Thermal conductance by Dirac fermions in a normal-insulator-superconductor junction of silicene

Abstract

We theoretically study the properties of thermal conductance in a normal-insulatorsuperconductor junction of silicene for both thin and thick barrier limit. We show that while thermal conductance displays the conventional exponential dependence on temperature, it manifests a nontrivial oscillatory dependence on the strength of the barrier region. The tunability of the thermal conductance by an external electric field is also investigated. Moreover, we explore the effect of doping concentration on thermal conductance. In the thin barrier limit, the period of oscillations of the thermal conductance as a function of the barrier strength comes out be π/2 when doping concentration in the normal silicene region is small. On the other hand, the period gradually converts to π with the enhancement of the doping concentration. Such change of periodicity of the thermal response with doping can be a possible probe to identify the crossover from specular to retro Andreev reflection in Dirac materials. In the thick barrier limit, thermal conductance exhibits oscillatory behavior as a function of barrier thickness d and barrier height V0 while the period of oscillation becomes V0 dependent. However, amplitude of the oscillations, unlike in tunneling conductance, gradually decays with the increase of barrier thickness for arbitrary height V0 in the highly doped regime. We discuss experimental relevance of our results.

Quantum charge pumping through resonant crossed Andreev reflection in a superconducting hybrid junction of silicene

“…Non-relativistic free electrons with energy E incident on a potential barrier with height V 0 are described by an exponentially decaying (non-oscillatory) wave function inside the barrier region if E < V 0 , since the dispersion relation is k ∼ √ E − V 0 . On the contrary, relativistic free electrons satisfies a dispersion k ∼ (E − V 0 ), consequently corresponding wave functions do not decay inside the barrier region 30,72,73 . Instead, the transmittance of the junction displays an oscillatory behavior as a function of the strength of the barrier.…”

“…Hence, the undamped oscillatory behavior of CAR is a direct manifestation of the relativistic low-energy Dirac fermions in silicene. The periodicity depends on the Fermi wave-length mismatch between the normal and superconducting region 30,31 .…”

“…Here, the superconducting region being located between 0 < x < L, while the insulating barriers are situated on its left, −d < x < 0, and on its right, L < x < L+d. The normal region of silicene occupies at the extreme left i.e., x < −d and extreme right ends, x > L + d. Here, superconductivity is assumed to be induced in the silicene sheet via the proximity effect, where a bulk s-wave superconductor is placed in close proximity to the sheet in the region 0 < x < L. The two insulating regions in silicene have gate tunable barriers of strength χ 1 and χ 2 in the thin barrier limit 30,31 . Two additional gate voltages G 1 and G 2 can tune the chemical potential in the left and right normal silicene regions respectively.…”

“…The silicene NISIN junction can be described by the Dirac Bogoliubov-de Gennes (DBdG) equation of the form 29,30 Ĥη ∆ 1…”

“…( 3)) by replacing µ m → µ m − V 0 (V 0 ) where V 0 and V 0 are the applied gate voltages at the left and right insulating regions respectively. We define dimensionless barrier strengths 30,31…”