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Electrical and thermal transport in a twisted heterostructure of transition metal dichalcogenide and CrI$_3$ connected to a superconductor

The broad tunability of the proximity exchange effect between transition-metal dichalcogenides (TMDCs) and chromium iodide (CrI$_3$) heterostructures offers intriguing possibilities for the use of TMDCs in two-dimensional magnetoelectrics. In this work, the influence of the twist angle and the gate electric field on the electric and thermal transport in a TMDC/CrI$_3$ junction is investigated using the Dirac -Bogoliubov-de Gennes equation. We show that significant amounts can be controlled by spin-splitting of band structures due to spin-orbit interaction, and that the exchange-splitting of bands arises from the proximity effect. The property of the Andreev reflection (AR) process is highly dependent on the spin valley polarized states due to spin-orbit coupling. Remarkably, perfect spin valley polarized AR is possible over a wide bias range by using a gate voltage to tune the local Fermi energy and varying the type of charge doping. The proposed structure with $p$-type doping is found to have larger spin valley polarized Andreev conductance and high thermal conductance. We further show that, depending on the TMDC material and chemical potential of the TMDC/CrI$_3$ layer, twisting can lead to suppression or a significant increase in Andreev conductance as well as enhancement of thermal conductance for chemical potentials smaller than that of the superconducting regime.