Usage Examples

Calculate the transmission spectrum and for each energy print out all k-dependent coefficients

# Define A,B directions of lattice
vector_a = [5.0, 0.0, 0.0]*Angstrom
vector_b = [0.0, 5.0, 0.0]*Angstrom

# Setup a device configuration
electrode = BulkConfiguration(
    bravais_lattice=UnitCell(vector_a, vector_b,
                             [0.0, 0.0, 9.0]*Angstrom),
    elements=[Lithium, Lithium, Lithium],
    cartesian_coordinates=[[ 2.5,  2.5,  1.5],
                           [ 2.5,  2.5,  4.5],
                           [ 2.5,  2.5,  7.5]]*Angstrom
    )
central_region = BulkConfiguration(
    bravais_lattice= UnitCell(vector_a, vector_b,
                             [0.0, 0.0, 22.0]*Angstrom),
    elements=[Lithium, Lithium, Lithium, Hydrogen, Hydrogen,
              Lithium, Lithium,  Lithium],
    cartesian_coordinates=[[  2.5,   2.5,   1.5],
                           [  2.5,   2.5,   4.5],
                           [  2.5,   2.5,   7.5],
                           [  2.5,   2.5,  10.5],
                           [  2.5,   2.5,  11.5],
                           [  2.5,   2.5,  14.5],
                           [  2.5,   2.5,  17.5],
                           [  2.5,   2.5,  20.5]]*Angstrom
     )
device_configuration = DeviceConfiguration(
    central_region,
    [electrode, electrode]
    )

# Setup calculators
numerical_accuracy_parameters = NumericalAccuracyParameters(
    k_point_sampling=(1, 1, 100))

electrode_calculator = HuckelCalculator(
    numerical_accuracy_parameters=numerical_accuracy_parameters)

calculator = DeviceHuckelCalculator(
    numerical_accuracy_parameters=numerical_accuracy_parameters,
    electrode_calculators=[electrode_calculator, electrode_calculator])

device_configuration.setCalculator(calculator)

# Calculate the transmission coefficient
transmission_spectrum = TransmissionSpectrum(
    configuration=device_configuration,
    energies=numpy.linspace(-0.5,0.5,10)*eV,
    kpoints=MonkhorstPackGrid(3,3,1),
    energy_zero_parameter=AverageFermiLevel,
    infinitesimal=1e-06*Units.eV
    )

# Print out all k-dependent transmission coefficients
data = transmission_spectrum.transmission()
energies = transmission_spectrum.energies()
kpoints = transmission_spectrum.kpoints()
spins = ["Up", "Down"]
print data
for s in range(len(data)):
    print "Transmission coefficient of spin component ", spins[s]
    for i in range(data[s].shape[0]):
        print 'Transmission at energy = %12.6f eV' % (energies[i].inUnitsOf(eV))
        print '      kx         ky      transmission '
        for j in range(data[s].shape[1]):
            print '%10.4f %10.4f %16.6e' % \
                  (kpoints[j][0],kpoints[j][1],data[s][i][j])
        print

li_h2_li_trans.py

Alternatively to using the MonkHorstPackGrid class, we can also specify the k-points and weights directly. The k-points are specified in units of the reciprocal lattice vectors. Note that except for the Gamma point, the k-points have weight 2 due to use of time-reversal symmetry, i.e .

transmission_spectrum = TransmissionSpectrum(
    configuration=device_configuration,
    energies=numpy.linspace(-0.5,0.5,10)*eV,
    kpoints=[[0.0,0.0,0.0],[0.0,1./3.,0.0],[1./3.,-1./3.,0.0],
    [1./3.,0.0,0.0],[1./3.,1./3.,0.0]],
    kpoints_weights = [1.,2.,2.,2.,2.],
    energy_zero_parameter=AverageFermiLevel,
    infinitesimal=1e-06*Units.eV
    )

li_h2_li_trans1.py

Calculate the current and differential conductance for different electron temperatures in the electrodes.

temp = numpy.linspace(0,1000,21)
currents = []
conductances = []

for t in temp:
    current = transmission_spectrum.current(
                   electrode_temperatures=[200.,t]*Kelvin
		   )
    currents.append(current)

    conductance = transmission_spectrum.conductance(
                       electrode_temperatures=[200.,t]*Kelvin
		       )
    conductances.append(conductance)

print "Temperature[Left]   Temperature[Right]"\
      + "     Current     Differential-Conductance "
output_format = "      %6.0f K          %6.0f K     %s  %s"
for i in range(len(temp)):
    print output_format%(200., temp[i], currents[i], conductances[i])

current.py

Notes

To export the data of a transmission spectrum, use the method nlsave.

Current

The current is calculated from the transmission coefficient using

where is the Fermi function, and and are the electron temperatures of the left and right electrodes, respectively. In this equation, is the transmission coefficient for spin .

A positive current flows when the applied voltage for the left electrode is higher than that of the right one. A positive current corresponds to the flow of charge from left to right, meaning that the flow of electrons is right-to-left, (since electrons have a negative charge).

The chemical potentials of the left electrode, and the right electrode, are defined relative to the Fermi level of the left electrode, and related to the applied bias through

The current is independent of how the voltages were applied, it only depends on their difference (i.e. on ).

The transmission spectrum itself is usually rather insensitive to the temperatures and used in the self-consistent calculation, but often depends strongly on the electrode voltages and . Thus, for an accurate estimate of the current, the transmission spectrum should be calculated self-consistently for each desired bias. It is, however, possible to obtain an approximation for the current at a different bias than that used in the self-consistent calculation, by giving a new set of voltages to the current() method. This will simply modify the integration range on the existing transmission spectrum. The temperature-dependence of the current can, on the other hand, in most cases be rather accurately estimated by only changing the electrode temperatures in the current calculation.

Differential conductance

The differential conductance is calculated from the transmission spectrum using

The coupling constants models how the transmission spectrum couples with the left and right electrode. For a molecule strongly bound to the left electrode we must have and .

A more accurate estimate of the differential conductance which avoid introducing is obtained by calculating the self-consistent current at a number of applied biases, , and performing numerical differentiation