SOL 102 › Unit 7

Unit 7 — Diffusion and Kinetics: Fick's Laws and Temperature Dependence

Big idea. Atoms in a solid move by random hops. The net result is diffusion: flow from high to low concentration, described by Fick's laws. Diffusion distance grows as \(\sqrt{Dt}\), and the diffusion coefficient \(D\) rises exponentially with temperature (Arrhenius). This is how emitters are formed (phosphorus diffused ~0.3–0.5 µm into silicon at ~800–900 °C) and why metals, which diffuse millions of times faster than dopants, contaminate whole wafers in seconds at high temperature.

Learning objectives

Why this matters

Tube diffusion (POCl₃, BBr₃), firing, hydrogenation, gettering, oxidation and TOPCon poly-Si anneals are diffusion problems. Process engineers tune time and temperature to hit junction depth and sheet resistance; contamination engineers worry about metals because of their huge diffusivities.

Random walk

At high temperature each atom occasionally hops to a neighbouring site — into a vacancy, or between interstitial positions. Hops are random in direction. After \(N\) hops of length \(a\), the average displacement is zero, but the mean-square displacement grows linearly:

\[\langle x^2 \rangle = N a^2 \quad\Rightarrow\quad x_\text{typical} \propto \sqrt{t}\]
Random walks. Left: six simulated 2-D walkers. Right: averaging 3000 walkers, the mean-square distance grows linearly with the number of hops — so typical distance grows with the square root of time. Simulated here.
Random walks. Left: six simulated 2-D walkers. Right: averaging 3000 walkers, the mean-square distance grows linearly with the number of hops — so typical distance grows with the square root of time. Simulated here.

Fick's laws

First law — flux is proportional to the concentration gradient:

\[J = -D\,\frac{\partial C}{\partial x}\]

In words: atoms flow downhill in concentration, faster for steeper gradients and larger D. \(D\) (cm²/s) is the diffusion coefficient.

Second law — how concentration changes in time:

\[\frac{\partial C}{\partial t} = D\,\frac{\partial^2 C}{\partial x^2}\]

In words: concentration rises where the profile curves upward (a dip fills in) and falls where it curves downward (a peak spreads). The characteristic diffusion length is \(\sqrt{Dt}\) (some texts use \(2\sqrt{Dt}\)).

Two classic solutions

Constant surface concentration (the surface held at \(C_s\) by a gas source, like POCl₃ pre-deposition):

\[C(x,t) = C_s\,\text{erfc}\!\left(\frac{x}{2\sqrt{Dt}}\right)\]

Limited source (a fixed dose \(Q\) per area driven in, no more supplied):

\[C(x,t) = \frac{Q}{\sqrt{\pi D t}}\,\exp\!\left(-\frac{x^2}{4Dt}\right)\]

The junction depth \(x_j\) is where the diffused dopant concentration equals the wafer's background doping of the opposite type.

Constant-source diffusion profiles after 5–60 min, with \(C_s = 10^{20}\) cm⁻³ and an assumed \(D = 3\times10^{-14}\) cm²/s. The junction forms where each curve crosses the background doping (red dashed). Illustrative values. Real POCl₃ emitters have a "kink-and-tail" shape because diffusion at very high P concentration is enhanced.
Constant-source diffusion profiles after 5–60 min, with \(C_s = 10^{20}\) cm⁻³ and an assumed \(D = 3\times10^{-14}\) cm²/s. The junction forms where each curve crosses the background doping (red dashed). Illustrative values. Real POCl₃ emitters have a "kink-and-tail" shape because diffusion at very high P concentration is enhanced.

Temperature: the Arrhenius law

Hops need to overcome an energy barrier, so the hop rate — and \(D\) — follow

\[D = D_0 \exp\!\left(-\frac{E_a}{k_B T}\right)\]

In words: every increase in temperature multiplies D by a large factor; the activation energy \(E_a\) sets how sensitive. Plot \(\ln D\) against \(1/T\): a straight line with slope \(-E_a/k_B\).

Arrhenius plot for boron and phosphorus (intrinsic values, Sze VLSI Technology: B \(D_0\) = 0.76 cm²/s, \(E_a\) = 3.46 eV; P \(D_0\) = 3.85 cm²/s, \(E_a\) = 3.66 eV) and interstitial iron (Istratov et al. 1999: \(D_0\) = 1.0 × 10⁻³ cm²/s, \(E_a\) = 0.67 eV). Iron diffuses ~10⁸ times faster than dopants at 1000 °C.
Arrhenius plot for boron and phosphorus (intrinsic values, Sze VLSI Technology: B \(D_0\) = 0.76 cm²/s, \(E_a\) = 3.46 eV; P \(D_0\) = 3.85 cm²/s, \(E_a\) = 3.66 eV) and interstitial iron (Istratov et al. 1999: \(D_0\) = 1.0 × 10⁻³ cm²/s, \(E_a\) = 0.67 eV). Iron diffuses ~10⁸ times faster than dopants at 1000 °C.

Worked example 1 — boron at 1000 °C

\(D = 0.76\exp\!\left(-\dfrac{3.46}{8.617\times10^{-5}\times1273}\right) = 1.5\times10^{-14}\) cm²/s. After 1 h: \(\sqrt{Dt} = \sqrt{1.5\times10^{-14}\times3600} = 7.4\times10^{-6}\) cm = 74 nm. Dopant diffusion is slow — microns take hours at ~1000 °C.

Worked example 2 — the intrinsic value underestimates real emitters

Phosphorus at 900 °C, intrinsic: \(D \approx 7\times10^{-16}\) cm²/s → after 30 min \(\sqrt{Dt} \approx 11\) nm. Yet industrial POCl₃ emitters reach junction depths of a few hundred nm in similar times. Why? At concentrations of 10²⁰ cm⁻³ and above, phosphorus diffuses via charged point defects much faster ("concentration-enhanced diffusion"), and the high-concentration region near the surface behaves differently (inactive P, the "kink"). Lesson: textbook intrinsic numbers are a starting point; real process models (and process engineers) calibrate to measured profiles (SOL 302).

Worked example 3 — iron races through a wafer

At 800 °C: \(D_\text{Fe} = 10^{-3}\exp(-0.67/(8.617\times10^{-5}\times1073)) = 7.1\times10^{-7}\) cm²/s. In just 60 s, \(\sqrt{Dt} = 65\) µm — half a wafer thickness. Even at 200 °C, \(D \approx 7\times10^{-11}\) cm²/s gives ~5 µm per hour. Metals picked up anywhere on a wafer spread everywhere during any thermal step — so cleanliness before furnaces is critical, and gettering layers can collect metals from the whole wafer.

Kinetics beyond diffusion

Chemical reaction rates follow the same Arrhenius form (\(k = A e^{-E_a/k_BT}\)). Rules of thumb you'll meet: thermal oxidation rates (Deal–Grove model: linear then parabolic growth, Unit 8), etch rates of KOH (strongly temperature-dependent — bath temperature is a tightly controlled parameter), and the "thermal budget" concept (time × temperature exposure) used to protect a-Si:H in HJT cells and to avoid oxygen precipitation in TOPCon wafers.

Common misconceptions

Misconceptions

  • "Twice the time, twice the depth." Depth goes as \(\sqrt{t}\): four times the time for twice the depth.
  • "A bit hotter is a bit faster." Arrhenius makes it exponential: for B near 1000 °C, +50 °C multiplies D by about 3.3 (review question 2).
  • "Atoms only diffuse in liquids and gases." Solids diffuse too — slowly, but furnaces make it practical.
  • "Dopant diffusion and contamination are separate problems." Same physics; metals just have much larger D.

What this means for manufacturing and the job

Key equations

Equation In words
\(J = -D\,\partial C/\partial x\) Fick I: flux down the gradient
\(\partial C/\partial t = D\,\partial^2 C/\partial x^2\) Fick II: profiles spread
\(C = C_s\,\text{erfc}(x/2\sqrt{Dt})\) constant-source profile
\(C = \frac{Q}{\sqrt{\pi Dt}}e^{-x^2/4Dt}\) limited-source (drive-in) profile
\(D = D_0 e^{-E_a/k_BT}\) Arrhenius temperature dependence

Review questions

1. A diffusion gives 0.3 µm in 20 min. How long for 0.6 µm at the same temperature?

Depth ∝ √t → 4 × 20 = 80 min.

2. Compute D for boron at 1050 °C and compare to 1000 °C.

D = 0.76 exp(−3.46/(8.617×10⁻⁵ × 1323)) ≈ 5.0 × 10⁻¹⁴ cm²/s — about 3.3× the 1000 °C value.

3. Two measurements: D = 1 × 10⁻¹³ cm²/s at 1100 K and 1 × 10⁻¹¹ at 1250 K. Activation energy?

Eₐ = k ln(D₂/D₁)/(1/T₁ − 1/T₂) = 8.617×10⁻⁵ × ln(100)/(1/1100 − 1/1250) = 8.617×10⁻⁵ × 4.605/1.091×10⁻⁴ ≈ 3.64 eV.

4. Why can a fingerprint on a wafer before a furnace step ruin the whole wafer?

It deposits metals (Fe, Cu, Na...) that diffuse micrometres to tens of micrometres in seconds to minutes at furnace temperatures, spreading lifetime-killing defects throughout the wafer.

5. Which profile (erfc or Gaussian) describes a drive-in after the source is removed?

Gaussian (limited source).

Sources

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