Calculating the transport properties of magnetic materials from first-principlesincluding thermal and alloy disorder, non-collinearity and spin-orbit coupling

Anton A. Starikov, Yi Liu,∗ Zhe Yuan,† and Paul J. Kelly‡

Faculty of Science and Technology and MESA+ Institute for Nanotechnology,University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

(Dated: June 17, 2021 version rv1d)

A density functional theory based two-terminal scattering formalism that includes spin-orbit cou-pling and spin non-collinearity is described. An implementation using tight-binding muffin-tinorbitals combined with extensive use of sparse matrix techniques allows a wide variety of inho-mogeneous structures to be flexibly modelled with various types of disorder including temperatureinduced lattice and spin disorder. The methodology is illustrated with calculations of the temper-ature dependent resistivity and magnetization damping for the important substitutional disorderedmagnetic alloy Permalloy (Py), Ni80Fe20. Comparison of calculated results with recent experimentalmeasurements of the damping (including its temperature dependence) indicates that the scatteringapproach captures the most important contributions to this important property.

PACS numbers: 71.15.Rf, 72.10.-d, 72.10.Bg, 72.10.Di, 72.25.-b, 72.25.Ba, 72.25.Rb

I. INTRODUCTION

As long as device dimensions were much larger thanthe spin-flip diffusion length of the constituent materi-als, the effect of the electron spin on transport proper-ties went largely undetected. When attention focussedon magnetic materials in thin film and multilayer form,new properties such as interface magnetic anisotropy andoscillatory exchange coupling emerged, culminating inthe discovery of giant magnetoresistance1,2 (GMR) al-most 30 years ago.3 This heralded the emergence of thefield of spintronics,4 which exploits the spin of electronsin addition to the charge used in conventional electron-ics, triggering a flood of new discoveries including tun-neling magnetoresistance (TMR),5,6 spin-transfer torque(STT),7–9 the spin Hall effect10,11 the spin Seebeck ef-fect, etc.12 Spin-dependent electron transport manifestsitself on microscopic length scales in magnetically in-homogeneous systems such as magnetic bilayers, multi-layers and magnetic textures where interface and finitesize effects are dominant. As important as the funda-mental physics of spin-dependent transport are the ap-plications that spintronics makes possible. The GMReffect allowed magnetic read heads to be miniaturisedand led to an explosion in the density of data that couldbe stored on a hard disk. The TMR effect in magnetictunnel junctions (MTJs) forms the basis for new formsof non-volatile storage, magnetic random access memo-ries (MRAM); MTJs are also used as sensor elements inread heads. STT makes it possible to write informationin MRAMs more efficiently leading to STT-RAMs13,14

or to make microwave frequency STT oscillators (STOs)where the injected spin forces a magnetisation to precesswith GHz frequency.8,15,16 Passage of a spin-polarisedcurrent can also cause a domain wall to move, which isthe principle behind a form of shift register called “race-track memory”.17,18

The search for new and improved kinds of magneticstorage provided another focus of attention in the field

of spintronics: magnetisation dynamics in response toexternal fields and currents in nanoscale systems.19 Thephysics of such devices involves two major contributions:(i) spin-dependent scattering of electrons in bulk ma-terials and at interfaces, and (ii) spin-non-conservingscattering of electrons because of spin-orbit coupling(SOC) when spin is no longer a good quantum num-ber, or because of magnetic disorder. A breakdownof spin-conservation is essential for spin-relaxation pro-cesses that are described with material dependent time-and length-scales, conventionally the Gilbert dampingparameter α and the spin-flip diffusion length lsf , respec-tively. Predicting and controlling these properties is veryimportant for understanding and designing new spin-tronic devices leading to numerous experimental20–27 andtheoretical28–31 material-dependent studies on the sub-ject. The development of a new theoretical framework32

for calculating magnetization damping and its implemen-tation in the framework of density functional theory33–37

has motivated systematic reinvestigation of the damp-ing in alloys38,39 and of the temperature dependenceof damping in permalloy40 allowing quantitative con-frontation of theory and experiment without invokingadjustable parameters such as the relaxation time in thetorque correlation method (TCM).29–31,41

In this paper we describe in detail a method we re-cently used to calculate the resistivity ρ, spin flip dif-fusion length (SDL), and Gilbert damping parameterfor Ni1−xFex substitutional alloys,33 the resistivity anddamping for the itinerant ferromagnets Fe, Co and Niwith thermal disorder,34 the resistance42 and anisotropicdamping43 of magnetic domain walls, the nonadia-batic STTs in ballistic systems,44 interface-enhanceddamping,45 thermal disorder effects in transport46 anda novel interface spin Hall effect.47 It extends earlierwork48–50 by including SOC and non-collinearity.

Central to the method is the scattering formalism51

for the conductance of a two-terminal device.52 The sys-tem under investigation is attached to reservoirs by semi-

arX

iv:1

805.

1006

2v1

[co

nd-m

at.m

es-h

all]

25

May

201

8

2

S R L

Z

FIG. 1. A sketch of a two-terminal configuration for the scat-tering problem. A gray scattering region (S) is sandwichedbetween left (L) and right (R) semi-infinite leads that havetranslational symmetry. A current flows along the directionof the z-axis.

infinite leads that support well defined scattering states(Fig. 1). For crystalline leads, these states are right- andleft-propagating Bloch states that are incident upon thescattering region and either reflected from it or transmit-ted through it. The probability amplitude that the νth

right-propagating state incident from the left lead withspin σ′ is scattered into the µth right-propagating statewith spin σ in the right lead defines the transmissionmatrix element tσσ

′

µν . The crystal momenta and bandindices of the scattering states are labelled by µ or ν.Similarly, the reflection matrix rσσ

′

µν can be defined forright-propagating states that are reflected back into left-propagating states in the left lead. Denoting leads as left(L) and right (R) for a two-terminal system, we have twotransmission matrices tLR and tRL, for electrons comingfrom left- and right-leads, and, similarly, two reflectionmatrices rLL and rRR. Together, they form the scatter-ing matrix S

S =

(rLL tRLtLR rRR

)(1)

that contains all the information needed to study a num-ber of important physical properties of the system. Thebest known such property is the conductance G that canbe expressed according to the Landauer-Buttiker formal-ism as52,53

G =e2

hTrtt† . (2)

The scattering formalism is not restricted to calculationsof the conductance but can provide us with useful in-formation about spin-dynamics and spin-relaxation pro-cesses in the scattering region. In particular, it can beused to calculate the Gilbert damping parameter α andthe spin-flip diffusion length lsf .

33,45,46

The present study is based upon a first-principles tight-binding (TB) linearized-muffin-tin-orbital (LMTO) im-plementation of the scattering formalism. TB-LMTOsform a minimal basis set54–56 that allows us to con-struct a highly efficient computational method, especiallywhen combined with sparse-matrix techniques.57,58 Incombination with the local spin-density approximation(LSDA) from density functional theory (DFT), it allows

us to study physical systems either entirely ab initio,i.e., without introducing any free parameters or in thecase of finite temperature transport, a minimal numberthereof. We extend earlier work49,50 by introducing non-collinear magnetism59 and spin-orbit interaction usingthe Pauli-Schrodinger Hamiltonian. We generalize thewave-function matching (WMF) technique introduced byAndo51 to eliminate the need for a principal layer decom-position and dispense altogether with partitioning thescattering region into layers. Compared to the widelyused recursive Green’s function method, the interactionrange in the leads and scattering region is arbitrarily longwithout loss of numerical stability and optimal use canbe made of sparse-matrix solvers which greatly improvesthe computational efficiency. Not having to divide thescattering geometry into “blocks” or “layers” results ingreater flexibility in applications to disorder.

We illustrate how this framework can be used toinvestigate spin-dependent transport in the diffusiveregime and spin-relaxation phenomena using recent de-velopments in scattering theory and spin-dynamics.32 InSect. II we outline the technical details of the method andillustrate it by applying it to the Ni80Fe20 alloy permal-loy in Sect. III. Some technical details of the SOC imple-mentation with LMTOs are given in a set of appendicesA. Appendix B contains a brief discussion of some lim-itations of the method as well as numerical tests aboutthe exchange-correlation functional in the LSDA and thebasis set of TB-LMTOs.

II. FORMALISM

To solve the scattering problem for the infinite systemdepicted in Fig. 1, we need to solve the single-particleSchrodinger equation

(H− EI) Ψ = 0, (3)

at some specified energy E, usually the Fermi energy. Weassume that the ground state charge and spin-densitiesand Kohn-Sham potentials for all atoms in the systemhave already been calculated self-consistently. Here, Ψis a vector of coefficients Ψi when the wave function Ψis expanded in some localised orbital basis (i ≡ Rlmσfor the MTOs we will use where R is an atom site indexand lmσ have their conventional orbital angular momen-tum and spin meaning; see Appendix A 1). H is theHamiltonian matrix in the localized orbital basis and asummation over i is implied in (3). Its sparsity is de-termined by the range of the localized orbitals which isminimal for TB-MTOs. The system in Fig. 1 is infi-nite so that the dimensions of the Hamiltonian H andunit matrix I in Eq. (3) are both infinite. By apply-ing the “wave-function matching” method51 the semi-infinite leads with full translational symmetry can be re-placed with appropriate boundary conditions in the formof energy-dependent embedding potentials on the bound-ary layers. This reduces the problem to a finite size and

3

results in a two-stage process for calculating the scatter-ing matrix. In the first stage, to be discussed in Sec. II A,eigenmodes um of the leads are calculated by solving theSchrodinger equation for each of the leads in turn tak-ing translational symmetry into account. By calculatingtheir wave vectors km and velocities vm, the eigenmodescan be classified as being either left-going um(−) or right-going um(+). They form a basis in which to expand anyleft- and right-going waves in the leads and their trans-formation under a layer translation in the leads is easilycalculated by using a generalization of Bloch’s theoremfor complex k.51

In the second stage, discussed in Sec. II B, these solu-tions from the first stage can be used to construct theenergy-dependent boundary conditions for the Hamilto-nian in the scattering region, which can be a slab of arandom alloy, a single interface, a multilayered struc-ture, a tunnel junction, a slab of thermally disorderedmaterial, etc. One then has to solve a system of linearequations (LEQs) with the original Hamiltonian modi-fied by incorporating the boundary conditions in the roleof a coefficient matrix to obtain the wave functions Ψthat provide all information about the scattering in thesystem and can be used to calculate the transmissionand reflection probability amplitudes, tµν and rµν , andmore. As a result of choosing a localized basis to mini-mize the hopping range, the Hamiltonian matrix is verysparse. This can be exploited by using efficient numeri-cal methods such as incomplete LU-factorisation (whichtakes into account the sparsity of the matrix) to solve theLEQs. The solution scales linearly with the extent of thescattering region in the transport direction.

Once the scattering matrix is known, we can extractthe resistivity and Gilbert damping parameter as dis-cussed in Sec. II C. The scattering formalism will be pre-sented in its general form not depending on details of theunderlying basis set and Hamiltonian; the most relevantaspects of the LMTOs used in the current implementa-tion are sketched in Appendix A 1. In Sec. II D we discussways of modelling different kinds of disorder using largesupercells transverse to the transport direction.

A. Eigenstates of ideal leads

We make use of an assumed two-dimensional (2D)translational symmetry in the plane perpendicular to thetransport direction to characterise states in this and thenext section with a lateral wave vector k‖ in the cor-responding two-dimensional Brillouin zone (2D BZ). Allvariables therefore have an implicit dependence on k‖that will be suppressed for simplicity. When we refer tothe number of atoms (orbitals) in a layer, we refer to thefinite number number of atoms (orbitals) in a transla-tional unit cell.

Because the ideal leads have translational symmetry inthe transport direction, they can be decomposed into aninfinite number of translationally invariant layers. When

Hi-N Hi-2 Hi-1 Hi

B1 11

Hi+2 Hi+N

B2 BN BN

†

B2†

B1†

Hi+1

FIG. 2. Hamiltonian matrix of an ideal quantum wire parti-tioned into slices determined by the translational symmetryof the leads. Hi ≡ Hi,i is the on-layer term of the Hamilto-

nian, Bl ≡ Hi,i+l and B†l ≡ Hi,i−l describe hopping to the

lth and −lth neighbouring layers, respectively.

the hopping range of the Hamiltonian of this systemis greater than the corresponding periodicity, i.e., whenhopping to layers beyond the nearest neighbouring layersis not negligible, the usual approach would be to increasethe layer thickness until only hopping between neighbour-ing layers occurs; these are called principal layers. Ingeneral the principal layer procedure results in increasedcomputational cost and decreased accuracy. To remedythis, we formulate the WFM method for arbitrary hop-ping range between layers, thus generalizing previous for-mulations of the WFM method.49–51,60

We start with an ideal wire with translational sym-metry (Fig. 2), in which every layer contains NO atomcentred orbitals and is coupled to some number (N) oflayers to its left and right. Then the Schrodinger equa-tion for the ith layer is given by

(EI−Hi) Ψi +

N∑l=1

(BlΨi+l + B†lΨi−l

)= 0 , (4)

where Hi ≡ Hi,i is the on-layer term of the Hamiltonian,

Bl ≡ Hi,i+l and B†l ≡ Hi,i−l describe hopping to the lth

and−lth neighbouring layers respectively, and Ψi+l is thewave function on the lth neighbouring layer. Taking intoaccount translational symmetry in the periodic crystal,the wave function on any arbitrary layer l is related tothe wave function on layer l − 1 by a generalized Blochfactor λ as

Ψl = λΨl−1 . (5)

Combining (4) and (5) leads to the generalised eigenvalueproblem of rank 2×N ×NO

(EI−H0) Ψ0 +

N−1∑l=1

(BlΨl+B†lΨ−l

)+ B†NΨ−N

=−λBNΨN−1, (6a)

Ψl = λΨl−1, ∀ l ∈ [−N + 1, N − 1], (6b)

where, without loss of generality, i = 0 has been as-sumed. Non-trivial solutions of (6) can be separated intotwo classes. The first class consists of N ×NO left-goingwaves, the second class of N × NO right-going waves.

4

HL,-N-2 HL,-N-1 HL,-N HL,0 HS HR,N HR,0 RRR NR HR,N+1 HR,N+2

FIG. 3. Geometry of the finite scattering problem comprising left (L), and right (R) leads sandwiching the scattering (S)region that is augmented by a finite number N of lead layers chosen to be sufficiently large so that there is no hopping fromthe scattering region proper to the (white) lead layers.

Each class can contain both propagating Bloch wavesand nonpropagating, evanescent waves. The correspond-ing Bloch factors are denoted by λ(+) and λ(−). Ofthese N×NO solutions, only NO Bloch factors λ(+) cor-respond to the translation of right-propagating waves toa neigbouring layer, the rest of the λ(+) factors describetranslations to more distant layers and do not provideany additional information; thus we have only NO uniquetranslation factors among the λ(+). Similarly, for left-propagating waves there are only NO unique translationsin the set of λ(−) factors. By using only the NO or-bitals belonging to the l = 0 layer with eigenvectors from(6) corresponding to the set of unique translation fac-tors λ(±), we construct normalised eigenvectors um(±)(≡ Ψl=0,m where l is the layer-index and m is the modeindex) and use these to form the NO ×NO matrices51

U(±) = (u1(±) · · · uNO (±)) . (7)

Any arbitrary wave function on the l = 0 layer can thenbe represented as a linear combination of left- and right-propagating waves

Ψ = Ψ(+) + Ψ(−) , (8)

and any left- or right-propagating wave can be expandedin terms of the eigenstates of the lead as

Ψ(±) = U(±)C(±) , (9)

where Cµ(±) is a vector of coefficients. We define theNO ×NO diagonal eigenvalue matrices by

Λ(±) = δnmλm(±) . (10)

Using the Bloch condition (5) and following Ando’s orig-inal procedure49–51,61 we define the translation matrices

F(±) = U(±)Λ(±)U−1(±) . (11)

The translation of the wave function on layer i over anarbitrary number of layers l is then given by

Ψi+l = Fl(+)Ψi(+) + Fl(−)Ψi(−) , (12)

allowing us to construct the full solution for the entirelead.

B. The scattering problem

The scattering region S is now inserted between theleft and right leads. It is important to emphasize thatwe do not define a layered structure inside the scatteringregion. Regardless of its size or contents, the scatteringregion acts as one large meta-layer. The resulting prob-lem is infinite but by making use of the translationalsymmetry in the leads and the solutions obtained in theprevious section, the leads can be incorporated in thescattering problem in the form of boundary conditionsimposed in the lead layers adjoining the scattering region.The system is partitioned as shown in Fig. 3 where theinfinite scattering geometry is truncated to include onlyN +1 (translationally invariant) lead layers on the leftand right in addition to the original (disordered) scat-tering region where N is the hopping range (in termsof number of layers) in the Hamiltonian describing theleads.

Specifically, we assume that the N+1 lead layers at-tached to the original scattering region on the left sideare indexed as −N, . . . , 0. Then the wave function inthe layer in the left lead with index −(N +Q), whereQ > 0 i.e., the wave function in the white layers onthe left in Fig. 3, can be related to the wave functionΨL,−N = ΨL,−N (+) + ΨL,−N (−), a superposition ofleft- and right-propagating waves, as

ΨL,−N−Q = ΨL,−N−Q(+) + ΨL,−N−Q(−)

= F−QL (+)ΨL,−N (+) + F−QL (−)ΨL,−N (−) (13)

=[F−QL (+)− F−QL (−)

]ΨL,−N (+) + F−QL (−)ΨL,−N ,

allowing us to express an arbitrary ΨL,−N−Q in terms ofΨL,−N and ΨL,−N (+). The set of Schrodinger equationsfor layers −N, . . . , 0 become

5

(EI−HL,n) ΨL,n +

−n∑l=1

BL,lΨL,n+l + BLS,nΨS +

N∑l=1

B†L,lΨL,n−l = 0 ∀ n ∈ [−N, 0], (14)

where BLS,n is the coupling between the lead layer on the left with index n (in the truncated transport geometry)to the (original) scattering region S, BL,l is the hopping to the l-th next layer in the left lead, and ΨS is the wave

function in the scattering region. By splitting the summation∑Nl=1 =

∑N+nl=1 +

∑Nl=N+n+1 in the last term on the

lhs and using (13) to eliminate ΨL,n (∀ n<−N), equation (14) can be transformed to

(EI−HL,n) ΨL,n +

−n∑l=1

BL,lΨL,n+l + BLS,nΨS +(1− δN,−n

)N+n∑l=1

B†L,lΨL,n−l +

N∑l=N+n+1

B†L,lF−l+N+nL (−)ΨL,−N

= −N∑

l=N+n+1

B†L,l

[F−l+N+nL (+)− F−l+N+n

L (−)]ΨL,−N (+) ∀n ∈ [−N, 0] . (15)

This effectively acts as a boundary condition for the left-hand side of (15) and removes a direct dependence on thewave functions to the left of the −N th layer.

Injection of electrons from the left electrode leads to only right-propagating waves on the right-hand side of thescattering region and the wave function in the layer with index N + Q can be related to the wave function in therightmost layer (N) within the truncated transport geometry as

ΨR,N+Q = FQR(+)ΨR,N (16)

allowing ΨR,N+Q to be eliminated from the Schrodinger equation

(EI−HR,n) ΨR,n +

n∑l=1

B†R,lΨR,n−l + B†RS,nΨS +(1− δN,n

)N−n∑l=1

BR,lΨR,n+l +

N∑l=N−n+1

BR,lFl−N+nR (+)ΨR,N = 0

∀n ∈ [0, N ] , (17)

where B†RS,n is the coupling between the scattering re-

gion and the lead layer with index n (in the truncatedtransport geometry), and BR,l is the hopping to the l-thnext layer in the right lead.

Combining the sets of equations (15) and (17) with theSchrodinger equation for the scattering region

(EI−HS) ΨS +

N−1∑l=0

[B†LS,lΨL,−l + BRS,lΨR,l

]= 0

(18)

results in a set of inhomogeneous LEQs. By assumingthat ΨL,−N (+) = UL(+) and solving the LEQs withmultiple right-hand sides in one go, we can obtain thewave functions in the system for electrons that are in-jected in all possible propagating modes of the left lead(i.e., modes with |λ| = 1).

Bloch states incident from the left and propagatingto the right are scattered by the breaking of translationsymmetry into left-going states on the left-hand side andright-going states on the right-hand side as

ΨL,−N (−) = UL(−) r , (19a)

ΨR,N (+) = ΨR,N = UR(+) t . (19b)

Once we know the set of wave functions Ψ for all incom-ing states from the left lead we can calculate the elements

of the matrices r = rµν with dimension ML×ML, whereML is the number of propagating modes in the left lead,and t = tµν with dimensions MR ×ML

r = U−1L (−)

[ΨL,−N −UL(+)

], (20a)

t = U−1R (+)ΨR,N . (20b)

The elements of the physical reflection and transmissionprobability amplitude matrices can be found by normal-ising with respect to the currents

rµν =

√vL,µ(−)

vL,ν(+)rµν ; tµν =

√vR,µ(+)

vL,ν(+)tµν (21)

where vR,L(±) are the group velocities of the eigenmodesin the left and right leads, which are determined using theexpressions derived in Appendix A 2

vν(±) =2a

~

N∑n=1

n Im[λnν (±)u†ν(±)Bnuν(±)

]. (22)

When SOC is included, spin is no longer a good quan-tum number and separating the equation of motion fordifferent spinors is not possible. Nevertheless, it willbe convenient for the purposes of analysing our results

6

to decompose the matrices rµν and tµν into the spin-projections rσσ′µν and tσσ′µν . This is discussed in Ap-pendix A 3.

When only nearest neighbour hopping is allowed (N =1), the expressions in sections II A and II B reduce toexpressions known from earlier work.49–51,61 For arbi-trary values of N they represent a generalised WFMtechnique that is similar to the widely used recursiveGreen’s function method. The advantage is that propertreatment of sparsity of the resulting LEQs allows us touse efficient sparse-matrix LEQ solvers which drasticallyimproves computational efficiency. Additionally, depar-ture from the recursive Green’s function method allowsus to describe the scattering region without introduc-ing “blocks” or “layers”. This eliminates numerical is-sues and simplifies application of the method in compli-cated, incommensurable systems. The equivalence of theWFM method with the Kubo-Greenwood formalism inthe linear-response regime was shown earlier.61

C. Extracting material-specific parameters fromthe scattering matrix

Once we know the scattering matrix (1) consisting ofthe rσσ′µν and tσσ′µν matrices calculated from the left andright-hand sides, we can use this to extract various bulk(and interface) parameters currently of interest in thefield of spintronics. We focus here on the bulk resistivityand Gilbert damping of the important Ni80Fe20 ferro-magnetic alloy, permalloy as illustrative examples.

1. Resistivity

The total resistance of a diffusive conductor (e.g. al-loy) of length L sandwiched between two identical ideal(ballistic) leads can be expressed as

1/G = 1/GSh +R , (23)

where G is the total conductance of the system andGSh =

(2e2/h

)N is the Sharvin conductance of each lead

with N conductance channels per spin. R, the resistanceof the scattering region corrected for the finite conduc-tance of the ballistic leads,49,62 has two contributions

R(L) = 2Ri +Rb(L) , (24)

where Ri is the resistance of a single alloy|lead inter-face, and Rb(L) is the bulk resistance of an alloy layer ofthickness L. For a sufficiently thick alloy layer, Ohmicbehaviour is recovered when Rb(L) ≈ ρL, where ρ is thebulk resistivity.

In materials whose SOC is weak, the transport of elec-trons is found to be well described by considering cur-rents of spin-up and spin-down electrons separately. Fora stack of materials comprising ferromagnetic (FM) andnonmagnetic (NM) metals, the resistance of each spin

channel is obtained by adding resistances in series, thetwo spin channels are then added in parallel accordingto the “two-current series-resistor” (2CSR) model.63–65

When a ferromagnetic alloy is sandwiched between non-magnetic leads, each spin species sees two spin-dependentinterface resistances Rσi and a spin-dependent bulk term:Rσ(L) = 2Rσi + ρσL. The total resistance that resultsfrom adding these terms in parallel can be written as

R(L) =2Ri

(β2 − 2βγ + 1

)1− γ2

+ ρL+

+4R2

i

(β2 − 1

)(β − γ)

2

(γ2 − 1)[

(γ2 − 1) ρL+ (β2 − 1) 2Ri] , (25)

in terms of the total interface resistance Ri given by

1

Ri=

1

R↑i+

1

R↓i, (26)

the corresponding interface spin asymmetry γ = (R↓i −R↑i )/(R

↑i + R↓i ), the total resistivity of the alloy ρ given

by

1

ρ=

1

ρ↑i+

1

ρ↓i, (27)

and the corresponding bulk spin asymmetry β = (ρ↓ −ρ↑)/(ρ↓ + ρ↑).

When the SOC can no longer be considered weak, eightparameters are used to describe the resistance of a diffu-sive NM|FM|NM system.66 Two parameters are requiredto describe the NM metal, a resistivity ρNM and a spin-flip diffusion length lNM. Three parameter are requiredto describe the FM metal: a resistivity ρσFM for eachspin channel as well as a spin-flip diffusion length lFM.And three parameters are required to describe the in-terface; an interface resistance Rσi for each spin channeland an interface spin-flip scattering parameter δ (alsocalled the spin memory loss parameter).67–69 The non-trivial evaluation of all of these parameters would go be-yond the present task of illustrating the use of the scat-tering formalism and will be the subject of a separatepublication.70 For the purpose of extracting a resistivityfrom a series of calculations of R(L) we will use (25) inthe form

R(L) = a+ ρL+ b/(ρL+ c) . (28)

For sufficiently thick slabs where the third term in (28)vanishes, ρ will be extracted from the slope of R(L).Otherwise all 4 independent parameters will be used toperform the fit. Both approaches will be examined inSec. III B.

2. Gilbert damping

The magnetisation dynamics of ferromagnets is com-monly described using the phenomenological Landau-

7

Lifshitz-Gilbert equation

dM

dt= −γM×Heff + M×

[G(M)

γM2s

· dMdt

], (29)

where Ms = |M| is the saturation magnetisation, G(M)is the Gilbert damping parameter (that is in general atensor) and the gyromagnetic ratio γ = gµB/~ is ex-pressed in terms of the Bohr magneton µB and the Landeg factor, which is approximately 2 for itinerant ferromag-nets. For a monodomain ferromagnetic layer sandwichedbetween nonmagnetic leads, NM|FM|NM, the energy dis-sipation due to Gilbert damping is

dE

dt=

∫V

d3rd

dt(Heff ·M)

=

∫V

d3rHeff ·dM

dt=

1

γ2

dm

dt· G(M) · dm

dt(30)

where m = M/Ms is the unit vector of the magnetisa-tion direction for the macrospin mode. By equating thisenergy loss to the energy flow into the leads71 associatedwith “spin-pumping”,72

IPumpE =

~4π

Tr

dS

dt

dS†

dt

=

~4π

Tr

dS

dm

dm

dt

dS†

dm

dm

dt

,

(31)

the elements of the tensor G were expressed in terms ofthe scattering matrix32

Gij(m) =γ2~4π

Re

Tr

[∂S

∂mi

∂S†

∂mj

]. (32)

Physically, energy is transferred from the slowly varyingspin degrees of freedom to the electronic orbital degreesof freedom where it is rapidly lost to the lattice (phonondegrees of freedom). Our calculations focus on the roleof elastic scattering as the rate-limiting first step.

To calculate the Gilbert damping tensor Gij using (32),we need to numerically differentiate the scattering ma-trices with respect to the magnetisation orientation m.Expressing this orientation in spherical coordinates (θ, φ)with the polar angle θ = 0 corresponding to the equilib-rium magnetization direction m, we vary the magnetisa-tion direction about θ = 0 to calculate the 2×2 dampingtensor in a plane orthogonal to m. Specifically, ∂S/∂mi

in (32) can be replaced by ∂S/∂ei, where ei are compo-nents of the Cartesian basis vectors in the plane orthogo-nal to m and φ0 defines the orientation of the coordinatesystem in this plane. Then the derivatives of the scatter-ing matrix can be approximated as

∂S

∂e1≈ S(∆θ, φ0)− S(∆θ, φ0 + π)

2∆θ, (33a)

∂S

∂e2≈ S(∆θ, φ0 + π/2)− S(∆θ, φ0 + 3π/2)

2∆θ, (33b)

where ∆θ is a small variation of the polar angle. Substi-tution of (33) into (32) yields four elements of the 3× 3

damping tensor for any particular orientation m of themagnetization. For cubic substitutional alloys the damp-ing can be assumed to be isotropic (see Appendix A 4), sowe limit ourselves to differentiating about a single orien-tation m and our primary interest will be in the diagonal

elements of G = Gii. When the damping is enhanced byFM|NM interfaces,22,72–74 the total damping of a ferro-magnetic slab of thickness L sandwiched between leads

can be written G(L) = Gif + Gb(L) where Gif is the in-terface damping enhancement and we express the bulkdamping in terms of the dimensionless Gilbert dampingparameter α as

Gb(L) = αγMs(L) = αγµsAL , (34)

where µs is the magnetisation density and A is the crosssection.

D. Modelling disorder

The structures used in spintronics studies are typicallystacked layers of magnetic and nonmagnetic materialsthat exhibit various types of disorder. The magneticmaterials themselves are frequently magnetic alloys likepermalloy that are intrinsically “chemically” disorderedand are chosen to have desirable magnetic properties.Even when two materials (like Fe and Cr) have the samecrystal structure (bcc) and are closely lattice matched, itis not possible to exclude intermixing at an interface andbecomes desirable to be able to model it.

Other important material combinations such aspermalloy and Pt have a large lattice mismatch. Forthin layers, this can be accommodated by straining ei-ther or both layers (pseudomorphic growth) but above acritical thickness these will relax to their preferred struc-tures with or without the formation of misfit dislocations.Since fully relaxed interface structures can only be mod-elled using lateral supercells,45,47 we apply the supercellapproach to all forms of disorder studied in this work.

Lastly, many experiments are performed at room andelevated temperatures making it desirable to take intoaccount temperature induced lattice and spin disorder.Our approach will be to model such thermal disorder inlarge lateral supercells. By doing so we will be able tomake contact with a large body of experiments that havebeen interpreted with phenomenological models65 thatassume a diffusive transport regime.

1. Chemical disorder (random alloys)

To study bulk alloys and interface mixing, we can cal-culate atomic sphere (AS) potentials self-consistently us-ing the coherent-potential approximation (CPA) imple-mented with MTOs75,76 or with periodic supercells.54–56

Transport calculations are performed with large lateralM × N supercells in which atomic sites are randomly

8

populated with AS potentials subject to the constraintimposed by the stoichiometry of the targeted experimen-tal system. This can be done either by enforcing thedesired stoichiometry layer by layer or globally.

For example, to simulate a (001) simple cubic A25B75

random alloy using an 8×8 lateral supercell, we can ran-domly assign 16 out of the 64 sites to A atoms and therest to B atoms maintaining the 25:75 stoichiometry inevery layer as sketched in Fig. 4. In the second case,we assign elements A and B randomly throughout thecomplete slab of material that is chemically disordered.In both cases, configuration averaging is carried out byrepeating the scattering calculations for a number of dif-ferent realisations of random disorder.

2. Positional (thermal) disorder

Thermal lattice disorder (or other kinds of positionaldisorder) can be modelled in lateral supercells by displac-ing atoms in the scattering region from their equilibriumpositions, denoted Ri, by a randomised displacement vec-tor ui for each atomic site resulting in the new set of

atomic coordinates Ri = Ri + ui (Fig. 5). The scat-tering matrices are then calculated for a number of suchdisordered configurations and the results averaged. Themain physical approximation which is invoked is the adi-abatic approximation. Though formally problematic fora metal with a gapless spectrum, within the frameworkof the lowest order variational approximation (LOVA) tothe Boltzmann equation, the adiabatic approximation isfound to describe the thermal and electrical transportproperties of transition metals very well.77

Different approaches to generating ui are possible,ranging from ab-initio molecular dynamics, through first-principles lattice dynamics,46 to parameterized Gaussiandisorder.34,46 The latter and simplest approach, that isadopted here, is based upon the harmonic approximationwhereby the energy cost of displacing atoms is quadraticin their displacements. Components of ui are then dis-tributed normally with a root-mean-square (rms) devia-tion ∆ that can be chosen in different ways. It can be

FIG. 4. Illustration of the configuration for a supercell withchemical disorder. Two arbitrary layers in an 8× 8 supercellare shown for an A25B75 alloy. Atoms of type A are blackand atoms of type B are grey.

FIG. 5. Schematic of frozen thermal lattice disorder. Atoms(gray balls) on an ideal lattice (left panel) are displaced fromtheir equilibrium positions by random vectors ui to form astatic configuration (right panel) for the electronic scatteringcalculation.

related to the temperature and extracted from experi-ment within the Debye model. Or it can be chosen toreproduce an experimental temperature dependent resis-tivity. As the particular method of generating displace-ments does not affect the implementation of the trans-port method, we will not concern ourselves overly withthe relationship between the displacements ui and thetemperature in this paper.

3. Non-collinear configurations and magnetic disorder

For collinear ferromagnets (or antiferromagnets), ther-mal spin disorder can be modelled by rotating magneticmoments in the scattering region away from their equi-librium orientations34 (Fig. 6). To lowest order in thepolar angles describing this orientation, the energy variesquadratically and temperature-induced spin disorder canbe modelled with Gaussian disorder. In a magnetic do-main wall separating domains in which the magnetiza-tion is collinear, the magnetization rotates continuouslyfrom one preferred orientation to the other. Addressingboth these problems, spin disorder and domain walls,requires an implementation of the scattering formalismwhereby atoms on different sites can have differently ori-ented magnetization directions. This is simply achievedin the atomic spheres approximation (ASA).78–80

FIG. 6. Schematic of frozen thermal spin disorder. Atomicmagnetic moments (arrows) of a ferromagnetically orderedsystem (left panel) are tilted by random polar angles θi(and azimuthal angles φi, not shown) to form a static spin-disordered configuration (right panel) for the electronic scat-tering calculation.

9

We assume that the spin quantisation axis σz of thecollinear system and the spatial z-axis are collinear withthe direction of transport. We then rotate the local mag-netic moment (exchange potential) on an arbitrary site(atomic sphere) by rotating all spin-dependent atomicparameters, which are 2×2 operators in spin-space [suchas the potential function Pα (A5) or SOC parameters(A13)], using the rotation operator

Rsi = exp( iσyθi

2

)exp

( iσzφi2

), (35)

where θi and φi are polar and azimuthal angles in the AS

on site Ri, while leaving spin-independent operators likethe structure constant matrix unchanged. The LMTOHamiltonian (A18) can then be constructed in the usualway using matrix operations on the modified operators.

By using a suitable distribution of spinor-rotationangles, we can simulate thermal disorder or orderedstructures like domain walls.34,42–44 For thermal disor-der, we can choose φi to be random while assuming aGaussian distribution for θi with an angular rms devi-ation ∆Θ related to the temperature using some modele.g. to reproduce an experimental temperature depen-dent resistivity34 or magnetization.46 Alternatively, wecan calculate the interatomic exchange interactions fromfirst-principles and use these to determine the magnondispersion relations. By occupying the magnon modesfor some chosen temperature, random sets of φi and θican be generated and used as input to a scattering cal-culation in a frozen-magnon approximation.46 Details ofhow θi, φi depend on temperature fall outside the scopeof this paper.

Although the above scheme is only applied to magneti-zation fluctuations about the global quantization axis inthis paper, we have applied it to nontrivial noncollinearmagnetizations such as spin spirals and magnetic domainwalls in references [42–44]. By explicitly taking the spa-tially varying magnetization into account, we calculatedthe domain wall resistance,42 the enhancement of theGilbert damping by noncollinearity,43 and the nonadi-abatic STT parameter44 in domain walls with differentprofiles. It could equally well be applied to study trans-port properties in spin glasses81 or amorphous magnets82

where the Gilbert damping is generally an anisotropictensor depending on the symmetry,83,84 as demonstratedfor magnetic domain walls.43

III. CALCULATIONS

The scattering calculations are carried out in two dis-tinct steps. In the first step semirelativistic85 AS poten-tials are calculated self-consistently for the atoms in thestructure we are interested in, starting with the calcula-tion of “bulk” potentials for the left and right leads. ASpotentials can be calculated self-consistently for the scat-tering region using the surface Green’s function (SGF)method86 or a supercell approach with a conventional

“bulk” band structure code.54–56 For substitutional ran-dom alloys, this is done very efficiently by combiningthe SGF method with the CPA.86 In the second step,the WFM method outlined in Sect. II is used to calcu-late the scattering matrix for the fully relativistic Pauli-Schrodinger Hamiltonian using a TB-MTO basis; for de-tails see Appendix A 1. In this step the scattering statesin the left and right leads are first determined follow-ing Sect. II A and then the scattering problem is solvedaccording to Sect. II B.

Although the calculations are entirely ab initio in thesense that the computational scheme does not containany free parameters, the results do depend on the nu-merical implementation that is necessarily approximate.In this section we discuss a number of relevant issues andillustrate some potential difficulties for a Cu|Py|Cu sys-tem consisting of a length L of permalloy sandwiched be-tween copper leads. Both fcc materials are chosen to havea (111) orientation in the transport direction. The lat-

tice constant of permalloy is taken to be aPy = 3.5412Aaccording to Vegard’s law. This is slightly smaller thanthe experimental lattice constant of Cu, aCu = 3.614A.As we will be interested only in the bulk properties ofthe alloy, we will choose the lattice constant of the cop-per leads to be equal to that of permalloy and ignorethe (small) modifications introduced into the electronicstructure of Cu that may result.

Should the lattice mismatch be important, as is thecase when modelling the interface properties of an A|Binterface between materials A and B, the lattice constantratio aA/aB can be approximated by the ratio of twointegers NA and NB such that NAaA ∼ NBaB . Whenthe A and B lattices are chosen to be aligned, this canresult in unfavourably large values of NA and NB . Bydropping the alignment condition, more flexibility canbe achieved by searching for lattice vectors in each latticewhose lengths match; in general this will require rotatingthe lattices with respect to one another. This approachmade it possible to study fully relaxed interfaces betweenpermalloy and the nonmagnetic materials Cu, Pd, Ta andPt.45,47

A. Modelling diffusive transport with supercells

The (lateral) supercell approach allows us to flexiblymodel interfaces between materials with different crys-tal structures and lattice constants by imposing a degreeof lattice periodicity in order to be able to use Bloch’stheorem. A substitutional alloy, or a crystalline materialat finite temperature has, however, no translational sym-metry and the supercell approach is formally only correctin the thermodynamic limit. Just as practical experiencehas shown that periodic boundary conditions can be veryeffectively used to model symmetry breaking surfaces, in-terfaces, impurities, etc. with very small supercells, wewill see that we can model diffusive transport with lateralsupercells of very modest size.

10

1. Γ-point calculations with large lateral supercells

By assuming lattice periodicity parallel to the inter-face, the wave functions can be characterized by a wavevector k‖ in a 2D BZ no matter how large the periodmight be. In the thermodynamic limit, this BZ becomesvanishingly small, the band dispersion becomes negligi-ble and neither BZ sampling nor configuration averag-ing over configurations of disorder should be necessary.We explore this limit in Fig. 7 where the dependence ofthe resistance of a Cu|Py|Cu system is shown as a func-tion of the Py slab thickness L for 5 × 5, 15 × 15 and25 × 25 supercells without SOC, i.e. examining the ma-jority and minority spin subsystems separately and ne-glecting the effects of periodicity entirely by using onlyk‖ = (0, 0) ≡ Γ.

The first feature we observe in Fig. 7 is a roughly two-order-of-magnitude difference between the resistances ofmajority and minority spins. Such a difference is qualita-tively consistent with what is known about the mean-freepaths of electrons in the two spin-channels.87 For minor-ity spins, this is extremely short with reported values inthe range 4 − 8 A while for majority spins it is muchlarger, in the range 50− 200 A.88,89 We can understandthis43 in terms of the energy bands that were calculatedfor fcc Fe and Ni using the AS potentials calculated self-consistently for permalloy with the CPA,86,90 shown inFig. 8. At the Fermi energy, the majority-spin bands forNi and Fe are almost identical so that in a disorderedalloy the majority-spin electrons see essentially the samepotentials on all lattice sites and are only very weaklyscattered by the randomly distributed Ni and Fe poten-tials. In contrast, the minority-spin bands are quite dif-ferent for Ni and Fe which can be understood in terms ofthe different exchange splittings; the magnetic momentscalculated for Ni and Fe in permalloy in the CPA are0.63 and 2.61µB , respectively. The random distribution

0.05

0.10

Rm

aj

0

10

20

Rm

in

5×5

0.65

0.70

0.75

Rm

aj

0

10

20

Rm

in

15×15

0 10 200.6

0.7

0.8

L [nm]

Rm

aj

0 10 200

10

20

L [nm]

Rm

in

25×25

MAJ MINN×N

FIG. 7. Resistance in fΩ m2 of a Cu|Py|Cu structure as a func-tion of the thickness L of Py for Γ-point calculations for 5×5(top), 15 × 15 (middle), and 25 × 25 (bottom) lateral super-cells, for majority (left column) and minority (right column)spins, with 10 random configurations of chemical disorder perthickness.

of Ni and Fe potentials in permalloy then leads to strongscattering of minority-spin electrons in transport. Thispicture is consistent with previous calculations of the re-sistivity and Bloch spectral function of permalloy.91–93

As we approach the diffusive limit, we expect to see alinear dependence of the resistance on the slab thickness.The minority spins clearly exhibit this behaviour evenfor the smallest supercell size studied, N × N = 5 × 5.Increasing N only reduces the spread between results fordifferent configurations of alloy disorder. For the major-ity spins the situation is rather different: the resistancedepends non-linearly on L with notable oscillations thatwe attribute to constructive and destructive interferenceof electron waves in the Fabry-Perot-like Cu|Py|Cu cav-ity. Only for thick Py or large supercells do the oscilla-tions vanish, restoring the expected linear dependence ofthe resistance for L > 10 nm.

In the case of a 5×5 supercell, with only 5 layers of Py(∼ 1 nm, the smallest Py slab thickness L in Fig. 7), thesystem size is already comparable to the mean free pathof electrons in the minority spin-channel, and we are inthe diffusive limit. For the majority-spin channel, thelateral dimensions of the slab only begin to approach thereported mean free path88,89 for a 25× 25 supercell andeven for such large supercells the resistance only showsdiffusive (linear) behaviour when the length of the slabis larger than the mean free path.

Once the system is large enough to approach the dif-fusive limit and the length dependence of the resistancebecomes linear, a resistivity ρΓ can be determined fromthe slope of R versus L. The dependence of ρΓ on thesupercell size N is shown in Fig. 9. One can see that inthe minority spin case the resistivity is converged witha negligible error bar to ρΓ

min ∼ 105µΩ cm when the lin-ear dimensions of the supercell are much larger than themean free path. For the majority spin case, as discussedabove, we are not yet in the diffusive limit and quan-

-9

-6

-3

0

3

E-E F (e

V)

Majority Spin Minority Spin

X Γ L

-9

-6

-3

0

3

E-E F (e

V)

X Γ L

Ni Ni

Fe Fe

FIG. 8. Band structures calculated with the Ni and FeAS potentials and Fermi energy that were calculated self-consistently for Ni80Fe20 using the coherent potential approx-imation. The same AS radii were used for Ni and Fe.

11

tum (interference) effects are still observable. For suffi-ciently large values of L we extract resistivity values oforder ρΓ

maj ∼ 0.5µΩ cm. Though the “error bar” that re-sults from configuration averaging is small, there is stilla strong dependence on the size of the lateral supercell.

2. Small supercell with integration over 2D BZ

Although Γ-point calculations with a large supercellmight be the most direct way of simulating a diffusivemedium, the computational cost is very high. We there-fore study the effect of improving the sampling of thewave functions by making use of Bloch’s theorem. Foran N×N lateral supercell, we calculate the transmissionusing a Q × Q set of k‖ points in the 2D BZ associ-ated with the supercell. The total transmission is givenby summation of partial transmissions. The same effectcould be obtained for a QN ×QN supercell with Γ pointsampling only (and with disorder in an N ×N unit cellartificially repeated Q×Q times).

The resistance of a Py slab is shown in Fig. 10 for a5× 5 supercell as a function of the thickness L for differ-ent Q × Q samplings of the 2D BZ. As discussed in theprevious section, for a 5× 5 supercell, especially for ma-jority spins, we were far from the diffusive limit and theΓ point picture was dominated by interference effects;these are still visible in the top left subplot in Fig. 10when only 4 × 4 k points are used for the BZ integra-tion. Nevertheless, even though the resistance displaysoscillatory behaviour for any individual k point, theseoscillations average out with increasing BZ sampling den-sity resulting in a substantially linear dependence visiblein the bottom left subplot of Fig. 10 for a sampling of256×256 k-points. For minority spins the picture is sim-pler, as we already approach the diffusive limit for a 5×5supercell and therefore even a quite small BZ sampling

0.00.5

1.01.5

2.0

ρm

aj

0 0.05 0.1 0.15 0.2100

110

120

130

1/N (SC size is N×N)

ρm

in

51020N =

MAJ

MIN

FIG. 9. Resistivity in µΩ cm of Py as a function of the super-cell size for Γ-point calculations: for majority (upper panel)and minority (lower panel) spins. The “error bars” measurethe spread on averaging different configurations of alloy dis-order.

results in a very linear dependence. Once the sampling isdense enough and we observe linear diffusive behaviour,we can extract values of the Py resistivity from the slopeof R(L). In the limit that L→ 0, the resistance does notvanish because of the interface resistance and the finite(Sharvin) conductance of the ideal leads.

To investigate the convergence of this procedure andcompare with the Γ-only limit, we plot in Fig. 11 theresistivity calculated for 5 × 5 and 15 × 15 supercells asa function of the equivalent BZ sampling NQ ×NQ to-gether with the results for the Γ-only calculations for anN ×N supercell from Fig. 9. For majority-spin electronsΓ-only calculations are dominated by interference effectsand far from convergence so an acceptable estimate ofthe resistivity could not be made. When the BZ sam-pling is increased, then the results for both 5 × 5 and15 × 15 supercells converge rapidly to the same value of0.57 ± 0.01µΩ cm for majority spin electrons, suggest-ing that this is the true calculated (albeit experimentallyunobservable because SOC has been omitted) resistivity.For minority spins, the BZ-integrated 15×15 supercell re-sult provides us with a converged value of 105±1µΩ cm,which is similar to the Γ-only result with a large super-cell. The converged value for a 5×5 supercell is 4% larger(109±1µΩ cm), which can be attributed to the error thatresults from the limited averaging of configuration spacepossible with a limited supercell size; for a mean-free-path of 1 nm, a volume containing only 5 × 5 × 5 atomsis sampled by a minority spin electron before undergoinga collision. For our present purposes, this error is notbig enough to justify the greater expense associated withlarger supercells and we will limit ourselves to the 5× 5supercell with 32 × 32 2D BZ sampling throughout therest of this paper. This is something which should be

0.70

0.75

0.80

Rm

aj

0

10

20

Rm

in

4×4

0.70

0.75

0.80

Rm

aj

0

10

20

Rm

in

48×48

0 10 20

0.70

0.75

0.80

L [nm]

Rm

aj

0 10 200

10

20

L [nm]

Rm

in

256×256

MAJ MINQ×Q

FIG. 10. Resistance in fΩ m2of a Py slab in a Cu|Py|Cuscattering geometry as a function of the slab thickness L for a5×5 supercell with different k-point samplings of the Brillouinzone. The normalised area of the 2D element used in the BZsummation is defined as ∆2k|| = ABZ/Q

2 where ABZ is thearea of the downfolded supercell BZ. Results are shown forQ = 4 (top), 48 (middle), and 256 (bottom) for majority (leftcolumn) and minority (right column) spins, with 6 randomconfigurations of chemical disorder for every value of L.

12

0.0

0.5

1.0

1.5ρ

maj

0 0.05 0.1 0.15 0.2100

110

120

130

1/(N×Q)

ρm

in51020NQ =

MAJMIN

15×155×5

5×5

15×15

FIG. 11. Resistivity in µΩ cm of Py as a function of equivalentk-point sampling (N × Q) in which the downfolded 2D BZwith area ABZ for an N × N supercell is sampled with BZelement ∆2k|| = ABZ/Q

2 for 5 × 5 (dash-dotted blue line)and 15 × 15 (solid red line) supercells. For comparison theresults of Γ-only calculations (Q = 1) with variable supercellsize are also shown (dashed black line). Results for majorityand minority spins are shown in the upper and lower panels,respectively. The converged values ρmaj = 0.57 µΩ cm andρmin = 105 − 109 µΩ cm are in very good agreement withthe calculated values in the literature91 around 0.6 and 100µΩ cm, respectively.

born in mind when comparing to experiment where cal-culations should be explicitly tested for convergence withrespect to lateral supercell size as well as BZ sampling.Additional tests of other aspects of the numerical imple-mentation of the method can be found in Appendix B.

B. Resistivity calculations with SOC

Fig. 12 shows the thickness dependence of the Py layerresistance where SOC was included with the magnetiza-tion perpendicular to the current direction (R⊥) and par-allel to it (R‖). What is most striking about these resultscompared to those without SOC in Fig. 10 is the nonlin-earity of R(L). When a current of electrons is injectedfrom the Cu lead on the left into disordered Py, they neednot scatter immediately at the interface but do so on alength scale measured in terms of the elastic mean freepath. However, we do not see any evidence for such an ef-fect in the absence of SOC (Fig. 10) and there is no goodreason why SOC should greatly alter this. When we in-clude SOC, spin is no longer a good quantum number andthe unpolarized current must adapt to the finite polariza-tion of Py. This it does asymptotically on a length scale

given by lPysf , the spin-flip diffusion length in Py which

was calculated in Ref. 33 to be ∼ 5.5 nm in good agree-ment with values determined experimentally.69,94 How-ever, in Fig. 12 R appears to be varying nonlinearly on

a length scale much larger than this value of lPysf . To

understand why, we must return to calculations for Py

0 20 40 60 80 100L [nm]

0

1

2

3

4

5

R (fΩ

m2 )

R||

R⊥

ρ||=2.70±0.02 µΩ cm

ρexp=4.2−4.8 µΩ cm (Poly.)ρ⊥=2.15±0.01 µΩ cm

FIG. 12. Resistance calculated for Cu|Py|Cu using a 5 × 5supercell with SOC as a function of the layer thickness Lwith the magnetization parallel to (R‖: dots, solid line) andperpendicular to (R⊥: squares, dashed line) the current di-rection. Calculated results are shown by symbols while thelines are fits using the 2CSR model. The fitted resistivities arenearly a factor of two smaller than experimental values in therange 4.2–4.8 µΩ cm measured for polycrystalline samples atlow temperature,95–98 where the presence of grain boundariescan increase the resistivity.99,100

without SOC.Within the 2CSR model the resistances of the individ-

ual spin channels are first determined and then addedin parallel to determine the total. These individual spinresistances are shown in Fig. 13 for the same system asstudied in Fig. 12 but with the SOC switched off. Themajority and minority spin resistances are perfectly lin-ear (except for a small mean-free-path effect showing upfor a Py thickness smaller than 2 nm in the majority spinchannel). The total resistance shown in the bottom panelexhibits a curvature that is absent in the individual spinchannels and in addition, the slope is about a factor offour smaller than with SOC included. We can under-stand the curvature from (25) or by (28); we only expectto observe the linear behavior characteristic of Ohm’slaw when the third term on the right hand side of theseequations is negligible compared to the other terms (orvanishes). This only happens when ρL Ri (or β = γ).

It is instructive to try and extract a value for the resis-tivity from Fig. 13(c) while pretending not to know theslopes of ρmaj = 0.57±0.01µΩ cm and ρmin = 109µΩ cmfrom Figs. 13(a) and 13(b) that when combined resultin ρnonrel = (ρ−1

maj + ρ−1min)−1 = 0.567 ± 0.009µΩ cm. We

can do this either by fitting the expression (28) to thecalculated data or by studying systems so long that thecontribution of the bulk resistivity to the total dominatesthe interface terms and can be extracted from the slopeof R(L). A linear fit of R(L) for L > 20 nm yields a totalresistivity of 1.08 ± 0.07µΩ cm which is almost twice aslarge as the true value, ρnonrel, demonstrating that thenon-linear contribution from the interface terms in (28)

13

0.6

0.7

0.8

0.9

R↑ (f

Ω m

2)

0

10

20

30

40

R↓ (f

Ω m

2)

0 5 10 15 20 25 30 35

0.5

1

1.5

R (f

Ω m

2)

(a)

(b)

(c)

MAJ

MIN

TOTAL

L [nm]

FIG. 13. Resistance calculated for Cu|Py|Cu for a 5×5 super-cell without SOC as a function of the Py slab thickness L for(a) majority spin electrons, (b) minority spin electrons and(c) total. The multiple symbols for a given length are resultsfor different configurations of disorder. Linear least-squaresfits for L > 20 nm are shown by solid lines, a non-linear leastsquares fit for total resistance is represented by a dashed line.

is still not negligible even though R(L) looks reasonablylinear. Estimating the resistivity with higher accuracyusing this approach requires calculations for much longersystems.

The alternative is to fit R(L) using (28). To ensure astable fitting procedure we use an iteratively re-weightedleast squares algorithm with bisquare weights.101 In thiscase the estimated total resistivity is 0.53 ± 0.05µΩ cm.Though this is in much better agreement with ρnonrel, anadditional error of 7% has nevertheless been introducedand the errorbar itself has increased by a factor 5.

Returning now to Fig. 12, we find that the R(L) datawith SOC shown as symbols can be fitted very well using(28) (solid and dashed lines) yielding resistivity values ofρ‖ = 2.70 ± 0.02µΩ cm and ρ⊥ = 2.15 ± 0.01µΩ cm.The average resistivity ρ = (ρ‖ + 2ρ⊥)/3 = 2.33 ±0.02µΩ cm is a factor four larger91,99 than ρnonrel =0.567±0.009µΩ cm. This compares reasonably well withCPA calculations performed within the Kubo-Greenwoodformalism92,93,99 but is almost a factor of two smallerthan experimental values in the range 4.2 − 4.8µΩ cmmeasured for polycrystalline samples.95–98 Note that theresistivity can be significantly enhanced by the grainboundaries.100 The magnetoresistance anisotropy valuewe estimate is (ρ‖ − ρ⊥)/ρ× 100% = 24± 1% that com-pares reasonably with experimental values in the range16− 18%95–97 and previous theoretical estimates.92,93,99

Our calculations confirm the overall picture that inspite of its smallness for 3d materials, SOC plays an es-sential role in determining the transport properties ofalloys when there is a very large difference between theresistivities of majority and minority spins in the absenceof SOC. When temperature-induced lattice and spin dis-

order are included below, the bulk resistivity will increaseand the curvature seen for low values of L decrease; it willturn out that low temperature Py is peculiarly difficultto describe accurately in our real space approach becauseof the large mismatch between the two spin channels.

Influence of the Interface

Determining an alloy resistivity from calculations thatinclude SOC using a 2CSR model that neglects it en-tirely is unsatisfactory. The 2CSR model does identifyan important issue however, namely the essential roleplayed by interfaces in the scattering formalism. Clearlythe interface is obscuring the character of the bulk prop-erty we want to study by introducing a number of ex-traneous effects: mean-free-path effects at the interface,spin-dependent interface resistances, interface and bulkspin flipping required to bring the unpolarized currentinjected from the Cu lead into equilibrium with the spin-polarized current in Py. Since the asymptotic resistiv-ity should be independent of the leads, ideally we wouldchoose lead materials that are perfectly matched to theproperties of the alloy we are studying with the same cur-rent polarization and minimal interface resistance. How-ever, we are constrained in our choice of lead material tochoose something with full lattice periodicity. We exam-ine a number of possibilities in this section.

We could choose leads to be ordered alloys with thesame chemical composition as Py. However, for an arbi-trary Ni1−xFex chemical composition this might requireusing impossibly large unit cells. Instead we adopt thevirtual crystal approximation (VCA)102 and constructartificial atoms with nuclear charge ZVCA = (1−x)ZNi +xZFe and a corresponding number of neutralizing valenceelectrons. A self-consistent calculation with this proce-dure for an fcc lattice with the same lattice constant as Pyresults in a magnetic moment of 1.06µB ; the correspond-ing bands are shown in Fig. 14 (top row). We see thatthe majority spin states (lhs) are free electron like whilethe minority spin d states (rhs) are partly occupied sowe expect a better matching of the electronic structuresat the interface that should result in a smaller interfaceresistance. The result of calculating R‖(L) using theseVCA leads is shown in Fig. 15 (black dots). The curva-ture for low values of L is strongly reduced compared toFig. 12 and a resistivity value of ρ‖ = 2.76± 0.01µΩ cmis directly extracted from the linear dependence.

The procedure can be further refined by noting that

the 1/L term in (25) vanishes if R↓i → ∞ i.e., γ = 1.This situation would correspond to using halfmetallic fer-romagnetic (HMF)103 leads. There is no need to actuallyuse a “real” HMF, we can construct one from Cu leadsby simply adding a strong (∼ 1 Ry) repulsive potentialto the minority spin Cu lead potentials to eliminate allminority spin states from the vicinity of the Fermi level(Fig. 14, lower panels). In this way only majority spinsare injected into Py whose low temperature polarization

14

-9

-6

-3

0

3E-

E F (eV)

Majority Spin Minority Spin

X Γ L-10

-5

0

5

10

E-E F (e

V)

X Γ L

VCA Fe20Ni80 VCA Fe20Ni80

HMF Cu HMF Cu

FIG. 14. Top row: self-consistent virtual crystal approxima-tion band structure for a nuclear charge ZVCA = (1−x)ZNi +xZFe. The majority spin states (lhs) are free electron like, theminority spin states have mainly 3d character (rhs). Bottomrow: majority (lhs) and minority (rhs) spin band structures ofCu in which a replusive constant potential of 1 Rydberg hasbeen added to the minority spin potential, the effect being toremove all minority spin states from the Fermi energy.

we calculated to be β ∼ 0.9.46 If we do this for the left-hand lead we obtain the results shown in Fig. 15 as reddots; constructing both leads in this way, we obtain theresults shown as blue dots. In both cases, we achievenearly ideal Ohmic behaviour. We attribute the smalldeviations from linearity for small values of L to spin-

flipping on a length scale of lPysf as the fully polarized

injected electrons equilibrate to β ∼ 0.9. The slopes areessentially identical within the error bars of the calcula-tion, consistent with the slope obtained with VCA leads

0 20 40 60 80 100L [nm]

0

1

2

3

4

5

6

R|| (f

Ω m

2 )

2 HMF Cu Leads1 HMF Cu LeadVCA Fe20Ni80 Leads

FIG. 15. Resistance calculated with SOC as a function of thePy slab thickness L. The black dots are calculated using theVCA Ni80Fe20 leads, while the red (blue) dots are obtainedusing Cu leads with one (both) of the Cu leads replaced bythe artificial HMF Cu. The solid lines are linear fits to thecalculated values.

10−5

10−4

10−3

10−2

10−1

0.096

0.098

0.100

0.102

0.104

G/(

γ µ

sA)

[nm

]

∆θ [π ]

FIG. 16. Gilbert damping of a 11.2 nm thick layer of Ni80Fe20alloy as a function of deviation of the polar angle ∆θ used fornumerical differentiation. Calculations are performed using a5× 5 supercell.

with ρ ∼ 2.8µΩ cm and only slightly larger than the valuewe extracted with unpolarized Cu leads, ρ ∼ 2.7µΩ cm.

All theoretical estimates of the permalloy resistivityare below the reported experimental range ρ = 4.2 −4.8µΩ cm.95–98 The discrepancy has been explained bynoting that theoretical calculations have been carriedout for monocrystalline, mono-domain permalloy, whileexperimental observations are made on polycrystallinesamples.99 In the latter case, additional scattering onthe grain boundaries increases the resistivity. We alsohave no information about the domain structure of theexperimental samples, but one can expect that for multi-domain samples the resistivity should further increasedue to the additional scattering involved.

C. Gilbert damping

The Gilbert damping can be calculated by numericallydifferentiating the scattering matrices with respect to themagnetization orientation as formulated in Sec. II C 2.

The value of G resulting from the differentiation maydepend on the choice of the finite but small value of ∆θchosen for the numerical procedure. An example is shownin Fig. 16 for Py in the Cu|Py|Cu system. One can see

that numerically G is very stable and does not dependon the choice of ∆θ for variation of the polar angle overa large range, indicating linear dependence of the scat-tering matrix on small variations of the magnetisationorientation.

When the thickness of the Py layer is increased, thetotal damping of the system grows proportionally, as an-ticipated in (34) and demonstrated in Fig. 17. More-over, the strict linearity and negligible variation for dif-ferent configurations of disorder indicate that the Gilbertdamping is very insensitive to details of how the randomchemical disorder in the alloy is modelled. The value of

α extracted from the slope of G(L)/(γµsA) is 0.0046 ±0.0002, which falls at the lower end of a range of val-ues, 0.004−0.013, reported in the literature for measure-ments at room temperature.22,24–27,39,40,73,105–114 Veryrecently, measurements were carried out as a functionof temperature from room temperature (RT) to low tem-

15

peratures, decomposing the damping into bulk and in-terface contributions.40 These yield a value for the bulkdamping of 0.0048 ± 0.0003 at 5 K in remarkably goodagreement with the value calculated above (that has beenconfirmed by subsequent coherent potential approxima-tion calculations36,37 within the error bars of the calcu-lations).

At room temperature, Zhao et al. reported a valueof α = 0.0055 ± 0.0003 that is at the low end of the0.004− 0.013 range measured previously. An even morerecent RT study of the Ni1−xFex alloys as a functionof x reported39 values of α in essentially perfect agree-ment over the full concentration range with the valuescalculated by Starikov et al. for T = 0.33 Schoen et al.attributed the better agreement they obtained with the-ory to corrections they made (i) for interface dampingenhancement22,23,73,115 and (ii) for radiative damping.For Py they reported a RT value of α = 0.0050. Takentogether with the Zhao et al. result, α = 0.0055±0.0003,this would seem to indicate a minimal temperature de-pendence of the magnetization damping that is strikingin view of a factor five increase in the resistivity of Pyover the same temperature range. To elucidate this (andthe apparent disagreement between the T = 0 calcula-tions of Starikov et al.33 and the older room temperatureexperiments), we undertook a study of the effect of ther-mally induced lattice and spin disorder on the resistivityand damping that will be the subject of the next section.

The ingredients contributing to the magnetizationdamping in the calculations are disorder and SOC; omit-ting either will lead to a vanishing bulk damping. Inthe next section we will examine what happens when we“tune” the amount of disorder. Before doing this, we in-vestigate how the damping depends on the magnitude ofthe SOC term in (A12) when we scale it with a parame-ter λ: Hso → λHso. From the results shown in Fig. 18 forPy it is clear that the damping scales quadratically withthe strength of the SOC. This is the scaling expectedfor the strong interband scattering limit of the torquecorrelation model. Though only strictly applicable toordered solids with well-defined band structures,29–31,41

0 5 10 15 20 250.0

0.1

0.2

L [nm]

G/(

γµ

sA

)[n

m]

αcalc= 0.0046 ± 0.0002αexp = 0.0048 ± 0.0003 (T= 5 K)

FIG. 17. Total damping of the Py slab as a function of itsthickness L.104 The error bars, which measure the spread overdifferent configurations of disorder for every thickness, are lessthan the marker size in this plot and thus nearly invisible.The damping extracted via a linear fitting (blue solid line),αcalc = 0.0046 ± 0.0002, is in excellent agreement with theexperimental value reported at low temperature.40

0

0.05

0.1

0.15

0.2

α

SOC multiplier

1 2 3 4 5 60

0.2

0.4

α1

/2

SOC multiplier

FIG. 18. Scaling of Gilbert damping with the SOC strength.The dots corresponds to the calculated results, and the solidline shows the quadratic fit. The quadratic dependence ofα on the SOC strength is in agreement with a recent CPAcalculation for Os-doped Py.36

the strong interband scattering limit is the appropriatelimit for a disordered alloy where momentum is not welldefined.

D. Thermal lattice and spin disorder

To resolve the discrepancy between room tempera-ture measurements22,24–27,39,73,105–114 and T = 0 calcu-lated values of α for the Ni1−xFex alloy system33 andbecause we are aware of only a single low temperaturemeasurement,40 we extend to Py the method introducedin Ref. 34 to study the temperature dependence of damp-ing in Fe, Co and Ni. Finite temperatures lead to dis-placements of the atoms from their equilibrium positions(lattice disorder) and to rotations of atomic magneticmoments away from their equilibrium orientations. Acorrect theoretical description of temperature effects insolids would begin with the fundamental lattice and spinexcitations (phonons and magnons, respectively) and theoccupancy of these excitations46 but this lies outside thescope of the present paper. Instead we apply a sim-pler scheme of uncorrelated Gaussian atomic and spindisplacements34 to study the effects of thermal latticeand spin disorder on the resistivity and damping of Py.

We describe lattice disorder in terms of independentrandom displacements ui of the NS atoms in the scat-tering region labelled i from their equilibrium positionsRi i.e., we describe the lattice as a collection of inde-pendent harmonic oscillators, see Fig. 5. The displace-ments ui are distributed normally with rms displace-ment ∆ =

√∑i u

2i /NS . As shown in Fig. 19, increas-

ing ∆ leads to increased scattering and increased resis-tivity. For ∆/a0 = 0.029 corresponding to a resistivityof 8.2µΩ cm, the resistance R(L) of the Cu|Py|Cu sys-tem (unpolarized Cu leads) is shown as a function ofL in Fig. 20. The increased bulk resistivity leads to a

16

substantial decrease of the curvature observed in Fig. 12underlining the peculiar difficulties presented by low tem-perature Py.

While a rms displacement of 4% (in units of the lat-tice parameter a0) is enough to cause an almost four-fold increase in the resistivity, the damping increases byonly ∼ 5%. Fig. 19 therefore indicates that while the re-sistivity might depend strongly on additional structuralsources of scattering in Py such as dislocations, grainboundaries etc., the damping is expected to be insensi-tive to this additional disorder. The resistivity estimatedtheoretically for crystalline Py at zero temperature canbe expected to be lower than the values determined ex-perimentally for polycrystalline samples.

The weak dependence of α on lattice disorder is consis-tent with the quadratic scaling of the damping with theSOC strength expected in the strong interband scatter-ing limit of the torque correlation model. The electronicstructure of Py strongly resembles that of clean Ni insofar as the majority spin d band is filled (seen clearlyin the VCA, Fig. 14); it is known from experiment20

and TCM calculations in the strong interband scatteringlimit29,31,41 that the damping parameter of Ni dependsonly weakly on the relaxation time in this limit.

Reverting to the ideal crystal structure, we next in-troduce a certain level of spin disorder, as sketched inFig. 6, by tilting the atomic moments randomly fromtheir equilibrium orientations through angles θi that areassumed to be distributed normally with a rms tilt an-gle ∆Θ =

√∑i θ

2i /NS . The results plotted in Fig. 21

show that the resistivity depends strongly on spin disor-der (suggesting that measured resistivity values shouldalso depend on the domain structure of the samples);for the range of ∆Θ shown, it increases by a factor ofalmost ten. Compared to the lattice disorder case, thedamping parameter also increases more strongly, by al-most 20%. A major factor in this increase is, however,

468

1012

ρ || [µΩ

cm

]

0 0.01 0.02 0.03 0.04

4.4

4.6

4.8

5.0

α [×

10−

3 ]

∆/a0

[10-

3 ]

FIG. 19. Resistivity ρ‖ (upper panel) and damping parameter(lower panel) in Ni80Fe20 alloy as a function of the rms dis-placment of atoms from their equilibrium positions (in unitsof the lattice constant a0).

the reduced effective magnetisation density µs in (34) as∆Θ increases. Note that these calculations assume thatthe external magnetic field is negligibly weak.

The qualitative picture sketched above can be im-proved by introducing quantitative measures for the rmsdisplacements and rotations in terms of the temperatureT . In the Debye model116 the mean-square displacementof the i-th atom from equilibrium 〈u2

i 〉 depends on T as

〈u2i (T )〉 = ∆2 =

3~2T

mkΘ2D

[Φ(ΘD

T

)+

ΘD

4T

], (36)

where ΘD is the Debye temperature and Φ(x) is the De-bye integral function

Φ(x) =1

x

∫ x

0

y

ey − 1dy . (37)

Expanding the integrand in (37) as a power series in y

leads to Φ(x) = 1 − x4 + x2

36 + ... so that in the hightemperature limit where x < 1, Φ(x) ' 1− x/4 and (36)reduces to the classical statistics116

∆2 =3~2T

mkΘ2D

. (38)

In the low temperature limit T ΘD(x 1), zero point

motion (zpm) is dominant and ∆2 = 3~2

4mkΘD. Because it

does not contribute to give a low temperature resistivity,we neglect the zpm at T = 0 but keep the complete Debyeintegral (37) for finite temperature calculation.

To map spin disorder onto temperature, we canuse a cubic spline to interpolate the experimentalmagnetisation117,118 at an arbitrary temperature46 or fitthe experimental data using some empirical analyticalfunction.119 We assume that the magnetisation at finite

0 20 40 60 80 100L [nm]

0

2

4

6

8

10

R|| (f

Ω m

2 )

∆/a0=0∆/a0=0.029

FIG. 20. Resistance calculated for Cu|Py|Cu with lattice dis-order corresponding to ∆/a0 = 0.029 in a 5× 5 supercell andincluding SOC as a function of the layer thickness L. Themagnetization is parallel to the current direction. The resultswithout lattice disorder are included for comparison. Thesolid lines illustrate the linear fits to the calculated values.

17

10

20

30ρ || [µ

Ω c

m]

0 0.05 0.1 0.15 0.2

4.5

5.0

5.5

α [×

10−

3 ]

∆Θ [π]

[10-

3 ]

FIG. 21. Resistivity (upper) and damping parameter (lower)in Ni80Fe20 alloy as a function of rms deviation of angle θibetween atomic magnetic moments and macrospin.

temperature can be defined in terms of the mean valueof the cosine of the tilting angle θ

M(T ) = M0〈cos θ〉 . (39)

For atomic tilting angles θi distributed according to theFisher distribution120 with probability density

P (θ, κ) =κ sin θ eκ cos θ

2 sinhκ, (40)

the expectation value of the z-projection of the local mag-netisation can be expressed via the distribution parame-ter κ as 〈cos θ〉 = cothκ− 1/κ which is just the Brillouinfunction for large J and κ ∼ 1/kBT . This results in atranscendental equation which relates temperature andmagnetic moment distribution

M(T )

M(0)= cothκ− 1

κ. (41)

The azimuthal angle φ in (35) defining the orientation ofthe projection of the magnetic moment in the xy planeis assumed to be distributed uniformly in [0, 2π] which isreasonable for an isotropic material like Py.

The thermally induced random field satisfies the fluc-tuation dissipation theorem, i.e. the time-averaged corre-lation function of the random field is proportional to theGilbert damping and depends on the temperature.121–123

In our modeling of the spin disorder, we do not explicitlyinvolve the random field but generate several snapshotsof magnetic configurations which are essentially “inde-pendent” of one another. So the implementation usingthe random distribution of the atomic magnetic moments(40) does not violate the fluctuation dissipation theorem.

The temperature dependent resistivity and dampingcalculated for Py using ΘD = 450K and the experi-mental magnetization118 are compared with experimentin Fig. 22. For temperatures around room-temperature,the model of independent harmonic oscillators describes

0

10

20

30

40

ρ− (µΩ

cm

)

0 100 200 300 400 500Temperature (K)

4.0

4.5

5.0

5.5

6.0

α (1

0-3)

ZhaoSchoenCalc.

Calc.

Expt.

FIG. 22. Resistivity (upper) and bulk damping (lower) ofNi80Fe20 as a function of the temperature. The red sym-bols are the calculated results (the red line is a guide to theeye). The asterisks (black) in the resistivity plot correspondto tabulated average experimental values from.124 The aster-isks (green) in the damping plot are the data of Zhao et al.40

The blue diamond is a room temperature measurement cor-rected for spin pumping and radiative contributions.39

phonon motion reasonably well and the phonon spectrumof transition metals is quite similar to the spectrum de-scribed by the Debye model. The agreement between thecalculated and experimentally observed values of ρ(T )124

and of α(T )40 is remarkably good. The results for thedamping confirm an earlier report of an at best weaktemperature dependence of α.125 A recent room temper-ature measurement39 of the damping containing correc-tions for spin pumping and radiative effects is in almostperfect agreement with our RT result. We note that themeasurements of Zhao et al.40 were corrected for spinpumping but not for radiative effects.

IV. CONCLUSION

We have developed a method to calculate the scat-tering matrix including SOC and non-collinearity andillustrated it here with calculations of the resistivityand Gilbert damping of permalloy. The very efficientimplementation with tight-binding muffin tin orbits al-lows it to be applied to a wide range of materials andsystems. In addition to zero-temperature calculationsfor ideal disordered alloys,33 the method can be usedto model temperature-induced disorder,34,46 for systemssuch as interfaces that are not periodic,45,47 or for non-collinear systems.42,43 Our results for disordered alloysare in agreement with or have been confirmed by otherestablished theoretical methods like CPA. Where com-parison can be made, our results are in good agreementwith experiment so they can be used to predict material

18

parameters.We acknowledge many useful discussions with Rien

Wesselink and Kriti Gupta. This work was financiallysupported by the “Nederlandse Organisatie voor Weten-schappelijk Onderzoek.” (NWO) through the researchprogramme of “Stichting voor Fundamenteel Onderzoekder Materie,” (FOM) and through the use of super-computer facilities of NWO “Exacte Wetenschappen”(Physical Sciences). It was also supported by EU Con-tract No. IST-033749 DynaMax, EU FP7 ContractNo. NMP3-SL-2009-233513 MONAMI, ICT Grant No.251759 MACALO, the Royal Netherlands Academy ofArts and Sciences (KNAW) and “NanoNed,” a nanotech-nology programme of the Dutch Ministry of EconomicAffairs.

Appendix A: Computational details

1. Linearised Muffin-Tin Orbitals (LMTOs) andthe Pauli-Schrodinger Hamiltonian

The practical implementation of the WFM methodpresented in this paper is based on LMTOs and ASA thatare described in detail elsewhere.54–56 Here we will focusonly on those aspects which are important for the presentversion of the scattering formalism. An earlier, non-relativisitic version of the method49,50 can be referredto for additional details regarding the use of muffin-tinorbitals in transport calculations.

In the ASA, muffin-tin spheres are expanded to fill thevolume of the solid.126 Considering only the l = 0, spher-ically symmetric component of the potential inside theresulting AS located at R, a solution φRl(E, rR) of theradial Schrodinger equation can be determined numeri-cally for energy E and angular momentum l resulting inthe partial wave56

φRL(E, rR) ≡ φRl(E, rR)Ylm(rR) , (A1)

with rR ≡ r−R, L ≡ lm labels the angular momentumand Ylm is a spherical (or real, cubic) harmonic. rR de-notes a unit vector and rR ≡ |r−R|. Later we can dropthe explicit R-dependence where it does not give rise toambiguity. In terms of the logarithmic derivative Dl ofφl(E, r) at r ≡ s (the AS radius)

Dl(E, s) ≡rφ′l(E, r)

φl(E, r)

∣∣∣∣r=s

≡ sφ′l(E, s)

φl(E, s), (A2)

we can define the energy-dependent potential function

P 0l (E) = 2(2l + 1)

(ws

)2l+1Dl(E) + l + 1

Dl(E)− l, (A3)

which in the ASA depends only on the potential andis independent of the crystal structure. When there ismore than one atom type, w is an average AS radius.In terms of the “canonical” structure constant matrix

S0R′L′,RL,56,126 which depends only on the positions of

the ions, we can define the Hermitian matrix at E = Eνas

h0(Eν) = −[P0(Eν)

]−1/2 [P0(Eν)− S0

] [P0(Eν)

]−1/2

,

(A4)where P 0

RL is an (m-independent) element of the diag-

onal matrix P0 and P0 is the energy-derivative of P0.In this expression only S0 is a non-diagonal matrix. Tofirst order in E − Eν , h0(Eν) is the Hamiltonian in theASA.54–56 The problem presented by the long range126

of the structure constant matrix S0 is resolved by intro-ducing a generalised representation characterised by a setof l-dependent screening parameters αl and defining so-called “screened” structure constants Sα and potentialfunctions Pα defined by

Pα = P0[1−αP0

]−1, (A5)

Sα = S0[1−αS0

]−1. (A6)

where α is a diagonal matrix with m-independent el-ements αl. For a suitable choice of screening param-eters, the range of Sα is essentially limited to thefirst- and second-nearest neighbours for close-packedstructures.54–56 In the screened representation, the two-center tight-binding matrix becomes

hα(Eν) = −[Pα(Eν)

]−1/2[Pα(Eν)− Sα

][Pα(Eν)

]−1/2

.

(A7)The energy-independent, linearised (at Eν) muffin-tin

orbitals for the AS located at R are defined56 as∣∣χαRL(Eν)⟩

=∣∣φRL(Eν)

⟩+∣∣φαR′L′(Eν)

⟩hαR′L′,RL(Eν) ,

(A8)where

∣∣φαR′L′(Eν)⟩

=1

Nαl (Eν)

∂[∣∣φR′L′(E)

⟩Nαl (E)

]∂E

∣∣∣∣∣E=Eν

,

(A9)with the normalization function

Nαl (Eν) =

[(s/2)Pαl (Eν)

]1/2. (A10)

For simplicity, we will later assume that the orbitals areconstructed at Eν and omit any explicit energy depen-dence. Now we can construct the energy independentHamiltonian matrix correct to second order in E − Eν

〈χα|H− EνI|χα〉 = hα + hαoαhα , (A11)

where oα = 〈φ|φα〉 = Nα/Nα is the so-called overlapmatrix. Equation (A11) shows that the hopping rangeof the LMTO Hamiltonian is double the hopping rangeof the screened structure constant matrix, defined by thethree-center integral hαoαhα. In a transport calculationdominated by what happens at the Fermi energy EF we

19

can choose Eν = EF and the second (three-centre) termin Eq. (A11) can be omitted.

We include the spin-orbit interaction in a perturbativeway by adding a Pauli term to the Hamiltonian

Hso =1

c2 r

dV (r)

drL · S . (A12)

In the LMTO basis set, the matrix elements of Hso aregiven by

〈χα|Hso|χα〉 = γ1 + γ2hα + hαγ+

2 + hαγ3hα,

where γ1, γ2 and γ3 are spin-orbit parameters for one-,two-, and three-center terms

γ1 =

⟨φ

∣∣∣∣ 1

c2 r

dV (r)

drL · S

∣∣∣∣φ⟩ = K⊗ ξ (A13a)

γ2 =

⟨φ

∣∣∣∣ 1

c2 r

dV (r)

drL · S

∣∣∣∣φα⟩ = K⊗ ξα (A13b)

γ3 =

⟨φα∣∣∣∣ 1

c2 r

dV (r)

drL · S

∣∣∣∣φα⟩ = K⊗ ξα (A13c)

and we introduce a matrix of coefficients

Klmσ,l′m′σ′ = 〈lmσ|L · S|l′m′σ′〉, (A14)

and set of SOC potential parameters

ξlσσ′ =

⟨φlσ(r)

∣∣∣∣∣ 1

c2 r

dV σσ′(r)

dr

∣∣∣∣∣φlσ′(r)

⟩(A15a)

ξαlσσ′ =

⟨φlσ(r)

∣∣∣∣∣ 1

c2 r

dV σσ′(r)

dr

∣∣∣∣∣ φαlσ′(r)

⟩(A15b)

ξαlσσ′ =

⟨φαlσ(r)

∣∣∣∣∣ 1

c2 r

dV σσ′(r)

dr

∣∣∣∣∣ φαlσ′(r)

⟩. (A15c)

The expressions for ξ and ξ can be slightly reworked bytaking (A9) into account

ξαlσσ′ = ξlσσ′ + ξlσσ′oαlσ′ (A16a)

ξαlσσ′ = ξlσσ′ + ξlσσ′ (oαlσ + oαlσ′) + ξlσσ′oαlσoαlσ′ , (A16b)

where

ξlσσ′ =

⟨φlσ(r)

∣∣∣∣∣ 1

c2 r

dV σσ′(r)

dr

∣∣∣∣∣ φlσ′(r)

⟩(A17a)

ξlσσ′ =

⟨φlσ(r)

∣∣∣∣∣ 1

c2 r

dV σσ′(r)

dr

∣∣∣∣∣ φl′σ′(r)

⟩. (A17b)

The parameters ξ, ξ and ξ can be obtained by radialintegration over the AS. Off-diagonal (in spin-space) el-

ements of V σσ′(r) are assumed to be the average of po-

tentials for different spins.127

Thus, the complete Pauli-Schrodinger Hamiltonianwill be

Hso =[EνI+γ1

]+[hα+γ2h

α+hαγ+2

]+[hα(oα+γ3)hα

],

(A18)

1.0

1.5

2.0

2.5

R|| (f

Ω m

2 )

w.o. three-center termswith three-center terms

0 5 10 15 20 25 30L [nm]

0.05

0.10

0.15

0.20

G/(γ

µsA)

(nm

)

(a)

(b)

FIG. 23. Effect of the three center terms in Hso on (a) thetotal resistance and (b) the total damping of Ni80Fe20 at T =0. Red open circles include three-center terms, black filledcircles omit them.

where we have grouped one-, two- and three-centre terms.Even when E = Eν , omitting the three-centre term in(A18) is formally less readily justified than when SOC isneglected because it introduces longer-range hopping inthe Hamiltonian matrix. The practical impact on theresistivity and damping of neglecting the three-centreterms is examined in Fig. 23 where no significant effectcan be seen while the computational cost is reduced bysome 70%.

2. Velocities

In this section we derive the expression (22) for thegroup velocities of eigenmodes in the ideal wire for thegeneralised WFM framework. The derivations are similarto previous work.60,61 The vectors um of (7) are solutionsof the polynomial equation of order 2N

λNHum+

N∑n=1

(λN+nm Bnum + λN−nm B†num

)= 0 . (A19)

Left-multiplying by u†m and differentiating with respectto energy leads to

d

dE

[λNmu†mHum+

N∑n=1

(λN+nm u†mBnum+λN−nm u†mB†num

)]

= λNm + 2ıλN−1m

dλmdE

N∑n=1

n Im(λnmu†mBnum

)= 0 ,

(A20)

which yield the following expression for dE/dλm:

dE

dλm= − 2ı

λm

N∑n=1

n Im(λnmu†mBnum

). (A21)

20

For propagating states, λm = eikma, km is real and a isthe thickness of the periodic lead layer so

dkmdE

=1

ıaλm

dλmdE

. (A22)

The standard definition of group velocity is υm = 1~dEdk ,

therefore, substituting Eq. (A21) and (A22), we obtain

υm =ıaλm~

dE

dλm=

2a

~

N∑n=1

n Im(λnmu†mBnum

). (A23)

3. Spin-projections of the scattering matrix

For convenience of interpretation it is useful to decom-pose the transmission and reflection matrices into spin-projected ones. Although spin is not a valid quantumnumber when SOC is included, we can still characterisestates in the leads as states which have a distinctive spinprojection onto the σz axis (σz = ± 1

2 ). For leads con-sisting of light non-magnetic metals, this is a reasonableapproximation and can be achieved in practice as follows:for every pair of spin-degenerate lead eigenmodes u1,u2

we can construct a new pair of orthogonal eigenmodesu′1,u

′2 by taking a linear superposition of the original

modes [u′1u′2

]=

[a11 a12

a21 a22

]×[

u1

u2

](A24)

and choosing the coefficients aij to maximize the σz =+ 1

2 component of u′1 and the σz = − 12 component of u′2.

We denote these new states as uσ+ and uσ−. This basisset transformation allows us to to operate with reason-ably well defined spin-projected scattering matrices. Forexample, the matrix rσσ

′

µν describes reflection from theνσ′ states into the µσ states in the same lead.

4. Reduction of the Gilbert damping tensor to ascalar

For a crystal with cubic symmetry and a collinear mag-netisation, the damping torque can in general be writtenas τ = m×(α·m). If the magnetisation m is taken to bealong the z-axis, then to leading order in the transversecomponents of the magnetisation (mx,my mz ∼ 1)the damping torque can be written in cartesian coordi-nates as

τx = −mz (αyxmx + αyymy) , (A25a)

τy = mz (αxxmx + αxymy) . (A25b)

Without loss of generality, we choose the momentary di-rection of magnetisation precession to be along the x-axisi.e., m = mx and |mz| = 1, as sketched in Fig. 24(a).Then τx = −αyxm and τy = αxxm. Keeping thesystem otherwise unchanged, we rotate the coordinate

axes through 90 clockwise about the z-axis as shown inFig. 24(b). In this case we can write the damping torqueτx′ = −αyym and τy′ = αxym. Comparing the com-ponents in Fig. 24(a) and (b), we find αxx = αyy andαxy = −αyx.

τ x

!m x

y

τ y

τ x

!m x

y

τ y

m = z m = −z

τ x

!m x

y

τ y

m = z(a) (c) (d)

τ ′y

!m

′x ′y

τ ′x

m = z(b)

FIG. 24. Geometry of damping torque exerted on magnetiza-tion. See text for detailed analysis.

If we reverse the magnetization in Fig. 24(a) but keepm unchanged, the damping torque should be reversedas plotted in Fig. 24(c). If we then rotate the system180 about the x-axis, it reproduces the configuration ofFig. 24(a) except that the component τx is inverted. Asa consequence, τx = −αyxm must be zero indicating thatthe off-diagonal elements αyx = −αxy = 0. In the samemanner, it can be proved that in a collinear magneticsystem with cubic symmetry, the Gilbert damping tensorreduces to a scalar, α = α1, where 1 is the 3 × 3 unitmatrix.

Appendix B: Limitations

A number of factors limit the application of the presentmethod. First and foremost, memory considerationslimit the size of system that can be addressed. A cal-culation for a single configuration of the longest scat-tering region shown in Fig. 12 and containing about15, 000 = 5×5×600 atoms requires about one hour on asupercomputer node with 32 cores and 256 GB memory,where the calculation parallelized perfectly over the twodimensional 32×32 k-point summation. The maximumlength of ∼105 nm places an upper limit on e.g. the spin-flip diffusion length that could be studied. For largersystems, the calculations need to be performed with amultithreading sparse matrix solver or simply with ex-tended memory. For metals, the computing time scaleslinearly with the length L of the scattering region andquadratically with the size of the lateral supercell. For alateral supercell containing M atoms a calculation for asingle k-point scales as M3. The BZ size scales as 1/Mso for constant sampling density of reciprocal space, thescaling goes as M2L. The upper limit of lateral supercellsize with SOC is in practice about 10×10 and 30 nm long;this scattering region also contains about 15,000 atoms.

A second important limitation is the ASA. This con-ventional description of the potential in combination withthe MTO scheme is usually very good for close-packedsystems; open systems can frequently be reasonably well

21

modelled by filling space with artificial “empty atomicspheres”,56,128 i.e. without nuclei inside. For lower sym-metry structures, where the spherically symmetric poten-tials around the nuclei are not sufficient to characterizethe real Kohn-Sham potential, the ASA breaks down. Inthese cases, the reliability and accuracy of transport cal-culations as currently implemented with MTO and ASAare limited by the ASA description of the Kohn-Shampotentials. Andersen has suggested ways of circumvent-ing this limitation without sacrificing the efficiency of theASA.129

The ultimate limitation is the DFT itself, or rather theapproximation to the exchange-correlation potential, thefunctional derivative of the exchange-correlation (XC)energy, that has to be chosen. Some of the uncertaintiesthat result from difference choices are discussed brieflynext.

Additional numerical tests

Although the formalism described in this paper is pa-rameter free, the practical implementation requires ap-proximating the XC energy of DFT130 as in the localdensity approximation131 where electron gas data havebeen parameterized by various workers.132–134 The nu-merical values of the resistivity and damping parameterwill depend on the parameterization chosen. Our imple-mentation with TB-MTOs requires truncating the orbitalangular momentum expansion at some maximum valueof the orbital angular momentum and this will also influ-ence the numerical results. For all calculations presentedin this paper, the von Barth-Hedin (vBH) XC potentialand an spd basis set were used. To demonstrate the un-certainty resulting from the somewhat arbitrary choice ofXC potential and basis sets, we show the results of calcu-lations for Cu|Py|Cu with different choices of potentialsand spd or spdf basis set in Table I.

The quantity most dependent on these choices is the

permalloy majority spin resistivity in the absence of SOC.The very weak scattering of majority spins makes thisvery sensitive to small details of the electronic structurewhich in turn depend strongly on the exchange splitting.Once SOC is included, the mean-free path is reduced andthe sensitivity of the resistivity and, especially, of theGilbert-damping parameter to these “technical” detailsbecomes acceptable.

Various forms of the XC potentials have been imple-mented in computer programs and examined for differ-ent physical and chemical quantities.135 For the transportproperties of magnetic metals and alloys that we studyin this paper, there is no clear evidence to show that oneXC potential is better than the others. The test for thebasis set using the vBH XC potential also indicates thatthe minimal spd basis is good enough for convergence, asinitially proposed by Lambrecht and Andersen.136 Theslight difference in the calculated resistivity and Gilbertdamping can be regarded as the “uncertainty” arising

TABLE I. Dependence of the atomic magnetic moment µs,non-relativistic resistivities ρmaj and ρmin, the relativistic re-sistivity ρ‖, and the Gilbert damping parameter for Py, onthe basis set and choice of exchange-correlation potential: vonBarth-Hedin (vBH),132 Perdew-Zunger (PZ)133 and Vosko-Wilk-Nusair (VWN).134 Calculations are performed with a5× 5 supercell and a k-point sampling grid of 32× 32 (equiv-alent to 160× 160 for a 1× 1 primitive interface cell). Resis-tivities are given in µΩ cm and magnetic moment is in Bohrmagneton µB per atom.

.

XC/Basis µs ρmaj ρmin ρ‖ α (10−3)

vBH/spd 1.025 0.57± 0.01 109± 1 2.7± 0.1 4.6± 0.2

vBH/spdf 1.001 0.67± 0.01 101± 1 2.6± 0.1 4.3± 0.2

PZ/spd 1.010 0.92± 0.01 108± 1 3.1± 0.1 4.7± 0.2

VWN/spd 1.022 0.60± 0.01 107± 1 2.8± 0.1 4.5± 0.2

from an arbitrary choice of XC potential.

∗ Present address: The Center for Advanced QuantumStudies and Department of Physics, Beijing Normal Uni-versity, 100875 Beijing, China

† Present address: The Center for Advanced QuantumStudies and Department of Physics, Beijing Normal Uni-versity, 100875 Beijing, China; Corresponding author:zyuan@bnu.edu.cn

‡ Corresponding author: P.J.Kelly@utwente.nl1 M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen

Van Dau, F. Petroff, P. Etienne, G. Creuzet,A. Friederich, and J. Chazelas, “Giant Magnetoresistanceof (001)Fe/(001)Cr Magnetic Superlattices,” Phys. Rev.Lett. 61, 2472–2475 (1988).

2 G. Binasch, P. Grunberg, F. Saurenbach, and W. Zinn,“Enhanced magnetoresistance in layered magnetic struc-

tures with antiferromaganetic interlayer exchange,” Phys.Rev. B 39, 4828–4830 (1989).

3 See the collections of articles in Ultrathin Magnetic Struc-tures I-IV, edited by J. A. C. Bland and B. Heinrich(Springer-Verlag, Berlin, 1994-2005); in Spin Current,Semiconductor Science and Technology, Vol. 17, edited byS. Maekawa, S. O. Valenzuela, E. Saitoh, and T. Kimura(Oxford Science Publications, Oxford, 2012).

4 S. D. Bader and S. S. P. Parkin, “Spintronics,” Annu.Rev. Condens. Matter Phys. 1, 71–88 (2010).

5 J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meser-vey, “Large Magnetoresistance at Room Temperature inFerromagnetic Thin Film Tunnel Junctions,” Phys. Rev.Lett. 74, 3273–3276 (1995).

6 S. Yuasa and D. D. Djayaprawira, “Giant tunnel mag-

22

netoresistance in magnetic tunnel junctions with a crys-talline MgO(001) barrier,” J. Phys. D: Appl. Phys. 40,R337–R354 (2007).

7 J. C. Slonczewski, “Current-driven excitation of mag-netic multilayers,” J. Magn. & Magn. Mater. 159, L1–L7(1996).

8 L. Berger, “Emission of spin waves by a magnetic multi-layer traversed by a current,” Phys. Rev. B 54, 9353–9358(1996).

9 D. C. Ralph and M. D. Stiles, “Spin transfer torques,” J.Magn. & Magn. Mater. 320, 1190–1216 (2008).

10 Axel Hoffmann, “Spin Hall effects in metals,” IEEE Trans.Mag. 49, 5172–5193 (2013).

11 Jairo Sinova, Sergio O. Valenzuela, J. Wunderlich, C. H.Back, and J. Jungwirth, “Spin Hall effects,” Rev. Mod.Phys. 87, 1213–1259 (2015).

12 See the collections of articles in Spin Current, Semi-conductor Science and Technology, Vol. 17, edited byS. Maekawa, S. O. Valenzuela, E. Saitoh, and T. Kimura(Oxford University Press, Oxford, 2012); Robert L.Stamps, Stephan Breitkreutz, Johan Akerman, Andrii V.Chumak, YoshiChika Otani, Gerrit E. W. Bauer, Jan-Ulrich Thiele, Martin Bowen, Sara A. Majetich, MathiasKlaui, Ioan Lucian Prejbeanu, Bernard Dieny, Nora M.Dempsey, and Burkard Hillebrands, “The 2014 mag-netism roadmap,” J. Phys. D: Appl. Phys. 47, 333001(2014).

13 J. Akerman, “Toward a universal memory,” Science 308,508–510 (2005).

14 Mark H Kryder and Chang Soo Kim, “After hard drives—what comes next?” IEEE Trans. Mag. 45, 3406–3413(2009).

15 A. Ruotolo, V. Cros, B. Georges, A. Dussaux, J. Grollier,C. Deranlot, R. Guillemet, K. Bouzehouane, S. Fusil, andA. Fert, “Phase-locking of magnetic vortices mediated byantivortices,” Nature Nanotechnology 4, 528–532 (2009).

16 A. Slavin, “Spin-torque oscillators get in phase,” NatureNanotechnology 4, 479–480 (2009).

17 Stuart S. P. Parkin, Masamitsu Hayashi, and LucThomas, “Magnetic domain-wall racetrack memory,” Sci-ence 320, 190–194 (2008).

18 Luc Thomas, Rai Moriya, Charles Rettner, and StuartS. P. Parkin, “Dynamics of magnetic domain walls undertheir own inertia,” Science 330, 1810–1813 (2010).

19 Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.Halperin, “Nonlocal magnetization dynamics in ferromag-netic nanostructures,” Rev. Mod. Phys. 77, 1375–1421(2005).

20 S. M. Bhagat and P. Lubitz, “Temperature variation offerromagnetic relaxation in the 3d transition metals,”Phys. Rev. B 10, 179–185 (1974).

21 B. Heinrich, D. J. Meredith, and J. F. Cochran,“Wave number and temperature-dependent Landau-Lifshitz damping in nickel,” J. Appl. Phys. 50, 7726–7728(1979).

22 S. Mizukami, Y. Ando, and T. Miyazaki, “Ferromagneticresonance linewidth for NM/80NiFe/NM films (NM=Cu,Ta, Pd and Pt),” J. Magn. & Magn. Mater. 226–230,1640 (2001).

23 S. Mizukami, Y. Ando, and T. Miyazaki, “Magneticrelaxation of normal-metal (NM)/80NiFe/NM films,” J.Magn. & Magn. Mater. 239, 42 (2002).

24 Liesbet Lagae, Roel Wirix-Speetjens, Wouter Eyckmans,Staf Borghs, and Jo de Boeck, “Increased Gilbert damp-

ing in spin valves and magnetic tunnel junctions,” J.Magn. & Magn. Mater. 286, 291–296 (2005).

25 J. O. Rantschler, B. B. Maranville, J. J. Mallett, P. Chen,R. D. McMichael, and W. F. Egelhoff, “Damping at nor-mal metal/permalloy interfaces,” IEEE Trans. Mag. 41,3523–3525 (2005).

26 N. Inaba, H. Asanuma, S. Igarashi, S. Mori, F. Kirino,K. Koike, and H. Morita, “Damping Constants of Ni-Fe and Ni-Co Alloy Thin Films,” IEEE Trans. Mag. 42,2372–2374 (2006).

27 Mikihiko Oogane, Takeshi Wakitani, Satoshi Yakata,Resul Yilgin, Yasuo Ando, Akimasa Sakuma, andTerunobu Miyazaki, “Magnetic damping in ferromagneticthin films,” Jpn. J. Appl. Phys. 45, 3889–3891 (2006).

28 V. Kambersky, J. F. Cochran, and J. M. Rudd,“Anisotropic low-temperature FMR linewidth in nickeland the theory of ‘anomalous’ damping,” J. Magn. &Magn. Mater. 104, 2089 (1992).

29 K. Gilmore, Y. U. Idzerda, and M. D. Stiles, “Identifi-cation of the Dominant Precession-Damping Mechanismin Fe, Co, and Ni by First-Principles Calculations,” Phys.Rev. Lett. 99, 027204 (2007).

30 K. Gilmore, Y. U. Idzerda, and M. D. Stiles, “Spin-orbitprecession damping in transition metal ferromagnets,” J.Appl. Phys. 103, 07D303 (2008).

31 V Kambersky, “Spin-orbital Gilbert damping in commonmagnetic metals,” Phys. Rev. B 76, 134416 (2007).

32 Arne Brataas, Yaroslav Tserkovnyak, and GerritE. W Bauer, “Scattering Theory of Gilbert Damping,”Phys. Rev. Lett. 101, 037207 (2008); A. Brataas,Y. Tserkovnyak, and G. E. W. Bauer, “Magnetizationdissipation in ferromagnets from scattering theory,” Phys.Rev. B 84, 054416 (2011).

33 A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak,and G. E. W. Bauer, “Unified First-Principles Study ofGilbert Damping, Spin-Flip Diffusion and Resistivity inTransition Metal Alloys,” Phys. Rev. Lett. 105, 236601(2010).

34 Yi Liu, Anton A. Starikov, Zhe Yuan, and Paul J. Kelly,“First-principles calculations of magnetization relaxationin pure Fe, Co, and Ni with frozen thermal lattice disor-der,” Phys. Rev. B 84, 014412 (2011).

35 H. Ebert, S. Mankovsky, D. Kodderitzsch, and P. J.Kelly, “Ab-initio calculation of the gilbert damping pa-rameter via linear response formalism,” Phys. Rev. Lett.107, 066603 (2011).

36 S. Mankovsky, D. Kodderitzsch, G. Woltersdorf, andH. Ebert, “First-principles calculation of the Gilbertdamping parameter via the linear response formalism withapplication to magnetic transition metals and alloys,”Phys. Rev. B 87, 014430 (2013).

37 I. Turek, J. Kudrnovsky, and V. Drchal, “Nonlocal torqueoperators in ab initio theory of the Gilbert damping inrandom ferromagnetic alloys,” Phys. Rev. B 92, 214407(2015).

38 Martin A.W. Schoen, Danny Thonig, Michael L. Schnei-der, T. J. Silva, Hans T. Nembach, Olle Eriksson, OlofKaris, and Justin M. Shaw, “Ultra-low magnetic damp-ing of a metallic ferromagnet,” Nature Physics 12, 839–842 (2016).

39 Martin A. W. Schoen, Juriaan Lucassen, Hans T. Nem-bach, Bert Koopmans, T. J. Silva, Christian H. Back,and Justin M. Shaw, “Magnetic properties of ultrathin 3dtransition-metal binary alloys. II. Experimental verifica-

23

tion of quantitative theories of damping and spin pump-ing,” Phys. Rev. B 95, 134411 (2017).

40 Yuelei Zhao, Qi Song, See-Hun Yang, Tang Su, Wei Yuan,Stuart S. P. Parkin, Jing Shi, and Wei Han, “Experi-mental Investigation of Temperature-Dependent GilbertDamping in Permalloy Thin Films,” Scientific Reports 6,22890 (2016).

41 V Kambersky, “On ferromagnetic resonance damping inmetals,” Czech. J. Phys. 26, 1366–1383 (1976).

42 Zhe Yuan, Yi Liu, Anton A. Starikov, Paul J. Kelly, andArne Brataas, “Spin-Orbit-Coupling-Induced Domain-Wall Resistance in Diffusive Ferromagnets,” Phys. Rev.Lett. 109, 267201 (2012).

43 Zhe Yuan, Kjetil M. D. Hals, Yi Liu, Anton A. Starikov,Arne Brataas, and Paul J. Kelly, “Gilbert Dampingin Noncollinear Ferromagnets,” Phys. Rev. Lett. 113,266603 (2014).

44 Zhe Yuan and Paul J. Kelly, “Spin-orbit-coupling inducedtorque in ballistic domain walls: Equivalence of charge-pumping and nonequilibrium magnetization formalisms,”Phys. Rev. B 93, 224415 (2016).

45 Yi Liu, Zhe Yuan, Rien J. H. Wesselink, Anton A.Starikov, and Paul J. Kelly, “Interface Enhancement ofGilbert Damping from First Principles,” Phys. Rev. Lett.113, 207202 (2014).

46 Y. Liu, Z. Yuan, R. J. H. Wesselink, A. A. Starikov,M. van Schilfgaarde, and P. J. Kelly, “Direct method forcalculating temperature-dependent transport properties,”Phys. Rev. B 91, 220405(R) (2015).

47 Lei Wang, R. J. H. Wesselink, Yi Liu, Zhe Yuan, Ke Xia,and Paul J. Kelly, “Giant Room Temperature InterfaceSpin Hall and Inverse Spin Hall Effects,” Phys. Rev. Lett.116, 196602 (2016).

48 K. Xia, P. J. Kelly, G. E. W. Bauer, I. Turek, J. Ku-drnovsky, and V. Drchal, “Interface resistance of dis-ordered magnetic multilayers,” Phys. Rev. B 63, 064407(2001).

49 K. Xia, M. Zwierzycki, M. Talanana, P. J. Kelly, andG. E. W. Bauer, “First-principles scattering matrices forspin-transport,” Phys. Rev. B 73, 064420 (2006).

50 M. Zwierzycki, P. A. Khomyakov, A. A. Starikov, K. Xia,M. Talanana, P. X. Xu, V. M. Karpan, I. Marushchenko,I. Turek, G. E. W. Bauer, G. Brocks, and P. J. Kelly,“Calculating scattering matrices by wave function match-ing,” Phys. Stat. Sol. B 245, 623–640 (2008).

51 T. Ando, “Quantum point contacts in magnetic fields,”Phys. Rev. B 44, 8017–8027 (1991).

52 S. Datta, Electronic Transport in Mesoscopic Systems(Cambridge University Press, Cambridge, 1995).

53 Y. Imry, Introduction to Mesoscopic Physics, 2nd ed. (Ox-ford University Press, Oxford, 2002).

54 O. K. Andersen and O. Jepsen, “Explicit, First-PrinciplesTight-Binding Theory,” Phys. Rev. Lett. 53, 2571–2574(1984).

55 O. K. Andersen, O. Jepsen, and D. Glotzel, “Canon-ical description of the band structures of metals in,”in Highlights of Condensed Matter Theory, InternationalSchool of Physics ‘Enrico Fermi’, Varenna, Italy, editedby F. Bassani, F. Fumi, and M. P. Tosi (North-Holland,Amsterdam, 1985) pp. 59–176.

56 O. K. Andersen, Z. Pawlowska, and O. Jepsen, “Illus-tration of the linear-muffin-tin-orbital tight-binding rep-resentation: Compact orbitals and charge density in Si,”Phys. Rev. B 34, 5253–5269 (1986).

57 P. R. Amestoy, I. S. Duff, J. Koster, and J. Y.L’Excellent, “A fully asynchronous multifrontal solver us-ing distributed dynamic scheduling,” SIAM J. MatrixAnal. Appl. 23, 15–41 (2001).

58 P. R. Amestoy, A. Guermouche, J. Y. L’Excellent, andS. Pralet, “Hybrid scheduling for the parallel solution oflinear systems,” Parallel Computing 32, 136–156 (2006).

59 Shuai Wang, Yuan Xu, and Ke Xia, “First-principlesstudy of spin-transfer torques in layered systems with non-collinear magnetization,” Phys. Rev. B 77, 184430 (2008).

60 P. A. Khomyakov and G. Brocks, “Real-space finite-difference method for conductance calculations,” Phys.Rev. B 70, 195402 (2004).

61 P. A. Khomyakov, G. Brocks, V. Karpan, M. Zwierzy-cki, and P. J. Kelly, “Conductance calculations for quan-tum wires and interfaces: mode matching and Green func-tions,” Phys. Rev. B 72, 035450 (2005).

62 Kees M. Schep, Jeroen B. A. N. van Hoof, Paul J. Kelly,Gerrit E. W. Bauer, and John E. Inglesfield, “Interfaceresistances of magnetic multilayers,” Phys. Rev. B 56,10805–10808 (1997).

63 S. F. Zhang and P. M. Levy, “Conductivity perpendicularto the plane of multilayered structures,” J. Appl. Phys.69, 4786–4788 (1991).

64 S.-F. Lee, W.P. Pratt Jr., Q. Yang, P. Holody, R. Loloee,P. A. Schroeder, and J. Bass, “Two-channel analysis ofCPP-MR data for Ag/Co and AgSn/Co multilayers,” J.Magn. & Magn. Mater. 118, L1–L5 (1993).

65 T. Valet and A. Fert, “Theory of the perpendicular mag-netoresistance in magnetic multilayers,” Phys. Rev. B 48,7099–7113 (1993).

66 J. Bass, “CPP magnetoresistance of magnetic multilayers:A critical review,” J. Magn. & Magn. Mater. 408, 244–320(2016).

67 Albert Fert and Shang-Fan Lee, “Theory of the bipolarspin switch,” Phys. Rev. B 53, 6554–6565 (1996).

68 K. Eid, D. Portner, J. A. Borchers, R. Loloee, M. A. Dar-wish, M. Tsoi, R. D. Slater, K. V. O’Donovan, H. Kurt,W. P. Pratt, Jr., and J. Bass, “Absence of mean-free-path effects in the current-perpendicular-to-plane magne-toresistance of magnetic multilayers,” Phys. Rev. B 65,054424 (2002).

69 J. Bass and W. P. Pratt Jr., “Spin-diffusion lengths inmetals and alloys, and spin-flipping at metal/metal in-terfaces: an experimentalist’s critical review,” J. Phys.:Condens. Matter 19, 183201 (2007).

70 Kriti Gupta, R. J. H. Wesselink, Z. Yuan, A. N. Other,Sum Wun Els, and Paul J. Kelly, “Calculating the tem-perature dependence of interface transport parametersfrom first-principles: Py|Pt,” to be published (2018).

71 J. E. Avron, A. Elgart, G. M. Graf, and L. Sadun, “Op-timal Quantum Pumps,” Phys. Rev. Lett. 87, 236601(2001); M. Moskalets and M. Buttiker, “Dissipation andnoise in adiabatic quantum pumps,” Phys. Rev. B 66,035306 (2002); “Floquet scattering theory of quantumpumps,” Phys. Rev. B 66, 205320 (2002).

72 Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, “En-hanced Gilbert Damping in Thin Ferromagnetic Films,”Phys. Rev. Lett. 88, 117601 (2002); “Spin pumping andmagnetization dynamics in metallic multilayers,” Phys.Rev. B 66, 224403 (2002).

73 Shigemi Mizukami, Yasuo Ando, and Terunobu Miyazaki,“The Study on Ferromagnetic Resonance Linewidth forNM/80NiFe/NM (NM = Cu, Ta, Pd and Pt) Films,” Jpn.

24

J. Appl. Phys. 40, 580–585 (2001).74 M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas,

and G. E. W. Bauer, “First-principles study of magneti-zation relaxation enhancement and spin transfer in thinmagnetic films,” Phys. Rev. B 71, 064420 (2005).

75 J Kudrnovsky and V Drchal, “Electronic structure of ran-dom alloys by the linear band-structure methods,” Phys.Rev. B 41, 7515–7528 (1990).

76 J. Kudrnovsky, V. Drchal, C. Blaas, P. Weinberger,I. Turek, and P. Bruno, “Ab-initio theory of the CPPmagnetoconductance,” Czech. J. Phys. 49, 1583–1589(1999).

77 S. Y. Savrasov and D. Y. Savrasov, “Electron-phonon in-teractions and related physical properties of metals fromlinear-response theory,” Phys. Rev. B 54, 16487–16501(1996).

78 L. M. Sandratskii and P. G. Guletskii, “Symmetrisedmethod for the calculation of the band structure of non-collinear magnets,” J. Phys. F: Met. Phys. 16, L43–L48(1986).

79 J. Kubler, K.-H. Hock, J. Sticht, and A. R. Williams,“Density functional theory of non-collinear magnetism,”J. Phys. F: Met. Phys. 18, 469–483 (1988).

80 L. M. Sandratskii, “Noncollinear magnetism in itinerant-electron systems: theory and applications,” Adv. Phys.47, 91–160 (1998).

81 Y. Niimi, M. Kimata, Y. Omori, B. Gu, T. Ziman,S. Maekawa, A. Fert, and Y. Otani, “Strong suppres-sion of the spin hall effect in the spin glass state,” Phys.Rev. Lett. 115, 196602 (2015).

82 Devin Wesenberg, Tao Liu, Davor Balzar, Mingzhong Wu,and Barry L. Zink, “Long-distance spin transport in adisordered magnetic insulator,” Nature Physics 13, 987–993 (2017).

83 K. M. D. Hals and A. Brataas, “Spin-orbit torques andanisotropic magnetization damping in skyrmion crystals,”Phys. Rev. B 89, 064426 (2014).

84 Yaroslav Tserkovnyak and Hector Ochoa, “Generalizedboundary conditions for spin transfer,” Phys. Rev. B 96,100402 (2017).

85 D. D. Koelling and B. N. Harmon, “A technique for rela-tivistic spin-polarised calculations,” J. Phys. C: Sol. StatePhys. 10, 3107–3114 (1977).

86 I. Turek, V. Drchal, J. Kudrnovsky, M. Sob, and P. Wein-berger, Electronic Structure of Disordered Alloys, Sur-faces and Interfaces (Kluwer, Boston-London-Dordrecht,1997).

87 B. Dieny, “Giant magnetoresistance in spin-valve multi-layers,” J. Magn. & Magn. Mater. 136, 335–359 (1994).

88 B. Dieny, “Quantitative interpretation of giant mag-netoresistance properties of permalloy-based spin-valvestructure,” Europhys. Lett. 17, 261–267 (1992).

89 Bruce A. Gurney, Virgil S. Speriosu, Jean-Pierre Nozieres,Harry Lefakis, Dennis R. Wilhoit, and Omar U. Need,“Direct measurement of spin-dependent conduction-electron mean free paths in ferromagnetic metals,” Phys.Rev. Lett. 71, 4023–4026 (1993).

90 P. Soven, “Coherent-potential model of substitutional dis-ordered alloys,” Phys. Rev. 156, 809–813 (1967).

91 J. Banhart, H. Ebert, and A. Vernes, “Applicabil-ity of the two-current model for systems with stronglyspin-dependent disorder,” Phys. Rev. B 56, 10165–10171(1997).

92 S. Khmelevskyi, K. Palotas, L. Szunyogh, and P. Wein-

berger, “Ab initio calculation of the anisotropic magne-toresistance in Ni1−cFec bulk alloys,” Phys. Rev. B 68,012402 (2003).

93 I. Turek, J. Kudrnovsky, and V. Drchal, “Ab initio theoryof galvanomagnetic phenomena in ferromagnetic metalsand disordered alloys,” Phys. Rev. B 86, 014405 (2012).

94 J. Bass and W. P. Pratt Jr., “Current-perpendicular(CPP) magnetoresistance in magnetic metallic multilay-ers,” J. Magn. & Magn. Mater. 200, 274–289 (1999).

95 J. Smit, “Magnetoresistance of ferromagnetic metals andalloys at low temperatures,” Physica 17, 612–627 (1951).

96 T. R. McGuire and R. I. Potter, “Anisotropic magnetore-sistance in ferromagnetic 3d alloys,” IEEE Trans. Mag.11, 1018–1038 (1975).

97 O. Jaoul, I. A. Campbell, and A. Fert, “Spontaneousresistivity anisotropy in Ni alloys,” J. Magn. & Magn.Mater. 5, 23–34 (1977).

98 M. C. Cadeville and B. Loegel, “On the transport prop-erties in concentrated Ni-Fe alloys at low temperatures,”J. Phys. F: Met. Phys. 3, L115–L119 (1973).

99 J. Banhart and H. Ebert, “First-principles theory ofspontaneous-resistance anisotropy and spontaneous Halleffect in disordered ferromagnetic alloys,” Europhys. Lett.32, 517–522 (1995).

100 B.-H. Zhou, Y. Xu, S. Wang, G. Zhou, and K. Xia, “Anab initio investigation on boundary resistance for mag-netic grains,” Sol. State Comm. 150, 1422–1424 (2010).

101 William Dumouchel and Fanny O’Brien, “Integrating arobust option into a multiple regression computing envi-ronment,” in Computing and graphics in statistics, editedby Andreas Buja and Paul A. Tukey (Springer-Verlag NewYork, Inc., New York, NY, USA, 1991) pp. 41–48.

102 L. Nordheim, “Electron theory of metals i,” Annalen derPhysik 9, 607–640 (1931); “On the electron theory ofmetals ii,” Annalen der Physik 9, 641–678 (1931).

103 R. A. de Groot, F. M. Mueller, P. G. van Engen, andK. H. J. Buschow, “New class of materials: Half-metallicferromagnets,” Phys. Rev. Lett. 50, 2024–2027 (1983).

104 The finite intercept at L = 0 in Fig. 17, Gi ≡ G(L =0) corresponds to an interface damping enhancementthat was observed for thin ferromagnetic films137–139

and NM|Py|NM layers.22,23,73,115 It was qualitativelyexplained in terms of how a precessing magnetization“pumps” the transverse component of magnetizationthrough the interface.72 The material parameter that de-scribes this “spin pumping” is the “mixing conductance”introduced to describe spin dependent transport betweennon-collinearly aligned elements of a magnetic circuit.140

To describe the interface damping enhancement quantita-tively, diffusive scattering and spin-flip scattering in theNM layer have to be described as well as interface spin-flipscattering and modification of the mixing conductance bySOC45 .

105 C. E. Patton, Z. Frait, and C. H. Wilts, “Frequency De-pendence of the Parallel and Perpendicular FerromagneticResonance Linewidth in Permalloy Films, 2-36 Ghz,” J.Appl. Phys. 46, 5002–5003 (1975).

106 W. Bailey, P. Kabos, F. Mancoff, and S. Russek, “Controlof magnetization dynamics in Ni81Fe19 thin films throughthe use of rare-earth dopants,” IEEE Trans. Mag. 37,1749–1754 (2001).

107 J. P. Nibarger, R. Lopusnik, Z. Celinski, and T. J. Silva,“Variation of magnetization and the Lande g factor with

25

thickness in Ni-Fe films,” Appl. Phys. Lett. 83, 93–95(2003).

108 S. Ingvarsson, G. Xiao, S. S. P. Parkin, and R. H.Koch, “Tunable magnetization damping in transitionmetal ternary alloys,” Appl. Phys. Lett. 85, 4995–4997(2004).

109 H Nakamura, Yasuo Ando, S Mizukami, and H Kub-ota, “Measurement of Magnetization Precession forNM/Ni80Fe20/NM (NM= Cu and Pt) Using Time-Resolved Kerr Effect,” Jpn. J. Appl. Phys. 43, L787–L789(2004).

110 R. Bonin, M. L. Schneider, T. J. Silva, and J. P. Ni-barger, “Dependence of magnetization dynamics on mag-netostriction in NiFe alloys,” J. Appl. Phys. 98, 123904(2005).

111 K. Kobayashi, Nabuyuki Inaba, N. Fujita, Y. Sudo,T. Tanaka, M. Ohtake, M. Futamoto, and F. Kirino,“Damping constants for permalloy single-crystal thinfilms,” IEEE Trans. Mag. 45, 2541–2544 (2009).

112 Justin M. Shaw, T. J. Silva, Michael L. Schneider, andRobert D. McMichael, “Spin dynamics and mode struc-ture in nanomagnet arrays: Effects of size and thicknesson linewidth and damping,” Phys. Rev. B 79, 184404(2009).

113 C. Luo, Z. Feng, Y. Fu, W. Zhang, P. K. J. Wong, Z. X.Kou, Y. Zhai, H. F. Ding, M. Farle, J. Du, and H. R.Zhai, “Enhancement of magnetization damping coefficientof permalloy thin films with dilute Nd dopants,” Phys.Rev. B 89, 184412 (2014).

114 Yuli Yin, Fan Pan, Martina Ahlberg, Mojtaba Ran-jbar, Philipp Durrenfeld, Afshin Houshang, MohammadHaidar, Lars Bergqvist, Ya Zhai, Randy K. Dumas, AnnaDelin, and Johan Akerman, “Tunable permalloy-basedfilms for magnonic devices,” Phys. Rev. B 92, 024427(2015).

115 S. Ingvarsson, L. Ritchie, X. Y. Liu, Gang Xiao, J. C.Slonczewski, P. L. Trouilloud, and R. H. Koch, “Role ofelectron scattering in the magnetization relaxation of thinNi81Fe19 films,” Phys. Rev. B 66, 214416 (2002).

116 Bertram Terence Martin Willis and Arthur WilliamPryor, Thermal vibrations in crystallography (CambridgeUniversity Press, London, New York, 1975).

117 R. J. Wakelin and E. L. Yates, “A Study of the Order-Disorder Transformation in Iron-Nickel Alloys in the Re-gion FeNi3,” Proc. Phys. Soc. B 66, 221–240 (1953).

118 R. Weber and P. E. Tannenwald, “Second-order exchangeinteractions from spin wave resonance,” J. Phys. Chem.Solids 24, 1357–1361 (1963); “Long-range exchange in-teractions from spin-wave resonance,” Phys. Rev. 140,498–506 (1965).

119 M. D. Kuz’min, “Shape of temperature dependence ofspontaneous magnetization of ferromagnets: Quantitativeanalysis,” Phys. Rev. Lett. 94, 107204 (2005).

120 R. Fisher, “Dispersion on a sphere,” Proc. Roy. Soc. A217, 295–305 (1953).

121 Jørn Foros, Arne Brataas, Yaroslav Tserkovnyak, andGerrit E. W. Bauer, “Current-induced noise and dampingin nonuniform ferromagnets,” Phys. Rev. B 78, 140402(2008).

122 Jørn Foros, , Arne Brataas, Gerrit E. W. Bauer, andYaroslav Tserkovnyak, “Noise and dissipation in magne-toelectronic nanostructures,” Phys. Rev. B 79, 214407(2009).

123 Jiang Xiao, Gerrit E. W Bauer, Ken-Chi Uchida, Eiji

Saitoh, and Sadamichi Maekawa, “Theory of magnon-driven spin seebeck effect,” Phys. Rev. B 81, 214418(2010).

124 C. Y. Ho, M. W. Ackerman, K. Y. Wu, T. N. Havill,R. H. Bogaard, R. A. Matula, S. G. Oh, and H. M.James, “Electrical resistivity of ten selected binary alloysystems,” J. Phys. Chem. Ref. Data 12, 183–322 (1983).

125 G. Counil, T. Devolder, J.-V Kim, P. Crozat, C. Chap-pert, S. Zoll, and R. Fournel, “Temperature depen-dences of the resistivity and the ferromagnetic resonancelinewidth in permalloy thin films,” IEEE Trans. Mag. 42,3323–3325 (2006).

126 O. K. Andersen, “Linear methods in band theory,” Phys.Rev. B 12, 3060–3083 (1975).

127 G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schu-urmans, “First-principles calculation of the magnetocrys-talline anisotropy energy of iron, cobalt and nickel,” Phys.Rev. B 41, 11919–11937 (1990).

128 D. Glotzel, B. Segall, and O. K. Andersen, “Self-consistent electronic structure of Si, Ge and Diamond bythe LMTO-ASA method,” Sol. State Comm. 36, 403–406(1980).

129 M. Zwierzycki and O. K. Andersen, “The overlappingmuffin-tin approximation,” Acta. Phys. Pol. A 115, 64–68(2009).

130 P. Hohenberg and W. Kohn, “Inhomogeneous electrongas,” Phys. Rev. 136, B864–B871 (1964).

131 W. Kohn and L. J. Sham, “Self-consistent equations in-cluding exchange and correlation effects,” Phys. Rev. 140,A1133–A1138 (1965).

132 U. von Barth and L. Hedin, “A local exchange-correlationpotential for the spin-polarized case: I,” J. Phys. C: Sol.State Phys. 5, 1629–1642 (1972).

133 J. P. Perdew and A. Zunger, “Self-interaction correctionto density-functional approximations for many-electronsystems,” Phys. Rev. B 23, 5048–5079 (1981).

134 S. H. Vosko, L. Wilk, and M. Nusair, “Accurate spin-dependent electron liquid correlation energies calcula-tions: a critical analysis,” Canadian Journal of Physics58, 1200–1211 (1980).

135 R. M. Martin, “Electronic structure: Basic theory andpractical methods,” (Cambridge University Press, Cam-bridge, United Kingdom, 2004) pp. 42–44.

136 W. R. L. Lambrecht and O. K. Andersen, “Minimal basissets in the linear muffin-tin orbital method: Applicationto the diamond-structure crystals C, Si, and Ge,” Phys.Rev. B 34, 2439–2449 (1986).

137 B. Heinrich, K. B. Urquhart, A. S. Arrott, J. F. Cochran,K. Myrtle, and S. T. Purcell, “Ferromagnetic-resonancestudy of ultrathin bcc Fe(100) films grown epitaxially onfcc Ag(100) substrates,” Phys. Rev. Lett. 59, 1756–1759(1987).

138 W. Platow, A. N. Anisimov, G. L. Dunifer, M. Farle,and K. Baberschke, “Correlations between ferromagnetic-resonance linewidths and sample quality in the study ofmetallic ultrathin films,” Phys. Rev. B 58, 5611 (1998).

139 R. Urban, G. Woltersdorf, and B. Heinrich, “Gilbertdamping in single and multilayer ultrathin films: Roleof interfaces in nonlocal spin dynamics,” Phys. Rev. Lett.87, 217204 (2001).

140 A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, “Finite-element theory of transport in ferromagnetic-normalmetal systems,” Phys. Rev. Lett. 84, 2481–2484 (2000);A. Brataas, Yu. V. Nazarov, and G. E. W. Bauer, “Spin-

26

transport in multi-terminal normal metal-ferromagnetic systems with non-collinear magnetizations,” Eur. Phys.J. B 22, 99–110 (2001).