New Polymorphs of 2D Indium Selenide with Enhanced Electronic Properties

Abstract

The 2D semiconductor indium selenide (InSe) has attracted significant interest due its unique electronic band structure, high electron mobility, and wide tunability of its band gap energy achieved by varying the layer thickness. All these features make 2D InSe a potential candidate for advanced electronic and optoelectronic applications. Here, the discovery of new polymorphs of InSe with enhanced electronic properties is reported. Using a global structure search that combines artificial swarm intelligence with first‐principles energetic calculations, polymorphs that consist of a centrosymmetric monolayer belonging to the point group D _3d are identified, distinct from well‐known polymorphs based on the D _3h monolayers that lack inversion symmetry. The new polymorphs are thermodynamically and kinetically stable, and exhibit a wider optical spectral response and larger electron mobilities compared to the known polymorphs. Opportunities to synthesize these newly discovered polymorphs and viable routes to identify them by X‐ray diffraction, Raman spectroscopy, and second harmonic generation experiments are discussed.

New polymorphs of 2D semiconductor indium selenide, which consist of the monolayer with different symmetry, are discovered by artificial intelligence‐based computational material design. The new polymorphs are thermodynamically and kinetically stable, and exhibit a wider optical spectral response and larger electron mobilities compared to the known polymorphs.

1. Introduction and Background

Atomically thin 2D layered van der Waals (vdW) semiconductors with high carrier mobility and tunable band gap energy hold promise for next‐generation nanoscale electronics and optoelectronics. In particular, InSe has attracted significant research interest.^{[ 1} , ² , ³ , ⁴ , ⁵ , ⁶ , ^{7 ]} Amongst the 2D layered semiconductors, it exhibits the highest room‐temperature electron mobility (≈10³ cm² V⁻¹ s⁻¹).^{[ 6} , ⁷ , ^{8 ]} This arises from the relatively small effective mass of the conduction band (CB) electrons and dispersive In‐s antibonding CB states.^{[ 9 ]} Also, it has a direct band gap of ≈1.25 eV in the bulk form,^{[ 10 ]} which increases by nearly 1 eV in monolayer InSe (2.1 eV)^{[ 11 ]} due to the cooperative effects of quantum confinement and interlayer coupling.^{[ 4} , ¹² , ¹³ , ¹⁴ , ¹⁵ , ^{16 ]} Despite its weak optical emission in thin layers^{[ 11} , ^{12 ]} and the low hole mobility,^{[ 1 ]} InSe has already found promising applications in field‐effect transistors,^{[ 1} , ^{7 ]} photodetectors,^{[ 3} , ¹⁷ , ^{18 ]} and image sensors.^{[ 19 ]}

Through mechanical exfoliation^{[ 3} , ^{12 ]} and epitaxial growth^{[ 20 ]} approaches, high‐quality InSe layers can be fabricated. To date, three bulk polymorphs (i.e., β, γ, and ε phases) were reported,^{[ 1} , ⁷ , ¹⁸ , ²¹ , ²² , ^{23 ]} which represent different stacking patterns of the common D _3h monolayer structure. In the monolayer In‐ and Se‐atoms are arranged in the sequence of [Se–In–In–Se], forming a graphene‐like honeycomb lattice but without inversion symmetry, in which vertically aligned In–Se atomic pairs occupy different sublattice sites. It is known that many 2D materials possess different structures of the monolayer, such as graphene and graphdiyne allotropes of monolayer carbon,^{[ 24} , ^{25 ]} trigonal prismatic (2H) and octahedral (1T) monolayers of transition metal dichacolcogenides,^{[ 26} , ^{27 ]} black phosphorene and blue phosphorene of monolayer phosphorus.^{[ 28} , ^{29 ]} Different monolayer structures endow 2D materials with variable properties useful for different applications. For example, the 2H‐monolayer derived phases of MoS₂ are promising semiconductors for electronic and optoelectronic devices,^{[ 30} , ³¹ , ³² , ³³ , ³⁴ , ³⁵ , ^{36 ]} whereas the 1T‐monolayer derived phases are conductive metals for electrocatalytic^{[ 37} , ^{38 ]} and energy storage^{[ 27 ]} applications. The polymorphic nature of the monolayers is clearly an enticing feature of many 2D materials. However, to date, only the D _3h monolayer structure was reported for InSe.^{[ 1} , ² , ³ , ⁴ , ⁵ , ⁶ , ⁷ , ⁸ , ⁹ , ¹⁰ , ¹¹ , ¹² , ¹⁷ , ¹⁸ , ¹⁹ , ²⁰ , ²¹ , ²² , ^{23 ]} Thus, it may be worthwhile to explore the existence of other InSe monolayers with potentially emergent properties.

Here, we report on a new monolayer polymorph of InSe. Through a global structure search study that combines artificial swarm intelligence with first‐principles energetic calculations, we identify a new monolayer polymorph that belongs to the point group D _3d with inversion symmetry, distinct from the known non‐centrosymmetric D _3h monolayer. It is thermodynamically comparable in energy with the D _3h, showing robust phonon and thermal stability, as well as kinetic stability with respect to a transformation to D _3h. Three bulk phases based on different stacks of the D _3d monolayer are predicted, one of which show an enhanced band gap tunability with varying layer thickness and higher electron mobility compared to the other phases. We discuss how the new phases could be identified by X‐ray diffraction (XRD), Raman spectroscopy, and second harmonic generation (SHG) measurements, thus opening realistic prospects for the experimental observation and investigation of the predicted polymorphs.

2. Results and Discussion

2.1. Searching for Stable Polymorphs of InSe with Swarm Intelligence Guided Structure Searches

We performed global structure searches at the chemical composition of In:Se = 1:1, which involves nearly 3600 structure points sampled from the free energy landscape. Figure 1a shows the evolution of the energy of the sampled structures as a function of search generation. The zoomed‐in low‐energy region is dominated by three types of polymorphs (green, red, and blue dots in Figure 1a), each of which is composed of the common monolayer structure (shown in corresponding colored box of Figure 1b). The higher‐energy structures consist of the monolayer in a modified version of the InSe structure (point group C _2h, Figure 1b, upper panel), a high‐pressure phase discovered experimentally.^{[ 39 ]} It is actually a non‐layered phase with strong covalent bonding. The two low‐energy types (blue and red dots in Figure 1a) have comparable energy within several meV per atom. The experimentally known non‐centrosymmetric D _3h monolayer constitutes the polymorphs shown in Figure 1b middle panel, corresponding to the β(D _3h),^{[ 40 ]} γ(D _3h),^{[ 41 ]} or ε(D _3h)^{[ 42 ]} phases with symmetries of P6₃/mmc, R3m, and $P\overline{6}m2$, respectively. The polymorphs shown in the bottom panel of Figure 1b are composed of a strikingly new monolayer with point group D _3d. Its graphene‐like honeycomb lattice is occupied by In–Se atomic pairs along the vertical direction with inversion symmetry, distinct from the vertically aligned In–Se atomic pairs with mirror symmetry and point group D _3h. The centrosymmetric feature of the D _3d monolayer endows its constituted polymorphs with emergent electronic and optical properties, as described below.

Figure 1.

a) Evolution of the energy of the predicted structures as a function search generation. The zoomed‐in low‐energy region is dominated by three types of polymorphs (shown in green, red, and blue dots). b) The monolayer structures of three types of polymorphs (C _2h, D _3h, and D _3d) shown in the corresponding colored boxes. c) Side views of the three energetically favorable bulk phases, named as δ(D _3d), ω(D _3d), and ϕ(D _3d), respectively. The rectangular frames with arrows are a visual aid to the particular stacking pattern of the layers.

By systematically analyzing the structure search results and additional calculations of candidate stacks based on the D _3d monolayer, we found three energetically favorable bulk phases, named as δ(D _3d), ω(D _3d), and ϕ(D _3d) (Figure 1c, with explicit structural information given in Table S1, Supporting Information). They are in space groups $P\overline{3}m1$, P6₃ mc, and $R\overline{3}m$, featuring vertical stacking patterns AA, AB (with adjacent monolayers in‐plane rotated by 180° and shifted by 1/3 unit), and ABC (with monolayers shifted by 1/3 unit), respectively. Comprehensive calculations using different vdW functionals indicate that they have comparable energies within 5 meV per atom (Figure S1, Supporting Information), of which the ϕ(D _3d) phase is energetically most favored. Comparing the new δ(D _3d), ω(D _3d), and ϕ(D _3d) phases with the experimentally known ones of the D _3h monolayer (β(D _3h), γ(D _3h), and ε(D _3h)), we found that the explicit energy differences are less than 8 meV per atom regardless of the specific vdW functional used. The energy difference is much smaller than that between the 2H and 1T phases of MoS₂ (≈70 meV per atom),^{[ 43 ]} or of that between black and blue phosphorene (≈14 meV per atom).^{[ 44 ]} This thermodynamically favorable condition implies that there is reasonable likelihood of synthesizing the newly found InSe polymorphs.

2.2. Robust Kinetic, Thermal, and Phonon Stability of the Newly Found Polymorphs

Even though one material phase is thermodynamically favored, its kinetic transformation into another favored phase may prohibit its stabilization. Kinetic stability of the new D _3d InSe monolayer is examined by calculating the transition barrier between D _3d and the known D _3h monolayer. By identifying feasible transition pathways involving a saddle‐point transition state, we obtained a quite large activation barrier of ≈150 meV per atom (Figure 2a). This indicates that the D _3d monolayer can be kinetically stabilized once it is synthesized. We further examined the thermal stability of the new D _3d monolayer using molecular dynamics simulations at 300 and 500 K (Figure 2b). All atoms vibrate around their equilibrium positions at both temperatures and the In–Se network of the monolayer is retained. The time‐dependent energy fluctuation at 500 K is expectedly larger, but the equilibrium structure remains anchored to the D _3d symmetry. Good thermal stability above room temperature indicates the possibility of growing D _3d monolayer based polymorphs via controlled high‐temperature methods, such as chemical vapor deposition that has been recently used to grow ultrathin InSe flakes.^{[ 45 ]} Finally, we examined the lattice dynamical stability by calculating the phonon spectrum of the D _3d monolayer. As shown in Figure 2c, the dynamical stability of the D _3d monolayer lattice is evidenced by the absence of imaginary phonon modes at 0 K. The phonon spectra of the δ(D _3d), ω(D _3d), and ϕ(D _3d) bulk polymorphs were also calculated (Figure 2d). All polymorphs demonstrate robust phonon stability. Although the phonon spectra of the three polymorphs look similar, the shear (in red) and breathing (in blue) mode branches show substantial differences, indicating different degrees of interlayer coupling, as discussed further below.

Figure 2.

a) Energy barrier and atomic structures during the transformation from the D _3d to the D _3h monolayer. The transition‐state (TS) structure is indicated. b) Fluctuations of the total potential energy of D _3d monolayer during the molecular dynamics simulation at 300 and 500 K, respectively. c) Phonon spectrum of the D _3d monolayer. d) Phonon spectra of δ(D _3d), ω(D _3d), and ϕ(D _3d). The shear and breathing mode branches are shown in red and blue, respectively.

2.3. Emergent Electronic Properties of the D _3d Monolayer Based Polymorphs

As shown in Figure 3a, the electronic band structure of the new D _3d monolayer show similar band‐edge states in proximity to the Γ point as for the known D _3h monolayer. The dispersive conduction band minimum is located at Γ. Two nearly degenerate weakly dispersed valence band maxima (VBM) are observed along the Γ–M and Γ–K directions. Thus, both D _3h and D _3d monolayers are indirect band gap semiconductors. Both conduction and valence band‐edge states are dominated by the Se‐p_z orbital. The band gap energy calculated with the hybrid functional approach^{[ 46 ]} is 2.30 eV, slightly smaller than the value of 2.39 eV for the D _3h monolayer. By stacking the D _3d monolayers into bulk δ(D _3d), ω(D _3d), and ϕ(D _3d) polymorphs, the band gap changes from indirect to direct and its value decreases (Figures S2–S4, Supporting Information). Figure 3b shows the evolution of the band gap energy with increasing number of layers for the three polymorphs and for the D _3h monolayer based β(D _3h) and γ(D _3h) polymorphs. To remedy the band gap underestimation issue of the density functional theory (DFT) calculations, we adopted a scissor operator with magnitude equal to the band gap energy difference between the DFT and hybrid functional calculations for the D _3d/D _3h monolayer to all the multiple‐layered cases. The resulting band gaps for the δ(D _3d), ω(D _3d), and ϕ(D _3d) phases are 0.81, 0.96, and 1.05 eV, respectively. Amongst all phases, bulk δ(D _3d) shows the smallest gap value, which is beneficial for applications that require an optical response in the infrared spectral range. Also, with varying the layer thickness the δ(D _3d) phase spans a wider energy range of energy gaps (≈1.5 eV) than the known β(D _3h) and γ(D _3h) polymorphs (≈1.2–1.3 eV). The decrease in the band gap from monolayer to multiple layers InSe is predominantly caused by the upward shift of the VBM (Figure S5, Supporting Information). Amongst all polymorphs, the δ(D _3d) phase shows the most dramatic change in the VBM energy, consistent with its largest band gap variation (Figure 3b). This originates from its shortest interlayer distance (8.27 Å, Figure 3c) associated with the D _3d symmetry and the specific layer stacking pattern. As a result, the stronger interlayer coupling occurs in the δ(D _3d) phase, which is evidenced by the stronger directional charge overlapping/chemical bonding between the Se‐atoms in adjacent layers (Figure 3c). This is responsible for the more pronounced upward shift of the VBM and thus the larger band gap change.

Figure 3.

a) Calculated band structures of the D _3d and D _3h monolayers. The band structures have been projected onto atomic orbitals with the blue color representing Se‐p_z and red representing Se‐p_{x / y}. b) Evolution of the band gap with increasing number of layers of the predicted three D _3d monolayer based polymorphs and two known D _3h monolayer based polymorphs. c) Interlayer differential charge densities of bilayer δ(D _3d), ω(D _3d), ϕ(D _3d), β(D _3h), and γ(D _3h), respectively. The isosurface value is set to 1 × 10⁻⁴ electrons per ų. The charge accumulation and depletion are shown in yellow and blue, respectively.

The calculated electron mobility of the new D _3d monolayer is 912 and 945 cm² V⁻¹ s⁻¹ along zigzag and armchair directions, respectively, higher than the values of 689 and 801 cm² V⁻¹ s⁻¹ of the D _3h monolayer calculated in the same approach (Table S2, Supporting Information). The increase in mobility originates primarily from the smaller electron effective masses (0.18 m ₀ and 0.19 m ₀) of the D _3d monolayer compared with those of the D _3h monolayer (0.20 m ₀ and 0.23 m ₀). With increasing number of layers, the δ(D _3d), ω(D _3d), and ϕ(D _3d) phases show remarkably enhanced electron mobility, reaching values of 9500–13 000 cm² V⁻¹ s⁻¹ and 10 000–14 000 cm² V⁻¹ s⁻¹ for six monolayers along zigzag and armchair directions, respectively (Figure 4 ). This resembles the behavior of the existing β(D _3h) and γ(D _3h) polymorphs,^{[ 4} , ^{47 ]} which we ascribe to the decreasing electron effective mass (Table S2, Supporting Information) and a likely increase in the carrier scattering time due to reduced electron–phonon coupling in the multiple layers. Here, the interlayer coupling or interaction plays an essential role. We note that for a given layer thickness the δ(D _3d), ω(D _3d), and ϕ(D _3d) phases exhibit a higher electron mobility than the β(D _3h) and γ(D _3h) phases. In particular, the δ(D _3d) phase, which has the strongest interlayer coupling, exhibits the highest electron mobility: for six monolayers, the mobility is enhanced by a factor of about 1.4 and 1.6 along zigzag and armchair directions with respect to the β(D _3h) phase. Therefore, compared to polymorphs based on the D _3h monolayer, polymorphs based on the D _3d monolayer have more favorable properties for electronic and optoelectronic applications.

Figure 4.

Evolution of electron mobilities along the a) armchair and b) zigzag directions with increasing number of layers for three D _3d monolayer based polymorphs and two known D _3h monolayer based polymorphs.

We note that the deformation potential theory used for carrier mobility calculations may lead to overestimation of mobility values, as it only considers the predominate scattering by the long wavelength longitudinal acoustic phonons and neglects the other important scatterings such as the one by the longitudinal optical (LO) phonons that has been demonstrated essential for polar 2D semiconductors.^{[ 48 ]} By fully considering all the electron–phonon coupling contributions, 2D semiconductor InSe has an optimal electronic structure feature for electron, which is favorable for high electron mobility.^{[ 49 ]} While such calculation is computationally challenging and beyond the scope of the current structure search study, we evaluated the electron–phonon coupling strength for the LO phonon by using the frozen‐phonon approach^{[ 50 ]} for both the known D _3h monolayer and the new D _3d monolayer. The results indicate a similar amount of the change in the CBM (with a difference less than 8 meV) with respect to the magnitude of the LO phonon at zone center (within 0.1 Å). This means that the scattering of conductive electron by the LO phonon in the two monolayers is similar. Therefore, while our calculated carrier mobility values may be overestimated, the demonstrated enhancement of the new D _3d monolayer based polymorphs compared with the known D _3h monolayer based ones is not affected.

2.4. Experimental Identification of the Predicted D _3d Monolayer Based Polymorphs

XRD,^{[ 7} , ¹² , ^{20 ]} Raman spectroscopy,^{[ 17} , ¹⁸ , ^{51 ]} and SHG measurements^{[ 22} , ^{45 ]} have been recently used to identify the D _3h monolayer based β(D _3h) and γ(D _3h) polymorphs. We suggest that the same characterization approaches can be adopted to identify the predicted D _3d monolayer based polymorphs and distinguish them from those based on the known D _3h monolayer. Figure 5a shows the simulated XRD patterns of δ(D _3d), ω(D _3d), and ϕ(D _3d) polymorphs, compared with the experimental powder diffraction file (PDF) of β(D _3h) and γ(D _3h). While all polymorphs show two common low‐angle peaks at about 10.6° and 21.2°, the δ(D _3d) and ω(D _3d) phases show a rather different peak distribution between 25° and 30° with respect to β(D _3h) and γ(D _3h). For the ϕ(D _3d) phase, the XRD pattern is quite similar to that of γ(D _3h), with a slight shift toward low‐angles. However, as shown in Figure 5b, the ϕ(D _3d) and γ(D _3h) phases can be distinguished by their Raman‐active phonon modes. The two phases have similar Raman modes in the low‐ and high‐frequency ranges; in contrast, between 150 and 200 cm⁻¹, they show distinct features: for γ(D _3h), there are two nearly degenerate E‐modes and one A ₁‐mode, but there is only one E _g‐mode in ϕ(D _3d). We note that the existence of the A ₁ mode in γ(D _3h) has not been always reported in InSe,^{[ 12} , ¹⁷ , ¹⁸ , ⁵¹ , ^{52 ]} which in retrospect may be a hint toward the identification of the ϕ(D _3d) phase. In addition, since the new D _3d monolayer has inversion symmetry, no SHG response should be observed, which is distinct from the SHG response of the non‐centrosymmetric D _3h monolayer.^{[ 53 ]} By stacking the D _3d monolayers into the δ(D _3d), ω(D _3d), and ϕ(D _3d) polymorphs, a SHG response may emerge. This would depend on the specific stacking pattern and number of layers (Table S3, Supporting Information), providing a means of identifying the predicted D _3d monolayer based polymorphs.

Figure 5.

a) XRD patterns of δ(D _3d), ω(D _3d), and ϕ(D _3d) predicted by theory. The reference PDF data of bulk β(D _3h) and γ(D _3h) are also shown for comparison. b) Raman spectra of ϕ(D _3d) and γ(D _3h) phases from DFT simulations (left panel). The zoomed‐in part of Raman spectra between 150 and 200 cm⁻¹, the inset shows the vibrational modes of the constituent D _3d/D _3h monolayer (right panel).

3. Conclusion

To conclude, we explored via swarm‐intelligence based computational structure searches the free energy landscape of 2D layered semiconductor InSe and discovered new polymorphs. In addition to the existing ambient and high‐pressure polymorphs, three yet to be reported polymorphs were identified, which consist of a new centrosymmetric monolayer with point group D _3d, distinct from the polymorphs built from noncentrosymmetric D _3h monolayer. The new D _3d monolayer based polymorphs show thermodynamic stability, comparable to that of known D _3h monolayer based ones, as indicated by the small energy difference (<10 meV per atom). The new polymorphs are kinetically stable against transformation to the known D _3h monolayer based polymorphs, demonstrate dynamical stability, and robust thermal stability at room and high temperatures. The D _3d monolayer based polymorphs have indirect gaps with band gap energies that vary widely from the monolayer to bulk by up to ≈1.5 eV, higher than in the D _3h monolayer based polymorphs calculated in the same way. In addition, for the same layer thickness the D _3d monolayer based polymorphs exhibit higher electron mobility, with an enhancement factor of up to ≈1.6. These properties arise from the stronger interlayer electronic coupling associated with the D _3d symmetry and the specific layer stacking pattern. Finally, the predicted new polymorphs can be distinguished from the known phases by XRD, Raman spectroscopy, and SHG measurements. Our prediction of new D _3d monolayer based InSe polymorphs with enhanced electronic properties offers prospects for further research on the synthesis and exploitation of new promising 2D materials with different polymorphic nature.

4. Computational Approaches

The search for the polymorphs of 2D InSe was carried out using a global minimum search of the free energy landscape with respect to structural variations by combining a particle swarm optimization algorithm with first‐principles energetic calculations.^{[ 54} , ^{55 ]} With this methodology one can find the ground‐state or metastable structures based on the known chemical composition, without relying on any prior known structural information. Its validity in crystal structure search has been demonstrated in a variety of material systems,^{[ 56} , ⁵⁷ , ⁵⁸ , ⁵⁹ , ⁶⁰ , ^{61 ]} including 2D layered materials.^{[ 62} , ⁶³ , ^{64 ]} The underlying first‐principles DFT calculations were carried out by using the plane‐wave pseudopotential method as implemented in Vienna Ab Initio Simulation Package.^{[ 65} , ^{66 ]} The electron–ion interactions were described by the projected augmented wave pseudopotentials^{[ 67} , ^{68 ]} with In‐5s²5p¹ and Se‐4s²4p⁴ treated as valence electrons. We used the generalized gradient approximation formulated by Perdew, Burke, and Ernzerhof^{[ 69 ]} as exchange‐correlation functional. We adopted a kinetic energy cutoff of 520 eV for wave‐function expansion and a k‐point mesh of 2π × 0.03 Å⁻¹ or less for Brillouin zone integration. A vacuum layer of more than 15 Å thickness was used in layer‐dependent calculations to isolate the InSe layer from its neighboring image. The structures (lattice parameters and atomic positions) were fully optimized including vdW interaction, until the residual forces were converged within 0.02 eV per Å. The optB86b‐vdW functional^{[ 70} , ^{71 ]} was adopted, that previously provided a good description of the structural properties of the known β, γ InSe phases.^{[ 4 ]} We employed the hybrid functional approach^{[ 46 ]} (with 25% exact Fock exchange) to remedy the band gap underestimation in DFT based calculations. The transition barriers between the monolayer polymorphs were calculated using the nudged elastic band method in conjunction with the climbing image method.^{[ 72} , ^{73 ]} Ab initio molecular dynamics simulations were performed at 300 and 500 K using the NVT ensemble and the temperature was controlled by using the Nosé–Hoover method.^{[ 74 ]} Phonon spectra were calculated using a finite‐difference supercell approach^{[ 75 ]} implemented in the Phonopy code.^{[ 76 ]} The layer‐dependent carrier mobility was evaluated within the deformation potential theory,^{[ 77} , ^{78 ]} using carrier effective mass along transport direction and in‐plane elastic modulus as inputs. A more detailed description of the computational procedures can be found in the Supporting Information.

Conflict of Interest

The authors declare no conflict of interest.

Supporting information

Supporting Information

Sun Y., Li Y., Li T., Biswas K., Patanè A., Zhang L., New Polymorphs of 2D Indium Selenide with Enhanced Electronic Properties. Adv. Funct. Mater. 2020, 30, 2001920 10.1002/adfm.202001920