Unifying recent experiments on spin-valley locking in TMDC quantum dots

Technical Summary: Unifying Recent Experiments on Spin-Valley Locking in TMDC Quantum Dots

Problem Statement
Transition metal dichalcogenide (TMDC) quantum dots (QDs) offer a promising platform for spin-valley (Kramers) qubits due to strong intrinsic spin-orbit coupling (SOC), which theoretically enables long spin-valley lifetimes. While recent experiments have provided evidence of spin-valley locking in the few-electron limit, critical parameters remain ambiguous. Specifically, the assignment of specific spin-valley eigenstates and the relative magnitudes of spin ($g_s$) and valley ($g_v$) g-factors have been uncertain. Furthermore, existing experimental data from different groups often report conflicting or incomplete effective g-factors, lacking a unified theoretical framework to reconcile these findings across different device geometries, layer numbers, and magnetic field orientations.

Methodology
The authors employ a unified theoretical framework combining quantum transport theory with numerical simulations to reverse-engineer experimental transport spectroscopy data.

• Model: The system is modeled as a 2D single quantum dot (SQD) with a hexagonal Brillouin zone. The Hamiltonian includes four discrete single-particle states representing electrons localized at the $K$ and $K'$ valleys with spin up/down.
• Key Parameters: The model incorporates intrinsic SOC ($\Delta_{so}$), inter-valley mixing ($t_v$), spin and valley Zeeman effects (governed by $g_s$ and $g_v$), and Coulomb repulsion ($U_C$).
• Simulation Technique: The authors utilize density matrix rate equations within the Fock space of the generalized Hamiltonian. They simulate both ground-state and excited-state transport spectroscopy, tracking transport channels ($\epsilon_{ij}^N$) between adjacent $N$-particle Fock spaces.
• Data Integration: The simulation platform is calibrated to reproduce data from three distinct recent experiments: Devidas et al. (2021), Kumar et al. (2023), and Krishnan et al. (2023). By fitting the simulated energy spectra to the experimental Coulomb diamonds and Zeeman splitting data under varying in-plane ($B_{\parallel}$) and out-of-plane ($B_{\perp}$) magnetic fields, the authors extract the underlying physical parameters.

Key Contributions and Results
The paper successfully unifies disparate experimental results by extracting a consistent set of parameters for MoS$_2$ quantum dots:

1. Parameter Extraction:

   • Devidas et al. (2021): By fitting excited state spectroscopy under in-plane fields, the authors extracted $\Delta_{so} = 100$ $\mu$eV, $t_v = 15$ $\mu$eV, and $g_s = 1.88$.
   • Kumar et al. (2023): By analyzing excited state spectroscopy under out-of-plane fields, the authors resolved the discrepancy in reported effective g-factors ($g_{\perp} = 5.2$ and $3.5$). They demonstrated that these values arise from different electron occupation numbers ($N=1, 2, 3$) in the Coulomb diamonds, yielding consistent intrinsic parameters: $g_v = 3.5$ and $g_s = 1.7$, with a lower bound for SOC of $\Delta_{so} \geq 1.52$ meV.
   • Krishnan et al. (2023): This dataset, featuring both in-plane and out-of-plane fields, allowed for the extraction of all four key parameters: $\Delta_{so} = 852$ $\mu$eV, $t_v = 65$ $\mu$eV, $g_s = 2.2$, and $g_v = 5$.

2. Unified Trends:

   • The analysis confirms that $\Delta_{so}$ decreases with the number of MoS$_2$ layers, consistent with theoretical expectations.
   • The study validates the existence of inter-valley mixing ($t_v$) and its role in anti-crossing energy states under out-of-plane magnetic fields.
   • The authors define threshold magnetic fields ($B_{t1}$ and $B_{t2}$) that delineate operating regions for spin-valley qubits, valley qubits, and spin qubits.

3. Comparison with Bilayer Graphene (BLG):

   • The authors compare MoS$_2$ with BLG platforms. They find that MoS$_2$ exhibits spin-valley locking that is one to two orders of magnitude stronger than BLG, characterized by a significantly higher threshold field $B_{t1}$ (2.9 T for MoS$_2$ vs. 76 mT for BLG).
   • The energy gap at the valley anti-crossing in MoS$_2$ ($\approx 130$ $\mu$eV) is an order of magnitude larger than in BLG, making valley qubit operations more feasible in TMDCs.

Significance and Claims
The paper claims to provide a "predictive modeling tool" for TMDC quantum dots at the few-electron limit. Its primary significance lies in:

• Unification: It weaves a connection between intrinsic material properties and experimental data under diverse conditions, resolving uncertainties regarding spin and valley g-factors in previous studies.
• Extraction Capability: It demonstrates that by tracking specific transport channels and magnetic field dependencies, one can deduce the degree of spin-valley locking, SOC strength, and inter-valley mixing.
• Future Milestone: The authors position their work as an important milestone toward the next challenge in the field: experimentally confirming valley-relaxation times using single-shot projective measurements. They argue that the larger energy gaps and stronger locking in TMDCs make them a more suitable platform for such measurements compared to BLG.

The authors maintain a modest tone regarding applications, focusing strictly on the validation of their theoretical framework against existing experimental data and its utility in guiding future transport spectroscopy experiments to measure relaxation times.