OpticalSpectrum

	OpticalSpectrum
MakeTrajectory	CurrentDensity

Name

OpticalSpectrum — Class for representing the optical spectrum for a configuraiton with an attached calculator.

Synopsis

Namespace: NanoLanguage

OpticalSpectrum(
configuration,
kpoints,
energies,
broadening,
bands_below_fermi_level,
bands_above_fermi_level
)

Description

Constructructor for the OpticalSpectrum object.

OpticalSpectrum Arguments

configuration

The configuration with an attached calculator.

Type: Bulkconfiguration

Default:


          None

kpoints

The k-points for which the transitions should be calculated.

Type: MonkhorstPackGrid

Default:


          MonkhorstPackGrid(nx,ny,nz), where nx, ny, nz is the sampling used for the selfconsistent calculation.

energies

The energies for which the polarizability should be calculated.

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

Default:


          numpy.linspace(0, 4, 101)*eV

broadening

The broadening parameter used for the polarizability.

Type: PhysicalQuantity with unit energy.

Default:


          0.1 eV

bands_below_fermi_level

The maxinum number of valence band states to include at each k-point.

Type: integer > 0

Default:

bands_above_fermi_level

The maximum number of conduction band states to include at each k-point.

Type: integer > 0

Default:

OpticalSpectrum Methods

A OpticalSpectrum object provides the following methods:

broadening(): Query function for the broadening.
energies(): Query function for the energies.
evaluateDielectricConstant(spin): Function for evaluating the real part of the dielectric constant using the Kubo-Greenwood formalism.

spin

The spin to calculate for.
Type: Spin.Up | Spin.Down | Spin.Sum
Default: Spin.Sum
evaluateImaginaryDielectricConstant(spin): Function for evaluating the imaginary part of the dielectric constant using the Kubo-Greenwood formalism.

spin

The spin to calculate for.
Type: Spin.Up | Spin.Down | Spin.Sum
Default: Spin.Sum
evaluateImaginaryPolarizability(spin): Function for evaluating the imaginary part of the polarizability using the Kubo-Greenwood formalism.

spin

The spin to calculate for.
Type: Spin.Up | Spin.Down | Spin.Sum
Default: Spin.Sum
evaluatePolarizability(spin): Function for evaluating the real part of the polarizability using the Kubo-Greenwood formalism.

spin

The spin to calculate for.
Type: Spin.Up | Spin.Down | Spin.Sum
Default: Spin.Sum
maxConductionStates(): Query function for the maximum number of conduction states to include.
maxValenceStates(): Query function for the maximum nunmber of valence states to include.
nlprint(stream): Print a string containing an ASCII table useful for plotting the optical spectrum.

stream

The stream the optical spectrum (susceptibility) should be written to.
Type: A stream that supports strings being written to using 'write'.
Default: sys.stdout

Usage Examples

Calculate the opticalspectrum of silicon and save it to a file. Print out the Dielectric Tensor.

# Set up silicon crystal
bulk_configuration = BulkConfiguration(
    bravais_lattice=FaceCenteredCubic(5.4306*Angstrom),
    elements=[Silicon, Silicon],
    cartesian_coordinates=[[ 0.     ,  0.     ,  0.     ],
                         [ 1.35765,  1.35765,  1.35765]]*Angstrom
    )

#setup calculator
numerical_accuracy_parameters = NumericalAccuracyParameters(
    k_point_sampling=(6, 6, 6),
    )

# setup DFT calculator using Meta-GGA
calculator = LCAOCalculator(
    basis_set=LDABasis.DoubleZetaDoublePolarized,
    exchange_correlation=MGGA.TB09LDA,
    numerical_accuracy_parameters=numerical_accuracy_parameters,
    )

bulk_configuration.setCalculator(calculator)
bulk_configuration.update()

# Optical spectrum
optical_spectrum = OpticalSpectrum(
    configuration=bulk_configuration,
    kpoints=MonkhorstPackGrid(15,15,15),
    energies=numpy.linspace(0,5,101)*eV,
    broadening=0.1*eV,
    max_valence=4,
    max_conduction=4,
    )

nlsave('si_optical.nc', optical_spectrum)

# Print out the dielectric constant (Experimental value is 11.9)
print 'Dielectric Tensor'
print optical_spectrum.evaluateDielectricConstant(energies=[0.0]*eV)[:,:,0]

si_optical.py

Notes

We use the Kubo-Greenwood formula to calculate the susceptibility tensor [38]

$\displaystyle \chi_{ij}(\omega) = - \frac{ e^2 \hbar^4}{m^2 \epsilon_0 V \omega^2} \sum_{nm} \frac{f(E_m)-f(E_n)}{E_{nm}-\hbar \omega - i \hbar \Gamma} \pi_{nm}^i \pi_{mn}^j,$

where $\pi_{nm}^i$ is the $i$ -component of the dipole matrix element between state $n$ and $m$ , $V$ the volume, $\Gamma$ the broadening, and $f$ the Fermi function.
The response coefficients, the dielectric constant $\epsilon_r$ , polarizability $\alpha$ , and optical conductivity $\sigma$ , are related to the susceptibility as

$\displaystyle \epsilon_r(\omega) = (1 + \chi(\omega) ),$

$\displaystyle \alpha(\omega) = V \epsilon_0 \chi(\omega),$

$\displaystyle \sigma(\omega) = - i \omega \epsilon_0 \chi(\omega).$

The derivation of the last relation can be found in Ref. [36].
The refractive index, $n$ , is related to the complex dielectric constant through

$\displaystyle n + i \kappa = \sqrt{\epsilon_r},$

here $\kappa$ is the extinction coefficient.

In terms of the real ( $\epsilon_1$ ) and complex parts ( $\epsilon_2$ ) of the dielectric constant

$\displaystyle n = \sqrt{\frac{\sqrt{\epsilon_1^2+\epsilon_2^2}+\epsilon_1}{2}},$

$\displaystyle \kappa = \sqrt{ \frac{ \sqrt{ \epsilon_1^2+ \epsilon_2^2}- \epsilon_1}{2}}.$
The optical absorption coefficient is related to the extinction coefficient through [37]

$\displaystyle \alpha_a = 2 \frac{\omega}{c} \kappa.$

The reflectivity is given by [35]

$\displaystyle r = \frac{(1-n)^2 + \kappa^2}{(1+n)^2 + \kappa^2}.$