Tensile behavior of gallium nitride monolayer via nonlinear molecular mechanics

European Journal of Mechanics A/Solids 65 (2017) 223e232

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Tensile behavior of gallium nitride monolayer via nonlinear molecular mechanics Georgios I. Giannopoulos a, b, *, Stylianos K. Georgantzinos c a

Materials Science Laboratory, Department of Mechanical Engineering, Technological Educational Institute of Western Greece, 26334 Patras, Greece Department of Materials Science, University of Patras, 26504 Patras, Greece c Machine Design Laboratory, Department of Mechanical Engineering and Aeronautics, University of Patras, Rio, 26500 Patras, Greece b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 28 June 2016 Received in revised form 19 April 2017 Accepted 30 April 2017 Available online 4 May 2017

The aim of the present study is the numerical prediction of the mechanical behavior of the hexagonal gallium nitride (GaN) monolayer. For this purpose, a nonlinear molecular mechanics (MM) method is developed and proposed which is based on the use of three-dimensional (3d) non-linear, spring-like, finite elements, capable of representing appropriately the interatomic interactions between the atoms of the nanomaterial. In order to establish the constitutive equations of the utilized finite elements, the Stillinger-Weber potential is adopted, which may approximate both two-body (2b) as well as three-body (3b) interatomic interactions via appropriate nonlinear functions of the bond length and the interbond angle, respectively. According to the proposed MM formulation, square-shaped, single-layered, hexagonal GaN sheets of several sizes are modelled and tested under tensional loadings in both in-plain directions to compute corresponding effective mechanical properties such as Young's modulus, Poisson's ratio, tensile strength, failure strain and fracture toughness. The numerical results are compared with other estimations which are available from the literature, where possible. © 2017 Elsevier Masson SAS. All rights reserved.

Keywords: Gallium nitride Fracture Elastic Stress-strain Nanomaterial Nanoribbon

1. Introduction Extensive interest has been recently devoted to the study of the GaN-based nanomaterials due to their remarkable electronic (Moradian et al., 2008; Wang et al., 2009), optic (Shokri and Ghorbani Avaresi, 2013) and electrochemical properties (Chakrapani, 2015). GaN, like carbon and boron nitride, is expected to be capable of forming interesting allotropes. GaN nanotubes (GaNNTs) have been already discovered and produced. In the near future, special GaN allotropes are expected to be key materials for high-power, high-frequency and electronic devices such as lightemitting diodes (LEDs). The synthesis of GaN based media has attracted the interest of many researchers due to their potential to be used in a variety of applications (Li et al., 2015). Hu et al. (2003) have achieved the creation of crystalline GaNNTs in bulk by a two-stage process based

* Corresponding author. Materials Science Laboratory, Department of Mechanical Engineering, Technological Educational Institute of Western Greece, 26334 Patras, Greece. E-mail address: [email protected] (G.I. Giannopoulos). http://dx.doi.org/10.1016/j.euromechsol.2017.04.010 0997-7538/© 2017 Elsevier Masson SAS. All rights reserved.

on the well-controllable amorphous gallium oxide nanotube conversion. Goldberger et al. (2003) have reported the successful formations of hexagonal cross-sections of single crystalline GaNNTs of various diameters utilizing an ‘epitaxial casting’ approach. Han et al. (1997) have prepared GaN nanorods through a carbon nanotube-confined reaction. Xing et al. (2009) have synthesized Zn-doped GaNNTs with zigzag morphology by a chemical vapor deposition method. Jiang et al. (2013) have prepared large-area porous single crystal GaN micro/nanotube arrays by a simple method using zinc oxide arrays as the template. Suresh et al (Suresh Kumar et al., 2008) have grown GaN nanocrystals on the tip of aligned carbon nanotubes (CNTs) substrate by chemical vapor transport method. Yan et al. (2009a) have demonstrated the growth of GaN nanowires by metal organic chemical vapor deposition with CNTs as templates. Finally, Lin et al. (2010) have demonstrated the first example of the use of NiI 2-filled CNTs for the synthesis of GaN nanowires. Many theoretical studies have been conducted in the effort to characterize the behavior of the GaN nanomaterials targeting mainly on their opto-electrochemical properties (Moradian et al., 2008; Wang et al., 2009; Shokri and Ghorbani Avaresi, 2013;

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Chakrapani, 2015) and their formation (Hao et al., 2006; Yan et al., 2009b). However, fewer are the theoretical approaches involving the contribution of their mechanical performance. Jeng et al. (2004) have adopted a classical molecular dynamics (MD) simulation with the realistic Tersoff many-body potential model to investigate the mechanical properties of GaNNTs. Wang et al. (2006, 2008a) have investigated the mechanical properties of wurtzite-type singlecrystalline GaNNTs under applied tensile strains via MD methods with a Stillinger-Weber potential while in a similar attempt they have studied the tensile behavior of GaN nanowires (Wang et al., 2007). Wang et al. (2008b) have investigated the tensile mechanical behavior of GaNNTs under combined tension-torsion using MD simulations with an empirical potential. More recently, Wang et al. (2008c) have investigated the buckling responses of singlecrystalline GaNNTs under torsion using an MD simulation based on a StillingereWeber potential. Zhang and Meguid (2015) have studied, analytically and numerically via MD simulations, the piezoelectric potential of strained intrinsic GaNNTs. Lastly, Kumar et al. (2015a) have studied the elastic properties of GaN nanotubes using the second generation REBO potential by Brenner and co-workers. Since the production of various GaN based nanostructures and especially GaNNTs has been already accomplished, the fabrication of single layers of hexagonal GaN is considered feasible. However, despite the fact that extensive theoretical research has been made on GaNNTs mainly by utilizing MD simulations, the hexagonal GaN monolayer has not been explored to the same extent. Peng et al. (2013) as well as Sahin et al. (Sahin et al., 2009). have investigated the mechanical properties of graphene-like hexagonal GaN nitride monolayer using first-principles calculations based on density functional theory (DFT). Sharma et al. (Sarma et al., 2013) have performed classical MD simulations by employing the Stillinger-Weber potential on a system of single layer nanosheet of GaN containing single and double atomic vacancy defects. Last but not least, Kumar et al. (2015b) have calculated the frequencies of various motions of hexagonal GaN sheets and GaNNTs using simple 6-exponential potential between gallium and nitrogen atoms. In the present study an atomistic finite element approach which is based on the use of 3d axial and torsional springs is developed for the simulation of the tensile response of hexagonal GaN monolayer. The tested GaN sheets are treated as frames of point and line spring-like, two-noded, finite elements of six translational degrees of freedom per node. In order to approximate the appearing forces and deformations within the nanostructure, a Stillinger-Weber potential (Stillinger and Weber, 1985; Lei et al., 2006) is adopted which describes effectively both 2b and 3b interatomic interactions. To describe appropriately the interatomic interactions, the one-to-one mapping is adopted within the atomic system. This leads to the use of a triple node at each atomic position which increases the computational cost. However, the present MM modelling technique provides realistic coupling of the degrees of freedom between different atomic positions while remains computationally attractive in comparison with other MD based schemes, since it does not require calculations over time. In addition, even though static MM methods do not include the effect of thermal motion, they have been proved to be efficient in estimating nanomechanical properties (Liu et al., 2008). Several sizes of almost squared-shaped GaN monolayers are tested under tensile loadings along both inplain directions in order to demonstrate the nanosize and chirality effect on their mechanical properties. Indicative solutions from other studies are presented for comparison reasons. The computational material characterization involves only perfect

sheets given that during nanostructural design, defect-free nanocomponents are naturally chosen despite the production difficulties and limitations. However, it should be mentioned that the proposed scheme may straightforwardly treat different kinds of defects due to its modelling simplicity.

2. Molecular mechanics formulation The static MM approaches are now well established (Liu et al., 2008). Furthermore, the MM methods have already proved their efficiency to simulate the mechanical behavior of different tabular (Zhang et al., 2002) as well as planar (Georgantzinos et al., 2011) nanomaterials regardless of the incorporated molecular potentials. Numerous comparisons with various experimental and other theoretical data have proved the stable performance of the MM methods (Liu et al., 2008; Zhang et al., 2002; Georgantzinos et al., 2011). However, concerning the hexagonal GaN monolayer limited theoretical works may be found in the literature (Peng et al., 2013; Sahin et al., 2009; Sarma et al., 2013), mainly due to the inefficient reported relevant interatomic potentials. In the present study a spring based MM method, which uses a Stillinger-Weber potential model parameterized for GaN interactions, is applied for the first time to provide new material property evidence for the GaN monolayer, giving prospects for further future evaluation.

2.1. Stillinger-Weber potential energy According to the Stillinger-Weber model (Stillinger and Weber, 1985), the total potential energy Utot within a hexagonal GaN monolayer may be expressed as a sum of 2b and 3b interatomic interaction terms as:

Utot ¼

X

Ur þ

X

Uq

(1)

where Ur is the potential energy between two bonded atoms i, p due to a change of the bond length rip from the equilibrium bond length r0 to the changed length r0 þ Dr while Uq is the potential energy between two linked bonds ip, iq, of the lengths rip and riq , respectively, due to a change of the interbond angle qpiq from the equilibrium inerbond angle q0 to the modified angle q0 þ Dq. The interatomic potentials Ur and Uq , respectively, may be expressed as (Lei et al., 2006):

2 Ur ¼ A4B

Uq ¼ l exp

s rip

3

!4

 15exp

gs rip  bs

þ

s

!

rip  bs

gs riq  bs

!  2 cos qpiq þ k

(2)

(3)

In the above equations the Ga-N bond parameters A, B, s, b, l and g are taken equal to 2:801 nN  nm, 0:694, 0:17 nm, 1:8, 10:342 nN  nm and 1:2, respectively (Lei et al., 2006). Finally the parameter k is set equal to 0:5 so that the potential favors bond configurations with the desired ideal hexagonal geometry. Fig. 1 illustrates the variation of the potential function Ur with respect to the bond length rip , which presents a minimum at the equilibrium length r0 ¼ 0:1949 nm. In addition, the variation of the potential term Uq with the interbond angle qpiq for rip ¼ riq ¼ r0 is presented in Fig. 2. As observed, the specific potential function is minimized at the equilibrium interbond angle values q0 ¼ ± 2p=3.

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225

Fig. 1. Potential energy of the 2b interatomic interaction versus the bond length.

Fig. 2. Potential energy of the 3b interatomic interaction versus the interbond angle.

2.2. Spring based finite element approach Let as assume the circled hexagonal single-layered GaN nanostructure of Fig. 3 which contains four bonded atoms which are described by the position nodes i, o, p and q. The system contains the three linked bonds io, ip and iq while is initially in equilibrium, i.e., rio ¼ rip ¼ riq ¼ r0 and qoip ¼ qpiq ¼ qqio ¼ q0 .

Now consider an external load which disturbs the equilibrium state and causes bond length changes Dr and interbond angle changes Dq to the system. Due to these changes axial forces Fr and torsional moments Mq are arisen which may be estimated by differentiating Equations (2) and (3) with respect to Dr ¼ rij  r0 and Dq ¼ qpiq  q0 , correspondingly:

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2 6 6 6 6 6 6 k2b cjz ¼ 6 6 6 6 4

k2b c

0

0

0

0

0

k2b j

0

0

0

0

0

k2b z

0

0

0

0

0

k2b rot c

0

0

0

0

0

k2b rot j

0

0

0

0

0

0

0

0

0

0

∞

0

0

0

0

∞

0

0

0

0

0

0

0

Fig. 3. Modelling the 2b and 3b interatomic interactions within GaN monolayer with nonlinear axial and torsional springs, respectively.

 Fr ðDrÞ ¼ As exp



2



s

B

K3b xyz

1

r0 þDr s 6 4 r0 þ Dr  bs ðr0 þ Dr  bsÞ2

3 

4Bs3 ðr0 þ DrÞ5

7 5

(4)

gs rip  bs

þ kÞsinðq0 þ DqÞ

" Kcjz ¼

k2b cjz

k2b cjz

k2b cjz

k2b cjz

0

∞

0

0

0

∞

¼

k3b xyz

k3b xyz

k3b xyz

k3b xyz

þ

gs riq  bs

k3b 6 x 6 0 6 6 6 0 ¼6 6 0 6 6 0 4 2

ðcosðq0 þ DqÞ (5)

# (6)

(7)

# (8)

0 k3b y

0 0

0 0

0 0

0 0 0

k3b z 0 0

0 k3b rotx 0

0 0 k3b roty

0

0

0

0

0

!

In order to treat the Equations (4) and (5) numerically, appropriate two-noded axial and torsional spring-like, 3d finite elements of six degrees of freedom per node i.e., three translations and three rotations, are adopted. Note that in order to simulate the response of the interbond angles qoip , qpiq and qqio in an independent manner three nodes of the same coordinates (triple node), denoted as i1 , i2 and i3 in Fig. 3, are introduced. Note that without using the triple node, the separate and simultaneous simulation of the 3b interatomic interactions between an atom and three surrounding ones would be impossible. Firstly, an axial, spring-like, line element denoted as 2b is utilized to describe the Equation (4). The line element 2b is used to interconnect the nodes i1 and o, i2 and p as well as i3 and q. Its stiffness matrix with respect to the local coordinate system (c; j; z) of Fig. 3 is given by:

2b

0

0

2

k3b xyz

Mq ðDqÞ ¼ 2l exp

0

07 7 7 07 7 7 07 7 7 07 7 7 07 5

Secondly, a torsional, spring-like, point element denoted as 3b is used to describe the Equation (5). The point element 3b is used to interconnect the nodes i1 and i2 , i2 and i3 as well as i3 and i1 . Its stiffness matrix according to the global coordinate systems (x; y; z) is written as:

"

4

7 0 7 7 7 0 7 7 0 7 7 7 0 7 5

k2b rot z 3

2

dFr ðDrÞ 6 dðDrÞ 6 6 6 0 6 6 6 0 ¼6 6 6 0 6 6 6 0 4

3

6 6 6 6 6 6 ¼6 6 6 6 6 6 4

∞

0

0

0

0

0

∞

0

0

0

0

0

∞

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

3 0 7 0 7 7 7 0 7 7 0 7 7 0 7 5

k3b rotz 3

7 7 7 7 0 7 7 7 7 0 7 7 7 0 7 vMq ðDqÞ 5 vðDqÞ 0

(9)

In the above equations the infinite symbol corresponds to a very high stiffness value which ensures straightness of the bonds and the cohesion of the triple node whereas allows convergence of the numerical solution. Finally, the diagonal element k2b rot c in Equation 3b (7) as well as the diagonal elements k3b rotx and kroty in Equation (9) have no effect on the computed in-plane deformations. However, the specific diagonal elements are set equal to zero in order to avoid convergence difficulties. Fig. 4 illustrates the model geometry of an almost square GaN monolayer of a length lx and ly along the x (armchair) and y (zigzag) direction, respectively, which is discretized with the aforementioned finite elements. After discretizing the given GaN sheet geometry, the elemental stiffness equations for every 2b and 3b finite elements are transformed to the global coordinate system. Then the final system of equations may be constructed according to the requirements of nodal equilibrium as follows:

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227

computational technique. It is convenient to define the size of the investigated GaN sheet in accordance with an average length defined as la ¼ ðlx þ ly Þ=2. The maximum tested average length is around 10 nm. The tensile strain-stress curves in both armchair (x) and zigzag (y) direction are computed by assuming a layer thickness of t ¼ 0:374 nm. The same value has been adopted elsewhere in MD simulations of the hexagonal GaN monolayer while is very close to the thickness t ¼ 0:34 nm which is usually assumed for graphene monolayer, CNTs and occasionally for single-walled GaNNTs (Jeng et al., 2004). During numerical tests the one edge of the GaN sheets is fully constrained, i.e., the three translational and three rotational degrees of freedom of each edge node are restricted. On the other hand, the opposite edge is uniformly deformed by an axial displacement. Finally, the degrees of freedom of the nodes which belong to the other two edges (parallel to the loading) are left free. Then, calculations of the corresponding normal stresses s and strains 3 may be respectively realized via the relationships:

sx ¼

Fig. 4. A spring based MM model of a hexagonal GaN monolayer.

KðUÞ U ¼ F

(10)

where KðUÞ, U and F are the assembled deformation-dependent stiffness matrix, displacement vector and force vector, respectively. By applying the known boundary conditions of the problem, i.e. support and load conditions, the last equation may be solved via standard Newton-Raphson iterative procedures which are required to treat the inherent nonlinearity of the global stiffness matrix KðUÞ due to the terms dFr ðDrÞ=dðDrÞ and vMq ðDqÞ=vðDqÞ of Equations (7) and (9).

3. Results and discussion Five different sizes of almost squared-shaped, i.e., lx zly , hexagonal GaN sheets are developed according to the proposed

3x

¼

Ry Rx ; sy ¼ ly t lx t

Dlx lx

;

3y

¼

Dly ly

(11)

(12)

where the parameters R and Dl denote the total reaction force at the constrained edge and displacement at the loaded or free edges, respectively. Figs. 5 and 6 illustrate the tensile stress-strain curves regarding the armchair (x) and zigzag (y) direction, respectively, for five different square sizes of the hexagonal GaN monolayer. The curves show a rather significant size effect which is a common phenomenon for hexagonal nanosheets (Georgantzinos et al., 2011) due to the absence of absolute structural and material continuity at nanoscale. The curves are slightly nonlinear while reveal a brittle nanomaterial behavior. The GaN sheets seem to present a superior mechanical strength along the zigzag direction. The best mechanical performance is observed for smaller sizes. However, the mechanical response of the squared-shaped GaN monolayer seems to

Fig. 5. Stress versus strain along the armchair (x) direction for several squared-shaped, hexagonal GaN monolayers.

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Fig. 6. Stress versus strain along the zigzag (y) direction for several squared-shaped, hexagonal GaN monolayers.

converge as its average length increases. The deformation just before the fracture initiation of the largest sheet being loaded along its armchair and zigzag direction is illustrated in Fig. 7(a) and (b), respectively. As observed, the more intense strains are located at the free edges near the support. At these locations, the first bond breaks start simultaneously from both free sides. Given the above stress-strain variations, significant elastic material properties may be estimated. The Young's modulus E and the Poisson's ratio n may be calculated by utilizing data points from the curves which correspond to small strains by the relationships:

Ex ¼

sx 3x

nyx ¼ 

for

3x 3y

3 x /0;

for

Ey ¼

3 x /0;

sy 3y

for

nxy ¼ 

3y 3x

3 y /0

for

3 y /0

(13)

(14)

The Young's modulus of the GaN monolayer along its both directions versus its average length is depicted in Fig. 8. The specific elastic property decreases as the sheet size increases. Small difference may be observed between Ex and Ey for all sheet sizes. Moreover, a higher nanomaterial stiffness is observed along the zigzag direction. Fig. 9, illustrates the size dependent variations regarding the Poisson's ratios. It becomes obvious that there is nyx < nxy especially for smaller monolayers. As the GaN sheet size increases, the Poisson's values seem to stabilize close to the value 0:5, fact that reveals an approximately incompressible nanomaterial behavior. Figs. 10 and 11 depict the tensile strength su and failure strain 3 f with respect to the GaN monolayer size. Both tensile strength and failure strain are defined by the peak points of the stress-strain curves presented in Figs. 5 and 6. The superiority mechanical performance regarding both material properties along the zigzag direction becomes evident. The size effect for both properties is more significant for smaller sizes. The higher the GaN sheet size the lower the tensile strength and failure strain. The tensile toughness values y of the almost squared-shaped hexagonal GaN sheets may be estimated via integration of their

tensile stress-strain curves, i.e.:

yx ¼

Z3 fx

sx d3 x

(15)

sy d3 y

(16)

0

yy ¼

Z3 fy 0

As Fig. 12 illustrates the tensile toughness of the investigated nanomaterial along its both in-pane directions versus its size. The tensile toughness of the GaN monolayer shows a descending behavior with respect to the average length la which is more intense for smaller sizes. When the GaN nanosheets are loaded at their armchair edges along their zigzag direction demonstrate a better ability to absorb energy. Table 1 presents some predictions concerning the hexagonal GaN monolayer which have arisen via the proposed numerical technique in contrast with other comparable theoretical

Fig. 7. Deformations just before fracture of the largest GaN investigated nanosheet under tension along its (a) armchair and (b) zigzag direction.

G.I. Giannopoulos, S.K. Georgantzinos / European Journal of Mechanics A/Solids 65 (2017) 223e232

229

Fig. 8. Young's modulus versus GaN monolayer size for the armchair and zigzag directions.

estimations (Peng et al., 2013; Sahin et al., 2009; Sarma et al., 2013). The properties which are have arisen via the present method and are included in Table 1 correspond to the largest GaN sheet and the armchair direction (x). Since there are limited works on hexagonal GaN monolayer, for qualitative reasons, some extra theoretical approximations corresponding to different GaN allotropes such as GaNNTs (Jeng et al., 2004) and GaN nanowires (Wang et al., 2008a) are also presented. To the author's best knowledge, there are no reported mechanical property measurements regarding hexagonal GaN yet. Thus, two experimental works on GaN nanowires based on transmission electron microscope (TEM) (Hung et al., 2005; Nam et al., 2006) are indicatively included in the table. It should be

mentioned that the MD based numerical predictions (Wang et al., 2008a) and the experimental measurements (Hung et al., 2005; Nam et al., 2006) correspond to GaN nanowires with wurtzite structure. In order to reach safe conclusions, comprehensive details and explanations are provided in the table for every property value set into contrast. It is observed that the present predictions, regarding Young's modulus and Poisson's ratio are in very good agreement with the first-principle DFT method (Peng et al., 2013; Sahin et al., 2009). In the MD based study (Sarma et al., 2013), various stress-strain curves of a hexagonal GaN sheet have been extracted with respect to the strain rate. The reported tensile curves (Sarma et al.,

Fig. 9. Poisson's ratio versus GaN monolayer size for the armchair and zigzag directions.

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Fig. 10. Tensile strength versus GaN monolayer size for the armchair and zigzag directions.

2013) have shown intense sensitivity on the adopted strain rate. However, convergence has been implied for low strain rates. Therefore, only the mechanical properties which have been computed using the three smaller strain rates (Sarma et al., 2013) are shown here. Note that the smallest strain rate, i.e., 0.1% ps1, has led to mechanical property estimations (Sarma et al., 2013) that are in relatively good agreement with the present ones. Finally, despite the different atomistic structure of GaN nanowires, the MD (Wang et al., 2008a) as well as the experimental (Hung et al., 2005; Nam et al., 2006) approaches produced similar Young's modulus with the present study.

4. Concluding remarks A simple, in computational terms, nonlinear MM method has been proposed for the analysis of the hexagonal GaN monolayer which takes into consideration the atomistic structure of the investigated nanomaterial while represents 2b and 3b interatomic interactions with appropriate axial and torsional spring-like finite elements, respectively. The presented method is theoretically grounded on Stillinger-Weber potential. Although the GaN monolayer has not been studied yet thoroughly, some indicative comparisons have demonstrated the predictive performance of the

Fig. 11. Failure strain versus GaN monolayer size for the armchair and zigzag directions.

G.I. Giannopoulos, S.K. Georgantzinos / European Journal of Mechanics A/Solids 65 (2017) 223e232

231

Fig. 12. Tensile toughness versus GaN monolayer size for the armchair and zigzag directions.

Table 1 Present mechanical property estimations in contrast with other indicative solutions from the literature. Nanomaterial:

Hexagonal GaN monolayer

GaNNT

GaN Nanowire

Method:

Present MM DFT (Peng et al., 2013)

DFT (Sahin et al., 2009)

MD (Sarma et al., 2013)

MD (Jeng et al., 2004)

MD (Wang et al., 2008a)

Experimental (Hung et al., 2005)

Experimental (Nam et al., 2006)

Modelling structure: Geometric aspects:

Nanosheet

Unit cell

Unit cell

Nanosheet

lx ¼ 10 nm ly ¼ 10 nm t ¼ 0.374 nm Armchair

3 Ga þ 3 N atoms t ¼ 0.374 nm e

lx ¼ 1.8 nm ly ¼ 1.8 nm t ¼ 0.374 nm e

lx ¼ 8.5 nm ly ¼ 8.5 nm t ¼ 0.374 nm Armchair

Zigzag single walled (9,0) dmean ¼ 0.81 nm l ¼ 11.8 nm t ¼ 0.34 nm Longitudinal

Hollow hexagonal cross section dinner ¼ 2.394 nm l ¼ 6.12 nm t ¼ 0.368 nm Longitudinal

Hollow hexagonal cross section dinner ¼ 20 nm douter ¼ 40 nm l ¼ 300 nm Longitudinal

Solid triangular cross section Triangle side ¼ 38 nm l ¼ 3200 nm Longitudinal

Room e

Zero e

e e

Room 0.89 0.4

Room 0.0125

Room e

Room e

Room e

254

237

411 302 173 223

283

223

227

0.43 e

0.48 e

e e e 0.24 19.1 20.4 20.6 e

e 49.2

0.24 e

e e

e

e

4.8

16.5

e

e

Loading direction: Temperature: Strain rate (% ps1):

Properties: E 240 (GPa) n 0.47 su 32.8 (GPa) 3 f (%) 15.8

7.6

0.1

15.2 e

proposed approach. The graphene-like GaN monolayer has be found to be brittle while its mechanical behavior, as expected, showed in-plane size dependence especially for dimensions below 10 nm. Furthermore, the nanomaterial has presented distinct in-plane orthotropy due its graphene-like nanostructural configuration. Specifically, it has been proved that the investigated nanosheet is rather stiffer, stronger and may absorb additional energy when loaded along its zigzag direction. Finally, a mean Poisson's ratio around 0.47 has been predicted which reveals an almost incompressible nanomaterial behavior.

Acknowledgement The authors wish to thank Ms. Ioanna Dourali and Mr. Vasileios

Gkouras for their help in some of the computations. References Chakrapani, V., 2015. Electrochemical transfer doping: a novel phenomenon seen in diamond, gallium nitride, and carbon nanotubes. ECS Trans. 66 (7), 29e37. Georgantzinos, S.K., Giannopoulos, G.I., Katsareas, D.E., Kakavas, P.A., Anifantis, N.K., 2011. Size-dependent non-linear mechanical properties of graphene nanoribbons. Comput. Mater. Sci. 50 (7), 2057e2062. Goldberger, J., He, R., Zhang, Y., Lee, S., Yan, H., Choi, H.-J., Yang, P., 2003. Singlecrystal gallium nitride nanotubes. Nature 422 (6932), 599e602. Han, W., Fan, S., Li, Q., Hu, Y., 1997. Synthesis of gallium nitride nanorods through a carbon nanotube- confined reaction. Science 277 (5330), 1287e1289. Hao, S., Zhou, G., Duan, W., Wu, J., Gu, B.-L., 2006. Hydrostatic-pressure-induced porous gallium nitride from nanotube bundles: an ab initio study. J. Chem. Phys. 125 (17), 174711. Hu, J., Bando, Y., Golberg, D., Liu, Q., 2003. Gallium nitride nanotubes by the conversion of gallium oxide nanotubes. Angew. Chem. - Int. Ed. 42 (30), 3493e3497.

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