tutorial5.rst

Superconductors: orbital vs. lattice degrees of freedom

This example deals with superconductivity on the level of the Bogoliubov-de Gennes (BdG) equation. In this framework, the Hamiltonian is given as

H = \begin{pmatrix} H_0 - \mu& \Delta\\ \Delta^\dagger&\mu-\mathcal{T}H\mathcal{T}^{-1}\end{pmatrix}

where H_0 is the Hamiltonian of the system without superconductivity, \mu the chemical potential, \Delta the superconducting order parameter, and \mathcal{T} the time-reversal operator. The BdG Hamiltonian introduces electron and hole degrees of freedom (an artificial doubling - be aware of the fact that electron and hole excitations are related!), which we now implement in Kwant.

For this we restrict ourselves to a simple spinless system without magnetic field, so that \Delta is just a number (which we choose real), and \mathcal{T}H\mathcal{T}^{-1}=H_0^*=H_0.

"Orbital description": using matrices

We begin by computing the band structure of a superconducting wire. The most natural way to implement the BdG Hamiltonian is by using a 2x2 matrix structure for all Hamiltonian matrix elements:

As you see, the example is syntactically equivalent to our :ref:`spin example <tutorial_spinorbit>`, the only difference is now that the Pauli matrices act in electron-hole space.

Computing the band structure then yields the result

We clearly observe the superconducting gap in the spectrum. That was easy, wasn't it?

"Lattice description": using different lattices

While it seems most natural to implement the BdG Hamiltonian using a 2x2 matrix structure for the matrix elements of the Hamiltonian, we run into a problem when we want to compute electronic transport in a system consisting of a normal and a superconducting lead: Since electrons and holes carry charge with opposite sign, we need to separate electron and hole degrees of freedom in the scattering matrix. In particular, the conductance of a N-S-junction is given as

G = \frac{e^2}{h} (N - R_\text{ee} + R_\text{he})\,,

where N is the number of channels in the normal lead, and R_\text{ee} the total probability of reflection from electrons to electrons in the normal lead, and R_\text{eh} the total probability of reflection from electrons to holes in the normal lead. However, the current version of Kwant does not allow for an easy and elegant partitioning of the scattering matrix in these two degrees of freedom [1].

In the following, we will circumvent this problem by introducing separate "leads" for electrons and holes, making use of different lattices. The system we consider consists of a normal lead on the left, a superconductor on the right, and a tunnel barrier in between:

As already mentioned above, we begin by introducing two different square lattices representing electron and hole degrees of freedom:

Note that since these two lattices have identical spatial parameters, the argument name to ~kwant.lattice.square has to be different. Any diagonal entry (kinetic energy, potentials, ...) in the BdG Hamiltonian corresponds to on-site energies or hoppings within the same lattice, whereas any off-diagonal entry (essentially, the superconducting order parameter \Delta) corresponds to a hopping between different lattices:

Note that the tunnel barrier is added by overwriting previously set on-site matrix elements.

Note further, that in the code above, the superconducting order parameter is nonzero only in a part of the scattering region. Consequently, we have added hoppings between electron and hole lattices only in this region, they remain uncoupled in the normal part. We use this fact to attach purely electron and hole leads (comprised of only electron or hole lattices) to the system:

This separation into two different leads allows us then later to compute the reflection probablities between electrons and holes explicitely.

On the superconducting side, we cannot do this separation, and can only define a single lead coupling electrons and holes (The += operator adds all the sites and hoppings present in one builder to another):

We now have on the left side two leads that are sitting in the same spatial position, but in different lattice spaces. This ensures that we can still attach all leads as before:

When computing the conductance, we can now extract reflection from electrons to electrons as smatrix.transmission(0, 0) (Don't get confused by the fact that it says transmission -- transmission into the same lead is reflection), and reflection from electrons to holes as smatrix.transmission(1, 0), by virtue of our electron and hole leads:

Note that smatrix.submatrix(0,0) returns the block concerning reflection within (electron) lead 0, and from its size we can extract the number of modes N.

Finally, for the default parameters, we obtain the following result:

We a see a conductance that is proportional to the square of the tunneling probability within the gap, and proportional to the tunneling probability above the gap. At the gap edge, we observe a resonant Andreev reflection.

Footnotes

[1]	Well, there is a not so elegant way to do it still. See the technical details