Boltzmann

boltzmann

FermiSurface(theta, fermi_wavevector, effective_mass, relaxation_time, c_axis_length)

Class to represent a Fermi surface and compute conductivity.

Examples:

>>> import quantalyze as qz
>>> theta = np.linspace(0, 2*np.pi, 100)
>>> fermi_wavevector = np.ones_like(theta) * 1e10  # Example Fermi wavevector
>>> effective_mass = np.ones_like(theta) * electron_mass  # Example effective mass
>>> relaxation_time = np.ones_like(theta) * 1e-14  # Example relaxation time
>>> c_axis_length = 1e-9  # Example c-axis length
>>>  
>>> fermi_surface = qz.FermiSurface(theta, fermi_wavevector, effective_mass, relaxation_time, c_axis_length)
>>> magnetic_field = 1.0  # Example magnetic field in Tesla
>>> sxx, sxy, syx, syy = fermi_surface.calculate_conductivity(magnetic_field)
>>> rxx = sxx / (sxx * syy - sxy * syx)

Source code in src/quantalyze/beta/boltzmann.py

def __init__(
    self,
    theta,
    fermi_wavevector,
    effective_mass,
    relaxation_time,
    c_axis_length,
    ):
    self.theta = theta
    self.fermi_wavevector = fermi_wavevector
    self.effective_mass = effective_mass
    self.relaxation_time = relaxation_time
    self.c_axis_length = c_axis_length

calculate_conductivity(magnetic_field, start_phi=1e-09, end_phi=2 * np.pi, phi_points=250, exponent_points=150)

Calculate the conductivity tensor components for the Fermi surface given a magnetic field.

Parameters:

Name	Type	Description	Default
magnetic_field		Magnetic field strength (in Tesla).	required
start_phi		Start of phi integration (in radians).	1e-09
end_phi		End of phi integration (in radians).	2 * pi
phi_points		Number of points for phi integration.	250
exponent_points		Number of points for exponent integration.	150

Returns:

Type	Description
	sxx, sxy, syx, syy: Conductivity tensor components.

Source code in src/quantalyze/beta/boltzmann.py

def calculate_conductivity(
    self,
    magnetic_field,
    start_phi=1e-9, # start of phi integration
    end_phi=2*np.pi, # end of phi integration
    phi_points=250, # number of points for phi integration
    exponent_points=150, # number of points for exponent integration
):
    """
    Calculate the conductivity tensor components for the Fermi surface given a magnetic field.

    Args:
        magnetic_field: Magnetic field strength (in Tesla).
        start_phi: Start of phi integration (in radians).
        end_phi: End of phi integration (in radians).
        phi_points: Number of points for phi integration.
        exponent_points: Number of points for exponent integration.

    Returns:
        sxx, sxy, syx, syy: Conductivity tensor components.
    """
    return _compute_conductivity_from_arrays(
        self.theta,
        self.fermi_wavevector,
        self.effective_mass,
        self.relaxation_time,
        self.c_axis_length,
        magnetic_field=magnetic_field,
        start_phi=start_phi,
        end_phi=end_phi,
        phi_points=phi_points,
        exponent_points=exponent_points,
    )

calculate_normal_vectors()

Calculate the normal vectors to the Fermi surface at each theta value.

Source code in src/quantalyze/beta/boltzmann.py

def calculate_normal_vectors(self):
    """
    Calculate the normal vectors to the Fermi surface at each theta value.
    """
    zeta = _calculate_zeta_array(self.theta, self.fermi_wavevector)
    normal_x = np.cos(self.theta - zeta)
    normal_y = np.sin(self.theta - zeta)
    return normal_x, normal_y

carrier_density()

Calculate the carrier density based on the Fermi volume.

Source code in src/quantalyze/beta/boltzmann.py

def carrier_density(self):
    """
    Calculate the carrier density based on the Fermi volume.
    """
    return self.fermi_volume() / (2 * np.pi)**3 * 2

cylotron_frequency(magnetic_field)

Compute the local cyclotron frequency for a given magnetic field at each theta value.

Source code in src/quantalyze/beta/boltzmann.py

def cylotron_frequency(self, magnetic_field):
    """
    Compute the local cyclotron frequency for a given magnetic field at each theta value.
    """
    return ELEMENTARY_CHARGE * magnetic_field / self.effective_mass

fermi_area()

Compute the area of the Fermi surface in k-space by integrating the Fermi wavevector over theta.

Source code in src/quantalyze/beta/boltzmann.py

def fermi_area(self):
    """
    Compute the area of the Fermi surface in k-space by integrating the Fermi wavevector over theta.
    """
    return self._integrate_fermi_wavevector()

fermi_volume()

Calculate the volume of the Fermi surface in k-space by integrating the Fermi wavevector and multiplying by the reciprocal lattice vector.

Source code in src/quantalyze/beta/boltzmann.py

def fermi_volume(self):
    """
    Calculate the volume of the Fermi surface in k-space by integrating the Fermi wavevector 
    and multiplying by the reciprocal lattice vector.
    """
    return self._integrate_fermi_wavevector() * self.reciprocal_lattice_vector()

fermi_wavevector_x()

Compute the x-component of the Fermi wavevector for each theta value.

Source code in src/quantalyze/beta/boltzmann.py

def fermi_wavevector_x(self):
    """
    Compute the x-component of the Fermi wavevector for each theta value.
    """
    return self.fermi_wavevector * np.cos(self.theta)

fermi_wavevector_y()

Compute the y-component of the Fermi wavevector for each theta value.

Source code in src/quantalyze/beta/boltzmann.py

def fermi_wavevector_y(self):
    """
    Compute the y-component of the Fermi wavevector for each theta value.
    """
    return self.fermi_wavevector * np.sin(self.theta)

mean_free_path()

Calculate the mean free path of electrons on the Fermi surface.

Source code in src/quantalyze/beta/boltzmann.py

def mean_free_path(self):
    """
    Calculate the mean free path of electrons on the Fermi surface.
    """
    return HBAR * self.fermi_wavevector * self.relaxation_time / self.effective_mass

omega_c_tau(magnetic_field)

Compute the product of the cyclotron frequency and relaxation time for a given magnetic field at each theta value.

Source code in src/quantalyze/beta/boltzmann.py

def omega_c_tau(self, magnetic_field):
    """
    Compute the product of the cyclotron frequency and relaxation time for a given magnetic field at each theta value.
    """
    return self.cylotron_frequency(magnetic_field) * self.relaxation_time

reciprocal_lattice_vector()

Compute the reciprocal c-axis lattice vector.

Source code in src/quantalyze/beta/boltzmann.py

def reciprocal_lattice_vector(self):
    """
    Compute the reciprocal c-axis lattice vector.
    """
    return 2 * np.pi / self.c_axis_length