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Today's Topics:
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
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Message: 2
Date: Fri, 17 Jul 2015 19:44:23 +0200
From: Joseph Weston <joseph.weston@cea.fr>
To: kwant-discuss@kwant-project.org
Subject: Re: [Kwant] Spin Currents using Greens Functions
Message-ID: <55A93EF7.6000606@cea.fr>
Content-Type: text/plain; charset="utf-8"
This time *with* the recipe attached
Joe
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Message: 3
Date: Fri, 17 Jul 2015 19:56:00 +0200
From: Anton Akhmerov <anton.akhmerov@gmail.com>
To: Joseph Weston <joseph.weston@cea.fr>
Cc: "kwant-discuss@kwant-project.org"
<kwant-discuss@kwant-project.org>
Subject: Re: [Kwant] Spin Currents using Greens Functions
Message-ID:
<CADr1H-8s-CswjQkZ-2FYpNg58hEy_BerzjF-waR49mwgWzPYKw@mail.gmail.com>
Content-Type: text/plain; charset=UTF-8
Hi Joe,
This is a nice hack! I once thought of a more systematic and cleaner
solution would be to produce the scattering matrix whose modes are
identical aside of spin degeneracy. This wouldn't rely on using the
slower and less stable Green's functions.
Forcing extra requirements on the modes in the lead is really quite
easy and can be preferable since you can probably save computational
costs also for the modes. For example in [1] we've done it for
time-reversal symmetry, and in [2] Michael used it to calculate
Andreev conductance in a neater way than what is suggested by the
Kwant tutorial.
I started implementing such leads once, but didn't have enough time to
come to a reasonable point.
Cheers,
Anton
[1]: http://arxiv.org/src/1408.1563v2/anc/trijunction.py (see class
TRIInfiniteSystem)
[2]: http://wwwhome.lorentz.leidenuniv.nl/~wimmer/tcscxviii/phlead.py
(was used for example in
http://nbviewer.ipython.org/url/wwwhome.lorentz.leidenuniv.nl/~wimmer/tcscxviii/Hybrid%20nanowire%20system.ipynb)
On Fri, Jul 17, 2015 at 7:44 PM, Joseph Weston <joseph.weston@cea.fr> wrote:
> This time *with* the recipe attached
>
> Joe
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