Chapter 4 : Excess Carriers in Semiconductors

Example number 4.1 , Page number 135

In [6]:
#importing module
import math
from __future__ import division

#Variable declaration
del_n0=10**16 #concentration of electrons in cm**-3
tau_n0=5 #excess carrier lifetime in micro-s
t=1 #time in micro-s

#Calculations
del_nt=del_n0*math.exp(-t/tau_n0)

#Result
print("excess electron concentration= %.2f*10**15 cm**-3" %(del_nt/10**15))

excess electron concentration= 8.19*10**15 cm**-3

Example number 4.2 , Page number 135

In [13]:
#importing module
import math
from __future__ import division

#Variable declaration
del_n0=10**16 #concentration of electrons in cm**-3
tau_n0=5 #excess carrier lifetime in s
tau_n01=5*10**-6 #excess carrier lifetime in micro-s
t=5 #in micro-s

#Calculations
del_nt=del_n0*math.exp(-t/tau_n0) #in cm**-3
Rn1=del_nt/tau_n01

#Result
print("Recombination rate= %.2f*10**21 cm**-3 s**-1" %(Rn1/10**21))

Recombination rate= 0.74*10**21 cm**-3 s**-1

Example number 4.3 , Page number 136

In [16]:
#importing module
import math
from __future__ import division

#Variable declaration
Nd=10**15 #dopant concentration in cm**-3
Na=0 #in cm**-3
tau_p0=10*10**-7 #in s
tau_n0=10*10**-7 #in s
ni=1.5*10**10 #in cm**-3
deln=10**14 #in cm**-3
delp=10**14 #in cm**-3
nt=1.5*10**15 #in cm**-3
pt=1.5*10**15 #in cm**-3
               
#Calculations
n0=Nd #in cm**-3
p0=ni**2/Nd #in cm**-3
n=n0+deln #in cm**-3
p=p0+delp #in cm**-3
R=((n*p)-ni**2)/(tau_n0*(n+p))

#Result
print("Recombination rate= %1.2f*10**19 cm**-3 s**-1" %(R/10**19))

Recombination rate= 9.17*10**19 cm**-3 s**-1

Example number 4.4 , Page number 146

In [17]:
#importing module
import math
from __future__ import division

#Variable declaration
n0=5*10**15 #carrier concentration in cm**-3
ni=10**10 #in cm**-3
p0=2*10**4 #in cm**-3
deln=5*10**13 #excess carriers in semiconductor in cm**-3
delp=5*10**13 #in cm**-3
Const=0.026 #constant value for kT/e in V

#Calculations
delE1=Const*math.log(n0/ni) 
delE2=Const*math.log((n0+deln)/ni)
delE3=Const*math.log((p0+delp)/ni)

#Result
print("1)\nposition of the Fermi level at thermal equilibrium= %0.4f eV\n" %delE1)
print("2)\nquasi-Fermi level for electrons in non-equilibrium= %0.4f eV\n" %delE2)
print("3)\nquasi-Fermi level for holes in non-equilibrium= %0.4f eV" %delE3)

1)
position of the Fermi level at thermal equilibrium= 0.3412 eV

2)
quasi-Fermi level for electrons in non-equilibrium= 0.3414 eV

3)
quasi-Fermi level for holes in non-equilibrium= 0.2214 eV

Example number 4.6 , Page number 147

In [18]:
#importing module
import math
from __future__ import division

#Variable declaration
l=1.8 #distance between plates in cm
E=3/2 #in V
t=0.6*10**-3 #time taken by the pulse in s 
del_t=236*10**-6 #pulse width in s

#Calculations
vd=l/t #in cm/s
myu_p=vd/E
Dp=(del_t*l)**2/(16*t**3)

#Result
print("1)\nHole mobility= %i cm**2/Vs\n" %myu_p)
print("2)\nDiffusion coefficient= %2.2f cm**2/s" %Dp)

1)
Hole mobility= 2000 cm**2/Vs

2)
Diffusion coefficient= 52.22 cm**2/s

Example number 4.7 , Page number 149

In [22]:
#importing module
import math
from __future__ import division

#Variable declaration
delp=4*10**14 #excess EHP in cm**-3
deln=4*10**14 #excess EHP in cm**-3
n0=10**15 #donor atoms in cm**-3
p0=0 #in cm**-3
t=0.5*10**-6 #hole-lifetime in s
myu_n=1200 #mobility of electron in cm**2/V*s
myu_p=400 #mobility of hole in cm**2/V*s
q=1.6*10**-19 #electron charge in eV
ni=1.5*10**10 #in cm**-3
Const=0.0259 #constant value for kT in eV

#Calculations
#a)
gop=delp/t

#b)
rho_0=(q*n0*myu_n)**-1 #Before illumination
n=n0+deln #in cm**-3
p=p0+delp #in cm**-3
rho=1/(q*((myu_n*n)+(myu_p*p)))#conductivity
rho1=q*myu_p*delp #in mho/cm
Pcond=(rho*rho1)*100

#c)
delE_e=Const*math.log(n/ni)
delE_h=Const*math.log(p/ni)

#Result
print("a)\n")
print("photo generation rate= %i*10**20 EHPs/cm**3s\n" %(gop/10**20))
print("b)\n")
print("resistivity before illumination= %1.2f ohm-cm\n" %rho_0)
print("resistvity after illumination= %1.3f ohm-cm\n" %rho)
print("percent of conductivity= %1.2f percent\n" %Pcond) #The answers vary due to round off error
print("c)\n")
print("quasi Fermi level due to electron=Efi+%0.3f eV\n" %delE_e)
print("quasi Fermi level due to holes=Efi-%0.3f eV\n" %delE_h)

a)

photo generation rate= 8*10**20 EHPs/cm**3s

b)

resistivity before illumination= 5.21 ohm-cm

resistvity after illumination= 3.397 ohm-cm

percent of conductivity= 8.70 percent

c)

quasi Fermi level due to electron=Efi+0.296 eV

quasi Fermi level due to holes=Efi-0.264 eV

Example number 4.8 , Page number 151

In [26]:
#importing module
import math
from __future__ import division

#Variable declaration
n0=10**16 #donor atoms in cm**-3
q=1.6*10**-19 #electron charge in J
ni=1.5*10**10 #in cm**-3
Nd=10**16 #Donors added to silicon to make it n-type) in cm**-3
GT=2.25*10**10 #Thermal generation rate of carriers under equilibrium cm**-3/s
gop=10**21 #in cm**-3/s
tau_n=10**-6 #in s
tau_t=2.5*10**-3 #transit time in s
V=1 #in V

#Calculations
#a)
alpha_r=GT/ni**2
tau_p=(alpha_r*n0)**-1
 
#b)
delp=gop*tau_n

#c)
delI=(q*V*gop*tau_n)/tau_t

#Result
print("a)\n")
print("lifetime of both type of carriers= %i micro-s\n" %(tau_p/10**-6))
print("b)\n")
print("excess carrier concentration= %i*10**15 cm**-3\n" %(delp/10**15))
print("c)\n")
print("Induced change in current= %.3f A" %delI)

a)

lifetime of both type of carriers= 1 micro-s

b)

excess carrier concentration= 1*10**15 cm**-3

c)

Induced change in current= 0.064 A

Example number 4.9 , Page number 151

In [30]:
#importing module
import math
from __future__ import division

#Variable declaration
E1000=8.48*10**5 #Current density for 1000 V in A/cm**2
delE=0.1 #in eV
q=1.6*10**-19 #electron charge in eV
ni=1.5*10**10 #in cm**-3
Nd=10**16 #Donors added to silicon to make it n-type) in cm**-3
gop=10**19 #in cm**-3/s
tau=10**-5 #in s
Const=0.0259 #constant value for kT in eV

#Calculations
#a)
E10000=E1000

#b)
n0=ni*math.exp(delE/Const)

#c)
deln=gop*tau #in cm**-3
n=n0 #in cm**-3
p=deln #in cm**-3s
delE_np=Const*math.log((n*p)/ni**2)

#Result
print("a)\n")
print("Current density for 1000V potential= %1.2f*10**5 A/cm**2\n" %(E10000/10**5))
print("b)\n")
print("Doping concentration= %1.1f*10**11 cm**-3\n" %(n0/10**11)) #The answer provided in the textbook is wrong"
print("c)\n")
print("Energy gap= %0.4f eV" %delE_np) #The answer provided in the textbook is wrong"

a)

Current density for 1000V potential= 8.48*10**5 A/cm**2

b)

Doping concentration= 7.1*10**11 cm**-3

c)

Energy gap= 0.3280 eV
In [ ]: