Past exam of the condensed matter physics major option course of the University of Oxford / 2018 / 2

The vectors from the B atom to its three nearest N neighbours can be chosen asIndeed ${\mathbf{n}}_1={\mathbf{d}}_N-{\mathbf{d}}_B$, ${\mathbf{n}}_2={\mathbf{n}}_1-{\mathbf{a}}_1$ and ${\mathbf{n}}_3={\mathbf{n}}_1-{\mathbf{a}}_2$. All three have length $a/\sqrt{3}$. Repeating the two-site basis over $\mathbf{R}=m{\mathbf{a}}_1+n{\mathbf{a}}_2$ produces the honeycomb pattern in the first panel of the original figure in solution.

Solving ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$ givesThe first Brillouin zone is the Wigner–Seitz hexagon of this reciprocal lattice. Its vertices are $(0,\pm 4\pi /(3a))$ and $(\pm 2\pi /(\sqrt{3}a),\pm 2\pi /(3a))$, with independent signs in the latter set. The centre is $\Gamma =(0,0)$; one inequivalent corner is $K=(0,4\pi /(3a))$, with $K^{\prime }=-K$. Corners related by a reciprocal lattice translation are equivalent, leaving two inequivalent valleys.

A linear combination of atomic orbitals expands a crystal wavefunction in localized orbitals on all sites,Here $\alpha$ labels the basis site, $\eta$ the retained atomic orbitals and $\mathbf{R}=m{\mathbf{a}}_1+n{\mathbf{a}}_2$. For the two-band model retain one relevant orbital on each B or N atom, usually the out-of-plane $p_z$ orbital, and neglect other orbital manifolds. Define normalized Bloch sums for $N_c$ cells,Then $\Psi ={\alpha }_B{\Phi }_B+{\alpha }_N{\Phi }_N$. Bloch periodicity puts the same two coefficients in every cell. We adopt the stated orthonormality, neglect longer-range hopping, and approximate the diagonal matrix elements by the supplied atomic energies ${ϵ}_B,{ϵ}_N$. More generally the crystal environment shifts these on-site energies, so equality with isolated-atom expectation values is an additional tight-binding approximation.

For equal real nearest-neighbour transfer integral $t_{\mathrm{nn}}$, this phase convention givesThe secular determinant is $({ϵ}_B-E)({ϵ}_N-E)-t_{\mathrm{nn}}^2|\gamma {|}^2=0$. ThusSince $({ϵ}_B-{ϵ}_N{)}^2/4=({ϵ}_B+{ϵ}_N{)}^2/4-{ϵ}_B{ϵ}_N$, this is the requested algebraic form. A hopping-sign convention or rephasing of the Bloch sums changes off-diagonal phases but not these eigenvalues. Explicitly,

At a valley $K$, $\gamma (K)=1+2\cos (2\pi /3)=0$. For h-BN the inequivalent sublattice energies therefore leave the direct two-band gapGraphene's equivalent carbon sites instead have ${ϵ}_B={ϵ}_N$ in this model, so the two bands touch at the valleys. Near a valley, $|\gamma (K+\mathbf{q})|\simeq (\sqrt{3}a/2)|\mathbf{q}|$: graphene has approximately linear, gapless Dirac cones while h-BN has gapped band edges. The model panel in the original figure in solution displays this distinction.

First, undoped h-BN is an electrical insulator with thermally suppressed carriers, whereas neutral graphene is a zero-gap semimetal whose transport is readily changed by gating. Disorder, temperature and contacts still affect its measured conductivity. Second, h-BN suppresses interband absorption below its large gap and is transparent over much of the visible range, whereas graphene permits low-energy interband absorption unless occupations block it. A quantitative optical threshold in real h-BN also involves additional bands and strong excitonic effects; a two-orbital model should not be presented as a numerical prediction of its complete optical spectrum.

Angle-resolved photoemission spectroscopy illuminates a surface with photons of known energy and measures the kinetic energy and emission direction of the escaping electrons. Referencing the initial binding energy to the Fermi level givesHere $\Phi$ is the sample/analyser work-function reference appropriate to the calibrated energy scale, and $\theta$ is the emission angle from the surface normal. Parallel crystal momentum is conserved modulo a surface reciprocal lattice vector; measuring two angular coordinates maps the two in-plane momentum components.

Sweeping angle and energy traces the occupied spectral weight and its dispersion. In a quasi-two-dimensional sheet, no detailed free-electron model of the perpendicular final-state momentum is required to map the in-plane bands. Peaks reflect the occupied single-particle spectral function, multiplied by Fermi occupation and photoemission matrix elements and broadened by finite lifetimes and resolution. Ordinary photoemission does not directly measure empty conduction bands; a gap may need complementary information. A clean surface, vacuum and charge control are especially important for an insulating specimen.