22.7: Behavior of the Chemical Potential

Last updated
Save as PDF

Page ID: 46337

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

The Fermi-Dirac distribution was introduced in the section The Fermi-Dirac Distribution. The relevant equation to describe the distribution is

\[f(\varepsilon)=\frac{1}{\exp \left((E-\mu) / k_{\mathrm{B}} T\right)+1}\]

so that for a chemical potential, μ, of 5 eV, the distribution takes the form

Fermi-Dirac distribution

as a function of temperature.

One feature that is very important about the Fermi-Dirac distribution is that it is symmetric about the chemical potential. Hence for a simple intrinsic semiconductor, which has equal numbers of electrons in the conduction band and holes in the valence band, and where the density of states is also symmetric about the centre of the band gap, the chemical potential must lie halfway between the valence band and the conductance band, regardless of the temperature, because each electron promoted to the conduction band leaves a hole in the valence band. This is shown in the band diagram below in which energy is plotted vertically against temperature horizontally.

Temperature dependence of an intrinsic semiconductor

[Note that if the density of states is not exactly symmetric about the centre of the band gap, then the chemical potential does not have to be exactly in the centre of the band gap. However, under such circumstances, it will still be extremely close to the centre of the band gap whatever the temperature, and for all practical purposes can be considered to be in the centre of the band gap.]

For an extrinsic semiconductor the situation is slightly more complicated. At absolute zero in an n-type semiconductor, the chemical potential must lie in the centre of the gap between the donor level and the bottom of the conduction band. At low temperatures in such a semiconductor there are more conduction electrons than there are holes. If the donor level is more than half full, the chemical potential must lie somewhere between the donor levels and the conduction band. At higher temperatures, when the donor level is completely depleted of electrons, and the contribution from intrinsic electrons to the overall electrical conductivity becomes more substantial, the chemical potential tends towards that for an intrinsic semiconductor, i.e., halfway between the conduction and valence bands, and therefore in the middle of the band gap.

Temperature dependence of a n-type semiconductor

For p-type semiconductors the behaviour is similar, but the other way around, i.e., the chemical potential starts midway between the valence band and the acceptor levels at absolute zeo and gradually increases in energy as the temperature increases, so that at high temperatures it too is in the middle of the band gap.

Temperature dependence of a p-type semiconductor