Thermoelectricity in 4-probe structure

Consider the simple 4-probe structure:

What is the thermovoltage induced between N2, N3 and S0, S1 when N2 and N3 are at different temperatures and their potentials float?

See also

example-thermo-4probe.py

Geometry specification

Geometry is specified exactly as in the other examples:

def get_geometry(phi):

    # 6 nodes and 5 wires
    g = u.Geometry(nnode=6, nwire=5)

    g.t_type = [u.NODE_CLEAN_S_TERMINAL, # 0
                u.NODE_CLEAN_S_TERMINAL, # 1
                u.NODE_CLEAN_N_TERMINAL, # 2
                u.NODE_CLEAN_N_TERMINAL, # 3
                u.NODE_CLEAN_NODE,       # 4
                u.NODE_CLEAN_NODE,       # 5
    ]

    g.t_delta[0:2] = 100
    g.t_phase[0] = -.5*phi
    g.t_phase[1] = +.5*phi
    g.t_inelastic = 1e-9

    g.w_type = u.WIRE_TYPE_N
    g.w_length = 1./3
    g.w_conductance = 1

    g.w_ends[0,:]  = [ 4, 0 ]
    g.w_ends[1,:]  = [ 1, 5 ]
    g.w_ends[2,:]  = [ 4, 2 ]
    g.w_ends[3,:]  = [ 3, 5 ]
    g.w_ends[4,:]  = [ 4, 5 ]
    return g

Note that wires connected to a node must either all end or start at the node.

Computing the thermovoltage

Solve the thermoelectric voltage between N2 and S, when there is a small temperature difference between N2 and N3.

def main():
    g = get_geometry(pi/2)

    file_name = 'thermo_4probe_spectral.h5'

    solver = u.CurrentSolver.resume(file_name, g)
    solver.set_solvers(kin_solver=u.KIN_SOLVER_BLOCK,
                       sp_solver=u.SP_SOLVER_TWPBVP)

    solver.solve_spectral_and_save_if_needed(file_name, calculate_G=True)

Here, because solving the spectral equations takes some time, we want to save the result to a file. If the file already exists, solving the equations is skipped, and old data is reused.

We also instruct the solver to compute the energy-dependent conductances between all terminals, which allows fast calculation of currents.

The following is a very straightforward calculation for the thermovoltage: we set the temperatures of the terminals explicitly, and adjust the potentials of N2 and N3 until no current enters them:

dT = 0.001

output = open('thermo_4probe.dat', 'w')
print >> output, "%% %14s %14s" % ("T (E_T)", "dV/dT (ueV/K)")

for T in logspace(log10(dT + 1e-4), log10(20), 100):
    # Compute the thermovoltage:
    g.t_mu = 0
    g.t_t = T
    g.t_t[2] += dT/2
    g.t_t[3] -= dT/2

    # Make the terminals 2, 3 to float
    # Currents entering them flow in wires 2, 3

    def zero_currents():
        Ic, Ie = solver.get_currents_from_G(w_jT=[2,3], w_jL=[])
        return [Ic[2], Ic[3]]

    def set_potentials(z):
        g.t_mu[2], g.t_mu[3] = z

    u.optimize_parameters_for([0,0], zero_currents, set_potentials)

    # Print the thermovoltage at terminal 2 in ueV/K
    print >> output, "  %14g %14g" % (T, g.t_mu[2] / dT * 86.17343)

The differential thermovoltage could have also been calculated directly from the energy-dependent conductances.

The result for the S-N voltage at phase difference between the S-terminals of course coincides with the published results: [VT04]

[VT04]

P. Virtanen, and T.T. Heikkilä, Physical Review Letters 92, 177004 (2004)