Intrinsic free charge carriers¶
Get a sense of these formulas by seeing the huge effect of temperature on semiconductor properties.
%matplotlib inline
import matplotlib as mpl
from pylab import *
#B_Si = (7.3e15)**2 #1.08e31 # K^-3 cm^-6 # ?? SedraSmith version
B_Si = (5.2e15)**2 # Razavi (2.1)
Eg_Si = 1.12 # eV (=1.602 x 10-19 J)
KB = 8.62e-5 # eV / K
def ni2(T, B=B_Si, Eg=Eg_Si):
return B * T**3 * exp(-Eg / (KB * T))
def ann_temp(T=25, text='Room temp'):
"""Helper to add an arrow to room temperature."""
ax = gca()
x,y = (T, sqrt(ni2(T+273.15)))
plot(x, y, 'o')
ax.annotate(text, xy=(x,y), xytext=(x, y/5),
arrowprops=dict(arrowstyle='->',))
# increase sizes for display
mpl.rcParams.update({'figure.figsize': (12.0, 8.0), 'font.size': 18,})
tempK = linspace(1, 400)
semilogy(tempK, sqrt(ni2(tempK)))
title('Thermally-generated free carrier density, $n_i$')
ylabel('carrier density ($cm^{-3}$)')
xlabel('Temperature (Kelvin)')
grid(True)
For a non-bogus formula, check that we get $n_i = 0$ at $T=0\,K$.
commercial =(0, 85)
industrial = (-40, 100)
automotive = (-40, 125)
military = (-55, 125)
for tr,name in ((military, 'Military'),
(automotive, 'Automotive'),
(industrial, 'Industrial'),
(commercial, 'Commercial'),):
tempC = arange(tr[0], tr[1], 1)
tempK = tempC + 272.15
semilogy(tempC, sqrt(ni2(tempK)),
label='%s (%i to %i)' % (name, tr[0], tr[1]),
linewidth=4)
legend(loc='upper left')
ann_temp(300-272)
ylabel('Carrier density, $n_i$ ($cm^{-3}$)')
xlabel('Temperature ($^\circ C$)')
title('Intrinsic free carrer density over temperature')
grid(True)
THIS IS A HUGE RANGE !!!
... and plain huge numbers
How do they compare to the number of atoms in the crystal?
hw02 question 1¶
Find the values of the intrinsic carrier concentration $n_i$ for silicon at -55, 0, 20, 75, and 125 $^\circ C$ in $\# / cm^3$.
At each temperature, what fraction of the atoms are ionized? (Silicon crystal has about $5 x 10^{22}$ $\text{atoms} / cm^3$).
print(' T n #/cm3 fraction free')
for t in (-55, 0, 20, 75, 125):
n = sqrt(ni2(t+273.15))
print('%3i %0.2e %.02e' % (t, n, n/5e22))
So, even at $125^\circ C$, only $1$ atom in $10^{10}$ is contributing a free electron. In this context, the numbers are pretty remarkable that such a small fraction of free electrons are responsible for all of semiconductor operation!
hw02 question 2¶
Calculate the value of $n_i$ for gallium arsenide (GaAs) at T=300K.
B_GaAs = (3.56e14)**2 # B or B^2 depends on which formula you are using! -> check!
Eg_GaAs = 1.42
ni = sqrt(ni2(300, B=B_GaAs, Eg=Eg_GaAs))
print('%.3g (cm^-3)' % ni)