Thermal transport in hard carbon prepared from C60 fullerene

Article abstract of DOI:10.1088/0953-8984/8/3/001

We report measurements of thermal conductivity in the 30-350 K range of hard fullerene-based carbon. The material has been prepared by pressure-temperature treatment of C₆₀ fullerene. It shows an unusual combination of high hardness and relatively high electrical conductivity. Its thermal conductivity is about 5.7 W mK^-1 at room temperature and decreases almost linearly in the investigated temperature range. Excellent linearity in a wide temperature range may arise from some sort of crystalline order in the transformed substance. Obtained data are compared with thermal properties of C₆₀, glassy carbon and diamond. The applicability of recent theoretical models for explanation of recorded data is discussed.

Full text of DOI:10.1088/0953-8984/8/3/001

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J. Phys.: Condens. Matter 8 (1996) L27–L32. Printed in the UK
LETTER TO THE EDITOR
Thermal transport in hard carbon prepared from C60
fullerene
K Biljakovi c´ †, A Smontara†, D Stare sˇ ini c´ †, D Paji c´ †, M E Kozlov‡§, M
Hirabayashi‡, M Tokumoto‡ and H Ihara‡
†
‡
Institute of Physics of the University, 41 001 Zagreb, PO Box 304, Croatia
Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba, Ibaraki 305, Japan
Received 16 October 1995
Abstract. We report measurements of thermal conductivity in the 30–350 K range of hard
fullerene-based carbon. The material has been prepared by pressure–temperature treatment of
C60 fullerene. It shows an unusual combination of high hardness and relatively high electrical
−
1
conductivity. Its thermal conductivity is about 5.7 W mK at room temperature and decreases
almost linearly in the investigated temperature range. Excellent linearity in a wide temperature
range may arise from some sort of crystalline order in the transformed substance. Obtained data
are compared with thermal properties of C60, glassy carbon and diamond. The applicability of
recent theoretical models for explanation of recorded data is discussed.
Investigation of thermal transport in various forms of carbon is of principal importance
from a theoretical as well as a practical point of view. The thermal conductivity of perfect
−1
crystalline diamond is λ = 3500 W mK at room temperature [1], the highest value known
for crystalline solids. Scattering of phonons on defects and imperfections of a crystal lattice
substantially decreases the thermal conductivity of the substance. That is why in amorphous
carbon the thermal conductivity has a smaller magnitude and normally lies in the range 2–
−
1
4
00 W mK [2]. Finally, C60 crystals or fullerites usually described as solids of the
Van der Waals type due to their molecular nature have very small thermal conductivity
−
1
(
λ = 0.4 W mK ) [3]. Detailed investigation of the thermal conductivity of C60 measured
under pressure enabled the mapping of its phase diagram and gave some new elements for
clarifying the possibility of polymerization of fullerene molecules [4].
Recently a transformation of C60 into a very hard form of carbon (HC) has been reported
[
7
5]. The material was obtained at relatively soft conditions (pressure, 3 GPa; temperature,
◦
00 C) in the form of bulk specimens. It shows the unusual combination of remarkable
−
2
micro-hardness (more than 4000 kg mm ) and substantial electrical conductivity (about
1
−
1
−3
00 ( cm) ). The estimated density of HC, ρ ≈ 1.9 g cm , is bigger than the density of
−3
−3
initial C60 (1.7 g cm ), but significantly lower than the density of diamond (3.5 g cm ). X-
ray diffraction study shows a lack of long-range crystalline order in the transformed material.
Films prepared from HC using laser sputtering in vacuum reveal, however, distinct evidence
of microcrystalline diamond [6]. Electron diffraction and Raman measurements of the films
allow us to suggest the existence of some crystalline order in the substance at least in a short-
range scale. It may be relevant to such a hypothetical form of carbon as hollow diamonds
[7], the material predicted theoretically, that comprises arrays of covalently interconnected
fullerene cages and should have an excellent thermal conductivity. As a continuation of the
§
On leave from the Institute of Semiconductors, Kiev, Ukraine.
0
953-8984/96/030027+06$19.50 ꢀc 1996 IOP Publishing Ltd
L27

L28
Letter to the Editor
characterization of HC we report data on thermal transport and compare them with those
for glassy carbon, artificial diamond and C60 crystals.
Thermal conductivity was measured by a steady-state comparative method in the
temperature range 30–300 K. It was performed relative to a constantan foil [8] and,
independently, relative to a crystal of artificial diamond (a.d.) of Ib type. The latter was
3
grown by Sumitomo Electric Industries, had dimensions 2 × 2 × 0.3 mm and orientation
(
100). For comparison measurements were carried out also on a sample of glassy carbon
3
fabricated by Nikon Carbon Ltd, company brand VCP-5, of a size 3 × 1 × 1 mm . The
sample of HC prepared from pure C60 as described previously [6] was cut in the form of
3
a baton (3.5 × 1 × 0.6 mm ). It was attached with GE varnish to the referent sample that
was also used as a thermal link between the sample and the heater. On the other side
the sample was glued directly to a thermal sink. The sample character was maintained
−4
in vacuum better than 10 Pa. The heat flow was estimated from the voltage difference
between two chromel wires (8 = 15 µm) point-soldered to the constantan foil, so the link
was also used as a thermocouple element. The thermal gradient in the sample was measured
by a thin chromel–constantan thermocouple (8 = 15 µm) glued to the sample with GE
varnish. The temperature difference measured on the sample was always smaller than 1 K
when constantan was used as reference. Due to the higher thermal conductivity of a.d.,
independently used as a reference, the measured temperature difference on the sample was
slightly larger depending on the temperature range. This was determined by the lowest
voltage difference of the thermocouple on the a.d. sample (∼1 µV) necessary to achieve
sufficient resolution. In order to minimize the radiation heat losses, the sample holder was
closely surrounded by the heat shield, kept at almost the same temperature. The cooling
−1
(
(
heating) rate was generally ∼5 K h or less. The relative accuracy of the measurement
1–2%) was much better than that for absolute values (about 20% due to uncertainty in
geometrical factors of the sample and the reference). Electrical resistivity was measured
independently on the same sample by a conventional four-probe method. Contacts were
made using conducting silver paste directly on the surface of the sample.
The thermal conductivity of HC at room temperature (R.T.) is about 5.7 W mK−1.
The value is much higher than for crystalline C60 [3], but significantly lower than for
diamond crystals [1]. The thermal conductivity of HC measured relative to an a.d. crystal
in the temperature range 120–290 K is more than two orders of magnitude lower and
it decreases monotonically towards lower temperatures as shown in figure 1. The inset
in figure 1 exhibits the temperature dependence of the thermal conductivity of the a.d.
−
1
(measured relative to constantan). The absolute value at R.T. does not exceed 700 W mK
indicating substantial amounts of impurities. The two sets of data for HC measured relative
to constantan and relative to a.d., regardless of very different measuring conditions, agree
very well. A small difference in the absolute values apparently originates from the estimation
of geometrical factors. The comparative measurement of the sample of glassy carbon shown
in figure 3 demonstrates similar temperature behaviour but a 50% smaller R.T. value than
for HC. The most intriguing fact is an almost perfect linear temperature dependence of the
thermal conductivity of HC in the whole temperature range and we are going to discuss
this.
Usually total thermal conductivity, λT , is considered as the sum of the lattice, λg,
and the electronic, λe, contributions: λT = λg + λe. As HC is electrically conducting,
an estimation of the possible contribution of free carriers becomes necessary. In ordinary
metals, it would suffice to evaluate λe at high temperatures from the Wiedemann–Franz
−
8
−2
law: λeρ/T = L0, where L0 = 2.45 × 10 W  K is the Lorentz number and ρ is the
dc resistivity. The electrical resistivity of our sample at room temperature is relatively low,

Letter to the Editor
L29
Figure 1. The temperature dependences of the thermal conductivity of HC measured relative
to constantan (
◦
) and relative to a crystal of a.d. (ꢀ). The inset shows data for a.d. measured
relative to constantan.
about 85 µ m. While cooling down to 40 K it increases by about 30% (figure 2). The
temperature dependence is in agreement with previously published data [5] and substantially
deviates from the activated regime typical for usual glassy carbon [9]. It resembles a
semimetallic behaviour rather than a semiconducting one [10]. Nevertheless, the small
overlap of the valence and conduction bands appearing in semimetals can be regarded as a
negative energy gap and some possible effects as in the heat conduction of semiconductors
should be taken into account. In addition to the electronic contribution (electrons and holes),
there is also so called bipolar diffusion originating from recombination of electrons and holes
[11]. The free carrier contribution can be treated through the Wiedemann–Franz law but
with a so called effective Lorentz number that can greatly exceed the above-given Lorentz
value. However, it seems to be unlikely that the effective Lorentz number in this almost
semimetallic material could surpass the value of 10 [12]. Even in this extreme condition the
free carrier contribution should be in the range of the absolute accuracy of our measurement.
The electronic contribution λe for our sample calculated using the simple Wiedemann–Franz
law and experimental resistivity data is shown in figure 2. Although the λe temperature
dependence is similar to that for the total thermal conductivity, its contribution to λT is less
than 1%.
Thermal conduction in HC originates mainly from lattice vibrations and it has
temperature behaviour very different from that of crystalline C60 [3]. They are compared
in figure 3. The conventional behaviour of λ(T ) of crystalline materials is a roughly
1/T decrease at higher temperatures (above the Debye temperature) and a peak at low
temperatures (not shown for C60 due to lack of low-temperature data in [3]). This is

L30
Letter to the Editor
Figure 2. The temperature dependences of the electrical resistivity of HC (ꢀ) and the estimated
electronic contribution to the thermal conductivity λe ( ).
◦
characteristic of heat carried by phonons with a mean free path limited by ordinary ‘umklapp’
scattering. On the other hand, liquids and amorphous materials represent systems in which
phonons are not well defined and they exhibit unconventional λ(T ). There are also some
other examples of unconventional behaviour of λ(T ) in the case when the unit cells become
very complicated. The lack of local periodicity may cause a change from crystalline
behaviour into the amorphous one as has been found for YB68 [13].
The typical behaviour of thermal conductivity for amorphous solids and certain
disordered crystals is illustrated in figure 3 by the example of SiO2 [14]. It follows
approximately the T 2 law at low temperatures (T < 1 K), has an almost temperature-
independent ‘plateau’ between 1 K and 10 K and increases with T at higher temperatures.
The low-temperature (T < 1 K) part of λ(T ) has been explained by a phenomenological
model in which the two-level-state (TLS) excitations [15] are the principal scatterers for
the phonons. TLSs are believed to arise from quantum tunnelling of atoms or molecules
but their microscopic origin is still unknown. For temperatures above the plateau, there are
still some controversies and open questions.
As mentioned, unlike crystalline insulators, where propagating short-wavelength
phonons scattered by anharmonic interactions at high temperatures cause a dependence
λ(T ) ∝ 1/T , noncrystalline solids seem to reveal a behaviour λ(T ) ∝ T in approximately
the same temperature range. The thermal conductivity for HC in the temperature range
(
1.00±0.03)
available fits very well a linear dependence λ(T ) = 0.022T
. The comparative
measurement of the sample of glassy carbon also shows the quasilinear behaviour
(
1.33±0.03)
in the temperature range 120–350 K. HC has a certain degree
λ(T ) = 0.002T

Letter to the Editor
L31
Figure 3. A comparative plot illustrating the conventional λ(T ) for a crystalline sample of C60
N, from [3]), and the unconventional behaviour of a typical amorphous material a-SiO2 (
from [14]) and our measured data for HC ( ) and glassy carbon (ꢀ).
(
•
,
◦
of amorphization as evidenced by very broad diffuse peaks obtained by x-ray diffraction
as well as by Raman scattering [5]. However, the material is not completely amorphous.
It has the sort of crystalline order that sometimes reveals itself in Raman and electron
diffraction measurements. This is probably the main reason that λ(T ) of this material fits
approximately the same frame as for SiO2 and glassy carbon above their plateau.
The problem of the heat transport above the plateau has been widely discussed in the
last decade (see review [16] and references therein). There have been two different concepts
for the explanation of the linear temperature dependence of λ(T ) above the plateau: the
‘
[
resonance’ interpretation [17] and the picture of an additional heat conduction channel
18]. In the first approach the anharmonic modes that can carry heat are supposed to be
introduced at a temperature above the plateau. They are scatterers for TLSs and as the
population of TLSs decreases linearly with T the scattering of anharmonic modes decreases
in the same manner, causing λ(T ) ∝ T . The second model [18] although originally based on
the fracton model should also be valid for more conventional localized vibrations [19]. The
recent simulations by means of the nonequilibrium molecular-dynamics (MD) method [16]
support the model of anharmonicity-induced hopping heat transfer in glasses. Presuming
that the aperiodicity of amorphous solids is their only common feature, the MD simulation
has demonstrated the presence of localized vibrational modes that are the source of the
strong anharmonicity in amorphous solids. It has been shown in addition that the influence
of quasiperiodicity on λ(T ) is the same as that for a simple aperiodicity.
To our knowledge HC is the unique material obeying a linear T dependence of the
thermal conductivity in such a wide temperature range. Unfortunately we did not reach
the plateau region that is necessary for the estimation of some parameters in the model of

L32
Letter to the Editor
anharmonicity-induced localized vibrational state hopping. However, it looks as if in this
specific material there is no departure of the thermal conductivity from linear dependence
upon temperature as observed in some amorphous systems. According to the same model
this means that there is no violation of the temperature-dependent condition on the hopping
lifetimes of the localized vibrational excitations that might cause the curling over of λ(T )
from its linear temperature dependence and this fact is related to the degree of anharmonicity
[18]. HC reveals some sort of crystalline order at least in the short-range scale. The
unusual mechanism of thermal transport in HC is presumably caused by the characteristic
microstructure (or mesostructure) of this material. It provides an anharmonic interaction that
is sufficient to invoke the hopping conductivity but not large enough to quench it. Further
structural investigation of HC should clarify this point.
In summary, the main characteristics of the thermal transport in the new material HC
have been discussed. A comparative study of the thermal conductivity in HC, crystalline
diamond, glassy carbon and molecular C60 crystals shows that it correlates with the density
of these compounds. Covalent linking of C60 cages is apparently responsible for the drastic
increase of the thermal conductivity in HC at 300 K compared with pristine C60.
A
3
substantial number of sp sites in its ‘polymerized’ structure causes the increase of the
thermal conductivity relatively to glassy carbon. However the disordered structure of HC
makes the room-temperature values much smaller than those in crystalline diamond. The
existence of the short-range crystalline order in HC can be related to the excellent linearity
of λ(T ) in a broad temperature range. This material seems to be very promising for the
verification of actual theories of thermal transport in amorphous solids.
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