Fick’s Laws

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. c0 Impurity concentration, c(x) c ( x, t ) D3t3 > D2t2 > D1t1 1 2 3 cB Distance from surface, x 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

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

9. Normalized distance from surface, Comparison of limited source and constant source models 1 10-1 exp(- ) 2 10-2 erfc( ) 10-3 Value of functions 10-4 10-5 10-6 0 0.5 1 1.5 2 2.5 3 3.5

10. Predep and Drive • Predeposition • 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

13. Temperature (o C) Temperature (o C) 1400 1300 1200 1100 1000 1200 1100 1000 900 800 700 10-9 10-4 10-10 10-5 10-11 Li 10-6 Fe Diffusion coefficient, D (cm2/sec) Diffusion coefficient, D (cm2/sec) Cu 10-12 10-7 Al Au Ga 10-13 In B,P 10-8 0.6 0.7 0.8 0.9 1.0 1.1 Sb As Temperature, 1000/T (K-1) 10-14 0.6 0.65 0.7 0.75 0.8 0.85 Temperature, 1000/T (K-1) Diffusion Of Impurities In Silicon • Arrhenius plots of diffusion in silicon

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

21. N = x 2 Dt ln 0 N j B N - = 1 x 2 Dt erfc B N j 0 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

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