Conducting Materials

Example number 6.1, Page number 170

In [1]:
#To calculate the Fermi energy

#importing modules
import math

#Variable declaration
m = 9.1*10**-31;          #mass of electron
vf = 1*10**6;             #Fermi velocity(m/s)
e = 1.6*10**-19;          #conversion factor from J to eV

#Calculation
EF = m*(vf**2)/(2*e);         #Fermi energy(eV)
EF=math.ceil(EF*10**2)/10**2;   #rounding off to 2 decimals

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

Fermi energy is 2.85 eV

Example number 6.2, Page number 170

In [2]:
#To calculate the Fermi energy

#importing modules
import math

#Variable declaration
EF0 = 7.04;               #Fermi energy at 0K(eV)
T = 300;                  #temperature(K)
k = 1.38*10**-23;         #boltzmann constant
e = 1.6*10**-19;          #conversion factor from J to eV

#Calculation
EF = EF0*(1-(((math.pi**2)/12)*(k*T/(EF0*e))**2));            #Fermi energy(eV)
EF=math.ceil(EF*10**5)/10**5;   #rounding off to 5 decimals

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

Fermi energy is 7.03993 eV

Example number 6.3, Page number 171

In [3]:
#To calculate the conductivity

#importing modules
import math

#Variable declaration
d = 2.7*10**3;          #density of Al(kg/m**3)
Mat = 27;               #atomic weight of Al
tow = 10**-14;          #relaxation time(sec)
Na = 6.022*10**23;       #avagadro constant
a = 3*10**3;             #number of free electrons per atom
e = 1.6*10**-19;         #charge of electron
me = 9.1*10**-31;        #mass of electron

#Calculation
n = d*Na*a/Mat;          #concentration of atoms(per m**3)
sigma = n*e**2*tow/me;          #conductivity(ohm m)
sigma = sigma*10**-7;
sigma=math.ceil(sigma*10**4)/10**4;   #rounding off to 4 decimals

#Result
print "conductivity of Al is",sigma,"*10**7 ohm m"

conductivity of Al is 5.0824 *10**7 ohm m

Example number 6.4, Page number 171

In [4]:
#To calculate the Lorentz number

#importing modules
import math

#Variable declaration
sigma = 5.87*10**7;         #electrical conductivity(per ohm m)
K = 390;                    #thermal conductivity(W/mK)
T = 20;                    #temperature(C)

#Calculation
T = T+273;                #temperature(K)
L = K/(sigma*T);          #Lorentz number(W ohm/K**2)

#Result
print "Lorentz number is",L,"W ohm/K**2"

Lorentz number is 2.26756051189e-08 W ohm/K**2

Example number 6.5, Page number 172

In [5]:
#To calculate the electrical conductivity

#importing modules
import math

#Variable declaration
d = 8900;          #density of Cu(kg/m**3)
Mat = 63.5;               #atomic weight of Cu
tow = 10**-14;          #relaxation time(sec)
Na = 6.022*10**23;       #avagadro constant
a = 1*10**3;             #number of free electrons per atom
e = 1.6*10**-19;         #charge of electron
me = 9.1*10**-31;        #mass of electron

#Calculation
n = d*Na*a/Mat;          #concentration of atoms(per m**3)
sigma = n*e**2*tow/me;          #electrical conductivity(ohm m)
sigma = sigma*10**-7;
sigma=math.ceil(sigma*10**4)/10**4;   #rounding off to 4 decimals

#Result
print "electrical conductivity is",sigma,"*10**7 ohm m"

electrical conductivity is 2.3745 *10**7 ohm m

Example number 6.6, Page number 172

In [9]:
#To calculate the relaxation time, mobility of electrons, average drift, velocity and mean free path

#importing modules
import math

#Variable declaration
rho = 1.54*10**-8;         #resistivity(ohm m)
EF = 5.5;                  #fermi energy(eV)
me = 9.1*10**-31;          #mass of electron
epsilon = 100;
e = 1.6*10**-19;           #charge of electron
n = 5.8*10**28;            #concentration of electrons(per m**3)

#Calculation
tow = me/(rho*n*e**2);        #relaxation time(sec)
mew = e*tow/me;               #mobility of electrons(m**2/Vs)
mew = mew*10**3;
vd = e*tow*epsilon/me;        #drift velocity(m/s)
vd=math.ceil(vd*10)/10;   #rounding off to 1 decimal
EF = EF*e;                    #fermi energy((J)
vF = math.sqrt(2*EF/me);           #fermi velocity(m/s)
vf = vF*10**-6;
vf=math.ceil(vf*10**3)/10**3;   #rounding off to 3 decimals
lamda_m = vF*tow;                  #mean free path(m)

#Result
print "relaxation time of electrons is",tow,"sec"
print "mobility of electrons is",mew,"*10**-3 m**2/Vs"
print "drift velocity of electrons is",vd,"m/s"
print "drift velocity given in the book is wrong"
print "fermi velocity of electrons is",vf,"*10**6 m/s"
print "mean free path is",lamda_m,"m"

relaxation time of electrons is 3.97972178683e-14 sec
mobility of electrons is 6.9973130318 *10**-3 m**2/Vs
drift velocity of electrons is 0.7 m/s
drift velocity given in the book is wrong
fermi velocity of electrons is 1.391 *10**6 m/s
mean free path is 5.53462691011e-08 m

Example number 6.7, Page number 174

In [10]:
#To calculate the thermal conductivity

#importing modules
import math

#Variable declaration
rho = 1.72*10**-8;          #electrical resistivity(ohm m)
L = 2.26*10**-8;            #Lorentz number(ohm W/K**2)
T = 27;                     #temperature(C)

#Calculation
T = T+273;                #temperature(K)
K = L*T/rho;          #thermal conductivity(W/mK)

#Result
print "thermal conductivity is",int(K),"W/mK"

thermal conductivity is 394 W/mK

Example number 6.8, Page number 174

In [11]:
#To calculate the Lorentz number

#importing modules
import math

#Variable declaration
sigma = 5.87*10**7;         #electrical conductivity(per ohm m)
K = 390;                    #thermal conductivity(W/mK)
T = 20;                    #temperature(C)

#Calculation
T = T+273;                #temperature(K)
L = K/(sigma*T);          #Lorentz number(W ohm/K**2)

#Result
print "Lorentz number is",L,"W ohm/K**2"

Lorentz number is 2.26756051189e-08 W ohm/K**2

Example number 6.9, Page number 174

In [12]:
#To calculate the value of F(E)

#importing modules
import math

#Variable declaration
E_EF = 0.01;               #energy(eV)
e = 1.6*10**-19;           #conversion factor from eV to J
T = 200;                    #temperature(K)
k = 1.38*10**-23;           #boltzmann constant(J/K)

#Calculation
E_EF = E_EF*e;            #energy(J)
A = E_EF/(k*T);
FofE = 1/(1+(math.exp(A)));           #value of F(E)
FofE=math.ceil(FofE*10**2)/10**2;   #rounding off to 2 decimals

#Result
print "value of F(E) is",FofE

value of F(E) is 0.36

Example number 6.10, Page number 175

In [13]:
#To calculate the electrical conductivity

#importing modules
import math

#Variable declaration
lamda = 4*10**-8;     #mean free path(m)
n = 8.4*10**28;          #density(per m**3)
vthbar = 1.6*10**6;        #average thermal velocity(m/s)
e = 1.6*10**-19;         #charge of electron(c)
m = 9.11*10**-31;        #mass of electron

#Calculation
sigma = n*e**2*lamda/(m*vthbar);        #electrical conductivity(ohm-1 m-1)
sigma = sigma*10**-7;
sigma=math.ceil(sigma*100)/100;   #rounding off to 2 decimals

#Result
print "electrical conductivity is",sigma,"*10**7 ohm-1 m-1"

electrical conductivity is 5.91 *10**7 ohm-1 m-1

Example number 6.11, Page number 176

In [16]:
#To calculate the electrical,thermal conductivity and Lorentz number

#importing modules
import math

#Variable declaration
tow = 10**-14;         #relaxation time(sec)
T = 300;               #temperature(K)
n = 6*10**28;          #electron concentration(per m**3)
e = 1.6*10**-19;         #charge of electron(c)
me = 9.1*10**-31;        #mass of electron
k = 1.38*10**-23;           #boltzmann constant(J/K)

#Calculation
sigma = n*e**2*tow/me;        #electrical conductivity(ohm-1 m-1)
sigmaa = sigma*10**-7;
sigmaa=math.ceil(sigmaa*100)/100;   #rounding off to 2 decimals
K = 3*n*(k**2)*tow*T/(2*me);           #thermal conductivity(W/mK)
K=math.ceil(K*10)/10;   #rounding off to 1 decimal
L = K/(sigma*T);         #Lorentz number(W ohm/K**2)

#Result
print "electrical conductivity is",sigmaa,"*10**7 ohm-1 m-1"
print "thermal conductivity is",K,"W/mK"
print "Lorentz number is",L,"W ohm/K**2"
print "answer for thermal conductivity and Lorentz number given in the book are wrong"

electrical conductivity is 1.69 *10**7 ohm-1 m-1
thermal conductivity is 56.6 W/mK
Lorentz number is 1.11775173611e-08 W ohm/K**2
answer for thermal conductivity and Lorentz number given in the book are wrong

Example number 6.12, Page number 177

In [17]:
#To calculate the relaxation time

#importing modules
import math

#Variable declaration
n = 5.8*10**28;          #electron concentration(per m**3)
e = 1.6*10**-19;         #charge of electron(c)
m = 9.1*10**-31;        #mass of electron
rho = 1.54*10**-8;       #resistivity of metal(ohm m)

#Calculation
tow = m/(n*rho*e**2);         #relaxation time(sec)

#Result
print "relaxation time is",tow,"sec"
print "answer given in the book is wrong"

relaxation time is 3.97972178683e-14 sec
answer given in the book is wrong

Example number 6.13, Page number 177

In [18]:
#To calculate the relaxation time, mobility of electrons, drift velocity 

#importing modules
import math

#Variable declaration
rho = 1.54*10**-8;         #resistivity(ohm m)
E = 1;                  #electric field(V/cm)
me = 9.1*10**-31;          #mass of electron
e = 1.6*10**-19;           #charge of electron
n = 5.8*10**28;            #concentration of electrons(per m**3)

#Calculation
E = E*10**2;            #electric field(V/m)
tow = me/(rho*n*e**2);        #relaxation time(sec)
vd = e*E*tow/me;        #drift velocity(m/s)
vd=math.ceil(vd*10)/10;   #rounding off to 1 decimal
mew = vd/E;               #mobility of electrons(m**2/Vs)
mew = mew*10**2;

#Result
print "relaxation time of electrons is",tow,"sec"
print "drift velocity of electrons is",vd,"m/s"
print "mobility of electrons is",mew,"*10**-2 m**2/Vs"

relaxation time of electrons is 3.97972178683e-14 sec
drift velocity of electrons is 0.7 m/s
mobility of electrons is 0.7 *10**-2 m**2/Vs

Example number 6.14, Page number 178

In [19]:
#To calculate the drift velocity 

#importing modules
import math

#Variable declaration
T = 300;                   #temperature(K)
l = 2;                     #length of wire(m)
R = 0.02;                  #resistance(ohm)
I = 15;                    #current(amp)
mew = 4.3*10**-3;          #mobility(m**2/Vs)

#Calculation
V = I*R;                   #voltage drop(V)
E = V/l;                   #electric field(V/m)
vd = mew*E;                #drift velocity(m/s)
vd = vd*10**3;
vd=math.ceil(vd*100)/100;   #rounding off to 2 decimals

#Result
print "drift velocity of electrons is",vd,"*10**-3 m/s"

drift velocity of electrons is 0.65 *10**-3 m/s

Example number 6.15, Page number 179

In [24]:
#To calculate the Fermi energy and Fermi temperature

#importing modules
import math

#Variable declaration
vf = 0.86*10**6;          #fermi velocity(m/s)
m = 9.1*10**-31;          #mass of electron(kg)
e = 1.6*10**-19;          #charge of electron(C)
k = 1.38*10**-23;         #boltzmann constant

#Calculation
EF = m*vf**2/(2*e);       #fermi energy(eV)
EF=math.ceil(EF*100)/100;   #rounding off to 2 decimals
TF = EF*e/k;                  #fermi temperature(K)

#Result
print "Fermi energy is",EF,"eV"
print "Fermi temperature is",int(TF),"K"
print "answer for fermi temperature given in the book is wrong due to rounding off the value of EF"

Fermi energy is 2.11 eV
Fermi temperature is 24463 K
answer for fermi temperature given in the book is wrong due to rounding off the value of EF

Example number 6.16, Page number 179

In [25]:
#To calculate the Fermi velocity

#importing modules
import math

#Variable declaration
TF = 2460;            #fermi temperature(K)
m = 9.11*10**-31;          #mass of electron(kg)
k = 1.38*10**-23;         #boltzmann constant

#Calculation
vF = math.sqrt(2*k*TF/m);          #fermi velocity(m/s)
vF = vF*10**-5;
vF=math.ceil(vF*10**3)/10**3;   #rounding off to 3 decimals

#Result
print "Fermi velocity is",vF,"*10**5 m/s"

Fermi velocity is 2.731 *10**5 m/s
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