Past exam of the physics A course of the University of Cambridge / 2019 / Paper 2 / Section C / C11

The quantum free electron model treats conduction Electrons as independent particles moving in an approximately constant potential, with parabolic dispersion relation $E={\hslash }^2{k}^2/(2m)$. Fermi-Dirac statistics successfully explains important features such as a Fermi energy and a small electronic heat capacity, but a constant potential gives no band gaps and cannot by itself explain why different band fillings produce metals, semiconductors and band insulators. It also misses lattice-dependent effective masses, detailed directional responses and the topology of real Fermi surfaces. Scattering and finite electrical resistivity require additional physics.

The nearly-free electron model introduces a weak periodic lattice potential. Bloch's theorem replaces individual free plane waves by Bloch states, and Fourier components of the potential mix plane waves whose wavevectors differ by a reciprocal lattice vector. Near degeneracy at a Brillouin zone boundary, this coupling opens a band gap; for a two-wave crossing its size is $2|V_G|$. Away from the crossing, energies remain close to free-particle values. Band curvature produces band effective masses, and band filling determines whether the Fermi energy lies inside an allowed band or a gap.

The improvement is the inclusion of lattice periodicity, rather than a switch from classical to quantum statistics: the quantum free electron model already uses Fermi-Dirac statistics. The strong delta barriers in the later numerical example are treated exactly by the Kronig-Penney model, rather than by assuming that their strength is perturbatively small.

A Bloch state on the loop has $\psi (x)=u(x)e^{i\varphi x}$, with $u(x+a)=u(x)$. Its circumference is $L=Na$, so $u(x+Na)=u(x)$. Single-valuedness after one complete circuit requires the periodic boundary conditionThereforeThe allowed Bloch wavevectors are spaced by $2\pi /(Na)$. Values of $\varphi$ differing by $2\pi /a$ label the same crystal momentum modulo a reciprocal lattice vector, so there are $N$ inequivalent choices in one Brillouin zone. In this notation $\varphi$ is the Bloch wavenumber, not a dimensionless phase; $\varphi a$ is dimensionless. Spin gives additional states but does not alter this condition.

Between delta barriers, $V=0$ and the Schrödinger equation gives ${\psi }^{\prime \prime }+k^2\psi =0$, with $k^2=2mE/{\hslash }^2$ for positive energy. Write the state and derivative as a two-component vector $\mathbf{v}=(\psi ,{\psi }^{\prime }{)}^T$. Free propagation across one interval of length $a$ isThe wavefunction is continuous at a delta barrier, while its derivative jumps by $g\psi$, with $g=2m\alpha /{\hslash }^2$. Thus the barrier matrix isBloch's theorem requires $\mathbf{v}(a^{+})=e^{i\varphi a}\mathbf{v}(0^{+})$. Both $D$ and $P$ have determinant one, so the two eigenvalues of $M$ are $e^{i\varphi a}$ and $e^{-i\varphi a}$. Taking its trace givesTherefore the Floquet discriminant of the delta-comb Kronig-Penney model yieldsThis trace argument is the one-cell transfer method for the Kronig-Penney model. The loop imposes the additional discrete values of $\varphi$ derived in part (b). Real $\varphi$ requires the right-hand side to lie in $[-1,1]$, giving allowed energy bands; values outside this range are band gaps. The strength $\alpha$ has units of energy times length, because a Dirac delta distribution has units of inverse length.

With length measured in units of $a=1$, plot the real functionOnly the portions with $|f(k)|\le 1$ can equal $\cos \varphi$, so the strip between the horizontal lines $+1$ and $-1$ identifies allowed energy bands. The band edges of a repulsive delta-comb potential have the following approximate first intervals:Each interval corresponds to energy $E={\hslash }^2{k}^2/(2m)$; the intervening intervals are forbidden. The exact upper endpoints follow from $f(j\pi )=(-1{)}^j$. The lower endpoints solve $f(k)=(-1{)}^{j-1}$ inside $((j-1)\pi ,j\pi )$. The divergence suggested by $\sin k/k$ at the origin is removable, and $f(0)=11>1$, so there is no zero-energy state in the repulsive model.

The periodic barriers produce separated energy bands and band gaps, unlike the continuous free-particle spectrum. A finite $N$-atom loop has discrete states within these bands. Whether the material conducts still depends on band filling: one Electron per primitive cell half-fills a spin-degenerate band, while two Electrons per cell can fill the first band and place the Fermi energy in a gap. Band gaps alone do not establish that every filling is insulating. At high $k$, the perturbing term $10\sin k/k$ becomes relatively small and the spectrum approaches free-electron behaviour in relative terms.