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Fick’s Laws Combining the continuity equation with the first law, we obtain Fick’s second law: c c D 2 t x 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 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 c D 2 0 x 2 c( x) a bx Solutions To Fick’s Second Law For a semi-infinite slab with a constant (infinite) supply of atoms at the surface x c( x, t ) co erfc 2 Dt The dose is Q cx, t dx 2c0 Dt 0 Solutions To Fick’s Second Law Complimentary error function (erfc) is defined as erfc(x) = 1 - erf(x) The error function is defined as erf ( z ) 2 exp d z 2 0 – 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 Impurity concentration, c(x) c0 c ( x, t ) D3t3 > D2t2 > D1t1 1 2 cB Distance from surface, x 3 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 Q c ( x, t ) e Dt Q c( x, t )dx 0 The surface concentration is given by: Q c(0, t ) Dt x2 4 Dt Solutions To Fick’s Second Law Limited source diffusion looks like Impurity concentration, c(x) c01 c ( x, t ) c02 D3t3 > D2t2 > D1t1 c03 1 2 cB Distance from surface, x 3 Comparison of limited source and constant source models 1 10-1 _ exp(-x 2 ) Value of functions 10-2 _ erfc( x) 10-3 10-4 10-5 10-6 0 0.5 1 _ 1.5_ Normalized distance from surface, x x 2 x 2 Dt 2.5 3 3.5 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. Diffusion Coefficient Probability of a jump is Pj Pv Pm e E f kT e Em kT Diffusion coefficient is proportional to jump probability D D0e E D kT 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 10.5 3.69 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 1400 1300 1200 1100 Temperature (o C) 1000 1200 1100 1000 900 10-4 800 700 10-10 Diffusion coefficient, D (cm2/sec) Diffusion coefficient, D (cm2/sec) 10-5 10-11 10-12 Al 10-13 In 0.7 10-7 10-8 0.6 As 0.65 Cu Ga Sb 0.6 Fe Au B,P 10-14 Li 10-6 0.75 0.7 0.8 0.9 Temperature, 1000/T 0.8 Temperature, 1000/T (K-1) 0.85 1.0 (K-1) 1.1 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 nand p-type dopants – Impurities will move as succeeding dopant or oxidation steps are performed The effective Dt product is ( Dt ) eff D1 (t1 t 2 ) D1t1 D1t 2 – No difference between diffusion in one step or in several steps at the same temperature If diffusions are done at different temperatures ( Dt ) eff D1t1 D2t 2 Successive Diffusions The effective Dt product is given by Dt eff Di ti i 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 Junction Formation Formation of a pn junction by diffusion Impurity Net impurity concentration |N(x) - NB | concentration N(x) N0 (log scale) N0 - NB p-type Gaussian diffusion (boron) n-type silicon (log scale) p-type region background NB n-type region xj Distance from surface, x xj Distance from surface, x Junction Formation The position of the junction for a limited source diffused impurity in a constant background is given by x j 2 Dt ln N 0 N B The position of the junction for a continuous source diffused impurity is given by 1 N x j 2 Dt erfc B N0 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 1 these parameters S 1 x xj q n( x) N B n( x)dx j 0 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 Example Design a B diffusion for a CMOS tub such that s=900/sq, xj=3m, and CB=11015/cc – First, we calculate the average conductivity 1 1 1 3 . 7 cm S x j 900/sq 3 104 cm – We cannot calculate n or because both are functions of depth – We assume that because the tubs are of moderate concentration and thus assume (for now) that the distribution will be Gaussian Therefore, we can use the P-type Gaussian Irvin curve to deduce that Example Reading from the p-type Gaussian Irvin’s curve, CS4x1017/cc This is well below the solid solubility limit for B in Si so we may conclude that it will be driven in from a fixed source provided either by ion implantation or possibly by solid state predeposition followed by an etch x 3 10 Dt 3.7 10 cm C 4 10 to be at the required In order for junction 4 ln the 4 ln C 10 depth, we can compute the Dt value from the Gaussian junction equation 4 2 2 j 17 S B 15 9 2 Example This value of Dt is the thermal budget for the process If this is done in one step at (for example) 1100 3.7 10 9 cm 2 -13cm2/s, the C wheretdriveDinfor is 1.5 x 10 B in Si 6 . 8 hrs 1.5 10 13 cm 2 /s drive-in time will be Q C (0, t ) Dt 4 1017 3.7 109 4.3 1013 cm- 2 Given Dt and the final surface concentration, we can estimate the dose Example Let us also look at doing it by predep from the solid state (as is done in the VT lab course) The text uses a predep temperature of 950 C In this case, we will make a glass-like oxide on the surface that will introduce the B at the solid solubility limit 2C Q Dt At 950 C, the solubility limit is 2.5x1020cm-3 and D=4.2x10-15 cm2/s S 4.3 10 t predep 20 2 . 5 10 13 2 2 1 2 4.2 1015 5.5 s Example This is a very short time and hard to control in a furnace; thus, we should do the pre-dep at lower temperatures In the VT lab, we use 830 – 860 C 14 9 Dt 2 . 3 10 Dt 3 . 7 10 predep drivein in? Does the predep affect the drive There is no affect on the thermal budget because it is done at such a “low” temperature DIFFUSION SYSTEMS Use open tube furnaces of the 3-Zone design Wafers are mounted in quartz boat in center zone Use solid, liquid or gaseous impurities for good reproducibility Use N2 or O2 as carrier gas to move impurity downstream to crystals Common gases are extremely toxic (AsH3 , PH3) SOLID-SOURCE DIFFUSION SYSTEMS Exhaust Platinum source boat Slices on carrier burn box and/or scrubber Valves and flow meters N2 O2 Quartz Quartz diffusion diffusion boat tube LIQUID-SOURCE DIFFUSION SYSTEMS Exhaust Slices on carrier Burn box and/or scrubber Valves and flow meters O2 N2 Quartz diffusion tube Liquid source Temperaturecontrolled bath GAS-SOURCE DIFFUSION SYSTEMS Exhaust Slices on carrier Burn box and/or scrubber Quartz diffusion tube Valves and flow meter To scrubber system N2 Dopant gas O2 Trap DIFFUSION SYSTEMS Al and Ga diffuse very rapidly in Si; B is the only p-dopant routinely used Sb, P, As are all used as n-dopants DIFFUSION SYSTEMS Typical reactions for solid impurities are: 2 CHO B2O3 6CO2 9H2O 3 3 B 9O2 900o C SiO2 4B 2B2O3 Si 4POCl3 302 2PO 2 5 6Cl2 SiO2 4P 2PO 2 5 5Si 2 As2O3 3Si 3SiO2 4 As 2SbO 3SiO2 4Sb 2 3 3Si PRODUCTION DIFFUSION FURNACES Commercial diffusion furnace showing the furnace with wafers (left) and gas control system (right). (Photo courtesy of Tystar Corp.) PRODUCTION DIFFUSION FURNACES Close-up of diffusion furnace with wafers. Rapid Thermal Annealing An alternative to the diffusion furnaces is the RTA or RTP furnace Rapid Thermal Anneling In this system, we try to heat the wafer quickly (but not so as to introduce fracture stresses) RTAs usually use infrared lamps and heat by radiation It is possible to ramp the wafer at 100 C /sec Such devices are used to diffuse shallow junctions and to anneal radiation damage In such a system, for the thermal conductivity of Si, a 12 in wafer can be Rapid Thermal Annealing Concentration-Dependent Diffusion If the concentration of the doping exceeds the intrinsic carrier concentration at the diffusion temperature, another effect occurs We have assumed that the diffusion coefficient, D, is independent of concentration This is not valid if the concentration of the diffusing species is greater than the intrinsic carrier concentration Concentration-Dependent Diffusion The concentration profiles for P in Si look more like the solid lines than the dashed line for high concentrations (see French et al) Concentration-Dependent Diffusion If we define the diffusivity to be a function of composition, then we can still use Fick’s law to describe the dopant diffusion Usually, we cannot directly integrate/solve the differential C equations eff C when D is a DA function of C t x x We thus must solve the equation Concentration-Dependent Diffusion It has been observed that the diffusion coefficient usually depends on concentration by either of the 2 following D (n / ni ) or D (n / ni ) relations Look, for example, at the diffusion of P in Si observed by French et al How do we obtain information about the Concentration-Dependent Diffusion B has two isotopes: B10 and B11 We create a wafer with a high concentration of one isotope (say B10) and then we diffuse the second isotope into this material We use SIMS to determine the concentration of B11 as a function of distance This gives us the diffusion of B as a function of the concentration of B Concentration-Dependent Diffusion We find that the diffusivity can usually be written in the form 2 eff A D for n n D D D ni ni 0 p p D D dopants D D and n-type ni ni eff A 0 2 Concentration-Dependent Diffusion The superscripts are chosen because we believe the interaction is between charged vacancies and the charged diffusing species DAeff D 0 D D For an n-type dopant in an intrinsic material, the diffusivity is D.E D D0 exp kT All of the various diffusivities are of the Concentration-Dependent Diffusion The values for all the pre-exponential factors and activation energies are known (see next Table) If we substitute into the expression for the n n effective diffusion 1 coefficient, we find n n 2 DAeff DA* i 1 i Concentration-Dependent Diffusion Concentration-Dependent Diffusion Expressed this way, is the linear variation with composition and is the quadratic variation Simulators like SUPREM include these effects and are capable of modeling very complex structures