Valley–Orbit Interaction in Quantum Devices

• Valley–orbit interaction is a coupling linking valley quantum numbers to orbital degrees of freedom in crystalline solids and engineered nanostructures.
• It arises from symmetry-breaking effects such as interface roughness, electrostatic gating, and moiré patterning, which modulate valley splitting and coherence.
• This phenomenon underpins gate-tunable quantum devices, valleytronics, and topological photonics, offering pathways for advanced quantum information processing.

Valley–orbit interaction (VOI) refers to a class of couplings in crystalline solids or engineered nanostructures that link the valley quantum degree of freedom—labeling inequivalent extrema in momentum space (e.g., K, K′ points in graphene or ±k₀ valleys in Si)—to the electronic or photonic orbital motion or other internal degrees of freedom. VOI is a fundamental phenomenon across multilayer two-dimensional materials, quantum dot heterostructures, and photonic crystals, and it underpins a spectrum of effects ranging from topological band structure to enhanced coherence properties in quantum devices.

1. Fundamental Mechanisms of Valley–Orbit Interaction

In semiconductors such as silicon or in gapped graphene, the valley index is an internal quantum number protected by crystal symmetry. VOI typically arises when spatial inhomogeneities—such as sharp interfaces, electrostatic gate potentials, or moiré superlattice modulations—break the perfect symmetry of the host crystal and generate valley-mixing terms. In single-electron systems, the generic effective-mass Hamiltonian in the two-valley basis is

$$H_{\text{valley-orbit}} = \frac{\Delta_v}{2}\tau_z + \Delta_{\text{vo}}\tau_x$$

where $\Delta_v$ is the valley splitting and $\Delta_{\text{vo}}$ is the valley–orbit coupling, with $\tau$ the valley Pauli matrices (Abraham et al., 2020, Dodson et al., 2021). In monolayer or bilayer graphene subject to a lateral electric field, the analogous term reads

$$H_{\rm VOI} = \tau\,\frac{\hbar}{4\,m^*\,\Delta}\,[\nabla V(\mathbf{r})\times\mathbf{p}]_z$$

establishing a direct analogy with the Rashba spin–orbit interaction but acting on the valley pseudospin (Lee et al., 2012, Wu et al., 2012).

In emergent moiré systems, such as hybrid excitons in twisted TMD bilayers, VOI is realized via long-range electron–hole exchange (Förster coupling), generating both valley-conserving and valley-flip hopping channels (Zheng et al., 2024).

2. Interface- and Disorder-Induced Valley–Orbit Coupling in Silicon Devices

The most prominent microscopic origin of VOI in silicon quantum dots is the interface between Si and SiO₂ or SiGe. The abrupt potential change at the interface produces a matrix element between the ±k₀ valley Bloch functions proportional to the local interface profile. Atomic-scale steps or roughness manifest as spatial variations: $$\Delta_D = |\Delta_D|e^{i\phi_D} \approx \Delta_0 \iint |\phi_D(x,y)|^2 e^{-2ik_0 \zeta(x, y)} dx\,dy$$ where $\zeta(x,y)$ encodes the interface height (Tariq et al., 2019, Zimmerman et al., 2016). Both the magnitude $|\Delta_D|$ and the phase $\phi_D$ are strongly modulated by sub-nm steps; single monolayer steps can reduce $|\Delta|$ by up to 75% and yield phase shifts up to nearly $\Delta_v$0, with two steps capable of cancelling the coupling entirely (Tariq et al., 2019).

Such interface-driven randomness directly impacts the tunnel coupling and exchange energy in double quantum dot architectures, causing highly sample- and device-dependent valley physics (Gamble et al., 2013, Tariq et al., 2021).

3. Manifestations and Measurement of Valley–Orbit Coupling

Valley–orbit coupling modifies both the energy spectrum and observable physical properties:

• In silicon donors (e.g., Mg₁⁰), valley–orbit splitting of the 1s manifold is directly measured by FTIR absorption and thermal activation to higher-energy orbitals (Abraham et al., 2020).
• In quantum dots, VOI leads to valley–orbit mixed eigenstates, nonzero electric dipole moments (rendering valleys sensitive to charge noise), and altered intervalley tunneling, as shown by configuration interaction (CI) and full CI approaches (Dodson et al., 2021, Yannouleas et al., 2022).
• Gate voltages or lateral displacements can be used to control the phase $\Delta_v$1, inducing coherent electrical rotations between valley eigenstates observable as Rabi oscillations in double-dot exchange spectroscopy (Wu et al., 2012, Zimmerman et al., 2016).
• In multi-electron dots, VOI is essentially a single-particle effect and not renormalized by electron–electron interactions, as established both theoretically and by addition-energy spectroscopy (Jiang et al., 2013).

4. Valley–Orbit Coupling and Topological Photonic/Moiré Systems

The interplay of valley and orbital (or orbital-like) degrees of freedom extends to synthetic and photonic lattices:

• In photonic valley crystals, inversion-symmetry breaking produces valley–OAM locking: the valley index (K, K′) corresponds to photonic Bloch modes carrying opposite orbital angular momentum $\Delta_v$2. An effective massive Dirac Hamiltonian links the valley pseudospin to the OAM eigenvalue, yielding topological edge modes characterized by a nonzero valley Chern number (Chen et al., 2016).
• In twisted bilayer TMDs configured as hybrid moiré exciton superlattices, Förster exchange produces strong, spatially nonlocal VOI. The unique aspect is the presence of both valley-conserving and valley-flip hopping amplitudes between traps separated by $\Delta_v$310 nm or more, even when single-particle kinetic propagation is locally quenched. The resultant tight-binding Hamiltonian includes both Peierls-like valley-dependent phases and pseudo-spin-orbit coupling terms (Pauli matrices $\Delta_v$4) with spatial anisotropy (angle-dependent hopping) (Zheng et al., 2024).

The combination of these interactions yields a nontrivial topological band structure with band inversion and edge modes. The system enters the quantum-valley-Hall regime when the ratio $\Delta_v$5 of staggered on-site energy to hopping satisfies a band inversion criterion, controlled by the strength of the Förster-induced VOI (Zheng et al., 2024).

5. Valley–Orbit Interaction in Graphene: Gate Control and Device Applications

In gapped graphene and related van der Waals structures:

• The VOI emerges as a first-order correction in the effective mass, proportional to lateral electric fields and tunable via top- and side-gate voltage (Lee et al., 2012, Wu et al., 2012). This enables coherent control of the valley index analogous to gate-controlled spin precession in Datta–Das FETs.
• Device architectures exploiting VOI include the valley FET, where the conductance is modulated by the gate-controlled valley-precession angle, and valley-pair qubits in coupled quantum dots, all realized in structures where intervalley scattering is suppressed by edge or substrate engineering (Wu et al., 2012).
• The proximity effect of TMD islands on graphene (e.g., via C₃ᵥ-symmetric quantum dot arrays) induces both sublattice-resolved staggered intrinsic spin–orbit and Rashba couplings. This generates controllable valley-selective gaps (valley–Zeeman effect) and enables gate-tunable switching between valley polarization states and localized valley-resolved bound modes. The strength of the induced parameters is further controlled by the twist angle in graphene–TMD heterostructures (Belayadi et al., 2024).

6. Topological and Quantum Information Implications

VOI has profound implications for qubit design, valleytronic devices, and topological states:

• In silicon qubits, disorder-induced VOI degrades valley-based qubit coherence by generating finite dipole moments and intervalley tunneling amplitudes; it can, however, be mitigated by operating at specific "sweet spots," reducing dot size, or interface engineering (Gamble et al., 2013, Mi et al., 2018).
• In valleytronic approaches, the gate-tunability and robustness of VOI allow for efficient, low-power switching, long coherence times, and optical-to-valley conversion suitable for quantum communications (Wu et al., 2012).
• In moiré hybrid exciton lattices, the competition between valley–orbit coupling and superlattice-induced mass terms produces a rich topological phase diagram, including quantum-valley-Hall phases with symmetry-protected dark exciton states (Zheng et al., 2024).
• Experimental detection of QVH edge modes and valley-dependent photoluminescence intensity, as well as dynamical signatures such as pump–probe detection of valley–orbit oscillations, are direct consequences of strong VOI.

7. Symmetry and Design Engineering of Valley–Orbit Couplings

The magnitude and qualitative properties of VOI are dictated by local and global symmetry breaking. Curvature in nanotubes (both carbon and silicon) robustly produces valley-contrasting spin–orbit coupling by breaking reflection and inversion symmetry, with coupling constants scaling as $\Delta_v$6 and adjustable by the chirality or the bending axis (Yamakage et al., 2023). In photonic and moiré metastructures, symmetry underpins both selection rules for valley–orbit couplings and the existence of protected topological bands (Chen et al., 2016, Zheng et al., 2024).

Careful engineering—controlling interface flatness, twist angle, electrostatic profile, and superlattice design—enables the tuning of valley–orbit interaction for specific applications ranging from topological photonic transport to robust quantum information processing.