Dear all, I have noticed that several people have posted on the mailing list asking how to calculate spin currents between two leads when an orbital representation is used for the spin degree of freedom (i.e. spin is implemented by 2x2 matrix elements for each site, as opposed to a separate lattice for each spin).
The arbitrary choice of spin quantization axis in the leads, in the case
where the lead Hamiltonian is spin-rotation invariant, renders direct
use of the scattering matrix cumbersome. The attached recipe
(spin_conductance.py) calculates the spin current aligned along the α
direction using the Greens functions and the Landauer formula:
G_{pq} = (e/h) Tr[ σ_{α} Γ_{q} G_{qp} Γ_{p} G^+_{qp} ]
where Γ_{q} is the coupling matrix to lead q ( = i[Σ - Σ^+] )
and G_{qp} is the submatrix of the system green's function
connecting sites which interfaces to leads q and p, σ_{α}
is the pauli matrix along direction α and Tr denotes the trace.
This was discussed in a non-mailing-list email thread with
Branislav Nikolic, Xavier Waintal and Christoph Groth but I
thought it would be useful to post the recipe here. The above relation
has been derived in reference [1].
Any thoughts/discussion welcome,
Joseph Weston
[1]: http://dx.doi.org/10.1103/PhysRevB.89.195418
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