Conduction band and valence band

The electron states of a crystal group into bands of allowed energy separated by gaps. In a semiconductor at zero temperature, the highest band that is completely filled is the valence band and the lowest band that is completely empty is the conduction band; the energy between the top of the valence band, $E_v$, and the bottom of the conduction band, $E_c$, is the bandgap $E_g$. A full band carries no current and neither does an empty one, so conduction requires electrons in the conduction band or vacancies, called holes, in the valence band. Typical gaps are 1.12 eV for silicon, 1.424 eV for GaAs and 0.75 eV for InGaAs lattice-matched to InP.

Electrons and holes

Exciting an electron across the gap, by heat, by a photon with energy above $E_g$ or by injection at a junction, leaves a hole behind. The hole behaves as a positive carrier with its own effective mass, set by the curvature of the valence band; electrons near the conduction-band minimum have a mass set by that band's curvature. In GaAs the electron mass is $0.067\,m_0$. The valence band of the common semiconductors has three subbands near its top: heavy holes (about $0.51\,m_0$ in GaAs), light holes ($0.082\,m_0$) and a split-off band lying 0.34 eV lower in GaAs. Heavy and light holes are degenerate at the band edge in bulk material; strain and quantum confinement in a quantum well separate them, which changes the gain and polarization of a laser.

Occupation and the Fermi level

Which states are occupied at temperature $T$ is described by the Fermi-Dirac distribution about the Fermi level $E_F$. In intrinsic silicon $E_F$ lies near midgap, 0.56 eV below $E_c$, and the occupation probability of a state at the band edge is about $\exp(-0.56/0.02585) = 3.9 \times 10^{-10}$; with the large number of available states this still yields an intrinsic density of about $10^{10}$ cm⁻³. When the Fermi level is several $k_BT$ below $E_c$, the electron density is

$$n = N_c \exp\!\left[-\frac{E_c - E_F}{k_BT}\right],$$

where $N_c$, the effective density of states of the conduction band, follows from the density of states and the effective mass:

$$N_c = 2\left(\frac{2\pi m_e^* k_BT}{h^2}\right)^{3/2}.$$

For GaAs at 300 K, $m_e^* = 0.067\,m_0$ gives $N_c = 4.4 \times 10^{17}$ cm⁻³, and the intrinsic density works out to about $2 \times 10^6$ cm⁻³; the 0.30 eV wider gap accounts for a factor of about 360 and the smaller density of states for the rest of the factor of several thousand below silicon. Doping moves $E_F$: donors raise it toward $E_c$ (in silicon with $10^{17}$ donors per cm³ it sits about 0.15 eV below $E_c$), and acceptors lower it toward $E_v$. At a p-n junction in equilibrium $E_F$ is flat, so the bands bend across the depletion region by the built-in voltage.

Direct and indirect gaps

The bands are functions of crystal momentum as well as energy. In GaAs, InP and InGaAs the conduction-band minimum and valence-band maximum both sit at zero momentum, a direct gap, and an electron can drop to the valence band by emitting a photon alone. In silicon and germanium the conduction-band minimum is at a different momentum, an indirect gap, and emission needs a phonon as well, which makes it slow and inefficient. The direct vs indirect bandgap entry compares the consequences for lasers and detectors.

Photon energies and the band edges

Because carriers relax within each band in less than a picosecond, much faster than they recombine, electrons collect near the bottom of the conduction band and holes near the top of the valence band before recombining. The emitted photon energy is therefore close to $E_g$. For bulk material the spontaneous emission spectrum peaks about $k_BT/2$ above the gap, 13 meV at 300 K: for GaAs that is 1.437 eV, or 863 nm against the 871 nm gap wavelength. The same edges set the long-wavelength limit of absorption, discussed under absorption edge. Under forward bias or optical pumping the single Fermi level splits into separate electron and hole quasi-Fermi levels; gain appears at photon energies between $E_g$ and their separation, and the spectrum shifts with carrier density as states higher in each band fill.

Band diagrams in devices

Device drawings show $E_c$ and $E_v$ against position. Band bending marks electric fields, steps mark the band offsets at a heterojunction, and flat regions mark neutral material. In a double-heterostructure laser the narrow-gap active layer forms a well in both bands, trapping electrons in the conduction band and holes in the valence band in the same layer. The diagram plots electron energy, so holes lower their energy by moving upward on it, collecting at local maxima of $E_v$ where electrons would collect at minima of $E_c$.

Common questions

What is the difference between the conduction band and the valence band?

The valence band is the highest band that is full in the ground state, formed from the bonding electron states; the conduction band is the next band up, empty in the ground state. Electrons conduct when promoted to the conduction band, and the vacancies they leave in the valence band conduct as holes.

Why does a semiconductor emit light at its bandgap energy?

Electrons and holes cool to the edges of their bands within a picosecond and recombine there, so the photon carries the energy difference between the two band edges, $E_g$ plus a few $k_BT$ at most.

Where is the Fermi level relative to the bands?

In an undoped semiconductor near the middle of the gap; in n-type material closer to the conduction band, and in p-type closer to the valence band. In heavily doped material it can move into a band.

References: S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley, 2007); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005); I. Vurgaftman, J. R. Meyer and L. R. Ram-Mohan, J. Appl. Phys. 89, 5815 (2001).