Valley-polarized and supercollimated electronic transport in an 8-Pmmn borophene superlattice

Analogous to real spins, valleys as carriers of information can play significant roles in physical properties of two-dimensional Dirac materials. On the other hand, utilizing external periodic potential is an efficient method to manipulate their band structures and transport properties. In this work, we investigate the valley dependent optics-like behaviors based on an 8-Pmmn borophene superlattice with the transfer matrix method and effective band approach. Firstly, it is found that the band structure is renormalized, more tilted Dirac cones are generated, and the group velocities are modified by the periodic potentials. Secondly, due to the exotic tilted Dirac cones in 8-Pmmn borophene, a perfect valley selected angle filter can be realized. The electrons with a specific incident angle can transmit completely in an energy window, which is flexibly tunable by changing the periodic potential. Thirdly, by using the Green’s function to simulate the time evolution of wave packets, electrons can be shown to propagate without any diffraction, valley electron beam supercollimation happens by modulating the potential parameters. Different from the graphene superlattice, the electron supercollimation here is valley dependent and can be used as a valley electron beam collimator. Fourthly, we can tune the polarization and supercollimation angles by changing the superlattice direction. These intriguing results in an 8-Pmmn borophene-based superlattice offer more opportunities in diverse electronic transport phenomena and may facilitate the devices applications in valleytronics and electron-optics.


Introduction
Special lattice structures in solids usually lead to interesting electronic behaviors [1]. The first fabrication of graphene with hexagonal honeycomb lattice in 2004 has excited great upsurge on the relativistic massless Dirac fermions, which possess peculiar physical properties and actual applications in many fields [2,3]. Afterwards, more and more concentrations have been paid on searching new materials with Dirac fermions, such as the heavy group-IV low-buckled monolayer materials, molybdenum disulfide, even some artificial optical lattices [4][5][6][7]. Nowadays, composed of the element with one electron less than carbon, boron-based various monolayer structures attract much attention. Several types of monolayer boron phases have been synthesized on Ag(111) [8][9][10][11][12]. Among most of the different forms, the 8-Pmmn borophene which has eight boron atoms in a unit cell was predicted to be the most stable structure [13]. The first-principles calculations and effective low-energy model show that it also hosts massless Dirac fermions, and more interestingly, its Dirac cones are tilted and anisotropic [14][15][16], different from the isotropic Dirac cones in graphene. Based on these unique characteristics, a series of intriguing physical phenomena has been confirmed, such as oblique Klein tunneling [17,18], anisotropic plasmons [19], excellent thermoelectric performance [20], anomalous Andreev reflection [21], and so on.  Different from that in graphene, the θ p here depends on both the wave vector and velocity.
Due to the anisotropy of the Dirac cones in 8-Pmmn borophene, the momenta and the group velocities of electrons or holes are noncollinear. The components of the group velocities in x and y directions are determined by the energy dispersion, The incident angle θ v is determined by the group velocity and written as We then consider a periodic potential which is applied on the 8-Pmmn borophene monolayer. The schematic diagram of the model is shown in figure 2, where the bottom is the profile of the periodic square barriers, d = d A + d B is the length of a unit cell. The Hamiltonian can be rewritten uniformly as where V(x) is a periodic function. It is noted that the unit cell length of the potential d is much larger than the lattice separation a. Considering the normalization condition, the scattering matrix element between different valleys can be written as ⟨±, k K ′ |V(x)|±, k K ⟩ = 1 2Nd (1 ± e i(θpK−θ pK ′ ) )´N d 0 V(x)e i(kK−k K ′ )·r dx, where r = (x, y). If we choose the parameters of the potential as d A = d B = 10 nm, V A = 50 meV, V B = 0, N = 10, the momentum difference between the two Dirac cones is given as |k K − k K ′ | = 1.28 Å −1 [16,20], the intervalley scattering matrix element can be estimated, and is very small as in the order of 10 −4 . On the other hand, the periodic potential is assumed to vary slowly on the atomic scale. Hence, the intervalley scattering can be ignored here [33,34]. Such a treatment is general and realistic, which has also been successfully used to study the n-p junctions and superlattices in monolayer and bilayer graphene [29,[35][36][37][38]. In the experimental realization, electron beam induced deposition, ripples, or gating under electrical field all have been used in graphene superlattices [39][40][41]. Although the 8-Pmmn borophene-based superlattice in experiment has not been reported, the techniques used in graphene offer possibility for the purpose. Especially, fabrication of periodically patterned gate electrodes is a common method, which can be expected to realize the borophene superlattice.
With the transfer matrix method (see appendix), the band structures of the superlattice based on 8-Pmmn borophene at different valleys with full numerical calculations are shown in figure 3. Compared the band structures with those in pristine 8-Pmmn borophene plotted in figure 1, they are modulated remarkably. For exhibiting the renormalized band structures more clearly, the sectional drawings at fixed k x or k y are shown in figures 3(c)-(f), not only the positions of the original Dirac cones are shifted up, but also more Dirac cones appear at the boundaries of the supercell Brillouin zone. As has been done, the transfer matrix method gives us the exact band structures in a periodic potential, however, its application is relatively limited. Therefore, we would use an alternative way to find the effective dispersion relation near any Dirac cone for extending our researches on other significant physical properties. In order to give explicit result analytically of the renormalized band structures, we start from the Hamiltonian in equation (8), and do a transformation H ′ = U † HU using a unitary matrix [36,37], which can be written as with For a reasonable derivation, we extract a constant potential is zero. In this case, α(x) behaves like a sawtooth function, and is guaranteed to be a periodic function satisfying α(x) = α(x + νd) with ν being an integer. After some calculations, the Hamiltonian is transformed into which is similar to that of graphene superlattices but with different matrix elements.
To obtain the correct eigen spectrum of H ′ , a plane wave expansion with an infinite number of reciprocal lattice vectors of the superlattice is needed. It is impossible to give an analytical solution of the whole superlattice spectrum accurately. From the full numerical calculations in figure 3, we can find that new tilted Dirac cones emerge around the wave vectors G l /2, where G l = lG 0 e x = l(2π/d)e x is the reciprocal lattice vector of the superlattice, and l is an integer. In the proper case, we are interested only in the quasiparticle states near the Dirac cones around the wave vector G l /2, i.e. k = ∆k + G l /2 and ∆k ≪ G 0 [36]. The back-diagonal terms containing ∂ y can be treated as a perturbation since G l is along x direction. The Hamiltonian H ′ can be written as H ′ = H ′ 0 + H ′ ′ , and the zero-order Hamiltonian is given as which has the eigenvalues When the wave vector locates near the minizone boundary G l /2, the main contribution to the dispersion is from the states whose energies are degenerate. A state with wave vector ∆k + G l /2 on one branch has a significant mixing with another state with wave vector ∆k − G l /2 on the other branch. According to the degenerate perturbation theory [42,43], the eigenvalues around the minizone boundaries can be obtained approximately from the equation below The back-diagonal terms are the scattering amplitude and its complex conjugate between the states |+, ∆k + G l /2⟩ and |−, ∆k − G l /2⟩, that is where and χ l,η (V)s are just the Fourier coefficients of e iα(x) and depend on the periodic potential. Then, equation (14) above can be rewritten detailedly as The eigen energies E can be derived analytically as It describes the dispersion with the wave vector very close to the minizone boundary G l /2 as well as −G l /2. From the derivation process, we can understand that when the Fermi wave vector approaches to G l /2, the corresponding dispersion relation plays the dominant role and is related to a certain χ l,η . The treatment here is consistent with that in [32], and it is easy to check that in the simpler situation our equation (18) can be returned to the dispersions around the Dirac cones in graphene superlattices. The results for the graphene superlattices have been directly used to investigate some physical properties such as the electrical conductivity and Zitterbewegung behavior [44,45]. From the dispersion, one can easily verify that multiple tilted Dirac cones emerge at different energies, where the energy separation between different Dirac cones in adjacent subbands are given as ℏv x G 0 /2. When the parameters are chosen as d A = d B = 10 nm, the energy separation is 0.089 eV. It is consistent with the numerical calculation, as marked in figures 4(a)-(d). Furthermore, the anisotropic velocity of the Dirac point is changed by χ l,η . Focusing on the transport properties of the low-energy electrons around the Dirac cones at G = 0, i.e. k = ∆k, the dispersion is given as The effective band parameter χ 0 can be derived as It is independent of the valley index, and χ 0 < 1 can be verified. Compared with the band structure in pristine 8-Pmmn borophene, the anisotropic velocity along k y in the superlattice is reduced. For contrasting the analytical results with those by full numerical calculations, we plot them with dashed and dotted lines in figures 3(c)-(f). It can be found that the analytical results in equation (19) has a good consistency with the numerical calculations in the low energy.
For showing the effect of the periodic potential on the boundary of the minizone clearly, the enlarged drawings of the band structures near the wave vector π/d with full numerical and proper analytical calculations are plotted in figures 4(a) and (b). The energy separation between the first and the second subbands at the minizone boundary can be given as where χ 1 can be derived as The energy separation around ±π/d depends on the wave vector and the external periodic potential. In particular, the gap opening is zero at the center of the minizone boundary. It is different from the conventional layer-structured one-dimensional superlattice, where the gap at the minizone boundary remains to be finite. The variations of the energy separation with wave vector at fixed k y = 0 and k x = 0 are shown in figures 4(c) and (d), respectively. When the wave vector is along the direction of the periodic potential, the gap is robust against the potential. As the dependencies of the velocity renormalization and strong anisotropy on k y , the gap is sensitive to the potential, which can be seen in figure 4(d). For simplification and highlighting main physical properties, we concentrate on the spectrum near the first Dirac cone in the following investigations. It is known that the energy spectrum of a superlattice denotes the transmitted states. In the small energy window around a Dirac cone, we can derive from the spectrum that the electrons with only specific angles are allowed for transport, resulting in the valley resolved transmission. In addition, from the analytical dispersion relation in equation (19), it can be found that for a special case of the renormalized band when χ 0 = 0, the group velocity of the quasi-particles is along specific direction no matter what the wave vector is. It is an important sign for realizing the electron beam supercollimation.

Valley selected transmission
To show the transport properties of the 8-Pmmn borophene-based superlattice, we deduce the transmission amplitude with the transfer matrix method [30], which can be given as where θ i p and θ e p are the azimuthal angles of the pseudospin vector of electrons at the incident and exit ends, respectively. M 11 , M 12 , M 21 , and M 22 are the components of the total transfer matrix M.
As a function of incident energy and angle, we present the contour plot of the transmission coefficient |t a | 2 in figure 5, where figures 5(a) and (b) correspond to the transmission of electrons on the K and K ′ valleys, respectively. It has been known that, in the graphene superlattice, due to the isotropic energy structure, the directions of the wave vector and the group velocity are collinear, the Klein tunneling happens at normal incidence and is valley-degenerate. However, in the 8-Pmmn borophene superlattice, the anisotropic dispersion makes the two directions noncollinear, and it supplies a good platform for the valley birefringence. From the expression of the transmission amplitude in equation (23), it can be confirmed that when k y = 0, i.e. θ i p = 0 and θ e p = 0, then the total transfer matrix becomes a unit matrix, its components are Hence, the transmission amplitude t a is always 1. Determined by the group velocity, the incident angle of the electrons in the Hence, the oblique Klein tunneling maintains in the periodic potential structure, and it is robust against not only the incident energy but also the width, the height and the number of barriers. Except the Klein tunneling, what we should emphasize more is there exist some other phenomena which can not be realized in the n-p junction as follows.
It has been researched and confirmed that massless Dirac cone is formed at the energy which corresponds to the zero averaged wave number in graphene superlattice [30]. According to the dispersion in equation (19), there also exist tilted Dirac cones at the energy It can be found that the filter angle is robust against the superlattice period, which is consistent with the analytical result above. However, the effect of the angle filter is influenced by the superlattice period. When the barrier width is modulated to d A = d B = 15 nm, the range of the angle around ±20.4 • is extended compared with the results in figures 5(c)-(e). This can be understood from the renormalized spectrum, when the barrier width is modulated from 7 nm to 15 nm, χ 0 is reduced, and the slope of the Dirac cone is decreased. For a certain energy interval, the filter effect becomes less obvious. In order to see the filter effect of the periodic potential, we give a schematic diagram to illustrate the special phenomenon in figure 5 It is obvious that the classical trajectories of electrons are analogous to the rays in geometrical optics. There are many similarities between electronics and optics. In the previous researches, the barrier in a p-n  junction is usually viewed as an interface, like different mediums in optics. All the phenomena such as electron focusing, Veselago lens, Goos-Hänchen displacement are based on the p-n junction [22,23,25]. Here, the 8-Pmmn borophene-based superlattice can be regarded as an electron angle filter, like an optical polarizer, which is shown schematically in figure 5(h). For the electrons in the K (K ′ ) valley, the superlattice behaves like a polarizing film with the polarization angle being +(−)20.4 • . It should be noted that even though the special angle is robust against the width, the height, and the number of the barriers, it is sensitive to the orientation of the superlattice, as will be exhibited further in section 5. More importantly, the incident energy window around (V A + V B )/2 is tunable by the external periodic potential.
In order to see the perfect valley polarized transmission in a borophene superlattice more clearly, a polar plot of the transmission with different numbers of barriers is shown in figure 6. The incident energy is chosen as E = 42 meV. Comparing the transmission of multiple barriers with those in one or two barriers, the valley electron beams are split completely in superlattice, the angles are focused around ±20.4 • . For the small numbers of potential barriers, the valleys are mixed at certain incident angles, the valley polarization is unperfect. With the number of the barriers increasing, the transmission spectrum of the superlattice tends to be a continuous one, hence, the transmission is robust against the barrier number. If we choose more period numbers, for example, N ⩾ 11, the transmission has few changes, which can be seen in figure 6.

Valley electron beam supercollimation
From equations (18) and (19), we can see when χ 0 = 0, that is, the potential satisfies the condition (V A − V B )d/ℏv x = 4mπ with m being an integer, the renormalized spectrum becomes For simplification, we set E = (V A + V B )/2 as the energy reference point in the above equation. In order to verify the results, the energy dispersions of the different valleys calculated by the transfer matrix method with full numerical calculation are shown in figure 7, which exhibit wedge-shaped structures. The result shows a dispersionless behavior along arctan(ηv t /v x ) direction, and is valley dependent. Obviously, the numerical results are consistent with the analytical expression in equation (24). The results are very different from that in the graphene superlattice [36,37,46,47], where the wedge-shaped band structure is dispersionless along the k y and valley degenerated. Hence, the 8-Pmmn borophene-based superlattice supplies a favorable opportunity for the valley electron beam supercollimation. It should be noted that if the incident energy is tuned to the vicinity of other Dirac cones, such as a higher energy near (V A + V B )/2 + ℏv x G 0 /2, the wedge-shaped band is formed when χ 1,η = 0. The electron beams supercollimation can also be found.
To exhibit the valley electron beam supercollimation in the 8-Pmmn borophene superlattice numerically, we investigate the dynamics of a wave packet based on the Green's function. Near the Dirac cone, the corresponding time-dependent eigenfunction is given as Here, tan θ p,χ0 = χ 0 v y k y /v x k x is related to the periodic potential and should not be confused with the polar angle θ p of pristine 8-Pmmn borophene in equation (3). With the help of the Green's function [48,49], the time evolution of an arbitrary state φ(r, t) can be obtained as where µ, ν =1, 2 are used to label the matrix elements, corresponding to the upper and lower components of the wave function φ(r, t). The Green's function has the standard form as follows, the components of the 2 × 2 Green's function matrix are enumerated as where E η (1) = ηℏv t k y and E(2) = ℏ (v x k x ) 2 + (χ 0 v y k y ) 2 . In order to present the time evolution of an arbitrary state, and to find out the reasons that the Gaussian wave packet can describe the localized quantum states roughly and any wave packets can be expressed by a superposition of finite Gaussian states, we then take the initial wave function to be a Gaussian wave packet with the width ∆ and initial center of wave vector k 0 , and send it from the left end towards the superlattice. At t = 0, the Gaussian wave packet was prepared as [49]  The width ∆ is supposed to be much larger than the lattice separation and φ(r, 0) is a smooth envelop function. Substituting equation (31) into equation (26), one obtains the time evolved wave function, which has the following form where ψ 1 (r, t) =´G 11 (r, r ′ , t)φ 11 (r, 0)dr ′ and ψ 2 (r, t) =´G 21 (r, r ′ , t)φ 11 (r, 0)dr ′ . After the integrations over the coordinates r ′ , the two components in the wave function can be obtained as We choose the initial central position of the wave packet r 0 = (0, 0). At t = 0, the wave packet mainly locates at the center position (0, 0), as shown in figure 8(a). By using the wave packet dynamics and the Green's function method, we demonstrate numerically the wave packet propagation in the pristine 8-Pmmn borophene and 8-Pmmn borophene-based superlattice in figures 8(b)-(e). The distance is in the unit of the wave packet width ∆, the time is in the unit of ∆/v F , and the wave vector is in the unit of 1/∆. We consider that the wave packet propagates in the region x > 0. The electron probability densities with different initial wave vectors are plotted. In figure 8(b), the initial wave vector is chosen as k 0y = 0, which corresponds to the direction of group velocity 20.4 • (−20.4 • ) for the electrons in the K(K ′ ) valley. We then change the wave vector to k 0x = k 0y in figure 8(d), the wave packet propagates in the direction of 48.2 • for the electrons in the K valley, and 9.5 • for those in the K ′ . Without the superlattice, the wave packet propagates along the direction of the group velocity like a cylindrical wave, and it spreads sideway. Although the valley electron beams are split, they are mixed in the space. However, in the superlattice, due to the wedge-shaped energy spectrum at special potential parameters, the Gaussian wave packet is guided to propagate along the direction of η arctan(v t /v x ), which is just ±20.4 • . The two directions are robust against the initial wave vector or propagation directions. More importantly, the electron wave packet propagates actually no diffraction in the one-dimensional periodic potential, and the shape remains unchanged as expected. The valley electron beams are supercollimated and split very well.

Angle tunable polarization and supercollimation
For the periodic potential along the x direction, shown in figure 2, the angle of polarization and supercollimation happens at ±20.4 • , which is fixed. It is possible to tune the angle by changing the direction of 8-Pmmn borophene or the periodic potential, as shown in figure 9(a). The coordinates x ′ -y ′ rotate an angle ϕ with respect to the coordinates x-y, ϕ can be viewed as a new superlattice direction. According to the transformation relation between the two bases, the wave vector components in the two coordinates systems can be written as The valley-dependent velocity components should be rewritten as In the process of transmission, k ′ y is a conserved quantity. From the discussion above, the angle polarization mainly caused by the Dirac cone at the energy E = (V A + V B /2) and the Klein tunneling. When the coordinates system rotates, the k ′ y = 0 ensures the Klein tunneling of electrons. Hence, the angle of polarization is θ ′ v = θ v − ϕ, which is obtained as where γ = v 2 x cos 2 ϕ + v 2 y sin 2 ϕ. The polarization angle as a function of the superlattice direction ϕ is shown in figure 9(b). When ϕ = 0, the angles for different valley just are ±20.4 • . By changing the direction of the periodic potential or the 8-Pmmn borophene, the angle can be modulated, and the valley electron beams remain to be split except for ϕ = ±π/2.
Utilizing the method, the angle of supercollimation can also be modulated. The effect of the supercollimation is mainly resulted from the wedged band structure, which is renormalized by the periodic potential. In the case of rotated periodic potential, the dispersion of the wedged conduction band can be rewritten as The direction of the supercollimation is θ ′ s = θ s − ϕ determined by the group velocity and obtained as The angle of supercollimation tuned by the superlattice direction is shown in figure 9(c). Obviously, the valley-dependent electron beams are always separated. When ϕ = 0, the electrons in different valleys are supercollimated to ±20.4 • , which are consistent with the results above. Moreover, we can find that when the superlattice rotates +(−)20.4 • , the electron beam from K(K ′ ) valley is collimated to the direction of 0 • . While the superlattice direction is −(+)69.5 • , the electron beam from K (K ′ ) valley is collimated to the direction of 90 • .

Conclusions
In summary, we have constructed an 8-Pmmn borophene superlattice by applying a periodic square potential and investigated the valley dependent transport properties and some special optics-like behaviors. By using the transfer matrix method and effective band approximation, it is found that the electron transmission can be angle-selected in a tunable energy window, and for the electrons in different valleys, the angles are completely distinguished. The superlattice structure serves like an angle filter, only the electrons with specific angle can penetrate. In addition, we have demonstrated that the valley electron beams can be supercollimated, which is robust against the incident directions. For a specific superlattice orientation with the 8-Pmmn borophene lattice, the electrons on the K valley are collimated into the direction of 20.4 • , while those on the K ′ valley are collimated into the direction of −20.4 • . Furthermore, the direction of valley polarization and supercollimation can be tuned by changing the superlattice direction. What we should notice is that if the tilted Dirac system degrades into v t = 0, the angle filter and the supercollimation happen at 0 • , the valley electrons beams are degenerate, valley polarized electronic transport cannot appear. Compared with the results in the graphene superlattice, our implementations in the borophene superlattice offers more opportunities to modulate the electronic transport and observe interesting physical properties in valleytronics and electron-optics.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Appendix. Transfer matrix method
Due to the translation invariance along the y direction, the two-component pesudospinor wave function can be written asΨ whereΨ A andΨ B are the enveloping functions for different sublattices. For the sake of simplification, we set (E − V(x))/ℏ = κv F . According to the Schrödinger equation, we can get By some substitutions in the above two equations, we obtain The solutions of the equation have the following forms The subscript j denotes the corresponding wave vector in the jth potential, as well as other quantities such as the position, the transfer matrix below. Then, by substituting equations (A6) and (A7) into equations (A2) and (A3), we can get the relations between the coefficients, which read c = v x q j + iv y k y ηκv F − v t k y a, (A8) We choose a basis ϕ(x) = (ϕ 1 , ϕ 2 ) T , and The wave function in equations (A6) and (A7) can be rewritten as where the matrix Υ j (E, k y ) is written as Inside the same barrier or well, the wave function Ψ A,B (x j−1 ) from the position x j−1 to x j−1 + ∆x can evolve into another form Ψ A,B (x j−1 + ∆x), which can also be expressed in terms of the basis, where Γ j (E, k y , ∆x) = cos(q j ∆x) i sin(q j ∆x) C D , with C = iv y k y ηκv F − v t k y cos(q j ∆x) + iv x q j ηκv F − v t k y sin(q j ∆x), Hence, the wave functions Ψ A,B (x j−1 ) and Ψ A,B (x j−1 + ∆x) can be connected by a transfer matrix T(E, k y , ∆x), where the transfer matrix is given as T j (E, k y , ∆x) = Γ j (E, k y , ∆x)Υ −1 j (E, k y , ∆x).
The equality det[T j ] = 1 can be verified. In fact, there are only two types of matrices, i.e. T a and T B , for T j . We assume Ψ A,B (x 0 ) are the wave functions at the incident end, the wave functions Ψ A,B (x) at an arbitrary position x can be related with Ψ A,B (x 0 ) by the total transfer matrix, The total transfer matrix M can be expressed as the sequential product of a series of transfer matrices, i.e.
M(E, k y , x) = T j (E, k y , ∆x)Π where d i is the width of the ith potential, x = ∆x + j−1 i=1 d i . The wave functions Ψ A,B (x 0 ) can be determined by matching the boundary conditions. According to the Bloch theory and utilizing the transfer matrix, the electronic dispersion of the 8-Pmmn borophene-based superlattice can be got from the relation 2 cos(β x d) = Tr(T A T B ). (A22)