-DFT calculations). A comparison of the lowest six
excitation energies with the CAM-B3LYP results obtained by
Kobayashi and Amos7gives an acceptable agreement (for R-
CV(1)-DFT using LDA: MAD = 0.29 eV , RMSD = 0.32 eV ,
BLYP: MAD = 0.22 eV , RMSD = 0.27 eV), and a similar pic-
ture is obtained when comparing to the GW-BSE (with full
diagonalization) results by Duchemin et al37(for R-CV(1)-
DFT using LDA: MAD = 0.29 eV , RMSD = 0.32 eV , BLYP:
MAD = 0.31 eV , RMSD = 0.35 eV).
V. CONCLUSION
With this work we could clearly show how CV( 1)-DFT
is able to reproduce a nice 1=Rlong-range behaviour for
charge-transfer excitations. We could also demonstrate how
orbital relaxation can play a significant role and R-CV( 1)-
DFT not only keeps the 1=Rlong-range behaviour but even,
using LDA as a functional, agrees nicely with the findings of
long-range corrected functionals. Applying the method on the
popular example for CT excitations, the zincbacteriochlorin-
bacteriochlorin complex ZnBC BC, we obtain the four
Q-bands with singlet excitation energies similar to the corre-
sponding ones for the isolated fragments as lowest excitations,
followed by two CT excitations. These findings are in agree-
ment with other methods obtained by Kobayashi and Amos7
and Duchemin et al.37
While for short distance intra-molecular CT excitations
the optimization of the transition matrix Uis beneficial,34
for medium and long-range distances it is possible that the
optimization leads to an unwanted mixing of transitions as
analysed for C 2H4C2F4.
Two different restrictions of the transition matrix opti-
mization are implemented and show that in general one can
prohibit the unwanted admixing of a different charge-transfer
excitation. But these restrictions depend highly on the mini-
mal overlap parameter, where a too small value can lead to
not restricting enough and a too large value results in block-
ing every possible optimization. As the optimal parameter
value is not known in advance, we recommend to use no
restriction, thus the general RSCF-CV( 1)-DFT, if there is no
risk of such an admixture of transitions and otherwise to use
R-CV(1)-DFT, where the transition matrix is not optimized.
ACKNOWLEDGMENTS
The authors would like to express their gratitude to the late
Professor Dr. Tom Ziegler for his unflinching support until his
untimely passing away. This work was supported by the Nat-
ural Sciences and Engineering Research Council of Canada
(NSERC) through Discovery Grant to T.Z. Further the authors
are grateful to Dr. Mykhaylo Krykunov and Professor Dr.
Dennis Salahub for helpful discussions. The computational
resources of WESTGRID were used for all calculations. F.S.
thanks the Eyes High program for the financial support.
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|
1.4897648.pdf | Absorption spectroscopy of isolated magnetic antivortices
Matthias Pues, Michael Martens, and Guido Meier
Citation: Journal of Applied Physics 116, 153903 (2014); doi: 10.1063/1.4897648
View online: http://dx.doi.org/10.1063/1.4897648
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/15?ver=pdfcov
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155.33.16.124 On: Wed, 26 Nov 2014 01:44:52Absorption spectroscopy of isolated magnetic antivortices
Matthias Pues,1,a)Michael Martens,1and Guido Meier1,2
1Institut f €ur Angewandte Physik und Zentrum f €ur Mikrostrukturforschung, Universit €at Hamburg, Jungiusstraße
11, 20355 Hamburg, Germany
2Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg,
Germany
(Received 27 August 2014; accepted 30 September 2014; published online 15 October 2014)
We present an analysis of the dynamics of isolated magnetic antivortices preformed by high
frequency absorption measurements from the linear via the non-linear to the switching regime.Static magnetic bias fields are used to deflect the antivortex out of the equilibrium position and the
shift of the resonance frequency of the gyrotropic eigenmode is observed. The results from the
absorption measurements for highly anisotropic annihilation fields of the antivortex are comparedwith magneto-resistance measurements and micromagnetic simulations.
VC 2014
AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4897648 ]
I. INTRODUCTION
Isolated vortices and antivortices have been studied as
potential candidates for fast non-volatile data-storage.1–3
These magnetic singularities can be found in thin-films of
soft ferromagnetic materials like permalloy (Ni 80Fe20). The
singularities are characterized by a special curling of the in-plane magnetization around a small core region, where the
magnetization tilts out-of-plane, either up or down. This so
called polarization p¼61 can be utilized to store a bit of
information.
In order to switch the polarization, several different
mechanisms for both vortices and antivortices have beendescribed. The gyrotropic eigenmode of the vortex or anti-
vortex, a circular motion of the core around the equilibrium
position, can be excited by oscillating external fields
4–7or
spin currents8,9either in a continuous or pulsed fashion.10,11
Once the core reaches a critical velocity, the radius of the
trajectory does no longer increase, but the magnetizationradially inwards of the core is deformed and a vortex-
antivortex pair with an opposite polarity is created. The orig-
inal core annihilates with its counterpart. In case of a vortex,it annihilates with the antivortex of the pair, and a single vor-
tex with opposite polarity remains. A corresponding process
for antivortices exists.
4,5,12,13Since both singularities are
involved in either switching process, a detailed knowledge
of the dynamics of vortices and antivortices is needed for a
complete understanding of the involved processes.
Vortices have been studies intensively, since they can be
found as the ground state in thin-film structures with proper
thickness and width that resemble a disc14,15closing the
magnetic flux. In contrast to this, experimental studies on
antivortices are more complicated as their magnetic structure
is characterized by four alternating poles giving rise to acomplex shape in which the antivortex can be stabi-
lized.
7,9,16,17In these thin-film structures, the antivortex is,
although stable, not easily recoverable once destroyed andmostly found as an as-grown state. Our method presented inRef. 17for a reliable antivortex nucleation in a specially
shaped structure enables an analysis of the antivortex dy-namics by means of high frequency (hf) absorption measure-
ments as well as magneto resistance (MR) measurements for
the static behavior of isolated antivortices.
II. SAMPLES AND MEASUREMENT TECHNIQUES
Samples are fabricated by electron-beam lithography
using a Zeiss Supra 55 SEM with a Raith lithography system
and lift-off processing on silicon substrates. For absorption
measurements, 85 u-shaped permalloy (Ni 80Fe20) elements
with a thickness of 50 nm and a wire width wof 1.1 lm are
deposited by thermal evaporation as well as the overlaying
80 nm thick copper stripline, capped by 5 nm gold, seeFig.1. For resistance measurements, a single permalloy ele-
ment is contacted by dc-magnetron sputtered gold leads.
Before deposition of the gold layer, the contact surface is
FIG. 1. Scanning-electron micrograph of several u-shaped permalloy micro-
structures overlaid by a copper stripline for high-frequency absorption meas-
urements. The surrounding Oersted-field of the high-frequency current is
schematically depicted. The inset in the upper left shows a magnetic force
micrograph with antivortices in each element, where the three-dimensional
depiction of the topography is overlaid by the color coded magnetic infor-mation.
17,18The inset in the lower right shows the in-plane magnetization
vector field of an antivortex.a)Electronic mail: Matthias.Pues@physik.uni-hamburg.de
0021-8979/2014/116(15)/153903/7/$30.00 VC2014 AIP Publishing LLC 116, 153903-1JOURNAL OF APPLIED PHYSICS 116, 153903 (2014)
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155.33.16.124 On: Wed, 26 Nov 2014 01:44:52cleaned in situ via radio frequency argon-plasma etching to
ensure a good electrical contact and a 3 nm adhesive alumi-
num layer is deposited without breaking the vacuum.
The absorption spectroscopy is performed in a serial
setup of an Agilent Technologies E8257D analog signal gen-
erator, the sample, and an Agilent E4418B power meter withan Agilent E9304A sensor.
6The electrical connection from
SMA connectors on a printed circuit board to the broadened
ends of the copper stripline is realized by aluminum wirebonding. The ground signals are closed on the printed circuit
board, since no impedance matched ground-signal-ground
layout is necessary within the frequency range used here. Atwo-dimensional electromagnet with two pairs of pole pieces
surrounding the sample, that is controlled via Hall sensor
feedback, ensures the precise remanence free control of theexternal in-plane field necessary for the generation of the
antivortices.
17The gyrotropic mode of the antivortices is
excited harmonically by the unidirectional alternatingOersted field of the stripline. At resonance, the antivortices
gyrate with a maximum radius around their equilibrium posi-
tion. The power absorption of the antivortex ensemble resultsin an increase of the total stripline impedance. This conver-
sion has been discussed in Ref. 6. In order to detect the small
signal DRcaused by the antivortex ensemble compared to
the resistance of the stripline ( R
SL/C2570X), a reference signal
has to be measured. This is done by saturating the magnet-
ization of all structures at 90 mT in x-direction and setting
the field back to the bias field used to deflect the antivortex
from its equilibrium position. Note, that for some experi-
ments, the bias field can be zero. Application of the satura-tion field ensures that a homogeneous magnetization is
present in the wire junctions of the structures. Antivortices
are then generated by a two dimensional field sequence thathas been presented in our previous work, see Ref. 17, and
the signal difference of both magnetic states is determined.
Thus for every frequency step, the antivortices are newlygenerated. This method is prone against temperature or pos-
sible other drifts of the resistance of the whole setup.
Moreover, possible magnetic state changes induced by thebias field have no effect on the following data points. Since
the method used to generate the antivortices also gives con-
trol over their orientation, antivortices with c¼/C01 are inves-
tigated in all measurements presented here. The
measurements are performed at T¼21.5
/C14C controlled with a
precision of 0.15 K.
We have performed complementary magneto resistance
measurements in lock-in technique. Figure 2(a) depicts a
microstructure with Au contacts. The antivortex state isinvestigated by comparison of the resistance in saturation
parallel to the current flow direction, indicated in Fig. 2(a),
and the resistance after antivortex generation and applicationof the bias in-plane field with angle Hfor each data point.
The magnetization pattern of the antivortex approximately
consists of four triangularly shaped uniformly magnetizeddomains, see Figs. 2(b) and2(c). For an estimation of the
MR signal of the antivortex state, the ratio of the domain
size parallel and perpendicular to the current flow is constantfor a core deflection in an external field. When we assume a
homogeneous current flow between the contact leads, aconstant magnetization within the wire arms, and no defor-
mation of the 90
/C14domain walls, the MR signal is expected to
be constant to a good approximation.
III. RESULTS AND DISCUSSION
The resonance frequency of the gyrotropic eigenmode
of isolated antivortices is determined by means of absorption
spectroscopy. Figure 3(b) shows a typical absorption spec-
trum for the antivortex ensemble described above with a res-
onance frequency of fres¼(169.460.3) MHz. This
frequency is approximately 40% lower than the resonancefrequency of a vortex confined in a square-shaped element
with the same film thickness, and edge length comparable to
the wire width.
6The lower frequency indicates a weaker
confining potential if the antivortex is considered as a rigid
quasiparticle as it is successfully done for vortices.
A variation of the excitation power and with that the
exciting Oersted field21is depicted in Fig. 3(a).22For low ex-
citation powers, the resonance frequency is rather constant.
In this regime (1), the gyration of the antivortex can bedescribed as a rigid quasiparticle in a nearly parabolic poten-
tial.
20If the excitation power is increased, the resonance fre-
quency drops significantly about 20% in regime (2). Inaddition to the frequency drop, an asymmetric absorption
curve can be observed in this non-linear regime, see Fig.
3(c). This asymmetry has also been reported for numerical
simulations and measurements of resonance curves of vorti-
ces by Drews et al. in Ref. 19. The core switching regime (3)
FIG. 2. (a) Scanning electron micrograph of a u-shaped structure with Au
contact leads for resistance measurements. (b) Scheme of a deflected anti-
vortex. The triangularly shaped domains are denoted as parallel or perpen-dicular to the current flow. (c) Combined atomic and magnetic force
micrograph of a contacted u-structure containing an antivortex with an ori-
entation of c¼/C01. For the MR measurements presented here, the structure
with two contacts shown in (a) is used.153903-2 Pues, Martens, and Meier J. Appl. Phys. 116, 153903 (2014)
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155.33.16.124 On: Wed, 26 Nov 2014 01:44:52is reached at l0Hhf¼3 mT, indicated by the characteristic
cone shaped signal23above 3 mT. The Oersted field to
induce antivortex core switching is about six times the field
strength needed to induce core switching in a similar setup
for vortices.6,23The continuous switching of the polarity
of the core is possible, since the linear excitation by the
stripline can couple to both the clockwise and the counter-
clockwise eigenmodes of the antivortex, in contrast to anexcitation by rotational Oersted fields
9or rotational spin
currents.7The absorption signal is reduced for strong excitation
fields above 3 mT near the resonance frequency, which can
be explained by the instability of the antivortex state, seeFig.3(d). If the antivortex switches its core polarization mul-
tiple times at high gyration radii, there could be a probability
that it moves away from the equilibrium position in the mid-dle of the wire junction into one of the arms and is destroyed
there. Since the absorption measurement is a time integrating
method, this process does not contribute to the ensemble sig-nal anymore. For vortices especially in discs, these consider-
ations are irrelevant, since the vortex state is the
energetically favored state. Consequently, even if a vortex isdestroyed by a strong excitation, it will recover shortly after
and will again contribute to the absorption signal.
To probe the confining potential of the antivortex, a
static in-plane field is applied in eight different directions,
H
ext(H), in order to deflect the antivortex from the center of
the wire junction. Since the generation process of the anti-vortices determines the orientation of all antivortices in the
ensemble, here c¼/C01, a single deflection direction of the
whole ensemble can be ensured. A detailed description ofthe shift of the energy minimum due to this Zeeman field for
vortices is presented in Ref. 24. For the absorption spectros-
copy, see Fig. 4(a), a low excitation field in the harmonic re-
gime of l
0Hhf¼0.3 mT is chosen. The resonance frequency
dependence of the static bias field Hext(H) shows a different
behavior for different field angles as well as a varying anni-hilation field H
an(H) of the antivortices. At the annihilation
field, the antivortex is pushed out of the wire junction and
consequently, the absorption signal vanishes. Three types offrequency shifts can be distinguished. The frequency shift la-
beled (I) exhibits a small drop in the resonance frequency of
6 MHz, which corresponds to about 4%, as well as adecrease in the absorption. The annihilation field is around
2 mT. The second type (II) is characterized by a similarly
small annihilation field, but shows a frequency increase ofabout 33 MHz (20%) with an increase of the absorption sig-
nal for bias field values close to the annihilation field. The
third type (III) has relatively high annihilation fields up to8 mT and the resonance frequency rises about 25 MHz
(15%). The small annihilation fields of antivortices in com-
parison to annihilation fields of vortices of about 35 mT(Ref. 6) indicate again the comparably very shallow poten-
tial, which confines the antivortex within the wire junction.
Moreover, the anisotropy of the annihilation fields and thefrequency shifts shows a strong influence of the wire arms
and the asymmetry of the structure on the antivortex.
To compare the drastic asymmetry of the annihilation
field for gyrating antivortices with the annihilation fields for
a static antivortex, magneto-resistance measurements are
performed, see Fig. 4(b). The annihilation fields are indicated
by abrupt resistance changes closest to zero fields in the MR
signal. The MR signal jumps from a nearly constant plateau,
as expected for the deflected antivortex state, to a MR signalindicating a magnetization diagonal, perpendicular, or paral-
lel to the current flow, see insets in Fig. 4(b). For some field
angles, the annihilation of the antivortex is followed by addi-tional step-like transitions most likely due to sudden depin-
ning processes of domain walls from the corners of the
FIG. 3. (a) Influence of the exciting field strength Hhfon the absorption
spectrum. Three regions can be distinguished: (1) linear gyrotropic motion,
(2) non-linear gyrotropic motion, and (3) continuous switching of the polar-
ity of the antivortices. In regime (3), the cone shaped absorption signal is
caused by continuous switching processes. Note the different field step sizes
in region (1) of about 0.05 mT compared to regions (2) and (3) with
0.28 mT. (b) Average of 17 frequency sweeps in the linear regime. The
absorption signal shows a Lorentzian shape caused by the resonant excita-tion of the gyrotropic eigenmode of the antivortex ensemble. (c) Absorption
signals for an increasing excitation fields in the non-linear regime of the
gyrotropic mode. The curves are offset successively by 50 m X. In order to
show the increasing asymmetry and red shift of the absorption curves,
19,20
the same Lorentzian fit at the lowest excitation field strength of 1.96 mT is
plotted as a solid line for all data curves. (d) Absorption signal from the core
switching regime. The solid line is a guide to the eye.153903-3 Pues, Martens, and Meier J. Appl. Phys. 116, 153903 (2014)
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155.33.16.124 On: Wed, 26 Nov 2014 01:44:52junction. As every data point represents a whole field sweep
starting after an antivortex nucleation from zero field to ei-ther positive or negative field values, the smooth field de-
pendence of the MR signal for each field angle indicates
both a successful antivortex nucleation for every data pointand a single path for the expulsion of the antivortex. The
annihilation fields range from 7.2 mT to 12.7 mT and show a
similar but much less distinct asymmetry as those derivedfrom the absorption measurements, see Fig. 5(a).
Analogous to the MR measurements, micromagnetic
simulations are performed with the OOMMF code,
25where
the deflection of the antivortex by a quasi static in-plane field
Hext(H) for eight different field angles is investigated. The
simulated element is mapped from a scanning electron micro-graph of a real element, ensuring the same dimensions and
edge roughness. Typical material parameters for permalloy
are used, a saturation magnetization of M
S¼8/C2105Am/C01,
and an exchange constant of A¼1.3/C210/C011Jm/C01. An artifi-
cially high Gilbert damping of a¼0.5 is used, as only thenew equilibrium position of the antivortex for the given bias
field is of interest here. Using a realistic damping value, the
step-like increase of the field would result in an unrealistic
spin-wave generation. The cell size of the simulation mesh ischosen to be 5 /C25/C225 nm
3.
The annihilation fields derived from these simulations
match the ones determined by the MR measurements, seeFig.5(a). Furthermore, the simulations give a deep insight in
the deflection behavior of the antivortex, as well as in the
annihilation process for different field directions. Figure 5(c)
shows the core deflection distance from its equilibrium posi-
tion for all fields with a minimal deflection of r
min¼142 nm
and a maximal deflection of rmax¼255 nm. A linear depend-
ence of the core deflection on the external field with a rate of
(14.660.7) nm/mT can be found up to about 5 mT,
FIG. 4. (a) Dependence of the antivortex resonance on a static bias field
Hext(H). (b) Magneto resistance of the wire junction of a u-shaped micro-
structure that contains an antivortex at zero field. The annihilation fields
of the antivortex vary strongly between the two experiments. The excited
antivortex is destroyed at lower static field strengths. In the absorption
spectra, three types (I, II, and III) of frequency shifts can be distinguished.FIG. 5. (a) Comparison of the annihilation fields of the deflected antivortexfor different in-plane field angles of the static field H
ext. Annihilation fields
from magneto-resistance measurements, micromagnetic simulations, and
high frequency absorption measurements are shown. (b) Simulated deflec-
tion from the equilibrium position of the core for bias fields with the indi-
cated angle. The different annihilation processes (V arm,VDW, and AV an) for
each angle are explained in the text. (c) Distance from the equilibrium posi-tion at zero field of the antivortex core depending on the bias field obtained
by micromagnetic simulations. (d) Sketch of a u-shaped structure. A deflec-
tion of the antivortex into one of the marked regions corresponds to the reso-
nance frequency shift types from Fig. 4(a).153903-4 Pues, Martens, and Meier J. Appl. Phys. 116, 153903 (2014)
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155.33.16.124 On: Wed, 26 Nov 2014 01:44:52indicating the parabolic confining potential. For higher field
strengths, the core deflection rate increases up to the annihi-
lation of the antivortex. This increase shows that the confin-ing potential becomes shallower once the antivortex is
pushed into the arms, which is contrary to the behavior of
isolated vortices. A vortex is confined in a closed microstruc-ture like a disc or a square, resulting in a deviation from the
parabolic confining potential near the boundary of the struc-
ture. Thus, a vortex needs much stronger fields to be pushedtowards the boundaries of the structure.
19,24,26In the graph
showing the core position for all field strengths and angles,
see Fig. 5(b), another effect of the open junction can be seen.
For field strengths causing a linear deflection rate up to
5 mT, the displacement of the core follows a certain angle.
This azimuth angle b0to the new equilibrium position of the
core, see Fig. 5(b), can be generally derived by
b0¼n(Hþp/C0U0)¼n(Hþp/C0cp/2) taking a rotational
symmetric potential and the Zeeman energy into account.Here, the antivortex with a winding number
27ofn¼/C01 and
the fixed orientation of c¼/C01 yields b0¼/C0H/C03p/2. At
certain field strengths, the core deflection deviates from thisdirection, mostly for the deflection into the arms. This can be
attributed to the depinning of the 90
/C14domain walls from the
corners of the wire junction, see Fig. 6(d), and a deformation
of the walls. Moreover, for high field strengths the magnet-
ization within the arms whose magnetization points antipar-
allel to the external field starts to bend in a zigzag fashion,leading to the deflection of the core away from the center of
the arm. This behavior is not described by a simple rigid qua-
siparticle model of the antivortex, where the antivortex isconfined in a parabolic potential. An extension of the para-
bolic potential to describe the deviation of the potential
towards the boundaries of the structure has been done forvortices in squares,
24but for antivortices this approach poses
several difficulties. For antivortices in u-shaped structures,
the potential is highly anisotropic, whereas for vortices insquares only two different field angles, towards the edge of
the square or diagonally have to be considered.
Similar shifts of the resonance frequency of the antivor-
tex can be found for adjacent field angles and considering
the deflection angle of the antivortex core caused by the
external field reveals the influence of the u-shaped structure
onto the confining potential, see Fig. 5(d), even though the
wire junction in which the antivortex is confined is com-
pletely rectangular. Thus, a much more complex potentialconfines the antivortex.
An analysis of the antivortex annihilation for each field
angleHreveals three different processes, as indicated in Fig.
5(b). In Fig. 6, two of these annihilation processes are exam-
plarily shown in the micromagnetic simulations for field
angles of H¼0
/C14in Figs. 6(a)–6(c) , and 90/C14in Figs.
6(d)–6(f) . At an external field of l0Hext(0/C14)¼11 mT, the
magnetization in the upper right curved wire folds, forming a
180/C14domain wall, see Fig. 6(a). From this wall, a vortex
FIG. 6. (a)–(c) Sequence of the simulated annihilation process of the antivortex with an orientation of c¼/C01 by a static external field of l0Hext¼11 mT in
x-direction. (a) A 180/C14domain wall forms in a curved segment of the u-shaped structure. (b) A vortex nucleates from the domain wall. (c) The upper 90/C14
domain wall of the antivortex detaches from the corner of the wire junction forming another vortex. This vortex moves to the center of the junction and b oth
the vortex and the antivortex are annihilated. (d)–(f) Sequence of the annihilation of the antivortex at l0Hext¼14 mT in y-direction. (d) The upper and lower
90/C14domain walls are no longer pinned at the respective corner and start moving into the arms. (e) The antivortex moves to the right corner and is annihilate d
there. Two edge defects are formed that move further into the wire. The numbers denote the winding number of the respective magnetization pattern.153903-5 Pues, Martens, and Meier J. Appl. Phys. 116, 153903 (2014)
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155.33.16.124 On: Wed, 26 Nov 2014 01:44:52nucleates and a domain that no longer points antiparallel to
the external field is created, see Fig. 6(b). This tilts the mag-
netization in the upper right arm and destabilizes the upper90
/C14domain wall of the antivortex. It detaches from the cor-
ner of the junction and another vortex nucleates from the for-
mer domain wall. The new vortex moves towards theantivortex and both are annihilated, see Fig. 6(c). This pro-
cess is labeled V
armin Fig. 5(b) since the annihilation of the
antivortex starts with the creation of a vortex in a wire arm.The second process, labeled V
DW, is similar, but a 90/C14do-
main wall of the antivortex detaches from the corner of the
junction via a vortex nucleation without a preceding vortexnucleation in a curvature of the structure. The last process,
AV
an, is only observed for a bias field in y-direction. The
upper and lower domain walls of the antivortex depin fromthe corners, but are still attached to the boundary of the struc-
ture, see Fig. 6(d). The domain walls and the antivortex start
moving to the right until the antivortex core reaches the rightcorner of the wire junction, see Fig. 6(e). Here, the antivortex
is annihilated and two edge defects
27form, which move fur-
ther along the inner boundary of the curved wire. Similarly,edge defects are generated, when a vortex nucleates in the
other processes described above. The magnetic texture of
these edge defects can be described by half integer windingnumbers n¼61/2. When taking all magnetic defects that
are generated or annihilated during the annihilation of the
antivortex into account, it can be observed that the sum ofthe winding number of the whole structure remains
n
sum¼/C01, the same as the initial antivortex, cf. Fig. 6. This
holds for all bias field angles up to the maximum fieldstrength of 15 mT in the simulation. The different annihila-
tion processes exhibit no special symmetry that can be found
in the u-shaped structure, cf. Figs. 5(b) and5(d), but give a
possible explanation of the anisotropy of the measured anni-
hilation fields. For most cases, the antivortex is not pushed to
the boundary of the wire junction and destroyed there, but itsmagnetic texture is distorted by changes of the magnetization
far from the antivortex. The annihilation processes for the
real element may differ from the ones observed in the simu-lations, however, a possible diversity in the annihilation
mechanism is revealed. Thus in the absorption and magneto-
resistance measurements, it cannot be distinguished, if theannihilation field of the antivortex is measured, or a field at
which the magnetization at some point in the u-shaped struc-
ture is reversed by the external field as in the V
armprocess.
The annihilation fields determined by MR measurements
and micromagnetic simulations for a static antivortex are in
good agreement, but the results from absorption spectroscopy,where the antivortex is gyrating at the resonance frequency ex-
hibit much smaller fields, see Fig. 5(a). This discrepancy may
not be attributed to high gyration radii and an expulsion of theantivortex from the decentered e quilibrium position at a certain
bias field. An estimation of the gyration radius r
gyrcan be done
by results from transmission X-r ay microscopy on antivortices
in elements with comp arable dimensions.7,9In these works,
maximal gyration radii of 95 nm to 115 nm measured at
140 MHz and 120 MHz, respectively, are measured before coreswitching starts. Assuming a similar switching threshold for the
here presented structures and a linear dependence of the gyrationradius on the excitation field H
hf, the radius rgyrcan be estimated
to be about 10 nm, a tenth of the maximal radius. Core switching
is reached at l0Hhf¼3.0 mT, cf. Fig 3(a), and 0.3 mT are used
as the excitation field strength for the measurements of the bias
field dependence on the resonance frequency, cf. Fig. 4(a).T h e
displacement for the lowest anni hilation field for the absorption
measurements with l0Han(180/C14)¼1.4 mT can be deduced from
the simulations, where the const ant deflection rate yields a dis-
placement of the equilibrium position for this field of about20 nm. The maximal displacemen t for a gyrating antivortex at
this position can thus be estimated to be under 30 nm away from
the center of the wire junction, w hereas it can be deflected up to
142 nm in the static case. Consequen tly, the reduced annihilation
fields for the deflection of an excite d antivortex cannot be attrib-
uted to the gyration radius of the antivortex, but rather to areduction of the activation fields of the above described proc-
esses of domain wall depinning and vortex nucleation by the
high frequency field. A similar reduction of the switching fieldof a nanoparticle by radio-frequency field pulses is described by
Thirion et al. in Ref. 28. However, the maximal gyration radius
of about 100 nm derived from the transmission X-ray micros-copy results
7,9before core switching s tarts and the minimal
deflection distance of 142 nm for the static case obtained by
micromagnetic simulations coul d explain the drastic decrease in
the absorption signal for increa sing excitation fields in the core
switching regime, cf. Fig. 3(a). It supports the above mentioned
hypothesis of an antivortex destruction for strong excitation fieldstrength and high gyration radii.
29
Another comparison of the antivortex and the vortex can
be made concerning the critical velocity needed for coreswitching. For vortices, a critical switching velocity
v
crit¼2prfreswas found to be 320 m/s by analytical and
micromagnetic calculations5for vortices in discs or 250 m/s
for vortices in squares by absorption measurements.13,23The
squares have comparable dimensions as the wire width and
thickness of the structures used to stabilize the antivortex.The vortices in these squares exhibit a resonance frequency
of 320 MHz and thus reach a gyration radius of about
124 nm at the critical velocity. The microscopy results forantivortices yield a much lower critical velocity of about
85 m/s at the comparable radii of 95 nm to 115 nm. The cause
of the low critical velocity needs to be investigated and if thecritical velocity is also applicable as a universal criterion for
the switching process of antivortices.
IV. CONCLUSIONS
The gyrotropic eigenmode of isolated antivortices has
been measured by high frequency absorption spectroscopy
for varying excitation field strengths in the linear, non-linear,
and core switching regimes. The behavior of isolated anti-vortices is similar to one of the excited vortices. When com-
paring antivortices in our structures with vortices in squares
of similar dimensions, the gyrotropic mode of the antivortexhas a lower resonance frequency. To induce core switching,
the antivortex needs to be exposed to a much stronger excita-
tion field than the vortex.
Deviations of the confining harmonic potential of the
antivortex due to a static in-plane field have been153903-6 Pues, Martens, and Meier J. Appl. Phys. 116, 153903 (2014)
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
155.33.16.124 On: Wed, 26 Nov 2014 01:44:52demonstrated by the shift of the resonance frequency in the
absorption measurements.
To complement the anisotropy of the annihilation fields
found in the absorption spectroscopy, magneto-resistance
measurements as well as micromagnetic simulations have
been performed. While the annihilation fields determinedfrom the MR measurements and the micromagnetic simula-
tions without an excitation of the antivortex match quite
well, the annihilation fields drop about a half for the absorp-tion spectroscopy, where the antivortex is excited. Moreover,
a much more distinct anisotropy of the annihilation can be
found for the excited state. It is demonstrated by means ofmicromagnetic simulations that for some field angles, the
annihilation of the antivortex is caused by a rotation of the
magnetization within an arm of the structure far away fromthe antivortex itself, thus destabilizing the antivortex.
These results show that a simple quasi particle descrip-
tion for the antivortex is no longer applicable for the case ofa strong excitation and a deflection of the equilibrium posi-
tion by a bias field exceeding about 5 mT for our structures,
but the change of the whole magnetization of the u-shaped
structure has to be taken into account.
In order to find an expansion for the confining harmonic
potential of isolated antivortices, further investigations areneeded. A direct depiction of the magnetization by transmis-
sion X-ray microscopy can show the deformation of the cir-
cular trajectory of the excited antivortex depending on thebias field and the annihilation process. The influence of the
u-shaped structure on the antivortex may also be investi-
gated by increasing the length of the straight wire junctionand the radius of the curved wire segments.
ACKNOWLEDGMENTS
We thank Ulrich Merkt for discussions, encouragement,
and continuous support and Michael Volkmann for superbtechnical assistance. Financial support by the Deutsche
Forschungsgemeinschaft via Sonderforschungsbereich 668 is
gratefully acknowledged.
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155.33.16.124 On: Wed, 26 Nov 2014 01:44:52 |
1.3587575.pdf | Injection locking of tunnel junction oscillators to a microwave current
M. Quinsat, J. F. Sierra, I. Firastrau, V. Tiberkevich, A. Slavin et al.
Citation: Appl. Phys. Lett. 98, 182503 (2011); doi: 10.1063/1.3587575
View online: http://dx.doi.org/10.1063/1.3587575
View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v98/i18
Published by the American Institute of Physics.
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Downloaded 18 Feb 2013 to 128.118.88.48. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsInjection locking of tunnel junction oscillators to a microwave current
M. Quinsat,1,2, a/H20850J. F . Sierra,2I. Firastrau,3V. Tiberkevich,4A. Slavin,4D. Gusakova,2
L. D. Buda-Prejbeanu,2M. Zarudniev,1J.-P . Michel,1,2U. Ebels,2B. Dieny,2M.-C. Cyrille,1
J. A. Katine,5D. Mauri,5and A. Zeltser5
1CEA-LETI, MINATEC-Campus, 17 Rue des Martyrs, 38054 Grenoble, France
2SPINTEC, UMR CEA/CNRS/UJF–Grenoble 1/Grenoble–INP, INAC, Grenoble F-38054, France
3Transilvania University of Brasov, 29 Boulevard Eroilor, R-500036 Brasov, Romania
4Department of Physics, Oakland University, Rochester, Michigan 48309 USA
5Hitachi Global Storage Technologies, 3403 Yerba Buena Road, San Jose, California 95135, USA
/H20849Received 16 December 2010; accepted 1 April 2011; published online 5 May 2011 /H20850
Injection locking of a spin transfer nano-oscillator, based on an in-plane magnetized magnetic tunnel
junction and generating the frequency f0, to an external signal of varying frequency feis studied
experimentally and with macrospin simulations. It is shown, that if the driving signal has the formof a microwave current, the locking effect is well-pronounced near f
e/H110612f0, but is almost completely
absent near fe/H11061f0, confirming predictions of analytical theory. It is also shown that noise plays an
important role in the locking process, causing the linewidth of the locked oscillation to substantiallyexceed that of the driving signal. © 2011 American Institute of Physics ./H20851doi:10.1063/1.3587575 /H20852
The recently developed spin-transfer nano-oscillators
/H20849STNOs /H20850/H20849Refs. 1and2/H20850could be of a considerable techno-
logical interest for use as active elements in integrated nano-electronic circuits if the generated microwave power is in-creased, e.g., by using magnetic tunnel junctions
3or/and the
generation linewidth is reduced, e.g., by using current-induced oscillations of a magnetic vortex.4An alternative
way to achieve these goals is to use phase-locked arrays ofSTNOs.5Phase locking of two STNOs based on magnetic
nanocontacts via spin wave coupling6,7and four STNOs
based on magnetic vortices8has been evidenced while mu-
tual phase locking of STNOs based on magnetic nanopillarsand coupled via the generated microwave current theoreti-cally considered in Refs. 9–11still needs to be experimen-
tally demonstrated. To optimize mutual phase-locking of ST-NOs in an array it is necessary to clearly understand theprocess of injection-locking of an STNO, having a free-running frequency f
0, to an external periodic signal of the
frequency fe.5This effect was demonstrated experimentally
for spin valve STNOs magnetized by an out-of-plane biasfield in the case of driving by an external microwave currentforf
e/H11061f0in Refs. 12and13and in the case of driving by a
microwave magnetic field for several rational values of
fe/f0.14The analytical theory of STNO injection locking pre-
sented in Ref. 14also predicts that in an in-plane magnetized
STNO driven by a microwave current /H20849when the direction of
the current spin-polarization is parallel to the axis of symme-try of the STNO trajectory determined by the direction of thebias magnetic field /H20850the locking effect at f
e=2nf0should be
substantially more pronounced, than at fe=/H208492n+1/H20850f0, where
nis a natural number.
In our current letter, we check this theoretical prediction
experimentally for an STNO based on a magnetic tunneljunction whose oscillating and polarizing layers are alignedantiparallel to each other. Our STNO devices are similar tothose used in Ref. 3, have the stack composition of IrMn/
CoFeB/Ru/CoFeB/MgO/CoFe/CoFeB and nominal resis-tance area product of 1 /H9024
/H9262m2.At the first stage, our experiments were performed by
injecting a dc current I dcinto the device to induce a self-
sustained oscillation and by tuning the in-plane magneticfield H so that the free-running STNO frequency is f
0
/H110155.0 GHz. The microwave signal was extracted from the
STNO via a bias tee and visualized on a spectrum analyzer.The resolution bandwidth was 3 MHz for scans in a wide
frequency band and was reduced to 200 kHz for more de-tailed analysis of the microwave emission peak. At the sec-ond stage, a microwave current of varying frequency f
eand
amplitude I rfwas added to the setup. The amplitudes I rfof
the driving rf current were estimated as in Ref. 15from the
measured power levels of the rf signal source, taking intoaccount reflections and the capacitance between the STNOtop and bottom electrodes. The frequency fand the linewidth
/H9004fof the externally driven STNO were extracted using a
Lorentzian fit of the voltage power spectral density /H20849PSD/H20850.I n
the following we show results for a nanopillar of 85 nmdiameter with oscillations that are stabilized at a bias field ofH
app=90 Oe /H20849applied along the easy axis /H20850and a dc current
of I dc=0.6 mA. At this current the resistance value in the
antiparallel state is 415 /H9024and the magnetoresistance is
50%. The dependence of f0on the dc current is shown in the
inset of Fig. 1while the dependence of f0with the bias field
follows a usual Kittel equation.
Figure 1shows the PSD map of the output voltage for
the STNO frequency fversus the driving frequency fe.I ti s
clear from Fig. 1that the frequency fof the driven STNO
follows the driving frequency feonly in the vicinity of the
point fe=2f0while very weak or even no locking is observed
near the points fe=f0orfe=3f0. The red dots at fe=f0are an
artifact of the measurement and are due to the signal of thedriving source that cannot be suppressed. The disappearanceof the generated power between 6 and 8 GHz might be re-lated to the presence of a secondary oscillation peak in theSTNO power spectrum.
16
When the amplitude of the driving signal increases from
Irf/Idc=0.3 /H20851Fig.2/H20849a/H20850/H20852to Irf/Idc=0.65 /H20851Fig.2/H20849b/H20850/H20852, the width of
the injection locking frequency range /H9254feincreases /H20849full
squares /H20850while the linewidth of the driven STNO oscillationa/H20850Electronic mail: michael.quinsat@cea.fr.APPLIED PHYSICS LETTERS 98, 182503 /H208492011 /H20850
0003-6951/2011/98 /H2084918/H20850/182503/3/$30.00 © 2011 American Institute of Physics 98, 182503-1
Downloaded 18 Feb 2013 to 128.118.88.48. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions/H20849empty squares /H20850varies inside the injection-locking range /H9254fe
and reaches a minimum value /H9004fminat its center. Here we
defined /H9254feas the interval inside which the oscillation line-
width is reduced to half of the difference between the free-running and the minimum value. With the further increase inthe driving amplitude I
rfthe injection locking range contin-
ues to increase, reaching /H9254fe=0.6 GHz at I rf/Idc=0.8 /H20851Fig.
2/H20849c/H20850/H20852while the minimum STNO linewidth /H9004fmincontinues to
decrease /H20851Fig.2/H20849d/H20850/H20852. Note, however, that even at a reasonably
large amplitude of the driving signal I rf/Idc=0.65 /H20851Fig.2/H20849b/H20850/H20852,when the frequency fof the driven STNO is locked to the
fe/2, the minimum STNO linewidth /H9004fminremains rather
large /H20849about 35 MHz /H20850. This corresponds to the improvement
by a factor of 7 as compared to the linewidth of the free-running STNO but is still much larger than the linewidth ofthe generator of the microwave driving signal /H20849on the order
of several hertz /H20850. These results indicate that despite a clear
“frequency-locking,” the oscillation phase is not stationaryand evolves in time, thus a “true” phase-locked state is notreached. Such a behavior of the driven STNO can be attrib-uted to the influence of noise.
12,17The external microwave
driving signal has to compete with noise, which results inphase slips and, thus, in fluctuations of the STNO phase.
In order to elucidate the role of noise in the injection
locking process we performed macrospin simulations usingthe Landau–Lifshitz–Gilbert–Slonczewski equation, wherewe included a white Gaussian thermal noise field
18,19corre-
sponding to the effective temperatures T=0, 50, and 400 K.The simulations were carried out for the in-plane-precessionmode of a planar-polarizer/planar-free-layer system and theSlonczewski spin torque term was used in the form given byEqs. /H208491/H20850–/H208495/H20850of Ref. 20. We also assumed that the spin po-
larization vector Pwas oriented in the plane of the STNO
free layer at the angle
/H9258pto the in-plane easy axis of this
layer. The total current I acting on the STNO was the sum ofthe dc current I
dcand the sinusoidal variable current I rfthat
represents the injected rf driving signal of the frequency fe;
I/H20849t/H20850=Idc+Irfcos/H208492/H9266fet/H20850. /H208491/H20850
The magnitudes of the bias current I dcand the bias magnetic
field H /H20849applied along the easy axis /H20850were chosen to make the
STNO free-running frequency equal to f0=5 GHz. From the
simulated time traces /H20849of 4/H9262s length /H20850we extracted the in-
plane magnetization component oriented perpendicular to thestatic equilibrium magnetization of the free layer and calcu-lated the oscillation frequency fof the driven STNO as a
function of the driving frequency f
efor T=0 K at two dif-
ferent angles /H9258p/H20849see Fig. 3/H20850.
From Fig. 3/H20849a/H20850it is clear that at /H9258p=0° the STNO is
locked to the external driving source only near fe=2nf0
while at /H9258p=15° locking also occurs near fe=/H208492n+1/H20850f0. This
result agrees with both the experimental data presented in
Fig.1and with the previously mentioned conclusions of the
analytical theory of STNO injection locking.14We also cal-
culated the mismatch fe−2fof the STNO frequency and the
STNO linewidth /H9004fat T=400 K for different values of the
ratio I rf/Idc. These results are presented in Figs. 4/H20849a/H20850and
4/H20849b/H20850, respectively.
It should be noted that the variation in /H9004facross the
injection locking range is qualitatively similar to that ob-served in the experiment /H20851see Figs. 2/H20849a/H20850and2/H20849b/H20850/H20852with a
gradual reduction from the free-running value toward a mini-mum value at the center. This behavior proves the importantrole of noise in the injection locking process. This role isfurther illustrated by Fig. 4/H20849d/H20850, where the minimum linewidth
as a function of the ratio I
rf/Idcis shown for two different
effective temperatures. Clearly for I rf/Idc/H110210.5,/H9004fminis
higher at T=400 K than at T=50 K, while for I rf/Idc/H110220.5
the linewidth at both temperatures is below the numericalresolution. While the effective temperature, and, therefore,the noise has a strong influence on the phase noise charac-teristics of the injection-locked state, the injection-locking51 0 1 54,04,55,05,56,0
0,2 0,4 0,64,95,05,1 0(GHz)
Current I (mA)
3 02 0
Source Frequency e(GHz)Frequency ( GHz)
02505007501000(nV²/Hz)
0
FIG. 1. /H20849Color online /H20850Experimental PSD map /H20849linear scale /H20850of the STNO
frequency fvs the frequency of the signal source feat the rf current
Irf/Idc=0.4. The positions of f0and its multiples are indicated. The dots near
f0andf0/2 are artifacts of the used measurement technique. Inset: depen-
dence of f0vs the bias current when Irf=0. f0demonstrated a Kittel-like
increase as a function of the bias magnetic field /H20849not shown here /H20850.
-1.5-1.0-0.50.00.51.01.5
0100200300400500
0100200300400500/g73e-2/g73(GHz)
9.5 10.0 10.5 11.0-1.5-1.0-0.50.00.51.01.5/g73e-2/g73(GHz)
/g73e(GHz)
Δ/g73(MHz) Δ/g73(MHz)
0.00 0.25 0.50 0.75 1.000.000.250.500.751.00δ/g73e(GHz)
IRF/IDC0.00 0.25 0.50 0.75 1.00050100150200250300
Δ/g73min(MHz)
IRF/IDC(a)
(b)
(c) (d)Δfmin
FIG. 2. /H20849Color online /H20850Experimental data on the STNO injection locking to
a double-frequency signal; /H20849a/H20850and /H20849b/H20850frequency mismatch fe−2fand
STNO linewidth /H9004fas functions of the driving frequency fefor two ampli-
tudes of the driving rf current; /H20849a/H20850Irf/Idc=0.3 and /H20849b/H20850Irf/Idc=0.65; The red
lines are a guide for the eye. /H20849c/H20850and/H20849d/H20850Injection locking range /H9254feand
minimum linewidth /H9004fminof the driven STNO as functions of the amplitude
Irfof the driving microwave current.182503-2 Quinsat et al. Appl. Phys. Lett. 98, 182503 /H208492011 /H20850
Downloaded 18 Feb 2013 to 128.118.88.48. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsrange /H9254feis only moderately affected by temperature, as
shown in Fig. 4/H20849c/H20850. The numerically obtained slight increase
in the /H9254fewith increasing temperature has been previously
reported in Ref. 21. In the simulations in Fig. 4for T /HS110050,/H9254fe
has been defined as in the experiment /H20849Fig.2/H20850while for T
=0 K /H20849with/H9004f=0/H20850,/H9254fecorresponds to the range, where fe
−2fis zero, similar to the definition in Ref. 14, where the
experiments have been conducted at T=4.2 K.
It should also be noted, that our numerical results dem-
onstrate only qualitative agreement with the experiment as,for example, in the numerical modeling at T=400 K weneed I
rf/Idc/H110150.25 to achieve /H9254fe=0.6 GHz /H20851see Fig. 4/H20849c/H20850/H20852,
while in the experiment the same result is obtained only atI
rf/Idc=0.7 /H20851see Fig. 2/H20849c/H20850/H20852. A similar picture is seen in thebehavior of the minimum STNO linewidth; the reduction in
/H9004fminby a factor of 7 in numerical modeling takes place at
Irf/Idc/H110150.3/H20851Fig. 4/H20849d/H20850/H20852, while the ratio of I rf/Idc=0.7 is
needed for a similar effect in the experiment /H20851Fig.2/H20849d/H20850/H20852. This
discrepancy might be explained by a possible overestimationof the rf current in our microwave measurements and by thefact that in the experiment our STNO is driven by the biascurrent that is just above the critical value. The critical valueof the bias current /H20849I
c=0.5 mA /H20850was estimated from the de-
pendences of linewidth on the bias current.5,16
In conclusion, we have demonstrated both experimen-
tally and numerically that in the case of injection locking ofan in-plane magnetized STNO to a driving signal in the formof a microwave current the injection locking takes place onlyin the vicinity of the point f
e=2f0. This fact is in agreement
with the analytical prediction made in Ref. 14. We have also
demonstrated that in the STNO, based on a magnetic tunneljunction, noise plays an important role in the injection-locking process, and the frequency-locking does not alwaysmean the exact “phase-locking” of the STNO to an externalsignal. A more detailed analysis of the phase noise PSD ofthe STNO, as presented in Refs. 22and23, could better
clarify and quantify the role of noise for different STNOconfigurations and excitations modes.
This work was supported in part by the French national
research agency /H20849ANR /H20850through Grant No. ANR-09-NANO-
037 and the Carnot-RF project and Nano2012 convention. J.F. Sierra acknowledges support from the FP7-People-2009-IEF Program No 252067. I. Firastrau acknowledges supportfrom the CNCSIS-UEFISCU, Project No. PN II-RU TE_77/2010. Oakland University group is supported by the NationalScience Foundation Grant No. ECCS-1001815, and by thegrants from U.S. Army TARDEC, RDECOM.
1S. I. Kiselev et al. ,Nature /H20849London /H20850425, 380 /H208492003 /H20850.
2W. H. Rippard et al. ,Phys. Rev. Lett. 92, 027201 /H208492004 /H20850.
3D. Houssameddine et al. ,Appl. Phys. Lett. 93, 022505 /H208492008 /H20850.
4V. S. Pribiag et al. ,Nat. Phys. 3,4 9 8 /H208492007 /H20850.
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6S. Kaka et al. ,Nature /H20849London /H20850437, 389 /H208492005 /H20850.
7F. B. Mancoff et al. ,Nature /H20849London /H20850437,3 9 3 /H208492005 /H20850.
8A. Ruotolo et al. ,Nat. Nanotechnol. 4, 528 /H208492009 /H20850.
9J. Grollier, V. Cros, and A. Fert, Phys. Rev. B 73, 060409 /H20849R/H20850/H208492006 /H20850.
10V. Tiberkevich et al. ,Appl. Phys. Lett. 95, 262505 /H208492009 /H20850.
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14S. Urazdhin et al. ,Phys. Rev. Lett. 105, 104101 /H208492010 /H20850.
15R. Lehndorff et al. ,Appl. Phys. Lett. 97, 142503 /H208492010 /H20850.
16See supplementary material at http://dx.doi.org/10.1063/1.3587575 for
multimode spectrum, /H9004fdependence versus Idcand the extraction of Irf.
17K. Kurokawa, IEEE Trans. Microwave Theory Tech. 16,2 3 4 /H208491968 /H20850.
18W. F. Brown, Phys. Rev. 130, 1677 /H208491963 /H20850.
19S. E. Russek, S. Kaka, W. H. Rippard, M. R. Pufall, and T. J. Silva, Phys.
Rev. B 71, 104425 /H208492005 /H20850.
20I. Firastrau, D. Gusakova, D. Houssameddine, U. Ebels, M.-C. Cyrille, B.
Delaet, B. Dieny, O. Redon, J.-C. Toussaint, and L. D. Buda-Prejbeanu,Phys. Rev. B 78, 024437 /H208492008 /H20850.
21M. d’Aquino, C. Serpico, R. Bonin, G. Bertotti, and I. D. Mayergoyz,
Phys. Rev. B 82, 064415 /H208492010 /H20850.
22T. J. Silva and M. Keller, IEEE Trans. Magn. 46, 3555 /H208492010 /H20850.
23M. Quinsat, D. Gusakova, J. F. Sierra, J. P. Michel, D. Houssameddine, B.
Delaet, M.-C. Cyrille, U. Ebels, B. Dieny, L. D. Buda-Prejbeanu, J. A.Katine, D. Mauri, A. Zeltser, M. Prigent, J.-C. Nallatamby, and R. Som-met, Appl. Phys. Lett. 97, 182507 /H208492010 /H20850.4.44.85.25.6/g73(GHz)
4 6 8 1 01 21 41 61 82 02 24.85.25.64/g73g3/g73g2/g73g
(b)θp=15°/g73(GHz)
/g73e(GHz)(a)θp=0°
/g73g
FIG. 3. Simulated dependence of the frequency fof the driven STNO on
the driving frequency fefor a polarizer oriented at /H20849a/H20850/H9258p=0° and /H20849b/H20850/H9258p
=15° for the following parameters: T=0 K, saturation magnetization Ms
=1000 emu /cm3, Gilbert damping constant /H9251=0.02, bias field H=400 Oe
/H20849bias field made the angle /H9252=1° with the plane of the free layer /H20850, and
Irf/Idc=0.6. The size of the free layer was 90 /H1100380/H110033.9 nm.
-2-1012(c)
(b) e-2 (GHz)(a)
01234δ e(GHz)
8 9 10 11 1201002003004005000.075
0.15
0.6Δ (MHz)
e(GHz)0.0 0.2 0.4 0.6 0.8 1.0050100150200(d)T = 400K
T=5 0 K
T=0 KΔ min(MHz)
IRF/IDC
FIG. 4. /H20849Color online /H20850/H20849a/H20850Simulated frequency mismatch and /H20849b/H20850linewidth
vs driving frequency fefor different ratios Irf/Idc=0.075, 0.15, and 0.6 with
Idc=15 mA. /H20849c/H20850Simulated synchronization range /H9254fefor T=0, 50, and 400
Ka n d /H20849d/H20850minimum linewidth /H9004fminvs Irf/Idcfor T=50 and 400 K.182503-3 Quinsat et al. Appl. Phys. Lett. 98, 182503 /H208492011 /H20850
Downloaded 18 Feb 2013 to 128.118.88.48. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions |
1.1447483.pdf | Smoothing of bit transition irregularity in coupled granular/continuous perpendicular
media
A. M. Goodman, S. J. Greaves, Y. Sonobe, H. Muraoka, and Y. Nakamura
Citation: Journal of Applied Physics 91, 8064 (2002); doi: 10.1063/1.1447483
View online: http://dx.doi.org/10.1063/1.1447483
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/91/10?ver=pdfcov
Published by the AIP Publishing
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130.70.241.163 On: Tue, 23 Dec 2014 02:35:43Smoothing of bit transition irregularity in coupled granular Õcontinuous
perpendicular media
A. M. Goodman and S. J. Greavesa)
RIEC, Tohoku University, Sendai 980-8577, Japan
Y. Sonobe
IBM, Almaden Research Center, San Jose, California 95120
H. Muraoka and Y. Nakamura
RIEC, Tohoku University, Sendai 980-8577, Japan
Coupled granular/continuous ~CGC!perpendicular media increases thermal stability without
compromising SNR. However, increasing thermal stability by using a continuous layer leads toconcerns regarding transition noise. Investigating these concerns, we examine bit transitionirregularity in CGC media using a 3D micromagnetic model with sub-grain discretization.Irregularity is introduced by writing tracks diagonally to the x-yaxes of the cubic computational
cell. Initially we model the granular layer as ideal, with exchange decoupled grains. Theinvestigation is then extended to a nonideal system by varying intergranular exchange coupling ina system of comprising clusters of 10 nm grains. The thickness of the granular layer, g, and the
continuous layer, c, are varied while maintaining a constant media thickness. For a range of
thickness ratio, R(5c/g), that depends on intergranular exchange coupling in the granular layer, it
isfoundthatbittransitionirregularityisreducedtolessthanthatoftheunderlyinggranularphysicalstructure. In CGC media with 50 nm grains, the irregularity is reduced to that of a granular mediawith ’10 nm grains. Thus, in addition to enhancing thermal stability, CGC media may provide a
way to reduce noise, thereby extending the limit to the areal density of conventional media.©2002 American Institute of Physics. @DOI: 10.1063/1.1447483 #
INTRODUCTION
The areal density of current hard disk drive technology,
based on longitudinal media, is fast approaching its theoret-ical limit. Possible alternatives are being investigated, ofwhich perpendicular and patterned media are the most prom-ising candidates. The theoretical limit to longitudinal andperpendicular media are around 100 Gbits/in
2and 400 Gbits/
in2, respectively. Patterned media has no fundamental limit
until beyond 1 Tbit/in2;1as a product, howerver, it is cur-
rently commercially unviable. Therefore, perpendicular me-dia is the only viable solution, but the time/cost to take thetechnology from laboratory demonstration to product com-bined with a short product lifetime, makes the transition arisky one.Thus, there is already a need to extend perpendicu-lar media beyond its current limit, in order to extend itsprojected product lifetime.
In principle, perpendicular media has several advantages
over longitudinal media, however, as a conventional media,its areal density remains limited by the random size and lo-cation of its grains, which may couple to form larger mag-netic switching units. The size of the switching unit defines:~a!the transition irregularity that causes noise, and ~b!the
thermal stability of the media. In decoupled granular media,where each grain switches individually, SNR and thermalstability criterion specify a critical minimum number ofgrains per bit, N
c, and a critical minimum grain size, Vc,
which limit the areal density.2In media where one or moregrains couple, an effective switching volume, Veff, and an
effective number of switching volumes per bit, N eff, may
instead be considered. In practice, perpendicular media ex-hibits poor SNR due to irregular bit transitions causing jitter,the origin of which depends on the type of perpendicularmedia used. In continuous media the irregularity results fromthe random strength and location of pinning sites, which pinthe domain walls. In granular media, irregularity is caused bygrains coupling together to form large magnetic clusters.
Understanding and controlling pinning strength and den-
sity in continuous media has proved difficult. However, byreducing the grain size and exchange interactions in granularmedia, the switching unit ideally becomes equal to a singlegrain whose volume is equal to the critical volume for ther-mal stability; then the magnetic irregularity is the same asthe physical irregularity, V
eff5Vgrain5Vc, and similarly for
the number of grains per bit, Neff5Ngrain5Nc.
Coupled continuous-granular ~CGC!perpendicular me-
dia consist of a layer of exchange-coupled grains exchangecoupled to a layer of exchange-decoupled grains. The result-ing media combine the desirable properties of granular andcontinuous perpendicular media: fine grains in the granular(G) layer induce dense pinning sites in the continuous ( C)
layer resulting in transitions with reduced noise, while lateralexchange interactions between grains in the Clayer com-
bined with coupling in the vertical direction between the C
andGlayers increase V
effat the bit center and improve ther-
mal stability.3,4Despite encouraging simulations that showa!Presently at Hoya, Tokyo.JOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 10 15 MAY 2002
8064 0021-8979/2002/91(10)/8064/3/$19.00 © 2002 American Institute of Physics
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130.70.241.163 On: Tue, 23 Dec 2014 02:35:43low noise, and experimental results that show improved ther-
mal stability without compromising SNR, concerns havebeen expressed regarding the noise properties of the media:The use of a continuous layer and enhanced thermal proper-ties would normally be expected to correlate with increasedtransition irregularity and greater media noise. In this workwe investigate these concerns by simulating the recording ofbit patterns in CGC media using a three-dimensional micro-magnetic model with sub-grain discretization.
MODEL
A head-media system comprising a ring head and CGC
perpendicular media was modeled. Shown in Fig. 1, the me-dia was divided into 5 layers, each 6 nm thick. Each layerwas divided into 10 nm square computational cells. The mo-tion of the magnetization of each cell was calculated usingthe Landau–Lifshitz–Gilbert equation of motion, includingterms for the exchange, anisotropy, demagnetizing, randomthermal and head fields. Apart from the exchange stiffnessparameter, A, parameters in the LLG equation were the same
for each cell: magnetization M
s5300 emu/cc, anisotropy
constant Ku5106erg/cm, gyro-magnetic ratio g51.76
3107rad/Oe s, damping factor a51.0, temperature T
5300 K. The exchange stiffness parameter, A, was used to
vary the magnetic structure of the media. In the verticalz-direction, between the layers, the exchange coupling was
1310
26erg/cm, making epitaxial columnar grains. In the
horizontal xandydirections, within the layers, the exchange
coupling was varied to define the layer type. Continuous lay-ers were assumed to be ideal, with horizontal coupling, be-tween cells A51310
26and no pinning sites. Granular lay-
ers were modeled as an assembly of identical exchangedecoupled ( A50 erg/cm !grains. Each grain was divided
into 5 3535 exchange coupled 10 nm cells. By varying the
exchange coupling between them, the size and the nature ofthe switching unit was varied. Irregularity along the bit tran-sition was simulated by writing tracks diagonally, at 45° tothe edge of the computational cell edge direction. The headfield was calculated using the Lindholm equations,
5using a
head width 540 nm, gap length 5100 nm and maximum
head field 56 kOe, at a flying height 520 nm. Bits were
written every 100 nm, corresponding to a linear density of254 kfci, at a frequency of 1760 Mbits/s.RESULTS AND DISCUSSION
Initially, bits were written into an ‘‘ideal’’CGC media in
which irregularity was due only to the finite size of thephysical grains in the granular layer. The added complexityof having to consider the effect of intergranular exchangeinteractions and head field gradient on the bit transition waseliminated by using a granular layer comprising exchange-decoupled grains that were large compared to the width ofthe head field distribution.
Figure 2 shows the effect of varying the thickness ratio,
R, of CGC media comprising 50 nm grains coupled to a
continuous layer.As Ris increased, the irregularity of the bit
transition is smoothed and its amplitude is decreased. In-creasingRsimultaneously increases the domain wall energy,
which is proportional to the thickness of the continuouslayer, and reduces the pinning energy, which is proportionalto the thickness of granular layer. In a CGC structure with alargeR, the energy of the domain wall is sufficient to cause
reversal of the granular layer. Thus, the smoothing effectobserved is caused by increasing the significance of the DWenergy, which minimizes the length of the domain wall toreduce the wall energy. The irregularity along the transitionin the cross track direction is reduced to less that than of theunderlying irregularity in the granular layer. The magnetiza-tion does not follow the irregular grain boundaries; rather,the domain wall in the continuous layer smoothes the under-lying irregularity in the granular layer.
These results show that it is possible to control the
smoothing effect by changing the domain wall energy andthe pinning strength by varying R. However, in real media,
the granular layer is not comprised of ideal segregated grainsas modeled, but small grains that couple to form large mag-netic clusters, which define the irregularity. The grain size ofperpendicular media may be reduced to around 10 nm with-out compromising uniaxial anisotropy, while measurementsof media noise indicate clusters comprise between 20 and 30physical grains. In order to simplify this problem we assumedecoupled clusters of a fixed size, comprised of grains be-tween which we assume an average exchange coupling pa-rameter,A. Since the exchange coupling will be between A
51310
26erg/cm for fully exchange coupled grains and A
50 erg/cm for fully decoupled grains, we use a range of
values between these two extremes. Clusters were defined byincluding exchange breaks every 50 nm. Thus, clusters of535 10 nm exchange coupled grains replace the 50 nm
grains used in the previous simulations.
FIG. 1. Media model and a typical bit transition.
FIG. 2. Bit patterns for 30 nm thick CGC media consisting of 50 nm grains
in the granular layer.8065 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Goodman et al.
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130.70.241.163 On: Tue, 23 Dec 2014 02:35:43Figure 3 shows the effect of varying the exchange cou-
pling between sub-grains for two systems with different val-ues ofR. In both cases, reducing the exchange coupling be-
tween the grains within the clusters causes a reduction in theirregularity of the transition. However, above and belowsome critical values, changes in Ahave no effect and aredependent on the value of R. For the case shown, the irregu-
larity provided by the granular layer becomes indistinguish-able for values of Abelow 5 37
27whenRis 1.5, whereas
for the case where R50.66, the sawtooth irregularity remains
clear for A513727. These results indicate that optimiza-
tion of the smoothing effect will require knowledge of thegranular layer.
CONCLUSIONS
Relative to the size of the physical grains, in CGC media
the size of the effective switching unit, Veff, at the bit center
is enlarged, while irregularity and therefore the size of theeffective switching unit at the transition is reduced. Thisvariation complements the nature of the dipolar interactions,which are destabilizing at the bit center and stabilizing at thetransitions. Thus, CGC media provides a way to reduce thenumber of grains per bit N
grain, and grain volume, Vgrain,t o
less than their critical values, NcandVc, without compro-
mising SNR or thermal stability, thereby extending the limitto the areal density and the product window of perpendicularmedia.
1R. L. White, J. Magn. Magn. Mater. 209,1~2000!.
2D. N. Lambeth, Vacuum 59,5 2 2 ~2000!.
3Y. Sonobe, D. Weller, Y. Ikeda, K. Takano, M. E. Schabes, G. Zeltzer, H.
Do, B. K. Yen, and M. E. Best, J. Magn. Magn. Mater. 235,4 2 4 ~2001!.
4S. J. Greaves, H. Muraoka, Y. Sonobe, M. Schabes, and Y. Nakamura, J.
Magn. Magn. Mater. 235,4 1 8 ~2001!.
5D.A. Lindholm, IEEE Trans. Magn. MAG-13 , 1460 ~1977!.
FIG. 3. The effect of varying the exchange coupling between 10 nm grains,
within 50 nm clusters.8066 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 Goodman et al.
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130.70.241.163 On: Tue, 23 Dec 2014 02:35:43 |
1.3662923.pdf | Normal modes of coupled vortex gyration in two spatially separated
magnetic nanodisks
Ki-Suk Lee, Hyunsung Jung, Dong-Soo Han, and Sang-Koog Kim
Citation: J. Appl. Phys. 110, 113903 (2011); doi: 10.1063/1.3662923
View online: http://dx.doi.org/10.1063/1.3662923
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v110/i11
Published by the AIP Publishing LLC.
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Downloaded 31 Aug 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsNormal modes of coupled vortex gyration in two spatially separated
magnetic nanodisks
Ki-Suk Lee, Hyunsung Jung, Dong-Soo Han, and Sang-Koog Kima)
National Creative Research Center for Spin Dynamics and Spin-Wave Devices, and Nanospinics Laboratory,
Research Institute of Advanced Materials, Department of Materials Science and Engineering, Seoul National
University, Seoul 151-744, South Korea
(Received 30 June 2011; accepted 17 October 2011; published online 1 December 2011)
We found from analytical derivations and micromagnetic numerical simulations that there exist
two distinct normal modes in apparently complex vortex gyrotropic motions in two dipolar-coupled
magnetic nanodisks. The normal modes have characteristic higher and lower single angular
eigenfrequencies with their own elliptical orbits elongated along the x(bonding axis) and yaxes,
respectively. The superposition of the two normal m odes results in coupled vortex gyrations, which
depend on the relative vortex-state configuration in a pair of dipolar-coupled disks. This normal-mode
representation is a simple means of understanding the observed complex vortex gyrations in two or
more dipolar-interacting disks of various vortex-state configurations. VC2011 American Institute of
Physics . [doi: 10.1063/1.3662923 ]
I. INTRODUCTION
The magnetic vortex, which is formed by out-of-plane
vortex-core magnetization together with in-plane curlingmagnetization,
1has nontrivial low-frequency translational
modes (in-plane orbital motion around its equilibrium posi-
tion) in micron-size or smaller magnetic dots.2–8This unique
dynamic characteristic of the magnetic vortex has attracted
growing interest in that, owing to persistent vortex-core os-
cillatory motion (i.e., vortex gyration) it can be implementedin nano-oscillators.
7,8As extension of intensive studies on
isolated single vortex-state dots, there have been studies on
dynamics of coupled vortex-state dots because, when spa-tially separated magnetic dots are sufficiently close to each
other, dipolar (magnetostatic) interaction affects the vortex
excitations, particularly, gyration of individual disks.
9–15A
common finding in earlier studies on the effect of neighbor-
ing disks’ dynamic dipolar interaction on vortex gyrations is
emergent frequency splitting. Shibata et al.9have analyzed
such frequency splitting in a pair of vortices as well as in a
two-dimensional array of same. Recently, several experi-
mental observations10–15on the gyrations of dipolar-coupled
vortices, for example, resonance-frequency broadening10in
arrays of disks, and asymmetric resonance-frequency split-
ting11in a pair of vortices, have been reported. Additionally,
vortex-core gyrations and their asymmetric eigenfrequency
splittings have been examined by the present authors14–16
and Vogel et al.17Nonetheless, a comprehensive understand-
ing of the fundamentals of dipolar-coupled gyrations remains
elusive.
In this article, we report on analytical derivations of the
normal modes and their dependences on the relative vortex-
state configuration in both disks. We also provide a simplemeans of understanding apparently complex coupled vortex
gyrations in terms of the superposition of the two normalmodes, which were also studied by micromagnetic numerical
simulations.
II. MICROMAGNETIC SIMULATIONS
In the present study, as part of our investigation of
coupled vortex gyrations, we conducted micromagnetic sim-
ulations of the magnetization dynamics in two identicalPermalloy (Py: Ni
81Fe19) disks of 2 R¼303 nm diameter,
L¼20 nm thickness, and 15 nm edge-to-edge interdistance.
We utilized the OOMMF code18that employs the Landau-
Lifshitz-Gilbert (LLG) equation.19The Py material parame-
ters were applied as described in Ref. 20. In the model, four
different relative vortex-state configurations were utilized, asshown in Fig. 1(a)and as represented by [ p
1,C1] along with
[p2,C2]¼[þ1,þ1], where p¼þ1(/C01) corresponds to the
upward (downward) core orientation, and C¼þ1(/C01), the
counter-clockwise (clockwise) in-plane curling magnetiza-
tion. The number in subscript indicates either disk 1 or disk
2. In order to excite all of the modes existing in the twodipolar-coupled disks, the vortex core only in disk 2 (the
right disk of each pair) was intendedly displaced to an initial
position, 69 nm in the þydirection by application of a 300
Oe field in the þxdirection locally,
21after which both disks
were relaxed.
Figures 1(b)–1(d) show the characteristic dynamics of
the coupled vortex gyrations for the indicated representative
configurations. In all of the cases, the common features were
the beating patterns of the oscillatory xandycomponents of
both vortex-core position vectors along with the crossovers
between the local maxima and minima of the modulation
envelopes between disk 1 and disk 2. In two of our earlierstudies,
14,15these patterns and crossovers were observed
experimentally in the case of [ p1,C1]¼[/C01,þ1] and
[p2,C2]¼[þ1,þ1], for example. The beating frequencies
[Fig. 1(b)], relative rotation senses and phase differences
[Fig. 1(c)] between disk 1 and disk 2, as well as the frequency
splitting [Fig. 1(d)], were in contrast with the vortex-statea)Author to whom all correspondence should be addressed. Electronic mail:
sangkoog@snu.ac.kr.
0021-8979/2011/110(11)/113903/5/$30.00 VC2011 American Institute of Physics 110, 113903-1JOURNAL OF APPLIED PHYSICS 110, 113903 (2011)
Downloaded 31 Aug 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsconfiguration in disk 1 with respect to that in disk 2, where
[p2,C2]¼[þ1,þ1] was maintained.
III. ANALYTICAL CALCULATIONS OF
NORMAL MODES
In order to fully understand such apparently complex
gyrations as found in those simulation results, we analyti-
cally derived the normal modes of different single eigenfre-quencies, which modes are nontrivial in the case of coupled
vortex oscillators in which two different dipolar-coupled vor-
tex cores gyrate. In the analytical derivations, we startedwith two coupled linearized Thiele’s equations,
22
/C0G1/C2_X1/C0^D_X1þ@WðX1;X2Þ=@X1¼0 (1a)
/C0G2/C2_X2/C0^D_X2þ@WðX1;X2Þ=@X2¼0; (1b)
where Xi¼ðxi;yiÞis the vortex-core position vector from
the center of disk i, and where i¼1 and 2, Gi¼/C0Gpi^zis
the gyrovector with constant G¼2pLM s=c>0 (the satura-
tion magnetization Msand the gyromagnetic ratio c), and
^D¼D^Iis the damping tensor with the identity matrix ^Iand
the damping constant D.6The total potential energy is given
asWðX1;X2Þ¼Wð0ÞþjðX2
1þX2
2ÞþWint, where Wð0Þis
the potential energy for Xi¼ð0;0Þ, the second term is that
for the shifted cores with the identical stiffness coefficient j
for the isolated disks,2andWintis the interaction energy for
both disks with displaced cores. Assuming a rigid vortex
model and also considering only the side-surface charges of
the two disks, Wintcan be written simply as C1C2(gxx1x2
–gyy1y2), as reported in Ref. 9, where gxandgyrepresent the
interaction strengths along the xandyaxes, respectively, and
are functions of the interdistance.9–11,15
In order to derive the analytical expression of the normal
modes of coupled vortex gyrations in a given system, weemployed coordinate transformations based on the in-phase
and out-of-phase relations between X1andX2along the x
andyaxes, respectively, as observed in our earlier work.15
Considering the symmetry of the two identical disks of a
given relative rotational sense of gyrotropic motions, i.e.,p
1p2, the two normal-mode coordinates can be expressed as
N¼(x1þx2,y1þp1p2y2) andX¼(x1–x2,y1–p1p2y2). The
product of p1andp2determines the phase relation in the y
component between the two disks for each mode. Through
the diagonalization of Eqs. (1a) and (1b) with respect to
the normal-mode coordinates, we can obtain these twouncoupled equations of vortex gyrotropic motion,
/C0Dp
1Gjj
/C0p1Gjj D/C20/C21
_Nþj1þCx 0
01 /C0Cy/C20/C21
N¼0 (2a)
/C0Dp 1Gjj
/C0p1Gjj D/C20/C21
_Xþj1/C0Cx 0
01 þCy/C20/C21
X¼0;(2b)
where Cx¼C1C2gx=jandCy¼C1C2p1p2gy=j. The general
solutions of Eqs. (2a) and (2b) are written simply as
N¼N0exp½/C0ið~xNtþuNÞ/C138andX¼X0exp½/C0ið~xXtþuXÞ/C138
with the corresponding amplitude vectors of N0¼ðN0x;N0yÞ
andX0¼ðX0x;X0yÞas well as the phase constants of uN
anduX. By inserting these general solutions into Eqs. (2a)
and(2b), we can obtain analytical expressions of the com-
plex angular frequencies ~xNand ~xXas well as the ratios of
N0y=N0xandX0y=X0x, for the normal modes. On the basis of
the relation between the ordinary and the normal-mode coor-
dinates, that is, X1¼1
2(NxþXx,NyþXy) and X2¼1
2(Nx
/C0Xx,p1p2Ny/C0p1p2Xy), the normal modes of coupled
vortex-core gyrations, in the ordi nary coordinates, can be derived
analytically: X1;N¼1
2N0exp½/C0ið~xNtþuNÞ/C138,X2;N¼1
2N0
0exp
½/C0ið~xNtþuNÞ/C138and X1;X¼1
2X0exp½/C0ið~xXtþuXÞ/C138,X2;X
¼1
2X0
0exp½/C0ið~xXtþuXÞ/C138with the corresponding angular
FIG. 1. (Color online) Representative
coupled vortex gyrations for the indi-
cated four different polarization ( p) and
chirality ( C) configurations in a pair of
vortex-state disks shown in Fig. 1(a). (a)
The streamlines with the small arrows
indicate the in-plane curling magnetiza-
tions, and the height displays the out-of-
plane magnetizations. (b) The xand y
components of the vortex-core positionvectors in both disks, as functions of
time. (c) Orbital trajectories of the
vortex-core gyrations during the time
period t¼0–5 ns. The open circles rep-
resent the initial core positions. (d) Fre-
quency spectra obtained from the data
shown in Fig. 1(b).113903-2 Lee et al. J. Appl. Phys. 110, 113903 (2011)
Downloaded 31 Aug 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionseigenfrequencies Re ð~xNÞand Re ð~xXÞ,w h e r e N0
0
¼ðN0x;p1p2N0yÞandX0
0¼/C0 ðX0x;p1p2X0yÞ. The general sol-
utions of Eqs. (1a)and(1b)can also be given, by the superposi-
t i o no ft h et w on o r m a lm o d e si nd i s k1a n dd i s k2 ,s u c ht h a t
X1¼X1;NþX1;XandX2¼X2;NþX2;X.
For the cases of Gjj/C29Djjandgx;gy/C28j,~xN, and ~xX
approximate to be ~xN/C25x0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þCx ðÞ 1/C0Cy/C0/C1q
þiD=G/C16/C17
and ~xX/C25x0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1/C0Cx ðÞ 1þCy/C0/C1q
þiD=G/C16/C17
with the angu-
lar eigenfrequency of vortex gyration in an isolated disk,
x0¼j=Gjj.2The angular eigenfrequencies of the uncoupled
NandXnormal modes are simply rewritten as Re ð~xNÞ
/C25x0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þC1C2gx=j ðÞ 1/C0C1C2p1p2gy=j/C0/C1q
and Re ð~xXÞ
/C25x0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1/C0C1C2gx=j ðÞ 1þC1C2p1p2gy=j/C0/C1q
, respectively.
Consequently, the angular frequency difference Dx, defined
as Re( ~xN)/C0Re( ~xX), is expressed as x0C1C2(gx/C0p1p2gy)/j,
which is determined by the value of C1C2p1p2. For the case
where the xaxis is the bonding axis, gy>gxalways holds,
such that Dx<0f o r C1C2p1p2¼þ1 and Dx>0
forC1C2p1p2¼–1. However, the magnitude of the angularfrequency splitting Dxjj is determined by only p1p2, and
Dxp1p2¼þ1/C12/C12/C12/C12¼x0ðgy/C0gxÞ=j<Dxp1p2¼/C01/C12/C12/C12/C12¼x0ðgyþgxÞ
=j, as confirmed by the simulation results shown in
Fig. 1(d), which are consistent with the analytical results
reported in Ref. 9.
The shapes of the orbital trajectory of the NandXmodes can be
estimated as N0y=N0x/C12/C12/C12/C12¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ðjþC1C2gxÞ=ðj/C0C1C2p1p2gyÞq
andX0y=X0x/C12/C12/C12/C12¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
ðj/C0C1C2gxÞ=ðjþC1C2p1p2gyÞq
. Thus,
the elongation axis and the degree (hereafter, “ellipticity”) of
elongations of the normal modes’ orbits vary according to
the combinations of C1C2¼61 and p1p2¼61 displayed in
Table I. The elongation axis of the normal mode that has the
higher angular eigenfrequency (shaded area) is always, for
all cases, the bonding ( x) axis.
IV. COMPARISON OF ANALYTICAL AND SIMULATION
RESULTS
To numerically calculate the NandXnormal modes
using the above analytical expressions, it is necessary toknow g
xandgyfor a given model system. These interaction
FIG. 2. (Color online) (a) Oscillatory x
andycomponents of the NandXmodes,
(b) their orbital trajectories, and (c) fre-
quency spectra obtained from the oscil-
lation of the xcomponents of the Nand
Xmodes for the four different configura-
tions of [ p1,C1] with respect to
[p2,C2]¼[þ1,þ1]. The solid lines and
open circles correspond to the analytical
calculations and the micromagnetic sim-
ulation results, respectively.TABLE I. The major (elongation) axis and the ellipticity (the ratio of the length of the major to that of the minor axis) of each mode for all combinations of
C1C2¼61 and p1p2¼61. The shaded area corresponds to the higher-frequency mode for the given C1C2andp1p2. The numbers in parentheses indicate the
numerical values of the ellipticity.
p1p2
þ1 /C01
Nmode Xmode Nmode Xmode
Major axis Ellipticity Major axis Ellipticity Major axis Ellipticity Major axis Ellipticity
C1C2þ1 yffiffiffiffiffiffiffiffiffiffiffiffiffij/C0gy
jþgxr
(0.930)xffiffiffiffiffiffiffiffiffiffiffiffiffij/C0gx
jþgyr
(0.934)xffiffiffiffiffiffiffiffiffiffiffiffiffi
jþgx
jþgys
(0.969)yffiffiffiffiffiffiffiffiffiffiffiffiffij/C0gy
j/C0gxr
(0.965)
/C01 xffiffiffiffiffiffiffiffiffiffiffiffiffij/C0gx
jþgyr
(0.934)yffiffiffiffiffiffiffiffiffiffiffiffiffij/C0gy
jþgxr
(0.930)yffiffiffiffiffiffiffiffiffiffiffiffiffij/C0gy
j/C0gxr
(0.965)xffiffiffiffiffiffiffiffiffiffiffiffiffi
jþgx
jþgys
(0.969)113903-3 Lee et al. J. Appl. Phys. 110, 113903 (2011)
Downloaded 31 Aug 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsstrengths can be simply estimated through the relations of
gx/j¼(|Dx/C0|þ|Dxþ|)/(2x0) and gy/j¼(|Dx/C0|/C0|Dxþ|)/
(2x0) for the case of C1C2¼þ1, where | Dxþ| and |Dx/C0| cor-
respond to the angular frequency splitting for the cases ofp
1p2¼þ1 and /C01, respectively. From the simulation results,
|Dxþ|¼2p/C2105 MHz and | Dx/C0|¼2p/C250 MHz, for the
case of C1C2¼þ1 [Fig. 1(d)] and the numerical values of
j¼3/C210/C03J/m2andx0¼2p/C2575 MHz for isolated Py
disks of the same geometry, we can extract the numerical
values of gx¼1.1/C210/C04andgy¼3.1/C210/C04J/m2(Ref. 23).
Using these above values, we calculated N¼N0
exp½/C0ið~xNtþuNÞ/C138andX¼X0exp½/C0ið~xXtþuXÞ/C138,where
the initial core displacements were set to ( x2,y2)¼(0, 69 nm)
and ( x1,y1)¼(0,C1/C212 nm) in order to obtain the same ini-
tial phases of the core positions as those used in the micro-
magnetic simulations.21Thex- and y-component oscillations
of the normal modes, that is, Nx,yandXx,y, their trajectories,
(Nx,Ny) and ( Xx,Xy) on the normal-mode coordinates, and
their frequency spectra, are plotted in Figs. 2(a),2(b), and
2(c), respectively, for all of the different combinations of
p1p2¼61 and C1C2¼61. The analytical calculations (solid
lines) are in quantitative agreement with those (open sym-bols) extracted from the simulation results plotted in Fig. 1.
These comparisons prove that complex coupled vortex-core
gyrations in two coupled oscillators such as those shown inFig.1can be predicted or interpreted simply in terms of the
superposition of the NandXmodes. Note that there appear
four different frequency spectra according to C
1C2p1p2,a s
shown in Fig. 2(c). These distinct spectra result from four
different dynamic dipolar interaction energies between two
neighboring disks, which are determined by the relative con-figuration of both the polarization and chirality between the
two disks.
15Furthermore, the individual contributions of the Nand
Xmodes to the gyration of each of disk 1 and 2 can be
decomposed into X1;N,X1;XandX2;N,X2;X, as shown in
Fig.3. The analytical calculations (solid lines) of X1;N,X1;X
andX2;N,X2;Xwere in excellent agreements with those
(symbols) obtained from the simulation results24through the
normal-to-ordinary coordinate transformation, as describedearlier. Since the superposition of the two normal modes
gives rise to the net coupled gyration of each disk (i.e.,
X
1¼X1;NþX1;Xand X2¼X2;NþX2;X), the contrasting
eigenfrequencies and phases between the NandXmodes,
which vary with both p1p2andC1C2, determine the modula-
tion frequency (see Fig. 1(b)) and the relative phase of the
vortex-core orbital trajectory (see Fig. 1(c)).
The physical origin of the above-noted frequency split-
ting and complex coupled vortex-core gyrations can beascribed to the breaking of the radial symmetry of the poten-
tial wells of decoupled disks, which is caused by dynami-
cally variable dipolar interaction between those disks, andwhich depends on the disk pair’s relative vortex-state config-
uration, as explained above.
V. CONCLUSION
In summary, we analytically derived two normal modes
of coupled vortex-core gyrations in two spatially separated
magnetic disks. Dipolar interaction between two such disks
breaks the radial symmetry of their potential energy, givingrise to two distinct normal modes, each with a characteristic
single eigenfrequency and an elliptical orbit. The frequency
splitting and the orbital shape vary with the relative vortex-state configuration. This work provides a simple but com-
plete means of understanding complex vortex gyrations in
FIG. 3. (Color online) Contributions of
theNandXmodes to each disk’s
vortex-core gyration, i.e., X1;N,X2;N,
X1;X,X2;X, for the four different config-
urations of [ p1,C1] with respect to
[p2,C2]¼[þ1,þ1]. The solid lines and
open circles correspond to the analytical
calculations and the micromagnetic sim-
ulation results, respectively.113903-4 Lee et al. J. Appl. Phys. 110, 113903 (2011)
Downloaded 31 Aug 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsdipolar-coupled vortex oscillators as well as offers the possi-
bility to describe collective vortex gyrations in arrays of
vortex-state disks based on a generalized normal-modeapproach.
ACKNOWLEDGMENTS
This work was supported by the Basic Science Research
Program through the National Research Foundation of Korea
(NRF) funded by the Ministry of Education, Science andTechnology (Grant No. 20110000441).
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18The version of the OOMMF code used is 1.2a4. See http://math.nist.gov/
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20We used the saturation magentization Ms¼8.6/C2105A/m, the exchange stiff-
ness Aex¼1.3/C210/C011J/m, the damping constant a¼0.01, and the gyroma-
gentic ratio c¼2.21/C2105m/As with zero magntocryst alline anisotropy. The
cell size used in the micromagnetic simulations was 3 /C23/C220 nm3.
21Although no external field was applied to the left disk (disk 1), its vortex
core was shifted to ( x1,y1)¼(0,C1/C212 nm), owing to a dipolar interaction
with the disk 2 surface and/or volume charges of the displaced vortex core
in disk 2.
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23To compare those values, we also extracted gx¼1.4/C210/C04and
gy¼4.0/C210/C04J/m2from the numerical integrations of Eq. (3) given in Ref.
9. These values are 30% larger than those obtained from our micromagnetic
simulations. This discrepancy may come from the assumptions used for the
derivation of Eq. (3) in Ref. 9; side-surface charges based on the rigid vortex
model only contribute to the dipolar interaction energy between the two disks.
In our micromagnetic simulations, not only the side-surface charges but also
the volume charges attribute to the dipolar interaction energy.
24The decomposition of the NandXmodes in each disk can be calculated
from the vortex-core position vectors via the relations ofX
1;N¼1
2x1þx2;y1þp1p2y2 ðÞ ,X1;X¼1
2x1/C0x2;y1/C0p1p2y2 ðÞ , and
X2;X¼/C01
2x1/C0x2;p1p2y1/C0y2 ðÞ ,X2;N¼1
2x1þx2;p1p2y1þy2 ðÞ , obtained
from the micromagnetic simulation data.113903-5 Lee et al. J. Appl. Phys. 110, 113903 (2011)
Downloaded 31 Aug 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions |
1.48030.pdf | AIP Conference Proceedings 333, 507 (1995); https://doi.org/10.1063/1.48030 333, 507
© 1995 American Institute of Physics.‘Operation and performance of a longitudinal
feedback system using digital signal
processing
Cite as: AIP Conference Proceedings 333, 507 (1995); https://doi.org/10.1063/1.48030
Published Online: 12 May 2008
D. Teytelman , J. Fox , H. Hindi , J. Hoeflich , I. Linscott , J. Olsen , G. Oxoby , L. Sapozhnikov , A. Drago , M.
Serio , W. Barry , J. Byrd , and J. Corlett
Operation and Performance of a Longitudinal Feed-
back System Using Digital Signal Processing
D. Teytelman, J. Fox, H. Hindi, J. Hoeflich, I. Linscott,
J. Olsen, G. Oxoby, L. Sapozhnikov
Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309
A. Drago, M. Serio
INFN Laboratori Nazionale, Frascati, Italy
W. Barry, J. Byrd, J. Corlett
Lawrence Berkeley Laboratory, 1 Cyclotron Road, Berkeley, CA 94720
Abstract
A programmable longitudinal feedback system using a parallel array of AT&T 1610
digital signal processors has been developed as a component of the PEP-II R&D pro-
gram. This system has been installed at the Advanced Light Source (LBL) and imple-
ments full speed bunch by bunch signal processing for storage rings with bunch
spacing of 4ns. Open and closed loop results showing the action of the feedback sys-
tem are presented, and the system is shown to damp coupled-bunch instabilities in the
ALS. A unified PC-based software environment for the feedback system operation is
also described.
INTRODUCTION
The PEP-II machine will require feedback to control multibunch instabilities
[1]. A longitudinal feedback system prototype has been installed and tested at the
Advanced Light Source at the Lawrence Berkeley Laboratory. This system uses a
bunch by bunch processing scheme and employs digital signal processing to calcu-
late a correction signal for each bunch. As shown in Fig. 1, signals from four but-
ton-type pickups are combined and fed to the stripline comb generator. The
generator produces an eight cycle burst at the sixth harmonic of the ring RF fre-
quency (2998 MHz). The resultant signal is phase detected, then digitized at the
bunch crossing rate. The detector is designed to have 400MHz bandwidth which
allows measurement of the each bunch's synchrotron motion independently for the
4 ns bunch spacing. A correction signal for each bunch is computed by a digital sig-
nal processing module and applied to the beam through a fast D/A, an output mod-
ulator, a power amplifier and a kicker structure [2, 3].
The signal processing is implemented by four AT&T 1610 processors operating
in parallel. These 16 bit processors are equipped with 16K of dual port memory on-
chip and allow 25 ns instruction cycle time for cached instructions [4]. The feed-
back algorithm is downloaded to the prototype through JTAG interface from a IBM
PC-compatible computer. This approach allows the user to quickly alter the feed-
back parameters as well as run a multitude of diagnostics, signal recorders and
*Work supported by the U.S. Department of Energy contract DE-AC03-76SF00515
© 1995 American Institute of Physics 5Q7
508 Longitudinal Feedback System
other programs.
As the synchrotron oscillation frequency in the ALS (10 kHz) is much less than
the revolution frequency (1.5 MHz) the processing is implemented as a downsam-
pled system, in which a correction signal for each bunch is computed once every n
revolutions, where n is a downsampling factor. The four-processor prototype sys-
tem allows control of up to 84 bunches when using a six-tap FIR filter algorithm.
The maximum number of bunches varies depending on the filter processing time
Figure 1. Block diagram of the longitudinal feedback system.
and fill pattern [5]. Design of the signal processing hardware and the front-end elec-
tronics has been addressed in the earlier publications [6]. Two important system
components; the QPSK (quad phase shift keyed) modulator and the support soft-
ware have not been described previously and are presented in the following sec-
tions.
QPSK MODULATOR OPERATION
The QPSK modulator function is implemented in the back-end signal process-
ing and is used to transfer the baseband computed correction signal into a modula-
tion on a kicker oscillator signal. The need for such a a modulator arises from the
design of the kicker structure, which produces a maximum in longitudinal imped-
ance at 1125 MHz, or 2.25 times the ring RF frequency (this choice minimizes the
impedance presented at the bunch crossing frequency and higher harmonics) [7].
The QPSK modulator is implemented using a 2 GHz bandwidth gilbert multiplier,
500 MHz ECL counter circuitry, and a passive 90 degree hybrid. The QPSK circuit
acts to shift the phase of an 1125 MHz carrier by -90 degrees every 2 ns to align the
kick phase for the next bucket. Figure 2 shows the QPSK modulated carrier wave-
D. Teytelman et al. 509
form as well as unmodulated 1125 MHz signal while Fig. 3 illustrates the resultant
carrier spectrum. Most of the power is at 1 GHz with a strong component at 1.25
Figure 2. Oscilloscope photograph of the 1125 MHz carrier and QPSK modulated 1125
MHz carrier. The two cursors are 2ns apart to show the spacing of two adjacent bunches
at 500 MHz.
Mkr l.OOOGHz
Ref Lvl -10. OdBm-10. 72dBm
6dB/ Atton 2QdB
Freq 1. 128GHz Span l.OGHz
ResBW 10MHz VidBW 300KHz SWP 20mS
LEVEL SPAN Rof Lvl -10. OdBm
KNOB 2 KNOB 1 KEYPAD Takbronix 2782
Figure 3. Spectrum of the QPSK modulated carrier.
GHz. Other spectral lines such as 0.75 GHz and 1.5 GHz fall outside the kicker
bandwidth and do not affect the beam. This QPSK modulated 1125MHz signal, if
applied to the beam, would produce a DC correction signal - the final function in
the QPSK modulator is an amplitude modulator which multiplies the QPSK'ed sig-
nal by the baseband correction signal from the output D/A. This modulation adjusts
the magnitude of the kicker drive signal every 4 ns to provide bunch by bunch cor-
rection signals (negative kicks require phase inversion of the kicker signal). The
510 Longitudinal Feedback System
resulting output spectrum for multi-bunch operation fills in the 250 MHz band-
width between 1000 and 1250 MHz and covers all coupled-bunch modes in the
storage ring. The circuitry as implemented has a 48 dB dynamic range and can be
operated at any RF/4 ring harmonic up to 2 GHz with the full 500 MHz QPSK
modulation rate.
UNIFIED SOFTWARE ENVIRONMENT
During the quick prototype development as the number and the sophistication of
the DSP programs grew, management of the many configurations and feedback fil-
ter programs became a serious concern.
To coordinate the development of various operational programs and accelerator
diagnostics a unified software environment has been created. This environment
uses a text-based parameter file to specify the operational modes of the quick proto-
type system. All of the variables for a given experimental configuration, such as the
machine revolution time, synchrotron frequency, filter gain, filter phase, etc. are
contained in the parameter file. The file is read in turn by a number of relatively
simple C programs which generate binary tables for downloading into the DSP
memory and include files for the assembly language DSP code. The DSP code and
tables are downloaded through the JTAG interface using the AT&T DSP1610
development system. All of these activities are coordinated by the UNIX make pro-
gram. Using file timestamps and dependencies defined in a makefile make program
ensures that tables and code downloaded to the DSP correspond to the variables in
the parameter file.
SYSTEM TESTS AT ALS
The prototype system including a high-gain longitudinal kicker has been
installed at the ALS and is being used to gain operational experience and to verify
the system design for the PEP-II system. Figure 4 shows the longitudinal transfer
function of a single bunch measured with no feedback, positive feedback, and neg-
ative feedback with two different loop gains. The action of the feedback system is
seen in the higher or lower Q of the synchrotron resonance for positive or negative
feedback respectively [8]. The graph shows that for a gain change of 8 (18dB) we
get a change in damping of about 15dB.
Presently the ALS kicker is driven by a 10W power amplifier. This power limits
the total current which can be controlled. It is interesting to note that relatively high
ring currents (up to 125mA) can be controlled with relatively low voltage correc-
tion kick as long as the feedback system is turned on during injection, and the injec-
tion process injects only a single bunch at a time. This injection method allows the
feedback system to damp the excitations caused by the injected bunch in the exist-
ing stored beam, and damp the resulting motion before the next injection cycle. If
D. Teytelman et al. 511
the feedback system is turned off for any substantial current (above 10mA) the
bunch motion becomes very large (greater than 10 degrees at the 500 MHz RF fre-
quency) and turning the feedback system back on does not control the synchrotron
motion. This happens because the feedback system saturates and cannot generate
enough voltage to control the large amplitude motion once it grows from the quies-
cent state. Figure 5 shows the bunch spectrum obtained from a BPM for 8 groups of
2 bunches equally spaced around the ring at 100mA. Data shows that the longitudi-
Single bunch frequency response
11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9
Figure 4. Single bunch transfer functions measured at the
ALS. The open loop synchrotron resonance at 11.5 kHz
can be damped or excited via negative or positive feed-
back respectively.
-60
-70
-80
-90
--110
-120
-130BPM spectrum: feedback on, off offset by +10bBm
Frequency, MHz
Figure 5. Spectrum of a BPM signal.
nal feedback suppresses the synchrotron oscillations (manifested as 10 kHz side-
bands) from -73dBm to the noise floor of spectrum analyzer, i.e. suppression of
50dB.
512 Longitudinal Feedback System
Since ALS is a light source machine is it possible to utilize optical diagnostics to
investigate the performance of the longitudinal feedback system. Experiments have
been conducted at the ALS to measure the optical spectrum with and without feed-
back. Figure 6 presents an undulator spectrum taken at a 108mA ring current with
84 bunch fill pattern. The feedback system increases the optical intensity by a factor
of 2.5 and narrows the peak width to almost 1/4 of that of the undamped system.
For synchrotron light users who are conducting narrowband spectroscopic mea-
surements such an improvement in machine performance is very desirable. It is
interesting to speculate on the bimodal structure visible in the "feedback off' spec-
trum which appears to be due to the coherent dipole mode longitudinal oscillations.
Undulator Spectrum - Feedback on (-).off(- -)
690 695 700 705 710
Energy (eV)
Figure 6. Undulator spectrum.
SUMMARY
The longitudinal bunch-by-bunch feedback system quick prototype is operated
at the ALS at Lawrence Berkeley Laboratory. It includes all of the subsystems
required for the PEP-II machine. The quick prototype system is used for algorithm
development and various accelerator measurements. Closed-loop feedback opera-
tion has been demonstrated and longitudinal instabilities have been controlled for
an 84 bunch fill pattern with 125mA ring current. We expect to be able to damp lon-
gitudinal motion at the 4QOmA design current when the high-power output ampli-
fier is installed. The information gained from the quick prototype system has been
incorporated in the PEP-II system design [9]. A complete PEP-II prototype for ALS
operations is in construction and should be installed and commissioned at the ALS
in early 1995.
D. Teytelman et al. 513
ACKNOWLEDGMENTS
The authors thank the ALS staff of LBL for their hospitality and interest in this
hardware development program and the SLAC PEP-II Group and Technical divi-
sion for their support. The authors particularly appreciate the help of Tony Warwick
for the optical spectrum measurement.
REFERENCES
1. "PEP-II, An Asymmetric B Factory - Design Update," Conceptual Design
Report Update, SLAC, 1992.
2. Pedersen, "Multi-bunch Feedback - Transverse, Longitudinal and RF Cavity
Feedback," Proceedings of the 1992 Factories with e+/e- Rings Workshop,
Benalmadena, Spain, November 1992.
3. Fox, et al., "Operation and Performance of a Longitudinal Damping System
Using Parallel Digital Signal Processing," Proceedings of the 1994 European
Particle Accelerator Conference, London, England.
4. "WE DSP1610 Digital Signal Processor Information Manual," AT&T Micro-
electronics Corporation, Allentown PA.
5. Hindi et al., "Down-Sampled Bunch by Bunch Feedback for PEP-II," B Fac-
tories: The State of Art in Accelerators, Detectors, and Physics, SLAC Report
400, p. 216.
6. Sapozhnikov, et al., "A Longitudinal Multi-Bunch Feedback System Using
Parallel Digital Signal Processing," Proceeding of the 1993 Beam Instrumen-
tation Workshop, Santa Fe, NM, AIP Conference Proceedings 319.
7. Corlett, et al., "Longitudinal and Transverse Feedback Kickers for the ALS,"
Proceedings of the 1994 European Particle Accelerator Conference, London,
England.
8. Hindi, et al., "Measurement of Multi-Bunch Transfer Functions Using Time-
Domain Data and Fourier Analysis," Proceedings of the 1993 Beam Instru-
mentation Workshop, Santa Fe, NM, AIP Conference Proceedings 319.
9. Oxoby, et al., "Bunch by Bunch Longitudinal Feedback System for PEP-II,"
Proceedings of the 1994 European Particle Accelerator Conference, London,
England.
|
1.4754805.pdf | Exchange coupled bilayer thin films with tilted out-of-plane anisotropy easy axis
A. Layadi
Citation: Journal of Applied Physics 112, 073901 (2012); doi: 10.1063/1.4754805
View online: http://dx.doi.org/10.1063/1.4754805
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/112/7?ver=pdfcov
Published by the AIP Publishing
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129.101.79.200 On: Wed, 20 Aug 2014 10:34:36Exchange coupled bilayer thin films with tilted out-of-plane anisotropy
easy axis
A. Layadi
LESIMS, D /C19epartement de Physique, Universit /C19e Ferhat Abbas, S /C19etif 19000, Algeria
(Received 24 May 2012; accepted 24 August 2012; published online 1 October 2012)
The ferromagnetic resonance (FMR) modes are worked out for the case of exchange coupled bilayer
thin films where the anisotropy axis in the fe rromagnetic film is tilted out of the plane. General
formulas are obtained for the mode position, fre quency and field linewidths, and intensity for an
arbitrary tilt angle. The analysi s is then applied for the in-plane, weak and strong perpendicular
anisotropies. Analytical expressions for the magn etization curve and the FMR modes are derived. It
will be shown how the exchange anisotropy field H E, the uniaxial anisotropy H K, and the magnetization
angle are related to the FMR spectrum characteris tics and how they can be found in a straightforward
manner. VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4754805 ]
I. INTRODUCTION
The interaction at the interface of a ferromagnetic thin
film (F) with an antiferromagnetic one (AF) gives rise to aunidirectional anisotropy called exchange anisotropy. This
anisotropy can be modelled as a magnetic field H
E, the
exchange anisotropy field. The phenomenon was firstobserved by Meiklejohn and Bean
1and has been thoroughly
studied lately.2–7
Several experimental techniques have been used to
detect the exchange anisotropy and measure H E. The hyster-
esis curve is the most common and most popular technique.
The shifted loop is an indication of the existence of exchangeanisotropy. The amount of shift is equal to H
E. It has been
found that the H Evalue seems to depend on the experimental
technique used in the investigation. This can be attributed tothe model used in analyzing the results. In the static AF
model, the AF moments are assumed to stay in a given direc-
tion (the unidirectional anisotropy easy axis) as the magnet-ization Mof the Flayer is rotated; initially, the exchange
field H
Edirection is taken to be along the easy axis of the F
layer (aligned exchange anisotropy). The off-alignedexchange anisotropy has been introduced to explain the
angular variation of H
E; in this situation, the unidirectional
anisotropy axis, i.e., the easy axis of the AF layer and the an-isotropy axis of the ferromagnetic layer are not parallel but
make an angle bknown as the off-alignment angle.
8–11In
this misalignment case, it is assumed that both axes remainin the film plane. The b¼90
/C14situation was encountered in
exchange coupled films, where for instance it was observed
that the Fe 3O4and CoO spins are perpendicular in exchange-
biased Fe 3O4/CoO supperlattices.12
In the present work, it will be assumed that the ferro-
magnetic film anisotropy axis is out-of-plane, while the anti-ferromagnetic spin direction (direction of H
E) remains in the
film, this kind of misalignment has not been worked out and
might lead to some interesting features in the ferromagneticresonance (FMR) spectra. Experimentally, it has been
reported that in some single ultra-thin films the magnetiza-
tion can be perpendicular to the film plane
13,14or may be
tilted.15These kinds of thin films are assumed to be part ofthe (F)/(AF) system under study here. In Sec. II, the equilib-
rium position of the (F) magnetization (the magnetization
curves) and the FMR modes (position, frequency and fieldlinewidths, and intensity) will be investigated for the more
general case, i.e., the ferromagnetic anisotropy axis is out-of-
plane and makes a dangle (the tilt angle) with the normal to
the plane. The analytical derivation will be carried out and
discussed for two cases of interest: (1) the in-plane anisot-
ropy axes ( d¼p/2) (Sec. III) and (2) the perpendicular ani-
sotropy axis ( d¼0) for weak and strong perpendicular
anisotropy (Sec. IV).
II. EQUILIBRIUM POSITIONS AND FMR MODES:
THE GENERAL CASE
In this section, the magnetization equilibrium positions
and the FMR modes (mode positions, linewidths, and inten-
sities) will be derived for an arbitrary out-of-plane easy axisdirection of the ferromagnetic thin film.
The two films (F)/(AF) are taken to be in the xy plane.
The exchange anisotropy field H
Eis oriented along the x-axis
while the anisotropy axis is out-of-plane making a dangle
with the z-axis. Without loss of generality, the anisotropy axis
is taken in the xz plane. The magnetization Mof the ferro-
magnetic layer is defined by the customary angles hand/.
The external applied magnetic field His defined by the angles
hHand/H. With all these considerations, the total free energy
system per unit volume can be explicitly written as
E¼/C0MH½sinhsinhHcosð//C0/HÞþcoshcoshH/C138
/C02pM2sin2h/C0MH Esinhcos/
þK/C20
sin2hðcos 2dþsin2dsin2/Þ/C01
2sin 2hsin 2dcos//C21
:
(1)
The first term corresponds to the interaction of Mwith the
external magnetic field (the Zeeman term); the second and
third terms are, respectively, the shape anisotropy and theinterfacial exchange anisotropy energies while the last term
accounts for the magneto-crystalline anisotropy energy with
0021-8979/2012/112(7)/073901/9/$30.00 VC2012 American Institute of Physics 112, 073901-1JOURNAL OF APPLIED PHYSICS 112, 073901 (2012)
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129.101.79.200 On: Wed, 20 Aug 2014 10:34:36an anisotropy axis tilted from the normal of the film by a d
angle.16In the more general case, the equilibrium positions, h
and/are given, respectively, by the following two equations
/C0H½coshsinhHcosð//C0/HÞ/C0sinhcoshH/C138
/C04pMsinhcosh/C0HEcoshcos/
þHK½sinhcoshðcos 2dþsin2dsin2/Þ
/C0cos 2hsindcosdcos//C138¼0 (2a)
and
HsinhHsinð//C0/HÞþHEsin/
þHKsindsin/½sinhsindco/þcoshcosd/C138¼0( 2 b )
where H K¼2K/M is the uniaxial anisotropy field.
The case to be addressed in Secs. III–Vis when the
magnetic field His applied in-plane, i.e, hH¼p/2, along the
x-axis, either in the forward direction (the same direction asthe exchange anisotropy field H
E) or in the reverse direction.
FMR is the absorption of energy by a ferromagnetic
sample subject to a steady field Hand to a variable field h.
As an experimental technique, it was used to investigate a
variety of phenomena and systems.17–29With the energy for-
mulation, the general resonance condition is given by thewell known formula
x
c/C18/C192
¼1
M2sin2h½EhhE///C0E2
h//C138: (3)
Here, xis the resonant (angular) frequency; cis the magne-
togyric ratio. E hh,E//, and E h/are the second derivatives of
the total energy with respect to the indicated variables and
are evaluated at the equilibrium position of themagnetization.
Upon substituting the different derivatives evaluated at
the equilibrium positions, one finds the following resonancecondition:
x
c/C18/C192
¼½Hsinhcosð//C0/HÞþHEsinhcos/
/C04pMcos2hþHKfðdÞ/C138
/C2½Hsinhcosð//C0/HÞþHEsinhcos/
/C04pMcos 2hþHKgðdÞ/C138 /C0H2
KpðdÞ; (4a)
where f(d),g(d), and p(d) are trigonometrical functions
depending on the different angles mainly the tilt angle d.
These are given by
fðdÞ¼cos2hðcos 2dþsin2dsin2/Þþsin2dcos 2/
þsinhcoshsin 2dcos/; (4b)
gðdÞ¼cos2hðcos2dþsin2dsin2/Þþsin2hsin2dcos/;(4c)
pðdÞ¼sin2dsin2/ðsindcoshcos//C0cosdsinhÞ2:(4d)
From Eq. (4a), the resonant field can be derived for a given
frequency, it is found to beHres¼/C0HEsinhcos/þHþðdÞþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2
c2þH2
/C0ðdÞþpðdÞH2
Kq
sinhcosð//C0/HÞ;
(5a)
where
H6ðdÞ¼1
2½4pMðcos2h6cos 2hÞ/C0HKðfðdÞ6gðdÞÞ/C138:(5b)
The field H 6(d) includes the shape and the magnetocrystal-
line anisotropies, it depends mainly on the tilt angle dwhich
is one of the parameters of interest in this work.
One may want also to derive the exchange anisotropy
field H E, knowing, from the experimental FMR spectrum,
the resonant field H resand the frequency; H Ewill be given
by the following formula:
HE¼/C0Hrescosð//C0/HÞ
cos/
þHþðdÞþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2
c2þH2
/C0ðdÞþpðdÞH2
Kq
sinhcos/: (5c)
The intrinsic linewidth is a fundamental property of the
material, it is related to damping. In terms of frequency (fora fixed dc field-variable frequency set-up), the linewidth is
given by the general formula
Dx¼ac
MEhhþE//
sin2h/C20/C21
: (6)
Here, ais the Gilbert damping coefficient.
In the present study and for the general case, the fre-
quency linewidth is found to be
Dx¼2ac½Hsinhcosð//C0/HÞþHEsinhcos//C0HþðdÞ/C138:
(7)
If one is using a variable dc magnetic field–fixed fre-
quency spectrometer, then the linewidth will be DH and is
found to be given by
DH¼2axHsinhcosð//C0/HÞþHEsinhcos//C0HþðdÞ
sinhcosð//C0/HÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2þc2½H2
/C0ðdÞþpðdÞH2
K/C138p :
(8)
If one substitutes the resonance field (Eq. (5a)) into DH
(Eq. (8)), then the field linewidth will reduce to the simpler
relation
DH¼2ax
csinhcosð//C0/HÞ: (9)
However, Eq. (8)might be useful as it gives explicitly the
dependence of the field linewidth on the different parametersof the system. For the saturated case, Eq. (9)will reduce to
the well known formula, DH¼2ax/c.
The mode intensity is also an important feature of a
FMR spectrum along with the mode positions and the mode073901-2 A. Layadi J. Appl. Phys. 112, 073901 (2012)
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
129.101.79.200 On: Wed, 20 Aug 2014 10:34:36linewidths. The absorbed power is related to the component
of the dynamic susceptibility vin the direction of the micro-
wave field h(taken here along the y-axis perpendicular to the
static magnetic field H). This susceptibility has a real part
(dispersive), v0, and an imaginary part (dissipative), v00. The
absorbed power is proportional to v00. The power per unit fre-
quency interval (or equivalently per unit field interval) is
P¼(1/2)xv00h2V, where V is the sample volume. P represents
the energy transferred from the microwave field to the sample.
The rf susceptibility components were derived following
the method described by Smith and Beljers30and widely
used in FMR studies of thin films and multilayers (see, forexample, Refs. 20,22,25, and 27). In this method, the equa-
tions coupling the excursions of the magnetization about the
equilibrium point ( DhandD/) are specified and written in amatrix form, from which the rf susceptibility tensor is
obtained. As mentioned above, of interest will be the
dynamic susceptibility vin the direction of the microwave
fieldh(the y-axis), i.e., the component v
yyof the susceptibil-
ity tensor (labelled here vfor simplicity, i.e, vyy¼v). This
component is found to be equal to
v¼c2ðEhhþixMa=cÞ
ðx2
r/C0x2ÞþixDx: (10a)
At resonance ( xr¼x), substituting E hh(the second deriva-
tive of the total energy with respect to h) and the frequency
linewidth Dxby their expressions, writing vasv¼v0/C0iv00
and taking the imaginary part, one will get
v00¼cM
2axHsinhcosð//C0/HÞþHEsinhcos//C04pMcos 2hþgðdÞHK
Hsinhcosð//C0/HÞþHEsinhcos//C0HþðdÞ/C20/C21
: (10b)
Alternatively, by using Eqs. (5a)and(10b) ,v00can be put in
the following form:
v00¼cM
2ax1þcH/C0ðdÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x2þc2H2
/C0ðdÞp"#
: (10c)
The imaginary part of the susceptibility v00represents the
amplitude of the absorption peak.28In a FMR spectrum, the
FMR intensity is defined as the area under the v00vs H field
curve (if one is using a resonant cavity with a fixed frequency
and variable dc field set-up) or the v00vs frequency curve (if one
is using a variable frequency s etup with a fixed dc field). Whenthe function cannot be easily integrated, one can make the
approximation that the area under the curve (the intensity I) is
proportional to the product of the imaginary part of the suscepti-bility at resonance v
00(x¼xr) by the linewidth, either the field
(intensity noted I Hin the subsequent analysis) or the frequency
linewidth (intensity noted I f) depending on the used setup.
The intensity I Hwill then be proportional to
v00(x¼xr).DH if one is using a variable field-fixed fre-
quency set-up; the proportionality factor containing severalparameters such that the sample volume and the strength of
the rf field is not shown in the formula since these parame-
ters do not depend on the magnetic properties of the sample.From Eqs. (9)and(10b) , the intensity I
Hwill be given by
IH¼MHsinhcosð//C0/HÞþHEsinhcos//C04pMcos 2hþgðdÞHK
sinhcosð//C0/HÞ½Hsinhcosð//C0/HÞþHEsinhcos//C0HþðdÞ/C138/C20/C21
: (11a)
Note that the intensity given above consists of the product of
the magnetization by a factor (called the ellipticity factor28),
which depends on different anisotropy fields and also on thetilt angle din the present case.
The intensity I
ffor a variable frequency-fixed field set-
up, i.e., v00(x¼xr).Dx, is given by the following relation,
when using Eqs. (7)and(10b) :
If¼c2M
x½Hsinhcosð//C0/HÞþHEsinhcos/
/C04pMcos 2hþgðdÞHK/C138: (11b)It is interesting to note that one can derive a value of the
damping constant a, by using the measurable quantities in a
FMR spectrum: the resonance frequency x, the correspond-
ing linewidth Dxand intensity I f, and also the magnetization
M and the magnetogyric ratio cwithout the need to know the
magnetization angles hand/or the tilt angle d. Indeed, by
combining Eqs. (4a),(7), and (11b) , one finds
a¼McðDxÞIf
xðI2
fþM2c2Þ: (12)073901-3 A. Layadi J. Appl. Phys. 112, 073901 (2012)
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129.101.79.200 On: Wed, 20 Aug 2014 10:34:36This universal formula allows to derive the damping constant
for any anisotropy and tilt angle values contrary to Eq. (7)
where equilibrium angles hand/, tilt angle d, and H Kvalues
are needed to derive a. Of course, these parameters will
affect the intensity and the linewidth.
One can see that the mode position [Eqs. (4a)–(4d),(5a),
and (5b)], linewidths [Eqs. (7)–(9)], and intensities [Eqs.
(10a) –(10c) ,(11a) ,a n d (11b) ] depend on the out-of-plane devi-
ation angle. There are an explicit dependence (see expressionsfand gin the equations) and also an implicit dependence
through the angles hand/, which depend on the out-of-plane
deviation angle (Eqs. (2a)and(2b)).
For an arbitrary dvalue (other than 0 and p/2), there is
no analytical formula giving the direction of the magnetiza-
tion and the saturation and switching fields. Equations (2a)
and(2b) have to be numerically solved. Then the resonance
relation, the linewidths, and the mode intensity can be found
from the above equations.
In the following, two cases of interest will be discussed:
The anisotropy easy axis is (1) in the plane of the film (in-plane
anisotropy) and (2) perpendic ular to the film plane with weak
and strong anisotropy. For each of these situations, analytical
expressions for the switching fields, the resonant relations, the
frequency and field linewidths, and the intensity will be derivedand discussed.
III. THE IN-PLANE ANISOTROPY CASE
Let us first recall the most usual case, when the ferro-
magnetic film is characterized by an in-plane anisotropy
axis. Even though this case is well known and also cannot
really fit in the out-of-plane scheme worked out here, themain results will be shown for two reasons: (1) to check that
the general relations of Sec. IIgive indeed the right results
for the in-plane anisotropy and (2) to compare this case withthe perpendicular one, investigated in Sec. IV, which has not
been worked out before.
The in-plane anisotropy axis corresponds to d¼p/2 in
the present theoretical analysis. In the following, Hwill be
taken in-plane along the anisotropy axis, H will be counted
positive if it is in the forward direction ( /
H¼0,Halong the
easy axis, the x-axis, and in the direction of HE), and nega-
tive if it is in the reverse direction ( /H¼p). The solutions of
Eqs. (2a)and(2b)will give h¼p/2 and
/¼0i f H >/C0HE/C0HK (13a)
and
/¼pif H </C0HEþHK: (13b)
Note that H Eand H Kare always counted positive. Thus when
the magnetic field is decreased from high positive values, M
will be in the forward direction, i.e., /¼0 (saturation in the
forward direction), down to a field equal to ( /C0HE/C0HK),
then the magnetization will switch to the opposite direction
(/¼p) and saturation in the reverse direction is achieved.
If the field is then increased from negative values, M
remains in the opposite direction ( /¼p)u pt oafi e l de q u a lt o
(/C0HEþHK). These magnetization rotations will give rise tothe shifted M-H loop as expected (see dashed line in Fig. 1)
with the shift equal to the exchange anisotropy field, H E,a n d
the curve width equal to 2H K.
The resonance relation for this in-plane anisotropy case is
found by setting d¼p/2 in Eq. (4a)and also h¼p=2. In this
case, and for /¼0o rp,E q s . (4b)–(4d) give f¼cos 2/¼1,
g¼cos2/¼1, and p¼0; then, Eq. (4a) will reduce to the
known relation:29
x
c/C18/C192
¼½Hcosð//C0/HÞþHEcos/þHKcos 2//C138
/C2½Hcosð//C0/HÞþHEcos/þ4pMþHKcos2//C138:
(14)
The dispersion curve, frequency vs applied field H, for
such a case is shown in Fig. 2, for increasing field (dashed
line) and decreasing field (dotted line), there is a hysteresis
phenomenon in the curve. At the critical field values ( /C0HE
/C0HK) and ( /C0HEþHK), the frequency vanishes and a jump inFIG. 1. Magnetization curve for exchange bilayer thin films. Dashed line:
in-plane anisotropy field ( d¼p/2, H K¼0.5 kOe). Solid line: perpendicular
anisotropy ( d¼0) and H K/C04pM<0( H K¼1 kOe). Dotted line: perpendic-
ular anisotropy ( d¼0) and H K/C04pM>0( H K¼7 kOe). Other parameters
used: 4 pM¼6 kG, H E¼0.3 kOe.
FIG. 2. Dispersion curve, frequency vs. applied magnetic field for a system
with in-plane anisotropy field ( d¼p/2) for increasing (dashed line) and
decreasing (dotted line) fields; and for a system with perpendicular anisot-
ropy ( d¼0) and H K/C04pM<0 (solid line). Parameters used: 4 pM¼6 kG,
HK¼1 kOe, H E¼0.3 kOe, c/2p¼2.8 GHz/kOe (g ¼2).073901-4 A. Layadi J. Appl. Phys. 112, 073901 (2012)
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129.101.79.200 On: Wed, 20 Aug 2014 10:34:36the frequency value is observed; the jump is equal to x¼2cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
HKðHKþ2pMÞp
for both field values. We also note that at
H¼/C0HE, the resonance frequency is the same for both mag-
netization directions and is equal to x¼cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
HKðHKþ4pMÞp
.
If one is using a fixed field–variable frequency set-up,
then the exchange anisotropy field H Ecan be expressed as a
function of the resonant frequency x(0) for Happlied in the
forward direction and x(p) for Hin the opposite direction
and with same magnitude (H is taken to be greater than thesaturation field in both directions), the following relation is
found:
H
E¼x2ð0Þ/C0x2ðpÞ
2c2ð2Hþ2HKþ4pMÞ: (15)
If one is using a fixed frequency-variable magnetic field,
then H Ecould be found by the following relation:
HE¼HRðpÞ/C0HRð0Þ
2; (16)
where H R(p) and H R(0) designate (in absolute values) the
resonant field in the reverse and forward directions, respec-tively, for the same frequency.
The frequency linewidth, Eq. (7), is given in this situa-
tion by
Dx¼ac½2ðHþH
EÞcos/þð2HKþ4pMÞ/C138: (17a)
After some transformations, Eq. (17a) can be written as
Dx¼caffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4x2
c2þð4pMÞ2s
: (17b)
The frequency linewidth vs applied field H curve is
shown in Fig. 3(a), for increasing field (dashed line) and
decreasing field (dotted line). Here too, there is a hysteresisphenomenon. Also, at the critical field values ( /C0H
E/C0HK)
and ( /C0HEþHK), a jump in the frequency linewidth is
observed; the jump value is equal to 4 acHKand is independ-
ent of the exchange anisotropy field H E. On the other hand,
the difference in the frequency linewidth between the for-
ward and the reverse directions (for the same applied fieldvalue insuring saturation in both directions) does depend on
H
Eand it is given by
Dxð0Þ/C0DxðpÞ¼4acHE: (18)
The crossing point between the increasing and decreasing
field curves, i.e., where the Dxis similar for both magnetiza-
tion directions occurs at H ¼/C0HEand the linewidth is equal
toDx¼ac½2HKþ4pM/C138.
The field linewidth, Eq. (9), is given by DH¼2ax
c.A sa
function of the applied field, the variation of the field line-
width is shown in Fig. 3(b).DH vanishes at the critical fields
as does the resonance frequency. For the same frequency,one notes that DH(0) ¼DH(p) contrary to the frequency line-
width where the difference in Dxbetween the two directions
is proportional to the exchange anisotropy field H
E. Also atH¼/C0HE, theDH is the same for both magnetization direc-
tion and is equal to DH¼2affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
HKðHKþ4pMÞp
.
The magnetic susceptibility is shown in Fig. 4for
increasing field (dashed line) and decreasing field (dottedline). There is a hysteresis phenomenon and at the critical
field values ( /C0H
E/C0HK) and ( /C0HEþHK), the v00(x¼xr)
value becomes infinite; recall that for these values the fre-quency vanishes. Note also that there is a large variation of
susceptibility in the vicinity of the critical fields.FIG. 3. Frequency (a) and field (b) linewidths vs. applied magnetic field for
a system with in-plane anisotropy field ( d¼p/2) for increasing (dashed line)
and decreasing (dotted line) fields; and for a system with perpendicular ani-
sotropy ( d¼0) and H K/C04pM<0 (solid line). Damping constant a¼0.01,
other parameters used as in Fig. 2.
FIG. 4. Imaginary part of the susceptibility, v00vs. applied magnetic field for
a system with in-plane anisotropy field ( d¼p/2) for increasing (dashed line)
and decreasing (dotted line) fields; and for a system with perpendicular ani-
sotropy ( d¼0) and H K/C04pM<0 (solid line). Damping constant a¼0.01,
other parameters used as in Fig. 2.073901-5 A. Layadi J. Appl. Phys. 112, 073901 (2012)
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129.101.79.200 On: Wed, 20 Aug 2014 10:34:36IV. THE PERPENDICULAR ANISOTROPY CASE
In this case, d¼0 in the above equations. Two cases have
to be considered: for (H K/C04pM)<0a n df o r( H K/C04pM)>0.
If (H K/C04pM) is negative, i.e., there is a positive uniax-
ial magnetocrystalline anisotropy (it favors the film perpen-
dicular direction) but it is not strong enough to overcome the
shape anisotropy. In the case, the magnetization remainsalways in the film plane ( h¼p/2) for all H. Also /¼0i f
H>/C0H
Eand/¼pif H</C0HE. These solutions will give
the M-H loop displayed in Fig. 1(solid line). It is a closed
shifted loop, once again the shift is equal to H E.
The resonance relation will be
x
c/C18/C192
¼½Hcosð//C0/HÞþHEcos//C138
/C2½Hcosð//C0/HÞþHEcos/þ4pM/C0HK/C138:(19)
The corresponding dispersion, xvs H, curve is shown in
Fig. 2(solid line). The curve is shifted and consists of two
branches which join at x¼0 corresponding to H ¼/C0HE,
this particular point leads to the experimental determinationof the exchange anisotropy field. Compare also the solid line
with the dashed and dotted lines; in both situations, the mag-
netization remains in the film plane, but the existence of asmall perpendicular anisotropy leads to the difference
between the two curves. At H ¼/C0H
E, while for the weak
perpendicular anisotropy, the resonance frequency is zero, itis equal to cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H
KðHKþ4pMÞp
for the in-plane anisotropy
case. For a given applied field and magnetization direction,
the resonance frequency for in-plane is higher than that forthe weak perpendicular one. Thus, the behaviour of the dis-
persion curve may reveal the out of plane anisotropy.
The frequency linewidth will be in this case,
Dx¼ac½2ðHþH
EÞcos//C0ðHK/C04pMÞ/C138: (20a)
In terms of frequency only, this linewidth can be put in
the following form:
Dx¼caffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
4x2
c2þðHK/C04pMÞ2s
: (20b)
The dependence of the frequency linewidth with H is
shown in Fig. 3(a) (solid line). The curve consists of two
straight lines joining at H ¼/C0HEwhere the value of Dxis
equal to Dx¼cað4pM/C0HKÞ. Note also that, even though
the frequency linewidth for the weak perpendicular anisot-
ropy is different from that of the in-plane one [compare
Eqs. (17) and(20a) ], the difference in the linewidth for the
forward and reverse directions is the same (given by Eq.
(18)) and is proportional to the exchange anisotropy field,
i.e.,Dxð0Þ/C0DxðpÞ¼ 4acHE.A tH ¼/C0HE,Dxfor this
weak perpendicular anisotropy is equal to ac½4pM/C02HK/C138
lower than that of the in-plane ( ac½2HKþ4pM/C138).
The field linewidth is given by the usual formula for sat-
urated sample, i.e., DH¼2ax
c. The variation of DH with the
applied field is shown in Fig. 3(b).DH is zero at H ¼/C0HEas
is the case for the frequency; it is also larger for in-planethan that for the weak perpendicular anisotropy for given
magnetization direction and applied field value.
The variation of the imaginary part of the susceptibility,
i.e., the amplitude of the absorption curve is shown in Fig. 4
(solid line). The value at resonance becomes infinite at
H¼/C0HEwhere the resonance frequency vanishes. This
behaviour is quite different for that of the in-plane anisotropy
where the magnetization is also in plane and also different
from the strong perpendicular anisotropy as will discussedlater.
The particular case where H
K/C04pM¼0 falls in this cat-
egory, i.e., the magnetization Mwill be in the film plane for
all H values. Indeed, in the absence of effective magnetic ani-
sotropy (the magnetocrystalline anisotropy compensating the
shape anisotropy), Mwill be in the same plane as the fields
(applied and exchange anisotropy). The equilibrium position
ofMis identical to the (H K/C04pM)<0 case, i.e., /¼0i f
H>/C0HEand/¼pif H</C0HE. The relations (Eqs. (19),
(20a) ,a n d (20b) ) hold true by putting H K¼4pM. Some of
the relations reduce to simple and interesting forms. Indeed,
the resonance relation (Eq. (19))w i l lg i v ex
c¼jHþHEj;
the frequency linewidth (Eq. (20b) )w i l lr e d u c es i m p l yt o
Dx¼2axwhile the imaginary part of the susceptibility will
bev00¼1
2cM
ax. The mode intensities will reduce to I H¼Ma n d
If¼cM, i.e., the ellipticity factor is equal to 1. Note once
again that the above values satisfy the more general relation
giving the damping constant (Eq. (12)).
If on the other hand (H K/C04pM) is positive, i.e., the
uniaxial magnetocrystalline anisotropy is positive and is
strong enough to overcome the effect of the shape anisot-ropy, then the solutions of the equilibrium conditions will be
h¼p=2;/¼0i f H >/C0H
EþðHK/C04pMÞ;(21a)
h¼p=2;/¼pif H </C0HE/C0ðHK/C04pMÞ;(21b)
and
sinh¼jHþHEj
HK/C04pMelsewhere : (21c)
In the last situation (Eq. (21c) ),/¼0 when H þHE>0 and
/¼pfor H þHE<0. Thus, the saturation field in the for-
ward direction is equal to H 1¼/C0HEþ(HK/C04pM), above
which the magnetization is along the magnetic field. The sat-uration field in the reverse direction is H
2¼/C0HE/C0(HK
/C04pM). Between these two fields, the saturation will rotate
out-of-plane; the corresponding M-H loop is shown in Fig. 1
(dotted line). It is a shifted hard axis-like loop. The shift (the
crossing of the curve with the H axis) is equal to H Eand the
interval between H 1and H 2is equal to 2(H K/C04pM).
The dispersion curve is plotted in Fig. 5. For the unsatu-
rated situation, Mmakes a hangle with the film normal and
Eq.(21c) holds. Upon substituting Eq. (21c) into Eq. (4a),
the resonance relation can be made in the following form:
x
c/C18/C192
¼ðHK/C04pMÞ2/C0ðHþHEÞ2: (22)
One can see, from the above equation and from the curve
(Fig. 5), that in this unsaturated state the dispersion curve073901-6 A. Layadi J. Appl. Phys. 112, 073901 (2012)
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129.101.79.200 On: Wed, 20 Aug 2014 10:34:36vanishes at H 1and H 2defined earlier has a maximum at
H¼/C0HEand the value of this maximum is equal to
xm¼c(HK/C04pM). Thus, the experimen tal determination of
H1,H2, the position, and the value of the curve maximum may
lead to the determination of (H K/C04pM), H E,andc.F u r t h e r -
more, one also notes that this c urve is a half ellipse; indeed,
Eq.(22)c a nb ew r i t t e ni nt h ef o r m
ðHþHEÞ2
ðHK/C04pMÞ2þx2
c2ðHK/C04pMÞ2¼1; (23)
which is the equation of an ellipse centred at H ¼/C0HEand
with half axes equal to (H K/C04pM) and c(HK/C04pM). Note
also that if one plots x=cvs H (see Fig. 5) instead of xvs H
(as done before), then the curve between H 1and H 2is a
circle centred at H ¼/C0HEwith radius equal to (H K/C04pM),
this can be seen easily from Eq. (23). In this case, the angle h
can be directly found from the curve; from a given field H,
the corresponding point is noted (point P in Fig. 5), the angle
between the line joining P to the centre of the circle and thevertical passing by the centre is just the angle h. This can be
shown by the use of Eqs. (21c) and(23). This geometrical
method may allow one to derive the magnetization angle atany applied field H in a straightforward manner from the dis-
persion curve.
Moreover, the area under the curve for this unsaturated
case can be evaluated and is found to be equal to A¼
cp
2
ðHK/C04pMÞ2. The area under the curve is independent of
the exchange anisotropy field and dependent only on theshape and uniaxial anisotropies and on the magnetogyric
ratio.
For the saturated case, h¼p/2, then f(0)¼0a n d g(0)¼/C01.
If one is using a fixed field, variable frequency set-up, then
applying the same field H (great er than the saturation field) in
the forward and in the reverse directions will lead to the determi-nation of H
Eas
HE¼x2ð0Þ/C0x2ðpÞ
2c2ð2H/C0HKþ4pMÞ: (24)
This relation is different from the one found for the in-plane
anisotropy case (Eq. (15)).The frequency linewidth can now be investigated. From
Eqs. (7)and(21c) , it is found that in the unsaturated region,
Dxis given by
Dx¼ac2ðHK/C04pMÞ/C0ðHþHEÞ2
HK/C04pM"#
if H 2<H<H1:
(25a)
While in the saturated region, it is given by the same formula
as Eq. (20a) . Note that in Eq. (25a) and in the following
equations, all expressions related to the unsaturated region
depend on the angle h. However, the angle hdoes not appear
explicitly in these equations but it is present through Eq.
(21c) ; in all expressions of susceptibilities and linewidths,
sinhis replaced by its value given by Eq. (21c) in the unsatu-
rated region.
The frequency linewidth can also be expressed in terms
of the resonant frequency, Eq. (25a) will reduce to
Dx¼ax2
cðHK/C04pMÞþcaðHK/C04pMÞif H 2<H<H1:
(25b)
In the saturated region, it is given by Eq. (20b) .
Thus when the system is saturated, the frequency line-
width is given by the same formulas (Eqs. (20a) and(20b) )
for weak [negative (H K/C04pM)] and strong perpendicular
[positive (H K/C04pM)] anisotropies. However numerically,
and for the same magnetic parameters except the anisotropy,
the frequency linewidth for the strong perpendicular anisot-ropy is lower than that corresponding to the weak one.
The variation of Dxwith the applied field H is shown in
Fig. 6(a). The two branches in the saturated regions are
straight lines with slope equal to 2 acand at H ¼0,Dx¼2ac
(2H
E/C0HKþ4pM). In the unsaturated region, it is a parabola
with a maximum equal to 2 ac(HK/C04pM), occurring at
H¼/C0HE. At the critical fields, Dxis equal to ac½HK/C04pM/C138.
Also note that the difference in the frequency linewidth
between the forward and the reverse directions (for the sameapplied field value insuring saturation in both directions) is
also given by Eq. (18); thus, the difference in frequency line-
width [ Dx(0)/C0Dx(p)] is the same for all three cases
(in-plane, weak, and strong perpendicular anisotropy) and is
proportional to the exchange anisotropy field H
E,e v e nt h o u g h
the linewidths for each cases are different.
If one is using a fixed frequency-variable magnetic field,
then the relation, Eq. (16), i.e., H R(p)/C0HR(0)¼2HEholds
here too and may be used for the determination of H E. The
field linewidth DH, in the unsaturated region, is found to be
DH¼2ax
cHK/C04pM
jHþHEjif H 2<H<H1: (26)
It reduces to the well known relation beyond the saturation
field, i.e., DH¼2ax
c. The field linewidth is plotted against
the applied field in Fig. 6(b). The behaviour is different from
the weak perpendicular anisotropy case. DH is infinite at
H¼/C0HEand vanishes at the critical fields where the reso-
nant frequency is also equal to zero.FIG. 5. Dispersion curve, reduced frequency ( x=c) vs. applied magnetic
field for a system with perpendicular anisotropy ( d¼0) and H K/C04pM>0
(HK¼7 kOe). Other parameters used: 4 pM¼6 kG, c/2p¼2.8 GHz/kOe
(g¼2) with H E¼0.3 kOe (solid line) and H E¼0 (dotted line).073901-7 A. Layadi J. Appl. Phys. 112, 073901 (2012)
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129.101.79.200 On: Wed, 20 Aug 2014 10:34:36The imaginary part of the susceptibility at resonance is
plotted against the applied field H in Fig. 7(a). The curve has
a maximum value at H ¼/C0HE. In the unsaturated region,
one also notes that v00is practically constant and equal to an
average value equal to v00¼M
2aðHK/C04pMÞover an applied field
range around the exchange anisotropy field H Evalue. The
mode intensity I His found after some calculations to be
given by
IH¼2MðHK/C04pMÞ
jHþHEjx2
x2þc2ðHK/C04pMÞ2"#
if H 2<H<H1: (27a)
The relation is valid for both magnetization directions.
While in the saturated region, it reads
IH¼2Mx2
x2þc2ðHþHEÞ2"#
: (27b)
It is easy to see, once again by the use of the resonant mode
given by Eq. (5a), that the intensities have the same value for
/¼0 and /¼p, i.e., I(0) ¼I(p). The variation of the mode
intensity I His shown in Fig. 7(b).
The mode intensity at variable frequency, I f, reduces to
If¼Mx
HK/C04pMif H 2<H<H1: (28a)At H ¼/C0HE, the maximum intensity occurs (see Fig. 7(c))
and it is given simply by I f¼cM. In the saturated region, the
mode intensity is found to be
If¼Mx
jHþHEj: (28b)
V. CONCLUSION
The effect of the out of plane anisotropy axis direction,
measured by the tilt angle d, in thin films with exchangeFIG. 7. Imaginary part of the susceptibility, v00, (a) and mode intensities for
variable field (b) and variable frequency (c) set-ups vs. applied magnetic field
for exchange bilayer thin films system with perpendicular anisotropy ( d¼0)
and H K/C04pM>0( H K¼7k O e ) .O t h e rp a r a m e t e r su s e da si nF i g . 6.FIG. 6. Frequency (a) and field (b) linewidths vs. applied magnetic field for
exchange bilayer thin films system with perpendicular anisotropy ( d¼0)
and H K/C04pM>0( H K¼7 kOe). Exchange anisotropy field H E¼0.3 kOe,
damping constant a¼0.01, other parameters used as in Fig. 5.073901-8 A. Layadi J. Appl. Phys. 112, 073901 (2012)
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129.101.79.200 On: Wed, 20 Aug 2014 10:34:36anisotropy has been investigat ed. Mode position, linewidth,
and intensity are derived for an arbitrary dvalue. A formula
giving the damping constant ais obtained in terms of the meas-
ured FMR spectrum characteristics regardless of the tilt angles
and the anisotropy and exchange field values. Asymptotic satu-
ration is observed for an arbitrary dvalue. The general analysis
is applied to the situations where analytic relations can be
obtained: in-plane anisotropy, weak and strong perpendicular
anisotropy. For these situati ons, analytical expressions have
been obtained for different switc hing fields. For strong perpen-
dicular anisotropy ( d¼0), the dispersion curve consists, in the
unsaturated region, of a shifted half ellipse and the magnetiza-
tion angle can be read in a str aightforward manner from the
dispersion curve. The main dif ferences in the magnetization
curve and in the FMR features between the three situations arehighlighted; some particular points in the FMR spectra are
discussed which may lead to distinguish between the types of
anisotropy and also to the determ ination of different parame-
ters. The derived relations might be, hopefully, useful for both
theoreticians and experimentalists.
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1.5089638.pdf | J. Phys. Chem. Ref. Data 48, 023102 (2019); https://doi.org/10.1063/1.5089638 48, 023102
© 2019 Author(s).Recommended Positron Scattering Cross
Sections for Atomic Systems
Cite as: J. Phys. Chem. Ref. Data 48, 023102 (2019); https://doi.org/10.1063/1.5089638
Submitted: 21 January 2019 . Accepted: 12 March 2019 . Published Online: 25 April 2019
Kuru Ratnavelu , Michael J. Brunger
, and Stephen J. Buckman
Recommended Positron Scattering Cross Sections
for Atomic Systems
Cite as: J. Phys. Chem. Ref. Data 48,023102 (2019); doi: 10.1063/1.5089638
Submitted: 21 January 2019 Accepted: 12 March 2019
Published Online: 25 April 2019
Kuru Ratnavelu,1Michael J. Brunger,2
and Stephen J. Buckman3
AFFILIATIONS
1Institute of Mathematical Sciences, University of Malaya, 50603, Kuala Lumpur, Malaysia and Institute for Mathematical Research,
Universiti Putra Malaysia, 43400 Serdang, Malaysia
2College of Science and Engineering, Flinders University, Bedford Park, Adelaide, South Australia 5042, Australia
3Research School of Physics and Engineering, The Australian National University, Canberra 0200, Australia
ABSTRACT
We present a critical analysis of available experimental and theoretical cross section data for positron scattering from atomic systems. From this
analysis, we present (where data are available) recommended cross sections for total scattering, positronium formation, inelastic scattering, an d
direct ionization processes. A complete bibliography of available measurement and theory is also presented.
Published by AIP Publishing on behalf of the National Institute of Standards and Technology. https://doi.org/10.1063/1.5089638
Key words: Positron scattering; Atoms; Recommended cross sections.
CONTENTS
1. Introduction ............................ 3
1.1. Background to the review ................ 3
1.2. Previous review articles ................. 3
1.3. Scope of this review .................... 4
2. Experimental Approaches .................... 4
2.1. Total scattering . ..................... 4
2.2. Positronium formation .................. 4
2.3. Inelastic scattering .................... 5
2.4. Direct ionization ..................... 5
3. Overview of Theoretical Methods ............... 5
3.1. Introduction . . . ..................... 5
3.2. CC methods . . . ..................... 5
3.2.1. CC or coupled-channel calculations .... 6
3.2.2. CCC method ................... 6
3.2.3. Coupled-channel optical (CCO) methods . 6
3.3. R-matrix ........................... 6
3.4. Relativistic optical potential calculations . . ..... 7
3.5. Other optical-model potential, Born and distorted-
wave methods . . ..................... 7
3.6. Many-body theory (MBT) calculations . . . ..... 8
3.7. Variational calculations ................. 8
4. Recommended Cross Sections for Atomic Species ..... 8
4.1. Atomic hydrogen (H) .................. 8
4.1.1. Total scattering ................. 9
4.1.2. Positronium formation ............. 1 04.1.3. Direct ionization ................ 1 0
4.2. Helium (He) . ....................... 1 1
4.2.1. Total scattering ................. 1 2
4.2.2. Positronium formation ............. 1 2
4.2.3. Electronic excitation .............. 1 2
4.2.4. Direct ionization ................ 1 2
4.3. Lithium (Li) . ....................... 1 3
4.3.1. Positronium formation ............. 1 3
4.4. Neon (Ne) . . ....................... 1 5
4.4.1. Total scattering ................. 1 5
4.4.2. Positronium formation ............. 1 5
4.4.3. Direct ionization ................ 1 5
4.5. Sodium (Na) . ....................... 1 6
4.5.1. Total scattering ................. 1 6
4.5.2. Positronium formation ............. 1 6
4.6. Magnesium (Mg) ..................... 1 7
4.6.1. Total scattering ................. 1 7
4.6.2. Positronium formation ............. 1 8
4.7. Argon (Ar) . . ....................... 1 8
4.7.1. Total scattering ................. 1 9
4.7.2. Positronium formation ............. 1 9
4.7.3. Electronic excitation .............. 1 9
4.7.4. Direct ionization ................ 2 0
4.8. Potassium (K) ....................... 2 0
4.8.1. Total scattering ................. 2 1
4.8.2. Positronium formation ............. 2 2
4.9. Krypton (Kr) . ....................... 2 2
J. Phys. Chem. Ref. Data 48,023102 (2019); doi: 10.1063/1.5089638 48,023102-1
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Chemical Reference DataARTICLE scitation.org/journal/jpr4.9.1. Total scattering ................. 2 2
4.9.2. Positronium formation ............. 2 3
4.9.3. Direct ionization ................ 2 4
4.10. Rubidium (Rb) . . ..................... 2 4
4.10.1. Total scattering ................. 2 5
4.10.2. Positronium formation ............. 2 5
4.11. Xenon (Xe) ......................... 2 5
4.11.1. Total scattering ................. 2 5
4.11.2. Positronium formation ............. 2 6
4.11.3. Direct ionization ................ 2 7
Acknowledgments ......................... 2 7
5. References ............................. 2 7
List of Tables
1. The TCS (in units of 10−16cm2) for positron scattering
from atomic hydrogen ..................... 8
2. Positronium formation cross section (in units of
10−16cm2)f o ra t o m i ch y d r o g e n ............. 9
3. The direct ionization cross section (in units of 10−16cm2)
for positron impact on atomic hydrogen .......... 9
4. The TCS (in units of 10−16cm2) for positron scattering
from helium ............................ 1 0
5. The positronium formation cross section (in units of
10−16cm2)f o r h e l i u m..................... 1 1
6. The cross section (in units of 10−16cm2) for positron
impact excitation of the 21S and 21P states of He ..... 1 2
7. The direct ionization cross section (in units of 10−16cm2)
for positron impact on helium ................ 1 3
8. The positronium formation cross section (in units of
10−16cm2)f o r l i t h i u m .................... 1 3
9. The TCS (in units of 10−16cm2) for positron scattering
from neon ............................. 1 4
10. The positronium formation cross section (in units of
10−16cm2)f o r n e o n...................... 1 5
11. The direct ionization cross section (in units of 10−16cm2)
for positron impact on neon .................. 1 6
12. The TCS (in units of 10−16cm2) for positron scattering
from sodium ........................... 1 6
13. The positronium formation cross section (in units of
10−16cm2)f o r s o d i u m .................... 1 7
14. The TCS (in units of 10−16cm2) for positron scattering
from Mg .............................. 1 7
15. The TCS (in units of 10−16cm2) for positron scattering
from argon ............................ 1 8
16. The positronium formation cross section (in units of
10−16cm2)f o r a r g o n ..................... 1 9
17. The cross section for positron impact excitation of the
3p54s levels in argon (in units of 10−16cm2) ....... 2 0
18. The direct ionization cross section (in units of 10−16cm2)
for positron impact on argon ................. 2 0
19. The TCS (in units of 10−16cm2) for positron scattering
from potassium .......................... 2 1
20. The positronium formation cross section (in units of
10−16cm2)f o r p o t a s s i u m ................... 2 1
21. The TCS (in units of 10−16cm2) for positron scattering
from Kr .............................. 2 222. The positronium formation cross section (in units of
10−16cm2)f o rK r ....................... 2 3
23. The direct ionization cross section (in units of 10−16cm2)
for positron impact on krypton ................ 2 3
24. The TCS (in units of 10−16cm2) for positron scattering
from rubidium . . . ....................... 2 4
25. The positronium formation cross section (in units of
10−16cm2)f o rR b ....................... 2 4
26. The TCS (in units of 10−16cm2) for positron scattering
from xenon (see text for details) ............... 2 5
27. The positronium formation cross section (in units of
10−16cm2)f o rX e ....................... 2 6
28. The direct ionization cross section (in units of 10−16cm2)
for positron impact on xenon ................. 2 7
List of Figures
1. The recommended total scattering cross section for H
(solid line), while the dashed lines represent the estimated
uncertainty limits of ±20% (see also Table 1) ....... 9
2. The recommended positronium formation cross section
for H (solid line) . . ....................... 9
3. The recommended direct ionization cross section for
positron impact on H (solid line) .............. 1 0
4. The recommended total positron scattering cross section
for He (solid line), while the dashed lines represent the
estimated uncertainty limits of ±10% (see also Table 4) . 11
5. The recommended total positronium formation cross
section for He (solid line) ................... 1 1
6. The recommended cross section for the excitation of
He 21S (solid line) . ....................... 1 2
7. The recommended cross section for the excitation of
He 21P (solid line) . ....................... 1 3
8. The recommended direct ionization cross section for
positron impact on He (solid line) .............. 1 3
9. The recommended positronium formation cross section
for Li (solid line) . . ....................... 1 4
10. The recommended total positron scattering cross section for
Ne (solid line), while the dashed lines represent the esti-
mated uncertainty limits of ±10% (see also Table 9) . . . 14
11. The recommended positronium formation cross section
for Ne (solid line) . ....................... 1 5
12. The recommended direct ionization cross section for
positron impact on Ne (solid line) .............. 1 6
13. The recommended TCS for positron scattering from Na
(solid line) ............................. 1 7
14. The recommended positronium formation cross section
for Na (solid line) . ....................... 1 7
15. The recommended TCS for positron scattering from Mg 18
16. The recommended total positron scattering cross section for
Ar (solid line), while the dashed lines represent the estimated
uncertainty limits of ±10% (see also Table 15) . . . . . . 19
17. The recommended positronium formation cross section
for Ar (solid line) . ....................... 2 0
18. The cross section for positron impact excitation of the
3p54s levels in argon (solid line) ............... 2 0
19. The recommended total direct ionization cross section for
positron impact on Ar (solid line) .............. 2 1
J. Phys. Chem. Ref. Data 48,023102 (2019); doi: 10.1063/1.5089638 48,023102-2
Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and
Chemical Reference DataARTICLE scitation.org/journal/jpr20. The recommended TCS for positron scattering from K
(solid line) ............................. 2 1
21. The recommended positronium formation cross section
for K (solid line) ......................... 2 2
22. The recommended total positron scattering cross section
for Kr (solid line), while the dashed lines represent the
estimated uncertainty limits of ±10% (see also Table 21) 22
23. The recommended positronium formation cross section
for Kr (solid line) . . . ..................... 2 3
24. The recommended direct ionization cross section for
positron impact on Kr (solid line) .............. 2 425. The recommended TCS for positron scattering from Rb
(solid line) ............................. 2 4
26. The recommended positronium formation cross section
for Rb (solid line) . ....................... 2 5
27. The recommended total positron scattering cross section
for Xe (solid line), while the dashed lines represent the
estimated uncertainty limits of ±10% (see also Table 26) 26
28. The recommended total positronium formation cross
section for Xe (solid line) ................... 2 6
29. The recommended direct ionization cross section for
positron impact on Xe (solid line) .............. 2 7
1. Introduction
1.1. Background to the review
Positron and electron (lepton) scattering from gas-phase atoms
and molecules are both mature experimental research fields which
provide data for fundamental tests of quantum-based scattering
calculations, as well as much-needed data for a host of applications in
technology, medicine, and the environment (e.g., Ref. 1). Indeed, for
electron interactions, the major motivation in recent years has been
the need for accurate and extensive cross section data, for all availableprocesses, in order to model the role of electron-driven chemistry in a
range of gaseous electronics environments such as lights and lasers,
plasma processing and deposition, medical plasmas, and environ-
mental or atmospheric applications. Another key area of growth and
need for electron-molecule scattering data has been in radiationdamage and dosimetry following the discovery that low energy
electrons can be a major cause of molecular damage in the body.
2
Thefield of positron interactions with atoms and molecules in the
gas phase presents considerably greater challenges, given the dif ficulty
in producing high flux, high energy resolution beams of positrons.
Indeed, conventional techniques using radioactive sources and metallic
moderators usually result in positron beam intensities, which are manyorders of magnitude lower than those obtainable with conventional
electron beam technology, and an energy resolution which is, at best,
about 150 meV.
3Notwithstanding these dif ficulties, many important
studies of positron –atom and positron –molecule interactions have
been performed over the past 40 years, yielding absolute cross sectionsfor a range of scattering processes [see, e.g., Ref. 4].
The past several decades have witnessed somewhat of a re-
naissance in the field of positron scattering with higher flux, higher
energy resolution beams becoming available as a result of higher
activity radioactive sources and the, realized, potential of even
higher flux beams from reactor-based sources. Perhaps the biggest
advance for normal laboratory-based studies has come as a result ofthe development of rare-gas-moderated, trap-based positron beams
and associated measurement techniques,
5,6which have achieved
higher fluxes and higher energy resolution than previous techniques.
The advent of this technology has enabled improvements in the
accuracy of absolute measurements and, with an energy resolution ofless than 50 meV readily achievable, it has opened up possibilities
for study of vibrational and electronic excitation [e.g., Refs. 7and8],
amongst other processes.
The other driver for this increased activity in positron scattering,
and the associated technology developments, has been the applicationsof positron interactions in medical science and nanomaterial analyses.
The key to these applications lies mainly in the formation and sub-
sequent annihilation of positronium —a short-lived electron –positron
pair, formed with high probability at energies below 100 eV, when apositron interacts with, and ionizes, an atom or molecule. Positrons arenow widely used in most major hospitals in the diagnostic technique
Positron Emission Tomography (PET), yet little is known of “positron
dosimetry ”or the interactions that a high energy positron undergoes in
the body when thermalizing, through scattering, from several hundredkeV to the low energies required for positronium formation and
subsequent annihilation. The role of positron and positronium
transport is not well understood in these environments, and manyrecent studies in this area have focused on interactions with biologicallyrelevant molecules.
9,10
1.2. Previous review articles
There have been a number of previous “review ”articles in-
volving cross sections for positron interactions with atoms and
molecules, and to the best of our knowledge, none of these have
provided tabulated cross section values or recommended cross sec-tions, with the notable exception of the recent review article by Chiariand Zecca,
11which we discuss below. However, they do provide an
excellent overall background to the field, including details of ex-
perimental and theoretical techniques —which we only consider
briefly in this article in order to provide overall context.
Thefirst substantive review of positron interactions was per-
formed by Grif fith and Heyland in 1978,12where current experi-
mental and theoretical techniques and results were discussed, but notabulated values were presented. Kauppila and Stein also reviewed the
current status of positron scattering in both 1982
13and 199014with
particular interest in comparing electron and positron scattering crosssections for similarities and differences. A similar approach wasadopted by Kimura and colleagues in their review.
4Charlton and
Humberston15provided a comprehensive discussion of all aspects of
positron and positronium physics in their book “Positron Physics ”in
2001, but did not provide tabulated values of cross sections. Surkoet al. reviewed the experimental and theoretical aspects of positron
scattering and annihilation in their 2005 review article,
16but they also
did not provide tabulated values or recommended cross sections.
In 2008, Laricchia and colleagues17reviewed the situation for
positron impact ionization of atoms and molecules and discussed the
level of agreement between experiments, and between experiment
and theory, but did not tabulate results. Recently, Danielson and
J. Phys. Chem. Ref. Data 48,023102 (2019); doi: 10.1063/1.5089638 48,023102-3
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Chemical Reference DataARTICLE scitation.org/journal/jprcolleagues reviewed trap-based techniques as applied to a range of
antimatter experiments.18
Finally, and of direct relevance to the present work, Chiari and
Zecca reviewed positron scattering by atomic targets.11They pro-
vided recommended, tabulated cross sections for total scattering in
the rare gases He, Ne, Ar, Kr, and Xe, and a recommended posi-
tronium formation and total ionization cross section for He. While
they discuss the relative merits of measurements of positronium
formation and ionization for Ne –Xe, they do not recommend cross
sections for these processes and gases, largely due to the signi ficant
spread in the published data. They also discuss measurements for
other atomic systems —H, the alkalis, and alkaline earth atoms. We
also note our sister publication to this work which concerned tab-
ulations of recommended cross sections for positron-molecule
scattering.19
1.3. Scope of this review
In this article, we are endeavoring to provide a comprehensive
collection and assessment of the available experimental data (cross
sections) for low- and intermediate-energy (0.1 eV –1 keV) positron
interactions with atoms. As mentioned above, in a previous article we
provided a similar collection of data for positron-molecule scattering.
This is not always an easy task when considering the available
published data, as the positron community has not been noted for
publishing tabulated values of measured cross sections, and this is
particularly the case amongst the earlier measurements. Where more
than one set of data is available for a particular target/scattering
process, we have also attempted to provide what we consider to be the
best “recommended ”cross section. This, of course, is a risky task
which is fraught with issues, not the least of which may be perceptions
of bias —we have tried our best to minimize any such perceptions and
hopefully give a clear explanation of any rationale that has been used
in selecting recommended values. Although we do not provide
tabulated values of theoretical calculations of positron scattering cross
sections, we do discuss and compare experiment and theory where it
is possible for a given target, and we often use theory in guiding our
determination of a “recommended ”cross section.
The recommended cross sections are presented as smooth
curves in the figures, with error estimates also provided as smooth
curves, and the corresponding absolute values for each atom are given
in tables in each section. It is hoped that in this fashion the data can be
useful for any modeling applications that require positron cross
sections or as a ready reference for new theory or experiment, with the
latter hopefully further re fining the “recommended ”sets.
This article is organized as follows: In Secs. 2and3, we give a brief
overview of the experimental and theoretical approaches, re-
spectively. Section 4provides data and evaluation for positron
scattering cross sections from atomic systems and these are presented
in tabular form, with an accompanying figure. Finally, we provide an
extensive list of references at the end of this paper.
2. Experimental Approaches
It is not our intention in this article to extensively review the
nature of the cross section measurements or the experimental ap-
paratus and techniques that have been used over the past (almost) 50
years to investigate positron interactions with atoms and molecules.
That has been done, and done well, in a number of previous reviewarticles4,11 –14and other major articles and books in the field.15–17
However, a brief summary of the various techniques that have been
used to measure the processes discussed in this article —total, posi-
tronium, ionization, and inelastic scattering cross sections —is rele-
vant, as most techniques have both advantages and drawbacks, and
these can be useful to keep in mind when assessing data for a
“recommended ”cross section. We will not discuss the rich collection
of work on positron sources, moderators, and detection schemes, but
we again refer the reader to previous studies (e.g., Refs. 15and16).
2.1. Total scattering
By far the most prevalent quantity measured for positron
scattering is the Total Cross Section (TCS), sometimes also called the
Grand Total Cross Section (GTCS), and it is a measure of the total
probability of scattering, irrespective of the process, energy loss, or
scattering angle. It is an important quantity as generally it can be
measured with high accuracy and often provides a “first point-of-
contact ”between experiment and theory.
The vast majority of total scattering measurements use the so-
called attenuation technique, where the attenuation (loss) of positrons
from a beam as it traverses a scattering cell containing the gas of
interest is measured. The Beer –Lambert law is then commonly used to
extract the TCS from the measured attenuation fraction, the length of
the scattering cell used (L), and the number density of the gas under
study (N). The TCS, usually labeled Q T, is given by
QT/equalsln/parenleftBiggI0
It/parenrightBigg1
NL,
where I0andItare the transmitted positron fluxes, with no gas in the
cell and with gas, respectively.
Recent applications of this technique have produced accurate
cross section measurements with absolute uncertainties as low as 3%.
However, there are a number of drawbacks to the attenuation
technique that need to be considered when assessing data, with
perhaps the most important of these relating to the effects of forward
scattering on the measurements. These effects arise because the ex-
periments are gas-dynamic, with the target gas (and positrons)
flowing into and out of the scattering cell through entrance and exit
apertures. The finite size of the exit aperture, in particular, means that
some forward scattered positrons will always be present in the
measured quantity IT, and as a consequence, this can result in a
measured cross section which is lower than the “real”value. We will
not discuss this particular issue further as it has been the subject of
much recent analysis and discussion [e.g., Ref. 20], but it is important
to note that it is thought to be one of the major reasons for some of the
significant discrepancies that exist amongst literature values for total
scattering cross sections. We do note that this effect can be a particular
problem for target atoms and molecules which have large dipole
polarizabilities and/or dipole moments for molecules, as this generally
translates into strong forward scattering.
2.2. Positronium formation
Positronium (Ps) formation is perhaps the major inelastic
process in low to intermediate energy (0 –100 eV) positron scattering
from most targets. It results in the loss of a positron from the incident
beam and the production of a positive ion and either two or three
J. Phys. Chem. Ref. Data 48,023102 (2019); doi: 10.1063/1.5089638 48,023102-4
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Chemical Reference DataARTICLE scitation.org/journal/jprgamma rays depending on the total spin of the positronium complex
before it annihilates. Given the range of reaction products, there are
also a range of techniques that have been used to measure, or estimate,
the Ps formation cross section. In summary, these are as follows:
Measuring the loss of positrons from the incident positron beam.
Coincident detection of the two or three gamma rays that result
from the annihilation of para-a n d ortho -positronium, respectively.
Techniques which measure both the total ionization cross section
(that is, direct ionization plus Ps formation) and the direct
ionization cross section in order to unravel the Ps formation
cross section.
These techniques have had varying degrees of success and accuracy,
although the best contemporary measurements typically have ab-
solute uncertainties of around 5%.
2.3. Inelastic scattering
There are relatively few measurements of inelastic scattering
cross sections following positron excitation, with the majority being
either the result of time-of- flight (ToF) experiments or, more recently,
experiments utilizing trap-based beams in high magnetic fields.
In the ToF experiments [e.g., Refs. 21and22), a pulsed positron
beam is used and inelastically scattered positrons are separated
temporally from those scattered elastically. Not surprisingly, these
experiments were particularly challenging, with low fluxes and dif-
ficult absolute normalization.
On the other hand, trap-based experiments have provided a
direct means to measure absolute, integral inelastic cross sections formany processes, including vibrational and electronic excitation. By
manipulating the magnetic field strengths between the scattering and
energy-analyzing regions in these experiments, inelastic processes can
be separated from elastic scattering, allowing the determination of
cross sections using the Beer –Lambert law.
23
2.4. Direct ionization
Given that there are two mechanisms which can lead to ioni-
zation by positron impact, positronium formation and direct ioni-
zation, techniques for measuring the direct ionization component
must effectively separate these two mechanisms.
Early measurements of direct ionization also used ToF tech-
niques to temporally separate positrons that had lost energy in an
ionization event (e.g., Ref. 24). Subsequent experiments have used
more sophisticated coincidence techniques, where scattered positrons
and positive ions are detected in coincidence (e.g., Ref. 25). Buffer gas
trap experiments have also served to improve the accuracy of direct
ionization measurements.23A comprehensive review of ionization
techniques and cross sections was given recently in Ref. 17and also
discussed in Ref. 11.
3. Overview of Theoretical Methods
3.1. Introduction
Theoretical approaches in positron scattering by atoms and
molecules have seen much progress since the early calculations by
Massey and co-workers (e.g., Refs. 26and27). However, even in the
simplest case of the positron-hydrogen atom scattering system, theearly theoretical methods were unable to treat the positronium
formation (Ps) channel, except by variational methods (Ref. 28and
references therein) which were limited for energies below the Ps
formation and ionization threshold.
The early calculations such as the close-coupling (CC) approaches
used the same computational codes as for the electron-atom case with a
simple change in sign for the positron case and the polarization po-
tential as well as ignoring exchange. However, these calculations
neglected the rearrangement channels for Ps formation. In the positron
case, the positron –electron correlations in the form of virtual and real
Ps formation require a much more complicated description to obtain
accurate results for various scattering parameters.
In the last thirty years, there has been tremendous advancement
of theoretical studies for positron-atom scattering, particularly in the
inclusion of the Ps effects correctly. Coupled with the emergence of
cheap and powerful computing resources, the tractability of various
positron scattering from the simplest H atom to larger inert atoms has
seen much success!
For much of the earlier and present state of theoretical methods
on positron-atom scattering, there is a wealth of information from a
number of previous reviews.11,13,15,16,29,30In particular, the recent
review by Kadyrov and Bray30gives a detailed overview of the
state-of-the-art in theoretical development. In the case of positron-
molecule scattering, the following reviews provide useful and current
information: Refs. 11,15,16, and 31–34.
This present theoretical overview will brie fly focus on these
advances and the state-of-the-art theoretical methods of the last
twenty years.
3.2. CC methods
CC or the coupled-channels method and its variants such as the
highly effective convergent close-coupling (CCC) and CC with
pseudostates methods are considered the most successful theoreticaltechniques to study positron scattering from atoms, especially
hydrogenic-type atoms at low to intermediate energies.
As noted above, the early idea of extending the basic single-
center CC formalism to the positron case, which only considered
changing the sign of the incident particle, was valid for energies wherePs formation is insigni ficant. Here, the CC method expands the total
wavefunction Ψ(r
1,r2)into an in finite number of orthogonal
eigenstates of the target atom ψα(r2), that is,
Ψ(r1,r2)/equals/C229
αFα(r1)ψα(r2), (1)
where r1and r2are the coordinates of the scattered positron and
atomic electron, respectively. The eigenstates have unknown scat-
tering coef ficients Fα(r1)which can, in principle, be obtained by
solving a set of coupled integro-differential equations.
However, at low and intermediate energies, the Ps formation
channel plays a signi ficant role in the scattering dynamics. Since the
late 1980s, the CC methods have been able to treat the Ps formationchannels for positron-atom scattering.
35–39
In the two-center CC formalism, the total wavefunction of the
positron-atom collision system can be expanded in terms of the
orthogonal eigenstates of the target atom ψαand the Ps state φβwith
the corresponding unknown scattering coef ficients FαandGβ
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αFα(r1)ψα(r2)+/C229
βGβ(R)φβ(s), (2)
where Ris the center of mass of the outgoing Ps atom and sis the
relative coordinate, while αandβrepresent the channels in the atom
and Ps, respectively. These calculations were denoted by the CC(m,n)
notation, where m is the number of atomic states in the expansion and
n is the number of Ps states used.
The challenge for the CC methods is in incorporating the
maximum number of physical channels that can be included but to
avoid weak convergence as the continuum channels are neglected.
Eventually, these neglected effects were addressed by the development
of CCC and to some extent earlier by the use of pseudostates and
optical potential approaches.
3.2.1. CC or coupled-channel calculations
Traditionally, CC methods and its variants have been extensively
used to study the electron (or positron) scattering on atoms.40–43
Among these older calculations, Ward et al.41–43used a 2-state, 4-
state, and 5-state CC (CC2, CC4, CC5) on positron scattering from Li,
Na, and K. McEachran et al.44had also reported a 5-state CC cal-
culation for positron scattering from Rb. We must also highlight a
multi-pseudostate CC work by Walters45who used the 1s, 2s, 2p
physical and 6 pseudostates of Fon et al.46to report positron scat-
tering by H atom at intermediate energies.
In parallel, we witnessed the first set of two-center CC calcu-
lations by Hewitt et al.47,48on Ps formation in positron-hydrogen
scattering. They were the first to demonstrate a realistic CC calcu-
lation with the inclusion of the Ps channels in the eigenfunction of the
total wavefunction. In this context, some early pioneering two-center
CC calculation studies of Basu et al. ,49,50Wakid and Labahn,51and
Abdel Raouf et al.52must be mentioned.
Subsequently, Mitroy37,53implemented the CC in momentum
space [denoted by CC(m,n), m is the number of physical and
pseudostates for the atomic channels and n is the number of physical
and pseudostates to represent the Ps channel] to obtain converged
cross sections for various physical parameters in the positron-H
system. Later, Mitroy and co-workers had also extended the
CC(m,n) to study positron-sodium scattering.54,55Unlike the re-
strictive number of channels used in the earlier studies,47–52the
CC(m,n) method allows for larger basis-state (using a L2formalism)
calculations such as the 31-state CC(28,3) work of Mitroy53for
positron-H atom scattering. The corresponding work for the R-
matrix approach will be discussed later.
3.2.2. CCC method
The CCC method is considered one of the most effective
methods in dealing with the issues of convergence and handling of the
neglected continuum states in the CC methods. It was developed by
Bray and Stelbovics56for handling the formidable electron-hydrogen
atom system. Essentially, the CCC uses square integrable ( L2) states
which allow for a large number of physical and continuum channels to
be used with ease in the eigenfunction expansion of the wavefunction.
These eigenstates were obtained by diagonalizing the target Hamil-
tonian in a large Laguerre or also Sturmian basis.Thefirst single-center CCC calculation on positron-hydrogen
atom was reported by Bray and Stelbovics in Refs. 57and58. Other
single-center CCC calculations have been comprehensively detailed
in the study of Kadyrov and Bray30and will not be mentioned here.
Eventually, Kadyrov and Bray39,59reported a two-center CCC
implementation in positron-hydrogen atom scattering. Using the
method of Mitroy,37they extended the CCC formalism of Bray and
Stelbovics56to calculate the total, elastic, break-up, ionization, and Ps
formation cross sections in the S-wave model.
Other two-center CCC studies include positron scattering by
helium,60lithium,61sodium,62magnesium,63and H 2.64Several
physical parameters such as TCS and the differential cross section
(DCS) for positron scattering by neon, argon, xenon, and krypton
were calculated using the single center CCC.65–67
3.2.3. Coupled-channel optical (CCO) methods
During the period spanning the 1960s –1990s, optical potential
methods had been useful to treat the neglected discrete or continuum
channels in a practically tractable calculation in electron-atom
physics. Its utility was seen by a number of researchers (McCar-
thy, Saha, and Stelbovics68and references therein).
The CCO ’s potential is derived from the Schr¨ odinger equation
using the Feshbach formalism.69Here, the reaction space is separated
into 2 spaces, P space and Q space. The P space consists of atomic states,
whereas the Q space consists of continuum and remaining discrete
states. The coupled-channel optical method (CCOM) of McCarthy and
Stelbovics70used ab initio complex-polarization potentials for the
continuum effects, and the remaining signi ficant discrete channels were
treated by second-order polarization potentials. Based on its success in e-
H systems (McCarthy and Stelbovics),70a simple extension was
implemented by Bransden et al.71for the positron-H system. Never-
theless, to be an effective method to treat the Ps formation, the optical
potential must also include the neglected Ps formation. McCarthy,Ratnavelu, and Zhou
72and McCarthy and Zhou73developed an
equivalent optical potential to allo w for these Ps formation channels.
In the late 1990s, Ratnavelu and Rajagopal74demonstrated an
optical potential method [CCO(m,n)], within the CC two-center
formalism of Mitroy,37that allowed for the continuum optical po-
tentials in the positron-atom channels. Using a small basis calculation
[CC(3,3) and CCO(3,3)], they reported ionization cross sections, Ps
formation cross sections, and TCSs that were in good qualitative and
reasonable quantitative agreement with the 31-state CC calculations
of Mitroy75and the 33-state R-matrix calculations of Kernoghan
et al.76Various implementations of the CCO(m,n) for positron-
hydrogenic atoms were also reported.77–82
In parallel, Zhou, McCarthy, and Ratnavelu83developed the
CCOM with a complex equivalent local potential, which treated the
neglected atomic states and allowed for the Ps formation channels.
In a series of calculations, Zhou and co-workers reported the CCOM
for positron-alkali as well as positron-helium and positron-
magnesium scattering.84–86
3.3.R-matrix
One of the techniques used in theoretical studies of atomic,
molecular, and nuclear processes is the R-matrix theory.87,88This
method was originally used to study the electron-atom collision
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of the R-matrix and its applications, the reader is referred to Ref. 89.
In the R-matrix approach, the con figuration space of the physical
system under study is divided into several parts and the system issolved separately in each of these domains. The wavefunction of the
scattering system is represented by two parts —the internal and the
external wavefunctions. The matching of these functions at the in-
ternal edge would give us the physical solutions ’that is needed to
generate the K-matrix.
90
Thefirst realistic R-matrix calculation that allowed for the Ps
channels was reported by Higgins et al.90in a study of positron-
hydrogen scattering. They used the intermediate energy R-matrix
(IERM) method with L2basis terms. Details of the development and
implementation of the continuum Ps channels in the expansion of the
total wavefunction were reported by Higgins and Burke.36,91These
allowed for overcoming convergence issues as well as to calculate the
Ps(1s) cross sections. Further work by Walters and co-workers had
extended this method to the positron-hydrogen, positron-alkali atom,
and positron-helium scattering systems.38,76,92 –95
A hybrid R-matrix96method for electron-impact ionization of
atoms and ions was also extended to positron impact ionization of
heavy noble gases.97This hybrid method used a first-order distorted
wave (DW) to represent the incident positron and the initial bound
state, and the physics of the residual ion and ejected electron was
treated by an R-matrix approach.
3.4. Relativistic optical potential calculations
Even with the advent of highly sophisticated CCC and CC
calculations, the role played by various perturbative methods in
positron scattering by atoms in recent years particularly in positron
scattering of inert gases is very signi ficant.65–67,98 –100
Chen et al.98proposed a relativistic optical potential (ROP)
method to study elastic electron and positron scattering from noble
gases. They derived a non-local ab initio absorption potential within
the Dirac relativistic formalism. Their imaginary part of the complex
optical potential allowed for the fluxes of the neglected inelastic
channels as well as the continuum channels. The earlier model used by
Bartschat et al.101,102had studied it in the non-relativistic formalism
and did not allow for the continuum channels.
Following Chen et al.,98the optical potential part of the coupled
equations can be written as
Uopt(x)/parenleftBiggF0(r)
G0(r)/parenrightBigg/equals/bracketleftBigUR
opt(r)−iUI
opt(r)/bracketrightBig/parenleftBiggF0(r)
G0(r)/parenrightBigg,
where F0(x)G0(x) are the elastic scattering functions, and UR
opt(r)is
the real part and UI
opt(r)is the imaginary part of the potential. The
real part of the optical potential is approximated by the local po-
larization potential based on the polarized orbital potential ofMcEachran and Stauffer.
103The polarization multipoles ( ν/equals0–7)
and dynamic distortion terms (up to 6 terms) as in the study of
McEachran and Stauffer104were used. The imaginary optical po-
tential contribution was handled using a Hulthen –Kohn prescription
that treats the complex part as a perturbation to reduce the tedious
iterative process that is otherwise needed.
Jones et al.65used the ROP in the study of positron scattering
from Ne and Ar to calculate the GTCS for Ne below the Ps threshold
and above the threshold. Their work was comparable with othertheories reported. In the Ar case, the ROP ’s GTCS showed a poorer
agreement with the experimental measurements. This was also re-
flected by other theories. The Ps formation cross sections also showed
poor agreement.
Machacek et al.66had reported the ROP calculations for low
energy calculations of positron scattering by xenon. We should note
that the ROP and the CCC did not allow for the two-center treatment
for handling the positron-atom scattering and were not able to de-
scribe the physics of the scattering at the Ps formation threshold such
as the Wigner cusps. The ROP work in the positron-Kr process also
did not show any improved results.67
In 2013, McEachran and Stauffer100reported an imple-
mentation of the ROP that allowed for the Ps formation in the
absorption channel following the procedures of Reid and Wahe-
dra.105,106The Ps formation cross sections for Ne, Ar, Kr, and Xe
were calculated. These cross sections gave better results than other
previous theoretical methods.
3.5. Other optical-model potential, Born
and distorted-wave methods
There have been other optical potential approaches that were
used to study positron scattering from atoms, such as the work of
Gianturco and Melissa.107They reported Ps formation cross sections
for positron scattering from Li, Na, and K. Their method used a global
modeling technique for the polarization potential, a generalized
damping function for the short-range effects, and a dispersion re-
lation for the absorption potential within a Feshbach formalism.
Reid and Wahedra105employed the parameter-free model
potentials to study positron-K and positron-Rb scattering. Their
method incorporated the absorption potential based on a quasi-freemodel of Reid and Wahedra
106and showed reasonable agreement
with the experimental TCS data.
Another optical potential method is due to Garcia and co-
workers (e.g., Ref. 108), where they implemented a version of the
quasi-free absorption potential109for positron scattering by using the
Reid and Wahedra prescription.106,110Furthermore, they proposed
anab initio absorption potential. In this approach, they derived the
potential of the excited bound states and continuum in a Dirac –Fock
formalism (Ref. 98and references therein). In their calculation for
Ar,108a total of 17 bound states and 36 continuum channels were
incorporated together with the inner-shell ionization. Perhaps the
most important of this groups ’work is that within the independent
atom method (IAM) and their screening-corrected additivity-rule
(SCAR) plus interference (I) terms approach (e.g., Refs. 111and112),
where their positron-atom optical model can be applied to molecular
systems. Indeed, as shown in our companion paper to this review,19
the IAM-SCAR+I approach to positron-molecule scattering has beenrelatively successful in giving a semi-quantitative description of these
scattering systems.
Recently, Bhatia
113had proposed a hybrid theory to calculate
accurate phase shifts, annihilation cross sections, and Ps formation
cross sections for positron-H scattering at energies below the ioni-
zation threshold. His calculated phase shifts provide lower bounds to
exact phase shifts.
There have been other theoretical methods that should be
mentioned, for completeness. Gien114 –118had used the modi fied
Glauber (MG) approximation in the model potential approach to
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allowed for the inclusion of core-exchange effects which simpli fied the
calculation of electron or positron scattering from hydrogenic atoms.
Other DW methods have been used extensively for positron-
atom scattering in the late 1980s (Ref. 119and references therein).
Pangantiwar and Srivastava120had applied the DW method to
positron-rubidium scattering. We also note the first Born approxi-
mation (FBA) and distorted wave Born approximation (DWBA)
calculations of Nahar and Wahedra.121,122They reported DCS and
integral cross sections (ICS) for Ps formation from Li and Na at
energies between 100 and 300 eV using both the FBA and DWBA
methods. Their work on elastic scattering of positrons from Ar atoms
at 3–300 eV needed model potentials for the lower partial-waves and
Born approximations. Their reported DCS at 100 –300 eV showed
limited agreement with normalized experimental data.122
Leet al.123implemented the hyperspherical close-coupling
(HSCC) method for the positron-Li and positron-Na scattering
systems. They extended the HSCC work on ion-atom scattering124
and considered the hyperspherical radius of the collision adiabaticallyfollowing the Born –Oppenheimer prescription. They also in-
corporated the positronic bound state effects using model potentials
as in the work of Ryzkhih et al.
125
Campeanu et al.126used the DW method to calculate the
ionization cross section for positron-H and positron-noble gas atom
scattering. They used the Coulomb plus plane waves with full energy
range (CPE) method, and the distorted CPE (DCPE) version to
calculate the scattering T-matrix. In particular, the DCPE4 model of
Campeanu et al.127gave results that looked quite promising. A newer
model DCPE5 was later proposed in 2002.128
3.6. Many-body theory (MBT) calculations
Green et al.129used the MBT framework, based on the Dyson
equation, to study positron scattering and annihilation by inert gasesbelow the Ps formation threshold. Details of the MBT formalism can
be found in the work of Green et al. and its associated references. In
particular, the MBT allowed for the electron-electron and electron-
positron correlations to be calculated via perturbative techniques (via
the Feynman diagrams). Additionally, the virtual Ps formation was
incorporated using the prescription of Gribakin and King.
130
3.7. Variational calculations
Variational techniques were employed by Hulthen131and
Kohn132to evaluate scattering phase shifts and were extensively used
in bound-state problems. In the 1960s, Schwartz133and Armstead134
had reported elaborate variational calculations on elastic positron-hydrogen scattering. Due to issues such as the non-boundedness of the
phase shifts at non-zero energies, this led to further work by others.
Bhatia et al.
135,136had applied the lower bound formalism of Gai-
litis137to obtain rigorous lower bound calculations of s- and p-wave
phase shifts for the positron-H case. These are considered to be exact.
Stein and Sternlicht138used the Kohn and Hulthen method to study
positron-H rearrangement collisions by extending it beyond the Ps
formation threshold. Humberston and co-workers139 –142and
Houston and Drachman143also reported accurate phase-shifts for s-,
p-, and d-waves, as well as the corresponding cross sections. Another
work by Humberston et al.144reported the “round cusp ”in the s-wavescattering cross section at threshold, in accord with Wigner ’st h r e s h o l d
theory.
There were some highly sophisticated variational calculations by
Humberston and van Reeth that studied positron scattering by heliumand hydrogen
145,146in the low-energy region. In the positron-
helium case, the variational K-matrix was calculated to energies
below the first excitation threshold. An accurate form of the helium
wavefunction, together with trial functions, were utilized with three
variants of the Kohn variational method being reported —Kohn,
inverse Kohn, and complex Kohn. These trial functions would allow
for the short-range effects. This work is considered as an important
benchmark below the Ore gap for positron –He interactions.
In the positron-H case, accurate cross sections were also re-
ported for the elastic scattering and Ps formation cross sections. These
calculations, which used elaborate trial functions, showed interesting
threshold structures due to the coupling between the Ps channels and
the elastic channel. The s-wave Wigner cusp was also observed in their
work.
4. Recommended Cross Sections for Atomic Species
4.1. Atomic hydrogen (H)
Atomic hydrogen (H) is a notoriously dif ficult target to prepare
for accurate quantitative scattering measurements in the laboratory.
To the best of our knowledge, there have only been a few experimental
determinations of absolute cross sections for positron scattering by
atomic hydrogen, and these include measurements of the total
scattering cross section,147 –149the positronium formation cross
section,148,150 –152and the direct ionization cross section (which
does not include Ps formation).151 –154
TABLE 1. The TCS (in units of 10−16cm2) for positron scattering from atomic hydrogen.
The estimated uncertainty is ±20% (see also Fig. 1 )
E0(eV) Recommended TCS ( 310−16cm2)
1.0 1.97
2.0 1.11
3.0 0.93
4.0 0.90
5.0 0.91
6.0 0.97
7.0 1.26
8.0 2.08
9.0 2.82
10.0 3.36
11.0 3.77
13.0 4.34
16.0 5.02
21.0 4.83
31.0 4.04
51.0 3.00
76.0 2.32
101.0 1.90
151.0 1.45
201.0 1.23
301.0 1.02
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The total scattering cross section measurements for H have been
done exclusively by the Wayne State group with their most recent
efforts,148,149representing their final, updated cross section. These
measurements were carried out using a gas cell and a molecular
hydrogen (Slevin) discharge as the source of the atomic target, and the
Beer–Lambert law was used in an otherwise conventional attenuation
experiment approach. The absolute normalization of the cross section
at a given energy was achieved by using the TCS for H 2at the same
energy, together with a range of other measured experimentalparameters. While there are no other experimental values with which
to compare, when compared (see, e.g., Ref. 16) with several state-of-
the-art theoretical approaches,56,75,95,142,155the agreement be-
tween experiment and theory is excellent at energies above about 8 eV.
The Wayne State group discusses possible forward scattering effects
in their measured cross sections and provides estimates of the extent
TABLE 2. Positronium formation cross section (in units of 10−16cm2) for atomic
hydrogen. The estimated uncertainty is ±30% (see also Fig. 2 )
E0(eV)Recommended positronium formation
cross section ( 310−16cm2)
7.0 0.568
8.0 1.08
9.0 1.68
10 1.94
11 2.36
12 2.77
13 2.93
16 2.93
18 2.70
20 2.45
25 1.94
30 1.48
40 0.91
50 0.56
75 0.14
100 0.035
FIG. 1. The recommended total scattering cross section for H (solid line), while the
dashed lines represent the estimated uncertainty limits of ±20% (see also Table 1 ).
FIG. 2. The recommended positronium formation cross section for H (solid line). The
dashed lines represent the estimated uncertainty limits of ±30% (see also Table 2 ).
TABLE 3. The direct ionization cross section (in units of 10−16cm2) for positron impact
on atomic hydrogen. The estimated uncertainty on these values is ±25% (see also
Fig. 3 )
E0(eV)Recommended direct ionization cross
section ( 310−16cm2)
13.6 0
15 0.07
20 0.23
25 0.38
30 0.55
35 0.68
40 0.75
50 0.85
60 0.88
70 0.84
80 0.80
100 0.71
125 0.61
150 0.50
175 0.41
200 0.36
300 0.27
400 0.23
500 0.20
700 0.16
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Chemical Reference DataARTICLE scitation.org/journal/jprthat these may affect the measured cross sections. We are of the view
that their low energy data, below 10 eV, considerably underestimate
the true cross section due to these effects. As a consequence, our
recommended cross section values at these lower energies, drawn
largely from theory, are signi ficantly higher than the measured ex-
perimental values. The recommended cross sections are given in
Table 1 and shown in Fig. 1 . We estimate that the uncertainty in these
cross section values is around ±20%, particularly at the lower energies.
4.1.2. Positronium formation
A variety of experimental techniques have been used to de-
termine the positronium formation cross section for atomichydrogen. A number of experiments in the Brookhaven –Bielefeld
collaboration were carried out during the 1990s,150 –152with final
values for the Ps formation cross section being provided by Ref. 152.
They used a crossed beam con figuration and ion detection scheme to
derive both Ps formation and impact ionization cross sections with
absolute normalization being provided via concurrent electron
ionization measurements which were normalised to earlier literature
values.156
A different range of techniques was employed by the Wayne
State group148to obtain the Ps formation cross sections. They
measured both annihilation gamma rays and the loss of transmitted
positrons in their scattering cell, in order to estimate the upper and
lower limits on the Ps formation cross section, respectively. The
absolute normalization relies implicitly on measurements of total
scattering for H and total and Ps formation for H 2(see the original
paper by Hoffman et al.284for details).
The Ps formation cross sections from these two groups provide a
challenge when assessing a recommended cross section. The earlier
results150,151favor a cross section with a peak amplitude around or
above 3 ˚A2, and the results of Ref. 151are largely in good agreement
with the later work from Wayne State group.148However, the more
recent result of the Bielefeld –Brookhaven collaboration,152which they
claim is an improved measurement to that of Ref. 151,i n d i c a t e sac r o s s
section with a lower peak magnitude —around 2 ˚A2. We can seek some
guidance in this case from theory where there are now many reasonably
reliable calculations of Ps formation. The majority of these predict a
cross section with a peak maximum of around 3 ˚A2, so we are inclined to
favor the data of Refs. 148and151, with the important caveat of a
conservative uncertainty estimate of ±30% on the recommended cross
sections. These values are tabulated in Table 2 and shown in Fig. 2 .
4.1.3. Direct ionization
For positron impact ionization, the results of Ref. 152 were
intended to supersede those of Refs. 151and153from the same group/
collaboration. These later results from the Bielefeld/Brookhaven col-
laboration are in good agreement with the results of the University
FIG. 3. The recommended direct ionization cross section for positron impact on H
(solid line). The dashed lines represent the estimated uncertainty limits of ±25%
(see also Table 3 ).
TABLE 4. The TCS (in units of 10−16cm2) for positron scattering from helium. The absolute error is estimated to be ±10% (see
also Fig. 4 )
Energy (eV) Recommended TCS ( 310−16cm2) Energy (eV) Recommended TCS ( 310−16cm2)
0.10 0.38 6.0 0.127
0.20 0.29 7.0 0.139
0.30 0.23 8.0 0.150
0.40 0.185 9.0 0.160
0.50 0.155 10 0.168
0.60 0.133 15 0.196
0.70 0.115 20 0.275
0.80 0.102 30 0.721
0.90 0.092 40 1.03
1.0 0.083 50 1.14
1.5 0.060 60 1.18
2.0 0.058 70 1.19
3.0 0.078 80 1.17
4.0 0.097 90 1.13
5.0 0.113 100 1.07
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Chemical Reference DataARTICLE scitation.org/journal/jprCollege London (UCL) group,154which were undertaken primarily to
validate the earlier measurements of Ref. 153, which were considerably
larger than most contemporary theoretical calculations of the ionizationprocess. Given the good agreement between the results of Refs. 152and
154, and between these results and contemporary theory,
58,75,76our
recommended cross section is largely based around these data. The
cross sections are tabulated in Table 3 and shown in Fig. 3 , with an
estimated uncertainty of ±25%.
4.2. Helium (He)
In rather stark contrast to atomic hydrogen, helium (He) has
perhaps been studied more than any other atomic system by lowand intermediate energy positron scattering. An example of this is
the more than 20 separate measurements of the total scattering
cross section for He157 –179spanning the period from the early
1970s until the present. For th e positronium formation cross
section, there are fewer independent measurements25,177,180 –186
and fewer still for electronic excitation22,187 –190and direct ion-
ization.25,189,191 –194The various cross section determinations for
total scattering, positronium formation, and direct ionization have
recently been assessed by Chiari and Zecca,11w h oa l s op r o p o s e d
“recommended cross sections ”for these three processes, and
w ew i l ld i s c u s si nd e t a i lo nt h e i ra s s e s s m e n t si nt h ef o l l o w i n g
sections (Secs. 4.2.1 –4.2.4 ).
FIG. 4. The recommended total positron scattering cross section for He (solid line),
while the dashed lines represent the estimated uncertainty limits of ±10% (see also
Table 4 ).
TABLE 5. The positronium formation cross section (in units of 10−16cm2) for helium. The absolute error is estimated to be
±15% (see also Fig. 5 )
E (eV)Recommended positronium formation
cross section ( 310−16cm2) E (eV)Recommended positronium formation
cross section ( 310−16cm2)
17.8 0 35 0.420
18.0 0.010 40 0.445
19.0 0.035 45 0.445
20 0.068 50 0.420
21 0.110 55 0.380
22 0.143 60 0.335
23 0.180 70 0.265
24 0.211 80 0.205
25 0.243 90 0.155
26 0.272 100 0.115
27 0.301 150 0.030
28 0.320
29 0.345
30 0.365
FIG. 5. The recommended total positronium formation cross section for He (solid
line). The dashed lines represent the estimated uncertainty limits of ±15% (see also
Table 5 ).
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Chemical Reference DataARTICLE scitation.org/journal/jpr4.2.1. Total scattering
Absolute total scattering measurements for positron interactions
with helium have been measured exte nsively since the 1970s, with the
bulk of measurements being complet ed before the turn of this century.
Comparisons of the various measurements can be found in a number of
recent papers [e.g., Refs. 11and176–179] and we will not repeat those
here. We also note the recent recommended TCS of Chiari and Zecca11
which they obtained by averaging a number of the results from morerecent determinations of the TCS, whilst ruling out some others that were
either too high or too low in magnitu de. In our view, another reasonable
gauge of the appropriate magnitude of the cross section, particularly at
energies below the Ps threshold at 17.8 eV, are the recent state-of-the-art
theoretical calculations (e.g., Refs. 146and195–198) which have been
shown to agree extremely well both amongst themselves and with the
most accurate measur ements (e.g., Refs. 174,176,a n d 179).
We do not see any need to greatly alter the recommended cross
section of Chiari and Zecca, with the possible exception of the low energy
(below 1 eV) values where we believe that the present theory is possibly
more accurate than the experiment —which is also limited to just a few
measurements in this energy region. We suggest therefore that the cross
section of Ref. 11should be about 5% higher at energies below about
1e V .O t h e r w i s e ,t h ev a l u e st h a tw er e c o m m e n da r et h o s ep r o p o s e db y
Chiari and Zecca. For completeness, we provide our full recommended
TCS in Table 4 and it is shown in Fig. 4 , where the error bounds, which
we conservatively assess to be ±10%, are also given. This is perhaps the
most accurately known positron scattering cross section —a benchmark.
4.2.2. Positronium formation
There have been a number of absolute measurements of the Ps
formation cross section.
25,177,180 –186At energies between the Ps
formation threshold (17.8 eV) and about 30 eV, the agreement between
the experimental values, particularly the most recent measure-
ments,177,186is excellent. At the peak in the cross section (35 –45 eV),
and for energies out to energies of about 100 eV, there are signi ficant
differences (30% –40%) between the various measured cross sections,
making the selection of a recommended cross section dif ficult.
However, we can also be guided, somewhat, in choosing a set of
recommended values by the weight of recent theoretical calculations
[e.g., Refs. 86,197,a n d 198] which tend to favor a lower energy, lower
magnitude peak cross section for the Ps formation channel. As a result
of these differences, our recommended cross section, which shows a
peak value of around 0.45 ˚A2at an energy in the region of 40 –45 eV,
has a conservatively estimated uncertainty of ±15%. These values are
given in Table 5 and shown in Fig. 5 .
4.2.3. Electronic excitation
There are only a few measurements of absolute cross sections for
electronic excitation of the helium atom by positron impact. These
include the earlier measurements of Coleman and colleagues,22,187
Sueoka and colleagues,188,189and the most recent data of Caradonna
et al.190These measurements are for the discrete excitation of the 21S
and 21P states of He and of the unresolved n /equals2 excitation. Caradonna
and co-workers also used their trap-based technique to measure the
total inelastic cross section for He which represents the sum of all
inelastic events, including ionization, but not including Ps formation.
The results of these investigations, including a comparison with pastand contemporary theory, are given in Ref. 190. The recommended
cross sections for the 21S and 21P states are given in Table 6 and are
illustrated in Fig. 6 and Fig. 7 , respectively. The estimated un-
certainties are ±25%.
4.2.4. Direct ionization
Direct ionization cross section measurements are available
from a number of experimental approaches —as discussed in Sec. 2.4.
The interplay of direct ionization, total ionization, and Ps formation
(which also leads to ionization) has also been used in some cases to
deduce either positronium formation or direct ionization crossTABLE 6. The cross section (in units of 10−16cm2) for positron impact excitation of the
21S and 21P states of He. The estimated uncertainty on these values is ±25% (see also
Figs. 6 and 7)
E0(eV)Recommended cross sections ( 310−16cm2)
21S21P
20.6 0 ...
21.0 0.003 ...
21.2 ... 0
22.0 0.011 0.0021
23.0 0.019 0.0066
24.0 0.027 0.0149
25.0 0.035 0.0232
26.0 0.042 0.0335
28.0 0.052 0.0522
30.0 0.058 0.0690
32 0.061 0.083334 0.062 0.094
36 0.060 0.103
38 0.057 0.109
40 0.054 0.112
FIG. 6. The recommended cross section for the excitation of He 21S (solid line). The
dashed lines represent the estimated uncertainty limits of ±25% (see also Table 6 ).
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Chemical Reference DataARTICLE scitation.org/journal/jprsections by subtraction of one or the other from the total ionization
measurements. The absolute direct ionization cross section for He has
been measured a number of times since the first investigations in the
mid 1980s,25,189,191 –194with the cross sections of Refs. 193and194
being renormalized by Ref. 186. The ionization cross sections have
been discussed extensively in the review articles of Laricchia and
colleagues17,26and by Chiari and Zecca,11the latter providing
recommended cross section values and uncertainties.The level of agreement between the various experimental cross
sections, and a number of theoretical approaches (see, e.g., Refs. 17and
146) is generally very good at energies from threshold up to 500 eV or
more, so there is no need for us to further adjust the recommended cross
section of Chiari and Zecca,11which we reproduce in Table 7 and show
inFig. 8 . The estimated uncertainties on these values are ±20%.
4.3. Lithium (Li)
4.3.1. Positronium formation
To the best of our knowledge, there is only one experimental
investigation of positron scattering from lithium (Li) and that is aTABLE 7. The direct ionization cross section (in units of 10−16cm2) for positron impact
on helium. The estimated uncertainty on these values is ±20% (see also Fig. 8 )
E0(eV)Recommended direct ionization cross
section ( 310−16cm2)
24.6 0
30 0.0215
40 0.124
50 0.255
60 0.369
70 0.450
80 0.500
90 0.528
100 0.540
150 0.506
200 0.446
300 0.351400 0.281
500 0.229
600 0.196
700 0.169
800 0.149
900 0.136
1000 0.119
FIG. 7. The recommended cross section for the excitation of He 21P (solid line). The
dashed lines represent the estimated uncertainty limits of ±25% (see also Table 6 ).
FIG. 8. The recommended direct ionization cross section for positron impact on He
(solid line). The dashed lines represent the estimated uncertainty limits of ±20%
(see also Table 7 ).
TABLE 8. The positronium formation cross section (in units of 10−16cm2) for lithium.
The absolute error is estimated to be ±25% (see also Fig. 9 )
E0(eV)Recommended positronium formation
cross section ( 310−16cm2)
0.1 19.6
0.2 28.7
0.3 34.2
0.5 39.2
0.8 41.9
1.0 42.1
1.5 41.4
2.0 38.7
3.0 33.3
4.0 27.0
5.0 20.8
7.5 12.5
10 8.2
15 3.5
20 1.8
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Chemical Reference DataARTICLE scitation.org/journal/jprmeasurement of the positronium for mation cross section by the Wayne
State group.199They measured what they consider to be a “lower limit ”
on the Ps formation cross section by detecting the yield of two-gamma-
ray coincidences arising from the d ecay of singlet positronium (see
Sec.2). Their measurements extend from 0.3 to 15.0 eV and we note that
they only quote statistical uncer tainties on the measurements. We
further note that with a direct ionization threshold of 5.39 eV, the Ps
formation channel for lithium is “open ”a t0e V .
We can also be guided in assessing a recommended cross
section by a signi ficant amount of theoretical activity for positron
scattering by lithium.48,61,80,93,123As a “one-electron atom ”
with a large dipole polarizability, which arises principally from theresonant 2s-2p transition, the lithium atom lends itself to a rea-
sonably accurate treatment by contemporary theoretical calcula-
tions, particularly CC approa ches. The most recent of these
approaches61is a CCC approach that also includes a two-center
e x p a n s i o ni nt h e final state, allowing, in principle, a more accurate
treatment of the Ps formation cross section as well as for other
scattering channels. A comparison of contemporary theory and the
experiment of Ref. 199can be found in Ref. 61. In contrast to many
other measurements of the Ps formation cross section, positronium
formation appears to be essentially exhausted by about 30 eV,
w h e r e a si nm a n yo t h e ra t o m sa n dm o l e c u l e s ,i tc a ns t i l lb es i g -
nificant above 50 –100 eV. The recommended cross section, based
FIG. 9. The recommended positronium formation cross section for Li (solid line). The
dashed lines represent the estimated uncertainty limits of ±25% (see also Table 8 ).
TABLE 9. The TCS (in units of 10−16cm2) for positron scattering from neon. The estimated uncertainty is ±10% (see also
Fig. 10 )
Energy (eV) Recommended TCS ( 310−16cm2) Energy (eV) Recommended TCS ( 310−16cm2)
0.25 0.274 7.0 0.752
0.30 0.229 8.0 0.784
0.40 0.180 9.0 0.809
0.50 0.164 10 0.831
0.60 0.155 15 1.04
0.70 0.156 20 1.40
0.80 0.161 30 1.71
0.90 0.170 40 1.87
1.0 0.184 50 1.90
1.5 0.265 60 1.94
2.0 0.329 70 1.95
3.0 0.466 80 1.95
4.0 0.569 90 1.95
5.0 0.651 100 1.91
6.0 0.710
FIG. 10. The recommended total positron scattering cross section for Ne (solid line),
while the dashed lines represent the estimated uncertainty limits of ±10% (see also
Table 9 ).
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Chemical Reference DataARTICLE scitation.org/journal/jpron both experiment and theory, is given in Table 8 and shown in
Fig. 9 . The estimated uncertainty is ±25%.
4.4. Neon (Ne)
There have been many studies of positron scattering from neon
(Ne), with measurements of the TCS,65,161,162,165,169 –173,200 –205
the positronium formation cross section,65,182,185,206 –209and the
direct ionization cross section26,189,191,192,208,210 –213being re-
ported. We also note several measurements207,208,214of the total
ionization cross section (direct ionization + positronium formation)
and a measurement of the direct double ionization cross section.215
There have also been a signi ficant number of theoretical calculations
of these various cross sections.65,97,100,126,129,197,216 –233
4.4.1. Total scattering
The total scattering measurements and calculations have been
discussed in some detail by Chiari and Zecca in their recent article.11
They also provided a recommended TCS based on what they per-ceived to be a reasonably good agreement amongst the bulk of the
(many) experimental measurements. We agree broadly with the
rationale they have proposed and also with the cross section they
recommend, and as there have not been further measurements since
this recommended data were published, we see no reason to add
further to this. There has, however, been an additional, and detailed,
MBT calculation by Gribakin and colleagues,
129which is also broadly
in agreement with the recommended cross section.
The recommended total positron scattering cross section for
neon is given in Table 9 and shown in Fig. 10 . The estimated un-
certainty on these cross section values is ±10%.
4.4.2. Positronium formation
There have been a number of measurements of positronium
formation in neon dating back to the early 1980s. The early re-
sults182,206appear to be superseded by higher quality results from the
past 15 years.65,208,209These results, and contemporary theory, werecompared and discussed by Chairi and Zecca in their review,11but they
did not assign a “recommended ”cross section for Ps formation in Ne.
The level of agreement between the three most recent measurements is
reasonably good across the whole energy range from threshold to 200
eV, although the best agreement is found in the near-threshold region.
The recommended positronium formation cross section for
neon is given in Table 10 and shown in Fig. 11 . The estimated un-
certainty on these cross section values is ±15%.
4.4.3. Direct ionization
The direct ionization cross section for neon has been reviewed in
the work of Laricchia et al.26and also recently assessed by Chiari and
Zecca,11but the latter chose not to provide a recommended cross
FIG. 11. The recommended positronium formation cross section for Ne (solid line).
The dashed lines represent the estimated uncertainty limits of ±15% (see also
Table 10 ).
TABLE 10. The positronium formation cross section (in units of 10−16cm2) for neon. The estimated uncertainty is ±15% (see
also Fig. 11 )
E (eV)Recommended positronium formation
cross section ( 310−16cm2) E (eV)Recommended positronium formation
cross section ( 310−16cm2)
14.76 0 40 0.45
15.0 0.09 50 0.38
16.0 0.17 60 0.33
17.0 0.23 70 0.27
18.0 0.28 80 0.23
20 0.38 90 0.20
22 0.44 100 0.17
24 0.47 125 0.10
26 0.49 150 0.055
28 0.50 175 0.018
30 0.50
32 0.49
35 0.48
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Chemical Reference DataARTICLE scitation.org/journal/jprsection, most likely because the spread in the available experimental
data is quite large, particularly in the vicinity of the cross section peak
at around 150 eV. On the other hand, the level of agreement between
the various experiments, and theory, between threshold (21.56 eV)and about 100 eV is reasonably good, the main exception to this being
the earliest result of Ref. 191, which is larger in magnitude than all
other results.
There are also several measurements of the total ionization cross
section, but rather than analyzing these, a recommended total ion-
ization cross section could be obtained by adding the Ps formation
and direct ionization cross sections.
The recommended direct ionization cross section for neon is
given in Table 11 and shown in Fig. 12 . The estimated uncertainty on
these cross section values is ±25%.
4.5. Sodium (Na)
Experimental measurements of positron scattering by sodium
(Na) are rather few, with the only processes studied being total
scattering
234,235and positronium formation,236,199and these
studies all emanated from the Wayne State group. There have,
however, been a number of theoretical calculations of positron-alkali
interactions (e.g., Refs. 41,48,54,95,123, and 237–239), and as was
the case with lithium, we can expect a reasonable level of accuracy
from these given the “one-electron ”nature of the target.
4.5.1. Total scattering
Total scattering measurements have been made in the energy
range from 3 to 102 eV234and 1 to 10 eV,235both experiments using
the attenuation method and the Beer –Lambert law to obtain absolute
cross sections. These authors discuss the potential effects of their
inability to discriminate between unscattered particles and forward
elastically scattered positrons, an effect which renders the measured
cross section lower than the true value (see, e.g., Ref. 20). These effects
were estimated to be as large as 40% at the lowest energy, reducing to
around 3% at 50 eV. Some effort was made41,48to calculate“effective ”TCSs using differential scattering cross sections from
theory to estimate the forward scattering correction. In general, the
agreement between the (adjusted) experimental values and calcula-
tions is reasonably good across the measured energy range. The
recommended total positron-sodium scattering cross section is
presented in Table 12 and shown in Fig. 13 . The estimated absolute
uncertainty on these values is 20%, which is possibly a little con-
servative at the higher energies.
4.5.2. Positronium formation
To the best of our knowledge, there have only been two mea-
surements of Ps formation for sodium, both by the Wayne State
group,236,199and these are for energies between 1.5 and 10 eV. These
are largely in agreement with each other, within experimental un-
certainty, and agree well with state-of-the-art theory for energies
greater than about 1 eV. However, the most recent experimental
determination199shows a completely different energy dependence toTABLE 11. The direct ionization cross section (in units of 10−16cm2) for positron impact
on neon. The estimated uncertainty on these values is ±25% (see also Fig. 12 )
E0(eV)Recommended direct ionization cross
section ( 310−16cm2)
21.6 0
25 0.042
30 0.113
40 0.275
50 0.40
75 0.65
100 0.77
125 0.80
150 0.79
200 0.75
300 0.67
500 0.53
750 0.39
1000 0.30
FIG. 12. The recommended direct ionization cross section for positron impact on Ne
(solid line). The dashed lines represent the estimated uncertainty limits of ±25%
(see also Table 11 ).
TABLE 12. The TCS (in units of 10−16cm2) for positron scattering from sodium. The
estimated uncertainty on these values is ±20% (see also Fig. 13 )
E0(eV) Recommended TCS ( 310−16cm2)
1.0 140
3.0 102
5.0 86
7.0 77
10 67
20 50
30 40
50 29
75 21
100 16
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Chemical Reference DataARTICLE scitation.org/journal/jprtheory below about 1 eV, with that experiment continuing to rise to a
value in excess of 80 ˚A2at 0.15 eV, while theory decreases in
magnitude at energies lower than 1 eV. Indeed, three independent
CC calculations show a maximum value of around 25 ˚A2at
1.5 eV.237,238,123This smaller, low energy cross section has also been
confirmed recently by a two-center, CCC calculation.239As a result,
we (cautiously) favor a smaller Ps formation cross section at low
energies, but also strongly suggest further experimental work is re-
quired in this energy range below about 3 eV. We also note that this
decreasing cross section at low energies is consistent with what is
observed in both experiment and theory for Li and K atoms.
The recommended Ps formation cross section for sodium is
given in Table 13 and shown in Fig. 14 , with the recommended
uncertainty on the cross section being 30%.
4.6. Magnesium (Mg)
There are only a few experimental measurements of positron
scattering from magnesium, which have been conducted by theWayne State group149,240,241and involved the measurement of the
total scattering cross section and the Ps formation cross section. To
the best of our knowledge, there are no measurements of the direct
ionization cross section. There have also been a number of theoretical
calculations which have provided comparison to the experimentalstudies.
63,85,242 –250
4.6.1. Total scattering
Total scattering measurements have been made in the energy
range from about 3 to 60 eV,149,240with the latter measurement
representing the final determination of this cross section by the
FIG. 13. The recommended TCS for positron scattering from Na (solid line). The
dashed lines represent the estimated uncertainty limits of ±20% (see also Table 12 ).
TABLE 13. The positronium formation cross section (in units of 10−16cm2) for sodium.
The estimated uncertainty on these values is ±30% (see also Fig. 14 )
E0(eV)Recommended positronium formation
cross section ( 310−16cm2)
0.15 25
0.50 30
1.0 36
1.5 39
2.0 40
3.0 37
5.0 28
10 15
20 5
FIG. 14. The recommended positronium formation cross section for Na (solid line).
The dashed lines represent the estimated uncertainty limits of ±30% (see also
Table 13 ).TABLE 14. The TCS (in units of 10−16cm2) for positron scattering from Mg. A
conservative estimate of the absolute error is ±20% (see also Fig. 15 )
E0(eV) Recommended TCS ( 310−16cm2)
0.01 265
0.05 391
0.1 971
0.15 1007
0.2 836
0.5 358
1 229
2 161.2
5 96.7
10 61.0
15 47.5
20 39.2
30 31.540 26.6
50 23.2
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Chemical Reference DataARTICLE scitation.org/journal/jprWayne State group. There have also been a number of theoretical
investigations, and indeed, one of the signi ficant and outstanding
issues, at least experimentally, is the prediction by theory of a very
large p-wave shape resonance in the elastic scattering cross section at
low energies. While there are some small differences in the position
and magnitude of this resonance, recent, accurate theoretical cal-
culations247 –250all agree as to the existence of this feature and, if
confirmed, it would represent one of the largest scattering resonances
in either electron or positron scattering —an interesting outcome
given the otherwise complete (detected) absence of positron scat-
tering resonances in most atomic and molecular scattering systems.Given this interest, the recommended TCS we provide is a
combination of both experiment and theory as we feel it is signi ficant
to highlight the existence of this resonance and its enormous, pre-
dicted magnitude. Hopefully, this will also provide stimulus forfurther experimentation.
The recommended cross section is listed in Table 14 and shown
inFig. 15 . That part of the cross section based on experiment and
theory is shown as the thick solid line, while that based on theory alone
(below 2 eV) is shown as the thick dashed line. The thin dashed lines
represent the estimated uncertainty at ±20%.
4.6.2. Positronium formation
There has only been one experimental measurement of the Ps
formation cross section for magnesium,
241and the authors claim this
to be a preliminary result. It actually comprises two measured cross
sections —an“upper level ”based on measurements of transmitted
positron intensities, and a “lower level ”estimate based on mea-
surements of decay of gamma rays. These differ in places by a factor of
three, and while there are several sophisticated theoretical calcula-
tions available for comparison,243,244,246,63they also show a sig-
nificant variation in the predicted cross section values. A comparison
of the experiment and theory can be found in the recent paper of
Utamuratov et al.63
Accordingly, we do not provide a “recommended ”cross section
for Ps formation in Mg and note that further experimental work
would be useful.
4.7. Argon (Ar)
Positron scattering from argon (Ar) has possibly received more
experimental and theoretical attention than any of the other heavy
rare gas atoms, no doubt due to the ready availability and use of argon
as a target gas. There have been a large number of total scattering cross
section measurements,65,161,162,165,171 –173,200 –202,204,251 –254as
well as measurements of the positronium formation cross
section,65,181,182,185,207 –209,236,255 –258electronic excitation,8and
FIG. 15. The recommended TCS for positron scattering from Mg. The solid line is
based on both experiment and theory, while the thick dashed lines are based ontheory alone (see text). The thin dashed lines represent the estimated uncertaintylimits of ±20% (see also Table 14 ).
TABLE 15. The TCS (in units of 10−16cm2) for positron scattering from argon. The estimated uncertainty is ±10% (see also
Fig. 16 )
Energy (eV) Recommended TCS ( 310−16cm2) Energy (eV) Recommended TCS ( 310−16cm2)
0.3 13.0 8 3.73
0.4 10.5 9 4.12
0.5 9.00 10 4.70
0.6 7.90 15 6.38
0.7 6.70 20 6.58
0.8 6.10 30 7.07
0.9 5.40 40 7.28
1.0 4.90 50 7.14
1.5 3.94 60 7.02
2 3.91 70 6.90
3 3.82 80 6.68
4 3.75 90 6.425 3.72 100 6.20
6 3.66
7 3.64
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Chemical Reference DataARTICLE scitation.org/journal/jprthe direct ionization cross section.189,191 –193,210,211,213There
have also been a considerable number of theoretical calculations
of these various proc-
esses.65,97,100,126,128,129,197,216,218,221 –227,229 –233,259 –265We
also note a previous cross section set for argon108which was de-
veloped to aid in the modeling of positron transport in argon, but
tabulated values were not presented.
4.7.1. Total scattering
Total scattering cross sections have been measured extensively,
and of all the rare gas atoms, the level of difference between the
measurements for argon is probably the greatest. This is particularly
the case at low energies, where there are differences in magnitude
between some of the measured cross sections of between 50% and100% at energies between 1 and 10 eV. It has been demonstrated that
much of this difference in magnitude could be due to the effects of
forward scattering.20
Chiari and Zecca11have recently reviewed the various TCS
measurements and have proposed a recommended cross section for
argon. We are largely in agreement with their assessment of the
available data, with the exception of the magnitude of the cross section
at the lowest energies. Below 1 eV, there are only a few reliable
measurements, but, more recently, accurate theoretical approaches
have emerged (e.g., Refs. 58and61) which predict a smaller cross
section at lower energies.
Thus our recommended TCS is identical to that of Chiari and Zecca
above 1 eV, but slightly lower in magnitude between 0.1 and 1.0 eV. The
recommended values are given in Table 15 and shown in Fig. 16 .T h e
estimated uncertainty on these cross section values is ±10%.
4.7.2. Positronium formation
The positronium formation cross section was also reviewed by
Chiari and Zecca, but they declined to propose a recommended cross
section for this process in argon. With a few possible exceptions, the
level of agreement between the various measurements of the Ps
formation cross section is reasonably good. The most signi ficant level
of disagreement between recent measurements ( ∼20%) occurs in the
region of the cross section maximum between about 15 and 40 eV.
Most of the earlier measurements from the 1980s and 1990s are larger
in magnitude across the whole energy range than the more recent
studies, and the weight of theoretical work also favors a lower
magnitude cross section across the whole energy range.
Our recommended positronium formation cross section is given
inTable 16 and shown in Fig. 17 . The estimated uncertainty on the
cross section values is ±15%.
4.7.3. Electronic excitation
There has been one measurement of electronic excitation in
argon by positron impact8by the San Diego group. They measured
the total excitation cross section for the components of the 3p54s
manifold in argon with total angular momentum J /equals1, namely, the
FIG. 16. The recommended total positron scattering cross section for Ar (solid line),
while the dashed lines represent the estimated uncertainty limits of ±10% (see also
Table 15 ).
TABLE 16. The positronium formation cross section (in units of 10−16cm2) for argon. The estimated uncertainty is ±15%
E (eV)Recommended positronium formation
cross section ( 310−16cm2) E (eV)Recommended positronium formation
cross section ( 310−16cm2)
8.95 0 25 2.65
10 0.95 30 2.53
11 1.47 40 2.23
12 1.93 50 1.75
13 2.26 60 1.32
14 2.52 70 0.98
15 2.68 80 0.68
16 2.77 90 0.46
17 2.80 100 0.29
18 2.79 125 0.05
19 2.78
20 2.76
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Chemical Reference DataARTICLE scitation.org/journal/jpr3p5(2P3/2,1/2 )4s levels from near threshold (11.63 eV) to 30 eV. We
summarize their results here by suggesting a recommended cross
section for the two combined excited states, noting their data show the
cross section for the 1/2 level to be about a factor of 3 –4 larger than
that for the 3/2 level.
The recommended cross section for the 3p54s excitation in
argon is given in Table 17 and shown in Fig. 18 . The estimated
uncertainty is ±15%.
4.7.4. Direct ionization
The direct ionization cross section has been measured by several
groups189,191 –193,210,211,213and has been discussed recently by
Chiari and Zecca11and Laricchia and colleagues,26and the level of
agreement between experimental measurements is relatively high.With the exception of one of the earlier measurements of direct
ionization,191which resulted in a much higher cross section, most of
the measurements and theory are in agreement across the whole
energy range, from threshold (15.75 eV) to 1000 eV, to within
about 20%.
The recommended direct ionization cross section is given in
Table 18 and shown in Fig. 19 . The estimated uncertainty on these
values is ±15%.
4.8. Potassium (K)
Investigations of positron scattering from potassium (K) consist
of just three experimental studies and again they are all by the Wayne
FIG. 17. The recommended positronium formation cross section for Ar (solid line).
The dashed lines represent the estimated uncertainty limits of ±15% (see also
Table 16 ).
TABLE 17. The cross section for positron impact excitation of the 3p54s levels in argon
(in units of 10−16cm2). The estimated uncertainty on these values is ±15% (see also
Fig. 18 )
E0(eV)Recommended excitation cross section
(310−16cm2)
12 0.112
13 0.39
14 0.49
15 0.40
16 0.43
18 0.35
20 0.36
22.5 0.37
25 0.51
27.5 0.53
30 0.58
FIG. 18. The cross section for positron impact excitation of the 3p54s levels in argon
(solid line). The dashed lines represent the estimated uncertainty limits of ±15%
(see also Table 17 ).
TABLE 18. The direct ionization cross section (in units of 10−16cm2) for positron
impact on argon. The estimated uncertainty on these values is ±15% (see also Fig. 19 )
E0(eV)Recommended direct ionization cross
section ( 310−16cm2)
15.75 0
20 0.26
30 0.99
50 2.31
75 2.83
100 2.96
150 2.77
200 2.46
300 1.95
400 1.58
500 1.34
750 0.91
1000 0.64
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Chemical Reference DataARTICLE scitation.org/journal/jprState group. The total scattering cross section has been measured by
Kwan et al.234at energies between 8 and 98 eV and by Parikh et al.266
from 1 to 102 eV. Positronium formation has been studied by Zhou
et al.236at energies between 1 and 100 eV. There have also been a
number of theoretical calculations of both the total scattering and the
Ps formation cross sections.41,43,48,81,84,92,267
4.8.1. Total scattering
The measured total scattering cross section for potassium234,266
shows similar behavior as a function of energy as that for lithium —it
exhibits a large, low energy peak (110 ˚A2at around 10 eV) before
decreasing in magnitude at both higher and lower energies. We note
that due to angular discrimination issues in the experiment, the
measured cross section at low energies likely underestimates the truevalue by a considerable amount. This has been discussed previously,
and indeed Kwan et al.234indicate that this effect may be as large as
14% at 10 eV, reducing to 2% at 50 eV. They placed an estimated
absolute uncertainty on their cross sections of 21%, not including the
possibility of forward scattering effects. Two CC calculations,81,92
both of which include elastic scattering and excitation of a number ofbound states, as well as Ps formation, reveal a TCS which is in good
agreement with the experiment, but only if the experimental values
are scaled upwards by a factor of 1.1 and further corrected at low
energies for forward scattering effects (see, for example, Fig. 7 of
Ref.92). Doing so moves the cross section peak closer to 150 ˚A
2in
magnitude.
Our recommended TCS for positron scattering from potassium
is given in Table 19 and shown in Fig. 20 . The estimated uncertainty
is 20%.FIG. 19. The recommended total direct ionization cross section for positron impact
on Ar (solid line). The dashed lines represent the estimated uncertainty limits of
±15% (see also Table 18 ).
TABLE 19. The TCS (in units of 10−16cm2) for positron scattering from potassium. The
estimated uncertainty is ±20% (see also Fig. 20 )
E0(eV) Recommended TCS ( 310−16cm2)
1.0 100
2.5 120
5.0 162
8.0 157
10 142
15 111
20 92
30 72
45 57
60 47
80 37
100 30
FIG. 20. The recommended TCS for positron scattering from K (solid line). The
dashed lines represent the estimated uncertainty limits of ±20% (see also Table 19 ).
TABLE 20. The positronium formation cross section (in units of 10−16cm2) for potassium.
The estimated uncertainty is ±30% (see also Fig. 21 )
E0(eV)Recommended positronium formation
cross section ( 310−16cm2)
1.0 10
1.5 16.7
2.0 21
3.0 27
5.0 34
7.5 31
10 23.8
15 14.5
30 5.4
50 2.2
100 1.3
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Chemical Reference DataARTICLE scitation.org/journal/jpr4.8.2. Positronium formation
The measured positronium formation cross section236consists
of both upper and lower limit estimates, as discussed previously in
Sec.2. The difference between these estimates is signi ficant (about a
factor of three) at low energies and it appears that modern theory
clearly favors the energy dependence and magnitude of the lower limit
measurement (see, e.g., Refs. 81and92). Given the expected accuracy
of these multi-con figuration CC calculations for one-electron sys-
tems, even for the dif ficult Ps formation cross section, we are inclined
to also favor the lower limit measurement for this cross section.
The recommended Ps formation cross section is given in
Table 20 and shown in Fig. 21 . The estimated uncertainty is ±30%.4.9. Krypton (Kr)
T h e r eh a v eb e e nm e a s u r e m e n t so ft h et o t a l
scattering,67,161,162,172,179,268 –271positronium forma-
tion,67,149,182,185,208,209,272and direct ionization193,209,212,213cross
sections for positron impact on krypton (Kr). There have also
been numerous theoretical calcu lations of these various pro-
cesses.67,97,100,126 –129,197,216,223,224,227,229,230,232,233,264,265,273 –275
4.9.1. Total scattering
TCS measurements for positron scattering from krypton date back
to the 1970s and there have been a reasonable number of subsequent
experimental determinations since then.67,161,162,172,179,268 –271The
FIG. 21. The recommended positronium formation cross section for K (solid line).
The dashed lines represent the estimated uncertainty limits of ±30% (see also
Table 20 ).
TABLE 21. The TCS (in units of 10−16cm2) for positron scattering from Kr. The estimated uncertainty on these values is ±10%
(see also Fig. 22 )
Energy (eV) Recommended TCS ( 310−16cm2) Energy (eV) Recommended TCS ( 310−16cm2)
0.2 67.2 6 6.71
0.3 43.8 7 7.15
0.4 31.8 8 8.14
0.5 24.2 9 9.09
0.6 19.4 10 9.73
0.7 16.4 15 10.9
0.8 14.2 20 11.3
0.9 12.5 30 11.5
1.0 11.2 40 11.4
1.5 8.97 50 11.1
2 8.32 60 10.9
3 7.67
4 7.23
5 6.88
FIG. 22. The recommended total positron scattering cross section for Kr (solid line),
while the dashed lines represent the estimated uncertainty limits of ±10% (see also
Table 21 ).
J. Phys. Chem. Ref. Data 48,023102 (2019); doi: 10.1063/1.5089638 48,023102-22
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Chemical Reference DataARTICLE scitation.org/journal/jprlevel of agreement between these experiments is mixed, with several
apparently suffering from the effects of insuf ficient discrimination
against forward scattering, which results in an anomalously low cross
section, particularly at low energies.
Chiari and Zecca11have recently reviewed the available TCS
data and have proposed a recommended TCS based on their analysis
and a comparison with theoretical predictions. Since their work, there
have been two other relevant determinations of this cross section, one
experimental179and one theoretical,129and these are also consistent
with the recommended values. Indeed, the latter calculation indicates
that the low energy cross section recommended by Chiari and Zecca,
which they speculated may be too low in magnitude, may in fact be a
reasonable estimate.Thus our recommended TCS is identical to that of Chiari and
Zecca. The recommended values are given in Table 21 and shown in
Fig. 22 . The estimated uncertainty on these cross section values
is±10%.
4.9.2. Positronium formation
There have been a number of measurements of the Ps formation
cross section for Kr,67,149,182,185,208,209,272and as was the case in
some of the lighter rare gases, the only signi ficant discrepancies
between these measurements occurs in the energy region around the
peak in the cross section, at around 15 –20 eV, where there are dif-
ferences between the various measurements of up to 20%. Chiari and
Zecca discussed these measurements but declined to recommend a PsTABLE 22. The positronium formation cross section (in units of 10−16cm2) for Kr. The estimated uncertainty on these values is
±15% (see also Fig. 23 )
E (eV)Recommended positronium formation
cross section ( 310−16cm2) E (eV)Recommended positronium formation
cross section ( 310−16cm2)
7.2 0 25 3.76
7.5 0.70 30 3.37
8 1.50 40 2.61
9 2.58 50 2.06
10 3.30 60 1.58
11 3.82 70 1.17
12 4.24 80 0.82
13 4.45 90 0.56
14 4.55 100 0.37
15 4.56 125 0.04
16 4.55
18 4.3820 4.21
FIG. 23. The recommended positronium formation cross section for Kr (solid line).
The dashed lines represent the estimated uncertainty limits of ±15% (see also
Table 22 ).TABLE 23. The direct ionization cross section (in units of 10−16cm2) for positron impact on
krypton. The estimated uncertainty on these values is ±20% (see also Fig. 24 )
E0(eV)Recommended direct ionization cross
section ( 310−16cm2)
14 0
16 0.11
18 0.25
20 0.4825 1.21
30 1.88
40 2.92
50 3.66
75 4.24
100 4.22
125 3.94
150 3.61
200 3.04
500 1.54
750 1.17
1000 0.95
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Chemical Reference DataARTICLE scitation.org/journal/jprformation cross section. The various theoretical calculations for this
process also show similar, if not larger, differences in this energy
range. On the other hand, the agreement between experiments at
near-threshold and higher energies is reasonably good.
Our recommended positronium formation cross section is given
inTable 22 and shown in Fig. 23 . The estimated uncertainty on the
cross section values is ±15%.
4.9.3. Direct ionization
There are only a few experimental measurements of the direct
ionization cross section by positron impact on krypton, with the ma-
jority from the UCL group193,212,213and one determination from the
University of California at San Diego (UCSD) group.209The agreement
b e t w e e nt h e s ec r o s ss e c t i o n si sr a t h e rg o o di nt h en e a r - t h r e s h o l dr e g i o n ,but, once again, the measurements diverge somewhat in the region from
about 50 eV up to the cross section maximum at around 100 eV. At the
maximum, the UCSD group predicts a cross section that is about 20%
higher than that of the UCL group.213The only available data above 100
eV are that of the UCL group and this indicates a finite ionization cross
section out to energies above 1000 eV.
These cross sections were also analyzed by Chiari and Zecca11
and Laricchia and colleagues,26but they did not a suggest recom-
mended cross section.
The recommended direct ionization cross section is given in
Table 23 and shown in Fig. 24 . The estimated uncertainty on these
values is ±20%.
4.10. Rubidium (Rb)
There is only one measurement each of the total scattering cross
section and positronium formation cross section for rubidium (Rb),
and these are from the Wayne State group.266,276There are also a
FIG. 24. The recommended direct ionization cross section for positron impact on Kr
(solid line). The dashed lines represent the estimated uncertainty limits of ±20%
(see also Table 23 ).
TABLE 24. The TCS (in units of 10−16cm2) for positron scattering from rubidium. The
estimated uncertainty in these values is ±25% (see also Fig. 25 )
E0(eV) Recommended TCS ( 310−16cm2)
1.0 108
2.0 124
3.0 148
5.0 177
6.0 180
7.0 163
15 136
20 115
30 88.5
50 62.5
75 45.0
100 35.0
FIG. 25. The recommended TCS for positron scattering from Rb (solid line). The
dashed lines represent the estimated uncertainty limits of ±25% (see also Table 24 ).
TABLE 25. The positronium formation cross section (in units of 10−16cm2) for Rb. The
estimated uncertainty in these values is ±30% (see also Fig. 26 )
E0(eV)Recommended positronium formation
cross section ( 310−16cm2)
1.0 12
2.0 21
3.0 30
4.0 37
5.0 39
7.5 31
10 22
15 12.5
20 6.5
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Chemical Reference DataARTICLE scitation.org/journal/jprnumber of theoretical calculations of these cross sec-
tions44,82,94,114,116,224,277using a variety of techniques including
the CC, Glauber, and polarized orbital approaches.
4.10.1. Total scattering
The total scattering cross section has been measured between
1 and 100 eV.266The measurements, as for potassium, exhibit a
strong cross section maximum at low energies, at around 5 eV in the
case of Rb. Kernoghan et al.94performed CC calculations for elastic
scattering and excitation of Ps (1s, 2s, 2p, 3s, 3p, and 3d) and Rb states
(5s, 5p, 6s, 6p, and 4d) and, by compiling these cross sections, also
determined a total scattering cross section for Rb. A similar approachwas more recently adopted by Chin et al.82Kernoghan et al. also
addressed the issue of forward angular discrimination in the exper-
imental cross sections by using their differential elastic scattering cross
sections to correct the experimental values for the experimentallyestimated missing angular ranges
266—23°at 2 eV reducing to less
than 9°above 30 eV. These corrected values, when scaled upward by a
further 5%, were found to be in very good agreement with the cal-
culated TCS (see Fig. 5 of Ref. 94).
Our recommended TCS for positron scattering from rubidium is
given in Table 24 and shown in Fig. 25 . The estimated uncertainty
is 25%.
4.10.2. Positronium formation
The positronium formation cross section has been measured by
Surdutovich et al.276at energies between 1 and 17 eV. There have also
been several calculations of the cross section for this channel (e.g.,
Refs. 82and94), which is “open ”and non-zero in magnitude at 0 eV.
Both theory and experiment indicate a cross section which peaks near
5 eV in energy and with a magnitude around 40 ˚A2, although there is a
reasonable level of uncertainty around this value.
The recommended Ps formation cross section is given in
Table 25 and shown in Fig. 26 . The estimated uncertainty is ±30%.
4.11. Xenon (Xe)
Positron scattering experiments for xenon have yielded mea-
surements of the total scattering cross
section,66,162,172,179,252,253,268 –270,278the positronium forma-
tion cross section,66,149,182,185,208,209,214,279and the direct ion-
ization cross section.209,212,213There have also been a signi ficant
number of theoretical calculations of positron-xenon
scattering.97,100,126 –129,197,223,227,229,230,255,280 –283
4.11.1. Total scattering
Total scattering cross section measurements for xenon extend
from recent years all the way back to the mid 1970s. As in the other
FIG. 26. The recommended positronium formation cross section for Rb (solid line).
The dashed lines represent the estimated uncertainty limits of ±30% (see also
Table 25 ).
TABLE 26. The TCS (in units of 10−16cm2) for positron scattering from xenon (see text for details). A conservative estimate of
the absolute error is ±10% (see also Fig. 27 )
Energy (eV) Recommended TCS ( 310−16cm2) Energy (eV) Recommended TCS ( 310−16cm2)
0.25 85.1 6 16.8
0.3 71.0 7 17.9
0.4 56.2 8 18.8
0.5 49.0 9 19.3
0.6 43.1 10 19.4
0.7 39.0 15 19.2
0.8 35.5 20 18.8
0.9 33.5 30 18.1
1 31.0 40 17.0
1.5 24.0 50 16.0
2 20.4 60 14.9
3 16.8
4 15.6
5 15.9
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Chemical Reference DataARTICLE scitation.org/journal/jprheavier rare gases, there appears to be a considerable spread in the
absolute values of the measurements, particularly at lower energies
where it is apparent that forward scattering effects are most likely
responsible for the majority of the differences.
The total scattering data were recently analyzed by Chiari and
Zecca,11and they provided a recommended cross section based on
their analysis. They comment that their recommended values below
1 eV may be too low due to forward scattering effects which are not
completely accounted for in the experiments, and a recent MBT
calculation129indicates this may in fact be the case. While further
experiment would be useful to verify this, we suggest that the values ofChiari and Zecca can probably be raised by around 10% for energies
below about 1 eV.
Thus, our recommended TCS is identical to that of Chiari
and Zecca, with the lower energy values increased by a further∼10%. These recommended values are given in Table 26 and shown
inFig. 27 . The estimated uncertainty on these cross section values
is±10%.
4.11.2. Positronium formation
There have been a number of absolute measurements of the Ps
formation cross section for Xe, dating back to the early 1980s. The
level of agreement amongst the various measurements is reasonably
good, with the cross section showing a maximum of just under 10 ˚A
2
at an energy of around 10 eV. The comparison between experiments,
and between experiment and theory, has been discussed in some
detail by Chiari and Zecca in their recent review,11who also point out,
as in the case of argon, that there remains some uncertainty around
the existence or otherwise of a second maxima in the Ps cross section
FIG. 27. The recommended total positron scattering cross section for Xe (solid line),
while the dashed lines represent the estimated uncertainty limits of ±10% (see also
Table 26 ).
TABLE 27. The positronium formation cross section (in units of 10−16cm2) for Xe. The estimated uncertainty on these values is
±15% (see also Fig. 28 )
E (eV)Recommended positronium formation
cross section ( 310−16cm2) E (eV)Recommended positronium formation
cross section ( 310−16cm2)
5.3 0 25 6.4
6 3.9 30 5.6
7 6.1 40 4.0
8 7.7 50 2.8
9 8.5 60 1.84
10 9.1 70 1.13
11 9.1 80 0.68
12 8.9 90 0.40
15 8.2 100 0.23
18 7.6
20 7.2
FIG. 28. The recommended total positronium formation cross section for Xe (solid
line). The dashed lines represent the estimated uncertainty limits of ±15% (see also
Table 27 ).
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Chemical Reference DataARTICLE scitation.org/journal/jprnear 20 eV. However, Chiari and Zecca did not provide a “recom-
mended ”cross section for Ps formation in Xe.
Our recommended positronium formation cross section is given
inTable 27 and shown in Fig. 28 . The estimated uncertainty on the
cross section values is +15%.
4.11.3. Direct ionization
There have been two experimental determinations of the
direct ionization cross section for Xe —by the UCL and San Diegogroups.209,212,213The measured cross sections are in reasonably
good agreement with each other across the energy range where they
overlap and they predict a cross section maximum of around 6 ˚A2at
about 100 eV. There is also a reasonably good agreement withtheory —particularly the two most recent calculations.
97,128
These cross sections were also analyzed by Chiari and Zecaa11
and Laricchia and colleagues,26but they did not a suggest a rec-
ommended cross section.
The recommended direct ionization cross section is given in
Table 28 and shown in Fig. 29 . The estimated uncertainty on these
values is ±15%.
Acknowledgments
We are grateful for the support of the Australian Research Council
(Grant Nos. DP140102854, DP150101521, and DP190100696) and our
respective institutions —The University of Malaya, Flinders University,
and the Australian National University.
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22P. G. Coleman and J. T. Hutton, Phys. Rev. Lett. 45, 2017 (1980).TABLE 28. The direct ionization cross section (in units of 10−16cm2) for positron
impact on xenon. The estimated uncertainty on these values is ±15% (see also Fig. 29 )
E0(eV)Recommended direct ionization cross
section ( 310−16cm2)
12.13 0
15 0.60
20 1.68
25 2.91
30 4.20
40 5.82
50 6.26
60 6.34
75 6.26
100 5.97
125 5.46
150 4.97200 4.08
500 2.13
750 1.52
1000 1.07
FIG. 29. The recommended direct ionization cross section for positron impact on Xe
(solid line). The dashed lines represent the estimated uncertainty limits of ±15%
(see also Table 28 ).
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Published by AIP Publishing on behalf of the National Institute of Standards and Technology.Journal of Physical and
Chemical Reference DataARTICLE scitation.org/journal/jpr |
1.3035316.pdf | Light and Sound for Engineers
R. C. Stanley Robert Lindsay ,
Citation: 22, (1969); doi: 10.1063/1.3035316
View online: http://dx.doi.org/10.1063/1.3035316
View Table of Contents: http://physicstoday.scitation.org/toc/pto/22/12
Published by the American Institute of Physics
PHYSICS TODAY
PLASMAS: PAGE 34
lECEMBER 1969 I 'ON 4!UJJ
QlVd9d
> n
UON
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PHYSICS TODAY . DECEMBER 1969
Industry StandardsThese general-purpose XY recorders set
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GRANVILLE-PHILLIPS COMPANYS675 EAST ARAPAHOE AVE. • BOULDER, COLORADO 80303, U.S.A.
Advancing Vacuum Technology
4 • DECEMBER 1969 • PHYSICS TODAY
VOL 22 NO 12
21 Nucleon-Nucleon Scattering Malcolm H. Mac Gregor
What similarities, what differences, can we find for the two-nucleon forces?
29 New Information Program for AIP
Arthur Herschman, Franz L. Alt and H. William Koch
Computer-organized files will help you find your way in the literature maze
34 Frontiers of Physics Today: Plasmas Harold Grad
Matter in this form shows an unrivaled range of parameters and phenomena
47 More About Tachyons Olexa-Myron Bilaniuk, Stephen L. Brown,
Bryce De Witt, William A. Newcomb, Mendel Sachs,
E. C. George Sudarshan, Shoichi Yoshikawa
Five readers take issue with the protagonists of faster-than-light particles
COVER: Solar prominence (above) and schematic confined plasma (below) con-
trast astrophysical plasma with our aspirations for controlled thermonuclear
power in the laboratory. Harold Grad examines the current state of plasma re-
search in his article on page 34. (Photo by Mt Wilson and Palomar Observatories.)
9 Letters
PhD employment • Lunar atmosphere modification
17 Phimsy
Painting competition at Iowa State University
69 Books
UFO's • Mechanics • Quantum theory • Astronomy
89 Meetings
Nonsuperconducting electron tunneling
95 We Hear That. . .
Van Vleck retires • Faculty changes • Amos deShalit dies at 42
103 Calendar
Partial listing contains new information about meetings
109 Annual Index
118 Guest Editorial
Is Your Research Moral?Arthur Schawlow55 Search and Discovery
Continuous-wave chemical laser requires no external energy
source • Cold octopole and hot Tokomak show long confinement
times • Air Force solar telescope and OSO-6 now observing the
sun • Dicke panel says US lags in radio-astronomy construction •
Measuring it better: a visit to Bureau International des Poids et
Mesure
63 State and Society
Metzner named assistant director of AIP publications • Fund of
Abdus Salam has first recipient • Dart, Moravcsik to evaluate
foreign graduate candidates • JILA has fellowships and asso-
ciateships for 1970-71 • AIP publishes guide to undergraduate
departments • Health Physics Society elects new officers • Nixon
names 12-man task force to review US science policy • APS ar-
ranges group flights • European Physical Society announces
division chairmen • AIP and society journals available in micro-
film
EDITORIAL STAFF R. Hobart Ellis Jr (editor), Theodora Johnides, Barbara G. Levi, Gloria B. Lubkin, Marian S. Rothen-
berg, Jill Russell, John T. Scott, Frederic Weiss (design)
ADVISORY COMMITTEE Dale T. Teaney (chairman), Solomon J. Buchsbaum, William W. Havens Jr, John N. Howard, Howard
J. Lewis, Robert S. Marvin, Paul M. Routly, Clifford E. Swartz
(Europe, Middle East, North Africa): $7.50; elsewhere: $5.50. Copy-
^ ,r>™ , 'of Physics. All right reserved.
ix weeks advance notice. Send
Department. Please include ad-
issues.
PHYSICS TODAY • DECEMBER 1969 • 5
high dispersion in a short path length,
minimal stray light
Seven light sources and a wide se-
lection of lenses and accessories make
the Bausch & Lomb Double Grating
Monochromator more versatile than
any other make.
Two Certified-Precision Gratings in
tandem are the heart of the high pre-
cision optical system. The two 1200
grooves/mm plane reflection gratings,
optimized for the 200nm region, cover
the wide, 190-825nm wavelength
range. Purging with dry nitrogen ex-
Double Grating Monochromatortends the lower range to 180nm.
Wavelength is displayed on a digital
counter. Highly accurate wavelength
calibration is easily accomplished.
Identical left and right side mount-
ing plates and a reversible optical sys-
tem allow interchangeable use of the
entrance and exit positions. Three
quickly selected fixed slits — 0.2nm,
0.5nm, and 2.0nm — assures utmost
precision in slit widths. A single lever
selects both entrance and exit slits
simultaneously.
Bausch & Lomb manufactures a com-
plete line of monochromators, includ-
ing the 250mm, 500mm and High In-
tensity models. Write for our brochure
33-2098, or we'll gladly arrange a
demonstration. Analytical Systems Divi-
sion, Bausch & Lomb, 20424 Linden
Avenue, Rochester, New York 14625.Member Societies
American Physical Society
Optical Society of America
Acoustical Society of America
Society of Rheology
American Association of
Physics Teachers
American Crystallographic Association
American Astronomical Society
The American Institute of Physics was
founded in 1931 as a federation of leading
societies in physics. It combines into one
operating agency those functions on behalf
of physics that can best be done by the so-
cieties jointly. Its purpose is the advance-
ment and diffusion of the knowledge of
physics and its applications to human wel-
fare. To this end the institute publishes for
itself and the societies 35 journals (includ-
ing translations) bulletins and programs;
promotes unity and effectiveness of effort
among all who are interested in physics,
renders numerous direct services to physi-
cists and the public and cooperates with
government agencies, national associations,
educational institutions, technical industries
and others in such manner as to realize the
opportunities and fulfill the responsibilities
of physics as an important and constructive
human activity.
Governing Board
Ralph A. Sawyer*, Chairman, H. William
Koch*, ex officio, Luis W. Alvarez, Ar-
nold Arons, Stanley S. Ballard*, John Bar-
deen, Robert T. Beyer, Joseph A. Burton,
H. Richard Crane, Herbert I. Fusfeld, Ron-
ald Geballe*, J. E. Goldman, Samuel A.
Goudsmit, William W. Havens Jr*, Ger-
ald Holton, W. Lewis Hyde, G. A. Jeffrey,
Karl G. Kessler, R. Bruce Lindsay*, Rob-
ert N. Little, Archie I. Mahan, G. C.
McVittie, Robert G. Sachs*, Frederick
Seitz, Thor L. Smith, Mary E. Warga*,
Wallace Waterfall, Albert E. Whitford,
Clarence Zener.
* executive committee
General Administration
H. William Koch, Director; Wallace Water-
fall, Secretary; Gerald F. Gilbert, Treasurer
and Controller; Lewis Slack, Associate Di-
rector, General Activities; Kathryn Setze,
Assistant Treasurer; Emily Wolf, Society
Services Manager; Dwight E. Gray, Wash-
ington Representative.
Directors of Professional Divisions
Hugh C. Wolfe, Publications; Arthur
Herschman, Information; Eugene H. Kone,
Public Relations; Charles Weiner, Physics
History; Arnold A. Strassenburg, Education
and Manpower; Harold L. Davis, Physics
Today.
Publishing Operations
A. W. Kenneth Metzner, Assistant Direc-
tor, Publications; David A. Howell, Edi-
torial Manager; Edward P. Greeley, Ad-
vertising Manager; John DiCaro, Sub-
scription Fulfillment.
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PHYSICS TODAY • DECEMBER 1969 • 7
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Losses to society
Although I have always admired your
editorial comment for its breadth of
vision and indeed its humanity, I must
take exception to your remarks in the
June issue concerning the current un-
employment among physics PhDs. I
can understand your irritation at sug-
gestions made by some of these people
that society owes them a living, but I
doubt if these complaints are typical,
and in any case, even if the unem-
ployed were to misdirect their criti-
cism, this misdirection would not mean
that the blame for their unfortunate
predicament was theirs alone. Thus,
although I agree that society at large
has more pressing concerns, I do sug-
gest that a large measure of respon-
sibility falls on the shoulders of the
academic physics community.
Briefly, I claim that the graduate
schools have no business turning out
more PhDs than can reasonably be
expected to find academic or other re-
search employment (allowing, natu-
rally, for the usual number of dropouts
and voluntary transfers to other fields).
Of course new PhDs "are equipped to
do other jobs" of value to society. The
point is that they were equipped for
this five or so years ago and would
have taken these jobs at that time if
that is what they had wanted. As it
is, feeling understandably frustrated,
they will presumably give less of
themselves in these jobs now than they
would have done originally. There are
two other losses which society suffers
through the undiscriminating admis-
sions policy of graduate schools: the
service of these students during the
period in which they were working for
their (useless) PhDs, and. the tax
money spent in producing an unem-
ployable elite.
And together with the loss to society
must be considered the great persona)
distress of the individuals concerned.
It is not simply a question of winding
up with $8 000 a year rather than
$14 000. Especially towards the end
of one's PhD work a fairly intimate
collaboration develops between student
and research director. As the two
sweat out their problems together, dis-
cuss them with other physicists, share
the same moments of frustration andLETTERS
satisfaction and travel together to
physics meetings, the student is led by
his professor into the circle to which
he aspires, namely, the world of phys-
ics research. At any point the associa-
tion can be terminated, should the
student prove inadequate, but if
awarding a PhD means anything at
all, it must surely be regarded as a
certificate of admission to this circle,
at least to the point of a few proba-
tionary years. Although there can be
no question of this admission consti-
tuting a formal contract, the new PhD
who finds himself rejected from this
community at the very moment of
acceptance must surely feel cruelly
betrayed. Obtaining his degree has
required not only several years of fi-
nancially unrewarding hard work but
also a considerable emotional dedica-
tion to physics in general and his field
in particular.
Confronted with this situation,
your "oldtime answer . . . physics is
tough .... If you want to work
with us despite the drawbacks we
will let you'* is a cynical irrelevancy.
For the point is that however good the
student is, we will only let him work
with us up to the completion of his
PhD. After that he can take his
chance on the market.
Probably my picture of the personal
relationships involved between student
and professor is somewhat idealized,
but only insofar as graduate students
have come to be regarded as slave
labor, engaged only to serve the ambi-
tions of expanding departments and
the careers of individual professors.
Therein, of course, lies the rub. As
long as there were jobs, no conflict of
interest arose and everyone was happy,
but in this new situation the only hon-
est thing to do is to reduce the output
of PhDs, either by reducing the intake
of graduate students or by raising the
requirements. (The latter solution
could take the form of demanding a
certain measure of competence and
experience in teaching, which would
be suitably remunerated.) If what-
ever solution is adopted involves a
cutback in research so be it: The loss
is not likely to be irreparable.
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PHYSICS TODAY • DECEMBER 1969
2:15 pm, positive ions;
2:16 pm, electrons
^"™- In between, somebody pushed a button on the
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And with appropriate choice of source and target the same
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> "Instant electrons" are a special feature of certain
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ZIPLETTERS
employ itself, I would certainly not
hold this to be true for the bachelor
or master degrees. On the contrary,
the programs for these courses have
been far too strongly oriented towards
the student who will eventually go
into research. Physics is an excellent
training for the mind, and it is one of
the scandals of our time that men in
public life are, for the most part,
scientifically illiterate. Physics de-
partments have been sadly derelict in
failing to develop rigorous under-
graduate programs for those who will
eventually do something else: eco-
nomics, law, sociology, politics, etc.
But once a man goes as far as the PhD
in physics, it must be assumed that
this is what he wants to do.
J. MICHAEL PEARSON
Universite de Montreal
Unethical promise of jobs
William Silvert writes of (1) an "em-
ployment crisis," and (2) of a crisis
... far deeper and more bitter than a
matter of jobs (PHYSICS TODAY, August,
page 9).
Regarding the employment crisis, it
is hardly reasonable to expect any
course of study to lead surely to well
paid permanent employment. No in-
stitution can properly hold out such a
promise to its students unless it has
the power to enforce it. Lacking this
power, such a promise is unethical.
Unemployment is common among ac-
tors, playwrights, musicians, poets and
composers, but they did not expect
their studies to guarantee jobs. They
studied for the love of the subject.
Beginning about 1950, many public
statements appeared that alleged a
"shortage" of scientific personnel—at
first, mainly of engineers. This pub-
licity began at about the time that the
defense contracting business started to
grow rapidly, on a cost-plus-fixed-fee
basis. One writer suggested that such
contractors made profits on the mere
buying and selling of technical labor,
the customer being the government.
This has not been proved and is not
provable, but it is a fair hypothesis.
The allegations of a "shortage" were
shown to be poorly justified, at best,
as long ago as 1957, when the Na-
tional Bureau of Economic Research
published its book-length study, The
Demand and Supply of Scientific Per-
sonnel. It is surprising that any high-
ly skilled group, such as physicists,should still believe official statements
from any source as to the demand for
its services, instead of drawing its con-
clusions independently from factual
sources.
Silvert's second remark suggests
deep and widespread disillusionment.
But it fails to advance reasons for this
second "crisis." In so failing, it be-
comes unscientific. This crisis clearly
exists, but it is a symptom. The dis-
ease seems to be hidden. This disease
is probably rooted in practices in in-
dustry and politics. Nobody seems to
know what they are.
Students, at least, are in a position
to search for the underlying disease,
and to try to explain it. I hope that
they will do so instead of merely
reacting to pronouncements from still
other sources. Persons in responsible
positions are likely to be under pres-
sure to protect and extend these posi-
tions as we all know, and so students
may properly question their motives.
What appears to be needed is the
clear application of the human brain
to the political problems that beset
young physicists. They, able to think
clearly, will always do better than spe-
cialists in the more pseudo sciences.
LAWRENCE FLEMING
Pasadena, California
Manpower contradictions
In your August issue there is an ap-
parent contradiction between the
letters of William Silvert and the reply
of Susanne Ellis, on the one hand, and
the reply by Hugh Wolfe to Robert C.
Johnson's letter on the other. The first
letters complain about lack of positions
for physicists. Wolfe complains of
staff losses and difficulties in recruiting
competent people. I have also heard
that the National Accelerator Labora-
tory encounters recruiting problems.
The resolution of the contradiction
might well lie in the areas of work for
which young physicists strive and the
editorial work that the American Insti-
tute of Physics can offer. However, it
would be good to have a more detailed
review of positions available and posi-
tions sought by applicants for jobs.
I know from first-hand information
that many of the smaller colleges are
eager to find good physics teachers,
and I think there are also some job
openings in national laboratories. On
the other hand, I also know of some
young physicists who had considerable
difficulty in locating positions to their
liking even though, in the cases about
PHYSICS TODAY . DECEMBER 1969 11
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ILETTERS
i
! which I know, they eventually suc-
iceeded. An article that would resolve
| the apparently conflicting statements
'of the letters would, I think, attract
considerable interest.
EUGENE P. WIGNER
Princeton University
Personal ivory towers
K^S a physicist turned engineer (by
jyhoice) I could not help commenting
^Vn two things in PHYSICS TODAY. First
he job shortage for PhDs. It exists
ecause some people got the idea the
IS owed them a personal ivory
)wer—equipped with secretaries,
^clinicians and an unlimited supply of
loney. Now the coach has turned
ito a pumpkin; the horses are mice,
id a cold cruel employer asks, "What
m you do for the corporation?" I
y it is just about time that Alice re-
Wijned from Wonderland.
^:to Second, the journals of the Ameri-
iwln Physical Society. They are excel-
*m it. Try submitting a paper to some
t ity?!gineering journal. Six months later
r#:nonestly) you get a letter saying,
Ve regret that the reviewer . . ."
STUART A. HOENIG
University of Arizona
I lunar atmosphere
lunar atmosphere (vacuum) is a
A puree that has become available to
\j inkind only within the last few
rs. It appears likely that studies of
'dual gas near the moon's surface
provide useful information con-
ting the history and composition of
body. It is possible that the
)n will find important use as a sup-
t for large infrared and ultraviolet
scopes, thermionic devices and
IT apparatus that requires high
ium for operation. Perhaps it is
|thwhile to point out that this envi-
nent may be changed appreciably
the process of lunar exploration
^i that in particular some consider-
.1 should be given to the effects of
;tion of large amounts of rocket
s into that environment.
typical manned landing module
it exhaust 5000 pounds of gases,
ly water and carbon dioxide in
ly equal molar amounts with mea-
Je amounts of heavier hydrocar-
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RCAPHYSICS TODAY • DECEMBER 1969 • 13
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LETTERS
the moon are 8.2 X 10° cm for water
and 3.4 X 10G cm for carbon dioxide.
These gases thus expand into effective
volumes of 3.1 x 1024 cm3 and 1.3
X 1024 cm3, respectively. To within
an order of magnitude, the pressure
rises to be expected due to ejection of
this amount of gas are 2 X 10~13
torr for water and 5 X 10~13 torr for
carbon dioxide. Pressures of this
magnitude are measurable with com-
mercially available equipment.
Simple estimates of typical escape
times for these gases indicate that
they will remain for at least several
thousand years. We may then expect
to modify the total lunar environment
irreversibly, and only partly predicta-
bly, each time a rocket lands there.
Only if the natural background pres-
sures of water and carbon dioxide are
several orders of magnitude larger
than the above values will our pertur-
bations of these quantities be unim-
portant.
JOHN O. S TONER JR
University of Arizona
, Emily Wolf and register
I enjoyed the article "The National
Register Looks at Manpower" in the
October PHYSICS TODAY. In one state-
ment, though, it is in error.
At the request of Henry A. Barton,
then director, and Wallace Waterfall,
then as now secretary of the American
, Institute of Physics, I organized the
register in November 1953. I em-
ployed Sylvia Barisch, your senior
author, in March 1954 as one of my
part-time coders. I remained in charge
of the register until 1960, when it was
transferred to the newly formed Ed-
f" ucation and Manpower Division. Mrs
"Barisch had been named supervisor in
* May 1959.
EMILY WOLF
American Institute of Physics
.CORRECTION: The editors apologize
for two typographical errors in Don
B. Lichtenberg's October review of
Paradoxes in the Theory of Rela-
tivity by Yakov P. Terletskii. The
word "comparable" was substituted
in the last sentence, which should
'have-read, ". . . the theory of rela-
tivity is compatible with dialectic
materialism." The first equation in
the fourth paragraph should have read
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16 • DECEMBER 1969 . PHYSICS TODAY
PHIMSY
Physicists can paint doors
PATHDAFRELDO was the name of the
project: "paint those damned freight
elevator doors." It started when Dan
J. Zaffarano, chairman at Iowa State,
decided that everything but the doors
was just fine in the new physics build-
ing. The contest he initiated stimu-
lated 85 entries from faculty, students
and employees. Six were chosen for
the six sets of doors. Lo, one of the
winning designs (entered with num-
"END ALL WAR" ceramic was first
member of a growing sculpture court.
SCRAPBOX SCULPTURE was a surrep-
titious graduate-student contribution.FISHING DRAGON won first prize for
Klaus Ruedenberg and daughter Ursula.
bers and not the artists' names) was
that of Zaffarano and his junior-high
daughter Elisa. Then at a "Slinga-
thon" the designs were transferred to
the doors.
First prize went to Klaus Rueden-
berg, professor of physics and chemis-
try, and his daughter Ursula for
"Charlie in Minnesota," a fantastic
dragon fishing with his tongue in a
fantastic fish pool. "Clyde" is a huge
psychedelic frog that opens his mouth
when the doors open. All other
entered designs are now framed and
decorate the building hallways.
Not only painted doors are part of
the art scene at Ames. A courtyard
between old and new parts of the
physics building will soon have a new
office complex on one of its other
sides. It became a sculpture garden
and acquired "End All War," a ce-
ramic showing red flames of war rising
from a green prairie and reducing
civilization to a black cinder. Chal-
lenged, some graduate students got
busy and surreptitiously put up a rival
on dedication morning. "From Chaos
New in nuclear power
"A tiny pellet, small enough to fit into
a thimble, can light your home for
three years," says a General Electric
ad that I hear repeatedly on my favor-
ite station, "the radio station of the
New York Times." That doesn't
sound much like huge cranes handling
giant fuel elements, heavy shippingZAFFARANO AND DAUGHTER made
one of six designs that went on doors.
to Beauty, Symbolizing the Wonder-
land Trip Toward the PhD" was an
abstraction made from the cryogenics
scrapbox with some round-the-clock
work. Chief solderer, it turned out,
was Durkee Richards, now a PhD
physicist at 3M Research.
"What is all this about?" I asked
Zaffarano. "I always thought you
physicists were dedicated specialists
quite unable to apply yourselves pro-
ductively to anything but the subjects
of your PhD theses." It seems I've
been given the wrong picture.
"Physicists are creative people
whose need for expression often tran-
scends even publication in The Physi-
cal Review" was Zaffarano's answer.
"Since our daily working environment
seems to consist mostly of black-
boards, vacuum pumps, dewars, mag-
nets and racks of modular electronics,
we thought it appropriate to observe
through our new windows that the
worlds of living things, abstract forms
and color also provide outlets for re-
search and creativity to intelligent
people."
containers to shield against radioactiv-
ity and pressure vessels weighing hun-
dreds of tons. You physicists at GE
must have come up with something
new in the way of tiny fuel pellets.
Somebody somewhere is confusing
me, and I hope that the confusion is
unintentional. •
PHYSICS TODAY DECEMBER 1969 17
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18 • DECEMBER 1969 • PHYSICS TODAY
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PHYSICS TODAY . DECEMBER 1969 19
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PHYSICS TODAY VOL 22 NO 12
NUCLEON-NUCLEON SCATTERINGAn atomic nucleus can be considered a set of two-nucleon systems.
What are the forces between these pairs?
How do protons and neutrons differ and how are they similar?
Studies during the past three decades have given some answers and indicated
which new kinds of experiments are likely to be most useful.
MALCOLM H. MAC GREGOR
DETERMINATION OF THE fundamental
law of force between two nucleons has
occupied many physicists for the past
three decades. Because the proton
and electron have obvious similarities
(elementarily, spin of 1/2, equal-but
opposite electric charge, Fermi statis-
tics, antiparticles) the derivation of a
nuclear "Coulomb's law" would seem
to be a just reward for working in this
area. As we have rather slowly and
laboriously learned, however, simplic-
ity appears to be inversely proportional
to some power of the coupling con-
stant.
Indirect means can be used to learn
about the forces between two nucleons.
An atomic nucleus, composed of pro-
tons and neutrons, can be reasonably
treated as a collection of interacting
two-nucleon systems. From the over-
all behavior of the nucleus, certain
properties of the nuclear force, such as
its range and the statistics it obeys,
can be adduced. Direct information,
however, is obtained only by scatter-
ing one nucleon off another. Analysis
of these scatterings will be the subject
of the present discussion.
Early history
The most distinctive feature of the nu-
clear force, its very short range, was
deduced by Ernest Rutherford in 1911.
Modem studies of nucleon-nucleon
interaction were initiated in 1932 when
the neutron was discovered1 and high-
voltage particle accelerators first pro-
duced nuclear reactions.2 Some of themost crucial discoveries were made
very early, as often happens. By
studying the binding energy of the
alpha-particle Eugene Wigner3 in 1933
confirmed that nuclear forces have a
short range and are very strong. Wer-
ner Heisenberg4 and Ettore Majorana5
pointed to the repulsive-core concept
when they invoked exchange forces to
explain the stability of nuclei against
collapse. In 1935 Hideki Yukawa6
predicted that the nuclear force should
be mediated by exchange of a virtual
meson with a mass of roughly 100
MeV.
Nucleon-nucleon scattering occurs
within the constraints imposed by in-
variance under time reversal and con-
servation of angular momentum and
parity. For a given total angular mo-
mentum J the proton-proton system
has five independent ways in which
the intrinsic spins and the orbital an-
gular momentum can couple together.
These alternatives are shown in figure
1. The two possibilities listed as am-
plitude 5 in figure 1 are equivalent;
one is the time-reverse of the other,
and we are assuming time-reversal in-
variance. The antisymmetry of the
proton-proton wave function when
combined with the conservation of
angular momentum and parity pre-
vents mixing of singlet (S = 0) and
triplet (S = 1) spin states.
If we assume that the proton and
neutron are isotopic states of the same
particle that differ only in the 7Z =
±1/2 projections of their isotopic spin,then the neutron-proton wave function
must be antisymmetric. In this case,
we have the same five scattering am-
plitudes in spin space as we did for
proton-proton scattering. Scattering,
however, now occurs in two isotopic
spin states (7=1 and I — 0); so the
direct product gives ten independent
neutron-proton scattering amplitudes.
The 7=1 amplitudes as measured in
proton-proton and neutron-proton
scattering should be identical in all
hut electromagnetic effects. This is
Malcolm H. Mac Gregor served in the
US Navy at the end of World War 2 be-
fore obtaining his BA (in mathematics),
MA and PhD (both in physics) at the
University of Michigan. In 1953 he
joined the Lawrence Radiation Labora-
tory at Livermore, where he is now a
research physicist. Mac Gregor's work
has included both experiment (beta de-
cay and neutron scattering) and theory
(nucleon-nucleon analysis). He is also
a graduate research adviser and occa-
sional lecturer at the University of Cali-
fornia, Berkeley, with research interest
in elementary-particle structure.
PHYSICS TODAY • DECEMBER 1969 • 21
the charge independence hypothesis.
The weaker charge-symmetry hypoth-
esis specifies that, apart from electro-
magnetic effects, proton-proton and
neutron-neutron forces are equal.
Nonconservation of isotopic spin is
indicated by the unequal masses of
charged and neutral pions and of the
proton and neutron, the nonlinearity
inherent in superimposing nuclear and
electromagnetic forces and the differ-
ing anomalous magnetic moments of
proton and neutron. Fortunately for
us here, these are all rather small ef-
fects. Assumption of charge inde-
pendence is, we shall see, indispensible
for analysis of existing neutron-proton
scattering data.
Because nucleon-nucleon scatter-
ing occurs simultaneously in five inde-
pendent spin states, it is necessary to
analyze five kinds of scattering experi-
ments at a given energy simultaneously
to determine the elastic-scattering
matrix at that energy. This fact makes
a definitive experimental determination
of the scattering amplitudes a formi-
dable task.
Observed spin and isospin dependence
Observations pointing both to the
greatest complication in the nuclear
force and to its greatest simplification
occurred in 1936. The deuteron is a
neutron and proton bound in a triplet
spin state. From the deuteron struc-
ture we can infer the cross-section
magnitude for triplet neutron-proton
scattering at zero energy. The ob-
served neutron-proton scattering,
which is 1/4 in the singlet state and
3/4 in the triplet state, is much larger
than this value. Thus, as Wigner7
pointed out, singlet neutron-proton
scattering must be much larger thanGREGORY BREIT (left) and EUGENE WIGNER. This famous resonance is shown
at the Gainesville, Florida international nucleon-nucleon conference in 1967. —FIG. 2
triplet neutron-proton scattering; this
work established that nuclear forces,
unlike simple Coulomb forces, are spin
dependent.
Further evidence for spin depen-
dence of the nuclear force was soon
forthcoming. In 1939, measurements
of the magnetic moment and electric
quadrupole moment of the deuteron8
showed that tensor forces, leading to a
D-state admixture in the S-wave
ground state, are present. Analysis of
high-energy proton-proton scattering
data by Kenneth Case and Abraham
Pais0 in 1950 showed that spin-orbit
components are also present in the
nuclear force. The conclusion that we
have today, which is borne out by
studies with a variety of potential
models, is that nature has taken full
advantage of the freedom in nucleon-
nucleon spin space afforded by the in-
variance principles; spin dependence
of nuclear force is as complicated asit is allowed to be.
The greatest and perhaps only sim-
plicity in nucleon-nucleon scattering
occurs in isotopic-spin space. In 1936,
Gregory Breit and Eugene Feenberg10
analyzed low-energy neutron-proton
and proton-proton scattering and
showed that the singlet-S nuclear
phase shift (the 7 = 1 scattering) is
the same for both processes to within
a few percent, thus experimentally es-
tablishing charge independence.
Many subsequent experiments during
the past third of a century have sub-
stantiated the usefulness of the charge-
independence approximation. It is in-
teresting that Breit and coworkers11
in 1968 were the first to introduce a
nucleon-nucleon phase-shift analysis
in which charge independence is no
longer strictly assumed. Breit's work
in nucleon-nucleon interactions has
spanned the entire modern develop-
ment of the subject (see figure 2).
Amplitude
1
2
3
4
5Spin S
0
1
1
1
1Orbital An£
Initial
1 = J
1 = J
I = J + 1
I=J-1
1 = J ± 1?ular Momentum 1
Final
1 = J
1 = J
I = J + 1
1 = J - 1
I = J +1
SPIN-SPACE NUCLEON-NUCLEON amplitudes for Jtotal angular momentum J. In-
trinsic spin and orbital angular momentum can couple in five ways. —FIG. 1Nucleon-nucleon amplitudes
As we have seen, the proton-proton
system has five complex amplitudes.
If we eliminate one overall phase fac-
tor, specification of these amplitudes
at one energy and angle requires nine
numbers and hence nine independent
experiments. If, however, measure-
ments are made over all angles from 0
deg to 90 deg at one energy, then uni-
tarity relations12 relating real and
imaginary parts of the amplitudes can
be formulated. The result is that, in
principle at least, five kinds of experi-
ments at one energy and all angles
suffice to specify the proton-proton
scattering matrix at that energy.
Because all experiments contain sta-
22 . DECEMBER 1969 . PHYSICS TODAY
over-tistical and other uncertainties,
specification of the scattering matrix is
desirable. If we do proton-proton
measurements at energies above the
pion-production threshold (280 Mev),
then the unitarity relations are lost;
so nine experiments are again required
to specify the elastic-scattering matrix.
In practice, inelastic effects are small
up to 450 MeV, and accurate phase-
shift analyses can be made up to that
energy.
The neutron-proton system has five
complex amplitudes for each of two
isotopic-spin amplitudes. Because
neutron-proton measurements from 0
deg to 90 deg and from 90 deg to 180
deg can be considered as independent
experiments, five measurements over
the angular range from 0 deg to 180
deg are enough to specify the neutron-
proton scattering matrix for energies
below 280 MeV. Unfortunately, neu-
tron-proton data are often of limited
statistical accuracy and often include
only a few scattering angles. Also,
neutron-proton experiments sometimes
involve deuterium as a neutron target.
MThis use makes a substantial and
somewhat controversial correction nec-
essary to remove the effect of the spec-
tator proton that is contained in the
deuteron.
Thus, except at lowest energies, at-
tempts to analyze neutron-proton data
by themselves13 have been unsuccess-
ful. If, however, proton-proton and
neutron-proton data at the same en-
ergy are available, then the proton-
proton data can be analyzed to give
the / = 1 amplitudes, charge inde-
pendence can be invoked to apply
these to the neutron-proton scattering
and the neutron-proton data can then
be analyzed to give the corresponding
7 = 0 amplitudes. An analysis of this
type was first published in 1961.
Neutron-neutron experiments are
difficult; so few of them have been
done. The main effort here has cen-
tered on the final-state neutron-neu-
tron interaction that is produced when
deuterium is bombarded with neu-
trons or pions. Results indicate agree-
ment to within 1% with the concept
of charge symmetry and to within a
A'
CKPfew percentage points with the con-
cept of charge independence.15
Nucleon-nucleon experiments
As we have seen, in the most general
case nine proton-proton experiments
are needed to specify the elastic-scat-
tering matrix at one energy and angle.
Not surprisingly, it turns out that nine
independent spin-space experiments
can be simultaneously defined.1216
Experiments were first done by scat-
tering protons once (o-), twice
(P, CNN, CK1,), and three times (D, DT,
R, R', A, A'). Figure 3 describes
these observables. The recent devel-
opment, however, of polarized proton
beams and polarized targets has en-
abled experimenters to reduce the
number of scatterings by one and to
improve greatly the accuracy and
comprehensiveness of the experiments.
This "second generation" of experi-
ments is just now starting to have an
important impact on nucleon-nucleon
work.
Fairly complete sets of nucleon-
nucleon data exist at 25, 50, 95, 142,
210, 330, 425 and 650 MeV. These
energies correspond, naturally enough,
to energies of existing cyclotrons. It
is interesting that, their primary mis-
sion of measuring nucleon-nucleon
scattering fulfilled, some of these cy-
clotrons are now being scrapped.
Plwse shift analyses
One major difficulty in analyzing nu-
cleon-nucleon data is that we have so
little theoretical guidance. Scattering
amplitudes are essentially unknown
functions of energy E and scattering
angle 0. The conventional way of
dealing with this situation is to ex-
pand the scattering amplitudes in
terms of angular-momentum states
a(£,0) = f(£) g(0)
The g(0) are known functions that de-
pend on the spin, orbital angular mo-
mentum and total angular momentum
(S, I, J) of the system. The i(E) are
unknown functions of energy and are
expressed in the following unitary form
where the phase shifts S(E) must of
course carry labels S, /, /.
The spectroscopic form for the
phase shifts17
POSSIBLE NUCLEON-NUCLEON EXPERIMENTS. Laboratory-frame diagrams
show polarization component to be measured; a dot indicates a vector out of the page,
and M indicates 90 deg precession in a magnetic field. —FIG. 3is probably the prevalent notation to-
day; notation used by the Yale group
is very similar.18 For the nuclear-bar
PHYSICS TODAY • DECEMBER 1969 • 23
phase shifts,17 as used for example in
our work at Livermore, the lowest
states of the proton-proton system are:
%, 3P0, *Flf 3PL,, e,, 3F2, .'. ., where
S, P, D, F, . . . correspond to / = 0, 1,
2, 3, . . ., and where ej is the mixing
parameter.
The phase-shift decomposition of
scattering amplitudes has several ad-
vantages: because a few low-/ phases
dominate the scattering the number of
free (phenomenological) phases can
be kept reasonably small; physical in-
formation can be inserted by using
effective-range low-energy limits for
S-waves; theory can be inserted by
calculating the small, high-/ phases
from the one-pion-exchange Feynman
diagram and the observed energy de-
pendence of the phase shifts can be
used to test theoretical models. The
outstanding disadvantage of phase-
shift formalism is that the equations
are nonlinear.
Calculation of phases directly from
experiment or of potentials directly
from phases has proved to be impos-
sible. It is necessary instead to go in
the other direction. This means that
we can determine a set of phase shifts
only by making least-squares fits to
the data, and we can determine pa-
rameters of a nuclear-force model only
by making least-squares fits to the
phases or to the data directly.
Phase-shift analyses can be made at
a single energy (actually a narrow
energy band), or over a whole range
of energies. For a range of energies
we must choose a set of parameters
that express the energy dependence of
the phase shifts, and these parameters
are then varied to minimize the least-
squares-sum x2. The only two groups
to carry out large-scale energy-de-
pendent analyses have been Breit's
Yale group and the Livermore group.
Energy-independent analyses have
been carried out at several labora-
tories.
To determine phase shifts one se-
lects a set of phases, calculates the
corresponding observables, determines
the least-square sum ^2 for a fit to the
data and then varies the phases to
minimize ^2. If the data are complete,
statistically accurate and self-consis-
tent, a unique solution (set of phases)
results. In a typical analysis, 1000 to
2000 data may be included in the ^2
sum. The variable parameters, which
include both phase-shift coefficients
and data-normalization constants, can
number 100 or more. Thus selection
of a method to minimize the param-eters is a nontrivial part of the prob-
lem.
Early computer problems at Liver-
more used the grid-search method, in
which one parameter at a time is
varied. Because the parameters are
highly correlated, this is a very in-
efficient method for a large problem.
An improved method, used in early
work at Yale,18 is the gradient search
in which all parameters are varied to-
gether but in an uncorrelated manner.
The most efficient method for large
problems is the matrix search,19 in
which all parameters are varied simul-
taneously along a correlated path in
parameter space.
Although the matrix search has been
used in other applications for a long
time, its first application to the nu-
cleon-nucleon problem was by Peter
Signell.20 The matrix search has an
0.1%-additional advantage; the error matrix
for the solution is automatically ob-
tained. At Livermore, a method of
matrix reduction devised by Richard
Amdt19 is used to split phase param-
eters and normalization constants into
a two-step minimization process. This
method lowers the dimensionality of
the matrices by almost a factor of 2
and greatly reduces computer storage
requirements.
Early phase-shift results
The first use of a "computer" to at-
tack the nucleon-nucleon problem was
in the work done by E. Clementel and
Claudio Villi21 in 1955. Their com-
puter was a set of mechanical arms
that could be set to give an analog
simulation of certain scattering-ampli-
tude functions. They were able to
show that, given only proton-proton
SQUARE OF COUPLING CONSTANT (g )
PROBABILITY FUNCTIONS for 310-MeV Stapp phase-shift solutions. The maximum
probability obtained for Stapp solutions 1 and 2 at g2 n 14 agrees with the g2 = 15
value obtained from pion-nucleon scattering analyses. —FIG. 4
24 • DECEMBER 1969 • PHYSICS TODAY
differential cross-section data, there
are four sets of P-phases for each value
of the S-phase (up to some maximal
value for S), and all give precisely the
same fit to the data. This work was
later adapted at Livermore22 for UNI-
VAC I, the world's first true electronic
computer.
Modem phase-shift analysis started
at Berkeley. In 1956 a group using
the 184-inch cyclotron completed mea-
surements of a, P, D, R and A at 315
MeV.23 Armed with these data, Henry
Stapp and his collaborators, who
had access to Livermore and Los Ala-
mos computers, did a proton-proton
phase-shift analysis. They used 14
free phases (S-H waves), set the re-
mainder equal to zero and found five
acceptable phase-shift solutions.
Following the lead of the Japanese
school,24 Michael Moravcsik25 and A.
F. Grashin20 independently proposed
that the Stapp analysis could be im-
proved by calculating the higher phase
shifts from one-pion exchange (OPE)
instead of just setting them equal to
zero. This reduced the number of
acceptable solutions to two, called
"Stapp solutions 1 and 2." In addi-
tion, the pion-nucleon coupling con-
stant, which enters into the calculation
of the OPE phases, was shown for the
first time to have a nucleon-nucleon
analysis value consistent with that ob-
tained from pion-nucleon analyses.
These results are illustrated in Figure
4. Subsequent analyses of Rochester
proton-proton data at 210 MeV27
showed that the Stapp-solution types
1 and 2 occurred there also. Later
analyses have shown that Stapp solu-
tion number 1 is the correct one.
The first energy-dependent analysis
of proton-proton scattering was car-
ried out by the Yale group,28 and was
soon followed by a similar analysis at
Livermore.29 Subsequent Yale anal-
yses14 included both proton-proton
and neutron-proton scattering.
Recent elastic-scattering studies
Analyses of energy-independent phase-
shifts have been carried out by groups
at Berkeley, CERN, Dubna, Harwell,
Kyoto, Livermore, and Michigan State.
All used essentially the same method
of analysis; differences in solutions
can be attributed to slightly different
choices of data or of the number of
phenomenological phases. The results
of these analyses are in general agree-
ment with each other.
The Yale group11 has carried out
energy-dependent phase-shift analyses<
<</> «P v ABC -
2- Continuum
3- Continuum
4- Continuum
K-K ContinuumElastic
\
ScatteringInelastic
/
-600 -500 -400 -300 -200 -100 0
REAL PART Re (T) (MeV)100 200 300 400
SINGULARITY STRUCTURE in the complex kinetic-energy plane for nucleon-nu-
cleon scattering amplitudes. Poles on negative real axis become cuts when a partial
wave projection is made. Left-hand singularities correspond to nuclear forces, and
right-hand singularities are unitarity cuts. Here T = 4KV2M. —FIG. 5
of proton-proton and neutron-proton
scattering from about 10 MeV to 350
MeV. We at Livermore have com-
pleted similar analyses from about 1
MeV to 450 MeV30 and additional
analyses extending to 750 MeV.3182
The Yale group, in choosing energy-
dependent forms for the phase shifts,
selected pure mathematical functions
with the requisite flexibility to fit their
data.
At Livermore functions that have a
singularity structure19 and threshold
behavior33 consistent with the dictates
of the Mandelstam representation were
used (see figure 5). In regions where
data are complete and accurate enough
to set limits on the solution, the Yale
and Livermore phase-shift values are
in reasonable agreement. This agree-
ment indicates that neither analysis is
appreciably form-limited and that en-
ergy dependences obtained for phase
shifts are reliable. As further confir-
mation the Livermore work also in-
cludes single-energy analyses at 25,
50, 95, 142, 210, 330, and 425 MeV.
The energy-dependent and energy-in-
dependent phase shifts are in agree-
ment; this agreement would not occur
if the energy-dependent forms were too
rigid.
These phase-shift results fulfill thelongstanding goal of obtaining a set
of nucleon-nucleon scattering ampli-
tudes that cover continuously the en-
tire elastic-scattering region. The final
Livermore analysis includes 1076 pro-
ton-proton data from 1 to 450 MeV
and 990 neutron-proton data from 0.5
to 425 MeV. 52 phenomenological
parameters representing 27 elastic
phases and one inelastic phase are
sufficient to give a statistically ac-
curate fit (^2 = 1.1 per data point) to
the entire collection of 2066 data span-
ning this energy range. Also, because
the parametrization is continuous and
mathematically well defined, the pa-
rameter error matrix gives statistically
determined uncertainties in the phases
and in all functions of the phases over
the energy range.
Remaining problems
There are still some difficulties with
phase-shift analyses, particularly with
the 7 = 0 amplitudes. At low energies
we expect from the sign of the deu-
teron quadrupole moment that the cx
coupling parameter should be posi-
tive.34 Also, the 1P1 phase shift might
be expected to approximate its OPE
value at low energies. Phase shift
analyses, however, often give anoma-
lous values below 50 MeV for these
PHYSICS TODAY • DECEMBER 1969 • 25
phases. The difficulty can be attrib-
uted to a lack of accurate neutron-
proton differential cross-section data at
low energies,30 but recent measure-
ments85 may remedy this deficiency.
Unfortunately, existing neutron-pro-
ton data below 50 MeV are not com-
pletely self-consistent. At energies
above 210 MeV, and particularly at
330 MeV, the neutron-proton data are
incomplete enough that an accurate /
= 0 matrix can not be defined. How-
ever, the latest triple-scattering neu-
tron-proton data at 425 MeV3e give
a well defined result for 7 = 0 ampli-
tudes at that energy. By adding these
data to the energy-dependent analysis,
it is possible to obtain reasonably re-
liable neutron-proton phase shifts at
330 MeV. This result illustrates one
of the virtues of an energy-dependent
analysis.
To achieve accurate fits to the data
below 10 MeV, one must apply vac-
uum-polarization corrections to the
proton-proton amplitudes and use sep-
arate 1S0 phases for the proton-proton
and neutron-proton systems. The
data are now so accurate that failure
of charge independence for the aS0
phase must be taken into account.11
The other phases do not yet require
this additional freedom.30
At energies above 280 MeV inelastic
effects should be considered. Up to
450 MeV, inelastic scattering is less
than 10% of elastic scattering. On
theoretical grounds it is reasonable to
attribute this small inelasticity entirely
to the 2Do phase shift. Inclusion of
an inelastic component in the 1D1»
phase does not appreciably lower x2,
but it gives slightly different and more
realistic phase-shift values.
To summarize the proton-proton sit-
uation, 1076 carefully selected pro-
ton-proton data form a set that spans
the 1-450 MeV region. This set
yields good statistical accuracy, rea-
sonable completeness at selected ener-
gies and self-consistency within the
data set. These data determine a
unique phase-shift solution; scattering
amplitudes are accurate to within a
few percent over the entire elastic en-
ergy range and up to about 450 MeV.
Restrictions imposed by fitting all of
these data simultaneously are stringent
enough that inconsistencies between
these data and any new measurements
can be promptly identified.87
The neutron-proton situation is not
so favorable: the 990 experimental
points form a set that spans the energy
region from 0.5 to 450 MeV, but al-though some selection has been made,
the remaining data are not completely
self-consistent. Also, statistical and
systematic uncertainties in some of the
data are quite large. The data are
nowhere complete and in many energy
regions are woefully incomplete.
Nevertheless, by combining the neu-
tron-proton data with proton-proton
data (or with the proton-proton
7=1 scattering matrix) and invoking
charge independence, we can obtain a
solution type that is reasonably well
delineated over most of this energy
region. Errors in the 7 = 0 phases
given by error matrices appear to be
realistic, although they must be used
with some reservations; an incomplete
data set can lead to actual errors much
larger than those predicted by the
standard statistical analysis, and sys-
tematic errors caused, for example, by
improper corrections for binding ef-
fects in the deuteron, would not be re-
flected in the error-matrix calculations.
Errors in energy-dependent phases
are given by the parameter error
matrix. These should be regarded as
the smallest possible errors and would
be the true errors if the energy-de-
pendent forms were correct. Errors
given by energy-independent analyses
should be regarded as the greatest
possible errors and would be the true
errors if experiments at one energy
were completely uncorrelated with ex-
periments at other energies. By carry-
ing out both types of analysis, we can
obtain bounds for the errors. Because
phase shifts are highly correlated, so
are the errors. To obtain accurate
statistical results in fitting to a model,
one must use the full error matrix; the
diagonal components are not sufficient.
Recent inelastic-scattering work
Inelastic corrections are small and
can be appropriately handled at ener-
gies up to 450 MeV. Few data exist
in the region between 450 and 600
MeV, but from 600 to 700 MeV quite
a complete proton-proton data set ex-
ists. Most of the data are from
Dubna,38 but substantial contributions
have been made at other laboratories,
such as Berkeley and Saclay, France.
The big difficulty at 650 MeV is that
the inelasticity is now roughly 40% of
the total scattering, and simple treat-
ment of inelastic phases does not suf-
fice.
Phase shift studies have been made
at 650 MeV,39 and solutions can be
obtained that give excellent fits to the
data. These solutions, however, in-volve a somewhat arbitrary handling
of the inelasticity; one must apportion
the inelasticity among a number of
phase shifts, and there is remarkably
little theoretical guidance as to just
how to do this. In studies at Liver-
more31 we tried many different models
for the inelasticity, and we obtained a
corresponding number of elastic phase-
shift solutions. Coupling between in-
elastic and elastic processes is strong.
Our conclusion at Livermore (to
which some of our colleagues do not
wholly subscribe40) is that a definitive
set of proton-proton phases at 650
MeV can not be obtained from present
data and the present state of inelastic-
scattering theory. Nine complete pro-
ton-proton experiments would in prin-
ciple define the proton-proton elastic-
scattering matrix at 650 MeV, but
these experiments do not yet all exist.
The data on inelastic scattering and
the theory to handle these data are
both very sketchy. High-intensity
cyclotrons planned for the Swiss Fed-
eral Institute of Technology, Zurich
and for Los Alamos should supply im-
portant new measurements in this en-
ergy region.
Any 650-MeV neutron-proton anal-
ysis, because it necessarily depends on
7=1 amplitudes obtained from pro-
ton-proton scattering and on 7 = 0 in-
elastic effects, is thus almost meaning-
less. Solutions can be obtained that
give precision fits to the data, and the
magnitudes of the large 7 = 0 phases
can be roughly determined. But small
uncertainties in 7 = 1 amplitudes be-
come large uncertainties in addition to
uncertainties for the 7 = 0 amplitudes.
As far as definitive phase-shift analy-
ses of the nucleon-nucleon system are
concerned, I feel that present experi-
mental and theoretical situations com-
bine to impose a sharp cutoff at 450
MeV; this is perhaps just a way of
saying that the opportunities exist at
higher energies.
Implications for theory
The outstanding theoretical success in
the nucleon-nucleon field in the last
decade has been the one-boson-ex-
change (OBE) model.41 The only
part of the nuclear force that can be
calculated unambiguously from field
theory is that due to exchange of a
single (virtual) particle, the pion. If,
however, we consider narrow reso-
nances in multipion states as "parti-
cles," then we can calculate their con-
tributions to the nuclear force. It is a
remarkable fact that if the pion, the
26 • DECEMBER 1969 • PHYSICS TODAY
100 200 300 0
ENERGY (MeV)100 200 300
P-WAVES as determined experimentally (error flags) and as calculated from one-boson
exchange, TT, />, w and a Born terms all make important contributions, and the sum
(heavy solid line) is in good qualitative agreement with experiment. —FIG. 6
p and OJ resonances and a strong scalar-
isoscalar interaction (taken for con-
venience to be the a resonance) are
treated in Born approximation, they
give phase-shift values for P-waves
and higher that are in good qualitative
agreement with experiment for proton-
proton and neutron-proton scattering
over the elastic-scattering range (see
figure 6). Furthermore, masses and
coupling constants that must be used
to obtain this good fit agree with
values that can be deduced from direct
measurements and other physical pro-
cesses.42
There are, however, definite limi-
tations to the one-boson-exchange
model; unitarity corrections to the
Born terms are small and unimportant
for the high-/ amplitudes and are large
and unbelievable for the lowest-/ am-
plitudes. Thus, although the lowest-order OBE model is a good one, it is
difficult to improve. The challenge
imposed on theorists by the OBE
model is to explain the existence of the
p and OJ resonances. Just why these
saturate their 2?r and 3?r quantum
states, and why they contribute so de-
cisively to the nuclear force, is in my
opinion the main question to be an-
swered by nucleon-nucleon theorists.
From nucleon-nucleon analyses, we
can not conclude much of anything
about the width of the a resonance.
A strong enhancement in this state,
however, is certainly required to fit
the data.
The challenge at higher energies is
to calculate, in a useful way, the pion-
production amplitudes. It is clear
from low-energy work that exchange
of a single virtual pion is the domi-
nant mechanism in all phases (evenincluding P-waves!) except S-waves.
At energies above 280 MeV we are in
the regime where a real pion is pro-
duced. One feels intuitively that
ability to handle the appearance of a
real pion from the virtual cloud sur-
rounding a nucleon would contribute
substantially to understanding the
properties of that cloud.
Another challenge, one that may
perhaps be studied at both lower and
higher energies, is to see what limits
nucleon-nucleon scattering data im-
pose on interaction at very short dis-
tances.43 The hard core has recently
become in many models a softer core,
and the nature of the core region is
important in nuclear-structure calcula-
tions. The extent to which measured
nucleon-nucleon amplitudes limit this
region, and the relevance of these am-
plitudes to phenomena like the non-
locality of the potential, remain fruit-
ful areas for investigation.
At the crossroads
In the 1930's broad features of nu-
cleon-nucleon interaction were de-
termined. The ensuing three decades
have seen this work extended experi-
mentally until now a reasonably com-
plete mapping has been obtained for
proton-proton and neutron-proton
scattering over the entire elastic en-
ergy region. Roughly speaking, this
mapping has an accuracy of perhaps
5% for proton-proton scattering and
10c/c for neutron-proton scattering.
This accuracy is good enough to im-
pose reasonable bounds on potential
models, and to make it appear un-
likely that any major surprises will
occur if these experiments are ex-
tended at the same level of sophistica-
tion.
The experimenter's choice is to re-
tire or to aim for the \c/c level. At
Harvard and Rochester the choice was
to retire. At Berkeley, Chicago,
Dubna, Los Alamos and Orsay, experi-
ments featuring polarized targets are
superseding older triple-scattering ex-
periments. At Saclay, a recent en-
trant into low-energy nucleon-nucleon
work, a high intensity polarized proton
ion source has been developed. Simi-
lar beams for low-energy measure-
ments have been developed at Berke-
ley and Los Alamos.
At VA -accuracy level, phase-shift
analyses must include careful correc-
tions for magnetic-moment effects, vac-
uum-polarization effects and manifest-
ations of charge-independence break-
down Theoretical models should be-
PHYSICS TODAY • DECEMBER 1969 • 27
gin to show some sorting of 2?r and 3?r
effects. Because experimental uncer-
tainties are magnified in analytically
continuing the scattering amplitudes
off the energy shell, improved ac-
curacy would permit a better deter-
mination of the usefulness of the boot-
strap concept in this area.
In the inelastic region, the right
turn at the crossroads would lead to
a double-barreled experimental-theo-
retical attack on the nucleon-nucleon
problem. Theorists must derive
models for production processes, tell
experimentalists just what kind of
pion-production experiments they
need to test the models, recheck their
models with the experiments and re-
peat the process. An on-line collabo-
ration is needed to get meaningful re-
sults in this difficult area. The plan-
ning groups at Ziirich and Los Alamos
see the need for this kind of close
collaboration between theory and ex-periment, and their new experimental
facilities will include, they hope, asso-
ciated theoretical groups. Dubna, and
other very high energy laboratories
have of course followed such as ap-
proach for years.
A new field of physics
One outcome of this work is the emer-
gence just now of a new field that we
might call intermediate-energy ele-
mentary-particle physics. High-en-
ergy physicists have remarkably little
interest in anything that happens be-
low a few GeV; nuclear physicists
have no reason to be interested in any-
thing higher than a couple of hundred
MeV. Thus physicists who wish to
work at 500 MeV find that they are
no longer welcome at the crowded
high-energy conferences, and they
can't understand what is going on at
the nuclear-physics conferences. Sothey have, in desperation, finally
started their own conferences.39-44
For nucleon-nucleon workers, this dif-
ficulty with energies is compounded
because the nucleon-nucleon field is
itself in the gray area between ele-
mentary-particle physics and nuclear
physics. Does it belong in volume 4 or
volume 5 of the Physical Review?
Development of polarized ion
sources, polarized targets and high-in-
tensity accelerators signals the begin-
ning of the next generation of nucleon-
nucleon and pion-nucleon experi-
ments. Workers in this field of inter-
mediate-energy physics will form a
more distinctive branch of physics than
was apparent in the past. If they suc-
ceed, however, in knocking down any
of the formidable barriers that limit
our present understanding, we can be
assured that the consequences will be
felt by their colleagues both above and
below.
References
1. I. Curie-Joliot, F. Joliot, Compt.
Rend. 194, 273 (1932); J. Chadwick,
Proc. Roy. Soc. (London) A136, 692
(1932).
2. J. D. Cockcroft, E. T. S. Walton,
Proc. Roy. Soc. (London) A136, 619
(1932).
3. E. P. Wigner, Phys. Rev. 43, 252
(1933).
4. W. Heisenberg, Z. Physik 77, 1
(1932).
5. E. Majorana, Z. Physik 82, 137
(1933).
6. H. Yukawa, Proc. Phys-Math. Soc.
(Japan) 17,48 (1935).
7. H. A. Bethe, R. F. Bacher, Rev. Mod.
Phys. 8, 193 (1936).
8. J. M. B. Kellogg, I. I. Rabi, N. F.
Ramsey, Jr, J. R. Zacharias, Phys.
Rev. 55, 318 (1939); 56, 728 (1939).
9. K. M. Case, A. Pais, Phys. Rev. 80,
203 (1950).
10. G. Breit, E. Feenberg, Phys. Rev. 50,
850(1936).
11. R. E. Seamon, K. A. Friedman, G.
Breit, R. D. Haracz, J. M. Holt, A.
Prakash, Phys. Rev. 165, 1579 (1968).
12. L. Puzikov, R. Ryndin, J. Smorodin-
skij, Nuclear Physics 3, 436 (1957).
13. M. H. Mac Gregor, R. A. Arndt, A. A.
Dubow, Phys. Rev. 135, B628 (1964).
14. M. Hull, K. Lassila, H. Ruppel, F.
McDonald, G. Breit, Phys. Rev. 122,
1606 (1961); M. H. Mac Gregor,
Phys. Rev. 123,2154 (1961).
15. I. Slausv Rev. Mod. Phys. 39, 575
(1967).
16. R. J. N. Phillips, Nucleon-Nucleon
Scattering Experiments, Harwell re-
port AERE-R3141 (1960).
17. H. P. Stapp, T. Ypsilantis, N. Metrop-
olis, Phys. Rev. 105, 302 (1957).18. G. Breit et al, Phys. Rev. 128, 826
and 830 (1962).
19. R. A. Arndt, M. H. Mac Gregor, Phys.
Rev. 141, 873 (1966); R. A. Arndt,
M. H. Mac Gregor, Methods in Com-
putational Physics, vol. 6, Academic
Press (1966).
20. P. Signell, N. R. Yoder, N. M. Mi-
skovsky, Phys. Rev. 133, B1495
(1964).
21. E. Clementel, C. Villi, Nuovo Ci-
mento [10] 2, 1165 (1965).
22. H. P. Noyes, M. H. Mac Gregor, Phys.
Rev. Ill, 223 (1958); M. H. Mac
Gregor, Phys. Rev. 113, 1559 (1959).
23. O. Chamberlain, E. Segre, R. D.
Tripp, C. Wiegand, T. Ypsilantis,
Phys. Rev. 105,288 (1957).
24. Prog. Theoret. Phys., Supplement 3,
(Kyoto) (1956).
25. P. Cziffra, M. H. Mac Gregor, M. J.
Moravcsik, H. P. Stapp, Phys. Rev.
114,880 (1959).
26. A. F. Grashin, Sow Phys.-JETP 9,
1223 (1959).
27. M. H. Mac Gregor, M. J. Moravcsik,
Phys. Rev. Lett. 4, 524 (1960).
28. Reported at the London "Few Nu-
cleon Conference," 1959.
29. H. P. Stapp, H. P. Noyes, M. J. Mor-
avcsik, Proceedings of the 1960 High
Energy Conferences at Rochester, p.
128; Proceedings of the 1962 High
Energy Conferences at CERN, p. 131.
30. M. H. Mac Gregor, R. A. Arndt, R. M.
Wright, Phys. Rev. 182, 1714, (1969).
31. M. H. Mac Gregor, R. A. Arndt, R. M.
Wright, Phys. Rev. 182, 1714 (1969).
32. M. H. Mac Gregor, R. M. Wright,
Phys. Rev. 173, 1272 (1968).
33. M. H. Mac Gregor, Phys. Rev. Lett.
12,403 (1964).34. G. Breit, R. D. Haracz, High Energy
Physics, (E. H. S. Burhop, ed.) vol.
1, p. 21, Academic Press Inc., New
York, (1967).
35. L. N. Rothenberg, T. G. Masterson,
Angular Distribution of 24-MeV Neu-
trons Scattered by Protons, Univ. of
Wise, abstract for APS Washington
meeting, April (1969).
36. S. C. Wright, D. Shawhan, L. Pon-
drom, S. Olsen, R. Handler, Phys.
Rev. 175, 1704 (1968).
37. M. H. Mac Gregor, R. M. Wright,
R. A. Arndt, Phys. Rev. Lett. 19,
1209 (1967); M. H. Mac Gregor,
R. A. Arndt, R. M. Wright, Phys.
Rev. 179, 1624 (1969).
38. J. Bystricky, J. Cech, Z. Janout, Yu.
M. Kazarinov, F. Lehar, L. B. Par-
fenov, Phys. Lett. 28B, 572 (1969).
39. Proceedings of the 1st International
Colloquium on Nucleon-Nucleon and
Pion-Nucleon Interactions, Dubna,
lune (1968) (in Russian).
40. S. I. Bilenkaya, G. Cozzika, F. Lehar,
Z. Janout, Phase-Shift Analysis of p-p
and n-p Elastic Scattering at 735
MeV, CERN preprint, June 1969.
41. N. Hoshizaki, S. Otsuki, W. Watari,
M. Yonezawa, Progr. Theoret. Phys.
(Kyoto) 27, 1199 (1962); R. A.
Bryan, C. R. Dismukes, W. Ramsay,
Nucl. Phys. 45, 353 (1963).
42. R. A. Arndt, R. A. Bryan, M. H. Mac
Gregor, Phys. Lett. 21, 314 (1966).
43. S. Stone III, Univ. of Calif, at
Berkeley thesis, A Eorm-Free, Semi-
phenomenological, Error-Bounded,
Potential Representation of Two-Nu-
cleon Experiments, July 1969.
44. A. E. S. Green, M. H. Mac Gregor,
R. Wilson, Rev. Mod. Phys. 39, 495
(1967). 0
28 . DECEMBER 1969 . PHYSICS TODAY
NEW INFORMATION
PROGRAM FOR AIP
How do you cope with the ever-increasing flood of literature?
A new computer-assisted system will offer new and better
ways of obtaining physics information. We seek your opinions.
ARTHUR HERSCHMAN, FRANZ L. ALT and H. WILLIAM KOCH
THE AMERICAN INSTITUTE OF PHYSICS,
with support from the National
Science Foundation, is currently en-
gaged in a major effort to develop and
implement a computer-assisted "Na-
tional Information System for
Physics." Designed by physicists, for
physicists, the new system is sched-
uled to begin pilot operations early
next year. We at AIP believe the sys-
tem to be urgently needed, but how
do you, the physicists, feel about it?
This article presents a description of
the main features of the new system so
that you can form an opinion on its
merits and its potential usefulness.
After you have read the article, we
hope to hear from you on this impor-
tant question: Do you feel there is
need for this program and is it aimed
in the right direction?
AIP responsibility
Because we believe that there is a
need for a physics-information system
and that AIP is the logical place for its
development, we have assumed the
responsibility and undertaken the for-
mulation and development of a new
system.
AIP was founded in 1931 as a fed-
eration of leading societies in physics
to serve those needs of the physics
community that could best be fulfilled
by the societies jointly. It presently
has seven member societies, whose
47 000 members are also institute
members, plus 19 affiliated societies
with an interest in physics, 150 corpo-
rate associates and a Society of Phys-
ics Students. The institute services
for this sizable community run the
gamut from publicizing physics and
physicists, strengthening educational
programs, documenting the history
and development of physics and rep-resenting physics nationally and inter-
nationally—to the largest single pub-
lishing effort for physics in the world.
AIP publishes 16 archival journals,
comprising 25% of the world's articles
in physics and translates 13 Russian
journals, for an additional 10%.
As a natural extension of its respon-
sibility and in accord with its mandate
to engage in activities "for the ad-
vancement and diffusion of the knowl-
edge of the science of physics," AIP,
with support from NSF, has been ac-
tively planning further information
services since mid 1966.1 These
plans led to our program for the de-
sign and development of a national in-
formation system for physics. At the
end of 1967, a new division was or-
ganized within AIP to handle the
project.2
The division has the assistance of a15-member advisory committee, which
was appointed by AIP member so-
cieties, of about 100 physicist-respon-
dents selected by the advisory com-
mittee and of liaison members from
other interested groups, both from re-
lated scientific societies (chemistry,
mathematics and engineering) and
from interested government agencies.
The results of this effort, a national
physics-information system, will be
ready for implementation during
1970. (A document describing the
proposed system was recently present-
ed to NSF in support of a request for
funding the pilot operations.)3
Why a new system?
A new system is needed to cope with
the exponential growth of physics lit-
erature, which has been doubling
about every seven and a half years. It
Arthur Herschman (left) has been director of the AIP information division since its
inception in 1967. A theoretical physicist, who received his PhD from Yale Uni-
versity in 1954, Herschman was formerly coeditor of The Physical Review.
Before becoming AIP director in 1966, H. William Koch (center) was chief of the
radiation-physics division at the National Bureau of Standards. Koch joined NBS
in 1949, after receiving his PhD from the University of Illinois, and worked in the
high-energy-radiation section until becoming division chief.
Franz L. Alt (right), who took his PhD in mathematics at the University of Vienna,
became deputy director of the information division after 19 years with the National
Bureau of Standards. At NBS he was assistant chief of the applied-mathematics
division and, later, area manager for information systems, design and research.
PHYSICS TODAY • DECEMBER 1969 • 29
is not that physicists are writing more,
but that more physicists are writing-
more physicists in every speciality. In
1968 alone, over 50 000 research pa-
pers were published in more than 500
journals. Finding information in the
traditional way, by scanning journals
and through formal and informal talks
at meetings, is no longer practical.
Although one may still keep up with
new developments of immediate inter-
est, it is almost impossible for any one
physicist to keep abreast of bordering
areas and related specialities with
which he should be familiar.
The tendency of the present proce-
dures to be designed for authors* con-
venience has also aggravated the
problem. Authors, not readers, deter-
mine when, where and how the infor-
mation is presented. As a result, pa-
pers on any given subject are dis-
persed over many journals, and a sin-
gle journal may contain, side by side,
papers on widely different subjects.
The reader is left to cope with the
flood as best he can, which all too
often results in information coming to
his attention too late for his need.
What is clearly required is a better
way to organize and manage the infor-
mation so it can be routed more accu-
rately and efficiently from author to
reader.
Computerized file
The only feasible way to organize and
manage a collection this large is by
computer—to have a computer record
of each new paper that allows a file to
be organized and searched on a cur-
rent basis according to physicists' in-
terests.
As presently conceived, the file
would initially contain records for
about one half of the world's physics-
journal articles, but would be expand-
ed to cover almost all journal litera-
ture, as well as nonjournal material, in
the not-too-distant future.
Every month, for AlP-published
journals during the prepublication
cycle and for other journals as they are
received, the information-division staff
will prepare, for each new paper, a
"record" that contains basic informa-
tion about the paper: author, journal,
title, abstract, citations (that is, refer-
ences to other literature), a list of
"key words" and a special "AIP classi-
fication number."
These records are then transcribed
onto magnetic tapes. Thus the cumu-
lative file of all such records consti-
tutes, in effect, a machine-searchable"physics almanac" that can be queried
for a multitude of purposes and pro-
duces a variety of services. Specially
formatted printed versions of all or
part of the file can be widely distrib-
uted for ready reference. The file it-
self could answer specific questions
both at AIP and at suitably equipped
subscribing institutions.
Each item in the article record rep-
resents a "handle" that can retrieve
the complete record. Thus one can
ask for all articles published in a given
journal or year or by a particular author
or institution; papers that contain cer-
tain specific words in their title or
abstract or that cite another paper orhave a number of citations in common
with a given paper; and finally, papers
about a particular kind of physics.
Classification scheme
The classification procedure that the
system as a whole will use was devel-
oped by AIP in cooperation with out-
side physicists specializing in various
branches. It is a procedure for writ-
ing a formatted statement of what,
objectively, a paper is about. The box
on page 31 is an example and shows
how the classification number is con-
structed using "verb expressions," rep-
resented as integers, followed by
"nouns," represented as decimal num-
"THE READER IS LEFT TO COPE with the flood as best he can ,
30 • DECEMBER 1969 . PHYSICS TODAY
bers. Typical verb expressions are:
"the subject of primary interest is . . ."
and "the method used is . . ." and "the
host or environment is ... ."
The more digits in a numerical
noun, the more specific its meaning.
For example ".2" means particle, ".28"
means hadron, ".282" means baryon,
".2821" means nucleon, ".28211"
means proton.
This particular way of "spelling"
nouns has the advantage of exhibiting
the word roots. Suppose, for in-
stance, you want papers on hadrons.
All nouns beginning with the digits
".28" belong to the class hadron.
Knowing this, a request can easily be
formulated. The same request, in
clear language, would require a speci-
fication of all words included in the
class hadron (meson, pion, kaon, bar-
von, hyperon and nucleon). In the
example (right) the title is less ex-
plicit, from the viewpoint of informa-
tion retrieval, than is the classification
number.
What the file will do
One of the principles underlying the
design of the system was that it should
evolve gradually; not only will its cov-
erage of physics literature be in-
creased step by step, but also its ser-
vices wall become more sophisticated
in stages. Thus we can improve the
system as we go along.
The services that will be offered
during 1970 and early 1971 are all
straightforward products of the infor-
mation file:
•Current Physics Titles (CPT), a
current-awareness journal, initially in
four sections that probably will be:
particle, field and nuclear physics;
atomic, molecular, chemical, plasma
and fluid physics; solid-state physics;
and optics, acoustics, astrophysics and
geophysics. We expect the sections
will be published every other week,
with each section representing a print-
out of the accumulated records since
the previous issue. The records will
be arranged, under a new system of
headings, in as many places as physi-
cists would expect to find them. The
journal will be produced through com-
puter-controlled photocomposition, so
that it will be of high typographic
quality.
•A series of specialized bibliogra-
phies in several of the narrower fields
of physics (updated periodically) as
well as indexes for the primary jour-
nals published by AIP
• Searchable Physics InformationCLASSIFICATION EXAMPLE
For the paper "Evidence of Quarks in Air-Shower Cores'
0.1; 1.271; 2.9534; 4.24; 6.29
0 The document type is ...
0.1 experimental;
1 The subject of primary interest is ...
1.2 particle physics,
1.27 more precisely, a particle property,
1.271 specifically, its existence;
2 The method used is ...
2.9 a technique,
2.95 more precisely, a particle technique,
2.953 still more precisely, a detection technique,
2.9534 specifically, track visualization;
4 The entity of primary interest is ...
4.2 a particle,
4.24 more precisely, a hypothetical particle;
6 The host or environment is ...
6.2 particles,
6.29 more precisely, cosmic rays.
Notices (SPIN), a magnetic-tape ser-
vice that will allow organizations with
adequate computer facilities to estab-
lish their own current file of physics
information. The tapes will be issued
monthly and will contain the records
accumulated since the previous issue.
The subscribing institution could use
its own search programs or specifically
designed AIP programs.
At a later date, the system will offer:
• File searches based on requests.
This service would be of particular
value to scientists writing reviews or
data compilations. Considering the
importance of this activity in evaluat-
ing-and distilling the literature into a
more meaningful and digestible form,
additional means for encouraging the
production of such articles are also
being planned.
• Lists of articles tailored to the
needs of groups of physicists working
in specialities and who do not have
local facilities for using the magnetic-
tape (SPIN) service, as well as proce-
dures for subdividing journals into
packages that would better suit the
needs of smaller interest groups.
• Microform copies of the primary
articles, as a backup to CPT and
SPIN.
Long-range prospects
We expect to improve the system on a
continuing basis, rendering services as
effectively, and as inexpensively, as
possible. As a long-range prospect,
we hope to offer a centralized service,
with decentralized satellites, thatwould cope with all the information
needs of the physics community as
well as those of the broader national
scientific and technical community for
physics information.
It would offer reference sendees
and would obtain copies of hard-to-get
material and refer questions it can not
answer to sources that could. The
system would also afford direct on-line
access to the computer file from
remote-access terminals in physics de-
partments and other institutions. The
centralized service also would have fa-
cilities for "scholars in residence," to
supply clerical and reference aid for
review writers.
Such a centralized facility would be
linked into a network of "information
centers" at various institutions and of
similar facilities for other disciplines
and for physics in other countries.
This organization, with its information
file and its broad spectrum of services,
linked into a network of other infor-
mation centers and services would
constitute the "National Information
System for Physics."
Value and cost
Each potential user must determine
for himself what a service like this is
worth. How many hours per week do
you spend looking for information?
How many hours would you save if
you only had to look through one
booklet, a short list or a response on
your computer terminal? How much
time would you save if the article was
in one small collection or a numbered
PHYSICS TODAY . DECEMBER 1969 • 31
entry on a reel of microfilm that your
librarian or secretary could copy?
These questions raise a number of
imponderables. To put them into
better perspective, consider that each
published article represents about
$60 000 worth of research investment;costs about $500 to be published and
will cost about $15 to process and
enter into the proposed physics-infor-
mation file. The distribution price for
listing that article in CPT will be only
a fraction of a cent per copy. IT, say,
one out of a hundred articles is inter-
u- 6
o
d 5
9 4
_lSystem development
Chemistry
Physics
Other disciplines
University based and otherB Operational support
LJ Research and studies
LJ Translations and international
1967 1968 1969
FISCAL YEAR1970 (estimate)
COST OF INFORMATION SYSTEMS. Chart compares NSF support of information
programs by type and discipline during 1967-70. —FIG. 1
Chemistry 1.81%
Psychology 0.40%
Average 0.16%
Physics and astronomy 0.14%
Mathematics 0.14%
Biological sciences 0.05%
Social sciences 0.03%
Engineering sciences 0.01%
INFORMATION VS RESEARCH COST. Chart depicts ratio of NSF support for
development of information systems to all federal support of research in the fiscal year
1968. The high percentage for chemistry is partly because most chemistry research is
privately financed. FIG. 2esting to you and one out of a thou-
sand important, would it be worth-
while for you to have it pinpointed?
Similar considerations apply to the
other services. Is a system that could
accomplish these things worth the
cost?
The initial cost of development and
pilot operation is being funded by the
NSF, as part of its nationwide support
of information services in scientific
disciplines. Figures 1 and 2 show
KTSF expenditures for these purposes,
both in absolute magnitude and in
relation to total research support. Is
it in the national interest to have the
NSF support these programs? In
1968 NSF spent about $14 million for
information activities in various scien-
tific disciplines—less than 10% of the
total was for physics. The improve-
ment of efficiency in physics research
and development activities is clearly
in the national interest. The saving of
a fraction of an hour per week by each
of the 30 000 physicists in the Nation-
al Register of Scientific and Technical
Personnel, not to mention all the other
users of physics information, would
more than make up for all of the costs
borne by the NSF.
In the future many of the operating
services are expected to be self-sup-
porting after the requested funding
period ends in 1973. Some of the
newer services would still need subsi-
dies, and funds would still be required
for further development of longer-
ranged projects. The rate of NSF
support, however, would probably de-
crease, and the additional cost would
be offset by the greater values of the
ultimate system.
A question
This program has been endorsed by
the AIP governing board, which rep-
resents the member societies. But
considering the magnitude of this un-
dertaking, we would like the addi-
tional opinions of individual physi-
cists: What do you think of our pro-
posed system? Please write us and
give us the benefits of your views and
ideas on this matter.
References
1. V. Z. Williams, E. Hutchisson, H. C.
Wolfe, PHYSICS TODAY, 19, no. 1, 45
(1966).
2. H. W. Koch, PHYSICS TODAY, 21, no. 4,
41 (1968).
3. A Program for a National Information
System for Physics, American Institute
of Physics, publication no. 1D69R
(August 1969). D
32 • DECEMBER 1969 . PHYSICS TODAY
100MHzball game.
And staying ahead
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Everywhere
you look, there's a
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In the compact multi-
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brawn, lots more brain that make everything
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And, hitting clean-up, our NIM-compatible
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Each subsystem accumulating and reading out
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PHYSICS TODAY DECEMBER 1969 33
PLASMASThis "fourth state of matter" offers an immense
variety of physical phenomena. Applications are
tentative, but surprisingly widespread.
HAROLD GRAD
THE TENOR OF OUR TIMES is receptive
to a very young science that claims
dominion over 99% of the matter in the
universe, proposes to fuel a cross-coun-
try auto trip with the deuterium from
one gallon of sea water, offers to re-
place the magic of catalysis in polymer
chemistry with precise knob turning,
promises to alleviate the pollution
problem by instant vaporization of
waste and garbage, ventures to propel
space ships and essays a role in cos-
mology. Even though these specific
future applications of plasma physics
are not proven, the potentialities of
plasma, the "fourth state" of matter,
are difficult to overstate.
Without regard to applications, the
wealth of physical phenomena en-
countered in the plasma state exceeds
the variety spanned by substances as
diverse as air, water, peanut butter and
superfluid helium. I will not presume
to give a balanced picture of this ex-
plosively developing field. Instead, I
present here some of the flavor of the
subject through a few topics of per-
sonal interest and familiarity, binding
it together by an overall evaluation of
where we are and where we may be
heading; at the conclusion are listed
some complementary articles of gen-
eral interest.
The frontiers of the subject are, in a
word, everywhere. Despite a phe-
nomenal growth in theoretical under-
standing and in experimental control
of plasmas, there are almost daily rev-
elations and discoveries of new and
unexpected fundamental insights, fre-
quently overturning our most cherished
beliefs.
In common with nuclear physics wehear echoes in plasmas; in common
with superfluid helium we observe not
only second sound but also third,
fourth, and more; in common with gas
dynamics, we find shock waves and
turbulence, both in bewildering vari-
ety; in common with crystallography
we find anisotropy, but in much more
exaggerated form; in common with all
other fields, plasmas display waves, but
in an unprecedented assortment of
types, packets and interactions.
One branch of physics, for example
superfluidity, is catalyzed by the dis-
covery of an unexpected natural phe-
nomenon. Another, say fundamental-
particle physics, explores unknown ter-
ritory simply "because it is there" and
will uncover unusual phenomena as a
matter of course. Plasma physics lies
closer, in spirit, to the latter. Unex-
pected and unfamiliar phenomena are
abundant, and each discovery opens a
new subfield. Yet no evident single
focus unifies the subject other than
our desire to discover what we can
about ionized and conducting matter.
Whether the conceptual unity hoped
for in fundamental-particle physics will
ever overtake plasma physics is doubt-
ful. Certainly the basic qualitative
principles that govern plasma behavior
are not yet established. Even so, the
initial dust cloud is beginning to settle,
goals are taking shape, measurements
are becoming reliable (see figure 1),
and practical means of answering ques-
tions are beginning to emerge.
What is a plasma?
A plasma is any electrically conducting
medium whose electrical properties are
sufficiently pronounced to react backon an external field. There is no end
of materials that fit this description.
Plasmas are found in the ionosphere, in
the solar wind, within the sun, and on
reentry from space; within the labora-
tory, we have hot hydrogen plasmas
and replicas of the sun, also relatively
cool gas discharges and alkali plasmas;
other plasmas occur in semiconductors,
in polymer chemistry, and in metals,
both liquid and solid. These diverse
substances are related by many quali-
tative and even some quantitative fea-
tures: plasma oscillations, Alfven
waves, the concept of magnetic flux
carried with the flow, and so on.
Nonetheless, even one of these distinct
types of plasmas possesses a vast range
of parameters and exhibits an awesome
variety of qualitatively different prop-
erties.
For example, the major experimental
programs that are currently considered
34 • DECEMBER 1969 • PHYSICS TODAY
to be directly relevant to the con-
trolled-thermonuclear goal deal with
hydrogen plasmas at temperatures
ranging over a factor of 103 and densi-
ties over 106 (see table on page 36).
The comparison of air with water,
which is only 103 times as dense, or
water with a white dwarf, which is
only 105 times as dense, or superfluid
helium with atmospheric helium,
which is only 102 as hot, leads us to ex-
pect similar large differences in the
properties of plasma states separated
by so many orders of magnitude.
One of the important plasma param-
eters is ft, the ratio of plasma to mag-
netic-field energy density. A factor of
105 separates the values of /? found in
hot-plasma research. Thus we can ex-
pect all theoretical and experimental
problems—orbits, equilibrium, diffu-
sion, stability, injection, heating, im-
purity control—to be five orders ofmagnitude apart. Different phenom-
ena dominate plasma behavior in
high- and low-/? plasmas; the tech-
nology, the diagnostic tools, the the-
oretical models, even the basic quali-
tative intuitions, are quite distinct.
Plasma parameters
A primitive but important clue to
the qualitative types of phenomena
that are likely to be encountered is
given by the values of key dimension-
less parameters. As a fluid, air be-
haves more like water, at similar Mach
and Reynolds numbers, than slowly
moving air behaves like hypersonic air.
Similarly, macroscopic MHD (see box
on page 37) theory may be adequate
for very different solar and laboratory
plasmas in comparable scaled param-
eter ranges. However, instead of two
basic parameters as in classical dis-
sipative fluid dynamics (Mach andReynolds), two in ordinary kinetic the-
ory (Mach and Knudsen) and two in
ideal MHD (Mach and Alfven) we
have seven or more in standard, fully
ionized, plasma physics.
Crudely subdividing the range of
each parameter into small, medium
and large, we can expect 32 = 9 quali-
tatively different regions to cover fluid
dynamics (potential, boundary-layer,
hypersonic, turbulent flow, and so on),
and 37 = 2187 regions to cover plasma
physics in a comparably crude way.
Entirely different physical phenomena
will arise depending on the relative
LASER INTERFEROGRAMS of
Scylla IV, showing plasma compression
and loss out of the ends, at 2.4,
3.6, 4.9 and 6.1 microsec. The number
of fringes is proportional
to density. —FIG. 1
PHYSICS TODAY • DECEMBER 1969 35
values of lengths such as the electron
radius, collision cross section, mean
distance between electrons, Debye
length, Larmor radius, mean free path.
Different phenomena will also arise
that depend on the frequencies ob-
tained by combining these lengths with
thermal speed, or Alfven speed, or
speed of light, not to speak of inter-
ference and resonances with each other
and with independent externally im-
posed geometrical lengths, excitation
frequencies, and speeds. The high
dimensionality of this parameter space
is the key without which we cannot be-
gin to understand the structure of
plasma physics. Our goal is not to find
one theory of plasma behavior but to
find very many theories of the behavior
of many different plasmas.
Medium versus geometry
A considerable amount is known about
shock waves. In ordinary kinetic
theory of an unionized gas a single
parameter (the shock strength or
Mach number) completely de-
termines the profile of a plane shock
wave. The corresponding steady
plane shock wave in a fully ionized
plasma takes six dimensionless param-
eters to specify its profile. In the
special case of a weak shock propa-
gating perpendicular to the magnetic
field, the profile depends only on /3 and
the Hall factor OJT, in addition to the
mass ratio a2 — m_/m + . One might
expect the mass ratio to be an unessen-
tial parameter, because it is always
small, but the limit as a2 approaches
zero is quite singular. It is most easily
Harold Grad, since 1958 director of the
Magneto-Fluid Dynamics division and
professor of mathematics at the
Courant Institute of Mathematical Sci-
ences, New York University, lists his re-
search interests as plasma physics and
controlled thermonuclear fusion, kinetic
theory, statistical mechanics and fluid
dynamics. He is retiring chairman of
the plasma-physics division of the
American Physical Society and a former
chairman (1963). of the division of
fluid dynamics.Typical
Device
2X (Lawrence Radiation
Laboratory) Mirror with
well
T3 (USSR)Tokomak
Scylla IV (Los Alamos)
Theta pinch
Focus (Los Alamos)
Coaxial gun*
DCX-2 (Oak Ridge)
Mirror
Stellarator C (Princeton)
Centaur (Culham)
Cusp-ended theta pinch
* Volume of plasma is about 0.01Hot Plasma Parameters
Density
(ions /cm3)
5 X 1013
5 X 10"
4 X 1016
2 X 101*
5 X 109
3 X 1013
1016
cm3.Ion temperature
(KeV or 107K)
8.0
0.5
5.0
6.0
500
0.15
0.3Confinement
time
1 millisec
20 millisec
3 microsec
0.1 microsec
0.5 sec
1 millisec
5 microsec0
0.1
0.002
0.8
high
0.001
0.0001
0.99 +
surveyed by taking /3 ~ a1 and OJT ~
as for a variety of values (positive and
negative) of r and s. Each r,s region
shown in figure 2 represents qualita-
tively different behavior; the shock
thickness is dominated by a different
dissipative mechanism such as ion vis-
cosity, electron Hall heat flow, and so
on. The transition regions, combining
two or more mechanisms, are more
complicated. To obtain all this infor-
mation must surely have taken scores
of man-years of calculation! Fortu-
nately this wealth of physical informa-
tion, representing the asymptotic solu-
tion of a pair of Boltzmann equations
(ions and electrons) and Maxwell's
equations, is given by an explicit,
though very complicated, algebraic for-
mula. Unfortunately we cannot ex-
pect other problems to yield explicit
solutions of such generality.
As we have mentioned, the number
of parameters for more general (finite
strength, oblique), but still classically
collision-dominated shock profiles goes
up from three to six. Only a very
small part of this parameter space has
been investigated. More seriously, en-
tirely new dissipative mechanisms, in-
volving a host of instabilities, turbu-
lence and so on, enter with greater
shock strength.
The shock problem is posed for an
infinite medium with no boundaries or
geometrical features; its complexity, in
different parameter ranges, arises en-
tirely from intrinsic plasma properties.
To isolate plasma from geometrical
complications we turn to the opposite
extreme of the simplest plasma model,
ideal static MHD. There is now only
a single plasma parameter, /? (instead
of up to seven with more realistic
models). But as soon as we try tocontain the plasma, geometrical com-
plications enter.
For example, consider containment
in a stellarator. In its more complex
forms this concept may involve sepa-
rate curved and straight sections, each
with a different helical winding, or
several superposed helical windings on
a common circular axis. The simplest
stellarator has a circular axis and a
single, symmetric, periodic helical
winding; to describe it requires four
lengths and three field parameters for
a total of at least six dimensionless
parameters. But from the limited
theory that is available, we find that
the simplest MHD model is sufficiently
fertile to reveal qualitatively different
physical behavior in different corners
of the six-dimensional parameter
space. Most of this space is still terra
incognita.
The shock and stellarator examples
just given illustrate plasma and geo-
metrical complexities respectively.
Some idea of the possible interplay
between physical and geometrical
effects in plasmas can be obtained by
a glance at classical fluid dynamics
where much more is known. One de-
scription of a fluid is by a dispersion
formula, say co2 = k2a2 where a2 =
dp/dp is the speed of sound. Hidden
in this trivial formula for an ordinary
gas are the theory of the organ pipe,
lift and drag, all of diffraction theory,
and the transition from wave to ray
optics. Spatial variation of a2, through
its dependence on density and tem-
perature, introduces refraction, tran-
sonic flows, shocks, implosions and ex-
plosions, all sorts of waves (gravity,
ship, and tidal), breakers and bores,
wakes, cavitation, bubbles, and so on.
Viscosity complicates the dispersion
36 • DECEMBER 1969 • PHYSICS TODAY
GLOSSARY
MHD—ideal, nondissipative, macro-
scopic magnetoflujd dynamics
guiding center-small Larmor radius
orbit (and collective plasma) ap-
proximation
/3=8TTP/H3— ratio of plasma to mag-
netic pressure (or enegy)
z-pinch—cylindrical plasma column
with /* and Be
0-pinch—cylindrical plasma column
with ld and Bz ("Pinch" originally
referred to a transient; now it refers
also to static equilibria.)
Q-machine—alkali plasma (originally
hoped to be "quiescent")
Tokomak—toroidal z-pinch; flux sur-
faces formed by plasma current
stellarator—toroidal; flux surfaces
formed by external windings, usually
helical
multipole—usually toroidal configura-
tion with internal conductors, either
supported or levitated
banana—drift surface (see figure 9b)
loss cone—part of phase space from
which an orbit will eventually escape
Bohm diffusion, DB —an arbitrary
•• ckT
umt' 1678-
1 formula only slightly, but it introduces
an assortment of boundary layers, sedi-
mentation, and all of meteorology and
oceanography. We see that almost
all the interesting qualitative physical
phenomena in a classical fluid are notvisible in the dispersion formula and
are found only in finite geometries with
boundaries.
In an infinite homogeneous plasma
the formula that describes propagation
of small-amplitude plane waves, cor-
responding to OJ2 = k2a2, has been
studied extensively but by no means
exhaustively. It is a transcendental
relation involving the ion and electron
velocity-distribution functions, and it
exhibits an infinite number of disper-
sive and anjsotropic modes as well as
many continua. In principle each
plasma mode could ramify as widely
as all of classical fluid dynamics in a
real geometry. Taking fluid dynamics
as our model, we expect that most of
the basic qualitative plasma phe-
nomena will be discovered only as non-
linear and finite geometry effects, not
directly visible in the dispersion for-
mula. Only in the simplest plasma
models, such as MHD, guiding center,
and magnetoionic theory, is there an
appreciable corpus of nonlinear- and
finite-geometry plasma effects. Some
geometrical effects appear as rather
direct generalizations of classical
effects; an example is the Fresnel
zones, which are essential to the de-
scription of excitation and detection of
ion-acoustic waves. Other effects
such as coupling of different linear
modes through boundary conditions or
through variable density of the me-
dium are more peculiar to plasmas.
We should not leave the impression
DISSIPATION MECHANISMS in a weak shock. Each region represents a different
dissination mechanism as a function of the values of p and wr relative to the mass
ratio From P. N. Hu, Phys. Fluids 9, 89 (1966). —FIG. 2that plasmas are always more complex
than their neutral counterparts. As a
possible counterexample we point to
fluid turbulence, which is a strongly
nonlinear and essentially three-dimen-
sional phenomenon, only slightly re-
lated to fluid instability. We can com-
pare it with the motion of a bouncing
ball on a cobblestone street and the
unrelated facts that the top of a stone
is unstable and a pot hole stable.
Plasma, with its greater variety of
waves and interactions and spectral
complexity, allows an entirely different
type of weak turbulence in which non-
linearity can be handled as a quasi-
linear perturbation. We also have
strong plasma turbulence, which is
likely to remain essentially empirical.
To return to the question "What is
a plasma?", we can only say that we
are just beginning to find out.
To catch a hot plasma
Although nature is always exceedingly
complex, physics gains its strength pre-
cisely by rejecting complexities as they
occur in nature in order to study se-
lected, isolated "basic" phenomena.
"The unreasonable effectiveness of
mathematics in the natural sciences"1
results from natural selection of iso-
lated phenomena—both experimental
and theoretical—as the subject matter
of science. The basic goal of experi-
mental plasma physics is the construc-
tion of experiments, each of which iso-
lates an individual phenomenon, in
enough variety to cover qualitatively
the entire field. This is a long-term
project, but the multiplicity of effects
is not the most serious road block.
Before we can study a plasma, we have
to catch one. For a hot plasma, this
stipulation conflicts with the best sci-
entific sequence. To create a hot
plasma and keep it away from the walls
long enough for study requires com-
plex experimental procedures and com-
plicated geometries that conflict with
the desire to isolate individual phe-
nomena. In a contained hot plasma
the scientific problems are presented
all at once rather than in sequence.
Analysis of complex systems is com-
monplace in engineering, but not when
the individual phenomena have not
yet been scientifically explored.
The relatively high degree of under-
standing of mirror plasma, compared
with toroidal plasmas, is probably a
result of the dominance, in mirrors, of
a single identifiable physical mecha-
nism. The mirror configuration is
characterized by extreme anisotropy
PHYSICS TODAY • DECEMBER 1969 37
and a pronounced loss cone (see figure
3). The two principal containment
problems are scattering into the loss
cone and plasma instability. Both the
basic scattering loss mechanism and
the basic loss-cone instability can be
studied analytically in an infinite
homogeneous medium without boun-
daries, and therefore with a high de-
gree of theoretical reliability. "Finite-
geometry" complications can be ad-
joined afterwards as relatively minor
peturbations of the basic phenomena
that do not drastically change the
qualitative picture.
Only recently have we discovered
that toroidal systems, in addition to
their own peculiarly toroidal difficul-
ties, possess most of the problems of
mirror machines. Loss cones emerge
in many forms in a toroidal system,
but less obtrusively than in a mirror.
Many classes of mirroring (trapped)
orbits appear with all their attendant
problems, but they do not dominate.
In all but the simplest toroidal
geometries, anisotropy appears in an
essential and complex way, but again
in a weakened form. In even the
geometrically simplest closed con-
figuration (Tokomak), the field
topology enters significantly. It ap-
pears more than likely that the reason
we do not yet understand the limita-
tions of toroidal confinement is that
there are so many comparable com-
peting effects, not that a single
elusive effect remains to be discovered.
Thus far every simple mechanism has
been proved to be inadequate. Syner-
gistic combinations are beginning to be
explored. Empirically the simplestconfiguration (Tokomak) is also the
most successful. Perhaps recent recog-
nition of the complexity of the toroidal-
containment problem will turn out to
be the single most important step
towards its ultimate resolution.
The rapid growth of technology and
empirical experience in containing hot
plasmas (see table on page 36) has
also made itself felt in accelerating the
discovery of basic physical phenomena.
For example, in a situation that is not
atypical of containment experiments,
the plasma found in the DCX-2 appa-
ratus is quite different from what is in-
jected, and it is contained by inci-
dental fine structure in the applied
mirror field. But the plasma is quite
uniform and has served as an excellent
medium for basic studies of finite Lar-
mor radius and extremely anisotropic
effects (including, ultimately, the dis-
covery of the mechanism for the origin
of the plasma).
There has also been a large recent
development of relatively low budget,
nonthermonuclear plasma experi-
ments insensitive to wall isolation.
From the point of view of basic phys-
ics, the two classes of plasmas are
sufficiently distinct to require pursuit
of both.
One example of a basic pheno-
menon not intimately tied to contain-
ment is the plasma echo (to be dis-
cussed later). Another example is
the interaction of plasma oscillations
and optical emissions. The fluctuat-
ing electron density modulates light
emission at the plasma frequency.
This effect, first observed in the iono-
sphere, can be used for atomic mea-
DENSITY CONTOURS in velocity space for a typical mirror-machine loss-cone
distribution. Darker color represents greater density. —FIG. 3suremenfs of excited-state lifetimes oi
for plasma diagnostics.
The relatively well developed field
of alkali plasmas is, in some respects,
a bridge between the physics of con-
tained and noncontained plasmas.
Where is the frontier?
The goal in plasma physics, as in the
study of any fluid such as air, water,
or liquid helium, is to understand and
to control it—to pump, to compress, to
heat and extract energy, to propagate
waves, to measure and, above all, to
keep it from leaking.
In view of the ramification of the
subject, it is not surprising to find that
progress is not uniform over all of
plasma physics. At the frontier, open
questions range from determination of
an equation of state in one type of
plasma to highly specialized effects
dominated by details of the geometry
and distribution function in another.
In highly condensed plasmas, such as
very high-pressure alkali (classical) or
solid-state (degenerate-electron) plas-
mas, elementary thermodynamic and
transport properties are the immediate
problem, both experimentally and
theoretically. In moderately dense
#-pinches, macroscopic equilibrium
and stability questions on a microsec-
ond time scale are the most urgent
present concern. In some well docu-
mented mirror-contained hot plasmas
the spectrum of identified phenomena
is much broader, and we have consist-
ent theoretical and experimental in-
formation about large classes of waves
and instabilities in strongly non-Max-
wellian plasmas on a relatively long
time scale. In at least one case, mea-
surements provide an essentially com-
plete ion-distribution function in veloc-
ity and physical space.
The one feature common to all ex-
perimental areas is the impressive im-
provement in reliability and flexibility
of diagnostic methods. By pushing
the state of the art in x-ray techniques,
in charge-exchange neutral measure-
ments, in Thomson scattering and in
laser holography, selective tools are
being developed to cover wide ranges
of density and temperature.
In problems where theory and ex-
periment make an attempt to converge
on an isolated phenomenon (for ex-
ample, the plasma echo, which is ap-
proximately one dimensional in an in-
finite homogeneous medium), there is
very good agreement between the two.
In containment geometries, where ex-
perimental flexibility is severely
38 DECEMBER 1969 • PHYSICS TODAY
MIRROR-MACHINE configuration, typical of open-ended devices. —FIG. 4
limited, the agreement is not nearly as
good.
The factors that limit currently
operating mirror machines and 0-
pinches are fairly well understood, but
scaling to new parameters is not at all
certain. In stellarators and Tokomaks
even the present limiting factors have
not been identified, and scaling is un-
known. Multipoles lie somewhere in
between.
The hallmark of a clean physics ex-
periment, isolation of a specific phe-
nomenon, is just as much a necessity
for an effective theory. Even in pure
theory it is not easy to isolate an effect
by fiat, just by adding or dropping a
term. As in experiment, the long-term
goal of plasma theory is to find and
develop an arsenal of models and diag-
nostic techniques that are capable of
separating out different effects. The
development of sophisticated theoreti-
cal diagnostics has lagged somewhat
behind that of experiment. This lag
may be because the tradition that
valid experimental results require great
care is not quite so widespread in
theoretical work.
Qualitative concepts
Because of the enormous complexity of
plasma physics, rough qualitative
models take on more than their usual
importance. Where do the qualitative,
intuitive concepts that bind a field
arise? If history is any guide, they do
not come from synthesis of masses of
experimental or theoretical data; nor
do they come from use of crude theo-
retical analyses of complicated prob-
lems (such as an analysis of a contain-
ment configuration). Rather they
arise from solutions of simple problems
that turn out to be more accurate and
more representative of the general case
than one could reasonably expect.
It is illuminating to consider fluid
dynamics as a prototype of a well de-veloped subject. The simplest fluid
model, incompressible irrotational flow,
is hardly realistic. But every aerody-
namicist expends considerable effort
to develop a strong intuition about this
nonexistent fluid. He describes actual
flows as deviations from this ideal (in
boundary layers, shocks, and so on).
Without a precise knowledge of these
deviations, potential flow has very little
value; with this knowledge, it is price-
less. Without the aid of the ideal
theory, the more exact viscous theory
would also have very little value, sim-
ply because of its complexity. Despite
great advances in theory and in com-
puting capability, we stilL solve the
full Navier-Stokes equations only in
elementary geometries. Fluid dy-
namics thus exhibits a complex, sym-
biotic relation among its subtheories,
with the whole greater than the sum
of its parts.
The ideal plasma concept of "frozen"
flux, carried with the plasma, has a
similar significance. Although it is al-
most always a poor approximation, one
can hardly cany on a sensible discus-
sion of plasma containment or motion
without this concept as the starting
point.
On the other hand, qualitative de-
scriptions do not always help. A
simple illustration of the pitfalls of
semantic analogy is given by the dia-
magnetic properties of a plasma.
This concept clearly requires quantita-
tive modification, because a plasma is
a rather complex substance. But the
extent of this modification is surprising.
The simplest evidence for the dia-
magnetic effect is the current carried
by the circular orbit of a charged par-
ticle in a uniform magnetic field.
Similarly, in a nonuniform but unidi-
rectional field, the macroscopic equilib-
rium pressure balance, p + B2/2 =
constant, indicates that the plasma de-
presses the field strength. In a moreCUSP GEOMETRY with opposed Helm-
holtz coils. This is the prototype mag-
netic-well geometry. —FIG. 5
complex geometry the orbits become
complicated, and the static pressure
balance is anisotropic. The guiding-
center approximation to the orbits is
closely tied to the diamagnetic image,
because it assigns to each particle a
constant (negative) magnetic moment.
The plasma current perpendicular to B
is the sum of a definitely diamagnetic
contribution from the magnetic-mo-
ment density and a current arising
from the drift of guiding centers across
the field. The latter component, fre-
quently called the "diamagnetic drift,"
can easily have a paramagnetic sign.
When paramagnetic it can even dom-
inate the contribution of the magnetic
moment and create a locally paramag-
netic region in the plasma.
One can be more precise with a
special class of guiding-center equilib-
ria that yields an exact mathematical
analog of a classical nonlinear mag-
netic medium (B is a function of H).
For this anisotropic equilibrium, the
pressure components, p'l and p±, are
constant on |B| contours. Taking /x =
BfH as the definition of permeability,
we find the criterion for a paramag-
netic region, \x > 1, to be dp^/dB >
0 (or P|| > p_j_). The alternative defi-
nition, /x = dB/dH, yields /.<, > 1
wherever dp±/dB < 0. These criteria
are not intuitively evident. But with
cither definition, locally paramagnetic
and diamagnetic regions can be found
easily.
More striking than the existence of
local paramagnetic regions is the
possibility of a fully self-consistent
plasma equilibrium that is globally
paramagnetic (for example, in a simple
mirror or cusp field, figures 4 and 5).
In other words, the inductance of the
PHYSICS TODAY • DECEMBER 196939
external coil is increased by introduc-
ing plasma. Because plasma current
along B complicates the interpretation
of the diamagnetic effect, I have given
only examples in which J^ = 0.
The special case quoted, in which
the plasma can be unambiguously
identified as diamagnetic or paramag-
netic, is also one in which stability is
easily determined. In contrast to the
classical result that a diamagnetic solid
is stable in a well, each of the four
combinations, diamagnetic or para-
magnetic plasma in a well or on a hill,
can be cither stable or unstable.
One plain conclusion is that, in
competition between the elegance and
simplicity of a concept and the com-
plexity possible in a plasma, complex-
ity can usually be expected to triumph.
Psychological roadblocks
Another example of an appealing
but somewhat specious qualitative con-
cept is that of a magnetic well. Basi-
cally, we expect a plasma (diamag-
netic!) to be stable in a well. The
original magnetic-well formulation
(1955), for a plasma with no internal
magnetic field, separated at a sharp
surface from a vacuum field, gave the
necessary and sufficient condition for
MHD stability that the magnitude of
B increase everywhere from the plasma
surface. An immediate consequence
was that no plasma with a smooth
boundary, mirror or toroidal, can be
stably contained; only the cusped ge-
ometries (for example, figure 5) are
stable. This qualitative stability
principle was dramatically demon-
strated experimentally, in 1960, by
applying cusped coils to the very un-
stable pinch.
A much more significant experiment
from the point of view of thermonu-
clear confinement, was M. S. Ioffc's
in 1962; he showed that cusped coils
reduced fluctuations and improved
containment in a mirror. But, al-
though this was a landmark from the
point of view of containment, the
physics was (and is) not clear. The
mirror-contained plasma is strongly
anisotropic, and its boundary is not a
flux surface. The mechanism for the
initial fluctuations and high loss rate
has not been definitely identified. Nor
do we know the reason for the im-
provement after application of the
well, because its imposition has impli-
cations for several classes of micro as
well as macro stability and also for the
equilibrium drift-surface topology.
Among the theoretical instabilitiesaffected by well-like field configuration
are interchange, drift, trapped particle,
resistive, local and modified negative
mass. Each one depends on a differ-
ent magnetic-field criterion. In addi-
tion, there are several related (but
different) well-like properties of the
field that have a bearing on the con-
tainment of individual orbits and phase
mixing of plasma imperfections rather
than on any collective property. And
finally there are cases where applica-
tion of a magnetic well is detrimental
for containment.
To summarize, the magnetic well
may be ten of the most important
plasma-containment concepts, but it is
not just one! It is one thing to syn-
thesize and coordinate; it is another to
obliterate essential differences.
Another example of a non-concept is
the term "Bohm diffusion." As a dif-
fusion coefficient, the Bohm value is
the product of thermal speed and Lar-
mor radius. The Bohm time can also
be obtained as the length of time re-
quired for a drifting ion or electron to
pass once around a minor circumfer-
ence. As the term is used, Bohm dif-
fusion does not refer to a phenomenon
or to a mechanism but to a natural
plasma time scale that can arise in
many ways, both collective and non-
collective. There are easily a dozen
different physical mechanisms that can
give rise to loss rates comparable to
Bohm. Their semantic synthesis into
a single concept is artificial and a
degradation of information.
Echoes, shocks, and phase mixing
Dissipation appears in a time-reversible
theory in the guise of phase mixing;
analytically, it is recognized as a con-
tinuous spectrum. The basic point is
that any finite or infinite discrete sum,
2#,,exp {UDJ) , oscillates indefinitely;
an integral, y*a(w)exp (iwt) dw, can,
however, decay. The most important
qualitative feature of a continuous
spectrum is that it preserves much of
the information fed into the system by
initial and boundary data and gives rise
to much more complex phenomena
than a discrete normal mode, which is
primarily a property of the medium.
The fact that "Landau damping'' is not
universally given by Landau's formula
and that the wave preserves initial and
boundary data has long been recog-
nized theoretically and has recently
come to the fore with experimental
observations of echoes and various
"ballistic" or free-flow effects.
Phase mixing with collisionlessdamping is not restricted to kinetic
models but is also found in macro-
scopic theory of Alfven waves and in
cold plasma and magnetoionic theory.
In ordinary air, a wall oscillating at
a fixed frequency o> gives rise to a dis-
turbance exp[io) (t—x/v) ]. Integrating
over a Maxwellian velocity distribution
gives a signal that damps approxi-
mately as exp[—(o>x)2/3]. This colli-
sionless decay has been experimentally
confirmed for high frequency waves
in argon. The damping is reduced at
lower frequencies, and when the
wavelength exceeds the mean free path
and collective behavior dominates over
ballistic, the wave eventually ap-
proaches an undamped ordinary sound
wave.
Exactly the same phenomenon holds
in a plasma, except that the collective
effect of the charge-separation field
enters much more strongly than that
of collisions—at the Debye length in-
stead of at the much larger mean free
path. Landau damping describes the
electrostatic modification of free-flow
collisionless damping and is valid for
wavelengths not smaller and not too
much larger than the Debye length.
Only within recent years has more
accurate theory delimited, and very
careful experiment been able to con-
firm Landau's more than 20-year-old
formula. At the same time, "non-
Landau" damping effects, such as
echoes and ballistic effects in ion-
acoustic and other waves, are also be-
ing observed.
To obtain a spatial echo, two parallel
grids are excited at different frequen-
cies. Any nonlinear coupling of the
two disturbances exp[ito(f—x/v)] and
exp[/a/(f—x'/v)] will produce a signal
exp[t(o>—*>')* + i(<a'xf—ax)/v]. The
phase mixing disappears and the mod-
ulated signal is regenerated as an echo
at a position such that o/x7—ux = 0.
Space-resolved electron-plasma ech-
oes have been observed, as have time-
resolved ion-acoustic echoes produced
by asynchronism from plasma gradi-
ents. The magnitude of the echo is
being used as a sensitive measure of
the collisional dissipation and of the
dissipation resulting from externally
imposed noise between initiation and
echo, as in the nuclear-magnetic-reso-
nance effect.
Collisionless shocks are also a mani-
festation of phase mixing, but in a
more complex nonlinear version. In
most laboratory shocks that are identi-
fied as collisionless, the dissipation is
presently attributed to an instability
40 • DECEMBER 1969 • PHYSICS TODAY
or to turbulence. In some of the more
elaborate theories, a structure involv-
ing two or more distinct instabilities in
sequence is invoked. For example, a
steep wavefront with large electron
current density induces a two-stream
instability. The instability produces
thermalization only in the direction of
the current; the resultant unstable
anisotropic distribution induces fur-
ther thermalization.
In principle, neither instability nor
turbulence is needed to effect irrevers-
ibility in a collisionless model. The
earliest collisionless shock models, an-
tedating experiment, were laminar;
they involved phase mixing of ion
orbits and "collisions" of particles with,
the electric and magnetic fields.
Although the first reliable collision-
less shock measurements were of the
earth's bow shock, laboratory experi-
ments have recently become quite re-
liable in several collisionless regimes.
Present understanding is largely em-
pirical, based on the introduction of
ad hoc "anomalous" collision frequen-
cies into a theoretical calculation to
achieve an experimental fit. Definite
identification of irreversibility mecha-
nisms remains open, because the most
reliable theory is for weak shocks
whereas most experimental data is for
strong shocks. The particular value of
a shock wave in heating a plasma is
that the plasma itself chooses which is
the most efficient irreversible mecha-
nism under the given circumstances.
Factors in containment
In the physics of hot plasmas the prob-
lem we must face first is containment.
This problem can be split into a study
of orbits, equilibrium, stability and
diffusion. They are all interrelated;in particular, knowledge of particle
orbits is used everywhere. But only
the most primitive approximations of
the highly developed orbit theory can
be used quantitatively in the more
difficult self-consistent plasma applica-
tion. Nevertheless, we shall see that
even the most sophisticated orbit re-
sults give very important qualitative
information about all these subjects.
The logical sequence for a study of
containment is that given above: first
individual orbits, then self-consistent
equilibrium, then stability and diffu-
sion. In particular, poor containment
can follow as easily from orbit and
equilibrium considerations as from in-
stability. This prescription has been
moderately well followed in mirrors,
taking into account the dominant role
of the loss cone. Although the em-
phasis in toroidal investigations has
long been on microinstability (in par-
ticular, drift and resistive), it has,
somewhat belatedly, become clear that
these mechanisms should defer in
priority to the more basic questions of
orbits, self-consistent equilibrium, and
macrostability, all of which are inade-
quately understood. Especially on
the present time scale of high-beta ex-
periments, it is quite unlikely that
microphenomena are important.
The distinction between "macro"
and "micro" instability is not a ques-
tion of the theoretical model but a dis-
tinction between an instability that
moves the plasma bodily to the wall
and one that exhibits small-scale fluc-
tuations. The containment effect of
the latter is usually described as "en-
hanced" diffusion. Either type of in-
stability can be tolerable or cata-
strophic, depending on the time scale.
The basic time of growth of a small-
ROTATIONAL TRANSFORM
PLASMA DENSITY in the Wendelstein 1=2 stellarator as a function of the rota-
tional transform. There are strong resonances where the field lines close after 3, 4,
5 ... 15 ... times the long way round the torus. —FIG. 6amplitude disturbance is rarely a mea-
sure of the importance of the instabil-
ity. The two-stream instability, for ex-
ample, exhibits extremely fast growth
but is self-limiting, saturating at a low
amplitude of fluctuation that preserves
the velocity profile at a marginally un-
stable shape.
Exactly the same distinction should
be made between a localized failure of
microequilibrium and global macro-
disequilibrium. For example, it was
the impossibility of gross pressure bal-
ance in the simplest toroidal field that
led to the invention of the original
figure-eight stellarator. But there are
also strictly local failures in maintain-
ing a self-consistent plasma-field equi-
librium. These failures can lead to
anomalously high currents, irreducible
fluctuations propagating as Alfven
waves, and enhanced losses.
Moreover, even in macroscopic
MHD stability theory, it is only by
making distinct separation between
local and global instability that we
can establish some points of contact
between theory and experiment.
The one property that is unique to
toroidal containment is closure (as dis-
tinguished from curvature, which can
be mimicked in open systems). Mag-
netic lines carry all sorts of informa-
tion: electrostatic potential, Alfven
waves, guiding-center orbits, and so
on. In an open system, information is
exchanged with the outside world in
both directions along a magnetic line,
whether intentionally or not. In a
toroidal system, the information re-
mains inside; there are specific plasma
complaints that are neither sensed nor
easily remedied. An example is reso-
nance effects in magnetic surfaces and
particle orbits, which have long been
known from experience with accelera-
tors. But recently collective plasma
closure effects have been predicted and
also observed. Figure 6 shows the ob-
served dependence of equilibrium
plasma density on rotational transform
in a carefully designed stellarator.
Distinct peaks are found for fields with
resonances up to the 15th order (that
is, a field line closes after 15 circuits
the long way round the torus). Be-
cause the mean free path is less than
one circuit of the torus, this observa-
tion can not be an orbit effect. A
possible explanation of this effect is in
terms of microequilibrium. Selective
mathematical diagnostic methods that
may allow comparison with experi-
ment are gradually being developed.
Instability has had a much more in-
PHYSICS TODAY • DECEMBER 1969 • 41
tensive development than equilibrium.
The proliferation over the years of
theoretical microinstabilities is itself
characteristics of an explosive insta-
bility. There are recent signs of satura-
tion. This can not be ascribed to the
hypothesis that most instabilities are
already known, because as we have al-
ready pointed out, only an infinitesimal
part of the totality of gross qualitative
plasma phenomena has yet been ex-
amined. What is a more likely ex-
planation of the microinstability slow-
down is discouragement, as only a
small fraction of the list of theoretical
instabilities has been identified experi-
mentally.
Equilibrium
The difficulty of attaining plasma
equilibrium can be seen by a glance
at figure 7, which shows a typi-
cal particle orbit in a stellarator mag-
netic field. As an indication of how
an equilibrium configuration might ap-
pear, recall that a fixed value of the
distribution function must be assigned
to each orbit in phase space (this re-
quirement is prior to any strictures ofself-consistency). It is clear that any
equilibrium distribution function is
very complicated, to say the least.
More careful study shows that it is,
in many cases, mathematically impos-
sible. Even a Maxwell demon could
not inject the plasma correctly. In a
real plasma we must expect "to find a
certain irreducible level of fluctuations
—independent of any question of sta-
bility. When we add self-consistency,
we find that the number of special situ
ations that allow time-independent so-
lutions is even more restricted.
There is, of course, no reason other
than mathematical convenience to look
for stationary states. But without this
convenience, the whole of conventional
stability theory, based on perturbations
about an assumed equilibrium, evap-
orates! When faced with the collapse
of a theory one usually argues that
something has been left out—finite
Larmor radius, Debye radius, resis-
tivity, and so on. But further study
in this case shows that the only chance
of resolving the crisis in containment
theory lies in using cruder rather than
more sophisticated models.
37
36
35
PARTICLE ORBITS in a stellarator field, calculated by computer. The numbers
are provided only to facilitate following the orbit sequence. From H. Gibson J. B.
Taylor, Phys. Fluids 10, 2653 (1967). —FIG. 7For some mathematical purposes it
is appropriate to consider rational
numbers as negligible, "of measure
zero" compared to irrationals. But for
many purposes the rationals must be
considered on a par with the irration-
als. For example, in a stellarator field
with shear (variable rotational trans-
form ), the volume occupied by rational
transform is finite. The variation of
rotational transform would look quali-
tatively as shown in figure 8, where the
flat stretches, of constant rational trans-
form, occupy finite regions. Within
these regions of constant transform,
the magnetic field exhibits islands,
ergodic regions, and all sorts of pathol-
ogy. (The magnetic field is as smooth
as you like—the pathology enters only
in answer to the delicate question of
what happens to a magnetic line if it is
followed forever.)
With axial symmetry the magnetic
field exhibits no such pathology.
There are no gaps in the flux surfaces,
and the rotational transform varies
smoothly. But particle orbits (which
can also be assigned a rotational trans-
form) will, even in the case of axial
symmetry, behave as in figure 8. In
other words any time-independent
equilibrium, in which a constant value
of the distribution function must be as-
signed to each orbit, will be pathologi-
cal.
Returning to an asymmetric geom-
etry (such as a stellarator), although
a field line can be very complex (for
example, ergodic) in a flat region, it is
contained forever between two legiti-
mate flux surfaces. This is not true of
orbits in the asymmetric geometry.
There are everywhere dense (possibly
thin) loss cones from which particles
can escape, given enough time. A true
equilibrium distribution would have
the value zero on this complex distrib-
uted loss cone. The same is true of
any asymmetric mirror machine.
In some cases the pathological
regions can be estimated to be very
thin. In this case, the additional re-
quirement of self-consistency of plasma
currents with magnetic field turns out
to be an independent source of path-
ology, namely very high current den-
sity in a new set of "flat" regions.
There is a large body of theory con-
cerning equilibrium, frequently pre-
senting "proofs" of the existence of
equilibria to all order in general ge-
ometries. Interpretation of such
formal analytic results needs great
care. They have the valuable property
of being blind to certain complicated
42 • DECEMBER 1969 • PHYSICS TODAY
phenomena. But they remain blind
whether the neglected phenomena are
negligible or dominant! We need a
careful mix of naive and sophisticated
calculations to extract the most useful
information-for example, how long
does it take for an approximate
equilibrium to break up? Some slight
progress is being made here.
More phase mixing
Most of the orbit and equilibrium
pathology disappears when we use the
guiding-center orbit approximation in
a mirror machine. Moreover, phase
mixing can sometimes be relied upon
to correct imperfections of symmetry
in the injected plasma. For example,
consider an axially symmetric mirror
machine. Because of field gradients,
a guiding-center orbit will drift
around a flux surface in a time com-
parable to the Bohm diffusion time.
It is easy to verify that the sense of
the drift reverses for particles that mir-
ror close to the center and those that
almost spill over. Phase-plane shear
(variable drift speed, see figure 9a)
provides phase mixing so that, if we
ignore collective effects, any asym-
metric plasma injection will be cor-
rected after a number of drift periods.
(More precisely, for low-energy non-
drifting orbits, there is a stationary
phase point, indicating poor mixing in
this part of phase space.)
If axial symmetry is disturbed, the
degenerate zero velocity curve in fig-
ure 9a will split, to form a drift pat-
tern like that in figure 9b. The con-
tours in figure 9a are level lines of a
volcano, which, in figure 9b, lies on
a slope. Although the shear is not
zero, it is small throughout the banana
region, and we can expect local errors
in plasma symmetry to die away rela-
tively slowly. During this process,
local fluctuations would be observed.
This poorly mixing region of phase
space is the same one that is influential
in creating drift instabilities, but the
present phenomenon is quite distinct—
in particular, it is noncollective. It is
interesting to note that there are some
magnetic-well configurations that do
not exhibit drift reversal and banana
regions.
In a mirror machine with multiple
mirrors, and in most toroidal devices,
there will be several trapped states, as
in figure 10, each generating its own
family of drift surfaces. These drift
surfaces have no relation to the flux
surfaces, even for vanishing Larmor
radius. Since a local maximum of BVOLUME
ROTATIONAL TRANSFORM shown schematically as a function of plasma volume.
In principle the flat portions occur at each rational value. FIG. 8
DRIFT CURVES in a mirror machine. The direction of drift reverses as the
turning point moves out. The zero-drift curve in a symmetric field (a) opens up
into a "banana" region with asymmetry (b). —FIG. 9
will vaiy from one magnetic line to
another, a drifting particle can spill
and change its trapped state (see fig-
ure 11). This change produces a ran-
dom walk among the drift "surfaces,"
which turn out not to be surfaces at
all, but to cover a finite volume of
phase space ergodically. Equilibra-
tion through phase mixing will prob-
ably be slow in such regions. An easy
estimate shows that drift surfaces or
drift volumes that touch a wall or a
loss cone give a direct loss rate com-
parable to Bohm, without the interven-
tion of scattering or fluctuations.
Diffusion
Diffusion is a very general term. It
describes a variety of dissipative mech-
anisms that allow violation of the ele-
mentary, perfectly conducting concept
of a plasma whose elements remain
fixed to given magnetic lines or flux
surfaces.
In the formulation as a random walk,
diffusion of particles is not self-con-
sistent. With an ambipolar calcula-
tion, it becomes slightly self-consistent.But to make it fully self-consistent in-
volves at least all the complexities of
the self-consistent-equilibrium prob-
lem that we have briefly outlined.
The diffusion problem is both quali-
tatively and quantitatively different in
different ranges of the collision fre-
quency y. If v is large, we have a ran-
dom walk of particles, localized in
physical space. With v smaller than
the Larmor frequency, there is a ran-
dom walk of guiding centers. If v be-
comes smaller than the "bounce" fre-
quency of reflection between mirrors,
collisions induce a random walk from
magnetic line to line. Still smaller v,
comparable to the drift frequency
around the machine, yields a random
walk of drift surfaces (qualitatively
similar to the noncollisional random
walk induced by changes in the
trapped state, figures 10, 11). The
first two cases are considered to be
classical because they can be treated
macroscopically, with a plasma resis-
tivity. Diffusion among magnetic lines
or drift surfaces is frequently termed
"anomalous," or nonclassical although
PHYSICS TODAY DECEMBER 1969 43
Magnetic field strength
TRAPPED STATES. Particles in different energy states can be trapped in different
mirrors in a complex mirror machine. —FIG. 10
it is a consequence of purely classical
orbits and Coulomb scattering.
But even the completely macro-
scopic resistive model of diffusion
(small mean free path) can exhibit
anomalies. Macroscopic plasma diffu-
sion across a field is a complex interac-
tion between two more basic types of
diffusion. In a mixture of neutral gases
we have a diffusion coefficient, Do.
Diffusion of a magnetic field through a
conducting solid is described by a co-
efficient DM = 1/fj^cr (<J is the conduc-
tivity) . The elementary kinetic-theory
formula, a ~ e2D0/kT, suggests that
Do and DM are essentially reciprocals!
For a plasma in a simple magnetic
field, the competition between these
two effects at different rates gives (at
low P) the classical diffusion coefficient
Dc ^ /?DM; (the competition is fierce,
and the combined diffusion equation
that results is unconventional, with a
nonlinear decay as \/t rather than ex-
ponential as in simple diffusion).
In more complex geometries we ex-
pect coupling between these two basic
rates. This coupling is only partially
accounted for in the standard theory
by the Pfirsch-Schluter factor. Be-
cause this factor diverges under the
same circumstances that lead to diffi-
culties with microequilibrium, we can
not consider the macroscopic "self-con-
sistent" theory to be definitive.
Some rough estimates have been
made of a slightly self-consistent model
with realistic guiding-center orbits. A
model that combines such drift sur-
face (banana) diffusion with realistic
self-consistency (as in the macroscopic
theory) appears far in the future.
Trends
Originating in discharge physics and
astrophysics, spurred mainly by thecontrolled thermonuclear program in
the past 15 years, plasma physics is
now branching into many new direc-
tions, meanwhile developing into a
recognized academic discipline.
We can grasp the significance of the
field of plasma physics only in the con-
text of its enormous phenomenological
variety and—especially for hot plasmas
—experimental difficulty. Growth,
measured both in achievable experi-
mental plasma parameters and in depth
of understanding, is either fabulous,
when compared to the state of the art
a few years ago, or negligible, when
compared to what visibly remains to
be done. Observable trends toward
simplicity in plasma experiments,
toward simpler theoretical models and
at the same time away from simplistic
theoretical explanations are both in the
right direction.
Both experimental and theoretical
techniques are becoming more specific
and more precise, more quantitative
and more professional. A portent is
the recent start, internationally, of seri-
ous engineering studies of hypothetical
operating thermonuclear reactors.
The significance is not that we can see
a target date, but that we can imagine
being caught short. The complexity
of plasma phenomena implies a con-
comitant large variety of options; with
some ingenuity, success is not in doubt.
But the time scale is not easily esti-
mated, because the scientific and tech-
nological problems that must be solved
are not yet fully formulated.
Plasma physics is, in a sense, the
union of three classical fields-fluid dy-
namics, kinetic theory, electromagnetic
theory. Although classical, these are
fields that have all seen profound ad-
vances in the past 20 years. In the
specialities of nonlinear waves and in-CHANGE OF TRAPPED STATE for a
drifting particle. The choice of second
state is essentially random. —FIG. 11
stabilities, there has been a gratifying
infusion into plasma physics of ideas
from electrical engineering. We
should hope and expect that in a sub-
ject as vast as plasma physics, similar
profit will ensue from interaction with
these and other scientific disciplines
with their divergent backgrounds,
techniques and viewpoints.
I am grateful for the unstinting assistance
and criticism of Albert A. Blank in pre-
paring this manuscript, and also to Harold
Weitzner, Herman Fostma, Robert Hirsch,
Nathan Marcuvitz, Norman Lazar and
Raul Stern for their valuable suggestions.
This work was supported by the US
Atomic Energy Commission under con-
tract AT-(30-1) 1480.
Bibliography
M. B. Gottlieb, "Plasmas," PHYSICS TODAY,
21, no. 5, page 46 (1968).
Perspectives on Controlled Thermonuclear
Research, R. L. Hirsch ed., TID-24804,
Oct. 1968.
A. S. Bishop, "Roadblocks in the Path of
Controlled Fusion," MATT-412, Princeton
Plasma Physics Laboratory, Jan. 1966.
G. Haerendel, R. Lust, "Artificial Plasma
Clouds in Space," Scientific American,
219, no. 11, page 80 (1968).
M. J. Lighthill, Waves in Fluids, Imperial
College of Science and Technology, May^
1965.
Reference k
1. E. Wigner, "The Unreasonable Effe§
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Sciences," Communications on Pure
and Applied Math., 13, 1(1960). 0
44 • DECEMBER 1969 • PHYSICS TODAY
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46 • DECEMBER 1969 • PHYSICS TODAY
MORE ABOUT TACHYONS
Not so fast! say critics of the May article
in which Bilaniuk and Sudarshan offered the arguments for
faster-than-light particles. Their letters
raise questions about causality and interactions. The
original authors contribute a reply.
OLEXA-MYRON BILANIUK, STEPHEN L BROWN, BRYCE DeWITT, WILLIAM
A. NEWCOMB, MENDEL SACHS, E. C. GEORGE SUDARSHAN, SHOICHI YOSHIKAWA
"Anything that is not forbidden is com-
pulsory," says Murray Gell-Mann's half-
facetious totalitarian principle. What
then about faster-than-light particles
called "tachyons"? In their May article1
Olexa-Myron Bilaniuk and E. C. George
Sudarshan argued that valid solutions
of Albert Einstein's relativity equations
describe such particles. Thus if Ein-
stein's equations are accurate descrip-
tions of the physical universe and if
solutions not forbidden are compulsory,
tachyons must exist.
The May article stirred up a flurry of
correspondence directed largely at two
questions: Are the tachyon solutions
valid? Do they have significance in our
real world? From those letters we have
chosen five that represent the principal
viewpoints. With them we publish
Bilaniuk's and Sudarshan's reply to
their commentators.
Real force, imaginary mass
The May article by Bilaniuk and Su-
darshan presented a very interesting
and provocative discussion of the pos-
sible existence of particles that can
travel faster than light. After pre-
senting their case, the authors pointed
to several objections that have been
raised against their proposal, and they
showed how their own viewpoint an-
swered these objections. Some fur-
ther objections that could be raised,
however, are not mentioned by the
authors. I should like to discuss them
in this letter.
The authors base their argument on
the relationships among energy, mo-mentum, mass and speed that follow
from the mechanics of particles in spe-
cial relativity theory. They point out
that since both energy and momentum
depend on the mass factor, Mo/ (1 —
i;2/c2)1/2, the conserved quantities
could remain real numbers if simul-
taneously v2/c2 > 1 and m0 is replaced
with the purely imaginary proper mass
im*. The argument is that since en-
ergy and momentum—not inertial mass
—are the observables, only these quan-
tities must have a description in terms
of real numbers.
A tacit assumption here is that the
appearance of inertial mass originates
in the expressions for energy, momen-
tum, etc. But this is not actually true,
according to the full meaning of rela-
tivity theory. For in Einstein's orig-
inal approach, special relativity is
only a special case of general relativ-
ity. (Indeed, the adjective "special"
implies this fact). In general relativ-
ity theory energy and momentum are
not defined quantities! The conser-
vation laws are in fact only the asymp-
totic features of the general formalism
in the limit of a local domain. How-
ever, inertial mass is defined here in
global terms. It relates to the metri-
cal field gv-" (x) through Einstein's
field equations. Thus inertia is a more
general property of matter than energy
or momentum. The inertia of mat-
ter appears in terms of a (continu-
ously distributed) field on the rightside of Einstein's equations. The met-
rical field solutions, g*v (x) appear on
the left side of these equations. Now
if the inertial mass of any bit of matter
(in the proper frame of reference)
should be represented by a purely
imaginary number, it would follow
that the corresponding metrical field
solution of Einstein's equations (in
the same frame of reference) must
also be represented by a set of imag-
inary numbers. But this would be
inadmissible for several reasons. One
important reason is that in the local
limit, the metric tensor must approach
the diagonal form (1,-1,-1,-1) that
characterizes special relativity theory.
The latter, of course, is a set of real
numbers. If gv-v is represented by a
set of purely imaginary numbers in
its global description, it could not
approach a set of real numbers in a
continuous fashion under any cir-
cumstances! Physically, the continual
approach of g^v toward the Lorentz
metric in the local domain corresponds
to the diminishing effect that one bit
of matter (in this case the tachyon)
would have on other matter.
The gist of this argument is that
the inertial mass term m0 derives from
a more primitive relation than the
expressions of energy and momentum
in special relativity. Once the general
relation that relates inertia to the
global features of a physical system
is found, one can take the asymptotic
PHYSICS TODAY • DECEMBER 1969 • 47
limit and derive the value for the
mass of a bit of matter in the local
domain. Only at this stage (in prin-
ciple) does one insert this parameter
in the energy and momentum expres-
sions. But the original general rela-
tion that identifies inertia with the
metrical field necessarily requires that
the proper mass be represented by a
purely real number. In this case, the
further requirement that the energy,
momentum, etc., be represented by
real numbers would not permit v/c to
be greater than unity.
One further argument against the
existence of tachyons has to do with
the fact that one does not measure
energy and momentum in any experi-
ment; one rather measures the energy
and momentum transfer, a change of
energy-momentum. But a change in
energy-momentum has to do with
force—the force that causes an inter-
action between matter and matter
and, in turn, relates to the correspond-
ing change of state of motion of the
interacting matter. Now if inertial
mass relates to a measure of the re-
sistance to the change in the state of
motion of matter and if we define the
force exerted by matter on matter(the momentum transfer that is
mutually exchanged) in terms of real
numbers, then the mass itself must
also be represented by a real number.
Otherwise an imaginary-mass particle
would not interact with a real-mass
particle. In particular, if one part of
this mutual interaction is a measuring
apparatus—which we have already
used to detect real-mass particles (for
example, a cloud chamber)—then it
should not be able to detect imaginary-
mass particles.
At the root of this objection is the
omission in the paper by Bilaniuk and
Sudarshan of discussion of interaction
between the tachyon and any other
matter. But it is essential in this
problem to introduce the description
of interaction because of the neces-
sary appearance of matter with real
mass to interact with the faster-than-
light particles. My argument above
implies that as soon as this interaction
is taken into account, the conclusion
is reached that (within the frame-
work of relativity theory) no matter
described by real mass could respond
in any way to the tachyon. From
this point of view, then, the tachyon
must remain in a theoretical domain
THE AUTHORS
Mendel Sachs, professor at the Stated
University of New York, Buffalo, is now
on leave at the Department of Applied
Mathematics and Theoretical Physics,
University of Cambridge, England.
William A. Newcomb, plasma physicist^ [
at Livermore, has BA and PhD degrees f
from Cornell and formerly worked at j
Project Matterhorn, Princeton.
M Shoichi Yoshikawa is at the plasma-
physics laboratory, Princeton. He has
a BS from the University of Tokyo and
a PhD from MIT.
Bryce DeWitt is a specialist in quan->
tized gravity and a professor at the
University of North Carolina. His three
degrees are from Harvard University.
^Stephen L. Brown, who does operations
research for the Stanford Research In-
stitute, got a PhD at Purdue as a high-
energy theorist.
Olexa-Myron Bilaniuk, professor at^
Swarthmore, was born in the Ukraine
and educated in Belgium and at the
University of Michigan.
^E. C. George Sudarshan came from In-
dia and the University of Madras to the
University of Rochester. He is now a
professor at University of Texas, Austin.that is beyond the domain of physics.
My argument has been based on a
look at the consistency of the tachyon
description within the theory of rela-
tivity. Therefore I do not at all dis-
agree with the attempt to find faster-
than-light particles. But I do dis-
agree with the authors' interpretation
of the results of such experimentation.
For if such particles should be found,
I should have to conclude (in contrast
with the authors' contention) that
the theory of relativity would have
been refuted.
MENDEL SACHS
State University of New York, Buffalo
Tachyonic Cerenkov radiation
I should like to raise one question in
connection with the recent article by
Bilaniuk and Sudarshan. The au-
thors alleged that a charged tachyon,
by the emission of Cerenkov radia-
tion, would ultimately enter a "trans-
cendent" state of infinite velocity or
zero energy. However, this would
not appear to be a relativistically in-
variant condition. An infinite-velocity
trajectory is one that is orthogonal
(in the space-time sense) to the time
axis of one's reference frame, and it
will not be orthogonal to the time axis ..
of another frame. How can this be ^
reconciled with the principle of rela-
tivity?
WILLIAM A. NEWCOMB
Lawrence Radiation Laboratory,
Livermore ,.
Violation of causality
The article by Bilaniuk and Sudarshan
is well written and the exposition of
tachyon theory is almost perfect.
This, however, permitted me to con-
ceive the following objection: If
tachyons are to be produced or ab-
sorbed by tardyons or luxons, the
causality principle is not upheld. My
objection does not exclude the possi-
bility that tachyons may interact with
other species in an uncontrolled
manner. (I will clarify the uncon-
trolled manner in the last paragraph.)
The causality principle is to be put
in the following form: If an event A
causes the event C at the same loca-
tion in a coordinate system S yet
earlier in time (figure 1), the causality
principle is violated. Whether the
event C is the emission of a tachyon
or absorption of a tachyon is imma-
terial. What I would like to point
out is that by transmitting a tachyon
48 • DECEMBER 1969 • PHYSICS TODAY
Light cone
CAUSALITY VIOLATION. Effect in frame S appears to precede cause in S through
signals to and from frame S' moving with respect to S. —FIG. 1it does not require sophisticated argu-
ments or the invocation of thermo-
dynamical irreversibility and quantum-
mechanical uncertainties to prove it.
First of all, if tachyons exist, they
must interact with normal matter. If
they interact with normal matter, it
must be possible, in principle, to
produce them in a beam. Moreover,
it must be possible to modulate this
beam at the source, and hence to
send a directed signal faster than
light. For purposes of the present
argument it is sufficient to represent
such a signal as a spacelike line in
spacetime. An actual signal would
be a striped ribbon since time is re-
quired both to emit it and to receive
it. But if emitter and receiver are
far enough apart, the width of the
ribbon can be neglected.
Let A and B be two observers, both
at rest in an inertial frame (x, t).
(We suppress coordinates y and z
for simplicity.) Let A emit a modu-
lated burst of tachyons at the space-
time event Z, as shown in figure 2.
Let this signal be received by B at
the event Y. Because Y is later than
Z, in the common inertial frame of
A and B, both observers agree that
A is the emitter and B the receiver,
and that positive energy has been
transmitted from A to B.
Now suppose a third observer D
at t = 0, the observer P in the co-
ordinate system S can induce the
emission of another tachyon at t =
"~*o (<0). This seems to me a very
clear case of the violation of causality.
It hardly requires any explanation.
I shall sketch the argument. Observer
P sends a tachyon at t = 0 to another
observer Q located at B on a moving
coordinate S'. Observer Q then finds
that a negative-energy tachyon is ab-
sorbed at B; that is, a positive-energy
tachyon is emitted in the negative x'
direction. As soon as he notes the
emission of this tachyon, he sends
another tachyon with a faster velocity
along the negative x' axis. This sec-
ond particle is then absorbed by an
absorber located at C. The observer
P finds that a positive tachyon was
emitted at C (t = —t0). Clearly the
emission of a tachyon at C was caused
by the decision of the observer at A
(t = 0). Hence, the causality prin-
ciple was violated.
If we can control the interaction
between a tachyon and other particlesin any way (such as blocking the
motion of a tachyon), we can violate
the causality principle. For example,
if we let observer P pass only a
tachyon with a specified velocity to
reach observer Q and if we let Q
allow the passage of only those
tachyons faster than the first tachyon
to reach P, eventually P finds a passing
of a tachyon earlier in time because
another tachyon with a specified
velocity passes later in time.
SHOICHI YOSHIKAWA
Princeton University
Reinterpretation won't work
Your authors Bilaniuk and Sudarshan
cannot get off the hook as easily as
they pretend they can in their article.
I refer to their claim that by reinter-
preting negative-energy tachyons
traveling backward in time as posi-
tive-energy tachyons traveling forward
in time they can avoid the causality
objections against the tachyon hy-
pothesis. This is simply not true, andIN ONE FRAME Y conies after X and Z.
Dots show tachyon signals. —FIG. 2
IN OTHER FRAME event
events X and Z.Y precedes
—FIG. 3
PHYSICS TODAY • DECEMBER 196949
is passing in the vicinity of B near Y,
with a relative velocity v (<c). In
figure 2 the world lines of B and D
are drawn as if they intersected at Y;
the intersection could actually take
place a little later. Suppose that
during the time of intersection (that
is, while they are fairly close to one
another) B transmits to D the infor-
mation he has received (by way of
the tachyon signal) from A, and
suppose this transmission takes place
by means of ordinary photons. Be-
cause photons are quite conventional
carriers of information, there will again
be no ambiguity about who is doing
the emitting and who the receiving.
On the other hand, by the relativity
principle, the laws of physics must
be the same for D as they are for A,
and hence he will be perfectly capable
of immediately sending back to A,
with an identical tachyon transmitter
of his own, the information he has
received from B.
Since the world lines of tachyons
are spacelike, there exists a range of
values for v, determined by the
tachyon velocity, for which the second
tachyon signal appears, from the point
of view of observers A and B, to propa-
gate into the past. Suppose v is in this
range. Then arguments will arise,
between A and B on the one hand,
and D on the other, about who is do-
ing the sending and who is doing the
receiving. To avoid such arguments
let us suppose that instead of sending
the tachyon signal to A, D sends it
instead to a fourth observer C who
happens to be at rest relative to D
but whose world line intersects that
of A at the moment of receipt of the
signal, denoted in the figures by X.
(Here again the intersection could
take place slightly later.)
Figure 3 shows the sequence of
events as viewed in the common in-
ertial frame of C and D, denoted by
(x',tf). Because event X is later than
Y in this frame, C and D agree that D
is the emitter and C the receiver.
Since the other observers, A and B, are
not involved in the transaction, their
views on the subject are irrelevant.
Finally, let C transmit to A by
means of photons, while the two are
close together (that is, in the vicinity
of X), the information he has received
from D. The net result is that A is
now in possession of information about
his own future, with all the paradoxes
that such knowledge entails.
I can think of only three ways to
avoid such paradoxes:1. Tachyons never exist other than
as virtual particles.
2. The universe as a whole is so
finely tuned (for example, by quan-
tum mechanical interference effects)
that whenever information is sent into
the past, as in the above example, it
is always wiped from the receiver's
memory in time to prevent paradoxes
from occurring.
3. Emission and absorption of
tachyons can take place only between
members of a restricted class of ob-
servers possessing velocities relative
to some preferred inertial frame (for
example, the frame of the "fixed"
stars, or some other cosmological
frame) less than some critical value.
None of these restrictions holds in
the scheme put forward by Bilaniuk
and Sudarshan.
BRYCE DEWITT
University of North Carolina
Why wait for light?
The article by Bilaniuk and Sudarshan
seems to me a remarkably clear ex-
position of the possibility of super-
luminal particles. In reading the ar-
ticle, I was struck by the practical
implications that such particles might
have. (I have not kept sufficiently
current with the research in the sub-
ject to know whether these implica-
tions have already been discussed.)
Briefly, the argument is as follows:
Class II particles (luxons) can be pro-
duced, modulated and detected by
tardyon observers. The tachyon prop-
erties discussed imply that similar
control could be exercised over Class
III particles (tachyons), especially
through the intermediation of luxons,
as in the Cerenkov-detection pro-
posal. Tachyons could therefore be
used for communication systems.
Such communication systems would
be useful only where ordinary electro-
magnetic radiation is too slow, as in
interstellar communication. Finally,
it would seem likely that any extra-
terrestrial life of high technology
would be aware of tachyons (if they
exist) and would use them for com-
munications instead of waiting cen-
turies for replies at the speed of light.
Perhaps, then, the Project OZMA con-
cept of monitoring electromagnetic
radiation for intelligible patterns will
turn out to have much less potential
for interstellar contact than a tachyon
monitoring system.
STEPHEN L. BROWN
Stanford Research InstituteThe rebuttal
We are gratified by the response of
so many physicists to our article.1
The comments published above con-
stitute only a small sample of the
letters, reports and preprints we have
received. Although we knew that sev-
eral points in our article needed
elaboration, that others were specula-
tive, and that a few were pure con-
jectures, yet we did not expect so
many physicists to take notice. After
all, there is little in that article that
we had not already said, for example,
in our paper "Meta-Relativity" pub-
lished in 1962 in the American Journal
of Physics.2 Then the reaction was en-
tirely positive. A very favorable com-
mentary by Angus Hurst on our
"Meta-Relativity" paper was published
in Mathematical Reviews.3 A team
of physicists at the Nobel Institute
in Stockholm undertook the first sys-
tematic search for faster-than-light
particles.4 Gerald Feinberg5 and
Arthur C. Clarke6 have given excel-
lent exposition of our ideas to a wider
audience. But because the causality
arguments remained unresolved and
because nothing at all was said about
tachyon interactions, such a favorable
reaction seemed almost too good to
be true. As Bryce DeWitt puts it, we
did not expect to "get off the hook
that easily."
After having studied the above
letters and all the other correspon-
dence quite carefully, we are now con-
vinced more than ever that our ex-
tension of the special theory of rela-
tivity to include superluminal par-
ticles (metarelativity) is viable and
that we can satisfactorily answer all
objections raised so far.
General relativity. Let us first deal
with the point questioned by Mendel
Sachs. He argues that our theory is
inconsistent with the general theory
of relativity. We disagree. We had
pointed out that for energy and mo-
mentum to be real, the proper mass
of a tachyon must be imaginary.
Sachs contends that an imaginary
proper mass raises difficulties regard-
ing gravitation because gravitation
couples to inertia. Let us recall that
the relativistic gravitational field is
50 • DECEMBER 1969 . PHYSICS TODAY
coupled to the density of energy and
momentum and not to the density of
proper mass. In the limit of slowly
moving tardyons (ordinary massive
particles) one can approximate the
relativistic interaction by a Newtonian
interaction using the proper mass den-
sity but only in this special case and
in this special limit. It just happens
that under these circumstances the
proper mass density and the energy
density are equal (apart from the c2
factor). As long as the energy and
momentum of tachyons are real (that
is, the proper mass is imaginary)
tachyons present no anomaly regard-
ing gravitational interactions in gen-
eral relativity theory.
Transcendent tachyons. William
Newcomb's question is quite intrigu-
ing. Indeed, a charged tachyon that
has reached its zero-energy "transcen-
dent" state in one frame still has some
energy left in some other frame moving
with a velocity w relative to the first;
hence in that frame the tachyon can
keep on radiating. This contradic-
tion can be resolved by recalling that
according to an observer in the second
frame the sign of the energy and the
direction of travel in time will be
reversed (in accordance with the
switching principle) when the tachyon
reaches a velocity c2/w relative to the
first frame. The events that lead to
a transcendent tachyon in one frame
look like a head-on collision and an-
nihilation of a tachyon and an anti-
tachyon in another. Thus Newcomb
is quite correct in pointing out that
the transcendent state would not be a
relativistically invariant condition.
There is nothing disquieting about this
because it is not the description of
events that must remain invariant
when we go from one frame to an-
other, only the laws that govern these
events.
Causality. As we pointed out in
our PHYSICS TODAY article,1 causality
objections against superluminal par-
ticles are by far the most subtle, and
much room for reflection remains in
this regard. The questions raised by
Shoichi Yoshikawa and DeWitt bear
this out. Both are refined versions of
earlier formulations of the causality
paradox. Yoshikawa follows closely
Richard Tolman's original arguments,7and DeWitt essentially parallels chap-
ter 28 of David Bohm's monograph on
relativity.8 Because the earlier pre-
sentations ignored the fact that a sig-
nal traveling backward in time carries
negative energy, they were incom-
plete and could be dismissed as such.
Yoshikawa and DeWitt, on the other
hand, do allow for the sequence re-
versal. They point out that in prin-
ciple the flow of information can be
opposite to the direction of travel of
a tachyon beam conveying the infor-
mation. This is a novel conclusion.
They show that if such counterdi-
rected information flow were indeed
possible, the closed causal loop would
be reestablished notwithstanding our
switching principle.
In devising gedanken experiments
on superluminal communication it is
necessary to take very careful account
of cosmological boundary conditions.
While assuming the existence of cer-
tain transmitters and receivers, we
may not at the same time ignore the
presence of other matter in the uni-
verse. In particular we have to make
certain assumptions regarding the
tachyon background. The simplest
assumption is that the number of
tachyons crisscrossing the universe is
finite. Moreover, we know that as
far as tardyons are concerned, this
situation would still hold for an ob-
server in a different inertial refer-
ence frame. Such would not be the
case for tachyons.
Preferred frame. To see why the
case is different with tachyons, con-
sider two pieces of equipment, one a
large emitter and the other a large
receiver. Let both be located in what
we shall call the "standard" frame
where the flux of tachyons coming
from distant regions of the universe is
finite. Under such circumstances,
however large the detector, the num-
ber of tachyons that it will detect per
unit time is finite. On the other hand,
the number of tachyons the large
emitter can emit is at our disposal
and can be made arbitrarily large. It
should be noted that as long as the
observations are made from the stan-
dard frame, the above situation holds
irrespective of whether the emitter-
receiver system is stationed in the
standard frame or whether it is car-
ried in a fast moving rocket. Further-
more the assumption that the number
of tachyons streaming into the stan-
dard frame from distant random
sources is finite implies that the num-
ber of tachyons within a certain mo-mentum range is also finite. We know
that corresponding to this momentum
range there exists a reference frame
in which the role of emitter and re-
ceiver for tachyons is interchanged.
An observer in that frame would find
that as far as he is concerned there is
a limit to the number of particles that
can be emitted within a velocity range
greater than a certain critical value
but that an arbitrarily large number
of such particles can be detected by
a suitable piece of apparatus.
Refutation. Let us assume now a
standard frame So in which the
tachyon background is zero. This will
simplify our arguments without any
essential loss of generality. While
the tachyon background in So is zero,
the observer Po can still emit any num-
ber of tachyons of any velocity v > c.
For another observer Px moving with
a velocity w < c relative to the stan-
dard frame this situation implies the
impossibility of his emitting tachyons
with a velocity greater than a certain
threshold velocity ux = c2/w. In-
stead he will see a flux of tachyons
with velocities u > ux streaming into
his receiver every time he activates it.
This is so because an arbitrary number
of tachyons can be emitted by Po.
Every time F± activates his receiver
(which is an emitter for Po), it will
register incoming tachyons. Con-
versely, F1 will not be able to use his
emitter (receiver of Po) for sending
tachyons with a velocity u > c2/w
towards distant regions of space be-
cause doing so would mean that ob-
server Po would register tachyons
coming from infinity every time he
opens his detector; such an action
is contrary to our assumption that
no tachyons from distant sources
exist for the observer in the standard
frame. In dealing with the causality
paradoxes it is not necessary to as-
sume that one of the observers is
at rest in the standard frame. But
by referring to this frame, we can
determine which of the signals of the
vicious causal cycle can not be sent.
In other words, irrespective of the
state of motion of the emitter, only
those signals that carry information
and energy in the same direction as
seen in the standard frame are pos-
sible. Under such circumstances no
causal loops could arise and no "anti-
telephone," such as proposed by
Gregory Benford, David Book and
William Newcomb,9 could be built.
The above suggested resolution of
the refined causality arguments cor-
PHYSICS TODAY • DECEMBER 1969 51
responds to the third way by which,
according to DeWitt, causality para-
doxes can be avoided. It is in no
ivay incompatible with our generaliza-
tion of the special theory of relativity.
However, a question that may be in
order is whether the assumption of
existence of a preferred frame, such
as So above, is consistent with the
postulates of special relativity. After
all, is not the exclusion of a pre-
ferred frame what relativity is all
about? No, it is not. The postulates
of special relativity require the laws
of physics, including the speed of
light, to be the same in all inertia]
frames. They do not preclude the
existence of cosmological boundary
conditions that permit us to single
out a particular local frame as a pre-
ferred reference system. For example,
the frame of reference in which the
cosmic 3-K black-body radiation is
isotropic could be considered a pre-
ferred frame that can be distinguished
from all other frames.
Other avenues. The approach we
suggest above is by no means the only
way by which hypothetical super-
luminal particles can be reconciled
with the logical requirements of the
causality principle. For example,
Raymond Fox, Charles G. Kuper
and Stephen G. Lipson10 attempt
to accomplish this by extending the
method of Arnold Sommerfeld and
Leon Brillouin.11 Another simple, if
somewhat brute force, solution is of-
fered by Ray Skinner12 who simply
postulates that negative-energy en-
ergy-momentum transfers must be un-
suitable for signaling.
Although it is not our feeling
that any radical changes in physical
concepts are necessary to accommo-
date the tachyon hypothesis, there are
some serious physicists who shrug off
the causality objection by simply say-
ing, "So what?" Roger G. Newton13
and Paul L. Csonka14 are doing pre-
cisely that. They feel that no pre-
cepts of logic would be violated if
the temporal order of cause and effect
were sometimes reversed. Which-
ever approach will ultimately prove
the best, we are convinced that
causality objections offer no compelling
grounds for desisting from further
theoretical and experimental work on
metarelativity.
Acausal experiments. This assertion
is particularly true of searchers for "sin-
gle events" for which the causality ar-
guments, such as raised above and else-
where,915 are irrelevant. An excel-lent example of this type of experi-
ment is the search for the reaction
p + p-*p + p + T (tachyon)
which Bogdan Maglic, James Norem,
Howard Brody and their collabora-
tors have told us they propose to carry
out at the Princeton-Penn accelerator.
In some other frame this reaction may
appear as p + p + T -> p + p. Since
data to be recorded by their missing-
mass technique16 pertain to tardyon
channels only, this type of experiment
would reveal the presence of tachyons
without forcing them to disclose the
direction of their path in time. [The
experimenters are placing their proton
detectors at 120 deg, whereas the
maximum angle for protons from the
p-fp->p-fp-fX (real-mass par-
ticle) reaction is 90 deg. Only
tachyons having a proper mass be-
tween 0.5t and 3.5t GeV could lead
to emission of protons in the 120 deg
direction.] Providing the experiment
is not thwarted by unexpected back-
ground problems, Maglic and his col-
laborators hope to be able to infer the
existence of tachyons even if the cross
section for their production is as small
as 10G times smaller than that for
the p + p -> p + p + 7T° reaction.
An earlier p + d -» He3 + X missing-
mass search for tachyons,16 also
initiated by Maglic, was inconclusive
because the cross section for produc-
tion of He3 turned out to be extremely
low (about 10"34 cm2 at 3 GeV).
Maglic holds out much more hope for
the p + p —» p + p + X reaction.
Other experiments unaffected by
causality objections include the bub-
ble-chamber search by Charles Baltay
and collaborators17 for the reactions
K- + p -» A + T and p- + p -> TT+
+ TT~ + T (we use p- for antiproton),
and the search for the reaction w~
+ p -» n + T that Michael Kreisler
tells us he is carrying out. In some
other frame these reactions may look
like K-+p + T-*A, p-+p + T
-> 7T+ + 7T-, and IT- + p + T -» n,
respectively.
Superluminal physics. We are very
much encouraged by imaginative sug-
gestions such as that of Stephen
Brown above and that of John W.
Rhee,18 but we prefer to withhold our
comment on them until tachyons ac-
tually have been detected and their
properties are better understood.
In conclusion we wish to say that
we are pleased to see our sentiments
echoed in a comment to us from
Iwo Bialynicki-Birula to the effect
that the concept of faster-than-lightparticles is not really that unorthodox.
He reminds us that all concepts of
nonlocal interactions in field theory
imply the existence of some agent
carrying the interaction over space-
like distances and thus nonlocal field
theories have implicitly assumed the
existence of some sort of superluminal
entity. Notwithstanding questions of
causality, we hope to have shown2
that the special theory of relativity
can be consistently generalized to ac-
commodate faster-than-light particles.
By way of encouragement to all
those working or contemplating work
in the field of superluminal physics
let us quote the adage coined by David
Farragut at Mobile Bay: "Damn the
torpedoes; full speed ahead!"
OLEXA-MYRON BILANIUK
Swarthmore College
E. C. GEORGE SUDARSHAN
University of Texas at Austin
References
1. O. M. Bilaniuk, E. C. G. Sudarshan,
PHYSICS TODAY 22, no. 5, 43 (1969).
2. O. M. P. Bilaniuk, V. K. Deshpande,
E. C. G. Sudarshan, Am. J. Phys. 30,
718 (1962).
3. C. A. Hurst, Math. Rev. 26, 667
(1963).
4. T. Alvager, J. Blomqvist, P. Erman,
1963 Annual Report of the Nobel
Research Institute, Stockholm, pp.
95-97.
5. G. Feinberg, Phys. Rev. 159, 1089
(1967).
6. A. C. Clark, The Promise of Space,
Harper & Row, New York (1968)
p. 299.
7. R. C. Tolman, The Theory of Rela-
tivity of Motion, University of Cali-
fornia Press, Berkeley (1917) p. 54.
8. D. Bohm, Special Theory of Rela-
tivity, W. A. Benjamin, New York,
1965, pp. 155-160.
9. G. A. Benford, D. L. Book, W. A.
Newcomb, Lawrence Radiation Lab-
oratory Report UCRL-71789, Liver-
more, (1969).
10. R. Fox, C. G. Kuper, S. G. Lipson,
Nature 223, 597 (1969).
11. A. Sommerfeld, Physics Z. 8, 841
(1907); L. Brillouin, Ann. Physik 44,
203 (1914).
12. R. Skinner, Relativity, Blaisdell Pub-
lishing Co, Waltham, Mass (1969)
p. 189.
13. R. G. Newton, Phys. Rev. 162, 1274
(1967).
14. P. L. Csonka, Phys. Rev. 180, 1266
(1969).
15. W. B. Rolnick, Phys. Rev. 85, 1105
(1969).
16. B. Maglic et al, Bull. Am. Phys. Soc.
14,840, (1969).
17. C. Baltay, G. Feinberg, N. Weh, R.
Linsker, US AEC Report NYO-
1932(2)-148(1969).
18. J. W. Rhee, Technical Report No. 70-
025, Center for Theoretical Physics,
University of Maryland (1969). 0
52 • DECEMBER 1969 • PHYSICS TODAY
Transmitting the changing scene
This girl's picture was produced
on a special Picturephone® system; it
will never look like this in your
home. The white areas mark the only
picture points which changed in 1/30
second (the duration of one video
frame). The remainder of the picture
was blanked out.
This emphasizes how
Picturephone use differs from ordi-
nary television: the Picturephone
camera usually points at a single
scene throughout a call and most of
the motion is confined to the
subject's lips and eyes. Everything
else—perhaps 90 percent of the
picture—remains stationary.
Frank W. Mounts of Bell Labora-
tories used this fact to design an
experimental video system that maymake it possible to transmit three
Picturephone calls over a channel
that otherwise could carry just one.
An ordinary Picturephone system
sends thirty complete pictures each
second. In Mounts' experimental
system, only changes from one
picture to the next are transmitted.
A complete picture (information
about dot positions and brightnesses)
is stored at both the transmitting and
receiving ends. As the camera's
electron beam scans the original
image, the brightness at each point is
compared with the stored value.
Whenever there is a significant
difference, the system updates the
stored frame and transmits the new
brightness level and dot position.
At the receiving end, as thepicture tube's beam arrives at each
point, the incoming information is
checked to see whether a picture-
point revision has arrived. If so, it is
displayed and stored.
Because some areas of the pictui
do not change, while others change
extensively, revised points may come
in bursts. Transmitter buffers smootf
the flow by reading the information
out onto the line at a constant rate.
This new technique, one of
several now being investigated at
Bell Laboratories, promises to help
keep transmission costs down when
the Picturephone service becomes
generally available.
From the Research and
Development Unit of
the Bell System— •*••••_
7 Bell Lab
Introducing
A NEW Ge(Li) STANDARD
Guaranteed Performance at 7.639 MeV
2-escape
peaks 1-escape
peaks7.639 MeV
FWHAA= 4.5 keV14.4 keVJB
tf«
HI
Neutron capture gamma rays from the 7.639 level in Fe 57
trio
ftr
Another standard in performance of Ge(Li) detec-
tors is set by Princeton Gamma-Tech. Now energy
resolution is guaranteed where you need it—at high
energies (7.639 MeV) as well as at low (1.33
MeV).
Our ultra-high-efficiency, high-resolution Ge(Li)
detectors are now guaranteed to have a system en-
ergy resolution of better than 6 KeV (FWHM) at
the 7.639 MeV Iron doublet. The typical perform-
ance of these detectors is illustrated in the spectrum
above—4.5 KeV (FWHM). The resolution at Co60
is 2.5 KeV FWHM, with a 25 cm. relative efficiency
of 8%.If you did not see Princeton Gamma-Tech's ultra-
high-efficienq', high-resolution Ge(Li) detectors
demonstrated at the New York and Washington
APS meetings, or wish further information, ask for
our latest inventory data sheets including spectra of
actual performance. These Ge(Li) detectors are
ready for immediate delivery.
PRINCETON,GAMMA-TECH-IS
Box 641, Princeton, N.J. 08540, U.S.A.
(609) 799-0345. Cable PRINGAMTEC.
SEARCH AND DISCOVERY
|Continuous-Wave Chemical Laser Requires No External Energy Source
Terrill A. Cool and Ronald R. Ste-
ens1 of Cornell University believe
ey have produced the first continu-
us wave all-chemical laser. In a pa-
• he delivered 26 Nov. at the Amer-
an Physical Society fluid dynamics
vision meeting in Norman, Okla-
homa, Cool told how they mixed com-
liiercially available bottled gases to
' get 1.06 X 104 nanometer emission
from carbon dioxide without any ex-
ternal energy source to initiate or sus-
tain lasing action. Maximal power
output was 8 watts; lasing continues
until the reactants are depleted (up to
several hours). The laser operates at
about 4% efficiency; Cool predicts
\5c/c efficiency with proper design
modifications. A typical electrically
excited CO2 laser has an overall effi-
ciency of about 8%.
Cool's mechanism for chemical
pumping of COL, involves a fluorine-
helium mixture, deuterium and nitric-
oxide gases as well as CO2. To ob-
tain fluorine atoms F2 and NO are
mixed:2
F2 + NO -> NOF + F
The flowing gas, which now contains
both F and F2, is mixed with deuteri-
um to produce vibrationally excited
deuterium fluoride in a chain
reaction3
F + D2 -> (DF)* + D
D (DF)* + F
The deuterium fluoride then transfers
vibrational-rotational energy to CO2,
pumping the CO2 to the upper laser
level8'4-5 from which it emits 1.06 X
104-nm radiation.
The reaction vessel is similar to one
used previously by Cool, Stephens and
Theodore J. Falk. F2 and NO are
mixed in an 11-mm bore quartz side-
arm; deuterium and carbon dioxide
are injected at the upstream end of a
9-mm bore Teflon tube. The reaction
time in this high-speed (600 m/sec)
flow is extremely rapid (100-200 mi-
crosec); Cool believes that most of the
laser output is from this portion of the
flow (see figure).
The Cornell results have shown that
a practical flow system is possible.Exhaust
2-meter radius mirror
Teflon tube
Pressure tap
NOGas injector for D2 and CO2
Sidearm
F2 and He
Inlets
CONTINUOUS-WAVE ALL-CHEMICAL LASER. Arrows show paths of reacting
gases. Lasing action occurs mainly in upstream portion of tube.
Because their system operates through
a collision mechanism and, unlike
some other chemical-laser systems, is
not limited to a maximal size, the Cor-
nell group believes it could be devel-
oped into a high-power laser. A con-
tinuous-wave chemical laser of this
type might be used in space.
Chemical lasers were first devel-
oped by Ceorge C. Pimentel and Je-
rome V. Kasper6 at the University of
California, Berkeley. The Berkeley
group, says Pimentel, has been using
pulsed chemical lasers to investigate
the role of vibrational and rotational
energy states in chemical-reaction dy-
namics. Other groups1-7 have re-
ported continuous chemically pumped
Cold Octopole and Hot Tokomak
Two results reported at the Dubna In-
ternational Symposium on Closed
Confinement Systems have excited fu-
sion physicists. The high temperature
and long confinement time that Lev
Artsimovich observed with Tokomak
(PHYSICS TODAY, June, page 54) have
been confirmed by a visiting British
team, and with the Gulf General
Atomic multipole Tihiro Ohkawa ob-lasers, but until now an external en-
ergy source has been required.
References
1. T. A. Cool, R. R. Stephens, J. Chem.
Phys. (to be published).
2. H. S. Johnston, H. J. Bertin Jr, J. Am.
Chem. Soc, 81, 6402 (1959).
3. T. A. Cool, T. J. Falk, R. R. Stephens,
Appl. Phys. Lett, (to be published).
4. R. W. F. Gross, J. Chem. Phys. 50,
1889 (1969).
5. H. L. Chen, J. C. Stephenson, C. B.
Moore, Chem. Phys. Lett. 2, 593
(1968).
6. J. V. Kasper, G. C. Pimentel, Phys.
Rev. Lett. 14, 352 (1965).
7. D. J. Spencer, T. A. Jacobs, H. Mir-
els, R. W. F. Gross, Internal. J.
Chem. Kin. (to be published).
Show Long Confinement Times
served very long confinement times.
In further experiments (which Ohkawa
reported at the November meeting of
the APS plasma-physics division in
Los Angeles) Ohkawa observed clas-
sical diffusion in a dilute cold plasma.
His collaborators were Masaji Yoshi-
kawa, Robert Kribel and A. A.
Schupp.
Ohkawa used an octopole, which
PHYSICS TODAY . DECEMBER 1969 • 55
SEARCH AND DISCOVERY
consists of four internal rings carrying
parallel currents in the toroidal direc-
tion. Just like all multipoles, the
device has axial symmetry about the
major axis of the torus. Ohkawa de-
signed the device to reduce losses to
the ring supports, one of the major
limitations in earlier octopoles; it has
a plasma volume of 10 000 liters.
In the first experiments Ohkawa
used a plasma density of 3 X 1010
particles/cm3; electron temperature
was about 5 eV. In the new experi-
ments Ohkawa pushed the density
higher (1011 particles /cm3) and the
temperature lower (a few eV), to a
regime where one should get classical
diffusion. Ohkawa did indeed ob-
serve classical diffusion for the first
150 millisec; then the behavior
smoothly changed and became Bohm-
like. His measured decay time of 200
millisec corresponds to about 300 times
the Bohm value.
Although the octopole confinement
is the longest observed in any toroidal
device, its plasma is cold and dilute
and not likely to be scaled up into areactor because of the interior rings.
(General Atomic plans to build a
Doublet device, in which internal
conductors are replaced by plasma
current.) However, because the oc-
topole plasma is well contained one
might now try to understand what
effects are responsible for the en-
hanced confinement and then apply
the knowledge to a geometry that is
more suitable for a fusion reactor.
The Tokomak plasma is already
nearly thermonuclear; it gives neu-
trons, it is hot and it is dense. At
Dubna N.J. Peacock and D. C. Robin-
son of Culham Laboratory and N.
Sammikov of the Kurchatov Institute
reported that Tokomak T-3 produced
in one mode of operation electron
temperatures of 900 ± 100 eV and
confinement times of about 25 milli-
sec with a density of 2 X 1013 par-
ticles/cm3. Earlier measurements by
Kurchatov had yielded 3 X 1013
particles/cm3 at 1000 eV and 20
millisec. The Culham-Kurchatov col-
laboration determined temperature
and density by analysis of Thomson
scattering from a pulsed ruby-laser
beam.
Air Force Solar Telescope and OSO-6 Now Observing the Sun
Two new devices are now observing
the sun—a solar vacuum-tower tele-
scope built by Air Force Cambridge
Research Laboratories and OSO (Or-
biting Solar Observatory)-6.
The solar telescope is 111 meters
AIR FORCE SOLAR TELESCOPE is
111 meters high. The optical system
is evacuated to 0.250 ton*.high and has a central core that con-
tains the entire optical system, which
is evacuated to 0.250 torr. Light en-
ters through a 76-cm aperture, passes
through a quartz window and is then
reflected by two flat mirrors to the 64-
inch (1.62-meter) focusing mirror
(focal length 55 meters) at the bot-
tom of the shaft. Theoretical resolv-
ing power is 0.2 sec of arc; so one
can expect to resolve fine details on
the solar disc.
Because the objective port is high
above most air turbulence and heat
currents that swirl up when the sun
heats the ground, and because the op-
tical system is evacuated, image sta-
bility is expected to be excellent.
Richard B. Dunn designed the system.
Located in the Sacramento Moun-
tains of New Mexico, the $3.3-million
instrument will be used to study solar
centers of activity—sunspots, magnetic
fields, flares and plage areas. One goal
is identification of precursors to solar
flares.
OSO-6 is returning data from seven
experiments. From its vantage point
above the atmosphere, it can study in
detail the ultraviolet and x-ray spectra
at any point in the solar disc. Its ex-
pected lifetime is six months.IN BRIEF
US and Soviet radio astronomers were
to collaborate this fall on the longest
baseline ever used for two-telescope
interferometry. Telescopes at
Green Bank, W. Va., and the
Crimean Astrophysical Observatoiy
near the Black Sea-9600 kilometers
apart—should provide a resolution of
0.0003 to 0.0005 seconds of arc at a
3-cm wavelength.
Construction has begun on an observa-
tory to house a 40-inch (101-cm)
astrometric telescope at the Fan
Mountain Observatory of the Uni-
versity of Virginia.
A two-year oceanographic study of the
central Mediterranean is taking
place. Geophysicists from the
Woods Hole Oceanographic Institu-
tion, the University of Bologna and
the University of Trieste are cooper-
ating in the project and expect to
obtain continuous reflection and re-
fraction data from the earth's crust
down to the Mohorivicic discontinu-
ity.
Dicke Panel Says US Lags in
Radio-Astronomy Construction
The National Science Foundation Ad-
Hoc Advisory Panel for Large Radio-
Astronomy Facilities, headed by Rob-
ert H. Dicke, has decried the lack
of US radio-astronomy construction.
The panel, originally convened in Au-
gust 1967 (PHYSICS TODAY, September
1967, page 71), met again to review
its original recommendations. In a re-
cently issued report the panel points
out that none of the suggestions made
two years ago has yet been imple-
mented. The US, it says, has stood
still while Germany, India, the Neth-
erlands and the UK have begun con-
struction on large radio telescopes,
several of which will soon be in opera-
tion.
Noting that discoveries since the
panel first met (pulsars, existence of in-
terstellar formaldehyde, ammonia and
water) have made construction of new
telescopes even more imperative now
than two years ago, the panel recom-
mends that:
• the 305-meter spherical-dish tele-
scope at Arecibo, Puerto Rico (PHYS-
ICS TODAY, April, page 65) be resur-
faced so that it can be useful for cen-
timeter-wave radio astronomy. Resur-
facing was urged two years ago as a
relatively inexpensive improvement.
• the Cal Tech proposal for con-
56 . DECEMBER 1969 • PHYSICS TODAY
struction of an eight-dish array at the
Owens Valley Observatory be ac-
cepted.
•construction of a fully steerable
134-meter radome-enclosed dish be
begun immediately, probably in the
dry southwestern portion of the US.
• construction of the Very Large
Array of 27 antennas, as proposed by
the National Radio Astronomy Obser-vatory, be begun immediately. This
array would produce up to three pic-
tures daily with a resolution of 1 sec of
arc, which is equal to that of optical
photographs.
• studies of methods for construc-
tion of very large steerable dishes be
continued. Emphasis should be on
design of an antenna useful at wave-
lengths as small as 3-6 mm.• support of university radio as-
tronomy be continued and improved.
• grants and contracts for US sup-
port of radio-astronomy installations
require not only that half the observ-
ing time be available to visitors, but
also that the installations be managed
to assure representation of national in-
terests and maximal usefulness to visi-
tors.
Measuring It Better: A Visit to Bureau International des Poids et Mesures
In an old house in Paris
All covered with vines
Lived twelve little girls
In two straight lines.
If you drive west from Paris toward
Versailles, you can easily pass through
the little town of Sevres without
knowing that in it is the International
Bureau of Weights and Measures.
Only when you turn through a narrow
arched gateway and climb a few hun-
dred yards through the woods to a
small clearing in the Pare de St Cloud
do you come to the little historic
manor, Pavilion de Breteuil.
The approach and the exterior
suggest an atmosphere like that of the
lines that open Ludwig Bemelmans's
"Madeleine in Paris/' Once, in fact, it
bad such an atmosphere. "Forty years
ago," Jean Terrien, the present direc-
tor told me on a recent visit, "Bureau
International des Poids et Mesures
had the feeling of an old lady. There
were few pieces of original research."Step inside, though, and you find a
different atmosphere. The neat labo-
ratories are making some of the most
careful measurements in the world.
The aim is to determine standard
values and best procedures to measure
them. Major concerns are length,
mass, time, acceleration of gravity,
electrical units, temperature, photom-
etry and ionizing radiation.
The main function of the bureau is
coordination of efforts everywhere to
define and measure quantities accu-
rately. Its small staff ("about 50 per-
sons including the gardener," said
Terrien) can not do such amounts of
work as go on at the US National Bu-
reau of Standards and the UK Nation-
al Physical Laboratory. But it does
much to test and compare the meth-
ods suggested by these and similar na-
tional laboratories. Moreover seven
international consultative commitees
based at BIPM make the most funda-
mental decisions required for coordi-
nation and cooperation. Their sevensubjects are electrical quantities, pho-
tometry, thermometry, ionizing radia-
tion, definition of the meter, definition
of the second, definition of units.
40 governments have signed the
"Convention du Metre," the 1875
treaty under which BIPM was born.
They meet at least every six years and
usually every four years in the Confer-
ence Generale des Poids et Mesures.
(Terrien shuddered at the thought
that BIPM might have become part of
the League of Nations or the United
Nations. As an organization fulfilling
a purpose, it is running more effec-
tively than those trying to find pur-
poses they can fulfill.) The 40 elect
an 18-member committee, which
operates BIPM and the seven con-
sultative committees.
The bureau is in no sense French al-
though it happens to have a French
home and a French director. Former
directors have been Swiss, Italian,
Norwegian and British. It does not
even function as a standards bureau
HISTORIC MANOR HOUSE in western outskirts of Paris is home for international
bureau that specializes in standard values an,d best procedures to measure them.DIRECTOR JEAN TERRIEN was for-
merly an opticist on staff of the bureau.
PHYSICS TODAY • DECEMBER 1969 57
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Neutron sources. From Po 210, Pu 238, Pu
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Beta and gamma sources. From a wide
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Threshold detectors. From PU 239, U 235,
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Non-radioactive target and secondary
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tainers. Radioisotopic heat sources. And
special shipping containers for sources.
Monsanto58 DECEMBER 1969 . PHYSICS TODAY
SEARCH ANDDISCOVERY
for France, which distributes standard-
ization work among several minis-
tries and only recently has moved to
coordinate the various efforts more
closely.
The BIPM staff has 12 physicists,
eight of them "pure" and four experi-
enced in the work of the bureau.
Working with them are 12 very skill-
ful senior technicians.
Length. "What measurements do
you consider particularly your own?"
I asked Terrien. "We make a special-
ty of length," he replied and described
the work that went into the redefini-
tion of the meter. When the change
from a standard bar to an optical
wavelength was proposed, the US sug-
gested a Hg198 line, the Germans
proposed Kr84 or Kr86 and the Rus-
sians preferred a standard based on
Cdm. Starting in 1955 Terrien,
who was not then director but an opti-
cist on the staff, spent three years
studying line shapes. He finally con-
cluded that the Kr86 transition 2p10
-> 5d5 (now the base of the defini-
tion) was best. It made a narrow
spectral line, and Terrien could ex-
plain its shape completely in terms of
Doppler shift, pressure broadening
and lifetimes of states.
Unfortunately the line is not entire-
ly symmetrical. To improve upon it
as a length standard two courses are
possible: One is to recognize the line
shape in a Michelson-interferometer
pattern and with it specify just whichpart of the line is the standard wave-
length. The other is to go to a laser
method. Lasers, to be sure, have the
difficulty that tuning can pull the oscil-
lation away from the natural wave-
length. To remedy it you can adjust
the laser to a natural absorption fre-
quency. Work is commencing on the
scheme. For example the helium-
neon laser has several coincidences
with iodine and methane absorptions.
Probably ten or 15 usable coincidences
are known now and 100 might ap-
pear with two or three years of work.
In the laboratory, I visited the com-
parator BIPM uses to compare stan-
dard bars with the krypton line.
Temperature of the room it stands in
is controlled to a few hundredths of a
degree, and temperature in its tank to
a thousandth. The operator sits next
door, directing a light beam along a
selected interference path and recog-
nizing scratches on the test bar by sig-
nals from optical scanning devices.
Time and gravity. Time is closely
related to length, or, if you prefer, it
has become the same quantity now
that measurements of optical frequen-
cies have become possible. The de-
velopment puts BIPM into a new busi-
ness. There are no time standards at
Sevres, but there exists the consulta-
tive committee on the second. "I am
learning now what I must know," says
Terrien as he discusses how BIPM
may get involved. He feels that with
recent improvements of technique the
present second based on a cesium
transition is the best unit, but the hy-
LENGTH COMPARATOR operates by remote control in constant-temperature envi-
ronment. With interferometry it compares standard bars with krypton-86 wavelength.drogen maser might produce a better
one. Laser standards are better in
principle, but accuracy with them is
not yet good enough to compete.
I stood at the spot where accelera-
tion of gravity is known to eight sig-
nificant figures. Changing elevation
by 2 cm changes the last figure, point-
ed out Terrien. So would a signifi-
cant amount of concrete construction
in the basement. Then we walked
next door where standard cells in tem-
perature-controlled oil baths and stan-
dard resistors offer the basis for elec-
trical measurements. Gravity and
electrical measurements are closely re-
lated. As you know better the weight
of a kilogram, you can measure more
accurately the forces between coils;
forces are related to the standard am-
pere, and so on.
A working group of the committee
on electrical units studies measure-
ments of the proton gyromagnetic
ratio. Well enough measured, it
might some day be a basis for better
electrical units.
Another new device that might
serve the same purpose uses the Jo-
sephson effect: A constant potential
appears across a narrow junction be-
tween superconducting metals when
they are driven with a fixed frequen-
cy.
Radiations. The newest section of
BIPM is devoted to ionizing radia-
tions. I saw x-ray and neutron genera-
tors, Co60 irradiators, free-air and
cavity ionization chambers, gamma
spectrometers and counting devices
for neutron and radioactivity sources.
Among unique accomplishments of
the section is an absolute alpha-parti-
cle spectrograph for maximal possible
accuracy. It uses a homogeneous
magnetic field that bends alphas
emerging from a slit through semi-
circles and causes them to focus on a
photographic plate. Results obtained
so far add at least one decimal place
to best former measurements. Hopes
are for detection of line shapes pro-
duced by interaction between alphas
and the electron clouds of the atoms
from which they come.
Terrien is a careful man whose man-
ner suggests the precision with which
French engineers design their cars and
vacuum tubes. He says his job makes
him travel too much in his efforts to
learn what he must know. Like phys-
icists of other times and places he and
BIPM appear to enjoy the challenge of
making discoveries by resolving the
next decimal. —RHE •
PHYSICS TODAY . DECEMBER 1969 • 59
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please contact:
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atPHYSIOLOGICAL GEOMAGNETIC
62 . DECEMBER 1969 . PHYSICS TODAYMaple Avenue, Pine Brook, N.J. 07058
Phone (201) 227-2000 • TWX: 710-734-4347
STATE AND SOCIETY
Metzner Named Assistant
Director of AIP Publications
A. W. K. Metzner was named to the
new position of assistant director of
publications for the American Insti-
tute of Physics, where he shares re-
sponsibility for all operations with
Hugli C. Wolfe, director of publica-
tions. Metzner will explore new com-
position techniques, particularly com-
puter-aided photocomposition and
METZNER
typewriter composition. He and the
editorial staff for The Physical Review
are located at the institute's newly ac-
quired 3000 square feet of space at
300 East 42nd Street, New York.
Metzner was formerly an editor of
The Physical Review and involved in
the typewriter composition of Section
1 at Brookhaven National Laboratory.
David Howell was also recently ap-
pointed as manager of the AIP edi-
torial section, within the publication
division, and replaces David Biesel.
Howell was formerly with the techni-
cal-information division at the Ameri-
can Institute for Aeronautics and As-
tronautics.
Fund of Abdus Salam
Has First Recipient
A Fund for Physics in Developing
Countries has been set up by AbdusSalam, director of the International
Centre for Theoretical Physics (PHYS-
ICS TODAY, Sept., page 77, 1968), with
the $30 000 he received as winner of
the 1968 Atoms for Peace Award.
The first fellowship recipient is A. Q.
Sarker, an East Pakistani physicist
who specializes in dispersion theory
and high-energy physics.
In an effort to thwart the brain
drain, the fund will help research
physicists, particularly theoretical
physicists, from the developing coun-
tries. First priority will be the award-
ing of fellowships to senior physicists,
permitting them to participate in the
Centre activities; preference will be
given to theorists from Pakistan. The
board of trustees consists of Paolo Bu-
dini, deputy director of the Centre;
P. T. Matthews of Imperial College,
London; I. H. Usmani, chairman of
the Pakistan Atomic Energy Commis-
sion; and Salam.
Support funds, which are invested
at 5.59c interest with the Italian
bank, Cassa di Disparmio di Trieste,
have come from the firms Messrs
Piaggio and Co, Genoa; Pirelli S.p.A.,
Rome; SNAM Progetti S.p.A., Milan;
and Tarbela Joint Venture, Milan—in
addition to the bank itself, which
made the first donation and agreed to
pay interest on its donation at 11%.
The appeal for donations was made
by N.A.M. Raza, former Pakistani am-
bassador to Italy.
Dart, Moravcsik to Evaluate
Foreign Graduate Candidates
How can universities evaluate a pros-
pective foreign graduate student with-
out seeing him? Francis Dart and
Michael Moravcsik, Univ. of Oregon,
aim to do something about the prob-
lem. With support from the universi-
ties of Oregon, Michigan, Pittsburgh
and California at Los Angeles they are
writing evaluations based on inter-
views in Korea, Hong Kong, Thailand,
Singapore, Malaysia, India and Pakis-
tan. Their month-long trip was de-
signed to include interviews with
about 150 students who want to study
advanced physics in the US. The visit
dates were 11 Oct.-19 Nov.
Each report will be available to any
school in which the student is inter-ested and any interested in him. The
four universities sponsoring the Phys-
ics Interviewing Project get the evalu-
ations first; two months later it be-
comes generally available. Dart and
Moravcsik only evaluate; they are not
involved in recruitment, admission,
scholarships and the like.
The two-man committee and their
sponsors view the project as an experi-
ment. Students admitted through it
will be surveyed during 1970-71 to
see whether personal interviews im-
prove selection.
JILA Has Fellowships and
Associateships for 1970-71
The Joint Institute for Laboratory As-
trophysics is soliciting applicants for
1970-71 visiting fellows and research
associates. "Jomt" means Bureau of
Standards plus University of Colorado,
and the institute is housed on the uni-
versity campus at Boulder. Its sub-
jects are theoretical astrophysics, low-
energy atomic physics and related
topics.
About ten stipends exist in each
category (associates get $11 000 plus
expenses), and other visitors are in-
vited to come with their own support.
One more fellowship is shared be-
tween JILA and the university Labo-
ratory for Atmospheric and Space
Physics. Fellows are expected to
come with extensive postdoctoral re-
search and have no obligations. As-
sociates are new PhDs and are ap-
pointed simultaneously to a university
department and JILA.
AIP Publishes Guide to
Undergraduate Departments
A guide to the physics departments at
622 US colleges and universities of-
fering undergraduate majors has been
published by the American Institute
of Physics. Another 207 schools that
offer majors but did not respond to
the AIP questionnaire are listed. The
176-page book includes information
on faculty, students, equipment, and
physics-major programs. Copies of
Student's Guide to Undergraduate
Physics Departments can be ordered
from AIP, 335 E. 45th St., New York,
N. Y. 10017. Price is $2.00 per copy
'PHYSICS TODAY • DECEMBER 1969 • 63
ell Laboratories, Eastman Kodak, Western
lectric, Texas Instruments, Gulf Research,
omsat, Hughes Research, Sprague
lectric, Signetics Division of Corning,
id other companies,
DS Alamos Scientific Laboratories,
ASA, Naval Applied Science Laboratories,
anscom Air Force Base, U.S. Bureau of
lines, and other government installations,
tanford Research, Catholic University,
olorado State, Mississippi State, Ball
tate, Rose Polytechnic, Carnegie Mellon
niversity, Oregon State, C.W. Post, State
niversity of N.Y., University of West Virginia
niversity of Missouri, and other universities-
Have all ordered our particle accelerators.
(How come no one knows our name?)
Because, until we reacquired the
total marketing responsibility for our
products from our good friends
(Picker Nuclear, if you must know),
you heard their name, not ours. Ac-
cordingly, we'd now like to introduce
ourselves as the independent com-
pany that designs and manufactures
the particle accelerators acquired by
the organizations listed above. The
knowledgeable organizations listed
above.
So: we are Accelerators, Inc., of
Austin, Texas, a major producer of
low energy particle acceleratorssince 1965. And, nowadays, in the
energy range in which we've con-
centrated we apparently supply more
particle accelerators than all of the
other accelerator manufacturers
combined. Rather gratifying, that.
But that's history. What can we do
for you now?
Whatever your interest, we can de-
sign and build the particle acceler-
ator that's specifically tailored to
your needs. Ion implantation. Neu-
tron activation analysis. Neutron
radiography. Teaching. Research.
Others? We'll work with you to pro-vide the particle accelerator that fits
your individual requirements. As we
did with Bell Laboratories, Eastman
Kodak, Western Electric, Texas In-
struments ...
Please now write for our catalogs
and/or tell us of your application.
Accelerators, Inc., 212 Industrial
Boulevard, Box 3293, Austin, Texas
78704 (phone 512-444-3639).
Accelerators Inc.
STATE ANDSOCIETY
prepaid or $2,50 if billed. Intended
to be particularly interesting to ad-
visers in high schools and colleges,
the book is a companion volume to
Graduate Programs in Physics and As-
tronomy.
Health Physics Society
Elects New Officers
The Health Physics Society has an-
nounced that officers for 1969-70 are:
J. Newell Stannard (University of
Rochester School of Medicine), presi-
dent; Claire C. Palmiter (Federal Ra-
diation Council), vice-president; Rob-
ertson J. Augustine (Bureau of Radio-
logical Health), secretary; Robert L.
Zimmerman (Nuclear Research Cen-
ter, Georgia Institute of Technology),
treasurer.
Nixon Names 12-Man Task Force
To Review US Science Policy
A second review of federal science
policy in the US has been ordered by
President Nixon. Ruben F. Mettler,
executive vice-president of TRW, Inc,
heads the 12-man group, which in-
cludes Charles H. Townes of the Uni-
versity of California at Berkeley, Alvin
M. Weinberg of Oak Ridge National
Laboratory and Philip Handler, presi-
dent of the National Academy of Sci-
ences.
APS Arranges Group Flights
To Europe and Japan in 1970
Group flights to Europe and Japan
timed to coincide with major inter-
national meeting in 1970 are being
arranged by the American Physical
Society for its members and also
their families. The schedule includes
spring and summer flights to London,
summer flights to Helsinki and Lei-
den, and a late summer flight to Tokyo.
Prices are substantially below com-
mercial rates. Details can be ob-
tained from the business manager of
the society at 335 E. 45th St., New
York, N. Y. 10017.
European Physical Society
Announces Division Chairmen
Chairmen have been named for the
first five divisions of the European
Physical Society. The EPS Council
has approved the divisions for two
years, after which it will review thesituation and make any changes it
feels appropriate. The divisions and
their chairmen are:
Atomic spectroscopy, Alfred Kast-
ler, Paris; Condensed matter, Samuel
F. Edwards, Manchester; Low-tem-
perature physics, Jan de Boer, Amster-
dam; plasma physics, Bo Lehnert,
Stockholm, and quantum electronics,
Klaus P. Meyer, Berne.
The council can approve additional
divisions on application of five or
more members.
AIP and Society Journals
Available in Microfilm
Microfilmed volumes of all American
Institute of Physics journals and some
member-society journals are available
as of 1 Jan. 1970. Supplied by Uni-
versity Microfilms Inc, they can be or-
dered from AIP at one cent per page
only after publication of a complete
journal volume. 1969 and later vol-
umes are in 16-mm reels; some earlier
volumes in 35 mm and others in 16
mm or microfiche; and Russian-trans-
lation volumes only in 16 mm. AIP
will charge the original subscription
price if the cost is higher than one
cent per page.
IN BRIEF
High isotopic-purity isotopes of U233,
U234 and Pu242 are being sold by
AEC, and CoG0 at high specific ac-
tivity (more than 200 curies per
gram) is available on loan from
AEC for heat sources.
State and local governments are get-
ting help in developing and plan-
ning science policies under a pro-
gram supported by the National
Science Foundation. Charles E.
Falk, NSF planning director, is in
charge.
The University of Miami has estab-
lished a solid-state laboratory with
$45 000 from the university research
council and $16 000 from the uni-
versity budget.
The American Nuclear Society has
added "organization members"
(paying $100 to $500 per year) to
its membership roster.
National Bureau of Standards has
completed eight years of construc-
tion at Gaithersburg, Md., with a
Fluid Mechanics Building, 20th
primary structure on the grounds.
Spectronics, Inc, is a new developerand manufacturer of optoelectronics
and infrared systems in Dallas, Tex.
G. W. Paxton is president.
An International Committee on Thin
Films with nine members from nine
countries has as chairman Klaus H.
Behrndt, NASA Electronics Re-
search Center, Cambridge, Mass.
Franklin A. Long, Cornell, is director
of a new interdisciplinary program
in science and technology. Starting
with a $140 000 National Science
Foundation grant, it will study prob-
lems of national and worldwide con-
cern: public policy, defense, food,
ecology, population, urbanization.
Science for the Blind, a nonprofit or-
ganization providing scientific mate-
rial on tape to about 500 persons,
urgently needs volunteer readers
with tape recorders for Science Re-
corded, which has been including
selections from PHYSICS TODAY for
more than five years. Volunteers
should contact: Mrs Donald A.
Duncan Jr, Science for the Blind,
221 Rock Hill Rd., Bala-Cynwyd,
Pa.
The American Society for Mass Spec-
trometry has grown out of Commit-
tee E-14 of the American Society
for Testing and Materials. J. L.
Franklin, Rice University, is presi-
dent.
A new national committee on material
sciences (metallurgy, chemistry and
solid-state physics) is headed by
Frank J. Blatt, Michigan State.
The Hospital Physicists' Association
(British) has transferred its secre-
tariat to headquarters of the Insti-
tute of Physics and the Physical So-
ciety. IPPS will carry on day-to-
day business, but the 25-year-old,
850-member association will still
control its own activities.
Sigvard Ekiund has been appointed to
his third four-year term as director
general of the International Atomic
Energy Agency.
43 persons have been named to a
Metric System Study Advisory Panel
by Secretary of Commerce Maurice
H. Stans. The panel will assist the
secretary, the director of the Na-
tional Bureau of Standards, and the
Bureau's Metric System Study
Group headed by Alvin McNish.
The Scientists' Institute for Public In-
formation has received a $210 000
grant from the Alfred P. Sloan
Foundation to help expand its efforts
to stimulate discussion of issues in-
volving science and technology.
Diffraction Limited of Bedford, Mass.,
PHYSICS TODAY • DECEMBER 1969 • 65
WHAT FREE-SPECTRAL RANGE DO YOU NEED?
The Model 242, a scanning Fabry-Perot Interferometer, provides a wide choice ...
an infinite number to be exact. Add to this such features as variable mirror
spacing and extreme stability and the 242's versatility becomes even more apparent.
To further enhance its appeal, Tropel's new PZM electromicrometers
have been incorporated as standard equipment. These micrometers permit
mirror adjustments to be made of accuracies of 10 7 radians. This means maximum
finesse can be achieved quickly and easily at any mirror separation. Finesse
greater than 225 has been observed with the new Model 242 with a 1 cm.
Piano cavity. Now super-resolution is a reality in the examination of mode structure
of CW or pulsed lasers, for Brillouin scatter studies or for other difficult
spectral analysis problems.
EXCLUSIVE FEATURES:
• Compactness and simplicity of design
• Extremely high degree of versatility which allows proper
selection of free-spectral range and resolution
• Unmatched stability permitting adjustment of the cavity
spacing while scanning
1 Choice of three mirror configurations ... piano, confocal,
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• Quick interchange of mirrors
• Remote cavity tuning
• Removable magnetic feet
• Usable in scanning configuration, photographic or visual
configuration, and ultra-narrow band filter configurationAVAILABLE WITH:
• Plano-Plano mirror combination for Piano cavity
• Piano-Spherical mirror combination (5cm radius or
2.5 radius) for bifocal cavity
1 Spherical-Spherical mirror combination (5cm radius or 2.5
radius) for confocal cavity
• Five spectral ranges (each range available in any mirror
configuration)
Cd UV 0.325M
Cd 0.43M to 0.46M
Ar 0.46/LI to 0.56M
HeNe 0.60M to 0.70M
IR 1.05M to 1.15M
• Manual mirror adjustments only
i Accessory systems equipment
For further
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contact IIUH>EL,,»c.Designers and Manufacturers of Precision Optical Systems and Instruments
52 WEST AVE., FAIRPORT, N. Y. 14450
PHONE: (716) 377-3200
66 • DECEMBER 1969 • PHYSICS TODAY
STATE ANDSOCIETY
has been sold for the second time in
a year. Last year (PHYSICS TODAY,
January, page 93) the Ealing Cor-
poration acquired the optical con-
cern. Now Sanders Associates of
Nashua, N. H., has bought it from
Ealing.
Ethel Snider was appointed to the new
position of administrative secretary
for both the American Crvstallo-
graphic Association and the Ameri-
can Association of Physicists in
Medicine, as of 1 Sept. Snider's
office is at the American Institute of
Physics, which has begun publish-
ing the Quarterly Bulletin of the
AAPM. ACA is a member society
of the AIP, and AAPM is an affili-
ated society.
NEW JOURNALS
Gordon and Breach is publishing four
new journals: Crystal Lattice De-
fects, a quarterly, with R. R. Hasi-
guti, University of Tokyo, as editor;
Geophysical Fluid Dynamics, a
quarterly with A. R. Robinson, Har-
vard, as editor; Modern Geology, a
quarterly, with Luciano B. Ronca,
Boeing Scientific Research Labora-
tories, as editor; Earth and Extra-
terrestrial Sciences: Conference Re-
ports and Professional Activities, to
be published irregularly, with A. G.
W. Cameron, Belfer Graduate
School of Yeshiva University, as
editor.
John G. Daunt is editor of the Journal
of Low Temperature Physics, a new
bimonthly from Plenum Publishing
Corp.
Atomic Data, a quarterly journal "de-
voted to compilations of experi-
mental and theoretical results in
atomic physics," had its first issue in
September. Katharine Way, Duke
University, is editor, and Academic
Press is publisher.
IKirT Nuclear Journal, published by
International Research and Technol-
ogy Corp in Washington, was
started this year to provide analysis
of developments in nuclear technol-
ogy and its impact on society.
Optics Communications, a new quar-
terly devoted to "rapid publication
of short contributions in the fields
of optics and interaction of light
with matter " is being published by
North-Holland Publishing Co. The
editor is Florin Abeles of the Labo-
ratoire d'Optique, Paris. •sta-bil-i-ty
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PHYSICS TODAY . DECEMBER 1969 • 67
FORMULATIONS OF CLASSICAL AND QUANTUM DYNAMICAL THEORY
by GERALD ROSEN, Department of Physics, Drexel Institute of Technology, Philadelphia,
Pennsylvania
This monograph reviews the mathematical structure within the logical relationships
between classical mechanics and quantum mechanics for nondissipative, closed physical
systems. Quantum mechanics is formulated according to Feynman, Schreodinger, and
Dirac. Up-to-date detail is given for the conceptually paramount Feynman passage
and "sum-over-histories" formulations for quantum mechanics. The mathematics
necessary for the understanding of this text are introduced in elementary terms, making
the work readily accessible to the reader. Recent quantum field theory applications of
functional differential operator formulations are also included.
December 1969, about 150 pp.
THE FUNDAMENTAL CONSTANTS AND QUANTUM ELECTRODYNAMICS
by B. N. TAYLOR, RCA Laboratories, Princeton, New Jersey. W. H. PARKER, Depart-
ment of Physics, University of California, Irvine, California; D. N. LANGENBERG, Department
of Physics and Laboratory for Research on the Structure of Matter, University of Pennsylvania,
Philadelphia, Pennsylvania.
This volume offers the most critical, comprehensive, and up-to-date analysis of theo-
retical and experimental information bearing on the fundamental physical constants.
This book makes available, in permanent monograph form, an original article appear-
ing in the July 1969 issue of Reviews of Modern Physics; it has been prepared in
cooperation with the American Physical Society in anticipation of the widespread inter-
est this material is certain to evoke among physicists in general as well as workers in the
precision measurements—fundamental constants fields.
December 1969, about 350 pp., $5.00
PHYSICAL ULTRASONICS
by ROBERT T. BEYER, Department of Physics, Brown University, Providence, Rhode Island
and STEPHEN V. LETCHER, Department of Physics, University of Rhode Island
Physical Ultrasonics is intended for graduate students and scientists who plan to apply
ultrasonic techniques to study the physical properties of solids, liquids and gases. A
thorough presentation is given of the generation, propagation and detection of ultra-
sonic waves with emphasis placed on the physical processes such as irreversible thermo-
dynamic treatment of relaxation theory, non linear effects, absorption in insulators, dis-
location damping and spin wave interaction.
November 1969, about 365pp., $18.50
RADIATION AND PROPAGATION OF ELECTROMAGNETIC WAVES
by GEORGE TYRAS, Cullen College of Engineering, University of Houston, Houston, Texas
This volume is intended for use in a two-semester graduate course in electrical engineer-
ing or electrophysics for students with only undergraduate preparation in electromag-
netic theory. It will also provide practicing engineers with a highly valuable reference
source. Topics covered include plane waves in anisotropic media and inhomogenous
media, spectral representation of elementary sources, field of a dipole in a stratified
medium, radiation in anisotropic plasma, axial currents and cylindrical boundaries,
diffraction by cylindrical structures, and aperatures on cylindrical structures.
1969, about 375 pp., $17.50
ACADEMIC PRESSNEW YORK AND LONDON
111 FIFTH AVENUE, NEW YORK, N. Y. 10003
68 • DECEMBER 1969 • PHYSICS TODAY
BOOKS
UFO's: fact or fiction?
SCIENTIFIC STUDY OF UNIDENTI-
FIED FLYING OBJECTS. E. U. Con-
don, scientific director; Daniel S. Gil-
mor, ed. E. P. Dutton, New York,
1969. Cloth $12.95, paper $1.95
ALIENS IN THE SKIES. By John
G. Fuller. 217 pp. Putnam, New
York, 1969. $5.95
UFO's? YES!: WHERE THE CON-
DON COMMITTEE WENT WRONG.
By David R. Saunders and R. Roger
Harkins. 256 pp. The New Ameri-
can Library, New York, 1969. $.95
by GERALD ROTHBERG
If I were asked for the most important
guideline in studying unidentified fly-
ing objects (UFO's), I would un-
doubtedly say, "Be skeptical of every-
thing!" I do mean everything, the
con as well as the pro of the UFO
controversy. Too many persons find
it impossible to delve into the subject
without eventually becoming overly
zealous supporters of their own points
of view. I like to believe this has not
yet happened to me, but so the reader
can judge I will first indicate my back-
ground in the subject.
In the summer of 1967 I worked
for the University of Colorado Un-
identified Flying Objects Project, di-
rected by Edward U. Condon. The
motivation was my belief that evidence
of extraterrestial intelligence (ETI),
if UFO's could provide it, would be
the most important discovery of all
time. The first difficulty, however,
is, "What constitutes evidence?" At
one extreme is Condon's attitude:1
"I won't believe in outerspace saucers
until I see one, touch one, get inside
one [and] haul it into a laboratory and
get some competent people to go
over it with me." At the opposite ex-
treme are the religious fanatics who
have gathered around some of the self-
proclaimed contactees.
This already delicate question of
evidence is further complicated by
economics. With its limited re-
sources, $526 000, the Colorado UFO
project produced a good, minimal ef-
fort. A thorough study would require
orders of magnitude more money.
For example, James E. McDonald,
meteorology professor and senior
physicist at the Institute of Atmo-spheric Physics, University of Arizona,
talks of an effort the size of the Na-
tional Aeronautics and Space Adminis-
tration. Faced with the economic im-
plications, I would be very confident
of my evidence before accepting ETI
as a reasonable working hypothesis for
recommending a large-scale investiga-
tion.
Almost all our information about
UFO's is from reports of visual sight-
ings. Some of these are truly startling
and mysterious. I am very unhappy
about these sightings, because it is so
easy to be deceived, and after my work
as a field investigator with the Colo-
rado project I am even more skeptical.
One night, for example, I was present
when about a dozen people in Harris-
burg, Pa., reported an object 1000
feet above the city, flashing red,
white and blue. It turned out to be
the star Capella, which was also later
responsible for a report of a flashing
object with projecting antennas and
a dome.
My reason for working with the
Colorado project was to attempt to
improve the objectivity of the data by
obtaining instrumental observations,
or combined instrumental and visual,with a team of scientists who were
prepared in advance to go into the
field with suitable instruments and
who had good mobility. This ap-
peared possible because there occa-
sionally is an outbreak of UFO activity
reported in a limited geographical
area. There happened to be such an
outbreak in the Harrisburg area that
summer and, after a month of prepara-
tion at project headquarters in Boul-
der, I spent a month there making
what I consider the best attempt so far
to see and record a UFO at first hand.
A brief description of this effort ap-
pears as case 27 in the UFO-project
report. I personally investigated
about 100 sightings and took 9000 pic-
tures with an all-sky camera, set up in
the center of activity, but never saw or
recorded a UFO. Three or four of the
sightings I investigated are as good as
some of the classic cases in the UFO
books, and it is this nagging residual
that keeps me from dismissing the
whole business as ridiculous. When
you arrive at the scene of a sighting
within 30 minutes and an otherwise
normal, respectable family tells you a
large luminous disk with a dome and
a flashing red light hovered 30 feet
LENTICULAR CLOUDS over Sao Paulo, Brazil. (Photo appears in the UFO-project
report and is reproduced courtesy of the Aerial Phenomena Research Organization.)
PHYSICS TODAY • DECEMBER 1969 69
over their heads, it is hard to believe
they did not see something real and
strange.
The report of the Colorado project,
Scientific Study of Unidentified Flying
Objects, has to be read by everyone
interested in the UFO question. It
is almost a thousand pages long, but
in the first reading many hundreds of
pages containing peripheral technical
information can be passed over.
Aliens in the Skies by John G. Fuller
is essentially just the transcript of the
"Symposium on Unidentified Flying
Objects" held on 29 July 1968, before
the House Committee on Science and
Astronautics. At this meeting six
reputable scientists, including Mc-
Donald and J. Allen Hynek, professor
and head of the astronomy depart-
ment at Northwestern University and
chief scientific consultant of UFO's to
the Air Force, presented a case for
the continued and expanded study of
UFO's in direct opposition to the
eventual recommendation of the
Colorado project.
Fuller has written two other books
on the subject, Incident at Exeter and
The Interrupted Journey, but this
latest book is definitely not of their
quality. Aside from a number of
nasty remarks about Condon and edi-
torial comments on the testimony,
there is nothing in this book that can
not be obtained from the printed
record available from the govern-
ment.2 Some of the most important
material at the symposium was pre-
pared documents submitted for in-clusion but not delivered orally at the
meetings. Most of this material is not
included in Fuller's book, which also
lacks a table of contents and does
not identify the congressmen who
participated.
David R. Saunders and R. Roger
Harkins's book, UFO's? Yes!: Where
the Condon Committee Went Wrong,
is meant to be read before reading the
project report itself because, like
Fuller's book, it attempts to question
the credibility of the report by ques-
tioning Condon's objectivity and that
of the project administrator, Robert J.
Low. In my opinion the report does
represent the thinking of a substantial
number of the senior staff, perhaps
even the majority, and therefore it
can not be faulted on grounds of bias.
I would have been less negative and
recommended a small continuing
study.
Although the report suggests that
the usual funding agencies accept re-
search proposals in this area, it ap-
pears very unlikely that the standard
machinery for processing proposals
will result in any grants. I would like
to know if anyone has submitted a
research proposal on UFO's. What-
ever one thinks about the controversy
between Saunders, on the one hand,
and Condon and Low, on the other,
which eventually reached the public in
Fuller's article in Look3 and resulted
in the dismissal of Saunders from the
project, this book and Fuller's are in
fact important complements to the
project report.
Reviewed in This Issue
69 GILMOR, ed.: Scientific Study of Unidentified Flying Objects
69 FULLER: Aliens in the Sky
69 SAUNDERS, HARKINS: UFO'S? Yes!: Where the Condon Committee Went
Wrong
71 DRAKE, DRADKIX, eds.: Mechanics in Sixteenth-Century Italy: Selections from
Tartaglia, Benedetti, Guido Ubaldo and Galileo
73 MOROZ: Physics of Planets
73 SLATER: Quantum Theory of Matter
75 STANLEY: Light and Sound for Engineers
75 SMART: Stellar Kinematics
75 MIHALAS: Galactic Astronomy
77 KILMISTER: Lagrangian Dynamics: An Introduction for Students
79 KLAUDER, SUDARSHAN: Fundamentals of Quantum Optics
81 EYRING, CHRISTENSEN, JOHNSTON, eds.: Annual Review of Physical Chemistry,
Vol. 19, 1968
83 Fox, MAYERS: Computing Methods for Scientists and EngineersHarkins was a reporter for the
Boulder Daily Camera during the
project. Saunders is a psychology
professor at the University of Colo-
rado and was one of the principal in-
vestigators of the project and its driv-
ing force. He put together a oatalog
of sightings that numbered roughly
2000 by the time I left the project.
The prospective reader then, if he is
not put off by the lurid title and book
oovers, will find a very readable ac-
count of the inner workings of the
project and the conflicts that finally re-
sulted in its schism.
But the book has more in it than
that. The title UFO's? Yes! means
that Saunders now believes there are
at least a small number of "real"
UFO's, that is, reports of UFO's that
lend themselves to thorough investiga-
tion and that have been investigated
and found inexplicable in terms of
known phenomena. This important
point is also made repeatedly in the
UFO symposium. Saunders also be-
lieves that ETI is the least implausi-
ble explanation of these real UFO's.
This is in marked contrast to the pro-
ject report, which plays down the few
unexplained sightings by burying them
in a mass of cases that were plausibly
explained. Of the 59 field investiga-
tions carried out by the project, none,
of course, conclusively support ETI,
but a few interesting cases remain
unexplained, and these should have
been prominently displayed to ensure
they would not be passed over. They
possibly contain the only worthwhile
information in the whole study. Ac-
tually the project case against UFO's
is much stronger than these numbers
indicate. Many sightings are dis-
cussed in other sections of the report
and satisfactorily explained, and per-
haps hundreds of other sightings,
most of which have also been ex-
plained, do not even appear; for ex-
ample, the 100 reported in Harris-
burg, and the numerous ones investi-
gated by telephone and discarded be-
fore field teams were sent.
Saunders discusses what he con-
siders the strongest evidence for real
UFO's and also describes his current
research on statistical and psychologi-
cal aspects of UFO's and possibilities
for future studies. There are also
descriptions of some other UFO sight-
ings that sometimes seem overdrawn
when compared with the descriptions
of them in the project report.
These three books should appeal
to a wide audience. The subject is
70 • DECEMBER 1969 • PHYSICS TODAY
inherently sensational and at times
the documents read like first-rate de-
tective stones, but the investigators'
scientific training also comes through
clearly. None of these books, though,
should be read without the others.
One annoying feature of the project
report is its deliberate obscurity in
witnesses' names and exact sight loca-
tions in the case studies. It is not
clear why this was done, because the
report does not do it consistently, and
in many other cases these identifica-
tions are made. Furthermore some
of these cases are classics in UFO
literature. This procedure makes it
difficult to compare the results of the
project's investigation of a UFO re-
port with the descriptions given in
the other books. Two of the more
difficult examples are case 5 of the
report, which appears on page 126 of
"UFO's? Yes!" and case 42, which ap-
pears on page 197. Paul Julian's
discussion of orthoteny, that is the
straight-line relationship among dif-
ferent UFO sightings, appears in the
report (section 6, chapter 10), but is
not listed in the index and is rele-
vant to Saunders's discussions.
The point of view of the project
report is that all but a small per-
centage of UFO reports can be rea-
sonably explained, including some
that seem very strange. Therefore
it is plausible that the residue of un-
explained reports could also be ex-
plained if more information were
available, and that the hypothesis of
ETI is unnecessary and unproductive.
Saunders, McDonald and others be-
lieve that among this residual are
cases that are demonstrably not caused
by known natural phenomena, and
that ETI is the most plausible hypoth-
esis. We now need some reputable
journal to recognize this legitimate
scientific controversy and to publish
analyses of UFO reports with the ETI
proponents also stating their results.
Who knows? They may just be right.
References
1. W. Rogers, Look, 31, 6, 76 (1967).
2. "Symposium on Unidentified Flying
Objects/' Publication PB 179541.
Clearinghouse for Federal Scientific
and Technical Information. US De-
partment of Commerce, Institute of
Applied Technology, Springfield, Va.
22151.
3. J. C. Fuller, Look, 32, 10, 58 (1968).
* * *
The reviewer is an associate professor of
physics at Stevens Institute of Technol-
ogy.Precursors of Galileo and
modern science
MECHANICS IN SIXTEENTH-CEN-
TURY ITALY: SELECTIONS FROM
TARTAGLIA, BENEDETTI, GUIDO
UBALDO & GALILEO. Translated
and annotated by Stillman Drake
and I. E. Drabkin. 428 pp. The
Univ. of Wisconsin Press, Madison,
Wisconsin, 1968. $12.50
by ROBERT S. SHANKLAND
This is a work of the very highest
scholarship and in the tradition of
Stillman Drake's other distinguished
works on Galileo and related subjects
in the history of science. The book
was prepared in collaboration with
the late I. E. Drabkin, and includes
introductions written by him for his
translations. This selection of writ-
ings covers a century that was the
final transition period leading from
medieval to modern science.
The emphasis at that time was
almost exclusively on mechanics, hy-
draulics and the related mathematics,
especially algebra, which was recently
introduced into Europe. It is also the
period when Aristotle's influence
steadily declined and Archimedes of
Syracuse's, whose works had recently
became available in a useful transla-
tion, became more and more domi-
nant. There is also evidence of Hero
of Alexandria's influence and faint
suggestions of ideas from Leonardo
da Vinci.
The editors have prepared a
splendid introduction that could hardly
be improved upon as model writing
in the treatment of the history of
science as a rigorous intellectual dis-
cipline. The excellent translations
present the works of Niccolo Tortag-
lia, Giovanni Benedetti, Guido Ubaldo
and an early hitherto unpublished
work of Galileo on motion, prepared
during his teaching days at Pisa.
Many of the subjects that the
physics student usually associates ex-
clusively with the name of Galileo
were considered in great detail by
some scientists during the 16th cen-
tury. Examples are the science of
weights that led to important appli-
cations in the balance and the in-
vestigations of levers and pulleys
that led to Fontana's success in erect-
ing the Egyptian obelisks in Rome.
Many military machines were studied
and perfected and also the screw of
Archimedes, which to this day plays
an important role in the agricultureof Egypt. Ballistics commanded
great attention, and also closely
studied before Galileo were falling
bodies and projectile motion, includ-
ing air resistance. During that time
scientists investigated many simple
and complex machines, both for their
inherent scientific interest and for
their great practicality in architecture,
especially as applied by Alberti, and
in shipbuilding and maritime equip-
ment, as shown above all in the great
arsenal at Venice.
This is a fascinating book that
clarifies the earlier scientific develop-
ments that made Galileo's great ad-
vances possible: It is also history in
TRAJECTORY DIAGRAM by Niccolo
Tartaglia, superimposed on a landscape,
as shown by Walther Ryff in Der geomet-
rischen Biixenmeisterey, in Der Archi-
tectiir . . . (Niirnburg, 1558). Photo
courtesy of Burndy Library.
a broader and deeper sense than sim-
ply a record of scientific progress.
There is a fine presentation through-
out of the groping and progress
needed to develop the scientific con-
cepts so essential to Galileo's syn-
thesis of mechanics. The literary
style is excellent, and the scholarship
is detailed and authoritative. The
book is certainly a contribution to our
PHYSICS TODAY . DECEMBER 1969 • 71
When you develop an instrument
capable of near-perfect voltage
measurements plus the capability to
measure charge, current, and
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You tell it like it is.
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The unit features a completely new
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- even at 100 volts. Stability? For an
electrometer — it's positively unreal.
OURSERange and accuracy:
Volts - 30 mV f .s. (lOfM V
resolution) to 100 V f.s., ± .3% to
± .1%, with > 1016 ft input Z.
Coulombs - 10"11 Coul. f.s. to
10-G Coul. f.s., ± .2% to ± .5%.
Amperes - 10~2 amps f.s. to 10-1-
amps f.s., ± .1% to ± .7%; Offset
< 5 x 10-15 amps; Input drop
100^ V, typical.
Ohms - 10n f.s. to 1014ft f.s.,
± .1% to ± .75%.
And that's like it is!
Or, for direct current measurement,
there's our new 706A Precision
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Output multiplier (xlO, x3, xl) allows
precise scaling of output current
from 10~6 to lO-is amps ( ± .1%
f.s. to ± .5% f.s.) Panel graphics
designed for error-free operation.
Also available is the 726A Pico-
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706A, featuring: the same accuracy,
built-in current suppression, internal/
external calibration capabilities,
automatic polarity display, and 200%
overrange for digital displays up to
2.999. Optional digital output also
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For complete specs., write or call
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Box 755, Goleta, California, 93017
(805) 967-0456.
^MjJ^r
LABORATORY PRODUCTS DIVISION
understanding the development ofMT WILSON AND PALOMAR OBSERVATORIES
science.
The reviewer is with Case Western Re-
serve University, where he is Ambrose
Swasey Professor of Physics.
Emphasis on hard facts
PHYSICS OF PLANETS. (NASA-
TT-F-515). By V. I. Moroz. 412
pp. NASA, Washington, DC, 1968.
$3.00
by ROMAN SMOLUCHOWSKI
There are few, if any, sciences that
stir the imagination more than astro-
physics. Even the length of articles
in the New York Times, which actually
is acquiring an enviable reputation
as a "science journal," shows that the
only peers of astrophysics in this re-
spect are genetics and other biosci-
ences. For the last ten years or so we
have been bombarded with spectacu-
lar discoveries concerning either re-
mote parts of the universe, which are
populated by such mysterious objects
as quasars, pulsars and John Wheeler's
"black holes," or concerning our own
familiar and much more easily identi-
fiable solar system and its planets.
Unfortunately there are no recent
books in English written on a reason-
ably advanced level dealing with
physics of all planets. Some do exist
on the popular side, such as the other-
wise excellent series published by the
National Aeronautics and Space Ad-
ministration and edited by C. M.
Michaux. Others encompass several
volumes each written by many au-
thors, which precludes continuity and
uniformity of level, and there are
also books that deal only with a few
planets, like the recent (1968) and
very good Introduction to Planetary
Physics by W. M. Kaula.
The author of Physics of Planets,
V. I. Moroz from the P. K. Sternberg
Astronomical Institute in Moscow, has
contributed widely to spectroscopic
observations of nearly all planets. His
present book is an excellent and com-
pact introduction to the whole field
of planetary physics. It starts with
a good summary of basic concepts,
tools and pertinent measurements, fol-
lowed by chapters dealing with
Mars, Venus, Mercury and the giant
planets. There are a large number
of illustrations, diagrams and over 600
references.JUPITER with red spot and shadow
of the satellite Ganymede above.
The tone of the book would appeal
to a skeptical observer; that is, the
primary effort is placed on facts and
on their evaluation, and only the most
acceptable theories are expounded in
some detail. This is a very welcome
feature in a field where the ratio of
hard facts to theories and hypotheses
is probably even lower than in bio-
sciences.
The main drawback is that the ref-
erences do not go beyond 1965, and
thus the book does not cover such ex-
citing observations as F. J. Low's mea-
surements of the thermal emission of
Jupiter, newer data on the nature of
the polar caps of Mars and of its sur-
face composition and the recent con-
troversy concerning the surface tern-
A partisan view
QUANTUM THEORY OF MATTER.
(2nd edition) By John C. Slater.
763 pp. McGraw-Hill, New York,
1968. $15.00
by PHILIP L. TAYLOR
It is probably true to say that the quan-
tum theory of matter is a subject that
has broadened rather than deepened in
the 18 years since the first edition of
this text was published. Our current
view of a crystal as a bestiary of ele-
mentary excitations has led to an un-
derstanding of many previously puz-
zling phenomena. On the other hand,
our present knowledge of atoms and
molecules, as well as of energy bands
in solids, owes more to large digital
computers that helped us develop con-
cepts formulated in the early days of
quantum mechanics.
In this new edition of his book, Johnperature of Venus. On the other
hand, the results obtained by Mariner
4 and the complex decametric- and
decimetric-radiation patterns of Jupi-
ter are discussed in considerable de-
tail. I was particularly impressed by
the space devoted to Jupiter's red
spot, to the famous "south tropical
disturbance" and to the atmospheres
of Jupiter and Mars. Many numeri-
cal data in the book are more up to
date than those in C. W. Allen's As-
trophysical Quantities, which was last
revised in 1962.
On the negative side, one has to
mention first the poor translation and
careless proofreading. For instance
"oblateness" is translated as "com-
pression," and a column in table 97
is titled "Ratio of Planet Mass to
Satellite Mass*' when it should be
"Ratio of the Mean Radius of the
Satellite Orbit to Planet Radius." As
a result the reader is told that Jupiter
is 2.5 times as heavy as its famous
fifth satellite. But a very valuable
feature of the book is that besides
references to Western literature there
are numerous references to Soviet
literature, which is so often unknown
to us. Altogether the book is useful
and should find a wide audience.
R. Smoluchowski is professor of solid state
sciences at Princeton University and has
been active in the part of astrophysics tlxat
deals with properties of condensed matter,
especially the surfaces and the interior of
the moon, Mars and Jupiter.
Slater has chosen not to follow the
path of diversification, but has instead
concentrated on enlarging his treat-
ment of the topics covered in the first
edition. Thus the first half of the
book represents an introduction to
quantum mechanics in the wave-me-
chanical-cum-historical tradition, and
the second half discusses the applica-
tion of the one- and two-electron
Schrodinger equation to a large variety
of molecules and solids. The discus-
sion of molecular orbitals is particularly
clear and extensive and includes de-
scriptions of the ammonia, ethylene
and benzene molecules. There are
ample instructive problems at the end
of each chapter.
Some readers may fault this text for
its failure to mention any aspect of
collective behavior or of those most ex-
PHYSICS TODAY . DECEMBER 1969 73
PSNS: The ideais involvement!Many students feel that sci-
ence has to be complex, unin-
telligible and uninteresting.
PSNS-a course designed
especially for nonscience
high school seniors and col-
lege freshmen, does away
with that idea.
This new program leads the
student to an understanding
of the nature of solid matter
through the close integration
of textbook and simple labo-
ratory experiments. It is builtaround the idea of active
involvement-showing stu-
dents, with everyday tools,
the basic concepts of
physical science.
Laboratory equipment, sup-
plied by Damon, is simple and
inexpensive. The text, An
Approach to Physical Sci-
ence, is developed around
experimentation and encour-
ages speculation, rather than
passive memorization.
Throughout, the programallows the student to be at
ease with science and to see
scientific concepts as prod-
ucts of human observation.
For more information write:
John Wiley & Sons, Inc.
605 Third Avenue, New York,
N.Y. 10016.
WILEY
(g) DAMON
ouch to I'htjsicul Science
74 • DECEMBER 1969 • PHYSICS TODAY
rating states of matter, the superfluid
land the superconductor. I would be
Fmore inclined to accept this work for
| what it is-a partisan view of the theory
' of matter-and forgive its author for
Lretaining the book's overly ambitious
•title. This new edition will be warmly
[welcomed by anyone who has enjoyed
f the earlier version, and will bear fur-
ther witness to Slater's qualities as one
of our most notable teachers.does have some attractive features that
would make it worthwhile for refer-
ence purposes, including a commenda-
ble neatness of organization, a clarity
of exposition that takes nothing for
granted and many meticulously drawn
diagrams. To the teacher of elemen-
tary physics it would provide a good
source of supplementary material, but
for the practically minded engineerand technician, who encounters prob-
lems involving light and sound, it has
enough useful information that can be
obtained quickly to make it a good
place to look first.
Robert Lindsay is a professor of physics
at Trinity College and has been teaching
physics to science and engineering majors
for 15 years.
Thilip Taylor is associate professor of
physics at Case Western Reserve Univer-
sity, and is the author of a forthcoming
text on the quantum theory of solids.
Waves and lines
LIGHT AND SOUND FOR ENGI-
NEERS. By R. C. Stanley. 344
pp. Hart Publishing Co., 1968.
$12.00
by ROBERT LINDSAY
This book, by a British author who is
lecturer in applied physics at Brighton
College of Technology, is an effort to
provide a broader and deeper exposi-
tion of sound and optics than the typi-
cal British engineering student might
be expected to obtain from his elemen-
tary physics course. The chapters de-
voted to geometrical optics give con-
siderable attention to such often by-
passed topics as thick lenses, aberra-
tions and photometry as well as ana-
lyzing in more than usual detail some
of the commonly encountered optical
instruments.
The chapters on physical optics em-
ploy standard approaches to interfer-
ence, diffraction and resolving power
with a theoretical development based
almost completely on the principles of
superposition and the Huygens tradi-
tion. No mention is made of recent
work in lasers and holography. The
chapters on sound include the de-
scription of techniques for measuring
the velocity of sound in solids, liquids
and gases, a thorough but elementary
treatment of the vibrating string and
several resonance situations and a sur-
vey of architectural acoustics and ul-
trasonics.
Most US engineering curricula re-
quire three or four semesters of ele-
mentary physics. Existing texts al-
ready treat these subjects at a reason-
able depth and it appears unlikely that
a book at this relatively low level
would be suitable as a regular text. ItTwo aids for galactic research
STELLAR KINEMATICS. By W. M.
Smart. 320 pp. Wiley, New York,
1968. $12.50
GALACTIC ASTRONOMY. By Dimi-
tri Mihalas, with collaboration of
Paul McRae Routly. 257 pp. W. H.
Freeman, San Francisco, Calif.,
1968. $10.00
by KENNETH YOSS
These two books are useful additions
to the sparse list in galactic research,
which is receiving more attention with
the recent availability of a new gen-
eration of modern observational
equipment. Proper appreciation and
analysis of the resulting data is es-
sential, and these two books should
aid in this increased activity.
Despite the first-glance similarity
(six of eight chapters in one are onthe same topics as six of 14 in the
other), the purposes are totally dif-
ferent, as are the levels of usefulness.
One is a textbook for a first course in
galactic structure, the other a detailed
mathematical explanation of well
known classical problems in stellar
kinematics.
W. M. Smart is well known for his
precise mathematical developments
concerning problems in galactic struc-
ture. His Spherical Astronomy and
Stellar Dynamics are classics, familiar
to and often used by researchers in
galactic structure. Stellar Kinematics
is limited to basic problems concern-
ing stellar motions, and many sections
are modifications from Stellar Dynam-
ics, which does not lessen its useful-
ness. Smart's attention to detail is
MEDIEVAL COSMOLOGY. Woodcut depicts traveler putting his head through the
vault of the sky to discover the complexities that move the stars. (Photo taken from
Knowledge and Wonder by Victor F. Weisskopf, Doubleday, 1966.)
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THE MANY-BODY PROBLEMMALLORCA INTERNATIONAL SCHOOL OF PHYSICS, AUGUST 1969
Director, L. M. Garrido, Professor of Theoretical Physics, University of Barcelona
Edited by A. Cruz, University of Zaragoza
and T. W. Preist, University of Exeter
In an attempt to encourage new research and to consolidate prog-
ress made, eminent physicists discuss numerous aspects of the
many-body problem. Invaluable as a state-of-the-art report on
this vital topic, the book features papers by L. J. Boya, E. R. C.
Caianiello, C. B. Dover, C. P. Enz, I. Fujiwara, L. van Hove, N. J.
Horing, P. C. Martin, W. Thirring, and E. J. Verboven.
333 PAGES NOVEMBER 1969 $15.00
ELEMENTARY EXCITATIONS IN SOLIDSPROCEEDINGS OF THE CORTINA LECTURES AND 4 LECTURES FROM
THE CONFERENCES ON LOCALIZED EXCITATION, BOTH HELD IN
MILAN
Edited by A. A. Maradlldin, Department of Physics, University of California at
Irvine
and G. F. Nardelli, Groppo Nazionale Struttura della Materia, C.N.R. and Physics
Institute, University of Milan, Italy
Reporting the latest advances in the field, this volume will be of
great value to solid state physicists and crystallographers.
CONTRIBUTORS: A. A. Maradudin, G. F. Nardelli, W. Ludwig, M. Bal-
kanski, M. F. Collins, A. J. Sievers, R. O. Pohl, R. J. Elliott, J. Callaway,
P. Resibois, E. Burstein, J. J. Hopfield, G. Baldini, I. P. Ipatova, A. A.
Klochikhin, R. F. Wallis, G. Chiarotti.
536 PAGES NOVEMBER 1969 $35.00
MOSSBAUER EFFECT METHODOLOGYSeries edited by Irwin J. Gruverman, Head, Special Sources Department,
New England Nuclear Corporation, Boston, Massachusetts
Volume 5
PROCEEDINGS OF THE FIFTH SYMPOSIUM ON MOSSBAUER EFFECT
METHODOLOGY, HELD IN NEW YORK, FEBRUARY, 1969
Presenting outstanding contributions on current developments,
the latest volume in this invaluable series includes reports on en-
vironmental control, new applications and methodology, and tech-
niques for measurements in radioactive materials. Interdis-
ciplinary in approach, this book discusses Mossbauer applica-
tions in such fields as metallurgy, mineralogy, and biology.
CONTENTS: Spectroscopy: Mossbauer effect studies of lattice dyna-
mic anisotropy and line asymmetry in semiconductor and organometallic
tin compounds, H. A. Stockier and H. Sano • Mossbauer spectroscopy
of inorganic antimony compounds, J. G. Stevens and L. H. Bowen •
Mossbauer spectroscorpy of organometallic compounds in noncrystalline
matrices, S. Chandra and R. H. Herber • Mossbauer effect studies on
Eu1*1 in mixed oxide structures, G. W. Dulaney and A. F. Clifford • Sys-
tematic interpretation of the isomer shifts in tin, antimony, tellurium,
iodine, and xenon, G. K. Sheney and S. L. Ruby • Mossbauer studies of
vitamin Bn and some related cobalamins, R. T. Mullen • Applications:
Polarization effects in Mossbauer absorption by single crystals, R. M.
Housley • Determination of zero point phonon parameters: Calibra-
tion of the second order Doppler shift, T. A. Kitchens, P. P. Craig, and
R. D. Taylor • The Mossbauer effect in microcrystals, D. Schroeder •
After-effects of Auger ionization following electron capture in cobalt com-
plexes, Amar Nath, M. E. Vin, P. Klein, W. Kundig, and D. Lichtenstein •
Methodology: Mossbauer effect in radioactive materials, A. J. F. Boyle
and G. J. Perlow • The Mossbauer effect: A new method for measuring
diffusion, J. G. Mullen and R. C. Knauer • Mossbauer spectrometry as
an instrumental technique for determinative minerajogy, C. L. Herzen-
berg • Mossbauer experiments with a He3/He4 dilution refrigerator, G.
M. Kalvius, T. E. Katila, and O. V. Lounasmaa.
APPROX. 267 PAGES JANUARY 1970 $19.50
RADAR CROSS SECTION HANDBOOKBy George T. Ruck, Sen/or Research Scientist
Donald E. Barrick, William D. Stuart, and Clarence K. Krichbaum,
Battelle Memorial Institute, Columbus, Ohio
In two volumes this extensive work is the first which attempts to
give radar cross section data and analytical techniques for all
radar targets for which information is available. With results
presented through curves, tables, and engineering equations,
the Handbook features a special chapter devoted to detailed de-
scription of theoretical techniques, and is invaluable as a ref-
erence for both scientists and students.
APPROX. 935 PAGES JANUARY 1970 2 VOLUMES, $75.00
consultants bureau/plenum press
Divisions of Plenum Publishing Corporation
114 FIFTH AVE., NEW YORK, NEW YORK 10011
76 . DECEMBER 1969 • PHYSICS TODAY
again evident, and this book should
prove invaluable to a worker coi>
cerned with proper procedure, the ef-
fect of observational errors and in-
complete sampling on the results.
This concern for detail is in vivid
contrast to his virtual omission of
modern interpretation of the observa-
tions. For example, solar motion is
treated in a classical manner; the
cause of variation in solar apex for
different stellar groups and the dis-
tinction between standard and basic
solar motion are not mentioned.
Chapters 3 and 4 are concerned with
star streaming, which has historical
and mathematical interest but little
immediate practical use. The next
chapter introduces the concept of el-
lipsoidal distribution of stellar-velocity
vectors, but his discussion of its fun-
damental cause, found in the final
chapter on galactic rotation, is very
brief. He avoids such interwoven
topics as stellar-density distribution
and galactic dynamics so limiting the
book to kinematic problems, as the
title indicates.
Dimitri Mihalas is known for his
fine work in stellar atmospheres, and
he is to be admired for his motivation
in writing Galactic Astronomy. A
first text in galactic structure is
needed, and this book goes far in fill-
ing the vacancy. It is regrettable that
the first modern text has not been
written by an experienced researcher
in the field, however, because so much
of its value depends on the proper
evaluation of available observational
data. Unlike Stellar Kinematics, dis-
cussion and interpretation of observa-
tions are included but at times should
be more extensive.
The first three chapters in Mihalas's
book are devoted to brief descriptive
topics found in elementary texts, but
these 45 pages should either have
been omitted entirely or significantly
expanded. The weakness of this sec-
tion is exemplified in the discussion
of eiTors in trigonometric parallaxes.
His explanation for negative parallaxes
is actually incorrect. It is regrettable
that he did not reference more au-
thoritative sources, such as Peter van
de Kamp's Principles of Astrometry
(W. H. Freeman), which deals in
detail with problems of this type.
From chapter 4 on, where he is con-
cerned with specific details of stellar
motions, galactic rotation and galactic
dynamics, there is little to criticize.
Mihalas writes well, and the book con-
tains sufficient detail to introducethe student to the concepts. Unlike
Smart, he effectively discusses the cur-
rently important problems of galactic
structure, such as interstellar absorp-
tion and the relation between stellar
populations (ages) and motions (ve-
locity ellipsoids). In many cases dia-
grams would have better conveyed
the concepts than the extensive tables,
most of which are unnecessary in a
book of this type.
The difference in detail in the two
books, which illustrates the basic dif-
ference in their purposes, is vividly
depicted in the respective chapters on
statistical parallax; Mihalas devotes
six pages to it (a good length for a
textbook), and Smart takes 36 pages.
Both books will remain useful for
some time: Stellar Kinematics be-
cause it represents a rigorous mathe-
matical approach to standard prob-
lems in the subject, quite independent
of the constant, but slow, improve-
ment and increase in observational
data; Galactic Astronomy because
it is presented in a readable form and
includes most major topics of interest
in the subject at a useful level for a
first text in the field.
Kenneth Yoss is an astronomy professor
at the University of Illinois Observatory,
Urbana, III
Beauty in the eye of the beholder
LAGRANGIAN DYNAMICS: AN IN-
TRODUCTION FOR STUDENTS. By
C. W. Kilmister. 136 pp. Plenum,
New York, 1968. $7.50
by GARRISON SPOSITO
In 1834, while in the process of de-
livering his own name onto the list of
the immortals in physics, Sir William
Rowan Hamilton wrote in celebration
of the men who had created analytical
mechanics. He singled out with obvi-
ous gratitude Comte Joseph Louis
Lagrange as one who had "perhaps
done more than any other analyst to
give extent and harmony to such de-
ductive researches, by showing that
the most varied consequences respect-
ing the motions of systems of bodies
may be derived from one radical for-
mula; the beauty of the method so
suiting the dignity of the results, as to
make his great work a kind of scientific
poem."
Hamilton's elegant praise has in no
sense become hyperbole with the pas-B
Technological
Injury:THE EFFECT OF
TECHNOLOGICAL ADVANCES
ON ENVIRONMENT, LIFE
AND SOCIETY
Edited by J. Rose
Technological advances in this cen-
tury have been of immense benefit
to mankind: they have also re-
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the very fabric of life and society.
Thus, the higher standard of living
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and an economy based on waste.
But man has a choice of keeping this
planet healthy or of dying with it.
This book is a collection of 15 chap-
ters contributed by experts in vari-
ous fields relating to the effect of
technology on environment, life and
society. The aim of this work is to
present to an intelligent public a
sober and fair account of the poten-
tial and actual dangers of techno-
logical advances. Technological In-
jury points out these dangers, im-
partially discusses their implica-
tions, and shows what steps should
be taken to counteract the existing
and potential effects. The contents
of this book are divided into 2 sec-
tions: POLLUTION OF THE EN-
VIRONMENT and EFFECTS ON SO-
CIETY AND LIFE. All who care
about the world they live in will
welcome this book.
-ORDER FORM-
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Texts from HVHey
THE ELEMENTS AND STRUCTURE OF THE
PHYSICAL SCIENCES
Second Edition
By J. A. RIPLEY, JR., Stanford University; and R. C. WHITTEN,
National Aeronautics and Space Administration.
Discusses the development of the underlying principles of the
physical sciences. 1969 Approx. 704 pages $11.50
QUANTUM MECHANICS
Second Edition
By EUGEN MERZBACHER, University of North Carolina,
Chapel Hill.
Revised and expanded, this new edition includes a thorough
treatment of second quantization and an introduction to the
quantum field theory of photons and electrons.
1969 Approx. 608 pages In press
THERMAL PHYSICS
By CHARLES KITTEL, University of California, Berkeley.
A new, modern, elementary approach to thermal physics based
on the methods of Gibbs. 1969 Approx. 448 pages $10.95OPTICS
By MILES V. KLEIN, University of Illinois.
An intermediate level text on classical geometrical and physi-
cal optics. 1969 In press
NUMBERS AND UNITS FOR PHYSICS
A Program for Self-Instruction
By ROBERT A. CARMAN, San Bernardino Valley College.
A programmed introduction to the quantitative language of
physical science; designed as a self-study supplement to
beginning courses. 1969 In press
ELEMENTARY RADIATION PHYSICS
By G. S. HURST, University of Kentucky; and J. E. TURNER,
Oak Ridge National Laboratory.
Explains basic atomic and nuclear physics, emphasizing as-
pects of importance in medicine and nuclear engineering.
1969 Approx. 326 pages In press
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78 • DECEMBER 1969 • PHYSICS TODAY
sage of time. In remarkable analogy
with its creator's undiminishing pres-
tige as succeeding revolutions racked
his adopted country, the Lagrangian
method has stood impervious to the
two great revolutions that have trans-
formed dynamics in this century. The
words written by Hamilton could in all
respects have been written as well by
Richard Feynman or Julian Schwinger.
It is no wonder then that one might
wish to include at least a peek at
Lagrangian dynamics in an advanced
undergraduate course on classical me-
chanics. The problem is that such a
peek has to be elementary but not
superficial, and that this condition is
difficult to meet in most textbooks
without their becoming impossibly
bulky. The solution to the dilemma,
according to C. W. Kilmister, mathe-
matics professor at King's College in
London, is to add to the reading list a
little volume such as his Lagrangian
Dynamics: An Introduction for Stu-
dents.
Kilmister's book contains six chap-
ters, of which the third through fifth
are involved directly with illustrations
of the Lagrangian method. To be
honest, one must say that these chap-
ters will be largely incomprehensible
to the reader who does not know fairly
well the calculus of variations and vec-
tor analysis. Moreover the reader
must have a feeling for, or at least a
great tolerance of, the dynamics of
rigid bodies, because the discussions
deal solely with macroscopic systems
subject to constraints.
In chapter 3, for example, we meet
the symmetric top, a hoop (inside of
which dangles a simple pendulum) a
bell and clapper slightly idealized and
a centrifugal governor. In the fourth
chapter, on small vibrations, we face
the double pendulum; in the fifth, on
impulsive forces, we observe a rhom-
bus of uniform rods collide with a wall.
The character of these applications will
likely preclude the use of the book by
anyone who believes heartily that the
notion of constraint is artificial in the
present milieu of dynamics.
It is probably not without some
value to remark that this book might
have a special appeal to professors or
students who prefer to see classical
mechanics as applied mathematics
rather than theoretical physics. The
tone of the book is decidedly mathe-
matical, and it achieves its finest form
with the statement, in chapter 2, that
"the reason why the anholonomic case
can arise is now simply that not allvector fields are families of normals to
hypersurfaces." In the same sense one
might add, with a twinkle in one's eye,
that the reason why aperiodic oscilla-
tions in three-space can arise is that
not all numbers are rational. Evi-
dently beauty is indeed in the eye of
the beholder.
An associate professor at Sonoma State
College, California, the receiver has
taught courses on analytical dynamics for
the past few years.
Highly coherent
FUNDAMENTALS OF QUANTUM OP-
TICS. By John R. Klauder and E.
C. G. Sudarshan. 279 pp. W. A.
Benjamin, New York, 1968. $13.50
by MARVIN M. MILLER
Since the publication in 1963 of a se-
ries of papers by R. J. Glauber, the
quantum theory of optical coherence
has become an active area of research.
However, with the notable exception of
Glauber's 1964 Les Houches lecture
notes, an authoritative account of the
many interesting developments in this
field has not been available in book
form. The appearance of a mono-
graph by two of the leading contribu-
tors in the field, J. R. Klauder and E.
C. G. Sudarshan, is especially timely
because of the importance of this re-
search, and the fruitful application of
the notion of coherent states to the
study of problems outside the domain
of quantum optics.
The first three chapters are devoted
to a concise review of selected topics
in classical-coherence theory and semi-
classical-counting statistics. Chapter 4
considers the physical origin and treat-
ment of coupled, nonlinear, partial dif-
ferential equations with stochastic-
driving terms, or stochastic-initial con-
ditions or both. Although such equa-
tions arise in many physical contexts,
this discussion has particular relevance
in quantum optics, in view of the suc-
cess of model-laser theories that de-
scribe the dynamics of the nonlinear
interaction between the laser systems
and reservoirs by means of fluctuation
equations with Markoffian noise-source
excitation.
Chapters 5 and 6 provide a lucid
exposition of some basic concepts of
abstract quantum mechanics and a
nonrelativistic analysis of the operator
equations of motion for the electro-Thinking about...
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PHYSICS TODAY • DECEMBER 1969 • 79
1970 GROUP FLIGHTS TO EUROPE
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80 • DECEMBER 1969 • PHYSICS TODAY
magnetic field. Chapter 7 is a detailed
account of the properties of the co-
herent states. Among the topics dis-
cussed are the completeness (appar-
ently first noted by John von Neu-
mann) and over completeness of these
states, their relationship to Segal-Barg-
mann Hilbert spaces of entire functions
and the differential-operator represen-
tation of the creation and annihilation
operators.
Especially noteworthy is the dis-
cussion in chapter 8 of a particular co-
herent-state representation of the den-
sity operator that specifies the statis-
tical state of the radiation field and is
known in the literature as the diagonal
or F representation. As Glauber has
remarked, the question of the gener-
ality of this representation "lies in
mathematically rather deep waters,"
and has provoked a fair amount of con-
troversy and confusion in the past.
The authors' rigorous formulation of
the optical-equivalence theorem, and
their discussion, particularly on page
192, of its physical implications should
A review of reviews
ANNUAL REVIEW OF PHYSICAL
CHEMISTRY, VOL. 19, 1968. H.
Eyring, C. J. Christensen, H. S.
Johnston, eds. 645 pp. Annual
Reviews, Palo Alto, California, 1968.
$6.50
by E. E. MUSCHLITZ, JR
Volume 19 of Annual Review of Phys-
ical Chemistry is the 1968 edition of
a long and successful series. The
current volume contains 20 articles
and presents the reviewer with a dif-
ficult task in doing justice to the ef-
forts of all the authors involved.
Physical chemistry includes a wide
variety of topics, and the breadth of
the subject is amply demonstrated by
individual review titles in the volume.
The reviews are well written and doc-
umented, most having 100 or more
references and many having over 200.
Periodic short reviews of progress in
active areas of research are of value
not only to the experienced investiga-
tor but also to the graduate student.
Especially for the latter, a good re-
view article should be instructive as
well as informative. Most of the re-
views in this volume have achieved
this objective.
A. N. Frumkin and N. M. Emanuel
of the USSR Academy of Sciences are
the authors of an interesting survey,prove valuable in illuminating the re-
lationship between the quantum and
semiclassical theories of optical coher-
ence.
The last two chapters deal with spe-
cial states of the radiation field and in-
tensity interferometry in quantum op-
tics. There is a discussion of various
laser models and J. P. Gordon's inter-
esting approach to the model devel-
oped by M. Lax is considered in some
detail.
The level is suitable for advanced
students and research workers in quan-
tum optics. It is written in a clear
style with a careful attention to math-
ematical and physical subtleties not
often considered in the literature, and
it is highly recommended to those who
wish an authoritative account of re-
cent work in this area.
The reviewer is assistant professor of elec-
trical engineering at Purdue University
specializing in quantum optics and elec-
tronics.
"Fifty Years of Soviet Physical Chem-
istry," which heads the list of articles.
This is followed by reviews on "Elec-
tric Paramagnetic Resonance" by Alan
Carrington and Geoffry Luckhurst;
"Fused Salts" by S. J. Yosin and H.
Reiss; "Electrochemistry" by Fred An-
son (perhaps too broad a field for a
short review article) and "Experi-
mental Inorganic Thermochemistry"
by W. N. Hubbard, P. A. G. O'Hare
and H. M. Feder. Recent develop-
ments, particularly new experimental
techniques, in studies of "Fast Reac-
tions in Solution" are described by
Edward Eyring and Bruce Bennion.
R. Henry and Michael Kasha have
written a penetrating review on "Ra-
diationless Molecular Electronic Tran-
sitions" in which they give a critical
historical summary of the theory of
these processes and develop the sta-
tionary-state approach to excited-state
interactions of Rhodes, Henry and
Kasha and of Jortner that eliminates
radiationless transitions. "Ligand
Substitution Dynamics" by Cooper
Langford and Thomas Stengle is the
next review, and it is followed by a
thorough analysis of recent theory and
experiment on "Vibrational and Rota-
tional Relaxation" by Roy Gordon,
William Klemperer and Jeffrey Stein-PHOTON
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82 . DECEMBER 1969 • PHYSICS TODAY
feld. These authors have included
several valuable tables collating the
various systems that have been stud-
ied with the experimental methods.
Only in recent years with the ad-
vent of ultrahigh-vacuum techniques
has it been possible to carry out sur-
face studies on clean single-crystal
surfaces. Gabor Somorjai reviews the
experiments on surface structure, sur-
face dynamics and gas-surface inter-
actions in his article on "Surface
Chemistry." Lewis Friedman's review
on "Ion-Molecule Reactions" empha-
sizes the new experimental techniques
with tandem mass spectrometers
(measurements of the product-ion en-
ergies and angular distributions have
recently been made for several reac-
tions), ion cyclotron resonance for
studies of reaction mechanisms and
photoionization as a means of produc-
ing reactant ions in known internal-
energy states.
The review "Mass Spectrometry" by
Kenneth Rinehart Jr and Thomas
Kinstle attempts to cover too large a
topic for a short review article. The
emphasis is on high-resolution mass
spectrometry and structure of organic
positive ions. Alan Haught writes a
very instructive review on "Lasers and
their Applications to Physical Chem-
Machine calculations
COMPUTING METHODS FOR SCIEN-
TISTS AND ENGINEERS. By L. Fox
and D. F. Mayers. 255 pp. Oxford
Univ. Press, New York, 1968. $6.25
by NORMAN A. BAILY
The authors, members of the Oxford
University Computing Laboratory,
state that the primary purpose of this
book is to enable its users to improve
their use of the computer and to ob-
tain more accurate and meaningful so-
lutions. If one restricts its application
to that of a handbook, it should have
no difficulty in achieving the authors'
aims. However, the mathematics are
complex enough that even physical sci-
entists who are primarily experimen-
talists would have to spend consider-
able time studying the suggested meth-
ods to determine the proper one for a
particular problem.
The field of automatic computation
is of prime importance in all branches
of science, and the book emphasizes
the proper selection of methods for
the numerical solution of many differ-
ent mathematical forms. The book,istry." This is followed by reviews on
"Gas Reactions Yielding Electronically
Excited Species" by B. A. Thrush,
"Statistical Mechanics—A Review of
Selected Rigorous Results" by Joel
Lebowits, "Vibrational Spectroscopy"
by Herbert Strauss and "Nuclear
Magnetic Resonance" by J. Jonas and
H. S. Gutowski.
D. W. Urry's review "Optical Ro-
tation" is centered on applications to
peptides and polypeptides. This is
followed by an article on "Quantita-
tive Conformational Analysis; Calcula-
tion Methods" by James Williams, Pe-
ter Stang and Paul Schleyer and one
on "He3—He4 Solutions" by Norman
Phillips.
The editors are to be congratulated
on their selection of authors for this
volume, for each is an acknowledged
expert in his field. These authors have
treated their subjects not only in a
comprehensive but also a critical fash-
ion. In the current era of a burgeon-
ing literature, good reviews such as
these are filling a role that is becom-
ing more and more essential.
E. E. Muschlitz Jr is a chemistry pro-
fessor and head of physical chemistry at
the University of Florida.
therefore, makes a very valuable con-
tribution because a vast majority of
computer users have not ordinarily
delved deeply into the problems dis-
cussed. Sections of the book are quite
sophisticated and possibly would be
difficult for the occasional machine
user to apply properly. It is specifi-
cally designed for persons thoroughly
familiar with computing but who per-
haps do not have either the training
or experience to obtain the best results.
In general, the book is an excellent re-
view of the methods for handling com-
mon difficulties.
Some of the more important topics
covered are: error analysis, floating-
point arithmetic, recurrence relations,
finite differences and the usual com-
mon operations such as polynomials,
matrices and numerical integration.
The authors have stressed the impor-
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induced stability and have treated fun-
damental theory where they felt that
it was not well known by most com-
puter users. The book does not em-CAMBRIDGE
UNIVERSITY PRESS
Elements of Advanced
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J. M. ZIMAN
In this newly published work, Pro-
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He gives a connected mathemat-
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unnecessary rigor. He explains in
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bols and concepts which frequently
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in solid state, nuclear, and high-
energy physics, and in theoretical
chemistry: field operators, propaga-
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ors, the S-matrix, irreducible repre-
sentations, continuous groups, etc.
$9.50
The Physics of Metals
Part 1: Electrons
Edited by J. M. ZIMAN
Part 1 of this two-volume work treats
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field (A. B. Pippard), surface and size
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Ziman), liquid metals (T. E. Faber),
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and special properties of transition
metals (J. Friedel).
Part 2, in preparation, is subtitled
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Parti: Electrons $14.50
CAMBRIDGE UNIVERSITY
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PHYSICS TODAY • DECEMBER 1969 83
mThe Lincoln Laboratory of the Massachusetts Institute of Technology
conducts research in selected areas of advanced electronics with
emphasis on applications to national defense and space exploration.
Radio Physics is a field of major interest. The program includes
radio propagation studies leading to systems for satellite and deep-
space communications, as well as investigations of the sun and
the planets, utilizing new techniques of radar astronomy. All
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phasize derivations but rather provides
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ror that might be caused by rounding
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Even though its treatment is limited,
the book provides a much needed
' compendium of computational meth-
ods applicable for the solution of many
common problems.
The reviewer is with the University of
California, San Diego, and is a machine
user both for the numerical evaluation of
theoretical expressions and for the practi-
| cal applications of radiation dosimetry.
NEW BOOKS
ELEMENTARY PARTICLES
Phenomenological Theories of High En-
ergy Scattering: An Experimental Eval-
uation. By Vernon D. Barger and David
B. Cline. 201 pp. W. A. Benjamin, New
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Theory and Phenomenology in Particle
I Physics, Part A and B. A. Zichichi, ed.
315 pp. Academic, New York, 1969.
$14.00
Springer Tracts in Modern Physics, Vol
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Solving SU(3), Charge-Current Algebra.
G. Hohler, ed. 146 pp. Springer-Verlag,
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NUCLEI
Springer Tracts in Modern Physics, Vol.
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G. Hohler, ed. 146 pp. Springer-Ver-
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Introduction to Nuclear Physics and
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Advances in Nuclear Physics, Vol 3.
Michel Baranger and Erich Vogt, eds.
480 pp. Plenum, New York, 1969.
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I> Induced Radioactivity. By Marcel Bar-
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ELECTRICITY AND MAGNETISM
Francis Bitter, Selected Papers and Com-
mentaries. T. Erber and C. M. Fowler,
eds. 551 pp. MIT Press, Cambridge,
* Mass., 1969. $20.00
' Fundamentals of Electrodynamics. By
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pp. Marcel Dekker, New York, 1969.
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FLUIDS, PLASMAS
Magnetodynamique des Fluides. (2ndedition). By Henri Cabannes. 289 pp.
Centre de Documentation Universitaire,
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Nonlinear Plasma Theory. By R. Z. Sag-
deev and A. A. Galeev. 122 pp. W. A.
Benjamin, New York, 1969. Cloth
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Flow Equations for Composite Gases.
J. M. Burgers ed. 332 pp. Academic
Press, New York, 1969. $18.50
SOLIDS
Semiconducting II-VI, IV-VI, and V-VI
Compounds. By N. Kh. Abrikosov, V. F.
Bankina, L. V. Poretskaya, L. E. Sheli-
mova, and E. V. Skudnova. (Trans, from
Russian) 252 pp. Plenum Press, New
York, 1969. $19.50
Tunneling In Solids: Solid State Physics
Supplement 10. C. B. Duke, ed. 353
pp. Academic Press, New York, 1969.
$16.00
Applied Solid State Science, Vol. 1: Ad-
vances In Applied Solid State Physics.
Raymond Wolfe, ed. 404 pp. Academic
Press, New York, 1969. $15.00
ASTRONOMY, SPACE, GEOPHYSICS
Geophysics and Space Data Bulletin, Vol.
6. Anne L. Carrigan, ed. 359 pp. US
Air Force, L. G. Hanscom Field, Mass.
Annual Review of Astronomy and Astro-
physics, Vol. 7. Leo Goldberg, ed. 717
pp. Annual Reviews, Palo Alto, Calif.,
1969. $8.50
Eclipse Phenomena in Astronomy. By F.
Link. 271 pp. Springer-Verlag, New
York, 1969. $19.50
BIOPHYSICS
Biology and the Physical Sciences. Sam-
uel Devons, ed. 379 pp. Columbia Univ.
Press, New York, 1969. $12.50
THEORY AND MATHEMATICAL PHYSICS
Springer Tracts in Modern Physics, Vol.
50; Current Algebra and Phenomenologi-
cal Lagrange Functions. (Papers from
1st International Summer School for The-
oretical Physics, Univ. of Karlsruhe, 22
July-Aug., 1968). G. Hohler, ed. 156
pp. Springer-Verlag, New York, 1969.
$11.00
Stochastic Theory and Cascade Processes.
By S. Kidambi Srinivasan. 216 pp.
American Elsevier, New York, 1969.
$12.50
Men of Physics: L. D. Landau, Vol. 2:
Thermodynamics, Plasma Physics and
Quantum Mechanics. By D. Ter Haar.
198 pp. Pergamon, New York, 1969.
Cloth $5.50, paper $3.25
Fruhgeschichte der Quantentheorie,
1899-1913. By A. Hermann. 181 pp.
Physik Verlag, Mosbach in Baden, 1969.
Quantum Chemistry: Elementary Prin-
ciples and Methods. By N. V. Riggs.
243 pp. Macmillan, Toronto, Canada.
1969. $9.95
Elements of Advanced Quantum Theory.
By J. M. Ziman. 269 pp. Cambridge
Univ. Press, New York, 1969. $9.50
Elements of Quantum Theory. By FrankOxford
The Collected Papers
of G. H. Hardy
INCLUDING JOINT PAPERS
WITHJ. E. LITTLEWOOD AND
OTHERS; VOLUMES III AND IV
Edited by a committee appointed by the London
Mathematical Society. The primary object of
these publications is to render more accessible
the papers of this great mathematician, which
in their original form appeared in many jour-
nals over a period of almost sixty years. The
editors have provided introductions to groups
of papers, and commentary where appro-
priate. To be completed in seven volumes.
Volumes III & IV, $14.75, each
Angular Momentum
SECOND EDITION
By DAVID MAURICE BRINK, Balliol College,
Oxford; and GEORGE RAYMOND SATCH-
LER, Oak Ridge National Laboratory. For the
second edition of this concise account of the
quantum theory of angular momentum, and
its basis in the symmetry properties of physical
laws, the authors have added a chapter on
graphical methods. (Oxford Library of Physical
Sciences.*) Paper, $3-50
High Voltage Technology
By L. L. ALSTON, Director of Electrical Re-
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introduction for graduate engineers and other
scientists, this book is based upon a high-volt-
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Thermal Neutron
Diffractions
Edited by B. T. M. WILLIS, University College,
Cardiff. In this book, based on papers pre-
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Neutron Diffractions, recent research on the
magnetic and nuclear elastic scattering of
thermal neutrons is reviewed by leading
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new information on the electronic and nuclear
charge distributions in solids, and the book
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NOW AVAOBLTA valuable reference work for
Solid State Theorists"Tables of Irreducible
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Introductory Probability Theory. By
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N. J., 1969. $6.95
Dispersion Relation Dynamics. By Hugh
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Linear Partial Differential Operators (3rd
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Quantum Mechanics with Applications.
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Thin-Film Transistors. By Andrew C.
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Digital Electronics for Scientists. By
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Hochspannungsmesstechnik, Massgerate
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Structures Technology for Large Radio
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Non-Destructive Testing Views, Reviews,
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MISCELLANY
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crow Press, Metuchen, N. J., 1968. $8.50MOIECUIAR
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PHYSICS TODAY • DECEMBER 1969 • 87
Why is the Jarrell-Ash
LASER RAMAN SYSTEM
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A. LOWEST SCATTERED LIGHT
The Jarrell-Ash Laser Raman System
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reversed from that of the top. Thiseliminates "double dispersion" and
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higher resolution with a narrow exit slit.
D. ACCEPTS NUMEROUS
COMMERCIAL LASERS
The Jarrell-Ash Laser Raman System can
utilize many of the newly developed
commercial lasers, e.g., the He-Ne, Kr,
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E. LARGEST SAMPLE CHAMBER
The sample chamber features a working
area 66 cm wide x 85 cm long x 60 cm
high to permit use of even large Dewars
for controlled temperature experiments.
F. FLEXIBLE OPTICAL PATH
The optical path offers high efficiency
and flexibility. The use of Brewster
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eliminates light loss due to reflection.
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Provisions are made to direct the laser
beam down through, straight through, or
up into the sample.G. POLARIZATION FEATURES
Polarization characteristics of the
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•DIVISION OF FISHER SCIENTIFIC CO.
Jarrell-Ash Division/Fisher Scientific Company, 590 Lincoln Street, Waltham, Mass. 02154
88 • DECEMBER 1969 • PHYSICS TODAY
MEETINGS
Normal-State Electron Tunneling Only Qualitatively Understood
From an experimentalist's point of
view, the field of electron tunneling
owes its present lively state to the dis-
covery of the p-n tunnel diode by Leo
Esaki in 1957 and of tunneling
through oxide layers by John C. Fisher
and Ivar Giaever in 1960. The results
in the oxide system became even more
remarkable when the electrodes
against the oxide were made super-
conducting by Giaever in 1960. In the
five years that followed, a happy com-
bination of theory and simple experi-
ments led to confirmation of the
Bardeen-Cooper-Schrieffer gap and
square-root singularity in the electron-
ic density of states, the fascination of
the Josephson effect and the measure-
ment of the details of the electron-
phonon interaction.
However, in the past two years, in-
terest has again cycled to p-n diodes,
metal-semiconductor conductor con-
tacts and metal-insulator-metal (M-I-
M) junctions in the normal state. As
a result a conference on nonsupercon-
ducting electron tunneling was held at
Prouts Neck, Maine, during 3-5 Sept.
The meeting was arranged in the style
of a Gordon conference with morning
and evening sessions. In keeping with
the Gordon conference tradition, no
further publication of proceedings is
contemplated. Previous conferences,
which considered tunneling into both
superconducting and normal elec-
trodes, were held at Philadelphia
(1961) and Ris0 (1967).
The result of this conference can be
summarized briefly: Tunneling in
normal systems, for experimentalists
and theorists alike, is in some trouble
unless one is satisfied with a purely
qualitative understanding of the field.
Remembering the successful applica-
tion of tunneling to superconductivity,
we may find this conclusion surpris-
ing. The origin of the difficulties of
the normal state was summarized by
Doug Scalapino (University of Cali-
fornia, Santa Barbara) in his impres-
sions at the end of the conference.
The superconducting experiments
probe properties of the electrodes over
distances comparable to the coherence
length, generally large enough to sam-
ple bulk effects (maybe not in thecase of transition metals and type-II
materials), whereas the normal-state
experiments are affected by the nature
of the tunnel barrier, sometimes only a
few atom layers thick, and by the
metal electrodes within a screening
length of the oxide-metal interface.
Thus tunneling has become a problem
of surface physics.
The first topic dealt with at the
conference was: How well can the
overall conductance-versus-voltage de-
pendences be explained by single-par-
ticle tunneling theory? Next, interac-
tions of the tunneling electron with
the oxide, or impurities or particles in
the oxide, led to discussion of "zero-
bias anomalies." Finally, observations
of interactions within the electrodes,
the many-body or self-energy effects,
were reported and the theory of these
effects received considerable discus-
sion.
Single-particle tunneling. The cal-
culation of single-particle tunneling
currents through a potential barrier re-
quires an exact knowledge of the bar-
rier potential as a function of distance.
As Gerald Mahan (University of Ore-
gon) pointed out in the opening talk,
this is poorly known in p-n diodes and
only guessed at in M-I-M junctions;
therefore, the metal-semiconductor
contact (Schottky barrier on degener-
ate material) has received the most at-
tention recently. He showed that a
calculation of the tunneling current
could be made, based on uniform
charge density in the depletion region,
which results in a parabolic potential
barrier. Experiments in which the
barrier height and thickness are deter-
mined by independent measurements
give an absolute conductance in "bet-
ter than an order of magnitude" agree-
ment with the calculation. The ex-
periments also show the correct volt-
age dependence of the conductance.
This agreement holds only when the
surface-barrier contacts are made by
cleaving the semiconductor in vacu-
um. Mahan's gloom with respect to
M-I-M junctions was questioned by
Carver Mead (Cal Tech) who pre-
sented a detailed investigation of
aluminum—aluminum-nitride metal
junctions. Combining capacitanceand current-voltage measurements on
a series of junctions with different ni-
tride thicknesses, he and collaborators
have determined the E versus k rela-
tionship for • the electron over the
whole of the forbidden gap of the in-
sulator. This result raised the inevita-
ble question: What "band structure"
can we associate with such thin layers,
and is the insulator crystalline or
amorphous? The extension of such
careful analysis to other systems will
be of interest.
For M-I-M systems, the tunneling
conductance at low voltages (less than
200 mV) is not constant but has a
roughly parabolic dependence on volt-
age. As reported by J. M. Rowell
(Bell Labs), calculations based on
simple trapezoidal barriers also show
that the minimal conductance only oc-
curs at V = 0 for symmetrical barriers.
However, no comparison of calculated
and measured conductance has been
made for junctions with barrier pa-
rameters determined independently.
Returning to an older barrier prob-
lem, phonon emission in p—n diodes,
Charlie Duke (University of Illinois,
Urbana) concluded that the theory of
Kleinman offers a good description of
the effect. New measurements of
such phonon-assisted tunneling in a
very wide gap material with complex
lattice dynamics, silicon carbide, were
reported by Phil Stiles (IBM).
Impurities. Although most oxide
junctions contain unknown impurities,
the addition of intentional impurities
to the barrier is a relatively recent de-
velopment. Two talks on the very in-
teresting effects of adding metallic
particles were given by Hansrudi Zel-
ler (GE) and John Lambe (Ford).
Although different in concept and re-
sults, the two experiments both raise a
puzzling question. In the work de-
scribed by Zeller (performed in collab-
oration with Giaever) the well known
agglomeration of very thin metal films
is used to introduce an array of parti-
cles (about 10 nm or less in diameter)
into the oxide of a tunnel junction.
The current flows as electrons tunnel
to the particles, localize, and then tun-
nel to the other electrode. However,
if a particle is about 5 nm in diameter,
PHYSICS TODAY • DECEMBER 1969 89
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with the Model 904 Rejector and
the Model 505 Restorer/Gate, both
peak and tail pile-up inspection,
Active/Passive dc restoration,
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fast signal component. Only those
fast signals which pass inspection
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Integrator/Clip 2 Output section of
the 501 for further shaping (Gaus-sian) prior to MCA analysis.
The system offers minimum
dead time in pile-up inspection and
does not needlessly reject tail pile-
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pulse height.
Additionally, locating the Active/
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90 • DECEMBER 1969 . PHYSICS TODAY
MEETI NGS
the addition of one electronic charge
requires a charging energy (Ar) of ap-
proximately 10 meV. For a single par-
ticle the junction conductance would
show a step at A(.. For a distribution
of particles the conductance rises
rapidly as a function of voltage; that is,
the resistance shows a strong peak at
V = 0. Zeller frequently pointed out
that this picture can be generalized to
explain all "zero-bias resistance peaks,"
by claiming that "states" exist in the
harrier with a density given by dG dV.
This explanation, of course, is possible,
but it appears dangerous to assume
that it is always correct, and hence to
lose interest in the problem. For ex-
ample, an alternative explanation of
the conductance dip near V = 0 in
metal—semiconductor contacts in-
volves the excitation of phonons in
the semiconductor depletion layer.
The theory of Duke and others was
compared to experiment by Tom Car-
mthers (University of Chicago) and,
although the energy range of the ob-
served structure is correct, a disagree-
ment in line-shape was apparent.
Further discussion of the various ex-
citation processes observable in
metal—semiconductor contacts was
given by Matthew Mikkor (Ford)
and William Thompson (IBM). The
possibility of observing organic impu-
rity vibrations was a point of disagree-
ment in these two talks.
Let us return now to the physics of
particles. A small globule brought
close to a metal electrode will, by tun-
neling, lose or gain electrons until its
highest filled electron level is within
the charging energy (Af.) of the Fermi
level in the electrode. In order to ex-
plain the data Zeller assumes that,
over all the particles of a given size,
the highest filled level is uniformly
distributed within —Ac to +AC of
the Fermi level. In other words,
there is no preferred alignment of the
particle level with the electrode, be-
cause partial electronic charge cannot
be exchanged. However, it is just
such an alignment that is essential to
the new work described by Lambe.
He and Bob Jaklevic (Ford) studied
the metal-oxide-particle-oxide-metal
system where one oxide is too thick to
permit tunneling. The properties of
the device are probed using capaci-
tance measurements, that is by mak-
ing electrons hop on and off the par-
ticle through the thinner oxide. The
resulting capacitance-voltage depen-dence, which shows symmetrical
structure about V = 0, is explained on
the basis of some degree of alignment
of particle "Fermi level" with that of
the electrode. Even more dramatical-
ly, if a voltage (or series of voltages)
is applied to the device at room tem-
perature and maintained during cool-
ing, then at low temperatures the ca-
pacitance-versus-voltage structure is
removed from V = 0 and shifted to the
"forming" voltage (or voltages). This
result implies that realignment of par-
ticle and electrode "Fermi levels" is
induced by the applied voltage. The
necessary transfer of partial charge to
the particle is achieved by "polariza-
tion" of the oxide. Although details
of this polarization were not under-
stood, results described by John Adler
(University of Alberta) may be rele-
vant. In a study of the excitations of
molecular impurities in aluminum-
oxide tunnel junctions he found that
the relative strengths of the various vi-
brational modes could be changed by
applying a voltage to the junction at
room temperature. If this change
implies a motion, or rotation, of polar-
ized molecules then it is equivalent to
rearrangement of charge in the oxide.
So far, all tunneling layers between
metal films have been thermally
grown oxides. However, Giaever de-
scribed his fabrication of junctions
using evaporated semiconductors such
as germanium, zinc sulfide and cadmi-
um sulfide. By oxidation, any pin-
holes in the semiconducting layer
were filled with oxide of the base
metal. That tunneling was taking
place through the semiconductor was
confirmed by observing conductance
structure at the correct energy for ex-
citation of LO phonons in the semi-
conductor. In the case of cadmium
sulfide Giaever showed that the tun-
neling characteristic could be changed
by shining light on the junction; a
"tunable tunneling matrix element."
Zero-bias anomalies. As mentioned
above, the question of zero-bias anom-
alies was discussed frequently at the
conference. One of the best under-
stood of these is the conductance-peak
anomaly. This anomaly is categorized
by a conductance obeying the law
log
where eV is the voltage, kBT the tem-
perature, and Eo a cut-off parameter.
The conductance is also strongly de-
pendent on magnetic field. An expla-
nation for this effect had been ad-150 KeV
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PHYSICS TODAY • DECEMBER 1969 • 91
CRYSTAL & ELECTRONIC PRODUCTS
Never heard of Harshaw's Crystal
& Electronic Products Department?
Neither has anybody else. Until now.
We used to call it the Crystal-Solid
State Department. But now that
we're headquartered in a brand new
plant at Solon, Ohio, we wanted an
accurate new name to match. But,
by any name, we stand for the ulti-
mate in products and service for our
customers.
Our new centralized facility, how-
ever, adds extra dimensions to our
well-known capabilities. Now that
we're centralized, it's even more
natural for you to think of us as
sole source for your projects.
We assume beginning-to-end re-
sponsibility. Including the manu-
facture of all components, assembly,
testing and a guarantee of the per-
formance of every Harshaw product
you buy.
As always, Harshaw quality con-trol is absolute and
conducted to your exact specifications. That
includes detectors and all downstream electronics.
Another bonus brought to you by our new
centralized facilities is the advantage of cross-
talk between disciplines which helps promote
even more advanced and effective products and
performance.
Our product line today incorporates the entire
line of our former affiliate, Hamner Electronics
Co., Inc., and further includes: Optical crystals
and materials for the IR/UV field. Nuclear
detectors. Nuclear electronics. Medical instru-
mentation. And microwave materials. For your
many needs, look to an old pro with a new name.
The Crystal & Electronic Products Department
of Harshaw.
Write or call for our complete catalog.
HarshawThe Harshaw Chemical Company. Division of Kewanee Oil
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Cochran Road. Solon. Ohio 44139 • Phone (216) 248 74OO
MEETl
vanced by Appelbaum and Anderson,
based on the electron spin-flip scatter-
ing of magnetic impurities in the bar-
rier region. David Losee and Edward
Wolf (Eastman Kodak) presented
data on a number of different
vacuum-cleaved degenerate semicon-
ductor Schottky-barrier junctions, in
which they ascribe the origin of the
magnetic impurities in their systems to
the neutral donors at the edge of the
depletion layer. They found good
agreement between their data and the
Appelbaum theory if they suitably ex-
tended the theory to include the life-
time broadening of the magnetic level
as well as a g-shift. Work on these
anomalies in metal-doped insulator-
metal junctions was reported by
Adrian Wyatt of Nottingham Univer-
sity and also Paul Nielsen of Chicago.
Many-body effects. The influence
of many-body effects on nonsupercon-
ducting electron tunneling generated
considerable discussion. In the past,
tunneling has been a powerful probe of
the many-body interactions in super-
conductors. This is because of the
strong momentum dependence of the
electron self-energy in a superconduc-
tor, which makes the structure seen in
the conductance large, and the super-
conductor's large coherence length,
which makes the superconducting
wave functions near the metal-oxide
interface only weakly dependent on
the details of this surface. Both these
effects no longer operate in nonsuper-
conducting tunneling. There the self-
energy is predominantly frequency de-
pendent, leading to small (1 % ) struc-
ture in the conductance. The effective
"coherence length" is the order of the
Fermi wavelength, so that the exact
form of the metal-oxide interface (or
semiconductor-Schottky barrier deple-
tion layer) has an important influence
of the structure one sees on the con-
ductance.
Craig Davis (Ford) presented
work, done in collaboration with
Duke, on the influence of the electron
self-energy, (resulting from the elec-
tron-optical phonon interaction in
semiconductors) on the conductance
of Schottky barriers. The self-energy
in this case is purely frequency depen-
dent. He emphasized that no struc-
ture in the conductance would be pre-
dicted unless the momentum depen-
dence of the tunneling matrix element
is taken into account. This momen-
tum dependence is uniformly ignoredin superconducting tunneling; so we
see again the important difference be-
tween the two types of tunneling.
The standard approach to tunneling
calculations, the tunneling Hamilto-
nian, came under attack in work pre-
sented by Joel Appelbaum and Bill
Brinkmann (Bell Labs). They ar-
gued that the tunneling Hamiltonian
predicts the incorrect form for the
transition matrix elements because it
first calculates the coupling between
the electrodes and then considers the
influence of the many-body effects.
To rectify this problem they proposed
a theory that considers the transition
rate between exact many-body states
of the electrodes. If the transition
rate is calculated by the WKB approx-
imation, they find they can recover
the conventional formula for the cur-
rent, but with the transition matrix
element replaced by one that is pre-
dominantly frequency dependent. In
general, they find that the current de-
pends on the electron Green's function
in the vicinity of the barrier. They
showed, for the particularly simple ex-
ample of the electron interacting with
magnetic impurities (zero-bias con-
ductance peak), that the size as well
as the sign of the zero-bias anomaly
depends on the relative position of the
impurity and the junction interface.
The theoreticians therefore con-
cluded that the surface can have a
profound influence on the self-energy
effects observed in the conductance of
metal-insulator-metal junctions. It
was also obvious that experimentally
great variations in junction properties
(presence of zero-bias conductance
peak, for example) are obtained by al-
tering oxidation procedures. This re-
sult indicates that, in future, tunneling
experiments must be increasingly tied
to surface studies of the metal elec-
trodes, with such tools as low-energy
electron diffraction, field emission,
Auger spectroscopy and optical stud-
ies.
The conference was sponsored by the
Ford Scientific Laboratory, by the Na-
tional Science Foundation, and by the Air
Force Office of Scientific Research. As
stated above, no proceedings are to be
published, but those interested in further
reading on the subject will find an excel-
lent up-to-date review in: C. B. Duke,
Tunneling in Solids, Academic Press, New
York (1969) (Solid State Physics, Sup-
plement 10).
J. A. APPELBAUM
J. M. ROWELL
Bell Telephone Laboratories
Murray Hill N.J. DXenon. We have it for you pure
and ultra pure. In a variety of
pressures and containers.
For this year's catalog, write:
Rare and Specialty Gases Dept.,
Airco Industrial Gases, 150 East
42nd Street, New York, N.Y. 10017.
PHYSICS TODAY • DECEMBER 1969 • 93
Not too many years ago, the word cryogenic was completely non-existant in the vernacu-
lar of the layman. Cold was somewhat taken for granted, being delivered daily to the
back porch in the form of large cubes at the much accepted and very seldom contemplated
temperature of 32°F.
The driver of the familiar horse drawn wagon was cutter, loader, delivery man and col-
lector; and so enjoyed the privilege and pride of ownership.
In servicing today's sophisticated technology, the need for "Ice" is being satisfied with
cryogenic fluids at temperatures as low as —456°F (LHe). The space age ice wagon, a
9,000 gallon vacuum/liquid nitrogen cooled tanker, travels thousands of miles delivering
to universities, research labs, and missile sites.
SPACE AGE ICE WAGONPresent day operations leave little room for the nostalgia of the earlier way of life, but
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The executives at GARDNER can recall putting their backs to building our first "ice
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we needed suitable transport for long distance bulk shipment of LHe and again relied on
"do it yourself" philosophy to get the job done. Our new engineering and manufacturing
facility at Bethlehem, Pennsylvania, is also a product of GARDNER people.
This is the way it is at GARDNER—In filling our own needs, we
have developed an overall resourcefulness that rubs off
on the products and services offered to our customers.
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2136 CITY LINE ROAD • LEHIGH VALLEY INDUSTRIAL PARK • BETHLEHEM, PENNSYLVANIA 18017 • PHONE (215) 264-4523
94 • DECEMBER 1969 • PHYSICS TODAY
WE HEAR THAT . . .
John H. Van Vleck
has retired from
Harvard Universi-
ty and is now Hol-
lis Professor of
Mathematics, Em-
Particular-
y known for
his work in magne-
VAN VLECK tism and quantum
theory of atomic structure, Van Vleck
was instrumental in creating the divi-
sion of engineering and applied phys-
ics at Harvard. In 1952 Van Vleck
was president of the American Physi-
cal Society, and he has served as vice-
president of both the American Acad-
emy of Arts and Sciences and the In-
ternational Union of Pure and Applied
Physics. Among the awards he has
received are the National Medal of
Science, the Michelson Award of the
Case Institute of Technology and the
Langmuir Prize of the American Phys-
ical Society.
i
Kenneth Fox has returned to the Uni-
i versity of Tennessee as assistant pro-
l| fessor after a two-year leave as a Na-
I tional Academy of Sciences senior
postdoctoral research associate. Fox
spent the two years at the Cal Tech
Jet Propulsion Laboratory.
Two Colgate University physicists,
James N. Lloyd and Charles H. Hol-
brow, are on leave for the current
academic year. Lloyd, an assistant
professor, is at the University of Mary-
land, and Holbrow, an associate pro-
fessor, is at Stanford.
Miles E. Anderson, professor of phys-
ics at North Texas State University,
has been appointed associate vice
president for academic affairs. James
R. Sybert is now chairman of the
physics department, H. James Mackey
has become professor, and James A.
Roberts and Thomas J. Gray are as-
sociate professors. R. Muthukrishnan
of Michigan State University has
joined the department as assistant pro-
fessor.
George C. Weiffenbach has been
named director of geoastronomy pro-
grams at the Smithsonian Astrophysi-
cal Observatory. In this new post,Weiffenbach will be responsible for
optical-laser tracking, long-base inter-
ferometry and maser-clock space ex-
periments. Weiffenbach was formerly
supervisor of the space research and
analysis branch of the applied physics
laboratory at Johns Hopkins. Harri-
son E. Radford has also joined the ob-
servatory staff. Radford was formerly
a molecular physicist with NBS, and
will establish a laboratory at the ob-
servatory to study molecules known to
exist in interstellar space.
John A. Da vies has been promoted to
associate professor in the physics de-
partment at Clark University. Wil-
liam R. Fehlner, formerly at the Uni-
versity of Illinois, is now assistant pro-
fessor at Clark.
Russell G. Groshans has been ap-
pointed staff engineer for product en-
gineering at RCA. Groshans, who
was a systems engineer at RCA,
Hightstown, had been a US Air Force
physicist until 1967.
Warren Proctor has been appointed
manager of market development labo-
ratories for the Varian analytical in-
strument division. Proctor was pro-
fessor of physics at the University of
Washington until he joined Varian in
1955.
James T. Shipman is the new physics
chairman at Ohio University, and
Roger W. Finlay and David S. Onley
were promoted to professor. Jacobo
Rapaport, who had been at Oak Ridge
National Laboratory, was appointed
associate professor.
State University of New York at Bing-
hamton has two new assistant pro-
fessors of physics. They are Noel Yeh,
formerly at Columbia, and Robert
Pompi, who had been a research as-
sociate at Binghamton.
Robert G. Breene Jr, has been ap-
pointed as professor and Mohindar S.
Seehra as assistant professor at West
Virginia University. Breene was for-
merly with Physical Studies, Inc., and
Seehra was at the University of Roch-
ester. The physics department also
announced that T. Tietz, chairman ofthe department of theoretical physics
at the University of Lodz, Poland, will
be visiting professor for 1969-70.
Frank J. Blatt is acting chairman of
the Michigan State University physics
department, succeeding Sherwood K.
Haynes who will remain in the depart-
ment. New members of the Michigan
faculty include B. Hobson Wildenthal,
formerly of Texas A & M University,
who will be associate professor and
William P. Pratt, formerly of Los Ala-
mos Scientific Laboratory, assistant
professor. Rubby Sherr of Princeton
and F. C. Barker of Australian Nation-
al University will spend this year as
visiting professors at Michigan State.
New director of the University of New
Hampshire Space-Science Center is
William R. Webber. Webber, for-
merly at the University of Minnesota,
succeeds Lawrence Cahill.
Langdon T. Crane has been named re-
search professor and director of the
Institute for Fluid Dynamics and Ap-
plied Mathematics of the University of
Maryland. Crane was formerly pro-
gram director for atomic and molecu-
lar physics at the National Science
Foundation. Frank W. S. Olver, for-
merly of the National Bureau of Stan-
dards, was also named research pro-
fessor at the institute, while R. Bruce
Kellogg, Herbert Lashinsky and
Thomas D. Wilkerson were promoted
to that position. James Yorke was
promoted to research associate profes-
sor.
Virgil B. Elings, on leave from the
University of California, Santa Bar-
bara, will spend this year as a senior
research scientist at Siginatron, Inc. in
Santa Barbara.
Yale University announces that Daniel
E. Rosner is associate professor of en-
gineering and applied science. Ros-
ner was formerly with the Aerochem
division of Sybron Corporation.
Florida State University announces
that Edward Desloge and Steve Ed-
wards have been promoted to profes-
sor and James Skofronick and Gerald
Speisman to associate professor. An-
PHYSICS TODAY • DECEMBER 1969 95
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96 • DECEMBER 1969 • PHYSICS TODAY
WE HEAR THAT
thony Colleraine, who was at the Uni-
versity of Maryland, has become assis-
tant professor at Florida State.
Piel of Scientific American
Wins Arches of Science Award
Gerard Piel, publisher of Scientific
American, is the 1969 recipient of the
Pacific Science Center Arches of
Science Award. The $25 000 awards
are given for contributions to the pub-
lic understanding of what science
means to man. In 1947, Piel, along
with Dennis Flanagan and Donald H.
Miller Jr, acquired Scientific American
and began to revitalize the magazine,
which now has a circulation of more
than 400 000.
Ronald J. Sladek is acting head of the
Purdue University physics depart-
ment. Sladek succeeds Richard W.
King who died 12 Aug. (PHYSICS
TODAY, October, page 105)
The physics department of the Uni-
versity of Virginia, Charlottesville,
will have a new chairman, Judah M.
Eisenberg, as of February. Robert V.
Coleman is acting chairman until
Eisenberg returns from his sabbatical
leave at Tel Aviv University. W. Dex-
ter Whitehead, former chairman of the
department, has become dean of the
Graduate School of Arts and Sciences
at the university, but will continue as
professor of physics and director of
the Center for Advanced Studies.
Several other changes have occurred
at the physics department: Vittorio
Celli was promoted to professor; Mi-
chael Coopersmith, formerly of Case-
Western Reserve University and Diet-
er Drechsel, formerly of the University
of Frankfurt, have been appointed as-
sociate professors; John Ruvalde, of
the University of Chicago and Ste-
phen Thornton of the University of
Wisconsin are new assistant profes-
sors.
Jerry L. Peacher and Alexander O.
Animalu are new assistant professors
at the University of Missouri, Rolla.
Stevens Institute of Technology has
promoted Earl L. Roller, a particle
physicist, to professor. Edward A.
Friedman, Bela M. Mecs and Norman
J. Horing have been promoted to asso-
ciate professor.The MIT instrumentation laboratory
will have a new director on 1 January,
Charles L. Miller, now head of the
civil-engineering department. The
present director, Charles Stark
Draper, will continue to serve as sen-
ior adviser and director of major
projects. Draper, who founded the
instrumentation laboratory, retired
from the MIT faculty two years ago.
The laboratory will be renamed the
Charles Stark Draper Laboratory in
recognition of his contributions.
William O. Statton has been promoted
to professor of materials science and
engineering at the University of Utah.
Robert S. Knox has
succeeded Morton
Kaplon as head of
the physics and as-
tronomy depart-
ment at the Uni-
versity of Roches-
ter. Kaplon will
i remain as profes-
KNOX sor. Joseph H.
Eberly and Thomas Ferbel were pro-
moted to associate professor, and G.
Badhwar and P. Slattery are now as-
sistant professors. John Krizan of the
University of Vermont is a visiting as-
sociate professor this year.
Michael J. Moravcsik has succeeded
Marvin D. Girardeau Jr as director of
the Institute of Theoretical Physics of
the University of Oregon. Girardeau
is on sabbatical leave.
Promotions in the physics department
at Indiana University include Walter
E. Bron and Guy T. Emery to profes-
sor and Delbert W. Devins and Rich-
ard M. Heinz to associate professor.
New assistant professors are Lloyd L.
Chase, Shu-Yuan Chu, A. W. Hendry
and Peter Schwandt.
Robert Vessot and Martin Levine
have joined a newly formed research
group at the Smithsonian Astrophysi-
cal Observatory to develop and adapt
hydrogen maser clocks for space and
geophysical applications. Vessot and
Levine were formerly at Hewlett-
Packard, Beverly, Mass.
New additions to the physics faculty
of Southern Illinois University at Ed-
wardsville are Hadi H. Aly, visiting
professor, from the American Univer-
sity of Beirut; Ika-Ju Kang, associate
professor, from Southern Illinois atCarbondale; Thomas O. Baldwin, as-
sociate professor, from Oak Ridge Na-
tional Laboratory; and Padmanabha
Narayanaswamy, assistant professor,
from the American University of
Beirut.
Irvin A. Miller, assistant professor of
physics, has been named acting dean
of the Drexel Institute of Technology
College of Science.
At the University of Wisconsin nu-
clear-engineering department Harold
K. Forsen has been promoted to pro-
fessor and John M. Donhowe and
Wesley K. Foell have been promoted
to associate professor. Charles W.
Maynard will spend this year on leave
at the Sandia Corporation.
New York University School of Medi-
cine announces the promotion of Ber-
nard Altshuler to professor of environ-
mental medicine.
The physics department at Illinois In-
stitute of Technology announces that
Porter Johnson, formerly of Case-
Western Reserve, is now assistant pro-
fessor, and Val R. Veirs, a recent
graduate of IIT, is visiting assistant
professor. Robert L. Warnock and
Thomas Erber have been promoted to
professor and Frederick J. Ernst to as-
sociate professor. A former visiting
associate professor, Fritz Herlach of
EURATOM, is now associate professor,
and a former visiting assistant profes-
sor, Cheuk-kin-Chau, is now assistant
professor.
New physics de-
partment chairman
at the State Univer-
sity of New York,
Stony Brook, is
Morton Hamer-
mesh. Hamermesh
had been head of
the school of
HAMERMESH physics and astron-
omy at the University of Minnesota
since 1965; Walter H. Johnson is now
acting chairman at Minnesota. Ham-
ermesh, a theoretical nuclear physi-
cist, is a fellow of the American Physi-
cal Society and was on the board of
trustees of Universities Research As-
sociates while at Minnesota.
Alexander J. Dessler has been appoint-
ed science adviser to the executive
secretary of the National Aeronautics
and Space Council; the council is a US
PHYSICS TODAY • DECEMBER 1969 97
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WE HEAR THAT
government advisory group headed by
Spiro Agnew. Dessler has been chair-
man of the Rice University space-
science department since 1963.
AEC Cites Three Men for
Outstanding Contributions
The Atomic Energy Commission has
awarded Lauriston S. Taylor, George
B. Darling and Paul M. Gross citations
for outstanding service to the national
atomic energy program. Taylor, spe-
cial assistant to the president of the
National Academy of Sciences, is hon-
ored for his work in radiation protec-
tion; Darling's award is for his studies
on the delayed effects of radiation on
men (he is director of the Atomic
Bomb Casualty Commission in Hiro-
shima); Gross, who helped organize
the Oak Ridge Institute of Nuclear
Studies and is president of Oak Ridge
Associated Universities, is being hon-
ored for his work at Oak Ridge.
Visiting professor at the University of
California, Riverside, this fall is Rich-
ard J. Eden of Cambridge University.
Kenneth C. Clark is the new program
director for aeronomy in the atmo-
spheric sciences section, division of
environmental sciences of the National
Science Foundation. Clark, a geo-
physicist, is on leave from the Univer-
sity of Washington.
William S. Porter has been promoted
to professor at Southern Connecticut
State College. John W. Snyder of
Ohio State University and Lee T.
Matthews of the University of Ver-
mont are new assistant professors.
L. Eric Cross, professor of electrical
engineering, and Heinz K. Henisch,
professor of physics, have been ap-
pointed associate directors of the Ma-
terials Research Laboratory at Penn-
sylvania State University.
Craig J. W. Gunsul, formerly at the
University of Delaware, has joined the
physics department at Whitman Col-
lege as assistant professor. James G.
Pengra, of Whitman, i§ currently vis-
iting at the Nuclear Research Center,
Georgia Institute of Technology.
Joseph W. Weinberg has been named
Kenan Professor of Physics at SyracuseUniversity. The Kenan professor-
ships, named for William R. Kenan Jr,
were established at five New York
universities to improve the quality of
undergraduate teaching. Weinberg, a
theoretical physicist, was at Case-
Western Reserve University before he
came to Syracuse.
Dame Kathleen Lonsdale, past presi-
dent of the International Union of
Crystallographers, is visiting professor
at the Ohio State University depart-
ment of mineralogy this fall.
Lonsdale is professor of chemistry at
the University of London.
DeShalit of Weizmann
Institute Dies at 42
It is our sad task to report that
Amos deShalit died of acute pancrea-
titis on 2 Sept. at the age of 42. His
untimely passing is a great loss to his
family, to the world of physics, to his
institute and country and to the entire
world. A brilliant physicist, deShalit
was one of the very few who are at
home with both experiment and theo-
ry. He was a brilliant administrator;
while he was head (1954-66), the nu-
clear physics department at the Weiz-
mann Institute, Rehovoth (Israel) de-
veloped into a leading center for the
study of nuclear and particle physics,
rivaled in its impact by only a handful
of other institutions. He was a brilli-
ant educator; since 1963 he had been
actively involved in improving science
education in Israel, particularly in the
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100 • DECEMBER 1969 • PHYSICS TODAY
WE HEAR THAT
secondary schools. This activity was
recently made formal by the creation
at Weizmann of a department of
science teaching that was headed by
deShalit. Deeply committed to his
country, he was much concerned with
the problems of the Arab population.
DeShalit's involvement in world af-
fairs motivated a great many of his ac-
tivities; he did much to bring togeth-
er physicists of all countries and politi-
cal persuasions and helped to organize
many international conferences. In
July he gave the summary talk at the
Heidelberg conference on heavy-ion
induced nuclear reactions, and he had
been scheduled to participate in a
round-table discussion of the future of
nuclear physics at an international
conference that took place late in
August.
He was one of the principal archi-
tects of the biennial international con-
ferences on nuclear and high-energy
physics; the proceedings of the most
recent of these conferences, held at
Columbia University, is to be dedicat-
ed to his memory. Involved for many
years with the problems of developing
countries, deShalit was a member of a
United Nations advisory committee.
At the time of his death he was host to
a conference at Rehovoth on "Science
and Education in Developing States"
and had been scheduled to address it.
| This partial list of his activities is a
pale reflection of his personal impact;
his greatest and unique contribution
came from his direct and indirect in-
fluence upon friends and colleagues
all over the world. Physics became
more interesting and exciting to every-
one who came in touch with him. His
presence made a discussion more fruit-
ful, a seminar more instructive, an ex-
periment more significant. He raised
questions and challenged ideas. He
brought life and excitement to phys-
ics; this was not only because of his
great insight, which enabled him to
point to the essential ideas and rela-
tions, but also because of his readiness
to listen and to follow the work of oth-
ers, his openness to questions, his in-
terest in any thought or idea, his en-
thusiasm for every new insight and his
ability to recognize the significance of
an idea.
So many friends had his help in de-
veloping their own ideas, help that he
gave freely and unsparingly. When he
visited laboratories, he left behind the
seeds of many successful theories andexperiments. His remarks and sugges-
tions spawned many papers.
DeShalit's own publications are in
nuclear physics, although his master's
thesis, done in 1949 under Giulio Ra-
cah's direction, was on the self-ener-
gy problem. An experimental thesis
in 1951 with Paul Scherrer at Zurich
began a series of experimental and
theoretical papers in which nuclear
structure was probed through electro-
magnetic and weak interactions.
DeShalit's fundamental contributions
to the understanding and exploitation
of the shell model culminated in 1962
with a seminal book that he wrote
with Igal Talmi, titled "Nuclear Shell
Theory."
He had been interested more re-
cently in the application of tools de-
veloped in elementary-particle physics
to studies of the nucleus. And, vice-
versa, he was using methods developed
for the shell model to extract from the
electromagnetic properties of elemen-
tary particles some of their underlying
structure. At the time of his death he
had just completed the first volume of
a two-volume book on nuclear theory
that he was writing with one of us.
He was scientific director of the
Weizmann Institute from 1962 to
1966 and director-general from 1966
to 1968.
He was a member of KTPAP, one
of the correspondents of Comments on
Nuclear and Particle Physics, and on
the editorial board of Nuclear Physics,
Annals of Physics, Nuclear Data and
Nuclear Instruments and Methods.
A member of the Israel National
Academy of Sciences and Humanities,
he received in 1964 the Israel Prize
for the Exact Sciences and in 1969
was elected a foreign member of the
American Academy of Arts and Sci-
ences. In recent years he was a visit-
ing professor at Stanford University
and the Massachusetts Institute of
Technology.
Amos deShalit is no longer among
us. We will miss his imaginative in-
sights and bold ideas. We will miss
the contagious pleasure he had in phys-
ics. We will miss his warm personal-
ity, his directness, his ability to create
bonds and bridges across political
chasms. Men like him are sorely need-
ed, and they are always in short sup-
ply; our world will be colder without
him.
HERMAN FESHBACH
VICTOR F. WEISSKOPF
Massachusetts Institute of
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102 • DECEMBER 1969 . PHYSICS TODAY
CALENDAR
This is a partial calendar comprising only notices received since last month. A
complete calendar is published every third month. Readers are referred to the
last one, published in October, if they wish a comprehensive listing of notices.
The January issue will contain the next complete calendar.
Information in the calendar is compiled from a file maintained in the PHYSICS TODAY
office. Readers are invited to write or telephone for general calendar information
beyond what we print. For complete information concerning an entry, readers are
advised to consult the contact and the original PHYSICS TODAY reference.
Abbreviations:
AAPT—American Association of Physics
Teachers
AAS—American Astronomical Society
ACA—American Crystallographic Assoc.
APS—American Physical Society
ASA—Acoustical Society of America
OSA—Optical Society of America
s OF R—Society of Rheology
AEC—US Atomic Energy Commission
AFCRL—Air Force Cambridge Research
Laboratories
Coding:
date subject • HOST • Location (Contact) [submission deadline] Physics Today ref.
• new listing • new informationANS—American Nuclear Society
AVS—American Vacuum Society
IAEA—International Atomic Energy
Agency
IEEE-Institute of Electrical and
Electronics Engineers
IPPS—The Institute of Physics and
The Physical Society
IUPAP—International Union of Pure
and Applied Physics
NBS—National Bureau of Standards
ORNL—Oak Ridge National Laboratory
DECEMBER 1969 MARCH 1970
10-12 • Holography and the Computer •
IBM • Houston (J. A. Jordan)
7/69
18-20 • Pulsars and High-Energy Activity
in Supernovae Remnants • AC-
CADEMIA INTERNAZIONALE DEI
LINCEI • Rome (B. Bertotti,
Laboratorio di Astrofisica, C. P.
67, Frascati (Rome), Italy) 12/69
19 • • N. Y. ACAD. sci. • 2 E. 63 St.,
N. Y. (J. Lebowitz, Belfer School,
N. Y., N. Y. 10033) 12/69
JANUARY 1970
19-23 • Electrochemistry D CORDON RE-
SEARCH CONFERENCES • Santa
Barbara, Calif. (Alexander M
Cruickshank, Pastore Chemical
Lab U. of Rhode Island, Kings-
ton, R. I-, 02881) 12/69
26-30 • Polymers • CORDON RESEARCH
CONFERENCE • Santa Barbara
Calif. (Alexander M. Cruickshank)
12/6911-13 • Scintillation and Semiconductor
Counters • IEEE, AEC, NBS •
Wash., D. C. (R. L. Chase, Brook-
haven Nat'l Labs., Upton, N. Y.
11973) 12/69
23-27 • Progress in Sodium-Cooled Fast-
Reactor Engineering • IAEA,
AEC • Monaco (John H. Kane,
Div. of Tech. Info., AEC, Wash.,
D. C. 20545) 12/69
Topics: Primary components, steam generators,safety technology, hydraulic and structural-coretechnology.
APRIL 1970
3, 4 • Midwest Theory Conference •
UNIV. OF NOTRE DAME • Notre
Dame, Indiana (W. D. McGlinn,
Dept. of Physics, U. of Notre
Dame, Notre Dame, Ind. 46556)
12/69
6-8 • Resonance in Conducting Mate-
rials • UNIV. OF WARWICK •
Univ. of Warwick, Coventry,
Partial calendar—see note at opening.SPECTROSCOPY
Cryo-Tip£l Refrigerators:
inexpensive solutions
to difficult
cryogenic interfaces.
• Temperatures down to 3.6° K
• Temperature control to ±0.1° K
• Uses gaseous, not liquid, helium.
• Wide variety of interfaces available.
A single Cryo-Tip? Refrigerator serves
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These refrigerators operate by the
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PHYSICS TODAY DECEMBER 1969 103
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104 • DECEMBER 1969 . PHYSICS TODAY
APRIL 1970
iWarwickshire, UK (R. Dupree,Univ. of Warwick) 12/69
27-29 • Frequency Control • ELEC-
TRONIC COMPONENTS LAB., US
ARMY ELECTRONICS COMMAND •
Fort Monmouth, N. J. (J. M.
Stanley, Electronic Components
Lab., Fort Monmouth, N. J.
07703) 12/69
MAY 1970
1,2 • Experimental Meson Spectros-
copy • UNIV. OF PA. • Phila.,
Pa. (Jules Halpern, Physics, U. of
Pa., Phila., Pa. 19104) 12/69
4, 5 • Transducers • IEEE • Gaithers-
burg, Md. (H. P. Kalmus, Harry
Diamond Labs., Dept. of the
Army, Wash., D.C.) 12/69
18-22 • Materials Symposium • us AIR-
FORCE, AMERICAN INST. OF AERO-
NAUTICS AND ASTRONAUTICS,
AMERICAN ORDINANCE ASSOC., SO-
CIETY OF AEROSPACE MATERIAL
AND PROCESS ENGINEERS • Miami
Beach, Fla. (Air Force Sympo-
sium '70, P.O. Box 38, Dayton,
Ohio 45420) 12/69
JUNE 1970
28-2 • • HEALTH PHYSICS SOCIETY
Chicago (R. F. Cowing) 7/69
JULY 1970
20-24 • Dielectric Materials, Measure-
ments and Applications • IEEE,
INSTITUTE OF ELECTRICAL ENGI-
NEERS (UK) • Univ. of Lancas-
ter, UK (1EE, Savoy Place, Lon-
don W.C. 2, UK) 12/69
21-24 • Nuclear and Space Radiation
Effects • IEEE • San Diego,
Calif. (R. Thatcher, Battelle Mem.
Inst., 505 King Ave., Columbus,
Ohio) 12/69
AUGUST 1970
11-15 • Magnetic Recording • HUNGAR-
IAN OPTICAL, ACOUSTICAL AND
CINEMATOGRAPHIC SOCIETY •
Budapest (M.J.K., Optical, Acous-
tical and Cinematographic Soci-
ety Budapest 5, Szabadsdg ter
17, Hungary) [2/70] 12/69
26-29 • Small-Angle X-ray Scattering D
ACA • Graz, Austria (O. Kratky,
Inst. for Physical Chemistry,
Univ. of Graz, Heinrichstrasse 28,
A 8010, Graz, Austria) 12/69
SEPTEMBER 1970
15-18 • Gas Discharges • IPPS, INSTI-
TUTE OF ELECTRICAL ENGINEERS
(UK) D London (IEE, Savoy
Place, London W.C. 2, UK) 12/69
Partial calendar—see note at opening.Model 1000 Current Integrator
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PHYSICS TODAY DECEMBER 1969 105
For magnetic
research
and testing
RFL Model 101
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In the lab or field, RFL's Model
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Use it for geophysical exploration,
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Weight: 11 lbs. Write for literature.
RFL Industries, Inc.
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TEL: 201-334-3100 / TWX: 710-987-8352 / CABLE: RADAIRCO, N. J.OCTOBER 1970
28-30 • Electron Devices • IEEE •
Wash., D. C. (IEEE, 345 E. 47th
St., N.Y.,N.Y. 10017) 12/69
NOVEMBER 1970
15-19 • Magnetism and Magnetic Mate-
rials • IEEE • Miami Beach,
Fla. (IEEE, 345 E. 47th St.,
N.Y.,N.Y. 10017) 12/69
NEW LISTING OF
SHORT COURSES AND SCHOOLS
12 JANUARY-10 APRIL
Theory of Imperfect Crystalline Solids
• INTERNATIONAL CENTRE FOR THEO-
RETICAL PHYSICS • Trieste, Italy
(Deputy Director, International Centre
for Theoretical Physics, P.O. Box 586,
1-34100 Trieste, Italy)
2-6 FEBRUARY
Quantum Electronics t • CONTINUING
EDUCATION IN ENGINEERING AND THE
COLLEGE OF ENGINEERING, UNIV. OF
CALIF., BERKELEY • San Francisco
(Continuing Education in Engineering,
Univ. of Calif. Extension, 2223 Fulton
St., Berkeley, Calif. 94720)
Topics: Q-switching and mode-locking oflasers, self-focusing and defocusing of laserbeams, laser deflection and modulation, gen-eration and propagation of ultrashort opticalpulses, far-infrared sources, nonlinear opticsand high-power lasers. Participants will in-clude J. Whinnery, S. E. Schwarz, R. Chiao,A. J. DeMaria, T. K. Gustafson, G. C. Pimen-tel and Y. R. Shen.
15-21 FEBRUARY
Gas Kinetics • DEPARTMENT OF CHEM-
ISTRY, UNIV. OF CALIF., IRVINE D Lake
Arrowhead, Calif. (D. L. Bunker, Dept.
of Chem., Univ. of Calif., Irvine, Calif.
92664)
Topics: Potential surfaces, cross sections
and rate constants, scattering theory, kinetic
spectroscopy, quenching reactivity of excited
molecular states, lasers, hot-atom chemistry,
ion-molecule reactions, molecular-beam stud-
ies of inelastic and reactive processes, tra-
jectory studies, chemical activation, energy
transfer and unimolecular reactions.
23 FEBRUARY-7 MARCH
New Developments in High-Energy
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106 . DECEMBER 1969 . PHYSICS TODAY
Are you a British scientist or engineer thinking of returning to work in research and
development in Britain ? You may find what you are looking for, without going home
first in the Scientific Civil Service, the United Kingdom Atomic Energy Authority, or
the Central Electricity Generating Board. And your family fares back to this job
may be paid.
The work is mainly applied, in that it has a practical end in view, but fundamental
research is frequently involved. There are likely to be openings in most branches of
the physical and engineering sciences, especially in mathematics and computing.
Career Appointments
are on offer mainly at the starting point of Scientific Officer and Senior Scientific
Officer (or their equivalents), for which candidates will most likely be between the
ages of 23 and 31.
Research Fellowships
are prestige awards, offered to scientists and engineers of exceptional ability, usually
for 2-3 years. Fellowships may lead to career appointments.
A Selection Board composed of practising research scientists and engineers,
drawn from R. & D. establishments in the three organisations, will be in:—
CANADA (OTTAWA):
interviews beginning in mid-January
1970. Last Day for the receipt of
applications: 5th December 1969.
If you are in Canada
please write to:
Mr. H. G. Sturman,
Senior UKAEA Representative
in Canada,
P.O. Box No. 1245,
Deep River, Ontario.U.S.A. (NEW YORK AND
SAN FRANCISCO):
interviews beginning in mid-March
1 970. Last Day for the receipt of
applications: 9th January 19.70.
If you are in the U.S.A.
please write to:
Dr. J. M. Lock, Director,
United Kingdom
Scientific Mission,
British Embassy,
Washington, DC. 20008.
Issued jointly by the Civil Service Commission, the U.K. Atomic Energy Authority
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SKOKIE ILLINOIS BOO7B
108 • DECEMBER 1969 • PHYSICS TODAY
ANNUAL INDEX
PHYSICS TODAY VOLUME 22 1969
KEY
BR
E
ER
L
MR
OBbook review
editorial
erratum
letter
meeting report
obituary
SUBJECT INDEX
ACCELERATORS
Argonne ZGS pulses. MAR 63
Canadian council weighs role on Batavia machine.
MAY 65
CERN-Serpukhov collaboration yields data on particle
yields. JUL 71
CERN storage rings in two years experimenters are
making plans. SEP 62
Electron cloud to produce highly stripped heavy ions.
MAY 58
First director of CERN 300-GeV accelerator. FEB 69
Giant scintillation counter is good for high energies.
MAY 58
Hermes II produces 150 000 amperes of 13-MeV
electrons. AUG 67
Indiana U builds 200-MeV sector-focused cyclotron.
JUL71
Intense MeV-electron beams and prospects for accel-
erators. JUN 59
Lamb-effect sources make better polarized ion beams.
JAN 67
LAMPF aims for 1972. users' group organizes. MAY
65
Matter meets antimatter in Akademgorodok. AUG 62
More electron rings LRL forms them. Dubna starts
extraction. APR 63
NAL plans for bubble chambers. JAN 64
Positron Beams. D E Yount. FEB 41
PPA proposes heavy-ion improvement program to
AEC. OCT59
Serpukhov data suggest asymptopia may be further
away than ever. OCT 57
Three Decades of Fast-Neutron Experiments. H H
Barschall. AUG 54
Tunnel dug for Stanford superconducting hnac. MAR
63
US groups may be able to work at Serpukhov. APR
79
Weisskopf panel reports on high-energy physics in
next decade. OCT 65
ACOUSTICS Acoustics. L L Beranek. NOV 47
Analysis of Musical-Instrument Tones. J. Risset and
M V. Mathews. FEB 23
Sound laboratory completed at NBS. JAN 87
Ultrasonic microscope may be more sensitive, nonde-
structive. AUG 66
AMERICAN ASSOCIATION OF
PHYSICS TEACHERS
AAPT should endorse AAUP statement on academic
freedom and tenure. (L) SEP 9
Book on demonstration experiments sponsored. MAY
66
Chicago meeting (see APS)
Humanists dissect morals, models, graphics of
ompttmonfor short teach.ng films. AUG 109Resource letters and reprint booklets, JAN 87
Winners of apparatus competition announced. APR 91
AMERICAN ASTRONOMICAL SOCIETY Fredrick
named AAS secretary; McVittie resigns. SEP 7 1
AMERICAN INSTITUTE OF PHYSICS
AIP discriminatory policies. (L) JUN 9
AIP in 1968 Expansion and Experimentation. J P.
Wiley Jr. JUN 43
AIP. societies seek ways to cut publishing costs. MAY
69
AIP and society journals in microfilm. DEC 65
APS. SPS councils. C0MPAS meet in Washington.
JUN 68
Corporate associates to discuss federal support. SEP
71
Feinberg, Commoner. Crane probe scientists- social
role. APR 80
Fewer prospective employers use AIP placement ser-
vice.JUL77
First three-day science-writers seminar held. FEB 7 1
Guide to undergraduate departments. DEC 63
History conference probes role of nuclear theorists,
JUL75
Information division asks $4 2 million over three years.
JUL74
Journal prices raised. NOV 67
Metzner new assistant director of publications. DEC
63
National Register. (L) JUL 19
The National Register Looks at Manpower. S Bansch
and T Johnides, OCT 48. (L) DEC 1 5
New Information Program for AIP. A Herschman. F.
Alt. H W. Koch. DEC 26
New policies for unsupported publishing. FEB 69. (L)
JUL 15
New York Times writer wins AIP-US Steel writing
award. MAY 67
1 969 governing board. MAY 67
North American faculty directory. NOV 67
Page charges. (L) MAR 15
Placement figures show tight physics job market. APR
83
References to unpublished works, (L) OCT 1 1
SATC0M challenges societies to improve their publi-
cations. AUG 75
Science writing award to Thorne. OCT 105
Study offers solutions to school science problems.
JUL74
Tax reform bill may limit scientific society activities.
OCT 67
An unemployment crisis. (L) FEB 13. JUL 9. AUG 9.
DEC 1 1. 13
US Steel. AIP add category to science-writing awards.
FEB 75
Visiting privileges for Americans and Australians. SEP
72
Work Complex Study. (L) MAY 1 3AMERICAN PHYSICAL SOCIETY
AAPT-APS Meeting Returns to New York. J P Wiley
Jr. JAN 57
Activist groups seek ways to bring science into poli-
tics. APR 79
Activists take ABM fight to Congress. White House,
JUN 69
Antiballistic missile system. APS in Washington dis-
cusses. JUL 99
APS scope of concern should include physicists. (L)
MAY 1 1
APS should endorse AAUP statement on academic
freedom and tenure. (L) SEP 9
APS. SPS Councils. C0MPAS meet in Washington.
JUN 68
Dissidents force a vote on 1970 Chicago meeting,
APR 85. (ER) SEP72
Exhibit, general sessions highlight Washington meet-
ing. APR 87
Group flights to Europe and Japan, DEC 65
New division will focus on cosmic radiations. JUN 73
New policies for unsupported publishing. FEB 69; (L)
JUL 15
1970 Chicago meeting (L) FEB 11. 69. MAR 65. (L)
APR 9. 79. (L) MAY 9. 65. (L) JUN 11. AUG 17.
SEP 15. OCT 1 1
1 932 APS meeting. APR 1 23
Petition for division on the problems of physics and
society. (L) JUN 15
Physical Review Letters cuts page budget 10%. MAY
71
SLAC and APS division publish new preprint list. JUN
73
Suggestions for a more relevant society. (L) JUN 1 5
Travel arrangements for meetings. (L) APR 1 1
An unemployment crisis. (L) FEB 13. JUL 9. AUG 9,
DEC 11.13
What Happened to My Paper?. S A Goudsmit. MAY
23. (L) AUG 15
AMERICAN PHYSICISTS ASSOCIATION Inception.
NOV 65
ASTRONOMY. SPACE AND GEOPHYSICS
Apollo 1 1 success brings astronomy down to earth.
SEP 65
Astronomy, progress summarized for New York State
section. JUN 89
British and Dutch build new radio telescopes. APR 63
Condon study rebuts UFOs. critics offer own version.
MAR 67
Continental Drift. D L Turcotte and E R Oxburgh.
APR 30. (L) AUG 11
Cornell facility to probe planets. MAR 63
Crab pulsar optically identified, other pulsars show
slowdown. MAR 60
Dicke panel says US lags in radio-astronomy con-
struction, DEC 56
PHYSICS TODAY • DECEMBER 1969 • 109
Doherty Foundation gift to Lamont Observatory, MAR
73
48-University consortium to coordinate space re-
search. OCT 69
Hot water source in space may act as maser, APR 63
Information from Deep-Space Tracking, P. M. Muller
and W L. Sjogren, JUL 46
Infrared background radiation higher than expected,
FEB 67
International space project will study solar processes.
NOV 59
Interstellar isotopic abundance of carbon agrees with
earth's, FEB 67
Interstellar medium contains ammonia, FEB 67
Interstellar medium has biological preservative. MAY
58
JILA has fellowships for 1970-1, DEC 63
Long-Baseline Interferometry, B F. Burke. JUL 54; (L)
NOV 1 1
Lunar atmosphere, (L) DEC 1 3
Mars Manners to study surface and atmosphere, MAY
59
NAS names Rubey director of lunar science institute,
JAN 85
Near-earth study program proposed for 1968-75.
MAR 61
Orbiting telescopes scan ultraviolet wavelengths. MAR
63
The Origin of the Elements, D. D. Clayton. MAY 28,
(L) OCT 15
Polar cap may have geoelectric field. APR 64
Presidential panel proposes new ocean resources
agency, MAR 73
Program is proposed for outer-planet trips in 70s.
OCT 59
Re Pulsars. S P. Maran and A G. W Cameron. (L)
JAN 1 1
Quasistel/ar Objects and Seyfert Galaxies, S. A Col-
gate. JAN 27
Sensors in the Deep Sea, D R. Caldwell. F. E Snod-
grass, and M H Wimbush, JUL 34
Short-period pulsar slows. FEB 67
Solar telescope and 0S0-6 observing the sun. DEC
56
Theorists offer explanation for pulsar speeding up. JUL
68
Vela pulsar slows, speeds up. and then slows down
again. JUN 63
A Visit to Arecibo finds a telescope seeking improve-
ment. APR 65
Weber reports 1660-Hz gravitational waves from
outer space, AUG 61
Where Do We Go From Here?. A. E. Ruark. SEP 25
X Rays from Crab have period of radio signals, JUL 68
ATOMIC ENERGY COMMISSION Six films available
from free-loan libraries. NOV 67
AWARDS
AEC honors Joliot, Halban, Kowarski. Perrin. FEB 1 13;
Anderson Award to Mills. SEP 115, ANS Special
Award to Ward. AUG 101; Arctowski Medal to Par-
ker and Wild. JUL 1 12; ASA Honors Waterfall. JUN
99; Atoms-for-Peace Award to Eisenhower, JUN
99. to seven scientists. JUL 111; Bingham Medal
to Encksen. JAN 137; Bonner Prize to Breit. APR
127; Buckley Solid-State Prize to Hopfield and
Thomas. MAR 115. Coblentz Award to Zerbi. MAR
117; Day Medal to Vine, MAR 117, Dunn Medal to
Lax, MAY 107, Franklin Institute Honors Theuerer,
Duwez, Berger, JAN 135; Gold Medal to Hunt.
MAY 107. Gray Medal to Spencer. AUG 101;
Guggenheim Award to Svestka, JAN 135. Harris
Medal to Radkowsky. MAY 109; High Polymer
Prize to Bunn. MAR 115; Hoover Medal to Seitz,
JAN 135. IPPS gives six prizes. MAY 109, Ives
Medal to Rank, AUG 101; Jansky Lectureship to
Shklovsky. JAN 137. Langmuir Prize to Slichter,
MAR 1 15; Lark-Horovitz prize to Spears. AUG 101,
Lawrence Award to Chew, Cromer, Hayes, Gelbard,
Nuckolls. JUN 99; Maryland Young Scientist Award
to Pugh. MAR 117, Materials-Technology Award to
Strnat. Olson. Hoffer, FEB 113; Michigan State
creates Osgood Award. MAY 109, Millikan Award
to Fowler. SEP 117; Mo. Science Educator Award
to Hilton. FEB 111; Montana State creates Johnson
Award. JAN 137, Navy Achievement Award to
Scanlon, JAN 139. Navy Civilian Service Award to
Karle, FEB 111; New Scientist Award to Joseph-
son. SEP 117; Nominations open for OSA Adolph
Lomb Medal, JAN 93; NYU Alumni Award to Pri-
makoff. JUN 99; Oppenheimer Prize to Dirac, APR
127; Penrose Medal to Wilson, MAR 117, Planck
Medal to Dyson. JUL 112. Quantum Molecular
Award to Levine, JAN 139, Research Corporation
Award to Gell-Mann, MAR 115, Richardson Medal
to Cary. MAR 117; Rosa Award to Memke. APR
128; Rumford Medal to Gabor, MAR 117; Salam
sets up fund, DEC 63; Science writing award to
Thorne, OCT 105; Spectroscopy Society creates
Burns Award. JAN 137. Stratton Award to Lide.
APR 128; Tillyer Medal to Riggs, MAR 117; Trent-
Crede Award to Vigness. MAY 107; US StandardsInstitute honor Wolfe. FEB 111; Vetlesen Prize to
Birch and Bullard. FEB 111; Warner Prize to Sar-
gent, APR 127
BIOPHYSICS Interstellar medium has biological pre-
servative, MAY 58
Re Magnetic Fields in Biology. A Kolm, (L) MAR 1 5
BUBBLE CHAMBERS CERN ultrasonic bubble cham-
ber. MAR 61
CHEMICAL PHYSICS Spectra suggest anomalous
water is a stable polymer of H2O, SEP 61
CRYOGENICS
Advances in Superconductivity, J Bardeen, OCT 40
Is there a new mechanism for superconductivity?. JAN
64
Josephson effect permits new look at fundamental
constants, AUG 66
Solid staters study fluctuations in superconductors,
MAY 57
Superconductivity, new materials and more applica-
tions. MAR 101. (L)SEP 1 1
CRYSTALLOGRAPHY: Crystal acts like a two-
dimensional antiferromagnet, JUL 69
Crystallographers elect Guinier as new president, OCT
67
Crystallographers Offer Meetings Within Meetings, W
C Hamilton. AUG 23
Crystallography. International Congress of. JUL 1 1 7
Crystals, S C. Abrahams, contributions by C. S. Barrett
and D Harker, AUG 30; (L) OCT 9
30 Years of Small-Angle X-Ray Scattering. A Guinier.
NOV 25
EDITORIALS
Better Teaching with Better Problems and Exams
(Guest Editorial). MAR 134, (L) OCT 1 1
D Phil, or D Phys.?. JAN 1 54, (L) APR 9. 1 1
Re In Politics. How Should We Do Our Thing?, (L)
JAN 9. FEB 1 1. MAR 1 1
Is Your Research Moral? (Guest Editorial) DEC 1 18
The Practical Need for Beauty. APR 144; (L) JUL 17
Reflections on the Moon, SEP 1 28
We Need an Informed Conscience. AUG 1 14
What Shall We Do for the Commission on College
Physics? (Guest Editorial), NOV 120
Who Finds the Job?. JUN 112; (L) SEP 9. NOV 9.11.
DEC 9
Who Pays the Bills?. MAY 124
EDUCATION
Academic freedom and tenure. (L) SEP 9
AEC will give used nuclear-studies equipment to
schools. SEP 72
All-girl physics course makes converts in Illinois. APR
80
Beams retires at Virginia. AUG 97
College Physics Commission reports results in
1966-68. JUN 71
Computers in Physics Instruction, G Schwarz, 0. M.
Kromhout, and S. Edwards. SEP 40
Decreasing physics enrollments, (L) MAR 1 3. JUL 9
Draft affects 12.6% of physics graduate students. SEP
71
Federally supported research in universities, AIP topic,
AUG 69
Foreign graduate candidates evaluated by Dart. Mor-
avcsik. DEC 63
Foreign scientists available under Fulbnght-Hays Act
JUN71
48-university consortium to coordinate space re-
search, OCT 69
The Graduate Student. Introduction, MAR 23
How Does He See Himself?: Cornell University. A
R Evans, MAR 25; University of Florida. J. and
M. Taube, MAR 26; Howard University. M. J.
Smith. MAR 28. University of Illinois. S. C Fain
Jr, MAR 29, University of New Mexico, B. D
Hansen III. MAR 30, City College, CUNY.J. Slev-
in, MAR 31; University of Pennsylvania, J. R
Powers, MAR 32; Northwestern University. J.
Oberteuffer, MAR 33.
Why Has He Changed?. J. C. Slater, MAR 35 (L)
JUL9
How Does He Fare in Britain?, C. C Butler. MAR
39;
What Does He Study?, A A Strassenburg and M.
T. Llano. MAR 45; (L) JUL9
Where Does He Come From? Where Does He Go?,
S D. Ellis. MAR 53; (L) AUG 9
IBO international physics syllabus and exam. JAN 85
To Joseph Henry, J A Wheeler. FEB 1 1 1
Lycommg College enjoys high-school physics day
FEB 71
Manpower studies show physics leveling off. SEP 72
Maryland plans to tram 23 Negro college teachers,
MAY 73
NSF grants to improve science teaching. FEB 70NSF physics section discusses support policies and
prospects, SEP 73
The physics dropout what turns him off?, OCT 67
RE: Physics and the Nation in a Crystal Ball, L M.
Branscomb. (L) JAN 9
Plodders are the backbone. (L) FEB 9
The Postdoctoral Research Associate-Instructor, A. E.
S Green. JUN 23
President underlines support for science. MAR 65
PSNS finds $60 000 surplus. JUL 79
Research stoppage focuses on national science goals,
APR81
Romanian physics education, (L) MAY 13
Specialized irrelevancy, (L) AUG 9
Student director becomes full-time visiting scientist.
JUL73
Study offers solutions to school science problems,
JUL74
Universities, Congress study institutional grant propos-
als. MAY 67
University scientists discuss government and science,
MAR 65
ELECTRICITY Electricity and Rain, J. D. Sartor, AUG
45
Grids instead of walls for electrogasdynamic genera-
tors, MAR 60
ELECTRONS. ATOMS AND MOLECULES Atomic
Physicists meet for Arnold Sommerfeld Centennial,
FEB 99
Atoms. V. W Hughes. FEB 33
Polarized beams show promise for atomic collision ex-
periments, NOV 87
Re Spectroscopy. Quantum Chemistry and Molecular
Physics, R. S. Mulliken. (L) JAN 9
ELEMENTARY PARTICLES AND FIELDS
Cascade particle completes octet, (L) APR 13
Check of T invariance in electromagnetic interaction.
APR 64
CP-violating decay of long-lived K meson, NOV 56
Elementary Particles, G Veneziano, SEP 31
Form Factors of Elementary Particles, R. Wilson, JAN
47
Fundamental particles at high energy, APR 11 5
Giant scintillation counter is good for high energies,
MAY 58
High-energy physics, theory falls behind experiments,
JAN 1 19
Is this a quark I see before me?, OCT 55
More About Tachyons. 0. M. Bilaniuk. S. L. Brown. B.
DeWitt, W. A. Newcomb. M. Sachs. E. C. G Sudar-
shan. S Yoshikawa. DEC 47
Nucleon-Nucleon Scattering, M. Mac Gregor. DEC 21
Particle physicists exchange facts, models and specu-
lations, NOV 93
Particles Beyond the Light Barrier, 0. Bilaniuk and E.
C G Sudarshan. MAY 43; (L) OCT 9
Regge-cut theory yields encouraging results, SEP 101
Serpukhov data suggest asymptopia may be further
away than ever, OCT 57
Symmetries and quarks raise more questions than so-
lutions, OCT 93
Veneziano representation excites strong-interaction
theorists, MAR 59
What Is the Point of So-Called "Axiomatic Field Theo-
ry"?. A S. Wightman. SEP 53
EUROPEAN PHYSICAL SOCIETY Announces division
chairmen, DEC 65
Growing society has 31 000 members, NOV 63
Members gather in Florence. JUN 67
FEDERATION OF AMERICAN SCIENTISTS: Sakharov
essay welcomed; Hollander to edit response. APR
80 '
FLUIDS and PLASMAS Artsimovieh Talks about Con-
trolled-Fusion Research, J. L. Tuck and G. B. Lub-
kin. JUN 54
Bernoulli theorem confirmed, MAR 63
Cold octopole and hot Tokomak show long confine-
ment times, DEC 55
Committee recommends magnetohydrodynamic study,
NOV 67
Hundred-joule lasers are producing high-temperature
plasmas. NOV 55
Plasmas, H. Grad. DEC 34
Tokomak proposals endorsed by AEC. AUG 69
US fusion experimenters want to try Tokomaks now,
JUL67
GOVERNMENT
Activists take ABM fight to Congress. White House.
JUN 69
AEC. Puerto Rico to study nuclear-energy center sites.
FEB 75
AEC will give used nuclear-studies equipment to
schools. SEP 72
110 • DECEMBER 1969 • PHYSICS TODAY
Antiballistic missile system. APS in Washington dis- Transition-radiation detector for high energy. NOV 59
cusses. JUL 3si
Batavia
mmittee °UtlmeS agenda f°r 91st Cong-Daddano seeks comment on unifying science activi-
ties. AUG 75
Dedication of solid-state building at Argonne JUL 73
Draft Draft Affects 12 6% of physics graduate stu-
dents. SEP 71. Nearly half of all graduate students
eligible for draft. MAR 65
DuBridge will be science adviser to Nixon, JAN 85
Federal Aid Budget cuts hurt many-but not as badly
as feared. JUN 67. Decline in federal support of re-
search documented. APR 93. Federally supported
research in universities. AIP topic. AUG 69. John-
son budget holds science to a "cost-of-living" raise.
MAR 65. LAMPF aims for 1972 users'group orga-
nizes. MAY 65. Nixon releases another $10 million
to NSF, MAR 65. Nixon's April budget revisions.
MAY 65. NSF announces plans for 1970 expendi-
ture limits. OCT 67. NSF appropriation approved.
AUG 69. NSF physics section discusses support
policies and prospects. SEP 73; President under-
lines support for science. MAR 65. Tax reform bill
may limit scientific • society activities. OCT 67,
Toward regionally relevant research. (L) FEB 9. Uni-
versities. Congress study institutional grant propos-
als. MAY 67; Weisskopf panel reports on high-
energy physics in next decade. OCT 65
House unit proposes steps to improve federal labs.
JAN 91
How the President gets his science advice A visit to
OST. AUG 70
IAEA seeks better ways to detect nuclear-material di-
version. AUG 69
National science board studies future goals for
upgraded NSF. FEB 69
Needs for a National Policy, E Q Daddano OCT 33
New leaders will overhaul US science policy for
1970s. FEB 73
Nixon intends to nominate Heffner deputy director of
OST. JUL 74
Nixon names task force to review science policy. DEC
65
Nuclear Diversion Safeguards 1 The IAEA Program.
B W Sharpe. NOV 33
Nuclear Diversion Safeguards 2 The US Program, W
A Higmbotham. NOV 40
Political storm breaks over appointment of NSF direc-
tor. JUN 67
Political upheaval causes cancellation of 1 1th Latin
American School of Physics. JUL 74
Presidential panel proposes new ocean resources
agency. MAR 73
Research stoppage focuses on national science goals.
APR 81
SATCOM challenges societies to improve their publi-
cations. AUG 75
Tokomak proposals endorsed by AEC. AUG 69
Two presidential science task forces will help Du-
Bridge. JAN 85
University scientists discuss government and science.
MAR 65
US ratification of nonprohferation treaty. APR 79
HISTORY AND PHILOSOPHY History conference
probes role of nuclear theorists. JUL 75
HUMOR A Philistine Asks for Equal Time. Sister J
Dilton. MAY 38
INDUSTRY Sociosystems laboratory explores urban
problems. JUN 73
INFORMATION AIP information division asks $4 2
million over three years. JUL 74
Evaluating published research results. (L) APR 1 5
Mmireview suggested. (L) OCT 13
New Information Program for AIP. A Herschman. F
Alt. H W Koch. DEC 26
SATCOM challenges societies to improve their publi-
cations. AUG 75
INSTITUTE OF PHYSICS AND THE PHYSICAL SOCI-
ETY Members approve royal-charter application.
FEB 70
New Council Officers. OCT 69
INSTRUMENTATION
Biomedical Applications of Holography. E J Feleppa.
JUL 25
Giant sc.nt.llat.on counter is good for high energies.
Information from Deep-Space Tracking. P M Muller
anH W L Sioaren. JUL 46
LongBasel.ne fnterferometry. B. F. Burke. JUL 54. (L)
Quan,um1elec,ron,cs conference in Japan. 1970. JUN
SenVors in the Deep Sea. 0 R Caldwell. F, E. Snod-
grass. andM H. Wimbush. JUL 34INTERNATIONAL ATOMIC ENERGY AGENCY IAEA
seeks better ways to detect nuclear material diver-
sion. AUG 69
International nuclear information system. OCT 71
Liechtenstein. Niger, and Zambia join. JAN 93
Nuclear Diversion Safeguards 1. The IAEA Program
B. W Sharpe. NOV 33
IUPAP The International Union of Pure and Applied
Physics. L Kerwin. MAY 53; (L) SEP 1 7
INTERVIEWS Edward Condon. MAR 66, Jesse W
Beams. AUG 97. Lev Artsimovich. JUN 54. Wayne
Gruner. et al. SEP 73
MAGNETISM Crystal acts like a two-dimensional an-
tiferromagnet. JUL 69
German National Magnet Lab will have 5-MW capaci-
ty. SEP 65
World's largest superconducting magnet. JAN 64
MANPOWER
Budget cuts hurt many—but not as badly as feared,
JUN 67
Draft affects 12 6% of physics graduate students. SEP
71
Fewer prospective employers use AIP placement ser-
vice. JUL 77
Median salary of US scientists in 1 968. MAR 7 1
The National Register -Looks at Manpower. S Barisch
and T Johnides, OCT 48. (L) DEC 1 5
Nearly half of all graduate students eligible for draft.
MAR 65
Placement figures show tight physics job market. APR
83
Relationship between academic training and job re-
quirements. (L) MAY 1 1
Studies show physics leveling off, SEP 72
An unemployment crisis. (L) FEB 13. JUL 9. AUG 9.
OCT 17. DEC 11. 13
MEETINGS
Amorphous semiconductors stimulate fundamental
and applied research. OCT 97
Astronomy, progress summarized for New York State
Section, JUN 89
Atomic physicists meet for Arnold Sommerfeld Cen-
tennial. FEB 99
Exact statistical mechanics at Irvine. APR 1 1 7
Fundamental particles at high energy. APR 1 1 5
Gordon Research Conferences. APR 133
High-energy physics, theory falls behind experiments.
JAN 1 19
Normal-state electron tunneling. DEC 89
Particle physicists exchange facts, models and specu-
lations. NOV 93
Polarized beams show promise for atomic collision ex-
periments. NOV 87
Regge-cut theory yields encouraging results. SEP 1 01
Semiconductor instabilities, interest grows in. AUG 89
Superconductivity, new materials and more applica-
tions. MAR 101 . (L) SEP 1 1
Symmetries and quarks raise more questions than so-
lutions. OCT 93
Thin-Film studies discussed in Boston, progress in.
AUG 91
NATIONAL ACADEMY OF SCIENCES
Academies of Science offer exchange visits to Ameri-
cans. SEP 71
Bromley heads physics survey committee. SEP 71
Philip Handler elected president. FEB 69
Report suggests regional problem-solving centers.
OCT 69
Rubey named director of Lunar Science Institute. JAN
85
NATIONAL BUREAU OF STANDARDS Astm honored
at dinner, NOV 103
Branscomb named head. JUL 74
Sound laboratory completed. JAN 87
NATIONAL SCIENCE FOUNDATION
Biologist named director. JUL 77
Dicke panel says US lags in radio-astronomy con-
struction. DEC 56
Grants to improve science teaching. FEB 70
Institutional grants will be computed differently. FEB
69
Median salary of US scientists in 1968. MAR 71
National science board studies future goals for
upgraded NSF. FEB 69
Needs for a National Policy. E Q Daddano. OCT 33
Nixon releases another $10 million to NSF, MAR 65
NSF appropriation approved. AUG 69
Physics section discusses support policies and pros-
pects. SEP 73
Plans for 1 970 expenditure limits. OCT 67
Political storm breaks over appointment of NSF direc-
tor. JUN 67SATCOM challenges societies to improve their publi-
cations. AUG 75
NUCLEAR RESEARCH
AEC. Puerto Rico to study nuclear-energy center sites
FEB 75
Berkeley group reports discovery of element 104, JUL
69
Electron cloud to produce highly stripped heavy ions,
MAY 58
History conference probes role of nuclear theorists,
JUL75
IAEA seeks better ways to detect nuclear-material di-
version. AUG 69
Isobanc Analog Resonances. W R. Coker and C F.
Moore. APR 53
Josephson effect permits new look at fundamental
constants, AUG 66
K-Mesic atoms indicate a nuclear neutron skin, OCT
57
Lamb-effect of sources make better polarized ion
beams. JAN 67
Lifetime of compound nucleus is measured by crystal
blocking. JUL 67
Re More Intense Thermal-Neutron Beams. We Need.
R M Brugger. (L) JUN 17
New insight is offered into the fission process, FEB
64. (L) JUN 9
Nuclear Diversion Safeguards 1 The IAEA Program,
B W Sharpe. NOV 33
Nuclear Diversion Safeguards 2 The US Program. W.
A Higmbotham. NOV 40
Nuclear Models. D R. Inglis. JUN 29
Nucleon-Nucleon Scattering, M Mac Gregor. DEC 21
Oak Ridge uses U233 as reactor fuel, MAR 63
The Origin of the Elements. D D Clayton. MAY 28;
(L)OCT 15
Polarized targets used to study spin effects, APR 64
Search for stable elements heavier than uranium, FEB
63
Three Decades of Fast-Neutron Experiments. H H.
Barschall, AUG 54
Tokomak proposals endorsed by AEC. AUG 69
Ultracold neutrons may redefine electnc-dipole-
moment value. NOV 56
US fusion experimenters want to try Tokomaks now,
JUL67
Variable moment of inertia for even-even nuclei. MAR
61
OBITUARIES
Carl E Adams. JAN 139. John G. Albright. MAY 1 13;
Sister Mary John Allard. APR 129; Leslie R. An-
ders. JUN 101. Gladys A Anslow. JUL 1 12. Walter
H Barkas. AUG 103, Arthur A Bless, JUN 100;
Frank P. Bowden. FEB 113. Janet H Clark. JUL
112. Amos deShalit. DEC 99. Warren DeSorbo.
APR 128; Ray L Edwards, AUG 103; Donald W.
Engelkemeir. SEP 117, Gordon Francis. MAR 119,
George Glockler. MAY 109, Nicholas Golovin. JUN
100, James H Harrold. JUL 1 12. Harvey C Hayes.
JAN 139. Harry H Hess. NOV 107. Egon A Hiede-
mann, MAY 111, Else Holm, JUN 101; Hilde Kall-
mann-BijI, MAR 119. Gunnar Kallen. APR 129;
Richard W King. OCT 105. Aleksandr A Lebedev.
JUL 112. Frank Matossi. APR 129. Alexander B.
McLay. JAN 139; Richard W Michie. AUG 103.
Raymond Morgan, MAY 113. W Adair Morrison,
MAY 111, George M Murphy, MAR 119; Richard
G. Nuckolls, MAY 111. Richard S Perkin. JUL 112,
Cecil F Powell. NOV 107. Fritz Reiche, MAR 119.
Jerzy Sawicki. FEB 113, Otto Stern. OCT 103; Ar-
thur Van Zee, JUL 1 12. Libor J Velinsky. FEB 1 13,
Kenichi Watanabe. NOV 107
OPTICAL SOCIETY OF AMERICA Qumn fills new
post of executive director, SEP 7 1
OPTICS
Biomedical Applications of Holography. E J Feleppa
JUL25
Coherent optics and holography, more applications for
JUL103
CW chemical laser with external source. DEC 55
Hundred-joule lasers are producing high-temperature
plasmas. NOV 55
Nonlinear Optics. J A Giordmaine. JAN 39
Penn. now has state registration of laser systems JUL
79
Sandia operates picosecond laser at 50-joule output
JUN 60
Ultrasonic microscope may be more sensitive, nonde-
structive. AUG 66
PHIMSY
American physicist on a coin. JUL 23, Change units
to solve problems. NOV 19; Chicago Tribune
speaks. FEB 21; A console for theater sound. MAY
19; The cost of calories. APR 19; D'Abro the
search goes on. JAN 17; Decimal angle units. MAY
PHYSICS TODAY • DECEMBER 1969 • 111
19; Decimal time is here. FEB 19; (L) MAY 17; De-
tective work with infrared, AUG 19, Does excite-
ment make you mean?, OCT 19; The earth is still
flat. MAR 19; The Fairbank Anti-Murphy Law, JAN
15; For everyone his own way, AUG 19. For the
man with no needs, JUN 19; The function is the
particle, AUG 21; Fusion power at last. APR 21,
Gamow gambols. FEB 19; The geography of parity.
APR 19, How about decimal time?. (L) FEB 19,
MAY 17. Integrated Circuit, A Mackay. OCT 19; 0
joyous need for jobs. OCT 19, Lasers in the kitchen,
FEB 21; To learn big, study small. APR 19, Lots of
Peltier devices. AUG 19; More about stamps, NOV
19. More fun with decimal places, AUG 19; More
physics philately, SEP 19; A name is a name is a
.... NOV 19; NBS metric wall chart. JUN 19,
Never trust anybody. JUN 21, New in nuclear
power, DEC 17; Nomenclature, nomenclature ....
NOV 19. Our German equivalent. JUN 19; Our
growing Gamow collection. JUL 21, A page is a
space is a page, JUN 19, Perchance to wake, MAR
19. Physicists can paint doors, DEC 17; Physicists
on coins, JAN 15; Physics has versatility, JUL 21;
Physics learns the hard sell, APR 19, Poems for
computers. NOV 21. A practical Peltier effect. FEB
19; Practical surface science. MAR 17; Pulsars in
poetry. FEB 19; Purcell on Dicke. FEB 19; Quarks
are up and down, APR 21; Rheology in poetry,
MAY 19, "Schlieren Effect," B Ahlborn, JUN 19,
Seen carbon 14 Lately7, MAR 17, Some attention
from outside. JUN 21, Some meetings are informa-
tive. JAN 15; "Sonnet on Maxwell's Equations," R
E. Swing, APR 21; "The Special Theory of Relativi-
ty," A Mackay. JAN 15; . . and stall on the ground.
MAR 19; That 1932 banquet picture. OCT 19. You
incredible witnesses, AUG 19; We take to the
air..., MAR 17, Weston Batavia Fermi machine.
AUG 21
PHYSICS TODAY AIP discriminatory policies. (L) JUN
9
Lack of discussion of the moral problem of working in
the national-security field. (L) MAR 1 1
No mention of an unemployment crisis. (L) FEB 1 3
PUBLISHING NEWS Benjamin monograph series in
paperback, JAN 87; Journal of Statistical Physics,
MAR 71; Minireview suggested. (L) OCT 13. North
American faculty directory, NOV 67. Preprints in
Particles and Fields, JUN 73; References to unpub-
lished works. (L) OCT 11; La Rivista del Nuovo
Cimento, JUN 71, Science Citation Index, (L) APR
1 5. Science writing award to Thorne, OCT 1 05
QUANTUM THEORY Mind Your k's and q's to simpli-
fy solid-state theory. FEB 64
An Operational Interpretation of Nonrelativistic Quan-
tum Mechanics. W E. Lamb Jr. APR 23; (L) OCT 9
Quantum electronics conference in Japan, 1970, JUN
107
Where Do We Go From Here?, A E Ruark. SEP 25
RELATIVITY Space, Time and Elementary Interactions
in Relativity. M. Sachs, FEB 51. (L) SEP 13. NOV
1 1
Where Do We Go From Here?, A E Ruark. SEP 25
SCIENCE AND SOCIETY Needs for a National Policy,
E Q Daddano. OCT 33
The Privilege of Being a Physicist, V. F. Weisskopf.
AUG 39
Report suggests regional problem-solving centers,
OCT 69
Sociosystems Laboratory Explores Urban Problems.
JUN 73
SOCIETY OF PHYSICS STUDENTS APS. SPS coun-
cils. COMPAS meet in Washington. JUN 68
7500 members in 365 chapters. MAY 73
SPS gives 9 undergraduate cash awards for research.
APR 89
Student director becomes full-time visiting scientist,
JUL73
SOLIDS
Advances in Superconductivity, J Bardeen. OCT 40
Amorphous semiconductors stimulate fundamental
and applied research, OCT 97
Crystal acts like a two-dimensional antiferromagnet,
JUL 69
Electrons in Metals. W. A. Harrison, OCT 23
Glassy semiconductors show switching and memory
effects, JAN 63
Mind Your k's and q's to simplify solid-state theory.
FEB 64
Normal-state electron tunneling, DEC 89
Ovshinsky effect. (L) MAR 9, JUL 1 1
Semiconductor instabilities, interest grows in, AUG 89
Solid staters study fluctuations in superconductors,
MAY 57
States of Aggregation. K Mendelssohn, APR 46Thin-Film studies discussed in Boston, progress in,
AUG 91
30 years of Small-Angle X-Ray Scattering. A. Guinier,
NOV 25
A visit to the semiconductor institute in Leningrad,
JAN 69
SOVIET UNION: Matter meets antimatter in Akadem-
gorodok. AUG 62
A visit to the semiconductor Institute in Leningrad,
JAN 69
STATISTICAL MECHANICS Exact statistical mechan-
ics at Irvine, APR 1 17
UNITS Redefinition of temperature, volt and gravity
standards, JUL 71
Units for Logarithmic Scales. C S McCamy. APR 42,
(L) JUL 19
Visit to Bureau International des Poids et Mesures,
DEC 57
AUTHOR INDEX
Aarons. J . (BR) MAR 91. FEB 87
Abrahams, S C , contributions by C. S Barrett and D
Harker, Crystals. AUG 30
Adomian. G , (L) APR 1 1
Agassi, J, (BR) SEP 95
Ahlborn, B., Schlieren Effect. JUN 19
Alt. F L (see A Herschman)
Alvarez. L W.. (L) APR 9
Amdur. I . (BR) JAN 1 1 1
Ashcroft. N . (BR) NOV 71
Atherton. D L (see V L Newhouse); (L) SEP 11
Baily, N. A.. (BR) MAY 89. JUL 94, DEC 83
Bahse, P. L. (BR) FEB 83
Ballard, S. S . (BR) APR 101
Bardeen, J . Advances in Superconductivity. OCT 40
Bansch, S and T Johnides, The National Register
Looks at Manpower, OCT 48
Barnard. A C L. and E A. Sallin. (L) OCT 9
Barrett. C. S. (see S. C Abrahams), (L) OCT 9
Barschall, H H., Three Decades of Fast-Neutron Ex-
periments. AUG 54
Bates. L F , (L) SEP 17
Bederson. B , (MR) Polarized Beams Show Promise
For Atomic Collision Experiments, NOV 87
Bederson. B . V W Hughes and L. Spruch. (BR) JAN
1 13
Beranek, L L. Acoustics, NOV 47
Bergmann, P G , (BR) MAY 95, 85. MAR 93. JUN
83; (L) JUL 17
Bernstein. B.. On Presenting the 1968 Bmgham
Award to Jerald L Encksen. MAY 1 9
Bernstein, J, (BR) OCT 83
Bilaniuk. 0, (BR) NOV 70
Bilaniuk. 0 and E C G Sudarshan, Particles Beyond
the Light Barrier, MAY 43
Bilaniuk, 0 . S L Brown, B DeWitt, W A. Newcomb.
M Sachs, E C G Sudarshan. S Yoshikawa. More
About Tachyons, DEC 47
Bhtzstem. W . (L) FEB 19. MAY 17
Bolton. John G , (L) JAN 1 1
Borcherds. P. H . (L) OCT 11
Borowitz. S.. (BR) NOV 75
Bradner. H. (BR) SEP 81
Branscomb. L , (BR) NOV 69
Brown, L M., (MR) Fundamental Particles at High En-
ergy. APR 115
Brown, S L. (see 0 Bilaniuk)
Brown, W S . (L) MAR 15
Brush, S G . (L) JAN 9, JUL9
Burke, B F , Long-Baseline Interferometry, JUL 54
Butler, C C , The Graduate Student How Does He
Fare in Britain?. MAR 39
Caldwell. D R . F E Snodgrass, and M. H. Wimbush.
Sensors in the Deep Sea. JUL 34
Callen, E and J B Goodenough. (L) MAY 13
Callen. E R . B T. Chertok. D. S. Falk, H Jehle. H P.
Kelly. R. H Parmenter, H E Stanley. (L) SEP 1 5
Camermi, U , (OB) Cecil F Powell. NOV 107
Cameron, A G W (See S P. Maran)
Campbell, J A., (MR) Symmetries and Quarks Raise
More Questions than Solutions, OCT 93
Canute V, (BR) OCT 78
Candes.J N ,(L) FEB 13
Cass.T. R.(BR) NOV 79
Chang. H . (BR) MAY 87
Chang, Howard H C . (BR) JAN 103
Chertok. B T (see E R Callen)
Chiu, H ,(BR) SEP 85
Chopra. K. L. (L) MAR 9Clayton, D. D.. The Origin of the Elements. MAY 28
Coker. W. R and C. F. Moore, Isobaric Analog Reso-
nances. APR 53
Colgate, S. A., Quasistellar Objects and Seyfen
Galaxies. JAN 27
Collier. R. J. (BR) JUN 75
Collins, K. E.. (L) JUL 15
Cook, W. R.,(L) AUG 11
Cox, E. F., (L) AUG 19
Cox. M E. (BR) APR 97
Craig, P.. (BR) JAN 97
Cranberg, L. (L) APR 15
Crane. D . (BR)OCT87
Crane. H R., (E) Better Teaching with Better Problems
and Exams. MAR 134
Daddano. Emilio Q., (L) JAN 9; Needs for a National
Policy, OCT 33
Dahl, P. F.. G H. Morgan, and W. B Sampson. (L)
SEP 1 1
Dauber. P.. A. H. Rosenfeld, G. R. Lynch, and C. G.
Wohl.(L) APR 13
DeWitt, B (see 0. Bilaniuk)
Dillon, Sister J, A Philistine Asks for Equal Time. MAY
38
Drake. W R.. (L) SEP 15
Edwards. S (see G Schwarz)
Ellis, R. H.. Jr. (E) D. Phil, or D. Phys.?. JAN 154; (E)
The Practical Need for Beauty. APR 144; (E) Re-
flections on the Moon. SEP 128; (E) We Need an
Informed Conscience. AUG 114. (E) Who Finds
the Job?. JUN 112. (E) Who Pays the Bills?. MAY
124
Ellis, S. D . The Graduate Student Where Does He
Come From? Where Does He Go?. MAR 53; (L)
MAY 13. AUG 11
Ellis. W N.,(L) FEB 9
Elsasser. W M (see D R Rodenhuis)
Epstein, K. J. (L) SEP 13
Ermenc. J J.. (BR) FEB 80
Evans. A R . The Graduate Student: Cornell Universi-
ty, MAR 25
Fain. S C . Jr. The Graduate Student University of Illi-
nois, MAR 29
Falk. D S (see E. R Callen)
Faust. W. L. (L) FEB 13
Feleppa, E J. Biomedical Applications of Holography.
JUL25
Feshbach. H and V F Weisskopf. (OB) DEC 101
Fishbane. P. M and L. M. Simmons Jr. (MR) Regge-
Cut Theory Yields Encouraging Results, SEP 101
Fleming. L, (L) DEC 11
Francombe, M (see H. H. Wieder)
Franken. P.. (MR) APS in Washington Discusses Anti-
ball istic Missile System. JUL 99
Freeman. I M . (BR) MAR 87
Fnedlander. M W . (BR) MAY 95
Gammel. J L, (BR) JUN 85. JUL 93. AUG 78
Garvin, D . (L) OCT 13
Geballe. R.. R. A. Sawyer, and E. L. Jossem. (E) What
Shall We Do for the Commission on College Phys-
ics?. NOV 120
Gillis. J .(BR) APR 109. JUN 76. OCT 83
Giordmaine. J A , Nonlinear Optics, JAN 39; (BR)
JUN 75
Goodenough. J. B. (see E. Callen)
Goudsmit. S. A.. What Happened to My Paper?, MAY
23
Goudsmit. S. A. and G. L Trigg, (L) MAR 9, JUL 13
Grad, H . Plasmas. DEC 34
Green. A E S . The Postdoctoral Research Associate-
Instructor, JUN 23
Greenberg. D F . (L) MAY 1 1
Greenberg. W M., (L) NOV 1 1
Guinier, A.. 30 Years of Small-Angle X-Ray Scattering.
NOV 25
Hambourger, P. D..(L) JUL 13
Hamilton. W C, Crystallographers Offer Meetings
Within Meetings. AUG 23
Hansen, B. D.. Ill, The Graduate Student University of
New Mexico. MAR 30
Harker. D. (see S. C Abrahams)
Harrison. W A.. Electrons in Metals. OCT 23
Haskell, R. E.. (MR) More Applications for Coherent
Optics and Holography. JUL 103
Hasted. J B , (BR) JUN 77
Havens, W W.. Jr. (U APR 13. MAY 9
Hayward. E. (BR) APR 107
Herring. C. (L) FEB 1 1
Herschman, A., F. L. Alt and H. W. Koch. New Infor-
mation Program for AIP, DEC 26
Hersh. H. N. (L) MAY 11, JUN 15
Higgins, R J. (BR) OCT 79
Higinbotham, W. A., Nuclear Diversion Safeguards: 2.
The US Program. NOV 40
Hilsum. C, (MR) Interest Grows in Semiconductor In-
stabilities, AUG 89
112 • DECEMBER 1969 • PHYSICS TODAY
Hoen.g. S A . (L) DEC 13
Hoffman. J G , (BR) NOV 79
Hollander. J M . (BR) MAY 77
Hotz. D F . (BR) JUL83
Huber. Peter J. (L) JAN 9
Hudson. R. P.. (L) SEP 17
Hughes. V. W Atoms, FEB 33; (see B. Bederson)
Hunter. G.T.(L) JUL 19
Inglis. DR.. Nuclear Models, J U N 29
Jamieson.C P.. (L) OCT 17
Jean. M (see A. Salam)
Jehle. H. (see E. R. Callen)
Johnides. T. (see S Barisch)
Johnson. R C . (L) AUG 15
Jossem. E. L (see R Geballe)
Keefe. D.. (BR) APR 95
Kelley. J B.. (BR) JAN 109. MAR 89 APR 109 OCT
79. NOV 73
Kelly. H P. (see E. R Callen)
Kerwin. J D . (L) NOV 9
Kerwin. L. The International Union of Pure and Ap-
plied Physics. MAY 53
King. L D. P.. (L) JUN 17
Kirk. T B W. (L) APR 9
Koch. H W (see A Herschman)
Kolin. A. (L) MAR 15
Koonce. C. S . (BR) AUG 83
Kromhout. 0 M (see G Schwarz)
Lamb. W. E . Jr, An Operational Interpretation of Non-
relativistic Quantum Mechanics, APR 23
Land. C. E..(L) JUN 11
Lande. A.. (L) NOV 1 1
Lasky. D M . (L) SEP 19
Lebowitz. J. L. (BR) JUL91
Lecomte. J .(L)SEP 17
Levine. H. B.. (BR) JUN 77
Levme. R . (L) NOV 9
Levinger. J. S . (L) JUL 1 1
Lewis. H W.. (L) JUL 1 1
Lichten. W.. (BR) JUL 91
Lichtenberg. D. B . (MR) Particle Physicists Exchange
Facts. Models and Speculations. NOV 93. (BR)
OCT 74. (ER) DEC 15
Liebhafsky. H A.. (BR) APR 1 1 1
Lillich. R. B . Phimsy. Phimsy. who are you?. NOV 1 9
Lindsay, B . (BR) MAR 85
Lindsay. R. B . (BR) MAY 97. SEP 83. DEC 85
Llano. M T (see A A Strassenburg)
Lockeretz, W . (L) SEP 9
Lubkin. G B (see J L Tuck)
Lynch. G R (see P. Dauber)
Mac Gregor. M.. Nucleon-Nucleon Scattering. DEC 21
Mackay. A.. The Special Theory of Relativity. JAN 15;
Integrated Circuit. OCT 19; Poems for Computers.
NOV 21
Malamud. H.. (BR) JUL 83
Maran. S P. and A G W Cameron. (L) JAN 1 1
Marton. L . (BR) MAR 76. JUL 86. AUG 81. OCT 76
Mathews. M V (see J. Risset)
Mattis. D C . (BR) JUL 84. AUG 85
Maxwell. E. and B. B Schwartz. (L) JUN 1 5
Mayer. M. E . (MR) Exact Statistical Mechanics at Ir-
vine. APR 1 17
Mayer. W.G.(BR) NOV 81
McCamy. C S . Units for Logarithmic Scales. APR 42
McCarthy. M F . (BR) APR 105
Mclnturff. A. D . (L) SEP 1 1
Mendelssohn. K . States of Aggregation. APR 46
Mermin. N. D . (BR) JUL 89
Merzbacher. E . (BR) APR 101
Mielczarek. E. V. (BR) MAR 77
Miller. M M.(BR) DEC 79
Moore. C F (see W R Coker)
Moravcsik. Michael J . (MR) Theory Falls Behind Ex-
periments in High-Energy Physics. JAN 1 1 9
Morgan. G. H (see P. F Dahl)
Muller. P M. and W L Sjogren. Information from
Deep-Space Tracking. JUL 46
Muschhtz. E. E. Jr. (BR) DEC 81
Newcomb. W A. (see 0. Bilamuk)
Newhouse. V. L. and D L Atherton. (MR) New Mate-
rials and More Applications for Superconductivity.
MAR 101
Nimeroff. I. (BR) JUN 81. JUL 85
Oberteuffer. J , The Graduate Student Northwestern
University. MAR 33
O'Brien. B.J.(BR) OCT 73
O'Connell. J . (BR) MAR 85
Ohphant M L. (BR) MAR 75
Olsen. L O..IUJUN 11
Oppenheimer. F. (BR) FEB 77
Orear.J. (DMAY9
Osgood T.H..(BR)FEB83
Ovshmsky.S R (UMAR9
Oxburgh E R (see D. L. Turcotte)Parmenter. R H. (see E. R Callen)
Pasachoff. J M . Twinkle. Twinkle. 1969, FEB 19
Paul. W.. (MR) Amorphous Semiconductors Stimulate
Fundamental and Applied Research. OCT 97
Pauling. L.. (L) JUN 9
Pearson. J M . (L) DEC 1 1
Percival. I. C and H H Stroke. (MR) Atomic Physi-
cists Meet for Arnold Sommerfeld Centennial. FEB
99
Perl. M L.. (L) MAR 1 1
Perry. J. A, Jr. (L) MAY 17
Pewitt. E. G . (BR) OCT 75
Plumb. H H . (BR) SEP 89
Pollack. G L. (BR) FEB 91
Pompi. R L. (MR) Progress in Astronomy Summa-
rized for New York State Section, JUN 89
Powers. J. R.. The Graduate Student University of
Pennsylvania. MAR 32
Ptak. R (see R Stoner)
Rabi. I I . (OB) Otto Stern. OCT 103
Rau. R R and N P Samios. (L) APR 15
Rice, S A . (BR) FEB 83. JUL94. OCT 77
Richards. W B . (I) APR 1 1
Rieckhoff, K E . (L) APR 9
Rindler. W , (BR) FEB 87
Risset. J and M. V Mathews, Analysis of Musical-
Instrument Tones. FEB 23
Robinson. C F . (L) AUG 9
Rodenhuis. D R and W M Elsasser. (BR) JUL 81
Romam. J . (BR) SEP 89. OCT 77
Rosenfeld. A. H (see P. Dauber)
Rothberg, G . (BR) AUG 83. DEC 69
Rowell. J M. (MR) Normal-State Electron Tunneling.
DEC 89
Ruark. A E.. Where Do We Go From Here?. SEP 25
Rudin. R. A. (L) JUL 9
Ryan. Ciaran, (BR) JAN 101
Sachs, M.. Space. Time and Elementary Interactions in
Relativity, FEB 51. (L) SEP 13. NOV 13. (see 0
Bilamuk)
Sachs. R G.. (BR) AUG 77. (ER) SEP 17
Salam. A and M Jean. (OB) Jerzy Sawicki, FEB 1 1 3
Sallm. E A (see A C L Barnard)
Samios. N P (see R R Rau)
Sampson. W. B (see P. F Dahl)
Sartor. J D.. Electricity and Ram. AUG 45
Sawyer. R A. (see R Geballe)
Schaefer, J . (L) JUL 11
Schawlow. A L. (E) Is Your Research Moral?, DEC
118
Schillaci. M. E.(L) OCT 1 1
Schlegel, R . (BR) APR 103
Schwartz. B. B (see E Maxwell)
Schwartz. C. (L) JUN 9
Schwartzmann, M J and M D Turner. Practitioner's
Lament. NOV 19
Schwarz. G.. 0 M Kromhout. and S Edwards. Com-
puters in Physics Instruction. SEP 40
Scott. T A.(BR) FEB 85
Shankland. R. S. (BR) MAY 98. OCT 85. DEC 7 1
Sharpe. B W. Nuclear Diversion Safeguards The
IAEA Program. NOV 33
Siegman. A. E . (L) FEB 17
Silverman. P. J.. (BR) JUN 81
Silverman. S . (L) OCT 15
Silvert. W . (L) AUG 9
Simmons. L M . Jr (see P. M. Fishbane)
Simpson. J A., (BR) SEP 87
Singer. S. F. (BR) MAR 77
Singleton, J. H.. (BR) AUG 79
Sjogren. W L (see P. M Muller)
Sklar, L, (BR) AUG 79
Slabmski. V. J . (L) MAY 19
Slater. J. C, The Graduate Student Why Has He
Changed?. MAR 35
Slevin. J . The Graduate Student City College. CUNY.
MAR 31
Smith. M J . The Graduate Student Howard Universi-
ty. MAR 28
Smoluchowski. R , (BR) DEC 73
Snodgrass. F E (See D. R Caldwell)
Snow. J A . (BR) JUL82
Sposito. G , (BR) JAN 107. MAR 83. APR 107. MAY
82, JUL84. SEP91. DEC 77
Spruch. L (see B Bederson)
Stanley, H E (see E R Callen)
Stoecklem. J D . (L) MAR 13
Stoner. J 0 . (L) DEC 15
Stoner, R and R Ptak, (L) JUL 9
Strassenburg. A A. (L) FEB 15, and M T Llano. The
Graduate Student What Does He Study? MAR
45
Straumams. M E.(BR)JUN79
Street, R E (BR) JUN 83
Stroke. H H (see I C Percival)
Sudarshan. E C G. (see 0 Bilamuk)
Swing. R E . Sonnet on Maxwell's Equations. APR 21
falbot. L. (BR)SEP85
Tanenbaum. B S . (BR) NOV 77Taube. J and M. Taube. The Graduate Student Uni-
versity of Florida. MAR 26
Taylor. P L. (BR) DEC 73
Terrell. J.. (L) NOV 1 1
Thomas, K M . (L) SEP 9
Thun. R. E..(L) JUN 13
Trammell. G. T. (L) OCT 9
Trigg. G. L. (see S. A Goudsmit)
Tuck. J. L. and G. B. Lubkin, Artsimovich Talks about
Controlled-Fusion Research. JUN 54
Turcotte. D L and E R Oxburgh. Continental Drift.
APR 30; (L) AUG 13
Turner. M D (see M J Schwartzmann)
Valk. H S . (BR) FEB 80, MAR 93. JUL93
Van Vleck, J H . (BR) JUL 86
Veneziano, G . Elementary Particles. SEP 31
Weber. J . (BR) FEB 81
Weber. R L . (BR) JUL 84. AUG 80
Weinstock. H ,(L) JUN 13
Weisberg. L R (L) OCT 1 1
Weiss. G . (BR) JUL95
Weisskopf. V F . The Privilege of Being a Physicist,
AUG 39. (see H Feshbach)
Weissman. S . (BR) MAY 93
Wheeler. J A , To Joseph Henry. FEB 1 1 1
Wickman. H H . (BR) JAN 99. JUL 87
Wieder, H H and M Francombe. (MR) Progress in
Thin-Film Studies Discussed in Boston. AUG 91
Wightman. A S . What Is the Point of So-Called "Axi-
omatic Field Theory•"?. SEP 53
Wigner. E . (BR) MAY 91. (L) DEC 1 3
Wiley, J P.. Jr, AAPT-APS Meeting Returns to New
York. JAN 57, A IP in 1968 Expansion and Ex-
perimentation. JUN 43
Williams. 0 W , (BR) JAN 107
Wilson. F L , (BR) JAN 103. FEB 1 1, JUL86
Wilson. R., Form Factors of Elementary Particles. JAN
47
Wimbush. M H (see D. R Caldwell)
Wohl. C G. (see P. Dauber)
Wolf. E. (L) DEC 15
Wolf, W. (BR) JUL 87
Wolfe, H C. (L) AUG 15
Wolfe. J. G.. (L) FEB 9
Wolfenstem. L..(L) JUL 17
Wortis. M.(BR) NOV 75
Yaes, R J ,(L) AUG 17
Yoshikawa, S (see 0 Bilamuk)
Yoss. K , (BR) DEC 75
Yount. D E. Positron Beams. FEB 41
Zernik. W. (L) FEB 13
Zimmermann. R E . (L) JUL 19
Zipin, R. B.. (BR) JAN 99. FEB 85. AUG 81. OCT 76.
NOV 77
BOOKS REVIEWED INDEX
ACOUSTICS Ingard, K U (see P M Morse)
Morse, P. M and K U Ingard, Theoretical Acoustics
(R S Shankland). MAY 98
ASTRONOMY. SPACE, GEOPHYSICS
Burbidge. G and M Burbidge. Quasi-Stellar Objects
(H Chiu). SEP 85
Burbidge. M (see G Burbidge)
Caputo, M . The Gravity Field of the Earth: From Clas-
sical and Modern Methods (J Gilds). JUN 76
Eisele, J. A.. Astrodynamics, Rockets. Satellites and
Space Travel An Introduction to Space Science
(R L Weber). JUL 84
Flugge. S . ed , Encyclopedia of Physics. Vol.
49/2. Geophysics III. Part II (J Aarons). FEB 87
Fuller, J G . Aliens in the Skies (G Rothberg). DEC 69
Gilmor. D S . ed . Scientific Study of Unidentified
Flying Objects (G Rothberg). DEC 69
Harkins. R. R (see D R Saunders)
Hess. W. N.. The Radiation Belt and Magnetosohere
IB J O'Brien). OCT 73
Kanamon, H (see H Takeuchi)
Kopal. Z . An Introduction to the Study of the Moon
(S. F Singer), MAR 77
Kopal. Z. Telescopes in Space (P G Bergmann).
MAR 93
Mihalas. D . Galactic Astronomy (K Yoss) DEC 75
Moroz. V I . Physics of Planets (R. Smoluchowski)
DEC 73
Saunders. D R and R R Harkins. UFO's? Yes'
Where the Condon Committee Went Wrong (G
Rothberg). DEC 69
Smart. W M . Stellar Kinematics (K Yoss), DEC 75
PHYSICS TODAY . DECEMBER 1969 • 113
Takeuchi, H . S. Uyeda. and H. Kanamon. Debate
About the Earth Approach to Geophysics
Through Analysis of Continental Drift (0 W. Wil-
liams), JAN 107
Uyeda, S. (see H. Takeuchi)
ATOMS. MOLECULES. CHEMICAL PHYSICS
Bates. D R and I Estermann. eds.. Advances in
Atomic and Molecular Physics. Vol. 4 (S. Borow-
itz). NOV75
Bederson, B and W L. Fite, eds.. Methods of Experi-
mental Physics. Vol 7A Atomic and Electron
Physics (J. B Hasted). JUN 77
Chu, B . Molecular Forces Based on the Baker Lec-
tures of Peter J W Debye (S. Weissman). MAY
93
Chnstensen. C J (see H Eyring)
Estermann. I (see D R Bates)
Eyring, H , C J Chnstensen, and H. S. Johnston, eds.,
Annual Review of Physical Chemistry, Vol 19.
1968 (E E Muschlitz), DEC 81
irabelinsku. I L. Molecular Scattering of Light (H. B.
Levine) JUN 77
Fite, W L (see B Bederson)
Hamilton, W C and J. A Ibers, Hydrogen Bonding in
Solids Methods of Molecular Structure Determi-
nation (J. G Hoffman) NOV 79
Hirshfelder. J 0 , ed.. Advances in Chemical Physics,
Vol 12 Intermolecular Forces (I Amdur), JAN
1 1 1
Hughes. V W and H L. Schultz. eds.. Methods of Ex-
perimental Physics. Vol. 4, Atomic and Electron
Physics. Part B Free Atoms (W. Lichten), JUL 9 1
Jenkins. R. and J. L. de Vries. Practical X-Ray Spec-
trometry (H. A. Liebhafsky). APR 1 1 1
Johnston, H S (see H. Eyring)
Lever. A B. P.. Inorganic Electronic Spectroscopy (S.
A. Rice). OCT77
McMillan. J A . Electron Paramagnetism (H H Wick-
man). JUL87
Melia. T. P., An Introduction to Masers and Lasers (R.
J Collier), (L) FEB 17
Schultz. H L (see V W Hughes)
de Vries. J L. (see R Jenkins)
BIOPHYSICS Sheppard. J J.. Jr. Human Color Per-
ception A Critical Study of the Experimental
Foundation (I. Nimeroff). JUN 81
CONFERENCE PROCEEDINGS
Bederson. B . V. Cohen. V. W. Hughes, and F. M J Pi-
chamck. eds.. Atomic Physics (B. Bederson. V W.
Hughes, and L. Spruch). JAN 1 1 3
Blinc. R . ed.. Magnetic Resonance and Relaxation (T.
A Scott), FEB 85
Cohen. V (see B Bederson)
Ehlers. J , ed.. Relativity Theory and Astrophysics. Part
1 Relativity and Cosmology (W. Rindler), FEB 87
Hughes, V. W. (see B Bederson)
Pichanick. F M J (see B. Bederson)
ELEMENTARY PARTICLES
de Beauregard. 0. C. Precis de M'e-
canique Quantique Relativiste (P. G Bergmann),
MAY 85
Kabir. P. K . The CP Puzzle Strange Decays of the
Neutral Kaon (R. G. Sachs). AUG 77; (L) SEP 1 7
Lurie. D.. Particles and Fields (J. Bernstein),
OCT83
Mattuck. R. D . A Guide to Feynman Diagrams in the
Many-Body Problem (H Chang). MAY 87
Rosenblatt. J. Particle Acceleration (N A Baily) MAY
89
Weissenberg. A 0.. Muons (J L. Gammel). JUL 93
Wilson, J G and S A. Wouthuysen. eds.. Progress in
Elementary Particle and Cosmic Ray Physics, Vol.
9{H. Valk). MAR 93
Wouthuysen. S A. (see J. G. Wilson)
FLUIDS. PLASMAS
Bekefi, G . Radiation Processes in Plasmas (H. H. C.
Chang), JAN 103
Betchov. R and W. 0 Criminale. Jr, Stability of Paral-
lel Flows (J Gilhs), OCT83
Cole. G H A.. An Introduction to the Statistical Theo-
ry of Classical Simple Dense Fluids (J. L. Lebo-
witz). JUL91
Criminale, W 0 . Jr (see R. Betchov)
Greenspan, H. P. The Theory of Rotating Fluids Cam-
bridge Monographs on Mechanics and Applied
Mathematics (D R Rodenhuis and W. M. Elsas-
ser).JUL81
Losev. S A. (see Ye V Stupochenko)
Osipov, A. I. (see Ye. V Stupochenko)
Rosa, R J.. Magnetohydrodynamic Energy Conversion
(J Kelley) NOV 73
Shidlovskiy, V. P., Introduction to the Dynamics of
Rarefied Gases (R. E. Street), JUN 83
Simon, A and W. B. Thompson, eds., Advances in
Plasma Physics Vol. 1 (B S. Tanenbaum), NOV
77Stupochenko, Ye. V., S. A. Losev, and A. I. Osipo
Relaxation in Shock Waves (E. Wigner). MAY 91
Thompson, W. B. (see A. Simon)
HEAT, THERMODYNAMICS. STATISTICAL PHYSICS:
Gray. P. (see S. A. Rice)
Rice. S. A. and P. Gray, The Statistical Mechanics of
Simple Liquids (J. L. Lebowitz). JUL 91
HISTORY. PHILOSOPHY
Bondi, H.. Assumption and Myth in Physical Theory (L.
Sklar). AUG 79
Childs, H , An American Genius: The Life of Ernest Or-
lando Lawrence (M. L. Oliphant). MAR 75
Davis. N. P., Lawrence & Oppenheimer (F. Oppen-
heimer). FEB 77; (ER) MAY 17
Drake. S. and I. E. Drabkin. eds.. Mechanics in Six-
teenth-Century Italy: Selections from Tartaglia.
Benedetti. Guido Ubaldo and Galileo (R. S. Shank-
land). DEC 71
Drabkin, I E. (see S. Drake)
Irving, D., The German Atomic Bomb: The History of
Nuclear Research in Nazi Germany (J. J. Ermenc).
FEB 80
Ludwig, G , Wave Mechanics (G Sposito), APR 107
Pfeiffer, A., Dialogues on Fundamental Questions of
Science and Philosophy (R. Schlegel). APR 103
Rigal. J. L. ed.. Le Temps et la Pens'ee Phy-
sique Contemporaine (L. Marton), OCT 76
Sakharov, A D.. Progress. Coexistence and Intellectual
Freedom (J. M. Hollander). MAY 77
Schonland. Sir B., The Atomists (1805-1933) (B
Lindsay), MAR 85
INSTRUMENTATION AND TECHNIQUES
Aleksandrov. Yu.. G. S. Voronov, V. M. Gorbunkov, N.
B. Delone. and Yu. I. Nechayev. Bubble Chambers
(E.G. Pewitt), OCT 75
Alston, L. L., ed., High-Voltage Technology (L. Mar-
ton), JUL 86
Bartee, E. M . Engineering Experimental Design Fun-
damentals (R. L. Weber), AUG 80
Chasmar, R. P. (see R. A. Smith)
Delone, N. B. (see Yu. Aleksandrov)
Dennis, N. T. M. and T. A Heppell, Vacuum System
Design (J. H. Singleton). AUG 79
Fox. L.. and D. F. Mayers, Computing Methods for Sci-
entists and Engineers (N. A. Baily) DEC 83
Gorbunkov, V. M. (see Yu. Aleksandrov)
Heard, H G., ed., Laser Parameter Measurements
Handbook (R. B. Zipin), OCT 76
Heppell, T. A. (see N. T. M. Dennis)
Jones. F. E. (see R. A Smith)
Kaufman, M . Giant Molecules The Technology of
Plastics. Fibers and Rubbers (R. B Zipin) NOV 77
Levi. L.. Applied Optics A Guide to Optical System
Design. Vol 1 (J. A. Giordmaine). JUN 75
Mayers. D. F. (see L. Fox)
Moss, H.. Narrow Angle Electron Guns and Cathode
Ray Tubes (J. A. Simpson). SEP 87
Nechayev, Yu. I. (see Yu. Aleksandrov)
Neubert. H. K. P., Strain Gauges Kinds and Uses (R.
B. Zipin). AUG 81
Shutt, R. P.. ed., Bubble and Spark Chambers Princi-
ples and Use, Vol. 1 and2(D. Keefe), APR 95
Skudrzyk, E . Simple and Complex Vibratory Systems
(G.Weiss). JUL 95
Smith, R. A., F. E. Jones, and R. P. Chasmar. The De-
tection and Measurement of Infra-Red Radiation
(2ndEdition) (H. Malamud). JUL 83
Thornton. P. R., Scanning Electron Microscopy Appli-
cations to Materials and Device Science (L. Mar-
ton), AUG 81
Voronov, G. S. (see Yu. Aleksandrov)
White, G. K, Experimental Techniques in Low-
Temperature Physics (2nd Edition) (H. H. Plumb).
SEP 89
Zijlstra. H., Experimental Methods in Magnetism, Part
J: Generation and Computation of Magnetic
Fields; Part 2 Measurement of Magnetic Quan-
tities (R. J. Higgms). OCT 79
NUCLEI
Baranger, M. and E. Vogt. eds.. Advances in Nuclear
Physics. Vol. 7 (E. Hayward). APR 107
Collard. H. R., L. R. B. Elton, and R. Hofstadter,
Landolt-Bornstein. Numerical Data and Functional
Relationships in Science and Technology. New
Series. Group 1. Vol. 2 Nuclear Radii (J. O'Con-
nell), MAR 85
Elton, L. R. B. (see H. R. Collard)
Gruverman, I. J.. ed., Mbssbauer Effect Meth-
odology, Vol. 3 (H. H. Wickman), JAN 99
Gurevich, I. I and L V. Tarasov. Low-Energy Neutron
Physics (R. S. Shankland), OCT 85
Hofstadter, R. (see H. R. Collard)
McCarthy, I. E., Introduction to Nuclear Theory (V.
Canuto), OCT 78
Migdal. A. B., Theory of Finite Fermi Systems and Ap-
plications to Atomic Nuclei (J. L Gammel) JUN
85
Tarasov, L. V. (see I I Gurevich)OPTICS
Brown, R., Lasers: Tools of Modern Technology (R. B.
Zipin) NOV 77
Fleury, P and J. Mathieu, Images Optiques {4th Edi-
tion) (J. Romain), SEP 89
Francon, M.. Optical Interferometry (S. S. Ballard),
APR 101
Goodman, J. W., Introduction to Fourier Optics (M. E.
Cox). APR 97
Klauder, J. R. and E. C. G. Sudarshan, Fundamentals
of Quantum Optics (M. M. Miller), DEC 79
Levine. A K.. Lasers, Vol. 2 (D. F. Hotz), JUL 83
Mathieu, J. (see P. Fleury)
Sudarshan, E. C. G. (see J. R. Klauder)
Wright, W. D.. The Rays Are Not Coloured: Essays on
the Science of Vision and Colour (I. Nimeroff).
JUL85
PHYSICS AND SOCIETY
Brooks. H.. The Government of Science (P. Craig).
JAN 97
Bube, R. H.. ed.. The Encounter Between Christianity
and Science (F. L. Wilson). JAN 1 03
Commission on Marine Science. Engineering and Re-
sources, Our Nation and the Sea (H. Bradner),
SEP81
Danhof. C. H , Government Contracting and Techno-
logical Change (J Agassi). SEP 95
Fachverband fur Strahlenschutz, Radiological Pro-
tection of the Public in a Nuclear Mass Disaster
(N. A. Baily). JUL 94
Feinberg, G . The Prometheus Project (L. Branscomb),
NOV 69
Leeds, M., ed.. Washington Colloquium on Science
and Society (Second Series) (M. W Friedlander).
MAY 95
Orlans, H., ed.. Science Policy and the University (P.
Craig). JAN 97
Seymour. S. F.. ed.. Washington Colloquium on
Science and Society (First Series) (M W Friedlan-
der). MAY 95
Wigner. E. P.. ed.. Who Speaks for Civil Defense? (L
Marton). MAR 76
Ziman. J M.. Public Knowledge: An Essay Concerning
the Social Dimension of Science (D. Crane), OCT
87
POPULARIZATIONS Bergmann. P G.. The Riddle of
Gravitation (J Weber). FEB 81
Koslow. A , ed.. The Changeless Order: The Physics of
Space, Time and Motion (E. V. Mielczarek), MAR
77
Shapley. H . Beyond the Observatory (M. F. McCar-
thy). APR 105
SOLIDS
Akhiezer. A. I.. V. G. Bar'yakhtar. and S. V. Peletmin-
skn. Spin Waves (M. Wortis). NOV 75
Alder, B., S. Fernbach and M Rothenberg, eds., Meth-
ods in Computational Physics, Vol 8: Energy
Bands of Solids (N. Ashcroft) NOV 71
Angus. W. R., J. Favede, J. Hoaru, and A. Pa-
cault, Landolt-Bdrnstein. Zahlenwerte und
Funktionen aus Physik. Chemie, Astronomie. Geo-
physik und Techmk. (6th edition) Vol. 2 Eigen-
schaften der Materie in ihren Aggregatzust-
anden. Part 10 Magnetische Eigenschaft-
en II (J. H Van Vleck), JUL 86
Bar'yakhtar. V G. (see A. I. Akhiezer)
Ehrenreich. H. (see F. Seitz)
Favede. J. (see W. R Angus)
Fernbach, S. (see B. Alder)
Gorter, C. J.. ed.. Progress in Low Temperature Phys-
ics. Vol 5 (G. Sposito), JAN 107
Hoaru. J. (see W. R. Angus)
Ibers. J A. (see W. C. Hamilton)
Kuper. C. G.. An Introduction to the Theory of Super-
conductivity (J. A. Snow), JUL 82
Long, D., Energy Bands in Semiconductors (D. C. Mat-
tis).JUL84
March. N. H., Liquid Metals (S. A. Rice). FEB 83
Mason. W. P., ed., Physical Acoustics. Principles and
Methods, Vol. 4, Parts A and B: Applications to
Quantum and Solid State Physics (W. G. Mayer).
NOV 81
VlcCreight. L R. (see H. W. Rauch)
Ovsienko. D. E.. ed., Growth and Imperfections of Me-
tallic Crystals (M. E. Straumanis), JUN 79
Pacault, A. (see W. R. Angus)
Peletminskii, S. V. (see A. I. Akhiezer)
Rauch, H. W., W. H. Sutton, and L. R. McCreight, Ce-
ramic Fibers and Fibrous Composite Materials (T.
R. Cass), NOV 79
Rothenberg, M. (see B. Alder)
Schieber, Michael M., Experimental Magnetochemis-
try: N on metallic Magnetic Materials, Vol. 8 (W.
Wolf), JUL 87
Seitz, F., D. Turnbull. and H. Ehrenreich, eds.. Solid
State Physics: Advances in Research and Applica-
tions, Vol.21 (D. C. Mattis), AUG 85
114 . DECEMBER 1969 • PHYSICS TODAY
T/' J3C, Qu,antum TheorV of Molecules and Solids.
D Mermrn)^ur89Se"/COn^CfO"' °nd MetalS (N
C73C"" °UantUm Theory of Matter (P. L Taylor).
Sutton. W. H.(seeH. W Rauch)Turnbull. D. (see F. Seitz)
TEXTBOOKS
Alonso. M. and E. J. Finn. Fundamental University
Physics. Vol. 3 Quantum and Statistical Physics
(F. L Wilson). JUL 86
Azaroff. L V.. Elements of X-Ray Crystallogra-
phy (R. B. Zipin). JAN 99
Barford. N C . Experimental Measurements Precision,
Error and Truth (J. B. Kelley). JAN 109
Battino. R and S E Wood, Thermodynamics An In-
troduction (J B Kelley). APR 109
Beiser. A , Modern Physics An Introductory Survey (T
H.Osgood). FEB83
Bernstein. J . Elementary Particles and Their Currents
(C. Ryan). JAN 101
Blatt. F J.. Physics of Electronic Conduction in Solids
(C S Koonce). AUG 83
Blokhintsev. D. I. Pnncipes Essentiels de la
Mecanique Quantique (R. B Lindsay)
SEP 83
Bonsenko. A. I. and I E. Tarapov. Vector and Tensor
Analysis with Applications (P. L. Balise). FEB 83
Brodkey. R. S . The Phenomena of Fluid Motions (L
Talbot). SEP 85
Cabannes, H.. General Mechanics (J E Romam) OCT
77
Chirgwin. B. H and C Plump'ton. Elementary Classical
Hydrodynamics (J B. Kelley). MAR 89
Feather. N.. Electricity and Matter (I. M Freeman).
MAR 87
Finn. E. J. (see M. Alonso)
Jackson. E. A.. Equilibrium Statistical Mechanics (G.
Sposito). MAR 83
Joseph. A. and D J Leahy. Programmed Physics, Part
4 Kinetic Theory and Thermodynamics; Part 5
Topics in Modern Physics (G L Pollack). FEB 91
Kawai. M. (see K. Kikuchi)
Kikuchi. K and M Kawai. Nuclear Matter and Nuclear
Reactions (J. L. Gammel). AUG 78
Lawden. D F.. The Mathematical Principles of Quan-
tum Mechanics (G Sposito). MAY 82
Leahy. D J. (see A. Joseph)
Levy, R A., ed . Principles of Solid State Physics (R J.
Collier). JUN 75
Plumpton. C. (see B. H. Chirgwin)
Richards. J. W.. Interpretation of Technical Data (J. B.
Kelley). JAN 109
Sakurai. J. J.. Advanced Quantum Mechanics (H. S.
Valk). FEB 80
Samarski. A A. (see A N. Tychonov)
Schwartz. H M . Introduction to Special Relativity (R.
B.Zipin). FEB 85
Stanley. R. C. Light and Sound for Engineers (R. Lind-
say). DEC 75
Tandberg-Hanssen. E., Solar Activity (J. Aarons). MAR
91
Tarapov. I E. (see A. I Bonsenko)
Tychonov. A. N and A. A Samarski. Partial Differen-
tial Equations of Mathematical Physics, Vol. 2 (J.
Gillis). APR 109
Wood, S E (see R Battino)
THEORY AND MATHEMATICAL PHYSICS
Beran. M. J , Statistical Continuum Theories (S A
Rice). JUL 94
Bialynicki-Birula. I and Bialynicka-Birula. Z. Quantum
Electrodynamics (0. M Bilaniuk) NOV 70
Birss. R. R.. Electric and Magnetic Forces (J B Kelley).
OCT 79
de Broglie. L. Ondes Electromagnetiques et
Photons (R. B Lindsay). MAY 97
Butkov. E.. Mathematical Physics (G Sposito). SEP 91
Cracknel. A. P.. Applied Group Theory (G Rothberg),
AUG 83
Flugge. S., Lehrbuch der Theoretischen Phys-
ik. Vol. 2: Klassische Physik I, Mechanik geordnet-
er und ungeordneter Bewegungen (P. G. Berg-
mann). JUN 83
French. A P.. Special Relativity The MIT Introductory
Physics Series (P. G Bergmann). MAY 95
Haray, F . ed.. Graph Theory and Theoretical Physics
(G. Sposito). JUL 84
Kilmister. C W . Lagrangian Dynamics: An Introduc-
tion for Students (G Sposito). DEC 77
Mehta, M. L. ed.. Random Matrices and the Statistical
Theory of Energy Levels (E Merzbacher). APR
101
Miller. W . Jr. Lie Theory and Special Functions (H S.
Valk). JUL 93
von Neumann. J , Mathematische Grundlagen der
Quantenmechamk (P. G Bergmann). MAY 85
Strauss H L., Quantum Mechanics: An Introduction
(P J Silverman). JUN 81
TprlPtskn Y P Paradoxes in the Theory of Relativity
(D.B L.chtenberg).OCT74;(ER)DEC15 DThe American Institute
of Physics
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Those who wish to make preliminary enquiries should write to the Chair-
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Victoria 3168, who will be glad to supply details of present facilities and of the
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tions for future developments.
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335 E. 45th St.,
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PHYSICS TODAY • DECEMBER 1969 • 117
GUEST EDITORIAL
Is Your Research Moral?
The author of this invited editorial is
chairman of physics at Stanford. He
has three degrees from Toronto, and
before going to Stanford he worked at
Research Enterprises, Columbia Uni-
versity and Bell Telephone Laboratories.
His research has been in rf, microwave
and optical spectroscopy, solid-state
physics and quantum electronics.lyrowadays science and scientists are
^ being attacked from all direc-
tions. On the one side there are those
in Congress, the military and the gen-
eral public who say that if science
were doing what is expected, it would
have won the war in Vietnam and pro-
duced horrible new weapons to terrify
all possible enemies. On the other
side many students and intellectuals,
even including some scientists, say
that if scientists were doing their job
properly, they should have produced
an end to poverty, racism, air pollu-
tion, overpopulation and war.
Strangely, both opposed groups of
critics argue that since science has not
performed according to their specifi-
cations, either scientists are misdirect-
ing their efforts or perhaps science is
irrelevant and has little to contribute
to the solution of important human
problems. Both groups equally fail to
understand the real nature of scientific
discovery and the ways in which sci-
entific knowledge eventually makes
possible the goals people desire.
A fuller understanding of nature is
esthetically pleasing and deeply satis-
fying, but its social significance comes
because it is also useful. It makes us
prophets so that we and our successors
can predict what can happen and caneven tell something of the conse-
quences that will follow if we make
.something occur. Part of the attrac-
tion of physics is that simple laws and
concepts have extremely far-reaching
consequences and apply in a very
wide range of situations. For just this
reason the ultimate applications of
physical discoveries are almost never
apparent at the beginning. We all
know something of the long history
extending from the nuclear atom and
the mass-energy equivalence to atomic
power. We who know such history
should recall it and tell it to those who
question the nature and utility of
science.
T et me give two examples from rel-
atively recent technological histo-
ry. The first I know personally, for
when Charles Townes and I were trying
in 1957 to see whether the maser prin-
ciple could yield a generator of coher-
ent radiation in or near the range of
visible wavelengths, we gave almost
no thought to applications. I had
never heard of a detached retina, and
yet one of the earliest applications of
lasers was for eye surgery to prevent
retinal detachment. Although lasers
are still quite primitive and many of
the more obvious applications remain
118 DECEMBER 1969 . PHYSICS TODAY
impractical, they have been applied to
a wide range of needs, most of which
could hardly have been foreseen ex-
cept by a person who specialized in
the particular area of application.
But if we had tried to attack these
needs head on, as might have been
done by a specialist in eye surgery, we
would never have been thinking about
stimulated emission from atomic sys-
tems.
Considerably more important conse-
quences have come from Felix Bloch's
discovery of the concept of energy
bands in solids and their influence on
conduction of electricity. In the 20
years after' Bloch's 1928 thesis, the
band ideas guided the whole develop-
ment of solid-state physics. And yet,
as late as 1953, 25 years after the dis-
covery, one could have said truthfully
that these ideas had not led to greatly
improved metals nor to any other im-
portant practical consequences. But a
year or so later, there began serious
applications of the transistor, a device
that really could not have been invent-
ed without the conceptual framework
of the band theory. Now, the impact
of the use of transistors and other semi-
conductor devices on human life is al-
ready enormous. To take a few ex-
amples, there are cardiac pacemakersand the ubiquitous transistor radio,
which is playing such an important
role in unifying some developing
countries. Without semiconductor
devices the entire space program
would be nearly impossible. It is
hard to conceive of either the human
aspects of space flight (such as envi-
ronmental and weather-observation
satellites), the scientific aspects (such
as astrophysical observatories and
moon landing probes) or the military
aspects without large-scale and light-
weight semiconductor computers. In
industry, it seems quite possible that
semiconductor logic will eliminate a
large part of the routine drudgery that
seemed for a while to be an inescap-
able consequence of mass production.
None of this could possibly have been
foreseen at the time of the original sci-
entific discovery. Yet from all our ex-
perience we should have faith that sci-
entific ideas do have consequences,
important consequences that greatly
increase the range of decisions that
man can make. It is the nature of
man to make choices and to master his
environment. With science and its
consequences we have the tools to
make decisions, good or bad. If we
sacrifice scientific research for imme-
diate social gains, we might have ashort-range benefit, but we are surely
mortgaging our future.
A fter applications of science become
apparent, the people and their
representatives must decide whether
the applications are good or bad. Here
scientists must play a part by sharing
knowledge of the possible courses and
their likely consequences. If the facts
are known, we can be optimistic that
the people will more often choose
courses to their own benefit than the
reverse. Every thinking scientist
must have faced this question and
concluded that, broadly, scientific
discoveries do eventually open up
badly needed alternatives from which
more good than evil will be extracted.
Whatever the grounds for such
faith, whether from a religious convic-
tion or from a knowledge of scientific
and technological history, we must put
these concerns aside when we con-
front the mysteries of the universe.
In the light of this belief that good
things do eventually come from new
knowledge, I am convinced that good
scientific research is a highly moral ac-
tivity. The only kind that is not moral
is that which can be characterized by
a phrase of Wolfgang Pauli's: "It
isn't even wrong."
—Arthur L. Schawlow
PHYSICS TODAY • DECEMBER 1969 • 119
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120 • DECEMBER 1969 . PHYSICS TODAY
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probability distribution
w,, = \cfe\*
Suppose that the eigenfunction <j>fg is
written in terms of certain real func-
tions R and S as
exp { -
where both R and S depend on the
indices / and g. The first step in the
measurement then involves converting
the wave function «HMM) of the sys-
tem into a new wave function $s(x)
given by
ts(x) = t(x,tu)
exp { iSfg(x) } (7)
by applying a pulse potential
Us(x,t)=-*Sft(x)8(t-til)
to the system. Then the probability
amplitude Cfg of equation 6 has the
value
Jfg=/«„(x)+s(x)dx
which is just the overlap integral be-
tween the new wave function ^g(x) of
the system and the real normalized
wave function Rfg. From the earlier
discussion of state preparation we know
how to find a potential Ufg(x) in which
Rfg(x) is an eigenfunction of energy E.
This potential is given by
Hence, the second stage of the mea-
surement process involves the sudden
application of the potential Ufg(x) and
removal of the potential V(x) from
the system with a wave function given
by equation 7. The next task is to find
the probability that the particle is in the
state of energy E. This could be ac-
complished by the usual kind of Stern-
Gerlach procedure. In this manner, we
find the desired probability that the
operators F, G have values Ff, Go for
the state \p (x,£M) of the system of inter-
est. It will be noted that the measure-
ment problem is now more complicated
than for Hamiltonian operators, as the
potentials U(x,t) to be applied depend
on the values of / and g, and hence a
series of measurements has to be made
for each set of /, g values.
Limitations
Some concluding comments are in
order.
• We have assumed that all classi-
cally describable potentials U(x,t) are
available to us experimentally. This is
quite similar and closely related to an
assumption made by Niels Bohr andLeon Rosenfeld13 in their discussion of
the measurement of electromagnetic
fields, that test bodies of very great
mass and charge density were available
to them, whose quantum-mechanical
fluctuations of position and momentum
could be neglected.
• Our discussion of measurement
considers the observation of any one
dynamical variable, A. Instead, we
could measure another one, B, for the
same state ^(.\\fM). For each operator
we could determine the dispersion
measure AA or AB. The product of the
fluctuation measures would obey the
Robertson14-Schr6dinger15 generaliza-
tion
AA- AB > 1/21<[ A,B]> |
of Heisenberg's uncertainty relations.
It should be noted, however, that such
uncertainty relations do not refer to
measurements, whether simultaneous
or not, of a pair of observables. In spe-
cial cases, involving commutability, it
might be possible to measure first one
observable and then another, but in
general the thoroughgoing measure-
ment of the first observable will so dis-
rupt phase relations that it will serve
no physical purpose to subsequently
measure a second observable on the
resulting mixture.
To measure simultaneously two non-
commuting observables A and B (for
example, x and p) one would have to
find a potential U(x,t) that was deter-
mined by both A and B. In general, I
do not believe that this can be done in
such a way that the desired informa-
tion emerges from the measurement.
One could, of course, form a single
Hermitian operator out of the two
Hermitian operators A and B. Some
examples are (AB + BA), -i(AB —
BA), A2B + BA\ ABA, etc. Any one of
these Hermitian operators could be
measured, as already indicated, but
this would not be the desired simul-
taneous measurement of A and B.
• It is possible to extend the methods
outlined so that measurements on
many-body systems can be made.
• I do not see how to apply proce-
dures of the kind outlined above to the
relativistic quantum domain, or to field
theory. In the absence of such gener-
alizations, it may well be doubted that
the story that I have given provides any
significant insight into the real mean-
ing of quantum mechanics. However,
it is true that almost all expositions of
quantum mechanics make use of the
fictional notion that some kinds of
measurements are possible. I havedescribed certain experimental procc
dures for making them. There may b
other ways. If they cannot be made i]
some fashion, either as I have sug
gested, or otherwise, then it appear
that our understanding of the meanim
of the quantum theory is correspond
ingly diminished, and it is only like!)
to be increased when a better theory oJ
measurement for the more general rela-
tivistic and field-theoretic cases can be
given. Of course, it may be that a sys-
tem of rules for calculation can exist,
despite the absence of an operational
interpretation of the kind I have
attempted. For the teaching of quan-
tum mechanics now, it is certainly a
convenient fiction to pretend that the
usual textbook assumptions about mea-
surement have a meaning, even if from
an operational point of view they do
not. The mathematical formulation of
quantum mechanics by Dirac beauti-
fully matches the assumed notion of
measurability. However, there is clear-
ly much more for us to learn.
This article is based on a lecture given 3
July 1968 at the 6th triennial Conference
of Physics Nohel Prize Winners held at
Lindau (Bodensee), West Germany. This
work was supported in part by the US
Air Force Office of Scientific Research.
References
1. W. Heisenberg, Physical Principles of
the Quantum Theory, University of|
Chicago Press (1930), p. 21. •
2. W. Pauli, "The General Principles of;
Wave Mechanics," in Handbuch der
Physik, vol. 24/1, Springer, Berlin
(1933), pp. 163-164.
3. E. Merzbacher, Quantum Mechanics,
Wiley, New York (1961), p. 158.
4. J. von Neumann, Nachr. Ges. Wiss.
Cottingen, p. 1 (1927); p. 245
(1927); p. 273 (1927).
5. L. D. Landau, Zeits. fur Physik 45,
430 (1927).
6. P. A. M. Dirac, Proc. Camb. Phil. Soc.
25, 62 (1929).
7. P. A. M. Dirac, Quantum Mechanics,
4th ed., Oxford University Press
(1958), p. 37.
8. Op. cit. ref. 2, pp. 164-166.
9. Op. cit. ref. 2, pp. 143-154.
10. D. Bohm, Quantum Theory, Prentice-
Hall, New York (1951), chap. 22.
11. K. Gottfried, Quantum Mechanics,
vol. 1, W. A. Benjamin, Inc., New
York (1966), chap. 4.
12. Op. cit. ref. 1, p. 32.
13. N. Bohr, L. Rosenfeld, Det. KgL
Dansk. Vid. Selskab 12, 8 (1933).
14. H. P. Robertson, Phys. Rev. 35, 667A
(1930).
15. E. Schrodinger, Sitzungsber. preuss.
Akad. Wiss, p. 296 (1930). •
28 • APRIL 1969 • PHYSICS TODAY
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1.5127766.pdf | AIP Advances 10, 015112 (2020); https://doi.org/10.1063/1.5127766 10, 015112
© 2020 Author(s).Synchronization and chaos in spin torque
oscillator with two free layers
Cite as: AIP Advances 10, 015112 (2020); https://doi.org/10.1063/1.5127766
Submitted: 12 September 2019 . Accepted: 05 December 2019 . Published Online: 07 January 2020
Tomohiro Taniguchi
COLLECTIONS
Paper published as part of the special topic on 64th Annual Conference on Magnetism and Magnetic Materials
Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials.
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Applied Physics Letters 114, 172403 (2019); https://doi.org/10.1063/1.5086476AIP Advances ARTICLE scitation.org/journal/adv
Synchronization and chaos in spin torque
oscillator with two free layers
Cite as: AIP Advances 10, 015112 (2020); doi: 10.1063/1.5127766
Presented: 7 November 2019 •Submitted: 12 September 2019 •
Accepted: 5 December 2019 •Published Online: 7 January 2020 •
Corrected: 13 January 2020
Tomohiro Taniguchia)
AFFILIATIONS
National Institute of Advanced Industrial Science and Technology (AIST), Spintronics Research Center, Tsukuba 305-8568, Japan
Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials.
a)Electronic mail: tomohiro-taniguchi@aist.go.jp
ABSTRACT
The magnetization dynamics in a spin torque oscillator (STO) consisting of two in-plane magnetized free layers is studied by solving the
Landau-Lifshitz-Gilbert equation and evaluating the Lyapunov exponent numerically. The phase diagrams of the oscillation frequencies of
the magnetizations and magnetoresistance and the maximum Lyapunov exponent are obtained from the numerical simulations. The phase
synchronization is found in the low current region, whereas the magnetizations oscillate with different frequencies in the middle current
region. On the other hand, positive Lyapunov exponents found in the high current region indicate the existence of chaos in the STO.
©2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5127766 .,s
I. INTRODUCTION
Spin torque oscillator (STO) is a nonlinear oscillator in
nanoscale, and generates an oscillating power having the frequency
on the order of gigahertz through giant or tunnel magnetoresis-
tance (MR) effect.1–3A conventional structure of STO consists of
three ferromagnets, called free, reference, and pinned layers, with a
nonmagnetic spacer between the free and reference layers. The mag-
netization direction in the reference layer is fixed by the pinned layer,
whereas that in the free layer can change its direction by applying
magnetic field and/or electric current.
Recently, however, another type of STO has been proposed for
a new scheme of magnetic recording,4namely microwave assisted
magnetization reversal (MAMR).5–16In MAMR, microwave mag-
netic field is emitted from an STO to a magnetic recording media
and induces an oscillation of the magnetization in a recording
bit, resulting in a reduction of a direct field for recording. At the
beginning of the study on MAMR, the STO consisted of an in-
plane magnetized free layer and perpendicularly magnetized ref-
erence and pinned layers.8,17–21The structure of such an STO
becomes, however, thick to make the magnetizations in the refer-
ence and pinned layers perpendicular.18,20The latest design of the
STO for MAMR consists of two in-plane magnetized ferromagnets
called field-generation layer and spin-injection layers.4The field-
generation layer acts as a microwave source for MAMR, whereas thespin-injection layer provides spin current into the field-generation
layer to excite an auto-oscillation of the magnetization. It should
be emphasized that this type of STO does not have a pinned layer
to make the recording head thin. Therefore, both two ferromag-
nets can be regarded as free layers. A coupled motion of two fer-
romagnets in nanostructured multilayers is a recent exciting topic in
magnetism.22For MAMR application, a theoretical study to clarify
the dynamical phase in this STO over a wide range of the electric
current is necessary, while an experimental work has been reported
recently.23
In this paper, a theoretical study on the magnetization dynam-
ics in an STO with two free layers is presented. We solve the Landau-
Lifshitz-Gilbert (LLG) equation numerically, and evaluate the Lya-
punov exponent to characterize the dynamical phase. A phase syn-
chronization is found in the low current region, whereas two mag-
netizations oscillate with different frequencies in the middle current
region. On the other hand, chaos is found in the high current region,
which is identified from positive Lyapunov exponents.
II. SYSTEM DESCRIPTION
The STO studied in this work consists of two ferromagnets, F k
(k= 1, 2), separated by a thin nonmagnetic spacer.23,24In the recent
experiment,23the F 1and F 2, corresponding to the field-generation
AIP Advances 10, 015112 (2020); doi: 10.1063/1.5127766 10, 015112-1
© Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv
and spin-injection layers, respectively, were CoFe and NiFe, whereas
the nonmagnetic spacer was Ag. The zaxis is perpendicular to the
film plane, whereas the xandyaxes lie in the plane. The unit vector
pointing in the magnetization direction of the F klayer is denoted as
mk. The dynamics of mkis described by the LLG equation,
dmk
dt=−γmk×Hk+αkmk×dmk
dt
−γ̵hpkj
2e(1 +p2
km1⋅m2)Mkdkmk×(m2×m1), (1)
whereγkandαkare the gyromagnetic ratio and the Gilbert damping
constant, respectively. The saturation magnetization and thickness
of the F klayer are denoted as Mkanddk. Since the magnetic record-
ing is achieved by applying a direct magnetic field generated in the
recording head close to the STO to the recording media, the magne-
tization dynamics in the STO is also affected by a direct field.23Thus,
an applied field should be taken into account in the magnetic field in
the LLG equation. The magnetic field consists of the applied field
Happlin the zdirection, the demagnetization field, and the dipole
field as
Hk=⎛
⎜⎜
⎝−4πMkNkxmkx−Hdkmk′x
−4πMkNkymky−Hdkmk′y
Happl−4πMkNkzmkz+ 2Hdkmk′z⎞
⎟⎟
⎠. (2)
The demagnetization coefficient Nki(i=x,y,z) and the dipole field
Hdkare evaluated from their analytical solutions as [( k,k′) = (1, 2)
or (2, 1)]25,26
Nkz=1
τk⎧⎪⎪⎪⎨⎪⎪⎪⎩4
3π−4
3π√
1 +τ2
k⎡⎢⎢⎢⎢⎢⎣τ2
kK⎛
⎜
⎝1√
1 +τ2
k⎞
⎟
⎠
+(1−τ2
k)E⎛
⎜
⎝1√
1 +τ2
k⎞
⎟
⎠⎤⎥⎥⎥⎥⎥⎦+τk⎫⎪⎪⎪⎬⎪⎪⎪⎭, (3)
Hdk=πMk′⎡⎢⎢⎢⎢⎢⎢⎢⎣dk
2+dN+dk′
√
r2+(dk
2+dN+dk′)2−dk
2+dN√
r2+(dk
2+dN)2⎤⎥⎥⎥⎥⎥⎥⎥⎦, (4)
whereτk=dk/(2r) with the radius randdNis the thickness of the
nonmagnet, whereas K(κ) and E(κ) are the first and second kinds
of complete elliptic integrals with the modulus κ. The last term in
Eq. (1) is the spin-transfer torque, where jis the current density
whereas pkcorresponds to the spin polarization. The positive cur-
rent is defined as the electrons flowing from the F 1to F 2layer. We
note that two magnetizations are coupled via the spin-transfer effect
and the dipole field.
The values of the parameters are derived from CoFe/Ag/NiFe
trilayer23asM1= 1720 emu/c.c., M2= 800 emu/c.c., α1= 0.006,α2
= 0.010, d1= 5 nm, d2= 3 nm, p1=p2= 0.3, andγ= 1.764 ×107
rad/(Oe s). The magnitude of the microwave magnetic field gener-
ated by a ferromagnet having the saturation magnetization as such
is on the order of 100 Oe,26which is sufficient to achieve MAMR
experimentally.10The thickness of the nonmagnet is 5 nm, whereas
the radius is 50 nm. The applied field is 8.0 kOe. Since the spacer
layer consists of a metal (Ag) in the experiment,23a large currentdensity on the order of 108A/cm2can be injected. The LLG equation
was solved by the 4th-order Runge-Kutta scheme with a constant
time step of Δt= 10−5ns for all simulation. In the present simu-
lation, we first solved the LLG equation without current to relax the
magnetizations to their energetically stable states. After that, the LLG
equation in the presence of a finite current was solved for time range
of 0.5μs to investigated the magnetization dynamics driven by spin-
transfer torque. The time necessary to reach a stable oscillation is,
typically, on the order of 1 ns.
The magnetization dynamics in an STO is detected through
the giant or tunnel magnetoresistance (MR) effect in the exper-
iment,23which depends on m1⋅m2.27,28On the other hand, the
microwave field required in MAMR reflects the oscillations of the
magnetizations, m1andm2. Therefore, we calculate the peak fre-
quencies of the Fourier spectra of m1x,m2x, and MR ≡m1⋅m2, in the
following.
III. SYNCHRONIZATION AND CHAOS
Figure 1(a) shows typical dynamics of m1(red) and m2(blue)
in a low current region. The auto-oscillations of the magnetizations
around the zaxis are excited in two ferromagnets. The time evolu-
tions of m1x(red dotted), m2x(blue dashed), and MR (black solid)
are also shown in Fig. 1(b). It can be seen that two magnetizations
oscillate with an identical frequency, i.e., a frequency synchroniza-
tion is excited. Since the relative angle between the magnetizations
is temporally constant in the synchronized state, the MR is also con-
stant. In this case, no oscillating signal will be detected through the
MR effect. In fact, the power spectrum density of an STO in a low
current region was found to be zero experimentally.23However, no
electric signal does not necessarily mean the absence of the auto-
oscillations of the magnetizations. We believe that the synchronized
oscillation is excited in the low current region, and it will be appli-
cable to MAMR application because the field-generation layer (F 1)
shows the oscillation, and therefore, emits microwave field.
One might consider that the microwave magnetic field gen-
erated outside the STO becomes zero because two magnetizations
oscillate with almost antiphase; see Fig. 1(b). It should be, how-
ever, noted that the magnitude of the magnetic field generated by
the oscillating magnetization is proportional to the saturation mag-
netization. Since the saturation magnetizations of two ferromagnets
in the present STO are largely different, the total microwave mag-
netic field remains finite even though two ferromagnets oscillate
with antiphase.
Figure 1(c) shows typical dynamics in the middle current
region. In this case, two magnetizations oscillate with different fre-
quencies. Therefore, the MR also shows an oscillation, where its fre-
quency is the difference of the frequencies in two magnetizations. It
should be noted that the oscillation of the MR can be detected exper-
imentally in this region. The oscillation frequency of the MR shows
redshift as discussed below, which is consistent with the experi-
ment.23It should be emphasized that this region is also applicable to
MAMR because the F 1layer shows an auto-oscillation with a unique
frequency.
A further increase of the applied current density leads to com-
plex dynamics of the magnetizations. Figures 1(d) and 1(e) respec-
tively show the dynamical trajectories of m1andm2in a high current
region ( j= 4.0 ×108A/cm2) after the magnetizations move to an
AIP Advances 10, 015112 (2020); doi: 10.1063/1.5127766 10, 015112-2
© Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv
FIG. 1 . (a) Dynamical trajectories of m1(red) and m2(blue) in a steady state at j= 0.1×108A/cm2. (b) Time evolutions of m1x(red dotted), m2x(blue dashed), and MR
(black solid) at j= 0.1×108A/cm2. (c) Time evolutions of m1x,m2x, and MR at j= 2.0×108A/cm2. Note that the range in the horizontal axis differs from that in (b). (d), (e)
Dynamical trajectories of m1(red) and m2(blue) at j= 4.0×108A/cm2, respectively. (f) The trajectories in the reduced phase space, ( mkx,mky) atj= 4.0×108A/cm2. (g)
Time evolution of MR at j= 4.0×108A/cm2. (h) Fourier spectra of | m1x| (red), | m2x| (blue), and |MR| (black) at j=−4.0×108A/cm2, respectively. (i) Time evolution of MR
atj=−4.0×108A/cm2.
attractor, where the data in last 10 ns are used for the plots. The
trajectories in the reduced phase space, ( mkx,mky), are also shown
in Fig. 1(f). The time evolution of MR is also shown in Fig. 1(g).
As can be seen in these figures, highly nonlinear dynamics appears
in two layers, and the MR does not show periodicity. We note that
highly nonlinear dynamics as such was found in STOs with two
ferromagnets in the previous works.29For example, Kudo et al.
performed numerical simulations of the LLG equation for two in-
plane magnetized ferromagnets with in-plane magnetic anisotropy,
where the ferromagnets are coupled via spin-transfer effect only,
and found chaotic dynamics of the magnetizations.29We identify
chaos in the present STO by evaluating Lyapunov exponent by using
the Shimada-Nagashima method,30,31where the Lyapunov exponentis defined as an average of instantaneous expansion rates of two
dynamical trajectories having different initial conditions at t=t0as
λ=lim
N→∞N
∑
n=11
NΔtlog∣ϵ+δ(t0+nΔt)
ϵ∣, (5)
whereϵis a perturbation applied to the STO at t=t0, whereasδ(t
+nΔt) is the expansion of the perturbation after time nΔt. In this
work, we introduce a four-dimensional phase space with the vari-
ables (θ1,φ1,θ2,φ2) defined as mk= (sinθkcosφk, sinθksinφk,
cosθk), and add the perturbation ϵ= 1.0 ×10−5rad to the phase
space. Then, the (maximum) Lyapunov exponent is obtained from
Eq. (5). For example, the Lyapunov exponent of the dynamics shown
AIP Advances 10, 015112 (2020); doi: 10.1063/1.5127766 10, 015112-3
© Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv
in Figs. 1(d)–1(f), where j= 4.0 ×108A/cm2, is a positive value of
6.69 GHz, indicating that the dynamics is chaos. We also note that
a large current does not necessarily guarantee chaos. For example,
the dynamical trajectories for the opposite current, j=−4.0×108
A/cm2, look similar to those shown in Figs. 1(d) and 1(e). In addi-
tion, the Fourier spectra of | m1x|, |m2x|, and |MR| at j=−4.0×108
A/cm2have multipeak over wide range of frequency, as shown in
Fig. 1(h). However, the time evolution of the MR shown in Fig. 1(i)
shows periodicity. In such a case, the Lyapunov exponent is zero.
It is considered that chaos is caused by the large spin-transfer
torque. Note that the spin-transfer torques act asymmetric to two
ferromagnets; for example, for a positive current, the spin-transfer
torque acting on the F 1layer prefers the antiparallel alignment of the
magnetizations, whereas that acting on the F 2layer prefers the par-
allel one. As a result, for a large current, the magnetizations cannot
stay in limit cycle oscillations, and chaos is excited. The thresh-
old current necessary to cause chaos increases with increasing the
field magnitude because the damping torques due to the field act
symmetric to the magnetizations.
IV. PHASE DIAGRAM
Here, let us summarize the magnetization dynamics studied in
Sec. III.
Figure 2(a) summarizes the current dependences of the oscil-
lation frequency of m1x(red square), m2x(blue triangle), and MR
≡m1⋅m2(black circle). Around zero current, two magnetizations
show synchronization, i.e., the oscillation frequencies of the mag-
netizations are identical. As a result, the MR does not show an
oscillation. Therefore, the magnetization oscillation will not be
detected by an experiment utilizing the MR effect. However, it
should be emphasized that this current region will be applicable
to MAMR because the magnetization oscillation is excited. In the
middle current region, two magnetizations oscillate with different
frequencies. When two magnetizations oscillate in a same direction
(clockwise or counterclockwise with respect to the zaxis), the fre-
quency of MR is the difference between those of two magnetizations.
Therefore, the frequency of MR decreases with increasing thecurrent in the positive current region. On the other hand, when two
magnetizations oscillate in the opposite direction, the frequency of
MR is the sum of those of two magnetizations, which can be found in
a narrow region of negative current. The frequency of the MR mainly
shows redshift, which is consistent with the experiment.23Similarly
to the small current region, the magnetization dynamics in the mid-
dle current region is also applicable to MAMR, where the oscillation
frequencies of two magnetizations are different, and therefore, the
MR shows an oscillation. On the other hand, complex dynamics are
found in the high current region, where the oscillation frequencies
ofmkand MR are not uniquely determined.
Figure 2(b) summarizes the Lyapunov exponent as a function of
the current density. The Lyapunov exponent in the low and middle
current regions are zero, indicating that the magnetization dynamics
are sustainable and periodic, as confirmed by the dynamical trajec-
tories in Figs. 1(a)–1(c). The positive Lyapunov exponents appear
in the high current region, indicating the existence of chaos in the
present STO. Interestingly, the Lyapunov exponent is always posi-
tive in the high positive current region, whereas it becomes either
positive or zero in the high negative current region. The abrupt
changes of the Lyapunov exponent between zero and positive in the
negative current region are similar to those often found in chaos sys-
tem, and indicate the appearance of multi-periodic or quasi-periodic
limit cycle.31Note that the magnetization dynamics in the high neg-
ative current region is highly nonlinear and complex, although the
dynamics is periodic, and thus, the Lyapunov exponent is zero, as
mentioned above and shown in Figs. 1(h) and 1(i). Therefore, an
oscillation frequency is not well-defined even in the region having
the zero Lyapunov exponent.
The results shown in Figs. 2(a) and 2(b) indicate that the
present STO is applicable to many kinds of practical devices. For
example, as repeated, the auto-oscillations of the magnetizations in
the low and middle current regions are applicable to MAMR, or
more widely, microwave generators. The wide frequency tunability
by the current found in Fig. 2(a) is an advantage for the applica-
tion of the microwave generator. We note that the previous experi-
ment on MAMR23focused on the negative current region, where the
electrons flow from the F 2to F 1layer. However, Fig. 2(a) indicates
FIG. 2 . (a) Current dependences of the oscillation frequencies of F 1(red square), F 2(blue triangle), and MR (black circle). The low (yellow-shaded) and middle (green-shaded)
current regions correspond to the synchronization and oscillations with different frequencies. The high (blue-shaded) current region corresponds to highly nonlinear dynamics.
(b) Lyapunov exponent as a function of the current density.
AIP Advances 10, 015112 (2020); doi: 10.1063/1.5127766 10, 015112-4
© Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv
that the positive current region might be suitable for MAMR because
it has a wide range of the current for the auto-oscillation.
The magnetization dynamics in the high current region is
not applicable to MAMR nor, more generally, microwave genera-
tor because the oscillation frequency of the magnetization is not a
unique value. However, the dynamics might be applicable to other
applications. For example, chaos having a positive Lyapunov expo-
nent might be used to random number generator. Also, the dynam-
ics between chaos and other dynamical phases will be of great
interest for brain-inspired computing,32,33such as reservoir com-
puting.34–41This is because highly nonlinear, not a simple auto-
oscillation found in the low current region, is necessary for the brain-
inspired computing, whereas chaos should be avoided to guarantee
the reproducibility of the computation against noise.
V. CONCLUSION
In conclusion, the magnetization dynamics in an STO with two
in-plane magnetized free layers was investigated by solving the LLG
equation numerically and evaluating the Lyapunov exponent. The
phase synchronization appears in the low current region, whereas
the magnetizations oscillate with different frequencies in the middle
current region. These dynamics will be applicable to MAMR. On the
other hand, the dynamics becomes highly nonlinear in the high cur-
rent region. The positive Lyapunov exponent found in this region
indicated the existence of chaos in the present STO.
ACKNOWLEDGMENTS
The author is thankful to Yuya Sakuraba, Weinan Zhou,
Nozomi Akashi, Kohei Nakajima, Hiroshi Tsukahara, Terufumi
Yamaguchi, and Sumito Tsunegi for valuable discussion. The author
is also grateful to Satoshi Iba, Aurelie Spiesser, Hiroki Maehara,
and Ai Emura for their support and encouragement. This paper
was based on the results obtained from a project (Innovative
AI Chips and Next-Generation Computing Technology Develop-
ment/(2) Development of next-generation computing technolo-
gies/Exploration of Neuromorphic Dynamics towards Future Sym-
biotic Society) commissioned by NEDO.
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© Author(s) 2020 |
1.4922868.pdf | Geometry effects on magnetization dynamics in circular cross-section wires
M. Sturma , J.-C. Toussaint, , and D. Gusakova,
Citation: Journal of Applied Physics 117, 243901 (2015); doi: 10.1063/1.4922868
View online: http://dx.doi.org/10.1063/1.4922868
View Table of Contents: http://aip.scitation.org/toc/jap/117/24
Published by the American Institute of PhysicsGeometry effects on magnetization dynamics in circular cross-section wires
M.Sturma,1,2J.-C. Toussaint,2,a)and D. Gusakova1,a)
1Univ. Grenoble Alpes, INAC-SPINTEC, F-38000 Grenoble, France; CNRS, SPINTEC, F-38000 Grenoble,
France; and CEA, INAC-SPINTEC, F-38000 Grenoble, France
2Univ. Grenoble Alpes, I. Neel, F-38000 Grenoble, France and CNRS, I. Neel, F-38000 Grenoble, France
(Received 9 February 2015; accepted 12 June 2015; published online 22 June 2015)
Three-dimensional magnetic memory design based on circular-cross section nanowires with modu-
lated diameter is the emerging field of spintronics. The consequences of the mutual interaction
between electron spins and local magnetic moments in such non-trivial geometries are still open to
debate. This paper describes the theoretical study of domain wall dynamics within such wires sub-jected to spin polarized current. We used our home-made finite element software to characterize
the variety of domain wall dynamical regimes observed for different constriction to wire diameter
ratios d/D. Also, we studied how sizeable geometry irregularities modify the internal micromag-
netic configuration and the electron spin spatial distribution in the system, the geometrical reasons
underlying the additional contribution to the system’s nonadiabaticity, and the specific domain wall
width oscillations inherent to fully three-dimensional systems.
VC2015 AIP Publishing LLC .
[http://dx.doi.org/10.1063/1.4922868 ]
I. INTRODUCTION
Non-trivial geometry of a magnetic structure subjected
to spin polarized current may strongly influence its dynamicsand the mutual interaction between electron spins and local
magnetic moments. Therefore, it is important to understand
how the different properties of a magnetic system in an ex-perimental framework can be changed through the modifica-
tion of geometry. This offers new challenges for current
spintronics theory and modelling. In particular, the accurate
mapping of any geometry and the self-consistent coupling of
spin transport and magnetization dynamics have become thetwo main requirements in this field of study. Consequently,
multipurpose micromagnetic finite element (FE) software is
attracting increasing attention. Its non-regular FE meshes
can accurately describe elliptical cross-section stacks,
notched circular shape nanowires, nanocontacts, and otherexperimental structures presenting crowding current effects.
Compared to regular geometries, sizeable irregularities
may strongly modify both the internal micromagnetic distribu-
tion of the system and spatial electron current distribution. Insuch cases, high current and magnetization gradients may
enhance the non-locality of spin transport effects and modify
the system properties. To quantify these phenomena, wemodel the current induced domain wall (DW) dynamics in a
three-dimensional notched metallic nanowire with a circular
cross-section [Fig. 1]. Such irregular geometries are of interest
for three-dimensional magnetic memory design based on self-
organized dense arrays of nanowires.
1–3Their modulated
diameter synthesized by electrodeposition in nanoporous
alumina templates2,4ensures DW position control within the
pinning centers by locally lowering DW magnetostatic
energy.5–8The diameters of such structures may approach
several tens of nanometers or less.9Although certain teams have studied current induced
DW dynamics in the presence of pinning centers theoreti-cally,
10–13the choice of specific geometries, as in Refs. 11
and13, lead to simplifications such as spatially uniform elec-
tron currents and simplified expressions for spin-transfer tor-que.
14,15In the case of the circular cross-section wires with a
modulated diameter studied here, these approximations may
be too drastic, particularly, when the diameter of the con-
stricted part becomes significantly smaller than that of thewider part.
II. MODEL
To understand the impact of geometry on spin transport
and magnetization dynamics, we used our home-made FE
software.16–18The advanced model implemented for this
study couples the Landau-Lifshitz-Gilbert equation (LLG)for magnetization dynamics
19in a micromagnetic approach
to diffusive transport equations.14,20–22In our approach, the
magnetic moments configuration takes into account the elec-tron spin distribution and vice-versa at every iteration. The
spatial distribution of spin-transfer torque is recalculated
from the spin distribution for every time step. It is subse-quently injected into a dynamical LLG equation that resetsthe new spin distribution for the next iteration. Several situa-tions, in which the feedback from the magnetization dynam-ics to the spin transport and back to the magnetizationdynamics can play an important role, have been discussedrecently.
23–25
The FE technique used in this paper is based on the
mathematically convergent integration scheme.17In particu-
lar, it uses specific tangent plane test functions for the FE
formulation imposed by the magnetization norm conserva-
tion and ensures the accurate description of the magnetiza-tion dynamics for realistic damping factors ( a/C2410
/C02). The
system’s behavior is described by variables: the unit vectoralong the local magnetization m¼M/M
Swith MSbeing thea)Authors to whom correspondence should be addressed. Electronic addresses:
jean-christophe.toussaint@neel.cnrs.fr and daria.gusakova@cea.fr.
0021-8979/2015/117(24)/243901/7/$30.00 VC2015 AIP Publishing LLC 117, 243901-1JOURNAL OF APPLIED PHYSICS 117, 243901 (2015)
saturation magnetization, the spin accumulation vector ns
(non-equilibrium carrier spin density) expressed in A/m, and
the scalar electrostatic potential uexpressed in V. The sys-
tem of coupled equations is written as follows:
@tm¼/C0c0ðm/C2Hef fÞþaðm/C2@tmÞþs/C01
sdM/C01
Sðns/C2mÞ;
(1)
X
k@kfC0@ku/C0beD0l/C01
Bðm/C1@knsÞg ¼ 0; (2)
X
k@kflBbC0e/C01ð@kuÞm/C0D0@knsg
¼/C0s/C01
sfns/C0s/C01
sdðns/C2mÞ: (3)
Equation (1)corresponds to the LLG equation augmented
with the spin-transfer torque term TSTin the non-local form
TST¼ssd/C01(ns/C2m), which introduces the mutual interac-
tion between the magnetization and the conducting electronspins. Equation (2)describes the charge current conserva-
tion. Finally, Eq. (3)sets the spin current conservation. The
index k¼x,y,z stands for the space coordinates with spatial
derivative @
k¼@/@k. We use the following notations for the
physical parameters and constants: c0for the gyromagnetic
ratio, Hefffor the effective field, which includes the
exchange and the demagnetizing fields, afor the Gilbert
damping factor, C0for the bulk conductivity, bfor the bulk
spin asymmetry, D0for the diffusion coefficient, efor the
electron charge, lBfor the Bohr magneton, ssdfor the s-d
exchange time linked to the exchange length lJ¼(D0ssd)1/2,
andssffor the spin-flip time linked to the spin-diffusion
length lsf¼(D0ssf)1/2. Since the time scales are of the order
of picoseconds for the spin accumulation and nanoseconds
for the magnetic moments,20we treated the spin accumula-
tion within the limit of long times and therefore neglectedits time derivative @
tnsin comparison to that of magnetic
moments @tm.
Figure 1shows the geometry studied: a cylindrical nano-
wire of diameter D¼20 nm with a circular cross-section
constriction of varying diameter d. The total length of the
simulated nanowire is L¼300 nm. The axis of the wire is
parallel to the zaxis. Furthermore, the corresponding mag-
netization component mzis considered as the longitudinal
magnetization, while the perpendicular component is the
transverse magnetization. The initial magnetic configurationcorresponds to a relaxed transverse tail-to-tail wall located at
the constriction center: z¼0. At lateral surfaces, the spin andelectron currents are tangent to it and we also used Brown
(Neumann) conditions for magnetization components. Weassume that spin accumulation tends to zero at the extremitiesn
s(zL)¼ns(zR)¼0 far from strong current and magnetization
gradients. The voltage applied at the extremities u(zL)¼0a n d
u(zR)¼þV0initiates the DW motion in the negative zdirec-
tion. The magnetic charges at the extremities are removednumerically to prevent magnetization reversal. The micro-magnetic parameters correspond to the permalloy materialwith M
S¼800/C2103A/m, c0¼2.21/C2105m/(A s), a¼0.02,
and the exchange constant Aex¼1/C210/C011J/m. The notched
wire is discretized into tetrahedrons, whose sizes do notexceed 2 nm. For the spin dependent transport parameters, weuse the following values: C
0¼4/C2106(1/(Xm),b¼0.7,
D0¼2.3/C210/C03(m2/s),lsf¼5 nm, and lJ¼1n m .
III. RESULTS AND DISCUSSION
A. Statics
Figures 2(a)and2(b) show the equilibrium distributions
of the longitudinal and transversal magnetizations. The blacklines highlight the averaged profiles along the wire axis(averaged over the sections perpendicular to the z-axis).
These distributions are obtained by the numerical relaxation
of the DW and correspond to the transverse-like DW. Notethat the wire geometry does not impose any lateral con-straints on the DW, contrary to the case of thin strips. TheDW distribution obtained is fully three-dimensional and weobserve that the presence of the geometrical constraint modi-fies the smooth distribution of a perfect wire. Additionally,Figs. 2(a)and2(b) show that the DW width shortens consid-
erably inside the pinning potential.
In order to characterize the full three-dimensional spatial
distributions of magnetic moments and electron spins, wecalculate corresponding standard deviations. The use of thismathematical quantity is the most appropriate in our case forseveral reasons. First, it allows us to estimate simultaneouslythe DW width and the width of electron spin distribution.Second, such DW width definition given below in this para-graph is quite general and does not impose any condition onthe DW shape and its dynamics. For example, the widelyused Thiele definition
26of the DW width corresponds to the
stationary DW displacement without changing its profile.These conditions do not hold in our case. In the following,we use the DW width term as that estimated from the calcula-
tion of the standard deviation of the square of longitudinal
magnetization r(m
z2). Note that, in the simplest case of the
one-dimensional Bloch distribution with mz¼tanh(z/DB) and
my¼cosh/C01(z/DB), the DW width DBis linked to r(mz2) via
r(mz2)¼(4DB/3Ltot)1/2, where Ltotis the total length of the
system and DB/C28Ltot. The low standard deviation value cor-
responds to the compact thin DW and the high value indi-cates a large DW. To obtain an approximation, we estimatethe following DW widths: /C248.9 nm for the perfect wire with
d¼D¼20 nm, /C244.2 nm for the notched wire with d¼12 nm
and/C242.6 nm for the notched wire with d¼10 nm. Thus, the
DW width decreases in the presence of the constriction. Its
structure becomes almost independent of the material param-
eters and is mostly determined by the constriction geometry
FIG. 1. Sketch of the notched cylindrical nanowire with varying constriction
diameter d.243901-2 Sturma, Toussaint, and Gusakova J. Appl. Phys. 117, 243901 (2015)for the small constriction diameters [Fig. 2(c)]. The latter
agrees with the theoretical prediction of Ref. 27.
The pinning potential in the presence of a notch gives rise
to a variation of the total energy of the system with the posi-
tion of the DW. In Fig. 2(d), we plot the total energy as a
function of the coordinate obtained from the free drift of the
DW towards the constriction in the absence of applied current.
During free drift towards the constriction, the width of theDW is modified non-monotonically, starting from the perfect
wire value [Fig. 2(e)]. The deepest pinning potential value and
smallest DW width value correspond to the most confined ge-ometry and the flat horizontal lines correspond to the perfect
wire. The balance between this geometry dependent pinningpotential and energy associated with the applied voltage deter-
mines the behavior of the DW.
B. Dynamics
Under the applied voltage, we observed different DW
behaviors which are summarized in a complex state diagramin Fig. 3. Depending on constriction diameter dand the
applied voltage value, several regimes can be distinguished:
(i) pinned DW, (ii) damped dynamical regime, (iii) damped-oscillating dynamical regime, and (iv) unstable dynamical
regime. The snapshots of micromagnetic configurations for
subsequent times as well as DW width time evolutions (i.e.,corresponding standard deviation r(m
z2)) are shown for three
dynamical regimes in Fig. 4. For the sake of convenience,
we use the normalized standard deviation rm¼r(mz2)/
rwire(mz2), where rwire(mz2) corresponds to the case of the
perfect wire. Thus, rm¼1 is a reference value. The per-
turbed system within irregular geometry evolves towards
this value far from the constriction. Note that we did notexpect to observe any dynamic modification of the DW
structure typical for the Walker breakdown with the collapse
of the micromagnetic structure. As shown in Ref. 28,f o r
example, in a perfect circular wire the transverse DW
behaves like a zero-mass micromagnetic object and its veloc-
ity is linearly dependent on the current density. In our caseof a notched wire, the DW should behave in the same man-
ner far from the constriction. Moreover, we work in a moder-
ate velocity range ( <400 m/s) and thus, below the magnonic
regime with a Cherenkov-like wave emission typical of
extremely high velocities.
29All the modifications of micro-
magnetic distribution are expected to be apparent in the vi-cinity of the pinning potential and vanish far from it.
Note that in experiments, the pinned versus dynamical
regimes could be distinguished, for example, by magneticforce microscopy after applying a current pulse. In the same
time, the differentiation between different types of dynami-
cal regimes is much more challenging to address. One possi-ble way to do so would be, for example, one-shot real-time
spin-motive force measurements. The details of such proce-
dure in a specific four-probe setup are given in Ref. 30.
FIG. 3. State diagram summarizing up different regimes obtained for differ-
ent constriction diameters dand applied voltage values.FIG. 2. (a) and (b) Equilibrium distribution of the longitudinal and trans-
verse magnetization as a function of zfor perfect and notched wires. The
solid black lines correspond to the averaged profiles. The vertical dashed
lines highlight the frontier of the constriction ( d¼10 nm). (c) Estimated
equilibrium DW width and standard deviation r(mz2) as a function of con-
striction diameter dfor two different materials: Py-like with MS¼800
/C2103A/m and Aex¼1/C210/C011J/m and Co-like with MS¼1400 /C2103A/m
andAex¼1.4/C210/C011J/m. (d) Pinning potential as a function of the distance
from the pinning center. (e) Normalized standard deviation rm¼r(mz2)/
rwire(mz2) as a function of the distance from the pinning center.243901-3 Sturma, Toussaint, and Gusakova J. Appl. Phys. 117, 243901 (2015)In the damped dynamical regime [Fig. 4(a)], the applied
voltage is strong enough to unpin the DW. The critical volt-age value needed to initiate DW movement is higher forsmaller diameters ddue to deeper pinning potential and
steeper slope [Fig. 3]. In this regime, the evolution of DW
width in time does not present any periodical behavior. Farfrom the constriction, the DW presents perfect wire behav-
ior. Under increasing voltage, this regime transforms either
into a damped-oscillating regime or an unstable dynamicalregime for particularly small diameters d.
The DW width changes periodically in damped-oscillating
dynamical regime [Fig. 4(b)]. The DW keeps its transverse-
like configuration and presents an oscillating forward move-
ment, remaining well localized in space. The decay of the
oscillation amplitude depends on the voltage applied. Thisdamped-oscillating regime is int ermediate between the damped
dynamical and unstable dynamical regimes.
Small diameters dand thus deep pinning potentials favor
the development of an unstable regime characterized by the
loss of the DW transverse-like configuration. The transverse-
like configuration reconstructs itself far from the constrictiononly for relatively small voltages [Fig. 4(c)]. Otherwise, the
behavior of the whole system is strongly perturbed and its
dynamics becomes chaotic. The latter must be avoided inpotential applications. The loss of the DW configuration andshape distortion result in a sharp rise of exchange energy.
Simultaneously, the evolution of DW width over time is
characterized by rough non-coherent behavior.
The evolution of the DW width in damped and damped-
oscillating regimes cannot be fitted with a simple mathemati-cal law due to the strong non-linearity of the coupled system.However, several similarities with the harmonic oscillatorresponse may exist on the external perturbation described by
Aexp(/C02pff
0t)sin(2 pfrtþw), where fis the attenuation (or
damping) ratio, f0is the eigen frequency, fr¼(1/C0f2)1/2f0is
the resonance frequency, wis the phase, and Ais the ampli-
tude. Harmonic oscillator theory presumes the existence of
different dynamical regimes depending on the attenuationratio value; for example, underdamped (0 <f<1) and over-
damped ( f>1) regimes. The latter is similar to the damped-
oscillating and damped regimes detected in our simulations.
We note that the attenuation ratio fin our simulations
depends on the balance between the Gilbert damping and driv-
ing force associated with spin-transfer torque. However, thedetermination of the exact relation between all the ingredients isnot trivial. Moreover, the accurate estimation of the attenuation
ratio and resonant frequency of a strongly perturbed non-linear
coupled system is not simple. Nevertheless, our simulationsshow that the attenuation ratio fincreases non-linearly with
increasing diameter d. For example, for the same damping con-
stant aand the same voltage 0.3 V, the estimation gives f¼0.2
ford¼10 nm and f¼0.3 for d¼12 nm. For the same geometry
(d¼12 nm), the rise in voltage from 0.3 V to 0.8 V results in the
decrease of the attenuation ratio from f¼0.3 to f¼0.1. Both
effects (the use of the large diameters dand low voltage values)
result in low spin-transfer torque efficiency and thus result inquicker relaxation of the perturbed system.
In the damped-oscillating regime for the same applied
voltage value, the initial swing amplitude increases withdecreasing diameter d[Fig. 5(a)]. The largest swing naturally
corresponds to the most confined and thus to the most per-
turbed configuration in comparison to the case of the perfectwire. We also observe that the initial swing amplitude variesslightly with the applied voltage [Fig. 5(b)]. Moreover, in
FIG. 4. Snapshots of the simulated DW configurations for subsequent instants and DW width time evolution for three dynamical regimes. The color scale b ar
represents the longitudinal magnetization amplitude and the white lines its isovalues. (a) Damped regime [ V¼0.1 V and d¼14 nm]. (b) Damped-oscillating re-
gime [ V¼0.3 V and d¼10 nm]. (c) Unstable regime [ V¼0.3 V and d¼6 nm].243901-4 Sturma, Toussaint, and Gusakova J. Appl. Phys. 117, 243901 (2015)this regime, we estimate the oscillation frequencies as being
close to that obtained by the relaxation of a perturbed mag-netic configuration in a perfect wire of the same diameter D
and material parameters in the absence of applied voltage
(f
r¼8.78 GHz). The whole coupled magnetization-spin sys-
tem seems to be dominated by a strongly perturbed magneticsubsystem response on the applied current.
Let us now study the details of the interaction between
the electron spin distribution and the magnetization distribu-tion in time in damped-oscillating regime. For the sake ofconvenience, we compare two normalized quantities: thestandard deviation of the square of longitudinal magnetiza-tionr
m¼r(mz2)/rwire(mz2) and the standard deviation of the
transverse spin accumulation rs¼r(jnstrj)/rwire(jnstrj), which
contribute to the spin-transfer torque term via the cross prod-uct (n
str/C2m).
Figure 6(a)depicts the time evolution of rm(t) and rs(t).
The initial time moment ( t¼0) corresponds to the DW
pinned in the center of the constriction. The solid horizontalblack line ( r¼1) corresponds to the case of the perfect wire,
for which the DW changes it position and orientation, whileits internal magnetic structure and thus the gradient of mag-netization remain unchanged. For the notched wire, r
m(t)
andrs(t) exhibit a damped-oscillating behavior around a
constant value. Both curves are in anti-phase far from theconstriction. The minima of r
m(t) corresponds to the com-
pact DW with high spatial gradients of magnetization. Thisresults in high spin polarization and gives rise to the maximaobserved for the corresponding r
s(t) curve. The grey arealimits the transition behavior in the vicinity of the constric-
tion. This area is smaller for a higher diameter dand thus for
less perturbed configurations.
We have also plotted the sn apshots of micromagnetic
configurations for successive times in Fig. 6(b).T h et i m e s
chosen correspond to the minima and maxima of rm(t)a n d
are highlighted in Fig. 6(a) by vertical arrows. The DW
exhibits oscillating forward movement and remains welllocalized in space. Its forward movement is accompanied
by the fan-like side oscillati ons of the transverse magnet-
ization in the x-y plane. This is shown schematically inFig. 6(c) for consecutive times s. Here, s
1ands5corre-
spond to the smallest DW width and minimum of rm, while
s3corresponds to the largest DW and maximum of rm.
These side oscillations of the t ransverse magnetization
m a yb eo b s e r v e di nf u l l yt h r ee-dimensional geometries.FIG. 5. Time evolution of the DW width (a) for applied voltage 0.3 V and
several constriction diameters d, (b) for d ¼12 nm and several values of
applied voltage.
FIG. 6. (a) Time evolution of the DW width and the width of spin accumula-tion distribution [ r
m¼r(mz2)/rwire(mz2) and rs¼r(jnstrj)/rwire(jnstrj)]. The
grey scale at the top of the graph corresponds to the distance travelled in
nm. (b) Snapshots of the simulated DW configurations at different instants.The corresponding times are indicated in (a) by vertical arrows. The
color scale bar represents the longitudinal magnetization amplitude and the
white lines its isovalues. In (a) and (b), the applied voltage is V
0¼0.3 V and
d¼10 nm. (c) Schematic illustration of the side DW excitation in the x-y
plane perpendicular to DW propagation for consecutive times s.243901-5 Sturma, Toussaint, and Gusakova J. Appl. Phys. 117, 243901 (2015)Indeed, in this case, the DW has an additional degree of
freedom in contrast to a conventional DW restrained withina thin strip. The x-y projection also reveals the preces-sional motion of the wall around the z-axis. The latter com-pares to Ref. 28.
The spin accumulations in the vicinity of the constric-
tion increase significantly [grey area on Fig. 6(a)]. This is
related to the electron current spatial distribution. Indeed, thenotched geometry not only defines the DW structure but also
it modifies the electron and spin current flows. In the pres-
ence of constriction, the electron current density is not ho-mogeneous and its amplitude in the center of the constrictiongrows drastically as diameter ddecreases [Fig. 7(c)].
Induced, current gradients contribute to modifying the spinaccumulation distribution. Thus, spin accumulation is pro-
portional to both the magnetization spatial gradient and cur-
rent spatial gradient. Figure 7(a) depicts the initial spatial
distribution of the transverse spin accumulation amplitudejn
strj. The peak of the spin accumulation observed in Fig.
7(a) results from the superposition of two contributions: the
first is proportional to the current gradient and the second isproportional to the magnetization gradient inside the con-
striction. Its amplitude decreases drastically as the cross-
section diameter dincreases and takes the smallest value for
the perfect wire with d¼D. It is possible to distinguish two
contributions for the subsequent time moment ( t
1>t0) [Fig.
7(b)], for which the DW travelled approximately the same
distance for all the geometries. Fig. 7(b) shows similar
shapes and amplitudes for the transversal spin accumulation
peak located around the DW center (right peak) for all the
geometries. The DW leaves the constriction and the wholesystem approaches the perfect wire behavior. The spin accu-mulation around this point is determined only by the magnet-ization gradient. In contrast, the amplitude of the left peaklocated at the constriction center varies greatly with the size
of the constriction. This residual spin accumulation is mostly
proportional to the spatial current gradient. Its amplitudevanishes rapidly as the cross-section of the constrictionincreases.High spin accumulation amplitude gives rise to high
DW driving force, thus to better current efficiency. The lattermay be quantified by calculating DW velocity. Figure 7(d)
shows the DW displacement as a function of time for differ-ent diameters dand the same applied voltage. In the case of
the prefect wire, the DW displacement is a linear function of
time and its slope corresponds to a DW velocity of 200 m/sfor the parameters chosen. This value compares with thatobtained in Ref. 28for the perfect wire and constant nona-
diabatic parameter b
NA¼(lJ/lsf)2¼0.04.14,15In contrast, in
the notched wire, the initial DW displacement deviates fromthe linear perfect wire behavior and recovers its position at a
certain distance from the constriction. In the vicinity of the
constriction, the steepness of the DW displacement sloperises dramatically when decreasing the constriction diameter.The latter corroborates the spin accumulation amplitudeplots in Fig. 7(a). The geometrical obstacle gives rise to the
additional nonadiabaticity mechanism, which amplifies theusual material dependent nonadiabaticity mechanism
reported, for example, in Refs. 23and24. In the structure
studied here, this geometrical contribution to the overall non-adiabaticity decays rapidly as a function of distance from theconstriction and has a weak influence on the resulting veloc-ity. Here, the effect of additional nonadiabaticity is visibleonly at very short times scales (less than 1 ns). After that the
behavior of the whole system is largely dominated by the
relaxation of the magnetic subsystem. But even in this case,the additional nonadiabaticity phenomenon should not beneglected. Indeed, its contribution may influence the deter-mination of the DW unpinning conditions or transition con-ditions between the different dynamical regimes essential forapplications. Thus, further systematic studies that take
account of different constriction shapes and material parame-
ters should be undertaken in order to optimize the workingconditions of potential memory devices.
Note that the use of the moderate wire diameters in the
present model ensured that the domain wall kept its internaltransverse-like wall structure throughout the simulation time.Larger diameters can lead to the so-called Bloch Point Wall
FIG. 7. (a) Amplitude of the transverse
spin accumulation distribution along
the wire axis at the initial time t0¼0
for two constriction sizes and for a per-
fect wire. The sketches in the upper part
helped us to visualize the DW position.
(b) The same as in (a) for subsequent
instant t1>t0. (c) Maximum electron
current density amplitude in the center
of the constriction. (d) DW displace-
ment as a function of time for several
constriction diameters dfor a perfect
wire. In (a), (b), and (c) the applied
voltage is V0¼0.3 V.243901-6 Sturma, Toussaint, and Gusakova J. Appl. Phys. 117, 243901 (2015)(BPW), whose dynamics are not sufficiently understood at
present. This is important to be able to distinguish the impact
of the additional current and magnetization gradients in theconfined system, to compare our results with the existing lit-
erature and avoid the artefacts related to the change of the
micromagnetic structure. Moreover, such system sizesensure reasonable computational times. Our on-going simu-lations for larger wire diameters and notches do not have a
qualitative impact on the results of the present paper involv-
ing wire to constriction diameter ratio D/d dependencies,
i.e., DW behavior in different dynamical regimes or addi-
tional nonadiabaticity mechanisms due to geometrical
reasons.
IV. CONCLUSION
In conclusion, we studied how sizeable geometry irregu-
larities may modify the internal micromagnetic configuration
and the electron spin spatial distribution in the system. Wereported the additional contribution to the system’s nonadia-
baticity due to geometrical reasons and found that this effect
to be significant over short time scales. The latter should betaken into account when determining DW unpinning condi-
tions. Moreover, we found that over long time scales the
relaxation of the magnetic subsystem had a predominanteffect on the whole system. The simulations allowed us to
summarize and characterize the variety of the DW dynamical
regimes observed for different constriction to wire diameterratios d/D. This study of different regimes revealed particular
DW width oscillations inherent to fully three-dimensional
systems.
We believe that our study will stimulate further investi-
gations into the mutual interaction between magnetic and
electron spin subsystems within complex geometries. Thisknowledge is crucial for opening new paths for magnetic
memory design. In addition to practical reasons, the study of
coupled dynamical phenomena within irregular geometriesprovides a unique opportunity for investigating the interac-
tions between physical phenomena and understanding how
an experimental issue may benefit from modifications in theproperties of systems due changes of geometry.
ACKNOWLEDGMENTS
We are grateful to O. Fruchart for a critical reading of
the manuscript. This work was funded by the AGIR Project(AGI14SMI15) and by the ANR Micro-MANIP (BLAN08-3
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1.2043236.pdf | Ultrafast direct writing scheme with unipolar field pulses for synthetic
antiferromagnetic magnetic random access memory cells
H. T. Nembach, C. Bayer, H. Schultheiss, M. C. Weber, P. Martin Pimentel, P. A. Beck, B. Leven, and B.
Hillebrands
Citation: Applied Physics Letters 87, 142503 (2005); doi: 10.1063/1.2043236
View online: http://dx.doi.org/10.1063/1.2043236
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/87/14?ver=pdfcov
Published by the AIP Publishing
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This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
140.254.87.149 On: Sun, 21 Dec 2014 14:43:11Ultrafast direct writing scheme with unipolar field pulses for synthetic
antiferromagnetic magnetic random access memory cells
H. T . Nembach,a/H20850C. Bayer, H. Schultheiss, M. C. Weber, P . Martin Pimentel,
P . A. Beck, B. Leven, and B. Hillebrands
Fachbereich Physik and Forschungsschwerpunkt MINAS, Technische Universität Kaiserslautern,
Erwin-Schrödinger-Strasse 56, 67663 Kaiserslautern, Germany
/H20849Received 15 April 2005; accepted 21 July 2005; published online 27 September 2005 /H20850
A writing scheme is presented for Savtchenko-type magnetic random access memory /H20849MRAM /H20850
cells, which allows for ultrafast direct writing with high stability against half select switching, usingtwo orthogonally oriented unipolar magnetic field pulses with time delay, which allows for
backswitching by reversing the temporal sequence of the two pulses. The numerical simulations arebased on the Stoner–Wohlfarth model and a Runge Kutta integration of the Landau–Lifshitz andGilbert equation. © 2005 American Institute of Physics ./H20851DOI: 10.1063/1.2043236 /H20852
The improvement of the operation of magnetic random
access memories /H20849MRAM /H20850is an important issue for future
data storage. MRAM is considered as a potential way torealize nonvolatile data storage with high long-time stability,fast access time, cost efficiency, and low power consump-tion. With increasing demand on the access time not only thequasistatic behavior of the MRAM storage cells is importantbut especially their dynamic magnetic properties. It is anultimate aim to reduce the access time down to the inverse ofthe clock frequency of the CPU. To achieve ultrafast mag-netic switching it is essential to develop elements withStoner–Wolfarth like magnetic properties, which show a welldefined coherent precessional switching behavior. Theswitching process needs to be extremely stable, in particularagainst half select switching, requiring insensitivity againstfield and parameter variations, which might exist due tomagnetic stray fields of neighboring elements as well asdue to cell-to-cell variations of material properties anddimensions.
Recently a new switching scheme has been introduced
for a 4 Mbit MRAM demonstrator using a toggle mode forswitching.
1,2In this case a synthetic antiferromagnet /H20849SAF /H20850
is chosen as the soft magnetic layer, which is the activeswitching part in these MRAM cells. Switching is initiatedby applying two orthogonally oriented, time delayed fieldpulses H
xandHyoriented in-plane at ±45° with respect to
the in-plane magnetic easy axis of the element, as displayedin Fig. 1. The underlying so-called Savtchenko switching
scheme
1,2is a quasistatic process, which is reported to show
good stability in a large field parameter range, in particularagainst half select switching. Disadvantages are the low
switching speed and that the switching process is a toggleprocess, i.e., for writing a bit a preread step is necessarywhich further increases the access time. Recent simulationsdemonstrate direct writing for precession-dominated switch-ing of SAF elements.
3
The main goal of our work reported in this paper is to
extend the Savtchenko-type MRAM scheme into the near-GHz switching regime and to allow for direct writing withunipolar field pulses, which allows for backswitching by re-versing the temporal sequence of the two pulses. We report anumerical study of the dynamic properties of circular shaped
thin film SAF elements with uniaxial in-plane anisotropy inthe geometry shown in Fig. 1. The simulations are based onthe Stoner–Wohlfarth macrospin approach and a Runge–Kutta integration of the Landau–Lifshitz and Gilbertequation.
4,5As the dimensions of the investigated nonbal-
anced SAF elements are in the micrometer range, each fer-romagnetic layer can be described by a single macrospin ingood approximation. Since we are considering a circularshaped thin film element the demagnetizing field can be as-
sumed to be homogeneous and described by a demagnetizing
tensor N
˜/H20849n/H20850/H20849Ref. 6 /H20850for the nth layer of the stack. The dimen-
sions of the nonbalanced shaped SAF are given by the radiusrand by the film thicknesses t
1andt2of the first and the
second ferromagnetic layer, respectively. The thickness ofthe nonmagnetic layer separating the two ferromagnetic lay-ers is indirectly considered by an effective coupling fieldstrength /H9011, which has been varied in the range 0 Oe /H33355/H9011
/H33355600 Oe.
The averaged dipolar field, which the magnetization of
layer mexerts on the magnetization of layer n, depends on
a/H20850Electronic mail: nembach@physik.uni-kl.de
FIG. 1. /H20849Color online. /H20850Schematic drawing of the geometry used in the
numerical simulations. A circular element of radius rwith uniaxial in-plane
anisotropy is employed. The red and the blue arrow indicate the magnetiza-tions M
1,2of the first and the second magnetic layer of the SAF, respec-
tively. The dashed lines mark the magnetic easy and the in-plane magnetichard axis of the element.APPLIED PHYSICS LETTERS 87, 142503 /H208492005 /H20850
0003-6951/2005/87 /H2084914/H20850/142503/3/$22.50 © 2005 American Institute of Physics 87, 142503-1
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
140.254.87.149 On: Sun, 21 Dec 2014 14:43:11the magnetization of layer m. For this highly symmetrical
case considered here, the dipolar field can be expressed in
the following way: Hdip/H20849n/H20850=D˜(m)M(m), with the diagonal ele-
ments of the tensor D˜as the only nonvanishing elements. A
uniaxial magnetocrystalline anisotropy is introduced by con-sidering an additional field H
uniwith the easy axis as shown
in Fig. 1. The Landau–Lifshitz and Gilbert equation of mo-tion is written for the case of two exchange coupled thinNi
81Fe19layers forming the SAF:
dM/H20849n/H20850
dt=− /H20841/H9253/H20841M/H20849n/H20850/H11003Heff/H20849n/H20850−/H20841/H9253/H20841/H9251
/H208491+/H92512/H20850MsM/H20849n/H20850
/H11003/H20849M/H20849n/H20850/H11003Heff/H20849n/H20850/H20850, /H208491/H20850
with n=1,2 the index of the layers, M/H20849n/H20850the magnetization of
each layer, Heff/H20849n/H20850the effective field, 4 /H9266Ms=10.8 kG the satu-
ration magnetization, /H9253=0.0176 Oe−1ns−1the gyromagnetic
factor and /H9251=0.008 the Gilbert damping factor.7,8The effec-
tive field for the two layers can be expressed as
Heff/H208491/H20850=Hpulse−N˜/H208491/H20850M/H208491/H20850−/H9011
Mst2
t1M/H208492/H20850+D˜/H208492/H20850M/H208492/H20850+Huni
/H208492/H20850
and
Heff/H208492/H20850=Hpulse−N˜/H208492/H20850M/H208492/H20850−/H9011
MsM/H208491/H20850+D˜/H208491/H20850M/H208491/H20850+Huni,/H208493/H20850
respectively. The first term on the right-hand side of Eqs. /H208492/H20850
and /H208493/H20850is the applied pulse field, the second term describes
the demagnetizing field, the third term considers the ex-change coupling between the two ferromagnetic layers givenby an effective exchange field /H9011in a mean field approach,
the fourth term is the dipolar interaction between the twolayers and the fifth term is an effective field representing theuniaxial anisotropy.
The pulse field H
pulseis chosen as a superposition of two
mutually orthogonal rectangular field pulses with the compo-nents H
xandHyaligned along the x- and y-axis displayed in
Fig. 1, pulse lengths TxandTyand a time separation of /H9004Txy.
Note, that the xy-coordinate system defined by the field pulse
directions is rotated by 45° in the film plane with respect tothe main coordinate system defined by the easy axis of thein-plane anisotropy /H20849see Fig. 1 /H20850.
In the widely used convention for the surface energy
density of the exchange coupling of two coupled ferromag-netic layers,
E=−J
1
MS2M/H208491/H20850M/H208492/H20850, /H208494/H20850
with J1/H110210 for antiferromagnetic coupling, the effective ex-
change field /H9011of Eqs. /H208492/H20850and /H208493/H20850corresponds to
/H9011=−J1
2MSt2. /H208495/H20850
The rectangular field pulses HxandHyare applied in the x-
andy-direction, respectively. The red and blue arrow in Fig.
1 indicate the magnetizations M1andM2of the two Ni 81Fe19
layers of the SAF element.
The switching behavior for a nonbalanced SAF element
with the dimensions r=400 nm and for different thicknesses
of the two Ni 81Fe19layers and different coupling strengthwas investigated. All simulations have been carried out with
a time resolution of 100 ps and for a time interval of 12 ns.
In Fig. 2 /H20849a/H20850an example of a direct writing scheme is
shown for a SAF element with t1=16 nm, t2=14 nm and a
coupling strength /H9011=350 Oe. The switching, respectively,
nonswitching behavior is plotted in a gray coded diagram asa function of the applied field pulse strengths H
xandHy. The
white areas indicate switching of the magnetization from theinitial state ‘1’ to the final state ‘0’ and the gray areas reflectnonswitching, respectively. The states ‘0’ and ‘1’ are definedin the following way: In both states the magnetizations ofeach layer lie along the easy axis of the anisotropy. In state‘1’M
1has a positive x component and M2has a negative x
component and vice versa in state ‘0.’
The field pulses with a length of Tx=Ty=0.8 ns are de-
layed against each other, i.e., the pulse Hxis applied at
T=0 ns, while the pulse Hyis applied at T=0.2 ns, i.e.,
/H9004Txy= +0.2 ns. The pulse sequence is sketched in the inset
of Fig. 2. This phase diagram reflects the high stability of theswitching process against half select switching, since nowhite regions appear along the pulse field axes, i.e., the mag-
FIG. 2. /H20849Color online. /H20850Diagram of the switching behavior of a nonbalanced
SAF element /H20849r=400 nm, t1=16 nm, t2=14 nm, coupling field strength
/H9011=350 Oe /H20850and uniaxial anisotropy field Huni=60Oe as a function of the
applied field pulse strengths Hx,Hy/H20849/H9011=350 Oe /H20850. White areas reflect
switching of the element, gray areas nonswitching. /H20849a/H20850The initial magneti-
zation state is chosen to be ‘1’ and is switched to ‘0.’ The pulse lengths areT
x=Ty=0.8 ns, the pulse delay /H9004Txy= +0.2 ns, which is sketched in the
inset. /H20849b/H20850Switching scheme with reversed field pulse sequence, i.e., /H9004Txy=
−0.2 ns describing the backswitching from ‘0’ to ‘1,’ which is the initial
state of the element in /H20849a/H20850./H20849c/H20850Time dependence of the normalized compo-
nent of the magnetization parallel to the easy axis of the circular element ofeach layer, M
1/H20849black curve /H20850,M2/H20849red curve /H20850, corresponding to a pulse
strength marked with a cross inside the red area.142503-2 Nembach et al. Appl. Phys. Lett. 87, 142503 /H208492005 /H20850
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
140.254.87.149 On: Sun, 21 Dec 2014 14:43:11netization cannot be switched by one field pulse alone.
Furthermore, it can be deduced that direct writing is pos-
sible, since the pattern shows no point symmetry. Thismeans, that the magnetization of the element does not switchin the same manner when reversing the polarity of the fieldpulses, i.e., no preread is necessary.
In the red marked area in quadrant I of Fig. 2 /H20849a/H20850, for both
positive field pulses a direct writing process from the initialmagnetization state ‘1’ to ‘0’ is possible. To reset the mag-netization state back to its initial state ‘1,’ it is not necessaryto change the field pulse polarity, but only the field pulsetiming sequence as demonstrated in Fig. 2 /H20849b/H20850. In analogy to
Fig. 2 /H20849a/H20850the same areas in quadrant I are marked. The ob-
tained white areas demonstrate in this case the backswitchingprocess from ‘0’ to the initial magnetization state ‘1’ with thesame unipolar pulses of reversed temporal order. There ispotential to increase the red area by further optimizing theparameters used in the simulation.
In detail, as can be seen in Fig. 2 /H20849c/H20850, the normalized
magnetization components of the two layers switch from 1 to−1 and from −1 to 1, respectively, followed by a few ringingperiods. Clearly a beating of the precessing magnetizationcan be observed, which is due to the coupling of the twolayers.
In conclusion, we demonstrate the feasibility of ultrafast
direct writing processes in the GHz regime applying uni-
polar field pulse sequences. Our numerical simulations yield
a high switching speed of about 2.5 ns which is a distinctimprovement of the switching speed presented in Ref. 1. Thechosen SAF system and measurement architecture provide
reliable and coherent precessional switching with very highstability against half select switching as well as against fieldand parameter variations. In particular, the new concept ofultrafast direct writing applying unipolar field pulse se-
quences provides a less power consuming process compared
to standard MRAM architectures, which is of fundamentalimportance for fast storage application concepts.
This work is supported by the European Commission
within the EU-RTN ULTRA-SWITCH /H20849HPRN-CT-2002-
00318 /H20850, the Studienstiftung des Deutschen Volkes /H20849C.B. /H20850and
the Graduiertenkolleg 792 /H20849M.C.W. /H20850of the Deutsche
Forschungsgemeinschaft. The authors would like to thankP. Candeloro for helpful discussions.
1B. N. Engel, J. Akerman, B. Butcher, R. W. Dave, M. Durlam, G.
Grynkewich, J. Janesky, S. V. Pietambaram, N. D. Rizzo, J. M. Saughter,K. Smith, J. J. Sun, and S. Tehrani, IEEE Trans. Magn. 41,1 3 2 /H208492005 /H20850.
2L. Savtchenko, A. A. Korkin, B. N. Engel, N. D. Rizzo, M. F. Deherrera,
and J. A. Janesky, U.S. Patent No. 6,545,906 B1, April 8 /H208492003 /H20850.
3J.-V. Kim, T. Devolder, and C. Chappert, Appl. Phys. Lett. 85, 4094,
/H208492004 /H20850.
4M. Bauer, J. Fassbender, B. Hillebrands, and R. L. Stamps, Phys. Rev. B
61, 3410 /H208492000 /H20850.
5J. Fassbender, Spin Dynamics in Confined Magnetic Structures II , edited
by B. Hillebrands and K. Ounadjela /H20849Springer, Berlin, 2003 /H20850.
6M. Hanson, C. Johansson, B. Nilsson, P. Isberg, and R. Wäppling, J. Appl.
Phys. 85,2 7 9 3 /H208491999 /H20850.
7C. E. Patton, Z. Frait, and C. H. Wilts, J. Appl. Phys. 46, 5002 /H208491975 /H20850.
8M. R. Freeman, W. Hiebert, and A. Stankiewicz, J. Appl. Phys. 83, 6217
/H208491998 /H20850.142503-3 Nembach et al. Appl. Phys. Lett. 87, 142503 /H208492005 /H20850
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
140.254.87.149 On: Sun, 21 Dec 2014 14:43:11 |
1.4953454.pdf | Platinum/yttrium iron garnet inverted structures for spin current transport
Mohammed Aldosary , Junxue Li , Chi Tang , Yadong Xu , Jian-Guo Zheng , Krassimir N. Bozhilov , and Jing Shi
Citation: Appl. Phys. Lett. 108, 242401 (2016); doi: 10.1063/1.4953454
View online: http://dx.doi.org/10.1063/1.4953454
View Table of Contents: http://aip.scitation.org/toc/apl/108/24
Published by the American Institute of Physics
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Mohammed Aldosary,1Junxue Li,1ChiTang,1Y adong Xu,1Jian-Guo Zheng,2
Krassimir N. Bozhilov,3and Jing Shi1
1Department of Physics and Astronomy and SHINES Energy Frontier Research Center,
University of California, Riverside, California 92521, USA
2Irvine Materials Research Institute, University of California, Irvine, California 92697, USA
3Central Facility for Advanced Microscopy and Microanalysis, University of California, Riverside,
California 92521, USA
(Received 24 March 2016; accepted 1 May 2016; published online 13 June 2016)
30-80 nm thick yttrium iron garnet (YIG) films are grown by pulsed laser deposition on a 5 nm
thick sputtered Pt atop gadolinium gallium garnet substrate (GGG) (110). Upon post-growth rapid
thermal annealing, single crystal YIG(110) emerges as if it were epitaxially grown on GGG(110)despite the presence of the intermediate Pt film. The YIG surface shows atomic steps with the root-
mean-square roughness of 0.12 nm on flat terraces. Both Pt/YIG and GGG/Pt interfaces are atomi-
cally sharp. The resulting YIG(110) films show clear in-plane uniaxial magnetic anisotropy with awell-defined easy axis along h001iand a peak-to-peak ferromagnetic resonance linewidth of 7.5 Oe
at 9.32 GHz, similar to YIG epitaxially grown on GGG. Both spin Hall magnetoresistance and
longitudinal spin Seebeck effects in the inverted bilayers indicate excellent Pt/YIG interfacequality. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4953454 ]
Magnetic garnets are important materials that offer
unique functionalities in a range of bulk and thin film deviceapplications requiring magnetic insulators.
1,2Among all
magnetic insulators, yttrium iron garnet (Y 3Fe5O12or YIG)
has been most extensively used in various high-frequencydevices such as microwave filters, oscillators, and Faradayrotators
3due to its attractive attributes including ultra-low
intrinsic Gilbert damping constant ( aas low as 3 /C210/C05),4
which is two orders of magnitude smaller than that of ferro-
magnetic metals, high Curie temperature (T C¼550 K), soft
magnetization behavior, large band gap ( /C242.85 eV),5and
relatively easy synthesis in single crystal form. These con-ventional applications demand bulk YIG crystals or micron-thick films grown by liquid phase epitaxy.
6For more recent
spintronic studies such as the spin Seebeck effect (SSE)7and
spin pumping,8submicron- or nanometer-thick films are typ-
ically grown by pulsed laser deposition (PLD) or sputtering.It has been shown that high-quality YIG films can be epitax-ially grown directly on gadolinium gallium garnet (GGG)substrates due to the same crystalline structure and a verysmall lattice mismatch of 0.057%.
9–11To form bilayers, a
thin polycrystalline metal layer is typically deposited on topof YIG by sputtering, which results in reasonably good inter-faces for spin current transport.
7,8,12For some studies such
as the magnon-mediated current drag,13,14sandwiches of
metal/YIG/metal are required, in which YIG needs to beboth magnetic and electrically insulating. However, high-quality bilayers of the reverse order, i.e., YIG on metal, arevery difficult to be fabricated. A main challenge is that theYIG growth requires high temperatures and an oxygen envi-ronment
15which can cause significant inter-diffusion, oxida-
tion of the metal layer, etc., and consequently lead to poorstructural and electrical properties in both metal and YIGlayers.
This letter reports controlled growth of high-quality sin-
gle crystal YIG thin films ranging from 30 to 80 nm inthickness on a 5 nm thick Pt layer atop Gd
3Ga5O12or GGG
(110) substrate. Combined with low-temperature growthwhich suppresses the inter-diffusion, subsequent rapid ther-mal annealing (RTA) and optimization of other growth pa-rameters result in well-defined magnetism, atomically sharpPt/YIG interface, and atomically flat YIG surface. In addi-tion, despite the intermediate Pt layer that has a drasticallydifferent crystal structure from the garnets, the top YIG layershows desired structural and magnetic properties as if it wereepitaxially grown on GGG (110).
5/C25m m
2of commercial GGG (110) single crystal sub-
strates are first cleaned in ultrasonic baths of acetone, isopro-pyl alcohol, then deionized water, and dried by pure nitrogengun. Subsequently, the substrates are annealed in a furnaceat 900
/C14Ci nO 2for 8 h which produces atomically flat sur-
face. Atomic force microscopy (AFM) is performed to trackthe surface morphology of the annealed substrates. Figure1(a) shows the 2 /C22lm
2AFM scan of an annealed GGG
(110) substrate. Flat atomic terraces are clearly present andseparated with a step height of 4.4 60.2 A˚which is equal to
1
4of the face diagonal of the GGG unit cell or the (220) inter-
planar distances of 4.4 A ˚of GGG. The 4.4 A ˚distance is the
separation between the GaO 6octahedral layers parallel to
(110) that might be defining the observed atomic step ledges.The root-mean-square (RMS) roughness on the terraces is/C240.74 A ˚. Then, the substrate is transferred into a sputtering
chamber with a base pressure of 5 /C210
/C08Torr for Pt deposi-
tion. DC magnetron sputtering is used with the Ar pressureof 5 mTorr and power of 37.5 W. The sputtering depositionrate is 0.76 A ˚/s, and sample holder rotation speed is 10 rpm.
After the 5 nm thick Pt deposition, the surface of the Pt filmis found to maintain the atomic terraces of the GGG (110)substrate, except that the RMS roughness on the Pt terracesis increased to 1.05 A ˚as shown in Figure 1(b). It is rather
surprising that the 5 nm thick Pt layer does not smear out theterraces separated by atomic distances given that the
0003-6951/2016/108(24)/242401/5/$30.00 Published by AIP Publishing. 108, 242401-1APPLIED PHYSICS LETTERS 108, 242401 (2016)
sputtering deposition is not particularly directional.
Strikingly, terraces are still present even in 20 nm thick Pt(not shown). The substrates are then put in a PLD chamber
which has a base pressure of 4 /C210
/C07Torr, and are slowly
heated to 450/C14C in high-purity oxygen with the pressure of
1.5 mTorr with 12 wt. % of ozone. The krypton fluoride(KrF) coherent excimer laser ( k¼248 nm, 25 ns/pulse) used
for deposition has a pulse energy of 165 mJ/pulse, and repe-
tition rate of 1 Hz. The deposition rate of /C251.16 A ˚/min is
achieved with a target to substrate distance of 6 cm. After
deposition, the YIG films are ex situ annealed at 850
/C14C for
200 s using rapid thermal annealing (RTA) under a steadyflow of pure oxygen. After RTA, the surface morphology is
examined by AFM again. Figure 1(c) shows the atomically
terraced surface of a 40 nm thick YIG film with RMS of1.24 A ˚on the terrace. In this study, the thickness of YIG
ranges from 30 to 80 nm and all samples exhibit clear atomic
terraces. Even though YIG is annealed at such a high temper-ature, with the short annealing time, the flat and smooth YIGsurface is maintained.
To track the structural properties of YIG, we use
RHEED to characterize the YIG surface at every step of the
process. Figure 1(d) shows the RHEED pattern of the as-
grown YIG surface. It clearly indicates the absence of anycrystalline order. After the ex situ RTA, the sample is intro-
duced back to the PLD chamber for RHEED measurements
again. A streaky and sharp RHEED pattern is recovered asdisplayed in Figure 1(e) which suggests a highly crystalline
order. This result is particularly interesting since it shows the
characteristic RHEED pattern of YIG grown on GGG.
10
To further confirm its crystalline structure, x-ray diffrac-
tion (XRD) using the Cu K a1line has been carried out over a
wide angle range (2 hfrom 10/C14to 90/C14) on the GGG/Pt/YIG
sample discussed in Figure 2(a). Because of the close match
in lattice constants between YIG and GGG substrate, weakYIG peaks are completely overlapped with strong peaks ofGGG so that they are indistinguishable. Three main Bragg
peaks of YIG and GGG are observed: 220, 440, and 660,
which suggests the (110) growth orientation of both YIG andGGG. No individual weak YIG peaks can be found. It is
striking that the YIG film adopts the crystallographic orienta-tion of GGG despite the intermediate Pt layer. By comparingwith the spectra of YIG grown directly on GGG, we canidentify a new peak (2 h/C2540.15
/C14) which is better seen in the
zoom-in view in the inset of Figure 2(a). We determine this
as the 111 peak of the 5 nm thick Pt film that suggests the(111) texture of the Pt layer. It is not clear whether the (111)texture in the intermediate Pt layer is required for YIG to de-velop the same crystallographic orientation as that of the
GGG substrate.
The locking of the (110) orientation in both YIG and
GGG is further investigated by the high-resolution transmis-
sion electron microscopy (HRTEM) in real space. Figure2(b) first reveals sharp and clean interfaces of Pt/YIG and
GGG/Pt. No amorphous phase or inclusions are visible atthese two interfaces. Furthermore, the (110) atomic planes of
YIG and GGG are parallel to each other and show very
closely matched inter-planar spacing. Despite the Pt layer inbetween, the crystallographic orientation of YIG is not inter-rupted as if it were epitaxially grown on GGG directly. Inthe selected area electron diffraction pattern shown in Figure
2(c), taken along the h112izone axis in garnet from an area
that includes all three phases, YIG and GGG diffractionspots overlap with each other, consistent with the XRDresults. There is minor splitting of the 110 type reflectionsfrom the two garnet phases due to a slight rotation of the two
garnet lattices of less than 0.5
/C14. Surprisingly, the diffraction
spots from the 5 nm Pt layer show a single crystal patternwith minor streaking parallel to 111 in Pt. The diffuse char-acter of the Pt reflections suggests that Pt is essentially a sin-
gle crystal consisting of small (few nanometers) structural
domains with minor misalignments. The contrast variation indifferent regions of Pt shown in Figure 2(b) is consistent
with such small structural domain misalignments in Pt crys-tal grain orientations. Furthermore, the 111 reciprocal vector
of Pt and the 110 reciprocal vector of YIG/GGG are both
perpendicular to the interfaces, indicating that the (111) Ptlayers are parallel to the (110) layers of both GGG and YIG.
FIG. 1. Surface characterization of YIG
thin film grown on GGG(110)/Pt (5 nm).
(a)–(c) 2 lm/C22lm AFM scans of
GGG(110) substrate, GGG(110)/Pt(5 nm),and GGG/Pt(5 nm)/YIG(40 nm), respec-
tively. RHEED patterns of as-grown (d)
and annealed (e) GGG(110)/Pt (5 nm)/
YIG(40 nm).242401-2 Aldosary et al. Appl. Phys. Lett. 108, 242401 (2016)Figure 2(d) is a HRTEM image with high magnification of
the three layers. It further reveals atomically sharp interfaces,
interlocked (110) crystallographic orientations betweenGGG and YIG, and single crystal (111)-oriented Pt.
To investigate the magnetic properties of the GGG/
Pt/YIG inverted heterostructure, vibrating sample magne-tometry (VSM) measurements are carried out at room tem-perature. As-grown YIG films do not show any well-defined
crystalline structure as indicated by the RHEED pattern. In
the meantime, the VSM measurements do not show any de-tectable magnetization signal. Upon RTA, single crystal YIGbecomes magnetic as shown by the hysteresis loops inFigure 3(a) for magnetic fields parallel and perpendicular to
the sample plane. GGG’s paramagnetic contribution has
been removed by subtracting the linear background from theraw data. The easy axis of all YIG films with different thick-nesses lies in the film plane due to the dominant shape ani-
sotropy. The coercivity falls in the range of 15–30 Oe for
different thicknesses, which is larger than the typical value(0.2–5 Oe)
9–11for YIG films grown on lattice-matched
GGG. The inset of Figure 3(a) shows a coercive field of
29 Oe for a 40 nm thick YIG film. The saturation magnetic
field in the perpendicular direction is /C241800 Oe, which cor-
responds well to 4 pMsfor bulk YIG crystals (1780 Oe).
Magnetic hysteresis loops are measured along differentdirections in the film plane. Figures 3(b) and3(c) show the
polar angular dependence of both the coercively field (H
c)
and squareness (M r/Ms), where M ris the remanence and M s
is the saturation magnetizations, respectively. In the filmplane, there is clear uniaxial magnetic anisotropy, with the
in-plane easy and hard axes situated along h001iatu¼145/C14
andh110iatu¼55/C14, respectively. This two-fold symmetry
indicates that the magneto-crystalline anisotropy is the main
source of the anisotropy since it coincides with the lattice
FIG. 2. Structure characterization of
GGG/Pt/YIG heterostructure. (a) XRD
of YIG film (40 nm) grown on
GGG(110)/Pt (5 nm). Inset: zoom-in
plot of Pt 111 peak (2 h¼40.15/C14). (b)
TEM image of GGG (110)/Pt (5 nm)/
YIG (110) (40 nm) heterostructure.
The h1/C2211iandh110idirections in
GGG are shown for reference. (c)
Selected area electron diffraction pat-
tern along ½/C22112/C138zone axis in GGG
obtained from an area containing all
three layers showing diffraction spots
of YIG, GGG, and Pt. The garnet
reflections are labeled with subscript“g” and Pt ones with “p.” (d) HRTEM
lattice image along the ½/C22112/C138zone axis
in garnet shows that (110) planes in
both YIG and GGG are parallel to the
interface with the Pt film, and the latter
is composed of nanometer size crystal-
line domains oriented with their (111)lattice planes parallel to the interface
as well. Slight bending and disruption
of the (111) lattice fringes between ad-
jacent Pt domains are visualized.
FIG. 3. Magnetic properties of GGG(110)/Pt(5 nm)/YIG (40 nm) (a) Roomtemperature normalized magnetic hysteresis loops of YIG (40 nm)/Pt
(5 nm)/GGG (110) with magnetic field applied in-plane and out-of-plane.
Inset: in-plane hysteresis loop at low fields. Polar plots of coercive field H
c
(b) and squareness M r/Ms(c) as the magnetic field His set in different orien-
tations in the (110) plane ( H//h112iat 0/C14). (d) FMR absorption derivative
spectrum of YIG/Pt/GGG at an excitation frequency of 9.32 GHz.Lorentzian fit (red line) shows a single peak with a peak-peak distance of
7.5 Oe.242401-3 Aldosary et al. Appl. Phys. Lett. 108, 242401 (2016)symmetry of (110) surface of the YIG films, which is also
consistent with the magnetic anisotropy property of YIG epi-
taxially grown on GGG (110).10
Ferromagnetic resonance (FMR) measurements of YIG
films are carried out using Bruker EMX EPR (Electron
Paramagnetic Resonance) spectrometer with an X-band
microwave cavity operated at the frequency of f ¼9.32 GHz.
A static magnetic field is applied parallel to the film plane.Figure 3(d) shows a single FMR peak profile in the absorp-
tion derivative. From the Lorentzian fit, the peak-peak line-
width ( DH
pp) and resonance frequency (H res) are 7.5 Oe and
2392 Oe, respectively. In literature, both the linewidth and
the saturation magnetization vary over some range depend-ing on the quality of YIG films. These values are comparable
with the reported values for epitaxial YIG films grown
directly on GGG.
9–11The FMR linewidth here seems to be
larger than what is reported in the best YIG films grown on
GGG. Considering the excellent film quality, it is reasonable
to assume that the same YIG would have similar FMR line-
width, e.g., 3 Oe. In the presence of Pt, increased damping inPt/YIG occurs due to spin pumping.
16,17This additional
damping can explain the observed FMR linewidth (7.5 Oe) if
a reasonable spin mixing conductance value of g"#
ef f
/C255/C21018m/C02is assumed.
The Pt layer underneath YIG allows for pure spin current
generation and detection just as when it is placed on top. It is
known that the interface quality is critical to the efficiency of
spin current transmission.18,19To characterize this property,
we perform spin Hall magnetoresistance (SMR) and SSE
measurements in GGG/Pt/YIG inverted heterostructures.
SMR is a transport phenomenon in bilayers of heavy
metal/magnetic insulator.12,20,21A charge current flowing in
the normal metal with strong spin-orbit coupling generates a
spin current orthogonal to the charge current via the spin
Hall effect. The reflection and absorption of this spin currentat the interface of the normal metal/magnetic insulator
depends on the orientation of the magnetization ( M) of the
magnetic insulator. Due to the spin transfer torque mecha-
nism, when Mis collinear with the spin polarization r,
reflection of the spin current is maximum. In contrast, when
Mis perpendicular to r, absorption is maximum; therefore, the
resistance of the normal metal is larger than that for Mkr
since the absorption behaves as an additional dissipation
channel. Metal/magnetic insulator interface quality affects
the SMR magnitude. As illustrated in Figure 4(a), we carry
out angle-dependent magnetoresistance (MR) measurements
by rotating a constant magnetic field in the xy-(H¼2000 Oe), xz- (H ¼1 T), or yz-plane (H ¼1 T), while
the current flows along the x-axis. The angular dependence
of the MR ratio,
Dq
q%ðÞ¼qangleðÞ /C0qðangle ¼p
2Þ
qðangle ¼p
2Þ/C2100, for Pt
film at room temperature is summarized in Figure 4(b).
According to the SMR theory,21the longitudinal resistivity
reads
q¼q0þq1m2
y; (1)
where q0andq1are magnetization-independent constants,
andmyis the y-component of the magnetization unit vector.
The red solid curves in Figure 4(b) can be well described byEquation (1). Here, the magnitude of SMR in xy- and yz-
scans is on the same order as that in normal YIG/Pt bilayer
systems. Therefore, we demonstrate that the SMR mecha-nism dominates in our devices, which indicates excellentinterface quality for spin current transport.
SSE, on the other hand, is related to the transmission of
thermally excited spin currents through the heavy metal/YIGinterface.
22–24As illustrated in Figure 4(c), we first deposit a
300 nm thick Al 2O3layer atop GGG(110)/Pt(5 nm)/
YIG(40 nm), and a top heater layer consisting of 5 nm Cr and50 nm Au. When an electrical current (50 mA) flows in theCr/Au layer, a temperature gradient is established along the
z-direction by Joule heating, which generates a spin current
in YIG. As the spin current enters the Pt layer, it is convertedinto a charge current or voltage due to the inverse spin Halleffect. A magnetic field is applied in the y-direction whilethe voltage is detected along the x-direction. In Figure 4(d),
we plot the field dependence of the normalized SSE signal at
300 K, which is consistent with the SSE magnitude reported
in YIG/Pt bilayers.
24Therefore, we have confirmed the
excellent interface quality for transmitting thermally excitedspin currents.
In summary, single crystal YIG thin films have been
grown on Pt film which is sputtered on GGG (110) substrate.RHEED and AFM show excellent YIG surface quality andmorphology. XRD and HRTEM further reveal an intriguingcrystal orientation locking between YIG and GGG as if no Ptwere present. These YIG films exhibit similar excellent mag-netic properties to those of the YIG films grown epitaxially
on GGG (110). Both SMR and SSE results confirm that the
superb structural and magnetic properties lead to excellentspin current transport properties.
We would like to thank Professor J. Garay and N. Amos
for the technical assistance and fruitful discussions. Bilayergrowth control, growth characterization, device fabrication
FIG. 4. SMR and longitudinal SSE of GGG(110)/Pt(5 nm)/YIG(40 nm). (a)
Illustrations of measurement geometry of SMR. a,b, and care angles
between Hand y, z, and z, axes, respectively. The magnitude of His
2000 Oe, 1 T, and 1 T for a-,b-, and c- scans, respectively. (b) Angular de-
pendence of SMR ratios for three measurement geometries at 300 K. (c) The
sample structure and measurement geometry of longitudinal SSE. The heater
current I is 50 mA and His applied along the y direction. All the thicknesses
are denoted in nanometers (nm). (d) Field dependence of room temperatureSSE signal, which is normalized by the heating power P and detecting length
L.242401-4 Aldosary et al. Appl. Phys. Lett. 108, 242401 (2016)and electrical transport measurements at UCR were
supported as part of the SHINES, an Energy FrontierResearch Center funded by the U.S. Department of Energy,Office of Science, Basic Energy Sciences under Award No.
SC0012670. Part of the transmission electron microscopy
was performed on a 300 kV FEI Titan Themis at the CentralFacility for Advanced Microscopy and Microanalysis at UCRiverside, supported by UCR campus funding. The TEMspecimen preparation was performed at the Irvine Materials
Research Institute (IMRI) at UC Irvine, using
instrumentation funded in part by the National ScienceFoundation Center for Chemistry at the Space-Time Limitunder Grant No. CHE-0802913.
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5.0025124.pdf | Appl. Phys. Lett. 117, 202401 (2020); https://doi.org/10.1063/5.0025124 117, 202401
© 2020 Author(s).Magnetic skyrmionium diode with a
magnetic anisotropy voltage gating
Cite as: Appl. Phys. Lett. 117, 202401 (2020); https://doi.org/10.1063/5.0025124
Submitted: 13 August 2020 . Accepted: 05 November 2020 . Published Online: 17 November 2020
Junlin Wang ,
Jing Xia ,
Xichao Zhang ,
Xiangyu Zheng , Guanqi Li , Li Chen ,
Yan Zhou ,
Jing Wu ,
Haihong Yin ,
Roy Chantrell , and Yongbing Xu
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Applied Physics Letters 117, 190501 (2020); https://doi.org/10.1063/5.0032368Magnetic skyrmionium diode with a magnetic
anisotropy voltage gating
Cite as: Appl. Phys. Lett. 117, 202401 (2020); doi: 10.1063/5.0025124
Submitted: 13 August 2020 .Accepted: 5 November 2020 .
Published Online: 17 November 2020
Junlin Wang,1,2
Jing Xia,3
Xichao Zhang,2,3
Xiangyu Zheng,1,2
Guanqi Li,2,4LiChen,5YanZhou,3,6,a)
Jing Wu,2,4
Haihong Yin,7
RoyChantrell,4
and Yongbing Xu1,2,a)
AFFILIATIONS
1Department of Electronic Engineering, University of York, York YO10 5DD, United Kingdom
2York-Nanjing International Center of Spintronics (YNICS), Nanjing University, Nanjing 210093, China
3School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China
4Department of Physics, University of York, York YO10 5DD, United Kingdom
5Faculty of Engineering, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, United Kingdom
6Key Laboratory of Magnetic Molecules and Magnetic Information Materials of Ministry of Education, Linfen 041004, China
7School of Information Science and Technology, Nantong University, Nantong 226019, China
a)Authors to whom correspondence should be addressed: zhouyan@cuhk.edu.cn andyongbing.xu@york.ac.uk
ABSTRACT
The magnetic skyrmionium can be seen as a coalition of two magnetic skyrmions with opposite topological charges and has potential
applications in next-generation spintronic devices. Here, we report the current-driven dynamics of a skyrmionium in a ferromagnetic nano-track with the voltage-controlled magnetic anisotropy. The pinning and depinning of a skyrmionium controlled by the voltage gate are inves-tigated. The current-driven skyrmionium can be used to mimic the skyrmionium diode effect in the nanotrack with a voltage gate. We have
further studied the skyrmionium dynamics in the nanotrack driven by a magnetic anisotropy gradient in the absence of spin current. The
performance of a single wedge-shaped voltage gate at different temperatures is studied. Our results may provide useful guidelines for thedesign of voltage-controlled and skyrmionium-based spintronic devices.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0025124
Magnetic skyrmions were predicted theoretically in 1989,
1and
since then, the creation, annihilation, and manipulation of magnetic
skyrmions have been widely investigated in the field of magnetism andspintronics.
2–11Magnetic skyrmions are particle-like nanoscale objects
and can be found in certain ferromagnetic bulk materials, thin films,and multilayers,
2–11where skyrmions are stabilized by a competition
between the Heisenberg exchange interaction, Dzyaloshinskii–Moriya
interaction (DMI), perpendicular magnetic anisotropy (PMA), andmagnetic field. One of the most important applications of magneticskyrmions is their use as information carriers in nanoscale spintronicdevices,
7–11where skyrmions can be driven by spin-transfer torques,
spin–orbit torques, and spin waves. The skyrmion-based devices could
have a lower power consumption or higher operation speed comparedwith the domain wall-based devices.
7–13
However, the skyrmion Hall effect (SkHE)14–16could be an
obstacle for the collimated transmission of skyrmions in narrow
nanoscale devices.17,18The SkHE has been observedexperimentally.15,16It is caused by the Magnus force acting on the
moving skyrmion and can lead to the destruction of skyrmions at
the device edges. One promising approach to avoid the SkHE is tocreate a topological spin texture with a zero net topological charge.For example, the synthetic antiferromagnetic bilayer skyrmion witha topological charge of Q¼0 is free from the SkHE.
17–20In this sys-
tem, the two exchange-coupled magnetic skyrmions in the top and
bottom layers have opposite Q, leading to a total topological charge
of zero.
On the other hand, a magnetic skyrmionium is also a topologi-
cal spin texture with Q¼0.21–39It has a doughnut-like out-of-plane
spin structure and can be seen as the combination of two skyrmions
with opposite Q. The magnetic skyrmionium can be generated by
ultra-fast laser pulses and has been observed experimentally38to be
stable for over 12 months. Due to the zero topological charge, themagnetic skyrmionium is free from the SkHE. The dynamics of the
skyrmionium have been studied theoretically
21–35and observed in
Appl. Phys. Lett. 117, 202401 (2020); doi: 10.1063/5.0025124 117, 202401-1
Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplexperiments,36–39showing the potential of using skyrmioniums in
next-generation spintronic devices.
In this work, we report a numerical study of current-induced sky-
rmionium dynamics in a ferromagnetic nanotrack with voltage-controlled perpendicular magnetic anisotropy (VCMA). The workperformance of a nanotrack with a single voltage gate at different tem-
peratures has also been studied, and the results show that the effect of
voltage gate could be affected by the thermal effect. Our results showthat a nanotrack with a voltage gate can be used to build askyrmionium-based diode and ratchet memory. We found that thevoltage gate-induced anisotropy gradient can realize the unidirectional
motion of the skyrmionium in the nanotrack. We also study the
dynamics of a skyrmionium driven by a PMA gradient, which hasbeen observed in experiments and theoretically.
34,40The simulation
results of the skyrmionium driven by the anisotropy gradient withoutcurrent demonstrate how the wedge shape voltage gate influences the
magnetic skyrmionium motion in the nanotrack. PMA-gradient-
induced skyrmionium motion can avoid the Joule heating effect,which may influence the stability of skyrmioniums. Our results areuseful for the design of the voltage-controlled skyrmionium diode andthe skyrmionium transport channel.
The simulation model is an ultra-thin ferromagnetic nanotrack
of 1000 /C2180/C20:4n m
3,a ss h o w ni n Fig. 1(a) , which has interface-
induced DMI and PMA. The mesh size is set as 2 /C22/C20:4n m3,
which is small enough to ensure the numerical accuracy of thesimulations. The micromagnetic simulations are performed using the
Object Oriented MicroMagnetic Framework (OOMMF) package.43
The dynamics of magnetization are governed by the Landau–Lifshitz–Gilbert (LLG) equation, written as
dm
dt¼/C0c0m/C2heffþam/C2dm
dt/C18/C19
/C0um/C2ðm/C2pÞ; (1)
where the third term represents the spin torque arising from a spin
polarized current. m¼M=MSis the reduced magnetization and MSis
the saturation magnetization. c0is the absolute value of the gyromag-
netic ratio and ais the damping coefficient. heffis the effective field,
including the contributions of Heisenberg exchange, DMI, PMA, and
demagnetization. The parameter uis equal to ðc0/C22hjhSHÞ=
ð2ael0MSÞ;/C22his the reduced Plank constant, jis the applied current
density, hSH¼0:08 is the spin Hall angle, eis the electron charge, l0
is the vacuum permeability constant, and ais the thickness of the
nanotrack. p¼/C0^yis the spin polarization direction. Other magnetic
material parameters are adopted from Ref. 28:MS¼580 kA m/C01,
ferromagnetic exchange constant A¼15 pJ m/C01, DMI constant
D¼3.5 mJ m/C02, PMA constant Ku¼0:8M Jm/C03,a n d a¼0:3.
In our simulations, the setups of the voltage gate are illustrated in
Fig. 1 , where the PMA constant controlled by the voltage gate is
defined as Kuv. First, in the study of the voltage-controlled pinning
and depinning effects, a single wedge-shaped voltage gate is placed inthe middle of and upon the ferromagnetic nanotrack [see Fig. 1(a) ],
which controls the PMA constant K
uvof the area underneath the
voltage gate. We model the voltage-controlled PMA constant KvðxÞas
a linear function of the longitudinal coordinate xand the default PMA
constant Kuas follows: KvðxÞ¼KuþðKuv/C0KuÞðx/C0x0Þ=lfor
x2½x0;x0þl/C138,w h e r e lis the length of the voltage gate, Kuvis the
maximum PMA induced by the voltage, and x0denotes the location of
the voltage gate.
Second, in order to study the voltage-gradient-induced skyrmio-
nium motion, as shown in Fig. 1(b) , a wedge-shaped voltage gate is
placed upon the whole ferromagnetic nanotrack, leading to varyingPMA K
vðxÞalong the xdirection. We again model the PMA constant
KvðxÞas a linear function, this time over the whole track, of the longi-
tudinal coordinate xand the default PMA constant Ku, specifically
KvðxÞ¼KuþðKuv/C0KuÞx=lforx2½x0;x0þl/C138.N o t et h a tw h e n
studying the skyrmionium driven by the VCMA gradient along the x
direction, no other external driving force, such as the spin current, is
applied. For the initial state of all simulations, a relaxed skyrmioniumis placed at the left or right end of the ferromagnetic nanotrack, whichis then driven by the spin current or VCMA gradient.
The pinning and depinning states of the current-driven skyrmio-
nium in the nanotrack with a local wedge-shaped voltage gate areshown in Fig. 2 . The effects of the voltage gate length land current
density jon the skyrmionium motion are given in Figs. 2(a) and2(b),
respectively. The relaxed skyrmionium is placed near the left end ofthe nanotrack as the initial state for Fig. 2(a) . When a driving current
is applied along the þxdirection in the heavy-metal layer, a damping-
like spin–orbit torque is generated to drive the magnetizationdynamics in the ferromagnetic nanotrack. Consequently, the skyrmio-nium moves toward the left side of the VCMA region. Due to the
VCMA in the nanotrack, only a current larger than a certain threshold
can drive the skyrmionium through the voltage-gated region from theleft to the right side. The reason is that the PMA gap in the ferromagnetic
FIG. 1. (a) Illustration of the skyrmionium-based device controlled by a gate volt-
age. The out-of-plane magnetization component (m z) is color coded: red means
mz¼þ 1, white means m z¼0, and blue means mz¼/C0 1. (b) The skyrmionium
driven by the voltage-controlled magnetic anisotropy (VCMA) gradient in a ferro-magnetic nanotrack.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 117, 202401 (2020); doi: 10.1063/5.0025124 117, 202401-2
Published under license by AIP Publishingnanotrack, which is defined as Kgap¼Kuv/C0Kuin this work, leads to
an energy barrier for the skyrmionium motion. The precise anisotropyprofile along the nanotrack is given in the supplementary material .
As shown in Fig. 2(a) ,aV C M Ar e g i o nw i t hal e n g t h lsmaller
than 125 nm can pin the skyrmionium when the driving current den-
sityjis smaller than 6 MA cm
/C02. When the length lis larger than
125 nm, a current-driven skyrmionium can pass the VCMA regionwith a current j/C214M A c m
/C02, which indicates that the pinning and
depinning states can be affected by the slope of the VCMA gradient.Namely, a longer VCMA gate length, and consequently reduced gradi-
ent, can lower the threshold depinning current density.
InFig. 2(b) , a relaxed skyrmionium is initially located near the
right end of the nanotrack and a driving current along the /C0xdirec-
tion is applied. The skyrmionium moves along the /C0xdirection and
toward the right side of the VCMA region. When the driving current
density jis smaller than 8 MA cm
/C02, the skyrmionium is pinned bythe VCMA region. The pinning and depinning states of the skyrmio-
nium are independent of the length of the VCMA region.
Compared with Fig. 2(a) , the right boundary of the VCMA
region induces a very high and sharp PMA energy barrier Kgap,
which is hard for the skyrmionium to overcome. The blue dashed box
inFig. 2 indicates the cases in which the skyrmionium displays
unidirectional motion along the nanotrack, which provide informationfor realizing a skyrmionium-based diode device.
InFigs. 2(c) and2(d), the effects of the different K
uvand current
density jare given, respectively. The pinning and depinning states are
sensitive to Kuvof the VCMA region. The parameters for the unidirec-
tional motion along the þxdirection have been marked in a blue
dashed box. When jKgapj/C210:10 MJ m/C03, the skyrmionium is difficult
to drive through the VCMA region because a larger Kgapleads to a
larger energy barrier. From Fig. 2 , under the same driving current den-
sity, a larger Kgapis more likely to result in the pinning of the
FIG. 2. The pinning and depinning states of an isolated skyrmionium driven by the spin current in a ferromagnetic nanotrack with a single wedge-shaped voltag e gate. The
solid red squares denote that the skyrmionium is pinned by the VCMA region, and the solid blue squares denote that the skyrmionium passes the VCMA regio n. The dotted
blue lines indicate areas of unidirectional skyrmionium motion. (a) The pinning and depinning states of a skyrmionium with various lengths lfrom 50 nm to 150 nm and various
driving current densities jfrom 2 MA cm/C02to 10 MA cm/C02. The driving current is applied along the þxdirection. Kuv¼0:85 MJ m/C03. (b) The driving current is applied along
the/C0xdirection. Kuv¼0:85 MJ m/C03. (c) The pinning and depinning states of a skyrmionium with various Kuvfrom 0.65 MJ m/C03to 0.90 MJ m/C03and various jfrom 2 MA cm/C02to
10 MA cm/C02. The driving current is applied along the þxdirection. l¼100 nm. (d) The driving current is applied along the /C0xdirection. l¼100 nm.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 117, 202401 (2020); doi: 10.1063/5.0025124 117, 202401-3
Published under license by AIP Publishingskyrmionium by the VCMA region. The simulation results obtained
with a smaller current density step are given in the supplementary
material .
The trajectories of a skyrmionium moving along the /C0xdirection
in the nanotrack with a single voltage gate are shown in Fig. 3 .T h e
driving current density in Fig. 3(a) is varied from 2 MA cm/C02to
10 MA cm/C02. It can be seen that when the driving current is lower
than the threshold value, the skyrmionium is pinned at the right
boundary of the VCMA region. When the current density increases,the pinned skyrmionium moves closer to the boundary until the
current density is large enough to drive the skyrmionium through the
VCMA region. The skyrmionium passing through the right boundary
of the VCMA region can be seen as the diode breakdown effect of this
skyrmionium-based device.
Furthermore, the slope-shaped VCMA region will enhance the
velocity of the skyrmionium on passing through the VCMA region. The
VCMA gradient can also cause a small deformation of the skyrmionium
because of the SkHE and varying PMA, which also affects the skyrmio-
nium trajectory. This phenomenon is demonstrated in Fig. 3(a) ,w h e n
the driving current density jis larger than 6 MA cm
/C02.
The trajectories of a skyrmionium driven by a current density of
6M Ac m/C02for different VCMA gradients are shown in Fig. 3(b) .
When Kgap<0 and the skyrmionium moves from the high PMA
region to low PMA region, the VCMA region could increase the
velocity of the skyrmionium and cause a deformation of the skyrmio-
nium. If the driving current density is lower than the threshold value,the skyrmionium is pinned inside the VCMA region. For the case of
K
gap>0, the skyrmionium is pinned at the right boundary of the
VCMA region if the driven current is lower than the threshold.
The spin configurations of a skyrmionium pinned in the nano-
track with Kgap>0a n d Kgap<0 are given in Fig. 4 , along with the
pinning position of the skyrmionium at different Kuv.I nFigs. 4(a) and
4(b), the skyrmionium can enter the VCMA region easily because the
Kuvis lower than Ku. After the skyrmionium has moved into theVCMA region, the skyrmionium will deform due to the decrease in
PMA and if the polarized current is not large enough, the skyrmio-
nium will be pinned at the left boundary of the VCMA region. For thecase of K
uv>KuinFigs. 4(c) and4(d), if the driven current density is
not large enough, the skyrmionium will be pinned at the right
FIG. 3. (a) The trajectory of a skyrmionium in the nanotrack driven by a current density jvarying from 2 MA cm/C02to 10 MA cm/C02.Kuv¼0:85 MJ m/C03and the voltage gate
area is located at x¼700 nm with l¼100 nm. (b) The trajectory of a skyrmionium in the nanotrack driven by a current density jof 6 MA cm/C02.Kuvvaries from 0.70 to 0.85 MJ m/C03,
and the voltage gate area is located at x¼700 nm with l¼100 nm.
FIG. 4. The spin configuration of an isolated skyrmionium driven by the spin current
as 6 MA cm/C02motion toward left in a ferromagnetic nanotrack with a single wedge-
shaped voltage gate. (a) The initial state of the isolated skyrmionium in a nanotrack
with a wedge-shaped voltage gate with a Kuvas 0.70 MJ m/C03. (b) The isolate sky-
rmionium pinned after entry into the VCMA region. (c) The initial state of the iso-lated skyrmionium in a nanotrack with a wedge-shaped voltage gate with a K
uvas
0.85 MJ m/C03. (d) The isolated skyrmionium pinned before entry the VCMA region.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 117, 202401 (2020); doi: 10.1063/5.0025124 117, 202401-4
Published under license by AIP Publishingboundary of the VCMA region. Then, the increased PMA and driving
current will induce deformation of the skyrmionium.
In addition, the spin configurations of a skyrmionium diode
device at different temperatures are given in Fig. 5 . The micromagnetic
simulation including the thermal effect is simulated by the OOMMFextensible solver (OXS) object.
41The time step in the simulation with
the thermal effect is fixed at 10 fs, and the temperature changes from50 K to 75 K. In Figs. 5(a) and5(b), the skyrmionium can be driven
through the VCMA region by the spin polarized current at 0 K.
However, the thermal fluctuation in the system will decrease the effec-
tive magnetic anisotropy, DMI, and exchange interaction, which willinduce a deformation of the skyrmionium as shown in Figs. 5(c) and
5(d).
42On the other hand, the energy barrier between KuvandKuin
the VCMA region will also be reduced, which will influence the sky-rmionium behavior in the VCMA region. In Figs. 5(e)–5(g) ,t h es k y -
rmionium is pinned by the VCMA region because the current density
is not large enough. Then in Fig. 5(h) , when the temperature is high
enough, the energy barrier between K
uvandKuwill reduce to a smaller
value, which makes the current density large enough to drive the sky-rmionium through the VCMA region. In Fig. S3 of the supplementary
material , the simulation results also show the same phenomenon with
al a r g e r K
uv. From the results in Figs. 5 and S3, it is found that the
skyrmionium-based diode is sensitive to the thermal effect. If the
applied thermal field is larger than a threshold, the reduced energy
barrier between Kuvand Kuwill break the unidirectional function of
the skyrmionium diode device. The simulation results show that theskyrmionium diode works well even under a weak thermal condition.If the thermal fluctuation is too strong, the unidirectional motionbehavior of the skyrmionium will disappear and the skyrmionium willbe deformed. These results show that the skyrmionium transmissionchannel can be controlled by the VCMA effect and can mimic the
field-effect transistor (FET) device.
The magnetic skyrmionium/skyrmion motion in a nanowire
induced by a current pulse are next studied and compared. The tra-jectories of the skyrmion/skyrmionium driven by a pulse current in ananotrack with the VCMA gate are given in Fig. 6 . When the current
pulse is on, the skyrmionium is driven by the current and movesthrough the VCMA region. But in this case, the driving current pulsetime and current density are not large enough to make the skyrmio-
nium move through the VCMA region. If the current pulse is
removed, the gradient of the VCMA gate will drive the skyrmioniumtoward the left side of the nanotrack. The magnetic skyrmionium isfree from the SkHE and has a linear trajectory as shown in Fig. 6(a) :
under the influence of a driving current, the skyrmionium moves ina straight line along the nanotrack. For the skyrmion case, the trajec-tory is much more complex as shown in Fig. 6(b) . When the current
pulse is on, the skyrmion has a velocity toward the top of the nano-
track, which comes from the SkHE until there is a balance betweenthe edge force and the SkHE. Under the action of these forces, theskyrmion moves along the track direction but with oscillatorymotion in the y-direction. Then, if the current pulse is turned off, theVCMA gradient of the gate will make the skyrmion move toward theleft side of the nanotrack. The skyrmion has a large velocity toward
the bottom of the nanowire, which is a combination of edge force
and the SkHE. From the simulation results, the trajectories of theskyrmionium and skyrmion in a nanowire with the VCMA gatedriven by the current pulse are significantly different. The current/current pulse-driven skyrmionium motion in the nanotrack with theVCMA gate is retained in the middle of the nanowire and avoidsdestruction at the edge. The trajectory shows that a skyrmion and
FIG. 5. The spin configurations of an isolated skyrmionium driven by the spin current j ¼5M Ac m/C02in a ferromagnetic nanotrack with a single wedge-shaped voltage gate
with a Kuv¼0.85 MJ m/C03at different temperatures. (a) The initial state of the skyrmionium moves toward the þxdirection at 0 ns under 0 K. (b) t ¼25 ns, T ¼0 K. (c)
t¼20 ns, T ¼50 K. (d) t ¼20 ns, T ¼75 K. (e) The initial state of the skyrmionium moves toward the /C0xdirection at 0 ns under 0 K. (f) t ¼20 ns, T ¼0 K. (g) t ¼20 ns,
T¼50 K. (h) t ¼20 ns, T ¼75 K.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 117, 202401 (2020); doi: 10.1063/5.0025124 117, 202401-5
Published under license by AIP Publishingskyrmionium can be driven by an anisotropy gradient without cur-
rent or other external driving force.
The motion of magnetic skyrmions driven by a PMA gradient
has been studied recently.40,44A skyrmion moves toward the area
with a lower PMA. Similar to a skyrmion, a skyrmionium also moves
toward the area with a lower PMA. From the simulation results in
Fig. 6 , the skyrmionium tends to move from the high-PMA region to
the lower-PMA region when no driving current is applied. In thiswork, we also study the skyrmionium motion driven by a VCMA
gradient [see Fig. 1(b) ]. The velocities of the skyrmionium driven by
different PMA gradients are given in Fig. S4, and the spin configura-
tion of skyrmionium driven by the VCMA gradient in the nanotrack
is given in Fig. S5 in the supplementary material . It can be seen that
the velocities of the skyrmionium at different PMA gradients have a
similar trend and depend on the amplitude of the PMA gradient.
The distortion of the skyrmionium induced by the anisotropy gradi-
ent may reduce the stability of the skyrmionium when it is close to
the sample edge.
In conclusion, we have studied the motion of a skyrmionium in
a ferromagnetic nanotrack with the PMA gradient controlled by a
gate voltage. Our simulation results show that the trajectory and
velocity of the skyrmionium can be controlled by a wedge-shaped
voltage gate. The unidirectional motion of the skyrmionium realized
by the VCMA effect can be used to build a skyrmionium-based one-
way information channel, that is, the skyrmionium diode. The
skyrmionium-based information channel can be controlled by the
VCMA effect and can mimic the FET function. A skyrmionium
driven by a current pulse in the nanotrack with a VCMA gate has a
different trajectory to that of a skyrmion, which shows that theskyrmionium-based information channel is free from the effects of
an edge defect. We further numerically demonstrated that the PMA
gradient can be used to drive the motion of a skyrmionium in ananotrack in the absence of a driving current. Our results, and the
basic principles demonstrated, are likely to prove useful for the
design and development of future skyrmionium-based information
storage and processing devices.
See the supplementary material for more results about the anisot-
ropy profile in the nanotrack, the simulation results obtained with a
smaller current density step, spin configurations of an isolated sky-
rmionium driven by the spin current in a ferromagnetic nanotrack
with a single wedge-shaped voltage gate at different temperatures, thevelocities of the skyrmionium driven by different PMA gradients, and
the spin configuration of the skyrmionium driven by the VCMA gra-
dient in the nanotrack.
This work was supported by the State Key Program for Basic
Research of China (Grant No. 2016YFA0300803), the National
Natural Science Foundation of China (Grant Nos. 61427812 and11574137), the Jiangsu Natural Science Foundation (Grant No.
BK20140054), the Jiangsu Shuangchuang Team Program, and the UK
EPSRC (No. EP/G010064/1). X.Z. acknowledges the support from theNational Natural Science Foundation of China (Grant No. 12004320)
and the Guangdong Basic and Applied Basic Research Foundation
(Grant No. 2019A1515110713). Y.Z. acknowledges the support fromthe President’s Fund of CUHKSZ, Longgang Key Laboratory of
Applied Spintronics, National Natural Science Foundation of China
(Grant Nos. 11974298 and 61961136006), Shenzhen Key LaboratoryProject (Grant No. ZDSYS201603311644527), and Shenzhen Peacock
Group Plan (Grant No. KQTD20180413181702403).
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding authors upon reasonable request.
FIG. 6. The trajectory of the magnetic skyrmionium and skyrmion driven by pulse current. The current duration is 2 ns, and there is a 2-ns delay between two curr ent pulses.
The skyrmionium and skyrmion move toward the right side of the nanotrack, and the total simulation time is 50 ns, which is indicated by the color bar. (a) The trajectory of the
magnetic skyrmionium in the nanotrack with the VCMA gate driven by current pulses. (b) The trajectory of the magnetic skyrmion in the nanotrack with th e VCMA gate driven
by current pulses.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 117, 202401 (2020); doi: 10.1063/5.0025124 117, 202401-6
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Published under license by AIP Publishing |
1.5006447.pdf | Micromagnetic simulation of the ground states of Ce-Fe-B amorphous nanodisks
D. Liu , G. Li , X. Zhao , J. F. Xiong , R. Li, T. Y. Zhao , F. X. Hu , J. R. Sun , and B. G. Shen
Citation: AIP Advances 8, 056011 (2018);
View online: https://doi.org/10.1063/1.5006447
View Table of Contents: http://aip.scitation.org/toc/adv/8/5
Published by the American Institute of Physics
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Micromagnetic simulation of the ground states of Ce-Fe-B
amorphous nanodisks
D. Liu,1,2G. Li,1,2X. Zhao,1,2J. F. Xiong,1,2R. Li,1,2T. Y. Zhao,1,2F. X. Hu,1,2
J. R. Sun,1,2and B. G. Shen1,2,a
1State Key Laboratory of Magnetism, Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, P . R. China
2University of Chinese Academy of Sciences, Beijing 100049, P . R. China
(Presented 10 November 2017; received 25 September 2017; accepted 23 October 2017;
published online 15 December 2017)
Using 3D micromagnetics package OOMMF, the ground states of Ce 2Fe14B amor-
phous nanodisks with different dimensions, initial magnetization states and magne-
tocrystalline anisotropy constants (K) in zero external field were investigated. The
simulations indicate that the disk size is the decisive factor in determining magnetic
configurations. A diagram is constructed to bring out the dependence of the different
equilibrium states on the disk thickness and diameter. When the ratio of thickness
(T) to diameter (D) is smaller than 1, the vortex state is energetically more favor-
able than other states and the eigenfrequency of vortex approximately proportional
to (T/D)1/2. A variety of magnetization distributions of ground states for different
anisotropy strengths is obtained. The result shows the magnetocrystalline anisotropy
not only shrinks or broadens the vortex core but also induces an out-of-plane mag-
netization component both at the edge and the center of disks. When the K strength
reaches a threshold value, there is a transition from vortex state to Bloch-type Skyrmion
state which suggests the possibility of Skyrmion in rare-earth materials. In addi-
tion, in the system with specific aspect ratio and low intrinsic anisotropy, the vortex
domain can always be sustained under various initial conditions. Meanwhile, the
existence of stable vortex domain is found by experimentation in amorphous Ce-Fe-B
ribbons which is in good agreement with the simulation result. © 2017 Author(s).
All article content, except where otherwise noted, is licensed under a Creative
Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
https://doi.org/10.1063/1.5006447
I. INTRODUCTION
RE-Fe-B ( RE= rare earth) permanent magnets are widely used because of their excellent mag-
netic performance at room temperature.1However, due to the high price and short supply of RE
metals, especially for Nd, Dy and Pr, many researchers have attempted to develop alternative and
economically more attractive permanent magnets. Among all the rare earth elements, cerium (Ce) is
the most abundant metallic element and its price is less than one-tenth of neodymium. Therefore, the
alloys related to Ce have been employed to prepare more economic magnetic materials.1,2However,
most of the investigations focused on Ce-Fe-B crystal alloys, the behavior of amorphous alloys was
seldom referred.
In this paper, magnetic ground states of Ce-Fe-B amorphous alloys were studied not only for the
basic investigation but also for their potential technological applications such as magnetic storage,
random access memory devices and medical applications.3In order to have a measurable study on
the effect of dimensions, initial magnetization states and magnetocrystalline anisotropy constants
on the magnetic ground state before the further experimental study, a micromagnetic simulation on
a magnetic microdisk was performed. To solve for the ground state, the Landau–Lifschitz–Gilbert
aCorresponding author: shenbaogen@yeah.net
2158-3226/2018/8(5)/056011/6 8, 056011-1 ©Author(s) 2017
056011-2 Liu et al. AIP Advances 8, 056011 (2018)
(LLG) dynamic equation as a function of time was used to find the magnetization which will produce
the lowest total energy. By studying the magnetic domain structure of Ce-Fe-B amorphous materials,
the new understandings gained could lead to new potential applications except for low-cost permanent
magnets.
II. SIMULATION METHOD
The Numerical method is widely used as a tool to describe the dynamics of magnetization
in ferromagnetic materials.4The total Gibbs free energy (E tot) consists of Zeeman energy (E ext),
magnetocrystalline anisotropy energy (E ani), exchange energy (E exc) and demagnetization energy
(Ed):
Etot=Eext+ E ani+ E exc+ E d. (1)
The ground magnetization configuration of the material is obtained by the LLG dynamic equation
as a function of time:5,6
dM
dt=
MHeff+
MMdM
dt, (2)
where Mis the magnetization, Heffis the effective field,
is the Landau–Lifshitz gyromagnetic ratio
and is the dimensionless damping coefficient. The effective field is the functional derivative of the
energy density which is defined as follows:
Heff=2A
0M2
sr2M Hd+2K1
0M2
s(Mˆu)ˆu, (3)
where ˆu is the unit vector along the anisotropy axis and Hdis the demagnetizing field.
Micromagnetic simulations were performed using the 3D micromagnetic package OOMMF7
and the simulation model was set to be a magnetic nanodisk. The material parameters for Ce 2Fe14B
amorphous alloys used in the calculation at room temperature are as follow:8the saturation magneti-
zation Ms = 9.3105A/m and the exchange integral constant A = 5 PJ/m. The sample was discretized
into cell sizes of 2.5 2.52.5 nm3, ensuring that for each mesh the average edge length of all the
tetrahedral elements was less than the exchange length of Ce 2Fe14B L ex=q
2A=
0M2
s
=3 nm.
Considering the simulating time and the precision, gyromagnetic ratio
= 2.211105mA-1s-1and
damping coefficient = 0.05 were chosen. The thickness, diameter and magnetocrystalline anisotropy
constant of each model will be given in response to the different situations below.
III. RESULTS AND DISCUSSION
By altering diameter (D) and thickness (T), the effects of different model dimensions on the
equilibrium state are examined which are shown in FIG. 1. The initial state in this study is set
to be a thermally neutralized state without any intrinsic anisotropy or external field. As such, the
calculated magnetic configuration is simply the result of the competition between the demagne-
tizing field and exchange energy. The former favors a closed flux arrangement which necessitates
a highly non-uniform spin arrangement due to the sample geometry while the latter favors uni-
form magnetization which inevitably generates magnetic poles at the sample surface. Depending
on the disk diameter and thickness, metastable magnetic configurations are observed including the
onion, vortex, and other different magnetic states. The largest disk diameter in this study is 250 nm
while the smallest is 5 nm, with different thickness-to-diameter ratio T/D ranging from 1/5 to
10/1. Specifically, when the T/D is much smaller than 1, the vortex state is energetically more
favorable than other states. While in some thicker disks, a much more pronounced magnetization
canting can be seen, because when the disk thickness becomes much smaller than the diameter,
the magnetization essentially aligns with the disk geometry so as not to lose too much exchange
energy, but to cancel the total dipole energy. Due to the directions of magnetic moment remain con-
fined in-plane, the angle between adjacent moments of the disk center becomes increasingly larger.
Therefore, the magnetization at the core of the vortex structure will become perpendicular to the
plane.056011-3 Liu et al. AIP Advances 8, 056011 (2018)
FIG. 1. The dependence of magnetization configuration of the equilibrium state on its geometrical parameters. The arrows
are the projection of the in-plane magnetization where the red and blue represent the direction of +z and –z, respectively.
The numerical simulation shows that the size of model has a great effect not only on the steady
state but also on the frequency response which is essentially useful for its potential applications in
high-density magnetic storage and spin electronic devices. The temporal evolution of the average
normalized magnetization component in x-direction (mx) is investigated by dynamic micromagnetic
simulation. It can represent that the vortex core moves in y-direction.9,10And the corresponding
resonance frequency !is obtained by Fourier transformation of the time-domain oscillation.11The
variation of the nanodisks resonance frequencies as a function of the ratio T/D is illustrated in FIG. 2
(the black square curve). The resonance frequency decreases abruptly as the disk ratio decreases
from 10/1 to 1/1. When the aspect ratio T/D = 1, the resonance frequency drops to a minimum
value of 0.43GHz. When the aspect ratio T/D 1/2, the resonance frequency decreases gradually
with the ratio reducing. The resonance frequency originates from the confinement of the vortex
core. A similar result was reported by Guslienko,12the eigenfrequency of magnetic disk depends
on the aspect ratio T/D, approximately proportional to (T/D)1/2if the aspect ratio is much less
than 1.
With a fixed model size in which the ratio of T/D is much less than 1 (T = 20 nm, D = 100 nm),
the influence of initial state on magnetization configuration of equilibrium state is investigated.
Starting with the magnetization in the random state, vortex state, x-direction, and z-direction, the
model relaxes in no external field or anisotropy, solving for their corresponding ground states. As one
can see in FIG. 3, the magnetization configuration of each system exhibits the vortex magnetization
distribution with different chirality. The positions of vortex center for four different initial states
FIG. 2. The spectra for resonance frequency of amorphous disks with different aspect ratios. The black squares are points
simulated numerically, the red line corresponds to fitting using the equation !(T/D)1/2.056011-4 Liu et al. AIP Advances 8, 056011 (2018)
FIG. 3. Magnetic configuration and the movements of vortex center for four different initial states: (a) thermal demagnetization
state, (b) vortex state, (c) in the x-direction and (d) in the z-direction on the left side and corresponding equilibrium states of
each system on the right side. The HSL color scale reflects the variation of a component of the magnetization in full orientation.
stable in the same place when the systems achieve their stability. In addition, each energy item
of each system becomes the same, which is E tot= 2.8810-18J, E ani= 7.2810-19J and E exc=
2.1510-18J, respectively. As one can conclude in this situation, the ground state of this system with
low anisotropy is a vortex magnetization distribution, unconnected to the initial magnetization. The
existence of magnetic vortex is found by experimentation investigations in amorphous Ce 14Fe80B6
ribbons which consonant with the simulation result.13
Compared to other parameters of magnetic materials, the degree of amorphization affected
magnetocrystalline anisotropy (K) most. To clarify the role of the anisotropy contribution, the
stable magnetic configuration for K = 0 J/m3is simulated first, and then by gradually varying
the K value how the anisotropy influences the magnetic configuration is studied. Applying the056011-5 Liu et al. AIP Advances 8, 056011 (2018)
FIG. 4. (a) - (k) Schematic representation of equilibrium states that form at H = 0 investigated under different values of K for D
=100 nm and T=20 nm. (l) The Bloch-type Skyrmion observed at K = 2.8 105J/m3. Mz is the z components of magnetization
and represented by regions in red (+z) and blue (-z).
same procedure of the energy minimization, the ground states diagram is obtained and presented
in FIG. 4. The initial distribution of the magnetic moments is random and the easy axis of the
anisotropy is +z.
From FIG. 4(a) to FIG. 4(d), the ground state of the system with low anisotropy keeps a vortex
magnetization distribution. Coincide with previous results, vortex state is a typical situation for soft
magnetic materials in symmetric structure taken exchange and magnetostatic interactions into account
only.14,15For K <2.5105J/m3, the size of the vortex core will increase with the increasing K until the
vortex state breaks into another phase for the strong anisotropy. The effect of changing anisotropy and
Dzyaloshinskii-Moriya (DM) interaction have a similar impact on the vortex in magnetic nanodisk.16
The magnetocrystalline anisotropy induces an out-of-plane magnetization component both at the
edge and the center of disks. There is a phase transition from the vortex state to the Bloch-type
Skyrmion when the K strength reaches a threshold value. The threshold value which depends on
the disk parameters in this simulation is 2.8 105J/m3(FIG. 4(l)). This simulation result indicates
that the Skyrmion state will appear spontaneously in the rare-earth amorphous material which agrees
well with theoretical prediction.17Spontaneous Skyrmion lattice ground states may exist generally
in condensed matter systems having chiral interactions without the assistance of external fields or
defects. Coincidentally, Montoya et al.18found that the stabilization of Skyrmions in amorphous
Fe/Gd is purely driven by tuning magnetic properties and film thickness and that no DMI is present
in these films. The results provide a guideline for engineering the formation and controllability
of Skyrmion phases in symmetrical structure. Moreover, as the value of K continues to increase,
the influence of K leads to an asymmetric deformation of the Bloch-type Skyrmion such that the
magnetization is orientated generally along the normal vector z.
IV. CONCLUSION
In conclusion, using the numerical micromagnetic model, magnetization ground states of Ce-
Fe-B amorphous disks for different aspect ratios T/D, initial magnetizations and magnetocrystalline056011-6 Liu et al. AIP Advances 8, 056011 (2018)
anisotropy constants are systematically investigated. The simulations show that the vortex state can
appear as a steady state for specific range of T/D which is significantly smaller than 1. In addition,
the system with low anisotropy favors the formation of vortex configuration under various initial
magnetization state, which will contribute to other applications except for hard magnets. A variety
of magnetization distributions of ground states for various anisotropy strengths is obtained. The
results indicate that the existence of the magnetocrystalline anisotropy not only shrinks or broadens
the vortex core but also induces an out-of-plane magnetization component both at the edge and the
center of disks, which cause the generation of Skyrmion. Therefore, fabricating RE-Fe-B magnets by
using Ce is highly beneficial to utilize the rare-earth resource in a balanced manner and manufacture
low-cost magnets.
ACKNOWLEDGMENTS
This work was supported by the National Key Research Program of China (Grant No.
2014CB643702, Grant No. 2016YFB0700903), the National Natural Science Foundation of China
(Grant No. 51590880) and the Knowledge Innovation Project of the Chinese Academy of Sciences
(Grant No. KJZD-EW-M05).
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1.2426381.pdf | Studies of the magnetization reversal process driven by an oscillating field
Hwee Kuan Lee and Zhimin Yuan
Citation: Journal of Applied Physics 101, 033903 (2007); doi: 10.1063/1.2426381
View online: http://dx.doi.org/10.1063/1.2426381
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/101/3?ver=pdfcov
Published by the AIP Publishing
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152.2.176.242 On: Mon, 01 Dec 2014 13:59:57Studies of the magnetization reversal process driven by an oscillating field
Hwee Kuan Leea/H20850and Zhimin Yuan
Data Storage Institute, 5 Engineering Drive 1, Singapore 117608
/H20849Received 8 August 2006; accepted 4 November 2006; published online 2 February 2007 /H20850
Magnetic recording based on coherent rotation is hitting its physical limitations. An alternative of
driving magnetization reversal using an oscillating field has been proposed. We performedsystematic studies of the magnetization reversal process in the presence of oscillating fields. Theoscillating field reduces the coercivity significantly, and for isolated particles, we observedconsistent results reported previously. For a circularly polarized field, we obtained a hysteresis loopthat exhibits a behavior that would not be observed in the absence of oscillating fields. © 2007
American Institute of Physics ./H20851DOI: 10.1063/1.2426381 /H20852
I. INTRODUCTION
Increase in areal density on magnetic recording had, in
the past, been achieved by reducing the dimensions of therecording devices, such as grain size of the thin film media.But thin film with small grains is subjected to the superpara-magnetic limit, and to circumvent this limit, materials withhigh anisotropy, such as L
10phase of FePt has been proposed
for media with areal density above 1 Tbits/in.2However,
current writer pole materials has insufficient msto generate
fields high enough to write on materials with such highanisotropies. In the Stoner-Wohlfarth model
1for single-
domain grains, the magnetic grains perform a coherent rota-tion upon an externally applied field. In this model, the re-quired external field to induce magnetization reversal isproportional to the strength of the grain’s anisotropy. Record-ing methods utilizing this simple Stoner-Wohlfarth’s switch-ing mechanism is hitting its limits, and novel-recordingmethods such as studies related to heat assisted recording,
2,3
composite media4,5and recording via nonlinear resonance6–8
have been investigated.
The phenomena of ferromagnetic resonance have been
used as an experimental tool, where it is often used as aprobe for measurements.
9–13Recently, it has been proposed
that resonance be used to induce magnetization reversal.6–8
This approach has profound implications for the magnetic
storage industry. Hence, the nonequilibrium system ofHeisenberg spins driven by an oscillating external field hasbeen extensively studied.
14–17However, extensive systematic
studies of magnetization reversal driven by an oscillatingfield have not been reported. We attempt to undertake such asystematic study in this paper.
We shall give a brief introduction of ferromagnetic reso-
nance in its simplest form to explain the basic physics, formore details, refer to Refs. 11and18. At this point, we
would like to caution the reader that ferromagnetic resonancein real material is much more complex than this simple ex-ample we use for illustrative purpose. When an isolated mag-netic moment m
/H6023is placed in a magnetic field, a torque is
exerted and the magnetic moment precess around the direc-tion of the field with an angular frequency /H9275/H6023=/H92530h/H11036eˆz./H92530is
the gyromagnetic ratio and h/H11036eˆzis the magnetic field. Here,
we do not assume other form of effective field such as thecrystallographic anisotropy field. If in addition, a circularly
polarized time varying field h/H6023/H20648/H20849t/H20850is applied on the xyplane.
Such that h/H6023/H20648/H20849t/H20850oscillates with an angular frequency /H9275/H6023
=/H92530h/H11036eˆz, then h/H6023/H20648/H20849t/H20850is “following” m/H6023. In the rotating frame of
reference, m/H6023would precess around h/H6023/H20648/H20849t/H20850. In the laboratory
frame of reference, m/H6023precesses around h/H11036eˆzas well as the
time varying field h/H6023/H20648/H20849t/H20850. The vector sum of two precessions
causes the magnetic moment m/H6023to make a spiral motion
about the z-axis with mzoscillating between ± /H20841m/H6023/H20841. This
simple example shows that an oscillating field h/H6023/H20648/H20849t/H20850can in-
duce magnetization reversal /H20849even when m/H6023is initially paral-
lel to h/H6023/H11036/H20850. In this paper, we performed systematic study of
resonance field on complex systems of interacting magneticgrains with uniaxial anisotropy on a triangular lattice. We
show numerically that a small h/H6023/H20648is sufficient to cause the
magnetic moment to switch from mz=/H20841m/H6023/H20841tomz=−/H20841m/H6023/H20841even if
h/H11036is much smaller than the anisotropy field.
II. MODEL
We performed our simulations on magnetic grains ar-
ranged in a two-dimensional triangular lattice with periodicboundary conditions. Grains posses uniaxial anisotropy per-pendicularly to the plane of the lattice and interact throughlong-range dipolar and nearest neighbor exchange interac-tions. The Hamiltonian is given by
H=−1
2Jms2v2/H20858
/H20855ij/H20856mˆi·mˆj−/H92620
2/H9266ms2v2/H20858
i/HS11005jmˆi·Dij·mˆj
−kuv/H20858
imˆiz2−/H92620vmsh/H6023ext/H20849t/H20850·/H20858
imˆi. /H208491/H20850
The first summation sums over six nearest neighbors’ grains,
Jis the intergrain exchange constant, msis the saturation
magnetization, vis the volume of each grain, and mˆiis the
unit magnetization of grain i. The second term sums over all
grains i/HS11005j,/H92620is the permeability of free space, and Dijis
the dipolar interaction matrix between grains iand j. The
third term sums over all grains, and kuis the uniaxial aniso-a/H20850Current address: Bioinformatics Institute, 30 Biopolis Street, #07-01, Ma-
trix, Singapore 138671; electronic mail: leehk@bii.a-star.edu.sgJOURNAL OF APPLIED PHYSICS 101, 033903 /H208492007 /H20850
0021-8979/2007/101 /H208493/H20850/033903/4/$23.00 © 2007 American Institute of Physics 101 , 033903-1
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152.2.176.242 On: Mon, 01 Dec 2014 13:59:57tropy constant. The last term represents the Zeeman energy
of the grains interacting with the time dependent external
field h/H6023ext/H20849t/H20850that consist of a constant perpendicular compo-
nent and a time varying parallel component. The time depen-
dent external field is given by
h/H6023ext/H20849t/H20850=h/H20648cos/H20849/H9275t/H20850eˆx+h/H11036eˆz. /H208492/H20850
/H9275is the angular frequency of the resonance field, h/H20648is the
amplitude of the oscillating parallel component, and h/H11036is
the constant perpendicular component.
Dynamics of the magnetization in each grain is governed
by the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation given in di-
mensionless units as19
dmˆi
d/H9270=− /H20849mˆi/H11003h/H6023i+/H9251mˆi/H11003mˆi/H11003h/H6023i/H20850. /H208493/H20850
/H9270is a dimensionless time unit that converts to real time unit
through /H9270=/H92530Hkt//H208491+/H92512/H20850./H92530is the gyromagnetic ratio and t
is the time unit in seconds. /H9251is the damping parameter speci-
fying the rate of energy dissipation. h/H6023i=−/H11612/H/H20849/H92620Hk/H20850+h/H6023this
the effective magnetic field for the ith grain normalized by
Hk, where Hkis given by Hk=2ku//H92620ms.h/H6023this a thermal field
use to incorporate random thermal effects. It is given by
h/H6023th=/H9257/H6023/H20881/H9251
1+/H92512kBT
vku1
/H9004/H9270, /H208494/H20850
where kBis the Boltzmann constant, Tis the temperature in
Kelvin, and /H9257/H6023is a random variable drawn from the standard
normal distribution. At this point, we would like to give acomment on the accuracy of Eq. /H208494/H20850. The fluctuation-
dissipation theorem does not apply strictly to a magneticsystem driven out-of-equilibrium. However, Eq. /H208493/H20850applies
for the case T=0.
14,20In our simulation, we check that the
thermal field at T=300 K is about 10% of the anisotropy
field. For the simulation parameters, we used specificationsconsistent with a media for 1 Tbits/in.
2. Values of relevant
parameters are given in Table I. The simulations are ran in
dimensionless units with effective fields normalized by Hk.
For the exchange interactions, we defined a dimensionlessexchange constant J
/H11569such that the exchange field between
two neighboring grains is given in units of Hk.J/H11569is related to
JbyJ/H11569=Jms2v//H208492ku/H20850. For the oscillating field, we defined a
dimensionless angular frequency by /H9275/H11569=/H9275/H208491+/H92512/H20850//H92530Hk.W e
performed our simulations to trace the hysteresis loops for
different oscillating field frequencies /H20849/H9275/H20850. The hysteresisloops are traced with a field step of 0.025 Hkwith 2 /H11003104
time steps for each field.
III. RESULTS AND DISCUSSIONS
We focus on hysteresis behavior of our model instead of
the dynamical order parameter used in most studies ondriven Heisenberg model.
15This is because our objective is
to be relevant to magnetic recording applications in whichthe temperature range is far below the Curie temperature.Hysteresis behavior is vital to magnetic recording applica-tion.
Figure 1shows the effect of resonance for a single grain.
Coercivity reduces to 0.57 H
kat/H9275/H11569=0.2. Cusps appear at
h/H11036/HK=0.06 for /H9275/H11569=0.8 and at h/H11036/Hk=0.26 for /H9275/H11569=0.6,
respectively. We traced the time evolution of mˆat these cusps
and found that the magnetization dynamics belong toP-modes of oscillations defined by Bertotti et al.
14Also note
that at /H9275/H11569=0.8, when h/H11036/Hk=0.06, the average magnetiza-
tion is 0.85, the magnetic grain precess at a Larmor fre-quency of
/H9275/H11569=0.85−0.06=0.79 which is approximately
equal to the oscillating field frequency. In Fig. 1, error bars,
plotted on one side of the hysteresis loop, were obtainedfrom simulating 1024 independent grains.
To show the dynamics of magnetization reversal, we plot
in Fig. 2the time series of m
zfor several grains. mzoscillates
several times between 0.75 and 1.0 before switching over.Once switched over, m
zremains at −1 with little fluctuations.
We have not observed any switching back from mz/H11015−1 to
mz/H110220. For these plots, a perpendicular field of h/H11036/Hk
=0.75 and field frequency of /H9275/H11569=0.2 is applied. Mean
switching time averaged over 128 independent grains is /H9270
=345±22 with standard deviation of /H9004/H9270=250. Using the pa-
rameters in Table Iconverting the mean switching time to
real time units gives ts=0.8 ns. To check the quantitative
influence of damping constant on switching dynamics, weperformed additional simulations for
/H9251=0.20 and /H9251=2 and
the mean switching times are 128±9 and 30.8±2.1, respec-tively. To be relevant to magnetic recording where the damp-ing constant is measured to be about
/H9251=0.02,21we will use
/H9251=0.02 for all subsequent simulations.TABLE I. Parameters used in numerical simulations for 1 Tbits/in.2mag-
netic recording media.
Saturation magnetization /H20849ms/H20850 106Am−1
Anisotropy constant /H20849ku/H20850 1.26/H11003106Jm−3
Volume /H20849v/H20850 2.165/H1100310−25m3
Lattice constant /H20849a/H20850 6/H1100310−9m
Temperature /H20849T/H20850 300 K
Damping constant /H20849/H9251/H20850 0.02
Time step /H20849/H9004/H9270/H20850 0.02
Resonance field amplitude /H20849h/H20648/Hk/H20850 0.05
FIG. 1. Hysteresis loops of isolated particles showing a maximum reduction
of coercivity at /H9275/H11569=0.2. Comparing to the coercivity of at constant field
/H20849Hc/H11015Hk/H20850, the coercivity reduces to Hc/H110150.6Hkat/H9275/H11569=0.2. At higher fre-
quencies, the oscillating field results in cusp at their respective resonancefields.033903-2 H. Lee and Z. Yuan J. Appl. Phys. 101 , 033903 /H208492007 /H20850
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152.2.176.242 On: Mon, 01 Dec 2014 13:59:57Figure 3shows the half hysteresis loops for arrays of
interacting particles with dipolar and nearest neighbor inter-grain exchanges /H20849J
/H11569=0.05 /H20850. Amplitude of the in-plane oscil-
lating is set at h/H20648/Hk=0.05. The hysteresis loops for 8 /H110038,
12/H1100312, and 16 /H1100316 matches to within error bars, suggesting
that there are no finite size effects. It has been known thatfinite size effects are indications of coherent rotation andsingle droplet nucleation
22–24whereas our results suggest
otherwise. Figure 3also shows that the effects of resonance
are strongest at /H9275/H11569=0.6 where the effective coercivity falls to
zero. Even at lower frequencies that the oscillating field re-duces the coercivity substantially. At
/H9275/H11569=0.2, the coercivity
is 0.25 Hkcompared to conventional perpendicular recording
without oscillating field where the coercivity is approxi-mately equal to H
k. For practical applications in magnetic
recording, an oscillating field with frequency of /H9275/H11569=0.2 may
be sufficient to increase the areal density significantly. If aanisotropy field of H
k=80 kOe is considered, an oscillating
field with amplitude of 4 kOe at frequency of f=45 GHz can
reduce the effective coercivity from 80 kOe to about 20 kOe.In addition, we performed the same set of simulations shownin Fig. 3, with a magnetic soft-underlayer /H20849SUL /H20850to mimic
perpendicular recording. The effect of SUL depends on theproximity of SUL to the recording layer. For our simulations,
the results with SUL does not differ from those shown in Fig.3to within error bars.
To study the effects of different oscillating waveforms,
we performed simulations of 12 /H1100312 arrays on circularly po-
larized and square waveforms. The time dependent externalfield for these waveforms are defined as
h
/H6023
extcir=h/H20648cos/H20849/H9275t/H20850eˆx+h/H20648sin/H20849/H9275t/H20850eˆy+h/H11036eˆz,
h/H6023
extsq=±h/H20648eˆx+h/H11036eˆz, /H208495/H20850
where for the square waveform h/H6023
extsq, the in-plane field oscil-
lates between + Aand − Awith frequency /H9275. Figure 4shows
the effects of different waveforms on the hysteresis loops atvarious frequencies. Effective coercivity reduces to zero for
/H9275=0.2, 0.4, and 0.6 for the square waveform. For the circu-
larly polarized waveform, our results show that mzfalls be-
low zero at positive external field for /H9275/H11569=0.6. This has not
been observed when there is no oscillating field /H20849when there
is no exchange bias /H20850. However, we cannot rule out such ob-
servations on physical grounds because unlike the sinusoidaland square waveform, the circularly polarized waveform hasa helicity that breaks the symmetry between + m
zand − mz.
When the Larmor frequency matches the external oscillatingfield frequency
/H9275, the system will experience resonance ef-
fect only if the effective field is positive and helicity of Lar-mor precession is the same as the helicity of the oscillatingfield. However, there is no resonance effect when the helicityof Larmor precession is opposite to that of the oscillatingfield. There are much implications of such abnormal behav-ior on application to magnetic recording. Further investiga-tion will be left for future studies.
IV. CONCLUSION
We systematically study the hysteresis behavior of mag-
netization under an oscillating field. For isolated particle, weshow that oscillating field does reduce the coercivity andobserved P-modes of oscillation reported in previous
publication.
14
FIG. 2. Time evolution of magnetization reversal for isolated particles with
/H9275/H11569=0.2 at h/H11036/Hk=0.75. Plots from several independent simulations are
plotted. The filled circle shows the mean switching time with error barsrepresenting the standard deviation of switching times obtained over 128simulations.
FIG. 3. Half hysteresis loops showing the combined effects of dipolar in-teractions and intergrain exchange coupling. The dimensionless exchangecoupling strength is J
/H11569=0.05. Coercivity falls to zero at /H9275/H11569=0.6. There is no
finite-size effects for lattice sizes 8 /H110038, 12/H1100312, and 16 /H1100316. Error bars for
larger system sizes are smaller due to self averaging.
FIG. 4. Hysteresis loops showing the effects of different oscillating fieldwaveforms on 12 /H1100312 interacting arrays. m
zreaches zero at positive exter-
nal field for the circularly polarize waveform. We propose that this is due todifferences in helicities of the driving field and Larmor precession.033903-3 H. Lee and Z. Yuan J. Appl. Phys. 101 , 033903 /H208492007 /H20850
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152.2.176.242 On: Mon, 01 Dec 2014 13:59:57For array of interacting particles, we show that the hys-
teresis loops are independent of system size, different fromsystems exhibiting coherent or single droplet nucleation.
22–24
For these arrays, the coercivities reduced to zero for a range
of oscillating field frequencies. Lastly, we found that the hys-teresis loop for circularly polarized oscillating field exhibitsa behavior that has not been observed in hysteresis loopswithout oscillating fields.
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152.2.176.242 On: Mon, 01 Dec 2014 13:59:57 |
1.4935437.pdf | Individual vortex nucleation/annihilation in ferromagnetic nanodots with broken
symmetry observed by micro-Hall magnetometry
T. Ščepka , T. Polakovič , J. Šoltýs , J. Tóbik , M. Kulich , R. Kúdela , J. Dérer , and V. Cambel
Citation: AIP Advances 5, 117205 (2015); doi: 10.1063/1.4935437
View online: http://dx.doi.org/10.1063/1.4935437
View Table of Contents: http://aip.scitation.org/toc/adv/5/11
Published by the American Institute of PhysicsAIP ADV ANCES 5, 117205 (2015)
Individual vortex nucleation/annihilation in ferromagnetic
nanodots with broken symmetry observed
by micro-Hall magnetometry
T. Ščepka, T. Polakovič, J. Šoltýs, J. Tóbik, M. Kulich, R. Kúdela, J. Dérer,
and V. Cambela
Institute of Electrical Engineering, Slovak Academy of Sciences, Dúbravská cesta 9,
Bratislava, 841 04 Slovak Republic
(Received 9 June 2015; accepted 28 October 2015; published online 4 November 2015)
We studied vortex nucleation /annihilation process and its temperature dependence
in micromagnetic objects with lowered symmetry using micro-Hall magnetometry.
Magnetization reversal curves were obtained for the Pacman-like nanodots placed
directly on Hall probes. Lowered symmetry of the object leads to good control of
its chirality. V ortex nucleation and annihilation fields strongly depend on the angle
of the external in-plane magnetic field with respect on the nanodot symmetry. The
micromagnetic simulations support the experimental results - the vortex nucleation
fields are controlled by local magnetization configurations present in the object (C-,
S-, and double S-states) for field just above vortex nucleation field. The experi-
ments also confirm that the vortex nucleation proceeds via thermal activation over
an energy barrier. C2015 Author(s). All article content, except where otherwise
noted, is licensed under a Creative Commons Attribution 3.0 Unported License.
[http: //dx.doi.org /10.1063 /1.4935437]
Magnetic nanoelements attract attention of researchers due to their interesting fundamental
properties and also the potential in novel high-density magnetic memories. Controlled manipulation
of magnetic domains and vortices in ferromagnet nanostructures have opened opportunities for
novel fast, high-density, and low-power memories, including race track memories, magnetic random
access memories, bit patterned media, and skyrmion memories.1–3Magnetic properties of the nano-
magnets are governed by magnetostatic and exchange energies, and are fundamentally influenced
by their shape.
Significant experimental and theoretical work have been devoted within last years to under-
stand magnetization reversal and vortex dynamics in submicron ferromagnetic disks.4–9A disk
is convenient high-symmetrical shape which simplifies both experimental and theoretical studies.
However, vortex chirality and polarity is di fficult to control in disks due to symmetry reasons.
Recently, a new prospective shape of a nanomagnet with broken symmetry was proposed
(“Pac-man like”, PL10–13). It shows easy control of vortex chirality /polarity in the in-plane magnetic
field for objects smaller than 100 nm.
V ortex nucleation was studied previously by micro-Hall technique in Ni8and Co individual
nanodots of cylindrical shape.6Both papers present influence of temperature on the nucleation and
annihilation fields with results not completely understood. The non-thermal dynamics of magnetic
vortices in micron-size Py disks was reported. The measurement has been done on ensemble of Py
dots by superconducting quantum interference device (SQUID). From the temperature dependence
of the time relaxation constant, the athermal behavior was found and interpreted as a quantum
depinning of vortex cores through the structural defects of the sample.7
Magnetization reversal can be observed experimentally on arrays of similar nanodots by tech-
niques like magneto-optical Kerr e ffect14or by SQUID.7The state of individual nanomagnet can be
explored non-invasively by scanning Hall probe microscopy or micro-Hall probe magnetometry.
aCorresponding author: vladimir.cambel@savba.sk
2158-3226/2015/5(11)/117205/7 5, 117205-1 ©Author(s) 2015
117205-2 Ščepka et al. AIP Advances 5, 117205 (2015)
In this letter, we present a temperature dependent study of the magnetization reversal in indi-
vidual submicron Permalloy (Py) PL nanodots using micro-Hall magnetometry. The technique is
used to gather quantitative information on stray magnetic field of individual ferromagnetic objects
placed directly on micro-Hall probes.15–17Changes of the stray field influence the magnetic-field
flux through the active area of the Hall sensor. Therefore, the Hall voltage directly shows the
magnetization changes of the PL nanodot including vortex nucleation and annihilation. We have
found (similarly to the Ref. 6) that the vortex nucleation field increases with temperature rapidly
for temperatures below ∼20 K, and only slightly for temperatures above ∼20 K. We also show that
for the PL nanomagnet the vortex nucleation field depends significantly on the angle of the external
field according its axis of symmetry.
It has to be stressed that lowered symmetry of the ferromagnet complicates the Hall voltage
interpretation as compared with highly symmetric objects like disks. In asymmetric objects, the
object shape and the vortex chirality and polarity can influence the Hall voltage signal significantly.
Therefore, in this work we combine the Hall probe measurement with micromagnetic simulations to
support our interpretation of the results obtained.
In previous calculations it was carried out that in small PL objects ( <100 nm) is the vortex nucle-
ation preceded by a C-state or an S-state formation.10Our simulations showed that much more compli-
cated picture with nucleation of several vortices appears for large objects ( d>500 nm). However,
smaller PL nanodots ( d<350 nm) gave much more reproducible results, which were supported by
presented experiments. Therefore, in this paper we concentrate on 310-nm object, for which simple
magnetic state with one vortex appears, and still reasonable large Hall voltage signal is obtained.6
In the case of 310-nm PL object we have identified 3 di fferent single-domain (SD) states for
fields closely above the vortex nucleation: C-state, S-state and double S-state (or 2S-state). The state
influences the vortex nucleation process, which is basically controlled by thermal activation over
an energy barrier. The process is influenced by the object asymmetry, which can be represented by
additional magnetic dipoles located within the object at its boundary imperfections. Such dipoles
change their value and direction during the magnetization reversal measurements. The charge rear-
rangement should be sensitive also to thermal processes and represents small energy barriers for the
vortex nucleation process.
Hall probes were fabricated from GaAs /AlGaAs heterostructure with a two-dimensional elect-
ron gas (2DEG) located 80 nm below the surface (electron density ∼7 x 1015m−2, mobility
∼6 m2/Vs at 77 K). Technology of the probe: First, standard optical lithography and wet etching
were used to define 15 µm Hall crosses and their leads. Then, electron beam lithography (EBL)
and wet etching were applied to lower the linear Hall probe dimensions to ∼1µm. Finally, Py PL
nanomagnets of the thickness of 38 nm were formed. The process consisted of EBL process on
thinned PMMA 950K resist (thickness 100 nm) with a dose of 110 µC/cm2at 10 kV , and e-beam
evaporation of Py layer at a pressure below 10−5Pa, followed by lift-o ff. Fig. 1(a) shows the SEM
FIG. 1. (a) SEM image of the Py nanomagnet placed asymmetrically on the active zone of the micro-HP. Bar corresponds to
300 nm. (b) Shape of the nanomagnet used for the micromagnetic simulations. Horizontal line is the line of its “symmetry”,
αis the angle between the line and applied field.117205-3 Ščepka et al. AIP Advances 5, 117205 (2015)
image of the finished micro-Hall probes with PL nanomagnet ( d=310 nm) located on the probe.
Only one PL nanodot was patterned per cross to evaluate single PL-dot properties.
Hysteresis loops were measured by applying an in-plane magnetic field and recording VH. The
nanomagnet was fabricated asymmetrically with respect to the active zone of the Hall cross in order
to improve the signal measured.6The main reason for asymmetric placement of the nanomagnet is
to obtain non-zero magnetic flux through the active zone of the Hall probe. In case of symmetric
placement, the Hall signal should be zero due to dipolar nature of the magnetic field. In our exper-
iments, best resolution was obtained in the Hall probe configuration (current flows from lead I +to
I-, Hall voltage is measured between leads V +and V- (Fig. 1(a)).
The measurements were carried out at temperatures 4 – 100 K in the physical property measure-
ment system (PPMS) using dc current (10 µA) across the Hall probe. Hall voltage was measured and
amplified by the PPMS electronics, thereby voltage noise lower than ∼20 nV was achieved in the best
case (integration time 10 s, T=30 K).
In the magnetization reversal measurements, a homogeneous in-plane magnetic field Hextwas
applied in parallel with the active area of the Hall probe, thus not contributing to the measured
signal. Hall voltage measured is proportional to the average magnetic flux through the active zone
of the probe generated by the PL nanodot. Sweeping amplitude of the external field was fixed to
2 T, for which positive and negative branches of the hysteresis loop gave the same fields Hnuc(Han).
This was not the case for much lower field amplitude (e.g. 200 mT), probably due to PL-boundary
imperfections - only high external magnetic field brings the object into identical SD states for both
field polarities.
We have analyzed the vortex nucleation /annihilation process for three di fferent field directions,
90◦, 120◦, and 160◦, according its “symmetry” axes (Fig. 1(b) shows that real nanomagnet is not
ideally symmetric due to edge imperfections). The angle-selection was based on the outputs of
micromagnetic simulations.
First, we have calculated the angular dependence of the single-domain (SD)-to-vortex (V) state
transitions, HnucandHan, for the PL nanodot shown in the Fig. 1(b). Dynamic behavior of the
vortex nucleation, propagation, and annihilation were evaluated by micromagnetic calculation using
OOMMF software package.18Parameters used in the simulation were: PL thickness t=38 nm,
diameter d=310 nm, Py exchange constant A=13×10−12J/m, saturated magnetization Ms=8.6
×105A/m, and damping parameter 0.5. The unit element size was fixed to 4 nm x 4 nm x 38 nm.
The software solves Landau-Lifshitz-Gilbert equation and simulates experiment at 0 K.
From the OOMMF data we have calculated also z-component of the PL stray magnetic field
10 nm below the PL nanomagnet, and also at the distance of the active area of the Hall sensor
(i.e. 80 nm). Magnetic stray fields are calculated as gradients of scalar potential:
H=-grad (Φm),
where the potential Φmis determined by the magnetization in a ferromagnetic object. Numerical
evaluation of the stray field of the object calls for spatial discretisation of the system. This is done
by splitting the object into rectangular cells as is done in the OOMMF simulation package. Under
assumption of uniform magnetization inside these cells, we calculate the values of stray fields by
methods proposed by either Newell et al.19or Abert et al.20
The OOMMF calculations have shown three basic di fferent configurations of local magnetiza-
tions in the PL nanomagnet. They are depicted in the Fig. 2, line 1, for external field value just above
the vortex nucleation field Hnuc- for field angle α=90◦(left column, C-state), for α=120◦(middle
column, S-state), and for α=160◦(right column, 2S-state). Red line follows the direction of local
magnetization lines. Lines 2 and 3 depict the map of relative values of the z-component of the
PL stray field in the active plane of the Hall probe before (line 2), and after (line 3) the vortex
nucleation, blue color is for the negative and red color for the positive z-component of the magnetic
field, respectively. The Hall voltage is proportional to the integral magnetic flux through its active
zone, i.e. through the the region above the dashed lines shown in the Figs. 2, second line. Overall
flux through the active area of the probe is much larger in the case of its asymmetrical position as
compared to its central position.117205-4 Ščepka et al. AIP Advances 5, 117205 (2015)
FIG. 2. Column A is for field angle α=90◦, column B for α=120◦, and column C for α=160◦. Field direction is indicated
also by the arrow in the line 2. Line 1: Field higher than Hnuc, 3 SD states shown, C-state for α=90◦; S-state for α=120◦,
and 2S-state for α=160◦.Line 2 :z-component of the stray field that corresponds to states shown in line 1, at the distance
of the HP’s active area. Dashed line represent Hall probe boundary, active zone is above it. Line 3 : Field lower than Hnuc,
z-component of the stray field at the distance of the HP. Stray field is strong at the PL’s opening and defines clearly vortex
chirality - vortex polarity is not so clear. Line 4: the same like line 3, but only 10 nm from the object – vortex chirality (CW,
CCW) and polarity (p +, p-) are clearly seen in the object.
Figs. 2, line 4, show the same fields components like in line 3, but at the distance of 10 nm only
from the PL object. Stray field at the PL’s opening is strong enough at the distance of 10 or 80 nm
and it defines clearly vortex chirality (clockwise, CW, for left column, and counter-clockwise,
CCW, for middle and right columns). On the other hand, vortex polarity can be much better recog-
nized at the distance of 10 nm (Figs. 2, line 4) as compared to the distance of 80 nm (Figs. 2, line 3).117205-5 Ščepka et al. AIP Advances 5, 117205 (2015)
V ortex state nucleated from C-state di ffers from the one from S-state in chirality, and vortex
states created from S-state and 2S-state di ffer in polarity. The chirality contribution to the Hall
signal is about 20%, meanwhile vortex polarity contribution is about 2% only for the distance
80 nm. The polarity signal is on the noise level in our experiment.
Now we discuss the influence of the PL shape, its location on the HP, and vortex chirality /polarity
on the magnetization reversal Hall signal. Figs. 3(a) and 3(b) show magnetization reversal curves for
field angle 90◦at 30 K. The signal obtained from HP di ffers significantly from the signal from disk.6In
both objects abrupt changes in the stray field correspond to significant changes of the magnetization
pattern (vortex nucleation and annihilation). The changes are directly connected with the step change
of the system energy or its redistribution between exchange and magnetostatic energies. On the other
FIG. 3. VHhysteresis loops for α=90◦(C-state), (a) is for time /field sequence 1-5 (Fig. 3(c)) – vortex is in this case expelled
oppositely to the PL opening (see also Fig. 3(d)). Green points in the Fig. 3(c) correspond to the vortex annihilation, red points
to the vortex nucleation, time scale is ∼1 hour. Fig. 3(b), right part of the graph, corresponds to the time /field sequences 4-6
(Fig. 3(c)), and left part of the graph corresponds to the time /field sequences 8-10 from Fig. 3(c). Fig. 3(d) shows vortex core
shift with external field (arrows out of the object) for CW chirality – the shift is perpendicular to the field direction change
and is opposite for CCW chirality or for opposite field as indicated by the arrow inside the object, but the same if both,
chirality and field direction are opposite.117205-6 Ščepka et al. AIP Advances 5, 117205 (2015)
hand, smooth changes of the signal can be attributed to the smooth transformation of the magnetization
field (shift of the vortex, C-state or S-state modification, etc.) in the ferromagnet.
As compared to signal from a disk,6Hall signal from the PL nanomagnet depends significantly
on vortex chirality (Fig. 3(a) and 3(b)), and can be easily controlled by the time sequence of the
applied field. Fig. 3(c) depicts the time sequence used to set desired chirality of the object. Chirality
setting in other shapes was also shown recently by other authors.21,22
Now we explain the hysteresis loops shown in the Fig. 3(a) and 3(b) according the time
sequence of the applied external magnetic field. Let us start CW chirality at zero field (point 1 in the
Fig. 3(c)). Then, positive field up to +2 T is applied (with detailed measurement only up to +10 mT,
shown in the figure 3(a)), and the vortex core is expelled from PL object at Han=88 mT at the right
side of the object (see Fig. 3(d)). Then is the field lowered, vortex nucleates at Hnuc=22.5 mT and
the chirality is set to CCW.10Negative field then expels the vortex core again to the right side due
to the CCW chirality, and vortex annihilates at Han=-88 mT. Then is the field increased, vortex
nucleates again at Hnuc=-22.5 mT with CW chirality.10Explained sequence (points 1-2-3-4-5 in
the Fig 3(c)) represents basic cyclic loop, for which is the vortex core expelled only to the right
boundary of the object (Fig. 3(d)). Fig. 3(a) shows described magnetization reversal loop obtained
for 100% runs with described magnetic-field time sequences, which proves that the vortex chirality
is perfectly controlled in our experiments. The physics behind such behavior is explained in Ref. 11.
To expel the vortex core to the opening of the PL object (PL left side, Fig. 3(d)), we have to
return from point 5 (zero field, Fig. 3(c)) to negative fields, point 6. The vortex with CW chirality
then annihilates at Han=-56 mT, and again nucleates at Hnuc=-22.5 mT with CW chirality when
the field is again increased. Such half-loop (points 5-6-7) is depicted in the left part of the Fig. 3(b).
Similar operation can be realized for the positive part of the loop (points 9-10-11, Fig. 3(b), right
part), for which we operate with CCW chirality, and vortex annihilates at 56 mT, and nucleates at
22.5 mT.
Fig. 4 shows the temperature dependence of the vortex nucleation field with field angle as a
parameter. The aim is to find the role of the magnetic state in the SD-V transitions. Therefore we
do not deal more with the vortex annihilation – magnetic state before vortex annihilation is for all 3
angles similar (vortex state). Each point of the magnetization reversals (except low temperature part
of the curve for angle 120◦) is a mean value of at least 8 measurements from its both, positive and
negative branches. Typically these values are distributed for each shown points (all temperatures,
field angles) within the interval ±0.5 mT, and the typical standard deviation is ∼0.15 mT. Similarly
to Ref. 6, the curves show two basic slopes – larger one for temperatures lower than 20 K, and small
one above 20 K. We assume that the large slopes are caused by presence of many shallow minima
of the total energy functional caused by edge corrugations as well as by granular structure of the
FIG. 4. Temperature dependence Hnucfor 3 angles of applied field. Slopes of each curve di ffer for T <20 K and T <20 K.
Interesting are details for field angle 120◦at temperatures <8 K. Full and empty triangles are for positive and negative
branches of the magnetization reversal, respectively. Each point is a mean value from 8 values of Hnucat least. For both
branches, two values of Hnucare possible.117205-7 Ščepka et al. AIP Advances 5, 117205 (2015)
Permalloy. At low temperatures, all small barriers presents obstacle for magnetization dynamics.
Thus at low temperatures the systems has to overcome many barriers which contribute significantly
to the final nucleation time, or at fixed rampage of the external field it is manifested as decrease
of the nucleation field. At higher temperatures the small barriers does not represent obstacle due to
the thermal energy of individual magnetic moments, and only the last barrier represents obstacle for
nucleation. Overcoming the last barrier the system lowers its energy significantly, and vortex state is
created.
Described mechanism needs further detailed analysis and modeling, which is over the scope of
this paper and will be published elsewhere.
Special attention has to be paid to field angle 120◦at low temperatures, for which nucleation
field shows sort of “digital” noise for both, negative and positive branches. The curve resembles two
metastable states in which the system can be trapped.
The appearance of two states (or the “digital” noise) can be explained by two chiralities
of the nanomagnet (CW or CCW). To confirm this assumption, we have analyzed the height of
Hall-voltage abrupt changes connected with the vortex nucleation. We have found that the heights
of the voltage jumps depend on the vortex nucleation field. If we select temperature 5 K, we have
for positive branch of the loop and vortex nucleation fields Hnuc=24 mT and Hnuc=32 mT (Fig. 4),
voltage jumps 4.5 ±0.1µV and 3.8 ±0.4µV, respectively. For negative branch corresponding voltage
jumps read 3.9 ±0.2µV and 2.9 ±0.25 µV. Based on our simulation, so high relative di fferences in
the Hall signal ( ∼25%) can be explained easily only by di fferent chirality of the final states obtained
(see also Fig. 2, line 3). It means that the ‘digital’ noise described for field angle 120◦at low
temperatures is a direct consequence of two chiralities generated in the nanomagnet.
In conclusion, we studied vortex nucleation /annihilation process and its temperature depen-
dence in Pacman-like nanoobjects using micro-Hall probe magnetometry. Chirality of the object can
be easily controlled due to its lowered symmetry. We show experimentally that vortex nucleation
field strongly depends on the angle of the external in-plane magnetic field. The experiments also
confirm that the vortex nucleation proceeds via thermal activation over an energy barrier.
This work has been supported by Slovak Grant Agency APVV , project APVV-0088-12 and
project APVV-0036-11, and by the Research & Development Operational Program funded by the
ERDF, “ CENTE 1 ”, ITMS code 26240120011 (0.5).
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1.1839632.pdf | Correlation of domain pattern and high-frequency response in pole-tip of inductive thin
film head
Dan Wei, Xuan Zhang, Guoguang Wu, Fulin Wei, and Zheng Yang
Citation: Journal of Applied Physics 97, 024501 (2005); doi: 10.1063/1.1839632
View online: http://dx.doi.org/10.1063/1.1839632
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/2?ver=pdfcov
Published by the AIP Publishing
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131.230.73.202 On: Fri, 19 Dec 2014 02:04:41Correlation of domain pattern and high-frequency response
in pole-tip of inductive thin film head
Dan Wei and Xuan Zhang
Key Laboratory of Advanced Materials, Department of Materials Science and Engineering,
Tsinghua University, Beijing 100084, People’s Republic of China
Guoguang Wu, Fulin Wei, and Zheng Yang
Research Institute of Magnetic Materials, Lanzhou University, Lanzhou 730000,People’s Republic of China
(Received 20 January 2004; accepted 2 November 2004; published online 22 December 2004 )
The high frequency response of the soft magnetic pole tip in the thin film inductive head is crucial
for the noise analysis of computer hard disk.Amicromagnetic model is established in a mesoscopicsoft magnetic thin film to analyze the domain pattern as well as the initial permeability in a widefrequency range from 10 MHz to 18 GHz. The simulated domain patterns in the pole tip, a vortex-and ladder-type, are chosen as the initial conditions for the high-frequency response studies. Thescaling law of the permeability is analyzed at different amplitudes of the alternating externalmagnetic field. It is found that in the ladder-type domains the high frequency response is muchbetter than that in the vortex-type domains, which agrees with experiment.Asimplified explanationof the simulation result is discussed based on the analysis of the nonlinear Landau–Liftshitzequation. © 2005 American Institute of Physics .[DOI: 10.1063/1.1839632 ]
I. INTRODUCTION
The invention of the thin film inductive head in 1978
was an important step in the evolution of the magnetic head,especially the write head.At present the areal recording den-sity of computer hard disk is above 100 Gb/in
2, and the size
of the pole tip in the thin film head is in the submicronregion. The frequency of the signals is around 1 GHz in thehard disk drive, to achieve high data rate and high storagedensity. Therefore, it is interesting to study the high-frequency response based on the analysis of the domainstructure in the pole tip.
1
The domain in the magnetic material is an old topic, but
still an important topic for the soft magnetic materials. Thedomain pattern depends on both the size and the intrinsicmagnetic properties. It has long been known that the insta-bility and noise in the read process of hard disk are associ-ated with domain wall motion.
2,3After extensive experiments
it was found that, in the thin film head’s pole-tip where thesignal flux density is largest, 180° walls were oriented par-allel or perpendicular to the direction of flux flow.
4In the
experiments of NiFe inductive thin film heads, a high-noisehead had a “vortex” modality of closure domains in the pole-tip; while a low-noise head had a “ladder” modality.
4,5
In this article, the correlation between the domain pat-
tern and the high frequency response in the pole tip softmagnetic layer will be studied by micromagnetic simulations
for the giant-magnetic saturation (GMS )thin film heads. The
Fe–X–N (X=Ta, Al, etc. )thin films have the proper perfor-
mance as the write head materials, such as high saturation,low coercivity, good mechanical property, and moderate con-ductivity. The Fe–Al–N film is chosen as the pole-tip mate-rials in this study because, up to now, Fe–Al–N films havebeen successfully fabricated and proven to satisfy various
requirements as potential candidates for thin-film headmaterials.
6–8
In Sec. II, the micromagnetic model of a soft mesoscopic
magnetic layer is briefly introduced. The vortex and laddermodalities of the domain structures in the pole tips are foundby the simulation. In Sec. III, the permeability of theFe–Al–N thin film is studied in a wide frequency rangefrom 10 MHz to 18 GHz, with respect to the vortex andladder domain patterns, respectively. Finally, a conclusion isgiven in Sec. IV.
II. MICROMAGNETIC MODEL AND DOMAINS
IN A POLE TIP
In the micromagnetic model of a soft magnetic layer, a
cluster is chosen as the basic magnetic unit instead of a nano-sized crystal grain.
6,9The pole tip is a mesoscopic Fe–Al–N
thin film of size 1.0 31.0mm2. The pole tip is divided into
16316 clusters which forms a square lattice. The size of
each cluster is 60 nm, the thickness of the film is 10 nm.The anisotropy fields of clusters are distributed as aripple structure with an angular distribution function asexps−
ausin2udaround the easy axis.6,9,10The orientation
parameter auis chosen as 8 in the simulation, and the easy
axis is along the horizontal x-axis in the pole tip, as shown in
Fig. 1. The saturation magnetization of Fe–Al–N is 4 pMs
=1.7 3104G, the anisotropy energy constant K1=4.06
3103erg/cm3, and the anisotropy field of a cluster Hk
=2K1/Ms=6 Oe. The exchange energy per unit area is 0.8
310−6erg/cm, thus the exchange length lex=˛A*/2K1is
100 nm.
The scaled Landau-Liftshitz-Gilbert equation is utilized
to solve the motion of the magnetic moments of clusters inan alternating external magnetic flux:JOURNAL OF APPLIED PHYSICS 97, 024501 (2005 )
0021-8979/2005/97 (2)/024501/4/$22.50 © 2005 American Institute of Physics 97, 024501-1
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
131.230.73.202 On: Fri, 19 Dec 2014 02:04:41dmˆi
dsntd=g
ns1+a2dsmˆi3Heffid−aSmˆi3dmˆi
dsntdD, s1d
where, nis the frequency of the external magnetic field in a
unit of MHz, while the time tis in the unit of microsecond.
The processing constant is g=17.6 rad/Oe/ ms, and ais a
dimensionless Landau damping constant. The effective mag-netic field H
effiin theith cluster is calculated by the variation
of the total energy with respect to the magnetic moment oftheith cluster. The total energy includes four energy items:
11
the Zeeman energy due to the external magnetic field; the
anisotropy energy; the exchange energy among neighboringclusters; the magnetostatic interaction energy among all clus-ters.The fast Fourier transform (FFT)technique is utilized in
this simulation. In a mesoscopic sized Fe–Al–N thin film,the sizes are of order microns or submicrons, hence the or-
dinary periodic boundary condition cannot be applied. Thetwo-dimensional FFT technique is slightly different with orwithout the periodic boundary condition; the rule is that thetotal energy, especially the magnetostatic interaction, shouldbe properly calculated with a given boundary condition.
The modality of domain structure depends on the initial
conditions of the clusters’ magnetization. In Figs. 1 (a)–1(c),
three different presetting conditions are given for the clustersin the pole tip, respectively. The simulation of the magneticvectors of clusters in the Fe–Al–N pole tip is carried outwith either one of the three presetting conditions. Two typesof domain patterns are obtained by micromagnetics: (1)a
“vortex-type” domain pattern, plotted in Fig. 2, is foundw.r.t. the initial condition given in Fig. 1 (a);(2)a “ladder-
type” domain pattern, shown in Fig. 3, is found w.r.t. theinitial condition given in Figs. 1 (b)or 1(c). Both the vortex-
type and the ladder-type domains agree with earlierexperiments.
4In Sec. III, the high frequency response of a
soft magnetic pole tip with either of the two most commonlyappeared domain patterns will be analyzed.
In the mesoscopic soft magnetic thin film, the existence
of the closure domains decreases the magnetostatic interac-tion among all clusters. If the magnetic poles exists in thefilm edge, the demagnetizing field will be very large, propor-tional to 4
pMs, as a result, the magnetization vectors are
parallel to the film edges to minimize the magnetostatic in-teraction. In the vortex-type domain pattern, the magnetiza-tion components forms four 90° domain walls, while in theladder-type domain pattern, the magnetization vectors areparallel to the horizontal 180° “ladder-type” domain wallwhere the magnetic poles do not appear.
III. RESONANCE IN A MESOSCOPIC FE–AL–N SOFT
MAGNETIC THIN FILM AND RESULT ANALYSIS
The magnetic resonance effects of soft magnetic materi-
als include the domain boundary resonance, the spin reso-nance, and the eddy current loss.
12In the low frequency re-
gion (tens of MHz or lower ), the domain wall resonance
occurs while in the high frequency region (hundreds of MHz
or higher ), the spin resonance and the eddy current losses are
significant, except that the eddy current losses becomeweaker when the film thickness decreases.
12,13In this work,
the eddy current losses are not considered because the film isvery thin. The effects of the domain boundary resonance andthe spin resonance are both included.
The frequency response of the permeability in a soft
magnetic pole tip is calculated based on the micromagneticmodel introduced above. In this simulation, the alternatingexternal magnetic field is applied in the vertical direction ofthe pole tip:
H
Wa=yˆH0sins2pf0tds 2d
where the amplitude H0is chosen as 12, 9, 6, 4.5, 3, and
1.5Hk, respectively. The alternating field is perpendicular to
the magnetization vectors in the ladder-type domain pattern,just as the flux flow in the actual pole tip of a thin film head.The average magnetization M
astdin the pole tip can be cal-
culated based on the simulated magnetization vectors by
Eq.(1).
FIG. 1. Three different initial conditions for the magnetization of clusters in
the pole-tip: (a)two ideal domains with antiparallel magnetization vectors
alongxdirection; (b)single ideal domain with magnetization vector along x
direction; (c)three ideal domains with antiparallel magnetization vectors
alongxdirection.
FIG. 2. Simulated vortex-type domain pattern, corresponding to the initial
condition given in Fig. 1 (a).
FIG. 3. Simulated ladder-type domain pattern, corresponding to the initial
condition given in Fig. 1 (b)or 1(c).024501-2 Wei et al. J. Appl. Phys. 97, 024501 (2005)
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131.230.73.202 On: Fri, 19 Dec 2014 02:04:41The time dependence of Mais not necessarily a simple
sine or cosine function. However, in the first order approxi-mation,M
astdcan be written in the form:
Mastd=MRsins2pf0td+MIcoss2pf0td. s3d
The real and the imaginary part of the susceptibility, x8and
x9, equal the ratios MR/H0andMI/H0, respectively.The real
and imaginary permeability m8andm9, where the initial per-
meability m=m8−jm9, are defined as 1+4 pMR/H0and
−4pMI/H0, respectively.6
The frequency range is wide, from 10 MHz to 18 GHz.
At each frequency, the average value of m8andm9are cal-
culated by an average of the simulated permeability in fivepole tips with five sets of the anisotropy orientation distribu-tion, respectively.
In Fig. 4, the high frequency response of a vortex-type
domain pattern is plotted where the Landau damping con-stant
a=0.02 (Ref. 11 )and the alternating field magnitude of
H0=12Hk. The real permeability m8is about 88 at low fre-
quencies. The rolloff frequency fhof the real permeability
m8, as well as the resonance frequency fpof the imaginary
permeability m9, are both around 2.5 GHz. The half value of
the peak m9is reached at a rise frequency frof 0.5 GHz.
In Fig. 5, the high frequency response of a ladder-type
domain pattern is plotted where the field magnitude H0
=12Hk. The real permeability m8is around 116 at low fre-
quency. The rolloff frequency fhis about 4.3 GHz. The peak
value 92 of the imaginary permeability m9occurs at a reso-
nance frequency fpof about 6.0 GHz, and the half of the
peak m9is reached at the rise frequency frof 1.0 GHz.Ascaling law of the initial permeability is studied versus
the external field. Table I gives the high frequency responseparameters with respect to the two kinds of domain patternsat a alternating field magnitude H
0=12, 9, 6, 4.5, 3, and
1.5Hk. It is found that the initial permeability of the ladder-
type domain pattern is higher than that of the vortex-typedomain. The high-frequency response curve of the ladder-type domain is smoother, indicating that the noise level ofladder-type domain pattern is much lower, which agrees withexperiments. The frequency range, which can be utilized inthe recording process, is much wider in the pole-tip with aladder-type domain. The
m8and m9increase with a lower
field magnitude H0, which is a phenomena of the scaling law
governed by the Landau–Liftshitz–Gilbert equation.
It is interesting to find the mechanism of the high fre-
quency response analytically, which will be a complementa-rity for the simulation result given above. The normalizedaverage magnetization along the alternating external fieldcan be expressed approximately as:
km
ystdl=m8H0sins2pf0td
4pMs−m9H0coss2pf0td
4pMs, s4d
where the stationary average kmys0dl;0 is implicit due to the
closure domain structure. To the first order approximation,
the total effective field acting on a magnetic moment in a
cluster equals H˜eff=H˜kxˆ+Hayˆ, whereH˜kis the effective an-
isotropy field after considering the mesoscopic size effect,andH
ais the alternative external field given in Eq. (2). From
the LLG equations, the time evolution of the magnetic vectorcomponents are:
FIG. 5. Simulated high frequency response in a mesoscopic Fe–Al–N thin
film with a ladder-type domain and a damping constant a=0.02. The real
and imaginary part of the permeability is displayed by the solid line and thedashed line, respectively.
FIG. 4. Simulated high frequency response in a mesoscopic Fe–Al–N thinfilm with a vortex-type domain and a damping constant
a=0.02. The real
and imaginary part of the permeability is displayed by the solid line and thedashed line, respectively.
TABLE I. The high frequency response parameters with respect to the two kinds of domain patterns at a alternating field magnitude H0=12,9,6,4.5,3,and
1.5Hk.
Parameter Vortex-type domain pattern Ladder-type domain pattern
Alternating field magnitude sHkd 12 9 6 4.5 3 1.5 12 9 6 4.5 3 1.5
Initial real permeability m8 88 80 103 114 100 115 116 123 135 135 121 148
Rolloff frequency fhsGHz d 2.5 3.1 4.1 3.3 4.5 3.8 4.3 6.0 6.5 6.8 7 4.3
Peak value of m9 65 88 100 113 118 123 92 130 169 155 162 175
Peak frequency fpsGHz d 2.5 3.5 4 4 4.5 4.5 6.0 7.5 7.0 7.5 7.5 7.5
Rise frequency frsGHz d 0.5 0.9 1.5 1.8 3 2.2 1.0 4.7 5.2 5.2 5.6 5.6024501-3 Wei et al. J. Appl. Phys. 97, 024501 (2005)
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131.230.73.202 On: Fri, 19 Dec 2014 02:04:41dmy
dt=gmzH˜k+agm’2Ha−agmxmyH˜k,
dmz
dt=−gH˜kmy+gmxHa+ofamzgHzg, s5d
wheremyis theycomponent of mˆalong the alternating
external field, mxis parallel to the effective anisotropy field
H˜kxˆ, andm’=˛m2
x+m2
zis thex-zcomponent. The normal-
ization condition mx2+my2+mz2=1 has been used in Eq. (5).
The effective anisotropy field H˜kdoes not equal to Hk
=2K/Ms=6 Oe; furthermore, H˜kcould be dependent on the
domain patterns in the pole tip.
At very low frequency such as tens of MHz, by applying
Eq.(2)and(4)into Eq. (5), it can be proved that the term
gmzH˜kis propotional to a huge parameter gH˜k/2pf0; there-
fore, to keep the balance of Eq. (5),m8andm9are enforced
to take the values:
m8=4pMs
H˜kkumxul,
m9<0. s6d
The dependence of low-frequency m8on the alternating
field amplitude H0, in the pole tip with the vortex- or the
ladder-type domain, can be explained by Eq. (6). The mag-
netization vectors of ladder-type domains are perpendicularto the alternating external field, so the value of km
xlof
ladder-type domains is almost 1.As listed inTable I, the ratio
of low-frequency permeability km8lw.r.t. the vortex type do-
main versus that w.r.t. the ladder type domain is about 0.7.
Therefore by Eq. (6), the value of kumxulof a vortex-type
domain is also around 0.7, which agrees with the simulated
domain patterns in Fig. 2.
When the magnitude of the alternating external field H0
decreases, the average magnetic moment component kumxul
perpendicular to the external field will increase. Following
Eq.(6), the real initial permeability m8is proportional to
kumxul, thus it will be higher for a lower H0. WhenH0tends
to zero, m8at low frequency will tend to the maximum value.
These analyses are roughly verified by the simulation resultsin Table I.The magnitude of low-frequency
m8is on the order of
102by simulation, thus the effective anisotropy field H˜kis on
the order of 102Oe for Fe–Al–N mesoscopic thin film,
which is much higher than the anisotropy field of a cluster
Hks0d=6 Oe. From Eqs. (5)and(6), the resonance frequency
is on the order of gH˜k,109Hz, which is also verified by the
simulation results in Table I.
IV. CONCLUSION
The correlation of the domain pattern and the high-
frequency response in a mesoscopic pole tip of the inductivewrite head is studied by micromagnetic simulations. The ini-tial real permeability
m8and the usable frequency range of a
thin film inductive head with a ladder-type domain in thepole tip is larger than that w.r.t. the vortex-type domain,which agrees well with the experiment. The low frequencypermeability is proportional to the averaged horizontal mag-netization magnitude. The resonance frequency is deter-mined by the effective anisotropy field in a mesoscopic poletip.
ACKNOWLEDGMENTS
This research was supported by the National Science
Foundation of China, Ministry of Education of P. R. China,Tsinghua University, and Lanzhou University.
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4IBM website: http://readrite.com/html/magbasic.html
5P. Kasiraj, R. M. Shelby, J. B. Best, and D. E. Horne, IEEE Trans. Magn.
22,8 3 7 (1986 ).
6D. Wei, F. Wei, and Z. Yang, J. Appl. Phys. 90,2 9 1 9 (2001 ).
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6677 (2003 ).
8F. Wei, D. Wu, D. Zheng, B. Ma, and Z. Yang, Proceedings, The Second
Magneto-Electronics International Symposium (1999 ),p .3 5 5 .
9D. Wei, C. K. Ong, and Z. Yang, J. Appl. Phys. 87, 3068 (2000 ).
10E. van de Riet and F. Roozeboom, J. Appl. Phys. 81, 350 (1997 ).
11X. Zhang and D. Wei, IEICE Trans. Electron. E85C,1 7 7 1 (1997 ).
12F. Brailsford, Physical Principle of Magnetism (Van Nostrand, Amster-
dam, 1966 ), Chap. 9.
13W. P. Jayasekara, J. A. Bain, and M. H. Kryder, IEEE Trans. Magn. 34,
1438 (1998 ).024501-4 Wei et al. J. Appl. Phys. 97, 024501 (2005)
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
131.230.73.202 On: Fri, 19 Dec 2014 02:04:41 |
5.0041072.pdf | Phys. Plasmas 28, 042301 (2021); https://doi.org/10.1063/5.0041072 28, 042301
© 2021 Author(s).Potential vorticity transport in weakly and
strongly magnetized plasmas
Cite as: Phys. Plasmas 28, 042301 (2021); https://doi.org/10.1063/5.0041072
Submitted: 21 December 2020 . Accepted: 09 March 2021 . Published Online: 07 April 2021
Chang-Chun Chen ,
Patrick H. Diamond ,
Rameswar Singh , and
Steven M. Tobias
COLLECTIONS
Paper published as part of the special topic on Papers from the 62nd Annual Meeting of the APS Division of Plasma
Physics
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Cite as: Phys. Plasmas 28, 042301 (2021); doi: 10.1063/5.0041072
Submitted: 21 December 2020 .Accepted: 9 March 2021 .
Published Online: 7 April 2021
Chang-Chun Chen,1,a)
Patrick H. Diamond,1,b)
Rameswar Singh,1
and Steven M. Tobias2
AFFILIATIONS
1)Department of Physics, University of California San Diego, La Jolla, California 92093, USA
2)Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Note: This paper is part of the Special Collection: Papers from the 62nd Annual Meeting of the APS Division of Plasma Physics.
a)Author to whom correspondence should be addressed: chc422@ucsd.edu
b)Electronic mail: pdiamond@ucsd.edu
ABSTRACT
Tangled magnetic fields, often coexisting with an ordered mean field, have a major impact on turbulence and momentum transport in many
plasmas, including those found in the solar tachocline and magnetic confinement devices. We present a novel mean field theory of potential
vorticity mixing in b-plane magnetohydrodynamic (MHD) and drift wave turbulence. Our results show that mean square stochastic fields
strongly reduce Reynolds stress coherence. This decoherence of potential vorticity flux due to stochastic field scattering leads to suppressionof momentum transport and zonal flow formation. A simple calculation suggests that the breaking of the shear-eddy tilting feedback loop bystochastic fields is the key underlying physics mechanism. A dimensionless parameter that quantifies the increment in power threshold is
identified and used to assess the impact of stochastic field on the L-H transition. We discuss a model of stochastic fields as a resisto-elastic
network.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0041072
I. INTRODUCTION
Momentum transport and the formation of sheared flows (i.e.,
zonal jets) are major research foci in quasi two-dimensional (2D) flu-ids
1,2and plasmas.3–7By “quasi 2D,” we mean systems with low effec-
tive Rossby number, in which dynamics in the third dimension is
constrained by, say, stratification or fast time averaging, due to small
electron inertia (as in magnetically confined plasmas). In such systems,Reynolds forces are equivalent to vorticity fluxes via the TaylorIdentity.
8For this and other reasons—the most fundamental being the
freezing-in law for fluid vorticity9—it is natural to describe such sys-
tems in terms of potential vorticity (PV). Generally, PV/C17f
¼fa/C1rw=q,w h e r e fais the absolute vorticity, wis a conserved scalar
field, and qis the fluid density. The advantage of a PV description of
the dynamics is that fis conserved along fluid particle trajectories, up
to dissipation, much like phase space density is conserved in the
Vlasov plasma. Examples of conserved PV are f¼by/C0r2w,w h e r e
bis the Rossy parameter and wis stream function for dynamics on
b-plane and PV¼ð1/C0q2
sr2Þjej/=Tþlnn0for the Hasegawa-
Mima system,10where /is electric potential and n0is a background
density. In such systems, momentum transport and flow formation
are determined by inhomogeneous PV mixing.11,12The mechanismfor PV mixing is closely related to the coherence and cross phase of
the vorticity flux. Mechanisms include viscous dissipation, wave-flowresonance, nonlinear mode interaction, and beat wave-flow interac-
tion, akin to nonlinear Landau damping.
13
Recently, the physics of PV transport in a disordered magnetic
field has emerged as a topic of interest in many contexts. One of theseis the solar tachocline,
7a weakly magnetized system, where momen-
tum transport (i.e., turbulent viscosity) is a candidate mechanism fordetermining the penetration of this layer and the flows within it. Thelatter is critically important to the solar dynamo.
4,14,15In this case, the
field is disordered16and confined (magneto-hydrostatically) to a thin
layer. The disordered magnetic field is amplified by high magneticReynolds number ( Rm) turbulent motions,
4,15pumped by convective
overshoot from the convective zone.17,18There is a weak mean toroidal
field, so magnetic perturbations are large. Another application, rele-vant to PV dynamics in a stochastic magnetic field, is to tokamaks(which are strongly magnetized), specifically those with stochasticityinduced by Resonant magnetic perturbations (RMPs).
19RMPs are
applied to the edge of tokamak plasma to mitigate Edge LocalizedModes (ELMs),
20,21which produce unacceptably high transient heat
loads on plasma-facing components. The “cost” of this benefit is an
Phys. Plasmas 28, 042301 (2021); doi: 10.1063/5.0041072 28, 042301-1
Published under license by AIP PublishingPhysics of Plasmas ARTICLE scitation.org/journal/phpincrease in the Low to High confinement mode transition (L-H transi-
tion) threshold power, as observed with RMPs.22–29Because several
studies suggest that the L-H transition is triggered by edge shear
flows,30–33this implies that the transition dynamics are modified by
the effects of stochastic fields on shear flow evolution. Indeed, analysissuggests that RMPs may “randomize” the edge layer. In this case, the
magnetic field is three dimensional (3D). Stochasticity results from
k/C1B¼0 resonance overlap, and field line separations diverge expo-
nentially. Hence, a key question is the effect of stochastic fields on self-
generated shear flows.
In both cases, the central question is one of phase—i.e., the effect
of the stochastic field on the coherence of fluctuating velocities, which
enters the Reynolds stress and PV. In physical terms, the disordered
field tends to couple energy from fluid motion to Alfv /C19enic and acoustic
waves, which radiate energy away and disperse wave packets. Ofcourse, Alfv /C19enic radiation is more effective in the case for low
b/C17p
plasma =pmag—the ratio of the plasma pressure to the magnetic
pressure—or for incompressible dynamics. The effect of this Alfv /C19enic
coupling is to induce the decoherence of the Reynolds stress (or vortic-
ity flux), thus reducing momentum transport and flow generation. In
this vein, we show that sufficiently strong coupling of drift waves to a
stochastic magnetic field can break the “shear-eddy tilting feedbackloop,” which underpins flow generation by modulational instability.
We note that the interaction of Alfv /C19en waves with a tangled magnetic
field differs from that of Alfv /C19en waves with an ordered field. Here, the
effect is to strongly couple the flow perturbations to an effective elastic
medium threaded by the chaotic field.
In this paper, we discuss the theory of PV mixing and zonal flow
generation in a disordered magnetic field, with special focus on appli-
cations to momentum transport in the solar tachocline and Reynolds
stress decoherence in the presence of a RMP-induced stochastic field.
Section IIaddresses a mean field theory for a tangled “in-plane” field
inb-plane magnetohydrodynamic (MHD),
34,35which is used to com-
pute the Reynolds force and magnetic drag in this weak mean field
(B0) system. The mean square stochastic magnetic field ( B2
st)w a s
shown to be the dominant element, controlling the coherence in the
PV flux and Reynolds force.7Of particular interest is the finding
that the Reynolds stress degrades for weak B0,a tal e v e lw e l lb e l o w
that required for Alfv /C19enization. It is also shown that the small-
scale field defines an effective Young’s modulus for elastic waves,
rather than a turbulent dissipation.7As a second application,
Sec.IIIpresents the study of Reynolds stress decoherence in toka-
mak edge turbulence. There, the stochastic field is 3D and is
induced by external RMP. Drift-Alfv /C19en wave propagation along
stochastic fields induces an ensemble averaged frequency shift that
breaks the “shear-eddy tilting feedback loop.” Reynolds stressdecoherence occurs for a modest level of stochasticity. The ratio of
the stochastic broadening effect to the natural linewidth defines a
critical parameter that determines the L-H transition power
threshold concomitant increment. With intrinsic toroidal rotation
in mind, we also explore the decoherence of the parallel Reynoldsstress. This is demonstrated to be weaker than for the previous
case since the signal propagation speed which enters parallel flow
dynamics is acoustic (not Alfv /C19enic). The interplay of symmetry
breaking, stochasticity, and residual stress is discussed. In Sec. IV,
w ed i s c u s st h ek e yfi n d i n go ft h i ss t u d ya n dp r o v i d es u g g e s t i o n sf o r
further research.II.b-PLANE MHD AND THE SOLAR TACHOCLINE
Stochastic fields are ubiquitous. One example is the tangled field
of the solar tachocline
7,36—a candidate site for the solar dynamo. The
solar tachocline is a thin strongly stratified layer between the radiationand convection zones, located at /C240:7 solar radius,
36where magnetic
fields are perturbed by “pumping” from the convection zone. Hence, a
model for strong perturbed magnetic fields is crucial for studying PVmixing and momentum transport in the solar tachocline. A study byTobias, Diamond, and Hughes 37 on b-plane MHD shows that a
modest mean field suppresses zonal flow formation and momentumtransport ( Fig. 1 ). Chen and Diamond 7 proposed that the effects of
suppression by random-fields are already substantial (even for weakB
0) on account of Reynolds stress decoherence. They discussed a
b-plane (quasi-2D) MHD model for the solar tachocline and studied
how the zonal flow is suppressed by random fields. We note that thedynamics of b-plane MHD are exceedingly complex. At small-scales,
it resembles MHD with a forward cascade and also supports large scaleRossby waves. Interactions of the latter tend to generate flows, as foran inverse cascade. In view of this multi-scale complexity, we followthe suggestion of Rechester and Rosenbluth
38and replace the full
problem by a more tractable one in which an ambient disordered field
is specified. We utilize a mean field theory which averages over the
small-scale field. Meso-scopic flow phenomena in this environmentare then examined.
A. Model setup
Theb-plane MHD system at high Rmwith weak mean field
supports a strong disordered magnetic field. Hence, analyzing thisproblem is a daunting task on account of the chaotic field and strongnon-linearity. Zel’dovich
39suggested the “whole” problem consists of
a random mix of two components: a weak, constant field ( B0) and a
random ensemble of magnetic “cells” ( Bst), for which the lines are
closed loops ( r/C1Bst¼0). Of course, the mean magnetic field B0
lines are closed toroidally. Assembling these two parts gives a field
FIG. 1. Scaling law for the transition between the forward cascades (circles) and
inverse cascades (plus signs) from Tobias et al.37B0is mean magnetic field and g
is the magnetic diffusivity. Colormaps are velocity intensity. Red indicates strong for-ward flows, while blue indicates strong backward flows. They show as mean mag-
netic field is strong enough, zonal flow generation stops and the system is fully
Alfv/C19enized. Reproduced with permission from Chen and Diamond, Astrophys. J.
892, 24 (2020), by permission of the AAS. Copyright 2020 The American
Astronomical Society.Physics of Plasmas ARTICLE scitation.org/journal/php
Phys. Plasmas 28, 042301 (2021); doi: 10.1063/5.0041072 28, 042301-2
Published under license by AIP Publishingconfiguration which may be thought of as randomly distributed
“cells” of various sizes, threaded by “sinews” of open lines ( Fig. 2 ).
Hence, the magnetic fields can be decomposed to B/C17B0þBst,
where B0is modest (i.e., jBstj>B0). This system with strong, tan-
gled field cannot be described by linear responses involving B0
only, and so it is not amenable to traditional quasilinear theory.
Linear closure theory allows analysis in a diffusive regime, where
fluid Kubo number40Kufluid <1 and magnetic Kubo number
Kumag<1. Here, the fluid Kubo number Kufluid/C17dl=D?,w h e r e
dlis the characteristic scattering length and Dis the eddy size. For
weak mean field, we have Kumag/C17lacjBst=B0j=D>1, rendering
standard closure method inapplicable. Here lacis magnetic auto-
correlation length and Dis eddy size. Hence, we employ the simpli-
fying assumption of lac!0s o Kumag’lacjBst=B0j=D<1. This
approximation allows us to peek at the mysteries of the strong per-
turbation regime by assuming fields with short correlation length.In a system with strong random fields ( B
st; such that ensemble
average of squared stochastic magnetic field B2
st>B2
0), this approx-
imation comes at the price of replacing the full b-plane MHD
problem with a model problem. Results for this model problem,where jB
stj>B0, are discussed in this section. Notice that in 3D
MHD, as for a tokamak, there are k/C1Bresonances. Stochastic fields
are due to overlapping of magnetic islands near the edge of toka-mak. The quasilinear closure works in tokamak since we have
jB
stj=B0’10/C03/C24/C04—the magnetic auto-correlation length lacis
proportional to RqandKumaghas a moderate value ( Kumag/C201).
Thus, for weak perturbation, the mean field method is still applica-
ble. Details are discussed in Sec. III.
B. Calculations and results
Following the argument above, a model which circumvents the
problem of simple quasi-linear theory for this highly disordered sys-
tem is presented. This is accomplished by considering the scale order-ing. In the two-scale average method proposed,
7an average over an
area is performed, with a scale (1 =kavg) larger than the scale of the sto-
chastic fields (1 =kst) but smaller than the magnetic Rhines scale41
(kMR) and Rossby wavelength ( kRossby ). This average is denoted as
/C22F/C17ÐdR2ÐdBst/C1PðBst;x;Bst;yÞ/C1F,w h e r e Fis arbitrary function, dR2
denotes integration over the region, and PðBst;x;Bst;yÞis probability distri-
bution function for the random fields. This random-field averageallows us to replace the total field due to MHD turbulence (something
difficult to calculate) by moments of a prescribed probability distribu-tion function (PDF) of the stochastic magnetic field. The latter canbe
calculated. Another average over zonal flow scales— k
zonal,d e n o t e da s
bracket average hi /C171
LÐ
dx1
TÐ
dt—is conducted. Hence, the scale
ordering for b-plane MHD is ultimately kst>kavg/H11407kMR/H11407kRossby >
kzonal(Fig. 3 ). They started with the vorticity equation and the induc-
tion equation:
@
@tþu/C1r/C18/C19
f/C0b@w
@x¼/C0B/C1 rðr2AÞ
l0qþ/C23r2f; (1)
@
@tA¼ðB/C1r Þwþgr2A; (2)
where Ais magnetic potential, wis the stream function, /C23is vis-
cosity, qis mass density, and gis the magnetic diffusivity. In the
b- p l a n em o d e l ,t h ex -a n dy - a x e sa r es e ti nt h el o n g i t u d i n a la n d
latitudinal direction, respectively. They employed periodical
boundary conditions—considering the b-plane in a domain
0/C20x;y/C202pusing pseudospectral methods.42This model,7
with its two-average method, allows insights into the physics of
how the evolution of zonal flows is suppressed by disordered
fields both via reduced PV flux ( C) and by an induced magnetic
drag, i.e.,
@
@thuxi¼h /C22Ci/C01
gl0qhB2
st;yihuxiþ/C23r2huxi: (3)
Here, huxiis the mean velocity in the zonal direction, and h/C22Ciis the
double-average PV flux. Here1
gl0qhB2
st;yii st h em a g n e t i cd r a g
coefficient.
First, stochastic fields suppress PV flux by reducing the PV diffu-
sivity ( DPV),w h e r e
/C22C¼/C0DPV@
@y/C22fþb/C18/C19
; (4)
where bis the Rossby parameter and the PV diffusivity can be
written as
FIG. 2. The large-scale magnetic field is distorted by the small-scale fields. The
system is the “soup” of cells threaded by sinews of open field lines. Reproducedwith permission from Chen and Diamond, Astrophys. J. 892, 24 (2020), by permis-
sion of the AAS. Copyright 2020 The American Astronomical Society.
FIG. 3. Length scale ordering. The smallest length scale is that of the random field
(lst). The random-field averaging region is larger than the length scale of random
fields but smaller than that of the Rossby waves. Reproduced with permission fromChen and Diamond, Astrophys. J. 892, 24 (2020), by permission of the AAS.
Copyright 2020 The American Astronomical Society.Physics of Plasmas ARTICLE scitation.org/journal/php
Phys. Plasmas 28, 042301 (2021); doi: 10.1063/5.0041072 28, 042301-3
Published under license by AIP PublishingDPV¼X
kj~uy;kj2/C2/C23k2þB2
0k2x
l0q !
gk2
x2þg2k4þB2
st;yk2
l0qgk2
x/C0B2
0k2x
l0q !
x
x2þg2k4!2
þ
/C23k2þB2
0k2x
l0q !
gk2
x2þg2k4þB2
st;yk2
l0qgk2!2: (5)
Equation (5)shows that strong mean square stochastic field ( B2
st)a c t s
to reduce the correlation of the vorticity flux, thus reducing PV mix-ing. This explains the Reynolds stress suppression observed in simula-
tion
7(Fig. 4 ). Note that this reduction in Reynolds stress sets in for
values of B0well below that required for Alfv /C19enization (i.e., Alfv /C19enic
equi-partition h~u2i’h ~B2i=l0q).
Second, magnetic drag physics is elucidated via the mean-field
dispersion relation for waves in an inertial frame ( b¼0), on scales
l/C29k/C01
avg,
xþiB2
st;yk2
y
l0qgk2þi/C23k2 !
ðxþigk2Þ¼B2
0k2x
l0q: (6)
The drag coefficient, v/C17B2
st;yk2
y
l0qgk2, emerges as approximately propor-
tional to an effectivespring constant
dissipation. The “dissipation” and “drag” effects
s u g g e s tt h a tm e a ns q u a r es t o c h a s t i cfi e l d s B2
stform an effective resisto-
e l a s t i cn e t w o r k ,i nw h i c ht h ed y n a m i c se v o l v e .T h efl u i dv e l o c i t yi sredistributed by the drag of small-scale stochastic fields. Ignoring vis-cosity ( /C23!0), we have
x
2þiðvþgk2Þ|fflfflfflfflffl{zfflfflfflfflffl}
dragþdissipationx/C0B2
st;yk2
y
l0qþB2
0k2x
l0q !
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
effective spring constant¼0: (7)
Note that this is effectively the dispersion relation of dissipative Alfv /C19en
waves, where the “stiffness” (or magnetic tension) is determined by
both the ordered and the mean square stochastic field ( B2
st). In practice,t h el a t t e ri sd o m i n a n ta s B2
st’RmB2
0andRm/C291. So, the ensemble
of Alfv /C19enic loops can be viewed as a network of springs ( Fig. 5 ). Fluid
couples to network elastic elements, thus exciting collective elastic
modes. The strong elasticity, due to Alfv /C19enic loops, increases the effec-
tive memory of the system, thus reducing mixing and transport andultimately causes Reynolds stress decoherence. The network is fractal
and is characterized by a “packing factor,” which determines the effec-
tive Young’s modulus. It is important to note that the ‘stochastic elasti-
cized’ effect is one of increased memory ( notone of enhanced
dissipation) as in the familiar cases of turbulent viscosity or resistivity.
C. Implications for the solar tachocline
The balance between Reynolds and Maxwell stress in a fully
Alfv/C19enized system where fluid and magnetic energy reach near equi-
partition is the conventional wisdom. Simulation results ( Fig. 4 ), how-
ever, show that Reynolds stress is suppressed by stochastic fields well
before the mean field is strong enough to fully Alfv /C19enize the system
(details are shown in Chen and Diamond7). These results suggest that
turbulent momentum transport in the tachocline is suppressed by the
enhanced memory of stochastically induced elasticity. This leaves no
viscous or mixing mechanism to oppose ‘burrowing’ of the tachocline
due to meridional cells driven by baroclinic torque rp/C2rq.
46This
finding suggests that the Spiegel and Zahn47scenario of burrowing
opposed by latitudinal viscous diffusion and the Gough and
McIntyre48suggestion of that PV mixing opposed burrowing both fail .
Finally, by process of elimination, the enhanced memory-induced sup-
pression of momentum transport allows the Gough and McIntyre48
suggestion that a residual fossil field in the radiation zone is what ulti-
mately limits tachocline burrowing.
III. DRIFT WAVE TURBULENCE IN A STOCHASTIC FILED
This section focuses on the effect of stochastic fields on zonal
flow suppression, such as in the case of RMPs at the edge of tokamak.
Experimental results show that pre-L-H transition Reynolds stress
FIG. 4. Average Reynolds stresses (orange line) and Maxwell stresses (blue line)
forb¼5,g¼10/C04from Chen and Diamond.7Full Alfv /C19enization happens when
jB0jis larger than jB0j¼ 10/C01. The yellow-shaded area is where zonal flows
cease to grow. This is where the random-field suppression on the growth of zonal
flow becomes noticeable. Reproduced with permission from Chen and Diamond,Astrophys. J. 892, 24 (2020), by permission of the AAS. Copyright 2020 The
American Astronomical Society.
FIG. 5. Site-Percolation Network. Schematic of the nodes-links-blobs model (or
SSdG model, see Refs. 43–45 ). This depicts the resisto-elastic medium formed by
small-scale stochastic fields. Reproduced with permission from Chen and Diamond,Astrophys. J. 892, 24 (2020), by permission of the AAS. Copyright 2020 The
American Astronomical Society.Physics of Plasmas ARTICLE scitation.org/journal/php
Phys. Plasmas 28, 042301 (2021); doi: 10.1063/5.0041072 28, 042301-4
Published under license by AIP Publishingbursts drop significantly when RMPs are applied to the edge of DIII-
D.49T h es t o c h a s t i cm a g n e t i cfi e l d sa r ef o r mt h ep o w e rt h r e s h o l df o r
L-H transition increases as the normalized intensity of radial RMPs
(dBr=B0) increases.22–29Here we aim to shed light on these two phe-
nomena and to address the more general question of Reynolds stressdecoherence in a stochastic magnetic field.
To begin, we explore the timescale ordering for the physics. We
construct a model in Cartesian (slab) coordinates— xis radial, yis
poloidal, and zis the toroidal directions, in which the mean toroidal
field lies ( Fig. 6 ). Hereafter, ?represents the x-a n d y-direction which
is perpendicular to parallel mean field (in z-direction). Considering a
generalized diffusivity ( D
0) and assuming modes are sufficiently
packed (P
k¼ðL
2pÞ3ÐdkkÐdk?),50we have
D0¼Re Cððð
dkxdkydkzð
dxk2
y
B2
0j/kxj2 i
x/C0vAkzþiDk2
?()
;
(8)
where Cis a parameter of integrals with dimension ½L3T/C138;vA
/C17B0=ffiffiffiffiffiffiffiffil0qpis Alfv /C19en speed,51and the Dis a spatial diffusivity under
the influence of stochastic field, defined as D/C17vADM. As discussed
below, vAappears as the characteristic velocity for signal propagation
along the stochastic field, since zonal flows follow from the need tomaintain
r/C1J¼0, in the face of ambipolarity breaking due to polari-
zation fluxes. Here DM’lacb2(hearafter b2/C17hB2
st;?i=B2
0)i st h e
stochastic magnetic diffusion, first derived by Rosenbluth et al.52
Here, the bracket average is a stochastic ensemble average hi
/C17ÐdR2ÐdBst/C1PðBst;x;Bst;yÞ/C1Fsimilar to the bar average in Sec. II B.
But here dR2is an averaging area (at scale 1 =kst)o v e r y-a n d z-direc-
tions.j/kxj2is the electric potential spectrum, such that
j/j2
kx¼/2
0S1ðk?ÞS2ðkzÞi
x/C0x2
0;k/C0iDxk; (9)
where x0;kis the centroid of the frequency spectrum, Dxis the natural
linewidth of potential field, and S1andS2are the k-spectrum of k?
and parallel kz, respectively. Performing the frequency integration, we
haveD0¼RefCððð
dkxdkydkz/2
0S1ðk?Þ/C1 (10)
S2ðkzÞk2
y
B2
0ð
dxi
ðx/C0x0;kÞ/C0iDxki
x/C0vAkzþiDk2
?/C26/C27 /C27
¼Re/C26
Cðð
dkxdky/2
0S1ðk?Þk2
y
B2
0
/C2ð
dkzS2ðkzÞ/C02pi
x0;k/C0vAkzþiDxkþiDk2
?/C27
: (11)
Now consider a Lorentzian kz-spectrum
S2ðkzÞ¼i
kz/C0kz;0þiDkz; (12)
where kz;0is the centroid and Dkzis the width of the spectrum. So we
have
D0¼Re/C26
Cðð
dkxdky/2
0S1ðk?Þk2
y
B2
0
/C2ð
dkzi
kz/C0kz;0þiDkz/C1/C02pi
x0;k/C0vAkzþiDxkþiDk2
?/C27
¼Re/C26
Cð2pÞ2ðð
dkxdky/2
0S1ðk?Þ
/C2k2
y
B2
0i
x0;k/C0vAk0;zþiDxkþiDkzvAþiDk2
?/C27
:
We do the kzintegral only since k/C1B0resonance defines the crit-
ical time scale in this system—the ordering of these broadenings
(DkzvA;Dxk,a n d Dk2
?) in the denominator is the key to quantifying
stochastic field effects. The first term, DkzvA, is the bandwidth of an
Alfv/C19en wave packet excited by drift-Alfv /C19en coupling. Here
vADkz/H11351vA=Rq,w h e r e Ri sm a j o rr a d i u sa n d q/C17rBt=RBpis the safety
factor. The bandwidth DkzvAis a measure of the dispersion rate of an
Alfv/C19en wave packet. The second term is the rate of nonlinear coupling
or mixing due to ambient electrostatic micro-instability Dxk’x/C3
¼khqsCs=Ln, where the x/C3is drift wave turbulence frequency, qsis
gyro-radius, Csis sound speed, and Lnis density scale length. Dxis
comparable to k2
?DGB,w h e r e DGB/C17x/C3=k2
?’q2
sCs=Lnis the gyro-
Bohm diffusivity (for khqs/C241). The third is the stochastic field scat-
tering rate Dk2
?’k2
?vADM. Ultimately, we will show that
k2
?vADM/H11407Dxk(orvADM>DGB) is required for Reynolds stress
decoherence ( Fig. 7 ). In practice, this occurs for k2
?vADM/H11407vAjDkkj,
i.e.,Kumag’1 is required. The condition k2
?vADM>Dxkrequires
that stochastic field broadening exceeds the natural turbulence line-
width,29so that k2
?vADM>Dx.S a t i s f y i n gt h i sr e q u i r e s
FIG. 6. Magnetic fields at the edge of tokamak. RMP-induced stochastic fields
(black loops) lie in radial ( x) and poloidal ( y) plane. Mean toroidal field is treading
through stochastic fields perpendicular in z-direction (blue arrows).
FIG. 7. Time scale ordering. We are interested in a regime where stochastic field
effect becomes noticeable, which requires Dx<Dk2
?. The comparison between
Alfv/C19enic dispersion rate vAjDkkjand stochastic broadening rate Dk2
?gives a mag-
netic Kubo number Kumag’1.Physics of Plasmas ARTICLE scitation.org/journal/php
Phys. Plasmas 28, 042301 (2021); doi: 10.1063/5.0041072 28, 042301-5
Published under license by AIP Publishingb2>ffiffiffibpq2
/C3/C15=q/C2410/C08,w h e r e lac’Rq;/C15/C17Ln=R/C2410/C02;b’10/C02,
and normalized gyro-radius q/C3/C17qs=Ln’10/C02/C24/C03. It is believed that
b2at the edge due to RMP is /C2410/C07for typical parameters; hence, the
stochastic broadening effect is likely sufficient to dephase the Reynoldsstress. Following from this condition, we propose a dimensionless
parameter a/C17b
2q=q2
/C3ffiffiffibp/C15—defined by the ratio k2
?vADM=Dxk—to
quantify the broadening effect. The increment in L-I and I-H power
thresholds as avaries are explored using a modified Kim-Diamond
L-H transition model53in Sec. IIIB. We also give a physical insight
into stress decoherence by showing how stochastic fields break the“shear-eddy tilting feedback loop,” which underpins zonal flow growthby modulational instability.
A. Model setup
In this Cartesian coordinate, a current flows in the toroidal direc-
tion, producing a mean poloidal field. In contrast to the tachocline,here the magnetic field is 3D, and stochasticity results from the overlap
of magnetic islands located at the resonant
k/C1B¼0 surfaces. The sto-
chasticity is attributed to the external RMP field and typically occurs ina layer around the separatrix. The distance between neighboring mag-netic field trajectories diverges exponentially as for a positive Lyapunovexponent. Stochastic fields due to RMPs resemble Zel’dovich “cells”
39
(Fig. 2 ), lying in the x–yplane with a mean toroidal field (on the
z-axis), threading through perpendicularly. Notice that we assume the
stochastic field is static. Of course, once overlap occurs, the coherentcharacter of the perturbations is lost, due to finite Kolmogorov-Sinaientropy (i.e., there exists a positive Lyapunov exponent for the field). Inthis case, the magnetic Kubo number is modest Ku
mag/H113511.
We start with four field equations as follows:
1. Vorticity evolution
@
@tfzþuy@
@yfzþuz@
@zfz¼1
qB0@
@zJzþ1
qBx;st@
@xJzþ2j
q@
@yp;
(13)
where fzis the vorticity, uyisE/C2Bshear flow, uzis intrinsic rotation,
andjis curvature. Notice that we only consider the vorticity in
z-direction so hereafter we define fz/C17ffor simplicity.
1. Induction evolution
@
@tAzþuy@
@yAz¼/C0Bx;st
B0@
@x//C0@
@z/þgr2Az; (14)
where /is electric potential field ( f/C17r ?/C2u?¼1
B0r2
?/).
1. Pressure evolution
@
@tpþðu/C1r Þp¼/C0cpðr /C1 uÞ; (15)
where cis the adiabatic index.
1. Parallel acceleration
@
@tuzþðu/C1r Þuz¼/C01
q@
@zp; (16)where pis pressure. Here we are interested in the simplest possible
problem-interaction between a wave spectrum and a zonal flow. We
later retain the minimal diamagnetic effect in the modified Kim-
Diamond model (see Sec. III B). This is presented in pressure gradient
evolution. A detailed study of diamagnetic effects will be added in
future work (Plasma Physics and Controlled Fusion, in preparation).
B. Calculation and results
We decompose the magnetic fields, magnetic potential, velocities, and
electrical potential
magnetic fields B¼ðBx;st;By;st;B0Þ
potential fields A¼/C01
2B0y;1
2B0x;~Aðx;yÞ/C18/C19
velocities u¼ð~ux;huyiþ~uy;huziþ~uzÞ
electric potential /¼h/iþ~/;8
>>>>>>><
>>>>>>>:(17)
where hu
yiis the mean poloidal flow and huziis the intrinsic rotation.
The tilde ~denotes the perturbations of the mean. Hence, from Eqs.
(13) and(14), we obtain (assume magnetic diffusivity ignorable, i.e.,
g!0)
ð/C0ixþhuyiikyÞ~/kxþvAikzþikxBx;st
B0þikyBy;st
B0/C18/C19
vA~Akx
¼~ux
k2@
@xr2h/iþ2j
qikyB0
/C0k2/C18/C19
~p;(18)
ð/C0ixþhuyiikyÞvA~AkxþvAikzþikxBx;st
B0þikyBy;st
B0/C18/C19
~/kx
¼/C0gk2vA~Akx’0: (19)
We define an Els €asser-like variable f6;kx/C17~/kx6vA~Akxand combine
Eqs.(18)and(19)to obtain
ð/C0ixþhuyiikyÞf6;kx6vAikzþikxBx;st
B0þikyBy;st
B0/C18/C19
f6;kx
¼~ux
k2@
@xr2h/iþ2j
qikyB0
/C0k2/C18/C19
~p/C17Sf; (20)
where Sfi st h es o u r c ef u n c t i o nf o r f6;kx.E q u a t i o n (20)is the evolution
equation for the Els €asser response to a vorticity perturbation. Note
that this response is defined by
1. Propagation along the total magnetic field, i.e.,
ikzþikxBx;st=B0þikyBy;st=B0. Note this includes propagation
along the wandering magnetic field component.
2. Advection by mean flow ikyhuyi.
3. Finite frequency ix.
From Eq. (20),w eh a v e f6;kx¼i
ðx/C0huyiky7vAkzÞþ7vAkjBj=B0/C2Sf.
The propagator can be written in integral form
i
ðx/C0huyiky7vAkzÞþ7vAkjBj=B0
¼ð
dseiðx/C0huyiky7vAkzÞsD
e7ivAÐ
ds0Bi;st
B0ki/C0/C1E
; (21)Physics of Plasmas ARTICLE scitation.org/journal/php
Phys. Plasmas 28, 042301 (2021); doi: 10.1063/5.0041072 28, 042301-6
Published under license by AIP Publishingwhere hirefers to an average over statistical distribution of Bst.H e n c e ,
the Els €asser response for f6;kxis obtained by integrating along trajec-
tories of total magnetic field lines (including perturbations), i.e.,
f6;kx¼ð
dseiðx/C0huyiky7vAkzÞshe7ivAÐ
ds0kxBx;st
B0þkyBy;st
B0/C0/C1
i/C2Sf:(22)
Integration along the perturbed field trajectory can be imple-
mented using the stochastic average over an scale (1 =kst), where the
bracket denotes an average over random radial excursions
dxi¼vAÐ
ds0Bi;st=B0such that
hi/C17ðð
i¼x;yddxiffiffiffiffiffiffiffiffiffiffipDisp e/C0dx2
i
Dis: (23)
Here, he7ivAÐ
ds0kxBx;st
B0þkyBy;st
B0/C0/C1
iis set by the diffusivity tensor
D¼v2
AÐ
ds00bi;stðs00Þbj;stðs00Þ,w h e r e iandjrepresent xorycomponent.
So we obtain
he7ivAÐ
ds0kxBx;st
B0þkyBy;st
B0/C0/C1
i’1/C0kiDijkjs’e/C0k/C1D/C1ks; (24)
where sis the decorrelation time due to field stochasticity, such that
s’Ðds00’lac=vA. We assume no correlation between x-a n d
y-direction of stochastic field (i.e., and hBx;stBy;sti¼0) and hBi;sti¼0.
Hence, only diagonal terms of Dsurvive (i.e., Dij¼dijvAlacb2
i).
A number of important comments are in order here. First,
D’vADM, indicating that vorticity response decorrelation occurs by
Alfv/C19enic pulse diffusion along wandering magnetic fields. This is a
consequence of the fact that PV (or polarization charge) perturbations
(which determine the PV or polarization charge flux—i.e., the
Reynolds force) are determined via r/C1J¼0, the characteristic signal
speed for which is vA.S e c o n d , vADMis actually independent of B 0and
is a set only by b2. To see this, observe that b2/C17hB2
sti=B2
0;vA
¼B0=ffiffiffiffiffiffiffiffil0qp,a n d lac¼Rq.T h u s , D/b2reflects the physics that
decorrelation occurs due to pulses traveling along stochastic fields,
only. In this respect, the result here closely resembles the 2D case (i.e.,
b-plane MHD) discussed in Sec. II.T h i r d , vAfor the mean field enters
only via the linear vorticity response—which is used to compute the
vorticity flux—and thus the Reynolds force.
N o ww eh a v et h ea v e r a g e dE l s €asser response
f6;kx¼i
ðx/C0huyiky7vAkzÞþiDk2/C2Sf; (25)
where Dk2¼Dxk2
xþDyk2
y.A n d ~/kx¼ðfþ;kxþf/C0;kxÞ=2y i e l d s
~f¼1
B0r2~/¼X
kxRe/C0k2
B0/C18/C191
2ðfþ;kxþf/C0;kxÞ/C20/C21
: (26)
We define Mf/C17ðfþ;kxþf/C0;kxÞ=2Sfis a propagator
Mf¼1
2/C18i
ðxsh/C0vAkzÞþiDk2þi
ðxshþvAkzÞþiDk2/C19
;(27)
where xsh/C17x/C0huyikyis the shear flow Doppler shifted frequency.
From Eq. (20), we have the fluctuating vorticity
~f¼1
B0r2~/¼X
kxRe M f/C0k2
B0Sf/C20/C21
: (28)Hence, the response of vorticity ( ~f)t ot h ev o r t i c i t yg r a d i e n ta n d
curvature term in the presence of stochastic fields is as follows:
~f¼X
kxRe M f/C0~ux;kx
B0/C18/C19@
@xr2h/i/C20/C21
þRe ik yMf2j
q~pkx/C20/C21
:(29)
The first term determines the diffusive flux of vorticity. The second
sets the residual stress that depends on the pressure perturbation and
the curvature of the mean magnetic field. Note that the residual stressis defined as a component of poloidal stress tensor that is neither pro-portional to flow nor flow shear.
54–56Here, it depends on ~pkxand
hence gives non-zero vorticity flux. We calculate the residual stressterm in Eq. (29) by using another set of Els €asser-like variables
g
6;kx/C17~pkx
qC2s6~uz;kx
Cs, derived from perturbation equations of Eqs. (15)
and(16)
ð/C0ixþhuyiikyÞ~p
qC2
sþCsikzþikjBj;st
B0/C18/C19~uz
Cs¼/C0~ux
qC2
s@
@xhpi/C17Sg;
(30)
ð/C0ixþhuyiikyÞ~uz
CsþCsikzþikjBj;st
B0/C18/C19~p
qC2
s¼0: (31)
Notice that Sg/C17/C0~ux
qC2s@
@xhpii st h es o u r c ef u n c t i o nf o r g6;kxsuch
that
g6;kx¼i
ðxsh7CskzÞþiDsk2/C2Sg; (32)
where Ds/C17CsDM(for pressure decorrelation rate sc¼lac=Cs)i st h e
diffusivity due to an acoustic signal propagating along stochastic fields.
To obtain ~pkx¼qC2
sðgþ;kxþgþ;kxÞ=2, we define a propagator
Mg/C17ðgþ;kxþg/C0;kxÞ=2Sg
Mg¼1
2/C18i
ðxsh/C0CskzÞþiDsk2þi
ðxshþCskzÞþiDsk2/C19
’i
xsh:
(33)
Notice that ~pis the pressure perturbation set by the acoustic coupling.
Hence, it has slower speed Cs/C28vA(orb/C281) as compared to
Alf/C19enic coupling. An ensemble average of total vorticity flux yields
h~ux~fi¼/C0X
kxj~ux;kxj2ReðMfÞ@
@xhfi
/C0X
kxj~ux;kxj2ReðikyMfMgÞ2j
q@
@xhpi/C20/C21
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Component of Residual Stress: (34)
Notice that Dsk2’CsDMk2. Hence, the broadening effect of random
acoustic wave propagation itself is negligible compared to the naturallinewidth since the plasma b/C281. Now, we have
h~u
x~fi¼/C0 DPV@
@xhfiþFresj@
@xhpi; (35)
where DPV/C17P
kxj~ux;kxj2ReðMfÞis PV diffusivity, and Fres
/C17P
kx2ky
xshqj~ux;kxj2ReðMfÞ’P
kx2ky
xshqDPV;kxis the residual stress.
Notice that there is no parity issue lurking in the term 2 ky=xshqsince
2ky=xshq/2k=y=k=yq/2=q(i.e., even) for kyhuyi/C28x’x/C3.B yPhysics of Plasmas ARTICLE scitation.org/journal/php
Phys. Plasmas 28, 042301 (2021); doi: 10.1063/5.0041072 28, 042301-7
Published under license by AIP Publishingusing the Taylor Identity,8we rewrite the PV flux as a Reynolds force
h~ux~fi¼@
@xh~ux~uyi.I nt h el i m i to ft h e DPVandFresslowly varying as
compared with vorticity hfiand pressure hpi, respectively, the poloidal
Reynolds force is
h~ux~uyi¼/C0 DPV@
@xhuyiþFresjhpi; (36)
where the effective viscosity is
DPV¼X
kxj~ux;kxj2 vAb2lack2
x2
shþðvAb2lack2Þ2: (37)
This indicates that both the PV diffusivity and residual stress (and thus
the Reynolds stress) are suppressed as the stochastic field intensity b2
increases ,s ot h a t vAb2lack2exceeds xsh.T h i sr e s u l ti sc o n s i s t e n tw i t h
our expectations based upon scaling and with the Reynolds stress burst
suppression in the presence of RMPs, observed in Kriete et al.49This
model is built on gyro-Bohm scaling and hence the stochastic dephas-ing effect is insensitive to the details of the turbulence mode (e.g., theion temperature gradient driven mode, trapped electron modes, etc.),within that broad class.
We propose that physical insight into the physics of Reynolds
stress decoherence can be obtained by considering the effect of a sto-chastic magnetic field on the “shear-eddy tilting feedback loop.” Recallthat the Reynolds stress is given by
h~u
x~uyi¼/C0X
kj~/kj2
B2
0hkykxi: (38)
Thus, a non-zero stress requires hkykxi6¼0, i.e., a spectrally averaged
wave vector component correlation. This in turn requires a spectralasymmetry. In the presence of a seed shear, k
xtends to align with ky,
producing correlation and hence hi6¼0(Fig. 8 ). To see this, observe
that Snell’s law states
dkx
dt¼/C0@ðx0;kþkyuyÞ
@x’0/C0@ðkyuyÞ
@x: (39)
So, to set a non-zero phase correlation hkykxi6¼0, we take
kx’kð0Þ
x/C0ky@huyi
@xsc,w h e r e scis a ray scattering time that limits ray
trajectory time integration. Ignoring kð0Þ
x,w et h e nfi n d
h~ux~uyi’0þX
kj~/kj2
B2
0k2
y@huyi
@xsc: (40)
Note that the existence of correlation is unambiguous, and the
Reynolds stress is manifestly non-zero. Here, eddy tilting (i.e., kxevolution) has aligned wave vector components. Once huxuyi6¼0,
flow evolution occurs due to momentum transport. Then, flow shear
amplification further amplifies the Reynolds stress, etc. This process
constitutes the “shear-eddy tilting feedback loop” and underpins mod-
ulational instability amplification of zonal shears. Central to shear-
eddy tilting feedback is the proportionality of stress cross-phase to
shear. However, in the presence of stochastic fields, the correlation
hkxkyiis altered. To see this, consider drift-Alf /C19en turbulence, for
which
x2/C0x/C3x/C0k2
kv2
A¼0: (41)
Letx0be the frequency of the drift wave roots. Now, let kk¼kð0Þ
k
þk?/C1ðBst;?=B0Þdue to stochastic field wandering, and dxthe corre-
sponding ensemble averaged correction to x0—i.e., x¼x0þdx.
After taking an ensemble average of random fields from Eq. (41),w e
obtain hdxi’v2
A/C16
2kkhBst;?i
B0/C1k?þh ðBst;?
B0/C1k?Þ2/C17
i=x0,w h e r e hBi;sti
¼0 so the first term vanishes. The ensemble averaged frequency shift
is then
hdxi’1
2v2
A
x0b2k2
?: (42)
Here, hx0i’x/C3, corresponding to the drift wave. Note that dx/
hB2
stiis independent of B0,e x c e p tf o r x0. Thus, in the presence of
shear flow, the Reynolds stress becomes
h~ux~uyi’X
kj~/kj2
B2
0k2
y@huyi
@xscþ1
2kyv2
Ak2?
x0@b2
@xsc/C18/C19
: (43)
This indicates that for@huyi
@x<v2
Ak2?
x0@b2
@x, the shear-eddy tilting feedback
loop is broken since the hkxkyicorrelation is no longer set by flow
shear. In practice, this requires b2/H1140710/C07, as deduced above.
We modify a well-known predator-prey model of the L-H transi-
tion, the Kim-Diamond model53to include the effects of stochastic
fields. The Kim-Diamond model is a zero-dimensional reduced model,
which evolves fluctuation energy, Reynolds stress-driven flow shear,
and the mean pressure gradient. As heat flux is increased, a transition
from L-mode to intermediate phase (I-phase) (dotted line in Fig. 9 )
and to H-mode (dashed line in Fig. 9 ) occurs. Here, we include the
principal stochastic field effect—Reynolds stress decoherence. This is
quantified by the dimensionless parameter a/C17qb2=ffiffiffibpq2
/C3/C15derived in
Sec.III. The aim is to explore the changes in L-H transition evolution
(i.e., power threshold increment) due to magnetic stochasticity.
This dimensionless parameter aquantifies the strength of stochas-
tic dephasing relative to turbulent decorrelation. As shown in theprevious paragraph, the E/C2Bshear feedback loop that forms the
zonal flow is broken by the stochastic fields. Hence, the modifica-
tion enters the shear decorrelation term i nt h et u r b u l e n c e( n)e v o -
lution, the corresponding term in the zonal flow energy ( v
2
ZF)
evolution, and the pressure gradient ( N) evolution. The third term
is smaller byffiffiffibp(i.e.,a!affiffiffibp) due to the fact that acoustic wave
scattering is what causes decoherence in the pressure evolution. A
factor 1 =ð1þcaÞcaptures the modification due to the effect of sto-
chastic suppression effect, where cis a constant. The modified
Kim-Diamond model becomesFIG. 8. Shear-eddy tilting feedback loop. The E/C2Bshear generates the hkxkyi
correlation and hence support the non-zero Reynolds stress. The Reynold stress,in turn, modifies the shear via momentum transport. Hence, the shear flow reinforcethe self-tilting.Physics of Plasmas ARTICLE scitation.org/journal/php
Phys. Plasmas 28, 042301 (2021); doi: 10.1063/5.0041072 28, 042301-8
Published under license by AIP Publishing@n
@t¼nN/C0 a1n2/C0a2/C18@huyi
@x/C192
n/C0a3v2
ZFn/C11
ð1þa4aÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Reynolds stress decoherence;(44)
@v2
ZF
@t¼a3v2
ZFn/C11
ð1þa4aÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Reynolds stress decoherence/C0b1v2
ZF; (45)
@N
@t¼/C0 c1nN/C11
1þa4affiffiffibp/C0/C1
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
turbulent diffusion of pressure/C0c2Nþ Q; (46)
where ai,bi,a n d ci(a1¼0:2;a2¼0:7;a3¼0:7;a4¼1;b1¼1:5;
c1¼1;c2¼0:5;ffiffiffibp¼0.05) are model-dependent coefficients, and
Qis the input power.
We fix all parameters but the aand find the L-I and I-H
power thresholds (hereafter defined as Qth;L/C0IandQth;I/C0Hrespec-
tively) increase in n,v2
ZF,a n d N, when aincreases (see Fig. 9 ).
Specifically, stochastic fields raise Qth;L/C0Iand Qth;I/C0H, linearly in
proportion to a(Fig. 10 ). This is a likely candidate to explain the
L-H power threshold increment in DIII-D.29Notice that in this
wave-zonal flow interaction problem, a possible effect of a meanshear would be to decorrelate the responses of PV and, hence, to
reduce velocity perturbations. The mean shear flow would thus
define a time scale k
hDhvE/C2Bi0(Dis the perturbation radial scale).
This would need to be compared to Dxk’x/C3¼khqsCs=Lnand
k2
?vADM.I fhvE/C2Bi0is weak, mean shear is irrelevant, and the story
here holds. If hvE/C2Bi0>Dxk, stochastic field scattering should be
compared to hvE/C2Bi0,n o tDxk. But if the mean shear is strong, the
discharge likely already is in the H-mode, and the point of this
paper is moot.
We are also interested in stochastic field effects on the toroidal
Reynolds stress h~ux~uzi, which determines intrinsic toroidal rotation.
Consider toroidal Eq. (16) with the stochastic fields effect
@
@z¼@
@zð0Þþb/C1r?.W eh a v e
@
@thuziþ@
@xh~ux~uzi¼/C01
q@
@xhb~pi: (47)
The second term on the LHS is the toroidal Reynolds force. The RHS
contains the hb~pi, the kinetic stress. Both of these terms can be
dephased by stochastic fields, but the dephasing of the former is of
primary importance. In the context of intrinsic rotation, we follow themethod for the derivation of decoherence of the poloidal residual
stress—i.e., using El €asser-like variables g
6;kx/C17~pkx
qC2s6~uz;kx
Csfrom Eqs.
(15) and(16). The only difference from the previous residual stress
calculation is the presence term of@
@xhuzi, and hence the source of
toroidal stress becomes Sg;6/C17/C0~ux;kx
qC2s@
@xhpi7~ux
Cs@
@xhuzi.W efi n d ~uz;kx
¼CsReðgþ;kx/C0g/C0;kxÞ=2 and define a “response” Rg/C17ðgþ;kx/C0
g/C0;kxÞ=2s u c ht h a t
Rg¼i
2Sg;þ
ðxsh/C0CskzÞþiDsk2/C0Sg;/C0
ðxshþCskzÞþiDsk2/C18/C19
:(48)
Noting that when@
@xhuzi¼0, we will have Sg;þ¼Sg;/C0¼Sgand
hence the propagator Rgreduces to MgSg[compare with Eq.(33) ].
Thus, the toroidal Reynold stress is
FIG. 9. Modified Kim-Diamond model. (a) Turbulent intensity n. The wiggles are the
limit cycle oscillations prior to the transition.57,58(b) Zonal flow energy v2
ZF. (c)
Pressure gradient Nevolution with increasing input power Q. Dotted lines indicate
L-I transitions (at power Qth;L/C0I), and dashed lines indicate I-H transitions (at power
Qth;I/C0H). As we increase the mean square stochastic field ( b2), i.e., from
b2=q2
/C3ffiffiffibp¼0 (blue) to 0.6 (green), L-I and I-H transitions power threshold
increase, i.e., from L-I power threshold Qth;L/C0I¼0:5 to 0.6 and from I-H power
threshold Qth;I/C0H¼1:20 to 1.41.Physics of Plasmas ARTICLE scitation.org/journal/php
Phys. Plasmas 28, 042301 (2021); doi: 10.1063/5.0041072 28, 042301-9
Published under license by AIP Publishingh~ux~uzi¼X
kxj~ux;kxj2
/C2/C02Dsk2
x2
shþð2Dsk2Þ2@huzi
@xþ/C02Dsk2
x2
shþð2Dsk2Þ2kz
xshq@hpi
@x"#
:
(49)
The first term on the RHS contains the turbulent viscosity (/C23turb),
which we define as
/C23turb/C17X
kxj~ux;kxj2 2Dsk2
x2
shþð2Dsk2Þ2
¼X
kxj~ux;kxj2 2Csb2lack2
x2
shþð2Csb2lack2Þ2: (50)
This turbulent viscosity has a form similar to DPVin Eq. (37).
However, decorrelation of /C23turbis set by Cswhile that of DPVis set byvA. Thus, decoherence effects here are weaker. The second term in Eq.
(49)contains the toroidal residual stress (Fz;res)
Fz;res/C17X
kx/C0kz
xshq/C18/C19
j~ux;kxj2ð2Dsk2Þ
x2
shþð2Dsk2Þ2/C24X
kx/C0kz
xshq/C23turb;kx:
(51)
Notice that non-zero value of Fz;resrequires symmetry breaking (i.e.,
hkzkyi6¼0) sincekz
xshq/kz
ky.T h u s , a symmetry breaking condition—
non-zero hkzkyi—must be met for finite residual toroidal residual stress
(Fz;res).Here,hkzkyimust now be calculated in the presence of the sto-
chastic field. The details of this calculation involve determining the
interplay of stochastic field effects with spectral shifts (i.e., symmetrybreaking by E/C2Bshear) and inhomogeneities (i.e., spectral symmetry
breaking by intensity gradient). This will involve competition between
the radial scale length of stochastic fields and the scales characteristicof the spectral shift (induced by E/C2Bshear) and the spectral intensity
gradient. This detailed technical study is left for a future publication.We rewrite the toroidal stress as
h~u
x~uzi¼/C0 /C23turb@
@xhuziþFz;res@
@xhpi; (52)
which has similar form to that of poloidal Reynolds stress in Eq. (36).
This shows that stochastic fields reduce the toroidal stress and hence
slow down the intrinsic rotation. However, from Eqs. (50) and(51),
the stochastic suppression effect on toroidal stress and residual stressdepends on C
sDM(notvADM), and so is weaker than for zonal flows.
IV. DISCUSSION
In general terms, we see that 42 years after the influential paper
by Rechester and Rosenbluth38the physics of plasma dynamics in a
stochastic magnetic field remains theoretically challenging and vital toboth astrophysical and magnetic fusion energy (MFE) plasma physics.Transport in a state of coexisting turbulence and stochastic magneticfield is a topic of intense interest. In this paper, we discussed aspects ofmomentum transport and zonal flow generation in two systems withlow effective Rossby number, where dynamics evolve in the presenceof a stochastic magnetic field.
The first system is the solar tachocline—with weak mean magne-
tization, strong magnetic perturbation, and b-plane MHD dynamics.
Here, a tangled magnetic network generated by fluid stretching at largeRmdefines an effective resisto-elastic medium in which PV transport
occurs. We show that coupling to bulk elastic waves, with frequencyx
2’B2
stk2=l0q, results in decoherence of the PV flux and Reynolds
force, thus limiting momentum transport. Moreover, this effect sets infor seed field energies well below that required for Alfv /C19enization.
Physically, the stress decoherence occurs via coupling of fluid energyto the elastic network of fields, where it is radiatively dissipated. One
implication of this prediction of quenched momentum transport is
that tachocline burrowing cannot be balanced by momentum trans-port. This bolsters the case for Gough and McIntyre’s suggestion
48
that a fossil magnetic field in radiation zone is what ultimately limitsmeridional cell burrowing.
The second system is the L-mode tokamak edge plasma in the
presence of a stochastic magnetic field induced by external RMP coils.Here, the system is 3D, and field lines wander due to islands overlap.
The magnetic Kubo number is modest. We showed that the “shear-FIG. 10. Power threshold increments ( Qth) in modified Kim-Diamond model. (a) L-I
transition power threshold increment. (b) I-H transition power threshold increment.
Mean-square stochastic fields ( b2) shift L-H and I-H transition thresholds to higher
power proportional to b2=q2
/C3ffiffiffibp.Physics of Plasmas ARTICLE scitation.org/journal/php
Phys. Plasmas 28, 042301 (2021); doi: 10.1063/5.0041072 28, 042301-10
Published under license by AIP Publishingeddy tilting feedback loop” is broken by a critical b2intensity and that
k2
?vADMcharacterizes the rate of stress decoherence. Note that the
Alfv/C19en speed follows from charge balance, which determines Reynolds
stress. A natural threshold condition for Reynolds stress decoherenceemerges as k
2
?vADM=Dx>1. In turn, we show that this defines a
dimensionless ratio a, which quantifies the effect on zonal flow excita-
tion, and thus power thresholds. a’1o c c u r sf o r b2’10/C07,c o n s i s -
tent with stochastic magnetic field intensities for which a significantincrement in power threshold occurs. Note that this scaling is some-what pessimistic (i.e., q
/C02
/C3).
This study has identified several topics for future work. These
include developing a magnetic stress–energy tensor evolution equationfor representing small-scale fields in real space. Fractal network modelsof small-scale magnetic field are promising in the context of intermit-tency. A better understanding of stochastic field effects on transport
forKu
mag/C211 is necessary as a complement to our Kumag/C201m o d e l -
based understanding. For MFE plasmas, a 1D model for the L-H tran-sition evolution is required. This study will introduce a new lengthscale (Jiang and Guo et al., in press), which quantifies the radial extent
of the stochastic region. Finally, the bursty character
49of pre-
transition Reynolds work suggests that a statistical approach to thetransition is required. The challenge here is to identify the physics ofthe noise and flow bursts, and how the presence of stochasticity
quenches them. The stochasticity-induced change in “shear-eddy tilt-
ing feedback loop” discussed herein is a likely candidate for thequenching of the noise and flow burst.
ACKNOWLEDGMENTS
We thank Lothar Schmitz, D. M. Kriete, G. R. McKee, Zhibin
Guo, Gyungjin Choi, Weixin Guo, and Min Jiang for helpful
discussions. We also acknowledge stimulating interactions with
participants of the 2019 Festival de Th /C19eorie and the 2021 KITP
program Staircase 21. KITP is supported in part by the NationalScience Foundation under Grant No. NSF PHY-1748958. This
research was supported by the U.S. Department of Energy, Office
of Science, Office of Fusion Energy Sciences, under Award No.DE-FG02–04ER54738.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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Published under license by AIP Publishing |
1.4975660.pdf | Phase-locking of multiple magnetic droplets by a microwave magnetic field
Chengjie Wang , Dun Xiao , Yan Zhou , J. Åkerman , and Yaowen Liu
Citation: AIP Advances 7, 056019 (2017); doi: 10.1063/1.4975660
View online: http://dx.doi.org/10.1063/1.4975660
View Table of Contents: http://aip.scitation.org/toc/adv/7/5
Published by the American Institute of PhysicsAIP ADV ANCES 7, 056019 (2017)
Phase-locking of multiple magnetic droplets
by a microwave magnetic field
Chengjie Wang,1Dun Xiao,1Yan Zhou,2J. Åkerman,3,4and Yaowen Liu1,a
1Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology,
School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
2Department of Physics, The University of Hong Kong, Hong Kong, China
3Materials Physics, School of Information and Communication Technology, KTH Royal Institute
of Technology, Electrum 229, 16440 Kista, Sweden
4Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden
(Presented 4 November 2016; received 23 September 2016; accepted 8 November 2016;
published online 1 February 2017)
Manipulating dissipative magnetic droplet is of great interest for both the fundamen-
tal and technological reasons due to its potential applications in the high frequency
spin-torque nano-oscillators. In this paper, a magnetic droplet pair localized in two
identical or non-identical nano-contacts in a magnetic thin film with perpendicu-
lar anisotropy can phase-lock into a single resonance state by using an oscillating
microwave magnetic field. This resonance state is a little away from the intrinsic
precession frequency of the magnetic droplets. We found that the phase-locking
frequency range increases with the increase of the microwave field strength. Fur-
thermore, multiple droplets with a random initial phase can also be synchronized
by a microwave field. © 2017 Author(s). All article content, except where oth-
erwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4975660]
I. INTRODUCTION
Spin-torque oscillators (STOs)1have attracted considerable attentions with the potential of
enabling novel spintronic devices for telecommunication and logic applications.2–7The STOs are typ-
ically fabricated in two different architectures: Nano-pillar1or nanoscale electrical contacts (NC)8to
ferromagnetic thin films with a free magnetic layer and a fixed spin polarizer layer. The spin-transfer
torque (STT) in such contacts can compensate the damping torque and excite steady state spin preces-
sion in the free layer at a threshold d.c. current. In free layers with perpendicular magnetic anisotropy
(PMA), the STT has been predicted to induce a localized oscillation mode—dissipative magnetic
droplet soliton.9,10Recent experiments have confirmed this type of localized oscillation soliton mode
generated in NC region,11–15which has been considered as a promising candidate for the STOs. In
such devices, the energy dissipation due to magnetic damping is compensated by the energy input
from the current-induced STT effect.16,17The typical droplet has a partially reversed magnetization
directly underneath the NC and all the spins at the NC perimeter rotate in phase around the film
normal with a large of precession angle, which can lead to an increase in the microwave output power
of NC-STOs by a factor of 40 compared to those of non-droplet counterparts.11,18However, this
enhanced power is still too weak ( 200 pW) for practical applications. Increasing output power of
STO is essential for successful adaptation of the STT excitation scheme for advanced microwave
oscillators. One promising approach to increasing the output power has been suggested by using the
phase-locking mode of an array of STOs through the synchronization technique.3,19–28This is a very
challenging issue for the droplet-based NC-STOs due to the strongly non-linear soliton property of
the magnetic droplets.
aCorresponding author. Email: yaowen@tongji.edu.cn
2158-3226/2017/7(5)/056019/6 7, 056019-1 ©Author(s) 2017
056019-2 Wang et al. AIP Advances 7, 056019 (2017)
To gain insight into the nature of droplet dynamics, micromagnetic simulations have been
powerful tools.9,29–32In this paper, by employing microwave (MW) magnetic fields, we show
that two magnetic droplets formed at identical or non-identical NCs can phase-lock into a sin-
gle resonance state over a frequency range close to the MW driving frequency. Furthermore,
multiple droplets distributed in a 4 4 spatial matrix can also synchronize into a phase-locking
state.
II. MODEL
As shown in Fig. 1(a), we consider a NC-STO geometry based on a pseudo spin-valve struc-
ture11,12patterned into a square shape with 512 512 nm2. The spin polarizer layer is assumed to be
magnetized along the + zdirection and the 5-nm thick free layer has perpendicular magnetic anisotropy
(PMA). The free layer has two NCs with a separation distance of 240 nm. Positive current is defined
as the flow of electrons from the free layer to the polarizer layer.
Micromagnetic modeling of the free layer was performed using the open-source simulation
software MuMax3,33which is based on the Landau-Lifshitz-Gilbert equation including the STT
term:16,34
dm
dt=
mHeff+ mdm
dt+aJm(mmf)
The magnetization m=M/MS,MSis the saturation magnetization. The first term describes the
spin precession, the second term is the Gilbert damping term, the third one is the Slonczewski STT
term16that only works on the NC region. aJis the STT strength. The effective magnetic field Heff
includes the exchange field, anisotropy field, demagnetization field (magnetic dipole interaction),
and external magnetic field. In this study, the external magnetic field contains a static magnetic
fieldH0applied in z-direction and a microwave (MW) magnetic field HMW=HMW,0sin(2fMWt)ˆey
applied in y-direction, where HMW,0is the field strength and fMWis the MW frequency. The following
material parameters measured on similar Co/Ni multilayers are used for the free layer:11,14Ms= 716
kA/m (saturation magnetization), Ku= 447 kJ/m3(magnetic anisotropy), A= 30 pJ/m (exchange
stiffness), =0.05 (Gilbert damping), P=0.5 (spin polarization). The applied current is 8 mA for
each NC except the case of specific notation in this study. The current-induced Oersted field is not
taken into account for most simulations. 0H0=0.8 T along z axis, which results in the Zeeman
precession frequency f0=
H22.5 GHz consequently. In order to reduce the influence of sample
boundary, a periodic boundary condition is used in both x-direction and y-direction. In this study, all
the simulations are performed at zero temperature.
FIG. 1. (a) Schematic diagram of NC-STOs. 'is the angle between the in-plane magnetization and x-axis. FFT spectra
calculated from <mx>of droplets for two identical NCs (b) and for two non-identical NCs (c). The initial frequencies for
NC1 and NC2 are indicated by the thin blue and red curves, respectively. The phase-locking frequency driven by the MW
field is given by the black curve. (d) Time dependent phase difference between the two droplets for identical and non-identical
cases, respectively.056019-3 Wang et al. AIP Advances 7, 056019 (2017)
III. SIMULATION RESULTS
A. Synchronization of two droplets at identical NCs
Figs. 1(b)–(d) show the typical dynamics of droplet pairs driven by a microwave magnetic field,
in which the ac MW magnetic field is applied along y-axis direction with the amplitude of 20 mT
and the frequency of 25.50 GHz. First, a droplet pair is generated at two identical NCs ( r1=r2
= 15 nm) by applying a current of 8 mA flowing in each NC. The details of creation process of
a droplet pair can be found in our previous study.32Simulation indicates that the magnetization
precession of the two droplets have almost same intrinsic frequency of 25.52 GHz, as the thin
curves shown in Fig. 1(b). Here the frequencies are calculated from time dependent mxusing the
fast Fourier transform (FFT) technique. In our simulations, we set the two droplets having different
initial magnetization phase ( '1,'2), where'is the angle of the in-plane magnetization component
of droplet at the NC circumference with the x-axis. The phase difference '= '1 '2 of the
droplet pair first will slightly increase to an antiphase magnetization precession state ( '180),
see Fig. 1(d). When an AC microwave magnetic field HMWis switched on at t= 100 ns, the droplet
pair with the antiphase state is quickly locked into an in-phase synchronization precession state
('0) within 2 ns (see movie S1 of the supplementary material), at which the two droplets start
to rotate with a frequency of 25.50 GHz same as that of the driving microwave field as shown in
Fig. 1(b).
We would like to point out that the appearance of antiphase precession state in double or
multiple NCs driven by STT effect is a normal feature, which has been suggested three possi-
ble reasons:23,24The dynamic dipole-dipole interaction (DDI), spin wave (SW), or the separa-
tion distance between the two droplets. In order to clarify which factor plays the role for the
antiphase state, we have carried out a series of simulations by switching on/off the DDI, SW,
and current-induced Oersted-field. Also the separation distance varies from 240 nm to 260 nm.
We find that without the microwave magnetic field the antiphase precession state is a favorite
state with the DDI and SW effect (not shown). This antiphase state only disappears at relative
large separation distance and with the Oersted field case. This result is consistent well with the
experiments.27
The phase-locking state of the droplet pair depends on the microwave driving field. Fig. 2(a)
shows the FFT output frequency of the droplet pair as a function of the microwave source, where the
driving source is fixed to be 20 mT and the frequency varies from 24 GHz to 27 GHz (correspondingly
to1.5 GHz away from the intrinsic frequency of a droplet). Fig. 2(b) shows the time dependent phase
difference of the two droplets. Note that the two droplets can quickly phase-lock into a synchronization
state when the source frequency changing from 24.8 GHz to 26.2 GHz. In this region, the two droplets
are locked into the frequency of driving source, resulting in a resonance state between the droplet
pair and the source. The frequency difference between the droplet and microwave source is smaller
than a specific value ( 0.7 GHz). This behavior is similar to that theoretically predicted by Slavin
and Tiberkevich,23in which the phase of magnetization is tuned by the combined effect from an
oscillating stray field and a spin wave. In addition, the NC2 droplet has a larger synchronization
range in frequency with the driving MW source, as shown in Fig. 2(a). Our simulations show that the
large microwave field may enlarge the droplet diameter somehow, resulting in one of the droplet is
larger than the other even for them generated at same size of NCs (see movie S2 of the supplementary
material). It is noticed that the intrinsic frequency of droplet decreases with the increase of droplet
diameter,9,32therefore, the window of frequency locking for NC2 is larger than that of NC1. However,
it is unclear why the droplets generated at NC1 and NC2 have the different response to the MW driving
source. For the relative small microwave driving source, this phenomenon will disappear. We would
like to point out that the phase-locking (PL) window in Fig. 2(a) is defined as the frequency region
both the two droplets having the same frequency as well as the same precession phase (or a fixed
phase difference).
The phase-locking window is also manipulated by the strength of driving magnetic fields. Fig. 2(c)
shows the phase-locking feature by tuning the microwave source strength from 2 mT to 20 mT.
Obviously, the stronger of driving source, the wider of phase-locking range. For the field strength
smaller than 2 mT, the phase-locking range is smaller than 200 MHz. Another important feature056019-4 Wang et al. AIP Advances 7, 056019 (2017)
FIG. 2. Phase-locking (PL) of a droplet pair at two identical NCs driven by a microwave magnetic field. (a) Frequency
analysis by FFT as a function of the driving source frequency. (b) The phase difference v.s. time. The curves are offset
vertically for clarity. The microwave driving source changes from 24 GHz to 27 GHz. (c) The dependence of phase-
locking region on the microwave strength. (d) The phase-locking difference f=fmax fmindepends on the microwave
strength.
shown in Fig. 2(c) is that the phase-locking range is asymmetry, showing the different response of the
droplet pair to high-frequency and low-frequency driving signals. The droplet pair prefers to resonate
with the low-frequency microwave fields. Fig. 2(d) summarizes the phase-locking difference fas a
function of the strength of microwave fields, where fis defined as the difference between the upper
and lower bounds of phase-locking frequency, f=fmax fmin. Note that, the flinearly increases
with the microwave strength for H<10mT. Interestingly, there is a pronounced jump between 10
and 18 mT, which may correspond to a new type of precession mode excitation. But the underlying
physics for this jump is unclear. After that, the fis saturated to be 1.8 GHz. This saturated frequency
interval is originated from the topological protection of droplet structure.
B. Synchronization of a droplet pair at non-identical NCs
In contrast, for a droplet pair formed at different size of NCs ( r1= 16.5 nm, r2= 15 nm),
the synchronization process demonstrates significant different behaviors. Firstly, the two droplets
have different intrinsic frequencies in absence of the MW magnetic fields, showing the big droplet
(r1= 16.5 nm) has a little lower intrinsic frequency of 25.38 GHz [Fig.1(c)]. This can be attributed to
the increased droplet size, the frequency decreases with the increase of radius size.9,32Secondly,
the transient phase difference 'before the synchronization state featured a drastic oscillation
between 180and 0, as shown in Fig. 1(d). However, when the MW driving field is switched
on at t= 100 ns, the magnetization precession of the two droplets synchronizes with each other very
quickly, with a same frequency (25.5 GHz) and a fixed phase difference '4.6. Thirdly, due to
the non-identical size, the phase-locking frequency window of the two droplets is shrunk a little
[see Fig. 3(a)]. Moreover, a nonzero stable 'value is observed for this phase-locking state, and
the'increases with the frequency of microwave increasing, as shown in Fig. 3(b). In addition,
we would like to point out that for the droplet pair at two non-identical NCs having too large dif-
ferent intrinsic frequencies (e.g. induced by big difference of NC size), the phase-locking will be
invalid.056019-5 Wang et al. AIP Advances 7, 056019 (2017)
FIG. 3. The phase-locking of a droplet pair at two non-identical NCs ( r1= 16.5 nm, r2= 15 nm) driven by a microwave
magnetic field of 20 mT. (a) Phase-locking frequency; (b) Phase difference '.
FIG. 4. Phase-locking of multiple droplets by microwave driving field, HMW,0=20 mT, fMW=25.5 GHz. Phase configuration
of the droplet matrix at (a) t=0 ns, the initial state; (b) t = 10 ns, without MW field; (c) t = 10 ns, with the MW field. The color
disk on the left represents the direction of the magnetization.
C. Synchronization of multiple droplets
Synchronization of multiple droplets is also available by using the microwave magnetic field.
Fig. 4 shows the simulation results for a matrix of 4 4 droplets performed with or without the
microwave driving source. The separation distance between the neighbor NCs is 240 nm. In this
simulation, a random initial droplet state is firstly generated at each NC by applied d.c. current. These
droplets have different initial phase angle ', see Fig. 4(a). Without the microwave magnetic field,
these droplets have never to be locked in phase state as the time increases, as shown in Fig. 4(b).
However, when the microwave field is switched on, we can see that all the droplets with the different
initial phase can be quickly synchronized into an in-phase state with a same phase-locking frequency
of 25.5 GHz, see Fig. 4(c) and movie S3 of the supplementary material.
IV. CONCLUSION
In summary, we show that droplet-based NC-STOs can be synchronized by using a microwave
field. The phase-locking range can be tuned by the microwave field strength, showing the range
increases with the field strength. An asymmetry phase-locking window is observed, because the
droplet pair prefers to synchronize with the relatively lower frequency of the microwave source,
compared with the intrinsic precession frequency of the droplet. Multiple droplets formed in a 4 4
NC matric can also be synchronized by the microwave magnetic field.
SUPPLEMENTARY MATERIAL
See the supplementary material for showing the process of the phase-locking two identical
droplets (movies S1 and S2) and multiple droplets (movie S3).056019-6 Wang et al. AIP Advances 7, 056019 (2017)
ACKNOWLEDGMENTS
This work is supported by the National Basic Research Program of China (2015CB921501)
and the National Natural Science Foundation of China (Grant No. 51471118, No.11274241). Y . Z.
acknowledges the support by National Natural Science Foundation of China (No. 1157040329), the
Seed Funding Program for Basic Research and Seed Funding Program for Applied Research from
the HKU, ITF Tier 3 funding (ITS/171/13, ITS/203/14), the RGC-GRF under Grant HKU 17210014.
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1.353851.pdf | Range of chaotic motion of a domain wall in a periodic drive field
R. A. Kosinski
Citation: Journal of Applied Physics 73, 320 (1993); doi: 10.1063/1.353851
View online: http://dx.doi.org/10.1063/1.353851
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/73/1?ver=pdfcov
Published by the AIP Publishing
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147.143.2.5 On: Sun, 21 Dec 2014 22:00:32Range of chaotic motion of a domain wall in a periodic drive field
FL A. Kosinski
Institute of Physics, Warsaw University of Technology, 00-662 Warsaw, Poland
(Received 13 January 1992; accepted for publication 16 September 1992)
The motion of a domain wall in a periodic drive field He sin wt is studied with Slonczewski’s
equations of motion using the fully implicit finite difference scheme. The motion is periodic,
quasiperiodic or chaotic, depending on the values of the frequency w, damping constant a, and
magnetic filed strength He, while the other parameters are held constant. For a given magnitude
of the magnetic field, there is a narrow frequency range in which the periodic motion extends to
low values of the damping constant. This may be due to spin-wave-like excitations appearing in
the sample. For low frequencies the border between the chaotic and periodic motion corresponds
to the Walker limit of stationary motion yielding a value of the damping constant according to
a = Hd2rM, where M is the magnetization.
I. INTRODUCTION
The domain wall in a magnetic material is a nonlinear
dynamical system for which numerical chaotic solutions of
the equation of motion were found. In some cases studied
the wall dynamics was based on the Landau-Lifshitz-
Gilbert equations. lP2 In other, Slonczewski’s equations of
wall motion3 valid for magnetic materials with a large
quality factor (Q= k;/2?rM2> 1, where Ir; is the uniaxial
anisotropy constant, and IV is the saturation magnetiza-
tion) were used.b8 It was found that the range of chaotic
motion depends on external parameters, such as the ap-
plied magnetic field, as well as on the parameters which
characterize the magnetic material. In the present article, a
periodic drive field, Hz=HO sin wt is considered. The
ranges of periodic, quasiperiodic, and chaotic motion of the
domain wall described by Slonczewski’s equations are
found as a function of the frequency w and the damping
parameter a of the magnetic material for three values of
HO.
II. EQUATIONS OF MOTION
The motion of a domain wall is determined by the
Landau-Lifshitz-Gilbert equation which describes the pre-
cession of magnetization M due to an effective field H,s in
a material with a damping constant a:
lSk= --“/lliP~H,~+ (a/M)M& (1)
Here y= 1.75 x lo7 Oe-* s-l is the gyromagnetic ratio and
the effective field H,, is the variational derivative of the
total energy of the wall W,,, which contains the contribu-
tions due to the exchange, uniaxial anisotropy, magneto-
static, and Zeeman energies:
6 Wt,t Hs=--w- (2)
We consider a section of a domain wall which lies in
the xz plane (see Fig. 1). The orientation of the magneti-
zation is given by the polar (0) and azimuthal (9) angles.
In order to simplify the problem, a Bloch-like form of the
polar angle 8 as a function of the coordinate y perpendic-
ular to the wall was assumed: 8(x,y) =2 arctan exp{b -q(x,t)]/A}. Here A is the Bloch width parameter, A=
a, with exchange constant A.9 Thus, the structure of
the wall is defined by the position of the Bloch surface3
q(x,t), and by the orientation of the magnetization at the
Bloch surface rp(x,t). Then, after integrating through the
wall thickness, Eq. ( 1) reduces to:3
24 a$ :=2=My sin[2(p---rp,)] --M %+a+, (3)
2
$=?‘H~+$$-2~~~A& sin[2(cp-up,)] -; 4,
(4)
where at is the angle between the tangent to the wall in the
x-y plane and the +x direction, and the dot over a symbol
denotes a time derivative. These equations were solved nu-
merically using finite differences with a fully implicit
scheme” and the numerical algorithm developed earlier
and described in Ref. 11. The accuracy of this algorithm
was checked in a number of tests, e.g., comparison with the
results obtained from a different numerical scheme pro-
posed by Matsuyama and Konishii2 and modified by Ze-
browski.13 The agreement between results obtained with
the present algorithm and those used in Ref. 13 was excel-
lent. A grid of N=52 points distributed uniformly along
the fragment of the wall of length L = 50A was used. (The
results obtained for N= 202 and N= 102 differ from those
obtained for iV=52 by less than 3%, therefore, to reduce
computer time, the latter value of N was chosen.) The
initial conditions q(x,O) =rp(x,O) =0 and the boundary
conditions aq/ax = &p/ax = 0 were used.
The material parameters used were 47~M = 140 G, A
=0.8 1 X 10M7 erg/cm, gyromagnetic ratio y= 1.75 X lo7
Oe-’ s-‘, and A=2.9 x 10V6 cm (these were the parame-
ters of a magnetic garnet sample investigated also in Ref.
9). The time step of the integration procedure was &=O. 1
ns.
In order to analyze the motion of the wall, the trajec-
tory of only two variables q(r) and @t was monitored. 4, @
are the instantaneous deviations of the variables q and y at
the midpoint of the wall from their averages over the
length of the wall. In such a subspace of the phase space of
320 J. Appl. Phys. 73 (I), 1 January 1993 0021-8979/93/010320-03$08.00 @I 1993 American Institute of Physics 320
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147.143.2.5 On: Sun, 21 Dec 2014 22:00:32; I
skt)/ f / ,/ $ -I--
z , ycxt1
/ ;
i 9 P I I
M Hz I I
2' -
0 LX
FIG. 1. A section of a domain wall of the length L considered in com-
putations is shown for the case of periodic motion. The position of the
Bloch surface q(x,t) and the azimuthal angle p(x,tj showed here, as well
as velocity v, are constant in each time for such a motion. (Only some
grid points with magnetization hl are shown.)
the wall, the motion of a wall with planar symmetry [i.e.,
q(x,t) =q(t) and p(x,t) =9(t)] corresponds to the point
attractor {g(t),@(t)} = {O,O}. When this symmetry is bro-
ken, the trajectories tend to other types of attractors, as we
shall see below.
HI. RESULTS AND DISCUSSION
Our computations were performed for 6 MHz<w< 10
GHz and 10m3<a<l. For Ho the values 4, 8,~and 12 Oe
were chosen. It was found that three types of wall motion
occur: periodic (P), quasiperiodic (Q), and chaotic (CH).
Periodic wall motion occurs at the high damping side of
the solid curves a, b, and c in Fig. 2 which correspond to
Ho=4, 8, and 12 Oe, respectively. The low frequency limit
of the quasiperiodic motion, for each of the above values of
Ho, is indicated in Fig. 2 by a dotted line at ~=@a,~,~
Chaotic motion was found below the curves formed by the
dotted and the solid lines. For a greater than a certain
value %,b,n only periodic motion of the wall occurs. On the
other hand, for sufficiently small values of a, periodic mo-
tion was not observed.
For a>a,,& (Fig: 2), the wail has planar symmetry in
the whole frequency range. At the low frequency end, the
movement of the wall may be treated as a sequence of
,~~J---c----y--~\l
10-3 10-P a, IO-la, c
a a,
FIG. 2. Ranges of periodic (P); quasiperiodic (Q), and chaotic (CH)
motion of the wall as functions of the drive field frequency w and damping
parameter of the material a. Curves (I, b, and c belong to H,=4.8, and 12
Oe, respectively. The values o.,J,~ mark the regions of extended periodic
motion. The values arr,6,.c correspond to the critical damping parameter
resulting from the Walker limit. motions following quasistatically the instantaneous value
of HZ(t). On the other hand, the maximum drive field
constant in times, for which the wall moves with a uniform
structure, is the Walker field Hw=2n-aM.‘4 Using the am-
plitudes Ho=4, 8, and 12 Oe for H, we obtain the values
a,-0.057, ab=0.114, and a,=0.17 which agree well with
those obtained numerically.
In the case of the periodic motion, the point attractor
in the subspace {c(t),@(t)} was observed for arbitrarily
long times, for instance for t> 120 ps. During this time
Iif I and I@(f) I were found to be smaller than lo-l2 (A
units and radians, respectively). This means that, in the
periodic case, numerical noise appearing in the computa-
tions had no influence on the symmetry of wall structure:
the Bloch surface of the wall remained flat and Walker-
type, and a constant and spatially uniform deflection of 9
was observed. For this type of motion, the energy delivered
to the sample can be dissipated with the uniform preces-
sion of the magnetization.3
For the cases-of the quasiperiodic and of the chaotic
motion 1 q(f) 1 and 1 G(t) I reached values significantly
larger than zero, distinctly greater in the case of chaotic
motion. The transients in these types of motion lasted from
300 ns to several thousands ns, depending on the values of
a and w. After the transient, for w > a&C the symmetry of
the wall was broken (this effect was initialized by the nu-
merical noise) and small distortions of the wall structure,
in the form of oscillations, were observed. These oscilla-
tions had an amplitude of lo” to 30” in the cp(x,t) variable
and some tenths parts of the value of A for ~(x,t); they
propagated along the wall. A part of the energy coming
from the external field was thus dissipated due to such
oscillations of the wall structure.
- The quasiperiodic character of wall motion was indi-
cated by characteristic features of the Poincare sections of
the trajectories c( 4)) .I5 These sections have the form of a
multiple ellipses in the {<,93 plane,8 which means ‘that
they lie on the surface of a multiple T2 torus.
For w <w4& strong deviations of the magnetization
from the planar symmetry were observed-the amplitudes
of c~(x) oscillations were of the order of rr, forming the
vertical Bloch lines (VBLs). The soliton like behavior of
these internal wall structures has been reported by many
authors (see, e.g., Refs. 16, 17). In our case the generation
of VBLs was not periodic in time, leading to a rather com-
plex chaotic attractor in the subspace {zfi.
In the case of Ho= 12 Oe, a number of VBLs were
present simultaneously in the wall and the observed angu-
lar span of such stacked VBLs reached 3~, while for the
smaller values of Ho it did not exceed 2~. Also, the distor-
tions of the surface of the Bloch wall q(x) (of the order of
5A) were greater than in the case of the smaller values of
H,.
The chaotic motion of the wall indicates that the en-
ergy delivered to the magnetic material by the external
field cannot be dissipated during translational wall motion
and due to the small oscillations of wall structure, as was in
the case of quasiperiodic motion. During chaotic motion
an additional, strongly nonuniform precession of the mag-
321 J. Appl. Phys., Vol. 73, No. 1, 1 January 1993 FL A. Kosinski 321
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147.143.2.5 On: Sun, 21 Dec 2014 22:00:32netization, connected with the appearance of VBLs, can
dissipate external field energy.3
For a narrow range of w, around the frequencies tiQ@
the regions of periodic wall motion extend to small values
of ct. The ratio of the frequencies, corresponding to the
centers of two of these regions is tidaa=2 which is equal
to the ratio of corresponding amplitudes of the drive fields
HdH,=2. For the case of the curves b and c, however, this
ratio equals tid@b= 1.3, while HJHb= 1.5. The inequality
between the ratios tidtib and I&./Hb may be due to above
described stacking of the VBLs which occurs in sufficiently
large drive fields, here at Ho= 12 Oe.
What is the physical origin of the frequencies w4b,$, the
most salient feature of Fig. l? They must be due to some
additional process of dissipation of the energy of the exter-
nal magnetic field.
It was foundi8’19 that two types of excitations appear in
wall motion. Wall type excitations are localized in the wall
and are connected with its deformations. The second are
spin-wave-like excitations similar to the spin waves that
form in a uniform ferromagnet. Such excitations would be
able to absorb a part of the energy of the external field
keeping the wall structure uniform. Using the results ob-
tained by Winter,” the lowest energy E. of the dispersion
relation E(k) of the spin-wave-like excitations was calcu-
lated. For a small damping and for 4?rMgp,> 2KS, E,-, is
equal to ,/%? ,/w = tiea, where K is the anisotropy
constant in the Hamiltonian (see E.d in Ref. 18), ps-the
Bohr magneton, g-the Lande factor. For the material pa-
rameters used here we obtain we,=6.7~ 10’ s-l, which is
in the same order of magnitude as the values ti,,@b,ti,- lo8
s-‘, obtained in our calculations (see Fig. 2). Thus, it
seems that the occurrence of the spin-wave-like excitations
propagating along the wall may be responsible for the ex-
tended regions of periodic wall motion in the range of
small damping; however the relation between their position
on the o axis and the amplitude of the drive field Ho is not
clear at the moment.
Note that an empirical relation
0=y (5)
describes, in a rather good approximation, the boundary of
the chaotic region.“) For a! -0 and Ho=4 Oe it gives woa
=0.7x 10’ s-l, which is to be compared with the value
w, = 1.23 X IO8 s - ’ in Fig. 2. Moreover, for w = 0, this yields
c&& as calculated from the Walker relation. Thus, relation
(5) suggests that the mechanism responsible for additional
energy dissipation has its origin’in the precession of q. It should be mentioned that in the present model of the
wall, chaotic motion may be connected only with the vari-
ation of the azimuthal angle Q, of magnetization along the
wall, because the Bloch-type distribution of polar angle
8(y) is assumed in Eqs. (3) and (4). Chaotic wall motion
may be connected also with the variation of the 0 angle
along the y axis*?z or with variations of both angles.
The character of the transitions between the different
types of wall motion within the accuracy of the computa-
tions [AT= f (0.5~2) ns for w= 1010+107, respectively]
seems to be sharp, however more exact examination of
these transitions may be interesting.
ACKNOWLEDGMENTS
The author acknowledges the hospitality of the Insti-
tute of Theoretical Physics of the ETH-Zurich, where most
of this work was performed. The author wishes to thank
Professor Dr. W. Baltensperger, Dr. A. Jaroszewicz, Dr. J.
Helman, and Dr. J. Zebrowski for helpful discussions and
a critical reading of the manuscript. In particular Dr. Hel-
man provided relation (5).
‘F. Waldner, J. Magn. Magn. Mater. 31, 1015 (1983).
‘H. Suhl and X. Y. Zhang, J. Appl. Phys. 61, 4216 (1987).
3A. P. Malozemoff and .I. C. Slonczewski, Magnetic Domain Walls in
Bubble Materials (Academic, New York, 1979).
4J. J. Zebrowski and A. Sukiennicki, Springer Proc. Phys. 25, 130
(1987).
‘J. J. Zebrowski, Phys. Scripta 38, 632 (1988).
‘R. A. Kosinski and A. Sukiennicki, Acta Phys. Polon. A 76, 309
(1989).
7R. A. Kosinski and A. Sukiennicki, J. Magn. Magn. Mater. 93, 128
(1991).
‘R. A. Kosinski and A. Sukiennicki, J. Magn. Magn. Mater. 104-107,
331 (1992).
‘For detailed descriptions of simplifying assumptions and the wall con-
figuration, see R. A. Kosinski, J. J. Zebrowski, and A. Sukiennicki, J.
Phys. D 22, 451 (1989).
“G E Forsythe and W. R. Wasow, Finite DlfirencesMethodsfor Partial
D~&rentiaI Equations (Wiley, New York, 1960).
“R. A. Kosinski and J. Engemann, J. Magn. Magn. Mater. 50, 229
(1985).
‘*K. Matsuama and S. Konishi, IEEE Trans. Magn. MAC-20, 1141
(1984).
13J. J. Zebrowski, Phys. Scripta 38, 632 (1988).
14N. L. Shrver and L. R. Walker. J. Auol. Phvs. 45, 5406 (1974).
“K. Geist and W. Lauterborn, Physica b 41,-I (1990).
“V. S. Gornakov, L. M. Dedukh, and V. I. Nikitienko, Sov. Phys. JETP
67, 570 (1988).
17A Sukiennicki, R. A. Kosinski, and J. J. Zebrowski, J. Phys. C 8, 1883
(l.988).
“J. M. Winter, Phys. Rev. 124, 452 (1961).
19J. E. Janak, Phys. Rev. 134, A411 (1963).
2oJ. S. Helman (private communication).
322 J. Appl. Phys., Vol. 73, No. 1, 1 January 1993 R. A. Kosinski 322
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147.143.2.5 On: Sun, 21 Dec 2014 22:00:32 |
1.4803127.pdf | A probabilistic model for the interaction of microwaves with 3-dimensional magnetic
opal nanocomposites
G. S. Makeeva, O. A. Golovanov, M. Pardavi-Horvath, and A. B. Rinkevich
Citation: Journal of Applied Physics 113, 173901 (2013); doi: 10.1063/1.4803127
View online: http://dx.doi.org/10.1063/1.4803127
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/113/17?ver=pdfcov
Published by the AIP Publishing
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130.209.6.50 On: Thu, 18 Dec 2014 22:59:35A probabilistic model for the interaction of microwaves with 3-dimensional
magnetic opal nanocomposites
G. S. Makeeva,1O. A. Golovanov,1M. Pardavi-Horvath,2and A. B. Rinkevich3
1Penza State University, Penza 440026, Russia
2ECE, The George Washington University, Washington, D.C. 20052, USA
3Institute of Metal Physics, Ural Division of Russian Academy of Science, Ekaterinburg 620990, Russia
(Received 23 November 2012; accepted 12 April 2013; published online 1 May 2013)
The complex diagonal and off-diagonal components of the effective permeability tensor were
calculated for the case of a realistic 3D opal, infiltrated with Ni 0.7Zn0.3Fe2O4nanoparticles. First,
an accurate electrodynamic effective medium permeability tensor approach is formulated. Next,
Maxwell‘s equations were solved rigorously for the case of an interacting systems of ferriteparticles in the opal matrix, having a normal distribution of the ferromagnetic resonance fields. The
method is demonstrated by calculating the bias field dependence of the components of the complex
permeability tensor at 26 GHz, and a good agreement with recent experimental data was obtained.
VC2013 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4803127 ]
I. INTRODUCTION
Magnetic nanocomposites, based on self-organized opal
nanosphere matrices, infiltrated with magnetic nanoparticles,are low loss, potentially high magnetization, magnetically
tunable materials with interesting and useful properties for
application in attenuators, phase shifters, filters and othermicrowave devices.
1Recently, microwave measurements
were performed on magnetic opals, yielding interesting data
to be interpreted in terms of interactions of microwaves withmagnetic nanosystems.
2
In order to understand, evaluate, interpret, and predict
the interaction of microwaves with 3-dimensional (3D) mag-netic nanostructures, a numerical model is required. In our
previous work, we developed a mathematical technique to
calculate the scattering of electromagnetic waves (EMWs)on periodic magnetic nanocomposites.
1However, to build a
more realistic and meaningful model, the real microstructure
of the magnetic opal nanocomposite should be takeninto account. The three dimensional opal network of
d¼100–250 nm SiO
2spheres has a regular network of tetra-
hedral and octahedral voids. Magnetic metal or ferrite nano-particles can be embedded into the inter-sphere voids via
several routes, including chemical precipitation.
3Typically,
the real structure of the ferrite filling in the voids has a ran-dom shape and size distribution of nanoparticles. The size of
the voids is related to the diameter of the silica spheres as
0.2dand 0.4 dfor tetrahedral and octahedral voids, respec-
tively. The voids are connected by narrow channels, devoid
of particles. According to TEM data, the size of the actual
magnetic nanoparticles in the voids varies from 5 to 60 nm.The particles might form loose aggregates, with individual
particles located very close to each other, at distances of few
nanometers or even less. The particles interact magnetostati-cally and, as a result, the theoretical investigation of these
interacting nanosystems is complicated by the distribution of
particle size, shape-dependent anisotropy, agglomeration ofthe particles, and the presence of exchange interactions
between them.The goal of the present paper is to construct a numerical
model to describe the electromagnetic properties of realisticopal nanocomposites, containing ferri- or ferro-magnetic
nanoparticles.
II. DETERMINISTIC ELECTRODYNAMIC MODEL
First, an effective medium approach is developed to
solve the problem of EMW interactions with a 3D magnetic
opal nanocomposite structure. The model is not a classical
mixing type formulation, it is a rigorous electrodynamicmodel to solve Maxwell‘s equations with electrodynamic
boundary conditions for the 3D magnetic opal configuration,
4
curlH¼e0e@E
@tþrE; (1)
curlE¼/C0@BðHÞ
@t; (2)
B¼Mþl0H; (3)
where EandHare the electric and magnetic field intensity
vectors, Mis the magnetization vector, Bis the magnetic
induction vector, ris the electrical conductivity, eis the rela-
tive dielectric constant, e0is the vacuum permittivity, l0is
the vacuum permeability.
Maxwell’s equations are complemented by the deter-
ministic Landau-Lifshitz equation of motion of the magnet-
ization vector, including the exchange term,5
@~M=@t¼/C0c½~M;~Hef f/C138/C0ða=MÞ½~M;@~M=@t/C138; (4)
where cis the gyromagnetic ratio, ais the Gilbert damping
constant, Heffis the effective (local) magnetic field acting on
M, including magnetostatic fields of external sources, crystal
anisotropy, shape-dependent dipolar interactions, and
exchange interactions with Hex¼(2A/l0Ms)DMthe effective
exchange field, where Ais the exchange constant, Msis the
saturation magnetization.
0021-8979/2013/113(17)/173901/6/$30.00 VC2013 AIP Publishing LLC 113, 173901-1JOURNAL OF APPLIED PHYSICS 113, 173901 (2013)
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130.209.6.50 On: Thu, 18 Dec 2014 22:59:35The numerical approach is based on the decomposition
of the nanocomposite into autonomous blocks with virtual
Floquet channels (FABs).4The domain of the 3D opal-based
magnetic nanocomposite is divided into FABs, shown in
Fig. 1, containing the SiO 2nanospheres and the magnetic
nanoparticles, filling the void regions of the opal structure(Fig. 1(b)).
For the calculations, we consider the elementary cell of
the 3D periodic nanostructure as the FAB (Fig. 1(c)). In con-
trast to our previous work,
6in this model instead of the com-
plete filling, we take into account a more realistic approach
ofNferromagnetic nanospheres, 1 /C20N/C205, filling the void
regions in each cell (FAB). A deterministic electrodynamic
model was developed and applied to the opal for several val-
ues of N. In each case, the number Nof spherical magnetic
nanoparticles, embedded into intersphere opal voids, is dif-
ferent, however, the diameter dof the magnetic nanospheres
is set the way that the filling factor p¼0.07 of the magnetic
component in the opal remains constant for all cases. The
cell is described by its FAB conductivity matrix Y, as in Ref.
4, taking into account electrodynamical boundary conditions,
the number of particles N, and assuming spherical shape of
the magnetic nanoparticles.
The electromagnetic wave (fields E,H; frequency x)
propagating in the 3D periodic nanostructure along axis f
(Fig. 1) is a superposition of inhomogeneous plane EMWs
having fields En(n,g),Hn(n,g) and propagation constants Cn,7
Cn¼/C23þ2pn
K;n¼0;61;62; :::;61; (5)
where C0¼/C23is the unknown propagation constant of the
fundamental wave ( n¼0); and Kis the cell periodicity along
the direction of propagation of the EMW.
For the 3D magnetic opal nanocomposite, we introduce
the effective permeability tensor with complex diagonal lR
and off-diagonal lR
acomponents, and the effective permit-
tivity eR. The components lRandlR
aof the effective perme-
ability tensor and the effective permittivity eRcan be
determined by solving the system of Eqs. (6)–(9), similar to
the case discussed in Refs. 1and6,Cþ
R¼xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e0l0eRðlRþlR
aÞq
; (6)
C/C0
R¼xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e0l0eRðlR/C0lR
aÞq
; (7)
CR
jj¼xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e0l0eRlR
zq
; (8)
CR
?¼xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e0eRl0ðlRÞ2/C0ðlR
aÞ2
lRs
; (9)
where Cþ
R,C/C0
Rare the propagation constants of clockwise
and counterclockwise polarized EMWs ( H0¼H0z);CR
||,CR
?
are the propagation constants of the ordinary and extraordi-
nary modes ( H0¼H0x) in the gyromagnetic medium.5
The dispersion relations of EMWs in a bulk gyromag-
netic medium (with parameters l,la, and e) form a system
of simultaneous equations, similar to Eqs. (6)–(9). In this
case, the system has a unique solution, and the values of the
three parameters l,la, and ecan be determined from the an-
alytical solution of this system of simultaneous equations,with one of the equations dropped. The uniqueness of the so-
lution is retained.
5
For the case under consideration, Eqs. (6)–(9)form a
system of quasi-simultaneous equations for the unknown
effective parameters lR,lR
a, and eRof the 3D magnetic opal
nanocomposite. The system of quasi-simultaneous Eqs.(3)–(6)can be solved if the following condition is satisfied:
ðC
R
?Þ2/C02ðCþ
RÞ2ðC/C0
RÞ2
ðCþ
RÞ2þðC/C0
RÞ2¼0: (10)
Due to the nature of the system of equations, the values of
parameters Cþ
R,C/C0
R,CR
||,CR
?, calculated from the characteris-
tic equation as in Ref. 6will satisfy condition (10) only
within a certain error limit D. Accordingly, the values of lR,
lR
a, and eRfound from the solution of Eqs. (6)–(9)also sat-
isfy condition (10)with a certain accuracy only.
The propagation constants C0of the fundamental modes
of EMWs propagating along direction zin a periodic 3D
nanostructure (Fig. 1) for transverse H0¼H0xand
FIG. 1. Model of the 3D opal-based
magnetic nanocomposite: (a) directionof propagating EMW of wave vector k;
(b) periodic 3D-nanostructure and orien-
tation of the DC magnetic field H
0; (c)
model of a cell of autonomous blocks
with Floquet channels ( FAB). 1—SiO 2
nanospheres; 2—void region, filled by
magnetic nanoparticles.173901-2 Makeeva et al. J. Appl. Phys. 113, 173901 (2013)
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130.209.6.50 On: Thu, 18 Dec 2014 22:59:35longitudinal H0¼H0zorientations of the DC bias magnetic
field were obtained from the characteristic equation as in
Ref. 6,
DðCnÞ¼j YAA/C0H/C01YBAþYABH/C0H/C01YBBHj¼0;(11)
where Cnare the unknown complex wave numbers. D(Cn)i s
the determinant of the matrix of the characteristic equation;
YAA,YAB,YBA,YBBare the blocks of conductivity matrix Y,
the indices Aare for a¼1, 2, 3; and Bfora¼4, 5, 6; His a
diagonal matrix having diagonal elements qi(lj)¼/C0idljCna
cosbi, where biare angles between the wave vector kand the
x, y, z axes. The Cnpropagation constants can be obtained
from here.
Substituting the computed values of the propagation
constants back into Eqs. (6)–(9)and solving the system of
equations, the complex diagonal lRand off-diagonal lR
a
components of the effective permeability tensor and the effec-
tive permittivity eRof the 3D opal magnetic nanocomposite
were calculated at microwave frequencies. The calculations
were performed for the case of a lossy magnetic opal com-posed of SiO
2nanospheres, r¼100 nm, er¼4.6/C0i4/C210/C04,
infiltrated with NiZn ferrite nanoparticles Ni 0.7Zn0.3Fe2O4
with 4 pMs¼5 kG, exchange constant A ¼2.2/C210/C09Oe cm2,
damping parameter a¼0.08, er¼9.5/C0i0.3. The same mate-
rial was used in the experiments of Ref. 2. The real and imagi-
nary parts of the complex diagonal lRand off-diagonal
lR
acomponents of the effective permeability tensor of the 3D
opal magnetic nanocomposite, depending on the relative value
of DC magnetic field H0rel¼(H0–H r)/Hr,w h e r e Hris ferro-
magnetic resonance field (FMR), at the frequency of f
¼26 GHz ( Hr¼9180 Oe) was calculated for different num-
bers of the magnetic nanoparticles in the voids, N¼1, 3, 4, 5,
having diameters d¼50, 35, 31, 29 nm, correspondingly. The
results are shown in Fig. 2. For each case (curves 1–4), the di-
ameter ddepends on the number Nof magnetic nanospheres
because the value of the filling factor pof the magnetic com-
ponent is kept constant at p¼0.07. As it follows from the
results of modeling, shown in Fig. 2(curves 1– 4) for a con-
stant filling factor, the effective permeability increases upon
reducing the diameter and increasing the number of particles.This may be the consequence of the competition of the domi-
nating interactions in the system. By enhancing the short-
range exchange coupling interaction and the weakening of thelong-range dipolar magnetostatic interaction between the
magnetic nanoparticles with decreasing particle size, the sys-
tem reaches the range of exchange length. Both the dipolarand exchange interactions affect the internal field and, conse-
quently, the FMR in the magnetic nanocomposites.
8The com-
posites with isolated larger particles display significanteffective permeability degradation (Fig. 2, compare curves 1
and 4). The interacting magnetic dipole field synchronized to
the magnetization precession causes the variation in the effec-tive permeability and the effective damping factor. The mag-
netic dipole interaction among nanoparticles depends on the
distance and the number of particles involved in the summa-tion.
9Upon reducing the size and the separation of neighbor-
ing magnetic nanospheres, the effect of the exchange
interaction between magnetic nanoparticles starts to dominate.There is an additional loss mechanism (Fig. 2, curve 4) due to
spin wave excitations of magnetic nanoparticles degenerate
with the homogeneous magnetization precession.10,11
The results, shown in Fig. 2, indicate that the interaction
field intricately influences the effective permeability and
FIG. 2. Real and imaginary parts of diagonal lR(a) and off-diagonal lR
a(b)
components of the effective permeability tensor depending on the relative
value of DC magnetic field H0rel¼(H0–Hr)/Hrfor the 3D opal magnetic
nanocomposite: f¼26 GHz. Curve 1— N¼1,d¼50 nm; 2— N¼3,
d¼35 nm; 3— N¼4,d¼31 nm; 4— N¼5,d¼29 nm. Circles mark experi-
mentally measured data from Ref. 2.
FIG. 3. Calculated imaginary part of the diagonal lRcomponent of the
effective permeability tensor of the 3D opal magnetic nanocomposite vs. the
DC magnetic field at f¼26 GHz; N¼5,d¼29 nm. Circles mark experimen-
tally measured data from Ref. 2.173901-3 Makeeva et al. J. Appl. Phys. 113, 173901 (2013)
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130.209.6.50 On: Thu, 18 Dec 2014 22:59:35effective damping factor through the real structure.9The
nonuniform internal field, and the spatial variation of themagnetic moments give rise to the variation in the effective
permeability. Thus the magnetic resonance can be modified
by the geometry of the filling via magnetic interactions.
12
Fig. 2(a) compares the results of calculation and the
experimentally measured data from Ref. 2. In the region of
FMR, the agreement of the effective medium deterministicmodel with the measured values is good for N¼5, diameter
ofd¼29 nm ferromagnetic nanospheres filling the voids in
each FAB cell in the opal matrix. The damping parameterwas assumed to be a¼0.08. However, below the FMR, the
deterministic model does not show such a good agreement
when applied to model the experimental data. The results ofcalculation of the imaginary parts of the complex diagonal
l
Rcomponent of the effective permeability tensor of the 3D
opal magnetic nanocomposite, depending on DC bias mag-netic field H
0, for N¼5, damping parameter a¼0.08, at the
frequency of f¼26 GHz and the experimentally measured
data from Ref. 2are shown in Fig. 3.
As it was stated before, there is an error Din the solu-
tion, due to a misalignment of the system of quasi-
simultaneous Eqs. (6)–(9)in satisfying condition (10) to
obtain Cþ
R,C/C0
R,CR
||,CR
?. The bias field dependence of
this error was calculated, and it is shown in Fig. 4. Outside a
narrow range around the resonance, the accuracy is betterthan 2%.
Based on this accuracy, it can be concluded that the pro-
posed deterministic model can be applied to calculate thecomponents l
R,lR
aof the effective permeability tensor and
the effective permittivity eRof 3D opal magnetic nanocom-
posites, a nanostructured gyromagnetic medium, in a waysimilar to the case of the effective medium approach in a
quasi-bulk continuum.
III. THE PROBABILISTIC MODEL
Experimental evidence shows that the FMR line shape
and the linewidth are influenced both by the random shape
and size distribution of magnetic nanoparticles and by therandom spatial distribution of particle clusters.
2In the fol-
lowing, a model is developed to account for the randomness
of the system, starting by using the deterministic electrody-namic model to evaluate the effective FMR linewidth DH
and interpret it in terms of an effective damping parametera(for our calculations in Sec. II.a¼0.08 was assumed).
Using this approach alone, based on the deterministic
Landau-Lifshitz equation (4)with the Gilbert form of the
magnetic damping term, the description of the damping
processes in these materials meets significant difficulties
when trying to incorporate several damping mechanisms.
8
There are two main contributions to the effective FMR
linewidth: intrinsic and extrinsic. The characteristic intrinsic
damping depends mainly on the electronic and crystallinestructure of the material. The extrinsic damping is due to
magnetic inhomogeneities, anisotropy dispersion, surface
and interface effects, and interparticle interactions. The FMRspectrum of magnetic nanocomposites can be interpreted as
the envelope of resonance curves arising from a large num-
ber of weakly interacting particles or clusters of magneticnanoparticles, each of which resonates in its effective mag-
netic field, composed of the applied field, the local magneto-
static field, interaction fields, and local randomly orientedmagnetic anisotropy fields.
13–15This is one of the reasons
why the Gilbert damping parameter for magnetic nanocom-
posites usually exceeds the bulk value.
A more accurate analysis of the high frequency mag-
netic properties of nanocomposites requires to consider the
effects of random size distribution of particles, the randomorientation of easy axes, deviations of particle shape from
spherical, as observed in real nanocomposite materials. That
is why it is necessary to develop a probabilistic model ofFMR of 3D magnetic opal-based nanocomposites.
We consider that the value of the FMR resonance field
H
rof an assembly of random size and shape nanoparticles is
determined by particle statistics, because for each particle its
resonance field depends on its shape and size.7It is proposed
that the resonance field is treated as a random quantity. ThenH
rof magnetic nanoparticles in any elementary cell of the
periodic 3D nanostructure has a normal distribution,
fðHrÞ¼1
rffiffiffiffiffiffi
2pp exp/C0ðHr/C0H0
rÞ2
2r2 !
; (12)
where f(Hr)is the probability density, Hr0is the expectation
value of the random quantity Hr,ris the standard deviation.
The magnetocrystalline anisotropy of randomly oriented
spherical particles is averaged out to zero. In this work, weconsider a model which is limited to small deviations of the
mean value r/C28H
r0and ( Hr0-3r)/C29Ha, where Hais the ani-
sotropy field, i.e., it is assumed that the random distributionof the FMR fields H
ris due to small deviations of particle
shape from spherical, and, consequently, to minor deviations
of demagnetizing factors of particles. That is why the negli-gible changes of randomly oriented fields H
r, due to surface
and shape-dependent anisotropy, may be not taken into
account. A random-number generator was used to simulatethe random quantity H
rwith normal distribution in Eq. (12).
In the following, we use the distribution of Hrfrom the simu-
lation to determine the random functions. First, using thedeterministic electrodynamic model, described in Sec. II, the
electromagnetic fields E,Hand propagation constants C
0of
EMWs were determined and the complex diagonal lRand
off-diagonal lR
acomponents of the effective permeability
FIG. 4. The relative error Dof effective parameters of the 3D opal magnetic
nanocomposite model depending on the bias field H0rel¼(H0–Hr)/Hr.173901-4 Makeeva et al. J. Appl. Phys. 113, 173901 (2013)
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130.209.6.50 On: Thu, 18 Dec 2014 22:59:35tensor and the effective permittivity eRof the 3D magnetic
opal nanocomposite were calculated by solving the system
of Eqs. (6)–(9). Next, this rigorous solution for fields E,H
was considered as an actual realization of a random distribu-
tion of fields, and the effective medium response was deter-
mined, i.e., random functions lR(Hr),lR
a(Hr), depending on
the random quantity Hr, were obtained. Then averaging over
an ensemble of realizations, i.e., the expectation values of
random quantities lR,lR
awere calculated by using the prob-
abilistic model. Using this numerical algorithm, the expecta-
tion values of random quantities RelR,ImlR,RelaR,Im
laR(the real and imaginary parts of complex diagonal lR
and off-diagonal lR
acomponents of the effective permeabil-
ity tensor of the 3D opal nanocomposites) depending on the
DC bias magnetic field H0were obtained at a frequency of
f¼26 GHz for a value of the damping parameter a¼0.03
and for the standard deviation rof random quantity Hr,a sa
parameter. The results are shown in Fig. 5(curves 1–4) for
number of particles N¼4 (diameter d¼31 nm). Other
parameters used in the calculations are the same as in Fig. 2.
Using the probabilistic algorithm, the influence of theGilbert damping parameter aof the magnetic nanoparticles
and the standard deviation rofH
ron the effective perme-
ability in the Ni 0.7Zn0.3Fe2O4infiltrated opal nanocomposites
as a function of the DC bias magnetic field were analyzed at
a frequency of f¼26 GHz.
Figs. 5and6illustrate the change of the FMR lineshape
upon changing the intrinsic damping afrom 0.03 to 0.08,
andr¼0, 535, 722, 895 Oe. The results of the probabilistic
model (Fig. 5) show that the measured large value of the
FMR linewidth cannot represent the intrinsic losses (curve 1
fora¼0.03, r¼0).
The inhomogeneous line-broadening contribution con-
tributes to the linewidth significantly, as expected. We note
that the FMR line shape changes to a more complex shape
for sufficiently large values of the effective linewidth DH.
The FMR linewidth is very sensitive to the details of the spa-
tial magnetic inhomogeneities, taken into account in the
standard deviation rof the random Hr, and it increasesmonotonously with the standard deviation r(curve 2, 3, 4
forr¼535, 722, 895 Oe). The calculated effective DH
(Fig. 5) is believed to be the result of inhomogeneity-related
processes which increase the FMR linewidth from the intrin-sic (bulk) value (curve 1 for r¼0).
The results of the simulation using the probabilistic
model show an agreement with the experimentally measureddata from Ref. 2if assume a standard deviation r¼722 Oe
fora¼0.03 (Fig. 5, curve 3) and, in contrast r¼0f o r
a¼0.08 (Fig. 6, curve 1). This suggests the fact that the typi-
cal, measured value of a¼0.08 already contains the inhomo-
geneity contribution, while the value of a¼0.03, assumed in
calculating Fig. 5, is closer to the real intrinsic value, and the
large standard deviation of H
rfrom the calculation is close
to its real value.
Fig.7illustrates the effect of separation of the measured
linewidth into the intrinsic and extrinsic components,
depending on DC bias magnetic field H0.
The agreement of the probabilistic model with the meas-
ured values is good for a damping parameter a¼0.03 and
r¼722 Oe, as it was shown in Fig. 5. Below the FMR, the
imaginary part of the complex diagonal lRcomponent of the
effective permeability tensor depends only on the intrinsic
FIG. 5. Calculated bias field dependence of the real and imaginary parts of
the diagonal lRcomponents of the effective permeability tensor of a 3D
magnetic opal nanocomposite at f¼26 GHz; for a¼0.03, N¼4,d¼31 nm,
Hr0¼9270 Oe.1 — r¼0; 2— r¼535 Oe; 3— r¼722 Oe; 4— r¼895 Oe.
Circles mark experimentally measured data from Ref. 2.
FIG. 6. Calculated bias field dependence of the real and imaginary parts of
the diagonal lRcomponents of the effective permeability tensor of a 3D
magnetic opal nanocomposite at f¼26 GHz for a¼0.08, N¼4,d¼31 nm,
Hr0¼9180 Oe. 1— r¼0; 2— r¼535 Oe; 3— r¼722 Oe; 4— r¼895 Oe.
Circles mark experimentally measured data from Ref. 2.
FIG. 7. Calculated bias field dependence of imaginary part of the lRdiago-
nal component of the effective permeability tensor of 3D magnetic opal
nanocomposite. For N¼4,d¼31 nm at f¼26 GHz; curve 1— a¼0.08,
r¼0,Hr0¼9,180 Oe; curve 2— a¼0.03, r¼722 Oe, Hr0¼9250 Oe.
Circles mark experimentally measured data from Ref. 2.173901-5 Makeeva et al. J. Appl. Phys. 113, 173901 (2013)
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130.209.6.50 On: Thu, 18 Dec 2014 22:59:35damping parameter aof magnetic nanoparticles and it is
negligibly low, when compared to the inhomogeneity
contribution.
IV. CONCLUSIONS
A reliable electrodynamic method, based on a probabil-
istic approach for numerical computation of electromagnetic
properties of realistic microwave 3D magnetic nanocompo-sites is developed. The method is demonstrated by calculat-
ing the real and imaginary parts of the complex components
of the effective permeability tensor of 3D opal magneticnanocomposites at microwave frequencies. The numerical
technique shows an agreement with recent experimental data
of waveguide measurements on NiZn ferrite infiltrated opalnanocomposites.
As it follows from the results of electrodynamic model-
ing for a constant filling factor, the effective permeabilityincreases upon reducing the diameter and increasing the
number of magnetic particles in the voids, due to the proxim-
ity effects, indicating a way to control the FMR in the 3Dopal magnetic nanocomposites. Using the probabilistic algo-
rithm, the influence of the intrinsic damping parameter aand
the standard deviation rof the random resonance field H
rof
magnetic nanoparticles on the effective permeability in the
Ni0.7Zn0.3Fe2O4infiltrated opal nanocomposites was ana-
lyzed at a frequency of f¼26 GHz as a function of the DC
bias magnetic field. The observed increase in FMR linewidth
in nanocomposites, as compared to bulk or thin film materi-
als, was modeled by an inhomogeneous line-broadeningcontribution due the standard deviation of Hrof magnetic
nanoparticles.
This work demonstrates that the effective permeability
and effective damping factor can be predicted and designed
taking into account the real structure of the magnetic nano-
composites opening an avenue to the CAD of magneticnanostructures.
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1.3133354.pdf | Microwave assisted magnetization reversal in composite media
Shaojing Li, Boris Livshitz, H. Neal Bertram, Manfred Schabes, Thomas Schrefl, Eric E. Fullerton, and Vitaliy
Lomakin
Citation: Applied Physics Letters 94, 202509 (2009); doi: 10.1063/1.3133354
View online: http://dx.doi.org/10.1063/1.3133354
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/94/20?ver=pdfcov
Published by the AIP Publishing
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138.251.14.35 On: Tue, 23 Dec 2014 05:25:19Microwave assisted magnetization reversal in composite media
Shaojing Li,1Boris Livshitz,1H. Neal Bertram,1,2Manfred Schabes,2Thomas Schrefl,3
Eric E. Fullerton,1and Vitaliy Lomakin1,a/H20850
1Department of Electrical and Computer Engineering and the Center for Magnetic Recording Research,
University of California, San Diego, California 92093, USA
2Hitachi San Jose Research Center, San Jose, California 95135, USA
3Department of Engineering Materials, University of Sheffield, Sheffield S10 2TN, United Kingdom
/H20849Received 8 April 2009; accepted 12 April 2009; published online 22 May 2009 /H20850
Magnetic reversal in exchange-coupled composite elements under microwave fields is characterized
by several unique properties including reduced reversal fields, microwave fields, microwaveresonant frequencies, and reduced sensitivity to anisotropy distributions as compared tohomogeneous elements. We find that reversal can occur in uniform and nonuniform regimes. Theuniform regime is characterized by coherent spin precession enhancement by the microwave field.In the nonuniform regime domain walls in the soft layer mediate reversal and under linearlypolarized microwave fields, can lead to a formation of localized reversal/nonreversal areas in the“applied field-frequency” phase plane. © 2009 American Institute of Physics .
/H20851DOI: 10.1063/1.3133354 /H20852
A major limitation to the continued evolution of high-
density magnetic recording is the superparamagnetic effect,which leads to spontaneous reversal when magnetic particlesbecome too small.
1,2Overcoming the superparamagnetic ef-
fect requires using materials with a very high anisotropy,which often translates into an excessively high reversalfields. Several methods including heat-, precessionalreversal-, and microwave-assisted magnetic recordingschemes have been proposed to solve this writabilityproblem.
3–11Microwave assisted magnetic recording
/H20849MAMR /H20850schemes allow for low reversal fields even for me-
dia with high anisotropy.6,9The reversal field reduction is
due to resonant energy pumping occurring when the micro-wave frequency matches the medium ferromagnetic reso-nance /H20849FMR /H20850frequency. MAMR relies on our ability to gen-
erate local microwave fields of sufficiently high frequencyand strength. Such microwave fields can potentially be gen-erated using spin-torque driven oscillators.
11,12Combined
with a conventional recording head, they can result in a sys-
tem that generates both switching fields and assisting localmicrowave fields. However, there are also several obstaclesthat may complicate practical implementations of MAMRschemes. For high anisotropy materials, the required micro-wave field strength and frequency may be very high. Anotherimportant potential problem is associated with inherent fluc-tuations of the medium anisotropy field. Such fluctuationslead to significant fluctuations of the FMR frequency andreversal field, which result in high bit error rates.
In this letter we describe MAMR mechanisms in com-
posite elements comprising exchange-coupled soft and hardsections under linearly polarized microwave field.
13–17Such
composite elements have been recently shown to be attrac-tive for magnetic recording due to their reversal and thermalstability properties.
13–19We show that composite elements
have several unique properties important for MAMR. Com-posite elements with high anisotropy hard sections can bereversed with low reversal fields, microwave fields, and mi-crowave frequencies. We demonstrate that reversal field de-
pendences in composite elements are different in the regimesof coherent and incoherent reversal and the reversal dynam-ics may exhibit surprising behaviors. In addition, we showthat fluctuations of the reversal fields caused by fluctuationsof the anisotropy field are substantially reduced compared tothose for homogeneous elements.
The elements investigated comprise exchange-coupled
soft /H20849top/H20850and hard /H20849bottom /H20850sections /H20849see the inset in Fig. 1/H20850.
The hard section has a vertical uniaxial anisotropy energy K
h
and size w,w,thin the x,y,zdimensions. The soft section
has a vanishing anisotropy and size w,w,ts. Both sections
have a damping constant /H9251, saturation magnetization Ms, and
exchange length lex=/H20881A/Mswhere Ais the exchange con-
stant. The sections are coupled ferromagnetically over theircommon interface with surface energy J
s. An external mag-
netic field simultaneously comprises a switching field and amicrowave field. The switching field is applied with an angle
a/H20850Electronic mail: vitaliy@ece.ucsd.edu.50 100 150051015202530Hr(kOe)
f(GHz)14 16 18 20 22 24051015202530Hr(kOe)
f(GHz)ts=1.5w
homogeneous
ts=0.75w
fmw(GHz) fmw(GHz)(a) (b)st
ht
wsJ
hardsoft
fmw, (GHz) fmw, (GHz)Hr,(kOe)
FIG. 1. Reversal field vs fmwfor different elements with HK=60 kOe,
/H9251=0.1, lex=1.6 w, and th=1.5 w. /H20849a/H20850 Hmw=0.05 HK=3 kOe,
ts=1.5 w;/H20849b/H20850Hmw=0.07 HK=4.2 kOe, ts=1.5 wfor the composite element,
and Hmw=0.14 HK=8.4 kOe, th=1.5 w/H20849where this the height /H20850for the homo-
geneous element. The shadowed areas represent the conditions under whichreversal occurs.APPLIED PHYSICS LETTERS 94, 202509 /H208492009 /H20850
0003-6951/2009/94 /H2084920/H20850/202509/3/$25.00 © 2009 American Institute of Physics 94, 202509-1
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138.251.14.35 On: Tue, 23 Dec 2014 05:25:1945° to the vertical /H20849z/H20850axis in the x-zplane and it has the time
dependence Hrerf/H208492t//H9270/H20850, where Hris the reversal field and /H9270
is the switching field rise time. The microwave field is ap-
plied along the xaxis and it has an amplitude Hmwand fre-
quency fmw. For a given Hmw, when the microwave fre-
quency matches a FMR frequency fmwresof the element the
reversal field Hrreaches its minimum Hrres.
Magnetization reversal is studied by numerically solving
the Landau–Lifshitz–Gilbert equation with discretizationchosen to obtain full convergence. For all presented results,A=10
−6erg /cm, /H9251=0.1, Ms=1250 emu /cm3, Js
=17 erg /cm2,/H9270=0.1 ns, and th=1.5 w. More simulations
with a wide range of /H9251,/H9270, and Jswere also pursued with
results qualitatively similar to those presented. All results arescalable with respect to the ratios M
s/HKand w/lex. Thermal
stability of all /H20849composite and homogeneous /H20850elements is de-
termined by the domain wall energy Edw=4w2/H20881AK hin the
hard section provided this larger than the domain wall length
tdw=4/H20881A/Kh.13,16The chosen parameters are representative
of materials that may be used for high-density recording me-dia/H20849e.g., FePt /H20850.
20For example, a medium comprising an ar-
ray of elements with pitch of 8 nm and w=5 nm th=1.5 w
/H110151.15 tdwwould result in a recording density of 10 Tbit /in2
with thermal barrier of around 100 kBT/H20849with T=400 K /H20850,
which was confirmed numerically via the elastic bandmethod.
21
First, we compare Hr,Hmw, and fmwfor composite and
homogenous elements. Figure 1depicts Hrversus fmwfor
different elements with HK=60 kOe. The reversal field de-
pendences for all elements exhibit deep minima. The homo-geneous element and composite element with a thin soft sec-tion exhibit a typical behavior attributed to MAMR, i.e.,resonant curves with deep minima are obtained and reversaloccurs for any values of H
agreater than the reversal field Hr
/H20849this is visualized by the shadowed areas in Fig. 1/H20850. For the
composite element with a thicker soft section, the behavior iscompletely different. For this case, reversal is only possiblein a certain areas in the H
a-fmwplane. Two areas are ob-
served. The top area is the same as that obtained without anymicrowave field. The bottom /H20849relatively small /H20850area only ex-
ists under microwave field and is related to resonance phe-nomena. Surprisingly, there is a gap between these two areasin which no reversal occurs.
From Fig. 1, for the homogeneous element, the minimal
reversal field is H
rres=0.19 HKand the corresponding fre-
quency is fmwres=105 GHz. The resonant frequencies scale
with the anisotropy, are very high, and may be hard to realizein practical systems. On the other hand, for the composite
elements, f
mwresdrops substantially. For example, for the ele-
ment with ts=1.5 w,fmwresis around 20 GHz. The reduction of
fmwresis accompanied with a significant reduction of Hrres, e.g.,
for the elements in Fig. 1,Hrrescan be below 0.09 HK. Another
important finding is that these reduced fmwresand Hrare ob-
tained for low microwave fields Hmw. For composite ele-
ments of ts=0.75 w, the microwave field is Hmw=0.07 HK; for
composite elements of ts=1.5 w, the microwave field is
Hmw=0.05 HK.These can be further reduced at a cost of some
increase of Hr. These low Hmwshould be compared to a
significantly larger Hmw=0.14 HKfor the homogeneous
element.
The obtained resonant behavior of Hrresis associated with
resonant effects. When the microwave frequency is near aFMR frequency, the system can efficiently absorb and accu-
mulate energy from the microwave field. For homogeneouselements, the FMR frequencies are determined mainly by theanisotropy field H
K. For composite elements, the FMR fre-
quencies are determined mostly by the external fields param-eters, the element material parameters H
Kand Js, and the
element geometrical parameters. The obtained FMR fre-quency reduction is due to the fact that the effective field inthe soft section of the elements is given only by the weak
external and coupling fields.
15Reversal in the soft section
assists reversal in the hard section thus reducing Hrand Hmw.
Depending on the thickness of the soft and hard sectionsreversal can occur in uniform or nonuniform regimes.
For thin elements, precession and reversal in both soft
and hard sections occurs coherently but the spin angle in thesections depends on coupling. Precession is first enhancedcoherently in the soft section leading to initiation of the softsection reversal. This assists reversal in the hard section thusleading to the reduction of the element’s reversal field. TheFMR frequency is reduced due to lower soft section effectivefield.
For thicker elements, precession and resonant reversal
occurs incoherently /H20849Fig.2/H20850. When the applied field and the
soft section thickness are such that a domain wall in the softsection cannot completely fit /H20851Fig. 2/H20849a/H20850/H20852, precession in the
top/H20849free/H20850end of the soft section is enhanced and its top part
is reversed. This reversal propagates from the top to the bot-tom end of the soft section and then it assists reversing thehard section. The associated required energy is low, hencethe significant reduction of the reversal field. The effectivefield in the top end of the soft section is low, which results ina significant reduction of the FMR frequency. When the ap-plied field and the soft section thickness are such that t
sis
sufficiently greater than the domain wall in the soft section,the mechanism of the reversal is very different /H20851Fig.2/H20849b/H20850/H20852.
First a domain wall is formed in the top part of the softsection and it starts propagating. However, the linearly po-larized microwave field affects differently the two sides ofthe domain wall /H20849since the considered linearly polarized mi-
crowave field contains two circular polarized fields with op-posite polarization sense /H20850. As a result the microwave field
cannot pump energy into the system anymore and the do-main wall stops at the top part of the soft section. If the fieldsH
mwand Haare removed at this stage, the domain wall
moves back and no reversal occurs. For sufficiently large Ha,
reversal occurs regardless of the presence of the microwavefield /H20851the upper reversal area for the composite elements in()a ()b
FIG. 2. /H20849Color online /H20850Schematic representation of the magnetization time
evolution the incoherent mode: . /H20849a/H20850for moderate soft-section thicknesses, a
domain wall is formed in the soft section assisting reversal; /H20849b/H20850for suffi-
ciently large soft section thicknesses, the domain wall stops at the top part ofthe soft section with no reversal.202509-2 Li et al. Appl. Phys. Lett. 94, 202509 /H208492009 /H20850
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138.251.14.35 On: Tue, 23 Dec 2014 05:25:19Fig.1/H20849a/H20850/H20852. This behavior explains the surprising gap of non-
reversal in Fig. 1/H20849a/H20850.
We also considered other angles of linearly polarized
microwave fields with respect to the easy axis and found thatthe size and shape of reversal/nonreversal areas can be modi-fied depending on the field angle, strength, and soft sectionlength. Under circularly polarized fields the phenomena ofthe reversal field, microwave field, and FMR frequency re-duction are preserved but the reversal/nonreversal areas areabsent, which is associated with a different magnetizationdynamics behavior.
MAMR performance may be restricted not only by the
limitation on maximally achievable head fields and micro-wave frequencies but also by deviations of the reversal fieldH
rcaused by random distributions of the element param-
eters. Among them, random distributions of the anisotropyfield H
Kcan have a crucial influence as they may lead to
significant deviations of fmwresand Hr. For homogeneous ele-
ments, deviations of fmwresscale linearly with deviations of HK.
Deviations of Hrcan be even more significant due to the
resonant nature of the MAMR reversal phenomena. Asshown next, composite elements allow significantly reducing
the deviations of f
mwresand Hr.
Figure 3compares the dependence of Hrversus fmwand
HKfor composite elements of different tsand a homogeneous
element. For the homogenous element /H20851shown in Fig. 3/H20849b/H20850/H20852,
fmwresis linearly proportional to HK, e.g., 10% deviations of HK
lead to about 10% deviations of fmwres. Deviations of Hraresubstantially more significant, e.g., 10% deviations of HK
lead to more than 50% deviations of Hr. This behavior may
lead to severe limitations on MAMR if homogeneous ele-ments are used. The situation is very different for composite
elements, where deviations of H
rand fmwresare substantially
reduced and the area of reversal of these two cases overlapwith each other for a major part on the phase graphs. For the
composite element with t
s=0.75 w, deviations of fmwresare only
3% for 10% deviations of HK, which represents a fivefold
improvement over the homogeneous element. For ts=1.5 w,
the reversal areas are slightly shifted but there is an overlap-
ping area where almost no dependence of fmwresand HronHK
is present. The reduction of the deviations of fmwreshas a
physical source similar to that leading to the reduction of fmwres
itself, i.e., fmwresare significantly affected by the soft section
where the field is mostly given by the external and exchangefields but not by H
K. This significant improvement correlates
with results obtained for conventional domain wall assistedreversal.
13Due to the potential improvements to bit error
rates, this weak sensitivity to the anisotropy field distributionis a crucial advantage of composite elements over homoge-neous elements.
In conclusion, we investigated reversal properties of
exchange-coupled composite elements. Composite elementsallow for a significant reduction of the reversal field, themicrowave field, and the FMR frequency as compared tohomogeneous elements. MAMR behaviors in the coherentand incoherent modes are completely different due to thephenomena associated with domain wall formation andpropagation. In addition, the reversal field for composite el-ements can be much less sensitive to the element anisotropyfield distributions, which is crucial to allow reducing bit er-ror rates.
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f(GHz)Hr(kOe )Hk=60 kOe
Hk=66 kOe
20 304050 100 20051020Hr(kOe)
f(GHz)Hk=60 kOe
Hk=66 kOe(a)
(b)
fmw,( GH z )Hr,( k O e ) Hr,( k O e )
fmw,( GH z )
FIG. 3. Reversal field vs fmwfor different HKfor composite and homoge-
neous elements. /H20849a/H20850Hmw=3 kOe, ts=1.5 w,th=1.5 w;/H20849b/H20850Hmw=8.4 kOe, th
=1.5 wfor the homogeneous element, and Hmw=4.2 kOe, ts=0.75 w,th
=1.5 wfor the composite element.202509-3 Li et al. Appl. Phys. Lett. 94, 202509 /H208492009 /H20850
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1.3026172.pdf | Thermal coercivity mechanism in Fe nanoribbons and stripes
F. Garcia-Sanchez and O. Chubykalo-Fesenko
Citation: Applied Physics Letters 93, 192508 (2008); doi: 10.1063/1.3026172
View online: http://dx.doi.org/10.1063/1.3026172
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82.46.164.173 On: Sun, 18 May 2014 19:02:48Thermal coercivity mechanism in Fe nanoribbons and stripes
F . Garcia-Sanchez and O. Chubykalo-Fesenkoa/H20850
Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain
/H20849Received 12 September 2008; accepted 23 October 2008; published online 11 November 2008 /H20850
We investigate the influence of thermally activated process on coercivity values of long Fe
nanostripes. By means of the Lagrangian multiplier technique and the micromagnetic approach, weevaluate energy barriers separating the two magnetization states of long Fe nanostripes, varyingtheir width from 30 to 250 nm. As the width of nanostripes decreases, the reversal time, evaluatedthrough the Arrhenius–Neel law, becomes comparable to the measurement time scale /H20849characteristic
for typical magnetometer /H20850for fields below the values obtained through zero-temperature
micromagnetic approach. We found appreciable variation of the coercivity due to thermal activationfor stripe widths below 100 nm. © 2008 American Institute of Physics ./H20851DOI: 10.1063/1.3026172 /H20852
Recent advances in lithography and self-assembled tech-
niques opened the possibility to prepare nanostripes, nanor-ibbons, and nanowires with the aim to study magnetizationdynamics in restricted geometry.
1–4Alternatively, ribbon-
shape objects can be created by extrusion technique,5al-
though with less control of particle orientations. Magneticnanowires /H20849stripes /H20850have important technological applica-
tions such as magnetic random access memory and morerecently domain-wall logic devices
6and “racetrack
memories.”7
It has been established that in thin nanostripes, the mag-
netization process occurs through nucleation and propagationof domain walls.
1,2Micromagnetic simulations have recom-
mended themselves as a useful technique, capable to get in-sight into dynamics and hysteretic processes in suchnanostructures.
8,9The coercive field of long magnetic stripes
and nanoribbons is a decreasing function of their width dueto the change of the character of the magnetization reversalmechanism.
9However, the coercivity values obtained
through micromagnetic simulations rarely coincide to thoseobtained experimentally because the possible defects are notaccounted for.
10Another possible explanations of the differ-
ences between experiments and micromagnetic simulationsis the possibility of thermal nucleation of domain wall.
In the present work, we evaluate within a micromagnetic
model the energy barriers corresponding to the thermally ac-tivated processes in long Fe nanostripes. From the energybarrier values, we estimate the thermal coercivity on themagnetometer measurement time scale and compare withzero-temperature micromagnetic coercivity calculations. Forthis purpose, we have considered Fe nanoelements havingdimensions in the range 4 nm /H20849thickness /H20850/H1100330–250
nm/H20849width /H20850/H11003400 nm /H20849length /H20850. We have checked that the ob-
served behavior does not change when the stripe length is
increased. The stripes were discretized into cubic elementswith 1.29 nm edge /H208491/10 of the exchange correlation length
of Fe /H9261
exFe=12.9 nm /H20850. The total system energy E, which
consisted of cubic anisotropy, Zeeman, exchange, and mag-netostatic terms, was minimized by integrating the Landau–Lifshitz–Gilbert /H20849LLG /H20850equation with large damping con-
stant. The field was applied along the stripe long axis /H20849X
direction /H20850. For the calculation of the magnetostatic potential,the dynamic alternating direction implicit /H20849DADI /H20850approach
was used.
11The considered values for the Fe intrinsic mag-
netic properties were taken from.12Two of the principle easy
axes of the cuadratic magnetic crystalline anisotropy wereconsidered in plane with an angle forming
/H9278=0° or 45° with
the long wire dimension, whereas the third axis /H20851/H20849001 /H20850Fe
direction /H20852was normal to the nanowire. The value /H9278=45°
corresponds to the case studied in Ref. 9but these orienta-
tions of the easy axes can be easily obtained by lithography.
The energy barrier values have been calculated using the
Lagrange multiplier technique.13In this approach, a suitable
constraint for the multidimensional magnetization distribu-tion is chosen and the total energy is minimized in the mul-tidimensional space. The use of the constraint effectivelyprojects the multidimensional configuration to one /H20849or sev-
eral /H20850“reaction-coordinates.” Since at the stationary points
the constraint vanishes, these points are the same for theconstrained and unconstrained systems. The choice of theconstraint is not trivial since the constrained system may nothave stationary points.
13In our case, there is a natural con-
straint, which uses the average magnetization component
/H20855mx/H20856=mx0, where /H20855mx/H20856=/H20858imxi/N,mxiis the x-component of
magnetization in the micromagnetic element iand Nis the
number of discretization units. In a simple situation of one
domain wall, which can be a minimum or a saddle pointconfiguration in a thin wire, this naturally describes its centerof mass. The use of the constraint means that an additional
energy term − /H9261/H20849/H20855m
x/H20856−mx0/H20850appears in the energy functional,
where /H9261is the Lagrangian multiplier. The corresponding ad-
ditional field is included in the integration of the LLG equa-tion. The set of the LLG equations is augmented by an ad-
ditional equation for the Lagrangian multiplier /H9261˙=
/H11509E//H11509/H9261.A s
a result of this conditional minimization procedure, we ob-tain the energy function E/H20849/H20855m
x/H20856/H20850 /H20849see, e.g., inset in Fig. 4/H20850
from which the energy barriers are evaluated.
Figure 1represents energy barriers EBfor zero applied
field as a function of the nanostripe width. We clearly ob-serve the competition between the magnetocrystalline aniso-tropy and magnetostatic energies, since the values obtainedfor
/H9278=0° are larger than those for /H9278=45°. Indeed, the mag-
netostatic energy produces an effect of additional shape an-isotropy parallel to the wire axis, which enforces one of theeasy axes directions in the former case and competes with itin the latter case. The minimum configurations /H20851see Figs.
a/H20850Electronic mail: oksana@icmm.csic.es.APPLIED PHYSICS LETTERS 93, 192508 /H208492008 /H20850
0003-6951/2008/93 /H2084919/H20850/192508/3/$23.00 © 2008 American Institute of Physics 93, 192508-1
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
82.46.164.173 On: Sun, 18 May 2014 19:02:482/H20849a/H20850and3/H20849a/H20850/H20852are characterized by the structures that are
created by the magnetostatic energy minimization at thenanostripe ends. Regarding the thermal switching mecha-nism, for
/H9278=0°, it always proceeds through a 180° vertical
domain wall, presented in Fig. 2/H20849b/H20850. However, for /H9278=45°,
this is only the case of narrow nanostripes. When the nanos-tripe width increases, there is a change of the character of thesaddle point for a width of approximately 75 nm, whichleads to the change of the slope in the dependence of theenergy barrier on the stripe width present in Fig. 1. This
change is related to the appearance of the structures thatminimize the magnetic charges at long edges of the wire/H20849m
x=0 at these borders, see Fig. 3/H20850. Note that a similar cross-
over between mechanisms has been reported in the coerciv-ity of these nanoribbons.9Consequently, for sufficiently wide
stripes, the saddle point configuration consists of the core-edge structure, as shown in Fig. 3/H20849b/H20850. The core structures has
magnetization pointing in one of the local directions of thebiaxial anisotropy, and a 90° domain wall forming the tran-sition between them. In the edge region, the magnetization ispointed parallel to the surface and the transition between thisregion and the core is also formed. The domain wall, corre-sponding to the saddle point, is not located in the center ofthe stripe. On the contrary, the domain wall is stabilized atthis position in a shallow energy minimum.
Figure 4represents energy barriers for nanoribbons with
different widths as a function of applied field for
/H9278=45°. The
inset in this figure shows the constrained system energy as afunction of the constraint variable /H20855m
x/H20856for 30 nm nanostripe.
At zero field, the saddle point position can be at any Xco-ordinate /H20849with the exception of being close to the stripe end /H20850.
The constrained minimization of the Zeeman energy deter-mines its exact position for H
app/HS110050. For applied fields close
to the coercive one, the saddle point is situated at one of thestripe ends. Consequently, the energy barriers of Fig. 4mea-
sure a field dependent energy for thermal nucleation of thedomain wall in the wire.
Several authors
14,15have found the applied field depen-
dence of the energy barrier value to be
EB=E0/H208731−H
Hc/H20874/H9253
, /H208491/H20850
where E0is the zero field energy barrier value and /H9253is the
scaling exponent. In our simulations for both easy axis ori-entations, the energy barrier values fit well to
/H9253=2. Note,
/CID2/CID3/CID4
/CID2/CID5/CID4
FIG. 2. /H20849Color online /H20850Plane view of /H20849a/H20850the minimum and /H20849b/H20850the saddle
point configurations for nanostripe with 40 nm width, Fe easy axis at 45° tothe long stripe axis and zero applied field.FIG. 1. /H20849Color online /H20850Energy barriers of elongated Fe nanoribbons as a
function of their width at zero applied field for two different orientations ofthe anisotropy axes with respect to the stripe long axis.
/CID2/CID3/CID4
/CID2/CID5/CID4
FIG. 3. /H20849Color online /H20850Plane view of /H20849a/H20850the minimum and /H20849b/H20850the saddle
point configurations for nanostripe with 162 nm width, Fe easy axes at 45°to the long stripe axis and zero applied field.
-1200 -1000 -800 -600 -400 -200 00200400
-1.0 -0.5 0.0 0.5 1.0010002000H=-400 OeE/KBTRoom
<mx>H=0H=-200 Oe
300 K, 0.1s30 nm
40 nm
50 nm
60 nm
75 nmEB/KBTRoom
H(Oe)<mx>
FIG. 4. Energy barriers in Fe nanoribbons as a function of applied field fornanostripes with Fe easy axes at 45° to the long stripe axis and differentwidths of the stripe. The solid line corresponds to typical magnetometermeasurements. The inset shows the system energy versus the constraintparameter /H20855m
x/H20856for a nanostripe with 30 nm width and for several applied
field values.192508-2 F . Garcia-Sanchez and O. Chubykalo-Fesenko Appl. Phys. Lett. 93, 192508 /H208492008 /H20850
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82.46.164.173 On: Sun, 18 May 2014 19:02:48however, that in a general situation this value could be even
field dependent.16
Within the nonthermal micromagnetism, the energy bar-
rier value vanishes for the field value corresponding to thecoercivity. However, in reality, thermal activation allows toovercome small energy barrier. As the absolute value of theapplied field increases, the reversal time corresponding to theenergy barrier separating the two equivalent magnetizationstates becomes comparable to the time scale of the measure-ment. In Fig. 4, the solid line shows the energy barrier value,
which would give the reversal time 0.1 s at T=300 K /H20849cor-
responding to typical magnetometer measurements condi-tions /H20850, evaluated from the Arrhenius–Néel formula
/H9270
=/H92700exp /H20849/H9004E/kBT/H20850with the attempt frequency f0=1 //H92700
=1010Hz. Below this limit, the magnetization process in the
stripes becomes thermally activated. Therefore, the field cor-responding to this energy barrier value gives the thermal co-ercivity value. This argument is equivalent to that of dy-namic coercivity according to the Sharrock law.
17Finally,
Fig.5compares these coercivity values with the static ones
/H20849T=0 K /H20850obtained through micromagnetic simulations of
nanostripes as a function of their widths. It can be clearly
observed that the thermally activated coercivity values at T
=300 K are smaller than the static micromagnetic values fornanostripe widths below 100 nm. For larger stripes, the ef-fect is negligible for considered conditions.
To conclude, in the studied stripes, the thermally acti-
vated coercivity values for typical magnetometer measure-ments at 300 K are smaller than those obtained through stan-dard micromagnetic approach for stripe widths below
100 nm. Consequently, in these cases, the micromagneticsimulations which are normally performed at T=0 or for
short time scale cannot reproduce the experimental results.Instead, the thermally activated coercivity should be takeninto account via the calculation of the energy barriers and theMonte Carlo algorithm.
16We have also studied thermally
activated reversal modes. Regarding the case of anisotropyaxis forming 45° with the long stripe dimension, there existtwo different possible thermal switching mechanisms forlarge and small aspect ratio nanostripes. For thin stripes, themechanism is a 180° tail-to-tail domain wall. In the case ofthick stripes, at the saddle point configuration, the nanostripeprefers to be divided into domains and the configuration cor-responds to a core-edge structure. Finally, we note that thestochastic nature of the magnetization process in nanostripescould be a limiting factor for many important applications.Stochastic behavior of magnetic walls has been experimen-tally observed in different situations.
3The present simula-
tions have been performed for idealized magnetic nanostruc-tures. The presence of different defects generally will makethe demagnetization process even more stochastic.
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1.4927769.pdf | Size dependence of spin-wave modes in Ni 80Fe20 nanodisks
P. Lupo , D. Kumar , and A. O. Adeyeye
Citation: AIP Advances 5, 077179 (2015); doi: 10.1063/1.4927769
View online: http://dx.doi.org/10.1063/1.4927769
View Table of Contents: http://aip.scitation.org/toc/adv/5/7
Published by the American Institute of PhysicsAIP ADV ANCES 5, 077179 (2015)
Size dependence of spin-wave modes
in Ni 80Fe20nanodisks
P . Lupo, D. Kumar, and A. O. Adeyeyea
Information Storage Materials Laboratory, Department of Electrical and Computer
Engineering, National University of Singapore, Singapore, 117576
(Received 9 June 2015; accepted 20 July 2015; published online 29 July 2015)
We investigate the radial and azimuthal spin-wave (SW) resonance modes in permal-
loy (Py: Ni 80Fe20) disks at zero external magnetic field, as function of disk diameter
and thickness, using broadband ferromagnetic resonance spectroscopy. We observed,
from both experimental and micromagnetic simulation results that the number of
SW absorption peaks increases with disk diameter. Numerically calculated SW mode
profiles revealed a characteristic minimum size, which does not scale proportionately
with the increasing disk diameter. We show that higher order modes could thus be
avoided with an appropriate choice of the disk diameter (smaller than the minimum
mode size). Moreover, based on the mode profiles, the existence of azimuthal SW
modes with even number of crests or troughs can be ruled out. These results could be
useful in enhancing our fundamental understanding as well as engineering of new
magnonic devices. C2015 Author(s). All article content, except where otherwise
noted, is licensed under a Creative Commons Attribution 3.0 Unported License.
[http: //dx.doi.org /10.1063 /1.4927769]
The understanding of magnetization dynamics in a confined nanomagnet is of fundamental
importance to foster the advancements in data storage and spintronics.1,2In the past decade, the
study of spin-waves (SWs)in magnetic nanostructures, has received a significant boost due to
advancements in nanofabrication and characterization techniques.3This is largely motivated by
the potential of SWs based devices to miniaturize microwave communication and achieve a more
energy e fficient computing.4
In soft ferromagnetic nanostructures, the ground state is determined by the competition be-
tween exchange and dipolar energies – and thus the shape of the elements. The e ffect of shape on
the dynamic behaviour of patterned magnetic elements, such as stripes,5wires,6triangles,7rectan-
gles,8and rings,9has been extensively investigated. In the case of nanodisks made of soft ferromag-
nets, for certain ranges of disk diameter and thickness,10–13a curling in-plane spin configuration,
known as the vortex state, is more energetically favoured than the formation of domain walls. The
spin dynamics in vortex state has been drawing particular interest from both phenomenological and
application perspectives; such as, high-speed information storage and processing.1,14–18
Following a simplified model, in the vortex core the exchange interaction forces the magne-
tization out-of-plane within a few exchange lengths of the center of the nanodisk, while the outer
region is characterized by an in-plane curling configuration.10,19Thus, a specific polarity (p)can
be assigned to the vortex where the magnetization at its core points either up ( p=1) or down
(p=−1). Similarly, a counterclockwise (CCW: c=1) or clockwise chirality (CW: c=−1) may
also be associated with the vortex depending upon the sense in-plane curling of magnetization away
from its core. The vortex core may exhibit a gyrotropic motion around its equilibrium position when
an external magnetic field or a spin polarized current is applied.1,20–22The gyrotropic motion of the
vortex core results in a gigahertz or a sub-gigahertz mode in the excitation spectrum of nanodisk.23
In addition, magnetostatic modes at higher frequency are also present. Based on the number of
aEmail: eleaao@nus.edu.sg
2158-3226/2015/5(7)/077179/7 5, 077179-1 ©Author(s) 2015
077179-2 Lupo, Kumar, and Adeyeye AIP Advances 5, 077179 (2015)
nodes along the radius of the disk, these modes can be assigned a radial mode number n.24These
radially quantized SWs move along the azimuthal direction. This is in contrast with the case of
perpendicularly magnetized nanodisks where only standing SW modes are observed.25The degen-
eracy between SWs moving in CCW, with m=1, and CW, with m=−1, is lifted by the presence
of the vortex core.26,27Here, mis known as the azimuthal mode number. Interestingly, Ho ffmann
et al.28have also shown that removing the nanodisk core restores this degeneracy.
Two theories have been developed to predict the magnitude of this splitting using di fferent
models; which rely, either on the interaction of azimuthal spin-waves (ASWs) with the static dipolar
and exchange fields,29or on their dynamic hybridization with the gyrotropic mode.30Furthermore,
the knowledge of mode profile22has been deemed necessary to accurately estimate the eigenfre-
quencies associated with these ASW modes.30However, the modeling and theoretical prediction of
the magnetization in the vortex state is still challenging due to the di fficulty of analytic calculation
and the presence of non-linear e ffects.18,31
In this paper, we used a combination of broadband ferromagnetic resonance (FMR) spectros-
copy and micromagnetic simulations to infer the existence and mode profiles of several radial and
ASW modes in nanodisks of diameter d=130 nm, 450 nm, and 1000 nm, and thickness L =20 nm
and 30 nm. We observed that scaling the diameter of the disks does not scale all the radial nodes in
a proportional manner. This made it possible to control the space of available quantized SW states
simply by engineering the dot aspect ratio – in particular its diameter. Using, simulations we were
also able to determine that size of the ASW is closely related to the exchange length of the magnetic
medium. We also found that SWs moving in the same azimuthal sense as the gyrotropic mode are
characterized by larger radial nodes.
Periodic arrays of Py nanodisks with diameter d=130 nm, 450 nm, and 1000 nm and lattice
constant a=250 nm, 930 nm, and 2000 nm, respectively, were fabricated on Si substrate over a large
area (4 mm x 4 mm) using deep ultraviolet lithography followed by electron beam evaporation and
lift-offprocesses. Details of the processing steps can be found elsewhere.32Py nanodisks with thick-
ness L=20 nm and 30 nm were deposited. A representative scanning electron microscopy (SEM)
micrograph of the 30 nm thick nanodisk array with a diameter of 1000 nm is shown in Fig. 1(a).
The collective magnetic behavior of the fabricated dot arrays were characterized using vibrat-
ing sample magnetometer (VSM) with the external magnetic field applied along the x-axis.
Figure 1(b) shows a representative hysteresis loop of the 30 nm thick dot array with diameter
d=1000 nm. The typical magnetic vortex ground state features, such as two triangular loops and a
negligible remanence, characterize the hysteresis loop.
The microwave absorption spectra of the dot arrays in the absence of an external in-plane mag-
netic field and at room temperature was measured using a vector network analyzer (VNA: model
Agilent E8363C) by sweeping the frequency from 50 MHz to 16 GHz. For FMR measurements a
coplanar waveguide (CPW) was fabricated using standard optic lithography followed by the deposi-
tion of Al 2O3(50 nm) /Ti (5 nm) /Au (150 nm) and then lift-o ff. The VNA is connected to the CPW
by a G-S-G-type microwave probe, the signal line is 40 µm wide and the gap with the ground line
is 25.5 µm. The samples were loaded on top of the CPW with the metallic surface in contact with it
(by flipping the sample). The microwave magnetic field hfproduced by the signal line is along the
Y-axis, while the static DC magnetic field Happis along the X-axis.
The simulations were performed using Object Oriented Micromagnetic Framework33(OOMMF)
which solves the Landau-Lifshitz-Gilbert equation to output magnetization M(τ,r)as a function
of timeτand position r=(x, y,z). An in-plane excitation signal of the form hx=h0sin(2πfc
(τ−τ0))/(2πfc(τ−τ0))34is used to trigger the magnetization dynamics. Here, h0is typically 10 Oe,
fcis 20 GHz, and τ0is 0.4 ns. Signals of this form have also been used to excite the magnetization
dynamics elsewhere.34The dynamics is simulated for 40.96 ns, while saving the data at every
20 ps. The saturation magnetization Msof 8.1×105A/m, exchange constant Aof 1.05×10−11J/m,
gyromagnetic ratio of 2 .21×105m/As, and damping constant of 0.008 are used during simula-
tions. The cell size used is 5 nm ×5 nm ×5 nm for the nanodisk with diameter d=1000 nm and
2.5 nm ×2.5 nm ×5 nm in all other cases. Di fferent components of magnetization are Fourier
transformed to obtain the energy spectral density (ESD) and the phase in frequency domain.34
Spatial summation operation before Fourier transform was done in cases where spatial mode profile077179-3 Lupo, Kumar, and Adeyeye AIP Advances 5, 077179 (2015)
FIG. 1. (a) SEM image of Ni 80Fe20dot array. The dot diameter d=1000 nm, lattice constant a=2000 nm, and thickness
L=30 nm; (b) Hysteresis loop of the dot array shown in Fig. 1(a) as measured by VSM. (c) Experimental, and (d) simulated
microwave absorption spectra in zero applied magnetic field for the dot arrays with thickness of L=20 nm, and di fferent
values of the dot diameter dfrom 130 to 1000 nm.
is not required. In some cases the ESD and the phase are represented by color saturation and hue
respectively. Results from simulations are presented on a logarithmic scale.
The experimental and simulated broadband microwave absorption spectra for the thickness
L=20 nm, and di fferent diameters are shown in Figs. 1(c)–1(d). The top panel in Fig. 1(c) shows
the absorption spectrum for the nanodisk with the smallest diameter ( i.e.,d=130 nm). Two ASW
absorption peaks are clearly identified at 9.8 GHz and 12.6 GHz, together with the gyrotropic peak.
Increasing the diameter up to 450 nm, the frequency of the first and the second ASW resonance
peaks monotonically decrease to 6 GHz and 7.5 GHz, respectively; and a new peak appears at
around 9.5 GHz. We note that the former sample ( i.e.,d=130 nm) has the higher filling fraction
(f f=π
4d2/p2) off f=0.21 compared with the latter ( i.e.,d=450 nm, f f=0.18).Thus, an insuf-
ficient amount of ferromagnetic material could not be the cause of the observed fewer peaks in the077179-4 Lupo, Kumar, and Adeyeye AIP Advances 5, 077179 (2015)
FMR spectrum for d=130 nm. A further increase of the nanodisk diameter to 1000 nm results in an
overall shift of the absorption frequencies to lower values and the formation of another pair of peaks
in the range between 4 to 10 GHz. Moreover, the presence of weaker pairs of peaks pattern at higher
frequencies can be seen in the inset.
These observations are better highlighted in the corresponding simulated absorption spectrum
shown in Fig. 1(d). For diameter d=130 nm, the two absorption peaks are well reproduced,
although there is a di fference in the absorption intensity. We also note that the gyrotropic peaks
shows a significant high intensity in all the simulated results. This is possibly induced by the
spatial summation operation, which causes some modes to have lower amplitude due to an artificial
phase cancellation.35Ford=450 nm, the simulated spectrum in Fig. 1(d) shows the presence of
additional pairs of peaks at higher frequencies with decreasing absorption intensities. The first pair
is clearly present in the experimental result shown in Fig. 1(c), and both the peaks’ positions are
well reproduced. For the second pair, only the first peak was clearly detected. This is possibly due
to the much lower intensity of the second peak, which may not be detected during the experiment
measurements. For d=1000 nm (Fig. 1(d)), additional pairs of peaks can be clearly seen together
with another overall lowering of the frequency in agreement with the experimental results.
Figure 2(a) shows the absorption spectrum for nanodisks with thickness L=30 nm with
smaller diameter d=130 nm. Two ASW absorption peaks are clearly present at 9.8 GHz and
13 GHz. This is similar to the spectrum at lower thickness L=20 nm seen in Fig. 1(c). When the
diameter is increased up to 450 nm (Fig. 2(b)), the frequency of the first and second resonance peak
decreases to 7 GHz and 8.5 GHz respectively and a new peak appears at around 10.8 GHz. Increas-
ing the diameter up to 1000 nm (Fig. 2(c)), the presence of several pairs of peak and an overall
lowering of the absorption frequencies becomes evident from 4 GHz to 10 GHz. For this sample
as well, the presence of weaker pairs of peaks is still evident at higher frequencies as shown in
the inset. Figure 2(d)-2(f) show the corresponding simulated absorption spectra for L=30 nm. For
d=130 nm, the two absorption peaks are well reproduced. For d=450 nm, the simulated spectrum
FIG. 2. (a) – (c) Experimental, and (d) – (f) simulated microwave absorption spectra in zero applied magnetic field for the
dot arrays with thickness of L=30 nm, and di fferent values of the dot diameter dfrom 130 nm to 1000 nm.077179-5 Lupo, Kumar, and Adeyeye AIP Advances 5, 077179 (2015)
in Fig. 2(e) shows the presence of few additional pairs of peaks with a decreasing absorption
intensity. Again, for the second pair, only the first peak was detected. For d=1000 nm (Fig. 2(f)),
new pairs of absorption peaks appear together with an overall lowering of the frequency spectrum in
agreement with the experimental results. Comparing both the experimental and simulation results,
we observed a strong correlation between the space of available ASW eigenstates and the disk
diameter. Depending upon the required resolution and sensitivity, some simulated modes may not
be detectable using a VNA-FMR setup. Moreover, at higher thicknesses, some thickness-related
gyrotropic modes may appear.21
Figures 3(a) to 3(d) show the simulated mode profiles for the gyrotropic and the ASW modes
extracted from the Fourier transform of the z-component of magnetization, of the 20 nm thick
dots as a function of di fferent diameters. Here, we have also simulated a nanodisk with diameter
d=250 nm, in order to closely track any changes in the mode profiles as the diameter increases.
These profiles appear to remain uniform along the dot thickness. In all cases, the gyrotropic mode
occupies a small region of space around the core radius. The phase changes by a total of 2 πradians
around the center of the disks. The sense of change of hue also indicates the CCW or CW sense
of gyration of the gyrotropic mode or the sense of circulation of the ASW modes. The rotation of
gyrotropic mode is found to be in the CW, CCW, CCW, and CCW senses for the nanodisks with
diameter 130 nm, 250 nm, 450 nm, and 1000 nm, respectively. As marked in Figs. 3(a) to 3(d),
this is in agreement with the observed polarities (p)of the vortices. For d=130 nm, a nodal ring
is seen around the vortex core for the two modes, which correspond to ASWs circulating in the
CCW ( m=1) and the CW ( m=−1) senses for the first and the second modes, respectively. The
radius of nodal ring is also larger in the latter case. Thus, we note that the loss of degeneracy in
the azimuthal modes is also accompanied by a change in shape of their mode profiles.30In partic-
ular, the azimuthal modes with the same sense of circulation as the vortex core gyration, feature
slightly larger nodal rings when compared to their chiral counterparts.
As the ASWs move around the center of the nanodot, a nodal point appears (at the center). Due
to such constrains – nodal rings and a node at the center, we infer that the mode profile of ASW
FIG. 3. (a) – (d) Simulated microwave absorption spectra in zero applied magnetic field for the dot arrays with thickness
ofL=20 nm, and (a) 130 nm, (b) 250 nm, (c) 450 nm, and (d) 1000 nm dot diameter. The ASW mode profile at di fferent
resonance frequencies are presented in insets. (e) – (g) Simulated ESD spectra as a function of saturation magnetization Ms
and exchange coe fficient A. Modes profiles associated with the first ASW mode (marked as W1) is shown in a column to the
right.077179-6 Lupo, Kumar, and Adeyeye AIP Advances 5, 077179 (2015)
modes will always be antisymmetric along the diameter of the nanodisk. This implies that cases
where phase changes by an even multiple of 2 π(even number of crests and troughs) around the
nanodisk are impossible. Accordingly, we did not notice any case where the phase changed in that
manner.
The radius of the greater nodal ring, with (n,pm )=(1,1), which is independent of diameter, is
measured to be around 20 nm. This does not appear to change when the disk diameter is scaled up to
d=250 nm, 450 nm, or 1000 nm. Higher order modes with additional nodal rings appear when the
diameter is increased. These additional rings are in the form of polygons as opposed to a circle. This
is possibly due to the gridding associated with finite di fference method based simulations involving
curved geometries.36Thus, hereafter, the radius of the nodal ring refers to the mean of the maximum
and the minimum distances from its centroid to its periphery. The radius of mode (2, 1) for diam-
eters d=250 nm, 450 nm, and 1000 nm are 0 .31d, 0.27d, and 0.26d, respectively. This leads us
to the conclusion that the mode profiles of the higher order modes do not scale down in a linear
fashion with diameter. In particular, the nodal ring’s radius approaches the radius of nanodisks for
smaller values of the diameter d. Such size dependence (or a lack of scale invariance), gives one
an opportunity to fabricate nanodisks, where the small disk radius rules out the possibility of higher
order modes. Thus, we established that the space of available states for the ASW modes can be
trimmed by reducing the diameter of the nanodisks. Even in the case of diameter d=1000 nm, a
limited number of SW eigenstates are shown ( c.f. Fig. 3(d)) to be available. As the frequency of the
modes reduced with the increasing diameter (and expanding space of states), no ASW modes were
observed above 16 GHz in any of the nanodisks studied.
In magnetic systems, the competition between the exchange and the dipolar interactions intro-
duces a characteristic length – known as the exchange length lex=
2A/µ0M2s– which results in
the loss of this scale invariance. Thus, we believe that the characteristic size of a given ASW mode
is also e ffected by this competition between the exchange and the dipolar interactions. In order
to verify this hypothesis, we have used micromagnetic simulations to evaluate how a change in
the exchange length a ffects the size of the ASW modes in a nanodisk of diameter d=1000 nm
and thickness L=30 nm. The normalized ESD spectrum for the case of exchange coe fficient
A=1.3×10−11J/m and saturation magnetization Ms=8.1×105A/m is presented in Fig. 3(e).
When we halve the saturation magnetization Msto 4.05×105A/m (in Fig. 3(f)) the spectrum shifts
towards lower frequency range and the number of the ASW modes is reduced as well. Finally,
we quadruple the exchange coe fficient Aup to J /m while keeping the saturation magnetization at
Ms=8.1×105A/m, as shown in Fig. 3(g). For these parameters, the spectrum is characterized by
the same number of modes as the previous case in Fig. 3(f), but they are shifted at higher frequency.
Furthermore, on the right of Figs. 3(e) to 3(g) are shown the corresponding profiles for the first
ASW mode, marked as W1 in each figure. Both halving Ms(in Fig. 3(f)), and quadrupling A(in
Fig. 3(g)) has the e ffect of doubling the exchange length, and as a result, even though the mode
frequencies are very di fferent the mode profile scales up in a similar manner. This proves the e ffect
of the competition between the exchange and the dipolar interactions on the number of ASW modes
in a nanodisk of soft ferromagnetic medium.
In conclusion, we have used FMR measurements to show that the space of available SW
eigenstates in a magnetic vortex of a given ferromagnetic medium can be engineered by changing
the diameter of the nanodisk. By analyzing the results of micromagnetic simulations, we are able
to determine the origin of this behavior in the scale dependence of the SW mode profiles, which
is introduced by the competition between dipolar and exchange interactions. These mode profiles
show a characteristic minimum size and do not scale proportionately with increasing the disk diam-
eter. Consequently, for a diameter smaller than the minimum mode size, it is possible to eliminate
the higher order SW modes. This allows for the design of systems with a finite number of quantized
eigenstates. In contrast, any number of higher harmonics is possible on an ideal string independent
of its length. Apart for providing some necessary phenomenological insights, the results presented
here may also aid the design of SW and magnetic vortex based devices.
This work was supported by National Research Foundation, Prime Minister’s O ffice, Singapore
under its Competitive Research Programme (CRP Award No. NRF-CRP 10-2012-03) and the077179-7 Lupo, Kumar, and Adeyeye AIP Advances 5, 077179 (2015)
SMF-NUS New Horizon Awards. The authors would also like to acknowledge Dr. N. Singh for his
assistance with template fabrication.
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1.114853.pdf | Laser deposition of diamondlike carbon films at high intensities
F. Qian, R. K. Singh, S. K. Dutta, and P. P. Pronko
Citation: Applied Physics Letters 67, 3120 (1995); doi: 10.1063/1.114853
View online: http://dx.doi.org/10.1063/1.114853
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/67/21?ver=pdfcov
Published by the AIP Publishing
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137.30.242.61 On: Wed, 10 Dec 2014 10:09:19Laser deposition of diamondlike carbon films at high intensities
F. Qiana)and R. K. Singh
Department of Materials Science and Engineering, The University of Florida, Gainesville, Florida 32611
S. K. Dutta and P. P. Pronko
Center for Ultrafast Optical Science, Department of Electrical Engineering and Computer Science,
The University of Michigan, Ann Arbor, Michigan 48109
~Received 3 January 1995; accepted for publication 15 September 1995 !
Unhydrogenated diamondlike carbon ~DLC!thin films have been deposited by laser ablation of
graphite, using a high power Ti: sapphire solid state laser system. DLC films were deposited ontosilicon substrates at room temperature with subpicosecond laser pulses, at peak intensities in the4310
14–531015W/cm2range. A variety of techniques, including scanning and transmission
electron microscopy ~SEM and TEM !, Raman spectroscopy, spectroscopic ellipsometry ~SE!, and
electron energy loss spectroscopy ~EELS !have been used to analyze the film quality. Smooth,
partially transparent films were produced, distinct from the graphite target. Sp3volume fractions
were found to be in the 50%–60% range, with Tauc band gaps ranging from 0.6 to 1.2 eV,depending on laser intensity. Kinetic energies carried by the carbon ions in the laser induced plasmawere measured through time-of-flight ~TOF!spectroscopy. Their most probable kinetic energies
were found to be in the 700–1000 eV range, increasing with laser intensity. © 1995 American
Institute of Physics.
The growth of hydrogen-free diamondlike carbon ~DLC!
films has attracted much interest due to the fact that thesefilms possess properties close or similar to that of diamond.These properties include transparency in the infrared ~IR!
and near-infrared range, high microhardness, high electricalresistivity, as well as excellent chemical inertness. A varietyof applications are anticipated for these films, primarily inthe microelectronics, optics, and tribology industries.
1–6
Recently, high quality DLC films with sp3volume frac-
tion higher than 70%, and ‘‘amorphic diamond’’ films withmicrohardness comparable to natural diamond, have beenproduced through pulsed laser depositions ~PLD!.
7,8It is sug-
gested that, as a kinetic condensation process, pulsed laserdeposited DLC film quality is closely related to its depositionparameters, among which the kinetic energies of the carbon
particles, as well as their charge states in the plasma, are twoof the most consequential factors.
9,10
Most successful PLD depositions of DLC thin films con-
ducted so far, employed either UV excimer ~KrF,ArF, XeCl !
or Nd:YAG lasers, all with pulse durations in the nanosecondrange. Depending on laser energy and spot size focused ontothe target, power densities in the 10
8–1011W/cm2range
were delivered. Previous experiments have shown that, for acertain laser system, the higher the laser intensity, the morediamondlike ~i.e., higher sp
3fraction !are these films.11,12
Until now, the highest intensity used to deposit DLC films
was 5 31011W/cm2carried out by Collins et al., with a
Nd:YAG laser.13They suggested that this high laser intensity
will give rise to a more highly ionized plasma, the plasmabeing expected to contain charged carbon particles with highkinetic energies, and consequently leads to a higher volumefraction of sp
3bonded carbon atoms.
It therefore would be interesting to study the effects ofeven higher laser intensity on the DLC film properties. How-
ever, intensities higher than 1012W/cm2are often not
achievable for nanosecond lasers, because substantiallyhigher laser energy is of limited availability while submicronbeam spots suffer from diffraction limitations. Instead ofnanosecond lasers, we used a high power solid-state Ti: sap-phire laser system, capable of producing laser pulses in thepicosecond and femtosecond range. With this laser systemwe are able to induce laser intensities in the 10
9–
1016W/cm2range.
The depositions were carried out with a chirped-pulse
amplified ~CPA!Ti:sapphire laser system developed at the
Center for Ultrafast Optical Science, University of Michigan.This system enables the generation of variable length laserpulses going as short as 70 fs ~FWHM !. The laser beam is
near-Gaussian shaped and centers at 780 nm wavelength.Details concerning the laser system are discussed else-where.
14
The deposition station consists of a high vacuum cham-
ber maintained by a cold-trapped oil diffusion pump. A ro-tating graphite target is placed at an angle of ;45° to the
incident laser beam. The separation between the target andsubstrate is 4 cm. The compressed laser pulse was measuredas 250 fs ~FWHM !, it was then delivered across ;5mo fa i r
and throug ha1c mS i O
2window into the vacuum chamber.
A plano-convex lens was then used to focus the beam ontothe target. The nonlinear refractive index contributions fromthe atmosphere and glass components were estimated to haveincreased the pulse duration by a factor of about 2 when thebeam reaches the target. Typical experimental conditionsused in this study are listed in Table I.
The DLC films deposited on silicon substrates at room
temperature are visually smooth and uniform with goldenbluish tint. They are virtually featureless under a scanningelectron microscope ~SEM!, similar to the DLC films pro-
duced by nanosecond KrF laser pulses ~248 nm !. Significant
a!Electronic mail: fqian@grove.ufl.edu
3120 Appl. Phys. Lett. 67(21), 20 November 1995 0003-6951/95/67(21)/3120/3/$6.00 © 1995 American Institute of Physics
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137.30.242.61 On: Wed, 10 Dec 2014 10:09:19increase of surface particle density as a result of longer
wavelength ~780 nm !, a common feature of PLD tech-
niques,15is not observed. Transmission electron microscopy
~TEM !indicated the films to be amorphous.
Sensitive to longitudinal and translational symmetry of
materials, Raman spectroscopy is widely used in thin-filmcharacterization. Raman spectra of the films deposited at dif-ferent density levels showed similar features: a predominant‘‘G’’ peak centered at about 1540 cm
21along with a broad
‘‘D’’ peak at a lower wavenumber, typical of room-temperature deposited DLC films. The probe power on thespecimen was 100 mW with anAr ion excitation wavelengthof 514 nm. Best fits are obtained using two damped har-monic oscillator functions. Parametric values used in the fit-ting procedure are summarized in Table II. The ‘‘D’’ peakcenters were found to decrease, while their full width at half-maximum ~FWHM !increased at higher laser intensities, in-
dicative of possible increasing bond-angle disorder in thefilms.
16Different intensities showed no obvious effects on
the ‘‘G’’ peaks.
The refractive indices (n)and extinction coefficients (k)
of the DLC films were measured with a variable angle spec-troscopic ellipsometer ~VASE !, in the 1.5–4.5 eVrange. Fig-
ure 1 shows nandkof sample A as a function of photon
energy. The refractive indices ndecrease as a function of
increasing photon energy, and its extinction coefficients k
showed a steep increase in the energy range at ;2.5 eV ~500
nm!and start to level off at higher energies, suggesting the
films are partially transparent in IR and near-IR range, andbecome absorbing at higher energies. Samples deposited atdifferent intensity levels showed similar trends.The effectiveoptical band gaps of the DLC films are obtained from theTauc relationship by extrapolating a plot of (
aE)0.5~ais the
absorption coefficient !as a function of photon energy E. For
samples A, B, and C, deposited at successively higher inten-sities, optical band gaps were found to be 1.2, 0.8, and 0.6eV, respectively.
Electron energy loss spectroscopy ~EELS !was em-ployed to quantify the sp
3volume fraction in the deposited
DLC films. Films were first deposited onto NaCl substratesat room temperature, with thickness of about 400 Å, andlater removed to make EELS samples. Shown in Figure 2 aretheK-shell edge EELS spectra of DLC samples deposited at
4310
14~sampleA !and 5 31015W/cm2~sample C !, along
with that of a reference graphite film ~assumed to contain
100%sp2bonding !and CVD diamond ~100%sp3!. Notice
the strong absorption peak at ;285 eV from the graphite
sample, caused by p!p*transitions, characteristic of sp2
bonding structure. On the other hand, spectra from the two
DLC samples are dominated by the s!s*peaks at around
289 eV, with only a small p*peak being present for each of
these samples. However, the distinct features found in CVDdiamond at above 290 eV are not observed in these DLCfilms. The sp
3/sp2ratio was extracted by normalizing the
area of the p*ands*peaks in the 280–320 eV range and
comparing this ratio to the value of graphite.17Thesp3frac-
tions for samples A and C are determined to be 60% and50%, respectively.
To better understand the mechanism of DLC film forma-
tion under high laser intensities, a multigrid TOF drift tubecoupled with a Faraday cup was used to measure the kineticenergies of ablated carbon ions. The experimental setup issimilar to that introduced by Demtro ¨der and Jantz.
18Assum-TABLE I. Deposition conditions for DLC films.
Laser source Ti:sapphire ~780 nm !
Repetition rate 10 HzPulse duration ;500 fs ~FWHM !
Laser energy 15–45 mJSpot size 50–100
mm
Peak power density 4 31014–531015W/cm2
Substrate Si, NaCl
Substrate temperature Room temperatureFilm thickness 2500–3000 Å
TABLE II. Raman fitting parameters of DLCs deposited at different densi-
ties.
Power density
~W/cm2!G~cm21)D ~cm21)
Center FWHM Center FWHM ID/IG
431014~sample A !1539 171 1369 351 0.82
831014~sample B !1538 179 1364 362 0.97
531015~sample C !1537 175 1363 376 0.95
FIG. 1. Refractive index nand extinction coefficient kof DLC deposited at
431014W/cm2.
FIG. 2.K-shell edge EELS of DLC, CVD diamond, and graphite.
3121 Appl. Phys. Lett., Vol. 67, No. 21, 20 November 1995 Qian et al.
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137.30.242.61 On: Wed, 10 Dec 2014 10:09:19ing the ablated carbon particles are atomic ions under these
high intensities, their median kinetic energies were found tobe in the 700–1000 eV range, increasing with laser intensity.These numbers are at least 10–20 times higher than whathave been observed with nanosecond laser pulses.
19Figure 3
shows the particle kinetic energy distribution at 4 31014
W/cm2. In this relatively high energy regime, subsurface
penetration and ion implantation will occur, resulting in pos-sible film structure rearrangement and disruption, which mayalso explain why the DLC films have somewhat lower sp
3
content as well as lower band gaps at higher intensities.
In conclusion, we have deposited diamondlike carbon
thin films with subpicosecond laser pulses, to study the ef-fects of high laser intensities on DLC film properties. Thefilms are smooth with few particles. They are amorphous,partially transparent, with optical band gaps varying from 0.6to 1.2 eV.The sp
3volume fraction was estimated to be in the
50%–60% range. It appeared that laser intensity of 4310
14W/cm2result in films with higher Sp3bond percent-
age and higher band gaps, while films made at intensities inthe 10
15W/cm2range are relatively more graphitic. We at-
tribute this to possible ion implantation damage introducedby the carbon particles, resulting from a more energetic hy-drodynamic plasma expansion under higher laser intensities.The authors gratefully acknowledge the contribution of
Dr.WillisWeber at Ford Research Center for the SE analysesand Dr. Niegel Browning at Oak Ridge National Lab for theEELS data. We also like to thank Dan Gorzan from XSIInstruments, Dr. Roy Clarke and Sangeeta Murugkar at theUniversity of Michigan, Donggu Lee and Don Gilbert at theUniversity of Florida for their help in this study. This workwas supported in part by the National Science Foundationthrough the Center of Ultrafast Optical Science under STCPHY 8920108.
1S. J. Rzad, S. M. Gasworth, C. W. Reed, and M. W. DeVre, 1992 IEEE
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~1992!.
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R. W. Collins, J. Appl. Phys. 71, 5675 ~1992!.
8C. B. Collins, F. Davanloo, T. J. Lee, J. H. You, and H. Park, Mater. Res.
Soc. Symp. Proc. 285, 547 ~1993!.
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1706 ~1991!.
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J. Appl. Phys. 67, 2081 ~1990!.
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Symp. Proc. 285, 557 ~1993!.
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Technol. B 11, 1936 ~1993!.
14J. Squire and G. Mourou, Laser Focus World, June ~1992!; J. Squire, F.
Salin, G. Mourou, and D. Harter, Opt. Lett. 16, 1965 ~1991!.
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Symp. Proc. 235, 879 ~1992!.
16K. Enke, Thin Solid Films 80, 227 ~1981!.
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18W. Demtro ¨der and W. Jantz, Plasma Phys. 12, 691 ~1970!.
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Phys.72, 3966 ~1992!.
FIG. 3. Kinetic energy distribution of carbon ions at 4 31014W/cm2.
3122 Appl. Phys. Lett., Vol. 67, No. 21, 20 November 1995 Qian et al.
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1.367716.pdf | Time-resolved scanning Kerr microscopy of ferromagnetic structures (invited)
M. R. Freeman, W. K. Hiebert, and A. Stankiewicz
Citation: Journal of Applied Physics 83, 6217 (1998); doi: 10.1063/1.367716
View online: http://dx.doi.org/10.1063/1.367716
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Published by the AIP Publishing
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130.209.6.50 On: Sun, 21 Dec 2014 00:39:08Magnetic Microscopy and Imaging I R. D. Gomez, Chairman
Time-resolved scanning Kerr microscopy of ferromagnetic structures
invited
M. R. Freeman, W. K. Hiebert, and A. Stankiewicz
Department of Physics, University of Alberta, Edmonton T6G 2J1, Canada
Time-resolved microscopy enables valuable new measurements of the dynamics of resonance and
relaxation in a range of magnetic systems. An overview of the scope of applications toferromagnetic microstructures is presented. These include observations of ferromagnetic resonanceand spatially nonuniform modes of oscillation, studies of magnetization reversal, andcharacterizations of the speed of magnetic recording devices. © 1998 American Institute of
Physics. @S0021-8979 ~98!36511-1 #
I. INTRODUCTION
Recently, novel experimental information concerning the
dynamics of a variety of magnetic systems has been obtainedusing picosecond time-resolved laser techniques combinedwith diffraction-limited optical microscopy. Ultrafast opticalmethods have been in use for some time in the extraction ofrelaxation and resonance information from magneticsystems.
1–4The addition of microscopic spatial resolution
powerfully extends the approach to much smaller specimens,enabling measurements of relaxation in micrometer-scalestructures, as well as some imaging of spatially nonuniformdynamics.
5,6In this paper we describe the study of ferromag-
netic dynamics in small permalloy structures using this tech-nique. A similar procedure applies to the time-domain char-acterization of devices such as high-speed magneticrecording heads.
In addition to the good spatial and temporal resolution
achieved using an optical technique, another key aspect isthe ability to perform vector measurements of the magneti-zation. The component of magnetization parallel to the wavevector of the incident light is resolved in the experiments. Avector measurement is not crucial to paramagnetic relaxationmeasurements, but is essential in order to perform magneticresonance and ferromagnetic relaxation studies. Controllingthe optical configuration, we are able to track dynamical ex-cursions of the magnetization in three dimensions throughpolar and longitudinal Kerr effect measurements.
II. EXPERIMENTAL DETAILS
An example of an experimental geometry for spatially
resolved time-domain ferromagnetic resonance measure-ments is shown in Fig. 1.
6The single turn lithographic coil is
patterned from a gold film with a titanium adhesion layer. Afast electrical pulse generated using a photoconductiveswitch propagates around the coil, inducing a transient mag-netic field at the sample, perpendicular to the substrate ~the
tipping pulse !. The rise time of this pulse at the sample is
limited to ;20 ps by dispersion of the coil leads, and thedecay time constant is ;500 ps. Peak tipping field ampli-
tudes are limited to ;30 Oe with this coil. The static mag-
netic field is applied in the plane of the sample, and the polarKerr effect is used to record the out-of-plane component of
the magnetization. Resonant precession of the magnetization~about the static field !induced by the tipping pulse is re-
flected in oscillations of the polar Kerr signal.
The permalloy films used in this work are sputter depos-
ited in a load-locked ultrahigh vacuum chamber pumped to abase pressure of ;5310
28Torr. A permanent magnet as-
sembly is used to apply an in situstatic field of approxi-
mately 150 Oe in the plane of the substrate to establish aneasy axis. The resulting films have low coercivity ~,2O ei n
the easy direction !, and low resistivity ( ;20
mVcm) indi-
cating very little oxidation. SIMS analysis of the films showthe composition to be 83% Ni 17% Fe ~by weight !, fairly
close to the 81/19 proportion of the target. Patterning of thefilms is accomplished through photolithography and wetchemical etching, yielding the very smooth edges with aslight undercut profile as seen in Fig. 1 ~b!.
FIG. 1. Electron micrographs of the 8 mm diameter permalloy disk sample.
Left panel: plan view, showing the surrounding lithographic gold coil. Rightpanel: tilted close-up view, clearly showing the clean edge of the disk.JOURNAL OF APPLIED PHYSICS VOLUME 83, NUMBER 11 1 JUNE 1998
6217 0021-8979/98/83(11)/6217/6/$15.00 © 1998 American Institute of Physics
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130.209.6.50 On: Sun, 21 Dec 2014 00:39:08III. RESULTS
A. Resonance
The modal resonance frequencies are determined by the
maxima in the power spectra of Fourier transformed time-domain data. The in-plane magnetic field dependence of thecharacteristic frequency is well described by the Kittel rela-tion for an infinite plane, as may be expected for such a large~on the scale of the domain wall width !disk. Modeling the
time-domain data in more detail through numerical integra-tion of the Landau–Lifshitz–Gilbert equation, we find verygood agreement.
6Data are shown in Fig. 2 for a range of
values of the in-plane field. The measurements ~shown by
the dotted lines in the figure !were made with the 0.7 mm
laser spot focused at the center of the particle. The t50
position is arbitrary, corresponding to the initial position ofthe delay line at a location yielding a reasonable baselinedetermination before onset of the signal. Using a pulse shapedetermined from higher field data by the procedure estab-lished in Ref. 7, the curves are all fit using the same set ofparameters. The results of the fit are shown by the solid lines.
Only the dimensionless Gilbert damping parameter is adjust-able, and we find
a50.008, in reasonable agreement with
earlier careful microwave measurements of Patton andco-workers.
8Note that these results represent the response of
the magnetization to a very broad-band excitation. The highfrequency components associated with the rising edge of thepulse excite the resonant oscillations, which then ring-downaccording to their intrinsic damping rate. Meanwhile themagnetization vector parametrically follows the slowly de-caying tail of the field pulse ~consisting of that part of the
spectrum of the broad-band excitation below the resonancefrequency !, giving rise to the offset of the centerline through
the envelope of the oscillations.
Time-resolved images of the magnetization across the
whole disk clearly show the presence of nonuniform modesof oscillation, however. At these dimensions we are in anintermediate size regime, where the modal frequencies arenot yet strongly influenced by size effects but the spatialresponse definitely is.
6Figure 3 shows a set of time-resolved
magnetic images of the particle. With the static bias field at500 Oe ~horizontal !, an image was taken for a series of time
delays corresponding to successive peaks in the oscillationsmeasured at the disk’s center. Strikingly apparent are theenhanced initial responses at the edges ~dark corresponds to
FIG. 2. Examples of the response to the pulsed field of the out-of-plane
component of magnetization, measured at the center of the disk by the polarKerr effect. The solid lines are fits to the data using the Landau–Lifshitz–Gilbert equation, using the same parameters for each value of the static field.
FIG. 3. A collection of full spatial images of the polar Kerr signal at timescorresponding to the successive peaks in the signal at the center of the diskin a static field of 500 Oe.6218 J. Appl. Phys., Vol. 83, No. 11, 1 June 1998 Freeman, Hiebert, and Stankiewicz
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130.209.6.50 On: Sun, 21 Dec 2014 00:39:08higher signal !, that subsequently seem to propagate toward
the center as a kind of shock wave. The effective velocity ison the order of 10
4m/s, much faster than a domain wall
velocity, for example. The observation of enhanced responseat the edges along the field direction is qualitatively consis-tent with the fact that in static equilibrium these edges arealready demagnetized, and should be able to respond to thetipping field more easily. In contrast, no such effects are seenat the edges one-quarter of a rotation around the disk fromthese points, where there are no free poles in the initial mag-netic configuration. In these data, a reflection in the electricalpulse induces a bit of additional structure after the fifth peak.At long time delays, the spatial mode of oscillation becomesmore uniform again. To highlight the nonuniformity, thegray scale in each image is adjusted to span between theminimum and maximum levels of the signal. In the laterframes ~8,9,10 !the signal becomes quite uniform across the
disk, with variations not much greater than the noise level inthe measurement. The scaling procedure then mainly high-lights the noise, which appears as graininess.
From these images we can conclude that we should in
fact expect discrepancies between the Landau–Lifshitz–Gilbert model and the time domain data for this sample.Consistently poorer agreement is found between the data andthe model in Fig. 2 over a time interval that begins coinci-dent with the arrival of the nonuniform response at the centerof the structure. A more detailed numerical model taking intoaccount the initial magnetic configuration of the disk seemsessential to further quantitative progress.
B. Reversal
In-plane dynamics of the magnetization, and most spe-
cifically those aspects related to magnetization reversal, arealso of great interest at the present time.
9–13In order to ad-
dress these questions using time-resolved microscopy, it isonly necessary to reconfigure the ‘‘vector geometry.’’ Wemeasure the in-plane components of the magnetization ~par-
allel and perpendicular to the static magnetic field !using the
longitudinal Kerr effect implemented in the traditional man-ner by masking half of the beam. While this is the simplestapproach, one must beware that a mix of polar and longitu-dinal signals is observed when there is also an out-of-planemagnetization present. In addition the effective numericalaperture in the masking direction is halved, resulting in anelongated focus and some loss of spatial resolution.
The other important geometric change is to place the
transient magnetic field in the plane of the sample. Placingthe sample directly on top of a current carrying transmissionline straightforwardly does this. A cross-sectional schematicof the arrangement used in this work is illustrated in Fig. 4.The transmission line is 300 nm by 40
mm gold, relatively
thick for high current carrying capacity, and broad for mag-netic field uniformity at the sample. The current pulses inthis case are generated by an avalanche transistor pulser ~Pi-
cosecond Pulse Labs Model 2000D !. An insulating spacer
~25 nm SiO
2!on top of the gold electrically insulates the
permalloy from the transmission line, and optimally posi-tions the magnetic sample in the in-plane field. The calcu-lated field distribution above the transmission line is shown
in Fig. 4 ~b!.
The permalloy samples for this study are rectangular
bars, again produced by sputtering in the UHV system fol-lowed by optical lithography and wet chemical etching. Ini-tial magnetic characterizations are performed optically, usinga small electromagnet to apply in-plane field to the sample.Spot measurements of the hysteresis are performed, with thelaser focused at specific locations. As these results arestrongly position dependent ~the bars are much easier to satu-
rate near the center than near the ends !we also use an im-
aging method to characterize the static magnetic propertiesof the samples. The synchronous response to low frequency~280 Hz !ac fields is measured, yielding a signal representing
the difference of the magnetization between positive andnegative fields. Magnetic images taken for different field am-plitudes show how the hysteresis varies as a function of po-sition. A typical result is shown in Fig. 5. This is the longi-
FIG. 4. ~a!Cross-sectional geometry showing the configuration for applying
transient fields in the plane of the sample. ~b!Calculated cross-sectional
transient field profile for the volume occupied by the samples.
FIG. 5. A longitudinal Kerr image showing the signal change due to mag-
netization reversal of the 10 34mm bar in a 630 Oe field switching at 280
Hz.6219 J. Appl. Phys., Vol. 83, No. 11, 1 June 1998 Freeman, Hiebert, and Stankiewicz
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130.209.6.50 On: Sun, 21 Dec 2014 00:39:08tudinal Kerr signal ~xcomponent of the magnetization,
parallel to the static field !from a 4 310mm particle for a
630 Oe square-wave field modulation, showing a uniformly
magnetized and saturated central region. Recording the dif-ferent Kerr components reveals information about the mag-netic anisotropy at the ends ~due mainly to demagnetizing
effects !. Such static images provide a direct point of com-
parison for the dynamical studies. A very detailed portrait ofstatic reversal in permalloy bars was developed much earlierthrough Bitter pattern images, which have higher spatialresolution than the present optical experiments.
14
A first glimpse at the dynamical information accessible
in time-resolved studies of magnetization reversal is seen inthe following data of the response of a nonuniformly mag-netized permalloy rectangle to a pulsed field in the plane ofthe particle. We examine a 20
mm36mm rectangle ~50 nm
thick!with the static bias magnetic field parallel to the long
direction. This bar is oriented transverse to the current flowdirection of the transmission line so that the transient mag-netic field is parallel ~or antiparallel !to the bias field. In a
field of 25 Oe, the static configuration of the particle corre-
sponds to two antialigned domains along most of the lengthof the bar, with the domain wall near the center, and someclosure domains at the ends. Time-resolved measurementsare then performed to study the approach to saturation in-duced by the ~0.4 ns rise time, 1.5 ns fall time, 10 ns dura-
tion, opposite polarity to the static field !pulsed magnetic
fields, starting from this simple demagnetized state.As a function of the amplitude of the pulsed field, two
distinct regimes are observed in the reversal dynamics of theinitially antialigned domain. At smaller pulse amplitudes~less than ;60 Oe !the reversal proceeds via uniform motion
of the central domain wall towards the edge of the bar. Char-acteristic data are shown in Fig. 6. Panel ~a!is a snapshot of
the magnetization change 5.5 ns after the onset of the pulse.The light area is the region that has been swept out by themoving domain wall. To see the progression in detail, thechange in the longitudinal Kerr signal is shown in two di-mensional images in Figs. 6 ~b!and 6 ~c!where the vertical
axis corresponds to position along a line section through thecenter of the bar ~in the short direction, y!, while the hori-
zontal axis is time. The signal growing with time in Fig. 6 ~b!
for a small tipping field arises from motion of the domainwall with nearly constant velocity. The initial displacementsof the wall are well below the spatial resolution of the mi-croscope, so the rate of reversal is extracted by integratingthe Kerr signal across the particle and plotting the integral asa function of time, as in Fig. 7 ~a!. Normalizing the curve by
the saturation signal divided by the width of the bar, theslope can be converted to a wall velocity.
Reversal rates as a function of pulsed field amplitude are
shown in Fig. 7 ~b!. A couple of points are of particular note.
First, we do not observe a linear region at low fields, indi-cating that the motion cannot properly be described using awall mobility. This may be indicating that we cannot ignore
FIG. 6. Changes of the xcomponent of magnetization ~larger signals shown
brighter !f o ra2 0 36mm bar in a 25 Oe static biasing field. ~a!Shift of the
domain wall after 5.5 ns in a small tipping field. ~b!and~c!are typical y-t
diagrams for small and large tipping fields, respectively, where a linearvertical spatial scan across the center of the bar is repeated for increasingdelay times. Note the change in time scale between ~b!and~c!.
FIG. 7. ~a!Wall shift as a function of time determined by integrating the
reversed component along yin data as shown in Fig. 6 ~b!.~b!The effective
reversal speed as a function of tipping field amplitude, determined from thelinear slope of curves as in ~a!.6220 J. Appl. Phys., Vol. 83, No. 11, 1 June 1998 Freeman, Hiebert, and Stankiewicz
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
130.209.6.50 On: Sun, 21 Dec 2014 00:39:08the influence of the closure domains at the ends of the bar on
the motion of the wall. Second, at pulse amplitudes above 65Oe, there is a nearly linear region, possibly preceeded by adiscontinuity in the curve. The images of the magnetizationare qualitatively different in this regime, with the reversalproceeding not by movement of the domain wall but ratherthrough a rotation process which propagates through the cen-ter of the domain after nucleating near the closure domainboundaries at the ends. This is clearly evident in Fig. 6 ~c!,
where change in the longitudinal Kerr signal is seen to beginnot at the wall but closer to the center of the domain, and todevelop symmetrically about this point. The distinct asym-metry arising from wall motion in Fig. 6 ~b!is notably absent.
Therefore the reversal rates found at higher fields in Fig. 7cannot correctly be interpreted as velocities because of thischange in reversal mechanism. Normalizing the data as de-scribed above, the maximum wall speed we find before thisphenomenological change is 650 m/s at 60 Oe.
A great richness of phenomena is observed upon pursuit
of these investigations to higher fields. Beginning from anequilibrium state in which the centers of the bars are uni-formly magnetized ~static field of 235 Oe !, we obtain a view
of the speed of reversal by recording the time evolution ofthe magnetization change with the laser focussed at the cen-ter of the structure. Results are shown in Fig. 8 for 4
mm
wide samples of three different lengths, 4, 10, and 20 mm.
The 10 ns duration transient field has amplitude 140 Oe. Therising and falling transitions are markedly asymmetric, par-ticularly for the longest and shortest samples. It is clear fromthe expanded view of the initial switch in the inset that re-versal at the center occurs more rapidly for the shorter bars.The rise time of the pulse itself starts to become significantin limiting the speed for the 4 34
mm structure.
Much more information is available in full time-resolved
images of the magnetization. We observe that the reversalprocess starts with a wave-like spatial oscillation of the in-plane magnetization. Figure 9 has panels showing the threecomponents of the magnetization ~long.x, long.y, polar !for
the 10 34
mm particle at t53 ns after the onset of the field
~see the abscissa of Fig. 8 !. The oscillation is initially quitesymmetric about the equilibrium ( x) direction, and shows up
most dramatically in the ycomponent. One direction then
grows at the expense of the other, culminating in reversal.The relative degree to which this behavior is related to the‘‘concertina’’ structure seen in static reversal,
14or is induced
by dynamics, remains to be sorted out. The wavelength ofthe spatial variation appears to be extremely stable and re-producible. At the same time these oscillations are clearlyobserved only on the rising edge of the pulses. Two effectsmay be at work to cause different behavior on the trailingedge. The time rate-of-change of the magnetic field is con-siderably less when the pulse shuts off relative to when itturns on, making it less effective at driving a dynamic insta-bility. In addition the switched state may be more uniformlymagnetized because of the unequal amplitudes of the staticand transient fields, although the bar may not have com-pletely relaxed into a switched equilibrium state by the endof the 10 ns pulse.
Strong dependences on sample size are also found in
imaging. Some results from a more detailed investigation ofmagnetization reversal in the longer bar are reported in acompanion paper.
15
FIG. 8. Time dependence of the xcomponent of magnetization measured at
sample center for three different structures, 4 34, 10 34, and 20 34mm
~light, medium, heavy line, respectively !.235 Oe static field, tipping field
amplitude 1140 Oe. The dashed line shows the shape of the pulse, and the
inset shows the rising edge response in more detail.
FIG. 9. The nonuniformity of response during the initial flip is shown in this
set of magnetic images taken for the 10 34mm bar att513 ns, under the
conditions of Fig. 8. ~a!Longitudinal ( x) component. ~b!Transverse ( y)
component. ~c!Polar (z) component.6221 J. Appl. Phys., Vol. 83, No. 11, 1 June 1998 Freeman, Hiebert, and Stankiewicz
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130.209.6.50 On: Sun, 21 Dec 2014 00:39:08IV. DISCUSSION
The present work only begins to scratch the surface of
what is possible with these pulsed optical methods. Manyother materials systems and geometries remain to be inves-tigated with our present set-up. Some of the choices will beguided by the desire for convergence between experimentand micromagnetic simulations, an obvious goal for the nearterm. The immediate goal is to study smaller particle sizes.
The same experimental techniques are well suited to the
characterization of devices and media for magnetic record-ing, particularly when high speed is an issue.
16,17The utility
of such measurements is further enhanced by time-domainmagneto-optical measurements of currents, for direct com-parison between the response ~e.g., of the magnetization at
the pole tips of a recording head !and the input ~e.g., the
drive signal to the coil !. A stroboscopic ‘‘movie’’ of the
magnetization at the air bearing surface of a write head inresponse to a dibit input can be viewed on the internet.
18
In terms of the experimental method, there are many
potential improvements that would further broaden the rangeof significant applications. Although the speed of the tech-nique is more than adequate for our present studies, this ismaterial dependent. Awschalom and co-workers
4have stud-
ied much faster dynamics in magnetic semiconductors usingoptical excitation of the sample. Substantial improvement ofthe speed of pulsed magnetic field generation is required be-fore our approach can enter the sub-picosecond regime. Ofeven more interest is increasing the amplitude of magneticfield pulses, to open the door to large tipping angle measure-ments and associated nonlinearities in resonance. Studies ofreversal in magnetically harder materials would also be ofinterest, as in the case of studies of high-speed switching inmedia by Doyle and co-workers.
19At present we are unable
to pass more than about 300 mA peak current through thephotoconductive switches. The corresponding current densi-ties are very high and may be near to intrinsic damagethresholds of the materials, but the fact that the quantumefficiency of the devices is very low suggests that there maystill be room for significant improvement here.
It is also imperative to improve spatial resolution in or-
der to investigate the detailed micromagnetic dynamics ofmuch smaller ferromagnetic particles. In the present experi-ments some information is already lost below the limit ofspatial resolution, especially near the edges of the particles.When we attempt to cleanly separate the vector componentsof the magnetization the demands on resolution increase stillfurther. Because of unwanted ‘‘clipping’’ of the beam whichoccurs as the focus spot scans over an edge, a small focusalso helps in keeping the entire laser spot on the particle as
close to the edge as possible. We have developed a solidimmersion lens capability for these experiments offering im-provements of a factor of three in spatial resolution over thatachieved here.
20This is probably good enough to explore
strong size effects and to be useful for the next few genera-tions of magnetic devices, but certainly not for superpara-magnetic particles or for such devices as seem certain toexist before the ‘‘endpoint’’ of magnetic recording isreached. In this regard the near-field and second harmonicgeneration methods of Silva, Rogers, and co-workers
21are
particularly exciting.
ACKNOWLEDGMENTS
The authors are indebted to the Alberta Microelectronics
Centre for access to their deposition and patterning facilities,and to Professor Abdul Elezzabi for his assistance during theearly stages of this project. We thank J. Giusti for pointingout Ref. 14. This work is supported by the Natural Sciencesand Engineering Research Council, Canada.
1D. D. Awschalom, J.-M. Halbout, S. von Molnar, T. Siegrist, and F.
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D. D. Awschalom, Phys. Rev. Lett. 76, 1948 ~1996!.
6W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, Phys. Rev. Lett. 79,
1134 ~1997!.
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8C. E. Patton, Z. Frait, and C. H. Wilts, J. Appl. Phys. 46, 5002 ~1975!.
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Thomas, B. Barbara, and D. Mailly, Phys. Rev. B 53, 3341 ~1996!.
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~1982!.
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18http://laser.phys.ualberta.ca/ ;freeman/maghead.mov
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Magn. Mater. 155,6~1996!.
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21T. J. Silva, T. M. Crawford, and C. T. Rogers ~these proceedings !.6222 J. Appl. Phys., Vol. 83, No. 11, 1 June 1998 Freeman, Hiebert, and Stankiewicz
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130.209.6.50 On: Sun, 21 Dec 2014 00:39:08 |
5.0029050.pdf | Appl. Phys. Rev. 8, 021305 (2021); https://doi.org/10.1063/5.0029050 8, 021305
© 2021 Author(s).Engineering plasmonic hot carrier dynamics
toward efficient photodetection
Cite as: Appl. Phys. Rev. 8, 021305 (2021); https://doi.org/10.1063/5.0029050
Submitted: 09 September 2020 . Accepted: 02 February 2021 . Published Online: 07 April 2021
Yisong Zhu , Hongxing Xu ,
Peng Yu , and
Zhiming Wang
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Cite as: Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050
Submitted: 9 September 2020 .Accepted: 2 February 2021 .
Published Online: 7 April 2021
Yisong Zhu,1Hongxing Xu,2Peng Yu,1,a)
and Zhiming Wang1,a)
AFFILIATIONS
1Institute of Fundamental and Frontier Science, University of Electronic Science and Technology of China, Chengdu 610054, China
2School of Physics and Technology, Center for Nanoscience and Nanotechnology, and Key Laboratory of Artificial Micro-
and Nano-structures of Ministry of Education, Wuhan University, Wuhan 430072, China
a)Authors to whom correspondence should be addressed: peng.yu@uestc.edu.cn andzhmwang@uestc.edu.cn
ABSTRACT
Nonradiative decay of surface plasmons (SPs) is usually considered an unwanted process. However, recent studies have proven that hot carriers
generated from nonradiative SP decay can be used for photodetection that circumvents the bandgap limitation in semiconductors. The majorproblem plaguing the plasmonic hot carrier photodetectors stems from the low quantum efficiency. In this review, we discuss recent progressof engineering plasmonic hot carrier dynamics and describe a host of plasmon-enhanced photodetectors, including optical antenna-based pho-
todetectors, planar photodetectors, photodetectors coupled with 2D materials, functionalized photodetectors, photodetectors for integrated
nanophotonics, and hot-hole photodetectors. Finally, we herein highlight some new directions in the plasmonic photodetection.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0029050
TABLE OF CONTENTS
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. ENGINEERING HOT CARRIER DYNAMICS . . . . . . . . 3
A. Boosting hot carrier generation . . . . . . . . . . . . . . . . 4
B. Preventing losses in hot carrier transport . . . . . . . . 7
C. Efficient hot carrier extraction . . . . . . . . . . . . . . . . . 14
III. PLASMON-ENHANCED HOT CARRIER
PHOTODETECTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
A. Plasmon-enhanced hot electron photodetector . . . 16
1. Optical antenna-based hot electron
photodetectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2. Planar hot electron photodetectors . . . . . . . . . . 20
3. Hot electron photodetection coupled
with low-dimension materials . . . . . . . . . . . . . . 22
4. Functionalized hot carrier devices . . . . . . . . . . . 24
B. Plasmon-enhanced hot carrier photodetectors
for integrated nanophotonics . . . . . . . . . . . . . . . . . . 28
C. Plasmon-enhanced hot hole photodetector . . . . . . 30
IV. CONCLUSION AND OUTLOOK . . . . . . . . . . . . . . . . . . 30AUTHORS’ CONTRIBUTIONS . . . . . . . . . . . . . . . . . . . . . . . . 33
I. INTRODUCTION
Surface plasmons (SPs), coherent electron oscillations in metals,
provide a novel means of enhancing light-matter interaction at thenanoscale.
1–3SPs have been harnessed for numerous applications,
including super light absorbers and their use for energy conversion,4–6
ultrafast light-emitting,7subwavelength light confinement within
ultrasmall mode volume,3,8ultra-compact lens and waveplates,9,10
ultrasensitive sensing,11,12and hot carrier generation for photodetec-
tion and photochemistry.6,13–18Following excitation, SPs can either
propagate on a planar metallic surface, i.e., surface plasmon polaritons(SPPs), or be confined to the particles, i.e., localized surface plasmons
(LSPs).
19,20However, most practical applications are plagued by plas-
monic loss—SPs dephase rapidly, transferring their energy to single-electron excitations. The plasmon decay process can be divided into
two categories: radiative SP decay and nonradiative SP decay.
21–23As
shown in Fig. 1(a) , the incoming light is partly re-emitted as scattering
photons and partly absorbed. The absorbed photons excite nonther-
mal energetic (hot) carriers in the metal particle.23The hot carrier
dynamics include plasmon excitation, Landau damping, carrier relaxa-
tion, and thermal dissipation, as shown in Fig. 1(b) .H o w e v e r ,i nm o s t
cases, nonradiative SP decay is considered to be a parasitic pro-cess.
13,24,25The optical absorption in metals from the nonradiative
decay process is inevitable, which results in reducing the performance
of plasmonic devices. For instance, nonradiative SP decay would limitthe propagation length of the plasmonic waveguide.
26Although much
effort was focused on avoiding or mitigating nonradiative SP decay,
such as by using highly doped semiconductors instead of metals,13
researchers now start to embrace the loss-enabled applications of
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-1
Published under license by AIP PublishingApplied Physics Reviews REVIEW scitation.org/journal/areplasmonics,27such as thermoplasmonics,5,11,28,29surface imaging,30,31
and hot carrier generation for photochemistry and photodetection.6,13–18
The hot carrier technology and science can be traced back to the
discovery of the photoelectric effect by Heinrich Hertz in 1887.32,33
The photoelectric effect is established in quantum mechanical proper-
ties of the electromagnetic field and only the photon with the excessenergy higher than the metal’s work function can eject an electronfrom the metal surface, regardless of the intensity of illumination.
These historical findings have established the initial cognition of hot
carrier effects that mine out its various wealth. They served a signifi-cant role in the development of quantum mechanics and offered excit-ing opportunities for critical research and applications with the rise ofplasmonics photonics. For example, photothermal heat generated
from hot carriers can be used as a heat source for cancer therapy or
solar desalination,
34–36and hot carrier injection can drive phase transi-
tions by electrically doping.37Moreover, due to the unique properties
of energetic hot carriers, plasmon-enhanced hot carrier photodetectionhas also attracted increasing attention.
13,38–40
Photodetectors are widely used in telecommunication,41,42imag-
ing,43surveillance,44and military purposes.45Conventionalphotodetection uses photoconductive,46photovoltaic,47–49and pyro-
electric schemes to convert light into a useful electronic signal.49–51
However, they cannot simultaneously meet the requirements of highsensitivity,
52fast speed,53,54broadband, and cryogen-free operation,51
especially for infrared photodetection. For example, the photoresponse
of a photoconductive photodetector is in the order of ms to s; the photo-
response of the pyroelectric detector is in the order of s; photoconduc-
tive and photovoltaic photodetectors are not able to detect light beyond
their bandgaps. Hot carriers generated from metallic structures can be
captured by an adjacent semiconductor to form photocurrent, offeringanother route to photodetection.
30In this case, the detection bandwidth
is usually determined by the height of the Schottky barrier, rather than
material bandgap, which results in additional energy harvest-
ing.13,25,38–40Moreover, conventional materials for photodetection usu-
ally rely on absorption, but they are inherently limited by the
absorption law exp( /C0ad). In contrast, enhancement of light absorption
resulting from plasmonic resonance can improve photoelectric conver-sion efficiency. Photoelectric conversion from hot carriers can compete
with processes, such as carrier relaxation and recombination, and the
transfer is fast (10–100 fs) and furious enough to effectively avoid
energy loss and long response time.
16Therefore, hot carrier photodetec-
tors enable substantial advantages of near-infrared (NIR) photodetec-
tion, additional bandwidth response, room-temperature, zero-bias
operation, and high tunability.39,55Besides, hot carrier photodetectors
have demonstrated novel functionalities, which are absent from conven-
tional semiconductor photodetectors, such as hot-electron nanoscopy,
sensing, and circularly polarized light (CPL) detection.31,56,57
Although hot carrier photodetectors have a series of advantages,
the quantum efficiencies of these devices are insufficient and limit theirpractical application due to poor light absorption, broad hot electron
energy distribution, and isotropic hot carrier momentum distribu-
tion.
58,59Each hot carrier excited by an electromagnetic wave is attrib-
uted to photon absorption, and SPs provide a fresh approach to
enhance light absorption and modify hot carrier momentum distribu-tion at the nanoscale. The excitation of SPs results in two types of
hot carriers: hot electrons or hot holes.
32Therefore, hot carrier photo-
detectors can be divided into hot electron- or hot hole-dominated
photodetectors, and their energy diagrams are shown in Fig. 2 .E v e n
though many studies were focused on plasmonic-enhanced hot
electron-dominated devices, the photoresponsivity and quantum effi-
ciency remain low, restricted by a high Schottky barrier, thermody-namic losses during hot carrier transport process, and imperfect
experimental fabrication. On the contrary, hot hole-dominated photo-
detectors can achieve a lower Schottky barrier resulting in significant
enhancement of responsivity; however, too low Schottky barriers are
not conducive to room-temperature operation due to a large dark cur-
rent. As one of the most important parameters of the photodetector,
dark current density influences the ability to distinguish the weak sig-nals from the noise. The relationship between dark current, Schottky
barrier height, and the operating temperature will be discussed in Sec.
III C on plasmon-enhanced hot hole photodetectors.
In this review, we focus on the recent developments of plasmon-
enhanced hot carrier photodetection, including hot electron and hot
hole photodetectors. First, we discuss the way to engineer dynamics of
the hot carriers, including generation, transport, and extraction.
Subsequently, the development of plasmon-enhanced hot electron
photodetectors will be summarized, including optical antenna-based
Radiative decay Non-radiative decay
d-band
Plasmon excitation
t = 0 s++h+h+
+–––
Landau damping
t = 1–100 fs
Population Population PopulationCarrier relaxation
t = 100 fs to 1 psThermal dissipation
t = 100 ps to 10 ps
EEE
EF EFEFhω
EF(a)
(b)
FIG. 1. Plasmon decay mechanism and hot carrier generation and relaxation in
metal nanoparticles (NPs). (a) Plasmon decay process: radiative decay and nonra-
diative decay. Here, the incoming light is partially re-emitted as a scattering photonand absorbed partially. The absorbed photons excite nonthermal energetic (hot)carriers in the metal nanoparticle. Adapted with permission from C. Clavero, Nat.
Photonics 8(2), 95–103 (2014). Copyright 2014 Springer Nature.
23(b) Process of
hot carrier dynamics in metal NP with corresponding time under the illumination.The hot carrier dynamics include plasmon excitation, Landau damping and carrierrelaxation as well as thermal dissipation. Adapted with permission from M. L.
Brongersma et al. , Nat. Nanotechnol. 10, 25 (2015). Copyright 2015 Springer
Nature.
32Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-2
Published under license by AIP Publishinghot electron photodetectors, planar hot electron photodetectors, 2D-
material-enhanced hot electron photodetectors, and functionalizedhot carrier photodetectors. We will then review the recent progress ofhot carrier photodetection for integrated nanophotonics, followed by
introducing hot hole-dominated photodetectors. Finally, future devel-
opment and perspectives are discussed.
II. ENGINEERING HOT CARRIER DYNAMICS
The process of hot carrier photoresponse is complicated, includ-
ing hot carrier generation and distribution, the transport of hot car-riers, and the extraction of hot carriers via an external circuit, asshown in Fig. 3 . Hot carriers generated from metallic nanostructures
can be injected into an adjacent semiconductor over the Schottky bar-
rier. Normally, the height of the Schottky barrier is lower than the
bandgap of semiconductor, and thus hot carrier photodetectors have
the advantages of NIR and below-band-gap photodetection.
25,38–40
The contact between semiconductors and metal can cause the energy
band bending forming a junction barrier. Different semiconductors
result in different types of Schottky barrier and hot carrier injection.
For instance, n-type silicon contacted with gold can form an n-typeSchottky with 0.8 eV, whereas for p-type silicon, it can form a p-type
Schottky barrier with 0.32 eV. The theoretical height of the Schottky
Efe–
h+EfEc
Eg
Ev
ΔEΔESchottky
junction
Schottky
junction
semiconductor semiconductorEc
Eg
Evhω
hωFIG. 2. Energy diagrams of the Schottky
barrier junction for the hot hole (left) and
hot electron (right) injection. The excitedhot holes and hot electrons are injectedinto the valence band and conduction
band of semiconductors, respectively.
Ekin
e–e–e–
h+e– e–e–
e– e–e–h+
h+
ΔEGeneration
A
A
kA
AExtractionmetal metal
LMFP˜L
metal metalTransport
Reach interface
Transport
Cross barrier(a) (b)
(c) (d)
semiconductor semiconductorsemiconductor semiconductorhω
FIG. 3. Process of hot carrier photoresponse from generation to extra circuit extraction. (a) Hot carrier generation process for intraband and interband tr ansitions. The hot car-
riers are generated by nonradiative plasmon decay through the intraband transition within the conduction band or interband transition, which is the transition between other
bands and the conduction band. (b) Process of hot carriers reaching the metal/semiconductor (M/S) interface through electron-electron and electro n-phonon scattering. The
electron-electron and electron-phonon scattering provide the momenta of hot carriers to reach the M/S interface. However, hot electron relaxation also occurs simultaneously
at this process leading to the transport loss. The transport loss is connected to the transport distance and the MFP of metals. (c) Process of hot carrie rs with excess energy
crossing the potential barrier. Only the part of hot carriers with appropriate momenta and sufficient kinetic energy can be emitted over the Schottky b arrier. (d) Process of hot
carriers collected by an external circuit. The hot carriers injected into the semiconductor would flow toward the counter electrode extracted by an ex ternal circuit.Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-3
Published under license by AIP Publishingbarrier can be calculated by DE¼W/C0vfor n-type silicon and
DE¼Eg/C0W/C0v ðÞ for p-type silicon, respectively. Here vandW
are the work function of metal and the electron affinity of semiconduc-tor, respectively, and E
gis the bandgap of the semiconductor. Different
junction barriers lead to hot carriers emitted into the conduction band
(hot electrons) or the valence band (hot holes) of semiconductors,respectively. Although the number of hot electrons and hot holes isconsistent, the fractions of over-barrier carriers to form photocurrentare different due to their different mean free path (MFP) and initial
energy distribution.
60To increase the responsivities for the hot carrier
photodetectors, one must optimize them from the perspective of carrierdynamics, that is, increasing hot carriers’ generation rate, avoidinglosses during the transport process, and collecting carriers efficiently. Int h i sr e g a r d ,w ew i l lfi r s ti n t r o d u c et h ew a yt oe n h a n c eh o tc a r r i e rg e n e r -
ation, and then we focus on hot carrier transport and the difference in
hot carrier extraction for both hot electrons and hot holes.
A. Boosting hot carrier generation
Hot carrier generation is the first step in hot carrier photodetec-
tion. Understanding initial hot carrier generation and distribution pro-cesses are critical to estimate the performance of hot carrier
photodetectors, as shown in Fig. 3(a) . For more discussion on the gener-
ation and distribution of hot electrons, see review in Ref. 16.H e r ew e
briefly discuss this process and focus on boosting carrier generation. Asshown in Fig. 1(b) , photoexcitation and relaxation of metallic NPs were
described by Brongersma et al. with a four-step model.
32First, the inci-
dent light can flow into the metallic particle, and the excitation of local-
ized surface plasmon resonance (LSPR) results in the enhancement of alight absorption cross-section larger than its physical size. The first stepof hot carrier generation is optical absorption in metals, and the absorp-tion distribution is proportional to the square of the local electric field
inside the metal, given by Q¼1/2/C2xIm(e)jE(x,y,z)j
2,w h e r eI m ðeÞ
andxrepresent the imaginary part of the metal permittivity and inci-
dent light frequency, respectively, and E(x, y, z )i st h el o c a le l e c t r i cfi e l d .
Bulk metallic structures such as planar metal surfaces cannot achievelight-harvesting effectively, and most of the light is reflected directly.
After that, plasmon resonances result in Landau damping in a short
timescale from 1 to 100 fs.
32Photons can be re-emitted by damping
radiatively, and the excitation of electrons in NPs results in the creationof hot electron-hole pairs via nonradiative plasmon decay. The propor-tion of these two decay mechanisms depends on plasmon radiation,
which can suppress subradiant (dark) plasmon modes. Electrons in
metals rise from the initial low energy level to a higher level by absorb-ing photon energy, leaving behind holes. In this condition, an absorbedphoton can only produce hot electron-hole pairs, and hot carriers arehighly nonthermal. The plasmonic electric field can induce electrons
changing from occupied to unoccupied states. And then, the distribu-
tion of hot carriers is determined by the shape of NP, the electronicstructure, the plasmon mode and so on. Landau damping is a basicphysical quantum mechanism, which is related to the imaginary part ofthe permittivity and photogenerated hot carrier optimization. In most
cases, the spatial distribution of hot-electron generation from photon
absorption can be written as: G¼ð1/C0P
rÞQ=/C22hx,w h e r e Pris the resis-
tive loss of absorbed photon energy, which is dissipated without hotelectron generation arising from the finite carrier lifetime. Due to thefeature shape and size on the nanostructures, the resistive loss ranges
from 10% to 40% at NIR wavelength for gold. Indeed, hot carriersgenerated in metals also need to undergo the transport loss and photo-
emission to form a photocurrent, which will be introduced in subse-
quent chapters. In a timescale from 100 fs to 1 ps, the energy of hot
carriers will go through the redistribution by electron-electron scattering
process. While the work functions of plasmonic metals are larger than
their LSPR energies, hot electrons will occupy negative energies ranging
from E
ftoEfþ/C22hxLSPRwithout a vacuum escape. Hot carrier relaxation
processes will quickly result in a large effective electron temperature sat-
isfying a Fermi–Dirac-like distribution. Meanwhile, the velocity of
lower-energy electrons decreases and the interactions of phonons
increase gradually in this step. This process can be described by a two-
temperature model, including effective electron temperature Teland lat-
tice temperature Tl.61The two temperatures are time-dependent and
will be equal eventually, resulting in hot carriers converted into heat.
Finally, photo-induced heat generated by hot carrier thermalization
transfers from the metallic structure to the surroundings in t ¼100 ps
to 1 ns. Only hot carriers can be injected into the semiconductor to
form photocurrent before inelastic relaxation and thermalization
through electron-electron and electron-phonon coupling.
Enhancing light absorption is the first step to boost hot carrier
generation.15,57,62Metamaterial perfect absorbers (MPAs) can achieve
unity light absorption—at resonant wavelengths, they produce strong
fields at surfaces due to impedance-matching with the free-space.
Although, lots of hot carrier photodetectors based on MPAs have beenproposed and demonstrated in recent years,
25,57due to their structural
difference, there is still no uniform method to calculate hot electron
generation, which can be used in any system. As shown in Fig. 4(a) ,
Wang et al. used a typically metal-insulator-metal (MIM) structure to
achieve a strong chiral effect in hot electron generation, and the hot
electron excitation is considered as a surface quantum effect from car-
rier scattering by the metal surface.15Using the quantum formalism
for describing hot carrier generation in a plasmonic structure, one can
calculate the rate of hot electron generation for arbitrary geometries,63
Rateover/C0barrier electron ¼1
42
p2e2E2
f
/C22h/C22hx/C0DE ðÞ
/C22hx4ð
SjEnormalj2dS; (1)
where DEis the potential barrier height; Enormal is the component of
the electric field normal to the surface, taken inside the metal; the inte-
gral is taken over the metal surface. This equation is most widely used
in assessing the generation rate of over-barrier hot electrons near the
surface of plasmonic nanostructures. In this formalism, considering
the electrons as quasi-free carriers, hot electrons moving parallel to the
interface will not be injected into the semiconductors. Therefore,
the ones with the electric field parallel to the interface cannot traverse
the surface to realize photoelectric conversion.58,59Although the equa-
tion has been used to estimate the performance of photochemistry, we
believe this formalism can also provide an insight for designing hot
carrier photodetectors. The chiral MPA can selectively absorb the
CPL—at 830 nm, the left-handed enantiomer absorbs /C2498% light
under left-handed circularly polarized light (LCP) while only absorbs
/C2420% light. The absorption difference originates from different elec-
tric field enhancement, as shown in Fig. 4(b) . And the over-barrier
electron generation rates for each component of MPA are plotted in
Fig. 4(b) ,g i v e nb yE q . (1). It demonstrates that light absorption has a
pronounced effect on hot carrier generation. The electric field can be
further enhanced by using plasmonic hot spots.64Figure 4(c) demon-
strates a normalized electromagnetic field map of the NP on a mirrorApplied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-4
Published under license by AIP PublishingE/E0
E0E/E0⎪E⎜(V/m)
LCP
LCP RCPLCP
RCP
TiO2
AuAu
Ag
Enormal
+
+–
–Auair
alumina1.0
0.80.60.40.2
0.0
012700 800 900 1000
5×10
17
4×1017
3×1017
2×1017
1×1017
3×1011
2×1011
1×1011
00Total
Wall
Base
BackplaneAg
Au
Cu
AlPtZrNTiN1100180
14
1210
8
62.5
21.5
1
0.54
2
0160
14012010080604020
1020 nmλ0 = 1210 nm
HCI 0.1 M, hνMolecule
Absorption
Wavelength (nm)
700 400 600 800 1000 1200 1400 800 900 1000 1100
Wavelength (nm)
Hot electrons
reduction1
3
S
SO
OOOO
SOO O
OCN
NHN
10
10 10OH
4NST 1
NST 2NST 3NST 4NST 5NST 6NST 7
NST 8
30nm NS2Wavelength (nm)
400 500 700 600 800 900
Wavelength (nm)t1
t2
t3z
y
x
6@830 nm
0
310
RateHigh-energy (1/s)Ratehigh-energy (×1013 s–1)
Ratehot electron (s–1)(a)
(b)
(e)
(f)I II III IV
E E E0
1 min 0 2 4 6 8 2 min 1 min 0 5 10 15 0 1 2 3 +20 4 1 min 0 1 2 3 4220
0220
0100
010
E(g)(c)
(d)
N
N
Cl–+ NH Cl–
HN HO
SO
10HNSNC
+FIG. 4. Hot carrier generation boosted by
light absorption and a strong electric field.(a) Schematic of chiral MPA (left) andabsorption spectra (right) under the illumi-
nation of RCP and LCP light. (b)
Simulated electric field distribution (left) ofchiral MPA for both LCP and RCP illumi-nation, at the wavelength of 830 nm, and
over-barrier electron generation rates
(right) for different components of MPAunder the illumination of LCP light.Reproduced with permission from W.
Wang et al. , ACS Photonics 6(12),
3241–3252 (2019). Copyright 2019American Chemical Society.
15(c) A theo-
retical normal electric field of the NP on
the mirror system. (d) Over-barrier elec-
tron generation rate for various NP on themirror system. Reproduced with permis-sion from T. Liu et al. , Faraday Discuss.
214, 199–213 (2019). Copyright 2019
Royal Society of Chemistry. Copyright2016 Wiley-VCH.
64(e) Scheme of hot
electron-assisted local surface chemistry
modification and Au NPs tracking process.
(f) Hot electron conversion map of Auantenna for under different irradiationtimes and polarization. Reproduced with
permission from E. Cort /C19es et al. , Nat.
Commun. 8(1), 14880 (2017). Copyright
2017 Springer Nature.
67(g) Color maps
(top) of the electric field normal to the sur-
face for the nanostar and the sphere.
Spectra (bottom) for the rates of genera-tion of over-barrier electrons for the nano-star with different spikes. Reproduced with
permission from X.-T. Kong et al. , Adv.
Opt. Mater. 5(15), 1600594 (2017).
Copyright 2016 Wiley-VCH.
68Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-5
Published under license by AIP Publishingsystem, with a typical pattern associated with the gap plasmon excita-
tion. The giant electric field enhancement leads to efficient hot electron
generation, as shown in Fig. 4(d) .P a r k et al. used perovskite deposi-
tion on a plasmonic nanodiode to improve hot electron generation
due to the absorption enhancement.65Also, amplified hot electron
generation of NP dimers with plasmonic hot spots was predicted and
experimentally demonstrated.66,67Emiliano et al. used localization of
electromagnetic fields in the bowtie and rod dimer to underpin the
fundamentals of hot electron generation.67Au NP was used as a track-
ing approach to monitoring local surface chemistry, which is related to
hot electrons, as shown in Fig. 4(e) . Hot carrier transport and spatial
distributions are mapped using hot electron conversion in Ag dimer
(bowtie and rod) for different illumination times and polarization, as
shown in Fig. 4(f) . In panels I and II, one can see Au NPs located at
the corner of the antenna (1 min illumination) while Au NPs locate at
central tips (2 min illumination), indicating that reactivity is the high-
est at sharp tips and lowest on flat planar sections of the structure. The
geometry and polarization-dependent reactivities are shown in panels
III and IV. Under parallel illumination, Au particles are always found
on top of the structure, without preference for the surrounding
regions, while under perpendicular illumination, no particles can be
detected around the antenna.
Size, shape, and material always matter to the hot electron gener-
ation. Small NPs are favored as efficient hot electron generators
because carrier distribution extends to larger energies and occupies the
whole region E F<e<EFþ/C22hx.63The field enhancement depends on
the shape of the metallic nanostructures. A nanocube is more efficient
for hot electron generation than a nanosphere and a slab.69As the hot
electron generation is dependent upon the electric field, complex
structures can generate hot electrons more efficiently due to larger fieldenhancement. Multispiked nanostars have strong electromagnetic
fields confined at their sharp tips that can generate large numbers of
hot electrons when compared with rods and spheres, as shown in Fig.
4(g).68Moreover, the hot electron generation is related to materials
selection, as shown in Fig. 4(d) . The conventional plasmonic materials,
such as Au, Ag, and Cu, show narrow and robust plasmon resonances
while the other plasmonic materials, such as Pt, TiN, and ZrN, dem-
onstrate weaker and broader resonances. Alloys outperform their con-
stituent metals in regard to the generation and lifetime of hot carriers.
Stofela et al. showed a 20-fold increase in the number of hot carriers
compared to pure Au due to the presence of hybridized d-band of
alloying Au with Pd near E F.70
The efficient hot electron generation that can contribute to the
photocurrent not only depends on the intrinsic properties of NPs but
also depends on external factors, such as interface condition, applied
bias, excitation power, incident angle, etc. Their effect on hot electron
generation at the single-particle level is promising for understanding
efficient hot electron generation. Zhu et al. mapped hot electron
response of individual gold NP on a TiO 2photoanode, as shown in
Fig. 5(a) .71The factors influencing hot electron generation are studied,
as shown in Figs. 5(b)–5(e) . One can see the dependence of photocur-
rent on structure interface, applied bias, excitation power, and incident
angle. The external factors, including excitation power, incident angle,
the structure interface, and applied bias, can affect hot electron genera-
tion and injection. The excitation power and incident angle are rele-
vant to hot electron generation, while the structure interface and
applied bias are relevant to hot electron injection. The increase of exci-
tation power provides more photons absorbed by gold NP to generate
hot electrons. Although the photocurrent is independent of incident
angle in this system, as shown in Fig. 5(e) , SPs are sensitive to the
CEglass
electrolyte TiO 2
ITO
glass WEAuNP0.2
0.0
–0.2
123
Particle #
Power (mW) Angle (degree)Applied bias (V)(a) (b) (c)
(f)(d) (e)Photocurrent (nA)1.0
0.5Photocurrent (nA)
1.0
0.5
0.0
04 8 1 2 1 6θPhotocurrent (nA)7
6
5
4
0Strong coupling between LSPs and SPPs
Radiative decayCoherent
energy exchange
2
SPPsLSPs
––– + + + ––– + + +Nonradia-
tive decay
34 5 6 7 83×10
2×10Photocurrent (nA)4 –0.02 0 0.02 0.04Objective
–131
4hω
FIG. 5. Hot carrier generation affected by external factors. (a) Schematic of the photoelectrochemical cell for detecting hot electron response from indiv idual gold NP . The pho-
toelectrochemical cell is designed for optical microscopic setup, including the gold NP-deposited TiO 2/ITO substrate as the working electrode and a Pt wire as the counter elec-
trode. Influence of various factors for hot electron response from individual gold NP, including (b) interface structure, (c) applied bias, (d) excit ation power, and (e) incident
angle. Here, the 532-nm pulsed laser was used in this system. The photocurrent is very sensitive to the interface structure, applied bias, and excitat ion power, and the photo-
current is independent of incident angle in this system. Reproduced with permission from H. Zhu et al. , Nano Lett. 20(4), 2423–2431 (2020). Copyright 2020 American
Chemical Society.71(f) The physical process of the strong coupling between LSPs and SPPs, leading to reabsorption of radiative energy of LSP. The process of “energy recy-
cle bin” is composed of capturing, storing, and delivering radiative energies of the LSPs. Reproduced with permission from H. Shan et al. , Light Sci. Appl. 8(1), 9 (2019).
Copyright 2019 Springer Nature.73Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-6
Published under license by AIP Publishingincident angle, affecting hot electron generation via the light absorp-
tion, especially for SPPs that only can be excited by a particular inci-
dent angle.72When Au NP was half-embedded into TiO 2film, as
shown in Fig. 5(b) , a larger and better contact increases the efficiency
of hot electron injection, whereas a negative response was measuredfor half-embedded Au NP in the TiO
2fi l md u et ot h eb l o c k a g eo fh o t
electron regeneration. The applied bias can tune the height of the
Schottky barrier and affect hot electron injection, and the photocur-
rent increases almost double with the applied bias increased from 0 to
0.05 V, as illustrated in Fig. 5(c) . A strong coupling region between
LSPs and SPPs has a synergetic influence on hot carrier generation. In
t h es t r o n gc o u p l i n gr e g i o n ,S h a n et al. use a nonradiative feature of
SPP, severing as an “energy recycle bin,” to reuse the radiative energy
of LSP to enhance hot carrier generation, as shown in Fig. 5(f) .73The
process of the strong coupling between LSPs and SPPs can be divided
into four steps. (1) The photons from the radiative decay of LSPs are
resorbed into SPPs by coherent energy exchange. (2) Afterward, both
intrinsic energies of the SPPs and reabsorbed energies from the radia-
tive decay of LSPs transfer to LSPs via coherent energy exchange. And
then, the LSPs undergo the radiative (3) and nonradiative (4) decays
once again to form an “energy recycle bin.” In this process, LSPs andSPPs can cooperate well to reuse the unavailable energies for hot car-
rier generation in the strong coupling region.
The energy distribution of plasmonic hot carriers is generally
narrow-band and dependent on size, material, shape, and adjacent
material, which has an effect on the carrier lifetime. A recent reviewon this topic can be found in Ref. 74. The hot carrier exhibits a nar-
rower energy distribution than the Drude electrons in bulk metals,
and the plasmonic nanostructure would generate more effective hot
carriers with higher average energy compared to those in bulk met-
als.
58,63Here, we would mention that the generated hot holes and elec-
trons are not uniformly distributed in the energy spectrum.60,75
Although, absorption of a photon generates only one hot electron-holepair, i.e., the number of generated hot electrons and holes are the
same; the energy of hot holes with energy /C21u
SBhave more propor-
tions than that of the hot electrons because holes left after photoexcita-
tion occupy preferentially high energy levels relative to E f.60
B. Preventing losses in hot carrier transport
It is worth noting that hot carriers generated in metals need to
transport to the metal/semiconductor interface and transport in the
semiconductor before being collected. Therefore, the possibility of hot
carriers reaching the interface is a significant factor for the responsivityassessment. The transport primarily includes two processes, hot carrier
transport to interface and cross the barrier, as shown in Figs. 3(b) and
3(c), and transport in the semiconductor. The transport loss is con-
nected to the transport distance and the MFP of metals. Since the
MFP is energy dependent, hot carriers in gold exhibit different MFP,
as shown in Fig. 6(a) , which illustrates that hot holes have a larger
MFP than that of hot electrons with E
c¼E/C0Ef/H113511.2 eV.60This dif-
ference mainly depends on the distinct energy losses for electrons or
holes in gold. The probability of hot carriers arriving at the metal/semiconductor interface without losing energy can be written as
76,77
PTrans¼1
2pðh2
h1exp/C0dðrÞ
lMFPcoshðÞ/C20/C21
dh; (2)where d(r) is the shortest distance from the position of hot carrier gen-
eration to the M/S interface, lMFPis the MFP of hot carriers, and his
the moving angle between the hot carrier diffusion direction and thenormal to the M/S interface, while h
1andh2denote the accepting
angle; that is to say, only hot carriers within the angles have the proba-
bility of reaching the M/S interface. Assuming that the initial hot car-rier momentum distribution is isotropic, the hot carrier diffusion
within the metals is also isotropic. Hot carriers will be dissipated via
thermalization in a short time; only hot carriers reaching the M/Sinterface before thermalization can be collected by an adjacent semi-
conductor to achieve responsivity. The probability of hot carriers is
given by Eq. (2), indicating that a larger MFP will lead to a larger
transport probability when the shortest distance is consistent.
Compared with hot electrons, hot holes generally have a larger MFP at
NIR wavelength, which means a larger transport probability for hotcarrier collection. Following this equation, only half of the hot carriers
will diffuse toward the M/S interface, and the probability is 0.5 for hot
carriers generated at the interface, which is not affected by the MFP.As for the photogenerated hot carriers away from the interface, they
would be thermalized via electron-electron and electron-phonon cou-
pling.
59,78–80In terms of hot electrons with high energies, the impor-
tant mechanism by which they reduce energy is electron-electron
scattering occurring in a very rapid timescale (10 /C24100 fs).16However,
electron-phonon scattering is related to the excited electrons withenergies closer to the Fermi level. The electrons clearly lose energy to
the phonon gas, heating the lattice in around 100 fs to 1 ps.
80To fur-
ther utilize hot carriers and reduce their energy loss, some strategieshave been adopted, such as double Schottky junctions and ultrathin-
film.
59,79,80Also, the hot carriers away from the interface can be
injected into semiconductors through multiple electron reflection. Hotcarriers will be reflected multiple times at the metal/semiconductor
interface or the metal/dielectric interface.
59,80In this condition, more
energetic hot carriers can be injected into the semiconductor, achiev-ing photoelectric conversion. Figures 6(b) and6(c)illustrate the reflec-
tion of hot carriers with a thin-film single-barrier Schottky
structure.
59,81The emission and reflection probabilities of hot carriers
from the M/S interface represent Pkand 1/C0Pk,r e s p e c t i v e l y ,w h i l et h e
emission and reflection probabilities from the metal/dielectric are 0
and 1, respectively. The energy of hot carriers will gradually reduce
after multiple electron reflection occurring in the metal film. The max-
imum number of hot carriers round trip can be written as
nMax¼lMFP
2t/C22hx
DE/C18/C19
; (3)
where tis the metal film thickness, and DEand /C22hxrepresent the
height of the Schottky barrier and the photon energy, respectively. Thetotal emission probability can be described as
59
PtE0ðÞ ¼ P0þ1/C0P0ðÞ P1þ1/C0P0ðÞ 1/C0P1ðÞ P2þ/C1/C1/C1þ PnYn/C01
k¼01/C0PkðÞ ;
(4)
where Pkis the emission probability of hot carrier with excess energy
during multiple electron reflections, which can be written as59,79
Pk¼1
21/C0ffiffiffiffiffiffi
DE
Eks0
@1
A; (5)Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-7
Published under license by AIP Publishingwhere Ekis the excess energy of a carrier ( Ek>DE), described as
Ek¼E0e/C02kt
lMFP: (6)
E0is the initial energy of hot carriers. Following this equation, the
emission is related to the height of the Schottky barrier. It can beattributed to hot carrier injection, which can only be achieved if the
energy of hot carriers is larger than the height of the Schottky barrier.
InFig. 6(d) , hot carriers generated in an ultrathin film can be injected
into the bottom or top semiconductor through multiple reflections at
the M/S interface. The maximum number of hot carriers round trip is
given byn
Max¼lMFP
t/C22hx
DE/C18/C19
: (7)
And the emission probability is twice that of a thin-film single-barrier
Schottky structure, given by
Pk¼1/C0ffiffiffiffiffiffi
DE
Eks0
@1
A: (8)
Also, SPs can be applied for the modulation of initial electron-
momentum distribution controlling hot carrier transport and
1 1–PK
prPK 0
025
0
0
01
1 (1–P0)(1–P1)1–P0
(1–P0)P1
(1–P0)(1–P1)(1–P2) (1–P0)(1–P1)P2
(1–P0)(1–P1)(1–P2)…(1–Pn–1)Pn…
…P0
1–P0
(1–P0)P1
(1–P0)(1–P1) )(1–P2) (1–Pn–1)Pn(1–P0)(1–P1) )(1–P2)(1–P0)(1–P1) P2(1–P0)(1–P1)P0150
–2 –1
E – Ef (eV)lMFP (nm)
012dielectrics
dielectricsPhenomenological models
Metal/semiconductor junction
metal
metal
metal metalsemiconductor
semiconductor(a) (b)
(c)
(d)Hot electron Hot hole
hωFIG. 6. Processes of hot carrier transport
in metals before injection. (a) Energy-
dependent hot carrier MFP calculated bythe first-principle. It demonstrates hotholes have a larger MFP than that of hot
electrons with E
c¼E/C0Ef/H113511.2 eV.
Reproduced with permission from Q. Sunet al. , ACS Omega 4(3), 6020–6027
(2019). Copyright 2019 American
Chemical Society.60(b) Schematic of the
phenomenological models of hot carriertransport in a single-barrier device. Thehot carrier emission and reflection proba-
bility at the M/S interface denote P
kand
1/C0Pk, respectively. (c) Transport proba-
bility of hot carrier as a function of carrierrefection number in the single-barrier junc-
tion (c) and double barrier junction (d). As
the number of round trips increases, theemission probability decreases gradually.And the maximum number of hot carrier
round trip is determined by the Schottky
barrier, the photon energy, and the MFP.Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-8
Published under license by AIP Publishingphotoemission.13The high electric field generated by SP can effectively
manipulate hot carrier distribution and the diffusion angle on trans-
port and extraction process.68
Due to the high carrier mobility of graphene, it enables ballistic
devices for ultrafast applications, meaning that it is possible to achieve
ballistic hot electron transport in graphene. Tse et al. theoretically
investigated the inelastic scattering rate and the carrier MFP for hot
electrons in graphene, taking into account electron-phonon andelectron-electron interactions.
82The calculated inelastic scattering rate
and the corresponding inelastic MPF are shown in Figs. 7(a) and7(b).
The inelastic scattering time sis/C2410/C02/C010/C01ps, and the mean free
path lis 10–102nm when electron density nis of 1012–1013cm/C02.T h e
MFP a larger transport probability, and most of the energetic electronscan finally escape from metal and then be collected. Moreover, the
injection of hot electrons into two-dimensional (2D) material has
attracted attention due to their unique optoelectronic properties. For
example, utilizing various traps at MoS
2interfaces, Wang et al. dem-
onstrated a photoresponsivity of 5.2 A/W at 1070 nm due to trap-
induced photoamplification induced by MoS 2.40Tanzid et al. com-
bined plasmonic hot carrier generation with free carrier absorption(FCA), achieving >1 A/W responsivities.
83FCA leads to a change in
Si carrier mobility, as shown in Fig. 7(c) . Carriers with a smaller effec-
tive mass can be generated by the transition from a heavy hole level to
a light hole and split-off hole level—the effective mass of the heavy
hole is 0.49 m 0while that of the light hole and split-off holes are 0.16
m0and 0.29 m 0, respectively. m 0denotes the invariant mass of theelectron about 9.1 /C210/C031kg. As a result, the device has higher carrier
mobility by changing the hole effective mass. The schematic of the
devices is shown in Fig. 7(d) , and the measured responsivities with
applied external bias across the device are shown in Fig. 7(e) .A t
1375 nm, the devices have more than 1 A/W responsivity, and an
equivalent noise power of 8.05 pW/ffiffiffiffiffiffi
Hzp
is observed.
As discussed above, hot holes generally have a larger MFP at NIR
wavelength and a lower barrier height that can be used for long-wavelength detection. A new concept based on a hot-cold hole energy
transfer mechanism was proposed for ultra-long wavelength hot car-
rier photodetection.
75The structure consists of three p-type regions,
an injector, absorber, and collector, as shown in Fig. 8(a) ,a n dt h ec o r -
responding band alignment is shown in Figs. 8(b) and8(c), for equilib-
rium and negative bias, respectively. The photoresponse is shown in
Fig. 8(d) . The detection bandwidth limit can be extended to 55 lm. It
can be explained by energy transfer from injected hot holes to the cold
holes in the absorber. Under photo-excitation, the injector can be seen
as a hot-hole reservoir that continuously provides hot holes for the
absorber, and the energy transfer occurs through a single hole–hole
scattering, leading to a redistribution of energies among the hot andcold holes.
75,84Therefore, the part of holes with the excess energies
(“hot” holes) can be excited by absorbing longer wavelength light and
cross over the barrier, thereby generating photocurrent, while the pho-
ton energy is much lower than the barrier height ( /C240.32 eV). Besides,
an optical excitation source is necessary for hot hole injection to tunethe energies of cold holes in this system. To verify the longer-
E (eV)Heavy
holes103 60
5
4
3
2
1
0
–400 –300 –200 –100 0 100 200 300 4000.06
0.04
0.02
0.0001234550
40
30
20
10
00 0.2 0.4 0.6 0.810(a) (b)
105
52
2 1 × 1012 cm–2n = 1 × 1012cm–2rs = 0.4rs = 0.8
1
E (eV)0 0.2
NIR Light
Metallic
GratingPlasmon-
induced
Hot Carriers
Responsivity (A/W)
Applied Bias (mV)PlasmonSurfacewhE
g0.4 0.6 0.8 1102
101Light
holes Split-off
bandX LLWave Vector
Free Carrier Absorption
Free Carrier AbsorptionNet Current
FlowP-type
Silicon
l (nm)τ–1(ps–1)
Γ(k)/EF
ξ(k)/EF
2μm(a)
(d) (e)(b) (c)
FIG. 7. Effects of the carrier mobility on device performances. Calculated inelastic scattering rates (a) and the corresponding inelastic MPF (b) as a funct ion of energy for differ-
ent carrier densities. The inset in (b) illustrates the damping rate divided by Fermi energy as a function of energy scaled by Fermi energy. Reproduced with permission from
W.-K. Tse et al. , Appl. Phys. Lett. 93(2), 023128 (2008). Copyright 2008 AIP Publishing LLC.82(c) Energy diagram of silicon showing the direct intraband transition in the
valence band corresponding to the absorption of free carriers. (d) Schematic of a photodetector with photocurrent generation through plasmonic hot carrier by metallic grating
and free carrier absorption in a heavily doped p-type semiconductor. (e) Measure responsivities of metallic grating with applied external bias at th e wavelength of 1375 nm. A
scanning electron microscope (SEM) image of Au grating is shown in the inset. Reproduced with permission from M. Tanzid et al. , ACS Photonics 5(9), 3472–3477 (2018).
Copyright 2018 American Chemical Society.83Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-9
Published under license by AIP Publishingwavelength response up to 55 lm, the author used an escape-cone
model to simulate the response spectra,85and the threshold energy
determines the cutoff wavelength of the responses reducing from 0.32
eV to 0 eV via tuning the hot-cold hole energy transfer. More impor-tantly, the hot carrier can divert their energy to the cold carrier rather
than energy wasting via heating up the lattice, thus improving the
energy efficiency of photodetectors. The lifetime of hot holes can befurther increased by using charge separation at the metal-
semiconductor interface.
Many hot carrier photodetectors have low external quantum effi-
ciencies (EQE) below 1%. One factor that can potentially limit the
EQE is the carrier injection efficiency. The hot carrier injection effi-ciency gis a crucial factor to determine device performances, whichdescribes the number of electrons with sufficient energy to overcome
the barrier,
39
g¼CF/C22hx/C0DEb ðÞ2
/C22hx; (9)
where C Fis the Fowler emission coefficient, DEbis the Schottky barrier
energy. Feng et al. improved the injection efficiency by connecting the
lower part of the top antennas to Si nanowire.87The detector demon-
strated a responsivity and detectivity of 94.5 mA/W and 4.38 /C21011
cm Hz1/2/W at 1.15 lmw i t ha nE Q Eo f /C2412%. Although Si is also
responsive from 0.9 to 1.1 lm, the fitting curves obtained by Eq. (9)
are in excellent agreement with the measured responsivity, proving
E (eV)z (μm)
VBEf
0.00.00.40.81.2
–0.4Infrared radiation
Experiment
AIAs-like
phonon
GaAs
phononSp1007
5.3 K, –0.06 V ModelSemi-insulating GaAs substratep+-GaAs (100 nm) Collector
Absorber
InjectorTi/Pt/AuTi/Pt/Au
p+-GaAs (700 nm)V
p+-GaAs absorberAl0.57Ga0.43As (400 nm)
AlxGa1–xAs (80 nm)
Wavelength ( μm)Photoresponse ( μA W–1)
wavelen gth (nm)injection efficiency
24 1 010
5
015
20 30 40 505500.00.20.40.63TiO2 TiO2
TiO2-Au
Al2O3-AuUV pump Visible pump
2
1
0
05
time (ps)ΔA (mOD)
10 15AuAu30 nm30 nm
substrate
data
Yφ=0.9eV
Yφ=1.2eV
650 700 750 600Injector
Graded
barrier
Without hot-hole effect
With hot-hole effectOOO
OConstant
barrierAbsorberCollector
δEνδEν(a) (b) (c)
(d) (e)
(g)(f)
FIG. 8. Hot carrier photodetection beyond the bandgap limit through hot–cold hole energy transfer mechanism and quantitative analysis of hot carrier injec tion efficiency. (a)
Schematic of hot-cold hole based photodetector formed by the p-type GaAs/Al xGa1–xAs structures. P-type GaAs was utilized as the injector, absorber, and collector;
AlxGa1–xAs was used to form the graded and constant barriers. (b) Calculated equilibrium valence-band alignment. The thick grey and dashed blue lines represe nt the equilib-
rium valence-band alignment with and without the image-force barrier lowering, respectively. (c) Calculated valence-band diagram under negative bias without (top) and with
(bottom) hot hole energy transfer. The barrier offset between the two barriers results in the energies of the hole on the injection side to be higher tha n that on the collection
side. (d) Comparison of photoresponse between experimental measure (red) and escape-cone model fit (dash) at 5.3 K. The longer-wavelength response c an be seen up to
55lm, which is much lower than the barrier height ( /C240.32 eV). Reproduced with permission from Y.-F. Lao et al. , Nat. Photonics 8(5), 412–418 (2014). Copyright 2014
Springer Nature.75(e) Side view of SEM of Al 2O3-Au NP stack on silicon. The circular white objects represent the layers of Au NP. (f) IR transient decay for the bare TiO 2film
under ultraviolet pumping (black) and transient absorption decay for TiO 2-Au (red), Al 2O3-Au (green), and bare TiO 2film (blue) under visible pumping. The IR transient absorp-
tion spectroscopy illustrates the presence of photogenerated free carriers to quantify the injection efficiency of electrons into the semiconducto r. (g) Calculated spectra of elec-
tron injection efficiencies for the barrier height of 0.9 eV (red) and 1.2 eV (blue). Black circulars are the injection efficiency for different pump wav elengths estimated by the IR
transient data. The error bar on the injection efficiency specifies the decay of the signal within the measurement response time of the pump-probe setup . Reproduced with per-
mission from D. C. Ratchford et al. , Nano Lett. 17(10), 6047–6055 (2017). Copyright 2017 American Chemical Society.86Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-10
Published under license by AIP Publishingthat the photocurrent mainly comes from plasmonic hot electron gen-
eration. Moreover, the measured responsivity is about two orders ofmagnitude higher than that of planar and Si nanowire detectors in thesame regime. A recent study showed that efficient carrier injection effi-ciency of 25% /C045% could be achieved in the Au-TiO
2system, as
shown in Fig. 8(e) .86Transient absorption spectroscopy (TAs) was
used to quantify the efficiency of the hot electron transfer in systems
composed of gold particles embedded in TiO 2and Al 2O3film, as
shown in Fig. 8(f) . The electron efficiency can be obtained from the
TAs, as plotted in Fig. 8(g) . The measured efficiency ranges from
/C2445% at 550 nm to /C2425% at 750 nm. Three reasons may be ascribed
to the high injection efficiency. First, NP has dimensions less thanMPF of electron-electron scattering; second, NPs are fully embeddedwithin the semiconductor, and all of them have an opportunity to
inject into the semiconductor; finally, momentum conservation
requirements are relieved by small NPs. Small NPs are not onlyfavored as efficient hot electron generators
63but also preferred for effi-
cient transfer.88Liuet al. observed an ultrafast hot electron transfer
process to CdS ( /C28140 fs).88A st h es i z eo fp l a s m o n i cN P sd e c r e a s e d
from 5.5 61.1 to 1.6 60.5 nm, the quantum efficiency increased
from/C241% to /C2418% due to enhanced hot electron generation and
transfer efficiency.
The conventional plasmon-induced hot electron transfer (PHET)
process is an indirect process—hot electrons generated can be partiallyquenched by electron-phonon and electron-electron scattering before
the injection. This process must compete with the electron-electron
scattering process, partially leading to a low efficiency of /C241%–2%.
74
On the other hand, plasmon-induced interfacial charge transfer transi-
tion (PICTT) is a highly efficient approach for hot electron transfer,which enables directly exciting the interfacial charge transfer transi-tion. Also, recent theory and experiments demonstrated that the time-scale of hot electron transfer is much shorter than that expected withthe indirect transfer mechanism.
74The two processes are compared in
Fig. 9 . The PICTT relieve the requirement of momentum conversion
and avoid competition with hot electron relaxation by directly excitingelectrons to the CB of the semiconductor.
89Wuet al. first proposedthe PICTT method and demonstrated efficient hot electron transfer
with >24% quantum efficiency.89Recently, Cresi et al. demonstrated
/C2416% hot electron injection efficiency via PICTT.90Using graphene-
WS 2heterostructure, Chen et al. proposed a 2 lphotothermionic
emission (2 lPTE) pathway to efficiently extracting quasi-thermalized
hot carriers before electron-hole coalescence sets in, and the measure-
ments showed /C2425 fs electron injection and /C2450% injection efficiency,
as shown in Figs. 10(a) and10(b) .91The excited thermalized electrons
showed one Fermi–Dirac distribution for 1 lphotothermionic emis-
sion (1 lPTE, i.e., hot electron has energy above potential barrier that
can transfer to semiconductor via thermionic emission) while quasi-thermalized electrons and holes showed two Fermi–Dirac distribution,as shown in Figs. 10(c) and10(f).lrepresents the chemical potential;
unlike 1 lPTE with one Fermi–Dirac distribution, 2 lPTE has a two
Fermi–Dirac distribution with separated chemical potentials for elec-
tron and hole, as plotted in Figs. 10(c) and10(f). The electron distribu-
tion above the Dirac point in graphene of the two modes is comparedinFigs. 10(d) and10(g) , indicating that the 2 lPTE model has a signifi-
cantly higher electron temperature and more hot electrons above theSchottky barrier. Therefore, the injection quantum efficiency of2lPTE reaches 50%, as compared in Figs. 10(e) and10(h) .A l s o ,e f fi -
cient hot electron transfer was observed by Shan et al. by using strong
coupling between LSPs and SPPs.
73Direct hot electron transfer from
Au grating and MoS 2was observed in a strong coupling region. The
transfer time is 40 fs with an EQE of 1.65%. A triple-channel hot elec-tron transport process was proposed by Li et al. ,a ss h o w ni n Figs.
11(a) and11(b) . Three channels were opened for hot electron trans-
port, including PHET, plasmon-induced resonant energy transfer, andd-band transitions at below 500 nm from Au film to Cu
2Os h e l l .92
After Landau damping, the relaxation time of the hot carrier is
around 100 fs to 1 ps for the formation of a Fermi–Dirac-like distribu-tion.
32A recent study from Memarzadeh et al. showed that a locally
confined electric field at the surface of the metal could slow the relaxa-
tion time of hot electrons due to modified hot carrier distribution and
electron temperature.93The experiment was investigated between the
hot carrier relaxation time and the characteristics of SPs on gold film
semiconductor
SP SPCB CB
VB VBEF EFsemiconductor metal metal
semiconductor
PHET PICTTmetal semiconductor metalFIG. 9. Comparison of hot carrier transfer
pathways between plasmon-induced hot
electron transfer (PHET), where the plas-
mon in metals decays into hot electron-hole pairs by Landau damping, followedby injecting hot electron into in the con-
duction band of the semiconductor, and
plasmon-induced interfacial charge trans-fer transition (PICTT), in which the plas-mon decays by directly generating an
electron in the conduction band of the
semiconductor and a hole in the metal.Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-11
Published under license by AIP Publishing(a)
(b)(c)
(f)(d)
(g)(e)
(h)
Gr
WS20.0
ee
e–h+
hμh*
μe*
f(ε) f(ε)g(ε) (×1012 cm–2 eV–1)ϕB0.2 0.4Energy (eV)
Photon energy (eV) 2μPTE1μPTE
μ
NphotonNphoton
Nphoton (cm–2)2
1
0
–1
–2
2
1
0
–1
–22.5
2.01.51.0
0.00.52.5
2.01.51.0
0.00.51.6
1.20.80.4
1.6
1.2
0.8
0.40.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 1002468 1 0
10
112 4 68 2 4 68
101250QY (%)
40
302010010
13
101124 6 8 24 6 8
10121013
FIG. 10. Highly efficient hot electron harvesting before electron-hole thermalization using 2 lPTE. (a) Scheme of hot electron transferring from graphene to WS 2. Graphene is
the component of hot electron generation, and WS 2is the electron-accepting component. The ultrafast hot electron injection from graphene to WS 2is about 25 fs. (b) Optical
image of graphene/WS 2heterostructure. The scale bar is 5 lm. A small height difference of approximately 0.4 nm between graphene and WS 2suggests a well-coupled inter-
face. Fermi–Dirac distributions for (c) conventional 1 lPTE and (f) proposed 2 lPTE, and the corresponding electron energy distributions above Dirac point in graphene for (d)
1lPTE and (g) 2 lPTE at three different photon density. Color map of quantum yield as a function of photon energy and density for (e) 1 lPTE and (h) 2 lPTE. Reproduced
with permission from Y. Chen et al. , Sci. Adv. 5(11), eaax9958 (2019). Copyright 2019 American Association for the Advancement of Science.91
(a) (b)
(c) (f) (d)
7601.5
1
0.5
0
1.4121
120.5
119.5
1191201.3
1.2
1.1–1.5×10–3
–1 –0.5
740Au20 nm 2 nm
MoSe2EC
Au hot
carriers 2.5x
carrier
lifetimeEF
EV
720
700
300 400 500 600 700 800(g) (e)
(h)
λ (nm)Pabs= 1/2 ωεI IEI2
IEI2
max / IEoI2
Pabs (mW)IEI2
Air
AuzPrism
θres
λ (nm)
τe-ph(ps)
ΔR/R
Te(K)Au filmVisible light
Au NPSPR excitation
Cu2O shelle–
e–h+
h+
Au film Au NPd-band transition
d-band
transitionValence Band
Cu2OConduction Band
RETDET
SPR
excitatione–
h+Fermi levelVac
730 750 770
FIG. 11. Effective strategies to suppress transport loss. (a) Schematic illustration of hot electron generation in Au@Cu 2O-Au film excited by SPRs and d-band transitions. Hot
electrons with sufficient energy can be directly injected into Cu 2O over the Schottky barrier. (b) Hot electron excitation and transfer in Au@Cu 2O-Au film. Triple-channel hot-
electron injection includes PHET, plasmon-induced resonant energy transfer, and d-band transitions at below 500 nm. Reproduced with permission from H. Li et al. , Nano
Energy 63, 103873 (2019). Copyright 2019 Elsevier Ltd.92(c) Schematic diagram of SPP excitation under the Kretschmann configuration. The Kretschmann configuration is
used to couple light to the propagating surface plasmons. (d) Contour plot of the reflectivity changes with the electron temperature at different inci dent wavelengths. The
change in reflectivity with the electron temperature is computed from the free electron model and transfer matrix methods. (e) Hot electron relaxatio n time and (f) calculated
field enhancement from 730 nm to 770 nm. Here, it illustrates that the electric field enhancement induced by the plasmon coupling can effectively prolon g the hot electron relax-
ation time under fixed absorbed power. Reproduced with permission from S. Memarzadeh et al. , Optica 7(6), 608–612 (2020). Copyright 2020 Optical Society of America.93
(g). Schematic illustration (top) and TEM image (bottom) of MoSe 2sheet decorated with Au NPs. The diameter of Au NPs varies from 5 to 10 nm. (h) Energy diagram formed
at the Au/MoSe 2interface and theoretical hot carrier distribution in gold. The charge-separated state at the Au/MoSe 2interface can increase the lifetime of hot holes about 2.5
times. Reproduced with permission from J. R. Dunklin et al. , ACS Photonics 7(1), 197–202 (2020). Copyright 2020 American Chemical Society.94Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-12
Published under license by AIP Publishingexcited under the Kretschmann configuration, as shown in Fig. 11(c) .
Transient reflectivity ( DR=R0) can be used to extract the electron tem-
perature under the intraband optical pumping. Figure 11(d) shows
changes in reflectivity with electron temperature over different wave-lengths, determined by the transfer matrix method. The enhanced hotcarrier relaxation time in the Au film can be confirmed by normalized
maximum intensity of electric field, as well as light absorption in Au
film, as shown in Figs. 11(e) and11(f). This study provides a solution
for increasing hot carrier relaxation time in photodetection devices.
Another factor in preventing carrier loss is increasing carrier life-
time. Hot electron lifetimes /C2422 times longer in MAPbI
3coupled with
the plasmonic Au were observed due to the slow hot electron cooling
time of the perovskite.65Also, the lifetime of hot holes can be increased
by using charge separation at the metal-semiconductor interface. Forexample, hot holes that transfer from the Au NP to the adjacent transi-tion metal dichalcogenide (TMD) nanosheets generate a charge-
separated state. It leads to a 2.5-fold lifetime increase, as shown in Figs.
11(g) and 11(h) .
94A threefold increase in the carrier lifetime was
observed in Au xPd1–xnoble-transition alloy when compared with that
of its pure counterpart.70Figure 12(a) shows how the alloy affects theshape and position of the d-band, and therefore, the lifetime and dis-
tribution of hot carriers. The Au d-band carriers cannot be excited
with NIR photons, limiting the generation rate of hot carriers; in pure
Pd, the d-band is accessible in the NIR while the carrier lifetime is
shorter due to rapid thermalization.70The alloy can combine their
advantages— d-bands hybridize and shift, changing the electronic den-
sity of states and the hot carrier lifetime under NIR excitation.
The hot carrier distribution is essential for understanding the
physical mechanism not only during the generation process but alsoduring transport. In other words, if the injected electrons in the semi-conductor are still hot due to a nonthermal steady-state distribution, it
is promising for photodetection. Cushing et al. studied the nonthermal
distribution of hot electrons in semiconductors injected from the plas-monic star, rod, and sphere.
95The predicted hot carrier distribution
injected in TiO 2for each nanostructure is shown in Figs. 12(b)–12(d) .
The plasmon decays into a carrier distribution in each nanostructure
with energy up to plasmon energy. Below 0.5 eV, the thermal and non-
thermal distributions overlap in population, while above 0.5 eV, thepopulation exists up to the plasmon energy at 2.25 (sphere), 1.90(rod), and 1.55 (star) eV, respectively, as compared in Fig. 12(e) .
(a)
1.6
6s5p
X1
0
–1
–2
2.5
2.0
1.5
1.0
0.5
0.0
0.00 0.05 0.10
Occupation (0 to 1)0.0 0.4 0.3 0.2 0.1 0.7 0.6 0.5
Occupation (0 to 1)0.0 0.4 0.3 0.2 0.1
Occupation (0 to 1)Energy (eV)2.5
2.0
1.5
1.0
0.5
0.0Energy (eV)2.5
2.0
1.5
1.0
0.5
0.0Energy (eV)Energy (eV)
5d4dEF
Energy (eV)
Interband not
excitedTunable NIR
interband viaband hybridizationNIR interband
but rapidthermalization1.2–0.8
Electronic Density of States
Plasmon Frequency
Non-thermal
DistributionSphere, Non-thermal
Distribution
Thermal
DistributionNon-thermal
DistributionRod, Non-thermal
Distribution
Rod
Thermal
DistributionNon-thermal
Distribution Star
Thermal
DistributionStar, Non-thermal
Distribution
SpherePlasmon Frequency
Plasmon Frequency0.5 ps
(b) (c) (d)1.5
0.52.5
2.0
1.0
0.0
0.0 0.2 0.4 0.6
Occupation (0 to 1)“Box” Extimate of
Carrier Distribution(e)
λ = 1550 nmAuAu AuxPd1–x Pdτe–ph
FIG. 12. Predicted hot carrier distribution model in different metals and shape. (a) Schematic of hot carrier distribution, lifetime, and the d-band position with the alloying of Au
modified Pd. The interband transition of hot carriers in Au cannot be excited upon illumination at the NIR regime. In pure Pd, although the interband tra nsition can be realized
in the NIR regime, the hot carrier lifetime is short due to the rapid thermalization. The alloying Au with Pd affects both the shape and position of d-band, influencing the elec-
tronic density of states and the hot carrier lifetime under NIR excitation. Reproduced with permission from S. K. F. Stofela et al. , Adv. Mater. 32(23), 1906478 (2020). Copyright
2020 Wiley-VCH.70(b)–(d) Hot carrier distribution in metals and thermal distribution for different Au structures. The hot carrier distribution prediction for each s hape is illustrated
by the dark dashed area in (b)–(d). Below 0.5 eV, the thermal and nonthermal distributions begin to overlap. (e) Injected hot electron distribution fo r various shapes in the semi-
conductors. A “box” distribution ranging from the Fermi level to the plasmon frequency is used for the state-filling contribution. Reproduced with pe rmission from S. K. Cushing
et al. , ACS Nano 12(7), 7117–7126 (2018). Copyright 2018 American Chemical Society.95Copyright 2018 American Chemical Society.Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-13
Published under license by AIP PublishingC. Efficient hot carrier extraction
The initial energy and momentum distribution are important
factors that influence the efficiency of hot carrier extraction. Flower’s
model is most widely used in the description of hot carrier extraction
and is a semiclassical model of internal photoemission of hot carriers
from metals.25,57,89The mode assumes that hot carriers are distributed
isotropically within the metal, and only the part located in the momen-tum cone can realize hot carrier injection due to the electronic
momentum conservation, as shown in Fig. 13(a) . The theoretical cal-
culation showed that the initial momentum distribution of the carriers
depends on both the crystallographic orientation of the metal and
plasmon polarization.
62Meanwhile, the momentum distribution mod-
ified by SP can further enhance the quantum yields through a geomet-
ric effect.66,69,96On the contrary, hot carriers with the momentum
parallel to the interface can hardly be collected by adjacent semicon-ductor over the Schottky barrier. To further increase the hot carrier
extraction efficiency, a 3D Schottky barrier or Omni–Schottky barrier
can be employed to increase the emission cone for hot electron injec-
tion, as illustrated in Fig. 13(b) .
97,98Recently, it is found that the M/S
interface roughening can achieve hot carrier injection with the parallelmomentum through reliving the momentum conversion slightly.
80
The internal photoemission based on the Fowler model is writtenas
13,89
g¼/C22hx/C0DEb ðÞ2=4EF/C22hx; (10)
where EFrepresents the metal Fermi energy. Different from the
electron-hole pair generation in semiconductors, the internal photo-
emission is related to the photon energy, and the higher photon energycan cause an increasing quantum yield. The responsivity calculated by
Flower’s model can be given by13,81
RðxÞ¼q/C2AðxÞ/C2g=/C22hx; (11)
where AðxÞis the light absorption in metals, and qis the elementary
charge. For plasmon-enhanced hot carrier photodetectors, their inter-
nal photoemission has the same functional form as described in Eq.
(10). And the internal photoemission can be re-written as Eq. (9).
Besides that, the momentum conversion can be relaxed if electrons are
only scattered at the M/S interface. Under this condition, the internal
photoemission can be calculated by g¼/C22hx/C0DEb ðÞ =2/C22hx.89
Although Flower’s model has been widely used in experimental
research, the theory is difficult to predict experimental results in some
novel hot carrier photodetectors due to the assumption of the isotropic
momentum distribution. To better understand the performance of hot
carrier photodetectors, a phenomenological internal quantum effi-
ciency (IQE) model has been proposed.81To calculate hot carrier
energy distribution, the electron density of states is approximated by afree electron gas model with a higher temperature compared to the
environment temperature exhibiting a broad energy distribution.
While hot carriers with sufficient energy above the Schottky barrier
can transport to the semiconductor, a broad energy distribution
becomes an obstacle for realizing efficient photoemission, which
results in many carriers with energies below the barrier. The hot car-
rier energy distribution can be calculated by the electron density of
states (EDOS) of metal, written as
76
DEðÞ¼qE/C0/C22hx ðÞ fE/C0/C22hx ðÞ qEðÞ 1/C0fEðÞ/C2/C3
; (12)
where Eis the energy of the excited electrons, qE/C0/C22hx ðÞ /qEðÞis the
parabolic electron density of states of the initial/final energy level, andfE/C0/C22hx ðÞ =fEðÞis the Fermi distribution for the initial/final energy
level. And the Fermi distribution can be written as
81,99
fEðÞ¼1
1þeE/C0EfðÞ =kT ðÞ: (13)
When we combine the hot carrier generation from the light absorp-
tion, the transport probability, and the emission probability, the pho-
tocurrent density can be described following the equation76,77,99
I¼qð/C22hx
DEDEðÞ/C2G/C2PTrans/C2PkdE
ð/C22hx
0DEðÞdE: (14)
In contrast to Flower’s model, the phenomenological IQE model is
more beneficial to estimate the impact of electric field enhancement and
hot carrier energy distribution on hot carrier extraction. Besides, the
geometric effect, including shape and size, also plays a vital role in hot
carrier extraction.100In the phenomenological IQE model, hot carrier
generation is derived from a surface-driven phenomenon, which means
that large momentum changes in the electron are required for the exci-
tation of electrons. Even though the phenomenon hardly exists in bulk
metals, the plasmonic enhanced light-matter interaction or carriers scat-tering at the interface leads to unconserved linear momentum, increas-
ing the high-energy hot carrier generation rate.
It was found that the linear momentum is not conserved, and the
normal electric field can generate more over-barrier electrons with a
(a)
(b)Ekx
kxkz
kzky
kyk
SiO2
SiAu
D
W
FIG. 13. Improving the efficiency of hot carrier extraction. Comparison of hot elec-
tron emission cone distributions between (a) 1D Schottky barrier interface and (b)3D Schottky barrier interface. 1D Schottky barrier interface only supports hot elec-tron transport through the bottom interface, while the embedded devices with 3D
Schottky barrier interface increase the probability that the momentum space of hot
electrons lies in the emission cone. Reproduced with permission from M. W. Knightet al. , Nano Lett. 13(4), 1687–1692 (2013). Copyright 2013 American Chemical
Society.
97Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-14
Published under license by AIP Publishinghigh energy state. Therefore, based on Flower’s model, some studies
also use the electric field normal to the M/S interface to modify the
total injection current calculation, given by57
I¼C/C1/C22hx/C0DE ðÞ2/C1ð
SjEnormalj2dS; (15)
where Cis a material constant depending on the barrier height and
the Fermi energy of metals, which is different from the Fowler emis-sion coefficient described in Eq. (9),a n d /C22hx/C0DE ðÞ
2stands for the
Flower’s law. As both hot electrons and hot holes are originated from
the oscillating electrons in the Drude model, the excitations of hot
holes can also be described by the quantum effect due to surface scat-tering. In the Drude model, hot carriers are located near the Fermilevel of metals. Hot (energetic) electrons occupy the energy intervalranging from E
ftoEfþ/C22hx, and hot (energetic) holes are located in
the interval from Ef/C0/C22hxtoEf. In contrast to thermalized carriers
from the photothermal effect, nonthermalized hot carriers have a
higher effective temperature from the photon temperature, written asT
Photon¼/C22hx=kB;101where kBis the Boltzmann constant. The energy of
thermalized carriers is lower than that of hot carriers, which is notbeneficial to achieve carrier extraction for high barriers.
101,102On the
contrary, thermalized carriers can contribute to the photoresponse
through the photothermal effect for low barriers. Moreover, the local-ized heating from hot carrier relaxation in plasmonic nanostructuresgives rise to the energy distribution change at the Schottky barrier, andthus leads to a change of the saturation current with the reverse bias.
81
The relationship of the temperature in metals and absorbed lightpower Q(r,t) can be written as
5
qrðÞCrðÞ@DTr;tðÞ
@t¼r/C1 krðÞrDTr;tðÞ/C0/C1þQr;tðÞ ; (16)
where qðrÞis the mass density, CrðÞis the thermal conductivity, and
krðÞis the specific heat capacity. In this equation, DTr;tðÞ is the local
increasing temperature given by DTr;tðÞ¼Tr;tðÞ/C0Texe.W h e r e
Texeis the ambient air temperature, tandrrepresent time and the
coordinate, respectively. And the photothermal photodetectors are
determined by the Seebeck effect following DU¼/C0S/C2DT.W h e r e
DUis the electric potential difference, and Sis the Seebeck coefficient,
following the Motto equation102
S¼/C0p2kBT
3edlnr
dE/C18/C19 /C12/C12/C12/C12
E¼Ef; (17)
where ris the materials’ electrical conductivity, Tis the absolute tem-
perature, and eis the elementary charge. It is worth noting that both
types of excited carriers can be employed in photon-to-electricity con-
version. The response times of thermal devices are usually measuredin the millisecond scale, while the speed of hot carrier photodetectioncan be very fast due to hot carrier relaxation time in the femtosecondscale.
101The performance of thermal detectors is primarily determined
by photothermal effect, and heat generation used for photodetection
via pyroelectric and thermoelectric effects is ascribed to thermalizedcarriers rather than hot carriers.
103–105For instance, heat generation
from the subwavelength absorber can diffuse into the pyroelectric film,generating the responsivity up to 0.18 V/W.
103Mauser et al. proposed
resonant thermoelectric nanophotonic structures that can produce
local heating in thermoelectric materials with the responsivity up to38V/W.104Although, in most cases, thermalized carriers are also con-
sidered as hot carriers, and the name of “warm” carriers are more suit-
able to distinguish nonthermalized hot carriers mainly discussed in
our article due to their lower energy and longer lifetimes.105,106In this
article, we primarily focus on nonthermalized hot carrier photodetec-
tion enhanced by SP.
Finally, the plasmonic structures must be electrically connected
with a semiconductor by using metallic contacts with low resistance.
In the case of hot electron collection, an electron can be directly
extracted in the conduction band of the semiconductor, while a hole
can be collected in metals. Many studies used indium tin oxide (ITO)
film as the conductive film in electrodes. The thickness of ITO influen-
ces the conductivity, which can improve the quantum efficiency of a
hot electron device. However, thick ITO seriously affects the light
transmittance, reducing hot carrier generation in plasmonic nano-
structures. For example, a 50-nm ITO film with excellent enough con-ductivity has only 92.5% light transmittance. Graphene can be used as
a transparent conductive electrode due to its atomic layer thickness
and unique mechanical, optical, and electronic properties. Hu et al.
showed that graphene electrodes could significantly enhance respon-
sivity of hot electron photodetectors when compared with ITO.
107
Asymmetric nanogap electrodes were proposed to improve the collec-tion efficiency by decreasing the microscale transmission length of the
hot electron to nanometer scale.
108
Although there are a series of theoretical advances of hot carrier
physics that have been made in recent years, much more fundamental
studies are required for optimizing hot carrier dynamics, including
generation, transport, and extraction process. The extraction efficiency
is affected by generation and transport. For instance, theoretical calcu-
lation showed that the net efficiency of carrier collection in a specific
geometry was dependent on the initial momentum distribution and
the subsequent transport of the carriers to the surface.62The principle
and the strategies of the plasmonic-enhanced hot carrier can be uti-
lized in any system where hot carriers are involved, and we believe
that the development of advanced strategies for plasmonic-induced
hot carrier dynamics is a crucial component for highly efficient hotcarrier photodetection.
Only if the extraction speed of hot carriers is faster than the ther-
malized rate through electron-electron coupling and electron-phonon
scattering can hot carriers be exported by external circuits utilizing for
optoelectronic devices. In this case, inhibition of plasmonic radiation
decay, reducing hot carrier transport loss, and increasing hot carrier
injection is the key to promoting the performance of hot carrier devi-ces.
73Therefore, much effort has been made to overcome these prob-
lems. Some effective approaches for enhancing the performance of hot
carrier detectors are shown in Table I . First, to suppress radiative
decay, which increases hot carrier generation, metamaterial perfect
absorbers (MPAs) have been proposed.15,25Hot spots excited by LSPR
are usually used to enhance light absorption. Plasmonic nanostructure
with enriched hot spots can realize light-trapping enhancement by
light diffusion and multiple-scattering.81Additionally, the electromag-
netic wave can be effectively confined between a distributed Bragg
reflector (DBR) and a metal film by Tamm plasmon resonance,76and
strong coupling between LSPs and SPPs is also an effective approach
to suppress radiative decay.73The radiative energy of LSPs can be
reused by SPPs via an “energy recycle bin” to generate hot carriers.
Second, the energies of hot carriers will be dissipated continuouslyApplied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-15
Published under license by AIP Publishingduring hot carrier transport. Hot carrier transport loss is usually
related to the lifetime of hot carriers in metals. Relaxation lifetimedenotes the inverse of scattering rate from phonon-electron andelectron-electron interaction. Recently, a perovskite-modified plas-
monic structure was used to prolong the lifetime of hot electrons, 22
times higher than that in gold.
65Besides that, when the relaxation is
constant, using an ultrathin film to reduce hot carrier transport dis-tance is also an effective approach to improve the device performance.An ultrathin nanostructure can also improve the efficiency of hot car-rier collection through multiple hot carrier reflections. Although FCA
in Si detector does not contribute photocurrent, it can increase the car-
rier mobility and reduce the device resistance.
83Moreover, hot-cold
hole energy transfer and 2 lPTE model are also significant methods to
reduce carrier loss.75,91Finally, to achieve photoelectric conversion,
hot carriers reaching the M/S interface need to cross over the barrier,resulting in hot carrier injection. However, mismatching momentum
is an impediment to hot carrier extraction. Adjacent semiconductors
can extract only hot carriers within the cone momentum for photonto electricity conversion. A 3D Schottky junction formed by plasmonicstructure inside the semiconductors can provide more effective hotelectron emissions due to more hot carrier momentum space and anew pathway for hot carrier transfer.
97As for the pyramid nanostruc-
ture, they can not only converge light on the apex to suppress radiative
decay but also relax the momentum mismatching between the semi-conductors and metals.
109,110In contrast to LSPRs, SPPs relax almost
nonradiatively, leading to higher photoelectric conversion efficiency.73
Consequently, plasmonic waveguides based on SPPs have more elec-tric field components normal to the M/S interface, increasing the pos-
sibility of hot carrier extraction.
13,42Different from the conventional
injection process based on PHET, PICTT provides an increased hotcarrier injection efficiency, up to 24%, by engineering the M/S inter-face.
89Indirect optical transitions exist on the rough metal films,
resulting in the local electric field with a strong spatial dependencethat breaks the translational invariance and provides a continuoussource of momentum.
111,112Therefore, the M/S interface roughening
can also relieve the momentum conversion and provide more injected
hot carriers.41Based on these methods, we will cover the advances of
plasmon-enhanced hot carrier photodetectors in Sec. III.
III. PLASMON-ENHANCED HOT CARRIER
PHOTODETECTORS
A. Plasmon-enhanced hot electron photodetector
1. Optical antenna-based hot electron photodetectors
In 2011, Knight et al. proposed a NIR photodetector composed
of metallic nanoantennas and silicon.39The plasmonic nanoantennas
achieved a significant light absorption enhancement and hot electron
injection, resulting in the responsivity below the bandgap of silicon. As
shown in Fig. 14(a) , the plasmonic antennas provide both longitudinal
and transverse plasmon resonances, which is determined by its geo-
metric effect. Additionally, the photodetector exhibits a polarization
dependence of photoresponse. The room temperature operation with-
out a bias voltage is also one of the important properties. Although
only 0.01% of absorbed photons in the optical antenna-based hot elec-
tron photodetectors are converted into photocurrent, a reverse bias of
1 V at the room temperature operation can increase the photocurrent
about 20 times in contrast to that without a bias voltage. The plas-
monic grating is another widely studied nanostructure that can realize
narrow-band hot electron detection, polarization dependence, and
extra bandwidth response. Figure 14(b) illustrates the schematic of
plasmonic grating Schottky photodetector.38Ar e s p o n s i v i t yo f0 . 6
mA/W without bias voltage has been achieved by the grating photode-
tector. An IQE of 0.2% is about 20 times that of plasmonic antennas.
SPs can propagate on both the upper and lower surface of the gating,
and the interslit gap in the grating can achieve destructive or construc-
tive interference of SPs, resulting in narrow-band resonance absorp-
tion. Hot electrons generated from the photon absorption in the
plasmonic gating can be injected into the semiconductor to form aTABLE I. Different methods for enhancing the performance of plasmonic hot carrier photodetector.
Approach Structure Method Ref.
Inhibition of radiation decay Metamaterial perfect absorber LSPR coupling with a Fabry–P /C19erot resonance 25
Au film/disordered silicon nanoholes Photonic/plasmonic scattering and LSPR 81
Plasmonic pyramid Trapping light on the nano apex 109,110
DBR/metal film Tamm plasmon resonance 76
Au grating/MoS 2/substrate sandwiched structure Strong coupling between LSPs and SPPs 73
Reducing transport loss A perovskite deposition Elongated hot carrier lifetime 65
Ultrathin metal film Multiple hot carrier reflection 59,80
3D Schottky barrier injunction Increasing hot carrier transport interfaces 97
Au grating/p-Si FCA in highly doped semiconductors 83
p-type GaAs/Al xGa1-xAs structures Hot-cold hole energy transfer 75
WS 2/graphene heterostructure 2 lPTE model 91
Increasing hot carrier injection 3D Schottky barrier injunction Providing more momentum space for injection 97
Plasmonic pyramid Relaxation of the carrier momentum mismatch 109,110
Plasmonic waveguide Producing more effective momentum for injection 42
CdSe/Au nanorod PICTT 89
Rough M/S interface Relieving the momentum conversion 41Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-16
Published under license by AIP Publishingphotocurrent. Moreover, the grating structure can effectively promote
electron transport. To manipulate the interslit distance, the responsiv-ity peak can be tuned by the plasmonic gating resonance. It is worth
noting that an ultrathin titanium adhesion layer was usually employed
in these devices that can reduce the Schottky barrier height ( /C240.5 eV),
increasing the internal photoemission efficiency.
25,39To further
enhance photoresponse, an active antenna with deep trench cavities/thin metal (DTTM) was proposed by Lin et al.
113The electric field
enhancement can be obtained by both SP resonance and 3D cavity
effect. More effective hot electrons can reach the metal/semiconduc-
tors interface over the Schottky barrier due to a shorter transport dis-
tance of hot electrons. The proposed active antenna structure withDTTM is illustrated in Figs. 14(c) and14(d) . The hot electron photo-
detector integrated with broadband MPAs was first demonstrated by
Liet al.
25Both LSPR and Fabry–P /C19erot resonance can be excited on the
upper and lower resonators leading to perfect light absorption, as
shown in Fig. 14(e) . The proposed MPAs hot electron photodetector
exhibited a photoresponsivity of 3.37 mA/W at 1250 nm. Differentfrom the plasmonic grating, hot electron photodetectors with DTTM
and MPAs exhibit a broadband and polarization-insensitive response.Moreover, broadband photodetectors have the capability of using a
single device for numerous applications such as data and transport
application. Above all, tailoring plasmonic properties can be employedto manage the response band, including narrow and broadband
response, and realize polarization detection without additional optical
elements; high electric field enhancement and the correct momentumcan also be introduced to meet requirements for various use.
Plasmonic pyramid nanostructure can produce a high electric
field enhancement to enhance the photoresponse of hot electron
detectors. Besides, it can relax the momentum mismatch from the con-
servation of electron linear momentum. Due to its large cross-section,light can be concentrated into the nano apex of the plasmonic pyra-mids, as illustrated in Fig. 15(a) .
109The nanostructures were prepared
by etching in KOH solution, which did not rely on using expensive
focused ion beam or electron beam lithography. Because of the largerefractive index difference between silicon and air, most of the light
(a)
laser φBenergy band diagram
100
10075
7550
5025
250 18090EC
EF
EV
Si (n-type) metal
ITO
contactOhmic
contact
Light
1250–1650 nmIncident light
zy
x2700
Wavelen gth (nm) Wavelen gth (nm)
AbsorptionResponsivity (nA mW–1)Current (% max)
Responsivity (mA/W)
Responsivity (nA mW–1)(b)
(c) (d)A
12504000600
500
400
300
200
100
0
3.5 11721log(Q) (W/m3)
03.0
2.52.0
1.5
0.5
0
1200 1250 1300 1350 1400 1450 15001.03500
2500
1500
500
10
5
010003000
2000
1300 1400 1500 1350 1450 1550Wavelength (nm)1200 1300 1500 1700 1400 1600Energy (eV)
1.03 0.95 0.82
D = 800 nm
W = 250 nm
T = 200 nmD = 850 nm
D = 900 nm
D = 950 nm
D = 1000 nm
D = 1050 nm
D = 1100 nm0.73 0.88 0.77
1600 1650(e)e–
FIG. 14. NIR optical antenna-based hot electron photodetectors. (a) Left: energy band diagram of plasmonic nanoantenna-semiconductor Schottky barrier. R ight: schematic of
the photodetector with optical antennas (top) and polarization response at the wavelength of 1500 nm (bottom) following a cos2ðhÞangular dependence. Reproduced with per-
mission from M. W. Knight et al. , Science 332(6030), 702 (2011). Copyright 2011 American Association for the Advancement of Science.39(b) Left: schematic of plasmonic
grating Schottky photodetector (top) and surface plasmons propagation on gratings (bottom). Right: response spectrum controlled by the interslit distance. The response can
be tuned in the NIR regime by controlling the interslit distance. Reproduced with permission from A. Sobhani et al. , Nat. Commun. 4(1), 1643 (2013). Copyright 2013 Springer
Nature.38(c) Schematic illustration of electrical configuration and photodetector with an active antenna with deep trench cavities/thin metal (DTTM). The re sonance wavelength
is determined by SP resonance and 3D cavity effect that can be tuned by modifying the geometry of the plasmonic nanostructures. (d) Responsivity spect ra of DTTM based
photodetector. The responsivity is two or three higher than that of plasmonic nanoantenna-based hot electron photodetector reported by Knight et al.39Reproduced with per-
mission from K.-T. Lin et al. , Nat. Commun. 5(1), 3288 (2014). Copyright 2014 Springer Nature.113(e) Absorption distribution (top) and responsivity spectra (bottom) of MPAs
based hot electron photodetector. The absorption (dashed line) and responsivity (solid line) contributions of the total, upper, and lower resonato rs correspond to red, green,
and blue, respectively. Reproduced with permission from W. Li et al. , Nano Lett. 14(6), 3510–3514 (2014). Copyright 2014 American Chemical Society.25Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-17
Published under license by AIP Publishingcannot escape from the pyramid, resulting in a large field intensity in
the apex [see Fig. 15(b) ] .T h ee l e c t r i cfi e l de n h a n c e m e n ti nt h ea p e x
was about 30 times that in the base of silicon. A responsivity of 5, 12,
and 30 mA/W was measured at the wavelength of 1064 nm, 1310 nm,
and 1550 nm, respectively, under the reverse bias of 0.1 V, and the
dark current is as low as /C24100 nA. Additionally, the plasmonic pyra-
mid decorated by metallic particles can also be utilized in enhancing
light trapping and localized electric fields. The metallic NPs can be
prepared by thermal dewetting, and the light can be trapped by the
pyramid and reflected multiple times between the particles, enhancing
light absorption in metals.110According to the control of the size of
metal particles, it can tune the hot electron generation rate and ener-
gies. It should be noted that the performance of plasmonic hot electron
photodetectors strongly depends on the precise fabrication precision
of subwavelength patterns, which increases the fabrication cost, and itis difficult for practical applications. In addition to the pyramid nano-
structure created by anisotropic chemical etching, there are many
novel strategies for large area and low-cost manufacturing. As illus-
trated in Fig. 15(c) ,W e n et al. designed a metal-semiconductor-metal
(MSM) absorber to increase the IQE by optimizing the electrical andoptical properties of hot electron photodetectors.
98A wide bandgap
semiconductor, Titanium dioxide (TiO 2), was employed to coat Au
NPs together with the underlying silicon. Hot electrons can be
accepted by both silicon and TiO 2, following the drift-diffusion carrier
transport framework. Under thermal equilibrium, a high electric field
will be distributed around Au NPs due to the built-in potential adja-
cent to metal-semiconductor contact. The electron affinity of TiO 2is
/C244.0 eV, which is similar to that of silicon, indicating that there is no
valence band offset between silicon and TiO 2. Hot electrons injected
into TiO 2can also transport to the cathode form photocurrents with
the drift field formed by the rear Schottky contact. Moreover, the
wide-bandgap TiO 2resulted in large energy offset in the valence band
as a hole blocking barrier, suppressing electron-hole recombination.
As illustrated in Fig. 15(d) , a large photoresponse would occur in the
wavelength shorter than 1200 nm, which can be mainly ascribed toelectron-hole generation from band-to-band transition in silicon. The
plasmonic absorber with TiO
2cavity enables the most outstanding
performance, in which Au NPs are sandwiched by two semiconduc-
tors, both of which are regarded as excellent hot electron acceptors.
On the contrary, only hot electrons generated near silicon can be
(a) (b)
500 nm
air
AirAI
AISU8
incident lightlight
generated
current
SU8SiSi200 nm
(c) (d)
Wavelength (nm)Wavelength (nm)Responsivity
(a.u.)Responsivity (mA/W)10EvDrift
Max
MinEmission1.5
–3.5Electron energy (eV)1.5
–3.5Electron energy (eV)Band diagram0 a.u. 1 0
Optimized PA (ITO cavity)
Opiminzed PA (TiO 2 cavity)
PA with small NPs (TiO 2 cavity)
Planner reference
Fowler modelSi response
Metal responseW/m31016
Silicon
AR coating
NIR illuminationMSM absorberEc
EvEc
TiO2
SiliconTiO2
Silicon
1
0.1
1100 1200 13001000 1400 1800
1400 1500 1600 1700 1800
FIG. 15. Plasmonic hot electron photodetector with the ability of large area and low-cost manufacturing. (a) Schematic of the plasmonic pyramid based hot ele ctron photode-
tector. Light is concentrated toward the nano apex of the plasmonic pyramids to generate hot electrons. (b) Calculated electric field distribution (l eft) and calculated distribution
(right) of hot electron generation in the plasmonic pyramid photodetector at 1300 nm. A strong electric field occurs at the nano apex, leading to a large number of hot electrons
at the interface. Reproduced with permission from B. Desiatov et al. , Optica 2(4), 335–338 (2015). Copyright 2015 The Optical Society.109(c) Left: schematic of metal-semicon-
ductor-metal (MSM) absorber integrated hot electron photodetector. The MSM absorber consists of random Au NPs, an electron-accepting semiconduct or, and an optically
thick Au reflector serving as the anode. Right: energy band diagrams of perfect absorber based hot electron device and planar reference structure (top ), and electric field under
thermal equilibrium (bottom). Hot electrons injected into TiO 2can flow freely toward silicon that serves as the cathode to contribute to the photocurrent. (d) Responsivity spectra
with different absorber structures. The response contribution from silicon and metal is shown in the inset. All devices exhibit a high response at the wavelength shorter than
1200 nm due to the electron-hole pair generation in the silicon. Reproduced with permission from L. Wen et al. , Laser Photonics Rev. 11(5), 1700059 (2017). Copyright 2017
Wiley-VCH.98Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-18
Published under license by AIP Publishingharvested to contribute to photocurrents in the ITO based plasmonic
detector.
Plasmonic hot electron photodetectors can also be utilized in the
visible regime by replacing silicon with the wide bandgap semiconduc-tors, such as TiO
2and ZnO. Similar to metallic NPs together with theunderlying silicon, TiO 2can also be employed in the visible regime,
exhibiting an external quantum efficiency (EQE) from 0.4% to
6.0%.114–116Recently, a broadband hot electron photodetector based
on metallic nanorod array has been demonstrated by Zhang et al. ,a s
shown in Fig. 16(a) .99The Schottky barrier formed by the Ag/TiO 2
(b)
123 4 5
(d)Top Metal Bottom MetalHot e-
Oxide Barrier(a)
(c)
(f) (e)Polarization angle (deg)Wavelength (nm)Photoresponsivity ( μA/W)
80yxzEEph
EFe e
e
EF EF
EFEphETEETM
k
I
V
A+
–
EF, tEF, beVapp
ϕB
ϕB ϕBIforwardIbackward
TMTiO2
TiO2SiO2AgAu
AuAI2O3
AI2O3TE70
60
5040
30
2010
0
0 40 80 120 160 200
Normalized
Absorption
large-area
0.01
012340.1110100
900 nm
experiment406 nm640 nm 70 mA/W
5 mm
500 nm1.0
0.5
Au ITOIncident
light
0Responsivity (nA/W)
Responsivity (mA/W)
Reverse bias (V)250
Exp
Calc 200
150
100
50
0400 500 600 700 800
0 bias
Au AITiO2Au AIReverse bias900 1000
FIG. 16. Visible optical antenna-based hot electron photodetectors. (a) Schematic of the metallic nanorod arrays absorber consisting of the metallic nanor od array. The hot
electron photodetector exhibits a broadband perfect absorption. Reproduced with permission from C. Zhang et al. , ACS Photonics 5(12), 5079–5085 (2018). Copyright 2018
American Chemical Society.99(b) Left: energy band diagram of metal-insulator-metal structure (MIM) diodes and hot carrier extraction process through a five-step model. In
contrast to the three-step model in the metal-semiconductor heterostructure, the two additional steps are the penetration of hot electrons across t he Al 2O3layer without inelastic
collisions and injection into the opposing electrode. Right: electrical configuration of MIM diodes formed by a wide bottom electrode and a series of n anoscale top electrodes.
(c) Polarization dependence responses of the proposed MIM diodes at 470 nm. Reproduced with permission from H. Chalabi et al. , Nano Lett. 14(3), 1374–1380 (2014).
Copyright 2014 American Chemical Society.121(d) Left: schematic of the metal-semiconductor-metal (MSM) photodetector with Silica nanocone array. The light irradiates the
gold film and generates hot electrons through the SP excitation. Right: experimental and calculated responsivity spectra (top), exhibiting a broadb and response from 500 to
900 nm, and energy diagram (bottom) of the proposed device without and with external bias. The energy diagram is modulated by the reverse bias, resulti ng in an increased
hot electron injection. Reproduced with permission from Z. Yang et al. , Photon. Res. 7(3), 294–299 (2019). Copyright 2019 The Optical Society.122(e) Absorption distribution of
ITO-insulator-metal device. Most of the light absorption occurs at the Au/insulator interface. Reproduced with permission from T. Gong et al. , Nano Lett. 15(1), 147–152 (2015).
Copyright 2015 American Chemical Society.72(f) Left: SEM and microscope images of ITO/TiO 2/Au plasmonic crystal photodetector. Right: the relationship between responsiv-
ity and reverse bias. A maximum responsivity of 70 mA/W is obtained at 640 nm under reverse bias. Reproduced with permission from F. P. Garc /C19ıa de Arquer et al. , ACS
Photonics 2(7), 950–957 (2015). Copyright 2015 American Chemical Society.123Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-19
Published under license by AIP Publishinginterface is /C240.9 eV. Compared with that composed of the gold, Ag
has a larger MFP ( /C2450 nm) in the visible wavelength, leading to the
incident photon-to-electron conversion efficiency enhanced by threetimes.
117Hot electrons can also penetrate the insulator, such as
Al2O3,to contribute to photocurrents. In contrast to the NIR regime,
the photons have higher energy at visible wavelength, which provideshigher quantum efficiencies for these devices. In the following para-graph, we will introduce some typical optical antenna-based hot elec-
tron photodetectors operating in the visible regime. Besides,
refractory plasmonic materials, such as titanium nitride (TiN), havealso been used for hot electron photodetection in the visible regime.In contrast to gold contact, the TiN-ZnO-TiN structure has a larger
photocurrent due to a lower barrier height.
118
The MIM diode can be used for hot electron photodetection.
Although MIM diodes can be regarded as nano-rectennas for infrared
detection and rectification, it is still challenging to be employed in thevisible regime due to the limitation of resistor-capacitor (RC) con-stant.
119,120Recent studies showed that plasmonic hot electron gener-
ated in metals could overcome this limitation.13The mechanism of
hot electron extraction in metal-semiconductor heterostructure can
be regarded as a three-step model, and the process in MIM diodescan be described as a five-step model, as shown in Fig. 16(b) .
121The
two additional steps are the penetration of hot electrons across the
Al2O3layer without inelastic collisions and hot electron injection
into the opposing electrode. Polarization dependence responseswere measured in the devices, which is similar to plasmonic nanoan-tennas operating in NIR regime, as illustrated in Fig. 16(c) . Also, the
metal-semiconductor-metal (MSM) structure can be fabricated for
hot electron photodetection in the visible wavelength, as shown inFig. 16(d) .
122The device was fabricated by sequentially deposing
aluminum (Al), TiO 2(semiconductor), and Au films. The absorp-
tion in the Al electrode is very low, and the Al-TiO 2contact is an
Ohmic contact. Therefore, the backward photocurrent created by
hot electrons in the Al electrode can be ignored. The calculated pho-toresponsivity is similar to that of an MS structure, as depicted byEq.(9). Combing with LSPRs and SPPs, a broadband response from
500 to 900 nm was obtained with a photoresponse of 180 lA/W at
620 nm. Only part of hot electrons with the excess energy higherthan the barrier height can contribute to the photocurrents withoutthe bias voltage, while the bending in the conduction of TiO
2
increases and the barrier height decreases under reverse bias, result-ing in a high fraction of hot electrons crossing over the barrier.
In a MIM or MSM configuration, the transparent conducting
oxides (TCO) and ITO have been widely used. Most of the light isabsorbed in the metallic structure rather than in TCO or ITO due totheir low optical absorption, leading to a higher asymmetric hot elec-tron generation. As a result, a larger net photocurrent can be achieved
by using ITO and TCO. As shown in Fig. 16(e) , the light is incidental
from the ITO side and mainly absorbed in metals near the Au/insula-tor interface.
72Absorption difference between the ITO and gold is a
majority factor for the net photocurrent generation.72The
photocurrent-voltage characteristic of the proposed device is summed
up into four parts arising from hot electron and hot hole from bothelectrodes, given by
IVðÞ¼IVðÞ
Au/C0ITO
e /C0IVðÞITO/C0Au
e þIVðÞITO/C0Au
h /C0IVðÞAu/C0ITO
e ;
(18)where the four components represent the directional flows of electron
and hole between the two electrodes. The net current is determined by
the applied voltage, barrier height, light absorption distribution,bandgap of the oxide, and so on. SEM and microscope images of ITO/
TiO
2/Au plasmonic crystal photodetector are shown in Fig. 16(f) .123
The large-scale photodetector was fabricated by nano-imprinting
lithography. According to geometric engineering, various intense col-
ors can be observed by the naked eyes due to the different diffractedorders. A strong plasmon resonance was excited in the Au/TiO
2inter-
face, leading to a maximum responsivity of 70 mA/W and the EQE of
12% under reverse bias.
2. Planar hot electron photodetectors
Fabricating complicated plasmonic nanostructures still faces the
challenge of low-cost and high-precision fabrication. Therefore,researchers have shifted their attention back to planar hot electron
photodetectors.
76,77,124,125The planar hot electron photodetectors
enable large-area and mass production. In 2016, a planar Au/ZnO/
TCO structure integrated with two DBRs had been proposed.77
Figure 17(a) illustrates the schematic diagram of the proposed device.
T h eM / S / T C Oi st h ec o r ee l e c t r i c a lc o m p o n e n tf o rl i g h t - h a r v e s t i n g
and conversion sandwiched between two DBRs; light can transmit
through the top DBR and be reflected by the bottom DBR. A strongelectric field enhancement can be achieved in the metal film though
the Fabry–P /C19erot (F–P) resonance.
77,126As illustrated in Fig. 17(b) ,
most of the light can be absorbed in the metal layer, about 92% in the
resonance frequency, and the adverse absorption in the ITO layer is
about 6.2%. For the reference device without two DBRs, most of thelight is reflected directly or transmitted through the device, leading to
a fairly low absorption for hot electron generation. The responsivity
spectrum is 239 nA/W at the resonance wavelength, an order of mag-
nitude larger than that in the control group ( /C2411 nA/W). The usage
of forward/reverse bias can lead to an increase/decrease in the photo-response for both devices.
Tamm plasmons (TPs) can be regarded as a novel type of SPs
formed at the interface between a DBR and a metal film.
76The surface
electromagnetic waves can propagate along the DBR/metal interface,
confined in the interface, leading to strong absorption in metals.127
Besides, TPs support surface waves with a dispersion that lies withinthe light cone. Therefore, hot electron photodetectors based on TPshave excellent potential for cost-effective photodetection. Figure 17(c)
illustrates the schematic diagram of the TP-based hot electron photo-
detector. The electrical configuration is composed of the M/S/M sand-
wich structure, and a DBR is integrated on the top metal layer to
excite the TPs.
76A narrow reflection dip can be obtained at the reso-
nance wavelength in Fig. 17(c) . The strongest electric field enhance-
ment at the resonance wavelength is located near the M/DBR
interface, as shown in Fig. 17(d) . The calculated unbiased photores-
ponsivity is 13.7 nA/mW, which is about two times larger than that of
the reference grating-based device. However, this system’s photodetec-
tor research is mainly focused on theoretical investigation. Wang et al.
has demonstrated the first experimental TPs photodetector, showing a
photoresponsivity of 8.26 nA/mW at the NIR regime, as shown inFigs. 17(e) and17(f).
128
Although perfect absorption can be achieved by TPs, the photo-
responsivity is still lower than that of plasmonic nanostructure-basedApplied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-20
Published under license by AIP Publishingdetectors, which can be attributed to a low probability of hot electrons
reaching the M/S interface and isotropic momentum distribution inthe metal layer.
13,80,129The dual cavity and the ultrathin-film double-
barrier hot-electron devices have been proposed to solve these prob-
lems.79,80As illustrated in Fig. 18(a) , the planar dual cavity hot electron
photodetector is composed of a DBR, two metal layers for opticalabsorption, two TiO
2layers for hot electron injection, and the silica
substrate. The electrical configuration and the proposed device’senergy diagram are shown in Fig. 18(a) . The bottom metal layer with a
double Schottky junction can further enhance the hot electron extrac-tion efficiency. In this system, the reverse photocurrent is not existent,
similar to the conventional hot electron photodetectors based on sili-
con. The absorption at the resonance wavelength can be analyzed bythe phase accumulation in the cavity. The phase accumulation ain the
top cavity is calculated by the equation
79
a¼c1þc2þ2b; (19)
where c1andc2are the phase shift from the reflection occurring in the
top metal layer/top-TiO 2interface and bottom metal layer/top-TiO 2
interface, respectively. And bis the round-trip phase accumulation in
the TiO 2layer. As for the bottom cavity, it has similar definitions
involving the bottom metal layer, bottom TiO 2layer, and the DBR.
According to the analysis of the phase accumulation, the resonanceoccurs when a¼0o r2 p. The proposed device exhibits a photorespon-
sivity of 2.0 mA/W at 950 nm with the three-fold enhancementcompared with that composed of a single-cavity, as shown in Fig.
18(b) .
79Additionally, another strategy to conquer these limitations is
to use an ultrathin double-barrier system. As shown in Fig. 18(c) ,a n
ultrathin 1-nm gold film is buried into the silicon layer covered with
two DBRs. A narrowband absorption can also be achieved in this sys-
tem, and a 30 folds enhancement of the local electric field can be
obtained in the Au film, as shown in Fig. 18(d) .80The hot electron
injection is much higher than the device with thick-film single-barrier
due to the emission across two barriers and multiple electron reflection
between the double M/S interface. Therefore, a maximum photores-ponsivity of 19.29 mA/W and EQE of 1.89% have been demonstrated
at the resonance wavelength, respectively. A detailed responsivity as
the dependence of the barrier height is further shown in Fig. 18(e) .
Although the interference theory can explain the light absorption
from the cavity resonance, the essence of these phenomena is the exci-tation of planar SPPs.
130Furthermore, hot electron devices of this sys-
tem can achieve a narrow band detection with the full width at half
maximum (FWHM) about a dozen nanometres, which is sharp
enough for sensing applications. The structure design can also obtain
multiband photodetection, and this property has attracted tremendous
attention due to its wide application value, including multicolor imag-
ing, medical treatment, and military application.131,132In contrast to
multicolor photodetection from a hetero-integrated semiconductor,
these devices have the advantage of arbitrary spectral selectivity to
manipulate the structure parameter.76,77However, a thick buffer layer
(b) (c)
+
–250
9
3520000 5 10 15600020406080100
700V–
+
800
20 25 30
1500
1000
500
Au00
0560
40
20
0
P (mW)
AMetal - AITOE 2 (a.u.)
6
3
0
1520 1550 1580 1610200
150
50
0
750 800 850 900 950100
(d)(a) (c)
(f) (e)Wavelength (nm)
Wavelength (nm)xzλ (nm)
λ = 1581 nm
λ = 1555 nm
λ = 1529 nm
R (%)
MSM DBRTPcavity enhancedM/S/TCO
w/o cavityT -BDR B-BDRBuffer
Substrate
Responsivity (nA/mW)Photoresponse (nA/mW)
z (nm)NDBR = 8Photocurrent (nA)⎜⎜
FIG. 17. Planar hot electron photodetectors achieved by TPs. (a) Schematic illustration and (b) responsivity of planar micro-cavity integrated hot electro n photodetector. It is
composed of silica substrate, the bottom and top DBRs excite the Fabry–P /C19erot (F–P) resonance and the core electrical component formed by the M/S/TCO stack. Reproduced
with permission from C. Zhang et al. , Nanoscale 8(19), 10323–10329 (2016). Copyright 2016 Royal Society of Chemistry.77(c) Reflection spectrum and schematic of TP based
hot electron photodetector. The device consists of eight pairs of DBR, the MSM stack, and silica substrate, and it enables a sharp reflection dip due to t he Tamm plasmons
(TPs) resonance. (d) Electric field distribution of the proposed device at the TP resonance. The strongest electric field occurs near the M/DBR interfa ce. Reproduced with per-
mission from C. Zhang et al. , ACS Nano 11(2), 1719–1727 (2017). Copyright 2017 American Chemical Society.76(e) Responsivity and absorption spectra of the experimental
TPs photodetector. The inset shows the photocurrent as a function of the incident power. (f) Schematic diagram (left) and SEM image (right) of the expe rimental TPs photode-
tector. Reproduced with permission from Z. Wang et al. , Nanoscale 11(37), 17407–17414 (2019). Copyright 2019 Royal Society of Chemistry.128Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-21
Published under license by AIP Publishingof micrometer thickness is often required to achieve multicolor photo-
detection, which increases the volume and complexity of the devices.133
It should be noted that the planar ultrathin perfect absorber can berealized by using high refractive index materials. And the multicolor
photodetection achieved by the planar metal-semiconductor-metal
(MSM) F–P cavity has been proposed.
129The usage of Molybdenum
disulfide (MoS 2) as a high refractive index material increases the proba-
bility of satisfying the F–P resonance, leading to a three-band response.
Therefore, we expect the system’s devices to be further optimized using
high/giant refractive index materials. Also, increasing electric field
intensity in the planar metal structure is another way to boost photo-
responsivity. Besides, in the latest research, a broadband response with
FWHM exceeding 240 nm has been proposed by using TiN film in the
TPs system, which is quite different from previous works and broadens
the application of planar hot electron photodetectors.134
3. Hot electron photodetection coupled
with low-dimension materials
SPs have been shown to enhance light and low-dimension mate-
rials interaction by the strong electric field enhancement. In this sub-
section, we will review hot electron photodetection using nanowires(NWs), quantum dots (QD), and 2D materials. NWs can be consid-
ered as an excellent component for hot electron collection due to their
outstanding electrical properties and large surface area. Figure 19(a)
illustrates the SEM of Au nanorod-ZnO nanowire hybrid hot electrondetector.
135The Au nanorods were deposited on the ZnO NW field
effect transistors (FETs) directly. Under the irradiation of 650 and850 nm, a 250-ms response has been demonstrated, which is about an
order of magnitude faster than that of bare ZnO NW. As shown in
Fig. 19(b) , a hybrid FET device exhibits a higher photoresponse than
that of bare ZnO NW FET. The photocurrent displays a sharp increaseafter illumination, and only 30% of the initial photocurrent was mea-
sured after interruption of illumination for 400 ms. Furthermore, Au
core-shell NWs and coaxial metal/semiconductor/metal single NWshave also been proposed to enhance photoresponse through hot elec-tron injection.
136,137Recently, plasmonic hot electron injection into Si
nanowire arrays has been proven to improve NIR photovoltaic perfor-
mance in the NIR region.138The illustration of the structure of a flexi-
ble hot electron device is shown in Fig. 19(c) .T h eP o l y ( 3 , 4 -
ethylenedioxythiophene) polystyrene sulfonate (PEDOT: PSS) orreduced graphene oxide (rGO) was employed as a hole-transportingagent, and the Ag nanostructures in ethanol were dropcasted into the
Si arrays. As shown in Fig. 19(d) , hot electrons generated in metals can
700 900 1000 1100 1200 800
400
100
75
2550
0
0.3
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.672
56
40
24
7
0.980
32
30
28
2414 24162.0
Single cavity
Double cavities
1.5
1.0
0.00.5
60
4020
0
0 1000 2000 3000 4000300
200100
0(a)
(c) (d) (e)(b)
DBR
Substrate
Si SiAuzTop Au
layerBottom
Au layerTop TiO2
layerA
Bottom
TiO2 layer
x
y
z
x yAu
TiO2
SiO2d1
Pα3 Pα2Pα1e–e–
d2
EF EF
Ev EvEc Ecd1
d2
Si
λ (nm)
ϕSB(eV)
ϕSB (eV)ΦB ΦBΦB
DBRDBR
Au
Responsivity (mA/W)
Rhigh-energy (%)Rtotal (mA/W)
z (nm)E 2 /2⎜⎜ E0⎜⎜
FIG. 18. Planar hot electron photodetection enhanced by the double-barrier junction and ultrathin film. (a) Schematic diagram (left) and electrical circuit setup (right) of planar
dual-cavity hot-electron photodetectors. The device consists of top and bottom Au layers as hot electron generators, two TiO 2layers for hot electron collection, a DBR, and the
silica substrate. The photon absorption within the top and bottom layers can be used to generate injected hot electrons to enhance the photoresponse. (b) Comparison of
responsivity of planar hot electron photodetector between a single cavity and double cavities. The response of double cavities is almost three times higher compared with that
of a single cavity due to three Schottky junctions and a stronger absorption. Reproduced with permission from W. Shao et al. , Nanoscale 11(3), 1396–1402 (2019). Copyright
2019 Royal Society of Chemistry.79(c) Schematic of planar hot electron photodetector with ultrathin Au film and double-barrier. The device effectively reduces hot electron
transport loss by utilizing an ultrathin Au film covered by two DBRs. (d) Electric field enhancement along z-direction at the resonance wavelength. The device enables a 30
folds enhancement of the local electric field in Au film. (e) Responsivity and hot electron generation rate as a function of the Schottky barrier. A lower barrier height allows
more hot electrons to be injected, leading to an increased responsivity. Reproduced with permission from C. Zhang et al. , Nano Energy 55, 164–172 (2019). Copyright 2019
Elsevier Ltd.80Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-22
Published under license by AIP Publishingcross over the Schottky barrier collected by the Si nanowires, while
hot hole can flow toward the ITO through the PEDOT: PSS.
Because of plasmonic hot electron injection and lower barrier formed
at the M/S interface, the current density and power conversion
efficiency have increased to 28% and 40%, respectively, as shown in
Fig. 19(e) . One can transfer this idea to hot electron photodetector
design.
Plasmonic hot electron photodetector incorporating QDs has
also been proposed by Lee et al.139Combining with the advantages ofplasmonic hot electron generated in Au nanostructures and optical
absorption of PbS quantum dots (QDs), the photocurrent can be
amplified about three times larger than that without PbS QDs. It
should be noted that too much hot electron with the mismatchmomentum is the leading cause of low internal photoemission. The
deposition of QDs on the plasmonic Au/TiO
2nanostructures leads to
the formation of a 3D Schottky barrier, increasing a new pathway ofhot electron transport, as shown in Fig. 19(f) .
97,139Hot electrons gen-
erated from light absorption in QDs can be transferred to the TiO 2,a s
(a)
(c) (d)
ITOAg/Al40
30
400 600Si NWs-PEDOT:PSS
5L PbS QDs on plasmonic Au
3L PbS QDs on plasmonic Au
1L PbS QDs on plasmonic Au
Plasmonic Au
Thin film Au (10 nm)Si NWs-Ag NPs-PEDOT:PSS
Si NWs-PEDOT:PSS-Ag NPs
Si NWs-PEDOT:PSS/Ag NPs
800 100020
10
12
8
4
0
1.5 2.0 2.5 3.00EQE (%)IPCE (%)ITO-PET
PEDOT
PSS
n-Si NWsAg
NPsn-Si NWs(e)
(f) (g) (h)(b)0.8
0.60.40.2
A 0.0
01 0Light onLight onLight off Light offAuHbNW at λ=650nm
AuHbNW at λ=850nm
FitZnO NW at λ=650nm
ZnO NW at λ=850nm
Fit
20 30 40 50 0 1 02 03 04 05 0
e–
hν = 2.0 eV
hν = 2.8 eVhν = 1.7 eV1μm
e–
e–e–e–
e–e– e–e–e–e–e–e–e–e–e–
e–e–
e–
e– e–e–e–
e–
e– e–
e–e–e–e–e–e–e–e–e–e–e–
EFEc
EvAuLight
kELight
kE
d-bandEg =
1.7 eV
PbS
QDsTiO2TiO2Plasmonic Au/TiO2
PbS QDs deposited on
plasmonic Au/TiO2
Plasmonic AuPbS QDsHot electron
Photocurrent (nA)0.04
0.03
0.020.010.00Photocurrent (nA)
Time (s)
Wavelength (nm)
Photo ener gy (eV)plasmonic
hot electronTime (s)
FIG. 19. Hot electron photodetection coupled with nanowire and QDs. (a) SEM image of Au nanorod-ZnO nanowire hybrid hot electron detector. Au nanorods distri bute ran-
domly on the surface of the ZnO nanowire. (b) Photoresponse of Au nanorod-ZnO nanowire hybrid and bare ZnO photodetectors at the wavelengths of 650 and 850 nm. In
contrast to bare ZnO photodetectors, a larger photoresponse and faster response speed can be attained by Au nanorod-ZnO nanowire hybrid photodetect or. Reproduced with
permission from A. Pescaglini et al. , Nano Lett. 14(11), 6202–6209 (2014). Copyright 2014 American Chemical Society.135(c) Structure of flexible hot electron device based on
Si nanowire arrays. Flexible characteristics display is shown in the inset. The flexible hot electron device is fabricated from 10 thick Si film by plasm a–reactive ion etching. (d)
Energy diagram and hot carrier transport process of flexible hot electron device. Plasmonic hot electrons can cross over the Schottky barrier injecte d into the Si nanowires,
while hot holes can flow toward the ITO through the PEDOT: PSS. (e) EQE of the flexible device with different combination approaches. The maximum EQE enha nced by Au/
Si nanowire junction achieves 40%. Reproduced with permission from D. Liu et al. , Angew. Chem. Int. Ed. 55(14), 4577–4581 (2016). Copyright 2016 Wiley-VCH.138(f)
Comparison of hot electron transport without (top) and with (bottom) PbS QDs. Plasmonic Au/TiO 2nanostructures without PbS QDs only allow hot electrons to be received
through the bottom interface, leading to low photoemission; on the other hand, the deposition of PbS QDs on Au/TiO 2leads to a new pathway for hot electron collection. (g)
Energy diagram of the QD deposited on plasmonic hot electron photodetector. Hot electrons generated in the PbS QDs can also flow into TiO 2to increase the photon-to-cur-
rent conversion efficiency (IPCE). (h) IPCE of plasmonic diode with different compositions. As the layers increases, the IPCE is improved significant ly. Reproduced with permis-
sion from C. Lee et al. , ACS Appl. Mater. Interfaces 10(5), 5081–5089 (2018). Copyright 2018 American Chemical Society.139Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-23
Published under license by AIP Publishingillustrated in Fig. 19(g) . As QDs layers increase, incident photon-to-
current conversion efficiency (IPCE) was improved dramatically in
Fig. 19(h) . On the other hand, size engineering and ligand effect on
QDs provide a means to control the bandgap so that the Schottky bar-
rier can be effectively manipulated, leading to a larger photocurrentincrease.
140
Due to the high electron mobility, broadband absorption, and
ultrafast response, 2D materials such as graphene and MoS 2are very
interesting for electronic and optoelectronic applications.141,142Even
though 2D materials have numerous remarkable properties, the light-
matter interaction in 2D materials is still limited by their atomic thick-
ness. For instance, the light absorption of single graphene is only
2.3%,143,144while only 5.3% of the light can be absorbed by single
MoS 2.145SPs can be regarded as an efficient pathway to enhance the
interaction between light and 2D materials due to high electric fieldconcentration. The physical mechanisms of plasmon-enhanced photo-
detection in 2D materials can be divided into two parts. First, a high
electric field induced by SPs promotes the electron-hole pair genera-
tion in 2D materials, and then hot electrons generated from SPs can
also cross over the Schottky barrier to form a photocurrent. Some
recent studies have proven that plasmonic hot electrons can rapidly
and effectively inject into 2D materials to achieve photon-to-electronconversion and other applications. In addition to energy conversion,
hot electron injection can also change the doping of 2D materials
resulting in the phase transitions and modulation of the optical
absorption spectrum.
Hot electrons generated in gapless graphene can flow into the
conduction band of the graphene directly without crossing the bar-
rier. Different from conventional semiconductors, hot electrons gen-
erated in isolated metal particles can be injected into graphene
directly without electrical connection, as shown in Figs. 20(a) and
20(b) .
146The existence of graphene broadens the SP resonance lead-
ing to a broader linewidth, as shown in Fig. 20(a) . The hot electron
transfer time and efficiency can be obtained by analyzing the plas-
mon width, and the average transfer time of hot electrons is 160 6
30 fs. On the other hand, graphene can also be regarded as an electri-
cally tunable plasmonic material due to its gate-voltage-dependent
optical conductivity.144Therefore, graphene can collect not only hot
electrons generated in metals but also generate hot electrons injectedinto adjacent semiconductors. As shown in Fig. 20(c) , a plasmonic
graphene-antenna photodetector has been demonstrated by Syed
Mubeen et al.
147Plasmonic antennas were sandwiched by two
monolayers of graphene, and a photocurrent enhancement of 800%
was measured, which is ascribed to plasmonic-induced carrier excita-
tion in graphene and hot electrons generated from the SPs decay
transferring from the metals to the graphene. Under the laser excita-tion of 785-nm laser, an antisymmetric photocurrent response was
measured, as illustrated in Fig. 11(d) . The photocurrent is deter-
mined by the Fano resonance from plasmonic nanostructures at
785 nm. The spectral sensitivity can be tuned by the geometry of the
plasmonic nanostructures, and photocurrent can be controlled andswitched by a gate bias. Recently, the photoresponsivity has been
confirmed to be controlled by the number of graphene sheets and
the excitation laser power.
145Hot carrier assisted photothermoelec-
tric (PTE) effect in graphene without phonon thermal transport can
also lead to high responsivity and the ultrafast speed detection due to
a weak electron-phonon interaction.102,148In contrast to graphene-based hot electron photodetectors,
MoS 2-based hot electron photodetectors exhibited a high photogain,
which has been demonstrated in recent reports.152In recent studies,
MoS 2has proven that it can be considered as an ideal hot electron
acceptor. Different from graphene, monolayer MoS 2exhibits a direct
bandgap of 1.8 eV. However, compared with monolayer MoS 2, due to
interlayer coupling, a lower Schottky barrier between Au and bilayer
MoS 2has been theoretically predicted.149,150,153Furthermore, because
of the indirect bandgap of multilayer MoS 2, a lower Schottky barrier is
caused by the band of R-point compared to that from K-point, as illus-
trated in Fig. 20(h) .150And then the interaction between sulfur atoms
and metal surface leads to hot electrons injected into the R-point con-
duction band more easily; this effect is not available in the bulk indi-
rect bandgap semiconductor such as silicon. A plasmonic hot electronphotodetector using bilayer MoS
2has been reported. Hot electrons
generated from the asymmetric structure lead to a sub-bandgap
response, and the responsivity can be tuned by source-drain bias volt-
age.40As source-drain voltage increases, a peak responsivity of 4.5A/
W was measured under the bias of 3V. Although the response speed is
relatively low due to the existence of carriers trapping, a large photo-
gain of 105has been measured, leading to the photoresponsivity of 5.2
A/W at 1070 nm. On the contrary, an ultrafast hot electron response
b a s e do nA un a n o a n t e n n a / M o S 2heterostructures has been proposed
and demonstrated by Yu et al.154Plasmonic hot electrons can be trans-
ferred from nanoantenna to MoS 2within 200 fs. In addition to con-
ventional metals, platinum can also be regarded as a novel plasmonic
material with a broad LSPR. The schematics of MoS 2-based hot elec-
tron photodetector with plasmonic Pt nanostrips is shown in Fig.
20(e) .149Pt has a higher probability of hot electron excitation due to
itsd-band close to the Fermi energy, in contrast to Al and Au. As the
excitation power increases, a linear increase of the photocurrent can
be observed, as shown in Fig. 20(f) . Its reproducibility at the wave-
length of 980 nm is shown in Fig. 20(g) . Although a low power sensi-
tivity of MoS 2has been proven in many reports, these results indicated
that the device exhibits a high-power sensitivity, and the response is
reproducible. The bandgap of bilayer MoS 2can be tailored by decorat-
ing it with plasmonic NPs due to localized strain, as shown in Fig.
20(i).151The plasmonic strain blue shifts the bandgap of bilayer MoS 2
with 32 times the enhanced photoresponse and immense hot electron
injection. In addition to MoS 2, other novel transition-metal dichalco-
genides (TMDs) can be widely studied for hot carrier photodetection,
such as MXene, Bornene, InSe, ReS 2, etc.
4. Functionalized hot carrier devices
Combining the excellent properties of SPs applied in various
nanophotonic devices, hot carrier photodetectors have proven thatthey can be used for novel functional photodetection, such as nano-
scale surface imaging, CPL detection, and direct wavelength determi-
nation. Those functionalities were reviewed in Ref. 13.I nt h i sr e g a r d ,
we will introduce hot electron photodetector with novel functionality
for plasmonic sensing, solar-blind UV detector, plasmon-modulated
FET photodetector, and NIR imaging. Due to the strong localized opti-
cal field, plasmonic sensors have the advantages of high sensitivity and
are label free.
155By changing the refractive index of the environment,
the excitation of the SP shifts is driven by the environment-plasmoninteraction. Despite these merits, the implantation of expensive andApplied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-24
Published under license by AIP Publishingcomplicated multiplex imaging equipment still limits their application.
Moreover, the operation bandwidth of the transducer still impedes the
application in extended wavelength ranges with less energy loss andrich molecular-fingerprint information. Previous works have proven
that the transmission of hot electrons from the metal surface to nearby
molecules can induce photocatalytic dissociation.
156Therefore, the
dissociative ability induced by hot electrons could be harvested forindirect nanoplasmonic sensing.
156,157For example, partial hot elec-
trons transfer into the antibonding orbital of H 2,leading to adsorbed
hydrogen atoms and metastable Au hydride, as shown in Fig. 21(a) .157
The different optical constant of metastable Au hydride changes the
LSPR position, which can be reflected by the transmission spectrum[seeFig. 21(b) ]. A multifunctional plasmonic structure with the capac-
ity for both photodetector and NO sensing has been demonstrated by
Narendar et al.158Hot electron excited by LSPR can interact with the
NO molecules or transfer into semiconductors through the barrier, as
illustrated in Fig. 21(c) . The increase of charge density from injected
hot electrons results in the enhanced interaction between ZnO and theNO molecules. Therefore, an increased photo-resistance was measuredthat can be used for NO gas sensing. The sensing response of the plas-
monic sensor with various concertations of NO gas is shown in Fig.
21(d) . The maximum response is at the wavelength of 550 nm, which
is ascribed to effective hot electron generation, increasing the resis-
tance of plasmonic sensor. Instead, hot electron injection through
e–Lasere–
e–
h+
k valley
0.5 eV270180100gold
nanorodgraphenegold nanorod
on quartz
gold nanorod
on grapheneelectron
transfer
450
300
150
0
0 100980 nm
12345678
9
10
11
12
13
14
200 300
Time (sec)
Time (sec)50
0
50
100Dimer
Heptamer90
0
Without strainPhotocurrent (nA)
40
3020
10
0
0 100OffOn980 nm
65098012501500
200Photocurrent (nA)Photocurrent (nA)
With strainΣ point has a lower
Energy, and is atomicorbital favoredhν
Φ
B ΦB(h) (i) (g)(a) (b)
(d)(f)
(c)(e)
Σ valley
MetalEFEF
EFMoS 2
MoS 2AuEcVG
Ec
Ev
Ev
FIG. 20. Hot electron photodetection coupled with 2D materials. (a) Plasmon linewidth of Au nanorod with and without graphene. By analyzing the plasmon linew idth, the aver-
age transfer time of hot electrons is 160 630 fs. (b) Hot electron transfer process from gold to graphene. Reproduced with permission from A. Hoggard et al. , ACS Nano
7(12), 11209–11217 (2013). Copyright 2013 American Chemical Society.146(c) Schematic representation of plasmonic graphene-antenna photodetector. The Au heptamers
are sandwiched between two monolayers of graphene. (d) Polarization dependence of the photocurrent of the proposed plasmonic graphene-antenna pho todetector under
750 nm. For the dimer antenna, the photocurrent strongly depends on the polarization of the incident laser. Reproduced with permission from Z. Fang et al. , Nano Lett 12(7),
3808–3813 (2012). Copyright 2012 American Chemical Society.147(e) Schematic diagram of plasmonic Pt nanostrip photodetector. Hot electrons generated from the excitation
of Pt d-band electrons can effectively transfer into the conduction band of the bilayer MoS 2. Detection sensitivity between time and different incident light intensity with the
applied bias of (f) 0.5 V and (g) 1 V. Linear dependence between photocurrent and incident light intensity is shown in the inset. Reproduced with permis sion from R. Kumar
et al. , Adv. Opt. Mater. 5(9), 1700009 (2017). Copyright 2017 Wiley-VCH.149(h) Schematic of hot electron injection from metal to MoS 2through R-point valley with lower
energy than K-point. Hot electron injection into R-point valley is more favorable than electron transfer through K-point due to the lower energy. Reproduced with permission
from Z. Li et al. , Nano Letters 15(6), 3977–3982 (2015). Copyright 2015 American Chemical Society.150(i) Energy band diagram of Au/MoS 2structure without (left) and with
(right) strain. The bandgap of MoS 2can be modified by the strain, promoting effectively hot electron injection. Reproduced with permission from P. Sriram et al. , Chem. Mater.
32(6), 2242–2252 (2020). Copyright 2020 American Chemical Society.151Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-25
Published under license by AIP PublishingSchottky junction is an effective routine to minimize the disjunct
filter-detector configuration.
Furthermore, plasmonic nanoantenna and silicon-gold core-shell
nanowire array have been theoretically proposed to be used for refractiveindex sensing, and signals of refractive index change can be measuredthrough hot carriers injected into the adjacent silicon.
159,160It is worth
noting that these devices are used not only as a sensing element but also
as an electrical component, resulting in dual functions of sensing and
photoelectric conversion. As shown in Fig. 21(e) , an ultrahigh figure-of-
merit (FOM) plasmonic sensor with direct electrical readout through hotelectron injection has been proposed and demonstrated by Wen et al.
56
This device is suitable for a sensing application and beneficial for direct
electrical reading without spectroscopy, as an external detector, and otheroptical elements. To investigate the relationship between the change ofphotoresponse and the environmental refractive index, the bulk refrac-tive index sensitivities of the first and second-order modes are 1084 and593 nm/RIU, respectively, while the power normalized electrical read-outsensitivity of the first- and second-order modes are 1326 and 3017 mA/W/C1RIU, respectively, as shown in Fig. 21(f) . The FOM was found up to
190, and the detection limit (DL) was down to /C2410
/C06RIU.
Solar-blind UV photodetection refers to the detection of UV light
in the range of 200 to 320 nm without responding to visible light, andthus have minimal background noise from solar radiation.
Conventional solar-blind UV detection schemes are based on wide
bandgap semiconductors, solar-blind band-pass UV filters coupled
with Si photodetectors, and photoemissive detectors based on UV
photocathodes.161However, they have intrinsic drawbacks. For exam-
ple, photoemissive detectors are bulky and fragile vacuum electronic
devices, and they need high operation voltages ( /C29100 V) to achieve a
high quantum efficiency. Wang et al. demonstrated a hot electron-
based solar-blind UV detector using a metal-oxide-semiconductor
structure, as shown in Fig. 22(a) .161A 3.8 eV potential barrier at the
interface can block electrons excited by visible photons while enabling
UV-excited hot electron to cross the barrier, as the band diagram
shown in Fig. 22(b) . The device demonstrated a responsivity of 29
mA/W, an IQE of /C2418%, and an EQE of 13.5% at k¼269 nm, 3–4
orders higher than that of the visible light responsivity, as shown in
Fig. 22(c) . The 10 times higher EQE than conventional hot electron
detectors can be attributed to photon management in metal absorbers
with a high density of states near the Fermi level that drastically
improve the quantum efficiency. The hot electron photodetector can
be used as FET.162The plasmon FET consists of a ZnO thin-film FET
structure, as shown in Fig. 22(d) . A heavily doped n-type Si substrate
serves as a back gate, and an n-type ZnO film deposited on thermally
004080
ONOFF2 ppm2 ppm2 ppm
Ec
Ev12004080120100200300400500
125SiAuH2
Hot e–Au
Substrate500–600 nm hν
75
50
25
750 800 850 950 900 1000 1100 105010052058
56545250
48
4658
56545250
48
46
540 560 580 600 620
1%
640
160200550 nm
655 nm
725 nm
500 1000 1500 2000 2500Wavelength (nm)No H2
H2 ON
H HHHH2 OFF% Transmittance
Au-ZnO(d)(b)
(f)(e)
(b)
Time (sec)ZnOAuLSPReNONO
NONO
OOOO
eeeeee
Response (%)
Responsivity (mA/W)Oblique incidence (20 deg)
hν >ESiWater
Alcoholhν
Δλ∼17 nm3017 mA/(W.RIU)
1326 mA/(W.RIU)
Δλ∼32 nm(a)
(c)
Wavelength (nm)
FIG. 21. Plasmonic hot carrier sensors. (a) Schematic illustration of hot carrier generation, H 2dissociation, and gold hydride. Plasmonic hot electrons induce H 2dissociation
and then lead to the formation of gold hydride, characterized by the change in transmittance. (b) Transmittance spectra of hot carrier sensor. The pre sence of gold hydride
reduces transmittance. The existence of H 2leads to a 1% reversible change in transmittance. Reproduced with permission from D. Sil et al. , ACS Nano 8(8), 7755–7762
(2014). Copyright 2014 American Chemical Society.157(c) Schematic diagram of hot electron transport and gas sensing mechanism. Plasmonic hot electrons are injected into
ZnO or interact with NO molecules; subsequently, the generated O 2molecules from the interaction between NO molecules and electrons are absorbed at the surface of ZnO
through the photoelectrons. (d) The sensing response of the plasmonic hot carrier sensor under NO atmosphere under the illumination of visible wavel engths. The charge den-
sity of ZnO is increased by hot electron injection, enhancing interaction with NO molecules, resulting in the increased photoresistance and sensing response. Reproduced with
permission from N. Gogurla et al. , Sci. Rep. 4(1), 6483 (2014). Copyright 2014 Springer Nature.158(e) Schematic of plasmonic hot electron sensor with electrical readout. It is
obtained by coating the shallow silicon nanotrenches with the Au film. (f) The responsivity of hot electron sensors in the water and ethanol solutions. The electrical readout sen-
sitivity is attained by comparing the change of photoresponse to the change of the environmental refractive index. Reproduced with permission from L . Wen et al. , ACS Nano
13(6), 6963–6972 (2019). Copyright 2019 American Chemical Society.56Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-26
Published under license by AIP Publishinggrown SiO 2serves as an active semiconductor channel with decorated
plasmonic NPs. The amplification and gate bias controlled photores-ponse can be explained by the energy band diagram, as shown in Fig.
22(e) . The generated hot electron contributed to the amplification ofthe drain current, and the spectral response as a function of wave-
length depends on the gate voltage bias, as shown in Fig. 22(f) .A
responsivity of 3A/W can be achieved under the saturation operationmode of plasmonic FET. If no gate voltage bias is applied, the
Ef Ef
MetalWhνX
ΦB
Va
hνe–
Oxide Semiconductore– e–UV LED
Si
Cu PlateSiO2SnLock-in
Amplifier
5 mm
450310–3
1.5 2.0Red Laser
UV LED
Fluoresent White (4100 K)
2.5 3.010–210–1100101102
2
1
0
550 500 600 700 650(a) (b) (c)
(d) (e)
(g) (h)(f)
Gate
Gate
1.6 600.0 0.00 0.82 Photocurrent (nA) (nA)SiO2Vg=0
n-ZnO
SiO2SiO2
Vg>0n-ZnOAu NP
Optical image
Spectral Response (A/W)Responsivity (mA/W)
Voltage (V)
λ = 1.105 μm 1.300 μm
10 μm1.350 μm 1.400 μm
(i) (ii) (iii) (iv)
(v) (vi) (vii) (viii)probes
Pt (absorbing contact)Al
n-SiSiNxQuartz
30 μmAu
Hot carrier
detectorCommercial
detector11
342Au NP
Au-ZnO
Schottky junction
Drain currentn-ChannelGate
bias
Wavelength (nm)Wavelength (nm)Gate Bias
Vg=20V
Vg=16V
Vg=12V
Vg=8V
Vg=4V
5000.000.020.040.060.080.100.120.140.16
550 600 650 700
FIG. 22. Functionalized hot carrier devices, including solar-blind UV photodetection, plasmon-modulated FET, and NIR imaging. (a) Schematics of photocur rent measurement
setup of hot electron-based solar-blind UV detector with a lock-in amplifier. The inset shows an exemplary sample with different device sizes. (b) Ene rgy diagram of the pro-
posed device. It also illustrates the process of hot electron generation under UV illumination, its penetration through the metal/oxide interfacia l barrier, and the impact ionization
process in semiconductors caused by the excess energy of the hot electron. (c) The responsivity of hot electron solar-blind UV detector under differe nt light source excitation.
With the increase of the bias voltage, the responsivity steadily increases under the UV excitation due to the internal photoemission of the UV-excite d hot electrons.
Reproduced with permission from Z. Wang et al. , ACS Photonics 5(10), 3989–3995 (2018). Copyright 2018 American Chemical Society.161(d) Top: schematic of plasmonic
hot electron FET. Bottom: illustration of hot electron transfer of the proposed device. Hot electrons transfer from plasmonic NPs to an n-type ZnO film and increase the drain
current. (e) Amplification mechanism and energy band bending of plasmonic FET without (top) and with (bottom) gate voltage. The gate voltage can not on ly lead to a thin
Schottky barrier to allow hot electrons tunneling from NPs to the ZnO channel but also create a drift field for hot electron migration. (f) The responsiv ity of the plasmonic FET
under different gate voltage bias. The inset shows the absorption spectrum of plasmonic FET. Reproduced with permission from H. Shokri Kojori et al. , Nano Lett. 16(1),
250–254 (2016). Copyright 2016 American Chemical Society.162(g) Schematic diagram of NIR-imaging hot electron photodetector consisting of Al top Ohmic contact, SiN x
anti-reflective coating, n-type Si as hot electron collector, and an ultrathin Pt absorbing layer for hot electron generation. (h) Optical (left) ima ge and the image obtained by hot
electron photodetector (right). Commercial Si detector is used as a reference. Hot electron photodetector can operate below the bandgap of Si and is s uperior to the conven-
tional Si photodetector for NIR imaging. Reproduced with permission from L. J. Krayer et al. , ACS Photonics 5(2), 306–311 (2018). Copyright 2018 American Chemical
Society.163Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-27
Published under license by AIP Publishingplasmon-induced hot electrons could migrate only with diffusion as
the drift current, while when positive gate voltage bias is applied, anelectron accumulation layer is formed that facilitates the electrons to
move to the other boundary where the FET channel is located, con-
tributing to the channel enhancement and increasing more drain cur-
rent flow. Krayer et al. demonstrated a NIR-imaging hot electron
photodetector, as shown in Fig. 22(g) . In the NIR imaging photodetec-
tor based on Pt/Si heterostructure, which outperforms commercial Si
devices because Pt achieves maximum absorption for thinner layers,
the probability for internal photoemission is potentially larger due tothe increased likelihood of hot carrier reflection from the back sur-
face.
163Figure 22(h) shows an optical image of the “object” used to
test the detector, corresponding images obtained by the hot carrierphotodetector consisting of 16 nm of Pt (i–iv) and images obtained by
commercial Si device (v–viii). It can be seen that the hot carrier device
has a detectable and reliable current signal for k>1.25lm, while the
signal of the commercial detector fades dramatically as the wavelength
increases beyond 1.1 lm.
B. Plasmon-enhanced hot carrier photodetectors
for integrated nanophotonics
Compared with electronics, the photon has the characteristics of
ultra-high speed, ultra-high parallelism, ultra-high bandwidth, ultra-
low transmission, and interactive power consumption. Therefore, theuse of integrated nanophotonics for information interaction and calcu-
lation is key to breaking the bottleneck of the integrated circuit.
Photodetection is one of the critical components in integrated nano-
photonics. However, photodetectors present ongoing challenges. For
example, most photodetectors for Si-photonics need additional Gelayers, which add system cost.
164With the development of silicon pho-
tonics, silicon has been successfully applied for light sources, low-loss
nanoscale waveguides, and high-speed modulators.13However, the
telecommunication response achieved by silicon-based on-chip photo-
detectors is still challenging due to its bandgap limitation, transpar-
ency in the NIR region, and bandwidth limitations. For instance, withincreasing bandwidth demands in an optical network, an optical fiber
network needs to process 100 Gbit/s data rate per channel in the near
further. Therefore, developing all-Si complementary metal-oxide semi-conductors (CMOS) compatible with photodetectors is essential for
the realization of on-chip integrated photonics.
SPP mode provides high electric field enhancement to increase
hot carrier generation on the interface between metal and dielectric.The nanoscale waveguide structure is still the main form of on-chip
hot carrier photodetectors. Compared with free-space plasmonic hot
electron photodetector, the waveguide geometry enables higher inter-nal photoemission, which can be ascribed to SPPs propagating along
with the metal/dielectric interface, and most of the hot carriers are
generated in the vicinity of the Schottky barrier. Also, the SPPs havemore electric field component normal to the interface, leading to hot
carriers generated with an effective momentum perpendicular to the
interface.
The SPP based Schottky photodetector can be formed by placing
metal stripes on silicon.
165The absorption in metals is originated from
SPP modes confined and localized to the M/S interface. The internal
photoresponsivity of 0.38 mA/W and 1.04 mA/W were measured at
the wavelength of 1280 nm for the Au and Al devices, respectively.
And the responsivity can be given by165RxðÞ¼1/C0e/C0al ðÞ ccg
/C22hx; (20)
where ccis the coupling efficiency of this arrangement, lis the length
of the on-chip photodetector, and ais the mode power attenuation.
The electric field is mainly confined to the first micrometer of thestripe for both metals. With the increase of optical wavelength, theconfinement of the electric field decreases gradually. Schottky contact
on local oxidation of silicon SPP waveguide was demonstrated.
42The
photoresponsivity is 0.25 and 13.3 mA/W at a wavelength of 1510 and1310 nm under a reverse bias of 0.1 V, respectively. The surface rough-ness can further optimize the structure at the M/S interface.
Afterward, the author demonstrated a responsivity of 12.5 mA/W at
the wavelength of 1.55 lm, which is about two orders of magnitude
higher than their previous reports.
41Kwon et al. proposed an on-chip
waveguide-integrated Schottky photodetector to enhance light absorp-
tion by using tapered metal nanobricks.166The tapered array struc-
tures with different block widths can gradually tailor the cut-offfrequencies and group velocities of the tightly confined plasmonicmodes for enhanced light absorption and suppressed reflection of the
photonic mode in the silicon waveguide, leading to a responsivity of
0.125 A/W at 1550 nm. Different from the Schottky interfaces on thesilicon waveguides, MIM diodes integrated on the waveguide can alsobe utilized in on-chip hot electron photodetection.
167Although the
sensitivity of the devices is very low, the semiconductor waveguides
are not necessary and thus reduce the complexity of the fabricationprocess, and the MIM diodes can be integrated into any type of wave-guide, including optical fiber and polymer waveguides. A silicon core
fiber can also be employed in on-chip hot electron photodetection, as
shown in Figs. 23(a) and23(b) .
168At the reverse of 0.45 V, the respon-
sivity of 0.226 mA/W operating in 1550 nm has been demonstrated, asshown in Fig. 23(c) . The proposed device can be integrated with other
devices flexibly, and the device scale can be further decreased.
Graphene can be utilized to enhance internal photoresponsivity byp l a c i n gi ta tt h eM / Si n t e r f a c e ,a ss h o w ni n Figs. 23(d) and23(e) .
169
With the reserve bias of 3V, the responsivity of 0.37 A/W at the wave-
length of 1550 nm has been demonstrated, and the avalanche photo-
gain is about 2. The improvement of photoresponsivity is attributed tothe combination of light confinement and hot carrier generated fromthe absorption in graphene. Moreover, hot carrier injection efficiency
through the graphene/semiconductor interface can be enhanced com-
pared to the M/S interface.
The detection speed is an essential parament of photodetection.
Due to the ultrashort relaxation time of hot carriers, some results pre-dict that an extreme detection speed of plasmonic hot electron photo-
detection can be achieved to overcome the speed limitation of
conventional photodetectors. Additionally, the speed is stronglydependent on the local electric field enhancement. On-chip opticalinterconnects are the key to high-performance computing systems. As
one of the most important components, high-speed on-chip hot car-
rier photodetectors have been widely investigated. The detection speedcan be improved by lowering the device capacitance, and the dark cur-rent can be lowered by the reduction of the M/S interface junction
area.
172The maximum responsivity of 4.5 mA/W was measured at the
wavelength of 1550 nm, and a transit-time-limited bandwidth of 1GHz has been demonstrated. By improving the structure via asymmet-ric M/S/M waveguides with a width less than 75 nm, the data reception
is up to 40 Gibt/s, and the responsivity of mA/W was measured at theApplied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-28
Published under license by AIP Publishingwavelength of 1.55 lm.173Salamin et al. proposed a 100 GHz band-
width hot carrier photodetector, with an IQE of 36% and 72 Gbit/sdata reception.
170The structure is shown in Fig. 23(f) ,w h i c hc o n s i s t s
of an MSM slot waveguide with a-Ge as the absorbing core and goldas plasmonic lateral claddings. Light is fed to the photoconductive
plasmonic detector via a Si access waveguide (indicated in pink) byevanescent coupling.
170The optical and direct current fields are con-
fined around the photoconductive Ge waveguide. The inset at the
(a)
(b)(c)
1.8
Experiment
Linear fit
LOCOS
Waveguide
DATA–REF (6.0 μW)
EF=0.1eV
EF=0.2eV
EF=0.3eV
EF=0.4eVDATA–SLG (1.5 μW)
DATA–SLG (6.0 μW)Ohmic
ContactSchottky
PhotodetectorSingle Layer
Graphene
20 μmSi-cored fiber SMF
0.226 mA/W
R-square = 0.971.6
1.4
1.2
1.0
0.8
0.60.40.20.0
8
IAu Au Au AuDC field Optical field
E ExGe Ge40
30
20
10yzMetal stripe
Vx6
1000
100
51 0 1 5 2 0 2 5 3 0 3 5 4 0 4 550 ×10–62
8
86
64
24
2
1
02468 1 0 001234567
0.4
0.3
0.2
0.1
–3 –2 –1 0(d)
(e)
(g)
(h) (j)(i)(f)
Responsivity (A/W)Photocurrent ( μA)
RPph (mA)
Pin (mW)Amplitude (a.u.)Reverse current ( μA)
Optical Input Power (mW)
Silica core
SiliconIncident lightInternal photoemission process
Silica cladding Electric are splicing Detection areaContact area
Voltage (V)
72 Gibit/s160 nmAu
AuElectron
HoleIIILight = 19.5 μW
EQE (%)Voltage (V)
FIG. 23. On-chip plasmonic hot carrier photodetector. (a) Optical image and (b) schematic diagram of hot electron photodetector composited with a silicon co re fiber. An elec-
tric arc splicing is used for the splicing of the silicon core fiber. A silver pad is deposited on the fiber to form an Ohmic contact, and then an Au layer as a d etection area is fab-
ricated on the fiber to form a Schottky barrier. (c) Measured photocurrent versus reverse bias from various optical power. The photocurrent increases approximately linearly
with the input power. Reproduced with permission from Y. P. Huang et al. , Appl. Phys. Lett. 106(19), 191106 (2015). Copyright 2015 AIP Publishing LLC.168(d) Illustration of
waveguide integrated Si/Graphene hot electron photodetector. Graphene is placed between the Si/metal interface. The Ohmic contact is formed by Al e vaporation followed by
lift-off and thermal alloying. (e) The responsivity of on-chip graphene-based hot electron photodetector for various reverse bias. Color solid lin es are fitted by thermionic-field
emission and avalanche multiplication processes to the bias-dependent responsivity. Reproduced with permission from I. Goykhman et al. , Nano Lett. 16(5), 3005–3013
(2016). Copyright 2016 American Chemical Society.169(f) schematic of hot carrier photodetector with a 100 GHz bandwidth. The top inset illustrates the simulated optical and
direct current field. And the bottom inset shows the energy band diagram under bias. (g) Bias-dependent photocurrent and internal quantum efficiency ( IQE) measurement. As
the bias voltage increases, the photocurrent strongly increases, and the extracted IQE rises to 36%. (h) Detected electrical eye diagram with a 72 Gbi t/s data reception. Such
superior performance comes from optical energy confinement in the photoconductive Ge waveguide, resulting in the shortest drift path and small resis tance capacitance prod-
uct. Reproduced with permission from Y. Salamin et al. , ACS Photonics 5(8), 3291–3297 (2018). Copyright 2018 American Chemical Society.170(i) Schematic diagram of gra-
phene hot electron bolometric photodetector. Si is employed as both a semiconductor ridge and a semiconductor buffer material. Graphene is located i n the propagation mode
with the maximum electric field. (j) Response spectra of hot electron bolometric photodetector with different Fermi energies. A responsivity of 1100 mA/W for Fermi energies of
0.1 eV can be attained under a low input power. Reproduced with permission from J. Gosciniak et al. , ACS Omega 5(24), 14711–14719 (2020). Copyright 2020 American
Chemical Society.171Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-29
Published under license by AIP Publishingbottom (ii) shows the schematic of the band diagram of the structure
under bias, indicating the generated hot carriers can be efficiently sep-arated and strongly accelerated by the applied field. Figure 23(g) shows
the photocurrent as a function of applied voltages and extracted IQE(up to 36%). The ability of the data process in the optical communica-
tion applications are demonstrated via a detected electrical eye dia-
gram with a line rate of 72 Gbit/s, as shown in Fig. 23(h) . The superior
performance can be attributed to optical energy confinement in thephotoconductive Ge waveguide, which enables the shortest drift pathsfor photogenerated carriers and a very small resistance-capacitance
product.
170Gosciniak et al. proposed a waveguide-integrated plas-
monic graphene photodetector based on the hot carrier photo-bolometric effect, which has a high responsivity of 1100 A/W, andhigh operating speed on the scale of hundreds of GHz, as shown inFigs. 23(i) and23(j).
171The author attributed the high performance to
band nonparabolicity of graphene. This effect allows for efficient
absorption in graphene over a short distance and, subsequently, a largechange of conductivity, indicating its application for high-speed com-munication systems.
C. Plasmon-enhanced hot hole photodetector
Compared with hot electrons above the Fermi level, hot holes are
mainly distributed at the upper edge of the noble metal d-band below
the Fermi level. In the NIR region, lower electron-electron and
electron-phonon scattering for hot holes lead to a larger MFP in con-trast to hot electrons.
60,174,175Because of the distinct difference in
intrinsic properties, the properties of plasmonic hot hole photodetec-tors are different from that of hot electron photodetectors in many
aspects, such as responsivity, dark current, and detectivity. Typically, a
low height Schottky barrier formed by metal/p-semiconductor contactprovides a more effective hot carrier injection.
60,83,176,177However, due
to their low Schottky barrier, the dark current of hot hole photodetec-tors is larger, given by J
d¼A/C3/C3T2expð/C0DEb=kBTÞ, and usually
operates under low temperature.60,80,177Where A/C3/C3denotes the effec-
tive Richardson constant, and Tis the operating temperature.
Although plasmonic hot hole photodetectors have a larger photores-ponse and broader detection band, a large dark current still limits theirresearch progress. A recent study showed that, by inserting a hexago-nal boron nitride (h-BN) insulating layer in MIM structure, the dark
current could be significantly reduced.
178Also, it should be noted that
plasmonic hot hole photodetectors have a coherent characteristic simi-lar to hot electron photodetector, including the spectral tunability andon-chip integration.
83Accordingly, only considering the influence of
dark current on the device, the detectivity reflecting the ability to dis-
tinguish weak signals from the noise is written as D/C3¼RðxÞ=ffiffiffiffiffiffiffiffi ffi2qJdp.
A higher Schottky barrier for Au/P-Si contact can be obtained by wetlithography, leading to an extremely low reverse leakage current at theexpense of bandwidth.
179In this regard, some typical plasmonic hot
hole photodetectors would be introduced.
Gold contact with doped p-type silicon can form a Schottky bar-
rier of 0.32 eV, smaller than conventional n-type contact, and aresponsivity of 13 mA/W at 1.55 lm was measured.
176The photores-
ponse of hot hole photodetector is one order of magnitude higher than
that of hot electron counterparts. An interfacial layer can be used toregulate the injection of hot carrier between the metals and p-type sili-con.
180The photoresponse of pSi/Au is compared with TiO 2–x.A s
shown in Fig. 24(a) ,t h eT i O 2–xlayer dramatically changes thephotoresponse between 1160 and 1400 nm by over one order of mag-
nitude. The existence of TiO 2–xassisted hot hole transport due to deep
level traps, as the energy diagram shows in Fig. 24(b) .T i Nc a na l s ob e
used in hot hole extraction, and a larger photocurrent was measured
due to higher light absorption and an ultrathin TiO 2/C0xinterfacial
interlayer formed between TiN and silicon.180An SrTiO 3interlayer
was employed to improve device stability, as shown in Fig. 24(c) .
Although a higher barrier is formed between the metals and p-type sil-icon with decreased photoresponse, dark current noise is an order of
magnitude lower by using a 5-nm SrTiO
3interlayer.181The responsiv-
ity noise as a function of the bias voltage is plotted in Fig. 24(d) .T h e
root mean square (RMS) of the responsivity is written as
RMS¼6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2
photoþr2
darkq
=Pin; (21)
where rphoto andrdarkrepresents the photoinduced current and dark
current, P inis the power intensity of the incident light. Moreover, the
SrTiO 3sample exhibits a stale on-off switch, as compared in Fig.
24(e) . Due to the perfect lattice matching formed by the epitaxial
growth of SrTiO 3, the device exhibited strong electrical stability of the
response under reverse bias as high as 100 V. The use of an interfacial
layer is an effective route to improve the performance of plasmonic
hot hole photodetectors. However, most of them require sacrificing
part of photoresponse by increasing the Schottky barrier height.
Therefore, balancing the relationship between the responsivity and the
detectivity is essential to take advantage of plasmonic hot hole photo-detectors, such as a longer MFP and a lighter effective mass in silicon.
Tagliabue et al. elucidated the relative advantages and limitations of
the hot hole and hot electron devices.
182They found that hot hole p-
type semiconductor was favored for plasmonic photodetection across
the visible and ultraviolet regimes. The schematic of the proposed hot
c a r r i e rp h o t o d e t e c t o r si ss h o w ni n Fig. 24(f) .B ye v a l u a t i n gt h eI Q Eo f
hot carrier devices, they revealed that the IQE of hot electron photode-
tector decreased upon exceeding the interband threshold of the metal.
In contrast, the IQE of hot hole devices increased as the favorable
energy distribution of d-band holes, as shown in Fig. 24(g) . Finally, the
performance comparison between hot electron and hot hole devices
are shown in Table II . Although hot hole photodetectors can realize a
higher responsivity, the relationship between detectivity and respon-sivity needs to be balanced.
The performance of a series of typical hot carrier photodetectors
is summarized in Table III . EQE is calculated by the ratio of collected
carriers to incident photons.
IV. CONCLUSION AND OUTLOOK
In conclusion, we have reviewed various approaches to engineer
hot carrier dynamics. Moreover, the recent development of selected
novel hot carrier photodetectors is discussed, including planar hot
electron photodetectors, hot electron photodetectors coupled with
low-dimension materials, and functional hot carrier photodetectors.
Plasmon-enhanced hot carrier photodetectors have made remarkable
progress in both theoretical and experimental research. So far, most of
them, however, still suffer from relatively low internal photoemission
efficiency when compared with commercial NIR photodetectors.
Emerging research has demonstrated plasmon-enhanced hot carrier
photodetector with responsivities >1 A/W, comparable with commer-
cial Si-Ge or InGaAs NIR photodetector.
83Their novel functionalitiesApplied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-30
Published under license by AIP Publishingare particularly attractive, such as CPL detection, nanoscale surface
imaging, and sensing, which does not exist in conventionalphotodetectors.
In general, to achieve practical applications of hot carrier photo-
detection, one needs to pay attention to two important aspects. First,
the performances of plasmonic hot carrier photodetectors strongly
depend on the process of hot carrier generation, transport, and extrac-tion. However, limited by the current understanding of hot carrierdynamics, achieving a high-performance hot carrier photodetector isstill challenging. Second, solutions for integration (with current
CMOS technology or next-generation integrated photonics), largescale production, dynamically tailored responsivity, and new function-alities need to be addressed to facilitate the development of the hot car-rier photodetectors to compete with current commercial systems forpractical applications. For future progress of hot carrier photodetec-tion, we anticipate the following directions will prevail.
Most existing research on hot carrier photodetection is based on
conventional plasmonic materials, such as gold and silver. However,
(a) (b)
(c)
(d)(e)101
10–7
10–8
10–9
10–10
10–11
02468 1 0100
10–1
10–2
1100 1200 1300 1400pSi/TiN
pSi/Au
pSi/TiO2 – x/Au
pSi/TiO2 – x/Ti
TiO 2–xEvac
EFTiO 2-xpSi
4.2 eV
0.85 eV
0.28 eV
metal trap states4.0 eV
0.9 eV0.15 eV
CNL
0.75 eV
0.65 eV0.2 eVTiN
1500
1.0
0.5
0.0
0.0 a p-Si 0.5 1.01600
(g) (f)Photo responsivity |PR| (mA/W)
Wavelength (nm)
With SrTiO3Photo current (a.u.)
Time (sec)
IQE IQECopper
GaN semiconductor1.0
0.5
0.0
0.0 0.5
Hot Hole Device
Hot Electron DeviceCu/p-GaN
Cu/n-GaN1.0Photo current (a.u.)
Time (sec)
Bias voltage (V)RMS (A/mW)
interband
1.6 2.2 2.8interbandW/O SrTiO3
Photon Ener gypSi
Ti
pSi
TiO 2–x
SrTiO3
(5 nm)Au
pSiAu
pSi
FIG. 24. Plasmon-enhanced hot hole photodetector. (a) Response spectra of hot hole photodetectors composited with different materials. The additional TiO 2–xfilm between
P-Si and Au dramatically enhances the photoresponse ranging from 1160 to 1400 nm by over one order of magnitude. (b) Energy diagram of the P-Si/TiO 2–x/metal structure.
Due to the deep level traps, the actual TiO 2–x/metal interface promotes hot hole transport in the sub-bandgap region. The inset shows hot carrier migration. Reproduced with
permission from N. A. G €usken et al. , ACS Photonics 6(4), 953–960 (2019). Copyright 2019 American Chemical Society.180(c) Schematic of plasmonic hot hole photodetector
with a SrTiO 3interlayer and electrical configuration. The layer is epitaxially grown on p-type silicon substrates, and then, the nanostructures are fabricated b y two-step electron
beam lithography. (d) RMS responsivity noise as a function of bias voltage. Four devices with nanograting arrays corresponding to the pitches of 360 ( blue), 380 (red), 400
(green), and 420 (yellow) nm are used to calculate the RMS responsivity noise. (e) Comparison of time-dependent photocurrent measurement with and wi thout a SrTiO 3inter-
layer. The device with SrTiO 3shows a more stable on-and-off switching in contrast to that without SrTiO 3. Reproduced with permission from T. Matsui et al. , Adv. Funct. Mater.
28(17), 1705829 (2018). Copyright 2018 Wiley-VCH.181(f) Schematic of Cu/GaN photodetector. An array of ultrathin Cu nanoantennas is fabricated on p-GaN for hot-hole col-
lection or n-GaN for hot electron collection. (g) Comparison of IQE between hot hole (top) and hot electron (bottom) devices. It illustrates that the t ransition from intraband to
interband excitations causes an abrupt decrease in IQE for hot-electron devices and an abrupt increase in IQE for hot-hole devices. Reproduced with p ermission from G.
Tagliabue et al. , ACS Nano 14(5), 5788–5797 (2020). Copyright 2020 American Chemical Society.182Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-31
Published under license by AIP Publishingthey have drawbacks, such as narrow plasmonic resonances, inability
to tailor material properties, and large work function, especially forgold. One can look into non-metallic materials—but they are plas-monic—for hot carrier photodetection, such as transitional metalnitrides, transitional metal carbides/borides, MXene, transitional metaloxides, or even highly doped semiconductors. For example, TiN alsohas a longer MFP, broadband absorption property, higher temperaturestability, and CMOS-compatibility; ultrafast hot electron transfers(<50 fs) were observed in non-noble metal plasmonic F and In
co-doped CdO nanocrystals.
185At present, few works are evaluating
hot carrier dynamics in non-metallic materials, not to mention theiruse as a hot carrier photodetector.
Dynamically tunable hot carrier photodetectors are promising
for practical applications. Bai et al. proposed phase-coupled simulta-
neous coherent perfect absorption to control hot-electron generationand photodetection.
186The peak of the spectrum of responsivity for
antisymmetric and symmetric incidences was switched to the samewavelength via altering the phase coupling only. Gao et al. controlled
the height of the Schottky by reducing the amount of chemisorbedoxygen using external ultraviolet light.
187Nevertheless, reports on
dynamically tunable hot carrier photodetectors are still very limited.Graphene, Dirac semimetal Cd
3As2, and ITO can be used as tunable
plasmonic materials controlled by the external field, which provideflexibility for broadband applications of photodetection.Interfacial states play an important role in hot carrier photodetec-
tion. For instance, PICTT can be influenced by interfacial states, such
as defects and imperfections at the interface, the strain of the metal
surface, or adsorption at the atomistic scale. Nonetheless, how to applythe interfacial states for charge transfer requires further investigation.Ultrafast spectroscopy is expected to apply to study hot carrier dynam-ics affected by interfacial states, including carrier lifetime and transfer.
So far, hot carriers are primarily generated within plasmonic
NPs, where hot carrier localization coincides spatially with positionoptical excitation. However, some applications, e.g., integrated nano-photonics, need delocalized hot carrier production to achieve a large-
distance spatial distribution. A recent study demonstrated a remote
generation of hot electrons by launching a propagative SP on a goldwaveguide.
188The remote hot carriers were generated at distances of
several micrometers from the excitation. It is also demonstrated that
remote spectroscopy enables the “copy and paste” of plasmonic hot
spots. The excitation-collection-separated enhanced spectroscopyusing a matched nanoantenna pair may be utilized to “copy and paste”hot carrier generation.
189
It should be noted that on-chip hot carrier photodetectors
with the feasibility of integrated nanophotonics exhibit outstand-
ing properties in both responsivity and detection speed. On-chipdetectors are crucial in optical communication systems. Since pho-todetectors for Si-photonics are facing bandwidth limitations, theTABLE II. Performance comparison between hot electron detector and hot hole detector. Performance comparison between hot electron detector and hot hole dete ctor.
Comparison Hot electron detector Hot hole detector
The Schottky barrier height Higher (Au/n-Si /C240.8 eV) Lower (Au/p-Si /C240.32 eV)
Generated carriers through the analysis
of initial energy distributionLess high energy carriers More high energy carriers
Carrier transport Shorter MFP ( /C2422 nm) Longer MFP ( /C2447 nm)
Responsivity Lower (1.72 mA/W@1200 nm) Higher (27.49 mA/W@1200 nm)Responsive bandwidth Shorter (cutoff at 1550 nm) Longer (cutoff at 3900 nm)Dark current Lower (3.56 /C210
–7A/cm2) Higher (11.32 A/cm2)
Detectivity Higher (5.095 /C2109Jones) Lower (1.444 /C2107Jones)
aThe performance comparison between hot electron detector and hot hole detector with the same structure, taken from Ref. 60.
TABLE III. Summary of the performance of typical hot carrier detectors.
Type and structureBarrier height
(eV)Responsivity
(mA/W)Dark current/
dark current density Detectivity EQE Ref
Au NPs/n-Si pyramid 0.8 8.71 @1200 nm 1.2 /C210–5A/cm–24.39/C2107Jones (cal.)a0.9% 110
Al/ n-Si pyramid N/A 30 @1064 nm 10–7A N/A 3.5% 109
Au grating/n-Si 0.5 0.6 @1450 nm N/A N/A 0.05% 183
TPs based Au/n-Si 0.8 1.72 @1200 nm 3.56 /C210–7A/cm25.095/C2109Jones 1.8% 60
Au/n-Si waveguide 0.76 0.38 @1280 nm 10 nA N/A 0.03% 38
TPs based Au/p-Si 0.32 27.49 @1200 nm 11.32 A/cm21.444/C2107Jones 2.84% 60
Au grating/p-Si 0.32 13 @1550 nm 11.32 A/cm2(cal.) 6.828 /C2107Jones (cal.)a1.04% 176
Au/p-Si waveguide 0.33 1 @1310 nm N/A N/A 0.09% 184
Au/oxide/Si 3.8 29.3 (4 V bias) N/A 101113.5 % (4 V bias) 161
aCal. denotes the results calculated from the relevant data in reference following the equation in this article.Applied Physics Reviews REVIEW scitation.org/journal/are
Appl. Phys. Rev. 8, 021305 (2021); doi: 10.1063/5.0029050 8, 021305-32
Published under license by AIP Publishingultrafast hot carrier dynamics provides a route to achieving high
bandwidth >100 GHz.170
Finally, we would like to point out that there are contradictions
among hot carrier research works. For example, the theory suggested
that small NPs are beneficial for generating more efficient hot elec-
trons with large energies and low transmission loss.63But, in contrast,
larger NPs generate more hot electrons and will give an overall large
photocurrent proven by an experiment, since the photocurrent is also
determined by surface.71Therefore, to engineer useful hot carrier devi-
ces, we need to consider the impact of optoelectronic conversion in all
aspects. Indeed, theoretical literature indicated that a hot electron pro-cess is inefficient. Further research on the mechanism of plasmon-
enhanced photodetection is desirable at the level of hot carrier dynam-
ics, such as directly probing the electron spatial, energy, and temporal
distributions, rather than attributing yielded photocurrent to putative
hot electrons.
190Although some models use classical theory to predict
hot electron dynamics, we consider the quantum theory plays an
important role. For instance, the term hot carriers with high-energy in
the nanostructure is derived from quantum transitions near the surfa-
ces. The momentum of hot carriers is not conserved due to the quan-
tum effect of dynamic scattering of electrons at the boundary.66Based
on the quantum theory developed recently, the number of hot carriers
with energies that approximate /C22hxnear the surface of the nanostruc-
ture is given by jEnormalj2=x4.191Additionally, hot carrier relaxation
through electron-phonon scattering is considered as an impediment
for a highly efficient hot carrier device; however, the quantum process
of phonon-assisted carrier generation and the transition has been pro-
posed in recent research.174Therefore, we believe that the advances in
quantum theory would bring many unexpected opportunities and evo-
lution to hot carrier photodetection.
AUTHORS’ CONTRIBUTIONS
All authors contributed equally to this manuscript. All authors
reviewed the final manuscript.
ACKNOWLEDGMENTS
The authors acknowledge the support from the National Key
Research and Development Program (No. 2019YFB2203400), the
“111 Project” (B20030); P. Y. was funded by China Postdoctoral
Science Foundation (2019M663467), NSFC (62005037) and
Sichuan Science and Technology Program (2020YJ0041).
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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Published under license by AIP Publishing |
5.0049804.pdf | J. Appl. Phys. 129, 153903 (2021); https://doi.org/10.1063/5.0049804 129, 153903
© 2021 Author(s).Increase of Gilbert damping in Permalloy
thin films due to heat-induced structural
changes
Cite as: J. Appl. Phys. 129, 153903 (2021); https://doi.org/10.1063/5.0049804
Submitted: 09 March 2021 . Accepted: 08 April 2021 . Published Online: 21 April 2021
Frank Schulz ,
Robert Lawitzki ,
Hubert Głowiński ,
Filip Lisiecki ,
Nick Träger , Piotr Kuświk , Eberhard
Goering , Gisela Schütz , and
Joachim Gräfe
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Cite as: J. Appl. Phys. 129, 153903 (2021); doi: 10.1063/5.0049804
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Submitted: 9 March 2021 · Accepted: 8 April 2021 ·
Published Online: 21 April 2021
Frank Schulz,1,a)
Robert Lawitzki,2
Hubert G łowiński,3
Filip Lisiecki,3
Nick Träger,1
Piotr Ku świk,3
Eberhard Goering,1Gisela Schütz,1and Joachim Gräfe1
AFFILIATIONS
1Max Planck Institute for Intelligent Systems, 70569 Stuttgart, Germany
2Department of Materials Science, University of Stuttgart, 70569 Stuttgart, Germany
3Institute of Molecular Physics, Polish Academy of Sciences, PL-60179 Pozna ń, Poland
a)Author to whom correspondence should be addressed: fschulz@is.mpg.de
ABSTRACT
Spin-wave based computing requires materials with low Gilbert damping, such as Ni 80Fe20(Permalloy) or yttrium iron garnet, in order
to allow for spin-wave propagation on a length scale comparable to the device size. Many devices, especially those that rely on spin –orbit
effects for operation, are subject to intense Joule heating, which can exacerbate electromigration and induce local phase changes. Here,the effect of annealing on the Gilbert damping coefficient αof 36 nm Py thin films grown on a Si substrate is examined. Ferromagnetic
resonance measurements, high resolution transmission electron microscopy, as well as energy dispersive x-ray spectroscopy have been
employed to determine αwhile also studying structural changes in the thin films. The Gilbert damping parameter was found to increase
sixfold when annealed at 350
/C14C, which was linked to the diffusion of Ni atoms into the Si substrate on a length scale of up to 50 nm.
The results demonstrate that magnonic devices have to be treated with caution when Joule heating occurs due to its detrimental effectson the magnonic properties, but the effect can potentially be exploited in the fabrication of magnonic devices by selectively modifyingthe magnonic damping locally.
© 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/ ).https://doi.org/10.1063/5.0049804
I. INTRODUCTION
The emerging field of magnonics has attracted great attention
due to the possibility of wave-based information processing, utiliz-
ing the amplitude, as well as the phase of spin waves, without thedrawback of heat production caused by moving electrons.
1–4This
may give spin-wave based computing an edge over conventionalCMOS technology for certain applications, such as neuromorphiccomputing and low energy consumption devices.
5,6
For the realization of those devices, materials with low Gilbert
damping are required, allowing for spin waves to be transmitted
coherently over distances that are comparable to the device size.
One common material that has been utilized for numerous
magnonic devices is Permalloy, an alloy consisting of approxi-
m a t e l y8 0 %N ia n d2 0 %F e .7–9It can be grown on Si substrates
using various techniques, such as magnetron and ion beamsputtering.
10,11When combining magnonics with spintronics, which offers
a large additional tool set for the control and manipulation of
spin waves, the required current densities are usually very high,
heating up the sample significantly.12This rise in temperature
can lead to irreversible structural changes within the magneticthin films rendering the device unusable, which will be discussedin detail in this work.
While the effects of vacuum annealing on magnonic devices
have not received much attention in the literature, which this workattempts to address, the structure of Ni thin films grown on a Si sub-strate has previously been studied by Julies et al.
13They observed the
formation of nickel silicides, with Ni –Si forming at approximately
350/C14C, well below the eutectic temperature14of the system. In their
work, experimental evidence pointed to the fact that Ni was themajor moving species during the growth of the silicides.
The effect of atomic composition on the magnetic properties
of homogeneous 3 dtransition-metal binary alloy thin films hasJournal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 129, 153903 (2021); doi: 10.1063/5.0049804 129, 153903-1
©A u t h o r ( s )2 0 2 1been studied by Schoen et al.15,16However, the link between struc-
tural changes and changes of the magnonic properties in thin films
has not been established yet, especially with regard to heat-inducedstructural phase transitions. In this article, we show how theGilbert damping coefficient αof Py thin films grown on Si is
affected by annealing at different temperatures, up to 350
/C14C using
ferromagnetic resonance (FMR) measurements. The same samples
are also analyzed using high resolution transmission electronmicroscopy (HR-TEM), combined with energy dispersive x-ray(EDX) spectroscopy, in order to obtain information about theirstructural properties and the diffusion of different atomic species
within the thin film system.
Although the observed effect seems undesirable at first glance,
it has potential applications in magnonics to locally modify thedamping of a device. This could be done using a focused laser spotor the tip of an atomic force microscope (AFM) probe to obtain
structures on a nanometer length scale, opening up many possibili-
ties for new magnonic devices that require alternating magnonicproperties, such as magnonic crystals.
17
II. EXPERIMENTAL DETAILS
As i n g l e1 0 /C210 mm2sample has been prepared by deposit-
ing 36 nm of Ni 80Fe20on a Si substrate using ion beam sputtering
with an Ar source. The process was carried out at room tempera-
ture, with a base pressure of 3 :9/C210/C07mbar and a working pres-
sure of 1 :53/C210/C04mbar. The sample was then cut into four
5/C25m m2pieces, which were subsequently annealed separately
for 60 min at Ta¼100, 200, 300, and 350/C14C in ultrahigh vacuum
(,4/C210/C07mbar). The unannealed sample that was measured as
deposited is labeled with Ta¼20/C14C.
The FMR measurements have been performed using a vector
network analyzer (VNA-FMR) and sweeping the external mag-netic field at fixed frequencies between 2 and 25 GHz. This was
done for all four samples before and after annealing. The real and
imaginary parts of S
12were then modeled using a single Lorentz
function for both parts. The g-factor and effective magnetizationM
effhave been extracted using the Kittel formula,18while the
damping parameter has been determined by a linear fit to the res-
onance linewidth over the frequency.19,20
For the preparation of a TEM specimen, the dual beam focused
ion beam (FIB) Nova 600 NanoLab by FEI was used to lift out thinlamellas and attach them to a copper grid using a Pt source.
TEM bright field images were recorded with a Philips CM-200
FEG TEM operated at 200 kV. Complementary EDX spectra wererecorded using an ultra-thin window EDX spectroscopy system fromEDAX to determine the samples ’composition. Elemental mappings
were collected with a probe size of /C253:5 nm, a step size of /C252n m ,
and a dwell time between 5 and 30 ms per pixel. Using these maps,
elemental cross sections were generated by averaging the counts overseveral pixels. In addition, the maps were also utilized to quantifythe elemental composition of different structural phases.
III. RESULTS AND DISCUSSION
Figure 1 shows the FMR linewidth ΔHas a function of reso-
nance frequency ffor the unannealed sample, as well as the four
different annealing temperatures. The plot shows the experimentaldata, which were acquired at certain fixed frequencies, as well as a
fit using a function of the form ΔH¼(4πfα)=(γμ
0)þΔH0, with
the Gilbert damping parameter α, the gyromagnetic ratio γ, and
the vacuum permeability μ0. The data set of the unannealed sample
and the ones annealed at 100 and 200/C14C show no differences
within the error margins, meaning that they are not discernible in
the plot. From the slope of the curve, we can determine the
damping of the sample, which is shown in Fig. 2 . The measure-
ments of the samples annealed at 300 and 350/C14C show a different
slope, as well as clear deviations from the linear behavior. Such
nonlinear behavior has been observed before in the context of
extrinsic contributions to the FMR response of ultrathin films andthe closely related two-magnon model of scattering.
21–24
Figure 2 shows how the Gilbert damping coefficient
changes when the thin films are annealed at different tempera-
tures Tafor 60 min.
The unannealed sample shows a low damping constant of
0.007 in accordance with literature values.16The samples main-
tain this low damping up to an annealing temperature of 200/C14C,
indicating that their structure is unchanged. When annealed at
300/C14C,αincreases drastically from 0.007 to 0.030. Higher
annealing temperatures increase αeven further, yielding a value
of 0.046 for Ta¼350/C14C. In addition to α, the FMR measure-
ments were used to determine the effective magnetization μ0Meff
of the samples, which is shown in Fig. 3 .μ0Meffshows a trend
opposite to the one of αat higher temperatures. While μ0Meff
also stays constant up to 200/C14C, it is then reduced significantly
for annealing temperatures of 300 and 350/C14C, with a stronger
decrease for higher temperatures.
Figure 4 shows bright field TEM images, together with the
respective EDX elemental map and a cross section of atomic
FIG. 1. FMR linewidth ΔHas a function of resonance frequency ffor different
annealing temperatures Ta, colors according to the legend. Squares represent
experimental data, and solid lines represent fits. Values for 20, 100, and 200/C14C
are too close to be discerned on this scale.Journal of
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©A u t h o r ( s )2 0 2 1composition as derived from the EDX map for samples annealed
at 100, 200, 300, and 350/C14C. The unannealed sample, which was
measured as deposited, looks identical to the one annealed at100
/C14C and was omitted here. The spatial resolution of the EDX
images is not sufficient to make individual grains visible, and thecolor distribution thus only indicates the presence of a certain
species within the layers and does not yield any information on
the grain size or inhomogeneities. The bright field image of thesample annealed at 100
/C14Ci n Fig. 4(a) shows three distinct
phases with small transition regions in between. These regionsare an artifact arising from the finite thickness of the sample,
combined with a small tilt of the sample with respect to the
cross-sectional plane. When looking at the samples in transmis-sion, the different layers then seemingly overlap. From top tobottom, the three phases correspond to Pt, Ni –F e ,a n dS i ,a sc a n
also be seen in the elemental map in the right part of the figure.
The Pt layer has been deposited during the preparation of the
TEM lamella as a protective layer. The respective cross sectionshows that the Ni –Fe layer consists of approximately 75% Ni and
20% Fe. For better quantification of the atomic composition, theEDX spectra were also averaged over each region individually,
resulting in improved statistics. Using this method, the Ni –Fe
layer was determined to contain 71% Ni, 22% Fe, 6% Si, and 1%P t .T h el a r g eS is i g n a li se x p e c t e dt oc o m ef r o mt h es c a t t e r i n go felectrons into the large adjacent Si substrate.
Figure 4(b) shows the TEM image and EDX mappings for a
sample annealed at 200
/C14C .T h eb r i g h tf i e l di m a g el o o k ss l i g h t l yd i f -
ferent, which can be attributed to a change in the contrast, focus, aswell as the lamella thickness. However, it does show the same kind ofstructure, which is also validated by the EDX mappings. The crosssection shows a composition of the Ni –Fe layer very similar to the
sample annealed at 100
/C14C. When averaging over the entire Ni –Fe
layer, we obtain values of 69% Ni, 22% Fe, 8% Si, and 1% Pt for theatomic composition. Since all four samples were cut from a singlelarge sample after ion beam sputtering, it can be expected that the
composition of the interlayer should not vary too much aside from
minor inhomogeneities over the sample area of 10 /C210 mm
2.
InFig. 4(c) ,w ec a ns e ead r a s t i cc h a n g ei nas a m p l es t r u c -
ture. When annealed at 300/C14C, a fourth phase forms between the
Ni–Fe and Si layers. The EDX spectra reveal that this fourth
phase consists of Ni and Si, confirming the results of a previous
study13that silicides form at these temperatures. It is evident that
Ni is the moving species in this case, migrating from the Ni –Fe
layer into the Si layer, leaving behind a Ni –Fe layer with a
lowered fraction of Ni. Averaging over the Ni –Fe layer, we find
that it now contains 52% Ni, 36% Fe, 11% Si, and 1% Pt. It
remains unclear whether the increased amount of Si is an experi-mental artifact or an actual indication of Si moving into the Ni –
Fe layer. When averaging over the newly formed Ni –Si layer, we
obtain values of 47% Ni, 3% Fe, 50% Si, and 0% Pt.
Figure 4(d) shows the results for the sample annealed at
350
/C14C. Again, the bright field image looks quite different due to
changed contrast, focus, and lamella thickness. However, the EDXmapping still shows the same trend as the one for the sample
annealed at 300
/C14C. Once gain, Ni atoms have migrated from the
Ni–Fe layer into the Si substrate, leaving behind a Ni –Fe layer with
altered atomic composition. When averaging over the differentregions, we find that the Ni –Fe layer contains 38% Ni, 38% Fe, 24%
Si and 0% Pt. The newly formed Ni –Si layer yields values of 55%
Ni, 1% Fe, 44% Si and 0% Pt. Although Ni is the species moving
into the Si substrate, Fe atoms also move in the opposite direction.This can be seen by looking at the change in the thickness of theNi–Fe layer when the sample is annealed at 300 or 350
/C14C. Without
annealing and up to temperatures of 200/C14C, the Ni –Fe layer was
measured to be 36 +1 nm in thickness without showing any tem-
perature dependence while also maintaining good homogeneity. At300
/C14C, the thickness of the Ni –Fe layer was reduced to 24 +2 nm,
FIG. 3. Effective magnetization μ0Meffas a function of annealing temperature
Tawith error bars. Colors chosen to match the other figures.
FIG. 2. Gilbert damping coefficient αas a function of annealing temperature Ta
with error bars. Colors chosen to match the other figures.Journal of
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©A u t h o r ( s )2 0 2 1and the newly formed Ni –Si layer has a thickness ranging from 29
to 53 nm. When annealed at 350/C14C, the Ni –Fe layer shrinks to a
size of 20 +3 nm, and the Ni –Si layer grows to thicknesses
ranging from 35 to 61 nm. This means that not only do Ni atomsmove into the Si layer, but Fe atoms are also moving away from the
Si/Ni –Fe interface, resulting in an Ni –Fe layer with reduced thick-
ness, which will inevitably change the crystallographic order of theNi–Fe layer, which is also recognizable in the bright field images,
where the samples annealed at 300 and 350
/C14C show less homoge-
neous Ni –Fe layers, indicating grain formation.
Figure 5 shows dark field images of the samples annealed at
100, 200, 300, and 350/C14C. Individual grains are visible in each
recorded picture. The decreasing size of the Ni –Fe layer, which was
already observed in the bright field images, is also evident in thedark field images, with layer thicknesses of approximately 35, 35, 25,
and 20 nm for annealing temperatures of 100, 200, 300, and 350
/C14C,
respectively. The sample annealed at 100/C14C shows grains that are
significantly smaller than the thickness of the Ni –Fe layer, and the
same thing is true for the sample annealed at 200/C14C. When
annealed at 300/C14C, however, the grains slightly increase in size, and
the Ni –Fe layer shrinks, resulting in grain sizes on the order of the
FIG. 4. TEM bright field images together with the respective EDX image and the cross section of atomic composition as derived from EDX spectra for samples anne aled
for 60 min at (a) 100/C14C, (b) 200/C14C, (c) 300/C14C, and (d) 350/C14C. Red denotes the signal coming from Pt, blue from Ni, yellow from Fe, and green from Si.
FIG. 5. TEM dark field images of the samples annealed for 60 min at (a)
100/C14C, (b) 200/C14C, (c) 300/C14C, and (d) 350/C14C. Light blue dashed lines mark
the boundaries of the Ni –Fe layer.Journal of
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©A u t h o r ( s )2 0 2 1layer thickness. This is even more pronounced in the dark field
image of the sample annealed at 350/C14C. The Ni –Si layer is not well
discernible in the dark field images due to the relative alignment ofthe detector, the Ni –Si layer, and the electron beam.
The results of the FMR measurements demonstrate that
devices with Ni
80Fe20thin films exposed to intense Joule heating
may lose their desired magnonic properties, such as low damping
and high effective magnetization. Combining the results from theFMR measurements and the TEM and EDX study, we can see thatthe increase in Gilbert damping as well as the decrease in μ
0Meff
correlate with structural changes in the thin film system, namely,
the migration of Ni atoms from the Ni –Fe layer into the Si sub-
strate and Fe atoms moving away from the Si/Ni –Fe interface. The
bright and dark field images both show that the Ni –Fe layer
decreases in size significantly for higher annealing temperatures,while the grain size slightly increases.
Earlier works have shown that Ni readily diffuses into an adja-
cent Si layer at temperatures above 200
/C14C, forming nickel silicides
in the process.13Julies et al. find that Ni 2Si is formed at tempera-
tures above 200/C14C, while increasing the temperature to 350/C14C
causes Ni –Si to form, which the TEM pictures confirm. This newly
formed Ni –Si could interact magnetically with the Ni –Fe, which in
turn could affect the damping of the system; however, in previousstudies, Ni –Si compounds have been found to be almost exclusively
non-magnetic,
25,26which makes this scenario highly unlikely.
Another way in which the Ni –Si layer could increase the damping
of the system is by means of spin pumping, where the precession ofmagnetization in the Ni –Fe layer is damped by the transfer of the
magnetic moment and energy to the itinerant charge carriers of theNi–Si layer.
27,28Further studies have to be conducted in order to
measure the magnitude of this effect.
Other works have investigated the effect of composition of 3 d
transition-metal binary alloys on their magnetic properties andfound that the effective magnetization of Ni –Fe alloys, also mea-
sured by FMR, decreases steadily as a function of Ni concentration
in the fcc regime.
15This trend is opposite to the one we observed
here. In a related work, the damping has been studied as a functionof the composition of Ni –Fe alloys.
16They also obtained a value of
0.0073 for αin an Ni 80Fe20alloy; however, their data showed an
increase in αfor higher amounts of Ni. In our samples, the
damping parameter increased drastically for higher annealing tem-
peratures, where the EDX data indicated Ni atoms migrating out ofthe Ni –Fe layer into the Si substrate, leaving behind a Ni –Fe layer
with significantly reduced Ni concentration. The opposite trends
forαmost likely result from the fact that Schoen et al. studied
homogeneous polycrystalline samples, while the present samplesexhibit grain formation, as can be seen in the TEM pictures. TheEDX measurements have indicated a migration of Ni atoms out ofthe Ni –Fe layer, which resulted in a layer with increased Fe
content, and since the annealing temperatures are well below the
eutectic temperature of the system, Ni and Fe will not form analloy, resulting in the formation of Fe grains.
29
These inhomogeneities contribute to the increase of Gilbert
damping as measured by FMR through a mechanism that can
either be described by the two-magnon model or the local reso-
nance model depending on the ratio of grain size and filmthickness.
21–24,30These effects arise due to the local variation of theanisotropy fields in inhomogeneous thin films with grains, and
the non-linearity in the frequency dependence of the FMR line-
width that is associated with this effect is evident in the presentFMR measurements. These mechanisms are not associated withthe coupling of the magnons to a thermal bath, meaning that the
scattering from the FMR mode ( k¼0) into higher order magnon
modes ( k=0) is reversible. However, these higher order modes
will still also dissipate to the lattice, contributing to the observedincrease in damping.
24,31
IV. SUMMARY
In conclusion, a more than sixfold increase of the Gilbert
damping has been observed in 36 nm thin Ni 80Fe20layers grown
on Si when the samples are annealed above 300/C14C, as well as a
reduction of the effective magnetization by 28%. These findingshave been linked to structural changes of the sample, studied byTEM and EDX measurements. These revealed that Ni atoms
migrate from the Ni –Fe layer into the Si substrate, forming a Ni –Si
layer of thicknesses up to 61 nm, while Fe atoms are pushed backfrom the Ni –Fe/Si interface. The changes of magnetic properties
after annealing have been attributed to the formation of Fe grains at
higher annealing temperatures due to the migration of Ni atoms
into the Si layer. These structural changes could result in anincreased damping parameter by means of two-magnon scatteringinduced viscous Gilbert damping. Thus, the results show that Pythin films can undergo irreversible changes that have fatal effects on
their magnonic properties and on the limit of their applicability in
magnonic devices, which should be taken into account whendesigning new devices. However, this seemingly unfavorable effectcan potentially be exploited in the fabrication of magnonic devicesto locally modify the damping of a device. This could be used, for
example, to induce a damping gradient to avoid unwanted reflection
of magnons in a magnon absorber.
ACKNOWLEDGMENTS
The authors want to thank Ulrike Eigenthaler for the prepara-
tion of the TEM lamellas.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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©A u t h o r ( s )2 0 2 1 |
1.2839382.pdf | Analyses on double resonance behavior in microwave magnetic permeability of
multiwalled carbon nanotube composites containing Ni catalyst
Fusheng Wen, Haibo Yi, Liang Qiao, Hong Zheng, Dong Zhou, and Fashen Li
Citation: Applied Physics Letters 92, 042507 (2008); doi: 10.1063/1.2839382
View online: http://dx.doi.org/10.1063/1.2839382
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/92/4?ver=pdfcov
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150.108.161.71 On: Thu, 31 Oct 2013 09:22:12Analyses on double resonance behavior in microwave magnetic
permeability of multiwalled carbon nanotube composites containingNi catalyst
Fusheng Wen, Haibo Yi, Liang Qiao, Hong Zheng, Dong Zhou, and Fashen Lia/H20850
Institute of Applied Magnetics, Key Laboratory of Magnetism and Magnetic Materials of Ministry
of Education, Lanzhou University, Lanzhou 730000, People’ s Republic of China
/H20849Received 4 December 2007; accepted 8 January 2008; published online 31 January 2008 /H20850
The double resonance behavior of microwave magnetic permeability has been observed for
multiwalled carbon nanotube composites containing Ni catalyst. One of them is due to the naturalresonance at 6.00 GHz and another is due to the exchange resonance at 10.11 GHz. The naturalresonance is dependent on magnetocrystalline anisotropy and shape anisotropy of Ni nanostickcatalyst and the calculated result of exchange resonance mode with a few modifications was closeto the experiment. It is believed that the coexistence of natural resonance and exchange resonanceis benefial to large bandwidth as a microwave absorber. © 2008 American Institute of Physics .
/H20851DOI: 10.1063/1.2839382 /H20852
Fine metal ferromagnetic nanoparticles has stimulated
intense research activities due to their potential applicationsfor microwave composite materials.
1In recent years, com-
posites with multiwalled carbon nanotubes /H20849MWCNTs /H20850/
magnetic nanoparticles /H20851Fe/H20849Ref.2/H20850or Ni /H20849Ref.3/H20850/H20852embedded
into a polymer host for microwave applications have at-tracted much scientific attention for their large dielectric loss/H20849owning to MWCNTs /H20850and high magnetic loss /H20849owning to
magnetic nanoparticles /H20850. Among the known composites of
this kind, MWCNTs with Ni is of special importance regard-ing that the nickel has a large resonance linewidth and may
be benefial for large bandwidth microwave absorbers.
4
Zhang et al. reported a single magnetic resonance peak in the
carbon-coated Ni nanocapsules and found that the small sizeeffect is the dominant factor to the resonance.3However, to
our knowledge, multiresonance phenomena have been ob-served in monodisperse ferromagnetic granular particles formicrowave application, such as CoNi and FeCoNi,
5and
could be well explained according to the exchange resonancemode proposed by Aharoni.6Generally speaking, there will
be two mechanisms making contributions to the permeabilitydispersion spectra of magnetic nanoparticles in gigahertzrange: natural resonance and exchange resonance. However,the contributions from these two components are hard toobserve at the same time in an experimental permeabilitydispersion spectrum and usually, only one dispersion isfound in the
/H9262/H11033-fspectrum. To know their individual contri-
bution, the permeability dispersion should be resolved. Inthis letter, we report the double resonance behavior ofMWCNTs containing Ni catalyst and apply the Landau–Lifshitz–Gilbert equation
7to fit the double resonance struc-
ture of the permeability spectra.
The MWCNTs were purchased from a vendor /H20849Nano-
techport, Inc.®, Shenzhen, China /H20850and were prepared by a
chemical vapor deposition method using Ni as the main cata-lyst. The crystal structure was examined using x-ray diffrac-tion /H20849XRD /H20850with Cu K
/H9251radiation on Philips X’perts diffrac-
tometer. The image of MWCNTs was taken by atransmission electronic microscope /H20849TEM /H20850. The compositesfor high-frequency magnetic property measurement were
prepared by epoxy resin with 50 wt %Ni/MWCNTs andwere pressed into toroidal shape /H20849
/H9272out: 7.00 mm, /H9272in:
3.04 mm /H20850. The scattering parameters /H20849S11,S21/H20850were mea-
sured on the toroidal-shape composites by a network ana-
lyzer /H20849Agilent Technologies E8363B /H20850in the range of
0.1–18 GHz. The relative complex permeability /H20849/H9262r/H20850was
determined from the scattering parameters.
Figure 1shows clearly the fcc features of the Ni nano-
particles linked to MWCNTs from XRD pattern, as well asthe typical fingerprint of a hexagonal graphite structure. Ac-cording to the Sherrer formula, the average crystal size of Ninanoparticles is deduced to be about 6 nm. More morphol-ogy information could be seen from the TEM of Ni nano-sticks embedded in MWCNTs, as presented in Fig. 2. The
outer diameter of MWCNTs is about 40 nm. Each MWCNTcarries a Ni nanostick at one of its ends. The nanosticks arein cylindrical shape with the length/diameter ratio of about3:1. Thus, the calculated demagnetization factor /H20849N
z/H20850is
0.1087 when the magnetic fields is along the length of the
nanostick, while it /H20849Nx=Ny/H20850is 0.4456 when the field is along
the diameter.8
a/H20850Electronic mail: lifs@lzu.edu.cn and wenfsh03@126.com.
FIG. 1. XRD pattern of MWCNTs with Ni nanostick catalyst.APPLIED PHYSICS LETTERS 92, 042507 /H208492008 /H20850
0003-6951/2008/92 /H208494/H20850/042507/3/$23.00 © 2008 American Institute of Physics 92, 042507-1
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150.108.161.71 On: Thu, 31 Oct 2013 09:22:12Figure 3shows the relative complex permeability of the
composites with 50 wt %. It reveals that the real part of per-meability /H20849
/H9262/H11032/H20850is about 1.2 from 0.1 to 3.08 GHz and de-
clines to around 0.55 at 18.00 GHz with increasing fre-
quency. As to the /H9262/H11033-fspectrum, clearly, two resonance
absorption peaks are observed: one is at 6.00 GHz and theother is at 10.11 GHz. The eddy current loss contribution tothe imaginary part permeability is related to thickness /H20849d/H20850
and the electric conductivity /H20849
/H9268/H20850of the composites:3
/H9262/H11033/H20849/H9262/H11032/H20850−2f−1=2/H9266/H92620d2/H9268where /H92620is the permeability of
vacuum. The calculated evolution /H9262/H11033/H20849/H9262/H11032/H20850−2f−1with fre-
quency is shown in Fig. 4and the double-peak character is
unambiguous. If the observed magnetic loss only resultsfrom eddy current loss, the value
/H9262/H11033/H20849/H9262/H11032/H20850−2f−1should be con-
stant with increasing frequency. As we can see, the difference
of/H9262/H11033/H20849/H9262/H11032/H20850−2f−1values is larger than 0.05 ns. Therefore, the
eddy current loss could also be precluded.
To further understand our experimental results, the reso-
nance spectrum will be fitted as the linear superposition oftwo overlapped peaks D
1andD1. Then, the resonance peaks
D1andD2could be separated from each other by fitting of
the Gilbert modification of Landau–Lifshitz equation7as
/H9262/H11032=B+/H20858
i2
Ii/H208511− /H20849f/fi/H208502/H208491−/H92512/H20850/H20852
/H208511− /H20849f/fi/H208502/H208491+/H92512/H20850/H208522+4/H9251i2/H20849f/fi/H208502, /H208491/H20850/H9262/H11033=/H20858
i2
Ii/H20849f/fi/H20850/H9251i/H208511+ /H20849f/fi/H208502/H208491+/H92512/H20850/H20852
/H208511− /H20849f/fi/H208502/H208491+/H92512/H20850/H208522+4/H9251i2/H20849f/fi/H208502, /H208492/H20850
with fthe frequency, fithe spin resonance frequency, /H9251ithe
damping constant, and Iithe intensity of the peak. The typi-
cal curve-fitting results are shown in Fig. 5. Firstly, the
double resonance peaks in the /H9262/H11033-fcurve was fitted, as
shown in Fig. 5/H20849a/H20850. Then the /H9262/H11032-fcurve was calculated using
obtained fitting parameters, as shown in Fig. 5/H20849b/H20850. All of the
fitting parameters are listed in Table I.
Actually, as pointed out by Kittle, the natural resonance
frequency depends strongly on the geometries of magnets.9
The Kittle equation is adopted to calculated the resonancefrequency of an isolated cylindrical magnet as f
R=/H92530/H20853/H20851Hk
+/H20849Nx−Nz/H20850Ms/H20852/H20851Hk+/H20849Ny−Nz/H20850Ms/H20852/H208541/2, where /H92530is the gyro-
magnetic ratio, Hkis the magnetocrystalline anisotropy field
of the magnet, Ni/H20849i=x,y,z/H20850is the demagnetization factor,
andMsdenotes the saturation magnetization. In the present
case,/H92530is chosen as 2.8 MHZ /Oe,Hkis 130 Oe, and Msis
6.09 T.10Moreover, Nx/H20849=Ny/H20850is 0.4456, Nzis 0.1087 for the
Ni nanostick, as calculated above. Thus, the calculated reso-
nance frequency fRis 6.11 GHz, which matches very satis-
factorily with our experimental data of the first peak in thespectrum. In addition, the value of damping constant
/H9251is
0.516, which agrees with the Ref. 5. Accordingly, the first
peak D1is mainly attributed to the natural resonance origi-
nated from the magnetocrystalline anisotropy and shape an-isotropy.
To our knowledge, multiresonance is a subject of con-
troversy in nanoparticles.1,5Among those modes which deal
with multiresonance, the most accepted one and may be rel-evant to the present observations is the exchange resonancemode placed forward by Aharoni.
6In our case, due to the
nanoscaled size of Ni in the sample, surface anisotropy andexchange energy caused by exchange effect between grainswould be evidenced. So, we assume that the modified Aha-roni’s method still works for our system.
11According to the
exchange resonance mode, the resonance frequencies are cal-culated by
/H9275//H92530=Hc+C/H9262kn2/D2Ms, /H208493/H20850
where Cis the exchange constant /H20849C=2/H1100310−7erg /cm/H20850,10
the eigenvalues /H9262knare the roots of the equation /H20849/H9262kn
=2.08 /H20850,6/H92530is the gyromagnetic ratio /H20849/H92530=3/H11003106Oe−1s−1/H20850,
FIG. 2. TEM image of MWCNTs with Ni nanostick catalyst.
FIG. 3. The relative complex permeability real part /H20849/H9262/H11032/H20850and imaginary part
/H20849/H9262/H11033/H20850of resin composite with 50 wt % vs frequency.
FIG. 4. Values of /H9262/H11033/H20849/H9262/H11032/H20850−2f−1for the composites vs frequency.042507-2 Wen et al. Appl. Phys. Lett. 92, 042507 /H208492008 /H20850
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150.108.161.71 On: Thu, 31 Oct 2013 09:22:12Dis the crystal size, and Hcis the coecivity /H20849Hc=50 Oe /H20850.
The calculated value of resonance frequency fRis
13.89 GHz, which accords with the fitting result /H20849fR
=11.00 GHz /H20850. Therefore, the modified exchange mode is
proven valid in explaining the second peak of D2in the spec-
trum.
Note that, as presented in Eq. /H208493/H20850, the resonance fre-
quency strongly depends on /H20849D2Ms/H20850−1, resulting from the
exchange energy contribution to the magnetization proces-
sion within the Ni nanosticks attributed to the surface aniso-tropy. The exchange energy will decrease with increasing D.
Thus, under the given conditions, the exchange resonancepeak could be detected at the crystal size of 6 nm. All theexperimental, fitting, and calculated parameters and resultsare listed in Table I.
The carbon-coated Ni nanocapsules only exhibit the
resonance at 5.5 GHz.
3In addition, our observations are also
different from Fiévet’s work on cobalt rich samples at similarparticle size, where more peaks were observed. The differ-ence may result from the nanoparticle size, geometry, and thelower anisotropy constant of Ni.
1,5It is believed that the size
and geometry of nanoparticles are the vital factors for theappearance of multiresonance phenomenon. Moreover, the
overlapping of natural resonance and exchange resonancemay be beneficial to the large bandwidth microwaveabsorber.
In conclusion, a double resonance behavior was detected
with MWCNTs containing Ni nanosticks catalyst. This be-havior depends on the mean crystal size and geometry. More-
over, the size effect can be qualitatively related to exchangeresonance mode. In addition the Ni/MWCNTs, experimentsgive complementary experimental results for a better inter-pretation of the spin dynamics within cylindrical particles.
This work was supported by the National Natural
Science Foundation of China under Grant Nos. 90505007and 10774061.
1P. Toneguzzo, G. Viau, O. Acher, F. Fiévet-Vincent, and F. Fiévet, Adv.
Mater. /H20849Weinheim, Ger. /H2085010, 1032 /H208491998 /H20850.
2R. C. Che, L. M. Peng, X. F. Duan, Q. Chen, and X. L. Liang, Adv. Mater.
/H20849Weinheim, Ger. /H2085016, 401 /H208492004 /H20850.
3X. F. Zhang, X. L. Dong, H. Huang, Y. Y. Liu, W. N. Wang, X. G. Zhu, B.
Lv, J. P. Lei, and C. G. Lee, Appl. Phys. Lett. 89, 053115 /H208492006 /H20850.
4J. S. S. Whiting, IEEE Trans. Magn. MAG-18 ,7 0 9 /H208491982 /H20850.
5P. Toneguzzo, G. Viau, O. Acher, F. Guillet, E. Bruneton, F.
Fiévet-Vincent, and F. Fiévet, J. Mater. Sci. 35, 3767 /H208492000 /H20850.
6A. Aharoni, J. Appl. Phys. 69, 7762 /H208491991 /H20850.
7A. Aharoni, Introduction to the Theory of Ferromagntism /H20849Clarendon,
Oxford, 1996 /H20850, Chap. 8, p. 181.
8S. Chikazumi, Physics of Ferromagnetism , 2nd ed. /H20849Oxford University
Press, Oxford, 1997 /H20850,p .1 2 .
9C. Kittel, Phys. Rev. 73, 155 /H208491948 /H20850.
10P. A. Voltatas, D. I. Fotiadis, and C. V. Massalas, J. Magn. Magn. Mater.
217,L 1 /H208492000 /H20850.
11L. J. Deng, P. H. Zhou, J. L. Xie, and L. Zhang, J. Appl. Phys. 101,
103916 /H208492007 /H20850.TABLE I. Fitting and calculated parameters for permeability dispersion
curves. fr/H20849exp /H20850,fR/H20849fit/H20850, and fR/H20849cal/H20850denote the frequencies at which the /H9262/H11033
values on experimental curves reach maximum for natural resonance and
exchange resonance. /H9251is the fitting damping constant.
fr/H20849exp /H20850/H20849GHz /H20850fR/H20849fit/H20850/H20849GHz /H20850fR/H20849cal/H20850/H20849GHz /H20850/H9251
Natural resonance 6.00 6.29 6.11 0.516
Exchange resonance 10.11 11.00 13.89 0.290
FIG. 5. Permeability dispersion spectra: /H20849a/H20850imaginary
part /H20849/H9262/H11033/H20850and /H20849b/H20850real part /H20849/H9262/H11032/H20850. Dotted and dashed lines
are calculation curves for D1andD2components.042507-3 Wen et al. Appl. Phys. Lett. 92, 042507 /H208492008 /H20850
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150.108.161.71 On: Thu, 31 Oct 2013 09:22:12 |
1.5050712.pdf | J. Appl. Phys. 125, 060901 (2019); https://doi.org/10.1063/1.5050712 125, 060901
© 2019 Author(s).Frontiers of magnetic force microscopy
Cite as: J. Appl. Phys. 125, 060901 (2019); https://doi.org/10.1063/1.5050712
Submitted: 02 August 2018 . Accepted: 12 January 2019 . Published Online: 08 February 2019
O. Kazakova
, R. Puttock
, C. Barton
, H. Corte-León
, M. Jaafar
, V. Neu
, and A. Asenjo
COLLECTIONS
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Frontiers of magnetic force microscopy
Cite as: J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712
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Submitted: 2 August 2018 · Accepted: 12 January 2019 ·
Published Online: 8 February 2019
O. Kazakova,1
R. Puttock,1,2
C. Barton,1
H. Corte-León,1
M. Jaafar,3
V. Neu,4
and A. Asenjo3
AFFILIATIONS
1National Physical Laboratory, Hampton Road, Teddington TW11 0LW, United Kingdom
2Department of Physics, Royal Holloway University of London, Egham TW20 0EX, United Kingdom
3CSIC, Campus Cantoblanco, 28049 Madrid, Spain
4Leibniz Institute for Solid State and Materials Research, Dresden 01069, Germany
ABSTRACT
Since it was first demonstrated in 1987, magnetic force microscopy (MFM) has become a truly widespread and commonly used
characterization technique that has been applied to a variety of research and industrial applications. Some of the main advan-
tages of the method includes its high spatial resolution (typically ∼50 nm), ability to work in variable temperature and applied
magnetic fields, versatility, and simplicity in operation, all without almost any need for sample preparation. However, for most
commercial systems, the technique has historically provided only qualitative information, and the number of available modeswas typically limited, thus not re flecting the experimental demands. Additionally, the range of samples under study was largely
restricted to “classic ”ferromagnetic samples (typically, thin films or patterned nanostructures). Throughout this Perspective
article, the recent progress and development of MFM is described, followed by a summary of the current state-of-the-art tech-niques and objects for study. Finally, the future of this fascinating field is discussed in the context of emerging instrumental and
material developments. Aspects including quantitative MFM, the accurate interpretation of the MFM images, new instrumenta-tion, probe-engineering alternatives, and applications of MFM to new (often interdisciplinary) areas of the materials science,
physics, and biology will be discussed. We first describe the physical principles of MFM, speci fically paying attention to common
artifacts frequently occurring in MFM measurements; then, we present a comprehensive review of the recent developments inthe MFM modes, instrumentation, and the main application areas; finally, the importance of the technique is speculated upon
for emerging or anticipated to emerge fields including skyrmions, 2D-materials, and topological insulators.
Published under license by AIP Publishing. https://doi.org/10.1063/1.5050712
I. INTRODUCTION
First demonstrated in 1987,1,2magnetic force microscopy
(MFM) is a well-established and widely used technique. Over thelast three decades, the method has been extensively used in avast number of applications, where the knowledge of the local
distribution of the magnetic properties of thin film materials
and their nanostructures is of paramount importance. Thisfunctional technique relies on quantifying the long-range mag-netostatic force between the magnetic sample and a magneti-cally coated probe positioned at a constant height over the
specimen surface. In its simplest form, the typical MFM proce-
dure involves two linear scans. First, the topography of thesurface is obtained by using tapping mode atomic force micros-copy (AFM) (i.e., exploiting van der Waals interactions between
the probe and sample). During the second scan, the probe is
lifted away from the sample [i.e., van der Waals interactions are
negligible and the probe experiences only long-range magnetic(and electrostatic) interactio ns] and the initial topography
profile is repeated at the constant lift scan height [ Fig. 1(a) ].
The knowledge and expertise accumulated in the initial
period of MFM development established a fundamental base for
the modern commercial MFM systems. However until recently,
unlike other functional scannin g probe microscopy (SPM) tech-
niques, commercial MFM systems have not demonstrated a
variety of modes and were used p rimarily on their own. At the
same time, the use of MFM was somewhat limited to “classic ”
ferromagnetic (FM) samples, although they were represented in
a variety of forms. Recently, the rise of novel materials, often
combining magnetic and other functional properties or demon-
strating complex forms of magnetism, such as multiferroics,
topological insulators (TIs), a nd magnetic semiconductors, has
stimulated a burst in the development of advanced MFM modes.
A number of methods have been developed to image
magnetic structures with different sensitivities and on manyJournal of
Applied PhysicsPERSPECTIVE scitation.org/journal/jap
J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-1
Published under license by AIP Publishing.lateral scales. These methods can be roughly divided into
beam- and scanning probe-based techniques. The former
involves a broad spectrum of physical principles of operation
(i.e., polarized light, x-rays, and electrons) and includes bothwell-established and novel techniques such as magneto-optical microscopy based on Kerr and Faraday effects,
3,4
Lorentz force microscopy,5scanning electron microscopy
(SEM) with polarization analysis,6,7and photoemission elec-
tron microscopy,8,9speci fically including x-ray magnetic
linear and circular dichroism microscopy.10
The latter group comprises a variety of magnetically sen-
sitive SPM-based techniques. One of the recent exciting
examples includes integration of nitrogen vacancy (NV) defect
centers with high-Q diamond mechanical oscillators, allowingrealization of a quantum qubit system with the advantages ofhighly coherent electron spin and narrow optical transitions,accompanied by nanometer scale resolution.
11,12Another
example is magnetic resonance force microscopy that suc-
ceeded in detecting single electrons and small nuclear spinensembles.
13,14Successful examples of mounting a magnetic
sensor on a scanning probe include Hall probe microscopy15
and superconducting quantum interference device (SQUID)microscopy.
16–18All these methods have both advantages and
drawbacks, as well as a different degree of application inresearch and industry. These techniques are, however,beyond the scope of this Perspective article, which will
entirely focus on MFM.
MFM has been most widely used for the local characteri-
zation of magnetic nanostructures and imaging the magneticfield distribution at the surface of magnetic materials.
1,2,19
Despite decades of advances in magnetic imaging,20obtaining
direct, uncoupled, and quantitative information with high
spatial resolution remains an outstanding challenge.
Among all methods for the observation of magnetic
domain structures, MFM is the most widely used, due to itshigh spatial resolution (down to ∼10 nm),
21sensitivity ( ∼10 pN),22
relative simplicity in sample preparation, capability to apply in
situmagnetic fields to study magnetization processes,23and its
ability to operate in different environments.24The MFM tech-
nique has been proven as an excellent characterization tool in
both fundamental research and industrial applications. For
comprehensive MFM reviews performed in the early days ofMFM, see Refs. 21,25,a n d 26.
The aim of this Perspective article is to analyze recent
progress in the development of MFM, present the current
state-of-the-art, and outline the future and perspective of this
fascinating field. Such emerging aspects as probe-engineering
alternatives, new instrumentation, quantitative measurements,the correct interpretation of the resulting MFM images, theloss of energy analysis and applications of MFM to new
emerging areas of the material science, physics and biology,
etc. are subjects of ongoing research that will be discussedin this work.
The article is organized as follows: Sec. IIdescribes the
physical principles of MFM, and common artifacts in MFM
measurement; the review (Sec. III) describes the recent devel-
opments in instrumentation and the main application areas;
FIG. 1. Schematics for different MFM modes. (a) Standard two-pass MFM: In
thefirst pass (left), the probe raster scans the surface, mapping the topography
of the sample by “tapping ”along the surface at its resonant frequency ( ω0); in
the second pass (right), the probe lifts a set distance away from the sample(h
lift) and maps the long-range interactions, via the phase change of the oscillat-
ing cantilever, at a constant probe –sample separation. (b) Frequency-modulated
Kelvin probe force microscopy-MFM: In addition to acquiring the sample topog-
raphy in the first-pass (left), the technique is sensitive to the probe –sample
contact potential difference (CPD) by monitoring the magnitude of the sidebandsof the probe ’s resonant peak induced from a modulated AC-voltage ( V
mod,fmod)
applied to the probe; the effects of the CPD are nulli fied in the second-pass
(right) by applying a DC-voltage of magnitude such that the sidebands are effec-tively reduced to zero (i.e., V
DC=VCPD). (c) Dynamic magnetoelectric force
microscopy: The first pass (left) is the same as in (a); in the second-pass (right)
the probe is not mechanically oscillated, instead a combined AC/DC bias is
applied to the sample base-electrode and the sample potential is electricallymodulated at the mechanical resonance of the cantilever ( ω
0). The resulting AC
magnetic field from the sample (from the linear magnetoelectric effect) induces
resonant motion of the magnetic probe. (d) Bimodal MFM: A single-pass tech-
nique where the probe is excited at two of its resonant frequencies ( ω1andω2),
each of these frequencies are sensitive to speci fic sample properties (e.g.,
short- and long-range probe sample interactions).Journal of
Applied PhysicsPERSPECTIVE scitation.org/journal/jap
J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-2
Published under license by AIP Publishing.finally, this Perspective article (Sec. IV) presents the emerging
trends in the field of MFM.
II. PRINCIPLES AND ARTIFACTS IN MFM
The long-range force interactions (i.e., force gradients)
between the magnetic probe and the magnetic sample in MFMare recorded and correlated in the second pass from the shiftin frequency ( Δω), amplitude ( ΔA), or phase ( Δ
f) from the initial
driven parameters (i.e., ω0,A0,a n df0, respectively) of the oscil-
lating cantilever. However, it is not possible to directly quantify
these tip-sample interactions without prior knowledge of theprobe properties and behavior. In the absence of any tip-sample interactions, the oscillating probe can be approximatedas a point-mass spring and thus can be de fined by a classic
non-linear, second order differential equation, i.e., from
Newton ’s second law of motion.
27From the possible recorded
data channels above, Δfof the cantilever is the most common
representation of magnetic contrast in the second-pass ofMFM; hence, it is useful to describe the relationship between
the phase in free space (
ff) and the excitation frequency ( ω)
without any externally acting forces as28,29
ff(ω)¼tan/C01 mωω0
Q(k/C0mω2)/C18/C19
, (1)
k¼mω2
0, (2)
where m,ω0,Q,a n d kare the point mass, resonant angular
frequency, quality factor, and the spring constant of the can-
tilever, respectively. When the probe is osc illated at ω=ω0,
Eq.(1)dictates ff(ω), which is equal toπ
2. If we introduce tip-
sample interactions ( Fts), this subtly changes the oscillation
and subsequently the instrument response. Assuming small
displacements ( z) with respect to the rest position ( z0)o ft h e
cantilever, the force can be described as follows after aTaylor expansion:
30
Fts/C25dFts(z)
dz/C12/C12/C12/C12
z¼z0z(t): (3)
Thus, the equation of motion is adapted to encompass the
force derivatives acting on the cantilever
F0cos ( ωt)¼mz00(t)þmω0
Qz0þ k/C0dFts
dz(z)/C20/C21
z(t)/C26/C27
:(4)
Here, a number of possible forces can be acting between
the probe and the sample simultaneously, including van der
Waals, magnetostatic, and electrostatic interactions. In order
to isolate solely the magnetic contrast, methods must be uti-lized to mitigate the parasitic signals (discussed in greaterdetail in Sec. III). In Eq. (4),F
0describes the amplitude of
the driving force, andmω0
Qrepresents the damping factor.
Equation (1)in the presence ofdFts
dzbecomes
f(ω)¼tan/C01 mωω0
QkþdFts
dz/C0mω2/C18/C190
BB@1
CCA, (5)and providing the probe is oscillated at ω
0anddFts
dz/C28k,E q . (2)
can be substituted into Eq. (5)and gives us the phase as a func-
tion ofdFts
dz
f(ω0)¼tan/C01k
QdFts
dz !
: (6)
Combining Eqs. (1)and (6)finally produces the approximate
relation between the ΔfanddFts
dz29
Δf(ω0)¼π
2/C0tan/C01k
QdFts
dz !
/C25Q
kdFts
dz: (7)
An understanding of how the cantilever resonant frequency
shifts from ω0is also desirable, as frequency-modulated
modes in MFM and other scanning probe techniques arebecoming more common. The Δωcan be detected by classi-
cal lock-in techniques and signal can be utilized for greater
parameter control, e.g., more controlled tip-sample distancecontrol (e.g., from capacitive coupling).
31Here, we will suc-
cinctly describe the relation of ΔωtodFts
dz.F r o mE q . (4),i ti s
possible to de fine the effective spring constant of the canti-
lever ( keff)a s30,32,33
keff¼k/C0dFts
dz(z)/C12/C12/C12/C12
z¼z0, (8)
where a positive (attractive) or negative (repulsive) force
gradient effectively leads to a softer or harder cantilever,respectively.
27,30This modi fication, hence, causes a shift in
ω0toω0
0in Eq. (2);t h u s ,
ω0
0¼k/C0dFts
dz
m ! 1
2
¼ω01/C0dFts
dz
k ! 1
2
: (9)
Assuming once again thatdFts
dz/C28k, a Taylor expansion can be
performed on Eq. (9)andΔωisfinally given by
Δω/C25/C0ω0
2kdFts
dz: (10)
Relating the calculated force gradients to quantitative
descriptions of a sample ’s magnetic parameters is a further
field of research, requiring an estimation of the MFM probe ’s
magnetic parameters from which to decouple from theacquired MFM dataset. A further discussion of how this isachieved is outlined in Sec. III B.
Despite the advantages highlighted throughout this
article, MFM is not without its limitations and errors.Just like other SPM techniques, MFM is susceptible toartifacts, which can perturb the measured image and,without careful handling, can lead to incorrect interpreta-
tion of the results. Many common SPM-based artifacts and
methods for reducing their effects are discussed else-where.
34Table I summarizes MFM-speci fic artifacts and
solutions to minimize their effects on recorded images. Anumber of these will be speci fically discussed throughout
the present work.Journal of
Applied PhysicsPERSPECTIVE scitation.org/journal/jap
J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-3
Published under license by AIP Publishing.Arguably the most important factor for accurately repre-
senting stray magnetic fields emanating from the measurand
is careful consideration of the probe and its own magneticand physical properties relative to the sample. The resolution
and sensitivity of MFM probes are primarily governed by the
tip’s shape and magnetic properties. However, as an MFM
image is a convolution of both the sample and the probe ’s
magnetic properties, it is imperative to also consider theinduced effects of the probe and sample ’s stray- field on each
other, as this can result in imaging artifacts, such as altering
the moment of either the sample or the probe.
35,36For
example, Fig. 2 shows that the magnetic state of a low coer-
civity Ni disk can be perturbed by the MFM probe with highermagnetic moment [standard moment (SM)] during data
acquisition, compared to the low moment (LM) probe [(a) and
(b), respectively].
Another common artifact in MFM data acquisition is
the effects of both induced electrostatic interactions between
the probe and the sample, and magnetic contamination.
There are a few examples of misinterpreted MFM images inliterature due to these parasitic artifacts, including proposedmagnetic highly ordered pyrolytic graphite (HOPG)
37(demon-
strated that observed contrast was due to electrostatics, i.e.,
not a magnetic origin, by Martínez-Martín et al.38) and room
temperature ferromagnetism in C 60polymers39,40[shown to
be Fe 3C contamination41,42and later retracted by (most) of
the original authors43]. Thus, for magnetic contamination, it
is vital to carefully monitor the magnetic history and exclude
exposure of magnetic materials and tools (e.g., catalysts,
tweezers, etc.) to the sample in fabrication/handling pro-cesses prior to the measurement. In the case of parasiticelectrostatics, it is crucial to be able to identify and nullifythe adverse artifacts. For this, it is primarily important to con-
sider the electrical grounding during the measurement, withTABLE I. Common limitations and errors in magnetic force microscopy.
Limitation Description Result on MFM image Method of compensating limitation Rel. Refs.
Coupled e-static and
magnetic signalse-static, frictional, and magnetic forces
all influence changes in probeoscillationImage contains contribution of all 3 signals Kelvin probe —MFM (KPFM-MFM)
Switching magnetization MFM (SM-MFM)Variable-field MFM (VF-MFM) 44–47
Sensitivity to acoustic
noise, air flow, and
vibrationsMFM (and SPM generally) are sensitive
to externally driven vibrationsNoise and artifacts due to external
influencesImage processing; vacuum operation;
vibration isolation tables, etc.32and 48
Magnetic impurities The probe is sensitive to artifacts,
which may present on/in the sample.False positives Careful sample preparation, handling, and
measurement procedures.49and 50
Probe ’s stray-field
distribution unknownThe exact magnetic distribution of
individual probes is not knownErrors in extracting meaningful quantitative
valuesModeling probe ’s magnetization. Probe
calibration51and 52
Resolution/sensitivitybalanceThe active magnetic volume is
proportional to sensitivity and inversely
proportional to resolutionImages from small force gradients will have
lower resolution.
Sensitivity requires a measurable interaction
force, which is proportional to theinteraction volumeResolution : Deconvolution processes;
ultra-sharp probes.
Sensitivity : suitable probe selection; in
vacuum measurement; optimized ext.variables ( T,B
ext)53–55
z-distance effects At larger z-separations, interaction
volume increases and signal strength
decreasesLower resolving power leads to errors in
lateral sizes. Calibration values vary as a
function of tip-sample distance.Modeling for tip-sample distance.
Controllable tip-sample distance between
calibration and test samples56and 57
FIG. 2. Imaging artifacts in MFM. MFM images of a nickel disk (diameter
800 nm and thickness 25 nm) measured with standard moment (SM) andlow moment (LM) probes. The disk ’s magnetization is perturbed by the strong
magnetic moment of the SM probe (a), but not by the LM probe (b). The line
profiles (green and blue lines) were obtained with LM (left vertical scale) and
SM (right vertical scale) commercial probes, respectively (c). Black solid linesshow the geometrical size of the Ni disk and red dashed lines mark theoutline of the vortex core measured by the LM probe. Reproduced with per-
mission from Wren et al. , Ultramicroscopy 179, 41 (2017). Copyright 2017
Elsevier.Journal of
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J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-4
Published under license by AIP Publishing.alternative active and passive methods for nullifying the
effects discussed later in Sec. III.
III. REVIEW
Here, we describe the recent developments in instru-
mentation, quantitative MFM (qMFM) modeling, and modernapplication areas of MFM. Speci fically, we address such areas
of instrumentational development as in- field and low/high
temperature MFM and discuss the compensation of electro-static signals and energy dissipation in MFM. We also brie fly
present different types of MFM, i.e., dynamic magneto-electric force microscopy (MeFM), bimodal MFM, and mag-
netic scanning gate microscopy (mSGM), as well as the devel-
opment of custom-designed MFM probes. The modernobjects of MFM studies discussed here include (ultra-)thinfilms with perpendicular magnetic anisotropy (PMA), arti ficial
spin ice as an example of patterned structures, magnetic
topological structures, multiferroic materials, and materials
for Life Science applications.
A. Advanced modes and instrumentation
Only a few years after the initial development of MFM,
different groups explored the power of MFM imaging within situ applied magnetic fields. Initially, custom-built approaches
(typically consisting of a system of coils or permanent magnets)were implemented in commercial MFM equipment with
in-plane (IP) or out-of-plane (OOP) field for maximum ampli-
tudes ranging from 300 to 500 mT. At that time, hot topicsincluded the evaluation of switching fields of sub-micron
magnetic patterns
58and the study of the magnetization
reversal both in thin films with perpendicular magnetic
anisotropy (PMA)59,60and in magnetic nanowires (NWs)61,62
(with OOP and IP fields, respectively). As the available range
of magnetic fields progressed (up to 1000 mT), it became pos-
sible to study the magnetization process in materials toutedfor magnetic recording media.
63,64Specialized custom MFM
systems with in field capabilities operating under extreme
conditions (7 T OOP at 7.5 K and UHV) were reported, e.g.,Kappenberger et al. ,
65and used currently for probing novel
nanomagnetism, e.g., exchange bias multilayers.66,67Moreover,
the application of vector magnetic fields in MFM was recently
demonstrated.21
In addition to the standard MFM images recorded at
fixed magnetic fields, two different groups developed in situ
MFM imaging in variable field, where the probe scans along
one spatial dimension, while the slow axis of the scan corre-
sponds to a gradual change of the magnetic field.21,68This
variable field MFM mode allows for the evaluation of the criti-
calfields in individual magnetic elements or the coercive field
of the MFM probes.
The in- field MFM technique provides a reliable under-
standing of the internal spin structure and its magnetizationreversal processes and has been recently applied to studies ofboth the domain con figuration and domain wall (DW) proper-
ties in various magnetic thin films and nano-objects.
69–71
For example, in- field MFM has been used to characterize thenovel spin con figuration and the magnetization mechanism in
cylindrical magnetic NWs, which are exempt of the Walker
breakdown limit that restricts the DW velocity.72,73The
in-field MFM technique is also paramount for studies of
the topologically protected magnetic states characterized bythe Dzyaloshinskii –Moriya interaction (DMI), e.g., magnetic
skyrmions, since this technique is being intensively used to
analyze their stability, nucleation, and propagation.
74–78
The combination of nanomagnetism and biomedical
applications has also been a hot topic in recent years, e.g., inapplication to studies of hyperthermia effect for cancer treat-ment. The study of individual magnetic nanoparticles (MNPs)
by in- field MFM allows for determination of the easy axis of
Fe
3–xO4MNPs79,80and the vortex state formation and annihi-
lation in individual 25 nm MNPs.67
Variable temperature MFM is another important topic for
MFM development. Low-temperature MFM has been utilized
to study superconducting flux vortices in Type II supercon-
ductors, where detailed information on the temperature andfield dependence
81of their properties can be obtained with
the high spatial resolution of the MFM. Understanding phe-
nomena such as flux creep and pinning82at the nanoscale is
important for technological applications such as high criticaltemperature (high- T
c) superconducting ceramics, where
creep can cause a reduction in the critical current andfields.
82,83Low-temperature MFM measurements in the
range of 7.6 –80 K have been used to image flux vortices in
YBa 2Cu3O7–x(YBCO) single crystal films.84In these experi-
ments, the authors used a bath cryostat with a variable tem-perature insert and a superconducting magnet. This systemallows for measurements with a temperature range of 6 –
400 K, ultra-high vacuum, and applied fields of 7 T. The same
authors also demonstrated how vortex bundles can bemanipulated and nucleated using the stray field from the
magnet probe.
85
While piezo excitation is the most common way to excite
AFM cantilevers, it is not speci fically advantageous in low
temperature systems, where instabilities originate from thethermal contraction of mechanical parts pressing the cantile-ver. In the past, the photothermal excitation of the cantilevers
using two laser sources was accepted to be the best alterna-
tive method. In this, one of the beams was focused at the endof the cantilever for de flection measurement, and the second
beam near the base of the cantilever for excitation.
86,87
Recently, a novel radiation pressure based cantilever
excitation method for imaging in dynamic AFM mode was
presented by Çelik et al .88In order to simplify the optical
design in cryogenic AFM/MFM, the authors explored the useof a single laser beam and fiber optic interferometry, both for
the excitation and detection of cantilever de flection in AFM
imaging. The high performance of the radiation pressure exci-
tation in AFM/MFM was demonstrated by magnetic domainsin Co/Pt multilayers and an Abrikosov vortex lattice inBSCCO(2212) single crystal at 4 K.
88
In addition to low-temperature measurements, it is also
possible to image magnetic phenomena and transitions that
occur at higher temperatures. Typically, these measurementsJournal of
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J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-5
Published under license by AIP Publishing.are performed using Peltier or, for higher temperatures,
resistive heaters, which can provide in situ measurement
environments from room temperature to 520 K. It has been
demonstrated that the temperature dependence of thedomain structure of FePt thin films can be imaged. This is
highly pertinent for future magnetic recording technologiessuch as heat assisted magnetic recording (HAMR),
89where
the energy required for magnetization reversal is reduced
through near- field laser heating.
FeRh undergoes a first order metamagnetic phase
transition from an antiferromagnet to a ferromagnet above acritical temperature of approximately 370 K, which is also
accompanied by an expansion of the crystal lattice and a
sharp drop in the electrical resistivity.
90It has been shown
that the control of the electrical resistivity in FeRh can beachieved via strain modulation of a (001) PMN-ZT piezoelec-tric substrate.
91This strain modi fies the relative contributions
to the total electrical resistivity by modifying the relative
volume of the antiferromagnetic and FM regions through thestrain induced phase transition. In this work, MFM was usedto investigate the first order metamagnetic phase transition
of FeRh at temperatures above and below the phase transi-
tion. It was found that the relative size of the FM domainsexpands rapidly through the phase transition and thenreduces in size upon cooling, highlighting the effectiveness ofMFM to gain insight to the magnetic landscape of complex
systems on micrometric length scales.
In MFM experiments such as those already discussed, it
is important to consider the electrostatic in fluences to the
MFM signal. Here, we discuss further the instrumental devel-opments and examples relevant to separation and compensa-
tion of electrostatic signals in MFM. At typical probe –sample
working distances, the magnetic and electrostatic interac-tions can have comparable values depending on the electricand magnetic properties of the system. An electrostatic con-tribution is present whenever the probe and sample exhibit
different work functions, which results in a contact potential
difference (CPD). Such electrostatic interaction can maskother long or short range interactions.
21,92In a homogeneous
sample, the CPD can be compensated by applying an appro-
priate bias voltage between the probe and the sample.
However, if the surface is composed of more than one mate-rial, this simple method is not applicable,
21since the CPD
values vary all over the surface. When a heterogeneoussample (e.g., nanostructures on a substrate) is studied, and
especially in the case of low magnetic moment materials, it is
necessary to consider this problem in order to prevent incor-rect image interpretation.
37
Thefirst method for separating both long range interac-
tions was proposed by Jaafar et al .45There, a combination
between Kelvin Probe Force Microscopy and MFM (KPFM/
MFM) was used to distinguish between electrostatic andmagnetic contributions [ Fig. 1(b) ]. The method records both
the CPD map and the real compensated MFM image, as itcancels the electrostatic interaction between the probe and
sample at every point of the image, thus obtaining a pure
magnetic signal.Angeloni et al.
44have also demonstrated the effect of
electrostatic tip-sample interactions in MFM, which limited
the accuracy of magnetic measurements at the nanometer
scale. They developed an alternative MFM-based approach, inwhich the two subsequent images of the same area were col-lected, one with the probe being magnetized and anotherwith a quasi-demagnetized probe. The MFM map of the true
signal is achieved by subtracting the images. Prior to mea-
surement, it is necessary to determine both the remanentsaturation and coercivity of the probe by imaging a referencesample with periodically patterned magnetic domains. Theauthors demonstrated the effectiveness of this technique by
characterizing the magnetization curves of individual MNPs.
93
The ability to distinguish magnetic and electrostatic
signals is still open for discussion. Recently, it has been pro-posed to perform electrostatic force microscopy (EFM) priorto MFM measurements to compare the respective images.
94
Alternatively, modifying the magnetic state of the sample withan external magnetic field was used to determine whether
the origin of the signal is magnetic. However, only by com-pensating the electrostatic interaction in each point, a true
MFM image (and in addition the CPD information) can be
obtained in real time.
45
Recently, a number of MFM-related techniques have
appeared, each of them designed to address a speci fics c i e n -
tific problem, thus having a somewhat narrower application
scope than standard MFM. One such specialized MFM-related
technique is magneto-electric force microscopy (MeFM).In this mode, the probe is not mechanically driven in thesecond-pass as in MFM. Instead, a combined AC/DC bias isapplied to the sample while the sample potential is electri-
cally modulated at the mechanical resonance of the cantile-
ver. The resulting AC magnetic field from the sample induces
the resonant motion of the magnetic probe
95,96[Fig. 1(c) ]. In
addition, the probe is electrically isolated and kept at a largeconstant tip-sample distance during imaging.
35This method
is of particular importance for materials exhibiting a strong
coupling and interdependence of electrical and magneticproperties and can be employed to detect the electricfield-induced magnetization.
MeFM has been used previously to decouple magnetic
and electrical effects in complex samples (e.g., 2D electronicliquids
35); visualize the magnetoelectric (ME) response from
domain patterns in hexagonal manganites95,96and antiferro-
magnetic 180° domains;97and estimate the upper limit of the
linear ME coef ficient of h-LuFeO 3.98Additionally, many con-
trolled experiments have been undertaken, e.g., a study ofMeFM performance in dependence on the modulation fre-quency,
96which showed that lower modulation frequency
produces a better signal-to-noise ratio (SNR). However, lower
modulation frequency requires longer averaging time to
obtain the intrinsic ME response.
Superior aspects and limitations of MeFM were recently
summarized by Schöenherr et al .97The advantages include:
(i) high lateral resolution with standard/specially formed
probes; (ii) ability to resolve and de fine the DW inclination;
(iii) low sensitivity to material inhomogeneities and thusJournal of
Applied PhysicsPERSPECTIVE scitation.org/journal/jap
J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-6
Published under license by AIP Publishing.reduced dependence on the associated scattering effect.
The current restrictions of MeFM are a relatively small output
response and low SNR. Nevertheless, the limited signal can be
improved by increasing the electric field, resulting in a larger
induced magnetic field; optimizing the temperature to maxi-
mize the ME effect response; optimizing the electrodes; theuse of probes with higher magnetic moments, leading to a
stronger force between the probe and the magnetoelectri-
cally induced magnetic field. The SNR can also be improved
by increasing the averaging time per data point or multiplemeasurements of the same area. It was thus compellinglydemonstrated that this advanced technique provides direct
visualization of the ME domains and DWs to open up a new
paradigm of explorations of emergent mesoscopic phenom-ena in materials with multiple coupled orders. It was pro-posed that the method is of utmost importance for exploringemergent phenomena at the mesoscopic scale such as ME
coupling in multiferroic domains and DWs, multiferroic sky-
rmions, or magnetic topological insulators.
Bimodal MFM belongs to the family of multi-frequency
SPM. One of the advantages of SPM is the simultaneous
detection of a variety of interactions at different probe –
sample separations. Multi-frequency SPM is a novel conceptthat has been developed in the last few years.
99These modes
are based on the consideration of the microcantilever-basedprobe as a mechanical system characterized by multiple reso-
nances and harmonics. Each of those frequencies is sensitive
to speci fic information on the sample properties. Appropriately
excited and decoded, those frequencies will provide completeinformation on the electronic and mechanical properties[Fig. 1(d) ].
For the particular case of MFM, the multi-frequency
techniques have become an active area of research. In 2009,Liet al. presented the bimodal AFM as a technique to simulta-
neously separate short- and long-range (topographic andmagnetic, respectively) forces in ferromagnetic samples.
100In
this work, the cantilever was driven at two flexural resonant
modes. Following this idea, Dietz et al. employed the bimodal
AFM to measure a nanomechanical effect that enables thedetection of ferritin molecules with high lateral resolution
and sensitivity.
101More recently, a non-contact bimodal MFM
technique operating in vacuum/UHV was developed.102In
this work, the higher-stiffness second mode is used to mapthe topography and the magnetic force is measured at first
oscillation mode, which is characterized by higher sensitivity.
The torsional resonance mode of cantilever oscillation was
employed to perform magnetic imaging without topography-related interference and to improve the lateral resolution.
103
Another alternative is to combine a mechanical (1st
mode) and electrical excitation (2nd mode) to drive a cantile-
ver. This approach has been explored in the literature to sep-
arate electrostatic and magnetic interactions38,45or as a tool
to control the probe –sample distance.31,104In a similar way, in
the secondary resonance MFM, the excitation of the probe isbimodal: the information from the first eigenmode (mechani-
cally excited) is used to obtain the topography but the higher
eigenmode is excited by an external magnetic field instead ofthe piezo. The long-range magnetic forces are separated from
short-range allowing a single-pass imaging of topography and
magnetic images with high sensitivity, which makes this
bimodal MFM technique a useful tool for the characterizationof samples with weak magnetic properties.
105
Another powerful tool for probing physical phenomena
in an MFM-related technique is the study of energy dissipa-
tion. In SPM, the dissipation of energy is evaluated by mea-
suring variations in the cantilever oscillation.106For MFM, the
pioneering work107uses these dissipative maps to distinguish
between Néel and Bloch DWs or identify pinning sites. It hassince been demonstrated that some instrumental artifacts
can affect the measured values.
108
Classical magnetic dissipation force microscopy (MDFM)
studies probe-induced magnetization changes in the sample,but recently the opposite effects have also been studied: thestrong probe –sample interaction where the stray field from
the sample induces changes in the magnetic state of the
probe [ Fig. 3(a) ]. Iglesias-Freire et al.
109demonstrated that
the magnetic switching at the apex of an MFM probe canproduce artifacts in MFM images and could be misinterpreted
as a true signal. The authors used this effect to obtain a 3D
map of the sample stray field [ Fig. 3(b) ]. More recently, Jaafar
et al.
110discussed a counterintuitive dependence of energy
dissipation on probe –sample distance for domains with mag-
netic moments parallel to the probe ’s magnetization. Thus,
for a large range of distances, the probe –sample separation is
directly proportional to the probe ’s oscillatory excitation
energy. The recorded dissipation values ( ∼fW) were in good
agreement with micromagnetic calculations, correspondingto the power losses caused by relatively small regions of
spins switching their magnetization. A high spatial resolution
(<8 nm) was achieved in the MDFM images; thus, MDFM hasbeen demonstrated to be a promising technique for MNPcharacterization.
109,111
As MDFM requires a strong probe –sample interaction,
which can be a limitation when measuring in high vacuum,
Zhao et al.31developed a frequency-modulated capacitive-
distance control method, which is valid even in the presenceof energy dissipative processes. Another proposed approach
for mapping energy dissipation is using drive amplitude
FIG. 3. Magnetic dissipative force microscopy. (a) Sketch of the dissipation
process associated to the variation of the stray field from the sample due to the
interaction with the probe. (b) Magnetic dissipation image corresponding to a Py
dot under in plane applied field of 60 mT . Reproduced with permission from
Iglesias-Freire et al ., Appl. Phys. Lett. 102, 022417 (2013). Copyright 2017 AIP
Publishing LLC.Journal of
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J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-7
Published under license by AIP Publishing.modulation AFM (DAM-AFM).112The method uses the mono-
tonicity of the dissipation force between the probe and the
sample to obtain stable images in all environments (e.g.,
vacuum or liquid suspension).24In DAM-AFM, the topography
map is obtained by using the dissipation of energy as thefeedback parameter while the frequency shift, controlled bythe phase-locked loop, provides information about the con-
servative interactions.
Magnetic scanning gate microscopy (mSGM), also known
as magnetoresistive sensitivity mapping, modi fies the electrical
properties of a device under applied voltage in proximity of thescanning MFM probe due to magnetoresistive effects ( Fig. 4 ).
From the applied potential difference across the device, elec-
trostatic interactions between the probe and the sampleheavily in fluence the acquired data. Thus, mSGM is often com-
bined with KPFM similar to the KPFM/MFM technique.
The modulated potential difference, induced by magne-
toresistive effects from the probe –sample interaction, can be
mapped by locking into the frequency of the MFM probe ’s
oscillation across the device with a lock-in ampli fier. Hence,
the noise generated by the frequencies of the sidebands
(from bias modulation on the probe) or the scan rate of the
probe across the sample can be signi ficantly suppressed
allowing for faster data-acquisition and greater SNR.
In the past, mSGM has been used to characterize giant
magnetoresistance (GMR) sensors and obtain sensitivity maps
to external magnetic fields. In particular, it has been applied
extensively to characterize hard disk drive reading heads.
113,114
Recently, it has been used for the characterization of L-shape
permalloy (Py) devices115,116and measure the probe stray field
using graphene Hall sensors.117–119For the former example,
devices with pinned DWs were scanned using non-magnetic
probes modi fied with a magnetic bead.116By monitoring the
resistance across the device, it is possible to estimate itssensing volume toward a speci fic magnetic bead (or any other
nanostructure on the probe). This approach enables testing
many devices with the same magnetic bead and thus allows
correlating results for the optimization of the sensing ele-ments. Other recent developments include using the probe ’sstray field to manipulate DWs; measuring electrical signals
originating from the anomalous Nernst and Hall effects as a
way of sensing the position of the DW inside of the nano-
structure;
120,121and writing magnetic landscapes with thermal
assistance for magnonic devices.122
Custom-made MFM probes have been developed to
improve the lateral resolution and sensitivity beyond the limit
of commercial MFM probes and also to facilitate quantitative
MFM (qMFM) studies, e.g., by increasing/reducing the coercivefield or modifying the stray field distribution and intensity
(Fig. 5 ). Three trends can be primarily identi fied: (i) customized
magnetic coatings , where the magnetic properties of the mate-
rial are varied; (ii) probes with magnetic adhered structures ,
such as Fe- filled carbon nanotubes (CNTs) or magnetic beads;
and (iii) MFM probes with fabricated nanostructures .
Among these three approaches, modifying the magnetic
coatings of an MFM probe is the most common as it does not
require extensive nanofabrication capabilities.
119,123,124Such
probes are characterized by the enhancement of the lateralresolution both in the topography and in the phase/fre-quency shift signal. This has been achieved, for instance, by
partially coating MFM probes,
123,125or depositing multiple
layers of magnetic material to be able to control high/lowmoment states and if necessary limit the eminent stray- field
to the probe ’s apex.
119,126The other advantage of customized
magnetic coating is a possibility to match the magnetic prop-
erties of the probe to that of the sample. For example, reduc-
ing the stray field produced by the probe reduces its
interaction with soft samples, or conversely increasing thecoercivity of the probe helps to image samples with strongstray fields.
A different approach is to adhere magnetic structures to
the apex of a non-magnetic probe, which was in many casesused in attempt to create a dipole-like
116,127–129or monopole-
like probe.130CNTs filled or coated with magnetic material
have been attached to the apex of standard AFM/MFM
probes131,132to improve the lateral resolution of the MFM
[Fig. 5(a) ]. However, this approach is usually at the cost of the
sensitivity, due to the small amount of magnetic material
FIG. 4. Schematics of magnetic scanning gate microscopy. Topography is scanned in Pass 1. An electrically connected, current-biased device is scanned by a m agneti-
cally coated probe, and the transverse voltage response at the resonant frequency of the probe is recorded as a function of the probe ’s position (Pass II). Typically, this
technique is combined with FM-KPFM, as the applied current gives rise to electrostatic artifacts.Journal of
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J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-8
Published under license by AIP Publishing.interacting with the sample. Another alternative is attaching
microscopic pieces of hard magnetic material133or magnetic
beads116,127–129,134,135[Figs. 5(b) and 5(c), respectively] to the
probe apex. In both cases, the typical size ( ∼1μm) is far larger
than a probe ’s apex diameter ( ∼30 nm), which could jeopar-
dize the lateral resolution.
The ability to fabricate nanostructures on the probe apex
not only opens the possibility for engineering the magnetic
properties by selecting the coating material, but also touse shape anisotropy as a way of governing magnetization.
The most common approach consists of using electrical
current to induce chemical deposition and hence achieve a
sharp apex.136,137However, the sharpness of the apex may
vary from probe to probe. Another approach uses focusedion beam (FIB) milling to sharpen probes into a needle withmagnetic coating only at the very end of the needle
25,138
[Fig. 5(d) ]. This approach has the advantage of producing
sharp probes with high lateral resolution, but with smallmagnetic moment. The last type of custom-made probesconsists of nanostructures built at the probe ’sa p e xt ou s e
shape anisotropy to constrain the magnetization and
produce a strong stray field just. For example, a V-shaped
magnetic nanostructure fabricated on one face of a non-magnetic probe was recently demonstrated [ Fig. 5(e) ]. Such
probes combine a low moment with high coercivity toreduce magnetic switching in the presence of strong stray
fields.
36A very recent work139combines all three strategies
byfirst developing a hard magnetic thin film architecture
most suitable for MFM on an appropriate flat substrate,
creating a nanostructure (slim triangular needle) from the
substrate film compound by FIB, and adhering this nano-
structure to a non-magnetic cantilever. In the above work, ahigh resolution MFM probe with unrivaled coercivity andthus stability against large magnetic fields has been fabri-
cated from a SmCo
5film grown epitaxially on MgO.
MFM probe characterization is a fundamental part of
the MFM experiments and particularly relevant for qMFMand in- field MFM. When assessing the suitability of an MFM
probe for an application, it is recommended to consider itsgeometry (e.g., by SEM);
119,121,140its coercive field (e.g., from
in-field MFM);131,141and its magnetization pro file (e.g., by
electron holography,119,121,140measurement of a reference
material,36,130,142or Hall sensors118,119,143–145).
B. Quantitative MFM modeling
Different approaches to qMFM have been developed in
the past two decades, which provide a quantitative descrip-tion of the magnetic probe. They range from simple point
probe approximations (PPA)
57to geometrical probe descrip-
tions146andfinally to parameter-free tip transfer function
(TTF) methods.25,52All approaches start from the correct
magnetostatic interaction between the probe ’s magnetization
and the sample ’ss t r a y field but use various degrees of simpli-
fication. For a linear oscillation regime and a negligible canti-
lever tilt, in the most general description, Δfis calculated
without any restrictions on the magnetization structure ~Mt(~r0)
within the probe as
Δf/difference@Fz
@z¼@2
@z2ððð
Vtip~Mt(~r0)/C1~Hs(~rþ~r0)dr03: (11)
Geometrical models often assume a simpli fied magnetization
structure for the probe, e.g., ~Mt(~r0)¼Mz,t, but attempt a real-
istic expression for its shape and volume. Equation (11)thus
FIG. 5. Examples of custom MFM probes. (a) Probe with a carbon nanotube
filled with a magnetic material. Reproduced with permission from Wolny
et al. ,J .A p p l .P h y s . 108, 1 (2010). Copyright 2010 AIP Publishing LLC. (b)
Probe with a magnetic disk on top of a FIB milled cylinder. Reproduced withpermission from Amos et al. ,J .A p p l .P h y s . 105, 07D526 (2009). Copyright
2009 AIP Publishing LLC. (c) Probe with a magnetic bead attached.
Reproduced with permission from Corte-León et al. , J. Magn. Magn. Mater.
400, 225 –229 (2016). Copyright 2016 Elsevier. (d) FIB sharpened probe.
Reproduced with permission from Belova et al. ,R e v .S c i .I n s t r u m . 83, 93711
(2012). Copyright 2012 AIP Publishing LLC. (e) Probe with a lithographically
patterned V-shaped nanostructure on one of the sides. Reproduced with
permission from Puttock et al. , IEEE Trans. Magn. 53,1–5 (2017). Copyright
2017 IEEE.Journal of
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J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-9
Published under license by AIP Publishing.reduces to
Δf/differenceMz,tððð
Vtip@2
@z2/C1Hz,s(~rþ~r0)dr03: (12)
In the PPA models, the magnetization is assumed to be con-
centrated in one point within the magnetic probe. In the caseof the point dipole approximation, Eq. (12)thus further simpli-
fies to
Δ
f/differencemz/C1@2Hz,s(~rþ~δ)
@z2: (13)
Here, ~δdescribes the position within the probe, where the
dipole moment mzis supposed to be located.
This disregards an important aspect of the magnetostatic
interaction: the interaction volume of a realistic 3D probewith the stray field of the sample ~H
swill depend on the size
and morphology of the measured domains. Thus, PPA models
should be applied only to samples with regular stray field pat-
terns. Recent works report on the moment quanti fication in
laterally con fined objects such as MNPs,93,147–149chains of
magnetosomes,150or patches of single molecular magnets
(SMM).151In most cases, the tip ’s point-probe parameters
were freely adjusted to allow a self-consistent data descrip-tion, but not determined from an independent sample. Onthe other hand, quantifying the signal of an individual nano-scale object is not easily done with other methods. Uhlig
et al.
127made use of the point probe character of an MNP by
picking up such a particle with a non-magnetic probe. By pre-paring such an MFM probe, the PPA model description ismore justi fied than for volume probes.
The TTF approach by Hug et al.
25calculates the force on
an MFM probe exerted by the stray field of a sample with per-
pendicular magnetization in Fourier space ( Fig. 6 ). By means
of a calibration measurement of a suitable reference sample,one derives a model-independent and parameter-freedescription of the probe ’s imaging properties. Considering
that even nominally identical probes (from the same manu-
facturer/batch) can result in large variance of the MFM con-
trast on an identical sample, this experimentally moreelaborate approach is thus judged to be of great importance.The researchers have successfully applied this approachto experimental means, e.g., for the quanti fication of non-
compensated moments in exchange-bias systems.
152Neu
et al. have followed this qMFM approach for, e.g., identifying
the vortex state in a magnetic nanowire,54calibrating custom-
made probes,119or quantifying the stray field in the corner of
an L-shaped Py structure.36A recent application of qMFM
quanti fies arti ficially patterned stray field landscapes in
CoFe/MnIr exchange bias layer systems.153Although success-
ful, this study also reveals the dif ficulties that arise with the
quanti fication of a complex multiscale domain pattern.
Reference samples and measurements need to cover a large
range of spatial frequencies to correctly calibrate the probe
for all relevant length scales. Due to the even larger complex-ity and multi-scale character of magnetic domains present inmodern permanent magnets,
154qMFM measurements have
not yet been performed on this important set of materials,
although it is expected that highly resolving and quantitativeMFM measurements can lead to a large improvement of theirunderstanding.
C. Modern objects of MFM studies
In this part, we discuss applications of MFM and the rele-
vant daughter techniques to modern areas of the physics andthe material science. It is noteworthy that such applicationsare very often quite challenging, dealing with extremely low
magnetic signals and requires the ability to distinguish the
magnetic response from the other components (i.e., electro-static contributions, magnetic contaminations, etc.).
We start this part of the Perspective article from
considering applications of MFM to thin films with PMA.
FIG. 6. Schematics for image-processing steps to acquire the real-space tip-transfer function (RS-TTF). The “real ”MFM image (top left) is used to generate an effective
surface charge pattern (bottom left) by binarizing the image and adding in magnetic or experimental parameters (i.e., DW-width, lever-canting, and Ms). The two images
are deconvolved in Fourier space by means of Wiener filtering to produce the stray- field derivative of the probe (top right). This can subsequently be used to produce cali-
brated/quantitative MFM measurements, as it can be deconvolved from the MFM image of a sample with unknown magnetic parameters.Journal of
Applied PhysicsPERSPECTIVE scitation.org/journal/jap
J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-10
Published under license by AIP Publishing.The interpretation of MFM images is most straightforward for
samples with a magnetization orientation perpendicular to
the imaged surface. Here, the stray fields produce a qualita-
tive MFM contrast that closely resembles the underlyingdomain structure. Hence, a wealth of MFM studies focus onthe imaging and interpretation of films with PMA, quanti fied
by the (perpendicular) uniaxial anisotropy constant K
u. In the
case of large PMA (ideally Ku/C29Kd¼1=2μ0M2
s), these band
domains can be approximated as being homogeneously mag-netized along the z-axis (colloquially up or down), forming a
domain morphology that depends among others on field
history, surface corrugation, and coercivity. Domain theory of
such high PMA films is well established and the knowledge of
M
s,Kuand the DW pro file allows a correct quantitative recon-
struction of the magnetic domain pattern (or, equivalently,the effective magnetic surface charge pattern) from a qualita-tive image. Thus, such films are well-suited as reference
samples for probe calibration (see Sec. III B).
Recent MFM work on films with PMA can be roughly seg-
regated into the following four groups. The first group deals
with films with large PMA, where the equilibrium domain
width can be used to judge the balance between the various
energy terms. For thin film systems with DMI, such compari-
sons between domain theory and observed domain widthsgathered great importance to conclude on the less accessibleDW energy.
155
The second group considers films with smaller PMA
(i.e., Ku<Kd), where the dominating shape anisotropy pulls
the magnetization vector into the film plane, but still the
presence of PMA can lead to a modulation of the magnetiza-tion vector perpendicular to the surface. These stripe
domains are again observable by MFM but the magnetization
possesses a complex depth dependent structure, which canonly be approximated by analytical theory and otherwiserequires micromagnetic calculations. A recent work demon-strates the in fluence of the weak PMA on the domain struc-
ture in soft magnetic NdCo
5films with anti-dot structure.156
Evaluating stripe domain patterns in a quantitative way has so
far not been accomplished to satisfaction. This is due to thelack of qMFM studies and the dif ficulties in theoretically
describing the magnetization pattern.
The third group includes samples, in which a layer
with PMA is exchange-coupled to a soft layer with in-planemagnetization, which are a subject of recent studies to obtaina microscopic view of how exchange-coupling occurs in
layers with orthogonal anisotropies, see, e.g., an example on
the [Co/Pd]/Py system in Ref. 157. The final group includes
laser-induced manipulation of a sample ’s magnetization state,
which can be imaged with high resolution by MFM and maygive insight into the origin of loss and sometimes also resto-
ration of magnetic order.
158
Beyond thin- films, another highly researched topic of
study is patterned magnetic media. Patterning FM materialsinto novel shapes and structures is of particular interest inapplications such as logic devices or novel magnetic record-
ing.
159As methods for patterning materials on the nanoscale
improve, as they have been consistently, ways to characterizethe new synthetic designs are required to measure the exotic
and useful properties they possess. MFM previously has been
highlighted as an important tool for understanding the mag-
netism within such structures, ranging from memory devices(e.g., bit-patterned media)
159to magnetic strips, and nanodot
and antidot arrays.160–163One of many modern examples of
magnetic patterned structures that are popularly researched
are arti ficial spin ice (ASI), which exhibit geometric frustra-
tion, ordering of effective magnetic charges, and a variety ofcollective dynamics.
164–166
ASI consists of lithographically patterned arrays of
nanoislands/NWs of different designs composed of in-plane
FM material, which are magnetically frustrated due to the
intrinsic geometric ordering to create two out-of-planeIsing-spins for each nanoisland.
167–169ASI have received
attention as the frustrated arrays can be controllably pinnedinto multiple stable/meta-stable states, priming them for
magnetic recording, logic devices, and experimental hot-beds
for understanding magnetic frustration in more complexsystems. In their ground-state, some of the most popularstructures in literature {e.g., squares and honeycomb lattices
[Figs. 7(a) –7(d), respectively]} obey the ice-rule,
167but can be
excited into higher energy states by external stimuli (e.g., byapplied field). Wang et al.
170demonstrated reading, writing,
and erasing of individual bits by applying in-plane field below
the nanoisland saturation- field and individually switching
nanoislands with an MFM probe, demonstrating great preci-
sion for single bit writing. Gartside et al.171similarly intro-
duced topological defect-driven magnetic writing on ASIusing the MFM.
Another extremely interesting example of recent
objects of MFM studies are magnetic topological structures.
Topological solitons, or defects, in magnetic materials haveprovided, and continue to provide, a rich plethora of phe-nomena to be studied for fundamental research
66,172–174and
future magnetic based technologies,175which rely on various
novel magnetic con figurations and architectures. Typically,
these defects in magnetic materials are manifested as magneticdomain-walls,
176,177vortices,178–183skyrmions,184,185or magnetic
bubbles.186Here, we focus solely on the use of MFM in observ-
ing and quantifying physical phenomena occurring in DWs and
vortices. MFM studies of skyrmions will be discussed in thePerspective section (Sec. IV C).
Magnetic domains and the walls that divide them are
determined by the subtle balance of the following main contri-
butions of micromagnetic energy: exchange interaction, mag-
netostatic, and magnetocrystalline energies.
177Understanding
DW motion and dynamics under the in fluence of an applied
stimulus such as magnetic field or spin polarized current
pulses can elucidate to the complex underlying magnetization
reversal processes and how DWs can be manipulated for use in
modern technologies. Here, MFM excels as a tool to investigatephenomena such as the domain structure in magnetic nano-patterned elements following the application of an externalstimulus in so-called quasi-static measurements. This is of high
importance for technological applications of DWs
187such as
that of the racetrack memory (RM).175RM offers a signi ficantJournal of
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J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-11
Published under license by AIP Publishing.gain over conventional magnetic storage devices and potential
silicon based logic circuitry in terms of performance.188Here,
spin polarized currents are used to generate spin torque trans-
fer189,190such that DWs in the racetrack can be moved along a
track, which extends into the third dimension175,191increasing
the effective bit density. Recent developments have moved tomore exotic phenomena to drive domain-wall motion where
spin-orbit torque (SOT) effects, such as the Rashba effect
192,193
and the spin Hall effect.194,195
MFM is also often used to investigate complex domain
type structures where the geometry, hence the magneto-static energy, of the material system starts to play a domi-
nant role.
196–198This alters the equilibrium con figuration
such that it becomes more complex than in the typical casesof Bloch or Néel type DWs in thin films. Examples of these
wall types include the transverse/asymmetric transverseDWs
196,197,199and single, as well as multiple, vortex
walls.196,197,200Understanding the internal structure of such
DW con fig u r a t i o n si si m p o r t a n tn o to n l yf o rs c i e n t i fici n t e r -
est but also for applications as the internal structurestrongly dictates the DW dynamics.
174Recently, Nguyen
et al. have demonstrated that in Py nanostrips with in plane
magnetization a so-called Landau DW exists.174This novelDW con figuration is described as a flux closure pattern that
resembles a Landau pattern; however, it is elongated andencircles a Bloch type wall. In this work, MFM was integral
in con firming the predicted domain con figurations obtained
by employing finite difference methods to solve the
Landau –Lifshitz –Gilbert equation.
201
Of particular interest is the case of cylindrical wire and
FM nanotube type geometries. Arrays of such wires have
potential in many advance technological areas, includingdata storage and information, energy, Life Science, and envi-ronmental sectors.
202Furthermore, numerical simulations
have predicted that the Walker breakdown limit in such 1D
nanostructures is topologically forbidden,203making them
extremely attractive for technological applications requiringDW displacement. In these geometries, a number of topologi-cal defects can be identi fied: transverse DWs; asymmetric
transverse DWs; and Bloch point walls, which are similar in
nature to vortex walls found in FM nanotubes.
204Due to its
high spatial resolution and sensitivity, MFM has been widelyused to study the domain con figurations of such wires.
For example, it has been shown that in Co NWs of dimensions45 nm in diameter and 10 μm in length, an alternating pattern
of vortex states is energetically favorable, offering an
FIG. 7. Artificial spin ice. Illustrations of the nanomagnet con figurations used to create arti ficial square (a) and kagome (c) spin ice, and their corresponding MFM images
[(b) and (d), respectively]. The black and white spots correspond to the magnetic poles of the islands. The arrows in (a) and (c) correspond to the magne tic moments
revealed by the MFM images. aindicates the lattice constant. Reproduced with permission from Zhang et al. , Nature 500, 553 (2013).317Copyright 2013 Springer Nature.Journal of
Applied PhysicsPERSPECTIVE scitation.org/journal/jap
J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-12
Published under license by AIP Publishing.interesting route toward novel spintronic memory devices.205
Similarly, the spin con figuration has also been investigated
in Co bamboo-like NWs with diameter modulation.202Using
MFM, it was demonstrated that, due to the competitionbetween the magnetocrystalline and shape anisotropy ener-gies, multi-vortex structures with alternating chirality form.Interestingly, as it was evidenced by MFM data, DW pinning
in the modulated diameters wires was avoided, in contrast to
other materials (FeCuCo) investigated in the same study.
In addition to DWs in thin films and nanowire type
topographies, con finement in nano-sized patterned elements
can lead to stabilization of vortex cores.
179These are of par-
ticular interest due to the potential they hold for future
microwave sources magnetic sensors and logic as well as innon-volatile memory applications.
206Vortex cores exist as a
thermally stable flux domain pattern that can be typically
characterized by in plane winding of the magnetization
around a perpendicularly magnetized core.182Vortex cores,
which can be as small as 10 nm181in size, possess polarity ±p
with respect the out-of-the-plane axis and a given chirality.MFM has proved to be an invaluable tool for the investigation
of vortex cores and indeed was used in the first observations
of vortex cores in patterned disks of Py.
178Additionally, MFM
has been utilized to explore the switching of vortex coresusing current driven vortex excitation via spin torque trans-fer
180paving the way for electrical control of magnetization in
logic devices.
Multiferroics are another modern class of material where
MFM and MeFM are used, often in conjunction with piezoforce microscopy (PFM) studies. The coupling between themagnetic and electric dipoles in multiferroics holds vast
promise for conceptually novel electronic devices and has
been widely explored in the last decade. ME phenomena havea profound and broad impact on diverse areas of materialsscience from multiferroic materials to topological insulators,where direct visualization of ME domains and DWs is of both
fundamental and practical importance. Speci fically, MFM has
been proven as an essential technique for studies of multifer-roic (in particular, ME) materials that exploit both FM and fer-roelectric (FE) properties.
MFM is typically used to reveal the microstructure of both
single-phased multiferroics and multiferroic composites, suchas detection of the strong magnetic contrast, visualization ofthe magnetic structure of grain boundaries, and reviewing theappearance of non-magnetic pores between the phases in
nanostructured ME materials.
207MFM imaging was used to
reveal the presence of magnetic domains being extended overseveral adjacent ferrite grains in BaTiO
3(Ni0.5Zn0.5)Fe2O4multi-
ferroics208and in BiFeO 3NWs.209In many cases, it was advan-
tageous to use extended modes of MFM, i.e., in- field MFM or
under the action of electrical poling.
Additionally, MFM was used to establish the nature and
overall contribution of the material properties originatingfrom magnetic and multiferroic defects. In the relaxor FEsingle-phase (BiFe
0.9Co0.1O3)0.4–(Bi 1/2K1/2TiO 3)0.6, CoFe 2O4
magnetic clusters with sizes 0.5 –1.5μm were identi fied using
MFM.210Such inclusions exhibit solely a magnetic dipolarresponse, indicating magnetization along the in-plane orien-
tation. On the other hand, a combination of MFM and PFM
showed that multiferroic clusters (unspeci fied in nature)
exhibit both FE and strong magnetic properties. It is expectedthat these findings will lead to new research in this novel
class of non-ergodic relaxor multiferroics, especially as thematerial is Pb-free and consists only of abundant elements.
210
The overall concept is ideal for an electrically controlled mag-netic nanodot storage device.
211
Local MFM studies were used to directly demonstrate
magnetization reversal under purely electrical control inanother BaTiO
3/Ni system, which is the overall goal in magne-
toelectrics.212The authors primarily used MFM to study a com-
mercially manufactured multilayer capacitor that displaysstrain-mediated coupling between magnetostrictive Ni elec-trodes and piezoelectric BaTiO
3-based dielectric layers. The
authors evidenced that the anisotropy field responsible for the
perpendicular magnetization could repeatedly be reversed by
the electrically-driven magnetic switching. The demonstrationof non-volatile magnetic switching via volatile FE switchingwas used to inspire the design of fatigue-free devices for
electric-write magnetic-read data storage.
212
Direct visualization of ME domains in multiferroics was
demonstrated using low temperature in situ MeFM from
lock-in detection of electrically-induced magnetization.The authors directly demonstrated the local intrinsic ME
response of multiferroic domains in hexagonal ErMnO
395and
YbMnO 3,96distinguishing contribution of six degenerate
states of the crystal lattice, which are locked to both FE andmagnetic DWs. Results were in excellent agreement with thesymmetry analysis, and a giant enhancement of the ME
response was observed in proximity of the critical tempera-
ture. This suggests that critical fluctuations of competing
orders may be harnessed for colossal electrically-inducedmagnetic responses ( Fig. 8 ). The use of cryogenic in- field
MFM was also demonstrated by Wang et al.
98Labyrinth-like
domains ( ∼1.8μm) in an h-LuFeO 3thin film were observed
after zero- field cooling below the Néel temperature, TN≈147 K,
where weak FM order with a canted moment exists. At 6 K,MFM images of the magnetization reversal process reveal a
typical domain behavior of a pinning-dominated hard magnet.
The temperature dependence of the domain contrast demon-strates that MFM is able to detect the domain contrast ofmagnets with miniscule magnetic moments ( ∼0.002 μ
B/f.u.).
Moving away from traditional applications in physics and
material science, MFM has lately gained a momentum for
studies of magnetic nanomaterials for Life Science applica-tions. There are a broad range of applications using magneticbeads and MNPs, including cell separation, bio-sensing, in vivo
imaging, magneto-thermal therapy, etc.
213,214Alternatively, the
use of elongated nanostructures such as magnetic cylindrical
NWs is of growing interest in different bio-magnetics appli-cations due to their high aspect ratio, anisotropic physicalproperties, and the possibility to work with different lengthscales.
215
A direct characterization of the magnetic properties of
individual beads and MNPs on nanoscale is possible byJournal of
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J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-13
Published under license by AIP Publishing.microscopy-based techniques such as MFM. For example,
MFM has recently been used to detect superparamagnetic
and low-coercivity79,80,148,216,217MNPs. Also, MFM has been
successfully employed to characterize MNPs inside biologicalsystems as vesicles (niosomes),
218virus-capsids,219or cells,220
where MFM images were used to evaluate the amount ofmagnetic material inside the different entities.
In addition, MFM has been used to investigate the mag-
netic properties of individual NWs
221and vortex-state dots163
for biomedical applications. Molecules, such as ferritin, havealso been characterized by liquid-MFM.
222,223However, it is
crucial to remember that MFM is sensitive to non-magnetic
(e.g., electrostatic) signals (see Sec. III A),44especially in the
case of biological systems, where the strength of the producedstray field are near the sensitivity limit of the technique.
In the case of biological systems, it is essential to
perform sample characterization in relevant environmental
conditions, e.g., physiological environment. For that reason,
non-standard methods such as bimodal,
101energy dissipa-
tion,111or AC- field modulated MFM224,225have been explored
in recent years. Another approach is the use of custom mag-
netic probes speci fically designed for biological applica-
tions.123,226However, since MFM has historically been applied
to the study of inorganic materials, the potential of MFM forbiological/biomedical applications is still under develop-ment.
227Recent studies have demonstrated essential MFM
capabilities (i.e., high enough lateral resolution and sensitivity)
for studies of individual MNPs in a liquid environment.24This
development opens new possibilities of studying magneticsystems in biologically relevant conditions.
IV. PERSPECTIVE OUTLOOK FOR MFM
The Perspective part of the paper presents the emerging
trends in the field of MFM concerning further development of
instrumentation (e.g., in combination with other SPM modesand radiation techniques), the wider applications of qMFMmeasurements, and application of MFM and its sister modes
to studies of advanced and emerging materials.A. Novel and multifunctional instrumentation
The MFM community incorporates a variety of users:
from beginners that demand a friendly and reliable interfaceto the highly specialized researchers that customize or even
build their own system. It should not be forgotten that the
majority of the commercial MFM users are interested inpushing the resolution and sensitivity limits of the technique.While commercial, off-the-shelf systems still remain a validindispensable tool for a routine inspection of magnetic prop-
erties of samples, modern challenges in both research and
industry demand development of new advanced MFM modes.To ful fill this need, the current research is targeted in differ-
ent directions, including the development of a new MFMinstrumentation and flexible software, novel types of MFM
probes (a key point still under development), and combination
of MFM with other techniques targeting complex materialproperties, which is a general trend to make the MFM com-patible with the simultaneous transport, thermal, or opticalcharacterization. Finally, there are several groups that push
the MFM technique to the limits of high speed scanning, fast
signal processing, and recording than allow exploring highfrequency processes.
Often the realistic experimental needs require measure-
ments in a speci fic, precisely de fined environment, e.g., tem-
perature (i.e., low, high, or variable), pressure, humidity, speci fic
gas atmosphere, vector magnetic field. Typically all these
options are not available commercially but rather custom-developed as a research tool (see Sec. III A). Another rapidly
filling niche is the development of custom-made magnetic
probes. While commercial suppliers usually offer magneticprobes of three main types (i.e., standard, low-/high-moment),the customized options provide a signi ficantly larger variety of
probes with properties targeted to a speci fic (sometimes very
narrow) application. The examples include the probes function-
alized with magnetic nanoparticles and microsized beads,Fe-filled CNTs, one-side coated switchable probes, lithographi-
cally modi fied probes (e.g., Fig. 5 ), etc. Another important
option is an ability to separate magnetic and electrostatic
FIG. 8. MeFM images and the magnetic field dependence of the MeFM signal. (a) –(f) The representative MeFM images taken at 2.8 K in various magnetic fields. All of
the images are in the same color scale. (g) Field dependence of the MeFM signal at 2.8, 4.0, 5.2, and 10 K, respectively. For details, see Ref. 95. Reproduced with per-
mission from Geng et al. , Nat. Mater. 13, 2 (2013). Copyright 2013 Springer Nature.Journal of
Applied PhysicsPERSPECTIVE scitation.org/journal/jap
J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-14
Published under license by AIP Publishing.signals and successfully eliminate the latter. This is an impera-
tive option for samples with low conductivity or electrically
biased devices.118,119,145,228One of the most promising trends is
the development of new multifunctional systems, allowingcombined mapping of magnetic and additional functional prop-erties (e.g., KPFM-MFM, MF-PFM, MF-SEM, etc.), or measure-ments of magnetic properties by different means [e.g., in
tandem MFM and magneto-optical Kerr effect microscopy
(MFM-MOKE)]. All these combined modes are currently avail-able only as custom-developed options (often due to a limitedsize of the specialized market). However, it might be expectedthat they will soon find their way to commercial options.
For the latter example, the combination of a MOKE
microscope and MFM provides an interesting and powerfultool to study novel magnetic materials,
229not only at different
length or time scales, dictated by the two methods, but it alsoallows for flexibility in terms of the magnetic sample to be
investigated in a highly ef ficient manner. For example, by uti-
lizing the MOKE, one can tune the domain pattern or magne-tization state of the sample and image within the, diffractionlimited, resolution of the microscope. Then, subsequent MFM
investigations can follow which would allow for higher spa-
tially resolved images to be taken. This combination has beenutilized to image the domain structure of NbFeB crystals.
230
In this work, two data analysis techniques were used to gainfurther insight to the magnetic structure, including surface
charge pattern and local susceptibility. This is achieved by
taking the difference and sum images, respectively, of twosubsequent scans with oppositely magnetized probes. Thisallows the general domain structure through charge contrastimages and also the variation in the sample permeability
through the susceptibility contrast images to be obtained.
Due to the depth sensitivity of the two techniques, compli-mentary information of the surface as well as the generalmagnetization structure within the domain can be investi-gated. Such functionality has signi ficant merits for topics that
are currently investigated and feature in this Perspective
section. For instance, systems hosting skyrmions or bubbledomains could be studied using this combined approach. Dueto the different skyrmion sizes possible, a cross-over between
the two techniques would be de fined: Kerr effect for a
general overview and location of optically resolvable featuresand MFM, which would be used as more local probe to inves-tigate the stray field signatures of the skyrmions. This is par-
ticularly interesting in terms of the field protocol used to
nucleate and annihilate skyrmions as it would allow for a
broad understanding of the regions of most interest in atimely fashion rather than searching within the field-of-view
of the SPM. Although the original combination of MFM andMOKE was published relatively long time ago,
229there is now
a clear industrial interest from the companies in resuming
this type of instrumentation on a commercial scale.
Multi-functional microscopes, with the ability to
combine data from different sources into a single image aswell as controllably and reproducibly modify the sample ’s
state, are becoming more and more ubiquitous. For instance,
quite recently, it has become p ossible to use an AFM insideof an SEM chamber to combine the two imaging tech-
niques
231,232or to perform nanofabrication with the
focused-ion beam (FIB).233This new instrument, called
AF-SEM, works in vacuum conditions and allows for largescanning areas and position ing the probe in ways that are
typically non-accessible to normal MFM. This is of interestwhen considering the possible shift from 2D fabrication
towards 3D magnetic nanostructures, since AF-SEM will
enable navigating complex samples and perform MFM ondifferent faces of a 3D structure.
205
Another interesting system, in particular for in-liquid
MFM,234is a combination of a SPM and an optical microscope
where the latter includes functionalities such as confocal- or
fluorescence microscopy. The combination of MFM with
these techniques will further enable a range of Life Sciencestudies (e.g., related to MNPs applications or combinedmagnetic and optical labeling).
150For example, in a typical
experiment where the cells are sensitive to light, proteins
marked both with MNPs and fluorescent markers are intro-
duced to the extracellular medium. Using either fluorescent
or confocal microscopy, it is possible to study the large scale
distribution and see if the MNPs are internalized by the cells,
while using the MFM, it is possible to detect individual MNPsand characterize their distribution at the nanoscale levelinside of the cell without having to expose the cell to highintensity light.
235,236
Apart from combining different imaging techniques, the
possibility of performing manipulation or modi fications on
the samples under study during imaging is a growing trendthat has seen big developments in the last couple of years.For example, some SPM systems now include a lithography
mode where the probe follows a custom-de fined path, while
ad eflection or bias voltage are applied to the probe.
122,171,237
Such experiments have been performed to move/capture
magnetic beads, to induce defects/nucleation sites in mag-netic films, and more recently, to print 3D nanostructures.
The possibility of inducing defects/nucleation sites in mag-
netic films and nanostructures has a wide range of applica-
tions, since the lithography mode allows direct introductionof desirable magnetic sites, while MFM enables imaging the
magnetization distribution and its consequent evolution.
Additionally, the possibility to manipulate magnetic beadsusing MFM enables single magnetic bead studies. Finally, the3D-printer AFM, which operates in-liquid and uses a hollowprobe to deposit materials,
238is a system that has so far dem-
onstrated rapid nanofabrication capabilities, without the need
of a clean-room or e-beam lithography, which are expensivefacilities that limit the access to nanofabrication. This is anexcellent system to be combined with MFM, since the 3Dprinter enables building magnetic nanostructures, and the
MFM allows imaging them to check if the magnetic nano-
structure behaves as expected.
The ability to perform real-time MFM is a desired func-
tion for researchers in micro-/nano-magnetics as it wouldcombine operational simplicity and availability of an SPM
system in non-specialized environment with the power to
map real (quasi-)dynamic effects, rather than “freeze-frames. ”Journal of
Applied PhysicsPERSPECTIVE scitation.org/journal/jap
J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-15
Published under license by AIP Publishing.Such advancements would allow for the study of effects
such as domain wall propagation/creep or magnetization
relaxation in MNPs with relative ease and high resolution.
High-speed AFM (HS-AFM) is not a new concept, with com-mercial systems capable to map topography with sub-100 mstime-resolution rather than minutes in standard AFM,
234but
speeds required to study a multitude of dynamic magnetic
effects (potentially μs resolution and below) appear to be
unlikely within the current scope of developments (as of timeof writing) for MFM.
Another interesting concept in MFM-instrumentation is
the ability to perform volumetric magnetic measurements.
Until recently, SPM was traditionally performed in zero- (e.g.,
noise measurements), one- (e.g., variably- field MFM), or most
typically two spatial dimensions in the sense that a change inphysical interaction between the probe and the sample isquanti fied within a de fined Cartesian coordinate above
the basal plane of the surface. More advanced SPM tech-
niques have extended into the third dimension by mappingchemical/physical properties with respect to physical matterinteractions (e.g., vibrational modes in chemical bonds by
tip/surface-enhanced Raman
239and scanning nano-IR
microscopy240)o r xyz(i.e., volumetric) data-acquisition (e.g.,
AFM force volume measurements241,242). Volumetric MFM can
be performed on commercial instruments as it is largelybased on the force volume methodology; the fast axis in such
measurements is the z-axis, thus the probe generates force
curves at each xy-pixel coordinate, mapping the phase
change as a function of z-displacement ( Fig. 9 ). Despite this,
MFM has instead largely stayed within a single spatial plane,despite the recent scienti fic drive towards “big data ”in other
areas.
243The likely causes for this shortfall thus far is large
data-sizes and lengthy acquisition times.
3D-data for MFM does not exclusively refer to the three
spatial dimensions, and there are many examples where the3rd variable is an alternative controllable property such as
temperature or applied field, which have been discussed
throughout the Review section. A recent example of acquiring3D matrices of MFM data is provided by the demonstration ofthe general-mode (G-Mode) SPM system.
243This system
samples the entire photodetector response of an SPM with a
MHz sampling rate, generating a three-dimensional datasets
(after post-segmentation). One interesting application forG-Mode is the identi fication and separation of magnetic and
electrostatic interactions in MFM.
244
However, volumetric MFM datasets remain uncommon,
despite improved data acquisition (in part developed from
the popularity of functions such as force-volume) and moreavailable tools/software for 3D-data visualization andanalysis.
245 –247Volumetric MFM is largely an under-
researched area in which, with further development of data-
handling practices, statistics, and with ut ilization of modern
techniques such as machine learning, interesting propertiescould be quanti fied at the nanoscale without specialized/
expensive equipment, e.g., 3D calibrated characterization ofMNP ’s stray field, magnetization dynamics with respect to
perturbing fields, or probe calibration/characterization.
The development of new MFM instrumentation goes
hand-in-hand with the development of new magnetic probesbuilt on demand (both custom-made and commercial).
Customized probes are used to perform very speci fic tasks and
push the limits of commercial MFM systems, e.g., to achieve ahigher resolution; reduce/increase probe –sample interaction;
or to be able to combine different scanning modes. Anexample of the latter is the use of a probe that is both magnetic
and conductive simultaneously, enabling the instantaneous
extraction of both magnetic and electrical signals.
5
Due to targeted speci ficity and high production costs,
the market for customized probes is often small; thus, manyof the proposed modi fications do not become available as
commercial products. However, occasionally some of the new
designs become commercially valid due to a reduction in fab-rication costs and growth of the market for MFM (and otherSPM techniques).
123The MFM probe with partial coating248is
an example of this, where only one side of the probe ’s tip is
coated with magnetic material; this reduces the magnetic
moment of the probe, achieving a higher spatial sensitivity.The magnetic coating of this probe model is deposited insuch a way as to prevent also coating the cantilever with mag-
netic material, reducing the cantilever-sample interaction.
Another example of custom probes entering the market is theMFM probe where the magnetic element is either inside or atthe end of a CNT attached to the probe ’s apex. These probes
are suitable for commercialization due to their apparent
advantages (very low magnetic moment, high spatial resolu-
tion, and extremely low probe –sample interaction), which are
becoming more and more critical in growing fields such as
bio-magnetism or magnetic topological structures (e.g.,skyrmions).
An emerging technology is the multifunctional nanoscale
sensor, which is able to detect several types of interactionsimultaneously, rather than being used only for a single appli-cation. Examples of this include the use of magnetic probes innear- field systems such as scanning near- field microscopy
249
or tip-enhanced Raman spectroscopy.250This multifunctional
approach would allow the production of probes to be more
FIG. 9. 3D-MFM. Schematic representation of volumetric MFM. The data are
acquired by z-axis orientated force curves at each xy-coordinated pixel across
the sample surface.Journal of
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J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-16
Published under license by AIP Publishing.cost-effective. For example, the single application probe used
in scanning thermal microscopy has very costly fabrication
steps, but the addition of a magnetic coating, to create a
multi-functional probe, would add only a relatively small costto the total amount. In addition to the economic advantage,multifunctional probes are able to signi ficantly reduce imaging
time as they are able to simultaneously extract several different
data types. It could be argued that the data quality would also
be increased as the necessity to locate the area of interest witha multiple probes would be eliminated.
Despite the limitations faced in making customized probe
models widely available, there are several examples of new
probe models being adopted by MFM probe suppliers. While
the probes with single functionality are expected to continueto dominate the MFM market in the short term, the multifunc-tional approach is expected to see greater success in themedium to long term, due to increased cost-effectiveness and
added probe functionalities that are advantageous to users.
B. Calibrated MFM
Typically, macroscopic magnetic field measurements
are traceable to nuclear magnetic resonance quantum stan-
dards and traceability chains to industry are already wellestablished. However, these calibration chains only relate tomeasurements of fields that are constant and homogeneous
over macroscopic volumes or surface areas down to the milli-
meter scale. At the same time, key international high-tech
industries such as magnetic sensor manufacturing, precisionposition control and sensing in information technology, con-sumable electronics, and Life Science, as well as in R&Drequire traceable and reliable measurements of magnetic
fields and flux densities on the micro- or nanometer scale,
e.g., for quantitative analysis and quality control. In order toaddress the gap between the technological capabilities andthe industrial needs, a collaborative European metrologicalproject (NanoMag; http://www.ptb.de/empir/nanomag.html )
has been established. The overall goal of this project is to
develop and provide coordinated and sustainable Europeanmetrology capabilities that extend reliable and traceable mea-surements of spatially resolved magnetic fields down to the
micrometer and nanometer length scale. Development of the
standards for traceable calibrations for MFM is one of the pri-marily goals of this project. The prime outcome of the projectis related to the development, comparison, and validation ofcalibration procedures for traceable quantitative MFM mea-
surements as well as establishing a high level of metrological
MFM capabilities across Europe.
Quantitative stray field measurements on the sub-50 nm
length scale, which can be easily achieved by qMFM, have amultitude of applications. One of the largest is the realization
of position control devices, which due to the much-reduced
length scale will find use in appliances, automotive, and con-
sumer electronics. Furthermore, tailored magnetic stray field
landscapes on the micrometer and nanometer length scaleallow controllable magnetic micro-bead and/or nanoparticle
manipulation and transport
220,251in future cost-ef ficientlab-on-a-chip devices for biological, chemical, medical, and
life science applications. Finally, a multitude of scienti fic
studies, which are already tackled by MFM (see Sec. III C ),
would bene fit from a quantitative analysis. We just mention
two large fields: (a) isolated nanoscopic object, in which size
and magnetic nature are not fully known [e.g., core-shell par-ticles (see Sec. III B) with a non-magnetic oxide shell, struc-
tured thin film elements with a magnetic dead layer] and
cannot satisfactorily be studied by global magnetometry, butcould be quanti fied microscopically. (b) Reconstructing the
magnetization state from stray field data is an ill-posed inver-
sion problem in magnetostatics. This is even more problem-
atic, when inhomogeneous magnetization structures or
magnetization textures are to be resolved. While qMFM maynot be able to unambiguously reconstruct such textures dueto fundamental limitations, it allows to decide between differ-ent hypothetical models. Thus, inhomogeneous magnetiza-
tion states, such as stripe domains in films with weak PMA
(Sec. III C) or skyrmions (Sec. IV C) can be identi fied and dis-
tinguished from band domains or bubble domains when MFMmeasurements are analyzed quantitatively.
With increasing automation of both measurement capa-
bilities and analysis procedures in modern AFM/MFM instru-mentation, qMFM based on the most versatile TTF approachwill become accessible for routine MFM experiments. Themost important requirements are the availability of appropri-
ate reference samples and of dedicated analysis software. Few
groups do already have these capabilities
25,54and they are
currently being evaluated and developed further for dissemi-nation to the public in the current European metrologyproject NanoMag ( http://www.ptb.de/empir/nanomag.html ),
including analysis software tools in the scanning force data
analysis package Gwyddion.
245A second requirement is the
availability of artifact-free, low noise, and reproducible MFMdata, which is aided by the improved stability and ease-of-operation in modern SPM-instrumentation.
Automation of measurement procedures (using scripting
and batch processing) will allow repeated measurements withunchanged parameters (for improved signal-to noise),repeated measurements with systematically changing param-
eters (e.g., varying lift height for con firming the correct decay
behavior of stray fields and thus excluding artifacts), and also
alternating measurements between reference sample and thesample of interest (to judge the stability or wear of theprobe ’s imaging properties during repeated use). Automation
of analysis procedures will easily allow for, e.g., drift correc-
tions, averaging, or more complex mathematical operations(filtering, deconvolution, etc.) of images, which finally results
in a quantitative evaluation of the MFM probe or the magneticsample under study.
We further describe a required standard procedure for
calibration. Prior to an automated quantitative measurementof a sample under study (i.e., measurand), MFM probe,reference sample, and measurement procedure have to beproperly selected to reveal the desired information. The main
characteristic of a reference sample is such that its domain or
stray field pattern can be quantitatively constructed from theJournal of
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J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-17
Published under license by AIP Publishing.measured MFM data without detailed knowledge of the yet to
be calibrated probe. A reference sample can be a thin film
with known PMA and saturation magnetization in a multi-
domain state (see Sec. III C), the stray field landscape of pat-
terned thin film elements in a single domain or the stray field
of current-carrying wire structures. Most important for thechoice of reference sample is that it covers all spatial frequen-
cies present in the studied sample.
A standard procedure for quantitative MFM is envisioned
as a flow diagram ( Fig. 10 ). An alternative to the final block
(red) is to develop a hypothetical surface charge/stray field
model of the sample and construct a theoretical MFM pattern
via convolution with the agreed TTF. The model should be
modified until suf ficient agreement with the experimental
MFM pattern is achieved.
C. Novel objects for MFM studies
We further discuss the application of MFM to studies of
advanced and emerging magnetic materials and structures,namely antiferromagnets, spin-caloritronic materials, sky-
rmions, topological insulators, 2D materials, and van der
Waals crystals as well as application of MFM to multidisciplin-
ary life Science and environmental studies, which are oftenbeyond a “traditional ”physics approach.
The applicability of MFM to characterize the stray mag-
netic fields from magnetic recording (MR) and logic devices is
historically well established in literature. In earlier studies,
Rugar et al.
32reviewed the application of MFM to longitudinal
recording media, and ever since there has been numerousstudies of MR by MFM along with a host of other techniques.However, the bit capacity for modern MR (e.g., those based
on perpendicular MR devices) has accelerated to the point
where they are almost beyond the limits of the spatial resolu-tion for standard MFM. As a consequence, MFM is currently aconfirmatory technique for characterizing stray fields in MR
devices industrially, used in tandem with other imaging
methods. Further development in MR is certainly going to
continue at pace, potentially circumventing the practicalityfor MFM imaging devices directly as it will not be able to fully
FIG. 10. Flowchart for the calibrated MFM process. Flow diagram of the standard measurement (left) and analysis (right) procedure which should be adopted for cali-
brated/quantitative measurements by MFM.Journal of
Applied PhysicsPERSPECTIVE scitation.org/journal/jap
J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-18
Published under license by AIP Publishing.resolve the features. However, MFM ’s simplicity and availabil-
ity means that, although it may not be used to characterize
MR devices directly as it has been used historically, it may
remain a popular tool for research in this area in other ways,as we have seen for heat- or microwave-assisted magneticrecording (HAMR and MAMR, respectively),
89,252,253which are
modern technologies set for commercial markets. Examples
of some creative studies into these devices with MFM
includes work by Chen et al., who used MFM and MOKE to
probe the erasure of the pre-recorded magnetic patterns as afunction of laser power,
254and to experimentally study a
novel bi-layered HAMR architecture that has one layer for
conventional MR and a dedicated servomechanism in the
underlayer.255More novel MR concepts, such as racetrack
memory (see Sec. III C ) shall remain a signi ficant research
topic as MFM offers the ability to image the domains in aquasi-dynamic state, and quality test the imperfections in
NWs, which currently limits the DW velocities in devices.
Antiferromagnetic materials are interesting for spintronic
applications due to the great variety of inherent phenomenathey possess.
256,257These include absence of stray fields due to
fully compensated magnetic moments, resilience to externally
applied fields, and faster spin dynamics than those of FM mate-
rials due to high magnetic resonance frequencies of the orderof THz. These properties make them attractive for applicationssuch as antiferromagnetic-based memory. It has recently been
demonstrated that current induced torques can be used to
shift the orientation of the Néel vector in CuMnAs,
258resulting
in the all electrical reading and writing of antiferromagneticrecording media. Indeed, thin films of Cr
2O3have been studied
due to their ME effect which can be signi ficantly enhanced
when the thickness dimensions are of the order of a few nano-
meters.259Here, MeFM has been extremely successful in iden-
tifying the antiferromagnetic domains in Cr 2O3. Furthermore,
the antiferromagnetic properties of Cr 2O3combined with its
ME effect can be used as an active exchange bias layer that can
be modi fied electrically which can manipulate the FM state of
exchange coupled magnetic layers.260
It is expected that both MFM and MeFM will be adopted
on a broader scale in order to understand better the local
magnetic properties of antiferromagnetic materials. The intrin-
sic properties and hence the functionality of such materials areextremely dependent on the local degree of disorder anddefects. The information gained by MFM and MeFM will beinvaluable for the miniaturization of current antiferromagnetic
based spintronic, multiferroic systems,
261and understanding of
the role defects play in these materials. This is evident inrecent investigations of multiferroic hexagonal rare earthmanganite where MeFM was used to observe ME domainson a micrometer scale.
95Here, it was evidenced, by observ-
ing a divergence in the ME effect near the tri-critical point
using MeFM, that an enhancement of the ME effect inh-ErMnO
3could be possible by utilizing critical fluctuations.
Combinations of MeFM and MFM at low temperatures areanticipated to play a crucial role in the understanding and
further development of multiferroic and antiferromagnetic
materials exhibiting ME coupling on the micro- and nanoscale.Further to the investigation of antiferromagnetic order
by MeFM,
35,97applications of MFM are likely to be employed
for studies of defects in antiferromagnetic materials such
as NiO. It has been shown that crystallographic defectscan exhibit signi ficantly different magnetic behavior to
that of the lattice, where MFM was used to visualize disloca-tions at the individual level.
262Moreover, it was found that
it was possible to create such dislocations in order to
generate high stability and high coercivity FM elementsembedded in an antiferromagnetic environment, where theferromagnetism arises due to the off-stoichiometry ofthe dislocations.
Spin caloritronics studies the combination of thermoelec-
tric properties and spintronics, i.e., heat currents and spin cur-rents.
263This combination potentially offers bene fits in
efficiency over traditional Seebeck effect based devices, such
as thermoelectric power generators264,265for energy harvesting
applications.266A particularly interesting system that is highly
studied in the field spin caloritronics is a thin film of heavy
metal exhibiting spin-orbit interaction on top of a FM insula-tor.
267Pt/YIG bilayers are popular candidates chosen to inves-
tigate phenomena such as spin pumping268–271where the FM
YIG is used to drive a spin current into the Pt, which isdetected via the inverse Spin Hall effect (ISHE), a manifestationof the spin-orbit interaction.
194,272–276These systems are also
used to observe the spin Seebeck effect (SSE),267,277,278where
temperature gradients are used to generate a thermally
induced spin voltage in the heavy metal layer, related to themagnetization dynamics of the magnet material in the thermalgradient. Again, the ISHE is used to as a spin current detectorto measure the magnitude of the conversion. Of particular
interest is the interface between the two layers where investi-
gations have shown that magnetic proximity effects could exist,which have driven intense discussion.
279–282Here, an induced
moment in the nonmagnetic heavy metal layer could convolutethe interpreted signal with additional effects such as the
anomalous Nernst effect. Further to this, recent x-ray magnetic
circular dichroism (XMCD) experiments have led to thedebate
283of the size of such an induced moment of Pt in Pt/
YIG samples. Polarized neutron re flectivity (PNR) is an
extremely sensitive technique which allows the layered mag-
netic structure of a material to be probed which has alsorevealed an induced magnetic moment at the Pt/YIG inter-face
284in these types of bilayer samples.
Previously, there has been little in the way of local scale
analysis/observation of the SSE in Pt/YIG type samples.
Local laser heating experiments have been used to observethe effect with a resolution of approximately 5 μm in Hall bar
type devices.
285Therefore, it is highly expected that MFM and
other relevant techniques (i.e., MFM + MOKE or MFM + SThM)
will be used to shed light on the complexity of this type of
materials and reveal new insights. Here, high spatial resolu-tion and sensitivity to the perpendicular field gradients could
potentially elucidate the magnetic properties and domainstructures close to the interface.
Skyrmions are chiral magnetic spin textures that are
non-trivial and topologically stable.
286,287Due to theseJournal of
Applied PhysicsPERSPECTIVE scitation.org/journal/jap
J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-19
Published under license by AIP Publishing.characteristics, they have been shown to demonstrate inter-
esting phenomena such as the skyrmion Hall effect173,288and
the topological Hall effect289 –292and therefore present an
interesting platform for investigation of emergent electro-magnetism associated with skyrmions. Figures 11(a) and11(b)
show example vector fields for Néel and Bloch skyrmions of
certain chirality, respectively, and the color scales depict the
z-component of the spin and the insets show a cross-
sectional dataset for each skyrmion highlighting the internalspin texture. Skyrmions are known to exist in bulk non-centrosymmetric chiral crystals
66,293 –295and also stabilized
in highly engineered thin- films comprising of FM/heavy
metal interfaces,296which can host skyrmions above room
temperature.297Due to the inherent or engineered inversion
asymmetry found in these lattices or layered interfaces, aDMI is induced,
291which contributes to the overall magnetic
ordering and tends to cant neighboring spins in favor of
pure parallel/antiparallel Heisenberg exchange interaction,
thus generating chiral spin structures. Due to their smallsize, theorized to range from 1 nm to 1 μmd e p e n d i n go nt h e
interplay of mechanisms that stabilize them
298and ability to
be generated and manipulated by SOT,76,297,299 –301it is
expected that skyrmions will give rise to a range of new sky-rmionic based logic and storage elements for future computertechnologies, which scale beyond dimensions predicted by
Moore ’sl a w .
287Among other imaging techniques,296MFM has
been used to image skyrmions and estimate the DMI value, as
it allows a relatively wide field of view and high resolution to
determine parameters such as the domain periodicity whichcan be used as an input parameter to numerically estimate theaverage DMI value.
302
Latest examples of qMFM have highlighted the possibility
to attain a deeper understanding of the nanoscale magneticcomplexity of skyrmions. Recent developments in implement-ing quantitative approaches have progressed the use of MFMin skyrmionic research from a simple imaging tool to an inte-
gral analysis procedure, which is the key to understanding
vital aspects of the magnetic characteristics of skyrmions.Yagil et al. have demonstrated that MFM can be used to study
the stray field pro file of skyrmions in sputtered Ir/Fe/Co/Pt
multilayers.
75By employing a closed expression from a multi-
pole expansion and a simulated stray field from the MFM
probe, it was demonstrated that fitting the experimental data
could reveal insights into the topological properties of theskyrmions. This approach allows for the determination of the
skyrmion texture and distinguish between Bloch and Néel
type skyrmions, demonstrating with reasonably certainty theprevailing nature of Néel-type skyrmions. Rather than using asimulated MFM probe, Yagil et al .
75utilized an alternative
approach that can be used to gain an insight into the magne-
tism on nanometer length scales. Ba ćaniet al. have recently
demonstrated through qMFM130,142that it is possible to quan-
tify the variation in DMI in sputtered Ir/Co/Pt multilayers tonanoscale precision.
302These observations elucidate the need
of the signi ficantly higher current densities required to initiate
skyrmionic motion in multilayered systems ( ∼1011Am−2)297
compared to those in bulk materials ( ∼106Am−2).298Here, the
authors used the TTF method to calibrate the instrumentsresponse, which is required when pushing the limits of themeasurement toward the resolution limit of the instrument.
This takes into account the physical characteristics of the can-
tilever, magnetic properties of the MFM probe, and also char-acteristics speci fic to the instrument such as the angle at
which cantilever is mounted into the system. This method
allowed observations of signi ficant inhomogeneity in the DMI
values of multilayers, revealing that variations up to 75% of theaverage value of the DMI can exist in spatial regions of ∼50 nm.
Thus, qMFM represents a considerable improvement in under-standing of inhomogeneity at a nanoscale level of precision.
The authors estimated that this corresponds to variations in
the Co layer thickness equal to ±1.2 monolayers, underlying thehigh level of control required to make skyrmion based memoryand logic a reality.
It has recently been demonstrated that not only can
MFM play a critical role in the determination of the properties
of skyrmions but it can also be used to manipulate the mag-netism in thin- films that exhibit DMI and generate skyrmions.
Zhang et al.
303showed that it is possible to use the stray mag-
netic field from an MFM probe to effectively slice the domain
structure in a sample that had an initial starting point in
the magnetostatic ground state and displayed a stripe-like
FIG. 11. Skyrmions. Vector fields for: (a) Néel and (b) Bloch skyrmions occurring
in multilayer systems exhibiting interfacial DMI and non-centrosymmetric crystals
with bulk DMI, respectively. The insets (top left in each panel) display cross-sectional spin con figurations through their skyrmion centers, highlighting the differ-
ences in the spin reorientation of the Néel and Bloch skyrmion. The color bars
represent the normal z-component of the magnetic moment within the skyrmion.Journal of
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J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-20
Published under license by AIP Publishing.domain pattern. By repeatedly scanning the surface, it was
possible to cut the stripe domains into skyrmions in the
absence of an applied magnetic field and at room temperature.
The TTF approach was used to calculate the stray field from
the different types of probes used in the experiment to under-stand the magnitude of the z-component of the magnetic at
the sample surface where the interaction occurs.
These examples capture the powerful way in which MFM
can be extended by incorporating quantitative methods, suchas the TTF approach, to extract information about a samplethat is otherwise dif ficult to achieve. It is expected that as
quantitative methods become widespread, a proliferation in
these types of insightful experiments will shed light into
emerging areas of magnetism at the nanoscale.
Topological insulators (TIs) are unique electronic materi-
als that, in addition to a bulk bandgap similar to an ordinaryinsulator, have protected conducting states on the edge or
surface that are possible due to the combination of spin-orbit
interactions and time-reversal symmetry. Besides a hugefundamental interest, ferromagnetic TIs hold a great promisefor applications in spintronics, metrology, and quantum com-
puting. However, due to the complexity of sample preparation
and cryogenic temperature of operation, so far, relativelylimited number of MFM studies have been reported for topo-logical insulators. Wang et al.
304have performed a systematic
in situ cryogenic MFM study of FM domains in both single-
crystal and thin- films samples of magnetic TIs, Cr-doped
(Bi0.1Sb0.9)2Te3. Bubble-like FM domains were observed in
both single crystals and thin films. In the latter, smaller
domain size ( ∼500 nm) with narrower DWs ( ∼150−300 nm)
were detected due to vertical con finement effect, suggesting
that thin films are more promising for visualization of chiral
edge states.304In a work by Niu et al. ,221cryogenic MFM was
used to study intrinsic ferromagnetism and quantum trans-port transition in individual Fe-doped Bi
2Se3topological insu-
lator NWs. The NW showed spontaneous magnetization with
aTcof 40 K. The intrinsic ferromagnetism and gapped topo-
logical surface states in individual NWs suggest a pathway forfuture memory and ME applications. As the research interestin the field will only grow in near future, the application of
advanced MFM modes (i.e., in- field low temperature MFM as
well as qMFM) is expected to accelerate to provide valuableinformation about these fascinating materials.
2D materials are another emerging class of modern arti fi-
cial materials with exceptionally rich fundamental properties.
Creating modern, smart materials with precise control over
their physical properties is crucial for a wide range of applica-tions and, as a trend, is most pronounced in the area of atomi-cally thin 2D materials and their heterostructures. Suchmaterials often possess unique and unexpected magnetic
properties and MFM is a well-suited tool to validate and study
them on nanoscale. For example, low-temperature in- field
MFM was applied to studies of ferro-/antiferromagnetic tran-sitions in a quasi-2D itinerant ferromagnet, Fe
3GeTe 2.305In the
local state, it was observed that the branching domain struc-
ture dynamically evolved into bubble domains as temperature
decreased from 210 to 150 K, demonstrating existence of twodistinct stable magnetic transitions and suggesting the exis-
tence of an instability in this temperature range.
In another recent study, the authors performed an MFM
study of a new material system, which comprises of the InSesemiconductor van der Waals crystal and FM Fe-islands.
306
In contrast to many traditional semiconductors, the elec-tronic properties of InSe are preserved after the incorpora-
tion of Fe. It was demonstrated that the formation of
crystalline Fe-clusters in InSe induces a uniaxial internal mag-netic field (∼1 T) perpendicular to the InSe layers. Thus, this
hybrid system, which consists of Fe-inclusions and a van derWaals crystal, enables the coexistence of magnetic and semi-
conducting properties within the same structure.
However, in a number of recent works on 2D materials,
MFM was used without applying the correct procedures andcontrol tests, which led to rushed and not experimentally jus-tified conclusions. For example, MFM was applied to charac-
terize the mechanically and liquid exfoliated single- and
few-layer MoS
2, graphene, and graphene oxide nanosheets.307
By the analysis of the phase and amplitude shifts, the authorsdemonstrated that the magnetic response of MoS
2and
graphene is dependent on the layer thickness. It was shown
that the mechanically and liquid exfoliated single-layer MoS 2
demonstrated the reverse magnetic signal. At the same time,it was shown that graphene and MoS
2flakes become non-
magnetic when they exceed a certain thickness. In this initial
work, the authors performed merely a simple MFM study and
the presence of electrostatic interaction was ruled out onlyon the basis of separate measurements on Fe
3O4and Au
nanoparticles rather than directly excluded by the means ofactive Kelvin compensation. No experiments with probe mag-
netization reversal were performed and no clear explanation
of the effect apart from a possible Li doping of MoS
2was pro-
vided. In the follow-up article by Li and Chen,308a more
methodical and careful experimental study was performed.It was found that the MFM response had signi ficant non-
magnetic contributions due to capacitive and electrostatic
interactions between the nanosheets of 2D materials andconductive cantilever tip, as demonstrated by EFM and SKPManalyses. In addition, the MFM signals of graphene and MoS
2
nanosheets were not responsive to reversed magneticmoment of the probe. Therefore, the observed MFM responsewas mainly originated from electrostatic artifacts and notcompelling enough to imply intrinsic magnetism in grapheneand MoS
2nanosheets.308
Similarly, MFM was used for studies of locally induced
magnetization in strained ReSe 2ribbons.309The authors
observed a big negative phase shift on top of a folded ribbon,which they attributed to strong attractive interactionbetween the ReSe
2wrinkles and the MFM probe. However,
in this case as well, the conclusions were drawn without a
convincing control experiment (i.e., reverse of the probemagnetization, use of non-magnetic metal coated probe,etc.). Similarly to what was discussed earlier, the field of 2D
materials in magnetism is extremely fast and successfully
growing. While magnetic properties of such materials were
somewhat late to be explored (primarily due to dif ficulty inJournal of
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Published under license by AIP Publishing.synthesis of ferromagnets in 2D state), very recent develop-
ment have demonstrated that this can be successfully over-
come, opening the way for advanced MFM studies. As in the
case of TI, it can be expected that such varieties of MFMmodes as in- field and low temperature MFM as well as qMFM
are to be applied.
Using HS-AFM, both the structure and dynamic pro-
cesses of biomolecules can be observed without disturbing
their function.
310The possibility to combine this technique
with MFM would open new opportunities of characterizationand manipulation of biological systems. Also, the combinationof AFM and inverted optical microscopy techniques, in partic-
ular Total Internal Re flection Fluorescence (TIRF) microscopy,
allows for simultaneous manipulation and imaging of samples,which can be applied for the measurement of mechanicalproperties of single proteins and the identi fication of speci fic
components in complex assemblies.
311For that reason, the
combination of MFM, capable of, e.g., detection of magnetic
labels, and these optical techniques opens the possibility ofnanomanipulation and simultaneous detection of differentproperties giving the chance to obtain information inaccessi-
ble with other techniques.
We further discuss the potential to use MFM in less tra-
ditional areas such as Life Science and environmental studies.In the case of in vivo applications, MNPs [i.e., superparamag-
netic iron oxides (SPIOs)] integrated into the material of a
mesh can be used, e.g., for the development of a surgical
mesh implant that is visible in magnetic resonance imaging.In order to get a high quality mesh, a narrow size distributionand homogenous spatial distribution, as well as a strong mag-netization of SPIOs within the filament of the mesh are
required. Slabu et al.
312used MFM to determine the bene ficial
properties for the assembly and imaging of the implant. Theseanalyses showed the feasibility of visualization of surgicalimplants with incorporated SPIOs and the in fluence of the
agglomeration of SPIOs on their magnetization and on a
homogenous spatial distribution within the polymer of the
mesh. The findings demonstrate that MFM is a very promising
tool for the characterization of surgical implants.
In addition to the traditional use of magnetic materials in
high-tech, advanced manufacturing, sensor, and biomedical
industries, they are also applied in geoscience, includingclimate change, pollution evolution, iron biomineralization,and diagenetic processes in sediments.
313Recently, the use of
magnetic micro- and nanoparticles has been proposed as a
crucial factor for water remediation314and oil recovery.315
MFM (together with other characterization techniques) has
been applied for a survey of different Fe-containing magneticcompounds targeting their use in environmental applications,such as in wastewater treatments and remediation, and
revealing their advantages and drawbacks.
316Due to its high
resolution and sensitivity, capability to study rough surfaces(i.e., topographic and magnetic signals separation), possibilityto detect simultaneously different interactions and proper-ties, and to operate under different ambience conditions and
magnetic fields, the MFM is a useful technique to perform
magnetic analysis of environmentally relevant systems.V. CONCLUSION
In the Review of the current state of the art, we
addressed the recent major developments in the field of
MFM, including a variety of the operational modes and new
trends in instrumentation, such as in- field and variable field
MFM, MFM under controllable temperature, electrostatic
compensation, energy dissipation, and MeFM. A variety of
specialized, custom-designed magnetic probes (one-side and
multilayer coated, functionalized with a MNPs, NWs of CNT
filled with magnetic materials, etc.) were presented. Special
attention was paid to commonly occurring artifacts in the
MFM images and the way to deal with them. Modern objectsof recent MFM studies were summarized, including objects
such as thin films with PMA, multiferroic materials, and mag-
netic topological structures.
In this Perspective article, we addressed the emerging
MFM trends, concerning further development of instrumen-
tation (e.g., in combination with other SPM modes and radia-
tion techniques) and software, routes toward calibrated MFM
imaging using either modeling approaches or physical means
of and application of MFM to studies of advanced and emerg-
ing materials.
While commercial, off-the-shelf MFM systems still
remain a valid indispensable tool for a routine inspection of
magnetic properties of samples, modern challenges in both
research and industry demand development of new advanced
MFM modes. To ful fill this need, the current research is tar-
geted in different directions including the development of a
new MFM instrumentation and flexible software; novel types
of MFM probes (a key point still under development); thedevelopment of multifunctional MFM, through combination
with other techniques; and targeting complex material prop-
erties, which is a general trend to make MFM compatible with
simultaneous transport, thermal, or optical characterization.
Another coming trend is the possibility to obtain volu-
metric MFM datasets (where the third dimension should be
understood in a broad sense, e.g., probe –sample separation,
magnetic or electrical field, etc.). This trend is well supported
by advances in Big Data acquisition and handling (in part
related to the popularity of force-volume functions) and avail-ability of tools/software for 3D-data visualization and analy-
sis. Further development of volumetric MFM (and other SPM
modes) is very closely aligned with the development of data-
handling practices, statistics, and utilization of machine learn-
ing and arti ficial intelligence. Following this trend, interesting
properties could be quanti fied on the nanoscale without spe-
cialized/expensive equipment, e.g., 3D calibrated characteri-
zation of a stray field emanating from a nano-object,
magnetization dynamics with respect to perturbing fields, or
probe calibration/characterization.
Calibrated MFM will remain an important topic for devel-
opment. While typically macroscopic magnetic field measure-
ments are traceable to nuclear magnetic resonance down to
the millimeter scale, here we outlined the need of such met-
rological procedures with respect to nanoscale characteriza-
tion as well as the development of capabilities that extendJournal of
Applied PhysicsPERSPECTIVE scitation.org/journal/jap
J. Appl. Phys. 125, 060901 (2019); doi: 10.1063/1.5050712 125, 060901-22
Published under license by AIP Publishing.reliable and traceable measurements of spatially resolved
magnetic fields down to the micrometer and nanometer
scale. We also described a standard procedure for MFM cali-
bration, which represents a comprehensive approach combin-ing the experimental measurements of the reference andmeasurand samples with the analytical procedure involvingimage deconvolution in Fourier space using appropriate noise
filters (e.g., Wiener Invert Filter) to reconstruct the tip trans-
fer function. We believe that with increasing automation ofboth measurement capabilities and analysis procedures inmodern AFM/MFM instrumentation, qMFM based on themost versatile TTF approach will soon become accessible for
routine MFM experiments. The most important requirements
along this route are the availability of appropriate referencesamples and of dedicated analysis software.
We further discussed the application of MFM to studies
of advanced and emerging magnetic materials and struc-
tures (often extremely demanding in terms of resolution,
sensitivity, and physical environment), namely, antiferro-magnets, spin-caloritronic materials, skyrmions, topologicalinsulators, 2D materials, and van der Waals crystals as well
as application of MFM to multidisciplinary Life Science and
environmental studies, which are often beyond a “tradi-
tional ”physics approach.
All these examples demonstrate why MFM remains a pow-
erful characterization tool. Equipped with novel modes and
additional functionalities, customized MFM is exceptionally
well-positioned to become an even more indispensable tech-nique, to be widely used in insightful experiments that willshed light in emerging areas of magnetism at the nanoscale.
ACKNOWLEDGMENTS
Dr. R. Schäfer and Professor R. Cowburn are thanked for
their useful insights into the perspectives of the MFM +MOKE multifunctional technique; R. Nevill is acknowledged
for assistance in the production of Figs. 1 and4; and S. Gorno
and K. Edmonds are thanked for their assistance in referencemanagement and general suggestions. O.K., R.P., C.B., H.C.,and V.N. acknowledge the financial support from the European
Metrology Programme for Innovation and Research (Grant
No. 15SIB06), NanoMag. M.J. and A.A. acknowledge the support
from the Spanish Ministerio de Economia y Competitividad(MINECO) under Project Nos. MAT2015-73775-JIN andMAT2016-76824-C3-1-R.
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Published under license by AIP Publishing. |
5.0023636.pdf | AIP Advances 10, 105202 (2020); https://doi.org/10.1063/5.0023636 10, 105202
© 2020 Author(s).Unusual behavior of coercivity in Hf/
GdFeCo bilayer with MgO cap layer by
electric current
Cite as: AIP Advances 10, 105202 (2020); https://doi.org/10.1063/5.0023636
Submitted: 01 August 2020 . Accepted: 14 September 2020 . Published Online: 01 October 2020
Ngo Trong Hai , Ivan Kindiak , Vladislav Yurlov , Ramesh Chandra Bhatt
, Chun-Ming Liao , Lin-Xiu Ye , Te-ho Wu
, K. A. Zvezdin
, and Jong-Ching Wu
COLLECTIONS
Paper published as part of the special topic on Chemical Physics , Energy , Fluids and Plasmas , Materials Science
and Mathematical Physics
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Unusual behavior of coercivity in Hf/GdFeCo
bilayer with MgO cap layer by electric current
Cite as: AIP Advances 10, 105202 (2020); doi: 10.1063/5.0023636
Submitted: 1 August 2020 •Accepted: 14 September 2020 •
Published Online: 1 October 2020
Ngo Trong Hai,1Ivan Kindiak,2Vladislav Yurlov,2Ramesh Chandra Bhatt,3
Chun-Ming Liao,3Lin-Xiu Ye,3
Te-ho Wu,3
K. A. Zvezdin,4
and Jong-Ching Wu1,a)
AFFILIATIONS
1Department of Physics, National Changhua University of Education, Changhua, Taiwan
2Moscow Institute of Physics and Technology, Institutskiy per. 9, 141700 Dolgoprudny, Russia
3Graduate School of Materials Science, National Yunlin University of Science and Technology, Douliu, Taiwan
4A. M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilova 38, 119991 Moscow, Russia
a)Author to whom correspondence should be addressed: phjcwu@cc.ncue.edu.tw
ABSTRACT
We investigate the Hf/GdFeCo bilayer with the MgO cap layer for both rare earth (RE)-rich and transition metal (TM)-rich configurations
of the ferrimagnetic sublattice in the presence of the perpendicular field. We study the coercivity using the anomalous Hall effect (AHE)
technique by multiple measurements on the same sample. In the first set of measurements and at low electric currents, coercivity sharply drops
because of the oxygen diffusion at the interface between MgO and GdFeCo when the AHE probe current is applied. During the subsequent
measurements on the RE-rich sample, we observe a moderate decrease in coercivity at low currents and the coercivity increases in a high
current range. Such nonlinear dependence of coercivity on electric current can be explained by the competing interplay of the spin–orbit
torque (SOT) and the Joule heating effects. On the other hand, for the TM-rich case, the SOT effect is observed over a widely applied current
range.
©2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0023636 .,s
Early investigations of current-induced torques on heavy
metal/ferromagnet/oxide (HM/FM/Ox) structures have revealed the
great promise of using in-plane current to manipulate magnetization
switching processes.1–7The Ox layer employed here might enhance
the perpendicular magnetic anisotropy (PMA) due to the hybridiza-
tion effects at the FM/Ox interface.8–10The heavy metals (Pt,11,12
Ta,13,14W,15,16Hf,17and so on18,19) with strong spin–orbit coupling
are placed under the FM layer to convert charge currents into spin
currents, in which accumulated spin polarization σat the interface
of HM/FM exerts a torque on the magnetic moments mof the FM
layer.
More recently, Kim et al. have investigated spin–orbit
torque (SOT) associated with heavy metal/ferrimagnetic/oxide
(HM/FI/Ox) order on Pt/GdFeCo/MgO multilayers using the spin-
torque ferromagnetic resonance (ST-FMR) technique.20According
to the report, on sputtering the MgO cap layer on top of GdFeCo, the
Gd atoms with strong oxygen-affinitive properties21,22tend to diffusetoward the MgO layer to form Gd 2O3. This suggestion is based on
the analysis of x-ray photoelectron spectroscopy (XPS) of blank film
GdFeCo/MgO layers, which shows the stronger peaks of Gd 3d adja-
cent to the MgO layer.20Kim et al. showed that HM/FI/Ox is quite
different from HM/FM/Ox. To date, it still remains under debate of
the magnetic properties making a difference in this structure. Moti-
vated by these concerns, in this paper, we report a study about the
magnetic properties of the Hf/GdFeCo/MgO heterostructure using
DC, in the presence of the out-of-plane magnetic field. The exper-
imental data exhibit that the Hf/GdFeCo bilayer capped by MgO
changes its magnetic properties during the early first stage of anoma-
lous Hall effect (AHE) measurements. We observe that the coercivity
sharply decreases in the first set of measurements at low currents. In
the second and third sets of measurements, we analyze data and find
out that for the rare earth (RE)-rich configuration, the SOT effect
leads to a moderate decrease in coercivity at low currents, while at
high currents, the Joule heating increases the coercive field. On the
AIP Advances 10, 105202 (2020); doi: 10.1063/5.0023636 10, 105202-1
© Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv
other hand, for the transition metal (TM)-rich composition, the SOT
effect dominates over a wide range of sending currents. These find-
ings are useful for the investigations of HM/FI bilayers with capping
Ox.
Multilayers of Hf(4 nm)/GdFeCo(7 nm)/MgO(4 nm) are
deposited by means of a high vacuum RF magnetron sputtering
method on thermally oxidized silicon substrates. The GdFeCo thin
film is grown by co-sputtering, in which the FeCo target is fixed
at 200 W DC power, while the Gd target is fixed at 70 W and 110
W DC power, respectively. The base pressure of the magnetron
sputtering chamber was 1.5 ×10−7Torr. In order to prevent the
drawback of losing PMA when annealed at high temperatures,23
no post-annealing is carried out for the GdFeCo alloy. By using
the X-Ray Fluorescence (XRF) spectroscopy technique, we deter-
mine that the Gd contents in the GdFeCo alloy corresponding to
the Gd growth power of 70 W and 110 W are 25% and 29%, respec-
tively. The magnetization and coercive fields of Hf/GdFeCo/MgO as
a function of the Gd content are measured, as shown in Fig. 1(a).
The graph reveals that depositing Gd at 70 W forms a TM-rich
GdFeCo, whereas Gd at 110 W exhibits RE-rich behavior. The hys-
teresis loops in the inset of Fig. 1(a) are measured by the Alternat-
ing Gradient Magnetometer (AGM) with the perpendicular field.
The square-shaped hysteresis loop exhibits high bulk PMA of the
film. The atomic force microscopy (AFM) is employed to charac-
terize the morphology of the blank thin film, yielding a surface
roughness of 0.7 ±0.04 nm. The Hall bar with a width of 10 μm
and length of 65 μm is fabricated as follows: (1) Standard pho-
tolithography is employed to define the outer electrodes; an ion-
beam sputtering system equipped with two ion guns is adopted
for etching 4 nm of MgO, and then 100 nm of Cu is deposited as
FIG. 1 . (a) Magnetization M sand coercive field H cof the Hf/GdFeCo/MgO tri-
layer as a function of the Gd content. Inset: out-of-plane hysteresis loops of Gd at
70 W and 110 W configurations, measured by an AGM. (b) Schematic illustration
of Hall bar devices and measurement setup. (c) Set of experimental AHE resis-
tance loops as a function of the perpendicular field under various input DCs. The
figure is data of the first set of measurements on the 110 W Gd configuration.probing pads. (2) Electron-beam lithography is used to define the
Hall bar shape, and an Ar ion-beam etching technique is then uti-
lized to fabricate the Hall bar on the trilayer film. In this work, all
measurements are performed at room temperature (300 K). The
experimental setup and Hall bar device schematic are shown in
Fig. 1(b).
We first examine the anomalous Hall effect (AHE) of the tri-
layer structure with the Gd component at 110 W. The AHE resis-
tance is measured as a function of the perpendicular field under
various input DCs. During the first set of measurements, we observe
that the hysteresis loops change depending on the values of input
currents, as shown in Fig. 1(c). The coercive field Hcextracted from
the AHE hysteresis loops is quantified as Hc=(Hc1−Hc2)/2, where
Hc1and Hc2are the left and right coercive fields, respectively. As
shown in Fig. 2(a), Hcdrops drastically in the first set of measure-
ments, from 71.6 Oe to 15.3 Oe, as current is increased from 10 μA
to 1.4 mA. Surprisingly, we recognize that the repeated measure-
ments give different results. In the second and third sets of mea-
surements, Hcjust decreases moderately from 17.5 Oe to 15.3 Oe
in the same current range. In addition, the hysteresis loops for the
current I≤1.4 mA in the first set of measurements exhibit the
rectangular loop (Z-type) with high squareness and sharp switching
behavior. Once the sense current increases to over 1.4 mA, the hys-
teresis shape changes to the round loop (R-type) and remains in this
shape for the second and third sets. The R-type shape implies that the
in-plane magnetic anisotropy component appears. Two additional
points are made about the data in Fig. 2(a). First, Hcremains stable
at about 14.4 Oe for currents from 2.5 mA to 3 mA. Second, as I
increases to 10 mA, Hclinearly enhances from 14.4 Oe to about 23.1
Oe. Interestingly, when carrying out the AHE measurements on the
70 W Gd configuration, the results we obtain are not the same as
those on the 110 W Gd configuration. In the second and third sets
of measurements for the TM-rich case, H cdecreases continuously
over a wide range of currents (0 mA <I<10 mA) [see Fig. 2(b)],
even though we still see the abnormal H cdrop in the first set of
measurements.
To gain more insight into the magnetization switching behav-
ior in the first set of measurements, we extract the nucleation field
HNand saturation field H sfrom the hysteresis loop. As illustrated in
the inset of Fig. 2(c), the magnetization starts switching at H Nand
then completely orients in the direction of the applied field at H s.
Therefore, the magnetization switching speed will be proportional
to the reciprocal of the field-difference η= 1/(H s−HN).24The plot
ofηwith respect to the sense current is shown in Fig. 2(c). For the
RE-rich case, when the current is within the range from 10 μA to
1.4 mA, η(I)shows stability around 0.27 Oe−1. However, as it passes
1.4 mA, η(I)drops steeply to 0.04 Oe−1. For the TM-rich case, η(I)
drops from 0.16 Oe−1to 0.03 Oe−1as current passes 3.4 mA.
It is interesting that we even observe the switching behavior
when the perpendicular external field H extis fixed. After the third
set of measurements, we examine AHE voltage as a function of
current on the 110 W Gd configuration. We first applied H ext=
−400 Oe to saturate the magnetic moment min the −z direction.
The external field is then reversed to various positive values in the
range from +5 Oe to +25 Oe (in the +z direction). Corresponding to
each H extvalue, the sweeping current is sent. Figure 3 shows that at
Hext= 14.3 Oe, when the current starts from 0, the Hall resistance
is 1.6 Ω. As the current reaches 2.94 mA, the switching behavior is
AIP Advances 10, 105202 (2020); doi: 10.1063/5.0023636 10, 105202-2
© Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv
FIG. 2 . Hcvariation as a function of input DC obtained from the AHE measurements for the (a) 110 W Gd configuration and (b) 70 W Gd configuration. (c) Plot of reciprocal
of different fields η= 1/(H s−HN) vs the input DC in the first set of measurements.
observed. That is, the Hall resistance shifts from 1.6 Ω to −11.9 Ω.
We then observe that at higher values of applied H ext, that is, 14.7 Oe,
15.2 Oe, 15.6 Oe, 16.3 Oe, and 16.8 Oe, respectively, the magneti-
zation in GdFeCo starts switching at smaller and smaller values of
sweeping current, that is, 2.2 mA, 1.7 mA, 1.2 mA, 0.7 mA, and
0.2 mA.
Let us now discuss the possible physical mechanism behind
the magnetization switching behavior of samples. There are several
effects that may lead to a drop of coercivity in the first set of mea-
surements. On sputtering the MgO cap layer on top of GdFeCo, a
thin Gd 2O3layer is formed at the interface of MgO/GdFeCo.20The
surface roughness examined by AFM [Fig. 4(a)] leads to our sugges-
tion of the Gd 2O3roughness, as illustrated in Fig. 4(b). Because RE
oxides such as Gd 2O3are known as ionic conductors with very high
O−2mobility,25,26one may expect while sending current, the electric
fieldEwould allow O2−ions from Gd 2O3to diffuse into the GdFeCo
region in the vicinity of the concave–convex barrier of Gd 2O3.25,27
The O2−diffusion mechanism has been explored by Bi et al. when
studying the Pt/Co/Gd 2O3structure in which the applied electric
fields (EFs) drive the O2−ion in the rare earth oxide Gd 2O3toward
FIG. 3 . AHE voltage as a function of current on the 110 W Gd configuration. For
each fixed H extvalue, the sweeping current is sent. The shift of Hall resistance is
observed in fields ranging from 14.3 Oe to 17 Oe. The inset shows a schematic of
the AHE resistance shift.Co, turning it into CoO x. Consequently, his group observed a large
change in coercivity.25The coercive field for our sample is written
asHc= 2ξK/m , where Kis the uniaxial magnetic anisotropy and
ξ<1 is the dimensionless factor that connects with inhomogeneous
switching. The O2−migration possibly combines with RE and TM
atoms, turning the upper side of the GdFeCo layer into GdO x, Fe xOy,
CoO x[see Fig. 4(b)], diluting the structural purity of the anti-parallel
RE–TM coupling moments. This is related to the increase in inho-
mogeneity of the sample ( ξdecrease ) and a simultaneous decrease
in magnetic anisotropy K(transitioning the hysteresis loop from
the square to the round shape). As a result, H cmight drop dras-
tically. It should be noted that the direct-current electric field Eis
in the horizontal direction. This drives the diffusion of O−2along
the roughness interface of Gd 2O3. Until some point in the measur-
ing process, all the concave–convex area in the vicinity of Gd 2O3
gradually changes into RE and TM oxides [see Fig. 4(b)]. Conse-
quently, the oxidation process comes to a stop, thus ending the sharp
reduction of Hc.
Another mechanism is involved in the moderate reduction of
coercivity, observed in the second and third sets of measurements
at low currents (from 10 μA to 2.5 mA) for the RE-rich case and
over a wide current range (from 10 μA to 10 mA) for the TM-rich
case [see Figs. 2(a) and 2(b)]. The H creduction here relates to the
SOT mechanism, i.e., to the increase in Slonczewski damping-like
torque (DLT) and transverse field-like torque (FLT). For zero cur-
rent and external field Hextapplied in the +z direction, the direction
of a net magnetization min the GdFeCo layer at the equilibrium
position is parallel to the sum of the anisotropic field and applied
field (in the +z direction). When the current flows through the +y
direction in the heavy metal Hf layer, it leads to an accumulation
of spin-polarized electrons σin the +x direction at the Hf/GdFeCo
interface, as illustrated in Fig. 5(a). The effective field Hdof DLT
τDL=(γ̵hJecDL/2eMstF)ˆm×(ˆm׈σ)is in the direction of current
andHfof FLT τF=(γ̵hJecFL/2eMstF)ˆm׈σis in the direction per-
pendicular to current [see Fig. 5(a)],28–31where γis the gyromagnetic
ratio, Jeis the current density, Msis the saturation magnetization, tF
is the thickness of the FM layer, and cDLandcFLare efficiencies cor-
responding to DLT τDLand FLT τFL, respectively. The torque τDL
tilts the magnetization mfrom the z direction toward the y direc-
tion in the y–z plane, while τFtiltsmtoward the x direction in
AIP Advances 10, 105202 (2020); doi: 10.1063/5.0023636 10, 105202-3
© Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv
FIG. 4 . (a) AFM-section analysis of
the surface morphology of the Hf
(4 nm)/GdFeCo (7 nm)/MgO (4 nm)
structure. (b) A schematic illustration of
Gd2O3roughness. The O2−migration
possibly turns the upper side of the
GdFeCo layer into GdO x, Fe xOy, CoO x.
the x–z plane. Therefore, these two torque components of SOT can
potentially assist the perpendicular field Hextin nucleating the
reverse magnetization mas crossing the x–y plane, as illustrated
in Fig. 5(b). Our analysis is consistent with the switching behavior
explored in Fig. 3. For I= 0.2 mA, the Oersted field is calculated to
be≈0.09 Oe, which could not contribute more than a trivial amount,
but we still see the shift of Hall resistance [see Fig. 3]. Therefore, the
decrease in H chere is mainly caused by the SOT effect.
On the contrary to the low current range, the increase in H cin
the high current range from 3 mA to 10 mA for the RE-rich sam-
ple is associated with the dominance of the Joule heating effect. The
change in the coercive field can, therefore, be written as
ΔHc(total)=ΔHc(soT)+ΔHc(Joule).
The effect of Joule heating on the variation of H cmight be
explained as follows: (i) When the induced current Iflows through
the Hall bar, the temperature in the GdFeCo layer rises as T
=T0+γRt
ρVcI2, where T0is the room temperature, γis a propor-
tional coefficient, Ris the longitudinal Hall bar resistance, cis the
specific heat, and Vis the volume of the longitudinal Hall bar. (ii)
As temperature changes, the magnetization mREof the sublattice Gd
andmTMof the sublattice FeCo vary, but with different ratios in the
anti-parallel relation.32As a result, the net magnetization mnet=mRE
+mTMchanges. The Zeeman energy coupled into the magnetization
by an external magnetic field is described as E =−μ0mnet⋅Hext.33
When Tapproaches the magnetization compensation temperature
Tcom,mnettends toward zero, thus necessitating the application of
FIG. 5 . (a) Current-induced effective field illustrations for SOTs τDLandτFand
corresponding effective fields HdandHfin the presence of the applied field Hext.
(b) As Hextreverses in the −z direction, two torque components assist in nucleating
mas crossing the x–y plane.larger and larger magnetic fields to generate sufficient energy to
reorient the magnetic moment mnet.34The experimental data carried
out by Ostler et al. have shown that the compensation tempera-
ture for the RE-rich sample with 29% of the Gd content (i.e., Gd at
110 W power) is 350 K.35According to Fig. 3 in their work, slight
changes in temperature above 300 K (our lab room temperature)
lead to a considerable increase in H c. Therefore, the dominance of
the Joule heating effect in this regime is likely because Tcomof ferri-
magnetic GdFeCo with Gd at 110 W is relatively close above room
temperature. On the contrary, in the case of Gd at 70 W power,
it is predominantly the SOT effect that causes the decrease in H c
over all values of current (0 mA <I<10 mA) in the second and
third sets of measurements [Fig. 2(b)]. That is because the TM-rich
composition with 25% Gd content (i.e., Gd at 70 W power) has
Tcomfar below the room temperature; thus, a slight rise in temper-
ature in the GdFeCo layer due to Joule heating would make only
trivial reductions in H c.35This implies that the TM-rich GdFeCo
alloy may possess superior properties in terms of utilizing the SOT
effect.
In summary, we have investigated the AHE of the Hf/GdFeCo/
MgO trilayer with various applied DCs and in the presence of the
out-of-plane external field. The effect of the MgO cap layer on
the magnetization of GdFeCo leads to an abnormal drop of H cin
the first stage of measurement. For the high RE-rich configuration,
the SOT effect on reduction of H cis observed at relatively small to
moderate currents, whereas at higher currents, the effect of Joule
heating dominates since the magnetization compensation point of
this configuration is close above the specimen temperature. For the
TM-rich GdFeCo alloy, on the contrary, the SOT effect on coercivity
is observed over a wide range of applied currents.
This research was supported by the Ministry of Science and
Technology, Taiwan (Grant No. MOST 108-2112-M-018-002), and
Russian Science Foundation (Grant No. 17-12-01333).
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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© Author(s) 2020 |
1.3056142.pdf | Spinwave propagation in lossless cylindrical magnonic waveguides
Haiwen Xi, Xiaobin Wang, Yuankai Zheng, and Pat J. Ryan
Citation: J. Appl. Phys. 105, 07A502 (2009); doi: 10.1063/1.3056142
View online: http://dx.doi.org/10.1063/1.3056142
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v105/i7
Published by the AIP Publishing LLC.
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Downloaded 05 Sep 2013 to 35.8.11.2. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsSpinwave propagation in lossless cylindrical magnonic waveguides
Haiwen Xi,a/H20850Xiaobin Wang, Yuankai Zheng, and Pat J. Ryan
Seagate Technology, 7801 Computer Avenue South, Bloomington, Minnesota 55435, USA
/H20849Presented 12 November 2008; received 13 September 2008; accepted 7 October 2008;
published online 2 February 2009 /H20850
Spinwave propagation in clad cylindrical magnonic waveguides is investigated under linear
approximation. With the assumption of no magnetic damping, characteristic equation to determinethe bound spinwave modes has been obtained based on the structural and magnetic properties of thewaveguides. The study is then applied to homogenous magnetic nanowires with no cladding.Spinwave characteristics and properties, such as the dispersion relationship and group velocity, canbe described analytically. © 2009 American Institute of Physics ./H20851DOI: 10.1063/1.3056142 /H20852
Recently, magnetic nanostructures have attracted consid-
erable attention due to the emerging research activity innanoscience and nanotechnology. From the fundamental
point of view, the magnetic structures with small lengthscales and low dimensionalities exhibit properties differentfrom those of bulk materials. The importance of the researchin this field also lies in the applications as data storage ele-ments for ultrahigh densities and possible logic devices forinformation process and transfer down to the nanometerscale.
1,2
Magnetic nanostructures include two-dimensional mag-
netic ultrathin films, one-dimensional nanowires andnanostripes,
3,4zero-dimensional ferromagnetic
nanoparticles5and nanodots,3and other confined structures.
Magnetic nanowires have potential applications in many ar-eas of advanced nanotechnology, one of which is for infor-mation transmission and processing proposed by Wang andco-workers.
6,7The basic idea is to transfer data by spinwave
buses, which are magnetic nanostripes or nanowires, alongwhich spinwaves can propagate. Compared to current nan-odevices based on complementary metal-oxide-semiconductor /H20849CMOS /H20850technology, the result of spinwave
buses is the reduction in power consumption and heat dissi-pation, and therefore the potential of smaller components.
7
Data processing can also be achieved by encoding data intothe phase of the spinwaves.
6–8
In our previous study,9the concept of spinwave buses is
generalized to be a magnetically nonuniform structure calledmagnonic waveguide, which is similar to the waveguide forelectromagnetic waves or lights. In this work, we focus onthe spinwave propagation in clad magnetic nanowires. Thecore and the cladding of the nanowires are with differentmagnetic properties such as saturation magnetization and ex-change coupling strength.
Figure 1shows a straight magnetic nanowire that con-
sists of a core and cladding with different magnetic materi-als. The cross section is circular so that it can be consideredas a cylindrical magnonic waveguide for spinwaves. An ex-ternal magnetic field is applied along the axial direction tostabilize the magnetizations. The magnetic moments of thecore and the cladding are exchange coupled at the interface.
Magnetization dynamics in a magnetic medium can be gen-erally described by the Landau–Lifshitz–Gilbert /H20849LLG /H20850
equation
/H11509M
/H11509t=−/H9253M/H11003Heff+/H9251M
Ms/H11003/H11509M
/H11509t, /H208491/H20850
where /H9253is the gyromagnetic ratio, /H9251is the Gilbert damping
constant, and Mdescribes the magnetization vector of the
magnetic medium with a saturation magnetization, Ms.Heff
is the total effective field for the magnetization, including
Heff=Ha+/H92542/H116122M−DI·M+Hmc. /H208492/H20850
The first term Hadescribes an external magnetic field.
The second term is the exchange coupling field characterized
by the exchange length, which is /H9254=/H208812A/Ms2, where Ais the
exchange constant. The third term is the demagnetizing field
with DIdenoting the demagnetizing factor tensor. The final
termHmcis the magnetocrystalline anisotropy field.
In this study, we shall consider a lossless cylindrical
magnonic waveguide with /H9251=0. Nonzero damping constant
will not alter the essence of the analysis. The magnetocrys-
a/H20850Electronic mail: haiwen.xi@seagate.com.z
y
xa
ρM2
δ2
n2M1
δ1
n1Haz
y
xa
ρM2
δ2
n2M1
δ1
n1Ha
FIG. 1. Schematic of a cylindrical magnonic waveguide comprising a core
and a cladding. The magnetic field is applied in the axial direction.JOURNAL OF APPLIED PHYSICS 105, 07A502 /H208492009 /H20850
0021-8979/2009/105 /H208497/H20850/07A502/3/$25.00 © 2009 American Institute of Physics 105 , 07A502-1
Downloaded 05 Sep 2013 to 35.8.11.2. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionstalline anisotropy field Hmcis neglected for simplicity. Be-
cause of the large shape anisotropy and the external magneticfield in the nanowire axis direction, the magnetization dy-namics is the precession with a small angle around the axisn=zˆ. Therefore, the magnetization can be written as
M=M
s/H20849n+m˜/H20850, /H208493/H20850
where m˜is the magnetization perturbation. Since the magni-
tude of the magnetization is assumed to be a constant, thesmall perturbation m
˜with /H20841m˜/H20841/H112701 is in the cross-section
plane, i.e., m˜/H11036n. Since nis a fixed vector, Eq. /H208491/H20850can then
be rewritten as
/H11509m˜
/H11509t=−/H9253/H20849n+m˜/H20850/H11003Heff. /H208494/H20850
Next we assume that the spatial magnetization variation
along the nanowire axis is much larger than the exchangelength and the diameter of the nanowire. The demagnetizingfactors of the magnetization at any given location of thenanowire are approximately zero, except for D
xx=Dyy=2/H9266.4
Therefore, the effective field is
Heff=Han−2/H9266Msm˜+Ms/H92542/H116122m˜. /H208495/H20850
Inserting Eq. /H208495/H20850into Eq. /H208494/H20850and using the linearization ap-
proach, we can obtain
/H11509m˜
/H11509t=−/H9253n/H11003/H20851−/H20849Ha+2/H9266Ms/H20850m˜+Ms/H92542/H116122m˜/H20852. /H208496/H20850
We then write the spinwave eigenmodes of the LLG
equation as
m˜/H11006/H20849/H9275,k/H20850=/H20849xˆ/H11006iyˆ/H20850u/H20849x,y/H20850exp /H20849−i/H9275t/H20850exp /H20849ik /H20648z/H20850, /H208497/H20850
in the xˆ-yˆ-zˆCartesian coordinates. These eigenmodes corre-
spond to the spinwave modes rotating clockwise /H20849/H11001/H20850and
counterclockwise /H20849-/H20850in circular precession mode with an an-
gular frequency /H9275and a wave vector k/H20841/H20841in the zˆ-direction, by
which the spinwave mode is called paraxial. idenotes the
imaginary unit /H20881−1. The generic magnetization dynamics
can then be expressed as a linear superposition of the eigen-modes, m
˜=a+m˜++a−m˜−.a/H11006are the amplitudes of the spin-
wave modes. Inserting Eq. /H208497/H20850into Eq. /H208496/H20850, we can readily
obtain
/H116122u/H20849x,y/H20850+/H20851n2/H20849/H9275//H9253−Ha−2/H9266Ms/H20850−k/H206482/H20852u/H20849x,y/H20850=0 . /H208498/H20850
Note that n=1 //H20881Ms/H9254=/H20881Ms/2Ais denoted as “magnetic re-
fractive index.”9Because of the cylindrical symmetry of the
magnetic nanowire, Eq. /H208498/H20850can be rewritten as
/H116122u/H20849/H9267,/H9272/H20850+/H20849n2c2−k/H206482/H20850u/H20849/H9267,/H9272/H20850=0 , /H208499/H20850
in cylindrical coordinates. Here we denote
c=/H20881/H9275//H9253−/H20849Ha+2/H9266Ms/H20850. /H2084910/H20850
The spinwave eigenmodes must be periodic in the polar
angle/H9278so that
u/H20849/H9267,/H9272/H20850=ul/H20849/H9267/H20850exp /H20849−il/H9272/H20850,l=0 ,/H110061,/H110062, ... . /H2084911/H20850
Therefore, the radial profile ul/H20849/H9267/H20850of the spinwave satisfiesd2ul
d/H92672+1
/H9267dul
d/H9267+/H20873n2c2−k/H206482−l2
/H92672/H20874ul=0 , /H2084912/H20850
which is the well-known Bessel equation of order l.
The existence of the core-cladding interface changes the
exchange coupling and effective fields in the cylindricalmagnonic waveguides. From previous studies,
9,10the core-
cladding boundary conditions for the spinwaves are
/H11509m˜1
/H11509/H9267=/H9264M2n12/H20849m˜2−m˜1/H20850/H20849 13/H20850
and
/H11509m˜2
/H11509/H9267=/H9264M1n22/H20849m˜2−m˜1/H20850. /H2084914/H20850
/H9264is the interface exchange coupling strength. m˜1andm˜2
are the spinwave components in the core and cladding, re-
spectively. From now on, we use the convention that the coreis denoted by subscript 1 while the cladding is denoted bysubscript 2. The boundary conditions are consistent withthose in Refs. 11and12where the interface anisotropy is
ignored.
By inserting expression /H208497/H20850to boundary conditions /H2084913/H20850
and /H2084914/H20850, the angular frequency
/H9275and the wave vector k/H20841/H20841
must be the same in the core and the cladding of the mag-
nonic waveguide. Furthermore, the radial profile ul/H20849/H9267/H20850of the
spinwave may be written as
d2ul
d/H92672+1
/H9267dul
d/H9267+/H20873k/H110362−l2
/H92672/H20874ul=0 , /H9267/H11021a, /H2084915/H20850
d2ul
d/H92672+1
/H9267dul
d/H9267−/H20873/H9260/H110362+l2
/H92672/H20874ul=0 , /H9267/H11022a, /H2084916/H20850
where
k/H110362=n12c12−k/H206482/H2084917/H20850
and
/H9260/H110362=k/H206482−n22c22. /H2084918/H20850
ais the radius of the core. Similar to light traveling in an
optic fiber,13the spinwaves in the clad cylindrical magnonic
waveguide should be bound so that the spinwave decaysaway in the radial direction in the cladding. Therefore, itrequires
/H9260/H11036be a positive number. k/H11036should be positive as
well. These set the condition for bound spinwaves in thecylindrical magnonic waveguides. Solutions of Eqs. /H2084917/H20850and
/H2084918/H20850are the family of Bessel functions. Note that the spin-
wave magnitude is finite everywhere in the core and claddingregions. We can write the radial profile of the spinwave as
u
l=/H20877AlJl/H20849k/H11036/H9267/H20850,/H9267/H11021a
BlKl/H20849/H9260/H11036/H9267/H20850,/H9267/H11022a,/H20878 /H2084919/H20850
where Jl/H20849x/H20850is the Bessel function of the first kind and order
l, and Kl/H20849x/H20850is the modified Bessel function of the second
kind and order l. Inserting Eq. /H2084919/H20850into boundary conditions
/H2084913/H20850and /H2084914/H20850, we can obtain07A502-2 Xi et al. J. Appl. Phys. 105 , 07A502 /H208492009 /H20850
Downloaded 05 Sep 2013 to 35.8.11.2. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions/H20873C11C12
C21C22/H20874/H20873Al
Bl/H20874=0 , /H2084920/H20850
where
C11=k/H11036aJl−1/H20849k/H11036a/H20850−/H20849l−a/H9264M2n12/H20850Jl/H20849k/H11036a/H20850,
C12=−a/H9264M2n12Kl/H20849/H9260/H11036a/H20850,
C21=a/H9264M1n22Jl/H20849k/H11036a/H20850,
and
C22=/H9260/H11036aKl−1/H20849/H9260/H11036a/H20850−/H20849l+a/H9264M1n22/H20850Kl/H20849/H9260/H11036a/H20850
are the elements of a 2 /H110032 matrix CI. Equation /H2084920/H20850immedi-
ately leads to
det /H20841CI/H20841=0 , /H2084921/H20850
for nonzero AlandBl. Recalling Eqs. /H2084910/H20850,/H2084917/H20850, and /H2084918/H20850,
this condition, called the characteristic equation, determinesthe wave vector k
/H20841/H20841and, therefore, the spinwave mode in the
cylindrical magnonic waveguide for a given angular fre-quency
/H9275. There can be multiple values for k/H20841/H20841. Equation /H2084921/H20850
cannot be solved analytically.
Let us consider spinwaves in a simple magnetic nano-
wire without magnetic cladding, i.e., M2=0. Thus, /H11509m˜1//H11509/H9267
=0 at /H9267=a, referring to Eq. /H2084913/H20850. It is not difficult to know
that k/H11036=/H9273lm/a,/H20849m=1, 2, ... /H20850, where /H9273lmis the root of
dJl/H20849x/H20850/dx. For instance, /H927311=1.841, /H927312=5.331, and /H927313
=8.536, etc. From Eq. /H2084917/H20850, the wave vector k/H20841/H20841can be ob-
tained to be
k/H20648=/H20881/H20873/H9275−/H9253/H20849Ha+2/H9266M1/H20850
/H9253M1/H208741
/H925412+/H9273lm21
a2, /H2084922/H20850
where l,m=0,1,2,.... W e define /H927300=0, of which the case
has been discussed in the previous study.13Therefore, the
wave vector of the spinwave modes is determined by themagnetic properties of the magnetic material and the cross-section size of the magnetic nanowire. Equation /H2084922/H20850can be
rewritten for the angular frequency, i.e.,/H9275=/H9253/H20849Ha+2/H9266M1+M1/H925412k/H206482/H20850−/H9253M1/H20849/H9273lm/H92541/a/H208502. /H2084923/H20850
It is noteworthy that the dispersion relationship of the spin-
wave in the magnetic nanowire is split into many subbandsaccording to the modes of the spinwaves.
To summarize, we have investigated the spinwave
propagation in magnonic waveguides based on the LLGequation under the linear approximation. The study is limitedto paraxial spinwaves. Under certain conditions related to themagnetic properties of the core and cladding, the spinwavesare bound in the magnonic waveguide, similar to the lighttraveling in optical fibers. Characteristic equation for thespinwave modes in the magnetic waveguides has been ob-tained from the core-cladding boundary conditions. In thesimple magnetic nanowires with no cladding, the spinwavemodes and associated dispersion relationship can be readilycharacterized. It should be noted that magnetic damping isnot taken into account in the study. Damping will cause thespinwave attenuation during propagation in magnonicwaveguides. It is believed that spinwave attenuation is re-lated to the wave vector and mode of the spinwaves.
10Fur-
ther study should consider the damping constant and applynumerical calculation on the formulas obtained in this studyand even micromagnetic simulation to better understand thecharacteristics of the spinwaves in the cylindrical magnonicwaveguides.
1J. G. Zhu, Y. Zheng, and G. A. Prinz, J. Appl. Phys. 87, 6668 /H208492000 /H20850.
2M. M. Miller, G. A. Prinz, S. F. Cheng, and S. Bounnak, Appl. Phys. Lett.
81, 2211 /H208492002 /H20850.
3R. Skomski, J. Phys.: Condens. Matter 15, R841 /H208492003 /H20850.
4L. Sun, Y. Hao, C. L. Chien, and P. C. Searson, IBM J. Res. Dev. 49,7 9
/H208492005 /H20850.
5S. Sun, C. B. Murray, D. Weller, L. Folks, and A. Moser, Science 287,
1989 /H208492000 /H20850.
6A. Khitun, R. Ostroumov, and K. L. Wang, Phys. Rev. A 64, 062304
/H208492001 /H20850.
7A. Khitun and K. L. Wang, Superlattices Microstruct. 38, 184 /H208492005 /H20850.
8T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps, and M.
P. Kostylev, Appl. Phys. Lett. 92, 022505 /H208492008 /H20850.
9H. Xi, X. Wang, Y. Zheng, and P. J. Ryan, J. Appl. Phys. 104, 063921
/H208492008 /H20850.
10H. Xi and S. Xue, J. Appl. Phys. 101, 123905 /H208492007 /H20850.
11M. Vohl, J. Barna ś, and P. Grünberg, Phys. Rev. B 39, 12003 /H208491989 /H20850and
references therein.
12A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves
/H20849CRC, Boca Raton, FL, 1996 /H20850, p. 186.
13B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics , 2nd Ed.
/H20849Wiley-Interscience, Hoboken, NJ, 2007 /H20850, p. 325.07A502-3 Xi et al. J. Appl. Phys. 105 , 07A502 /H208492009 /H20850
Downloaded 05 Sep 2013 to 35.8.11.2. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions |
1.1534386.pdf | Temperature and field dependence of high-frequency magnetic noise in spin valve
devices
N. Stutzke, S. L. Burkett, and S. E. Russek
Citation: Applied Physics Letters 82, 91 (2003); doi: 10.1063/1.1534386
View online: http://dx.doi.org/10.1063/1.1534386
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On: Thu, 13 Nov 2014 17:23:21Temperature and field dependence of high-frequency magnetic noise
in spin valve devices
N. Stutzke
Boise State University, Boise, Idaho 83725
S. L. Burkett
University of Arkansas, Fayetteville, Arkansas 72701
S. E. Russeka)
National Institute of Standards and Technology, Boulder, Colorado 80305
~Received 21 October 2002; accepted 6 November 2002 !
The high-frequency noise of micrometer-dimension spin valve devices has been measured as a
function of applied field and temperature.The data are well fit with single-domain noise models thatpredict that the noise power is proportional to the imaginary part of the transverse magneticsusceptibility. The fits to the susceptibility yield the ferromagnetic resonance ~FMR !frequency and
the magnetic damping parameter. The resonant frequency increases, from 2.1 to 3.2 GHz, as thelongitudinal field varies from 22 to 4 mT and increases from 2.2 to 3.3 GHz as the temperature
decreases from 400 to 100 K. The shift in the FMR frequency with temperature is larger than thatexpected from the temperature dependence of the saturation magnetization, indicating that othertemperature-dependent anisotropy energies are present, in addition to the dominant magnetostaticenergies. The measured magnetic damping parameter
adecreases from 0.016 to 0.006 as the
temperature decreases from 400 to 100 K. The value of the damping parameter shows a peak as afunction of longitudinal bias field, indicating that there is no strict correlation between the dampingparameter and the resonant frequency. © 2003 American Institute of Physics.
@DOI: 10.1063/1.1534386 #
Advanced data storage applications require magnetic de-
vices to have submicrometer dimensions and operate at highfrequencies in the gigahertz regime. It has been recentlypointed out that high-frequency thermal fluctuations of themagnetization in giant magnetoresistive devices, which scaleinversely with the device volume, will become significant inthe next generation of recording read heads and will providea fundamental limitation on the ability to scale down thedevice size and increase the operating frequency.
1,2High-
frequency magnetic noise, in addition to being of practicalconcern in device operation, provides a powerful method tocharacterize the dynamic modes in small magnetic struc-tures. Mode frequencies and linewidths ~or equivalently,
damping parameters !can be determined over a wide range of
applied fields and temperatures. Here, we present the tem-perature and field dependence of the high-frequency mag-netic noise in spin valve devices that show single-domainbehavior and whose noise spectrum can be fit with simplesingle-domain models.
The device structures consisted of Ta (5 nm)–Ni
0.8Fe0.2
(5nm)–Co 0.9Fe0.1~1n m!–Cu (2.7 nm)–Co 0.9Fe0.1~2.5
nm!–Ru (0.6 nm)–Co 0.9Fe0.1(1.5 nm)–Ir 0.2Mn0.8~10
nm!–Ta~5n m !multilayers sputtered on oxidized ~100!Si
substrates. The films were deposited in a 15 mT field to setthe pinned direction of the fixed layer. The fixed layer~CoFe–Ru–CoFe !was a low-moment synthetic antiferro-
magnet. The wafer-level magnetoresistance ratio, R
AP
2RP/RP, was 7.8%, where RAPandRPare the resistanceswith the free and fixed layers antiparallel and parallel. The
wafers were patterned to form spinvalve devices with dimen-sions of 1
mmb y3 mm. The devices studied here have the
pinned-layer magnetization oriented perpendicular to theeasy axis, which is along the long dimension of the device.The devices were contacted with high-bandwidth transmis-sion lines and used overlapping electrodes. The data pre-sented here are from a device whose resistance, includinglead and contact resistance, was 20.2 Vin the parallel state,
and the change in resistance from parallel to antiparallelmagnetization states was 1.0 V.The free-layer magnetization
switched between its two easy-axis states consistently at alongitudinal field of 2.2 mT. The magnetic noise was evalu-ated from the measured voltage noise spectrum by subtract-ing a reference spectrum in which the free-layer magnetiza-tion was saturated by applying a large magnetic field alongthe hard axis.
The noise spectra from thermal fluctuations of the mag-
netization can be related to the imaginary part of the trans-verse susceptibility by the fluctuation-dissipation theorem
2,3
Vn~f!5IDRAkBT
2pfm0Ms2Vxt9~f!, ~1!
whereVnis the voltage noise spectrum, fis the frequency, I
is the current through the device, DRis the change in resis-
tance from the parallel to antiparallel magnetization state, T
is the device temperature, Msis the saturation magnetization
of the free layer, and Vis the free-layer volume. The trans-
verse susceptibility xt(f) is the ratio of the hard-axis mag-
netization, My, to the applied hard-axis field. The suscepti-
bility can be determined from the linearized Landau–Lifshitza!Author to whom correspondence should be addressed; electronic mail:
russek@boulder.nist.govAPPLIED PHYSICS LETTERS VOLUME 82, NUMBER 1 6 JANUARY 2003
91 0003-6951/2003/82(1)/91/3/$20.00 © 2003 American Institute of Physics
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On: Thu, 13 Nov 2014 17:23:21equation for a thin-film single-domain element and, for ap-
plied and anisotropy fields much less than Ms, is given by4,5
x9~f!5f
g8am0Ms
3Hm02Ms21f2
g82
Sm02~Hk1Hl!Ms2f2
g82D2
1Sf
g8am0MsD2J,
~2!
where g8is the gyromagnetic ratio divided by 2 p~28 GHz/
T!,Hkis the in-plane anisotropy field, Hlis the longitudinal
bias field that is applied along the easy axis of the device,and
ais the Gilbert damping parameter. The susceptibility
shows a resonance behavior. The imaginary part of the sus-ceptibility, which describes energy loss from the magneticsystem, has a peak at the ferromagnetic resonance frequency,
f
r5g8m0A(Hk1Hl)Ms, with the peak width proportional to
the damping parameter a. The saturation magnetization of
the free layer was measured to be 775 kA/m at 300 K. Thisvalue is lower than predicted from the bulk magnetizationvalues, due in part to a magnetically dead layer at the Ta–NiFe interface.
6The anisotropy field Hkis due predomi-
nantly to magnetostatic shape anisotropy. The measuredroom-temperature value of the low-frequency anisotropyfield, as determined from the slope of the hard-axis magne-toresistance curve, was DR/2(dR/dH
t)21510.6 mT,
whereas the calculated value for the magnetostatic anisot-ropy, assuming uniform magnetization, was 8.0 mT. Otherenergy terms enter into the measured anisotropy field, suchas the magnetostatic coupling to the pinned layer and anyinduced anisotropy energies. These terms are expected tocontribute 0.5–2 mTto the anisotropy field at room tempera-ture.
The measured noise spectra for various temperatures and
longitudinal bias fields are shown in Figs. 1 ~a!and 1 ~b!. Theresonance is clearly seen, with the resonant frequency in-
creasing as the temperature decreases or as the longitudinalbias field increases. The longitudinal-field data are similar tostandard ferromagnetic resonance ~FMR !measurements for
which the resonant frequency increases and the amplitude ofthe resonance peak decreases as the stiffness field increases.Comparison of the temperature-dependent data and thelongitudinal-field data shows that decreasing the temperaturefrom 400 to 100 K is roughly equivalent to increasing thestiffness field by 6 mT.
The data were fit using Eqs. ~1!and~2!to determine the
resonant frequencies, anisotropy fields, and damping param-eters.Asample fit is shown in Fig. 2. Here the H
k,a, and an
overall scale factor, C, were allowed to vary.All other quan-
tities were determined experimentally. Hkdetermines the
resonant frequency, adetermines the width of the resonance,
andCdetermines the overall scale. The scale factor is pre-
dicted by Eq. ~1!to beC51. However, the experimentally
determined scale factor is expected to be less than 1 sincethere are additional high-frequency attenuations of the noisesignal due to losses in the microwave circuit. For the dataanalyzed here, Cvaried between 0.1 and 0.4. The fits to the
temperature-dependent noise spectra used a temperature-dependent saturated magnetization measured by a supercon-ducting quantum interference device magnetometer. Themagnetization measurements, shown in Fig. 3, indicate thatthe magnetization changes only by 10% over the relevanttemperature range. However, the coupling field, determinedby measuring the shift in the free layer M–Hloop, increases
from 0.5 to 1.6 mT as the temperature decreases from 400 to100 K.
The results of the fitting all the resonance curves are
shown in Fig. 4. Figure 4 ~a!shows the dependence of the
resonant frequency and damping parameter on temperature.The resonant frequency increases from 2.2 to 3.3 GHz as thetemperature decreases from 400 to 100 K, indicating an in-crease of the stiffening fields with decreasing temperature.The in-plane anisotropy field increases from 6.9 to 13.6 mTas the temperature decreases from 400 to 100 K. The calcu-lated contribution to the increase in in-plane anisotropy, dueto the effect of increasing M
son the magnetostatic shape
anisotropy, is only 0.7 mT.The measured damping parameterdecreases from 0.016 to 0.006 as the temperature decreases
FIG. 1. Voltage noise spectra of a 1 mm33mm spin valve device for ~a!a
series of substrate temperatures with no applied longitudinal field, and ~b!a
series of longitudinal bias fields at room temperature. Both sets of curves aresimilar, with the resonant frequency increasing and the noise amplitude de-creasing as the stiffness fields increase.
FIG. 2. Fit of the noise spectrum of a spin valve device at 400 K with a biascurrent of 5 mA. The parameters indicated with an asterisk were allowed tovary. The other parameters were taken from experimental measurements.92 Appl. Phys. Lett., Vol. 82, No. 1, 6 January 2003 Stutzke, Burkett, and Russek
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On: Thu, 13 Nov 2014 17:23:21from 400 to 100 K. The room-temperature values of the
damping parameter are slightly less then the values ( a
>0.02 to 0.03 !obtained by directly measuring the high-
frequency susceptibility of similar spinvalve devices.7The
rms variation in the magnetization angle urmsfor the noise
measurements can be estimated from the fluctuation-dissipation theorem,
3which yields a value of
urms5KMy
MsL
rms5AkBT
m0MsHkV’0.15° to 0.37°. ~3!
These values of magnetization motion are considerably less
than those used to directly measure the device susceptibilityand are similar to the values used in standard FMR measure-ments. The temperature dependence of the damping coeffi-cient is considerably larger than that observed by FMR forsheet NiFe films.
8
The dependence of the resonant frequency and damping
parameter on longitudinal field is shown in Fig. 4 ~b!. The
measured resonance frequency increases as a function of lon-gitudinal field in a manner consistent with a constant anisot-ropy field. The average anisotropy field, determined from thefits to the data in Fig. 1, is
m0Hk59.0 mT, and the maximum
deviation from the mean value is 0.9 mT. The measured an-isotropy field is in reasonable agreement with the low-frequency value, given the uncertainty in determining thefree-layer moment and volume. The damping parametershows a peak as a function of longitudinal field. At largepositive fields there is a decrease in the damping parameterthat is consistent with observed behavior in single-layersheet films.
9The decrease in damping parameter for small
and negative fields does not correlate with any observablefeature in the longitudinal magnetoresistive response. Theresistance has no large variations as the longitudinal field isvaried until the switching threshold of 2.2 mT, which indi-cates that there is no large change in the micromagneticstructure as the longitudinal field is varied.
The increase in coupling field as the temperature de-
creases has been explained by assuming that there is a cou-pling component due to magnetostatic interactions arisingfrom surface roughness ~which is proportional to M
s) and a
component due to a temperature-dependent exchangecoupling.
10Another possibility, which is more consistent
with the observed temperature dependence of the dampingparameter, is that there are thermal fluctuations of the mag-netization at the interfaces of the ferromagnetic layers. Thefluctuations suppress the magnetostatic coupling at highertemperatures and provide an additional energy-loss mecha-nism.The micromagnetic fluctuations will depend on appliedfields and may account for the observed peak in the dampingparameter at small longitudinal fields.
The authors acknowledge the support of the NIST Nano-
technology Initiative and the DARPA Spintronics program.
1N. Smith and P. Arnett, Appl. Phys. Lett. 78, 1448 ~2001!.
2N. Smith, J. Appl. Phys. 90, 5768 ~2001!.
3L. D. Landau and E. M. Lifshitz, Statistical Physics ~Pergamon, New
York, 1980 !, Chap. 12.
4C. E. Patton, in Magnetic Oxides , edited by D. J. Craik ~Wiley, NewYork,
1975!.
5J. Huijbregtse, F. Roozeboom, J. Sietsma, J. Donkers, T. Kuiper, and E.
van de Riet, J. Appl. Phys. 83, 1569 ~1998!.
6M. Kowalewski, W. H. Butler, N. Moghadam, G. M. Stocks, T. C.
Schulthess, A. S. Arrott, T. Zhu, J. Drewes, R. R. Katti, M. T. McClure,and O. Escorcia, J. Appl. Phys. 87,5 7 3 2 ~2000!.
7S. E. Russek and S. Kaka, IEEE Trans. Magn. 36, 2560 ~2000!.
8R. D. McMichael, C. G. Lee, M. D. Stiles, F. G. Serpa, P. J. Chen, and W.
F. Egelhoff, Jr., J. Appl. Phys. 87, 6406 ~2000!.
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7849 ~1999!.
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~2002!.
FIG. 3. Magnetic moment as a function of applied field for a coupon from
the spin valve wafer. The field was applied parallel to the fixed-layer mag-netization. The hysteresis loops are not centered around zero moment be-causeofthefixedmomentofthepinnedlayer.Theinsetshowsthemeasuredcoupling field and relative free layer saturation magnetization calculatedfrom the hysteresis loops.
FIG. 4. The resonant frequencies and damping parameters determined fromfitting the noise data for ~a!the temperature-dependent data with no applied
longitudinal field, and for ~b!the longitudinal field dependent data at 300 K.93 Appl. Phys. Lett., Vol. 82, No. 1, 6 January 2003 Stutzke, Burkett, and Russek
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1.2204033.pdf | Photoionization cross section and angular distribution calculations of carbon
tetrafluoride
D. Toffoli, M. Stener, G. Fronzoni, and P. Decleva
Citation: The Journal of Chemical Physics 124, 214313 (2006); doi: 10.1063/1.2204033
View online: http://dx.doi.org/10.1063/1.2204033
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/124/21?ver=pdfcov
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130.88.90.140 On: Wed, 26 Nov 2014 17:06:28Photoionization cross section and angular distribution calculations
of carbon tetrafluoride
D. Toffolia/H20850
Dipartimento di Scienze Chimiche, Universitá degli Studi di Trieste, Via L. Giorgieri 1, I-34127 Trieste,
Italy and INFM DEMOCRITOS, National Simulation Center, I-34127 Trieste, Italy
M. Stener, G. Fronzoni, and P . Decleva
Dipartimento di Scienze Chimiche, Universitá degli Studi di Trieste, Via L. Giorgieri 1, I-34127 Trieste,Italy; INFM DEMOCRITOS, National Simulation Center, Trieste, Italy; and ConsorzioInteruniversitario Nazionale per la Scienza e Tecnologia dei Materiali, INSTM, Unitá di Trieste,I-34127 Trieste, Italy
/H20849Received 6 January 2006; accepted 18 April 2006; published online 7 June 2006 /H20850
Correlation in the photoionization dynamics of carbon tetrafluoride is studied in the framework of
the time-dependent density-functional theory /H20849TDDFT /H20850approach by employing a multicentric basis
set expansion of the scattering wave function linear combination of atomic orbitals /H20849LCAO /H20850
TDDFT. Results obtained with the statistical average of orbital potentials and LB94exchange-correlation /H20849xc/H20850potentials are compared with photoabsorption, photoionization, and
electron-scattering experiments as well as with past theoretical calculations. Inadequacies in boththeV
xcparametrizations employed have been suggested from the analysis of the intensity plots for
theD˜2A1ionization. The formation of resonant scattering states in selected continuum channels has
been studied through the analysis of the dipole-prepared scattering wave function; our findings arethen compared with results of electron-scattering calculations. Overall, the LCAO-TDDFT resultshighlight the effectiveness of the approach for the calculation of the unbound spectrum of fairlylarge molecules. © 2006 American Institute of Physics ./H20851DOI: 10.1063/1.2204033 /H20852
I. INTRODUCTION
The photoionization of tetrafluoromethane /H20849CF4/H20850is of
both fundamental and practical interest. It is one of the sim-
plest nonhydrogenic polyatomic molecules commonly usedas a source of fluorine atoms in the plasma dry etching ofsemiconductor materials and films.
1The characterization of
the scattering dynamics of CF 4is also a stimulating and quite
prolific field of research since highly symmetric moleculessurrounded by strong negative fluorine ligands provide someof the clearest examples of resonance phenomena.
2
The experimental work on CF 4has covered its whole
electronic spectrum, employing a variety of techniques, in-cluding absorption, fluorescence, photoionization, and scat-tering experiments /H20849Refs. 3–38and references therein /H20850. Ear-
lier calculations of the continuum spectrum employed themultiple-scattering /H20849MS-X
/H9251/H20850formalism,39,40while the
Schwinger variational iterative method41,42/H20849SVIM /H20850has been
recently used for obtaining ab initio cross sections and an-
gular distribution data for the valence43a n dC1 s/H20849Ref. 44/H20850
ionizations at the frozen core Hartree-Fock /H20849FCHF /H20850level.
While SVIM results agree nicely with the experiment forcore ionization,
44the accord is only qualitative for the ion-
izations out of the valence orbitals,43and the interpretation of
the experimental data11,30still remains an open issue.
We have recently proposed a novel approach to the study
of molecular photoionization in the framework of the time-dependent density-functional theory /H20849TDDFT /H20850and based ona multicentric basis set expansion of the scattering wave
function linear combination of atomic orbitals /H20849LCAO /H20850
TDDFT.
45While the implementation takes advantage of the
fundamental computational economy of DFT compared to ab
initio methods of similar quality, the multicentric nature of
the basis set employed enhances the convergence propertiesof the partial-wave expansion, overcoming thus the typicalbottleneck of single-center-expansion /H20849SCE /H20850-based methods
when applied to fairly large molecular systems with heavyoff-center nuclei. Additionally, certain classes of many-bodyeffects are either phenomenologically /H20849correlation in the ini-
tial and final target states /H20850or explicitly /H20849interchannel cou-
pling between single-hole main-line states /H20850included in the
TDDFT approach.
46
In this paper, the LCAO-TDDFT approach is applied to
the study of the photoionization dynamics of the CF 4mol-
ecule. Both valence and core ionizations are considered. In-terchannel coupling effects between main-line channels areassessed through a comparison of the TDDFT results withsingle-channel calculations, and shape-resonant states in se-lected photoionization continua are analyzed. Correlation ef-fects phenomenologically included in the formalism throughthe exchange-correlation functional in the standard Kohn-Sham /H20849KS/H20850scheme,
47have been studied by using both the
van Leeuwen-Baerends48/H20849LB94 /H20850and the statistical average
of orbital potential49,50/H20849SAOP /H20850approximations to the true
exchange-correlation /H20849xc/H20850functional.
The organization of the paper is as follows. In Secs. II
and III we will briefly outline our theoretical approach anda/H20850Electronic mail: toffoli@univ.trieste.itTHE JOURNAL OF CHEMICAL PHYSICS 124, 214313 /H208492006 /H20850
0021-9606/2006/124 /H2084921/H20850/214313/10/$23.00 © 2006 American Institute of Physics 124, 214313-1
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130.88.90.140 On: Wed, 26 Nov 2014 17:06:28the computational details. In Sec. IV we will discuss our
results, and a comparison with previous investigations willbe made. Our conclusions and perspectives are then summa-rized in the final section.
II. THEORETICAL METHOD
The LCAO-TDDFT algorithm employed for the resolu-
tion of the scattering equations in a B-spline finite basis set
has been fully described in earlier publications45,51to which
the reader is referred for a broad discussion. Here we will
only present the basic equations that need to be solved46and
give a short account of the multicentric basis set employed.
Within the adiabatic local density approximation
/H20849ALDA /H20850to the xc response kernel,52the response of the
molecular system interacting with a weakly time-dependent
electromagnetic field of frequency /H9275is modeled through the
introduction of an effective time-dependent potentialV
SCF/H20849r,/H9275/H20850:
VSCF/H20849r,/H9275/H20850=VExt/H20849r,/H9275/H20850+/H20885/H9254/H9267/H20849r/H11032,/H9275/H20850
/H20841r−r/H11032/H20841dr/H11032
+/H20879/H11509Vxc
/H11509/H9267/H20879
g.s./H9267/H20849r/H20850/H9254/H9267/H20849r,/H9275/H20850/H11013VExt/H20849r,/H9275/H20850
+/H20885K/H20849r,r/H11032/H20850/H9254/H9267/H20849r/H11032,/H9275/H20850dr/H11032, /H208491/H20850
which in turn determines the time-dependent molecular den-
sity response /H9254/H9267/H20849r,/H9275/H20850through the KS noninteracting re-
sponse function, /H9273KS/H20849r,r/H11032,/H9275/H20850:
/H9254/H9267/H20849r,/H9275/H20850=/H20885/H9273KS/H20849r,r/H11032;/H9275/H20850VSCF/H20849r/H11032,/H9275/H20850dr/H11032. /H208492/H20850
In Eq. /H208491/H20850VSCFis evaluated to the first order in the per-
turbing /H20849dipole /H20850potential, VExt. Our implementation of the
TDDFT eqs. /H208491/H20850and /H208492/H20850takes advantage of a noniterative
algorithm,45,53where the integral equation
VSCF/H20849r,/H9275/H20850=VExt/H20849r,/H9275/H20850+/H20885/H20885 K/H20849r,r/H11032/H20850/H9273KS/H20849r/H11032,r/H11033,/H9275/H20850
/H11003VSCF/H20849r/H11033,/H9275/H20850dr/H11032dr/H11033 /H208493/H20850
is solved by representing the kernel of the integral operators
in a LCAO-type composite basis set.45,51The LCAO basis
set consists in a large SCE located on a chosen origin /H20849usu-
ally the center of mass of the molecule /H20850,
/H9273nlhp/H9262,SCE=1
rBn/H20849r/H20850/H20858
mblmhp/H9262YlmR/H20849/H9258,/H9278/H20850/H110131
rBn/H20849r/H20850Xlhp/H9262/H20849/H9258,/H9278/H20850,
/H208494/H20850
and supplemented by functions of the same type, located on
the off-center arbitrary positions j,
/H9273nlhp/H9262,i=/H20858
j/H33528Qi1
rjBn/H20849rj/H20850/H20858
mblmh,jp/H9262YlmR/H20849/H9258j,/H9278j/H20850. /H208495/H20850In Eq. /H208495/H20850, index iruns over the nonequivalent nuclei, j
runs over the set of equivalent nuclei, Qi, and gives the ori-
gin of the off-center spherical coordinates /H20849rj,/H9258j,/H9278j/H20850. The
sets of coefficients blmhp/H9262andblmh,jp/H9262define the unitary transfor-
mations between real spherical harmonics YlmR/H20849/H9258,/H9278/H20850and the
symmetry adapted angular basis sets54which transform as
the/H9262th element of the pth irreducible representation /H20849IR/H20850of
the molecular point group. Bnis the nth spline monodimen-
sional function.55The Bsplines are built over the radial in -
terval /H208510,RmaxSCE/H20852for the set /H9273nlhp/H9262,SCE, and over the intervals
/H208510,Rmaxi/H20852for the off-center functions /H9273nlhp/H9262,i.
Then, we solve with respect to the unknown VSCFthe
algebraic linear system:
/H20849K/H9273−1/H20850VSCF=−VExt, /H208496/H20850
and photoionization dynamical quantities are extracted with
standard formulas by using VSCFin place of the dipole
operator.46
Shape-resonance phenomena in selected continuum
channels have been investigated by inspecting the dipole-
prepared scattering wave function obtained by applying a
unitary transformation to the degenerate set of partial-waveopen channels in such a way that only one partial-wave com-ponent of the transformed set carries all the intensity of theprocess.
56The method has already been successfully applied
to the analysis of the mechanisms giving rise to shape reso-
nances in several molecular systems, such as transition metalcomplexes
56and fullerene endohedral compounds.57
III. COMPUTATIONAL DETAILS
The ground state electron density of carbon tetrafluoride
at the equilibrium geometry, with a C–F bond length of1.323 Å /H20849Ref. 58/H20850and a T
dpoint-group symmetry, was cal-
culated with the ADFpackage59,60employing an all-electron
double- /H9256plus polarization /H20849DZP /H20850basis set of Slater-type or-
bitals taken from the ADF database. Such a density is thenused to build the h
KSHamiltonian. Calculations have been
done with both the LB94 /H20849Ref. 48/H20850and SAOP /H20849Ref. 49and
50/H20850Vxcpotentials.
The fixed-nuclei /H20849FN/H20850photoionization calculations used
a SCE up to lmaxSCE=12 for expanding the bound molecular and
continuum orbitals, with the SCE placed on the carbon atom.B-spline functions of the order 10 were employed in the
calculation, defined over a linear radial grid with a step sizeof 0.20 a.u. and extending up to R
maxSCE=20 a.u.. The B-spline
basis set on the off-center nuclei /H20849i.e., the fluorine centers /H20850is
defined over a linear radial grid with a step size of 0.20 a.u.up to a value of R
maxF=1.6 a.u.. The maximum angular mo-
mentum lmaxFemployed in the off-center expansion was lmaxF
=2.
For valence ionizations we performed a seven-channel
calculation by including all the target states with a singlehole in a valence orbital. For the fluorine 1 sionization only
the two target states with a single hole in the F 1 slevel were
included in a two-channel calculation, whereas single-channel calculations were done for the C 1 sionization.
For a meaningful comparison with the various experi-
mental data sharp Feshbach resonances clustering the various214313-2 Toffoli et al. J. Chem. Phys. 124, 214313 /H208492006 /H20850
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130.88.90.140 On: Wed, 26 Nov 2014 17:06:28ionic thresholds have been convoluted with Gaussian func-
tions. Photoelectron kinetic energies were converted to pho-ton energies by using the experimental ionization potentials/H20849IP’s/H20850.
13,61IV. RESULTS AND DISCUSSION
The ground state electronic configuration of CF 4can be
written as
/H208497/H20850
Outer valence orbitals 4 a1and 3 t2are/H9268bonding formed
by a superposition of C 2 s/2pa n dF2 s/2patomic orbitals
whereas, the three outermost orbitals 1 e,4t2, and 1 t1are
halogen lone-pair porbitals. Inner valence states 3 a1and 2 t2
have primaril yaF2 satomic parentage. The virtual valence
orbitals are the corresponding /H9268*/H20849C–F /H20850.
Outer valence Green’s function /H20849OVGF /H20850method and
third-order algebraic-diagrammatic construction approxima-tions /H20849ADC /H208493/H20850/H20850calculations
30predict spectroscopic factors
close to unity for the outer valence ionizations, while strong
correlation effects shape the photoelectron spectrum in theregion of the inner valence 2 t
2and 3 a1ionizations where the
main-line intensities are spread over a manifold of satellitestates.
30The experimental vertical IPs of CF 4are reported in
Table Iand compared with the ADF IP results, obtained with
both SAOP and LB94 xc functionals. As a rule, DFT valenceIPs obtained with the SAOP xc functional are in rather betteragreement with the experiment than the LB94 ones,
49,50al-
though the latter compare fairly well with the experimentaldata in the whole spectrum. Since among the xc potentialswith a correct asymptotic behavior the LB94 one is the onlyroutinely used for photoionization calculations,
62we tested
the performances of the SAOP xc potential against the LB94
one. Note that correlation-polarization enters into both the
/H9273KSkernel, which is built from the KS orbitals and energies,
and the xc screening kernel of Eq. /H208491/H20850. It is, nonetheless,
suggested63that the largest source of errors comes from theapproximation to the static Vxcpotential rather than from the
ALDA approximation. We plotted the total TDDFT photo-ionization cross sections, from the threshold /H20849at 16.29 eV /H20850up
to about 150 eV of photon energy, in the upper panel of Fig.1and made a comparison with the total photoabsorption data
of Au et al.
38Overall, one sees that the experimental spec -
trum is fairly well reproduced with both the SAOP and LB94V
xcparametrizations, notably the strong near- threshold
modulations extending up to about 25 eV and due, in thecalculated profiles, to autoionizations, and the underlyingbroad maximum at about 30 eV ascribed to scattering reso-
nances in the /H9255eand/H9255t
2continua of the X˜2T1,A˜2T1,C˜2T2,
and D˜2T1target states /H20849vide infra /H20850. Nonetheless, a closer
inspection of the theoretical results suggests that the LB94scattering potential proves to be more attractive than the
TABLE I. CF4vertical ionization potential energies.
Orbital ionization ADF SAOPa/H20849eV/H20850ADF LB94a/H20849eV/H20850Expt.b/H20849eV/H20850
1a1−1/H20849F1s/H20850 668.0 692.2 695.0c
1t2−1/H20849F1s/H20850 668.0 692.2 695.0c
2a1−1/H20849C1s/H20850 283.9 301.1 301.8c
3a1−1/H20849F˜2A1/H20850 41.2 41.4 43.81d
2t2−1/H20849E˜2T2/H20850 38.4 38.6 40.30d
4a1−1/H20849D˜2A1/H20850 24.4 25.0 25.11
3t2−1/H20849C˜2T2/H20850 22.0 22.8 22.04
1e−1/H20849B˜2E/H20850 18.6 19.2 18.54
4t2−1/H20849A˜2T2/H20850 18.0 18.7 17.51
1t1−1/H20849X˜2T1/H20850 16.8 17.5 16.29
a−/H9280KS/H20849present work /H20850obtained with a Slater-type orbital /H20849STO /H20850double- /H9256plus
polarization basis set /H20849see text for details /H20850.
bReference 13.
cReference 61.
dTaken as the maxima of the correlation bands seen in the photoelectron
spectra /H20849Ref. 30/H20850.
FIG. 1. Upper panel: LCAO TDDFT total photoionization cross section of
CF4obtained with both the LB94 and SAOP xc potentials. Lower panel:
LCAO TDDFT and KS total photoionization cross sections of CF4obtained
with the SAOP xc potential. The experimental photoabsorption data aretaken from Ref. 38. The theoretical profiles have been convoluted with a
Gaussian line shape of 1.0 eV FWHM.214313-3 Photoionization of carbon tetrafluoride J. Chem. Phys. 124, 214313 /H208492006 /H20850
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130.88.90.140 On: Wed, 26 Nov 2014 17:06:28SAOP one /H20849with a mean shift of about 1.2 eV of the LB94
profile toward the threshold compared to the SAOP one /H20850, and
the energy positions of the experimental features are betterreproduced with the SAOP V
xcparametrization. Note further
that the two theoretical profiles nearly superimpose forhigher excitation energies because the fast escaping photo-electron is less sensitive to the finer details of the effectivepotential in the molecular region. A comparison of the twosets of DFT results for the individual orbital ionizations re-veals a trend similar to that described above. In the followingwe therefore compare the literature data with our TDDFTand KS results obtained with the SAOP V
xcpotential.
In the lower panel of Fig. 1the KS SAOP total photo-
ionization cross section result is compared with both the TD-DFT SAOP one and the experimental data.
38The KS ap -
proach is a single-channel formalism, and a comparison withthe TDDFT results allows us to assess the extent of inter-channel coupling effects. These prove to be sizeable in thewhole spectrum; the oscillator strength is somewhat trans-ferred closer to the threshold, and the decaying rate of thecross section is reduced for higher photon energies, in rathergood agreement with the experiment.
A. Valence ionizations
Partial cross sections for the X˜2T1,A˜2T2, and B˜2Eionic
states of CF 4are reported in Fig. 2where TDDFT and KS
results are compared with the experimental data11,14,30and
with the MS-X /H9251results of Stephens et al.39The valence
shell photoionization dynamics of CF 4has been recently in-
vestigated by Nascimento et al.43at the FCHF level of the
theory, and their results are included in the figure. Focusing
for the moment on the DFT KS data, an interesting trendalong the three ionizations can be pointed out. A broad spec-tral feature peaking at about 30 eV of photon energy in the
X
˜2T1continuum spectrum shifts at a somewhat higher en-
ergy which loses intensity in the A˜2T2continua and almost
disappears in the 1 e−1ionization cross section. For photon
energies above the 4 a1−1ionization /H20849at about 25 eV, see Table
I/H20850the KS profiles agree nicely with the experimental data.
Furthermore, the sharp prominent near-threshold peaks cal-culated by Stephens et al.
39/H20849see Fig. 2/H20850and interpreted as
due to a resonant trapping in the /H9268*/H20849C–F /H20850/H9255t2and/H9255a1con-
tinua are located in the discrete part of the spectrum in our
KS results, in agreement with both experiments30,36and re -
cent ab initio single-channel calculations.40,43We further per -
formed bound-state TDDFT calculations with the ADFcode
and using different basis sets from standard all-electrondouble- and triple-
/H9256plus polarizations /H20849DZP and TZP /H20850to a
quadruple- /H9256plus polarization with diffuse functions added
on the carbon center. Excitations to the 5 a1and 5 t2virtual
valence /H9268*/H20849C–F /H20850orbitals were predicted to occur below the
threshold for every valence target state; thus any shape reso-
nance in the ionization continua of CF 4would be due to a
nonvalence state.
At this point few remarks on the sizeable discrepancies
we find when comparing our single-channel KS results withthose of Stephens et al.
39and Nascimento et al.43are worth
to do. While the set of approximations included in the MS-X/H9251formalism, i.e., the muffin-tin approximation and the
rather crude treatment of the exchange part of the effectivepotential, could be invoked as a principal source of disagree-ment with our results, the differences in magnitude and shapebetween our KS cross sections and the FCHF ones of Ref. 43
are quite surprising because additional correlation-polarization effects were included in the FCHF treatment ofRef.43through the use of the Padial-Norcross potential. The
FCHF approach is expected to provide an adequate descrip-
tion of the outer valence molecular photoionization dynam-ics, and the energy positions of one-particle resonances areusually fairly accurately predicted.
64On the other hand, one
notes that ab initio cross sections obtained in the length and
velocity forms of the dipole operator are similar in shape butgreatly differ in magnitude, especially for the 1 t
1−1,4t2−1, and
3t2−1/H20849see Fig. 3/H20850ionizations. Furthermore, the FCHF profiles
are, in many circumstances, shifted to higher photon energieswhen compared with the KS ones /H20849striking examples are the
FIG. 2. LCAO TDDFT and KS partial cross sections for the three outermost
valence ionizations of CF4, obtained with the SAOP xc potential. The TD-
DFT profiles have been convoluted with a Gaussian line shape of 0.5 eVFWHM. Also included are the MS-X
/H9251results from Ref. 39and the SVIM
calculations from Ref. 43. The experimental data are taken from Refs. 11,
14, and 30.214313-4 Toffoli et al. J. Chem. Phys. 124, 214313 /H208492006 /H20850
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130.88.90.140 On: Wed, 26 Nov 2014 17:06:28photoionization events leading to the X˜2T1andD˜2A1target
states, and several of the FCHF computed resonant states donot match the energy position of experimental features byseveral eV.
43The use of the SAOP and LB94 xc potentials
provides a partial screening of the hole formed during the
photoionization process, while the screening is completelylacking in a frozen core formalism; this could partially ex-plain energy shifts between KS and FCHF profiles. More-over, the FCHF results fail to predict the correct symmetry ofresonant states characterized in previous ab initio electron-
molecule scattering calculations
65,66and confirmed by our
DFT calculations /H20849vide infra /H20850. One could speculate that defi-
ciencies in the FCHF potential prove unusually large forCF
4, but also accurate experimental cross section measure-
ments would be needed for a complete assessment of theperformances of the KS and FCHF methods for this particu-lar case.
None of the previous single-channel calculations
39,40,43
could give a convincing account of the strong modulations of
the experimental data in the near-threshold range of the 1 t1−1,
4t2−1, and 1 e−1ionizations.11,14,30In valence photoionization,
the interchannel coupling between open and closed channels
gives rise to autoionization features as a results of the non-radiative decay of excited target states in the underlyingcontinua.
67The interchannel coupling is incorporated in the
TDDFT formalism,46and strong autoionization features are
predicted to occur in the near-threshold region for the threeoutermost valence ionizations /H20851note that the TDDFT results
have been convoluted with a Gaussian line shape of 0.5 eVfull width at half maximum /H20849FWHM /H20850in order to account for
the vibrational broadening and the experimentalresolution
30/H20852. It is noteworthy that the TDDFT results seem
to account fairly well for some of the excursions of the ex-
perimental data. We stress that care must be paid in interpret-ing experimentally observed strong near-threshold modula-tions on the basis of single-particle theories, since near-threshold resonances can exhibit a multielectron nature. Thesame physical effect /H20849i.e., autoionization /H20850is responsible for
the spectral features at about 40 eV in the TDDFT profiles,converging to the inner valence ionization thresholds. Theresonances computed at 40 eV are probably washed out bydissociation processes associated with strongly antibondingresonant states. Nuclear motion effects are also likely re-sponsible for the broadening of experimental resonances atabout 20 eV, but one further notes that the quality of theexperimental data is poor, and more accurate experimentalmeasurements would be needed for a sound evaluation of theperformance of our LCAO-TDDFT approach.
The partial cross section for the ionizations leading to
theC
˜2T2andD˜2A1excited target states are reported in the
upper and lower panel of Fig. 3respectively, and compared
with the experimental data of Holland et al.30and of Carlson
et al.11The theoretical predictions of Stephens et al.39and of
Nascimento et al.43are also included in the figure. The KS
profiles display pronounced maxima below 40 eV of photon
energy due to enhancements in the dipole matrix elementsfor the /H9255eand/H9255t
2continua. While DFT KS results agree
qualitatively with the experimental data, they invariably failfor a quantitative account, and the intensity of the ionizationprocesses is overestimated. Previous theoreticalcalculations
39,43also lack quantitative agreement with the ex -
perimental data. The FCHF data of Ref. 43, in the velocity
form of the dipole operator, are in fair agreement with the
experimental findings for the ionization leading to the D˜2A1
excited target state /H20849lower panel of Fig. 3/H20850, but the resonant
behavior predicted for the C˜2T2channels is greatly overes-
timated and somewhat shifted to lower photon energies. TD-DFT quantitatively recovers the discrepancies found at the
KS level for the ionization leading to the C˜2T2state but does
not improve over the KS predictions for the ionization to the
D˜2A1state. The huge overestimate of the intensity by the
DFT results should be connected with deficiencies in the xc
part of the effective potential for the D˜2A1ionization, quite
commonly observed for the photoionization out of molecularorbitals with a 2 satomic parentage.
68
Both KS and TDDFT partial cross section profiles for
the photoionization leading to the X˜2T1,A˜2T2,C˜2T2, and
D˜2A1target states of Figs. 2and3display modulations in
the 30–40 eV photon energy range, suggesting the occur-
rence of shape resonances. By inspection of the symmetry-resolved partial cross sections one then determines the sym-metry of the resonant state, and from the analysis of thecorresponding scattering wave functions useful insights onthe electronic structure of the molecule could be obtained.
FIG. 3. LCAO TDDFT and KS partial cross sections for the ionizations
leading to the C˜2T2/H20849upper panel /H20850andD˜2A1/H20849lower panel /H20850ionic states of
CF4+. The TDDFT profiles have been convoluted with a Gaussian line shape
of 0.5 eV FWHM. Also included are the MS-X /H9251results from Ref. 39and
the SVIM calculations from Ref. 43. The experimental data are taken from
Refs. 11and30.214313-5 Photoionization of carbon tetrafluoride J. Chem. Phys. 124, 214313 /H208492006 /H20850
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130.88.90.140 On: Wed, 26 Nov 2014 17:06:28The symmetry-resolved KS partial cross sections for the ion-
ization leading to the X˜2T1,A˜2T2, and C˜2T2target states are
plotted in Fig. 4. Obviously, due to dipole selection rules,
only the /H9255t2continuum could be reached from the 4 a1−1or-
bital ionization which is therefore omitted in the figure.
Modulations in the X˜2T1,A˜2T2, and C˜2T2partial cross sec-
tions are associated with enhancements in the dipole matrixelements with the /H9255econtinuum. The corresponding KS
dipole-prepared scattering wave functions at the peak energypositions /H20851with photoelectron kinetic energy /H20849KE/H20850of 11.0,
17.9, 12.3, and 12.4 eV for the ionizations leading to the
X
˜2T1,A˜2T2,C˜2T2, and D˜2A1states, respectively /H20852were
plotted and analyzed. Because we found similar contourshapes of the /H9255eresonant wave function for the 1 t
1,4t2, and
3t2ionizations, only that for the 1 t1−1continuum is plotted inthe left panels of Fig. 5. Contour plots of the /H9255t2resonant
continuum, corresponding to the peak energy position of the
resonance in the D˜2A1channel are presented in the right
panels of the same figure. Three different molecular planeswere selected for plotting: a fourfold symmetry axis definethezaxis of the reference system whose origin coincides
with the carbon atom, whereas two fluorine atoms are con-tained in the plane passing through the yaxis and the x-z
direction. Remaining cuts were made along the x-yandx-z
planes. An inspection of the /H9255econtour plots reveals a con-
tinuum state of predominant dcharacter at the carbon center,
but slightly distorted by the ligand’s cage. An examination ofthe contour plots for the /H9255t
2continuum wave function reveals
a dominant pcharacter at the carbon center /H20849corresponding
t oa2 s→/H9255ptransition /H20850with dpartial-wave contributions and
an antibonding character along the C–F bonds /H20849lowest right
panel /H20850. While the /H9255t2shape resonance in the 4 a1−1channel
was also characterized by Nascimento et al. ,43resonant en -
hancements in the X˜2T1,A˜2T2, and C˜2T2contina were as-
signed to a1ort2symmetries. Our assignment is however,
consistent with the results of ab initio electron-scattering
calculations;65,66these calculations found a broad scattering
resonance of the /H9255esymmetry at about 26 eV, thus shifted by
FIG. 4. LCAO TDDFT and KS partial cross sections for the ionizations
leading to the X˜2T1/H20849upper panel /H20850,A˜2T2/H20849central panel /H20850, and C˜2T2/H20849lower
panel /H20850ionic states of CF4+, together with the KS symmetry-resolved contri-
butions. The TDDFT profiles have been convoluted with a Gaussian lineshape of 0.5 eV FWHM.
FIG. 5. Left panels: Contour plots of the final continuum orbitals for the
1t1−1→/H9255eionization channel at the given photoelectron kinetic energy. Right
panels: Contour plots of the final continuum orbitals for the 4 a1−1→/H9255t2ion-
ization channel at the given photoelectron kinetic energy. Solid line, positivecontributions; dashed line, negative contributions; dash-dotted line, zerocontributions /H20849nodal line /H20850. Axes are explicitly given in the figure. Closed
squares in the lower panels denote atomic /H20849C and F /H20850positions.214313-6 Toffoli et al. J. Chem. Phys. 124, 214313 /H208492006 /H20850
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130.88.90.140 On: Wed, 26 Nov 2014 17:06:28about 10−15 eV to higher KE’s when compared with our
photoionization results. This energy shift is typically ob-served when comparing energy positions of shape-resonantstates found in scattering and photoionization experiments.
2
At2scattering resonance located at about 29 eV of KE was
also characterized in one of these theoretical studies,66and
its position correlated fairly well with the energy position of
the resonant state found in the 4 a1−1ionization.
The KS and TDDFT asymmetry parameter profiles for
all valence ionizations are reported in Fig. 6. A quantitative
agreement is found when comparing the TDDFT predictionswith the experimental data.
11,30MS-X /H9251results of Ref. 14are
also reported in the figure, whereas the FCHF results of Ref.
43are not reported for the sake of clarity. The TDDFT pro-
files show complicated features in the near-threshold and in-termediate energy range due to the formation of either reso-nant scattering states or autoionizing states, as discussedabove. The account of the linear density response of the sys-tem does not change appreciably the KS predictions for thethree outermost valence ionizations; on the contrary, a quan-titative agreement with the experiment is obtained only at theTDDFT level for the remaining valence ionizations. For thesake of completeness, the DFT asymmetry parameter profilesfor the inner valence ionizations are plotted in the lowest
right panel of Fig. 6and compared with the recent experi-
mental data of Holland et al.
30The theoretical results have
been obtained by averaging the /H9252’s for the photoionization
leading to the E˜2T2and F˜2A1target states by the corre-
sponding cross sections. As expected, a comparison with theexperimental data is not satisfactory because of the strongdepartures from the quasiparticle picture of the ionizationthat occur in this spectral region.
30
The branching ratio /H20849BR/H20850profiles, defined as the ratio
between the intensity in a given spectral region and the totalspectral intensity,
30up to 110 eV of photon energy, are plot -
ted in Fig. 7. The agreement between our DFT results and the
experimental data14,30could be considered excellent in the
near-threshold range, both for the spectral positions and in-
tensity of the observed fluctuations. For higher excitationenergies, i.e., above 60 eV, there are localized discrepancies
between theory and experiments, notably for the X˜2T1and
B˜2Etarget states, that would be partially attributed to an
experimental uncertainty and to our theoretical model em-ployed. We should note that in deriving their experimental
data for the production of the X˜2T1andA˜2T2ionic states,
FIG. 6. LCAO TDDFT and KS asym-
metry parameter profiles for the va-lence ionizations, obtained with theSAOP xc potential. The TDDFT pro-files have been convoluted with aGaussian line shape of 0.5 eV FWHM.Also included are the MS-X
/H9251results
from Ref. 39. The experimental data
are taken from Refs. 11and30.214313-7 Photoionization of carbon tetrafluoride J. Chem. Phys. 124, 214313 /H208492006 /H20850
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130.88.90.140 On: Wed, 26 Nov 2014 17:06:28Yates et al. neglected the weak contribution of the intensity
arising from the F 2 s/H20849inner valence states /H20850ionizations.
Therefore their data do not compare well with the corre-sponding data of Holland et al.
30for photon energies higher
than /H1101160 eV. Because the physical information embodied in
the branching ratio profiles is equivalent to that provided bythe absolute measurements, the disagreement found for the
ionization leading to the D˜2A1state for excitation energies
up to the opening of the threshold of the inner valence statesis not surprising.
B. Core ionizations
Inner-shell photoionization dynamics of carbon tet-
rafluoride have been investigated by several experimental-ists. Truesdale et al.
12measured photoionization asymmetry
parameters and partial cross sections for the C 1 sionization
and suggested the presence of a low-energy shape resonanceat about 315 eV. Electron-energy loss spectroscopy /H20849EELS /H20850
measurements
31assessed the occurrence of shake-up and
shake-off processes in the C 1 scontinuum, whereas Zhang
et al.34provided assignments for most of the observed dis -
crete excitations below the C and F Kedges. High resolution
electron yield spectra in the vicinity of the C 1 sand F 1 sthresholds have recently appeared.22Furthermore, there are
an extensive bibliography and a database available.18
We plot in Fig. 8DFT partial cross sections and asym-
metry parameter profiles for the C 1 sphotoionization and
compare our results with the SVIM calculation of Natalense
et al.44and with available experimental data.12,18Because of
the large energy separation with the valence photoionization
channels, an interchannel coupling is not expected to signifi-cantly change single-channel predictions. Therefore single-channel calculations have been performed in this spectralregion. Furthermore, in view of the very tiny differences be-tween single-channel TDDFT and KS predictions, we willfocus on the comparison between the LCAO-TDDFT resultsobtained with the SAOP and LB94 xc potentials. The TD-DFT single-channel results agree nicely with the ab initio
SVIM results of Natalense et al.
44The partial cross section
displays an oscillating behavior persisting 100 eV above the
threshold. The theoretical data compare favorably with theexperimental total absorption cross section of Hitchcock andMancini
18especially in the region of the shape resonance at
about 315 eV that would correspond to the resonant state
characterized in the D˜2A1ionization. We, furthermore, sug-
gest that the bump visible in the experimental data at about320 eV is connected with shake-up channels not included in
FIG. 7. LCAO TDDFT and KS
branching ratios for the valence ion-izations, obtained with the SAOP xcpotential. The theoretical profiles havebeen convoluted with a Gaussian lineshape of 0.5 eV FWHM. The experi-mental data are taken from Refs. 14
and30.214313-8 Toffoli et al. J. Chem. Phys. 124, 214313 /H208492006 /H20850
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130.88.90.140 On: Wed, 26 Nov 2014 17:06:28our theoretical method. The asymmetry parameter is also
characterized by strong near-threshold modulations suggest-ing the formation of a scattering resonant state. One lastcomment on the LCAO-TDDFT single-channel results is im-portant. While the TDDFT SAOP results closely follow therise of the experimental points toward the threshold, theLB94 profile does not show any inflection but a rapid de-crease, as much as the SVIM profile.
44Actual differences in
the DFT effective scattering potentials show up preferentially
in the partial cross section data, since TDDFT SAOP andLB94 asymmetry parameter profiles are almost identical inthe whole energy range explored and are in overall goodagreement with both the ab initio SVIM results
44and the
experimental data.12
DFT partial cross sections and asymmetry parameters for
the photoionization out of the deepest F 1 sorbitals are plot-
ted in Fig. 9. We checked that the interchannel mixing be-
tween the 1 a1−1and 1 t2−1ionization channels do not alter the
KS independent-particle predictions and presented a com-parison between the LCAO-TDDFT results obtained with theSAOP and LB94 parametrizations. Only tiny differences be-tween the LB94 and SAOP results can be pointed out, butstill a slightly better agreement with the experimental data ofRef. 18is obtained when the SAOP parametrization is used,
whereas the theoretical profiles nearly superimpose forhigher excitation energies. A broad enhancement at about10 eV of the photoelectron kinetic energy corresponds to ascattering resonance in the 1 t
1→/H9255econtinuum, similar to
that characterized in the valence ionizations. The asymmetryparameter profiles show a hint of the presence of a resonanttrapping at very low kinetic energies, and damped oscilla-tions persist even at the higher excitation energies.
V. CONCLUSION
This paper provides a broad discussion of valence and
inner-shell photoionizations from CF 4. Near-threshold dy-
namics of valence ionizations is characterized by the pres-ence of strong autoionizing excited states, whose presence isalso suggested by the experimental data. Our TDDFT predic-tions agree satisfactorily with the experimental measure-ments, and the need for a balanced inclusion of electron cor-relation effects for a quantitative account of the scatteringdynamics of this relatively simple molecular system isstressed. The formation resonant scattering states in selectedcontinuum channels has been analyzed with the aid ofdipole-prepared scattering wave functions; our findings fit
well with results of electron-scattering calculations. In theLCAO-TDDFT calculations we employed two different V
xc
parametrizations, the LB94 and the SAOP one, whose per-
formances have been compared. Overall, a clear trendemerges in that a more accurate description of the short-
FIG. 8. Upper panel: Single-channel LCAO TDDFT C 1 spartial cross
sections. Lower panel: Single-channel LCAO TDDFT C 1 sasymmetry pa-
rameter profiles. Also reported are the SVIM results of Natalense et al. /H20849Ref.
44/H20850and the experimental data from Refs. 12and18.
FIG. 9. Upper panel: Two-channel LCAO TDDFT F 1 spartial cross sec-
tions. Lower panel: Two-channel LCAO TDDFT F 1 sasymmetry parameter
profiles. Also reported are experimental cross section data from Ref. 18.214313-9 Photoionization of carbon tetrafluoride J. Chem. Phys. 124, 214313 /H208492006 /H20850
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130.88.90.140 On: Wed, 26 Nov 2014 17:06:28range effective scattering potential is provided by the SAOP
parametrization, whereas the LB94 potential proves to beslightly too attractive. Inadequacies in both the V
xcparam-
etrizations employed have been highlighted from the analysis
of the intensity plots for the D˜2A1ionization. Overall, it is
suggested that developing a database would be helpful inassessing the performances of the SAOP xc potential forDFT-based scattering calculations over the more dated andstandard LB94 choice.
ACKNOWLEDGMENTS
The authors gratefully acknowledge grants from MIUR
/H20849Programmi di Ricerca di Interesse Nazionale COFIN and
FIRB /H20850of Italy for the support of this research. One of the
authors /H20849D.T. /H20850also acknowledges the CNR-INFM DEM-
OCRITOS for a postdoctoral fellowship.
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1.2384842.pdf | Experimental study of A0 and T1 modes of the concert harp
J-L. Le Carroua/H20850and F. Gautier
Laboratoire d’Acoustique de l’Université du Maine, UMR-CNRS 6613, Université du Maine,
Avenue Olivier Messiaen, 72085 Le Mans Cedex 09, France
E. Foltête
Institut FEMTO ST, Université de Franche-Comté, Laboratoire de Mécanique Appliquée,24 rue de l’Epitaphe, 25000 Besançon, France
/H20849Received 20 January 2006; revised 5 October 2006; accepted 5 October 2006 /H20850
String instruments are usually composed of a set of strings, a soundboard, and a soundbox with
sound holes, which is generally designed to increase the sound level by using the acousticresonances of the cavity. In the case of the harp, the soundbox and especially the sound holes areprimarily designed to allow access to the strings for their mounting. An experimental modalanalysis, associated to measurements of the acoustic velocity in the holes, shows the importance oftwo particular modes labeled A0 and T1 as it was done for the guitar and the violin. Their modeshapes involve coupled motions of the soundboard’s bending and of the oscillations of the airpistons located in the sound holes. The A0 mode is found above the frequency of the lowestacoustically significant structural mode T1. Thus, the instrument does not really take advantage ofthe soundbox resonance to increase its radiated sound in low frequencies. However, contribution ofmode A0 is clearly visible in the response of the instrument, confirming the importance of thecoupling between the soundboard and the cavity. © 2007 Acoustical Society of America.
/H20851DOI: 10.1121/1.2384842 /H20852
PACS number /H20849s/H20850: 43.75.Gh /H20851NHF /H20852 Pages: 559–567
I. INTRODUCTION
The harp is one of the oldest string instruments. Its evo-
lution from the prehistoric instrument to the modern concertharp led to the elaboration of constitutive elements, whichare designed to efficiently radiate the sound. The modernconcert harp is composed of a set of strings directly con-nected to a long thin flat soundboard attached to a fairly solidsoundbox with several sound holes. These three elements arecoupled in a complex manner and are attached to a base, apillar, and an arm as shown in Fig. 1.
In a string instrument, the mechanical characteristics of
the strings define the note to be played and the soundboard isdesigned as a sound radiator. Unfortunately, this sound radia-tor is not efficient in the low-frequency range when theacoustic wave length is greater than the size of the sound-board. An acoustical resonator, called the soundbox, is gen-erally added in order to increase the sound level. The firstacoustic resonance of the cavity can be used to reinforce thesound radiation of the instrument. This effect is used in thedesign of bass-reflex enclosures. The acoustical resonator hasbeen the subject of many studies
1on the guitar and on the
violin: the acoustic motion inside the cavity interacts withthe motion of the soundboard to produce two coupled modes.The first mode is called the plate mode and is associated with
a strong bending motion of the soundboard. In the case of theguitar
2and of the violin,3this mode is commonly labeled T1.
The second mode is called the Helmholtz mode or A0 airmode and corresponds to a strong motion of an air pistonlocated in the hole. The A0 air mode contributes to a signifi-
cant increase of the sound radiation in the low frequencyrange. In order to well understand this low-frequency behav-ior of the guitar or of the violin, simple discrete models
4–6
have been carried out. The parameters of these models canbe obtained from transfer functions measurements on theseinstruments.
In the case of the harp, the cavity and holes’ sizes and
shapes are not particularly designed to amplify the sound inthe low-frequency range. One of the main reasons for thechoice of sizes and locations of the holes is the facility forstring mounting. The acoustic role of the holes is not wellunderstood because the harp, and especially the soundbox,has not been the subject of many vibroacoustic studies.
The first study
7was carried out on the small harp of
Scotland. Modal analysis has been performed on the sound-board at different steps of its manufacturing. Air resonanceswere also investigated in the soundbox alone by burying itinto sand in order to damp wall vibrations. No evidence ofthe presence of a Helmholtz resonance was found. Moreover,in playing configuration, the relationship between vibrationmodes of the instrument and radiated sound was not investi-gated. This study was later carried out on a Spanish harp ofthe baroque period,
8close in size to the current concert harp.
In this study, it was found that vibroacoustic interactionsbetween soundboard vibrations and the acoustic motions ofthe air cavity lead to two coupled modes /H20849112 and 146 Hz /H20850
having similar shapes and corresponding to A0 and T1modes, respectively. This kind of result was also found on anunstrung Salvi Orchestra Concert Harp
9by using holo-
graphic interferometric analysis of the soundbox. The author
of this last study identifies A0 and T1 modes by measuringa/H20850Electronic mail: jean-loic.le_carrou@univ-lemans.fr
J. Acoust. Soc. Am. 121 /H208491/H20850, January 2007 © 2007 Acoustical Society of America 559 0001-4966/2007/121 /H208491/H20850/559/9/$23.00
Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termsthe changes of the structural response when sound holes are
closed. For the Salvi Orchestra’s soundbox, another study10
has confirmed that if wall vibrations are damped by sand, theHelmholtz mode is clearly present in the acoustic response.A semi-empirical formula was proposed to predict its eigen-frequency. Another conclusion by Bell
9is that one of the two
coupled modes’ /H20849A0 and T1 /H20850presence in the response weak-
ens when the soundboard is stressed by the strings. Thus, fora strung harp, the A0 mode is particularly difficult to identify.The reason for that is not clear. This difficulty was alsopointed out on a Celtic harp.
11
The aim of this paper is to identify the A0 air mode for
the concert harp and to investigate the importance of its con-tribution to the instrument’s response. For this purpose, the
paper is divided into two parts. A study of the response func-tions of the instrument is first performed through the experi-mental modal analysis of the instrument’s body and throughan investigation of the acoustic field in the cavity. Then, theidentification of the A0 and T1 modes is achieved by study-ing a modified instrument.
II. EXPERIMENTAL STUDY OF THE CONCERT HARP
A. Experimental procedure
The vibroacoustic behavior of a concert harp is experi-
mentally investigated. All measurements are performed onanAtlantide Prestige concert harp lent by a French harp
maker, Camac Harps. A schematic diagram is proposed inFig. 2 with the principal dimensions of the instrument. Thesoundbox of the studied concert harp consists of a 6-mm-thick semi-conical shell with a total volume of the enclosedair of 0.029 m
3. On the back of the soundbox, there are five
elliptical sound holes whose dimensions are shown in TableI. The concert harp is studied in playing configuration: allstrings are mounted and tuned. For these measurements,strings are damped with paper to prevent their vibrationwhile keeping the static deformation and load imposed bythem on the soundboard. So, the string modes, including
FIG. 1. /H20849Color online /H20850Experimental setup.
FIG. 2. Schematic diagram with di-
mensions of the Atlantide Prestige
concert harp. The locations of twocharacteristic points 34 and 23 andhole number are also shown.TABLE I. Dimensions of the five elliptical sound holes. The two dimen-
sions correspond to the major axis and minor axis of each ellipse.
No. Major axis /H20849cm /H20850 Minor axis /H20849cm /H20850
1 16.6 4.8
2 17.2 5.63 17.7 74 18.1 85 18.5 9.3
560 J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Le Carrou et al. : Experimental study of the concert harp
Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termssympathetic modes,12are highly damped and are not evident
in the instrument’s response.
The experimental setup is shown in Fig. 1: the instru-
ment is excited by a shaker driven by a white noise con-nected via a rod, through sound hole 4, to the back of thesoundboard. The excitation force Fis measured with an ap-
propriate force sensor directly glued to the back of sound-board. The excitation point is labeled 34, as shown in Fig. 2,and is located between the Ab and the Bb string /H20849respective
fundamental frequencies at 103.8 and 116.5 Hz /H20850attachment
points. The vibratory velocity w˙is measured with a laser
vibrometer. The acoustic velocity Vin the middle of the
sound holes is measured with an intensity probe. The farfield acoustic pressure Pis measured with microphones
placed around the concert harp. Frequency response func-tions /H20849FRFs /H20850H=w˙/F,H
V=V/F, and HP=P/Fare then com-
puted by a standard analyzer.
B. Experimental modal analysis of the instrument’s
body
The identification of structural modes of the soundbox in
the low-frequency range is carried out by modal testing:eigenfrequencies, mode shapes, and damping parameters canbe extracted from response functions measured at differentpoints of the structure. The experimental mesh is composedof 60 points on the soundboard and of 18 points on thecurved surface at the back of the harp, as shown on eachmodal shape in Fig. 4. The laser vibrometer is adjusted tomeasure the normal velocity on the soundboard. For eachpoint on the curved surface, both the velocity along the zaxis
and along the xaxis, defined in Fig. 1, are measured. Mea-
surements are performed at each mesh point in the frequencyrange 0–300 Hz. A typical example of the measured fre-quency response functions is shown in Fig. 3.The modal identification is carried out using the least
square complex exponential method
13implemented in the
LMS software. Only six consecutive modes in the frequencyrange 24–181 Hz are identified because of the high modaldensity above 181 Hz as shown by the typical measurementat point 23 /H20849H
23=w˙23/F/H20850in Fig. 3. In this figure the synthe-
sized response function /H20849Hˆ23/H20850and the least square error /H9280,
defined by
/H9280=/H20841Hˆ23−H23/H208412
/H20841Hˆ23/H208412, /H208491/H20850
are plotted in order to validate the modal identification. Ac-
cording to this indicator /H9280, a good agreement between the
measurement and the model can be found. Parameters ob-tained from this modal analysis are shown in Fig. 4. Thefollowing conclusions can be drawn for each identifiedmode.
14
/H20849i/H20850 Mode 1 has no nodes on its mode shape: the modal
displacement is close to a global motion of the bodydepending on its connections to the arm and to thebottom of the pillar.
/H20849ii/H20850 Modes 2 and 3 have common characteristics: The
axial profiles of soundbox’s displacements are similarto the first two mode shapes of a simply supportedfree beam. Note that as for mode 1, the shapes ofmodes 2 and 3 do not induce a change in the volumeof the cavity: a weak coupling of these modes withthe fluid inside the cavity can be expected.
/H20849iii/H20850Modes 4 and 6 have very similar mode shapes. The
soundboard’s displacement field corresponds to thefirst bending mode of a quasi-clamped plate. A slight
FIG. 3. Measured FRF w˙23/F, synthe-
sized FRF w˙23/F, and least square er-
ror/H9280are shown versus frequency /H20849Ref.
1 dB: 5 /H1100310−8ms−1N−1/H20850. The grayed
area corresponds to the frequencyrange in which modes have been iden-tified. Numbers associated to verticallines indicate the modal frequenciesgiven in Fig. 4.
J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Le Carrou et al. : Experimental study of the concert harp 561
Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termsbreathing motion of the soundbox is also observed.
Shapes of modes 4 and 6 lead to an important changein the volume of the cavity.
/H20849iv/H20850Mode 5 is a pitch mode. In the measured response
functions, this mode is not clearly present. It is actu-ally not well excited since the shaker is connectedclose to the central line of the soundboard, which ex-actly corresponds to its nodal line. Since the stringsare also attached on this nodal line, the role of thismode is not important when the instrument is played.
For this reason, it will not be considered afterwards.
The two modes 4 and 6, which have similar shapes, have
also been found on an unstrung concert harp
9and on a strung
Spanish harp.8However, when the harp is strung it seems
difficult9to extract these two similar mode shapes. Neverthe-
less, in our study, these two modes were found in the playingconfiguration. Moreover, it should be noticed that the dis-
FIG. 4. Eigenfrequencies, damping
coefficients and mode shapes of iden-tified modes.
562 J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Le Carrou et al. : Experimental study of the concert harp
Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termsplacements of the soundbox have the same order of magni-
tude as those of the soundboard; this is unexpected becausethe cavity seems to be much more rigid than the soundboard.Such a result was already mentioned for a Celtic harp.
15
C. Analysis of the acoustic response functions of the
instrument
In order to characterize the acoustic field inside the
soundbox, the acoustic velocity in each hole has been mea-sured in the low-frequency range 50–300 Hz, as shown inFig. 5. Measurements are performed using the two micro-phones of an acoustic intensity probe. After an accurate cali-bration of the microphones, the acoustic velocity can becomputed from the pressure measured at two points close toeach other. Each sound hole can be described as an air pistonof which the velocity is measured. It is found that the lowerthe hole is, the higher the magnitude of the velocity of thepiston will be. Since the level of acoustic velocity for theupper hole 1 is far smaller than that of the four others, itcannot be considered as significant and will be ignored after-wards. So, in the studied frequency range the four other airpistons are found to be in phase below f
A/H20849=250 Hz /H20850and are
no longer above. Thus, these measurements show that the
acoustic field inside the cavity is mostly governed by the firstacoustical mode below f
A=250 Hz. Above this particular fre-
quency, other acoustical modes like longitudinal or pipemodes are present.
The mobility at excitation point 34 of the soundboard
w˙
34/Fis also plotted in Fig. 5. Its phase can be compared
with the Vi/Fphase, Vibeing the acoustic velocity measured
in hole i. It is found that below a second characteristic fre-
quency, fB/H20849=160 Hz /H20850, the soundboard and all air pistons are
in phase. Above this frequency fB, but below fA, the phase
difference between FRF w˙34/Fand FRF Vi/Fincreases from
0° to 180°. This shows that in the frequency range fA–fB, thesoundboard and the air pistons are out-of-phase. These par-
ticular phase relationships are schematically represented inFig. 5 by arrows in harp drawings /H20849a/H20850and /H20849b/H20850. The direction
and length of the arrows that are plotted in these diagramsrepresent the phase and the magnitude of the velocity of thesoundboard and of the air pistons below and above f
B.This
result had already been found on another Camac concert harpin a previous paper
16where the characteristic frequency fB
was found to equal 175 Hz.
In order to find out the implication of the acoustic field
inside the soundbox on the acoustic far field of the instru-ment, we investigate the acoustic pressure around the concertharp. The pressure is measured in an anechoic room by 32microphones regularly placed around the harp on a 2.35 mradius circle at 1.2 m in height. In the frequency range50–220 Hz, the directivity patterns are found to be nondirec-tional, as shown in Fig. 6 for two selected frequencies cor-responding to the eigenfrequencies of modes 4 and 6. Theacoustic transfer function P
C/Fmeasured in front of the harp
/H20849at the point labeled C defined in Fig. 6 /H20850is also shown in Fig.
7. As for afterwards measurements, the shaker used for theexcitation is connected exactly on the central line of thesoundboard and the acoustic effect of the pitch mode is thencanceled out. In Fig. 7, we note that for a same force appliedby the shaker, the acoustic pressure is much more importantin the range 140–230 Hz than in the rest of the studied fre-quency range. Therefore, in a playing configuration, the harpseems to radiate the sound more efficiently in the range140–230 Hz. The first two important peaks of acoustic pres-sure correspond to the eigenfrequencies of modes 4 and 6.Moreover, modes whose eigenfrequencies are above 200 Hzcannot be individually distinguished and their contribution tothe response below 200 Hz is probably not negligible.
To conclude, six structural modes have been identified in
the low-frequency range. Among these six modes, two play
FIG. 5. /H20849Color online /H20850Magnitude and
phase of FRF Vi/F/H20849Vi: acoustic velo-
vity in hole i/H20850and of w˙34/F/H20849w˙34:v e -
locity at point 34 on the soundboard /H20850.
Note that magnitude scales for thesetwo kinds of FRF are different. Num-bers indicate the modal frequenciesgiven in Fig. 4.
J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Le Carrou et al. : Experimental study of the concert harp 563
Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termsan important part in the sound radiated by the concert harp.
They lead to a strong acoustic radiation, associated to a non-directional directivity pattern.
III. IDENTIFICATION OF A0 AND T1 MODES OF THE
CONCERT HARP
A. Frequency response functions of a modified
instrument
Considering only frequency response measurements on
the instrument’s body, modes 4 and 6 have similar modeshapes /H20849see Fig. 4 /H20850. However, the air piston motions are dif-
ferent for these two modes. To identify the nature /H20849A0 or T1 /H20850
of modes 4 and 6, the study of frequency response functionsof a slightly modified instrument is performed. Mobilitiesmeasured at point 34 on the soundboard for three different
configurations are compared with the normal configuration.The amplitude and frequency shifts of peaks are shown inFig. 8 and in Table II.
The first modification consists of closing the sound holes
of the concert harp as shown in Fig. 9 and labeled /H208491/H20850. Those
are closed by using stoppers made with small tar plates. This
configuration prevents all fluid motions inside the soundholes. This modification has heavily affected the instrument.Eigenfrequencies of modes 1, 2, and 3 undergo a shift ofapproximately −2 Hz due to the additional mass loading in-duced by the stoppers. Two additional peaks are seen below200 Hz and in the rest of the frequency range the level islower than in the normal configuration. The peak for mode 6
FIG. 6. Directivity patterns for modes
4 and 6 in two configurations: normaland with all holes closed by stoppers.Points A and B correspond to the bot-tom of the pillar and to the top of thesoundboard respectively. Point C is themeasurement point in front of theharp.
FIG. 7. Frequency response functionat point C defined in Fig. 6. Numbersindicate the modal frequencies givenin Fig. 4.
564 J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Le Carrou et al. : Experimental study of the concert harp
Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termsis no longer distinct. This result is also confirmed by the
acoustic pressure measurement in front of the harp as shownin Fig. 7 when the sound holes are closed. This observationcan be interpreted by the fact that the resonance of the opencavity does not exist anymore /H20849see Sec. II B /H20850.
The second modification /H208492/H20850consists of inserting a
2-cm-high chimney in the lower hole of the harp as shown inFig. 9. This change induces an increase of the mass of thefirst air piston. Only two structural modes are affected:modes 4 and 6. This proves that these two modes are coupledto the fluid inside the cavity. The other modes are weaklycoupled to the air cavity and do not participate in the acous-tic response function as shown in Fig. 7.
The third modification /H208493/H20850consists of adding a mass
/H20849m=200 g /H20850on both sides of the central line of the sound-board as shown in Fig. 9. All eigenfrequencies of structural
modes are lowered but modes 4 and 6 more than modes 1, 2,and 3. This is probably due to the fact that the mass is lo-cated on the maximum displacement area of these twomodes.
B. Discussion
The most important effects of the modifications /H208491/H20850,/H208492/H20850,
and /H208493/H20850on modes 4 and 6 can be summarized as follows: on
one hand, when sound holes are closed, mode 6 disappears.On the other hand, when the mass of the soundboard is in-creased, the eigenfrequency of mode 4 is lowered whereasthe eigenfrequency of mode 6 is nearly stable. When themass of the air pistons is increased, the eigenfrequency ofmode 6 is lowered whereas the eigenfrequency of mode 4undergoes smaller modifications. By considering these ex-perimental results, it can be concluded that modes 4 and 6involve a coupling between the bending motion of the sound-board mode and the oscillation of the air piston. These twomodes can respectively be labeled, with the common nota-tion, T1 and A0.
The fact that the A0 mode is present in the instrument’s
response clearly depends on the modal density and on thedamping coefficients of the acoustical and structural modes.For some configurations, these parameters are such that thecontribution of the A0 mode can be a minor one.
17In our
configuration, although sound holes are designed to ease the
FIG. 8. Magnitude of the FRF w˙34/F
on the soundboard depending on fourconfigurations: normal, mass added onthe soundboard, chimney inserted inthe lower hole, and with all holesclosed. Numbers associated to verticallines indicate the modal frequenciesgiven in Fig. 4.
TABLE II. Resonance frequencies for the first six modes according to four
configurations of the instrument: normal /H20849f/H20850, mass loaded on the soundboard
/H20849fm/H20850, chimney inserted in the lower hole /H20849fc/H20850, and holes closed /H20849fcl/H20850.
Resonance frequencies /H20849Hz /H20850 Deviations /H20849Hz /H20850
Modes ffm fc fcl f−fm f−fc f−fcl
1 61.5 60 61.5 60 1.5 0 1.5
2 84.5 82 84.5 81.5 2.5 0 33 124.5 123.5 124.5 120.5 1 0 44 153.5 148.5 152 156 5 1.5 2.56 172 168.5 169 … 3.5 3 …
J. Acoust. Soc. Am., Vol. 121, No. 1, January 2007 Le Carrou et al. : Experimental study of the concert harp 565
Downloaded 11 Oct 2013 to 128.197.27.9. Redistribution subject to ASA license or copyright; see http://asadl.org/termsstring mountings, they are found to have a significant influ-
ence on the vibroacoustic response of the concert harp. Thisis confirmed by the measurements of the far field acousticpressure performed with opened and closed sound holes, asshown in directivity patterns /H20849Fig. 6 /H20850and in the frequency
response functions /H20849Fig. 7 /H20850.
Contrary to the violin and to the guitar, the A0 mode is
above rather than below the frequency of the lowest acous-tically significant structural mode T1. The concert harp doesnot take advantage of the soundbox resonance to increase itssound radiated in low frequencies, below the T1 mode.
Since these two modes, T1 and A0, are dominant in the
low-frequency range, the response of the instrument can beapproximated by using a two degrees of freedom oscillatormodel in which one degree is due to the soundboard and theother one to the fluid inside the cavity as it was done for theguitar.
4
IV. CONCLUSION
This paper deals with the vibroacoustic behavior of a
concert harp in the low-frequency range. The nature of themodes of the soundbox coupled to the internal fluid is inves-tigated.
A classic experimental modal analysis has permitted the
identification of six modes in the frequency range24–181 Hz. Since the modal density increases with the fre-quency, mode identification at higher frequencies was notpossible. Among the six identified modes, four correspond toglobal motions of the soundbox, which do not induce achange in the volume of the cavity and are thus weaklycoupled to the internal acoustic field. These modes, whichmostly depend on the characteristics of the connection of thesoundbox to the arm and to the bottom of the pilar, lead to
the weakening of the acoustic radiation. The two remainingmodes, called T1 and A0, play an important acoustic role andhave the following characteristics. /H208491/H20850They are associated to
coupled motions of the bending vibration of the soundboardand to the oscillations of the air pistons located in the soundholes. They correspond to the first two modes of a Helmholtzresonator with yielding walls. The labels T1 and A0 wereused for the guitar and the violin for which this Helmholtzeffect is known. /H208492/H20850Modes T1 and A0 lead to important
acoustic radiation: the acoustic pressure radiated by the harptakes high values in the range 140–230 Hz and the first twopeaks of the pressure amplitude correspond to the resonancefrequencies of T1 and A0. /H208493/H20850The mode shapes of T1 and A0
are such that the displacement of the air pistons located inthe five holes are all in phase. For T1, the displacements ofthese pistons are approximately in phase with the bendingdisplacement of the soundboard. For A0, these motions areapproximately out of phase. /H208494/H20850Contrary to the violin and to
the guitar, the A0 mode is above rather than below the fre-quency of the lowest acoustically significant structural modeT1. Thus, the concert harp does not take advantage of thesoundbox resonance to increase its sound radiated in lowfrequencies. However, the study reveals the importance ofthe contribution of mode A0 in the response of the instru-ment, confirming the importance of the coupling between thesoundboard and the cavity. This result is valid for the studiedharp: Atlantide Prestige concert harp. Future works may con-
cern others harps with different characteristics on which theeigenfrequencies of modes A0 and T1 depend: cavity vol-ume, sound holes sizes, and soundboard material.
ACKNOWLEDGMENTS
The authors acknowledge financial support from the
CNRS and the Région des Pays de la Loire for Jean-Loïc LeCarrou’s PhD scholarship. They also acknowledge the instru-ment maker CAMAC Harps for the lending of the concertharp and the CTTM /H20849Le Mans Centre for Technology Trans-
fer/H20850where measurements were performed.
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/H208493/H20850with mass added on the soundboard.
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