Nanomechanical characterization of the Kondo charge dynamics in a carbon nanotube

K. J. G. 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 [6–8].

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 [11–14] 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 [17–21],

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 [22–24] and strong

interaction between single electron tunneling and vibra-

tional motion [25–28]. 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 [3–5, 17–21, 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

ῶ (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

ῶ (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

)

2mω

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

ῶ (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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