† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11704190, 11874221, and 11504240) and the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20171030).
We numerically investigate the valley-polarized current in symmetric and asymmetric zigzag graphene nanoribbons (ZGNRs) by the adiabatic pump, and the effect of spatial symmetry is considered by introducing different pumping regions. It is found that pumping potentials with the symmetry Vp(x, y) = Vp(−x, y) can generate the largest valley-polarized current. The valley-polarized currents
Valleytronics has attracted much attention in the field of condensed matter physics, which aims to manipulate and control the valley degree of freedom of electrons in addition to charge and spin in order to carry and process information.[1–14] In general, a local minimum (maximum) of the conduction (valence) band is referred to as the valley. Recently, valleytronics has been widely enabled with the rapid development in fabricating the hexagonal two-dimensional (2D) materials, such as graphene and monolayer transition metal dichalcogenides (TMDCs) whose electronic properties can be characterized by two inequivalent valleys at the band edge, namely, the K and K′ points in the first Brillouin zone.[1,2,6,12] In particular, a valley filter was first proposed in a ballistic point contact with one-dimensional (1D) zigzag graphene nanoribbons (ZGNRs) and the valley polarity can be inverted by a gate voltage on the point contact region.[1] The valley Hall effect that electrons in different valleys flow to opposite directions driven by an in-plain electric field was reported in graphene superlattices with broken inversion symmetry.[2,6]
One of the fundamental goals in valleytronics is the generation of pure valley current and valley-polarized current in 2D materials. Tarucha and coworkers detected the pure valley current in bilayer graphene by inducing the Berry curvature using a perpendicular electric field to break the spatial inversion symmetry.[15] The photoinduced valley-polarized current has been theoretically proposed in the monolayer TMDC by external electric field.[16,17] Moreover, the quantum parametric pump is an efficient way to generate dc currents through nanostructure by periodically varying two or more system parameters.[18–22] It has been widely used in many 2D materials.[23–27] A quantized charge pump based on massive Dirac electrons has been realized in graphene theoretically which is originated from the topologically protected edge states.[28] Recently, the pure bulk valley current was generated through pumping in a graphene mechanical resonator with cyclic deformations in the suspended region.[29] A pure valley current was also proposed by quantum pumping in a strain-engineered graphene in the presence of electric potential barriers and ferromagnetic fields.[30]
In this paper, the valley-polarized current in ZGNRs based devices generated by the adiabatic pump is studied. In order to study the effect of spatial symmetry of the pumping potential on the valley-polarized current, different pumping regions are considered. It is found that the pumped current induced by the pumping potentials with the symmetry Vp(x, y) = Vp(−x, y) exhibits the largest value. We also find the pumping current with Vp(x, y) = Vp(x, −y) decreases with the increase of pumping amplitude while that with Vp(x, y) = Vp(−x, −y) increases with the increasing pumping amplitude. Moreover, the dephasing effect from the electron–phonon coupling is studied for the valley-polarized pumping current by using the B¨uttiker dephasing scheme.
Previous theoretical studies on 1D graphene nanoribbons have shown that for armchair nanoribbons, each propagating mode is constructed by the mixed states of both valleys, while the lowest propagating mode of ZGNRs is only contributed by a single valley, which means that the valley index is not well defined for the armchair graphene nanoribbons.[1] Therefore, ZGNRs are chosen to study the valley polarization of current in our present work. In the tight-binding approximation, the Hamiltonian of ZGNRs can be written as (here we set ħ = e = 1 for simplicity),
For the adiabatic electron pump, the average current flowing through lead α due to the slow variation of system parameter Vk in one period is given by[15]
By using the variable substitution, the integration of Eq. (
By combining Eqs. (
In order to study the dephasing effect from the electron–phonon coupling, fictitious voltage probes are introduced into the graphene nanoribbons to model the influence of phase relaxing scattering proposed by B¨uttiker, which can be described by an on-site self-energy term
It is reported that the symmetric and asymmetric ZGNRs exhibit completely different transport behaviors under bias although they have similar band structure.[35] In our calculation, both the symmetric ZGNRs with even numbers of zigzag carbon chains and asymmetric ZGNRs with odd numbers of zigzag carbon chains are considered. The width and length of the symmetric (asymmetric) ZGNRs based device are 4.1 nm (3.9 nm) and 14.8 nm, respectively and the energy of the first subband of our proposed ZGNRs ranges from about −0.6 eV to 0.6 eV. In order to study the spatial symmetry of the pumping potential, four different pumping regions with indices k = 1,2,3,4 are considered, as shown in Fig.
Figure
In order to study the even–odd effect on pumping current of ZGNRs, the valley-polarized pumping current of asymmetric ZGNRs as a function of the Fermi level with V0 = 0.2 V and Vp = 50 mV are calculated and plotted in Fig.
We then examine the relation between the valley-polarized pumping current of symmetric ZGNRs and the phase difference ϕkk′ of the pumping potentials at Ef = -0.076 eV. As shown in Fig.
The effect of the pumping amplitude Vp on the valley-polarized pumping current is then studied. We calculate the pumping current of symmetric ZGNRs with different pumping potentials and a static potential V0 = 0.2 eV at the resonant energy Ef = −0.076 eV, as plotted in Fig.
In order to consider the electron–phonon coupling in the ZGNRs, the dephasing effect is introduced in the parametric pump devices described in Fig.
In conclusion, we have studied the valley-polarized current in symmetric and asymmetric ZGNRs based pump devices. Resonance characteristics are clearly observed for the pumping current near the resonant energy in the static transmission spectrum. Different pumping regions are considered in order to study the effect of spatial symmetry on the valley-polarized current. It is found that the valley-polarized current
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