Atomic-like behaviors and orbital-related Tomonaga-Luttinger liquid in peapod quantum dots
ABSTRACT We report encapsulated C60 molecules on electron transport in carbon-nanotube peapod quantum dots. We find atomic-like behaviors with doubly degenerate electronic levels, which exist only around ground states, by single electron spectroscopy measured at low back-gate voltages (Vbg's). Correlation with presence of nearly free electrons (NFEs) unique to the peapods is discussed. In contrast, we find that encapsulated C60 molecules do not affect to single charging effect. Moreover, we find anomalously high values of powers observed in power laws in conductance versus energy relationships, which are strongly associated with the doubly degenerate levels. It is revealed that the powers originate from Tomonaga-Luttinger liquids via the occupied doubly degenerate levels. Encapsulated C60 molecules do not eliminate a ballistic charge transport in single-walled nanotubes.
- ); they reported an α = 1.66, and interpreted by correlated sequential tunneling . Since they, however, integrated G0 over the entire Vbg regions, their α value cannot be compared with our results In addition, some possibilities for the origin of power laws other than TLLs have been discussed. 166801..
Atomic-like behaviors and orbital-related Tomonaga-Luttinger liquid
in peapod quantum dots
J.Mizubayashi1, 5, J.Haruyama1, 5, I.Takesue1, 5, T.Okazaki2, 5, H.Shinohara3, 5, Y.Harada4, 5,
1Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 229-8558 Japan
2National Institute of Advanced Industrial Science and Technology, Tsukuba, 305-8565, Japan
3Nagoya University, Furo-cho, Chigusa, Nagoya 464-8602 Japan
4Fujitsu Laboratory, 10-1 Wakamiya, Morinosato, Atsugi, Kanagawa 243-0197 Japan,
5JST-CREST, 4-1-8 Hon-machi, Kawaguchi, Saitama 332-0012 Japan
We report ballistic charge transport phenomena observed in carbon-nanotube peapod
quantum dots. We find atomic-like behaviors (shell filling) sensitive to applied back-gate
voltages (Vbg) by single electron spectroscopy. Doubly degenerate electronic levels are found
only around ground states at very low Vbg. Those correlations with presence of nearly free
electron states unique to the peapods are discussed. Moreover, we find power laws in
conductance versus energy relationships with anomalously high values of power, which are
strongly associated with shell filling to the doubly degenerate levels. It is investigated that the
powers originate from Tomonaga-Luttinger liquid via the occupied doubly degenerate levels.
These results imply that a ballistic charge transport is still preserved at low Vbg regions in
peapod quantum dots in spite of presence of the encapsulated C60 molecules.
Nano-peapods, which are single-walled carbon nanotubes (SWNTs) encapsulating a series of
fullerenes, such as C60, C70, and Gd@C82 (C82encapsulating Gd) in their inner space [1, 2], have
recently attracted considerable attention. This is because their remarkable nanostructures yield
exotic electronic states, charge transports, and one-dimensional (1D) quantum phenomena.
However, there are still a few reliable reports that reported those electronic states and quantum
From theoretical viewpoints, in C60@(n,n) peapods that are arm-chair type SWNTs
encapsulating C60 molecules, it has been predicted that electrons that were transferred from the
SWNT accumulated in the space between the C60 molecules and SWNTs, forming the so-called
nearly free electron (NFE) states . Hybridization of these NFE states with the π and σ orbitals of
C60 introduced four asymmetric subbands including the approximately doubly degenerate ground
states in the C60@(10,10) peapod in contradiction to the two subbands in conventional SWNTs [3,
Measurements of semiconductive peapods encapsulating a series of Gd@C82 by a scanning
tunnel microscope revealed that a conduction band was periodically modulated around Gd@C82 in
a real space due to the hybridization of orbitals between the SWNT and Gd@C82 . Moreover,
electrical measurements of peapods encapsulating C60 and Gd@C82 indicated the possibility of the
presence of variable range hopping . Refs. [3 – 6] at least suggested the presence 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
liquid (TLL) [7 – 10], and atomic-like behaviors (shell filling) as quantum dots [11 – 14]. For
instance, the behavior of TLL, 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 carbon nanotubes (CNs) [7 – 10]. The reported 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 interaction in CNs. When CNs act as quantum dots,
electron can be placed on the quantized electronic levels in the dots one by one due to single
charging effect. This effect has caused atomic-like behaviors in CN quantum dots [7 - 14], such as
even-odd effect, shell-filling in two spin-degenerate electronic states, and Kondo effect. How such
phenomena are affected by encapsulating a series of fullerenes, however, has not yet been
investigated in any carbon nano-peapods to date.
For the present study, field-effect transistors (FETs) using peapods encapsulating C60 molecules
as the channel were fabricated. An SEM top view indicated that the FETs included two bundles of
peapods  as the channels. The number of peapods included in one bundle was estimated to be
approximately 20 from measurements by the SEM, AFM, and TEM. Since the observed differential
conductance was largely independent of the change in back gate voltage (Vbg), metallic transport in
the present peapod was suggested .
First, the measurement results by single electron spectroscopy are shown in Fig.1(a) and Fig.2.
Figure 1(a) shows Coulomb diamonds observed in the Vbg region < +2V. The four diamonds, a
sequence of one large diamond (shown as n = 4) followed by three smaller ones (shown as n = 1 –
3), are observable. This sequence indicates possible presence of atomic-like behaviors with the
doubly degenerate electronic levels only at ground states, based on previous reports of the four fold
diamonds in SWNT  and multi-walled CN (MWNT) quantum dots . However, the observed
sequence of these diamonds was only one set only at 0V < Vbg < +2V. This result is very different
from those periodically observed over wide ranges of Vbg in refs.[13, 14]
In conventional CN quantum dots, such one-set degenerate levels cannot exist, 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 like the four
fold diamonds as observed in refs.  and .
Hence, in order to confirm presence of doubly degenerate levels for Fig.1(a), we have
investigated the Vbg shift of the linear-response conductance peaks (i.e., shown by arrows in
Fig.1(a)) as a function of magnetic field, B, perpendicular to the tube axis. The result, Fig.1(b),
reveals that adjacent peaks shift in opposite directions. 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.1(c), we plot addition energy, which was deduced from
the separation of adjacent peaks involving electrons on the same orbital in Fig.1(b) (i.e., peaks 1
and 2, 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 + gLµBB, where µB is the Bohr magneton and gL is the Lande factor, and gives
gL = 1.96. This value of gL is approximately consistent with those mentioned in ref.. Therefore,
we conclude that Fig.1(a) indicates presence of doubly degenerate electronic levels existing only at
ground states and presence of atomic-like behaviors (shell filling).
We interpret that these one-set doubly degenerate levels originate from the encapsulated C60
molecules and NFE states unique to peapods from the following six reasons;
1. Our empty SWNTs quantum dots, in which C60 molecules are not encapsulated with the
same structures as those used for the present peapods, exhibited just even-odd effects (i.e.,
non-degenerate discrete levels) as shown in Fig.2(a) with ΔE = ∼3 m eV, which is consistent with
the previous studies in SWNT quantum dots (e.g., a ∆E of ~5 meV for a tube length of 100 ∼ 200
nm) and the relationship ΔE=hvF/2L (L and vF are the tube length and Fermi velocity,
respectively). This obviously stresses that the observed doubly degenerate levels are unique to
peapod quantum dots in our case in contradiction to refs.[13,14] and, hence, the encapsulated C60
molecules yield the degenerate levels.
2. The observed atomic-like behaviors were very sensitive to applied Vbg as follows. The
doubly degenerated levels disappeared and only even-odd effect appeared in the Vbg region higher
than that in Fig.1(a) (i.e., at +2V < Vbg < +5V) as shown in Fig.2(b). Moreover, at Vbg > +5 V, this
even-odd effect disappeared and only conventional Coulomb diamonds appeared. Furthermore, in
entire -Vbg region, no atomic-like behavior has been detected and only Coulomb diamonds
appeared. Because we have confirmed this tendency in three samples at least, these are not
accidental results like stochastic Coulomb diamonds reported in some previous papers.
3. The value of Uc (the single-electron charging energy of the system) is approximately 6 ~
10 meV in the small diamonds in Fig.1(a). This value for the present peapod with a length of 500
nm is approximately consistent with expectations based on a previous study of SWNT bundles (e.g.
a Uc of ~25 meV for a tube length of 100–200 nm) . Hence, this result stresses that the
encapsulated C60 molecules do not contribute to the effective electrostatic capacitance (Ce) for the
single charging effect of the peapod quantum dot in this low Vbg region < +2V.
4. In contrast, the sizes of Coulomb diamonds in Fig.2(b) become smaller than those in
Fig.1(a). This means a decrease in Uc to ~2 meV from 6 ~ 10 meV due to an increase in Ce and that
the encapsulated C60 molecules are electrostatically coupled with the SWNT in parallel for Ce in
this Vbg region; +2V < Vbg < +5V. The value of capacitance of C60 molecules are estimated to be
three ~ five times larger than that in the SWNT.
5. In contradiction to these results, it is a well known fast that conventional CN quantum dots
can place many electrons on those many quantized electronic levels one by one via single electron
tunneling and show periodical atomic-like features over wide Vbg regions, because the shape of
dots are independent of Vbg and Vbg just changes positions of chemical potentials in the CN
quantum dots unlike most of semiconductor quantum dots.
6. Ref. predicted that the doubly degenerate subbands existing only at the ground states
originated from the hybridization of orbitals in C60 molecules and NFE states in bulk of
C60@(10,10) peapod as mentioned in introduction. In the case of quantum dot structure, however,
these subbands should result in quantized electronic levels. In this sense, presence of the one-set
doubly degenerate electronic levels observed only around ground states in Fig.1(a) is qualitatively
relevant, because of non-modulation of C60 molecules by very low Vbg (< +2V).
Based on the 1st term, the 2nd – 5th terms stress that the applied high Vbg > 2V modulates the
encapsulated C60 molecules and change the shapes of Coulomb diamonds in the present peapod
quantum dots. If the encapsulated C60 molecules have no chemical bonds to the SWNT and can
freely rotate inside the SWNT, this modulation cannot occur. Hence, this means presence of
chemical bonds between C60 molecules and SWNT and, therefore, also presence of NFEs even in
the present peapod quantum dot as ref  predicted in bulk of peapods. This implies that Vbg
modulates not only C60 molecules but also NFEs at Vbg > +2V. Consequently, the doubly
degenerate levels observed in the Vbg < +2V can be attributed to non-modulated NFEs. This is
qualitatively consistent with ref. as explained in the 6th term.
When applied Vbg increases to the region in +2V < Vbg < +5V, electrons start to be
accumulated on the SWNT. Because the NFEs had been formed by electron transfer from the
SWNT, this accumulation indicates that NFEs are transferred back to the SWNT, resulting in those
depletions. This depletion of NFEs modulates electronic levels, leading to elimination of one-set
doubly degenerate levels. However, non-degenerate electronic levels and atomic-like behaviors still
survive under coupling with C60 molecules for single electron charging effect, resulting in Fig.2(b).
Further increase in Vbg to the region > +5V makes the NFEs entirely deplete and hybrid states
disappear. This leads to disappearance of atomic-like behaviors. Because applying –Vbg causes
electron depletion in the SWNT and electron transfer from the SWNT to NFEs, absence of any
atomic-like behaviors in the entire –Vbg region indicates that such an electron transfer are sensitive
much more than those in +Vbg region.
Consequently, it can be confirmed that peapod quantum dots can preserve a ballistic charge
transport in spite of presence of the encapsulated C60 molecules and NFEs, when a small +Vbg (e.g.,
Vbg < +5V in the present case) was applied. Moreover, it should be noticed that the applied low
+Vbg (e.g., Vbg < +2V in the present case) play a major role for oreserving NFEs without
modulation and observation of atomic-like behaviors as similar as those in bulk peapods.
Next, behaviors of orbital-related TLL states are discussed. Figure 3 shows the
double-logarithmic plot of differential conductance divided by Tα as a function of eV/kBT
measured at Vbg = +0.4V for three different temperatures. All data collapse on a single universal
value with showing a saturation at eV/kBT < hvF/L. This result stresses presence of TLL states in
the present peapod , but showing a larger value of α. Figure 4 shows the relationships of
differential conductance (dIsd/dVsd) to source-drain voltage (Vsd) on doubly logarithmic scales for
one - Vbg and three + Vbg regions. In the –Vbg region, any differential conductance did not follow a
linear relationship as shown in Fig.4(a). On the contrary, saliently linear relationships with different
α values are observable in the +Vbg region. The behaviors are classified into three regions (Fig.4
(b) - (d)) as mentioned in the figure captions, showing anomalously large values of α (1.6 < α <
The summary of values of α observed in all the Vbg region included in Fig.4(b) - (d) are shown
in Fig.4(e). The differences in tendencies of α among the three regions are apparent in this figure.
Moreover, the values of α observed in empty SWNTs used for Fig.1(b) 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(a) is unique to peapods.
The presence of power laws has been discussed as evidence for TLLs in CNs [7 – 10], as
mentioned in the introduction. The values of α were very sensitive to the boundary conditions
between the metal electrodes and CNs , namely the tunneling density of state; such as αbulk= ∼
0.3 and αend =2 αbulk for the tunneling from an Au electrode to the bulk and to the end of CNs
within the large-channel number TLL states, respectively . The formulas of α for each tunneling
were also given by αbulk = (g–1 + g – 2)/8 and αend = (g–1 – 1)/4. However, it should be noted that
even the maximum value of α reported in CNs to date is approximately 1.25, except for refs.[9, 16].
Therefore, we imply that the α values of 1.6 ∼ 12 observed in Figs.3 and 4 are anomalously large in
comparison with the α values reported thus far in conventional TLLs . The junction structures in
this study, in which the ends of the peapod bundles were placed under an Au electrode, should have
shown a maximum αend of only ~0.6. In fact, the empty SWNTs have exhibited α = ~0.8 even at
the maximum case as explained for Fig.4(e) above.
Here, it should be noted that the power laws shown in Fig.4 (b) – (d) exist at each Vbg in the
gray areas, which are just the nearest outside regions of the Coulomb diamonds shown in Fig.1(a).
The three +Vbg regions shown in Fig. 4(b) - (d) and Fig.4(e) are in good agreement with the three
Vbg regions classified by the boundaries of diamonds and, hence, the number of electrons confined
in the dot as shown in Fig.1(a). These results clearly indicate that the power law behaviors with the
large values of α are strongly associated with the number of (partially) occupied electronic levels,
N, in each diamond
The correlation of power laws and α (TLLs) with the electronic-state filling effect (i.e., orbital
filling effect) in CNs has not yet been reported in previous studies. Only a single study ,
however, predicted that a small g and large α could be obtained from the large N in peapods. The
theory predicted g = (1+2Nvq/πh vF)-1/2 for armchair CNs, where N and vq are the number of
(partially) occupied symmetric subbands with degenerate Fermi vector waves and the same band
width, 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 by replacing N
to the number of electronic states and using the same value of vq. The value of g = 0.135 is
obtained from αend = (g–1 – 1)/4  using α = 1.6 that is observed in region I (N=2). The value of
vq can be estimated by substituting these values of g and N in g = (1 + 2(N/2)vq/πh vF)–1/2.
Then, g = 0.11 and 0.099 are respectively obtained for N = 3 and N = 4 by substituting the
estimated value of vq in g = (1 + 2(N/2)vq/πh vF)–1/2. The value of g = 0.11 for N = 3 is
approximately in good agreement with g = 0.082 estimated from αend = (g–1 – 1)/4 by using α = 2.8
that is observed in the portion of region II with low Vsd values.
On the other hand, this g value is irrelevant to α = 8 ~ 10 that is observed in the portion of
region II with high Vsd values. Moreover, the g value of 0.099 leads to α = 2.28 for N = 4, which is
significantly less than the values of α > 10 observed in the region III at high Vsd values. 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 Vsds, this is reasonable. When different values
of vq are used for large N, α = 10 (for N = 3) and 12 (for N = 4) could be obtained from the values
of 2vq/πh vF = 1160 and 1250, respectively, g = (1 + 2(N/2)vq/πh vF)–1/2 , and αend = (g–1 – 1)/4.
Consequently, the theory  is quantitatively relevant when N = 2 and 3 (at lower Vsd) under
the same value of vq and 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.4(c) is attributed to change in vq due to
increase in Vsd. Therefore, we conclude that the power laws with large values of α (1.6 < α < 12)
can be attributed to the TLL via the occupied doubly degenerated electronic levels, which are
located near the ground states unique to the peapod quantum dots. Here, presence of TLL states
means also presence of a ballistic charge transport. Because power laws were not detected in –Vbg
region, this is consistent with absence of the atomic-like behaviors in –Vbg region. Further
investigation is, however, required to reconfirm the values of vq. As shown in Fig.4(c)(d), the Vsd
regions exhibiting power laws are very narrow at Vsd’s > 10 mV, i.e., at best half decade and the
high values of power do not become constant in the region III. Hence, other interpretation may be
possible like .
The observations reported in this study strongly suggest that the 1D quantum phenomena
observable in peapods are very exotic and sensitive to the encapsulated C60 molecules, NFE states,
applied Vbg, and shell filling via the NFE states. Further investigation is required in order to
develop a comprehensive understanding of these phenomena (e.g., changing the number of
encapsulated C60 molecules). .
We acknowledge T.Nakanishi, S.Tarucha, W.Izumida, and M.Thorwart for valuable discussions
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possibilities for the origin of power laws other than TLLs have been discussed, e.g., a
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20. The metallic behavior and the tube diameter (~1.6 nm) confirmed by TEM indicate the
possibility of a C60@(10,10) as used in ref.. Based on this, Gmax was estimated to be ∼40
×[4(2e2/h) ≈ 640] µS for of our FET. Since the actually total Gmax observed here was,
however, as low as ~10 µS, a large contact resistance (of the order of MΩ) at the
electrode/peapods interface is estimated, thus resulting in peapod quantum dots.
21. If the capacitance of peapods is ∼600 times smaller than those in MWNTs due to presence of
C60 molecules connected to a 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 electromagnetic
environment shown for MWNTs  may explain the α = 8 ∼ 12, because α 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 (≈ 1nH/µm), C is the external electrostatic capacitance (≈ 30 aF/µm), and RQ
(=h/e2) is the quantum resistance.
Fig.1: (a) Coulomb diamonds (white regions surrounded by the dotted lines) observed in a
peapod quantum dot at Vbg < +2V and at T = 1.5 K. The z-axis is the differential conductance with
the magnitudes of which are indicated on the right side. n indicates the number of electrons
confined in each diamond. Vbg regions I, II, and III well correspond to those in Fig.4(e).
(b) Vbg shift of conductance peak positions (at Vbg = 0.11 V, 0.26 V, 0.53 V, and 0.77 V around
Vsd = 0V shown by arrows in (a)) in Fig.1(a) as a function of magnetic field, B.
(c) Addition energy obtained from each peak pair in (b) versus B.
Fig.2: (a) Coulomb diamonds observed around ground states in an empty SWNT quantum dot
(i.e., without encapsulating C60 molecules) with the same structures as that used for Fig.1.
(b) Typical Coulomb diamonds measured at +5V > Vbg > +2V and at T = 1.5 K in the peapod
used for Fig.1
Fig.3: The double-logarithmic plot of differential conductance (G = dIsd/dVsd) divided by Tα as a
function of eV/kBT measured at Vbg = +0.4V for three different temperatures.
Fig.4: Relationships of dIsd/dVsd to source-drain voltage (eVsd /k >> T = 1.5 K measured) on
doubly logarithmic scales for four different Vbg regions. These power laws primarily appear just
the nearest outside regions of diamonds (i.e., in the gray areas as shown by arrow) of Fig.1(a).
Only the power law in the low Vsd region at Vbg = 1 V appears in the n = 4 diamond. The liner
lines were obtained from accurate data fitting including measurement points as many as possible
and values of power α were exactly estimated. (a): For –Vbg region. No dI/dV follows a linear
relationship. (b) - (d): For three +Vbg regions. (b): The linearities with 1.6 < α < 2 are observable
only at Vsd < 0.01 V. (c): Two linear relationships with different α ranges (i.e., α = 2 ∼ 3 and α = 8
∼ 10 for Vsd < 0.01 V and Vsd > 0.02 V, respectively) are observable. (d): The linearities with α =
10 ∼ 12 are observable only at Vsd > 0.01 V.
(e) Dependence of power α on different Vbg values, estimated from Fig.4(b)-(d), in present
peapod and empty SWNT quantum dots. Three Vbg regions ((I): Vbg < 0.8 V, (II): 0.8 V < Vbg <
1.8 V, and (III): 1.8 V < Vbg corresponding to Fig.4 (b), (c), and (d), respectively) are evident.
Several values of α were added in addition to Fig.4(d) only in region III.
Magnetic field [T]
Peak position; Vbg [V]
Magnetic field [T]
Addition energy [meV]
-6 -6-6 -6 -6-6-6-6
-4 -4-4 -4-4 -4-4-4
-2-2-2 -2-2 -2-2-2
GT -α (au)
T = 1.5 K
T = 5 K
T = 10 K
dI/dV [nS] 1000
dI/dV [nS] 1000
dI/dV [nS] 1000
dI/dV [nS] 1000
Power α α
× ×× ×× × ×
The values of power α α