Quasi-exact solutions for guided modes in two-dimensional materials with tilted Dirac cones

Abstract

We show that if the solutions to the (2+1)-dimensional massless Dirac equation for a given one-dimensional (1D) potential are known, then they can be used to obtain the eigenvalues and eigenfunctions for the same potential, orientated at an arbitrary angle, in a 2D Dirac material possessing tilted, anisotropic Dirac cones. This simple set of transformations enables all the exact and quasi-exact solutions associated with 1D quantum wells in graphene to be applied to the confinement problem in tilted Dirac materials such as 8-Pmmn borophene. We also show that smooth electron waveguides in tilted Dirac materials can be used to manipulate the degree of valley polarization of quasiparticles travelling along a particular direction of the channel. We examine the particular case of the hyperbolic secant potential to model realistic top-gated structures for valleytronic applications.

Optical absorption in two-dimensional materials with tilted Dirac cones

“…With the recent burst of interest in massless tilted Dirac cone materials there have been several theoretical works investigating the valley-dependent transport of carriers traversing gated junctions, waveguides and external fields [34][35][36][37][38]. Combining these transport techniques with the optical spatial separation of valley carriers proposed in our work could enable the design of valleytronic components such as valley filters and switches in gapless materials.…”

Bipolar electron waveguides in two-dimensional materials with tilted Dirac cones

“…In practical experiments, it is not yet possible to select the exact angle between the top gate and the crystallographic orientation. Although we will consider the specific case of aligning the waveguide along the y-direction, it should be emphasized that one can derive spectra for any waveguide orientation by applying the mapping method outlined in [26].…”

“…Furthermore, if a bound-state solution exists for a given E and Δ, then a solution also exists for E and −Δ. Therefore, the eigenvalue spectrum of the s = -1 valley can be obtained by reflecting the eigenvalue spectrum of the s = 1 valley about k y = 0 [26].…”

“…and optical absorption [14], to now incorporate a tilt term. Transport across quasi-1D heterostructures [72][73][74][75] in tilted Dirac materials has also been investigated, as has the alternative geometry: propagating fully confined modes along electrostatic potentials [26].…”

“…For the other two types, the equienergy contours are open: in a type-II cone the equienergy contours are hyperbolic, while in type-III they are parabolic [8,9]. These various cone geometries lead to remarkably different physical properties [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. In certain materials, the Dirac cones can be naturally tilted due to the material's inherent crystal structure and properties, and there is a plethora of theoretical materials that have been predicted to possess tilted Dirac cones [7,[30][31][32][33][34][35][36][37][38][39][40][41][42][43].…”