Letter pubs.acs.org/NanoLett
Instability and Spontaneous Reconstruction of Few-Monolayer Thick GaN Graphitic Structures A. V. Kolobov,*,† P. Fons,† J. Tominaga,† B. Hyot,‡ and B. André‡ †
Nanoelectronics Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Higashi, Tsukuba 305-8565, Japan ‡ Université Grenoble Alpes, CEA, LETI, MINATEC campus, F38054 Grenoble, France
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S Supporting Information *
ABSTRACT: Two-dimensional (2D) semiconductors are a very hot topic in solid state science and technology. In addition to van der Waals solids that can be easily formed into 2D layers, it was argued that single layers of nominally 3D tetrahedrally bonded semiconductors, such as GaN or ZnO, also become flat in the monolayer limit; the planar structure was also proposed for few-layers of such materials. In this work, using first-principles calculations, we demonstrate that contrary to the existing consensus the graphitic structure of few-layer GaN is unstable and spontaneously reconstructs into a structure that remains hexagonal in plane but with covalent interlayer bonds that form alternating octagonal and square (8|4 Haeckelite) rings with pronounced in-plane anisotropy. Of special interest is the transformation of the band gap from indirect in planar GaN toward direct in the Haeckelite phase, making Haeckelite few-layer GaN an appealing material for flexible nano-optoelectronics KEYWORDS: few-monolayer GaN structure, geometry optimization, phonon dispersion, instability of graphitic phase, Haeckelite phase, indirect-to-direct gap transformation
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monolayer TMDCs possess a direct gap5,6 and extraordinary large exciton and trion binding energies,7−9 which makes them efficient competitors for conventional III−V optoelectronics. The ultimately thin layers of TMDC make these materials extremely energy efficient and suitable for transparent and flexible optoelectronics.10 The progress achieved with layered semiconductors has generated increased interest in atomically thin layers of conventional III−V semiconductors as a new class of 2D materials. These materials are fundamentally different from the easily cleaved vdW solids. Because these materials in their bulk form are so-called tetrahedrally bonded semiconductors, with all atoms being sp3 hybridized, the surfaces contain a large number of dangling bonds making them unstable. In addition, due to the different electronegativity of the constituent species, in binary crystals the two opposing surfaces associated with “cleavage” into hexagonal layers (0001 and 0001̅) are polar, that is, formed by cations and anions. They are type III surfaces according to Tasker’s classification11 and are intrinsically unstable due to the divergence of the surface energy. Several mechanisms have been proposed for the stabilization of such surfaces, such as vacancy formation, surface reconstruction, and charge transfer from the cation surface to the anion surface.12−14 An alternative mechanism for stabilization of monolayer wurtzite structures is the formation of planar
he III−V compound semiconductors, such as GaAs, AlAs, InAs, InP, and GaN and their ternary and quaternary alloys, combine elements in columns III and V of the Periodic Table. They possess direct gaps with the resulting ability to efficiently emit and detect light, which makes them ideal for uses in lasers, light-emitting diodes, and detectors for optical communications, instrumentation, and sensing. In 2000, the Nobel prize for physics was awarded “for developing III−V semiconductor heterostructures used in high-speed- and optoelectronics”.1 Transistors based on III−V materials are at the heart of many high-speed and high-frequency electronic systems and III−V complementary metal oxide semiconductor technology is a mainstream part of semiconductor research.2 For reviews of other optoelectronic applications of III−V semiconductors see, for example, refs 3 and 4. In most existing applications, III−V semiconductors are used in their 3D, or bulk, form with the characteristic dimensions of several nanometers, that is, ∼10 or more lattice constants. At the same time, following the success of graphene, the search for other two-dimensional (2D) materials has acquired momentum and transition-metal dichalcogenides (TMDC) with the generic formula of MX2 (M = Mo, W; X = S, Se, Te) have emerged as very promising materials. These materials possess a layered structure, where ∼6 Å thick X−M−X triple layers (usualy referred to as monolayers, ML), consisting of a metal plane sandwiched between two chalcogen planes, are held together by weak van der Waals (vdW) forces. Similar to graphene, mono- and few layer TMDC structures can be easily prepared. While bulk TMDCs are indirect gap materials, © 2016 American Chemical Society
Received: March 23, 2016 Revised: July 4, 2016 Published: July 7, 2016 4849
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of planar structures, for example, the maximum number of GaN layers up to which the graphitic structure is stable increased from 10 in unstrained conditions to 16 for 5% tensile strain.19 The stability of ultrathin diamond layers was studied20 also using the GGA approximation. It was found that a monolayer diamond layer became flat as graphene. The same trend was observed up to five atomic layers, where flat graphene layers were formed. Structures thicker than 12 atomic planes retained the diamond structure. For intermediate thicknesses, the internal layers were diamond-like sp3-hybridized, while the outer layers were graphene-like.20 It is also interesting to note that when the separation between two graphene layers stacked directly on top of each other was decreased to 1.56 Å, strong covalent bonds were formed between the layers.21 It should be noted here that bulk GaN with the wurtzite structure is a direct-gap semiconductor with a wide gap, which determines its huge potential for optoelectronics (the Nobel prize for physics was awarded in 2014 for successful use of “the difficult-to-handle semiconductor GaN to create efficient blue light-emitting diodes”22). At the same time, graphitic GaN was predicted to be an indirect-gap material,17,18 which significantly limits its application potential. We note here that while the formation of stable planar fewlayer structures for naturally layered materials such as graphite or h-BN can be easily understood; for III−V materials, the situation is more complex. While the formation of the flat outermost layers does serve to minimize surface dipoles, thus decreasing the energy of the system, the absence of covalent bonds between the layers is certainly energetically unfavorable, suggesting that the planar geometry may not be the most stable phase for few-layer structures and calling for further studies of this issue. Experimentally, little is known about the structure of atomically thin III−V layers, which is a direct consequence of their bulk structure: the 3D nature of covalent bonds does not allow to exfoliate a bulk crystal, similar to graphene or TMDCs and layers grown on substrates are necessarily affected by the presence (and the structure) of the latter. There are limited reports on the growth of nanosheets of tetrahedrally bonded semiconductors. For ZnO, the formation of planar layers was observed experimentally for growth on Ag(111), with the
graphene-like structures. In this case, the cations and anions are arranged in a trigonal-planar configuration, which serves to remove the surface dipole moment and consequently to stabilize the film.15 It was argued that planar structures are energetically more favorable also for few-layer structures, namely, up to 6 planar layers of GaN are preferred to 12 bilayers in the wurtzite phase or up to 10 planar layers of ZnO are preferred to 18 bilayers in the wurtzite phase.15 This is illustrated in Figure 1a,b, which shows a 3 ML GaN slab in the wurtzite and planar configurations
Figure 1. Three ML GaN structure: (a) the bulk wurtzite phase and (b) the proposed stable graphitic phase.
Monolayer honeycomb structures of group-IV and III−V semiconductors were subsequently studied16,17 using density functional theory, where it was demonstrated that for group-IV elements only C forms planar layers. For the case of both Si and Ge, planar honeycomb structures were found not to be the lowest energy configurations and transformed into low-buckled (puckered) structures.16 For the III−V layers, some of them (e.g., BN, AlN, and GaN) were found to form planar layers, while others (e.g., GaAs and InAs) formed puckered structures (the local density approximation (LDA) was used).17 Bilayer and trilayer GaN structures with different stacking patterns were studied using the generalized gradient approximation (GGA), where additionally the effect of vdW interlayer interaction was accounted for using the DFT-D2 method. The inclusion of the vdW correction generally resulted in a shorter interlayer separation but did not change the structure in general. In both cases, it was concluded that “both bi- and trilayers prefer a planar configuration rather than a buckled bulklike configuration”.18 It was further proposed that application of epitaxial strain could increase the stability range
Figure 2. (a) The starting wurtzite and relaxed planar structures of 1-ML GaN, demonstrating that the energetically most favorable structure for 1 ML GaN is planar. The \ sign represents an unstable structure and the parabola with horns pointing upward (∪) represents a structure in a local minimum (b) The relaxed structure of 5 ML GaN remains in the wurtzite phase despite some asymmetry in the bond lengths. Both results are in agreement with earlier work. 4850
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Figure 3. Three-dimensional views of the optimized structures for 2 ML (a), 3 ML (b), and 4 ML (c) GaN slabs. Panels (d−f) show the 4 ML structure in three specific projections.
Also in agreement with previous reports is the stable wurtzite structure for 5 ML GaN (10 atomic planes if the wurtzite phase is considered) as well as for thicker slabs. As reported earlier,20 we also observed that the degree of buckling decreases for the outermost layers and the interplanar spacing slightly increases toward the surfaces (Figure 2b). What is strikingly different from earlier reports, however, is the fully relaxed structures of 2−4 ML slabs. While several previous studies concluded that few-layer GaN also acquires the graphitic phase,15,18,19 our results presented below demonstrate that this is not the case. Figure 3a−c shows 3D views of the fully relaxed structures for 2−4 ML thick GaN as well as the 4 ML structure viewed along specific crystallographic directions (Figure 3d−f). One can clearly see the existence of interlayer Ga−N bonds in all these structures with the 8|4 bonding motifs being common for these three thicknesses. There are several aspects to be noted about the obtained 8|4 bonded structures. First, the number of interlayer bonds is the same between the wurtzite and the 8|4 structure. The shortest Ga−N interatomic distance in the out-of-plane direction in the 8|4 structure is 2.17 Å, that is, longer than the Ga−N covalent bond length in the bulk wurtzite phase (1.98 Å) but significantly shorter than the interlayer distance between planar Ga−N layers (ca. 2.4−2.50 Å) reported in ref 18 and also for the partially relaxed structures obtained in our simulations. Second, the outermost atomic planes are nearly flat, that is, the surface dipoles are minimized as required for structure stabilization. Third, while in the wurtzite phase all interlayer bonds are polarized in the same direction, in the 8|4 phase the neighboring Ga−N interlayer bonds have inverted polarization directions and the overall polarization arising from the interlayer bonds is zero. The flatness of the outermost atomic layers and the net zero polarization resulting from the interlayer bonds ensure minimization of the surface dipoles and stabilize the structure. The formation of the 8|4 zeolitic motifs for a structure that is expected to be hexagonal is not totally unexpected. Such motifs have been reported to form at grain boundaries of graphene and other 2D materials35 and continuous lattices formed from 8|4 motifs were also found to be stable.34,36,37 Such motifs are ideally suitable for compositions where only heteropolar bonds are expected. Furthermore, zeolitic structures were found to be
transition to the bulk wurtzite structure in the 3−4 ML coverage range.23 It is not clear if the smaller critical thickness observed experimentally (compared to a theoretical prediction of 10 layers15) was caused by the interaction with the substrate. It is also not clear to what extent the presence (and the structure) of the substrate affects the structure of few-layer III− V semiconductors. Recent progress in GaN epitaxial growth on graphene,24−32 a perfectly flat material that does not possess any dangling bonds thus minimizing the chemical interaction between the substrate and the overlayer, opens some interesting opportunities in this direction. Wherever reported, however, the thickness of GaN exceeded the critical thickness and the structure of the grown GaN was wurtzite.25,27−30 An interesting approach to fabricate few-layer nanosheets of GaN was proposed in ref 33. The authors obtained few-layer GaN by heating in ammonia at ca. 650 °C few-layer flakes of GaS and/or GaSe. Because the starting materials possess mica-like morphology, nanosheets could be easily fabricated by micromechanical cleavage. GaN (6−10 layer) was produced using this method, whose structure was wurtzite. As recently noted,34 “monolayer and few layer materials made of GaN and other III−V semiconductors are now a challenge for the experimentalist”. In this work, we revisit the issue of structural stability of GaN in the few-layer limit using rigorous first-principles density functional simulations, based on both geometry optimization and phonon simulations, and demonstrate that the current consensus is incorrect and the graphitic phase in the few-layer limit is not the lowest energy state. The details of the calculations are provided in Methods. We start our simulations by calculating the structures of a single layer and of rather thick slabs. For monolayer GaN, the relaxed structure is planar and the obtained Ga−N bond length is 1.86 Å (vs 1.75 Å in the wurtzite phase), which is in good agreement with previous theoretical work17,18 and slightly shorter than the experimental value of the bulk phase (1.95 Å). An energy reduction of 0.38 eV per Ga−N formula unit was observed as a result of the layer planarization (Figure 2a). We note that the stability of monolayer graphitic GaN was confirmed by the phonon calculations reported in17,34 and also by our simulations (Supporting Information Figure 1S). 4851
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Figure 4. Phonon dispersion curves for 3 ML GaN in the graphitic phase (left) and in the Haeckelite phase (right). The corresponding Brillouin zones are shown above the plots. The presence of imaginary modes in the graphitic phase unambiguously demonstrates that this phase is unstable.
similar to the previous work18,19 where the effect of vdW interaction on few-layer (graphitic) GaN was also rather small. To make sure that the obtained result is not an artifact of the functional used, we additionally recalculated the energies of the Haeckelite phase described above and a hypothetical graphitic phase obtained as a transient phase in our structural relaxation using the solid-state functional PBEsol: the order of phase stability remained unchanged Following the structure optimization, we further performed phonon calculations in order to check the stability of the obtained phases. First, we checked the stability of the few-layer graphitic phase. These results are exemplified by the phonon dispersion curves of the 3 ML slab (Figure 4, left). The presence of imaginary frequencies (visualized as negative frequencies in Figure 4) at finite-k unambiguously demonstrates that the graphitic phase is unstable; the location of the imaginary modes suggests that the true stable phase has a double unit cell size. On the other hand, the similar dispersion curves calculated for the Haeckelite structure (Figure 4, right) do not have any imaginary modes, clearly demonstrating that the Haeckelite phase is stable. We argue that the stability of the obtained Haeckelite phase in the few-layer limit is determined by the combination of the following factors. The formation of (additional) interlayer bonds makes this structure energetically more favorable than the planar graphitic structure. At the same time, the outermost atomic planes in the Haeckelite phase remain nearly flat, thus minimizing the surface dipoles associated with the presence of oppositely charged cations and anions. Finally, the net zero moment associated with interlayer bonds in the Haeckelite phase also leads to structure stabilization. It may be informative to compare the phase stability in the few-layer limit and in bulk phases. The results are shown in
energetically competitive with wutzite nanostructures for ZnO,38 another material that forms a planar graphitic phase in the monolayer limit.15,23 Hence, the formation of the 8|4 phase may be a general feature of few-layer materials that tend to minimize the energy associated with with polar surfaces by acquiring the graphitic phase in the monolayer limit. It may be also interesting to note that the electronic properties significantly differ between the hexagonal and 8|4 phases of the same material.36 Sometimes, the 8|4 bonding pattern is referred to as Haeckelite. 37,39 The term was originally proposed for monolayers of carbon-based structures and later also used for other 2D materials. In what follows, we shall also use this term. We note that the term Haeckelite was recently used for description of hypothetical GaN monolayers with the 8|4 bonding motif and the corresponding nanotubes.34 What is fundamentally different between the cited work34 and the present study, is that in ref 34 the hypothetical 8|4 atomic bonding was in plane and it was argued that such a phase should be structurally (meta)stable, if realized, while the energy of this phase was somewhat larger than that of the graphitic phase. In the present work, we demonstrate that this phase actually is energetically the most favorable for a certain range of thicknesses but the 8|4 bonding pattern is not located in plane but in the direction of the growth, that it, perpendicular to individual GaN layers. We note that qualitatively the results do not depend on the inclusion of vdW correction in DFT simulations. We performed such simulations both with and without vdW correction (Grimme’s DFT-2D correction was used as implemented in CASTEP) and obtained the same results in terms of the structure (with slightly different bond lengths). This result is 4852
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simulation cells used is not explicitly mentioned in previous publications.15,18,19 Figure 1 of ref 18, which shows different stacking sequences studied, suggests that the authors may have used a reasonably large simulation cell that should have been able to detect the distortion, but this is not obvious. On the other hand, the figure may only have been used to visualize different stacking orders. Because the formation of the Haeckelite phase requires doubling of the cell size, the possible use of a smaller 1 × 1 cell would have inhibited this reconstruction (and would only identify the graphitic phase as energetically more favorable than the wurtzite phase in the few-layer limit). It should also be noted that in none of the previous publications was the stability of few-layer structures verified by phonon calculations. Another possibility is associated with the fact that a flat structure represents a saddle point with respect to a distorted phase and it is well-known that saddle points may be confused with local minima in geometry optimization. This situation is nicely discussed in a classical textbook on density functional calculations40 for the example of H adsorption on copper. “The force in the plane of the surface on an H atom that lies precisely above a surface Cu atom must be precisely zero, by symmetry. This means that during the optimization calculation, the H atom will stay exactly above the Cu atom even if its energy could be lowered by moving away from this symmetric location”. This symmetry-induced trap can be avoided “by deliberately breaking the symmetry of the atom’s coordinates by moving the H atom a small amount in an arbitrary direction that does not coincide with one of the symmetry directions”. Another example is the high-temperature Cmcm phase of SnSe, which is a saddle point between two lower-symmetry Pnma minima. If the structure is relaxed maitaining the symmetry, it remains in the Cmcm phase, whereas if the symmetry is turned off, it relaxes to the Pnma phase.41 It is thus possible that in the earlier simulations, even if large enough simulations cells were used, the structure was trapped at the saddle point. (See also Figure 2S in Supporting Information.) We attempted to reproduce the results of ref 18 using the same method as was used previously (i.e., we used VASP and exactly the same simulation parameters and conversion criteria that were used in ref 18) but starting with the Haeckelite phase; the Haeckelite structure remained unchanged. We further followed the algorithm suggested in ref 40 and additionally performed the calculation starting with a structure that was essentially flat but breaking the symmetry by
Figure 5. One can clearly see that for the bulk, the Haeckelite phase has a higher energy (0.09 eV if normalized to the formula
Figure 5. Comparison of the order of phase stability in the bulk phase (left) and in the few-layer limit (right), exemplified by the 3 ML structure. The vertical axes show relative energies of different phases normalized per formula unit. The \ sign represents the unstable structure, the parabolas with horns pointing upward (∪) represent structures in a local minimum, and the parabolas with horns pointing downward (∩) represent the structures at a saddle point. A change in the phase stability order is indicated by colored solid lines.
unit) than the wurtzite phase, which is understandable because the 4-fold coordinated Ga and N atoms in the Haeckelite phase significantly deviate from the ideal tetrahedral configuration that is present in the wurtzite phase and required by sp3 hybridization, which generate stresses and increases the total energy of the system. The hypothetical bulk graphitic phase has the highest energy. (The latter was simulated starting with the structural parameters reported in ref 18 and imposing the symmetry of the graphitic phase, while both the atomic positions and lattice constants were allowed to vary). This phase stability order can be compared with the different stability order in the few-layer limit shown in the same figure. The observed reversal of the stability order between the bulk and few-layer phases demonstrates the crucial role of surface energies in stabilizing the nanostructured forms. An obvious question that arises is why the Haeckelite structure reported here was not detected in earlier studies, which in essence were very similar to the present work. We can consider several possible reasons. First, the size of the
Figure 6. (a) CDD isosurfaces and Mulliken charges for the bulk wurtzite phase of GaN. (b,c) CDD isosurfaces Mulliken charges for 2 ML GaN, respectively, in the unstable wurtzite phase and the stable Haeckelite phase. 4853
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same time, this difference becomes smaller than the thermal energy at room temperature in the Haeckelite phase (Figure 7, right), that is, the material effectively acquires a direct gap, which makes it very attractive for optoelectronic applications. While the level of theory used typically underestimates the bandgap, it has generally given the correct description of the band “directness” for monolayer materials such as transitionmetal dichalcogenides.43 One might argue that the use of the GW level of theory is more accurate and should be used to verify the band structure, but the situation is actually more complicated. Thus, for monolayer MoS2 both indirect44 and direct45 band gaps were reported using GW. Hence, while such simulation may shed additional light on the band structure of few-layer GaN, growth of few-layer GaN and experimental verification of its band structure are crucial to provide a definitive answer. The strong in-plane anisotropy of 2D GaN is an additional parameter to control its properties. We also note the possibility of using the Haeckelite phase of few-layer GaN as a building block of engineered 2D structures. As noted in a recent editorial,46 “one of the most exciting frontiers in 2D materials is stacking them”. While at present, such structures mainly pivot around graphene, h-BN, and TMDCs, recently the range of materials has been expanded to include other 2D materials such as organohalide perovskites.47 The direct-gap few-layer Haeckelite GaN may offer new perspectives for 2D materials design. Also, experimentally grown few-layer structures are typically supported by substrates and the substrate-layer interaction may have a strong effect on the phase stability of the grown layers. Work in these directions is currently underway. In conclusion, we have demonstrated using both geometry optimization and phonon simulations that, different from the current consensus, the few-layer free-standing GaN layers planar graphitic structure is not energetically the most favorable phase and is in fact unstable. Our results show that the energy can be further minimized by the formation of interlayer bonds while preserving the nearly zero net polarization of the surfaces. The obtained stable GaN structure in the few-monolayer limit effectively possesses a direct gap, which is a key requirement for applications in optoelectronics and which was not satisfied by the planar structures. Recent progress in the epitaxial growth of GaN on graphene,24−32 and also the initial success of using 2D TMDCs as substrates for GaN growth48 where the interaction with the substrate material is minimized due to the surfaces being chemically passive, have great promise in growing such structures experimentally in the near future. The possibility of integrating the flat Haeckelite few-layers into artificially engineered 2D structures is also very appealing. Here we would like to specifically mention a very recent work brought to our attention on the graphene stabilization of two 2D GaN49 where it was shown that the structure of few-monolayer thick GaN grown using a migration enhanced encapsulated growth technique is different from reported theoretical predictions and is not flat. The observed bond polarity inversion in the 2D GaN49 compared to the bulk wurtzite phase may well be indicative of the formation of the 8| 4 phase described in this work. Methods. Density functional calculations were carried out at 0 K using the plane-wave codes CASTEP.50 Ultrasoft pseudopotentials were used for Ga and N atoms. The Ga and N pseudopotential included the Ga 3d104s24p1 and N 2s22p3 as valence electrons, respectively. The exchange term was
introducing slight distortions toward the Haeckelite phase; again, with exactly the same simulation parameters as those used in ref 18. The structure readily changed to the Haeckelite phase. On the basis of the discussion above, we argue that the previous studies missed the Haeckelite phase either due to the use of a small calculation cell or by terminating the simulation with the structure trapped at a saddle point before the true stable phase was reached. The bottom line of this discussion is that when investigating the formation of any new phases obtained from geometry optimization, the phase stability must be checked using phonon dispersion calculations. An interesting question is also the driving force for the layer flattening. In previous work, it was argued that it is the disappearance of the surface momentum that makes the transformation energetically favorable.14,15 While we completely agree with this statement, we would like to note another factor that acts in favor of the flattening of a few-layer structure. Figure 6 compares charge density difference (CDD) isosurfaces for the bulk wurtzite phase (panel a) and a 2 ML slab in the wurtzite phase (panel b). As the name implies, CDD represents the difference in electron density between atoms in the structure under investigation and isolated pseudo atoms. Hence, the appearance of a CDD cloud between two interacting atoms is a signature of a covalent bond. For homopolar bonds, the CDD cloud would be located midway between the atoms,42 while for heteropolar bonds the CDD cloud is shifted toward the atom with larger electronegativity. One immediately notices that while in the bulk phase all Ga−N bonds are equivalent (and strongly polarized), in the 2 ML structure the interlayer covalent bonds are much weaker than the “in-plane” bonds, as evidenced by significantly smaller isosurfaces for the former. We propose that this bonding energy hierarchy, which is a consequence of the 2D nature of the slab only, is also a factor leading to the instability of the wurtzite phase in a few-layer limit. It may be also interesting to note that the isosurfaces are very similar for the in-plane and interlayer bonds in the Haeckelite phase (panel c). Of special interest is the band gap evolution between the planar and Haeckelite phases. The planar GaN structure is an indirect-gap semiconductor with the difference between the direct and indirect gaps of ca. 0.3 eV (Figure 7, left), which limits the application potential of planar GaN structures. At the
Figure 7. Simulated band structures for 3 ML thick GaN in the planar (left) and Hackaelite (right) phases. While the band gap is indirect for the planar phase, it changes toward direct in the Haeckelite phase. 4854
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Letter
Nano Letters
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evaluated using the generalized gradient approximation (GGA) and the PBE functional51 as implemented in CASTEP. To make sure the results were not affected by the choice of the functional, the structures were also optimized using a solid-state potential PBEsol (which has not affected the results). A planewave cutoff energy of 295 eV and a 4 × 4 × 1 Monkhorst−Pack grid were used. The simultaneous convergence criteria were set to the following values: energy, 5 × 10−6 eV/atom; max. force, 0.01 eV/Å; max. stress, 0.02 GPa; max. displacement, 5 × 10−4 Å. The Broyden−Fletcher−Goldfarb−Shanno (BFGS) optimization algorithm was used. Grimme’s DFT-D2 method52 was used to account for vdW interactions. The structural relaxation was performed under conditions when both the atomic position within the cell and the cell size were allowed to vary. Confirmation of earlier results as well as phonon calculations were carried out using the PBE functional51 within the densityfunctional code VASP.53−56 A plane wave basis set was used within the framework of the (PAW) projector augmented wave method.57 The Ga and N PAW pseudopotentials included the 4s2 4p1 and the 4s2 4p3 electrons as valence states, respectively. Convergence calculations of the plane wave basis set size confirmed that the large cutoff energy used of 600 eV led to less than 1 meV/atom from the fully converged value. In order to reduce possible FFT aliasing errors, a value of 1200 eV (a value exactly twice that of the plane wave cutoff energy) was used for the augmented charge grid. PAW calculations were carried out in reciprocal space. Additional convergence calculations with a sub-meV/atom convergence criteria led to the choice of a total of 113 irreducible k-points being used (a 15 × 15 × 1 Monkhorst−Pack grid) for integration in the Brillioun zone. Gaussian smearing was used for the occupations of the bands with a smearing parameter of 0.05 eV. Phonon dispersion calculations were carried out using the phonopy package58 with the force constants for the dynamical matrix computed using density-functional perturbation theory within VASP. In order to minimize the “interaction” between the neighboring simulation cells, which may serve to induce (or remove) new periodicity, for example, the use of a 1 × 1 cell would miss a reconstruction that requires doubling of the lattice parameters, we have chosen 2 × 2 supercells to perform structural relaxation. A vacuum layer of 20 Å was used to separate the slabs with a convergence of less than 1 meV/atom from the fully converged result.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.6b01225. Figure of phonon dispersion curve calculated for 1ML graphitic phase and figure of the Structural evolution during the structure optimization at 0 K.(PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
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DOI: 10.1021/acs.nanolett.6b01225 Nano Lett. 2016, 16, 4849−4856
Letter
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DOI: 10.1021/acs.nanolett.6b01225 Nano Lett. 2016, 16, 4849−4856
