Diffusion
Impurity Diffusion
• Fundamental process step for
microelectronics
– Controls majority carrier type
– Controls semiconductor resistivity
• We want Substitutional diffusion
– Needed to provide carriers
III
IV
Silicon Dopant Types
• N-type (electron donor)
V
– P, As, Sb
• P-type (hole donor)
–B
– (Al+Ga have high diffusion
constants/don’t mask well)
Sb
Methods for Doping Silicon
• Diffusion
• Ion-Implantation
• Combinations of the above
Diffusion
Fick’s First Law
Particle flux J is proportional to the negative
of the gradient of the particle concentration
J  D
N
x
D = diffusion coefficient

• Same mathematical
“model” as
oxidation model
Diffusion
Continuity Equation for Particle Flux :
Rate of increase of concentration is equal to the
negative of the divergence of the particle flux
Fick’s Second Law
N
J

t
x
(in one dimension)
Fick's Second Law of Diffusion :
Combine First Law with Continuity Eqn.
N
 2N
D
t
x 2
D assumed to be independent of concentration!

• We use this because we are in
a non-steady state situation,
dopants continually diffuse
• Dose (Q) = Impurities/cm^2
Constant Source Diffusion
Complementary Error Function Profiles
erfcz  1 erf z
erf z 
2

z
 expx  dx
2
Concentrat ion :
 x 
N  x, t   N 0 erfc

 2 Dt 
0

Total Dose :

Q   N  x, t dt  2 N 0
0
Dt

N 0  Surface Concentrat ion
D  Diffusion Coefficient
erfc = Complement ary Error Function
• Solve PDE with boundary
conditions (No=const)
• Dose changes over time
• Furnace/chamber/etc
Limited Source Diffusion
Gaussian Profiles
Initial Impulse with Dose Q
Concentration :
  x 2 
  x 2 
Q
N x,t   N 0 exp


exp


 


 2 Dt  
 Dt
 2 Dt  

N 0  Surface Concentration N 0 
Q
Dt
D  Diffusion Coefficient
Gaussian Profile

• Solve PDE with boundary
condition (Impulse dose at
surf)
• Source never is
replenished
• Area under each curve
(dose) is constant
Diffusion
Profile Comparison
Complementary Error Function and
Gaussian Profiles are Similar in Shape
erfcz  1 erf z
erf z 

2

z
 expx  dx
2
0
Diffusion Coefficients
Substitutional
Diffusers
Interstitial
Diffusers
Diffusion Coefficients
 E 
D  DO exp A 
 kT 
Arrhenius Relationship
E A  activation energy
k = Boltzmann' s constant = 1.38 x10 -23 J/K
T = absolute temperature

• Dt product is the measure of driving force in the
diffusion
– D is proportional to Temp
– Time (t)
– Increase either of these or both and you will change
the diffusion parameters
• At high concentrations (~ni) diffusion constant
becomes dependant on concentration
Two-step Diffusion Process
• Short, high concentration
constant source pre-diffusion
approximates impulse dose at
surface
• Longer “drive in” step diffuses
impurities into lattice
• If Dt for drive in >> Dt for
predeposition
– Final profile will be Gaussian - - MOST CASES
• If Dt for drive in << Dt for
predeposition
– Final profile will be Erfc fn.
Successive Diffusions
• Successive diffusions using different times and
temperatures
• Any process which involves high temperatures
also affect this
• Final result depends upon the total Dt product
• This (Dt)tot is plugged into the equation to
determine final distribution
Dt tot   Di t i
i

Diffusion
Solid Solubility Limits
• There is a limit to the amount of a
given impurity that can be “dissolved”
in silicon (the Solid Solubility Limit)
• At high concentrations, all of the
impurities introduced into silicon will
not be electrically active
Diffusion
p-n Junction Formation
x j  Metallurgi cal Junction Depth
Gaussian Profile :
 N0 

x j  2 Dt ln
 NB 
 N0 

Error Function profile : x j  2 Dt erfc 
 NB 
-1
• P-n junction occurs where the net
impurity concentration is = 0
• P doping cancels n doping/ etc.
• Set N(xj)=0
• Solve equations for xj
Lateral Diffusion
Under Mask Edge
Original Mask
Concentration Dependent
Diffusion
Second Law of Diffusion
N 
N
 Dx 
t x
x
Profiles More Abrupt at High Concentrations

Concentration Dependent
Diffusion
• Phosphorus diffusion is more
complex, includes a “Kink”
which makes it harder to use in
actual devices
• Arsenic used instead
Diffusion
Resistivity vs. Doping


   1  qn n   p p
1
n  type :   qn N D  N A 
1
p  type :   q p N A  N D 
1

Resistors
Sheet Resistance
A W t
  L 
 L 
R     RS  
 t W 
W 
RS 

t
= Sheet Resistance [Ohms per Square]
 L 
  Number of Squares of Material
W 

Resistors
Counting Squares
• Top and Side Views of Two
Resistors of Different Size
• Resistors Have Same
Value of Resistance
• Each Resistor is 7 sq in
Length
• Each End Contributes
Approximately 0.65 sq
• Total for Each is 8.3 sq
Figure 4.14
Resistors
Contact and Corner Contributions
• Effective Square
Contributions of Various
Resistor End and Corner
Configurations
Figure 4.15
Sheet Resistance
Irvin’s Curves

1

RS 


xj
1
xj
1
xj

  x dx
•
0
1
xj
  x dx
0
Irvin Evaluated this Integral and
Published a Set of Normalized
Curves Plot Surface
Concentration Versus Average
Resistivity
  RS x j
1
x j

RS   qN x dx


0

•

Four Sets of Curves
– n-type and p-type
– Gaussian and erfc
Two Step Diffusion
Sheet Resistance - Predep Step
Initial Profile
N o  1.1x10 20 /cm 3
N B  3x1016 /cm 3
x j  0.0587 m
p  type erfc profile
RS x j  50  - m

RS 

32  - m
 850 /Square
0.0587 m
Two Step Diffusion
Sheet Resistance - Drive-in Step
Final Profile
N o  1.1x1018 /cm 3
N B  3x1016 /cm 3
x j  2.73 m
p  type Gaussian profile
RS x j  700  - m

RS 

700  - m
 260  /Square
2.73 m
Doping Systems
• Spin on
– Glass containing the dopant impurity
• Not as uniform of a doping
• Furnaces (3 zone)
– Source material
• Liquid, Solid, Gas
– Boron
• Gas/solids react to supply impurities on surf
– Phosphorus
• Gas/solids react to supply impurities on surf
– Arsenic
• Hard to make high concentrations with furnace methods –
Use ion implantation