Diffusion Furnace

1. Theory

• Definition: A diffusion furnace dopes silicon by holding wafers on a quartz boat in a tube furnace with a dopant source, so dopant atoms enter the surface and spread inward by thermally activated diffusion; BCl3 boron diffusion runs at 800 to 1100 ${}^{\circ }\mathrm{C}$ (PV-Manufacturing; Stanford; Illinois).
• Two steps: Predeposition at solid solubility gives an erfc profile with a set dose; drive-in without a source spreads that dose into a Gaussian (Stanford).
• Diffusivity: $D$ follows an Arrhenius law and rises at high doping through charged point defects (Stanford).
• Point defects: Oxidation and implant damage change the interstitial population and so the diffusivity of boron and phosphorus (Stanford).

2. Working Principle

• Source: Nitrogen bubbled through liquid POCl3 carries it into the tube, where it reacts with oxygen to phosphorus pentoxide; boron uses BCl3 with oxygen (PV-Manufacturing).
• Predeposition: Phosphorus pentoxide deposits as a phosphosilicate glass while the surface is held near solid solubility (PV-Manufacturing; Stanford).
• Drive-in: The pentoxide is reduced by silicon to phosphorus, which diffuses to the target depth; the glass is then removed in dilute HF for 1 to 2 min (PV-Manufacturing; Stanford).
• Check: Sheet resistance by four-point probe (Stanford; Four-Point Probe page).

3. Applications

• Wells and deep junctions: Long drive-ins such as the handout’s 3 $\mu \mathrm{m}$ CMOS well (Stanford).
• Solar cell emitters: POCl3 tube diffusion of silicon solar cells (PV-Manufacturing).
• Bipolar bases and emitters: Classic predeposition and drive-in sequences (Doolittle).

Diffusion Solutions

Name	Formula	Related Formulae	Variables/Constants
Fick’s laws	$F=-D\frac{\partial C}{\partial x},\frac{\partial C}{\partial t}=\frac{\partial }{\partial x}\left(D\frac{\partial C}{\partial x}\right)$	$\frac{\partial C}{\partial t}=D\frac{{\partial }^{2}C}{\partial x^2}\left(D\text{constant}\right)$	$F:dopantflux$ $C:concentration$ $\begin{array}{r}D:diffusivity;Stanfordhandoutequations\\ 4to6\end{array}$ $\begin{array}{r}(myreading:constantDholdsonlyfor\\ intrinsicdoping,sohigh-concentration\\ profilesneedthecharged-defectterms\\ below)\end{array}$
Predeposition	$C\left(x,t\right)=C_S\mathrm{erfc}\left(\frac{x}{2\sqrt{Dt}}\right)$	$Q=2C_S\sqrt{\frac{Dt}{\pi }}$	$\begin{array}{r}{C}_S:surfaceconcentrationheldatsolid\\ solubility;Stanfordequations10to12,\\ boronsolubility2.5\times 10^{20}\mathrm{c}{\mathrm{m}}^{-3}at950{}^{\circ }C;\\ Illinois:unlimitedsourceinexcessof\\ solubility,N_0=N_{sl};Doolittle:boron\\ solubility5\times 10^{20}\mathrm{c}{\mathrm{m}}^{-3}at1200{}^{\circ }C\end{array}$ $\begin{array}{r}(computed:the4.3\times 10^{13}\mathrm{c}{\mathrm{m}}^{-2}welldoseneeds\\ only5.6\mathrm{s}ofpredepositionat950{}^{\circ }C,\\ matchingthehandou{t}^{\prime }s5.5\mathrm{s})\end{array}$
Drive-in	$C\left(x,t\right)=\frac{Q}{\sqrt{\pi Dt}}\exp \left(-\frac{x^2}{4Dt}\right)$	$Dt=\frac{x_j^2}{4\ln \left(C_S/C_B\right)}$	$Q:fixeddose$ $x_j:junctiondepth$ $\begin{array}{r}{C}_B:backgrounddoping;Stanfordequation7\\ andworkedexample(x_j=3\mu \mathrm{m},C_B=1\times 10^{15}\\ \mathrm{c}{\mathrm{m}}^{-3},C_S=4\times 10^{17}\mathrm{c}{\mathrm{m}}^{-3}givesDt=3.7\times 10^{-9}c{m}^2,\\ 2.5\times 10^4\mathrm{s}=6.8hat1100{}^{\circ }CwithD=\\ 1.5\times 10^{-13}\mathrm{c}{\mathrm{m}}^2/s,Q=4.3\times 10^{13}\mathrm{c}{\mathrm{m}}^{-2});validwhen\\ predepositionDtismuchsmallerthan\\ drive-inDt\end{array}$ $\begin{array}{r}(computed:Dt=3.76\times 10^{-9}c{m}^2,t=2.50\times 10^4\\ \mathrm{s},Q=4.34\times 10^{13}\mathrm{c}{\mathrm{m}}^{-2},allmatching)\end{array}$

Diffusivity

Name	Formula	Related Formulae	Variables/Constants
Intrinsic diffusivity	$D=D_0\exp \left(-\frac{E_A}{k_{B}T}\right)$	$B:1.0,3.5;P:4.70,3.68;As:9.17,3.99;Sb:4.58,3.88$	$D_{0}in\mathrm{c}{\mathrm{m}}^2/s,E_{A}in\mathrm{eV}$ $\begin{array}{r}{k}_B:Boltzmannconstant;Stanfordequation\\ 13table,alsoSiself-diffusion560and\\ 4.76,In1.2and3.5;Doolittlehomework\\ usesB0.037and3.46,P3.9and3.66,As\\ 0.066and3.44,giving{D}_{As}=4.32\times 10^{-14}\mathrm{c}{\mathrm{m}}^2/s\\ at1150{}^{\circ }C\end{array}$ $\begin{array}{r}(computedat1100{}^{\circ }C:B1.43\times 10^{-13},P\\ 1.46\times 10^{-13},As2.08\times 10^{-14},Sb2.63\times 10^{-14}\mathrm{c}{\mathrm{m}}^2/s)\end{array}$
Charged-defect diffusivity	$D_{eff}=D^0+D^{-}\frac{n}{n_i}+D^{=}{\left(\frac{n}{n_i}\right)}^2$	$D_{eff,p}=D^0+D^{+}\frac{p}{n_i}$	$\begin{array}{r}{D}^0,D^{+},D^{-},D^{=}:diffusivitiesthrough\\ neutralandchargedpointdefects\end{array}$ $n,p:carrierconcentrations$ $\begin{array}{r}{n}_i:intrinsicconcentrationatprocess\\ temperature;Stanfordequation19\\ (componenttablepartlygarbledin\\ extraction;assignmentsinferred),n_{i}very\\ largeatprocesstemperatures\end{array}$ $\begin{array}{r}(myreading:D^{-}and{D}^{=}makearsenicand\\ phosphorusdiffusefasterathighdoping,\\ flatteningtheprofiletop)\end{array}$
Field enhancement	$J=-hD\frac{\partial C}{\partial x}$	$h=1+\frac{C}{\sqrt{C^2+4n_i^2}}\le 2$	$\begin{array}{r}h:field-enhancementfactorfromthe\\ built-infieldoftheionizeddopant;\\ Stanfordequations16and17;segregation\\ coefficientk=C_S/C_{L}andinterfacefluxin\\ equations20and21\end{array}$ $\begin{array}{r}(computed:h=1.0atCmuchlessthan{n}_i,\\ 1.71atC=2n_{i}andcloseto2atCmuch\\ greaterthan{n}_i)\end{array}$

Layer Properties and Defects

Name	Formula	Related Formulae	Variables/Constants
Sheet resistance	$R_S=\frac{\rho }{x_j}=\frac{1}{{\int }_0^{x_j}q\mu n\left(x\right)dx}$	$\bar{\sigma }=\frac{1}{R_S{x}_j}$	$R_{S}in\Omega /\square$ $\begin{array}{r}\mu :mobility;Stanfordequations1to3,\\ evaluatednumericallybyIrvinfor\\ analyticalprofiles,measuredwitha\\ four-pointprobe;example900\Omega /\square and3\mu \mathrm{m}\\ give3.7S/cm,readonIrvi{n}^{\prime }scurveas{C}_S\\ about4\times 10^{17}\mathrm{c}{\mathrm{m}}^{-3}\end{array}$ $(computed:1/(900x3\times 10^{-4})=3.70S/cm)$
Point-defect coupling	$f_I/f_V:Si0.6/0.4,B1.0/0,P1.0/0$	$f_I/f_V:As0.4/0.6,Sb0.02/0.98$	$\begin{array}{r}{f}_I,f_V:fractionsofdiffusionthrough\\ interstitialsandvacancies;Stanford:\\ oxidation-enhancedandoxidation-retarded\\ diffusionfrominterstitialandvacancy\\ injection,fractionsaslisted\end{array}$ $\begin{array}{r}(myreading:oxidationinjects\\ interstitials,soboronandphosphorus\\ speedupwhileantimonyslows)\end{array}$
Interface dose loss	$\frac{\Delta Q}{Q}=\frac{Q_0-Q_f}{Q_0}$	$Q_0=1\times 10^{15}\to Q_f=6.8\times 10^{14}\mathrm{c}{\mathrm{m}}^{-2}$	$\begin{array}{r}{Q}_0,Q_f:dosebeforeandafteranneal;\\ Stanford:uptoabout50percentofa\\ shallowdosecanbelosttointerfaces,\\ oneexperimentreporting40percentloss\\ in30\mathrm{s}at1050{}^{\circ }C\end{array}$ $\begin{array}{r}(computed:thequoteddosesgivea32per\\ centloss,not40percent)\end{array}$

Important Questions

Ques: How do predeposition and drive-in combine to set a junction?

Predeposition holds the surface at solid solubility, giving $C_S\mathrm{erfc}(x/2\sqrt{Dt})$ and a dose $Q=2C_S\sqrt{Dt/\pi }$; drive-in then spreads that fixed dose into a Gaussian $Q/\sqrt{\pi Dt}\exp (-x^2/4Dt)$ (Stanford). The Stanford well example targets 900 ohm per square and 3 $\mu \mathrm{m}$ on a $10^{15}$ $\mathrm{c}{\mathrm{m}}^{-3}$ substrate. Irvin’s curve gives a surface concentration of $4\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$, so $Dt=x_j^2/4\ln (C_S/C_B)$ = $3.76\times 10^{-9}$ cm${}^2$ and the drive-in takes 6.8 $\mathrm{h}$ at 1100 ${}^{\circ }\mathrm{C}$, with a dose of $4.3\times 10^{13}$ $\mathrm{c}{\mathrm{m}}^{-2}$ (computed, matching the handout). The same dose needs only 5.6 $\mathrm{s}$ of predeposition at 950 ${}^{\circ }\mathrm{C}$ (computed). Such a short step is hard to control in a furnace, which is why the handout notes the dose is easily implanted instead (my reading). Reference: J. D. Plummer, M. D. Deal and P. B. Griffin, EE 212 class handout 12, chapter 7 ‘Dopant Diffusion’ (Stanford, 1999), hosted on the University of Washington EE 528 site (fetch-tool summary).

Ques: Which boron diffusivity parameters should be used?

Both sets appear. The Stanford handout lists an intrinsic boron diffusivity with $D_0$ = 1.0 $\mathrm{c}{\mathrm{m}}^2/s$ and 3.5 $\mathrm{eV}$, while Doolittle’s homework uses 0.037 $\mathrm{c}{\mathrm{m}}^2/s$ and 3.46 $\mathrm{eV}$. At 1100 ${}^{\circ }\mathrm{C}$ they give $1.4\times 10^{-13}$ and $7.4\times 10^{-15}$ $\mathrm{c}{\mathrm{m}}^2/s$, a factor of 19 (computed). Doolittle’s values are close to the neutral-defect term in Stanford’s own component table, about 0.05 against 0.95 for the positive-defect term, both at 3.5 $\mathrm{eV}$, though those column assignments were partly garbled in extraction. Stanford’s total intrinsic value is therefore more likely correct for lightly doped boron, and its 1100 ${}^{\circ }\mathrm{C}$ value reproduces the handout’s worked example (my reading). For phosphorus the two sets agree within 3 per cent (computed), because the neutral term dominates (computed from Stanford’s component table). Arsenic differs by about 1.3 to 1.6 times. Reference: J. D. Plummer, M. D. Deal and P. B. Griffin, EE 212 class handout 12, chapter 7 ‘Dopant Diffusion’ (Stanford, 1999), hosted on the University of Washington EE 528 site (fetch-tool summary); W. A. Doolittle, ECE 6450 Homework 2 solution, Georgia Institute of Technology (fetch-tool summary).

Ques: Why does diffusion speed up at high doping?

Because charged point defects become more numerous. The effective diffusivity of an n-type dopant is $D^0+D^{-}(n/n_i)+D^{=}(n/n_i{)}^2$, so when the electron concentration exceeds the intrinsic value at process temperature the charged terms grow (Stanford). In addition, the built-in field of the ionized dopants pushes them outward with a factor $h=1+C/\sqrt{C^2+4n_i^2}$, which reaches 1.71 at $C=2n_i$ and approaches 2 at high doping (Stanford; computed). Together these flatten the top of high-concentration arsenic and phosphorus profiles and steepen their fronts (my reading). Since $n_i$ is very large at process temperatures, many moderately doped profiles still diffuse intrinsically, and constant-$D$ solutions remain useful there (Stanford; my reading). Reference: J. D. Plummer, M. D. Deal and P. B. Griffin, EE 212 class handout 12, chapter 7 ‘Dopant Diffusion’ (Stanford, 1999), hosted on the University of Washington EE 528 site (fetch-tool summary).

Ques: How do oxidation and interfaces change the dose and profile?

Oxidation injects silicon interstitials, so dopants that diffuse by interstitials speed up. The Stanford handout lists interstitial fractions of 1.0 for boron and phosphorus, 0.4 for arsenic and 0.02 for antimony, and attributes oxidation-enhanced and retarded diffusion to interstitial and vacancy injection, so boron and phosphorus speed up during oxidation while antimony slows (Stanford; my reading). Interfaces also take up dopant: the handout says up to about 50 per cent of a shallow dose can be lost, and quotes 40 per cent loss in 30 $\mathrm{s}$ at 1050 ${}^{\circ }\mathrm{C}$. Its own numbers, $10^{15}$ to $6.8\times 10^{14}$ $\mathrm{c}{\mathrm{m}}^{-2}$, correspond to 32 per cent (computed), so the 40 per cent figure is more likely a rounded or different-sample value and 32 per cent fits the stated doses (my reading). Reference: J. D. Plummer, M. D. Deal and P. B. Griffin, EE 212 class handout 12, chapter 7 ‘Dopant Diffusion’ (Stanford, 1999), hosted on the University of Washington EE 528 site (fetch-tool summary).

Ques: How is a diffused layer checked after the furnace?

By sheet resistance and junction depth. The sheet resistance of a non-uniform layer is $1/{\int }_0^{x_j}q\mu n(x)dx$, which Irvin evaluated numerically for erfc and Gaussian profiles so that a measured value and junction depth give the surface concentration (Stanford). For 900 ohm per square and 3 $\mu \mathrm{m}$ the average conductivity is 3.70 S/cm (computed), read as a surface concentration of about $4\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$. Sheet resistance is measured with a four-point probe (Stanford; Four-Point Probe page). In solar-cell tube diffusion the POCl3 source leaves a phosphosilicate glass that is removed in dilute HF before measurement (PV-Manufacturing). The Illinois notes stress that predeposition fixes the surface at the solid solubility limit, so surface concentration is set by temperature, while dose is set by time (Illinois; my reading). Reference: J. D. Plummer, M. D. Deal and P. B. Griffin, EE 212 class handout 12, chapter 7 ‘Dopant Diffusion’ (Stanford, 1999), hosted on the University of Washington EE 528 site (fetch-tool summary); Four-Point Probe page of this vault; University of Illinois ECE 444 fabrication equations, ‘Predeposition’ page (fetch-tool summary); PV-Manufacturing.org, ‘Tube diffusion’ (POCl3 and BCl3 tube diffusion for solar cells) (fetch-tool summary).

References

1. J. D. Plummer, M. D. Deal and P. B. Griffin, EE 212 Handout 12, “Chapter 7: Dopant Diffusion,” Stanford University, 1999. Available: https://dunham.ece.uw.edu/ee528/notes/Chapter7.pdf
2. W. A. Doolittle, ECE 6450 Homework 2 Solution, Georgia Institute of Technology. Available: https://alan.ece.gatech.edu/ECE6450/Homeworksolutions/Homework2Solution.pdf
3. PV-Manufacturing.org, “Tube diffusion.” Available: https://pv-manufacturing.org/solar-cell-manufacturing/tube-diffusion/
4. Four-Point Probe page of this vault.
5. University of Illinois ECE 444, “Predeposition,” fabrication equations page. Available: https://fabweb.ece.illinois.edu/gt/equations/predep.aspx

Related pages in this vault: Basic Semiconductor Equations, Beamline Ion Implanter, Thermal Oxidation Furnace, Four-Point Probe, Secondary Ion Mass Spectrometer (SIMS and ToF-SIMS), Spreading Resistance Profiler (SRP), Spin-On and Gas-Source Doping System, Furnace Anneal System, RCA Wet Bench, LPCVD Furnace.