Strain Modulation of Electronic Properties in Monolayer SnP 2 S 6 and GeP 2 S 6

: In recent years, two-dimensional (2D) materials have attracted signiﬁcant attention due to their distinctive properties, including exceptional mechanical ﬂexibility and tunable electronic properties. Via the ﬁrst-principles calculation, we investigate the effect of strain on the electronic properties of monolayer SnP 2 S 6 and GeP 2 S 6 . We ﬁnd that monolayer SnP 2 S 6 is an indirect bandgap semiconductor, while monolayer GeP 2 S 6 is a direct bandgap semiconductor. Notably, under uniform biaxial strains, SnP 2 S 6 undergoes an indirect-to-direct bandgap transition at 4.0% biaxial compressive strains, while GeP 2 S 6 exhibits a direct-to-indirect transition at 2.0% biaxial tensile strain. The changes in the conduction band edge can be attributed to the high-symmetry point Γ being more sensitive to strain than K. Thus, the relocation of the conduction band and valence band edges in monolayer SnP 2 S 6 and GeP 2 S 6 induces a direct-to-indirect and indirect-to-direct bandgap transition, respectively. Consequently, the strain is an effective band engineering scheme which is crucial for the design and development of next-generation nanoelectronic and optoelectronic devices.


Introduction
Two-dimensional (2D) materials, such as graphene [1,2], have attracted significant interest because of their exotic physical properties and potential application in nanoelectronic and optoelectronic devices [3].However, graphene is a semimetal with zero bandgaps [4], and the absence of a bandgap will limit its application in nanoscale optoelectronic and cutting-edge ultra-fast electronic devices of the next generation [5].Hence, the exploration of 2D materials with appropriate bandgaps is of great importance for device applications.The transition metal dichalcogenides (TMDs) [6], hexagonal boron nitride (b-BN) [7], germanene (2D germanium) [8], and metal halogenides [9,10], among others, have garnered substantial interest in the realms of materials science, microelectronics, physics, and related fields.Among these encouraging candidates, 2D nanoporous metal chalcogen phosphates MP 2 S 6 (M = metal, X = S, Se) including SnP 2 S 6 and GeP 2 S 6 have also captured considerable attention because of their moderate-to-wide bandgaps ranging from 1.2 eV to 3.5 eV [11].Similar to TMDCs, SnP 2 S 6 and GeP 2 S 6 share properties of typical 2D materials [12], with a weak van der Waals interlayer interaction [13], a high surface-to-volume ratio and excellent optical properties [14].Moreover, metal chalcogen phosphates MP 2 X 6 have rich properties, including topological magnetism [15], ferroelectric ordering [16], photocatalytic properties [17], and H 2 storage and Li intercalation for batteries [11].For example, monolayer SnP 2 S 6 is an indirect bandgap semiconductor with high carrier mobility and with a bandgap of 1.1 eV, while monolayer GeP 2 S 6 is a direct bandgap semiconductor with a bandgap of 1.06 eV, which shows promising potential in electronic and photoelectronic applications [18,19].Furthermore, SnP 2 S 6 and GeP 2 S 6 are 2D porous materials that are important as membranes, adsorbents, catalysts, and in other chemical applications.The novel physical properties of 2D nanoporous metal chalcogen phosphate materials have the potential to expand the range of applications in nanoelectronics [11].He et al. reported that SnP 2 S 6 exhibits nonlinear optical characteristics with significant nonresonant second harmonic generation, which lays the foundation for developing novel optoelectronic devices [20].
The development of 2D materials has attracted considerable attention owing to its unique attributes, including tunable electronic properties.Strain is a promising avenue for tuning the electric properties of two-dimensional materials [21,22].Theoretical studies have shown that transition metal disulfide compounds (TMDs) are sensitive to strain [23], and strain engineering can shift the conduction band minimum (CBM) and valence band maximum (VBM) [23][24][25][26].It has been reported that the application of tensile strain can lead to a narrowing of the bandgap in MoS 2 in a monolayer system [27,28].Moreover, the directto-indirect bandgap transition can occur in monolayer MoS 2 with 6% tensile strain [29].The strain-tuned electronic properties in 2D materials have significant implications for the development of next-generation nanoelectronic and optoelectronic devices [30][31][32].Furthermore, studies have also investigated the effects of applying pressure to layered SnP 2 S 6 and GeP 2 S 6 .The transition from a semiconductor to a metal has been reported in layered SnP 2 S 6 and GeP 2 S 6 [33].We will now shift our research focus towards investigating the effects of biaxial strain on monolayer SnP 2 S 6 and GeP 2 S 6 .The metal chalcogen phosphates SnP 2 S 6 and GeP 2 S 6 with intrinsic nanoporous structures possess excellent mechanical performance, and are expected to be able to effectively tune the electronic properties under strain [34].
In this study, we investigated the ground-state monolayers of SnP 2 S 6 and GeP 2 S 6 , revealing that monolayer SnP 2 S 6 and GeP 2 S 6 are thermodynamically stable.Monolayer SnP 2 S 6 and GeP 2 S 6 are indirect bandgap semiconductors with a 1.35 eV gap and a direct bandgap semiconductor with a gap of 1.06 eV in equilibrium, respectively.Additionally, we modulated their electronic structures by applying strain from 6% compression to 6% tension.When a 4% biaxial compression (BC) strain was applied to the monolayer SnP 2 S 6 , it transformed from an indirect bandgap semiconductor to a direct bandgap semiconductor.Similarly, when a 2% biaxial tension (BT) strain was applied to the monolayer GeP 2 S 6 , it transitioned from a direct bandgap semiconductor to an indirect bandgap semiconductor.These results provide valuable insights into the strain engineering in tunning the electronic structures of the monolayer systems.

Result and Discussion
SnP 2 S 6 and GeP 2 S 6 are members of the family of novel 2D metal thiophosphates.These monolayers are characterized by space group P312 and contain one metal cation (Sn or Ge) and one anionic [P 2 S 6 ] 4-unit, as illustrated in Figure 1a,b [34].The metal cation undergoes coordination with six sulfur (S) atoms, thereby forming a hexahedral structure, whereas each phosphorus (P) atom is coordinated with three sulfur atoms in a tetrahedral arrangement, as depicted in Figure 1b.The monolayer SnP 2 S 6 and GeP 2 S 6 exhibit optimized lattice parameters of 6.13 Å and 5.99 Å, respectively, which is consistent with the previous result [35].The optimized lattice parameters reflect the equilibrium distances between the atoms in the monolayers, providing insights into their structural stability and interatomic bonding characteristics.Undoubtedly, the stability of materials is a crucial consideration in materials science and applications.Therefore, we employed the finite displacement method to calculate the phonon spectra of monolayer SnP 2 S 6 and monolayer GeP 2 S 6 using a 4 × 4 × 1 supercell via phonopy [36].Notably, as shown in Figure 2, no imaginary phonon frequencies were found throughout the entire Brillouin zone for monolayer GeP 2 S 6 and the imaginary frequency near Γis negligible for monolayer SnP 2 S 6 , indicating that both structures are stable.In fact, monolayer SnP 2 S 6 has been experimentally synthesized via chemical vapor transport technique along with the mechanical exfoliation method and used for preparing optoelectronic devices [37].
imaginary phonon frequencies were found throughout the entire Brillouin zone for monolayer GeP2S6 and the imaginary frequency near Γis negligible for monolayer SnP2S6, indicating that both structures are stable.In fact, monolayer SnP2S6 has been experimentally synthesized via chemical vapor transport technique along with the mechanical exfoliation method and used for preparing optoelectronic devices [37].Biaxial strain is employed in this study to investigate the impact of strain on the electronic properties of monolayer SnP2S6 and GeP2S6.Biaxial strain is defined as follows: where a0 represents the optimized lattice constant when the structure is unstrained and Δa denotes the variation in the lattice constant after the application of a specific strain in the xy plane.Electronic property calculations are conducted over a range of δ values spanning from −6% to +6%, representing the percentage change in the lattice constant.The negative value of δ indicates compressive strain, where the lattice is compressed along the xy plane.Conversely, the positive sign refers to tensile strain, where the lattice is stretched imaginary phonon frequencies were found throughout the entire Brillouin zone for monolayer GeP2S6 and the imaginary frequency near Γis negligible for monolayer SnP2S6, indicating that both structures are stable.In fact, monolayer SnP2S6 has been experimentally synthesized via chemical vapor transport technique along with the mechanical exfoliation method and used for preparing optoelectronic devices [37].Biaxial strain is employed in this study to investigate the impact of strain on the electronic properties of monolayer SnP2S6 and GeP2S6.Biaxial strain is defined as follows: where a0 represents the optimized lattice constant when the structure is unstrained and Δa denotes the variation in the lattice constant after the application of a specific strain in the xy plane.Electronic property calculations are conducted over a range of δ values spanning from −6% to +6%, representing the percentage change in the lattice constant.The negative value of δ indicates compressive strain, where the lattice is compressed along the xy plane.Conversely, the positive sign refers to tensile strain, where the lattice is stretched Biaxial strain is employed in this study to investigate the impact of strain on the electronic properties of monolayer SnP 2 S 6 and GeP 2 S 6 .Biaxial strain is defined as follows: where a 0 represents the optimized lattice constant when the structure is unstrained and ∆a denotes the variation in the lattice constant after the application of a specific strain in the xy plane.Electronic property calculations are conducted over a range of δ values spanning from −6% to +6%, representing the percentage change in the lattice constant.The negative value of δ indicates compressive strain, where the lattice is compressed along the xy plane.Conversely, the positive sign refers to tensile strain, where the lattice is stretched along the xy plane.By systematically varying the strain within this range, we were able to investigate the influence of strain on the electronic properties of the monolayers.This analysis provides valuable insights into the strain-dependent behavior of the materials, including changes in band structure, bandgap, and other electronic characteristics.The comprehensive range of strain values allows us to explore the full spectrum of electronic responses in SnP 2 S 6 and GeP 2 S 6 , enhancing our understanding of their potential applications in strain-engineered devices and electronic systems.
The electronic band structures of monolayer SnP 2 S 6 and GeP 2 S 6 structures under biaxial strain are investigated.Our calculations reveal that the total energy of the system exhibits a weak dependence on the applied biaxial strain, as depicted in Figure 3.The calculated total energy for the relaxed monolayer structures of SnP 2 S 6 and GeP 2 S 6 is determined to be −42.6808eV and −42.4794806 eV, respectively.The maximum change in total energy is only around 0.5 eV and 0.6 eV within the considered strain range.While the impact of strain on total energy may appear to be minimal, it plays a significant role in determining the material properties.Even slight changes in total energy can result in noticeable modifications to the electronic band structure, bandgap, and other electronic characteristics of the monolayers.Thus, comprehending the influence of strain on total energy is vital for the precise prediction and control of the electronic behavior exhibited by SnP 2 S 6 and GeP 2 S 6 under varying strain conditions.
along the xy plane.By systematically varying the strain within this range, we were able to investigate the influence of strain on the electronic properties of the monolayers.This analysis provides valuable insights into the strain-dependent behavior of the materials, including changes in band structure, bandgap, and other electronic characteristics.The comprehensive range of strain values allows us to explore the full spectrum of electronic responses in SnP2S6 and GeP2S6, enhancing our understanding of their potential applications in strain-engineered devices and electronic systems.
The electronic band structures of monolayer SnP2S6 and GeP2S6 structures under biaxial strain are investigated.Our calculations reveal that the total energy of the system exhibits a weak dependence on the applied biaxial strain, as depicted in Figure 3.The calculated total energy for the relaxed monolayer structures of SnP2S6 and GeP2S6 is determined to be −42.6808eV and −42.4794806 eV, respectively.The maximum change in total energy is only around 0.5 eV and 0.6 eV within the considered strain range.While the impact of strain on total energy may appear to be minimal, it plays a significant role in determining the material properties.Even slight changes in total energy can result in noticeable modifications to the electronic band structure, bandgap, and other electronic characteristics of the monolayers.Thus, comprehending the influence of strain on total energy is vital for the precise prediction and control of the electronic behavior exhibited by SnP2S6 and GeP2S6 under varying strain conditions.The electronic band structure of the unstrained SnP2S6 is presented in Figure 4a, where the VBM is located at the K point, while the CBM is located at the Γ point, with an indirect bandgap of 1.346 eV, which is consistent with previous calculation results (1.35 eV) [34].The contribution of different orbitals to the VBM and CBM are investigated by calculating the orbitals' resolved density of states.S-p orbitals contribute mostly to the VBM, while Sn-s orbitals have the highest contribution to the CBM, along with a minor contribution from S-p orbitals, as illustrated in Figure 4a.To further explore the effect of homogeneous biaxial strain on the electronic structure of SnP2S6, we examined the impact of a 4.0% BC strain.Under this strain condition, a notable shift in the CBM is observed from the Γ point to the K point, while the VBM remains localized at the K point.This shift results in an intriguing indirect-to-direct bandgap transition, as depicted in Figure 4b.The findings emphasize the significant role of strain engineering in modifying the electronic properties of SnP2S6.By manipulating the strain, it becomes possible to tailor the band structure and control the bandgap characteristics of the material.This newfound capability holds immense promise for the development of strain-tunable electronic devices, as well as for applications in fields such as optoelectronics and energy harvesting.Importantly, the information obtained from this study contributes to a deeper understanding of the underlying mechanisms governing the electronic behavior of SnP2S6 under strain, providing valuable insights for future research endeavors in this field.The electronic band structure of the unstrained SnP 2 S 6 is presented in Figure 4a, where the VBM is located at the K point, while the CBM is located at the Γ point, with an indirect bandgap of 1.346 eV, which is consistent with previous calculation results (1.35 eV) [34].The contribution of different orbitals to the VBM and CBM are investigated by calculating the orbitals' resolved density of states.S-p orbitals contribute mostly to the VBM, while Sn-s orbitals have the highest contribution to the CBM, along with a minor contribution from S-p orbitals, as illustrated in Figure 4a.To further explore the effect of homogeneous biaxial strain on the electronic structure of SnP 2 S 6 , we examined the impact of a 4.0% BC strain.Under this strain condition, a notable shift in the CBM is observed from the Γ point to the K point, while the VBM remains localized at the K point.This shift results in an intriguing indirect-to-direct bandgap transition, as depicted in Figure 4b.The findings emphasize the significant role of strain engineering in modifying the electronic properties of SnP 2 S 6 .By manipulating the strain, it becomes possible to tailor the band structure and control the bandgap characteristics of the material.This newfound capability holds immense promise for the development of strain-tunable electronic devices, as well as for applications in fields such as optoelectronics and energy harvesting.Importantly, the information obtained from this study contributes to a deeper understanding of the underlying mechanisms governing the electronic behavior of SnP 2 S 6 under strain, providing valuable insights for future research endeavors in this field.
Similarly, we investigated the electronic structure of the unstrained GeP 2 S 6 and strained system, as illustrated in Figure 4c,d.It is worth noting that the VBM and CBM of the unstrained GeP 2 S 6 monolayer are both located at the K point, indicating a direct bandgap semiconductor with a bandgap of 1.06 eV, which is consistent with the results of previous studies employing the same methodology (1.06 eV) [19].As depicted in Figure 4c, the majority of the orbital contribution to the VBM results from S-p orbitals, whereas Ge-s orbital dominates the CBM along with a minor contribution from S-p orbitals.Under the influence of a 2.0% BT strain, the CBM experienced a significant shift from the K point to the Γ point, while the VBM remained unchanged at the K point.This strain-induced phenomenon led to an intriguing transition from a direct bandgap to an indirect bandgap in the material.By manipulating the strain, researchers can fine-tune the electronic properties of GeP 2 S 6 , enabling the design and development of strain-tailored devices with enhanced performance.In contrast, molybdenum disulfide (MoS 2 ) undergoes a semiconductor-tometal transition under significantly large strains [38], exhibiting a pronounced shift in the valence band maximum from the high-symmetry point K to the Γ point when the tensile strain reaches 6%.This transition leads to a transformation of the initial direct bandgap into an indirect one [24].Therefore, compared to MoS 2 , both systems investigated in this study exhibit a stronger strain-tuning effect under a relatively small strain, which holds significant implications for researchers studying two-dimensional and nanoscale electronic materials.
strained system, as illustrated in Figure 4c,d.It is worth noting that the VBM and CBM of the unstrained GeP2S6 monolayer are both located at the K point, indicating a direct bandgap semiconductor with a bandgap of 1.06 eV, which is consistent with the results of previous studies employing the same methodology (1.06 eV) [19].As depicted in Figure 4c, the majority of the orbital contribution to the VBM results from S-p orbitals, whereas Ge-s orbital dominates the CBM along with a minor contribution from S-p orbitals.Under the influence of a 2.0% BT strain, the CBM experienced a significant shift from the K point to the Γ point, while the VBM remained unchanged at the K point.This strain-induced phenomenon led to an intriguing transition from a direct bandgap to an indirect bandgap in the material.By manipulating the strain, researchers can fine-tune the electronic properties of GeP2S6, enabling the design and development of strain-tailored devices with enhanced performance.In contrast, molybdenum disulfide (MoS2) undergoes a semiconductor-to-metal transition under significantly large strains [38], exhibiting a pronounced shift in the valence band maximum from the high-symmetry point K to the Γ point when the tensile strain reaches 6%.This transition leads to a transformation of the initial direct bandgap into an indirect one [24].Therefore, compared to MoS2, both systems investigated in this study exhibit a stronger strain-tuning effect under a relatively small strain, which holds significant implications for researchers studying two-dimensional and nanoscale electronic materials.In order to gain deeper insights into the orbital contributions to the band structure, we conducted an analysis of the projected band structure in the absence of strain.Our calculations reveal distinct orbital contributions to the conduction and valence band edges in both materials.It is evident that the conduction band edge of system SnP2S6 is primarily contributed by Sn-s orbitals, with additional small contributions from S-pz orbitals and a small contribution from S-px/py orbitals, while the valence band edge is predominantly attributed to the S-px orbital with a small contribution from the S-py orbital in Figure 5a-d.In order to gain deeper insights into the orbital contributions to the band structure, we conducted an analysis of the projected band structure in the absence of strain.Our calculations reveal distinct orbital contributions to the conduction and valence band edges in both materials.It is evident that the conduction band edge of system SnP 2 S 6 is primarily contributed by Sn-s orbitals, with additional small contributions from S-p z orbitals and a small contribution from S-p x /p y orbitals, while the valence band edge is predominantly attributed to the S-p x orbital with a small contribution from the S-p y orbital in Figure 5a-d.Similarly, the conduction band edge of the GeP 2 S 6 monolayer is primarily influenced by the Ge-s orbital, while the valence band edge is predominantly affected by the S-p x orbital with a small contribution from S-p z orbitals, as depicted in Figure 6a-d.This analysis provides valuable insights into the orbital characteristics responsible for the electronic properties of SnP 2 S 6 and GeP 2 S 6 .Understanding the specific orbital contributions aids the elucidation of the underlying mechanisms that govern the material's electronic behavior and facilitates the design and optimization of electronic devices based on these materials.
Similarly, the conduction band edge of the GeP2S6 monolayer is primarily influenced by the Ge-s orbital, while the valence band edge is predominantly affected by the S-px orbital with a small contribution from S-pz orbitals, as depicted in Figure 6a-d.This analysis provides valuable insights into the orbital characteristics responsible for the electronic properties of SnP2S6 and GeP2S6.Understanding the specific orbital contributions aids the elucidation of the underlying mechanisms that govern the material's electronic behavior and facilitates the design and optimization of electronic devices based on these materials.Similarly, the conduction band edge of the GeP2S6 monolayer is primarily influenced by the Ge-s orbital, while the valence band edge is predominantly affected by the S-px orbital with a small contribution from S-pz orbitals, as depicted in Figure 6a-d.This analysis provides valuable insights into the orbital characteristics responsible for the electronic properties of SnP2S6 and GeP2S6.Understanding the specific orbital contributions aids the elucidation of the underlying mechanisms that govern the material's electronic behavior and facilitates the design and optimization of electronic devices based on these materials.According to Figure 4a-d, the conversion from an indirect to direct bandgap for monolayer SnP 2 S 6 and the transformation from a direct to indirect bandgap for monolayer GeP 2 S 6 is primarily caused by the shift in the conduction band minimum.To further investigate this phenomenon, we plotted the variation in the energy of the conduction band edge state at the Γ point (CB-Γ) and at the K point (CB-K) as a function of strain as depicted in Figure 7a,b.It is evident that both systems exhibit an approximately linear decrease in energy with increasing tensile strain and an increase in energy with increasing compressive strain, for both the Γ and K points.The high-symmetry point Γ exhibits a higher rate under strain as compared to the K point, indicating the band edge near Γ point is more sensitive to strain.Hence, the positional relationship the energy bands at Γ and K points reverses under certain strain.In the case of system SnP 2 S 6 , when the applied strain is less than −4%, E CB-Γ is smaller than E CB-K located at K.However, when the applied strain exceeds −4%, E CB-Γ becomes larger than E CB-K , and the CBM shifts to the Γ point.Conversely, in the case of GeP 2 S 6 , when the strain is less than 1%, E CB-Γ is larger than E CB-K , and the CBM is located at the K point.However, when the strain exceeds 2%, E CB-Γ becomes smaller than E CB-K , and the CBM shifts to the Γ point.These findings indicate that the strain-induced changes in the position of the CBM play a crucial role in determining the bandgap characteristics of SnP 2 S 6 and GeP 2 S 6 .The precise control of strain allows for the manipulation of the electronic properties of these materials, providing opportunities for tailoring their behavior and optimizing their performance in various electronic applications.
According to Figure 4a-d, the conversion from an indirect to direct bandgap for monolayer SnP2S6 and the transformation from a direct to indirect bandgap for monolayer is primarily caused by the shift in the conduction band minimum.To further investigate this phenomenon, we plotted the variation in the energy of the conduction band edge state at the Γ point (CB-Γ) and at the K point (CB-K) as a function of strain as depicted in Figure 7a,b.It is evident that both systems exhibit an approximately linear decrease in energy with increasing tensile strain and an increase in energy with increasing compressive strain, for both the Γ and K points.The high-symmetry point Γ exhibits a higher rate under strain as compared to the K point, indicating the band edge near Γ point is more sensitive to strain.Hence, the positional relationship the energy bands at Γ and K points reverses under certain strain.In the case of system SnP2S6, when the applied strain is less than −4%, ECB-Γ is smaller than ECB-K located at K.However, when the applied strain exceeds −4%, ECB-Γ becomes larger than ECB-K, and the CBM shifts to the Γ point.Conversely, in the case of GeP2S6, when the strain is less than 1%, ECB-Γ is larger than ECB-K, and the CBM is located at the K point.However, when the strain exceeds 2%, ECB-Γ becomes smaller than ECB-K, and the CBM shifts to the Γ point.These findings indicate that the strain-induced changes in the position of the CBM play a crucial role in determining the bandgap characteristics of SnP2S6 and GeP2S6.The precise control of strain allows for the manipulation of the electronic properties of these materials, providing opportunities for tailoring their behavior and optimizing their performance in various electronic applications.Figure 7 reveals that the two systems have the same energy level shift tendency.The energy level of the Γ point and K point for the conduction band in both materials increases with increasing compressive strain and decreases with increasing tensile strain.Furthermore, the energy level of the Γ point for the conduction band in both materials is more sensitive than that of the K point.The difference is that the energy band types of monolayer SnP2S6 and monolayer GeP2S6 at zero strain are inconsistent.Monolayer SnP2S6 with an indirect bandgap will transform into a direct bandgap semiconductor; since the Γ point is more sensitive, the energy of the Γ point rises faster than that of the K point, and the energy positions are reversed at 2% compressive strain.However, monolayer GeP2S6 with a direct bandgap will transform into an indirect bandgap semiconductor since the energy of the Γ point drops faster than that of the K point, and the energy positions are reversed at 4% tensile strain.Furthermore, we believe these insights can be applied to other 2D materials because other 2D systems also have similar phenomenon with different strain sensitivities at different k points.Therefore, these insights provide crucial references and guidance for researchers exploring strain modulation effects in other two-dimensional materials.The energy level of the Γ point and K point for the conduction band in both materials increases with increasing compressive strain and decreases with increasing tensile strain.Furthermore, the energy level of the Γ point for the conduction band in both materials is more sensitive than that of the K point.The difference is that the energy band types of monolayer SnP 2 S 6 and monolayer GeP 2 S 6 at zero strain are inconsistent.Monolayer SnP 2 S 6 with an indirect bandgap will transform into a direct bandgap semiconductor; since the Γ point is more sensitive, the energy of the Γ point rises faster than that of the K point, and the energy positions are reversed at 2% compressive strain.However, monolayer GeP 2 S 6 with a direct bandgap will transform into an indirect bandgap semiconductor since the energy of the Γ point drops faster than that of the K point, and the energy positions are reversed at 4% tensile strain.Furthermore, we believe these insights can be applied to other 2D materials because other 2D systems also have similar phenomenon with different strain sensitivities at different k points.Therefore, these insights provide crucial references and guidance for researchers exploring strain modulation effects in other two-dimensional materials.
Furthermore, we investigated the modulation of the bandgap as a function of strain along distinct symmetry axes, namely K-K and K-Γ.Analyzing the bandgap behavior, we observed that in the case of SnP 2 S 6 , when the strain is less than −4%, the bandgaps along the K-K and K-Γ directions exhibit remarkable proximity, suggesting a nearly degenerate nature.This finding implies that the band structure is relatively insensitive to strain in this range.However, as the strain exceeds −4%, an intriguing trend emerges.The disparity between the bandgaps along K-K and K-Γ becomes increasingly prominent, as depicted in Figure 8a.Specifically, we found that the bandgap undergoes only subtle changes under compressive but exhibits significant variations under tensile strain.This straininduced modification in the bandgap characteristics highlights the potential for tailoring the electronic properties of SnP 2 S 6 through strain engineering.On the other hand, our investigation of monolayer GeP 2 S 6 revealed a different pattern.Under compressive strain, the distinction between the bandgaps along the K-K and K-Γ directions is relatively more pronounced compared to that of SnP 2 S 6 .This implies that the band structure of GeP 2 S 6 is more sensitive to compressive strain, leading to a significant difference in the bandgap behavior between these two symmetry directions.Figure 8b illustrates the substantial changes in the bandgap under both compressive and tensile strains, further highlighting the potential for strain engineering in manipulating the electronic properties of GeP 2 S 6 .By precisely controlling the strain, researchers can fine-tune the bandgap characteristics of SnP 2 S 6 and GeP 2 S 6 , enabling the development of tailored electronic devices with enhanced performance.The ability to manipulate the band structure through strain engineering opens up new avenues for exploring novel device concepts and applications.Furthermore, our study contributes to the broader understanding of strain-dependent phenomena in 2D materials.By unraveling the intricate relationship between strain and bandgap variations, we gain deeper insights into the underlying mechanisms governing the electronic behavior of these materials.This knowledge serves as a valuable foundation for future research endeavors in the field of strain engineering and paves the way for the discovery and optimization of other 2D materials with tailored electronic properties.The distinct variations observed under different strain conditions offer exciting possibilities for bandgap engineering and the design of advanced electronic and optoelectronic devices.
observed that in the case of SnP2S6, when the strain is less than −4%, the bandgaps along the K-K and K-Γ directions exhibit remarkable proximity, suggesting a nearly degenerate nature.This finding implies that the band structure is relatively insensitive to strain in this range.However, as the strain exceeds −4%, an intriguing trend emerges.The disparity between the bandgaps along K-K and K-Γ becomes increasingly prominent, as depicted in Figure 8a.Specifically, we found that the bandgap undergoes only subtle changes under compressive strain but exhibits significant variations under tensile strain.This straininduced modification in the bandgap characteristics highlights the potential for tailoring the electronic properties of SnP2S6 through strain engineering.On the other hand, our investigation of monolayer GeP2S6 revealed a different pattern.Under compressive strain, the distinction between the bandgaps along the K-K and K-Γ directions is relatively more pronounced compared to that of SnP2S6.This implies that the band structure of GeP2S6 is more sensitive to compressive strain, leading to a significant difference in the bandgap behavior between these two symmetry directions.Figure 8b illustrates the substantial changes in the bandgap under both compressive and tensile strains, further highlighting the potential for strain engineering in manipulating the electronic properties of GeP2S6.By precisely controlling the strain, researchers can fine-tune the bandgap characteristics of SnP2S6 and GeP2S6, enabling the development of tailored electronic devices with enhanced performance.The ability to manipulate the band structure through strain engineering opens up new avenues for exploring novel device concepts and applications.Furthermore, our study contributes to the broader understanding of strain-dependent phenomena in 2D materials.By unraveling the intricate relationship between strain and bandgap variations, we gain deeper insights into the underlying mechanisms governing the electronic behavior of these materials.This knowledge serves as a valuable foundation for future research endeavors in the field of strain engineering and paves the way for the discovery and optimization of other 2D materials with tailored electronic properties.The distinct variations observed under different strain conditions offer exciting possibilities for bandgap engineering and the design of advanced electronic and optoelectronic devices.In addition, we have also investigated the electronic structures considering spin-orbit coupling (SOC).When SOC is included as shown in Figure 9, the band splitting is observed for some bands, especially at high-symmetry points in both systems.However, no noticeable change in the electronic properties near the band edges is observed.The band structures with SOC are highly similar to those without SOC.Specifically, an indirect bandgap semiconductor with a bandgap of 1.35 eV remains for SnP2S6 with SOC, while In addition, we have also investigated the electronic structures considering spinorbit coupling (SOC).When SOC is included as shown in Figure 9, the band splitting is observed for some bands, especially at high-symmetry points in both systems.However, no noticeable change in the electronic properties near the band edges is observed.The band structures with SOC are highly similar to those without SOC.Specifically, an indirect bandgap semiconductor with a bandgap of 1.35 eV remains for SnP 2 S 6 with SOC, while for GeP 2 S 6 with SOC, a direct bandgap semiconductor with a bandgap of 1.06 eV remains.Moreover, the positions of the conduction band minimum and valence band maximum remain the same as those without SOC.Based on these results, we conclude that the inclusion of SOC does not significantly affect the electronic structures as discussed in the previous sections.Therefore, SOC was not considered in the previous analysis.
for GeP2S6 with SOC, a direct bandgap semiconductor with a bandgap of 1.06 eV remains.Moreover, the positions of the conduction minimum and valence band maximum remain the same as those without SOC.Based on these results, we conclude that the inclusion of SOC does not significantly affect the electronic structures as discussed in the previous sections.Therefore, SOC was not considered in the previous analysis.

Calculation Method
Density functional theory (DFT) [39] calculation was performed to investigate the electronic properties of monolayer metal chalcogen phosphates of SnP2S6 and GeP2S6 using the Vienna ab initio simulation package (VASP) [40].The projector-augmented wave method was adopted with a cutoff energy of 450 eV [40].The Perdew-Burke-Ernzerhof (PBE) [41] exchange-correlation function, based on the generalized gradient approximation (GGA), was employed in this study.To minimize the number of interactions between vertically periodic layers, a vacuum layer of 25 Å was set in the z direction for the monolayer structures.A 15 × 15 × 1 Monkhorst-Pack k-grid mesh was employed for Brillouin zone sampling.The phonon spectra of monolayer SnP2S6 and monolayer GeP2S6 were calculated using the finite displacement method with a low repetition rate of a 4 × 4 × 1 supercell.The atomic positions were iteratively optimized until the magnitude of the Hellmann-Feynman forces acting on all atoms was below 0.01 eV/Å, ensuring the convergence of the system.

Conclusions
In summary, we have conducted a comprehensive investigation on the electronic properties of monolayer SnP2S6 and GeP2S6 under biaxial strain.The results demonstrate that the monolayer SnP2S6 with an indirect bandgap of 1.346 eV undergoes a transition from an indirect to a direct bandgap under 4.0% uniform BC strain.Additionally, we report a direct-to-indirect bandgap transition in the monolayer semiconductor GeP2S6 under a uniform BT strain of 2.0%.The transition occurs because the high-symmetry point Γ is more sensitive to strain than K. Overall, our findings highlight the significant role of strain engineering in modulating the electronic properties of monolayer SnP2S6 and GeP2S6.The ability to induce bandgap transitions through strain opens up avenues for designing innovative devices with enhanced performance and functionality.Further exploration of strain effects and the development of strain engineering techniques will undoubtedly contribute to the advancement of next-generation electronic and optoelectronic devices based on these materials.

Calculation Method
Density functional theory (DFT) [39] calculation was performed to investigate the electronic properties of monolayer metal chalcogen phosphates of SnP 2 S 6 and GeP 2 S 6 using the Vienna ab initio simulation package (VASP) [40].The projector-augmented wave method was adopted with a cutoff energy of 450 eV [40].The Perdew-Burke-Ernzerhof (PBE) [41] exchange-correlation function, based on the generalized gradient approximation (GGA), was employed in this study.To minimize the number of interactions between vertically periodic layers, a vacuum layer of 25 Å was set in the z direction for the monolayer structures.A 15 × 15 × 1 Monkhorst-Pack k-grid mesh was employed for Brillouin zone sampling.The phonon spectra of monolayer SnP 2 S 6 and monolayer GeP 2 S 6 were calculated using the finite displacement method with a low repetition rate of a 4 × 4 × 1 supercell.The atomic positions were iteratively optimized until the magnitude of the Hellmann-Feynman forces acting on all atoms was below 0.01 eV/Å, ensuring the convergence of the system.

Conclusions
In summary, we have conducted a comprehensive investigation on the electronic properties of monolayer SnP 2 S 6 and GeP 2 S 6 under biaxial strain.The results demonstrate that the monolayer SnP 2 S 6 with an indirect bandgap of 1.346 eV undergoes a transition from an indirect to a direct bandgap under 4.0% uniform BC strain.Additionally, we report a direct-to-indirect bandgap transition in the monolayer semiconductor GeP 2 S 6 under a uniform BT strain of 2.0%.The transition occurs because the high-symmetry point Γ is more sensitive to strain than K. Overall, our findings highlight the significant role of strain engineering in modulating the electronic properties of monolayer SnP 2 S 6 and GeP 2 S 6 .The ability to induce bandgap transitions through strain opens up avenues for designing innovative devices with enhanced performance and functionality.Further exploration of strain effects and the development of strain engineering techniques will undoubtedly contribute to the advancement of next-generation electronic and optoelectronic devices based on these materials.

Figure 1 .
Figure 1.(a) Top and (b) side views of the monolayer MP2S6 are shown in the figure.The M (M = Sn, Ge) atoms, P atoms, and S atoms are represented by blue, green, and yellow spheres, respectively.

Figure 1 .
Figure 1.(a) Top and (b) side views of the monolayer MP 2 S 6 are shown in the figure.The M (M = Sn, Ge) atoms, P atoms, and S atoms are represented by blue, green, and yellow spheres, respectively.

Figure 1 .
Figure 1.(a) Top and (b) side views of the monolayer MP2S6 are shown in the figure.The M (M = Sn, Ge) atoms, P atoms, and S atoms are represented by blue, green, and yellow spheres, respectively.

Figure 5 .
Figure 5.The orbital-resolved band structure of SnP2S6 under unstrained conditions for the Sn-s (a), S-px (b), S-py (c), and S-pz (d) orbitals is as follows: Red circles represent the corresponding VBM and CBM in the band structures.

Figure 6 .
Figure 6.The orbital-resolved band structure of GeP2S6 under unstrained conditions for the Ges (a), S-px (b), S-py (c), and S-pz (d) orbitals is as follows: Red circles represent the corresponding VBM and CBM in the band structures.

Figure 5 .
Figure 5.The orbital-resolved band structure of SnP 2 S 6 under unstrained conditions for the Sn-s (a), S-px (b), S-py (c), and S-pz (d) orbitals is as follows: Red circles represent the corresponding VBM and CBM in the band structures.

Figure 5 .
Figure 5.The orbital-resolved band structure of SnP2S6 under unstrained conditions for the Sn-s (a), S-px (b), S-py (c), and S-pz (d) orbitals is as follows: Red circles represent the corresponding VBM and CBM in the band structures.

Figure 6 .
Figure 6.The orbital-resolved band structure of GeP2S6 under unstrained conditions for the Ges (a), S-px (b), S-py (c), and S-pz (d) orbitals is as follows: Red circles represent the corresponding VBM and CBM in the band structures.

Figure 6 .
Figure 6.The orbital-resolved band structure of GeP 2 S 6 under unstrained conditions for the Ge-s (a), S-px (b), S-py (c), and S-pz (d) orbitals is as follows: Red circles represent the corresponding VBM and CBM in the band structures.

Figure 7 .
Figure 7.The variation in CBM energy at the Γ point (CB-Γ) and at the K point (CB-K) with strain for SnP2S6 (a) and GeP2S6 (b).

Figure 7 .
Figure 7.The variation in CBM energy at the Γ point (CB-Γ) and at the K point (CB-K) with strain for SnP 2 S 6 (a) and GeP 2 S 6 (b).

Figure 7
Figure 7 reveals that the two systems have the same energy level shift tendency.The energy level of the Γ point and K point for the conduction band in both materials increases with increasing compressive strain and decreases with increasing tensile strain.Furthermore, the energy level of the Γ point for the conduction band in both materials is more sensitive than that of the K point.The difference is that the energy band types of monolayer SnP 2 S 6 and monolayer GeP 2 S 6 at zero strain are inconsistent.Monolayer SnP 2 S 6 with an indirect bandgap will transform into a direct bandgap semiconductor; since the Γ point is more sensitive, the energy of the Γ point rises faster than that of the K point, and the energy positions are reversed at 2% compressive strain.However, monolayer GeP 2 S 6 with a direct bandgap will transform into an indirect bandgap semiconductor since the energy of the Γ point drops faster than that of the K point, and the energy positions are reversed at 4% tensile strain.Furthermore, we believe these insights can be applied to other 2D materials because other 2D systems also have similar phenomenon with different strain sensitivities at different k points.Therefore, these insights provide crucial references and guidance for researchers exploring strain modulation effects in other two-dimensional materials.Furthermore, we investigated the modulation of the bandgap as a function of strain along distinct symmetry axes, namely K-K and K-Γ.Analyzing the bandgap behavior, we observed that in the case of SnP 2 S 6 , when the strain is less than −4%, the bandgaps along the K-K and K-Γ directions exhibit remarkable proximity, suggesting a nearly degenerate nature.This finding implies that the band structure is relatively insensitive to strain in this range.However, as the strain exceeds −4%, an intriguing trend emerges.The disparity between the bandgaps along K-K and K-Γ becomes increasingly prominent, as depicted in Figure8a.Specifically, we found that the bandgap undergoes only subtle changes under

Figure 8 .
Figure 8.(a) Variations in bandgap along the high-symmetry directions K-K and K-Γ in SnP2S6.(b) Bandgap fluctuations along the high-symmetry directions K-K and K-Γ in GeP2S6.

Figure 8 .
Figure 8.(a) Variations in bandgap along the high-symmetry directions K-K and K-Γ in SnP 2 S 6 .(b) Bandgap fluctuations along the high-symmetry directions K-K and K-Γ in GeP 2 S 6 .