Valley-Exchange Correlation Physics

• Valley-exchange correlation physics is the study of how multivalley electronic wavefunctions interact via Coulomb exchange and correlation effects in semiconductors.
• It reveals valley-dependent exchange splitting and anisotropy in quantum dots, influencing excitonic fine structure and spin–valley interactions critical for quantum technologies.
• Advanced techniques such as STM imaging and microwave spectroscopy enable precise mapping and control of valley-induced quantum phenomena.

Valley-exchange correlation physics encompasses the interplay between electronic wavefunction structure in momentum (valley) space and exchange-correlation effects arising from Coulomb interactions, atomic-scale symmetry, and quantum confinement. It is central to the quantum many-body physics of multivalley semiconductors (e.g., Si, PbX, TMDs, graphene) and governs a wide range of quantum phenomena from the fine structure of excitons to the controllability of spin-qubit interactions and the emergence of novel correlated ground states. The "valley" degree of freedom, associated with distinct Brillouin-zone extrema, acts as a pseudospin that can participate in, or mediate, exchange and correlation processes, giving rise to physics far richer than in single-valley electronic systems.

1. Valley Structure and Exchange in Multivalley Materials

In canonical multivalley systems, the conduction- and/or valence-band extrema occur at multiple symmetry-inequivalent $k$-points (often labelled as valleys such as $K/K'$ in TMDs or $L$ in PbX). Each valley hosts a (pseudo)spin degeneracy, leading to an enhanced Hilbert space for low-energy quasiparticles. In Si, the six-fold conduction-band minima are along $\langle100\rangle$; in PbX quantum dots (QDs), conduction and valence bands have extrema at four $L$ points with twofold Kramers degeneracy; in monolayer TMDs, $K$ and $-K$ valleys carry locked spin-$1/2$ projections.

The presence of multiple valleys causes the exchange interaction to be valley-dependent. Exchange splitting—ubiquitous in quantum dots (PbS, PbSe), donor arrays (Si:P), or excitons (TMDs)—depends not only on spatial and spin configurations, but also on valley composition, phases, and couplings. The valley degree of freedom can also participate in nontrivial correlations, e.g., $\mathbb{Z}_2$ gauge structures in silicon quantum dots (Woods, 3 Mar 2025), valley-mixed collective modes in twisted bilayer graphene (Wolf et al., 2020), or valley-driven topological transitions in ferrovalley systems (Hu et al., 2020).

2. Valley-Resolved Exchange Hamiltonians and Anisotropy

The valley structure introduces both intravalley and intervalley exchange channels, with critical consequences for quasiparticle spectra and dynamics.

• Lead Chalcogenide Quantum Dots (PbS, PbSe):

The effective intravalley exchange Hamiltonian for the ground exciton state, in the valley-localized basis $|c,\mu,\eta\rangle \otimes |v,\mu,\eta'\rangle$ (with $\mu$ labeling $L$-points), reads (Avdeev et al., 2023):

$$\hat H_{\mathrm{exch}} = K_s\,\frac{\mathbb{1}}{2} - K_t\,\frac{\sigma_x^*\,\sigma_x+\sigma_y^*\,\sigma_y}{6} - K_l\,\frac{\sigma_z^*\,\sigma_z}{6}$$

where $\sigma_i^*$ acts on the valence pseudospin, $\sigma_i$ on the conduction pseudospin, and $(K_s,K_t,K_l)$ transform according to $D_{3d}$ irreps.

• Silicon Donor and Quantum Dot Systems:

The exchange Hamiltonian incorporates valley indices in superexchange pathways (Salfi et al., 2017):

$$J(\mathbf{R}) \simeq \sum_{\mu,\nu} |S_{\mathrm{D}\mu;\mathrm{Q}\nu}(\mathbf{R})|^2 E_{\mu\nu}(\mathbf{R})$$

where $S_{\mathrm{D}\mu;\mathrm{Q}\nu}$ is the valley-resolved overlap. Valley filtering, e.g., selecting $\pm z$ valleys in an interface quantum dot, suppresses exchange oscillations and endows the donor–QD system with nearly monotonic, compensable $J$.

• Determinants of Exchange Anisotropy:

In PbS QDs, the exchange anisotropy parameter $\eta_X(D)=[K_\ell(D)-K_t(D)]/K_s(D)$ is large and negative, approaching $-0.78$ for large $D$, while in PbSe QDs, it is positive and smaller ($+0.69$), reflecting the underlying momentum matrix element anisotropy. This leads to large fine-structure splittings in PbS (up to $>50\mu$eV in small QDs), but nearly degenerate levels in sufficiently large PbSe QDs (Avdeev et al., 2023).

3. Valley-Exchange Correlation: Beyond Single-Particle Exchange

Valley-exchange correlation encompasses collective, nonlocal, or many-body effects beyond simple two-particle exchange.

• Negative Exchange via Valley-Phase-Induced $\mathbb{Z}_2$ Gauge Fields:

In Si dot arrays, valley phase differences between dots, $\phi_i - \phi_j$, modulate the effective tunneling and, in closed loop geometries (e.g., triangles), impart a $\mathbb{Z}_2$ gauge field. The ground-valley tunneling acquires a sign, and the total flux around a plaquette $$\Phi_C = \prod_{\langle ij\rangle \in C}\mathrm{sgn}[t'_{ij}]$$ can be nontrivial, allowing the exchange $J$ to become negative even without a magnetic field (Woods, 3 Mar 2025). This breaks the constraints of conventional antiferromagnetic Heisenberg systems (Nagaoka’s theorem), directly influencing quantum magnetism.

• Spin-Valley Entanglement and Fidelities:

In exchange-coupled Si quantum dots, finite valley splittings and nontrivial valley-phase relations cause spin-valley leakage. Two-qubit operations are resilient only for $\Delta_v/J_0 \gtrsim 10$ and $\phi_i \approx \phi_j$; in larger arrays, residual entanglement accumulates and reduces overall gate fidelity (Buterakos et al., 2021).

• Collective Valley-Pseudospin Ordering:

In twisted bilayer graphene encapsulated by magnetic van der Waals materials, strong correlations in a quasi-flat band regime lead to spontaneous valley spiral states described by an effective easy-plane pseudospin-1/2 Heisenberg model with DM and anisotropy terms (Wolf et al., 2020). The interlayer bias can tune the valley–orbit coupling, controlling the ground state between spiral and valley-polarized phases.

4. Valley-Exchange Physics in Excitonic and Optical Phenomena

Valley-exchange correlation governs the coupling between exciton states in different valleys, with prominent manifestations in two-dimensional materials and nanocrystals:

• Long-Range Exchange and Valley Depolarization:

In monolayer MoS$_2$ and TMDs, the intervalley long-range electron–hole exchange interaction mixes bright A excitons at $K$ and $K'$, leading to valley depolarization through the Maialle–Silva–Sham mechanism (Yu et al., 2013). The effective Hamiltonian for the valley-pseudospin takes the form:

$$H_{\mathrm{exch}}(\mathbf{P}) = \Omega_x(\mathbf{P})s_x + \Omega_y(\mathbf{P})s_y$$

where $\Omega$ is set by the long-range exchange and depends linearly on the exciton COM momentum $|\mathbf{P}|$. For large momentum, depolarization is ultrafast (tens of fs); for thermalized excitons, residual polarization survives on $\sim$ps timescales.

• Angle-Resolved Probing of Valley-Exchange Coupling:

The intervalley exchange interaction $J(\mathbf{k})$ in TMDs causes a nonanalytic linear-in-$k$ splitting of exciton branches, observable via angle-resolved PL (Thompson et al., 2021). Emission intensity as a function of emission angle and polarization reveals fingerprints of valley-exchange—for example, "teardrop" angular intensity profiles for longitudinal versus transverse branches, and B-field induced mixing effects shifting the relative intensities between circular polarizations.

• Exchange-Correlation in 2D Antiferromagnets:

In MnPS$_3$, strong spin–valley coupling, inverted valence-band dispersion, and AFM order yield massive ($\sim$0.5–0.7 eV) bright–dark exciton exchange splitting, an order of magnitude larger than in TMDCs. The excitonic fine structure and valley relaxation pathways are determined by the interplay between exchange splitting, spontaneous valley polarization, and magnetic order (Wang et al., 21 Apr 2025).

5. Techniques for Probing and Engineering Valley-Exchange Correlations

Advanced experimental and computational techniques have enabled the detection, control, and engineering of valley-exchange physics:

• Mapping Valley Coupling and Phase:

Microwave spectroscopy of Si triple quantum dots yields both intra- and inter-valley tunnel couplings ($t_{ij}, t_{ij}'$). The inter-valley tunneling ratios $t_{12}'/t_{12} = 0.90$, $t_{23}'/t_{23} = 0.56$ correspond to substantial spatial variation in valley phase differences ($\delta\phi_{12} \approx 84^\circ$, $\delta\phi_{23} \approx 58^\circ$), demonstrating the critical role of atomic-scale disorder (Borjans et al., 2021).

• STM Imaging/Valley Filtering:

STM techniques resolve the valley interference in real space and Fourier space, and "valley filtering" in hybrid donor–QD systems can effectively suppress exchange oscillations arising from x/y valleys, stabilizing $J$ for quantum information tasks (Salfi et al., 2017, Voisin et al., 2021).

• Optical Manipulation in 2D Materials:

Higher-order optical vortex beams allow direct tuning of the exciton COM momentum $\mathbf{Q}$ to paper the $Q$-dependence of the valley-exchange-induced depolarization, with $Q^2$ scaling of polarization relaxation rates observable in steady-state PL (Pattanayak et al., 2022).

6. Impacts and Outlook

Valley-exchange correlation physics is central for several applications and phenomena:

• Quantum Information: Design of robust exchange gates and scalable qubit arrays requires engineered valley splitting, tuned phase coherence, and understanding of valley-induced error mechanisms (Buterakos et al., 2021, Tariq et al., 2021).
• Quantum Magnetism and Topological Phases: Valley degrees of freedom enable negative exchange, $\mathbb{Z}_2$ gauge fields, controllable Nagaoka violations, and field-tunable easy-plane or spiral ground states (Woods, 3 Mar 2025, Wolf et al., 2020).
• Optoelectronics and Valleytronics: Valley-exchange determines the polarization, coherence, and relaxation of excitonic emission in TMDs and 2D magnets, underpinning valleytronic device concepts and quantum optical control (Yu et al., 2013, Thompson et al., 2021, Wang et al., 21 Apr 2025).
• Materials-by-Design: Ferrovalley and half-valley-metal states, driven by strong exchange–correlation effects, exhibit full valley and spin polarization with tunable topological responses (quantized conductivity, Chern number) (Hu et al., 2020).

Valley-exchange correlation remains a frontier for discoveries in solid-state quantum information, 2D materials science, and collective electronic phenomena, with ongoing research focused on engineering valley mixing, suppressing decoherence, and realizing new correlated ground states.