Valley Optimization in Quantum & Energy Systems

• Energy Valley Optimization is the targeted engineering of local band minima to enhance quantum device performance and grid stability.
• It encompasses techniques like variational design for valley splitting in silicon heterostructures and electrically induced valley polarization in 2D materials.
• In power systems, it optimizes valley-filling schedules to balance load, reduce costs, and mitigate grid fluctuations.

Energy valley optimization encompasses a spectrum of optimization strategies across condensed matter physics, quantum electronics, and energy systems engineering. In electronic materials, "valley" denotes local minima in the conduction or valence band structure associated with distinct momentum states leveraged for information processing (“valleytronics”) or energy filtering. In complex energy systems and grid scheduling, "valley filling" refers to reshaping temporal load or energy flow profiles to minimize operational costs and grid instability. Across these areas, optimization techniques are developed to systematically manipulate, exploit, or stabilize valley-dependent phenomena via algorithmic, structural, or functional design.

1. Optimization of Valley Splitting in Quantum Materials

In silicon-based quantum computing, the valley degree of freedom in conduction band minima introduces nearly degenerate valley states whose splitting, termed "valley splitting," must be reliably controlled for robust qubit operation. The design challenge centers on maximizing and stabilizing the intervalley coupling under complex constraints imposed by heterostructure composition, interface roughness, and external fields.

A systematized variational optimization for the Si/SiGe quantum well heterostructures defines the valley splitting as $E_\text{VS}=2|\Delta|$, with $\Delta$ the intervalley coupling. The optimization functional $J[x,\psi_0,\chi,E_0,\kappa]$ incorporates (i) objectives—maximization of deterministic splitting, minimization of disorder-induced fluctuation, or their robustness ratio, (ii) state constraints from the effective-mass Schrödinger equation for the envelope function $\psi_0(z)$, and (iii) realistic profile limitations such as Ge budget and spectral low-pass filtering to preclude atomic-scale, technologically unfabricable oscillations (Thayil et al., 19 Dec 2025).

The approach produces optimal profiles such as the "modulated wiggle well," which features Ge concentration oscillations at a period resonant with the valley momentum difference, modulated in amplitude to coincide with the quantum dot’s electronic density maxima. This results in deterministic valley splittings exceeding $0.7$ meV with low variance and allows tunability from $\sim0.2$ meV to above $1$ meV via the applied vertical electric field. Classic heterostructure designs (Ge spikes, uniformly distributed Ge, sharp interface wells) are derived as limiting cases of the framework.

2. Valley Optimization in Valleytronic Devices

In Weyl and graphene-like two-dimensional materials, energy valleys offer discrete quantum numbers that can be manipulated for information encoding and thermoelectric enhancement. Optimization here entails achieving selective control or filtering of valley populations.

For tilted Weyl semimetals, electrically tunable valley polarization is theoretically realized in a gate-defined p–n–p junction with both an energy-dispersion tilt ($w_y$), and a localized magnetic field $B$. The optimization condition aligns the electric-field-induced valley shift $\chi w_y(E_F-U_0)/(\hbar v_F^2)$ with the Lorentz shift $\Delta$0, leading to nearly perfect valley polarization ($\Delta$1) upon electrical tuning of the barrier height $\Delta$2. Optimal device parameters are set by the resonant cancellation or amplification of these shifts, barrier width $\Delta$3, and field strengths, yielding electrically switchable valley filters suitable for valleytronic applications (Yesilyurt et al., 2017).

In strained honeycomb lattices, energy valleys coupled with spin degree of freedom (spin-valley locking) are optimized for thermoelectric energy conversion. Analytical expressions for the Seebeck coefficient and the thermoelectric figure of merit ($\Delta$4) are maximized as explicit functions of the gap $\Delta$5, Fermi energy $\Delta$6, and deformation parameters. The optimal relation $\Delta$7 with $\Delta$8 yields maximum $\Delta$9 for $J[x,\psi_0,\chi,E_0,\kappa]$0, with dynamic tunability achieved by strain $J[x,\psi_0,\chi,E_0,\kappa]$1 and perpendicular electric field $J[x,\psi_0,\chi,E_0,\kappa]$2 (Sengupta et al., 2019).

3. Energy Valley Filling in Power Systems Optimization

In energy systems, "valley filling" signifies the optimization of temporal energy flows to smooth demand profiles, thus minimizing generation costs and volatility in grid operations. Techniques target the temporal valleys—periods of low demand or energy price—to shift flexible loads (e.g., EV charging, thermal storage charging) into these intervals.

For electric vehicle fleet charging, the global joint optimization of power flow and individual charging constraints is addressed by decomposing the problem using convex duality. The optimal schedule is characterized as a valley-filling profile, which fills demand troughs with flexible loads and thereby reduces cost and grid stress. Decentralized algorithms dynamically track this profile with near-optimality ($J[x,\psi_0,\chi,E_0,\kappa]$3 relative error compared to the offline optimum) (Chen et al., 2012).

More generally, in electricity–heat systems integrating renewables and multiple thermal/electric storage devices, dynamic optimization is formulated as minimizing total system cost and a specifically penalized grid-purchase fluctuation term

$J[x,\psi_0,\chi,E_0,\kappa]$4

where $J[x,\psi_0,\chi,E_0,\kappa]$5 is the net grid purchase. An improved reinforcement learning algorithm (PVTD3) introduces an explicit penalty on grid-power variations to enforce valley filling. PVTD3 achieves up to 13.6% cost savings at high renewable penetration and reduces grid purchase fluctuation amplitude by $J[x,\psi_0,\chi,E_0,\kappa]$6 relative to standard TD3, while maintaining storage parameters within operational limits (Ye et al., 19 Nov 2025).

4. Mathematical and Algorithmic Formulations

Optimization in these domains is formalized through variational and control-theoretic problem statements. In quantum structures, the cost functionals include both deterministic intervalley couplings and statistical disorder measures, solved using finite-difference discretization and L-BFGS minimization under normalization, budget, and spectral constraints (Thayil et al., 19 Dec 2025).

For energy systems, the multi-period objective functions include energy cost, storage state penalties, and explicit fluctuation penalties, solved via model predictive control or advanced actor-critic reinforcement learning. In PVTD3, the Markov Decision Process state aggregates all relevant generation, load, and storage variables; the reward incorporates cost, state deviation, and grid fluctuation penalties. Training leverages scenario randomization and uncertainty modeling for robustness testing (Ye et al., 19 Nov 2025).

In valleytronic materials, band-structure optimization targets device-level figures of merit (valley polarization, thermoelectric $J[x,\psi_0,\chi,E_0,\kappa]$7) as analytical functions of tunable fields, geometric parameters, and materials properties, with optimization proceeding by extremizing these expressions under experimental constraints (Yesilyurt et al., 2017, Sengupta et al., 2019).

5. Comparative Performance and Experimental Feasibility

Performance outcomes across these domains are quantified by deterministic figures of merit (e.g., valley splitting, $J[x,\psi_0,\chi,E_0,\kappa]$8, valley polarization $J[x,\psi_0,\chi,E_0,\kappa]$9), operational cost reductions, and stability of device/system responses to randomness or uncertainty.

Domain	Optimized Metric	Tunability (Control)	Peak Performance	Reference
Si/SiGe quantum wells	Valley splitting	Electric field, Ge profile	$\psi_0(z)$0–$\psi_0(z)$1 meV (controllable)	(Thayil et al., 19 Dec 2025)
Tilted Weyl semimetal filter	Valley polarization $\psi_0(z)$2	Tilt $\psi_0(z)$3, gate $\psi_0(z)$4, $\psi_0(z)$5	$\psi_0(z)$6 (switchable)	(Yesilyurt et al., 2017)
Spin-valley thermoelectric	$\psi_0(z)$7, $\psi_0(z)$8	Strain $\psi_0(z)$9, $0.7$0	$0.7$1 up to $0.7$2	(Sengupta et al., 2019)
Smart grid/EV, e-heat systems	Cost, grid fluctuation	Load scheduling, storage	13.6% cost, 12.8% fluctuation reduction	(Ye et al., 19 Nov 2025, Chen et al., 2012)

Operational feasibility is routinely enforced via realistic constraints (epitaxial limitations, budget, operation ranges) and validated for a range of operating scenarios and uncertainties (e.g., ±30% load/renewable volatility in energy systems, disorder variance in quantum wells). Achievable and controllable performance metrics are explicitly traced to physically implementable device and system parameters.

6. Special Cases and Design Maps

Systematic optimization frameworks can recover existing heuristic designs as special cases by modulating the optimization objective or imposing additional constraints. In quantum wells, Ge spike and uniform Ge profiles, as well as conventional wiggle wells, are all derivable as limiting cases of the general variational protocol. The deterministic splitting versus disorder fluctuation map categorizes designs by both mean performance and variance, enabling targeted trade-off selection (Thayil et al., 19 Dec 2025). In energy scheduling, valley-filling penalties continuously interpolate between pure cost minimization and grid-stability regimes (Ye et al., 19 Nov 2025).

A plausible implication is that universal, tunable optimization frameworks enable the identification and realization of both known and novel architectures in valley-dependent devices and energy systems, supporting robust, scenario-specific engineering for future quantum and smart grid technologies.