Exploring the Electronic Landscape of Two-Dimensional Tin Monoxide: Layer Thickness and Crystallographic Symmetry Effects

Abstract

The ability to precisely control the electronic bandgap is crucial for tailoring two-dimensional (2D) materials for optoelectronic applications. In this work, we systematically investigate the electronic structure of 2D tin monoxide (SnO) across various layer thicknesses (monolayer to tetralayer) and crystallographic symmetries using first-principles calculations. Our results reveal a strong dependence of the bandgap on the number of layers, which decreases dramatically from 3.94 eV in the monolayer to nearly metallic in the tetralayer. Furthermore, different space group symmetries are found to significantly influence the bandgap, providing an additional degree of freedom for property tuning. This bandgap engineering is quantitatively linked to enhanced interlayer electronic coupling, as evidenced by a progressive increase in interlayer charge transfer with layer count. Our findings establish a clear structure–property relationship and offer practical guidance for designing SnO-based devices in flexible electronics and tunable optoelectronics.

1. Introduction

Over the past decade, research on two-dimensional (2D) materials, sparked by the exceptional properties of graphene, has surged due to their potential in next-generation electronics [1,2,3]. This exploration has expanded the family of 2D materials to include silicene [4], germanene [5], boron nitride [6], pentagraphene [7], transition metal dichalcogenides [8], and metal oxides [9,10]. The synthesis of these materials often relies on top-down exfoliation or bottom-up growth on substrates. Concurrently, computational methods have played a pivotal role in accelerating the discovery of novel 2D materials, such as borophene and specific metal oxides, by predicting structures that were later verified experimentally [11,12,13,14,15,16,17,18]. This synergy between theoretical and experimental approaches forms the cornerstone of identifying 2D materials with tailored properties for electronics, optics, and quantum devices.

Among the diverse 2D materials, tin monoxide (SnO) stands out due to its unique electronic structure, which is influenced by its layered architecture and the presence of lone-pair electrons [19,20,21,22,23,24,25,26,27,28,29,30]. The exploration of SnO has been facilitated by the successful experimental synthesis of several stable phases. Most notably, the common blue-black $\mathrm{\alpha }$-SnO (tetragonal, P4/nmm) has been produced via wafer-scale growth [31]. Additionally, a red SnO phase with distinct symmetry has also been reported [32]. These experimentally accessible structures, along with theoretically predicted stable monolayers (e.g., with Pmmn symmetry [22,26]), provide a rich foundation for investigating the property space of layered SnO. Pioneering studies by Li et al. [33], Mubeen et al. [19] and Hu et al. [20] revealed the tunability of SnO’s bandgap with layer thickness, highlighting its promise for applications in optoelectronics and thin-film transistors. These works, along with the seminal study by Ogo et al. [21], have established SnO as a viable channel material, though challenges in optimizing device performance remain.

The understanding of SnO’s crystallographic behavior has evolved significantly through rigorous investigations. Watson’s study [25] challenged conventional wisdom by highlighting the role of lone pairs in its bonding mechanism and electronic properties. This perspective was further refined by Hunanyan et al. [26], who explored various SnO polymorphs and assessed their stability, revealing the potential for novel structures with intriguing properties.

Recent advances in synthesis have enabled detailed studies of 2D SnO. The mechanical exfoliation technique demonstrated by Kun et al. [29] has facilitated the creation of 2D SnO layers, allowing the investigation of their properties free from bulk effects. Meanwhile, Kripalani et al. [30] highlighted the remarkable mechanical properties of ultrathin SnO, such as superplasticity, which has profound implications for flexible electronics.

While previous research has laid a solid foundation, a comprehensive understanding of how different crystallographic symmetries (space groups) influence the electronic properties of layered SnO remains elusive. This gap motivates our in-depth investigation.

In this work, we employ density functional theory (DFT) to systematically explore the electronic properties of 2D SnO, focusing on the effects of layer thickness (from monolayer to tetralayer) and crystallographic symmetry. Our study expands beyond traditional SnO structures by examining a wide range of layered configurations, aiming to establish a unified design principle for bandgap engineering. We demonstrate that each incremental change in layer count or symmetry significantly modulates SnO’s electronic behavior. The computational methodology is detailed in Section 3, our results and discussion are presented in Section 2, and the conclusions are summarized in Section 4.

2. Results and Discussions

2.1. Structural Survey of SnO Polymorphs

To identify candidate structures for our study, we initiated a comprehensive survey of tin–oxygen compounds using the Materials Project database [34] and its PhaseDiagram module [35]. The thermodynamic stability of various SnO_x phases was assessed through the construction of a binary phase diagram based on DFT-calculated formation energies, as shown in Figure 1a. Phases located below the convex hull are thermodynamically stable, while those above it are metastable or unstable. Focusing on the Sn:O = 1:1 composition (Figure 1b), we identified several SnO polymorphs with distinct formation energies, which serve as the starting point for our investigation into their two-dimensional derivatives.

SnO crystallizes in three primary structure types: tetragonal, orthorhombic, and triclinic. The tetragonal phase includes three variants (mp-2097, mp-1065811, mp-999142), with mp-2097 representing the common blue-black SnO. The orthorhombic family encompasses mp-545820, mp-545552, and mp-1078644; the latter is recognized as the red SnO phase [32], while mp-545552 corresponds to a high-pressure form. Notably, prior theoretical studies [22,26] have identified a stable 2D structure in the orthorhombic family with space group Pmmn (No. 59). In contrast, the triclinic SnO structures (e.g., mp-1245076, mp-1245329) are non-layered and thus unsuitable for exfoliation.

While the exploration of SnO under pressure has revealed other polymorphs [28,36], experimental reports of 2D SnO are currently limited to structures related to mp-2097, synthesized via wafer-scale growth [31] or mechanical exfoliation [29]. Theoretically, high-throughput computational screening [37] has predicted two stable 2D SnO phases with Pmmn and P4/nmm symmetries, with phonon calculations confirming the thermodynamic stability of Pmmn over P4/nmm [22,26]. Building on this foundation, we systematically examine seven potential quasi van der Waals SnO configurations, including six derived from the phase diagram and the theoretically favored SnO-1L-Pmmn structure.

2.2. Crystal Structures of SnO Polymorphs and Their Layered Derivatives

The structural diversity of the investigated SnO systems is summarized in

Table 1. Structural properties of studied crystal and layered SnO. Labels * in columns 3, 4, and 5 mark the monoclinic structures.

	Orthorhombic				Tetragonal
Crystal	Notation	Pmn21-SnO	Cmc21-SnO	Cmce-SnO	P4/mmm-SnO	$\alpha$-SnO	P4/nmm-SnO
	International symbol	Pmn21	Cmc21	Cmce	P4/mmm	P4/nmm	P4/nmm
	Space group number	31	36	64	123	129	129
	Material ID	mp-545552	mp-1078644	mp-545820	mp-1065811	mp-2097	mp-999142
	Wyckoff position for Sn2+	2a	2a	4f	1a	2c	2a
	Experimental realization	Yes; SnO at high pressure	Yes Red SnO	No	No	Yes; Black-blue SnO	No
Monolayer	SnO-1L-Pmmn (Pmmn; No. 59)	Pmn21-SnO-1L (Pmn21; No. 31)	Cmc21-SnO-1L * (Cm; No. 8)	Cmce-SnO-1L (Aem2; No. 39)	P4/mmm-SnO-1L (P4/mmm; No. 123)	$\alpha$-SnO-1L (P4/nmm; No. 129)	P4/nmm-SnO-1L (P4/nmm; No. 129)
Bilayer	SnO-2L-Pmm2 (Pmm2; No. 25)	Pmn21-SnO-2L (Pmn21; No. 31)	Cmc21-SnO-2L * (Cm; No. 8)	Cmce-SnO-2L (Aem2; No. 39)	P4/mmm-SnO-2L (P4/mmm; No. 123)	$\alpha$-SnO-2L (P4/nmm; No. 129)	P4/nmm-SnO-2L (P4/nmm; No. 129)
Trilayer	SnO-3L-Pmm2 (Pmm2; No. 25)	Pmn21-SnO-3L * (Pm; No. 6)	Cmc21-SnO-3L * (Cm; No. 8)	Cmce-SnO-3L * (Cm; No. 8)	P4/mmm-SnO-3L (P4/mmm; No. 123)	$\alpha$-SnO-3L (P4mm; No. 99)	P4/nmm-SnO-3L (P4mm; No. 99)
Tetralayer	SnO-4L-Pmm2 (Pmm2; No. 25)	Pmn21-SnO-4L * (Pm; No. 6)	Cmc21-SnO-4L * (Cm; No. 8)	Cmce-SnO-4L * (Cm; No. 8)	P4/mmm-SnO-4L (P4/mmm; No. 123)	$\alpha$-SnO-4L (P4mm; No. 99)	P4/nmm-SnO-4L (P4mm; No. 99)

and visually represented in Figure 2. Table 1 systematically categorizes both three-dimensional crystalline structures and their corresponding two-dimensional layered derivatives, organized by crystal system (orthorhombic and tetragonal) and layer thickness (monolayer to tetralayer).

In this table, the “Experimental realization” status is explicitly noted for the parent bulk crystals: an entry of “Yes” indicates that the phase has been reported in experimental literature, while “No” denotes a theoretically predicted structure not yet synthesized in bulk form. The inclusion of the latter group is essential for a systematic and comprehensive exploration of how crystallographic symmetry governs the electronic properties across the full spectrum of possible SnO polymorphs.

A key observation from Table 1 is the structural continuity between bulk crystals and their layered counterparts. Most layered architectures retain the crystallographic symmetry of their parent bulk structures, with the notable exception of monoclinic structures (marked with asterisks in columns 3–5), which exhibit symmetry reduction in their layered forms. The table also highlights subtle structural distinctions: for instance, $\alpha$-SnO and P4/nmm-SnO share the same space group (P4/nmm, No. 129) but differ in their Wyckoff positions, leading to distinct atomic arrangements and material properties despite their apparent symmetry similarity.

Figure 2 provides complementary visual evidence of this structural evolution, showing both top and side views of all configurations. The systematic progression from monolayer to tetralayer structures demonstrates how layer stacking modifies the overall architecture while largely preserving the characteristic features of each polymorph.

All structures presented in Figure 2 have been fully relaxed. The detailed structural parameters and electronic properties for $\alpha$-SnO across different layer thicknesses are quantified in

Table 2. Structural properties for crystal and layered $\alpha$-SnO up to 4 layers.

		Layers				Crystal
	Notation	$\mathsf{\alpha }$ -SnO-1L	$\mathsf{\alpha }$ -SnO-2L	$\mathsf{\alpha }$ -SnO-3L	$\mathsf{\alpha }$ -SnO-4L	$\mathsf{\alpha }$ -SnO
Lattice	a(Å) = b(Å)	3.76511	3.77593	3.78001	3.78060	3.84082
	c(Å)	15	25	35	45	4.91413
Intra-layer	Sn-O(Å)	2.24114	2.24057	2.24211	2.24192	2.25009
	Sn-Sn(Å)	3.60594	3.59585	3.59502	3.59495	3.84082
	O-O(Å)	2.66234	2.66999	2.67287	2.67329	2.71587
Interlayer	O-O(Å)	-	4.96604	4.91053	4.90900	4.91413
	Sn-Sn(Å)	-	4.97051	4.92445	4.92280	3.73840
Electronic properties	Band character	Indirect	Indirect	Indirect	Indirect	Indirect
	PBE bandgap (eV)	3.0144	0.9572	0.3686	0.1432	0.04
	HSE06 bandgap (eV)	3.9430	1.5290	0.9391	0.1440	0.4818

. A clear trend emerges: lattice parameters expand gradually with increasing layer count, while bandgaps decrease substantially. The HSE06 functional, which incorporates exact Fock exchange, yields larger bandgaps than PBE, as expected. Our calculated HSE06 bandgaps for $\alpha$-SnO-1L (3.94 eV) and bulk $\alpha$-SnO (0.48 eV) show excellent agreement with previous theoretical (3.92 eV and 0.45 eV) [30] and experimental (3.95 eV and 0.64 eV) [29] values, validating our computational approach.

Notably, the P4/mmm and P4/nmm structures (columns 5 and 7 in Figure 2) exhibit minimal layer corrugation and the smallest bandgaps among all configurations, indicating metallic character. Complete structural and electronic data for all configurations are provided in Section SI of the Supporting Information.

2.3. Electronic Structure and Stability

The thermodynamic stability and electronic properties of the investigated SnO structures are summarized in Figure 3. The cohesive energy, calculated as $E_{\mathrm{coh}}=(E_{{\mathrm{Sn}}_n{\mathrm{O}}_m}-n{E}_{\mathrm{Sn}}-m{E}_{\mathrm{O}})/(n+m)$ where $E_{{\mathrm{Sn}}_n{\mathrm{O}}_m}$ is the total energy of each configuration and $E_{\mathrm{Sn}}/E_{\mathrm{O}}$ are atomic energies, provides a crucial metric for assessing structural stability (Figure 3a). More negative values indicate higher stability, suggesting greater likelihood of experimental realization. Our calculations reveal that metallic phases (P4/mmm-SnO and P4/nmm-SnO) and their layered derivatives exhibit relatively higher cohesive energies, rendering them less stable compared to other configurations. In contrast, $\alpha$-SnO and SnO-nL-Pmmn (n = 1–4) structures demonstrate notably lower cohesive energies, identifying them as the most promising candidates for stable 2D configurations. The relative thermodynamic stability indicated by the cohesive energy is further supported by prior reports on the dynamical stability of key structures. For instance, the Pmmn monolayer [22,26] and the $\mathrm{\alpha }$-SnO polymorph family [33] have been confirmed to be dynamically stable.

Figure 3b displays the HSE06-calculated bandgaps, revealing two dominant trends in SnO’s electronic structure. First, a pronounced layer-dependent behavior: bandgaps systematically decrease with increasing layer count across all structural families. This reduction can be attributed to enhanced interlayer electronic coupling, which modifies the band edge positions through increased wavefunction overlap between adjacent layers. Second, a strong symmetry dependence: different space groups significantly modulate the electronic properties. While $\alpha$-SnO-1L and SnO-1L-Pmmn exhibit wide bandgaps (>3.5 eV) characteristic of semiconductors, the Cmce-SnO family shows substantial bandgap narrowing towards narrow-gap semiconducting behavior. The Pmn2₁-SnO and Cmc2₁-SnO families maintain moderate semiconducting character, whereas the P4/mmm-SnO and P4/nmm-SnO families approach metallic conductivity with negligible bandgaps. The corresponding PBE bandgaps, which are systematically smaller due to the well-known bandgap underestimation of standard DFT, are provided in the Supporting Information for completeness.

The strong reduction of bandgap with increasing layer number, as reported for the specific case of $\mathrm{\alpha }$-SnO (P4/nmm) [33], is confirmed here to be a general trend across all quasi-2D SnO polymorphs studied (Figure 3b). A comparative analysis of our comprehensive dataset, however, reveals that crystallographic symmetry imposes an even greater modulation on the electronic structure. Two distinct regimes are evident. Within a given structural family, the bandgap reduction is predominantly and monotonically driven by increasing layer number. However, when comparing across different polymorphs at the same layer count, crystallographic symmetry emerges as the decisive factor, setting the absolute electronic baseline. This is clearly demonstrated by the monolayer data (Figure 3b), where the bandgap spans a range of nearly 4 eV—a variation larger than that induced by layering in any single family. Consequently, these two parameters define a two-dimensional design space for 2D SnO: symmetry selection provides a “coarse-tuning” mechanism for the fundamental electronic character, while subsequent control of layer thickness enables continuous “fine-tuning” within the selected material system.

To gain deeper insight into the electronic structure evolution, we examine the band dispersions of monolayer SnO configurations derived from different crystalline prototypes (Figure 4). A detailed comparative analysis reveals that all monolayers except P4/mmm-SnO-1L exhibit indirect bandgap character, with the valence band maximum (VBM) and conduction band minimum (CBM) occurring at different k-points in the Brillouin zone. This indirect nature has important implications for carrier dynamics and optoelectronic performance, as it typically leads to longer carrier lifetimes due to the requirement of phonon-assisted optical transitions. The significant variation in both bandgap magnitude and band dispersion characteristics across different structural families underscores the critical role of crystallographic symmetry in determining the electronic properties of 2D SnO. The wide range of bandgaps accessible through layer and symmetry control—spanning from deep UV (3.94 eV) to near-infrared (0.03 eV)—suggests SnO’s potential for broadband optoelectronic applications. Specifically, the wide-gap monolayers ($\alpha$-SnO-1L and SnO-1L-Pmmn) could serve as deep-UV photodetectors or transparent conductive layers, while the narrow-gap multilayers approach the ideal bandgap range for thin-film photovoltaics and infrared optoelectronics.

2.4. Layer-Dependent Electronic Evolution in SnO-nL-Pmmn

To elucidate the mechanism behind the layer-dependent bandgap reduction observed in Figure 3, we focus on the SnO-nL-Pmmn family as a representative system. Figure 5 depicts the electronic band structures for SnO-1L-Pmmn (structure (a-i) in Figure 2) and its layered counterparts. Figure 5 depicts the HSE06-calculated band structures from monolayer to tetralayer, clearly demonstrating the progressive closure of the bandgap with increasing layer count. The monolayer exhibits a substantial indirect bandgap of 3.81 eV, which rapidly decreases to 1.46 eV (bilayer), 0.33 eV (trilayer), and ultimately to a near-metallic 0.03 eV in the tetralayer. This systematic reduction arises from the evolution of both valence and conduction band edges: interlayer interactions cause a pronounced upward shift in the valence band maximum and a concurrent downward shift in the conduction band minimum, effectively narrowing the bandgap through enhanced interlayer electronic coupling. For a detailed description of the electronic band structures of other configurations obtained from PBE calculations, please refer to Section SII of the Supporting Information. Band structures from HSE06 calculations can be found in Section SIII of the Supporting Information. This focused analysis on the SnO-nL-Pmmn family complements the broader survey of monolayer band structures in Figure 4, providing a detailed view of layer-dependent effects in one of the most stable configurations identified in our study.

The fundamental driving force behind this electronic structure evolution is directly visualized through charge density analysis. Figure 6 provides a comprehensive picture of the layer-dependent charge redistribution. The first two rows display the charge distributions at the VBM and CBM for the monolayer, showing the characteristic orbital contributions that define the band edges. More importantly, the charge density difference (CDD) plots, defined as $\Delta \rho ={\rho }_{\mathrm{total}}-{\sum }_i{\rho }_i$ (where ${\rho }_i$ represents the charge density of isolated individual layers), reveal the net charge redistribution due to interlayer interactions.

The CDD analysis demonstrates a progressive enhancement of interlayer charge transfer with increasing layer count. Bader charge analysis quantifies this trend: the interlayer charge transfer increases from 0.011 e in the bilayer to 0.014 e in the trilayer and 0.019 e in the tetralayer. This systematic increase in charge transfer, coupled with the expanding regions of charge accumulation (yellow isosurfaces) in the interlayer regions, provides direct evidence of strengthened interlayer electronic coupling. The growing charge cloud in the central regions of multilayer configurations indicates enhanced wavefunction overlap between layers, which facilitates the bandgap narrowing observed in both PBE and HSE06 calculations.

This combined analysis of band structures and charge distributions establishes a coherent physical picture: the enhanced interlayer coupling, quantified by increasing charge transfer, modifies the electronic structure through wavefunction hybridization across layers. This leads to band renormalization where the valence band maximum rises and conduction band minimum descends, systematically reducing the bandgap as layer count increases. The excellent agreement between our HSE06 bandgaps and experimental values, coupled with this mechanistic understanding of the layer-dependent behavior, provides a solid foundation for designing SnO-based devices with tailored electronic properties. For complete band structures and charge density analyses of all SnO families, see Sections SII–SIV of the Supporting Information.

3. Methodology

First-principles calculations based on density functional theory (DFT) were performed using the Vienna ab initio simulation package (VASP) [38,39]. The electron–ion interactions were described by the projector-augmented wave (PAW) method [40], and the exchange-correlation functional was treated within both the generalized gradient approximation (GGA) using the Perdew–Burke–Ernzerhof (PBE) formulation [41] and the hybrid HSE06 functional [42]. To accurately describe the interlayer interactions in the layered SnO structures, the van der Waals (vdW) corrections were incorporated using the DFT-D3 method with Becke–Johnson damping as proposed by Grimme [43,44].

A plane-wave kinetic energy cutoff of 520 eV was employed for the basis set. The Brillouin zone integration was sampled using $\mathrm{\Gamma }$-centered k-point meshes with a grid spacing of less than 0.03 Å⁻¹, specifically $6\times 6\times 5$ for bulk phases (corresponding to a spacing of ∼0.02 Å⁻¹) and $9\times 9\times 1$ for all layered structures (corresponding to a spacing of ∼0.015 Å⁻¹), ensuring energy convergence. For the HSE06 calculations, single-point energy calculations were performed on the PBE-optimized structures. The electronic self-consistent field iteration was considered converged when the total energy change between successive steps was below ${10}^{-6}$ eV. All atomic positions and lattice parameters were fully relaxed until the Hellmann–Feynman forces on each atom were less than 0.01 eV/Å.

To model the two-dimensional systems, periodic boundary conditions were applied in the in-plane directions (x and y), while a vacuum layer of 15 Åwas introduced along the z-direction to eliminate spurious interactions between periodic images. This vacuum size was confirmed to be sufficient by testing that the total energy variation was less than 1 meV/atom when increased further.

Additionally, to gain deeper insight into the orbital contributions to the electronic structure, projected density of states (PDOS) analyses were performed for all configurations. and can be found in Section SV of the Supporting Information. These PDOS calculations were carried out using the QuantumATK software (W-2024.09) package [45], which employs localized basis sets and is highly efficient for such post-processing analysis.

4. Conclusions

In summary, this systematic first-principles study demonstrates that the electronic structure of two-dimensional SnO can be effectively engineered through both layer thickness and crystallographic symmetry. We have quantified a pronounced bandgap reduction from 3.94 eV in the monolayer to nearly 0 eV in the tetralayer, a trend validated by HSE06 calculations. Furthermore, distinct space groups are shown to significantly modulate the electronic properties, offering an alternative strategy for bandgap tuning. The underlying mechanism for the layer-dependent behavior is attributed to progressively enhanced interlayer electronic coupling, directly evidenced by the increasing interlayer charge transfer revealed through Bader analysis. These findings establish clear design principles for manipulating the electronic landscape of 2D SnO, providing valuable guidance for its application in future optoelectronic and quantum devices.