Editing Fick's laws of diffusion (section)

=== Semiconductor fabrication applications ===
The [[semiconductor]] is a collective term for a series of devices. It mainly includes three categories：two-terminal devices, three-terminal devices, and four-terminal devices. The combination of the semiconductors is called an integrated circuit.

The relationship between Fick's law and semiconductors: the principle of the semiconductor is transferring chemicals or dopants from a layer to a layer. Fick's law can be used to control and predict the diffusion by knowing how much the concentration of the dopants or chemicals move per meter and second through mathematics.

Therefore, different types and levels of semiconductors can be fabricated.

[[Integrated circuit]] fabrication technologies, model processes like CVD, thermal oxidation, wet oxidation, doping, etc. use diffusion equations obtained from Fick's law.

==== 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).

==== Invalidity of Fickian diffusion ====
Even though Fickian diffusion has been used to model diffusion processes in semiconductor manufacturing (including CVD reactors) in early days, it often fails to validate the diffusion in advanced semiconductor nodes (< 90&nbsp;nm). This mostly stems from the inability of Fickian diffusion to model diffusion processes accurately at molecular level and smaller. In advanced semiconductor manufacturing, it is important to understand the movement at atomic scales, which is failed by continuum diffusion. Today, most semiconductor manufacturers use [[random walk]] to study and model diffusion processes. This allows us to study the effects of diffusion in a discrete manner to understand the movement of individual atoms, molecules, plasma etc.

In such a process, the movements of diffusing species (atoms, molecules, plasma etc.) are treated as a discrete entity, following a random walk through the CVD reactor, boundary layer, material structures etc. Sometimes, the movements might follow a biased-random walk depending on the processing conditions. Statistical analysis is done to understand variation/stochasticity arising from the random walk of the species, which in-turn affects the overall process and electrical variations.