Chapter 1 : Semiconductor Materials

Example : 1.1 - Page No : 15

In [1]:
from __future__ import division
from math import exp
#Given data
E_G = 0.72 # in eV
E_F = (1/2)*E_G # in eV
k = 8.61*10**-5 # in eV/K
T = 300 # in K 
# The fraction of the total number of electrons 
n_C_by_n = 1/( 1 + (exp((E_G-E_F)/(k*T))) ) 
print "The fraction of the total number of electrons = %0.2e" %n_C_by_n

The fraction of the total number of electrons = 8.85e-07

Example : 1.2 - Page No : 20

In [3]:
 #Given data
n_i = 1.4*10**18 # in /m**3
N_D = 1.4*10**24 # in /m**3
n = N_D # in /m**3
p = (n_i**2)/n # in /m**3
# Ratio of electron to hole concentation,
ratio = n/p 
print "Ratio of electron to hole concentration = %0.1e" %ratio

Ratio of electron to hole concentration = 1.0e+12

Example : 1.3 - Page No : 22

In [4]:
 #Given data
e = 1.6*10**-19 # in C
m = 9.1*10**-31 # in kg
miu_e = 7.04 * 10**-3 # in m**2/V-s
n = 5.8*10**28 # in /m**3
torque = (miu_e/e)*m # in sec
print "The relaxation time = %0.3e second " %torque
sigma = n*e*miu_e 
rho = 1/sigma  # in ohm-m
print "The resistivity of conductor = %0.3e ohm-m " %rho

The relaxation time = 4.004e-14 second 
The resistivity of conductor = 1.531e-08 ohm-m

Example : 1.4 - Page No : 22

In [5]:
 #Given data
e = 1.601*10**-19 # in C
m = 9.107 * 10**-31 # in kg
E = 100 # in V/m
n = 6*10**28 # in /m**3
rho = 1.5*10**-8 # in ohm-m
sigma = 1/rho 
torque = (sigma*m)/(n*(e**2)) # in second
print "The relaxation time = %0.3e second " %torque
v = ((e*E)/m)*torque # in m/s
print "The drift velocity = %0.3f m/s " %v

The relaxation time = 3.948e-14 second 
The drift velocity = 0.694 m/s

Example : 1.5 - Page No : 22

In [6]:
from numpy import pi
#Given data
d = 2 # in mm
d = d * 10**-3 # in m
sigma = 5.8*10**7 # in S/m
miu_e = 0.0032 # in m**2/V-s
E = 20 # in mV/m 
E = E * 10**-3 # in V/m
e = 1.6*10**-19 # in C
n = sigma/(e*miu_e) # in /m**3
print "The charge density of free electrons = %0.3e /m**3 " %n
J = sigma*E # in A/m**2
print "The current density = %0.2e A/m**2 " %J
I = J * ( (pi*(d**2))/4 ) # in A
print "The current flowing in the wire = %0.3f A " %I
v = miu_e*E # in m/s
print "The electron drift velocity = %0.1e m/s " %v

The charge density of free electrons = 1.133e+29 /m**3 
The current density = 1.16e+06 A/m**2 
The current flowing in the wire = 3.644 A 
The electron drift velocity = 6.4e-05 m/s

Example : 1.6 - Page No : 25

In [7]:
 #Given data
l = 1 # in cm
l = l * 10**-2 # in m
A = 1 # in mm**2
A = A * 10**-6 # in m**2
R = 100 # in ohm
rho = (R*A)/l # in ohm-m
sigma = 1/rho 
e = 1.6*10**-19 # in C
miu_e = 1350 # in cm**2/V-s
miu_e = miu_e * 10**-4 # in m**2/V-s
n = sigma/(e*miu_e) # in /m**3
print "The dopant density = %0.2e /m**3" %n

# Note: The unit of the answer is wrong because 0.0463*10**23/m**3 = 4.63*10**21/m**3, not in /cm**3

The dopant density = 4.63e+21 /m**3

Example : 1.7 - Page No : 26

In [8]:
 #Given data
R = 1 # in k ohm
R = R * 10**3 # in ohm
L = 400 # in µm
L = L * 10**-6 # in m
W = 20 # in µm
W = W * 10**-6 # in m
a = L*W # in m**2
l = 4 # in mm
l = l * 10**-3 # in m
rho_i = (R*a)/l # in ohm-m
sigma_i = 1/rho_i # in S/m
e = 1.6*10**-19 # in C
miu_h = 480 # in cm**2/V-s
miu_h = miu_h * 10**-4 # in m**2/V-s
# sigma_i = p*e*miu_h 
p = sigma_i/(e*miu_h) # in /m**3
print "The concentration of acceptor atom = %0.2e /m**3 " %p

The concentration of acceptor atom = 6.51e+22 /m**3

Example : 1.8 - Page No : 26

In [9]:
 #Given data
rho = 0.5 # in ohm-m
J = 100 # in A/m**2
miu_e = 0.4 # in m**2/V-s
e = 1.6*10**-19 # in C
sigma = 1/rho 
E = J/sigma 
v = miu_e*E # in m/s
print "The drift velocity = %0.f m/s " %v
D = 10 # distance of travel in µm
D = D * 10**-6 # in m
# Time taken by electron
t= D/v # time taken in second 
print "The time taken = %0.1e second " %t

The drift velocity = 20 m/s 
The time taken = 5.0e-07 second

Example : 1.9 - Page No : 26

In [10]:
 #Given data
rho = 0.039 # in ohm-cm
sigma_n = 1/rho # in mho/cm
miu_e = 3600 # in cm**2/V-s
e = 1.602*10**-19 # in C
# sigma_n = n*e*miu_e = N_D*e*miu_e 
N_D = sigma_n/(e*miu_e) # in /cm**3
n = N_D # in /cm**3
print "The electrons density = %0.2e per cm**3 " %n
n_i = 2.5*10**13 # in /cm**3
p = (n_i**2)/n # in /cm**3
print "The hole density = %0.1e per cm**3 " %p

The electrons density = 4.45e+16 per cm**3 
The hole density = 1.4e+10 per cm**3

Example : 1.10 - Page No : 26

In [11]:
 #Given data
rho_i = 0.47 # in ohm-m
sigma_i = 1/rho_i # in S/m
miu_e = 0.39 # in m**2/V-s
miu_h = 0.19 # in m**2/V-s
e = 1.6*10**-19 # in C
#sigma_i = n_i*e*(miu_e+miu_h) 
n_i = sigma_i/( e*(miu_e+miu_h) ) # in /m**3
print "The density of electrons = %0.3e per m**3 " %n_i
E = 10**4 
v_n = miu_e*E # in m/s
print "The drift velocity for electrons = %0.f m/s " %v_n
v_h = miu_h*E # in m/s
print "The drift velocity for holes = %0.f m/s " %v_h

The density of electrons = 2.293e+19 per m**3 
The drift velocity for electrons = 3900 m/s 
The drift velocity for holes = 1900 m/s

Example : 1.11 - Page No : 27

In [12]:
 #Given data
rho = 3000 # in ohm-m
n = 1.1*10**6 # in /m**3
e = 1.6*10**-19 # in C
#miu_e = 3*miu_h   (i)
# miu_e+miu_h = 1/(rho*e*n)    (ii)
# From eq (i) and (ii)
miu_h = (1/(rho*e*n))/4 # in m**2/V-s
print "The holes mobility = %0.3e m**2/V-s " %miu_h
miu_e = 3*miu_h # in m**2/V-s
print "The electron mobility = %0.2e m**2/V-s " %miu_e

# Note: The calculated value of hole mobility is wrong .

The holes mobility = 4.735e+08 m**2/V-s 
The electron mobility = 1.42e+09 m**2/V-s

Example : 1.12 - Page No : 27

In [13]:
 #Given data
n_i = 2.5*10**13 # in /cm**3
miu_e = 3800 #in cm**2/V-s
miu_h = 1800 # in m**2/V-s
e = 1.6*10**-19 # in C
sigma_i = n_i*e*(miu_e+miu_h) # in (ohm-cm)**-1
print "The intrinsic conductivity = %0.4f (ohm-cm)**-1 " %sigma_i
n = 4.4*10**22 
impurity = 10**-7 
N_D = n*impurity # in /cm**3
n = N_D # in /cm**3
p = (n_i**2)/N_D # in holes/cm**3
sigma_n = e*N_D*miu_e # in (ohm-cm)**-1
print "The conductivity in N-type Ge semiconductor = %0.2f (ohm-cm)**-1 " %sigma_n

The intrinsic conductivity = 0.0224 (ohm-cm)**-1 
The conductivity in N-type Ge semiconductor = 2.68 (ohm-cm)**-1

Example : 1.13 - Page No : 27

In [14]:
 #Given data
e = 1.6*10**-19 # in C
miu_e = 0.38 # in m**2/V-s
miu_h = 0.18 # in m**2/V-s
V = 10 # in V
l = 25 # in mm
l = l * 10**-3 # in m
w = 4 # in mm
w = w * 10**-3 # in m
t= 1.5*10**-3 # in m
E = V/l # in V/m
v_e = miu_e*E # in m/s
print "The electron drift velocity = %0.f m/s " %v_e
v_h = miu_h*E # in m/s
print "The hole drift velocity = %0.f m/s " %v_h
n_i = 2.5*10**19 # in /m**2
sigma_i = n_i*e*(miu_e+miu_h) # in (ohm-cm)**-1
print "The interinsic conductivity of Ge = %0.2f (ohm-cm)**-1 " %sigma_i
A = w*t # in m**2
I = sigma_i*E*A # in A
I = I * 10**3 # in mA
print "The total current = %0.3f mA " %I

The electron drift velocity = 152 m/s 
The hole drift velocity = 72 m/s 
The interinsic conductivity of Ge = 2.24 (ohm-cm)**-1 
The total current = 5.376 mA

Example : 1.14 - Page No : 28

In [15]:
 #Given data
I_electrons = 3/4 
I_holes= 1/4 
v_h = 1 
v_e = 3 
ratio = (I_electrons/I_holes)*(v_h/v_e) 
print "Ratio of electrons to holes = %0.f" %ratio 

Ratio of electrons to holes = 1

Example : 1.15 - Page No : 28

In [16]:
 #Given data
miu_e = 0.17 # in m**2/V-s
miu_h = 0.025 # in m**2/V-s
e = 1.602*10**-19 # in C
T = 27 # in degree C
T = T + 273 # in K
kdas = 1.38*10**-23 # in J/K
De = miu_e*( (kdas*T)/e ) # in m**-2/s
De = De * 10**4 # in cm**2/s
print "The diffusion coefficients of electrons = %0.2f cm**2/s" %De
Dh = miu_h*( (kdas*T)/e ) # in m**2/s
Dh = Dh * 10**4 # in cm**2/s
print "The diffusion coefficients of holes = %0.2f cm**2/s" %Dh

The diffusion coefficients of electrons = 43.93 cm**2/s
The diffusion coefficients of holes = 6.46 cm**2/s

Example : 1.16 - Page No : 28

In [17]:
 #Given data
N = 3*10**25 # in /m**3
e = 1.602*10**-19 # in C
E_G = 1.1 # in eV
E_G = E_G*e # in J
kdas = 1.38*10**-23 # in J/K
T = 300 # in K
miu_e = 0.14 # in m**2/V-s
miu_h = 0.05 # in m**2/V-s
n_i = N*(exp((-E_G)/(2*kdas*T))) # in /m**3
print "The interinsic carrier concentration = %0.3e /m**3 " %n_i
sigma = n_i*e*(miu_e+miu_h) # in S/m
print "The conductivity of silicon = %0.3e S/m " %sigma

The interinsic carrier concentration = 1.715e+16 /m**3 
The conductivity of silicon = 5.219e-04 S/m

Example : 1.17 - Page No : 28

In [18]:
 #Given data
Je = 360 # in A/cm**2
T = 300 # in K
d = 1.5 # in mm
d = d * 10**-1 # in cm
e = 1.6*10**-19 # in C
delta = 2*10**18-5*10**17 # assumed
dnBYdx = delta/d 
De = Je/(e*dnBYdx) # in cm**2/s
V_T = T/11600 
miu_e = De/V_T # in cm**2/V-s
print "The mobility of electrons = %0.f cm**2/V-s " %miu_e

The mobility of electrons = 8700 cm**2/V-s

Example : 1.18 - Page No : 30

In [19]:
 #Given data
E_CminusE_F = 0.24 # in eV
T = 300 # in K
T1 = 350 # in K
# E_CminusE_F = K*T*log(n_c/N_D) (i)
# E_CminusE_F1 =K*T1*log(n_C/N_D) (ii)
# From eq(i) and (ii)
E_CminusE_F1 = E_CminusE_F*(T1/T) # in eV
print "The new position of the Fermi level lies ",round(E_CminusE_F1,2)," eV below the conduction band"

The new position of the Fermi level lies  0.28  eV below the conduction band

Example : 1.19 - Page No : 30

In [20]:
from math import log 
#Given data
E_FminusE_V = 0.39 # in eV
kT = 0.026 # in ev
#N_A1 = n_V * (%e**(-E_FminusE_V)/kT)       (i)
# N_A2=3*N_A1=n_V * (%e**(-E_F2minusE_V)/kT)    (ii)
#From eq(i) and (ii)
E_F2minusE_V = kT*(15-log(3)) # in eV
print "The new position of fermi level = %0.2f eV " %E_F2minusE_V

The new position of fermi level = 0.36 eV