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University – HochiMinh City, Viet Nam b Computational Materials Physics Research Group & Faculty of Applied Sciences, Ton Duc Thang University, 19 Nguyen Huu Tho Street, Tan Phong Ward,

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Ironene – A new 2D material

Vo Van Hoanga, Vuong Phu Taia, Tran Ky Thinha, Nguyen Hoang Giangb,⇑, Le Ngoc Quic

a

Comp Physics Lab, Ho Chi Minh City Univ of Technology, Vietnam Natl University – HochiMinh City, Viet Nam

b

Computational Materials Physics Research Group & Faculty of Applied Sciences, Ton Duc Thang University, 19 Nguyen Huu Tho Street, Tan Phong Ward, District 7,

Ho Chi Minh City, Viet Nam

c

Hung Thuan High School, Can Tho City, Viet Nam

a r t i c l e i n f o

Article history:

Received 12 May 2016

Received in revised form 24 August 2016

Accepted 3 September 2016

Keywords:

2D iron

Ironene

2D metal

Solidification of 2D liquid

a b s t r a c t

Discovery of 2D iron with a square lattice structure suspended in pores of graphene sheet by experiment (Zhao et al., 2014) has stimulated the researches related to 2D iron and other 2D metals by both exper-iments and computer simulations in general However, our understanding of structure and thermody-namics of 2D iron is completely lacking since main attention has focused on its thermal stability, magnetic behaviors and/or possibility of applications in practice A comprehensive molecular dynamics (MD) simulation of structure and thermodynamics of 2D liquid and crystalline Fe including ‘a natural for-mation’ of 2D Fe from the liquid state is done in the present work We find that 2D Fe with a triangle lat-tice structure spontaneously forms from the liquid state instead of that with a square latlat-tice structure although a set of atomic potentials for Fe have been used in MD simulation Both structure and thermo-dynamics of 2D liquid and crystalline Fe are close to those found by DFT calculations or experiments We find that crystallization of 2D liquid Fe exhibits a first-order-like phase transition behavior and it follows classical nucleation theory

Ó 2016 Published by Elsevier B.V

1 Introduction

It is well-known that the bond between atoms in metals is

mediated by conduction electrons which can move in any

direc-tion, i.e the system has a tendency to form 3D structure rather

than 2D sheet Therefore, the formation of a free-standing 2D metal

seems to be impossible However, the situation has changed due to

the recent discovery of 2D iron with a square lattice structure

sus-pended in graphene pores via in situ low-voltage

aberration-corrected TEM and supporting image simulation[1] This is not a

free-standing 2D iron sheet and the role of graphene pores for

the formation of 2D iron cannot be ignored Indeed, the dangling

edge C atoms of pores in graphene are highly reactive, and

there-fore, mobile Fe atoms have a tendency to bond to these C atoms

Then, these Fe atoms bond to the other Fe atoms around the edge

leading to the formation of 2D Fe sheet in the pores of graphene

Moreover, it is found by DFT calculations that atomic magnetic

moment of 2D Fe monolayer is of around 3.1lBwhich is much

higher than 2.2lBof the bulk counterpart[1] It promises possible

applications of this material for magnetic nano-electronic devices

such as magnetic recording media [1] In contrast, formation of

2D iron with a triangle lattice structure supported by graphene edges has been found by both experiment and computer simula-tion[2] 2D iron sheet is called ‘ironene’[2] Here-and-after we also call it ironene It raises a question about the most stable structure

of a free-standing 2D iron: square or triangle lattice? Subsequent investigations by both experiments and computer simulations for this 2D metal can be found[3–5], including Fe-C layers with differ-ent Fe/C ratios[5]and monolayer pyrite (FeS2)[6] In particular, electronic structure and magnetic behaviors of graphene edge sup-ported ironene are studied by DFT calculations, which are found to

be different from those of 3D counterpart[3] Note that the DFT-optimized model of constrained ironene has a triangle lattice not

a square one unlike that found in Ref.[1] Similarly, via DFT calcu-lations it is found that free-standing monolayer Fe with a triangle lattice structure is more stable compared to that with both square and honeycomb ones[4] However, embedded Fe membranes in graphene perforations can be more stable in a square lattice config-uration compared to that with a triangle one It indicates an impor-tant role of the graphene in the formation of Fe membranes with different atomic structures[4] In addition, also via DFT calcula-tions stability of 2D Fe-C sheets with various Fe/C ratios suspended

in graphene pores is systematically studied in order to highlight the situation[5] It is found that embedded Fe1C1 in graphene pores with a square lattice structure is formed instead of a pure

Fe monolayer[5] It is suggested that square lattice in graphene http://dx.doi.org/10.1016/j.commatsci.2016.09.011

0927-0256/Ó 2016 Published by Elsevier B.V.

⇑Corresponding author.

E-mail address: nguyenhoanggiang@tdt.edu.vn (N.H Giang).

Contents lists available atScienceDirect Computational Materials Science

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m m a t s c i

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pores observed previously in TEM image by Zhao et al may be a

mixture of Fe1C1and Fe2C2instead of pure Fe monolayer[1,5] It

is noted that C atoms near Fe ones cannot be ‘seen’ in TEM images

because of a large different contrast of atoms of two elements[5]

In contrast, it is found that monolayer FeS2 with several atomic

thicknesses constructed by cleaving from the bulk exhibits a

square lattice structure and advanced magnetic behaviors[6] A

large number of studies related to the Fe monolayer supported

on various substrates can be found (see for example[7,8])

How-ever, it is not a real 2D iron due to strong substrate effects and it

maybe a multilayer, not a single one Therefore, it is out of scope

of our paper and we do not pay more attention on Fe monolayer

on substrate in the present work

Besides ironene, 2D membranes of various metals or alloys have

been under much attention due to their enormous importance in

science and technology (see for example [9–18]) In particular,

ultrathin Rh nanosheets with the thickness less than 4 Å

contain-ing some planar 2D Rh monolayers have been found by experiment

and DFT calculations[9] The existence of 2D liquid Au membrane

suspended in graphene pores has been studied based on quantum

MD and density-functional-tight-binding (DFTB) methods [10]

Planar stability of Au membrane is suggested due to relativistic

effects and existence of 2D liquid Au membrane shows an extreme

fluxionality of metal nanostructures in general[10] DFT

calcula-tions and ab initio MD simulation also predict the stability of

free-standing 2D solid Ag and Au monolayers which exhibit a

hexagonal close-packed atomic structure[13,14] It is found that

2D solid Ag monolayer is stable in ab MD simulations for 10 ps

up to 800 K while Au monolayer is stable for the same annealing

time up to a much higher temperature of 1400 K[14] Similarly,

the early melting stages of free-standing Pt, Ag, Au and Cu

mono-layers have been studied based on quantum calculation methods

[17] These four monolayers can form stable quasi-2D liquid layers

with a significant amount of out-of-plane motion and in-plane

dif-fusion up to 2300–2400 K, 1050 K, 1600 K and 1320–1400 K,

respectively[17] In addition, properties of free-standing 2D

cop-per monolayers have been recently studied [18] Based on the

results described above, one important point should be

empha-sized that the transition metal atoms prefer being in

close-packed atomic configuration with hexa-coordination in 2D space

It is contrary to the honeycomb structure of prototypical graphene

with tri-coordination Thermal stability of 2D planar monolayers of

various alloys also has been found by the quantum calculation

methods Planar 2D hyper-coordinate Cu2Si, Cu2Ge, Ni2Ge, Ni2Si,

Cu2P, Cu2As alloys have been found[11,12,15,16] Due to difficulty

of stabilization of planar hyper-coordinate atomic configurations,

2D materials with hyper-coordinate structure are rarely found

Therefore, existence of planar hyper-coordinate 2D materials

pre-dicted by quantum calculations is of great interest Indeed,

two-dimensional Cu2Si monolayer with planar hexa-coordinate Cu

and Si bonding is found to be stable for short annealing up to

1200 K and it is a non-magnetic alloy[11] This material is metallic

and in this alloy, each Si atom is coordinated to six Cu atoms while

each Cu atom is coordinated to three Cu and three Si ones It is

found that this planar Cu2Si monolayer has a strong chemical

bonding and high in-plane stiffness [11] Similarly, planar 2D

hyper-coordinate Cu2Ge has been found and this 2D monolayer

is also stable for 10 ps of annealing up to 1200 K[12] This is the

first stable planar hexa-coordinate germanium material in 2D

space and its structure or chemical bonding are similar to those

found for Cu2Si given above [11] Existence of 2D

hyper-coordinate crystalline planar Ni2Ge or quasi-planar Ni2Si has been

found by quantum calculations[15] Planar Ni2Ge is stable up to

1500 K while quasi-planar Ni2Si is stable to around 900 K It is

found that planar Ni2Ge and quasi-planar Ni2Si are more stable

than germanene and silicene, respectively [15] Other new 2D

materials such as Cu2P, Cu2As have been found by quantum meth-ods[16] The former is found to be slightly buckled while the latter

is true planar 2D and both are diamagnetic 2D materials[16] It is clear that the binary 2D materials mentioned above have planar or quasi-planar hyper-coordinate motifs, i.e some have exactly pla-nar while other have slightly buckled structure Interestingly, while chemical bondings of Cu2Si and Cu2Ge are similar each to other, Ni2Si and Ni2Ge have quite different chemical bondings In general, the works related to various monolayers with planar and/or quasi-planar hexa-coordination mentioned above open a new branch of hyper-coordinated 2D materials for study

It is clear, predictions of the existence of various 2D metals or alloys by quantum methods such DFT or ab initio MD are more reli-able compared to those found by classical MD However, using quantum methods requires a large computation time and there-fore, the models used for quantum calculations are rather small

of around tens atoms (i.e mostly 64 atoms[9–17]) Although exis-tence of ironene containing tens of atoms has been found by both experiment and DFT calculations, atomic structure of a free-standing ironene has been under debate[1–4] Therefore, it is of great interest to carry out a comprehensive MD simulation of structure and thermodynamics of ironene models containing thou-sands atoms formed from 2D liquid Fe This is an alternative choice

to gain more detailed information of this important 2D material It motivates us to carry out the MD study in this direction

2 Calculations

MD simulations have been carried out in 2D square models con-taining 6400 iron atoms interacted via the EAM potential[19,20] EAM potentials have been widely used for simulations of metals since these potentials describe well interaction in metals and we

do not pause here for more discussion Initial 2D iron atomic figurations with a square lattice structure and with a lattice con-stant equal to that found by DFT calculation (2.35 Å [1]) have been relaxed for 105 MD steps at 50 K before heating to 4300 K

at heating rate of 1011K/s and at zero pressure in order to get 2D liquid configuration Models obtained at 4300 K are relaxed for

105 MD steps before cooling down to 300 K Periodic boundary conditions (PBCs) are applied in the x and y Cartesian directions while z¼ 0 is kept for all simulation procedure (models are in strictly 2D space), i.e we use NPT zero pressure ensemble for heat-ing procedure However, for coolheat-ing process PBCs are applied only

in the x direction while a fixed with reflection behavior boundary is used for y direction NVT ensemble simulation is used for further simulation including relaxation for 105 MD steps at 4300 K and cooling down to 300 K at the cooling rate of 2 1010K=s As a result, the final configurations are obtained in the form of nanorib-bons instead of 2D infinite sheets Final models obtained at 300 K have been relaxed for 105MD steps at this temperature before car-rying out further structural analysis

The Verlet algorithm and time step of 1.0 fs are used Tempera-ture is corrected via simple velocity rescaling LAMMPS software is used for MD simulations[21] ISAACS software is used for calculat-ing rcalculat-ing statistics[22] For calculations of rings, the ‘Guttmann’ rule

is applied[22] VMD software is used for 2D visualization of atomic configurations[23] The cutoff radius of 3.30 Å is taken in order to calculate coordination number, bond-angle and interatomic dis-tance distributions in the system This cutoff radius is equal to the position of the first minimum after the first peak in radial dis-tribution function (RDF) of models obtained at 300 K Note that we employ EAM potential implemented in the LAMMPS software that describes well both structure and thermodynamics of liquid and amorphous Fe thin films[24]

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3 Results and discussion

3.1 Thermodynamics and evolution of structure upon cooling from the

melt

Temperature dependence of total energy per atom and heat

capacity of the system upon cooling from the melt can be seen in

Fig 1 Total energy curve has two linear parts: the high

tempera-ture one is related to the liquid state of the system while the low

temperature part is related to the solid state A sudden-like change

between two linear parts is related to the solidification of the

sys-tem which exhibits a first-order-like phase transition behavior In

contrast, heat capacity has a sharp peak at around TX¼ 2640 K

which can be considered as a crystallization temperature of the

system Note that experimental melting temperature of 3D bulk

iron is Tm¼ 1811 K [25] It is clear that due to constraint in a

strictly 2D space of the simulation in the present work, freezing

of 2D liquid iron occurs at temperature much higher than that of

3D counterpart On the other hand, the starting point of deviation

from the linearity of the low temperature part of total energy can

be considered as temperature of final freezing of 2D liquid iron

(Tf ¼ 2200 K) We will use this temperature for defining of

solid-like atoms occurred during cooling process and we will return to

this problem later May be due to finite size and free edge (in the

y direction) effects, freezing of the system does not occur at a

cer-tain temperature It lasts over a cercer-tain temperature region (see

Fig 1) Total energy per atom for model obtained at 300 K is equal

to3.13 eV/at which is close to the binding energy of Fe

mono-layer with a triangle lattice structure found by DFT calculation

for the bond length of 2.45 Å, which is of around2.95 eV/at[4]

Note that the heat capacity is found approximately via the simple

relation: CV¼D E

D T,DE is the discrepancy of total energy between T1

and T2on cooling Heat capacity of 2D iron model at 300 K is equal

to 19.24 J

mol:Kwhich is not far from the value 25.10 J

mol:Kfor the bulk crystalline Fe obtained experimentally at 300 K and at pressure of

100 kPa[25]

Evolution of structure of the system upon cooling from the melt

can be seen inFig 2 One can see that at 4300 K, RDF of the system

exhibits a liquid-like behavior, i.e it has only two peaks at short

distances and the height of the peaks is rather small At

tempera-ture lower than the freezing one (TX¼ 2640 K), additional peaks

at intermediate and far distances occur indicated solidification of

the system At 300 K, RDF has separated peaks pointed out a high

degree of crystallinity in the system (Fig 2) Indeed, diffraction pat-tern of the atomic configuration obtained at 300 K exhibits a well-ordered crystalline behavior with 6-fold symmetry (Fig 3) This means that 2D iron obtained by cooling from the melt should have

a triangle lattice structure instead of a square lattice one Detailed information of structure of our 2D iron is given below

3.2 Structural properties of 2D iron obtained at 300 K Final atomic configurations obtained at 300 K are relaxed at this temperature for 105MD steps before carrying out further struc-tural analysis We find that 96% Fe atoms in the models have coor-dination number Z¼ 6 while around 4% have Z ¼ 5, 4, 3 (see Fig 4) These under-coordinated atoms are mainly related to the edge atoms in the y Cartesian direction (see below) These dangling bonds at the edge are more reactive sites for attraction of impuri-ties which may lead to the modification of atomic and electronic structure of ironene nanoribbons like that found for graphene and silicene nanoribbons[26–28] In contrast, almost 100% atoms

in the 2D iron models are involving into 3-fold rings (see the inset

ofFig 4) Concerning on the rings differed from 3-fold, we find only two 6-fold rings, i.e their fraction is too small compared to that of

Fig 1 Temperature dependence of total energy per atom and heat capacity of

models (the inset) upon cooling from liquid to solid state The dot line is total

Fig 2 Evolution of RDF upon cooling from the melt The bold line is for T = 2600 K which is close to the crystallization temperature T X = 2640 K.

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3-fold ones and their fraction cannot be visible inFig 4 Atoms

with Z–6 and rings differed from 3-fold plus the dangling bonds

at the edge can be considered as structural defects in ironene

nanoribbons In general, structural defects of 2D materials are

more reactive sites which may play an important role in

perfor-mance of various physico-chemical behaviors of 2D materials

[26–28] On the other hand, small fraction of structural defects

found for ironene models indicates a relatively homogeneous

structure of the obtained 2D crystals

In addition, we find that bond-angle distribution in the system

is relatively narrow which has a sharp peak at around 60°, i.e the

angle of an equilateral triangle (Fig 5) However, the distribution of

bond-angle ranged from around 50° to 70° indicates a certain

degree of the distorted structure of 2D models obtained by cooling

from the melt (Fig 5) Moreover, we also find a relatively narrow

interatomic distance distribution as shown in the inset ofFig 5

The distribution has a sharp peak at around 2.45 Å which very

close to the values of 2.41 Å and 2.44 Å found by DFT calculations

for 2D iron with a triangle lattice structure[4,5] Experimentally

found that the lattice constant of 2D iron with a square lattice

structure is 2.65 Å, while DFT calculations show that the most

stable lattice constant of 2D iron with a square lattice structure

is of around 2.35 Å[1] This means that our mean lattice constant

of 2.45 Å lies between these values In addition, distribution of interatomic distance in our 2D iron is ranged from around 2.20 Å

to around 2.75 Å indicated the existence of structural defects in models including distorted triangles, rings differed from 3-fold, the dangling bonds at the edge (seeFig 6) It is essential to note that the nearest interatomic distance in bcc 3D iron is 2.48 Å[25]

In order to get more detailed information of structure of 2D iron, we also calculate local and global bond-orientation orders [29–31] The local bond-orientation order, U6ðri

!

Þ, measures the degree of 6-fold-orientation ordering as follows:

U6ðri

!

Þ ¼ 1 nðiÞ

Xn ðiÞ j¼1

wherehijis the angle of the bond between particles i and j and an arbitrary but fixed reference axis, the sum over j is calculated over all nðiÞ nearest-neighbors The global bond-orientation order,W6, is calculated via averaging over all atoms in the system (N):

W6¼N1X

N

i ¼1

U6ðri

!

For a perfect triangle lattice structureW6¼ 1:0 and for a full disordered state W6 is equal to zero As shown in the inset of Fig 7, in the high temperature region (T> TX) the value ofW6is almost equal to zero indicated a strong disordered structure of the liquid state However, it has a sudden increase at around the freezing point exhibited a first-order-like phase transition At

300 K,W6is almost equal to 1.0 meaning that a well-ordered 2D crystal is formed (see the inset of Fig 7) We must choose a critical-like value for W6 in order to define solid-like atoms occurred in the system upon cooling from the melt It is well-known that at a freezing point, a significant amount of atoms in the system remain in the liquid state It ranges from 25% to 50%

of total number of atoms in the system (see for example [32– 34]) Therefore, it is not a good choice if one takes the value for

W6at a freezing point (TX¼ 2640 K) as a critical value for defining

of solid-like atoms An appropriate choice is the valueW6¼ 0:74 for Tf ¼ 2200 K (see Fig 1 and discussion given there for

Tf¼ 2200 K) If atom has local bond-orientation order U> 0:74,

it is considered as solid-like one As shown inFigs 6 and 7, below

TX¼ 2640 K almost all atoms in the system become solid-like On the other hand, via coloring of atoms with different local bond-orientation-orders we find some important points:

 Structure of the obtained 2D crystal is not perfect but it is rela-tively homogeneous since most atoms in the system have

U62 ½0:9—1:0Þ It indicates a high degree of crystallinity with

a triangle lattice structure

 Atoms with the same or close local bond-orientation-order have

a tendency to aggregate together into clusters which may lead

to ‘static heterogeneity’ of 2D crystals, i.e crystals containing clusters/domains with various bond-orientation orders This tendency may be cooling rate dependent (and/or depending

on the synthesis method)

 Main structural defects of the bulk 2D iron are single vacancies (SV) with a small fraction For ironene nanoribbons, there is a significant amount of the dangling bonds and under-coordinated atoms at the free edge in addition to vacancies (Fig 6)

There is no information about structural defects in ironene in order to compare and discuss However, 2D crystals with Lennard-Jones (LJ) interatomic potential also have a triangle lattice structure like that found for 2D iron in the present work Therefore, one can take the data for 2D crystals with LJ potential for

Fig 4 Coordination number and ring distributions (inset) in model obtained at

T = 300 K.

Fig 5 Bond-angle and interatomic distance distributions (inset) in model obtained

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discussion Indeed, the main structural defects in 2D crystals with

LJ potential are also vacancies and their behaviors have been under

much attention (see for example[35,36]) It is found that SVs are

the most mobile and two SVs have a tendency to coalesce into

one di-vacancy in order to lower energy [37] In addition, SVs

may transform semi-metallic silicene into metallic one or

vacan-cies may induce a small band gap in silicene[37] Important effects

of SVs on behaviors of 2D iron can be suggested, however, main

effects of SVs in 2D iron maybe those on the thermal stability

and on magnetic behaviors of material Indeed, DFT calculations

show that ferromagnetism can be introduced in graphene by

add-ing vacancies[38,39] Therefore, significant effects of vacancies on

magnetic behaviors of ironene can be suggested.Fig 6shows that

free edge of 2D iron exhibits a more complicated type although

fraction of the zig-zag edge dominates The type of the edge may

have a strong effect on behaviors of 2D materials including

elec-tronic structure like that found for other 2D materials[26–28]

3.3 Atomic mechanism of solidification

In order to highlight atomic mechanism of solidification of the

system upon cooling from the melt, we present temperature

dependence of fraction of solid-like atoms occurred during cooling

including 2D visualization (Figs 7 and 8) We find that fraction of

solid-like atoms is small and almost constant in the high

tempera-ture region (Fig 7) This means that these atoms maybe not real

solid-like in the high temperature region since their lifetime is

short, i.e the frequent transformation from solid-like atoms into

liquid-like ones and vice versa should frequently occur However,

fraction of solid-like atoms suddenly increases at around the

freez-ing point and reaches almost 1.0 at 300 K (Fig 7) This confirms

again a first-order behavior of crystallization of 2D liquid iron Note

that it is very difficult for experimentalists to observe the phase

transitions in 2D materials using traditional calorimetric methods

Therefore, our MD simulation provides a deeper understanding of

the problem Note that at a freezing point, fraction of solid-like

atoms in the system is of around 0.59 which is close to the range

from 0.50 to 0.56 found for simple 2D system in[34]

On the other hand, we find that solid-like atoms occur almost

homogeneously in the system and they have a tendency to

aggre-gate into local clusters (Fig 8a) Solidification proceeds further

with cooling via occurrence/growth of solid-like clusters with a tri-angle lattice structure following classical theory of nucleation However, occurrence/growth of solid-like clusters does not pro-ceed by the same manner throughout the model due to free edge effects in the y direction (Fig 8b) Free edge effects on structure and thermodynamics of 2D iron are out of scope of the paper

In addition, we find no evidence of the formation of an interme-diate phase during crystallization of 2D liquid iron It may be due

to a finite size of the models used in the present work It is essential

to note that we have employed the same simulation procedure for all potentials for Fe and/or Fe based alloys implemented in LAMMPS software [21] and final 2D iron with a triangle lattice structure is formed (not shown) On the other hand, we also find relaxation induced square lattice? triangle lattice transition in 2D iron even at very low temperature of 50 K That is, if initial atomic configurations of 2D iron with a square lattice structure (the lattice constant of 2.35 Å) are relaxed at 50 K using all inter-atomic potentials for Fe or Fe based alloys implemented in LAMMPS, square lattice eventually transforms into triangle one

Fig 6 2D visualization of atomic configuration obtained at T = 300 K Atoms of different local bond-orientation orders (U6 ) are colored as follows: cyan forU6 2 ½0:9—1:0Þ, pink forU6 2 ½0:8—0:9Þ, blue forU6 2 ½0:7—0:8Þ, red forU6 2 ½0:6—0:7Þ, gray forU6 2 ½0:5—0:6Þ, yellow forU6 2 ½0:4—0:5Þ, orange forU6 2 ½0:3—0:4Þ, tan forU6 2 ½0:2—0:3Þ, silver forU6 2 ½0:1—0:2Þ, green forU6 2 ½0:0—0:1Þ (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig 7 Temperature dependence of the fraction of solid-like atoms occurred upon cooling from the melt (N S =N) and global bond-orientation-order (inset).

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like that found in[5] This means that free-standing 2D iron with a

triangle lattice structure may be the most stable form compared to

those with square or other lattice ones

Finally, it is of great interest to carry out stress analysis of the

obtained ironene nanoribbon to clarify the edge effects on stress

distribution or warping, scrolling of nanoribbon in general Such

calculations should be done in 3D space It is found that depending

on the type of the edge termination, the bonding configurations at

the edges of graphene nanoribbons can be different from those

found in the interior (or in the bulk) If the atomic bonds at the

edges are shorter or longer than those found in the bulk graphene,

the edges should be under the state of compressive or tensile

stres-ses[40] Edge stresses can have a strong effect on morphology of

graphene nanoribbons leading to warping and rippling of

nanorib-bons for reduction of the edge energy at the cost of deformation of

the ‘bulk’ sheet[40] It is found that compressive edge stresses

cause out-of-plane warping of graphene sheet and morphology of

warped sheets depends strongly on their size/shape and on

magni-tude of the edge stresses[40] It leads to strong effects on

elec-tronic structure of 2D material since elecelec-tronic structure of

graphene can be strongly altered by both strain and curvature

[41] Total energy of graphene sheets with compressive edge

stres-ses can be reduced by stretching of the atomic bonds by

out-of-plane movement of the atoms leading to the warping and rippling

of graphene sheets[40] On the other hand, depending on the size

and shape of the sheets warping can be localized in the boundary

region or can influence the entire morphology of nanoribbons[40] Overall, similar edge stress effects on morphology and various behaviors of Fe nanoribbons including magnetic ones can be sug-gested However, it is out of scope the present paper

4 Conclusions

A comprehensive MD simulation of the formation of free-standing 2D iron from the liquid state has been carried out and some conclusions can be drawn as follows:

 Free-standing 2D iron with a triangle lattice structure sponta-neously forms from the liquid state using EAM potential (and/

or all potentials for Fe or Fe-based alloys implemented in LAMMPS software[20]) Our MD simulation confirms again that 2D iron with a triangle lattice structure maybe more stable compared to that with a square lattice It is unlike that found experimentally or by DFT calculation for 2D iron suspended in graphene pores [1] Note that triangle lattice or hexagonal close-packed structure of ironene found in the present work is

in good accordance with that found for various 2D pure metals including Fe, Au, Ag[2–5,13,14]

 2D iron formed ‘naturally’ from the liquid state has nearly homogeneous and well-ordered triangle structure However, a slightly distorted structure of the models should be mentioned including a relatively narrow interatomic distance and bond angle distributions compared to those of the equilateral triangle lattice structure

 Structural behaviors of 2D iron with a triangle lattice structure including interatomic distance are close to those found for 2D and 3D iron The main structural defects of 2D iron are single vacancies like that found for a triangle lattice structure of 2D Lennard-Jones crystals

 Crystallization of 2D liquid Fe exhibits a first-order behavior of phase transition Both binding energy and heat capacity of 2D iron at 300 K have a reasonable value compared to those found for 2D and 3D counterparts

Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant 103.01-2014.86

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(a) T = 2700 K

(b) T = 2600 K

Fig 8 2D visualization of configuration of solid-like atoms in models obtained at

temperatures above/below T X = 2640 K.

Trang 7

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