kinetics of atomic ordering in metal-doped graphene

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Kinetics of atomic ordering in metal-doped graphene

Taras M. Radchenko*, Valentyn A. TatarenkoDepartment of Solid State Theory, Institute for Metal Physics, N.A.S.U., 36 Acad. Vernadsky Boulevard, UA-03680 Kyiv-142, Ukraine

a r t i c l e i n f o

Article history:Received 11 February 2009Received in revised form16 May 2009Accepted 26 May 2009Available online 6 June 2009

Keywords:Graphene-type latticeMetal-doped grapheneOrdering kinetics

a b s t r a c t

Possible stably ordered substitutional structures based on a graphene-type crystal lattice are considered.A kinetic model of atomic ordering in metal-doped graphene with stoichiometric (1/8, 1/4, 1/2) andnonstoichiometric compositions is developed. Inasmuch as the intrasublattice and intersublattice‘interchange’ (‘mixing’) energies are competitively different for graphene-based lattice, kinetic curves ofthe long-range order (LRO) parameters may be nonmonotonic for the structures described by two orthree LRO parameters.

� 2009 Elsevier Masson SAS. All rights reserved.

1. Introduction

Grapheneda one-atom-thick layer modification of carbonrecently discovered in the free state [1,2]dis a hot-topic object inboth materials science and condensed matter physics, where it isa popular model system for investigations. Graphene is the basicstructural element of graphite lattice. Crystal lattice of graphene isa nanoscale structure (so-called two-dimensional carbon), whereatoms are distributed over the vertexes of regular hexagons, asshown in Fig. 1a. Due to a high mechanical strength, hardness,heat conductivity [3], and electrical conductivity, graphene isa perspective material for a wide application in the differentfields: from nanoelectronics (graphene will be its basis [4,5],substituting silicon in the integrated circuit chips) to coating ofairliner fuselage.

Probably, graphene doping with metal (Me) atoms mayimprove some of its physical properties (for a wider range ofapplication). Particularly, the doping with a metal changes theband structure (which strongly depends on atomic order) and,consequently, improves an electrical conductivity of graphene[2,6–20].

This work is focused on the construction of both statisticalthermodynamic and kinetic models for long-range atomic order ina metal-doped graphene, i.e. in a two-dimensional substitutionalsolid solution C–Me based on a graphene-type crystal lattice.

2. Model

Since the graphene-type lattice is a two-dimensional ‘honey-comb’-crystal lattice consisting of two interpenetrating hexagonalsublattices, as shown in Fig. 1a, its reciprocal lattice is a two-dimen-sional hexagonal lattice as well (Fig. 1b). Nearest-neighbour distance(between C atoms) in hexagons (Fig. 1a) is a0 z 0.142 nm [6]. It isconveniently to consider a graphene-type lattice as consisting of twointerpenetrating hexagonal sublattices displaced with respect to eachother by the vector h¼ a1/3þ 2a2/3, where a1 and a2 are the funda-mental translation vectors of a lattice along the [1 0] and [0 1] direc-tions, respectively, in the oblique system of coordinates (see Fig. 1a).Lattice translation parameter is a¼ ja1j ¼ ja2j ¼

ffiffiffi3p

a0 y 0.246 nm. Asshown in Fig.1a, each ABCD primitive unit cell contains two sites. Eachlattice site location can be described by a sum of two vectors:Rþ hq¼ r (Fig. 1a). Vector R denotes an ‘origin’ position of the unitcell. Vector hq denotes the distance of a given site with respect to theunit cell ‘origin’, and q subscript numerates the sublattice (q¼ 1, 2).The radius-vector R is related to the basis vectors as R¼ n1a1þ n2a2,where n1, n2 are integers.

Let us consider possible stably ordered (super)structures of two-dimensional substitutional C–Me solid solution based on the gra-phene-type lattice with superstructural stoichiometric C3Me, C7Me,CMe compositions (Figs. 2–4). Single-site occupation-probabilityfunctions for these (super)structures are obtained using the staticconcentration waves’ approach [21] and presented in Table 1, wherePq(R) is the probability to find a Me atom at the (q, R) site, i.e. at thesite of q-th sublattice within the unit cell with ‘origin’ R and ha

s

(s¼ 0, 1 or 2) are the LRO parameters (a index denotes their totalnumber for a given structure; a¼ I, II or III).

* Corresponding author. Tel.: þ380 44 424 12 21; fax: þ380 44 424 25 61.E-mail address: [email protected] (T.M. Radchenko).

Contents lists available at ScienceDirect

Solid State Sciences

journal homepage: www.elsevier .com/locate/ssscie

1293-2558/$ – see front matter � 2009 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.solidstatesciences.2009.05.027

Solid State Sciences 12 (2010) 204–209

https://www.researchgate.net/publication/236390552_Theory_of_Structural_Transformation_in_Solids_Ch_11?el=1_x_8&enrichId=rgreq-3754e5ab-a91d-4dc0-8372-5d6f11db5d0d&enrichSource=Y292ZXJQYWdlOzI0MTQ3MDg3NjtBUzoxMDE5OTMzNTE3NDU1NTJAMTQwMTMyODUxODk2OQ==


Fig. 1. The real-space lattice of graphene (a) and its reciprocal space (b). Here, (a) ABCD is a primitive unit cell, a1 and a2 are the basis translation vectors of the lattice, a is the latticetranslation parameter, a0 is a distance between the nearest-neighbour sites, circles denote the first three coordination shells with respect to the ‘origin’ (at site A) of the obliquecoordinate system; (b) the first Brillouin zone (G, M, K are its high-symmetry points), a*

1 and a*2dthe basis translation vectors of reciprocal lattice.

Fig. 2. Primitive unit cells of graphene-based C3Me superstructures described by (a) one, (b) two or (c) three LRO parameters.

T.M. Radchenko, V.A. Tatarenko / Solid State Sciences 12 (2010) 204–209 205


Interatomic interactions in C–Me lattice can be taken intoconsideration by means of the ‘mixing’ (‘interchange’) energies [21]:

wpqðR � R0ÞhWCCpq ðR � R0Þ þWMeMe

pq ðR � R0Þ � 2WCMepq ðR � R0Þ:

Here, p and q subscripts number the sublattices, where corre-sponding atoms can be distributed; WCC

pq ðR � R0Þ, WMeMEpq ðR � R0Þ,

and WCMepq ðR � R0Þ are the pair-wise interaction energies of C–C,

Me–Me, C–Me pairs of atoms, respectively, located at the sites of p-th and q-th (p, q¼ 1, 2) sublattices within the unit cells with origins(‘zero’ sites) at R and R0 sites.

For a statistical-thermodynamic description of the arbitrary-range interatomic interactions (i.e. in all coordination shells), it isconvenient to apply the Fourier transformations for the elements ofthe ‘mixing’-energy matrix [21],

k ~wpqðkÞkh

~w11ðkÞ ~w12ðkÞ~w*

12ðkÞ ~w11ðkÞ

!; where

~wpqðkÞhX

R

wpqðR � R0Þe�ik,ðR�R0Þ:

Here, k is a wave vector of a two-dimensional reciprocalspace (Fig. 1b), which ‘generates’ corresponding (super)structure;~w*

12ðkÞ is a complex conjugate to ~w12ðkÞ. Writing Hermitian

Fig. 3. Primitive unit cell of graphene-based C7Me-superstructure described by threeLRO parameters.

Fig. 4. Primitive unit cells of graphene-based CMe-superstructures described by one LRO parameter.

T.M. Radchenko, V.A. Tatarenko / Solid State Sciences 12 (2010) 204–209206




Table 1Single-site occupation probabilities for dopant Me atoms in graphene-based (super)structures.

(Super)structure Probability function

C3Me (Fig. 2a)�

P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

4hI2

h� 1�1

�cosðpn1Þ þ

�11

�cosðpn2Þ þ

� 1�1

�cosðpðn1 � n2ÞÞ

i

C3Me (Fig. 2b)�

P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

4

hhII

1

�11

�cosðpn1Þ þ hII

2

�11


1

�11


i,�

P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

4

hhII

1

�11


1

� 1�1


2

� 1�1


i,

�P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

4

hhII

2

� 1�1


1

� 1�1


1

�11


i

C3Me (Fig. 2c)�

P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

4 hIII0

� 1�1

�þ 1

4

hhIII

1

�11

�þ hIII

2

� 1�1

�icosðpðn1 � n2ÞÞ,�

P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

4hIII0

� 1�1

�þ 1

4

hhIII

1

�11

�þ hIII

2

� 1�1

�icosðpn1Þ,�

P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

4hIII0

� 1�1

�þ 1

4

hhIII

1

� 1�1

�þ hIII

2

�11

�icosðpn2Þ

C7Me (Fig. 3)�

P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

8hIII0

� 1�1

�þ 1

8hIII1

h�11

�cosðpn1Þ þ

� 1�1

�cosðpn2Þ þ

�11


i

þ18

hIII2

h� 1�1

�cosðpn1Þ þ

�11

�cosðpn2Þ þ

� 1�1


i

CMe (Fig. 4a)�

P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

2hI1

�11

�cosðpðn1 � n2ÞÞ,�

P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

2hI1

�11

�cosðpn1Þ;

�P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

2hI1

� 1�1

�cosðpn2Þ

CMe (Fig. 4b)�

P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

2hI2

� 1�1

�cosðpðn1 � n2ÞÞ,�

P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

2hI2

� 1�1

�cosðpn1Þ;

� P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

2hI2

�11

�cosðpn2Þ

CMe (Fig. 4c)�

P1ðRÞP2ðRÞ

�¼ c

�11

�þ 1

2hI0

� 1�1

�

Table 2Kinetic equations for LRO parameter(s) of different graphene-based (super)structures.

(Super)structure Differential kinetics equation

C3Me (Fig. 2a) dhI2

dt� ¼ ð1� cÞ"

hI2

T�� lnðcþ 3hI

2=4Þð1� cþ hI2=4Þ

ð1� c� 3hI2=4Þðc� hI

2=4Þ

#

C3Me (Fig. 2b) dhII1

dt� ¼ cð1� cÞ"

hII1

T�� lnðcþ hII

1=2þ hII2=4Þð1� cþ hII

1=2� hII2=4Þ

ðc� hII1=2þ hII

2=4Þð1� c� hII1=2� hII

2=4Þ

#,

dhII2


hII2

T�� ln

ðcþ hII1=2þ hII

2=4Þðc� hII1=2þ hII

2=4Þð1� cþ hII2=4Þ2

ð1� c� hII1=2� hII

2=4Þð1� cþ hII1=2� hII

2=4Þðc� hII2=4Þ2

#

C3Me, (Fig. 2c) dhIII0


hIII0

T�� ln

ðcþ ðhIII

0 þ hIII1 þ hIII

2 Þ=4Þðcþ ðhIII0 � hIII

1 � hIII2 Þ=4Þ

ðcþ ð�hIII0 þ hIII

1 � hIII2 Þ=4Þðcþ ð�hIII

0 � hIII1 þ hIII

2 Þ=4Þ�ð1� c� ð�hIII

0 þ hIII1 � hIII

2 Þ=4Þð1� c� ð�hIII0 � hIII

1 þ hIII2 Þ=4Þ

ð1� c� ðhIII0 þ hIII

1 þ hIII2 Þ=4Þð1� c� ðhIII

0 � hIII1 � hIII

2 Þ=4Þ

!#,

dhIII1


hIII1

T�� ln

ðcþ ðhIII

0 þ hIII1 þ hIII

2 Þ=4Þðcþ ð�hIII0 þ hIII

1 � hIII2 Þ=4Þ

ðcþ ðhIII0 � hIII


0 � hIII1 þ hIII

2 Þ=4Þ�ð1� c� ðhIII


2 Þ=4Þð1� c� ð�hIII0 � hIII

1 þ hIII2 Þ=4Þ


1 þ hIII2 Þ=4Þð1� c� ð�hIII

0 þ hIII1 � hIII

2 Þ=4Þ

!#

dhIII2


hIII2

T�� ln

ðcþ ðhIII

0 þ hIII1 þ hIII

2 Þ=4Þðcþ ð�hIII0 � hIII

1 þ hIII2 Þ=4Þ



0 þ hIII1 � hIII

2 Þ=4Þ�ð1� c� ðhIII


2 Þ=4Þð1� c� ð�hIII0 þ hIII

1 � hIII2 Þ=4Þ


1 þ hIII2 Þ=4Þð1� c� ð�hIII

0 � hIII1 þ hIII

2 Þ=4Þ

!#

C7Me (Fig. 3) dhIII0


hIII0

T�� ln

ðcþ ðhIII

0 þ 3hIII1 þ 3hIII

2 Þ=8Þðcþ ðhIII0 � hIII

1 � hIII2 Þ=8Þ3

ð1� c� ðhIII0 þ 3hIII

1 þ 3hIII2 Þ=8Þð1� c� ðhIII


2 Þ=8Þ3�ð1� c� ð�hIII

0 � 3hIII1 þ 3hIII

2 Þ=8Þð1� c� ð�hIII0 þ hIII

1 � hIII2 Þ=8Þ3

ðcþ ð�hIII0 � 3hIII

1 þ 3hIII2 Þ=8Þðcþ ð�hIII

0 þ hIII1 � hIII

2 Þ=8Þ3

!#,

dhIII1


hIII1

T�� ln

ðcþ ðhIII


2 Þ=8Þðcþ ð�hIII0 þ hIII

1 � hIII2 Þ=8Þ

ðcþ ð�hIII0 � 3hIII

1 þ 3hIII2 Þ=8Þðcþ ðhIII


2 Þ=8Þ�ð1� c� ð�hIII

0 � 3hIII1 þ 3hIII

2 Þ=8Þð1� c� ðhIII0 � hIII

1 � hIII2 Þ=8Þ

ð1� c� ðhIII0 þ 3hIII

1 þ 3hIII2 Þ=8Þð1� c� ð�hIII

0 þ hIII1 � hIII

2 Þ=8Þ

!#,

dhIII2


hIII2

T�� ln

ðcþ ðhIII


2 Þ=8Þðcþ ð�hIII0 � 3hIII

1 þ 3hIII2 Þ=8Þ



0 þ hIII1 � hIII

2 Þ=8Þ�

ð1� c� ðhIII0 � hIII

1 � hIII2 Þ=8Þð1� c� ð�hIII

0 þ hIII1 � hIII

2 Þ=8Þð1� c� ðhIII


2 Þ=8Þð1� c� ð�hIII0 � 3hIII

1 þ 3hIII2 Þ=8Þ

!#

CMe (Fig. 4a) dhI1


hI1

T�� ln

ðcþ hI


ðc� hI1=2Þð1� c� hI

1=2Þ

!#

CMe (Fig. 4b) dhI2


hI2

T�� ln

ðcþ hI



2=2Þ

!#

CMe (Fig. 4c) dhI0


hI0

T�� ln

ðcþ hI



0=2Þ

!#



‘mixing’-energy matrix, the symmetry relations, ~w11ðkÞ ¼ ~w22ðkÞand ~w21ðkÞ ¼ ~w*

12ðkÞ, are taken into account.The ‘mixing’ energies and corresponding eigenvalues of the

‘mixing’-energy matrix k~wpqðkÞk, l1ðkÞ ¼ ~w11ðkÞ þ j ~w12ðkÞj, andl2ðkÞ ¼ ~w11ðkÞ � j~w12ðkÞj, entering into expressions for the config-urational free energy, which can be obtained within the self-consis-tent field approximation [21], define the statistical thermodynamicsand kinetic behaviour of the doped graphene-based structures.

Substitution of the functions from Table 1 into the configura-tional Helmholtz free-energy functional [21],

Fy12

X2

p;q¼1

XR;R0

wpqðR � R0ÞPpðRÞPqðR0Þ þ kBTX2

q¼1

�XR0

�PqðR0Þln PqðR0Þ þ

�1� PqðR0Þ

ln�1� PqðR0Þ

(T is temperature, kB is the Boltzmann constant), and simplemathematical transformations yields expressions [22] for theconfigurational free energy (per atom) of the (super)structures,which are ‘generated’ high-symmetry point wave vectors andstable against the antiphase shifts (Figs. 2–4).

Within the model for the long-range atomic order kinetics, weconsider the case of exchange (‘ring’) diffusion mechanism [21–34]‘governing’ atomic ordering in a two-dimensional binarysolution basedon the graphene-type lattice. To investigate the ordering kinetics of Cand Me atoms over the sites of this lattice, a model based on the Ons-ager-type microdiffusion master equation [21–34] was applied:

dPapðR; tÞdt

z� 1kBT

X2

q¼1

XR0

Xb¼C;Me

cacbLabpqðR � R0Þ dF

dPbq ðR0; tÞ

;

where t is time, ca (cb) is the relative fraction of a-kind (b-kind)atoms, Lab

pqðR � R0Þ is a matrix of kinetic coefficients whoseelements represent probabilities of elementary exchange-diffusionjumps of a pair of a and b atoms at the r site of p-th sublattice and r0

site of q-th sublattice, respectively (a, b¼ C, Me).Applying the conservation condition for the atoms of each kinds,

the fact that each lattice site is definitely occupied by one of theatoms composing the binary solution, and the Fourier trans-formation for the last equation (in details, see Ref. [22]), we deriveddifferential equations (see Table 2) for the kinetics of LRO parame-ter(s) of the (super)structures shown in Figs. 2–4. In Table 2, reducedtime t* is defined by the Onsager-type kinetic coefficients, t*h~LðkÞt,where ~LðkÞ is the Fourier-transform of a certain concentration-dependent combination of fLab

pqðR � R0 Þg, and reduced temperature

T* depends on the ‘mixing’ energies as follows: T*¼ kBT/jl2(k)j.

3. Results and conclusions

Curves in Figs. 5 and 6 represent numerical solutions of differ-ential kinetics equations for LRO of stoichiometric (Fig. 5; [22]) andnonstoichiometric (Fig. 6) C1�cMec structures (c is atomic fractionof doping Me-component) at the reduced temperature T*¼ 0.1.

Slight (Figs. 5a and 6a,d) and significant (Fig. 5d and e) non-monotonies of the evolution of the LRO parameters are caused bythe competitive difference of intrasublattice and intersublattice‘mixing’ energies.

As shown in Fig. 5a–c, an initial value of the LRO parameter (e.g.,ha

sðt� ¼ 0Þ ¼ 0:1 or hasðt� ¼ 0Þ ¼ 0:3) does not affect its end

(‘equilibrium’) value for a given structure. As expected, this value isthe same at other equal conditions.

Kinetic curves for C1�cMec structures with deviation ofDc¼�0.01 from stoichiometry are represented in Fig. 6. It is inter-esting that, for equiatomic (CMe) (super)structures, decrease andincrease of atomic fraction of dopant do not affect the relaxation

kinetics of LRO parameter: its instantaneous (and ‘equilibrium’)values are equal in Fig. 6c and f. However, for two other (super)-structures (C3Me and C7Me), decreasing and increasing of the rela-tive concentration of dopant with respect to its stoichiometry affectkinetic process differently. Decrease in this case reduces instanta-neous (and ‘equilibrium’) LRO parameter values, and increase of thedopant concentration may elevate instantaneous (and ‘equilibrium’)LRO-parameter values (see Fig. 6a, b, d, and e in comparison withFig. 5a and b).

In closing, it is important to note that possible interstitial(super)structures based on a graphene-type lattice will be consid-ered as well in another article.

Acknowledgement

This work was supported by the Grant of the National Academyof Sciences of Ukraine within the framework of research project foryoung scientists, which is gratefully acknowledged.

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kinetics of atomic ordering in metal-doped graphene

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