Quickstart Example: Free Electrons

First, import the module (and anything else you need).

import elecboltz
import numpy as np

For a minimal example, let’s calculate the conductivity of a simple free electron system. The dispersion relation for free electrons is given by

\[E(\mathbf{k}) = \frac{\hbar^2 ||\mathbf{k}||^2}{2m^*}.\]

For our simple example, ignore the units. Just note that energy units are in milli electronvolts (meV) and distance units are in angstroms (Å). We can define the dispersion as:

dispersion = "kx**2 + ky**2 + kz**2"

For this to generate a unit sphere for the Fermi surface in k-space, we need to set the chemical potential to 1 and the unit cell dimentions to \(2\pi\) in all three directions.

mu = 1.0
cell = (2 * np.pi, 2 * np.pi, 2 * np.pi)

We’re now ready to create the band structure object.

band = elecboltz.BandStructure(
    dispersion=dispersion, chemical_potential=mu, unit_cell=cell,
    periodic=False)

Since we’re dealing with a simple sphere, the periodic boundary conditions are not needed. Next, we can run the discretization to create the mesh used by the solver.

band.discretize()

Now we can create the conductivity calculator object. Let’s just use a scattering rate of 1 and a field of magnitude 1 in the z direction. Note that the scattering rate is in units of THz (or 1/ps) and the magnetic field is in units of Tesla.

conductivity = elecboltz.Conductivity(
    band=band, scattering_rate=1.0, field=(0, 0, 1.0))

Finally, we can calculate the conductivity tensor.

sigma = conductivity.calculate()

That’s it! This is the basic workflow for using elecBoltz. Of course, you can do much more; you can define more complex band structures with various parameters; set the resolution; and control which axes have periodic boundary conditions;

dispersion = "- 2*t * (cos(a*kx)+cos(b*ky))" \
             "- 4*tp * cos(a*kx)*cos(b*ky)" \
             "- 2*tpp * (cos(2*a*kx)+cos(2*b*ky))" \
             "- 2*tz * (cos(a*kx)-cos(b*ky))**2" \
             " * cos(a*kx/2)*cos(b*ky/2)*cos(c*kz/2)"

band = elecboltz.BandStructure(
    dispersion=dispersion, chemical_potential=mu, unit_cell=cell,
    band_params={'t': 1, 'tp':-0.13642799, 'tpp':0.06816836, 'tz':0.06512192},
    resolution=[41, 41, 21], periodic=2)
    # periodic in axis 2, which is the z axis (0 is x, 1 is y, 2 is z)

you can have any arbitrary function as the scattering rate;

def scattering_rate(kx, ky, kz):
    phi = np.atan2(ky, kx)
    return 1.0 + 0.1 * np.abs(np.cos(2*phi))**2

conductivity = elecboltz.Conductivity(
    band=band, scattering_rate=scattering_rate, field=(0, 0, 1.0))

and even set a frequency for the fields, which gives you optical conductivity.

conductivity = elecboltz.Conductivity(
    band=band, scattering_rate=1.0, field=(0, 0, 1.0), frequency=1.0)