Anomalous Coulomb diamonds and power-law behavior sensitive to back-gate voltages in carbon nanoscale peapod quantum dots
ABSTRACT We report anomalous charging effect of single electrons (Coulomb diamonds) observed in carbon nanoscale peapod quantum dots that encapsulate a series of C60 molecules. We find that behaviors of diamonds are anomalously sensitive to back-gate voltages (V_bg), exhibiting two evidently different V_bg regions and a large polarity on V_bg. In particular, we find only a sequence of one large diamond followed by three smaller ones existing around ground state. Magnetic-field dependence indicates the presence of shell filling by spin singlet to doubly degenerate electronic levels for these. The encapsulated-C60 molecules indirectly affect this shell filling at low V_bg possibly via nearly free electrons. In contrast, they act as individual quantum dots coupled in series in high V_bg region. It directly contributes to highly overlapped very large diamonds. Moreover, we report power-law behaviors on conductance versus energy relationships observed in the same carbon nanoscale peapods. We find that the values of powers are also highly sensitive to applied V_bg with three different regions and anomalously high at high source-drain voltages. Because the power laws are found at voltages, which are the nearest outside of the above-mentioned fourfold Coulomb diamonds, correlation of the anomalous powers with orbital-related Tomonaga-Luttinger liquid is discussed. PHYSICAL REVIEW B. v.75, n.20, 2007, p.205431-1-205431-7
- SourceAvailable from: Junji Haruyama[Show abstract] [Hide abstract]
ABSTRACT: We report findings on the asymmetrical current properties on both the source-drain and back-gate voltage (VBG) dependence (unconventional ambipolar behavior) found in a double-walled carbon nanotube (DWNT) field-effect transistor, which has electrode contacts to different layers. We also find Coulomb oscillations with a large charging energy observable only in +VBG region at low temperature. As origins for these phenomena, we discuss the possible presence of outer p- and inner n-type semiconducting layers, a corresponding interlayer nano-p-n junction, and a small quantum dot region in the inner n-layer exposed from the outer layer. Annealing of the DWNT in air atmosphere after synthesis allows change in only outer layer to p-type, remaining n-type behavior in the inner layer.Applied Physics Letters 04/2009; 94(14):143104-143104-3. · 3.52 Impact Factor
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ABSTRACT: We have measured systematic repetitions of avoided crossings in low temperature three-terminal transport through a carbon nanotube with encapsulated C60 molecules. We show that this is a general effect of the hybridization of a host quantum dot with an impurity. The well-defined nanotube allows identification of the properties of the impurity, which we suggest to be a chain of C60 molecules inside the nanotube. This electronic coupling between the two subsystems opens the interesting and potentially useful possibility of contacting the encapsulated molecules via the tube. Comment: 6 pages, 3 figuresPhysical review. B, Condensed matter 02/2010; · 3.77 Impact Factor
Anomalous Coulomb diamonds and power-law behavior sensitive to back-gate voltages in carbon
nanoscale peapod quantum dots
J. Mizubayashi,1,2J. Haruyama,1,2I. Takesue,1,2T. Okazaki,3,2H. Shinohara,4,2Y. Harada,5,2and Y. Awano5,2
1Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 229-8558, Japan
2JST-CREST, 4-1-8 Hon-machi, Kawaguchi, Saitama 332-0012, Japan
3National Institute of Advanced Industrial Science and Technology, Tsukuba 305-8565, Japan
4Nagoya University, Furo-cho, Chigusa, Nagoya 464-8602, Japan
5Fujitsu Laboratory, 10-1 Wakamiya, Morinosato, Atsugi, Kanagawa 243-0197, Japan
?Received 21 February 2006; revised manuscript received 14 March 2007; published 21 May 2007?
We report anomalous charging effect of single electrons ?Coulomb diamonds? observed in carbon nanoscale
peapod quantum dots that encapsulate a series of C60molecules. We find that behaviors of diamonds are
anomalously sensitive to back-gate voltages ?Vbg?, exhibiting two evidently different Vbgregions and a large
polarity on Vbg. In particular, we find only a sequence of one large diamond followed by three smaller ones
existing around ground state. Magnetic-field dependence indicates the presence of shell filling by spin singlet
to doubly degenerate electronic levels for these. The encapsulated-C60molecules indirectly affect this shell
filling at low Vbgpossibly via nearly free electrons. In contrast, they act as individual quantum dots coupled in
series in high Vbgregion. It directly contributes to highly overlapped very large diamonds. Moreover, we report
power-law behaviors on conductance versus energy relationships observed in the same carbon nanoscale
peapods. We find that the values of powers are also highly sensitive to applied Vbgwith three different regions
and anomalously high at high source-drain voltages. Because the power laws are found at voltages, which are
the nearest outside of the above-mentioned fourfold Coulomb diamonds, correlation of the anomalous powers
with orbital-related Tomonaga-Luttinger liquid is discussed.
DOI: 10.1103/PhysRevB.75.205431PACS number?s?: 73.63.?b, 71.10.Pm, 73.22.Lp, 73.23.?b
Carbon nanoscale peapods, which are single-walled car-
bon nanotubes ?SWNTs? encapsulating a series of fullerenes
such as C60, C70, and Gd@C82?C82encapsulating Gd? mol-
ecules in their inner space,1,2have recently attracted consid-
erable attention. This is because their unique nanostructures
are expected to yield exotic electronic states, quantum charge
?spin? transports, and one-dimensional ?1D? quantum phe-
nomena. There are, however, still a few reliable reports that
experimentally reported such electronic states and quantum
From theoretical viewpoints, in C60@?n,n? peapods that
are armchair-type SWNTs encapsulating C60molecules, it
has been predicted that electrons that were transferred from
the SWNT accumulated in the space between the C60mol-
ecules and SWNTs, forming the so-called nearly free-
electron ?NFE? states.3Hybridization of these NFE states
with the ? and ? orbitals of C60molecules introduced four
asymmetric subbands including the approximately doubly
degenerate ground states in the C60@?10,10? peapod in con-
tradiction to the two subbands in conventional SWNTs.3,4
Measurements of semiconductive peapods encapsulating
a series of Gd@C82by a scanning tunnel microscope re-
vealed that a conduction band was periodically modulated
around Gd@C82in a real space due to the hybridization of
orbitals between the SWNT and Gd@C82.5Moreover, elec-
trical measurements of peapods encapsulating C60 and
Gd@C82indicated the possibility of the presence of variable
range hopping.6References 3–6 at least suggested the pres-
ence of charge transfer and orbital hybridization between the
encapsulated fullerenes and SWNTs.
On the other hand, it is well known that SWNTs are
within a 1D ballistic charge transport regime and exhibit a
variety of quantum effects, such as quantized energy levels,
Tomonaga-Luttinger liquids ?TLLs?, and shell ?orbital? fill-
ing ?atomiclike behaviors? as quantum dots.7–10For instance,
when carbon nanotubes ?CNs? act as quantum dots, electron
can be placed on the quantized electronic levels ?orbitals or
shells? in the dots one by one via single-electron charging
effect. This effect has caused shell filling in CN quantum
dots,7–10such as even-odd effect, shell filling in two spin-
degenerate electronic states, and Kondo effect. In particular,
two different types of shell filling have been experimentally
reported, that is, antiparallel spins ?spin singlet? and parallel
spins ?spin triplet?. Reference 22 proposed a theoretical
model that explained the results by taking into consideration
several families of single-electron states and Coulomb repul-
sion. Our present system is analogous to this model.
Moreover, the behavior of TLLs, which is a collective
phenomenon arising from electron-electron interaction in 1D
conductors, has been identified by observing power laws in
relationships of conductance vs energy in CNs.12–15The re-
ported correlation exponent g, which denotes the strength of
an electron-electron interaction, was as low as ?0.2. This
implied the presence of a strong repulsive Coulomb interac-
tion existing in CNs. How such phenomena are affected by
encapsulating a series of fullerenes, however, has not yet
been investigated in any carbon nanoscale peapods to date.
For present study, we report finding anomalous behaviors
of Coulomb diamonds observed in carbon nanoscale peapods
quantum dots that encapsulate a series of C60molecules.
First, we find that behaviors of diamonds are anomalously
sensitive to back-gate voltages ?Vbg?, exhibiting two evi-
PHYSICAL REVIEW B 75, 205431 ?2007?
©2007 The American Physical Society205431-1
dently different Vbgregions ?i.e., ?1? small diamond region
for −1.7 V?Vbg?+1.7 V, and ?2? large diamond region for
Vbg?−1.7 V and +1.7 V?Vbg? and a large polarity on Vbg.
In particular, we find only a sequence of one large diamond
followed by three smaller ones existing only around ground
state in +Vbgregion. Magnetic-field dependence indicates the
presence of shell filling to doubly degenerate electronic lev-
els by spin singlet for these. The size of diamond indicates
that this is independent of the encapsulated-C60molecules,
basically in this low Vbgregion. However, they might indi-
rectly affect this shell filling via nearly free electrons, which
accumulate on the space between C60’s and SWNT by elec-
tron transfer from the SWNT. In contrast, we find that the
encapsulated-C60molecules directly contribute to overlapped
very large diamonds by acting as individual quantum dots
coupled in series in high Vbgregion.
Next, we report finding the presence of power laws in
conductance vs energy relationships, which is also highly
sensitive to Vbgshowing three Vbgregions in different power-
law behaviors ?i.e., ?1? Vbg?0.8 V, ?2? 0.8 V?Vbg?1.7 V,
and ?3? 1.7 V?Vbg?. Anomalously high values of powers ?
?8? are also found for Vbg’s ?0.8 V in high Vsdregions. The
power laws are found at voltages, which are the nearest out-
side of the above-mentioned fourfold Coulomb diamonds.
Because these fourfold diamonds mean single-electron filling
to doubly degenerate orbitals in the peapod quantum dot,
correlation of the anomalous powers with orbital-related
TLL states is discussed.
II. ANOMALOUS COULOMB DIAMONDS SENSITIVE TO
BACK-GATE VOLTAGES AND SHELL-FILLING
For the present study, field-effect transistors ?FETs?, using
peapods encapsulating C60molecules as the channel, were
fabricated. A scanning electron microscopy ?SEM? top view
indicated that the FETs included two bundles of peapods2as
the channels. The number of peapods included in one bundle
was estimated to be approximately 20 from atomic force mi-
croscopy ?AFM?, and transmission electron microscopy
?TEM? observations. Since change in the observed differen-
tial conductance was largely independent of the change in
A. Anomalously Vbg-sensitive Coulomb diamonds
The measurement results by single-electron spectroscopy
are shown in Fig. 1: ?a? for −4 V?Vbg?+4 V and ?b? ex-
pansion of ?a? for 0 V?Vbg?+2 V. Figure 1?a? indicates
that ?1? the sizes of Coulomb diamonds are highly sensitive
to applied Vbgand ?2? they have significant polarities on
applied Vbg?i.e., asymmetric features between +Vbgand −Vbg
ranges?. From the first viewpoint, the sizes of diamonds can
be evidently classified to the following two Vbgregions: ?1?
small diamond region for −1.7 V?Vbg?+1.7 V, and ?2?
large diamond region for Vbg?−1.7 V and +1.7 V?Vbg. In
the first Vbgregion, the sizes of diamonds are smaller than
?10 mV except for those appearing at Vbg=+1 V ?noted as
n=4 in Fig. 1?b??, whereas in the second Vbgregion, they are
approximately 20 mV except for a few diamonds at Vbg?
−1.7 V and larger than 40 mV at Vbg?+1.7 V.
From the second viewpoint, the sizes of diamonds are
very asymmetric as mentioned above. Moreover, periodicity
of diamonds is quite different between −Vbgand +Vbgranges
in the first Vbgregion. A sequence of closed one large dia-
mond ?noted as n=4? followed by three closed smaller ones
?noted as n=1–3 in Fig. 1?b?? is observable for 0 V?Vbg
?+1.7 V, whereas they cannot be observed for −1.7 V
?Vbg?0 V. Although some similar-size diamonds are ob-
servable for −0.5 V?Vbg?0 V, they are unclosed and other
diamonds for −1.7 V?Vbg?−0.5 V become much smaller
as Vbgvalue decreases. In the second Vbgregion, the dia-
monds are highly overlapped and cannot be separated. In
particular, the overlapping is very significant for +1.7 V
It is a well-known fact that conventional CN quantum
dots can place many electrons on those many quantized elec-
tronic levels one by one via single-electron tunneling and
that shell-filling effects are mostly independent of Vbg, be-
cause the shape of dots is independent of Vbg. Vbgjust
changes the positions of chemical potentials in the conven-
tional CN quantum dots, unlike most cases of semiconductor
quantum dots. Only the case of impurity- or defect-induced
many small quantum dots, which were connected in series,
showed diamonds with inhomogeneous sizes, resulting in
stochastic Coulomb diamonds. However, no such behaviors
have been observed in our empty SWNT quantum dots ?i.e.,
without C60’s?. Therefore, the above-mentioned anomalously
high Vbg-sensitive Coulomb diamonds indicate strong asso-
ciation with contribution of the encapsulated-C60molecules.
Here, the value of Uc=e2/2Ceff?the single-electron charg-
ing energy of the system; Ceffis the effective capacitance for
single charging effect21? should be approximately 6–10 meV
in the case of empty and high-quality SWNTs quantum dots
?i.e., without C60’s, defects, and impurities?, if one follows
the expectations based on only the length of present carbon
peapod of 500 nm following a previous study of SWNT
bundles ?e.g., a Uc of ?25 meV for a tube length of
100–200 nm.7The values of Uc??10 meV? in the first Vbg
region mentioned above are in approximately good agree-
ment with this estimation of Uc=6–10 meV. This strongly
indicates no contribution of the encapsulated-C60molecules
at least to Ceffof Ucand only the SWNT acts as a quantum
dot. Hence, single electrons flow only through the SWNT in
this low Vbgrange around 0 V and cannot flow into the C60
B. Shell filling to doubly degenerated electronic levels at
In particular, the period of Coulomb diamonds observed
for 0 V?Vbg?+1.7 V, as shown in Fig. 1?b?, can be inter-
preted as the four diamonds, a sequence of one large dia-
mond ?noted as n=4? followed by three smaller ones ?noted
as n=1–3?, as mentioned above. This sequence indicates the
possible presence of shell-filling effect with the doubly de-
generate electronic levels only at ground states, based on
MIZUBAYASHI et al.
PHYSICAL REVIEW B 75, 205431 ?2007?
previous reports of the fourfold diamonds in SWNT ?Ref. 9?
and multiwalled CN ?MWNT? quantum dots.10However, the
observed sequence of these diamonds is only one set for
0 V?Vbg?+1.7 V, because appearance of much larger dia-
monds obstructs the observation of further sets of such se-
quence. This result is very anomalous as compared with
those periodically observed over wide ranges of Vbgin Refs.
9 and 10 In conventional CN quantum dots, such one-set
degenerate levels cannot be found, because individual non-
degenerate electronic level is formed only from quantization
of two subbands existing in bulk of a SWNT, while only in
some cases, all levels are doubly degenerate, such as the
fourfold diamonds, as observed in Refs. 9 and 10.
Hence, in order to confirm the presence of shell-filling
effects and doubly degenerate levels for Fig. 1?b?, we have
investigated the Vbgshift of the linear-response conductance
peaks ?i.e., shown by arrows in Fig. 1?b?? as a function of
magnetic field B perpendicular to the tube axis. The result,
Fig. 2?a?, reveals that adjacent peaks shift in opposite direc-
tions. This is a behavior of spin singlet state whose spins
alternate as S=0→1/2→0... and exist on the same orbital
state, unlike a spin triplet state formed by Hund’s rule. In
Fig. 2?b?, we plot additional energy, which was deduced
from the separation of adjacent peaks involving electrons on
the same orbital in Fig. 2?a? ?i.e., peaks 1 and 2, and peaks 3
and 4?, as a function of magnetic field B. A dashed line
shows the result of the best fit of the data to Uc+gL?BB,
n = 5n = 5
FIG. 1. ?Color online? ?a? Coulomb diamonds observed in a carbon nanoscale peapod quantum dot at T=1.5 K. The z axis is the
differential conductance with the magnitudes of which are indicated on the right side. Dotted lines at Vbg=±1.7 V indicate boundaries for
two different Vbgregions for small and large diamonds. ?b? Expansion of Coulomb diamonds ?red regions surrounded by the dotted lines? for
0 V?Vbg?+1.7 V in ?a?. n indicates the number of electrons confined in peapod quantum dots for each diamond. The dotted diamonds are
guides to the eye. Arrows mean the Vbgpoints for Fig. 2.
Magnetic field [T]
Magnetic field [T]
Peak Position: Vbg[V]
Addition energy [meV]
Magnetic field [T]
Addition Energy [meV]
FIG. 2. ?a? Vbgshift of conductance peak positions ?at Vbg
=0.11, 0.26, 0.53, and 0.77 around Vsd=0 V shown by arrows in
Fig. 1?b?? in Fig. 1?b? as a function of magnetic field B. ?b? Addi-
tional energy obtained from each peak pair in Fig. 2?a? versus B.
ANOMALOUS COULOMB DIAMONDS AND POWER-LAW…
PHYSICAL REVIEW B 75, 205431 ?2007?
where ?Bis the Bohr magneton and gLis the Lande factor,
and gives gL=1.96. This value of gLis approximately con-
sistent with those mentioned in Ref. 10. Therefore, we con-
clude that Fig. 1?b? indicates the presence of doubly degen-
erate electronic levels existing only at ground states and the
presence of a shell filling to such levels by spin singlet.
Because of the absence of interaction between the C60
molecules and the SWNT as mentioned above, these doubly
degenerate electronic levels are not directly associated with
the encapsulated-C60molecules. Therefore, this does not cor-
respond to the model predicted in Ref. 22. These, however,
may be indirectly associated with the C60molecules via
NFEs and doubly degenerate subbands due to the NFE states
Reference 3 predicted that the doubly degenerate sub-
bands, which exist only at the ground states, originated from
the hybridization of orbitals of the encapsulated-C60mol-
ecules and NFE states in bulk of C60@?10,10? peapod as
mentioned in the Introduction. In contrast, in the case of
conventional quantum dot structure, this degeneration is
solved and the subbands are quantized to many electronic
levels ?shells?. However, because the channel length of
present peapod FET is as long as 500 nm, the level spacing
?E=h?F/2L ?L and ?Fare the tube length and Fermi veloc-
ity, respectively? should be as small as ?3 meV even in
nondegenerate levels. In such a case, the doubly degenerate
subbands around the ground state may still remain, resulting
in formation of approximately double-degenerate electronic
levels around Vbg=0 V, which was observed in Fig. 1?b?,
even in the quantum dot.
In contrast, this shell-filling effect was not observable for
−1.7 V?Vbg?0 V as mentioned above. Moreover, the sizes
of unclosed diamonds are inhomogeneous without specified
periodicities in this Vbgrange. These behaviors are analogous
to stochastic Coulomb diamonds, which have been reported
in impurity- or defect-induced small quantum dots connected
in series, although the number of quantum dots is very small
in the present case because the size of diamonds is as small
as less than 10 mV.
These significantly different behaviors of diamonds for
±Vbgrange evidently support the correlation of the doubly
degenerate levels and shell filling with NFE states mentioned
above. This is because NEF states can be formed by electron
transfer from SWNT to the space between the encapsulated
C60’s and the SWNT as mentioned in the Introduction3and
applying +Vbginduces this electron transfer as well as the
single-electron injection from source electrode. In contrast,
applying −Vbgobstructs these electron transfers, resulting in
the above-mentioned stochastic diamonds in the low −Vbg
range. This also implies a possibility of the presence of some
defect-induced small quantum dots in the SWNT for yielding
these stochastic diamonds, because no electron transfer from
the SWNT exists and, thus, such electrons in the SWNT may
be injected into the small dots.
C. Interplay between encapsulated-C60molecules and SWNTs
at high Vbg
On the other hand, the sizes of diamonds in the second
Vbgregion are much larger than those in the first Vbgregion
showing heavy overlapping and, thus, this means the values
of Ceffare smaller than those in the first Vbgregion. This
implies disappearance of the shell filling and the connection
of a few dots mentioned above in the first ±Vbgregions. In
conventional Coulomb diamonds, such large and unclosed
diamonds have been interpreted by a series connection of
defect- or impurity-induced many small dots. However, be-
cause such large diamonds have not been found in our empty
SWNT quantum dots as mentioned earlier, they do not cor-
respond to the present case. This indicates a possibility that
individual encapsulated-C60molecules behave as individual
small quantum dots and they are electrostatically coupled to
each other, because applying high +Vbgstrongly induces the
above-mentioned electron transfer from SWNT to the space
between C60’s and SWNT for yielding NFEs. The induced
transfer results in excess accumulation of the NEFs and the
excess NFEs can make single-electron tunneling into indi-
vidual C60molecules. In such a case, electron will flow only
through the encapsulated-C60molecules connected in series
under applied Vsd, because C60molecules are fully encapsu-
lated into the inner space of the SWNT and neighboring
C60’s face each other via tunnel junction in most parts of the
If there is no such a coupling and encapsulated-C60mol-
ecules act as independent quantum dots, our system corre-
sponds to the model proposed in Ref. 22. This will lead to
shell filling by parallel spins ?spin triplet?, because of the
presence of Tomonaga-Luttinger liquid ?Coulomb repulsion?
as mentioned in the next section.
The values of Uc’s of ?20 meV for Vbg?−1.7 V and
?40 meV for +1.7 V?Vbgin the second Vbgregion are at
least two and four times larger than those for the first Vbg
region ??10 meV?. Hence, the values of the total capaci-
tance of C60molecules, which are coupled with SWNTs, can
be estimated to be the value of 1/3 capacitance of SWNTs
?CSW? for +1.7 V?Vbg. In contrast, it can be estimated to be
the same value as ?CSW? for Vbg?−1.7 V. This indicates the
presence of different origins to yield effective capacitance
Cefffor Ucfor ±Vbg. As mentioned above, electron transfer
from SWNT to the C60’s could not occur in the −Vbgrange.
Hence, this large value of Ceffmay be attributed to single
charging effect of defect or impurity-induced small quantum
dots as well as that for −1.7 V?Vbg?0 V, although large
−Vbgvalue induces charging effect, resulting in the larger
In conclusion, our argument implies that the anomalously
high Vbg-sensitive Coulomb diamonds are attributed to the
following: ?1? shell filling to doubly degenerate electronic
levels associated with NFE states in low +Vbgregion with no
direct interaction between the encapsulated-C60molecules
and the SWNT and ?2? single-electron injection from the
SWNT into the C60molecules, which act as individual quan-
tum dots coupled in series, in high +Vbgrange. Moreover, we
argued that diamonds in −Vbgregion originate from defect-
or impurity-induced small quantum dots. In order to further
clarify these arguments, more investigation is expected, such
as by changing the number of encapsulated-C60molecules.
MIZUBAYASHI et al.
PHYSICAL REVIEW B 75, 205431 ?2007?
III. ANOMALOUS POWER-LAW BEHAVIORS AND
ORBITAL-RELATED TOMONAGA-LUTTINGER LIQUID
A. Anomalous power-law behaviors
Figure 3 shows the double-logarithmic plot of differential
conductance divided by T?as a function of eV/kBT measured
at Vbg=+0.4 V for three different temperatures. All data col-
lapse on a single universal value showing saturation at
eV/kBT?hvF/L. These results are qualitatively consistent
with those in previous reports of TLL states in CNs.13
Figure 4 shows the relationships of differential conduc-
tance ?dIsd/dVsd? to Vsdon doubly logarithmic scales for one
−Vbgand three +Vbgregions. In the −Vbgregion, any differ-
ential conductance did not follow a linear relationship, as
shown in Fig. 4?a?. On the contrary, saliently linear relation-
ships with different ? values are observable in the +Vbgre-
gion, although the Vsdregions with exhibiting power laws are
narrow at Vsd’s ?10 mV in Figs. 4?c? and 4?d? ?e.g., half
decade?. The behaviors are classified into the following three
Vbgregions; ?1? Vbg?0.8 V, the linearities with 1.6???2
are observable only at Vsd?0.01 V ?Fig. 4?b??; ?2? 0.8 V
?Vbg?1.7 V; two linear relationships with different ?
ranges ?i.e., ?=2–3 and ?=8–10 for Vsd?0.01 V and Vsd
?0.02 V, respectively? are observable ?Fig. 4?c??, and 3.
1.7 V?Vbg, the linearities with ?=10–12 are observable
only at Vsd?0.01 V ?Fig. 4?d??.
The summary of values of ? observed in all the Vbgregion
included in Figs. 4?b?–4?d? is shown in Fig. 5. The differ-
ences in tendencies of ? among the three regions are appar-
ent in this figure. Moreover, the values of ? observed in
empty SWNTs are also shown in this figure. All the values
are less than 1, which is consistent with previous reports of
TLLs in SWNTs. This implies that Fig. 4 is unique to pea-
B. Correlation with possibly orbital-related Tomonaga-
The presence of power laws has been discussed as evi-
dence for TLLs in CNs,12–15as mentioned in the Introduc-
T = 1.5 K
T = 5 K
T = 10 K
GT- α[arb. Units]
FIG. 3. The double-logarithmic plot of differential conductance
?G=dIsd/dVsd? divided by T?as a function of eV/kBT measured at
Vbg=+0.4 V for three different temperatures.
FIG. 4. ?Color online? Relationships of dIsd/dVsdto source-drain
voltage ?eVsd/k?T=1.5 K measured? on doubly logarithmic scales
for four different Vbgregions; ?a? Vbg?0 V, ?b? 0 V?Vbg?0.8 V,
?c? 0.8 V?Vbg?1.7 V, and ?d? 1.7 V?Vbg. These power laws pri-
marily appear just at the nearest-outside regions of fourfold Cou-
lomb diamonds in Fig. 1?b?. Only the power law in the low Vsd
region at Vbg=1 V appears inside of the large Coulomb diamond in
fourfold diamonds. The liner lines were obtained from accurate data
fitting including measurement points as many as possible and val-
ues of power ? were exactly estimated.
Power α α
The values of power
FIG. 5. ?Color online? Dependence of power ? on different Vbg
values, estimated from Figs. 4?b?–4?d?, in both the present peapod
and the empty SWNT quantum dots. Three different Vbgregions
??1? Vbg?0.8 V, ?2? 0.8 V?Vbg?1.7 V, and ?3? 1.7 V?Vbgcor-
responding to Figs. 4?b?–4?d?, respectively? separated by dotted
lines are evident. Several values of ? were added in addition to Fig.
4?d? only in region 3.
ANOMALOUS COULOMB DIAMONDS AND POWER-LAW…
PHYSICAL REVIEW B 75, 205431 ?2007?
tion. The values of ? were very sensitive to the boundary
conditions between the metal electrodes and CNs,13namely,
the tunneling density of state, such as ?bulk=?0.3 and ?end
=2?bulkfor the tunneling from a Au electrode to the bulk and
to the end of CNs within the large-channel number TLL
states, respectively.13The formulas of ? for each tunneling
were also given by ?bulk=?g−1+g−2?/8 and ?end=?g−1
−1?/4. However, it should be noted that even the maximum
value of ? reported in previous CNs to date is approximately
1.25, except for Refs. 14 and 17. Therefore, we imply that
the ? values of 1.6–12 observed in Figs. 3 and 4 are anoma-
lously large in comparison with the ? values reported thus
far in conventional TLLs.14The junction structures in this
study, in which the ends of the peapod bundles were placed
under a Au electrode, should have shown a maximum ?endof
only ?0.6. In fact, the empty SWNTs have exhibited ?=
?0.8 even at the maximum case, as explained for Fig. 5
Here, it should be noticed that the power laws, as shown
in Figs. 4?b?–4?d?, were found at the Vbg’s in the nearest-
outside voltage regions of the Coulomb diamonds, as shown
in Fig. 1 of Sec. II. Region 1 ?Vbg?0.8 V? for Fig. 4?b?
mostly agrees with the Vbgregion including three small dia-
monds ?n=1,2,3 in Fig. 1?b??, whereas region 2 ?0.8 V
?Vbg?1.7 V? for Fig. 4?c? agrees with the Vbgregion in-
cluding one large diamond and one small diamond ?n=4, 5
in Fig. 1?b??. Region 3 ?Vbg?1.7 V? for Fig. 4?d? agrees
with the Vbgregion, showing very large diamonds without
shell filling. Moreover, no shell-filling effect was observed in
−Vbgregion in Fig. 1?a?. This agrees with the presence of no
power laws in Fig. 4?a?.
Each Coulomb diamond region for this shell ?orbital?-
filling region meant the presence of electrons, which were
placed one by one on electron orbitals via single-electron
charging effect, in the peapod quantum dot. Hence, the ob-
served power laws sensitive to Vbgare strongly associated
with the number of such electrons ?n? in the peapod quantum
dot and the number of ?partially? occupied electronic levels
?N?; i.e., n=1,2,3 ?N=1,2? for Vbg?0.8 V and n=4,5 ?N
=2,3? for 0.8 V?Vbg?1.7 V.
The correlation of power laws, the values of ?, and TLL
states with the electronic-level filling effect ?i.e., orbital-
filling effect? in CNs have not yet been reported in previous
studies. Only a single study,13however, predicted that a
small g and large ? could be obtained from the large N in
peapods. The theory predicted g=?1+2Nvq/??vF?−1/2for
armchair CNs, where N and vqare the number of ?partially?
occupied symmetric subbands with degenerate Fermi vector
waves and the same bandwidth, and the electron-electron
interaction matrix element, respectively. If the subbands are
asymmetric and each of them crosses the Fermi level only
once, N can be replaced by N/2. This holds true for the
subbands of the C60@?10,10? peapod in this study.
We quantitatively examine the validity of this theory for
the present measurement results by replacing N to the num-
ber of ?partially? occupied electronic levels ?orbitals? and us-
ing the same value of vq. The value of g=0.135 is obtained
from ?end=?g−1−1?/4 ?Ref. 13? using ?=1.6 that was ob-
served in region 1 ?N=1,2? in Figs. 4?b? and 5. The value of
vq can be estimated by substituting these values of g
=0.135 and N=2 in g=?1+2?N/2?vq/??vF?−1/2.18Then, g
=0.11 and g=0.099 are, respectively, obtained for N=3 and
N=4 by substituting this estimated value of vqin g=?1
+2?N/2?vq/??vF?−1/2. The value of g=0.11 obtained for N
=3 is in approximately good agreement with g=0.082 esti-
mated from ?end=?g−1−1?/4 by using ?=2.8 that was ob-
served in the portion of region 2 with low Vsdvalues in Figs.
4?c? and 5.
On the other hand, this g value is irrelevant to ?=8–10
that is observed in the portion of region 2 with high Vsd
values in Figs. 4?c? and 5. Moreover, the g value of 0.099
leads to ?=2.96 for N=4, which is significantly less than the
values of ??10 observed in region 3 at high Vsdvalues in
Figs. 4?d? and 5. These indicate that different values of vq
should be used for the case of higher Vsd. Because strength of
electron-electron interaction varies from low to high Vsd’s,
this is reasonable. When different values of vqare used for
large N, ?=10 ?for N=3? and 12 ?for N=4? could be ob-
tained from the values of 2 vq/??vF=1160 and 1250, re-
spectively, g=?1+2?N/2?vq/??vF?−1/2, and ?end=?g−1−1?/4.
Consequently, the theory18is quantitatively relevant when
N=2 and 3 ?at lower Vsd? under the same value of vqand
N=3 ?at higher Vsd? and 4 under the larger values of Vq. This
indicates that the presence of two power laws observed in
Fig. 2?c? is attributed to change in vqdue to increase in Vsd.
Therefore, we conclude that the power laws with large values
of ??1.6???12? can be attributed to the TLL via the occu-
pied doubly degenerated electronic levels, which are located
near the ground states unique to the peapod quantum dots.
However, the high Vsdregion for power laws in region 2
does not locate at the nearest outside of Coulomb diamonds.
In addition, power laws observed in region 3 are not consis-
tent with no shell-filling effect reported in. ?Ref. 16? These
may indicate that such power laws are not associated with
orbital-related TLLs. Therefore, further investigation is re-
quired to reconfirm relevance of the values of vqand com-
prehensive understanding of these power-law behaviors.
Hence, other interpretation should be discussed described
as follows. If the capacitance of peapods is ?600 times
smaller than those in MWNTs due to the presence of C60
molecules electrostatically coupled with the SWNT in series
and the value of N is as large as 10–20 like those in MWNTs,
the large-channel TLL model coupled with external electro-
magnetic environment shown for MWNTs ?Refs. 13, 19, and
20? may explain the ?=8–12, because ? in conventional
MWNTs is given by 2R/RQ=2?L/C?1/2/RQ?0.44, where L
is the kinetic inductance given by RQ/2NvF??1 nH/?m?, C
is the external electrostatic capacitance ??30 aF/?m?, and
RQ?=h/e2? is the quantum resistance. This may correspond
to the high values of ? in region 3, because we reported that
the encapsulated-C60molecules were not electrostatically
coupled with the SWNT at Vbg?+1.7 V ?i.e., in regions
1 and 2?, while the coupling occurred at Vbg?+1.7 V
In conclusion, we reported anomalous behaviors of Cou-
lomb diamonds observed in carbon nanoscale peapod quan-
MIZUBAYASHI et al.
PHYSICAL REVIEW B 75, 205431 ?2007?
tum dots that encapsulated a series of C60molecules. We
found that behaviors of diamonds were anomalously sensi-
tive to Vbg, exhibiting two evidently different Vbgregions
?i.e., ?1? small diamond region for −1.7 V?Vbg?+1.7 V,
and ?2? large diamond region for Vbg?−1.7 V and +1.7 V
?Vbg? and a large polarity on Vbg. In particular, we found
only a sequence of one large diamond followed by three
smaller ones existing around ground state. Magnetic-field de-
pendence indicated the presence of shell filling to doubly
degenerate electronic levels by spin singlet state for these.
The size of diamond indicated that this was independent of
the encapsulated-C60molecules. However, they might indi-
rectly affect this shell filling via NFEs, which accumulate on
the space between C60’s and SWNT by electron transfer from
the SWNT. In contrast, we found that the encapsulated-C60
molecules directly contributed to overlapped very large dia-
monds by acting as individual quantum dots coupled in se-
ries in high Vbgregion.
Moreover, we reported the power-law behaviors on con-
ductance vs energy relationships observed in the same car-
bon nanoscale peapods. We found that the values of powers
? were highly sensitive to Vbgalso showing three different
?3? +1.7 V?Vbg? and anomalously high ???8? at high Vsd
voltages. Because the power laws were found at the nearest-
outside voltages of the above-mentioned fourfold Coulomb
diamonds, the correlation of the anomalous powers with
orbital-related TLL states was discussed. Further investiga-
tion is required in order to develop a comprehensive under-
standing of these phenomena ?e.g., changing the number of
−1.7 V?Vbg?+1.7 V,
We acknowledge T. Nakanishi, S. Tarucha, W. Izumida, P.
E. Lindelof, and M. Thorwart for fruitful discussions.
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ANOMALOUS COULOMB DIAMONDS AND POWER-LAW…
PHYSICAL REVIEW B 75, 205431 ?2007?