Pseudospin collapse, multidirectional supercollimations, and all-electrons transmission and reflection in irradiated 8-Pmmn borophene

Propagation of ballistic electrons shows various optical-like phenomena. Here, we demonstrate a flexible method to modulate the band structure and manipulate the electron beams propagation in 8-Pmmn borophene by an off-resonant linearly polarized light. It is proposed to form fully tunable anisotropic dispersion by changing the polarization direction of the off-resonant light in an experimentally controllable way. Accompanied with it, the pseudospin symmetry of the electronic state in 8-Pmmn borophene collapses from a helical form into x or y direction, which undergoes a dramatic alteration. As a result of the wedge-shaped dispersions, the electron wave packet can be guided to propagate with undistorted shape along different directions, multidirectional electron supercollimations are exhibited in the system. Moreover, by constructing the optical sensing n–p and n–p–n junctions, interesting transport phenomena such as all-electrons Klein tunneling and omnidirectional reflection are realized by modulating the illumination parameters of the off-resonant light, both of them are independent of the incident energy and wave vector. It is expected that the peculiar transport properties in 8-Pmmn borophene modified by the off-resonant light field can offer more opportunities for device applications in valleytronics and electron-optics.


Introduction
Although electrons and photons are intrinsically different, importing useful concepts in optics to electronics and performing similar applications have been actively pursued in recent years [1][2][3][4][5].The electron-optics which originates from the wave-particle duality has led to outstanding applications such as electron microscope [6], quantum information processing [7].Nowadays, many electronic analogues of optics such as focusing [3], superlenses [8,9], guiding [10], and birefringence [11] have been achieved in two-dimensional materials.By manipulating the external gates and forming a p-n junction, electrons are refracted in different regions following the Snell's law, like the light transmitting in the mediums with different refractive indices.Quantum interference, Klein tunneling, negative refraction have been observed in experiment [12][13][14].Among most of the optical-like phenomena, the collimation of an electron beam, i.e. the quasi-one-dimensional motion of electrons is still a long-standing goal which has potential applications in quantum devices or ultracompact integrated circuits.It has been demonstrated that electron beam supercollimation can be achieved in a graphene superlattice by an external periodic or disorder potentials [15,16].However, due to the integrated gating potentials, the direction of the supercollimation is difficult to be changed in experiment.
The 8-Pmmn borophene, as one of the most stable predicted structures of borophene, also hosts massless Dirac Fermions and high mobility [17][18][19][20].Different from that in graphene, the low-energy effective band structure shows that the Dirac cones in the 8-Pmmn borophene are anisotropic and tilted [21].On account of the anisotropy, some unusual electronic properties in 8-Pmmn borophene have been investigated, such as Weiss oscillation [22], anomalous caustics [23], optical conductivity [24,25], anomalous Andreev reflection [26].Interestingly, the tilted directions are different for K D and K ′ D valleys, the valley degeneracy is broken due to the opposite chirality in 8-Pmmn borophene.It provides an ideal platform to study the valley dependent transport properties.Gate-tunable valley filtering effect has been reported in a magnetic-electric barrier [27].Most recently, valley dependent oblique Klein tunneling has been found in the n-p or n-p-n junction [28][29][30], and the angle of Klein tunneling can be changed by adjusting the direction of the junction.Aiming at the tunneling phenomena, the modulated method is not very convenient obviously.Seeking for some other approaches which can be used to tune the anisotropy and Klein tunneling flexibly and explore new physical properties in the 8-Pmmn borophene are our research emphases.
It has been known that manipulating external fields is a very effective and convenient approach to change the band structure and electronic properties.As one of the most common and useful means, the light can induce various physical properties, which have been researched widely.Photoinduced quantum Hall insulators without Landau levels can be found in graphene via the application of an off-resonant circularly polarized light [31].A linearly polarized light applied to the Weyl semimetal can induce controllable Weyl points [32].Anisotropic Andreev reflection in a superconducting junction and ballistic transport properties in potential barrier have been reported in an 8-Pmmn borophene with the polarization of the off-resonant linearly polarized light along x or y direction [33,34].In order to realize the electron beam propagation in a quasi one-dimensional motion, here we consider the flexible method to realize the electron beam supercollimation in an 8-Pmmn borophene by irradiating an off-resonant linearly polarized light.Under the modulation of the light field, the group velocity of the two-dimensional chiral electronic state can be controlled without additional structural design.Although the modulated measure by the off-resonant linearly polarized light is the same as previous researches, the physical properties in our work are totally different.Firstly, it is found that due to the tunable anisotropic dispersion, the pseudospin of the particles in the 8-Pmmn borophene undergoes a significant change, the helical pseudospin can collapse into two specific directions, the left and right or the up and down.Secondly, taking advantage of the peculiar wedge-shaped band structure, the group velocities of all the electrons are along ±20.4 • or 90 • , the electron wave packet can be guided to propagate virtually undistorted along certain directions.Thirdly, different from traditional Klein tunneling, there exist two interesting tunneling phenomena in the off-resonant light modified n-p and n-p-n junctions, i.e. all-electrons Klein tunneling and omnidirectional reflection, both of which are robust against the incident energy and wave vector.We use the term 'all-electrons transmission' instead of 'electron transmission' for emphasizing the peculiar transport property, which is a special kind of electron transmission.
The organization of the paper is as follows.In section 2, we propose the off-resonant linearly polarized light modulated model based on the 8-Pmmn borophene, and give necessary formulas for the effective Floquet Hamiltonian and time evolution of the wave functions with wave packet dynamics.In section 3, the main results including the anisotropic band structure, pseudospin texture, tunable electron beams supercollimation, and all-electrons transmission and omnidirectional reflection are exhibited.Finally, a summary is given in section 4.

Floquet Hamiltonian of the irradiated 8-Pmmn borophene
The low-energy effective Hamiltonian of the pristine 8-Pmmn borophene can be obtained as [35,36] where v x = 0.86v F , v y = 0.69v F , and v t = 0.32v F with v F = 10 6 m s −1 being the Fermi velocity, η = +(−) corresponds to the K D (K ′ D ) valley index.The sublattice pseudospin is described by the Pauli matrix σ = (σ x , σ y ), σ 0 is the unit matrix.The single-electron Hamiltonian in equation (1) can be used to describe the transport properties for the low-energy electrons very well [26,28], so that the Coulomb interaction is usually disregarded as we assume.Then, we consider the effect of an off-resonant linearly polarized light with arbitrary polarization direction, which is applied perpendicularly to the plane of the 8-Pmmn borophene.The time-dependent vector potential of the linearly polarized light can be written as A(t) = A 0 cos(ωt)(cos ξ, sin ξ), with A 0 = E0 ω , E 0 being the amplitude, ω being the frequency, and ξ denoting the polarization direction of the light.When ξ = 0 (π/2), the direction of the polarization is along x (y) axis.Considering the vector potential and using the canonical substitution, the Hamiltonian in equation ( 1) is modified into New J. Phys.25 (2023) 113002 When the parameter of the light satisfies the condition of off-resonant, i.e. the energy of the light hω is much larger than the bandwidth of the 8-Pmmn borophene, it does not excite electrons directly and instead modifies the band structures through virtual photon absorption and emission processes [31,37].According to the Floquet theory, the effective Hamiltonian can be established as [38][39][40][41] where is the discrete Fourier component.After the deduction, we can obtain The effective Hamiltonian in equation ( 3) can be rewritten concisely as in which the terms v 1 , v 2 , v 3 , and v 4 have the forms in the following The parameter γ = (eE 0 v F /hω 2 ) 2 is related to the off-resonant light.Via the Schrödinger equation H eff Ψ = EΨ, we obtain the modified dispersion, which is The eigenstates can be written as where the angle θ satisfies the relation tan θ = v3ky+v4kx v1kx+v2ky .It is obvious to check that by setting A 0 = 0, the dispersion degrades into that of pristine 8-Pmmn borophene.Moreover, if A 0 ̸ = 0 and ξ = 0, we can find respectively, the anisotropy of the band structure is modified only in v x .Note that the effect of the off-resonant linearly polarized light on the band structure is completely different from that of the off-resonant circularly polarized light, which modifies the band gap and the topological physical properties of the Dirac system [31,42].
Before the discussion of peculiar transport properties, we give an estimate of the irradiation parameter of the off-resonant light.According to the requirement of the Floquet theory, the energy of the irradiated light should be much larger than the bandwidth of the borophene.However, the precise bandwidth cannot be concluded from the first principles calculation because of the discrepancy of the role of p x and p z orbits in forming the Dirac cone at present [17][18][19]34].If a high evaluation of the hopping integral is taken as a criteria, the requirement for the off-resonant light is very strict.We then choose a low evaluation of the hopping integral in 8-Pmmn borophene, which is about 0.5 eV via the Slater-Koster tight-binding method.The energy of the irradiated light can be selected as hω = 2 eV, which is about four times larger than the hopping integral.For the two special cases , the amplitude of the light E 0 is estimated as 0.7 × 10 10 V m−1 and 0.88 × 10 10 V m−1.The effective Hamiltonian in equation ( 3) requires the high order of the parameter A = eE 0 a/hω is far less than 1 where a ∼ 1.62 Å is the lattice constant.Here A 4 ≃ 0.09 satisfies the condition of the off-resonant light, and the required frequency and magnitude of the light can be reached in current laser experiment.In addition, the time independent effective Hamiltonian in equation ( 3) is obtained by using the average time Hamiltonian scheme, the effect of the light-matter coupling is observed at multiples of cycle time of the light [31,34].When the borophene is irradiated by the off-resonant light, there is no any direct excitation.It is different from another type of light called on-resonant light, the photon energy of which is within the bandwidth of the borophene.

Wave packet dynamics with Green's function method
Wave packet dynamics are usually used to investigate the evolution of wave function with time.In particular, the interesting zitterbewegung effect has been researched in various systems [43][44][45].In this subsection, in order to exhibit the effect of the electron beam supercollimation, we extend the theoretical framework of the wave packet dynamics to the light field-modulated borophene system with Green's function.The time dependent eigen function of the Hamiltonian in equation ( 6) is given as At an arbitrary time t > 0, according to the Green's function, the wave packet Ψ µ (r, t) has the form where G µν is the component of the 2 × 2 Green's function matrix and µ, ν = 1, 2 are the matrix indices.The matrix component G µν is defined as After some algebraic calculation, each component can be written as where E 1 = ηhv t k y , and Both are the parts of the energy E η,± , and satisfy the equation E η,± = E 1 ± E 2 .Any wave packet can be written as a superposition of a finite number of Gaussian wave packets.For simplification, the initial wave packet at t = 0 is chosen as a Gaussian form, which is localized at the coordinates origin.It can be written as where d is the width of the wave packet, k 0 is the initial momentum, c 1 and c 2 denote the pseudospin polarizations of the injected wave packet.By substituting equation (19) into equation ( 14), the wave function Ψ(r, t) at a later time can be written as where the parameters Φ 1,2,3,4 (r, t) have the forms in the following and Φ 4 (r, t) = Φ 1 (r, t).After the integration of r ′ , we have

Tunable anisotropy of the band structure
In order to show the effect of the off-resonant linearly polarized light on the band structure of 8-Pmmn borophene, we plot the 3D dispersion and Fermi surface in figure 1 with different illumination parameters.Without the light, the band structure of the pristine 8-Pmmn borophene exhibits obvious anisotropy, and the band tilts in opposite directions for different valleys.The Fermi surfaces of K D and K ′ D valleys have an offset along k y direction, as shown in figure 1(a).If we apply an off-resonant linearly polarized light with the illumination parameter γ = 0.3v 2 F /v 2 y and the polarization direction ξ = π/4, the band structures and the constant energy surface with different valleys are shown in figure 1(b).Compared with that in pristine 8-Pmmn borophene, the anisotropy of the energy band is modulated effectively.It can be found that the Fermi surface is squashed, and inclines to the −k x direction.When the illumination parameters of the light are changed further, such as γ = v 2 F /v 2 x , and the polarization direction is along the x axis (ξ = 0), the band structure has a significant change.The band structures with different valleys and the Fermi surfaces are shown in figure 3(c).The band becomes to be wedge-shaped, and the constant energy surface is furcate.According to the dispersion in equation (11), the dispersion in the special case becomes into the following form The direction of the velocity can be given as θ v = ± arctan(ηv t /v x ) = ±20.4• .In the conduction band, no matter what the energy and wavevector are, the velocities of the electrons are fixed, the electrons on K D (K ′ D ) valley move along the direction of 20.4 • (−20.4 • ) angle to the x axis.However, it is found further that if the off-resonant linearly polarized light is modulated to γ = v 2 F /v 2 y and ξ = π/2, the energy structure has another significant change, which becomes dispersionless along the k x direction.According to equation (11), the dispersion becomes into The band structure is also turned to be wedge-shaped, but the Fermi surfaces are flat along the wave vector k y , as shown in figures 1(d1) and (d2).Regardless of the wave vector k x and energy, the particles move along the k y direction.The velocities of the particles on different valleys are different.The band structures in the last two cases are very peculiar, and rarely appear in other two-dimensional materials as we have known.By utilizing the off-resonant linearly polarized light, the band structure of the pristine 8-Pmmn borophene is modulated, and the control methods by changing the laser intensity, the frequency, and the polarization direction are very flexible.Such variations of the nontrivial energy spectra in 8-Pmmn would lead to peculiar electronic behaviors, we will demonstrate them in the following subsections.Moreover, in the 8-Pmmn borophene, the energy is linear with the wave vector.When a parabolic term is included in the Hamiltonian, a semi-Dirac point arises, where the quasiparticles behave as massless particles in one direction and as massive ones in the perpendicular direction.The interesting dispersion has been found in many systems such as a graphene-like structure [46], multilayer VO 2 /TiO 2 nanostructure [47,48], a three-dimensional topological insulator modulated by a spiral magnetization superlattice [49], black phosphorus [50], tunable optical lattice [51], and even polariton honeycomb lattices [52].Although the semi-Dirac points arise in many systems, but the forms of the Hamiltonian are different, the effect of the off-resonant linearly polarized light and the conditions for forming wedge-shaped band structure are various.To see the effect of the off-resonant linearly polarized light on a semi-Dirac system, we take a graphene-like structure as an example.After the derivation, it can be found that the band can also become a wedge-shaped under a specific light parameter, but the direction is only along x axis.The derivation process of the renormalized band about a semi-Dirac system is supplemented in the appendix.

Pseudospin texture
Except the anisotropic band structure, the pseudospin symmetry of the electronic states also undergoes a dramatic alteration.The pseudospin vector of the 8-Pmmn borophene can be defined as s = (s x , s y ) [28].The two components of the pseudospin can be given as and The azimuthal angle of the pseudospin can be expressed as tan θ s = s y /s x = v3ky+v4kx v1kx+v2ky , which just corresponds to the angle θ in equation (12).For exhibiting the variation of the pseudospin in the 8-Pmmn borophene, the pseudospin textures of the electrons in the conduction with different illumination parameters are plotted in figure 2. The pseudospins of the hole in the valence band are opposite to those of the electrons in the conduction band.For the Dirac cone-shaped dispersion in pristine 8-Pmmn borophene, the pseudospin vector radiates outward, and undergoes a continuous rotation, as shown in figure 2(a).When the off-resonant light is applied and the parameters are chosen as same as those in figure 2(b), the pseudospin texture has a small deformation.The orientation of the pseudospin is sensitive to the wave vector.From figures 2(a) and (b), it can be found that when we consider an n-p junction based on the irradiated borophene, the pseudospin conservation only appears when k y = 0, which is usually connected with the Klein tunneling phenomenon.Due to the anisotropy of the band structure, when k y = 0, the corresponding incident angle of the group velocity can be derived as From the pseudospin texture, it is obvious to find that the incident angle of the oblique Klein tunneling can be tuned under the modulation of the linearly polarized light, such as when γ = 0.3v 2 F /v 2 y , ξ = π/4, the Klein tunneling happens at 36.78 • (−6.13 • ) for the electrons on the angle in the above equation degenerates into ηv t /v x = ±20.4• , which is consistent with the previous investigation [28].
More interestingly, for one of the special cases where the illumination parameters are tuned to γ = v 2 F /v 2 x and ξ = 0, the low-energy pseudospin distribution is changed dramatically.It collapses into a forward or a backward state along the k x axis.All the electrons satisfy the conservation of pseudospin, the incident angles of the Klein tunneling maintain to be ±20.4• no matter what the wave vector is.However, for the case γ = v 2 F /v 2 y and ξ = π/2, the pseudospin then collapses into an upward or a downward state along the k y axis, as shown in figure 2(d).For this unique situation, there is no pseudospin conservation, we can infer that the Klein tunneling is forbidden completely.

Multidirectional electron beams supercollimations
Now we exhibit the time evolution of a Gaussian wave packet in the 8-Pmmn borophene modulated by the off-resonant linearly polarized light.In the calculation, we use the reduced unit for simplification.The distance is in the unit of the wave packet width d, the wave vector is in the unit of 1/d, the time is in the unit of d/v F , and h = 1.The initial wave vector is chosen as k y0 = 0 and k x0 = 1.2.At t = 0, the initial Gaussian packet is located at the center position r = (0, 0), which is shown in figure 3(a F /v 2 y and ξ = π/4, the propagation direction of the wave packet is modulated due to the anisotropic band structure, but the diffusion is also obvious.Interestingly, if the illumination parameter is tuned to be γ = v 2 F /v 2 x and the polarized direction is along x axis (ξ = 0), as a result of the wedged band structure, the direction of group velocity for K D (K ′ D ) valley is always along 20.4 • (−20.4 • ) and it is robust against the wave vector.Regardless of the initial propagation direction, the wave packet can propagate with undistorted shape and maintain good collimation.Furthermore, if the off-resonant linearly polarized light is modulated to the parameter γ = v F /v 2 y and the polarized direction of the light along y axis, the group velocity of the electron at the Dirac cone is along the y axis, the velocity along the x direction vanishes, which can be seen from the dispersion in equation (29).Hence, the supercollimation also exists, and the direction of the collimation is changed into y axis, as shown in figure 3(e).From the results above, we can find that by utilizing the off-resonant linearly polarized light and tuning the illumination parameter, valley dependent electron beam supercollimator can be obtained, and the collimated direction can be changed.

All-electrons transmission and omnidirectional reflection in optical sensing n-p and n-p-n junctions
In order to exhibit more interesting physical properties about the wedge-shaped band structure and verify the Klein tunneling about the model, we propose a two-terminal n-p junction which is irradiated by the off-resonant linearly polarized light, as shown in figure 4(a).For the n-p junction, the potential is V(x) = V 0 Θ(x) with Θ(x) being the Heaviside step function.In experiment, the potential in p region can be adjusted by doping or applying a gate voltage.The electrons from the left end can be injected with arbitrary initial wave vectors.The wave functions in the two regions can be written as where the angles θ i , θ r , and θ t satisfy the equations tan , and tan θ t = v3ky+v4kxt v1kxt+v2ky , respectively.When the incident energy E is given, the incident and reflected wave vectors k xi and k xr satisfy the dispersion in equation (11), while the transmitted wave vector k xt can be obtained from the dispersion ηhv t k y ± h√ (v where k y is a conserved quality.Notice that due to the anisotropy, θ i , θ r , and θ t are not the incident, reflected, and transmitted angles, which are determined by the azimuthal angles of the group velocities.With the continuity condition of the wave functions at x = 0, the following equations are obtained as Considering the conservation of probability current, the transmitted probability should be given as T = |t| 2 cos θt cos θ i .The transmission and reflection coefficients can be analyzed as follows To exhibit the interesting tunneling behaviors under the modulation of light, we then concentrate on two cases: (1) the light fields in the left and right regions represented by the symbols L L and L R are same; (2) the light field in the right region is applied, and in the left region is absent L L = 0.In the first case, when the parameters of the light are tuned to the same as those in figure 1(c), the angles θ i and θ t are always 0. No matter what the wave vector is, the incident angles for all the electrons on K D (K ′ D ) valley are +(−)20.4• , the transmission coefficient remains to be one.The all-electrons Klein tunneling happens and it is robust against the wave vector and energy, a schematic of perfect valley electron beam splitting is plotted in figure 4(c).It should be noted that the special tunneling phenomenon is different from the previously reported Klein tunneling (the electrons at normal incidence are transmitted perfectly) [53] and super-Klein tunneling (the electrons at any incident angles are transmitted perfectly) [54].Here, all the electrons propagate along ±20.4 • and are transmitted completely.Under another special parameters of the light field γ = v 2 F /v 2 y and ξ = π/2, the velocities of all the electrons are along the y direction, there are no electrons to penetrate.The transmission coefficient as a function of the wave vector is shown in figure 4(b).We can find that the numerical calculation is consistent with the analysis of the pseudospin texture in section 3.2.In the second case, the light field is only applied in the N region, the incident angle of the electrons in P region can be arbitrary.For the special illumination parameters as those in figure 1(c), the angle θ t remains to be 0, the transmission coefficient is deduced as T = 1 − tan 2 (θ i /2).Only when θ i = 0, Klein tunneling happens, at other incident angles, the transmission coefficient is not 1.However, for the special illumination parameters as those in figure 1(d), the angle θ t is always π/2.From equation (37), we can find that the transmission coefficient remains to be zero.No matter what the incident angles and energies are, all electrons are forbidden, as the schematic shown in figure 4(e).The all-electrons Klein tunneling is also different from the oblique Klein tunneling reported previously [28].These peculiar transport behaviors including the omnidirectional reflection result from the pseudospin collapse under the modulation of the linearly polarized light.They are very interesting and rarely found in other systems.In addition, we can infer that if the shape of the interface is changed, the electrons can be screened absolutely by utilizing the off-resonant linearly polarized light.We can find that besides the traditional modulation measures to generate band gap, the electron can also be forbidden by forming special dispersion with semimetal.
To show the robust properties of the all-electrons Klein tunneling against the width and height of the barrier, we further consider the transmission of an n-p-n junction.The schematic is shown in figure 5(a), where the gate voltage V 0 is applied in the middle region, and the light is irradiated on the whole n-p-n junction for simplification.The forms of the wave functions in the left and right regions are the same as those in equations ( 33) and ( 34), but the wave vectors k xi , k xr and k xt meet the dispersion In the middle region, the wave function can be written as where the wave vectors q xi and q xr satisfy the dispersion x and ξ = 0, e iθ 1i = −e iθ1r = e iθ2r = −e iθ 2i = 1.Hence, the transmission coefficient T in equation ( 40) is always 1.However, for the parameters γ = v 2 F /v 2 y and ξ = π/2, e iθ 1i = e iθ1r = i, the transmission coefficient T in equation (40) remains to be 0. The all-electrons transmission and reflection are maintained in the n-p-n junction no matter what the width and the height of the barrier are.For other illumination parameters such as γ = 0.3v 2 F /v 2 y and ξ = π/4, we can see from the figures that the Klein tunneling at k y = 0 is also robust against the height and width of the barrier.

Conclusions
In summary, we have investigated the anomalous electronic transport properties in the 8-Pmmn borophene under the modulation of an off-resonant linearly polarized light.It is found that the anisotropy of the band structure is tuned effectively by changing the amplitude, the frequency, and the polarization direction of the light.The cone-shaped dispersion of the borophene can be renormalized into two wedged structures with different illumination parameters.Not only the band structure but also the pseudospin texture undergoes a dramatic alternation.The intrinsic helical nature of the Dirac cones in pristine 8-Pmmn borophene can collapse into two special directions.Utilizing the special properties, the electron beam shows a ballistic propagation without any diffraction.Valley electron beams splitter, multidirectional supercollimators, and all-electrons transmission and omnidirectional reflection all can be realized in this system.Although the band structure is anisotropic in pristine 8-Pmmn borophene, it cannot be modulated.By applying the linearly polarized light, all the new physics properties above which cannot be realized in the pristine system are attributed to the modulation of the off-resonant linearly polarized light.Beyond providing a convenient way to construct novel quasi-one-dimensional electronic states by the off-resonant linearly polarized light, our findings are expected to be useful in exploring potential applications in Dirac-like materials especially in electron-optics.

Figure 2 .
Figure 2. Pseudospin textures of the conduction band electrons in the 8-Pmmn borophene with different illumination parameters.The arrows denote the magnitude and direction of the pseudospin projection in the kx-ky plane.The parameters in (a)-(d) are the same as those in figures 1(a)-(d), respectively.

Figure 3 .
Figure 3.The wave packet evolution with time under different illumination parameters.(a) The electron probability density for the initial Gaussian wave packet at t = 0. (b)-(e) The electron probability density at time t = 15 in the position coordinates.The corresponding illumination parameters are given in the figures.The angle θv denotes the propagation direction of the wave packet with respect to the x axis.
).With the method of Green's function, we study the wave packet evolution with time.The probability densities of the electrons at time t = 15 are plotted in figures 3(b)-(e) for different illumination parameters γ = 0.3v 2 F /v 2 y and ξ = π/4, γ = v 2 F /v 2 x and ξ = 0, γ = v 2 F /v 2 y and ξ = π/2, respectively.In the absence of the light, the wave packet splits into two parts along x axis with different directions.The center of the wave packet with K D (K ′ D ) valley propagates along the angle θ v = 20.4• (θ v = −20.4• ), but the wave packets have a large extension, and both have a large mixture.When the illumination parameters are changed into γ = 0.3v 2

Figure 4 .
Figure 4. (a) Schematic of an off-resonant linearly polarized light modulated n-p junction, where the left and right light fields are represented by the symbols LP and LN.(b) and (d) The transmission of the electrons on KD and K ′ D valleys through the junction as a function of the incident angle under different illumination parameters.(c) and (e) The corresponding schematics of the valley electron beam splitter and omnidirectional reflection.

2 F /v 2
According to the continuity condition of the wave functions at the interfaces,Ψ L (0) = Ψ M (0) and Ψ M (d) = Ψ R (d).The transmission amplitude t can be analyzed ast = ( e iθ 1i − e iθ1r ) ( e iθ 2i − e iθ2r ) (e iθ 1i − e iθ 2i ) (e iθ1r − e iθ2r ) e iq xi d − (e iθ1r − e iθ 2i ) (e iθ 1i − e iθ2r ) e iqxrd .(40)The potentials in the left and right regions are the same, so the transmission coefficient can be written as T = |t| 2 .The transmission coefficients for the electrons on K D valley as a function of the wave vector k y with different barrier widths and heights are shown in figures 5(b) and (c), respectively.The transmission coefficient for the electrons on K ′ D valley is symmetric about the k y = 0.For the special parameters γ = v

Figure 5 .
Figure 5. (a) Schematic of an n-p-n junction based on the 8-Pmmn borophene, where the whole region is irradiated by the same off-resonant linearly polarized light.The height and width of the barrier are V0 and d, respectively.(b) The transmission coefficient as a function of the wave vector ky with different illumination parameters and barrier widths.The height of the barrier is given as V0 = 0.2 eV.The incident energy is E = 0.05 eV, and η = 1.(c) The transmission coefficient as a function of the wave vector ky with different illumination parameters and barrier heights.The width of the barrier is d = 80 nm.The other parameters are the same as those in (b).