TA

QSa

b

c

d

e

f

g

a

ARRA

KSMEFH

1

bcdAadsWTt

h2

Applied Materials Today 7 (2017) 169–178

Contents lists available at ScienceDirect

Applied Materials Today

j ourna l h o mepage: www.elsev ier .com/ locate /apmt

heoretical prediction of a graphene-like structure of indium nitride: promising excellent material for optoelectronics

ing Penga,b,c,∗, Xin Suna, Han Wangd, Yunbo Yangd, Xiaodong Wene,f, Chen Huangg, Sheng Liub,uvranu Dea

Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USAResearch Center of Electronic Manufacturing and Packaging Integration, School of Power and Mechanical Engineering, Wuhan University, Wuhan 430072, ChinaNuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USADepartment of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, NY 12180, USAInstitute of Coal Chemistry, Chinese Academy of Sciences, Taiyuan 030001, ChinaSynfuels China Co. Ltd., Huairou, Beijing 101407, ChinaDepartment of Scientific Computing, Florida State University, Tallahassee, FL 32306, USA

r t i c l e i n f o

rticle history:eceived 26 October 2016eceived in revised form 18 February 2017ccepted 5 March 2017

eywords:tabilityechanical properties

lectronic propertiesirst-principles calculationsigh-order elastic constants

a b s t r a c t

Indium nitride is a very important solid state light material. Here we theoretically predict a planargraphene-like structure namely g-InN monolayer. We investigate its the mechanical, dynamical andthermal stabilities and structural, electronic, thermal, and mechanical properties via first-principles cal-culations. The electronic band structure shows that the g-InN monolayer possesses a direct band gap of0.57 eV. The analysis of its partial electron density of state reveals that the Nitrogen 2pz electrons andIndium 5s electrons are critical in forming this electronic band gap. The phonon band structure indicatesthat the plane structure is dynamically stable. Ab initio molecular dynamics simulations imply its thermalstability up to 600 K. We obtained the Helmholtz free energy, entropy, and heat capacity at constant vol-ume of g-InN monolayers as a function of temperature up to 600 K. The Debye temperature is 290.92 K.The g-InN has a low in-plane stiffness, and a high Poisson ratio of 0.586. The potential profiles and the

stress–strain curves indicate that the free standing g-InN monolayers can sustain large tensile strains,up to 0.13, 0.21, and 0.15 for armchair, zigzag, and biaxial deformations, respectively. Our results implythat g-InN monolayers are excellent optoelectronics materials and mechanically, dynamically, and ther-modynamically stable, suggesting that it is promising to fabricate it in near future. Our predicted elasticlimits could provide a safe-guide for strain-engineering the g-InN based electronic devices.

© 2017 Elsevier Ltd. All rights reserved.

. Introduction

Group III-nitride (BN, AlN, GaN, InN, TlN) semiconductors haveeen intensively studied for a long time because of their appli-ation in semiconductor devices, such as blue/UV light emittingiodes (LEDs), laser diodes and high-frequency transistors [1–5].mong them, InN exhibits unique electronic and optical propertiesnd thus has been a key material for optical and high temperatureevice applications. In most of the reports InN has been grown onapphire (0001) substrates and it is an n-type semiconductor [6,7].

urtzite InN possesses a direct band gaps of about 0.7 eV [8,9].he band gaps of hexagonal InN are weakly dependent on the elec-ron concentrations and close to 0.65 eV [10]. Alloyed with large

∗ Corresponding author.E-mail address: qpeng.org@gmail.com (Q. Peng).

ttp://dx.doi.org/10.1016/j.apmt.2017.03.001352-9407/© 2017 Elsevier Ltd. All rights reserved.

band gap material GaN, InxGa1−xN has a tunable band gap [11,12]between 3.46 and 1.9 eV [13] depending on the alloy composition.As a result, Ga-rich InxGa1−xN alloys are important materials used ingreen and blue light emitting diodes (LEDs) commercial production.Although it is not the equilibrium crystal structure, Zinc-blende InNis also observed and fabricated [14]. InN is measured to possessvery high electron mobility as 2700 cm2/Vs at room temperature[15]. InN is reported to have very high electron peak velocity of2 × 108 cm/s [16], which is larger than other III–V compounds, andit enables InN to be a promising material for high-speed field-effecttransistors [17]. Besides, heterojunctions made of InN and its alloysare very important for making quantum well LED and laser [18].

Due to the large lattice constant and thermal coefficients mis-

match between substrate materials and InN, the epitaxially grownInN is subject to high defects (such as dislocations, twins, andstacking faults) and misfit concentration [3]. However, nanoscaleInN could grow as nano-crystal, which has fewer defects [19,20].

170 Q. Peng et al. / Applied Materials

Fig. 1. Geometry ofg-InN monolayers. The proposed graphene-like planar struc-ture of g-InN are displayed in (a) overview, (b) armchair-side-view, and (c)zr

StstmmdI[

lbspbtepots[

acmedmapctTte

mesh are used for phonon calculations.

igzag-side-view. The In and N atoms are depicted by the large and small balls,espectively.

ingle-crystal Wurtzite type Indium nitride nano-wires and nano-ubes have been fabricated [21–24] and similar to BN nano wires, aemiconductor-to-metal transition is observed at 80 K [21]. Thisransition is attributed to a donor level below conduction band

inimum (CBM). Other nano-structure of InN is also successfullyade, strong aligned InN nano- and micro-scale rods were pro-

uced with chemical vapor deposition (CVD) [25], and branchednN nano-structures were grown with hydride vapor phase epitaxy26].

Despite the extensive efforts, the fabrication of mono-atomic-ayer graphene-like hexagonal InN monolayers (g-InN) has not yeteen reported to the authors’ best knowledge, though the atomistictructure of g-InN as depicted in Fig. 1 were proposed using first-rinciples calculations [4,5]. Motivated by the success of grapheneut non-successful fabrication of g-InN, We carry out computa-ionally investigation on the mechanical stabilities to ensure itsxistence, and the structural, electronic, thermal, and mechanicalroperties of g-InN monolayers. We focus on the planar structuref g-InN in this study because it is more interesting in applica-ions as buckled structure can be suppressed under constraintsuch as substrate, sandwiching, or heterogeneous multi-layers27–29].

Mechanical forces are much more tangible, reliable, and widelypplicable than other stimuli to materials. Macroscopic mechani-al stimuli are used for control of molecular systems and molecularachines, in addition to micro-device fabrication [30]. The knowl-

dge of mechanical properties of a material [31,32] is the base forevelopment of mechanically responsive nanomaterials which willake a significant contribution to improvements in our lifestyles

nd our society [30,33,34]. It is desirable to know the mechanicalroperties of g-InN monolayers as it will be different from its bulkounterpart [35]. We use density functional theory (DFT) calcula-ions to model their responses under various mechanical loadings.

he strain energy potential profile, the stress–strain relationships,he high order elastic constants, and the pressure dependent prop-rties are studied.

Today 7 (2017) 169–178

Combining the first-principles calculations with elastic theory,we make a link between continuum formulation and atomisticmodeling. Therefore, our results could also be useful in finiteelement modeling of the multi-scale calculations for mechanicalproperties of g-InN monolayers. As a part of the effort searching forflexible electronics, we have previously examined other graphene-like compounds, such as g-BN, g-GaN, g-TiN, g-ZnO, g-ZnS, g-GeC,g-MoS2, g-C6O, g-Boron, etc. [36–48]. Using the same method,we examine the mechanical stability and mechanical propertiesincluding high-order elastic constants up to fifth orders. Our resultsare compared to that of graphene and graphene-like boron nitridemonolayer (g-BN). The results of graphene [38] and g-BN [49] werereported previously. They serve as references to g-InN monolayersbecause they have similar structures. Another goal of this reportis the propose the existence of g-InN, though it has not been fab-ricated yet. To that end, we systematically studied the dynamicalstabilities and thermal stabilities in addition to the mechanical sta-bilities, which is prominent in the similar work. Furthermore, westudied the electronic properties and thermal properties besidesmechanical properties. Our results shows that the g-InN is quitestable with a direct band gap of 0.57 eV, indicating that it is verypromising to fabricate it in the near future for optoelectronics. Theorganization of this paper is as follows. Section 2 presents the com-putational details of DFT calculations. The results and analysis arein Section 3, followed by conclusions in Section 4.

2. Computation method

We perform the first-principles calculations with density-functional theory (DFT) for the total energies of the system, forceson each atom, and stresses of the simulation box under the desireddeformation configurations. The stress–strain relationship of g-InNmonolayers then are obtained. The first-principles calculations areappropriate and required to capture the quantum confinementin the third dimension in the atom-thick 2D materials, especiallyunder extremely large mechanical loading. DFT calculations werecarried out with the Vienna Ab-initio Simulation Package (VASP)[50] which is based on Kohn–Sham Density Functional Theory [51]with the generalized gradient approximations as parameterizedby Perdew, Burke, and Ernzerhof for exchange-correlation func-tions [52]. The electrons explicitly included in the calculations arethe (4d105s25p1) electrons for indium atoms and (2s22p3) elec-trons for nitrogen atoms. The core electrons are replaced by theprojector augmented wave and pseudo-potential approach [53]. Aplane-wave cutoff of 600 eV is used in all the calculations. The con-vergence of the total energy and forces is 10−5 eV and 10−3 eV/A,respectively. The calculations are performed at zero Kelvin withinthe frame of DFT. The irreducible Brillouin Zone was sampled witha Gamma-centered 24 × 24 × 1 k-mesh. There was a 15 A thick vac-uum region to reduce the inter-layer interaction to model the singlelayer system. The van der Waals interactions are described by theDFT-D3 method [54] throughout this study.

The atomic structures of all the deformed and undeformed con-figurations were obtained by fully relaxing 6-atom-unit cell whereall atoms were placed on one plane. The simulation invokes periodicboundary conditions for the two in-plane directions. The phonondispersion curves and thermal properties are calculated using thefinite displacement method implemented in the phonopy package[55]. A super cell containing 3 × 3 ×1 unit cell (54 atoms in total) areused to avoid significant artificial image interactions due to finitedisplacement for the force constants calculations. A 31 × 31 × 1 q-

To eliminate the artificial effect of the out-of-plane thick-ness of the simulation box on the stress, we used the secondPiola–Kirchhoff stress [49] to express the 2D forces per length with

Q. Peng et al. / Applied Materials Today 7 (2017) 169–178 171

F entioo The fir

ud

gcmapliw[

wawsdt

3

3

capawidy

Ta6a

flNoatasl

ments are compared in Fig. 4(a). It is clear that the valence electronsare mainly from nitrogen atoms. The local DOS plot in Fig. 4(b)implies that the p electrons are the majority of valence electrons. sand p electrons have the same population in conduction bands just

ig. 2. Atomic structure and orientation. (a) Atomic structure of g-InN in the convf the system, the atom species, and the In-N bond length 2.074 A are depicted. (b)

nits of N/m. The Lagrangian strain [56] was used in this study,efined as � = � + 1/2�2, where � is the engineering strain [49].

For high order elastic constants up to fifth order in thisraphene-like structure, there are only 14 independent elasticonstants that need to be explicitly considered due to the sym-etries of the atomic lattice point group D6h which consists of

sixfold rotational axis and six mirror planes [49]. The 14 inde-endent elastic constants of g-InN monolayers are determined by

east-squares fit to the stress–strain results from DFT calculationsn two steps, detailed in our previous work [49], which have been

ell used to explore the mechanical properties of 2D materials36,37,57,38,58,39,40,59,41–48].

Ab initio molecular dynamics (AIMD) simulations are carried outith Canonical ensemble (or NVT ensemble) where the volume V

re fixed. A super cell is built with 5 × 5 ×1 6-atom-unitcells, ofhich the total atom number is N = 150. The Nose–Hoover thermo-

tat is used to control the frequency of the temperature oscillationsuring the simulations. The time step is 1 fs and the total simulationime is 2 ps.

. Results and analysis

.1. Atomic structure

We consider a conventional unit cell as shown in Fig. 2 whichontains three nitrogen atoms and three indium atoms forming

coplanar hexagonal ring with periodic boundary conditions inlane directions only. The armchair direction is the direction of antom to one of its nearest neighbors in the planar honeycomb net-ork and was set to be parallel to the x axis, which is denoted “1”

n superscript of strains and stresses, as shown in Fig. 2. The zigzagirection is perpendicular to the armchair direction and was set as-axis, which is denoted “2” in superscript of strains and stresses.

We first optimize the atomistic structures for g-InN monolayers.he initial configuration is generated by placing 3 indium atomsnd 3 nitrogen atoms alternatively on the same plane, forming the-atom-unit cell. The relaxed structure shows that the six atomsre still coplanar, as shown in Fig. 2.

The most energetically favorable structure is set as the strain-ree structure, whose In–N bond length is 2.074 A (Fig. 2), mucharger than the corresponding B–N bonds in g-BN (1.52 A). The–In–N and In–N–In bond angles are 120◦. The lattice constantr the second nearest neighbor distance is 3.585 A. Our resultgree with a previous DFT study [4]. It is worth noting the that

he lattice constants of Wurtzite structure of bulk InN is a = 3.540nd c = 5.704 A in experimental measurement [60]. This strain-freetructure is then set as the reference when mechanical strain isoaded.

nal unit cell (6 atoms) in the undeformed reference configuration. The orientationsst Brillouin zone and high symmetrical k points.

3.2. Electronic band structure and density of state

We study the electronic band structure of unstrained monolayerg-InN. All the bands are shifted so that the Fermi energy are set to bezero. We only illustrate those bands within 4 eV around the Fermienergy, as in Fig. 3. The first Brillouin zone and high symmetricalk points of �, X, and M are shown in Fig. 2. We observed a directband gap of 0.57 eV at � point. The total density of of states (DOS)is also plotted in Fig. 3 aligned with the band structure for betterview. As a comparison, the band gap of bulk InN is 0.65–0.8 eV byexperiments [9,10] and 0.4–0.8 eV [4,61–64] by first-principles cal-culations. It is worth noting that the indium atoms have heavy 5delectrons which contribute the spin orbit coupling (SOC). However,the contribution should be smaller than 0.089 eV which is the SOCin g-TlN with 6d electrons [65]. Therefore, the contribution of SOCto the total band gap could be ignored. It is well known that theGGA-PBE underestimates the electronic band gap. Therefore, ourresult 0.57 eV gives a lower boundary of the band gap of g-InN andthe direct band gap is still valid.

To gain a better insight of how the electrons of individual ele-ments contribute to the band structures, we have studied theirprojected DOS (namely PDOS). The contribution from the two ele-

Fig. 3. Electronic structure.The electronic band structure (left) and the correspond-ing total electronic density of states (right) of a g-InN monolayer predicted fromGGA-PBE with vdW-D3 calculations. The Fermi energy are set to be zero.

172 Q. Peng et al. / Applied Materials Today 7 (2017) 169–178

F jectedc ic ban

asTmpfcf

3

tisntpdtvvaaa

ltaain(atfiiea

results agree with a previous DFT prediction [4]. The correspond-ing total density of states of these phonon modes confirm such anobservation, as illustrated by the gray line and area in right panelof Fig. 5. We further analyze the projected phonon density of states

Fig. 5. Phonon. Phonon band structure (left) and the corresponding total phonondensity of state (right, InN) for a g-InN monolayer predicted from GGA-PBE withvdW-D3 calculations. The ground state monolayer g-InN is thermodynamically sta-

ig. 4. Projected electronic DOS. Electronic density of state (DOS) of g-InN as proomponents. The N 2pz electrons and In 5s electrons are critical in forming electron

bove the band gap. The analysis of the DOS of nitrogen in Fig. 4(c)hows that p electrons of nitrogen are outweighed than s electrons.his means that the s electrons in low part of conduction bands areainly from Indium atoms. Further study of the components of the

electrons of nitrogen reveals that the pz electrons are responsibleor the upper part of the valence bands. From these analysis, we canonclude that the N 2pz electrons and In 5s electrons are critical inorming band gap.

.3. Phonon modes

The lattice vibrations play important role in the crystals proper-ies. The dynamics instability of a crystal structure is marked by themaginary frequencies of phonon modes. A crystal is dynamicallytable if its potential energy always increases against any combi-ations of atomic displacements. In the harmonic approximation,his is equivalent to the condition that all phonons have real andositive frequencies. Therefore, a imaginary frequency indicatesynamical instability of a system, which means that the correc-ive atomic displacements could reduce the potential energy in theicinity of the equilibrium atomic positions. Imaginary mode pro-ides useful information to study displacive phase transition, as

given structure having imaginary phonon modes can be led tolternative structures through continuous atomic displacementsnd lattice deformations.

To investigate the dynamics instability of the g-InN mono-ayers, we calculate its phonon dispersion curves. The results ofhe phonon band structure and its corresponding density of statesre shown in Fig. 5 for an strain-free monolayer of g-InN. First ofll, there is no imaginary frequencies of all phonon modes, whichndicates that the monolayer g-InN is dynamically stable. It is worthoting that there is a negligible amount of imaginary frequencies<50 Hz) around gamma point which is numerical artifact due to rel-tively small simulation box in this supper-cell method limited byhe computing resources. That all phonon modes are positive con-

rms that the atomic structures are in an energy-minimum position

n a configurational space. The phonon band structure is a directvidence that the g-InN monolayer are stable in finite temperature,nd it should be fabricated experimentally.

to (a) individual elements; (b) s, p, d orbitals; (c) for nitrogen’ orbitals; (d) three pd gap.

The frequencies of all the phonon modes are no more than21.8 THz. The three acoustic phonon bands with lowest frequen-cies around � points reflect the long-wave phase motions. Theyare longitudinal acoustic (LA), transverse acoustic (TA), and flexuralacoustic (ZA) modes. These three bands show a linear dependenceon the wave vectors around the zone center. There are 15 opti-cal phonon modes with much higher frequencies. There is a largephonon band gap of 9.3 THz lies between 7.6 and 16.9 THz. Our

ble. The projected phonon density of states on Indium atoms (In) and Nitrogen atoms(N) are depicted in red and green line, respectively on right panel. Indium (Nitrogen)atoms contribute more on lower (upper) optic phonon modes. (For interpretation ofthe references to color in this figure legend, the reader is referred to the web versionof this article.)

terials Today 7 (2017) 169–178 173

opfga7stbmw

3

tucvff

F

wcrt

C

T

S

latovohdtioaAaap

3

eatAo

Fig. 6. Thermal properties. The Helmholtz free energy FA , entropy S, and heat

Q. Peng et al. / Applied Ma

n each atoms. It is interesting that the upper branches of optichonon modes that are above the phonon band gap (or those with arequency larger than 16.9 THz) are contributed more by the Nitro-en atoms. On the contrary, the lower branch of optic phonon thatre below the phonon band gap (or those with a frequency less than.6 THz) depends mainly on the Indium atoms. This could be under-tood by the fact that an Indium atom has a mass 8.2 times largerhan that of a Nitrogen atom. We could predict that the phononand gap will be smaller in BN, AlN, and GaN, but larger in TlNonolayers. This could be useful in designing phononic materialsith 2D materials.

.4. Thermal properties

Once phonon frequencies over Brillouin zone are known, fromhe canonical distribution in statistical mechanics for phononsnder the harmonic approximation, the energy of phonon systeman be obtained [55]. The thermal properties of solids at constantolume can be calculated from their phonon density of states as aunction of frequencies. The phonon contribution to the Helmholtzree energy FA is given by [55]:

A = 12

∑q,�

�ωq,� + kBT∑q,�

ln[1 − exp(−�ωq,�/kBT)], (1)

here kB and � are the Boltzmann constant and the reduced Planckonstant, respectively, q and � are the wave vector and band index,espectively, ωq,� is the phonon frequency at q and �, and T is theemperature. The heat capacity CV at constant volume is given by

V =∑q,�

kB

(�ωq,�

kBT

)2 exp(−�ωq,�/kBT)

[exp(−�ωq,�/kBT) − 1]2(2)

he entropy S is calculated as the relationship of

= −kB

∑q,�

{ln[1 − exp(−�ωq,�/kBT)] − 1

T

�ωq,�

exp(−�ωq,�/kBT) − 1

}.

(3)

With the phonon density of states plotted in Fig. 5, We calcu-ated the Helmholtz free energy FA, entropy S, and heat capacity CV

t constant volume of g-InN monolayers as a function of tempera-ure, as shown in Fig. 6. It is known that the melting temperaturef a bulk InN is 1320 K. The quasi-harmonic approximation isalid in general up to 2/3 of melting temperature. Therefore, wenly plot their thermal properties up to 600 K within the quasi-armonic approximation. We notice that the Helmholtz free energyecreases monotonically with respect to temperature, opposed torends of entropy and heat capacity. These normal thermal behav-ors imply the validity of our calculations. At a room temperaturef 300 K, the Helmholtz free energy, entropy, and heat capacityre −2.977 kJ/mol, 197.17 J/Kmol, and 120.57 J/K mol, respectively.ccording to the Debye approximation, the Debye frequency can bechieved from the specific heat capacity. The Debye frequency of

g-InN monolayer is 6.06 THz and the corresponding Debye tem-erature is 290.92 K.

.5. Thermal stability

Molecular dynamics simulations are a common approach toxamine the thermal stabilities of a material. Limited by the avail-

bility of the reliable empirical force field of InN, we investigate thehermal stabilities using the accurate and computing demandingIMD simulations [66]. We study the thermo-dynamical behaviorsf g-InN monolayers at finite temperature T = 600 K. The snapshot

capacity CV at constant volume of g-InN monolayers are calculated as a functionof temperature with harmonic approximation. Physical units are shown with labelsof the physical properties, and the value of the vertical axis is shared by them.

of the structure of g-InN after 2.0 ps equilibrium at 600 K are dis-played in Fig. 7. There are small wiggles due to the out-of-planevibration modes at finite temperatures. However, no broken bondsare observed, which implies that this g-InN structure is thermody-namically stable under finite temperatures, up to 600 K.

3.6. Strain energy profile

To obtain the strain-energy profile, we need to have mechani-cal loading on the system, both elongation or compression. Whenstrain is applied, the system will be disturbed away from the equi-librium state. All the atoms of the system are allowed full freedomof motion. A quasi-Newton algorithm is used to relax all atoms intoequilibrium positions within the deformed unit cell that yields theminimum total energy for the imposed strain state of the supercell. Since the configuration energy of the strain-free configura-tion is the minima of the potential well, any strain will increasethe system’s energy. By applying different amounts of strain alongdifferent directions, the potential well can be explored.

In this study, Lagrangian strains ranging from −0.1 to 0.3 areconsidered with an increment of 0.01 in each step for all threedeformation modes. It is important to include the compressivestrains since they are believed to be the cause of the rippling ofthe free standing atomic sheet [67]. Because it was reported thata graphene sheet experiences biaxial compression after thermalannealing [68], which could also happen with g-InN monolayers.We selected such an asymmetrical strain range (−0.1 ≤ � ≤ 0.3)owing to the non-symmetric mechanical responses of material. Thelower and upper boundary of the strain are consequent upon itsmechanical instability to the compressive and the tensile strains.

The strain energy is defined as Es = (Etot − E0)/n, where Etot is thetotal energy of the strained system, E0 is the total energy of thestrain-free system, and n = 6 is the number of atoms in the unitcell. The results of Es of the g-InN monolayer are plotted in Fig. 8aas a function of strain in three modes of uniaxial armchair, uni-axial zigzag, and biaxial deformation. We have also compared thepotential profile of g-InN with that of g-BN (Fig. 8b). It is clear thatEs is anisotropic with strain direction. Es is non-symmetrical for

compression (� < 0) and tension (� > 0) for all three modes. Thisnon-symmetry indicates the anharmonicity of the g-InN monolayerstructures, and it is common in large deformation of condensedmatters. It is worth noting that for small strain, there is a harmonic

174 Q. Peng et al. / Applied Materials Today 7 (2017) 169–178

F mond and N

rcah

Fs

ig. 7. Thermal stability. ab initio molecular dynamics simulations show that g-InNisplayed in (a) overview, (b) armchair-side-view, and (c) zigzag-side-view. The In

egion, where the Es is a quadratic function of applied strain. As aonsequence, the stresses as the derivatives of the strain energiesre linearly increasing with the increase of the applied strains in thearmonic region. From the plots of the strain energy profile, one can

ig. 8. Strain energy. Energy-strain responses for armchair, zigzag, and biaxialtrains of g-InN (top), compared with g-BN (bottom) [49].

olayers are stable at T = 600 K. The snapshot of the structure of g-InN after 2.0 ps are atoms are depicted by the large and small balls, respectively.

tell that the harmonic region can be taken between −0.02 < � < 0.02in the g-InN monolayers.

When the strain is larger, the linear stress–strain relationship isinvalid. The range of these strain is anharmonic region. The mainfeature of the anharmonic region is that the higher order terms arenot negligible. With even larger loading of strains, the systems willundergo irreversible structural changes. Then the system enters aplastic region where it may fail into parts. The maximum strain inthe anharmonic region is the ultimate strain. The summation of thecritical tensile strain and critical compressive strain is the stableregion of that deformation. The width of the stable region definesthe opening width of the potential energy well (Fig. 8). In general,the opening width and depth of the potential energy well �s reflectsthe flexibility and strength of a nano structure, respectively. Theaverage width of the stable regions of the three deformation modes(i.e., the opening width of the potential energy wells) is a reason-ably good scale for the mechanical stabilities of the nano structures.As a result, from the point view of potential energy, we concludethat g-InN is mechanically stable. However, it is less stable than g-BN, because both the depth and width of the potential energy wellare smaller than that of g-BN. Besides the scale for he mechanicalstabilities, strain energy profile can be used to estimate the range oflattice mismatch feasible for epitaxial growth of the 2D materialson the substrates [5].

3.7. Stress–strain curves

The stresses are the derivatives of the total energies with respectto the strains. We plot the stress–strain relationship in Fig. 9 alongthe three modes of uniaxial strains along the armchair direction(mode a), uniaxial strains along the zigzag direction (mode z), andbiaxial strains (mode b). The material behaves in an asymmet-ric manner with respect to compressive and tensile strains. Withincreasing strains, the In–N bonds are stretched and eventually rup-ture. When strain is applied in the mode a, the bonds of thoseparallel in this direction are more severely stretched than those

in other directions. Under the deformation mode z, in which thestrain is applied perpendicularly to the armchair direction, there isno bond parallel to this direction. The two In–N bonds at an inclineto the zigzag direction with an angle of about 30◦ are more severely

Q. Peng et al. / Applied Materials Today 7 (2017) 169–178 175

F mchair (top), zigzag (middle), and biaxial (bottom) strains, compared with that of g-BN( of DFT calculations (“DFT”) to continuum elastic theory.

ssiuao

3

eIfmtroi

sot23scla

[ldpGibc

Table 1Elastic constants. Nonzero independent components for the second order elasticconstants (SOEC), third order elastic constants (TOEC), fourth order elastic constants(FOEC), and fifth order elastic constants (FFOEC) tensor components (in units ofN/m), Poisson’s ration � and in-plane stiffness Ys of g-InN, g-BN, and graphene fromDFT calculations.

g-InN g-BN Graphene

a (A) 3.585 2.512 2.468

Ys (N/m) 62.0 278.3 340.8

� 0.586 0.225 0.178

SOECC11 94.4 293.2 352.0C12 55.4 66.1 62.6

TOECC111 −716.7 −2513.6 −3089.7C112 −461.0 −425.0 −453.8C222 −609.1 −2284.2 −2928.1

FOEC

C1111 2978 16547 21927C1112 3307 2609 2731C1122 2514 2215 3888C2222 −1745 12288 18779

FFOEC

C11111 3382 −65265 −118791C11112 −591 −8454 −19173C11122 −17843 −28556 −15863

ig. 9. Stress–strain relationship. Stress–strain responses of g-InN (left) under arright). �1 (�2) denotes the x (y) component of stress. “Cont” stands for the fitting

tretched than those in the armchair direction. Under the ultimatetrain in mode z, which is 0.26, the two In–N bonds that are at anncline to the armchair direction are observed to rupture. Under theltimate strain in mode b, �b

u = 0.18, the two In–N bonds that aret an incline to the zigzag direction with an angle of about 30◦ arebserved to rupture.

.8. Elastic constants

The elastic constants are essential parameters to describe thelasticity of materials. The 14 independent elastic constants of g-nN are determined by a least-square fit to the stress–strain resultsrom DFT calculations [49]. The second order elastic constants

odel the linear elastic response. The higher (>2) order elas-ic constants are important to characterize the nonlinear elasticesponse of g-InN using a continuum description. These can bebtained using a least square fit of the DFT data and are reportedn Table 1. Corresponding values for graphene are also shown.

Our results of in-plane stiffness of g-InN is Ys = 62.0 N/m, muchmaller than that of g-BN and graphene. In order to interpolateur results in terms of standard units of GPa, we need the size ofhe out-of-plane dimension, which however is not well defined inD material. Taking the interlayer vdW distance of 5.7 A from itsD layered Wurtzite-type structure [60], our result of the in-planetiffness is 109 GPa, less than the bulk modulus (125 GPa) of its bulkounterpart [60]. This could be a consequence of the stretch of itsattice constants. The Poisson’s ratio of g-InN is � = 0.586, about 2.6nd 3.3 times that of g-BN and graphene, respectively.

Higher order (>2) elastic constants are important quantities69] in studying the nonlinear elasticity, harmonic generation,attice defects, phase transitions, strain softening, temperatureependence of elastic constants, phonon–phonon interactions,hoton–phonon interactions, thermal expansion (through the

runeisen parameter), echo phenomena, and so on [38]. Exper-

mentally the higher order elastic constants can be determinedy measuring the changes of sound velocities under the appli-ation of hydrostatic and uniaxial stresses [70]. As it is more

C12222 −9615 −36955 −27463C22222 40801 −100469 −134752

convenient to apply pressure in experiment, the full stress–strainrelationship described by the higher order elastic constants arecritical. Explicitly, when pressure is applied, the pressure depend-ent second-order elastic moduli can be obtained from the higherorder elastic continuum description [49]. The third-order elasticconstants are important in understanding the nonlinear elasticity ofmaterials, such as changes in acoustic velocities due to finite strain.As a consequence, nano devices (such as nano surface acoustic wavesensors and nano waveguides) could be synthesized by introducing

local strain [71,72].

These 14 components of the elastic constants in Table 1 canbe divided into two classes. One is the normal terms, which are

176 Q. Peng et al. / Applied Materials Today 7 (2017) 169–178

Fig. 10. Order effect. Predicted stress–strain responses of biaxial deformation ofgp

CCstntttSsvbFFIem

petaotsom0fp

3

cfplpietaE

the stress–strain curves and the positive ultimate tensile stressesindicate that this structure is mechanical stable. Our results showthat the g-InN monolayers are stable under various strains.

Table 2Elastic limits. Ultimate strengths (�a

m , �zm , �b

m) in units of N/m and ultimate strains(�a

m , �zm , �b

m) under uniaxial strain (armchair and zigzag) and biaxial from DFT cal-culations, compared with g-BN and graphene.

g-InN g-BNa Grapheneb

�am 8.0 23.6 28.6

�am 0.18 0.18 0.19

�zm 8.0 26.3 30.4

�zm 0.21 0.26 0.23

b

-InN monolayer from different orders (second, third, fourth, and fifth order) com-ared to that from the DFT calculations (circle line).

11, C22, C111, C222, and so on. The other is shear terms, which are12, C112, C1112, and so on. The normal terms are in charge of thetrength of the material, and the shear terms reflects more abouthe flexibility. The comparison of the data in Table 1 reveals that theormal terms of g-InN monolayers are smaller in magnitude thanhose of g-BN and graphene, consistent with the conclusion thathe g-InN is “softer”. However, the shear terms do not show a clearrend. The g-InN monolayers exhibit instability under large tension.tress–strain curves in the previous section show that they mightoften when the strain is larger than the ultimate strain. From theiew of electron bonding, this is due to the bond weakening andreaking. This softening behavior is determined by the TOECs andFOECs in the continuum aspect. The negative values of TOECs andFOECs ensure the softening of g-InN monolayer under large strain.t should be pointed out that the tensile strength could be an over-stimate of the onset of the instability due to the limitation of theodel [73].The importance of the high order elastic constants can be

erceived when we considering their contribution to the nonlin-ar elasticity. When we only consider the second order elasticity,he stress varies with strain linearly. Take the biaxial deformations an example. As illustrated in Fig. 10, the linear behaviors arenly valid within a small strain range, about 0.02, �h = 0.02. Withhe knowledge of the elastic constants up to the third order, thetress–strain curve can be accurately predicted within the rangef � ≤ 0.06. Using the elastic constants up to the fourth order, theechanical behaviors can be well treated up to a strain as large as

.1. For the strains beyond 0.1, the fifth order elastic are requiredor accurate modeling. The analysis of the uniaxial deformationsrovides similar results.

.9. Pressure effect on the elastic moduli

As a demonstration of the usage of the high order elasticonstants, we predicted the pressure effect on the elastic modulirom these elastic constants (Table 1). The pressure effect on thehotoluminescence and emission energies of the InN/GaN super

attices has been reported [74]. The nonlinear elasticity and theressure dependence of elastic constants have been investigated

n bulk III-N compounds [75]. However, the pressure effect on the

lastic moduli is still unknown in g-InN monolayers. We calculatehe effect of the second-order elastic moduli on the pressure pcting in the plane of g-InN from the third-order elastic moduli.xplicitly, when pressure is applied, the pressure dependent

Fig. 11. Pressure effect. Second-order elastic moduli, in-plane Young’s modulus,and Poisson ratio as a function of the pressure predicted from the high order elasticconstants.

second-order elastic moduli ( ˜C11, ˜C12, ˜C22) can be obtained fromC11, C12, C22, C111, C112, C222, Ys, �, as detailed in Ref. [71].

The second-order elastic moduli of g-InN are seen to increaselinearly with the applied pressure (Fig. 11). Poisson’s ratio alsoincreases monotonically with increasing pressure. ˜C11 is not sym-metrical to ˜C22 any more; only when P = 0, ˜C11 = ˜C22 = C11. Thisanisotropy could be the outcome of anharmonicity.

3.10. Mechanical instabilities

The mechanical instabilities are the most important in thereally applications and material or system designs. The ultimatestrains are determined as the corresponding strain of the ultimatestress, which is the maxima of the stress–strain curve. The ulti-mate strengths and strains corresponding to the different strainconditions are summarized in Table 2, compared with that of g-BN and graphene. since they have similar structure and they areclose to each other in the periodic table. The g-InN sheet behavesin an asymmetric manner with respect to compressive and tensilestrains. With increasing strains, the B–B bonds are stretched andeventually rupture. The positive slopes of the stress–strain curvesand the positive ultimate tensile stresses indicate that this structureis mechanical stable.

The g-InN sheet behaves in an asymmetric manner with respectto compressive and tensile strains. With increasing strains, the In–Nbonds are stretched and eventually rupture. The positive slope of

�m 8.0 27.8 32.1�b

m 0.15 0.24 0.23

a Ref. [38].b Ref. [49].

terials

tme(

bscvpp

4

fpcemqiaba

tlNib

fmdpcr

tlPpgaHtfioaip

usps

C

A

c

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Q. Peng et al. / Applied Ma

From the view of energy, the mechanical instabilities are relatedo the energy barriers, over which the material fails. In our ideal

odel, this energy barrier is the binding energy. Therefore, we canstimate the binding energy of 1.3 eV from the strain-energy plotFig. 8) with �b

m = 0.15.Note that the softening of the g-InN monolayers under strains

eyond the ultimate strains only occurs for ideal conditions. Theystems under this circumstance are in a meta-stable state, whichan be easily destroyed by long wavelength perturbations andacancy defects, as well as high temperature effects, and enter alastic state [76]. Thus only the data within the ultimate strain hashysical meaning.

. Conclusions

In summary, our first-principles calculations based on densityunctional theory shed light on the stabilities and properties of theroposed g-InN. For stabilities, the main concern is the mechani-al, dynamical, and thermal stabilities, which could be theoreticallyxamined by the mechanical loading, phonon band structure, andolecular dynamics simulations, respectively. The imaginary fre-

uencies are absent from the phonon band structure of g-InN,ndicating that the planar structure of g-InN is dynamically stable.b initio molecular dynamics simulations illustrate that no brokenonds are observed under thermo-dynamical vibrations at temper-ture of 600 K, implying its thermal stability.

A direct band gap of 0.57 eV is predicted which indicates thathe excellent photoelectronic performance is still kept in the mono-ayer structures. The partial electron DOS analysis shows that theitrogen 2pz electrons and Indium 5s electrons are critical in form-

ng this band gap. This knowledge could be useful in tuning theand gap and the photoluminescence of this material.

From the phonon density of states, we obtained the Helmholtzree energy, entropy, and heat capacity at constant volume of g-InN

onolayers as a function of temperature up to 600 K using latticeynamics within quasi-harmonic approximation. At a room tem-erature of 300 K, the Helmholtz free energy, entropy, and heatapacity are −2.977 kJ/mol, 197.17 J/Kmol, and 120.57 J/K J/kmol,espectively. The Debye temperature is 290.92 K.

By applying various mechanical strains, we examined the elas-ic and non-elastic mechanical properties of g-InN. We find that aow in-plane stiffness, about 18% of that of graphene, and a highoisson ratio of 0.586, 3.3 times that of graphene. The potentialrofiles and the stress–strain curves indicate that the free standing-InN monolayers can sustain large tensile strains, up to 0.13, 0.21,nd 0.15 for armchair, zigzag, and biaxial deformations, respectively.owever, both the strength and flexibility are reduced compared

o graphene-like boron nitride monolayers. The third, fourth, andfth order elastic constants are indispensable for accurate modelingf the mechanical properties under strains larger than 0.02, 0.06,nd 0.10 respectively. The second order elastic constants, includ-ng in-plane stiffness, are predicted to monotonically increase withressure, while the trend in the Poisson ratio is reversed.

Our results imply that g-InN monolayers are mechanically stablender various large strains, in addition to dynamical and thermaltability, suggesting the existence of such a planar structure at tem-eratures up to 600 K. The elastic limits provide a safe-guide fortrain-engineering the g-InN based electronics.

ompeting financial interests

The authors declare no competing financial interests.

uthors’ contributions

Q.P., S.L., and S.D. designated the research topic. Q.P. and X.S.arried out the calculations. Q.P., H.W., Y.Y., C.H., and X.W. did the

[

[

Today 7 (2017) 169–178 177

analysis. Q.P. wrote the main manuscript text and designed thefigures. All authors discussed the results and commented on themanuscript.

Acknowledgements

The authors would like to acknowledge the generous financialsupport from the Defense Threat Reduction Agency (DTRA) Grant# BRBAA08-C-2-0130 and # HDTRA1-13-1-0025.

References

[1] O. Ambacher, Growth and applications of group III-nitrides, J. Phys. D 31 (20)(1998) 2653.

[2] A.G. Bhuiyan, A. Hashimoto, A. Yamamoto, Indium nitride (InN): a review ongrowth, characterization, and properties, J. Appl. Phys. 94 (5) (2003)2779–2808.

[3] S.C. Jain, M. Willander, J. Narayan, R.V. Overstraeten, III-nitrides: growth,characterization, and properties, J. Appl. Phys. 87 (3) (2000) 965–1006.

[4] H.L. Zhuang, A.K. Singh, R.G. Hennig, Computational discovery of single-layerIII–V materials, Phys. Rev. B 87 (2013) 165415.

[5] A.K. Singh, H.L. Zhuang, R.G. Hennig, Ab initio synthesis of single-layer III–Vmaterials, Phys. Rev. B 89 (2014) 245431.

[6] T. Inushima, M. Higashiwaki, T. Matsui, Optical properties of Si-doped InNgrown on sapphire (0001), Phys. Rev. B 68 (2003) 235204.

[7] G. Kaczmarczyk, A. Kaschner, S. Reich, A. Hoffmann, C. Thomsen, D.J. As, A.P.Lima, D. Schikora, K. Lischka, R. Averbeck, H. Riechert, Lattice dynamics ofhexagonal and cubic InN: Raman-scattering experiments and calculations,Appl. Phys. Lett. 76 (15) (2000) 2122–2124.

[8] T. Matsuoka, H. Okamoto, M. Nakao, H. Harima, E. Kurimoto, Optical bandgapenergy of wurtzite InN, Appl. Phys. Lett. 81 (7) (2002) 1246–1248.

[9] J. Wu, W. Walukiewicz, K.M. Yu, J.W. Ager, E.E. Haller, H. Lu, W.J. Schaff, Y.Saito, Y. Nanishi, Unusual properties of the fundamental band gap of InN,Appl. Phys. Lett. 80 (21) (2002) 3967–3969.

10] V.Y. Davydov, A.A. Klochikhin, V.V. Emtsev, S.V. Ivanov, V.V. Vekshin, F.Bechstedt, J. Furthmller, H. Harima, A.V. Mudryi, A. Hashimoto, A. Yamamoto,J. Aderhold, J. Graul, E.E. Haller, Band gap of InN and In-rich InxGa1−xN alloys(0.36 < x < 1), Phys. Status Solidi (B) 230 (2) (2002) R4–R6.

11] M.D. McCluskey, C.G. Van de Walle, C.P. Master, L.T. Romano, N.M. Johnson,Large band gap bowing of InxGa1−xN alloys, Appl. Phys. Lett. 72 (21) (1998)2725–2726.

12] B.-T. Liou, C.-Y. Lin, S.-H. Yen, Y.-K. Kuo, First-principles calculation for bowingparameter of wurtzite InxGa1−xN, Opt. Commun. 249 (13) (2005) 217–223.

13] K. Osamura, S. Naka, Y. Murakami, Preparation and optical properties ofGa1−xInxN thin films, J. Appl. Phys. 46 (8) (1975) 3432–3437.

14] M. Oseki, K. Okubo, A. Kobayashi, J. Ohta, H. Fujioka, Field-effect transistorsbased on cubic indium nitride, Sci. Rep. 4 (2014) 3951.

15] T.L. Tansley, C.P. Foley, Electron mobility in indium nitride, Electron. Lett. 20(25–2) (1984) 1066–1068.

16] K.T. Tsen, C. Poweleit, D.K. Ferry, H. Lu, W.J. Schaff, Observation of largeelectron drift velocities in InN by ultrafast Raman spectroscopy, Appl. Phys.Lett. 86 (22) (2005) 222103.

17] M. Oseki, K. Okubo, A. Kobayashi, J. Ohta, H. Fujioka, Field-effect transistorsbased on cubic indium nitride, Sci. Rep. 4 (2014) 3951.

18] S. Nakamura, The roles of structural imperfections in InGaN-based bluelight-emitting diodes and laser diodes, Science 281 (5379) (1998) 956–961.

19] R. Calarco, InN nanowires: growth and optoelectronic properties, Materials 5(11) (2012) 2137–2150.

20] J. Kamimura, M. Ramsteiner, U. Jahn, C.-Y.J. Lu, A. Kikuchi, K. Kishino, H.Riechert, High-quality cubic and hexagonal InN crystals studied bymicro-Raman scattering and electron backscatter diffraction, J. Phys. D: Appl.Phys. 49 (15) (2016) 155106.

21] L. Huang, D. Li, P. Chang, S. Chu, H. Bozler, I.S. Beloborodov, J.G. Lu, Quantumtransport in indium nitride nanowires, Phys. Rev. B 83 (24) (2011).

22] T. Richter, H. Lueth, T.H. Schaepers, R. Meijers, K. Jeganathan, S.E. Hernandez,R. Calarco, M. Marso, Electrical transport properties of single undoped andn-type doped InN nanowires, Nanotechnology 20 (40) (2009) 405206.

23] L.W. Yin, Y. Bando, D. Golberg, M.S. Li, Growth of single-crystal indium nitridenanotubes and nanowires by a controlled-carbonitridation reaction route,Adv. Mater. 16 (20) (2004) 1833.

24] Y. Cho, S. Sadofev, S. Fernandez-Garrido, R. Calarco, H. Riechert, Z. Galazka, R.Uecker, O. Brandt, Impact of substrate nitridation on the growth of InN onIn2O3(111) by plasma-assisted molecular beam epitaxy, Appl. Surf. Sci. 369(2016) 159–162.

25] B.S. Simpkins, A.D. Kansal, P.E. Pehrsson, Induced epitaxy for growth ofaligned indium nitride nano- and microrods, Cryst. Growth Des. 10 (9) (2010)3887–3891.

26] I. Jung, S. Lee, D. Shin, H. Park, M.Y. Ju, C. Kim, Regularly branched InNnanostructures: zinc-blende nanocore and polytypic transition, J. Appl.Crystallogr. 45 (3) (2012) 503–506.

27] A.K. Geim, I.V. Grigorieva, Van der Waals heterostructures, Nature 499 (7459)(2013) 419–425, http://dx.doi.org/10.1038/nature12385.

1 terials

[

[

[

[[

[

[

[[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[75] S.P. Lepkowski, J.A. Majewski, G. Jurczak, Nonlinear elasticity in III-N

78 Q. Peng et al. / Applied Ma

28] Y. Lei, F. Chen, R. Li, J. Xu, A facile solvothermal method to producegraphene-ZnS composites for superior photoelectric applications, Appl. Surf.Sci. 308 (2014) 206–210.

29] A.K. Kole, P. Kumbhakar, T. Ganguly, Observations of unusual temperaturedependent photoluminescence anti-quenching in two-dimensionalnanosheets of ZnS/ZnO composites and polarization dependentphotoluminescence enhancement in fungi-like ZnO nanostructures, J. Appl.Phys. 115 (22) (2014) 224306.

30] K. Ariga, T. Mori, J.P. Hill, Mechanical control of nanomaterials andnanosystems, Adv. Mater. 24 (2) (2012) 158–176.

31] W. Soboyejo, Mechanical Properties of Engineered Materials, CRC Press, 2002.32] J.F. Nye, Physical Properties of Crystals, Oxford Science Publications, Oxford,

1995.33] Y. Dai, S. Li, H. Gao, W. Wang, Q. Sun, Q. Peng, C. Gui, Z. Qian, S. Liu, Stress

evolution in AlN and GaN grown on Si(111): experiments and theoreticalmodeling, J. Mater. Sci. 27 (2) (2015) 2004–2013.

34] Y. Dai, W. Wang, C. Gui, X. Wen, Q. Peng, S. Liu, A first-principles study of themechanical properties of AlN with Raman verification, Comput. Mater. Sci.112 (2016) 342–346.

35] Q. Peng, S. De, Elastic limit of silicane, Nanoscale 6 (2014) 12071–12079.36] Q. Peng, C. Liang, W. Ji, S. De, A first principles investigation of the mechanical

properties of g-TlN, Model. Numer. Simul. Mater. Sci. 2 (2012) 76–84.37] Q. Peng, C. Liang, W. Ji, S. De, A first principles investigation of the mechanical

properties of g-ZnO: the graphene-like hexagonal zinc oxide monolayer,Comput. Mater. Sci. 68 (2013) 320–324.

38] Q. Peng, C. Liang, W. Ji, S. De, A theoretical analysis of the effect of thehydrogenation of graphene to graphane on its mechanical properties, Phys.Chem. Chem. Phys. 15 (2013) 2003–2011.

39] Q. Peng, C. Liang, W. Ji, S. De, A first-principles study of the mechanicalproperties of g-GeC, Mech. Mater. 64 (2013) 135–141.

40] Q. Peng, X.-J. Chen, W. Ji, S. De, Chemically tuning mechanics of graphene byBN, Adv. Eng. Mater. 15 (2013) 718–727.

41] Q. Peng, Z. Chen, S. De, A density functional theory study of the mechanicalproperties of graphane with van der Waals corrections, Mech. Adv. Mater.Struct. 22 (2015) 717–721.

42] Q. Peng, X. Wen, S. De, Mechanical stabilities of silicene, RSC Adv. 3 (2013)13772–13781.

43] Q. Peng, S. De, Outstanding mechanical properties of monolayer MoS2 and itsapplication in elastic energy storage, Phys. Chem. Chem. Phys. 15 (44) (2013)19427–19437.

44] Q. Peng, S. De, Mechanical properties and instabilities of ordered grapheneoxide C6O monolayer, RSC Adv. 3 (46) (2013) 24337–24344.

45] Q. Peng, L. Han, X. Wen, S. Liu, Z. Chen, J. Lian, S. De, Mechanical propertiesand stabilities of ˛-boron monolayers, Phys. Chem. Chem. Phys. 17 (2015)2160–2168.

46] Q. Peng, L. Han, X. Wen, S. Liu, Z. Chen, J. Lian, S. De, Mechanical propertiesand stabilities of g-ZnS monolayers, RSC Adv. 5 (2015) 11240–11247.

47] Q. Peng, A.K. Dearden, X.-J. Chen, C. Huang, X. Wen, S. De, Peculiar pressureeffect on Poisson ratio of graphone as a strain damper, Nanoscale 7 (2015)9975–9979.

48] Q. Peng, L. Han, J. Lian, X. Wen, S. Liu, Z. Chen, N. Koratkar, S. De, Mechanicaldegradation of graphene by epoxidation: insights from first-principlescalculations, Phys. Chem. Chem. Phys. 17 (2015) 19484–19490.

49] Q. Peng, W. Ji, S. De, Mechanical properties of the hexagonal boron nitridemonolayer: ab initio study, Comput. Mater. Sci. 56 (2012) 11–17.

50] G. Kresse, J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev.

B 47 (1993) 558.

51] P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (3B)(1964) B864.

52] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximationmade simple, Phys. Rev. Lett. 77 (1996) 3865–3868.

[

Today 7 (2017) 169–178

53] R.O. Jones, O. Gunnarsson, The density functional formalism, its applicationsand prospects, Rev. Mod. Phys. 61 (3) (1989) 689–746.

54] S. Grimme, J. Antony, S. Ehrlich, H. Krieg, A consistent and accurate ab initioparametrization of density functional dispersion correction (DFT-D) for the 94elements H–Pu, J. Chem. Phys. 132 (15) (2010) 154104.

55] A. Togo, I. Tanaka, First principles phonon calculations in materials science,Scr. Mater. 108 (2015) 1–5.

56] K. Brugger, Thermodynamic definition of higher order elastic coefficients,Phys. Rev. A 133 (6A) (1964) 1611.

57] Q. Peng, W. Ji, S. De, First-principles study of the effects of mechanical strainson the radiation hardness of hexagonal boron nitride monolayers, Nanoscale5 (2013) 695–703.

58] Q. Peng, X.-J. Chen, S. Liu, S. De, Mechanical stabilities and properties ofgraphene-like aluminum nitride predicted from first-principles calculations,RSC Adv. 3 (2013) 7083–7092.

59] Q. Peng, C. Liang, W. Ji, S. De, Mechanical properties of g-GaN: a firstprinciples study, Appl. Phys. A 13 (2013) 483–490.

60] M. Ueno, M. Yoshida, A. Onodera, O. Shimomura, K. Takemura, Stability of thewurtzite-type structure under high pressure: GaN and InN, Phys. Rev. B 49(1994) 14–21.

61] R. Ahmed, H. Akbarzadeh, F. e Aleem, A first principle study of band structureof III-nitride compounds, Physica B 370 (2005) 52–60.

62] P. Carrier, S.-H. Wei, Theoretical study of the band-gap anomaly of InN, J.Appl. Phys. 97 (3) (2005) 033707.

63] T.K. Maurya, S. Kumar, S. Auluck, Ab-initio study of electronic and opticalproperties of InN in wurtzite and cubic phases, Opt. Commun. 283 (23) (2010)4655–4661.

64] Q. Yan, E. Kioupakis, D. Jena, C.G. Van de Walle, First-principles study ofhigh-field-related electronic behavior of group-III nitrides, Phys. Rev. B 90(2014) 121201.

65] X. Li, Y. Dai, Y. Ma, W. Wei, L. Yu, B. Huang, Prediction of large-gap quantumspin hall insulator and Rashba–Dresselhaus effect in two-dimensional g-TlA(A = N, P, As, and Sb) monolayer films, Nano Res. 8 (9) (2015)2954–2962.

66] Q. Sun, Y. Dai, Y. Ma, T. Jing, W. Wei, B. Huang, Ab initio prediction andcharacterization of Mo2C monolayer as anodes for lithium-ion andsodium-ion batteries, J. Phys. Chem. Lett. 7 (6) (2016) 937–943.

67] E. Cerda, L. Mahadevan, Geometry and physics of wrinkling, Phys. Rev. Lett. 90(2003) 074302.

68] W. Bao, F. Miao, Z. Chen, H. Zhang, W. Jang, C. Dames, C.N. Lau, Controlledripple texturing of suspended graphene and ultrathin graphite membranes,Nat. Nanotechnol. 4 (9) (2009) 562–566.

69] Y. Hiki, Higher-order elastic-constants of solids, Annu. Rev. Mater. Sci. 11(1981) 51.

70] K. Brugger, Determination of 3rd-order elastic coefficients in crystals, J. Appl.Phys. 36 (3P1) (1965) 768.

71] Q. Peng, A.R. Zamiri, W. Ji, S. De, Elastic properties of hybrid graphene/boronnitride monolayer, Acta Mech. 223 (2012) 2591–2596.

72] Q. Peng, W. Ji, S. De, Mechanical properties of graphene monolayer: afirst-principles study, Phys. Chem. Chem. Phys. 14 (2012) 13385–13391.

73] T. Li, Ideal strength and phonon instability in single-layer MoS2, Phys. Rev. B85 (2012) 235407.

74] G. Staszczak, I. Gorczyca, T. Suski, X.Q. Wang, N.E. Christensen, A. Svane, E.Dimakis, T.D. Moustakas, Photoluminescence and pressure effects in shortperiod InN/nGaN superlattices, J. Appl. Phys. 113 (12) (2013) 123101.

compounds: ab initio calculations, Phys. Rev. B 72 (2005) 245201.76] M. Topsakal, S. Cahangirov, S. Ciraci, The response of mechanical and

electronic properties of graphane to the elastic strain, Appl. Phys. Lett. 96 (9)(2010) 091912.