Quantum Drift-Diffusion Equations for a Two-Dimensional Electron Gas with Spin-Orbit Interaction

Formal derivation of quantum drift-diffusion equations with spin-orbit interaction

“…where Tr is the operator trace, log is the operator logarithm, and H ε is the Hamiltonian (2). Note that the operator logarithm is well defined for positive definite density operators.…”

“…Identifying the Pauli coefficients, we infer that a (1) 0 = 0 and a (0) = n/n 0 . For M (2) , we use (42) in ⟨M (2) ⟩ = 0, and insert the expressions for the partial derivatives of M. A tedious but elementary computation leads to 1 8 ∆a…”

“…(2) 0 n + n 0 a (2) , which in fact determines a (2) . A straightforward but again tedious computation shows that the first term equals g (3) (1) ε=0 = n 0 12 ∆a…”

“…Order two. The differential equation reads here as ∂ β g (2) = h 0 σ 0 # 0 g (2) + h 0 σ 0 # 1 g (1) + h 0 σ 0 # 2 g (0) + (h 1 • σ)# 0 g (1) + (h 1 • σ)# 1 g (0) , with initial condition g (2) (0) = 0. The first term on the right-hand side contains the unknown, while the others are known from the preceding orders.…”

“…Therefore, the differential equation for g (2) becomes ∂ β g (2) = h 0 σ 0 g (2) + βg (0) 1 4 ∆a 0 − β 8 p T (∇ ⊗ ∇a 0 )p − |∇a 0 | 2 + |h 1 | 2 σ 0 for β > 0 with initial datum g (2) (0) = 0. Duhamel's formula leads to (39).…”