Spin-transfer torque in magnetic tunnel junctions: Scattering theory

Spin-transfer torque in magnetic tunnel junctions: Scattering theory

Jiang Xiao and Gerrit E. W. Bauer

Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands

Arne Brataas

Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 共Received 29 March 2008; revised manuscript received 15 May 2008; published 12 June 2008兲

We study the bias-dependent spin-transfer torque in magnetic tunnel junctions in the Stoner model by scattering theory. We show that the in-plane共Slonczewski type兲 torque vanishes and subsequently reverses its direction when the bias voltage becomes larger or the barrier wider than material and device-dependent critical values. We are able to reproduce the magnitude and the bias dependence of measured in-plane and out-of-plane torques using realistic parameters. The condition for the vanishing torque is summarized by a phase diagram depending on the applied bias and barrier width, which is explained in terms of an interface spin polarization and the electron focusing by the barrier. Quantum size effects in the spin-transfer torque are predicted as a function of the thickness of a normal-metal layer inserted between the ferromagnet and tunnel barrier. DOI:10.1103/PhysRevB.77.224419 PACS number共s兲: 73.63.⫺b, 85.75.⫺d, 72.25.⫺b, 73.40.Gk

I. INTRODUCTION

Magnetic tunnel junctions共MTJ兲 are layered structures in which an insulating tunnel barrier 共I兲 separates two ferro-magnetic layers共F兲.1,2_{The interplay between electronic} cur-rents and an order-parameter difference, i.e., magnetizations rotated away from the equilibrium configurations, is the magnetic equivalent to the Josephson effect in superconduc-tivity. The MTJ with a thin normal-metal insertion layer is the only magnetoelectronic structure that shows quantum size effects on electron transport.3 _{MTJs based on epitaxial} MgO barriers4,5 _{are used in the magnetic random-access} memory 共MRAM兲 devices that are operated by the spin-transfer torque.6,7 _{MTJs have been studied vigorously,} ini-tially focusing on the tunnel magnetoresistance共TMR兲.3,8–10 More recently, the focus shifted to spin-transfer torque and current-induced magnetization switching.11–24 _{On the} theo-retical front, spin-transfer effects have been studied exten-sively in metallic spin valve structures based on various models.25–30 _{For tunneling structures, such studies are still} relatively scarce.15,17–20

Here we report a model study of the spin-transfer torque in magnetic tunnel junctions. Since the ferromagnets are separated by tunnel barriers, we cannot use theories existing for metallic structures that are mostly based on semiclassical methods.25,31_{Instead, we chose a fully quantum-mechanical} treatment of transport through the tunnel barrier by scattering theory. The high quality of MgO tunnel junctions and the prominence of quantum oscillations observed in ferromagnetic-normal metal-insulator-ferromagnetic 共FNIF兲 structures共even for alumina barriers兲 provide the motivation to concentrate on ballistic structures in which the transverse Bloch vector is conserved during transport. We qualitatively 共and even quantitatively兲 confirm the results in Refs.15,19, and20. However, our model is simpler and physically more transparent than the tight-binding method used in Ref. 15

and the numerical studies of Refs.19and20. We are able to reproduce simultaneously both the in-plane and out-of-plane torque experimental data using realistic material and device parameters, in contrast to a fit based on the tight-binding

model.22 _{We also show finite zero-bias out-of-plane torque} for asymmetric structures. Scattering theory enables us to distill a clear physical picture of the peculiarities of the spin-transfer torque in MTJs, which allows us to understand why and when the torque goes to zero. The torque zero-crossing condition can be summarized by a phase diagram spanned by the applied bias and barrier width parameters. With our ap-proach we can go beyond the ferromagnetic-insulator-ferromagnetic 共FIF兲 MTJ and study the effect of a normal-metal insertion共FINF structures兲. Quantum size oscillations in the torque are predicted as a function of the thickness of the N insertion layer.

This paper is organized as follows: Section II introduces the FIF and FNIF structures and the scattering theory. In Sec. III, approximations are introduced in order to derive analytic expressions. Section IV presents our main results. Section V compares our model with experimental results. Sections VI and VII contain a brief discussion and summary, respec-tively. Two appendices are attached at the end.

II. STRUCTURE AND METHOD

We consider multilayers as shown in Fig. 1共a兲in which two semi-infinite F leads关F共L兲 and F共R兲兴 are connected by an insulating layer 共I兲 of width d and a nonmagnetic metal layer 共N兲 of width a. The magnetization direction of

F共L兲/F共R兲, m1/m2共兩m1兩=兩m2兩=1兲, is treated as fixed/free. This structure reduces to a conventional FIF MTJ when a = 0.

Let A, B, C, D, C

⬘

, D

⬘

, E, and F be the spin-dependent

amplitudes A†_=共A

↑

†_{, A}

↓

†_{兲 of flux-normalized spinor wave} functions at specific points. The scattering states can be ex-pressed in terms of two incoming waves A and F, such as:

C

⬘

= sˆC⬘AA + sˆC⬘FF, 共1兲

where sˆC⬘Aand sˆC⬘Fare 2⫻2 matrices in spin space that can

be constructed by concatenating the scattering matrices of region S1,2and of the insulating layer bulk 共see Fig.1兲. To

leading order of the transmission 共tb兲 through the bulk I

共similar expansions hold for sˆD⬘A and sˆD⬘Fas well兲:

sˆC_⬘A=关共1 − rb

⬘

rˆ2兲−1tb共1 − rˆ1

⬘

rb兲−1兴tˆ1, 共2a兲

sˆC_⬘F=关共1 − rb

⬘

rˆ2兲−1rb

⬘

+ tb共1 − rˆ1

⬘

rb兲−1rˆ1

⬘

tb

⬘

共1 − rˆ2rb

⬘

兲−1兴tˆ2

⬘

, 共2b兲 where tˆ1,2/rˆ1,2are the 2⫻2 transmission/reflection matrices for S1,2 共see Fig. 1兲 and tb/rb are the spin-independent

transmission/reflection coefficient for the insulating bulk ma-terial that are proportional to the unit matrix in spin space and therefore without hat. The primed and unprimed versions indicate scattering of electrons impinging from the left and right, respectively. The reflection coefficient rb is due to the

impurity scattering inside the bulk insulator, which, as tb,

contains an exponential decay factor representing evanescent states in I. For this reason the magnitude of rbis comparable

to or much smaller than that of tbdepending on the density of

the impurities. All scattering matrices are matrices in k space defined by the propagating states of left and right leads in the energy window available for transport, labeled by their trans-verse wave vectors in the leads: q , q

⬘

共the band index is suppressed兲 at a given energy.

An applied bias voltage V drives a 共conserved兲 charge current Jcand 共spatially dependent兲 spin current Js through

the device. At zero temperature, the charge current reads,

Jc= 1 共2␲兲3

冕

dE

兺

q,q⬘ jc共q,q

⬘

兲, jc=4e បTr␴关Im共sˆEAsˆEA † fL− sˆEFsˆEF † fR兲兴. 共3兲

where Tr_␴关¯兴 denotes the spin trace and the summation is over all the transverse modes at energy E. fL= fL共E兲 and fR

= fR共E+eV兲 are 共zero temperature兲 Fermi–Dirac electron

dis-tribution functions in the left and right reservoirs. We are therefore disregarding any spin accumulation in the

ferro-magnet, which is valid for tunnel junctions of current interest in which the spin-flip rate in the ferromagnet is larger than the tunnel rate. The scattering matrices depend on V by the bias-induced potential profile. The spin current, or the angular-momentum current, at the left side of the I/F共R兲 interface 共within I兲 reads

Js= 1 共2␲兲3

冕

dE

兺

q,q⬘ js共E,q,q

⬘

兲, js= 2 Tr␴关␴ˆ Im共sˆC⬘AsˆD⬘A † fL− sˆC⬘FsˆD⬘F † fR兲兴. 共4兲

Since the spin current deep in the ferromagnetic 共FM兲 lead is longitudinal to the magnetization, the torque N acting on F共R兲 is equal to the transverse component of the incom-ing spin current that is absorbed at the interface:6,25,26,28

N = Js−共Js· m2兲m2= N储+ N⬜, 共5兲 with the in-plane共Slonczewski兲 torque N储⬀m2⫻共m1⫻m2兲 and out-of-plane 共fieldlike兲 torque N_⬜⬀m1⫻m2. Similarly n = js−共js· m2兲m2= n储+ n⬜.

At low bias, the nonequilibrium part of the spin current is proportional to the bias voltage Js− Js

0

= GsV, where Js

0 is the equilibrium spin current that is related to interlayer exchange coupling at equilibrium and Gsis the spin conductance:

Gs= 1 共2␲兲2

兺

q,q⬘ E=EF gs共k,k

⬘

兲, gs= e ␲Tr␴关␴ˆ Im共sˆC⬘AsˆD⬘A † + sˆC⬘FsˆD⬘F † _{兲兴, 共6兲}

where the scattering matrices are evaluated at zero bias V = 0 and the summation is over transport channels at the Fermi energy. We define the linear-response torkance30 _T = N/V, and T=Gs−共Gs· m2兲m2= T储+ T⬜ and ␶= gs

−共gs· m2兲m2=␶储+␶_⬜.

III. APPROXIMATIONS

We assume in the following that the spin is conserved during the scattering. Then tˆifor Si共i=1,2, similar for rˆi兲 is

diagonal when choosing mi as spin-quantization axis:

Ex-panded in Pauli matrices␴ˆ =共␴ˆx,␴ˆy,␴ˆz兲, tˆi= ti

+ + ti

−_{␴ˆ · m}

i, with ti⫾=共ti↑⫾ti↓兲/2. ti␴共␴=↑ ,↓兲 is the transmission amplitude for

spin ␴ for spin-quantization axes mi through the scattering

region Si. In the absence of impurities共rb= 0兲 and to leading

order of tb: sˆC_⬘A= tb共t1 + + t₁−␴ˆ · m₁兲, 共7a兲 sˆD⬘A=共r2+tbt+1+ r2−tbt1−m1· m2兲 +␴ˆ ·共r2−tbt1+m1+ r2+tbt1−m2 − ir₂−tbt₁−m1⫻ m2兲, 共7b兲 Similar expansions hold for sˆC⬘F and sˆD⬘F.

Next, we adopt the free-electron approximation tailored for transition-metal ferromagnets.8 _{We assume spherical} Fermi surfaces for spin-up and spin-down electrons关in both

A B E F C C' D D' EF Ub _eV
0 -eV
x (a) (b) F(L) N I F(R) Ub EF S1 θ S2 tb -a 0 d m₁ m2

FIG. 1. 共Color online兲 共a兲 FNIF heterostructure, in which S1,2

indicate two different interface scattering regions;共b兲 The potential profiles共at positive bias兲 for majority and minority electron spins in F are shown by solid and dashed lines, respectively. The exchange splitting is⌬ and the tunnel barrier has height U_b relative to the Fermi energy EF. The applied bias V pictured in共b兲 corresponds to a net electron共particle兲 flow from right to left.

F共L兲 and F共R兲兴 with Fermi wave vectors kF↑=

冑

2mEF/ប2and k_F↓=

冑

2m共EF−⌬兲/ប2, with an effective electron mass m in F.

Electrons in N are assumed to be ideally matched with the majority electrons in F共kF= kF↑兲. The effective electron mass

in the tunnel barrier is assumed to be mb=␤m. Figure 1共b兲

shows the adopted potential profile with barrier height Ub.

We assume an applied potential bias that is smaller than the barrier height 兩eV兩⬍Ubthat fully drops over the tunnel

bar-rier. Positive bias corresponds to charge current flow from left to right and electron particle flow from right to left. As in Refs. 15,18, and19, we assume that energy and transverse wave vector q are conserved, thus disregard any impurity/ interface roughness scattering. All scattering matrices then become diagonal in k space.

In the free-electron model, the flux-normalized wave functions in N and F are:

␺_⫾N =e ⫾ikxx

冑k

x in N and ␺_⫾␴ =e ⫾ikx␴x

冑

kx␴ in F, 共8兲 Using the WKB approximation,18 _{the wave function in the} tunneling barrier is ␺⫾b = e⫾兰0 x_␬共w兲dw

冑

⫾i␬共x兲 in I, 共9兲 with ␬共x兲 =

冑

2mb ប2

冉

Ub+ EF− E − eV x d

冊

+ q 2_{, 共10兲} where E is the energy of the electron. The WKB approxima-tion is valid when the potential profile varies slowly in space within the tunneling barrier, i.e.,␬

⬘

共x兲Ⰶ␬2共x兲. The transmis-sion coefficient through I reads tb= exp关−兰0

d_{␬共w兲dw兴.}

For finite bias, from Eqs.共4兲, 共7a兲, and 共7b兲,

n储= tb 2 T₁−T₂+共fL− fR兲m2⫻ 共m1⫻ m2兲, 共11a兲 n_⬜= 2tb 2_Re共T 1 − r₂−fL+ T₂−r

⬘

₁−fR兲m1⫻ m2, 共11b兲 n_⬜0 = 2tb2Re共T1−r2−+ T2−r1

⬘

−兲f0m1⫻ m2, 共11c兲 where Ti +

=兩ti↑兩2+兩ti↓兩2is the average transmission probability

for scattering region Si, Ti−= piTi+=兩ti↑兩2−兩ti↓兩2 with

polariza-tion pi= Ti

−_/T

i

+

, and f0is the equilibrium distribution function at zero bias. n_⬜in Eq. 共11b兲 includes both the equilibrium

and nonequilibrium contribution to the out-of-plane torque. The former 关n_⬜0 in Eq. 共11c兲兴 is related with the nonlocal

interlayer exchange coupling.32 The nonequilibrium contri-bution is therefore n_⬜

⬘

= n_⬜− n_⬜0. The optical theorem 2 Im共r_1,2⫾兲=T_1,2⫾ 共see Appendix A兲 is used in the derivation of Eq.共11a兲 to get rid of all internal reflection in I. In Eq. 共11a兲,

we observe that the in-plane torque is caused by the polar-ization of the current at the left interface that is expressed by

T₁−. The subsequent absorption of the spin current by the second magnet is governed by the geometrical projection expressed by the vector product and the total transparency of the second interface T₂+. It follows from Eq.共11b兲 that the out

out-of-plane torque has a very different origin. It does not depend directly on the difference of the electron distributions

on both sides of the junctions, but consists of two indepen-dent contributions from both reservoirs. Each contribution consists of the spin polarization of the first interface, but is sensitive to the phase of the reflection coefficient of the sec-ond interface. The out-of-plane torque can be interpreted as the net spin created at one interface that while reflected at the second interface briefly precesses in the exchange field of the second ferromagnet.

With vanishing bias, tb= exp共−␬d兲 with ␬ =

冑

2mbUb/ប2+ q2. By Eqs. 共6兲, 共7a兲, and 共7b兲,

␶=␶储= e 2␲e −2␬d_T 1 − T₂+m2⫻ 共m1⫻ m2兲, 共12兲 and ␶_⬜= 0. For reference, the conductance within the same theoretical framework is given by:33

gc= e2 2he −2␬d_共T 1 + T₂++ T₁−T₂−m1· m2兲. 共13兲 The vector product 兩m2⫻共m1⫻m2兲兩=sin␪ in Eq. 共12兲 and m1· m2= cos␪ in Eq. 共13兲, leading to the well-known geometrical dependence of the angular transport properties of tunnel junctions.6_{The vanishing of the out-of-plane torque}

␶⬜= 0, is a rather general result that holds for symmetric tunneling junctions and spin valves in the linear-response regime.15 _{We consider a symmetric system with an applied} voltage −V/2 to the left and a voltage V/2 to the right res-ervoir. To the second order in the bias voltage, the spin cur-rent in the spacer between the ferromagnets can be expanded as

Is=关A1m1+ B1m2+ C1m1⫻ m2兴V

+关A2m1+ B2m2+ C2m1⫻ m2兴V2. 共14兲 When applying the mirror operation left↔right 共1↔2, −V/2↔V/2,Is↔−Is兲 symmetry requires that

− Is=关A1m2+ B1m1+ C1m2⫻ m1兴共− V兲

+关A2m2+ B2m1+ C2m2⫻ m1兴共− V兲2, 共15兲 which should be identical to Eq. 共14兲. Therefore A1 = B1, C1= 0 , A2= −B2, whereas C2 is not restricted. Then the torque on m2is

N = m₂⫻ 共Is⫻ m2兲 = 关A1V − A2V2兴m2⫻ 共m1⫻ m2兲

+ C2V2m1⫻ m2. 共16兲

This proves that, for symmetric systems, the out-of-plane torkance 共⬀2C2V兲 vanishes at V=0. It also shows that be-yond linear response, there are quadratic 共in bias兲 contribu-tions to both the in-plane and out-of-plane torques. The ar-gument does not hold for asymmetric tunneling systems. An experimental zero-bias out-of-plane torkance should there-fore provide interesting information on MTJ asymmetries.

IV. RESULTS

In this section, we discuss three different structures:共a兲 A symmetric FIF magnetic tunneling junction,共b兲 an asymmet-ric FIF structure in which the left and right FM layers have

different exchange splitting, and 共c兲 an FNIF structure in which a nonmagnetic layer is inserted between the insulator barrier layer and one of the ferromagnetic layers.

A. Symmetric ferromagnetic-insulator-ferromagnetic For a symmetric Fe/MgO/Fe MTJ: k_F↑= 1.09 Å−1 _{and k}

F ↓

= 0.42 Å−1 for Fe,8 and Ub⯝1.0–1.2 eV and ␤= mb/m

= 0.4m for MgO.4,34–36 _{This implies E}

F⯝4.5 eV, ⌬

⯝3.8 eV⬇0.85EF, and Ub⬇0.25EF. For an FIF structure

共a=0兲, both S1 and S2contain only a single interface. Using the potential profile in Fig.1共b兲, we have

t₁␴=2

冑

ik1 ␴_{␬共0兲/␤} k₁␴+ i␬共0兲/␤, t2 ␴₌2

冑

ik2␴␬共d兲/␤ k₂␴+ i␬共d兲/␤, 共17a兲 r₁

⬘

␴=− k1 ␴_{+ i␬共0兲/␤} k₁␴+ i␬共0兲/␤ , r2 ␴₌− k2␴+ i␬共d兲/␤ k₂␴+ i␬共d兲/␤ , 共17b兲 where␴=↑ ,↓ and k₁↑2+ q2_{= 2mE/ប}2_{, k}

1

↓2_{+ q}2_{= 2m共E−⌬兲/ប}2_.

k₂␴ are defined similarly with E replaced by E + eV. We set

t_1,2␴ = 0 when Im共k_1,2␴ 兲⫽0.

Figure2 shows the computed bias dependence of the in-plane torque N储 共left兲 and the nonequilibrium part 共i.e., not

containing the equilibrium interlayer exchange coupling兲 of the out-of-plane fieldlike torque N_⬜

⬘

共V兲=N_⬜共V兲−N_⬜共0兲 共right兲 at various exchange splittings for mb= 0.4m at d = 1.0 nm. The equilibrium exchange coupling gives rise to an effective magnetic field in the LLG equations that we do not explicitly discuss. The main features of these curves are:

共1兲 the in-plane torque has both linear and parabolic contri-butions, and 共2兲 the fieldlike torque is paraboliclike. These plots are very similar to the corresponding plots by Theodo-nis et al.,15_{meaning that the band-structure effects caused by} the tight-binding approximation are not important. The in-plane torque at negative bias and small positive bias is “nor-mal,” but changes sign at higher positive bias, where normal means that the direction of the torque in FIF is the same as the torque in metallic spin valves predicted,6_{i.e., the torque} curve appears in the second and fourth quadrants in the left panel of Fig.2. When the torque curve is found in the first or third quadrant, we say that the torque is reversed. In the normal region, the positive bias 共electron flow from right to left兲 favors the antiparallel configuration and a negative one the parallel configuration. In the reversed region, on the other hand, a current polarity that stabilizes the parallel configura-tion in the normal region has the opposite effect.

The zero crossing of the in-plane torque in Fig.2 can be traced to the sign change of T₁− in Eq. 共11a兲, i.e., the sign

change of the polarization of S₁ 关the F共L兲/I interface兴p₁ = T₁−/T₁+⬍0.37 _{The polarization p}

1⬀␬2共0兲−k↑k↓, which can take any sign depending on parameters chosen12_共see Appen-dix B for a more detailed discussion of this point兲. The van-ishing torque phenomenon becomes more transparent with-out an effective-mass mismatch, i.e., for ␤= 1 instead of ␤ = 0.4 used in Fig.2. The polarization vanishes when

␬2_{共0兲 − k}

↑k↓= 0. 共18兲

Since␬共x=0兲 共near left interface兲 increases and k_␴decreases with q, Eq.共18兲 is fulfilled at a certain critical value qc. The

−0.2 −0.1

0

0.1

0.2

−2

0

2

4

6x 10

−5

in−plane

bias voltage: eV/E

_F

N

||

(E

F

k

F 2

/8

π

3

)

−0.2 −0.1

0

0.1

0.2

−2

−1

0

x 10

−5

out−of−plane

bias voltage: eV/E

_F

N’

⊥

(E

F

k

F 2

/8

π

3

)

−0.2 0 0.2 −6 −4 −2 0x 10 −5 bias voltage N⊥

∆ = 0.85 E

_F

∆ = 0.75 E

_F

∆ = 0.65 E

_F

FIG. 2. 共Color online兲 The magnitude of N储共left兲 and N_⬜⬘ 共right兲 acting on the right magnetization in Fig.1vs applied bias eV for␤ = 0.4 and d = 1.0 nm. Inset figure shows the out-of-plane torque including the equilibrium contribution at zero bias.

latter increases with the electron energy E, because␬共0兲 de-creases and k_␴ increases with E. This can be seen clearly from the following equation:

qc2= 2m ប2

冋

E − 共EF+ Ub兲2 2共EF+ Ub兲 − ⌬

册

. 共19兲

which implies that at the Fermi energy qc is well defined

共qc2⬎0兲 only when Ub2⬍EF共EF−⌬兲.

When q_c2⬍0 at low bias, there is no polarization sign change for any q, and the 共in-plane兲 torque behaves nor-mally. When the potential profile becomes distorted by an applied bias as in Fig.3共b兲, the electrons injected from the right lead have a maximum energy E = EF+ eV. When the

applied bias V is large enough, we reach the regime qc

2_⬎0, and a polarization sign change of the left interface comes into play. As qc increases further, more and more electron

contribute to the opposite torque. When V is large enough, the total torque changes sign as seen in Fig.2. On the other hand, when the applied bias is negative关see Fig. 3共a兲兴, the transport is dominated by the electron injected from the left lead. The electron energy and effective barrier height at the left interface do not change with applied bias, which means

T₁− 共and so the polarization兲 does not change either.

There-fore, we do not see a zero crossing 共on the right magnetiza-tion兲 at negative bias in Fig.2.

We can analogously understand the dependence of the in-plane torque on the barrier width d. We know that there is a polarization sign change for q⬍qc if Ub is not too high

共such that qc

2_{⬎0兲. In tunneling junction, the transport is} dominated by electrons with small q because of the focusing effect due to the exponential extinction factor in tb. When the

electron with q values smaller than qcdominates, the torque

reverses sign. At a critical barrier width dc, the contributions

from q⬍qccancel those from q⬎qc, and the torque or

tor-kance vanishes: N储共dc兲=T储共dc兲=0, whereas it has opposite

direction for d⬎dc. The left panel of Fig.4 shows dcvs Ub

in linear response for various values of the barrier effective mass. dc increases with Ub simply because the polarization

sign change behavior is less prominent at higher barrier heights共qcdecreases with Ub兲.

The right panel of Fig.4shows the bias dependence of dc

at barrier height Ub= 0.25EF for various barrier effective

masses. Because both the positive and negative biases reduce the average barrier height, tbincreases with兩V兩. However, the

applied bias is very ineffective in changing the focusing be-havior, i.e., the transmission is hardly less focused by the reduced average barrier height. This is very unlike the geo-metrical barrier width d, to which, focusing very sensitively, larger共smaller兲 d means more 共less兲 focusing. Rather, a

posi-tive bias enhances the polarization sign change behavior, which leads to a smaller critical barrier width dc. Since there

is no polarization sign change 共qc2⬍0兲 at zero or negative

bias for ␤= 0.7 and 0.4, a torque zero crossing is not ob-served. For ␤= 1.0, polarization could change sign at zero bias, hence the torque zero crossing is also observed at nega-tive bias. As mentioned before, neganega-tive bias does not change T₁− or polarization of the left interface, but it does change T₂+. At negative bias, the barrier height at the right interface is reduced by 兩eV兩, which leads to the decrease of

T₂+ at small q, thus the polarization sign change behavior is weakened because of the smaller product T₁−T₂+ at small q where the product is negative. Weaker polarization sign changes, then requires larger dc for torque zero crossing at

negative bias and we see dcincreasing with negative bias for ␤= 1.0. For comparison, a critical barrier width for the sign change of TMR is also calculated for␤= 0.4, and is shown as the solid black curve with “+” symbol in the right panel of Fig. 4. Since the TMR is symmetric in the applied bias for the symmetric structures, the curve is also symmetric.

B. Asymmetric ferromagnetic-insulator-ferromagnetic As discussed at the end of Sec. III, the zero-bias 共linear-response兲 out-of-plane torkance does not vanish for asym-metric structures. Figure5 shows both the in-plane and out-of-plane torkances for the right FM layer at zero bias 共V = 0兲 for an asymmetric FIF structure, in which the left and the right FM layers have different exchange splitting: ⌬1 = 0.85EF for the left and ⌬2 for the right, where the latter varies from 0 to EF. From Fig.5, we can see that the

out-of-plane torkance is generally nonzero for asymmetric struc-tures when ⌬₂⫽⌬₁, and it vanishes when the right layer becomes nonmagnetic 共⌬2= 0兲 or when the structure be-comes symmetric 共⌬2=⌬1兲. The in-plane torkance in Fig.5 decreases with ⌬2 simply because the average transmission probability through the right interface T₂+decreases.

C. Ferromagnetic-normal metal-insulator-ferromagnetic An FNIF structure, with a nonmagnetic layer of width a between one of the F layers and the insulating I layer, has

e e (b) (a) Ub Ub EF EF EF + |eV | EF + |eV | |eV | |eV | T− 1 T2+ T1− T2+

negative bias: V < 0 positive bias: V > 0

FIG. 3.共Color online兲 Positive and negative bias situations.

0 0.2 0.4 0 0.5 1 1.5 2 2.5 barrier height: U b/EF critical barrier width: dc (nm) −0.2 −0.1 0 0.1 0.20 0.5 1 1.5 2 2.5

bias voltage: eV/E

F critical barrier width: dc (nm) β = 1.0 β = 0.7 β = 0.4

FIG. 4. 共Color online兲 Critical barrier width dcvs barrier height

U_bat linear response共left兲 and applied bias V 共right兲. ⌬=0.85E_F. Critical width for TMR at ␤=0.4 共thin solid black line with “+” symbols兲 is also shown in the right panel.

never been studied in the regime of spin-transfer torque. Such an asymmetric FNIF device can be operated in two nonequivalent modes 共as in any nonsymmetric MTJ兲: The left F is static and the right F共F˜兲 is free 共mode 1: FNIF˜兲 and vice versa 共mode 2: F˜NIF兲. Equations 共11a兲–共11c兲 and 共12兲

apply to mode 1, and apply to mode 2 with subscripts 1 and 2 exchanged. The a dependence of the in-plane torkance共in linear response兲 is shown in Fig.6. The sign of the in-plane torkance can be controlled by a in mode 1, but not in mode 2. This sign is determined by the sign of T₁−. In mode 1, T₁−共a兲 covers region F共L兲-N-I and its sign can be modulated by the

N insertion layer width a. However, in mode 2, T₁− covers

I-F共R兲, which is independent of a, therefore the sign is

un-changed. The a dependence of the in-plane torkance in mode 2 comes from T₂+共a兲, which is always positive. Due to the aliasing effect caused by discrete thickness of the N layer,38 the period of the quantum oscillation in Fig. 6 should be about ␲/兩kF−␲/␭兩 instead of ␲/kF⬇3 Å shown in the

fig-ure, where ␭ is the monolayer thickness for N layer. The asymmetry in FNIF structure shall also give rise to a finite linear-response out-of-plane torkance, which shows similar oscillations as the in-plane torkance in Fig. 6. The magnitude of the zero-bias out-of-plane torkance could be comparable to the in-plane counterpart.

V. COMPARISON WITH EXPERIMENTS

Using realistic material parameters and the geometry pa-rameters provided by Ref.22, we are able to reproduce, even the absolute scale, the experimental data from Ref. 22 as shown in Fig.7, which includes the bias dependence of the in-plane torque N储共V兲 and the nonequilibrium part of the

out-of-plane torque N_⬜

⬘

共V兲=N_⬜共V兲−N_⬜共0兲 and the correspond-ing torkance. The experimental data for the torkance in the bottom panels of Fig.7are adapted from Fig. S3共d兲 共␤_ST,FT

⬘

兲 and Fig. 2 共I-V data兲 of Ref. 22 by 共dN储,⬜/dV兲/sin␪ =共dI/dV兲共dN储,⬜/dI兲/sin␪=共dI/dV兲␤ST,FT

⬘

. Our model ap-pears to have a problem with the upturn of the torque at higher positive bias. The fit in Ref. 22 based on the tight-binding model of Ref.15is slightly better in this respect for a rather large exchange splitting. However, the out-plane torque is poorly reproduced for the same parameter set. In contrast, we succeed with a single set of 共realistic兲 param-eters to reproduce both in-plane and out-of-plane torques 共torkance兲. The resistance for this particular device in our model is R共␪= 137°兲⬇150 ⍀, which is consistent with the experimental values共⬃200 ⍀兲. However, the TMR value in our calculation共about 15%兲 is considerably smaller than that in the experiment. We believe that the spin-transfer torque is better represented by the free-electron model than the TMR because TMR⬀p1p2, whereas torque on m2⬀p1. If the po-larization p1, p2 at the interfaces are underestimated by a factor of ␩, the TMR value is too small by a factor of ␩2_, whereas the spin-transfer torque is affected only by a factor of ␩ 共see also Sec. VI兲. In addition to this, the TMR value depends sensitively on the exchange splitting ⌬. For in-stance, the TMR value increases from 15% to 30% when⌬ increases from 0.87EF to 0.9EF,

Another set of experimental data is shown in Fig. 8

adopted from Fig. 3共a兲 of Ref.21. The experimental data are now the in-plane 共red squares兲 and out-of-plane 共blue

dia-0 0.2 0.4 0.6 0.8 1 −1 1 3 5 7x 10 −4 ∆₂/E F linear−response torkance (ek F 2/8 π 3) out−of−plane: T_⊥ in−plane: T_|| ∆₁= 0.85 E_F ∆₂=∆₁

FIG. 5. Linear-response torkance vs exchange splitting⌬₂in the right FM共⌬1= 0.85EF, Ub= 0.25EF, d = 1.0 nm兲.

0 0.5 1.0 1.5 2.0 −3 −1.5 0 1.5x 10 −3

N insertion layer width: a (nm)

torkance: T|| (e kF 2/8 π 3) mode 1 mode 2

FIG. 6. 共Color online兲 Linear-response in-plane torkance vs nonmagnetic insertion layer width a 共⌬=0.85EF, Ub= 0.25EF, d = 1.0 nm兲. −1 0 1 2 3 AN || (10 −19 J) in−plane −0.2 −0.1 0 AN ⊥ (10 −19 )J ) out−of−plane −0.4 −0.2 0 0.2 0.4−3 0 3

bias voltage: V (Volts)

A (dN ⊥ /dV)/sin( θ) (10 −19 C ) −0.4 −0.2 0 0.2 0.4 −20 −10 0

bias voltage: V (Volts)

A (dN || /dV) /sin( θ) (10 −19 C)

FIG. 7. 共Color online兲 Fitting 共solid curve兲 of the experimental data 共dots兲 from Ref.22. Top: in-plane torque AN储共left兲 and out-of-plane torque AN_⬜ 共right兲; Bottom: in-plane torkance A共dN储/dV兲/sin␪ 共left兲 and out-of-plane torkance A共dN_⬜⬘/dV兲/sin␪ 共right兲. The following parameters are used in all fittings: EF = 4.5 eV, ⌬=0.87E_F⬇3.9 eV, U_b= 0.23E_F⬇1.0 eV, ␤=0.36; d = 1 nm, cross-section area A = 70 nm⫻250 nm and␪=137° from Ref.22.

monds兲 torkance. We fit the out-of-plane and the linear part of the in-plane torkance again on absolute scale. Theory con-tains a quadratic component of the in-plane torque, which does not show up in this experiment. The resistance for this particular device in our model is R共␪= 71°兲⬇4 k⍀, consis-tent with the experimental value 共⬃3.5 k⍀兲, whereas our TMR, 5%, again is too small. Note that the torkances are much smaller than in Ref.22because of the thicker barrier and the smaller cross-section area, as reflected by the higher resistance.

Deac et al.23_{also measured the in-plane and out-of-plane} torque in a MgO based tunneling junction. The out-of-plane torque in this experiment agrees well with other experiments and theory. However, the in-plane torque depends paraboli-cally on the applied bias共AV2for positive bias and −BV2for negative bias, where A and B are positive constants兲, which is quite different from both Refs.21and22. A voltage noise measurements done by Petit et al.16 _{also suggest a linear} out-of-plane torkance 共or parabolic out-of-plane torque兲, which is about 20% of the in-plane counterpart. Hence all experiments and theories appear to agree on the out-of-plane torque, whereas consensus about the in-plane torque has not been reached yet.

VI. DISCUSSION

Because of the high quality of epitaxial MgO tunnel layers,4,5 _{we ignored interface roughness and barrier} disor-der. The main effect of the geometric interface roughness is to reduce the nominal thickness of the barrier.39 _Impurity states in the barrier generally increase tunneling because of the opening of additional tunneling channels with lower bar-rier height Ub

⬘

⬍Ub. Impurities states also weaken the

spdependent effects when spin flip is involved. In general, in-terface roughness and disorder can be important quantitatively, but have been shown not to qualitatively change the features predicted by a ballistic model.40–43

The free-electron Stoner model is only a poor representa-tion of the real electronic structure of transirepresenta-tion metals for

the tunneling problem: It fails to properly reproduce the nearly half-metallic nature of transition-metal ferromagnets based MgO tunnel junctions, that is caused by the symmetry of the bands at the Fermi energy,44 _{leading to the} underesti-mated TMR ratios by our model noted above. On the other hand, the band-structure calculations by Heiliger and Stiles17 show that the free-electron model can perform quite well as far as the torque is concerned. We explained this in Sec. IV by its dependence on only one interface polarization leading to a better performance of a model that is not accurate in this respect. The band-structure calculations in Ref. 17 indicate that the torque is strongly localized to a few monolayers which is in support of our simple model.

The issue of the wave-function symmetry should also be considered when a normal metal is inserted. When the elec-trons with wave vector normal to the interface dominate, the normal metal 共Cr兲 is actually a potential barrier for the Fe majority spins,45 _{rather than a potential well as assumed} here.

In contrast to metallic spin valve structures, in which the out-of-plane torque is generally less than 10% of the in-plane counterpart, the out-of-plane contribution has been found quite large in tunneling junctions共a 30% contribution at high bias is measured in Ref.21兲. Close to the zero crossing of the

in-plane torque at positive bias the out-of-pane torque should become dominant.

An experimental “phase diagram” that can be compared with Fig. 4 would constitute a stringent test of our predic-tions. Since the barrier height and the effective electron mass in the barrier cannot be controlled, we suggest measuring the torques systematically for several MTJ structures with differ-ent barrier width共otherwise identical兲 to test the red curve in the right panel of Fig. 4. The zero crossing of the in-plane torque is predicted to occur at voltages that are too high for the experiments in Refs.21–23. For wider tunneling barriers it should happen at smaller voltages.

VII. SUMMARY

To summarize, scattering theory of transport is used to calculate the spin-transfer torque for a magnetic multilayer structure at finite bias. The experimental spin-torque data 共both in-plane and out-of-plane兲 can be reproduced using re-alistic parameters in our model. The spin-transfer torque on a given layer may change sign for only one bias polarity. The torque zero crossing is caused by the combined effects of the polarization sign change at the FI interface and the focusing effect of the barrier. The bias voltage required for torque zero-crossing decreases as a function of the barrier width. The out-of-plane torkance at zero bias vanishes for symmet-ric FIF structures, but remains finite for asymmetsymmet-ric struc-tures. In FNIF structure we find on top of the previously reported oscillating TMR 共Ref. 3兲 and charge pumping

voltage33_{that the spin-transfer torque also oscillates and may} change sign with the N layer thickness.

ACKNOWLEDGMENTS

We acknowledge M. D. Stiles, C. Heiliger, and A. Deac

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.3 0 0.3 0.6 0.9

bias voltage: V (Volts)

A (dN/dV)/sin( θ) (10 − 19 C) in−plane (exp) out−of−plane (exp) in−plane (theory) out−of−plane (theory)

FIG. 8. 共Color online兲 Fitting 共curves兲 of the experimental data 共dots兲 from Ref. 21. The following parameters are used in both fittings: E_F= 4.5 eV, ⌬=0.85E_F⬇3.8 eV, U_b= 0.25E_F⬇1.1 eV, ␤=0.43; d=1.25 nm, cross-section area A=50 nm⫻100 nm, and ␪=71° from Ref. 21共notice the sign convention in Ref.21 is op-posite to that in Ref.22兲.

for helpful discussions. This work is supported by EC Con-tract No. IST-033749 “DynaMax.”

APPENDIX A: OPTICAL THEOREM IN TUNNEL JUNCTION

The scattering matrices in Eq. 共2a兲 and 共2b兲 include the

reflection amplitude inside the insulator共rˆ₁

⬘

and rˆ2兲. Expres-sions become more transparent when the reflection ampli-tudes are replaced by the transmission probabilities, how-ever. This can be achieved by the optical theorem tailored for tunnel junction that reflects current conservation. Note that the equivalent statement in metallic systems is the well-known relation 兩r兩2_+兩t兩2_{= 1 that follows from the unitary of} the scattering matrix.

The optical theorem for light is derived from conservation of energy, whereas in electronic transport it is based on con-servation of charge. We consider here the nonstandard situa-tion of the interface between a metal and a tunneling barrier, for which the unitary of the scattering matrix cannot be in-voked without some care. Let us consider a nonmagnetic

IXN structure, where X could be basically anything. We

as-sume flux-normalized plane waves eik_nx_/

冑

_k

nin N with mode

index n, and exponential solutions e⫾␬mx/

冑

␬m_{in I with mode} index m. The electron with wave function e−␬px/

冑

␬p _{in I is} reflected with amplitude rmp, and transmitted into N with

amplitude tnp. Then the wave functions in I and N are:

␺Im = e −␬px

冑␬p

␦mp+ rmp e␬mx

冑

␬m, ␺N n = tnp eiknx

冑kn

. 共A1兲

The current in I is given by the imaginary part of the reflec-tion amplitudes since

II= ប m

兺

m Im共␺I m_ⴱ ⳵x␺Im_{兲 =}2ប mIm共rpp兲, 共A2兲

The current in N reads

IN= ប m

兺

n Im共␺Nnⴱ⳵x␺Nn兲 =ប m

兺

n 兩tnp兩2. 共A3兲

By current conservation: II= IN, we have

2 Im共rpp兲 =

兺

n

兩tnp兩2⬅

兺

n

Tnp. 共A4兲

This relation reduces to

2 Im共rpp兲 = 兩tpp兩2⬅ Tpp, 共A5兲

for the ballistic model used in the text.

APPENDIX B: POLARIZATION SIGN CHANGE

For a better understanding of the polarization sign change, let us inspect the simple potential barrier depicted in Fig. 9

共thick solid line兲, ignoring the spin dependence for the mo-ment. The barrier width is d and the relative barrier height

Ub=ប2_␬2_{/2m. As seen in Fig.9, U}

b is measured relative to

the longitudinal electron energy in the barrier E − E_⬜= E储

+ V, where E is the total electron energy, E储 and E⬜are the longitudinal共normal to the interfaces兲 and transverse kinetic energies, and V is the band edge. By solving this standard quantum-mechanical exercise, we find the transmission prob-ability through the barrier

T = 1 1 +共E储+Ub兲2 4E储Ub sinh 2_共␬d兲⬇ 16E储Ub 共E储+ Ub兲2 e−2␬d, 共B1兲

where the approximation is accurate when ␬dⰇ1. Equation

共B1兲 shows that for a fixed Ub 共or ␬兲, T is maximal when E储/Ub= 1共see Fig.10兲, where E储 can be tuned by changing

the band edge V.

In an FIF MTJ, for electrons in F with the same total energy E = E储↑/↓+ V↑/↓+ E⬜ and the same transverse energy E_⬜, the relative barrier height 关Ub in Eq.共B1兲兴 is the same

for spin-up and spin-down electrons. However, the band edges 共V_↑/↓兲 are spin dependent as indicated by the solid 共spin up兲 and dashed 共spin down兲 lines in Fig.9. The longi-tudinal kinetic energies关E储in Eq.共B1兲兴 is larger for spin-up

than spin-down electrons. In our Stoner model we typically find E_储↓/Ub⬃1, whereas E储↑/Ub⬎2. According to Eq. 共B1兲

and Fig.10, we find for the parameters in our Stoner model the surprising result that T_↑⬍T_↓. In general, electrons close to the Fermi energy with small transverse wave vectors q show this inverted polarization sign change behavior. For large q, spin-up electrons tend to have higher transmission again, and the polarization becomes positive.

d

U

b

_E

⊥

E



T

1

T

2

V

↑

V

↓

FIG. 9. 共Color online兲 Potential barrier. 0 1 10

0.01

E

||

/U

b

T

FIG. 10. T vs. E储/Ub 共Ub= 0.25EF, E = EF, ␤=0.4, and d = 1 nm兲.

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