Dopant diffusion (diffusion coefficient, junction depth) calculator
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KEY POINTS
Constant-surface predeposition gives a Cs·erfc(x/2√(Dt)) profile and fixed-dose drive-in gives a Gaussian; in both, depth is set by the diffusion length 2√(Dt). The diffusion coefficient D = D0·exp(−Ea/kT) depends exponentially on temperature, rising one to two orders of magnitude per 100 °C. Junction depth is where the dopant concentration equals the substrate concentration.
Inputs
Analytical solutions with intrinsic diffusion coefficients (Sze). Oxidation-enhanced diffusion (OED) and transient enhanced diffusion (TED) from implant damage are not included. Modern processes are simulated with TCAD tools such as Sentaurus Process; this calculator is for estimates and teaching.
Concentration profile
Diffusion coefficient by temperature (cm²/s)
| Temperature | B | P | As | Sb | 1-hour diffusion length of Boron (B) |
|---|---|---|---|---|---|
| 900 °C | 1.04×10⁻¹⁵ | 7.28×10⁻¹⁶ | 1.63×10⁻¹⁶ | 4.47×10⁻¹⁷ | 0.039 µm |
| 950 °C | 4.21×10⁻¹⁵ | 3.20×10⁻¹⁵ | 6.87×10⁻¹⁶ | 1.96×10⁻¹⁶ | 0.078 µm |
| 1000 °C | 1.53×10⁻¹⁴ | 1.25×10⁻¹⁴ | 2.59×10⁻¹⁵ | 7.62×10⁻¹⁶ | 0.148 µm |
| 1050 °C | 5.03×10⁻¹⁴ | 4.41×10⁻¹⁴ | 8.82×10⁻¹⁵ | 2.68×10⁻¹⁵ | 0.269 µm |
| 1100 °C | 1.52×10⁻¹³ | 1.42×10⁻¹³ | 2.75×10⁻¹⁴ | 8.59×10⁻¹⁵ | 0.468 µm |
| 1150 °C | 4.25×10⁻¹³ | 4.21×10⁻¹³ | 7.91×10⁻¹⁴ | 2.54×10⁻¹⁴ | 0.782 µm |
Erfc and Gaussian profiles
Diffusing from a dopant gas or doped oxide while holding the surface concentration constant produces a complementary error function (erfc) profile, and the total amount of dopant grows over time (predeposition). Removing the source and annealing at high temperature then spreads that fixed amount deeper into a Gaussian profile (drive-in). In both cases progress is set by the square root of Dt, the diffusion length 2√(Dt).
The diffusion coefficient changes exponentially with temperature, rising by one to two orders of magnitude for every 100 °C. Scaled transistors need shallow junctions, so the dose is placed precisely by ion implantation and then only activated with anneals that keep Dt tiny, such as spike anneals and flash lamp anneals.
FAQ
What is the difference between predeposition and drive-in?
Predeposition holds the surface concentration constant so the dose grows over time (erfc profile). Drive-in removes the source and spreads a fixed dose deeper (Gaussian profile).
Why do modern processes avoid long diffusions?
Scaled transistors need shallow junctions, so dopants are placed by ion implantation and activated with very short anneals such as spike or flash anneals that keep Dt small.
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Last updated: October 23, 2026