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Spatial variation of the valence band and conduction band edges associated with a varying electric potential. Near a semiconductor boundary it can deplete mobile charge carriers or create a semiconductor inversion layer.

A band gap is an energy interval between adjacent energy bands containing no allowed bulk state.

A band offset is an interface discontinuity in a conduction band or valence band edge.^[ref]

For a chosen two-material example, let the central material A have $E_{c,A}=1$ and $E_{v,A}=0$, and the surrounding material B have $E_{c,B}=1.6$ and $E_{v,B}=-0.4$, in stipulated energy units. Define positive confinement barriers by $\Delta E_c=E_{c,B}-E_{c,A}$ and $\Delta E_v=E_{v,A}-E_{v,B}$. Then $\Delta E_c=0.6$, $\Delta E_v=0.4$, and the band gaps obeyThis identity determines the sum of the band offsets; a band gap difference alone cannot determine their partition. The chosen numbers illustrate the definition and do not specify a measured material pair.

State counting with total degeneracy $g$ gives a constant two-dimensional density of states $gm/(2\pi {\hslash }^2)$ above a parabolic conduction band edge. Integrating the Boltzmann occupation yields the displayed areal carrier density. For one valley and two spin states the coefficient is $m{k}_{B}T/(\pi {\hslash }^2)$. Applying the same count to holes gives $np=N_c{N}_v{e}^{-E_g/(k_{B}T)}$ and the intrinsic density $n_i=\sqrt{N_c{N}_v}{e}^{-E_g/(2k_{B}T)}\propto T{e}^{-E_g/(2k_{B}T)}$, distinct from the three-dimensional $T^{3/2}$ prefactor.

An electron occupying an extended state in a conduction band. Its response to an applied electric field depends on its band group velocity, occupation and scattering. Merely counting such electrons does not by itself determine electrical conductivity.

A semiconductor whose mobile charge carrier population requires Fermi-Dirac statistics rather than a dilute Maxwell-Boltzmann distribution. In a strongly degenerate n-type semiconductor, the Fermi level lies inside the conduction band; in a strongly degenerate p-type semiconductor, it lies inside the valence band. The occupied and empty quantum states on opposite sides of a tunnel diode can then overlap in energy. Heavy semiconductor doping both changes these occupations and narrows the depletion region; a narrow barrier and accessible final quantum states serve different roles.

A band gap whose conduction band minimum and valence band maximum occur at the same crystal momentum, allowing an optical transition without a momentum-balancing phonon.

The number of dopant sites per volume. For a roughly homogeneous three-dimensional distribution, the characteristic site separation is of order $n^{-1/3}$. This counts introduced sites rather than automatically counting free conduction band electrons: semiconductor carrier freeze-out or compensation can make the mobile carrier density much smaller.

An electron hole is an unoccupied electron quantum state in an otherwise nearly full energy band, often the valence band. “Filled” refers to occupation of the allowed quantum states, not to filling all space with a continuous fluid of electrons. Each state, including its spin label, can hold at most one electron by the Pauli exclusion principle. The almost full energy band and its few missing occupants are therefore simpler to describe as a background plus a few electron holes. The electron hole is a quasiparticle, not a Positron inserted into the crystal.

Removing an electron of electric charge $-e$ changes the background's electric charge by $+e$. Thus an electron hole carries positive electric charge relative to that background. Promoting an electron into the conduction band creates an electron hole without changing the whole crystal's electric charge: $(-e)+(+e)=0$. An unbound pair still consists of two separate charge carriers; only a bound state of their relative motion is an exciton.

The local cartoon explains why a vacancy can move: when a neighboring electron fills it, another vacancy appears where that electron was. The vacancy has moved opposite to that particular transfer of an electron. In an extended Bloch state, use crystal momentum instead of imagining a permanently empty atomic site. For a completely filled energy band the summed electric current vanishes. Omitting one electron with group velocity ${\mathbf{v}}_v$ removes its contribution $-e{\mathbf{v}}_v$, leaving a contribution $+e{\mathbf{v}}_v$. This fixes both the positive electric charge and the velocity convention of the electron hole.

More precisely, near a parabolic valence band maximum, set its energy $E_v(0)$ as reference and define the electron hole wave vector by ${\mathbf{k}}_h=-{\mathbf{k}}_v$. ThenThe occupied electron branch has negative curvature there, while the electron hole branch has positive effective mass $m_h^{\ast }$. The velocity equality compares the electron hole with the missing Bloch state's group velocity; it does not contradict the opposite vacancy motion in a local transfer cartoon. The sign conventions are listed in Phys 460 Lecture 17.

Symmetry of the excitation spectrum under exchange of conduction band electrons and valence band holes, after choosing a central zero of energy. In a two-band model with equal effective masses and equal degeneracy factors, charge neutrality in an intrinsic semiconductor fixes the chemical potential at that central energy. Indeed the Fermi-Dirac distribution obeys $1-f(-E;\mu =0)=f(E;\mu =0)$, so excitation populations are equal at every positive energy.

An exciton is a bound state of an electron and an electron hole. Their opposite electric charges attract through the Coulomb potential, producing a neutral quasiparticle with internal relative motion and motion of its center of mass. In a semiconductor, promoting an electron from the valence band into the conduction band leaves an electron hole; the pair can occupy either an exciton bound state or an unbound scattering state. Merely creating the two charge carriers does not establish binding. A neutral exciton can transport excitation energy without transporting net electric charge. Coupling this material excitation strongly to a photon can form a polariton.

The striking optical consequence is that an exciton optical transition can absorb a photon with less energy than the threshold for creating unbound electrons and electron holes. The difference is the exciton binding energy, not an extra one-electron energy band inside the band gap. The Wannier-Mott exciton supplies a useful large-radius model; the Frenkel exciton describes the opposite, tightly localized limit.^[ref]

A zone-centre effective-mass description retaining the s-like conduction band, heavy-hole and light-hole branches of the p-like $j=3/2$ valence manifold, and the spin-orbit-split $j=1/2$ valence branch. Each named branch includes spin multiplicity; four bands does not mean four individual quantum states.

A shallow donor impurity in an isotropic semiconductor can be approximated by an electron of effective mass $m^{\ast }$ in a Coulomb potential screened by relative permittivity ${ϵ}_r$. Its ground-state binding energy and radius areThe donor level lies $E_B$ below the conduction band. These are magnitudes: the bound electron's energy relative to the band edge is negative. Band anisotropy, short-distance impurity structure and screening by mobile carriers can invalidate the simple Bohr model rescaling.

An impurity level is an electronic energy level associated with a dopant or defect in a semiconductor. A shallow donor impurity introduces a level just below the conduction band, and a shallow acceptor impurity introduces one just above the valence band. Under electrostatic band bending, these levels shift with the local band edges in the same material; their occupation is determined relative to the equilibrium chemical potential.

An indirect band gap separates a valence band maximum and a conduction band minimum at different values of crystal momentum. A near-edge radiative transition then usually needs a phonon or another momentum-transfer process in addition to the photon, whose momentum is small on the Brillouin zone scale. A direct band gap has the extrema at the same crystal momentum and more readily supports efficient band-edge light emission. A material can have direct transitions at higher energy while its smallest gap is indirect.

Quantum tunnelling between different energy bands, including transfer between the valence band and conduction band in a tunnel diode.

For nondegenerate conduction bands and valence bands, charge neutrality gives $N_c{e}^{-(E_c-{\mu }_i)/(k_{B}T)}=N_v{e}^{-({\mu }_i-E_v)/(k_{B}T)}$. Taking logarithms yieldsFor single spin-degenerate parabolic extrema this becomes $(E_c+E_v)/2+(3k_{B}T/4)\ln (m_h^{\ast }/m_e^{\ast })$. Valley multiplicities or additional hole bands belong in $N_c,N_v$ and change the logarithm. The intrinsic chemical potential lies exactly at midgap only when the two effective density of states values agree.

For an ideal MOSFET with a nondegenerate p-type semiconductor of acceptor density $N_A$, define ${\varphi }_F=(k_{B}T/e)\ln (N_A/n_i)>0$, where $n_i$ is the intrinsic carrier concentration. A positive surface electric potential ${\psi }_s$ lowers the local conduction band and valence band by $e{\psi }_s$. The surface electron density is $n_s=(n_i^2/N_A)\exp (e{\psi }_s/k_{B}T)$. Equality $n_s=N_A$ gives the conventional onset of strong inversion at ${\psi }_s=2{\varphi }_F$. The earlier crossover $n_s=p_s=n_i$ occurs at ${\psi }_s={\varphi }_F$. Neither condition requires the Fermi level to enter the conduction band.

For a representative unstrained zinc blende structure III-V semiconductor such as gallium arsenide, take the valence-band maximum at the Gamma point as zero. The near-centre ordering is an s-like ${\Gamma }_{6c}$ conduction band doublet above a p-like ${\Gamma }_{8v}$ valence quartet. Spin-orbit interaction puts the p-like ${\Gamma }_{7v}$ doublet below the quartet by ${\Delta }_{\mathrm{so}}$. These are the zone-centre bands of a zinc-blende III-V semiconductor.

The $j=3/2$ quartet produces the heavy-hole and light-hole bands. Along a chosen symmetry direction, their components have $|m_j|=3/2$ and $1/2$ respectively; their different curvatures define different positive hole effective masses. They are degenerate at zero wavevector in the unstrained cubic crystal, but separate away from it. The $j=1/2$ doublet is the split-off valence band. To leading isotropic order,Valence-band curvature is negative for electrons; the corresponding hole masses are positive. Real valence bands warp with direction, so these are illustrative effective-mass parabolas rather than an exact spherical band structure.

The k·p perturbation theory couples the s-like and p-like states through momentum matrix elements. It controls the conduction-band mass, nonparabolicity and mixing of hole characters. Strain or quantum confinement can split the heavy/light degeneracy even at zero in-plane wavevector. The lack of inversion symmetry in zinc blende also permits spin splitting away from the zone centre; time-reversal symmetry still protects the zone-centre doublets.

The s–p electric-dipole transitions near $\Gamma$ are optically important and produce structures at $E_g$ and $E_g+{\Delta }_{\mathrm{so}}$ with polarization-dependent weights. For direct band gap compounds the conduction minimum is also at $\Gamma$, allowing near-edge absorption and emission without a momentum-supplying phonon. Not every III-V compound is direct-gap, and wurtzite materials have an additional crystal-field splitting; the cubic scheme should not be imposed on every III-V semiconductor. ^[ref]

In a weakly doped n-type semiconductor, the mobile electrons occupy states near the bottom of the conduction band. Their local band curvature is positive, so their band effective mass is positive and approximately constant sufficiently near that minimum. For the usual $\beta <0$ convention, expand about $k=0$:Stronger overlap gives larger $|\beta |$, greater curvature and a smaller mass within this model. Positive $\beta$ simply places the minimum at the zone boundary and again gives a positive edge mass. A negative electron mass near a band maximum instead motivates the positive-mass electron hole description; it is not the usual conduction-edge situation for an n-type semiconductor. No numerical comparison with $m_e$ follows without values of $a$ and $\beta$.