Chapter 9: Semiconductors

Example 9.1, Page number 9.11

In [1]:
#Variable declaration
ni = 2.37*10**19     #intrinsic carrier density(m^-3)
ue = 0.38            #electron mobility(m^2/V-s)
uh = 0.18            #hole mobility(m^2/V-s)
e = 1.6*10**-19      #charge on electron(C)

#Calculations
sigma_i = ni*e*(ue+uh) #(1/ohm-m)
p = 1/sigma_i

#Result
print "Resistivity =",round(p,3),"m"

Resistivity = 0.471 m

Example 9.2, Page number 9.12

In [2]:
import math

#Variable declaration
Eg = 1.12           #bandgap(eV)
k = 1.38*10**-23    #Boltzman constant(J/K)
T = 300             #Temperature(K)
mh = 0.28           #Effective Mass of the hole(kg)
me = 0.12           #Effective Mass of the hole(kg)
e = 1.6*10**-19     #charge on electron(C)

#Calculation
Ef = (Eg/2)+3/4*k*T*(math.log(mh/me))/e

#Result 
print "The position of the Fermi level is at",round(Ef,2),"from the top of valence band" 

The position of the Fermi level is at 0.56 from the top of valence band

Example 9.3, Page number 9.12

In [3]:
from math import pi, exp

#Variable declaration
m = 9.109*10**-31   #mass of an electron(kg)
k = 1.38*10**-23    #Boltzman constant(J/K)
T = 300             #Temperature(K)
h = 6.626*10**-34   #Planck's constant
Eg = 0.7            #bandgap(eV)
e = 1.6*10**-19     #charge on electron(C)

#Calculation
C = (((2*pi*m*k)/h**2)**(3./2.))  
T1 = T**(3./2.)
E = exp((-Eg*e)/(2*k*T))
ni = 2*C*T1*E

#Result
print "Concentration of intrinsic charge carriers =",round((ni/1E+18),2),"*10**18/m^3"

Concentration of intrinsic charge carriers = 33.48 *10**18/m^3

Example 9.4, Page number

In [4]:
#Variable declaration
ni = 2.4*10**19      #intrinsic carrier density(m^-3)
ue = 0.39            #electron mobility(m^2/V-s)
uh = 0.19            #hole mobility(m^2/V-s)
e = 1.6*10**-19      #charge on electron(C)

#Calculations
sigma_i = ni*e*(ue+uh) #(1/ohm-m)
p = 1/sigma_i

#Result
print "Resistivity =",round(p,3),"m"

Resistivity = 0.449 m

Example 9.5, Page number 9.13

In [6]:
#Variable declaration
ni = 2.5*10**19      #intrinsic carrier density(m^-3)
ue = 0.39            #electron mobility(m^2/V-s)
uh = 0.19            #hole mobility(m^2/V-s)
e = 1.6*10**-19      #charge on electron(C)
l = 1*10**-2         #length of rod(m)
A = 10**-3*10**-3    #area(m^2)

#Calculations
sigma = ni*e*(ue+uh) #(1/ohm-m)
R = 1/(sigma*A)

#Result
print "Resistivity =",round(R,3),"Ohms"

Resistivity = 431034.483 Ohms

Example 9.6, Page number 9.14

In [7]:
from math import pi, exp

#Variable declaration
ue = 0.48            #electron mobility(m^2/V-s)
uh = 0.013           #hole mobility(m^2/V-s)
Eg = 1.1             #bandgap(eV)
T = 300              #assumption - Temperature(K)
h = 6.626*10**-34    #Planck's constant
e = 1.6*10**-19      #charge on electron(C)
k = 1.38*10**-23     #Boltzman constant(J/K)
m = 9.1*10**-31      #mass of an electron(kg)

#Calculation
C = 2*(((2*pi*m*k)/h**2))**(3./2.)
ni = C*T**(3./2.)*exp((-Eg*e)/(2*k*T))
sigma_i =  ni*e*(ue+uh)

#Result
print "Conductivity=",round((sigma_i/1E-3),3),"*10^-3/ohm-m"

Conductivity= 1.159 *10^-3/ohm-m

Example 9.7, Page number 9.15

In [8]:
from math import pi, exp

#Variable declaration
ue = 0.4             #electron mobility(m^2/V-s)
uh = 0.2             #hole mobility(m^2/V-s)
Eg = 0.7             #bandgap(eV)
T = 300              #assumption - Temperature(K)
h = 6.626*10**-34    #Planck's constant
e = 1.6*10**-19      #charge on electron(C)
k = 1.38*10**-23     #Boltzman constant(J/K)
m = 9.1*10**-31      #mass of an electron(kg)

#Calculation
C = 2*(((2*pi*m*k)/h**2))**(3./2.)
ni = C*T**(3./2.)*exp((-Eg*e)/(2*k*T))
sigma_i =  ni*e*(ue+uh)

#Result
print "Intrinsic carrier density =",round((ni/1E+19),2),"*10^19 per m^3"
print "Conductivity=",round(sigma_i,2),"/ohm-m"

Intrinsic carrier density = 3.34 *10^19 per m^3
Conductivity= 3.21 /ohm-m

Example 9.8, Page number 9.15

In [9]:
#Variable declaration
ue = 0.36            #electron mobility(m^2/V-s)
uh = 0.17            #hole mobility(m^2/V-s)
P = 2.12             #resistivity(ohm-m)
e = 1.6*10**-19      #charge on electron(C)
k = 1.38*10**-23     #Boltzman constant(J/K)
m = 9.1*10**-31      #mass of an electron(kg)
h = 6.626*10**-34    #Planck's constant
T = 300              #assumption - Temperature(K)

#Calculations
sigma = 1/P
ni = sigma/(e*(ue+uh))
C = 2*(((2*pi*m*k)/h**2))**(3./2.)
Eg = ((2*k*T)/e)*math.log(C*(T**(3./2.))/ni)

#Result
print "Forbidden energy gap =",round(Eg,3),"eV"

Forbidden energy gap = 0.793 eV

Example 9.9, Page number 9.16

In [10]:
from math import log10

#Variable declaration
p1 = 2        #resistivity(ohm-m)
p2 = 4.5      #resistivity(ohm-m)
T1 = 20.+273  #Temperature(K)
T2 = 32.+273  #temperature(K)
k = 1.38*10**-23     #Boltzman constant(J/K)

#Calculations
dy = log10(p2)-log10(p1)
dx = (1/T1)-(1/T2)
dy_by_dx = dy/dx
Eg = (2*k*dy_by_dx)/e

#Result
print "Energy band gap =",round(Eg,3),"eV"

Energy band gap = 0.452 eV

Example 9.10, Page number 9.16

In [11]:
from math import log

#Variable declaration
e = 1.602*10**-19    #charge on electron(C)
k = 1.38*10**-23     #Boltzman constant(J/K)
Eg = 1*e             #bandgap(J)

#Calculations
'''At T = 0K
(Ev+0.5)=(Ec+Ev)/2  -----(1)

Let at temperature T, fermi level be shited by 10%
(Ev+06) = (Ec+Ev)/2 +(3kT*ln(4))/4  ----(2)

Subtracting (1) from (2), we get the following expression'''

T = (4*e/10)/(3*k*log(4))

#Result
print "Temperature =",round(T,2),"K"

Temperature = 1116.52 K

Example 9.11, Page number 9.17

In [12]:
#Variable declaration
Na = 5*10**23    #no. of atoms of boron
Nd = 3*10**23    #no. of atoms of arsenic
ni = 2*10**16    #intrinsic charge carriers(/m^3)

#Calculations
p = (2*(Na-Nd))/2  #hole concentration(/m^3)
n = ni**2/p        #electron concentration(/m^3)

#Result
print "Hole concentration =",round((p/1E+23),2),"*10^23 per m^3"
print "Electron concentration =",round((n/1E+9),2),"*10^9 per m^3"

Hole concentration = 2.0 *10^23 per m^3
Electron concentration = 2.0 *10^9 per m^3

Example 9.12, Page number 9.18

In [13]:
#Variable declaration
ue = 0.13            #electron mobility(m^2/V-s)
uh = 0.05            #hole mobility(m^2/V-s)
e = 1.602*10**-19    #charge on electron(C)
ni = 1.5*10**16      #intrinsic charge carriers(/m^3) 


#Calculations
#Part a
sigma = ni*e*(ue+uh)  #conductivity(1/ohm-m)

#Part b
w = 28.1                  #atomic weight of Si
den = 2.33*10**3          #density of Si(kg/m^3)
n = (den*6.02*10**26)/w   #no. of atoms of silicon
#Since one donor type impurity atom is added in 10^8 Si atoms, 
Nd = n/10**8
p = ni**2/Nd
sigma_ex = Nd*e*ue        #(per ohm-m)

#Part c
Na = Nd         #Since one acceptor type impurity atom is added in 10^8 Si atoms
n2 = ni**2/Na
sigma_ax = Na*e*uh         #(per ohm-m)

#Results
print "a)Conductivity =",round((sigma/1E-3),3),"*10^-3 per ohm-m"
print "b)Conductivity if donor type impurity is added =",round(sigma_ex,2),"per ohm-m"
print "c)Conductivity if acceptor type impurity is added =",round(sigma_ax,2),"per ohm-m"

a)Conductivity = 0.433 *10^-3 per ohm-m
b)Conductivity if donor type impurity is added = 10.4 per ohm-m
c)Conductivity if acceptor type impurity is added = 4.0 per ohm-m

Example 9.13, Page number 9.20

In [14]:
from math import log

#Variable declaration
ue = 0.135            #electron mobility(m^2/V-s)
uh = 0.048            #hole mobility(m^2/V-s)
e = 1.602*10**-19     #charge on electron(C)
ni = 1.5*10**16       #intrinsic charge carriers(atoms/m^3)
k = 1.38*10**-23      #Boltzman constant(J/K)
T = 300               #assumption - Temperature(K)
Nd = 10**23           #doping concentration(atoms/m^3)

#Calculations
sigma = ni*e*(ue+uh)  #conductivity of intrinsic Si

p = ni**2/Nd          #hole concentration

sigma_ex = Nd*e*ue    #conductivity at equilibrium
F = (3*k*T)/(4*e)*log(ue/uh)  #position of Fermi level

#Results
print "Conductivity of intrinsic Si is",round((sigma/1E-3),4),"*10^-3 per ohm-m"
print "Hole concentration at equilibrium is",round((Nd/1E+23)),"*10^23 per m^3"
print "Conductivity at equilibrium is",round((sigma_ex/1E+3),2),"*10^3 per m^3"
print "Fermi level will be",round(F,2),"eV above intrinsic level" 

Conductivity of intrinsic Si is 0.4397 *10^-3 per ohm-m
Hole concentration at equilibrium is 1.0 *10^23 per m^3
Conductivity at equilibrium is 2.16 *10^3 per m^3
Fermi level will be 0.02 eV above intrinsic level

Example 9.14, Page number 9.35

In [15]:
#Variable declaration
ue = 0.19             #electron mobility(m^2/V-s)
e = 1.602*10**-19     #charge on electron(C)
T = 300               #Temperature(K)

#Calculation
Dn = (ue*k*T)/e

#Result
print "Diffusion co-efficient =",round((Dn/1E-4),2),"*10^-4 m^2/s"

Diffusion co-efficient = 49.1 *10^-4 m^2/s

Example 9.15, Page number 9.45

In [6]:
#Variable declaration
Rh = 3.66*10**-4   #Hall coefficient
I = 10**-2         #current(A)
B = 0.5            #magnetic field intensity(wb/m^2)
t = 1.*10**-3      #thickness of plate(m)

#Calculations
Vh = (Rh*I*B)/t

#Result
print "Hall coefficient =",(Vh/1E-3),"mV"

Hall coefficient = 1.83 mV

Example 9.16, Page number 9.46

In [7]:
#Variable declaration
Vy = 37*10**-6      #voltage(V)
t = 10**-3          #thickness of crystal(m)
Bz = 0.5            #magnetic field intensity(Wb/m^2)
Ix = 20*10**-3      #current(A)

#Calculations
Vh = (Vy*t)/(Ix*Bz)

#Result
print "Hall coefficient =",(Vh/1E-6),"*10^-6 m^3/C"

Hall coefficient = 3.7 *10^-6 m^3/C

Example 9.17, Page number 9.46

In [8]:
#Variable declaration
Rh = 7.35*10**-5   #Hall coefficient(m^3/C)
e = 1.6*10**-19    #charge on electron(C)
sigma = 200        #conductivity(/ohm-m)
n = 8.023*10**22   #Avogadro's number

#Calculations
n = 1/(Rh*e)

u = sigma/(n*e)

#Results
print "Density =",round((n/1E+22),3),"*10^22 m^3"
print "Conductivity =",round((u/1E-3),2),"*10^-3 m^2/V-s"

Density = 8.503 *10^22 m^3
Conductivity = 14.7 *10^-3 m^2/V-s

Example 9.18, Page number 9.47

In [9]:
#Variable declaration
I = 50           #current(A)
B = 1.5          #magnetic field intensity(T)
n = 8.4*10**28   #free electron concentration in copper(electron/m^3)
t = 0.5*10**-2   #thickness of slab(m)

#Calculation
Vh = (I*B)/(n*e*t)

#Result
print "The magnitude of Hall voltage is",round((Vh/1E-6),3),"*10^-6 V"

The magnitude of Hall voltage is 1.116 *10^-6 V

Example 9.19, Page number 9.48

In [10]:
#Variable declaration
Rh = 3.66*10**-4   #Hall coefficient(m^3/C)
e = 1.6*10**-19    #charge on electron(C)
Pn = 8.93*10**-3   #resistivity(ohm-m)

#Calculation
n = 1/(Rh*e)

ue = Rh/Pn

#Result
print "n =",round((n/1E+22),3),"*10^22/m^3"
print "u =",round(ue,3),"m^2/V-s"

n = 1.708 *10^22/m^3
u = 0.041 m^2/V-s