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.


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