= Solution
<Bloch's theorem> states that an eigenfunction in a periodic potential can be written $\psi_{\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{\mathbf k}(\mathbf r)$, with $u_{\mathbf k}$ lattice-periodic. In the <orthogonal tight-binding approximation>, choose localized orthonormal orbitals $|j\rangle$ with on-site matrix element $E_0$ and real nearest-neighbour <hopping integral> $t$. The Bloch sum $|k\rangle=N^{-1/2}\sum_j e^{ikja}|j\rangle$ is an eigenstate because each neighbour shift contributes a phase $e^{\pm ika}$. Thus
$$
\boxed{E(k)=E_0+t(e^{ika}+e^{-ika})=E_0+2t\cos ka.}
$$
The <simple-cubic nearest-neighbour tight-binding band> has six neighbours, giving $E(\mathbf k)=E_0+2t[\cos(k_xa)+\cos(k_ya)+\cos(k_za)]$. The cosine sum ranges from $-3$ to $3$, so the positive <bandwidth> is $W=12|t|$. The stated hopping formula is negative when $E_1>0$, so the printed $12t$ must be read as a bandwidth magnitude.
Near the zone centre, $E=E_0+6t-ta^2k^2+O(k^4)$. To leading order the constant-energy surfaces are spheres, and comparison with $E_\Gamma+\hbar^2k^2/(2m^*)$ gives
$$
\boxed{m^*=-\frac{\hbar^2}{2ta^2}=\frac{\hbar^2e^{a/a_1}}{4E_1a^2(1+a/a_1)}.}
$$
This is the <effective mass for overlap-weighted exponential hopping>. With $s=a/a_1$, its logarithmic derivative is $d\ln m^*/ds=1-2/s-1/(1+s)$. The positive stationary point is $s=1+\sqrt3$, and it is a minimum. The formula grows at both very small and very large $s$; the small-separation divergence is a mathematical extrapolation into a regime where localized tight binding itself becomes unreliable.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2007-tight-binding-mass.png]
{title=Competing orbital hopping and geometric curvature give an effective-mass minimum}
{description=The mass is plotted in units $\hbar^2/(4E_1a_1^2)$ against $a/a_1$. The marked minimum is at $1+\sqrt3$; the formula's small-spacing continuation should not be treated as a quantitative strong-overlap model.}
For the bandwidth, $W=24E_1(1+s)e^{-s}$ and $d\ln W/d\ln a=-s^2/(1+s)$. At $s=1$, a fractional spacing change $-0.01$ therefore gives
$$
\boxed{\frac{\Delta W}{W}\simeq+0.005=+0.50\%.}
$$
The exact ratio $W(0.99a_1)/W(a_1)=(1.99/2)e^{0.01}=1.004999916\ldots$ checks the first-order result. Compression strengthens the hopping and widens the band.
Back to article page
