Charged-state dynamics in Kelvin probe force microscopy

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by  Martin Ondráček

Charged-state dynamics in Kelvin probe force microscopy
Martin Ondr´aˇcek1, Prokop Hapala1, and Pavel Jel´ınek1
1Institute of Physics of the Czech Academy of Sciences,
Cukrovarnicka´ 10/112, 162 00 Prague, Czech Republic
(Dated: January 28, 2016)
WepresentanumericalmodelwhichallowsustostudytheKelvinforceprobemicroscopyresponse
tothechargeswitchinginquantumdotsatvarioustimescales. Themodelprovidesmoreinsightinto
6 thebehavioroffrequencyshiftanddissipatedenergyunderdifferentscanningconditionsmeasuring
1 atemporarilychargedquantumdotonsurface. Namely,weanalyzethedependenceofthefrequency
0 shift, its fluctuation and of the dissipated energy, on the resonance frequency of tip and electron
2 tunnelingratesbetweentip–quantumdotandquantumdot–sample. Wediscusstwocomplementary
approaches to simulating the charge dynamics, a stochastic and a deterministic one. In addition,
n
a we derive analytic formulas valid for small amplitudes, describing relations between the frequency
J shift,dissipatedenergy,andthecharacteristic ratesdrivingthecharginganddischargingprocesses.

7
2
I. INTRODUCTION

]
l
l Furtherdevelopmentofelectronicdevicesandtheirperformancestronglydependsonourabilitytocharacterizeand
a
h control charge distribution down to atomic scale. From this perspective, scanning tunneling microscopy (STM) [1]
- andatomicforcemicroscopy(AFM)[2]areveryimportanttoolsthatallowimagingandmanipulationofsingleatoms
s
onsurfaces. Namely,recentdevelopmentofdynamicalatomicforcemicroscopy(dAFM)[3–5]providesapowerfultool
e
m to probe the local structure [6–9] and chemical composition [10, 11] of surfaces with a resolutionreaching the atomic
scale on semiconductor [12], metallic [13] and also insulating [14] surfaces. Kelvin Probe Force Microscopy (KPFM),
.
t a technique derived from dAFM, can be used to probe local variations of local work functions down to nanometer
a
m scale [15]. It has been demonstrated that KPFMis able to reachthe atomic [16–18]and sub molecular [19] resolution
too. Thestraightforwardinterpretationintermsoflocalcontactpotentialdifferenceisnomorepossibleattheatomic
-
d scale though [20–23].
n Recent development of scanning probe technique (see e.g. [13, 24, 25]) allowed simultaneous acquisition of AFM
o andSTM channels. This broughtnew possibilities not only foradvancedcharacterizationofsurfaces[26, 27]but also
c
tostudyaninfluenceofthetunnelingcurrentondetectedforces[28]. ThelatterreliesonaproperdecouplingofAFM
[
and STM channels [29]. Kelvin probe force microscopy and spectroscopy (KPFM/KPFS) complemented with STM

1 channel have the potential to become a convenient tool for studying the dynamics of charging and discharging local
v structures here referred as “quantum dots” (QD). These QDs are thought of as objects capable of storing electric
2
charge and thus being able to switch between two or more distinguishable charge states (for instance charged and
9
neutral). Theymayhavethe formofnanoclusters(likeInAsclustersonInPinRef. [30–32]),moleculesorevensingle
2
7 atoms (like Au atoms on NaCl in [33, 34])
0 Such QDs are of considerable interest because of their potential applications in nanoelectronics [35]. Above ref-
. erenced examples have been indeed successfully probed using KPFM in two seminal papers [30, 34], which paved a
1
route towards a new concept of controlling charge on atomic scale. However, the characteristic time scales of this
0
6 quantum-dot charge-statedynamics can span many orders of magnitude depending on the details of the studied sys-
1 tem as well as on the immediate position of the KPFM scanning probe. Consequently, the time scales of the charge
: dynamics relate in different ways to the characteristic time scales of the measuring instrument, like the cantilever
v
oscillationperiodT , reactiontime ofthe electronicfeedback ofthe AFM machine,acquisitiontime fora single point
i o
X in the measurementetc. The manifestation ofphenomena relatedto the switching in the KPFMdata will depend on
r the ratio of the various time constants involved. This fact may complicate interpretation of the experimental results.
a
TheaimofthispaperistodevelopanumericalmodelwhichallowsustosimulatetheKPFMresponsetothecharge
switching in QD at various time scales. While gating effect of the biased AFM tip was crucial for the explanation
of charging in the above quoted experiments on Au atoms or InAs nanostructures [30, 33, 34, 36], our focus in this
paper will be on cases where an electron tunneling from the tip has the decisive effect. We should note that the
model could be easily adapted to the charge gating processes too, but this is beyond the scope of the paper. We
develop and compare two complementary approaches,stochastic and deterministic, to simulate the charge dynamics.
Such insight into the response of oscillating probe driven by the charging processes helps in better understanding
the complex behavior observed in KPFM experiments. In addition, we will establish a relatively simple analytic
formulas providing, in the limit of small amplitudes, relations between frequency shift ∆f, dissipated energy E ,
diss
and characteristic tunneling rates (ν , ν ) driving the charging processes.
1 2
2
II. MODEL
Theoutlineofthemodelwepropose(sketchedinFig1(a))isasfollows. ConsideraQD(beitanatom,molecule,or
a nanostructure)ona surface. Assume the dotcanacceptor releaseanelectron,thus switching betweentwo possible
charge states. The AFM probe is sensitive to the charge state of the QD due to the electrostatic component of force
between the chargeddot and the probe apex. Severaltime constants or rate parameters determine the characteristic
time scales of the model:
Tunneling rate between the AFM tip and the QD on the surface ν .
1
•
Discharging rate of the charged dot by tunneling to a conductive substrate ν .
2
•
Oscillation period of the AFM tip T =1/f , where f means resonant oscillation frequency .
o 0 0
•
Measuring time t .
m
•
Furthermore,thetunnelingprobabilitybetweenthetipandthedotdependsontipposition(z)andonappliedvoltage
bias (V).
A. Forces
We aimtocreateamodelassimple aspossiblebutstillcapturingthe essentialfeaturesofchargedynamicscoupled
to tip oscillations. The force between the AFM tip and the surface consists of the following components:
F derived from the Lenard-Jones potential between the QD and the probe apex (represents the short-range
LJ
•
forces independent from charge and bias: Pauli repulsion and the local part of van der Waals interaction).
(RLJ )12 (RLJ )6
F (z)=12ELJ min min , (1)
LJ min z zLJ 13 − z zLJ 7!
− 0 − 0
(cid:0) (cid:1) (cid:0) (cid:1)
F derived from the Hamaker model for a spherical tip of radius R and planar surface [37] to describe
Ham tip
•
the long-range part of van der Waals (or, more precisely, London dispersion) interaction. The Hamaker model
approximates the tip with a homogeneous sphere and the surface and substrate beneath it by a homogeneous
half-space delimited by the surface plane.
A R
H tip
F (z)= . (2)
Ham −6(z zHam)2
− 0
The electrostatic part F +F . We describe the long-range part of the electrostatic force F as the force
cap Q cap
•
betweentwoelectrodesofacapacitor,oneofwhichisplanarandrepresentsthesurfaceandotheroneisspherical
and represents the tip [38]. The same geometry, that is the same tip radius and distance from the surface, is
assumedasinthe Hamakermodelusedabove. Tocalculatethe localpartofthe electrostaticforceF ,we treat
Q
the charge of the QD as a point charge of the size corresponding to one electron. The point charge q may be
present or absent, depending on the immediate “charge state”. If absent, the local electrostatic contribution is
assumed to be zero. If present, we assume an interaction of the point charge with (i) an electric field from the
tip and (ii) an other static point charge Q positioned on the very end of the AFM tip. The electric field
apex
is assumed to depend linearly on the voltage bias and be inversely proportional to the tip distance from the
QD (as if the field were homogeneous) while the apex charge is assumed to be characteristic for the tip and
independent from the bias.
F (z,V)= πǫ0Rt2ip(V −VCPD)2 πǫ0Rtip(V −VCPD)2, (3)
cap −(z zHam)(R +z zHam) ≈− z zHam
− 0 tip − 0 − 0
where ǫ =8.854187817 10−12 F/m is the vacuum permittivity, and
0
×
e2 qQ e q(V V )
F (z,V)= 0 apex + 0 − CPD , (4)
Q 4πǫ (z zLJ)2 2 z zLJ
0 − 0 − 0
where e = 1.6021765 10−19 C is the elementary charge. The charges q and Q are expressed here in the
0 apex
×
unit of the electron charge (negative elementary charge), so they are themselves dimensionless quantities.
3
FIG.1: (a)Schematicsketchofthemodelusedinthispaper. Themodelconsistsessentially ofatiponacantileverandaQD
onasurface. Thegeometryofthemodelisinfact treated aspurelyone-dimensional,i.e. onlythez coordinateisrelevant. (b)
Model of the tunneling process between the tip and the QD. The tunneling from the tip to the dot is assumed to be elastic
and two localized states are assumed to take part in it, one at the tip termination and theother one in thedot, resulting in a
Gaussian-shaped resonance.
The total force by which the surface with the QD pushes the tip is then F(z,V)=F (z)+F (z)+F (z,V)+
LJ Ham cap
F (z,V).
Q
Themechanicaldynamicsofthetipischaracterizedbyits(undamped)resonantoscillationfrequencyf ,cantilever
0
stiffness k, and quality factor Q. For the instantaneous vertical position z(t) of the tip, the equation of motion
d2z 2πf dz
+ 0 +(2πf )2(z z )=F(z) (5)
dt2 Q dt 0 − eq
holds, where z is the equilibrium position of the tip determined by the piezolectric positioning system of the AFM
eq
cantilever and F is the total force applied on the oscillating tip by its interaction with the surface below it. The
quality factor Q and resonance frequency (in the absence of damping) f characterize the cantilever.
0
B. Model for the electronic states of the tip apex and on the surface
The electronic state at the QD, to which or from which the electrons are tunneling when the charge state changes,
is characterized by its mean energy ǫ and a broadening w around this mean energy ǫ . The broadening w
dot dot dot dot
is a consequence of an interaction with the substrate. We take the shape of the energy broadening is described by
Gaussian distribution (but the exact shape is not crucial for the effects described in this paper) then the density of
state of QD ρ holds:
dot
2
1 ǫ ǫ
dot
ρ = exp − . (6)
dot
√πw w
dot (cid:18) dot (cid:19)
Twomodelsoftheelectronicstructureofthe tipwereimplemented: i)eitherthetipcarriesasingleelectronicstate
capable of participation in the tunneling (this model may representa tip with a localized state, like a dangling bond,
on its terminal apex); or ii) the tip hosts a continuum of electronic states with a constant density of states near the
Fermilevel(a metallic tip). Inthis paper,we discuss onlythe localized-statetip modelasthe continuummodelgives
very similar results (see discussion later). In the case of the tip model with the localized apex state, we describe the
corresponding density of states on the tip ρ by a Gaussian distribution, similarly to the QD state:
tip
4
1 ǫ ǫ 2
tip
ρ = exp − . (7)
tip
√πw w
tip (cid:18) tip (cid:19)
Fermi-Dirac distribution of occupancy probabilities is invoked to account for a finite temperature of the tip T.
Temperature effects on the substrate are disregarded. The tunneling probability per unit time from the tip to the
QD (ν ) is assumed to exponentially decay with tip distance from the surface. Further, this tunneling probability
1
ν resonantly depends on voltage bias through the Gaussian densities of states on the dot and tip in a way that
1
corresponds to the assumption of exclusively elastic tunneling. Taking all these conditions in account, the tunneling
probability from the tip to QD is expressed as:
∞
ν (z,V)=ν exp( βz)N dǫρ (ǫ)ρ (ǫ e V)f (ǫ e V,T). (8)
1 0 norm dot tip 0 FD 0
− − −
Z0
Fig.1(b)displaysaschemeofthetunnelingprocess. Inthemodelpresentedhere,wedonotconsiderthepossibility
oftunnelingintheoppositedirection,i.e.fromthedottothetip. Whilethismayseemtobeanunrealisticassumption,
astheprobabilityofelastictunnelingshouldbethesameinbothdirection,itmayinfactbeareasonableapproximation
−
forcasessimilartoAuonNaCl. ThenegativeAu ion,oncecreated,getsstabilizedbyinteractionwiththesubstrate
and does not change back to its original neutral state at the same tip bias [33]. In our model, the charged dot can
resume its neutral state by giving up an electron to the substrate. The probability of this event in a unit of time is
a constant parameter (ν ) of the model. The normalization factor N in Eq. (8) is chosen in such a way that the
2 norm
tunneling rateatV =(ǫ ǫ )/e andz =0isν =ν . The prefactorν isaz-andV-independentparameter,which
2 1 0 1 0 0
−
characterizes the overall feasibility of tunneling from the tip to the dot.
C. Stochastic vs deterministic model
We compare two approaches to simulating the charge dynamics, to which we will refer as a stochastic and a
deterministic approach. In both cases the probabilities of electron tunneling from the tip to the dot (ν dt) and from
1
the dot to the substrate (ν dt) are evaluated in each time step for the present position of the tip. In the stochastic
2
case, they are treated as genuine probabilities and a pseudorandom number generator is used to decide whether the
charge state of the QD should be changed or not based on these probabilities and the pseudorandom value. Only
charges0or1 (meaningno oroneextraelectron)wereallowedinthe stochasticapproach. When0,itchangesto 1in
the next step with probability ν dt; when 1, it changes to 0 with probability of ν dt. In the deterministic approach,
1 2
the tunneling ratesν andν areinterpretedasa chargeflow. A fractionalchargebetween0and1 isallowedandthe
1 2
current value of charge q(t) is updated in every time step according to the rate equation
q(t+dt)=q(t)+ν (z(t),V)(1 q(t))dt ν q(t)dt. (9)
1 2
− −
This finite-step evolution of q(t) corresponds, in the dt 0 limit, to the differential equation (in simplified notation
→
omitting the functional dependences)
dq
=ν (1 q) ν q. (10)
1 2
dt − −
All simulations of the tip dynamics were carried out using following parameters: z = 4.0 ˚A, f = 46.858 kHz,
eq 0
k =3681N/m,Q=2438andasmallamplitudeofA=0.1˚A.Thecurvesweresampledwiththestepof∆V =0.02V.
The bias voltage V was kept constant for 1000 oscillationperiods of the cantilever to measure one point on a ∆f(V)
curve. Following values were chosen for the parameters that determine individual force components in our model:
ELJ =0.2eV,RLJ =3˚A, zLJ = 3.2˚A,R =400nm, A =0.029eV, zHam = 4.8˚A, V =0, Q = 0.5.
Thmeindecay constamnitn for tunn0eling−probabilittyipwas β=2.3 ˚AH−1. This choice0 of par−ameters iCsPmDostly arabpeitxrary−but
should be quite realistic, as it was motivated by several (unpublished) STM/AFM measurements carried out in our
department’s lab by our experimentalist colleagues.
III. RESULTS
Fig.2 presentsresultsofthe stochasticmodelfor variouscharginganddischargingratesdeterminedby parameters
ν and ν . The values of ν are indicated indirectly, in terms of ν∗ =ν exp( βz ) 10−4ν . Such rescaling by the
0 2 0 1 0 − eq ≈ 0
exponential factor facilitates a direct comparison between the different time scales.
5
FIG. 2: (a,e) ∆f and Ediss for ν1∗ = ν0exp(−βzeq) = 108 s−1, ν2 = 105 s−1 and f0 = 4.6858×104 Hz in a wider range of
voltagetoillustratethetypicalshapeofwholemeasured curvesincludingthepartsawayfrom theresonance. (b-d)Aseriesof
simulated∆f(V)measurementsfordifferenttunnelingrateprefactorsν0(specificforeachpanel,expressedasν1∗),anddifferent
QD discharging rates ν2 (encoded by the color of the corresponding curve), with the same f0. (f-h) Corresponding series of
simulated resultsforthedissipated energyperoscillation (or, equivalently,powerneededtodrivetheoscillation) asafunction
of voltage bias. (b,f) ν1∗ = 106 s−1, (c,g) ν1∗ = 107 s−1, (d,h) ν1∗ = 108 s−1, ν2 ranges from 103 s−1 (red), through 104 s−1
(violet), 105 s−1 (dark blue),106 s−1 (light blue),107 s−1 (dark green), 108 s−1 (light green) to 109 s−1 (orange).
The figure displays a set of curves that represent the simulated frequency shift ∆f and the dissipated energy E
diss
as a function of voltage bias V. Occurrence of the resonant tunneling though a localized QD electronic state ǫ
dot
introduces a variation of the frequency shift ∆f at the corresponding voltage, as seen in Fig. 2(a). This effect is
accompanied with an appearance of a pronounced signal in the dissipated energy E channel. Here we should note
diss
thatsimilarphenomenaatbiasvoltagesnearthethresholdfortunnelingarealsoobservedwiththemetallic-tipmodel,
which is not discussed here any further. The only difference between results obtained with a localized-state and a
metallic tip is that in the latter case the systemstays in the chargedstate for higher voltagesas a consequence of the
continuum density of tip states ρ .
tip
Both the variation of the frequency shift ∆f and the dissipated energy E is directly related to the charging
diss
process occurring during the resonant tunneling. In our model, the local electronic states in the QD and on the tip
apex are chosen so that a resonance for tunneling occurs around the bias of +0.5 V. The relevant parameters were:
the dot state energy E =0.5 V with respect to the substrate Fermi level, the apex state energy E =0 (right a
dot tip
the Fermi level of the tip), and Gaussian smearing for both states was given by the full width at half maximum of
0.03 eV (corresponding to w =w 0.05 eV). As expected, the QD stays discharged (neutral) most of the time
dot tip
≈
for out-of-resonance values of voltage bias. In the neutral state, the ∆f(V) dependence follows a parabolic curve as
knownfrom traditional KPFMmeasurements, see Fig. 2(a) dashed line. Note that the dissipation signalis negligible
Fig. 2(e).
The situation changes when the tunneling junction is brought to the resonance (V around 0.5 V) and the tip is
sufficiently close. Provided the prefactor ν of Eq (8) is sufficiently large too, the tunneling rate between the tip and
0
the QD becomes much larger than the discharging rate, ν ν . Consequently, the QD will be found in the charged
1 2
≫
state most of the time. In such a case, the parabolic curve just shifts downwards(or upwards)by a constant amount
of ∆f with respect to the neutral case, depending on the sign of an extra Coulomb force F . Here we consider the
Q
attractive interaction F so the frequency shift ∆f is more negative.
Q
Onecanexpectthatthesystemoscillatesbetweentwo(neutralandcharged)statesrepresentedbyidealizedKelvin
parabolas,oneforthecharged(dotline)stateandtheotherfortheneutralone(dashedline). However,thesimulations
revealthatthefrequencyshift doesnothaveto liealwaysbetweenthe twoparabolas,seeFig.2(b-d). Inotherwords,
the changes of the frequency shift with respect to the neutral state sometimes tend to “overshoot” the curve that
6
FIG. 3: Comparison of results between the stochastic and the deterministic approach. Simulated frequency shift ∆f(V) (a-c)
and dissipated energy Ediss (d-f) are shown for different numbersof oscillation periods over which the result is averaged. The
numberof periods taken intotheaveraging isthetotal numberof periods spend on measuring, viz. (a,d) 10 periods, (b,e) 100
periodsand(c,f)1000 periods. Thecolorcodeforcurvesshowingthestochasticsimulation isthesameasinFig. 2,thecurves
corresponding to the deterministic simulation are all depicted as dashed black lines. Comparison to the stochastic results can
be used to identify curves corresponding to individualν2 valuesin thedeterministic case.
would correspond to the ∆f(V) on a fully charged state. We observe this effect, in particular, when the charging
anddischargingprobabilitiesν ,ν areofcomparablemagnitudeandthe QDthereforeoftenchangesits chargestate.
1 2
Such a situation occurs in two different regimes. First, for high resonant tunneling rates ν , at the “edges” of the
0
resonance, where the high resonant tunneling rate ν becomes partially compensated by being slightly off-resonance.
0
Thusitmakesthe actualtunneling rateν fromthe tipcomparabletothe dischargingrateν . Second,ithappens for
1 2
moderateresonanttunnelingratesν justatthecenteroftheresonance. Theregionsoftheovershootpartiallyoverlap
0
with regions of large dissipation, cf. Fig. 2(b-d) and Fig. 2(f-h). Nevertheless the dissipation tends to increase for
slowercharginganddischargingrates(closertothetiposcillationfrequencyf )whileforfastcharginganddischarging
0
rates, the dissipation is small even at voltages corresponding to the ∆f overshoot.
Now, we compare the stochastic vs deterministic approach, shown in Fig. 3, to get more insight into the effect of
randomness in the charging on AFM dynamics. The outcome of the stochastic simulation crucially depends on the
number ofoscillationsspend ateachpointof the measurement. In other words,it depends onthe time which ittakes
to measure one point of the ∆f(V) or E plot. The longer this measuring time, the better the stochastic results
diss
averageto suppress fluctuations andapproachthe mean value expected for the givenpoint. Fig. 3 demonstrates that
the stochastic simulation agrees perfectly with the deterministic one if the measuring time is long enough, with the
exceptionoftheorangecurvesrepresentingν =109s−1. Inthiscase,thedischargingtosubstrateissohighlyprobable
2
that the mean life time of the charged state is comparable to the simulation step dt = 0.0001/f 2 10−9 s−1,
0
≈ ×
which distorts the results of the simulations. Even for the shortest measuring time shown, t = 10/f (10 periods),
m 0
the frequency shift and dissipation calculated in the stochastic way tend to fluctuate around the values given by
the deterministic simulation. However, the fluctuations are large in that case, so they almost mask the functional
dependence. The region of large dissipation at bias near but slightly off the resonance is also a region of largest
fluctuations both of the frequency shift ∆f and of the dissipated energy E .
diss
IV. THEORETICAL ANALYSIS
First, we will analyze the effect of the (dis)charging process of the QD under the tip on the measured frequency
shift∆f. Inthisparagraph,weprovideanintuitivequalitativeexplanationofrelationbetweenfrequencyshift∆f(V)
and the (de)charging rates ν ,ν . A detailed derivation of quantitative formulas for the frequency shift in a small-
1 2
amplitude limit can be found in Appendix A. If the system stood in the neutralstate indefinitely, the frequency shift
∆f(V) would follow one Kelvin parabola as sweeping the voltage bias through a finite range; see dashed lines in
Fig. 2(b-e). Similarly for the chargedstate, the system follows a different parabola—see dotted lines in Fig. 2(b-e)—
which is rigidly shifted accordingto the neutral case. Now, for a suitable voltage,a resonance between the electronic
7
FIG. 4: Illustration of the relation between charge switching and energy dissipation. Panels from bottom to top: Vertical
position ofthetip∆z asafunctionoftimet;charge-dependentforcecomponentFQ(t);totalforceF(t)actingbetweenthetip
and the sample (dashed line shows thecharge-independent part); and theenergy Ediss dissipated since t=0 (dEdiss=Fdz).
state on the tip apex and the electronic state of the QD is established. This initiates the electron transfer from the
oscillating tip into the dot.
If the rate of QD charging ν is comparable to the rate of discharging ν , the average charge on the QD will be
1 2
something between 0 and 1 electron. This mean value of the charge corresponds accordingly to a frequency shift ∆f
somewhere in between the two above mentioned parabolas. However, the mean charge does not yet explain why the
frequency shift sometimes falls outside the area between the two parabolas. In particular, it goes below the lower
parabola that corresponds to the fully charged state, as seen in Fig. 2(b-e).
Tounderstandthis extrafrequencyshift, correlationbetweenthe dynamicsofthe chargeandtiposcillationshasto
be taken into account. As the tunneling is more probable when the tip comes closer to the QD during its oscillation,
the charge of the QD will be on average closer to 1 when the tip goes through its lower positions. Conversely, the
charge will on average be closer to 0 when the tip is in its upper positions. This variation of charge during the tip
oscillation creates an extra effective gradient of the electrostatic force between the tip and the sample, on top of
the usual distance dependence arising e.g. from the 1/r2 factor in the Coulomb law. Namely, the frequency shift
∆f derived within the approximation of small oscillation amplitude are given by expression (see also Eqs. (A15) in
Apendix A):
f ∂F z ,q¯(z ,V) ∂F z ,q¯(z ,V) βν ν (z ,V)
0 eq eq eq eq 2 1 eq
∆f(z ,V)= + , (11)
eq 2k − (cid:0) ∂zeq (cid:1) (cid:0) ∂q¯ (cid:1) ν2+ν1(zeq,V) 2+(2πf0)2!
whereAistheamplitudeoftheoscillation,βthedecayfactorofthetunneli(cid:0)ngprobabilityw(cid:1)ithdistance,f aresonance
0
frequencyofthecantilever,ν (z ,V)themeantunnelingrateandν the(constant)dischargingratetothesubstrate.
1 eq 2
The first term on the right-handside of Eq. (11) is a contribution to the frequency shift which includes all charge-
independentforcesaswellasthecharge-dependentforcecomponentevaluatedforthetime-averagedvalueofthecharge
q¯. The second term corresponds to the modification of the frequency shift by the charge dynamics. We can see that
therelativefrequencyshift∆f/f correspondingtothissecondtermtendstobemaximalwhenν (z ,V) ν f .
0 1 eq 2 0
≈ ≫
In such instances, peaks on the ∆f(V) curve can be expected to appear.
Similarly, correlation between the temporal charging and probe dynamics is also manifested by appearance of
enhanced signal in the energy dissipation channel[30, 31] E . It means that energy has to be supplied to the
diss
cantilever (or sometimes retrieved from it, if the dissipation is negative) in order to maintain constant amplitude A
of the oscillation. From the results shown in Fig. 3 we can see that the dissipation signal appears at the “edges” of
the resonance.
Because of the stochastic nature of the charge dynamics, random fluctuations in the measured values of both the
frequency shift andenergydissipationhaveto be expected, as exemplified by the resultsof the stochastic simulations
8
plotted in Fig. 3. Such fluctuations are particularly pronounced when each measurement involves only few oscilla-
tion cycles. Quantitative expressions for this fluctuations will be derived for the small-amplitude approximation in
Appendix C. Interesting observation can be made if we express the magnitude of frequency-shift fluctuations as the
relative root mean square deviation from the mean expectation value of ∆f. This quantity turns out to be propor-
tional (as a function of the three time scales f , ν and ν ) to the square root of the expectation value of the energy
0 0 2
dissipation E :
diss
∆f ∂F/∂q
δ = E . (12)
diss
f √2NπkA2√β | |
(cid:18) (cid:19) p
p
SeeAppendixCforthedetailedderivationofEq.(12)andforthedefinitionofallquantitiesthatappearinit. Eq.(12)
suggests that one should expect large fluctuations in the measured value of ∆f whenever the dissipated energy E
diss
isalsolarge. Indeed,sucheffectisobservedinournumericalsimulations,seeFig.3. Theinstabilitiesofthefrequency
canbe understoodasconsequenceofthe frequentabruptchangesofthe forceF whichthe tip experiencesduringits
Q
oscillation. Based on on Eq. (12) together with the analysis of E that will follow, we realize that large frequency-
diss
shiftfluctuationsareexpectedtoappearundertheconditionν ν f . Moreover,thepresenceoflargefluctuation
1 2 0
≈ ≈
in the frequency shift channel is accompanied by large fluctuations of the dissipated energy E too.
diss
To get more insight into the origin and character of the dissipation signal E during the (de)charging process,
diss
let us analyze the dynamics of the probe driven by time-dependent Coulomb force F . Fig. 4 illustrates correlation
Q
between the dissipation signal E and time-dependent force F during two tip oscillation periods. The cosine
diss Q
function in the bottom panel represents the immediate position ∆z(t) = z(t) z of the tip with respect to its
eq
−
equilibrium position z . The second panel from the bottom shows what the time development of the Coulomb force
eq
F (t) may look like. The force F (t) directly relates to the charge q(t) of the QD. By definition, the Coulomb force
Q Q
equals zero (F =0) for q =0 and it jumps to a non-zero value when the state of the QD switches from the neutral
Q
to the chargedone. The charge-dependent component F is a sizable contribution to the total force F, shown in the
Q
second panel of Fig. 4 from the top. All other components of F besides F are assumed to be conservative forces.
Q
The dissipated energy E is tied to the mechanical work consumed by the tip; it can be calculated by integrating
diss
the total force over the path given by ∆z(t). Alternatively, because all forces except F are conservative, E can
Q diss
also be calculated by integrating F dz alone, as indicated in the top panel of Fig. 4.
Q
First, let us discuss the originof the strong fluctuations observedin the dissipation signal E , which is especially
diss
pronouncedwhenthe measuringtime equalsonlyfew oscillationperiods. Thedissipationcanbe understoodinterms
of the Coulomb force F affecting the tip dynamics. Importantly, F does not change sign, being always attractive
Q Q
in our case. Consequently, it accelerates the probe when the probe is approaching the surface but slows it down
when it is retracting. In general, the occurrence of the F is not synchronized with the motion of the probe. The
Q
accelerating and decelerating effects cancel each other if the force F acts during the whole oscillation period, i.e.
Q
the case when ν ν . Similarly, there is no substantial net effect if F is almost zero because ν ν or if
1 2 Q 2 1
≫ ≫
ν ,ν f . In the last case, the QD may be either charged or neutral but tends to stay in the same state for the
1 2 0
≪
whole oscillation period. The effect also diminishes when the measuring time becomes sufficiently large to average
outthis (de)accelerationovermany periods. However,under certainconditions the netactiononthe probemotionis
notcompletely compensatedandthe probe becomes acceleratedor damped, respectively. Consequently,the feedback
loop has to take an appropriated action to correct the oscillation amplitude A. This gives rise to sudden fluctuation
of the dissipated energy E . This effect is accentuated when the frequency of the charging ν and discharging ν
diss 1 2
processes are comparable to the resonant oscillation frequency f , cf. Eq. (C20).
0
Secondly,the presenceof positivemeandissipationsignalE is relatedto a phasedelay ofthe CoulombforceF
diss Q
with respect to the tip oscillations. As we mentioned before, the tunneling rate ν of an electron from the tip to the
1
QD is given by Eq. (8), which depends exponentially on the z-distance. Therefore the charging process occurs more
frequently when the tip is closer to the QD. The actual charging tends to happen only some time after the tunneling
conditions become favorable for it. So even though the switching of F is random, it tends to be partially correlated
Q
with ∆z, but with some delay behind it. This lag of the charge behind tip oscillations ensures that the Coulomb
force F resulting from the charged state acts on the tip more often when the tip goes up than when it goes down.
Q
One effect of F on the probe motion, either accelerationor damping, thus prevails over the other. This gives rise to
Q
non-conservativeforce componentintroducing non-zerodissipatedenergy E . In particular,if the Coulombforceis
diss
attractive,asweconsiderinourexample,thereispositivedissipation. Fig.4,toppanel,showsatypicalcaseinwhich
there is a total energy loss (positive dissipation) over two oscillation cycles, although there is negative dissipation
during the first cycle.
We should note, the dissipation signal diminishes if the force F acts during the whole oscillation period, as is the
Q
case when ν ν , or if the phase shift between F and z is negligible because ν f . On the other hand, the
1 2 Q 1 0
≫ ≫
signal is maximal when both tunneling rates and the oscillation frequency are comparable, ν ν f . Then the
1 2 0
≈ ≈
force F switches frequently during one oscillation cycle. To justify the condition ν ν f for the maximum
Q 1 2 0
≈ ≈
9
dissipation signal more rigorously, we derived an analytic expression for the dissipated energy E (for details see
diss
Appendix B):
∂F 2π2A2βf ν¯ ν
0 1 2
E = ; (13)
diss −∂q (ν¯ +ν )[(ν¯ +ν )2+(2πf )2]
1 2 1 2 0
from the expression above, we see that the dissipated energy is proportional to the force derivative with respect to
charge. From a detailed analysis we can also see that the dissipated energy tends to be large when ν¯ ν f ,
1 2 0
≈ ≈
where ν¯ =ν (z ,V).
1 1 eq
Fromthediscussionabove,wecandeducethatthecharacteristicshapeandobservedinstabilitiesinKelvinparabola
encode temporal information about the charge states. Thus it can be seen as a complementary tool to pump-probe
STM experiments [39], but providing only qualitative information about characteristic tunneling rates ν ,ν , (i.e.
1 2
lifetime of generated charge state). In principle, we have established a set of three equations (11,13,12), which could
be employed to determine e.g. the characteristic rates ν ,ν . Nevertheless, this is not immediately possible, because
1 2
the equations contain more unknown variables such as β or derivatives of force. On the other hand, some of these
parameters could be perhaps estimated from independent measurements on given QD system (e.g. β from current
measurement). However, more elaboration on the strategy is beyond the scope of this paper.
V. CONCLUSIONS
In conclusion, we discussed in detail the temporal response of a dynamical AFM probe to charge-state switching
in QDs at different time scales. We presented numerical simulations that captured the coupled dynamics of both
the switching charge states and the oscillating probe. We tested two complementary approaches to the simulation:
a stochastic (based on pseudo-random decisions at each step) and a deterministic one (based on numerical solution
of differential equations for mean values). The analysis reveals that the presence of the resonance tunneling between
the tip and QD (ν ) and between the QD and the substrate (ν ) gives rise to the instabilities in frequency shift ∆f
1 2
and the enhanced dissipated energy E under certain conditions.
diss
Inaddition,wederivedapproximateanalyticformulasforthe frequencyshiftandthe dissipatedenergyinthelimit
ofsmallamplitudes. Theseformulasallowsustorelatethe frequencyshift∆f,its fluctuationδ ∆f anddissipation
f
Ediss to the characteristic rate parameters that control the charging and discharging process(cid:16), i.e(cid:17). to the electron
tunneling rates ν (tip–QD) and ν (QD–substrate). Firstly, we found that the observed frequency shift ∆f can be
1 2
much larger than frequency shift corresponding to the permanently charged QD. This effect is maximized when the
tunneling rates ν and ν are of comparable magnitude. Secondly, the dissipated energy E and the frequency-
1 2 diss
shift fluctuations δ ∆f are enhanced under the condition ν ν f . Thirdly, the frequency-shift fluctuation
f 1 ≈ 2 ≈ 0
magnitude δ ∆f i(cid:16)s pro(cid:17)portionalto the square root of the expectation value of the energy dissipationE . Finally
f diss
we discussed(cid:16)how(cid:17)the characteristic shape and observed instabilities in Kelvin parabolas encode information about
temporal variations of QD charge states. We believe that these features can be, in principle, exploited in future
researchto obtainmorequantitativeinformationconcerningthe dynamicalpropertiesofchargeableQDsfromKelvin
probe measurements.
VI. ACKNOWLEDGEMENT
WethankM.Sˇvec,J.Berger,andJ.Reppforvaluablediscussions. ThisworkwasfinanciallysupportedbyaCzech
Science Foundation grant no. 14-02079S.
Appendix A: Frequency shift
We are now going to demonstrate the origin of the “frequency shift overshoot” and of the dissipation signal by
derivinganalyticformulasforboththefrequencyshiftandenergydissipationundercertainapproximations. Ourgoal
isnotfindingcompletelygeneralanalyticformulas,whichwouldbe abletoreplacethenumericalsimulation. Instead,
we wantto understandqualitatively the influence of the three time scalesf , ν and ν onthe measurement. We will
0 1 2
assumeacaseinwhichthedeterministicmodelisa“goodenough”description. Thereforewedisregardthestochastic
nature of the charging process. We will consider only the small amplitude limit of the cantilever oscillations, so that
we can restrictthe changesof the short-rangeforce andtunneling probabilities to the first orderin Taylorexpansion.
10
It means that we only consider constant terms and terms linear in the position z. With this approximation, tip
oscillation is well described by a sinusoidal function
∆z(t) Acos(2πft)=ARe[exp(2πift)] (A1)
≈
as a function of time t (where ∆z =z z ) We have arbitrarily chosen t=0 in such a way that ∆z(t) is the cosine
eq
−
function with a zero phase shift. In what follows, the phase of other periodically oscillating quantities will be given
relative with respect to the phase of ∆z(t). We will look for a harmonic solution to describe the temporary changes
of the charge q(t) too:
q(t)=q¯+A cos(2πft+φ )=q¯+Re[qˆexp(2πift)]. (A2)
q q
In the second form of the above expression, we have introduced the complex amplitude qˆ = A exp(2πiφ ). The
q q
complex formalism will be more convenient for the next steps of the derivation than working with sine and cosine
functions. The z-dependence of the tunneling rate, Eq. (8), can be rewritten as
ν (z)=ν¯ exp( β∆z), (A3)
1 1
−
where ν¯ =ν (z ). In the small amplitude approximation, the z-dependence of ν can be linearized as
1 1 eq 1
ν =ν¯ (1 β∆z). (A4)
1 1
−
With ∆z given by Eq. (A1), Eq. (A4) can be rewritten as
ν =ν¯ (1 βARe[exp(2πift)]) (A5)
1 1
−
andifwethensubstituteEq.(A2)andEq.(A5)intoEq.(10)whileneglectingatermproportionaltoA A (justified
q
×
in the small amplitude limit), we get
Re[2πifqˆexp(2πift)]=ν¯ (1 βARe[exp(2πift)])(1 q¯) ν¯ Re[qˆexp(2πift)] ν (q¯+Re[qˆexp(2πift)]). (A6)
1 1 2
− − − −
After rearrangement,Eq. (A6) becomes
Re[(2πif +ν¯ +ν )qˆexp(2πift)]= βAν¯ (1 q¯)Re[exp(2πift)]+ν¯ (ν¯ +ν )q¯ (A7)
1 2 1 1 1 2
− − −
The last equation will be satisfied for arbitrary t if
ν¯
1
q¯= (A8)
ν¯ +ν
1 2
and
ν¯ ν
1 2
qˆ= βA . (A9)
− (ν¯ +ν )(ν¯ +ν +2πif)
1 2 1 2
The time dependence of the charge q(t) in the small amplitude approximation can be thus obtained by substituting
the expressions Eq. (A8) and Eq. (A9) into Eq. (A2).
The frequency shift measured in AFM is given by [40]
f
0
∆f = F∆z , (A10)
−kA2h i
assuming ∆f f and thus f f . The angle brackets denote simultaneous temporal and ensemble averaging.
0 0
≪ ≈
Showing only the time averaging explicitly, we can write
f2 To f2 To
∆f = 0 F(t)∆z(t)dt= 0 F(t)cos(2πf t)dt, (A11)
−kA2 −kA 0
Z0 Z0
where T = 1/f is the period of the cantilever oscillation. As the interaction force F(t) felt by the oscillating tip
o 0
depends on the tip position z(t) and on the quantum-dot charge q(t). For small amplitudes, we can linearize F(z,q)
and write (retaining the f f approximationfrom now on)
0
≈
∂F(z =z ,q¯) ∂F(z ,q =q¯)
eq eq
F(t)= ∆z(t)+ Re[qˆexp(2πif t)]. (A12)
0
∂z ∂q