Directions
This simulation shows dopant concentration versus depth in a silicon wafer for the two-step diffusion process used to form p–n junctions. In the first step (predeposition), the surface is held at a constant dopant concentration Cs, which gives a complementary error function (erfc) profile. In the second step (drive-in), the dopant source is removed and the fixed amount of dopant introduced during predeposition diffuses deeper into the wafer, which gives an approximately Gaussian profile.
Select the dopant (boron or phosphorus) and set the temperature and time for each step with the sliders. The wafer has a uniform background dopant concentration CB of the opposite type. The junction depth xj is where the diffused dopant concentration equals CB; it is marked on the plot with a dashed vertical line.
Check "exact drive-in" to compare the Gaussian approximation with the exact solution for diffusion from the predeposited erfc profile. The Gaussian is accurate when the drive-in diffusion length is much larger than the predeposition diffusion length . Concentration is plotted on a log scale.
Details
Dopant diffusion in silicon follows Fick's second law with a constant diffusivity:
The diffusivity depends on temperature through an Arrhenius expression:
where for boron D0 = 0.76 cm2/s and Ea = 3.46 eV, and for phosphorus D0 = 3.85 cm2/s and Ea = 3.66 eV (intrinsic diffusivities); kB = 8.617 × 10−5 eV/K and T is in kelvin.
Predeposition (constant surface concentration Cs, initially undoped by this species, semi-infinite wafer):
The dose (amount of dopant per unit area) introduced during predeposition is:
Drive-in (no flux through the surface, so the dose Q is conserved). If the predeposited dopant is treated as a thin layer at the surface (a delta function), the solution is a Gaussian:
The exact drive-in solution starts from the erfc profile and uses the method of images for the zero-flux surface:
This integral is evaluated numerically. It reduces to the Gaussian when ≫ .
Junction depth: the junction forms where the diffused dopant concentration equals the background concentration CB, C(xj) = CB. After predeposition:
After drive-in (Gaussian):
For the exact solution, xj is found numerically. The model neglects concentration-dependent diffusivity at high doping, oxidation-enhanced diffusion, segregation into a growing oxide, and diffusion that occurs during the drive-in temperature ramp.
Nomenclature
About
This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.