Graphene

Graphene is a single sheet of $sp^2$-bonded carbon whose low-energy bands meet at two Dirac valleys. Gates, substrates and strain can shift those bands, open a gap, mix valleys or change their geometry. To understand a measured change, start with the operator that the perturbation adds to the Hamiltonian. A shifted cone, a massive cone and a Landau-level spectrum are physically different outcomes.

This page connects those operators to material design. It also distinguishes graphene’s integer Hall effect in a real magnetic field from a zero-field Chern phase, and explains why strain can create Landau-like levels while preserving time reversal.

Required background. Graphene and Dirac Materials supplies the honeycomb, valley and sublattice-pseudospin baseline. Two-Dimensional Materials supplies the roles of dielectric surroundings, layers and contacts. Tight-Binding Models supplies hopping matrices and their Bloch representation, which connect physical perturbations to low-energy operators.

Graphene as a two-dimensional electronic material

The honeycomb construction and ideal-cone derivations are in Graphene and Dirac Materials. Here we use that basis to ask what a gate, substrate or deformation changes. The Graphene Dirac Model provides a compact formula reference.

Fix the basis before naming a perturbation

Initially suppress real spin; each displayed two-component block acts on sublattice amplitudes. Let $\tau=\pm1$ label the two valleys, let $\mathbf p=-i\hbar\nabla$ measure momentum from the appropriate valley center, and let $\sigma_i$ act on the $A/B$ sublattice spinor. One convenient convention is

$$\boldsymbol{\sigma}_\tau \equiv (\tau\sigma_x,\sigma_y), \qquad H_\tau^{(0)} = v_F\boldsymbol{\sigma}_\tau\cdot\mathbf p.$$

Changing valley, unit-cell, Bloch-phase, or sublattice conventions can move signs among $\tau$, $\sigma_x$, and $\sigma_y$. Physical spectra and gauge-invariant response do not depend on that bookkeeping, but a quoted mass sign or pseudogauge field does. Every effective Hamiltonian should therefore state its spinor and valley convention.

Write $V_\Sigma(\mathbf r)\equiv V_{\mathrm{ext}}(\mathbf r)+V_s(\mathbf r)$ and define $\Pi_{\tau,j}\equiv p_j+eA_j+\tau eA_{s,j}$. A useful engineering expansion is

$$H_\tau ={} V_\Sigma\mathbb I + \frac12\sum_{i,j=x,y} \sigma_{\tau,i}\left\{v_{ij}(\mathbf r),\Pi_{\tau,j}\right\} + \left( \Delta_S+\tau\Delta_H \right)\sigma_z + H_{\mathrm{so}}+H_{\mathrm{iv}}+\cdots .$$

Here $e>0$, so $\mathbf p+e\mathbf A$ is the electromagnetic minimal coupling for an electron. The strain field $\mathbf A_s$ is written in the same units as a vector potential. The real velocity tensor $v_{ij}(\mathbf r)$ allows anisotropy and spatial variation; $\{A,B\}=AB+BA$. In the ordinary flat-measure spinor inner product, the displayed anticommutator makes the kinetic term Hermitian on compatible boundary conditions. For each Cartesian pair it equals $\sigma_{\tau,i}[v_{ij}\Pi_{\tau,j}-(i\hbar/2)\partial_jv_{ij}]$. The velocity-gradient connection is required when $v_{ij}$ varies, as in de Juan, Mañes, and Vozmediano’s strain construction; for constant $v_{ij}$ the familiar product is recovered. $H_{\mathrm{iv}}$ mixes valleys and cannot be represented inside one fixed-$\tau$ block.

Added term	Source and leading effect
$V\mathbb I$	A smooth gate or deformation potential shifts energy without opening a local Dirac mass.
$\Delta_S\sigma_z$	Sublattice inequivalence opens a time-reversal-symmetric mass.
$\tau\Delta_H\sigma_z$	A time-reversal-breaking Haldane mass makes the two valley Chern contributions add.
$e\mathbf A$	A real magnetic field couples with the same charge sign in both valleys.
$\tau e\mathbf A_s$	Smooth bond strain shifts valleys in opposite directions; a spatial curl produces pseudoflux.
$H_{\mathrm{iv}}$	Atomic defects or armchair edges can mix valleys.
$H_{\mathrm{so}}$	Intrinsic or proximity spin–orbit coupling requires adding real-spin components to the basis.

Several microscopic changes can produce the same operator, and an interface usually produces more than one. This table identifies possible effects; it does not uniquely reconstruct a device from one spectrum.

Pseudospin describes sublattice coherence

For an ideal band with sign $s=\pm1$,

$$\left\langle \boldsymbol{\sigma}_\tau \right\rangle = s\,\widehat{\mathbf p}.$$

The expectation value records sublattice phase coherence. A scalar potential tests spinor overlap differently from a sublattice mass or intervalley defect; this is why the ideal suppression of exact intravalley backscattering does not imply immunity to resistance.

The pseudospin-overlap derivation makes this concrete: smooth scalar disorder cannot connect ideal opposite-momentum states within one valley as efficiently as a mass or an atomic-scale valley-mixing defect. The protection depends on the operator, not just on a carrier being called a Dirac fermion.

Two different Hall effects

The unconventional integer sequence uses a real field

For a perpendicular real magnetic field $B\ne0$, the ideal monolayer Dirac levels are

$$E_n = \operatorname{sgn}(n) v_F \sqrt{2e\hbar|B|\,|n|}, \qquad n\in\mathbb Z,$$

with a zero-energy level and approximate fourfold spin–valley degeneracy before Zeeman, interaction, substrate, or lattice-scale splittings are resolved (Novoselov and collaborators, 2005; Zhang and collaborators, 2005). The corresponding ideal filling sequence is

$$\begin{aligned} \lvert\nu\rvert&=4\left(N+\frac12\right),\\ N&=0,1,2,\ldots,\\ \nu&=\pm2,\pm6,\pm10,\ldots, \end{aligned}$$

where $\nu=n_{\mathrm{net}}h/(e|B|)$ uses signed excess electron density $n_{\mathrm{net}}$ relative to neutrality. In the volume tensor convention, $\sigma_{xy}=-\operatorname{sgn}(B)\nu e^2/h$; some experimental plots instead define a signed Hall plateau label by $\sigma_{xy}/(e^2/h)$. The offset is historically called an anomalous or half-integer quantum Hall effect, but the measured total Hall conductance is still an integer multiple of $e^2/h$. Plateau formation still requires localization and edge transport; the single-particle ladder alone is insufficient.

A zero-field anomalous Hall phase is a different claim

The modern phrase quantum anomalous Hall effect usually means a quantized Hall response at zero applied magnetic field, supported by a nonzero Chern number and broken time reversal. For one real-spin species in the two-valley Haldane/Semenoff lattice model, take nonzero Dirac masses $m_K=\Delta_S+\Delta_H$ and $m_{K'}=\Delta_S-\Delta_H$, with no additional band inversions. The magnitude of the occupied-band Chern number is (Haldane, 1988)

$$|C| = \frac12 \left| \operatorname{sgn}m_K - \operatorname{sgn}m_{K'} \right|.$$

The overall sign depends on orientation and Hamiltonian conventions. Equal-sign sublattice masses give cancelling valley contributions and $C=0$; opposite-sign masses can give $|C|=1$. These numbers are per spin: two identical spin copies double the charge Chern number, while opposite spin Chern numbers can cancel it. When either mass vanishes the insulating Chern number is not defined at that transition. A gap at neutrality alone therefore does not establish a Chern phase. Hall quantization, edge transport and the broken time-reversal symmetry must agree with the same physical model.

This distinction resolves an ambiguity in the older phrase “anomalous quantum Hall effect in graphene”: it may refer to the unconventional real-field sequence rather than a zero-field Chern phase.

Strain acts differently from a magnetic field

Bond deformation enters through geometry and hopping

Let $\mathbf u(\mathbf r)$ be the in-plane displacement and $h(\mathbf r)$ the out-of-plane height. To leading nonlinear order, the membrane strain tensor is

$$u_{ij} = \frac12 \left( \partial_i u_j + \partial_j u_i + \partial_i h\,\partial_j h \right).$$

If the three nearest-neighbor bonds change by $\delta\ell_n$, a common linearized hopping model is

$$t_n \simeq t_0 \left( 1-\beta\frac{\delta\ell_n}{a_{\mathrm{cc}}} \right),$$

where $a_{\mathrm{cc}}$ is the carbon–carbon distance and $\beta=-\partial\ln t/\partial\ln a_{\mathrm{cc}}$ is a dimensionless hopping-strain parameter. Its fitted value depends on the microscopic model and strain range.

For an axis convention with $x$ along a chosen zigzag direction, the leading valley-odd vector potential may be written

$$\begin{aligned} A_{s,x} &= \frac{\hbar\beta} {2ea_{\mathrm{cc}}} \left( u_{xx}-u_{yy} \right), \\ A_{s,y} &= -\frac{\hbar\beta} {ea_{\mathrm{cc}}} u_{xy}. \end{aligned}$$

Rotating the crystal axes rotates the tensor combination, and changing the valley or Bloch convention can reverse signs. Some authors define a wave-vector field $\boldsymbol{\mathcal A}_s=e\mathbf A_s/\hbar$ or use the graphene lattice constant instead of $a_{\mathrm{cc}}$; numerical comparisons must reconcile those conventions first.

Strain also generates a scalar deformation potential,

$$V_s \simeq g_D \left( u_{xx}+u_{yy} \right),$$

and generally changes $v_{ij}$. In inhomogeneous or curved sheets, geometric connection terms and higher gradients can matter. The pseudogauge-only Hamiltonian is a leading approximation, not the complete continuum theory.

Uniform strain shifts a cone; gradients create pseudoflux

Define

$$B_s = \partial_xA_{s,y} - \partial_yA_{s,x}.$$

A spatially uniform strain tensor gives a constant $\mathbf A_s$ and therefore $B_s=0$ in the bulk. In this weak-strain, separated-valley approximation it moves the valley centers and changes velocities, without producing pseudo-Landau levels. Sufficiently strong uniform hopping anisotropy can instead merge the two Dirac points and open a gap, beyond this local cone-shift approximation (Pereira, Castro Neto and Peres, 2009). Nonuniform strain, curvature-induced bond modulation, or boundary relaxation is required for nonzero pseudoflux.

Because the Hamiltonian contains $\tau e\mathbf A_s$, the effective field is opposite in the two valleys:

$$B_{\mathrm{eff},\tau} = \tau B_s.$$

Time reversal exchanges the valleys and reverses the effective field, so the complete strained system can preserve time reversal even though each valley separately resembles a charged Dirac particle in a magnetic field. An equilibrium state that respects time reversal has no intrinsic charge Hall conductivity from this pseudofield. Equal integrated valley numbers alone do not guarantee that an arbitrary nonequilibrium distribution respects time reversal.

Pseudo-Landau levels are local, valley paired, and conditional

For the massless ladder below, assume an approximately constant isotropic $v_F$, negligible mass terms and a slowly varying scalar potential. For the lowest levels, a useful sufficient condition is $|\nabla V_\Sigma|\ll e v_F|B_s|$: the scalar energy change across an orbit must be small compared with its level spacing. A strong scalar gradient can alter or destroy the ladder even when $B_s$ is uniform (Castro, Cazalilla and Vozmediano, 2017).

Also require $B_s$ to vary slowly on the magnetic length

$$\ell_{B_s} = \sqrt{\frac{\hbar}{e|B_s|}},$$

the local spectrum can approximate

$$E_{n,\tau}-V_\Sigma \simeq \operatorname{sgn}(n) v_F \sqrt{ 2e\hbar|B_s|\,|n| }.$$

The level energies depend on $|B_s|$, while their valley and sublattice wave functions remember the sign reversal. In the simplest convention, the pseudo-$n=0$ states from the two valleys can occupy the same physical sublattice, unlike the opposite-sublattice pattern for a uniform real field.

A controlled local-Landau description needs a scale window such as

$$a_{\mathrm{cc}} \ll \ell_{B_s} \ll L_{\mathrm{strain}},$$

Here $L_{\mathrm{strain}}$ is the variation scale of the effective field. For high index, the larger orbit scale $\sqrt{|n|}\ell_{B_s}$ must also fit inside the region with nearly constant fields and velocity. Atomic reconstruction invalidates the smooth-valley expansion at the lower end; rapid field variation destroys a local Landau ladder at the upper end.

Ideal massless levels for a positive real field (top) or a positive strain pseudofield (bottom), with no scalar shift or spin splitting. Energies are in units $E_1=v_F\sqrt{2e\hbar|B|}$. Both cases give the same $\operatorname{sgn}(n)\sqrt{|n|}$ ladder, but the $n=0$ wave functions differ: in the stated $(A,B)$ basis, the real-field valleys occupy opposite sublattices, while the strain-field valleys both occupy $B$. Reversing the field exchanges the sublattice labels. A spectral ladder alone therefore cannot distinguish the two mechanisms.

Identifying a pseudofield in spectroscopy

Levy and collaborators (2010) used scanning tunnelling spectroscopy on graphene nanobubbles and reported peaks consistent with a Dirac $\sqrt{|n|}$ pseudo-Landau sequence and very large local effective fields. The quoted field is inferred from a local spectral model; it is not a statement that a uniform hundreds-of-tesla electromagnetic flux threads the whole device.

Compare a measured ladder with a strain reconstruction, its predicted spatial curl, and the sublattice pattern. Check how peaks change with tip height, gate voltage and position; confinement and scalar-potential resonances can also create local peaks. Raman shifts and polarization help map strain and doping but do not by themselves measure pseudoflux.

Substrates, contacts and a measured gap

There is no substrate-free device Hamiltonian

Suspended graphene approaches a mechanically isolated sheet but still has strain, adsorbates, contacts, gates, and edge boundary conditions. Encapsulated graphene trades exposed-surface disorder for a designed dielectric and crystallographic environment. Neither geometry is simply “pristine.”

Hexagonal boron nitride can improve charge homogeneity and mobility (Dean and collaborators, 2010). Near crystallographic alignment it can also produce periodic scalar and sublattice potentials. Woods and collaborators (2014) found that graphene can relax into approximately commensurate regions separated by strained walls. Thus the same supporting material can be a dielectric, a source of a mass, and a moiré partner.

A metallic gate changes both density and image-charge screening. A magnetic or strong-spin–orbit neighbor can add exchange or spin-dependent terms; a metal contact can add local doping, strain and valley mixing. The heterostructure article develops these interface effects.

A measured gap needs an operator diagnosis

A neutrality-region suppression can arise from a sublattice mass, interaction order, finite-size confinement, localization, contact barriers, or inhomogeneous puddles. Transport activation measures a mobility or charge gap only under stated assumptions; scanning tunnelling measures a local spectral gap; capacitance measures compressibility; optical response probes neutral transitions.

Combining structural registry, symmetry tests and complementary probes helps identify the cause. For example, a sublattice mass predicts a spectral gap and a specific sublattice polarization, whereas a poorly transmitting contact can suppress conductance without opening a bulk band gap.

From two Dirac layers to moiré bands

Twist introduces a new momentum and length scale

For two graphene layers with relative angle $\theta$, let

$$K = \frac{4\pi}{3a}, \qquad k_\theta = 2K\sin\frac{\theta}{2},$$

where $a$ is the graphene lattice constant. Moiré Superlattices derives the ideal period $L_M$, the cell area $A_M$, and their mismatch and heterostrain corrections. For graphene’s fourfold spin–valley manifold, the density interval corresponding to four carriers per cell is

$$n_s = \frac{4}{A_M}.$$

These are geometric calibrations. Lattice relaxation, heterostrain, spatial twist disorder, and unequal lattice constants modify the local pattern and can broaden any density inferred from a single nominal angle.

Interlayer tunnelling competes with Dirac kinetic energy

A compact coupling ratio is

$$\alpha = \frac{w} {\hbar v_Fk_\theta},$$

where $w$ is an appropriate interlayer tunnelling scale. Small twist reduces $\hbar v_Fk_\theta$ and makes interlayer hybridization nonperturbative. This is the kinematic origin of strong velocity renormalization and narrow moiré bands in the continuum model of Bistritzer and MacDonald (2011).

The number $\alpha$ is not by itself a universal magic-angle criterion. Relaxation distinguishes local stacking regions and modifies tunnelling harmonics; heterostrain breaks rotational symmetry; remote hoppings and particle–hole asymmetry reshape bands; screening and gates change interactions; and disorder sets whether a narrow band is experimentally coherent.

Twisted Bilayer Graphene introduces the layer spinor and tunnelling matrices, then develops the resulting bands. Graphene supplies the valleys and sublattices; the new translation scale reorganizes them into minibands. Observing a moiré wavelength or a narrow band alone does not establish an interacting insulator or superconductor.

Exercises

Exercise 1: classify four perturbations

Consider $V_0\mathbb I$, $\Delta_S\sigma_z$, $\tau\Delta_H\sigma_z$, and a smooth constant $\mathbf A_s$. Which terms open a Dirac gap, which can support a zero-field Chern phase, and which merely shift energy or momentum?

Solution

$V_0\mathbb I$ shifts both bands and does not gap the cone. A constant $\mathbf A_s$ shifts the two valley centers in opposite directions and has zero bulk curl, so it also does not gap or Landau-quantize the ideal cone.

Both mass terms anticommute with the kinetic $\sigma_x$ and $\sigma_y$ matrices and open a gap. The sublattice mass has equal signs in the two valley blocks; opposite valley chiralities then cancel, giving total $C=0$ when time reversal is preserved. The Haldane-type mass changes sign between valleys, allowing the two half-integer valley contributions to add to $|C|=1$ per spin in the stated two-band lattice model.

With a spin-independent Haldane mass, two spin copies give $|C|=2$. A gap measurement alone cannot identify which mass is present.

Exercise 2: uniform strain versus a strain gradient

Using

$$\begin{aligned} A_{s,x} &= \frac{\hbar\beta} {2ea_{\mathrm{cc}}} (u_{xx}-u_{yy}), \\[4pt] A_{s,y} &= -\frac{\hbar\beta} {ea_{\mathrm{cc}}} u_{xy}. \end{aligned}$$

show that a constant uniaxial strain has zero $B_s$. Then take $u_{xx}=gy$, $u_{yy}=u_{xy}=0$ and find $B_s$.

Solution

If every strain component is constant, both components of $\mathbf A_s$ are constant, so

$$B_s = \partial_xA_{s,y} - \partial_yA_{s,x} = 0.$$

For $u_{xx}=gy$,

$$A_{s,x} = \frac{\hbar\beta g} {2ea_{\mathrm{cc}}}y, \qquad A_{s,y} = 0,$$

and therefore

$$B_s = -\frac{\hbar\beta g} {2ea_{\mathrm{cc}}}.$$

The sign reverses if the valley or crystallographic-axis convention is reversed. A compatible displacement is $u_x=gxy$, $u_y=-gx^2/2$, $h=0$: it gives $u_{xx}=gy$ and $u_{yy}=u_{xy}=0$ to linear order. The result applies in a region with $|gx|,|gy|\ll1$. Thus this is a realizable local strain field, not just an arbitrary assignment of tensor components.

Exercise 3: pseudo-Landau energy scale

Take $v_F=1.00\times10^6\,\mathrm{m\,s^{-1}}$ and a locally uniform $|B_s|=50\,\mathrm T$. Assume the total scalar potential $V_\Sigma=V_{\mathrm{ext}}+V_s$ varies slowly enough for the local ladder approximation. Estimate the $n=1$ pseudo-Landau energy relative to this local potential.

Solution

The first Dirac level has magnitude

$$|E_1-V_\Sigma| = v_F \sqrt{2e\hbar|B_s|}.$$

For the stated velocity,

$$\begin{aligned} |E_1-V_\Sigma| &\simeq (36.3\,\mathrm{meV}) \sqrt{|B_s|[\mathrm T]} \\ &\simeq 0.257\,\mathrm{eV}. \end{aligned}$$

The magnetic length is about $3.63\,\mathrm{nm}$. The strained region must be much wider than this, and $B_s$ must vary slowly over it, to support the local ladder approximation. The scalar-gradient condition $|\nabla V_\Sigma|\ll e v_F|B_s|$ concerns the total potential, including gates and contacts. Subtracting only the strain-induced $V_s$ would leave any external scalar shift in the quoted energy. A peak at $0.257\,\mathrm{eV}$ alone does not identify its mechanism.

Exercise 4: real and pseudomagnetic time reversal

Explain why a real perpendicular field can produce a net charge Hall response while a static strain pseudofield does not do so when the state and its momentum distributions respect time reversal.

Solution

A real vector potential enters as $\mathbf p+e\mathbf A$ in both valleys. Time reversal reverses $B$ but the laboratory field is held fixed, so the Hamiltonian is not time-reversal invariant. The valley contributions can therefore add to a charge Hall conductivity.

The strain field enters as $\mathbf p+\tau e\mathbf A_s$. Time reversal sends $\tau\to-\tau$ and reverses the effective field seen by the low-energy block. The complete two-valley Hamiltonian can remain invariant. For time-reversed momentum distributions, opposite transverse valley responses cancel in charge current, although a valley counterflow or valley-resolved local spectrum can remain.

Valley imbalance or an additional time-reversal-breaking perturbation can prevent the cancellation, but then that extra ingredient must be measured.

Exercise 5: twist-angle uncertainty

At small angle, $L_M\simeq a/\theta$ and $n_s=4/A_M\propto L_M^{-2}$. A device has nominal $\theta=1.20^\circ\pm0.03^\circ$. Estimate the fractional uncertainties in $L_M$ and $n_s$ from angle uncertainty alone.

Solution

Logarithmic differentiation gives

$$\frac{\delta L_M}{L_M} \simeq -\frac{\delta\theta}{\theta}, \qquad \frac{\delta n_s}{n_s} \simeq 2\frac{\delta\theta}{\theta}.$$

The magnitude of the angle uncertainty is

$$\frac{0.03^\circ}{1.20^\circ} = 0.025.$$

Thus $L_M$ has about $2.5\%$ fractional uncertainty and $n_s$ about $5\%$, before heterostrain, relaxation, density calibration, and spatial angle disorder are included.

Exercise 6: distinguish two explanations of a nanobubble spectrum

Scanning tunnelling spectroscopy on a graphene nanobubble shows several peaks that fit $E_n-E_0\propto\operatorname{sgn}(n)\sqrt{|n|}$. Describe measurements that would distinguish strain-induced pseudo-Landau levels from confinement resonances or a real magnetic field.

Solution

A strong case would include atomic topography or diffraction from which a strain tensor can be reconstructed; an explicit bond-to-gauge conversion with $\beta$, axes, and valley convention stated; a spatial map showing that the fitted $B_s$ follows the reconstructed strain curl; peak energies and degeneracies consistent across several indices; sublattice or valley behavior compatible with a pseudofield; and stability against tip height, gate voltage, and fitting window.

Controls should exclude real magnetic contamination, quantum-dot confinement, standing waves, scalar-potential resonances, moiré minibands, and ordinary disorder peaks. The $\sqrt{|n|}$ pattern becomes more informative when the same strain model predicts its spatial evolution and sublattice structure.

References

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• F. de Juan, J. L. Mañes, and M. A. H. Vozmediano, “Gauge Fields from Strain in Graphene,” Physical Review B 87, 165131 (2013), doi:10.1103/PhysRevB.87.165131.
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• C. R. Woods et al., “Commensurate–Incommensurate Transition in Graphene on Hexagonal Boron Nitride,” Nature Physics 10, 451–456 (2014), doi:10.1038/nphys2954.
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