Chapter 11 - Conductors, Semiconductors and Superconductors

Example 3 - pg 357

In [1]:
#pg 357
#calculate the Fermi energy
#Given :
n =8.48*10**28; # number of conduction electrons / m**3
#calculations
Ef = 3.65*10**-19*(n**0.6667);#Fermi energy in eV
#results
print "Fermi energy (eV) =  ",round(Ef,2)

Fermi energy (eV) =   7.06

Example 4 - pg 358

In [2]:
#pg 358
#calculate the Fraction of electrons which are excited
#Given :
Ef = 7.04 ; # Ef for copper in eV
kT = 0.026; # kT value at room temperature in eV
F = (3/2.)*(0.026/7.04); # Fraction of electrons 
#results
print "Fraction of electrons which are excited are",round(F,4),"or",round(F*100,2),"percentage."

Fraction of electrons which are excited are 0.0055 or 0.55 percentage.

Example 6 - pg 370

In [3]:
#pg 370
#calculate the resistivity of Ge, Si
#Given :
ni1 = 2.5*10**19; # per m**3 for Ge
ni2 = 1.5*10**16; # per m**3 for Si
mu_e1 = 0.38; # mobility of free electrons for Ge in m**2/Vs
mu_h1 = 0.18; #mobility of holes for Ge in m**2/Vs
mu_e2 = 0.13;#mobility of free electrons for Si in m**2/Vs
mu_h2 = 0.05;#mobility of holes for Si in m**2/Vs
e = 1.6*10**-19; # charge of an electron in C
#calculations
sigma1 = ni1*e*(mu_e1 + mu_h1); # intrinsic conductivity in mho m**-1 for Ge
sigma2 = ni2*e*(mu_e2 + mu_h2);# intrinsic conductivity in mho m**-1 for Si
rho1 = 1/sigma1; #intrinsic resistivity in ohm m for Ge
rho2 = 1/sigma2;#intrinsic resistivity in ohm m for Si
#results
print "Resistivity of Ge (ohm m) =  ",round(rho1,3)
print "Resistivity of Si",round(rho2*10**-3,3),"x 10^3 ohm m"

Resistivity of Ge (ohm m) =   0.446
Resistivity of Si 2.315 x 10^3 ohm m

Example 7 - pg 370

In [5]:
#pg 370
#calculate the vaue of F1, F2, F3
#Given :
#Fraction F = n/N
import math
from math import exp
Eg = 0.72; # Energy gap in eV
k = 0.026/300;# kT value at 300 K , so k = kT/T
T1 = 30.; # Temperature in K
T2 = 300.; #Temperature in K
T3 = 1210.;#Temperature in K
#Fraction of electrons : n/N = exp(-Eg/2*k*T)
#calculations
F1 = exp(-Eg/(2*k*T1));
F2 = exp(-Eg/(2*k*T2));
F3 = exp(-Eg/(2*k*T3));
#results
print " For 30 K , n/N  =",round(F1*10**61,2)," x 10^-61"
print " For 300 K , n/N  =",round(F2*10**7,1)," x 10^-7"
print " For 1210 K , n/N  = ",round(F3,4)

 For 30 K , n/N  = 7.36  x 10^-61
 For 300 K , n/N  = 9.7  x 10^-7
 For 1210 K , n/N  =  0.0323

Example 8 - pg 371

In [6]:
#pg 371
#calculate the n/N ratio for Germanium, Silicon and diamond
#Given :
import math
from math import exp
Eg1= 0.72; #Energy gap for Germanium in eV
Eg2= 1.10; #Energy gap for Silicon in eV
Eg3= 5.6; #Energy gap for diamond in eV
#Fraction of electron : n/N = exp(-Eg/(2*k*T)) , k*T = 0.026 eV
#calculations
F1 = exp(-Eg1/(2*0.026)); # For Germanium
F2 = exp(-Eg2/(2*0.026)); # For Silicon
F3 = exp(-Eg3/(2*0.026)); # For diamond
#results
print "For Germanium , n/N =",round(F1*10**7,1),"x 10**-7"
print "For Silicon , n/N =",round(F2*10**10,1),"x 10**-10"
print "For diamond, n/N =",round(F3*10**47,1),"x 10**-47"

For Germanium , n/N = 9.7 x 10**-7
For Silicon , n/N = 6.5 x 10**-10
For diamond, n/N = 1.7 x 10**-47

Example 9 - pg 371

In [7]:
#pg 371
#calculate the Temperature
#Given :
D = 5*10**28; # density of atoms in silicon per m**3
C = 2.0*10**8; #donor concentration
#calculations
ND = D/C; # donor atoms density per m**3
# ND = 4.82*10**21*T**(3/2)
T = (ND/(4.82*10**21))**(2./3.);
#results
print "Temperature (K) = ",round(T,2)

Temperature (K) =  0.14

Example 10 - pg 372

In [9]:
#pg 372
#calculate the Nd and n,p
#Given :
import math
from math import exp
Ecd = 0.045; # Ec-Ed in eV
Ecf = 0.035; # Ec-Ef in eV
Efd = 0.01;# Ef-Ed in eV
Ev = 0; # in eV
Ef = 1.065; # in eV
me = 9.1*10**-31;# electron mass in kg
m_e = 0.31*me; # free electron mass
m_h = 0.38*me;# hole mass
kT = 0.026; # kT value at room temperature
h = 6.625*10**-34; # planck's constant in Js
#calculations
Nc = 2*((2*math.pi*m_e*kT*1.6*10**-19)/(h**2))**(3/2.); # per m**3
Nv = 2*((2*math.pi*m_h*kT*1.6*10**-19)/(h**2))**(3/2.); # per m**3
#(a)
# Nc*exp[-(Ec-Ef)/kT] = Nd*[1 - 1/(1+ exp[(Ed-Ef)/kT])]
#Ed - Ef = -(Ef-Ed) = - Efd
Nd = (Nc*exp(-Ecf/kT))/(1 - (1/(1+exp(-Efd/kT)))); # per m**3
#(b)
Nd_plus = Nd*(1 - (1/(1 + exp(-Efd/kT)))); # per m**3
#(c)
n = Nc*exp(-Ecf/kT); # per m**3
#(d)
p = Nv*exp((Ev-Ef)/kT);# per m**3
#results
print "Nd =",round(Nd*10**-24,1),"x 10^24 / m^3"
print "Nd_plus =",round(Nd_plus*10**-24,2),"x 10^24 / m^3"
print "n =",round(n*10**-24,2),"x 10^24 / m^3"
print "p =",round(p*10**-6,1),"x 10^6 / m^3"

Nd = 2.8 x 10^24 / m^3
Nd_plus = 1.13 x 10^24 / m^3
n = 1.13 x 10^24 / m^3
p = 9.6 x 10^6 / m^3

Example 11 - pg 374

In [14]:
#pg 374
#calculate the Major and Minor carrier concentration, resistivity
#Given :
ni = 1.5*10**16; # ni for Si in m**-3
mue = 0.135; # mobility of free electrons in m**2/Vs
muh = 0.048; # mobility of holes in m**2/Vs
Nd = 10**21; # phosphorus atoms/m**3
e = 1.6*10**-19;# charge of an electron in C
#calculations
#(a) 
n = Nd; # electrons/m**3
#(b)
p = ni**2/Nd; # holes/m**3
#(c)
sigma = e*(n*mue + p*muh); # conductivity in mho m**-1
rho = 1/sigma; # resistivity in ohm m
#results
print "Major carrier concentration  =",n*10**-21,"x 10^21 electrons/m^3"
print "Minor carrier concentration  =",p*10**-11,"x 10^11 holes/m^3"
print "Resistivity (ohm m) = ",round(rho,3)

Major carrier concentration  = 1.0 x 10^21 electrons/m^3
Minor carrier concentration  = 2.25 x 10^11 holes/m^3
Resistivity (ohm m) =  0.046

Example 12 - pg 375

In [15]:
#pg 375
#Compare the electrical conductivity of silicon 473 and silicon 300 K
import math
#Given :
Eg = 1.1;# Energy gap in eV
T1 = 300. ;# Temperature in K
T2 = 473.; # Temperature in K (273+ 200 = 473 K)
k = 8.62*10**-5 ; # in eV
#calculations
# sigma = A*exp(-Eg/(2*k*T))
#Ratio = sigma_473/sigma_300
Ratio = math.exp((-Eg/(2*k))*((1/T2)-(1/T1)));
#results
print "Thus, sigma_473 is",round(Ratio,0),"times  sigma_300"

Thus, sigma_473 is 2390.0 times  sigma_300

Example 13 - pg 387

In [12]:
#pg 387
#calculate the wavelength of Ge, Si and GaAs
#Given :
Eg1 = 0.72; # Energy gap for Ge in eV
Eg2 = 1.1; # Energy gap for Si in eV
Eg3 = 1.32; # Energy gap for GaAs in eV
# lambda = c/v = (c*h)/Eg   or lambda(A) = 12422/Eg (eV)
#calculations
lambda1 = 12422/Eg1; # wavelength in A  (Ge)
lambda2 = 12422/Eg2; # wavelength in A  (Si)
lambda3 = 12422/Eg3; # wavelength in A  (GaAs)
#results
print "Wavelength for Ge (A) = ",round(lambda1,1)
print "Wavelength for Si (A) = ",round(lambda2,1)
print "Wavelength for GaAs (A) = ",round(lambda3,2)

Wavelength for Ge (A) =  17252.8
Wavelength for Si (A) =  11292.7
Wavelength for GaAs (A) =  9410.61

Example 14 - pg 388

In [13]:
#pg 388
#calculate the times conductivity increased
#Given :
sigma = 4*10**-4; # conductivity at room temperature in ohm**-1 m**-1
M = 28.1; # atomic weight in kg/kmole
d = 2330; # density in kg/m**3
dop = 10**8 ;# doping per 10**8 silicon atoms
e = 1.6*10**-19; # charge of an electron in C
mue = 0.135; # mobility of free electrons for silicon in m**2/Vs
Na = 6.023*10**26 ; # Avagadro's constant in atoms/kmole
#calculations
N = (d*Na)/M; #atoms/m**3
Nd = N/dop; # per m**3
n = Nd; # electron concentration / m**3
sigma1 = n*e*mue; # conductivity in ohm**-1 m**-1
t = sigma1/sigma; # number of times the conductivity  increased
#results
print "Conductivity increased",round(t,0), "times"
print 'Result obtained differs from that in textbook, because approximate value for sigma1 was considered.'

Conductivity increased 26968.0 times
Result obtained differs from that in textbook, because approximate value for sigma1 was considered.