Supplementary Material
An Analytical Model for Calculating Thermal Properties of 2D Nanomaterials
Te-Huan Liu, Chun-Wei Pao and Chien-Cheng Chang
SI. Derivations of heat capacity and thermal conductance
We will derive the heat capacity and thermal conductance for ZA mode in this section. Derivations
for other modes are similar.
A. Heat capacity
Consider the dispersion relation of ZA mode, q=(ω/vZA)0.5. The heat capacity of ZA mode is
defined as
CZA 
1
V

ZA
0

f BE
D   d 
T
(S1)
where ωZA is the characteristic frequency of ZA mode. By following ref. S1, we consider that
graphene is isotropic in thermal transport. The constant energy surface in momentum space is a shell
of n-sphere with radius qZA. Thus the phonon density of states (PDOS) can be evaluated as follows
 As
qdq

 2
D   d   
 V q 2 dq

 2 2
for 2D
(S2)
for 3D
where As is the surface area in real lattice. Substitution of Eq. (S2) into (S1) leads to the heat capacity
is of the form
CZA 
1
2 
ZA
0

f BE dq
q
d
T d
(S3)
Taking the derivative dω/dq=2vZAq, we can recast Eq. (S3) into, by assuming vZA~constant
CZA 
f BE
1  1  ZA
d

 0 
4  vZA 
T
(S4)
Carry out the derivative of the Bose-Einstein distribution with respect to T. Let us set ξ=ħω/kBT=θ/T
and ξZA=ħωZA/kBT =θZA/T, then Eq. (S4) becomes
CZA 
k B2T
4 
 1  ZA  2 e
d


2
 vZA  0  e  1
(S5)
The integral is referred to as the characteristic integral f2(2,ξZA), and then we arrive at the final
expression for heat capacity of ZA mode
kB2T  f 2  2,  ZA  
CZA 


4   vZA

(S6)
B. Thermal conductance
Next we consider the thermal conductance, g=Δq̇ /ΔT, where q̇ is the heat flux which can be
calculated from Landauer formula.S2 For an isotropic medium, the heat flux transport from one heat
reservoir to another is given by
qZA 
 d 
1 ZA
  f BE 
 D   d 

0
V
 dq 
(S7)
The phonon wave velocity v is replaced by dω/dq. When the temperature difference between the two
heat reservoirs is quite small, the thermal conductance can be rewritten as a differential form: g=
∂q̇ /∂T, i.e. that is
g ZA 
f  d 
1 ZA
  BE 
 D   d 

V 0
T  dq 
(S8)
The averaged transmissivity is obtained by integrating η over the half-space 2D solid angle.S3,S4
Substituting Eq. (S2) into (S8), we obtain
g ZA 
1
2  
2
ZA
0

f BE  d 

 qdq
T  dq 
(S9)
Again, let us set ξ=ħω/kBT and ξZA=θZA/T; Eq. (S9) thereby yields the final form for the thermal
conductance
g ZA 
kB2.5T 1.5  f 2.5  2,  ZA  


0.5
2 2 1.5 
vZA

(S10)
The contribution to the heat capacity and thermal conductance from other modes can be obtained
similarly. It is noticed that for the bulk modes we should consider the 3D PDOS.
SII. Analytical analysis for characteristic integral fn
The n-dimensional characteristic integrals and Debye functions are respectively defined as
f n  s,  p     n e  e  1 d
p
s
(S11)
0
and
Dn  s,  p  
n

n
p

p
0
 n  e  1 d
s
(S12)
where p denotes the polarization. These expressions are useful in deriving analytical formulas for the
phonon heat capacity and phonon total energy associated with s=2. In this letter, we shall need to
deal with f2, f2.5, f3, and f4. For temperatures are much less than the characteristic temperature, T<<θp,
the upper limit of integration ξp can be regarded as infinity. Under this condition, the definite
integrals
are
approximately
constants,
which
are
given
by
fn(2,∞)=nΓ(n)ζ(n),
where

n
  n    x n1e x dx and   n   1 i are the gamma function and Riemann zeta function,

0
i 1
respectively.
Consider graphene for an example. This condition is satisfied when the temperature is below 50 K.
The values of fn(2,∞) for n=2, 2.5, 3, 4 are listed in Table SI. The heat capacity and thermal
conductance then assume, respectively, the simple analytical forms
C T  
and
3.6061kB3T 2  1
1   kB2T  1  2 2 kB4T 3  1
2 


 2


 3  3 
2
2 
3
   vLA vTA  12   vZA 
15
 vLB vTB 
(S13)
g T  
3.6061kB3T 2  1
1  2.2291kB2.5T 1.5  1   2 kB4T 3  1
2 

 2 


 0.5  
2 2
2 1.5
3  2
   vLA vTA 


 vZA  15
 vLB vTB 
(S14)
On the other hand, when the temperature is beyond 50 K, the characteristic integral will decrease
rapidly with increasing T; Eqs. (S13) and (S14) are no longer applicable. Instead, the characteristic
integrals can be written in terms of the Debye functionsS5
f n  2,  p  
p
 Dn 1,  p   Dn  2,  p  

n 
(S15)
In order to obtain series forms for the integer and non-integer n-dimensional Debye functions, we use
the binomial expansions
a  b
s

   1 Bi  s  a s i bi
i
(S16)
i 0
where Bi(s) are the binomial functions
 s!
 i ! s  i  !

Bi  s   
i
  1   i  s 
 i !   s 

for integer n
(S17)
for non-integer n
Therefore, the integer and non-integer n-dimensional Debye functions may take the asymptotic
formsS6
Dn  s,  p  
n
p
  1 B  s 
i
i
i
  n  1,  i  s   p 
i  s 
n 1
(S18)

where   ,     x 1e x dx is the incomplete gamma function, and can be evaluated by
0
downward recurrence, which is given byS7
   1,    e    
  ,   

Using Eqs. (S15) and (S18), we obtain the analytical forms of heat capacity and thermal
conductance for higher temperatures
(S19)
C T  
k B3T 2  D3 1,  LA   D3  2,  LA  D3 1, TA   D3  2, TA  



2
2
2 2 
vLA
vTA

2
k T  D 1,  ZA   D2  2,  ZA  
 B  2

4  
vZA


(S20)
k B4T 3  D4 1,  LB   D4  2,  LB  2  D4 1, TB   D4  2, TB   



3
3
2 2 3 
vLB
vTB

and
g T  
k B3T 2  D3 1,  LA   D3  2,  LA  D3 1, TA   D3  2, TA  



2 2 2 
vLA
vTA

k 2.5T 1.5  D 1,  ZA   D2.5  2,  ZA  
 B 2 1.5  2.5

0.5
2

vZA


(S21)
k B4T 3  D4 1,  LB   D4  2,  LB  2  D4 1, TB   D4  2, TB   



2
2
4 2 3 
vLB
vTB

where Dn(s,ξp) can be estimated by using Eq. (S18) and (S19). Fig. S1 compares the results between
direct evaluations of f2.5(2, ξp) with use of Simpson's 1/3 rule and the asymptotic results for the right
hand side of Eq. (S15). It is shown that the errors of using the asymptotic forms are, respectively,
2.49% and 0.04% of the summation from 5 and 100 series terms.
SIII. Thermal conductivity and scattering mechanisms
The characteristic integrals in the semi-analytical expressions of thermal conductivity, such as Eqs.
(6) and (7), contain the phonon relaxation time τ. In this letter, we consider the phonon-boundary,
phonon-isotope, and phonon-phonon (including umklapp and normal process) scatterings. (1) The
phonon-boundary scattering is given byS8
v 1  p  q  

l 1  p  q  
 B1  
(S22)
where l is the size of graphene sheet; p(q) is the specularity of its boundary. Although the scattering
from the boundary is in general partially diffusive, here we suppose that the boundary of graphene
sheet is extremely rough and its specularity is approximately zero. The latter assumption is known as
the Casimir limit.S9 (2) The averaged atomic mass of carbon is set to m=12.01 amu in whole
theoretical analyses and MD simulations, hence we should consider the phonon-isotope scattering,
which is given byS10,S11
 I1 
3 4
 v3
(S23)
where Ω is the effective volume of carbon atom. μ refers to the mass difference coefficient, which is
defined as     i 1  mi m  , where εi and mi is the natural abundances and atomic mass of ith
2
i
isotope of carbon, respectively. (3) For the phonon-phonon scatterings, we use Callaway's approach
to describe the mechanisms.S12 There are two processes in this approach: the umklapp and normal
process, which are given byS11,S13,S14
 p 3T

1
U
 2 2 e
~ p
m p v 2
T
(S24)
and
 k B3  p2 2 3
T

 m 2v5
1
N ~  4 2
 k B  p 4
T

 m 3v 5
for longitudinal waves
(S25)
for transverse waves
where m is the averaged mass of a single atom, and γp refers to the Grüneisen parameter for each
polarization. The values for longitudinal and transverse modes are usually taken to be 2 and 0.67,
respectively. In particular, the Grüneisen parameter for ZA mode is negative, and is approximately
−1.5.S8,S15-S17
The thermal conductivity of a phononic system is given by

p
1
C p v 2p p d 


3 p 0
(S26)
where p denotes the polarizations. We can also write down the thermal conductivity by the effective
phonon mean free path (PMFP) Λeff

 eff
3

p
p
0
C p v p d
(S27)
Therefore, the right hand side of Eqs. (S26) and (S27) can defines the effective PMEP
 eff 

p
0
p

p
C p v 2p p d
p
0
C p v p d
(S28)
The numerator of Eq. (S28) is calculated by Eqs. (6) and (7), and the dominator can be derived in an
analytical form

p
p
0
k B3T 2  f3  2,  LA  f3  2, TA   k B2.5T 1.5  f 2.5  2,  ZA  
C p v p d 




1.5 
0.5
6 2  vLA
vTA
vZA
 6  

4 3
k T  f  2,  LB  2 f 4  2, TB  
 B 3 4


6  vLB
vTB

(S29)
Hence the effective PMFP Λeff modeled from Eqs. (S22) to (S25) can be defined; the results for Λeff
are shown in Fig. S3. It is found that the PMFP is approximately equal to the sheet size at very low
temperatures, and decreases rapidly while the temperature increases. At 300 K, Λeff is estimated to be
from 6366 (including all scattering) to 7899 (excluding phonon-isotope scattering) Å for a 10 μm
graphene sheet. In addition, κ/Λeff can be calculated analytically by substituting Eq. (S29) into (S27),
whose value is 7.4×109 W/m2K at 300 K. By multiplying the experimental PMFP at room
temperature, Λeff=7000 Å, we can readily obtain the thermal conductivity κ=5180 W/mK.
SIV. MD simulations
In this letter, all the MD simulations are performed with LAMMPS (large-scale atomic/molecular
massively parallel simulator).S18 The optimized Tersoff empirical potentialS19,S20 is implemented,
which can accurately reproduce the phonon band structure of graphene.
A. Phonon band structure
In our model, we need to input the wave velocities and characteristic temperatures of guided and
bulk waves. For guided waves, they are extracted from the phonon dispersion curves. The dispersion
relations can be obtained by using the "FixPhonon" packageS21 in LAMMPS. We used a supercell
containing 900 (30×30) primitive cells of which the lattice constant is 2.492 Å.S20 The system
temperature is set to 300 K with time step 0.5 fs. The simulations are performed on NVE ensemble
for 5.0 ns after the system reaches thermal equilibrium, and the dynamic matrix is obtained by
averaging the entire MD simulation. The phonon band structure is shown in Fig. S3. It is observed
that most of the contribution to heat transport comes from the long-wavelength modes. The velocities
of guided waves can be evaluated by v=dω/dq|q→0; and the characteristic temperatures can be
calculated from θp=ħωp/kB. It is noted that there are two high-symmetry directions in reciprocal
space of graphene: Γ→M and Γ→K, and we obtain the wave velocities and characteristic
temperatures by averaging them over the two directions. The velocities obtained for graphene are:
vLA=2.34×104 m/s, vTA=1.58×104 m/s and vZA=6.25×10-7 m2/s; the characteristic temperatures for
graphene are: θLA=1780 K, θTA=1400 K and θZA=780 K.
B. Elastic constants
For bulk waves, the LB and TB modes have the following velocities
vLB 
  2

(S30)


(S31)
and
vTB 
The elastic constants can be extracted from the slopes of stress-strain curves. In order to obtain such
relations, the investigated sheet of graphene which contains 5488 atoms (117.6×118.8 Å2), is subject
to uniaxial or biaxial deformation with a strain rate of 0.001% per ps. This simulation is performed
on NVE ensemble associated with the Langevin thermostat at 300 K with a time step 0.5 fs. The
i
atomic-level stress  
of carbon atom i is computed by the virial stress formula
i
 

1 
1
i i i
ij ij 
 m v v   r F 
i  i
2 i j i

(S32)
where α and β are the directions in Cartesian coordinate. rij, vij and Fij respectively refer to the
distance, velocity and interatomic force between the atoms i and j. The total stress is obtained by
i
summing up the  
in whole system, and taking average the summation every 50000 MD time
steps. The stress-strain relations are shown in Fig. S4. In order to obtain the elastic constants, we first
investigate the stiffness tensor of graphene
 c11 c12 0 
c  
c11 0 
sym
c66 
(S33)
where c11=λ+2μ, c12=λ and c66=(c11－c12)/2=μ. Thus we can determine each elastic constants by
evaluating the slopes of the stress-strain curves: c11=987 GPa, c12=126 GPa and thereby c66=431
GPa. Hence the velocities of LB and TB modes can be calculated from Eqs. (S30) and (S31):
vLB=21.1×104 m/s and vTB=14.0×104 m/s, respectively. In addition, the characteristic temperature is
determined as the Debye temperature of graphene, θLB=θTB=2100 K.
C. Thermal conductivity
We use the NEMD method to compute the thermal conductivity of graphene.S22,S23 The geometry is
illustrated in the inset of Fig. S5. The atoms in gray regions are fixed and do not participate in the
integration of equation of motion. The adjacent red and blue regions refer to the hot and cold
reservoirs, which are placed under the Nosé-Hoover thermostat with temperatures TH and TC,
respectively. The intermediate region is governed by Newton 2nd law only, and thereby builds up the
temperature gradient with the two adjacent reservoirs.
The resulting heat flux is generated by the hot reservoir and absorbed by the cold. The equations of
motion for Nosé-Hoover thermostat are given by
dp i
 Fi   p i
dt
and
(S34)
d
1 T t  
 2 
 1
dt  MD  T0

(S35)
where τMD is the damping time of energy in MD simulation. T0 is the temperature of reservoir (TH or
TC), and T(t) is the instantaneous temperature of system. pi and χ refer to the momentum and friction
coefficient, respectively, and their product J=χpi is the heat flux
J  3 NkBT t 
(S36)
where N is the number of atoms in hot or cold reservoir. The heat fluxes from hot and cold reservoir
should be equal: |JH|=|JC|. Therefore, the heat flux can be calculated by J=(JH－JC)/2, and their error
can be estimated from ΔJ=(JH+JC)/2. The temperatures of hot and cold reservoirs are set to
TH,L=(1±ϕ)T, where we choose ϕ=0.1 throughout the present study. The time step and τMD are set to
0.5 and 1000 fs, respectively, and the total MD simulation time is 10 ns. We calculate J by averaging
every instantaneous heat flux from 5 to 10 ns, and therefore the thermal conductivity is obtained by
Fourier’s law κ=Jdx/AcdT, where Ac is the cross section to the heat flux, and dT/dx is the temperature
gradient along the transport direction. Fig. S5 shows the constructed temperature gradient for a 1.0
μm graphene from the NEMD method, and the calculated thermal conductivities are listed in Table
SII.
Figure captions
Fig. S1. f2.5(2,ξZA) for different approximate algorithm. The blue line represents the Simpson’s 1/3
rule; the dotted red and green lines are the series expansion with 100 and 5 series.
Fig. S2. The effective PMFP for a 10 μm graphene.
Fig. S3. Phonon dispersion relation and density of states of graphene
Fig. S4. Stress-strain relation of graphene. The blue and red lines represent the graphene under
uniaxial and biaxial tensile loading, respectively.
Fig. S5. The constructed temperature gradient for graphene with l=1.0 μm. The inset shows the
geometry and atom allocating of the NEMD method.S22,S23
Tables
TABLE SI. The values of fn(2,∞)
n
Γ(n)
ζ(n)
fn(2,∞)
2
2.5
3
4
1
π2/6
1.3415
1.2021
π4/90
π2/3
4.4582
7.2123
4π4/15
3π0.5/4
2
6
TABLE SII. The thermal conductivities calculated from NEMD method
l (μm)
T (K)
κ (W/mK)
Error (%)
0.03
0.05
0.07
0.1
0.2
0.4
0.6
1.0
300
300
300
300
300
300
300
300
77.7
223.7
363.9
515.4
856.8
1309.3
1596.9
1980.7
1.2
2.3
2.9
2.9
5.8
2.0
1.9
5.9
1.0
1.0
400
500
1599.8
1273.9
9.9
7.2
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