2.6. 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.

2.6.1. “Orbital description”: using matrices

See also

The complete source code of this example can be found in tutorial/superconductor_band_structure.py

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:

tau_x = tinyarray.array([[0, 1], [1, 0]])
tau_z = tinyarray.array([[1, 0], [0, -1]])


def make_lead(a=1, t=1.0, mu=0.7, Delta=0.1, W=10):
    # Start with an empty lead with a single square lattice
    lat = kwant.lattice.square(a)

    sym_lead = kwant.TranslationalSymmetry((-a, 0))
    lead = kwant.Builder(sym_lead)

    # build up one unit cell of the lead, and add the hoppings
    # to the next unit cell
    for j in range(W):
        lead[lat(0, j)] = (4 * t - mu) * tau_z + Delta * tau_x

        if j > 0:
            lead[lat(0, j), lat(0, j - 1)] = -t * tau_z

        lead[lat(1, j), lat(0, j)] = -t * tau_z

    return lead

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

Computing the band structure then yields the result

../_images/superconductor_band_structure_result.png

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

2.6.2. “Lattice description”: using different lattices

See also

The complete source code of this example can be found in tutorial/superconductor_transport.py

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:

../_images/superconductor_transport_sketch.svg

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

def make_system(a=1, W=10, L=10, barrier=1.5, barrierpos=(3, 4),
                mu=0.4, Delta=0.1, Deltapos=4, t=1.0):
    # Start with an empty tight-binding system and two square lattices,
    # corresponding to electron and hole degree of freedom
    lat_e = kwant.lattice.square(a, name='e')
    lat_h = kwant.lattice.square(a, name='h')

Note that since these two lattices have identical spatial parameters, the argument name to 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:

    sys = kwant.Builder()

    #### Define the scattering region. ####
    sys[(lat_e(x, y) for x in range(L) for y in range(W))] = 4 * t - mu
    sys[(lat_h(x, y) for x in range(L) for y in range(W))] = mu - 4 * t

    # the tunnel barrier
    sys[(lat_e(x, y) for x in range(barrierpos[0], barrierpos[1])
         for y in range(W))] = 4 * t + barrier - mu
    sys[(lat_h(x, y) for x in range(barrierpos[0], barrierpos[1])
         for y in range(W))] = mu - 4 * t - barrier

    # hoppings for both electrons and holes
    sys[lat_e.neighbors()] = -t
    sys[lat_h.neighbors()] = t

    # Superconducting order parameter enters as hopping between
    # electrons and holes
    sys[((lat_e(x, y), lat_h(x, y)) for x in range(Deltapos, L)
         for y in range(W))] = Delta

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:

    # Symmetry for the left leads.
    sym_left = kwant.TranslationalSymmetry((-a, 0))

    # left electron lead
    lead0 = kwant.Builder(sym_left)
    lead0[(lat_e(0, j) for j in range(W))] = 4 * t - mu
    lead0[lat_e.neighbors()] = -t

    # left hole lead
    lead1 = kwant.Builder(sym_left)
    lead1[(lat_h(0, j) for j in range(W))] = mu - 4 * t
    lead1[lat_h.neighbors()] = t

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):

    sym_right = kwant.TranslationalSymmetry((a, 0))
    lead2 = kwant.Builder(sym_right)
    lead2 += lead0
    lead2 += lead1
    lead2[((lat_e(0, j), lat_h(0, j)) for j in range(W))] = Delta

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:

    sys.attach_lead(lead0)
    sys.attach_lead(lead1)
    sys.attach_lead(lead2)

    return sys

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:

def plot_conductance(sys, energies):
    # Compute conductance
    data = []
    for energy in energies:
        smatrix = kwant.smatrix(sys, energy)
        # Conductance is N - R_ee + R_he
        data.append(smatrix.submatrix(0, 0).shape[0] -
                    smatrix.transmission(0, 0) +
                    smatrix.transmission(1, 0))

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:

../_images/superconductor_transport_result.png

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.

Technical details
  • If you are only interested in particle (thermal) currents you do not need to define separate electron and hole leads. In this case, you do not need to distinguish them. Still, separating the leads into electron and hole degrees of freedom makes the lead calculation in the solving phase more efficient.
  • It is in fact possible to separate electron and hole degrees of freedom in the scattering matrix, even if one uses matrices for these degrees of freedom. In the solve step, smatrix returns an array containing the transverse wave functions of the lead modes (in SMatrix.lead_info. By inspecting the wave functions, electron and hole wave functions can be distinguished (they only have entries in either the electron part or the hole part. If you encounter modes with entries in both parts, you hit a very unlikely situation in which the standard procedure to compute the modes gave you a superposition of electron and hole modes. That is still OK for computing particle current, but not for electrical current).

Footnotes

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