Supporting Online Material for
Temperature dependent elastic constants and ultimate strength of graphene
and graphyne
Tianjiao Shaoa, b, Bin Wena, 1, Roderick Melnikc,d, Shan Yaob, Yoshiyuki Kawazoee, Yongjun Tiana
aState
Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, China
bSchool
of Materials Science and Engineering, Dalian University of Technology, Dalian 116023, China
cM2NeT
Lab, Wilfrid Laurier University, Waterloo,75 University Ave. West, Ontario, Canada N2L 3C5
dIkerbasque,
eInstitute
Basque Foundation for Science and BCAM, Bilbao 48011, Spain
for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
This file includes:
Computational Methods
References
1
Authors to whom any correspondence should be addressed
E-mail address: wenbin@ysu.edu.cn (Bin Wen), Tel: 086-335-8568761
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Supporting Online Material
Computational Methods
In our previous work, we have developed a new methodology for determining lattice geometries
and temperature dependent elastic constants at high temperature for arbitrary symmetry three
dimensional crystals [1]. The computation can be implemented by employing the first principles
combined with the quasi harmonic approximation (QHA). We further extend this theory and
methodology to predict the lattice geometries and temperature dependent elastic constants (TDEC)
for two dimensional lattices.
A. Temperature dependent geometry optimization of two dimensional lattices
Based on the crystallographic theory [2], the Gibbs energy is a function of the lattice lengths,
lattice angles and all free crystallographic coordinates of the atoms in non-fixed Wyckoff positions
and so forth. In this work, we suppose that free crystallographic coordinates of the atoms in
non-fixed Wyckoff positions are fixed, then, configuration tensor X is only a function of lattice
lengths a, b; lattice angle of the two dimensional lattice. According to thermodynamics arguments
[3, 4], non-equilibrium Gibbs energy for a specific phase can be written as
G[ X (a, b,  ); P, T ]  E[ X (a, b,  )]  PV [ X (a, b,  )]  Avib [ X (a, b,  ); T ] .
where
X ( a , b,  )
represents
the
crystal
configuration
tensor
(1)
under
pressure
P
and
temperature T . G[ X (a, b,  ); P, T ] is the Gibbs energy for the configuration tensor X (a, b,  ) .
E[ X (a, b,  )] and V [ X (a, b,  )] is the total energy and volume of the unit cell for crystal
configuration tensor X (a, b,  ) , respectively. Avib [ X (a, b,  ); T ] stands for vibrational Helmholtz
free energy.
According to Eq. (1), the Gibbs energy is given as an energy surface in terms of a multi-variable
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function of a, b and  . To obtain the Gibbs energy at equilibrium, full minimization of Gibbs energy
is required. One big challenge in this full minimization lies exactly with the fact that the Gibbs
function is a multi-variable function, including a, b and  . In our previous work [1], we have
proposed a scheme to address this issue by deriving the Gibbs energy in one variable , allowing us
to determine the anisotropy thermal expansion of this crystal and to obtain the linear directional
thermal expansion coefficient.
By applying linear algebra and tensor analysis arguments [5], deformed two dimensional lattice
configuration tensor can be expressed as the product of 4-dimensional deformation tensor and the
initial configuration tensor, so that the deformed configuration tensor is given by,
 e e    1 0 
X  X 0  1 2 
,
 e3 e4   0   1
e
where X 0 is the initial configuration tensor,  1
 e3
(2)
e2 
 is a normal deformation tensor and  is
e4 
the deformation strain.
According to Wang’s work [6], if the applied deformation matrix is symmetric, then the
deformed configuration tensor can be further rewritten as
e
X  X0  1
 e2
e2    1 0 
.

e3   0   1
(3)
Then, the Gibbs energy is given as a unique function of the deformation strain  , and the
non-equilibrium Gibbs energy is expressed as follows
G[ X (a, b,  ); P, T ]  E[ X ( )]  PV [ X ( )]  Avib [ X ( ); T ] ,
(4)
where E[ X ( )] is the total energy of the specific deformed configuration, Avib [ X ( ); T ] is the
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vibrational Helmholtz free energy, which can be calculated from phonon density of states by using
the DFT-QHA.

Avib [ X ( ); T ]   [
0
1
  kT ln(1  e 
2
 / k BT
)]g[ X ( );  ]d  ,
where  is the phonon frequency, T is the temperature, k and
(5)
are the Boltzmann constant
and the reduced Planck constant, respectively.
The Gibbs energy at equilibrium, which is only related to the parameter of temperature T , can
be evaluated as the minimum of the non-equilibrium Gibbs energy with respect to the deformation
strain  ,
G ( P0,T ; P, T )  min E[ X ( )]  PV [ X ( )]  Avib [ X ( ); T ] .

(6)
When the deformation mode is fixed, the deformation strain  P0,T at equilibrium along this
deformation mode can be obtained, which is a function of lattice parameters a, b and  . To find the
lattice parameter a, b and  for a P1 symmetry two dimensional lattice, a system of three equations
is required to be solved simultaneously,
 F1 (a, b,  , 1,0P ,T )  0,

0
 F2 (a, b,  ,  2, P ,T )  0,
 F (a, b,  ,  0 )  0,
3, P ,T
 3
(7)
where F1 , F2 and F3 represents the multi-variable function including variables of lattice
parameters a , b ,  and equilibrium strain i0,P,T . By solving the system of Eq. (7), three
equilibrium deformation strains 1,0P,T , 2,0 P,T and 3,0 P ,T can be obtained, the lattice parameter of a ,
b and  can be determined.
Graphene and graphyne calculated in this work both belong to the hexagonal symmetry. To
 0 
obtain the lattice geometry at a given temperature, the deformation tensor  1
 is chosen to
 0 1
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implement the lattice geometry optimization at this temperature. In this case, the Eq. (6) can be cast
in the following form to obtain the lattice geometry at temperature T ,
G (1,0P,T ; P, T )  min E[ X (1 )]  PV [ X (1 )]  Avib [ X (1 ); T ] .
1
(8)
We solve the Eq. (8) to obtain the deformation strain 1, 0P ,T at equilibrium at given temperature
T . Then the lattice length for hexagonal graphene or graphyne can be obtained as
aP,T  (1  1,0P,T )a0 ,
(9)
where a0 represents the optimized hexagonal two-dimensional lattice parameter a at 0 K, which can
be determined from the first-principles calculation; aP ,T is the lattice parameter a at equilibrium at
given temperature T and pressure P .
B. Temperature dependent elastic constants of two dimensional lattices
Based on our developed methodology for calculating the TDEC for three dimensional solid [1],
we have derived the Helmholtz free energy for evaluating the temperature dependent elastic
constants
F[ X (a, b,  ); P, T ]  E[ X ( )]  Avib [ X ( ); T ] ,
(10)
where E[ X ( )] is the total energy for the specific deformed configuration, Avib [ X ( ); T ] is the
vibrational Helmholtz free energy, which can be calculated by first-principles approaches combined
with QHA model.
Since all the stress components out of the two-dimensional plane equal to zero, for the two
dimensional lattice the dimension of elastic constant matrix is reduced to 3 and is given by,
 c11

C   c12
c
 16
c12
c22
c26
c16 

c26  .
c66 
(11)
On the basis of continuum elasticity theory [7, 8], the elastic constants are given by the second
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derivative of the Helmholtz free energy with respect to the deformation strain tensor. For evaluating
the 6 independent elastic constants cijP ,T (i, j=1, 2 or 6), a system of equations that correspond to the
6 independent deformation modes are required to solve.
P ,T
P ,T
 F1 ( D1P ,T , c11P ,T , c12P ,T , c22
, c16P ,T , c26
, c66P ,T )  0

P ,T
P ,T
P ,T
P ,T
P ,T
P ,T
P ,T
 F2 ( D2 , c11 , c12 , c22 , c16 , c26 , c66 )  0
P ,T
P ,T
 F3 ( D3P ,T , c11P ,T , c12P ,T , c22
, c16P ,T , c26
, c66P ,T )  0
,

P ,T
P ,T
P ,T
P ,T
P ,T
P ,T
P ,T
F
(
D
,
c
,
c
,
c
,
c
,
c
,
c
)

0
4
4
11
12
22
16
26
66

P ,T
P ,T
 F5 ( D5P ,T , c11P ,T , c12P ,T , c22
, c16P ,T , c26
, c66P ,T )  0

P ,T
P ,T
P ,T
P ,T
P ,T
P ,T
P ,T
 F6 ( D6 , c11 , c12 , c22 , c16 , c26 , c66 )  0
(12)
where DiP ,T is the second order derivative of the Helmholtz free energy with respect to the
deformation strain under the i-th deformation mode.
For the hexagonal graphene and graphyne lattices, due to the intrinsic symmetry, the elastic
constant matrix is simplified to
 c11

C   c12
0

c12
c11
0


0
.
(c11  c12 ) / 2 
0
(13)
 0 
 0 
Based on the Eq. (9) and Eq. (11), we choose the 
 as the deformation
 and 
0  
 0 0
modes to implement calculations for the elastic constants for c11P ,T and c12P ,T . For the two
deformation modes, the second order derivative of the Helmholtz free energy with respect to the
deformation strain corresponds to the independent elastic constants or their linear combination. The
corresponding relationship is given in Table 1. After is c11P ,T and c12P ,T calculated, in the case of small
deformation applied on hexagonal graphene and graphyne, Young’s modulus is in-plane isotropic and
is given by following,
E   c112  c12 2  / c11
C. Temperature dependent ultimate strength of two dimensional lattices
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(14)
In recent years, ultimate or ideal strengths have been calculated for various three dimensional
crystals by using the first principle method [9, 10]. Most results have been obtained for three
dimensional crystal’s ideal strength at 0 K. To shed light on temperature’s influence in graphene and
graphyne and subsequent limitations on their tensile mechanical properties, we extend our
methodology to evaluate the TDUS for these materials.
In order to evaluate the ultimate strength, we first need to find out the relationship for the lattice
configuration under deformation strain versus the strain . We obtain the equilibrium lattice
configuration X 0P ,T at temperature T and stretch the unit cell in each step with a constant increase
of  (in our calculations  =0.01 for the hexagonal graphene lattice and 0.02 for the hexagonal
graphyne lattice). Then, we fix the length along the direction of tension, relax the unit cell to ensure
that the other intrinsic stress perpendicular to the direction of tension is zero. Based on the above two
steps, we obtain the configuration tensor X ( ) P ,T for the two-dimensional lattice under the tensile
strain  at given temperature T .
Recall that the Helmholtz free energy for the configuration lattice X ( ) P ,T is given by Eq. (10).
To calculate the stress versus the strain, we differentiate the Helmholtz free energy with respect to
the strain [10],
 tensile ( , P, T ) 
1   dF ( , P, T )
V ( ) P ,T
d
(15)
where  tensile ( , P, T ) is the tensile stress of the lattice under the tensile strain of  , V ( ) P ,T is the
volume of the lattice under tensile strain of  , F ( , P, T ) is the Helmholtz free energy of the
deformed lattice, P is the pressure and T is the temperature. Then, along a specific tensile
deformation direction, the tensile stress  tensile ( , P, T ) with respect to the tensile strain  at given
pressure P and given temperature T can be calculated. Since ultimate strength is defined as the
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maximum stress that any material will withstand before destruction, by plotting the stress-strain
curve, we determine the yielding point as the ultimate strength.
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REFERENCES
[1] T. Shao, B. Wen, R. Melnik, S. Yao, Y. Kawazoe, Y. Tian, Journal of Applied Physics 111,
083525 (2012).
[2] J. F. Smith, JPEDAV 25, 405 (2004).
[3] A. A. Maradudin, E. W. Montroll, G. H. Weiss, I. P. Ipatova, Theory of Lattice Dynamics in the
Harmonic Approximation (Academic, New York, 1971).
[4] K. Parlinski, Journal of Physics: Conference Series 92, 012009 (2007).
[5] I. V. Oseledets, D. V. Savostyanov, E. E. Tyrtyshnikov, Computing 85, 169 (2009).
[6] J. H. Wang, J. Li, S. Yip, S. Phillpot and D. Wolf, Physical Review B 52, 12627 (1995).
[7] J. J. Zhao, J. M. Winey, and Y. M. Gupta, Physical Review B 75, 094105 (2007).
[8] D. C. Wallace, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic, New York,
1970), Vol. 25, p. 301.
[9] D. Roundy, C. R. Krenn, M. L. Cohen and J. W. Morris, Jr, Physical Review Letters 82, 2713
(1999).
[10] W. Luo, D. Roundy, Marvin L. Cohen and J. W. Morris, Jr, Physical Review B 66, 094110
(2002).
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Table 1 Deformation tensors used to calculate the elastic constants for hexagonal graphene lattice
and hexagonal graphyne lattice. A linear combination of elastic constants (LCEC) equals to the
second order strain derivatives of the Helmholtz free energy under the corresponding deformation
modes (or deformation tensors).
Type of
Deformation
LCEC
crystals lattice
Hexagonal
symmetry
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tensors


0


0
0

0
c11
0 

 
2c11-2c12