Intrinsic Fracture Nonreciprocity in 2D Heterostructures

• Intrinsic fracture nonreciprocity is the directional asymmetry in crack resistance in 2D heterostructures caused by prestrain differences from lattice mismatch.
• It arises from the interplay between atomic-scale electronic structure and mesoscale crack behavior, analyzed using DFT and ML-MD simulations.
• Universal exponential bond scaling and linear toughness mismatch relations quantify the nonreciprocity, guiding strain-engineered nanodevice design.

Intrinsic fracture nonreciprocity denotes a fundamental, direction-dependent asymmetry in the resistance to crack propagation within a single, coherent two-dimensional (2D) heterostructure. Unlike classical homogeneous crystals or lattice-matched heterojunctions, which exhibit identical crack-tip bond-breaking thresholds regardless of crack direction across the interface, lattice-mismatched 2D heterostructures display a pronounced ‘diode-like’ fracture response. This behavior is governed by asymmetric prestrain states at the crack tip, imprinted by the intrinsic lattice mismatch between the component crystals. The phenomenon ultimately arises from the coupling between atomic-scale electronic structure and mesoscale fracture processes, and is quantitatively described by universal scaling relations between bond-center charge density and bond strain.

1. Definition and Mechanistic Origin

Intrinsic fracture nonreciprocity is defined as the directional asymmetry in crack resistance (toughness) that appears when a crack propagates across a lattice-mismatched interface in a coherent 2D heterostructure. This arises when two 2D crystals with distinct lattice constants, joined with perfect registry (i.e., with no misfit defects), establish an interfacial lattice mismatch quantified by $\delta = (a_{\max} - a_{\min})/a_{\max}$, where $a_{\max}$ and $a_{\min}$ are the principal lattice constants of the respective crystals.

At any given interface crack-tip, one side (originating from the smaller-$a$ crystal) is under tensile prestrain, corresponding to weakened bonds, while the opposite side (from the larger-$a$ crystal) is subject to compressive prestrain, corresponding to strengthened bonds. This asymmetry causes the direction of crack advance to significantly influence the threshold for bond rupture and thus the overall fracture resistance—a property absent in reciprocal (homogeneous or lattice-matched) systems.

2. Atomistic Bond-Rupture Mechanism

The microscopic origin of fracture nonreciprocity is the interplay between prestrain fields and electronic charge redistribution at the crack-tip. Each metal-chalcogen (M–X) bond (e.g., M = Mo, W; X = S, Se, Te) exhibits a strongly polar covalent electron density profile along its bond dissociation path. Under external tensile loading, the minimum electron charge density at the bond center, $\rho(\ell)$, decreases as the bond length $\ell$ increases, with bond rupture occurring when this density falls below a material-specific threshold.

In a heterostructure, the built-in prestrain shifts the starting bond length from its equilibrium value $\ell_0$ to $\ell_0(1 + \varepsilon_{\text{pre}})$ even before any external loading. As a consequence:

• The tensile-prestrained (smaller-$a$) side possesses elongated bonds; therefore, less additional external strain is required for $\rho(\ell)$ to reach the rupture threshold.
• The compressively prestrained (larger-$a$) side exhibits contracted bonds; thus, more external strain is necessary to induce bond rupture, raising the local toughness.

This microscopically bidirectional response at the crack tip imparts a marked nonreciprocity to the macroscopic fracture behavior.

3. Universal Exponential Scaling of Bond Breakage

Bond rupture in both pristine monolayers and pre-cracked heterostructures conforms to a universal exponential scaling law linking normalized bond-center charge density to bond strain. For bond strain $\varepsilon_b = (\ell - \ell_0)/\ell_0$, the normalized minimum charge density $p(\varepsilon_b) = \rho(\ell)/\rho_0$ follows:

$p(\varepsilon_b) = A \exp(\alpha \varepsilon_b)$

where $A \approx 1$ and $\alpha \approx -4.43$ are empirically universal across transition-metal dichalcogenides (TMDs) and independent of metal-chalcogen species, local atomic environment, or bonding chemistry. Bond rupture is triggered when $p(\varepsilon_b)$ drops to a critical value $p_c$ set by the peak stress point for the pristine system. This exponential law enables direct prediction of bond failure thresholds purely from the instantaneous bond strain and charge density, without requiring full-scale fracture simulations or dependence on detailed local chemistry.

4. Quantification and Tuning of Nonreciprocity

Directional asymmetry in toughness is quantified by the fracture toughness mismatch ratio:

$K_m = \frac{K_{\max} - K_{\min}}{K_{\max}}$

where $K_{\max}$ and $K_{\min}$ are the toughnesses associated with crack propagation in the tougher and weaker directions, respectively. Systematic studies show that for TMD heterostructures with lattice mismatch varied over $0\% < \delta < 10\%$, all systems with $\delta > 0$ exhibit pronounced nonreciprocity, whereas $\delta = 0$ yields reciprocal behavior. Empirically, the relationship is nearly linear:

$K_m \approx 4.9\,\delta$

This scaling achieves $K_m \approx 0.49$ (a 49% asymmetry) at $\delta = 0.10$. Thus, the degree of nonreciprocity is a directly controllable function of the interfacial lattice mismatch, enabling the design of ‘fracture diodes’ analogous to electronic rectifiers, where the structural mismatch acts as a built-in mechanical bias.

A further quantitative connection is established by defining the bond-strain variation rate at the crack tip:

$$B_v = \frac{\varepsilon^{\rm het}_b - \varepsilon^{\rm TMD}_b}{\varepsilon^{\rm TMD}_b}$$

and expressing the corresponding toughness difference (relative to the pristine monolayer) as:

$$K_v = \frac{K_{\rm het} - K_{\rm TMD}}{K_{\rm TMD}} = C_0 + C_1 B_v$$

with fit constants $C_0 \approx 0.10$ and $C_1 \approx -6.5$. This demonstrates that the prestrain field’s impact on the crack-tip bond rupture threshold, via the universal $p$–$\varepsilon_b$ law, quantitatively governs either toughening (for $B_v < 0$) or weakening ($B_v > 0$) depending on the crack propagation direction.

5. Computational and Methodological Framework

The analysis of intrinsic fracture nonreciprocity employs a dual-computational approach integrating density-functional theory (DFT) and machine-learning molecular dynamics (ML-MD):

• DFT simulations utilize the Vienna Ab-Initio Simulation Package (VASP), treating exchange-correlation with PBE/GGA and PAW pseudopotentials. Supercells sized $\sim29$ Å (zigzag) $\times$ $58$ Å (armchair), containing mirrored pre-cracks of $13$ Å length along the interface, are subject to uniaxial tension via $1\%$ incremental cell-length scaling and full ionic relaxation.
• ML-MD leverages the MACE equivariant message-passing neural network, fine-tuned on 993 DFT-labeled structures (MoS$_2$, MoSe$_2$, and their heterostructures, both defective and pristine). Energy/force/stress RMSEs are maintained below 5 meV/atom, 102 meV/Å, and 0.2 N/m, respectively. LAMMPS-based MD simulations are performed at 300 K, with NPT equilibration (100 ps), followed by NVT uniaxial loading at 0.3 ns$^{-1}$ for 1 ns with 1 fs timesteps. Statistical robustness is ensured by five independent replicas per system.

This approach enables both ab initio accuracy in bond rupture characterization and statistical sampling of fracture pathways at finite temperature and time scales.

6. Universality across Two-Dimensional Lattice Symmetries

Intrinsic fracture nonreciprocity is not limited to any specific material class or lattice symmetry. Studies encompassing all four 2D crystal families—hexagonal, square, rectangular, and oblique—demonstrate the generality of the phenomenon. Representative examples include:

Lattice Type	Heterostructure Example	Lattice Mismatch δ (%)
Hexagonal	InS–GaSe	2.0
Square	KBr–KCl	4.6
Rectangular	SiTe–GeTe	3.1
Oblique	BSe–BS	6.4

All systems with $\delta > 0$ exhibit directional toughness asymmetry, $K_m$ scaling with $\delta$ similarly to TMDs, with crack-tip bond strains obeying the universal $p(\varepsilon_b)$ exponential law and $K_v$–$B_v$ relations. This establishes that intrinsic fracture nonreciprocity is a fundamental and robust feature of interfacial-strain phenomena in two-dimensional materials, independent of material chemistry or lattice symmetry.

7. Implications for Strain Engineering and Nanodevice Design

The quantitative understanding of intrinsic fracture nonreciprocity enables rational design principles for damage-tolerant nanostructures via interface strain engineering:

• Select pairs of 2D crystals with a desired lattice mismatch $\delta$ to achieve target magnitudes of directional toughening; higher $\delta$ yields greater $K_m$ and thus stronger nonreciprocity.
• Orient cracks intentionally to propagate toward the pre-compressed side when enhanced fracture resistance is required, or toward the pre-tensile side where guided, sacrificial failure is beneficial.
• Utilize the universal exponential $p$–$\varepsilon_b$ law to predict bond rupture thresholds under any combination of prestrain and applied loading, removing the necessity for replicated full-scale fracture simulations.
• Apply these design principles to 2D heterostructures of any symmetry class, facilitating the creation of “mechanical diodes” in which atomic-scale electronic structure is coupled directly to macroscopic fracture toughness.

Lattice mismatch in coherent 2D interfaces thus provides an intrinsic, tunable route to directional fracture control, bridging electronic structure with mesoscale materials engineering and enabling the development of strain-engineered, damage-tolerant nanodevices.