Chapter1:ELECTRONS IN SOLIDS

Ex1.1:pg-06

In [6]:
A1=27#initializing value of atomic mass of alluminium
AV = 6.023*10**23 #initializing value of avagadro number
N = 13 #initializing value of number of electrons of alluminium per atom
P1 = 2.7 #initializing value of density of alluminium
E1=AV*(N*P1/A1)
print"Electrons density of alluminium,n(Al)= ","{:.2e}".format(E1)," cm**-3"
A2=12 #initializing value of atomic mass of carbon
N1 = 6 #initializing value of number of electrons of carbon per atom
P2 = 3.515 #initializing value of density of carbon
E2=AV*(N1*P2/A2)
print"Electrons density of carbon,n(C)= ","{:.3e}".format(E2)," cm**-3"
A3=28 #initializing value of atomic mass of silicon
N2 = 14 #initializing value of number of electrons of silicon per atom
P3 = 2.33 #gcm**-3, initializing value of density of silicon
E3=AV*(N2*P3/A3)
print"Electrons density of silicon,n(Si)=","{:.2e}".format(E3)," cm**-3"
#using Drudes approach
print"using Drudes approach"
Zc1=3 ##initializing value of valence electron of alluminium atom
E4=AV*(Zc1*P1/A1)
print"Electrons density of alluminium,n(Al)=","{:.1e}".format(E4)," cm**-3"
Zc2=4 #initializing value of valence electron of carbon atom
E5=AV*(Zc2*P2/A2)
print"Electrons density of carbon,n(C)=","{:.2e}".format(E5)," cm**-3"
Zc3=4 #initializing value of valence electron of silicon atom
E6=AV*(Zc3*P3/A3)
print"Electrons density of silicon,n(Si)=","{:.1e}".format(E6)," cm**-3"

Electrons density of alluminium,n(Al)=  7.83e+23  cm**-3
Electrons density of carbon,n(C)=  1.059e+24  cm**-3
Electrons density of silicon,n(Si)= 7.02e+23  cm**-3
using Drudes approach
Electrons density of alluminium,n(Al)= 1.8e+23  cm**-3
Electrons density of carbon,n(C)= 7.06e+23  cm**-3
Electrons density of silicon,n(Si)= 2.0e+23  cm**-3

Ex1.2:pg-13

In [9]:
# silicon has diomond structure which is made up of FCC lattice 
N=4.0 #initializing value of number of points per cube of volume
print"N=4"
A = 5.43*10**-8 #"cm**-1" #initializing value of lattice constant of silicon
D = 2.0 #initializing value of number of silicon atoms per lattice point
E1 = N*D/A**3
print"number density of silicon,N(Si)= ","{:.3e}".format(E1)," atomscm**-3"
#for gallium in GaAs there is 1 Ga atom and 1 As atom as per lattice point , it also has fcc structure
A1 = 5.65*10**-8 #initializing value of lattice constant of gallium
D1 = 1.0 #initializing value of number of gallium atoms per lattice point
E2 = N*D1/A1**3
print"number density of gallium atoms,N(Ga)=""{:.2e}".format(E2)," atomscm**-3"

N=4
number density of silicon,N(Si)=  4.997e+22  atomscm**-3
number density of gallium atoms,N(Ga)=2.22e+22  atomscm**-3

Ex1.3:pg-14

In [11]:
# silicon has diomond structure which is made up of FCC lattice 

N=4.0 #initializing value of number of points per cube of volume
A = 5.43*10**-8 #"cm**-3" #initializing value of lattice constant of silicon
D = 2.0 #initializing value of number of silicon atoms per lattice point
E1 = N*D/A**3
print"number density of silicon,Nsi = ""{:.2e}".format(E1)," atomscm**-3"

#for gallium in GaAs there is 1 Ga atom and 1 As atom as per lattice point , it also has fcc structure

A1 = 5.65*10**-8 #initializing value of lattice constant of gallium
D1 = 1.0 #initializing value of number of gallium atoms per lattice point
E2 = N*D1/A1**3
print"number density of gallium atoms,NGa= ""{:.2e}".format(E2)," atomscm**-3"

# using above answer in following part
S1=10*10**-12 #initializing value of dimensions of silicon transistor
N1 = (E1*S1)
print"number Si atom in silicon transistor,N(Si)= ","{:.2e}".format(N1)," atoms"
S2 = 200*10*5*10**(-12) #" cm**3", #initializing value of dimensions of GaAs semiconductor laser
N2 = (E2*S2)
print"number of Ga atom in GaAs semiconductor,N(Ga)= ","{:.2e}".format(N2)," atoms"

number density of silicon,Nsi = 5.00e+22  atomscm**-3
number density of gallium atoms,NGa= 2.22e+22  atomscm**-3
number Si atom in silicon transistor,N(Si)=  5.00e+11  atoms
number of Ga atom in GaAs semiconductor,N(Ga)=  2.22e+14  atoms

Ex1.4:pg-15

In [13]:
# In the (001) surface the top atoms are either Ga or As
#A square of area a**2 has 4 atoms on the edges of square shared by 4 other square and 1 atom in centre

N=2.0 #initializing value of total number of atoms per square
a = 5.65*10**-8 #"cm**-1", #initializing value of lattice constant of gallium
SD = N/(a**2)
print"surface density of Ga,N(Ga)= ","{:.3e}".format(SD),"cm**-2"

surface density of Ga,N(Ga)=  6.265e+14 cm**-2

Ex1.5:pg-15

In [24]:
a = 5.65*10**-8 #initializing value of lattice constant of gallium
A = a/2
print"monolayer distance in the (001) direction,(A(ml)=","{:.3e}".format(A)," cm**-1"

monolayer distance in the (001) direction,(A(ml)= 2.825e-08  cm**-1

Ex1.6:pg-22

In [25]:
h=6.6*10**-34    #"Js", # plancks constant
c = 3*10**8 #"m/s", # velocity of light
E1 = 1.6*10**-19 #"J", #initializing value of energy of photon
L1 = h*c/E1
print"wavelengh of photon,L(ph)= ""{:.2e}".format(L1)," m"
E2 = 1.6*10**-19 #"J", #initializing value of energy of electron
mo = 9.1*10**-31 #"kg", #initializing value of mass of electron
L2 = h/sqrt(2*mo*E2)
print"wavelengh of electron,L(e)= ""{:.2e}".format(L2)," m"
m=1.0/1824 #initializing value of ratio of mass of electron to mass of neutron
L3 = L2*sqrt(m)
print"wavelengh of neutron,L(n)=""{:.2e}".format(L3)," m"

wavelengh of photon,L(ph)= 1.24e-06  m
wavelengh of electron,L(e)= 1.22e-09  m
wavelengh of neutron,L(n)=2.86e-11  m

Ex1.7:pg-25

In [28]:
import math
h=1.05*10**-34    #initializing value of reduced plancks constant or dirac constant or h-bar
m = 9.1*10**-31 #initializing value of mass of electron
E = 0.1 #initializing value of energy of electron
N = (sqrt(2)*(m)**(3.0/2))/((math.pi)**2*(h)**3)
print"density of states in 3D is ,N(E)= ","{:.2e}".format(N),"E**1/2 J**-1m**-3"

#Expressing E in eV and the density of states in commonly used units of eV**-1cm**-3
N1 = 6.8*10**21*sqrt(E)
print"density of states in 3D is ,N(E)= ""{:.2e}".format(N1),"eV**-1cm**-3"

density of states in 3D is ,N(E)=  1.07e+56 E**1/2 J**-1m**-3
density of states in 3D is ,N(E)= 2.15e+21 eV**-1cm**-3

Ex1.8:pg-25

In [1]:
h=1.05*10**-34     #initializing value of reduced plancks constant or dirac constant or h-bar
m = 9.1*10**-31 #initializing value of mass of electron
E = 2.0 #initializing value of energy of electron
#N = (sqrt(2)*(m)**(3.0/2))/((math.pi)**2*(h)**3)
#Expressing E in eV and the density of states in commonly used units of eV**-1cm**-3
N1 = 6.8*10**21*sqrt(E-2.0)
print"density of states in 3D is ,N(E)= ",round(N1,2),"eV**-1cm**-3"

density of states in 3D is ,N(E)=  0.0 eV**-1cm**-3

Ex1.9:pg-38

In [32]:
h=1.05*10**-34 #initializing value of reduced plancks constant or dirac constant or h-bar
m = 9.1*10**-31 #initializing value of mass of electron
n = 10**28 #initializing value of mass of electron
E = (3*(math.pi)**(2)*n)**(2/3)*(h**2/(2*m))
print"The fermi energy at 0K is ,E[F]= ","{:.2e}".format(E),"J"
Ef= E/(1.6*10**(-19))
print"The fermi energy at 0K in eV is ,E[F] = ","{:.2e}".format(Ef),"eV"
# Answer givenin the textbook is wrong

The fermi energy at 0K is ,E[F]=  6.06e-39 J
The fermi energy at 0K in eV is ,E[F] =  3.79e-20 eV

Ex1.10:pg-39

In [35]:
print"for temperature T1=77K"
kBT1=0.0067 #initializing value of multiplication of boltzmann constant and temperature T1
n1 = 10**19 #initializing value of density of electron
Nc1 = 3.34*10**18#initializing value of effective density of electron
print"Nc1 = 3.34*10**18 cm**-3"
Ef1= kBT1*((log(n1/Nc1)))
print"The fermi level at 77K (using boltzmann static) is ,Ef1(B)= ","{:.2e}".format(Ef1),"eV"
Ef2= kBT1*((log(n1/Nc1))+(1.0/sqrt(8))*(n1/Nc1))
print"The fermi level at 77K (using Joyce-Dixon static) is ,Ef1(J)= ","{:.2e}".format(Ef2),"eV"
print"for temperature T2=300K"
kBT2=0.026 #initializing value of multiplication of boltzmann constant and temperature T2
Nc2 = 2.56*10**19 #initializing value of effective density of electron
print"Nc2 = 2.56*10**19 cm**-3"
Ef3= kBT2*((log(n1/Nc2)))
print"The fermi level at 300K (using boltzmann static) is ,Ef2(B)= ","{:.2e}".format(Ef3),"eV"
Ef4= kBT2*((log(n1/Nc2))+(1.0/sqrt(8))*(n1/Nc2))
print"The fermi level at 300K (using Joyce-Dixon static) is ,Ef2(J)= ","{:.2e}".format(Ef4),"eV"

for temperature T1=77K
Nc1 = 3.34*10**18 cm**-3
The fermi level at 77K (using boltzmann static) is ,Ef1(B)=  7.35e-03 eV
The fermi level at 77K (using Joyce-Dixon static) is ,Ef1(J)=  1.44e-02 eV
for temperature T2=300K
Nc2 = 2.56*10**19 cm**-3
The fermi level at 300K (using boltzmann static) is ,Ef2(B)=  -2.44e-02 eV
The fermi level at 300K (using Joyce-Dixon static) is ,Ef2(J)=  -2.08e-02 eV