Chapter 4 : Bonding the Band Theory of Solids and Statistics¶

Example 4.6 Page No : 154¶

In [1]:
import math

# Given
E_FO = 7.0          # in eV
e = 1.6*10**-19     # in coulombs
E_FO *= e           # in Joules
me = 9.1*10**-31    # in Kg

# Calculations and Results
v_f = math.sqrt(2*E_FO/me)
print("Speed of the conduction electrons in m/s is {0:.4g}".format(v_f))

Speed of the conduction electrons in m/s is 1.569e+06


Example 4.7 Page No : 159¶

In [2]:
# Given
e = 1.6 * 10 ** -19     # in coulombs
Eg = 1.1                # in eV
Eg *= e                 # in Joules
h = 6.6 * 10 ** -34     # in Js
c = 3 * 10 ** 8         # in m/s

# Calculations and Results
lambda_val = h * c / Eg
print("Wavelength of light that can be absorbed by an Si photo-detector"
" at Eg=1.1 eV in micro meter is {0:.4g}".format(lambda_val * 10 ** 6))
print("Hence the light of wavelength 1.31 micro meter and 1.55 micro meter will not "
"be absorbed by Si and thus cannot be detected by detector")

Wavelength of light that can be absorbed by an Si photo-detector at Eg=1.1 eV in micro meter is 1.125
Hence the light of wavelength 1.31 micro meter and 1.55 micro meter will not be absorbed by Si and thus cannot be detected by detector


Example 4.8 Page No : 162¶

In [3]:
import math

# Given
e = 1.6*10**-19     # in coulombs
h = 6.626*10**-34   # in Js
me = 9.1*10**-31    # in Kg
# let x=k*T
x = 0.026           # in eV
E = 5.0               # in ev
E *= e              # in Joules

# Calculations and Results
g_E = (8*math.pi*math.sqrt(2))*(me/h**2)**(3./2)*math.sqrt(E)  # in J**-1*m**-3
# conversion of units
g_E = g_E*10**-6*e  # in eV**-1 cm**-3
print("density of states at the center of the band in cm**-3*J**-1 is {0:.4g}".format(g_E))
# part(b)
n_E = g_E*x         # in cm**-3
print("No.of states per unit volume within kT about the center in cm**-3 is {0:.4g}".format(n_E))
# part(c)
# Given
E = 0.026           # in eV
E *= e              # in joules

# Calculations and Results
g_E = (8*math.pi*math.sqrt(2))*(me/h**2)**(3./2)*math.sqrt(E)  # in J**-1*m**-3
# conversion of units
g_E = g_E*10**-6*e  # in eV**-1 cm**-3
print("density of states at at kT above the band in cm**-3*J**-1 is {0:.4g}".format(g_E))
# part(d)
n_E = g_E*x         # in cm**-3
print(" No.of states per unit volume within kT about the center in cm**-3 is {0:.4g}".format(n_E))
# solved using the values taken from the solution of textbook

density of states at the center of the band in cm**-3*J**-1 is 1.518e+22
No.of states per unit volume within kT about the center in cm**-3 is 3.946e+20
density of states at at kT above the band in cm**-3*J**-1 is 1.095e+21
No.of states per unit volume within kT about the center in cm**-3 is 2.846e+19


Example 4.9 Page No : 165¶

In [4]:
import math

# Given
e = 1.6*10**-19     # in coulombs
h = 6.626*10**-34   # in Js
me = 9.1*10**-31    # in Kg
d = 10.5            # in g/cm
Mat = 107.9         # g/mol
NA = 6.023*10**23   # mol**-1
E_ctr = 5.0           # in ev
E_ctr *= e          # in Joules

# Calculations and Results
S_band = 2*(16*math.pi*math.sqrt(2)/3)*(me/h**2)**(3./2)*E_ctr**(3./2)  # in states m**-3
# conversion of units
S_band *= 10**-6    # in states cm**-3
print("No. of states in the band in states cm**-3 is {0:.4g}".format(S_band))
n_Ag = d*NA/Mat
print("No.of atoms per unit volume in silver in atoms per cm3 is {0:.4g}".format(n_Ag))

No. of states in the band in states cm**-3 is 1.012e+23
No.of atoms per unit volume in silver in atoms per cm3 is 5.861e+22


Example 4.10 Page No : 169¶

In [5]:
import math
# Given
e = 1.6*10**-19     # in coulombs
h = 6.626*10**-34   # in Js
me = 9.1*10**-31    # in Kg
d = 8.96            # in g/cm
Mat = 63.5          # g/ mol
NA = 6.023*10**23   # mol**-1
n = d*NA/Mat        # in cm**-3
n *= 10**6          # in m**-3

# Calculations and Results
E_FO = (h**2/(8*me))*(3*n/math.pi)**(2./3)   # in J
E_FO /= e           # in eV
print("Fermi energy at 0 Kelvin in eV is {0:.4f}".format(E_FO))
E_FO = (h**2/(8*me))*(3*n/math.pi)**(2./3)   # in J
v_e = math.sqrt(6*E_FO/(5*me))
print("Average speed of conduction electrons in m/s is {0:.4g}".format(v_e))

Fermi energy at 0 Kelvin in eV is 7.0653
Average speed of conduction electrons in m/s is 1.221e+06


Example 4.11 Page No : 174¶

In [6]:
# Given
e = 1.6*10**-19     # in coulombs
me = 9.1*10**-31    # in Kg
u_d = 43*10**-4     # in cm2/V/s
v_e = 1.22*10**6    # in m/s

# Calculations and Results
T = u_d*me/e
l_e = v_e*T
print("Mean free path of electrons in meters is {0:.4g}".format(l_e))

Mean free path of electrons in meters is 2.984e-08


Example 4.13 Page No : 178¶

In [7]:
import math
# Given
e = 1.6*10**-19   # in coulombs
T = 373.0         # in kelvin
To = 273.0        # in kelvin
k = 1.38*10**-23  # in m2 kg /k/s2
# from table 4.3
E_FAO = 11.6      # in eV
E_FAO *= e        # in J
x_A = 2.78
E_FBO = 7.01      # in eV
E_FBO *= e        # in J
x_B = -1.79

# Calculations and Results
# Mott jones Equation
V_AB = (-math.pi**2*k**2/(6*e))*((x_A/E_FAO)-(x_B/E_FBO))*(T**2-To**2)
print("EMF in micro volts available from Al and Cu thermocouple with the given respective"
" temperatures at the junctions is {0:.4f}".format(V_AB*10**6))

EMF in micro volts available from Al and Cu thermocouple with the given respective temperatures at the junctions is -391.2988


Example 4.14 Page No : 182¶

In [8]:
import math
# Given
phi = 2.6           # in eV
e = 1.6*10**-19     # in coulombs
phi *= e            # in Joules
Be = 3*10**4        # schottky coefficient in A/m2/K2
T = 1600.0            # in degree celsius
T += 273.0            # in Kelvin
k = 1.38*10**-23    # m2 kg s-2 K-1
d = 2*10**-3        # in m
l = 4*10**-2        # in in m

# Calculations and Results
# Richardson-Dushman Equation
J = Be*T**2*math.exp(-phi/(k*T))
A = math.pi*d*l
I = J*A
print("Saturation current in Amperes if the tube is operated at 1873 kelvin is {0:.4g}".format(I))

Saturation current in Amperes if the tube is operated at 1873 kelvin is 2.708