Direct and indirect band gaps

Implications for light absorption

The exact reverse of radiative recombination is light absorption. For the same reason as above, light with a photon energy close to the band gap can penetrate much farther before being absorbed in an indirect band gap material than a direct band gap one (at least insofar as the light absorption is due to exciting electrons across the band gap).

This fact is very important for photovoltaics (solar cells). Crystalline silicon is the most common solar-cell substrate material, despite the fact that it is indirect-gap and therefore does not absorb light very well. As such, they are typically hundreds of microns thick; thinner wafers would allow much of the light (particularly in longer wavelengths) to simply pass through. By comparison, thin-film solar cells are made of direct band gap materials (such as amorphous silicon, CdTe, CIGS or CZTS), which absorb the light in a much thinner region, and consequently can be made with a very thin active layer (often less than 1 micron thick).

The absorption spectrum of an indirect band gap material usually depends more on temperature than that of a direct material, because at low temperatures there are fewer phonons, and therefore it is less likely that a photon and phonon can be simultaneously absorbed to create an indirect transition. For example, silicon is opaque to visible light at room temperature, but transparent to red light at liquid helium temperatures, because red photons can only be absorbed in an indirect transition.

Formula for absorption

A common and simple method for determining whether a band gap is direct or indirect uses absorption spectroscopy. By plotting certain powers of the absorption coefficient against photon energy, one can normally tell both what value the band gap is, and whether or not it is direct.

For a direct band gap, the absorption coefficient ${\displaystyle \alpha }$ is related to light frequency according to the following formula:

${\displaystyle \alpha \approx A^{\ast }\sqrt{h\nu -E_{\text{g}}}}$, with ${\displaystyle A^{\ast }=\frac{q^2{x}_{vc}^2(2m_{\text{r}}{)}^{3/2}}{{\lambda }_0{ϵ}_0{\hslash }^{3}n}}$

where:

• ${\displaystyle \alpha }$ is the absorption coefficient, a function of light frequency
• ${\displaystyle \nu }$ is light frequency
• ${\displaystyle h}$ is the Planck constant (${\displaystyle h\nu }$ is the energy of a photon with frequency ${\displaystyle \nu }$)
• ${\displaystyle \hslash }$ is the reduced Planck constant (${\displaystyle \hslash =h/2\pi }$)
• ${\displaystyle E_{\text{g}}}$ is the band gap energy
• ${\displaystyle A^{\ast }}$ is a certain constant, with formula above
• ${\displaystyle m_{\text{r}}=\frac{m_{\text{h}}^{\ast }{m}_{\text{e}}^{\ast }}{m_{\text{h}}^{\ast }+m_{\text{e}}^{\ast }}}$, where ${\displaystyle m_{\text{e}}^{\ast }}$ and ${\displaystyle m_{\text{h}}^{\ast }}$ are the effective masses of the electron and hole, respectively (${\displaystyle m_{\text{r}}}$ is called a "reduced mass")
• ${\displaystyle q}$ is the elementary charge
• ${\displaystyle n}$ is the (real) index of refraction
• ${\displaystyle {ϵ}_0}$ is the vacuum permittivity
• ${\displaystyle {\lambda }_0}$ is the vacuum wavelength for light of frequency ${\displaystyle \nu }$
• ${\displaystyle x_{vc}}$ is a "matrix element", with units of length and typical value the same order of magnitude as the lattice constant.

This formula is valid only for light with photon energy larger, but not too much larger, than the band gap (more specifically, this formula assumes the bands are approximately parabolic), and ignores all other sources of absorption other than the band-to-band absorption in question, as well as the electrical attraction between the newly created electron and hole (see exciton). It is also invalid in the case that the direct transition is forbidden, or in the case that many of the valence band states are empty or conduction band states are full.

On the other hand, for an indirect band gap, the formula is:

${\displaystyle \alpha \propto \frac{(h\nu -E_{\text{g}}+E_{\text{p}}{)}^2}{\exp (\frac{E_{\text{p}}}{kT})-1}+\frac{(h\nu -E_{\text{g}}-E_{\text{p}}{)}^2}{1-\exp (-\frac{E_{\text{p}}}{kT})}}$

where:

• ${\displaystyle E_{\text{p}}}$ is the energy of the phonon that assists in the transition
• ${\displaystyle k}$ is the Boltzmann constant
• ${\displaystyle T}$ is the thermodynamic temperature

This formula involves the same approximations mentioned above.

Therefore, if a plot of ${\displaystyle h\nu }$ versus ${\displaystyle {\alpha }^2}$ forms a straight line, it can normally be inferred that there is a direct band gap, measurable by extrapolating the straight line to the ${\displaystyle \alpha =0}$ axis. On the other hand, if a plot of ${\displaystyle h\nu }$ versus ${\displaystyle {\alpha }^{1/2}}$ forms a straight line, it can normally be inferred that there is an indirect band gap, measurable by extrapolating the straight line to the ${\displaystyle \alpha =0}$ axis (assuming ${\displaystyle E_{\text{p}}\approx 0}$).

Other aspects

In some materials with an indirect gap, the value of the gap is negative. The top of the valence band is higher than the bottom of the conduction band in energy. Such materials are known as semimetals.