{{{!highlight python #!/usr/bin/env python

from pylab import *

T = 300 # K kB = 0.02585 / 300 # eV/K EG = 1.170 - 4.730e-4 * T ** 2 / (T + 636) # pages 69,70 mnr = 1.028 + 6.11e-4 * T - 3.09e-7 * T ** 2 # mnstar / m_e -- page 55 mpr = 0.610 + 7.83e-4 * T - 4.46e-7 * T ** 2 # mpstar / m_e -- page 55

# What do we need to plot? # e distribution function, edf = gc(E)*f(E) -- for E >= Ec # h distribution function, hdf = gv(E)*(1-f(E)) -- for E <= Ec # Define gcp(x) = gc(x+Ec). # Then, # gc(E)*f(E) = gcp(x)*f(x+Ec) # where x = E - Ec >= 0 is the electron excitation energy. # Likewise, define gvp(x) = gv(Ev-x) # gv(E)*(1-f(E)) = gvp(x)*(1-f(Ev-x)) # where x = Ev - E >= 0 is the hole excitation energy.

# x is in unit of eV -- it is positive and used both for e and h. # We are taking it to go to 7 kB T. x = linspace (0, 5, 100) * kB * T

# Check that they are reasonable. print 'mnr =', mnr print 'mpr =', mpr

# These are from Eqs. T2.6a,b with x = E-Ec or Ev-E. # If x is in unit of eV, then gcp, gvp are in unit of # eV ** 0.5 * eV ** 1.5 / eV ** 3 / Angstrom ** 3 # = 1 / eV / Angstrom ** 3 gcp = sqrt(2. * x) * (mnr * 0.511e6) ** 1.5 / pi ** 2 / 1973 ** 3 gvp = sqrt(2. * x) * (mpr * 0.511e6) ** 1.5 / pi ** 2 / 1973 ** 3

# As given in the problem. Ev = 0. Ec = EG EF = 0.75 * EG

# Multiply by f(x+Ec) and 1-f(Ev-x) respectively, as shown above. edf = gcp / (exp ((x + Ec - EF) / kB / T) + 1) hdf = gvp / (exp ((x - Ev + EF) / kB / T) + 1)

# Finally, convert the unit to 1 / eV / cm ** 3 edf *= 1.e24 hdf *= 1.e24

# Check the orders of magnitude. print 'max (edf) / max (hdf) = %g' % (max (edf) / max (hdf))

# However, the hole counts are very low -- blow them up. # Based on what we learnd just now. hdf *= 5.e9

# All hard compuations now done -- plot them! plot (x / kB / T, edf, label = 'Electron') plot (x / kB / T, hdf, label = r'Hole ($\times$ $5\times10^9$)') grid ()

# matplotlib comes with its own LaTeX parser -- cool! xlabel (r'Carrier Excitation Energy (in unit of kT)') ylabel ('Carrier Distribution (1/(eV cm$^3$))') title ('Carrier Distribution Functions at T = 300 K') legend ()

# Sort of equivalent to "print" of matlab. "print" is a python keyword. # Comment out one of these lines to get a print-out in batch mode. #savefig ('E2p3.svg') #savefig ('E2p3.eps') savefig ('P2p9.png') }}}