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Non-equilibrium grain boundaries with excess energy in graphene
Accepted Manuscript Non-Equilibrium grain boundaries with excess energy in graphene A.E. Romanov, A.L. Kolesnikova, T.S. Orlova, I. Hussainova, V.E. Bougrov, R.Z. Valiev PII: DOI: Reference:
S0008-6223(14)00905-1 http://dx.doi.org/10.1016/j.carbon.2014.09.053 CARBON 9350
To appear in:
Carbon
Received Date: Accepted Date:
25 June 2014 17 September 2014
Please cite this article as: Romanov, A.E., Kolesnikova, A.L., Orlova, T.S., Hussainova, I., Bougrov, V.E., Valiev, R.Z., Non-Equilibrium grain boundaries with excess energy in graphene, Carbon (2014), doi: http://dx.doi.org/ 10.1016/j.carbon.2014.09.053
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NON-EQUILIBRIUM GRAIN BOUNDARIES WITH EXCESS ENERGY IN GRAPHENE A.E. Romanov1,2, A.L. Kolesnikova1,3, T.S. Orlova1,2*, I. Hussainova4, V.E. Bougrov1, R.Z. Valiev1,5 1
ITMO University, Kronverskiy av. 49, St. Petersburg, 197101, Russia
2
Ioffe Physical-Technical Institute, Polytechnicheskaya st. 26, St. Petersburg, 194021, Russia
3
Institute of Problems of Mechanical Engineering, Bolshoj pr. 61, V.O., St. Petersburg, 199178,
Russia 4
Tallinn University of Technology, Ehitajate tee 5, Tallinn, 19086, Estonia
5
Ufa State Aviation Technical University, K. Marx Street 12, Ufa, 450000, Russia
Abstract. Exploring an approach to grain boundary (GB) mesoscopic structure in 3D polycrystals, we develop a model description for equilibrium and non-equilibrium states of GBs in graphene. Non-equilibrium states of GBs in graphene are analyzed for the first time. On the base of this model, dependence of total energy for symmetric GBs on the misorientation angle, , in graphene was calculated. It was found that for the same average misorientation angle the energy in a non-equilibrium state of a GB in graphene can significantly exceed the energy in the equilibrium state. In graphene, the GB energy depends to a greater extent on the deviation from the equilibrium state than on the misorientation angle. It is proposed to account for the presence of non-equilibrium GBs in the explanations of unusual physical properties of graphene.
*Corresponding author. Tel: +7 812 2927157 E-mail address: [email protected]
1. Introduction
Outstanding electronic and mechanical properties of graphene make it a promising material for a wide range of potential applications ranging from nano-electronics and photonics to advanced composites’ nano-fillers. Many applications require fabrication of large-scale graphene sheets, which usually represent polycrystalline layers with numerous grain boundaries (GBs) [1]. Grain boundaries significantly alter physical and mechanical properties of the pristine graphene [1-6]. For example, study by scanning tunneling microscopy (STM) evidenced an order of magnitude lower conductivity within the grain boundary as compared to the defect-free graphene lattice [2]. These STM findings are in agreement with electrical transport measurements across an individual GB and in intra-grains forming the GB [4-6]. On the other hand, graphene GBs can exhibit unique electronic properties, such as ideal 1 D conductivity of some special GBs [3] and inversion of conductivity into n-type at GB in the p-doped graphene [2] that opens potential for new applications exploiting the graphene GBs. It is generally accepted that a contribution of each given grain boundary to the material properties is related to its excess energy per unit area compared to defect-free grain. It is also well-known that the energy of the GB depends on its structure [7,8]. Therefore, studying the structure of GBs and their energy characteristics is of great interest in the direction of exploration of graphene unique properties. Theoretical description of the real GBs structure in graphene is also a challenge for fundamental scientific standpoint because graphene gives practically the first possibility to observe GBs and study their performance as 1D objects in 2D systems. Several research groups have reported on experimental study of GB structure in graphene by the transmission electron microscopy (TEM) [9-11] and STM [2,12]. It was found that GBs tend to be composed mainly of 5- and 7-member carbon atom rings [9,12] in contrast to the ideal honeycomb lattice of graphene consisting of only 6-member rings. In addition, some other atomic configurations were also detected in the arrangement of GB line, for example, distorted hexagons [1,11] and the chains of 8- and 5-member carbon atom rings [3,13]. The later configuration is typical for so-called domain boundaries when there is no misorientation between adjusting grains (domains). The experimentally investigated GBs in polycrystalline CVD (chemical vapor deposition) graphene have tilt character, they are usually neither straight, non periodic and contain a significant amount of disorder [9-11,14,15]. Theoretically, structure and formation energy of variety of the GBs in graphene were investigated by classical simulation techniques using Tersoff- or Brenner-type potentials [16,17] or semi-empirical density-functional tight–binding (DFTB) method [18,19]. Computer simulations of GBs in graphene were based on the energy minimization analysis and, eventually, 2
revealed GBs as classes of periodic grain boundary dislocation walls. In fact, mainly GBs with locally minimal energies were theoretically considered defining such GBs as equilibrium objects. For series of periodic grain boundaries containing mainly uniformly distributed pentagons and heptagons (or uniformly distributed dislocations), the energy versus misorientation angle dependences, , were calculated in [16,17,19-21]. However, real structure of GBs in CVD graphene is in greater extent controlled by kinetic processes of graphene grain nucleation and growth. Graphene GBs can be considered as kinetically “frozen” objects and hence they are strikingly improbable objects from energy minimization viewpoint [22]. As it was identified for metals and alloys (3D polycrystals), the grain boundaries could be in equilibrium or non-equilibrium states with an excess energy depending on the processing conditions [23-25]. Structure of the equilibrium GB in a specific range of misorientations may be described as uniformly distributed arrays of dislocations [26]: such a GB possesses minimal energy for a given misorientation angle. The structure of non-equilibrium GB in metals and alloys consists of non-uniformly distributed dislocations or disclinations and results in storage of an additional energy in the boundary when compared to the energy in equilibrium state for the same angle of misorientation [26,27]. The non-equilibrium grain boundary energy becomes less sensitive to the misorientation angle [27]. Grain boundaries in the two different states are responsible for significantly different mechanical and some physical properties of metals and alloys. For example, non-equilibrium GBs, which are characteristic of ultra-fine grained metals processed by severe plastic deformation methods, provide enhanced strength and outstanding plasticity of these materials [23-25]. In the present study, we develop mesoscopic model description for grain boundary structure in graphene (2D system) and utilize the proposed approach for the analysis of energy of equilibrium and non-equilibrium graphene GB states. Non-equilibrium GBs in graphene are considered for the first time. The proposed mesoscopic model description of GBs in graphene is based on the previously developed approach in the description of non-equilibrium grain boundaries in metals (3D polycrystals) [26,27]. The mesoscopic description of GB structures in graphene allows one to systematically describe symmetric GBs in equilibrium or nonequilibrium states for any misorientation angle and to obtain a simple algorithm for calculation of their energy. The mesoscopic structural modeling of graphene grain boundaries can serve as a starting point for describing their unique physical properties.
3
2. Background. Mesoscopic disclination-structural unit model for grain-boundaries in 3D polycrystals
For description of non-equilibrium state of GBs in metals the disclination-structural unit (DSU) multiscale model was proposed 25 years ago combining both the atomistic (structural unit model) and the mesoscopic (disclination model) scales [26,27]. According to the approach developed for metals [7,28], each favored GB consists of only one structural element (unit), each random GB with a misorientation angle consists of only two structural units belonging to two nearest favored GBs, numerated as “r” and “r+1”, with misorientations and (r=1,2…): (Fig. 1).
Fig. 1. Geometrical model of the tilt grain boundary (GB) with misorientation angle as disclination dipole wall. The nearest favored grain boundaries GBr and GBr+1 with misorientations and are shown. Disclination strength is designated as ; and are the lengths of GBr and GBr+1 structural units A and B, respectively; , are the disclination coordinates; H is the period of GB structure.
4
Since the junctions of structural elements with different misorientations represent disclinations [29], structure of the GB is described as a disclination dipole wall (DDW) (Fig. 1), in which the arm of the dipoles depends on , and it is equal or multiples to or , where and are the lengths of structural elements A and B corresponding to GBr and GBr+1, respectively [26]. In an arbitrary GB the dipoles could be distributed non-uniformly, but periodically with one period , where m and n are the numbers of structural elements forming the period. In the disclination-structural unit model, the average misorientation angle can be determined with quite good accuracy from a simple expression [30]:
ಲ ೝ ಳ ೝశభ ಲ ಳ
(1)
.
In the framework of DSU model, the total energy of the GB can be calculated on the base of three terms [26]: .
(2)
Here is the GB interface (surface) energy, is the elastic energy of the disclination ensemble, and is the energy of disclination cores. The elastic energy of disclination assembling may be calculated as the sum of energies of composing DDWs accounting for disclination interactions. It was shown that is minimal when the structural units (and also disclination dipoles) are distributed along the GB uniformly [27]. Such configuration of the GB with minimal energy refers to the equilibrium state of GB. One of the advantages of the DSU model of GBs [26,27] is a possibility to describe the non-equilibrium state of GB as non-uniformly distributed disclination dipoles [27] and compute the energy and mechanical stresses of such type of GBs. At the atomic level, using molecular dynamics simulation techniques, computing GB energy is difficult and for complex boundary structures is even impossible so far, hence consideration of GBs by simulation methods has been restricted mainly by equilibrium GBs. In the present work, the non-equilibrium GBs in graphene are described on the base of DSU approach previously developed for metals [26,27]. The elastic stresses and boundary energy are estimated taken into consideration the possible structural configurations of the equilibrium and non-equilibrium GBs.
5
3. Results and Discussion
3.1. Disclination-structural unit model of equilibrium and non-equilibrium grain boundaries in graphene
We apply the DSU model developed for metals [26,27] to symmetric GBs in graphene. In two dimensional materials such as graphene, GBs are 1D-interfaces between two grains (crystals) with different crystallographic orientations. In the ideal honeycomb lattice of graphene, the length of arbitrary translation vector , can be easily found using a special basis , (Fig. 2), see, for example, [21]: |, | √ ,
(3)
where √3 0.246 , is a hexagon side.
Fig. 2. Coordinate basis , and arbitrary vector , in graphene lattice. Grain boundary between left (L) and right (R) grains is geometrically possible when the lengths of the translation vectors in the both grains oriented along the GB are equal, i.e. | , | | , |. GB between grains is designed as , | , . For symmetric GB and . The angle between vector , and vector can be determined from [31]: arcsin &
√ √ మ మ
'
.
(4)
The GB misorientation angle can be defined by [31]:
6
( arcsin )
√ ೃ మ మ ೃ ೃ ೃ ೃ
* ( arcsin )
√ ಽ ಽమ ಽ ಽ ಽమ
*.
(5)
Analysing the variety of GBs simulated in [16,19] one can distinguish several structural units from which we can construct all favored and any arbitrary symmetrical GBs in graphene. The structural units A, B (B’), and C are shown in Fig. 3. The favored GBs constructed only by one type unit A, B (B’), or C have misorientation angle 0 , 21.8 , and 60 , respectively [16,19]. The lengths of units A, B (B’), and C are 3, √21, and √3, correspondingly. The length of the structural unit B (B’) is calculated using the Eq.(3) taking into account that the units B (B’) compose the favored GB (2,1)|(1,2) with misorientation angle 21.8 , see, Eq.(5).
Fig. 3. Structural units of graphene grain boundaries (GBs). A is the structural unit of GB with misorientation angle 0 (perfect crystal); B and B’ are the equivalent structural units of favored GB with 21.8 , C is the structural unit of favored GB with 60 (perfect crystal); dA, dB, and dC are the lengths of A, B, and C along the favored GBs.
Using the structural units (A, B (B’), and C) of the favored GBs in graphene, any geometrical configuration for arbitrary GB can be constructed. First, we built GBs with the uniform distribution of structural elements (equilibrium GBs) and the misorientation in the range 0 – 600 which is divided by the favored GBs into two intervals 0 21.8 and 21.8 60 . In Fig. 4 some symmetrical GBs of graphene are demonstrated. It should be noted that in graphene the favored GB with 21.8 can be constructed with two equivalent units B or B’ with the same misorientation and length (Fig. 4). An arbitrary equilibrium GB is constructed by two structural units A and B, or B’ and C which are uniformly distributed. We believe that the structural units (A, B (B’), and C) cover all possible symmetric GB structural motives in 7
graphene because they permit us to build all known GB structures found by MD method [16, 19]. Basing on the ideas presented in [27] we construct some non-equilibrium GBs with different misorientations in graphene. For a chosen average misorientation angle non-equilibrium state is achieved in two different ways. First, a deviation from equilibrium state could be reached by increasing the period H of grain boundary structure keeping the same types and numbers of structural units (Fig. 5,a). Second, a deviation from the uniform distribution of structural elements is caused by rearrangement of the structural units appropriately (Fig. 5,b).
Fig. 4. Equilibrium symmetrical tilt GBs in graphene, composed with structural units A, B (B’), and C. Misorientation angles 0 , 21.8 , and 60 correspond to favored GBs: GB1, GB2, and GB3. The formula for GB period H is given, for example, AAB ( 9.4 ), B’B’C ( 27.8 ) etc. In the frame of the disclination-structural unit approach one can determine the disclination content of GBs in graphene. Examples of disclination-structural unit description of equilibrium and non-equilibrium GBs from the both intervals 0 21.8 and 21.8 60 in graphene are given in Fig. 6. 8
Fig. 5. Simple examples of equilibrium and non-equilibrium grain boundaries with misorientation angles (a) 13.2 and (b) 35.6 .
Fig. 6. Disclination-structural unit (DSU) model of equilibrium and non-equilibrium GBs in graphene. The GB misorientation angle is in the range: (a) 0 13.2 21.8 ; (b) 21.8 35.6 60 . 9
3.2. Energies of equilibrium and non-equilibrium grain boundaries in graphene In the framework of DSU model the total energy of GBs in graphene is calculated from Eq. (2) taking into account 2D hexagonal structure of graphene lattice and 1D nature of GBs in graphene. For GBs with misorientation angle 0 0 (1 1,2): the interface energy per length unit can be written as following:
ಲ భ ಳ మ ಲ ಳ ಳ మ య ಳ
for GBs with 0 0 0 21.8 ,
(6a)
for GBs with 21.8 0 0 60 ,
(6b)
where m and n are the numbers of the structural units in the period (or ) of GB; dA, dB, and dC are the lengths of structural units A, B, and C (see Fig. 3); 2 , 2 and 2 are the interface energies per unit length of the favored GBs (Fig. 4). Graphene containing GBs with misorientation angles 0 and 60 are the perfect crystals (Fig. 4), therefore 2 2 0. Interface energy 2 .!బ . It is well-known that hexagonal crystalline plane is an isotropic plane, so the elastic energy of GB Eelastic , see Eq. (2), can be calculated in the framework of linear isotropic elasticity. The energy is the energy of disclination ensemble in 2D medium and it can be estimated on the base of the elastic fields of the wedge disclinations or the edge dislocations perpendicular to the free surfaces of an infinitesimally thin plate [32,33]. It was demonstrated in [32] that in an infinite thin film plane strain state characteristic for edge dislocations and wedge disclinations transforms to plane stress state. In a result, the stresses of the periodically distributed disclination dipoles (see Fig. 1) take the following form:
3""
#$%&' 5
5
(
4 ln &
)*+, "-.)*+ $/-./- .0& )*+, "-.)*+ $/-./- &
"- +12, ")*+, "-.)*+ $/-./- &
(
'+
"- +12, ")*+, "-.)*+ $/-./- .0&
7,
10
(7a)
3//
#$%&' 5 (
5(
4 ln &
)*+, "-.)*+ $/-./- .0& )*+, "-.)*+ $/-./- &
"- +12, ")*+, "-.)*+ $/-./- &
'-
"- +12, ")*+, "-.)*+ $/-./- .0&
7,
(7b)
333 0, 3"/
(7c)
#$%&'"(
3"3 0,
4
+12 $/-./- & )*+, "-.)*+ $/-./- &
(
+12 $/-./- .0& )*+, "-.)*+ $/-./- .0&
7,
(7d)
33/ 0,
(7e,f)
where disclination strength ∆ ( (1 1,2); 9: ;9/ , :$& ; $& / , (0, $& ) are the disclination coordinates (Fig. 1), = is the arm of dipole in the wall; H is a period of DDW, > is shear modulus; ν is Poisson’s ratio. Taking into account Eqs. (7) the energy can be represented [27] for definiteness in the case of ? by the following expression:
.
>1 @
)A= B B CAD : ( : =E ( 2AD : ( : E AD : ( : ( =EF* 32; 4 4
1 1,2,
(8)
where = ; / for GBs with 0 21.8 and = ; / for GBs with 21.8 60 . Functional A is taken as [34]:
6
AG 2 H sin I I H
5 5 )*+, 5.)*+
.
(9)
For easy of calculation A can be rewritten: AG 2C(Li D2 . E ( Li D2 E 2K3F,
(10)
6 . where polylogarithm Li L ∑6 4 L / , and Riemann zeta-function K3 ∑ 4 .
For graphene hexagonal lattice we can write the last term in Eq.(2) as following [35]:
11
7$%&8#9మ ∆ మ ;(య <
.
(11)
Here is a disclination core parameter, and N . Parameter can be taken in the range of 1-3 [26]. The value of affects the magnitude of GB energy and can be considered as a fitting parameter of the disclination structural unit model. In Fig. 7, the calculated with the help of Eqs.(2,8,11) grain boundary energies E are plotted as the function of misorientation angle for the cases of equilibrium (filled blue circles) and non-equilibrium (green triangles) GBs. The calculations utilize the following set of parameters: interface energy .!బ 3.44 eV/nm [19], shear modulus > 80.15 N/m [36], the Poisson’s ratio @ 0.416 [36], lattice parameter 0.246 nm [21]; disclination core parameter 1. This value for was taken to approximately fit the energies of GBs calculated by different methods [19] in the interval of misorientations 12 20 . The elastic modules > and @ were chosen from a set of elastic modules of graphene found by several authors and analyzed in detail in [37]. Note that the elastic modules we use have been calculated by computer simulation with application of Brenner-type II potential [36]. It has been shown that these modules are in good agreement with those calculated by other methods, see [37].
Fig. 7. Grain boundary energy versus misorientation angle in graphene for equilibrium (filled blue circles) and non-equilibrium (green triangles) GB. Plots are calculated at the following parameters: interface energy E21.80 = 3.44 eV/nm [19], shear modulus > 80.15 N/m [36], the Poisson’s ratio @ 0.416 [36], lattice parameter 0.246 nm [21]; disclination core parameter 1. Dashed line and empty red circles correspond to results from Ref. [16] and [19], respectively.
12
In the case of equilibrium GBs, grain boundary energy was calculated for different GB misorientations in the whole interval of GB misorientation angles 0 0 0 60 (Fig. 7). There is a satisfactory agreement between the -dependence obtained in the present work for the case of equilibrium GBs (filled blue circles) at 0 0 0 32.2 and the previous results [16,19]. This proves the validity of the DSU approach for characterization of GBs in graphene. In the interval 32.2 0 0 60 our -dependence differs substantially from the plot given in Ref. [16] (Fig. 7). Unlike the nearly inverted parabolic -curve in [16] our results demonstrate the -dependence which is far from to be bell-shaped: there are two cusps on curve at misorientation angles of 21.80 and 32.20 and the energy values in the interval 32.2 60 are larger those in the interval 0 32.2 . However, these findings are consistent with other data sets [17,38] demonstrating similar asymmetric behavior of dependence for GB in graphene. The two cusps on curve at misorientation angles of 21.790 and 32.210 were shown in Ref. [17] on studying the GB structures and energies by atomistic simulations. Yazev and Louie [38] investigated symmetrical periodic GBs in graphene based on dislocation theories and performed the first-principle calculation of the grain boundary energies, which also demonstrated the presence of such cusps on the dependence and clearly larger energies in the interval 32.2 50 compared to those in the interval 10 32.2 . In the structure-unit grain boundary description (Fig. 4) it is evident that despite the GBs with misorientation angles 0 and 60 belong to the perfect crystal, they consist of different structural units. Namely, GB with 0 is constructed by the structural units A oriented parallel to the armchair graphene direction [1], whereas GB with 60 consists of the structural units C oriented parallel to zigzag graphene direction [1]. In addition, the elements
A and C are of different lengths. As a result, the GBs with misorientation angle from the interval 0 21.8 , which are constructed by A and B units, and the GBs with misorientation angle (60 ( ) from the interval 21.8 60 , which are constructed by B and C units, are structurally different, hence they should have different structure-dependent energy characteristics. Available literature data [1,39,40] also indicate that for about the same misorientation angle, properties of GB depend on its crystallographic orientation in pristine (defect-free) graphene. It was shown by MD simulations that symmetric tilt GB between two zigzag-oriented domains in graphene bicrystal is more efficient for heat transmitting [39] and have higher strength [40] than such GB between two armchair-oriented domains with about the same misorientation. Moreover, strength of pristine graphene stretching in zigzag direction is higher
13
than in armchair direction [40]. The graphene sheet’s overall electronic and magnetic properties are strongly dependent on the type (armchair or zigzag) of graphene sheet edges [41]. With use of the disclination-structural unit approach we also calculated energy of nonequilibrium GBs for some misorientations and different extent of structure disordering (green triangles in Fig. 7). Energies of non-equilibrium grain boundary states were calculated for the following misorientation angles: 11 , 13.2 , 32.2 . Non-equilibrium states of GBs for a given misorientation angle were built by changing the order of structural units and (or) by increasing the GB period. All of these transformations lead to a reduction of uniformity of the structural unit distribution in the grain boundary period. For 11 the equilibrium grain boundary ABABA became a non-equilibrium GB under permutation of structural units in the period (grain boundary 3А2В) or increase of the period (grain boundary 6A4B); for 13.2 the equilibrium grain boundary AB transformed into non-equilibrium boundaries 4A4B and 5A5B; for the misorientation angle 32.2 the grain boundary B'C transformed in nonequilibrium boundary 2B'2C. It is evident that the total grain boundary energy in a nonequilibrium state can greatly exceed that of the equilibrium state for the same misorientation angle. Simple explanation of this fact can be found in the increase of disclination dipole arm (and therefore the energy) for GBs with longer periods. Another important finding is the absence of a pronounced dependence of non-equilibrium boundary energy on the misorientation angle. Indeed, our calculations show that in the case of the non-equilibrium state of the grain boundaries in graphene, the energy of low- and mediumangle grain boundaries can be as high as the energy of high-angle grain boundary (Fig. 7). This well agrees with experimental results [2] according to which electronic properties of GBs in graphene are robust against their misorientation angle. In order to illustrate the universal nature and robustness of the observed electronic behavior of GBs in CDV graphene, Topaszto et al. [2] demonstrated similarity of the conductivity map and charge doping profiles for three distinct GBs with tilt angles of 270, 210 and 120 comprising the same triple junction. This most probably suggests that grain boundaries in the CVD polycrystalline graphene are in non-equilibrium state providing less sensitivity of GB properties to the misorientation angle. The considered nonuniform structure of non-equilibrium GBs in graphene corresponds in more extent to experimentally observed GBs containing disorder [9-11] than the simulated GB models obtained with the GB energy minimization [16,17,19]. The DSU model of non-equilibrium GBs in graphene can be proposed for description of outstanding properties of graphene GBs.
14
4. Conclusions
In the present work, the structure of symmetric grain boundaries (GBs) in graphene has been studied on the base of disclination-structural unit (DSU) approach originally developed for metals. This approach allows considering both equilibrium (energy-minimized for a given misorientation angle) as well as non-equilibrium states of the GBs with the same given misorientation. The non-equlibrium GBs in graphene have been analyzed for the first time. In accordance with the DSU approach, non-equlibrium GB structures in graphene were introduced as non-uniform distribution of structural units, which can be described at mesoscale as disordered network of disclination dipoles. The elastic stresses and energy in the whole interval of GB misorientation angles 0 0 0 60 have been calculated. The obtained dependence for equilibrium GBs is in good agreement with similar dependences found by direct simulation methods for the misorientation angle range of 0 0 0 32.2 . The obtained - dependence in the whole interval 0 0 0 60 is substantially nonbell-shaped that is associated with structural differences of the GBs with и (60 ( ) and agrees well with literature data on orientation-dependent properties of a GB in graphene lattice. It has been demonstrated that the energy of non-equilibrium GB in graphene can substantially exceed the energy of equilibrium GB with the same misorientation angle. The energy of non-equilibrium GB is highly affected by degree of a structural unit distribution irregularity and to much lesser extent by average misorientation angle in the GB. The proposed model of non-equilibrium state of GBs in graphene is along with the experimental findings, demonstrating substantial disordering in GB atomic structures and robustness of GB electronic properties against the misorientation angles. Therefore, we suggest that CBs in CVD polycrystalline graphene could be in non-equilibrium states depending on the processing regimes. The proposed disclination-structural unit model of non-equilibrium GBs in graphene can serve a key structural platform for the description of physical and mechanical properties of graphene GBs, including unique misorientation angle-independent electronic properties of GBs in CVD grown graphene. Experimental studies of GBs in graphene also evidence on the formation of irregular structures, see for example [9,10]. Meandering of grain boundaries at micro- and nanoscale is related to the formation of facets with different crystallographic orientation. This can be typical non-symmetric grain boundaries. Then, each facet can be a section of a symmetric boundary. The energy of such a boundary can be either higher or lower of the straight linear boundary with the same in average orientation. In this case, the facet junctions will also demonstrate
15
disclination counterparts. The consideration of these complex configurations in the framework of the disclination approach is the subject of the ongoing research.
16
References
[1] Biro LP, Lambin Ph. Grain boundaries in graphene grown by chemical vapor deposition. New J Phys 2013; 15:1-38. [2] Tapaszto L, Nemes-Incze P, Dobrik G, K. Yoo Jae, Hwang C, Biro LP. Mapping the electronic properties of individual graphene grain boundaries. Appl Phys Lett 2012; 100(5):053114-1-4. [3] Lahiri J, Lin Y, Bozkurt P, Oleynik I I, Batzill M. An extended defect in graphene as a metallic wire. Nature Nanotechnol 2010; 5(5):326-9. [4] Kim P. Graphene: Across the border. Nat Mater 2010; 9(10):792-3. [5] Yu Q, Jauregui LA, Wu W, Colby R, Tian J, Su Z, et al. Control and characterization of individual grains and grain boundaries in graphene grown by chemical vapour deposition. Nat Mater 2011; 10(6):443-9. [6] Jauregui LA, Cao H, Wu W, Yu Q, Chen YP. Electronic properties of grains and grain boundaries in graphene grown by chemical vapor deposition. Solid State Commun 2011; 151(16):1100-4. [7] Sutton AP, Balluffi RW. Interfaces in crystalline materials. Oxford: Oxford University Press. 1995. [8] Bulatov VV, Reed BW, Kumar M. Grain boundary energy function for fcc metals. Acta Materialia 2014; 65:161-75. [9] Huang PY, Ruiz-Vargas CS, van der Zande AM, Whitney WS, Levendorf MP, Kevek JW, et al. Grains and grain boundaries in single-layer graphene atomic patchwork quilts. Nature 2011; 469(7330):389-92. [10] Kim K, Lee Z, Regan W, Kisielowski C, Crommie MF, Zett A. Grain boundary mapping in polycrystalline graphene. ACS Nano 2011; 5(3):2142-6. [11] Kim KS, Zhao Y, Jang H, Lee SY, Kim JM, Kim KS, et al. Large-scale pattern growth of graphene films for stretchable transparent electrodes. Nature 2009; 457(7230):706-10. [12] Simonis P, Goffaux C, Thiry PA, Biro LP, Lambin P, Meunier V. STM study of a grain boundary in graphite. Surf. Sci 2002; 511(1-3):319-22. [13] Pelc M, Chico L, Ayuela A, Jaskolski W. Grain boundaries with octagonal defects in graphene nanoribbons and nanotubes. Phys. Rev. B 2013; 87:165427-1-7. [14] Gao L, Guest JR, Guisinger NP. Epitaxial Graphene on Cu(111). Nano Lett 2010; 10(9):3512-6. [15] Zhang Y, Gao T, Gao Y, Xie S, Ji Q, Yan K, et al. Defect-like structures of graphene on copper foils for strain relief investigated by high-resolution scanning tunneling microscopy. ACS Nano 2011; 5(5):4014-22. 17
[16] Liu Y, Yakobson BI. Cones, pringles, and grain boundary landscapes in graphene topology. Nano Lett 2011; 10:2178-83. [17] Liu T-H, Gajewski G, Pao C-W, Chang C-C. Structure, energy, and structural transformations of graphene grain boundaries from atomistic simulations. Carbon 2011; 49(7):2306–17. [18] Malola S, Hakkinen H, Koskinen P. Structural, chemical, and dynamical trends in graphene grain boundaries. Phys Rev B 2010; 81(16):165447-1-6. [19] Zhang J, Zhao J. Structures and electronic properties of symmetric and nonsymmetric graphene grain boundaries. Carbon 2013; 55(1):151-9. [20] Yazyev OV. Polycrystalline graphene: Atomic structure, energetics and transport properties. Solid State Commun 2012; 152(15):1431-6. [21] Yazyev OV, Louie SG. Electronic transport in polycrystalline graphene. Nat Mater 2010; 9(10):806-9. [22] Yakobson BI and Ding F. Observational geology of graphene, at the nanoscale. ACS NANO 2011; 5(3):1569-74. [23] Valiev RZ, Islamgaliev RK, Alexandrov IV. Bulk nanostructured materials from severe plastic deformation. Prog. Mater. Sci. 2000; 45:103-89. [24] McFadden SX, Mishra RS, Valiev RZ, Zhilyaev AP, Mukherjee AK. Low-temperature superplasticity in nanostructured nickel and metal alloys. Nature 1999; 398(6729):684-6. [25] Valiev RZ. Nanostructured alloys: Large tensile elongation. Nat Mater 2013; 2(4):289-91. [26] Gertsman VV, Nazarov AA, Romanov AE, Valiev RZ, Vladimirov VI. Disclinationstructural unit model of grain boundaries. Phil Mag A 1989; 59(5):1113-8. [27] Nazarov AA, Romanov AE, Valiev RZ. On the structure, stress fields and energy of nonequilibrium grain boundaries. Acta Metal Mater 1993; 41(4):1033-40. [28] Sutton AP, Vitek V. On the structure of tilt grain boundaries in cubic metals I. Symmetrical tilt boundaries. Phil Trans R Soc London A 1983; 309(1506):1-36. [29] Romanov AE, Kolesnikova AL. Application of disclination concept to solid structures. Prog Mater Sci 2009; 54(6):740-69. [30] Nazarov AA, Romanov AE. On average misorientation angle of general tilt boundaries. Phil Mag Lett 1989; 60(5):187-93. [31] Wu J-T, Shi X-H, Wei Y-J. Tunable band structures of polycrystalline graphene by external and mismatch strains. Acta Mechanica Sinica 2012; 28 (6):1539-44. [32] Kolesnikova AL, Romanov AE. Circular dislocation-disclination loops and their application to boundary problem solution in the theory of defects. Preprint No 1019. Leningrad: PhysTech Inst. 1986 (in Russian). 18
[33] Kolesnikova AL, Romanov AE. Dislocation and disclination loops in the virtual-defect method. Phys Solid State 2003; 45(9):1706-18. [34] Shih KK, Li JCM. Energy of grain boundaries between cusp misorientations. Surf Sci 1975; 50(1):109-24. [35] Vladimirov VI, Gertsman VYu, Nazarov AA, Romanov AE. Energy of grain boundaries in disclination model. Preprint No 1150. Leningrad: Phys-Tech Inst. 1987 (in Russian). [36] Reddy CD, Rajendran S, Liew KM. Equilibrium configuration and continuum elastic properties of finite sized graphene. Nanotechnology 2006; 17(3):864-70. [37] Berinskii IE, Krivtsov AM. On using many-particle interatomic potentials to compute elastic properties of graphene and diamond. Mech Solids 2010; 45(6):815-34. [38] Yazyev OV and Louie SG. Topological defects in graphene: Dislocations and grain boundaries. Phys Rev B 2010; 81:195420-1-7. [39] Cao A, Qua J. Kapitza conductance of symmetric tilt grain boundaries in graphene. J Appl Phys 2012; 111(5):053529-1-9. [40] Wei Y, Wu J, Yin H, Shi X, Yang R, Dresselhaus M. The nature of strength enhancement and weakening by pentagon–heptagon defects in graphene. Nat Mater 2012; 11(9):759-63. [41] Terrones H, Lv R, Terrones M, Dresselhaus MS. The role of defects and doping in 2D graphene sheets and 1D nanoribbons. Rep Prog Phys 2012; 75(6):062501-1-30.
19
S0008-6223(14)00905-1 http://dx.doi.org/10.1016/j.carbon.2014.09.053 CARBON 9350
To appear in:
Carbon
Received Date: Accepted Date:
25 June 2014 17 September 2014
Please cite this article as: Romanov, A.E., Kolesnikova, A.L., Orlova, T.S., Hussainova, I., Bougrov, V.E., Valiev, R.Z., Non-Equilibrium grain boundaries with excess energy in graphene, Carbon (2014), doi: http://dx.doi.org/ 10.1016/j.carbon.2014.09.053
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NON-EQUILIBRIUM GRAIN BOUNDARIES WITH EXCESS ENERGY IN GRAPHENE A.E. Romanov1,2, A.L. Kolesnikova1,3, T.S. Orlova1,2*, I. Hussainova4, V.E. Bougrov1, R.Z. Valiev1,5 1
ITMO University, Kronverskiy av. 49, St. Petersburg, 197101, Russia
2
Ioffe Physical-Technical Institute, Polytechnicheskaya st. 26, St. Petersburg, 194021, Russia
3
Institute of Problems of Mechanical Engineering, Bolshoj pr. 61, V.O., St. Petersburg, 199178,
Russia 4
Tallinn University of Technology, Ehitajate tee 5, Tallinn, 19086, Estonia
5
Ufa State Aviation Technical University, K. Marx Street 12, Ufa, 450000, Russia
Abstract. Exploring an approach to grain boundary (GB) mesoscopic structure in 3D polycrystals, we develop a model description for equilibrium and non-equilibrium states of GBs in graphene. Non-equilibrium states of GBs in graphene are analyzed for the first time. On the base of this model, dependence of total energy for symmetric GBs on the misorientation angle, , in graphene was calculated. It was found that for the same average misorientation angle the energy in a non-equilibrium state of a GB in graphene can significantly exceed the energy in the equilibrium state. In graphene, the GB energy depends to a greater extent on the deviation from the equilibrium state than on the misorientation angle. It is proposed to account for the presence of non-equilibrium GBs in the explanations of unusual physical properties of graphene.
*Corresponding author. Tel: +7 812 2927157 E-mail address: [email protected]
1. Introduction
Outstanding electronic and mechanical properties of graphene make it a promising material for a wide range of potential applications ranging from nano-electronics and photonics to advanced composites’ nano-fillers. Many applications require fabrication of large-scale graphene sheets, which usually represent polycrystalline layers with numerous grain boundaries (GBs) [1]. Grain boundaries significantly alter physical and mechanical properties of the pristine graphene [1-6]. For example, study by scanning tunneling microscopy (STM) evidenced an order of magnitude lower conductivity within the grain boundary as compared to the defect-free graphene lattice [2]. These STM findings are in agreement with electrical transport measurements across an individual GB and in intra-grains forming the GB [4-6]. On the other hand, graphene GBs can exhibit unique electronic properties, such as ideal 1 D conductivity of some special GBs [3] and inversion of conductivity into n-type at GB in the p-doped graphene [2] that opens potential for new applications exploiting the graphene GBs. It is generally accepted that a contribution of each given grain boundary to the material properties is related to its excess energy per unit area compared to defect-free grain. It is also well-known that the energy of the GB depends on its structure [7,8]. Therefore, studying the structure of GBs and their energy characteristics is of great interest in the direction of exploration of graphene unique properties. Theoretical description of the real GBs structure in graphene is also a challenge for fundamental scientific standpoint because graphene gives practically the first possibility to observe GBs and study their performance as 1D objects in 2D systems. Several research groups have reported on experimental study of GB structure in graphene by the transmission electron microscopy (TEM) [9-11] and STM [2,12]. It was found that GBs tend to be composed mainly of 5- and 7-member carbon atom rings [9,12] in contrast to the ideal honeycomb lattice of graphene consisting of only 6-member rings. In addition, some other atomic configurations were also detected in the arrangement of GB line, for example, distorted hexagons [1,11] and the chains of 8- and 5-member carbon atom rings [3,13]. The later configuration is typical for so-called domain boundaries when there is no misorientation between adjusting grains (domains). The experimentally investigated GBs in polycrystalline CVD (chemical vapor deposition) graphene have tilt character, they are usually neither straight, non periodic and contain a significant amount of disorder [9-11,14,15]. Theoretically, structure and formation energy of variety of the GBs in graphene were investigated by classical simulation techniques using Tersoff- or Brenner-type potentials [16,17] or semi-empirical density-functional tight–binding (DFTB) method [18,19]. Computer simulations of GBs in graphene were based on the energy minimization analysis and, eventually, 2
revealed GBs as classes of periodic grain boundary dislocation walls. In fact, mainly GBs with locally minimal energies were theoretically considered defining such GBs as equilibrium objects. For series of periodic grain boundaries containing mainly uniformly distributed pentagons and heptagons (or uniformly distributed dislocations), the energy versus misorientation angle dependences, , were calculated in [16,17,19-21]. However, real structure of GBs in CVD graphene is in greater extent controlled by kinetic processes of graphene grain nucleation and growth. Graphene GBs can be considered as kinetically “frozen” objects and hence they are strikingly improbable objects from energy minimization viewpoint [22]. As it was identified for metals and alloys (3D polycrystals), the grain boundaries could be in equilibrium or non-equilibrium states with an excess energy depending on the processing conditions [23-25]. Structure of the equilibrium GB in a specific range of misorientations may be described as uniformly distributed arrays of dislocations [26]: such a GB possesses minimal energy for a given misorientation angle. The structure of non-equilibrium GB in metals and alloys consists of non-uniformly distributed dislocations or disclinations and results in storage of an additional energy in the boundary when compared to the energy in equilibrium state for the same angle of misorientation [26,27]. The non-equilibrium grain boundary energy becomes less sensitive to the misorientation angle [27]. Grain boundaries in the two different states are responsible for significantly different mechanical and some physical properties of metals and alloys. For example, non-equilibrium GBs, which are characteristic of ultra-fine grained metals processed by severe plastic deformation methods, provide enhanced strength and outstanding plasticity of these materials [23-25]. In the present study, we develop mesoscopic model description for grain boundary structure in graphene (2D system) and utilize the proposed approach for the analysis of energy of equilibrium and non-equilibrium graphene GB states. Non-equilibrium GBs in graphene are considered for the first time. The proposed mesoscopic model description of GBs in graphene is based on the previously developed approach in the description of non-equilibrium grain boundaries in metals (3D polycrystals) [26,27]. The mesoscopic description of GB structures in graphene allows one to systematically describe symmetric GBs in equilibrium or nonequilibrium states for any misorientation angle and to obtain a simple algorithm for calculation of their energy. The mesoscopic structural modeling of graphene grain boundaries can serve as a starting point for describing their unique physical properties.
3
2. Background. Mesoscopic disclination-structural unit model for grain-boundaries in 3D polycrystals
For description of non-equilibrium state of GBs in metals the disclination-structural unit (DSU) multiscale model was proposed 25 years ago combining both the atomistic (structural unit model) and the mesoscopic (disclination model) scales [26,27]. According to the approach developed for metals [7,28], each favored GB consists of only one structural element (unit), each random GB with a misorientation angle consists of only two structural units belonging to two nearest favored GBs, numerated as “r” and “r+1”, with misorientations and (r=1,2…): (Fig. 1).
Fig. 1. Geometrical model of the tilt grain boundary (GB) with misorientation angle as disclination dipole wall. The nearest favored grain boundaries GBr and GBr+1 with misorientations and are shown. Disclination strength is designated as ; and are the lengths of GBr and GBr+1 structural units A and B, respectively; , are the disclination coordinates; H is the period of GB structure.
4
Since the junctions of structural elements with different misorientations represent disclinations [29], structure of the GB is described as a disclination dipole wall (DDW) (Fig. 1), in which the arm of the dipoles depends on , and it is equal or multiples to or , where and are the lengths of structural elements A and B corresponding to GBr and GBr+1, respectively [26]. In an arbitrary GB the dipoles could be distributed non-uniformly, but periodically with one period , where m and n are the numbers of structural elements forming the period. In the disclination-structural unit model, the average misorientation angle can be determined with quite good accuracy from a simple expression [30]:
ಲ ೝ ಳ ೝశభ ಲ ಳ
(1)
.
In the framework of DSU model, the total energy of the GB can be calculated on the base of three terms [26]: .
(2)
Here is the GB interface (surface) energy, is the elastic energy of the disclination ensemble, and is the energy of disclination cores. The elastic energy of disclination assembling may be calculated as the sum of energies of composing DDWs accounting for disclination interactions. It was shown that is minimal when the structural units (and also disclination dipoles) are distributed along the GB uniformly [27]. Such configuration of the GB with minimal energy refers to the equilibrium state of GB. One of the advantages of the DSU model of GBs [26,27] is a possibility to describe the non-equilibrium state of GB as non-uniformly distributed disclination dipoles [27] and compute the energy and mechanical stresses of such type of GBs. At the atomic level, using molecular dynamics simulation techniques, computing GB energy is difficult and for complex boundary structures is even impossible so far, hence consideration of GBs by simulation methods has been restricted mainly by equilibrium GBs. In the present work, the non-equilibrium GBs in graphene are described on the base of DSU approach previously developed for metals [26,27]. The elastic stresses and boundary energy are estimated taken into consideration the possible structural configurations of the equilibrium and non-equilibrium GBs.
5
3. Results and Discussion
3.1. Disclination-structural unit model of equilibrium and non-equilibrium grain boundaries in graphene
We apply the DSU model developed for metals [26,27] to symmetric GBs in graphene. In two dimensional materials such as graphene, GBs are 1D-interfaces between two grains (crystals) with different crystallographic orientations. In the ideal honeycomb lattice of graphene, the length of arbitrary translation vector , can be easily found using a special basis , (Fig. 2), see, for example, [21]: |, | √ ,
(3)
where √3 0.246 , is a hexagon side.
Fig. 2. Coordinate basis , and arbitrary vector , in graphene lattice. Grain boundary between left (L) and right (R) grains is geometrically possible when the lengths of the translation vectors in the both grains oriented along the GB are equal, i.e. | , | | , |. GB between grains is designed as , | , . For symmetric GB and . The angle between vector , and vector can be determined from [31]: arcsin &
√ √ మ మ
'
.
(4)
The GB misorientation angle can be defined by [31]:
6
( arcsin )
√ ೃ మ మ ೃ ೃ ೃ ೃ
* ( arcsin )
√ ಽ ಽమ ಽ ಽ ಽమ
*.
(5)
Analysing the variety of GBs simulated in [16,19] one can distinguish several structural units from which we can construct all favored and any arbitrary symmetrical GBs in graphene. The structural units A, B (B’), and C are shown in Fig. 3. The favored GBs constructed only by one type unit A, B (B’), or C have misorientation angle 0 , 21.8 , and 60 , respectively [16,19]. The lengths of units A, B (B’), and C are 3, √21, and √3, correspondingly. The length of the structural unit B (B’) is calculated using the Eq.(3) taking into account that the units B (B’) compose the favored GB (2,1)|(1,2) with misorientation angle 21.8 , see, Eq.(5).
Fig. 3. Structural units of graphene grain boundaries (GBs). A is the structural unit of GB with misorientation angle 0 (perfect crystal); B and B’ are the equivalent structural units of favored GB with 21.8 , C is the structural unit of favored GB with 60 (perfect crystal); dA, dB, and dC are the lengths of A, B, and C along the favored GBs.
Using the structural units (A, B (B’), and C) of the favored GBs in graphene, any geometrical configuration for arbitrary GB can be constructed. First, we built GBs with the uniform distribution of structural elements (equilibrium GBs) and the misorientation in the range 0 – 600 which is divided by the favored GBs into two intervals 0 21.8 and 21.8 60 . In Fig. 4 some symmetrical GBs of graphene are demonstrated. It should be noted that in graphene the favored GB with 21.8 can be constructed with two equivalent units B or B’ with the same misorientation and length (Fig. 4). An arbitrary equilibrium GB is constructed by two structural units A and B, or B’ and C which are uniformly distributed. We believe that the structural units (A, B (B’), and C) cover all possible symmetric GB structural motives in 7
graphene because they permit us to build all known GB structures found by MD method [16, 19]. Basing on the ideas presented in [27] we construct some non-equilibrium GBs with different misorientations in graphene. For a chosen average misorientation angle non-equilibrium state is achieved in two different ways. First, a deviation from equilibrium state could be reached by increasing the period H of grain boundary structure keeping the same types and numbers of structural units (Fig. 5,a). Second, a deviation from the uniform distribution of structural elements is caused by rearrangement of the structural units appropriately (Fig. 5,b).
Fig. 4. Equilibrium symmetrical tilt GBs in graphene, composed with structural units A, B (B’), and C. Misorientation angles 0 , 21.8 , and 60 correspond to favored GBs: GB1, GB2, and GB3. The formula for GB period H is given, for example, AAB ( 9.4 ), B’B’C ( 27.8 ) etc. In the frame of the disclination-structural unit approach one can determine the disclination content of GBs in graphene. Examples of disclination-structural unit description of equilibrium and non-equilibrium GBs from the both intervals 0 21.8 and 21.8 60 in graphene are given in Fig. 6. 8
Fig. 5. Simple examples of equilibrium and non-equilibrium grain boundaries with misorientation angles (a) 13.2 and (b) 35.6 .
Fig. 6. Disclination-structural unit (DSU) model of equilibrium and non-equilibrium GBs in graphene. The GB misorientation angle is in the range: (a) 0 13.2 21.8 ; (b) 21.8 35.6 60 . 9
3.2. Energies of equilibrium and non-equilibrium grain boundaries in graphene In the framework of DSU model the total energy of GBs in graphene is calculated from Eq. (2) taking into account 2D hexagonal structure of graphene lattice and 1D nature of GBs in graphene. For GBs with misorientation angle 0 0 (1 1,2): the interface energy per length unit can be written as following:
ಲ భ ಳ మ ಲ ಳ ಳ మ య ಳ
for GBs with 0 0 0 21.8 ,
(6a)
for GBs with 21.8 0 0 60 ,
(6b)
where m and n are the numbers of the structural units in the period (or ) of GB; dA, dB, and dC are the lengths of structural units A, B, and C (see Fig. 3); 2 , 2 and 2 are the interface energies per unit length of the favored GBs (Fig. 4). Graphene containing GBs with misorientation angles 0 and 60 are the perfect crystals (Fig. 4), therefore 2 2 0. Interface energy 2 .!బ . It is well-known that hexagonal crystalline plane is an isotropic plane, so the elastic energy of GB Eelastic , see Eq. (2), can be calculated in the framework of linear isotropic elasticity. The energy is the energy of disclination ensemble in 2D medium and it can be estimated on the base of the elastic fields of the wedge disclinations or the edge dislocations perpendicular to the free surfaces of an infinitesimally thin plate [32,33]. It was demonstrated in [32] that in an infinite thin film plane strain state characteristic for edge dislocations and wedge disclinations transforms to plane stress state. In a result, the stresses of the periodically distributed disclination dipoles (see Fig. 1) take the following form:
3""
#$%&' 5
5
(
4 ln &
)*+, "-.)*+ $/-./- .0& )*+, "-.)*+ $/-./- &
"- +12, ")*+, "-.)*+ $/-./- &
(
'+
"- +12, ")*+, "-.)*+ $/-./- .0&
7,
10
(7a)
3//
#$%&' 5 (
5(
4 ln &
)*+, "-.)*+ $/-./- .0& )*+, "-.)*+ $/-./- &
"- +12, ")*+, "-.)*+ $/-./- &
'-
"- +12, ")*+, "-.)*+ $/-./- .0&
7,
(7b)
333 0, 3"/
(7c)
#$%&'"(
3"3 0,
4
+12 $/-./- & )*+, "-.)*+ $/-./- &
(
+12 $/-./- .0& )*+, "-.)*+ $/-./- .0&
7,
(7d)
33/ 0,
(7e,f)
where disclination strength ∆ ( (1 1,2); 9: ;9/ , :$& ; $& / , (0, $& ) are the disclination coordinates (Fig. 1), = is the arm of dipole in the wall; H is a period of DDW, > is shear modulus; ν is Poisson’s ratio. Taking into account Eqs. (7) the energy can be represented [27] for definiteness in the case of ? by the following expression:
.
>1 @
)A= B B CAD : ( : =E ( 2AD : ( : E AD : ( : ( =EF* 32; 4 4
1 1,2,
(8)
where = ; / for GBs with 0 21.8 and = ; / for GBs with 21.8 60 . Functional A is taken as [34]:
6
AG 2 H sin I I H
5 5 )*+, 5.)*+
.
(9)
For easy of calculation A can be rewritten: AG 2C(Li D2 . E ( Li D2 E 2K3F,
(10)
6 . where polylogarithm Li L ∑6 4 L / , and Riemann zeta-function K3 ∑ 4 .
For graphene hexagonal lattice we can write the last term in Eq.(2) as following [35]:
11
7$%&8#9మ ∆ మ ;(య <
.
(11)
Here is a disclination core parameter, and N . Parameter can be taken in the range of 1-3 [26]. The value of affects the magnitude of GB energy and can be considered as a fitting parameter of the disclination structural unit model. In Fig. 7, the calculated with the help of Eqs.(2,8,11) grain boundary energies E are plotted as the function of misorientation angle for the cases of equilibrium (filled blue circles) and non-equilibrium (green triangles) GBs. The calculations utilize the following set of parameters: interface energy .!బ 3.44 eV/nm [19], shear modulus > 80.15 N/m [36], the Poisson’s ratio @ 0.416 [36], lattice parameter 0.246 nm [21]; disclination core parameter 1. This value for was taken to approximately fit the energies of GBs calculated by different methods [19] in the interval of misorientations 12 20 . The elastic modules > and @ were chosen from a set of elastic modules of graphene found by several authors and analyzed in detail in [37]. Note that the elastic modules we use have been calculated by computer simulation with application of Brenner-type II potential [36]. It has been shown that these modules are in good agreement with those calculated by other methods, see [37].
Fig. 7. Grain boundary energy versus misorientation angle in graphene for equilibrium (filled blue circles) and non-equilibrium (green triangles) GB. Plots are calculated at the following parameters: interface energy E21.80 = 3.44 eV/nm [19], shear modulus > 80.15 N/m [36], the Poisson’s ratio @ 0.416 [36], lattice parameter 0.246 nm [21]; disclination core parameter 1. Dashed line and empty red circles correspond to results from Ref. [16] and [19], respectively.
12
In the case of equilibrium GBs, grain boundary energy was calculated for different GB misorientations in the whole interval of GB misorientation angles 0 0 0 60 (Fig. 7). There is a satisfactory agreement between the -dependence obtained in the present work for the case of equilibrium GBs (filled blue circles) at 0 0 0 32.2 and the previous results [16,19]. This proves the validity of the DSU approach for characterization of GBs in graphene. In the interval 32.2 0 0 60 our -dependence differs substantially from the plot given in Ref. [16] (Fig. 7). Unlike the nearly inverted parabolic -curve in [16] our results demonstrate the -dependence which is far from to be bell-shaped: there are two cusps on curve at misorientation angles of 21.80 and 32.20 and the energy values in the interval 32.2 60 are larger those in the interval 0 32.2 . However, these findings are consistent with other data sets [17,38] demonstrating similar asymmetric behavior of dependence for GB in graphene. The two cusps on curve at misorientation angles of 21.790 and 32.210 were shown in Ref. [17] on studying the GB structures and energies by atomistic simulations. Yazev and Louie [38] investigated symmetrical periodic GBs in graphene based on dislocation theories and performed the first-principle calculation of the grain boundary energies, which also demonstrated the presence of such cusps on the dependence and clearly larger energies in the interval 32.2 50 compared to those in the interval 10 32.2 . In the structure-unit grain boundary description (Fig. 4) it is evident that despite the GBs with misorientation angles 0 and 60 belong to the perfect crystal, they consist of different structural units. Namely, GB with 0 is constructed by the structural units A oriented parallel to the armchair graphene direction [1], whereas GB with 60 consists of the structural units C oriented parallel to zigzag graphene direction [1]. In addition, the elements
A and C are of different lengths. As a result, the GBs with misorientation angle from the interval 0 21.8 , which are constructed by A and B units, and the GBs with misorientation angle (60 ( ) from the interval 21.8 60 , which are constructed by B and C units, are structurally different, hence they should have different structure-dependent energy characteristics. Available literature data [1,39,40] also indicate that for about the same misorientation angle, properties of GB depend on its crystallographic orientation in pristine (defect-free) graphene. It was shown by MD simulations that symmetric tilt GB between two zigzag-oriented domains in graphene bicrystal is more efficient for heat transmitting [39] and have higher strength [40] than such GB between two armchair-oriented domains with about the same misorientation. Moreover, strength of pristine graphene stretching in zigzag direction is higher
13
than in armchair direction [40]. The graphene sheet’s overall electronic and magnetic properties are strongly dependent on the type (armchair or zigzag) of graphene sheet edges [41]. With use of the disclination-structural unit approach we also calculated energy of nonequilibrium GBs for some misorientations and different extent of structure disordering (green triangles in Fig. 7). Energies of non-equilibrium grain boundary states were calculated for the following misorientation angles: 11 , 13.2 , 32.2 . Non-equilibrium states of GBs for a given misorientation angle were built by changing the order of structural units and (or) by increasing the GB period. All of these transformations lead to a reduction of uniformity of the structural unit distribution in the grain boundary period. For 11 the equilibrium grain boundary ABABA became a non-equilibrium GB under permutation of structural units in the period (grain boundary 3А2В) or increase of the period (grain boundary 6A4B); for 13.2 the equilibrium grain boundary AB transformed into non-equilibrium boundaries 4A4B and 5A5B; for the misorientation angle 32.2 the grain boundary B'C transformed in nonequilibrium boundary 2B'2C. It is evident that the total grain boundary energy in a nonequilibrium state can greatly exceed that of the equilibrium state for the same misorientation angle. Simple explanation of this fact can be found in the increase of disclination dipole arm (and therefore the energy) for GBs with longer periods. Another important finding is the absence of a pronounced dependence of non-equilibrium boundary energy on the misorientation angle. Indeed, our calculations show that in the case of the non-equilibrium state of the grain boundaries in graphene, the energy of low- and mediumangle grain boundaries can be as high as the energy of high-angle grain boundary (Fig. 7). This well agrees with experimental results [2] according to which electronic properties of GBs in graphene are robust against their misorientation angle. In order to illustrate the universal nature and robustness of the observed electronic behavior of GBs in CDV graphene, Topaszto et al. [2] demonstrated similarity of the conductivity map and charge doping profiles for three distinct GBs with tilt angles of 270, 210 and 120 comprising the same triple junction. This most probably suggests that grain boundaries in the CVD polycrystalline graphene are in non-equilibrium state providing less sensitivity of GB properties to the misorientation angle. The considered nonuniform structure of non-equilibrium GBs in graphene corresponds in more extent to experimentally observed GBs containing disorder [9-11] than the simulated GB models obtained with the GB energy minimization [16,17,19]. The DSU model of non-equilibrium GBs in graphene can be proposed for description of outstanding properties of graphene GBs.
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4. Conclusions
In the present work, the structure of symmetric grain boundaries (GBs) in graphene has been studied on the base of disclination-structural unit (DSU) approach originally developed for metals. This approach allows considering both equilibrium (energy-minimized for a given misorientation angle) as well as non-equilibrium states of the GBs with the same given misorientation. The non-equlibrium GBs in graphene have been analyzed for the first time. In accordance with the DSU approach, non-equlibrium GB structures in graphene were introduced as non-uniform distribution of structural units, which can be described at mesoscale as disordered network of disclination dipoles. The elastic stresses and energy in the whole interval of GB misorientation angles 0 0 0 60 have been calculated. The obtained dependence for equilibrium GBs is in good agreement with similar dependences found by direct simulation methods for the misorientation angle range of 0 0 0 32.2 . The obtained - dependence in the whole interval 0 0 0 60 is substantially nonbell-shaped that is associated with structural differences of the GBs with и (60 ( ) and agrees well with literature data on orientation-dependent properties of a GB in graphene lattice. It has been demonstrated that the energy of non-equilibrium GB in graphene can substantially exceed the energy of equilibrium GB with the same misorientation angle. The energy of non-equilibrium GB is highly affected by degree of a structural unit distribution irregularity and to much lesser extent by average misorientation angle in the GB. The proposed model of non-equilibrium state of GBs in graphene is along with the experimental findings, demonstrating substantial disordering in GB atomic structures and robustness of GB electronic properties against the misorientation angles. Therefore, we suggest that CBs in CVD polycrystalline graphene could be in non-equilibrium states depending on the processing regimes. The proposed disclination-structural unit model of non-equilibrium GBs in graphene can serve a key structural platform for the description of physical and mechanical properties of graphene GBs, including unique misorientation angle-independent electronic properties of GBs in CVD grown graphene. Experimental studies of GBs in graphene also evidence on the formation of irregular structures, see for example [9,10]. Meandering of grain boundaries at micro- and nanoscale is related to the formation of facets with different crystallographic orientation. This can be typical non-symmetric grain boundaries. Then, each facet can be a section of a symmetric boundary. The energy of such a boundary can be either higher or lower of the straight linear boundary with the same in average orientation. In this case, the facet junctions will also demonstrate
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disclination counterparts. The consideration of these complex configurations in the framework of the disclination approach is the subject of the ongoing research.
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References
[1] Biro LP, Lambin Ph. Grain boundaries in graphene grown by chemical vapor deposition. New J Phys 2013; 15:1-38. [2] Tapaszto L, Nemes-Incze P, Dobrik G, K. Yoo Jae, Hwang C, Biro LP. Mapping the electronic properties of individual graphene grain boundaries. Appl Phys Lett 2012; 100(5):053114-1-4. [3] Lahiri J, Lin Y, Bozkurt P, Oleynik I I, Batzill M. An extended defect in graphene as a metallic wire. Nature Nanotechnol 2010; 5(5):326-9. [4] Kim P. Graphene: Across the border. Nat Mater 2010; 9(10):792-3. [5] Yu Q, Jauregui LA, Wu W, Colby R, Tian J, Su Z, et al. Control and characterization of individual grains and grain boundaries in graphene grown by chemical vapour deposition. Nat Mater 2011; 10(6):443-9. [6] Jauregui LA, Cao H, Wu W, Yu Q, Chen YP. Electronic properties of grains and grain boundaries in graphene grown by chemical vapor deposition. Solid State Commun 2011; 151(16):1100-4. [7] Sutton AP, Balluffi RW. Interfaces in crystalline materials. Oxford: Oxford University Press. 1995. [8] Bulatov VV, Reed BW, Kumar M. Grain boundary energy function for fcc metals. Acta Materialia 2014; 65:161-75. [9] Huang PY, Ruiz-Vargas CS, van der Zande AM, Whitney WS, Levendorf MP, Kevek JW, et al. Grains and grain boundaries in single-layer graphene atomic patchwork quilts. Nature 2011; 469(7330):389-92. [10] Kim K, Lee Z, Regan W, Kisielowski C, Crommie MF, Zett A. Grain boundary mapping in polycrystalline graphene. ACS Nano 2011; 5(3):2142-6. [11] Kim KS, Zhao Y, Jang H, Lee SY, Kim JM, Kim KS, et al. Large-scale pattern growth of graphene films for stretchable transparent electrodes. Nature 2009; 457(7230):706-10. [12] Simonis P, Goffaux C, Thiry PA, Biro LP, Lambin P, Meunier V. STM study of a grain boundary in graphite. Surf. Sci 2002; 511(1-3):319-22. [13] Pelc M, Chico L, Ayuela A, Jaskolski W. Grain boundaries with octagonal defects in graphene nanoribbons and nanotubes. Phys. Rev. B 2013; 87:165427-1-7. [14] Gao L, Guest JR, Guisinger NP. Epitaxial Graphene on Cu(111). Nano Lett 2010; 10(9):3512-6. [15] Zhang Y, Gao T, Gao Y, Xie S, Ji Q, Yan K, et al. Defect-like structures of graphene on copper foils for strain relief investigated by high-resolution scanning tunneling microscopy. ACS Nano 2011; 5(5):4014-22. 17
[16] Liu Y, Yakobson BI. Cones, pringles, and grain boundary landscapes in graphene topology. Nano Lett 2011; 10:2178-83. [17] Liu T-H, Gajewski G, Pao C-W, Chang C-C. Structure, energy, and structural transformations of graphene grain boundaries from atomistic simulations. Carbon 2011; 49(7):2306–17. [18] Malola S, Hakkinen H, Koskinen P. Structural, chemical, and dynamical trends in graphene grain boundaries. Phys Rev B 2010; 81(16):165447-1-6. [19] Zhang J, Zhao J. Structures and electronic properties of symmetric and nonsymmetric graphene grain boundaries. Carbon 2013; 55(1):151-9. [20] Yazyev OV. Polycrystalline graphene: Atomic structure, energetics and transport properties. Solid State Commun 2012; 152(15):1431-6. [21] Yazyev OV, Louie SG. Electronic transport in polycrystalline graphene. Nat Mater 2010; 9(10):806-9. [22] Yakobson BI and Ding F. Observational geology of graphene, at the nanoscale. ACS NANO 2011; 5(3):1569-74. [23] Valiev RZ, Islamgaliev RK, Alexandrov IV. Bulk nanostructured materials from severe plastic deformation. Prog. Mater. Sci. 2000; 45:103-89. [24] McFadden SX, Mishra RS, Valiev RZ, Zhilyaev AP, Mukherjee AK. Low-temperature superplasticity in nanostructured nickel and metal alloys. Nature 1999; 398(6729):684-6. [25] Valiev RZ. Nanostructured alloys: Large tensile elongation. Nat Mater 2013; 2(4):289-91. [26] Gertsman VV, Nazarov AA, Romanov AE, Valiev RZ, Vladimirov VI. Disclinationstructural unit model of grain boundaries. Phil Mag A 1989; 59(5):1113-8. [27] Nazarov AA, Romanov AE, Valiev RZ. On the structure, stress fields and energy of nonequilibrium grain boundaries. Acta Metal Mater 1993; 41(4):1033-40. [28] Sutton AP, Vitek V. On the structure of tilt grain boundaries in cubic metals I. Symmetrical tilt boundaries. Phil Trans R Soc London A 1983; 309(1506):1-36. [29] Romanov AE, Kolesnikova AL. Application of disclination concept to solid structures. Prog Mater Sci 2009; 54(6):740-69. [30] Nazarov AA, Romanov AE. On average misorientation angle of general tilt boundaries. Phil Mag Lett 1989; 60(5):187-93. [31] Wu J-T, Shi X-H, Wei Y-J. Tunable band structures of polycrystalline graphene by external and mismatch strains. Acta Mechanica Sinica 2012; 28 (6):1539-44. [32] Kolesnikova AL, Romanov AE. Circular dislocation-disclination loops and their application to boundary problem solution in the theory of defects. Preprint No 1019. Leningrad: PhysTech Inst. 1986 (in Russian). 18
[33] Kolesnikova AL, Romanov AE. Dislocation and disclination loops in the virtual-defect method. Phys Solid State 2003; 45(9):1706-18. [34] Shih KK, Li JCM. Energy of grain boundaries between cusp misorientations. Surf Sci 1975; 50(1):109-24. [35] Vladimirov VI, Gertsman VYu, Nazarov AA, Romanov AE. Energy of grain boundaries in disclination model. Preprint No 1150. Leningrad: Phys-Tech Inst. 1987 (in Russian). [36] Reddy CD, Rajendran S, Liew KM. Equilibrium configuration and continuum elastic properties of finite sized graphene. Nanotechnology 2006; 17(3):864-70. [37] Berinskii IE, Krivtsov AM. On using many-particle interatomic potentials to compute elastic properties of graphene and diamond. Mech Solids 2010; 45(6):815-34. [38] Yazyev OV and Louie SG. Topological defects in graphene: Dislocations and grain boundaries. Phys Rev B 2010; 81:195420-1-7. [39] Cao A, Qua J. Kapitza conductance of symmetric tilt grain boundaries in graphene. J Appl Phys 2012; 111(5):053529-1-9. [40] Wei Y, Wu J, Yin H, Shi X, Yang R, Dresselhaus M. The nature of strength enhancement and weakening by pentagon–heptagon defects in graphene. Nat Mater 2012; 11(9):759-63. [41] Terrones H, Lv R, Terrones M, Dresselhaus MS. The role of defects and doping in 2D graphene sheets and 1D nanoribbons. Rep Prog Phys 2012; 75(6):062501-1-30.
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