---
title: Trigonal Warping in 2D Materials
url: https://www.emergentmind.com/topics/trigonal-warping
type: topic
---

# Trigonal Warping in 2D Materials

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Trigonal warping is the deformation of a constant-energy contour from a circular or otherwise azimuthally symmetric form into a threefold-symmetric triangular or trigonal shape in momentum space. In the literature summarized here, it appears as the leading mechanism by which an approximate continuous rotational symmetry is reduced to the physically relevant \(C_3\) symmetry, and it is realized through several distinct microscopic channels: higher-order lattice corrections beyond the Dirac approximation in monolayer graphene, skew interlayer hopping in bilayer graphene and graphite, remote-band coupling in monolayer \(\mathrm{MoS}_2\), moiré-periodic sublattice asymmetry on incommensurate substrates, and momentum-dependent spin-orbit or pairing vertices in spintronic and superconducting settings [1409.0130, 1304.4084, 1111.0399, 2508.09387].

## 1. Definition and symmetry content

In monolayer graphene beyond the Dirac-point approximation, trigonal warping is the deformation of the Fermi contour around each Dirac point from a circle into a trigonal shape as the Fermi energy increases away from the low-energy linear regime. The nearest-neighbor tight-binding spectrum is anisotropic in \(k\)-space, and the paper on anisotropic resistivity divides the behavior into a low-energy isotropic Dirac regime, an intermediate-energy trigonal-warping regime for \(0<\epsilon_F<t\), and a high-energy connected-Fermi-curve regime for \(\epsilon_F>t\), with \(t=2.7\,\mathrm{eV}\) as the relevant scale [1409.0130].

In Bernal bilayer graphene and graphite, the same designation refers to the threefold distortion generated by skew interlayer hopping, conventionally \(\gamma_3\). In that setting, trigonal warping breaks the simplest isotropic low-energy structure near a valley and can split a quadratic touching into one central Dirac cone plus three satellite cones [1911.12853, 1106.3401]. In monolayer \(\mathrm{MoS}_2\), it is the reduction of an almost rotationally invariant low-energy theory to the actual threefold crystal symmetry, with the characteristic angular dependence \(q^3 \cos 3\phi\) generated by remote-band coupling [1304.4084]. In moiré graphene on incommensurate substrates, the same threefold distortion appears in the renormalized Dirac cone and breaks the effective time reversal symmetry in a single valley while preserving full graphene time-reversal symmetry through opposite warping in opposite valleys [1111.0399].

A common misconception is that trigonal warping is merely a geometric correction to equal-energy contours. The cited work consistently assigns it broader significance: it changes velocity distributions, scattering amplitudes, Berry-phase structure, optical selection rules, and the location of band inversions. This suggests that the term is best understood as a symmetry-lowering perturbation with direct consequences for transport, topology, spectroscopy, and confinement rather than as a purely kinematic contour distortion.

## 2. Microscopic origins and representative Hamiltonians

The microscopic origin of trigonal warping is material dependent, but the formal structure is recurrent: the perturbation enters as an anisotropic momentum-dependent term with angular harmonic \(e^{\pm 3 i \phi}\), or equivalently as a cubic invariant such as \(k_x^3-3k_xk_y^2\).

For monolayer graphene, the full tight-binding band energy already contains the anisotropy from which trigonal warping emerges:
\[
\epsilon_{k\lambda} =\lambda t\left(1+4\cos\!\left(\frac{3a}{2}k_x\right) \cos\!\left(\frac{\sqrt{3}a}{2}k_y\right) +4\cos^2\!\left(\frac{\sqrt{3}a}{2}k_y\right)\right)^{1/2},
\]
with \(a=1.42\,\text{\AA}\) and \(\lambda=\pm1\) [1409.0130]. In Bernal bilayer graphene, the standard low-energy mechanism is the skew interlayer hopping usually denoted \(\gamma_3\) or \(v_3\), which appears as the linear term competing with the quadratic bilayer term in effective two-band models. In the bilayer-quantum-dot literature this is written as
\[
H^{\rm eff}_\xi = \begin{pmatrix} U/2 & -\dfrac{(\pi_\xi^\dagger)^2}{2m} + \xi v_3 \pi_\xi \\
-\dfrac{\pi_\xi^2}{2m} + \xi v_3 \pi_\xi^\dagger & -U/2 \end{pmatrix} + V_{\rm conf}(\mathbf r),
\]
which makes explicit that the quadratic piece gives the usual low-energy bilayer dispersion while the linear \(v_3\) term is the trigonal-warping term [2501.07107]. In graphite, the \(\tilde{\gamma}_3 v k_{\pm}/\gamma_0\) matrix elements are isolated as the trigonal-warping perturbation in the Slonczewski–Weiss–McClure model [1106.3401].

In monolayer \(\mathrm{MoS}_2\), the symmetry-based \( \mathbf{k}\cdot\mathbf{p} \) reduction identifies
\[
H_{3w}=\kappa \begin{pmatrix} 0 & (q_{+})^2 \\ (q_{-})^2 & 0 \end{pmatrix}
\]
as the term responsible for trigonal warping, while \(H_{cub}\) is needed for a quantitative fit to the valence band away from \(K\) [1304.4084]. In graphene with Rashba spin-orbit coupling, the momentum-dependent Rashba correction produces a reduced low-energy Hamiltonian
\[
H_{w}=-\frac{\sqrt{3}}{2}a\lambda(k_x \sigma_y+k_y \sigma_x) +\gamma \begin{pmatrix} 0 & ik_{-}^{2}\\ -ik_{+}^{2} & 0 \end{pmatrix},
\]
and the paper identifies the \(k_\pm^2\) structure as the origin of trigonal warping in that system [1710.08669]. In anomalous Hall Dirac materials, the warping perturbation is introduced directly into the \(d\)-vector as quadratic terms,
\[
\mathbf{d} = (\chi vp_x + \lambda (p_y^2 - p_x^2),\; vp_y + 2 \chi \lambda p_x p_y ,\; \Delta),
\]
with \(\lambda\) controlling the strength of warping [2207.08432].

These Hamiltonians are not formally identical, but they share two technical features. First, trigonal warping is tied to terms that are subleading with respect to the isotropic Dirac or quadratic bilayer piece near the high-symmetry point. Second, those terms are symmetry allowed and become decisive once states away from the immediate valley center are involved. The papers on bilayer graphene, van der Waals multilayers, and superconducting single-valley systems all emphasize precisely that point [1612.08893, 2508.09387].

## 3. Band reconstruction, satellite Dirac points, and topological consequences

The most characteristic band-structure consequence of trigonal warping is the reconstruction of a single isotropic low-energy valley into a central cone plus three symmetry-related off-center structures. In Bernal bilayer graphene this is the familiar splitting of the low-energy quadratic touching into four Dirac points; in twisted double bilayer graphene it reappears as one central moiré Dirac cone plus three satellite Dirac cones around \(K_M\) [1512.06556, 2010.06619].

In twisted double bilayer graphene, the satellite cones are the key ingredient behind two successive field-tuned topological transitions. The direct bandgap closes twice as the perpendicular electric field is varied, with valley-Chern-number changes of \(-3\) and \(+1\). The paper attributes the lower direct-gap-closing line to the three satellite Dirac points, which are the moiré descendants of the trigonal-warped bilayer dispersion [2010.06619]. A closely related logic appears in anomalous Hall Dirac materials: trigonal warping generates three additional Dirac cones around the original one, and the combined contribution of the central cone and the three secondary cones changes the total Chern number from \(\pm \frac{1}{2}\) to \(\mp 1\) for a single valley sector [2207.08432].

In Bernal bilayer graphene with exchange field, interlayer potential difference, and Rashba spin-orbit coupling, trigonal warping reshapes the phase diagram by inducing extra band inversion points at momentum further away from high-symmetric point. The result is the emergence of valley-polarized QAHE with high Chern numbers ranging from \(\mathcal{C}=-7\) to \(7\), while the conventional QVHE and QAHE regions shrink [1612.08893]. In gated bilayer silicene, the analogous skew interlayer hopping \(t_3\) produces several critical electric fields where the gap closes due to the trigonal warping effect, and the Dirac cone is split into three cones where the band touches the Fermi level [1204.3971].

Trigonal warping can also preserve topological quantities while altering geometric spectra. In the entanglement-spectrum analysis of Bernal bilayer graphene, tracing out one layer yields an entanglement spectrum that is geometrically different from the monolayer energy spectrum, with zero entanglement-energy line segments connecting central and satellite points, yet Berry-phase-type contributions associated with the valleys still match those of monolayer graphene in the appropriate sense [1512.06556]. This suggests that trigonal warping redistributes topological content in momentum space even when the global invariant is unchanged.

A further misconception is that all topological phase transitions driven by warping occur exactly at \(K\) or \(K'\). Several of the cited papers explicitly contradict that simplification. Gap closings can occur on the \(M-K'\) line, on the \(K'-\Gamma\) line, or at three symmetry-related momenta away from the valley center, and those off-center closures are precisely what force Chern-number changes by three or six in multilayer systems [1612.08893, 1204.3971].

## 4. Anisotropic transport, scattering, and quasiparticle interference

Transport papers consistently show that trigonal warping enters observables through both band anisotropy and scattering structure. In monolayer graphene beyond the Dirac approximation, the conductivity anisotropy is controlled not only by the noncircular Fermi contour and anisotropic velocities but also by the pseudospin-dependent scattering matrix. The paper therefore distinguishes a trigonal-warping regime for \(0<\epsilon_F<t\) from a connected-Fermi-curve regime for \(\epsilon_F>t\), and notes that at high energies the anisotropy generated by the scattering matrix is more effective than the band anisotropy [1409.0130].

For sharp graphene \(P\)–\(N\) junctions, trigonal warping qualitatively changes Klein tunneling. The exact tight-binding mode-matching analysis shows that perfect transmission at normal incidence disappears beyond the linear Dirac regime and is replaced by perfect transmission at oblique incidence, with the effect interpreted as a consequence of generalized pseudospin conservation in the tight-binding model [1801.08647]. In a different ballistic setting, weak trigonal warping combined with a weak magnetic field and strong collimation shifts the guiding centre coordinate in a valley-dependent way, enabling spatial valley separation with the real-space splitting
\[
x_+ - x_- \approx 8 u \frac{\hbar k_0}{eB},
\]
where \(u=ka/4\) and \(k_0=\sqrt{\pi n}\) [1909.12479].

Scanning-tunneling and QPI studies provide a direct view of the warping orientation. Gate-tunable STM on Bernal bilayer graphene/hBN, combined with tight-binding-based joint density of states simulations, identifies the low-energy trigonal warping orientation as the case corresponding to \(\gamma_3<0\) in the McCann–Koshino convention, inverted relative to the high-energy anisotropy seen in ARPES [1911.12853]. Within a low-energy description, perturbative inclusion of trigonal warping in bilayer graphene yields real-space impurity-induced LDOS corrections of the form
\[
\delta \rho \sim \frac{\alpha + \beta \gamma_{03}\sin 3\phi_r}{r}e^{-\Omega r},
\]
and Fourier-space anisotropy proportional to \(\sin 3\phi_p\), in good agreement with experiment and semi-analytical \(T\)-matrix results [2105.01068].

Transport along topological domain walls in biased bilayer graphene is another case where trigonal warping is not a small correction. The isotropic approximation gives the expected pair of gap-crossing branches, but the inclusion of warping makes the 1D dispersion strongly orientation dependent and can produce three or even four zero-energy crossings per valley per spin while preserving the topological lower bound of two [1012.4318]. This suggests that in directional or interface-based transport, the role of trigonal warping is to reorganize how conserved longitudinal momentum samples a threefold-anisotropic two-dimensional band structure.

## 5. Optical, magneto-optical, and spin-resolved manifestations

Optical response is one of the clearest arenas in which trigonal warping acts as a symmetry selector. In graphite under quantizing magnetic fields, the dominant optical transitions obey \(\Delta n=1\), but the \(\gamma_3\) trigonal-warping term generates second-order Landau-level shifts and additional weak \(\Delta n=2\) transitions. Those weak transitions leave fingerprints in the conductivity, Kerr rotation, and reflectivity, and the perturbative treatment is justified for \(B>1\ \text{T approximately}\) [1106.3401].

In monolayer \(\mathrm{MoS}_2\), trigonal warping is the microscopic mechanism by which the low-energy continuum theory becomes compatible with finite low-energy even-order harmonic generation. The isotropic part of the theory already permits odd-order responses, whereas second- and fourth-order responses vanish without warping. The paper therefore explains why, at a pump photon energy of \(0.8\) eV, the measured third harmonic is \(\sim 30\times\) stronger than the second harmonic and the fourth harmonic is of the same order as the second: THG is governed mainly by the large isotropic Hamiltonian, while SHG and FHG are controlled by the much smaller trigonal-warping correction [1608.04101].

Trigonal warping can also reshape spin textures rather than only orbital contours. In graphene with Rashba SOC, it gives rise to a terraced spin texture in the low-energy bands, with three broad plateaus in \(\theta_s(\theta_k)\) and a winding number of \(2\) rather than \(1\). The same plateau structure underlies strongly spin-selective transmission through a Rashba barrier and nearly perfect spin polarization achievable electrically when only one concave segment contributes [1710.08669]. In anomalous Hall Dirac materials, trigonal warping activates higher-order Hall responses that do not exist in a rotationally symmetric conventional Dirac material, including a non-zero second order Hall current even in the absence of Berry curvature dipole [2207.08432].

A more recent extension places the same logic in superconductivity. In single-valley superconductors, trigonal warping produces an inversion-even velocity component \(\tilde{\mathbf v}_{\mathbf k}\) and thereby breaks the antisymmetry between the velocities of the two electrons in a Cooper pair. The result is that Bardasis–Schrieffer modes, clapping modes, and the quasiparticle threshold at \(2\Delta\) become visible in the longitudinal and Hall optical responses at \(\mathbf q=0\), with the selection rule that modes whose angular momenta differ from that of the condensate by two units are optically brightened [2508.09387]. This is a particularly clear example of trigonal warping acting as a symmetry-breaking optical vertex rather than merely as a distortion of the normal-state Fermi surface.

## 6. Quantum dots, moiré systems, and semiclassical wave mechanics

Confinement experiments show that trigonal warping can reorganize shell structure itself. In circular few-electron bilayer graphene quantum dots, transport reveals bunching of conductance resonances in groups of \(4\), \(8\), and \(12\), interpreted as filling a fourfold-degenerate \(s\)-shell, an eightfold-degenerate \(p\)-shell, and then a shell dominated by a threefold minivalley structure induced by trigonal warping [2011.07951]. A later experiment demonstrates switchable electron shell structure under electrical gating: at small perpendicular electric field, the lowest \(s\)-shell is sequentially filled with two spin-up and two spin-down electrons of opposite valleys, whereas at larger field an additional three-fold minivalley degeneracy is generated so that the \(s\)-shell can be filled with \(12\) electrons with the first or last \(6\) having the same spin polarization, changing the filling sequence from “\(2+2+4+4\)” to “\(6+6\)” [2501.07107].

The physical mechanism in these dot experiments is explicit. A perpendicular electric field creates interlayer asymmetry and opens a tunable gap in bilayer graphene; in the presence of skew interlayer hopping, the gapped band structure develops stronger trigonal warping near the band edge; and once the depth of the three local minivalleys becomes comparable to or larger than the orbital level spacing, the lowest shell effectively acquires an extra threefold degeneracy [2501.07107]. In the reported device, previous work is cited as showing minivalley depths of a few meV, larger than the extracted first shell spacing \(\Delta E_{\mathrm{sh}1} \approx 0.53~\text{meV}\), which is why even the lowest \(s\)-shell is affected.

Moiré systems realize a different but closely related regime. For graphene on incommensurate substrates such as hBN or Ir(111), the small lattice mismatch prevents a uniform mass term from opening a gap at the original Dirac point. Instead, the system exhibits a renormalized Dirac dispersion with trigonal warping and a new set of massless Dirac fermion replicas at the corners of the superlattice Brillouin zone, with group velocity asymptotically equal to one-half that of pristine graphene in the weak-coupling limit [1111.0399]. This is a case in which a slowly varying moiré-periodic sublattice asymmetry produces valley-contrasting warping and secondary Dirac structures without eliminating the primary Dirac cone.

At the semiclassical level, small trigonal warping can leave the energy spectrum unchanged while strongly altering wavefunctions. For bound states of graphene’s 2D Dirac operator in a magnetic field, the trigonal-warping correction enters the transport equation through the cubic harmonic \(2\gamma(\mathbf p_1^3-3\mathbf p_1\mathbf p_2^2)\), or equivalently through \(\cos 3\varphi\) and \(\sin 3\varphi\) terms in polar form. The paper proves that the semiclassical bound-state spectrum remains unchanged at the order considered because the trigonal term averages to zero over the invariant torus, yet the asymptotic eigenfunctions are significantly affected and their density plots acquire the broken-symmetry structure that could be observed with scanning tunneling microscopy [2403.15748].

Taken together, these confinement and moiré results show that trigonal warping is capable of changing the effective degeneracy, orbital sequence, spatial profile, and valley structure of low-energy states even when it does not produce a leading-order shift of the corresponding eigenvalues. That distinction recurs throughout the broader literature: trigonal warping is often most visible not as a uniform shift of all energies, but as a reorganization of how low-energy states are distributed in momentum space, real space, and symmetry space.

Source: https://www.emergentmind.com/topics/trigonal-warping