Mode-resolved transmission functions: an individual Caroli formula

Efficient manipulation of energy at the nanoscale is crucial for advancements in modern computing, energy harvesting, and thermal management. Specifically, controlling quasiparticle currents is critical to these ongoing technological revolutions. This work introduces a novel and physically consistent approach for computing polarization-resolved transmission functions, a crucial element in understanding and controlling energy transport across interfaces. We show that this new method, unlike several previously derived formulations, consistently yields physically meaningful results by addressing the origin of unphysical behavior in other methods. We demonstrate that while multiple decompositions of the transmission function are possible, only there is a unique and unambiguous route to obtaining physically meaningful results. We highlight and critique the arbitrary nature of these alternative decompositions and their associated failures. While developed within the framework of phonon transport, the individual Caroli formula is general and applicable to other fermionic and bosonic quasiparticles, including electrons, and to internal degrees of freedom such as spin and orbital polarization. Through a comparative analysis using a simple model system, we validate the accuracy and reliability of the individual Caroli formula in capturing polarization-specific transmission properties. This new method provides a more accurate understanding of both phonon and electron transport, offering novel ways for optimization of thermoelectric devices and energy-efficient computing technologies.