Electric and thermal spin transfer torques across ferromagnetic/normal/ferromagnetic graphene junctions

We investigate the spin transfer torque (STT) driven by electric bias voltages across and temperature gradients through ferromagnetic/normal/ferromagnetic graphene junctions. Due to the unique band structure of the ferromagnetic graphene, there exists two transport regimes: the electron to electron (I) and hole to electron (II) transport. The electric STTs originated from the two regimes have opposite sign and can be reduced by the competition between the two transport processes. On the contrary, the thermal STTs originated from the transport regimes I and II have the same sign and are enhanced when the two regimes coexist. Remarkably, the thermal STT is comparable with the electric STT. Furthermore, the electric and thermal counterpart can be manipulated by the Fermi level. The controllable STT reported here makes the ferromagnetic graphene junction ideal for future spintronics applications.

On the other hand, in magnetic materials spin polarized currents can be created by thermal currents [26][27][28][29][30][31]. The transfer of spin angular momentum through this process has been named thermal STT [26,27]. The thermal STT in metallic spin valves was theoretically predicted [27] and then experimentally observed [28]. The thermal STT in MgO based magnetic tunnel junctions has also been extensively explored [26,[32][33][34]. In order to be sufficient to switch the magnetic configurations, very thin barriers of 3 MgO atomic layers are needed [26,33,34]. Heiliger et al [34] pointed out that even for such a thin barrier the thermal STT is still too small for reorienting the magnetic order and further optimization is needed. Thus it is highly challenging to induce magnetic tunnel junction switching by virtue of the thermal STT. Unlike the MgO based magnetic tunnel junctions, in graphene junctions the Klein tunneling process appears, and the exponential decay of the current with barrier thickness is absent. Therefore, it is expected Figure 1. Schematic plot of a F 1 /N/F 2 junction. Here the magnetization direction of the F 1 is along z axis, while that of the F 2 along the z axis, deviates the z axis by an angle α. τ x is along x direction in the x y z coordinate frame. that a large thermal STT can be observed in magnetic graphene junction. However, the thermal STT in magnetic graphene junctions has not been explored.
In this work, the electric and thermal STTs through ferromagnetic/normal/ferromagnetic (F 1 /N/F 2 ) graphene junctions are studied. We compare the electric STT with the thermal STT. Due to the unique band structure of the ferromagnetic graphene, there exists two transport regimes: the electron to electron and hole to electron transport. The coexistence of the two regimes can reduce the electric STT but enhance the thermal STT. Remarkably, the thermal STT is comparable with the electric STT. Furthermore, the STT can be manipulated by the Fermi level.

Model and formulation
We consider a two-dimensional F 1 /N/F 2 graphene junction under the electric bias V b or the temperature gradient ΔT, as shown in figure 1. Here the temperature gradient is defined as ΔT = T L − T R , where T L (T R ) is the temperature in the left (right) lead. The interfaces between the ferromagnetic/normal regions locate at x = 0 and x = L with the normal region width L. In the F 1 a graphene monolayer sheet in the xy plane is grown on a substrate such as SiC that provides a staggered potential λ v for graphene due to the breaking of sublattice symmetry induced by the graphene-substrate interaction [35]. The ferromagnetic electrodes deposited on top of the graphene sheet induce the finite exchange fields with the strength M L and M R in the F 1 and F 2 , respectively [22,25,29,30,36]. The translational symmetry along the y direction is preserved so that the electron momentum k y is a good quantum number. The Hamiltonian of the present junction is expressed as Here c β+ m,k y ,σ (c β m,k y ,σ ) with β = A, B and σ σ = ↑, ↓ stands for the creation (annihilation) operator of an electron with spin σ at site m. ε m is the on-site energy, which is set as ε 0 in the F 2 and zero in the F 1  M R in the F 1 and F 2 , respectively. The magnetization direction of the F 1 is along z axis, while that of the F 2 is along (sinα, 0, cosα). The last term is the staggered potential with λ v induced by the graphene-substrate interaction, where μ m is 1 (−1) for A (B) sites. We assume that λ v is only finite in the F 1 .
The STT is studied by the non-equilibrium Green's function method. The retarded Green's function G r is calculated by using the iteration technique developed by Lopez-Sancho et al [37]. The in-plane STT τ x acting on the F 2 can be obtained as [23][24][25] Here [9] is along the x direction in the x y z coordinate frame (figure 1).
with the Fermi level E F stands for the Fermi distribution function in the L (R) lead. For the STT driven by the electric bias, the bias voltages are V L = 0 and V R = V b for the left and right leads, respectively. In this work like the previous works [23][24][25] we do not consider the out-plane torque τ y . This is because τ y show a similar feature but its magnitude is very small, so we neglect its effect.

Electric spin-transfer torque
The angular dependence of the electric STT τ x at various Fermi levels is investigated in figure 2. For simplicity, we set t = 1 and e = 1. In the numerical calculations, the parameters λ v = h = 0.005, ε 0 = −0.2, and N = 4 are taken. τ x is in units of eV throughout this work. The curves of τ x versus α essentially follow the usual sinusoidal behavior. For the parameters taken here, when E F increases, τ x first decreases, then reverses its sign and increases successively. This can be understood by the two transport regimes I and II, as shown in figure 2(b). Depending on E F and V b the transport is divided into electron-to-electron I and hole-to-electron II regimes, where the incident currents from the F 1 are initially spin polarized along z and −z directions, respectively. Thus the STTs originated from the two regimes have the opposite sign and can cancel each other out, leading to the nonmonotonic behavior of τ x . We also confirm this by analyzing the energy dependent torque τ x (E), as shown in the inset of figure 3. For zero E F τ x only arises from the hole to electron transport. When E F is finite, for E F < V b the electron-to-electron transport also contributes to τ x . Thus τ x decreases due to the competition between the transport processes in the I and II regimes. For example, τ x (E) at V b = 2E F is plotted in the inset of figure 3. In this case the transport window is |E| < E F . The energy dependent torques in the I (0 < E < E F ) and II ( −E F < E < 0) regimes have the opposite sign, but the difference of the Fermi distribution functions f L − f R is an even function of E, so τ x from the two regimes almost cancels each other out, leading to a vanishing τ x . On the other hand, when E F exceeds V b , τ x only comes from the I regime and varies slightly with E F . We display the bias V b dependence of τ x at various α in figure 3. For the negative V b the current only comes from the electron to electron transport (regime I marked in figure 2(b)). With increasing |V b |, the spin polarized currents acting on the F 2 increases, so τ x monotonically increases with |V b |. On the other hand, for the positive bias τ x exhibits a nonmonotonic bias dependence. τ x first increases and then decreases with V b . When V b is further enhanced, τ x reverses its sign and increases monotonically. We can  explain this nonmonotonic behavior by analyzing the two transport processes indicated in figure 2(b). For V b < E F with increasing V b the spin polarized current from the I regime acting on the F 2 increases, resulting in the enhancement of τ x . When V b exceeds E F , the hole-to-electron transport also appears, so because of the competition between the transport processes in the I and II regimes τ x decreases with V b . When V b is further enhanced, τ x is determined by the II regime and increases.

Thermal spin-transfer torque
Next, the thermal STT τ x as a function of α at various Fermi levels is shown in figure 4. Here the current is only driven by the temperature gradient. Similar to the electric STT, the angular dependence of the thermal STT is a sine dependence. The amplitude of τ x decreases with E F . For large E F τ x becomes very small and even reverses its sign. The behavior of τ x can be illustrated from the following fact. At zero E F the spin up current from the electron to electron transport and the spin down one from the hole to electron transport counterpropagate, so the thermal STTs from the regimes I and II marked in figure 2(b) have the same sign, and can be enhanced when the two regimes are involved. This is different from τ x driven by V b , where the electric STTs from the two regimes have the opposite sign and are reduced by the competition between the two transport processes. We also confirm this feature by studying τ x (E) and f L − f R , as shown in the inset of figure 4. τ x (E) and f L − f R in the I (E > 0) and II (E < 0) regimes have the opposite sign, so τ x (E)(f L − f R ) has the same sign in the regimes I and II and the coexistence of the two transport processes can enhance τ x . When E F is finite, for E > 0τ x (E) has the same sign for each energy, but f L − f R antisymmetric with respect to E F , so part of τ x from the electron current cancels that from the hole current, leading to the decrease of τ x . For large E F τ x from the electron and hole currents almost cancels each other out, leading to a small τ x .
In figure 5(a) τ x versus ΔT at k B T = 0.001 and various α is plotted. Because at zero E F the incident current is enhanced by ΔT, τ x increases monotonically with ΔT. At fixed ΔT when α sweeps from zero to π/2, τ x increases with α, which is consistent with figure 4. In particular, it is worthy noted that the thermal STT is comparable with the electric STT. In addition, we also study τ x versus ΔT at large E F (not shown), where the features of the STT are quite similar to these in figure 5(a), except that the magnitude of the STT is very small. In the discussion above we focus on comparing the electric STT with the thermal STT, so we do not consider the bias voltages and temperature gradients simultaneously. If both the bias voltages and temperature gradients are considered, the electric STT and thermal STT occur at the same time. In figure 5(b) we study τ x versus ΔT at various V b . With the increase of ΔT, due to the competition between the electric STT and its thermal counterpart the STT first decreases. When ΔT is further enhanced, the thermal STT begins to play a dominant role, so the STT increases monotonically with ΔT. We can also confirm the feature of the STT by analyzing τ x (E) and f L − f R . τ x (E) is almost antisymmetrical about E = 0 (see the inset of figure 3). As shown in the inset of figure 5(b) in the presence of V b f L − f R is redistributed and shifts to negative E and the antisymmetry about E = 0 does not hold, so τ x (E)( f L − f R ) is redistributed and its energy integration τ x depends on V b and ΔT.

Summary
In summary, we have theoretically investigated the electric and thermal STT across F 1 /N/F 2 graphene junctions. We compare the electric STT with the thermal STT. It is found that there exists two transport regimes I and II, where the currents originate from the electron to electron (I) and hole to electron (II) transport, respectively. The electric STTs originated from the two regimes have opposite sign and are reduced when both regimes are involved, which leads to a nonmonotonic bias dependence. However, the thermal STTs originated from the two regimes have the same sign and are enhanced by the coexistence of both regimes. Remarkably, the thermal STT is comparable with the electric STT. Furthermore, the electric and thermal STT can be manipulated by the Fermi level. The controllable STT obtained here suggests the ferromagnetic graphene junction ideal for future spintronics applications.