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,
Trang 1Ironene – 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
Trang 2pores 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]
Trang 33 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.
Trang 43-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
Trang 5discussion 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).
Trang 6like 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
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Fig 8 2D visualization of configuration of solid-like atoms in models obtained at
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