The Bound State In Tilted Dirac Material

Dirac material is a new kind of emerging materials which has the low-energy fermionic quasi-particles described by relativistic Dirac-like equations.The electrons in Dirac materials indicate linear energy momentum relativistic dispersion,which is accurately described by the massless Dirac Hamiltonian.This unique electronic structure leads to many exceptional physical properties and offers a broad perspective in nanoscale electronic devices.Recently,searching for new states of matter and unusual quasi-particles in Dirac materials attracts increased interests in condensed matter physics.Atomic collapse is a relativistic quantum effect in atomic physics.Namely,when nucleus Z of a super-heavy atomic nucleus surpasses a certain threshold Z_c,the strong Coulomb field will cause electron falling toward the nucleus,as well as a positron escaping to infinity.However,it is hard to observe the phenomena in traditional atomic physics because of the very rigid condition,whichZ_c at least 170 is required.Fortunately,the presence of Dirac materials provided a new platform to study quantum electrodynamics in condensed matter physics.Electrons behave as massless relativistic Dirac particles in Dirac materials,which can be used to investigate the atomic collapse states in a strong Coulomb electric field.The solution of Dirac equations in Coulomb potentials is a history issue in quantum physics.In real materials,the Coulomb potential is usually used to describe the interaction between an electron and a hole.The strength of the Coulomb potential can be controlled by having multiple charged impurities or changing the dielectric environment.With the most recent development in low-dimensional materials,the Coulomb problem in reduced dimensions attracts significant interests.Especially,experiments with electrons confined in a plane led to considerations of the Coulomb problem in two dimensions.Nevertheless,the one-dimensional Coulomb problem is also an important and sometimes controversial issue.Searching for new states of matter and unusual quasi-particles in Dirac materials attracts a significant interest in condensed matter physics.Here we investigate the bound state formation in tilted Dirac materials in the presence of the one-and two-dimensional Coulomb potentials.Based on the exact solution of the Dirac equation,we obtain the energy spectrum of the bound states and the corresponding wave function in the one-dimensional case.It is shown that the tilted field facilitates lowing the bound state energy and reducing the energy spacing.For the problem of one-dimensional Coulomb potential in a quasi-relativistic system with a tilted magnetic field,we find that there are some critical band gaps for the loss of low-energy states.Namely,atomic collapse can be observed in Dirac materials.Most notably,it is found that the bound states are absent for strong enough tilted field,such as in type-II Weyl semimetals.For type-II Weyl semimetals,the lower energy bound states dive into the continuum below the band gap and the atomic collapse appears,irrelevant to the strength of the Coulomb potential.For the two-dimensional case,we perform the second-order perturbation theory to study the effect of the tilted field.