Novel carbon materials: new tunneling systems (Review Article)
This review covers recent achievements in the studies of quantum properties of the novel carbon materials (fullerite C₆₀ and bundles of single-walled nanotubes (SWNT)) saturated with such light-mass species as helium isotopes, the homonuclear molecular hydrogens, and neon. It is shown that even some...
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irk-123456789-1184462017-05-31T03:03:54Z Novel carbon materials: new tunneling systems (Review Article) Strzhemechny, M.A. Dolbin, A.V. 9th International Conference on Cryocrystals and Quantum Crystals This review covers recent achievements in the studies of quantum properties of the novel carbon materials (fullerite C₆₀ and bundles of single-walled nanotubes (SWNT)) saturated with such light-mass species as helium isotopes, the homonuclear molecular hydrogens, and neon. It is shown that even some heavy dopants demon-strate kinetic phenomena, in which coherent effects play an essential role. Two theoretical concepts are surveyed which have been suggested for the explanation of the anomalous phenomena in saturation kinetics and linear thermal expansion of doped C60. Most unusual effects have been also observed in the low-temperature radial ex-pansion of bundles of single-walled carbon nanotubes saturated with the helium isotopes. First, it was shown that low-temperature radial expansion of pure SWNT is negative, i.e., a nanotube shrinks with warming. Second, sa-turation of SWNT bundles with the helium isotopes entails a huge increase of the negative expansion effect, when the dopant is He. So far, no detailed physical picture has been put forward. It is worth mentioning that the dynamics of a single helium atom on an isolated nanotube corresponds to that of a tight-bound quasiparticle with a band width of about 10 K. 2013 Article Novel carbon materials: new tunneling systems (Review Article) / M.A. Strzhemechny, A.V. Dolbin // Физика низких температур. — 2013. — Т. 39, № 5. — С. 531–540. — Бібліогр.: 46 назв. — англ. 0132-6414 PACS: 81.05.U–, 61.72.jj, 65.40.–b, 85.40.Ry http://dspace.nbuv.gov.ua/handle/123456789/118446 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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9th International Conference on Cryocrystals and Quantum Crystals 9th International Conference on Cryocrystals and Quantum Crystals Strzhemechny, M.A. Dolbin, A.V. Novel carbon materials: new tunneling systems (Review Article) Физика низких температур |
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This review covers recent achievements in the studies of quantum properties of the novel carbon materials (fullerite C₆₀ and bundles of single-walled nanotubes (SWNT)) saturated with such light-mass species as helium isotopes, the homonuclear molecular hydrogens, and neon. It is shown that even some heavy dopants demon-strate kinetic phenomena, in which coherent effects play an essential role. Two theoretical concepts are surveyed which have been suggested for the explanation of the anomalous phenomena in saturation kinetics and linear thermal expansion of doped C60. Most unusual effects have been also observed in the low-temperature radial ex-pansion of bundles of single-walled carbon nanotubes saturated with the helium isotopes. First, it was shown that low-temperature radial expansion of pure SWNT is negative, i.e., a nanotube shrinks with warming. Second, sa-turation of SWNT bundles with the helium isotopes entails a huge increase of the negative expansion effect, when the dopant is He. So far, no detailed physical picture has been put forward. It is worth mentioning that the dynamics of a single helium atom on an isolated nanotube corresponds to that of a tight-bound quasiparticle with a band width of about 10 K. |
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Strzhemechny, M.A. Dolbin, A.V. |
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Strzhemechny, M.A. Dolbin, A.V. |
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Strzhemechny, M.A. |
| title |
Novel carbon materials: new tunneling systems (Review Article) |
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Novel carbon materials: new tunneling systems (Review Article) |
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Novel carbon materials: new tunneling systems (Review Article) |
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Novel carbon materials: new tunneling systems (Review Article) |
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Novel carbon materials: new tunneling systems (Review Article) |
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novel carbon materials: new tunneling systems (review article) |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Novel carbon materials: new tunneling systems (Review Article) / M.A. Strzhemechny, A.V. Dolbin // Физика низких температур. — 2013. — Т. 39, № 5. — С. 531–540. — Бібліогр.: 46 назв. — англ. |
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Физика низких температур |
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1837087541174468608 |
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© M.A. Strzhemechny and A.V. Dolbin, 2013
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5, pp. 531–540
Novel carbon materials: new tunneling systems
(Review Article)
M.A. Strzhemechny and A.V. Dolbin
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: strzhemechny@ilt.kharkov.ua
Received February 1, 2013
This review covers recent achievements in the studies of quantum properties of the novel carbon materials
(fullerite C60 and bundles of single-walled nanotubes (SWNT)) saturated with such light-mass species as helium
isotopes, the homonuclear molecular hydrogens, and neon. It is shown that even some heavy dopants demon-
strate kinetic phenomena, in which coherent effects play an essential role. Two theoretical concepts are surveyed
which have been suggested for the explanation of the anomalous phenomena in saturation kinetics and linear
thermal expansion of doped C60. Most unusual effects have been also observed in the low-temperature radial ex-
pansion of bundles of single-walled carbon nanotubes saturated with the helium isotopes. First, it was shown that
low-temperature radial expansion of pure SWNT is negative, i.e., a nanotube shrinks with warming. Second, sa-
turation of SWNT bundles with the helium isotopes entails a huge increase of the negative expansion effect,
when the dopant is He. So far, no detailed physical picture has been put forward. It is worth mentioning that the
dynamics of a single helium atom on an isolated nanotube corresponds to that of a tight-bound quasiparticle with
a band width of about 10 K.
PACS: 81.05.U– Carbon/carbon-based materials;
61.72.jj Interstisials;
65.40.–b Thermal properties of crystalline solids;
85.40.Ry Impurity doping, diffusion and ion implantation technology.
Keywords: C60, bundles of carbon nanotubes, interstitial doping, thermal expansion, tunneling, negative
expansivity.
Contents
1. C60 fullerite properties ........................................................................................................................ 531
2. Experimental methods and samples .................................................................................................... 532
3. Thermal expansion of pure and doped C60 at low temperatures .......................................................... 533
4. Diffusion of light-mass particles in C60 .............................................................................................. 535
5. Radial expansion of carbon nanotubes ................................................................................................ 537
Conclusions ............................................................................................................................................. 538
Appendix A: Grüneisen parameter .......................................................................................................... 538
References ............................................................................................................................................... 539
To Prof. V.G. Manzhelli, a good collegua and a great teacher.
Although the time elapsed since the discovery of the
new carbon nanomaterials is rather short the Nobel Prize
Committee had twice recognized [1,2] their striking novel-
ty and importance for fundamental science. Researchers
attacked these materials from every imaginable angle and,
seemingly, after 20 years of these efforts really nothing
new could be found in the properties of these materials.
However, it turned out that the new is always at hand. This
brief review presents unusual results recently obtained for
C60 and single-walled carbon nanotubes doped with light-
mass species like helium and hydrogen isotopes. All expe-
rimental results have been obtained at the Verkin Institute
for Low Temperature Physics and Engineering.
1. C60 fullerite properties
For easier understanding we give below a short com-
pendium of the properties of the C60 which we will be
mentioning in the review. At room temperature C60 adopts
a fcc structure [3], determined mostly by van der Waals
forces. At room temperature the lattice parameter is
14.16 Å and the nearest neighbor distance is close to 10 Å;
the molecules rotate more or less freely.
M.A. Strzhemechny and A.V. Dolbin
532 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5
Near 260 K, molecular rotations cease to be completely
chaotic because the anisotropic interactions become com-
parable with the kinetic energy of rotation. Below 260 K a
(partial) orientational ordering sets in, the molecules exe-
cuting thermally activated jump-like rotations around cer-
tain axes [4]. As the temperature is decreased, the rotation-
al jumps become progressively slower to stop completely
at approximately 100 K. Below this point one deals with a
new state which was termed “orientational glass”. The rea-
son for this is as follows. The fullerene molecule C60 is
highly symmetric. In particular, it has many 3-folds axes,
which go through two opposite hexagons. The molecular
array between 100 and 260 K is such [5] that one of the
double bonds in one molecule faces either a pentagon or a
hexagon of the respective neighbor molecule as shown in
Fig. 1. That means that in the former case the double bond,
an entity with an elevated electron density, is closermost to
a pentagon, which consists only of single bounds, that is,
an entity with a depleted electron density. Evidently, the
latter case is preferable as far as the electrostatic energy is
concerned. Thus, when a molecule rotates through /6 it
goes from one potential well to a shallower or deeper one.
The important moment is that the energy gain in Fig. 2
by more than an order of magnitude exceeds the potential
barrier (at least, E1). The quantity controls the thermo-
dynamic probabilities while the barrier height determines
the characteristic time of events. If becomes much
longer than the typical experimental time, the correspond-
ing events, even urged by the thermodynamics, could hard-
ly happen. Because of that, as the temperature goes below
approximately 100 K the molecules cease to rotate com-
pletely and the fullerite is said to be in the state of orienta-
tional glass. The fraction of energetically unfavorable mu-
tual pairwise molecular orientations is frozen [6] at
roughly 16%.
The large size of the molecules that constitute the lattice
of the molecular crystal fullerite C60 explains why the in-
terstitial voids are sufficiently roomy to easily accommo-
date atoms and smaller molecules. The diameters of these
voids (cf. Fig. 3) amount to 4.2 and 2.2 Å for octahedral
and tetrahedral voids, respectively.
2. Experimental methods and samples
The experimental techniques employed in the studies
under survey were high-precision dilatometry, high-
precision low-pressure measurements, and powder x-ray
diffraction. Every of these techniques necessitated rather
involved special sample preparation procedures, which had
first to be invented and then put into life. The high-precision
dilatometry setup published in sufficient detail elsewhere
[7,8] is unique as far as its accuracy of 2 10
–9
cm is con-
cerned. Since the quantity measured is the uniaxial elonga-
tion caused by warming or cooling, the structure of the
solid under study should be preferably cubic, which luckily
Fig. 1. A double bond (elevated electron density), marked with a
tiny circle, of any molecule faces either a hexagon (b) (elevated
electron density) or a pentagon (a) (lower electron density). The
latter mutual orientation is energetically preferable.
Fig. 2. A sketch of the electrostatic interaction energy of two
neighboring C60 molecule as a function of the rotation angle of
one of these molecules.
Fig. 3. Positions of the voids in the fullerite lattice.
Novel carbon materials: new tunneling systems
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5 533
is so with fullerite C60. The best accuracy is reached for
long enough samples (a few centimeters). No single crystal
of C60 of that size exists, therefore a powder C60 was com-
pressed under not too severe conditions (pressures below
1 MPa). Since C60 saturated with various dopant species
was to be studied, the samples were produced first by satu-
ration (in Australia, gas pressure of about 200 MPa and
temperature of 575 C), then compressed (in Sweden) and
shipped to Ukraine for final investigation.
A still more stringent requirements apply to SWNT
samples to be investigated using the dilatometric method.
Usually the nanotube samples are bundles, in which the
number and types of nanotubes may differ significantly. To
tackle the task, nanotube samples consisting of unknown
number of bundles were compressed to a cylinder (Fig. 4),
in which all nanotubes, however randomly oriented, lay
flat with each axis (albeit, arbitrarily twisted in plane) pre-
dominantly at the same particular distance from bottom.
Such a configuration allowed measurement of the radial
expansion coefficient [9]. Another technique that proved
very efficient for measurements of saturation levels to high
accuracy was high-precision low-pressure monitoring [10].
The x-ray diffraction studies were carried out in a standard
way, the distinguishing feature being that saturation with
light-mass dopants (He, H2) was affected in situ at a con-
stant low pressure (about 1 bar), which allowed accurate
measurement of variations of the lattice parameter of C60
in time and subsequent evaluation of the respective diffu-
sion coefficients. Molecular hydrogen was also stuffed at
elevated (temperatures up to 300 C) and pressures (up to
30 kbar) in a special saturator.
3. Thermal expansion of pure and doped C60 at low
temperatures
Very first measurements [8] of nominally pure C60
yielded negative linear expansion coefficients below 5 K,
which served as a big incentive to begin systematic studies
as well as to improve accuracy. Since at low temperatures
C60 is in the orientational glass state, the first explanation
[11] was based on the assumption that the states responsible
for negative are tunnel rotations of C60 molecules.
More accurate later dilatometric investigations showed
that pure C60 does not feature negative expansivity. The
peculiarity of low-temperature dilatometry measurements
consists in the fact that only the initial warm-up run very
often yields negative expansion coefficients. The next ones
(after the first warm-up and cool-down runs) do not result
in negative expansivities. But no matter what, every time
dependence of the sample elongation on warmup consists
of two contributions of different signs as depicted in Fig. 5,
which indicates that there are two competing mechanisms,
one of which is due to phonons while the other is presuma-
bly related to the dopant.
When studying thermal expansion of fullerite, in which
octahedral voids were filled with heavy rare gas atoms like
Ar or Xe, it was found [11,12] that the low-temperature
expansion coefficients, if measured during warmup, are
much lower than those measured during cooldown. In oth-
er words, a clearly detectable hysteresis has been docu-
mented as depicted in Fig. 6. The following explanation
has been suggested. The low-temperature hysteretic phe-
nomena were treated as manifestations of a first-order
phase transition between several orientational glass states,
which phenomenon is known as “polyamorphism”. Transi-
tions between those orientational-glass states are poly-
amorphic transformations, which can be of first order.
However, it should be mentioned here that x-ray diffrac-
tion experiments [13] actually on the same sample revealed
an incomparably larger hysteresis in the temperature de-
pendence of the lattice parameter, spanning a range from
150 K across the phase transition point at 260 K to about
Fig. 4. A view of the sample of single-walled carbon nanotube
bundles compacted so that axes of all nanotubes are perpendicu-
lar to the cylinder axis.
Fig. 5. Time dependence of the elongation ∆L of a C60–Xe0.3
sample upon heat pulse input, showing two thermal expansion
contributions of opposite signs: experimental data (1); positive
(2) and negative (3) contributions.
M.A. Strzhemechny and A.V. Dolbin
534 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5
Fig. 7. Four lowest energy levels as a function of perturbation
energy parameter , the parameter at the octahedral invariant.
The dash curves are absent in Devonshire’s theory.
300 K. On the other hand, since the temperatures are low
enough to exclude thermal activation, the next assumption
was that the rotations, which take place between orienta-
tional glass states, should have tunnel nature, which was
also substantiated by negative expansivities. The relevant
theory was developed by Bakai [8,11,14]. Later, Loktev
and coworkers within a 2D model [15,16] showed that
small-angle tunnel rotations are feasible in irregular areas,
like dislocations or grain boundaries. An important consid-
eration in the above approach was the following. A suffi-
ciently large dopant particle expands the lattice, thereby
lowering the energy barriers, which hamper molecular ro-
tations. As a consequence, tunnel rotations become more
likely bringing about negative expansion and making wider
the temperature range of negative .
A different approach has been recently suggested [17]
in which the negative expansion was ascribed entirely to
dopants. The basis for this approach was two experimental
facts: first, pure C60 does not show negative expansion
and, second, this effect has been observed [18] for more
than ten dopant types, both molecular and atomic. In the
case of a molecular dopant, the negative expansion could
be easily explained by referring to Devonshire’s model of a
linear molecule in octahedral fields [19]. Yet, a similar
effect was observed with rare gas dopants, even so heavy
as xenon. This seemingly troublesome issue was explained
[17] as being due to the fact that the first excited state of an
atom in a three-dimensional well is a triplet, which is de-
generate if the well is isotropic. But even in this case, the
first triple-degenerate excited state can be represented as a
set of appropriate combinations of three spherical harmon-
ics, which corresponds to quantum rotation states. The
potential energy U of a Xe atom in an octahedral cage
was evaluated using the known [20] Lennard-Jones poten-
tial between C and Xe atoms. The angular dependent part
of the potential energy was truncated up to the rank-4 oc-
tahedral invariant: 0 4 4( , ) ( ) ( ) ( ),U r U r U r Iw w where
r is the distance of the Xe atom from the void center; w
designates angular variables; 0( )U r is the isotropic part of
the potential, which for Xe is very close to the harmonic
one; 4( )I w is the rank-4 octahedral invariant; is the
parameter, which for the particular case of Xe is known but
was varied to obtain a general dependence. Account for the
anisotropy of a real octahedral well results in a tunneling-
related splitting (Fig. 7) of the energy spectrum resem-
bling that of the Devonshire model. Thus, the negative
contribution to the expansion coefficient can come from
the transitions between levels of the manifold formed due
to tunneling (the lower four curves) and, thus, are charac-
terized by a negative Grüneisen parameter. On the other
hand, transitions between different manifolds are charac-
terized by positive Grüneisen parameters, providing an
additional contribution to the positive expansivity as a
function of temperature in the form of broad maxima.
Introduction of methane and deuteromethane molecules
(for octahedral void occupancies about 50%) brought about
a substantial (about 30%) decrease of thermal expansivities
(Fig. 8) compared to pure C60, whereas rare gas doping
led to higher values as shown in Fig. 6. This finding
was unexpected, since the van der Waals radii of both me-
thane and deuteromethane compares with octahedral void
size. It was known [21] that methane doping at low tem-
peratures entails a measurable (about 0.12%) lattice expan-
sion, which decreases the rotation barriers and thereby fa-
cilitates rotational tunneling. Therefore, the decrease of the
Fig. 6. Temperature dependence of the thermal expansion coeffi-
cient in the following experiments: warmup of C60–Xe0.3 (solid
curve 1); cooldown of C60–Xe0.3 (solid curve 2). Dash curve 3 is
for pure C60.
Novel carbon materials: new tunneling systems
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5 535
thermal expansion in C60 doped with the methanes can be
related with tunnel molecular rotations, which in turn (see
also Appendix) ensures a negative contribution to the
thermal expansion. A stronger effect of CH4 can be ex-
plained by its more pronounced quantum nature compared
to the heavier isotope with its larger moment of inertia.
Under warmup within the range 4 to 5.5 K both systems
(CH4)0.5–C60 and (CD4)0.5–C60 showed peaks in the tem-
perature dependence of (Fig. 9). These features have
been tentatively explained [22] as being due to a polymor-
phic orientational phase transformation point present within
this temperature range. Evidence for this conclusion comes
from sharp peaks in the temperature dependence of the cha-
racteristic times of the positive contribution to . Such
peaks were observed (Fig. 10) for dopants characterized by a
strong noncentral interaction with C60 molecules.
4. Diffusion of light-mass particles in C60
Previous investigations of absorption rates of various
gases in C60 have been carried out at sufficiently high tem-
peratures (from room to 300 C), for otherwise the process
would have been too slow and the resulting saturation le-
vels, too low, at least from the technologic standpoint. Ac-
curate measurements which can result in reasonably precise
diffusion rates are long to measure, especially if the experi-
ment is performed at low pressures to avoid overstuffing
problems. It was established [25] that He infusion at room
temperature into a C60 powder with an average grain size of
about 9.2 m is a two-stage process as shown in Fig. 11.
Knowing the grain size and deducing the diffusion time
from a fit to the relevant diffusion theory curve, the diffu-
sion coefficient was evaluated to be D 7.5 10
–14
cm
2
/s
for the process of octa-void filling. Filling of tetrahedral
voids occurs much slower during characteristic times which
are an order of magnitude longer than during octa-filling.
Fig. 8. “Cooldown” temperature dependence of the linear thermal
expansion coefficient of fullerite C60 doped with methane and
deuteromethane.
Fig. 9. Part of the “warmup” temperature dependence of the ther-
mal expansion coefficient of fullerite C60 doped with methane
(curve 1) and deuteromethane (curve 2). Curve 3 is for pure C60.
Fig. 10. Temperature dependence of the characteristic times for the
positive contribution to thermal expansivity [22–24] in C60 doped
with CO and the methanes: (CO)0.9–C60 (circles); (CH4)0.5–C60
(squares); (CD4)0.5–C60 (crosses).
Fig. 11. The C60 lattice parameter variation in time [28]. The
brown vertical line demarcates the regions with prevailing O-void
(left) and T-void (right) filling.
M.A. Strzhemechny and A.V. Dolbin
536 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5
The process of He diffusion in fullerite C60 over the
wide temperature range from 2 to 300 K was studied [26]
using the pressure monitoring technique mentioned in
Sec. 2. The quantity measured was the pressure P in the
work cell as a function of time t since the beginning of
saturation or degassing. An example of experimental runs
at three temperatures is shown in Fig. 12. The total amount
of the He gas introduced before saturation was reasonably
small, the helium to carbon atomic ratio not exceeding 0.1.
It should be noted that the values obtained for degassing
and saturation did not differ significantly. The pressure vs.
time curves, like those shown in Fig. 12, were fitted to a
sum of two exponentials with two characteristic times to
be determined from fitting. These two sets ( )T were
treated as describing the kinetics of filling octahedral and
tetrahedral voids with He atoms. Taking the typical grain
size r to be 1 m, it was easy [26] to evaluate the diffu-
sion coefficients from Einstein’s relation 2= /6D r for
both helium isotopes as depicted in Fig. 13.
It is worthwhile to compare the room-temperature dif-
fusion coefficients derived from pressure monitoring (see
Fig. 13) D = 8 10
–12
cm
2
/s and from x-ray diffraction [28]
D = (2.8 0.8) 10
–14
cm
2
/s, i.e., the former to latter D
value ratio is 285 while the corresponding grain size ratio
squared used in estimates is 85, i.e., we come to an almost
3-fold difference, which is quite acceptable for diffusivity
evaluations.
Since the general picture resembles very much corres-
ponding curves for the quantum diffusion coefficient of
3
He impurities in solid
4
He in the regime of ultimately low
3
He fractions [27,28] (see also Fig. 14), we recapitulate the
main points of this phenomenon and corresponding con-
siderations. At temperatures close to the melting point the
diffusion is of purely thermally activated nature. If the
3
He
fraction is not low enough (as 1X in Fig. 14), the deforma-
tion-related interaction between impurity atoms is suffi-
ciently strong to render D temperature independent below
a certain temperature. In this regime, 1/D x at low tem-
peratures [29]. In strongly diluted helium isotope solid
mixtures, the interaction between impurities is sufficiently
weak, which allows phonons to come into play (concentra-
tion 2X and 3X in Fig. 14), and here 9.D T The pho-
non contribution to the scattering cross-section diminishes
steeply as the temperature goes down and at a sufficiently
low T the main role goes to the inter-impurity interac-
tions. In this region, again, 1/ .D x
Applying similar reasoning to what is depicted in
Fig. 13, one can surmise that there is a clear analogy be-
tween the two phenomena under discussion. At higher
temperatures (above roughly 100 K) the diffusion of He
isotopes in C60 is controlled by thermal activation. Below
approximately 100 K the diffusion coefficient goes up with
decreasing temperature. It should be remarked here that
this turning point is very close to the temperature gT
where the orientational glass state completes its formation.
Fig. 12. Time dependence of the He gas pressure in the sample
chamber during degassing for three typical temperatures.
Fig. 13. Diffusion coefficients for the octahedral and tetrahedral
voids for two helium isotopes. The circles are for octahedral
voids (empty symbols for
4
He, solid symbols for
3
He), the
squares are for tetrahedral voids (empty symbols for
4
He; solid
symbols for
3
He).
Fig. 14. Quantum diffusion of
3
He in solid
4
He for a range of
3
He
fractions.
Novel carbon materials: new tunneling systems
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5 537
Below the “turning point” near gT the diffusion coeffi-
cient increases quite fast as the temperature goes down,
which is very possibly due to the diminishing role of pho-
nons with decreasing temperature. As will be argued be-
low, the fact that gT is a “turning point” is not fortuitous.
Although the energetics of the He atom in the lattice
voids of C60 was calculated in sufficient detail [30] in the
atom-atom approximation with the relevant constant bor-
rowed from data on He scattering on the graphite surface
[31], no consistent theory is available so far.
There is clear evidence (see Fig. 15) of a quantum cha-
racter of the diffusion of other light-mass particles such as
hydrogen molecules and neon atoms [32]. These impurities
fill only octahedral voids. Just like with the helium iso-
topes, increased diffusivities have been obtained for these
dopants at temperatures below the orientational vitrifica-
tion point .gT By contrast, the diffusion coefficient of
“classical” heavier particles like argon [33] goes down with
decreasing temperature over the entire temperature range of
experiment (58 to 290 K), which is evidence that the diffu-
sion is a thermally activated process. Yet, below gT an es-
sential decrease (from 410 K at higher temperatures to about
45 K at lower temperatures) of the activation energy occurs.
This is another confirmation of the cardinal influence of
abrupt changes in the rotational dynamics of C60 molecules
on the diffusion of dopant particles in fullerite.
Now concerning the role of the orientational glassifica-
tion point gT in dopant transport over the lattice voids. In
the case of a coherent (tunneling) process in a molecular
crystal with internal degrees of freedom, two types of dis-
order, static and dynamic, control the situation. Static dis-
order can be due to any deviation from the ideal periodicity
that shapes the dispersion law of diffusing particles as well
as due to other tunneling particles, because, as scatterers,
they are practically immobile. Dynamic disorder stems
from time variation of the surrounding of tunneling par-
ticles. Usually, the main source of time-dependent varia-
tions are all kinds of phonons. However, in the case of C60
there are unusual variations associated with molecular rota-
tions, which results in specific effects inherent only in C60.
Thus, the sharp changes in the rotational dynamics across
the phase transition at 260 K brings about a sharp response
in the electrical conductivity [34]. In addition, when the
temperature grows up across the orientational glassifica-
tion point gT and molecular rotations are getting “defro-
zen”, the photoluminescence emission starts to fade be-
cause rotational motion breaks up coherent transport of
triplet excitions moving to traps [35]. Summing up, one
may surmise that the turnover of the diffusion coefficient at
gT as shown in Fig. 13 occurs because below that point the
dynamic disorder stemming from molecular jump-like rota-
tions ceases to exist and only phonons remain in force, their
contribution diminishing fast with decreasing temperatures.
The temperature-independent part of the ( )D T dependence
at lowermost temperatures can be explained either by re-
maining static disorder of the orientational glass state or by
the interaction between diffusing He atoms.
5. Radial expansion of carbon nanotubes
Among neat carbon nanotubes [36,37] it is the single-
walled species that attract most attention of researches be-
cause they are least rigid and, as such, promise to support
novel quantum properties in these hybrids of one- and two-
dimensional systems.
As explained in Sec. 2, the cylinder-shaped compacted
sample, depicted in Fig. 4, ensures to a high degree of re-
liability that the quantity to be measured is the total of the
radial expansion of the nanotubes compressed in the sam-
ple. According to Schelling and Keblinski [38], owing to
the fact that a nanotube, as a wrapped-up graphite plane,
has a specific low-frequency phonon dispersion law, which
must entail a negative Grüneisen parameter. The dilatome-
tric measurements on a non-doped NT sample corroborated
[39] those predictions: below 6 K down to 2 K the thermal
expansion turned out to be negative (black curve in
Fig. 16). Saturation of bundles of single-walled carbon
nanotube with various sufficiently heavy particles like Xe,
O2, N2, etc. showed [40] that retained its negativity but
the region of negative values shrank towards lower ,T as
demonstrated in Fig. 16. The explanation is that the Xe
atoms physisorbed on the outer surfaces of SWNT bundles
suppress the bending phonon modes which are responsible
for negative values. Even doping NT bundles with hy-
drogen fits the picture.
When helium isotopes were used as dopants, the tem-
perature dependence of the thermal expansion coefficient
changed drastically. In Fig. 17 we show the radial expan-
sivities of the nanotube ensemble in bundles doped with
Fig. 15. Temperature dependence of the diffusion coefficients
over octahedral voids of Ar (red downward triangles) [33],
4
He
(blue empty circles [26]),
3
He (green solid circles [26]), H2 (emp-
ty upright triangles [32]), and Ne (magenta stars [32]) in double
logarithmic scale. Inset: data for argon doping within for a shorter
T range, showing a sharp kink in the ( )D T dependence.
M.A. Strzhemechny and A.V. Dolbin
538 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5
helium isotopes in comparison with the effect caused by
xenon and molecular hydrogen; the He to C atomic ratio
upon completion of the saturation process was about 9.4%.
The effect of
4
He doping proved [41] to be appreciable, to
say the least: the lowermost negative value in doped
C60 exceeded by more than an order of magnitude the val-
ue for the pure NT sample. Yet, the effect of the
3
He dop-
ing [42] was absolutely miraculous, the minimum value
was roughly one hundred (!) times larger than for doping
with the heavier helium isotope.
Considering the overall wealth of the experimental find-
ings we conclude that the most likely (and which, if not)
reason behind the phenomena documented in expansivity
measurements on the helium-doped sample is some quan-
tum-mechanical mechanism, which is still unknown. Even
plausible, however radical, suggestion are lacking. To get
insight into the problem, the nature of helium atom dynam-
ics on the surface of the specific single-walled nanotube,
for which the interaction energy of a He atom with its sur-
face was known [43], has been investigated theoretically
[44]. It was shown that helium isotopes tunnel over the
nanotube surface as a light-bound quasi-particle within a
band 10 K (for
4
He) or 14 K (for
3
He) wide.
Conclusions
In this short overview of recent findings in experiments
on the saturation kinetics of the novel carbon nanomate-
rials, C60 fullerite and single-walled nanotube bundles, we
have shown that there is a wealth of new phenomena
which could not be treated otherwise but as based on
tunneling. In particular, below approximately 100 K the
diffusion coefficients of the helium isotopes increase with
decreasing temperature to become almost temperature in-
dependent roughly below 10 K, the entire situation resem-
bling the behavior of the diffusion coefficient of
3
He in
solid
3
He. Bundles of single-walled carbon nanotubes,
when doped with
4
He and, especially,
3
He demonstrate
negative radial expansion, which is also a sure signature of
some tunnel mechanisms. The authors hope that the nu-
merous aspects of the phenomena surveyed will serve a
stimulus for theorists to develop models and approaches
for a better understanding of the underlying physics.
Appendix A: Grüneisen parameter
When the contribution of the several subsystems can be
treated as independent, the thermal expansion coefficient
can be represented in the form
ph ph tun tun= ,Tk
C C
V
(A.1)
where T is the isothermal compressibility; phC and tunC
are the heat capacities due to phonons and tunneling states,
respectively; ph and tun are the Grüneisen parameters
for phonons and tunneling states, which are defined as
ln
= .
ln
E
V
(A.2)
Here E is the characteristic energy level difference for
the corresponding subsystem. Thus, in rare gas solids the
attractive energy (proportional to
2 )V determines the
phonon thermodynamics, which results in typical values of
close to 2. The tunneling energy splitting E can be
represented [45] in the form
exp ( )/ ,E U V (A.3)
where is an expression of dimension “energy”. It can be
easily shown that the Grüneisen parameter for tunneling
states as defined in Eq. (A.3) is negative and can reach
values as high as a few hundreds.
Fig. 16. Radial thermal expansion coefficients of pure and Xe-
doped SWNT bundles at low temperatures.
Fig. 17. Effect of doping on the thermal expansion of SWNT
bundles: low-T region. Curve 1 is for
4
He; curve 2 is for H2;
curve 3 is for Xe; curve 4 is for pure NTs. Solid circles are
values for complete
3
He doping; empty circles are for the sample
after a ceratin part of dopant has been removed. Inset: a blowup
of the 2 to 4.5 K range, showing in detail the negative values
of
4
He-doped NTs.
Novel carbon materials: new tunneling systems
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5 539
Using high-precision experimental data on radial expan-
sivities and heat capacities, obtained on the same compacted
sample, the respective Grüneisen coefficient was estimated
[46]. It turned out to be strongly temperature dependent,
which is quite understandable, negative and equal to roughly
–10 at 2 K, changing sign at approximately 6 K, and level-
ling fast off at a value of 4 at higher temperatures.
Acknowledgment
The authors express their gratitude to V.G. Manzhelii
for his interest in this review and valuable remarks. Our
thanks are also due to Razet Basnukayeva for helping with
manuscript preparation.
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