Solid state diffusion is a straight forward process and the typical method for introducing dopant atoms into semiconductors. In silicon solar cell processing starting substrates are typically uniformly doped with boron giving a p-type base. The n-type emitter layer is formed through phosphorous doping (see Doping).

### Calculation of Diffusion Profiles (Ghandi[1])

In its simplest form the diffusion process follows Fick's law:

where j is the flux density (atoms cm^{-2}), D is the diffusion coefficient (cm^{2}^{ } s^{-1}), N is the concentration volume (atoms cm^{-3 }) and x is the distance (cm).

The profiles can then be calculated for specific cases. Typical cases are an unlimited source such as heating a wafer in the presence of a phosphorus saturated carrier gas and then turning off the source and driving in the phosphorous atoms on the surface.

#### Diffusion from an Unlimited Source

Diffusions from an unlimited source commonly produce a shallow junction with a very high surface concentration of phosphorus atoms. The diffusion is described by the complementary error function.

where N_{0}_{ } is the impurity concentration at the surface (atoms cm^{-3 }), D is the diffusivity (cm^{2}^{ } s^{-1 }), x is the depth (cm)and t is the time (sec). A simple one-step diffusion is useful where there is no surface passivation of the device.

#### Diffusion from a limited source

Diffusions often consist of a two step process: a short pre-deposition as outlined above, followed by a longer drive in at a higher temperature to provide a deep lightly doped emitter. A simplified analysis of the drive-in assumes that it is at a higher temperature and that the dopant atoms incorporated in the pre-deposition simply redistribute. The final profile is a Gaussian and is described by:

where Q_{0}_{ } is atoms introduced in the pre-deposition (atoms cm^{-2 }), D is the diffusivity (cm^{2}^{ } s^{-1 }), x is the depth (cm) and t is the time (s)

Second order effects cause deviations from the simple models [2] and computer simulations are employed.

- 1. . {VLSI} Fabrication Principles: Silicon and Gallium Arsenide, 2nd Edition. 2nd ed. Wiley-Interscience; 1994.
- 2. . Solar Cells: From Basic to Advanced Systems. New York: McGraw-Hill; 1983.