Past exam of the quantum condensed matter course of the University of Cambridge / 2021 / Section B / 4

For a Hamiltonian with a lattice-periodic potential $V(\mathbf{r}+\mathbf{R})=V(\mathbf{r})$, its simultaneous energy eigenstates and lattice-translation eigenstates can be chosen in the Bloch wave formEquivalently, ${\psi }_{n\mathbf{k}}(\mathbf{r}+\mathbf{R})=e^{i\mathbf{k}\cdot \mathbf{R}}{\psi }_{n\mathbf{k}}(\mathbf{r})$. The index $n$ labels an energy band; crystal momentum $\hslash \mathbf{k}$ is defined modulo a reciprocal lattice vector. These are the content of Bloch's theorem.

Use one orthonormalized localized atomic orbital $|j⟩$ at each position $ja$. In the nearest-neighbour tight-binding model, translation invariance gives the on-site energy and real hopping integralThe Bloch state isSince $H|j⟩=E_0|j⟩+t|j-1⟩+t|j+1⟩$, shifting the two summation indices givesThe hopping integral is an energy, and its sign sets whether $k=0$ is a minimum or maximum. $E_0$ is the diagonal matrix element in the crystal Hamiltonian, which can differ from an isolated-atom energy.

The orthogonal tight-binding approximation matters. If raw neighbouring orbitals retain a nonzero orbital overlap $s$ in their normalization, projection instead gives a generalized eigenvalue problem and the nonorthogonal nearest-neighbour tight-binding band $E(k)=[E_0+2h\cos (ka)]/[1+2s\cos (ka)]$. The stated cosine band follows after orthonormalization and nearest-neighbour truncation, or after neglecting that overlap metric while retaining the hopping it motivates.

A site in a primitive cubic lattice has six nearest neighbours, at $\pm a\hat{\mathbf{x}}$, $\pm a\hat{\mathbf{y}}$ and $\pm a\hat{\mathbf{z}}$. Repeating the Bloch state calculation with equal hopping integrals $t$ along all six bonds givesThe cosine sum ranges from $-3$ to $3$ within the first Brillouin zone. ThereforeFor $t<0$ the minimum is at the zone centre; for $t>0$ the maximum is there.

For $|k_{i}a|\ll 1$, expand each cosine to second order:To this order a fixed energy fixes $k_x^2+k_y^2+k_z^2$, so the constant-energy surfaces are spheres. The band effective mass tensor satisfiesFor bonding hopping $t<0$ this is a positive electron mass at the band bottom. If $t>0$, $k=0$ is a maximum with negative electronic mass, or positive electron hole mass ${\hslash }^2/(2t{a}^2)$. The spherical shape is a quadratic approximation; cubic anisotropy appears in the fourth-order terms.

Taking $E_1>0$ so that the zone centre is a minimum, the effective mass for exponentially decaying tight-binding hopping isIts logarithmic derivative isThe curve decreases for $a<2a_1$, has a unique minimum at $a=2a_1$, and increases for $a>2a_1$:The formal limits are an $a^{-2}$ divergence as $a\to 0$ and an exponential divergence as $a\to \infty$. The small-separation extrapolation eventually invalidates the localized-orbital tight-binding model; the sketch describes the specified formula.

The electronic bandwidth is $W(a)=24E_1{e}^{-a/a_1}$. With $a$ reduced from $a_1$ to $0.99a_1$,Thus the energy band widens by approximately 1.005 percent, or 1 percent to first order. Compression increases the orbital coupling.