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Appl. Phys. Lett. 101, 113111 (2012)
of the thermal reservoirs, and Tph(w) is the transmission func-
tion of the phonons. In the limit of small temperature differ-
ence AT, the phonon thermal conductance Kph(T) = J(T)/AT
is given by
T)T 1 nd(h) t h() h [n(T, c )
Kph(T) =2i d (ho)w) h 0 .
(2)
The transmission functions Tph (w) for phonons can be cal-
culated using a real-space Green's function approach, similar
to the one used for the electron transport across nanoscale
interfaces,s8 where the main ingredients are (i) the diagonal
matrix Mph corresponding to the masses of the atoms and (ii)
the matrix Kph(r) of the interatomic force constants (IFCs)
in real space.20'21 The thermal resistance, R(T), is then
obtained by inverting Eq. (2) and normalizing to the inter-
face area. In the present treatment, IFCs are calculated fully
from first principles22 within DFPT15,23 that allows unequiv-
ocal consideration of the microscopic geometry as well as
chemical and electronic modification at the interface without
resorting to phenomenological or ad hoc models. For sim-
plicity, we restrict our analysis only to the harmonic contri-
bution to lattice dynamics. The inclusion of three phonon
scattering effects, although obtainable from DFPT,15 is out-
side of the scope of this investigation.23
The device geometry considered for the evaluation of
the interfacial thermal resistance is comprised of a hetero-
structure containing an interface between different materials
connected to two thermal baths (reservoirs) at two different
temperatures, TL,R (see Figure S1 in supplemental mate-
rial24). In this configuration and in the absence of inelastic
scattering, the thermal resistance measured across the sys-
tem, R(T), coincides with the interfacial thermal resistance
Rin (T), i.e., the Kapitza resistance through the graphene/
dielectric interface. For more details about the method see,
for instance, Ref. 11 and references therein.
To understand thermal transport at multilayer graphene
(G) and dielectric interfaces, we have considered two differ-
ent classes of substrates, hexagonal BN and wurtzite SiC.
Both share a planar hexagonal symmetry in the direction
perpendicular to the interface, but they have a different
interlayer coupling in the direction parallel to transport: one
is a 2-D layered (h-BN) and the other an intrinsically 3-D
system (wurtzite SiC). See Figures S2 and S3 in supplemen-
tary material24 for details on the geometry of the bulk and
interface structures.
In the case of the h-BN substrate, the structure of the
interface is planar without any buffer layer, so the model sys-
tem for the transport calculation is a slab with eight h-BN
planes and eight graphene layers. As for SiC, we focus on
multilayer graphene grown on the Si-face of 2 H-SiC(0001)-
[v x- 3cos(30 )] that is characterized by a carbon buffer
layer bound to the SiC surface and an electron-doped gra-
phene layer on top. More specifically, we consider both the
native (SiC) and the hydrogenated case (H:SiC), where H
atoms passivate Si dangling bonds (lonely atoms) in the
interface layer,25 to investigate how the coupling of the sub-
strate with the graphene layers affects the phonon transport.
In this case, the model system for the interface is comprised
by a SiC slab (eight planes, 2 x 2 lateral periodicity), the
9 8 8
- h-BN - G - SiC
7
6
8 7
E
S5
r RT . RT 4RT
,4 -
7 -
3I-
6 -
(a) i (b) (c)
150 300 150 300 150 300
temperature (K)
FIG. 1. The intrinsic contact thermal resistance for (a) h-BN, (b) G, and (c)
SiC in the (0001) direction. Vertical dashed lines mark the R values at room
temperature.
carbon buffer layer (with or without hydrogenation), and
three graphene layers26 (see Fig. S3 in supplementary mate-
rial for details24).
We first discuss the characterization of the bulk systems
and their intrinsic thermal contact resistance a la Landauer.
The preliminary calculations of phonon dispersion and
transmittance for the bulk systems can be found in the
supplementary material.24 Using Eq. (2), we calculate
the contact thermal resistance of the bulk h-BN, "bulk
graphene" and bulk SiC as a function of temperature in the
range 50-400 K, as shown in Fig. 1. The thermal resistances
hold a 1/T dependence with room temperature values of
7.2 x 10-10 m2K/W and 6.3 x 10-10 m2 K/W for the case
of bulk h-BN and graphene, respectively. These numbers are
consistent with the expectation of a small out-of-plane trans-
mission in these layered systems, in good agreement with
available experimental results.27'28 Similar results are found
for bulk SiC, with the important difference that now the con-
tact resistance at room temperature is 3.5 x 10-10 m2K/W,
i.e., almost half that of layered systems.
The same calculation route has been carried out for the
whole interface structure of the G/h-BN system. The interfa-
cial thermal resistances are plotted in Fig. 2. The relatively
low (albeit still one order of magnitude larger than the bulk
59 390 750
- G/h-BN - G/SiC - G/H:SiC
58
740
57 380
E 56 730
70 55 370
- RT RT 720 - ,RT
C54 -
53 360 710
(a) (b) (c) i)
150 300 150 300 150 300
temperature (K)
FIG. 2. The interfacial thermal resistance for (a) G/h-BN, (b) G/SiC, and (c)
G/H:SiC. Vertical dashed lines mark the Rin values at room temperature.
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113111-2 Mao et al.
