1. Fick’s Laws
Combining the continuity equation with the first law, we obtain Fick’s second law:

2. Solutions to Fick’s Laws depend on the boundary conditions.
Assumptions
D is independent of concentration
Semiconductor is a semi-infinite slab with either
Continuous supply of impurities that can move into wafer
Fixed supply of impurities that can be depleted

3. Solutions To Fick’s Second Law
The simplest solution is at steady state and there is no variation of the concentration with time
Concentration of diffusing impurities is linear over distance
This was the solution for the flow of oxygen from the surface to the Si/SiO2 interface in the last chapter

4. Solutions To Fick’s Second Law
For a semi-infinite slab with a constant (infinite) supply of atoms at the surface
The dose is

5. Solutions To Fick’s Second Law
Complimentary error function (erfc) is defined as erfc(x) = 1 - erf(x)
The error function is defined as
This is a tabulated function. There are several approximations. It can be found as a built-in function in MatLab, MathCad, and Mathematica

6. Solutions To Fick’s Second Law
This solution models short diffusions from a gas-phase or liquid phase source
Typical solutions have the following shape c0 cB
Distance from surface, x 1 2 3
D3t3 > D2t2 > D1t1
Impurity concentration, c(x)
c ( x, t )

7. Solutions To Fick’s Second Law
Constant source diffusion has a solution of the form
Here, Q is the does or the total number of dopant atoms diffused into the Si
The surface concentration is given by:

8. Solutions To Fick’s Second Law
Limited source diffusion looks like
c ( x, t )
c01
c02
c03 cB 1 2 3
Distance from surface, x
D3t3 > D2t2 > D1t1
Impurity concentration, c(x)

9. Comparison of limited source and constant source models1
10-1
exp( ) 2
10-2
erfc( )
Value of functions
10-3
10-4
10-5
10-6
Normalized distance from surface,

10. Predep and Drive Predeposition Drive A Dteff is not used in this case.
Usually a short diffusion using a constant source
Drive
A limited source diffusion
The diffusion dose is generally the dopants introduced into the semiconductor during the predep
A Dteff is not used in this case.

11. Diffusion Coefficient
Probability of a jump is
Diffusion coefficient is proportional to jump probability

12. Diffusion Coefficient
Typical diffusion coefficients in silicon
Element
Do (cm2/s)
ED (eV) B
10.5
3.69 Al
8.00
3.47 Ga
3.60
3.51 In
16.5
3.90 P As
0.32
3.56 Sb
5.60
3.95

13. Diffusion Of Impurities In Silicon
Arrhenius plots of diffusion in silicon
Temperature (o C)
10-9
10-10
10-11
10-12
10-13
10-14
Temperature, 1000/T (K-1) Al Ga
B,P In As Sb
Diffusion coefficient, D (cm2/sec)
10-4
10-5
10-6
10-7
10-8 Li Cu Fe Au

14. Diffusion Of Impurities In Silicon
The intrinsic carrier concentration in Si is about 7 x 1018/cm3 at 1000 oC
If NA and ND are <ni, the material will behave as if it were intrinsic; there are many practical situations where this is a good assumption

15. Diffusion Of Impurities In Silicon
Dopants cluster into “fast” diffusers (P, B, In) and “slow” diffusers (As, Sb)
As we develop shallow junction devices, slow diffusers are becoming very important
B is the only p-type dopant that has a high solubility; therefore, it is very hard to make shallow p-type junctions with this fast diffuser

16. Limitations of Theory
Theories given here break down at high concentrations of dopants
ND or NA >> ni at diffusion temperature
If there are different species of the same atom diffuse into the semiconductor
Multiple diffusion fronts
Example: P in Si
Diffusion mechanism are different
Example: Zn in GaAs
Surface pile-up vs. segregation
B and P in Si

17. Successive Diffusions
To create devices, successive diffusions of n- and p-type dopants
Impurities will move as succeeding dopant or oxidation steps are performed
The effective Dt product is
No difference between diffusion in one step or in several steps at the same temperature
If diffusions are done at different temperatures

18. Successive Diffusions
The effective Dt product is given by
Di and ti are the diffusion coefficient and time for ith step
Assuming that the diffusion constant is only a function of temperature.
The same type of diffusion is conducted (constant or limited source)

19. Junction Formation
When diffuse n- and p-type materials, we create a pn junction
When ND = NA , the semiconductor material is compensated and we create a metallurgical junction
At metallurgical junction the material behaves intrinsic
Calculate the position of the metallurgical junction for those systems for which our analytical model is a good fit

20. p-type Gaussian diffusion
Junction Formation
Formation of a pn junction by diffusion
Impurity
concentration
N(x) N0 NB
(log scale) xj
p-type Gaussian diffusion
(boron)
n-type silicon
background
Distance from surface, x
Net impurity
|N(x) - NB |
N0 - NB
p-type
region
n-type region

21. Junction Formation
The position of the junction for a limited source diffused impurity in a constant background is given by
The position of the junction for a continuous source diffused impurity is given by x Dt N j B = 2 ln x Dt N j B = - 2 1
erfc

22. Junction Formation
Junction Depth
Lateral Diffusion

23. Design and Evaluation
There are three parameters that define a diffused region
The surface concentration
The junction depth
The sheet resistance
These parameters are not independent
Irvin developed a relationship that describes these parameters

24. Irvin’s Curves
In designing processes, we need to use all available data
We need to determine if one of the analytic solutions applies
For example,
If the surface concentration is near the solubility limit, the continuous (erf) solution may be applied
If we have a low surface concentration, the limited source (Gaussian) solution may be applied

25. Irvin’s Curves
If we describe the dopant profile by either the Gaussian or the erf model
The surface concentration becomes a parameter in this integration
By rearranging the variables, we find that the surface concentration and the product of sheet resistance and the junction depth are related by the definite integral of the profile
There are four separate curves to be evaluated
one pair using either the Gaussian or the erf function, and the other pair for n- or p-type materials because the mobility is different for electrons and holes

26. Irvin’s Curves

27. Irvin’s Curves
An alternative way of presenting the data may be found if we set eff=1/sxj