Anderson Insulators

An Anderson insulator has available single-particle states near the chemical potential, but those states are spatially localized and cannot sustain bulk diffusion at zero temperature. It can therefore be spectrally gapless and thermodynamically compressible while remaining electrically insulating. The obstruction is not absence of states; it is absence of extended transport paths.

At strictly zero temperature, a closed noninteracting Anderson insulator has vanishing dc conductivity in the thermodynamic limit. At nonzero temperature, localized carriers can move by exchanging energy with phonons or another environment. The resulting hopping conductivity is not a failure of localization. It is a dissipative transport channel between localized orbitals.

We first treat noninteracting localized orbitals coupled to an equilibrated phonon bath, then add long-range interactions. The interaction-induced Coulomb gap is extra physics; it is not required for Anderson localization itself.

Required background. Disorder in Quantum Matter supplies microscopic disorder and scattering scales. Anderson Localization supplies exponentially localized states and suppressed coherent transmission. Scaling Theory of Localization supplies the size-dependent conductance flow and localization length used to identify the insulating regime.

Localized charge transport and insulating regimes

Here $\xi$ is the amplitude decay length of an orbital, $\rho(E)$ is the density of states per spatial volume and energy, and localized-site energies $\varepsilon_i$ are measured from the chemical potential $\mu$. Internal degeneracies are included in $\rho$. Conductivity is measured in the low-field Ohmic regime.

The canonical stretched-exponential form is

$$\sigma(T) = \sigma_0(T) \exp \left[ -\left( \frac{T_0}{T} \right)^p \right].$$

The exponent $p$ is only one part of a mechanism claim. Dimensionality, density of states, localization length, dielectric screening, prefactor, electric-field range, and temperature window must agree as well.

Insulation without a Band Gap

Density of states is not mobility

For extended diffusive carriers, the Einstein relation is

$$\sigma = e^2\rho(E_{\mathrm F})D.$$

An Anderson insulator can have

$$\rho(E_{\mathrm F})>0, \qquad D(E_{\mathrm F})=0,$$

and therefore

$$\sigma_{\mathrm{dc}}(0)=0.$$

The density of states counts available eigenvalues. The diffusion coefficient asks whether a wave packet spreads without bound. These are logically independent. A smooth tunneling spectrum does not prove metallic transport, and a divergent resistance does not prove a hard spectral gap.

For a sample of length $L$ with localized states at the Fermi energy, define a transport localization length $\xi_{\mathrm{tr}}$ through the typical coherent conductance:

$$\begin{aligned} g_{\mathrm{typ}}(L) &\equiv \exp\!\left[ \overline{\ln g(L)} \right], \\ \overline{\ln g(L)} &\sim -\frac{2L}{\xi_{\mathrm{tr}}}. \end{aligned}$$

Some authors absorb the factor of $2$ into $\xi_{\mathrm{tr}}$. Moreover, this estimator need not equal the orbital-tail length $\xi$ in a finite or multichannel geometry. The finite density of localized levels can still contribute to compressibility, heat capacity, optical absorption, local spectroscopy, and polarization. Those responses do not require dc charge to cross an infinite sample.

Distinguishing insulating mechanisms

Insulating regime	Low-energy obstruction to charge transport
Band insulator	A hard single-particle gap
Anderson insulator	Localized states with vanishing long-distance diffusion
Mott insulator	Interaction-induced cost to add or remove charge
Interacting localized insulator	Localization with a possible soft Coulomb gap
Topological insulator	Insulating bulk; suitable boundaries can conduct

Real materials need not occupy one pure limit. Disorder can localize Hubbard-band states, interactions can reshape an Anderson transition, and granular films can combine island charging with intergrain hopping. Spectroscopy, spatial localization and charge response help distinguish these possibilities; a resistance curve alone usually cannot.

Zero temperature and finite temperature are different problems

Static disorder creates localized eigenstates but does not itself move an electron between two nondegenerate localized orbitals. A hop from energy $\varepsilon_i$ to $\varepsilon_j$ must exchange

$$\Delta\varepsilon = \varepsilon_j-\varepsilon_i$$

with a bath, another electron, a photon, or a time-dependent field. In the standard low-field hopping problem, phonons provide this energy exchange and enforce detailed balance.

Consequently:

• localization determines exponentially small spatial overlap;
• the bath determines energy exchange and occupation kinetics;
• percolation through the random rate network determines macroscopic conductivity.

Calling all three ingredients “Anderson localization” hides the transport physics. Conversely, observing finite $\sigma(T>0)$ does not establish extended eigenstates.

Localized Single-Particle States

Orbital overlap

A localized orbital centered at $\mathbf R_i$ has the asymptotic form

$$\lvert\psi_i(\mathbf r)\rvert \sim \xi^{-d/2} \exp \left( -\frac{ \lvert\mathbf r-\mathbf R_i\rvert }{ \xi } \right).$$

The tails determine exponentially small tunneling or bath-induced transition matrix elements. For exact orthogonal eigenstates, their inner product vanishes and the static Hamiltonian is diagonal; $t_{ij}$ below denotes a tunneling matrix element in a localized-site description, not that inner product. A phonon perturbation can have off-diagonal matrix elements between the exact eigenstates. For separation

$$r_{ij} = \lvert\mathbf R_i-\mathbf R_j\rvert$$

therefore scales as

$$\lvert t_{ij}\rvert \sim t_0 \exp \left( -\frac{r_{ij}}{\xi} \right).$$

A transition rate contains the squared matrix element:

$$\Gamma_{ij} \propto \exp \left( -\frac{2r_{ij}}{\xi} \right).$$

This exponential distance penalty competes with the energetic penalty for finding an accessible final state.

Miller–Abrahams rates

For dilute carriers coupled to an equilibrated phonon bath, the exponential part of the Miller–Abrahams (1960) rate can be written

$$\Gamma_{i\to j} =\Gamma_0 e^{-2r_{ij}/\xi} e^{-u_{ij}/(k_{\mathrm B}T)}.$$

Here $u_{ij}=\max(\varepsilon_j-\varepsilon_i,0)$ is the uphill energy cost. The simplified $\Gamma_0$ suppresses algebraic energy and phonon-spectrum factors. An uphill hop must absorb a phonon and is thermally suppressed. A downhill hop can emit one. The rate ratio obeys detailed balance:

$$\frac{ \Gamma_{i\to j} }{ \Gamma_{j\to i} } = \exp \left[ -\frac{ \varepsilon_j-\varepsilon_i }{ k_{\mathrm B}T } \right].$$

Fermi occupation factors must be included when sites are not dilute. Correlated occupations and multiparticle hops can invalidate an independent-resistor reduction.

For independent fermionic occupations in linear response, including the occupied-initial and empty-final factors gives a pair resistance with exponential part

$$R_{ij}\sim R_0 e^{2r_{ij}/\xi+E_{ij}/(k_{\mathrm B}T)},$$

where

$$2E_{ij}=\lvert\varepsilon_i\rvert+\lvert\varepsilon_j\rvert +\lvert\varepsilon_i-\varepsilon_j\rvert.$$

Site energies are measured from $\mu$. The macroscopic resistance is not the average of $R_{ij}$. Current chooses a percolating backbone of comparatively favorable links, and its largest indispensable resistance sets the critical exponential scale, as developed by Ambegaokar, Halperin and Langer (1971).

Nearest-neighbor activation

If transport is restricted to a characteristic separation $a_{\mathrm{hop}}$ and a fixed energy barrier $E_a$, then

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

This Arrhenius law has $p=1$. It can describe activation to a mobility edge, nearest-neighbor hopping within an impurity band, contact injection, a band gap, or a polaron barrier. The same exponent does not identify which energy is being paid.

Variable-range hopping becomes favorable when a carrier can travel farther in space to find a state closer in energy, thereby reducing the thermal cost enough to compensate for weaker overlap.

Variable-Range Hopping

The Mott optimization

Assume a constant localized density of states

$$\rho(E) \simeq \rho_0$$

within the active energy window. A region of radius $R$ and energy half-width of order $\varepsilon$ contains

$$\mathcal N(R,\varepsilon) \sim \rho_0\varepsilon R^d$$

candidate states. Requiring at least one accessible state gives

$$\varepsilon(R) \sim \frac{1}{ \rho_0R^d }.$$

The logarithmic hopping cost is therefore

$$X(R) \sim \frac{2R}{\xi} + \frac{1}{ \rho_0k_{\mathrm B}TR^d }.$$

The first term increases with distance; the second decreases because a larger volume offers a closer energy match. Differentiating,

$$\frac{dX}{dR} = \frac{2}{\xi} - \frac{d}{ \rho_0k_{\mathrm B}TR^{d+1} },$$

so the optimum satisfies

$$R_\ast^{d+1} \sim \frac{ d\xi }{ 2\rho_0k_{\mathrm B}T }.$$

Percolation theory fixes numerical constants omitted by this one-hop estimate. The resulting Mott law is

$$\sigma(T) = \sigma_{\mathrm M}(T) \exp \left[ -\left( \frac{T_{\mathrm M}}{T} \right)^{1/(d+1)} \right],$$

with

$$T_{\mathrm M} = \frac{ C_d }{ k_{\mathrm B}\rho_0\xi^d }.$$

The dimension-dependent constant $C_d$ depends on the percolation model, density-of-states convention, and localization profile.

The canonical exponents are

$$\begin{array}{c|ccc} d&1&2&3\\ \hline p_{\mathrm M}&1/2&1/3&1/4. \end{array}$$

Strict one-dimensional networks can be controlled by rare breaks and acquire important finite-size or logarithmic corrections, so $p=1/2$ should not be fitted mechanically.

The upper panel plots a two-dimensional example with $r=R/\xi$: the distance cost is $2r$, the energy cost is $8/r^2$, and their sum is minimized at $r=2$. At that point the costs are $4+2=6$. The lower panel compares a constant density of states with linear (2D) and quadratic (3D) Coulomb-gap forms; their vertical normalizations are arbitrary. These illustrate the assumptions entering Mott and Efros–Shklovskii hopping.

Optimal distance and energy

At scaling accuracy,

$$\frac{R_\ast}{\xi} \sim \left( \frac{T_{\mathrm M}}{T} \right)^{1/(d+1)}$$

and

$$\frac{\varepsilon_\ast}{k_{\mathrm B}T} \sim \left( \frac{T_{\mathrm M}}{T} \right)^{1/(d+1)}.$$

Cooling therefore lengthens the typical hop while narrowing its energy window. The phrase “variable range” refers to this temperature-dependent spatial optimization.

The variable-range regime requires $R_\ast\gg\xi$, an optimal energy inside a constant-DOS window, and a sample much larger than the critical hopping network. Linear response additionally requires the field work over an optimal hop to be small:

$$eER_\ast \ll k_{\mathrm B}T$$

at the level of the standard rate model. When this condition fails, field-assisted hopping and electron heating can mimic a different temperature exponent.

Crossovers

As temperature falls, nearest-neighbor activation may give way to Mott hopping and then interaction-modified hopping. The sequence is not automatic. A finite sample departs from the bulk network law when the hopping or percolation correlation length approaches a device dimension; this does not by itself require resistance saturation. A nearby gate can screen interactions; a changing dielectric constant can shift $T_0$; and electron–phonon decoupling can replace the electron temperature by a bias-dependent value.

Interactions and the Coulomb Gap

Stability of localized charges

For localized electrons with long-range repulsion, let $\kappa$ be the relative dielectric permittivity, so the interaction is $e^2/(4\pi\epsilon_0\kappa r)$. Write $U_{ij}=e^2/(4\pi\epsilon_0\kappa r_{ij})$ and $\delta n_i=n_i-\bar n_i$, where $\bar n_i$ is the neutralizing background. A schematic site Hamiltonian is

$$H=\sum_i\varepsilon_i^{(0)}n_i +\frac12\sum_{i\ne j}U_{ij}\delta n_i\delta n_j.$$

Let $\widetilde\varepsilon_i<0$ be the interaction-renormalized energy of an occupied site and $\widetilde\varepsilon_j>0$ that of an empty site, measured from $\mu$. Moving the electron from $i$ to $j$ costs

$$\Delta E_{ij} = \widetilde\varepsilon_j -\widetilde\varepsilon_i - \frac{ e^2 }{ 4\pi\epsilon_0\kappa r_{ij} }.$$

The final term is the attraction between the created electron and hole. Stability of the ground state requires

$$\Delta E_{ij}\ge0$$

for every occupied–empty pair. Two sites extremely close to the chemical potential therefore cannot also be arbitrarily close in space.

Soft density-of-states suppression

Let $n(\varepsilon)$ be the concentration of sites within energy $\varepsilon$ of $\mu$. Their typical spacing is

$$r(\varepsilon) \sim \left[ n(\varepsilon) \right]^{-1/d}.$$

A homogeneous exclusion-volume estimate based on this pair stability gives, at scaling accuracy,

$$\varepsilon \gtrsim \frac{ e^2 }{ 4\pi\epsilon_0\kappa r(\varepsilon) }.$$

Hence

$$n(\varepsilon) \lesssim \left( \frac{ 4\pi\epsilon_0\kappa\varepsilon }{ e^2 } \right)^d,$$

For a smooth power-law density of states, the corresponding scaling constraint is

$$\rho(E) \lesssim A_d \left( \frac{ 4\pi\epsilon_0\kappa }{ e^2 } \right)^d \lvert E-\mu\rvert^{d-1},$$

where $A_d$ is a geometry-dependent constant. This is a power-counting argument, not differentiation of an arbitrary inequality. The standard self-consistent Efros–Shklovskii (1975) picture saturates the power law for an unscreened $1/r$ interaction in two and three dimensions. It predicts

$$\begin{aligned} \rho_{2\mathrm D}(E) &\propto \lvert E-\mu\rvert, \\ \rho_{3\mathrm D}(E) &\propto \lvert E-\mu\rvert^2. \end{aligned}$$

This gap concerns single-electron addition and removal energies; neutral electron–hole excitations can remain at low energy. It is a soft gap: the density of states vanishes at one energy but is nonzero arbitrarily nearby. It is not a hard band gap, a superconducting gap, or the charging gap of one isolated island. Finite temperature, finite size, metallic screening, and nonequilibrium occupations round the ideal asymptote.

Efros–Shklovskii hopping

More generally, suppose $s>-1$ and a power law holds throughout the hopping window:

$$\rho(E) \propto \lvert E-\mu\rvert^s.$$

Counting available sites gives

$$R^d\varepsilon^{s+1} \sim \text{constant}.$$

Optimizing the distance and energy costs then yields

$$p = \frac{ s+1 }{ s+d+1 }.$$

For a Coulomb gap, $s=d-1$, so

$$p_{\mathrm{ES}} = \frac12$$

in both two and three dimensions. The conductivity becomes

$$\sigma(T) = \sigma_{\mathrm{ES}}(T) \exp \left[ -\left( \frac{T_{\mathrm{ES}}}{T} \right)^{1/2} \right],$$

where

$$T_{\mathrm{ES}} = C_{\mathrm{ES}} \frac{ e^2 }{ 4\pi\epsilon_0\kappa k_{\mathrm B}\xi }.$$

The numerical constant $C_{\mathrm{ES}}$ is network- and convention-dependent. Unlike $T_{\mathrm M}$, the leading Efros–Shklovskii scale depends directly on dielectric screening and $\xi$, not on a constant $\rho_0$ that the Coulomb gap has already invalidated.

Screening and interaction limits

A metallic gate, large polarizability, mobile carriers, or finite screening length cuts off the $1/r$ interaction. At energies below the corresponding screening scale, the Coulomb gap can fill or change form and Mott-like hopping may re-emerge. A fit that assigns $p=1/2$ while ignoring a gate closer than the inferred hopping length is physically incomplete.

Interactions can also produce glassy relaxation, occupation correlations, multiparticle hops, and history dependence. Those phenomena motivate the term electron glass, but slow relaxation is not required by the definition of a noninteracting Anderson insulator. Many-Body Localization Preview treats the distinct question of memory and entanglement in an isolated interacting quantum system.

Identifying the Transport Law

Reduced activation energy

Several stretched exponentials can look nearly linear over a narrow plotting window. For

$$R(T) = R_0 \exp \left[ \left( \frac{T_0}{T} \right)^p \right],$$

define

$$W(T) \equiv - \frac{d\ln R}{d\ln T}.$$

If the prefactor is constant,

$$W(T) = p \left( \frac{T_0}{T} \right)^p$$

and

$$\frac{d\ln W}{d\ln T} = -p.$$

A log–log plot of $W$ can therefore estimate $p$ without choosing in advance among $T^{-1}$, $T^{-1/2}$, $T^{-1/3}$, and $T^{-1/4}$ axes. Numerical differentiation amplifies noise, and a prefactor

$$\sigma_0(T)\propto T^m$$

gives $W=m+p(T_0/T)^p$. Its logarithmic slope need not equal $-p$ over the fitted interval. Smoothing and fit-window choices should therefore be tested against the original resistance data, with uncertainty from the differentiation included.

Comparing the fitted scale with other measurements

A fitted scale becomes informative when it agrees with quantities measured separately:

Fitted behavior	Independent comparison
Activation energy $k_{\mathrm B}T_a$	A spectral edge or known barrier
Mott scale $T_{\mathrm M}$	Localized density of states and $\xi$
Efros–Shklovskii scale $T_{\mathrm{ES}}$	Dielectric screening and $\xi$
Departure from a bulk law	Device dimensions, contacts and percolation length
Bias dependence	Field work $eER_\ast$ and electron temperature

Magnetoresistance can provide additional evidence, but hopping magnetoresistance has several competing mechanisms: wavefunction shrinkage, interference among alternative paths, spin blockade, and field-dependent density of states. Its sign is not a unique label.

A fitted $p=1/2$ is not unique to a Coulomb gap: one-dimensional Mott scaling and other mechanisms can share it. In a gated sample, compare the gate distance with the inferred hop length before assuming an unscreened interaction. Check current dependence, contacts and leakage before assigning low-temperature deviations to a new transport law.

Exercises

1. Detailed balance for a hop

Let $\varepsilon_j-\varepsilon_i=\Delta>0$. Use the Miller–Abrahams rates to show detailed balance between $i$ and $j$.

Solution

The uphill rate is

$$\Gamma_{i\to j} = \Gamma_0 e^{-2r_{ij}/\xi} e^{-\Delta/(k_{\mathrm B}T)}.$$

The reverse hop is downhill:

$$\Gamma_{j\to i} = \Gamma_0 e^{-2r_{ij}/\xi}.$$

Therefore

$$\frac{\Gamma_{i\to j}}{\Gamma_{j\to i}} = e^{-\Delta/(k_{\mathrm B}T)},$$

which is the required Boltzmann ratio.

2. Optimal-hop scale in two dimensions

Use the one-hop optimization $X(R)=2R/\xi+A/R^2$. At $T_{\mathrm M}=900\ \mathrm K$ and $T=4.0\ \mathrm K$, suppose its optimized cost is $X_\ast=(T_{\mathrm M}/T)^{1/3}$. Find $R_\ast/\xi$ and the resistance factor $e^{X_\ast}$.

Solution

The stationary condition gives $A/R_\ast^2=R_\ast/\xi$. Consequently,

$$X_\ast=\frac{3R_\ast}{\xi}.$$

The total optimized cost is $X_\ast=225^{1/3}\simeq6.08$, so

$$\frac{R_\ast}{\xi}\simeq2.03, \qquad e^{X_\ast}\simeq4.4\times10^2.$$

The distance cost alone is $2R_\ast/\xi\simeq4.05$; the remaining $2.03$ is the energy cost. Thus the hop distance and total cost share a temperature exponent but have different coefficients. This is a one-hop illustration; a percolation calculation determines the quantitative network scales. Here $R_\ast$ is only a few localization lengths, so the deep variable-range limit is at best marginal.

3. A general soft-gap exponent

Suppose $\rho(E)\propto\lvert E-\mu\rvert^s$. Derive

$$p = \frac{s+1}{s+d+1}.$$

Then evaluate it for a Coulomb gap.

Solution

The number of sites within radius $R$ and energy width $\varepsilon$ scales as

$$\mathcal N \sim R^d\varepsilon^{s+1}.$$

Setting $\mathcal N\sim1$ gives

$$\varepsilon(R) \propto R^{-d/(s+1)}.$$

The hopping exponent has the form

$$X(R) \sim R + \frac{ R^{-d/(s+1)} }{ T },$$

with fixed microscopic factors suppressed. Minimization gives

$$R_\ast \propto T^{-(s+1)/(s+d+1)}.$$

Both optimized costs have that temperature power, so

$$p = \frac{s+1}{s+d+1}.$$

For an unscreened Coulomb gap, $s=d-1$, hence

$$p = \frac{d}{2d} = \frac12.$$

4. Efros–Shklovskii temperature

For a three-dimensional illustrative network, take $\kappa=12$, $\xi=10\ \mathrm{nm}$, and $C_{\mathrm{ES}}=2.8$. Use

$$\frac{e^2}{4\pi\epsilon_0} \simeq 1.440\ \mathrm{eV\,nm}$$

to estimate $T_{\mathrm{ES}}$. Then find the hopping exponent at $T=4.0\ \mathrm K$.

Solution

The interaction energy at $\xi$ is

$$\frac{ e^2 }{ 4\pi\epsilon_0\kappa\xi } = \frac{ 1.440\ \mathrm{eV\,nm} }{ 12\times10\ \mathrm{nm} } = 0.0120\ \mathrm{eV}.$$

Using $1\ \mathrm{eV}/k_{\mathrm B}\simeq11604.5\ \mathrm K$,

$$T_{\mathrm{ES}} \simeq 390\ \mathrm K.$$

At $4.0\ \mathrm K$,

$$\left( \frac{T_{\mathrm{ES}}}{T} \right)^{1/2} \simeq \left( \frac{390}{4.0} \right)^{1/2} \simeq 9.87.$$

5. A temperature-dependent prefactor

Suppose $\sigma(T)=C T^m\exp[-(T_0/T)^p]$. Derive the slope of $\ln W$ versus $\ln T$ and evaluate it for $m=1$, $p=1/3$ at a temperature where $(T_0/T)^p=6$.

Solution

Set $Y=(T_0/T)^p$. Since $R\propto T^{-m}e^Y$,

$$W=m+pY,\qquad \frac{dW}{d\ln T}=-p^2Y.$$

Where $W>0$, the logarithmic slope is

$$\frac{d\ln W}{d\ln T} =-\frac{p^2Y}{m+pY}.$$

For the stated values, $W=3$ and the slope is $-2/9$, not $-1/3$. Ignoring the prefactor would underestimate the exponent in this example. The slope approaches $-p$ only when $pY$ dominates $m$.

Related mechanisms

• Mobility Edges distinguishes activation into extended states from hopping within localized states.
• Coulomb Blockade treats charging of a finite island, a different origin of suppressed conductance.
• Many-Body Localization Preview considers isolated interacting dynamics rather than bath-assisted hopping.

References

• V. Ambegaokar, B. I. Halperin, and J. S. Langer, “Hopping Conductivity in Disordered Systems,” Physical Review B 4, 2612–2620 (1971), doi:10.1103/PhysRevB.4.2612. Formulates the critical-path and percolation treatment of the hopping network.
• A. L. Efros and B. I. Shklovskii, “Coulomb Gap and Low Temperature Conductivity of Disordered Systems,” Journal of Physics C: Solid State Physics 8, L49–L51 (1975), doi:10.1088/0022-3719/8/4/003. Derives the soft interaction gap and the $p=1/2$ hopping law.
• P. A. Lee and T. V. Ramakrishnan, “Disordered Electronic Systems,” Reviews of Modern Physics 57, 287–337 (1985), doi:10.1103/RevModPhys.57.287. Reviews localization, interactions, and insulating transport.
• A. Miller and E. Abrahams, “Impurity Conduction at Low Concentrations,” Physical Review 120, 745–755 (1960), doi:10.1103/PhysRev.120.745. Derives phonon-assisted localized-site rates and the equivalent resistor network.
• N. F. Mott, “Conduction in Glasses Containing Transition Metal Ions,” Journal of Non-Crystalline Solids 1, 1–17 (1968), doi:10.1016/0022-3093(68)90002-1. Develops the distance–energy optimization behind variable-range hopping.
• M. E. Raikh, J. Czingon, Q. Ye, F. Koch, W. Schoepe, and K. Ploog, “Mechanisms of Magnetoresistance in Variable-Range-Hopping Transport for Two-Dimensional Electron Systems,” Physical Review B 45, 6015–6025 (1992), doi:10.1103/PhysRevB.45.6015. Demonstrates the nonuniqueness of hopping magnetoresistance mechanisms.