Uncovering New Buckled Structures of Bilayer GaN: A First-Principles

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C: Physical Processes in Nanomaterials and Nanostructures

Uncovering New Buckled Structures of Bilayer GaN: A First-Principles Study Anh Khoa Augustin Lu, Tomoe Yayama, Tetsuya Morishita, Michelle Jeanette Sapountzis Spencer, and Takeshi Nakanishi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b09973 • Publication Date (Web): 31 Dec 2018 Downloaded from http://pubs.acs.org on January 2, 2019

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The Journal of Physical Chemistry

Uncovering New Buckled Structures of Bilayer GaN: a First-principles Study Anh Khoa Augustin Lu,

∗,†

Tomoe Yayama,

Spencer,

¶

‡

Tetsuya Morishita,

and Takeshi Nakanishi

∗,‡,†

Michelle J. S.

†

†Mathematics for Advanced Materials Open Innovation Laboratory (MathAM-OIL), National Institute of Advanced Industrial Science and Technology (AIST), c/o Advanced Institute for Materials Research (AIMR), Tohoku University, 211 Katahira, Aoba, 9808577 Sendai, Japan

‡Research Center for Computational Design of Advanced Functional Materials (CD-FMat), National Institute of Advanced Industrial Science and Technology (AIST), 3058568 Tsukuba, Japan

¶School of Science, RMIT University, GPO Box 2476, Melbourne, Victoria 3001, Australia E-mail: [email protected]; [email protected]

1

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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract New structures of bilayer GaN displaying buckling are revealed, and their structural and electronic properties are studied using rst-principles calculations. Layered GaN is a promising two-dimensional material of the III-V family with a sizable band gap. The impact of buckling and the dierence in atomic structure of the dierent models lead to signicant changes in the electronic structure, including an indirect-to-direct band gap transition. Due to the small energy dierence between dierent models, it can be expected that at room temperature, regions of the material may have dierent structures that coexist. The possibility to tailor the structural and electronic properties of bilayer GaN by applying an external strain is also explored. Under tensile strain, all structures tend to become at while under bi-axial compressive strain, buckled structures are favored, with an indirect-to-direct band gap transition.

Introduction Two-dimensional materials have become an attractive research eld since the experimental realization of graphene in 2004. 1 Following this breakthrough, other materials have emerged, such as two-dimensional transition metal dichalcogenides 2,3 (MoS2 , MoSe2 , WSe2 , ...) and materials with a honeycomb-like atomic structure similar to graphene, such as silicene, 46 germanene, 7,8 stanene 9 and phosphorene. 10 New phenomena have been predicted and are being explored, such as transitions to a topological insulating phase. 11 The eect of buckling in Xenes (X = Si, Sn, P,...) has also been the subject to much interest. 12 In recent years, 2D materials have been studied as the potential building blocks for a broad range of applications in elds such as bioengineering, 13 nanoelectronics, 14 optoelectronics, 15 spintronics or structural materials. 16 III-V materials are composed of atoms of the third and fth columns of the periodic table and over the last few decades, have been intensively studied for a broad range of applications in optoelectronics, owing to their direct band gap, and in electronics, owing to their high 2


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carrier mobilities. Among these materials, bulk GaN has been the subject of numerous studies for power electronics and optoelectronic applications. 17 In recent years, the possibility of engineering 2D GaN has been demonstrated experimentally via graphene encapsulation. 18 Several methods have been developed in an attempt to synthesize 2D GaN, including spin coating, 19 molecular beam epitaxy (MBE) 20 and chemical vapor deposition (CVD). 21 Monolayer and few-layer GaN have also been theoretically studied in recent years to evaluate their electronic properties 22,23 and magnetic properties. 24,25 However, few-layer GaN can potentially undergo a surface reconstruction, similar to other two-dimensional materials. For instance, previous ab-initio molecular dynamics simulations have highlighted the occurrence of surface reconstructions in silicene. 26 Such surface reconstructions can have a signicant impact on the properties of the system. Therefore, it is crucial to properly understand this aspect in order to understand how it aects the properties of the system and how it may also be exploited. Although GaN surfaces may be passivated by hydrogen atoms, 27 we focus on pristine bilayer GaN in this study to comprehend the relaxation mechanisms in bilayer GaN and their impact on the properties. In this work, using rst-principles calculations, we discover alternative structures of bilayer GaN that have buckling. These structures are dierent from the ones obtained by slicing two layers from a bulk wurtzite GaN structure or from the at bilayer structure. Our calculations demonstrate that these newly discovered structures are more stable than both the at bilayer GaN and the bulk cleaved structure. In addition, we study the impact that buckling has on the electronic properties of bilayer GaN, which reveals an indirect-todirect band gap transition. Finally, we investigate the impact of strain on the structure and properties of these 2D systems.

3



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Theoretical Methods Density functional theory In this study, we used the density functional theory (DFT) 28,29 to study the structural and electronic properties of bilayer GaN. We used the Perdew-Burke-Ernzerhof (PBE) exchangecorrelation functional, 30 with corrections for the van der Waals interactions as introduced by Grimme 31 (DFT-D2) and implemented in the Quantum Espresso software package. 32 The behavior of the core electrons was approximated using the Projector-Augmented Wave (PAW) pseudopotentials. 33 The cuto for the kinetic energy was set to 70 Ry the k-point sampling was done by using a Monkhorst-Pack grid 34 with a density of 10 points per Å−1 . For the primitive cell, this corresponds to an 18 × 18 × 1 grid. A vacuum region of at least 15 Å was set in the direction perpendicular to the layers to ensure that they did not interact with their periodic images. For each structure, the in-plane cell parameters and atomic positions were relaxed until all components of all forces were smaller than 10−7 a.u. 3

(or 5 × 10−6 eV /Å) and the pressure was lower than 10−4 kbar (or 6.3 × 10−8 eV /Å ). Since at layered GaN has a hexagonal structure, the electronic properties were evaluated along symmetry segments in the Brillouin zone (BZ) including the Γ(0, 0, 0), K(1/3, 1/3, 0) and

M (1/2, 0, 0) points. In contrast, the buckled structures of bilayer GaN determined in this study adopt an orthorhombic unit cell. Therefore, in that case, a dierent symmetry path in the Brillouin zone was studied, including the symmetry points Γ(0, 0, 0), X(1/2, 0, 0),

Y (0, 1/2, 0) and S(1/2, 1/2, 0). The top of the valence band (TVB) was chosen as the reference energy level. For each conguration, the TVB and the bottom of the conduction band (BCB) were computed, then eective mass ts were performed around these extrema using a 21 point path in reciprocal space with a k-point spacing smaller than 0.001 Å

−1

to

avoid non-parabolic eects. The eective masses m∗ were computed as the inverse of the second derivative of the energy dispersion prole, as written in Eq. (1), where m∗e (m∗h ) denotes the electron (hole) eective mass. 4


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m∗e/h

2

= ±¯ h

∂2 E ∂k 2

−1

(1)

Construction and characterization of structure models of bilayer GaN We rst considered dierent possible stacking patterns of at bilayer GaN, labeled AA1, AA2, AB1, AB2 and AB2', in a similar way to what had been done in previous studies of hexagonal boron nitride (h-BN). 35,36 Among them, we determined that the AA2 stacking pattern was the most stable (see the Supporting Information). Hence, this structure is considered as the reference one for the at bilayer GaN. Therefore, in the following, the AA2 stacking alignment will be referred to as the 'at bilayer', unless stated otherwise.

Inner sub-layers

0

1

0

0

1

0

Outer sub-layers dO

dI

z

dI = inner interlayer distance dO = outer interlayer distance z = buckling amplitude Figure 1: Schematic view of the inner and outer sub-layers in our models of buckled GaN (red and blue dashed lines), generated using the GDIS software. 37 Ga and N atoms are colored in teal and purple, respectively. The example shows the Opt-100 model. The denitions of the inner and outer interlayer distances are illustrated as well as the buckling height ∆z . We built atomic congurations based on 2 × 2 × 1, 3 × 3 × 1 and 4 × 4 × 1 super cells 5



of bilayer GaN, having dierent periodic patterns based on the AA2 stacking pattern of at bilayer GaN. For structures with buckling, each GaN layer can be divided into two sub-layers: one inner sub-layer, and one outer sub-layer. In the particular case of a at structure, all atoms are considered as belonging to one of the the inner sub-layers (top or bottom). Each structure was characterized by a sequence of digits {xi }i=1,...,n composed of zeros and/or ones. The following rules were used to characterize each structure by a unique label Opt-{x1 x2 ...xn } dened by the following rules (as illustrated in Fig. 1): 1. The length of the codeword n corresponds to the period of the pattern. E.g. in Fig. 1, n = 3. 2. For each atom i along the pattern (1 ≤ i ≤ n), we dene each digit of xi of the label as follows: if the atom is located in an inner sub-layer, then xi is attributed a 0. Such an atom is therefore called an

inner atom.

if the atom is located in an outer sub-layer, then xi is attributed a 1. Such an atom is therefore called an

outer atom.

This case occurs in buckled structures.

3. To attribute unique codewords, we write the cyclic conguration that maximize the codeword binary number. E.g. in Fig. 1, 010, 100 and 001 are the possible labels, and

100 is the one that maximizes the number represented by the codeword. As an example, the structure with a period of n = 3, built from a 3 × 3 × 1 super cell, with an alternation of two inner atoms (i.e. located in the inner sub-layers) and one outer one (i.e. located in the outer sub-layers) is labeled by Opt-100, as illustrated by Fig. 1. In Fig. 2, a schematic view of the models of at and buckled GaN proposed in this work is presented. The at structure is labeled as Opt-0 (Fig. 2a, n = 1). The buckled structure with a period length of (n = 2) is labeled as Opt-10 (Fig. 2b). It should be noted that this structure diers from the one obtained by cutting w-GaN into a bilayer structure. 6


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(a) Opt-0 (n=1, at)

(b) Opt-10 (n=2)

(c) Opt-100 (n=3)

(d) Opt-110 (n=3)

(e) Opt-1000 (n=4)

(f) Opt-1100 (n=4)

Figure 2: Schematic view of several structures of at and buckled structures of bilayer GaN (with n = 1, 2, 3, 4), built from the primitive cell, 2 × 2 × 1, 3 × 3 × 1 and 4 × 4 × 1 super cell. Ga and N atoms are colored in teal and purple, respectively. In the Opt-10 structure, each sub-layer is composed of 50% Ga atoms and 50% N atoms, while in the later case, each sub-layer is exclusively made of only one type of atoms (see the Supporting Information). In our calculations, the former was found to be stable while the latter was found to be unstable. The structures for (n = 3) are denoted by Opt-100 and Opt-110 (Fig. 2c and 2d). The structures for (n = 4), Opt-1000 and Opt-1100, are respectively illustrated in Fig. 2e and Fig. 2f. Note that larger structures (5 × 5 × 1, 6 × 6 × 1, etc.) could also be constructed but we restricted our study to the 2 × 2 × 1, 3 × 3 × 1 and 4 × 4 × 1 super cells to keep the computational burden at a reasonable level, since our objective was to understand the impact of buckling on the properties of bilayer GaN. To characterize the atomic structures of bilayer GaN, the following three denitions are introduced (and illustrated in Fig. 1): 1.

Inner interlayer distance dI

: Distance between the inner sub-layers (top and bot-

tom) of the structure. 2.

Outer interlayer distance

dO : Distance between the outer sub-layers (top and

bottom) of the structure. 7



3.

Buckling height ∆z :

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Distance in the direction perpendicular to the layers between

the outer and inner sub-layers. It is dened as ∆z = (dO − dI )/2. This quantity is zero for at structures, where by convention, we consider that dO = dI . Finally, to analyze the nature and the strength of the interaction between the GaN layers, we also evaluated the interlayer interaction energy Einterlayer (in units of eV/atom), dened in Eq. (2), where Ebilayer is the energy of the full bilayer system, Etop and Ebottom are the energies of the top and bottom layers alone, and natoms is the number of atoms in the periodic cell of the bilayer system. A negative value indicates that the full system is more stable than the separate layers.

Einterlayer =

Ebilayer − (Etop + Ebottom ) (eV /atom) natoms

(2)

We also calculated the cohesive energy (in eV per GaN dimer) as dened in Eq. 3, where

Ecoh is the cohesive energy, Etot is the total energy of the system, N is the number of Ga (or N) atoms, and EGa (resp. EN ) is the energy of an isolated Ga (resp. N) atom.

Ecoh = Etot /NGaN − EGa − EN (eV /GaN )

(3)

Although the initial atomic congurations of our structures were based on hexagonal super cells, it turned out that after geometry relaxation the symmetry of the buckled structures was no longer hexagonal. Therefore, for each system size, we analyzed the symmetry of the relaxed structures and the isolated the unit cell. It turned out that smaller, orthorhombic unit cells could be obtained for these structures. For instance, the number of atoms per cell is reduced from 16 to 8 for the Opt-10 model, from 36 to 24 for the Opt-100 model and from 64 to 16 for the Opt-1100 model. From the primitive cells of at monolayer and bilayer GaN, orthorhombic super cells were also built to make the comparison easier with respect to buckled structures and to apply dierent types of strain, as explained in the next subsection. We illustrate the shape of the unit cells in Fig 3. From now on and unless stated 8


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Flat bilayer (hexagonal)

Opt-10 (orthorhombic)

b=b* b b*

a=a*

a=a* Figure 3: Top view of the unit cells of the at bilayer GaN (left) and Opt-10 (right) models. The rst cell is hexagonal while the second one is orthorhombic. The lattice parameters of the unit cells are a and b in the in-plane directions, with a ≤ b. The equivalents lengths respective to the primitive cell of GaN (a∗ and b∗ ) will be preferred to allow direct comparison between dierent models. Note that a∗ and b∗ are not necessarily the same (see Table 1).

9



otherwise, the term lattice parameter depicts the values of a∗ or b∗ . In the next section, we will therefore present the results for hexagonal cells of at mono- and bilayer GaN, and for orthorhombic cells of the buckled bilayer structures.

Strain

Parallel Perpendicular

0 0 1 0 0 1

0 0 1 0 0 1

Perpendicular

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Parallel

Figure 4: Illustration of the directions along which strain is applied on the periodic cell. A bi-axial strain is applied by stretching both in-plane directions with the same ratio. A uni-axial strain is applied by stretching the structure in the direction perpendicular to the buckling lines (blue). This example shows the Opt-100 model. To examine the impact of strain on the properties of the dierent GaN geometries, we considered the cases of bi-axial strain and uni-axial strain. Since the buckled structures appeared to have an orthorhombic unit cell (with a rectangular in-plane section) and two nonequivalent directions, we can dene two directions for the uni-axial strain, one perpendicular and one parallel. The reference direction is the direction of the lines formed by the outer sub-layers of GaN in the buckled congurations (as shown in Fig. 4). We also considered 10


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the case of uni-axial perpendicular strain since it is the one that would allow control over the buckling height. For the at structures, we built rectangular super-cells in order to apply the uni-axial strain. The strain value was studied from -10 % (compressive) to +10 % (tensile), with steps of 1 %, where 0 % represents the relaxed structure. For each model, we started from the zero-stress relaxed structure, then we altered the lattice parameters to apply the desired strain and nally, we relaxed the atomic positions.

Results and discussion This section is divided into two parts. First, we discuss the structure and properties of bilayer GaN, with and without buckling. Then, we analyze the impact of strain on the structural and electronic properties of the most stable structures.

Properties of buckled bilayer GaN Table 1: Summary of the structural and electronic properties of layered GaN. Results for w-GaN are also displayed. The values of a∗ and b∗ are also indicated to allow comparison between the structures. Model Lattice parameter a (Å) Lattice parameter b (Å) a∗ (Å) b∗ (Å) Ratio of inner atoms (%) Ratio of outer atoms (%) Inner interlayer distance dI (Å) Outer interlayer distance dO (Å) buckling height ∆z (Å) Intralayer bond lengths (Å) Energy (eV/atom) Relative energy (eV/atom) Ecoh (eV/GaN) Band gap - PBE (eV) Gap type me (×m0 ) ([100]) mh (×m0 )

Bulk wurtzite 3.22 3.22 3.22 3.22 / / 1.98 3.28 0.65 1.97 -2082.225 -0.079 -11.851 1.65 Direct 0.15 0.15a,lh /2.21a,hh

Monolayer 3.22 3.22 3.22 3.22 100 0 / / 0.0 1.86 -2081.991 0.155 -11.383 2.103 Indirect 0.22 1.22a /0.30b

AA1 3.22 3.22 3.22 3.22 100 0 3.63 3.63 0.0 1.86 -2082.028 0.118 -11.456 1.502 Indirect 0.20 1.50a

AA2 3.27 3.27 3.27 3.27 100 0 2.48 2.48 0.0 1.89 -2082.134 0.012 -11.669 1.925 Indirect 0.20 1.08a

AB1 3.23 3.23 3.23 3.23 100 0 2.95 2.95 0.0 1.86 -2082.092 0.054 -11.584 1.625 Indirect 0.20 3.39a

Opt-10 3.25 5.61 3.25 3.24 50 50 2.30 2.90 0.30 1.85/1.89/1.93 -2082.137 0.009 -11.675 1.808 Indirect 0.22 0.07c

Opt-100 5.61 9.75 3.24 3.25 67 33 2.24 3.14 0.45 1.85/1.89/1.92/1.93 -2082.146 0.000 -11.693 1.950 Indirect 0.21 4.05d

Opt-1100 5.62 6.47 3.25 3.24 50 50 2.21 3.11 0.45 1.86/1.89/1.93 -2082.145 0.001 -11.691 1.917 Direct 0.20 3.40a

light-holes, hh heavy-holes Direction along which the hole eective mass is calculated: a K − Γ, b K − M ,c X − S , d Γ − S. lh

The calculated results are summarized in Table 1. The results of calculations performed on bulk wurtzite GaN (w-GaN) are also included in the table for comparison. The calculated 11



lattice parameters of w-GaN are a = b = 3.22 Å and c = 5.26 Å, a slight overestimation compared to the reported experimental value (aexp = bexp = 3.19 Å and cexp = 5.19 Å), 38 as expected for a GGA-type exchange correlation functional. The energy calculations suggest that the Opt-100 and Opt-1100 structures are the most stable, with a small dierence in energy of only 0.001 eV/atom. The Opt-110 and Opt1000 models were found to be less stable than the Opt-100 and Opt-1100 ones, respectively. Therefore, for n = 3 and n = 4, we focused on the Opt-100 and Opt-1100 models, respectively. The most stable model with a period of n = 2 was the Opt-10 structure, with an energy 0.009 eV/atom higher than the Opt-100 structure but 0.003 eV/atom lower than the at bilayer one. It should be noted that all buckled structures presented in this table have a lower energy than the at congurations, showing that buckling leads to more stable structures. The bilayer GaN model based on a slice of w-GaN was found to be unstable. We also built buckled structures analogous to the Opt-10, Opt-100, Opt-110, Opt-1000 and Opt-1100, based on the AA1 and AB1 stacking patterns. These structures were found to be unstable and relaxed back to the at (AA1 or AB1) conguration. The lattice parameter increases from a monolayer GaN to a bilayer structure. When buckling occurs, a slight decrease in the lattice parameter is observed, accompanied by a reduction of the interlayer distance from around 2.5 Å in the at case to 2.2-2.3 Å in the buckled cases. In this case, the symmetry of the system is modied and the two in-plane lattice parameters in the two inplane directions become slightly dierent (a∗ 6= b∗ ). It should be noted that these structures, as well as the at ones, have symmetry inversion, which is absent in the wurtzite structure. Interestingly, in the buckled structures, the inner interlayer distance dI is within the range of bonded atoms, though larger than the sp3 bonds found in w-GaN, while the outer interlayer distance dO is around 3 Å, which is in the range generally found for the interlayer distance in van der Waals hetero-structures. Therefore, the layers are partially bonded. Therefore, since the inner atoms of one layer are also bonded to inner atoms of the other layer, their coordination number (fourfold) diers from that of the outer atoms (threefold). 12


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It can be seen in Table 1 that the intralayer bond lengths are also impacted by buckling. The bulk wurtzite structure (w-GaN) has sp3 bonded atoms, with a calculated bond length of 1.97 Å. This bond length decreases for at mono- and bilayer structures (1.86 Å and 1.89 Å, respectively). In these structures, the atomic arrangements are no longer sp3 -type tetrahedral ones (with 109 deg. angles), but sp2 -type hexagonal ones (with 120 deg. angles). Therefore, the nature of the in-plane bonding mechanism diers from the bulk wurtzite case. This is also revealed later by the density of states proles (Fig. 5). In fact, sp3 bonds are partially recovered for the inner atoms after buckling in the Opt-10, Opt-100 and Opt1100 structures, as the in-plane (inner) atoms, become fourfold-coordinated. As a result, the bond length between these atoms is closer to the one found in w-GaN (1.93 Å). An important dierence between the Opt-10, Opt-100 and Opt-1100 models with buckled structures based on the wurtzite structure of GaN is that each of the (four) sub-layers along the out-of-plane axis (i.e. two sub-layers for the top layer and two sub-layers for the bottom one) are made of both Ga and N atoms. In previous studies, 18 each sub-layer was composed of only one type of atom, as in w-GaN (see the Supporting Information). The atomic compositions of the sub-layers therefore dier for each type of buckled conguration (see Table 1). Upon buckling, dierent values for the intralayer bond lengths occur, as there are dierent types of bonds. For instance, for the Opt-10 model, three dierent values occur, for the innerinner bond (1.93 Å), the inner-outer bond (1.89 Å) and the outer-outer bond (1.85 Å). For the Opt-100 model, in addition to the inner-outer bond (1.89 Å) and the outer-outer bond (1.85 Å), the inner-inner bonds exist in two nonequivalent directions, yielding two distinct values (1.92 Å and 1.93 Å). Finally, for the Opt-1100 model, we have three values, for the inner-inner bond (1.93 Å), the inner-outer bond (1.89 Å) and the outer-outer bond (1.86 Å). The electronic structure calculations showed that the band gap of monolayer GaN is around 2.1 eV (indirect) while the band gaps of the bilayer structures only slightly vary around 1.8 and 1.9 eV. This is larger than the band gap of w-GaN, evaluated to be 1.65 eV (direct at Γ), and an underestimation from the experimental value of 3.39 eV. 38 The band 13



2.5 Density of states (arb. units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

2.0 1.5 1.0

2.0

Flat monolayer Flat bilayer Opt-10 Opt-100 Opt-1100

Zoom a ound the TVB

1.5 1.0 0.5 0.0 −1.5

−1.0

−0.5

0.0

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0.5

0.5 0.0

−6

−4

−2 0 Energy (eV)

2

4

Figure 5: Density of states of the monolayer, at bilayer, Opt-10 bilayer, Opt-100 bilayer and Opt-1100 bilayer GaN, normalized with respect to the number of atoms per periodic cell. The inset shows a magnication around the TVB. structures calculated using PBE for the at structures are presented in Figs. 6a and 6b, plotted alongside the projected density of states proles. In both cases, the band gap is found to be indirect, with the bottom of the conduction band (BCB) being located at Γ. The top of the valence band (TVB) is located at K for the monolayer structure and along the

K − Γ symmetry line for the at bilayer structure. To dierentiate the contributions from dierent orbitals to each point of the band structure, we projected the wave functions onto pseudo-atomic wave functions and colored the corresponding point in a color that reects the weight of each orbital. We plotted the projected density of states (PDOS) of each system using the same color convention. It can then be seen that the BCB arises from the 2s orbitals of N and the 4s orbitals of Ga, while the main contribution to the TVB comes from the 2pz orbitals of N. The local maximum of the valence band in Γ mainly consists of the N atom 2px and 2py orbitals. Since the 2pz orbitals are the ones oriented out-of-plane, they are the most impacted by the change from a monolayer to a bilayer structure, as seen in both the PDOS 14


6

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2

−2

−2

4

4

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−4

Γ

M

K

Symmetry points

Γ

6

Energy (eV)

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Energy (eV)

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(a) Monolayer

Energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Energy (eV)

Page 15 of 28

Γ

X

S

Y

Symmetry points

Γ

−6

pz px+py s Total

0

10

20

Density of states

(e) Opt-1100 Figure 6: Band structure (left) and projected density of states (right) of dierent structures of (a) monolayer, (b) at bilayer, (c) Opt-10, (d) Opt-100 and (e) Opt-1100 GaN. For each eigenvalue shown in the band structure, a projection onto pseudo-orbitals is performed and a color is attributed, representing the weights of the orbitals. The same color scheme is used for both graphs for each orbital of interest.

15



proles and the highest valence bands shown in red. The 3d states from Ga lie deep in the valence bands, around 13 eV below the Fermi level (not shown). For the buckled structures (Opt-10, Opt-100 and Opt-1100), the symmetry points in the Brillouin zone dier due to the change from a (deformed) hexagonal to a rectangular periodic cell. It can be seen that the PDOS for the pz orbitals are more strongly aected by the structural dierences than the s and in-plane px /py ones. Almost at valence bands associated with the pz orbitals are found in the buckled structures, as can be seen in Figs. 6c, 6d and 6e. This is consistent with the fact that main dierence between the structures lies in the interlayer interactions, and in the proportion of inner and outer atoms. Interestingly, the Opt-1100 displays a direct band gap (Fig. 6e) while the other buckled structures show an indirect one (Figs. 6c and 6d). It should also be noted that despite the apparent presence of dangling bonds at the surface of bilayer GaN, no surface states were found inside the band gap in all the presented band structures. This indicates that the under coordinated atoms are saturated to some extent by the charge transfer between Ga and N atoms. In terms of electron eective mass, it can be seen that the value for layered GaN is higher than for w-GaN. Nevertheless, it is only marginally impacted by buckling, an observation consistent with the fact that the BCB is dominated by s states of Ga and N. The hole eective mass, on the other hand, shows large variations depending on the atomic structure. For w-GaN, the TVB is degenerate at Γ, yielding the existence of light- and heavy-hole eective masses. For the layered structures, the TVB is no longer found at Γ nor degenerate. By comparing the density of states (DOS) proles (Fig. 5), it can be seen that the DOS around the TVB is signicantly impacted by the structural changes in the GaN layers. In particular, the peak located around the TVB is lower in intensity in the buckled structures than in the at one. Also, the Opt-100 geometry displays a splitting of this peak into two visible spikes, as illustrated in Fig. 5. The dierence in nature for the the in-plane and out-of-plane bonds hints at a dierence in behavior under stress, and at the possibility to tune the location of the TVB by a modulation of the applied stress. The presence of nearly at valence bands in the band structures of the 16


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Opt-10 and Opt-100 models are also an interesting feature for potential applications that would use magnetism. This could be achieved by doping buckled GaN via substitution or by creating defects in an analogous manner to what has been demonstrated in other 2D materials, 39 or by electrostatic doping, obtained by employing an electrolytic gate similar to what has been demonstrated previously for graphene. 40

On the impact of strain To evaluate the impact of strain on the structure and properties of bilayer GaN, we considered the separate cases of bi-axial and uni-axial strain applied on these structure (schematically illustrated in Fig. 4). To compare the results, we plot the properties with respect to the lattice parameter a∗ for the bi-axial case, and with respect to the variable lattice parameter

b∗ for the uni-axial case. It is noteworthy that for a tensile strain, all congurations tend to become at. For instance, in the cases where a∗ > 3.35 Å, no buckled structure can stably exists and it always transformed to the at bilayer structure. During the geometry relaxation, the initial structure may undergo a transition to another structure. To highlight this transition in the plots, we use two dierent symbols, representing buckled nal structures with full circles and solid lines and at nal structures with crosses and dashed lines in all the gures presented in this subsection. The evolution of the energy with respect to the applied strain is shown in Fig. 7a. To allow a direct comparison between the structures, the x-axis represents the lattice parameter

a∗ . Since the same ratio is applied on both lattice parameters a∗ and b∗ in the case of bi-axial strain, it can be seen for large values of the lattice parameter (tensile strain), that all bilayer structures become at and therefore they become equivalent in terms of total energy (Fig. 7a), interlayer interaction energies (Fig. 7c) and band gap (Fig. 7d). Under tensile strain, for a∗ > 3.35 Å, the most stable structure is the at one (crosses and dashed lines in the right region of Fig. 7a). Under compressive strain (left region), the buckled structures are favored. This trend is also visible by considering the buckling height in the structures (Fig. 17



1.4

Fla bilayer Op -10

−2082.02 −2082.06

1.0

−2082.08 −2082.10

0.8 0.6

−2082.12

0.4

−2082.14

0.2

−2082.16

2.9

3.0

3.1

3.2

3.3

Flat bilayer OptΔ10 OptΔ100 OptΔ1100

1.2

Opt-100 Opt-1100

−2082.04

Δz (Å)

Energy (eV/a om)

−2082.00

3.4

Lattice parame er (Å)

3.5

0.0 2.9

3.6

3.0

(a) Energy

3.1

3.2

3.3

3.4

Lattice parameter (Å)

3.5

3.6

(b) Buckling height

0.0

3.0

Flat bila er Opt-10

−0.1

Opt-100 Opt-1100

−0.2


2.5

Band gap (eV)

Interaction energ (eV/atom)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 28

2.0 1.5

−0.3

1.0

−0.4

0.5

−0.5 2.9

3.0

3.1

3.2

3.3

3.4


3.5

0.0 2.9

3.6

3.0

(c) Interlayer interaction energy

3.1

3.2

3.3

3.4


3.5

3.6

(d) Band gap

Figure 7: Evolution of (a) the energy per atom with respect to the lattice parameter a∗ , (b) the buckling height, (c) the interlayer interaction energy and (d) the band gap for the case of bi-axial strain. Circles and solid lines indicate that the structure adopts a buckled structure under strain. Crosses and dashed lines indicate a at structure. The equilibrium lattice parameter can be found in Table 1.

18


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7b). Interestingly, under a compressive stress beyond 4% (a∗ < 3.12 Å), the at bilayer structure relaxes into the Opt-10 one. Overall, under no strain, the Opt-100 structure is the most stable, then for a∗ < 3.11 Å, the Opt-1100 structure becomes the most stable. The structures relax via an increase of the buckling height, which is visible in the left region of Fig. 7b. Under compressive strain, we observe two concurrent mechanisms to relax the atomic structures. First, an increase in the inner interlayer distance dI . Second, the occurrence of buckling (∆z > 0), where some of the interlayer bonds are broken. In that respect, the Opt-1100 structure shows the lowest energy under compressive bi-axial and uni-axial (perpendicular) strain. To gain a better insight into the interlayer interaction, we evaluated the interlayer interaction energy (dened in Eq. (2)) as a function of the applied strain, as illustrated in Fig. 7c. On the one hand, under tensile strain, only the interlayer interaction for the at structure is obtained, since no buckled structure is stable under the tensile condition. On the other hand, under compressive strain, the buckling allows the system to remain in a more stable state, with a stronger interaction between the layers. Opt-100 has 67 % of inner atoms while Opt-10 and Opt-1100 have only 50 %, so the former is expected to display a stronger interlayer interaction than the latter. This is illustrated by the lower interaction energy observed in Fig. 7c. The evolution of the band gap as a function of bi-axial strain is presented in Fig. 7d, together with the results for a monolayer. Two general trends are observed. First, under tensile strain, the band gap decreases and becomes indirect for all structures, which all become at and equivalent to each other. Second, when a compressive strain is applied, the band gap rst increases, then decreases. In the at bilayer, Opt-10 and Opt-100, this corresponds to a crossing in the TVB between the pz states and the px /py ones. Interestingly, for a compressive strain beyond 5 %, all structures display a direct band gap. A summary of the results for a 6 % bi-axial compressive strain is presented in Table 2. It can be seen that 19



Page 20 of 28

Table 2: Summary of the properties under a bi-axial compressive strain of 6 %. The directions along which the eective masses are computed are also indicated in the rst column. No results are shown for the at bilayer case, since the structure was found to be unstable under those conditions. Model Lattice parameter a (Å) Lattice parameter b (Å) a∗ (Å) b∗ (Å) Inner interlayer distance dI (Å) Outer interlayer distance dO (Å) Buckling height ∆z (Å) Intralayer bond lengths (Å) Energy (eV/atom) Relative energy (eV/atom) Band gap - PBE (eV) Gap type me (×m0 ) ([100]/[010]/[110]) mh (×m0 )[100] mh (×m0 )[010] mh (×m0 )[110]

Opt-10 3.06 5.27 3.06 3.04 2.17 3.51 0.67 1.81/1.85/1.91 -2082.030 0.035 1.767 Direct 0.29/0.32/0.30 0.25 1.37 0.31

Opt-100 5.28 9.16 3.05 3.05 2.17 3.87 0.85 1.83/1.84/1.87/1.89 -2082.042 0.024 2.014 Direct 0.28/0.30/0.29 1.22 0.29 0.68

Opt-1100 5.29 6.08 3.05 3.04 2.10 5.17 1.04 1.83/1.87/1.92 -2082.066 0.000 2.048 Direct 0.25/0.28/0.26 1.09 1.76 1.30

under compressive strain, the buckling height ∆z increases sharply. Under this condition, all three models showed a direct band gap, in the range of 1.7-2.1 eV. This feature is interesting as it hints at an ability to possibly control the electronic properties of bilayer GaN using an external force, switching from an indirect to a direct band gap depending on the applied stress. The impact of a uni-axial strain perpendicular to the outer atom lines is also studied. Fig. 8a shows the evolution of the energy with respect to the variable lattice parameter b∗ , while a∗ is kept constant. The same trend as in the bi-axial strain case (Fig. 7a) is observed, with all structures becoming at for a tensile strain while under an increasing compressive strain, Opt-100 is the most stable structure, then for b∗ < 3.21 Å, Opt-1100 becomes the most stable structure. Nevertheless, since the strain is only applied in the direction that is expected to modulate the degree of buckling, the energies calculated under strain remain lower than that under a bi-axial strain, for the same percentage of applied strain. The trend 20


Page 21 of 28

observed in the energy curves is correlated with the increase in buckling height, as shown in Fig. 8b, which shows the evolution of the buckling height with respect to the variable lattice parameter for the uni-axial case. Interestingly, for the Opt-10 structure, the change in energy and in band gap under a uni-axial strain is much less pronounced than under a bi-axial one as illustrated by Figs. 8c and 8d, which respectively show the evolution of the interlayer interaction energy and the band gap with respect to the lattice parameter. 1.4

Fla bilayer Op -10

−2082.02 −2082.06

1.0

−2082.08 −2082.10

0.8 0.6

−2082.12

0.4

−2082.14

0.2

−2082.16

2.9

3.0

3.1

3.2

3.3

Flat bilayer OptΔ10 OptΔ100 OptΔ1100

1.2

Opt-100 Opt-1100

−2082.04

Δz (Å)

Energy (eV/a om)

−2082.00

3.4

Lattice parame er (Å)

3.5

0.0 2.9

3.6

3.0

(a) Energy

3.1

3.2

3.3

3.4


3.5

3.6

3.5

3.6

(b) Buckling height 3.0

0.0

2.5

−0.1

Band gap (eV)

Interaction energ (eV/atom)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.0

−0.2

1.5

−0.3

1.0

Flat bila er Opt-10

−0.4 −0.5 2.9

Opt-100 Opt-1100 3.0

3.1

0.5 3.2

3.3

3.4


3.5

0.0 2.9

3.6


3.0

(c) Interlayer interaction energy

3.1

3.2

3.3

3.4


(d) Band gap

Figure 8: Evolution of (a) the energy per atom with respect to the lattice parameter b∗ , (b) the buckling height, (c) the interlayer interaction energy and (d) the band gap for the case of bi-axial strain. Circles and solid lines indicate that the structure adopts a buckled structure under strain. Crosses and dashed lines indicate a at structure. The equilibrium lattice parameter can be found in Table 1. The results for structures under a uni-axial strain of 6% (perpendicular to the outer atom lines for buckled structures) are presented in Table 3.

21



Page 22 of 28

Table 3: Summary of the properties under a compressive uni-axial perpendicular strain of 6 %. Model Lattice parameter a (Å) Lattice parameter b (Å) a∗ (Å) b∗ (Å) Inner interlayer distance dI (Å) Outer interlayer distance dO (Å) buckling height ∆z (Å) Energy (eV/atom) Relative energy (eV/atom) Band gap - PBE (eV) Gap type

Flat bilayer 3.27 3.07 3.27 3.07 2.83 2.83 0.0 -2083.085 0.023 2.183 Direct

Opt-10 3.25 5.27 3.25 3.04 2.21 3.24 0.51 -2082.099 0.009 1.915 Indirect

Opt-100 5.61 9.16 3.24 3.05 2.22 3.48 0.63 -2082.099 0.009 1.991 Direct

Opt-1100 5.62 6.08 3.25 3.04 2.14 3.71 0.78 -2082.108 0.000 1.956 Direct

Conclusions Using rst-principles calculations, new models of buckled bilayer GaN were built and studied. These structures were shown to be more stable than the at bilayer GaN, with two of them showing the lowest energies (namely Opt-100 and Opt-1100). The structural changes induced by buckling allow relaxation of the structures to lower-energy states and have a signicant impact on the interlayer interactions. As a result, the out-of-plane orbitals, namely the 2pz orbitals of N, are strongly impacted, as seen at the top of the valence band of the electronic structure. Interestingly, in the Opt-1100 case, this leads to an indirect-to-direct band gap transition. The presence of a nearly at valence band could be exploited for applications using magnetic properties. It is interesting to note that the structures discovered here displayed surface reconstructions that only involve displacement of the atoms perpendicular to the surface, resulting in the surface buckling. It is also important to mention that the dierence in energy between the dierent structures studied in this work is relatively small (in the range of 1-10 meV). Therefore, at room temperature, several regions with the dierent geometries are likely to coexist, leading to more complex structural and electronic properties. Such a case, as well as the case of H-passivated GaN, should be investigated in further studies to

22


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better understand the properties of layered GaN. In addition, particular attention should be paid to the interaction between GaN and a substrate or an intermediate layer (such as graphene or h-BN) to evaluate the impact on the structural properties and, as a matter of fact, on the electronic and other properties. Applying a bi-axial or uni-axial stress modulates the buckling height as well as changes the electronic properties. In particular, for bi-axial strain, an indirect-to-direct band gap transition has been unveiled under application of a compressive stress. This observation opens the door to mechanically tailoring the properties of few-layer GaN by the application of stress in one or both in-plane directions.

Acknowledgement This work has been supported in part by MEXT Grants-in-Aid for Scientic Research (JPSJ KAKENHI Grants No. JP16K05412), and JST-CREST (JPMJCR18T1).

Supporting Information Available The Supporting Information is available free of charge on the ACS Publications Website at DOI: xxxx/xxxxxxxx Schematic view of bulk wurtzite GaN. Comparison of dierent stacking patterns of at bilayer GaN.

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Graphical TOC Entry Bulk GaN

Bilayer GaN

Ga N

28


Page 28 of 28

Uncovering New Buckled Structures of Bilayer GaN: A First-Principles

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