Spin-orbit coupling in magnetic tunnel junctions

Anisotropic tunneling magnetoresistance and tunneling anisotropic magnetoresistance:

Spin-orbit coupling in magnetic tunnel junctions

A. Matos-Abiague and J. Fabian

Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany 共Received 22 July 2008; revised manuscript received 14 January 2009; published 6 April 2009兲 The effects of the spin-orbit coupling 共SOC兲 on the tunneling magnetoresistance of ferromagnet/

semiconductor/normal-metal tunnel junctions are investigated. Analytical expressions for the tunneling aniso- tropic magnetoresistance共TAMR兲are derived within an approximation in which the dependence of the mag- netoresistance on the magnetization orientation in the ferromagnet originates from the interference between Bychkov-Rashba and Dresselhaus SOCs that appear at junction interfaces and in the tunneling region. We also investigate the TAMR effect in ferromagnet/semiconductor/ferromagnet tunnel junctions. The conventional tunneling magnetoresistance 共TMR兲 measures the difference between the magnetoresistance in parallel and antiparallel configurations. We show that in ferromagnet/semiconductor/ferromagnet heterostructures, because of the SOC effects, the conventional TMR becomes anisotropic—we refer to it as the anisotropic tunneling magnetoresistance共ATMR兲. The ATMR describes the changes in the TMR when the axis along which the parallel and antiparallel configurations are defined is rotated with respect to a crystallographic reference axis.

Within the proposed model, depending on the magnetization directions in the ferromagnets, the interplay of Bychkov-Rashba and Dresselhaus SOCs produces differences between the rates of transmitted and reflected spins at the ferromagnet/semiconductor interfaces, which results in an anisotropic local density of states at the Fermi surface and in the TAMR and ATMR effects. Model calculations for Fe/GaAs/Fe tunnel junctions are presented. Finally, based on rather general symmetry considerations, we deduce the form of the magnetoresis- tance dependence on the absolute orientations of the magnetizations in the ferromagnets.

DOI:10.1103/PhysRevB.79.155303 PACS number共s兲: 73.43.Jn, 72.25.Dc, 73.43.Qt

I. INTRODUCTION

The tunneling magnetoresistance 共TMR兲 effect is ob- served in ferromagnet/insulator/ferromagnet heterojunctions, in which the magnetoresistance exhibits a strong dependence on the relative magnetization directions in the two ferromag- netic layers and on their spin polarizations.^1–6 Because of this peculiarly strong asymmetric behavior of the magnetore- sistance, TMR devices find multiple uses ranging from magnetic sensors to magnetic random access memory applications.^4,5

Beyond the conventional TMR effect, it has been ob- served that the magnetoresistance in magnetic tunnel junc- tions共MTJs兲may also depend on the orientation of the mag- netizations in the ferromagnetic leads with respect to the crystallographic axes.^7–12This phenomenon is called the tun- neling anisotropic magnetoresistance共TAMR兲effect. It is re- markable that TAMR is present even in MTJs in which only one of the electrodes is magnetic and the conventional TMR is absent.^9,13 Thus, in contrast to the conventional TMR- based devices, which require two magnetic layers for their operation, TAMR-based devices can operate with a single magnetic lead, opening new possibilities and functionalities for the operation of spintronic devices. The TAMR may also affect the spin injection from a ferromagnet into a nonmag- netic semiconductor. Therefore, in order to correctly interpret the results of spin injection experiments in a spin-valve con- figuration, it is essential to understand the nature, properties, and origin of the TAMR effect.

Depending on the specific configuration considered, dif- ferent authors have used different expressions for quantify- ing TAMR 共see, for example, Refs. 6, 9, 12, 14, and 15兲.

However, the phenomenon is indistinctly referred to as TAMR. In order to clearly distinguish between these differ- ent definitions, we classify them in what we call hereout-of- plane and in-plane TAMR. The out-of-plane TAMR in a MTJ with a single magnetic layer refers to the changes in the tunneling magnetoresistance when the magnetization is ro- tated within a plane perpendicular to the ferromagnetic layer.

The situation is illustrated in Fig.1共a兲, where the unit vector ndenotes the magnetization direction measured with respect to the film normal direction and R is the tunneling magne- toresistance. The reference crystallographic axis denoted by 关x兴and the film normal direction define the plane in which the magnetization is rotated. On the other hand, the in-plane TAMR关see Fig.1共b兲兴refers to the changes in the tunneling magnetoresistance when the in-plane magnetization direction n, defined with respect to a fixed reference axis关x兴, is rotated in the plane of the ferromagnetic layer. In both cases, the out-of-plane and the in-plane, the TAMR coefficient is deter- mined by

TAMR_关x兴共␾兲=R共␾兲−R共0兲

R共0兲 . 共1兲

While this definition isformally the same for both the out- of-plane and in-plane configurations, the meaning of the angle ␾is different in each case. In fact, the two configura- tions correspond to different physical situations. While in the out-of-plane configuration the tunneling magnetoresistance changes due to the different orientations of the magnetization with respect to the direction of the current flow, the situation becomes more subtle in the in-plane configuration, where the 1098-0121/2009/79共15兲/155303共19兲 155303-1 ©2009 The American Physical Society

magnetization remains always perpendicular to the current direction.

The TAMR is also present in MTJs with two ferromag- netic leads. In such systems, the tunneling magnetoresistance R共␪^,␾兲depends on the magnetization directions, n_l andn_r, of the ferromagnetic layers共see Fig.2兲. In the conventional TMR, the tunneling magnetoresistance depends only on the relative directions 共i.e., R is only a function of ␪兲. In the TAMR, however, the tunneling magnetoresistance depends

on the magnetization directions relative to the crystallo- graphic axes共i.e.,Rbecomes a function of both␪and␾兲. In the out-of-plane configuration the magnetizations nl and nr

of the corresponding ferromagnetic leads are rotated in the plane defined by the reference axis关x兴and the normal to the layers direction关see Fig.2共a兲兴, while in the in-plane configu- rationnlandnrare rotated in the plane of the layers关see Fig.

2共a兲兴. The definition of the TAMR in Eq. 共1兲 can be gener- alized to the case of MTJs with two ferromagnetic leads as follows:

TAMR_关x兴共␪^,␾兲=R共␪^,␾兲−R共␪,0兲

R共␪,0兲 . 共2兲 Here as in Eq.共1兲the meaning of the angles␪and␾depends on the specific configuration considered. Note also that there is no ambiguity in the notation used in Eqs. 共1兲and共2兲: the number of arguments in the TAMR coefficient关one argument 共␾兲in the case of MTJs with a single magnetic electrode and two arguments共␪and␾兲in the case of MTJs with two mag- netic leads兴 eliminates any possible confusion between the two cases.

Another interesting effect may emerge in MTJs with two magnetic leads when the conventional TMR becomes aniso- tropic. The conventional TMR is defined as

TMR =R_AP−R_P

R_P , 共3兲

whereR_P共RAP兲is the magnetoresistance measured when the magnetization of the ferromagnetic layers are parallel 共P兲 关antiparallel共AP兲兴. Allowing for a dependence of the tunnel- ing magnetoresistance on the specific axis along which the parallel and antiparallel configurations are defined, the aniso- tropic tunneling magnetoresistance 共ATMR兲 can be defined as

ATMR_关x兴共␾兲=R_AP共␾兲−R_P共␾兲

R_P共␾兲 , 共4兲 where the resistances in parallel and antiparallel configura- tions are, respectively, R_P共␾兲=R共0 ,␾兲andR_AP共␾兲=R共␲^,␾兲 关here we have use the same notation for the resistance R共␪^,␾兲as in Fig.2兴.

A summary of the abbreviations used throughout the pa- per is given in TableI.

Most of the early experiments on the magnetization direc- tion dependence of the tunneling magnetoresistance were performed in MTJs with GaMnAs ferromagnetic leads. The in-plane ATMR was experimentally observed in GaMnAs/

AlAs/GaMnAs tunnel junctions,^7,8 where in-plane ATMR ratios ATMR_关100兴共0兲⬇75% and ATMR_关100兴共−␲/4兲⬇30%

were found 关here we have used the notations introduced in Eqs. 共2兲 and 共4兲兴. A similar but larger effect in GaMnAs/

GaAs/GaMnA heterostructures was reported in Ref.16. The first experimental observation of the TAMR in MTJs with a single magnetic layer was done in共Ga,Mn兲As/AlOx/Au het- erojunctions, in which an in-plane TAMR ratio of about TAMR_关110兴共␲/2兲⬇2.7% was found.⁹ Theoretical investiga- tions considering the out-of-plane^12,17 and in-plane¹⁷ con- figurations in GaMnAs based MTJs in which both electrodes R(φ)

φ ⁿ

[x]

Ferromagnet

(a)out-of-planeconfiguration

[x]

Ferromagnet nφ

R(φ) (b)in-planeconfiguration

FIG. 1.共Color online兲Schematics of the configurations used for measuring the TAMR in MTJs in which one of the leads is ferro- magnetic. 共a兲 Out-of-plane configuration. 共b兲 In-plane configura- tion. The vectornindicates the magnetization orientation, while关x兴 refers to a reference crystallographic axis. The tunneling magne- toresistanceR共␾兲depends on the magnetization direction specified by the angle␾.

R(θ,φ) φ

n_l

[x]

Ferromagnet

(a)out-of-planeconfiguration

n_r θ

Ferromagnet

[x]

Ferromagnet n_lφ

R(θ,φ) (b)in-planeconfiguration

Ferromagnet n_r

θ

FIG. 2.共Color online兲Schematics of the configurations used for measuring the TAMR in MTJs in which both leads are ferromag- netic. 共a兲 Out-of-plane configuration. 共b兲 In-plane configuration.

The vectorsn_l and n_rdetermine the magnetization orientation in each of the leads and关x兴denotes a reference crystallographic axis.

The tunneling magnetoresistance R共␪,␾兲 depends on the absolute magnetization directionsn_landn_r.

are ferromagnetic have also been reported. Experimentally, tunnel junctions such as 共Ga,Mn兲As/GaAs/共Ga,Mn兲As and 共Ga,Mn兲As/ZnSe/共Ga,Mn兲As have been used for measuring the in-plane TAMR.^10,11 In the case of 共Ga,Mn兲As/ZnSe/

共Ga,Mn兲As, the in-plane TAMR ratio TAMR_关110兴共0 ,␲/2兲 was found to decrease with increasing temperature, from about 10% at 2 K to 8.5% at 20 K.¹¹ This temperature de- pendence of the in-plane TAMR is more dramatic in the case of共Ga,Mn兲As/GaAs/共Ga,Mn兲As, for which a TAMR ratio of the order of a few hundred percent at 4 K was amplified to 150 000%at 1.7 K.¹⁰This huge amplification of the in-plane TAMR was suggested to originate from the opening of the Efros-Shklovskii gap¹⁸ at the Fermi energy when crossing the metal-insulator transition.¹⁰Measurements of the TAMR in p⁺-共Ga, Mn兲As/n⁺-GaAs Esaki diode devices have also been reported.^19,20In addition to the investigations involving vertical tunneling devices the TAMR has also been studied in break junctions,^21,22 nanoconstrictions,^20,23 and nanocontacts.²⁴

Beyond the area of currently low Curie temperature ferromagnetic semiconductors, the TAMR has recently been investigated both theoretically and experimentally in tunnel junctions such as CoFe/MgO/CoFe and CoFe/ Al₂O₃/CoFe,²⁵ Fe/GaAs/Au,^6,13 and multilayer- 共Co/Pt兲/AlOx/Pt structures.¹⁵Experimental investigations in Co/AlOx/Au 共Ref. 26兲 and theoretical calculations in Fe/MgO/Fe,¹⁴ Fe共001兲/vacuum/bcc-Cu共001兲,²⁷ CoPt structures,²⁸ and in layered bimetallic nanostructures of the type Mn/W共001兲²⁹have also been reported.

In TableII we show some of the previous investigations in MTJs with metallic ferromagnets specifying the used con- figuration.

Role of spin-orbit coupling

Most of the theoretical investigations have been devoted to the out-of-plane TAMR in MTJs with isolating barriers.9,14,15,25,27 For the case of asymmetric structures the Bychkov-Rashba-type SOC due to the strong electric field across the ferromagnet/insulator interface has been identified as being the responsible mechanism for the out-of-plane TAMR. An intuitive simple picture of the role of the SOC is explained as follows. Consider, for simplicity, a MTJ with a single ferromagnetic lead, as the one sketched in Fig. 1共a兲.

The potential gradient along the growth direction generates the effective Bychkov-Rashba spin-orbit coupling 共SOC兲 field共SOCF兲,

w_BR=共−␣^ky,␣^kx,0兲, 共5兲 where ␣ is the Bychkov-Rashba SOC parameter and k_储

=共kx,ky兲 is the wave vector in the plane of the layers. The observation that w_BR lies in a plane parallel to the layers suggests that the situation in which the magnetization direc- tionnand the Bychkov-Rashba SOCFw_BRare coplanar共i.e., when ␾⁼␲/2兲differs from the case in which ␾⫽␲/2 关see Fig.1共a兲兴. In fact, the effective Bychkov-Rashba SOC shifts the electron energy by²⁷

TABLE I. Summary of used abbreviations sorted by alphabetic order.

ALDOS Anisotropic local density of states ATMR Anisotropic tunneling magnetoresistance

DDFM Dirac-delta function model

F Ferromagnet

LDOS Local density of states

MTJ Magnetic tunnel junction

NM Normal metal

S Semiconductor

SOC Spin-orbit coupling

SOCF Spin-orbit coupling field

SSOM Slonczewski spin-orbit model

TAMR Tunneling anisotropic magnetoresistance TASP Tunneling anisotropic spin polarization

TMR Tunneling magnetoresistance

TABLE II. Some typical theoretical共theor兲and experimental共expt兲values of the TAMR and ATMR in MTJs with metallic ferromagnets for different configurations. Note that the notation used here for the TAMR and ATMR关see Eqs.共1兲–共4兲兴may differ from the ones in the original references.

System Reported quantities Typical values Ref.

Fe共001兲/vacuum/Cu共001兲 out-of-plane TAMR_关100兴共␾=␲/2兲 20%^a共theor兲 27 in-plane TAMR_关110兴共␾= −␲/4兲 10%^a共theor兲

Fe/GaAs/Au in-plane TAMR_关110兴共␾兲 −0.4%^b共expt and theor兲 6and13

CoFe/MgO/CoFe out-of-plane differential conductance 25

CoFe/Al₂O₃/CoFe

共Co/Pt兲/AlO_x/Pt out-of-plane TAMR_关x兴共␾兲 −12.5%^c共expt兲 15 Fe/FeO/MgO/FeO/Fe out-of-plane ATMR_关100兴共␾= 0兲 4205%共theor兲 14

out-of-plane ATMR_关100兴共␾=␲/2兲 2956%共theor兲 out-of-plane TAMR_关100兴共␪= 0 ,␾=␲/2兲 44%共theor兲

aTaken at a bias voltage of about −50 mV.

bTaken at␾=␲/2 and bias voltage −90 mV.

cTaken at␾=␲/2 and bias voltage −5 mV.

⌬E_↑,↓= ⫾w_BR·n, 共6兲 where↑ and↓ refer to the up- and down-spin channels, re- spectively. In this way, the energy bands become anisotropic with respect to the magnetization direction n. For example,

⌬E共k储兲vanishes whennis normal to the ferromagnetic layer 关this corresponds to␾^{= 0 ,}␲^{in Fig.} 1共a兲兴, while at nonvan- ishingk_储,⌬E共k储兲remains finite for any other value of␾^{. The} final outcome is a␾-dependent tunneling magnetoresistance.

Furthermore, from Eq.共6兲 关see also Fig.1共a兲兴it follows that the situations at ␾, −␾, and␾+␲are physically equivalent but differ from the case at ␾⁺␲/2. This implies that the tunneling magnetoresistance must obey the relations R共␾兲

=R共−␾兲=R共␾⁺␲兲⫽R共␾⁺␲/2兲, i.e., the tunneling magne- toresistance exhibits a twofold symmetric anisotropy as a function of ␾. The twofold symmetry of the out-of-plane TAMR has been experimentally observed.^15,25

Recent experimental investigations of the in-plane TAMR in MTJs with a single ferromagnetic lead have revealed that similarly to the out-of-plane, the in-plane TAMR also exhib- its a twofold symmetry^9,13,15as a function of␾共recall, how- ever, that the meaning of the angle␾is different for the two configurations, as schematically shown in Fig. 1兲. One may naively think that, as for the out-of-plane TAMR, the twofold symmetry of the in-plane TAMR originates from the Bychkov-Rashba SOC. However, a simple symmetry analy- sis shows that in the case of the in-plane TAMR the linear Bychkov-Rashba SOC alone is not sufficient for generating a twofold symmetry. The main observation here is that in such a case the effective Bychkov-Rashba SOCF is invariant un- der in-plane rotations关see Fig.4共b兲兴. Therefore, the orienta- tion of the Bychkov-Rashba SOC vector field relative to the in-plane magnetization is independent of ␾, and no aniso- tropy is obtained. The situation may be different if higher orders of the Bychkov-Rashba SOC become relevant. In such a case the C_4v symmetry of this SOCF will lead to a fourfold symmetric in-plane TAMR.

In order to explain the twofold symmetry of the in-plane TAMR experimentally observed in共Ga,Mn兲As/AlOx/Au het- erojunctions the existence of an uniaxial strain was assumed.⁹Surprisingly, recent experiments have shown that in epitaxial Fe/GaAs/Au tunnel junctions the in-plane TAMR exhibits also a twofold symmetry.¹³ Because of the high quality matching at the epitaxial Fe/GaAs interface strain effects are unlikely to play a sizable role. What is then the mechanism leading to the observed twofold symmetry of the in-plane TAMR in Fe/GaAs/Au tunnel junctions? We have proposed共see Refs.6 and13兲that the twofold symmetry of the in-plane TAMR originates from the interference of the Bychkov-Rashba and Dresselhaus SOCs. The presence of GaAs, a zinc-blende semiconductor, as the barrier material plays here a decisive role. Zinc-blende semiconductors are noncentrosymmetric. Therefore, the so-called Dresselhaus SOC, which originates from the bulk inversion asymmetry 共BIA兲 of the semiconductor is intrinsically present in such materials. Thus, although the Bychkov-Rashba-type SOC is present in asymmetric MTJs, the Dresselhaus SOC may or not be present, in dependence on the symmetry of the con- stituent materials. This produces qualitative differences be-

tween Fe/GaAs 共or, in general, ferromagnet/zinc-blende semiconductor兲 based MTJs and other structures in which the Dresselhaus SOC is absent 关e.g., Fe共001兲/vacuum/bcc- Cu共001兲in Ref.27兴. In fact, the interference of the Bychkov- Rashba and Dresselhaus interactions leads to a net aniso- tropic SOC with a C_2v symmetry which reflects the symmetry of the Fa/GaAs interface.^6,13 This observation is crucial because, as shown below, the C_2v symmetry of the SOCF共which is the underlying structure symmetry兲is trans- ferred into the tunneling magnetoresistance and results in a twofold symmetric in-plane TAMR. Based on these symme- try considerations we can conclude that the in-plane TAMR will exhibit a twofold symmetry in ferromagnet/

semiconductor 共F/S兲 based MTJs, while a fourfold symme- try in ferromagnet/insulator 共F/I兲 based tunnel junctions 关e.g., Fe/MgO/Fe, Fe共001兲/vacuum/bcc-Cu共001兲, etc.兴is ex- pected 共as long as the ferromagnet has cubic symmetry and additional uniaxial strain effects are negligible兲.

Motivated by the recent measurements of the in-plane TAMR in Fe/GaAs/Au MTJs 共Fig. 3兲, we shall focus our discussion on the case of the in-plane TAMR in ferromagnet/

semiconductor/normal-metal共F/S/NM兲and in ferromagnet/

semiconductor/ferromagnet 共F/S/F兲 MTJs. Here and in what follows, when referring to a semiconductor 共ferromag- net兲, a zinc-blende semiconductor共cubic metallic ferromag- net兲is meant. Until now, all the theoretical investigations of the TAMR关with the exception of Refs.6 and13兴have been based on first-principles calculations. We believe that, as a complement and inspiration to the first-principles approach, model calculations can be of great interest for a better under- standing of the phenomenology of the TAMR effect. Because of the highly complicated band structure of F/S based het- erojunctions, model calculations in such systems may appear as an oversimplified picture. However the relative simplicity and regularity of the angular dependence of the in-plane TAMR experimentally observed in Fe/GaAs/Au MTJs indi- cates that the TAMR may not depend strongly on the specific details of the complicated band structure, suggesting that some of the observed features of the TAMR effect could be understood on the basis of a simple minimal model. Thus, the model here proposed is not intended to completely de- scribe all the details of the TAMR effect but to offer the simplest view which, by incorporating the spin-orbit interac- tion as the relevant physical mechanism, is capable of repro- ducing the main traits observed in the in-plane TAMR.

E_F V0

Fe Au

GaAs

z zl zr

z

n

φ Fe

GaAs Au

z_l zr [001]

[110]

[-110]

(a) (b)

FIG. 3. 共Color online兲 共a兲 Schematics of a Fe/GaAs/Au MTJ.

The magnetization direction in the ferromagnet is specified by the vectorn.共b兲Schematics of the potential profile of the heterojunc- tion along the关001兴direction.

As mentioned above, in our model the twofold symmetry of the in-plane TAMR in F/S based MTJs originates from the interference of Dresselhaus and Bychkov-Rashba-type SOCs.^6,13Such interference effects have already been inves- tigated in lateral transport in two-dimensional 共2D兲electron systems,^30–32in spin relaxation in quantum wells³³and quan- tum dots,³⁴ or in 2D plasmons.³⁵ The symmetry, which is imprinted in the tunneling probability becomes apparent when a magnetic moment is present. Our main results are as follows: 共i兲finding analytical expressions for evaluating the in-plane TAMR in bothF/S/NM andF/S/FMTJs,共ii兲pre- diction and evaluation of the in-plane ATMR inF/S/Fhet- erojunctions, and共iii兲derivation of a simple phenomenologi- cal relation describing the dependence of the tunneling magnetoresistance on the absolute orientation of the in-plane magnetization共s兲of the ferromagnet共s兲.

Since we are especially interested in the study of the in- plane TAMR and in-plane ATMR, and for the sake of brev- ity, in what follows we will refer to them as the TAMR and ATMR effects.

The paper is organized as follows. In Sec. IIwe present the theoretical model describing the tunneling through a MTJ. In a first approximation we consider the case of an infinitesimally thin barrier共Sec.II A兲, while the finite spatial extension of the potential barrier is incorporated in a more sophisticated approach discussed in Sec.II B. Detailed solu- tions and tunneling properties within these approximations are given in Appendixes A and B, respectively. In Sec.IIIwe discuss the TAMR in bothF/S/NM共Sec.III A兲andF/S/F 共Sec. III B兲MTJs. The ATMR inF/S/F tunnel junctions is investigated in Sec.IV, where specific calculations for model Fe/GaAs/Fe MTJs are presented. In Sec. V we develop a phenomenological model for determining the dependence of the TAMR and ATMR on the absolute orientation共s兲 of the magnetization共s兲 in the ferromagnetic lead共s兲. Finally, con- clusions are given in Sec.VI.

II. THEORETICAL MODEL

Consider an F/S/F tunnel heterojunction in which the semiconductor lacks bulk inversion symmetry; zinc-blende semiconductors are typical examples. The bulk inversion asymmetry of the semiconductor together with the structure inversion asymmetry共for the case of asymmetric junctions兲 of the heterojunction give rise to the Dresselhaus^6,36–38 and Bychkov-Rashba^6,38,39 SOCs, respectively. The interference of these two spin-orbit interactions leads to a net anisotropic SOC with aC_2vsymmetry which is imprinted onto the tun- neling magnetoresistance as the electrons pass through the semiconductor barrier. This was discussed in some details in Refs. 6 and 13 for the case of F/S/NM tunnel junctions.

Here we generalize the model proposed in Refs.6and13to the case of F/S/F tunnel junctions. For such structures our model predicts the coexistence of both the TAMR and ATMR phenomena.

We consider an F/S/F tunnel junction grown in the z

=关001兴 direction, where the zinc-blende semiconductor forms a barrier of width d between the left and right ferro- magnetic electrodes. At first we discuss a simplified model

for very thin barriers. In that case the barrier can be approxi- mated by a Dirac-delta function and the SOC reduced to the plane of the barrier. In what follows we will refer to this model as the Dirac-delta function model共DDFM兲. A second model in which Slonczewski’s proposal^3,6 for ferromagnet/

insulator/ferromagnet tunnel junctions is generalized to the case of ferromagnet/semiconductor/ferromagnet junctions by including the Bychkov-Rashba and Dresselhaus SOCs will be referred to as the Slonczewski spin-orbit model共SSOM兲.

In both the DDFM and the SSOM we assume coherent tun- neling and a lattice mismatch small enough for the strain effects to be negligible. The fourfold anisotropy of the cubic metallic ferromagnet共s兲is assumed to be much smaller than the tunneling anisotropic effects 共this is particularly true in Fe/GaAs-based MTJs, where TAMR measurements have shown no trace of any fourfold anisotropy兲.¹³

A. DDFM

We consider here the case of a very thin tunneling barrier.

Assuming that the in-plane wave vector k储 is conserved throughout the heterostructure, one can decouple the motion along the growth direction共z兲from the other spatial degrees of freedom. The effective model Hamiltonian describing the tunneling across the heterojunction reads as

H=H₀+HZ+H_SO. 共7兲 Here

H₀= − ប² 2m₀

d²

dz²+V₀d␦共z兲, 共8兲 withm₀as the bare electron mass andV₀anddas the height and width, respectively, of the actual potential barrier 关here modeled with a Dirac-delta function ␦共z兲兴 along the growth direction 共z=关001兴兲 of the heterostructure.

The spin splitting due to the exchange field in the left 共z⬍0兲and right共z⬎0兲ferromagnetic regions is given by

HZ= −⌰共−z兲⌬l

2 nl·␴^{−⌰共z兲⌬r}

2 nr·␴^. 共9兲 Here⌬land⌬rrepresent the exchange energy in the left and right ferromagnets, respectively, and ⌰共z兲 is the Heaviside step function. The components of the vector ␴are the Pauli matrices, andnj=共cos␪j, sin␪j, 0兲, withj=l,r, is a unit vec- tor defining the in-plane magnetization direction in the left 共j=l兲and right共j=r兲ferromagnets with respect to the关100兴 crystallographic direction. The Zeeman splitting in the semi- conductor can be neglected.

In recent experiments with Fe/GaAs/Au tunnel junctions,¹³ the reference axis was taken as the关110兴 direc- tion. Therefore, it is convenient to express the magnetization direction relative to the关110兴 axis by introducing the angle shifting ␾j=␪j−␲/4 共j=l,r兲. One can then write nj

=关cos共␾j+␲/4兲, sin共␾j+␲/4兲, 0兴with␾j giving the magne- tization direction in the left共j=l兲and right共j=r兲 ferromag- nets with respect to the关110兴crystallographic direction.

Within the DDFM, the spin-orbit interaction throughout the semiconductor barrier 共including the interfaces兲 can be written as

H_SO=共w·␴兲␦共z兲, 共10兲 with the effective SOCF

w=共−␣^{¯ k}y+^{¯ k}␥ x, ␣^{¯ k}x−^{¯ k}␥ y, 0兲. 共11兲 Here ␣^{¯ and ¯}␥ represent effective values of the Bychkov- Rashba and linearized Dresselhaus parameters, respectively, andkx andky refer to thex andy components of the wave vectork. In terms of the usual Dresselhaus parameter␥^{, the} linearized Dresselhaus parameter can be approximated⁶ as

␥

¯⬇␥Q, where Q= 2m₀V₀d/ប² stands for the strength of the effective wave vector in the barrier.

In a real situation the barrier width is finite and there actually are two contributions of the interface Bychkov- Rashba SOC. Therefore there are two interface Bychkov- Rashba parameters ␣l and ␣r corresponding to the left and right interfaces, respectively. In the DDFM the two interfaces merge to form the Dirac-delta barrier with an effective Bychkov-Rashba parameter ␣^¯⬇具␣l␦共z兲−␣r␦共z−d兲典0. Here the Dirac-delta functions account for the interface Bychkov- Rashba SOC in the real finite barrier with left and right in- terfaces at z= 0 and z=d, respectively. The average 具. . .典0

refers to space and momentum averages with respect to the unperturbed states共i.e., in the absence of SOC兲at the Fermi energy. Since the parameters ␣l and ␣r are not known for F/Sand semiconductor/normal-metal共S/NM兲interfaces, we still have to consider␣^¯ as a phenomenological parameter.

The scattering states in the left 共z⬍0兲 and right 共z⬎0兲 ferromagnetic regions are given by

⌿_␴^共l兲=e^ik␴z␹_␴^共l兲

冑

k_␴ +r_␴,␴e^−ik␴z␹_␴^共l兲+r_␴,−␴e^−ik−␴z␹_−␴^共l兲, 共12兲 and

⌿_␴^共r兲=t_␴,␴e^i␬␴z␹_␴^共r兲+t␴,−␴e^i␬−␴z␹_−␴^共r兲, 共13兲 respectively. Here we have introduced the wave-vector com- ponents

k_␴=

冑

^2mប²⁰

冉

^E+␴⌬2l

冊

^{−k储2, 共14兲}

and

␬_␴=

冑

^2m_ប²⁰

冉

^E+␴⌬2r

冊

^{−k储2, 共15兲}

withk_储=

冑

k_x²+k²_ydenoting the length of the wave-vector com- ponent corresponding to the free motion in thex-yplane. The spinors

␹_␴^共j兲=

冑

¹2

冉

␴^{ei共␾1j+␲/4兲}

冊

^{共j=l,r兲, 共16兲}

correspond to a spin parallel共␴⁼↑兲or antiparallel共␴⁼↓兲to the magnetization direction nj=关cos共␾j+␲/4兲, sin共␾j

+␲/4兲, 0兴 in the left共j=l兲and right共j=r兲ferromagnets.

Here and in what follows, we consider a free-electron behavior for the in-plane wave vector k_储, which is a simpli- fied view of the actual band structure. Such an approxima- tion may not be appropriate for describing systems or phe-

nomena in which the contribution of states with large k_储 is relevant. For the system considered here we expect that the main contribution to TAMR comes from the states with small k_储. This expectation is based on the fact that the experimen- tally observed magnetization-direction dependence of the TAMR in Fe/GaAs based MTJs exhibits a twofold symmetry with only two lobes 关see Figs. 2共a兲 and 2共b兲 of Ref. 13兴, indicating that only the lowest nonvanishing order in the SOCF is relevant. If, for example, higher orders in the SOCF terms 共which are also higher ink_xandk_y兲were relevant for the transmissivity, the corresponding symmetry of the SOCF would lead to a twofold symmetric TAMR but with four lobes instead of the only two that are observed experimen- tally. The absence of traces of higher order in the SOCF terms in the observed TAMR suggests that the states with largek储 do not play a significant role.

The reflection and transmission coefficients can be found by imposing appropriate boundary conditions and solving the corresponding system of linear equations共for details see Ap- pendix A兲. The transmissivity of an incoming spin-␴^particle can then be evaluated from the relation

T_␴共E,k储兲= Re关␬␴共兩t_␴,␴兩²+␬−␴兩t_␴,−␴兩²兲兴. 共17兲 Explicit analytical expressions for the transmission coeffi- cients共t_␴,␴ andt_␴,−␴兲are given in Appendix A.

B. SSOM

We present a generalization of the Slonczewski model³ for ferromagnet/insulator/ferromagnet tunnel junctions to the case in which the insulator barrier is replaced by a zinc- blende semiconductor. Unlike in the DDFM, now the spatial extension of the potential barrier is taken into account. The model Hamiltonian is

H=H₀+H_Z+H_BR+H_D, 共18兲 where

H₀= −ប²

2 ⵜ

冋

^m1共z兲ⵜ

册

^{+V0⌰共z兲⌰共d−z兲. 共19兲}

The electron effective mass m共z兲is assumed to bem=mcin the central 共semiconductor兲 region andm=m₀ in the ferro- magnets. The exchange splitting in the ferromagnets is now given by

HZ= −⌰共−z兲⌬l

2 nl·␴^{−⌰共z−d兲⌬r}

2 nr·␴^. 共20兲 The Dresselhaus SOC can be written as6,37,38,40,41

HD=共kx␴x−ky␴y兲⳵

⳵^z

冉

^␥共z兲⳵^⳵z

冊

^{, 共21兲}

wherexandycorrespond to the关100兴and关010兴directions, respectively. The Dresselhaus parameter ␥共z兲 has a finite value ␥ in the semiconductor region, where the bulk inver- sion asymmetry is present and vanishes elsewhere. Note that because of the steplike spatial dependence of ␥共z兲, the Dresselhaus SOC关Eq. 共21兲兴implicitly includes both the in- terface and bulk contributions.^6,37,41

The Bychkov-Rashba SOC is given by⁴²

H_BR=关␣l␦共z−zl兲−␣r␦共z−zr兲兴共kx␴y−ky␴x兲, 共22兲 and arises due to the F/S interface inversion asymmetry.⁶ Here ␣l 共␣r兲 denotes the SOC strength at the left 共right兲 interfacezl= 0共zr=d兲. For the small voltages considered here 共up to a hundred mV兲, the Bychkov-Rashba SOC inside the semiconductor can be neglected.

The z components of the scattering states in the left and right ferromagnets have the same form as in Eqs. 共12兲 and 共13兲, respectively.

In the central共semiconductor兲region共0⬍z⬍d兲we have

⌿_␴^共c兲=

兺

i=⫾共A_␴,ie^qiz+B_␴,ie^−qiz兲␹i共c兲, 共23兲

␹_⫾^共c兲=

冑

¹2

冉

⫾¹e^i␰

冊

^{. 共24兲}

The angle ␰ is defined through the relation tan共␰兲= −ky/kx. We have also used the notation

q_⫾= q₀

冑

^1⫿

冉

^2mប^c2␥k储

冊

^{2, 共25兲}

where

q₀=

冑

^2mc共Vប^02−E兲+k_储² 共26兲 is the length of the z component of the wave vector in the barrier in the absence of SOC.

The expansion coefficients in Eqs.共12兲,共13兲, and共23兲can be found by applying appropriate matching conditions at each interface and by solving the corresponding system of linear equations关for details, see Eq.共A35兲兴. Once the wave function is determined, the particle transmissivity can be cal- culated from Eq. 共17兲. Approximate analytical expressions for the transmission coefficients t_␴,␴ and t_␴,−␴ are given in Appendix B.

III. TAMR

The magnetoresistance of a tunnel junction can be ob- tained by evaluating the current through the device or the conductance. The current flowing along the heterojunction is given by

I= e 共2␲兲³ប

兺

␴=↑,↓

冕

^{dEd2k储T␴共E,k储兲关fl共E兲−fr共E兲兴, 共27兲}

wherefl共E兲andfr共E兲are the Fermi-Dirac distributions with chemical potentials␮l and␮r in the left and right共metallic or ferromagnetic兲 leads, respectively. For the case of zero temperature and small voltages, the Fermi-Dirac distribu- tions can be expanded in powers of the voltageV_bias. To first order in V_biasone obtains fl共E兲−fr共E兲⬇␦共E−EF兲Vbias, with

␦共x兲as the Dirac-delta function andEFas the Fermi energy.

One then obtains the following approximate expression for the conductance:

G=_␴=↑,↓

兺

^{G␴, G␴=}_共2␲^e2_兲³_ប

冕

^{d2k储T␴共EF,k储兲. 共28兲}

We note that although similar, the expression above differs from the linear-response conductance.

In our model, the transmissivityT_␴共EF,k储兲depends on the voltage via the voltage dependence of the Bychkov-Rashba parameter ␣^¯.43 Recent first-principles calculations⁴⁴ have shown that the SOCF is different for different bands; there- fore the effective value of␣^¯ is energy dependent. By apply- ing an external voltage the energy window relevant for tun- neling can be changed, resulting in voltage-dependent values of␣^¯ and, therefore, in a voltage dependence of the transmis- sivity. Consequently, the conductance in Eq. 共28兲 depends, parametrically, on the applied voltage.

A. TAMR in ferromagnet/semiconductor/normal-metal tunnel junctions

The tunneling properties ofF/S/NM junctions can be ob- tained as a limit case of the models proposed in Sec. IIfor F/S/F tunnel junctions by taking ␾l=␾r=␾ ^and ⌬r as the Zeeman splitting in the normal-metal region. In the present casel,c, andrrefer to the ferromagnetic共left兲, semiconduc- tor共central兲, and normal-metal共right兲regions, respectively.

The TAMR in F/S/NM tunnel junctions refers to the changes in the tunneling magnetoresistance共R兲 when vary- ing the magnetization directionnlof the magnetic layer with respect to a fixed axis. Here we assume the 关110兴crystallo- graphic direction as the reference axis. The TAMR is then given by⁶

TAMR_关110兴共␾兲=R共␾兲−R共0兲

R共0兲 =G共0兲−G共␾兲

G共␾兲 . 共29兲 Since in a F/S/NM tunnel junction only one electrode is magnetic, the conventional TMR effect is absent.

An alternative to the magnetoresistance, which refers to the charge transport, is the spin-polarization efficiency of the transmission characterized by the tunneling spin polarization,^4,5

P=I_↑−I_↓

I , 共30兲

where I_␴ is the charge current corresponding to the spin-␴ channel andIis the total current. The changes in the tunnel- ing spin polarization when the magnetization of the ferro- magnet is rotated in plane can then be characterized by the tunneling anisotropic spin polarization共TASP兲, which is de- fined as⁶

TASP_关110兴共␾兲=P共0兲−P共␾兲

P共␾兲 . 共31兲 Taking into account that the Zeeman splitting in the normal metal is small we can approximate ␬_␴⬇␬−␴ and the total conductance can be written as the sum of isotropic共G^iso兲and anisotropic关G^aniso共␾兲兴 contributions共for details see Appen- dix A兲. It follows then from Eq.共29兲that the TAMR is given by

TAMR_关110兴共␾兲=G^aniso共0兲−G^aniso共␾兲

G^iso+G^aniso共␾兲 . 共32兲 For junctions in which the Bychkov-Rashba and Dresselhaus SOCs can be considered as small perturbations, the aniso- tropy is small and G^aniso共␾兲ⰆG^iso. In addition, the SOC ef- fects on the isotropic part of the conductance is also small andG^iso⬇G^共0兲 共hereG^共0兲 denotes the conductance in the ab- sence of the SOC兲. The TAMR can then be approximated as

TAMR_关110兴共␾兲 ⬇G^aniso共0兲−G^aniso共␾兲

G^共0兲 . 共33兲 The substitution of Eq.共A17兲into Eq.共33兲leads to

TAMR_关110兴共␾兲 ⬇e²

h

具g2↑k_储²典_↑+具g2↓k_储²典_↓

G^共0兲 ␭_␣␭_␥关cos共2␾兲− 1兴.

共34兲 Here we have introduced the dimensionless SOC parameters

␭_␣= 2m₀␣^¯/ប² and ␭_␥= 2m₀␥^¯/ប². The functions g_2↑ and g_2↓

are given by Eq.共A18兲.

The expression above gives the angular dependence of the TAMR and is consistent with the angular dependence experi- mentally observed in Fa/GaAs/Au tunnel junctions.¹³It also suggests that the inversion 共change in sign兲 of the TAMR 共such an inversion has been experimentally observed¹³兲may originate from bias-induced changes in the product ␭_␣␭_␥

⬀␣^¯␥^¯. In general, the bias dependence of the Dresselhaus parameter in semiconductors is weak, while the Bychkov- Rashba parameter can be tuned by varying the voltage. Thus, we consider in our model that the possible change in sign of the TAMR_关110兴⬀␭_␣␭_␥is determined by bias-induced changes in the sign of the effective Bychkov-Rashba parameter ␣^¯. Furthermore, one can see from Eq.共34兲that the amplitude of the TAMR is governed by the product ␭_␣␭_␥⬀␣^¯␥^{¯ and the} averages 具g_2␴k²_储典_␴ 共␴=↑,↓兲. When ␣^¯␥^¯= 0, the twofold TAMR is suppressed共the suppression of the TAMR was also observed in Ref.13兲, i.e., as long as other anisotropic effects such as uniaxial strain are not present, the Bychkov-Rashba 共or Dresselhaus兲 SOC alone cannot explain the experimen- tally observedC_2vsymmetry of the TAMR. The TAMR van- ishes also if the spin polarization of both electrodes becomes sufficiently small. In such a casek_F,↑⬇k_F,↓andg_2␴vanishes 关see Eq. 共A18兲兴, resulting in the suppression of the TAMR.

On the contrary, Eq.共34兲predicts an increase in the TAMR amplitude for F/S/NM tunnel junctions whose constituents exhibit large values of␣^¯¯␥as well as a large spin polarization in the magnetic electrode.

A simple intuitive explanation of the origin of the uniaxial anisotropy of the TAMR can be obtained by investigating the dependence of the effective SOCFw共k储兲 关see Eq.共11兲兴, i.e., the effective magnetic field that the spins feel when travers- ing the semiconducting barrier. A schematics of the aniso- tropy of the SOCF w共k_储兲 is shown in Fig. 4共a兲, where the thin arrows represent a vector plot of w共k储兲, while the solid line is a polar plot of the field amplitude 兩w共k储兲兩for a fixed value of k_储=兩k储兩. The SOCF is oriented in the 关110兴 共关−110兴兲 direction at the points of low共high兲SOCF, where the field amplitude 兩w兩 reaches a minimum 共maximum兲.

When the magnetization in the ferromagnet points along the 关−110兴direction, the direction of the highest SOCF is paral- lel to the incident, majority spins which are then easily trans- mitted through the barrier. On the other hand, for a magne- tization direction关110兴, the highest SOCF is perpendicular to the incident spins 共to both the majority and minority spins兲 and the transmission becomes less favorable than in the case the magnetization is in the关−110兴direction. This spin-orbit- induced difference in the tunneling transmissivities depend- ing on the magnetization direction results in the uniaxial an- isotropy of the TAMR.⁴⁵We remark that this effect relies on the uniaxial anisotropy of the SOCF amplitude兩w兩, which is

kx

ky

[110]

[-110]

k

_x

k

_y

kx

ky

[-110] k

_y

[110]

k

_x

αααα ≠≠≠≠ 0, γγγγ = 0

αααα ≠≠≠≠ 0, γγγγ ≠≠≠≠ 0

(a)

(b)

FIG. 4. 共Color online兲 共a兲 Schematics of the anisotropy of the spin-orbit coupling fieldw共k_储兲when both the Bychkov-Rashba and Dresselhaus SOCs are present 共␣,␥⫽0兲. Thin arrows represent a vector plot of the SOCF w. The solid line is a polar plot of the SOCF strength兩w共k_储兲兩for a fixed value ofk_储=兩k_储兩. When the mag- netization of the ferromagnet points along the关−110兴direction关see thick blue 共black兲 arrow兴, the direction of the strongest SOCF is parallel to the incident majority spins which easily tunnel through the barrier. On the contrary, for a magnetization direction关110兴 关see thick green 共gray兲 arrow兴, the strongest SOCF is perpendicular to the incident spins and the tunneling becomes less favorable. The net result is a spin-valve effect whose efficiency depends upon the ab- solute orientation of the magnetization and gives rise to the in-plane TAMR.共b兲 Same as in共a兲 but in the absence of Dresselhaus SOC 共␣⫽0 , ␥= 0兲. In this case the amplitude of the SOCF兩w共k_储兲兩be- comes isotropic and the in-plane TAMR is suppressed.

a consequence of the interference of the Bychkov-Rashba and Dresselhaus SOC. In systems such as Fe共001兲/vacuum/

bcc-Cu共001兲, in which the Dresselhaus SOC is absent, the linear inkxandkyBychkov-Rashba terms lead to an isotropic SOCF amplitude关see Fig.4共b兲兴and the effect is absent. Thus for such systems no in-plane TAMR is expected to occur unless higher-order SOC terms become relevant. However, even in such a case the TAMR will exhibit a fourfold sym- metry compatible with the C_4v symmetry of the general Bychkov-Rashba SOCF.

The magnetization-direction dependence of the transmis- sion and reflection of the incident spins should be reflected in the local density of states at the interfaces of the barrier.

Within the DDFM the left共F/S兲and right共S/NM兲interfaces are merged into a single plane and one cannot distinguish between them. A more detailed view of the role of the inter- faces requires the use of the SSOM. It turns out共this will be shown later in this section兲 that the F/S interface plays a major role in the TAMR phenomenon while theS/NM inter- face appears irrelevant. This is intuitively expected since the exchange splitting in the ferromagnet is much larger than the Zeeman splitting in the normal metal. Consequently, the spin-valve effect at the F/S interface is much stronger than in theS/NM interface.

The local density of states reflects also the uniaxial aniso- tropy of the TAMR with respect to the magnetization orien- tation in the ferromagnet. In fact, one can introduce the an- isotropic local density of states 共ALDOS兲 through the definition

ALDOS_关110兴共z,␾兲=LDOS共z,0兲− LDOS共z,␾兲 LDOS共z,␾兲 , 共35兲 where

LDOS共z,␾兲=_␴=

兺

↑,↓

冕

共2^dk␲^储兲²兩⌿_␴共z,␾^,k_␴F兲兩² 共36兲 is the total local density of states at positionzand evaluated at the Fermi surface determined by the Fermi wave vectors

k_␴F=

冑

^2m_ប²⁰

冉

^EF+␴⌬2l

冊

^{−k2储. 共37兲}

Since we are interested only in propagating states, we may restrict the possible values ofk储to the interval关0 ,k_max^␴ 兴, with k_max^␴ given by Eq. 共A16兲. Since the spin splitting in the normal-metal region is negligibly small,␬_␴⬇␬_−␴. It follows from Eqs.共13兲and共17兲that

T_␴共E,k储兲⬀兩⌿_␴^共r兲兩², 共38兲 and, therefore, the conductance 关see Eq. 共28兲兴 is related to the LDOS atz=d asG_␴共␾兲⬀LDOS共d,␾兲. One then obtains that

TAMR_关110兴共␾兲 ⬇ALDOS_关110兴共z=d,␾兲. 共39兲

For a numerical illustration we consider an epitaxial Fe/

GaAs/Au heterojunction similar to that used in the experi- mental observations reported in Ref. 13. We use the value m_c= 0.067 m₀ for the electron effective mass in the central

共GaAs兲region. The barrier width and height共measured from the Fermi energy兲 are, respectively, d= 80 Å and Vc

= 0.75 eV, corresponding to the experimental samples in Ref. 13. For the Fe layer a Stoner model with the majority- and minority-spin channels having Fermi momenta kF↑

= 1.05⫻10⁸ cm⁻¹ andkF↓= 0.44⫻10⁸ cm⁻¹,⁴⁶ respectively, is assumed. The Fermi momentum in Au is taken as ␬F

= 1.2⫻10⁸ cm⁻¹.⁴⁷ We consider the case of relatively weak magnetic fields 共specifically, B= 0.5 T兲. At high magnetic fields, say, several Tesla, our model is invalid as it does not include cyclotron effects which become relevant when the cyclotron radius approaches the barrier width.

The Dresselhaus spin-orbit parameter in GaAs is ␥

⬇24 eV ų.^6,38,40On the other hand, the values of the inter- face Bychkov-Rashba parameters␣land␣r关see Eq.共22兲兴are not known for metal-semiconductor interfaces. Due to the complexity of the problem, a theoretical estimation of such parameters requires first-principles calculations including the band-structure details of the involved materials,⁴⁴ which are beyond the scope of the present paper. Here we assume ␣l

and ␣r as phenomenological parameters which must be un- derstood as the values of the interface Bychkov-Rashba pa- rameters at theF/SandS/NM interfaces, respectively, aver- aged over all the relevant bands contributing to the transport across the corresponding interfaces. In order to investigate how does the degree of anisotropy depend on these two pa- rameters we performed calculations of the ratioR_关1^¯_10兴/R_关110兴 共which is a measure of the degree of anisotropy¹³兲as a func- tion of␣land␣rby using the spin-orbit Slonczewski model described in Sec.II B. The results are shown in Fig.5, where one can appreciate that the size of this ratio 共and, conse- quently, of the TAMR兲 is dominated by␣l. This is because the Zeeman splitting in Au is very small compared to the exchange splitting in Fe and, consequently, the spin flips mainly when crossing theF/S interface. Then, since the val- ues of the TAMR are not very sensitive to the changes in␣r

we can set this parameter, without loss of generality, to zero.

This leaves␣las a single fitting parameter when comparing to experiment. The values of the phenomenological param- eter ␣l were determined in Refs. 6 and 13 by fitting the theory to the experimental value of the ratio R_关−110兴/R_关110兴,

-40 -20 0 20 40

-40 -20 0 20 40

α αα

α_l(eV Ų) αααα r(eVÅ2 )

0.9950 0.9970 0.9990 1.001 1.003 1.005

R_[-110]/ R_[110]

FIG. 5. 共Color兲Values of the ratioR_关1¯10兴/R_关110兴as a function of the interface Bychkov-Rashba parameters␣land␣r.