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The nearly-free electron model expands the Bloch state in plane waves,All basis functions have the same translation eigenvalues because $e^{i\mathbf{G}\cdot \mathbf{R}}=1$. A weak periodic potential mixes them most strongly near Bragg degeneracies and opens band gaps. Delocalized $s,p$ conduction states of simple metals, such as sodium and aluminium, are typical applications; the approximation need not describe tightly bound core states.

At a valley $K$, $\gamma (K)=1+2\cos (2\pi /3)=0$. For h-BN the inequivalent sublattice energies therefore leave the direct two-band gapGraphene's equivalent carbon sites instead have ${ϵ}_B={ϵ}_N$ in this model, so the two bands touch at the valleys. Near a valley, $|\gamma (K+\mathbf{q})|\simeq (\sqrt{3}a/2)|\mathbf{q}|$: graphene has approximately linear, gapless Dirac cones while h-BN has gapped band edges. The model panel in the original figure in solution displays this distinction.

First, undoped h-BN is an electrical insulator with thermally suppressed carriers, whereas neutral graphene is a zero-gap semimetal whose transport is readily changed by gating. Disorder, temperature and contacts still affect its measured conductivity. Second, h-BN suppresses interband absorption below its large gap and is transparent over much of the visible range, whereas graphene permits low-energy interband absorption unless occupations block it. A quantitative optical threshold in real h-BN also involves additional bands and strong excitonic effects; a two-orbital model should not be presented as a numerical prediction of its complete optical spectrum.

The lower curve is a valence band and the upper curve is a conduction band, separated by an indirect band gap. The readable upper curve has a minimum just inside the X boundary and slightly below the plotted chemical potential. It is therefore partially occupied, while the visible valence states lie below the chemical potential and are filled at low temperature. The sample is electron-doped, with electron pockets around the six symmetry-related conduction minima of silicon.

The full high-symmetry path and part of the lower-left dispersion are absent from the held image. Source gap: those missing path labels and curve segments cannot be read or reconstructed from this copy. The identification of the conventional six silicon valleys and the valence maximum at $\Gamma$ uses the usual silicon band model, rather than purported readings of missing pixels.

Two methods for calculating such an electronic band structure are density functional theory, solving a self-consistent periodic electronic problem, and a tight-binding model, diagonalizing a hopping Hamiltonian built from localized atomic orbitals. Neither is a measurement of the plotted dispersion relation.

In a conventional superconductor, an attractive pairing interaction forms Cooper pairs and a phase-coherent condensate. The elementary fermionic excitations have a superconducting energy gap $\Delta$; breaking a pair into two excitations costs at least $2\Delta$. The density of thermally excited quasiparticles is exponentially small for $k_{B}T\ll \Delta$. Nevertheless the vanishing DC electrical resistivity below the critical temperature is a property of the coherent supercurrent, not a claim that all thermal quasiparticles disappear. Below the relevant critical-current and critical-field limits, a homogeneous bulk sample carries dissipationless current; above the superconducting transition the normal resistivity returns.

In a semiconductor, the band gap separates filled valence states from empty conduction states because of the periodic crystal potential and orbital hybridization. Thermally exciting carriers across the gap produces both conduction electrons and valence holes. For an intrinsic sample,Thus its electrical resistivity generally rises rapidly on cooling, with an activated factor $e^{E_g/(2k_{B}T)}$ up to mobility and density-of-states prefactors. This differs from the zero-resistivity condensate response. Dopant freeze-out and extrinsic-carrier regimes require their own activation scales; the intrinsic formula is not universal for every doped sample.

Let $u=V_{10}<0$ andAt $X=(\pi /a,0)$ the two degenerate plane waves have momenta $(\pi /a,0)$ and $(-\pi /a,0)$ and coupling $u$. ThusTheir symmetric and antisymmetric combinations are the lower and upper states respectively. The local band gap is $2|u|$.

A double heterostructure laser places a lower-band gap active layer between wider-band gap layers with suitable type-I band alignment. The band offsets confine injected electrons and electron holes in the same small volume, increasing their overlap and reducing escape. Usually the active layer also has larger refractive index, so the same sandwich forms an optical waveguide. Optical confinement increases the gain-overlap factor $\Gamma$, while carrier confinement lowers the injection needed to reach the required quasi-Fermi-level splitting. These two distinct mechanisms reduce threshold current and heating. A band gap difference alone does not establish the sign of both confinement barriers; suitable band offsets are essential.

A quasi-Fermi level is the chemical potential parameter of an approximately thermalized carrier population within one energy band. Rapid intraband scattering can establish a Fermi-Dirac distribution in each energy band before slower interband recombination equilibrates their populations. Thus separate $E_{\mathrm{Fc}}$ and $E_{\mathrm{Fv}}$ describe a pumped population inversion even though the entire semiconductor is far from thermal equilibrium.

At a common positive temperature, the Fermi-Dirac distribution is strictly decreasing in $E-E_F$. ConsequentlyThe joint density of states is nonzero only above the band gap, soIn particular the quasi-Fermi-level splitting must exceed $E_g$. The upper boundary is transparency, where stimulated emission balances absorption; the lower boundary has zero joint density of states in this ideal three-dimensional parabolic model. The strict occupation comparison uses $T>0$; at exactly zero temperature occupations become step functions, and their supports should be checked explicitly, as in the next part.

A double heterostructure laser sandwiches a narrow-band gap active layer between wider-band gap cladding layers. In a type-I band alignment, the conduction band edge is lower and the valence band edge higher in the active layer. Electrons therefore see an energy well, and electron holes also see a well because their excitation energy increases when the valence band edge decreases. Both carrier populations remain concentrated where stimulated emission occurs.

The active layer also normally has the higher refractive index, forming an optical waveguide. The optical normal mode overlaps the injected carriers strongly, and the wider-band gap cladding has less interband absorption at the emitted photon energy. Carrier confinement reduces leakage, while optical confinement reduces the required pump current and hence heating.

Writing $\Gamma$ for the fraction of the optical normal mode overlapping the active layer, the threshold condition is $\Gamma g={\alpha }_i+{\alpha }_m$, with internal and mirror losses ${\alpha }_i,{\alpha }_m$. A larger overlap and a higher carrier number density at a given current both lower the threshold current. The figure shows composition-induced offsets; a working junction also has band bending under bias, which is omitted to make the confinement mechanism visible.

In crystal polymorphism, the same chemical composition has different crystal structures; for an chemical element, this is a form of allotropy. Diamond has a three-dimensional tetrahedral network of covalent bonds, giving high hardness and a wide band gap. Graphite has strongly bonded planar sheets, with weaker interactions between sheets. Sliding between sheets makes it a useful solid lubricant; mobile electronic states give good electrical conduction within the sheets. Their different properties come from different bonding and structures, despite both being carbon.

For potential period $a$, a Bloch state has the form ${\psi }_k(x)=e^{ikx}{u}_k(x)$, with $u_k(x+a)=u_k(x)$, or equivalently ${\psi }_k(x+a)=e^{ika}{\psi }_k(x)$. Quasimomentum is defined modulo the reciprocal-lattice vector $2\pi /a$. A Brillouin zone is one fundamental interval for that identification, usually $[-\pi /a,\pi /a)$. Here $a=2\pi$, so it is $[-1/2,1/2)$.

For small $V_0>0$, the first free-particle degeneracy occurs at the zone edge $k=1/2$, between plane waves $e^{ix/2}$ and $e^{-ix/2}$ of energy $1/4$. In this two-dimensional subspace, the diagonal matrix elements of the potential vanish. Its Fourier coefficient at wave number one is $-V_0/2$; the $\cos 2x$ term cannot connect these two waves at first order. Thus degenerate perturbation theory gives the matrixwith energies $1/4\pm V_0/2+O(V_0^2)$. Near the edge the two uncoupled energies have opposite slopes, and diagonalization gives the avoided crossing, so the lowest band gap is

For large $V_0$, expand about each deepest minimum $x=2\pi j$:The local quantum harmonic oscillator $-d^2/d{x}^2+(5V_0/2)x^2$ has levels $(2n+1)\sqrt{5V_0/2}$. Its localization width is $O(V_0^{-1/4})$, so the quartic term changes these low levels by $O(1)$. Tunnelling between successive deep wells gives exponentially narrow energy bands. The secondary minima at $x=(2j+1)\pi$ have potential zero, and their levels lie far above the first two deep-well levels, whose energies are $-2V_0+O(\sqrt{V_0})$. Therefore the gap between the first and second bands is the first oscillator spacing to leading order:The $O(1)$ anharmonic shifts and exponentially small bandwidths do not change this leading estimate.

Bloch theorem states that energy eigenfunctions in a Bravais lattice potential may be chosen asThe wavevector is defined modulo a reciprocal lattice vector, so one can restrict it to the first Brillouin zone. This follows because lattice translations commute with the Hamiltonian and have simultaneous eigenvalues $e^{i\mathbf{k}\cdot \mathbf{l}}$.

In one dimension write $V(x)={\sum }_G{V}_G{e}^{iGx}$ with $G=2\pi j/a$ and expand a Bloch state as ${\psi }_k={\sum }_G{c}_G{e}^{i(k+G)x}$. The Schrödinger equation becomesFor zero potential, the parabolas $E_G^0(k)={\hslash }^2(k+G{)}^2/(2m)$ fold into the Brillouin zone. A weak potential shifts a nondegenerate level by $V_0$ at first order, with second-order correction ${\sum }_{G\ne 0}|V_G{|}^2/(E_0^0-E_G^0)$ where the denominators stay away from zero. This expansion fails near crossings, where degenerate perturbation theory is required.

At the first zone boundary, put $G=2\pi /a$ and $k=G/2+q$. The nearly degenerate waves with momenta $\hslash k$ and $\hslash (k-G)$ are coupled by $V_G$. Retaining their two-dimensional subspace givesDiagonalizing producesAt $q=0$ there is a gap $\boxed{\Delta E=2|V_G|}$: the two standing-wave combinations sample the periodic potential differently. Similar Bragg degeneracies occur at $k=j\pi /a$, with gaps controlled to first order by the corresponding Fourier components. If that component vanishes, a gap need not open at first order. Between the gaps the continuous energy eigenvalues form energy bands, periodically represented in the reduced-zone scheme. For the first zone edge, the curvature of the upper and lower branches gives a minimum and a maximum respectively for a sufficiently weak nonzero potential, so the avoided crossing is a genuine local forbidden energy interval.

At zero temperature electrons occupy states up to the Fermi energy. The Fermi surface is the set $E_n(\mathbf{k})=E_F$ separating occupied from unoccupied states within partially filled bands (Fermi points in one dimension). A partially filled band supplies arbitrarily nearby empty states, allowing occupations to respond to a weak electric field and conduct. Completely filled bands give no net current from such a redistribution; if the next empty band is separated by a positive band gap, the system is a band insulator. Band overlap can instead yield a metal even at an integer band filling. At finite temperature an insulator can conduct through thermally excited electrons and holes.

Label the two atoms in primitive cell $j$ by $u_{2j}$ and $u_{2j+1}$, with $j$ understood modulo $N$ because of the periodic boundary condition. Choose the spring between them to have constant $\lambda$. Newton's second law gives

For a longitudinal normal mode, putPeriodicity requires $e^{2iNka}=1$, so $k=\pi r/(Na)$. Since the primitive-cell length is $2a$, wavevectors differing by $\pi /a$ describe the same mode, and the first Brillouin zone may be chosen asThe amplitudes obeySetting its determinant to zero gives the alternating-spring chain dispersion relationThe lower sign is the acoustic phonon branch and the upper sign is the optical phonon branch.

At the edge of the Brillouin zone,so the frequency band gap isIt vanishes at $\alpha =1$, when the apparent two-atom primitive cell can be reduced to one atom.

Near the centre of the Brillouin zone, a Taylor expansion givesandThe group velocity of the acoustic branch at long wavelength is the speed of sound,

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.

The potential has period $2\pi$, so the first Brillouin zone is $-1/2\le k<1/2$. At its boundary the free-particle statesare degenerate, with energyThe relevant Fourier coefficient is supplied by $V_0\cos x$:The term $-V_0\cos 2x$ changes wavevector by two and has no matrix element within this degenerate pair. Thus degenerate perturbation theory gives the matrixand the two band-edge energiesThe nearly-free electron model therefore predicts the lowest band gap

From the illuminated side downward, the p-type gallium arsenide cap provides a low-resistance route to the front metal contact. It should remain beneath the contact grid but be removed from the optical aperture to reduce parasitic absorption. The wider-band gap p-type aluminium gallium arsenide window transmits much of the useful light and reduces front-surface recombination of carriers in the adjacent absorber. The optical-window and reduced-recombination rationale has a primary experimental basis in early heteroface devices.^[ref]

The p-type gallium arsenide emitter and n-type gallium arsenide base form the p-n junction. Absorbed photons generate electrons and electron holes; diffusion and the junction's electric field direct them toward the appropriate contacts. The n-type aluminium gallium arsenide back-surface-field layer provides a barrier to minority holes moving toward the substrate and reduces rear recombination. The n-type gallium arsenide substrate supports the epitaxial layers and conducts majority carriers to the rear contact.

The front metal must collect current with limited shading and the back metal provides the return contact. Interpreting the patterned hatched region as the optical antireflection coating, it reduces reflected light over the exposed window. Carrier selectivity, low recombination, low series resistance and low shading must work together; the equilibrium junction field alone is not a source of electrical energy.

The emission energies are approximatelyDraw the Al-rich interface as a wider-gap barrier between these two emitting regions: a raised conduction-band edge impedes electron transfer, and a lowered valence-band edge impedes hole transfer. The following is a consistent straddling-barrier schematic, with the shell having the smaller optical gap.

The two emission wavelengths determine optical transition energies, not unique conduction- and valence-band offsets. Exciton binding, confinement and strain can shift emission relative to the bulk band gap. The stated barrier supports inhibited charge exchange; its separate electron and hole barrier heights need additional band-alignment data.

Two applications are ultraviolet photodetection and ultraviolet/near-band-edge light emission. The bulk edge corresponds approximately toAbove-gap absorption creates carriers that can be collected in a photodetector. Measure dark current and wavelength-dependent response to separate photocarriers from heating or leakage. An optically or electrically excited emitter instead uses electron-hole recombination. quantum confinement in sufficiently small dots can shift absorption and emission above the bulk band gap. Surface states can produce broad defect luminescence or nonradiative loss, so a direct bulk gap does not guarantee an efficient, narrow-band emitter.

For standard deviations of position and momentum,Confinement to a region of size $R$ requires a momentum spread of order $\hslash /R$, hence kinetic energy of order ${\hslash }^2/(2m^{\ast }{R}^2)$. A sufficiently small semiconducting nanocrystal therefore has discrete energy levels and an optical edge shifted above the bulk band gap.

An illustrative strong-confinement model iswhere $\alpha$ is a geometry-dependent number of order unity. The positive confinement contribution grows as $R^{-2}$; electron-hole Coulomb interaction contributes an attractive term of order $-R^{-1}$. The uncertainty principle motivates the scale, not the exact coefficient or oscillator strength. Strong size dependence occurs near the exciton Bohr radius. Finite barriers, dielectric mismatch and surface states modify this model, while size distributions broaden ensemble spectra.

An exciton is a correlated electron–hole pair with a full wavefunction $\Psi ({\mathbf{r}}_e,{\mathbf{r}}_h)$. In strong confinement, a useful leading approximation is a product of lowest electron and hole envelope functions, followed by a Coulomb correction. A spherical hard-wall ground envelope is nodeless, proportional to $\sin (\pi r/R)/r$, and finite at the origin. Finite barriers produce exterior tails.

For a core-shell semiconductor nanocrystal with type-I band alignment, both conduction- and valence-band offsets confine the carriers mainly to the core. Their envelopes penetrate the shell but overlap less with external traps. Chemical surface passivation and spatial separation from defects can suppress nonradiative losses and increase photoluminescence quantum yield. Shell thickness, interfacial quality and strain must be controlled; a defective shell can introduce losses.

A larger shell band gap alone does not guarantee type-I alignment. A staggered type-II band alignment can move one carrier toward the shell, reduce electron–hole overlap and alter lifetime and emission energy. The sketch illustrates the conditional type-I case. In CdSe/ZnS experiments, optical spectra were combined with structural and compositional measurements to test shell-growth interpretations.^[ref]

Three point-defect structures are a Stone-Wales defect, a reconstructed graphene monovacancy and a reconstructed graphene divacancy. The original sketch gives ring inventories rather than fitted atomic coordinates.

A Stone-Wales defect rotates one C–C bond, converting four hexagons into a 5–7–7–5 group without changing the number of atoms. A graphene monovacancy removes one carbon; a common reconstruction gives a 5–9 group with residual under-coordination. A graphene divacancy removes two adjacent carbons; a 5–8–5 reconstruction can eliminate dangling bonds. Other divacancy reconstructions exist, so atom loss does not uniquely select one topology.

Irradiation, energetic processing, etching and incomplete growth can create vacancies; thermally or beam-activated bond rearrangement can change their structures. Electron microscopy itself can create defects, so compare exposures before attributing a time sequence to unperturbed material.

Pentagons favor positive curvature and heptagons negative curvature. Such defects can bend tubes, help form caps or junctions, or change a local wrapping relationship. They scatter carriers and alter the local density of states; vacancies can add localized states and chemical reactivity. Whether a clean band gap appears depends on defect identity, concentration and arrangement, not merely on the presence of a defect.