Basic diffusion mechanisms and profiles for dopants and impurities into semiconductors are based on a group of equations known as Fick's Laws.
Fick's first law for one-dimensional diffusion is known as
where
'F' is the flux defined as the number of dopant atoms passing through a unit area in a unit of time.
'C' is the dopant concentration per unit volume, and
'D' is the
diffusion coefficient or diffusivity of the semiconductor.
Fick's second law is the continuity equation defined below with the first law substituted appropriately
This equation can be simplified when the diffusion coefficient
'D' is independent of the doping concentration
'C' as follows
If the diffusion coefficient is not independent, the continuity equation becomes very difficult to solve even when basic conditions are applied.
The calculator and graph associated with this page are based on independent diffusion coefficients and low doping concentrations.
The diffusion coefficient can usually be expressed as
where
'Ea' is the activation energy measured in 'eV' of the dopant or impurity being diffused and
'D0' is the diffusion coefficient extrapolated to infinite temperature.
The activation energy is dependent on both the element being diffused and the specific semiconductor that it is being diffused into.
Fick's second law or the continuity equation can be solved for a variety of different initial and boundary conditions. Each solution represents a different
diffusion profile. Two of the most common profiles are available as part of this calculator.
Constant-Surface-Concentration Diffusion
The initial condition for this profile is a dopant concentration of zero inside the semiconductor substrate. For this profile the dopant concentration
at the surface of the semiconductor remains constant, and the concentration eventually goes to zero at some point in the semiconductor.
These conditions can be summarized mathematically as
The solution to Fick's second law meeting these conditions is
where
erfc is the complementary error function. This calculator is equipped to calculate
dopant levels and impurity concentrations at specific depths in various semiconductors for a variety of conditions. It is also capable of
graphing the overall diffusion profile.
Constant-Total-Dopant Diffusion
The initial condition for this profile is the same as the constant-surface profile in that there exists a zero impurity concentration in the semiconductor.
However, the boundary conditions for this profile are different. For this profile, a constant or fixed amount of impurities or dopants is deposited on
the surface of the semiconductor rather than maintaining a set surface concentration as in the previous profile. These boundary conditions can be summarized
mathematically as
Given these conditions, the solution to Fick's second law, the continuity equation, or the diffusion equation is
This profile is a Gaussian distribution and is the other diffusion profile supported by this calculator and graph.
Extrinsic vs. Intrinsic Diffusion
The profiles calculated here are based on intrinsic diffusion. Intrinsic diffusion is represented by constant diffusivities, and doping concentrations being
less than then intrinsic-carrier concentration at the diffusion temperature. The intrinsic-carrier concentration for silicon at 1000°C is 5 X 10
18 cm
-3
and 5 X 10
17 cm
-3 for gallium arsenide. If the dopant-impurity concentration exceeds the intrinsic-carrier concentration at the diffusion temperature, then
the diffusion becomes extrinsic, and the profiles become more complicated.
This calculator does not verify whether the diffusion conditions inputted meet the constraint of
intrinsic diffusion. It is assumed the user is knowledgeable as to whether or not the conditions meet the intrinsic diffusion requirement. This calculator is entirely theoretical
and predicts the diffusion profiles mathematically; however, some diffusion profiles may not be experimentally possible depending on the impurity, the semiconductor, and other diffusion parameters such as temperature.
References
H.C. Casey, and G.L. Pearson, "Diffusion in Semiconductors," in J.H. Crawford, and L.M. Slifkin, Eds.,
Point Defects in Solids, Vol. 2, Plenum, New York, 1975.
S.M. Sze,
Semiconductor Devices: Physics and Technology, 2
nd Ed., Wiley, New York, 2002.
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