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Energy gap

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In solid-state physics, an energy gap or band gap is an energy range in a solid where no electron states exist, i.e. an energy range where the density of states vanishes.

Especially in condensed matter physics, an energy gap is often known more abstractly as a spectral gap, a term which need not be specific to electrons or solids.

Band gap

[edit]

If an energy gap exists in the band structure of a material, it is called band gap. The physical properties of semiconductors are to a large extent determined by their band gaps, but also for insulators and metals the band structure—and thus any possible band gaps—govern their electronic properties.^[1]^[2]

Superconductors

[edit]

For superconductors the energy gap is a region of suppressed density of states around the Fermi energy, with the size of the energy gap much smaller than the energy scale of the band structure. The superconducting energy gap is a key aspect in the theoretical description of superconductivity and thus features prominently in BCS theory. Here, the size of the energy gap indicates the energy gain for two electrons upon formation of a Cooper pair.^[1]^[2]^[3] If a conventional superconducting material is cooled from its metallic state (at higher temperatures) into the superconducting state, then the superconducting energy gap is absent above the critical temperature $T_{\rm {c}}$ , it starts to open upon entering the superconducting state at $T_{\rm {c}}$ , and it grows upon further cooling. BCS theory predicts that the size $\Delta$ of the superconducting energy gap for conventional superconductors at zero temperature scales with their critical temperature $T_{\rm {c}}$ :^[3] $\Delta (T=0)=1.764\,k_{\rm {B}}T_{\rm {c}}$ (with Boltzmann constant $k_{\rm {B}}$ ).

Pseudogap

[edit]

If the density of states is suppressed near the Fermi energy but does not fully vanish, then this suppression is called pseudogap. Pseudogaps are experimentally observed in a variety of material classes; a prominent example are the cuprate high-temperature superconductors.^[4]

Hard gap vs. soft gap

[edit]

If the density of states vanishes over an extended energy range, then this is called a hard gap. If instead the density of states exactly vanishes only for a single energy value (while being suppressed, but not vanishing for nearby energy values), then this is called a soft gap. A prototypical example of a soft gap is the Coulomb gap that exists in localized electron states with Coulomb interaction.^[5]

References

[edit]

^ ^a ^b Neil N. Ashcroft; N. David Mermin (1976). Solid State Physics. Saunders College. ISBN 0-03-083993-9.
^ ^a ^b Charles Kittel (1996). Introduction to Solid State Physics (7th ed.). John Wiley & Sons. ISBN 0-471-11181-3.
^ ^a ^b Michael Tinkham (1996). Introduction to Superconductivity (2nd ed.). McGraw-Hill. ISBN 0-07-064878-6.
^ Timusk, Tom; Statt, Bryan (1999). "The pseudogap in high-temperature superconductors: an experimental survey". Rep. Prog. Phys. 62 (1): 61–122. arXiv:cond-mat/9905219. Bibcode:1999RPPh...62...61T. doi:10.1088/0034-4885/62/1/002. S2CID 17302108.
^ Efros, A.L.; Shklovskii, B.I. (1975). "Coulomb gap and low temperature conductivity of disordered systems". J. Phys. C: Solid State Phys. 8 (4): L49 – L51. Bibcode:1975JPhC....8L..49E. doi:10.1088/0022-3719/8/4/003.

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In condensed matter physics, an energy gap is a range of energies in which no electronic or quasiparticle states can exist, often referred to more abstractly as a spectral gap. While most commonly referring to the electronic band gap in solids—the minimum energy difference between the top of the valence band and the bottom of the conduction band, representing a forbidden energy range where no electron states exist—the term also applies to other phenomena such as the superconducting energy gap and pseudogaps, discussed in subsequent sections.^[1]^[2]^[3] This band gap fundamentally determines the material's electrical and optical properties, classifying solids as metals (overlapping bands, no gap), semiconductors (band gaps typically ~0.1–4 eV, including wide-bandgap types up to ~6 eV), or insulators (gaps typically >4 eV).^[4]^[2]^[5] In semiconductors, the band gap governs conductivity by controlling the thermal excitation of electrons from the valence band to the conduction band, with smaller gaps enabling higher carrier concentrations and thus greater electrical response, especially as temperature increases.^[4]^[2] Optically, the band gap dictates light absorption and emission wavelengths, as photons with energy matching or exceeding the gap can excite electrons, influencing material color—for instance, silicon (1.1 eV gap) appears dark due to visible light absorption, while wider-gap materials like gallium nitride (3.4 eV) transmit visible light and emit in the ultraviolet.^[2] Band gaps are categorized as direct (valence band maximum and conduction band minimum at the same momentum, enabling efficient photon interactions) or indirect (requiring phonon assistance for momentum conservation, reducing optical efficiency), with direct-gap materials like gallium arsenide (1.42 eV) excelling in light-emitting applications.^[3] The significance of the energy gap extends to diverse technologies, including optoelectronics where it enables devices such as LEDs, lasers, and solar cells by tuning emission or absorption spectra, and power electronics where wide-bandgap semiconductors like silicon carbide (3.2 eV) support high-voltage operation with low losses.^[3]^[2] Bandgap engineering—through alloying, strain, or nanostructuring—allows precise control over these properties, as seen in III-V compounds for infrared detectors or ultraviolet emitters.^[2]

Fundamentals

Definition

In quantum mechanics, an energy gap, also referred to as a spectral gap, denotes a range of energies within a system's spectrum where no quantum states are available, thereby prohibiting electron or other particle transitions across that interval. This absence of states arises from the discrete nature of quantum energy levels, distinguishing it from classical continuous spectra and influencing properties such as stability and excitation thresholds in physical systems.^[6] Mathematically, the energy gap

\Delta E

is quantified as the difference between the energy of the lowest unoccupied quantum state and the highest occupied state; for instance, in molecular systems, it is expressed as

\Delta E = E_{\text{LUMO}} - E_{\text{HOMO}}

, where

E_{\text{LUMO}}

is the energy of the lowest unoccupied molecular orbital and

E_{\text{HOMO}}

is that of the highest occupied molecular orbital.^[7] This representation captures the minimal energy required to promote an electron from a filled to an empty state, a fundamental concept underlying optical and electrical responses in various materials.^[8] The notion of an energy gap is most prominently applied in condensed matter physics to describe forbidden energy regions in electronic structures.^[9] It also extends to atomic, molecular, and nuclear physics, where discrete energy levels similarly produce gaps between allowed states, though the scales and mechanisms differ across these domains.^[10] In solid-state contexts, a specific manifestation is the band gap, representing the separation between valence and conduction bands in materials like semiconductors.^[11]

Historical Development

The concept of the energy gap in solid-state physics originated in the late 1920s amid efforts to apply quantum mechanics to electron behavior in crystals. Felix Bloch, in his 1928 doctoral thesis, developed the periodic potential model, demonstrating how electrons in a crystal lattice form extended wavefunctions that result in allowed energy bands separated by forbidden gaps, a result encapsulated in Bloch's theorem. This foundational work shifted understanding from free-electron models to periodic structures, laying the groundwork for band theory.^[12] In the early 1930s, Alan Wilson advanced Bloch's ideas by applying band theory to classify materials as metals, insulators, or semiconductors based on the positioning of the Fermi level relative to these energy gaps. Wilson's 1931 paper introduced the notion that a significant band gap in insulators prevents electron excitation to conduction bands at typical temperatures, while narrower gaps in semiconductors allow thermal activation, thus explaining diverse electrical conductivities. These influential contributions by Bloch and Wilson established the energy gap as a central parameter in solid-state physics, influencing subsequent theoretical developments. Following World War II, the 1950s saw intensified experimental and theoretical focus on semiconductors, where researchers like John Bardeen connected band gaps directly to device performance. Bardeen's work at Bell Laboratories and later at the University of Illinois explored surface states and impurity effects on band gaps, elucidating how controlled doping modulates the gap to enable transistor operation, a breakthrough recognized in the 1956 Nobel Prize. This era bridged abstract band theory with practical electronics, highlighting the energy gap's role in conductivity control.^[13] The energy gap concept expanded into superconductivity in 1957 through the Bardeen-Cooper-Schrieffer (BCS) theory, which described a pairing mechanism opening an excitation gap in the superconducting state below the critical temperature.^[14] Decades later, in the late 1980s, the discovery of high-temperature cuprate superconductors revealed pseudogap features—partial gaps in the normal state—first evidenced by nuclear magnetic resonance studies showing suppressed density of states. These milestones underscored the energy gap's versatility across condensed matter phases.

Band Gap in Solids

Characteristics in Semiconductors and Insulators

In solids, the energy gap, denoted as

E_g

, represents the energy difference between the maximum of the valence band (

E_v

) and the minimum of the conduction band (

E_c

), expressed as

E_g = E_c - E_v

.^[15] This gap determines the material's ability to conduct electricity by separating filled electron states from empty ones.^[16] Materials are classified based on the magnitude of

E_g

: metals exhibit

E_g \approx 0

eV with overlapping valence and conduction bands, enabling free electron movement; semiconductors have

E_g

ranging from approximately 0.1 to 4 eV, allowing partial thermal excitation of electrons across the gap for moderate conductivity; and insulators possess

E_g > 4

eV, preventing conduction at room temperature due to insufficient thermal energy to bridge the gap.^[17] In semiconductors, this intermediate gap facilitates tunable electrical properties through doping or temperature changes, while in insulators, the large gap ensures high resistivity.^[18] The band gap in semiconductors and insulators exhibits temperature dependence, generally decreasing with increasing temperature primarily due to lattice expansion from anharmonic vibrations, which alters interatomic distances and band overlaps.^[19] This behavior is empirically modeled using relations like the Varshni equation, which fits experimental data for various materials.^[20] Phonon interactions also contribute to this reduction, enhancing electron-phonon coupling at higher temperatures.^[21] Representative examples illustrate these characteristics: silicon, a quintessential semiconductor, has

E_g \approx 1.1

eV at room temperature, enabling its widespread use in electronics.^[22] In contrast, diamond serves as a classic insulator with

E_g \approx 5.5

eV, reflecting its exceptional hardness and thermal stability but poor electrical conductivity.^[22]

Direct and Indirect Band Gaps

In solid-state physics, the nature of the band gap in semiconductors is classified as direct or indirect depending on the momentum-space alignment of the valence band maximum (VBM) and conduction band minimum (CBM) within the Brillouin zone. A direct band gap arises when the VBM and CBM occur at the same wavevector k, permitting vertical electronic transitions that inherently conserve crystal momentum as dictated by the selection rules for optical processes.^[23] This configuration allows electrons to move between bands solely through interaction with photons, without requiring additional momentum exchange. In contrast, an indirect band gap features the VBM and CBM at distinct k points, such that a pure electronic transition across the gap would violate momentum conservation.^[24] To enable such transitions, electrons must couple with lattice vibrations, or phonons, to bridge the momentum mismatch, typically denoted as Δ**k** ≈ k_phonon.^[25] This phonon-mediated involvement introduces non-radiative pathways and reduces the overall transition probability, distinguishing indirect gaps from their direct counterparts.^[26] The distinction profoundly influences optical absorption and emission efficiencies in these materials. Direct band gap semiconductors facilitate strong light-matter coupling, enabling efficient photon absorption above the band gap energy E_g and radiative recombination, which is ideal for optoelectronic devices like LEDs and lasers. For instance, gallium arsenide (GaAs) possesses a direct band gap of approximately 1.43 eV, allowing for high-efficiency light emission in the near-infrared spectrum.^[23] Indirect band gap materials, however, exhibit weaker optical responses because phonon assistance is required for both absorption and emission, often leading to predominant non-radiative decay and lower quantum yields.^[25] Silicon (Si) and germanium (Ge) serve as representative examples, with indirect band gaps of about 1.1 eV and 0.67 eV, respectively, which contribute to their limited direct optical activity despite widespread use in electronics.^[27]^[28] Phonon-assisted processes in indirect band gaps typically involve either the absorption of a phonon (for upward transitions) or its emission (for downward ones), with the phonon energy matching the difference in k-space.^[29] These second-order perturbations, described within Fermi's golden rule framework, result in transition rates that are suppressed by factors of 10³ to 10⁵ relative to direct processes due to the need for virtual intermediate states.^[26] Consequently, indirect semiconductors like Si and Ge rely on indirect optical absorption edges that tail into lower energies, but with significantly reduced coefficients compared to direct materials.

Superconducting Energy Gap

BCS Theory and Gap Formation

In the Bardeen-Cooper-Schrieffer (BCS) theory, the superconducting energy gap arises from the attractive interaction between electrons mediated by lattice vibrations, or phonons, which overcomes the repulsive Coulomb forces at low temperatures. This interaction leads to the formation of Cooper pairs—bound states of two electrons with opposite momenta and spins near the Fermi level

E_F

—resulting in a collective condensation that lowers the ground-state energy of the system. The pairing opens an energy gap of magnitude

2\Delta

in the electronic density of states around

E_F

, prohibiting low-energy single-particle excitations and enabling zero-resistance current flow.^[14] The magnitude of the superconducting gap

\Delta

is determined self-consistently through the BCS gap equation, which balances the pairing potential against the density of states at the Fermi level. At zero temperature, for a weak-coupling superconductor with phonon-mediated attraction limited by the Debye frequency

\omega_D

, the gap is given by

\Delta(0) \approx 2 \hbar \omega_D \exp\left(-\frac{1}{N(0) V}\right),

where

N(0)

is the single-spin density of states at

E_F

and

V

is the effective pairing interaction strength. This exponential dependence highlights the sensitivity of the gap to the pairing mechanism, with typical values of

\Delta

on the order of millielectronvolts for conventional superconductors.^[14]^[30] In the superconducting state, the elementary excitations are Bogoliubov quasiparticles rather than free electrons, with a dispersion relation that reflects the paired nature of the ground state:

E(\mathbf{k}) = \sqrt{\varepsilon(\mathbf{k})^2 + \Delta^2},

E(k)=ε(k)2+Δ2

, where $\varepsilon(\mathbf{k})$ is the normal-state kinetic energy relative to $E_F$ . The minimum excitation energy is $\Delta$ , confirming the gap's role in stabilizing the condensate. Coherence factors $u_{\mathbf{k}}^2 = \frac{1}{2} \left(1 + \frac{\varepsilon(\mathbf{k})}{E(\mathbf{k})}\right)$ and $v_{\mathbf{k}}^2 = \frac{1}{2} \left(1 - \frac{\varepsilon(\mathbf{k})}{E(\mathbf{k})}\right)$ (with $u_{\mathbf{k}}^2 + v_{\mathbf{k}}^2 = 1$ ) describe the amplitude of electron-like and hole-like components in these quasiparticles, essential for processes like tunneling and electromagnetic response.^[14] For conventional superconductors described by BCS theory, the pairing exhibits isotropic s-wave symmetry, meaning the gap $\Delta$ is independent of momentum direction on the Fermi surface, leading to a fully gapped spectrum without nodes. This symmetry aligns with the even-parity spin-singlet nature of Cooper pairs and is a hallmark of phonon-mediated superconductivity in materials like elemental metals.^[14]

Relation to Critical Temperature

In the weak-coupling limit of BCS theory, the superconducting energy gap at absolute zero temperature, Δ(0), is related to the critical temperature T_c by the approximate relation Δ(0) ≈ 1.764 k_B T_c, where k_B is the Boltzmann constant; this value emerges from the numerical solution of the BCS gap equation under assumptions of weak electron-phonon coupling (λ ≪ 1) and temperatures much lower than the Debye temperature.^[31] The temperature dependence of the energy gap, Δ(T), is determined by solving the full BCS integral gap equation, which takes the form Δ(T) = V ∫_0^{ℏω_D} dξ N(ξ) (Δ(T) / (2 E(ξ, T))) tanh(β E(ξ, T) / 2), where V is the pairing interaction strength, N(ξ) is the density of states, E(ξ, T) = sqrt(ξ^2 + Δ(T)^2), β = 1/(k_B T), and ℏω_D is the Debye energy; near T_c, this can be approximated by the empirical form Δ(T) ≈ Δ(0) tanh[1.74 √(T_c / T - 1)], capturing the rapid closure of the gap as temperature approaches T_c.^[14] Experimental validation of these relations primarily comes from tunneling spectroscopy in superconductor-insulator-normal metal (SIN) or superconductor-insulator-superconductor (SIS) junctions, where the differential conductance exhibits peaks at voltages corresponding to ±2Δ/e, allowing direct measurement of the gap; for instance, in low-T_c elemental metals like aluminum (T_c ≈ 1.2 K), tunneling measurements yield Δ(0) ≈ 0.18 meV at low temperatures, closely matching the BCS prediction of ≈ 0.182 meV and confirming the weak-coupling behavior.^[32] Deviations from the weak-coupling BCS ratio occur in strong-coupling superconductors (λ > 1), where electron-phonon interactions are more pronounced, leading to enhanced pairing and larger gap values; in such cases, the ratio Δ(0)/k_B T_c increases beyond 1.764, reaching approximately 2.2 in materials like lead (T_c ≈ 7.2 K, Δ(0) ≈ 1.4 meV), as described by Eliashberg theory extensions to BCS.^[33]

Pseudogap Phenomena

Definition and Contexts

The pseudogap refers to a partial suppression of the density of states near the Fermi energy (

E_F

) in the normal state of certain superconductors, occurring above the critical temperature

T_c

where long-range superconducting coherence is absent, in contrast to the full vanishing of states characteristic of a true energy gap.^[34] This phenomenon manifests as an incomplete depletion rather than a complete gap, allowing residual low-energy excitations while significantly reducing spectral weight around

E_F

.^[35] Unlike the superconducting gap that emerges below

T_c

due to coherent pairing, the pseudogap persists in the pseudogap phase up to a higher temperature scale

T^*

.^[36] The pseudogap is most prominently observed in underdoped high-temperature cuprate superconductors, such as YBa

_2

_3

_{7-\delta}

(YBCO), where it appears for doping levels below optimal values, leading to a dome-shaped phase diagram with

T_c

suppression.^[34] It also occurs in heavy-fermion superconductors like CeCoIn

_5

, where quantum critical fluctuations near

E_F

contribute to the pseudogap formation in the normal state.^[36] Similar signatures are found in organic superconductors, including

\kappa

-(BEDT-TTF)

_2

Cu[N(CN)

_2

]Br, particularly in the crossover regime toward Bose-Einstein condensation of pairs.^[37] These contexts highlight the pseudogap's role in strongly correlated electron systems beyond conventional BCS superconductivity. Theoretical origins of the pseudogap include the formation of preformed Cooper pairs in the normal state, where local pairing occurs without global phase coherence, as proposed in models for cuprates and organics.^[35] Alternative explanations involve competing orders, such as stripe-like charge density waves or spin density waves, which suppress states near

E_F

by introducing inhomogeneous electronic textures. The microscopic origin of the pseudogap remains a subject of ongoing debate, with no consensus as of 2025.^[38] The energy scale of the pseudogap typically spans hundreds of meV, setting it apart from the smaller superconducting gap below

T_c

.^[38] Experimentally, the pseudogap is detected through angle-resolved photoemission spectroscopy (ARPES), which reveals "Fermi arcs"—open segments of the Fermi surface along nodal directions—indicating anisotropic suppression of quasiparticle coherence in the antinodal regions.^[39] Specific heat measurements show anomalies, such as a reduced linear term

\gamma T

and suppressed Sommerfeld coefficient

\gamma

above

T_c

, reflecting the diminished density of low-energy states.^[38] These signatures collectively distinguish the pseudogap as a precursor to superconductivity in underdoped regimes.^[37]

Distinctions from Conventional Gaps

Pseudogaps in cuprate superconductors display a pronounced anisotropic structure, often exhibiting d-wave-like symmetry with nodal regions where the gap vanishes, in stark contrast to the isotropic band gaps typical in conventional semiconductors and insulators, as well as the s-wave superconducting gaps in BCS theory.^[40]^[41] This anisotropy arises primarily along the antinodal directions in momentum space, leading to Fermi arcs rather than a complete closure of the Fermi surface.^[42] Furthermore, unlike conventional gaps that fully suppress the density of states (DOS) at the Fermi level, pseudogaps result in only partial DOS depletion, allowing residual low-energy excitations that prevent a true insulating behavior.^[43]^[44] In terms of symmetry and pairing mechanisms, pseudogaps do not break time-reversal symmetry, as evidenced by magneto-optical measurements showing no significant internal magnetic fields in the pseudogap phase of La-based cuprates.^[45] This absence distinguishes them from states involving static order parameters that could induce such breaking. Instead, pseudogaps are often attributed to fluctuating electron pairs—such as those in a pair density wave state—that lack the long-range phase coherence essential for true superconductivity, resulting in precursor pairing without global order.^[46]^[47] The implications of these distinctions are profound for phase competition in cuprates: the pseudogap suppresses the onset of superconductivity by competing for electronic states in the underdoped regime, thereby lowering the critical temperature Tc away from optimal doping and shaping the characteristic dome in the temperature-doping phase diagram.^[48]^[49] Recent post-2020 investigations, including 2022 analyses of transport properties, have revealed that the pseudogap evolves into the strange metal phase through a quantum critical crossover, where critical fluctuations at the pseudogap endpoint drive non-Fermi liquid behavior and linear-in-temperature resistivity.^[50]^[51]

Hard and Soft Gaps

Definitions and Properties

In condensed matter physics, a hard gap describes a complete suppression of the density of states (DOS) over a finite energy interval Δ centered at the Fermi energy

E_F

, such that the DOS

\rho(E) = 0

for

|E - E_F| < \Delta/2

. This results in an absolute absence of available electronic states within that range, prohibiting low-energy excitations. Hard gaps typically originate from the periodic ordering in clean crystalline systems, where translational symmetry enforces band separation, as seen in insulators, or from electron pairing mechanisms in superconductors that reorder the quasiparticle spectrum.^[52] In contrast, a soft gap features a DOS that vanishes precisely at the Fermi level (

E = 0

) but increases continuously for small deviations, without a strict forbidden energy window. The DOS near

E = 0

often exhibits a power-law behavior,

\rho(E) \sim |E|^\alpha

with

\alpha > 0

, reflecting a gradual restoration of states rather than abrupt prohibition. Soft gaps arise primarily from disorder or strong electron-electron interactions that preferentially deplete states at the Fermi level, such as in amorphous or interacting electron systems, leading to enhanced localization or correlation effects without the rigidity of periodic lattices.^[53]^[54] A distinguishing property of hard gaps is their robustness in ordered environments, where symmetry protects the zero-DOS region, enabling clear spectroscopic signatures like sharp absorption edges. Soft gaps, however, are more susceptible to perturbations, with their exponent

\alpha

tunable by disorder strength or interaction parameters, and they can manifest through singular behaviors like the Dyson singularity in one-dimensional disordered chains, where random hopping induces anomalous low-energy scaling in the DOS.^[55]^[56] In both cases, these gaps profoundly influence transport and thermodynamic responses by altering the availability of states near

E_F

. The pseudogap in certain correlated materials can resemble a partial soft gap through momentum-dependent suppression, but it differs in its inhomogeneous spatial character.^[53]

Examples in Disordered Materials

In disordered doped semiconductors, electron-electron Coulomb repulsion induces a soft gap known as the Efros-Shklovskii Coulomb gap in the density of states near the Fermi level, suppressing states such that the density of states follows

\rho(E) \sim |E|^{d-1}

d

dimensions. This gap arises from the long-range nature of the Coulomb interaction among localized electrons, leading to a quadratic suppression in three dimensions (

\rho(E) \sim E^2

) without fully depleting the spectrum. In strongly correlated systems, the Mott gap exemplifies a hard gap in insulators like transition metal oxides (e.g., NiO), where on-site Coulomb repulsion

U

between electrons in partially filled

d

-orbitals overwhelms kinetic energy, opening a charge excitation gap of order

U

and prohibiting metallic conduction. Recent studies have also identified soft gaps in twisted bilayer graphene at magic angles, where electron interactions flatten bands and induce interaction-driven suppressions in the density of states near charge neutrality, manifesting as correlated insulating phases.^[57] These gaps profoundly influence charge transport in disordered materials; soft gaps, such as the Efros-Shklovskii type, promote variable-range hopping conduction at low temperatures, where carriers hop over longer distances to minimize energy barriers within the softened spectrum.

Applications and Modern Advances

In Optoelectronics

In optoelectronics, the energy gap, particularly the band gap in semiconductors, fundamentally governs the interaction between materials and light, determining the wavelengths at which absorption and emission occur. The absorption edge corresponds to the band gap energy

E_g

, where photons with energy

h\nu \geq E_g

can excite electrons from the valence to the conduction band, enabling processes like photodetection and light emission; the cutoff wavelength is given by

\lambda = \frac{hc}{E_g}

, with

h

as Planck's constant and

c

the speed of light. For instance, gallium nitride (GaN), with a direct band gap of 3.4 eV, absorbs in the ultraviolet and emits blue light around 365 nm, making it essential for high-efficiency blue light-emitting diodes (LEDs) that revolutionized solid-state lighting.^[58]^[59] In photovoltaics, the band gap sets the balance between capturing a broad solar spectrum and minimizing thermal losses, as outlined by the Shockley-Queisser limit, which predicts a maximum single-junction efficiency of approximately 33% for an optimal

E_g \approx 1.3

eV under standard solar illumination. Materials like silicon (

E_g = 1.1

eV) approach but fall short of this limit due to their slightly suboptimal gap, achieving practical efficiencies around 25-27%. To overcome this, tandem solar cells stack absorbers with progressively narrower band gaps—such as a wide-gap top cell (e.g., 1.7 eV perovskite) over a narrower bottom cell (e.g., 1.1 eV silicon)—to sequentially harvest high- and low-energy photons across the full solar spectrum, potentially reaching efficiencies over 45%.^[60]^[61] Semiconductor lasers and detectors also rely on band gap engineering, with direct band gap materials like gallium arsenide (GaAs,

E_g = 1.42

eV) preferred for low-threshold lasers due to efficient radiative recombination that minimizes non-radiative losses and enables stimulated emission at near-infrared wavelengths. In contrast, indirect band gap semiconductors such as silicon or germanium are suited for infrared detectors, where phonon-assisted transitions allow absorption of longer wavelengths (e.g., up to 1.1 μm in silicon) despite lower quantum efficiency, supporting applications in thermal imaging. A pivotal historical advancement occurred in 1962, when Robert N. Hall at General Electric demonstrated the first continuous-wave semiconductor laser using a GaAs p-n junction, operating at 77 K and emitting at 840 nm, which laid the foundation for modern diode lasers.^[62]^[63]^[64]

In Quantum and 2D Materials

In two-dimensional (2D) materials, the energy gap exhibits remarkable tunability through mechanical strain and electrostatic gating, enabling tailored electronic properties for quantum applications. Monolayer molybdenum disulfide (MoS₂) possesses a direct bandgap of approximately 1.8 eV, which can be engineered via strain, as experimentally demonstrated in suspended devices where the gap decreases by roughly 100 meV per 1% strain.^[65] Electric gating in MoS₂ further modulates the effective gap by altering carrier density and inducing dielectric screening, with shifts up to several hundred meV observed in field-effect configurations.^[66] These techniques, advanced in studies from 2020 onward, facilitate on-demand control of excitonic and transport properties in 2D semiconductors. A prominent recent development involves magic-angle twisted bilayer graphene, where specific twist angles near 1.1° produce flat bands with near-zero energy gaps, leading to strongly correlated electron states; 2023 theoretical and experimental work confirmed bandwidths approaching zero in these configurations, enhancing phenomena like unconventional superconductivity.^[67] In quantum dots, size-dependent bandgap tuning follows the relation

E_g = E_{\text{bulk}} + \frac{\hbar^2 \pi^2}{2 m^* r^2},

derived from particle-in-a-box confinement, allowing gaps to increase from bulk values (e.g., ~1.74 eV for CdSe) by up to 1 eV for radii below 5 nm, as verified in colloidal syntheses for optoelectronic integration. Topological insulators represent a cornerstone of quantum technologies, with materials like bismuth selenide (Bi₂Se₃) featuring a bulk bandgap of approximately 0.3 eV that insulates the interior while supporting spin-helical surface states protected by time-reversal symmetry.^[68] This gapped bulk ensures dissipationless edge conduction, crucial for spintronics. In gapped superconductors, Majorana zero modes emerge at defects or interfaces, forming nonlocal pairs that enable topological qubits; recent 2024 realizations in striped 2D inhomogeneous superconductors demonstrated these modes delocalized over separated sites, robust against local perturbations.^[69] Exploiting these gapped phases advances quantum computing, where 2D heterostructures provide topological protection for qubits, as in twisted bilayer systems that stabilize Majorana-based encodings with coherence times extended by gap-induced isolation from decoherence.^[70] In photovoltaics, halide perovskites leverage defect-tolerant bandgaps around 1.5–1.7 eV, where shallow defects minimize nonradiative recombination, yielding power conversion efficiencies exceeding 25% in 2023–2025 devices through compositional tuning for enhanced stability under operational stresses; as of November 2025, certified single-junction efficiencies have reached 27.2% and flexible tandems 33.6%.^[71]^[72]^[73] In disordered 2D systems, pseudogap effects from localization can subtly influence these tunable gaps, broadening applications in noisy quantum environments.^[74]

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