Quasi-bound states and resonant skew scattering in two-dimensional materials with a Mexican-hat dispersion
Introduction
Mexican-hat shaped dispersion of electronic spectrum is a relatively common property of many two-dimensional (2D) materials such as topological insulators in which the Mexican-hat dispersion naturally appears as a consequence of the band inversion [1], bilayer graphene [2], [3], monolayers of group III–IV chalcogenides [4], [5], [6]. The main interest to the Mexican-hat dispersion is usually attracted due to a Van Hove singularity of density of states close to the band edge, which opens the way for many striking effects caused by electron correlations, including the formation of a stable ferromagnetic phase [7], [8], stimulation of electron pairing [9], [10], and dramatic changes in the spectrum of the bound state in the attractive potential [11], [12]. Recent experiments reveal sharp peaks in optical conductivity due to this feature of the density of states [13].
In this paper, we show that there is another feature of the Mexican-hat dispersion which also leads to nontrivial effects. This feature consists in the fact that the sign of the effective mass in the momentum space near the central extremum is opposite to the sign of the mass outside this region. For definiteness, consider a Mexican-hat dispersion in the conduction band. The effective mass of the electrons with the wave vectors near the central maximum is negative, while outside this region the effective mass is positive. Therefore, it can be expected that a negatively charged defect that normally repels band electrons will attract electrons with momenta near the center of the Mexican hat and thereby create a quasi-bound state against the background of a continuum of band states. We show that these states do exist and study their properties in the case when the Mexican-hat dispersion is due to the hybridization of electron-like and hole-like band states described within the frame of the Bernevig–Hughes–Zhang model [14].
The energy of quasi-bound states lies above the central maximum of the Mexican hat, where they form resonances of the local density of states. The key role in the mechanism of the formation of quasi-bound states is played by the hybridization of the electron and hole bands. In particular, the hybridization parameter largely determines the width of the resonance. But besides this, the resonance width also depends on some overlap integral of the localized component of the wave function and the wave function of the continuum states near the defect. Because of this, the quasi-bound states have an interesting feature: under certain conditions, the resonance width can vanish and the quasi-bound state turns into a bound state in a continuum.
An interesting question is about the possible manifestations of quasi-bound states in the experiment. To this end, let us study how the quasi-bound state manifests itself in the process of scattering of band electrons. It turns out that the quasi-bound state strongly enhances the spin-dependent skew scattering of electrons with energies close to resonance ones, and as the energy changes, the skewness angle changes from 0 to . In addition, in a certain energy range, a nontrivial effect of spin-dependent scattering suppression occurs in the direction opposite to the skewness angle for each spin.
Section snippets
Quasi-bound states in a repulsive potential
As a model of Mexican-hat dispersion we use four-band model of Bernevig, Hughes, and Zhang (BHZ) in which such a dispersion arises due to inversion and hybridization of the electron- and hole-like bands. For simplicity the model is supposed to be symmetric with respect to the electron and hole bands. The hybridization is conveniently described by a dimensionless parameter , where , , and are standard parameters of the BHZ model [14]. is the mass term, the parameter describing
Electron scattering by defects with a resonant state
In this section we study the scattering of band electrons by a point-like defect with a repulsive potential that creates a quasi-bound state. Interest in this problem stems from our expectation that it is precisely in scattering that the quasi-bound states under study can manifest themselves most clearly.
The problem is solved on the basis of the Hamiltonian (1) for spin-up states. According to the standard scattering theory [19], the wave function should asymptotically, at , be the sum of an
Discussion and concluding remarks
We have shown that in materials with a Mexican-hat dispersion, defects with a localized repulsive potential create specific quasi-bound states, which lead to the enhanced skew scattering of electrons with very nontrivial polar diagram and energy dependence. The mechanism of the formation of the quasi-bound states is due to the feature of the Mexican-hat dispersion, which has remained unexplored until now. It consists in the fact that the sign of the effective mass in the region of -space near
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was carried out in the framework of the state task for the Kotelnikov Institute of Radio Engineering and Electronics.
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2023, Physics Letters, Section A: General, Atomic and Solid State Physics