Chapter 10: Transport Properties of Semiconductors

Example 10.1, page no-267

In [3]:
# intrinsic properties
import math

#Variable declaration
T=300                          # Temperature
mue=0.4                        # Electron mobility 
muh=0.2                        # Hole mobility 
e=1.6*10**-19                  # electronic charge
eg=0.7*e                       # Band gap
m=9.1*10**-31                  # Mass of electron
me=0.55                        # electron effective mass 
mh=0.37                        # hole effective 
h=6.626*10**-34                # Planck's constant
k=1.38*10**-23                 # Boltzmann's constant

#Calculations
ni=2*(2*math.pi*k*T/(h**2))**(1.5)
ni=ni*(m**1.5)*(mh*me)**(3.0/4.0)
ni=ni*math.e**(-eg/(k*T))
sig=ni*e*(mue+muh)
rho=1/sig

# Result
print("\nThe intrinsic concentration ni=%.3f *10^13 /m^3"%(ni*10**-13))
print("\nIntrinsic Conductivity,Sigma =%.3f *10^-6 per m^3\n\nIntrinsic Resistivity, rho = %.2f*10^6 Ohm-m"%(sig*10**6,rho*10**-6))

The intrinsic concentration ni=1.352 *10^13 /m^3

Intrinsic Conductivity,Sigma =1.298 *10^-6 per m^3

Intrinsic Resistivity, rho = 0.77*10^6 Ohm-m

Example 10.2, page no-268

In [4]:
# Fermi energy

import math
# variable declaration
ni=1.45*10**10                  # intrinsic concentration
nd=10**16                       # donor concentration
k=1.38*10**-23                  # Boltzmann's constant
T=300                           # Temperature
e=1.6*10**-19                   # electronic charge

#Calculations
Ef=k*T*math.log(nd/ni)
Ef=Ef/e

#Result
print("The Fermi energy with respect to Ef in intrinsic Si = %.3f eV"%Ef)

The Fermi energy with respect to Ef in intrinsic Si = 0.348 eV

Example 10.3, page no-269

In [5]:
# conductivity of intrinsic

import math
#Variable declarations
ni=2.5*10**19                    # intrinsic concentration
mue=0.39                         # electron mobility 
muh=0.19                         # hole mobility 
l=10**-2                         # length of rod
e=1.6*10**-19                    # charge of an electron

# Calculations
sig=ni*e*(mue+muh)
R=l/(sig*10**-6)

#Result
print("The conductivity of intrinsic Ge is %.2f /ohm-m\nThe Resistance is %.0f"%(sig,R))

The conductivity of intrinsic Ge is 2.32 /ohm-m
The Resistance is 4310

Example 10.4, page no-269

In [6]:
# conductivity of intrinsic Ge

import math
#Variable declaration
ni=1.5*10**16                      # intrinsic concentration
mue=0.13                           # electron mobility 
muh=0.05                           # hole mobility 
e=1.6*10**-19                      # electronic charge

#Calculations
sig=ni*e*(mue+muh)

#Result
print("The conductivity of intrinsic Ge is %.2f *10^-4 /ohm-m"%(sig*10**4))

The conductivity of intrinsic Ge is 4.32 *10^-4 /ohm-m

Example 10.5, page no-270

In [8]:
# intrinsic conductivity and resistivity

import math

#variable declaration
ni=2.15*10**13                      # intrinsic concentration
mue=3900                            # electron mobility 
muh=1900                            # hole concentration
e=1.6*10**-19                       # electronic charge

#calculation
sig=ni*e*(mue+muh)
r=1/sig

# Result
print("The conductivity of intrinsic Ge is %.2f *10^-2 /ohm-cm\nThe intrinsic resistivity is %.0f Ohm-cm"%(sig*10**2,r))
#answers in the book is wrong

The conductivity of intrinsic Ge is 2.00 *10^-2 /ohm-cm
The intrinsic resistivity is 50 Ohm-cm

Example 10.6, page no-270

In [10]:
# intrinsic conductivity and resistivity

import math
# variable declaration
ni=2.1*10**19                     # intrinsic concentration
mue=0.4                           # electron mobility 
muh=0.2                           # hole mobility
e=1.6*10**-19                     # electronic charge
p=4.5*10**23                      # boron density 

# Calculation
sig=ni*e*(mue+muh)
r=p*e*muh

#Result
print("The conductivity of intrinsic Ge is %.3f *10^-2 /ohm-cm\nThe intrinsic resistivity is %.2f *10^4  per ohm-m"%(sig,r*10**-4))

The conductivity of intrinsic Ge is 2.016 *10^-2 /ohm-cm
The intrinsic resistivity is 1.44 *10^4  per ohm-m

Example 10.7, page no-271

In [13]:
# intrinsic conductivity and resistivity

import math
# variable declaration
n=5*10**28                       # Atomic concentration
ni=1.45*10**13                   # intrinsic concentration
mue=1.35                         # electron mobility
muh=0.45                         # hole mobility
e=1.6*10**-19                    # electronic charge
p=4.5*10**23                     # boron density

# calculation
sig=ni*e*(mue+muh)
rho=1/sig
r=rho*10**12
nd=n/10**9
p=(ni**2)/nd
sig2=nd*e*mue

#Result
print("\nThe intrinsic conductivity is %.2f *10^-6 /ohm-cm\n\nThe intrinsic resistivity is %.2f *10^-5 Ohm-m\n\nResistance = %.2f*10^7 Ohm\n\nDonar concentration is %.0f*10^19\n\nConcentration of hole is %.1f*10^6 m^-3\n\nConductivity = %.1f per ohm-m"%(sig*10**6,rho*10**-5,r*10**-17,nd*10**-19,p*10**-6,sig2))

The intrinsic conductivity is 4.18 *10^-6 /ohm-cm

The intrinsic resistivity is 2.39 *10^-5 Ohm-m

Resistance = 2.39*10^7 Ohm

Donar concentration is 5*10^19

Concentration of hole is 4.2*10^6 m^-3

Conductivity = 10.8 per ohm-m

Example 10.8, page no-272

In [4]:
# Conductivity, Intrinsic carrier concentration and band gap of Ge

import math
# Variable declaration
T=300                         # Temperature
rho=2.12                      # Resistivity
mue=0.36                      # Electron mobility
muh=0.17                      # Hole mobility
e=1.6*10**-19                 # electronic charge
m=9.1*10**-31                 # mass of electron
h=6.626*10**-34               # Planck's constant
k=1.38*10**-23                # Boltzmann's constant

# Calculations
sig=1/rho
ni=sig/(e*(muh+mue))
Nc=2*(2*math.pi*k*T/h**(2))**(1.5)
Nc=Nc*(0.5*m)**(1.5)
Nv=2*(2*math.pi*k*T/h**(2))**(1.5)
Nv=Nv*(0.37*m)**(1.5)
eg=2*k*T*math.log(math.sqrt(Nc*Nv)/ni)
eg=eg/e

# Result
print("\nConductivity = %.6f per Ohm-m\nIntrinsic carrier concentration, ni=%.5f*10^18"%(sig,ni*10**-18))
print("\nNc=%.3f*10^24\nNv=%.3f*10^24"%(Nc*10**-24,Nv*10**-24))
print("\nThe band gap of Ge is %.3f eV"%eg)

Conductivity = 0.471698 per Ohm-m
Intrinsic carrier concentration, ni=5.56248*10^18

Nc=8.852*10^24
Nv=5.635*10^24

The band gap of Ge is 0.727 eV

Example 10.9, page no-273

In [9]:
# carrier concentration 

import math
# Variable declaration
e=1.6*10**-19                   # Electronic charge
m=9.1*10**-31                   # Mass of electron
h=6.62*10**-34                 # Planck's constant
k=1.38*10**-23                  # Boltzmann's constant
eg=0.7*e                        # Band gap energy
T=300                           # Temperature

#Calculations
ni=2*(2*3.14*m*k*T/(h**(2)))**(1.5)                      # math.pi= 3.14
ni=ni*math.e**(-eg/(2*k*T))

#Result
print("The carrier concentration of an intrinsic semiconductor is = %.2f*10^18 per m^3"%(ni*10**-18))

The carrier concentration of an intrinsic semiconductor is = 33.49*10^18 per m^3

Example 10.10, page no-273

In [12]:
# Carrier concentration

import math
# Variable declaration
e=1.6*10**-19                         # Electronic charge
m=9.1*10**-31                         # mass of electron
h=6.626*10**-34                       # planck's constant
k=1.38*10**-23                        # Boltzmann's constant
eg=1.1*e                              # Energy gap
mue=0.48                              # Mobility of electron
muh=0.013                             # Mobility of hole
T=300                                 # temperature

#Calculations
ni=2*(2*math.pi*m*k*T/(h**(2)))**(1.5)
ni=ni*math.e**(-eg/(2*k*T))
sig=ni*e*(mue+muh)

#Result
print("\nThe carrier concentration of an intrinsic semiconductor is = %.2f*10^16 per m^3\nThe electrical conductiivity of Si is %.2f*10^-3 per Ohm-m"%(ni*10**-16,sig*10**3))

The carrier concentration of an intrinsic semiconductor is = 1.47*10^16 per m^3
The electrical conductiivity of Si is 1.16*10^-3 per Ohm-m

Example 10.11, page no-275

In [14]:
# Fermi energy of Si

import math
# Variable declaration
e=1.6*10**-19               # ELectronic charge
eg=1.12                     # Band gap
me=0.12                     # Effective mass of electron
mh=0.28                     # Effective mass of hole
T=300                       # Temperature 
k=1.38*10**-23              # Boltzmann's constant

# Calculations
ef=(eg/2)+(3*k*T/(4*e))*math.log(mh/me)

# Result
print("The Fermi energy of Si at 300 K is %.3f eV"%ef)

The Fermi energy of Si at 300 K is 0.576 eV

Example 10.12, page no-275

In [16]:
# Fermi level shift

import math
# Variable declaration
e=1.6*10**-19             # Electronic charge
eg=1*e                     # Energy gap
k=1.38*10**-23             # Boltzmann's constant
m=4.0                      # hole to elctron mass ratio

# calculations
T=0.1*e*4/(3*k*math.log(m))

# Result
print("Temperature at which Fermi level is shifted 10%% is %.f K"%T)
# Answer in the book is wrong

Temperature at which Fermi level is shifted 10% is 1115 K

Example 10.13, page no-276

In [19]:
# conductivity of Ge

import math
# variable declaration
e=1.6*10**-19               # electronic charge
ni=2.4*10**19               # intrinsic concentration
mue=0.39                    # Electron mobility 
muh=0.19                    # hole mobility

# caclualtions
sig=ni*e*(mue+muh)

#Result
print("The conductivity of Ge at 300 K is %.2f per Ohm-m"%(math.floor(sig*100)/100))

The conductivity of Ge at 300 K is 2.22 per Ohm-m

Example 10.14, page no-277

In [20]:
# Fermi energy level position

import math
# variable declaration
e=1.6*10**-19               # electronic charge
T1=300                      # Lower Temperature 
T2=330                      # Higher Temperature
eg=0.3                      # Fermi level posiion at lower temperature

# Calculations
eg2=eg*T2/T1

#Result
print("E_c-E_f330=%.2f eV\n\nAt 330 K, the Fermi energy level lies %.2f eV, bellow the conduction band."%(eg2,eg2))

E_c-E_f330=0.33 eV

At 330 K, the Fermi energy level lies 0.33 eV, bellow the conduction band.

Example 10.15, page no-277

In [27]:
# conductivity of Ge

import math
# Variable declaration
e=1.6*10**-19                       # Charge of electron
eg=0.72*e                           # Energy gap
t1=293.0                            # lower temperature
t2=313.0                            # higher temperature
k=1.38*10**-23                      # Boltzmann's constant

# calculations
sig1=2
n=((t2/t1)**(3.0/2.0))*math.e**((eg/(2*k))*((1/t1)-(1/t2)))
sig2=sig1*n

#Result
print("The conductivity of Ge at 40°C is %.3f per Ohm-m"%sig2)

The conductivity of Ge at 40°C is 5.487 per Ohm-m

Example 10.16, page no-278

In [28]:
# intrinsic concentration of Si

import math
# Variable declaration
e=1.6*10**-19              # electronic charge
m=9.1*10**-31              # mass of electron 
mm=0.31*m                  # effective mass of electron
h=6.626*10**-34            # Planck's constant
k=1.38*10**-23             # Boltzmann's constant  
eg=1.1*e                   # Energy gap
T=300                      # Temperature

# Calculations
ni=2*(2*math.pi*mm*k*T/(h**(2)))**(1.5)
ni=ni*math.e**(-eg/(2*k*T))

#Result
print("The intrinsic concentration of Si at %d K is %.4f * 10^15 electrons per m^3"%(T,ni*10**-15))

The intrinsic concentration of Si at 300 K is 2.5367 * 10^15 electrons per m^3

Example 10.17, page no-279

In [29]:
# drift mobility
import math
# Variable declaration
hc=0.55*10**-10               # Hall coefficient of Cu (modulus)
cc=5.9*10**7                  # Conductivity of Cu 
T=300                         # Temperature

#Calculations
dm=hc*cc

#Result
print("The drift mobility is given by mu_d = %.1f * 10^-3 m^2/V-s"%(dm*10**3))

The drift mobility is given by mu_d = 3.2 * 10^-3 m^2/V-s

Example 10.18, page no-279

In [33]:
# concentration and averrage o of electron contributed per atom

import math
#Variable declaration
sig=5.9*10**7                     # Resistivity
e=1.6*10**-19                     # electronic charge
mu=3.2*10**-3                     # electron drift mobility 
d=8900                            # Density
avg=6.022*10**23                  # Avogadro's number
awt=63.5                          # Atomic weight

#calculations
ni=sig/(e*mu)                     
n=avg*d*1000/awt
k=ni/n

#Result
print("Concentration of free electron in pure Cu is %.2f*10^28\nThe average number of electrons contributed per Cu atom is %.2f i.e. %.0f"%(n*10**-28,math.floor(k*100)/100,k))

Concentration of free electron in pure Cu is 8.44*10^28
The average number of electrons contributed per Cu atom is 1.36 i.e. 1

Example 10.19, page no-280

In [34]:
# mobility of the Ge

import math
# Variable declaration
i=5*10**-3                     # current through the specimen
v=1.35                         # voltage across the specimen   
l=0.01                         # length of the sample
b=5*10**-3                     # Breadth of the sample     
t=10**-3                       # Thickness of the sample
a=5*10**-6                     # Area of the sample
vy=20*10**-3                   # Hall voltage
H=0.45                         # Magnetic field

# Calculations
rho=v*a/(l*i)
Ey=vy/t
j=i/a
k=Ey/(H*j)
Rh=3*math.pi*k/8
mu=Rh/rho

#Result
print("The mobility of the Ge sample is %.2f m^2/V-s"%mu)

The mobility of the Ge sample is 0.39 m^2/V-s

Example 10.20, page no-282

In [35]:
#Hall potential difference

import math
#Variable declaration
I=200                            # current flowing
H=1.5                            # Applied magnetic field
n=8.4*10**28                     # no of electrons per unit volume
d=1.0*10**-3                     # thickness of the strip
e=1.6*10**-19                    # electronic charge

# calculations
v=I*H/(n*d*e)

# Result
print("The Hall potential difference appearance between the ship is %.0f µv"%(v*10**6))

The Hall potential difference appearance between the ship is 22 µv

Example 10.21, page no-283

In [36]:
#carrier concentration and mobility of Si

import math
#Variable declaration
rh=3.66*10**-4              # Hall coefficient of specimen
rho=8.93*10**-3             # resistivity of thespecimen
e=1.6*10**-19               # electronic charge

#calculations
ni=1/(rh*e)
muh=rh/rho

#Result
print("the carrier concentration of Si doped specimen is %.3f *10^22 m^-3"%(ni*10**-22))
print("\n The mobility of Si doped specimen is %.5f m^2/V-s"%muh)

the carrier concentration of Si doped specimen is 1.708 *10^22 m^-3

 The mobility of Si doped specimen is 0.04099 m^2/V-s

Example 10.22, page no-283

In [37]:
# #carrier concentration and  electron mobility

import math
#Variable declaration
Rh=3.66*10**-11                     # Hall coefficient
sig=112*10**7                       # Conductivity
e=1.6*10**-19                       # electronic charge

# Calculations
n=3*math.pi/(8*Rh*e)
mu=sig/(n*e)

# Result
print("\nThe concentration of electrons is %.0f*10^29 m^-3\nthe electron mobility at room temperature = %.3f m^2/V-s"%(n*10**-29,mu))

The concentration of electrons is 2*10^29 m^-3
the electron mobility at room temperature = 0.035 m^2/V-s

Example 10.23, page no-284

In [39]:
# Hall voltage

import math
# Variable declaration
I=50                      # Current
B=1.5                     # Magnetic field 
t=0.5*10**-2              # Thickness of the slab
e=1.6*10**-19             # Electronic charge
d=2*10**-2                # Width of the slab  
N=8.4*10**28              # Concentration of electron

# Calculations
v=B*I/(N*e*d)

# Result
print("The Hall voltage is %.2f *10^-7 V"%(v*10**7))

The Hall voltage is 2.79 *10^-7 V

Example 10.24, page no-284

In [40]:
# relaxation time of electrons in metal
import math
# Variable declaration
rho=1.54*10**-8                 # resistivity of metal
ni=5.8*10**28                   # carrier concentration
m=9.1*10**-31                   # mass of an electron
e=1.6*10**-19                   # electronic charge

# Calculations
tau=m/(rho*ni*(e**2))

#Result
print("The relaxation time of electrons in metal is %.2f*10^-14 s"%(tau*10**14))

The relaxation time of electrons in metal is 3.98*10^-14 s

Example 10.25, page no-285

In [41]:
# mobility of electrons

import math
# variable declaration
sig=6.22*10**7                    # conductivity of metal
n=5.9*10**28                      # carrier concentration of electron
e=1.6*10**-19                     # electronic charge

#calculation
mu=sig/(n*e)

# Result
print("The mobility of electrons in Si is %.2f*10^-3 m^2/V-s"%(mu*10**3))

The mobility of electrons in Si is 6.59*10^-3 m^2/V-s

Example 10.26, page no-285

In [42]:
# mobility of the electrons

import math
# Variable declaration
rho=0.1                   # resistivity of metal
ni=10**20                 # carrier concentration of electron 
vd=1                      # drift velocity      
e=1.6*10**-19             # electronic charge

# calculations
mu=1/(rho*ni*e)
E=vd/mu

# Result
print("\nThe mobility of the electrons in material is %.3f m^2/V-s\nThe electric field is %.1f V/m"%(mu,E))

The mobility of the electrons in material is 0.625 m^2/V-s
The electric field is 1.6 V/m

Example 10.27, page no-286

In [43]:
# mobility of electrons

import math
#variable declaration
sig=6.22*10**7              # conductivity of metal
n=5.9*10**28                #carrier concentration of electron  
e=1.6*10**-19               # electronic charge

# calculations
mu=sig/(n*e)

# Result
print("The mobility of electrons in silver is %.2f*10^-3 m^2/V-s"%(mu*10**3))

The mobility of electrons in silver is 6.59*10^-3 m^2/V-s

Example 10.28, page no-286

In [44]:
# mobility of the electrons

import math
# Variable declaration
rho=0.1                   # resistivity of metal
ni=10**20                 # carrier concentration of electron 
vd=1                      # drift velocity      
e=1.6*10**-19             # electronic charge

# calculations
mu=1/(rho*ni*e)
E=vd/mu

# Result
print("\nThe mobility of the electrons in material is %.3f m^2/V-s\nThe electric field is %.1f V/m"%(mu,E))

The mobility of the electrons in material is 0.625 m^2/V-s
The electric field is 1.6 V/m

Example 10.29, page no-287

In [45]:
# relaxation time, mobility and conductivity

import math
#variable declaration
avg=6.023*10**23                    # Avogadro's number
m=9.1*10**-31                       # mass of electron
e=1.6*10**-19                       # charge of an electron   
d=8.92*10**3                        # density of copper 
rho=1.73*10**-8                     # resistivity of copper
z=63.5                              # Atomic weight of copper

# Calculations
n=avg*d/z
sig=1/rho
tau=sig*m/(n*(e**2))
mu=sig/(e*n)

#Result
print("\nThe relaxation time is %.2f *10^-11 s\nThe mobility of electrons in copper is %.2f m^2/V-s"%(tau*10**11,mu))
print("The conductivity of coppper is %.2f * 10^7 per Ohm-m\n"%(sig*10**-7))

The relaxation time is 2.43 *10^-11 s
The mobility of electrons in copper is 4.27 m^2/V-s
The conductivity of coppper is 5.78 * 10^7 per Ohm-m

Example 10.30, page no-288

In [46]:
# mobility of electrons and drift velocity

import math
#variable declaration
rho=1.54*10**-8               # resistivity of silver
E=100                         # electric field along the wire
ni=5.8*10**28                 # carrier concentration of electron
e=1.6*10**-19                 # electronic charge

# calculations
mu=1/(rho*ni*e)
vd=mu*E

#Result
print("The mobility of electrons in silver is %.4f*10^-3 m^2/V-s\nThe drift velocity id %.5f m/s"%(mu*10**3,vd))

The mobility of electrons in silver is 6.9973*10^-3 m^2/V-s
The drift velocity id 0.69973 m/s

Example 10.31, page no-288

In [50]:
# relaxation time for electrons

import math
#variable declaration
rho=1.43*10**-8               # resistivity of metal
ni=6.5*10**28                 # carrier concentration of electron
e=1.6*10**-19                 # electronic charge
m=9.1*10**-31                 # mass of an electron

# calculations
tau=m/(rho*ni*e**2)

# Result
print("The relaxation time for electrons in the metal is %.2f *10^-14 s"%(math.ceil(tau*10**16)/100))

The relaxation time for electrons in the metal is 3.83 *10^-14 s

Example 10.32, page no-289

In [51]:
# concentration, mobility  and velocity of electron

import math
#variable declaration
R=60                      # resistance of aluminium
rho=2.7*10**-8            # resistivity of aluminium
i=15                      # current in the wire
l=5                       # length of the aluminium wire
m=3                       # number of free electron per atom   
e=1.6*10**-19             # electronic charge   
d=2.7*10**3               # density of aluminium
awt=26.98                 # Atomic weight of aluminium
avg=6.023*10**23          # Avogadro's number

# calculations
n=m*avg*1000*d/awt
mu=1/(rho*n*e)
vd=mu*i*R*10**-3/l

# Result
print("Free electron concentration is %.3f * 10^29"%(n*10**-29))
print("\nThe mobility of electron in aluminium is %.4f*10^-3 m^2/v-s"%(mu*10**3))
print("\nThe drift velocity of the electron in Al is %.1f*10^-4 m/s"%(vd*10**4))

Free electron concentration is 1.808 * 10^29

The mobility of electron in aluminium is 1.2801*10^-3 m^2/v-s

The drift velocity of the electron in Al is 2.3*10^-4 m/s

Example 10.33, page no-290

In [52]:
# Thermal and drift velocity of electron in copper

import math
# variable declaration
R=0.02                    # resistance of the Cu
i=15                      # current in the wire
mu=4.3*10**-3             # mobility of the free electron  
l=2                       # length of the Cu wire
k=1.38*10**-23            # Boltzmann's constant   
m=9.1*10**-31             # mass of electron
T=300                     # temperature 

# Calculations
v=i*R          
E=v/l
vd=E*mu
vth=math.sqrt(3*k*T/m)

# Result
print("\nThe thermal velocity of the free electrons in copper is %.3f mm/s"%(vth*10**-5))
print("The drift velocity of electrons in copper is %.3f mm/s"%(vd*10**3))

The thermal velocity of the free electrons in copper is 1.168 mm/s
The drift velocity of electrons in copper is 0.645 mm/s