# Solid State Diffusion

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 phosphorus doping (see Doping). Solid state diffusion. Heating the wafer at a high temperature in an atmosphere containing dopant atoms causes some of the atoms to be incorporated into the top surface of the wafer.

### Calculation of Diffusion Profiles (Ghandi1)

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 (cm2 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 phosphorus 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 N0 is the impurity concentration at the surface (atoms cm-3 ), D is the diffusivity (cm2 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:

$$N(x, t)=\frac{Q}{\sqrt{\pi D t}} \exp \left(-\frac{x^{2}}{4 D t}\right)$$

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

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Doping profiles resulting from a phosphorus pre deposition step plus a high temperature drive in. The calculations assume that the drive in temperature is greater than the pre deposition temperature. The calculations are approximate and do not include second-order effects such as the "kink".3