Chapter 2: Semiconductor Fundamentals

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2.9. Continuity equation

2.9.1. Derivation
2.9.2. The diffusion equation
2.9.3. Steady state solution to the diffusion equation

2.9.1. Derivation

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The continuity equation describes a basic concept, namely that a change in carrier density over time is due to the difference between the incoming and outgoing flux of carriers plus the generation and minus the recombination. The flow of carriers and recombination and generation rates are illustrated with Figure 2.9.1.

Figure 2.9.1 :Electron currents and possible recombination and generation processes

The rate of change of the carriers between x and x + dx equals the difference between the incoming flux and the outgoing flux plus the generation and minus the recombination:

(2.9.1)

where n(x,t) is the carrier density, A is the area, Gn(x,t) is the generation rate and Rn(x,t) is the recombination rate. Using a Taylor series expansion,

(2.9.2)

this equation can be formulated as a function of the derivative of the current:

(2.9.3)

and similarly for holes one finds:

(2.9.4)

A solution to these equations can be obtained by substituting the expression for the electron and hole current, (2.7.31) and (2.7.32). This then yields two partial differential equations as a function of the electron density, the hole density and the electric field. The electric field itself is obtained from Gauss’s law.

(2.9.5)
(2.9.6)

A generalization in three dimensions yields the following continuity equations for electrons and holes:

(2.9.7)
(2.9.8)

2.9.2. The diffusion equation

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In the quasi-neutral region – a region containing mobile carriers, where the electric field is small - the current is due to diffusion only. In addition, we can use the simple recombination model for the net recombination rate since the recombination rates depend only on the minority carrier density. This leads to the time-dependent diffusion equations for electrons in p-type material and for holes in n-type material:

(2.9.9)
(2.9.10)

2.9.3. Steady state solution to the diffusion equation

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In steady state, the partial derivatives with respect to time are zero, yielding:

(2.9.11)
(2.9.12)

The general solution to these second order differential equations are:

(2.9.13)
(2.9.14)

where Ln and Lp are the diffusion lengths given by:

(2.9.15)
(2.9.16)

The diffusion constants, Dn and Dp, are obtained using the Einstein relations (2.7.29) and (2.7.30). The diffusion equations can also be written as a function of the excess>/i> carrier densities, dn and dp, which are related to the total carrier densities, n and p, and the thermal equilibrium densities, n0 and p0, by:

(2.9.17)
(2.9.18)

yielding:

(2.9.19)
(2.9.20)

The diffusion equation will be used to calculate the diffusion current in p-n junctions and bipolar transistors.

Boulder, December 2004