Theoretical investigations of valley-selective quantum confinement in two-dimensional materials such as MoS2 and WSe2 have emerged as a critical area of research in condensed matter physics. These materials exhibit unique electronic and optical properties due to their atomically thin structures and strong spin-orbit coupling. The valleys, located at the K and K' points of the Brillouin zone, serve as discrete energy minima that can be selectively populated and manipulated. Quantum confinement effects further modify these valley properties, enabling precise control over valley polarization and coherence. This article explores ab initio calculations of valley-dependent optical selection rules, valley lifetimes under spatial confinement, and the influence of external electric and magnetic fields on valley dynamics.
In monolayer transition metal dichalcogenides (TMDCs), the broken inversion symmetry and strong spin-orbit coupling lead to valley-dependent optical selection rules. Circularly polarized light selectively excites carriers in either the K or K' valley, a phenomenon confirmed by first-principles density functional theory (DFT) calculations. These calculations reveal that the optical transitions at the K and K' valleys are governed by opposite helicities due to time-reversal symmetry. For instance, in MoS2, right-handed circularly polarized light couples exclusively to the K valley, while left-handed polarization excites the K' valley. The degree of valley polarization is influenced by the strength of quantum confinement, which can be tuned by reducing the lateral dimensions of the material. Ab initio studies demonstrate that in nanoribbons or quantum dots of MoS2, the valley-selective optical transitions remain robust, though the exact energy separation between valleys may shift due to edge effects or strain.
Valley lifetimes, a key parameter for valleytronic applications, are strongly affected by spatial confinement. Theoretical investigations using many-body perturbation theory within the GW approximation and Bethe-Salpeter equation (BSE) frameworks show that electron-phonon interactions and intervalley scattering mechanisms dominate valley depolarization. In confined systems, such as MoS2 quantum dots, the valley lifetime is prolonged due to suppressed phonon-mediated intervalley scattering. For example, calculations predict that in a 3 nm MoS2 quantum dot, the valley lifetime can exceed 10 ps at room temperature, compared to a few picoseconds in extended monolayers. The reduction in scattering phase space under confinement plays a crucial role in enhancing valley coherence. Additionally, defects and edge states introduce new scattering channels, which can either shorten or extend valley lifetimes depending on their symmetry and energetic alignment.
External electric and magnetic fields provide additional knobs to tune valley-selective quantum confinement. Ab initio simulations reveal that perpendicular electric fields break the degeneracy between the K and K' valleys through the Stark effect. In bilayer WSe2, for instance, an electric field of 0.5 V/nm induces a valley splitting of approximately 20 meV, as predicted by DFT calculations. This splitting can be further modulated by quantum confinement, with narrower nanostructures exhibiting larger Stark shifts due to enhanced field localization. Magnetic fields, on the other hand, couple to the valley pseudospin via the Zeeman effect. Theoretical models based on k·p perturbation theory show that a magnetic field of 10 Tesla can induce a valley Zeeman splitting of around 0.5 meV in MoS2 monolayers. In confined systems, the interplay between quantum confinement and magnetic field effects leads to non-trivial modifications of the valley polarization dynamics.
The influence of quantum confinement on excitonic effects in TMDCs is another critical aspect explored by theoretical studies. Excitons in these materials exhibit binding energies on the order of hundreds of meV due to reduced dielectric screening. Confinement enhances these effects, with ab initio calculations predicting exciton binding energies exceeding 500 meV in MoS2 nanoribbons narrower than 2 nm. The valley-selective optical response of these confined excitons is further modulated by their spatial localization, which alters the overlap between electron and hole wavefunctions. Theoretical frameworks combining DFT with configuration interaction methods have been employed to unravel the excitonic fine structure in quantum dots, revealing dark states that can impact valley polarization lifetimes.
Strain engineering represents another dimension for controlling valley properties in confined systems. First-principles calculations demonstrate that uniaxial strain can tune the relative energy of the K and K' valleys, enabling strain-induced valley polarization. For example, a 2% tensile strain along the armchair direction in WSe2 monolayers shifts the K valley energy by approximately 15 meV relative to the K' valley. In quantum dots or nanoribbons, the strain distribution becomes inhomogeneous, leading to spatially varying valley polarization. Finite-element modeling coupled with tight-binding approximations has been used to predict these effects, showing that strain gradients can generate valley-selective carrier accumulation regions.
Theoretical studies have also addressed the role of many-body interactions in valley-selective quantum confinement. Electron-electron interactions, treated within the Hartree-Fock or DFT+U frameworks, are found to renormalize the valley splitting in confined systems. For instance, in doped MoS2 quantum dots, Coulomb interactions can enhance the valley polarization by suppressing intervalley exchange processes. Similarly, electron-hole interactions in confined excitonic systems lead to binding energy renormalization, which varies between the K and K' valleys due to their different effective masses. These effects are captured by advanced theoretical methods such as quantum Monte Carlo or time-dependent DFT, providing insights into the many-body physics of valleytronic nanostructures.
The impact of substrate interactions on valley properties in confined TMDCs has also been investigated theoretically. Dielectric screening from substrates modifies the Coulomb interaction and exciton binding energies, which in turn affects valley-selective optical transitions. Ab initio calculations incorporating substrate effects via model dielectric functions show that high-k dielectrics like hexagonal boron nitride (hBN) can reduce exciton binding energies while preserving valley polarization. In quantum dots, the substrate-induced dielectric screening is non-uniform, leading to position-dependent valley splittings that can be mapped using electrostatic models.
Temperature-dependent effects on valley dynamics in confined systems are another area of theoretical exploration. Electron-phonon coupling calculations using density functional perturbation theory (DFPT) reveal that optical phonon modes dominate intervalley scattering at elevated temperatures. In quantum-confined structures, the phonon density of states is quantized, leading to discrete scattering channels that can be engineered to minimize valley depolarization. Theoretical predictions indicate that below 50 K, valley lifetimes in MoS2 nanoribbons can exceed 100 ps due to frozen phonon modes.
Theoretical frameworks have also been developed to describe valley transport in confined geometries. Boltzmann transport equations incorporating valley-dependent scattering rates predict that edge states in nanoribbons can facilitate valley-polarized currents. Quantum kinetic models further show that external electric fields can drive valley Hall effects in confined systems, with the magnitude of the response depending on the interplay between confinement-induced level quantization and Berry curvature effects.
In summary, theoretical investigations of valley-selective quantum confinement in 2D materials have provided profound insights into the manipulation and control of valley degrees of freedom. Ab initio calculations have elucidated the optical selection rules, valley lifetimes, and external field responses in spatially confined systems. These studies establish a foundation for designing nanoscale valleytronic devices where quantum confinement serves as a powerful tool to tailor valley properties. Future theoretical work may explore more complex nanostructured geometries, heterostructures, and non-equilibrium valley dynamics to further advance this field.