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Valence and conduction bands of electrons in graphene. The two bands touch each other in six points of the Brillouin zone, called Dirac points.
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We present an exact algebraic solution of a single graphene plane in transverse electric and perpendicular magnetic fields. The method presented gives both the eigenvalues and the eigenfunctions of the graphene plane. It is shown that the eigenstates of the problem can be cast in terms of coherent states, which appears in a natural way from the for...
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Multicritical behavior of interacting fermions in graphene's honeycomb lattice is presented. In particular, we considered the spin triplet insulating orders, where the spin rotational symmetry of the order parameter is explicitly broken. By casting the problem in terms of Gross-Neveu-Yukawa theory, we show that such symmetry-breaking terms are irre...
The computational research that will be presented compares the coherent states of multiple layer graphene versus the coherent states of lithium ions diffused within this multilayer graphene. Unlike the prevailing research on graphene coherent states that uses an external magnetic vector potential, this research will utilize an electric vector poten...
We construct the coherent states for charge carriers in a graphene layer immersed in crossed external electric and magnetic fields. For that purpose, we solve the Dirac-Weyl equation in a Landau-like gauge avoiding applying techniques of special relativity, and thus we identify the appropriate raising and lowering operators associated to the system...
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... The presence in (1) of a term proportional to the unity matrix σ 0 prevents this decoupling (an analogous problem appears in the presence of the mass term proportional to σ 3 ). Different methods have been used [12][13][14][15][16][17][18][19][20][21][22][23] to study such two-dimensional Dirac equations, mainly with strong restrictions on the conditions of the problem. There are problems with only electrostatic or only magnetic fields, as well as the problems with different specific one-dimensional ansatzes for external fields depending on a variable x 1 or radial variable r. ...
... Thus, Equations (6) and (7) are reduced to the system of Equations (12) and (17) in the four possible variants above. We have not yet considered the matrix differential Equation (8). ...
... Equations (10) and (11) are fulfilled automatically. Equations (12), (14), (15) and (17) ...
It is known that the excitations in graphene-like materials in external electromagnetic field are described by solutions of a massless two-dimensional Dirac equation which includes both Hermitian off-diagonal matrix and scalar potentials. Up to now, such two-component wave functions were calculated for different forms of external potentials, though as a rule depending on only one spatial variable. Here, we shall find analytically the solutions for a wide class of combinations of matrix and scalar external potentials which physically correspond to applied mutually orthogonal magnetic and longitudinal electrostatic fields, both depending really on two spatial variables. The main tool for this progress is provided by supersymmetrical (SUSY) intertwining relations, specifically, by their most general—asymmetrical—form proposed recently by the authors. This SUSY-like method is applied in two steps, similar to the second order factorizable (reducible) SUSY transformations in ordinary quantum mechanics.
... [20,32,33]). Supersymmetric nature of the (2+1) Dirac Hamiltonian in the presence of electric and magnetic fields has been reported in [32,[34][35][36]. Supersymmetric QM leads to correct results for the considered problem. ...
... An important problem is a description of a graphene electron in crossed uniform electric and magnetic fields. In this case, the contraction or collapse of Landau levels takes place [34][35][36][37][38][39][40]. Similar effects occur in a uniform electric field and a pseudo-mangetic field induced by a strain [41][42][43] and in uniform magnetic and radial electric fields [44]. ...
The relativistic Foldy-Wouthuysen transformation is used for an advanced description of planar graphene electrons in external fields and free (2+1)-space. It is shown that the initial Dirac equation should by based on the usual $(4\times4)$ Dirac matrices but not on the reduction of matrix dimensions and the use of $(2\times2)$ Pauli matrices. Nevertheless, the both approaches agree with the experimental data on graphene electrons in a uniform magnetic field. The pseudospin of graphene electrons is not the one-value spin and takes the values $\pm1/2$. The exact Foldy-Wouthuysen Hamiltonian of a graphene electron in uniform and nonuniform magnetic fields is derived. The exact energy spectrum agreeing with the experiment and exact Foldy-Wouthuysen wave eigenfunctions are obtained. These eigenfunctions describe multiwave (structured) states in the (2+1)-space. It is proven that the Hermite-Gauss beams exist even in the free space. In the multiwave Hermite-Gauss states, graphene electrons acquire nonzero effective masses dependent on a quantum number and move with group velocities which are less than the Fermi velocity. Graphene electrons in a static electric field also can exist in the multiwave Hermite-Gauss states defining non-spreading coherent beams. These beams can be accelerated and decelerated.
... The cost-effectiveness, thermodynamic conductivity, chemical stability, large surface area, and desired charge mobility are among the properties that make graphene suitable for photocatalytic applications. As the energy gap between the conduction band and valence band is zero for graphene at the Dirac points [98], band gap broadening becomes essential for the improvement in catalytic performance. The broadening of bandgap can be accomplished through structural tuning by the insertion of functional groups or heteroatoms [92]. ...
... After discovery of graphene an attention to 2 + 1relativistic fermions in crossed fields was brought because of the spectacular phenomenon of Landau level collapse. As the dimensionless parameter β = cE/(v F H) reaches its critical value, |β c | = 1, the Landau level staircase merges [9,10]. Here H is a magnetic field H applied perpendicular to the sheet of graphene, E is an applied in-plane electric field E, v F is the Fermi velocity and CGS units are used. ...
... The spectrum of an infinite graphene's sheet in presence of crossed uniform electric and magnetic fields was investigated analytically in Ref. [9] by means of a "Lorenz boost" transformation that eliminates the electric field and thus reduces the problem of finding spectrum to the known one. The same problem was addressed in Ref. [10] using algebraic methods. The influence of a Hall electric field on the Hall conductivity in graphene was analytically studied in Ref. [13] using the spectrum and wave functions found in Refs. ...
... The influence of a Hall electric field on the Hall conductivity in graphene was analytically studied in Ref. [13] using the spectrum and wave functions found in Refs. [9,10]. Another possibility to realize the Landau-level collapse would be by generating strain induced either pseudomagnetic or electric fields suggested in Refs. ...
Herein, the edge states in a finite‐width graphene ribbon and a semi‐infinite geometry subject to a perpendicular magnetic field and an in‐plane electric field, applied perpendicular to a zigzag edge, are explored. To accomplish this, a combination of analytic and numerical methods within the framework of low‐energy effective theory is employed. Both the gapless and gapped Dirac fermions in graphene are considered. It is found that a surface mode localized at the zigzag edge remains dispersionless even in the presence of electric field. This is shown analytically by employing Darwin's expansion of the parabolic cylinder functions of large order and argument.
... An important relativistic effect of the Dirac system occurs when an electric field is applied perpendicularly to the magnetic field: It has been shown that, at a critical value of the electric field, the LLs collapse and the spectrum becomes continuus again. The collapse of LLs was first predicted to occur in graphene [11][12][13] and the possible experimental confirmations were analyzed in Refs. 14,15 . ...
The collapse of Landau levels under an electric field perpendicular to the magnetic field is one of the distinctive features of Dirac materials. So is the coupling of lattice deformations to the electronic degrees of freedom in the form of gauge fields which allows the formation of pseudo-Landau levels from strain. We analyze the collapse of Landau levels induced by strain on realistic Weyl semimetals hosting anisotropic, tilted Weyl cones in momentum space. We perform first-principles calculations, to establish the conditions on the external strain for the collapse of Landau levels in TaAs which can be experimentally accessed.
... However, the experimental exploration of the relativistic Landau levels, in contrast to the nonrelativistic ones, in condensed matter systems became possible almost 80 years later after the discovery of graphene [4,5]. Naturally, the most exciting are the properties of the relativistic Landau levels that do not have their counterparts for standard electron systems and among them is the Landau-level collapse phenomenon predicted in Ref. [6] (see also Ref. [7]) and observed experimentally in Refs. [8,9]. ...
... For = 0, the spectrum Eqs. (3) reduces the spectrum obtained by an exact solution of the problem [6,7]. The Landau level collapse occurring at |β| = 1 can be viewed as a transition from the closed elliptic quasiparticle orbits for |β| < 1 (|v 0 | < v F ) to open hyperbolic orbits for |β| > 1 (|v 0 | > v F ) [13]. ...
... Comparing the system of Eq. (13) with the corresponding system describing the Dirac fermions in the uniform magnetic field and constant electric field in the x direction [6,7] (see also Ref. [23]), one can see that the latter contains only two matricesB +Cx. The problem in the crossed uniform fields in the Cartesian coordinates is exactly solvable by diagonalizing the matrixC, while the present problem with the radial electric field cannot be solved analytically. ...
It is known that in two-dimensional relativistic Dirac systems placed in orthogonal uniform magnetic and electric fields, the Landau levels collapse as the applied in-plane electric field reaches a critical value ±Ec. We study this phenomenon for a distinct field configuration with in-plane constant radial electric field. The Dirac equation for this configuration does not allow analytical solutions in terms of known special functions. The results are obtained by using both the WKB approximation and the exact diagonalization and shooting methods. It is shown that the collapse occurs for positive values of the total angular momentum quantum number, the hole (electron)-like Landau levels collapse as the electric field reaches the value +(−)Ec/2. The investigation of the Landau level collapse in the case of gapped graphene shows a number of distinctive features in comparison with the gapless case.
... 7(a) and 7(b)]. Such pseudo-Landau levels are symmetric with respect to the Brillouin zone center because of the time-reversal symmetry, and thus are fundamentally different from the ordinary Landau levels that produce quantum Hall effects [52,53]. ...
... (30)] used to model the spinorbit coupling, the hopping parameters in Eqs. (51) are chosen to be purely real and exponentially varying as t 1,2 (y) = t exp g 1 − 1 4 (1 + λy) 2 + 3 4 , t 3 (y) = t exp(−gλy), (52) where t ∈ [0.02t, 0.2t] is the next-nearest-neighbor hopping in the absence of strain [60]. Following the procedure we have formulated in Sec. ...
Nonuniform elastic strain is known to induce pseudo-Landau levels in Dirac materials. But these pseudo-Landau levels are hardly resolvable in an analytic fashion when the strain is strong because of the emerging complicated space dependence in both the strain-modulated Fermi velocity and the strain-induced pseudomagnetic field. We here analytically characterize the solution to the pseudo-Landau levels in strongly bent graphene nanoribbons by treating the effects of the nonuniform Fermi velocity and pseudomagnetic field on equal footing. The analytic solution is detectable through angle-resolved photoemission spectroscopy and allows quantitative comparison between theories and various experimental signatures of transport, such as the Shubnikov-de Haas oscillation in the complete absence of magnetic fields and the negative strain-resistivity resulting from the valley anomaly. The analytic solution can be generalized to various Dirac materials and will shed light on the related experimental explorations and straintronics applications.
... Quantized Landau levels (LLs) have been obtained through diagonalizing similar Hamiltonian in previous works [40,[49][50][51][52][53][54][55][56][57], for which an appropriate hyperbolic or Lorentz boost transformation is found to be very helpful. ...
... Note that, the electron motion based on Eq. (2) can be arranged into quantized LLs only within the so-called magnetic regime [58,59]. This condition has been revealed as either β E x < 1 [40,[53][54][55]57] or β ω y < 1 [49,51,56] in previous works. In contrast, |β| = |β w y + β E x | < 1 is expected here to support the quantized LLs. ...
In view of searching the signature of the celebrated chiral anomaly (CA) in Weyl semimetals (WSMs) in ongoing experiments, quantum oscillation in linear response regime has been considered as an important signature in the magneto transports in WSMs, due to its unique relation to CA. Investigating the nonlinear planar effects (NPEs) starting from the semiclassical regime to the ultra-quantum limit within the framework of Boltzmann transport theory incorporating Landau quantization, we here propose the quantum oscillations in NPEs can serve as a robust signature of CA in WSMs. By obtaining analytical expressions, we show that the quantum oscillations of the nonlinear effects exhibit two different period scales in 1/B (B is the magnetic field) compared to the linear responses where only one period scale exists. We find that these quantum oscillations in NPEs are attributed to the deviation of chiral chemical potential (CCP), which is proportional to the finite band tilt as well as transverse electric field and therefore, directly linked to CA in WSMs. In addition, we also show that the CA-induced nonlinear magneto conductivity is linear and independent in the magnetic field in the semiclassical and ultraquantum regimes, respectively. We conclude that the proposed behaviors of NPEs in different regimes uniquely signify the existence of CA and therefore, can serve as a probe of identifying CA in WSMs in experiments.
... While Eq. (7) is derived from a nearest-neighbor model, we generalize it in Sec. SVI of Ref. [42] to incorporate various realistic effects such as the Semenoff mass [44][45][46][47], the spin-orbit coupling [48][49][50], the electric fields [51][52][53][54][55][56][57][58], and the nextnearest-neighbor hoppings [59]. ...
Nonuniform elastic strain is known to induce pseudo-Landau levels in Dirac materials. But these pseudo-Landau levels are hardly resolvable in an analytic fashion when the strain is strong, because of the emerging complicated space dependence in both the strain-modulated Fermi velocity and the strain-induced pseudomagnetic field. We analytically characterize the solution to the pseudo-Landau levels in experimentally accessible strongly bent graphene nanoribbons, by treating the effects of the nonuniform Fermi velocity and pseudomagnetic field on equal footing. The analytic solution is detectable through the angle-resolved photoemission spectroscopy (ARPES) and allows quantitative comparison between theories and various transport experiments, such as the Shubnikov-de Haas oscillation in the complete absence of magnetic fields and the negative strain-resistivity resulting from the valley anomaly. The analytic solution can be generalized to twisted two-dimensional materials and topological materials and will shed a new light on the related experimental explorations and straintronics applications.
... Graphene is generally described by massless Dirac fermions [1], nevertheless different techniques have been developed for nanotechnological applications and for exploring non-trivial topological properties, to generate a gap at the Dirac points [2][3][4] so to include a mass term in the Dirac-Weyl Hamiltonian, which describes the low-energy physics in graphene [1,[5][6][7]. Mass terms, confining scalar potentials or magnetic fields can spoil the simple linear dispersion of the original massless Dirac fermions [8][9][10][11][12][13][14][15]. ...
We calculate the spectrum of massive Dirac fermions in graphene in the presence of an inhomogeneous magnetic field modeled by a step function. We find an analytical universal relation between the bandwidths and the propagating velocities of the modes at the border of the magnetic region, showing how, by tuning the mass term, one can control the speed of these traveling edge states.
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