arXiv:1108.2251v2 [cond-mat.mtrl-sci] 31 Aug 2011 1 Irreversible Thermodynamics of Transport across Interfaces Matthew R. Sears and Wayne M. Saslow Abstract: With spintronics applications in mind, we use irreversible thermodynamics to derive the rates of entropy production and heating near an interface when heat current, electric current, and spin current cross it. Associated with these currents are apparent discontinuities in temperature (ΔT ), electrochemical potential (Δ˜ µ), and spin-dependent “magnetoelectrochemical potential” (Δ¯ µ ↑,↓ ). This work applies to magnetic semiconductors and insulators as well as metals, due to the inclusion of the chemical potential µ, which usually is neglected in works on interfacial thermodynamic transport. We also discuss the (non-obvious) distinction between entropy production and heat production. Heat current and electric current are conserved, but spin current is not, so it necessitates a somewhat different treatment. At low temperatures or for large differences in material properties, the surface heating rate dominates the bulk heating rate near the surface. We also consider the case, noted by Rashba, where bulk spin currents occur in equilibrium. Although a surface spin current (in A/m 2 ) should yield about the same rate of heating as an equal surface electric current, production of such a spin current requires a relatively large “magnetization potential” difference across the interface. PACS Nos.: 05.70.Ln,05.70.Np,67.40.Pm,73.40.Cg R´ esum´ e : Avec applications dans l’esprit de spintronics, nous employons la thermodynamique irr´ eversible ` a obtenir les taux de production d’entropie et de chauffage ` a proximit´ e d’une interface lorsque la chaleur actuelle, le courant ´ electrique, et courant de spin la traverser. Associ´ es ` a ces courants sont discontinuit´ es apparentes de la temp´ erature (ΔT ), potentiel ´ electrochimique (Δ˜ µ), et d´ ependant du spin potentiel “magneto´ electrochimique” (Δ¯ µ ↑,↓ ). Ce travail s’applique ` a semi-conducteurs magn´ etiques et isolants ainsi que des m´ etaux, due ` a l’inclusion de la potentiel chimique µ, ce qui est g´ en´ eralement n´ eglig´ ee dans les travaux sur les transports thermodynamique interfaciale. Nous discutons ´ egalement de la distinction (non ´ evidente) entre la production d’entropie et la production de chaleur. Chaleur actuelle et le courant ´ electrique sont conserv´ es, mais n’est pas courant de spin, il n´ ecessite un traitement quelque peu diff´ erent. A basse temp´ erature, ou pour de grandes diff´ erences dans les propri´ et´ es du mat´ eriau, la vitesse de chauffage de surface domine la vitesse de chauffage en vrac pr´ es de la surface. Nous consid´ erons ´ egalement le cas, a not´ e par Rashba, o´ u les courants de spin en vrac se produire ` a l’´ equilibre. Mˆ eme si un courant de spin de surface (en A/m 2 ) devrait donner environ le mˆ eme taux de chauffage d’une surface ´ egale de courant ´ electrique, la production d’un tel courant de spin n´ ecessite un potentiel relativement important “aimantation diff´ erence” entre l’interface. 1. Introduction It is well-known that apparent voltage and temperature dis- continuities, determined by extrapolation from the bulk, ap- pear at interfaces in the presence of heat or electric current. For small currents, these discontinuities are proportional to the heat or electric current. For heat current, the coefficient of proportionality is known as the thermal boundary resistance, and was first studied at low temperatures by Kapitza for the solid–liquid 4 He interface [1–4]. For electric current, the co- efficient of proportionality is known as the surface resistance, or specific resistance [5]. In principle, there can also be off- diagonal terms, corresponding to a discontinuity in the tem- perature causing an electric current [6]. There also are more recently, spin-dependent conduction effects across surfaces, as studied, for example, in [6–8]. Johnson and Silsbee [6] studied the surface and bulk trans- port coefficients for these phenomena, but without considering Matthew R. Sears and Wayne M. Saslow . 1 Department of Physics, Texas A&M University, College Station, TX 77843-4242 1 Corresponding author (e-mail: wsaslow@tamu.edu). details of the non-conservation of the spin current due to spin- flip processes, and did not study the rate of heating near the sur- face. The present work considers these non-conservation phe- nomena, which require more refined considerations than when they are not present. (The terms “magnetoelectrochemical po- tential” and “magnetization potential” that appear in the theory were first employed in [6]; they are made more precise below.) The present work also includes the chemical potential µ of the charge-carriers, neglected by [6], which considered metals. The present results are more general; µ is negligible (compared to electrical potential energy) for metals but not necessarily for semiconductors or insulators, where small changes in carrier density can have a large effect on µ. The present work also considers the conditions under which bulk heating dominates surface heating, and vice-versa. It also explicitly considers the near-surface region. Note that heat- ing implies entropy production, but not the converse; Sect. IIB presents some considerations on this matter. For specificity, note that if there is an extrapolated tempera- ture discontinuity ΔT across a solid-solid interface, the energy flux (and heat flux) are given by |j ε | = h K |ΔT |, (1) where h K ≥ 0 is the Kapitza, or thermal boundary, conduc- Can. J. Phys. 99: 1–10 (2018) DOI: 10.1139/Zxx-xxx c  2018 NRC Canada