Klein Tunneling through Double Barrier in ABC‐Trilayer Graphene

Abstract

Klein tunneling and conductance for Dirac fermions in ABC-stacked trilayer graphene through symmetric and asymmetric double potential barrier are investigated. This was done by using the continuum model of two and six-bands. The numerical results show that the transport is sensitive to the height, the width, and the distance between the two barriers. It is found that the Klein paradox at normal incidence (k y = 0) and resonant features at k y ≠ 0 in the transmission result from resonant electron states in the wells or hole states in the barriers. It is shown that such features strongly influence the ballistic conductance of the structures. IntroductionGenerally, graphene [1][2][3][4] is a 2D lattice of carbon atoms arranged in hexagonal geometry. Its stacking can be realized in different methods to engineer multi-layered graphene showing various physical properties. Typical examples of stacking includes order, Bernal (AB), and rhombohedral stacking (ABC). [5][6][7][8][9][10] In the first all carbon atoms of each layer are well-aligned, while the second and third have cycle periods composed, respectively, of two and three layers of non-aligned graphene. It was shown that the band structure, Klein tunneling, band gap, transport and optical properties of graphene depend on the way how its layers are stacked [5,9,[11][12][13][14][15][16][17][18][19][20][21][22] and the applied external sources. [5,9,19,[23][24][25][26][27][28][29][30][31][32][33] Recently, trilayer graphene (TLG) has attracted more attention. [15,19,26,29,31,[34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49] It has two distinct allotropes: the Bernal (ABA) and rhombohedral (ABC) stackings. ABA has atoms of the top layer lie exactly on top of the bottom layer. It possess a dispersion relation as summation of the linear and quadratic dispersions corresponding to the single layer and bilayer, respectively. It has no opening gap under applied external electric field. [50] As for ABC, the atoms of one of the sublattices of topmost layer lie above the center of the hexagons of bottom layer.

Transport properties in ABC-ABA-ABC trilayer graphene junctions

“…For V 0 = 0 in figure 5, the transmission and reflection channels exhibit symmetry with respect to k y = 0, akin to the cases of unbiased ABC-TLG [46] and unbiased AB-BLG [18,27] tunneling. However, for V 0 = 1.5γ 1 in figure 6, the scattered transmission channels, i.e., T i j with i ≠ j, and all reflection channels are no longer symmetric with respect to k y = 0, similar to the outcome for the double barrier in biased ABC-TLG [37]. This asymmetrical behavior can be attributed to the non-alignment of ABC-TLG's bands outside and ABA-TLG's bands inside the ABC-ABA-ABC junction due to the presence of a potential barrier [56].…”

“…These wave vectors play a crucial role in comprehending the distinct propagation modes across our junction. Specifically, for [37,46], there is only one mode of propagation corresponding to the wave vector k 1 . However, for g > ¢ E…”

“…1 , three propagation modes emerge, associated with the wave vectors k 1 , k 2 , and k 3 , as depicted in figure 3 (for further details, refer to [37,46]).…”

“…where Ψ represents the spinor solutions in regions (I, III), and a  is a 6 × 6 diagonal matrix with the diagonal elements being the three Pauli matrices σ x . Utilizing equation (33), we deduce the incident J inc , reflected J ref , and transmitted J tra currents that lead to [37,46]…”

“…Furthermore, AA-BLG exhibits Klein tunneling similar to MLG, while AB-BLG demonstrates perfect reflection [6,20,[22][23][24][25][26][27]. Trilayer graphene comes in three distinct configurations: ABC (rhombohedral), ABA (bernal), and AAA stackings [28][29][30][31][32][33][34][35][36][37][38][39]. The ABC stacking displays a cubic dispersion with conduction and valence bands touching at low energy, while the ABA stacking exhibits both MLG linear bands and BLG parabolic bands [28,31,33,40].…”