Dear Joe, Can we use this method to calculate spin currents between two leads for a three dimensional system? Spin is implemented by 2x2 matrix elements for each site, but other couplings (ie. couplings between layers) are also included. For example, we have a bilayer graphene; for each layer, we have 2x2 matrix elements for each site, but we also have a coupling between the two layers. Can we use this method for this system? Best wishes, Kwok-Long Lee On Sat, Jul 18, 2015 at 8:02 PM, <kwant-discuss-request@kwant-project.org> wrote:

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1. Spin Currents using Greens Functions (Joseph Weston) 2. Re: Spin Currents using Greens Functions (Joseph Weston) 3. Re: Spin Currents using Greens Functions (Anton Akhmerov)

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Message: 1 Date: Fri, 17 Jul 2015 19:42:01 +0200 From: Joseph Weston <joseph.weston@cea.fr> To: kwant-discuss@kwant-project.org Subject: [Kwant] Spin Currents using Greens Functions Message-ID: <55A93E69.80602@cea.fr> Content-Type: text/plain; charset="utf-8"

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