1. IntroductionThe concept of “spin transfer torque” was first introduced in 1996 independently by Slonczewski and Magn[1] and Berger.[2] Spin transfer means that whenever a current of polarized electrons enters the ferromagnetic state, there appears a transfer of angular momentum between the electrons and the magnetic moment of the ferromagnetic area. In other words, the spin torque transfer occurs when the flow of transfer of spin angular momentum is not constant. For example, a spin current, which is polarized by the filtration of a thin magnetic layer, faces a second polarization by another thin magnetic layer, in which its magnetic moment direction is not the same as the first one. In this process the second magnetic layer will definitely absorb part of the spin angular momentum carried by the electron.
In Ref. [3], authors have investigated and reported spin-current densities and spin transfer torques of the general non-magnetic structure and ferromagnetic structure. The results indicate that only the transverse components of the spin-current play a role in the spin transfer torque, while the longitudinal component of spin current does not show any effect. Here, the longitudinal component of current means that the current whose electrons have spin directions the same as the directions of the magnetizations of the ferromagnetic and the transverse components of the current includes the current of electrons whose spins are normal to the magnetization direction (see Fig. 1).
The spin–orbit coupling in semiconductor low-dimensional system plays an important role in developing the spintronics, which exploits electron spin in order to store information.[4–10] The prediction theory of spin transfer torque is created with harnessing the spin–orbit coupling first reported in Ref. [11]. Then experimentally for the first time they were observed and reported in Ref. [12]. In Ref. [13] authors have shown the effect of Rashba interaction using the Boltzmann transport theory that a maximum efficiency of spin torque transfer can be achieved by optimally changing the parameter of the Rashba interaction. Also, authors of Ref. [14] have reported spin transfer torque due to the non-uniform Rashba interaction. In the present paper, considering the effect of Rashba in the semiconductor, we investigate spin-current density and spin transfer torque in semiconductor Rashba/ferromagnetic junction for two free single electron and distribution of electrons models.
The rest of the present paper is organized as follows. In Section 2, we summarize the model, Hamiltonian and formalism. More details are given in the supplementary material. In Section 3, we present our numerical calculations. Finally, conclusion is given in Section 4.
2. Theory and modelThe model considered here, which also was used in Ref. [15], is the Rashba semiconductor/ferromagnetic junction, in which the magnetization in the ferromagnetic layer is considered in ẑ direction. The Hamiltonian is given as follows:
 | (1) |
where 2Δ is the exchange split and Hamiltonian in the Rashba semiconductor can be described as follows:
 | (2) |
where
α is the effective mass and
σ = (σ
x, σ
y, σ
z) is the Pauli vector matrix. Assuming that the electrical field is
𝐸 = (0,0,
Ez), we will have
 | (3) |
where Δ
R = 〈
αEz〉(
kx − i
ky). In the above equation 〈
α Ez〉 is the value of expectation on the smallest subband with positive
E0 energy and normally, its observed value is of the order of 10
−11 eV · m.
[6,16] Eigenstates for moving on the surface with wave vector of
k = (
kx,ky) and
s = ±1 are given as
 | (4) |
where
Nks is the normalization factor. Regarding the above equation for eigenstates, it is obvious that split subband by Rashba does not have spin polarization and then the electron energy dispersion can read as
 | (5) |
in which
and
kR = 〈
α Ez〉
m/
ħ2 is Rashba-dependent wave vectors. In the following, by using a rotation matrix
 | (6) |
with
ϕ = −
π/2,
θ =
π/2 and applying it to Eq. (
3), we will have new following Hamiltonian in the Rashba semiconductor
 | (7) |
Comparing Eq. (7) and Eq. (1), the Rashba semiconductor can be considered to be the same as a pseudo-ferromagnet in which its magnetization is along the ẑ direction. Now, the wave vector in a Rashba semiconductor[12] can be recast as follows:
 | (8) |
The wave vector in ferromagnetic layer can be defined as follows:[11]
 | (9) |
In the above two equations, 
is the Rashba spin wave vector which shows that up and down spins in semiconductor region are different as much as kR. Also the wave vector in this region in the presence and absence of Rashba spin wave vector is defined as follows:
 | (10) |
This equation indicates that up and down spins in semiconductor move at different velocities in the propagation direction 
. In the ferromagnet the velocity operator is always diagonal in spin space
 | (11) |
 | (12) |
In these equations q(ky,kz) and R(y,z) are the wave vector and position vector, respectively. Reflection and transmission coefficients based on the boundary conditions, that is to say, wave function continuity and its first derivative at the interface:
 | (13) |
 | (14) |
are as follows:
 | (15) |
 | (16) |
The possibilities of reflection and transmission are respectively as follows:
 | (17) |
 | (18) |
In general, when a polarized spin current with kx wave vector in a semiconductor layer enters into the ferromagnetic layer there is a possibility that by the interface it undergoes spin filtering and thus in ferromagnetic layer the up and down spins feel different wave vectors (
respectively).[3] There are three effective major 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 
is not real and positive. So, this quantity can be written as follows:
 | (19) |
This relationship indicates that during the reflection, Δϕ phase is added to azimuthal angleϕ that shows the direction of incident electrons. In other words, the spin direction of electron changes at the interface during reflection. This is completely a quantum phenomenon and cannot imagine a classical counterpart for that. Finally, the third factor in the development of spin transfer torque is the spin precession. In the ferromagnetic layer, as 
, the spatially-varying phase factors which appear in the transmitted transverse spin currents is in a way that their outcome creates a spatial precession.
 | (20) |
The angular momentum is conserved in the system. This is true if the FM and electron spin in the Rashba semiconductor are uncoupled (i.e., in the absence of s–d exchange coupling), and the conservation holds only for the FM layer, not for the whole system. So, due to the conservation of angular momentum, the spin transfer torque on the studied material can be achieved which is conducive to the calculation of pure spin current flow from this object’s surfaces.[17]
 | (21) |
However, it should be noted that the spin-current density Q is a tensor, which includes spatial and spin parts. Point-wise multiplier of 
operates only on the spatial part and does not act on the spin part. This relationship lies in the fact that if we consider a rectangular surface (Gauss surface) at the interface, the divergence theorem implies that a spin transfer torque of induced current is applied to the interface magnetization.
In the following, the formulation provided above will be investigated for the single-electron model and the distribution model of electrons.
2.1. Spin current for single electronTo investigate spin torque transfer in this model, first we define the spin current density as follows[3]
 | (22) |
where
Qij (
r) is a tensor quantity. Left subscript belongs to spin space and right subscript belongs to real space. The incident wave functions, reflection and transmission which are described by Eqs.
11) and (
12) can be written as a linear combination of spin up and spin down components, i.e.,
ψ =
ψ↑ +
ψ↓. Now, by inserting the wave functions in Eq. (
22), current components will be as follows. The incident current components are
 | (23) |
the reflected currents are
 | (24) |
the transmitted currents are
 | (25) |
Now, substituting the obtained currents into Eq. (21), for the torque we have
 | (26) |
2.2. Spin current for distribution of electronsGenerally, in the description of transport, it is necessary to consider the effect of quantum mechanics coherence of all the electrons with different special modes. However, for the spin torque transfer model, up to now, experiments have shown[18–20] that it is not necessary to consider the coherence between electrons and different wave vectors of Fermi sources. This is equivalent to the use of semi-classical theory that only note the coherence between up and down spin state at any point of k in the Fermi sea.
In this model, the formula for calculating density of incident, reflected and transmitted currents are defined respectively as follows:[3]
 | (27) |
 | (28) |
 | (29) |
where
f (
k, r) is the electron distribution function and can be defined as follows:
[3]
 | (30) |
where
f↑(↓)(
k, r) is the occupying function for up (down) spin. The operator
U(
k, r) is the rotation matrix that has the following form:
 | (31) |
For simplicity, the dependence of k and r on angles θ and ϕ have been removed. Now, with the help of Eqs. (27)–(29), the spin-current density components for θ = π/2, ϕ = −π/2 at the interface between Rashba ferromagnetic and semiconductor, in the place of r = 0 are as follows:
 | (32) |
 | (33) |
 | (34) |
 | (35) |
More details of calculations are given in the supplementary document.
3. Numerical results and discussionIn this section, we investigate the spin wave vector, reflection, and transmission coefficients for single-electron model, then the current density of the electrons in distribution model will be studied.
Figure 2 shows the spin wave vectors in the semiconductor and ferromagnetic layers. As can be seen, the Rashba effect in semiconductor makes the wave vectors for up and down spin differ, and this difference is such that the up and down spin wave vectors have the same magnitude, but they, for the non-Rashba, are separated from each other as much as the added amount of Rashba semiconductor. The difference in way of the spin wave vector detachment in Rashba semiconductor and ferromagnetic areas are clearly illustrated in Fig. 2.
Figure 3 displays the plots of transmission and reflection probabilities for up and down spin in the absence and presence of the Rashba interaction versus q. As can be seen in the figure, the possibility of the reflection in the absence of the Rashba interaction for down spin in q = 0 is close to zero and in q ≃ 0.5 the reflection is one. While the possibility of the reflection for up spin in q = 0 is almost zero and the closer to the Fermi wave vector (kF = 1) the q is, the more complete the reflection will be. Considering the Rashba interaction the reflection possibilities for up spin and down spin have similar trends to those in the non-Rashba case. The reason is that the wave vector increases in the semiconductor Rashba, with considering the Rashba interaction. In general, as the difference between the Rashba semiconductor wave vectors 
and the ferromagnetic spin wave vector 
increases the reflection increases as well.
Figure 4(b) shows the phase difference of an electron due to spin rotation which is experienced at the time of reflection. In Fig. 4(b), at q = 0 we have 
and 
. Thus, the wave vector difference of Δk is kF as can be seen in the figure. Also, the wave vector difference of Δk is 1.3, for q = 0.5kF. In fact, by reducing q to zero, the Fermi sphere difference for up and down spins increases, as a result of Δk reducing.
Figure 4(a) shows the wave vector difference of a transmitted electron. As can be seen in the figure, by increasing the parameter q, the phase difference between the two cases, that is, in the presence of the Rashba interaction and the absence of Rashba interaction, reduces. In the absence of Rashba interaction for q = kF = 1, the phase difference becomes zero, which means that the electrons around the Fermi surface with the same incident spin direction are reflected. Meanwhile, in the presence of the Rashba interaction, a non-zero phase difference in q = kF = 1 is observed, even for a small value of kR = 0.5. In general, by increasing the Rashba interaction, the phase difference of the reflected electrons compared to that of the incident electrons experience further changes around the Fermi surface.
Figure 5 shows the spin torque transfers as a function of distance from the interface of two environments in the presence and absence of Rashba interaction for a single electron model. As can be seen in the figure, in the absence of Rashba interaction, the spin torque only contains the component of transverse spin current. While considering the Rashba interaction, the torque, in addition to the components of the transverse, also will have longitudinal component. Moreover, the influences of Rashba interaction on the transverse components of the spin torque transfer are different.
Figure 6 shows the densities of spin-current for distribution of electrons as a function of distance from the interface of two environments for some parameters. Clearly, the decay of the current density can be seen by getting the distance from the interface. However, this decay is slightly small in the presence of Rashba interaction. In the absence of Rashba interaction Fermi wave vectors for up and down spins are coincided. Thus, the Fermi sphere in the semiconductor is placed between Fermi spheres of up and down spins in ferromagnetic. In Fig. 6(b) Fermi sphere of semiconductor is equal to Fermi sphere of spin majority in ferromagnetic and both are larger than Fermi sphere of spin minority. In Fig. 6(c) Fermi spheres of spin majority and minority are slightly larger and smaller than Fermi sphere in semiconductor, respectively. Consideration of the Rashba effect causes the Fermi sphere and wave vector difference in a semiconductor 
with wave vector spin (majority and minority) in ferromagnetic 
to increase. The difference is less for up spin than for down spin, as a result the chance of pass for that will increase. Thus, it can be concluded that in the transmitted current, in this structure, up spins play a major role in transmitting current.
