Thermodynamic Locking Criterion (I_lock)

• Thermodynamic Locking Criterion is defined as the ratio of directional energy gains to isotropic energetic costs, predicting a phase transition when I_lock exceeds unity.
• It integrates fundamental parameters from quasi–vdW epitaxy and mode-locked laser models to determine when a system transitions from a free, degenerate state to a macroscopically ordered one.
• The framework supports a two-tiered computational approach that couples semi-empirical predictors with rigorous DFT calculations, enabling accurate material screening and phase control.

The thermodynamic locking criterion, denoted $I_{\rm lock}$, is a rigorous quantitative measure that determines whether a physical system will undergo a transition from a freely fluctuating or degenerate phase to a macroscopically ordered, "locked" phase. The concept has independent foundations in distinct branches of condensed matter and nonlinear optics, including the theory of quasi–van der Waals (vdW) epitaxy in layered heterostructures and the statistical mechanics of mode-locked lasers. In both domains, $I_{\rm lock}$ prescribes the threshold at which the free energy gain from directional interactions overcomes the isotropic or entropic costs, yielding a predictive and experimentally validated criterion for orientation selection or phase coherence in complex systems (Liang et al., 26 Dec 2025, Antenucci et al., 2014).

1. Mathematical Formulation of the Thermodynamic Locking Criterion

In quasi-vdW epitaxy, the thermodynamic locking criterion separates "free" (rotation-permissive) and "locked" (orientation-constrained) growth regimes. The precise form of $I_{\rm lock}$ depends on the system geometry:

For 3D-on-2D (orientation locking):

$$I_{\rm lock}^{(3D/2D)} = \frac{\Delta Y_{es} + \Delta Y_{chem}}{\Delta Y_{sur} + \Delta E_{strain} + \Delta Y_{vdW}}$$

where:

• $\Delta Y_{es}$ = electrostatic coupling energy gain,
• $\Delta Y_{chem}$ = chemical bonding energy gain,
• $\Delta Y_{sur}$ = surface energy penalty (non-ideal facet),
• $\Delta E_{strain}$ = elastic strain energy,
• $\Delta Y_{vdW}$ = vdW adhesion energy difference.

For 2D-on-3D (rotational locking):

$$I_{\rm lock}^{(2D/3D)} = \frac{|\Delta\gamma(0)|}{\Delta Y_{es} + \Delta Y_{chem}}$$

where $|\Delta\gamma(0)|$ is the amplitude of the vdW binding energy ripple with respect to in-plane rotation.

In both cases, the locking transition is predicted to occur when: $I_{\rm lock} > 1$

Within the statistical mechanics of mode-locking lasers, an analogous criterion is formulated in terms of the pump intensity and system nonlinearities: $$I_{\rm lock} = P_c \sqrt{\frac{J\,\epsilon^2}{k_B T}}$$ where $J$ is the net nonlinear coupling strength, $\epsilon$ is the per-mode energy, $k_B$ is Boltzmann's constant, $T$ is the effective noise temperature, and $P_c \approx 1.567$ is a universal threshold in the mean-field regime (Antenucci et al., 2014).

2. Physical Interpretation of Constituent Terms

All forms of $I_{\rm lock}$ operationalize the balance between anisotropic interfacial gains and isotropic energetic costs:

• Electrostatic Energy Gain ($\Delta Y_{es}$): Results from directional interfacial dipoles or net charges, favoring registry between film and substrate.
• Chemical Bonding Gain ($\Delta Y_{chem}$): Quantifies the strength of new local chemical bonds in the locked configuration.
• Surface Energy Penalty ($\Delta Y_{sur}$): Indicates the energetic disadvantage of exposing a high-energy facet unavoidable in some locked arrangements.
• Strain Energy ($\Delta E_{strain}$): Accounts for elastic penalties imposed by forced lattice matching; critical in rigid 3D films.
• vdW Adhesion ($\Delta Y_{vdW}$): Small, isotropic component; generally negative but favors neither orientation.
• Rotational vdW Ripple ($\Delta\gamma(0)$): Tiny, orientation-dependent modulation in 2D/3D lattices that resists in-plane locking.

In laser models, the analogous parameters encode nonlinear coupling robustness ($J$), per-mode stored energy ($\epsilon$), and spontaneous emission bath ($T$).

3. Thermodynamic Derivation and Underlying Assumptions

The locking criterion emerges from a minimization of total free energy, $G=U-TS$, between locked and unlocked (free) states. In quasi-vdW systems, the change in Gibbs free energy simplifies to

$$\Delta G = G_{\rm locked} - G_{\rm free} \approx \Delta Y_{sur} + \Delta E_{strain} + \Delta Y_{int}$$

with $\Delta Y_{int}$ combining isotropic vdW and anisotropic directional gains. The critical ratio $I_{\rm lock}$ arises naturally from this decomposition (Liang et al., 26 Dec 2025).

Key assumptions include:

• Thermodynamic equilibrium (sufficiently slow growth or slow mode dynamics).
• Neglect of configurational entropy in all but rotational or surface terms.
• Explicit computation of all energies via first-principles DFT with dispersion corrections (PBE+D3), basis-set counterpoise corrections, and large supercells (≤5% mismatch) in the case of heterostructures.
• Canonical ensemble and spherical-spin constraints in the laser mapping; large-$N$ mean-field limit and frequency-matching quartets for network topology in theory (Antenucci et al., 2014).

4. Locking Thresholds and Predictive Power

The strict threshold $I_{\rm lock}=1$ demarcates qualitative behavior:

$I_{\rm lock}$ Value	Predicted Regime	System Behavior
$<1$	Free/degenerate	Lowest-energy facet or continuous in-plane rotation
$>1$	Locked/orientation-selected	Fixed orientation, discrete domains, phase coherence

Empirically, all "locked" systems (e.g., STO(111)/mica, MoS$_2$/sapphire, Fe$_4$N(111)/MoS$_2$) present $I_{\rm lock}>1$; all known "free" cases (STO(001)/HOPG, MoS$_2$/STO(001), Fe$_4$N(001)/mica) exhibit $I_{\rm lock}<1$ (Liang et al., 26 Dec 2025).

In mode-locked lasers, $I_{\rm lock}$ as the critical pump intensity $P_c$ robustly marks the transition between continuous wave and phase-coherent, mode-locked states, with the universality of $P_c$ preserved across realistic interaction networks (Antenucci et al., 2014).

5. Integration with Fast-Screening Predictors

The computational cost of evaluating all $I_{\rm lock}$ terms motivates a two-tiered workflow:

• Tier 1: Semi-empirical predictive index ($I_{\rm pre}$), rapidly estimated from surface-potential steps ($P_{\rm coupling}$) and adsorbate affinity ($C_{\rm affinity}$):

$$I_{\rm pre} = a\,P_{\rm coupling} + b\,C_{\rm affinity}\,, \quad \text{with } a:b = 1:4$$

Material pairs with $I_{\rm pre} \gtrsim 20$ are strong candidates for $I_{\rm lock}>1$ and thus locking.

• Tier 2: Rigorous $I_{\rm lock}$ determination with explicit DFT or ensemble simulations.

This pipeline underpins unified phase diagrams displaying $I_{\rm lock}$ and $I_{\rm pre}$ for diverse 2D–3D and 3D–2D combinations, establishing robust, quantitative correspondence to experimental reality (Liang et al., 26 Dec 2025).

6. Illustrative Case Studies

Empirical and first-principles data confirm the predictive capability of $I_{\rm lock}$:

• STO(111)/mica (locked): $\Delta Y_{es}+\Delta Y_{chem}\approx0.334$ eV/Å$^2$; Denominator $\approx$0.313 eV/Å$^2$; $I_{\rm lock}^{(3D/2D)}\approx2.3-3.0 \gg 1$.
• STO(001)/HOPG (free): $\Delta Y_{es}\approx0$, $\Delta Y_{chem}\approx0$; $I_{\rm lock}\approx0$.
• MoS$_2$/sapphire (locked 2D/3D): $\Delta Y_{es}+\Delta Y_{chem}\approx73$ meV, $\Delta\gamma(0)\approx69$ meV; $I_{\rm lock}^{(2D/3D)}\approx1.06>1$.
• MoS$_2$/STO(001) (free): $I_{\rm lock}\approx0.2<1$.
• Fe$_4$N(111)/MoS$_2$ (locked): $I_{\rm lock}\approx1.12>1$.

All systems predicted to be locked by $I_{\rm lock}$ exhibit strict orientation registry or restricted domain multiplicity in experiment; those below threshold remain degenerate or randomly oriented (Liang et al., 26 Dec 2025).

In mode-locked laser systems, simulated energy per mode and phase coherence order parameters undergo a marked jump at pump parameter $P_c\approx1.56$, consistent across fully connected and dilute frequency-comb networks (Antenucci et al., 2014).

7. Significance and Scope of the Criterion

The thermodynamic locking criterion $I_{\rm lock}$ provides a unified theoretical foundation for understanding the emergence of discrete orientation locking and phase coherence in systems governed by competing isotropic and directional energy scales. It resolves longstanding contradictions in the interpretation of quasi-vdW epitaxy, reconciling rotation-free locking with the ostensibly weak vdW paradigm (Liang et al., 26 Dec 2025). In nonlinear optical systems, $I_{\rm lock}$ serves as a universal, microscopic threshold for the onset of mode coherence and pulse formation (Antenucci et al., 2014). The framework is extensible to new material pairings and frequency-comb architectures, prescribing a general roadmap for the quantitative prediction and control of locked versus free regimes in layered heterostructures and multimode nonlinear cavities.