Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0038 Acta Polytechnica 62(1):38–49, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague ABOUT THE TIME EVOLUTION OF COHERENT ELECTRON STATES IN MONOLAYERS OF BORON ALLOTROPES Erik Díaz-Bautista Instituto Politécnico Nacional, Unidad Profesional Interdisciplinaria de Ingeniería Campus Hidalgo, Departamento de Formación Básica Disciplinaria, Ciudad del Conocimiento y la Cultura, Carretera Pachuca-Actopan km 1+500, San Agustín Tlaxiaca, 42162 Hidalgo, México correspondence: ediazba@ipn.mx Abstract. In this paper, we theoretically analyze the massless Dirac fermion dynamics in two- dimensional monolayers of boron allotropes, 8B and 2BH − pmmn borophene, interacting with external electric and magnetic fields. We study the effect of the Dirac cone tilt in these materials, which is known as valley index, through the time evolution of probability density of coherent electron states as well as their phase-space representation obtained via the Wigner function. Our results show that the time evolution of the coherent electron states in these materials is valley dependent, which is reinforced in the presence of external electric fields. Keywords: Tilted Dirac cones, anisotropic Dirac materials, borophene, coherent states, Wigner function. 1. Introduction Coherent states (CSs) are minimal uncertainty states [1, 2], so that they are considered the most classical states in quantum mechanics. For this rea- son, they arise in multiple branches of physics, mainly in quantum optics [2] and information processes [3, 4]. Although the wave function provides interesting fea- tures about any state, its experimental realization in several quantum systems [3] requires a different approach. In this sense, the Wigner function (WF) constitutes one of the most important theoretical tools for describing quantum systems in the phase-space representation. The WF for a bidimensional system is a quasi-probability distribution defined as [5–7] W (r,p) = 1 (2π)2 ∫ ∞ −∞ ei p·q Ψ ( r − q 2 ) Ψ† ( r + q 2 ) dq, (1) where Ψ(r) is the wave function, r = (x, y) and p = (px, py) are two-dimensional vectors representing the classical position and momentum values in phase space, respectively; and q = (q1, q2) is a position vec- tor needed in the integration process. In contrast with the probability density of any quantum state, the WF can take negative values, which indicates the nonclassicality of a state and it is a sign of quantum- ness [3, 8, 9]. Despite this, it has been implemented in quantum optics [3, 4, 9–13], and recently also applied in condensed matter for studying electron dynamics in two-dimensional materials, particularly in graphene under the presence of electromagnetic fields [3, 14– 26] as well as in strained honeycomb lattices with dispersive pseudo-Landau-levels [27]. Following this trend, and since the number of two- dimensional materials has been increasing recently, our aim is to provide an adequate description in phase space of the physics of certain quantum macro- scopic phenomena, and their semi-classical represen- tation, that occurs in condensed matter systems in the context of valleytronics [28–30]. In this emerg- ing research area, two-dimensional materials such as 8 − pmmn borophene [31–35], strained graphene [36], Weyl semimetals [37–40], and the organic compound α-(BEDT-TTF)2I3 [41–44], characterize due to the presence of anisotropy and tilted Dirac cones at their low-energy band structure [31, 32, 40, 43, 45–53]. The anisotropy and tilt can be intrinsic, as occurs in borophene [31–34] and phosphorene [54, 55], or induced by strain-engineering and external electric fields, as observed in graphene [20, 36, 56–72]. In this work, we will focus on two-dimensional mono- layers of boron allotropes, 8B and 2BH − pmmn borophene [73], which have recently attracted at- tention due to the boron capacity of flexible bond- ing [47, 74, 75]. The geometry of two-dimensional boron-based Dirac cone materials is much more com- plicated than that of the pristine honeycomb structure of graphene and, as a consequence, their electronic and transport properties are valley dependent due to the tilting of Dirac cones. These features moti- vate the research of unusual effects by the intrinsic Dirac cone tilt under the presence of external elec- tric and magnetic fields. For instance, the study of 8 − pmmn borophene conductivity in the presence of crossed electric and magnetic fields exhibits a clear valley-dependence in magnetotransport properties and polarization currents [33, 34]. It exists the possibility of the realization of coherent electron states in the laboratory and the development of electron quantum optics [15, 76, 77] due to the recent advances in the experimental reconstruction of the WF of electronic systems in quantum tomography experiments. 38 https://doi.org/10.14311/AP.2022.62.0038 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 62 no. 1/2022 Coherent electron states in monolayers of boron allotropes Thus, this paper is organized as follows. In Sec- tion 2, we describe the Dirac Hamiltonian of mono- layers of boron allotropes at low-energy regime in the presence of external electric and magnetic fields, and obtain the corresponding Landau states and energy spectra. In Section 3, we discuss the construction of the matrix ladder operators associated to the physical system and also construct the corresponding coher- ent electron states as eigenstates of the annihilation operator. We also study the time evolution of the probability density and the corresponding phase-space representation. In Section 4, we present our conclu- sions. 2. Electron dynamics in monolayers of boron allotropes 8 − pmmn borophene [31–34] and other two- dimensional monolayers of boron allotropes, such as 8B and 2BH − pmmn borophene [73], present tilted Dirac cones at their low electronic band structure, so that their electronic properties are described by the continuous Dirac Hamiltonian H = ν (vtσ0py + vxσxpx + vyσypy), (2) where the matrices σx,y are the Pauli matrices, while σ0 is the identity matrix. The quantity ν, known as valley index, allows us to transit from valley K (ν = 1) to valley K’ (ν = −1). The terms vx and vy are the anisotropic Fermi velocities and vt is the velocity that quantifies the tilting of the Dirac cone. These velocities depend on the material. For instance, the velocity values {vx, vy, vt} in multiples of the Fermi velocity vF = 1 for three different allotropes of boron are shown in Table 1. Boron allotrope vx vy vt 8 − pmmn 0.86 0.69 0.32 8B − pmmn 0.534 0.785 -0.345 2BH − pmmn 0.77 1.348 -0.386 Table 1. The velocities vx, vy and vt in units of the Fermi velocity vF = 106 m/s, for three boron allotropes in the low-energy single-particle effective model. 2.1. Effective Dirac-Weyl Hamiltonian Now, let us consider massless Dirac fermions in a two- dimensional boron monolayer under the presence of an in-plane electric field E = E x̂ and a perpendicular magnetic field B = Bẑ. These fields are included in the Hamiltonian in Eq. (2) through the scalar and vector potentials U = −x E , A = xBŷ, (3) to obtain the following eigenvalue equation in natural units (e = −1 and ℏ = 1) [35, 42, 43]: H ′Ψ̄(r) = (ν [vxσxpx + (vtσ0 + vyσy)(py + xB)] + xEσ0) Ψ̄(r) = EΨ̄(r). (4) Taking advantage of the translational invariance along the y-axis, so that Ψ̄(r) = exp(ikyy)Ψ(x), the eigen- value equation in Eq. (4) becomes:[( E − xc E v′ F ) σ0 + i∂xσx − ( xc ωB + 2kc y ) 2 σy ] Ψ(x) = 0, (5) where E = E− ν vtky, E = (E + ν vtB) √ vx/vy, ωB = 2B, xc = x √ vy/vx, kc y = ky √ vy/vx, and v′ F = √ vxvy is an effective Fermi velocity. By introducing the parameter βν and the dimensionless quantity ξ, βν = E v′ FB = vd vy + ν vt vy , ξ = √ ωB 2 ( xc + 2kc y ωB ) , (6) with vd = E/B being the drift velocity, Eq. (5) can be rewritten as[( ϵ0 − √ ωB 2 βνξ ) σ0+i √ ωB 2 d dξ σx− √ ωB 2 ξσy ] Ψ(ξ) = 0, (7) where ϵ0 = E/v′ F + kc yβν = (E + kyvd)/v′ F. 2.1.1. Energy spectrum In order to find the solutions of the initial problem, we proceed as follows [78, 79]. Multiplying by −iσx to the left of Eq. (7), we get:[√ ωB 2 d dξ σ0 − i ( ϵ0 − √ ωB 2 βνξ ) σxσ0 + i √ ωB 2 ξσxσy ] Ψ(ξ) = 0. (8) Differentiating the above expression with respect to ξ, we obtain the following equation[( d2 dξ2 + (√ 2 ωB ϵ0 − βνξ )2 − ξ2 ) σ0 + K ] Ψ(ξ) = 0, (9) where K = i (σxβν + σxσy) is a complex symmetric matrix. The solutions of Eq. (9) can be expressed as Ψ(ξ)η = ψη(ξ)χη, where χη fulfills the eigenvalue equation Kχη = ηχη, and ψη(ξ) is a scalar function that satisfies the differential equation[ d2 dξ2 + (√ 2 ωB ϵ0 − βνξ )2 − ξ2 + η ] ψη(ξ) = 0. (10) In order to simplify the above equation, the variable ζ is defined as ζ = ξ(1 − β2 ν)1/4 + √ 2 ωB βν ϵ0 (1 − β2 ν)3/4 , (11) 39 Erik Díaz-Bautista Acta Polytechnica (a) ν = 1 (b) ν = −1 0.00 0.25 0.50 0.75 1.00­2.0 ­1.0 0.0 1.0 2.0 E  [e V ] n = 0 n = ± 1 n = ± 2 n = ± 3 n = ± 4 0.0 0.1 0.2 0.3 0.4 ­1.5 ­1.0 ­0.5 0.0 0.5 1.0 1.5 E  [e V ] n = 0 n = ± 1 n = ± 2 n = ± 3 n = ± 4 (c) ν = 1 (d) ν = −1 0.0 0.5 1.0 1.5 ­3.0 ­2.0 ­1.0 0.0 1.0 2.0 3.0 E  [e V ] n = 0 n = ± 1 n = ± 2 n = ± 3 n = ± 4 0.0 0.2 0.4 0.6 0.8 1.0 ­2.0 ­1.0 0.0 1.0 2.0 E  [e V ] n = 0 n = ± 1 n = ± 2 n = ± 3 n = ± 4 Figure 1. Energy spectrum in Eq. (15) with ky = 0 and B = 1 as a function of the electric field E for 8B − pmmn borophene (a, b) and 2BH − pmmn (c, d) in each Dirac point (ν = ±1). where βν must fulfill the condition |βν | < 1 for keeping real values of ζ. Hence, we obtain the Weber equation[ d2 dζ2 − ζ2 + 2ϵ2 0 ωB(1 − β2 ν)3/2 + η (1 − β2 ν)1/2 ] ψη(ζ) = 0. (12) On the other hand, the eigenvalues η of the matrix K turn out to be σ(K) = {ηk = (−1)k(1 − β2 ν)1/2} with k = 1, 2, while the corresponding normalized eigenvectors are given by χη1 = 1√ 2 ( √ C+ −i √ C− ) , χη2 = 1√ 2 ( − √ C− i √ C+ ) , (13) where C± = 1± √ 1 − β2 ν . Substituting the eigenvalues ηk in Eq. (12) and taking ψη(ζ) = exp ( −ζ2/2 ) fη(ζ), one gets the following ODE: f ′′ η (ζ) − 2 ζf ′ η(ζ) = ( 1 − (−1)k − 2ϵ2 0/ωB (1 − β2 ν)3/2 ) fη(ζ), (14) with k = 1, 2. By solving the above ODE, the energy spectrum turns out to be [80] (see Figure 1): En,ky = sgn(n)√vxvy(1−β2 ν)3/4 √ ωB|n|−ky vd. (15) The Landau energy levels in Eq. (15) depend on valleys K and K’ [43] via the amount βν in Eq. (6), which indicates whether the orbits are closed (|βν | < 1) or opened (|βν | ≥ 1) [43, 80, 81]. Also, the critical values of vc d = Ec for which βν = 1 depend on each tilted anisotropic Dirac material (see again Figure 1). For instance, in Table 2 we summarize the critical values Ec in valleys K and K’ for three monolayers of boron allotropes. Borophene monolayer Ec (K) Ec (K’) 8 − pmmn 0.37 1.01 8B − pmmn 1.13 0.44 2BH − pmmn 1.734 0.962 Table 2. The electric field critical value Ec = (vy − νvt)B for three boron allotropes according to the valley index ν. The data for 8 − pmmn borophene were obtained from [35]. 40 vol. 62 no. 1/2022 Coherent electron states in monolayers of boron allotropes Finally, the average velocity in the y-direction is given by [82, 83] ⟨vy⟩ = ∂En,ky ∂ky = −vd = [ E × B B2 ] y , (16) that means the Dirac fermions move with an average velocity vd in the negative y-direction. 2.1.2. Eigenstates The eigenstates of the Hamiltonian in Eq. (4) can be written as Ψn(x) = [(1 − δ0n)χη1ψn−1(x) + λχη2ψn(x)]√ 2(1−δ0n) = MΦn(x), (17) where δmn denotes the Kronecker delta, the band index λ indicates the conduction (λ = 1) or (λ = −1) valence band, and M = √ 1 2 ( √ C+ i √ C− −i √ C− √ C+ ) , (18a) Φn(x) = 1√ 2(1−δ0n) ( (1 − δ0n)ψn−1(x) iλψn(x) ) . (18b) The components of the pseudo-spinor Φn(x) are given by the wave functions [79] ψn(ζn) = (1 − β2 ν)1/8 √ n! ( ωBvy 2πvx )1/4 Dn( √ 2 ζn), (19) where Dn(·) are the parabolic cylinder functions with n a non-negative integer, and the quantity ζn is given by Eq. (11) with ϵ0 = (En,ky + kyvd)/√vxvy. 3. Coherent electron states We start to obtain the CSs by first considering the set of ladder operators A± given in [35] and acting on the Hilbert basis Φn(x) = M−1Ψn(x), namely, A+Φn(ζn) = √ 2(1−δ0n) √ n+ 1 Φn+1(ζn+1), (20a) A−Φn(ζn) = √ 2(1−δ1n)√nΦn−1(ζn−1), (20b) and whose commutation relation reads [A−,A+]Φn(x) = c(n)Φn(x), c(n) =  1, n = 0, 3, n = 1, 2, n > 1. (21) Now, we define the CSs as eigenstates of the anni- hilation operator A−: A−Φz(x) = zΦz(x), z ∈ C, (22) with complex eigenvalue, where Φz(x) = ∞∑ n=0 anΦn(x). (23) Using Eq. (20a), the explicit expression for the CSs is given by Φz(x) = Nα [ Φ0(x) + ∞∑ n=1 √ 2αn √ n! Φn(x) ] , (24) where N −2 α = 2 exp ( |α|2 ) − 1 and α = z/ √ 2 = |α| exp (iφ). Here, the physical meaning of |α| is that it is the oscillation amplitude while the phase angle φ is identical to the angular rotation in the classical motion. It is worth to mention that the procedure described allows to obtain the so-called Barut-Girrardello CSs. However, this is not the only way to build CSs. In [27], the displacement-operator method has been implemented in order to construct such states in other honeycomb lattices. Finally, defining the matrix operators B− ≡ MA−M−1, B+ ≡ MA+M−1, (25) whose actions on the Landau states Ψn(x) in Eq. (17) reads as B+Ψn(ζn) = √ 2(1−δ0n) √ n+ 1 Ψn+1(ζn+1), (26a) B−Ψn(ζn) = √ 2(1−δ1n)√nΨn−1(ζn−1), (26b) it is possible to verify that the states Ψα(x) = MΦz(x) are eigenfunctions of the annihilation operator B− with the same eigenvalue z. Therefore, the states Ψ̄α(r) = exp (ikyy) Ψα(x) are the coherent electron states of the system. In addition, the commutation relation in Eq. (21) is also fulfilled writing B± and Ψn instead of A± and Φn, respectively. 3.1. Overcompleteness and resolution to the identity The CSs satisfy the following relation |⟨Ψ̄α′ |Ψ̄α⟩| = ∣∣∣∣∣ 2 exp(α′∗α) − 1√ (2 exp(|α|2) − 1)(2 exp(|α′|2) − 1) ∣∣∣∣∣ ̸= δ(α′ − α). (27) Since the coherent electron states are not orthogonal for α ̸= α′, we say the set of such states is overcom- plete. Besides, these CSs satisfy the following relation that can be considered as an unusual resolution to the identity: |Ψ̄0⟩⟨Ψ̄0| 2 + ∫ C dρ(α) π |Ψ̄α⟩⟨Ψ̄α| = I+, (28) where I+ denotes the identity operator in the Hilbert space for Landau states in the conduction band, and dρ(α) is a positive measure defined as dρ(α) = 2 exp ( |α|2 ) − 1 2 exp (−|α|2) |α| d|α| dφ. (29) 41 Erik Díaz-Bautista Acta Polytechnica 0 5 10 15 20 25 30 35 n 0.00 0.03 0.05 0.08 0.10 0.13 0.15 0.18 0.20 P (n ) = 4 = 8 = 12 = 16 = 20 Figure 2. Occupation number distribution Pα(n) in Eq. (30) for the coherent electron states Ψ̄α for different values of µ = |α|2. 3.2. Occupation number distribution The CSs follow a Poisson-like distribution with mean µ = |α|2, according to the occupation number distri- bution Pα(n) = |⟨Ψ̄n|Ψ̄α⟩| = 1 2 exp (µ) − 1 { 1, n = 0, 2µn n! , n > 0, (30) which gives the probability of a CS of being in an Landau state Ψ̄n (see Figure 2). 3.3. Mean energy value On the another hand, the expectation value of the energy in the CS basis is given by ⟨H⟩α = N 2 α [ kyvd ( 1 − 2 exp ( |α|2 )) + 2√ vxvy ωB × (1 − β2 ν)3/4 ∞∑ n=1 sgn(n) |α|2n n! √ |n| ] . (31) The mean group velocity of the CSs is obtained as ⟨vy⟩α = ∂⟨H⟩α ∂ky = −vd, (32) which agrees with Eq. (16). 3.4. Time evolution of the wave packet Now, let us consider the time-evolution operator U(t) = exp(−iHt) applied on the expansion of CSs in terms of Landau states Ψ̄n(r). Hence, the time- dependent coherent electron states are: Ψ̄α(r, t) = Nα exp (ikyy)M ( ψα,1(x, t) i λψα,2(x, t) ) , (33) where ψα,1(x, t) = ∞∑ n=1 αne−iEnt √ n! ψn−1(x), (34a) ψα,2(x, t) = ∞∑ n=0 αne−iEnt √ n! ψn(x). (34b) The time-dependent probability density |Ψ̄α(r, t)|2 is |Ψ̄α(r, t)|2 = N 2 α { |ψα,1(x, t)|2 + |ψα,2(x, t)|2 − 2λβνℜ [ ψ∗ α,1(x, t)ψα,2(x, t) ] } , (35) where ℜ(z) denotes the real part of a complex number z. Figure 3 shows the time evolution of the probability distribution of the CSs for 8B and 2BH − pmmn borophene. We can see that in both cases, the density probability in valley K evolves faster than that in valley K’. This means that Dirac fermions take less time to complete a loop around the equilibrium point. The function |Ψ̄α(r, t)|2 shows maximum values close to the turning points in the x-axis. Besides, with the values chosen for the parameters α = 4i, ky = 0 and B = 1, the probability density of the CSs with ν = −1 for 8B − pmmn borophene shows a different behavior in time in comparison to the other cases. This is related to the fact for the electric field considered (E = 0.25), the corresponding energy spectrum is near to collapse, indicating the classical orbits that the charge carriers in valley K’ follow are more open compared with those in K for 8B − pmmn borophene, and even for those ones in 2BH − pmmn borophene. 3.5. Obtaining of the time-dependent Wigner function for coherent electron states To calculate the Wigner matrix (WM) [84] for the coherent states in Eq. (33) in valleys K and K’, we substitute them into the integral matrix representation in Eq. (1) to get [35]: Wα(r,p) = MWα(r,p)M†. (36) Thus, the trace of this matrix provides us an expres- sion of the time-dependent WF of the coherent states: Tr[Wα(r,p, t)] = N 2 α δ (py − ky) { W11(χ, t) +W22(χ, t) − 2λβνℜ[W12(χ, t)] } , (37) where the components W11, W22 and W12 are the cor- responding Wigner functions of the terms in Eq. (34) and their product, and that in general involve sums 42 vol. 62 no. 1/2022 Coherent electron states in monolayers of boron allotropes (a) ν = 1 (b) ν = −1 10 5 0 5 10 x 0 20 40 60 80 100 t 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15 10 5 0 5 x 0 50 100 150 200 t 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (c) ν = 1 (d) ν = −1 10 5 0 5 10 x 0 20 40 60 t 0.00 0.12 0.24 0.36 0.48 0.60 0.72 0.84 0.96 10 5 0 5 x 0 20 40 60 80 t 0.000 0.135 0.270 0.405 0.540 0.675 0.810 0.945 1.080 1.215 Figure 3. Time evolution of the probability density |Ψα(r, t)|2 in Eq. (35) with ky = 0, B = 1, α = 4i, E = 0.25 and λ = 1 for 8B − pmmn borophene (a, b) and 2BH − pmmn borophene (c, d) in each Dirac point (ν = ±1). of functions of the form (see [35] for more details) Wu,v(χn,m) = 1 π exp ( −1 2 |χn,m|2 + i(ζn − ζm)s ) ×  (−1)u √ u! v!χ v−u n,mL v−u u ( |χn,m|2 ) , if u ≤ v, (−1)v √ v! u!χ ∗u−v n,m Lu−v v ( |χn,m|2 ) , if u ≥ v, (38) with Lm n (·) denoting the associated Laguerre polyno- mials, and χn,m = ζn + ζm√ 2 + i √ 2 s, χn ≡ χn,n, (39a) s = ( 1 − β2 ν )−1/4 √ 2vx ωBvy px. (39b) The time evolution of the WM trace for the CSs is shown in Figures 4 and 5 for 8B and 2BH − pmmn borophene, respectively. In both cases, we observe that the WF in valley K propagates faster than in valley K’. As the state evolves in time, the trace of the WM takes negative values, which is an indication of the increasing of the CS quantumness and also of the uncertainty relations, as is discussed in [35]. For larger times, the WM traces become identical to that of the Landau state with n equal to the integer part of |α|2, in agreement to the number occupation distribution in Eq. (30). 3.5.1. Period of motion Now, in order to provide an approximate period τ for the CSs, we proceed as follows [85]. First, we calculate the mean energy ⟨H⟩α for the CSs Ψ̄α(r). Then, setting the eigenvalue z, we compute the energy interval in which ⟨H⟩α lies, namely, Ej,ky < ⟨H⟩α < Ej+1,ky . Thus, the approximate period is determined as: τ = 2π ∆E = 2π Ej+1,ky − Ej,ky , (40) that will be different for each valley since the energy spectrum depends on the tilting parameter ν. For instance, for the CSs with α = 4i and the same values used in Figures 3, 4 and 5, we have E15 < ⟨H⟩α < E16. Note that ⟨H⟩α is bounded by the Landau level with n = |α|2 = 16. Thus, the respective periods are reported in Table 3. The period τ in Eq. (40) increases as ∆E → 0 close to the electric field critical value Ec, since a Dirac fermion takes a longer time to complete a loop an opened orbit (see red and blue curves in Figure 6). In contrast, the orbits are closed for more separated energy levels resulting in a shorter period τ (see green curve in Figure 6). 43 Erik Díaz-Bautista Acta Polytechnica (a) t = 0, ν = 1 (b) t = 0, ν = −1 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.000 0.036 0.072 0.108 0.144 0.180 0.216 0.252 0.288 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.000 0.036 0.072 0.108 0.144 0.180 0.216 0.252 0.288 (c) t = 15, ν = 1 (d) t = 15, ν = −1 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.000 0.036 0.072 0.108 0.144 0.180 0.216 0.252 0.288 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.000 0.036 0.072 0.108 0.144 0.180 0.216 0.252 0.288 (e) t = 30, ν = 1 (f) t = 30, ν = −1 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.072 0.036 0.000 0.036 0.072 0.108 0.144 0.180 0.216 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.06 0.03 0.00 0.03 0.06 0.09 0.12 0.15 0.18 (g) t = 45, ν = 1 (h) t = 45, ν = −1 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.081 0.054 0.027 0.000 0.027 0.054 0.081 0.108 0.135 0.162 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.036 0.000 0.036 0.072 0.108 0.144 0.180 0.216 0.252 Figure 4. Time evolution of the trace of the Wigner matrix Wα(r,p) in Eq. (37) for different values of t in each Dirac point (ν = ±1) of 8B − pmmn borophene. B = 1, ky = 0, α = 4i, {vx, vy, vt} = {0.534, 0.785,−0.345}, E = 0.25, and λ = 1. In the figure labels, lB = 1/ √ B. 44 vol. 62 no. 1/2022 Coherent electron states in monolayers of boron allotropes (a) t = 0, ν = 1 (b) t = 0, ν = −1 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.000 0.032 0.064 0.096 0.128 0.160 0.192 0.224 0.256 0.288 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.000 0.032 0.064 0.096 0.128 0.160 0.192 0.224 0.256 0.288 (c) t = 15, ν = 1 (d) t = 15, ν = −1 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.072 0.036 0.000 0.036 0.072 0.108 0.144 0.180 0.216 0.252 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.072 0.036 0.000 0.036 0.072 0.108 0.144 0.180 0.216 (e) t = 30, ν = 1 (f) t = 30, ν = −1 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.081 0.054 0.027 0.000 0.027 0.054 0.081 0.108 0.135 0.162 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.072 0.036 0.000 0.036 0.072 0.108 0.144 0.180 0.216 0.252 (g) t = 45, ν = 1 (h) t = 45, ν = −1 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.075 0.050 0.025 0.000 0.025 0.050 0.075 0.100 0.125 ­15 ­10 ­5 0 5 10 15 x [lB] ­15 ­10 ­5 0 5 10 15 p x  [ /l B ] 0.075 0.050 0.025 0.000 0.025 0.050 0.075 0.100 0.125 0.150 Figure 5. Time evolution of the trace of the Wigner matrix Wα(r,p) in Eq. (37) for different values of t in each Dirac point (ν = ±1) of 2BH − pmmn borophene. B = 1, ky = 0, α = 4i, {vx, vy, vt} = {0.77, 1.348,−0.380}, E = 0.25, and λ = 1. In the figure labels, lB = 1/ √ B. 45 Erik Díaz-Bautista Acta Polytechnica Figure 6. A generic honeycomb lattice interacts with crossed electric and magnetic fields directed along the x- and z-directions, respectively. In absence of electric field E, a non-relativistic classical charged particle performs a circular trajectory (green curve), while if the strength E increases, the trajectory becomes into a trochoid (red and blue curves). The drift velocity vd = E × B/B2 is directed along the y-direction. Borophene monolayer τ+ (K) τ− (K’) 8 − pmmn 34.165π 14.566π 8B − pmmn 17.388π 32.642π 2BH − pmmn 11.012π 13.200π Table 3. Valley-dependent period τ for the CSs with eigenvalue α = 4i = z/ √ 2 in three boron allotropes. The data for 8 − pmmn borophene were obtained from [35]. 3.6. Discussion Anisotropic and tilted Dirac cone materials, such as 8B and 2BH − pmmn borophene, possess valley- dependent electronic properties under the interaction with crossed electric and magnetic fields. The effective Hamiltonian depends on two anisotropic velocities and one tilt velocity (see Eq. (2)). In the case in which these materials interact with external crossed electric and magnetic fields (see Eq. (4)), it is possible to ob- tain the solutions to the physical problem in a simple algebraic way (see Eq. (15) and (17)). We have constructed the CSs Ψ̄α(r) as a linear combination of the Landau eigenfunctions Ψ̄(r) of the Hamiltonian in Eq. (4), that also be eigenfunction of a matrix ladder operator B− with a complex eigenvalue z. As the coherent electron states evolve in time, their probability density clearly shows maximum values only around the turning points in the x-axis, in which momentarily the velocity of charge carriers reduces. In turn, the emergence of negative values in the trace of the Wigner matrix for longer times is related to the increasing uncertainties of the position and mo- mentum, as is studied in [35], in agreement to the probability distribution Eq. (30). Also, the increasing of the electric field to a critical value can delay the time-evolution of CSs in one of the valleys, allowing us to distinguish the Dirac fermions of one valley from those of another (see Figures 3, 4 and 5). In Figure 6, we have showed the classical picture of a (valley-independent) non-relativistic charge carrier that follows a closed trajectory in the xy-plane in presence of an external magnetic field B along the z-direction. When an electric field E is applied along the x-axis, the trajectory becomes into a trochoid with velocity vd directed to the y-direction. In contrast, we observe that in the quantum picture of our problem, due to the Lorentz transformation into the reference frame and the energy [43], as well as the valley in- dex, there is a factor (1 − β2 ν)3/4 that modifies the spacing between two adjacent Landau levels, and as a consequence also the period of motion in Eq. (40). 4. Conclusions We studied the dynamics of massless Dirac fermions in two bidimensional monolayers of boron allotropes un- der the interaction with crossed external electric and magnetic fields. We analyzed the effect of the Dirac cone tilt in the time evolution of probability density of coherent electron states as well as the corresponding Wigner function. We conclude that the time evolu- tion of the coherent electron states in these materials is valley dependent, and the presence of an in-plane external electric field reinforces such a dependency. We consider that the findings here presented may contribute to the understanding of the effects of the tilting of the Dirac cones 8B and 2BH − pmmn borophene on the charge carrier dynamics under the interaction of electromagnetic fields, with which these materials could be considered as viable val- ley splitters in experimental applications. Besides, the coherent state description developed through the phase-space representation may provide a satisfac- tory semi-classical description of similar quantum valley-dependent phenomena that occur in other tilted anisotropic Dirac materials interacting with external electromagnetic fields. Acknowledgements The author acknowledges T. Stegmann and Y. Betancur- Ocampo for their invaluable comments to improve this work, as well as the financial support from CONACYT Project FORDECYT-PRONACES/61533/2020 and SIP- IPN research grant 20210317. References [1] E. Schrödinger. Der stetige Übergang von der Mikro- zur Makromechanik. Naturwissenschaften 14(28):664– 666, 1926. https://doi.org/10.1007/BF01507634. 46 https://doi.org/10.1007/BF01507634 vol. 62 no. 1/2022 Coherent electron states in monolayers of boron allotropes [2] R. J. Glauber. Coherent and incoherent states of the radiation field. Physical Review 131:2766–2788, 1963. https://doi.org/10.1103/PhysRev.131.2766. [3] J. Weinbub, D. K. Ferry. Recent advances in Wigner function approaches. Applied Physics Reviews 5(4):041104, 2018. https://doi.org/10.1063/1.5046663. [4] C. Gerry, P. Knight, C. C. Gerry. Introductory Quantum Optics. Cambridge University Press, 2010. ISBN 052152735X. [5] E. Wigner. On the quantum correction for thermodynamic equilibrium. Physical Review 40(5):749– 759, 1932. https://doi.org/10.1103/physrev.40.749. [6] M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner. Distribution functions in physics: Fundamentals. Physics Reports 106(3):121–167, 1984. https://doi.org/10.1016/0370-1573(84)90160-1. [7] A. Kenfack, K. Zyczkowski. Negativity of the Wigner function as an indicator of non-classicality. Journal of Optics B: Quantum and Semiclassical Optics 6(10):396–404, 2004. https://doi.org/10.1088/1464-4266/6/10/003. [8] D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani. Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum. Physical Review Letters 70(9):1244–1247, 1993. https://doi.org/10.1103/physrevlett.70.1244. [9] C. Baune, J. Fiurášek, R. Schnabel. Negative Wigner function at telecommunication wavelength from homodyne detection. Physical Review A 95(6):061802, 2017. https://doi.org/10.1103/physreva.95.061802. [10] H.-W. Lee. Theory and application of the quantum phase-space distribution functions. Physics Reports 259(3):147–211, 1995. https://doi.org/10.1016/0370-1573(95)00007-4. [11] W. B. Case. Wigner functions and Weyl transforms for pedestrians. American Journal of Physics 76(10):937– 946, 2008. https://doi.org/10.1119/1.2957889. [12] R. P. Rundle, P. W. Mills, T. Tilma, et al. Simple procedure for phase-space measurement and entanglement validation. Physical Review A 96(2):022117, 2017. https://doi.org/10.1103/physreva.96.022117. [13] M. V. Berry. Semi-classical mechanics in phase space: A study of Wigner’s function. Philosophical Transactions of the Royal Society of London Series A, Mathematical and Physical Sciences 287(1343):237–271, 1977. https://doi.org/10.1098/rsta.1977.0145. [14] C. Jacoboni, P. Bordone. The Wigner-function approach to non-equilibrium electron transport. Reports on Progress in Physics 67(7):1033, 2004. https://doi.org/10.1088/0034-4885/67/7/r01. [15] C. Bäuerle, D. C. Glattli, T. Meunier, et al. Coherent control of single electrons: a review of current progress. Reports on Progress in Physics 81(5):056503, 2018. https://doi.org/10.1088/1361-6633/aaa98a. [16] E. Colomés, Z. Zhan, X. Oriols. Comparing Wigner, Husimi and Bohmian distributions: which one is a true probability distribution in phase space? Journal of Computational Electronics 14(4):894–906, 2015. https://doi.org/10.1007/s10825-015-0737-6. [17] D. J. Mason, M. F. Borunda, E. J. Heller. Revealing the flux: Using processed Husimi maps to visualize dynamics of bound systems and mesoscopic transport. Physical Review B 91(16):165405, 2015. https://doi.org/10.1103/physrevb.91.165405. [18] C. M. Carmesin, P. Kling, E. Giese, et al. Quantum and classical phase-space dynamics of a free-electron laser. Physical Review Research 2(2):023027, 2020. https://doi.org/10.1103/physrevresearch.2.023027. [19] O. Morandi, F. Schürrer. Wigner model for quantum transport in graphene. Journal of Physics A: Mathematical and Theoretical 44(26):265301, 2011. https://doi.org/10.1088/1751-8113/44/26/265301. [20] E. Díaz-Bautista, Y. Betancur-Ocampo. Phase-space representation of Landau and electron coherent states for uniaxially strained graphene. Physical Review B 101(12):125402, 2020. https://doi.org/10.1103/physrevb.101.125402. [21] E. Díaz-Bautista, Y. Concha-Sánchez, A. Raya. Barut–Girardello coherent states for anisotropic 2D-Dirac materials. Journal of Physics: Condensed Matter 31(43):435702, 2019. https://doi.org/10.1088/1361-648x/ab2d18. [22] E. Díaz-Bautista, D. J. Fernández. Graphene coherent states. The European Physical Journal Plus 132(11):499, 2017. https://doi.org/10.1140/epjp/i2017-11794-y. [23] D. J. Mason, M. F. Borunda, E. J. Heller. Semiclassical deconstruction of quantum states in graphene. Physical Review B 88(16):165421, 2013. https://doi.org/10.1103/physrevb.88.165421. [24] G. J. Iafrate, V. N. Sokolov, J. B. Krieger. Quantum transport and the Wigner distribution function for Bloch electrons in spatially homogeneous electric and magnetic fields. Physical Review B 96(14):144303, 2017. https://doi.org/10.1103/physrevb.96.144303. [25] P. Ghosh, P. Roy. Quasi coherent state of the Dirac oscillator. Journal of Modern Optics 68(1):56–62, 2021. https://doi.org/10.1080/09500340.2021.1876261. [26] D. K. Ferry, I. Welland. Relativistic Wigner functions in transition metal dichalcogenides. Journal of Computational Electronics 17(1):110–117, 2017. https://doi.org/10.1007/s10825-017-1094-4. [27] E. Díaz-Bautista, M. Oliva-Leyva. Coherent states for dispersive pseudo-Landau-levels in strained honeycomb lattices. The European Physical Journal Plus 136(7):765, 2021. https://doi.org/10.1140/epjp/s13360-021-01753-w. [28] J. R. Schaibley, H. Yu, G. Clark, et al. Valleytronics in 2D materials. Nature Reviews Materials 1(11):16055, 2016. https://doi.org/10.1038/natrevmats.2016.55. [29] A. Kundu, H. A. Fertig, B. Seradjeh. Floquet-engineered valleytronics in Dirac systems. Physical Review Letters 116:016802, 2016. https://doi.org/10.1103/PhysRevLett.116.016802. 47 https://doi.org/10.1103/PhysRev.131.2766 https://doi.org/10.1063/1.5046663 https://doi.org/10.1103/physrev.40.749 https://doi.org/10.1016/0370-1573(84)90160-1 https://doi.org/10.1088/1464-4266/6/10/003 https://doi.org/10.1103/physrevlett.70.1244 https://doi.org/10.1103/physreva.95.061802 https://doi.org/10.1016/0370-1573(95)00007-4 https://doi.org/10.1119/1.2957889 https://doi.org/10.1103/physreva.96.022117 https://doi.org/10.1098/rsta.1977.0145 https://doi.org/10.1088/0034-4885/67/7/r01 https://doi.org/10.1088/1361-6633/aaa98a https://doi.org/10.1007/s10825-015-0737-6 https://doi.org/10.1103/physrevb.91.165405 https://doi.org/10.1103/physrevresearch.2.023027 https://doi.org/10.1088/1751-8113/44/26/265301 https://doi.org/10.1103/physrevb.101.125402 https://doi.org/10.1088/1361-648x/ab2d18 https://doi.org/10.1140/epjp/i2017-11794-y https://doi.org/10.1103/physrevb.88.165421 https://doi.org/10.1103/physrevb.96.144303 https://doi.org/10.1080/09500340.2021.1876261 https://doi.org/10.1007/s10825-017-1094-4 https://doi.org/10.1140/epjp/s13360-021-01753-w https://doi.org/10.1038/natrevmats.2016.55 https://doi.org/10.1103/PhysRevLett.116.016802 Erik Díaz-Bautista Acta Polytechnica [30] Y. S. Ang, S. A. Yang, C. Zhang, et al. Valleytronics in merging Dirac cones: All-electric-controlled valley filter, valve, and universal reversible logic gate. Physical Review B 96:245410, 2017. https://doi.org/10.1103/PhysRevB.96.245410. [31] A. Lopez-Bezanilla, P. B. Littlewood. Electronic properties of 8−Pmmn borophene. Physical Review B 93(24):241405, 2016. https://doi.org/10.1103/physrevb.93.241405. [32] A. D. Zabolotskiy, Y. E. Lozovik. Strain-induced pseudomagnetic field in the Dirac semimetal borophene. Physical Review B 94(16):165403, 2016. https://doi.org/10.1103/physrevb.94.165403. [33] SK Firoz Islam, A. M. Jayannavar. Signature of tilted Dirac cones in weiss oscillations of 8 − Pmmn borophene. Physical Review B 96(23):235405, 2017. https://doi.org/10.1103/physrevb.96.235405. [34] SK Firoz Islam. Magnetotransport properties of 8-Pmmn borophene: effects of Hall field and strain. Journal of Physics: Condensed Matter 30(27):275301, 2018. https://doi.org/10.1088/1361-648x/aac8b3. [35] Y. Betancur-Ocampo, E. Díaz-Bautista, T. Stegmann. Valley-dependent time evolution of coherent electron states in tilted anisotropic Dirac materials. Physical Review B 105(4):045401, 2022. https://doi.org/10.1103/PhysRevB.105.045401. [36] M. Assili, S. Haddad, W. Kang. Electric field-induced valley degeneracy lifting in uniaxial strained graphene: Evidence from magnetophonon resonance. Physical Review B 91(11):115422, 2015. https://doi.org/10.1103/physrevb.91.115422. [37] D. Sabsovich, T. Meng, D. I. Pikulin, et al. Pseudo field effects in type II Weyl semimetals: new probes for over tilted cones. Journal of Physics: Condensed Matter 32(48):484002, 2020. https://doi.org/10.1088/1361-648x/abaa7e. [38] K. Das, A. Agarwal. Linear magnetochiral transport in tilted type-I and type-II Weyl semimetals. Physical Review B 99(8):085405, 2019. https://doi.org/10.1103/physrevb.99.085405. [39] A. Menon, B. Basu. Anomalous Hall transport in tilted multi-Weyl semimetals. Journal of Physics: Condensed Matter 33(4):045602, 2020. https://doi.org/10.1088/1361-648x/abb9b8. [40] P. P. Ferreira, A. L. R. Manesco, T. T. Dorini, et al. Strain engineering the topological type-II Dirac semimetal NiTe2. Physical Review B 103(12):125134, 2021. https://doi.org/10.1103/physrevb.103.125134. [41] J. Sári, M. O. Goerbig, C. Tőke. Magneto-optics of quasirelativistic electrons in graphene with an inplane electric field and in tilted Dirac cones in α-(BEDT TTF)2I3. Physical Review B 92(3):035306, 2015. https://doi.org/10.1103/physrevb.92.035306. [42] M. O. Goerbig, J.-N. Fuchs, G. Montambaux, F. Piéchon. Tilted anisotropic Dirac cones in quinoid-type graphene and α-(BEDT-TTF)2I3. Physical Review B 78(4):045415, 2008. https://doi.org/10.1103/physrevb.78.045415. [43] M. O. Goerbig, J.-N. Fuchs, G. Montambaux, F. Piéchon. Electric-field–induced lifting of the valley degeneracy in α-(BEDT-TTF)2I3 Dirac-like Landau levels. EPL Europhysics Letters 85(5):57005, 2009. https://doi.org/10.1209/0295-5075/85/57005. [44] T. Morinari, T. Himura, T. Tohyama. Possible verification of tilted anisotropic Dirac cone in α-(BEDT-TTF)2I3 using interlayer magnetoresistance. Journal of the Physical Society of Japan 78(2):023704, 2009. https://doi.org/10.1143/jpsj.78.023704. [45] X.-F. Zhou, X. Dong, A. R. Oganov, et al. Semimetallic two-dimensional boron allotrope with massless Dirac fermions. Physical Review Letters 112(8):085502, 2014. https://doi.org/10.1103/physrevlett.112.085502. [46] A. J. Mannix, X.-F. Zhou, B. Kiraly, et al. Synthesis of borophenes: Anisotropic, two-dimensional boron polymorphs. Science 350(6267):1513–1516, 2015. https://doi.org/10.1126/science.aad1080. [47] B. Feng, J. Zhang, Q. Zhong, et al. Experimental realization of two-dimensional boron sheets. Nature Chemistry 8(6):563–568, 2016. https://doi.org/10.1038/nchem.2491. [48] W. Li, L. Kong, C. Chen, et al. Experimental realization of honeycomb borophene. Science Bulletin 63(5):282–286, 2018. https://doi.org/10.1016/j.scib.2018.02.006. [49] Z.-Q. Wang, T.-Y. Lü, H.-Q. Wang, et al. Review of borophene and its potential applications. Frontiers of Physics 14(3):33403, 2019. https://doi.org/10.1007/s11467-019-0884-5. [50] S.-H. Zhang, W. Yang. Oblique Klein tunneling in 8-Pmmn borophene p− n junctions. Physical Review B 97(23):235440, 2018. https://doi.org/10.1103/physrevb.97.235440. [51] S.-H. Zhang, W. Yang. Anomalous caustics and Veselago focusing in 8-pmmn borophene p–n junctions with arbitrary junction directions. New Journal of Physics 21(10):103052, 2019. https://doi.org/10.1088/1367-2630/ab4d8f. [52] N. Tajima, K. Kajita. Experimental study of organic zero-gap conductor α-(BEDT-TTF)2I3. Science and Technology of Advanced Materials 10(2):024308, 2009. https://doi.org/10.1088/1468-6996/10/2/024308. [53] N. P. Armitage, E. J. Mele, A. Vishwanath. Weyl and Dirac semimetals in three-dimensional solids. Reviews of Modern Physics 90(1):015001, 2018. https://doi.org/10.1103/revmodphys.90.015001. [54] Y. Betancur-Ocampo, F. Leyvraz, T. Stegmann. Electron optics in phosphorene pn junctions: Negative reflection and anti-super-Klein tunneling. Nano Lett 19(11):7760–7769, 2019. https://doi.org/10.1021/acs.nanolett.9b02720. [55] Y. Betancur-Ocampo, E. Paredes-Rocha, T. Stegmann. Phosphorene pnp junctions as perfect electron waveguides. Journal of Applied Physics 128(11):114303, 2020. https://doi.org/10.1063/5.0019215. [56] N. Levy, S. A. Burke, K. L. Meaker, et al. Strain- induced pseudo-magnetic fields greater than 300 tesla in graphene nanobubbles. Science 329(5991):544–547, 2010. https://doi.org/10.1126/science.1191700.48 https://doi.org/10.1103/PhysRevB.96.245410 https://doi.org/10.1103/physrevb.93.241405 https://doi.org/10.1103/physrevb.94.165403 https://doi.org/10.1103/physrevb.96.235405 https://doi.org/10.1088/1361-648x/aac8b3 https://doi.org/10.1103/PhysRevB.105.045401 https://doi.org/10.1103/physrevb.91.115422 https://doi.org/10.1088/1361-648x/abaa7e https://doi.org/10.1103/physrevb.99.085405 https://doi.org/10.1088/1361-648x/abb9b8 https://doi.org/10.1103/physrevb.103.125134 https://doi.org/10.1103/physrevb.92.035306 https://doi.org/10.1103/physrevb.78.045415 https://doi.org/10.1209/0295-5075/85/57005 https://doi.org/10.1143/jpsj.78.023704 https://doi.org/10.1103/physrevlett.112.085502 https://doi.org/10.1126/science.aad1080 https://doi.org/10.1038/nchem.2491 https://doi.org/10.1016/j.scib.2018.02.006 https://doi.org/10.1007/s11467-019-0884-5 https://doi.org/10.1103/physrevb.97.235440 https://doi.org/10.1088/1367-2630/ab4d8f https://doi.org/10.1088/1468-6996/10/2/024308 https://doi.org/10.1103/revmodphys.90.015001 https://doi.org/10.1021/acs.nanolett.9b02720 https://doi.org/10.1063/5.0019215 https://doi.org/10.1126/science.1191700 vol. 62 no. 1/2022 Coherent electron states in monolayers of boron allotropes [57] F. Guinea, M. I. Katsnelson, A. K. Geim. Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering. Nature Physics 6(1):30–33, 2009. https://doi.org/10.1038/nphys1420. [58] S. H. R. Sena, J. M. Pereira Jr, G. A. Farias, et al. The electronic properties of graphene and graphene ribbons under simple shear strain. Journal of Physics: Condensed Matter 24(37):375301, 2012. https://doi.org/10.1088/0953-8984/24/37/375301. [59] P. Ghosh, P. Roy. Collapse of Landau levels in graphene under uniaxial strain. Materials Research Express 6(12):125603, 2019. https://doi.org/10.1088/2053-1591/ab52ad. [60] V. M. Pereira, A. H. Castro Neto. Strain engineering of graphene’s electronic structure. Physical Review Letters 103(4):046801, 2009. https://doi.org/10.1103/physrevlett.103.046801. [61] V. M. Pereira, A. H. Castro Neto, N. M. R. Peres. Tight-binding approach to uniaxial strain in graphene. Physical Review B 80(4):045401, 2009. https://doi.org/10.1103/physrevb.80.045401. [62] G. Cocco, E. Cadelano, L. Colombo. Gap opening in graphene by shear strain. Physical Review B 81(24):241412, 2010. https://doi.org/10.1103/physrevb.81.241412. [63] F. M. D. Pellegrino, G. G. N. Angilella, R. Pucci. Strain effect on the optical conductivity of graphene. Physical Review B 81(3):035411, 2010. https://doi.org/10.1103/physrevb.81.035411. [64] H. Rostami, R. Asgari. Electronic ground-state properties of strained graphene. Physical Review B 86(15):155435, 2012. https://doi.org/10.1103/physrevb.86.155435. [65] S. Barraza-Lopez, A. A. Pacheco Sanjuan, Z. Wang, M. Vanević. Strain-engineering of graphene’s electronic structure beyond continuum elasticity. Solid State Communications 166:70–75, 2013. https://doi.org/10.1016/j.ssc.2013.05.002. [66] G. G. Naumis, S. Barraza-Lopez, M. Oliva-Leyva, H. Terrones. Electronic and optical properties of strained graphene and other strained 2D materials: a review. Reports on Progress in Physics 80(9):096501, 2017. https://doi.org/10.1088/1361-6633/aa74ef. [67] D.-N. Le, V.-H. Le, P. Roy. Graphene under uniaxial inhomogeneous strain and an external electric field: Landau levels, electronic, magnetic and optical properties. The European Physical Journal B 93(8):158, 2020. https://doi.org/10.1140/epjb/e2020-10222-3. [68] S. M. Cunha, D. R. da Costa, L. C. Felix, et al. Electronic and transport properties of anisotropic semiconductor quantum wires. Physical Review B 102(4):045427, 2020. https://doi.org/10.1103/physrevb.102.045427. [69] T. Stegmann, N. Szpak. Current flow paths in deformed graphene: from quantum transport to classical trajectories in curved space. New Journal of Physics 18(5):053016, 2016. https://doi.org/10.1088/1367-2630/18/5/053016. [70] T. Stegmann, N. Szpak. Current splitting and valley polarization in elastically deformed graphene. 2D Materials 6(1):015024, 2019. https://doi.org/10.1088/2053-1583/aaea8d. [71] Y. Betancur-Ocampo. Partial positive refraction in asymmetric Veselago lenses of uniaxially strained graphene. Physical Review B 98(20):205421, 2018. https://doi.org/10.1103/physrevb.98.205421. [72] Y. Betancur-Ocampo, P. Majari, D. Espitia, et al. Anomalous floquet tunneling in uniaxially strained graphene. Physical Review B 103(15):155433, 2021. https://doi.org/10.1103/physrevb.103.155433. [73] T. Cheng, H. Lang, Z. Li, et al. Anisotropic carrier mobility in two-dimensional materials with tilted Dirac cones: theory and application. Physical Chemistry Chemical Physics 19:23942–23950, 2017. https://doi.org/10.1039/C7CP03736H. [74] B. Feng, J. Zhang, R.-Y. Liu, et al. Direct evidence of metallic bands in a monolayer boron sheet. Physical Review B 94:041408, 2016. https://doi.org/10.1103/PhysRevB.94.041408. [75] Y. Zhao, X. Li, J. Liu, et al. A new anisotropic Dirac cone material: A B2S honeycomb monolayer. The Journal of Physical Chemistry Letters 9(7):1815–1820, 2018. https://doi.org/10.1021/acs.jpclett.8b00616. [76] D. Ferraro, B. Roussel, C. Cabart, et al. Real-time decoherence of Landau and Levitov quasiparticles in quantum hall edge channels. Physical Review Letters 113(16):166403, 2014. https://doi.org/10.1103/physrevlett.113.166403. [77] T. Jullien, P. Roulleau, B. Roche, et al. Quantum tomography of an electron. Nature 514(7524):603–607, 2014. https://doi.org/10.1038/nature13821. [78] J. Oertel. Solutions of the Dirac equation in spacetime-dependent electric fields. Master’s thesis, University of Duisburg-Essen, Freiberg, Germany, 2014. [79] M. Castillo-Celeita, E. Díaz-Bautista, M. Oliva-Leyva. Coherent states for graphene under the interaction of crossed electric and magnetic fields. Annals of Physics 421:168287, 2020. https://doi.org/10.1016/j.aop.2020.168287. [80] V. Lukose, R. Shankar, G. Baskaran. Novel electric field effects on Landau levels in graphene. Physical Review Letters 98(11):116802, 2007. https://doi.org/10.1103/physrevlett.98.116802. [81] N. Gu, M. Rudner, A. Young, et al. Collapse of Landau levels in gated graphene structures. Physical Review Letters 106(6):066601, 2011. https://doi.org/10.1103/physrevlett.106.066601. [82] J. M. Ziman. Principles of the Theory of Solids. Cambridge University Press, 2nd edn., 1972. https://doi.org/10.1017/CBO9781139644075. [83] T. Huang, R. Chen, T. Ma, et al. Electronic Bloch oscillation in a pristine monolayer graphene. Physics Letters A 382(42-43):3086–3089, 2018. https://doi.org/10.1016/j.physleta.2018.07.035. [84] M. Kai, W. Jian-Hua, Y. Yi. Wigner function for the Dirac oscillator in spinor space. Chinese Physics C 35(1):11–15, 2011. https://doi.org/10.1088/1674-1137/35/1/003. [85] D. J. Fernández, D. I. Martínez-Moreno. Bilayer graphene coherent states. The European Physical Journal Plus volume 135(9):739, 2020. https://doi.org/10.1140/epjp/s13360-020-00746-5. 49 https://doi.org/10.1038/nphys1420 https://doi.org/10.1088/0953-8984/24/37/375301 https://doi.org/10.1088/2053-1591/ab52ad https://doi.org/10.1103/physrevlett.103.046801 https://doi.org/10.1103/physrevb.80.045401 https://doi.org/10.1103/physrevb.81.241412 https://doi.org/10.1103/physrevb.81.035411 https://doi.org/10.1103/physrevb.86.155435 https://doi.org/10.1016/j.ssc.2013.05.002 https://doi.org/10.1088/1361-6633/aa74ef https://doi.org/10.1140/epjb/e2020-10222-3 https://doi.org/10.1103/physrevb.102.045427 https://doi.org/10.1088/1367-2630/18/5/053016 https://doi.org/10.1088/2053-1583/aaea8d https://doi.org/10.1103/physrevb.98.205421 https://doi.org/10.1103/physrevb.103.155433 https://doi.org/10.1039/C7CP03736H https://doi.org/10.1103/PhysRevB.94.041408 https://doi.org/10.1021/acs.jpclett.8b00616 https://doi.org/10.1103/physrevlett.113.166403 https://doi.org/10.1038/nature13821 https://doi.org/10.1016/j.aop.2020.168287 https://doi.org/10.1103/physrevlett.98.116802 https://doi.org/10.1103/physrevlett.106.066601 https://doi.org/10.1017/CBO9781139644075 https://doi.org/10.1016/j.physleta.2018.07.035 https://doi.org/10.1088/1674-1137/35/1/003 https://doi.org/10.1140/epjp/s13360-020-00746-5 Acta Polytechnica 62(1):38–49, 2022 1 Introduction 2 Electron dynamics in monolayers of boron allotropes 2.1 Effective Dirac-Weyl Hamiltonian 2.1.1 Energy spectrum 2.1.2 Eigenstates 3 Coherent electron states 3.1 Overcompleteness and resolution to the identity 3.2 Occupation number distribution 3.3 Mean energy value 3.4 Time evolution of the wave packet 3.5 Obtaining of the time-dependent Wigner function for coherent electron states 3.5.1 Period of motion 3.6 Discussion 4 Conclusions Acknowledgements References