# Chapter2 : Energy Bands and Charge Carriers in semiconductor¶

### Example 2.1 Page No.58¶

In [9]:
#Example 2.1
#Find probability of an electronic state

#Given
dE1=0.1             #eV
dE2=-0.1           #eV
k=8.61*10**-5     #Boltzman constant
T=300             #K

#Calcualtion
import math
FE1=1/(1+math.exp(dE1/(k*T)))
FE2=1/(1+math.exp(dE2/(k*T)))

#Result
print"Probability when the energy of the state is above 0.1 eV",round(FE1,2)
print"Probability when the energy of the state is below 0.1 eV",round(FE2,2)

Probability when the energy of the state is above 0.1 eV 0.02
Probability when the energy of the state is below 0.1 eV 0.98


### Example 2.2 Page No. 58¶

In [2]:
#Calculate the temprature at which there is 1 percent probability
#that a state of 0.30 eV below the fermi energy level will not contain electrons.
import math
#Exa 2.2
Ef=6.25         #EV  fermi energy level
dE=-0.30        #eV
k=8.61*10**-5    #Boltzman constant
fE=0.99

#calculation
#From the probability formula fE=1/(1+math.exp(dE/(k*T)))
T=(dE)/(k*math.log(1/fE-1))

#result
print"The Temprature is",round(T,1),"K"

The Temprature is 758.3 K


### Example 2.3 Page No. 64¶

In [1]:
#Example 2.3
#Determine the fraction of total no. of electron

#Given
Eg=0.72            #eV
Ef=0.5*Eg
dE=Eg-Ef           #eV
k=8.61*10**-5     #Boltzman constant
T=300             #K

#Calcualtion
import math
N=1/(1+math.exp(dE/(k*T)))

#Result
print"the fraction of total no. of electron is ",round(N,9)

the fraction of total no. of electron is  8.85e-07


### Example 2.4 Page No. 64¶

In [22]:
#Example 2.4
#Calculate the wave length
import math
#Given
E=300*1.602*10**-19            #eV Energy
m=9.108*10**-31                #kg, mass of electron
h=6.626*10**-34                #Planck constant

#Calculation
v=math.sqrt(2*E/m)
lam=h*v/E

#Result
print"The wavwlength is",round(lam*10**10,3),"A"

The wavwlength is 1.416 A


### Example 2.5 Page No. 70¶

In [25]:
#Exa 2.5
#Find the ratio of electron to hole concentration ratio

#given data
ni=1.4*10**18			#in atoms/m**3
Nd=1.4*10**24			#in atoms/m**3
n=Nd				#in atoms/m**3

#Calculation
p=ni**2/n			#in atoms/m**3
ratio=n/p			#unitless

#Result
print"Ratio of electron to hole concentration : ",round(ratio,2)

Ratio of electron to hole concentration :  1e+12


### Example 2.7 Page no 74¶

In [35]:
#Example 2.7
#Calculate the magnitude of current

#Given
n=10**24          #Electron density
e=1.6*10**-19     #Electron charge
v=0.015           #m/s drift velocity
A=10**-4            #m**2 area

#Calculation
I=n*e*v/A

#Result
print"The magnitude of current is",round(I/10**8,2),"A"

The magnitude of current is 0.24 A


### Example 2.8 Page No. 74¶

In [32]:
#Exa 2.8
#calculate (i) Relaxation time (ii)Resistivity of conductor (iii) velocity of electron

#given data
Ef=5.5			#in eV
MUe=7.04*10**-3		#in m**2/V-s
n=5.8*10**28		#in m**-3
e=1.6*10**-19		#constant
m=9.1*10**-31		#in Kg

#calculation
#part (i)
import math
tau=MUe*m/e		#in sec
rho=1/(n*e*MUe)		#in ohm-m
vF=math.sqrt(2*Ef*1.6*10**-19/m)

#Result
print"Relaxation time in sec : ",tau,"s"
print"Resistivity of conductor in ohm-m : ",round(rho,11),"ohm m"
print"velocity of electron with fermi energy is ",round(vF,0),"m/s"

Relaxation time in sec :  4.004e-14 s
Resistivity of conductor in ohm-m :  1.531e-08 ohm m
velocity of electron with fermi energy is  1390707.0 m/s


### Example 2.9 Page No. 75¶

In [51]:
#Example 2.9
#Find (i)the valence electrons per unit volume (ii) mobility

#Given
rho=1.73*10**-8       #resistivity
Tav=2.42*10**-14      #Average Time
e=1.6*10**-19		#constant
m=9.1*10**-31		#in Kg

#Calculation
n=m/(e**2*Tav*rho)
mu=(e*Tav)/m

#Result
print"NO. of free electrons are",round(n,-26)
print"mobility of  electrons is",round(mu,3),"m**2/Vs"

NO. of free electrons are 8.49e+28
mobility of  electrons is 0.004 m**2/Vs


### Example 2.10 page No. 75¶

In [57]:
#Example 2.10
#calculate Relaxation time and drift velocity

Ef=100			#in V/m Applied electric field
n=6*10**28		#in m**-3
e=1.6*10**-19		#constant electronic charge
m=9.1*10**-31		#in Kg  mass of electron
rho=1.5*10**-8      #Density

#calculation
import math
tau=m/(n*e**2*rho)		#in sec
vF=e*Ef*tau/m

#Result
print"Relaxation time in sec : ",round(tau,16),"s"
print"velocity of electron with fermi energy is ",round(vF,1),"m/s"

Relaxation time in sec :  3.95e-14 s
velocity of electron with fermi energy is  0.7 m/s


### Example 2.11 Page No.75¶

In [69]:
#Exampl 2.11
#Determine charge density, current density ,Current flowing in the wire, Electron drift velocity

#Given
d=0.002             #m, diameter of pipe
s=5.8*10**7         #Conductivity S/m
mu=0.0032            #m**2/Vs, Electron mobility
e=1.6*10**-19		#constant electronic charge
m=9.1*10**-31		#in Kg  mass of electron
E=0.02               #V/m Electric field

#Calculation
import math
#From eq 2.62
n=s/(e*mu)
J=s*E
I=J*(math.pi*d**2/4.0)
v=mu*E

#Result
print"Charge density is",round(n,-26),"m**-3"
print"current density is",round(J,6),"A/m**2"
print"curret flowing is",round(I,3),"A"
print"electron drift velocityis",round(v,6),"m/s"

Charge density is 1.133e+29 m**-3
current density is 1160000.0 A/m**2
curret flowing is 3.644 A
electron drift velocityis 6.4e-05 m/s


### Example 2.12 page no 76¶

In [156]:
#example 2.12
#calculate the drift velocity and time

#Given
rho=0.5        #ohm-m Resistivity
J=100          #A/m**2 Current density
mue=0.4        #m**2/Vs Electron mobility
d=10*10**-6    #m distance

#calculation
Ve=mue*J*rho
t=d/Ve

#Result
print"The drift velocity is ",Ve,"m/s"
print"Time taken by the electron is",round(t,8),"s"

The drift velocity is  20.0 m/s
Time taken by the electron is 5e-07 s


### Example 2.13 Page No.76¶

In [76]:
#Example 2.13
#Calculate drift velocity and time

#Given
e=1.6*10**-19		#constant electronic charge
m=9.1*10**-31		#in Kg  mass of electron
rho=0.039             #ohm-cm resistivity
mu=3600               #cm**2/Vs  Carrier mobility
ni=2.5*10**13

#Calculation
Nd=(1/(rho*e*mu))
n=Nd
p=(ni**2/n)

#Result
print"Concentration of electron is",round(n,-14),"/cm**3"
print"Concentration of holes is",round(p,0),"/cm**3"

Concentration of electron is 4.45e+16 /cm**3
Concentration of holes is 14040000000.0 /cm**3


### Example 2.14 page No 76¶

In [4]:
#Example 2.14
#Determine concentration of holes and electrons

#Given
rho=5.32              #kg/m**3, density
Aw=72.6               #kg/K kmol atomic weight
ni=2.5*10**13
di=10**8              #Donor impurity
e=1.6*10**-19         #Electronic charge
mue=0.38              #m**/Vs
muh=0.18               #m**/Vs

#CAlculation
N=6.023*10**23*rho/Aw       #No 0f germanium atoms per cm**3
Nd=N/di
n=Nd
p=(ni**2/n)
s=n*e*mue*10**4

#Result
print"Concentration of electrons is",round(n,-12),"atoms/cm**3"
print"Concentration of holes is",round(p,-10),"atoms/cm**3"

if n > p:

print"Conductivity of N-type germanium",round(s*100,1),"/ohm-m"
else:
print "Calculate p-type germanium conductivity"

Concentration of electrons is 4.41e+14 atoms/cm**3
Concentration of holes is 1.42e+12 atoms/cm**3
Conductivity of N-type germanium 26.8 /ohm-m


### Example 2.15 Page no.79¶

In [5]:
#Example 2.15
#Calculate the density and drift velocity

#Given
e=1.6*10**-19         #Electronic charge
mue=0.39              #m**/Vs
muh=0.19               #m**/Vs
rhoi=0.47              #ohm-m, intrinsic resistivity
E=10**4                #Electric field

#Calculation
sigmai=1/rhoi
ni=sigmai/(e*(mue+muh))
Vn=mue*E
Vh=muh*E

#Result
print"Density of electrons is",round(ni,-17),"/m**3"
print"Drift velocity for electrons",round(Vn,0),"m/s"
print"Drift velocity for holes",round(Vh,0),"m/s"

Density of electrons is 2.29e+19 /m**3
Drift velocity for electrons 3900.0 m/s
Drift velocity for holes 1900.0 m/s


### Example 2.16 page No.80¶

In [10]:
#Example 2.16
#Calculate conductivity

#Given
i=10**7            #IMpurity in Ge atom
ni=2.5*10**13      #/cm**3
N=4.4*10**22       #No. of atoms of Ge
mue=3800            #cm**2/Vs
muh=1800            #cm**2/Vs
e=1.6*10**-19       #Electronic charge
E=400               #Electric field

#Calculation
sigmai=ni*e*(mue+muh)
Nd=N/i
n=Nd
p=ni**2/(Nd)
sigman=e*Nd*mue

print"Intrinsic conductivity of Ge is ",sigmai,"ohm-cm**-1"
print"Conductivity of N type Ge semiconductor is",round(sigman,2),"ohm-cm**-1"

Intrinsic conductivity of Ge is  0.0224 ohm-cm**-1
Conductivity of N type Ge semiconductor is 2.68 ohm-cm**-1


### Example 2.17 Page No. 80¶

In [8]:
#Example 2.17
#(i)Electron drift velocity & hole drift velocity .
#(ii)Intrinsic Conductivity of Ge,(iii)The total current .

#Given
V=10              #Volt
l=0.025           #m, length
w=0.004           #m width
t=0.0015         #m thickness

ni=2.5*10**19      #/cm**3
mue=0.38               #m**2/Vs
muh=0.18               #m**2/Vs
e=1.6*10**-19           #Electronic charge
E=400              #Electric field

#Calculation
E=V/l
Ve=mue*E
Vh=muh*E
sigmai=ni*e*(mue+muh)
I=sigmai*E*w*t

#Result
print"(i)Electron drift velocity is ",Ve,"m/s"
print"   hole drift velocity is ",Vh,"m/s"
print"(ii)Intrinsic Conductivity of Ge is",sigmai,"ohm-m**-1"
print"(iii)The total current is ",I*1000,"mA"

(i)Electron drift velocity is  152.0 m/s
hole drift velocity is  72.0 m/s
(ii)Intrinsic Conductivity of Ge is 2.24 ohm-m**-1
(iii)The total current is  5.376 mA


### Example 2.18 page no.80¶

In [11]:
#Example 2.18
#What is ratio of electrons to holes

#Given
Ie=3/4.0           #Current due to electron
Ih=1-Ie            #Current due to holes
Vh=1               #Hole velocity
Ve=3                #Electron velocity 3 times the hole velocity

#ccalculation
R=(Ie*Vh/(Ih*Ve))

#Result
print"The ratio of electrons to holes drift velocity is ",R

The ratio of electrons to holes drift velocity is  1.0


### Example 2.19 Page No.81¶

In [17]:
#Exa 2.19
#Find the diffusion coefficients of electrons and holes

#given data
e=1.6*10**-19			#in coulamb
T=300				#in Kelvin
MUh=0.025			#in m**2/V-s
MUe=0.17			#in m**2/V-s
k=1.38*10**-23			#in J/K
De=MUe*k*T/e			#in cm**2/s
Dh=MUh*k*T/e			#in cm**2/s

#Result
print"Diffusion constant of electron is ",round(De*10000,2),"(in cm**2/s)"
print"Diffusion constant of hole is ",round(Dh*10000,2),"(in cm**2/s)"

Diffusion constant of electron is  43.99 (in cm**2/s)
Diffusion constant of hole is  6.47 (in cm**2/s)


### Example 2.20 Page no. 81¶

In [4]:
#Example 2.20
#Find intrinsic carries cncentration and conductivity
import math
#Given
N=3*10**25          #No of atoms
e=1.6*10**-19
Eg=1.1*e         #eV
k=1.38*10**-23   #j/k  boltzman's constant
T=300            #K
mue=0.14
muh=0.05

#Calculation
ni=N*math.exp(-Eg/(2*k*T))
sigma=ni*e*(mue+muh)

#Result
print"The intrinsic carries concentration is ",round(ni,-14),"/m**3"
print"The conductivity of Si is ",round(sigma,5),"S/m"

The intrinsic carries concentration is  1.76e+16 /m**3
The conductivity of Si is  0.00054 S/m


### Example 2.21 Page No.84¶

In [44]:
#Example 2.21
#Find the effective density

#Given
a=1.5        #a=me/mo
T=300     #K

#calculation
#from eq. 2.29
Nc=4.82*10**21*(a)**(1.5)*T**(1.5)

#Result
print"The effective density is",round(Nc,-23),"/m**3"

The effective density is 4.6e+25 /m**3


### Example 2.22 page no. 84¶

In [55]:
#Example 2.22
#Calculate the intrinsic concentration

#Given
a=0.07          #a=me/mo
b=0.4            #b=mh/mo
T=300           #K
Eg=0.7          #eV
k=8.62*10**-5   # Boltzman constant

#calculation
import math
#From eq 2.101
ni=math.sqrt(2.33*10**43*(a*b)**(1.5)*T**3*math.exp(-Eg/(k*T)))

#Result
print"The intrinsic concentration of charge carrier is",round(ni,-16),"/m**3"

The intrinsic concentration of charge carrier is 2.27e+18 /m**3


### Example 2.23 Page no. 85¶

In [3]:
#Example 2.23
#Find the value of absolute temprature

#Given
C=5*10**28            #atom/m**3, concentration of Si atoms
DL=2*10**8            #Doping level
m=1
me=m
#calculation
Nd=C/DL
nc=Nd
T=((nc/(4.82*10**21))*(m/me)**(1.5))**(2/3.0)

#Result
print"The absolute temprature is",round(T,2),"K"

The absolute temprature is 0.14 K


### Example 2.24 Page No. 85¶

In [110]:
#Example 2.24
#Determine the effective density

#Given
T1=300.0          #K temprature
T2=400.0
k=1.38*10**-23  #J/k
m=1.25*9.107*10**-31
h=6.625*10**-34
dE=0.3           #eV
k_=8.62*10**-5

#calculation
import math
nc1=2*(2*math.pi*m*k*T1/(h**2))**(1.5)
n1=nc1*math.exp(-(0.3/(k_*T1)))

nc2=2*(2*math.pi*m*k*T2/(h**2))**(1.5)
n2=nc2*math.exp(-(0.3/(k_*T2)))

#result
print"The effective density at temprature 300 K is",round(n1,-19),"/m**3"
print"The effective density at temprature 400 K is",round(n2,-19),"/m**3"

The effective density at temprature 300 K is 3.2e+20 /m**3
The effective density at temprature 400 K is 8.98e+21 /m**3


### Example 2.25 Page no.86¶

In [2]:
#example 2.25
#determine the position of intrinsic fermi level
import math
#Given
T=300.0
k=8.62*10**-5  #J/k
m=9.107*10**-31
me=0.6*m
mh=0.4*m

#calculation
dE=-3*k*T*math.log((me/mh)**(1))/4.0   #dE=Ef-Emidgap

#Result
print"The position of fermi level is",round(dE,4),"eV"

The position of fermi level is -0.0079 eV


### Example 2.26 Page no 86¶

In [131]:
#example 2.26
#determine the position of intrinsic fermi level

#Given
T=300.0
Eg=0.72      #eV Energy gap
k=8.62*10**-5  #J/k
me=1
mh=5.0

#calculation
#from   Ef=Ec-kTlog(nc/Nd)
import math
dE=(Eg/2.0)-3*k*T*math.log(me/mh)/4.0 #dE=Ef-Emidgap

#Result
print"The position of fermi level is",round(dE,4),"eV"

The position of fermi level is 0.3912 eV


### Example 2.27 Page no 87¶

In [134]:
#example 2.27
#determine the position of intrinsic fermi level

#Given
T1=300.0
T2=350
Eg=0.24        #eV Energy gap

#calculation
#from   Ef=Ev+kTlog(nc/Nd)
import math
dE=(T2/T1)*Eg

#Result
print"The position of fermi level is",round(dE,4),"eV"

The position of fermi level is 0.28 eV


### Example 2.28 Page no.88¶

In [133]:
#Example 2.28
#determine the position of intrinsic fermi level

#Given
T1=300.0
T2=400
Eg=0.27        #eV Energy gap

#calculation
import math
dE=(T2/T1)*Eg

#Result
print"The position of fermi level is",round(dE,4),"eV"

The position of fermi level is 0.36 eV


### Example 2.29 page no.88¶

In [137]:
##Example 2.29
#determine the position of intrinsic fermi level

#Given
dE1=0.3        #eV Energy gap
kT=0.026       #eV

#calculation
import math
x=math.exp(-dE1/kT)    #x=Nd/nc
y=5           #y=Nd2/Nd1
dE2=-math.log(y)*kT+dE1

#Result
print"The position of fermi level is",round(dE2,3),"eV"

The position of fermi level is 0.258 eV


### Example 2.30 Page no.89¶

In [143]:
##Example 2.30
#determine the position of intrinsic fermi level

#Given
dE1=0.39        #eV Energy gap
kT=0.026        #eV

#calculation
import math
x=math.exp(-dE1/kT)    #x=NA1/nV
y=3           #y=NA2/NA1
dE2=((dE1/kT)-math.log(y))*kT

#Result
print"The position of fermi level is",round(dE2,2),"eV"

The position of fermi level is 0.36 eV


### Example 2.31 Page no.91¶

In [9]:
#example 2.31
#Determine electron density and mobility

#Given
rho=1       #ohm-m Resistivity
Rh=100.0       #cm**3/coulomb
e=1.6*10**-19

#calculation
con=1/rho   #Conductivity
R=1/Rh      #Charge density
ED=R*10**6/e
mu=con/(R*10**6)

#Result
print"The electron density is",ED,"/m**3"
print"The mobility is %.e"%mu,"/m**3"

The electron density is 6.25e+22 /m**3
The mobility is 1e-04 /m**3


### Example 2.32 Page no. 92¶

In [146]:
#Example 2.32
#Calculate Hall Voltage

#Given
w=0.1      #m width
t=0.01     #m thickness
F=0.6       #T, field
Rh=3.8*10**-4  #Hall Coefficient
I=10        #mA

#calculation
Vh=(Rh*F*I/w)

#Result
print"Hall Voltage is",Vh*1000,"micro V"

Hall Voltage is 22.8 micro V


### Example 2.33 Page No. 92¶

In [19]:
#Exa 2.33
#What is magnitude of Hall Voltage

#given data
e=1.6*10**-19			#in coulamb
ND=10**17			#in cm**-3
Bz=0.1				#in Wb/m**2
w=4				#in mm
d=4				#in mm
Ex=5				#in V/cm
MUe=3800			#in cm**2/V-s

#calculation
v=MUe*Ex			#in cm/s
v=v*10**-2			#in m/s
VH=Bz*v*d			#in mV

#Result
print"Magnitude of hall voltage is",VH,"mV"

Magnitude of hall voltage is 76.0 mV


### Example 2.34 Page No.92¶

In [22]:
#Exa 2.34
#What is magnitude of hall voltage

#given data
e=1.6*10**-19			#in coulamb
ND=10**21			#in m**-3
Bz=0.2				#in T
d=4				#in mm
d=d*10**-3			#in meter
J=600				#in A/m**2
n=ND				#in m**-3

#calculation
#formula : VH*w/(B*I)=1/(n*e)
VH=Bz*J*d/(n*e)			#in V

#Result
print"Magnitude of hall voltage is ",VH*10**3,"mV"

Magnitude of hall voltage is  3.0 mV


### Example 2.35 Page No.¶

In [169]:
#Exa 2.35
#Calculate hall angle

#given data
e=1.6*10**-19			#in coulamb
rho=0.00912			#in ohm-m
B=0.48				#in Wb/m**2
RH=3.55*10**-4			#in m**3-coulamb**-1
SIGMA=1/rho			#in (ohm=m)**-1

#calculation
import math
THETAh=math.atan(SIGMA*B*RH)	#in Degree

#result
print"Hall angle is",round(THETAh*180/3.14,4),"degree"

Hall angle is 1.0709 degree