File:Mott Seebeck silicon.svg
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Summary[edit]
DescriptionMott Seebeck silicon.svg |
English: w:Seebeck coefficient of silicon at 300K, using Mott approximations (charge carrier diffusion). |
Date | |
Source | Own work |
Author | Nanite |
SVG development InfoField | This plot was created with Matplotlib. |
Source code InfoField | Python code#!/usr/bin/env python3
# -*- coding: utf-8 -*-
from scipy.special import gamma
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
k = 1.380649e-23 # J/K
e = 1.602176634e-19 # C
eV = e # J
me = 9.10938356e-31 # kg
h = 6.62607015e-34 # J.s
def calc_A(mass, mobility, a, kT):
# calculate A based on density-independent parameters.
# both conductivity and carrier density depend exponentially on
# chemical potential; their ratio however,
# sigma / n = mobility * e
# has no dependence on chemical potential. So:
# A (kT)^a Gamma(a+1) / N = mobility*e
# where N is concentration coefficient:
# N = 2 * (2*pi*mass*kT/h^2)^1.5
return 2*e*mobility * (2*np.pi*mass)**1.5/(h**3 * gamma(a+1)) * kT**(1.5-a)
T = 300 # K
kT = k*T
# material params for Silicon at 300 K
# band energies
EV = 0*eV
EC = 1.124*eV
EF = np.arange(EV + 1*kT, EC - 1*kT, 0.001*eV)
# effective masses for density of states
m_C = 1.09*me
m_V = 1.15*me
# concentration coeffs (m^-3)
conc_C = 2 * (2*np.pi*m_C*kT/h**2)**1.5
conc_V = 2 * (2*np.pi*m_V*kT/h**2)**1.5
# mobilities
mobility_C = 0.140 # m^2/V/s
mobility_V = 0.045 # m^2/V/s
# calculate conductivity prefactor given known mobility
B_C = mobility_C * conc_C * e
B_V = mobility_V * conc_V * e
# scattering mechanism (acoustic phonon = 1.0)
a_C = 1.
a_V = 1.
# Calculate derived functions
cond_C = B_C * np.exp((EF - EC)/kT)
cond_V = B_V * np.exp((EV - EF)/kT)
n_C = conc_C * np.exp((EF - EC)/kT)
n_V = conc_V * np.exp((EV - EF)/kT)
seeb_C = (-k/e) * (1 + a_C + (EC - EF)/kT)
seeb_V = (+k/e) * (1 + a_V + (EF - EV)/kT)
cond = cond_C + cond_V
seeb = (cond_C * seeb_C + cond_V * seeb_V) / (cond_C + cond_V)
# plotting stuff
# data too near to band edges is bad, make it dashed.
leadin = (EF < EV + 4*kT)
leadout = (EF > EC - 4*kT)
main = ~(leadin | leadout)
midgap = EF[np.argmax(seeb < 0)] # zero crossing point of seebeck
# midgap = 0.5 * (EV + EC) # halfway
plt.close('all')
fig = plt.figure()
axl = plt.axes()
fig.set_size_inches(4,3)
plt.xlim(EV - 3*kT, EC + 3*kT)
plt.xticks([EV, EC],
[r"$E_V$", r"$E_C$"])
plt.subplots_adjust(0.15,0.17,0.82,0.98)
axl.plot(EF[main], seeb[main]*1000, color='k', lw=1.5)
axl.plot(EF[leadout], seeb[leadout]*1000, color='k', ls='dashed', lw=1)
axl.plot(EF[leadin], seeb[leadin]*1000, color='k', ls='dashed', lw=1)
axl.set_xlabel(r"Fermi level $\mu$")
axl.set_ylabel(r"Seebeck coefficient $S$ (mV/K)")
#plt.axvline(EV)
#plt.axvline(EC)
#plt.axvline(cross)
anty = -1.99
axl.annotate('',
xy=(midgap-2*kT, anty), xytext=(midgap+2*kT, anty), xycoords='data', textcoords='data',
arrowprops=dict(arrowstyle='<->', shrinkA=0, shrinkB=0, linewidth=0.5))
axl.annotate(r'$4kT$',
verticalalignment='center',
xy=(midgap+2*kT, anty), xytext=(1, 0), xycoords='data', textcoords='offset points')
axl.set_ylim(-2.25, 2.05)
axl.set_yticks([-2,-1,0,1,2])
# draw conductivity content with right axis
axr = axl.twinx()
rcolor='tab:blue'
axr.semilogy(EF[main], cond[main], color=rcolor, lw=1.5, alpha=0.7)
axr.semilogy(EF[leadout], cond[leadout], color=rcolor, ls='dashed', lw=1, alpha=0.7)
axr.semilogy(EF[leadin], cond[leadin], color=rcolor, ls='dashed', lw=1, alpha=0.7)
axr.set_ylabel(r'Conductivity $\sigma$ (S/m)', color=rcolor) # we already handled the x-label with ax1
axr.tick_params(axis='y', labelcolor=rcolor)
axr.set_ylim(0.1e-4,2e6)
axr.set_yticks([1e-4,1e-2,1,1e2,1e4])
fig.savefig('seeb.svg')
|
Licensing[edit]
This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. | |
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse |
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 23:08, 6 April 2019 | 360 × 270 (43 KB) | Nanite (talk | contribs) | {{Information |description ={{en|1=w:Seebeck coefficient of silicon at 300K, using Mott approximations (charge carrier diffusion).}} |date =2019-04-06 |source ={{own}} |author =User:Nanite }} |
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