Current induced torques and interfacial spin-orbit coupling: Semiclassical Modeling

0.66 MB·English

by  Paul M. Haney

Current induced torques and interfacial spin-orbit coupling: Semiclassical Modeling
Paul M. Haney,1 Hyun-Woo Lee,2 Kyung-Jin Lee,3,4,1,5 Aur´elien Manchon,6 and M. D. Stiles1
1Center for Nanoscale Science and Technology, National Institute of
Standards and Technology, Gaithersburg, Maryland 20899-6202, USA
2PCTP and Department of Physics, Pohang University of Science and Technology, Kyungbuk 790-784, Korea
3Korea University, Department of Material Science & Engineerin, Seoul 136701, South Korea
4Korea Institute of Science and Technology, Seoul 136-791, Korea
5Univeristy of Maryland, Maryland Nanocenter, College Pk, MD 20742 USA
6Core Labs, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia
3
1 In bilayer nanowires consisting of a ferromagnetic layer and a non-magnetic layer with strong
0 spin-orbit coupling, currents create torques on the magnetization beyond those found in simple
2 ferromagnetic nanowires. The resulting magnetic dynamics appear to require torques that can be

n separated into two terms, damping-like and field-like. The damping-like torqueis typically derived
a from models describing the bulk spin Hall effect and the spin transfer torque, and the field-like
J torque is typically derived from a Rashba model describing interfacial spin-orbit coupling. We
8 deriveamodelbasedontheBoltzmannequationthatunifiestheseapproaches. Wealsoconsideran
1 approximation to the Boltzmann equation, the drift-diffusion model, that qualitatively reproduces
thebehavior,butquantitativelyfailstoreproducetheresults. WeshowthattheBoltzmannequation

] withphysicallyreasonableparameterscanmatchthetorquesforanyparticularsample,butinsome
ci cases, it fails to describe theexperimentally observed thickness dependences.
s
-
l
r I. INTRODUCTION in the material with strong spin-orbit coupling and the
mt mixingconductance. Theotherpicture21–24 assumestwo
dimensionaltransportthatcanbedescribedbyaRashba
. Spintronic applications like spin-transfer-torque mag-
t model, similar to those used to describe spin-orbit cou-
a neticrandomaccessmemory(STT-MRAM)ormagnetic
m domain wall-based devices require advances in materials pling in two-dimensional electron gases.25 The Rashba
model gives direct coupling between the magnetization
- to reachtheir full potential. The goalofimprovingthese
d materials has led to the study of bilayers consisting of and the flowing current. Both models give qualitatively
n similar results,that is torques alongthe M (j ˆz) and
ferromagneticlayersandnon-magneticlayerswithstrong × ×
o M [M (j zˆ)] directions, where M is the magneti-
spin orbit coupling. Recent measurements on such sys-
c zati×on, j i×s the×in-planecurrentdensity andthe interface
[ tems have demonstrated efficient switching of magnetic
tunnel junctions,1 like those used in STT-MRAM, and normal is in the zˆ direction. We refer to the first torque
1 efficient current-drivendomain wall motion.2 asafield-liketorquebecauseithasthesameformaspre-
v cessional torque around an effective field in the j ˆz
3 There are a number of physical processes3 in these direction. The second torque has the same form− a×s a
1 systems that contribute to the magnetization dynam-
dampingtorquetowardafieldinthatsamedirectionand
5 icsasdescribedbytheLandau-Lifshitz-Gilbertequation.
we refer to it as a damping-like torque.26
4 These include the typical micromagnetic contributions,
1. like interatomic exchange, magnetostatic interactions, Both models have strengths and weaknesses. The
0 magnetocrystalline anisotropy, and damping, as well as Rashba model treats the strong spin-orbit coupling at
3 theadiabaticandnon-adiabaticspintransfertorques4–11 theinterfacesbetweenthematerialsbuttreatsthetrans-
1 that are typically added to account for the coupling be- port as two-dimensional. The layer thicknesses are usu-
v: tween the magnetization and the electrical current flow- ally comparable to mean free paths and spin diffusion
i ing through it. In the bilayers of interest here, there lengths requiring a three dimensional description of the
X
areadditionalcontributionsthat havereceivedextensive transport. On the other hand, the spin-Hall-effect spin-
r attention. These arise from the spin-orbit coupling in transfer-torque model treats the three dimensional as-
a
thenon-magneticlayerandfromtheenhancedspin-orbit pectofthetransport,butignoresanycontributionsfrom
coupling at the interfaces between layers. the modification of the spin-orbit coupling near the in-
terface. The non-magnetic layer and the magnetic layer
These additional contributions have been modeled in
terms of two different pictures. One picture12 assumes affect the electronic structure of each other close to the
interfaceandthe interactioncansignificantlychangethe
that the layers are thick and the two layers have their
spin-orbit coupling there. In particular, the proximity
bulk properties. A current flowing through the non-
to the ferromagnetcan induce a moment in the material
magnetic layer with strong spin-orbit coupling generates
with strongspin-orbitcoupling giving a thin layerwhere
a spin current perpendicular to the interface (the spin
Halleffect.13–16)Whenthis spincurrentimpingesonthe the magnetism and the spin-orbit coupling coexist.27
interface,thereisaspintransfertorque17–20 onthemag- Attempts to develop predictive models face the com-
netizationofthemagneticlayer. Thedetailsofthetorque plicationthattheexperimentalstructuresdeviatesignifi-
inthispicturearedeterminedbythebulkspinHallangle cantlyfromtheidealstructurestreatedtheoretically. Ex-
2
perimentalindications28,29thatinterfacesofCogrownon FM
Pt have different properties than interfaces of Pt grown
on Co argue strongly that the details of the interface
NM +z
structure are both nontrivial and important. Unfortu- k +y, αRPHR
nately, the interfaces are not well enough characterized
+x, j
to know what types of disorder might be present. There
may be significant interdiffusion at the interfaces be-
cause, for example, Pt alloyswith Co in the bulk. There
isalsosignificantlatticemismatchbetweenthematerials.
FIG. 1: (color online) Schematic of a bilayer structure that
This mismatchcouldpromotethickness fluctuations and
consists of a ferromagnetic layer and a non-magnetic metal
dislocation formation. Without measurements of atomic layer with strong spin-orbit coupling. The spin Hall effect
scale structure of the experimental samples, it is impos- in the non-magnetic layer bends electron trajectories and in-
sible to know how important such defects are to the be- jects electrons with proper spin direction into the adjacent
havior of the system. magnetic layer, thereby generating the damping like torque.
Motivated by the uncertainty in the details of the ex- In this illustration, the spin Hall angle for the non-magnetic
perimental structures and the goal of incorporating the layerisassumedtobepositive. Thefigurealsohighlightsthe
interface region between the two layers, where the Rashba
strengthsofexistingmodels,wedevelopsimplesemiclas-
spin-orbit coupling may havesizable magnitude.
sical models for these systems. One approach is based
on the Boltzmann equation and the other on the drift-
diffusion equation. Both capture the essential physics of
cappinglayer,seeFig.1,althoughthedetailsofeachdif-
the models thathavebeen usedso farandprovidea test
fer from experiment to experiment. The authors of the
for whether a model based on bulk properties and en-
first set of experiments interpret their results in terms
hanced spin-orbit coupling at the interface can account
of a dominant damping like torque that they attribute
for the experimental behavior. We find that these mod-
to the spin Hall effect and as subsequent spin transfer
els are general enough to reproduce the torques mea-
torque. Ontheotherhand,the authorsofthe secondset
sured in any single sample for reasonable values of the
of experiments interpret their results in terms of an im-
parameters, but not all samples with a single parame-
portantfieldliketorquethattheyattributetointerfacial
terization. In Sec. II we summarize the experimental
spin orbit coupling.
evidence for the different interpretations. We also sum-
In the case of the spin Hall effect, electron trajecto-
marize the micromagnetic simulations that provide the
riesarepreferentiallyscatteredindifferentdirectionsde-
basis for these interpretations. We briefly present the
pending on their spin directions. For instance, in a non-
semiclassicaltransportmodelsinSec.III andthenapply
magnetic material with positive spin Hall angle, elec-
them in Sec. IV to the model for a bulk spin Hall cur-
trons are scattered more strongly into directions such
rent in the non-magnetic layer leading to a spin transfer
that (v v ) S is positive, where v is the electron
torque at the interface with the ferromagnet. In Sec. V i × f · i(f)
velocity before (after) the scattering and S is the elec-
we add in the interfacial spin orbit coupling and show
tronspin direction. Thus in the bilayersystemin Fig. 1,
thatthis capturesthe importantphysicsthatis included
thespinHalleffectinthenon-magneticlayerinjectselec-
in the Rashba-model approaches.
trons with spin along +yˆ directioninto a magnetic layer
if j is along +xˆ direction and the spin Hall angle is pos-
itive. As it does for perpendicular flow of electrons in
II. EXPERIMENTAL RESULTS AND multilayersystems, this spincurrentcausesspintransfer
THEORETICAL IMPLICATIONS torques,17–20 the injected electrons giving rise to a spin
torque along the direction M (M m), where m is
− × ×
Recent experiments on multi-layer structures report thedirectionofthemagneticmomentcarriedbyinjected
evidence for current-induced torques due to spin-orbit electrons and points along yˆ direction since m and S
−
coupling. These torques are reported to be large enough are anti-parallel. This torque amounts to the damping-
to modify the magnetization dynamics and may be uti- liketorque M [M (j ˆz)]arisingfromthe spinHall
− × × ×
lizedtofacilitatespintronicapplications. Variousexperi- effect.
ments reportconflictingresultsforthe size anddirection Liu et al. examine various magnetization dynam-
of the torque. Many of these values are inferred from ics caused by the spin Hall effect. In Pt/Py bilayer,30
measurements of the resulting dynamics in conjunction they use the spin-torque ferromagnetic resonance tech-
with simulations. This section aims to summarize the nique to show that the spin Hall effect is strong enough
evidence from experiments, in conjunction with simula- to cause magnetic precession. Through the resonance
tions, for different forms of the torque. line shape analysis, they quantify the magnitude of
The recentgrowthofinterestin bilayersystems began the spin-Hall-effect-induceddamping like torque andthe
withaseriesofexperimentsbyLiuetal.1,30–32andMiron torque due to the Oersted field. They estimate the spin
et al.2,33,34 Both sets of experiments treat systems of a Hall angle of Pt to be about +0.076, which is about
substrate, non-magnetic layer, ferromagnetic layer, and two orders of magnitude larger than the correspond-
3
ing value in n-doped GaAs.35,36 This demonstrates that 0.2 eVnm,39 which is comparable to α = 0.3 eVnm
R
the spin Hall effect can be a realistic tool to enhance reported for Bi/Ag(111) surface40 and comparable to
spin torque efficiency. In their subsequent work1 for the a recent first principles calculation result for a Pt/Co
Ta/Co Fe B bilayer,theydemonstratethatthespin- bilayer.41
40 40 20
Hall-induceddamping-like torquecanswitch the magne- Further evidence for a field-like torque is found in
tization in a reliable and efficient way, facilitating the measurements2 ofcurrent-drivendomainwall(DW) mo-
development of magnetic memory and nonvolatile spin tion in the same system. As the driving current density
logic technologies. From three different measurements goesup,thedomainwallvelocityv increasestospeeds
DW
(spin-torqueferromagneticresonance,current-dependent upto 400m/s,morethanthreetimesfasterthanprevi-
anomalousHalleffect,thresholdcurrentdensityformag- ously≈measured current-driven domain wall velocities.42
netization switching), they estimate the spin Hall angle The authors claim that this velocity is twice as large
ofTatobe 0.12to 0.15. Thisangleisofoppositesign as the rate v = j(Pgµ )/(2eM ) of the spin angu-
comparedto−Pt but −mostnotably, largerby abouta fac- lar momentums tran|s|fer, wBhere g(S 2) is the gyromag-
tor of two. In their still later work,31 they demonstrate netic ratio. Even up to such high≈domain wall speeds,
that the magnetizationswitching canbe achievedby the the domain walls apparently did not undergo structural
spin Hall effect in Pt(2 nm)/Co(0.6 nm)/AlOx systems instability (Walker breakdown.43,44) According to con-
as well, although the spin Hall angle in Pt is smaller. ventional theories6,8 of current-driven domain wall mo-
Most recently, they have determined an even larger spin tion, domain wall velocities above v are possible below
s
Hall angle in W.37 the breakdown current density j . However, as the ra-
W
Ontheotherhand,Mironet al.2,33,34 exploreeffectsof tio v /v increases, the breakdown current density de-
DW s
the Rashba spin-orbitcoupling with focus onthe system creases such that conventional theories cannot explain
Pt(3 nm)/Co(0.6 nm)/AlO (2 nm). The layer structure the velocities measured in this experiment.2
x
of the film breaks inversion symmetry and gives rise to As a possible explanation, the authors suggest2 that
perpendicular magnetic anisotropy strong enough that the current-induced effective field H could increase the
R
thegroundstateforthemagnetizationisoutofplane. In breakdown current density. Consider the two low en-
these experiments, the nominally symmetric controlsys- ergy structures [Figs. 2(a) and (b)] of the Bloch domain
temPt(3nm)/Co(0.6nm)/Pt(3nm),showsmuchweaker wall. These walls differ from each other because they
effects,leadingtothe inferencethatthebrokeninversion have opposite magnetization directions at the domain
symmetry for the layer plays a crucial role. wall center. Since the two directions are parallel and
Miron et al interpret their results in terms of a large anti-parallelto H , the effective field either stabilizes or
R
field-liketorqueasexpectedfromtheRashbamodel25 for destabilizes the corresponding Bloch domain wall struc-
interfacial spin-orbit coupling. Calculations21–24 predict tures [Fig. 2(d)]. Recalling that the conventional spin
an effective field38 torques produced by the current tend to shift8 φ away
α from the low energy values (φ = 0 and π), the deeper
µ0HR R P(ˆz j), (1) energyvalleyimpliesanenhancedthresholdcurrentden-
≈ 2µ M ×
B s sitytoescapethevalley. SincetheWalkerbreakdownoc-
when a system is subject to the Rashba spin-orbit cou- curswhenφcannotremainstationary,theWalkerbreak-
plingoftheformα (k zˆ) σ. HereM isthesaturation downthresholdcurrentdensitybecomesthelargerofthe
R s
magnetization, P is th×e spi·n polarization, and µ is the two, which is larger than the Walker breakdown thresh-
B
Bohr magneton. The effective field HR generates the old value without HR.
field-like torque γM H M (ˆz j), where γ is Another interesting feature of the experiment2 is that
R
the gyromagneti−c ratio×. ∝− × × contrarytopreviousexperiments,42,45,46whichreportdo-
In Ref. 33, the authors measure the reversal of the mainwallmotionalongtheelectronflowdirection,Ref.2
perpendicular magnetization. They find that transverse reports domain wall motion against the electron flow.
in-plane magneticfields enhancethe nucleationandthat Within the scope of the conventional theories,6,8 the re-
thisenhancementismodifiedbycurrentsflowingthrough verseddomainwallmotionimpliesnegativePβ,whereP
the film. In-plane currents j flowing through the sys- isthe polarizationofthe currentandβ is the dimension-
tem enhance (suppress) the nucleation of reversed mag- less coefficient of the non-adiabatic spin transfer torque.
netic domains when the direction zˆ j is parallel (anti- Allofthequalitativefeatureoftheexperiment2wouldbe
parallel) to externally applied in-pla×ne magnetic fields. explained if there were an HR that gives Walker break-
In the nominally symmetric control sample, the effect down threshold enhancement, negative Pβ for reversed
of in-plane current is much weaker. The authors inter- motion,and β/α >1forvelocityenhancement(αisthe
| |
pret the effect of in-plane currents as evidence for the Gilbert damping constant).
current-induced effective field along zˆ j, as predicted Further measurements34 on the same Pt(3
×
theoreticallyforsystemswithRashba-likespinorbitcou- nm)/Co(0.6 nm)/AlO (1.6 nm) system reveal a
x
pling. Experimentally,33 the magnitude of the effective problem in this simple theoretical picture. Based on
field is proportional to j with a proportionality constant measurements of bipolar switching in a tilted field,
(1.0 0.1) 10−12 T/(A m−2). Thisvalueimpliesα the authors conclude that the switching is due the
R
± × · ≈
4
y (c)
(a) φ gr 300
e
n
E
200
φ
(b) z j y −π 0 π 2π (d) )s/m 100
gre ( y
nE tic 0
o
−π 0 π 2π φ lev
W -100
D
-200
FIG. 2: (color online) Two possible structures of the Bloch
domain wall. In (a), the magnetization at the center of the
domainwallpointsalongˆz×jandin(b),itpointsalong−ˆz×j. -300
Panel (c) shows the domain wall energy as a function of the
-4 -2 0 2 4
domain wall tilting angle φ without the effective field HR.
j (1012 Am2)
Panel (d) shows the energy with the effective field assuming
that αRP is positive and HR prefers φ=0.
FIG. 3: (color online) Terminal domain wall velocity vDW
as a function of the current density j = jxˆ. The dotted
additional presence of a damping-like torque along linegivestheprediction oftheLandau-Lifshitz-Gilbert equa-
M [M (ˆz j)]. The damping-like torque is either tion when the current generates only the conventional (adi-
par×allel or×anti×-parallel to the nonadiabatic STT at the abatic and nonadiabatic) spin transfer torques and the solid
domain wall center and thus modifies the domain wall line the prediction when the current also generates field-like
velocity.47,48 A recent theoretical study47 demonstrates and damping-like torques. For this calculation, we consider
a nanostrip of length × width × thickness = 2000 nm ×
that high v against electron flow direction is possible
DW 200nm×5nm,withperpendicularmagneticanisotropyand
(Fig.3)evenwhenboth P andβ arepositive andβ/α is
the materials parameters as follows: γ/(2π) = 28 GHz/T,
smaller than1, if both the field-like torque −M×(ˆz×j) Ms = 1 × 106 A/m, Aex = 1.3 × 10−11 J/m, P = 0.7,
and the damping-like torque M × [M × (ˆz × j)] are Ku = 1.5 × 106 J/m3, α = 0.5, β = 0.25, and αR =
sufficiently large. 0.7×10−10 eVm. For the conventional calculation (dotted
Miron et al.34 estimate that the spin Hall curve), the non-linearities are due to Walker breakdown. In
contribution31 to the damping-like torque is not the solid curve, the small blip at a small negative current is
strong enough to explain the bipolar switching and con- caused by the chirality switching of domain wall due to the
spin-orbit-related field-like torque.
clude that the damping-like torque arises mainly from
theRashbaspin-orbitcoupling,basedontheobservation
that the efficiency of the bi-polar switching increases
with the magnetic anisotropyof the cobaltlayerand the ter conclusion, they report that the ratio between the
oxidation of the aluminum layer. Calculations3,47,49,50 current-inducedeffectivefieldalongˆz jandthein-plane
×
confirmthattheRashbaspin-orbitcouplingcangiverise current density is more than 75 times smaller than the
to the damping-like torque. Note however that the spin corresponding ratio in Ref. 33. A third experiment51 on
Hall contribution and the Rashba spin-orbit coupling a similar system Pt(3.0 nm)/Co(0.6 nm)/AlOx(1.8 nm)
contributiontothe damping-liketorquehaveexactlythe found that this ratio is about 3.4 times smaller than the
same structure, making the distinction difficult. In this ratio reported in Ref. 33.
regard, recent micromagnetic calculations47 show that Experiments52 that measure the field-like torque on
the spin Hall effect and the Rashba spin-orbit coupling Ta(1.0 nm)/Co Fe B (1.0 nm)/MgO(2.0 nm) find
40 40 20
contributions give qualitatively different domain wall thattheratiooftheeffectivefieldtothecurrentisabout
motion if there is no field-like torque as might be 23 % of the corresponding ratio in Pt/Co/AlO as re-
x
expected if the spin Hall effect is the dominantsource of ported in Ref. 33. Interestingly, the current-induced ef-
theeffect. Withoutthefield-liketorque,thedomainwall fectivefieldintheTa/CoFeB/MgOsystemisalong ˆz j
− ×
motion at high current densities is along the electron and thus opposite to the corresponding direction in the
flow direction (if βP > 0) whereas with the strong Pt/Co/AlO system if the Ta and MgO layers in the
x
field-like torque, the domain wall motion against the Ta/CoFeB/MgO system are matched with Pt and AlO
x
electron flow is possible even for βP > 0 (solid line in layers in the Pt/Co/AlO system, respectively. It was
x
Fig. 3). also reported that the torque to current ratio decreases
In contrast to Miron et al.’s interpretation,34 Liu by more than an order of magnitude when the thickness
et al.31 conclude that the damping-like torque in of the Co40Fe40B20 layer in the Ta/CoFeB/MgO system
Pt(2.0 nm)/Co(0.6 nm)/Al(1.6 nm) arises mainly from increases slightly from 1.0 nm to 1.2 nm.
the spin Hall effect in Pt and the Rashba spin-orbitcou- The thickness dependence was investigated
pling contribution is negligible. As evidence for the lat- systematically53 for the two wedge systems,
5
Ta(d )/Co Fe B (1 nm)/MgO(2 nm) and ley and Barna´s60 is the simplest model that describes
Ta 20 60 20
Ta(1 nm)/Co Fe B (t )/MgO(2 nm). When current-in-the-plane (CIP) giant-magnetoresistance
20 60 20 CoFeB
the Talayerthickness d changesby 1 nm, the effective (GMR). In the Boltzmann equation, the variables are
Ta
field along the ˆz j direction changes its magnitude by the distribution functions. The distribution function
nearlytwoorders×ofmagnitude. Forsmalld <0.6nm, accounts for electrons moving in all directions even
Ta
the sign of the effective field is opposite to∼that in thoughthetotalcurrentonlypointsinasingledirection.
the larger d regime and agrees with the sign of the This generality allows the approach to describe the flow
Ta
Pt/Co/AlO system.33 The damping-like effective field ofspinsbetweenthelayerseventhoughthecurrentflows
x
along the (ˆz j) M direction is also sensitive to d in the plane of the interfaces.
Ta
× ×
with the sign change at dTa 0.5 nm. The sign change The drift-diffusion approach of Valet and Fert61
≈
of the damping-like effective fields is interpreted as is based on integrating the distribution function in
an evidence of competition between Rashba spin-orbit the Boltzmann equation to derive transport equations
coupling and spin Hall effect.1 that depend on the densities and currents. It has
Measurements of current-driven domain wall motion had wide success describing current-perpendicular-to-
in multi-layer structures containing non-magnetic heavy the-plane(CPP)GMR,butdoesnotdescribeCIPGMR.
metal layers and ferromagnetic layers but without ox- It fails because it does not describe the flow of spin cur-
ide layers29,54 are reported. The motion is also very rents between the layers when the current flows in the
sensitive to layer thicknesses.29 The domain wall veloc- plane of the layers. However, this limitation may be less
ity can be up to almost 1 km/s in certain multi-layer importantin the bilayer systems of interesthere. In ma-
structures with the Ptlayerthickness ofabout 1 nm but terials with strong spin-orbit coupling, like Pt, spin cur-
changes very quickly as the Pt thickness varies. Inter- rentsdoflowperpendiculartothechargecurrentbecause
estingly, very high domain wall speed ( 1 km/s) are of the spin Hall effect62 so that the drift-diffusion ap-
≈
observed only when the domain wall moves against the proachdoes qualitatively describe the physics. However,
electronflow,implyingthattheoriginofthereverseddo- we compare the two approaches below, and show that
mainwallmotionisprobablycorrelatedwiththemecha- the drift-diffusion approach differs quantitatively from
nism behind very high domain wall speed. Several other the Boltzmann equation for similar reasons to its quali-
experiments28,55–57 alsoreportreverseddomainwallmo- tative failure for CIP GMR.
tioninultrathinmulti-layersystemscontainingPtlayers.
Reference63describesthematrixBoltzmannequation
The analysis of domain wall motion described above
we use in this paper. It is a generalization of the model
considersfour currentinducedtorques,the the adiabatic used to describe CIP GMR,60 and is based on a very
andnon-adiabaticspin-transfertorquesandtwotorques,
simplified model for the electronic structure. We treat
damping-like and field-like, that depend on the layer
all Fermi surfaces as spherical and as having the same
structure of the device. The first two torques depend on
Fermi wavevector. This approach ignores the details of
the gradientofthe magnetizationbut are determined by
theFermisurfaces,whichareundoubtedly importantfor
bulkproperties. Thelasttwoareindependentofthegra-
specific systems, particularly when including spin-orbit
dient of the magnetization. Other possibilities that de-
coupling. However, the scattering mechanisms are both
pend on the gradient of the magnetization and the layer
unknown and uncharacterized, so we feel that it is ap-
structure are allowed by symmetry3 and may be impor-
propriate to consider models in which scattering rates
tant for the dynamics. In addition, recent calculations58
and other physical processes are parameterized and the
suggest that a current-independent torque due to the
details of the electronic structure are neglected for sim-
Dzyaloshinskii-Moriyainteraction59 mightprovideanal-
plicity. Byperformingappropriateintegralsoverthedis-
ternate mechanism for stabilizing a moving domain wall
tributionfunction,theBoltzmannequationcanbetrans-
above the nominal Walker-breakdown field. In this pa-
formed into a drift-diffusion equation like that given in
perweonlyconsiderthecurrent-inducedtorquesthatare
Ref. 64. The parameterized processes in the Boltzmann
independent of the gradient of the magnetization.
equation then have a simple connection to those in the
drift-diffusion equation.
One such process that we include through parameter-
III. SEMICLASSICAL MODELS
ization is the spin-dependent conductivity in the ferro-
magnetic layer. We model the spin dependence by using
To explore possible mechanisms for the torques oper- spin-dependent scattering rates. Such spin-dependent
ative in these systems, we develop semiclassical mod- scatteringisphysicallysensible asitis believedtoplay a
els that allow for easy exploration of parameter space. biggerroleinthepolarizationofthecurrentinthesema-
We use a Boltzmann equation approach and the sim- terials than the electronic structure itself. For example,
pler drift-diffusion approach. The Boltzmann equation in Co and Ni, the current is expected to be dominated
is better suited to describe in-plane transport but the by majority carriers,even though there are more minor-
drift-diffusion approach is simpler and provides a useful ity carriers at the Fermi surface. In the drift diffusion
language to describe the physics. model, the spin-dependent scattering leads to a differ-
TheBoltzmannequationapproachdevelopedbyCam- ent conductivity for the majority electrons σ↑ than the
6
minority electrons σ↓. This difference is parameterized continuity equations in the ferromagnet are
in terms of the spin polarization of the current, defined
through P = (σ↑ σ↓)/(σ↑ +σ↓). The drift-diffusion ∇ j = 0 (4)
− ·
transport equations in the ferromagnet are 1 1
Q = (s Mˆ) s
i ij j j
∇ −τ × − τ
j = σ∇µ Pσ∇(Mˆ µs), (2) ex sf
1
¯h − ¯h· Mˆ (s Mˆ) (5)
Qij = 2eMˆjPσ∇iµ− 2eσ∇iµsj, (3) −τdp h × × ij
wherethespinaccumulationsisproportionaltothespin
wherejisthechargecurrentdensity,andQisthetensor
chemical potential s = µs and the precession time
spin current density where the first index is the spatial Ns
τ = ¯h/∆ is related to the exchange splitting between
component and the second index the spin component. ex
the magnetization and the spin accumulation. Repeated
µ is the electrochemical potential, such that negatively
indices are summed over. The first term on the right
charged electrons diffuse against the gradient giving an
hand side of Eq. (5) is the precession in the exchange
overallpositivesignforthefirstterminEq.(2). Similarly
field, the second term is the spin flip scattering that re-
µs is the spin chemicalpotential,which is a vectoralong
duces all components of the spin accumulation, and the
the direction ofthe spin accumulation,and the unit vec-
last term is the dephasing that reduces only the parts of
tor Mˆ is the direction of the magnetization. The minus
the spin accumulation transverse to the magnetization.
signinthesecondtermofEq.(2)arisesbecausemajority
Settingthetransversespinaccumulationtozero,asdone
electronspins are alignedopposite to the magnetization.
inearlierBoltzmannequationcalculations63 andinmag-
These two signs are typical of possible sources of confu-
netoelectroniccircuittheory,67isequivalenttotakingthe
sion in this subject matter. They arise because with the
limit that the dephasing time goes to zero.
charge on the electron is negative and angular momenta
Both the size of the spin Hall effect and its underly-
and moments are in opposite directions.
ingmechanismarecontroversial. Thetheoryforthespin
The equality of the Fermi surfaces would also allow
HalleffectisrelatedtothatfortheanomalousHalleffect,
for perfect transmission of electrons across the interface asubjectthathasbeencontroversialfordecades.69 Mea-
between the materials. In the Boltzmann equation, we
surements of the spin Hall angle (the ratio of spin Hall
include spin-dependent reflection by the addition of a
and charge conductivities) for various materials span a
spin-dependent sheet potential (delta function) at the
rangeofvalues. Partofthevariationmayresultfromthe
interface. Choosing the strength of this delta function
sensitivity of the extraction of the spin Hall angle from
allows us to tune the spin-dependent interface resistance
experimental data to other material parameters needed
to any arbitrary value.66 In the drift-diffusion approach, to model the experiments.32 Measurements1 show that
the spin-dependent reflection becomes a spin-dependent
thespinHalleffectinTaisbiggerandoftheoppositesign
interfaceresistanceorconductance asused inthe closely of that in Pt, in agreement with previous calculations.70
related circuit theory.67
Theagreementbetweenthesetrendsintheoryandexper-
While the overallstructure ofthe Boltzmannequation iment argues for an intrinsic origin of the effect. How-
approachis the same as that published earlier, there are ever,thecalculatedspinHallconductivityforPtappears
some modifications. One difference is the treatment of to be approximately an order of magnitude too small in
dephasing. In a ferromagnet, spins on different parts of comparison to the measured value. Spin Hall angles of
the Fermi surface precess at different rates and travel approximately the right order of magnitude have been
with different velocities. These differences, combined computed71 forthe extrinsiccontributionsofvariousim-
with scattering between different parts of the Fermi sur- purities in Cu and Au.
face, cause the precessing spins to rapidly become out of With this uncertainty in the mechanism for the spin
phase with each other.68 In Ref. 63, the transverse spin Hall effect, we use the form of scattering appropriate for
accumulation and current are forced to zero in the fer- the extrinsic skew scattering contribution for computa-
romagnetto accountfor this dephasing of the transverse tionalsimplicity. In the Boltzmannequation, we include
spinpopulation. Here,weallowfortransversespinaccu- skew scattering as described by Engel et al.36 but gen-
mulation in the ferromagnetbut build in rapid spin pre- eralize their results to include scattering that leads to
cessionandexplicitlyaccountfortheprocessesthatcause the inverse spin Hall effect in addition to the scattering
dephasing. The simplified model of the Fermi surfaces that gives rise to the spin Hall effect. These scattering
that we use can lead to underestimation of dephasing termsconnectthe currentwithaperpendicularspincur-
processes. We have tested this approximationby adding rent and vice versa. Both our approach and the earlier
anexplicit dephasing term. While sucha termquantita- work36 neglect the scattering processes that couple spin
tivelychangesthe spinaccumulationinthe ferromagnet, currentstospincurrentsmovinginotherdirections. Such
we find that it does not change the calculated torques. process contribute to spin relaxation, which we include
The same dephasing process is absent in a drift- as a phenomenological spin flip scattering process. De-
diffusionmodelbutcanbeincludedbyaddinganexplicit tails of the scattering and the default parameterswe use
dephasing term. In this approximation, the steady-state are given in Appendix A.
7
Inthenon-magneticmaterial,theexplicitformsofthe where the interface is in the ˆz direction at z = 0, u
0
charge and spin currents in the drift-diffusion approxi- is the spin-independent part of the potential, u is the
ex
mation we use are64 spin-dependent part of the potential that gives rise to
spin-dependentreflection,u istheRashbacontribution,
R
j = σ∇µ σSH(∇ µs), (6) with k being the wave vector of an electron scattering
− ×
¯h ¯h from the interface, k is the Fermi wave vector, and m
Q = σ µs σ ǫ µ. (7) F
ij −2e ∇i j − 2e SH ijk∇k is the electron mass. This additional term captures the
form of spin-orbit coupling that is allowed for the sim-
where σ is the conductivity, σSH is the spin Hall con- ple electronic structure assumed here. For more realistic
ductivity coupling the spin and charge currents to the band structures, the form would be much more compli-
charge and spin potentials, and ǫijk is the Levi-Civita cated. Unfortunately, it is difficult to compare uR with
symbol. As with the Boltzmann equation, we neglect a the α used in previous publications. Doing so requires
R
term corresponding to the spin Hall effect coupling the a procedure for reducing the Hamiltonian for a three di-
spin current to the spin potential, assuming that it is mensional system to one for a two-dimensional system.
small.
In Eq. (10), the last two terms can be combined to
In both models, the torque on the magnetization is give a wave vector dependent field direction uˆ(k) and
given by the torque between the magnetization and the strengthu (k)suchthatu (k)uˆ(k)=u mˆ +u kˆ zˆ.
eff eff ex R
spin accumulation Withrespecttothisdirection,themajorityandmino×rity
transmission and reflection amplitudes are
γ γ
T= M s+ M (M s), (8)
τexMs × τdpMs × × ikz/kF
T = (11)
ik /k (u u )
’where the gyromagneticratioγ =gµB/¯hconvertsfrom z F− 0± eff
u u
angular momentum (spin density) to magnetization (so R = 0± eff (12)
T is a term in the Landau-Lifshitz-Gilbert equation26). ikz/kF (u0 ueff)
− ±
The second term, which is not presentin the Boltzmann
Since both the magnitude and phase of the transmission
equationcalculations,capturesthe torquedue tothe de-
and reflection amplitudes are different for the majority
phasing of the electron spins as they precess in the ex-
and minority spin components, an electron spin oriented
change field.
alongsomearbitrarydirectionundergoesafiniterotation
If there is no coupling of angular momentum into the when transmitted or reflected. A part of the torque on
lattice (spin-orbit coupling or spin-flip scattering) it is the electron spin is due to the coupling to the exchange
straightforward to relate this torque, Eq. (8), to the di- field and a part due to the spin-orbit coupling (Rashba
vergence of the spin current. In the ferromagnet, where contribution). The reactiontorqueonthe magnetization
there is spin-flip scattering but no other spin-orbit cou- can be computed from the exchange coupling between
pling, the corrections to torque being simply the diver- the spin density at the interface and the exchange field
gence of the spin currentcanbe found fromEq. (5). For
the componentsperpendicular( )tothemagnetization, γ ¯hk u
⊥ T=δ(z) F ex s M (13)
the torque is M m ×
s (cid:18) (cid:19)
T = γ(1 β)(∇ Q) +γβMˆ (∇ Q), where the spin density s is calculated from the incoming
− − · ⊥ × ·
τ 1 τ τ wave function and the transmission amplitudes. Since
ex ex ex
β = , ξ = + , (9)
τ 1+ξ2 τ τ the potential is proportional to a delta function, the
sf dp sf
torquedensitydivergesbutisfinitewhenintegratedover
where(∇ Q) = Mˆ [Mˆ (∇Q)]isthecomponentof a finite thickness.
· ⊥ − × × ·
thedivergenceofthespincurrentthatisperpendicularto Thetreatmentweusefortheinterfacialspin-orbitcou-
the magnetization. Equation (9) indicates that the spin plinginthe Boltzmannequationdoes notgeneralizeeas-
torqueisingeneralnotsimplygivenbythedivergenceof ily to the drift-diffusion equation because there are not
the spin current but possesses an additional component any wave vector dependent quantities in that model. It
Mˆ ∇ Q arising from the presence of spin relaxation. maybe possibleto define ageneralizationofthe conduc-
× ·
For the parameter set used here, β is negligible. tance matrix used in the magnetoelectronic circuit the-
ory. In typical usage, the longitudinal spin components
An important difference with the previously
coupletoeachotherandthe transversespincomponents
published63 formalism for the Boltzmann equation
coupletoeachother,butthe longitudinalandtransverse
is the inclusion of spin-orbit coupling at the interface.
spin components do not couple. With the Rashba inter-
This is done by including an additional term in the
action included, all spin components would couple.
interface potential
The Boltzmann equation based approach differs quite
¯h2k significantlyfromtheapproachusedinwhichthesystem
V(r)= Fδ(z) u0+uexσ mˆ +uRσ (kˆ ˆz) ,(10) ismodeledwithatwodimensionalRashbamodel. Inour
m · · ×
h i
8

Boltzmann equation approach, electron spins get kicked 4 Q s
whenthey passthroughthe interface,butthey spend no zy y
Q s
time “in” the interface. In the Rashba model, the entire zx x
)
system is the interface so the electrons (and spins) are m j
n x
“in” the interface at all times. In this case, there is a (
spin accumulation that builds up in the interface. This z Qxz
spinaccumulationgivesrisetothestrongfield-liketorque -6 a b c
found in these models. In spite of this difference, we
4
find that the both approaches give qualitatively similar
torques.
)
Theremainingdifferencewiththepreviouslypublished m
formalismfortheBoltzmannequationisthatthebound- n
(

aryconditionsaredifferent. Thepreviousversiontreated z
perpendiculartransportandherewetreatin-planetrans- d e f
-6
port. Here, the outer boundariesare perfectly reflecting,

either diffusely, specularly, or somewhere in between. 4
×10 ×5
)
m
IV. SPIN HALL EFFECT PLUS SPIN n
(
TRANSFER TORQUE z
-6 g h i
In this Section, we describe the behavior of the model
0 5 -0.1 0.0 -0.2 0.0
in the absence of spin-orbit coupling at the interface. In
Scaled currents and spin densities
this limit, the spin Hall effect in the non-magnetic layer
generatesaspin Hallcurrentthatpropagatesperpendic-
ular to the interface with spins pointed perpendicular to FIG. 4: (color online) Currents, spin currents, and spin ac-
both the interface normal and the direction of the cur- cumulations. The left panels (a, d, and g) show the current
rent. When this spin current hits the interface with the density (heavy lines) jx, which is flowing in the plane of the
ferromagnet, angular momentum is transfered from the sample and the spin current Qxz (lighter lines), flowing in
the x-direction with spins aligned with the magnetization in
flowing spins to the magnetization as is typical for spin
transfer torques in magnetic multilayers.17,18,20 the z-direction. The dotted lines indicate the bulk values.
Allcurrentsandspincurrentsaredimensionless; currentsare
This process is shown in Fig. 4 based on calculations
scaled by the bulk current in the non-magnet and spin cur-
done with the Boltzmann equation described in Sec III.
rentsarescaledbythebulkcurrentinthenon-magnetandan
Parameter choices are given in Table I in Appendix A. additionalfactorofh¯/2e. Thespindensitiesarescaledbythe
Fig. 4 shows the currents, spin currents, and spin den- samefactortwofactorsandvF. Themiddlepanels(b,e,and
sities for a 4 nm ferromagnetic layer coupled to a 6 nm h)showthespincurrents,Qzy (heavylines)andQzx (lighter
non-magnetic layer with the interface at z = 0. Panel lines), flowing perpendicular to the layers (z-direction) with
(a) shows the distribution through the thickness of the spins pointing perpendicular to the magnetization, i.e. the
films of the current flowing in the plane of the film (in x- and y-directions. The right panels (c, f, and i) show the
the x-direction). The current is greater in the ferromag- accumulation of spin perpendicular to the magnetization, sy
netic layer because it has a higher conductivity than the (heavy lines) and sx (lighter lines). The top panels (a, b,
c) are for the case in which there is no interfacial spin orbit
non-magnetic layer for this choice of parameters. The
coupling, the bottom panels (g, h, i) for the case with inter-
current is suppressed close to the outer boundaries be-
facial spin-orbit coupling uR = 0.04 and no spin Hall effect
causeweassumethatthescatteringfromthoseinterfaces in the non-magnet, and the middle panels (d, e, f) for the
is completely diffuse. In fact, the ferromagnetic layer is case when both are present. In panels (h) and (i), the spin
not thick compared to the mean free paths, so the cur- accumulations havebeen scaled bytheindicated factors.
rentis suppressedthroughthe thicknessofthe film. The
spin current with spins aligned with the magnetization
(z-direction) and moving in the plane also reduced from discussed below.
the bulk value, in fact more so than the current, so the Panel (b) of Fig. 4 shows the two components of the
polarizationofthecurrentisreducedfromthebulkvalue. spincurrentwithspinsalignedperpendiculartothemag-
At the interface between the two materials, the current netization and moving perpendicular to the plane of the
is enhanced in the lower conductivity layer due to elec- film. In the non-magnetic layer, this is due to the spin
trons entering from the higher conductivity layer, and Hall effect. The spin current is zero at the lower bound-
the current is reduced in the higher conductivity layer. ary,whichisbothimpenetrableandhasnospin-flipscat-
This modification of the current near the interface is tering. It increasesto closeto its bulk value atthe inter-
not captured by a drift-diffusion model. It is one source face between the non-magnet and the ferromagnet. In-
of the quantitative disagreement between the models as sidethe non-magneticlayer,thespincurrentis acompe-
9
1.5 0.0
τ
f
y
sit
n
aled current de 1.0 tFM= 4 nm tFM= 0.6tF nMm= 40 nm caled torque λFM = 5.5 nm
Sc s τ λFM = 11 nm
d
0.5 λ = 22 nm
-0.05 FM
0 5 10 15 20 0 5 10 15 20
t
NM t
NM
FIG. 5: (color online) Average current density in the non-
FIG. 6: (color online) Torques as a function of nonmagnetic
magnetic layer. For three thicknesses of the ferromagnetic
layerthickness. ThesolidcurvesarethefullBoltzmannequa-
layer, the average current density in the non-magnetic layer
tion calculation and the dashed curves give the analytic ap-
isshownscaledbythebulkvalueasafunctionofthethickness
proximation based on the drift-diffusion model and the cir-
of the non-magnetic layer.
cuittheory(AppendixB).Themorenegativecurvesshowthe
field-liketorqueτf andthoseclosertozeroshowthedamping-
like torqueτd. For both torques, the Boltzmann results have
tition between the spin Hall current and a diffusive spin been calculated for three different mean free paths (labelled
current from the spin accumulation, seen in Panel (c), on the damping-liketorques) in the ferromagnet.
that builds up due to the impenetrability of the outer
interface. Panels (d-i) show calculations with interfacial
spin-orbitcoupling included andare discussedin Sec. V. thespintransfertorqueisdeterminedsolelybythetrans-
verse spin current,68 just outside the magnetic layer.
Atandneartheinterface,thisspincurrentisconverted
Since neither the majority transmission probability is
into aspintransfer torqueonthe ferromagnet. We write
zero nor the minority reflection probability is one, some
the interfacial torque in the form
of the transverse spin current is reflected. The reflected
gµ j spin current is seen in the reduction of the transverse
T = δ(z) B 0 τ Mˆ (Mˆ yˆ)+τ Mˆ yˆ ,(14)
2e d × × f × spincurrentclosetotheinterface. Someofthetransverse
h i spin current is absorbed right at the interface and some
where δ(z) localizes the torque to the interface at z = is transmitted into the ferromagnet. In the ferromagnet,
0. The dimensionless coefficients τd and τf characterize spincomponentstransversetothemagnetizationrapidly
the “damping-like”and“field-like”contributionsrespec- precess as they traverse the layer, as seen in the oscilla-
tively. Othertermsarepossible,aswediscussinRef.27, tions in panel (b). Further, different parts of the Fermi
but in the three-dimensional transport calculations we surface precess at different rates so they dephase as the
findtheseothertermstobenegligiblefortheparameters traverse the layer as can be seen by the decay of the
we consider. transverse spin current in the ferromagnet in panel (b).
The prefactor in Eq.(14) is based on j0, which is the The dominant spin transfer torque arises from the ab-
“bulk” current density in the non-magnetic layer, that sorption of the incident transverse spin current either at
is j0 = σNE where E is the applied electric field. This the interface or in the ferromagnet. However, not all
choice seems to be that typically made in analyses of of the current is absorbed, and some is rotated into the
experiments even though the total current is all that x-component of the spin current on reflection. The non-
is directly measurable. The rest of the factors convert zero reflection reduces the damping like torque and the
from current density to magnetization torque density, rotation gives rise to a small field-like torque. These
M˙ . This choice makes sense in analyzing experiments in torques are shown in Fig. 6 as a function of the thick-
terms of the spin Halleffect because the torque is driven ness of the non-magnetic layer.
by the current density in the non-magnet. However, for Figure 6 shows the damping-like and field-like torques
thin films, there canbe important correctionsdue to the calculatedwithboththeBoltzmannequationapproach65
outer boundaries and the interface with the ferromag- and the drift-diffusion approach. The drift-diffusion ap-
net. These corrections are shown in Fig. 5 for a variety proachgivesananalytic result,Eq.(B2)in Appendix B.
of thicknesses for the two layers. The average current That result is based on the drift-diffusion model de-
density is reduced by the diffuse scattering assumed at scribedinSec.IIIandmagnetoelectroniccircuittheory67
the outer boundary of the layer, but is increased by the for transport across the interface. Both approaches give
(assumed)higherconductivityofthe ferromagneticlayer the same behavior as a function of the thickness of the
when that layer is thick enough. non-magneticlayer. Becausethe spinHallcurrentinthe
Forthis model, withnointerfacialspin-orbitcoupling, non-magnetic layer is suppressed when the layer is thin,
10
as seen in Fig. 4, the torque is reduced when the layer V. INCLUSION OF INTERFACIAL RASHBA
thickness is less than a few spin diffusion lengths, which COUPLING
for this set of parameters is ℓsf =2.5 nm.
N
The results in Ref. 27 show that there is an interfacial
For thick layers,the value saturates,but does not sat-
region with significant spin orbit coupling and exchange
urate to the spin Hall angle θ as might be expected.
SH splitting. In this section, we model that overlap region
Eq.(B2)showsthatforthedrift-diffusionmodel,thesat-
by adding a Rashba term to the energy at the interface,
urationvaluedependsontheratio,ℓsfRe[g↑↓]/σ . When
N N seeEq.(10). We findthatthis additionaltermprimarily
this ratio is small, the saturation value is reduced from
leads to a field-like torque and that as long as it is not
θ andtheBoltzmanncalculationandthedrift-diffusion
SH too strong it does not significantly modify the torques
calculation saturate to different values. When that ra-
due to the spin Hall effect.
tio is large, the drift-diffusion and Boltzmann equation
Previously, this region has been treated by two-
results agree. However, a large value of this ratio is not
dimensional calculations in which the electronic struc-
physicallyrealisticforsystemswithstrongspin-orbitcou-
ture is modified by the competition between the Rashba
pling. The mixing conductance depends mainly on the
interaction and the exchange interaction.21–24 Typically,
area of the Fermi surface in the non-magnet (as a re-
the Rashba interaction and the exchange interaction are
minder our calculations assume the same Fermi surface
takentobeverydifferentinmagnitudesothattheFermi
for all materials), but does so in the same way that the
surfaces remain essentially circular. However, the spin
conductivity does (see the expression in Table I, so it
eigendirectionsontheFermisurfacesaremodifiedsothat
is difficult to increase the ratio by changing the mixing
thenon-equilibriumoccupationduetoacurrentflowgive
conductance. It is possible to decrease the conductivity
rise to a net spin accumulation that is not aligned with
by increasing the non-spin-flip scattering, but this also
the magnetization. This nettransversespin density gen-
decreases the spin-diffusion length. For the default pa-
erates a field-like exchange torque on the magnetization.
rameters we consider, see Table I, the value of this ratio
In the Boltzmann equation approach that we use, the
is about 0.3.
Rashba interactionmodifies the boundary conditions for
Eq. (B2) also shows that the torque calculated with the distribution functions at the interface. The result-
the drift-diffusion approachis independent of the details ing torque is very similar to what is found in the two-
oftheferromagneticlayer,dependingonlyonthe mixing dimensionalcalculations. Dependingonthedetailsofthe
conductance. The results for the Boltzmann equation, parameters, the transmission probability is, on average,
for which we do not have analytic results, do depend on eithergreaterorlesserforspinsalignedwithˆz jthanfor
×
thedetailsoftheferromagneticlayerasseeninFig.6for those in other directions. The spin density at the inter-
differentvaluesofthemeanfreepathintheferromagnet. face is determined by the transmission probabilities and
When the mean free path is long so that the conductiv- the incident fluxes. The bias in the transmission proba-
ity in the ferromagnetic layer is much greater than that bilities favors a net spin polarization aligned with ˆz j,
×
in the non-magnetic layer, the current near the inter- verysimilartothebehaviorfoundinthetwo-dimensional
face in the non-magnet is increased (see Fig. 4) giving treatments. Then, through Eq. (13), there is a field-like
a greater spin Hall current. Another difference between torque.
the approaches is that the only length scale for varia- The effect of introducing the Rashba-termon the cur-
tion in the drift-diffusion approach is the spin-diffusion rent distribution is shown in Fig. 4. Unfortunately, the
length. There are many more length scales in the Boltz- spindensitiesattheinterfaceareobscuredbytheapprox-
mannequationapproach(seeTableI)andtheseturnout imations of the Boltzmann equation. In this approach,
to play a non-negligible role when ℓsfRe[g↑↓]/σ is not we assume that electrons on different parts of the Fermi
N N
large. The deviation between the results of the Boltz- surface are incoherent with each other. However, the
mann equation calculations and those found from the matchingconditions forthe distributionfunctions across
drift-diffusion equation should provide a note of caution the interface are found through coherent scattering cal-
for the extraction of physical parameters, like the spin culations. Once the scattering states are used to con-
Hall angle, from comparisons between experiment and struct the matching conditions, the coherence between
the drift-diffusion equation. the incoming and outgoing states is neglected. As a re-
sult, for each electron the spin density at the interface is
We conclude that without additional spin-orbit cou- equal to the incident amplitude times either the trans-
pling at the interface between the two materials, three- mission probability T 2 or 1+R2, which are the same
| | | |
dimensional transport models predict a torque that is because the wave function is continuous across the in-
predominantly damping-like but has a minor field-like terface. However, immediately outside the interface, the
contribution. In the parameter range we have studied, incident and reflected states are no longer treated as co-
the torque is always well described by the combination herent, on one side of the interface the spin density is
of these two forms. The drift-diffusion approachqualita- proportionalto T 2andontheother1+ R2. Sincethere
| | | |
tively captures the physics but can quantitatively fail in are electrons incident from both sides, the spin density
physically relevant parameter regimes. at the interface is not equalto the spin density oneither