Surface states of a system of dirac fermions: A minimal model

Abstract

A brief survey is given of theoretical works on surface states (SSs) in Dirac materials. Within the formalism of envelope wave functions and boundary conditions for these functions, a minimal model is formulated that analytically describes surface and edge states of various (topological and nontopological) types in several systems with Dirac fermions (DFs). The applicability conditions of this model are discussed. I. THE ENVELOPE-FUNCTION METHOD. INTRODUCTION TO THE HISTORY OF THE PROBLEMBy the middle of the 20-th century, the problem arose of explaining and predicting the electronic properties of semiconductors in external fields. Various versions of the single-band effective-mass method were developed. The Luttinger-Kohn envelope-function (EF) method 1 based on a generalization of the kp-approach turned out to be very convenient. The formalism of EFs admitted a natural multiband generalization. Such a generalization was made by Keldysh in his theory of deep impurities 2. In [2], Keldysh also noticed that, under certain conditions, EFs in a narrow-band semiconductor obey an effective Dirac equation (a system of four first-order differential equations). It is interesting that, in III-V semiconductors, this situation occurs under reversal of the sign of the strong spin-orbit splitting of the valence band. The idea of inversion of bands in crystals with strong spinorbit interaction will be addressed below in this paper. In the same year as [2], it was shown [3] that the spectrum of electrons and holes in bismuth near the L points of the Brillouin zone should be described by an anisotropic Dirac Hamiltonian.According to modern terminology, bismuth can be considered as a Dirac material, and the first Dirac material at that. These materials include graphene, bismuth-antimony alloys, lead chalcogenides, 2D and 3D topological insulators, Dirac semimetals, Weyl semimetals, and many other materials. One-particle excitations in these materials are called massless (for gapless materials) or massive Dirac fermions (DFs).The energy spectrum E(p) of free DFs in the threedimensional isotropic case is analogous to the spectrum of a relativistic electron:Here m is the electron effective mass, which is usually one or two orders of magnitude less than the mass of a free electron in vacuum, and c is the effective velocity of light. This velocity is two orders of magnitude less than the velocity of light in vacuum; therefore, real Fermi excitations are, in fact, non-relativistic. However, relativistic corrections due to spin-orbit interaction in crystals with close bands may be rather large, which contributes to the formation of DFs. A finite mass in the spectrum of relativistic particles corresponds to a finite width of the forbidden band according to the well-known formulaThe 1960s marked the emergence of the physics of 2D electron systems. Quantum size effect was started to be used to realize the 3D → 2D transition. The conventional theory of size quantization is based on the single-band effective mass approximation with zero bou...

Robustness and universality of surface states in Dirac materials

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Virial theorem, boundary conditions, and pressure for massless Dirac electrons

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Kagome lattice from an exciton-polariton perspective

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