5.4 Diffusion Current

In the previous section we discussed drift current，which flows when an electric field is applied and which follows Ohm’s law. Ohmic behavior is observed in metals and semiconductors and is probably familiar from direct experience. In semiconductors，an additional important component of current can flow if a spatial variation of carrier energies or densities exists within the material. This component of current is called diffusion current. Diffusion current is generally not an important consideration in metals because metals have very high conductivities. The lower conductivity and the possibility of nonuniform densities of carriers and of carrier energies，however，often makes diffusion an important process affecting current flow in semiconductors.

To understand the origin of diffusion current，we consider the hypothetical case of an n-type semiconductor with an electron density that varies only in one dimension（Fig.5.7）. We assume that the semiconductor is at a uniform temperature so that the average energy of electrons does not vary with x；only the density n（x）varies. We consider the number of electrons crossing the plane at x=0 per unit time per unit area. Because they are at finite temperature，the electrons have random thermal motion along the single dimension，but we assume that no electric fields are applied. On the average，the electrons crossing the plane x=0 from the left in Fig.5.7 start at approximately x=-λ after a collision where λ is the mean-free path of an electron，given by λ=vthτcn. The average rate（per unit area）of electrons crossing the plane x=0 from the left，therefore，depends on the density of electrons that started at x=－λ and is

The factor（1/2）appears because half of the electrons travel to the left and half travel to the right after a collision at x=-λ . Similarly，the rate at which electrons cross the plane x=0 from the right is given by

so that the net rate or flux of particle flow per unit area from the left（denoted F）is

Approximating the densities at x=±λ by the first two terms of a Taylor-series expansion，we find

Because each electron carries a charge-q，the particle flow corresponds to a current

We see that diffusion current is proportional to the spatial derivative of the electron density and arises because of the random thermal motion of charged particles in a concentration gradient. For an electron density that increases with increasing x，the gradient is positive，as is the current. Because we expect electrons to flow from the higher density region at the right to the lower density region at the left and current flows in the direction opposite to that of the electrons，the direction of current indicated by Equation（5.10）is physically reasonable.

We can write Equation（5.10）in a more useful form by applying the theorem for the equipartition of energy to this one-dimensional case. This allows us to write

We now use the relationship l=vthτcn to write Equation（5.10）in the form

The quantity in parentheses on the right side of Equation（5.12）is defined as the diffusion coefficientDn and our short derivation has established that

Equation（5.13）is known as the Einstein relation. It relates the two important coefficients that characterize free-carrier transport by drift and by diffusion in a solid. Its validity can be established rigorously by considering the statistical mechanics of solids in detail. Our derivation is aimed at intuition，not physical exactness.

If a field is present，drift，as well as diffusion，occurs. The total current is then the sum of the drift and diffusion currents.

where Ex is the component of the electric field in the x-direction.

A similar expression can be found for the hole diffusion current so that the total hole current is written as

where the negative sign arises because of the positive charge of a hole. The Einstein relation ［Equation（5.13）］ also relates Dp to μp.

Our discussion of diffusion current has been framed in terms of nonuniform carrier concentrations. Although this is the most frequent situation encountered in device analysis and Equations（5.14）and（5.15）are usually adequate，diffusion also occurs if carriers are equal in density in a semiconductor but are more energetic in one region than in another. In this case，other formulations must be used.

We now have four possible independent current mechanisms in a semiconductor. These components are electron drift and diffusion currents and hole drift and diffusion currents. The total current density is the sum of these four components. Fortunately in most situations，we will only need to consider one term at any one time at a particular point in a semiconductor.

Words and Expressions

perturbation n. 动摇，混乱，激励

disturb vt. 扰乱

trajectory n. 运动轨迹

hypothetical adj. 假设的，假定的

flux n. 通量

Taylor-series expansion 泰勒级数展开

rigorously adv. 严厉地，严格地

intuition n. 直觉，直观

Glossary of Important Term

mobility 迁移率

two-dimensional electron gases 二维电子气

drift current 漂移电流

diffusion current 扩散电流

phonon or lattice scattering 晶格（声子）散射

ionized impurity scattering 电离杂质散射

drift-saturation velocity 饱和漂移速度

negative differential resistivity（NDR）负微分电阻率

mean-free path 平均自由程

diffusion coefficient 扩散系数

Einstein relation 爱因斯坦关系

uniform carrier concentrations 均匀掺杂

nonuniform carrier concentrations 非均匀掺杂

Exercises

1. Define carrier mobility. What is the unit of mobility？

2. Explain the temperature dependence of mobility. Why is the carrier mobility a function of the ionized impurity concentrations？