--- title: Graphene and Dirac Materials module: Nanostructures moduleNumber: 11 lessonNumber: 4 order: 1104 summary: > Graphene is one atomic layer of carbon on a honeycomb lattice. A tight-binding calculation on its two-atom basis gives valence and conduction bands that touch at the corners of the Brillouin zone, where the dispersion is linear and the electrons behave as massless two-dimensional Dirac particles with a fixed speed. This lesson derives the Dirac cones, the Berry phase of pi and the sublattice chirality, the anomalous half-integer quantum Hall effect that follows, and how opening a gap in a Dirac cone points toward topological insulators. topics: [Nanostructures] draft: false sources: - book: Castro Neto et al. ref: "Rev. Mod. Phys. 81, 109 (2009) — The electronic properties of graphene" - book: Kittel ref: "Ch. 18 — Nanostructures; graphene" - book: Simon ref: "Ch. 11 — tight binding on the honeycomb lattice" --- 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](/condensed-matter/band-theory/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 $\pi$ 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 $$ \vec a_1 = \frac{a}{2}(3, \sqrt3), \quad \vec a_2 = \frac{a}{2}(3, -\sqrt3), \qquad \vec\delta_1 = \frac{a}{2}(1, \sqrt3),\ \vec\delta_2 = \frac{a}{2}(1, -\sqrt3),\ \vec\delta_3 = a(-1, 0), $$ with $a = 0.142\ \text{nm}$ the carbon–carbon bond length. The lattice constant is $\lvert\vec a_1\rvert = \sqrt3\,a = 0.246\ \text{nm}$. 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 $$ \vec K = \frac{2\pi}{3a}\left(1, \frac{1}{\sqrt3}\right), \qquad \vec K' = \frac{2\pi}{3a}\left(1, -\frac{1}{\sqrt3}\right). $$ These two corners are where the bands touch. $$ % caption: 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. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % ---- real-space honeycomb ring ---- \begin{scope} % ring atoms: A at 30,150,270 ; B at 90,210,330 (radius 0.95) \coordinate (A0) at (0.823,0.475); \coordinate (B1) at (0,0.95); \coordinate (A2) at (-0.823,0.475); \coordinate (B3) at (-0.823,-0.475); \coordinate (A4) at (0,-0.95); \coordinate (B5) at (0.823,-0.475); % outward B atoms from each A \coordinate (Bo0) at (1.646,0.95); \coordinate (Bo2) at (-1.646,0.95); \coordinate (Bo4) at (0,-1.9); % ring bonds \draw[black, thick] (A0)--(B1)--(A2)--(B3)--(A4)--(B5)--(A0); % outward bonds \draw[black, thick] (A0)--(Bo0); \draw[black, thick] (A2)--(Bo2); \draw[black, thick] (A4)--(Bo4); % A atoms filled \foreach \p in {A0,A2,A4}{ \fill[acc] (\p) circle (3.2pt); } % B atoms open \foreach \p in {B1,B3,B5,Bo0,Bo2,Bo4}{ \draw[acc, thick, fill=white] (\p) circle (3.2pt); } \node[acc, anchor=south west] at (A0) {$A$}; \node[acc, anchor=south] at (B1) {$B$}; % lattice vectors \draw[black, ->] (A4) -- ($(A4)+(1.234,0.712)$); \node[black, anchor=west] at ($(A4)+(1.234,0.712)$) {$a_1$}; \node[black, anchor=north] at (0,-2.1) {honeycomb lattice}; \end{scope} % ---- reciprocal-space Brillouin zone ---- \begin{scope}[xshift=6.2cm] % hexagon corners (pointy-top), radius 1.3 \foreach \ang in {30,90,150,210,270,330}{ \coordinate (h\ang) at (\ang:1.3); } \draw[acc, very thick] (h30)--(h90)--(h150)--(h210)--(h270)--(h330)--cycle; % center Gamma \fill[black] (0,0) circle (1.6pt); \node[black, anchor=north east] at (0,0) {$G$}; % K and Kp on alternating corners \fill[acc] (h30) circle (2.4pt); \node[acc, anchor=west] at (h30) {$K$}; \fill[acc] (h90) circle (2.4pt); \node[acc, anchor=south] at (h90) {$K_2$}; % M edge midpoint \fill[black] ($(h30)!0.5!(h90)$) circle (1.6pt); \node[black, anchor=west] at ($(h30)!0.5!(h90)$) {$M$}; \node[black, anchor=north] at (0,-1.55) {Brillouin zone}; \end{scope} \end{tikzpicture} $$ ## Tight-binding bands and Dirac points Keep one $p_z$ orbital per atom and allow hopping with amplitude $t \approx 2.8\ \text{eV}$ between nearest neighbors. The Bloch Hamiltonian couples the two sublattices through the structure factor $f(\vec k) = \sum_{j} e^{i\vec k\cdot\vec\delta_j}$, giving a $2\times2$ matrix whose eigenvalues are $E_\pm(\vec k) = \pm t\,\lvert f(\vec k)\rvert$. Carrying out the sum, $$ E_\pm(\vec k) = \pm t\sqrt{3 + 2\cos(\sqrt3\,k_y a) + 4\cos\!\left(\tfrac{\sqrt3}{2}k_y a\right)\cos\!\left(\tfrac{3}{2}k_x a\right)}. $$ The lower band $E_-$ (valence) and the upper band $E_+$ (conduction) are mirror images across $E = 0$. With one $\pi$ electron per carbon atom the valence band is exactly full, and the Fermi level sits at $E = 0$. The two bands meet where $f(\vec k) = 0$, which happens precisely at the zone corners $\vec K$ and $\vec K'$: 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, $\vec k = \vec K + \vec q$ with $q$ small. The structure factor becomes linear in $\vec q$, and the effective Hamiltonian near $K$ is $$ H_K = \hbar v_F\,(\sigma_x q_x + \sigma_y q_y) = \hbar v_F\,\vec\sigma\cdot\vec q, \qquad v_F = \frac{3 t a}{2\hbar} \approx 1.0\times10^6\ \text{m/s}, $$ where the Pauli matrices $\vec\sigma$ 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 $$ E_\pm(\vec q) = \pm\hbar v_F\,\lvert\vec q\rvert, $$ 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 $v_F = \lvert\nabla_{\vec q}E\rvert/\hbar$ is constant, independent of energy. > **Definition (Dirac point).** A Dirac point is a wavevector at which two bands > touch with a linear (conical) dispersion in every direction. Near it the > low-energy excitations obey a massless Dirac equation, with a fixed velocity in > place of a curvature-dependent effective mass. $$ % caption: The valence and conduction bands touch at a Dirac point with a linear % dispersion, forming a double cone; the electron velocity is the constant slope % of the cone rather than a mass-dependent curvature. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black, ->] (0,-2.4) -- (0,2.6) node[left] {energy $E$}; \draw[black, ->] (-2.6,0) -- (2.8,0) node[below] {wavevector $q$}; % upper cone (conduction) \draw[acc, very thick] (0,0) -- (-1.5,2.2); \draw[acc, very thick] (0,0) -- (1.5,2.2); \draw[acc, very thick] (0,2.2) ellipse (1.5 and 0.32); \node[acc, anchor=west] at (1.55,1.9) {conduction}; % lower cone (valence) \draw[acc, very thick] (0,0) -- (-1.5,-2.2); \draw[acc, very thick] (0,0) -- (1.5,-2.2); \draw[acc, very thick, densely dashed] (0,-2.2) ellipse (1.5 and 0.32); \node[acc, anchor=west] at (1.55,-1.9) {valence}; \fill[black] (0,0) circle (2pt); \node[black, anchor=south west] at (0.08,0.06) {Dirac point}; \end{tikzpicture} $$ ## Chirality and the Berry phase The two-component eigenstates of $H_K = \hbar v_F\vec\sigma\cdot\vec q$ are pseudospinors whose "spin" is the sublattice degree of freedom. For the conduction band the pseudospin points along $\vec q$, 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 $2\pi$ and its pseudospin through $2\pi$ as well, so the wavefunction acquires a geometric **Berry phase** $$ \gamma = \oint \vec{\mathcal{A}}\cdot\d\vec q = \pi. $$ A Berry phase of $\pi$ is the graphene analogue of the Berry curvature that [quantized the Hall conductance](/condensed-matter/nanostructures/fractional-quantum-hall-and-topology). It suppresses backscattering — reversing $\vec q$ 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 $\hbar v_F\vec\sigma\cdot\vec q$ in a field yields $$ E_n = \operatorname{sgn}(n)\,v_F\sqrt{2 e\hbar B\,\lvert n\rvert}, \qquad n = 0, \pm1, \pm2, \dots $$ Two features distinguish this from the parabolic result $E_n = \hbar\omega_c (n+\tfrac12)$. First, the levels scale as $\sqrt{\lvert n\rvert}$ and as $\sqrt{B}$, 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 $$ \sigma_{xy} = \pm 4\left(N + \tfrac{1}{2}\right)\frac{e^2}{h} = \pm 2, \pm 6, \pm 10, \dots \ \frac{e^2}{h}. $$ The plateaus occur at **half-integer** multiples of $4e^2/h$, 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 $\pi$, and its observation was the definitive proof that graphene's carriers are massless Dirac fermions. $$ % caption: Graphene's Hall conductance jumps by 4 e-squared over h between % plateaus but lands at half-integer positions plus or minus 2, 6, 10; the large % step across zero comes from the shared n equals 0 Landau level. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black, ->] (-3.4,0) -- (3.6,0) node[below] {carrier density $n$}; \draw[black, ->] (0,-3.2) -- (0,3.2) node[left] {conductance $\frac{e^2}{h}$}; % staircase: plateaus at 2,6,10 and -2,-6,-10, steps of 4, crossing zero with a big step \draw[acc, very thick] (-3.2,-2.6) -- (-2.2,-2.6) (-2.2,-2.6) -- (-2.2,-1.5) -- (-1.1,-1.5) (-1.1,-1.5) -- (-1.1,-0.5) -- (0,-0.5) (0,-0.5) -- (0,0.5) -- (1.1,0.5) (1.1,0.5) -- (1.1,1.5) -- (2.2,1.5) (2.2,1.5) -- (2.2,2.6) -- (3.2,2.6); % plateau labels (electron side only; hole side symmetric, noted in caption) \node[acc, anchor=south] at (1.6,0.5) {$+2$}; \node[acc, anchor=south] at (2.7,1.5) {$+6$}; \node[black, anchor=north east] at (-1.1,-0.5) {hole side}; \draw[black, dashed] (0,0.5) -- (0,-0.5); \end{tikzpicture} $$ ## 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 $m\sigma_z$ appears in the Hamiltonian, whose spectrum $$ E_\pm(\vec q) = \pm\sqrt{(\hbar v_F q)^2 + m^2} $$ opens a gap $2\lvert m\rvert$ 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. $$ % caption: A sublattice-symmetry-breaking mass term opens a gap at the Dirac % point, turning the gapless cone (left) into a gapped hyperbolic dispersion % (right); a topological version keeps gapless states on the boundary. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % ---- gapless ---- \begin{scope} \draw[black, ->] (0,-2.0) -- (0,2.2) node[left] {$E$}; \draw[black, ->] (-1.8,0) -- (1.9,0) node[below] {$q$}; \draw[acc, very thick] (-1.5,1.7) -- (0,0) -- (1.5,1.7); \draw[acc, very thick] (-1.5,-1.7) -- (0,0) -- (1.5,-1.7); \node[black, anchor=south] at (0,2.25) {gapless}; \end{scope} % ---- gapped ---- \begin{scope}[xshift=5.4cm] \draw[black, ->] (0,-2.0) -- (0,2.2) node[left] {$E$}; \draw[black, ->] (-1.8,0) -- (1.9,0) node[below] {$q$}; \draw[acc, very thick, domain=-1.5:1.5, samples=60, variable=\x] plot ({\x},{sqrt(\x*\x+0.36)}); \draw[acc, very thick, domain=-1.5:1.5, samples=60, variable=\x] plot ({\x},{-sqrt(\x*\x+0.36)}); \draw[black, <->] (0.25,0.6) -- (0.25,-0.6); \node[black, anchor=west] at (0.3,0) {gap}; \node[black, anchor=south] at (0,2.25) {gapped}; \end{scope} \end{tikzpicture} $$ 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.[^cn] [^cn]: Castro Neto, Guinea, Peres, Novoselov, and Geim, "The electronic properties of graphene," Rev. Mod. Phys. 81, 109 (2009), [arxiv.org/abs/0709.1163](https://arxiv.org/abs/0709.1163).