Signatures of d-wave symmetry on thermal Dirac fermions in graphene-based F|I|d junctions

Abstract

We study theoretically behavior of thermal massless Dirac fermions inside graphene-based Ferromagnetic |Insulator| d-wave/s-wave superconductor (F|I|d and F|I|S) junctions in the ballistic regime. We employ Dirac–Bogoliubov–de Genne formalism and use Dirac–BdG wave functions within the ferromagnetic, normal, and superconducting regions and appropriate boundary conditions, to derive Andreev and normal reflection coefficients. By employing the obtained Andreev and normal reflection coefficients, thin barrier approximation and also appropriate values of parameters, we investigate characteristics of heat current through the F|I|d and F|I|S junctions. We find that for s-wave superconductors, thermal conductance (Γ) oscillates sinusoidally versus strength of barrier in the thin barrier approximation. The finding persists for all values of α, orientation of d-wave superconductor crystal in the k-space, for values of α very close to π/4, magnitude of the oscillations suppress completely. By increasing temperature, the thermal conductance increases exponentially for small values of α and for larger values of superconductor crystal orientation angle the thermal conductance modifies and interestingly for α=π/4 exhibits linear behavior, i.e., Γ∝T which is similar to Wiedemann–Franz law for metals in low temperatures. While increasing the temperature move minimum value of the thermal conductance with respect to strength of magnetization texture h/Δ0 towards smaller values of h/Δ0, increasing the superconductor crystal orientation angle, α from 0 to π/4 enhances whole values of the thermal conductance and consequently the minimum value move towards up. At last we suggest an experimental setup for verifying these findings.