# Chapter 8:Mechanism of Conduction in Semiconductors¶

## Example 8.1,Page No:8.13¶

In :
import math

#variable declaration
Ephoton = 1.5;               # energy of photon in eV
Eg      = 1.4;               # energy gap in eV
m       = 9.1*10**-31;       # mass of electron in kg
e       = 1.6*10**-19;       #charge of electron in coulombs
me_GaAs = 0.07;              #times of electro mass in kilograms
mh_GaAs = 0.068;             #times of electro mass in kilograms

# Calculations
Eke     = Ephoton - Eg;                    #energy on eV
pe      = math.sqrt(2*m*me_GaAs*Eke*e)     # momentum of electrons  in kg m/s
ph      = math.sqrt(2*m*mh_GaAs*Eke*e)     # momentum of electrons in kg m/s

# Result
print'Kinetic Energy = %3.1f'%Eke,'eV';
print'Momentum of electrons = %3.1e'%pe,'kg m/s';
print'Momentum of holes = %3.1e'%ph,'kg m/s';

Kinetic Energy = 0.1 eV
Momentum of electrons = 4.5e-26 kg m/s
Momentum of holes = 4.4e-26 kg m/s


## Example 8.2,Page No:8.27¶

In :
import math

#variable declaration
T1  = 300;               # temperature in kelvin
nv  = 1.04*10**19;       #in cm**-3
T2  = 400;               #temperature in K
fl  = 0.25;              #fermi level position in eV

#Calculations
Nv  = (1.04*10**19)*(T2/float(T1))**(3/float(2));           #Nv at 400 k in cm**-3
kT  = (0.0259)*(T2/float(T1));                              #kT in eV
po  = Nv*math.exp(-(fl)/float(kT));                         #hole oncentration in cm**-3

# Result
print'Thermal equilibrium hole concentration = %3.2e '%po,'cm**-3';
print'Note: Calculation mistake in textbook Nv is not multiplied by exponentiation';

Thermal equilibrium hole concentration = 1.15e+16  cm**-3
Note: Calculation mistake in textbook Nv is not multiplied by exponentiation


## Example 8.3,Page No:8.27¶

In :
import math

#variable declaration
Nc  = 3.8*10**17;            #constant in cm**-3
Nv  = 6.5*10**18;            #constant in cm**-3
Eg  = 1.42;                  # band gap energy in eV
KT1 = 0.03885;               # kt value at 450K
T1  = 300;                   #temperature in K
T2  = 450;                   #temperature in K

# calculation
n1i  = math.sqrt(Nc*Nv*math.exp(-Eg/float(0.0259)));               #intrinsic carrier concentration in cm**-3
n2i  = math.sqrt(Nc*Nv*((T2/float(T1))**3) *math.exp(-Eg/float(KT1)));     # intrinsic carrier conc at 450K in cm**-3

# Result
print'Intrinsic Carrier Concentration at 300K = %3.2e'%n1i,'cm**-3';
print'Intrinsic Carrier Concentration at 300K = %3.2e'%n2i,'cm**-3';
print' Note : Calculation mistake in textbook in finding carrier conc. at 450K';

Intrinsic Carrier Concentration at 300K = 1.95e+06 cm**-3
Intrinsic Carrier Concentration at 300K = 3.34e+10 cm**-3
Note : Calculation mistake in textbook in finding carrier conc. at 450K


## Example 8.4,Page No:8.28¶

In :
import math

#variable declaration
mh  = 0.56;             #masses interms of m0
me  = 1.08;             #masses interms of m0
t   = 27;               #temperature in °C
k   = 8.62*10**-5;

# Calculations
T   = t+273;                                         #temperature in K
fl  = (3/float(4))*k*T*math.log(mh/float(me));          #position of fermi level in eV

#result
print'The position of Fermi level with respect to middle of the bandgap is %3.1f'%(fl*10**3),'meV';

The position of Fermi level with respect to middle of the bandgap is -12.7 meV


## Example 8.5,Page No:8.30¶

In :
import math

#variable declaration
mo  = 9.11*10**-31;          #mass of electron inkilograms
e   = 1.6*10**-19;           # charge of electron in coulombs
er  = 13.2;                  #relative permitivity in F/m
eo  = 8.85*10**-12;          # permitivity in F/m
h   = 6.63*10**-34;          # plancks constant J.s

# Calculations
me  = 0.067*mo;
E   = (me*(e**4))/float((8*(eo*er)**2)*(h**2)*e);         #energy in eV

# Result
print'Donor binding energy = %3.4f'%E,'eV';

Donor binding energy = 0.0052 eV


## Example 8.6,Page No:8.30¶

In :
import math

#variable declaration
no      = 10**17;            # doping carrier conc
ni      = 1.5*10**10;        #intrinsic concentration
kT      = 0.0259

#Calculations
po      = (ni**2)/float(no);             #Equlibrium hole concentration in cm**-3
fl      = kT*math.log10(no/float(ni));   #Position of fermi energy level in eV

#Result
print'Equlibrium hole concentration = %3.2e'%po,'cm**-3';
print'Position of fermi energy level = %3.3f'%fl,'eV';

Equlibrium hole concentration = 2.25e+03 cm**-3
Position of fermi energy level = 0.177 eV


## Example 8.7,Page No:8.33¶

In :
import math

#variable declaration
k   = 8.62*10**-5;            #in eV/K
Eg  = 1.10;                   #energy in eV
t1 = 200;                     #temperature in °C
t2 = 27;                      #temperature in °C
psi = 2.3*10**3;

# Calculations
# sigma = sigmao*exp(-Eg/(2kT))
# k     = sigma_473/sigma_300;

t3        = t1+273;                                 #temperature in K
t4        = t2+273;                                 #temperature in K
k1        = math.exp((-Eg)/float(2*k*t3));          #electrical conductivity in cm**-1.m**-1
k2        = math.exp((-Eg)/float(2*k*t4));          #lectrical conductivity in cm**-1.m**-1
k         = k1/float(k2);
pm        = k/float(psi);

#Result

print'electrical conductivity of pure silicon =%3.2e'%k,'ohm**-1.m**-1';
print'Note:calculation mistake in electrical conductivity,and units of conductivity';


electrical conductivity of pure silicon =2.39e+03 ohm**-1.m**-1
Note:calculation mistake in electrical conductivity,and units of conductivity


## Example 8.8,Page No:8.33¶

In :
import math

#variable declaration
ni  = 2.5*10**19;            # carrier density in per m**3
q   = 1.6*10**-19;           # charge of electron in coulombs
un  = 0.35;                 #mobility of electrons in m**2/V-s
up  = 0.15;                 #mobility of electrons in m**2/V-s

# Calculations
sigma = ni*q*(un + up);         #conductivity in per Ω-m
p     = 1/float(sigma);                #resistivity in Ω-m

# Result
print'Resistivity = %3.1f'%p,'Ω-m';

Resistivity = 0.5 Ω-m


## Example 8.9,Page No:8.33¶

In :
import math

#variable declaration
p  = 3.16*10**3;            # resistivity Ω-m
e  = 1.6*10**-19;           # charge of electron in coulombs
ue = 0.14;                 #mobility of electrons in m**2/V-s
uh = 0.05;                  #mobility of holes in m**2/V-s

# Calculations

n  = 1/float((p*e)*(ue + uh));           #carrier density  in m**-3

# Result
print'Intrinsic Carrier Concentration = %3.2e'%n,'m**-3';

Intrinsic Carrier Concentration = 1.04e+16 m**-3


## Example 8.10,Page No:8.34¶

In :
import math

#variable declaration
p   = 5.32*10**3;        #density of germanium
Nav = 6.023*10**26;      # Avagadros number
AW  = 72.59;            # atomic wt
ni  = 1.5*10**19;        # carrier density
ue  = 0.36;
uh  = 0.18;
e   = 1.6*10**-19;

# calculations
N   = (p*Nav)/float(AW);            # no of germanium atoms per unit volume
Nd  = N*10**-6 ;                   # no of pentavalent impurity atoms/m**3
f   = Nd/float(ni);
nh  = (ni**2)/float(Nd);           # hole concentration
sigma = e*((Nd*ue)+(nh*uh));

#Result
print'The factor by which the majority conc. is more than the intrinsic carrier conc = %d'%f;
print'Hole concentration = %3.1e'%nh,'m**-3';
print'Conductivity = %d'%sigma,'ohm**-1 m**-1';

The factor by which the majority conc. is more than the intrinsic carrier conc = 2942
Hole concentration = 5.1e+15 m**-3
Conductivity = 2542 ohm**-1 m**-1


## Example 8.11,Page No:8.34¶

In :
import math

#variable declaration
p   = 5*10**-3;          # resistivity in Ω-m
ue  = 0.3;              # electron mobility m**2/volt-s
uh  = 0.1;              # hole mobility m**2/volt-s
e   = 1.6*10**-19        # charge of electron in coulombs

# calculations
sigma   = 1/float(p);                        # conductivity in per Ω -m
n       = sigma/float(e*(ue + uh));          # carrier density per m**3

#Result
print'Carrier Density = %3.1e'%n,'m**-3';

Carrier Density = 3.1e+21 m**-3


## Example 8.12,Page No:8.35¶

In :
import math

#variable declaration
Jd  = 500;                   # current density A/m**2
p   = 0.05;                  # resistivity in Ω-m
l   = 100*10**-6;            #  travel length m
ue  = 0.4;                   # electron mobility m**2/Vs
e   = 1.6*10**-19;            # charge of electron in coulombs

# Calculations
ne  = 1/float(p*e*ue);            #in per m**3
vd  = Jd/float(ne*e);            #drift velocity in m/s
t   = l/float(vd);                 #time teken in s

#result
print'Drift velocity = %d'%vd,'m/s';
print' time = %3.0e'%t,'s';

Drift velocity = 10 m/s
time = 1e-05 s


## Example 8.13,Page No:8.35¶

In :
import math

#variable declaration

#psi1 is increased by 30%, psi1/ps2 is 130/100
a    = 1.3;                  #ratio of psi1/psi2
K    = 8.82*10**-5;              #constant in eV/K
Eg   = 0.719;                  #band gap in eV/K
T    = 300;                      #temperature in K

#calculation
d=1/float((1/float(T))-((2*K/float(Eg))*math.log(1.3)));
dT=d-T;                                             #temperature rise in  K

#result
print'temperature rise is of = %3.2f'%dT,'K';

temperature rise is of = 5.91 K


## Example 8.14,Page No:8.39¶

In :
import math

#variable declaration
v   = 5;            # voltage in volts
r   = 10;           # resistance in k-ohm
J   = 60;           # current density in A/cm**2
E   = 100;          # electric field in V.m**-1
Nd  = 5*10**15;      # in cm**-3
up  = 410;          # approx hole mobility cm**2/V-s
Na  = 1.25*10**16;   # approx in cm**-3
e   = 1.6*10**-19;   # charge of electron in coulombs

#Calculations
I       = v/float(r);              # total current A
A       = I/float(J);               # cross sectional area cm**2
L       = v/float(E)                 # length of resistor cm
sigma   = L/float(r*A);        #conductivity in (Ω-cm)**-1
sigma_comp = e*up*(Na - Nd);        #conductivity in (Ω-cm)**-1

# Result
print'Conductivity of the compensated p-type semiconductor is %3.3f'%sigma_comp;

Conductivity of the compensated p-type semiconductor is 0.492


## Example 8.15,Page No:8.39¶

In :
import math

#variable declaration
e   = 1.6*10**-19;       # charge of electron in coulombs
Dn  = 250;              # electron diffusion co-efficient cm**2/s
n1  = 10**18;             # electron conc. in cm**-3
n2  = 7*10**17;         # electron conc. in cm**-3
dx  = 0.10;              # distance in cm

# Calculations
Jdiff   = e*Dn*((n1-n2)/float(dx));    #diffusion current density A/cm**2

#Result
print'Diffusion Current Density = %d '%Jdiff,'A/cm**2';

Diffusion Current Density = 120  A/cm**2


## Example 8.16,Page No:8.43¶

In :
import math

# Variable declaration
e   = 1.6*10**-19;         # charge of electron in coulombs
Eg  = 0.75;                #bandgap energy eV
c   = 3*10**8;             # velocity of light in m
h   = 6.62*10**-34;        # plancks constant in J.s

# Calculations
lamda   = (h*c)/float(Eg*e);     # wavelength in Å

#Result
print'Wavelength at which Ge starts to absorb light = %d '%(lamda*10**10),'Å';

Wavelength at which Ge starts to absorb light = 16550  Å


## Example 8.17,Page No:8.43¶

In :
# import math

#variable declaration
Eg        = 1.35*1.6*10**-19;           #energy in eV
h         = 6.63*10**-34;               #plancks constant in J.s
c         = 3*10**8;                    #velocity in m

#calculation
lamda     = (h*c)/float(Eg);             #wavelength in m

#result
print'cutoff wavelength =%3.2f '%(lamda*10**6),'um';


cutoff wavelength =0.92  um


## Example 8.18,Page No:8.43¶

In :
import math

#variable declaration
h     = 6.62*10**-34            # plancks constant J.s
c     = 3*10**8;                # velocity of light in m
lamda = 1771*10**-9;            # wavelengthg in m
e     = 1.6*10**-19             # charge of electron in coulombs

# Calculations
Eg  = (h*c)/float(lamda*e);      #bandgap energy eV

#Result
print'bandgap energy = %3.3f'%Eg,'eV';

bandgap energy = 0.701 eV


## Example 8.19,Page No:8.45¶

In :
import math

#variable declaration
Nd  = 10**21;             # donar density per in  m**3
H   = 0.6;                # magnetic field in T
J   = 500;                # current density A/m**2
d   = 3*10**-3;           # width in m
e   = 1.6*10**-19         # charge of electron coulombs

#Calculations
Ey  = (J*H)/float(Nd*e);      # field in V/m
vh  = Ey*d;                   # hall voltage V

#Result
print'Hall Voltage = %3.1f '%(vh*10**3),'mV';

Hall Voltage = 5.6  mV


## Example 8.20,Page No:8.46¶

In :
import math

#variable declaration
e   = 1.6*10**-19;        # charge of electron in coulomb
Rh  = -0.0125;          # hall co-efficient
ue  = 0.36;             # electron mobility
E   = 80;               # electric field

# Calculations
n   = -1/float(Rh*e);
J   = n*e*ue*E          # current density in Ampere/m**2

# Result
print'Current density = %d '%J,'Ampere/m**2';

Current density = 2304  Ampere/m**2


## Example 8.21,Page No:8.46¶

In :
import math

#variable declaration
p   = 0.00893;                # resistivity  in ohm-m
Hz  = 0.5;                    # field in weber/m**2
Rh  = 3.66*10**-4;             # hall co-efficient hall coefficient in m**3

# Calculations

u      = Rh/float(p);                                   #mobility of charge cerrier in m**2*(V**-1)*s**-1
theta_h = (math.atan(u*Hz))*(180/float(math.pi));      # hall angle in degrees

# Result
print'Hall angle = %3.4f '%theta_h,'°';

Hall angle = 1.1740  °