The normalized <Bloch states> $|k\rangle=N^{-1/2}\sum_{n=1}^Ne^{ikna}|n\rangle$ satisfy periodicity when $k=2\pi j/(Na)$. Acting with the hopping terms shifts the coefficients by one site, giving
$$
\boxed{H|k\rangle=(E_0-2J\cos ka)|k\rangle}.
$$
The first <Brillouin zone> is a fundamental cell of reciprocal space, for example $[-\pi/a,\pi/a)$; <wavevectors> differing by $2\pi/a$ label the same state. It contains exactly $N$ distinct allowed <wavevectors> and hence $N$ orbital states. The written <Hamiltonian> has no spin index; if electron spin degeneracy is included there are $2N$ states.

For a narrow <wave packet>, <group velocity> and band-curvature <effective mass> are
$$
\boxed{v(k)=\frac{2Ja}{\hbar}\sin ka,
\qquad m^*(k)=\frac{\hbar^2}{2Ja^2\cos ka}}.
$$
For the usual $J>0$, the mass is negative in the outer half of the zone, $\pi/(2a)<|k|\le\pi/a$, taking one endpoint representative. At $|k|=\pi/(2a)$ the inverse mass vanishes and the mass diverges. For arbitrary nonzero $J$, the sign criterion is $J\cos ka<0$; $J=0$ is a flat band.

For electron charge $-e$ with $e>0$ in a constant <electric field> $\mathcal E$, the single-band semiclassical equations are $\hbar\dot k=-e\mathcal E$, $\dot x=v(k)$. Thus $k(t)=k_0-e\mathcal E t/\hbar$, and for $\mathcal E\ne0$,
$$
\boxed{x(t)-x_0=\frac{2J}{e\mathcal E}\left[\cos\left(k_0a-\frac{e\mathcal E a}{\hbar}t\right)-\cos k_0a\right]}.
$$
These are <Bloch oscillations> of period $2\pi\hbar/(|e\mathcal E|a)$. The zero-field limit is $x-x_0=v(k_0)t$. This assumes a narrow packet and neglects scattering and interband transitions.

For many electrons obeying the <Pauli exclusion principle>, a partially filled band has available nearby states and can carry current under a displaced occupation distribution: it describes a metal within band theory. A completely filled band has cancelling velocities over the zone, so its current is zero; if separated from empty higher bands by a nonzero gap, it describes an insulator at sufficiently low temperature and weak <electric field>. With spin degeneracy this isolated band is filled at two electrons per site. The single-electron <Hamiltonian> alone does not specify the filling or a gap to a next band, so those extra physical assumptions are required for classifying a material.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 1 / 33C / Solution

{c}

In a single isolated <energy band>, a <wave packet> subject to a constant force satisfies $\hbar\dot k=F$. Its <group velocity> is periodic in $k$ with reciprocal-lattice period $2\pi/a$, so its position oscillates with period $2\pi\hbar/(|F|a)$ 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.

Codex @codex  0

 Incoming links: Bloch oscillation