Valley controlled spin-transfer torque in ferromagnetic graphene junctions

The presence of the valley degree of freedom in graphene leads to the valleytronics, in which information is encoded by the valley quantum number of the electron. We propose a valley controlled spin-transfer torque (STT) in graphene-based normal/normal/ferromagnetic junctions with the normal lead irradiated by the off-resonant circularly polarized light. The interplay of the spin–orbit interaction and the staggered potential in the central normal part results in the coupling between the valley and spin degrees of freedom, so a valley dependent spin polarized current can be demonstrated, which can exert a valley controlled STT on the ferromagnetic lead. The amplitude of the STT can be manipulated by the intensity of the light, the Fermi energy and the magnetization direction of the ferromagnetic lead. This valley controlled STT may find potential application in future valleytronics and spintronics.


Introduction
Over the past decade, motivated by the development of spintronics, the presence of the valley degree of freedom in graphene leads to the valleytronics [1,2], whose goal is to manipulate the valley degree of freedom and search for its potential applications in semiconductor technologies and quantum information. Much substantial progress in valleytronics has been made recently, such as quantum valley Hall effect [3][4][5][6], valley polarization controlled by circularly polarized light in molybdenum disulfide [7], and valley and spin currents in silicene junctions [8][9][10]. Interestingly, if the intrinsic spin-orbit interaction and the staggered potential coexist in graphene-like materials, the band structure becomes spin-valley coupling, so one can control the spin polarized current by the valley degree of freedom [8][9][10][11].
Recently, on the other hand, the spin-transfer torque (STT) has also attracted much attention [12][13][14][15][16][17][18][19]. When a spin polarized current is injected into a ferromagnetic layer with a magnetization misaligned to the spin polarization of the current, it can transfer spin angular momentum to the ferromagnetic layer, and hence exerts a torque on the magnetic moments of the ferromagnetic layer, which may change the magnetization orientation of the ferromagnetic layer. The STT effect has the potential application in random access memory, which offers an all-electrical read and write process. The STT was theoretically predicted by Slonczewski [12] and Berger [13], and then it has been extensively confirmed experimentally [14]. Recently, the STT in ferromagnetic graphene junctions has also been reported. Yokoyama and Linder [15] demonstrated that both the magnitude and the sign of the STT can be controlled by means of the gate voltage in a bulk ferromagnetic/normal/ferromagnetic graphene junction. Then Ding et al [16] investigated theoretically the effect of strain on the STT in a zigzag-edged graphene nanoribbon spin-valve device. Later Zhang et al [17] studied the helical spin polarized current induced STT in graphene-based normal/topological insulator/ferromagnetic junctions. Although the STT in ferromagnetic graphene junctions has been investigated in [15][16][17], the effect of the valley degree of freedom on the STT is not explored. If the coupling between the valley and spin degrees of freedom exists, a valley controlled spin polarized current is generated. Naturally, it is interesting to discuss the effect of the valley degree of freedom on the STT in a ferromagnetic graphene junction. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
In this work, we predict a valley controlled STT in graphene-based normal/normal/ferromagnetic (N 1 /N 2 /F) junctions, where the N 1 is irradiated by the off-resonant circularly polarized light. In the N 1 , the valley polarization can be modulated by the interaction between the staggered potential and the light. While in the N 2 , the interplay of the spin-orbit interaction and the staggered potential leads to the coupling between the valley and spin degrees of freedom. In this case one can control the valley dependent spin polarized current by the valley degree of freedom, so a valley controlled STT is demonstrated. The influences of the intensity of the light, the Fermi level and Rashba spin-orbit interaction on the STT are discussed.
The rest of this paper is organized as follows. In section 2 we give the Hamiltonian of the N 1 /N 2 /F junctions, discretize the Hamiltonian in the basis x k i y ñ Ä ñ | | , and then present the formula of the valley dependent STT by the Keldysh non-equilibrium Green's function method. In section 3, numerical results and detailed discussions are demonstrated. Finally, in section 4 we summarize the main conclusions of this work.

Model and formulation
We consider a graphene-based two dimensional N 1 /N 2 /F junction (see figure 1(a)) with the interfaces located at x=0 and x=L, where L is the width of the N 2 . The N 1 (x<0) is irradiated by off-resonant circularly polarized light that requires ÿω ? γ in principle, where ω is the frequency of light and γ is the nearest-neighboring hopping energy in graphene [20][21][22][23][24]. In the central part N 2 the spin-orbit interaction and the staggered potential are considered, which leads to the coupling between the valley and spin degrees of freedom. The ferromagnetic electrode deposited on top of the graphene sheet (x>L) induces a finite exchange field h h cos , 0, sin q q = ( ) [8,15,25], where h is the magnitude of exchange field and q describes the direction angle of the magnetization. In the low-energy approximation, the Hamiltonian of the present junctions is expressed as [8,9,21,26] H Here 1 1 h = + -( )represents the K (K′) valley with v F the velocity of electrons. τ j and σ j ( j=x, y, z, 0) are the Pauli matrices and unit matrices in valley and spin space, respectively. z z so hl s t Ä is intrinsic spin-orbit coupling term in graphene. As indicated in Kane and Mele investigation [26], graphene will be driven into topological phase when this term in graphene is enhanced. In general, the spin-orbit interaction is very weak in graphene. However, many works suggested that spin-orbit interaction can be enhanced by the substrates or adatom deposition [27][28][29][30], which results in the nontrivial topological phase. λ ω is related to the intensity of the off-resonant circularly polarized light and provides an additional site energy of a sublattice in the N 1 . By using the Floquet theory [20][21][22][23][24] λ ω can be written as Iv 8  Here the staggered potential induced by the substrates is assumed to be finite in the N 1 and N 2 but zero in the F. (b) The band structure of the N 1 /N 2 /F junctions. In the N 1 , the spin is degenerate, and due to the interaction between the staggered potential (SP) and the light, a valley dependent band gap exists. In the N 2 , the interplay of the spin-orbit interaction (SOC) and the SP leads to the coupling between the valley and spin degrees of freedom. While in the F the valley is degenerate.
α;1/137. The sign and strength of λ ω can be tuned by the handedness and the intensity of the circularly polarized light. It is noted that for the system under the off-resonant circularly polarized light, light does not directly excite electrons but effectively modifies the electron band structures through virtual photon absorption processes [31]. In this case the systems would not heat up by the light. λ v is the staggered potential induced by the substrates [32,33], which is assumed to be finite in the N 1 and N 2 but zero in the F. By solving equation (1), the dispersion relation of the N 1 for the electrons can be written as where n=+(−) represents the conduction (valence) band, and k k k It is noted that for the electrons in the η valley there exists an energy gap E 2 |, which can be tuned by the parameters λ v and λ ω . The energy ε of the incident electron should be satisfy E 2 g e > h | | to generate propagating incident modes in the η valley of the normal lead.
Due to the translation invariant along the y-axis, the transversal wave vector k y of the incident electron must be conserved. Following the [34], we can discretize the Hamiltonian is the mesh spacing along the x direction (i0 for the left lead, 1iN for the central part, and iN+1 for the right lead). H i,i is given as The valley dependent STT are studied by the non-equilibrium Green's function method. The influence of the two semi-infinite leads can be treated by using the iteration technique [36][37][38]. The retard Green's function of the central part N 2 can be calculated by the following expression: where μ F is the Fermi level. By using equation (6), we can obtain the total STT Rx

Results and discussions
Before presenting the numerical results for the valley dependent STT in the N 1 /N 2 /F junctions, we first make a physical analysis for the valley dependent STT. As shown in equation (2), in the N 1 the spin is degenerate and the valley polarization can be manipulated by stagger potential and the off-resonant circularly polarized light. While from equation (1) the dispersion relation of the N 2 can be written as where n = + -( )corresponds to the conduction (valence) band. Therefore, the interplay of the spin-orbit interaction λ so and the stagger potential λ v in the N 2 leads to the coupling between the valley and spin degrees of freedom. When the electrons travel across the N 2 , the currents become spin polarized and have different spin polarization in the K and K′ valleys, so the torques for the K and K′ valleys exerted on the F may have different amplitude and sign.
In what follows we show some numerical results for the valley dependent STT in the N 1 /N 2 /F junctions. Figure 2(a) shows the valley dependent STT as a function of the magnetization direction of the F. Here the light is not considered, so that due to the valley degeneracy in the N 1 , the current injected into the junction from the N 1 is valley unpolarized. As seen in figure 1(b), the solid (dashed) lines in the N 2 correspond to the spin-up (spindown) channel. When the electrons travel across the N 2 , only spin-up (spin-down) electrons in the K (K′) valley can arrive at the N 2 /F interface (see figure 1(b)), so the torques for the K and K′ valleys have opposite sign. As seen in figure 2(a) figure 2(b), the effect of the light on Rx t is discussed. When λ ω is finite, because the relationship of is absent, the period of τ Rx becomes 2π instead of π. With increase of λ ω the amplitude of τ Rx first increases and then begins to decrease, so in order to get a large τ Rx , one should choose an appropriate parameter λ ω . We can understand this behavior as follows. For a finite positive λ ω , because the density of states of the incident electrons in the K valley nonmonotonically depends on λ ω , the current coming from the K valley first increases and then decreases with λ ω , leading to a nonmonotonic dependence of K Rx t on λ ω . While for the parameters taken here the density of states of the incident electrons in the K′ valley monotonically decreases with λ ω and becomes zero for large λ ω , so K Rx t ¢ decreases and becomes zero for large λ ω . As discussed above, the off-resonant circularly polarized light strongly influences on the valley dependent STT, so it is interesting to analyze the effect of λ ω on the valley dependent STT in detail. Figure 3(a) displays the valley dependent STT as a function of λ ω at θ=π /4 (black lines), π /2 (read lines) and 3π /4 (blue lines). As shown in equation (2), the band structure of the η valley in the N 1 has an band gap E g h , which can be tuned by λ ω .
should be satisfied to generate propagating incident modes in the N 1 , so for the parameters μ F =30 meV and λ ω =20 meV taken here, K Rx t nonmonotonically depends on λ ω in the regime of  figure 3(a)). On the other hand, for the other q , the symmetry of K )is broken, τ Rx is not an odd function of λ ω any more. In order to explain the behavior of Rx t h , we divide Rx t h into two parts:   , one observes the STTs at λ ω =−10 meV and λ ω =10 meV have the same amplitude but opposite sign, thus the sign of the STT can be controlled by handedness of the light.
In addition, due to the presence of the structure inversion asymmetry in the z direction, the Rashba spinorbit interaction ) may appear in the N 2 . In figure 6 we study the effect of the Rashba spin-orbit interaction on τ Rx at θ=π /2. As shown in figure 6 τ Rx is an odd function of λ ω and decreases with λ. This is because without λ the current can arrive at the N 2 /F interface with the spin polarization direction along z (for the K valley) or −z (for the K′ valley) axis, which is perpendicular to the magnetization direction of the F. However, when λ is finite, the electrons precess in the process of traveling across the N 2 , so the spin polarization direction deviates from the z axis, which results in the decrease of τ Rx with λ. Therefore in order to obtain a large τ Rx , a small λ is needed.
Last we will comment on the experimental feasibility of our results. The staggered potential can be induced by the substrate. Actually, in experiment, the gap induced by the SiC substrate can range from several meV to 0.26 eV [32,33]. A strong spin-orbit interaction can be induced by the substrates or adatom deposition [27][28][29][30]. For example, Kou et al reported a large intrinsic spin-orbit interaction is generated in Bi 2 Se 3 /graphene/Bi 2 Se 3 [28] or BiTeI/graphene/BiTeI [29] quantum well structure. In the N 2 we can consider a Bi 2 Se 3 /graphene/SiC or BiTeI/graphene/SiC quantum well structure, where the staggered potential and spin-orbit interaction are induced by the proximity effect, respectively. In fact, we can also use silicene or stanene as a central part, which has strong intrinsic spin-orbit interaction but weak Rashba spin-orbit interaction. The off-resonant circularly polarized light requires w g  in principle, so the frequency should be larger than 3500 THz [20]. When one takes the lowest frequency of light, ω≈3500 THz, and 0.0026 0.39 l » w | | -eV [20] is obtained when graphene is irradiated by an ultrashort pulse [39] with the range of laser intensity from 10 10 to 1.5 ×10 12 W cm −2 . Here the spin is degenerate in the normal lead, the spin polarization of the current originates from the spin-valley coupling in the central part, so a large central region width L is needed. For large L (L>50 nm), the contribution of evanescent states to the currents is completely suppressed, and τ Rx is nearly independent of L (not shown here). A real N 1 /N 2 /F junction also inevitably contains impurities or atomic defects in the bulk. As pointed in [40,41], to realize the valley controlled STT, the defect ratio cannot exceed 8%, otherwise the valley controlled STT effect is broken by defect states. It should be pointed out that the amplitude of τ Rx obtained here is comparable to the STT in a previous work, where a ferromagnetic/normal/ferromagnetic junction is investigated [15]. However, unlike [15], the spin polarized currents in this work originates from the coupling between the valley and spin degrees of freedom, so the STT reported here does not require additional ferromagnetic layer with fixed magnetization.

Summary
In summary, we study the valley dependent STT in N 1 /N 2 /F junctions. The N 1 is irradiated by the off-resonant circularly polarized light, and the valley polarization can be modulated by the interaction between the staggered potential and the light. While in the N 2 , due to the interplay of the spin-orbit interaction and the staggered potential induced by the substrate, the band structure is spin-valley coupling, so one can control the valley dependent spin polarized current by the valley degree of freedom, which exerts a valley controlled STT on the F. The effects of the intensity of the light, the Fermi level and Rashba spin-orbit interaction on the STT are investigated. The valley controlled STT reported here suggests the ferromagnetic graphene junction ideal for very efficient magnetization manipulation of magnetic materials without external magnetics fields.