lbteint module¶

This module, lbteint.py contains the integral solvers for the linearized Boltzmann Transport equations.

Contains the routines to perform the Boltzmann transport integrals.

lbteint.analytic_k_space_energy(kx, ky, kz, effmass, e_shift)¶

Returns the parabolic energy dispersion.

Parameters:	transport : object A Transport() object kx : float The $k_x$ in cartesian coordinates. ky : float The $k_y$ in cartesian coordinates. kz : float The $k_z$ in cartesian coordinates. effmass : ndarray Dimension: (3) The effective mass along $k_x$ , $k_y$ and $k_z$ , respectively.
Returns:	float The energy value in eV. Warning This routine only accepts the diagonal elements of the effective mass tensor

lbteint.analytic_k_space_integrand(kz, ky, kx, eta, beta, effmass, e0, i, l, m)¶

Returns the integrand for the anlytic reciprocal space integration of the transport tensor.

Parameters:

Parameters:	kz : float The $k_z$ in cartesian coordinates. ky : float The $k_y$ in cartesian coordinates. kx : float The $k_x$ in cartesian coordinates. eta : float The reduced chemical potential. beta : float The $\\beta$ factor, $\\beta=(k_bT)^{-1}$ in eV. effmass : float The effective mass in units of the free electron mass. e0 : float The energy shift, e.g. $E=\\hbar^2 k^2/2m + E_0$ , where $E_0$ is the energy shift in eV. i : int The order of the transport tensor. l : {0,1,2} The first index of the transport tensor. m : {0,1,2} The second index of the transport tensor.
Returns:	float The integrand value.

kz : float: The $k_z$ in cartesian coordinates.
ky : float: The $k_y$ in cartesian coordinates.
kx : float: The $k_x$ in cartesian coordinates.
eta : float: The reduced chemical potential.
beta : float: The $\\beta$ factor, $\\beta=(k_bT)^{-1}$ in eV.
effmass : float: The effective mass in units of the free electron mass.
e0 : float: The energy shift, e.g. $E=\\hbar^2 k^2/2m + E_0$ , where $E_0$ is the energy shift in eV.
i : int: The order of the transport tensor.
l : {0,1,2}: The first index of the transport tensor.
m : {0,1,2}: The second index of the transport tensor.

Returns:

float: The integrand value.

lbteint.analytic_k_space_velocity(kx, ky, kz, effmass, i)¶

Returns the parabolic velocity dispersion.

Parameters:	kx : float The $k_x$ in cartesian coordinates. ky : float The $k_y$ in cartesian coordinates. kz : float The $k_z$ in cartesian coordinates. effmass : ndarray Dimension: (3) The effective mass along $k_x$ , $k_y$ and $k_z$ , respectively. i : {0,1,2} The direction to evaluate the velocity (0 is along $k_x$ etc.).
Returns:	float The velocity in eVAA. Warning This routine only accepts the diagonal elements of the effective mass tensor. The $\\hbar/m_e$ factor is not returned and need to be introduced externally.

lbteint.concatenate_integrand(energies, velocities, scatter, spin_fact, chempot, beta, order)¶: Concatenates the integrand in the Boltzmann transport integral.

lbteint.concatenate_integrand_band(energies, velocities, tau, spin_fact, chempot, beta, order)¶: Concatenates the integrand in the Boltzmann transport integral and sums the bands.

lbteint.fermiintclosed(order, eta, spin_fact)¶: Returns the value of the closed expressions for the Fermi integrals.

lbteint.integrandpar(eps, transport, w0, eta, beta, energy_trans, effmass, i)¶

Returns the integrand used in the analytic energy integration of the transport coefficients in integrate_e()

Parameters:

Parameters:	eps : float The reduced carrier energy. transport : object A Transport() object. w0 : ndarray Dimension: (12) Contains the scattering rate prefactor for the different scattering mechanisms in units of inverse fs. eta : float The reduced chemical potential. beta : float The $\\beta$ factor in eV. energy_trans : ndarray Dimension: (12) Contains the energy transitions (that is added to the energy in $\\tau=\\tau_0E^{r-1/2}$ , typically, $E=E+\\hbar \\omega$ , where $\\hbar \\omega$ is the size of the energy transition. Set it to zero for the non-relevant scattering mechanisms. effmass : float The effective mass in units of the electron mass. i : int The order of the transport integral to be evaluated.
Returns:	float The integrand value.

eps : float: The reduced carrier energy.
transport : object: A Transport() object.
w0 : ndarray: Dimension: (12)

Contains the scattering rate prefactor for the different scattering mechanisms in units of inverse fs.
eta : float: The reduced chemical potential.
beta : float: The $\\beta$ factor in eV.
energy_trans : ndarray: Dimension: (12)

Contains the energy transitions (that is added to the energy in $\\tau=\\tau_0E^{r-1/2}$ , typically, $E=E+\\hbar \\omega$ , where $\\hbar \\omega$ is the size of the energy transition. Set it to zero for the non-relevant scattering mechanisms.
effmass : float: The effective mass in units of the electron mass.
i : int: The order of the transport integral to be evaluated.

Returns:

float: The integrand value.

Notes

The total scattering is calculated based on the well known scattering models for parabolic energy dispersions $\\tau=\\tau_0\\epsilon^{r-1/2}$ , where $r$ is the scattering factor.

lbteint.integrandpardos(eps, transport, w0, eta, beta, energy_trans, effmass, i)¶

The integrand for the density of states integral over energy.

Parameters:

Parameters:	eps : float The reduced carrier energy transport : object A Transport() object w0 : ndarray Dimension: (12) Contains the scattering rate prefactor for the different scattering mechanisms in units of inverse fs. Not used in this routine, but it needs the dummy from the call argument. eta : float The reduced chemical potential beta : float The $\\beta$ factor in eV. Not used in this routine, but it needs the dummy from the call argument. energy_trans : ndarray Dimension: (12) Contains the energy transitions (that is added to the energy in $\\tau=\\tau_0E^{r-1/2}$ , typically, $E=E+\\hbar \\omega$ , where $\\hbar \\omega$ is the size of the energy transition. Set it to zero for the non-relevant scattering mechanisms. Not used in this routine, but it needs the dummy from the call argument. effmass : float The effective mass in units of the electron mass. Not used in this routine, but it needs the dummy from the call argument. i : int The order of the transport integral to be evaluated. Not used in this routine, but it needs the dummy from the call argument.
Returns:	float The integrand value for the density of states.

eps : float: The reduced carrier energy
transport : object: A Transport() object
w0 : ndarray: Dimension: (12)

Contains the scattering rate prefactor for the different scattering mechanisms in units of inverse fs. Not used in this routine, but it needs the dummy from the call argument.
eta : float: The reduced chemical potential
beta : float: The $\\beta$ factor in eV. Not used in this routine, but it needs the dummy from the call argument.
energy_trans : ndarray: Dimension: (12)

Contains the energy transitions (that is added to the energy in $\\tau=\\tau_0E^{r-1/2}$ , typically, $E=E+\\hbar \\omega$ , where $\\hbar \\omega$ is the size of the energy transition. Set it to zero for the non-relevant scattering mechanisms. Not used in this routine, but it needs the dummy from the call argument.
effmass : float: The effective mass in units of the electron mass. Not used in this routine, but it needs the dummy from the call argument.
i : int: The order of the transport integral to be evaluated. Not used in this routine, but it needs the dummy from the call argument.

Returns:

float: The integrand value for the density of states.

Notes

Calculates the density of states integrand

System Message: WARNING/2 (\\frac{\\epsilon^{1/2}}{1+\\exp{(\\epsilon-\\eta)}})

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lbteint.integrandpart2(eps, transport, w0, eta, beta, energy_trans, effmass, i)¶

Returns the integrand used in the analytic energy integration of the transport distribution function with a quadratic $\\tau$ term

Parameters:

Parameters:	eps : float The reduced carrier energy. transport : object A Transport() object. w0 : ndarray Dimension: (12) Contains the scattering rate prefactor for the different scattering mechanisms in units of inverse fs. eta : float The reduced chemical potential. beta : float The $\\beta$ factor in eV. energy_trans : ndarray Dimension: (12) Contains the energy transitions (that is added to the energy in $\\tau=\\tau_0E^{r-1/2}$ , typically, $E=E+\\hbar \\omega$ , where $\\hbar \\omega$ is the size of the energy transition. Set it to zero for the non-relevant scattering mechanisms. effmass : float The effective mass in units of the electron mass. i : int The order of the transport integral to be evaluated.
Returns:	float The integrand value.

eps : float: The reduced carrier energy.
transport : object: A Transport() object.
w0 : ndarray: Dimension: (12)

Contains the scattering rate prefactor for the different scattering mechanisms in units of inverse fs.
eta : float: The reduced chemical potential.
beta : float: The $\\beta$ factor in eV.
energy_trans : ndarray: Dimension: (12)

Contains the energy transitions (that is added to the energy in $\\tau=\\tau_0E^{r-1/2}$ , typically, $E=E+\\hbar \\omega$ , where $\\hbar \\omega$ is the size of the energy transition. Set it to zero for the non-relevant scattering mechanisms.
effmass : float: The effective mass in units of the electron mass.
i : int: The order of the transport integral to be evaluated.

Returns:

float: The integrand value.

Notes

The total scattering is calculated based on the well known scattering models for parabolic energy dispersions $\\tau=\\tau_0\\epsilon^{r-1/2}$ , where $r$ is the scattering factor. Difference from integrandpar(): here tau^2 is used in the integrand (for the calculation of the Hall factor).

lbteint.scipy_e_integrals(transport, integrand, e_min, e_max, w0, eta, beta, energy_trans, effmass, order, spin_fact, method='quad')¶

Calculates the one dimensional energy integrals.

Uses the SciPy function scipy.integrate.quad().

Parameters:

Parameters:	transport : object A Transport() object integrand : {“normal”,”hall”,”dos”} Selects the type of integrand to be used. “normal” selects `integrandpar()`. “hall” selects `integrandpart2()`, “dos” selects `integrandpardos()`. e_min : float The lower integration limit in eV. e_max : float The higher integration limit in eV. w0 : ndarray Dimension: (12) Contains the scattering rate prefactor (inverse of relaxation time) for the different scattering mechanisms in units of inverse fs. eta : float The reduced chemical potential. beta : float The $\\beta$ factor, $(\\mathrm{k_b}T)^{-1}$ in eV. effmass : ndarray Dimension: (3) The effective mass along the three reciprocal unit vectors in units of the free electron mass. e0 : float The energy shift, e.g. $E=\\hbar^2 k^2/2m + E_0$ , where $E_0$ is the energy shift in eV. i : int The order of the transport tensor. l : {0,1,2} The first index of the transport tensor. m : {0,1,2} The second index of the transport tensor. spin_fact : int The spin degeneracy factor. 1 for non-spin degeneracy, 2 for spin degeneracy. method : {“quad”}, optional The SciPy three dimensional integration method using `scipy.integrate.quad()`.
Returns:	float The resulting integral over energy.

transport : object: A Transport() object
integrand : {“normal”,”hall”,”dos”}: Selects the type of integrand to be used. “normal” selects integrandpar(). “hall” selects integrandpart2(), “dos” selects integrandpardos().
e_min : float: The lower integration limit in eV.
e_max : float: The higher integration limit in eV.
w0 : ndarray: Dimension: (12)

Contains the scattering rate prefactor (inverse of relaxation time) for the different scattering mechanisms in units of inverse fs.
eta : float: The reduced chemical potential.
beta : float: The $\\beta$ factor, $(\\mathrm{k_b}T)^{-1}$ in eV.
effmass : ndarray: Dimension: (3)

The effective mass along the three reciprocal unit vectors in units of the free electron mass.
e0 : float: The energy shift, e.g. $E=\\hbar^2 k^2/2m + E_0$ , where $E_0$ is the energy shift in eV.
i : int: The order of the transport tensor.
l : {0,1,2}: The first index of the transport tensor.
m : {0,1,2}: The second index of the transport tensor.
spin_fact : int: The spin degeneracy factor. 1 for non-spin degeneracy, 2 for spin degeneracy.
method : {“quad”}, optional: The SciPy three dimensional integration method using scipy.integrate.quad().

Returns:

float: The resulting integral over energy.

lbteint.scipy_k_integrals(eta, beta, effmass, e0, i, l, m, method='tplquad')¶

Calculates the three dimensional wave vector integrals.

Uses the SciPy function scipy.integrate.tplquad().

Parameters:

Parameters:	eta : float The reduced chemical potential beta : float The $\\beta$ factor, $(\\mathrm{k_b}T)^{-1}$ in eV. effmass : ndarray Dimension: (3) The effective mass along the three reciprocal unit vectors in units of the free electron mass. e0 : float The energy shift, e.g. $E=\\hbar^2 k^2/2m + E_0$ , where $E_0$ is the energy shift in eV. i : int The order of the transport tensor. l : {0,1,2} The first index of the transport tensor m : {0,1,2} The second index of the transport tensor method : {“tplquad”}, optional The SciPy three dimensional integration method using `scipy.integrate.tplquad()`.
Returns:	float The resulting integral over the wave vectors.

eta : float: The reduced chemical potential
beta : float: The $\\beta$ factor, $(\\mathrm{k_b}T)^{-1}$ in eV.
effmass : ndarray: Dimension: (3)

The effective mass along the three reciprocal unit vectors in units of the free electron mass.
e0 : float: The energy shift, e.g. $E=\\hbar^2 k^2/2m + E_0$ , where $E_0$ is the energy shift in eV.
i : int: The order of the transport tensor.
l : {0,1,2}: The first index of the transport tensor
m : {0,1,2}: The second index of the transport tensor
method : {“tplquad”}, optional: The SciPy three dimensional integration method using scipy.integrate.tplquad().

Returns:

float: The resulting integral over the wave vectors.