Metals, Insulators, and Semiconductors

Metals, insulators, semiconductors and semimetals describe how low-energy charge moves through a material. Band filling provides a useful first classification: a dispersive partially filled band typically gives a metal, while a positive global gap between filled and empty bands gives a band insulator. A semiconductor is a gapped material whose carriers can be usefully controlled by temperature, doping, light or gates; a semimetal has band touching or small electron and hole pockets.

Interactions, disorder and topology refine that picture. An interaction-driven Mott insulator has a charge gap, an Anderson insulator can retain states at the chemical potential but lack diffusion, and a topological insulator can conduct along its boundary while its bulk remains insulating. The classification therefore requires both the excitation spectrum and the transport mechanism.

Required background. Band Theory Overview supplies band occupation, chemical potential, and the independent-particle/interacting boundary used here.

Helpful background. Band Gaps distinguishes global band, charge, optical, transport, and mobility gaps. Density of States supplies normalized spectral counting, while Fermi Surface supplies the geometry of partially filled bands.

Charge excitations, localization and transport

Ask three separate questions.

Are there low-energy charged excitations?

A positive charge gap means that adding or removing charge costs finite energy. In an independent-particle crystal this often comes from a band gap, but interactions can create a charge gap without a band-insulator filling.

Are the available states extended?

A state can exist at the chemical potential and still be spatially localized. A nonzero density of states is therefore not sufficient for metallic dc transport.

What response is being called insulating?

Longitudinal electrical conduction, Hall response, thermal transport, optical absorption, compressibility, and boundary conduction probe different structures. A sample can have:

• vanishing bulk longitudinal conductivity;
• quantized transverse conductivity;
• conducting boundary states;
• a finite optical threshold;
• neutral gapless excitations;
• thermally activated but nonzero conductivity at finite temperature.

One adjective cannot encode all of these facts.

Band Filling Criterion

For independent-electron bands with all internal labels included in $n$, the electron count per primitive cell at zero temperature is

$$\nu = \frac{\Omega_c}{(2\pi)^d} \sum_n \int_{\mathrm{BZ}} d^dk\, \Theta \left[ \mu-\varepsilon_n(\mathbf k) \right].$$

A completely filled nondegenerate band contributes one electron per cell. If the occupied bands are separated everywhere from the empty bands,

$$E_g = \min_{\mathbf k,c} \varepsilon_c(\mathbf k) - \max_{\mathbf k,v} \varepsilon_v(\mathbf k) > 0,$$

the independent-particle ground state is a band insulator.

A regular crossing of a dispersive extended band supplies a Fermi surface and a metallic band response. Partial filling of an exactly flat band needs separate analysis: its group velocity vanishes, so filling alone does not establish conduction. If nominal valence and conduction bands touch or overlap only weakly, the model can instead describe a semimetal.

Filling is necessary, not sufficient

The elementary criterion assumes:

• a meaningful primitive cell and translation symmetry;
• an independent-electron description for the displayed occupation formula;
• extended states near the chemical potential;
• the correct magnetic and structural order;
• correct degeneracy counting;
• no interaction-driven obstruction to charge motion.

Interactions can redistribute physical electron occupation across a broad spectral function; counting only coherent quasiparticle peaks is then insufficient. The classification can change without changing the electron count.

Odd filling warning

With spin degeneracy and no symmetry breaking, a displayed band commonly holds two electrons per cell. An odd electron count then forces partial filling in the simplest model. Yet an enlarged unit cell can make the new filling even, magnetism can lift spin degeneracy, nonsymmorphic symmetry can enforce larger connected band groups, and strong repulsion can produce a Mott state. “Odd means metal” is a first diagnostic, not a universal theorem.

Metals

A conventional metal has gapless charged excitations and extended states at the chemical potential. In a Fermi liquid, low-energy excitations are quasiparticles near one or more Fermi-surface sheets:

$$\varepsilon_n(\mathbf k_{\mathrm F}) = \mu.$$

The Fermi Surface page develops the material geometry, pockets, velocities, and experimental reconstruction.

Density of states is not the definition

An ordinary three-dimensional metal usually has

$$\rho(\mu)>0.$$

But this is not a universal criterion. A nodal semimetal can have $\rho(\mu)=0$ while remaining gapless, and an Anderson insulator can have $\rho(\mu)>0$ while lacking diffusion.

Clean and dirty limits

A perfect periodic noninteracting metal does not relax its intraband current. Its conductivity contains a zero-frequency Drude delta function rather than a finite ordinary resistivity. Disorder, boundaries and processes that transfer momentum to the lattice or an external bath can broaden that peak. Electron–electron Umklapp can also relax the current. The mere presence of collisions or fluctuating fields does not guarantee a finite resistivity: conserved quantities and their overlap with the current matter.

Drude Theory distinguishes the finite-relaxation peak from the clean delta-function limit and states which scattering and multiband assumptions are needed to infer a resistivity.

Good metals, bad metals, and strange metals

In a good quasiparticle metal, a scattering rate is small compared with the excitation energy over an appropriate low-energy window. A bad metal can exceed simple mean-free-path criteria while still conducting. A strange metal has transport or spectral behavior not organized by standard long-lived Fermi-liquid quasiparticles.

These names describe regimes, not new electron-count rules. A broad spectral function can make a sharp band crossing or Fermi surface operationally ambiguous even when the system is plainly conducting.

A superconductor is not merely a perfect metal

A superconductor has phase coherence, superfluid stiffness and a Meissner response. A normal clean metal can also have a Drude delta function, so infinite dc conductivity alone does not establish superconductivity. Many superconductors have a gapped electron spectrum; others have nodes. When classifying a superconducting material as a metal or insulator, specify whether its normal state is meant.

Band Insulators

In an ordinary band insulator, all bands below the chemical potential are completely filled and all bands above it are empty. The many-electron ground state of the independent-particle model is a filled-band Slater determinant.

At zero temperature and in the absence of in-gap states, creating separated charge carriers requires at least the fundamental gap:

$$E_g = E_c-E_v.$$

The bulk carrier density vanishes at $T=0$, and low-field longitudinal dc conductivity vanishes in the thermodynamic limit.

Full bands are not empty physics

The equilibrium group velocities cancel across a full ordinary band:

$$\int_{\mathrm{BZ}} d^dk\, \nabla_{\mathbf k} \varepsilon_n(\mathbf k) = 0.$$

Nevertheless, filled bands determine:

• dielectric polarization;
• interband optical absorption;
• orbital magnetization;
• virtual-transition contributions to response;
• Berry phases and topological invariants;
• bound excitons after electron–hole interactions are included.

“No free carriers” must not be replaced by “no quantum response.”

Finite-temperature conduction

Thermal excitation, impurities, and defects can produce carriers. A measured conductivity often follows an approximate activated form,

$$\sigma(T) \sim \sigma_0(T) \exp \left( -\frac{E_a}{k_{\mathrm B}T} \right),$$

over a restricted temperature interval. The activation energy $E_a$ need not equal $E_g/2$. It can reflect donor or acceptor ionization, hopping between localized states, a mobility edge, contact barriers, or a temperature-dependent mobility.

Semiconductors

An intrinsic semiconductor and an ordinary band insulator are not separated by a sharp zero-temperature phase transition. Semiconductor is a practical category: the gap and materials chemistry permit controlled carrier creation and useful device behavior.

Doping and gates control the carrier count; light creates electron–hole pairs; strain, confinement and interfaces can change the band edges.

Intrinsic carrier statistics

Near nondegenerate parabolic edges in three dimensions,

$$\varepsilon_c(\mathbf k) \simeq E_c + \frac{\hbar^2q^2}{2m_c^*}$$

and

$$\varepsilon_v(\mathbf k) \simeq E_v - \frac{\hbar^2q^2}{2m_v^*}.$$

In the nondegenerate Boltzmann regime,

$$n = N_c \exp \left[ -\frac{E_c-\mu}{k_{\mathrm B}T} \right]$$

and

$$p = N_v \exp \left[ -\frac{\mu-E_v}{k_{\mathrm B}T} \right].$$

For isotropic valleys,

$$N_c = g_c \left( \frac{ m_c^*k_{\mathrm B}T }{ 2\pi\hbar^2 } \right)^{3/2}$$

and

$$N_v = g_v \left( \frac{ m_v^*k_{\mathrm B}T }{ 2\pi\hbar^2 } \right)^{3/2},$$

Here $N_c$ and $N_v$ are effective densities of available states per physical volume, not numbers of primitive cells. The masses refer to one valley; $g_c$ and $g_v$ explicitly count its equivalent spin and valley copies. For an ellipsoidal valley, replace its isotropic mass by the density-of-states mass $(m_1m_2m_3)^{1/3}$. Do not absorb valley multiplicity into that mass and also count it in $g$.

Charge neutrality in an intrinsic crystal gives $n=p=n_i$, so

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

The intrinsic chemical potential is

$$\mu_i = \frac{E_c+E_v}{2} + \frac{k_{\mathrm B}T}{2} \ln \left( \frac{N_v}{N_c} \right).$$

It lies at exact midgap only when the effective state counts are equal.

These formulas assume parabolic edges, three dimensions, dilute nondegenerate carriers, thermal equilibrium, and no important impurity or excitonic correction.

Electrons, holes, and conductivity

A hole describes a missing electron in an otherwise filled band. Its positive charge and velocity give a convenient account of the valence-band current.

The low-field conductivity is approximately

$$\sigma = e \left( n\mu_e^{\mathrm{mob}} + p\mu_h^{\mathrm{mob}} \right),$$

where $\mu_e^{\mathrm{mob}}$ and $\mu_h^{\mathrm{mob}}$ are positive carrier mobilities. The chemical potential $\mu$ and a mobility are different quantities despite the conventional reuse of the same Greek letter.

The electron and hole densities alone do not determine the Hall coefficient in a multicarrier system; mobility and anisotropy also enter.

Doping regimes

A doped semiconductor commonly passes through three regimes as temperature rises:

1. Freeze-out: many dopants are neutral because their bound carriers are not ionized.
2. Extrinsic regime: ionized dopants dominate the carrier count.
3. Intrinsic regime: thermally generated electron–hole pairs dominate.

Donors introduce electron states near a conduction edge; acceptors introduce hole-producing states near a valence edge. Real dopants also bring disorder, screening, lattice relaxation, degeneracy, and possible impurity-band formation. “Move the Fermi level” is useful shorthand, not a complete microscopic theory of doping.

Optical and transport gaps

The fundamental charge gap, direct transition threshold, optical gap, and transport activation energy need not coincide. Momentum conservation, selection rules, phonon assistance, exciton binding, disorder, and contacts separate them.

Use Band Gaps to identify the direct, global, charge, optical, transport, or mobility object and its probe assumptions. Semiconductor laser gain and population inversion belong to Semiconductor Lasers Overview.

Semimetals

A semimetal has low-energy electron and hole states but much less phase space than an ordinary metal. Two common mechanisms are distinct.

Band overlap

If the conduction minimum lies below the valence maximum,

$$E_g = E_c-E_v < 0,$$

small electron and hole pockets coexist. Charge neutrality may produce a compensated semimetal with equal total electron and hole pocket volumes after degeneracy factors are included.

The density of states at $\mu$ is typically nonzero but small. Transport can be highly sensitive to mobility imbalance, magnetic field, and compensation.

Nodal touching

Valence and conduction bands can meet at points or lines without overlapping. For an isotropic three-dimensional Dirac point,

$$\varepsilon_\pm(\mathbf k) = \pm \hbar v|\mathbf k|.$$

Let $g$ count cones and internal copies within one energy branch; it does not count the positive and negative branches twice. The density of states per volume is

$$\rho(E) = \frac{ g|E|^2 }{ 2\pi^2(\hbar v)^3 }.$$

At exact neutrality,

$$\rho(0) = 0,$$

yet the system is gapless. For $\mu>0$, the electron density is

$$n = \frac{ g\mu^3 }{ 6\pi^2(\hbar v)^3 }.$$

Dimensionality and dispersion change these power laws. Symmetry or topology can protect some nodes, but not every semimetal is Dirac, Weyl, nodal-line, or topological.

Semimetal is not a synonym for narrow-gap semiconductor

A narrow-gap semiconductor has $E_g>0$ and no zero-temperature bulk carriers in the ideal intrinsic limit. A semimetal has $E_g\le0$ in the band description. Disorder, finite temperature, and experimental resolution can make the practical distinction difficult near a tiny gap or overlap.

Gaps and localized states

A gap in the single-electron spectrum alone does not distinguish band and Mott mechanisms. Conversely, an Anderson insulator can have states at the chemical potential but no diffusion. The final panel sketches a time-reversal-protected boundary crossing inside a bulk gap. These are qualitative spectra; the energy zero is the chemical potential. Boundary protection requires the relevant symmetry and a sufficiently isolated boundary.

Mott Insulators

A partially filled independent-electron band ordinarily supplies mobile charge, but interactions can instead produce an incompressible state at commensurate filling. This Mott mechanism is distinct from band filling, Anderson localization, and symmetry-induced band reconstruction. Magnetic order commonly coexists with it but is not its definition.

The atomic Hubbard example below makes the addition and removal costs explicit. Mott Insulators explains how bandwidth, filling and ligand charge transfer modify this picture, and how experiments distinguish it from a gap produced primarily by symmetry breaking.

Anderson Insulators

Disorder can leave a finite density of one-particle states at the chemical potential while localizing those states, so the zero-temperature dc conductivity vanishes without a hard spectral gap. The obstruction is absence of long-distance diffusion, not absence of available energies. This separates an Anderson insulator from an ideal band insulator and from an interaction-driven Mott insulator.

Wavefunction decay and participation ratios quantify Anderson localization. At nonzero temperature, a bath can assist hopping between localized states; Anderson Insulators develops the corresponding transport laws.

In three dimensions a Mobility Edge can separate localized and extended states at different energies even while the density of states remains smooth. Finite size, dephasing, interactions, contacts, and hopping all complicate the strict zero-temperature limit, so an insulating temperature dependence alone does not uniquely establish Anderson localization.

Topological Insulators

A noninteracting topological insulator is a band insulator whose occupied Bloch-state subspace cannot be continuously deformed to an ordinary atomic insulator while preserving the relevant gap and symmetries.

The bulk energy spectrum can look fully gapped in both cases. The distinction resides in the global geometry of the occupied eigenvectors.

For a two-dimensional Chern insulator, the occupied-band Chern number determines the quantized Hall response. With $\mathcal A=i\langle u|\nabla_{\mathbf k}u\rangle$, electron charge $-e$ and $J_x=\sigma_{xy}E_y$,

$$\sigma_{xy}=-C\frac{e^2}{h}.$$

Chern Numbers in Band Theory derives this relation in these conventions. Spinful time-reversal-invariant band insulators with $\Theta^2=-1$ in two and three dimensions instead admit $\mathbb Z_2$ indices while their total Chern response vanishes. A nontrivial index can enforce boundary states when the protecting symmetry is preserved.

Boundary conduction does not make the bulk metallic. Conversely, a surface can be disordered, doped, hybridized in a thin film, or gapped by symmetry breaking without erasing every bulk topological distinction.

Topological Insulators explains helical edges, surface Dirac cones and the material conditions for observing them.

Quantum Hall Insulators

In an integer quantum Hall plateau, the two-dimensional bulk has a spectral or mobility gap for longitudinal transport. For electrons in $B_z>0$, let $\nu>0$ be the integer plateau index. In the same tensor convention,

$$\sigma_{xx} \longrightarrow 0$$

and

$$\sigma_{xy} = - \nu \frac{e^2}{h}.$$

Chiral edge states carry boundary current. Calling the state simply “insulating” hides the quantized transverse response; calling it simply “conducting” hides the gapped bulk. The conductivity tensor and geometry must be stated; Integer Quantum Hall Effect develops the filling, localization, and boundary interpretation.

Fractional quantum Hall states go further: interactions and topological order produce fractionalized excitations and cannot be reduced to occupied noninteracting electron bands.

Comparing Mechanisms

State	Spectral or charge structure	State character near $\mu$	Zero-temperature bulk longitudinal response
Fermi-liquid metal	gapless Fermi surface	extended quasiparticles	metallic; ideal clean model has Drude weight
Band insulator	positive band and charge gap	no bulk states at $\mu$	insulating
Intrinsic semiconductor	positive controllable band gap	thermally activated carriers at $T>0$	insulating at ideal $T=0$
Overlap semimetal	electron and hole pockets	extended, low density	metallic or semimetallic
Nodal semimetal	point or line touching	extended but vanishing phase space possible	gapless; response is dimension and disorder dependent
Mott insulator	interaction-driven charge gap	correlated spectral weight away from $\mu$	insulating
Anderson insulator	no necessary spectral gap	localized	insulating
Topological band insulator	bulk band gap plus nontrivial occupied subspace	bulk gapped, boundary states possible	bulk insulating; boundary or Hall response can remain

Comparing transport with spectroscopy

Transport tests whether charge can travel across the sample. ARPES and tunneling test momentum-resolved and local spectral weight, respectively. Their disagreement can be informative: a finite local density of states together with vanishing diffusion is consistent with localization. Activated resistivity supplies evidence for an energy scale, but that scale may come from a mobility edge, dopant ionization or contacts rather than a fundamental band gap.

Optical absorption creates neutral excitations. Selection rules, phonons and exciton binding can move its threshold away from the charge gap; Band Gaps compares these quantities explicitly. Heat capacity and magnetic response can reveal neutral excitations even when charge transport is insulating. Thickness dependence and boundary-sensitive probes help separate bulk from surface conduction. A useful classification explains these measurements together, including temperature, disorder and geometry.

Exercises

Exercise 1: intrinsic density and chemical potential

Derive $n_i$ and $\mu_i$ from the nondegenerate electron and hole densities.

Solution

Set $n=p$:

$$N_c \exp \left[ -\frac{E_c-\mu_i}{k_{\mathrm B}T} \right] = N_v \exp \left[ -\frac{\mu_i-E_v}{k_{\mathrm B}T} \right].$$

Taking logarithms and solving for $\mu_i$ gives

$$\mu_i = \frac{E_c+E_v}{2} + \frac{k_{\mathrm B}T}{2} \ln \left( \frac{N_v}{N_c} \right).$$

Multiplying the two densities,

$$np = N_cN_v \exp \left( -\frac{E_c-E_v}{k_{\mathrm B}T} \right).$$

Since $n=p=n_i$ and $E_g=E_c-E_v$,

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

Exercise 2: activation slope

Assume three-dimensional parabolic band edges, a fixed gap and fixed masses. Both carrier mobilities scale as $T^{-r}$. Find the local activation energy

$$E_{\mathrm{app}}(T)=-k_{\mathrm B}\frac{d\ln\sigma}{d(1/T)}.$$

Evaluate it at $T=300\,\mathrm K$ for $E_g=0.40\,\mathrm{eV}$ and constant mobilities. Use $k_{\mathrm B}=8.61733\times10^{-5}\,\mathrm{eV/K}$.

Solution

The effective state densities give $n_i\propto T^{3/2}e^{-E_g/(2k_{\mathrm B}T)}$. Therefore

$$\ln\sigma=\text{constant}+ \left(\frac32-r\right)\ln T-\frac{E_g}{2k_{\mathrm B}T}.$$

Differentiating, using $d\ln T/d(1/T)=-T$, yields

$$E_{\mathrm{app}}(T)=\frac{E_g}{2} +\left(\frac32-r\right)k_{\mathrm B}T.$$

For $r=0$ at 300 K, $E_{\mathrm{app}}\simeq0.2388\,\mathrm{eV}$, rather than $0.2000\,\mathrm{eV}$. Dividing the conductivity by its known $T^{3/2-r}$ prefactor would recover the latter slope. A straight-line fit over a finite interval averages a temperature-dependent local slope. Gap variation, impurity ionization, hopping or contacts introduce further changes.

Exercise 3: atomic Mott gap

At $t=0$, consider one Hubbard site per cell at one electron per site. Show that the charge gap is $U$.

Solution

At one electron per site there is no double occupancy, so choose the reference interaction energy as zero.

Removing one electron creates an empty site and costs no interaction energy:

$$E_0(N-1)-E_0(N) = 0.$$

Adding one electron forces one site to be doubly occupied and costs

$$E_0(N+1)-E_0(N) = U.$$

Therefore

$$\begin{aligned} \Delta_c &= \left[ E_0(N+1)-E_0(N) \right] - \left[ E_0(N)-E_0(N-1) \right] \\ &= U. \end{aligned}$$

The zero removal interaction energy is not a zero grand-canonical excitation cost. For $0<\mu<U$, the equilibrium one-electron state has removal cost $\mu$ and addition cost $U-\mu$, both positive. Their sum is $U$. Finite hopping changes the thresholds and spreads the atomic spectral peaks into bands; the charge gap is no longer exactly $U$.

Exercise 4: inverse participation ratio

Evaluate the scaling of $\operatorname{IPR}=\sum_i|\psi_i|^4$ for a normalized state: (a) a state uniform over $N$ sites and (b) a state uniform over $M$ sites with $M$ fixed as $N\to\infty$.

Solution

For a state uniform over $N$ sites,

$$|\psi_i|^2 = \frac{1}{N}.$$

Hence

$$\operatorname{IPR} = N \left( \frac{1}{N} \right)^2 = \frac{1}{N} \longrightarrow 0.$$

For a state uniform over only $M$ sites,

$$\operatorname{IPR} = M \left( \frac{1}{M} \right)^2 = \frac{1}{M}.$$

If $M$ remains fixed, the IPR remains finite as the system grows. This measures spatial concentration; by itself it does not establish an entire sample’s conductivity or distinguish all critical and extended states.

References

• P. W. Anderson, “Absence of Diffusion in Certain Random Lattices,” Physical Review 109, 1492–1505 (1958), doi:10.1103/PhysRev.109.1492.
• N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976), Chapters 1–2, 12–13, and 28.
• M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Reviews of Modern Physics 82, 3045–3067 (2010), doi:10.1103/RevModPhys.82.3045.
• C. L. Kane and E. J. Mele, “$Z_2$ Topological Order and the Quantum Spin Hall Effect,” Physical Review Letters 95, 146802 (2005), doi:10.1103/PhysRevLett.95.146802.
• C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005), Chapters 6–8.
• W. Kohn, “Theory of the Insulating State,” Physical Review 133, A171–A181 (1964), doi:10.1103/PhysRev.133.A171.
• N. F. Mott, Metal-Insulator Transitions, 2nd ed. (Taylor & Francis, 1990).
• S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley, 2006), Chapters 1–2, doi:10.1002/0470068329.
• D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall Conductance in a Two-Dimensional Periodic Potential,” Physical Review Letters 49, 405–408 (1982), doi:10.1103/PhysRevLett.49.405.
• P. Y. Yu and M. Cardona, Fundamentals of Semiconductors, 4th ed. (Springer, 2010), Chapters 2–4.