Spin-valley filter effect and Seebeck effect in a silicene based antiferromagnetic/ferromagnetic junction

The presence of the coupled spin and valley degrees of freedom makes silicene an important material for spintronics and valleytronics. Here we report a spin-valley filter effect in a silicene based antiferromagnetic/ferromagnetic junction. It is found that at zero Fermi level a valley locked bipolar spin filter effect is observed, where in a broad gate voltage range in one valley one spin (the other spin) electrons contribute to the current under the positive (negative) bias, but in the other valley the transport is forbidden. At the finite Fermi level a valley locked fully spin-polarized current can exist under both the positive and negative biases. Furthermore, at the high Fermi level by reversing the bias direction, the spin filter effect can switch to the valley filter effect. In addition, by changing the sign of the Fermi level, the spin polarization direction of the current can be reversed. If a temperature bias is applied, the spin-dependent Seebeck effect (SSE) always exists. With increasing the temperature bias, the system undergoes three regions: valley locked SSE, normal SSE and valley Seebeck effect. Moreover, by tuning the interlayer electric field, three phases: thermally induced valley locked spin filter effect, valley Seebeck effect and valley mixed Seebeck effect are observed.


Introduction
Silicene, the counterpart of graphene for silicon, has been successfully synthesized in the laboratory recently [1][2][3]. Similar to the graphene, conduction and valence bands in silicene form two inequivalent valleys K and K′ at the corner of the Brillouin zone [4]. The valley degree of freedom is similar to the spin one and provides another means to control electrons, leading to the so-called valleytronics [5][6][7][8][9][10][11][12][13]. Unlike graphene, silicene has an observable intrinsic spin-orbit interaction [14,15]. The band structure of silicene is spin and valley coupled, making silicene an important material for valleytronics and spintronics [2,3,[15][16][17][18][19]. The buckled structure allows us to control the bulk band gap of the Dirac electrons by applying external fields such as electric field [20], ferromagnetic/antiferromagnetic exchange field [15,18,19]. Although the spin-valley polarized current in silicene has been studied recently, most studies were focused on the conductance at zero bias, and very little attention [21] has been paid to the effect of the electric bias on the spin-valley resolved currents.
Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. triggered and controlled by an interlayer electric field, can be engineered near room temperature in a ferromagnetic/antiferromagnetic junction based on heavy group-IV monolayers. But they focused on the Seebeck magnetoresistance effect and did not discuss the spin-valley polarized currents in detail. Thus the detail studies of the thermally spin-valley polarized currents in a silicene based ferromagnetic/antiferromagnetic junction are still lacking.
Motivated by the works mentioned above, we study the spin-valley resolved currents in a silicene based antiferromagnetic/ferromagnetic (AF/F) junction under the electric bias and temperature bias, respectively. We first consider the current driven by the electric bias. It is found the behavior of the current is sensitive to the Fermi level. A valley locked (bipolar) filter effect can be seen at the low Fermi level. Furthermore, at the high Fermi level a valley filter effect can be demonstrated. Then the current induced by the temperature bias is discussed. With increasing the temperature bias, the system undergoes three regions: valley locked SSE, normal SSE and valley Seebeck effect. Moreover, by tuning the interlayer electric field, three phases: thermally induced valley locked spin filter effect, valley Seebeck effect and valley mixed Seebeck effect are observed.
The manuscript is organized as follows. We first introduce the system Hamiltonian and obtain the expression of the spin-valley resolved currents in section 2. Next we show the results of the spin-valley resolved currents under the electric bias and discuss the spin-valley filter effect in the AF/F junction in section 3. The spin-valley Seebeck effect is investigated in section 4. Finally, in section 5, we present a conclusion.

Model and formulation
We consider a silicene-based two dimensional AF/F junction with the interface located at x=0 under the electric bias V b or temperature bias T T T silicene is parallel to the x-y plane. The low energy effective Hamiltonian of such a structure can be expressed as [4,18] H H eV is the Hamiltonian without external fields, where λ so is the spin-orbit coupling strength. η=+(−) represents the K (K′) valley, σ=+(−) denotes the spin indices and v F is the velocity of electrons. In the second term lE v z l = with half of the interlayer distance l is the on-site potential difference between A and B sublattices, which can be efficiently tuned by the interlayer electric field E z .
x Q( ) is the Heaviside function. The third term corresponds to the AF exchange field with strength λ AF . The fourth term denotes the gate voltage V g in the AF lead. h in the fifth term stands for the ferromagnetic exchange field strength. The F and AF exchange fields can be induced by putting EuO, EuS or YIG on the top or bottom of a silicene sheet [18,42]. The last term indicates the longitudinal electric bias with strength V b . We set 1  = , v F =1 and e=1 for the brevity of notation. The eigenvalues of the Hamiltonian (1) are given by , which can be tuned by the parameters λ v and λ AF . In order to generate a finite current, the propagating modes in the leads should be generated, which requires that the energy E of the incident carriers should be satisfy Consider an electron with the energy E incident from the AF lead on the AF/F interface at an angle θ to the interface normal. With general solutions of equation (1), the wave functions in the AF and F leads are given by ) . Here the wave vectors are ) . From the wave function continuity at the interface we can obtain the transmission coefficient t. Then by using t the transmission probability for spin σ and valley η carriers is given Once the transmission probability T ησ is obtained, the current can be written as [34,35] with W the width of the silicene sheet is the carrier density of

Spin-valley filter effect
The gate voltage dependence of spin-valley resolved currents under zero Fermi level is plotted in figure 1(a) Next, spin-valley resolved currents versus V g at finite E F (E F =λ so ) is given in figure 1(b). Unlike figure 1(a), the valley-locked bipolar spin filter effect is destroyed. Here only spin up currents are allowed to transport. For the positive (negative) bias in the range of −1.0<V g /λ so <2.5 (−0.5<V g /λ so <3.0) I K¢ is nonzero, leading to a valley locked spin filter effect. The behavior of this effect can be understood from the band structures of this AF/F junction depicted in figure 2(c). In the AF the spin degree of freedom is degenerate, but the valley degeneracy is broken. The band structure of the K′ valley is gapless, while a band gap exists for the K valley. In the F for the low V g only spin up channel is open (see shaded region in figure 2(c)). Because of the specific spin-valley band-matching tunneling mechanism, a finite I K¢ presents. For high V g the electrons in the K valley begin to contribute to I K .
In figure 1(c) we further study the spin-valley resolved currents at E F =2λ so . Similar to the low Fermi level case (E F =0 or λ so ), under the negative bias only spin up currents flow, thus the spin filter effect still holds. However, unlike the valley locked (bipolar) spin filter effect at (E F =0) E F =λ so , under the positive bias there exhibits a valley filter effect instead of spin filter effect, where at the positive V g the currents in the K valley are allowed to flow but the transport in the K′ valley is blocked. In this case by reversing the bias direction the spin filter effect can switch to the valley filter effect. We can explain these phenomena as follows. Under the negative bias only the spin up channel in the F is open, leading to the spin filter effect. Different from the negative bias case, under the positive bias both spin up and spin down electrons contribute to the current, which destroys the spin filter effect. When V g is positive, the region responsible for the electron transport lies in the band gap of the K valley, thus a valley filter effect is seen. However, for the negative V g the region responsible for the electron transport lies in the conduction band of the K and K′ valleys, therefore the currents in both valleys are finite.
Since the spin-valley resolved currents are sensitive to the Fermi level, it is necessary to investigate the dependence of the currents on the Fermi level. As shown in figure 3, for low Fermi level at V g =0 a valley locked spin filter effect is found, where for the positive Fermi level I K¢ is finite, while by reversing the sign of the Fermi level I K¢ converts into I K¢ . For high Fermi level, the currents in the K valley appear. Therefore we can obtain a valley locked fully spin polarized current with its spin direction depending on the sign of the Fermi level. It is noted that this valley locked spin filter effect can be also found in the system under the negative bias (not shown here). For the low V g we can see the valley locked spin filter effect too.

Spin-valley Seebeck effect
Now we turn our attention to the effect of the temperature bias ΔT=T L −T R on the spin-valley resolved currents. As shown in figure 4(a) olds, leading to a pure spin (valley) current. This is different from the system under the electric bias, where the valley (spin) current accompanying with a finite charge current is generated. Next, in figure 4(b) thermally induced spin and valley currents through the AF/F junction are given. Here λ v in the left AF is fixed at λ vL =0, while that in the right F is set to be λ vR =0.5λ so . It is found that I  and I  flow in opposite direction, so SSE always exists. With increasing ΔT, three regions: I, valley locked SSE, where I K¢ and I K¢ counterpropagate along the temperature bias direction (see the inset of figure 4(c)), while no carriers from the K valley are excited; II, normal SSE, where I  and I  propagate in opposite direction but both I K and I K¢ flow in the same direction; III,  Last, it is necessary to discuss the robustness of this spin-valley filter effect and valley Seebeck effect on some realistic situations. First, we consider the bias V b changes abruptly across the interface. In fact because our results depend on the band structures of the AF and F leads, we can obtain the same results in the AF/barrier/F junction, where the bias voltage drop linearly in the middle normal barrier region [44,45]. Second, a real silicene sample inevitably contains atomic defects in the bulk. As discussed in [36][37][38][39] the features reported here can be held when the defect ratio is less than 7.5%. Third, due to the weak Rashba spin-orbit interaction, we neglect its influence on the carriers transport. Last, although silicene is considered here, the results can be observed in other heavy group-IV monolayers, such as germanene and stanene. In germanene and stanene based junction a large ferromagnetic exchange field and higher bias are required.

Summary
In summary, we study the spin-valley resolved currents under the electric bias and the temperature bias, respectively. For zero Fermi level in a broad gate voltage range due to specific spin-valley band-matching tunneling mechanism I K¢ (I K¢ ) flows under the positive (negative) bias, but in the K valley the transport is   figure 4(d) in the AF the black (blue) lines correspond to the K (K′) valley, while in the F the red (blue) lines correspond to the spin up (spin down) channel. The other parameters are λ vL =0 , λ AF =λ so , λ vR =h=0.5λ so , and T R =5 K. the sign of the Fermi level, the spin polarization direction of the current can be reversed. When the system is under a temperature bias, SSE always exists. With increasing the temperature bias, the system undergoes three regions: valley locked SSE, normal SSE and valley Seebeck effect. In addition, by increasing λ vL , three phases: thermally induced valley locked spin filter effect, valley Seebeck effect and valley mixed Seebeck effect are shown. Although the silicene-based AF/F junction is investigated here, the results can be observed in other heavy group-IV monolayers, such as germanene and stanene. Our results open opportunities for fabricating valleytronics and spin caloritronics devices based on silicene.
Note added. After we finish our manuscript, we became aware of a related work by Zhai et al [46]. In their work they found a valley-mediated and electrically switched bipolar-unipolar transition of the spin-diode effect in Heavy Group-IV Monolayers. But they did not consider the effect of the gate voltage and the Fermi level on the valley locked spin filter effect and the spin-valley Seebeck effect, which is focused on in our present work.