Farzad Mahfouzi, Branislav K. Nikolic
Experiments observing spin density and spin currents in spintronic devices measure only the nonequilibrium contributions to these quantities. On the other hand, theoretical approaches to calculate these quantities operate with both the nonequilibrium (carried by electrons around the Fermi surface) and equilibrium (carried by the Fermi sea electrons) contributions to them. Thus, an unambiguous procedure should remove the equilibrium contributions, thereby rendering the nonequilibrium ones which are measurable and satisfy the gauge-invariant condition according to which expectation values of physical quantities should not change when electric potential everywhere is shifted by a constant amount. Using the framework of nonequilibrium Green functions, we delineate such procedure which yields the proper gauge-invariant nonequilibrium density matrix in the linear-response and elastic transport regime for current-carrying steady state of an open quantum system connected to two macroscopic reservoirs. Its usage is illustrated by computing: (i) conventional spin-transfer torque (STT) in asymmetric F/I/F magnetic tunnel junctions (MTJs); (ii) unconventional STT in asymmetric N/I/F semi-MTJs with the strong Rashba spin-orbit coupling (SOC) at the I/F interface and injected current perpendicular to that plane; and (iii) current-driven spin density within a clean ferromagnetic Rashba spin-split two-dimensional electron gas (2DEG) which generates SO torque in laterally patterned N/F/I heterostructures when such 2DEG is located at the N/F interface and injected charge current flows parallel to the plane. We contrast these results with improper expressions for the density matrix often found in the literature which arbitrarily mix nonequilibrium and equilibrium expectation values due to a violation of gauge invariance.
View original:
http://arxiv.org/abs/1305.3180
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