# Chapter 4: Metal Semiconductors Contacts¶

## Example 1, Page 57¶

In :
#Variable declaration
Qm=4.55#Qm=work function of tungsten in eV
X=4.01#X=electron affinity of silicon in eV

#Calculations&Results
eQb=(Qm-X)#eQb=barrier height as seen from the metal
print "eQb=%.2f eV"%eQb
a=0.21#a=(Ec-Ef)=forbidden gap in eV
eVbi=eQb-a#eVbi=barrier height from semiconductor side
print "eVbi=%.2f eV"%eVbi
Es=11.7*8.854*(10**-12)#Es=permittivity of semiconductor;11.7=dielectric constant of silicon
e=1.6*10**(-19)#e=charge of an electron
Nd=10**22#Nd=donor concentration in m^-3
W=((2*Es*eVbi)/(e*Nd))**(1./2)#W=width of the depletion region
print "W=%.3e m"%W
Fm=((e*Nd*W)/Es)#Fm=maximum electric field in V/m
print "Fm= %.1e V/m"%Fm

eQb=0.54 eV
eVbi=0.33 eV
W=2.067e-07 m
Fm= 3.2e+06 V/m


## Example 2, Page 58¶

In :
#Variable declaration
# as per given data barrier height =Vbi=intercept on Vr axis=0.4 V
Es=11.7*8.854*(10**-12)#Es=permittivity of semiconductor;11.7=dielectric constant of silicon
e=1.6*10**(-19)#e=charge of an electron
m=4.4*10**(15)#m=slope of (1/C^2) vs Vr plot of a Schottky contact in(cm^4)(F^-2)(V^-1)

#Calculations
#m=2/(e*Es*Nd)
Nd=(2*10**8)/(e*Es*m)#Nd=donor concentration in silicon in m^-3

#Result
print "Nd = %.2e m**-3"%Nd

Nd = 2.74e+21 m**-3


## Example 3, Page 58¶

In :
import math

#Variable declaration
e=1.6*10**-19#e=charge of an electron in C
Fa=7.*10**6#Fa=reverse bias field in V/m

#Calculations&Results
Es=13.1*8.854*10**-12#(Es/Eo)=13.1;Eo=8.854*10^-12
dQ=((e*Fa)/(4*math.pi*Es))**(1./2)#dQ=barrier lowering in V
print "dQ= %.4f V"%dQ
Xm=(dQ)/(2*Fa)#Xm=position of the maximum barrier height
print "Xm= %.2e m"%Xm

dQ= 0.0277 V
Xm= 1.98e-09 m


## Example 4, Page 58¶

In :
import math

#Variable declaration
#Js=A*(T**2)*math.exp(-((e*Qbn)/(kB*T)))
kB=1.38*10**(-23)#kB=Boltzmann's constant in J/K
T=300#T=temperature in Kelvin
e=1.6*10**-19#e=charge of an electron in C
Js=6*10**-5#Js=emission current density in A/cm^2

#Calculations
Qbn=0.668#Qbn=barrier height in V
A=(Js/(T**2))*math.exp((e*Qbn)/(kB*T))#A=Richardson constant

#Result
print "A=%.1f (cm^-2)(K^-2)"%A

A=108.6 (cm^-2)(K^-2)


## Example 5, Page 58¶

In :
import math

#Variable declaration
e=1.6*10**-19#e=charge of an electron in C
V=0.32#V =applied forward bias in V
kB=1.38*10**(-23)#kB=Boltzmann's constant in J/K
T=300#T=Temperature in Kelvin
Js=0.61#Js=reverse saturation current density in A/m^2

#Calculations&Results
J=Js*(math.exp((e*V)/(kB*T))-1)#J=current density in A/m^2
print "J=%.2e A/m^2"%J
A=4*10**-8#A=cross sectional area in m^2
I=(J*A)*10**3#I=current
print "I=%.2f mA"%I

J=1.43e+05 A/m^2
I=5.73 mA