Electrically tunable MoSe$_2$/WSe$_2$ heterostructure-based quantum dot

Technical Summary: Electrically Tunable MoSe2/WSe2 Heterostructure-based Quantum Dot

Problem Statement
Transition metal dichalcogenide (TMDC) monolayers offer rich physics, including spin-valley locking, but their electronic properties are often limited to single-layer constraints. Van der Waals (vdW) heterostructures, such as MoSe2/WSe2, provide a platform to engineer electronic properties beyond monolayers, particularly for quantum information applications requiring the initialization and control of individual quantum states. While optical methods can populate specific valleys, electrical control over valley degrees of freedom in TMDC-based quantum dots (QDs) remains a critical area of investigation. Previous theoretical works have largely focused on K-valley states; however, the potential to control different valley minima (specifically K and Q valleys) using external electric fields in MoSe2/WSe2 heterostructures requires a comprehensive theoretical framework. A significant gap exists in atomistic-level models that describe interactions between two different TMDC monolayers in a full orbital basis, accounting for both metal ($d$) and chalcogen ($p$) orbitals.

Methodology
The authors employ a multi-stage theoretical approach combining density functional theory (DFT) and an ab initio-based tight-binding (TB) model:

1. DFT Analysis: The electronic structure of the MoSe2/WSe2 heterostructure (AB stacking) is determined using DFT. This establishes the band alignment, spin-orbit coupling (SOC) gaps, and the energetic proximity of conduction band (CB) minima at the K and Q points, as well as valence band (VB) maxima at K and $\Gamma$ points. The study analyzes layer, spin, and orbital contributions using Kohn-Sham wavefunctions.
2. Tight-Binding Model Construction: To facilitate the study of nanostructures, the authors construct a full orbital TB model. The basis includes metal $d$ orbitals ($m = \{0, \pm 1, \pm 2\}$) and chalcogen $p$ orbitals ($m = \{0, \pm 1\}$) for both even and odd subspaces relative to the metal planes. The model incorporates spin-orbit coupling and interlayer interactions up to next-nearest neighbors (NNN), including metal-metal, chalcogen-chalcogen, and metal-chalcogen couplings.
3. Parameterization: The TB parameters (Slater-Koster integrals and SOC strengths) are fitted to the DFT results using a differential evolution method. The loss function minimizes energy differences between TB and DFT while enforcing correct spin ordering.
4. Electric Field Simulation: A vertical electric field ($E_z$) is introduced into the TB model via a potential term that accounts for dielectric screening and the potential drop across the heterostructure layers.
5. Quantum Dot Modeling: A laterally gated quantum dot is modeled within a computational rhombus using the derived TB Hamiltonian. The QD is defined by a lateral Gaussian confining potential. The Schrödinger equation is solved in the basis of bulk Bloch functions to determine the QD energy spectrum and wavefunctions under varying vertical electric fields.

Key Contributions and Results

• Electronic Structure Characterization: The DFT analysis confirms a type-II band alignment where the CB minimum originates from the MoSe2 layer and the VB maximum from the WSe2 layer. Crucially, the CB minima at the K and Q points are found to be energetically close (separated by only 1.8 meV), while VB maxima at K and $\Gamma$ are separated by 35.5 meV. The study details the orbital composition, noting that K-valley states are strongly layer-localized, whereas Q-valley states exhibit significant interlayer delocalization.
• Validation of the TB Model: The constructed TB model successfully reproduces the DFT electronic structure, including the type-II alignment, spin-layer ordering, and the degeneracy of K and Q valleys. It captures the microscopic details of spin and layer contributions, including the "dark" nature of direct K-K transitions and "bright" indirect K-Q transitions due to spin selection rules.
• Electric Field Control of Valleys: The application of a vertical electric field ($E_z$) is shown to tune the energetic ordering of the valleys.
  • Negative $E_z$: Favors the K valley for the CB minimum.
  • Positive $E_z$: Favors the Q valley for the CB minimum.
  • The field also modulates the layer character of the states; for instance, the K-point CB state, initially localized in MoSe2, can become strongly admixed with the WSe2 layer at high fields. The Q valley states remain delocalized but their spin-splitting characteristics evolve with the field.
• Quantum Dot Spectral Tuning: The study demonstrates that the valley character of the low-energy states in a laterally gated QD can be selectively controlled by the vertical electric field.
  • K-valley Configuration: Under a negative field, the lowest QD shell is 2-fold degenerate (corresponding to the K and K' valleys).
  • Q-valley Configuration: Under a positive field, the lowest QD shell becomes 6-fold degenerate (corresponding to the six non-equivalent Q points surrounding the K points).
  • The wavefunctions transition from being confined within K/K' valleys to forming superpositions across the Q valleys. Higher energy shells exhibit alternating degeneracies (2-fold vs. 6-fold) corresponding to the underlying valley character of the bulk bands.

Significance
The paper establishes a theoretical framework for understanding and controlling valley degrees of freedom in MoSe2/WSe2 heterostructures. By demonstrating that a vertical electric field can switch the ground state of a quantum dot between K-valley (2-fold) and Q-valley (6-fold) configurations, the work highlights the potential for electrical control over valley occupancy. This tunability is presented as a mechanism for engineering specific degeneracies and wavefunction characteristics in QDs, which is essential for the development of valley-based qubits and other quantum information platforms. The authors note that while previous works focused on K-valley states, their model provides a pathway to access and control the Q-valley physics, which has been less explored in the context of electrical gating in these heterostructures.