Bipolar and unipolar valley filter effects in graphene-based P/N junction

We use the spin and valley degrees of freedom to design the bipolar and unipolar valley filter effects based on the graphene-based P/N junction. When the modified Haldane model and staggered potential are applied on the region P, while the off-resonant circularly polarized light and staggered ferromagnetic exchange field are applied on the region N, the unipolar valley filter effect emerges with the unidirectional spin–valley current. The direction and type of the unidirectional spin–valley current depend on the phase of the modified Haldane model and the direction of polarized light. Other types of the bipolar valley filter effects are also reported, such as the valley-mixed bipolar spin filter effect, valley-mixed bipolar filter effect, valley-locked bipolar spin filter effect and valley-locked bipolar filter effect. These bipolar filter effects have the similarity that the spin–valley currents flow bidirectionally. These types of the unipolar and bipolar valley filter effects can be also mutually switched by modulating the external fields. Moreover, these unipolar and bipolar valley filter effects are robust against a weak temperature. This work reveals that the graphene-based junction has the potential applications in designing the valley filter device and improving the reprogrammable spin logic.


Introduction
Graphene, a two dimensional honeycomb crystal of the carbon atoms, has many remarkable features, such as electric transport, optical and topological properties [1][2][3][4]. In analogy to the spin degree of freedom leading to the emergence of spintronics [5][6][7], the valley degree of freedom is utilized to the application of valleytronics [8,9], where the bits of information is stored and manipulated. In the momentum space, the two degenerate and inequivalent valley indexes refer to the local maximum of the valence band and minimum of the conduction band in the first Brillouin zone [10][11][12][13]. The intervalley scattering is generally not considered due to the large separation between the two valleys [14][15][16][17]. In order to improve the functionalities in valleytronics, it has been realized in experiment that the electric, magnetic and optical approaches have been proposed to manipulate the valley degree of freedom [18][19][20]. Some theoretical models are also reported to manipulate the valley degree of freedom. For instance, an off-resonance circularly polarized light is applied on the epitaxial graphene to turn on or off a nonequilibrium valley current [21]. The strain-induced inhomogeneous pseudomagnetic fields acting oppositely on the two valleys are used to generate the valley polarized current [22,23].
In recent years, the valley caloritronics driven by a temperature difference has attracted much attention due to the presence of the valley degree of freedom [16,[24][25][26]. It is found that a longitudinal spin Seebeck effect can be driven near room temperature [26]. A valley Seebeck effect is also found, where the currents flow in the opposite direction with different valleys [16]. Then the valley-locked spin-dependent Seebeck effect (VL-SSE), where the carriers from only one valley could be excited by a thermal gradient, is put forward with the help of Seebeck effects above. Based on these types of the Seebeck effects, the valley filter effects under the bias are naturally proposed in ferromagnetic/antiferromagnetic junction, such as the bipolar-unipolar transition of the spin-diode effect [27] essentially regarded as a valley filter effect and a spin-valley filter effect [25].
The ideal of the valley filter effect has emerged [23,25,[28][29][30]. But there is still a huge space to explore the bipolar and unipolar valley filter effects. For instance, how many types of the bipolar and unipolar valley filter effects exist? How many spin-valley current forms are there for each bipolar or unipolar valley filter effect? Do the bipolar and unipolar filter effects exist whether in graphene or not? What a surprise is that we find four types of the bipolar valley filter effects exist such as valley-mixed bipolar spin filter effect, valley-mixed bipolar filter effect, valley-locked bipolar spin filter effect and valley-locked bipolar filter effect. And each bipolar filter effect has abundant spin-valley current forms. We only find one type of the unipolar filter effect, but eight spin-valley current forms of this effect have been found. We here suggest four types of the bipolar valley filter effect and one type of the unipolar filter effect in a graphene-based P/N junction on the basic of filling a vacancy of the valley filter effects. These bipolar and unipolar valley filter effects have different spin-valley current forms, which depend on the direction and type of external fields. For instance, the unipolar valley filter effect has eight spin-valley current forms consisting of its directions and types, which depends on the direction of polarized light and the phase of the modified Haldane model [31,32]. Therefore, our systematic and comprehensive findings of the valley filter effects have potential applications in improvement in future spintronics, valleytronics and reprogrammable spin logic.

Model and method
Our proposed graphene-based P/N junction is plotted in figure 1(a), the regions P and N are covered by the left and right charge batteries with different chemical potentials [33]. In the absence of the charge battery, the chemical potentials are both zeros. As we know, the chemical potential in the graphene without external field is zero. Actually, the external field proposed in our manuscript just modulate valence and conduction bands. And the electrochemical potential has the same effect on the spin-up and spin-down electrons [33].
With the modified Haldane model [31,32], the low-energy effective Hamiltonian of graphene without polarized light is expressed as The first term denotes the massless Dirac fermion, and the Fermi velocity reads ν F = √ 3 at/2 , where a is the lattice constant and t is the nearest-neighboring hopping energy. For simplicity, we set = v F = 1. η = ±1 stand for the valleys K and K , respectively. The second term is the staggered potential, which can be induced by an h-BN substrate [34,35] in graphene-based P/N junction. The third and fourth term are the ferromagnetic and anti-ferromagnetic exchange interactions. The details of experimental realization are described in the appendix A. The Pauli matrices τ i and σ i , with the index i = x, y, z, describe the sublattice pseudospin and electron spin, respectively. The fifth term is the modified Haldane model, t a 2 = −3 √ 3t 2 sin φ and t b 2 = −3t 2 cos φ. And the hopping t 2 is a next-nearest-neighbor interaction with a complex value t 2 e −iυ ij φ due to a periodic magnetic flux density, and a phase 2π(2φ a + φ b )/φ 0 where φ a is the flux in the region a and φ b is the flux in the region b. In addition, φ 0 = h/e is the flux quantum. The details of experimentally realizing the modified Haldane model are also presented in the appendix A. The last term is the staggered spin-orbit coupling [36][37][38]. And we assume the spin-orbit couplings between the nearest neighbors vanish following the reference [38].
When the monolayer graphene is exposed to a beam of off-resonant circularly polarized light of where the signs ξ = ±1 denote the right and left circularly polarized light, respectively, the Hamiltonian (1) can be rewritten as where F = ξ(eAv F ) 2 / ω. The detailed derivation of the last term in Hamiltonian (3) is presented in the appendix B. The corresponding Dirac equation is expressed as By solving equation (4), the corresponding eigenvalues read where the signs ± respectively denote the conduction and valence bands, and we define the variable S = 1 and −1 denoting the spin-up and spin-down modes, respectively. The corresponding wave functions of the graphene-based P/N monolayer, utilized to calculate the transmission probability, are expressed as where The x-direction wavevectors are given as k x = k cos θ i and q x = q cos θ t in the regions P and N, respectively. 2 are the wavevectors, where the external fields M P and M P1 applied on the region P are defined as M P = Sλ FM + ηt a 2 + t b 2 + Sηλ I and M P1 = Δ + Sλ AF + ηF, respectively. In the same definition, M N = Sλ FM + ηt a 2 + t b 2 + Sηλ I and M N1 = Δ + Sλ AF + ηF are defined as the external fields applied on the region N. According to the conservation component in the y direction, the transmitted angle θ t can be obtained by k sin (θ i ) = q sin (θ t ), θ t = arcsin k sin (θ i ) /q if incident and transmitted states both from either the conduction or valence bands; and θ t = π − arcsin k sin (θ i ) /q if incident and transmitted states from different bands.
The coefficients r and t can be obtained by matching the wave function at the interface x = 0 of the graphene-based P/N junction. The angle-dependent transmission coefficient is expressed as in which the real and imaginary θ t cases are both considered. For the real θ t case, equation (6) can be induced as According to the generalized Landauer-Büttiker transport approach, the spin-valley dependent currents read where N 0 = W π λ with the supposed spin-orbit coupling λ = 0.0039eV in order to make the factor k λ dimensionless, W is the width of graphene monolayer. In addition, We here define a constant I 0 = e h N 0 utilized as the reduced value of I S η .

Results and discussion
In the following, we present the bipolar and unipolar valley filter effects based on the graphene-based P/N junction shown in figure 1.

Valley-mixed bipolar spin filter effect
We consider a graphene-based P/N junction, where the ferromagnetic exchange field and staggered potential are presented in the region P, while the high-frequency circularly polarized light and antiferromagnetic exchange field are presented in the region N. The charge battery U [33] is used to modulate the chemical potential μ R in the region N depicted in figure 1. To better understand the reason why the VMB-SFE happens, we need to induce equation (11) as We also investigate how the temperature effects the VMB-SFE, it is shown that the VMB-SFE is robust against a weak temperature, the detail of which is shown in figure D1 in the appendix D.

Valley-mixed bipolar filter effect
We consider staggered spin-orbit coupling and potential in the region P, while antiferromagnetic exchange field and staggered potential are applied on the region N. We find the valley-mixed bipolar filter effect (VMB-FE) in the interval of −0.1 eV U 0.1 eV, where the spin-down valley current I ↓ K flowing from the left to the right for U < 0, and the spin-down valley current I ↓ K flowing from the right to the left for U > 0 ( figure 3(a)).
For  As we reverse the direction of the staggered potential in the region N, the band structure in figure 3(d) is switched into the band structure in figure 3(e). The corresponding spin-valley currents are depicted in figure 3(b), which is also called VMB-FE. These spin-valley current forms of the VMB-FE are summarized in table C2 of the appendix C. We also find that the VMB-FE is robust against a weak temperature not shown here, which is analogy to the VMB-SFE.  figure 4(e) as we modulate the direction of circularly polarized light from the right to the left, and the corresponding spin-valley currents shown in figure 4(b) are also called as the VLB-SFE. These spin-valley current forms of the VLB-SFE are summarized in table C3 of the appendix C. Moreover, the VLB-SFE is also robust against a weak temperature not shown here.

Valley-locked bipolar filter effect
The VLB-SFE shown in figure 4(a) will be switched into the valley-locked bipolar filter effect (VLB-FE) as we use the antiferromagnetic exchange field and circularly polarized light instead of the ferromagnetic exchange field and staggered potential in the region P, where in the interval −0.1 eV U 0.1 eV the same type of the spin-valley current I s η moving from the left to the right for U < 0 and moving from the right to  figure 5(b) is also the VLB-FE. We also further modulate the direction of the light applied on the region P to change the spin-valley current forms, it is shown that the spin-valley currents I ↓ K , I ↓ K in the region of −0.1 eV U 0.1 eV are also the VLB-FE. These spin-valley current forms of the VLB-FE are summarized in table C4 of the appendix C. We also find that the VLB-FE is robust against a weak temperature not shown here.

Valley unipolar filter effect
The VMB-SFE shown in figure 2(a) will be switched into the valley unipolar filter effect (VU-FE) shown in figures 6(a) and (b) as we consider the modified Haldane mode instead of the ferromagnetic exchange field in the region P, where in the interval of −0.1 eV U 0.1 eV the same type of the spin-valley current I s η moving from the left to right for U < 0 or moving from the right to the left for U > 0. In figure 6(a), only the spin-valley current I ↑ K moving from the left to the right for U < 0 in the interval of −0.1 eV U 0.1 eV. The induced formula I S η = e h E T S η (E) Δf ΔE is used to explain the VU-FE. For the region of −0.1 eV U < 0, the condition of T ↑ K = 0 under E < 0 is valid due to the specific spin-matching tunneling shown in figures 6(c) and (e) , and the condition of Δf = 0 under E μ L or E μ R and Δf > 0 under μ R < E < μ L is also valid. Therefore, the spin-valley current I ↑ K is positive in the region of −0.1 eV U < 0 shown in figure 6(a). For the region of 0 U 0 · 1eV, the condition of T ↑ K = 0 under 0 < E 0.1 eV due to the gap shown in figures 6(c) and (e) is valid, and the condition of Δf = 0 under E μ L is also valid. Thus, the spin-valley current I ↑ K is zero shown in figure 6(a). When we modulate the phase φ = −π/6 instead of the phase φ = 5π/6, the band structure in figure 6 (c) will be

Conclusion
In summary, we systematically and comprehensively investigate four types of the bipolar valley filter effects in the graphene-based P/N junction using the Landauer's formalism. Each type of the bipolar valley filter effect has abundant spin-valley current forms consisting of its direction and type, which depends on the direction of polarized light and another external fields. The unipolar valley filter effect having eight spin-valley current forms is also found in addition to the bipolar valley filter effects. These types of the bipolar and unipolar valley filter effects can also be mutually switched by changing the types of external fields. Moreover, the bipolar and unipolar valley filter effects are robust against a weak temperature. These results suggest that the graphene-based P/N junction is a perfect choice to design valley filter devices, where the four types of spin-valley current are independently separated. In addition, the switches between these types of the valley filter effects are also important for the improvement of reprogrammable spin logic based on the spin and valley degrees of freedom.
where F = ξ(eAv F ) 2 / ω is related to the amplitude of light field, which arises from the second term in equation (2).

Appendix C. Tables for different filter effect
In all the tables below, we consider the bias voltage region of U ∈ [−0.1, 0.1]. The units of all quantities except φ in the tables are set as eV. In the spin-valley current part, the sign + denotes the current propagating from the left to the right while the sign-denotes the current propagating from the right to the left; 0 denotes there is no corresponding current in the considered bias region.

Appendix D. Temperature on these valley effects
Here, we give two typical examples to discuss how the temperature affects the bipolar and unipolar valley filter effects. T L and T R are the temperature parameters in the regions P and N, respectively. In figures D1(a) and (b), the VMB-SFE is almost not broken in the region of k B T L 0.01eV, but when k B T L > 0.01eV, this effect are obviously broken shown in figures D1(c) and (d). We also investigate the effect of the temperature on the VU-FE. In the region of k B T L 0.01eV, the unipolar filter effect is almost steady shown in In summary, these filter effects almost keep steady for the region of k B T L 0.01eV, while for the region of k B T L > 0.01eV it will be obviously broken.