**Growth strategy and Stability of 2D QLDS-GaTe single crystal.** In crystallography, atoms in crystal configurations are predisposed to a highly dense arrangement, which can mitigate entropy and augment structural stability. This effect becomes particularly evident in layered materials propagated on a traditionally vdW substrate. In such systems, interatomic forces generally manifest a symmetric distribution within a 2D plane parallel to the substrate (Fig. 1a), thereby leading to a random distribution of growth orientations on the substrate (Supplementary Fig. 1). The traditional growth mode of 2D materials with the best stable layered configuration is represented by planar growth mode.

To break the uniformity of interaction strength in the aforementioned 2D plane, the interaction-strength-modulation growth (ISMG) strategy has been proposed. This strategy aims to disrupt the uniform interaction and augment the interaction strength between the substrate and the crystal in a direction orthogonal to the plane, leading to a phenomenon known as skewed growth (Fig. 1b). A liquid metal was chosen as the growth substrate owing to its inherently flat surface, significant capacity to accommodate additional elements and well-matched interactions with skewed growth mode22–25. The crystalline attribute, as evidenced by the temperature-dependent X-ray diffraction (XRD) diagram (Supplementary Fig. 2), suggests that Ga substrate becomes a highly interactive substrate. This substrate facilitates significant charge interactions with the material nucleating on its surface and disrupts the conventional planar growth mode of the materials (Supplementary Fig. 3). Simultaneously, the as-nucleated material induces the precursor atoms dissolved in liquid metal to accumulate on their interface since the layering effect25–27, further promoting the crystalline process (Supplementary Fig. 4). Therefore, ISMG strategy with liquid metal can reduce the formation energy across an expansive chemical potential window tuned by temperatures (Supplementary Fig. 5–7) and facilitates an inclined crystal growth with a well-aligned growth orientation (Supplementary Fig. 8).

The skewed growth mechanism engenders the formation of domino-structured material, characterized by a distinct interaction paradigm. In this model, the force orthogonal to the 2D growth plane constitutes a synergistic blend of vdW forces and covalent bonds. This unique amalgamation results in a force that surpasses that in layered materials, yet falls short of that in non-layered materials (Fig. 1c). Consequently, we categorize this kind of material as a quasi-layered substance. As an exemplar, the 2D QLDS-GaTe unveils a uniquely skewed growth structure, diverging from the substrate orientation by approximately 25°. This deviation indicates a cooperative interplay of van der Waals forces and covalent bonds, neither of which aligns perpendicularly to the 2D growth plane. Significantly, the impact of a densely packed arrangement persists in this class of quasi-layered materials. To ensure structural stability, an evident enhancement in interlayer coupling transpires, contributing to entropy minimization and structural fortification. Therefore, the interlayer interaction strength exceeds that of the vdW forces, as evidenced by the existence of high electron density between layers shown in the differential charge density plot (Fig. 1d).

Furthermore, to corroborate the structural stability of 2D QLDS-GaTe at ambient conditions, we conducted calculations of the phonon dispersion curves at a temperature of 300 K, as shown in Fig. 1e. The resultant phonon spectrum for the 2D QLDS-GaTe demonstrated that all phonon branches remained positive across the entirety of the Brillouin zone, thereby affirming its structural stability at room temperature. Molecular dynamics simulations performed at a temperature of 300 K manifested no discernible energy drift or structural dissociation (Fig. 1f, Supplementary Fig. 9), highlighting the substantial stability of 2D QLDS-GaTe. Collectively, these findings robustly substantiate that the 2D QLDS-GaTe retains its post-synthesis stability on a liquid metal substrate.

**Structure characterization of 2D QLDS-GaTe single crystal.** As delineated in Fig. 2a, the 2D domino-structured material needs the amplification of interlayer forces to ensure structural stability, a process that concurrently enables entropy minimization and structural reinforcement. Drawing an analogy to domino tiles, wider inter-tile spacing is required when larger tiles are inclined at an identical angle, while their smaller counterparts necessitate narrower spacing. The correlation between the lattice constant and thickness is depicted in Fig. 2b. On approaching the 2D limit (~ 1 nm), there is a marked contraction in the lattice constant in the stacking direction, with the maximum shrinkage extending to as much as 10.8%. This emphatically underscores the remarkable potential inherent in the ISMG strategy. The approach facilitates the synthesis of materials that can approach the 2D limit of 1.2 nm (Supplementary Fig. 10). Moreover, it yields 2D QLDS-GaTe specimens of diverse thicknesses, thereby presenting a compelling platform for investigating interlayer coupling phenomena.

To scrutinize the enhancement of interlayer coupling effects during the transition to two-dimensionality in QLDS-GaTe, samples of disparate thicknesses were examined. All samples demonstrated a single set of diffraction structures in their Fast Fourier Transform (FFT) patterns (Fig. 2c, 2f), confirming the robust crystallinity across QLDS-GaTe samples of diverse thicknesses synthesized via the ISMG strategy. These FFT patterns align closely with the calculated simulated FFT diffraction structures (Supplementary Fig. 11), corroborating the synthesis of GaTe with a domino structure on the (− 101) plane.

Notably, the interlayer (101) spacing in the high-resolution transmission electron microscopy (HRTEM) images of samples with various thicknesses were delineated (Fig. 2d, 2g), measuring 7.33 Å for the thin sample and 8.06 Å for the thicker one. These results harmonize with the simulated HRTEM images of 1.2 nm and 10 nm thick samples (Fig. 2e, 2h), further affirming the domino structure of GaTe (− 101) plane. The lattice constant in the material undergoes a 7.7% contraction during the two-dimensionalization process, a value nearing the theoretical limit of 10.8%. Simultaneously, energy-dispersive X-ray spectroscopy (EDS) elemental mapping confirmed uniform and strong signals of Ga and Te elements throughout the crystal (Fig. 2i), highlighting the aptness of 2D QLDS-GaTe as a platform for probing interlayer coupling effects.

**Investigation of SHG in 2D QLDS-GaTe structure.** SHG characterization serves as an effective method for identifying material symmetry in nonlinear optics28. The SHG investigation scheme applied to the 2D QLDS-GaTe structure elucidates its structural symmetry (Fig. 3a). A femtosecond laser with a steady output power of 1 mW was employed to incite the SHG signal. Remarkably, the 2D QLDS-GaTe structure exhibited a substantial nonlinear optical response within a range of 570 nm to 670 nm, corresponding to an excitation wavelength of 1140 nm to 1340 nm (Fig. 3b). Subsequent power-dependent investigations were carried out on the SHG intensity at a fixed wavelength. The spectrum presented in Fig. 3c attests to the significant increase in SHG intensity with a tuning power range from 0.355 mW to 1.014 mW. The extracted power-dependent SHG intensity is displayed in Fig. 3d, revealing a double logarithmic relationship between the SHG intensity and laser power. The slope of the fitted line is 1.72, aligning with the theoretical value of 229, as predicted by the electric dipole approximation. Intriguingly, a marked increase in the SHG response signal is observed as the thickness of the material decreases. As shown in Fig. 3e and 3f, a six-fold increase in SHG response signal intensity is observed when the thickness is reduced to 4 nm from 20.5 nm. The calculated second-order nonlinear optical susceptibility is found to be 394.3 pm V− 1, considered as an impressively substantial value, highlighting the superior nonlinear optical property of the 2D QLDS-GaTe single crystal.

We ascribe the anomalous intensification in the SHG to a restructuring in the band architecture (Supplementary Fig. 12), incited by alterations in the bonding mechanism of the material. Density functional theory (DFT) calculations reveal that the bonding mechanism of 2D QLDS-GaTe significantly differs from that of the bulk structure (Supplementary Fig. 13), due to the reconstruction of the surface atoms of 2D QLDS-GaTe, thereby introducing new bonding states. Hence, we categorize the bonding arrangement in 2D QLDS-GaTe as an amalgamation of surface and inner bonding states. The inner bonding state, akin to the bulk structure, manifests virtually identical orbital hybridization and bond strength (Supplementary Fig. 14–17, Table S1, S2). On the contrary, the interactions of reconstructed surface atoms significantly deviate from the bulk structure (Supplementary Fig. 18–19, Table S3, S4). Taking Ga atoms as a case in point, they manifest a mixed state of sp2 and sp3 hybridization on the surface, while those in the bulk structure retain an exclusive sp3 hybridization. Consequently, as the thickness decreases, the proportion of the surface charge state in the overall charge state gradually increases, leading to changes in the band structure. This is the underlying reason for the significant enhancement of the SHG signal as the thickness decreases.

**HER performance of 2D QLDS-GaTe single crystal.** Previous research underscores the superior catalytic performance of unsaturated edges in layered materials, evidenced by 2D MoS2 catalysis (Supplementary Fig. 20). These edge states favor catalytic reactions due to their unique charge characteristics. Remarkably, for domino-structured materials, a skewed growth pattern projects the unsaturated edges of the layered structure onto the surface of the domino-structured material, thereby not only augmenting the so-called edge area but also endowing the material with a novel density of states (DOS), as depicted in Fig. 4b. This transformation is coined as the 'Region Fold' theory. In domino-structured materials, the reactive edge states project onto larger reactive surfaces, while inactive regions fold into non-reactive areas. Guided by this concept, more pronounced surface charge polarization occurs, with the surface acting as both a charge donor and acceptor (Fig. 4c, Supplementary Fig. 21–23), facilitating catalytic reactions. Gibbs free energy profiles in Fig. 4d affirm this theory. The unsaturated Ga atoms on the surface of the 2D QLDS-GaTe structure favor the stabilization of H*, while the highly active Te sites promote the conversion of H* to H2 (Fig. 4d, Supplementary Fig. 24–27). Moreover, the reactivity of Te atoms surpasses that of layered GaTe (Supplementary Fig. 28, 29), with Δ*G** concentrated in the range of − 0.5 eV to 0.5 eV, which is highly conducive to the HER process. This further validates the applicability of the Region Fold theory in domino-structure materials.

Therefore, to evaluate the HER activity of the 2D QLDS-GaTe single crystal, a micro-electrochemical setup was constructed with great ohmic contact (Fig. 4a, Supplementary Fig. 30). Electrochemical assessment using linear sweep voltammetry (LSV) revealed a superior catalytic performance of the 2D QLDS-GaTe single crystal in comparison to both polycrystalline and layered GaTe materials (Fig. 4e, Supplementary Fig. 31). Notably, the overpotential required for the 2D QLDS-GaTe to achieve a current density of 10 mA cm2 was merely 41 mV, significantly less than the 355 mV and 250 mV required for the polycrystalline and layered GaTe materials, respectively. The 2D QLDS-GaTe exhibited an impressive current density of 3100 mA cm2 at an overpotential of 0.31 V, nearly 100 times that of layered GaTe under similar conditions. The Tafel slope, a valuable metric of catalytic performance, was considerably lower for the 2D QLDS-GaTe (73 mV dec− 1), when compared to those of the other two GaTe materials (115 mV dec− 1 and 229 mV dec− 1), indicating enhanced reaction kinetics (Fig. 4f).

On the microscopic level, the interaction between the nonbonded states from Te (5p) and H (1s) triggers the formation of a Te − H σ-bond upon H-adsorption (Fig. 4g, 4h, Supplementary Fig. 32, Tables S5, S6). The crystal orbital Hamilton population (COHP) results underscore the superior capacity of 2D QLDS-GaTe for H* capture, while the layered GaTe shows a relatively weaker interaction with a partially filled σ* antibonding orbital. This discrepancy, further corroborated by the natural population analysis (NPA) charge of H, accounts for the high Δ*G** inherent to layered GaTe. In 2D QLDS-GaTe, Ga atoms linked to Te compete with H for electron donation (Supplementary Fig. 33), resulting in the weakening of the original Te–Ga bonds. Under these circumstances, the substrate can undergo minor lattice contraction to revert to its original stable bonding structure, which leads to the shortening of the Te–Ga distance and consequently increases (reduces) the electron contribution from Ga (Te). This sequence of events results in a smaller Δ*G** during the HER reaction, which is a key factor underlying the superior HER performance of 2D QLDS-GaTe.