Band filling (Burstein-Moss shift)

Band filling is what happens to the optical properties of a semiconductor when free carriers occupy the lowest states of a band. An electron in the valence band can absorb a photon only if the final state in the conduction band is empty; once electrons fill the bottom of the conduction band up to a Fermi level $E_F$ above the edge, photons with energy just above the bandgap can no longer be absorbed, and the absorption edge moves to higher energy. For heavy doping this is called the Burstein-Moss shift, after the 1954 explanations of the wide optical gap of n-type InSb. In n-type GaAs it is about 54 meV at 10¹⁸ cm⁻³ and several hundred meV at 10¹⁹–10²⁰ cm⁻³; the same physics, with carriers injected rather than doped, sets the gain spectrum and much of the refractive-index change in diode lasers and amplifiers.

Size of the shift

The effect becomes large once the semiconductor is degenerate, when the carrier density exceeds the effective density of states, about 4.7 × 10¹⁷ cm⁻³ for the GaAs conduction band at 300 K. For a parabolic band at low temperature the electrons fill a sphere in k-space up to the Fermi wavevector $k_F = (3\pi^2 n)^{1/3}$, and the shift of the edge is

$$\Delta E_{BM} = \frac{\hbar^2 (3\pi^2 n)^{2/3}}{2 m_e^*}.$$

With $m_e^* = 0.067\,m_0$ for GaAs this gives 54 meV at $n$ = 10¹⁸ cm⁻³, 0.16 eV at 5 × 10¹⁸ cm⁻³ and 0.40 eV at 2 × 10¹⁹ cm⁻³, the same estimate quoted under density of states. The shift is large in GaAs and InSb because a light effective mass means few states per unit energy, so a given density fills the band to a higher energy. A vertical transition at $k_F$ also starts below the valence-band maximum, which multiplies the shift by $(1 + m_e^*/m_h^*)$, about 13 % more for heavy holes in GaAs.

Measured shifts are smaller than this estimate. The conduction band flattens at high energy (nonparabolicity), so it holds more states than the parabola assumes, and many-body exchange and correlation shrink the gap itself. This bandgap renormalization grows roughly as $n^{1/3}$ and works in the opposite direction, so the observed edge is the net of the two.

Band filling under injection

In a forward-biased laser or LED, electrons and holes fill both bands at once, each up to its own quasi-Fermi level. For photon energies between $E_g$ and the quasi-Fermi level separation $\Delta F$, more states are filled at the upper level than at the lower, and absorption turns into optical gain: the Bernard-Duraffourg condition $E_g < h\nu < \Delta F$. As the injected density rises, $\Delta F$ grows and the gain spectrum widens toward higher photon energy, so the gain peak moves to shorter wavelength. This is the blue shift of the gain peak with current below threshold and of LED emission with drive current; above threshold the carrier density is clamped and the shift stops.

Electro-refraction

Removing absorption near the edge also changes the refractive index, because the real and imaginary parts of the dielectric function are linked by the Kramers-Kronig relations. Bleaching absorption just above the gap lowers the index at photon energies below it. Bennett, Soref and del Alamo found that in InP, GaAs and InGaAsP near the band edge this band-filling term is the largest of the carrier-induced index changes, larger than the free-carrier (plasma) term and partly offset by bandgap renormalization. Far below the gap, as at 1550 nm in silicon, the free-carrier term described under the plasma dispersion effect dominates. The band-filling index change is the main reason gain and index move together in diode lasers, which the linewidth enhancement factor expresses as a single ratio.

Measurement and pitfalls

The shift is measured from absorption or transmission spectra of doped layers, or from photoluminescence, whose high-energy edge follows the Fermi level. The absorption edge of a degenerate sample is not sharp, because the Fermi distribution is smeared over a few $k_BT$, with $k_BT$ = 26 meV at 300 K, and papers define the edge by different conventions, which can change a reported shift by tens of meV. In a laser, self-heating also moves the gap, so the gain-peak shift with current is measured in short pulses.

Common questions

Why do transparent conducting oxides have a wider optical gap than the undoped material?

Their free-electron densities, of order 10²⁰–10²¹ cm⁻³, fill the conduction band far above its edge. Absorption begins only at the Fermi level, which moves the onset of absorption further into the ultraviolet.

Is the Burstein-Moss shift the same as band filling?

They describe one mechanism. "Burstein-Moss shift" usually means the edge shift from doping; "band filling" also covers injected or photoexcited carriers.

Does band filling raise or lower the refractive index?

It lowers the index below the gap, in the same direction as the free-carrier contribution.

References: E. Burstein, "Anomalous optical absorption limit in InSb," Phys. Rev. 93, 632 (1954); T. S. Moss, "The interpretation of the properties of indium antimonide," Proc. Phys. Soc. B 67, 775 (1954); B. R. Bennett, R. A. Soref and J. A. del Alamo, "Carrier-induced change in refractive index of InP, GaAs, and InGaAsP," IEEE J. Quantum Electron. 26, 113 (1990); L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).