Controllable valley filter in graphene topological line defect with magnetic field

The extended line defect of graphene is an extraordinary candidate in valleytronics while the high valley polarization can only occur for electrons with high incidence angles which brings about tremendous challenges to experimental realization. In this paper, we propose a novel quantum mechanism to filter one conical valley state in the line defect of graphene by applying a local magnetic field. It is found that due to the movement of the Dirac points, the transmission profiles of the two valleys are shifted along the injection-angle axis at the same pace, resulting in the peak transmission of one valley state being reduced drastically while remaining unaffected for the other valley state, which induces nearly perfect valley polarization. The valley polarization effect can occur for all the incident angle and plays a key role in graphene valleytronics.


Introduction
Graphene, a single atomic layer of graphite, exhibits peculiar electronic structure [1,2]. The valence and conduction bands of graphene touch each other at the two inequivalent corners of the hexagonal Brillouin zone, the K and K ′ points. The lowenergy electron around the two points can be described by the massless Dirac equation. The two Dirac points, called K and K ′ valleys, were suggested as an information carrier and played an important role in the recently emerged valleytronics [4][5][6].
The polycrystalline which is composed of different singlecrystal grains separated by grain boundaries, always appear in two dimension (2D) samples synthesized at large scale, such as graphene [7,8] and MoS 2 [9]. The polycrystalline has signi cant applications in valleytronics and has received considerable attention in recent years [10][11][12]. Nguyen found that the two valleys can be separated in different directions in the presence of strain in polycrystalline graphene [10]. Obviously, this is the inevitable consequence when the two Dirac points move in opposite directions in the momentum space under strain. However, an interesting question arises naturally: what will happen when the two Dirac points move in the same direction in the momentum space in such polycrystalline graphene?
One particular type of polycrystalline graphene, as learned from electrons, is the extended line defect containing carbon atom pentagons and octagons. Many peculiar electronic characteristics have been explored theoretically due to the unique structure of this defect [11,[13][14][15][16][17]. Furthermore, such extended linear defect has been experimentally observed on a graphene layer grown on a metallic substrate [18]. Gunlycke and White pointed out that a valley polarization of nearly 100% can be achieved by scattering off a line defect [11], attracting intense interest in the physics community. However, perfect valley polarization can only appear for the electrons with large incident angles, and the ef ciency is reduced or even completely disappears when an electron is transmitted through the line defect perpendicularly. This effect indicates that the electron must always follow the direction of the line defect to maintain a high valley polarization, bringing about major challenges for the experimental studies of this phenomenon. Not con ned to the line defect, the dilemma also exists in other polycrystalline graphene [10] and this is also why the strain should be taken into account.
As an effective means of eld regulation and control of valley polarization, the function of the magnetic eld has been widely explored. For instance, a quantizing magnetic eld generating the valley-polarized quantum Hall state in the graphene p-n junction has been investigated in theoretical studies [19][20][21][22][23][24] and con rmed in experiment with a high magnetic eld [25]. Of course, the valley-polarized Landau level can be created under an even stronger magneitc eld [26]. In addition, the magnetic eld can also be used to realize the valley polarization by introducing the vector potential to break the symmetry of the two valleys [27,28] or coupling with the orbital magnetic moment in a system with broken inversion symmetry to generate the valley-dependent Zeeman interaction [29]. However, a surprising function related to the magnetic eld is rarely mentioned: the transmission pro les are de ected to one side along the incidence angle axis due to movement of the Dirac points in the magnetic eld, and the peak transmission is shifted to the left or right of the normal axis depending on the direction of the magnetic eld [30][31][32]. It is expected that this phenomenon will signi cantly in uence the transmission characteristics of the valley states in the line defect because the transmission coef cient is dependent of the incidence angle, the peak transmission of one valley is situated at one end of the transmission image while situated at the opposite end for the other valley, and the transmission enhances (weakens) from 0(1) to 1(0) as the angle varies through the transmission image for the two valleys [11,19]. Currently, the research focus of experimental [33] and theoretical [16,34,35] attention in this eld are mainly on the quantum Hall effect (QHE) [33][34][35] and the QH boundary states [16] under a perpendicular magnetic eld, whereas the manipulation of the valley state with the magnetic eld is still in its infancy.
In this paper, we propose a simple technique to ltrate one valley state in the line defect of graphene with a magnetic eld. Interestingly, it is found that, in the presence of a perpendicular magnetic eld, the transmission curves of the two valleys shift rightward together with the same pace, resulting in the peak transmission of the K valley with unit transmission shifts along the incidence angle axis, while that of the K ′ valley diminishes rapidly. Therefore, the transmission of the K ′ valley is enormously restrained, and nearly 100% valley polarization can be realized. The physical mechanism of this phenomenon is that due to the movement of the two Dirac points in the momentum space, the eject angle of the K valley state through the scattering region can always maintain the preferred angle (−π/2), whereas the outgoing angle of the K ′ valley is far from the preferred angle (π/2) due to the momentum conservation. This scheme provide a reliable way to realize a controllable valley polarization in experiment with a weak magnetic eld. Moreover, we also provide a convenient technique to adjust the incidence direction of the valley electrons by magnetic eld, similar as the light in optics.

Model
We model the electronic structure of the graphene line defect shown in gure 1 by a nearest-neighbor tight-binding Hamiltonian [17,36,37]: where c † i and c † i y ,γ/δ represent the electron creation operator at site i and the line defect, respectively. Here, ǫ i in the rst term of equation (1) is the on-site energy (i.e., the energy of the Dirac point), which can be controlled experimentally by the gate voltage, and w i is the random disorder potential uniformly distributed in the interval w i ∈ [−W/2, W/2]t, with W being the disorder strength. The second term in equation (1) represents the nearest-neighbor interaction in pristine graphene, the third term denotes the interaction between the two atoms in the line defect and the fourth one is the hopping term between the defect atom and its nearest neighbor in pristine graphene. t, τ 1 and τ 2 represent different nearest-neighbor hopping energies, as shown in gure 1(c), and · · · runs over all the nearestneighbor hopping sites. This type of defect conserves the coordination number, which implies variations less than 5% in t, suggesting that t values could be considered nearly identical and it is reasonable to set τ 1 ≈ τ 2 ≈ t (for which t ≈ 3.1 eV) [17,38]. In the presence of a perpendicular magnetic eld B, a Peierls phase factor φ i j = 2π j i A · d l/φ 0 should be added in the hopping interactions with the vector potential (A x , A y ) = (0, Bx), where φ 0 = /e, and the magnetic length . It is supposed that the magnetic eld is applied adjacent to the line defect with a certain width L in units of √ 3a/2, where a(a = 0.246 nm) is the graphene lattice constant, as shown in gure 1(a).
The line defect of graphene is shown in gure 1(a), which extends immensely along the y direction. The translational symmetry of the lattice structure along the y direction indicates that k y is a conserved quantity and that the creation (annihilation) operators can be rewritten as follows, according to the Fourier transformation: Then, the Hamiltonian matrix in equation (1) is decoupled into H = k y H k y , and H k y can be described in the following form: , andx represents the unit length between the neighboring supercells at the graphene part. Here, i represents the position of a supercell, γ takes the integer number from 1 to 4 denoting the different columns in a supercell, and 1/2 in c † k y ,i,γ,1/2 corresponds to the up/down site in the same column in gure 1(c).
We also explore the non-equilibrium Green's function technique to calculate the transmission coef cient [37], where G r (G a ) is the retarded (advanced) Green's function related to the line-defect Hamiltonian and Γ L/R = i(Σ r L/R − Σ a L/R ) with the retarded/advanced self-energy Σ r L/R /Σ a L/R of the left (right) lead. Note that here the left/right lead is represented by a semi-in nite quasi-one dimensional graphene lattice sketched in gure 1(c), and the scattering region includes the line defect and the B region with width L, as shown in gure 1(a). The two Dirac points, K and K ′ , are located at [0, ±π/3a] for graphene with a line defect. Therefore, the momentum can be expanded at k x = q x and k y = q y ± π/3a where q x (q y ) represents the group velocity of electrons along the x (y) direction. Combining this with the linear dispersion relation of the Dirac electrons E = √ 3q , the transmission coef cients can be calculated as a function of the electron incident angle α with α = arctan(q y /q x ).
We would now like to brie y illustrate the shift of the peak transmission of the two valleys under the B-eld, as shown in gure 1(b). In the presence of a perpendicular B-eld, the two Dirac points are shifted along the k y -axis by an amount of ∆k = eBL/ [30,31]. It is well known that the transmission of the K(K ′ ) valley adopts the peak value T K (T K ′ ) ≈ 1 as the scattering angle is about −π/2(π/2), whereas it will decrease when the scattering angle is away from −π/2(π/2). It is clear that when an electron from the K ′ valley injects with an incidence angle π/2, the actual outgoing angle α 1 through the B-eld will be less than π/2 according to the conservation of k y , as shown in gure 1(a) and the lower panel of gure 1(b). Hence, the transmission of K ′ valley through the scattering region will be greatly restrained which can be depicted visually by the line width variation of the blue arrow shown in gure 1(b). However, the situation is completely different for the K valley. It is shown in the upper panel of gure 1(b) that regardless of the magnitude of ∆k, there always exists such an incident angle that makes the outgoing angle through the B-eld equal to −π/2 due to the conservation of k y . Therefore, the peak transmission of the K valley T K = 1 can always survive as ∆k increases, as depicted by the red arrow with no variation in line width. According to the conservation of k y , q sin α = q sin(−π/2) + ∆k, one can obtain the relation between α and ∆k for the peak transmission T K = 1:

Numerical results
In the following, we will numerically investigate the shift of the transmission curves and the valley polarization phenomenon in the line defect of graphene under a perpendicular magnetic eld. In the calculations of the transmission coef cients, we set τ 2 = τ 1 = t = 1 as the energy unit, the Fermi energy E f = 0.01t, the width of the B-eld L = 174 and the disorder average is carried out over 4000 sample con gurations. In gure 2, the transmission coef cient is mapped as a function of the electron's incident angle α for different magnetic elds. As B = 0 (the blue line), the transmission coefcient of K/K ′ valley T K /T K ′ varies slowly from 1/0 to 0/1 with α, and the peak transmission T K /T K ′ ≈ 1 appears at α ≈ − π 2 / π 2 , as shown in gures 2(a)/(b). Clearly, the valley polarization can only appear for high incident angles. When a Beld is applied, the transmission pro les combined with the peak transmission of the two valleys are shifted along the incidence angle axis from negative angles to positive angles with the same pace, and the deviation continues as the B-eld strengthens, as depicted by the curves with different colors in gure 2. Consequently, the peak transmission of the K valley T K = 1 is maintained well, while that of the K ′ valley weakens and even disappears as B increases. For instance, the maximum transmission coef cient of the K valley with T K = 1 appears at α = −0.04 π when B = 1 T, while this position moves to α = 0.25 π as B = 2.1 T and even up to α = 0.5 π as B = 2.31 T. This is because as an electron injects with an angle α = −0.04 π(0.5 π), it will eject from the scattering region with an angle −π/2 at B = 1T(2.31 T). As B further strengthens, the peak transmission of the K valley state still has a considerable magnitude which can reach 0.96 as B = 2.35 T, as shown in gure 2(a). However, the peak transmission of the K ′ valley decreases rapidly with increasing B, and the maximum value found at B = 2.1 T is less than 0.04, which indicates a nearly 100% valley polarization. Therefore, the nearly 100% valley polarization can always appear in a certain range of the magnetic eld (2.1-2.35 T). This conclusion is consistent with the analytical results presented above.
To further interpret the shift of the transmission curve and the peak transmission in a B-eld, we plot the transmission coef cients of the two valleys versus the incident angle α and B-eld, as shown in gure 3. The relation between α and B (or ∆k) according to equation (6) as T K = 1 is also plotted, as depicted by the blue line in gure 3(a). It is shown that as B strengthens, the critical incident angle α corresponding to T K = 1 gradually varies along the incident angle axis, which has the same variation tendency with the line obtained from equation (6). However, as α > 0.2 π it requires a higher magnetic eld to realize the peak transmission T K = 1 compared with the theoretical analysis. This clearly indicates that the shift of the peak transmission and the valley polarization should be induced by the movement of the Dirac points in the B-eld.
As mentioned above, nearly 100% valley polarization can always occur when the peak transmission of the K valley T K = 1 appears as α = π/2. Since the nearly 100% valley polarization is related to the movement of the Dirac points which is dependent of the magnetic eld B and the width of the scattering region L, the valley polarization should occur in a certain range of B and L. In gure 4, we present the critical magnetic eld B c and the corresponding width L to realize T K = 1 as α = π/2 (the blue line) for different energies. It is  In fact, the nearly 100% valley polarization can always occur when the magnetic eld B is in such special range not con ned to B c . In the insert of gure 4(b), we select a position randomly in these ranges (the red circle between the two dashed lines) and plot the the transmission curves of the two valleys. It is found that the peak transmission of the K valley remains at 1 while it is less than 0.04 for the K ′ valley which indicates a nearly 100% valley polarization.
To conveniently detect the valley polarized current in the experiment, we present the ranges of B and L to realize the nearly 100% valley polarization with T K = 1 as α approaches π/2 at different energies, as shown in table 1. It is found that at a lower energy, a weak magnetic eld is required to generate the valley polarized current. In the experiment, it is easy to apply a magnetic eld up to approximately 3T, and the scheme is promising to be realized in a real experiment for at least E 0.01t. During an experiment, one can analyze the valley-polarized electrical currents from the left  to right lead with an experimentally measurable quantity such as the conductance, which can be given according to the Landauer-Büttiker formula −π/2 T K/K ′ cos αdα, where L y = 0.492 nm is the sample size along the y direction, υ F ≈ 1.0 × 10 6 m s −1 is the Fermi velocity, = h/2π is the reduced Planck constant with h = 4.135 667 43 × 10 15 eV s, and E is the on-site energy of the incident electrons. For a homogeneous system where the electrons in the whole device have the same on-site energy, the conductance may be weak, approximately G K ≈ 5.0 × 10 −3 e 2 /h at E = 0.01t, which seems dif cult to detect. However, as the electrons' on-site energy in the left electrode E L is raised, the valley polarization effect is essentially unaffected due to the conservation of k y while the conductance can be increased dozens of times, which can be detectable in a real experiment. For instance, as the on-site energy in the left electrode is raised to E L = 0.2 t (or even higher), the maximal value of T K can still remain at a large magnitude (about 0.9), while that of T K ′ is still less than 0.05, as shown in gure 5, which manifests a perfect valley polarization. However, π/2 −π/2 T K cos αdα remains almost unchanged compared with the E L = 0.01t curve, and the magnitude of G K can be up to the order of 0.1e 2 /h, which can be detectable in practical experiments.
In a real device, the disorder is always present due to the existence of adatoms or vacancies. To simulate a more realistic line defect, a random potential with a uniform distribution in the range [−W/2, W/2]t, with W being the disorder strength, should be introduced on the scattering region to mimic a generic short-range disorder. In gure 6, we calculate the transmission coef cients of the two valleys for different disorder strengths. For weak disorder (W < 0.25), T K changes only slightly compared with the clean sample. As W increases, the maximal value of T K decreases, while it still has a considerable magnitude at strong disorder strength (W = 0.65), as shown in gure 6(a). The disorder does not have a substantial in uence on the transmission of the K ′ valley, as shown in gure 6(b). Thus, the pure valley current can be detected in a real experiment.

Conclusion
In summary, we propose a new approach for achieving valley polarization in a graphene line defect by application of a magnetic eld. In contrast to the previous techniques to achieve the valley polarization with a magnetic eld, we consider a special transmission characteristic of graphene whereby the transmission curves shift along the incident angle axis in a magnetic eld. The valley polarization can reach nearly 100% in a certain range of magnetic eld. Superior to other schemes, it does not require a high magnetic eld to create quantum Hall states, and the manufacturing technology is relatively simple and feasible. Moreover, theoretical work suggests that such an extended line defect can be produced in suspended graphene in a controlled manner and a predetermined location even without a catalyzing metal substrate [40]. We hope that our nding may provide a route for designing valleytronic devices with graphene.