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Cite this: RSC Advances, 2013, 3,7083
Mechanical stabilities and properties of graphene-likealuminum nitride predicted from first-principlescalculations
Received 12th January 2013,Accepted 26th February 2013
DOI: 10.1039/c3ra40841h
www.rsc.org/advances
Qing Peng,*a Xiao-Jia Chen,b Sheng Liucd and Suvranu Dea
A graphene-like hexagonal aluminum nitride monolayer (g-AlN) is a promising nanoscale optoelectronic
material. We investigate its mechanical stability and properties using first-principles plane-wave
calculations based on density-functional theory, and find that it is mechanically stable under various
strain directions and loads. g-AlN can sustain larger uniaxial and smaller biaxial strains than g-BN before it
ruptures. The third, fourth, and fifth-order elastic constants are essential for accurately modeling the
mechanical properties under strains larger than 0.02, 0.06, and 0.12 respectively. The second-order elastic
constants, including in-plane stiffness, are predicted to monotonically increase with pressure while the
Poisson ratio monotonically decreases with increasing pressure. g-AlN’s tunable sound velocities have
promising applications in nano waveguides and surface acoustic wave sensors.
1 Introduction
The fruitful study and applications of graphene have triggereda new era of two-dimensional (2D) nanomaterials research.1–4
The graphene analogue of BN (g-BN, or white graphene), aninsulating material that serves as an excellent dielectricsubstrate for graphene electronics, was exfoliated recentlyand was the subject of extensive studies with promisingapplications in electronics and energy storage.5–7 Besides thenanosheet structure, copious research was carried out on otherBN nanostructures such as nanotubes, nanoshells,nanosheets, antidots, nanoribbons, bilayers of graphene andg-BN, quantum dots and nanorods of graphene embedded ing-BN, and a hybrid graphene/g-BN monolayer.8–10 As aconsequence, group III-nitrides have raised a lot of inter-est.11–16
Aluminum nitride has zinc-blende17 (z) and hexagonal18,19
(h) bulk crystal structures. The unusual combination of highthermal conductivity while remaining a strong dielectricmakes AlN a critical advanced material for many futureapplications in optics, lighting, electronics and green environ-mental technologies. The hexagonal aluminum nitride mono-layer (g-AlN) is a graphene-like 2D material and has attracted
considerable interest due to its promising applications inoptoelectronics and energy engineering.20 g-AlN is different toh-AlN which is a bulk structure, as it is only one atom thick. Incontrast to AlN nanowires21,22 and nanoribbons,23 g-AlN hasnot yet been fabricated. However, the theoretical study of theg-AlN can expand the range of possible applications of g-BN orgraphene, and open new opportunities for miniaturization inengineering functional nano-devices and interconnects bychemical modification. In addition, taking silicene as anexample, the theoretical predictions24 could be confirmed byexperimental observations25 in only a few years. The theore-tical prediction of the structural and electronic properties ofg-AlN as well as its linear elastic constants were reportedrecently.20,26 The ab initio calculation of phonon dispersionsindicates that g-AlN is dynamically stable. However, itsmechanical stabilities are still unknown. In addition, aspointed out later in this study, the linear behaviors are onlyvalid within a small strain range. The high order non-linearelastic constants are indispensable when describing themechanical behavior of g-AlN, which is critically importantin its synthesis, by mechanical exfoliation for example.
Due to the geometry confinement of the third dimension,2D materials possess distinct properties compared to their 3Dbulk structures, including high specific surface areas. As aresult, 2D materials in general are important in variousapplications, such as optoelectronics, spintronics, catalysis,chemical and biological sensors, supercapacitors, solar cells,lithium ion batteries, mass sensors, and even DNA sequen-cing.27 In particular, with a predicted wide band gap (indirectband gap of 3.08–5.57 eV),26 g-AlN is a good candidate for thefabrication of energy-efficient reliable long-lifetime blue and
aDepartment of Mechanical, Aerospace and Nuclear Engineering, Rensselaer
Polytechnic Institute, Troy, NY 12180, USA. E-mail: [email protected];
Tel: 1-518-279-6669bKey Laboratory of Materials Physics, Institute of Solid State Physics, Chinese
Academy of Sciences, Hefei 230031, ChinacInstitute for Microsystems, School of Mechanical Engineering, Huazhong University
of Science & Technology, Wuhan, China 430074dWuhan National Lab of Optoelectronics, Huazhong University of Science &
Technology, Wuhan, China 430074
RSC Advances
PAPER
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ultraviolet light emitting diodes, sustainable water/air purifi-cation, biomedical systems and photovoltaic applications.28,29
It’s also applicable for the design and preparation of multi-layer infrared (IR) anti-reflection and protection coatings ofwindows.
The mechanical properties are critical with regard to thepractical applications when designing parts or structuresusing g-AlN. Strain engineering is a common, important andeffective approach for tailoring the functional and structuralproperties of nanomaterials,30 including strain-based electro-nics,31 due to its relative simplicity of control and ‘‘clean’’nature, without needing to change the chemistry of theproduct. One can expect that the properties of g-AlN will beaffected by applied strain. In addition, g-AlN is vulnerable tobeing strained with or without intent because of its mona-tomic thickness.32 For example, there are strains because of amismatch of lattice constants or surface corrugation of thesubstrate,33,34 which could be as large as 23.4%.35 Due to anextremely thin structure, the material properties are subject tolocal and global deformations.36 It is critically important todevelop an understanding of local strain and deformation ofsuch 2D materials.37 Furthermore, a clear understanding ofthe stress–strain relationship is a great help for its potentialapplication in strain sensors.38 Therefore, knowledge of themechanical properties of g-AlN is greatly desired.
Depending on the loading, the mechanical properties aredivided into four strain domains: linear elastic, nonlinearelastic, plastic, and fracture. Materials in first two straindomains are reversible, i.e., they can restore to equilibriumstatus after the release of the load. On the contrary, the lasttwo domains are non-reversible. Defects are nucleated andaccumulated with the increase of the strain, until rupture. Asin graphene, the nonlinear mechanical properties are promi-nent in g-AlN since it remains elastic until the intrinsicstrength is reached.39,40 Thus it is of great interest to examinethe nonlinear elastic properties of g-AlN, which are necessaryfor understanding the strength and reliability of structuresand devices made of g-AlN.
Extensive efforts have been made to link the first-principlescalculations with the elastic continuum theory for two-dimensional materials.26,40–42 Several previous studies haveshown that 2D monolayers present a large nonlinear elasticdeformation during tensile strain up to the ultimate strengthof the material, followed by a strain softening untilfracture.40,43,44 We expect that g-AlN would behave in a similarmanner. Under large deformation, the strain energy densityneeds to be expanded as a function of strain in a Taylor seriesto include quadratic and higher order terms. The higher orderterms account for both nonlinearity and strain softening of theelastic deformation. They can also express other anharmonicproperties of 2D nanostructures including phenomena such asthermal expansion, phonon–phonon interaction, etc.39
The goals of this paper are to examine the stability of theatomic structure of a graphene-like hexagonal AlN monolayerand its mechanical behavior at large strains, and to find anaccurate continuum description of its elastic properties from
ab initio density functional theory calculations. The totalenergies of the system, the forces on each atom, and thestresses on the simulation boxes are directly obtained fromDFT calculations. The response of g-AlN under nonlineardeformation and fracture is studied, including ultimatestrength and ultimate strain. The high order elastic constantsare obtained by fitting the stress–strain curves to analyticalstress–strain relationships that belong to the continuumformulation.45 We compared this proposed new material withwell known 2D materials such as g-BN, graphene, andgraphyne. Based on our results of the high order elasticconstants, the pressure dependence properties, such as soundvelocities and the second-order elastic constants, including thein-plane stiffness, are predicted. Our results for the continuumformulation could also be useful in finite element modeling ofthe multiscale calculations for the mechanical properties ofg-AlN at the continuum level. The remainder of the paper isorganized as follows. Section 2 presents the computationalmethod, including the computational details of the DFTcalculations and the basic nonlinear elastic theory applied to2D hexagonal structures. The results and analysis are insection 3, followed by the conclusions in section 4.
2 Density functional theory calculations
We considered a conventional unit cell containing 6 atoms (3aluminum atoms and 3 nitrogen atoms) with periodicboundary conditions (Fig. 1). The 6-atom conventional unitcell was chosen to capture the ‘‘soft mode’’, which is aparticular normal mode exhibiting an anomalous reduction inits characteristic frequency and leading to mechanicalinstability. This soft mode is a key factor in limiting thestrength of monolayer materials, and only occurs in unit cellswith hexagonal rings.46 For a direct point to point comparison,we also calculated the mechanical properties of g-BN45 in thesame environment as g-AlN.
The total energies of the system, forces on each atom,stresses, and stress–strain relationships of g-AlN under thedesired deformation configurations were characterized viafirst-principles calculations with density-functional theory(DFT). DFT calculations were carried out with the Vienna Ab
Fig. 1 Atomic structure of g-AlN in the conventional unit cell (6 atoms) in theundeformed reference configuration.
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initio Simulation Package (VASP)47 which is based on theKohn–Sham Density Functional Theory (KS-DFT)48 with thegeneralized gradient approximations as parameterized byPerdew, Burke, and Ernzerhof (PBE) for exchange–correlationfunctions.49 The electrons explicitly included in the calcula-tions are the (3s23p1) electrons for Al, the (2s22p1) electrons forboron and the (2s22p3) electrons for nitrogen. The coreelectrons (1s2 for nitrogen and 1s22s22p6 for Al) were replacedby the projector augmented wave (PAW) approach.50 A plane-wave cutoff of 600 eV was used in all the calculations. Thecalculations were performed at zero temperature.
The criterion to stop the relaxation of the electronic degreesof freedom was set to be when the total energy change issmaller than 1026 eV. The optimized atomic geometry wasachieved through minimizing the Hellmann–Feynman forcesacting on each atom until the maximum forces on the ionswere smaller than 0.001 eV Å21. The atomic structures of allthe deformed and undeformed configurations were obtainedby fully relaxing a 6-atom unit cell where all the atoms wereinitially placed in one plane. The simulation invoked periodicboundary conditions for the two in-plane directions.
The irreducible Brillouin zone was sampled with aC-centered 25 6 25 6 1 k-mesh. Such a large k-mesh wasused to reduce the numerical errors caused by the strain of thesystems. The initial charge densities were taken as a super-position of atomic charge densities. There was a 15 Å thickvacuum region to reduce inter-layer interactions to model thesingle layer system. To eliminate the artificial effect of the out-of-plane thickness of the simulation box on the stress, we usedthe second Piola–Kirchhoff stress45 to express the 2D forcesper length with units of N m21.
For a general deformation state, the number of indepen-dent components of the second-, third-, fourth-, and fifth-order elastic tensors are 21, 56, 126, and 252 respectively.However, there are only fourteen independent elastic con-stants that need to be explicitly considered due to thesymmetries of that atomic lattice point group D6h whichconsists of a six-fold rotational axis and six mirror planes.40
The fourteen independent elastic constants of g-AlN weredetermined by a least-squares fit to the stress–strain resultsfrom the DFT based first-principles studies in two steps,detailed in our previous work on g-BN,45 which had been usedto explore the mechanical properties of other 2D materi-als.12,43,44,51,52 A brief overview is that, in the first step, we useda least-squares fit of five stress–strain responses. Five relation-ships between stress and strain were necessary because thereare five independent fifth-order elastic constants (FFOEC). Weobtained the stress–strain relationships by simulating thefollowing deformation states: uniaxial strain in the zigzagdirection (zigzag); uniaxial strain in the armchair direction(armchair); and equibiaxial strain (biaxial). For each deforma-tion direction, there are two components of the stresses. As aresult, there are five independent stress–strain relationshipssince the two components of stress in the biaxial strain areidentical. From the first step, the components of the second-order elastic constants (SOEC), the third-order elastic con-
stants (TOEC), and the fourth-order elastic constants (FOEC)were over-determined (i.e., the number of linearly independentvariables was greater than the number of constraints), and thefifth-order elastic constants were well-determined (the numberof linearly independent variables was equal to the number ofconstraints). Under such circumstances, a second step isneeded: a least-squares solution to these over- and well-determined linear equations.
3 Results and analysis
3.1 Atomic structure
We first optimized the equilibrium lattice constant for g-AlN.The total energy as a function of lattice spacing was obtainedby specifying nine lattice constants varying from 2.8 Å to 3.4 Å,with full relaxations of all the atoms. A least-squares fit of theenergies versus lattice constants with a fourth-order polyno-mial function yielded the equilibrium lattice constant as a =3.127 Å, which agrees with previous first-principles calculationresults of 3.09 Å26 and 3.15 Å.20
The most energetically favorable structure was set as thestrain-free structure in this study and the atomic structure, aswell as the conventional cell is shown in Fig. 1. Specifically, thebond length of the Al–N bond is 1.805 Å, which is 0.355 Å (or25%) longer than the bond length of the B–N bond in g-BN.The N–Al–N and Al–N–Al angles are 120u and all atoms arewithin one plane. Our resulting atomic structure is in goodagreement with previous DFT calculations for pristineg-AlN,20,26 and hydrogen passivated flakes.15,53,54 The furtherstudies in the following subsections imply that this theoreti-cally predicted structure is mechanically stable.
3.2 Strain energy
When the strains were applied, all the atoms were allowed fullfreedom of motion. A quasi-Newton algorithm was used torelax all atoms into equilibrium positions within the deformedunit cell that yielded the minimum total energy for theimposed strain state of the super cell.
Both compression and tension were considered withLagrangian strains ranging from 20.1 to 0.4 with anincrement of 0.01 in each step for all three deformationmodes. It is important to include the compressive strains sincethey are believed to be the cause of the rippling of the freestanding atom-thick sheet.32 It was observed that a graphenesheet experiences biaxial compression after thermal anneal-ing,31 which could also happen with g-AlN. Such an asymme-trical range was chosen due to the non-symmetric mechanicalresponses of material, as well as its mechanical instability,55 tothe compressive and tensile strains.
We define strain energy per atom as Es = (Etot 2 E0)/n, whereEtot is the total energy of the strained system, E0 is the totalenergy of the strain-free system, and n = 6 is the number ofatoms in the unit cell. This size-independent quantity is usedfor comparison between different systems. Fig. 2 shows the Es
of g-AlN as a function of strain in uniaxial armchair, uniaxialzigzag, and equibiaxial deformation. Es is seen to beanisotropic with strain direction. Es is non-symmetrical for
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compression (g , 0) and tension (g > 0) for all three modes.This non-symmetry indicates the anharmonicity of the g-AlNstructure.
The harmonic region where the Es is a quadratic function ofthe applied strain can be observed between 20.02 , g , 0.02.The stresses, which are derivatives of the strain energies,linearly increase with the increase of the applied strains in theharmonic region. The anharmonic region is the range of strainwhere the linear stress–strain relationship is invalid andhigher order terms are not negligible. With an even largerloading of strain, the system will undergo irreversiblestructural changes, and the system is then in the plasticregion where it may fail. The maximum strain in theanharmonic region is the critical strain. For all threedirections, the critical strains are not spotted in the testingrange. The ultimate strains are determined as the correspond-ing strain of the ultimate stress, which is the maximum of thestress–strain curve, as discussed in the following section.
It is worth noting that in general compressive strain willcause buckling of the free-standing thin films, membranes,plates, and nanosheets.32 The critical compressive strain forbuckling instability is much less than the critical tensile strainfor fracture, for example, 0.0001% versus 2% in graphene
sheets.55 However, buckling can be suppressed by applyingconstraints, such as embedding (0.7%),56 placing on asubstrate (0.4% before heating),31 thermal cycling on a SiO2
substrate (0.05%)57 or a BN substrate (0.6%),58 and sandwich-ing.59 Thus, although the buckling relaxation modes are notconsidered in this study due to the unit cell limit, our study ofcompressive strains is important in understanding themechanics of these non-buckling applications.
3.3 Stress–strain curves
The second P–K stress versus Lagrangian strain for uniaxialstrains along the armchair and zigzag directions, as well asbiaxial strains are shown in Fig. 3. The stresses are thederivatives of the strain energies with respect to the strains.The ultimate strength is the maximum stress that a materialcan withstand while being stretched, and the correspondingstrain is the ultimate strain. Under ideal conditions, thecritical strain is larger than the ultimate strain. The systems ofperfect g-AlN under strains beyond the ultimate strains are in ametastable state, which can be easily destroyed by longwavelength perturbations, vacancy defects and high tempera-ture effects.60 The ultimate strain is determined by theintrinsic bond strength and acts as a lower limit of the criticalstrain. Thus it has a practical meaning in consideration of itsapplications.
The values of the ultimate strengths and strains correspond-ing to the different strain conditions are summarized inTable 1, compared with those of g-BN, graphene, andgraphyne. The material behaves in an asymmetric mannerwith respect to compressive and tensile strains. With increas-ing strain, the Al–N bonds are stretched and eventually break.When the strain is applied in the armchair direction, thebonds which are parallel to this direction are more severelystretched than those in other directions. The ultimate strain inarmchair deformation is 0.22, larger than that of g-BN,graphene, and graphyne. Under zigzag deformation, in whichthe strain is applied perpendicular to the armchair direction,there are no bonds parallel to this direction. The bondsinclined to the zigzag direction with an angle of 30u are moreseverely stretched than those in the armchair direction. Theultimate strain in this zigzag deformation is 0.27, larger thanthat of g-BN, graphene, and graphyne. At this ultimate strain,the bonds that are at an incline to the armchair directionappear to be broken (Fig. 3, middle panel). Under the biaxialdeformation, the ultimate strain is gb
m = 0.21, which is smallerthan that of g-BN and graphene, but bigger than that ofgraphyne. As such ultimate strain is applied, all the Al–Nbonds are observed to be broken (Fig. 3, bottom).
Our results for the positive ultimate strengths as well as theultimate strains along the three deformation directions implythat the g-AlN structure is mechanically stable. Our resultagrees with a previous study of the phonon calculations whereimaginary frequencies near the C point are absent.26
It should be noted that the softening of the perfect g-AlNunder strains beyond the ultimate strain only occurs underideal conditions. The systems in these circumstances are in ametastable state, which can be easily destroyed by longwavelength perturbations, vacancy defects and high tempera-ture effects, and so enter a plastic state.60 Thus only the data
Fig. 2 Energy-strain responses for uniaxial strain in armchair and zigzagdirections, and equibiaxial strains of g-AlN (top), compared with g-BN (bottom).
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for strain values lower than the ultimate strain have physicalmeaning and were used in determining the high order elasticconstants in the following subsection.
Compared to g-BN (right of Fig. 3), the stresses in g-AlNrespond in a similar fashion to the strain as those in g-BN, butto a much smaller degree. This could be attributed to theweaker chemical bonds in g-AlN than in g-BN.
Fig. 3 Stress–strain responses of g-AlN (left) under the armchair, zigzag, and biaxial strain, compared with g-BN (right). S1 (S2) denotes the x (y) component of thestress. ‘‘Cont’’ stands for the fitting of DFT calculations (‘‘DFT’’) to continuum elastic theory.
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3.4 Elastic constants
The elastic constants are critical parameters in finite elementanalysis models for the mechanical properties of materials.Our results for these elastic constants provide an accuratecontinuum description of the elastic properties of g-AlN fromab initio density functional theory calculations. They aresuitable for incorporation into numerical methods such asthe finite element technique.
The second-order elastic constants model the linear elasticresponse. The higher (>2) order elastic constants are impor-tant for characterizing the nonlinear elastic response of g-AlNusing a continuum description. These can be obtained using aleast squares fit of the DFT data and are reported in Table 2.The corresponding values for graphene are also shown.
The in-plane Young’s modulus Ys and Poisson’s ratio n maybe obtained from the following relationships: Ys = (C11
2 2
C122)/C11 and n = C12/C11. We have Ys = 135.7 N m21 and n =
0.366, which agrees with a previous study.26 The in-planestiffness of g-AlN is very small compared to g-BN (49%) andgraphene (40%), but comparable to graphyne. The reductionof the in-plane stiffness from g-BN to g-AlN is a result of theweakened Al–N bond compared to the B–N bond in g-BN. All
other things being equal, bond length is inversely related tobond strength and the bond dissociation energy, as a strongerbond will be shorter. Considering the bond length, in g-AlNthe bond length of Al–N is 1.805 Å, about 25 percent largerthan the B–N bond length in g-BN (1.45 Å). The bonds can beviewed as being stretched by the replacement of boron atomswith aluminum atoms with reference to g-BN. These stretchedbonds are weaker than the un-stretched ones, resulting in areduction of the mechanical strength.
It is worth noting that our results for the positive second-order elastic constants as well as the in-plane Young’smodulus and Poisson ratio indicate that the g-AlN structureis mechanically stable. This result is consistent with the energystudy in the previous section and agrees with a previous studyof the phonon calculations where imaginary frequencies nearthe C point are absent.26
The stress–strain curves in the previous section show thatthe structure will soften when the strain is larger than theultimate strain. From the point of view of electron bonding,this is due to the bond weakening and breaking. Thissoftening behavior is determined by the TOECs and FFOECsin the continuum aspect. The negative values of TOECs andFFOECs ensure the softening of the g-AlN monolayer underlarge strain.
The hydrostatic terms (C11, C22, C111, C222, and so on) ofg-AlN monolayers are smaller than those of g-BN andgraphene, consistent with the conclusion that the g-AlN is‘‘softer’’. The shear terms (C12, C112, C1122, etc.) are in generalsmaller than those of g-BN and graphene, which contributes tothe high compressibility of g-AlN. Compared to graphene,graphyne, and g-BN, one can conclude that the mechanicalbehavior of g-AlN is similar to graphyne, and that it is muchsofter than graphene and g-BN.
The high order elastic constants are strongly related to theanharmonic properties, including thermal expansion, thermoelastic constants, and thermal conductivity. With higher orderelastic constants, we can easily study the pressure effect on thesecond-order elastic moduli, generalized Gruneisen para-meters, and equations of state. In addition, using the higherorder elastic continuum description, one can ascertain themechanical behavior under various loading conditions, forexample, under applied stresses rather than strains asdemonstrated in a previous study.40 Furthermore, the highorder elastic constants are important in understanding thenonlinear elasticity of materials such as changes in acousticvelocities due to finite strain.
A good way to check the importance of the high order elasticconstants is to consider the case when they are missing. Withthe elastic constants, the stress–strain response can bepredicted from elastic theory.45 When we only consider thesecond-order elasticity, the stress varies linearly with strain.Take the biaxial deformation as an example. As illustrated inFig. 4, the linear behaviors are only valid within a small strainrange, about 20.02 ¡ g ¡ 0.02, the same result as obtainedfrom the energy versus strain curves in Fig. 2. With knowledgeof the elastic constants up to the third-order, the stress–straincurve can be accurately predicted within the range of 20.06 ¡
g ¡ 0.06. Using the elastic constants up to the fourth-order,the mechanical behavior can be well treated up to a strain as
Table 1 Ultimate strengths (Sam , Sz
m , Sbm) in units of N m21 and ultimate strains
(gam , gz
m , gbm) under uniaxial strain (armchair and zigzag) and biaxial strain from
DFT calculations, compared with g-BN, graphene, and graphyne
g-AlN g-BN45 Graphene40 Graphyne43
Sam 16.2 23.6 29.5 17.8
gam 0.22 0.18 0.19 0.20
Sam 15.9 26.3 31.4 18.8
gzm 0.27 0.26 0.24 0.20
Sbm 14.8 27.8 33.1 20.64
gbm 0.21 0.24 0.24 0.18
Table 2 Nonzero independent components for the SOEC, TOEC, FOEC, andFFOEC tensor components, Poisson’s ratio n and in-plane stiffness Ys of g-AlNfrom DFT calculations, compared with g-BN, graphene, and graphyne
g-AlN g-BN45 Graphenea Graphyne43
a 3.127 2.512 2.446 6.889Ys 135.7 278.3 348 162.1n 0.366 0.225 0.169 0.429C11 156.7 293.2 358.1 198.7C12 57.4 66.1 60.4 85.3C111 21265.7 22513.6 22817 2890.9C112 2328.3 2425.0 2337.1 2872.6C222 2968.9 22284.2 22693.3 21264.2C1111 8753 16 547 13 416.2 27966C1112 699 2609 759 4395C1122 4604 2215 2582.8 8662C2222 2447 12 288 10 358.9 1154C11111 246 941 265265 231 383.8 89 000C11112 22801 28454 288.4 210 393C11122 213 643 228 556 212 960.5 226 725C12222 218 237 236 955 213 046.6 215 495C22222 23855 2100 469 233 446.7 214 262
a A 2-atom unit cell was used along with fitting data beyond ultimatestrains in ref. 40.
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large as 0.12. For strain values beyond 0.12, the fifth-orderelastic constant is required for accurate modeling. An analysisof the uniaxial deformations comes to the same results.Further analysis on g-BN (Fig. 4 bottom) also confirms theresults.
Our results illustrate that the monatomic layer structurespossess different mechanical behaviors in contrast to the bulkor multi-layered structures, where the second-order elasticconstants are sufficient in most cases. The second-orderelastic constants are relatively easy to calculate from the strainenergy curves,60,61 however, they are not sufficient formonatomic layer structures. The high order elastic constantsare required for an accurate description of the mechanicalbehavior of monatomic layer structures since they arevulnerable to strain due to their geometry confinements.
Our results for the mechanical properties of g-AlN arelimited to zero temperature due to the current DFT calcula-tions. Once a finite temperature is considered, the thermalexpansions and dynamics will in general reduce the interac-tions between atoms. As a result, the longitudinal mode elasticconstants will decrease with respect to the temperature of the
system. The variation of shear mode elastic constants shouldbe more complex in responding to the temperature. Athorough study would be interesting, but is, however, beyondthe scope of this study.
3.5 Pressure effect on the elastic moduli
With third-order elastic moduli, we can study the effect ofpressure on the second-order elastic moduli, where thepressure p acts in the plane of g-AlN. Explicitly, when pressureis applied, the pressure dependent second-order elasticmoduli (C11, C12, C22) can be obtained from C11, C12, C22,C111, C112, C222, Ys, and n as:
~C11~C11{(C111zC112)1{n
Ys
p, (1)
~C22~C11{C2221{n
Ysp (2)
~C12~C12{C1121{n
Ysp (3)
The second-order elastic moduli of g-AlN are seen toincrease linearly with the applied pressure (Fig. 5). However,Poisson’s ratio decreases monotonically with the increase ofpressure. C11 is asymmetrical to C22 unlike in the zero pressurecase. C11 = C22 = C11 only occurs when the pressure is zero.This anisotropy could be the outcome of anharmonicity.
Compared to g-BN (bottom of Fig. 5), the second-orderelastic moduli and Poisson’s ratio in g-AlN are more sensitiveto the in-plane pressure. This could be attributed to the Al-Nbonds being weaker than the B–N bonds.
3.6 Pressure effect on the velocities of sound
In the g-AlN monolayer, there are non-zero in-plane Young’smoduli and shear deformations. Hence it is possible togenerate sound waves with different velocities depending onthe deformation mode. Sound waves generating biaxialdeformations (compressions) are compressional or p-waves.Sound waves generating shear deformations are shear ors-waves. The velocities of these two types of sound wave arecalculated from the second-order elastic moduli and massdensity (rm) using the following relationships:
vp~
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
~Y s(1{~n)
rm(1z~n)(1{2~n)
s
, (4)
vs~
ffiffiffiffiffiffiffiffi
~C12
rm
s
: (5)
The dependence of np and ns on pressure (biaxial stress) isplotted in Fig. 6. The minimum (12 km s21) of the np curveoccurs at an in-plane pressure of 24 N m21. However, ns
monotonically increases with an increase in pressure. Thus vp
and vs can be tuned by introducing biaxial strain through thestress–strain relationship shown in Fig. 3c.
Fig. 4 The predicted stress–strain responses from different orders: second-,third-, fourth-, and fifth-order, and compared to the DFT calculations in thebiaxial deformation of g-AlN (top) and g-BN (bottom).
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The compressional to shear wave velocity ratio (np/ns) is avery useful parameter in the determination of a material’smechanical properties. It depends only on the Poisson’s ratioas
vp
vs~
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
~n(1z
~n2
1{2~n)
r
: (6)
The ratio of np/ns monotonically decreases with increasingpressure as shown in Fig. 6. It converges to a value of 2.0 atpositive pressure.
As shown in Fig. 6, a sound velocity gradient could beachieved by introducing stress into a g-AlN monolayer. Such asound velocity gradient could lead to the refraction of soundwavefronts in the direction of lower sound speed, causing thesound rays to follow a curved path.62 The radius of thecurvature of the sound path is inversely proportional to thegradient. Also a negative sound speed gradient could beachieved by a negative strain gradient. This tunable soundvelocity gradient can be used to form a sound frequency andranging channel, which is the functional mechanism of
waveguides and surface acoustic wave (SAW) sensors.35,63,64
Compared to g-BN (bottom of Fig. 6), the sound velocities ing-AlN are more sensitive to the in-plane pressure. Thus, g-AlN-based nano-devices for use as SAW sensors, filters, andwaveguides may be synthesized using local strains for nextgeneration electronics.
4 Conclusions
In summary, we studied the mechanical response of g-AlNunder various strains using DFT based first-principles calcula-tions. We also calculated the mechanical properties of g-BN inthe same environment for a direct comparison. It is observedthat g-AlN exhibits a nonlinear elastic deformation up to anultimate strain, which is 0.22, 0.27, and 0.21 for armchair,zigzag, and biaxial directions, respectively. The deformationand failure behavior and the ultimate strength are anisotropic.It has a relatively low in-plane stiffness (135.7 N m21) and alarge Poisson ratio compared to g-BN and graphene.Compared to g-BN, g-AlN has 49% in-plane stiffness, 67%,
Fig. 6 p-Wave and s-wave velocities, and compressional to shear wave velocityratio np/ns as a function of in-plane pressure for g-AlN (top), compared withg-BN (bottom).
Fig. 5 Second-order elastic moduli and Poisson ratio as a function of pressurefor g-AlN (top), compared with g-BN (bottom).
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60%, and 53% ultimate strengths in armchair, zigzag, andbiaxial strains respectively, and a Poisson’s ratio 1.6 timeslarger. It was found that g-AlN can sustain larger uniaxial andsmaller biaxial strain before it ruptures.
The nonlinear elasticity of g-AlN was investigated. Wefound an accurate continuum description of the elasticproperties of g-AlN by explicitly determining the fourteenindependent components of the high order (up to fifth-order)elastic constants from the fitting of the stress–strain curvesobtained from DFT calculations. This data is useful to developa continuum description which is suitable for incorporationinto a finite element analysis model for g-AlN’s applications atlarge scale. We also find that the harmonic elastic constantsare only valid within a small range of 20.02 ¡ g ¡ 0.02. Withthe knowledge of the elastic constants up to the third-order,the stress–strain curve can be accurately predicted within therange of 20.06 ¡ g ¡ 0.06. Using the elastic constants up tothe fourth-order, the mechanical behavior can be accuratelypredicted up to a strain as large as 0.12. For the strains beyond0.12, the fifth-order elastic constants are required for accuratemodeling. The high order elastic constants reflect the highorder nonlinear bond strength under large strains.
The second-order elastic constants including the in-planestiffness are predicted to monotonically increase with pressurewhile Poisson’s ratio monotonically decreases with increasingpressure. The sound velocity of a compressional wave has aminimum at an in-plane pressure of 24 N m21 while that ofthe shear wave monotonically increases with pressure. Theratio of np/ns monotonically decreases with the increase ofpressure and converges to a value of 2.0 at positive pressure.Our results of the positive ultimate strengths and strains,second-order elastic constants and the in-plane Young’smodulus indicate that the graphene-like structure of hexago-nal AlN monolayers is mechanically stable. The property ofhaving tunable sound velocities has promising applications innano waveguides and surface acoustic wave sensors.
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.
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