A numerical method to solve the Boltzmann equation for a spin valve

0.29 MB·English

by  Jiang Xiao

EPJ manuscript No.
(will be inserted by the editor)
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A numerical method to solve the Boltzmann equation for a spin

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valve
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J

6 Jiang Xiao1, A. Zangwill1, and M. D. Stiles2
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1 School of Physics, Georgia Instituteof Technology, Atlanta, GA 30332-0430

] 2 Electron Physics Group, National Instituteof Standards and Technology, Gaithersburg, MD 20899-8412
l
l
a
h Received: date/ Revised version: date
-
s
e Abstract. Wepresentanumericalalgorithm tosolvetheBoltzmannequationfortheelectrondistribution
m function in magnetic multilayerheterostructures with non-collinear magnetizations. Thesolution is based
on a scattering matrix formalism for layers that are translationally invariant in plane so that properties
.
t onlyvaryperpendiculartotheplanes.Physicalquantitieslikespindensity,spincurrent,andspin-transfer
a
torque are calculated directly from the distribution function. We illustrate our solution method with a
m
systematic study of the spin-transfer torque in a spin valve as a function of its geometry. The results
- agree with a hybrid circuit theory developed by Slonczewski for geometries typical of those measured
d
experimentally.
n
o
c PACS. XX.XX.XX No PACS code given
[

1
1 Introduction cluded in such an approach does not seem to play a role
v
5 in experimental results.
8 ThesemiclassicalBoltzmannequationisausefulcom-
3 Magneticmultilayerstructureshaveattractedagreatdeal promise between these extremes. In such an approach,
1 ofexperimentalandtheoreticalattention.Onemotivation the scattering is treated semiclassically. For transport in
0 forthesestudiesisthepotentialapplicationofsuchstruc- collinear magnetic systems, it has been used in two ways.
7
turesfordatastorageandspindependenttransistors.One Superlattices that havemean free paths longerthan layer
0
special effect in a magnetic multilayer that can induce thicknessescanbetreatedasartificialbulkmaterials[23,24].
/
t magnetic reversal or magnetic dynamics is called spin- Thisapproachretainsthecoherentmultiplescatteringbe-
a
m transfer. In the ten years since its theoretical prediction, tween the interfaces. The other approach is to solve the
[1,2] spin-transfer has been studied extensively both ex- Boltzmann equation within layers and join the solutions
-
d perimentally [3,4,5,6,7]andtheoretically.[8,9,10,11,12,13] throughboundaryconditionsattheinterface.Here,wede-
n One fundamental issue is how to reliably calculate the scribeageneralizationofthelatterapproachtotreatnon-
o spin-transfer torque in spin valve systems. To solve this collinearmagnetizations.Inthisapproach,theBoltzmann
c problem, different approaches have been developed, in- equationtracks individual electrons throughthe distribu-
v: cludingtheBoltzmannequation[14,15],microscopicquan- tionfunction,butignoresthecoherentmultiplescattering.
i tum mechanics [16], drift-diffusion theory [17,18,7,19,20], It is easier to treat defect scattering in such an approach
X
and circuit theory [21,22]. thanitisinafullycoherentcalculation.Theadvantageof
r Each approach has its own advantages and disadvan- the Boltzmanncalculationisthatit issimply computable
a
tages. Simple theories, for example circuit theory or the and includes the essential physics. One disadvantage is
drift-diffusion approach, treat the transport in terms of that it cannot give analytical results. The neglect of co-
densities and current densities and do not track individ- herent multiple scattering between interfaces is both an
ualelectrons.Suchmethodshavetheadvantagethatthey advantageandadisadvantage.Thegreatersimplicitythat
can give analytic results in some limits and the disad- results in the calculation is an advantage for the metallic
vantagethattheyleaveoutsomeessentialphysics.Oneof devices that have been measured to date because the co-
themajorapproximationsofthesetheoriesisthattheyig- herentscatteringdoesnotappeartoplayarole.However,
nore the differences between electrons propagating in dif- this neglect would be a disadvantage in devices in which
ferent directions. Fully quantum mechanical calculations such effects were important.
trackalloftheelectronsandallofthecoherentscattering Slonczewski has developed a hybrid approach com-
processeslikecoherentmultiplescatteringbetweenlayers. bining aspects of circuit theory with a simplified Boltz-
However,such calculations are quite time consuming and mannequationtogiveananalyticexpressionforthespin-
the coherentmultiple scatteringbetweenlayersthat isin- transfer torque [25]. This approach gives much more ac-
2 Jiang Xiao et al.: A numerical method to solve theBoltzmann equation for a spin valve
curate results than the drift-diffusion method for typical where the integration over k′ is now a two-dimensional
device geometries. However, it breaks down when layer integral restricted to the Fermi surface. The scattering
thicknesses become longer than are typical. rate Pk,k′ has been rescaled. A delta function has been
In this paper, we describe a numerical algorithm that factored out of each term. In the second term on the left
solves the Boltzmann equation for the electron distribu- hand side, this delta function comes from ∂f /∂ǫ.
0
tionfunctioninamagneticmultilayersystemusingascat- For spin dependent magnetic materials, one needs a
tering matrix formalism. The spin-transfer properties are spin dependent distribution function as well as a spin de-
calculated from the resulting distribution function. We pendent Boltzmann equation. When there is a natural
compare our Boltzmann results to the results from more quantizationaxis,i.e.,allelectronspinsareeitherparallel
approximatemethods.ThesecomparisonsshowSlonczewski’s(σ =↑)oranti-parallel(σ =↓)to the axis,anelectrondis-
hybrid theory is highly accurate for typical experimental tribution function is separated into the distribution func-
structures,butdrift-diffusionshowssystematicdeviations. tions for spin-up and spin-down electrons: f↑(r,k) and
Wehavereportedresultsofcalculationsusingthemethods f↓(r,k)
described in the present paper in earlier papers [14,15].
Thepaperisorganizedasfollows:Section2givesagen- ∂fσ(k)
eralized spin-dependent Boltzmann equation appropriate vkσ· ∂r −eE·vkσ = dk′Pkσ,k′[fσ(k′)−fσ(k)]
for ferromagnetic systems; Section 3 describes in detail ZFS
the algorithm that solves the Boltzmann equation in a + dk′Pksf,k′[fσ′(k′)−fσ(k)],
spin valve magnetic multilayer system; Section 4 shows ZFS
the results of the Boltzmann method developed in Sec- (4)
tion 3 applied to a spin valve. Section 5 gives a summary.
Readers who are not interested in the formalism can skip The r dependence of the distribution function has been
directly to Section 4. suppressed, and σ =↑, ↓ and σ 6= σ′. Compared with
Eq.(3),Eq.(4)hasanadditionalspinflipscatteringterm
ontherighthandsidebecausethedistributionfunctionin
2 Generalized matrix Boltzmann equation Eq. (3) includes both spin types, while the one in Eq. (4)
is foronly one spin type.Similar to the definitionof Pk,k′
In non-magnetic materials, the spin independent Boltz- inEq.(3),Pkσ,k′ andPksf,k′ aretheprobabilitiesofelectron
mann equation is scattering from state k′ to state k without and with spin
flip.WeassumethatPsf isthesameforspinflipinboth
∂g(r,k) ∂g(r,k) k,k′
vk· ∂r −eE·vk ∂ǫk = dk′Pk,k′[g(r,k′)−g(r,k)], direIcntiaonnso,nu-mpatogndeotw,tnheorredisownnontoatuupra.lquantizationaxis,
Z
(1) so it is convenient to use the axis of neighboring fer-
where g(r,k) is an electron distribution function that de- romagnetic layers if the magnetizations of those layers
pendsonthespatialcoordinaterandelectronwave-vector are collinear. In this case, the scattering Pk,k′ is spin-
k. vk is the electron velocity, ǫk its energy, E is the elec- independent, so the substitutions f0 = 1(f↑ + f↓) and
tric field, and we have ignoredthe magnetic field. Pk,k′ is fz = 1(f↑−f↓) give the pair of equation2s
theprobabilityofelectronscatteringfromstatek′tostate 2
k. The Boltzmann equation is valid only when the bulk
∂f0(k)
pfarcoepseorrtibesouvnardyarsileosw;llay.teIrtwcaendneostcrbibeeuhseodwftooruasberbupoutnindtaerry- vk· ∂r −eE·vk = ZFSdk′Pk,k′[f0(k′)−f0(k)]
conditions to relate solutions of the Boltzmann equation
in different regions across the interfaces. + dk′Pksf,k′[f0(k′)−f0(k)],
ZFS
Transportinmetalsisdominatedbytheelectronsnear ∂fz(k)
the Fermi energy. The occupancy of states far from the vk· ∂r = dk′Pk,k′[fz(k′)−fz(k)]
Fermi energy does not change and those states do not ZFS
clionnetarriibzuedteBtoolttzhmeatnrannsepqouratt.ioTnhiisnswuhggicehsttshtehediustsreiboufttiohne − dk′Pksf,k′[fz(k′)+fz(k)],
ZFS
function is assumed to have the form (5)
g(r,k)=f0(k)+f(r,k)δ(EF−ǫk), (2) In the first equation, spin flip scattering acts as another
form of non-flip scattering as far as the number accumu-
where f is the equilibrium distribution function. The
0
lation is concerned. In the second equation, the electric
delta function restricts the wave vector to the Fermi sur-
field plays no role because it does not couple to the spin
face, |k|=k for free electrons. With this approximation
F
accumulation.In the systems of interesthere, the magne-
for the distribution function, the Boltzmann equationbe-
tizations are not collinear, so the spin axis in the distri-
comes
butionfunctioncanvarywithbothrandk.However,the
vk· ∂f(∂rr,k) −eE·vk = dk′Pk,k′[f(r,k′)−f(r,k)], gEeqn.e(r5a)liizsarteiopnliciastsetdrafoigrhetafochrwoafrtdh.eTohtheesredcoirnedcteioqnusatiniosnpiinn
ZFS
(3) space, with z → x, y. The generalization can be derived
Jiang Xiao et al.: A numerical method to solve theBoltzmann equation for a spin valve 3
fixed free H bulk layer in the spin valve; (3.4) construct the interface
Reservoir lead M FM Spacer θFmM lead Reservoir scscionangtntleeecrstiyntsghtememab-tuwrliikxdeafonsrdceaaitntcteherriinnfatgceemrfasacctaertiixtne;rt(ih3ne.g6s)mpdianettrveiarcmlevsein;ine(t3to.h5ae)
x, J boundary conditions and apply them to the system-wide
Fig.1.Schematicviewofaperpendicularspinvalvestructure. scatteringmatrixtocalculatethedistributionfunctionex-
pansion coefficients; (3.7) calculate the distribution func-
tion values within the spin valve; and (3.8) calculate the
from a matrix form of the Boltzmann equation in terms spin density (spin accumulation), spin current, and spin-
of the matrix distribution function transfer torque using the distribution function.
fˆ(r,k)= f0σ +fxσ +fyσ +fzσ .
0 x y z
f↑(r,k) 0
= Uˆ(r,k) Uˆ†(r,k) (6) 3.1 Discretization of the Boltzmann equation
0 f↓(r,k)
(cid:20) (cid:21)
where σ ,σ and σ are Pauli spin matrices, σ is a 2×2 To discretize the Boltzmann equation, we need a numer-
x y z 0
identity matrix, and Uˆ is a unitary rotation matrix. The ical mesh for the electron wave-vector k that can accu-
rately and simultaneously describe all Fermi surfaces for
rotation matrix allows spin direction to point in an arbi-
both spin types and for all layers. A simple method for
trarydirectionovertheFermisurfaceandasafunctionof
choosing a mesh is as follows. Choose a uniform mesh in
position.
the direction parallel to the interfaces (perpendicular to
In strong ferromagnets,the rotation matrix in Eq. (6)
is independent of the position on the Fermi surface or k x): kj. For each material, there could be several points
k
asinEq.(4).Thisconstraintisaconsequenceofthe large on the Fermi surface that have the same kj. We label
k
difference in the Fermi surfaces for majority and minor- their longitudinal wave-vectors kn. A complete mesh for
ityelectronsinstrongferromagnets.Theconstraintarises x
the Fermi surface is k = (kn,kj). The mesh weights are
becauseanytransversespinaccumulationinthe electrons i x k
determined by the area on each Fermi surface associated
near the Fermi surface rapidly dissipates. The transverse
withkn.Thismeshmayconvergeslowly,soitmaybenec-
spinsprecessinthelargeexchangefieldanddosoatdiffer- x
ent rates because of the complicated Fermi surfaces. The essaryto refinethe kj meshtoinclude ahigherdensity of
k
precessingtransversecomponentsrapidlydephasewithre- points. Let us assume one such mesh has sampling points
spect to each other, leaving no net transverse moment. {k }N with weighting factor {w }N. Using this mesh, the
i 1 i 1
integrationforanycontinuousfunctionh(k)onthe Fermi
surface can be discretized as:
3 Formal technique
N
Figure 1 shows a schematic picture of a spin valve — h(k)dk= wih(ki). (7)
a magnetic multilayer structure where a thin-film non- ZFS i=1
X
magnet (spacer) is sandwiched between two thin-film fer-
romagnet (FM) layers. The latter connect to reservoirs We discretize the integrations in Eq. (4) using this
with non-magnetic leads. The main purpose of this paper mesh.Assuming the systemisone dimensionalinxdirec-
istosolvetheBoltzmannequationnumericallyforthedis- tion, we have a discretized form of the Boltzmann equa-
tributionfunctioninsuchaspinvalvestructure.Inreality, tion:
thecrosssectionaldimensionsofthesestructuresaremuch ∂fiσ −eE = [Vˆ−1Bˆ]σσ′fσ′, (8)
largerthan typicallayerthicknesses so that the transport ∂x x ij j
j,σ′
in the interior of the sample is more important than that X
nearthe edges.The reservoirshavemuchlargercrosssec- where the subscript i and j mean that k is evaluated at
tional areas and hence smaller resistances than the active k or k , for instance fσ = fσ(k ). The velocity matrix
i j i i
structuressothatthetransportislargelyperpendicularto Vˆ and the scattering matrix Bˆ are 2N ×2N matrices (N
the layers.This combinationof features suggests a simple from k index, 2 from spin index) with matrix elements
i
model in which the layers are treated as translationally
invariantin plane so that properties only vary perpendic- Vσσ′ = vσxδσσ′, (9)
ular to the planes. Electrons move in all three directions, ij i ij
bi.ue.tftˆ(hre,kne)t=cufˆr(rxe,nkt)a.nd all variationis in the x-direction, Biσjσ′ = wjPiσjδσσ′ − δτiσjσσ′ +wjPisjf(1−δσσ′)− δτiσjsσf′,
i i
Thecalculationproceedsineightsteps:(3.1)discretize
tchael mspeisnhdoefptehnedeFnetrmBiosltuzrmfaacnen; (e3q.2u)atsioolnveutshinegdaiscnruemtiezreid- whereδiσjσ′ =δijδσσ′,1/τiσ = jwjPiσj,isthespin-dependent
non-spin-flipscattering rate,and1/τsf = w Psf is the
Boltzmann equation for the eigensolutions in the non- P i j j ij
magnetic and ferromagnetic bulk; (3.3) use the eigenso- spinflip scattering rate.See Eq.(4). The matrixBˆ is sin-
P
lutions to construct the layer scattering matrix for each gular because Bσσ′ =0.
j,σ′ ij
P
4 Jiang Xiao et al.: A numerical method to solve theBoltzmann equation for a spin valve
3.2 Solution of the Boltzmann equation in the bulk thatarenecessarynearinterfaces.Thesesolutionsarenot
allowed in a uniform bulk because they diverge in one
Inthissection,webrieflysummarizetheresultsofRef.[26] direction. Together, these solutions describe arbitrary so-
andthengeneralizethemtotreatsystemswithnon-collinear lutions of the Boltzmann equation in each layer.
magnetizations. According to Ref. [26], the Boltzmann Both the particular solution F and the homogeneous
0
equation Eq. (8) can be solved by breaking it into a par- solutionF arecurrentcarryingbecauseofthek -dependence
2 i
ticular equation of their summation terms. But, they describe current as-
sociatedwithdifferent processes.F describesthe current
0
[Vˆ−1Bˆ]σijσ′fjσ′ =−eEx, (10) due to the electric field, and F2 describes the current due
j,σ′ to density gradients. Formally, F2 =A(Exx−F0), where
X
A is a constant, so F plus the electric field E canbe in-
0 x
and a homogeneous equation, terchangedwithF .Computationally,sincewe workwith
2
uniform current, we solve everything with E = 0 and
x
δσσ′ ∂ −[Vˆ−1Bˆ]σσ′ fσ′ =0. (11) no F0 and work with F2. Once we find the coefficient of
ij ∂x ij j F , the physical solution is the corresponding F and E .
j,σ′(cid:26) (cid:27) 2 0 x
X Thatis,wesolvetheproblemasiftheuniformcurrentwas
exclusivelydue todiffusion withno electricfieldandthen
The solution to the particular equation Eq. (10) is
reinterpret the charge accumulation as an electric poten-
Fσ(x,k )=−eE [Bˆ−1Vˆ]σσ′. (12) tial and set the charge accumulation to zero. This inter-
0 i x ij pretation makes sense due to the short screening length
j,σ′
X in metals.
Due to the singularity of Bˆ, Bˆ−1Vˆ is not the inverse of There is a natural quantization axis in a ferromagnet
defined by its magnetization. The distribution function is
Vˆ−1Bˆ.Instead,the inversematrixisdefinedasinSection
described by the eigensolutions in Eq. (15):
2.9ofRef.[27]usingasingularvaluedecomposition.Since
the constantvectorisperpendicular tothe nullspace,the fˆ(x,k )= f↑σ +f↓σ
i ↑ ↓
solutionis welldefined. Physically,the solutionis the dis-
2N
tribution function of the current in bulk material. Since fσ = ασFσ(x,k )≡Fσ ·ασ.
this distribution is well defined physically, we expect it n n i
n=1
to be mathematically as well, if we have formulated the X
(16)
problem correctly.
Most solutions to the homogeneous equation Eq. (11) Here, σ = (1/2)(I+σ ), σ = (1/2)(I−σ ), and the ασ
↑ z ↓ z n
vary exponentially in space: are the expansion coefficients.
Inanon-magnet,thereisnonaturalquantizationaxis,
Fσ(x,k )=gσ(k )eλnx with n∈[3,2N], (13)
n i n i so we write the distribution function as in Eq. (6). We
construct a different basis set Fτ, τ =0,x,y,z from F↑,↓
where gσ(k ) = giσ and λ are the n-th eigenvector and n n
n i n n for the non-magnet. First, we know that the eigenvectors
eigenvalue of Vˆ−1Bˆ Fσ(x,k ) in Eq. (13) and Eq. (15) break into separate
n i
eigenvectors for charge transport with F↑ = F↓ and spin
[Vˆ−1Bˆ]iσ,jσ′gnjσ′ =λngniσ. (14) transport with Fn↑ = −Fn↓. For instance,nthe eingenvectors
j,σ′ with n=1 and 2 correspond to charge transport because
X
F↑ = F↓ . In general, half of the eigenvectors (assume
See the Appendix in Ref. [26] for more details. Since the 1,2 1,2
for the first half: n ∈ [1,N]) corresponds to the charge
bulk is translationally invariantin x direction, half of the
transport; the other half (for the second half: n ∈ [N +
eigenvaluesλ arepositiveandhalfarenegative.Thema-
n 1,2N]) is for the spin transport. Therefore, the basis set
trix Vˆ−1Bˆ is defective, which means that the degenerate
in a non-magnet can be constructed as:
zeroeigenvalueonly hasoneeigenvector.Becauseitisde-
fective, the homogeneous equation has two solutions that 1
Fτ=0(x,k )= F↑(x,k )+F↓(x,k ) n∈[1,N],
do not have exponential form n i 2 n i n i
(17a)
(cid:2) (cid:3)
FF1σσ((xx,,kki))== x1Fσ(x,k )+ [Bˆ−1Vˆ]σσ′. (15) Fnτ=x,y,z(x,ki)= 21 Fn↑+N(x,ki)−Fn↓+N(x,ki) n∈[1,N].
2 i 1 i ij
h i (17b)
j,σ′
X
Thus, in the non-magnet layers, the general solution for
which can be verified by plugging them back in Eq. (11).
the distribution function is
Physically,F isthesolutiondescribingauniformshift
1
of the chemical potential and F2 describes a current car- fˆ(x,ki)= f0σ0+fxσx+fyσx+fzσz
rying solution having a spatially varying density and a
N
associated uniform diffusion current. The rest of eigenso- fs = αsFs(x,k )≡Fs·αs, (18)
lutions, Eq. (12) and Eq. (13), are exponential solutions n n i
n=1
X
Jiang Xiao et al.: A numerical method to solve theBoltzmann equation for a spin valve 5
where s = 0,x,y,z, and αs are expansion coefficients. 3.4 Interface scattering matrix
n
Eq.(17)tells us thatfx, fy,andfz sharethe samesetof
Using the layer scattering matrix, we are able to relate
eigenvectors.ThisisbecausetheBoltzmannsolutiondoes
the distribution function values at the two sides of a bulk
not depend on the choice of spin quantization axis in a
layer. Next, we find an interface scattering matrix which
non-magnet.
connects the distribution functions across an interface.
Rightattheinterface,theBoltzmannequationisnotvalid
3.3 Layer scattering matrix because the material properties vary rapidly. The distri-
bution functions across the interface are related through
The eigensolutions are used to constructa scattering ma- the scattering matrix for the electron wave-functions at
trix which relates the boundary values of each layer. Fig- the interface.
ure 2 shows a schematic picture of a spin valve, where Like the layer scattering matrix in Eq. (19), the in-
lead/FM/spacer/FM/lead are labeled layer 1 to layer 5, terface scattering matrix R relates the incoming to the
and x denotes the x coordinate of each interface. outgoing distribution functions. If the interface is specu-
0,1,2,3,4,5
InEq.(16)andEq.(18),weexpandthedistributionfunc- lar,when electrons scatter fromthe interface,the compo-
tionineachlayerusingasetofeigensolutions.Fromthese nent of the wave vector that is parallel to the interface
expansions we construct the (layer) scattering matrix for is conserved. In this case, the interface scattering matrix
layerm,S ,whichrelatesthe incomingelectrondistribu- is block diagonal, with non-zero elements only for those
m
tionfunctionsfin andtheoutgoingonesfout atthe states with the same parallel wave vector. On the other
m,L/R m,L/R
left (L) and right (R) sides of the layer m (Figure 2): hand,defectscatteringattheinterfacecouplesstateswith
differentparallelwavevectorsandtheinterfacescattering
fout fin matrixbecomesdense.Inthepresentwork,weneglectde-
m,L =S m,L . (19)
fout m fin fect scattering at the interfaces for simplicity. Interfacial
(cid:20) m,R(cid:21) (cid:20) m,R(cid:21)
defect scattering has been considered by several authors
Here, the matrices in the ferromagnet (m = 2, 4) and [28,29,30]. Xia et al. [30] treated interdiffusion at Co/Cu
non-magnet(m= 1, 3, 5) are respectively interfaceswithfirstprinciplescalculationsandfoundthat
such defects caused only minor changes in the average
fmin,L T = fmin,,L↑,fmin,,L↓ transport properties for these interfaces.
(cid:2)fmin,L(cid:3)T =hfmin,,L0,fmin,,Lxi,fmin,,Ly,fmin,,Lz . (20) andCaonfesridroemr aangniestol(aNteMd/iFnMter)faacnedbcehtowoeseentahensopni-nmqaugannet-
and fin,↑ =(cid:2)f↑(x(cid:3) ,hk+), and fin,↑ = f↑(ix ,k−), etc. tfeizrarotimonaganxeits. TtohebeinptearrfaalcleelbteotwtheeenmLaegandet1izaantidonFMof2thine
m,L m m−1 i m,R m m i
k± istheelectrons’wave-vectorontheFermisurfacewith Figure 2 is such an interface if those two materials were
i
the (+/-) superscript indicating whether the electorn is extended to infinity. Suppose the wave-function for an
right(+)orleft(-)going.Thesuperscripts“in”and“out” electron on the Fermi surface in the corresponding NM is
denote electrons going into or out of the layer. The defi- φ(ki), which is orthonormal to other states on the Fermi
nitioninEq.(20)generalizesinastraightforwardmanner surface: hφ(ki)|φ(kj)i = δij. The wave-function for an
for “R” and “out.” electron on the Fermi surface in the FM is ψσ(ki), which
FromEq.(16),wecanexpressthedistributionfunction is also orthonormal to other states on the Fermi surface:
as fmσ(x,ki) = Fσ(x,ki)·αs in the ferromagnetic layers hψσ(ki)|ψσ′(kj)i=δijδσσ′.
(m = 2, 4), then In the non-magnet, the wave-function for an electron
movingtowardtheinterfacewithspinpointinginanarbi-
F↑(x ,k−) trary direction is written as a linear combination of spin-
m−1 i
fout F↓(x ,k−) up and spin-down components:
(cid:20)fmmou,,RLt (cid:21) = FFF↑(↓↑x((xxmmm−,,1kk,k+i+i+i)))·α≡Fomut·α Φin =(cid:20)abφφ((kk+i+i ))(cid:21), (23)
m−1 i where a and b are the coefficients of the up and down
fin F↓(x ,k+)
m,L =  m−1 i ·α≡Fin ·α, (21) spinorcomponents.This incident state is scatteredatthe
fin F↑(x ,k−) m
(cid:20) m,R(cid:21)  F↓(xm,k−i )  NM/FM interface, and the scattered states are
 m i  aRNN,↑φ(k−)
TwhheerreefoFrimen, aEnqd. (F1om9u)timarpelibeostthha2tNth×e2laNyesrqsucaartetemriantgrimceas-. Φref = j "bRiNijjN,↓φ(k−jj )# for x<0,
X
trix for ferromagnetic layers (m = 2, 4) is aTNF,↑ψ (k+)
Φ = ij ↑ j for x>0, (24)
Sm =Fomut· Fimn −1. (22) tr Xj "bTiNjF,↓ψ↓(k+j )#
The layerscatteringmatrix for(cid:2)non(cid:3)-magneticlayers(m= where RNN,σ and TNF,σ are the reflection and transmis-
ij ij
1, 3, 5) is constructed in a similar way using Eq. (18) as sionamplitudes forelectronfromk to k forspin-upand
i j
the expansion and Eq. (20) for m = 1, 3, 5. spin-down electrons: σ =↑,↓.
6 Jiang Xiao et al.: A numerical method to solve theBoltzmann equation for a spin valve
S R S R S R S R S
1 1 2 2 3 3 4 4 5
Lead 1 FM 2 SP 3 FM 4 Lead 5
m
M
oir f1in,L f1o,uRt f2in,L f2o,uRt f3in,L f3o,uRt f4in,L f4o,uRt f5in,L f5o,uRt Re
v s
r e
se z’(cid:1) rv
Re f1o,uLt f1in,R f2o,uLt z f2in,R f3o,uLt f3in,R f4o,uLt f4in,R f5o,uLt f5in,R oir
x x’(cid:1)
x
x x x x x x
0 1 2 3 4 5
Fig.2. Scatteringmatrices.Thecenterofthefigureliststheincomingandoutgoingdistributionfunctionsateachofthelayers.
The top lists thelayer, Sn, and interface, Rn scattering matrices that relate thedistribution functions.
The 2×2 matrix distribution function is then defined as
by the outer product of the spinor coefficients. For in-
f0(x ,k−) f0(x ,k+)
stance, for the incident state N 0 i N 0 i
fx(x ,k−) fx(x ,k+)
fˆin =(cid:20)abaa∗∗ abbb∗∗(cid:21). (25) (cid:20)ffNFiinn(cid:21)≡fffNNNFzy↑(((xxx0000,,,kkk+i−i−ii )))=RfffNNNFzy↑(((xxx0000,,,kkk−i+i+ii )))≡R(cid:20)ffNFoouutt(cid:21),
f↓(x ,k+) f↓(x ,k−)
Straightforwardalgebra reveals  F 0 i   F 0 i 
    (29)
where R is the interface scattering matrix for the distri-
fˆref(k−j ) =RNijN†fˆin(k+i )RNijN bution functions across the interface, and the matrix ele-
ments in R are obtained from the scattering matrices R
fˆ (k+) =TNF†fˆ (k+)TNF, (26)
tr j ij in i ij andT inEq.(27).Note the apparentreversalof“in”and
“out”inEq.(29)comparedtothatininEq.(19).Thedi-
wherethereflectionmatrix(NMtoNM)andtransmission rections in and out are defined with respect to the layers,
matrix (NM to FM) are rather than the interfaces and the incoming distribution
foroneofthelayersisthe outgoingdistributionforoneof
the interfaces.
RNN,↑ 0
RNN = ij
ij " 0 RiNjN,↓#
3.5 System scattering matrix
TNF,↑ 0
TNF = ij . (27)
ij " 0 TiNjF,↓# WithallthelayerandinterfacescatteringmatricesS1,2,3,4,5
and R (see Figure 2), we are ready to construct a
1,2,3,4
system-widescatteringmatrixTthatrelatestheincoming
Considering the electrons incident onto the interface
andoutgoingdistributionfunctionsneartheleftandright
frombothsidesoftheinterface,thescatteringrelationship reservoirs.The scattering matrix T is obtained by joining
Eq. (26) becomes
all the layer scattering matrices and interface scattering
matricesinorder:T=S −R −S −R −S −R −S −
1 1 2 2 3 3 4
fˆN(k−j )= RNijN†fˆN(k+i )RNijN+ TFijN†fˆF(k−i )TFijN, R4−ThSe5.joining procedure is as follows. First we join S
i i 1
X X (28a) to R1 (see Figure 2), where S1 is the layer scattering ma-
trix that covers the interval [x+,x−] (layer 1), and R is
fˆF(k+j )= RFijF†fˆF(k−i )RFijF+ TNijF†fˆN(k+i )TNijF. the interface scattering matrix0tha1t covers [x−1,x+1] (1the
i i interface at x ):
X X 1
(28b)
fout SLL SLR fin
1,L = 1 1 1,L
fout SRL SRb fin
The matrix forms of the distribution functions fˆ in (cid:20) 1,R(cid:21) (cid:20) 1 1 (cid:21)(cid:20) 1,R(cid:21)
Eq. (28) are expanded using σ in the FM or Pauli ma- fin RLL RLR fout
↑,↓ 1,R = 1 1 1,R . (30)
trices σ in the NM as in Eq. (6). We represent the fin RRL RRb fout
0,x,y,z (cid:20) 2,L(cid:21) (cid:20) 1 1 (cid:21)(cid:20) 2,L (cid:21)
distributionfunctionsbytheirexpansioncomponentsf
↑,↓
intheFMandf intheNMasinEq.(20).Afterthis Here we have subdivided the distribution vectors (dis-
0,x,y,z
transformation,the scatteringformula Eq.(28) is written cretized functions) into two subvectors for the values on
Jiang Xiao et al.: A numerical method to solve theBoltzmann equation for a spin valve 7
theleft(L)andright(R).TheSandRmatricesarecorre- with the rotated scattering matrix
spondingly subdivided into four submatrices labelled LL,
1 0 1 0
LR, RL, and RR, where, for example, the LR submatrix T′ = T . (36)
connects incoming values on the right to outgoing values lifi 0 Uˆ lifi 0Uˆ†
(cid:20) (cid:21) (cid:20) (cid:21)
on the left. We denote the joint scattering matrix by T ,
li T′ is joined with S using the same procedure as in
where the subscript “l” denotes a lead, “i” denotes an lifi 3
Eq. (32) and becomes T . Continuing by joining T
NM/FMorFM/NMinterface,“f”denotesaferromagnet, lifin lifin
and“n”denotes the non-magneticspacerlayer.T covers to R3,S4,R4 and S5 gives a system wide scattering ma-
li trix T = T that relates the distribution functions
[x+,x+] and relates fin/out and fin/out: lifinifil
0 1 1,L 2,L near the left reservoirand the right reservoir:
(cid:20)ff12oin,,uLLt(cid:21)=Tli(cid:20)ff21oin,,uLLt(cid:21). (31) (cid:20)ff51′((xx50,,kk+i−i ))(cid:21)≡(cid:20)ff5o1o,u,RuLtt′(cid:21)=T(cid:20)ff51iinn,,RL′ (cid:21)≡T(cid:20)ff51′((xx50,,kk−i+i ))(cid:21).
Byeliminatingtheintermediatedistributionfunctionsfin/out (37)
1,R Eq. (37) is a condition on the boundary values of the dis-
in Eq. (30), we have
tribution functions for an arbitrary solution of the Boltz-
TLL TLR mann equation in the multilayer.
Tli = TRL TRR , (32) Inthe casethatthe twoleads (layer1andlayer5)are
(cid:20) (cid:21) semi-infinite, we need a system scattering matrix T that
where covers the interval [x−,x+]:
1 4
TLL = SLL+SLRRLL(1−SRRRLL)−1SRL f (x ,k−) f (x ,k+)
1 1 1 1 1 1 1 1 i =T 1 1 i . (38)
TLR = SL1R[1+RL1L(1−SR1RRL1L)−1SR1R]RL1R (cid:20)f5′(x4,k+i )(cid:21) (cid:20)f5′(x4,k−i )(cid:21)
TRL = RRL(1−SRRRLL)−1SRL Eq. (37) and Eq. (38) each have a total of 8N un-
1 1 1 1
TRR = RRR+RRL(1−SRRRLL)−1SRRRLR. (33) knowncoefficientsinthe expansionsoff1 andf5 butonly
1 1 1 1 1 1
4N equations.Tosolvethisproblem,westudythebound-
The scattering matrices described here and the method ary conditions, which apply restrictions on the distribu-
of joining them is more complicated than approaches us- tion functions f and f , and so reduce the number of
1 5
ing transfer matrices, which relate the boundary values unknowns in the expansions.
fromone sideto the otherratherthanoutgoingto incom-
ingboundaryvalues.However,transfermatrixapproaches
become unstable as layers become thick and the present 3.6 Boundary conditions
method does not.
UsingthesameprocedureasabovewecanjoinT with By examining the properties of the leads and reservoirs,
li
S2andthenwithR2tohaveascatteringmatrixTlifiwhich we can restrict the form of the distribution functions f1
covers [x+0,x+2]. We continue to construct Tlif, Tlifi, Tlifin, andf5 in the leads,suchthat the number ofunknownco-
T ,T ,T ,andT .Whencomingtothethe efficientsintheexpansionsequalsthenumberofequations
lifini lifinif lifinifi lifinifil
spacer layer,the eigensolutions and the scattering matrix in Eq. (37) or Eq. (38). Thus, we can uniquely determine
S usez′-axisinsteadofthe z-axis(whichisusedforT ) the distributionfunctions fromEq.(37) andEq.(38).We
3 lifi
asthespinquantizationaxis(seeFigure2).Therefore,we treat semi-infinite leads and finite leads differently, and
have to make an axis rotation at the spacer layer when separate the discussion into two cases: 1) the leads are
joining T with S . semi-infinite: the left/right lead extends to the left/right
lifi 3
The distribution functions that T and S relate are: infinitely, and 2) both leads have finite length before con-
lifi 3
necting to reservoirs.
fout fin
1,L = T 1,L
fin lifi fout
(cid:20) 3,L(cid:21) (cid:20) 3,L (cid:21) 3.6.1 Semi-infinite leads
fout′ fin′
"f33o,,uRLt′#= S3"f33in,,RL′ #, (34) If the leads are semi-infinite, the distribution function f1
in the left lead (layer 1) includes only the exponential
Theprimeddistributionfunctionsarewrittenusingthez′- eigensolutions with λ >0. Then from Eq. (18), we have
n
axisasthespinquantizationaxis.TojoinT withS ,we
lifi 3
write f3in,L/out in terms of f3in,L/out′: f3in,L/out = Uˆ†f3in,L/out′. Uˆ f0(x,k )=α0F0(x,k )+F0(x,k )+ N α0F0(x,k ),
1 i 1 1 i 2 i n n i
is the component representation [Eq. (20)] of the unitary
n=3
rotation matrix Uˆ in Eq. (6): fˆ′ = UˆfˆUˆ† ⇒ f′ = Uˆf. λXn>0
Then (39a)
fout fin fout fin N
Uˆ†1f,Lin′ =Tlifi Uˆ†f1,oLut′ =⇒ f1i,nL′ =T′lifi fo1u,Lt′ , f1x,y,z(x,ki)= αxn,y,zFnx,y,z(x,ki). (39b)
(cid:20) 3,L(cid:21) (cid:20) 3,L (cid:21) (cid:20) 3,L(cid:21) (cid:20) 3,L (cid:21) n=1
(35) λnX+N>0
8 Jiang Xiao et al.: A numerical method to solve theBoltzmann equation for a spin valve
Wesetα02 =1tofixthecurrent,becauseF2 isthecurrent SL SR
carryingterm.Duetothex-translationalinvarianceofthe
Lead 1 FM 2 SP 3 FM 4 Lead 5
ekpspxnoipsoniowtnitnvreeanαnt0ais,anxpl,doyer,zihtg,aienlrnfestsoaoplrtueaectl,tinoiNvenges/la,y2t.hifvaoTelrfhefeooarfrcehtfcohhcreeoaemrtighgpeeeonrtnevreaananlutrseeposfo2rλN0t,nxaua,nnryde-, Reservoir ff11ion,,uLLt z x ff33oin,u,RRt z’(cid:1) x’(cid:1) ff55oin,u,RRt Reservoir
n
and z. x0 x1 x2 x3 x4 x5 x
Similarly,thedistributionfunctionf′ intherightlead
5
(layer5)includesonlytheexponentialeigensolutionswith Fig. 3. Back-propagation matrices.
λ <0:
n
N To determine the form of the distribution functions
f′0(x,k )=β0F0(x,k )+ β0F0(x,k ), (40a) near the reservoirs, let us first use F0,x,y,z to denote the
5 i 2 2 i n n i p
n=3 eigensolutions with positive eigenvalues: λp > 0; and use
λXn<0 F0,x,y,z to denote the eigensolutions with negative eigen-
q
N values: λ <0. We construct a new set of basis functions
q
f5′τ(x,ki)= βnτFnτ(x,ki). (40b) G0n,x,y,z using linear combinations of the old basis func-
n=1 tions F0,x,y,z such that,
X n
λn+N<0
G0,x,y,z =F0,x,y,z, (41a)
where τ =x,y,z.The choice β0 =0 sets the potentialat 1,2 1,2
1
the rightleadto be zero.Eq.(40) alsohas 2N unknowns, G0,x,y,z(x ,k+)=0, (41b)
p 0 i
βτ, τ =0,x,y,z, in total.
n G0,x,y,z(x ,k−)=0. (41c)
Plugging Eq. (39) and Eq. (40) into Eq. (38), we have q 5 i
4N equations with 4N unknowns, the expansion coeffi-
Essentially, G is obtained by using the linear combi-
p
cients αn’s and βn’s can be solved. This gives the distri- nationof F (x ,k+) and F (x ,k+) to cancelF (x ,k+),
bution functions in the left and right leads. The distri- 1 0 i q 0 i p 0 i
andsimilarlyG isobtainedbyusingF (x ,k−)andF (x ,k−)
butionfunctionvaluesinside the spinvalvearecalculated q 1 5 i p 5 i
to cancel F (x ,k−).
fromtheboundaryvaluesusingtheappropriatescattering q 5 i
G andG formthe basis forthe electronsthathave
matrices, as will be shown later. 1,2 p
bulk-likeright-goingbehavioratx=x ,andG andG
0 1,2 q
form the basis for electrons that have bulk-like left-going
3.6.2 Finite leads behavioratx=x .Thedistributionfunctionsthatsatisfy
5
the requirements (1) and (2) in the leads are constructed
In actual spin valve samples, the leads are usually short, as the following:
after which the sample connects to essentially bulk ma-
terial, here referred to as reservoirs. Electrons that enter f0(x,k )=α0G0(x,k )+G0(x,k )+ α0G0(x,k ),
1 i 1 1 i 2 i p p i
the reservoir are much less likely to scatter back into the p
X
sample than to stay in the reservoir until they are ther- (42a)
malized. This means that the reservoirsbehave as perfect
fx,y,z(x,k )= αx,y,zGx,y,z(x,k ), (42b)
absorbers. The distribution of the electrons coming out 1 i p p i
of the reservoir is characteristic of the bulk, independent Xp
of the distribution of electrons coming in. The connec- f′0(x,k )=β0G0(x,k )+ β0G0(x,k ), (42c)
5 i 2 2 i q q i
tion between the reservoirs and leads has been studied in
q
X
more detail by Berger [31] and Hamerle et al. [32]. When
f′x,y,z(x,k )= βx,y,zGx,y,z(x,k ). (42d)
the leads have finite length and are connected to electron 5 i q q i
reservoirs, the exponential eigensolutions have no singu- Xq
laritiesandallofthemshouldbeincluded.Insuchacase,
In these equations, α0 = 1 fixes the current, and β0 = 0
the form of the distribution functions is constrained by 2 1
fixes the chemical potential at the right boundary to be
the following properties of a reservoir: the electrons leav-
zero. Since the indexes p and q each take (N −2)/2 val-
ingthereservoirhaveabulk-likedistributionfunctionand
ues for the 0-component and N/2 values for the {x,y,z}-
the electrons with arbitrary distribution function can be
components, the total number of unknown coefficients in
absorbedbythereservoir.Basedonthesetwopropertiesof
Eq.(42)is4N.Therefore,bypluggingEq.(42)intoEq.(37),
reservoirs,we propose that the distribution function near
the coefficients αs,βs can be determined.
the reservoirs should satisfy: (1) for the electrons going n n
from the reservoir to the lead, the distribution function
is bulk-like, namely fin and fin′ are spin independent
1,L 5,R 3.7 System distribution function
and have contributions only from F0 and F0 in Eq. (17);
1 2
(2) for the electrons going from the lead to the reservoir, Oncewehavethedistributionfunctionvaluesatthebound-
thedistributionfunctionhaswhateverstructureitwishes, aries,eithernearthelead/FMinterfaceforthesemi-infinite
namelyfout andfout′ havecontributionsfromallF0,x,y,z. lead case or near the reservoir for the finite lead case, we
1,L 5,R n
Jiang Xiao et al.: A numerical method to solve theBoltzmann equation for a spin valve 9
can calculate the distribution function values everywhere 4 Applications
inside the spin valve using the scattering matrices. For
example,assumethe scatteringmatricesSL coversthein-
terval [x+,x−] and SR covers the interval [x−,x−]: In this section, we apply our numerical method to calcu-
0 3 3 5
latethespin-transfertorqueactingonthe rightferromag-
fout SL SL fin netic layer of a model spin valve. We compare the results
fo1u,Lt′ = SLLL SLLR f1in,L′ with equivalent calculations using the drift-diffusion ap-
(cid:20) 3,R (cid:21) (cid:20) RL RR(cid:21)(cid:20) 3,R(cid:21) proachandSlonczewski’shybridtheory.First,wedescribe
fin′ SR SR fout′ the approximations we make to simplify the calculation
"f5o3,u,RRt′# =(cid:20)SRRLLL SRRLRR(cid:21)" f35i,n,RR′ #. (43) so as to focus on the differences between the different ap-
proaches.
in/out′
f can be solved from Eq. (43), but the equations
3,R
in Eq. (43) are redundant, so we choose the half of the
equations that use the incoming boundary values rather
4.1 Approximations and their rationale
than the outgoing ones, i.e.,
fout′ =SL fin +SL fin′, (44a)
3,R RL 1,L RR 3,R Comparedwiththedrift-diffusionmethodandcircuitthe-
fin′ =SR fout′ +SR fin′. (44b) ory,themostimportantfeatureoftheBoltzmannmethod
3,R LL 3,R LR 5,R
is its treatment of electrons in the bulk moving in dif-
Fromthese equations,wecancalculatefin/out′.Similarly, ferent directions. In circuit theory, the average over the
3,R Fermi surface, the spin accumulation, is used to charac-
wecancalculatethe distributionfunctionvalue elsewhere terizethe electronsinside thatnode (herealayer).Inthis
using a different pair of scattering matrices SL and SR. treatment all of the electrons inside a node are effectively
aligned.Inthedriftdiffusionapproximation,thedistribu-
tionismodeledbyitsfirstmoment,thespinaccumulation,
3.8 Transport properties andsecondmomentwithrespecttovelocity,thespincur-
rent. This allows for greater flexibility in describing the
distribution function, but clearly if higher moments are
With the distribution functions in hand, it is straightfor-
important,neglectingthemwillleadtoerrors.InaBoltz-
wardtocalculatetransportpropertiesh(k)byintegrating
mannequationtreatmentofthetransportthedistribution
over the whole Fermi surface:
function is allowed its full flexibility.
Tostudythedifferentapproachesweconsiderasimple
h= h(k)dk‘. (45)
model in which we ignore the actual shape and/or size of
ZFS
the Fermi surfaces and assume that the Fermi surfaces in
Using Eq. (7), the integrations for spin density and spin both the non-magnet and the ferromagnet (both spin-up
current are discretized as and spin-down) are perfectly spherical and are the same
size. We use different mean free paths to distinguish the
N differences between the electrons in the non-magnet and
spin density: nsm(x)= wifms(x,ki), (46a) the ferromagnet: lN for non-magnet, lF↑,↓ for spin-up and
Xi=1 spin-down electrons in ferromagnet. The choice of iden-
N tical and spherical Fermi surfaces makes finding a wave
spin current: jsm(x)= wivifms(x,ki), (46b) vector mesh particularly simple. Since the problem is az-
i=1 imuthally symmetric, there are no contributions to the
X
distribution function that are not azimuthally symmet-
where s = 0,x,y,z for m = 1,3,5 (non-magnetic layers),
ric so that only polar variation need be considered. We
and s =↑,↓ for m = 2,4 (ferromagnetic layers). The spin
choose Gauss-Legendre sampling for the polar direction
current at x=x− is written in the x′-z′ frame:
3 with typically 40 mesh points.
For the interface scatteringcoefficientsin Eq.(27),we
Q(x−)=jx(x−)xˆ′+jz(x−)ˆz′, (47)
3 3 3 3 3 treat the case of ideal interfaces with no defect scatter-
ing.Consideran electronwith wave-vectork incident on
where the jz(x−) is the longitudinal piece parallel to the x
3 3 an interface. With the Fermi surface assumption made in
rightFM layer’smagnetization(z′ direction), andj3x(x−3) the previousparagraph,the wave-vectorsfor the reflected
is the piece perpendicular to z′-axis. From Ref. [11] we and transmitted electrons are −k and k , respectively.
x x
know that the perpendicular spin current is absorbed at The matrix elements of the reflection and transmission
the NM/FM interface, therefore the spin-transfer torque matrices R and T in Eq. (27) are calculated in Ref. [33].
acting on the right FM layer is Toallowforfiniteandspin-dependentinterfaceresistance
in the equal-Fermi-surface model, we assume δ-function
Nst =j3x(x−3)xˆ′. (48) scattering at the interface to give the following transmis-
10 Jiang Xiao et al.: A numerical method to solve theBoltzmann equation for a spin valve
Table 1. Material parameters usedintheBoltzmann calcula- TheparametersA,B,andq± arecalculatedusingthema-
tion. terialparametersandgeometriesasshowninRef.[15].An
equivalent spin-transfer torque formula was obtained by
Parameter Material Value Units Reference
Manschot, et al., [36] independently.
l Cu 110 nm [14] As discussed above in Sec. 4.1, the drift diffusion ap-
lsf Cu 450 nm [34] proximation assumes that the distribution function has a
l↑ Co 16.25 nm [14]
simple form consisting of a uniform expansion and a con-
l↓ Co 6.01 nm [14]
tribution proportional to the velocity. Thus, we can ex-
lsf Co 59 nm [35] pect that the drift-diffusion approximation breaks down
α↑ Co/Cu 0.051 [14] when the variation of the distribution function over the
α↓ Co/Cu 0.393 [14] Fermi surface is more complicated. There are three sit-
uations where more complicated behavior is introduced.
The first is when the transmission through the interface
depends strongly on wave vector as is typically the case
sion and reflection probabilities:
[37]. In the immediate vicinity of the interface, the wave-
α vector dependence of the transmission gives the distribu-
|RiNjN,σ|2 =|RiFjF,σ|2 = α +σ(kx)2δkix,−kjx, (49a) tion function a complicated variation over the Fermi sur-
σ i
face. This variation includes contributions that decay on
(kx)2
|TiNjF,σ|2 =|TiFjN,σ|2 = ασ+i(kix)2δkix,kjx, (49b) tfahceesoradreersoefptahreatmedeabnyfrmeeorpeatthh,anseethEisq.le(n1g3t)h.,Ifththeesitnrtoenrg-
variation decays between the interfaces, and the two ap-
where σ =↑ or ↓, and kx is a discretization of k . The
i x proaches can be brought into agreement through an ap-
parameter α is proportional to the square root of the
σ propriate choice of an effective interface resistance in the
strengthoftheδ-function-likeinterfacepotential.Thiscan
driftdiffusionapproach.However,whentheinterfacesare
bereadofffromthehorizontalaxisinFig.1ofRef.[33]us-
closer, the interaction of these exponential contributions
ingexperimentalspindependentinterfaceresistancedata.
between interfaces complicate the transport. Evaluating
We also use the relaxation-time approximation: Piσj = the importance of these effects requires a calculation us-
Pσ =AFS/τσ and Pisjf =Psf =AFS/τsf, where AFS is the ing realistic band structures, which is beyond the present
area of the Fermi surface. In this limit, the currentcarry- calculations.We insteadevaluatethe othertwosituations
ing eigensolution F2σ in Eq. (15) reduces to F2σ(x,ki) = where such difficult calculations are not necessary.
x+vxlσ/v , where lσ is the mean free path for different
i F Thesecondsituationinwhichthedistributionfunction
spins and v is the Fermi velocity.
F has a complicatedangulardependence is when the spacer
Using the algorithmdescribedabovein Sec. 2 and the
layeristhincomparedtoitsmeanfreepathandthe mag-
approximations discussed above (equal, spherical Fermi
netizations are not collinear. Figure 4 compares the an-
surfaces), we calculated the spin-transfer torque for the
gular dependence of the torque calculated with the drift
spin valve shown in Figure 1. For the rest of this paper,
diffusionapproximationtothetorquescalculatedwiththe
weassumethe non-magneticandferromagneticlayersare
Boltzmann equation. In these calculations, the reflection
composedofCuandCo,respectively.Theresultsarequite
at the interfaces has been set to zero so that the compli-
similar if Cu and Co are replaced by other non-magnetic
cations described in the previous paragraph do not play
and ferromagnetic metal. The input values in the Boltz-
a role. Inset (b) in Figure 4 shows that the torques agree
mann calculation are listed in Table 1.
when the spacer layer is very thick, and inset (a) shows
that when they are thin, there are significant differences.
The main panel shows the variation of the maximum of
4.2 Results and Comparisons thetorquecurvesasafunctionofthickness.Thedifference
between the curves gives the corrections due to the com-
In this section we test the drift-diffusion approach and plicated angular dependence of the distribution function.
Slonczewski’shybridtheorybycomparingtheresultswith
The torques decrease with thickness for two reasons.
thosefoundwiththeBoltzmannequation.Thespin-transfer
The large length scale decay is set by the spin diffusion
torque acting on the right ferromagnet layer (m layer in
length. When the layer is thicker than its spin diffusion
Figure 1) in a spin valve can generally be written in the
length, spin-flip scattering leads to a significant decrease
following form [1]:
in the polarization of the current that crosses from one
sidetotheother.Forspacerlayersthinnerthantheirspin
~I
NR(θ)=η(θ) mˆ ×(mˆ ×Mˆ), (50) diffusion length, but longer than their mean free paths,
st 2e thepolarizationofthecurrentdependsontheratioofthe
effective polarized resistance to the effective unpolarized
where the cross product has magnitude of sinθ. In Slon-
resistance. For these structures, in which the ferromag-
czewski’s hybrid theory, [15,25]
netic layers are thicker than their spin diffusion length,
q q the polarization of the current decays roughly like one
+ −
η(θ)= + . (51) over the thickness of the spacer layer.
A+Bcosθ A−Bcosθ