DopingMVP
Diffusion profile (educational)
Visualize how a dopant concentration spreads with depth as diffusion coefficient and time change.
Inputs
Educational model — not a production simulation
Uses a single constant-source (erfc) diffusion profile. Real diffusion involves concentration-dependent coefficients, multiple steps, and later thermal budget that this does not capture.
Characteristic diffusion depth (2√(D·t))
84.85 nm
Depth where C/Cs ≈ 0.16
Interpretation
With a fixed surface concentration, the dopant profile follows C(x) = Cs·erfc(x / (2√(D·t))). The quantity 2√(D·t) sets the characteristic depth — raising the diffusion coefficient (hotter) or the time pushes dopant deeper and makes the profile more graded. This is why thermal budget is carefully controlled: heat that activates dopants also moves them.
Formula
C(x) = Cs · erfc( x / (2√(D·t)) )
Educational erfc profile.
Variables
| Symbol | Variable | Unit |
|---|---|---|
| C(x) | Concentration at depth x | same as Cs |
| Cs | Surface concentration | m⁻³ |
| D | Diffusion coefficient | m²/s |
| t | Diffusion time | s |
| x | Depth | m |
Assumptions & validity
This tool assumes:
- Constant surface concentration (predeposition-style erfc profile).
- Constant, concentration-independent diffusion coefficient.
- A single diffusion step — not a full process simulation.
Worked example
D = 1×10⁻¹⁴ cm²/s, t = 30 min.
2√(D·t) ≈ 85 nm — the characteristic diffusion depth.