Band Gap

A band gap is an interval of energy containing no allowed electronic states in the band structure of a material. In a semiconductor or band insulator, the term usually denotes the separation between the highest occupied valence-band states and the lowest unoccupied conduction-band states in the ideal, zero-temperature description. Its magnitude helps determine how readily a material can generate mobile charge carriers and which wavelengths of light it can absorb or emit. A band gap is an energy difference, not a physical space between atoms. (ocw.mit.edu)

Physical origin

An isolated atom has discrete electronic energy levels. When many atoms form a solid, interactions among their atomic orbitals produce large numbers of closely spaced levels that form energy bands. Some energy intervals fall within these bands; others lie between them and contain no allowed states. The Pauli exclusion principle governs how electrons occupy the available states, but the existence and size of a gap depend on the electronic structure and interactions within the solid. (ocw.mit.edu)

In a crystal, the periodic arrangement of atoms makes the potential experienced by an electron periodic. Within a one-electron description based on quantum mechanics, Bloch’s theorem expresses electronic states as waves modulated by functions with the crystal’s periodicity. Their energies form branches $E_n(\mathbf{k})$, where $n$ labels the band and $\mathbf{k}$ is the crystal wave vector. Gaps arise between branches when the ranges of allowed energies do not overlap. (live.ocw.mit.edu)

For an ordinary semiconductor, the band-gap energy is

where $E_{\mathrm{CBM}}$ is the conduction-band minimum and $E_{\mathrm{VBM}}$ is the valence-band maximum. These extrema need not occur at the same wave vector. Band gaps are commonly expressed in electronvolts (eV). (ocw.mit.edu)

Electrical significance

In the elementary band picture, a completely filled band cannot support ordinary electrical conduction by simply redistributing its electrons among nearby states. Conduction becomes possible when electrons occupy a partially filled band or when excitations create electrons in the conduction band and vacancies in the valence band. A valence-band vacancy behaves as a positively charged hole. (ocw.mit.edu)

This picture distinguishes three broad cases:

• Metals: a band is partially occupied, or occupied and unoccupied bands overlap, leaving low-energy electronic states available for conduction.
• Semiconductors: a gap separates valence and conduction bands, but thermal excitation, illumination, or doping can produce appreciable carrier populations.
• Band insulators: a filled valence band is separated from the conduction band by a gap that makes carrier generation difficult under the conditions considered.

There is no universal numerical boundary between a semiconductor and an insulator; wide-gap materials can still function as semiconductors when suitable carriers are introduced. (ocw.mit.edu)

For an intrinsic semiconductor in thermal equilibrium, under the nondegenerate approximation,

where $n_i$ is the intrinsic carrier concentration, $N_c$ and $N_v$ are effective densities of states, $k_{\mathrm B}$ is the Boltzmann constant, and $T$ is absolute temperature. The exponential dependence means that a modest change in gap can strongly alter the carrier concentration. (ioffe.ru)

Semiconductor doping introduces donor or acceptor states and changes carrier populations. Ordinary dilute doping does not simply replace the host material’s band gap with the much smaller energy required to ionize a dopant. At high doping concentrations, however, carrier interactions can modify the band edges and narrow the electronic gap. (ocw.mit.edu)

Direct and indirect gaps

A direct band gap has its conduction-band minimum and valence-band maximum at the same crystal wave vector. An indirect band gap has these extrema at different wave vectors. The distinction concerns crystal momentum, not the spatial direction in which an electron travels. (ocw.mit.edu)

An optical photon carries little momentum compared with typical differences between electronic wave vectors in a crystal. Consequently, a direct interband optical transition is approximately vertical on an energy-versus-wave-vector diagram. A transition across an indirect gap normally also requires a phonon, a quantum of lattice vibration, to supply or remove the required crystal momentum. (ocw.mit.edu)

Direct-gap materials generally have stronger absorption near the band edge and more efficient band-edge radiative recombination than indirect-gap materials. They are therefore particularly useful for light-emitting devices. Indirect-gap materials can nevertheless absorb light and support photovoltaic operation; their weaker near-edge absorption often requires greater absorber thickness or improved light trapping. Optical selection rules also matter: a direct gap does not guarantee that the lowest-energy transition is strongly allowed. (live.ocw.mit.edu)

Representative room-temperature values include:

These are approximate values; temperature, composition, and crystal structure must be specified when comparing precise measurements. (doi.org)

Electronic and optical gaps

The fundamental electronic gap concerns the energy cost of adding and removing an electron. In a many-electron formulation,

where $E(N)$ is the ground-state energy of a system containing $N$ electrons. This is distinct from an optical excitation, which rearranges electrons without changing the total electron number. (pubmed.ncbi.nlm.nih.gov)

An optical gap describes the onset of specified optical excitations. An excited electron and the hole it leaves behind can attract one another and form an exciton. A bound exciton can absorb at an energy below the continuum threshold for creating an unbound electron–hole pair. In a simple description, the exciton transition energy is the corresponding electronic gap minus the exciton binding energy. (nvlpubs.nist.gov)

The relation between gap energy and a characteristic optical wavelength follows from

where $h$ is the Planck constant and $c$ is the speed of light. This relation converts an energy into a wavelength; it does not imply that every material has a sharp absorption threshold exactly at $E_g$. Excitons, phonon-assisted transitions, defects, and absorption tails can alter the observed edge. (ocw.mit.edu)

Factors that change the gap

A band gap is not necessarily a fixed property independent of a material’s environment or preparation.

Temperature and pressure. Many conventional semiconductor gaps decrease as temperature rises. A widely used empirical representation is the Varshni relation,

with material-dependent parameters $a$ and $b$. Pressure can shift different conduction-band valleys by different amounts, potentially changing which valley defines the lowest gap. (ioffe.ru)

Composition. Alloying semiconductors changes their electronic structure. In aluminium gallium arsenide, for example, increasing aluminium content changes both the gap magnitude and whether the lowest gap is direct or indirect. (ioffe.ru)

Size and dimensionality. Confinement in sufficiently small semiconductor structures changes their electronic energy levels. The excitation gap often increases as a nanocrystal becomes smaller, although surface states, boundary conditions, and material-specific electronic structure can complicate this trend. Atomically thin materials can also change from indirect to direct gaps as their layer number decreases. (arxiv.org)

Defects and disorder. Impurities, vacancies, and surfaces can introduce states within the host gap. Such states may enable lower-energy absorption or carrier trapping without eliminating the separation between the underlying bulk band edges. (nist.gov)

Measurement and calculation

Optical spectroscopy can estimate a gap from an absorption edge, but interpretation requires a model of the relevant transitions. A Tauc plot extrapolates a transformed absorption spectrum to estimate an optical gap. The original method was developed for amorphous semiconductors; extensions to crystalline materials require appropriate assumptions about band shape and transition type. Excitonic resonances, overlapping transitions, strong defect absorption, and dimensional confinement can invalidate a simple extrapolation. (doi.org)

Computational band gaps also depend on what quantity and approximation are used. Common local and semilocal implementations of density-functional theory often underestimate experimental fundamental gaps. Hybrid functionals and many-body approaches can improve predictions, but their results depend on the treatment of electronic interactions and numerical convergence. A calculated orbital-energy gap should therefore not automatically be identified with a measured optical gap. (authors.library.caltech.edu)

Technological applications

In solar cells, the gap helps determine which photons can generate charge carriers. Photons below the relevant absorption threshold are not absorbed through the intended interband process, while excess energy from higher-energy photons commonly becomes heat as carriers relax. Gap selection therefore involves a trade-off between absorbing more photons and retaining more energy per generated carrier. (ocw.mit.edu)

In light-emitting diodes and semiconductor lasers, electronic transition energies help set the emission wavelength. Direct-gap materials are especially important because they support efficient radiative recombination. In power electronics, wide-band-gap materials such as silicon carbide and gallium nitride enable devices designed for high voltages and elevated operating temperatures. Their performance also depends on defects, carrier transport, contacts, and device architecture, rather than on gap magnitude alone. (ocw.mit.edu)

Historical development and limits of the band picture

The concept emerged from the quantum theory of electrons in solids. Alan H. Wilson’s 1931 paper The Theory of Electronic Semi-Conductors developed a band-based account of semiconductor behavior, helping connect allowed electronic energies with electrical conduction. (ethw.org)

The elementary distinction between filled and empty bands is not sufficient for every insulating material. In strongly interacting systems, electron–electron interactions and charge-transfer processes can be essential to the origin of an insulating gap. Nickel oxide, for example, exhibits mixed charge-transfer and Mott–Hubbard character rather than being adequately described as a simple noninteracting band insulator. (nist.gov)

References

1. Lecture #13, Band Theory of Solidsocw.mit.edu
2. Part II, Lesson 6: From Atoms to Solidslive.ocw.mit.edu
3. Semiconductorsocw.mit.edu
4. 772/SMA5111 — Compound Semiconductors, Lecture 15ocw.mit.edu
5. Lecture 8 — Materials Parameterslive.ocw.mit.edu
6. Lecture 8 — The Solar Cell Lectureocw.mit.edu
7. Charge Transportocw.mit.edu
8. Optical Properties and Electronic Density of Statesnvlpubs.nist.gov
9. Some Fundamental Issues in Ground-State Density Functional Theory: A Guide for the Perplexedpubmed.ncbi.nlm.nih.gov
10. Resolution of the Band Gap Prediction Problem for Materials Designauthors.library.caltech.edu