Following the isolation of graphene, the discovery of monolayer Transition Metal Dichalcogenides (TMDs) revolutionized 2D quantum materials. Featuring intrinsic direct bandgaps, broken spatial inversion symmetry, and enormous spin-orbit coupling, monolayer TMDs introduce an entirely new degree of freedom for information processing: the valley pseudospin.
1. Crystal Structure and Indirect-to-Direct Bandgap Crossover
Group VI transition metal dichalcogenides ((MX_2), where (M = ext{Mo}, ext{W}) and (X = ext{S}, ext{Se}, ext{Te})) in the bulk form layered hexagonal crystals with weak interlayer van der Waals bonding. In bulk form, TMDs are indirect bandgap semiconductors with conduction band minima between (Gamma) and (K), and valence band maximum at (Gamma).
When mechanically exfoliated down to a single monolayer, interlayer quantum hybridization of the out-of-plane (d_{z^2}) orbitals is completely eliminated. As a result, the valence band at (Gamma) plummets, while the localized in-plane (d_{x^2-y^2}, d_{xy}) orbitals at the Brillouin zone corners (K) and (K’) remain unaffected. Consequently, monolayer TMDs undergo a fundamental indirect-to-direct bandgap crossover, transforming into direct bandgap semiconductors at the high-symmetry points (K) and (K’) with strong optical photoluminescence in the visible and near-infrared.
2. Inversion Symmetry Breaking and Valley-Spin Coupling
Unlike bilayer graphene, monolayer TMDs belong to the non-centrosymmetric (D_{3h}) point group: spatial inversion symmetry (mathcal{P}) is explicitly broken because the transition metal atom (M) and chalcogen pairs (X_2) occupy different sublattices.
At the two inequivalent Brillouin zone corners (K) and (K’ = -K), time-reversal symmetry (mathcal{T}) connects the two degenerate energy valleys:
[
mathcal{T} E(K, uparrow) = E(K’, downarrow)
]
Because inversion symmetry is broken ((mathcal{P} E(mathbf{k}, s)
eq E(-mathbf{k}, s))), atomic spin-orbit coupling from the heavy transition metal (d)-orbitals induces massive valence band splitting ((Delta_{ ext{SO}} approx 150 ext{ meV}) for ( ext{MoS}_2), (Delta_{ ext{SO}} approx 450 ext{ meV}) for ( ext{WSe}_2)). This establishes complete spin-valley locking:
[
ext{Valley } K implies ext{Spin } uparrow, quad ext{Valley } K’ implies ext{Spin } downarrow
]
Because the spin orientation is clamped to the valley index, intervalley scattering requires a simultaneous spin flip and a large momentum transfer (Delta mathbf{k} = K – K’), rendering valley polarized states exceptionally robust against thermal decoherence.
3. Valleytronics and Circular Dichroism Selection Rules
The topological character of the Bloch wavefunctions in monolayer TMDs is captured by the non-zero Berry curvature (boldsymbol{Omega}(mathbf{k}) =
abla_{mathbf{k}} imes langle u_{mathbf{k}} | i
abla_{mathbf{k}} | u_{mathbf{k}}
angle). Time-reversal symmetry dictates that the Berry curvature must be opposite in the two valleys:
[
Omega_z(K) = -Omega_z(K’)
]
Coupling light to these valleys obeys strict optical circular dichroism selection rules:
- Right-circularly polarized photons ((sigma^+)) selectively excite interband optical transitions exclusively at the (K) valley.
- Left-circularly polarized photons ((sigma^-)) selectively excite transitions exclusively at the (K’) valley.
Furthermore, under an in-plane electric field (mathbf{E}), carriers acquire a transverse anomalous velocity (mathbf{v}_{ ext{anom}} = -rac{e}{hbar} mathbf{E} imes boldsymbol{Omega}(mathbf{k})), driving electrons in the (K) valley to one edge of the sample and electrons in the (K’) valley to the opposite edge—manifesting the Valley Hall Effect.
4. Moiré Superlattices and Correlated Flat Bands
When two monolayer 2D crystals are stacked with a small twist angle ( heta ll 1^circ) (e.g., twisted bilayer graphene or ( ext{MoS}_2 / ext{WSe}_2) heterobilayers), the slight spatial interference between their atomic lattices creates a periodic moiré superlattice with an enormous supercell period:
[
lambda_{ ext{moir’e}} approx rac{a}{sqrt{ heta^2 + delta^2}}
]
This long-range periodic potential reconstructs the electronic band structure, folding the original Brillouin zone into a miniature moiré Brillouin zone. At specific “magic” twist angles (( heta approx 1.1^circ) in graphene), the Fermi velocity drops to zero, and the kinetic energy dispersion flattens into isolated ultra-narrow flat bands with bandwidth (W lesssim 10 ext{ meV}).
When the on-site Coulomb repulsion (U) dominates over the flattened bandwidth ((U gg W)), electronic correlations trigger emergent quantum phases: Mott insulators, unconventional high-temperature superconductivity, and fractional Chern topological states.
