Band Gaps

A gap is an energy interval or excitation threshold only after the object, state, momentum constraint, charge sector, and limiting procedure have been specified. A direct gap in an auxiliary one-particle Hamiltonian, the exact energy required to add and remove charge, a neutral optical onset, an Arrhenius slope, and an interval containing only localized states can all be reported in electronvolts. They are not interchangeable measurements of one hidden number.

The central task is to distinguish three operations: comparing one-electron energies, changing the total particle number, and exciting the material at fixed particle number. Transport introduces a further question: can the available states carry charge across the sample? The definitions below connect these operations without identifying them prematurely.

Required background. Band Theory Overview supplies Bloch-band filling, valence and conduction manifolds, and the independent-particle/interacting boundary used throughout this page.

Helpful background. Spectral Functions supplies the addition/removal spectrum and the distinction between a sharp quasiparticle and incoherent weight. Raman and Optical Spectroscopy supplies the measurement boundary between an intrinsic response and a recorded optical onset. Metals, Insulators, and Semiconductors supplies mechanism-aware material classification.

Spectral, charge and optical gaps

A useful comparison starts with the physical quantity, the state of the sample and the excitation that a probe creates. For example, an optical photon approximately preserves crystal momentum, whereas adding an electron need not. The same energy unit does not make the resulting thresholds equal.

Gap object	Defining operation	Sector	Central warning
minimum direct band gap	smallest lower–upper band separation at one common $\mathbf k$	auxiliary or declared one-particle bands	can remain positive in an overlap semimetal
global band gap	upper-band minimum minus lower-band maximum, with independent momenta	auxiliary or declared one-particle bands	a high-symmetry path does not establish it
interacting charge gap	second particle-number difference of $E_0(N)$	charged many-body states	need not have sharp quasiparticle edges; pairing requires additional charge-sector checks
quasiparticle gap	separation of coherent addition and removal poles or narrow peaks	one-particle spectral function	may be undefined when edge weight is incoherent
Kohn–Sham gap	occupied–empty separation of the auxiliary Kohn–Sham operator	auxiliary ground-state system	differs from the exact charge gap by a derivative discontinuity
neutral or optical gap	lowest neutral excitation, optionally restricted to optically active states	fixed particle number	dark states, excitons, phonons, and matrix elements alter the onset
transport activation scale	exponent extracted from a stated conductivity or resistivity law	response	the convention may differ by a factor of two
mobility gap	energy interval without extended states	localization	localized spectral weight can remain inside it

The notation below reserves $E_g^{\mathrm{band}}$ for a global one-particle band separation and $\Delta_{\mathrm{ch}}$ for the full interacting charge gap. This avoids using fundamental gap for two different objects. When quoting literature that uses other notation, identify the defining operation before comparing numbers.

Direct and Global Band Separations

Consider a periodic one-particle Hamiltonian, or specified coherent quasiparticle dispersions, with lower band group $v$ and upper group $c$. The labels denote band groups: if their energy ranges overlap, they are not everywhere occupied and empty. At each crystal momentum define

$$\varepsilon_v^{\max}(\mathbf k) = \max_{n\in v}\varepsilon_{n\mathbf k}, \qquad \varepsilon_c^{\min}(\mathbf k) = \min_{m\in c}\varepsilon_{m\mathbf k}.$$

The vertical separation at fixed momentum is

$$\Delta_{\mathrm{dir}}(\mathbf k) = \varepsilon_c^{\min}(\mathbf k) - \varepsilon_v^{\max}(\mathbf k),$$

and the minimum direct gap is

$$E_g^{\mathrm{dir}} = \min_{\mathbf k\in\mathrm{BZ}} \Delta_{\mathrm{dir}}(\mathbf k).$$

The global band gap allows the two extrema to occur at unrelated momenta:

$$E_g^{\mathrm{band}} = \min_{\mathbf k} \varepsilon_c^{\min}(\mathbf k) - \max_{\mathbf k} \varepsilon_v^{\max}(\mathbf k).$$

Because the minimum direct gap compares a restricted set of pairs,

$$E_g^{\mathrm{band}} \le E_g^{\mathrm{dir}}.$$

These definitions, including multiband and degeneracy qualifications, are standard in the treatments by Ashcroft and Mermin, Cohen and Louie, and Yu and Cardona. They require all relevant bands throughout the Brillouin zone.

Direct, indirect, and overlapping cases

When the valence-band maximum and conduction-band minimum occur at the same momentum, the positive global gap is direct and $E_g^{\mathrm{band}}=E_g^{\mathrm{dir}}$. When the extrema occur at different momenta, a positive global gap is indirect and is strictly smaller than the minimum vertical separation in the generic case.

A third possibility is often missed:

$$E_g^{\mathrm{dir}}>0, \qquad E_g^{\mathrm{band}}<0.$$

The two manifolds are separated at every fixed $\mathbf k$, so the lower-band projector can remain isolated and smooth, but their energy ranges overlap. At the nominal insulating filling the true independent-electron ground state then transfers occupation from high-energy valence states into low-energy conduction states, producing electron and hole pockets. This is an overlap semimetal, not a band insulator. An isolated band subspace and an insulating energy interval answer different questions.

At $E_g^{\mathrm{band}}=0$, specify whether there is an isolated touching, an extended nodal set, or merely equality of extrema at different momenta. The local dispersion and symmetry decide the low-energy phase space; the number zero alone does not.

Parallel bands with a positive direct gap and indirect overlap

Let $k\in[-\pi/a,\pi/a]$ and consider two noninteracting bands

$$\varepsilon_v(k) = -\frac{\Delta}{2} +2t\cos(ka)$$

and

$$\varepsilon_c(k) = +\frac{\Delta}{2} +2t\cos(ka),$$

with $t>0$ and $\Delta>0$. Their separation at every common momentum is

$$\varepsilon_c(k)-\varepsilon_v(k)=\Delta,$$

so

$$E_g^{\mathrm{dir}}=\Delta.$$

The valence maximum occurs at $k=0$, while the conduction minimum occurs at $k=\pi/a$. Therefore

$$E_g^{\mathrm{band}} = \left( \frac{\Delta}{2}-2t \right) - \left( -\frac{\Delta}{2}+2t \right) = \Delta-4t.$$

For $\Delta>4t$ the system has a positive indirect band gap. At $\Delta=4t$ the indirect gap closes. For $0<\Delta<4t$, every vertical separation remains positive, yet the energy ranges overlap and the nominal insulating filling produces electron and hole pockets. This simple example separates a direct projector gap from a global insulating gap.

A full-zone claim, not a path-plot claim

A conventional band plot samples selected lines and can miss an off-path extremum, pocket or node. Find both extrema over the full Brillouin zone, including every band and internal sector that can enter the relevant energy window. Refine the momentum mesh and basis until the extrema and their difference converge; use the actual structural and magnetic unit cell.

Band Structure Workflows explains interpolation and convergence. A gap that closes and reopens may motivate a topological calculation, but the energies alone do not establish a change of invariant.

In the illustrated bands, the global gap is $1E_0$ but the vertical electron–hole continuum begins at $3E_0$, where $E_0$ is an energy unit. The hypothetical neutral spectrum below illustrates a bright state at $2.4E_0$, bound by $0.6E_0$ relative to that same-momentum continuum. Its energy can exceed the global gap. The lower dark state does not set the absorption onset in the indicated channel. The neutral levels are illustrative, not calculated from the drawn dispersions.

Charged Excitations and Quasiparticle Edges

Band eigenvalues cease to be the primary definition when interactions are essential. Hold the external potential, nuclear geometry, boundary conditions, and other control parameters fixed, and let $E_0(N)$ be the exact ground-state energy with $N$ particles. Define the addition and removal chemical potentials

$$\mu^+(N) = E_0(N+1)-E_0(N)$$

and

$$\mu^-(N) = E_0(N)-E_0(N-1).$$

The full charge gap, also called the interacting fundamental gap, is

$$\begin{aligned} \Delta_{\mathrm{ch}} &= \mu^+(N)-\mu^-(N) \\ &= E_0(N+1)+E_0(N-1)-2E_0(N). \end{aligned}$$

With the conventional positive ionization energy $I=E_0(N-1)-E_0(N)$ and electron affinity $A=E_0(N)-E_0(N+1)$,

$$\Delta_{\mathrm{ch}}=I-A.$$

This is a difference of canonical ground-state energies and remains meaningful when no band labels the low-energy states. Some many-body literature uses $\Delta_c$ for half this quantity, especially under particle–hole symmetry. A numerical value without the full-gap or half-gap convention is incomplete.

For a noninteracting band insulator in the thermodynamic limit, $\Delta_{\mathrm{ch}}=E_g^{\mathrm{band}}$. Interactions can instead open a Mott charge gap, renormalize an existing band gap, or transfer edge weight into continua. The equality to a fixed-band separation is then not available. Martin, Reining, and Ceperley give a systematic many-body treatment of these particle-addition and removal energies.

Single-electron gaps and pair motion

For an ordinary band or Mott insulator, a positive thermodynamic-limit addition/removal gap describes a finite interval of chemical potential over which the particle density remains fixed. In a paired system, however, changing particle number by one can break a pair while changing it by two costs much less. The $N\pm1$ difference alone then does not establish an incompressible insulator. Compare pair sectors and the lower envelope of $E_0(M)-\mu M$ over all $M$, rather than only adjacent sectors.

This distinction appears in spin-gapped systems with gapless charge motion, such as the Luther–Emery liquid discussed by Seidel and Lee. Exercise 4 gives a simple energy function that isolates the even–odd effect. Here $\Delta_{\mathrm{ch}}$ denotes the displayed single-electron addition/removal difference; claims about insulation require the additional charge-response information.

Spectral gap versus quasiparticle gap

The zero-temperature one-particle spectral function has removal support at energies associated with transitions from $N$ to $N-1$ particles and addition support associated with transitions from $N$ to $N+1$. The absence of support around the chemical potential defines a one-particle spectral gap for the chosen operator and thermodynamic state.

Spectral matrix elements matter: the lowest $N\pm1$ state may have zero overlap with the chosen electron operator. The minimum excitation cost inferred from that operator can therefore exceed the corresponding total-energy threshold. Restricting momentum, spin or orbital channel makes that possibility especially important.

A quasiparticle gap is more restrictive. It is the separation between the lowest coherent addition and removal poles, or between sufficiently narrow peaks that can be followed as quasiparticle bands. The Green-function framework introduced by Hedin and the semiconductor calculations of Hybertsen and Louie motivate the $GW$ approximation to such addition and removal energies. The review by Onida, Reining, and Rubio explains why the electron self-energy and the electron–hole response solve different excitation problems.

Three distinctions follow.

• A charge gap can exist even if the thresholds are broad continua and no long-lived quasiparticle is present.
• A sharp occupied removal edge alone does not determine the full gap; the addition edge is also required on the same energy reference.
• A computed $GW$ gap is a method-labelled approximation to quasiparticle energies. It is not, by its name alone, an exact charge gap or a measured optical onset.

Spectral Functions derives the spectral weights, poles and continua and explains their relation to measured intensity.

Finite systems and charged supercells

For a finite isolated sample, $E_0(N\pm1)$ can include physical charging energy. In a periodic charged-cell calculation it can also include the chosen neutralizing background, image interactions, potential alignment, and finite-$\mathbf k$ sampling. In exact diagonalization, a level spacing or a shell effect can mimic a gap. A thermodynamic charge-gap claim therefore needs either a size extrapolation or a documented correction scheme appropriate to the geometry and electrostatics.

Do not relax the ions in one charge sector while holding them fixed in another unless an adiabatic or relaxed-ion gap is explicitly intended. A vertical electronic gap and a relaxed addition energy include different lattice physics.

Kohn–Sham Gaps and the Derivative Discontinuity

Kohn–Sham density-functional theory replaces the interacting ground-state problem by an auxiliary noninteracting system constructed to reproduce the ground-state density. Kohn and Sham derived the self-consistent auxiliary equations; the complete spectrum of their operator was not introduced as an exact spectrum of charged excitations.

For the exact multiplicative Kohn–Sham potential, define the global occupied–empty gap

$$\Delta_{\mathrm{KS}} = \varepsilon_{\mathrm{LUMO}}^{\mathrm{KS}} - \varepsilon_{\mathrm{HOMO}}^{\mathrm{KS}},$$

with the corresponding full-zone extrema in a periodic solid. For the usual normal electronic system whose ensemble energy interpolates between stable adjacent integer sectors, exact ground-state DFT gives the decomposition

$$\Delta_{\mathrm{ch}} = \Delta_{\mathrm{KS}} + \Delta_{\mathrm{xc}},$$

where $\Delta_{\mathrm{xc}}$ is the constant discontinuous shift of the exact exchange-correlation potential as the particle number crosses an integer. Perdew, Parr, Levy, and Balduz established the piecewise-linear energy and integer discontinuity; Perdew and Levy and, independently, Sham and Schlüter connected that discontinuity to the energy gap.

This exact relation is a warning about identification, not a universal recipe for correcting an approximate calculation.

What the decomposition does and does not say

• A Kohn–Sham gap is a well-defined eigenvalue separation of a declared auxiliary operator. It is not generally the exact charge gap.
• Common local and semilocal approximations do not reproduce the exact discontinuity and can have self-interaction, delocalization, and correlation errors. Nevertheless, “DFT always underestimates gaps” is not a theorem about every functional, material, geometry, or reported gap.
• Hybrid and other generalized Kohn–Sham methods use nonmultiplicative operators. Their eigenvalue gaps can absorb part of what appears as a discontinuity in multiplicative Kohn–Sham theory, so the simple equation above cannot be transferred term by term without defining the formalism.
• A scissor shift, tuned hybrid, Hubbard correction, or $GW$ calculation is a method and parameter choice. Agreement with one measured onset is not an independent validation if that onset was used to tune the correction.

Neutral and Optical Thresholds

A neutral excitation remains in the $N$-particle Hilbert space. For a unique ground state its lowest excitation threshold is

$$\Delta_{\mathrm{neutral}} = \inf_{S:E_S>E_0} \left[ E_S(N)-E_0(N) \right].$$

For a degenerate ground-state subspace, define the excitation gap above that subspace; a transition between degenerate ground states is not such an excitation. Gapless modes give a vanishing infimum in the thermodynamic limit. The lowest neutral excitation may be a spin excitation, a dark exciton, a finite-momentum state, a collective mode, or a symmetry-forbidden transition. It need not appear in ordinary long-wavelength absorption.

At zero temperature, for incident polarization $\hat{\mathbf e}$ and photon momentum negligible on the Brillouin-zone scale, an idealized optical threshold can be written

$$E_g^{\mathrm{opt}} = \inf_{ \substack{S:E_S>E_0\\ \langle S|\hat{\mathbf e}\!\cdot\!\hat{\mathbf j}|0\rangle\ne0} } \left[ E_S(N)-E_0(N) \right].$$

The operator, polarization, geometry, and assistance mechanism are part of the definition; the condition $E_S>E_0$ excludes an elastic ground-state contribution. At nonzero temperature, transitions from thermally occupied states and weak spectral tails can replace a sharp threshold by an operational onset. In an indirect semiconductor, a phonon can supply the missing crystal momentum; the onset then contains both electronic and phonon energies and a temperature-dependent occupation factor. Elliott’s analysis remains the classic account of allowed and forbidden direct-transition exciton series and their continuum line shapes; an indirect phonon-assisted onset needs a separate electron–phonon response model.

A phonon can lower or raise the photon threshold

Take a clean indirect semiconductor with free-pair gap $E_g^{\mathrm{ind}}>\hbar\Omega$, and approximate the relevant phonon energy by a constant $\hbar\Omega$ near the momentum needed to connect the band extrema. Treat the valence band as filled and the conduction band as empty, neglect excitons and disorder, and assume the transitions are allowed. Energy conservation gives

$$\begin{aligned} \text{phonon absorption:}\quad E_\gamma+\hbar\Omega&=E_g^{\mathrm{ind}}, &E_{\gamma,\mathrm{abs}}&=E_g^{\mathrm{ind}}-\hbar\Omega,\\ \text{phonon emission:}\quad E_\gamma&=E_g^{\mathrm{ind}}+\hbar\Omega, &E_{\gamma,\mathrm{em}}&=E_g^{\mathrm{ind}}+\hbar\Omega. \end{aligned}$$

The photon is absorbed in both processes; the labels describe the phonon. Squaring the phonon ladder-operator matrix elements gives occupation factors $n_{\mathrm B}(\Omega)$ for absorption and $n_{\mathrm B}(\Omega)+1$ for emission, where

$$n_{\mathrm B}(\Omega)=\frac{1}{e^{\hbar\Omega/(k_{\mathrm B}T)}-1}.$$

These signs and factors appear explicitly in the phonon-assisted response of Noffsinger and collaborators. At $T=0$, there is no phonon to absorb, but spontaneous emission remains possible: the onset of this phonon-assisted channel is $E_g^{\mathrm{ind}}+\hbar\Omega$. At nonzero temperature the lower branch can appear with a small thermal weight. Neither branch changes the electronic gap. Actual intensity also depends on electronic phase space and matrix elements; phonon dispersion, additional branches, and bound excitons require extending the model.

Exciton binding is a difference between matched thresholds

Let $E_X$ be the excitation energy of a specified bound electron–hole state, measured from $E_0(N)$, and let $E_{\mathrm{cont}}$ be the free electron–hole continuum threshold in the same material, environment, momentum sector, geometry, and temperature. Its binding energy is

$$E_b^X = E_{\mathrm{cont}}-E_X.$$

For coherent quasiparticle bands, an electron–hole pair with total crystal momentum $\mathbf Q$ has continuum threshold

$$E_{\mathrm{cont}}(\mathbf Q)= \min_{\mathbf k,c,v} \left[\varepsilon_c^{\mathrm{QP}}(\mathbf k+\mathbf Q) -\varepsilon_v^{\mathrm{QP}}(\mathbf k)\right].$$

The unrestricted quasiparticle gap minimizes this expression over $\mathbf Q$ as well. Optical absorption without assistance instead samples $\mathbf Q\simeq0$. In an indirect material these thresholds differ. Thus the appropriate relation is

$$E_b^X(\mathbf Q)=E_{\mathrm{cont}}(\mathbf Q)-E_X(\mathbf Q).$$

The familiar shorthand $E_g^{\mathrm{opt}}\simeq\Delta_{\mathrm{QP}}-E_b^X$ requires both $E_{\mathrm{cont}}(0)=\Delta_{\mathrm{QP}}$ and a lowest exciton visible in the optical channel. The lowest exciton may be dark, spin forbidden, momentum indirect, or hidden by disorder and resolution. Conversely, an Urbach tail or defect transition can put weak absorption below the clean continuum without defining the intrinsic band gap.

Rohlfing and Louie show how quasiparticle energies and the electron–hole Bethe–Salpeter problem combine in a first-principles optical spectrum. Excitons in Materials develops the internal wavefunction, screening kernel, Coulomb-series limits and material taxonomy.

Absorption, emission and photoemission

• Absorption and ellipsometry constrain optically active neutral response after propagation, polarization, and line-shape modeling.
• Photoluminescence records radiative recombination after carriers and excitons have relaxed. Stokes shifts, localization, defects, and nonequilibrium populations can move its peak below an absorption edge.
• ARPES samples occupied removal spectral weight. It does not see the unoccupied addition edge needed for a full charge or quasiparticle gap.
• Inverse photoemission samples addition weight, but combining it with ARPES requires aligned energy zeros, compatible surfaces, and comparable state preparation.
• STS measures a tunneling-matrix-weighted local spectrum at a surface. Tip-induced band bending, charging, surface states, and the setpoint can move or obscure apparent edges.

Damascelli, Hussain, and Shen review the ARPES matrix-element and surface boundary; Tersoff and Hamann give the standard tunneling connection behind the STS qualification. The corresponding ARPES and STM/STS pages explain how to model those measurements.

Transport Activation and Mobility Gaps

Transport does not measure an energy spectrum without a kinetic model. If a conductivity is fitted to

$$\sigma(T) = \sigma_0(T) \exp\!\left( -\frac{E_A}{k_{\mathrm B}T} \right),$$

then $E_A$ is the activation energy of that fit. Its physical interpretation depends on the carrier source, chemical potential, mobility, contacts, temperature interval, and competing channels. A straight segment in an Arrhenius plot is evidence for an exponential scale, not by itself a band gap.

The intrinsic-semiconductor factor of two

For a clean, nondegenerate intrinsic semiconductor in equilibrium, let $E_g^{\mathrm{free}}(T)$ denote the thermally renormalized threshold for creating a free electron–hole pair. Then

$$n_i = \sqrt{N_cN_v} \exp\!\left( -\frac{E_g^{\mathrm{free}}(T)}{2k_{\mathrm B}T} \right),$$

and

$$\sigma = e \left( n\mu_e+p\mu_h \right).$$

At $n=p=n_i$, the prefactor multiplying the exponential is

$$\sigma_{\mathrm{pref}}(T) = e\sqrt{N_cN_v}\,(\mu_e+\mu_h).$$

The exact local slope of an Arrhenius plot within this intrinsic model is

$$\begin{aligned} E_A^{\mathrm{loc}}(T) &:=-k_{\mathrm B} \frac{d\ln\sigma}{d(1/T)} \\ &=\frac12\left[ E_g^{\mathrm{free}}(T) -T\frac{dE_g^{\mathrm{free}}}{dT} \right] +k_{\mathrm B}T^2 \frac{d\ln\sigma_{\mathrm{pref}}}{dT}. \end{aligned}$$

A constant fitted $E_A$ approximates this local slope only over a window where the slope varies little. If intrinsic carriers dominate, the free-carrier threshold is nearly temperature independent, and the prefactor’s logarithmic derivative is negligible on the gap scale, then

$$E_A \simeq \frac{E_g^{\mathrm{free}}(T)}{2}.$$

Sze and Ng and Yu and Cardona develop this semiconductor limit. Some authors report the fitted $E_A$; others call $2E_A$ the transport gap. The fit equation must accompany the number. A temperature-dependent gap requires the derivative term above: for example, $E_g^{\mathrm{free}}(T)=E_0-aT$ with constant prefactor gives $E_A^{\mathrm{loc}}=E_0/2$, not $E_g^{\mathrm{free}}(T)/2$. Temperature-dependent semiconductor gaps are documented by Varshni. If the nearly constant free-carrier threshold is also matched to the interacting charge threshold, one may further write $E_A\simeq\Delta_{\mathrm{ch}}/2$.

The factor of two fails when the chemical potential is pinned away from midgap, one carrier dominates, dopants are being ionized, contacts or grain boundaries control the resistance, polarons activate, parallel surface channels conduct, or the prefactor changes appreciably. A crossover between two channels can even imitate a limited Arrhenius window.

Mobility edges and localized spectral weight

In a one-particle disordered problem, a mobility edge separates localized from extended states in the thermodynamic limit. If two such edges bound an interval containing no extended states,

$$E_-^{\mathrm{mob}} < E < E_+^{\mathrm{mob}},$$

one may define the full mobility-gap width

$$\Delta_{\mathrm{mob}} = E_+^{\mathrm{mob}}-E_-^{\mathrm{mob}}.$$

The density of states can remain nonzero throughout this interval because the states are localized. If simple activation from a chemical potential inside the interval to the nearest extended state controls transport, the one-sided scale is

$$E_A^{\mathrm{edge}} = \min\!\left( E_+^{\mathrm{mob}}-\mu, \mu-E_-^{\mathrm{mob}} \right).$$

It is generally neither the full mobility gap nor half of it. Mott introduced the mobility-edge language for disordered structures; Anderson’s localization mechanism, the scaling theory of Abrahams and collaborators, and the review by Evers and Mirlin establish why dimension, symmetry class, size scaling, and wavefunction statistics matter.

Hopping among localized states can produce finite-temperature conduction even inside a mobility gap. Mott variable-range hopping has the idealized form

$$\sigma(T) \sim \exp\!\left[ -\left( \frac{T_0}{T} \right)^{1/(d+1)} \right],$$

while the Coulomb-gap analysis of Efros and Shklovskii gives exponent $1/2$ under its own assumptions. Neither law defines a hard spectral gap. Mobility Edges explains localization diagnostics and critical scaling; Anderson Insulators explains hopping transport and the Coulomb gap.

Calculating a gap and measuring a threshold

A model diagonalization or Kohn–Sham calculation gives eigenvalue separations. Total-energy differences compare charged sectors; a self-energy approximation such as $GW$ estimates one-electron addition/removal energies. A Bethe–Salpeter calculation includes the electron–hole interaction and optical matrix elements. Localization-length or conductance scaling tests whether states are extended. These methods address different questions even when applied to the same sample.

Complementary measurements can connect the questions. Energy-aligned photoemission and inverse photoemission constrain removal and addition spectra; absorption then tests neutral excitations; transport asks whether excited charge becomes mobile. Match temperature, structure, doping and surface sensitivity before subtracting measured energies. Electron–phonon coupling and thermal expansion shift thresholds with temperature, as reviewed by Giustino, so a frozen-ion zero-temperature calculation need not equal a warm-sample onset.

Band overlap and optical momentum

Return to the parallel-band example. Its direct and global gaps differ because the electron and hole can carry different momenta. Keeping their total crystal momentum fixed makes that distinction quantitative.

For the insulating case $\Delta>4t$, a pair with momentum $Q$ has

$$\begin{aligned} E_{\mathrm{cont}}(Q) &=\min_k\{\Delta+2t[\cos((k+Q)a)-\cos(ka)]\}\\ &=\Delta-4t\left|\sin\frac{Qa}{2}\right|. \end{aligned}$$

Thus the unrestricted threshold is $\Delta-4t$, while the vertical continuum starts at $\Delta$. With $\Delta=3E_0$ and $t=E_0/2$, these are $1E_0$ and $3E_0$. An optically active $Q=0$ exciton at $2.4E_0$ would be bound by $0.6E_0$ relative to its own continuum, even though its energy exceeds the global gap. The figure shows this distinction. No binding energy follows from subtracting $2.4E_0$ from the unrelated $1E_0$ threshold.

Exercises

1. The many-body charge gap

A finite cluster has ground-state energies $E_0(N-1)=-97.8\,\mathrm{eV}$, $E_0(N)=-100.0\,\mathrm{eV}$, and $E_0(N+1)=-101.1\,\mathrm{eV}$. Compute $\mu^-$, $\mu^+$, and the full charge gap. What additional step is needed for a bulk claim?

Solution

The removal and addition chemical potentials are

$$\mu^- = E_0(N)-E_0(N-1) = -2.2\,\mathrm{eV}$$

and

$$\mu^+ = E_0(N+1)-E_0(N) = -1.1\,\mathrm{eV}.$$

Therefore

$$\Delta_{\mathrm{ch}} = \mu^+-\mu^- = 1.1\,\mathrm{eV}.$$

The signs of the individual chemical potentials depend on the energy zero; their difference does not. A bulk claim still requires size and boundary scaling or controlled corrections for charging and finite-cell effects.

2. Kohn–Sham and optical thresholds

An exact multiplicative Kohn–Sham description has $\Delta_{\mathrm{KS}}=1.1\,\mathrm{eV}$ and $\Delta_{\mathrm{xc}}=0.7\,\mathrm{eV}$. The minimum band gap is direct, and a bright exciton lies $0.2\,\mathrm{eV}$ below the matched zero-momentum quasiparticle continuum. Estimate the charge gap and bright optical threshold under these assumptions.

Solution

The exact DFT decomposition gives

$$\Delta_{\mathrm{ch}} = 1.1\,\mathrm{eV} + 0.7\,\mathrm{eV} = 1.8\,\mathrm{eV}.$$

If the coherent quasiparticle continuum begins at the same threshold and the specified exciton is the lowest optically active state, then

$$E_g^{\mathrm{opt}} = 1.8\,\mathrm{eV} - 0.2\,\mathrm{eV} = 1.6\,\mathrm{eV}.$$

The last step would fail for a mismatched continuum, dark lowest exciton, indirect transition without accounting for its momentum and phonon energy, or different temperature and environment.

3. Mobility width and nearest-edge activation

Localized states occupy the interval from $E_-^{\mathrm{mob}}=-0.6\,\mathrm{eV}$ to $E_+^{\mathrm{mob}}=0.4\,\mathrm{eV}$, and $\mu=0.1\,\mathrm{eV}$. Find the full mobility gap and the nearest-edge activation scale. Does a nonzero density of states at $\mu$ contradict either answer?

Solution

The full interval has width

$$\Delta_{\mathrm{mob}} = 0.4-(-0.6) = 1.0\,\mathrm{eV}.$$

The distances from the chemical potential to the upper and lower mobility edges are $0.3\,\mathrm{eV}$ and $0.7\,\mathrm{eV}$, respectively. Thus

$$E_A^{\mathrm{edge}} = 0.3\,\mathrm{eV}.$$

A nonzero density of localized states at $\mu$ is compatible with both statements. The mobility gap excludes extended states, not spectral weight.

4. A single-electron gap with compressible pair motion

Consider the illustrative finite-volume energy function

$$E_0(N)=\frac{N^2}{2\kappa V} +\Delta_p\frac{1-(-1)^N}{2}, \qquad \kappa>0,\quad\Delta_p>0.$$

At even $N$, compare the second differences for $N\pm1$ and $N\pm2$. Does the first difference alone prove an insulating state as $V\to\infty$ at fixed density?

Solution

The odd sectors pay the additional pairing cost, giving

$$E_0(N+1)+E_0(N-1)-2E_0(N) =\frac{1}{\kappa V}+2\Delta_p.$$

The neighboring even sectors do not:

$$E_0(N+2)+E_0(N-2)-2E_0(N) =\frac{4}{\kappa V}\longrightarrow0.$$

Minimizing $E_0(N)-\mu N$ over even $N$ gives a staircase whose density steps $2/V$ vanish in the bulk limit. Its envelope obeys $n=\kappa\mu$, hence $\partial n/\partial\mu=\kappa$. A finite single-electron gap coexists with a compressible charge sector in this example. The energy function demonstrates a logical distinction; it is not a microscopic derivation of a particular superconductor’s transport.

5. Two phonon-assisted thresholds

In the preceding one-phonon model, hold $E_g^{\mathrm{ind}}=1.10\,\mathrm{eV}$ and $\hbar\Omega=0.040\,\mathrm{eV}$ fixed. Find the two photon thresholds and the ratio of their phonon occupation factors at $300\,\mathrm K$ and $77\,\mathrm K$, using $k_{\mathrm B}=8.61733\times10^{-5}\,\mathrm{eV/K}$. Which branch survives at zero temperature? Does a lower onset establish an exciton binding energy?

Solution

The phonon-absorption threshold is $1.06\,\mathrm{eV}$; the phonon-emission threshold is $1.14\,\mathrm{eV}$. Their occupation-factor ratio is

$$\frac{n_{\mathrm B}}{n_{\mathrm B}+1} =e^{-\hbar\Omega/(k_{\mathrm B}T)} \simeq \begin{cases} 0.213,&T=300\,\mathrm K,\\ 0.00241,&T=77\,\mathrm K. \end{cases}$$

This is an intensity ratio only after matching the electronic excess energy above each threshold and equating the remaining matrix-element and phase-space factors. Comparing the two processes at one photon energy generally does not meet those conditions. At $T=0$ the absorption factor vanishes and the emission factor tends to one, leaving the $1.14\,\mathrm{eV}$ onset. The $1.06\,\mathrm{eV}$ thermal branch arises without any bound electron–hole state: subtracting it from $1.10\,\mathrm{eV}$ does not measure an exciton binding energy.

References

• E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions,” Physical Review Letters 42, 673–676 (1979).
• P. W. Anderson, “Absence of Diffusion in Certain Random Lattices,” Physical Review 109, 1492–1505 (1958).
• N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976), Chapters 7–12 and 28.
• M. L. Cohen and S. G. Louie, Fundamentals of Condensed Matter Physics, Cambridge University Press (2016).
• A. Damascelli, Z. Hussain, and Z.-X. Shen, “Angle-Resolved Photoemission Studies of the Cuprate Superconductors,” Reviews of Modern Physics 75, 473–541 (2003).
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