Nanomechanical characterization of the Kondo charge dynamics in a carbon nanotube
K. J. G. otz, D. R. Schmid, F. J. Schupp, P. L. Stiller, Ch. Strunk, and A. K. H¨uttel
Institute for Experimental and Applied Physics, University of Regensburg,
Universit¨atsstr. 31, 93053 Regensburg, Germany
(Dated: April 23, 2018)
Using the transversal vibration resonance of a suspended carbon nanotube as charge detector for
its embedded quantum dot, we investigate the case of strong Kondo correlations between a quantum
dot and its leads. We demonstrate that even when large Kondo conductance is carried at odd electron
number, the charging behaviour remains similar between odd and even quantum dot occupation.
While the Kondo conductance is caused by higher order processes, a sequential tunneling only model
can describe the time-averaged charge. The gate potentials of maximum current and fastest charge
increase display a characteristic relative shift, which is suppressed at increased temperature. These
observations agree very well with models for Kondo-correlated quantum dots.
The Kondo effect [1] is a striking manifestation of elec-
tronic correlations. In semiconductor quantum dots as
Coulomb blockade systems [2], in its most prevalent type
it expresses itself as a distinct zero-bias maximum of dif-
ferential conductance at odd electronic occupation [3
5]. In spite of this strong impact on electronic charge
transport, the degeneracy central to its formation is then
given by the spin states of an unpaired electron: for the
SU(2) spin Kondo effect, below a characteristic tempera-
ture T
K
, exchange coupling between a localized electron
and conduction band charges leads to the formation of
the Kondo resonance at the Fermi level. A question that
lends itself immediately is how the strongly enhanced
Kondo conductance within Coulomb blockade relates to
the precise charge trapped within the quantum dot and
its evolution as a function of applied gate voltage [68].
Suspended carbon nanotube quantum dots provide ex-
traordinarily clean and controllable mesoscopic model
systems [9, 10], where transport spectra from single- and
few electron physics [1114] all the way to open systems
and electronic Fabry-Perot interferometry [15, 16] can be
analyzed. Also regarding Kondo phenomena a wide range
of experimental work on carbon nanotubes exists [1721],
making use of the well-characterized electronic structure.
Then again, as nano-electromechanical systems, carbon
nanotubes have shown at cryogenic temperatures exceed-
ingly high mechanical quality factors [2224] and strong
interaction between single electron tunneling and vibra-
tional motion [2528]. The detection of the transver-
sal vibration frequency of a carbon nanotube provides a
powerful means to measure the charge on its embedded
quantum dot [25, 28].
In this Letter, we investigate the parameter region
of strong Kondo correlations between a suspended nan-
otube quantum dot and its metallic leads [35, 1721, 29].
We measure the gate voltage dependence of the time-
averaged charge e hNi(V
g
) on the quantum dot. The ob-
served typical asymmetry in conductance between odd
and even occupation states, indicating SU(2) Kondo be-
havior, is clearly absent in the gate-dependent trapped
charge. This shows that the current is carried by higher-
order processes leading to only virtual occupation on the
quantum dot, while the time-averaged charge remains
determined by the first-order processes of sequential tun-
neling. In addition, we observe a distinct gate voltage
offset between charging of the quantum dot and the cur-
rent maximum, which is suppressed at increasing tem-
perature. Our results agree very well with theoretical
studies of Kondo-correlated quantum dots [6, 30].
Device characterization– Figure 1(a) displays a
sketch of our device structure; a table of fabrication pa-
rameters can be found in the Supplement [31]. On a
highly p
++
doped Si substrate with thermally grown
SiO
2
on top, electrode patterns were defined via elec-
tron beam lithography and metal evaporation. The
metal layer directly serves as etch mask for subsequent
anisotropic dry etching of the oxide, generating deep
trenches between the electrodes. As last steps, growth
catalyst was locally deposited and carbon nanotubes were
grown via chemical vapor deposition [32].
Electronic transport measurements were performed in
a dilution refrigerator at T
MC
25 mK. The measure-
ment setup combines dc current measurement as required
for Coulomb blockade transport spectroscopy [33] with
radio-frequency irradiation using an antenna several mil-
limeters from the device [22, 23, 25]. As can be seen
from the differential conductance in linear response in
Fig. 1(b,c), in both devices close to a small band gap
Coulomb blockade and sequential tunneling dominates.
For larger positive gate voltages V
g
the transparency of
the tunneling barriers increases. This leads to a crossover
towards regular Kondo enhancement of the conductance
[4, 17]; the clear two-fold pattern in Fig. 1(b) indicates
approximate SU(2) Kondo behaviour [34].
Mechanical resonance detection– With a radio-
frequency signal applied at the mechanical resonance f
0
of its transversal vibration mode, the nanotube is driven
into motion, leading to a change in detected dc current
[22, 23, 25, 28]: the capacitance between back gate and
nanotube C
g
is modulated with the deflection, broaden-
ing the Coulomb oscillations in a slow dc measurement,
as shown in Fig. 1(d). This enables us to detect f
0
and its
2
2
dI/dV (e /h)
N
8
0.5 1.0
device B
242 270
5.55.0
(V)V
g
10
0
-1
-2
-3
10
10
10
(c)
V (V)
g
2
dI/dV (e /h)
1.0 2.0
24168
N
10
0
-1
-2
-3
-4
10
10
10
10
(a) (b)
antenna
metal
CNT
++
p Si
SiO
2
C
g
(d)
(e)
(f)
V
g
I
I
V
g
V
g
f
0
+ + + +
+ +
F
F
device A
30
FIG. 1. (a) Sketch of the device geometry (not to scale).
See the Supplement [31] for a table of the device proper-
ties. (b,c) Low-bias differential conductance dI/dV
sd
of car-
bon nanotube devices A and B as a function of applied back
gate voltage V
g
. N indicates the number of trapped electrons.
The transition from strong Coulomb blockade (left edge) to
strongly Kondo enhanced transport is visible in both cases.
(d) At resonant driving a nanotube vibrates strongly, leading
to a fast oscillation of C
g
and, averaged over the vibration, a
broadening of Coulomb oscillations. (e) Typical gate voltage
evolution of the transversal vibration resonance frequency and
the current in the Coulomb blockade regime, see [25] and the
text. (f) Principle of electrostatic vibration softening: when
a vibrating capacitor at constant voltage adapts its charge to
the momentary position, an electrostatic force opposite to the
mechanical restoring force occurs. This results in an effective
smaller spring constant and resonance frequency.
dependence on the back gate voltage V
g
in the dc-current.
Figure 1(e) sketches a typical evolution of the reso-
nance frequency with increasing positive gate voltage in
the strong Coulomb blockade regime [25]. The continu-
ous increase of the gate charge and the discrete increase
of the quantum dot charge both contribute via mechan-
ical tension to f
0
, as continuous increase and step func-
tion, respectively. Further, when the electronic tunnel
rates are large compared to f
0
, near charge degeneracy
points the charge on the quantum dot can adapt (by a
fraction of an elementary charge) to the momentary po-
sition within a vibration cycle. The vibration mode is
electrostatically softened [25, 35], cf. Fig. 1(f), propor-
tional to hN i/∂V
g
. Thus, resonance frequency minima
indicate the increase of the quantum dot charge ehNi at
the charge degeneracy points [25, 35], and hN i(V
g
) can
be calculated from the frequency evolution f
0
(V
g
).
Figure 2(a) shows a measurement of the vibration-
induced signal in the Kondo regime. For different gate
voltages V
g
the time-averaged dc current I(V
g
, f) is
510
500
495
f (MHz)
490
-0.2
0.0
0.2
I-I (nA)
(b)
(a) (c)
(d)
0
|I| (nA)
15
42
44
N
= 40
I-I
(nA)
0 0.4
4.05 4.10 4.15 4.20 4.25 4.30
V (V)
g
-3
0
3
6
(10 /s)
0
FIG. 2. (a) Current through the quantum dot, as function
of gate voltage V
g
and rf driving frequency f, with the mean
current I(V
g
) of each frequency trace subtracted; nominal rf
generator power 25 dBm, bias voltage V
sd
= 0.1 mV. (b)
|I(V
g
)| at off-resonant driving frequency f = 492 MHz. Kondo
enhanced conductance occurs at odd electron numbers. (c)
Example trace I(V
g
, f) I(V
g
) from (a) at V
g
= 4.1 V. The
effect of the mechanical resonance on the time-averaged dc
current is clearly visible. (d) Extracted resonance frequency
shift ˜ω
0
(V
g
) = 2πf
0
(V
g
) (a + bV
g
) with respect to a lin-
ear background; see the Supplement [31] for the detailed fit
parameters. Device A.
recorded while sweeping the driving signal frequency f.
In Fig. 2(a) (and Fig. 2(c), which displays a trace cut
from Fig. 2(a)), the mean value I(V
g
) of each frequency
sweep has been subtracted for better contrast. The vibra-
tion resonance becomes clearly visible as a diagonal fea-
ture. To evaluate its detailed evolution, we extract f
0
(V
g
)
and plot it in Fig. 2(d) as ˜ω
0
(V
g
) = 2πf
0
(V
g
) (a + bV
g
),
i.e., with a linear background subtracted [36]. Every
single-electron addition into the dot exhibits a distinct
dip. While the off-resonant dc current I(V
g
), Fig. 2(b),
clearly shows Kondo zero bias conductance anomalies at
odd quantum dot charge [4], this odd-even electron num-
ber asymmetry is barely visible in the evolution of the
resonance frequency [37].
Evolution with V
g
In Fig. 3(a), we show a detail of the
resonance frequency evolution from Fig. 2(d), accompa-
nied by the current I(V
g
) in Fig. 3(b). To model it, we
reduce the quantum dot to two non-degenerate Lorentz-
broadened levels, separated by a capacitive addition en-
ergy U > Γ, without taking any higher order tunneling
effects into account. We only consider the case of lin-
ear response, i.e., eV
sd
~Γ; in addition, for the large
transparency of the contact barriers present at N 41
and for electron temperatures of roughly T . 50 mK, we
neglect the thermal broadening of the Fermi distribution
in the contacts. Then, hNi(V
g
) is only smeared out by
3
-2
0
2
0
5
10
15
4.08 4.1 4.12 4.14 4.16 4.18
40
41
42
6
(10 /s)
0
N
|I| (nA)
(a)
(b)
(c)
V (V)
g
ΔV
g,right
-ΔV
g,left
|t|
0
1
FIG. 3. Analysis of the Kondo regime around hN i = 41;
V
sd
= 0.1 mV. (a) Data points: resonance frequency shift
˜ω
0
(V
g
), cf. Fig. 2(d). Solid line: curve fit assuming subse-
quent occupation of two non-degenerate levels, see the text.
(b) Data points: simultaneously measured off-resonant cur-
rent |I|(V
g
). Solid red line: sequential tunneling current ac-
cording to the fit model from (a). Dashed gray line: T = 0,
V
sd
= 0 Fermi liquid model transmission |t| derived from
hNi(V
g
) via Friedel’s sum rule, see the text; right axis. (c)
Time-averaged quantum dot occupation hNi(V
g
) derived from
the fit in (a).
the lifetime broadening Γ of the quantum dot states. The
tunnel barrier transmittances between dot and leads are
assumed to be energy-independent and equal; the tun-
nel rates Γ
±
1/2
onto and off the quantum dot levels are
obtained by integrating over the density of states on the
dot.
We use this model to fit the functional dependence of
the resonance frequency to the data in Fig. 3. Following
[25, 28], the decrease of the resonance frequency at finite
single electron tunneling (cf. Fig. 1(e)) is given by
ω
0
=
V
g
(V
g
V
CNT
)
2
0
C
Σ
dC
g
dz
2
1
e
C
g
hNi
V
g
, (1)
with V
CNT
= (C
g
V
g
e hNi)/C
Σ
as the voltage on the
CNT, m the nanotube mass, and ω
0
= 2πf
0
. The gate
and total capacitances C
g
and C
Σ
are extracted from
Coulomb blockade measurements. Since we do not know
the precise position of our CNT, we treat the capacitive
displacement sensitivity dC
g
/dz, where z is the deflec-
tion of the nanotube, as a free parameter. A detailed
discussion of the fit procedure, a table of the device pa-
rameters entering the calculation, and the resulting fit
parameters can be found in the Supplement [31]. Note
that the relevant gate dependent term in Eq. (1) is the
quantum capacitance, i.e., the derivative of the charge
occupation, hN i/∂V
g
, also called compressibility in [8].
Our simplified model reproduces the functional depen-
dence of the resonance frequency in Fig. 3(a) very well.
The result can be used to derive the expected sequential-
tunneling current from our model and the time-averaged
charge evolution hNi(V
g
) in the quantum dot, see the
solid lines in Fig. 3. While Kondo processes absent in
our model strongly contribute to electronic transport,
they do not significantly influence the time-averaged oc-
cupation of the quantum dot and thereby the mechanical
resonance. This is in excellent agreement with results by
Sprinzak et al. [7], combining a quantum point contact
as charge detector [38, 39] with a gate-defined quantum
dot, as well as recent data analyzing the charge compress-
ibility of a quantum dot by means of a coupled coplanar
waveguide cavity, see [8]. The suppression of quantum
dot charging by Coulomb blockade is independent of the
Kondo enhanced conductance via virtual occupation.
Gate potential of current and compressibility maxima–
In a naive analogy, one would expect that in the Kondo
case, as in the case of strong Coulomb blockade [8, 25, 28],
the increase of the time-averaged charge on the quantum
dot takes place predominantly at the gate voltage of the
current maxima. The data points of Fig. 3(b) show the
current I(V
g
) at fixed bias, recorded simultaneously with
the mechanical resonance frequency, Fig. 3(a). Com-
paring the extrema of the resonance frequency ˜ω
0
(V
g
),
Fig. 3(a), and the current |I|(V
g
), Fig. 3(b), distinct shifts
V
g,left
and V
g,right
are observed, see the green arrows.
In experimental literature, a temperature-induced shift
of the current maximum due to Kondo correlations has
already been reported in the first publications [3]. In the
data of Sprinzak et al., [7], a systematic shift between
current and quantum capacitance extrema similar to our
observations is visible (though not discussed). This con-
firms that the phenomenon is intrinsic to the Kondo ef-
fect in a quantum dot, independent of the experimental
realization. Early calculations by Wingreen and Meir,
[30], using the noncrossing approximation in the Ander-
son model, have already predicted a temperature depen-
dent shift of the current maximum position (see Fig. 6
and Fig. 7(a) in [30]).
Temperature dependence– Figure 4 illustrates the sup-
pression of correlation effects at elevated temperature. In
the region of the figure we obtain Kondo temperatures in
the range 1 K . T
K
. 5 K. While the large dot-lead cou-
pling strongly distorts the stability diagram at base tem-
perature [19] [40], see Fig. 4(a), at T & 5 K in Fig. 4(b)
regular, thermally broadened Coulomb blockade oscilla-
tions reemerge. Figures 4(c-e) display both extracted
mechanical resonance frequency and measured dc current
for (c) T = 15 mK, (d) T = 0.7 K, and (e) T = 5 K. With
increasing temperature the mechanical resonance broad-
ens [22] and the determination of the resonance frequency
becomes more challenging. At the same time, the current
evolves from a complex, Kondo- and level renormaliza-
tion dominated behavior to broadened but regular and
4
V (mV)
sd
-2
2
5750 5800
V (mV)
g
(a)
15 mK
1
-1
(b)
-1
-2
2
0
1
5.4 K
5750 5800
V (mV)
g
0.4
0.3
0.2
2
G (e /h)
|I|(nA)
1
0.0
0.2
-0.2
2
0
6
(10 /s)
0
V (mV)
g
(c)
15 mK
5800
5760
ΔV
g,left
ΔV
g,right
(d)
700 mK
5800
5760
ΔV
g,left
ΔV
g,right
(e)
5.1 K
5800
5760
FIG. 4. (a), (b) Differential conductance of device B at (a)
base temperature T = 15 mK and (b) T = 5.4 K. While trans-
port in the millikelvin regime is dominated by higher order
effects, above T = 5 K regular, strongly broadened Coulomb
blockade oscillations emerge. (c–e) Combined plots of me-
chanical resonance shift ˜ω
0
(V
g
) and dc current |I|(V
g
), for
V
sd
= 0.1 mV and (c) T = 15 mK, (d) 0.7 K, (e) 5.1 K.
(b)
ΔV (mV)
g, right
T (K)T (K)
(a)
ΔV (mV)
g, left
0
1
2
0.01 0.1 1 10
0
2
4
6
0.01 0.1 1 10
FIG. 5. Temperature dependence of (a) V
g,left
and
(b) V
g,right
, the shift between mechanical resonance fre-
quency minimum and dc current maximum, for the oscilla-
tions marked in Fig. 4. The solid lines correspond to a loga-
rithmic fit, with the T = 15 mK point omitted [30].
in the plotted range nearly bias-independent Coulomb
blockade oscillations.
As expected, at higher temperature charging and cur-
rent maxima coincide better. This is quantified in
Fig. 5, where the relative shifts in gate voltage V
g,left
and V
g,right
between resonance frequency minimum
and current maximum are plotted for two exemplary
Coulomb oscillations marked in Fig. 4. Starting from
about 1.5 mV, respectively 5.2 mV, the peak shifts de-
crease with increasing temperature asymptotically to-
wards zero. The straight lines in the figure, fits excluding
the T = 15 mK point due to likely saturation there, corre-
spond to the typical logarithmic scaling present in Kondo
phenomena and predicted for the peak shift [30] and are
consistent with the data.
Relation to the transmission phase– In an early the-
oretical work on Kondo physics, Gerland et al. [6] dis-
cuss the electronic transmission phase of a Kondo quan-
tum dot, a topic of intense attention over the previ-
ous decades. Friedel’s sum rule [41, 42] intrinsically
relates the transmission phase to the number of elec-
tronic states below the Fermi energy and thereby the
time-averaged occupation. This means that we can di-
rectly compare the combined Figs. 3(c) and 3(d) of [6]
(transmission magnitude and phase) with our data of
Figs. 3(b) and 3(c) here (current and time-averaged occu-
pation). Indeed, a highly similar functional dependence
is visible; see the Supplement [31] for a detailed compar-
ison. With this background and based on Fermi-liquid
theory of the SU(2) Kondo effect, the dashed gray line
in Fig. 3(b) plots the transmission amplitude evolution
|t(V
g
)| = sin(π hNi(V
g
)/2) of the quantum dot expected
for V
sd
= T = 0. This clearly demonstrates the Kondo
ridge as well as the distinct shift between large transmis-
sion magnitude and maximum slope of the transmission
phase. The deviations in current behaviour I(V
g
) may be
due to the finite temperature and bias, and/or indicate
an experimental situation more complex than the SU(2)
Kondo effect.
Conclusion– We use the mechanical resonance fre-
quency of a suspended carbon nanotube to trace the aver-
age electronic occupation of a strongly Kondo-correlated
quantum dot embedded in the nanotube. We show that
sequential tunneling alone already provides a good model
for the average charge hNi(V
g
) and the mechanical res-
onance frequency ω
0
(V
g
). While dominant for electronic
transport (conductance), the influence of Kondo correla-
tions on the time averaged charge and thereby the me-
chanical system is small in the chosen parameter regime.
We observe a distinct shift in gate voltage of the cur-
rent maxima, relative to the maxima of the charge com-
pressibility hNi/∂V
g
, effectively distorting the Coulomb
blockade regions. This shift decays with increasing tem-
perature, a clear signature that it is caused by the Kondo
correlations. Our results are in excellent agreement with
theoretical modelling [6, 30].
Future work, applying our highly versatile sensing
method to higher harmonic modes of the vibration, may
address the parameter region f
mech
> k
B
T
K
[43, 44],
or even the charge distribution along the carbon nan-
otube axis via a spatially modulated electron-vibration
coupling [35]. Kondo phenomena in carbon nanotubes
beyond the SU(2) spin Kondo effect provide further sys-
tems of obvious experimental and theoretical interest.
The authors acknowledge financial support by the
Deutsche Forschungsgemeinschaft (Emmy Noether grant
Hu 1808/1, GRK 1570, SFB 689) and by the Studien-
5
stiftung des deutschen Volkes. We thank J. von Delft,
J. Kern, A. Donarini, M. Marga´nska, and M. Grifoni for
insightful discussions.
andreas.huettel@ur.de
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