---
title: Star-of-David Charge-Density-Wave
url: https://www.emergentmind.com/topics/star-of-david-charge-density-wave
type: topic
---

# Star-of-David Charge-Density-Wave

Searching arXiv for recent and foundational papers on star-of-David charge-density-wave states across transition-metal dichalcogenides and kagome metals.
The star-of-David charge-density-wave is a commensurate charge–lattice reconstruction in which atomic displacements and charge or bond modulations organize into a six-pointed-cluster motif. In layered \(1T\) transition-metal dichalcogenides, the canonical form is a \(\sqrt{13}\times\sqrt{13}\) supercell containing one central transition-metal atom, six inner-ring atoms, and six outer-ring atoms; in kagome metals it usually appears as a \(2\times2\) or \(2\times2\times2\) triple-\(Q\) breathing pattern centered on kagome hexagons, with a closely related inverse star-of-David or tri-hexagonal counterpart. Across these materials, the motif is not a single electronic phase but a structural archetype that can accompany Mott localization, correlated metallicity, unusual ferromagnetism, orbital-flux order, or superconductivity [1803.10727] [2103.07912] [2201.06477].

## 1. Structural motif and order-parameter geometry

In the triangular \(1T\) dichalcogenide lattice, the standard star-of-David reconstruction is generated from primitive vectors
\[
a_1=a(1,0),\qquad a_2=a\left(\tfrac12,\tfrac{\sqrt3}{2}\right),
\]
with a commensurate supercell such as
\[
A_1=3a_1+a_2,\qquad A_2=-a_1+4a_2,\qquad |A_i|=\sqrt{13}\,a.
\]
In monolayer \(1T\)-NbSe\(_2\), each supercell contains thirteen Nb atoms reorganized into a central Nb surrounded by two concentric rings of six Nb each; the six nearest Nb are displaced radially inwards by \(\simeq 0.10\) Å and the outer six by \(\simeq 0.04\) Å, while the Se sublattice relaxes so that the local symmetry is reduced from \(D_{3d}\) to \(C_3\). Single-layer \(1T\)-NbS\(_2\) realizes the same \(\sqrt{13}\times\sqrt{13}\) geometry, again as a 13-atom cluster with radial inward displacements of the two shells [1803.10727] [1902.00234].

The kagome realization is structurally different but order-parameter equivalent in the sense of a triple-\(Q\) commensurate modulation. In \(A\)V\(_3\)Sb\(_5\), the ordering vectors are the three \(M\)-point wavevectors,
\[
Q_1=\left(\tfrac12,0,0\right),\quad
Q_2=\left(0,\tfrac12,0\right),\quad
Q_3=\left(\tfrac12,\tfrac12,0\right)
\]
or symmetry-equivalent conventions, and the charge or bond modulation is written as
\[
\delta\rho(r)=2\,\mathrm{Re}\!\left[\Delta_1 e^{iQ_1\cdot r}+\Delta_2 e^{iQ_2\cdot r}+\Delta_3 e^{iQ_3\cdot r}\right].
\]
The star-of-David corresponds to \(\Delta_1=\Delta_2=\Delta_3\) up to common phase. In real space, six V atoms around a kagome hexagon move inward, while the complementary inverse star-of-David or tri-hexagonal pattern reverses the breathing sense. In a single kagome layer, some analyses assign the inverse star-of-David to the fully symmetric \(A_{1g}\) channel of \(D_{6h}\), whereas three-dimensional \(\pi\)-shifted stacking lowers the crystal symmetry to \(D_{2h}\) or \(C_2\) [2103.07912] [2106.10150] [2503.07442].

## 2. Microscopic mechanisms

No single mechanism explains all star-of-David CDWs. In monolayer \(1T\)-NbSe\(_2\), the transition from the metallic undistorted \(1T\) lattice to the \(\sqrt{13}\times\sqrt{13}\) commensurate phase is driven by the interplay of Fermi-surface nesting, strong Nb–Se covalency, and moderate Hubbard interactions. Density-functional perturbation theory finds a pronounced softening of an in-plane Nb–Se bond-stretching phonon along \(\Gamma\)–\(M\), while the bare susceptibility \(\chi_0(q)\) peaks at \(|q|\approx 0.27|b_1|\); the incommensurate instability then locks onto the nearest commensurate \(\sqrt{13}\times\sqrt{13}\) vector through higher-order free-energy terms [1803.10727].

Bulk \(1T\)-NbS\(_2\) shows a different balance. There the low-temperature ground state is also a \(\sqrt{13}\times\sqrt{13}\) star-of-David CCDW, but the calculated bare susceptibility has no sharp peak at the measured ordering vector. Instead, the phonon linewidth and Eliashberg analysis identify a momentum-selective, strongly softened acoustic branch around \(q_{\mathrm{CDW}}\), with \(\lambda\approx 1.55\) in the undistorted lattice. In that case the principal CDW driver is strong, \(q\)-dependent electron–phonon coupling rather than simple nesting [2004.01839].

In antiferromagnetic FeGe, the experimentally observed \(2\times2\) charge order is described as an interaction-assisted phonon-instability scenario. DFT+\(U\) with \(U\approx 4.1\) eV pushes Fe-\(3d\) flat bands away from \(E_F\), enhances Ge-\(4p\) weight at \(E_F\), and softens a nearly flat optical phonon branch at \(q\approx M\), which becomes imaginary for \(U\gtrsim 3.1\) eV. The generalized susceptibility does not diverge even up to \(U=4\) eV, so electron–electron interactions alone do not drive the CDW; instead they renormalize the electronic structure so as to increase the electron–phonon coupling to the soft \(M\)-phonon [2303.02824].

In kagome metals, several mechanisms are explicitly in play. ARPES on KV\(_3\)Sb\(_5\) identifies a CDW gap tied to inter-saddle-point scattering at the kagome \(M\)-point van Hove singularities, while Landau and DFT analyses find unstable phonon modes at \(M\), \(L\), and \(U\) with strong electronic-temperature dependence, consistent with an electronically driven but lattice-coupled instability. Other studies emphasize V \(3d\)–Sb \(5p\) orbital hybridization as the direct mediator of the CDW structural transition, or nearest-neighbor and interlayer Coulomb interactions that first generate charge bond order and then induce star-of-David or inverse star-of-David distortions through lattice coupling [2206.02315] [2107.04546] [2210.12934] [2302.10178].

## 3. Transition-metal dichalcogenide realizations

Monolayer \(1T\)-NbSe\(_2\) is the clearest SoD–Mott case in the set considered here. The undistorted \(t_{2g}\) manifold spans \(\sim 3\) eV and is metallic, but the nonmagnetic star-of-David distortion produces a very narrow band at \(E_F\) of width \(W_0\approx 30\) meV. Spin-polarized GGA opens a small \(\Delta_{\mathrm{GGA}}\approx 20\) meV gap with \(S=\tfrac12\) per star, and GGA+\(U\) with \(U_{\mathrm{Nb}}=3.02\) eV gives \(\Delta_{\mathrm{GGA}+U}\approx 0.3\) eV, in much better agreement with the experimentally observed \(\sim 0.4\) eV insulating behavior. Energetically, the \(\sqrt{13}\times\sqrt{13}\) phase gains \(\Delta E_{\mathrm{SoD}}\approx 69\) meV/NbSe\(_2\), larger than the \(4\times4\) CDW gain of \(\approx 49\) meV/NbSe\(_2\) [1803.10727].

Single-layer \(1T\)-NbS\(_2\) is closely analogous structurally but somewhat different electronically. Its \(\sqrt{13}\times\sqrt{13}\) reconstruction yields an ultraflat band of bandwidth \(W\approx 0.016\) eV, isolated by \(\gtrsim 0.2\) eV from all other bands. Spin-polarized GGA stabilizes a ferrimagnetic insulating state with \(\Delta E_g^{\mathrm{GGA}}\approx 0.15\) eV and a \(0.21\,\mu_B\) moment localized on the central Nb, while GGA+\(U\) with \(U=2.87\) eV enhances the central moment to \(0.41\,\mu_B\) and produces a fundamental gap \(\simeq 0.35\) eV [1902.00234].

Not all SoD dichalcogenides are Mott insulators. Monolayer \(1T\)-NbTe\(_2\) develops an unusual \(\sqrt{19}\times\sqrt{19}R\,23.4^\circ\) star-of-David lattice, characterized as a sparsely occupied SoD pattern. ARPES shows a partial CDW gap \(\Delta_{\mathrm{CDW}}\simeq 0.5\) eV on the nested \(K\)-centered pocket, but ungapped \(\Gamma\)-centered pockets remain, so the system stays metallic and shows no signature of a Mott gap. This is explicitly contrasted with monolayer \(1T\)-NbSe\(_2\) and bulk \(1T\)-TaS\(_2\), where the \(\sqrt{13}\times\sqrt{13}\) SoD pattern supports a Mott phase [2212.12853].

Monolayer \(1T\)-VTe\(_2\) exhibits multimorphism rather than a unique SoD phase. STM and STS identify a metallic \((4\times4)\) CDW with 13 V atoms condensed into a typical star-of-David cluster, and a gapped \((2\sqrt3\times2\sqrt3)\) CDW with truncated-triangle clusters and a hard gap of \(\Delta\approx 40\) meV. DFT+\(U\) with \(U=2.0\) eV reproduces both the metallic and gapped phases. The CDW-driven reorganization weakens ferromagnetic superexchange, strengthens antiferromagnetic exchange, and suppresses long-range magnetic order [1912.01336].

In \(1T\)-TaSe\(_2\), the equilibrium star-of-David phase is a \(\sqrt{13}\times\sqrt{13}R13.9^\circ\) commensurate CDW below \(T_{\mathrm{CCDW}}\approx 473\) K. The six inner-ring Ta atoms move inward by \(\Delta r_{\mathrm{inner}}\approx 0.045\) Å, the six outer-ring Ta atoms by \(\Delta r_{\mathrm{outer}}\approx 0.020\) Å, and AA stacking enhances interlayer coupling and stabilizes the high-temperature commensurate order [2602.11582].

## 4. Kagome and related realizations

In KV\(_3\)Sb\(_5\), STM and X-ray diffraction show a \(2\times2\) in-plane superstructure below \(T_{\mathrm{CDW}}\approx 78\) K. Within each \(2\times2\) cell, six V atoms around one kagome-hexagon center move inward by \(\delta\approx 0.03\)–\(0.05\) Å and the remaining six move outward by roughly the same amount. Optical spectroscopy finds suppression of low-energy conductivity up to \(\sim 0.1\) eV and a new peak at \(\sim 0.17\) eV, giving \(2\Delta_{\mathrm{CDW}}\approx 59\) meV. Uykur et al. also report strong phonon anomalies, including softening of the \(E_{1u}\) mode at \(188\) cm\(^{-1}\) and a linewidth described by an electron–phonon form with \(\gamma_0\approx 52\) cm\(^{-1}\). DFT-relaxed \(2\times2\) supercells find nearly degenerate star-of-David and tri-hexagon solutions, both remaining metallic [2103.07912].

The wider \(A\)V\(_3\)Sb\(_5\) family does not yet have a single universally accepted three-dimensional CDW structure. Several works argue for inverse star-of-David or tri-hexagonal distortions. ARPES on KV\(_3\)Sb\(_5\) finds that the low-temperature band reconstruction is better captured by the inverse star-of-David pattern, with a strongly anisotropic gap reaching \(\Delta_{\max}\approx 80\) meV on the SP1 saddle-point band at \(M\) and small three-dimensional pockets near \(K\) only at \(k_z\approx 0\). Temperature-dependent X-ray absorption and first-principles energetics in CsV\(_3\)Sb\(_5\) identify the inverse-star-of-David as the preferred reconstruction, with \(E_{\mathrm{ISD}}-E_{\mathrm{pristine}}\approx -120\) meV/f.u. and \(E_{\mathrm{SOD}}-E_{\mathrm{pristine}}\approx -90\) meV/f.u., and interpret V \(3d\)–Sb \(5p\) orbital hybridization as the microscopic driving force [2206.02315] [2210.12934].

Other measurements support more complicated coexistence or stacking scenarios. ARPES combined with DFT has been interpreted as evidence that AV\(_3\)Sb\(_5\) hosts intrinsic coexistence of star-of-David and tri-hexagonal distortions, naturally leading to \(2\times2\times2\) or \(2\times2\times4\) order and two distinct splitting scales at the same momentum. Polarization-resolved Raman, X-ray, and DFT studies on CsV\(_3\)Sb\(_5\) describe a \(2\times2\times4\) structure containing one inverse-star-of-David layer and three consecutive star-of-David layers, with a \(U_1\) distortion as the primary order parameter and \(L_2^-\) and \(M_1^+\) distortions as secondary. Time-resolved reflectivity on CsV\(_3\)Sb\(_5\) further finds that close phonon pairs near \(1.3\) THz and \(3.1\) THz arise from coexistence of star-of-David and inverse star-of-David distortions combined with six-fold rotational symmetry breaking [2201.06477] [2201.05188] [2503.07442].

NMR and NQR provide a different bulk-sensitive perspective. In CsV\(_3\)Sb\(_5\), \({}^{51}\)V NMR and \({}^{121/123}\)Sb NQR detect a first-order commensurate transition below \(T_{\mathrm{CDW}}\sim 98\) K and report that the observed charge modulation is of star-of-David pattern, not inverse star-of-David, with an additional weaker charge modulation appearing below \(T^\ast\sim 40\) K. This result is one reason structural identification remains controversial [2108.10263].

Antiferromagnetic FeGe extends the SoD concept beyond the nonmagnetic kagome metals. There the \(2\times2\) CDW organizes six Fe sites into a real-space star-of-David bond pattern, and in the CDW phase the ground-state current density exhibits a star-of-David loop with amplitude \(|\langle J\rangle|\approx 3.6\times10^{-7}\) A together with counter-circulating second-neighbor loops of \(\approx 2.8\times10^{-7}\) A [2303.02824].

## 5. Electronic reconstruction, Mottness, magnetism, and topology

A recurrent misconception is that a star-of-David CDW necessarily implies Mottness. In fact, the consequences range from Mott insulating to metallic. Monolayer \(1T\)-NbSe\(_2\) is explicitly a SoD Mott insulator: in a three-band Wannier DMFT treatment at \(T=300\) K, the effective interaction on the central “type I” Wannier function is \(U_{\mathrm{eff}}(\mathrm{type\ I})\approx 0.9\) eV, its occupancy is pushed from \(1.18\rightarrow 1.00\), and the gap between the upper Hubbard band and the Se–Nb valence bands is reproduced. Single-layer \(1T\)-NbS\(_2\) likewise develops a spin-\(\tfrac12\) insulating state. By contrast, monolayer \(1T\)-NbTe\(_2\) remains a correlated metal with no signature of Mott gap, and the kagome metals remain metallic despite clear SoD- or ISD-like reconstruction [1803.10727] [1902.00234] [2212.12853] [2103.07912].

The SoD motif can also host unconventional magnetism. In monolayer \(1T\)-NbSe\(_2\), mapping total-energy differences to
\[
H=-\sum_{\langle i,j\rangle}J_{ij}\,\mathbf S_i\!\cdot\!\mathbf S_j
\]
gives \(J_1\approx +2.4\) K in GGA and \(+4.8\) K in GGA+\(U\), with \(J_2\approx +0.1\) K and \(+0.04\) K, respectively. The positive nearest-neighbor sign is interpreted as a hallmark of flat-band ferromagnetism rather than conventional superexchange. Single-layer \(1T\)-NbS\(_2\) similarly yields ferromagnetic inter-star couplings \(J_1=9.5\) K and \(J_2=0.4\) K. Conversely, in monolayer VTe\(_2\), cluster formation pushes \(J_{\mathrm{tot}}=J_D+J_S\) toward zero or negative values and suppresses the long-range ferromagnetism predicted for the undistorted lattice [1803.10727] [1902.00234] [1912.01336].

Bulk \(1T\)-NbS\(_2\) illustrates a different electronic consequence: the CCDW phase opens an in-plane direct gap of \(\approx 0.40\) eV, but the nearly flat band just below \(E_F\) retains substantial dispersion along \(\Gamma\)–\(A\), producing one-dimensional metallic behavior along the stacking direction. The top valence state is dominated by the central Nb \(4d_{z^2-r^2}\) orbital, with real-space charge strongly localized on the central Nb of each star. Under pressure, the CCDW is suppressed by \(\sim 30\) GPa, \(\lambda\) falls to \(\approx 0.82\), and the Allen–Dynes estimate yields a peak \(T_c\approx 13\) K [2004.01839].

In kagome and antiferromagnetic kagome systems, the reconstructed state can be topological or orbital-current bearing. In FeGe, the orbital flux through the central hexagon is \(\Phi_{\mathrm{hex}}\approx +0.15\,\Phi_0\), while the small triangles carry \(\Phi_\Delta\approx -0.12\,\Phi_0\). The CDW phase then hosts Weyl points protected by \(M_xT\) symmetry and a nearly flat Chern band near \(E_F\), with robust edge modes in the partial bulk gap. In KV\(_3\)Sb\(_5\), the CDW gap has periodicity of the undistorted Brillouin zone along \(k_z\), again showing that the reconstructed low-energy electronic structure, rather than the high-symmetry lattice, is the correct basis for discussing superconductivity and anomalous transport [2303.02824] [2206.02315].

## 6. Ultrafast control, metastability, and unresolved questions

The star-of-David lattice is not only an equilibrium order. In \(1T\)-TaSe\(_2\), femtosecond pumping with \(\hbar\omega_{\mathrm{pump}}=1.6\) eV and \(\tau_p\simeq 35\) fs can coherently over-drive the equilibrium \(\sqrt{13}\times\sqrt{13}\) mode into an inverted CDW with the same periodicity but opposite sign of the displacement field. The critical fluence is \(F_c\simeq 0.61\) mJ/cm\(^2\); the amplitude mode passes through the transient normal state at \(t\sim 150\)–\(350\) fs and overshoots into the inverted regime at \(t\sim 350\)–\(450\) fs. tr-ARPES and TDDFT show that the inverted state has a higher density of states at \(E_F\) than even the normal metallic state, enhanced metallicity, and altered electron–phonon couplings [2011.07623].

Photoinduced metastability is also seen in kagome metals. In vanadium kagome compounds, femtosecond time-resolved X-ray scattering identifies a coherent phonon at \(\approx 1.3\) THz tied to the CDW reflection and shows that an out-of-plane Cs mode is frustrated in the CDW phase. Photoexcitation relieves that frustration, producing a metastable CDW with \(\tau_{\mathrm{rise}}\approx 100\) fs, \(\tau_{\mathrm{decay}}\approx 300\) fs, and a \(>2\) ns lifetime before full thermal recovery. This suggests that phononic frustration, not only static lattice energetics, is part of the star-of-David problem in kagome metals [2406.06913].

The principal unresolved issue is structural identification in AV\(_3\)Sb\(_5\). The literature summarized here contains mutually inconsistent but individually well-supported assignments: star-of-David from bulk NMR/NQR; inverse star-of-David from XAS, DFT, and some ARPES analyses; coexistence of star-of-David and tri-hexagonal distortions from ARPES plus DFT; and several \(\pi\)-shifted \(2\times2\times2\) or \(2\times2\times4\) stacking patterns from Landau, Raman, X-ray, and effective-Hamiltonian approaches [2108.10263] [2210.12934] [2201.06477] [2302.10178]. A plausible implication is that the phrase “star-of-David charge-density-wave” now denotes a family of closely competing triple-\(Q\) states whose selection depends sensitively on interlayer coupling, orbital hybridization, electron–phonon coupling, and electronic correlations.

Across both dichalcogenides and kagome materials, the star-of-David motif therefore functions less as a unique phase label than as a geometrically recognizable endpoint of several different instability channels. Its importance lies precisely in this variability: the same cluster geometry can support a Hubbard-driven gap, a correlated metal, flat-band ferromagnetism, orbital-flux order, or a laser-created hidden state, depending on how lattice distortion, band topology, and many-body interactions are combined [1803.10727] [2212.12853] [2303.02824] [2011.07623].

Source: https://www.emergentmind.com/topics/star-of-david-charge-density-wave