Current Density in Semiconductors

This is a continuation from the previous tutorial - carrier recombination.

An electric current in a semiconductor results from the flow of electrons and holes. The current density flowing in a semiconductor is the current flowing through a unit cross-sectional area of the semiconductor; its unit is amperes per square meter.

There are two mechanisms that can cause the flow of electrons and holes: drift, in the presence of an electric field, and diffusion, in the presence of a spatial gradient in the carrier concentration.

The electron current density, $$\pmb{J}_\text{e}$$, and the hole current density, $$\pmb{J}_\text{h}$$, can be expressed as

$\tag{12-58}\pmb{J}_\text{e}=e\mu_\text{e}n\pmb{E}_\text{e}+eD_\text{e}\boldsymbol{\nabla}{n}$

$\tag{12-59}\pmb{J}_\text{h}=e\mu_\text{h}p\pmb{E}_\text{h}-eD_\text{h}\boldsymbol{\nabla}{p}$

respectively, and the total current density is the sum of the two:

$\tag{12-60}\pmb{J}=\pmb{J}_\text{e}+\pmb{J}_\text{h}$

In (12-58) and (12-59), $$e$$ is the electronic charge; $$\mu_\text{e}$$ and $$\mu_\text{h}$$ are the electron and hole mobilities, respectively; $$\pmb{E}_\text{e}$$ and $$\pmb{E}_\text{h}$$ are the electric fields seen by the electrons and holes, respectively; and $$D_\text{e}$$ and $$D_\text{h}$$ are the diffusion coefficients of electrons and holes, respectively.

The electron and hole mobilities strongly depend on temperature, the type of semiconductor, and the impurities and defects in the semiconductor. They generally decrease with increasing concentration of impurities and defects. For most semiconductors of interest, the electron mobility is larger than the hole mobility.

At $$300\text{ K}$$, $$\mu_\text{e}=1350\text{ cm}^2\text{ V}^{-1}\text{ s}^{-1}$$ and $$\mu_\text{h}=480\text{ cm}^2\text{ V}^{-1}\text{ s}^{-1}$$ for intrinsic Si, $$\mu_\text{e}=3900\text{ cm}^2\text{ V}^{-1}\text{ s}^{-1}$$ and $$\mu_\text{h}=1900\text{ cm}^2\text{ V}^{-1}\text{ s}^{-1}$$ for intrinsic Ge, and $$\mu_\text{e}=8500\text{ cm}^2\text{ V}^{-1}\text{ s}^{-1}$$ and $$\mu_\text{h}=400\text{ cm}^2\text{ V}^{-1}\text{ s}^{-1}$$ for intrinsic GaAs.

For nondegenerate semiconductors, the diffusion coefficients are related to the mobilities by the following Einstein relations:

$\tag{12-61}D_\text{e}=\frac{k_\text{B}T}{e}\mu_\text{e}\qquad\text{and}\qquad{D}_\text{h}=\frac{k_\text{B}T}{e}\mu_\text{h}$

From these relations, we have, at $$300\text{ K}$$, $$D_\text{e}=35\text{ cm}^2\text{ s}^{-1}$$ and $$D_\text{h}=12.5\text{ cm}^2\text{ s}^{-1}$$ for intrinsic Si, $$D_\text{e}=100\text{ cm}^2\text{ s}^{-1}$$ and $$D_\text{h}=50\text{ cm}^2\text{ s}^{-1}$$ for intrinsic Ge, $$D_\text{e}=220\text{ cm}^2\text{ s}^{-1}$$ and $$D_\text{h}=10\text{ cm}^2\text{ s}^{-1}$$ for intrinsic GaAs.

For degenerate semiconductors, (12-61) is not valid, and the Einstein relations between the diffusion coefficients and the mobilities must be generalized by taking into account the Fermi integrals.

The mobilities and diffusion coefficients, as well as the effective masses, of electrons and holes for intrinsic Si, Ge, and GaAs are summarized in Table 12-2 below.

The electric fields seen by electrons and holes can be expressed in terms of the gradients in the conduction- and valence-band edges, respectively:

$\tag{12-62}\pmb{E}_\text{e}=\frac{\boldsymbol{\nabla}E_\text{c}}{e}\qquad\text{and}\qquad\pmb{E}_\text{h}=\frac{\boldsymbol{\nabla}E_\text{v}}{e}$

In a homogeneous semiconductor where the conduction- and valence-band edges are parallel to each other, $$\pmb{E}_\text{e}$$ and $$\pmb{E}_\text{h}$$ are generally the same. However, in an inhomogeneous semiconductor, such as in a graded-gap superlattice where the conduction-band edge is not parallel to the valence-band edge, $$\pmb{E}_\text{e}$$ and $$\pmb{E}_\text{h}$$ can be quite different.

Using (12-62), the drift components of the electron and hole current densities can be expressed, respectively, as

$\tag{12-63}\pmb{J}_\text{e}^\text{drift}=e\mu_\text{e}n\pmb{E}_\text{e}=\mu_\text{e}n\boldsymbol{\nabla}E_\text{c}$

$\tag{12-64}\pmb{J}_\text{h}^\text{drift}=e\mu_\text{h}p\pmb{E}_\text{h}=\mu_\text{h}p\boldsymbol{\nabla}E_\text{v}$

Using (12-44) and (12-45) [refer to the electron and hole concentrations tutorial] for the electron and hole concentrations and the relations between the diffusion coefficients and the mobilities of the carriers, the diffusion components of the electron and hole current densities can be expressed, respectively, as

$\tag{12-65}\pmb{J}_\text{e}^\text{diffusion}=eD_\text{e}\boldsymbol{\nabla}n=\mu_\text{e}n\boldsymbol{\nabla}E_\text{Fc}-\mu_\text{e}n\boldsymbol{\nabla}E_\text{c}$

$\tag{12-66}\pmb{J}_\text{h}^\text{diffusion}=-eD_\text{h}\boldsymbol{\nabla}p=\mu_\text{h}p\boldsymbol{\nabla}E_\text{Fv}-\mu_\text{h}p\boldsymbol{\nabla}E_\text{v}$

By combining the drift and diffusion components, we find that the total electron and hole current densities can be simply expressed in terms of the gradients in the quasi-Fermi levels:

$\tag{12-67}\pmb{J}_\text{e}=\mu_\text{e}n\boldsymbol{\nabla}E_\text{Fc}$

$\tag{12-68}\pmb{J}_\text{h}=\mu_\text{h}p\boldsymbol{\nabla}E_\text{Fv}$

Consequently, the total current density can be expressed as

$\tag{12-69}\pmb{J}=\pmb{J}_\text{e}+\pmb{J}_\text{h}=\mu_\text{e}n\boldsymbol{\nabla}E_\text{Fc}+\mu_\text{h}p\boldsymbol{\nabla}E_\text{Fv}$

The relations in (12-67) - (12-69) are quite general. They are valid for both nondegenerate and degenerate semiconductors, which can be either homogeneous or inhomogeneous.

Note, however, that (12-61) is valid only for nondegenerate semiconductors.

Some very important conclusions can be drawn from the relation in (12-69).

A semiconductor in thermal equilibrium carries no net electric current, meaning that $$\pmb{J}=0$$ in thermal equilibrium. We also know that when a semiconductor is in thermal equilibrium, the electrons and holes in it are characterized by a common Fermi level: $$E_\text{F}=E_\text{Fc}=E_\text{Fv}$$. From (12-69), we find that these two facts indicate that $$\boldsymbol{\nabla}E_\text{F}=0$$ when a semiconductor is in thermal equilibrium.

Consequently, a semiconductor in thermal equilibrium is characterized by a single, constant Fermi level throughout its entire volume no matter whether the semiconductor is homogeneous or inhomogeneous and regardless of the detailed structures in the semiconductor.

On the other hand, when a semiconductor carriers an electric current, it must be in a quasi-equilibrium state with separate quasi-Fermi levels: $$E_\text{Fc}\ne{E}_\text{Fv}$$. Furthermore, these quasi-Fermi levels must not be constant in space but must have nonvanishing spatial gradients in order to support an electric current.

Conductivity

The electric conductivity, $$\sigma$$, of a material is the proportionality constant between the current density and the electric field.

In a semiconductor, the conductivity is contributed by both electrons and holes. Only the drift current has to be considered because the the diffusion current is not generated by an electric field.

For a homogeneous semiconductor, $$\pmb{E}_\text{e}=\pmb{E}_\text{h}=\pmb{E}$$. Taking $$\boldsymbol{\nabla}n=\boldsymbol{\nabla}p=0$$ to eliminate the diffusion current, we find from (12-58) - (12-60) that $$\pmb{J}=e(\mu_\text{e}n+\mu_\text{h}p)\pmb{E}=\sigma\pmb{E}$$ for a homogeneous semiconductor.

We thus find the following relation for the conductivity of a semiconductor:

$\tag{12-70}\sigma=e(\mu_\text{e}n+\mu_\text{h}p)$

which is measured per ohm per meter but is usually quoted per ohm per centimeter.

The conductivity of a semiconductor in thermal equilibrium is $$\sigma_0=e(\mu_\text{e}n_0+\mu_\text{h}p_0)$$, often known as the dark conductivity; that of an intrinsic semiconductor is $$\sigma_\text{i}=e(\mu_\text{e}+\mu_\text{h})n_\text{i}$$, known as the intrinsic conductivity.

The resistivity of a semiconductor is simply the inverse of its conductivity: $$\rho=1/\sigma$$, in ohm-meters but usually also given in ohm-centimeters.

As can be seen from (12-70), the conductivity of a semiconductor increases with increasing carrier concentrations. For semiconductors with low impurity concentrations, $$\mu_\text{e}$$ and $$\mu_\text{h}$$ vary little with the impurity concentration; therefore, the conductivity increases with doping density.

The conductivity does not continue to increase linearly with doping density at high impurity concentrations because the mobilities decrease at high impurity concentrations.

Because $$\mu_\text{e}\gt\mu_\text{h}$$ for most semiconductors of interest, an n-type semiconductor generally has a higher conductivity than a p-type one of the same impurity concentration.

The conductivity of a given semiconductor is a strong function of temperature because carrier concentrations and carrier mobilities are both sensitive to tempearture.

Example 12-7

Find the intrinsic conductivity and the intrinsic resistivity of GaAs at $$300\text{ K}$$.

We find from Example 12-2 [refer to the electron and hole concentrations tutorial] that $$n_\text{i}=2.33\times10^{12}\text{ m}^{-3}$$ for GaAs at $$300\text{ K}$$. From Table 12-2, $$\mu_\text{e}=8500\text{ cm}^2\text{ V}^{-1}\text{ s}^{-1}=0.85\text{ m}^2\text{ V}^{-1}\text{ s}^{-1}$$ and $$\mu_\text{h}=400\text{ cm}^2\text{ V}^{-1}\text{ s}^{-1}=0.04\text{ m}^2\text{ V}^{-1}\text{ s}^{-1}$$. Therefore, the intrinsic conductivity is

\begin{align}\sigma_\text{i}&=1.6\times10^{-19}\times(0.85+0.04)\times2.33\times10^{12}\text{ }{\Omega}^{-1}\text{ m}^{-1}\\&=3.32\times10^{-7}\text{ }{\Omega}^{-1}\text{ m}^{-1}\\&=3.32\times10^{-9}\text{ }{\Omega}^{-1}\text{ cm}^{-1}\end{align}

The intrinsic resistivity is

$\rho_\text{i}=\frac{1}{\sigma_\text{i}}=3.01\times10^6\text{ }\Omega\text{ m}=3.01\times10^8\text{ }\Omega\text{ cm}$

Though GaAs is a semiconductor, intrinsic GaAs is often considered to be semi-insulating because of its high resistivity.

The next tutorial covers the topic of semiconductor junctions