Graphene and Dirac Materials

Graphene is a single sheet of carbon atoms arranged on a honeycomb lattice, the two-dimensional building block of graphite. Isolated in 2004, it is the cleanest realization of a lattice whose band structure has valence and conduction bands meeting at isolated points rather than overlapping or leaving a gap. Near those points the energy is linear in wavevector, so the low-energy electrons obey a two-dimensional Dirac equation for massless particles moving at a fixed speed $v_{F} \approx c /300$ . The tight-binding method applied to the honeycomb's two-atom basis produces this structure directly, and the resulting massless-Dirac electrons carry a Berry phase of $π$ that reshapes graphene's quantum Hall effect and links it to topological insulators.

The honeycomb lattice

The honeycomb is not a Bravais lattice: no single set of primitive translations maps every atom onto an equivalent one. It is a triangular Bravais lattice with a two-atom basis, the sublattices $A$ and $B$ . The primitive vectors and the three nearest-neighbor vectors from an $A$ atom to its $B$ neighbors are

a_{1} = \frac{a}{2} (3, 3), a_{2} = \frac{a}{2} (3, - 3), δ_{1} = \frac{a}{2} (1, 3), δ_{2} = \frac{a}{2} (1, - 3), δ_{3} = a (- 1, 0),

with $a = 0.142 nm$ the carbon–carbon bond length. The lattice constant is $∣ a_{1} ∣ = 3 a = 0.246 nm$ . Every $A$ atom has three $B$ neighbors and no $A$ neighbors; the two sublattices interpenetrate.

The reciprocal lattice is again triangular, and the first Brillouin zone is a regular hexagon. Its six corners fall into two inequivalent classes, labeled $K$ and $K^{'}$ , connected to their own kind by reciprocal-lattice vectors but not to each other. Their positions are

K = \frac{2 π}{3 a} (1, \frac{1}{3}), K^{'} = \frac{2 π}{3 a} (1, - \frac{1}{3}) .

These two corners are where the bands touch.

Graphene's honeycomb lattice has two interpenetrating sublattices A (filled) and B (open), with each A atom bonded to three B neighbors; the first Brillouin zone is a hexagon whose two inequivalent corners (labeled K and K_2 here, the Dirac points K and K') host the band touchings.

Tight-binding bands and Dirac points

Keep one $p_{z}$ orbital per atom and allow hopping with amplitude $t \approx 2.8 eV$ between nearest neighbors. The Bloch Hamiltonian couples the two sublattices through the structure factor $f (k) = \sum_{j} e^{i k \cdot δ_{j}}$ , giving a $2 \times 2$ matrix whose eigenvalues are $E_{\pm} (k) = \pm t ∣ f (k)∣$ . Carrying out the sum,

E_{\pm} (k) = \pm t 3 + 2 cos (3 k_{y} a) + 4 cos (\frac{3}{2} k_{y} a) cos (\frac{3}{2} k_{x} a) .

The lower band $E_{-}$ (valence) and the upper band $E_{+}$ (conduction) are mirror images across $E = 0$ . With one $π$ electron per carbon atom the valence band is exactly full, and the Fermi level sits at $E = 0$ . The two bands meet where $f (k) = 0$ , which happens precisely at the zone corners $K$ and $K^{'}$ : the gap is zero at those points and nowhere else. Graphene is therefore a semimetal — a zero-gap semiconductor — with a pointlike Fermi surface.

The massless Dirac dispersion

Expand the Hamiltonian about a corner, $k = K + q$ with $q$ small. The structure factor becomes linear in $q$ , and the effective Hamiltonian near $K$ is

H_{K} = ℏ v_{F} (σ_{x} q_{x} + σ_{y} q_{y}) = ℏ v_{F} σ \cdot q, v_{F} = \frac{3 t a}{2ℏ} \approx 1.0 \times 1 0^{6} m/s,

where the Pauli matrices $σ$ act on the sublattice ( $A$ / $B$ ) index. This is the two-dimensional Dirac Hamiltonian for a massless particle, with the Fermi velocity $v_{F}$ playing the role of the speed of light. Its eigenvalues are

E_{\pm} (q) = \pm ℏ v_{F} ∣ q ∣,

a pair of cones — the Dirac cones — touching at the point $E = 0$ . The dispersion is linear rather than parabolic, so near the Dirac point the electrons have no effective mass; their group speed $v_{F} = ∣ \nabla_{q} E ∣ /ℏ$ is constant, independent of energy.

The valence and conduction bands touch at a Dirac point with a linear dispersion, forming a double cone; the electron velocity is the constant slope of the cone rather than a mass-dependent curvature.

Chirality and the Berry phase

The two-component eigenstates of $H_{K} = ℏ v_{F} σ \cdot q$ are pseudospinors whose spin is the sublattice degree of freedom. For the conduction band the pseudospin points along $q$ , and for the valence band opposite to it; the pseudospin is locked to the momentum direction. This locking is chirality. Carrying an electron once around the Dirac point rotates its momentum through $2 π$ and its pseudospin through $2 π$ as well, so the wavefunction acquires a geometric Berry phase

γ = \oint A \cdot d q = π .

A Berry phase of $π$ is the graphene analogue of the Berry curvature that quantized the Hall conductance. It suppresses backscattering — reversing $q$ would require flipping the pseudospin, which a smooth potential cannot do — and it shifts the Landau-level spectrum, producing an anomalous quantum Hall effect.

The anomalous quantum Hall effect

In a perpendicular field the Dirac Hamiltonian gives Landau levels unlike those of an ordinary 2D gas. Quantizing $ℏ v_{F} σ \cdot q$ in a field yields

E_{n} = sgn (n) v_{F} 2 e ℏ B ∣ n ∣, n = 0, \pm 1, \pm 2, \dots

Two features distinguish this from the parabolic result $E_{n} = ℏ ω_{c} (n + \frac{1}{2})$ . First, the levels scale as $∣ n ∣$ and as $B$ , not linearly, so they crowd together at high $n$ . Second, there is a level exactly at $E_{0} = 0$ , sitting at the Dirac point and shared equally by electrons and holes. Because each level carries a fourfold degeneracy (two spin, two valleys $K$ and $K^{'}$ ), and the zero level is half electron and half hole, the Hall conductance is quantized at

σ_{x y} = \pm 4 (N + \frac{1}{2}) \frac{e ^{2}}{h} = \pm 2, \pm 6, \pm 10, \dots \frac{e ^{2}}{h} .

The plateaus occur at half-integer multiples of $4 e^{2} / h$ , offset from the integer sequence of an ordinary 2DEG. This half-integer shift is a direct fingerprint of the $E_{0} = 0$ Landau level and the Berry phase $π$ , and its observation was the definitive proof that graphene's carriers are massless Dirac fermions.

Graphene's Hall conductance jumps by 4 e-squared over h between plateaus but lands at half-integer positions plus or minus 2, 6, 10; the large step across zero comes from the shared n equals 0 Landau level.

Opening a gap: toward topological insulators

The Dirac cones are protected only as long as the two sublattices are equivalent. Break that symmetry — put the $A$ and $B$ atoms in different environments, as in hexagonal boron nitride — and a mass term $m σ_{z}$ appears in the Hamiltonian, whose spectrum

E_{\pm} (q) = \pm (ℏ v_{F} q)^{2} + m^{2}

opens a gap $2 ∣ m ∣$ at the former Dirac point. The cone becomes a hyperbola. How the gap is opened matters: a staggered sublattice potential gives an ordinary insulator, but a spin–orbit mass that has opposite sign at $K$ and $K^{'}$ (the Kane–Mele mechanism) produces a topological insulator. Such a material is insulating in the bulk yet carries gapless Dirac states on its edges or surfaces, protected by time-reversal symmetry and characterized by the same kind of topological invariant as the quantum Hall conductance. The gapped Dirac cone of a topological insulator's surface is the direct descendant of graphene's gapless one.

A sublattice-symmetry-breaking mass term opens a gap at the Dirac point, turning the gapless cone (left) into a gapped hyperbolic dispersion (right); a topological version keeps gapless states on the boundary.

The honeycomb lattice thus closes the module where the free-electron continuum opened it: a two-dimensional crystal whose electrons are relativistic in form, whose Berry phase reshapes the quantum Hall effect of the previous two lessons, and whose gapped cousins are the topological insulators that now organize much of condensed-matter physics.¹

Castro Neto, Guinea, Peres, Novoselov, and Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009), arxiv.org/abs/0709.1163. ↩