Supplemental material
Zhiyong Wei,1, 2 Yunfei Chen,1 and Chris Dames2*
1. Jiangsu Key Laboratory for Design & Manufacture of Micro/Nano Biomedical Instruments and Department of
Mechanical Engineering, Southeast University, Nanjing 210096, People’s Republic of China
2. Department of Mechanical Engineering, University of California at Berkeley, California 94720-1740, USA
* email: cdames@berkeley.edu
The most interesting observations from the main text are the “off-diagonal” calculations
of Figs. 1c and 3c, which show that increasing the IP coupling strength IP substantially reduces
CP, in part by reducing HCP. Here we present additional calculations to verify that this basic
trend is not an artifact of the MD simulations’ finite sample size or classical thermal statistics.
First we consider the effects of finite simulation size, which are well known to reduce the
calculated thermal conductivity compared to the intrinsic value for an infinite sample [S1].
Figure S1 confirms that for the baseline scenario (IP=CP=1), the calculated CP is virtually
independent of the IP sample size, for IP sizes ranging from 12.39 nm2 (used for Figs. 1c and 1d)
to 31.64 nm2. Similarly, Fig. S2 shows the effect of the CP sample size (number of layers) on
1
the calculated CP. As expected, the absolute magnitude of CP does depend on the number of
CP layers, consistent with other work [S1] and indicating that some of the mean free paths are
larger than the sample thickness. However, for all thicknesses the dominant trend remains that
increasing the IP coupling reduces CP, with power law exponents within 10% of the nominal
value of -0.70 reported in the main text, thus confirming the observation of Fig. 1c.
MD simulations are inherently based on classical thermal statistics, and therefore are only
strictly appropriate at high temperatures. As one indicator of the transition between classical and
quantum regimes, consider the “knee” defined by the intersection of power-law extrapolations of
the DuLong-Petit and Debye heat capacity laws. This occurs at Tknee /  D   5 / 4 4 
1/ 3
 0.234 .
Calculations of thermal conductivity are expected to be somewhat more forgiving than this
simple criterion, because for transport and phonon-phonon scattering the acoustic modes are
most relevant, thus shifting emphasis to smaller energies and lower characteristic temperatures.
For the layered graphite-like material used in this work, lattice dynamics calculations at
IP=CP=1 give characteristic temperatures   max / kB of around 2120 and 110 K for acoustic
modes in the IP and CP directions, respectively. Thus, although MD simulations of related
graphitic materials are also routinely performed around 300 K [S2-S3], this is only expected to
be strictly correct for the CP modes. Although a quantum correction [S4] is sometimes used, it
cannot fix the mode-wise relative misallocation of the phonon population caused by Maxwellian
2
rather than Bose-Einstein statistics, and it has also been reported [S5] that such a quantum
correction does not bring about better agreement with experiments.
To check whether the major trend of Fig. 1c might be an artifact of having used classical
rather than quantum statistics, we repeated the calculation at higher MD temperatures of 500 and
800 K [S6]. As shown in Fig. S3, the basic trend of negative correlation between IP and CP is
also observed at higher T, although it is also becoming substantially weaker. In the harmonic
approximation and with classical statistics, the power law in H CP   IP m should be independent
of T, so at first it appears that the phonon focusing phenomena discussed in the main text cannot
explain the evolution in  CP (  IP ) with T seen in Fig. S3. However, we now show that the
weakening in the  CP   IP n power law at high T can still be understood largely in terms of the
phonon dispersion relation, by accounting for higher order anharmonic effects.
The key is the impact of IP on the CP thermal expansion and thus vCP. Stronger IP
bonding (Tersoff potential) reduces the amplitude of the out-of-plane thermal vibrations of the
graphene-like sheets, which reduces the average CP layer spacing c.
Then, due to the
anharmonicity of the CP potential, this reduced c increases the average CP stiffness.
For
example, our MD relaxation process using NPT simulations showed that increasing χIP from 0.2
to 5.0 (while holding χCP constant at 1.0) reduces the equilibrium interlayer spacing at 300 K by
about 3%, from c=3.50 Å to 3.40 Å, while the C-C bond length remains almost constant (a=1.46
 0.003 Å). LD calculations confirm that this increases the CP frequencies: at 300 K the
3
maximum LA frequency in the CP direction (i.e. at the A point), is found to increase weakly as
max, LA,CP   IP0.07 .
Importantly, this anharmonic effect is stronger at higher T, with
max, LA,CP   IP0.23 at 800 K. Increasing these CP frequencies and velocities should also increase
HCP and ultimately CP. Thus, this higher-order anharmonic effect is found to couple  and  in
orthogonal directions via the dispersion relation and phonon irradiation, but with a positive
correlation: that is, opposite of the primary phonon focusing mechanism presented in the main
text. Furthermore, this correction becomes more important at higher T.
To quantify the expected correction to H CP (  IP ) caused by this anharmonic effect we
apply an analytical result from an anisotropic Debye model for phonon focusing [S7]. In limit of
v IP  vCP and classical statistics, that model predicts
H CP 
2
vCP
.
v IP
(S1)
0
0
12
 CP
When anharmonic effects are neglected, v IP   1IP2  CP
and vCP   IP
, and thus for fixed CP
we expect H CP   IP0.5 . Accounting for the additional thermal expansion effects just described
12
0.23 1 2
 CP at 800 K. Substituting these
for max, CP suggests vCP   IP0.07  CP
at 300 K and vCP   IP
into Eq. (S1) yields H CP (300 K )   IP0.36 and H CP (800 K )   IP0.04 . The main point is that these
anharmonic corrections are expected to weaken the H CP (  IP ) power law by a greater amount at
0.32
high T, here by a factor of  IP
at 800 K compared to 300 K. Comparison with Fig. S3 suggests
4
that this correction to HCP is the primary reason for the weakening of the  CP (  IP ) exponent by a
0.40
factor of  IP
as T increases from 300 K to 800 K.
Summarizing this discussion, the temperature dependence of the  CP (  IP ) curves in Fig.
(S3) can be understood as primarily caused by anharmonic corrections to the phonon irradiation,
rather than indicating overwhelming artifacts from classical thermal statistics. Therefore we
believe that the primary conclusion of Figs. 1(c) and 3(d) of the main text holds, namely, that
increasing the IP coupling strength tends to reduce the CP irradiation and CP thermal
conductivity.
References
[S1] P. K. Schelling, S. R. Phillpot, and P. Keblinski, Phys. Rev. B 65, 144306 (2002).
[S2] W. J. Evans, L. Hu, and P. Keblinski, Appl. Phys. Lett. 96, 203112(2010)
[S3] S. Chien, Y. Yang, and C. Chen, Appl. Phys. Lett. 98, 033107 (2011)
[S4] Y. H. Lee, R. Biswas, C. M. Soukoulis, C. Z. Wang, C. T. Chan, and K. M. Ho, Phys. Rev.
B 43, 6573 (1991).
[S5] J. E. Turney, A. J. H. McGaughey, and C. H. Amon, Phys. Rev. B 79, 224305 (2009).
[S6] Simulations attempted at higher T were unsuccessful because the layers were found to
spontaneously separate during the initial NPT step.
[S7] Z. Chen, and C. Dames, submitted.
5
0.7
40-layer at T=300 K
CP [W/m K]
0.6
0.5
0.4
10
15
20
25
30
35
2
IP size [nm ]
Fig. S1. (Color online.) Confirmation that the calculated CP thermal conductivity is insensitive to the IP
simulation size. All the IP sizes are approximately square.
6
2
20-layer
40-layer
60-layer
1.6
20-layer fitting: CP  -0.76
IP
CP [W/m K]
40-layer fitting: CP  -0.70
IP
60-layer fitting: CP  -0.67
IP
1.2
0.8
0.4
0
0
2
4
6
IP
Fig. S2. (Color online.) Confirmation that the dominant trend of Fig. 1c is insensitive to the number of
layers. Although the absolute magnitude of the thermal conductivity increases with the number of layers,
the most important feature of negative correlation between CP and IP remains in all cases, with a power
law of approximately -0.7. The simulation temperature is 300 K, CP=1, and the IP size is about
3.50×3.54 nm2.
7
1.6
300K
500K
800K
300K fitting: CP  -0.70
IP
1.2
CP [W/m K]
500K fitting: CP  -0.57
IP
800K fitting: CP  -0.30
IP
0.8
0.4
0
0
2
4
6
IP
Fig. S3. (Color online.) The effect of simulation temperature on the dominant trend of Fig. 1c. CP = 1
and the simulation dimensions are 40 layers and 3.50×3.54 nm2. As described in the text, the weakening
n
of the  IP
power law trends can be largely understood through anharmonic thermal expansion effects on
HCP.
8