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

transportobject: A Transport() object
kxfloat: The $k_x$ in cartesian coordinates.
kyfloat: The $k_y$ in cartesian coordinates.
kzfloat: The $k_z$ in cartesian coordinates.
effmassndarray: 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

kzfloat: The $k_z$ in cartesian coordinates.
kyfloat: The $k_y$ in cartesian coordinates.
kxfloat: The $k_x$ in cartesian coordinates.
etafloat: The reduced chemical potential.
betafloat: The $\\beta$ factor, $\\beta=(k_bT)^{-1}$ in eV.
effmassfloat: The effective mass in units of the free electron mass.
e0float: The energy shift, e.g. $E=\\hbar^2 k^2/2m + E_0$ , where $E_0$ is the energy shift in eV.
iint: 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

kxfloat: The $k_x$ in cartesian coordinates.
kyfloat: The $k_y$ in cartesian coordinates.
kzfloat: The $k_z$ in cartesian coordinates.
effmassndarray: 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

epsfloat: The reduced carrier energy.
transportobject: A Transport() object.
w0ndarray: Dimension: (12)

Contains the scattering rate prefactor for the different scattering mechanisms in units of inverse fs.
etafloat: The reduced chemical potential.
betafloat: The $\\beta$ factor in eV.
energy_transndarray: 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.
effmassfloat: The effective mass in units of the electron mass.
iint: 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

epsfloat: The reduced carrier energy
transportobject: A Transport() object
w0ndarray: 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.
etafloat: The reduced chemical potential
betafloat: The $\\beta$ factor in eV. Not used in this routine, but it needs the dummy from the call argument.
energy_transndarray: 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.
effmassfloat: The effective mass in units of the electron mass. Not used in this routine, but it needs the dummy from the call argument.
iint: 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)}} )

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017-04-15> Babel <3.15> and hyphenation patterns for 84 language(s) loaded. (/home/efl/opt/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/home/efl/opt/texlive/texmf-dist/tex/latex/base/size12.clo)) (/home/efl/opt/texlive/texmf-dist/tex/latex/base/inputenc.sty (/home/efl/opt/texlive/texmf-dist/tex/latex/base/utf8.def (/home/efl/opt/texlive/texmf-dist/tex/latex/base/t1enc.dfu) (/home/efl/opt/texlive/texmf-dist/tex/latex/base/ot1enc.dfu) (/home/efl/opt/texlive/texmf-dist/tex/latex/base/omsenc.dfu))) (/home/efl/opt/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/home/efl/opt/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/home/efl/opt/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/home/efl/opt/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/home/efl/opt/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/home/efl/opt/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/home/efl/opt/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/home/efl/opt/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/home/efl/opt/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/home/efl/opt/texlive/texmf-dist/tex/latex/tools/bm.sty) (./math.aux) (/home/efl/opt/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/home/efl/opt/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Missing } inserted. <inserted text> } l.14 ...2}}{1+\\exp{(\\epsilon-\\eta)}}\end{split} ! Missing } inserted. <inserted text> } l.14 ...2}}{1+\\exp{(\\epsilon-\\eta)}}\end{split} ! Extra }, or forgotten $. <argument> ...n^{1/2}}{1+\\exp{(\\epsilon-\\eta)}} \end {split} l.14 ...2}}{1+\\exp{(\\epsilon-\\eta)}}\end{split} ! Missing { inserted. <inserted text> { l.14 ...2}}{1+\\exp{(\\epsilon-\\eta)}}\end{split} ! Missing } inserted. <inserted text> } l.14 ...2}}{1+\\exp{(\\epsilon-\\eta)}}\end{split} ! Missing } inserted. <inserted text> } l.14 ...2}}{1+\\exp{(\\epsilon-\\eta)}}\end{split} ! Extra }, or forgotten $. <argument> ...n^{1/2}}{1+\\exp{(\\epsilon-\\eta)}} \end {split} l.14 ...2}}{1+\\exp{(\\epsilon-\\eta)}}\end{split} ! Missing { inserted. <inserted text> { l.14 ...2}}{1+\\exp{(\\epsilon-\\eta)}}\end{split} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 532 bytes). Transcript written on math.log.

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

epsfloat: The reduced carrier energy.
transportobject: A Transport() object.
w0ndarray: Dimension: (12)

Contains the scattering rate prefactor for the different scattering mechanisms in units of inverse fs.
etafloat: The reduced chemical potential.
betafloat: The $\\beta$ factor in eV.
energy_transndarray: 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.
effmassfloat: The effective mass in units of the electron mass.
iint: 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

transportobject: 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_minfloat: The lower integration limit in eV.
e_maxfloat: The higher integration limit in eV.
w0ndarray: Dimension: (12)

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

The effective mass along the three reciprocal unit vectors in units of the free electron mass.
e0float: The energy shift, e.g. $E=\\hbar^2 k^2/2m + E_0$ , where $E_0$ is the energy shift in eV.
iint: 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_factint: 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

etafloat: The reduced chemical potential
betafloat: The $\\beta$ factor, $(\\mathrm{k_b}T)^{-1}$ in eV.
effmassndarray: Dimension: (3)

The effective mass along the three reciprocal unit vectors in units of the free electron mass.
e0float: The energy shift, e.g. $E=\\hbar^2 k^2/2m + E_0$ , where $E_0$ is the energy shift in eV.
iint: 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.