Landau levels in uniaxially strained graphene: A geometrical approach

Abstract

The effect of strain on the Landau levels (LLs) spectra in graphene is studied, using an effective Dirac-like Hamiltonian which includes the distortion in the Dirac cones, anisotropy and spatial-dependence of the Fermi velocity induced by the lattice change through a renormalized linear momentum. We propose a geometrical approach to obtain the electron's wave-function and the LLs in graphene from the Sturm-Liouville theory, using the minimal substitution method. The coefficients of the renormalized linear momentum are fitted to the energy bands, which are obtained from a Density Functional Theory (DFT) calculation. In particular, we evaluate the case of Dirac cones with an ellipsoidal transversal section resulting from uniaxially strained graphene along the armchair (AC) and zig-zag (ZZ) directions. We found that uniaxial strain in graphene induces a contraction of the LLs spectra for both strain directions. Also, is evaluated the contribution of the tilting of Dirac cone axis resulting from the uniaxial deformations to the contraction of the LLs spectra.

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.…”

“…which is obtained from the Dirac points equation 3 j t j e −i K D · δ j = 0, it is possible to show that It is important to mention that the parameters a and b can be related with the extremal angle of the elliptical Dirac cones [49]. This shows clearly that the physical properties of strained graphene can be modulated by the geometrical parameters of the Dirac cone and therefore of the strain using the relation (11).…”

“…Furthermore, the quantity √ ab is always less than one for positive deformations, (see Fig. 2(b)), causing the contraction of LLs spectra [49,91]. In constrast, for compressing graphene lattice the quantity √ ab increases and we can expand the LLs spectra.…”

“…where ω ζ is a frequency defined by ω ζ = 2eB ζ . Since this problem is very similar to solve the quantum harmonic oscillator, the LLs spectra is straightforwardly obtained [49]…”

“…For these motivations, we propose the study of quasi-probability distributions for describing the time evolution of electron dynamics in uniaxially strained graphene under a uniform and perpendicular magnetic field. We start developing an effective Weyl-Dirac model based [49]. Such parameters allow us to modulate successfully the shape of the WF through the tensile parameter.…”