Semiconducting Materials

Example number 7.1, Page number 208

In [1]:
#To calculate the approximate donor binding energy

#importing modules
import math

#Variable declaration
me = 9.11*10**-31;          #mass of electron(kg)
epsilon_r = 13.2;  
epsilon0 = 8.85*10**-12;
h = 6.63*10**-34;
e = 1.6*10**-19;          #charge of electron(C)

#Calculation
m_nc = 0.067*me;
E = m_nc*e**4/(8*(epsilon0*epsilon_r*h)**2);       #energy(J)
E = E/e;                #energy(eV)
E = math.ceil(E*10**5)/10**5;   #rounding off to 5 decimals
E_meV = E*10**3;        #energy(meV)

#Result
print "donor binding energy is",E,"eV or",E_meV,"meV"

donor binding energy is 0.00521 eV or 5.21 meV

Example number 7.2, Page number 208

In [2]:
#To calculate the equilibrium hole concentration

#importing modules
import math
import numpy as np

#Variable declaration
Nd = 10**16;         #donor concentration(atoms/cm**3)
ni = 1.5*10**10;        #concentration(per cm**3)
T = 300;                #temperature(K)
kT = 0.0259;

#Calculation
n0 = Nd;           #for Nd>>ni, assume n0=Nd
p0 = ni**2/n0;        #equilibrium hole concentration(per cm**3)
p0 = p0*10**-4;
EF_Ei = kT*np.log(n0/ni);
EF_Ei = math.ceil(EF_Ei*10**4)/10**4;   #rounding off to 4 decimals


#Result
print "equilibrium hole concentration is",p0,"*10**4 per cm**3"
print "value of EF-Ei is",EF_Ei,"eV"

equilibrium hole concentration is 2.25 *10**4 per cm**3
value of EF-Ei is 0.3474 eV

Example number 7.3, Page number 209

In [3]:
#To calculate the resistivity of sample

#importing modules
import math

#Variable declaration
e = 1.6*10**-19;          #charge of electron(C)
Nd = 10**14;              #donor density(atoms/cm**3)
mew_n = 3900;

#Calculation
n = Nd;
sigma = n*e*mew_n;        #conductivity(ohm-1 cm-1)
rho = 1/sigma;            #resistivity(ohm cm)
rho = math.ceil(rho*100)/100;   #rounding off to 2 decimals


#Result
print "resistivity of sample is",rho,"ohm cm"

resistivity of sample is 16.03 ohm cm

Example number 7.4, Page number 209

In [4]:
#To calculate the resistivity, Hall coefficient and Hall voltage

#importing modules
import math

#Variable declaration
e = 1.6*10**-19;          #charge of electron(C)
n0 = 5*10**16;              #donor density(atoms/cm**3)
mew_0 = 800;
Ix = 2;             #current(mA)
Bz = 5*10**-5;
d = 200;            #thickness(micrometre)

#Calculation
Ix = Ix*10**-3;          #current(A)
d = d*10**-4;            #thickness(m)
sigma = e*n0*mew_0;        #conductivity(ohm-1 cm-1)
rho = 1/sigma;            #resistivity(ohm cm)
rho = math.ceil(rho*10**4)/10**4;   #rounding off to 4 decimals
RH = -1/(e*n0);          #Hall coefficient(cm**3/C)
VH = Ix*Bz*RH/d;         #Hall voltage(V)
VH = VH*10**5;


#Result
print "resistivity of sample is",rho,"ohm cm"
print "Hall coefficient is",RH,"cm**3/C"
print "Hall voltage is",VH,"*10**-5 V"

resistivity of sample is 0.1563 ohm cm
Hall coefficient is -125.0 cm**3/C
Hall voltage is -62.5 *10**-5 V

Example number 7.5, Page number 210

In [19]:
#To calculate the intrinsic carrier concentration, intrinsic conductivity and resistivity

#importing modules
import math
from __future__ import division

#Variable declaration
T = 300;          #temperature(K)
mew_n = 0.4;      #electron mobility(m**2/Vs)
mew_p = 0.2;      #hole mobility(m**2/Vs)
Eg = 0.7;         #band gap(eV)
me = 9.11*10**-31;       #mass of electron(kg)
k = 1.38*10**-23;        #boltzmann constant
T = 300;                 #temperature(K)
h = 6.625*10**-34;
kT = 0.0259;
e = 1.6*10**-19;          #charge of electron(C)

#Calculation
mn_star = 0.55*me;           #electron effective mass(kg)
mp_star = 0.37*me;           #hole effective mass(kg)
a = (2*math.pi*k*T/(h**2))**(3/2);
b = (mn_star*mp_star)**(3/4);
c = math.exp(-Eg/(2*kT));
ni = 2*a*b*c;     #intrinsic concentration(per m**3)
sigma = ni*e*(mew_n+mew_p);               #intrinsic conductivity(per ohm m)
sigma = math.ceil(sigma*10**4)/10**4;   #rounding off to 4 decimals
rho = 1/sigma;                            #intrinsic resistivity(ohm m)
rho = math.ceil(rho*10**4)/10**4;   #rounding off to 4 decimals

#Result
print "intrinsic concentration is",ni,"per m**3"
print "intrinsic conductivity is",sigma,"per ohm m"
print "intrinsic resistivity is",rho,"ohm m"
print "answers given in the book are wrong"

intrinsic concentration is 1.02825111151e+19 per m**3
intrinsic conductivity is 0.9872 per ohm m
intrinsic resistivity is 1.013 ohm m
answers given in the book are wrong

Example number 7.6, Page number 211

In [24]:
#To calculate the Fermi energy

#importing modules
import math
import numpy as np
from __future__ import division

#Variable declaration
Nd = 10**16;           #donor concentration(per cm**3)
ni = 1.45*10**10;         #concentration(per cm**3)
kT = 0.0259;

#Calculation
#ni = Nc*math.exp(-(Ec-Ei)/kT)
#Nd = Nc*(math.exp(-(Ec-Efd)/kT)
#dividing Nd/ni we get 
EFd_Ei = kT*np.log(Nd/ni);
EFd_Ei = math.ceil(EFd_Ei*10**4)/10**4;   #rounding off to 4 decimals

#Result
print "Fermi energy is",EFd_Ei,"eV"

Fermi energy is 0.3482 eV

Example number 7.7, Page number 211

In [20]:
#To calculate the resistance

#The given information in the question is not sufficient to solve the entire problem. And the problem is completely wrong in the book

Example number 7.8, Page number 212

In [38]:
#To calculate the forbidden energy gap

#importing modules
import math
import numpy as np
from __future__ import division

#Variable declaration
T = 300;          #temperature(K)
mew_n = 0.36;      #electron mobility(m**2/Vs)
mew_p = 0.17;      #hole mobility(m**2/Vs)
rho = 2.12;         #resistivity(ohm m)
me = 9.11*10**-31;       #mass of electron(kg)
kT = 0.0259;
h = 6.625*10**-34;
k = 1.38*10**-23;        #boltzmann constant
e = 1.6*10**-19;          #charge of electron(C)

#Calculation
mn_star = 0.55*me;           #electron effective mass(kg)
mp_star = 0.37*me;           #hole effective mass(kg)
sigma = 1/rho;               #conductivity(per ohm m)
sigma = math.ceil(sigma*10**3)/10**3;   #rounding off to 3 decimals
ni = sigma/(e*(mew_n+mew_p));          #concentration of electrons(per m**3)
a = (2*math.pi*kT/(h**2))**(3/2);
Nc = 2*a*(mn_star**(3/2)); 
Nv = 2*a*(mp_star**(3/2)); 
b = (Nc*Nv)**(1/2);
Eg = 2*kT*np.log(b/ni);

#Result
print "forbidden energy gap is",Eg,"eV"
print "answer given in the book is wrong"

forbidden energy gap is 4.09465494989 eV
answer given in the book is wrong

Example number 7.9, Page number 213

In [10]:
#To calculate the conductivity

#importing modules
import math

#Variable declaration
ni = 2.4*10**19;       #concentration(per m**3)
mew_n = 0.39;      #electron mobility(m**2/Vs)
mew_p = 0.19;      #hole mobility(m**2/Vs)
e = 1.6*10**-19;          #charge of electron(C)

#Calculation
sigma = ni*e*(mew_n+mew_p);         #conductivity(per ohm m)
sigma = math.ceil(sigma*10**3)/10**3;   #rounding off to 3 decimals

#Result
print "conductivity of sample is",sigma,"ohm-1 m-1"

conductivity of sample is 2.228 ohm-1 m-1

Example number 7.10, Page number 214

In [39]:
#To calculate the new position of Fermi level

#importing modules
import math
from __future__ import division

#Variable declaration
Ec = 0.3;          #initial position(eV)
T1 = 300;           #initial temperature(K)
T2 = 330;           #increased temperature

#Calculation
#Ec/T1 = Ec_EF330/T2
Ec_EF330 = Ec*T2/T1;        #new position of Fermi level(eV)

#Result
print "new position of Fermi level is",Ec_EF330,"eV"

new position of Fermi level is 0.33 eV

Example number 7.11, Page number 214

In [41]:
#To calculate the concentration in conduction band

#importing modules
import math
from __future__ import division

#Variable declaration
k = 1.38*10**-23;         #boltzmann constant
T = 300;           #temperature(K)
me = 9.1*10**-31;        #mass of electron(kg)
h = 6.63*10**-34;        #planck's constant
Ec_Ev = 1.1;             #energy gap(eV)
e = 1.6*10**-19;         #charge of electron(C)

#Calculation
me_star = 0.31*me;
A = (2*math.pi*k*T*me_star/(h**2))**(3/2);
B = math.exp(-(Ec_Ev*e)/(2*k*T));
ni = A*B;                  #concentration in conduction band(per m**3)

#Result
print "intrinsic electron concentration is",ni,"per m**3"
print "answer given in the book is wrong"

intrinsic electron concentration is 1.26605935487e+15 per m**3
answer given in the book is wrong

Example number 7.12, Page number 214

In [14]:
#To calculate the drift mobility of electrons

#importing modules
import math

#Variable declaration
RH = 0.55*10**-10;          #Hall coefficient(m**3/As)
sigma = 5.9*10**7;          #conductivity(ohm-1 m-1)

#Calculation
mew = RH*sigma;              #drift mobility(m**2/Vs)
mew = mew*10**3;
mew = math.ceil(mew*10**2)/10**2;   #rounding off to 2 decimals

#Result
print "drift mobility of electrons is",mew,"*10**-3 m**2/Vs"

drift mobility of electrons is 3.25 *10**-3 m**2/Vs

Example number 7.13, Page number 215

In [1]:
#To calculate the concentration and average number of electrons

#importing modules
import math
from __future__ import division

#Variable declaration
A = 6.022*10**23;           #avagadro constant
d = 8.96*10**-9;            #density(kg/m**3)
n = 9.932*10**14;           #no. of free electrons per atom
sigma = 5.9*10**7;          #conductivity(ohm-1 m-1)
e = 1.6*10**-19;            #electron charge(C)
mew = 3.2*10**-3;           #drift mobility(m**2/Vs)
w = 63.5;                   #atomic weight of Cu(kg)

#Calculation
ni = sigma/(mew*e);         #conductivity(per m**3)
N = A*d*n/w;                #concentration of free electrons in pure Cu
AN = ni/N;                  #average number of electrons contributed per Cu atom

#Result
print "concentration of free electrons in pure Cu is",N,"per m**3"
print "average number of electrons contributed per Cu atom is",int(AN)

concentration of free electrons in pure Cu is 8.43940339906e+28 per m**3
average number of electrons contributed per Cu atom is 1

Example number 7.14, Page number 215

In [16]:
#To calculate the charge carrier density and electron mobility

#importing modules
import math
from __future__ import division

#Variable declaration
RH = 3.66*10**-11;          #hall coefficient(m**3/As)
e = 1.6*10**-19;            #electron charge(C)
sigma = 112*10**7;          #conductivity(ohm-1 m-1)

#Calculation
n = 1/(e*RH);               #charge carrier density(per m**3)
mew_n = sigma/(n*e);        #electron mobility(m**2/As)
mew_n = math.ceil(mew_n*10**3)/10**3;   #rounding off to 3 decimals

#Result
print "charge carrier density is",n,"per m**3"
print "electron mobility is",mew_n,"m**2/As"
print "answers given in the book are wrong"

charge carrier density is 1.70765027322e+29 per m**3
electron mobility is 0.041 m**2/As
answers given in the book are wrong

Example number 7.15, Page number 216

In [17]:
#To calculate the magnitude of Hall voltage

#importing modules
import math
from __future__ import division

#Variable declaration
e = 1.6*10**-19;            #electron charge(C)
B = 1.5;                    #magnetic field(T)
I = 50;                     #current(Amp)
n = 8.4*10**28;             #free electron concentration(per m**3)
d = 0.2;                    #thickness of slab(cm)

#Calculation
d = d*10**-2;               #thickness of slab(m)
VH = B*I/(n*e*d);           #hall voltage(V)

#Result
print "magnitude of Hall voltage is",VH,"V"
print "answer given in the book is wrong"

magnitude of Hall voltage is 2.79017857143e-06 V
answer given in the book is wrong

Example number 7.16, Page number 216

In [18]:
#To calculate the resistance of intrinsic Ge rod

#importing modules
import math
from __future__ import division

#Variable declaration
e = 1.6*10**-19;            #electron charge(C)
n = 2.5*10**19;             #free electron concentration(per m**3)
mew_n = 0.39;               #electron mobility(m**2/Vs)
mew_p = 0.19;               #hole mobility(m**2/Vs)
L = 1;                      #length(cm)
w = 1;                      #width(mm)
t = 1;                      #thickness(mm)

#Calculation
L = L*10**-2;               #length(m)
w = w*10**-3;               #width(m)
t = t*10**-3;               #thickness(m)
A = w*t;                    #area(m**2)
sigma = n*e*(mew_n+mew_p);         #conductivity(ohm-1 m-1)
R = L/(sigma*A);                   #resistance(ohm)

#Result
print "resistance of intrinsic Ge rod is",int(R),"ohm"

resistance of intrinsic Ge rod is 4310 ohm

Example number 7.17, Page number 216

In [19]:
#To determine the position of Fermi level

#importing modules
import math
import numpy as np
from __future__ import division

#Variable declaration
e = 1.6*10**-19;            #electron charge(C)
Eg = 1.12;                  #band gap(eV)
me = 1;
mn_star = 0.12*me;               #electron mobility(m**2/Vs)
mp_star = 0.28*me;               #hole mobility(m**2/Vs)
k = 1.38*10**-23;                #boltzmann constant
T = 300;                         #temperature

#Calculation
a = mp_star/mn_star;
EF = (Eg/2)+((3*k*T/(4*e))*np.log(a));
EF = math.ceil(EF*10**3)/10**3;   #rounding off to 3 decimals

#Result
print "position of Fermi level is",EF,"eV"

position of Fermi level is 0.577 eV

Example number 7.18, Page number 217

In [20]:
#To calculate the electrical conductivity

#importing modules
import math

#Variable declaration
e = 1.6*10**-19;            #electron charge(C)
ni = 1.5*10**16;            #intrinsic carrier density(per m**3)
mew_n = 0.13;               #electron mobility(m**2/Vs)
mew_p = 0.05;               #hole mobility(m**2/Vs)

#Calculation
sigma = ni*e*(mew_n+mew_p);        #electrical conductivity
sigma = sigma*10**4;

#Result
print "electrical conductivity is",sigma,"*10**-4 ohm-1 m-1"

electrical conductivity is 4.32 *10**-4 ohm-1 m-1

Example number 7.19, Page number 217

In [6]:
#To calculate the intrinsic resistivity

#importing modules
import math
from __future__ import division

#Variable declaration
e = 1.6*10**-19;            #electron charge(C)
ni = 2.15*10**-13;            #intrinsic carrier density(per cm**3)
mew_n = 3900;               #electron mobility(cm**2/Vs)
mew_p = 1900;               #hole mobility(cm**2/Vs)

#Calculation
sigmai = ni*e*(mew_n+mew_p);        #electrical conductivity(ohm-1 cm-1)
rhoi = 1/sigmai;                    #intrinsic resistivity(ohm cm)

#Result
print "intrinsic resistivity is",rhoi,"ohm cm"
print "answer given in the book is wrong"

intrinsic resistivity is 5.01202886929e+27 ohm cm
answer given in the book is wrong

Example number 7.20, Page number 217

In [22]:
#To calculate the electrical conductivity

#importing modules
import math

#Variable declaration
e = 1.6*10**-19;            #electron charge(C)
ni = 2.1*10**19;            #intrinsic carrier density(per m**3)
mew_n = 0.4;               #electron mobility(m**2/Vs)
mew_p = 0.2;               #hole mobility(m**2/Vs)

#Calculation
sigma = ni*e*(mew_n+mew_p);        #electrical conductivity

#Result
print "intrinsic resistivity is",sigma,"ohm-1 m-1"

intrinsic resistivity is 2.016 ohm-1 m-1

Example number 7.21, Page number 218

In [23]:
#To calculate the Hall coefficient and electron mobility

#importing modules
import math

#Variable declaration
e = 1.6*10**-19;            #electron charge(C)
V = 1.35;                   #voltage supply(V)
I = 5;               #current(mA)
b = 5;               #breadth(mm)
d = 1;               #thickness(mm)
L = 1;               #length(cm)
H = 0.45;            #magnetic field(Wb/m**2)
Vy =20;              #Hall voltage(mV)

#Calculation
Vy = Vy*10**-3;       #Hall voltage(V)
L = L*10**-2;         #length(m)
d = d*10**-3;         #thickness(m)
b = b*10**-3;         #breadth(m)
I = I*10**-3;         #current(A)
R = V/I;              #resistance(ohm)
A = b*d;              #area(m**2)
rho = R*A/L;          #resistivity(ohm m)
Ey = Vy/d;            #Hall field(V/m)
Jx = I/A;           
a = Ey/(H*Jx);         #current density(m**3/C).Here a is 1/ne 
RH = a;               #Hall coefficient(m**3/C)
RH = math.ceil(RH*10**4)/10**4;   #rounding off to 4 decimals
mew_n = RH/rho;       #electron mobility(m**2/Vs)
mew_n = math.ceil(mew_n*10**2)/10**2;   #rounding off to 2 decimals

#Result
print "Hall coefficient is",RH,"m**3/C"
print "electron mobility is",mew_n,"m**2/Vs"

Hall coefficient is 0.0445 m**3/C
electron mobility is 0.33 m**2/Vs

Example number 7.22, Page number 219

In [24]:
#To calculate the Hall potential difference

#importing modules
import math

#Variable declaration
e = 1.6*10**-19;            #electron charge(C)
Ix = 200;               #current(A)
Bz = 1.5;               #magnetic field(Wb/m**2)
p = 8.4*10**28;         #electron concentration(per m**3)
d = 1;               #thickness(mm)

#Calculation
d = d*10**-3;         #thickness(m)
VH = Ix*Bz/(e*p*d);         #Hall potential(V)
VH = VH*10**6;              #Hall potential(micro V)

#Result
print "Hall potential is",int(VH),"micro V"

Hall potential is 22 micro V
In [ ]: