Energy band

An energy band is a continuous branch $E_j(k)$ of the dispersion relation of a periodic quantum system.

The range of energy within one electronic energy band, obtained by varying its wavevector. This differs from an interband band gap.

The electronic bandwidth is the difference between the maximum and minimum of one energy band over the Brillouin zone. It has units of energy, unlike a frequency bandwidth in signal processing. A nearest-neighbour tight-binding model on a primitive cubic lattice with equal real hopping integrals $t$ has $E(\mathbf{k})=E_0+2t{\sum }_{i=1}^3\cos (k_{i}a)$ and $W=12|t|$.

A conical touching of two energy bands, with locally linear dispersion relation. The nearest-neighbour model of graphene supplies an explicit example.

A region of momentum space surrounding a Dirac cone. The two inequivalent Dirac valleys in graphene provide a familiar example; a valley labels a region and is not the electron spin.

A region of wave vector space around a low-energy band feature, such as a band minimum or a Dirac point.

The number of symmetry-related equivalent electronic valleys contributing to a specified low-energy energy band. It multiplies the single-valley density of states, independently of spin degeneracy. Equal electron and hole effective masses alone do not ensure equal total density of states values unless their valley degeneracies and spin degeneracies also agree.

A spatially localized electronic level produced by a defect or foreign atom in a crystalline potential. Such levels can lie in a host band gap, where no extended host state has the same energy, while impurity resonances can also occur inside an energy band. Dilute localized levels are nearly independent; increasing overlap can broaden them into an impurity band. The location and existence of a bound level depend on the impurity potential.

A localized orbital with two electron spin states has energies $E_0=0$, $E_1=ϵ$ and $E_2=2ϵ+U$. Its zero-temperature equilibrium occupation minimizes ${\mathcal{E}}_N=E_N-\mu N$, where $\mu$ is the chemical potential. For repulsion $U>0$, the occupation is successively $0,1,2$ as $\mu$ crosses $ϵ$ and $ϵ+U$. For attraction $U<0$, the singly occupied state is never the minimum: the occupation jumps from $0$ to $2$ at $\mu =ϵ+U/2$. Exactly at a crossing the ground-state occupation is not unique.

At positive temperature, its grand canonical partition function is $\Xi =1+2e^{-\beta (ϵ-\mu )}+e^{-\beta (2ϵ+U-2\mu )}$, where $\beta =1/(k_{B}T)$. The expected value of the occupation follows by differentiating: $⟨N⟩=k_{B}T{\partial }_{\mu }\ln \Xi$. The factor two counts the two singly occupied microstates, not an additional pair state.

In one dimension the band effective mass is defined by $1/m^{\ast }(k)=E^{\prime \prime }(k)/{\hslash }^2$. Together with $v=E^{\prime }(k)/\hslash$ and $\hslash \dot{k}=F$, this gives $\dot{v}=F/m^{\ast }$. Negative band curvature produces negative effective mass for the electron wave packet; near a filled-band maximum the equivalent missing-electron description uses positively responding holes. In several dimensions the inverse mass is the Hessian ${\partial }_i{\partial }_{j}E/{\hslash }^2$.

The curvature tensor of a nondegenerate energy band, determining the linear relation between force and acceleration in semiclassical band dynamics. From group velocity $v_i={\hslash }^{-1}{\partial }_{k_i}E$ and $\hslash {\dot{k}}_j=F_j$, one obtains ${\dot{v}}_i=(M^{-1}{)}_{ij}{F}_j$. Near a parabolic minimum, $E-E_0={\hslash }^2\delta {\mathbf{k}}^T{M}^{-1}\delta \mathbf{k}/2$. The principal values are reciprocal curvature effective masses; a cyclotron experiment measures a direction-dependent combination rather than every tensor entry separately.

For an isotropic valence-band maximum, the electron curvature is negative. The missing-electron excitation has positive charge and positive mass $m_h^{\ast }={\hslash }^2/|E\prime \prime |$. Its cyclotron angular frequency is $eB/m_h^{\ast }$. An anisotropic orbit requires the corresponding cyclotron mass rather than an arbitrary single directional curvature.

Take the nearest-neighbour hopping $t(a)=-2E_1(1+a/a_1)e^{-a/a_1}$, with $E_1,a_1>0$, in a simple-cubic nearest-neighbour tight-binding band. Substituting into $m^{\ast }=-{\hslash }^2/(2t{a}^2)$ gives the displayed result. With $s=a/a_1$, its logarithmic derivative is $1-2/s-1/(1+s)$, which vanishes only at $s=1+\sqrt{3}>0$. The mass diverges at both positive-domain ends in this formula, so that point is its unique minimum. Its bandwidth has derivative $d\ln W/d\ln a=-s^2/(1+s)$: a small compression increases $W$. The small-$a$ continuation lies beyond reliable localized-orbital tight binding.

For a semiclassical wave packet in an isolated energy band, $v_i={\hslash }^{-1}{\partial }_{k_i}\epsilon$ and $\hslash {\dot{k}}_j=F_j$ imply ${\dot{v}}_i={\sum }_j(m^{\ast -1}{)}_{ij}{F}_j$. The inverse effective-mass tensor is the Hessian matrix of the dispersion divided by ${\hslash }^2$. Along a unit vector $\mathbf{e}$, $1/m_{\mathbf{e}}^{\ast }={\sum }_{ij}{e}_i(m^{\ast -1}{)}_{ij}{e}_j$. The Laplacian is its trace: ${\hslash }^{-2}{\nabla }_k^2\epsilon =\mathrm{tr}(m^{\ast -1})$. Thus in $d$ isotropic dimensions the scalar mass is $d{\hslash }^2/({\nabla }_k^2\epsilon )$, rather than ${\hslash }^2/({\nabla }_k^2\epsilon )$.

In a one-dimensional two-wave nearly-free electron model, set ${\epsilon }_B={\hslash }^2{G}^2/(8m)$ and let the coupling magnitude be $U>0$. Near the Bragg plane,Expansion at $q=0$ gives $E_{\pm }={\epsilon }_B\pm U+[{\hslash }^2/(2m)](1\pm 2{\epsilon }_B/U)q^2+O(q^4)$, henceThe lower edge has negative electronic curvature, and the upper edge positive curvature. The last expression is a leading weak-potential approximation, not the exact curvature of the two-wave dispersion.

Band curvature is the Hessian matrix of an energy band as a function of wavevector. Dividing it by ${\hslash }^2$ gives the inverse band effective mass tensor. At a one-dimensional band minimum the curvature is positive, whereas at a maximum it is negative. Near a weak two-wave avoided crossing with reciprocal wavenumber $G$ and coupling magnitude $U$, the two edge curvatures are $E_{\pm }^{\prime \prime }={\hslash }^2/m\pm {\hslash }^4{G}^2/(4m^{2}U)$. This is a crystal response coefficient, not a change in the bare electron mass.

For $t(a)=-2E_1{e}^{-a/a_1}$ with $E_1,a_1>0$, a nearest-neighbour cubic tight-binding model has zone-centre band effective massIts logarithmic derivative is $1/a_1-2/a$. The unique minimum occurs at $a=2a_1$ and has value ${\hslash }^2\exp (2)/(16E_1{a}_1^2)$. The explicit $a^2$ in band curvature competes with the exponentially weakening hopping integral. Extrapolation to vanishing atomic separation lies outside the localized-orbital approximation.

For a closed semiclassical orbit with area $S(E)$ in a plane of wave vector space, the cyclotron effective mass is defined by the displayed derivative. It need not equal the local curvature-based band effective mass.

For a positive definite parabolic effective mass tensor $M$, the Lorentz force equation is $M\dot{\mathbf{v}}=e\mathbf{B}\times \mathbf{v}$. Its nonzero eigenfrequencies are $\pm eB/m_c$, giving the displayed mass. With principal values $(m_t,m_t,m_l)$, a field along the long valley axis gives $m_c=m_t$; a perpendicular field gives $\sqrt{m_t{m}_l}$. For equally oriented cubic valleys viewed along $[111]$, all valleys give $m_c=\sqrt{3m_t^2{m}_l/(2m_t+m_l)}$.

Oscillations of electrical resistance as a magnetic field moves Landau levels through the Fermi energy.

A finite-temperature measurement averages the oscillatory response over the derivative of the Fermi-Dirac distribution. For the fundamental harmonic and a locally constant cyclotron effective mass, this gives $R_T=X/\sinh X$ with $X=2{\pi }^2{k}_{B}T/(\hslash {\omega }_c)$. The oscillations become small when thermal smearing substantially exceeds their Landau-level spacing.

The Dingle factor describes the damping of quantum oscillations by Landau-level broadening. For a Lorentzian half-width $\hslash /(2{\tau }_q)$, the fundamental harmonic has $R_D=\exp [-\pi /({\omega }_c{\tau }_q)]$. Since ${\omega }_c$ decreases with field, the amplitude decays at large inverse field. The lifetime here is the quantum lifetime, not automatically the mobility-derived transport time.

An allowed energy band is an interval of energies supporting extended Bloch states. Adjacent bands may be separated by a band gap.

A band gap is an energy interval between adjacent energy bands containing no allowed bulk state.

A direct band gap compares two energy bands at the same crystal momentum. A global gap instead requires the lowest energy of the upper band to exceed the highest energy of the lower band, allowing the two extrema to occur at different momenta. For example, the periodic toy bands $E_1(k)=A\cos k$ and $E_2(k)=A\cos k+\delta$, with $A>0$ and $0<\delta <2A$, have positive direct separation $\delta$ everywhere but global overlap $2A-\delta$. A positive direct splitting at each Brillouin zone boundary therefore does not alone establish an insulating filled band. At a suitable electron count the overlap can produce both electron and hole pockets.

An indirect band gap separates a valence band maximum and a conduction band minimum at different values of crystal momentum. A near-edge radiative transition then usually needs a phonon or another momentum-transfer process in addition to the photon, whose momentum is small on the Brillouin zone scale. A direct band gap has the extrema at the same crystal momentum and more readily supports efficient band-edge light emission. A material can have direct transitions at higher energy while its smallest gap is indirect.

For $-V_0(\cos Qx+\cos Qy)$, two-dimensional bands are sums of one-dimensional bands. At first order, the first-band maximum is $E_Q/2-V_0$ at the corner, while the second-band minimum is $E_Q/4+V_0/2$ at an edge centre. Hence ${\Delta }_{\mathrm{ind}}\simeq 3V_0/2-E_Q/4$, giving a rough onset $V_0\sim E_Q/6$. Higher plane waves shift this estimate; insulating filling must be checked separately.

A band gap whose conduction band minimum and valence band maximum occur at the same crystal momentum, allowing an optical transition without a momentum-balancing phonon.

An avoided crossing occurs when two parameter-dependent eigenvalues approach one another but an interaction prevents equality. Near a two-level crossing, a nonzero off-diagonal matrix element opens a positive minimum gap.

For spin-one-half electrons, one nondegenerate orbital band contains two one-particle states per primitive cell, one for each spin. A partially filled band has nearby empty states into which an electric field can move electrons and is ordinarily conducting. A completely filled band carries no net current under a small field because all crystal momenta are occupied symmetrically.

With spin degeneracy two, one full square-zone band has density $Q^2/(2{\pi }^2)$. Thus $n_e-n_h=n-Q^2/(2{\pi }^2)$. If $Q=\alpha \sqrt{2\pi n}$, compensation requires $\alpha =\sqrt{\pi }$ and the real-space cell contains two electrons. A global gap is also necessary for an insulator.

A band insulator has an integer number of completely filled energy bands, with the Fermi energy inside a positive band gap. At zero temperature, no arbitrarily low-energy charged excitation is then available.

In an independent-electron crystal with two spin states and no spin splitting, each completely filled energy band contains two electrons per primitive cell. An odd electron count per primitive cell therefore cannot fill an integer number of such bands. Enlarging the actual primitive cell, breaking the relevant degeneracy or including correlated-electron physics changes the premises. Even filling is necessary in this model but not sufficient: bands can overlap.

A Kondo insulator has a bulk insulating state associated with mixing between relatively localized correlated electronic states and itinerant conduction states. A renormalized two-band picture captures a narrow hybridization gap, although an uncorrelated orbital-mixing model does not explain the formation and temperature dependence of the coherent low-energy state. Samarium hexaboride is an example.

If the minimum of the next energy band lies at or below the maximum of the nominally filled band, electrons can lower their energy by occupying the next band and leaving holes in the first. The resulting partially filled bands prevent a band-insulating state.

Suppose a model replaces a set of free-electron boundary energies $E^{(0)}$ by lower and upper edges $E^{(0)}-V$ and $E^{(0)}+V$, with $V>0$. If these are the relevant global band edges, the indirect band gap is $(E_{\min }^{(0)}+V)-(E_{\max }^{(0)}-V)$, giving the displayed expression. A positive direct gap $2V$ at each crossing need not be enough: a band insulator needs $2V>E_{\max }^{(0)}-E_{\min }^{(0)}$. This is a band-edge estimate, not a general three-dimensional nearly-free-electron theorem; multiple Bragg planes at a corner can require a larger coupling matrix.