ComplexBandstructure

	ComplexBandstructure
Bandstructure	TotalEnergy

Name

ComplexBandstructure — Class for representing the complex band structure for a given configuration and calculator.

Synopsis

Namespace: NanoLanguage

ComplexBandstructure(
configuration,
energies,
k_point,
energy_zero_parameter
)

Description

Constructor for the ComplexBandstructure object.

ComplexBandstructure Arguments

configuration

The bulk configuration with attached calculator for which the complex bandstructure should be calculated.

Type: A BulkConfiguration.

Default:


          None

energies

The energies for which the complex bandstructure should be calculated.

Type: List containing elements of the type PhysicalQuantity of with energy unit.

Default:


          numpy.linspace(-2, 2, 101)*eV

k_point

The k-point (in the A-B plane) for which the complex bandstructure should be calculated.

Type: Tuple of length 2

Default:


          (0, 0)

energy_zero_parameter

Specifies the choice for the energy zero.

Type: FermiLevel | AbsoluteEnergy.

Default:


          FermiLevel.

ComplexBandstructure Methods

A ComplexBandstructure object provides the following methods:

energies(): Query function for the energies
energyZero(): Return the energy zero
evaluate(spin): Method for obtaining the complex bandstructure k vectors, i.e. the touple (k_propagating, k_decaying)

spin

The spin to calcualte for.
Type: Spin.Up | Spin.Down
Default: Spin.Up
fermiLevel(): Return the Fermi level in absolute energy
fermiTemperature(): Return the Fermi_temperatures
kPoint(): Query function for the k_point
nlprint(stream): Print a string containing an ASCII table useful for plotting the complex bandstructure.

stream

The stream the complex bandstructure should be written to.
Type: A stream that supports strings being written to using 'write'.
Default: sys.stdout
repetition(): Return the repetition used
spins(): Query function for the spins

Usage Examples

Calculate the complex bandstructure of silicon in the (100) direction and save it to a file

# Set up si (100) surface
bulk_configuration = BulkConfiguration(
    bravais_lattice=SimpleTetragonal(3.84*Angstrom, 5.43*Angstrom),
    elements=[Silicon, Silicon, Silicon, Silicon],
    fractional_coordinates=[[0.0, 0.0, 0.00],
                            [0.5, 0.0, 0.25],
                            [0.5, 0.5, 0.50],
                            [0.0, 0.5, 0.75]]
    )

# Define calculator
calculator = SlaterKosterCalculator(
    basis_set=Vogl.Si_Basis,
    numerical_accuracy_parameters=NumericalAccuracyParameters(
                k_point_sampling=(4, 4, 2))
     )

bulk_configuration.setCalculator(calculator)

# Calculate and save the complex bandstructure
complex_bandstructure = ComplexBandstructure(
    configuration=bulk_configuration,
    energies=numpy.linspace(-2,2,501)*eV,
    )
nlsave('si_complex_band.nc', complex_bandstructure)
nlprint(complex_bandstructure)

si_complex_band.py

Complex bandstructure for Silicon in the (100) direction. The right part of the plots illustrates the real bands, while the left hand side of the plot shows the complex bands.

Figure 10: Complex bandstructure for Silicon in the (100) direction. The right part of the plots illustrates the real bands, while the left hand side of the plot shows the complex bands.

Notes

The complex bandstructure is obtained by for a specified energy, $E$ and parallel k-point $k_x, k_y$ finding the eigenstates, $\psi_{n k}$ , with no imposed boundary conditions in the z direction,

$\displaystyle H \psi_{n k} = E \psi_{n k}.$

We may write the eigenstates as

$\displaystyle \psi_{n k}({\bf r}) = u_k({\bf r}) e^{i k z},$

where $u_k$ is a periodic function and $k$ is a complex number.

To calculate the complex bandstructure you must have a BulkConfiguration that could be used as a valid electrode in a DeviceConfiguration. This means that the geometry must be sufficiently long in the z-direction(typically > 7 Å), such that there are only matrix elements with atoms in the nearest cell along the z-direction.
In case the cell is too small in the z-direction, the structure will automatically be repeated along the z-direction. The repetition will give rise to zone folding of the calculated bands.
To print the data of a complex bandstructure, use the method nlprint.


Bandstructure		TotalEnergy