Supersymmetric quantum electronic states in graphene under uniaxial strain

Abstract

We study uniaxially strained graphene under the influence of non-uniform magnetic fields perpendicular to the material sample with a coordinate independent strain tensor. For that purpose, we solve the Dirac equation with anisotropic Fermi velocity and explore the conditions upon which such an equation possesses a supersymmetric structure in the quantum mechanical sense through examples. Working in a Laudau-like gauge, wave functions and energy eigenvalues are found analytically in terms of the magnetic field intensity, the anisotropy scales and other relevant parameters that shape the magnetic field profiles.

Phase-space representation of Landau and electron coherent states for uniaxially strained graphene

“…By using tight-binding approach to nearest neighbors, it was possible to derive an effective Dirac-Weyl model, where we obtained Landau levels spectra and their corresponding wave functions. This effective model has a direct connection with other methods frequently used for studying strained materials such as the geometrical approach [49], where an analytical expression of the geometrical parameters a and b as a function of the strain tensor components is found, as well as connection with the supersymmetric potential model [86]. Employing a generalized annihilation operator, we can build the electron coherent states from Landau ones and demonstrate that stretching along the zigzag direction favors the obtention of electron coherent states.…”

“…This effective one-dimensional potential has been discussed in a supersymmetric point of view [86] and we show how this potential through the parameter ζ is tuned with the tensile strain in Fig. 2(c).…”

Dirac electron in graphene with magnetic fields arising from first-order intertwining operators

“…The first-order intertwining method presented here generalizes the shape invariant technique that different authors have used previously to describe the behavior of an electron near to the Dirac points in a graphene layer with applied external magnetic fields [23][24][25][26][27][28]. Similarly as Figure 7: Second intertwining for the Morse potential with k = 6α, ν 1 = − 3 2 , ν 2 = − 1 2 , 1 = − 1 2 E − 1 = − 11α 2 2 , 2 = −E − 1 = −11α 2 : (a) generated potential V 2 (x, 2 ) (continuous line) and initial oneṼ 1 (x, 1 ) (dashed line), with energy levels E n (x)| 2 for the ground state (GS, blue) and the excited states n = 1, 2, 3 (red, green, purple); (d) current density for the excited states with the same colors that in (c).…”

Trends in Supersymmetric Quantum Mechanics

“…In addition, the coherent state methods have been started to be applied recently to graphene subject to static homogeneous magnetic fields [179]. As can be seen, the SUSY methods applied to Dirac materials is a very active field which surely will continue its development in the near future [151][152][153][154][155][156][157][158][159][160][161].…”

“…Recently, the SUSY methods started to be used also in the study of Dirac electrons in graphene and some of its allotropes, when external electric or magnetic fields are applied [151][152][153][154][155][156][157][158][159][160][161]. It is worth to mention as well some systems in optics, since there is a well-known correspondence between Schrödinger equation and Maxwell equations in the paraxial approximation, which makes that the SUSY methods can be applied directly in some areas of optics [162][163][164][165][166][167][168][169].…”