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 V_p(x, y) = V_p(−x, y) exhibits the largest value. We also find the pumping current with V_p(x, y) = V_p(x, −y) decreases with the increase of pumping amplitude while that with V_p(x, y) = V_p(−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), 1 where c_i ( ) annihilates (creates) an electron on site i of ZGNRs. 〈 · 〉 refers to the nearest-neighboring sites and t₀ is the nearest neighbor hopping energy which is set to be 2.7 eV.^[31] V_k(t) = V₀ + V_p cos(ωt + ϕ_k) with V₀ and V_p the amplitudes of the static and pumping potential, respectively. Here, ω is the pumping frequency and ϕ_k is the initial phase of the time-varying potential. Two pumping potentials with different initial phases should be introduced in order to generate the pumping current and the potentials are applied in different regions k (k = 1,2,3,4) as shown in Fig. 1(a).

Fig. 1.

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Fig. 1. (color online) (a) Schematic diagram of ZGNRs with two semi-infinite leads (orange shadow). Four pumping regions are represented by blue and green shadows in the central region. The origin of the axis is set to be the center of ZGNRs. (b) Band structure of ZGNRs with a width of 4.1 nm. The first subbands are highlighted with orange color and the arrows for different valley indices K and K′ denote the moving directions of electrons.

For the adiabatic electron pump, the average current flowing through lead α due to the slow variation of system parameter V_k in one period is given by^[15] 2 where τ = 2π/ω is the variation period of potential V_k and dQ_α is the pumped charge passing through lead α due to the change of system parameters which can be given by^[17,32] 3 Here, f is the Fermi–Dirac distribution and V_t = ∑_k V_k(t). The instantaneous retarded Green’s function G^r in real space is defined as 4 with Σ^r the self-energy due to the leads. is the line-width function of lead α.

By using the variable substitution, the integration of Eq. (2) can be presented as, 5 Therefore, the integration does not involve the frequency ω and we can choose ω = 1 in our calculation for simplicity.

By combining Eqs. (2) and (3), the pumping current can be written as^[17] 6 From the band structure of ZGNRs as plotted in Fig. 1(b), it has been reported that the momentum and valley index of electrons in the first subband are locked together which is independent of the ribbon width. As a result, the left-moving electrons in ZGNRs has valley index K while the right-moving electrons have valley index K′. Therefore, the pumping current with the pumping potential in the first subband of ZGNRs is fully valley polarized. The positive pumping current in the left (right) lead is a valley current with K (K′) index while a negative pumping current in the left (right) lead is contributed by the valley index K′ (K).

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 ,^[33,34] 7 where Γ_d denotes the on-site dephasing strength. Then the corresponding retarded Green’s function including the dephasing effect can be written as, 8

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. 1(a). The size of each pump region is set to be 1.2 nm × 5.9 nm and 1.0 nm × 5.9 nm for the symmetric and asymmetric ZGNRs, respectively.

Figure 2 presents the valley-polarized pumping current as well as the static transmission coefficient of symmetric ZGNRs as a function of the Fermi level E_f with a static potential V₀ = 0.2 V and a pumping potential V_p = 50 mV. Here the subscript kk′ denotes different pumping regions. We fix one region of k = 1 and change the other region with k′ = 2,3,4 to study the effect of spatial symmetry on the pumping current. The phase difference of two different pumping regions is set to be ϕ_kk′ = π/2. It is found that the transmission spectrum of ZGNRs with static pumping potentials shows two major dips which indicate that the transport process is dominated by the quantum resonant effect. As a result, the pumped current shows resonance characteristics near the resonant energy in the static transmission coefficient. For instance, all valley-polarized currents with different pumping regions present a peak at −0.076 eV where a sharp dip of transmission coefficient occurs. Such a resonance-assisted behavior of pumping current is a generic property of electron pump.^[36] Moreover, the valley-polarized current with pumping potential V_p(x, y) = V_p(−x, y) shows large values in the whole energy domain and the pumped current changes its sign at about −0.16 eV due to the transmission dip, while the valley-polarized current with the pumping potential V_p(x, y) = V_p(−x, −y) is relatively small and a broad peak is found around −0.16 eV without sign change. The pumped current with the pumping potential V_p(x, y) = V_p(x, −y) is smaller than although their transmission spectrum are almost the same.

Fig. 2.

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	Fig. 2. (color online) Valley-polarized pumping current and transmission coefficient of symmetric ZGNRs as a function of the Fermi level with a static potential V0 = 0.2 V and a pumping potential Vp = 50 mV applied on (a) region 1 and region 2, (b) region 1 and region 3, and (c) region 1 and region 4. The initial phase is set to be ϕ1 = 0 and ϕk = 2,3,4 = π/2.

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 V_p = 50 mV are calculated and plotted in Fig. 3. The valley-polarized current of asymmetric ZGNRs is similar to that of the symmetric case. For the valley-polarized current of asymmetric ZGNRs with the pumping potential V_p(x, y) = V_p(x, −y) and of asymmetric ZGNRs with the pumping potential V_p(x, y) = V_p(−x, −y), they become much larger than those of symmetric ZGNRs. This is in agreement with the very small saturated current in the symmetric ZGNRs due to the presence of conductance gap around the Fermi level.^[35] Especially, it is found that a sign change occurs at E_f = −0.185 eV for the valley-polarized pumping current .

Fig. 3.

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	Fig. 3. (color online) Valley-polarized pumping current of asymmetric ZGNRs as a function of the Fermi level with a static potential V0 = 0.2 V and a pumping potential Vp = 50 mV applied on different pumping regions. The initial phase is set to be ϕ1 = 0 and ϕk = 2,3,4 = π/2.

We then examine the relation between the valley-polarized pumping current of symmetric ZGNRs and the phase difference ϕ_kk′ of the pumping potentials at E_f = -0.076 eV. As shown in Fig. 4, a sinusoidal behavior is observed for with the pumping potential V_p(x, y) = V_p(−x, −y) while it slightly destroyed for with the potential V_p(x, y) = V_p(x, −y). The sinusoidal behavior cannot hold for which keeps almost zero within the phase difference of [−π, −(3/4)π] and [(3/4)π, π]. In previous studies, it is reported that the pumping current is proportional to the phase difference, namely, while this relation dose not hold for large pumping amplitude. In order to demonstrate this relation, we calculate the pumping current with a small pumping amplitude V_p = 10 meV, as plotted in the inset of Fig. 4. It is found that shows the sinusoidal behavior under a small pumping amplitude. Moreover, a general antisymmetry relation between the pumping current and the phase difference is found for all pumping regions, namely, I(ϕ_1k) = −I(−ϕ_1k). We also find that the valley-polarized current vanishes when the phase difference is nπ since two pumping parameters vary simultaneously to enclose a line at ϕ_1k = nπ. The valley polarization of the pumped current defined as is then calculated for , as shown in Fig. 4. The positive (negative) pumping current in the left lead presents a completely valley polarization with P = 1 (−1) since it is fully generated by the valley K (K′). Therefore, the pumping current exhibits a purely valley-polarized current I_K′ within the phase difference of [−π, 0) while it is fully contributed by the K valley within the phase difference of (0, π].

Fig. 4.

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Fig. 4. (color online) Valley-polarized pumping current of symmetric ZGNRs as a function of the phase difference ϕ1k with a static potential V0 = 0.2 V and a pumping potential Vp = 50 mV at Ef = −0.076 eV for different spatial symmetries. A factor of 100 and 10 is multiplied to and for illustration, respectively. The dashed blue line denotes the valley polarization of pumping current as a function of the phase difference. Inset: Valley-polarized pumping current as a function of the phase difference ϕ12 with a static potential V0 = 0.2 V and a pumping potential Vp = 10 mV at Ef = −0.076 eV.

The effect of the pumping amplitude V_p 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 E_f = −0.076 eV, as plotted in Fig. 5. The phase difference between two pumping potentials is set to be π/2. It is found that the valley-polarized current with pumping potential V_p(x, y) = V_p(−x, y) keeps almost zero when the pumping amplitude is smaller than 0.03 V. It starts to increase rapidly when V_p exceeds 0.03 V and reaches the maximum value at V_p = 0.044 V. Finally, it decreases gradually with the increase of pumping potential. For the pumping current , it first decreases slowly with the increasing pumping potential and then decreases linearly when the pumping amplitude exceeds about 0.56 V. The valley-polarized current increases almost linearly with the increase of V_p. In all cases, we find that the sign of the valley-polarized current does not change by the amplitude of pumping potentials.

Fig. 5.

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	Fig. 5. (color online) Valley-polarized pumping current as a function of the pumping potential Vp with a static potential V0 = 0.2 V and a phase difference of π/2 at Ef = −0.076 eV for different spatial symmetries. A factor of 100 and 10 is multiplied to and for illustration,respectively.

In order to consider the electron–phonon coupling in the ZGNRs, the dephasing effect is introduced in the parametric pump devices described in Fig. 2. Figure 6 presents the valley-polarized pumping current of symmetric ZGNRs with different dephasing strengths at the Fermi energy of −0.076 eV. We employ the B¨uttiker dephasing scheme in our present work and the dephasing is modeled by a single parameter Γ_d which can be considered as the average inelastic broadening parameter to describe the phase-relaxation effects. It is found that the valley-polarized current decreases as the dephasing strength increases and changes the sign at Γ_d = 0.1 eV while increases with the increase of Γ_d. For the pumping current with the pumping potential symmetry V_p(x, y) = V_p(−x, −y), it first increases and reaches the maximum at Γ_d = 0.06 eV. Then decreases to the minimum at Γ_d = 0.12 eV and it finally starts to increase with the increasing dephasing strength.

Fig. 6.

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	Fig. 6. (color online) Valley-polarized pumping current of symmetric ZGNRs as a function of the dephasing strength Γd with a static potential V0 = 0.2 V and a pumping potential Vp = 50 mV at Ef = −0.076 for different spatial symmetries. The phase difference is set to be π/2.

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 generated by pumping potentials V_p(x, y) = V_p(−x, y) is the largest one. Moreover, the valley-polarized currents with the pumping potential symmetry V_p(x, y) = V_p(x, −y) and V_p(x, y) = V_p(−x, −y) of symmetric ZGNRs are much smaller than those of asymmetric ZGNRs. The pumping current of symmetric ZGNRs decreases with the increase of pumping amplitude V_p while of symmetric ZGNRs increases with the increasing V_p. The dephasing effect is also studied in symmetric ZGNRs within the Büttiker dephasing scheme and we find that increases with the increasing dephasing strength Γ_d while decreases as Γ_d increases. Our theoretical results are helpful to generate and control valley-polarized currents in graphene-based valleytronic devices.