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1 Spin Transfer Torque in the Seiconductor/agnetic Structure in the Presence of ashba Effect Javad Vahedi * and Sahar Ghasab Satoory Departent of Physics, Islaic Aad University, Sari Branch, Sari, Iran * Eail: javahedi@gail.co Abstract Spin transfer torque in agnetic structures occurs when the transverse coponent of the spin current that flows fro the nonagnetic ediu to ferroagnetic ediu are absorbed by the interface. In this paper, considering the ashba effect on seiconductor region, we have discussed the spin transfer torque in seiconductor/ferroagnetic structure and obtained the coponents of spin-current density for two odels: (I)-single electron and (II)- the distribution of electrons. We have shown that whatever the difference between eri surfaces in seiconductor and eri spheres for the up and down spins in ferroagnetic increase, the transission probability decreases. The obtained results for the values used in this article illustrate that ashba effect increases the difference between a eri sphere in seiconductors and eri sphere for the up and down spins in ferroagnetic. The results also show that the ashba effect, brings an additional contribution to the coponents of spin transfer torque, which is not eist in the absence of the ashba interaction. Moreover, the ashba ter has also different effects on the transverse coponents of the spin torque transfer. Keywords: Spin transfer torque; ashba interaction; agnetic 1-Introduction The concept of "spin transfer torque" was first introduced in 1996 independently by Sloncewsi [1] and Berger []. Spin transfer eans that, whenever a current of polaried electrons enters the ferroagnetic, there is a transfer of angular oentu between the electrons and the agnetic oent of the ferroagnetic area. In other words, the spin torque transfer occurs when the flow of transfer of spin angular oentu is not constant. or eaple, a spin current, which is polaried by filtration of a thin agnetic layer, faces a second polariation by another thin agnetic layer, which its agnetic oent direction is not alie the first one. During this process the second agnetic layer will definitely absorbs part of the spin angular oentu carried by the electron. In the paper [3], authors have investigated and reported spin-current density and spin transfer torque on the general structure of a non-agnetic/ferroagnetic. The results indicate that only the transverse coponents of the spin-current play role on the spin transfer torque, while the longitudinal coponent of spin current does not show any effect. Here, the longitudinal coponent of current eans the current which its electrons spin direction is in the sae direction of the agnetiation of the ferroagnetic and the transverse coponents of the current include current of electrons that their spin is noral to the agnetiation direction (See figure 1). The spin-orbit coupling in seiconductor low-diensional systes plays an iportant role in the developent of spintronic, which eploits electrons spin in order to store inforation [4, 5]. The prediction theory of spin transfer torque creates with harnessing the spin-orbit coupling first reported in reference [7]. And then eperientally for the first tie were observed and reported in reference [8]. In ef. [9] Authors have

2 shown the effect of ashba interaction using the Boltann transport theory that a aiu efficiency of spin torque transfer can be achieved by optially changing paraeter of the ashba interaction. Also, authors of article [10] have reported spin transfer torque due to the non-unifor ashba interaction. In this paper, considering the effect of ashba in the seiconductor, we have investigated spincurrent density and spin transfer torque in seiconductor ashba/ferroagnetic junction for two free single electron and distribution of electrons odels. The paper is organied as follows. In section, we suarie the odel, Hailtonian and foralis. More details are given in the suppleentary aterial. In section 3, we present our results by nuerical calculations. inally, conclusion is given in section 4. - Theory and odel The odel considered is the ashba/ferroagnetic seiconductor junction, in which the agnetiation in the ferroagnetic layer has been considered in ẑ direction. The Hailtonian is given as follows: 0 (1) H 0 here is the echange split and Hailtonian in the ashba seiconductor can be described as follows: H i E. () where is the effective ass and, y, is the Paoli vector atri. Assuing that electrical field is as E (0,0,E ), we will have: E0 ( ) E ( ) i y y H E ( i ) E0 ( ) y y i i which E ( i y ). In the above equation E is the aount of epectation on the sallest subband with positive E 0 energy and norally, its 11 observed aount is order of 10 ev [6, 11]. Eigenstates for oving on the surface with wave vector of,y and s 1 is given as: y is i i. r r N e (4) s s 1 where N is the noraliation factor. s egarding the above equation for eigenstates, it is obvious that split subband by ashba does not have spin polariation and then the electron energy dispersion can read as: E E (5) 0 [( s ) ] s in which and E wave vectors. In the following by using a rotation atri: i cos e sin U, (6) i e sin cos with /, / and applying it to the Eq.(3), we will have new following Hailtonian in the ashba seiconductor: / is ashba dependent y (3)

3 p E U (, ) HU (, ) 0 0 p E (7) Coparing Eq.(7) and Eq.(1), the ashba seiconductor can be considered sae as a pseudo-ferroagnetic which its agnetiation is along the ẑ direction. Now, the wave vector in a ashba seiconductor [1] can be recast as follows: ; 1 q s q s s And the wave vector in ferroagnetic layer can be defined [8]:, q ( ) q (9) s In the two above equations, is the ashba spin wave vector which shows that up and down spins in seiconductor region are different as uch as. Also the wave vector in this region in the presence and absence of ashba is defined as follows, q q, q q, respectively., and are spin dependent wave vector and eri wave vector in ferroagnetic. In order to calculate the charge current density and spin current density on the interface of agnetic and ashba seiconductor, it is needed to introduce velocity operator in two areas. Assuing the propagation direction in ˆ ais, the velocity operator in this direction is defined as follows: H v p p (10) 0 p 0 this equation indicates that up and down spins in seiconductor ove at different velocities in the propagation direction ˆ. (8) Now, in order to obtain transission probability and eventually density of transitted currents, it is needed to define wave functions in ferroagnetic and ashba seiconductor regions. Therefore, we consider the wave functions for the agnetic/ashba seiconductor junction (agnetiation in the agnetic area is considered in ẑ direction) in three diensions as follows: i i iq. ashba () e r e e (11) S 1 i iq. err () te e S 1 i i iq. e r e e i iq. ashba () err () t e e (1) In these equation q (, ) and (y, ) are the wave vector and position vector, respectively. eflection and transission coefficients based on the boundary conditions, that is to say, wave function continuity and its first derivative at the interface: s (13) ( 0) ( 0) ashba y ( i ) s ashba 0 0 are as follows: q t q ; q q r q q q if q q q q i q if q q i q (14) (15) (16) And the possibility of reflection and transission: q r q (17)

4 q q T (18) t q In general, when a polaried spin current with wave vector in a seiconductor layer gets into agnetic layer there is a possibility that by the interface, it undergoes under spin filtering and thus in agnetic layer, up and down spins feel different wave vector (, respectively) [3]. There are three effective ajor factors contributing to the torque. The first factor is the spin filtering. This occurs when reflection probabilities are spin dependent. The second factor is the spin rotation. The spin rotation occurs when the quantity of * r r is not real and positive. So, this quantity can be written as follows: * * i (19) r r r r e This relationship indicates that during the reflection, phase is added to aiuthally angle that shows the direction of incident electrons. In other words, the spin direction of electrons changes at the interface during reflection. This is copletely a quantu phenoenon and cannot iagine a classical counterpart for that. inally, the third factor in the developent of spin transfer torque is spin precession. In the agnetic layer, as, the spatially-varying phase factors which appear in the transitted transverse spin currents, is in a way that their outcoe creates a spatial precession. (0) Due to the conservation of angular oentu, the spin transfer torque on the igure-1 Setch of the transverse and longitudinal coponents of spin-current density. In this figure, Q and Q spin currents that are called the transverse coponents and spin current coponents. Q and Q y, respectively, are Q longitudinal studied aterial can be achieved getting help of calculation of pure spin current flow fro this object surfaces [13]. N ˆ ST d n. Q pillbo surfaces (1) 3 d r Q pillbo volue However, it should be noted that a spincurrent density Q is a tensor, which includes spatial and spin parts. Point wise ultiply of nq ˆ. operates only on the spatial part and does not wor on the spin part. This relationship is ade of the fact that, if we consider a rectangular surface (Gauss surface) at the interface, the divergence theore iplies to apply a spin transfer torque of induced current to the interface agnetiation. In the following, above provided forulation will be investigated for the single-electron odel and the distribution odel of electrons. -1 Spin current for single electron To investigate spin torque transfer in this odel, first we define the spin current density as follows [3]:

5 (r) e (r) s v ˆ i i (r) Q i where i Qij e (r) v ˆ i i (r) () r is a tensor quantity. Left inde belongs to spin space and right inde belongs to real space. The incident wave functions, reflection and transission which are described by equations (11) and (1) can be written as a linear cobination of spin up and spin down coponents. Now, by inserting the wave functions in equation (), current coponents will be as follows. or incident current coponent: in in Q Q cos( ) (3) in in Q Qy(r) sin( ) in Q(r) The reflected current: ref ref ( ) i Q e Q rre ( ) i i e rr e e ref ref i ( ) Q Qy I rre (4) i ( ) i rr I e e Q r r ref The transitted current are tr tr i( ) Q Q e t t e ( ) tr tr i( ) Q Qy I t t e ( ) tr Q t t (5) Now, substituting the obtained currents in Eq.(1), for the torque we have: ( ) i N cos ( ) e st rre i ( ) e tt e ( ) ˆ i( ) sin () I rre i ( ) I tt e ( ) ˆ y ( ) ˆ r r (6). Spin current for distribution of electrons Generally, in the description of transport, it is necessary to consider the effect of quantu echanics coherence between all the electrons with different special odes. However, for the spin torque transfer odel, up to now, eperients show [14, 15, 16] that it is not necessary to consider the coherence between electrons and different wave vectors of eri sources. This is equivalent to the use of sei-classical theory that only note the coherence between up and down spin state at any point of in the eri sea. In this odel, the forula for calculating related to density of incident, reflected and transitted currents are defined as follows [3]: d 3 in in Q (, ) ( ) ri f r I V ( ) v 0 3 ref d ref Q r f(, r) V () ( ) v 0 d 3 trans trans Q rt f r T V ( ) v 0 (, ) ( ) (7) (8) (9)

6 which f (, r) is electron distribution function and would be defined as follows [3]: f (, r) 0 f ( r, ) Ur (, ) U( r, ) (30) 0 f (, r) In which f (,r) ( ) is occupying function for up (down) spins. The operator U (,r) is the rotation atri that has the following for: i i cos( /) e sin( /) e U(, r) (31) i i sin( /) e cos( /) e And for sae of siplicity, the dependence of and r to angles and have been reoved. Now, with the help of Eqs.7, 8 and 9 the spin-current density coponents for /, /at the interface between ashba ferroagnetic and seiconductor, in the place of r 0 are as following. in 1 Q dq q sin( ) 8 ref 1 dqq * Q r r sin( ) 8 trans 1 dqq * v v Q t t sin [( )] ( ) 4 v trans 1 dqq * v v Qy t t cos[ ( )]( ) 4 v (3) (33) (34) (35) More details of calculations are given in the suppleentary docuent. 3-Nuerical results and discussion In this section, we will investigate the spin wave vector, reflection and transission coefficients for single-electron odel, then the current density of the electron in distribution odel will be studied. igure : (Color online) spin wave vector for the up and down spins in ferroagnetic (left panel), spin wave vector in ashba seiconductor (right panel) versus / q for and 0.5. igure- shows spin wave vector in the seiconductor and ferroagnetic layers. As it can be seen, the ashba effect in seiconductors differs the wave vector for up and down spin, and this difference is such that the up and down spin wave vectors have the sae aount, but each one to the non-ashba are separated fro each other as uch as applied aount of ashba. The difference in way of the spin wave vector detachent in ashba seiconductor and ferroagnetic areas are clearly illustrated in igure. In igure-3 we have displayed transission and reflections probabilities for up and down spin in the absence and presence of ashba interaction as a function of q. As it can be seen, possibility of reflection in the absence of ashba for down spin in q 0 is close to ero and in q 0.5 reflection is one. While possibility of reflection for up spin in q 0 is alost ero and as the closer q is to the eri wave vector ( 1), the coplete reflection we will have. /

7 igure-3 (Color online) eflection and transission possibility for up and down spins as function of q for and 0.5. / Considering the ashba interaction, the possibility reflection for up and down spin shows siilar trend as the non-ashba case. The reason is that wave vector increases in the seiconductor ashba, considering the ashba. In general, as the difference between the ashba seiconductor wave vectors ( ) and the ferroagnetic spin wave vector ( ) increases, the reflection increases as well. Left panel in figure 4 presents phase difference of an electron due to spin rotation which eperiences at the tie of reflection. In the left panel of figure-4, at q 0 we have 1.5 and 0.5. Thus, wave vector difference of is as it can be seen in the figure. Also, the wave vector difference of is 1.3, forq 0.5. In fact, by reducing q toward ero, the eri sphere difference for up and down spin increases, as a result reduces. In the right panel of figure 4, we have illustrated wave vector difference of a transitted electron. As it can be seen, by increasing q, phase difference in the presence and absence of ashba reduces. In / igur-4 (Color online) Left panel shows the phase acquired by an electron because its spin rotates upon reflection and right panel shows wave vector difference for a transitted electron a way that in the absence of ashba forq 1, phase difference becoes ero, which eans that the electrons around the eri surface with the sae incident spin direction are reflected. While in the presence of ashba, a non-ero phase difference in q 1 is observed, even for sall aounts of 0.5. In general, by increasing ashba, phase difference of the reflected electrons copared to the incident electrons eperience further changes around the eri surface. In figure-5 we have plotted spin torque transfer in the presence and absence of ashba interaction for a single electron odel as function of distance fro the interface of two environents. As one can see in the figure, in the absence of ashba, the spin torque only contains coponents of transverse spin current. While considering the ashba, torque, in addition to the coponents of the transverse, also will have longitudinal coponent. Moreover, the influence of ashba interaction on the transverse coponents of the spin torque transfer is different. igure-6 shows density of spin-current for distribution of electrons as function of

8 igure-5: (Color online)spin torque in the absence and presence of ashba for a free single electron odel. 0.5, 1.5 / / distance fro the interface of two environents for soe paraeters. Clearly, decay of the current density can be seen by getting distance fro the interface. However, this decay is slightly sall in the presence of ashba interaction. In the absence of ashba, eri wave vector for up and down spins are coincided. Thus, eri sphere in seiconductor is placed between eri spheres of up and down spins in ferroagnetic. In figure6-b, eri sphere of seiconductor is equal to eri sphere of spin ajority in ferroagnetic and both are larger than eri sphere of spin inority. In figure 6-c eri spheres of ajority and inority are slightly larger and saller than eri sphere in seiconductor. Considering the ashba effect leads the eri sphere and wave vector difference in a seiconductor ( ) with wave vector spin (ajority and inority) in ferroagnetic ( ) to be increased. The difference is less for up spin than down spin, as a result the chance of passing for that will increase. Thus, it can be concluded that in the transitted current, in igure-6: (Color online)transitted spin current as a function of distance fro the interface for the eri energy of 1. a) / 0.5, / 1.5, b) / 0.5, / 1.0, c) / 0.9, / 1.1 this structure, up spins play a ajor role in the transitting current. 4-Conclusion In this article we have discussed the theory of the spin transfer torque in the ashba seiconductor/agnetic junction and obtained the coponents of spin-current density for two odels: (I)- single electron odel and (II)- the distribution of electrons. In the single electron odel, the spin torque calculation, Eq.(6), shows that considering the ashba effect in seiconductor and being different the eri wave vector for up and down spins in ferroagnetic region we will have all three coponents of torque. While in the absence of ashba and by taing, the spin torque will be only included the transverse coponents. In the distribution odel of electrons, it can be concluded fro the results of current density that the ore difference between the eri sphere in seiconductor with eri spheres

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