Editing Fick's laws of diffusion (section)

==== CVD method of fabricate semiconductor ====
The wafer is a kind of semiconductor whose silicon substrate is coated with a layer of CVD-created polymer chain and films. This film contains n-type and p-type dopants and takes responsibility for dopant conductions. The principle of CVD relies on the gas phase and gas-solid chemical reaction to create thin films.

The viscous flow regime of CVD is driven by a pressure gradient. CVD also includes a diffusion component distinct from the surface diffusion of adatoms. In CVD, reactants and products must also diffuse through a boundary layer of stagnant gas that exists next to the substrate. The total number of steps required for CVD film growth are gas phase diffusion of reactants through the boundary layer, adsorption and surface diffusion of adatoms, reactions on the substrate, and gas phase diffusion of products away through the boundary layer.

The velocity profile for gas flow is:
<math display="block">\delta(x) = \left( \frac{5x}{\mathrm{Re}^{1/2}} \right) \mathrm{Re}=\frac{v\rho L}{\eta}, </math>
where:
* <math>\delta</math> is the thickness,
* <math>\mathrm{Re}</math> is the Reynolds number,
* {{mvar|x}} is the length of the substrate,
* {{math|1=''v'' = 0}} at any surface,
* <math>\eta</math> is viscosity,
* <math>\rho</math> is density.

Integrated the {{mvar|x}} from {{math|0}} to {{mvar|L}}, it gives the average thickness:
<math display="block">\delta = \frac{10L}{3\mathrm{Re}^{1/2}} . </math>

To keep the reaction balanced, reactants must diffuse through the stagnant boundary layer to reach the substrate. So a thin boundary layer is desirable. According to the equations, increasing vo would result in more wasted reactants. The reactants will not reach the substrate uniformly if the flow becomes turbulent. Another option is to switch to a new carrier gas with lower viscosity or density.

The Fick's first law describes diffusion through the boundary layer. As a function of pressure (''p'') and temperature (''T'') in a gas, diffusion is determined.

<math display="block">D = D_0 \left(\frac{p_0}{p}\right) \left(\frac{T}{T_0}\right)^{3/2} , </math>
where:
* <math>p_0</math> is the standard pressure,
* <math>T_0</math> is the standard temperature,
* <math>D_0</math> is the standard diffusitivity.

The equation tells that increasing the temperature or decreasing the pressure can increase the diffusivity.

Fick's first law predicts the flux of the reactants to the substrate and product away from the substrate:
<math display="block">J = -D_i \left ( \frac{dc_i}{dx} \right ) , </math>
where:
* <math>x</math> is the thickness <math>\delta</math>,
* <math>dc_i</math> is the first reactant's concentration.

In ideal gas law <math>pV = nRT</math>, the concentration of the gas is expressed by partial pressure.

<math display="block">J = - D_i \left ( \frac{p_i-p_0}{\delta RT} \right ) , </math>
where
* <math>R</math> is the gas constant,
* <math>\frac{p_i-p_0}{\delta}</math> is the partial pressure gradient.

As a result, Fick's first law tells us we can use a partial pressure gradient to control the diffusivity and control the growth of thin films of semiconductors.

In many realistic situations, the simple Fick's law is not an adequate formulation for the semiconductor problem. It only applies to certain conditions, for example, given the semiconductor boundary conditions: constant source concentration diffusion, limited source concentration, or moving boundary diffusion (where junction depth keeps moving into the substrate).