Understanding spin Hall effect in two-dimensional fermionic systems with generic spin–orbit interaction in III–V heterostructures

• Spin–orbit force responsible for spin Hall effect has been derived.
• The spin–orbit force has been cast in a Lorentz force-like form.
• Relation among spin Hall conductivity, flux quanta and spin–orbit field is derived.

We delve into spin Hall effect in generic spin–orbit coupled two-dimensional fermionic systems. We derive analytically the spin–orbit force responsible for the spin Hall effect, and find that it has ‘Lorentz force’-like form. We also derive the pseudo-magnetic field responsible for this force. We establish the relation between the spin Hall conductivity, flux quanta and this pseudo-magnetic field, similar to the one between charge Hall conductivity, flux quanta and the external field. We also present an exact closed-form expression of the spin Hall conductivity in a generic spin–orbit coupled system.