PRL 104, 017203 (2010)

PHYSICAL REVIEW LETTERS

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8 JANUARY 2010

Strong Electromechanical Coupling of an Atomic Force Microscope Cantilever to a Quantum Dot
Steven D. Bennett, Lynda Cockins, Yoichi Miyahara, Peter Grütter, and Aashish A. Clerk
Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8(Received 6 October 2009; published 8 January 2010)
We present theoretical and experimental results on the mechanical damping of an atomic force
microscope cantilever strongly coupled to a self-assembled InAs quantum dot. When the cantilever
oscillation amplitude is large, its motion dominates the charge dynamics of the dot which in turn leads tononlinear, amplitude-dependent damping of the cantilever. We observe highly asymmetric line shapes of
Coulomb blockade peaks in the damping that reflect the degeneracy of energy levels on the dot.
Furthermore, we predict that excited state spectroscopy is possible by studying the damping versus
oscillation amplitude, in analogy with varying the amplitude of an ac gate voltage.
DOI: 10.1103/PhysRevLett.104.017203

PACS numbers: 85.85.+j, 73.22.f, 85.35.Gv

Coupling a mechanical object to quantum electronics isa means to probe both the mechanics and the electronics
with extreme sensitivity. It has been predicted that theelectronics may be used to measure the quantum natureof the mechanical object [1], and the reverse—using mechanics to measure quantum effects in mesoscopic electronics—was recently demonstrated with superconducting
qubits [2]. Electromechanical systems that have attracted
considerable attention recently include quantum shuttles[3], and mechanics coupled to single electron transistors(SETs) [4,5] or tunnel junctions [6,7]. In most systems
studied, the interaction between the electronic and mechanical components is weak.
In this Letter we study strong coupling effects, boththeoretically and experimentally, in an electromechanicalsystem consisting of a quantum dot capacitively coupled to
an atomic force microscope (AFM) cantilever. Electrons
tunneling on and off the dot effectively damp the cantilever, and this damping exhibits Coulomb blockade peaks asa function of applied voltage similar to those well known inthe dot conductance [8–10]. It has long been predicted thatlevel degeneracy on the dot leads to line shape asymmetryof Coulomb blockade peaks in the conductance [11].
Recently, we observed corresponding temperaturedependent peak shifts in the damping at weak coupling[10], but the line shape asymmetry was far too small to bemeasured. However, by driving the cantilever to large
oscillation amplitudes we enter a regime of strong coupling: within one period, the cantilever motion is large
enough to swing the dot completely on or off of a charge
degeneracy point. This leads to a dramatic enhancement of
the line shape asymmetry that is much greater than expected from simply extrapolating the weak coupling theory; it is a nonadiabatic effect that stems from the similar
time scales for dynamics of the cantilever and the dot.
Furthermore, we predict that by measuring the damping
versus applied voltage and oscillation amplitude, strong
coupling provides a way to perform excited state spectroscopy on the dot. Note that very different strong coupling
0031-9007=10=104(1)=017203(4)effects unrelated to degeneracy were recently reported for adriven carbon nanotube coupled to an embedded dot[12,13].
Our results show that an AFM operated at large oscillation amplitudes may be used to study degeneracy andlevel spacing, so-called shell structure, in confined electronic systems. The particular systems studied here, selfassembled quantum dots, are candidates for applications
such as quantum information processing, and measuring
their shell structure has attracted extensive research effort[14–17]. Our technique allows the level degeneracy to beread off from a single sweep of damping versus applied
voltage, and offers the practical advantage of being able toaddress many dots one by one without the need for electrical contacts.
Setup.—The mechanical oscillator is an AFM cantileverwith resonant frequency !0 =2 ¼ 166 kHz, spring constant k0 ¼ 48 N=m, and intrinsic quality factor of typicallyQ0  2  105 . It is driven on resonance in self-oscillation
mode at constant amplitude and mean tip-sample gap of
19 nm [10]. The cantilever is coated with a 10 nm Ti
adhesion layer and a 20 nm Pt layer to ensure goodelectrical conductivity at low temperature. All data inthis Letter were collected at 5 K. The sample is grown bychemical beam epitaxy, with the relevant features being
uncapped InAs dots on top of a 20 nm InP tunnel barrier
and a 10 nm InGaAs two-dimensional electron gas (2DEG)

FIG. 1 (color online). (a) Schematic of the setup.
(b) Equivalent circuit diagram, where Ctip depends on x.

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Ó 2010 The American Physical Society

PRL 104, 017203 (2010)

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PHYSICAL REVIEW LETTERS

which acts as a back electrode. For full sample details, see
Ref. [10]. The voltage VB is applied to the 2DEG layer,with the cantilever electrically grounded (see Fig. 1). The
potential drop between the 2DEG and the dot is VB ,where  ¼ Ctip =C is extracted from the experiment andC ¼ Ctip þ C2DEG is the total dot capacitance. The dotcantilever coupling arises through Ctip , which depends on
the tip position x. Electrons tunnel between the 2DEG and
the dot when VB is sufficient to lift Coulomb blockade,while tunneling between the dot and tip is negligible due tothe much larger tunnel barrier height of the vacuum gap.
The fluctuating charge on the dot results in both damping
and a resonance frequency shift ! of the cantilever; in thelimit of weak coupling these are well described by linear
response [18]. Here we focus on the damping, which isprovided in addition to the frequency shift by a phaselocked loop frequency detector and automatic gain controller [10].
Model.—For small cantilever motion compared to thetip-dot separation, Ctip is linear in x and we can write thecharging Hamiltonian of the dot as [19][23,24]. The probabilities P0 ðx; uÞ and P1 ðx; uÞ to find theoscillator at position x and velocity u with 0 or 1 electrons
on the dot satisfy master equations,@t P1 ðx; uÞ ¼ L1 P1 þ þ ðxÞP0   ðxÞP1 ;(4)@t hxi ¼ hui;

(6)@t hP1 i ¼ þ ðhxiÞhP0 i   ðhxiÞhP1 i:

(7)

(1)

(2)

where  is the tunneling rate to a single particle state and fis the Fermi function. The asymmetry between adding andremoving electrons is the root of the asymmetry in
Coulomb blockade peaks [11].
We describe the coupled system using a master equation
for the charge on the dot combined with a Fokker-Planck
equation for the phase space distribution of the oscillator

(5)



F þ AhP1 i
 hxi  0 hui;k0

@t hui ¼ !20

where n is the number of electrons on the dot, N ¼
VB Ctip =e is the dimensionless gate voltage, and EC ¼e2 =2C is the charging energy [20]. H C;0 is the oscillatorindependent part of H C , and H osc modifies the oscillator potential. The last term describes dot-cantilever coupling with strength A ¼ ð2EC VB =eÞð1  Þ@Ctip =@x;
from this term we see that the dot charge exerts a force
An on the oscillator.
We focus on the voltage range where 0 or 1 extra
electrons reside on the dot with other charge states prohibited by Coulomb blockade; it is simple to generalize this ton or n þ 1. The tunneling rates are calculated using
Fermi’s golden rule and accounting for the degeneracy of
single particle levels on the dot. For a shell of degeneracy 
occupied by nshell electrons, there are þ ¼   nshellways to add an electron, and once it has been added thereare  ¼ nshell þ 1 ways to remove it. The extra energy
with 1 electron on the dot is EðxÞ ¼ 2EC ð1=2  N Þ  Ax.
In the classical oscillator limit, @!0  kB T [21,22], thisresults in x-dependent rates þ ( ) to add (remove) anelectron, ðxÞ ¼  f½EðxÞ;(3)

where Ln ¼ !20 ðx  xn  F =k0 Þ@u  u@x þ 0 @u u describes a driven, damped harmonic oscillator and xn ¼
An=k0 is the equilibrium position with n electrons on thedot. The damping 0 is intrinsic to the oscillator without
coupling to the dot, and F is the external driving force.
While it is straightforward to simulate the master equations directly, we gain further insight by focusing on thesimpler dynamics of system averages. Following Ref. [25],we make the approximation that averages of products may
be factorized, justified by comparison with full simulations, and use Eqs. (3) and (4) to obtain coupled equations
for the average quantities,

H C ¼ EC ½ðn  N Þ2  ð1 þ C2DEG =Ctip ÞN 2 ’ H C;0 þ H osc  Anx;@t P0 ðx; uÞ ¼ L0 P0 þ  ðxÞP1  þ ðxÞP0 ;We seek a solution where the cantilever oscillates at constant amplitude a, as in experiment, such that hxðtÞi ¼
a cosð!0 tÞ. Ignoring the frequency shift due to tunneling(since !  !0 ) and assuming that the total damping issmall (0 þ 1  !0 , justified self-consistently), we find
100

(b)

80

60
0.8

0.5

0.2

10

1

5

0
1

0
0
1

40

0

20

0

(a)

c

1

0.8

(c)

d

0.6

(d)

(e)

0

1
e

0.4

0.2

0

FIG. 2 (color online). (a) First damping peak calculated fromsimulation (dots) and semianalytic theory (solid lines). The
green dashed line is from linear response. N is plotted inreverse for consistency with experiment. (b) Adiabatic approximation (dash-dotted), semianalytic theory (solid), and simulation(dots) for oscillation amplitude a ¼ 1 nm. (c)–(e) Average dot
charge versus time (solid) for a ¼ 1 nm, at voltages marked in(a). Cantilever position is also shown (thin dashed) as a reference. We took 2EC ¼ 31 meV, !0 = ¼ 1, and A ¼
10 meV=nm. Other parameters were taken from the experiment(see setup).

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PRL 104, 017203 (2010)that the effective, amplitude-dependent damping due totunneling is given by [25]
1 ¼

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PHYSICAL REVIEW LETTERS!20 A Z 2=!0
dt sinð!0 tÞhP1 ðtÞi;k0 a 0(8)

and obtain constant amplitude oscillations for F ¼
ak0 =!0 ð0 þ 1 Þ sinð!0 tÞ. Equation (8) connects thedamping to the out-of-phase part of the average dot charge,hnðtÞi ¼ hP1 ðtÞi, with respect to hxðtÞi. Note that the fluctuating dot charge causes damping even at weak coupling,
and this is measurable using a sufficiently high quality
oscillator [10], but Eq. (8) remains valid at strong coupling.
It also reduces our calculation of 1 to solving Eq. (7) for
hP1 ðtÞi numerically, a much easier task than directly simulating Eqs. (3) and (4).
Coulomb blockade peaks occur in the damping versus
applied voltage at charge degeneracy points, where the dot
energy is equal with either 0 or 1 electrons and charge
fluctuations are maximal [10]. Figure 2(a) shows the first
damping peak for several oscillation amplitudes, calculated from Eq. (8) and from direct simulation of Eqs. (3)and (4) following Ref. [24]. We assume the level structureof a cylindrical dot, with a twofold degenerate s shell and afourfold degenerate p shell. The simulated damping is welldescribed by linear response (green dashed) at weak coupling; note that even this peak is slightly asymmetric as
expected. As the oscillation amplitude is increased, thepeak becomes broadened and highly asymmetric. Notethat the line shape at strong coupling is completely
missed by an adiabatic approximation, where one assumes
that the oscillator motion is much slower than tunneling[see Fig. 2(b)]. On the other hand, our semianalytic theory[i.e., Eq. (8)] agrees very well with the full simulation, so
we use it to understand why the line shape is so highly
asymmetric at strong coupling.
Asymmetric line shape.—The asymmetric line shape of
Coulomb blockade peaks is a result of the asymmetry
between adding or removing electrons to or from a degenerate shell on the dot [see Eq. (2)]. Consider the voltage
points c and e on either side of the peak in Fig. 2(a), equal
distances from its center such that the largest amplitude
motion (broadest peak) swings N onto the charge degeneracy point. A tunneling event near N ¼ 1=2 is twice as
likely to occur when starting from point e, where the dot is
initially empty. This is because the rate to tunnel onto thedot when N ¼ 1=2 is þ  2fð0Þ ¼  (for the first peakin the twofold degenerate s shell), while the rate to tunnel
off is only   ½1  fð0Þ ¼ =2. This asymmetry isapparent in the time dependence of hnðtÞi at three voltage
points, shown in Figs. 2(c)–2(e). Tunneling is more likely
starting from point e, and this leads to increased damping.
The situation is reversed in the second peak where there is
only one way to add an electron to the half full s shell, but
two ways to remove one once it is full. Thus the line shapeasymmetry is a way to read off the shell degeneracy from asingle VB sweep: each peak is skewed away from the centerof its shell.
While a similar argument leads to a very slightly asymmetric line shape at weak coupling, the asymmetry atstrong coupling is much greater than one obtains by extending the weak coupling result to large oscillation amplitudes [see Fig. 2(b)]. For sufficiently strong coupling A,
the change in gate voltage due to the oscillator motion
dominates over the thermal broadening from the Fermi
distribution of electrons in the back electrode, i.e., Aa 
kB T. Thus the harmonic distribution of the oscillator position PðxÞ, peaked at the turning points of its motion, causes
the most asymmetric tunneling rates,  ðaÞ at the oscillator extrema, to become especially important and dramatically increases the line shape asymmetry.
Here we point out the importance of relative time scales.
For a slow oscillator, !0  , the adiabatic approximation
is valid since the dot charge quickly equilibrates in response to the slow cantilever motion. In this case thedamping is simply given by a weighted average of thelinear response result taken over the oscillating gate voltage, and the line shape asymmetry remains immeasurably
small [see Fig. 2(b)]. In the opposite limit, !0  , the dot
charge cannot respond to the rapid oscillator motion anddamping is suppressed. This is seen from Eq. (8): hP1 ðtÞi isroughly constant over one period of a fast oscillator and the
dot-induced damping becomes vanishingly small. We
study the regime !0  , where the interplay between
the two time scales leads to maximal and highly asymmetric damping.
Measured damping.—The experimentally measured
cantilever damping is compared with theory in Fig. 3. In(a) and (b) we fit the first two Coulomb blockade peaks forthree oscillation amplitudes using Eq. (8). For each peak,
we used a single fit parameter A to fit the damping at all
three amplitudes, obtaining the values given in the caption,
20

(a)

(b)

20

30

(c)

20
10

15

10
0
−10

10

0

10

0
30

(d)

20

5

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0

−10

0

10

−60 −40 −20 0 20

FIG. 3 (color online). (a) Experiment (solid) and theory(dashed) for the first damping peak. We converted eVB to E
using  ¼ 0:04 extracted at weak coupling. One fit value A ¼
7:8 meV=nm produced the theory curves for all three amplitudes. (b) Second peak with A ¼ 9:2 meV=nm. (c) Measured
damping over the p shell; theory shown in (d) with A ¼
11 meV=nm. Other parameters were taken from experiment as
described in the text.

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PRL 104, 017203 (2010)

PHYSICAL REVIEW LETTERS

3
2.5
2
1.5
1
0.5
−30

−20

−10

0

10

20

FIG. 4 (color online). Differential damping versus applied
voltage (converted to E), and oscillation amplitude a. For large
amplitudes, a peak appears on the line E ¼ Aa  sp (whitedashed line). Inset: cut along black dashed line. Parameters are
the same as in Fig. 2.

in good agreement with those obtained at weak coupling on
the same dot. We took 2EC ¼ 31 meV,  ¼ 0:04, and  ¼
70, 90 kHz for peaks 1, 2; these values were extracted at
weak coupling, with the rates  obtained from the on-peakratio !=1 [10]. Note that in our system, the damping is
dominated by tunneling for voltages near a charge degeneracy point, where we find up to 1 =0  20 for small
oscillations. This compares well with 1 =0  50 for
SETs [5], 35 for nanotubes [12], and 1 for previous AFM
experiments [8,9].
In Figs. 3(c) and 3(d) we show the measured and theoretical damping versus applied voltage over the entire p
shell. This is calculated by extending our derivation of
Eq. (8) to allow up to four electrons to occupy the fourfold
degenerate p shell. We find qualitative agreement even in
the crude approximation of constant EC , and using single
values of A and  over the entire shell [26]. We took 2EC ¼
20 meV for the p shell (estimated from the peak spacing),
and roughly aligned the peaks by adjusting the p shell level
splitting phenomenologically. Once this was done, a single
set of parameters was used to produce the damping spectra
at all three amplitudes. Most importantly, in both theory
and experiment the four peaks in the p shell become five atlarge amplitudes, with peaks emerging between the charge
degeneracy points. This is consistent with our theory
above: at large amplitudes, the oscillator distributionPðxÞ overwhelms temperature and makes the tunnelingrates at its extrema most important. Thus, at very large
amplitudes where Aa ¼ EC , tunneling is maximal when
the applied voltage is at the midpoint between two degeneracy points.
Excited state spectroscopy.—Theoretically, our setup
can be used to do excited state spectroscopy on the dot,
since the oscillator is directly analogous to an ac gate
voltage [27]. When the oscillator motion is large enoughto allow tunneling to excited states, the damping increases.

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This occurs when the change in energy due to the oscillator
is equal to the energy spacing, or Aa  sp . At large
amplitudes we expect a jump in 1 at the applied voltage
where E ¼ Aa  sp , and a peak in @1 =@E. This defines
a line versus E and a, shown in Fig. 4. Measuring the slope
and intercept of this line in experiment would directly
provide A and sp .
Conclusions.—We have demonstrated that oscillation
amplitude is a useful new axis to exploit in using AFMs
to probe quantum electronic systems. The implications
extend beyond quantum dots to other confined electronicsystems that can be placed on an insulating surface with aback electrode. In particular, we envision using an AFM tomeasure the level structure of single molecules.
This work was supported by NSERC, FQRNT, and
CIFAR.

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