The discovery and synthesis of new two-dimensional (2D) materials with hexagonal lattice structures, such as silicene,^[1–6] have greatly advanced the study of quantum physics, and can help overcome the difficulties associated with graphene to enable future applications in the next-generation electronic devices.^[7–22] Silicene, a silicon analogue of graphene, is a monolayer of silicon atoms instead of carbon atoms assembling in a honeycomb lattice and has been extensively explored both theoretically^[23–26] and experimentally.^[27,28] Silicene inherits some properties from graphene, such as electrons in silicene are linearly dispersed within the low-energy regime. Besides, silicene introduces advanced and distinguishing features thanks to its prominent buckled structure that the two inequivalent sublattices A and B are not coplanar, which allows us to adjust the Dirac mass by a perpendicular electric field (E-field)^[25,29–32] and brings some unique electronic transport properties.^[18,33–35] Therefore, the manipulation of Dirac electrons in the silicene systems will also be of great importance.

It is worthy of note that in the early years, those works on the physics of valleytronics^[36–39] have aroused considerable interest among researchers. The key issue in the exploitation of valley degree of freedom of electrons is the generation of pure valley current, which consists of two oppositely flowing distinct currents originated from the two inequivalent conical (Dirac) points K and K′ valleys in Brillouin zone,^[30,40–63] resembling the pure spin current.^{[7,8,24,63–83]}

On the other hand, quantum pumping^[58,84–90] is a useful technique which can generate dc currents without bias voltage. It uses two pumping parameters as ac signals to modify the electron transport of nanostructures and further alter the scattering characteristics of the device. When two ac signals are out of phase and their periods are longer than the time required for the electrons to traverse the device, a nonzero pump current is generated.^[86,91] Recently, Jiang et al. proposed to generate a pure valley current based on the quantum pumping effect by employing the lattice strain.^[40] However, that approach is technically complicated. In this regard, it is extremely desirable to have an alternative, which minimizes the technical and experimental requirements by using the time-dependent strain and E-field to directly generate the valley pump currents.

In this paper, we propose an uncomplicated method to generate pure valley pump currents in a silicene-based two-parameter pump device, in which the pumping sources are the time-dependent lattice strain^[92,93] and staggered potential generated by the time-dependent lattice deformation and perpendicular E-field, respectively. It is found that the strain and staggered potential are necessary for breaking the degeneracy of the K and K′ valleys and leading to a pure valley pump current. Due to the quantum interference effect, the valley pumped current can be modulated by controlling the pump parameters. Besides, when the Fermi energy lies in the bandgap opened by the pumping potentials, the pumped currents are maximized and quantized.

In this work, we consider a pump device based on silicene as schematically shown in Fig. 1. The silicene sheet is in the xy plane, in the spirit of an adiabatic quantum pump, two pumping regions characterized by two oscillating pumping potentials A(t) and Δ(t) = E_z(t)l generated from the time-dependent longitudinal strain and perpendicular E-field E_z(t) respectively are introduced, with l being the buckling height between sublattices A and B.^[95] The whole pump device is connected with the outside by the left (L) and right (R) leads without any bias. The lengths of both pumping regions are L along the current direction (the x axis). The transverse direction (the y axis) is assumed homogeneous so the system observes translational symmetry.

Fig. 1.

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Fig. 1. Schematic of the proposed silicene-based two-terminal pump device. The silicene sheet is in the xy plane; the two pumping sources, with L in length, are the strain A = A0 cos ωt and staggered potential Δ = Δ0 cos(ωt + ϕ) with a phase lag ϕ and a distance L0 between them. The device is connected to electrodes by the left (L) and right (R) leads, the pumped current is assumed along the x axis.

We shall analyze the character of the valley pump current in the adiabatic pumping process.^[94] To this end, we employ the low-energy continuum Hamiltonian^{[20,59–63,95,96]} of silicene, then we solve the time-dependent formulation using scattering matrix. The silicene system can be described by the following continuum model:

In the zero-temperature and adiabatic limit, the pumped current can be evaluated directly by adopting the BTP formula^[99]

With the aim of figuring out the scattering coefficients, the following combining method is preferred. The formulae of two required coefficients, r_± and , in two individual scattering events at A and Δ potentials are

The above three wavefunctions are respectively in the first three regions on the left of the device. The momenta of electrons in the normal and potential regions are given by and , respectively (ħv_F = 1). r₂ and t₂ at the second pumping potential Δ can also be obtained in a similar way. and can be determined afterwards by considering the injection from the right, conjugate to the process of Eq. (7).

Armed with the groundwork above, we can straightforwardly calculate the valley-resolved pumped current I_LK(K′) flowing through the L lead of the proposed pump device according to Eq. (3). Since the quantized pumped currents are the focus, which come up when the Fermi energy situates in the bandgap opened by A and Δ, the transverse momentum k_y does not quantitatively affect the results,^[98] we can just consider a pure one-dimensional case, k_y = 0.

Firstly, we plot the valley-resolved pumped current I_LK(K′) as a function of the pumping phase ϕ in Fig. 2. One can see that the pumped currents for the two valleys have the same magnitude but opposite directions, this implies that a pure valley current can be pumped.

Fig. 2.

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	Fig. 2. Valley-resolved pumped current ILK(K′) as a function of ϕ. Other parameters are E = 0.01 eV, A0 = 1 eV, Δ0 = 1 eV, L0 = 0, and L = 10a with a being the lattice constant.

Most importantly, I_LK(K′) deviates from the supposed sine-function behavior I ∼ sin ϕ of an adiabatic quantum parameter pump because the obtained pumped current is quantized as the Fermi energy |E| ≪ A₀ and Δ₀. Therefore, they finally show I_LK(K′) = ±e/T. The physics inside the quantization is the TIS^[98] forming between two time-evolving pumping potentials. Either strain or staggered potential will generate a QVHI phase in the pumping region,^[18] but this phase has no boundary or edge state. Thus the interface state with zero energy will emerge in the joint undoubtedly when sgn(A) = −sgn(Δ). This TIS, together with the Fermi energy situated in the bandgap, proves the quantization of the valley pump current.

Here comes to mind that the pumping action is a quantum interference effect in nature, the outcomes can be inevitably tuned by the dynamic phase of electrons φ^[83] in Eq. (6). Thus, one can also utilize either Fermi energy E or the length L₀ of the central region to control the pumping output.

Then we present the valley currents as a function of E and L₀ in Figs. 3 and 4, respectively. They both validate that the pumping currents are maximized and quantized when the Fermi energy E lies in the bandgap. It is also easy to get that the symmetry property of the structure plays an important role in obtaining the pure valley current in the whole energy region. When the two pumping potentials are symmetric about the central line of the structure, the Hamiltonians of the two valleys deliver the relationship I_K = −I_K′. So that one can obtain nonzero valley current^[40] I_v = I_K − I_K′. Similar to Fig. 2, I_L exhibits an abrupt reversal between the two quantized values ±e/T in Fig. 4. They confirm the effect of the dynamic phase of electrons in modulating I_L.^[98]

Fig. 3.

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	Fig. 3. Valley-resolved pumped current ILK(K′) as a function of the Fermi energy E. The lengths of the pump regions are L = 6a. Other parameters are A0 = 1 eV, Δ0 = 1 eV, and L0 = 0.

Fig. 4.

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	Fig. 4. Valley-resolved pumped current ILK(K′) as a function of the length of the central normal region L0. The lengths of the pump regions are L = 10a. Other parameters are E = 0.01 eV, A0 = 1 eV, and Δ0 = 1 eV.

Note that it is not strange to achieve valley pumped currents in devices, but the proposed model can yield an optimal pumping effect of valley pump currents, i.e., the valley pumped current is maximized for each transport model. Moreover, as discussed in Ref. [98], a larger transverse momentum k_y makes the propagating energy smaller and more likely to be located in the bandgap, so the quantization of pumping currents is more prone to appear. Here, we have discussed the ideal valley pumped currents in a silicene-based device by using the time-dependent elastic deformation and E-field which induce strain and staggered potential, respectively. Furthermore, these results also hold true in devices based on other 2D materials. However, the transport direction is crucial since the bandgap which is necessary for the quantized pump currents cannot exist along all directions. This is also the reason why we consider a one-dimensional pump device, only the perpendicular incident electrons encounter the energy gap due to the lattice deformation.

In summary, we have proposed to generate pure valley pump currents in a silicene-based device by cyclic modulation of time-dependent strain and staggered potential. We calculated the pumped currents in the one-dimensional case by setting the transverse momentum to zero and found them quantized when the Fermi energy resides in the energy gap opened by the pumping potentials. Furthermore, a pure valley pump current can be obtained by tuning the relevant parameters such as the phase lag and the dynamic phase of electrons. Our findings are helpful for the generation of pure valley pump currents in the Dirac-electron systems, the proposed structure can be applied in the realization of silicene valleytronic devices.