Bloch theorem

In a periodic potential, energy eigenstates have the form $e^{ik\cdot x}{u}_k(x)$ with lattice-periodic $u_k$.

In a single isolated energy band, a wave packet subject to a constant force satisfies $\hslash \dot{k}=F$. Its group velocity is periodic in $k$ with reciprocal-lattice period $2\pi /a$, so its position oscillates with period $2\pi \hslash /(|F|a)$ when its mean velocity over a band is zero. For $E(k)=E_0-2J\cos ka$, integrating the sinusoidal velocity gives the oscillation explicitly. This ideal description neglects scattering and interband tunnelling.

The lattice translations commute with one another and with a lattice-periodic Hamiltonian. They can therefore be diagonalized simultaneously within each energy eigenspace. Every unitary character of a Bravais lattice has the form $r\mapsto e^{-ik\cdot r}$, with $k$ defined modulo the reciprocal lattice. Hence $T_r{\psi }_k=e^{-ik\cdot r}{\psi }_k$, so $u_k(x)=e^{-ik\cdot x}{\psi }_k(x)$ is lattice-periodic.

A Bloch state is an energy eigenstate of a lattice-periodic Hamiltonian that is simultaneously an eigenstate of every lattice translation. It can be written as ${\psi }_k(x)=e^{ik\cdot x}{u}_k(x)$ with lattice-periodic $u_k$.

Crystal momentum is the translation quantum number $\hslash k$ of a Bloch state. Since $k$ and $k+G$ give the same translation character whenever $G$ belongs to the reciprocal lattice, crystal momentum is defined modulo reciprocal-lattice vectors.

A periodic potential satisfies $V(x+a)=V(x)$ for some lattice period $a$. Its Fourier coefficients couple plane waves whose wavevectors differ by reciprocal-lattice vectors.

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

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 ${\partial }_i{\partial }_{j}E/{\hslash }^2$.

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.

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.

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.

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.

The nearly-free electron model treats a weak periodic potential by degenerate perturbation theory. At a Bragg crossing, a Fourier coefficient $V_n$ couples the two degenerate plane waves and opens an energy gap of width $2|V_n|$.

Forwrite $k=n\pi /a+\kappa$. Near the $n$th Bragg crossing,At $\kappa =0$, the two branches differ by $2|V_n|$.

For a potential of period $a$, the Floquet matrix $M(E)$ maps Cauchy data through one cell:For a real Schrodinger equation, $\det M=1$.

The Floquet multipliers solveBounded Bloch waves occur when $|\mathrm{tr}M|\le 2$, with $\mu =e^{\pm iKa}$. Band edges satisfy $\mathrm{tr}M=\pm 2$; outside the bands the multipliers are real reciprocal numbers and one solution grows exponentially.

The delta-comb Kronig-Penney potential is

For $E={\hslash }^2{k}^2/(2m)$ and $\gamma =m{V}_0/({\hslash }^{2}k)$,Thus the allowed bands satisfy $|\cos (ka)+\gamma \sin (ka)|\le 1$.

Forthe wavefunction is continuous across every lattice point and obeysAt negative energy $E=-{\hslash }^2{\mu }^2/(2m)$, an allowed energy band satisfiesAt positive energy $E={\hslash }^2{k}^2/(2m)$, it satisfies

With well strength $\lambda$ and spacing $a$, negative-energy Bloch states obey $\tanh y\le 2y/(\lambda a)\le \coth y$, where $y=\kappa a/2$. There is one negative band: it reaches zero when $\lambda a\le 2$ and is separated from zero when $\lambda a>2$.

If one symmetric cell has reflection and transmission amplitudes $r,t$, thenAt a periodic or antiperiodic band edge, with $\mathrm{tr}M=2\sigma$ and $\sigma =\pm 1$, setting $z=e^{-ika}$ gives

A tight-binding model expands a particle's state in localized orbitals and represents tunnelling between sites by off-diagonal Hamiltonian matrix elements.

For equivalent minima with nearest-neighbour hopping magnitude $\Delta$, paths with $r$ right steps and $s$ left steps have $r-s=n-m$. Summing $(\Delta \tau /\hslash {)}^{r+s}/(r!s!)$ and applying the Fourier representation of a Kronecker delta gives the displayed modified Bessel function kernel in a normalized localized-site basis. Its Fourier exponent gives the tight-binding model energy $E(\theta )=E_{\mathrm{well}}-2\Delta \cos \theta$. Position-endpoint kernels can have an additional common local-wavefunction prefactor.

Near a nondegenerate minimum $x_0$, a smooth potential satisfiesThe localized levels are therefore approximately those of a harmonic oscillator of frequency $\sqrt{V^{\prime \prime }(x_0)/m}$; tunnelling broadens each level into a narrow band.

For $N$ identical sites of spacing $a$ with periodic boundary conditions, translation eigenstates have coefficientsThus $k=2\pi j/(Na)$ modulo the reciprocal-lattice period $2\pi /a$.

For hopping amplitude $-\lambda$ from each lattice site to its neighbours at $\pm a_1$ and $\pm a_2$, a Bloch state has energyThe band ranges from $E_0-4|\lambda |$ to $E_0+4|\lambda |$ and therefore has width $8|\lambda |$.

The first Brillouin zone is a fundamental cell in reciprocal space. For a one-dimensional lattice of spacing $a$, one conventional choice is

A Bragg point is a crystal momentum at which two plane waves differing by a reciprocal lattice vector are degenerate. A periodic potential mixes the waves there, producing an avoided crossing and opening a band gap.

The reciprocal lattice of a two-dimensional triangular Bravais lattice is triangular, so its Wigner-Seitz cell is a regular hexagon. Opposite boundary points are identified modulo reciprocal-lattice translations, and the six corners fall into two inequivalent classes of three corners each, conventionally denoted $K$ and $K^{\prime }$.