###### user

Summarize the paper in 3 sentences.

**AI:**The paper explores the concept of spin currents in spin-orbit coupled systems and develops formulas for calculating intrinsic proper spin current (IPSC). The authors derive expressions for the total spin current in crystals with inversion symmetry and show that the conventional spin current is physically meaningless. They also calculate the spin conductivity in two different systems and discuss their findings.

###### user

What is the spin-Hall torque and what do the authors mean by "proper" in their title?

**AI:**The spin-Hall torque is a torque induced by the spin current in spin torque devices, generated by a transverse electric field creating a perpendicular spin-current. The authors use the term "proper" in their title to distinguish their calculation from those that employ the conventional spin current definition.

## Authors

Hong Liu, James H. Cullen, Dimitrie Culcer

## Abstract

Spin currents driven by spin-orbit coupling are key to spin torque devices, but determining the proper spin current is highly non-trivial. Here we derive a general quantum-mechanical formula for the intrinsic proper spin current showing that it is a topological quantity, and can be finite even in the gap. We determine the spin-Hall current due to the bulk states of topological insulators both deep in the bulk, where the system is unmagnetized, and near the interface, where a proximity-induced magnetization is present, as well as for low-dimensional spin-3/2 hole systems.