# Chapter 9: Semiconductors¶

## Example 9.1, Page number 9.11¶

In :
#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 :
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 :
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 :
#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 :
#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 :
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 :
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 :
#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 :
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 :
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 :
#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 :
#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 :
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 :
#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 :
#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 :
#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 :
#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 :
#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 :
#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