Anomalous caustics and Veselago focusing in 8-Pmmn borophene p–n junctions with arbitrary junction directions

Abstract

Negative refraction usually demands complex structure engineering while it is very natural for massless Dirac fermions (MDFs) across the p-n junction (PNJ), this leads to Dirac electron optics. The emergent Dirac materials may exhibit hitherto unidentified phenomenon due to their nontrivial band structures in contrast to the isotropic MDFs in graphene. Here, as a specific example, we explore the negative refraction induced caustics and Veselago focusing of tilted MDFs across 8-Pmmn borophene PNJs. To this aim, we develop a technique to effectively construct the electronic Green's function (GF) in PNJs with arbitrary junction directions. Based on analytical discussions and numerical calculations, we demonstrate the strong dependence of interference pattern on the junction direction. As the junction direction perpendicular to the tilt direction, Veselago focusing or normal caustics (similar to that in graphene) appears resting on the doping configuration of the PNJs, otherwise anomalous caustics (different from that in graphene) occurs which is manipulated by the junction direction and the doping configuration. Finally, the developed GF technique is generally promising to uncover the unique transport of emergent MDFs, and the discovered anomalous caustics makes tilted MDFs potential applications in Dirac electron optics.

Valley-dependent time evolution of coherent electron states in tilted anisotropic Dirac materials

“…It is important to note that the Landau level spectra (8) depends on the valleys K and K'. The factor (9) indicates whether the orbits are closed when |β ν | < 1 or opened for |β ν | ≥ 1 [11,73,74]. The transition from closed to opened orbits appears for the collapse of Landau levels at the critical value of β ν = ±1, where classically the electrons travel on a straight line.…”

“…Most of the two-dimensional materials discovered so far, such as borophene, strained graphene, Weyl semimetals, and the organic conductor α-(BEDT-TTF) 2 I 3 present anisotropy and tilted Dirac cones in their low energy band structure [1][2][3][4][5][6][7][8][9][10][11][12]. The intrinsic anisotropy of some materials like phosphorene has shown intriguing phenomena such as negative reflection, antisuper-Klein tunneling, and perfect electronic waveguides [13,14].…”

“…Meanwhile, the anisotropy induced by strainengineering in graphene has led to the possibility of obtaining pseudo-magnetic fields [15,16], the tuning of electronic and optical properties [17][18][19][20][21][22][23][24][25][26][27], valley polarized currents [28,29], and anomalous tunneling [30,31]. The electronic and transport properties are valley-dependent due to the tilting of Dirac cones, as shown in the relative spacing of Landau levels [11,32,33], pseudo-magnetic fields [4], and other physical properties [8,9,17,[34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. These singular features motivate the investigation of unusual effects by the intrinsic Dirac cone tilt.…”

Phosphorene pnp junctions as perfect electron waveguides

“…The ballistic electron propagation in 2D materials allows the observation of analogies from optics [42,[48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66]. For example, it has been shown that an electron beam, which hits the interface of a pn junction, is refracted similar to a light beam at the interface of two different media.…”

Unusual phase transition of layer-stacked borophene under pressure

“…The maximum Fermi velocity of 8-P mmn borophene along the chain direction is nearly 1.5 times that of graphene [9]. Such unique band structure stimulated further study of the electronic properties of 8-P mmn borophene, such as anisotropic plasmons, magnetotransport properties, anomalous Klein tunneling, valley-dependent electron retroreflection, Veselago focusing, Ruderman-Kittel-Kasuya-Yosida exchange interaction, metal-insulator transition, and the photoinduced Hall effect [11][12][13][14][15][16][17][18][19][20]. Most recently, a variety of exotic electronic states, including flat-band phases, superconductivity, magnetism, and the Chern insulators phases, were discovered moiré superlattice systems based on multilayer graphene [21][22][23][24][25].…”