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

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

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

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

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

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

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 )

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:

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)

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

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.

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

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

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

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

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

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

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

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)

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

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

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

Junction Formation Junction Depth Lateral Diffusion

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

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

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

Irvin’s Curves

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

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