# Chapter 7 : Metal - Semiconductor Junction and Hetero - junctions¶

## Example number 7.1 , Page number 266¶

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

#Variable declaration
Nd=5*10**16 #Doping level of n-type silicon in cm**-3
Nc=2.8*10**19 #in cm**-3
e=1.6*10**-19 #in J
phi_B0=1.09 #in eV
epsilon_r=11.7 #in F/cm
epsilon_0=8.85*10**-14 #in F/cm
Const=0.026 #constant for kT/e in V

#Calculations
phi_n=Const*math.log(Nc/Nd) #in eV
Vbi=(phi_B0-phi_n) #in eV
xn=((2*epsilon_r*epsilon_0*Vbi)/(e*Nd))**0.5
Emax=(e*Nd*xn)/(epsilon_r*epsilon_0)

#Result
print("a) Ideal Schottky barrier height= %0.3f eV\n" %phi_n)
print("b) Built-in potential barrier= %0.3f V\n" %Vbi)
print("c) Space charge width at zero bias= %1.3f*10**-4 cm\n" %(xn/10**-4))
print("d) maximum electric field= %.2f*10**4 V cm**-1" %round(Emax/10**4,2))#answer vary due to sound-off error

a) Ideal Schottky barrier height= 0.165 eV

b) Built-in potential barrier= 0.925 V

c) Space charge width at zero bias= 0.155*10**-4 cm

d) maximum electric field= 11.96*10**4 V cm**-1


## Example number 7.2 , Page number 267¶

In [19]:
#importing module
import math
from __future__ import division

#Variable declaration
Nd=2.01*10**7 #Doping level of n-type silicon in cm**-3
Nc=2.8*10**19 #in cm**-3
e=1.6*10**-19 #in J
epsilon_r=11.7 #in F/cm
epsilon_0=8.85*10**-14 #in F/cm
slope=6*10**13
Vbi=0.45 #in V
Const=0.026 #constant for kT/e in V

#Calculations
Nd=2/(e*epsilon_r*epsilon_0*slope) #in cm**-3
phi_n=Const*math.log(Nc/Nd) #in V
phi_Bn=Vbi+phi_n

#Result
print("Actual barrier height= %0.3f V" %phi_Bn)

Actual barrier height= 0.578 V


## Example number 7.3 , Page number 267¶

In [25]:
#importing module
import math
from __future__ import division

#Variable declaration
E=10**4 #Electric field in V/cm
e=1.6*10**-19 #in J
epsilon_r=11.7 #in F/cm
epsilon_0=8.85*10**-14 #in F/cm

#Calculations
del_phi=math.sqrt((e*E)/(4*math.pi*epsilon_r*epsilon_0))
xm=math.sqrt(e/(16*math.pi*epsilon_r*epsilon_0*E))

#Result
print("Schottkybarrier-lowering for Si-metal contact= %0.3f V\n" %del_phi)
print("maximum barrier height= %1.2f*10**-7 cm" %(xm/10**-7))

Schottkybarrier-lowering for Si-metal contact= 0.011 V

maximum barrier height= 5.54*10**-7 cm


## Example number 7.4 , Page number 268¶

In [28]:
#importing module
import math
from __future__ import division

#Variable declaration
A=114 #effective Richardson constant A/K**2*cm**2
e=1.6*10**-19 #in J
T=300 #in K
phi_Bn=0.82 #in eV
const=0.026 #value for kT/e in V

#Calculations
J0=A*T**2*math.exp(-(phi_Bn/const))

#Result
print("Reverse saturation current density= %1.2f*10**-7 A/cm**2" %(J0/10**-7))

Reverse saturation current density= 2.06*10**-7 A/cm**2


## Example number 7.5 , Page number 272¶

In [29]:
#importing module
import math
from __future__ import division

#Variable declaration
xGe=4.13  #in eV
xGaAs=4.07 #in eV
Eg_Ge=0.7 #in eV
Eg_GaAs=1.45 #in eV

#Calculations
delE_c=xGe-xGaAs
delE_v=(Eg_GaAs-Eg_Ge)-delE_c

#Result
print("Conduction band= %1.2f eV\n" %delE_c)
print("Valence band= %1.2f eV" %delE_v)

Conduction band= 0.06 eV

Valence band= 0.69 eV


## Example number 7.6 , Page number 272¶

In [35]:
#importing module
import math
from __future__ import division

#Variable declaration
Nd=3*10**15 #Doping level of n-type silicon in cm**-3
Nc=2.8*10**19 #in cm**-3
e=1.6*10**-19 #in J
phi_m=4.5 #work function for chromium in eV
epsilon_si=11.7 #in F/cm
epsilon_0=8.854*10**-14 #in F/cm
xsi=4.01 #electron affinity for Si in eV
Vbi=5 #reverse bias voltage in V
VR=0 #in V

#Calculations
phi_B=phi_m-xsi #in eV
xn=((2*epsilon_si*epsilon_0*(Vbi+VR))/(e*Nd))**0.5 #in cm
Emax=(e*Nd*xn)/(epsilon_si*epsilon_0)
CJ=((e*epsilon_si*epsilon_0*Nd)/(2*(Vbi+VR)))**0.5

#Result
print("a)\n")
print("ideal schottky barrier height= %1.2f ev\n" %phi_B)
print("b)\n")
print("peak electric field= %1.2f*10**4 V/cm\n" %(Emax/10**4))
print("c)\n")
print("depletion layer capacitance per unit area= %1.2f*10**-9 F/cm**2" %(CJ/10**-9)) #The answer provided in the textbook is wrong

a)

ideal schottky barrier height= 0.49 ev

b)

peak electric field= 6.81*10**4 V/cm

c)

depletion layer capacitance per unit area= 7.05*10**-9 F/cm**2


## Example number 7.9 , Page number 272¶

In [37]:
#importing module
import math
from __future__ import division

#Variable declaration
phi_m=4.3 #work function in eV
xsi=4 #electron affinity in eV
p0=10**17 #in cm**-3
Na=10**17 #in cm**-3
ni=1.5*10**10 #in cm**-3
delE_fc=0.957 #in eV
Const=0.0259 #constant value for kT in eV

#Calculations
delE_if=Const*math.log(p0/ni)

#a) Before contact
phi_s=xsi+delE_fc

#b) After contact
phi_B=phi_m-xsi
eV0=phi_s-phi_m

#Result
print("Energy state difference= %.3f eV\n" %delE_if)
print(" a)phi_s= %.3f eV\n" %phi_s)
print(" b)Forward Bias (phi_B)= %.1f eV\n" %phi_B)
print("   eV0= %.3f eV" %eV0) #The answer provided in the textbook is wrong

Energy state difference= 0.407 eV

a)phi_s= 4.957 eV

b)Forward Bias (phi_B)= 0.3 eV

eV0= 0.657 eV


## Example number 7.9 , Page number 272¶

In [41]:
#importing module
import math
from __future__ import division

#Variable declaration
ni=1.5*10**10 #in cm**-3
delE_iF=0.0259 #in eV
delE_cF=0.29 #in eV
phi_G=4.8 #in eV
impurity_conc=9.9*10**14 #in cm**-3
affinity=0.55 #in eV
Const=0.0259 #constant value for kT in eV
x=4.05 #electron affinity for silicon in eV

#Calculations
#a)
n0=ni*math.exp(delE_iF/Const) #in cm**-3
phi_s=x+delE_cF

#b)
Ptype_conc=impurity_conc-n0 #net concentration of p-type on B side in cm**-3
delE_iF_Bside=Const*math.log(Ptype_conc/ni) #in eV
phi_s_Bside=x+delE_iF_Bside+affinity

#d)
ni1=8*10**12 #increased ni in cm**-3
delE_iF1=Const*math.log(n0/ni1) #in eV
phi_s1=x+(affinity-delE_iF1)
eV0=phi_s-phi_m

#Result
print("electron doping concentration = %.1f*10**10 cm**-3\n" %(n0/10**10)) #The answer provided in the textbook is wrong
print("workfuntion of the semiconductor = %.2f eV\n" %phi_s)
print("workfuntion of the semiconductor on B side = %.2f eV\n" %phi_s_Bside) #The answer provided in the textbook is wrong
print("workfuntion of the semiconductor at 400K = %.2f eV " %phi_s1) #The answer provided in the textbook is wrong

electron doping concentration = 4.1*10**10 cm**-3

workfuntion of the semiconductor = 4.34 eV

workfuntion of the semiconductor on B side = 4.89 eV

workfuntion of the semiconductor at 400K = 4.74 eV

In [ ]: