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

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Solutions to Fick’s Laws depend on the boundary conditions. Solutions to Fick’s Laws depend on the boundary conditions. Assumptions 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

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Solutions To Fick’s Second Law The simplest solution is at steady state and there is no variation of the concentration with time 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/SiO 2 interface in the last chapter This was the solution for the flow of oxygen from the surface to the Si/SiO 2 interface in the last chapter

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

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Solutions To Fick’s Second Law Complimentary error function (erfc) is defined as erfc(x) = 1 - erf(x) Complimentary error function (erfc) is defined as erfc(x) = 1 - erf(x) The error function is defined as 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

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Solutions To Fick’s Second Law This solution models short diffusions from a gas-phase or liquid phase source This solution models short diffusions from a gas-phase or liquid phase source Typical solutions have the following shape Typical solutions have the following shape c0c0 cBcB Distance from surface, x 1 2 3 D 3 t 3 > D 2 t 2 > D 1 t 1 Impurity concentration, c(x) c ( x, t )

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

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Solutions To Fick’s Second Law Limited source diffusion looks like Limited source diffusion looks like c ( x, t ) c 01 c 02 c 03 cBcB 12 3 Distance from surface, x D 3 t 3 > D 2 t 2 > D 1 t 1 Impurity concentration, c(x)

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

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

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

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Diffusion Coefficient Typical diffusion coefficients in silicon Typical diffusion coefficients in silicon Element D o (cm 2 /s) E D (eV) B10.53.69 Al8.003.47 Ga3.603.51 In16.53.90 P10.53.69 As0.323.56 Sb5.603.95

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Diffusion Of Impurities In Silicon Arrhenius plots of diffusion in silicon Arrhenius plots of diffusion in silicon 1400 1300 1200 1100 1000 Temperature ( o C) 10 -9 10 -10 10 -11 10 -12 10 -13 10 -14 0.6 0.65 0.7 0.75 0.8 0.85 Temperature, 1000/T (K -1 ) Al Ga B,P In As Sb Diffusion coefficient, D (cm 2 /sec) 10 -4 10 -5 10 -6 10 -7 10 -8 0.6 0.7 0.8 0.9 1.0 1.1 Temperature, 1000/T (K -1 ) 1200 1100 1000 900 800 700 Temperature ( o C) Diffusion coefficient, D (cm 2 /sec) Li Cu Fe Au

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Diffusion Of Impurities In Silicon The intrinsic carrier concentration in Si is about 7 x 10 18 /cm 3 at 1000 o C The intrinsic carrier concentration in Si is about 7 x 10 18 /cm 3 at 1000 o C –If N A and N D are

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Diffusion Of Impurities In Silicon Dopants cluster into “fast” diffusers (P, B, In) and “slow” diffusers (As, Sb) 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

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Limitations of Theory Theories given here break down at high concentrations of dopants Theories given here break down at high concentrations of dopants –N D or N A >> n i at diffusion temperature If there are different species of the same atom diffuse into the semiconductor 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

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Successive Diffusions To create devices, successive diffusions of n- and p-type dopants 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 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 If diffusions are done at different temperatures

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Successive Diffusions The effective Dt product is given by The effective Dt product is given by D i and t i are the diffusion coefficient and time for i th step –Assuming that the diffusion constant is only a function of temperature. –The same type of diffusion is conducted (constant or limited source)

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Junction Formation When diffuse n- and p-type materials, we create a pn junction When diffuse n- and p-type materials, we create a pn junction –When N D = N A, 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

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Junction Formation Formation of a pn junction by diffusion Formation of a pn junction by diffusion Impurity concentration N(x) N0N0 NBNB (log scale ) xjxj p-type Gaussian diffusion (boron) n-type silicon background Distance from surface, x Net impurity concentration |N(x) - N B | N 0 - N B p-type region n-type region xjxj Distance from surface, x (log scale)

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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 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 The position of the junction for a continuous source diffused impurity is given by xDt N N j B 2 0 ln xDt N N j B 2 1 0 erfc

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Junction Formation Junction DepthLateral Diffusion

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Design and Evaluation There are three parameters that define a diffused region 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 developed a relationship that describes these parameters

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Irvin’s Curves In designing processes, we need to use all available data 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

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Irvin’s Curves If we describe the dopant profile by either the Gaussian or the erf model 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 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

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Irvin’s Curves

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An alternative way of presenting the data may be found if we set eff =1/ s x j An alternative way of presenting the data may be found if we set eff =1/ s x j

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