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Angle-resolved photoemission spectroscopy illuminates a surface with photons of known energy and measures the kinetic energy and emission direction of the escaping electrons. Referencing the initial binding energy to the Fermi level givesHere $\Phi$ is the sample/analyser work-function reference appropriate to the calibrated energy scale, and $\theta$ is the emission angle from the surface normal. Parallel crystal momentum is conserved modulo a surface reciprocal lattice vector; measuring two angular coordinates maps the two in-plane momentum components.

Sweeping angle and energy traces the occupied spectral weight and its dispersion. In a quasi-two-dimensional sheet, no detailed free-electron model of the perpendicular final-state momentum is required to map the in-plane bands. Peaks reflect the occupied single-particle spectral function, multiplied by Fermi occupation and photoemission matrix elements and broadened by finite lifetimes and resolution. Ordinary photoemission does not directly measure empty conduction bands; a gap may need complementary information. A clean surface, vacuum and charge control are especially important for an insulating specimen.

For ordinary one-phonon infrared optical absorption, both an energy and a momentum selection rule must hold. The optical photon has negligible crystal momentum on the scale of the Brillouin zone, so crystal momentum conservation selects $\mathbf{q}\simeq 0$ modulo a genuine reciprocal lattice vector. In addition, the mode must change the macroscopic electric polarization, with a component along the incident polarization of an electromagnetic wave.

In terms of atomic displacements ${\mathbf{u}}_j$ and Born effective charge tensors $Z_j^{\ast }$, the relevant derivative is proportional to ${\sum }_j{Z}_j^{\ast }{\mathbf{u}}_j$ per primitive unit cell. Thus an optical phonon at $\mathbf{q}=0$ is not automatically infrared active: its dipole contributions may cancel. Disorder-assisted and multiphonon processes relax these simple first-order selection rules.

For a vertical interband transition, the crystal momentum is effectively unchanged because the photon momentum is small. A stimulated downward transition needs an occupied upper state and an empty lower state, with factor $f_c(E_2)[1-f_v(E_1)]$; upward photon absorption has factor $f_v(E_1)[1-f_c(E_2)]$. The two matrix elements agree by microscopic reversibility. Their difference is thereforeA common slowly varying transition-strength factor is implicit. Broadening replaces the sharp transition by a convolution; multiple valence bands contribute separate terms.

For one isotropic pair of parabolic energy bands, writeThus $E_2-E_1=E_g+{\hslash }^2{k}^2/(2m_r)$. Including the usual twofold spin degeneracy, the number of paired states per volume inside a sphere of radius $k$ is $k^3/(3{\pi }^2)$. Differentiating with respect to transition energy $E$ gives the three-dimensional parabolic interband joint density of statesFor density per angular frequency, $g_j^{(\omega )}(\omega )=\hslash g_j^{(E)}(\hslash \omega )$. Stating the density convention avoids a missing factor of $\hslash$; other degeneracies multiply the formula.

At a common carrier temperature, the Fermi-Dirac distribution is a strictly decreasing function of $E-F$. The direct-gap semiconductor gain condition follows:The lower inequality denotes a transition above the ideal band edge, where the displayed joint density of states is nonzero. Thus quasi-Fermi-level splitting must exceed the transition photon energy. Equality is transparency, rather than net amplification; at zero temperature an allowed transition must also lie within the occupied-electron and empty-valence windows.

The single-particle Hamiltonian $H_0=-{\hslash }^2{\partial }_x^2/(2m)+V(x)$ has a periodic potential $V(x+d)=V(x)$. Define the unitary spatial translation operator $T_{d}f(x)=f(x+d)$. Its derivatives commute with translation, and the potential's periodicity gives $H_0{T}_d=T_d{H}_0$.

Each energy eigenspace is therefore preserved by $T_d$, so within it one may choose simultaneous energy and translation eigenstates. On a large periodic interval of $M$ cells, $T_d^M=1$ and its eigenvalues are phases $\lambda =e^{ikd}$, with $k=2\pi j/(Md)$; the infinite-volume limit gives continuous quasimomentum modulo $2\pi /d$. For a simultaneous state in band $j$, this provesThen $u_{jk}(x)=e^{-ikx}{\psi }_{jk}(x)$ obeys $u_{jk}(x+d)=u_{jk}(x)$, which constructs the usual Bloch state rather than assuming its form. The label $j$ distinguishes energy bands at the same crystal momentum $\hslash k$. In a degenerate energy eigenspace an arbitrary superposition need not itself have one translation phase; the result refers to the simultaneous Bloch states.

A periodic lattice Hamiltonian commutes with lattice translations. The Bloch theorem therefore classifies its states by characters of the translation group:where $u$ is lattice periodic. For each crystal momentum $\mathbf{k}$, solving on a primitive cell with the corresponding quasiperiodic boundary condition gives discrete eigenvalues indexed by $n$. Their continuous dependence on $\mathbf{k}$ produces the energy bands $E_n(\mathbf{k})$ as $\mathbf{k}$ ranges through the Brillouin zone. Bragg mixing opens gaps at suitable degeneracies, although bands can overlap in three dimensions and gaps are not guaranteed between every pair.

The quantum numbers are band index, wavevector modulo reciprocal lattice vectors, and spin where spin is conserved. In a finite crystal the allowed wavevectors are discrete; in the thermodynamic limit they become dense and the band description emerges. Spin-orbit coupling can replace a separately conserved spin label by an appropriate internal band label.

Dilute randomly placed impurities break exact translation symmetry, so crystal momentum is no longer an exact quantum number. Most extended states retain approximately the host band structure but acquire impurity scattering and disorder broadening. An impurity potential can also produce impurity levels in a crystal, often in a band gap: levels associated with a donor impurity lie near the conduction-band edge and can supply electrons, while those associated with an acceptor impurity lie near the valence-band edge and can create holes. The impurity's species and potential determine whether such levels exist and how deep they are; not every impurity necessarily makes a gap state. At low density these levels are nearly isolated, while overlap at increasing density can broaden them into an impurity band. Added or removed carriers shift the chemical potential as well.

Bloch theorem states that the energy eigenstates of a Hamiltonian invariant under translations by a Bravais lattice can be chosen as Bloch statesIndeed, the unitary operators $T_r$ commute with one another because $T_r{T}_s=T_{r+s}=T_s{T}_r$, and they commute with $H$ by hypothesis. The simultaneous diagonalization theorem therefore lets us diagonalize all translations within each energy eigenspace. Their eigenvalues form a unitary character $\chi$ of the additive lattice:Writing $\chi (a_j)=e^{-ik\cdot a_j}$ on a primitive basis gives $\chi (r)=e^{-ik\cdot r}$. In position space, with $(T_r\psi )(x)=\psi (x-r)$, this implies ${\psi }_k(x+r)=e^{ik\cdot r}{\psi }_k(x)$. Hence $u_k(x)=e^{-ik\cdot x}{\psi }_k(x)$ is lattice-periodic. Adding a reciprocal lattice vector to $k$ leaves the character unchanged, so the crystal momentum lies in a Brillouin zone.

Forthe equations $a_i\cdot b_j=2\pi {\delta }_{ij}$ give the reciprocal basisThe reciprocal lattice is triangular, so the first Brillouin zone is its Wigner-Seitz cell, a regular hexagon. Its six corners areReciprocal-lattice translations identify these corners in two classes of three. Representatives areFor example, $(b_1-b_2)/3=K^{\prime }-b_2$, while the other equivalences follow by symmetry and reciprocal translations. This gives the requested sketch: a regular hexagon with the vertical edge from $K^{\prime }$ to $K$ at $k_x=2\pi /(\sqrt{3}a)$ and alternating $K,K^{\prime }$ corner classes.

For the tight-binding model, introduce the normalized Bloch sumEach hop by $\pm a_j$ multiplies this state by $e^{\mp ik\cdot a_j}$. Thus the two-direction nearest-neighbour tight-binding dispersion isAlong the boundary edge from $K^{\prime }$ to $K$, $k_x=2\pi /(\sqrt{3}a)$ and $|k_y|\le 2\pi /(3a)$, soFor $\lambda >0$, it rises from $E_0+2\lambda$ at either corner to $E_0+4\lambda$ at the midpoint; for $\lambda <0$ the ordering reverses. Globally the two cosines in the first expression can simultaneously equal $1$ or $-1$, and therefore

Each orbital band contains two one-electron states per lattice site because an electron has two spin states. By band filling, a valency of one leaves this band half-filled and the material conducts, whereas a valency of two fills it. A filled band can be insulating only if it is separated from every empty band by a positive band gap.

The first band's maximum is $E_0+4|\lambda |$. If the second band's minimum is $E_0+\Delta$, a gap exists precisely when $\Delta >4|\lambda |$. Consequently, among the nontrivial fillings described here,For valency one the first band is partially filled. If $\Delta \le 4|\lambda |$, the bands overlap or touch, so even at valency two the overlapping energy bands prevent a band insulator.

For elastic scattering, $E_f=E_i$ and ${\mathbf{k}}_f-{\mathbf{k}}_i=\mathbf{G}$, where $\mathbf{G}$ is a reciprocal lattice vector. In the ideal static crystal, coherent scattering obeys this lattice selection rule; the crystal as a whole can take the mechanical recoil with negligible energy. crystal momentum is defined modulo $\hslash \mathbf{G}$, not as unconstrained free-particle momentum.

For annihilation of a phonon $(\mathbf{q},\omega )$,For phonon creation,The signs follow whether the neutron gains or loses the phonon's energy and crystal momentum. These are one-phonon coherent inelastic neutron scattering conditions, with $\mathbf{G}$ accounting for lattice periodicity.

A three-phonon normal process conserves crystal wavevector without a reciprocal-lattice shift, ${\mathbf{k}}_1+{\mathbf{k}}_2={\mathbf{k}}_3$. An Umklapp process instead hasThe crystal can absorb the reciprocal-lattice momentum; after reduction to the first Brillouin zone the phonon drift transporting heat is relaxed. A normal process redistributes phonons while preserving their total crystal momentum, so it is not by itself the same resistive mechanism. energy must also be conserved in either process.

At low temperature, large-wavevector phonons are rare, suppressing Umklapp scattering. Boundary and defect scattering then limit the mean free path; with a nearly constant boundary-limited path, ${\kappa }_{\mathrm{ph}}\simeq (C_{\mathrm{ph}}/V)v\ell /3$ grows as $T^3$. As temperature increases, heat capacity rises but Umklapp scattering shortens $\ell$, often producing a electrical conductivity maximum. At sufficiently high temperature a simple anharmonic model has $C_{\mathrm{ph}}$ nearly constant and ${\tau }_U^{-1}\propto T$, giving ${\kappa }_{\mathrm{ph}}\propto T^{-1}$. These are regime estimates, not a universal law for every material or size.

In an electrical insulator, lattice vibrations usually carry most heat. A small temperature gradient produces an imbalance between phonons moving in opposite directions. In an isotropic kinetic approximation,where $c_V$ is heat capacity per unit volume, $v_s$ an appropriate acoustic speed and ${\ell }_{\mathrm{tr}}$ the transport mean free path. The factor one third is the angular average of the squared direction cosine. With several branches and frequency-dependent scattering, replace this estimate by the corresponding weighted mode integral. A short lifetime matters for conductivity only insofar as it relaxes heat current.

For a clean three-dimensional crystal at $T\ll {\Theta }_D$, the Debye heat capacity varies as $T^3$. If diffuse boundary scattering fixes ${\ell }_{\mathrm{tr}}$ near a specimen dimension, then ${\kappa }_{\mathrm{ph}}\propto T^3$. The size dependence is a test of that boundary-limited regime. Specular surfaces, defects and small specimens can change the simple law; at ballistic scales a local conductivity description itself needs care.

As temperature rises, more phonons are excited, but phonon-phonon scattering shortens their transport path. Normal processes conserve total crystal momentum, whereas an Umklapp process satisfies ${\mathbf{q}}_1+{\mathbf{q}}_2-{\mathbf{q}}_3=\mathbf{G}\ne 0$ with reciprocal vector $\mathbf{G}$ and can relax directed heat flow. Low-temperature Umklapp scattering is strongly suppressed; its increasing importance often produces a conductivity maximum followed by a decline. Isotope and impurity scattering lower or broaden that maximum.

At sufficiently high temperature, $c_V$ approaches its classical constant while a typical anharmonic Umklapp rate grows approximately as $T$. Under that further approximation,Thus a clean crystalline insulator commonly shows a low-temperature rise, a peak, then a high-temperature fall. This is not a universal curve for every insulator: glasses, strong disorder, multiple scattering mechanisms, radiative heat transfer and structural transitions can give different behavior.

A steady-state bar experiment supplies a concrete test. Inject known heating power, measure the small temperature difference between two separated sensors and infer $\kappa =PL/(A\Delta T)$ after correcting for heat leaks. Keep the gradient small enough that a single representative temperature is meaningful, calibrate sensor contact effects and subtract radiative/background losses. Repeat with different widths or isotope/defect content to distinguish boundary limitation from intrinsic scattering. A power-dependent result may reflect a changing temperature profile rather than a new microscopic law.

A phonon is one quantum of a crystal's vibrational normal modes. In the harmonic approximation, each mode is a quantum harmonic oscillator with energy levels $(n+1/2)\hslash \omega$. A phonon carries excitation energy $\hslash \omega$ and crystal momentum $\hslash q$ modulo a reciprocal lattice vector. It is a collective lattice excitation, not one atom travelling along the chain. Phonons obey Bose-Einstein statistics; their number need not be conserved in thermal equilibrium.

A semiconductor has a filled valence band and an empty conduction band at zero temperature in the intrinsic idealisation, separated by a modest band gap. Thermal excitation or deliberately introduced impurities can create mobile electrons and electron holes. Its conductivity is consequently very sensitive to temperature and doping. A metal has available electronic states at the occupied energy boundary, through a partly filled band or overlapping bands, and need not thermally excite carriers across a gap.

For a direct band gap, the conduction-band minimum and valence-band maximum occur at the same crystal momentum. An electron can recombine with a hole near these extrema and emit a photon while conserving crystal momentum, since optical photon momentum is small on the lattice scale. For an indirect band gap, the extrema lie at different crystal momenta. A band-edge radiative transition then ordinarily needs a phonon as well, making it less probable and allowing nonradiative processes to compete more effectively.

gallium arsenide has a direct fundamental gap, while silicon has an indirect one. This makes efficient band-edge emission and hence a light-emitting diode easier to obtain from gallium arsenide. It does not mean silicon has no optical transitions or that every direct-gap sample emits efficiently: defects, surfaces and carrier confinement also matter.

For a Bravais lattice with primitive translation vectors ${\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3$, define the reciprocal lattice by vectors $\mathbf{G}$ satisfying $e^{i\mathbf{G}\cdot \mathbf{R}}=1$ for every direct-lattice translation $\mathbf{R}$. A convenient basis iswith cyclic formulas for ${\mathbf{b}}_2,{\mathbf{b}}_3$. Then ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$ and $\mathbf{G}=h{\mathbf{b}}_1+k{\mathbf{b}}_2+l{\mathbf{b}}_3$ for integers $h,k,l$. The reciprocal primitive volume is $(2\pi {)}^3/V_{\mathrm{cell}}$.

These vectors are the spatial Fourier series frequencies of a periodic lattice potential or density. Elastic coherent crystal diffraction occurs when the scattering wavevector transfer equals a reciprocal lattice vector, with intensities additionally controlled by the basis crystal lattice structure factor. This is the reciprocal-space version of Bragg's law. A lattice of points and its basis should not be confused: a basis can make allowed reciprocal-lattice diffraction intensities vanish.

The first Brillouin zone is the Wigner-Seitz cell around the reciprocal origin: it consists of points closer to $\mathbf{G}=0$ than to any other reciprocal point. Its faces lie on the perpendicular bisectorsTo see the condition, equate $|\mathbf{k}{|}^2$ and $|\mathbf{k}-\mathbf{G}{|}^2$. These are also free-electron Bragg-degeneracy planes, where a periodic potential can open a band gap. Higher zones classify regions beyond successive Bragg planes; the reduced-zone description folds them into the first zone while retaining a band label.

In one dimension with spacing $a$, reciprocal points occur at $2\pi j/a$, and the first zone is $[-\pi /a,\pi /a]$. A square lattice of spacing $a$ has reciprocal basis $2\pi \hat{\mathbf{x}}/a,2\pi \hat{\mathbf{y}}/a$, and a square first zone with edges at $k_x,k_y=\pm \pi /a$.

The reciprocal of a face-centered cubic lattice is a body-centered cubic lattice and vice versa, with the scale fixed by the primitive-vector definition. In a Bloch state, wavevectors differing by $\mathbf{G}$ describe equivalent crystal momenta modulo the lattice periodicity. A crystal with $N$ primitive cells has $N$ allowed wavevectors in one Brillouin zone, hence $N$ states in each band per spin. This state count links reciprocal geometry to band filling, metallic conduction and band insulating behaviour. Reciprocal space also labels phonons and makes conservation of crystal momentum modulo $\hslash \mathbf{G}$ transparent.

A Bloch state in a crystal experiences a periodic lattice potential. Its band dispersion relation $ϵ(k)$ encodes the lattice response, so an effective mass allows a wavepacket's acceleration under a weak applied force to be described without explicitly following the microscopic periodic forces.

In one dimension the wavepacket group velocity and semiclassical equation for crystal momentum areDifferentiating the first equation with respect to time givesDefining $F=m^{\ast }a$ yieldsThis band effective mass is controlled by band curvature, not the slope. A band minimum has positive $m^{\ast }$; a band maximum has negative electron $m^{\ast }$ and is often more conveniently described with positive-mass electron holes. In three dimensions the inverse effective mass is generally a tensor, $(m^{\ast -1}{)}_{ij}={\hslash }^{-2}{\partial }_{k_i}{\partial }_{k_j}ϵ$. The scalar formula assumes motion along a specified direction in an isolated band, weak fields and negligible interband transitions; the bare electron mass has not physically changed.

A Bloch state on the loop has $\psi (x)=u(x)e^{i\varphi x}$, with $u(x+a)=u(x)$. Its circumference is $L=Na$, so $u(x+Na)=u(x)$. Single-valuedness after one complete circuit requires the periodic boundary conditionThereforeThe allowed Bloch wavevectors are spaced by $2\pi /(Na)$. Values of $\varphi$ differing by $2\pi /a$ label the same crystal momentum modulo a reciprocal lattice vector, so there are $N$ inequivalent choices in one Brillouin zone. In this notation $\varphi$ is the Bloch wavenumber, not a dimensionless phase; $\varphi a$ is dimensionless. Spin gives additional states but does not alter this condition.

Let $N=L/a$ be the integer number of primitive unit cells. A Bloch state can be written as ${\psi }_k(x)=e^{ikx}{u}_k(x)$, with $u_k(x+a)=u_k(x)$. The periodic boundary condition ${\psi }_k(x+L)={\psi }_k(x)$ therefore requiresThe spacing of allowed wavevectors is $\Delta k=2\pi /L$. The first Brillouin zone has width $2\pi /a$, soFor an exact endpoint convention take a half-open Brillouin zone, such as $[-\pi /a,\pi /a)$. Its two endpoints differ by a reciprocal lattice vector and must not both be counted as distinct values of crystal momentum. There are exactly $N$ inequivalent integers $j$ modulo $N$, irrespective of whether $N$ is even or odd. This is state counting in a Bloch band: one allowed wavevector per primitive unit cell per energy band. A separate twofold spin multiplicity gives $2N$ electronic states per energy band, without doubling the number of wavevectors.

The crystal has periodic potential $V(x+a)=V(x)$. By Bloch's theorem, an energy eigenstate may be chosen asExpand the periodic factor in a Fourier series:Multiplication by $e^{ikx}$ gives the stated plane-wave expansion. Here $x$ is position, $a$ is the primitive unit cell length, $G_1$ is the primitive reciprocal lattice wavenumber, $n$ runs over every integer, and $C_{k,n}$ are complex numbers specifying the component wave amplitudes. The wavevector $k$ labels the Bloch state modulo $G_1$; $\hslash k$ is its crystal momentum, rather than generally its measured free-space momentum.

The component $e^{i(k+n{G}_1)x}$ is a plane wave of wavevector $k+n{G}_1$. All these components acquire the same phase factor $e^{ika}$ under $x\mapsto x+a$, since $e^{in{G}_{1}a}=1$. Their mixing therefore respects the crystal's translation symmetry. The coefficients and energy eigenvalue follow from the Schrödinger equation; the expression is not a claim that the coefficients are equal or that only finitely many are nonzero. The energy band index is suppressed in the notation, so different energy bands at a common $k$ generally have different coefficient sets.

In an isolated energy band, semiclassical motion satisfies $\hslash \dot{k}=F$ and $v={\hslash }^{-1}dE/dk$. The band is periodic in crystal momentum, so a constant force sweeps the state through equivalent points of the Brillouin zone. Its velocity repeats rather than increasing without bound. For an electron in a constant electric field, $F=-e\mathcal{E}$, and the Bloch oscillation period isFor $E(k)=E_0-2J\cos ka$, $v=(2Ja/\hslash )\sin ka$ and integration gives a bounded position oscillation of amplitude $2J/|F|$ about its mean. A filled band has no corresponding net transport oscillation after summing uniformly over all $k$.

Observation requires scattering times longer than $T_B$ and negligible interband tunnelling. Ordinary crystals often scatter too rapidly before a full zone traversal; larger-period structures and clean atom-optical lattices help. In the cold-atom Bloch oscillation experiment, accelerating a standing-wave lattice supplied a controlled force, and releasing the atoms at different times revealed an oscillating momentum distribution.^[ref]

For a lattice-periodic potential, Bloch's theorem permits energy eigenfunctions ${\psi }_{\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}}{u}_{\mathbf{k}}(\mathbf{r})$, where $u_{\mathbf{k}}(\mathbf{r}+\mathbf{R})=u_{\mathbf{k}}(\mathbf{r})$ for every lattice translation $\mathbf{R}$. The crystal momentum is $\hslash \mathbf{k}$, defined modulo a reciprocal lattice vector. A reciprocal vector $\mathbf{G}$ satisfies $\mathbf{G}\cdot \mathbf{R}\in 2\pi \mathbb{Z}$, so $e^{i\mathbf{G}\cdot \mathbf{r}}$ is lattice-periodic.

Expand the Bloch wave as plane waves in one momentum class, $\psi ={\sum }_{\mathbf{q}}{c}_{\mathbf{q}}{e}^{i\mathbf{q}\cdot \mathbf{r}}$ with $\mathbf{q}=\mathbf{k}+{\mathbf{G}}^{\prime }$. Substitution into the Schrödinger equation, using $U={\sum }_{\mathbf{G}}{U}_{\mathbf{G}}{e}^{i\mathbf{G}\cdot \mathbf{r}}$, gives a kinetic coefficient ${\hslash }^2{q}^2{c}_{\mathbf{q}}/(2m)$ and potential coefficient ${\sum }_{\mathbf{G}}{U}_{\mathbf{G}}{c}_{\mathbf{q}-\mathbf{G}}$. Independence of the plane waves therefore givesThis uses $\mathbf{q}$ for the paper's running plane-wave index, to distinguish it from the fixed reduced Bloch wavevector.

For $G=2\pi /a$, the sine potential has coefficients $U_G=-iV$, $U_{-G}=iV$. At the Brillouin zone boundary, the free states $e^{i\pi x/a}$ and $e^{-i\pi x/a}$ are degenerate. Keeping these two resonant waves in the nearly-free electron model givesThe determinant is $(\beta -E{)}^2-V^2$, so the two energies are $E_{\pm }=\beta \pm |V|$, separated by $2|V|$. With $V>0$, the lower combination has coefficients $(1,-i)/\sqrt{2}$ and density proportional to $1-\sin Gx$; the upper has $(1,i)/\sqrt{2}$ and density $1+\sin Gx$. The lower wave places less weight where the potential is positive. Coupling to nonresonant plane waves gives higher-order corrections beyond this two-wave approximation.

The first Brillouin zone of a primitive cubic lattice is the cube $|k_i|\le \pi /a$. Its boundary along $(100)$ is the face center $(\pi /a,0,0)$, while along $(111)$ it is the corner $(\pi /a,\pi /a,\pi /a)$. ThereforeA spin-degenerate band accommodates two electrons per cell. To be a band insulator, the filled first band must be separated from the second at every wavevector; a direct local splitting alone is insufficient. Under the paper's uniform-boundary-gap estimate, take the lower edge as $E^{(0)}-V$ and the upper edge as $E^{(0)}+V$. The highest lower edge is $3\beta -V$ and the lowest upper edge is $\beta +V$. Thus the indirect gap in a uniformly split zone-boundary model isEquality gives touching rather than a positive gap; for smaller $V$, overlapping energy bands prevent a band insulator. This final threshold is the stated uniform-splitting band-edge estimate. A quantitative three-dimensional potential mixes several plane waves at corners and requires its full band structure rather than assuming the two-wave sine calculation remains exact there.

Let the missing electron occupy the Bloch state of crystal momentum $\hslash {\mathbf{k}}_e$ and energy $E_e({\mathbf{k}}_e)$. Relative to the filled band, define ${\mathbf{k}}_h=-{\mathbf{k}}_e$ modulo a reciprocal-lattice vector and $E_h({\mathbf{k}}_h)=C-E_e(-{\mathbf{k}}_h)$, with a fixed reference $C$. ThenThe velocity equality follows by differentiating the hole energy with respect to ${\mathbf{k}}_h$; two minus signs cancel. The signed band effective mass is therefore opposite, becoming a positive hole mass near an electron-band maximum. The hole electric charge is $+e$ when the electron charge is $-e$. These momenta are crystal momenta, and the electron mass being compared is the signed band mass, not the bare electron mass. The positive hole current equals the current lost by removing that electron.