# Diffusion basics

The diffusion of dopants into silicon via high-temperature thermal processes is one method in which silicon wafers are doped with extrinsic elements such as boron or phosphorous. During a diffusion process, extrinsic elements are introduced, commonly in a gaseous or liquid phase, at high temperature and come into contact with the silicon wafer. A commercial diffusion process may consist of one or two steps including, a deposition step in which the dopant source is supplied into the furnace and a drive-in step, in which the source is cut-off and no further dopants are introduced into the furnace. The deposition process, if performed in a two-step process, is typically at a lower temperature than the drive-in process. During this step, the target material is coated in dopant source to supply the surface with a uniform diffusion source. The point of the drive‑in step is to provide the thermal energy for the dopants to diffuse into the material. A two-step process is potentially more time and resource consuming than a single step process. However, the separation of the drive-in and deposition processes allows for greater control of the resulting diffused atom profile and hence the overall performance of the device.

The depth and the profile of the dopants can be estimated using the Fick’s law. Solid solubility determines the dopant concentration at the surface by pre-deposition process.

Atomic diffusion in solid state materials such as silicon is driven by Fick’s law. Atoms diffuse from the high concentration regions to low concentration regions. The concentration of the dopants, C depends on the depth x, and the time, t i.e., C(x,t).

The number of atoms that diffuse as a unit area in unit time is determined using Fick’s 1st law, $F=-D\frac{\partial C}{\partial x}$

where F is the atomic flux, D is the diffusivity and is the concentration gradient. The diffusivity is a function of temperature which can be expressed in Arrhenius form, $D=D_{0}exp^{-\frac{E_{a}}{kT}}$

where D0 is the pre-exponential constant, Ea is the activation energy, k is the Boltzmann constant and T is the temperature. The diffusion length, λ is the distance at which the concentration of the diffusing atoms drops to a lower value. it can be calculated based on diffusivity and the time: $\lambda=\sqrt{4Dt}$

Fick’s 2nd law is used to determine the dopant concentration in a cross-sectional area as a function of time. The change in concentration with time is proportional to the diffusion flux, $\frac{\partial C(x,t)}{\partial t}=-\frac{\partial F}{\partial x}$

Which can be expressed as, $\frac{\partial C(t)}{\partial t}=-\frac{\partial}{\partial x}(D\frac{\partial C}{\partial x})$

If the diffusivity (D) is constant, it can also be re-written as: $\frac{\partial C(x,t)}{\partial t}=-D\frac{\partial^{2} C}{\partial x^{2}}$

At steady stated, therefore and C(x,t) = a + bx, which is a linear concentration profile over distance that is independent of time. This would be an example of thermal oxidation of silicon.

The diffusion process is schematically shown in the animation below.