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functions.py
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# -*- coding: utf-8 -*-
"""
Created on Mon Aug 7 11:14:14 2023
@author: viola
"""
# import necessary modules
import numpy as np
import mpmath
from scipy.integrate import quad
# useful elementary functions to recall later
plog = np.frompyfunc(mpmath.polylog, 2, 1)
def log1(x):
"""
function to calculate the formula log(1 + exp(-x)),
if x is an array
Parameters
----------
x : TYPE float
DESCRIPTION variable that in later calculations will be given
by the difference between the chemical potential and the energy gap
Returns
-------
calculated function : TYPE float
DESCRIPTION calculated value for the log(1 + exp(-x)) function
"""
return np.log(1 + np.exp(-x))
def exp1(x):
"""
function to calculate the formula [exp(x) + 1],
if x is an array
Parameters
----------
x : TYPE float
DESCRIPTION variable that in later calculations will be given
by the difference between the chemical potential and the energy gap
Returns
-------
calculated function : TYPE float
DESCRIPTION calculated value for the [exp(x) + 1] function
"""
return np.exp(x) + 1
# F functions for conduction band
def F0c(delta,
eta):
"""
function to calculate the F_0 function for the conduction band
Parameters
----------
delta : TYPE float
DESCRIPTION energy gap value to be entered in the function
eta : TYPE float
DESCRIPTION chemical potential value to be entered in the function
Returns
-------
calculated function : TYPE float
DESCRIPTION calculated value for the F_0 function for the conduction band
"""
return np.exp(eta - delta)/exp1(eta - delta)
def F1c(delta,
eta):
"""
function to calculate the F_1 function for the conduction band
Parameters
----------
delta : TYPE float
DESCRIPTION energy gap value to be entered in the function
eta : TYPE float
DESCRIPTION chemical potential value to be entered in the function
Returns
-------
calculated function : TYPE float
DESCRIPTION calculated value for the F_1 function for the conduction band
"""
exp = (eta - delta)/exp1(eta - delta) # 1st term (exponential dependency)
log = log1(eta - delta) # 2nd term (logarithmic dependency)
return exp + log
def F2c(delta,
eta):
"""
function to calculate the F_2 function for the conduction band
Parameters
----------
delta : TYPE float
DESCRIPTION energy gap value to be entered in the function
eta : TYPE float
DESCRIPTION chemical potential value to be entered in the function
Returns
-------
calculated function : TYPE float
DESCRIPTION calculated value for the F_2 function for the conduction band
"""
A = (np.pi**2)/3 # 1st term (constant)
exp = -((eta - delta)**2)/exp1(eta - delta) # 2nd term (exponential dependency)
log = -2*(eta - delta)*log1(eta - delta) # 3rd term (logarithmic dependency)
polylog = 2*plog(2, -np.exp( -(eta - delta))) # 4th term (polylogarithmic)
return A + exp + log + polylog
def F3c(delta,
eta):
"""
function to calculate theF_3 function for the conduction band
Parameters
----------
delta : TYPE float
DESCRIPTION energy gap value to be entered in the function
eta : TYPE float
DESCRIPTION chemical potential value to be entered in the function
Returns
-------
calculated function : TYPE float
DESCRIPTION calculated value for the F_3 function for the conduction band
"""
exp = ((eta - delta)**3)/exp1(eta - delta) # 1st term (exponential dependency)
log = 3*((eta - delta)**2)*log1(eta - delta) # 2nd term (logarithmic dependency)
polylog_1 = -6*(eta - delta)*plog(2, - np.exp( - (eta - delta))) # 3rd term (1st polylogarithmic term)
polylog_2 = -6*plog(3, - np.exp( - (eta - delta))) # 4th term (2nd polylogarithmic term)
return 2*(exp + log + polylog_1 + polylog_2)
# F functions for valence band
def F0v(delta,
eta):
"""
function to calculate the F_0 function for the valence band
Parameters
----------
delta : TYPE float
DESCRIPTION energy gap value to be entered in the function
eta : TYPE float
DESCRIPTION chemical potential value to be entered in the function
Returns
-------
calculated function : TYPE float
DESCRIPTION calculated value for the F_0 function for the valence band
"""
return 1/exp1(eta + delta)
def F1v(delta,
eta):
"""
function to calculate the F_1 function for the valence band
Parameters
----------
delta : TYPE float
DESCRIPTION energy gap value to be entered in the function
eta : TYPE float
DESCRIPTION chemical potential value to be entered in the function
Returns
-------
calculated function : TYPE float
DESCRIPTION calculated value for the F_1 function for the valence band
"""
exp = -(eta + delta)/exp1(eta + delta) # 1st term (exponential dependency)
log = -log1(eta + delta) # 2nd term (logarithmic dependency)
return exp + log
def F2v(delta,
eta):
"""
function to calculate the F_2 function for the valence band
Parameters
----------
delta : TYPE float
DESCRIPTION energy gap value to be entered in the function
eta : TYPE float
DESCRIPTION chemical potential value to be entered in the function
Returns
-------
calculated function : TYPE float
DESCRIPTION calculated value for the F_2 function for the valence band
"""
exp = ((eta + delta)**2)/exp1(eta + delta) # 1st term (exponential dependency)
log = 2*(eta + delta)*log1(eta + delta) # 2nd term (logarithmic dependency)
polylog = -2*plog(2, -np.exp(-(eta + delta))) # 3rd term (polylogarithmic term)
return exp + log + polylog
def F3v(delta,
eta):
"""
function to calculate the F_3 function for the valence band
Parameters
----------
delta : TYPE float
DESCRIPTION energy gap value to be entered in the function
eta : TYPE float
DESCRIPTION chemical potential value to be entered in the function
Returns
-------
calculated function : TYPE float
DESCRIPTION calculated value for the F_3 function for the valence band
"""
exp = ((eta + delta)**3)/exp1(eta + delta) # 1st term (exponential dependency)
log = 3*((eta + delta)**2)*log1(eta + delta) # 2nd term (logarithmic dependency)
polylog_1 = -6*(eta + delta)*plog(2, -np.exp( -(eta + delta))) # 3rd term (1st polylogarithmic term)
polylog_2 = -6*plog(3, -np.exp(-(eta + delta))) # 4th term (2nd polylogarithmic term)
return 2*(exp + log + polylog_1 + polylog_2)
# integrand to compute G functions
def func_Gi(x,
i,
delta,
eta):
"""
function to calculate the G_i function integrand
Parameters
----------
x : TYPE nd.ndarray
DESCRIPTION variable with respect to which the integration will be done,
corresponding to the difference between the energy level and chemical potential
i : TYPE int
DESCRIPTION index parameter to be entered in the function,
indicating the corresponding transport integral
delta : TYPE float
DESCRIPTION energy gap value to be entered in the function
eta : TYPE float
DESCRIPTION chemical potential value to be entered in the function
Returns
-------
calculated function : TYPE float
DESCRIPTION calculated value for the G_i function integrand
"""
if x == -eta:
f=ZeroDivisionError
else:
f = ((delta**2)/((x + eta)**2))*((x**i)*(np.exp(x))/(exp1(x)**2))
return f
# G function for conduction band
def Gic(func_Gi,
i,
delta,
eta):
"""
function to calculate the G_i function for the conduction band
Parameters
----------
func_Gi : TYPE function
DESCRIPTION integrand function
i : TYPE int
DESCRIPTION index parameter to be entered in the function,
indicating the corresponding transport integral
delta : TYPE float
DESCRIPTION energy gap value to be entered in the function
eta : TYPE float
DESCRIPTION chemical potential value to be entered in the function
Returns
-------
calculated function : TYPE float
DESCRIPTION calculated value for the G_i function for the conduction band
"""
integral, error = quad(func_Gi, delta - eta, 300, (i, delta, eta), points=[-eta])
return integral
# G function for valence band
def Giv(func_Gi,
i,
delta,
eta):
"""
function to calculate the G_i function for the valence band
Parameters
----------
func_Gi : TYPE function
DESCRIPTION integrand function
i : TYPE int
DESCRIPTION index parameter to be entered in the function,
indicating the corresponding transport integral
delta : TYPE float
DESCRIPTION energy gap value to be entered in the function
eta : TYPE float
DESCRIPTION chemical potential value to be entered in the function
Returns
-------
calculated function : TYPE float
DESCRIPTION calculated value for the G_i function for the conduction band
"""
integral, error = quad(func_Gi, -300, -(delta + eta), (i, delta, eta), points=[-eta])
return integral