A sharp peak in the number of states electrons can occupy at one energy, found where a band flattens or has a saddle point – rising one way and falling the other, like a mountain pass. Many electrons then share almost the same energy, so when the reaches the peak they are especially ready to order, into magnetism, a or . In a two-dimensional layer the peak is particularly sharp, and twisting two layers, or heavy can move it to the Fermi level.
As the site uses it
The kagome band structure supplies Dirac points, van Hove singularities and flat bands in one lattice, and in AV3Sb5 the van Hove points sit close enough to the Fermi level for nesting to matter.
In the monolayer the valence band maximum moves off Γ into a ring, producing a van Hove singularity near the band edge; predictions of magnetism and enhanced thermoelectric response are therefore sensitive to spin–orbit coupling, strain and functional.
Where a band has a saddle point the count of states spikes. In graphene, in a simple tight-binding model, the pockets of electrons around the K corners meet at the M points, about 2.8 eV from the Dirac point; twisting two layers creates a pair of peaks much closer in, which move towards zero energy as the twist angle shrinks.
Where states pile up
The counts how many electron states lie in each slice of energy. Where a band is steep, a slice of energy holds few states; where it is flat, many. At a saddle point – a place in where the band rises in one direction and falls in another, like a mountain pass – the band is flat in every direction at once, and the count shoots up into a peak.
Léon Van Hove showed in 1953 that such points cannot be avoided: every crystal has them, and what they do depends only on the number of dimensions. In two dimensions a saddle point gives a peak that grows without limit, logarithmically; in three the count itself stays finite and only its slope jumps (Van Hove, 1953). He was thinking of the vibrations of atoms, but the same holds for electrons – one reason flat layers are so prone to instabilities when the Fermi level sits on such a peak.
In graphene and its twisted stacks
In graphene the saddle points sit at the M points of the Brillouin zone, roughly one hopping energy – about 2.8 eV – from the (Castro Neto and colleagues, 2009), far beyond the reach of a gate. Only chemistry gets there: with ytterbium beneath it and potassium on top, graphene on silicon carbide has been doped to 5.5 × 1014 electrons per square centimetre, past the singularity, and its changed from two small electron pockets to one giant hole pocket – a Lifshitz transition (Rosenzweig and colleagues, 2020).
Twisting is the easier route. Two graphene layers at a twist have a pair of van Hove singularities much closer to the Dirac point, seen by as two pronounced peaks that move towards the Fermi level as the angle shrinks (Li and colleagues, 2010). Near the of about 1.1° each peak is at its narrowest; doped near half filling of the band, the peak that reaches the Fermi level splits by up to 7.5 meV, a gap opened by the interactions between electrons (Kerelsky and colleagues, 2019).
Why the peak matters
A peak at the Fermi level means many electrons with nearly the same energy and little kinetic energy to lose, so even modest interactions between them can reorganise the whole system. Depending on the shape of the Fermi surface and on the interactions, the result can be magnetism, a density wave or superconductivity (Li and colleagues, 2010) – which is why moving a van Hove singularity to the Fermi level, by doping, gating, or twisting, is a standard strategy in the search for new phases. , whose lattice builds such saddle points into its bands, are studied for the same reason.
The peaks are also directly visible. A scanning tunnelling microscope sees them as maxima in the conductance at one spot, as flat stretches of band, and optical spectroscopy as strong absorption at the energy that connects two of them.
For specialists
A singularity in the density of states at a critical point of the , where the group velocity vanishes. In 2D a saddle point gives a logarithmic divergence, as Van Hove showed in 1953 for crystal vibrations, while in 3D the density of states stays finite and only its slope diverges. In graphene the saddle points at M lie about t ≈ 2.8 eV from the Dirac point; doping epitaxial graphene to 5.5 × 1014 cm−2 carries the Fermi level past them, in a Lifshitz transition from two electron pockets to one large hole pocket. Twisting two graphene layers creates a pair of low-energy van Hove singularities that move towards the Fermi level as the angle shrinks; near 1.1° each is at its narrowest, and doped to the Fermi level one splits by a correlation gap of up to 7.5 meV.
Moiré provide a physical realization of the model. Bandwidth-tuned Mott transitions and complex mimicking those of high-temperature superconductors have been demonstrated near a weak Mott insulating state at half-band-filling. The platform also provides an opportunity to explore…
Moiré semiconductors provide a physical realization of the Hubbard model. Bandwidth-tuned Mott transitions and complex phase diagrams mimicking those of high-temperature superconductors have been demonstrated near a weak Mott insulating state at half-band-filling. The platform also provides an opportunity to explore…
Tuning the chemical potential of a solid to the vicinity of a Van Hove singularity (VHS) is a well-established route to discovering emergent quantum phases. In graphene, the use of electron-donating metal layers has recently emerged as a method to dope the chemical potential to the nearest… [Phys. Rev. B 114…