Abstract
We report a theoretical study on the valleyfilter and valleyvalve effects in the monolayer graphene system by using electrostatic potentials, which are assumed to be electrically controllable. Based on a lattice model, we find that a single extremely strong electrostaticpotential barrier, with its strength exceeding the hopping energy of electrons, will significantly block one valley but allow the opposite valley flowing in the system, and this is dependent on the sign of the potential barrier as well as the flowing direction of electrons. In a valleyvalve device composed of two independent potential barriers, the valleyvalve efficiency can even amount to 100% that the electronic current is entirely prohibited or allowed by reversing the sign of one of potential barriers. The physics origin is attributed to the valley mixing effect in the strong potential barrier region. Our findings provide a simple electric way of controlling the valley transport in the monolayer graphene system.
Introduction
Recently, the valley transport in 2D graphenelike materials has attracted much attention of researchers, because it is expected that the valley degree of freedom of electrons can exert the same effect as the electron spin in carrying and manipulating information^{1,2,3}. This newly rising discipline is referred to as the valleytronics in a much similar way to spintronics. In graphene, the valley degree of freedom comes from the fact that the six corners of the hexagonal Brillouin zone are divided into two inequivalent groups, labeled as the K or K′ valley. These two valleys are related by the timereversal symmetry and can be transformed into each other by spatial inversion operation. Due to much momentum difference between the two valleys, the intervalley scattering is suppressed^{4,5,6,7} in clean graphene samples and valley is largely a conserved quantum number in electron transports.
There were tremendous works devoted to the valleytronics field, especially, after several research groups have measured and confirmed valley currents driven by the Valley Hall effect in the monolayer^{8} or bilayer graphene^{9, 10} system. At present, the production and measurement of an imbalance of valley carriers are still the principal tasks in this field, since the valleytronics is still in its infancy. A lot of proposals in the literature were put forward to generate valleypolarized currents by using graphene nanoribbon/nanoconstriction^{11, 12}, electromagnetic or optical field^{13,14,15,16,17,18,19,20,21}, and line defects^{22,23,24}, as well as lattice strain^{25,26,27,28,29,30,31,32,33,34}.
Since graphene has excellent flexibility and the lattice deformation can bring about the opposite pseudogauge potential or magnetic field for two valleys^{34}, the lattice strain is an ideal method to affecting the valleydependent transport of electrons. For example, Settnes et al.^{35} has recently proposed to use the nanobubbletype lattice deformation in graphene to filtrate and split valleys, and showed that some concrete lattice deformation makes valley polarization of electrons quite high or valley completely splitted in real space. Milovanović and Peeters^{36} studied the same straininduced bump structure in a graphene ribbon system and found an effective valley filter phenomenon under some special parameters. Certainly, the accurate control of strains in graphene is a great challenge in order to obtain a special pseudomagnetic field. The ideal way of controlling the valley degree of freedom in experiment should be an electric one like that in spintronics.
Given the fact that the valley in graphene is defined in the momentum space with the electron energy around the Dirac points, one can see that the valley definition is no longer valid if the electron energy is far from the Dirac points. i.e., the valley will be severely mixed when the electron transport occurs at this energy level. According to this inference, we study the possible valleyfilter and valleyvalve effect in monolayer graphene modulated by extremely strong electrostaticpotential barriers, which are assumed to be constructed by gate voltage. It is shown that the potential barrier almost blocks one valley but allows the opposite valley passing through, and the filtering efficiency is quite high even amounting to 100%. The filtered valley is dependent on the sign of the potential barrier and the transport direction of electrons. It is also found that the two opposite potential barriers can bring about an efficient valleyvalve effect similar to the GMR effect in the spintronics field. This method by using the electrostatic potentials to controlling valleys does not involve any external magnetic field or material, and is favorable to experiment observation.
Model and Method
We first consider a simple twoterminal device in Fig. 1(a), where an electrostaticpotential barrier V _{0} is constructed in pristine graphene and connects directly with the left and right graphene leads. The barrier length is L and its shape is set as a rectangle one, which does not bring any qualitative effect on our results. Since we focus on the valley filter and valve effects, which are induced by the possible valleymixing effect in the strong potential barrier, a lattice model is employed to describe the system
where the first term stands for pristine graphene, 〈ij〉 denotes the nearest neighboring sites, \({c}_{i}^{\dagger }({c}_{i})\) is the creation (annihilation) operator at the site i, and V _{0} is the onsite energy representing the barrier region, which is assumed controllable by gate voltages. The spin degree of freedom of electrons is omitted here and the graphene leads are absent of any interaction of electrons. The valleydependent transmission T ^{ττ′} at the Fermi energy E is given by
where τ, τ′(=K, K′) is the K or K′ valley index. T ^{ττ′} stands for the transmission coefficient of the τ′ valley electrons in the left lead that are transformed into the τ valley in the right graphene lead. \({G}^{r}={[E{H}_{c}{{\rm{\Sigma }}}_{L}^{r}{{\rm{\Sigma }}}_{R}^{r}]}^{1}\) is the retarded Green’s function of the device, and H _{ c } is the Hamiltonian of the barrier region, \({{\rm{\Sigma }}}_{L(R)}^{r}\) is the left (right) selfenergies of graphene leads with \({{\rm{\Sigma }}}_{L,R}^{r}={{\rm{\Sigma }}}_{L,R}^{rK^{\prime} }+{{\rm{\Sigma }}}_{L,R}^{rK}\) and \({{\rm{\Gamma }}}_{L,R}^{\tau }=i({{\rm{\Sigma }}}_{L,R}^{r\tau }{[{{\rm{\Sigma }}}_{L,R}^{r\tau }]}^{\dagger })\), i.e., the left and right selfenergies consists of the K and K′ dependent selfenergy \({{\rm{\Sigma }}}_{L,R}^{rK}\) and \({{\rm{\Sigma }}}_{L,R}^{rK^{\prime} }\). The Green function \({G}_{LR}^{r}\) is calculated by usual recursion method, while the selfenergy of the graphene lead can be constituted by the left or right propagating electron eigenfunctions at the Fermi energy. It is assumed that H _{0} is the Hamiltonian of a unit cell of the uniform graphene lead, and the hopping matrix between the neighboring cells is H _{ LR } as well as its hermite conjugate H _{ RL } = [H _{ LR }]^{†}. Then the eigenfunction χ〉 satisfies
where H(k) is the Bloch Hamiltonian, k is the wavevector, and a is the lattice constant. The energy dispersion of electrons in the graphene lead is worked out by diagonalizing H(k). One can follow the method in ref. 37 to obtain the leftgoing (+) or rightgoing (−) valleydependent wavefunction \({\chi }_{m}^{\tau }(\pm )\rangle \) at the Fermi energy E with its corresponding wavevector \({k}_{m}^{\tau }(\pm )\), where m = (1, …, M) and M is the matrix dimension of the unit cell H(k). Afterwards, the left and right propagation matrices F ^{τ}(±) can be directly constructed as
where \({U}^{\tau }(\pm )=({\chi }_{1}^{\tau }(\pm ),\ldots ,{\chi }_{M}^{\tau }(\pm ))\) is a matrix from the eigenfunction \({\chi }_{m}^{\tau }(\pm )\rangle \) at the Fermi energy E. According to these propagation matrix, one can build directly the valleydependent selfenergies of the left and right leads as
and
The electron transporting is assumed along the zigzag edge of graphene as shown in Fig. 1(a), because in this case, the wavefunctions of electrons are clearly valleyseparated, i.e., the propagating wavevectors of two valleys are different and one can easily construct the valleydependent selfenergy of leads. Notice that we take an periodic boundary condition along the transverse direction in order to avoid the zeroedge state in the zigzag nanoribbon system and simulate a very large graphene system. The energy band of the graphene lead is shown in Fig. 1(c), where the two Dirac points explicitly denotes two valleys at K = 2π/3a and K′ = −2π/3a. At E > t, there is no clear definition of valley and thus, the valleydependent scattering will occur. The energy diagram of the armchairedge graphene is also plotted in Fig. 1(d), where the valley is shown degenerate around Dirac points. This implicates that the valley transport should heavily depend on the propagation direction of electrons in the graphene lattice, e.g., along the zigzagedge or the armchair edge. The Kvalley (K′valley) transmission of electrons is defined as T ^{K} = ∑_{ τ } T ^{Kτ} (T ^{K′} = ∑_{ τ } T ^{K′τ}), so the valley filtering efficiency can be represented by the dimensionless transmission as
A valleyvalve device similar to the spinvalve one is also considered as schematically shown in Fig. 1(b), where two opposite potential barriers are put onto the monolayer graphene, which may be regarded as an antiparallel configuration. Similarly, the device is termed as the parallel configuration when the two potential barriers have the same sign. The valleyvalve efficiency (vve) is defined as the difference between the conductances of these two configurations.
where T _{ p,ap } = T ^{K} + T ^{K′} is the total transmission of electrons in the parallel (antiparallel) structure of the valleyvalve device. Certainly, vve should critically depend on the efficiency of the valley filtering effect in a single barrier region which is relied on V _{0}.
Results and Discussions
In our calculations, we take the hopping energy t = 1 as the energy unit, the temperature as zero T = 0 K, and the Fermi energy as E = 0.1t. The transverse width of the device is set as 2048 atoms in a unit cell amounting to 220 nm or so with the lattice constant a = 2.44 Å.
We first present the electron transmission in the simple twoterminal device in Fig. 2(a), where T ^{K} and T ^{K′} is plotted as a function of the barrier strength V _{0}. It is shown that the electrostatic potential \({V}_{0}/t\mathop{ < }\limits_{ \tilde {}}0.2\) near the Dirac point E = 0 does not lead to serious valley splitting of the electron transmissions. When V _{0} is far from the Dirac point significantly, the valley splitting becomes conspicuous. At \({V}_{0}/t\mathop{ > }\limits_{ \tilde {}}1\), one valley transmission exceeds plumb the other one. As a result, the device functions as a valley filter and this is clearly shown in Fig. 2(b). The valley filter efficiency can amount to as high as \({\eta }_{K({K}^{^{\prime} })}\sim \mathrm{94 \% }\). The clear oscillations come from the resonant transmission of electrons through the rectangle potential barrier V _{0} region.
In Fig. 2, the efficient valley filtering effect becomes evident just at \({V}_{0}/t\mathop{ > }\limits_{ \tilde {}}1\) and keeps almost unchanged afterwards. This stems from the fact that when the electron energy meets E > t in Fig. 1(c), the valley degree is no longer valid, and more importantly, the transport modes are shrunk by half because of no valley degeneracy. This immediately indicates that one valley species of electrons incident from leads will be blocked, which is certainly determined by the wavevector match. Therefore, for the electrons incident from the opposite graphene leads, the opposite valley is blocked. This is also the requirement of the timereversal symmetry. In addition, for a weak V _{0}, there is no valley (mode) shrinkage so as to no valley filter effect.
Since our studied system has the timereversal symmetry, we have \({T}^{K}({V}_{0},E)={T}^{{K}^{^{\prime} }}({V}_{0},E)\) and T ^{ττ′} = T ^{τ′τ}. The curves are not rigorously symmetric upon V _{0} = 0 in Fig. 2, because the Fermi energy E = 0.1t is fixed in numerics. Certainly, for the strong barrier case \({V}_{0}\gg E\), the symmetry is recovered. Moreover, the opposite sign of potentia barrier V _{0} will lead to opposite filtering effect. In other words, reversing the sign of V _{0} will lead to the opposite valley filter effect. Additionally, the K and K′ valley electrons have the opposite wave vectors, so the leftgoing K′ valley would be blocked by the potential barrier V _{0} if the rightgoing K valley is prohibited by the same barrier potential.
The single potential barrier V _{0} can filtrate valley and the efficiency is quite high, but it is not perfect, since the intervalley scattering due to V _{0} cannot be depressed entirely and the symmetry of T ^{ττ′} = T ^{τ′τ} persists. A superlattice structure consisting of multiple V _{0} barriers should in principal enhance the efficiency further. In Fig. 3, the two and fourbarrier superlattice devices resembling that in Fig. 1(b) are studied and η _{ K(K′)} is shown. One can see that the overall efficiency η is enhanced and the maximum efficiency even arrives at 100% in the case of the fourbarrier device in Fig. 3(b). This is attributed to the resonant tunneling effect. Certainly, the dips in those curves are also strengthened. The curves are not as smooth as ones in Fig. 2, because several different oscillating periodicities superposition together from either the barrier length, L, or the distance between the two barriers, L _{0}.
Based on the above results, it is naturally envisaged that two opposite barriers consisting in a doublebarrier device should block the current totally, which is similar to the GMR effect. This opposite barrier structure can be dubbed as the antiparallel configuration, while the two same barriers V _{0} is the parallel configuration. The latter structure shall allow one valley electrons flowing and thus, its conductance should be much sizable in comparison to the former one. Similar to the spinvalve effect, we calculate the conductances of these two configurations and present the valleyvalve efficiency vve in Fig. 4. It is shown that the transmission T _{ p } of the parallelconfiguration device is mostly larger than that of the antiparallel one (T _{ ap }) in Fig. 4(a). At some resonances, however, T _{ ap } exceeds T _{ p }, and the valleyvalve effect is not ideal as expected. This is clearly shown in Fig. 4(b). The exact reason is the coherent transport and the resonant tunneling, which accounts for such oscillating behaviors.
In a realistic device, there generally exist some interactions like disorders and electronphonon coupling, which will smear the phasecoherent transport and may improve the valleyvalve effect provided that the intervalley scattering does not severely occur in the process of the incoherent scattering. In our calculations, we simply consider a zero distance between two opposite barriers of Fig. 1(b) (L _{0} = 0) in order to diminish the resonant tunneling effect. The results are shown in Fig. 5, where the parameters are the same as those in Fig. 4 except for L _{0} = 0. It is shown that T _{ p } > T _{ ap } is valid in nearly all V _{0} range (\({V}_{0}/t\mathop{ > }\limits_{ \tilde {}}1\)), T _{ ap } are entirely suppressed in Fig. 5(a), which in turn leads to a saturated valleyvalve efficiency vve = 1 in Fig. 5(b).
We also consider a fourbarrier superlattice structure as that studied in Fig. 3(b), which is actually a parallel structure. While the antiparallel configuration is defined as V _{0}/−V _{0}/V _{0}/−V _{0} with an alternative V _{0} serial. Both nonzero and zero L _{0} cases are calculated and shown in Fig. 6(a,b). Similarly, the efficiency vve shows a strong resonant tunneling feature when L _{0} is nonzero in Fig. 6(a), i.e., the conductance of the parallel structure does not always exceed the antiparallel one and the peaks and dips are strengthened. Whereas for the case of L _{0} = 0, a saturated value vve = 1 appears again at \({V}_{0}/t\mathop{ > }\limits_{ \tilde {}}1.1\) in Fig. 6(b), i.e., the current is totally blocked in the antiparallel structure, which is the same as that in Fig. 5(b) again.
In above numerics, the rectangle profile of the potential barrier was employed, however, other continuous and smooth functions of V _{0} were also computed but showed no qualitative influence on our obtained results. Actually, a real factor influencing the filtering effect is the transport direction of electrons. As stated earlier, the electron transport is assumed along the zigzag edge and there is a valley (mode) shrinkage phenomenon at E > t as shown in Fig. 1(c), which is necessary to bring about the valley filter and valve effect. When the transport direction along the armchair edge is considered, the valley is almost degenerate whatever V _{0} is taken, and there shall be no valleyfiltering or valleyvalve effect. Since the doping level of pristine graphene was proved to be changed easily by gate voltages^{38}, it is not difficult to observe our proposal of the valleyfilter or valleyvalve effect.
Conclusion
In summary, we have proposed a simple method to filter valley in the monolayer graphene system by introducing an extremely strong potential barrier. It is shown that the potential barrier can block one valley but allow the opposite valley tunneling through it, which is dependent on the sign of the barrier as well as the current direction. The valley filtering efficiency can be further enhanced in a multibarrier superlattice structure. The valleyvalve effect was also studied and the device conductance can be significantly controlled by reversing the barrier sign, when the distance between barriers are short enough for suppressing the resonant tunneling effect. These findings are dependent on the electron transport being along the zigzag edge direction of the graphene lattice. Our findings may shed light on fully electric controlling of the valley transport in graphene.
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Acknowledgements
The work described in this paper is supported by the National Natural Science Foundation of China (NSFC, Grant Nos 11204187, and 11574045).
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School of Physics, Southeast University, Nanjing, 210096, China
 Juan Juan Wang
 , Su Liu
 & Jun Wang
Department of Physics, South University of Science and Technology of China, Shenzhen, 518055, China
 JunFeng Liu
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Contributions
J.F.L. and J.W. conceived the study. J.J.W. and S.L. performed the numerical calculations. J.J.W. wrote the main manuscript text. All authors contributed to discussion and reviewed the manuscript.
Competing Interests
The authors declare that they have no competing interests.
Corresponding authors
Correspondence to Jun Wang or JunFeng Liu.
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