Abstract
Frequency modulated Kelvin probe force microscopy (FMKFM) is the method of choice for high resolution measurements of local surface potentials, yet on coarse topographic structures most researchers revert to amplitude modulated liftmode techniques for better stability. This approach inevitably translates into lower lateral resolution and pronounced capacitive averaging of the locally measured contact potential difference. Furthermore, local changes in the strength of the electrostatic interaction between tip and surface easily lead to topography crosstalk seen in the surface potential. To take full advantage of the superior resolution of FMKFM while maintaining robust topography feedback and minimal crosstalk, we introduce a novel FMKFM controller based on a Kalman filter and direct demodulation of sidebands. We discuss the origin of sidebands in FMKFM irrespective of the cantilever quality factor and how direct sideband demodulation enables robust amplitude modulated topography feedback. Finally, we demonstrate our singlescan FMKFM technique on an active nanoelectronic device consisting of a 70 nm diameter InAs nanowire contacted by a pair of 120 nm thick electrodes.
Introduction
Device performance of current nanoelectronic devices, and even more so of potential future generations including nanowires or molecular junctions, critically depends on transport properties varying on a length scale of a few nanometres only in the active channel or at electrode interfaces. Methods for local electronic characterisation, providing accurate measurements with nanometre spatial resolution, are in very high demand, but have been lagging behind the technological requirements.
Kelvin probe force microscopy (KFM) is an established technique that allows for the mapping of local electrostatic potentials with an atomic force microscope (AFM) [13]. In contrast to electrostatic force microscopy (EFM), which measures merely the effect of electrostatic forces on the oscillation of the tip, a feedback loop nullifies the electric field by adjusting a bias voltage between tip and sample. Hence, Kelvin probe force microscopy is able to quantify the local contact potential difference (CPD), U_{lcpd}, which contains contributions, e.g., from the difference in work function between the AFM tip and structures on the sample, dopants and trapped charges in the device, or voltages applied to electrodes.
For electronic devices on the nanoscale, KFM measurements provide a unique tool to shed light upon a variety of otherwise inaccessible properties. For example, with a constant current passing through a twoterminal device, the potential drop at the contacts directly relates to the contact resistance. To extract contact resistance through traditional fourpoint measurements becomes increasingly difficult for scaled devices, in which the contact length is comparable to the device length. Recently, KFM has been used to extract the surface state density and Schottky depletion region in semiconductor nanowires [4,5] or to determine the mean free path in carbon nanotubes [6]. KFM also allows one to determine intrinsic doping of twodimensional crystals such as graphene [7,8], where surface potential and electronic properties depend on the number of layers.
KFM has found widespread use in both vacuum and ambient environments. Most commercial instruments for operation in air include a scan mode based on amplitude modulation KFM (AMKFM). In this mode, the feedback loop nullifies the cantilever oscillation that is excited by a modulated electrostatic force. Hence, the KFM image is a map of voltages required to compensate the electrostatic force at every point of the scanned field. However, since cantilever and AFM tip are extended objects, this voltage does not necessarily correspond to the local contact potential difference, U_{lcpd}, but represents a weighted average over the potentials present on the entire sample surface [9]. For AMKFM, the weights are determined by the capacitance gradient, C', between the probe and the sample. Due to the long range electrostatic force, even parts far from the surface, such as the cantilever beam, can account for a significant fraction of the signal, limiting the spatial resolution and accuracy of the measurement. Within nanoscale devices, for example, electrode potentials may completely overshadow the channel [10].
Known approaches to increase spatial resolution and accuracy of surface potential measurements include deconvolution techniques [11,12] or the use of slightly blunt tips supported by a cantilever of minimal surface area [9]. However, deconvolution techniques require a detailed model of the AFM tip to be accurate and usually neglect the sample topography [12], whereas blunt tips inevitably reduce topography resolution on threedimensional structures.
KFM measurements are further complicated by a strong dependence of the detected signal on the tip–sample distance. In the often employed liftmode schemes, each line is scanned twice: first to acquire topography, and subsequently to retrace the scanned line at a small distance, Δz, above the surface to perform KFM measurements. This enables tuning the ac modulation frequency for KFM to resonance to enhance the signal, and, at the same time, to reduce the contribution of van der Waals forces to the total force measured and compensated. The scan at elevated height, however, reduces lateral resolution and accuracy of the KFM data as we will detail below. To minimise such lateral averaging, singlescan methods are preferred, performing topography and KFM measurements simultaneously. An additional benefit of singlescan AFM and KFM is the inherent suppression of electrostatically induced topography artefacts present in noncompensated topography scans [13,14]. In AMKFM, singlescan methods can be implemented taking advantage of multiple eigenmodes of the cantilever, using one mode for topography and another for KFM. Nevertheless, the averaging effect of the cantilever beam remains (see below in Figure 1).
An alternative approach typically applied in vacuum is based on frequency modulation [15]. To this end, the frequency of the cantilever is usually tracked by a phaselocked loop (PLL). Its output signal, the frequency shift Δf, exhibits a frequency component at the electrostatic modulation frequency, which is nullified by the Kelvin feedback loop. Frequency modulated KFM (FMKFM) [16,17] thus provides a map of potentials required to minimise the electrostatic force gradient, proportional to Δf for small mechanical amplitudes, at every point during the scan. As a consequence, the contributions from different parts of the sample and the probe to the measured signal are weighted by the secondorder capacitance gradient, C'', which effectively eliminates the averaging contribution of the cantilever beam as we explain in the following.
Figure 1 shows a model calculation using typical cantilever and interaction parameters, summarising how much tip apex, cone, and beam of an AFM cantilever probe contribute to the measured KFM signal in AM and FM operation. Shown are the percentages of the contributions and corresponding weighting factors C' and C'' for AM and FM, respectively. To this end, we applied an analytic model of the electrostatic tip–sample interaction force [18] to the approximate geometry of a typically used cantilever (Olympus AC160), and we calculated C' and C'' as a function of tip–sample separation for different oscillation amplitudes (see Supporting Information File 1 for details). While tip apex and cone clearly dominate the FMKFM signal, opening the avenue to high resolution quantitative imaging, the cantilever beam at a distance of 14 μm dominates the AMKFM signal even close to the sample, which is the main reason for the notoriously low lateral resolution and poor potential accuracy in this mode. When comparing AM and FM modes, one should note that in liftmode AMKFM the cantilever is not oscillating anymore when the electrostatic forces are nullified, whereas the mechanical oscillation remains in multifrequency AMKFM and FMKFM. Hence, for liftmode the case A → 0 should be considered, whereas in singlescan modes the oscillation applied for tracking topography remains. For best sensitivity and minimal spatial averaging, AM and FM modes need to be operated very close to the surface.
While the FMKFM approach is clearly superior in terms of signal composition, several issues complicate its use in practice. First, it is often performed together with frequency modulated topography feedback that employs a PLL to determine Δf. The nonmonotonous tip–sample interaction, by which Δf can change its slope between netattractive and netrepulsive forces, can complicate stable operation of the topography feedback and may ultimately render PLL and amplitude controller unstable. On samples with coarse topography and steep features, maintaining stable FM topography feedback demands careful selection of operating parameters and slow scanning speeds. Furthermore, the choice of suitable bandwidths for topography and KFM feedback is more involved in traditional FMAFM/FMKFM implementations. For example, when Δf is used as an input to the lockin amplifier detecting the electrostatic modulation, the PLL bandwidth must be wide enough to include the modulation frequency. Yet, it should be kept as small as possible for stable PLL operation and maximum noise rejection [19]. Finally, the pronounced distance dependence of C'' for apex and cone, as depicted in Figure 1, makes operation close to the surface more challenging, since small errors of the topography feedback produce marked changes of the effective Kelvin feedback gain. Similarly, when the tip encounters steep edges in topography, C'' may increase due to a larger effective tip–sample capacitor area, further complicating stable feedback operation. The distance dependence is less pronounced at larger distances employed in liftmode FMKFM [10], but in addition to reduced lateral resolution, large modulation voltages are required due to weaker signals [20], which may induce band bending. Furthermore, when scanning across insulating parts of devices, such as gate oxides, not only the local dielectric constant changes, but because of their thickness also a limit is put on the minimum approachable distance in Figure 1. As a result, deliberately slow feedback settings to ensure stable operation are common practice.
In this paper, we describe a practical approach to FMKFM providing solutions to these issues. We remove the interdependence of topography and KFM feedbacks by focusing on the information contained in the sidebands produced by the electrostatic modulation [20]. Employing a commercially available lockin amplifier, we detect these sidebands directly. Thus, with frequency modulated distance feedback, the PLL bandwidth can be restricted to the topography only. We further demonstrate the advantage of combining FMKFM with amplitude modulated AFM (AMAFM) for tracking topography of highly structured surfaces with small amplitudes and netattractive interaction in air. Since the oscillation amplitude decreases monotonically with distance, no special precautions are required to ensure feedback stability. Finally, we introduce an improved Kelvin feedback loop based on stochastic optimal control that continuously adjusts its sensitivity to local changes in C'', thereby reducing the risk of feedback instabilities and topography crosstalk on difficult heterogeneous samples.
Theory
The origin of sidebands
The cantilever motion and the origin of sidebands are understood from a damped harmonic oscillator driven by an external drive, a(t), and perturbed by the tip–sample interaction force ,
where z(t) is the cantilever deflection, ω_{0} the eigenfrequency, k the spring constant, and Q the quality factor of the cantilever. For an oscillation with amplitude A and drive frequency ω_{d} ≈ ω_{0}, the interaction force can be approximated to
where z_{0} is the mean tip position, and and are the effective force and force gradient, respectively. Explicit expressions for the effective force and force gradient, averaged over the oscillation period, T ≈ 2π/ω_{d}, are [21]
and
The effective force and force gradient, as introduced here, allow one to describe the motion of the tip in the nonlinear force field close to the sample with the model of a perturbed harmonic oscillator, provided the oscillation remains approximately harmonic with constant amplitude [22].
With a small perturbation, k_{ts} << k, the resonant frequency of the cantilever changes from ω_{0} to ω_{0} + Δω with Δω/ω_{0} = −k_{ts}/2k [15]. Accordingly, a modulation of the force gradient, e.g., by an oscillating electric field, will cause a frequency modulation of the resonance. A modulation at a single frequency ω_{m} will produce sidebands at integer multiples of the modulation frequency, that is, cantilever oscillations at ω_{0} ± ω_{m}, ω_{0} ± 2ω_{m} and so on.
For the derivation of the sideband signals and their respective amplitudes, we assume a modulation of the force gradient at the frequency ω_{m}: . Note that the effective force gradient as calculated above, Equation 3, is valid for ω_{m} << ω_{d}.
Then, by Fourier transformation of the equation of motion, Equation 1, we arrive at
with
where and are the Fourier transformed deflection and drive, respectively, and G(ω) is the complex transfer function of the damped harmonic oscillator.
Equation 4 and Equation 5 present an iterative scheme to determine the spectral components of the cantilever oscillation, where in each step on the left hand side of Equation 4 is refined by the expressions on the right hand side. Starting from an oscillator at rest, is the carrier oscillation due to the external drive, as in the unperturbed system. Spectral components at ω ± ω_{m} emerge in the next iteration step,
This is the fundamental pair of sidebands of the force modulated damped harmonic oscillator. With << k, the higher order sidebands arising in the subsequent iterations are usually negligible.
Equation 6 also describes the sideband amplitude transfer function when the expression is evaluated close to the sideband frequencies. With the substitution ω − ω_{d} → ω, we find
with the approximation
where, in the latter approximation, we consider only the dominant term for a drive close to the eigenfrequency, i.e., ω_{d} ≈ ω_{0} and ω << ω_{0}, and ω_{c} = ω_{0}/2Q is the cantilever bandwidth.
For modulation frequencies well beyond the cantilever bandwidth, G(ω_{0} ± ω_{m}) ≈ −iω_{0}/2ω_{m}, and the amplitude of each sideband is , where A is the carrier amplitude. The latter expression also follows immediately from a narrowband frequency modulation of a carrier oscillation at ω_{d}. With a carrier amplitude A and the peak frequency deviation , a frequency modulation at ω_{m} produces two sidebands with amplitudes βA/2, where is the modulation index [23]. Under ultrahigh vacuum conditions, large Q factors typically cause negligible cantilever bandwidths, making this approximation valid, e.g., for finding the noise power spectral density of the frequency shift signal in FMAFM [24]. When the narrowband conditions are not met (β >> 1), the iterative scheme for the sideband amplitudes in Equation 4 and Equation 5 still approaches the Bessel functions describing the sideband amplitudes in a general frequency modulation for ω_{m} >> ω_{c} (see Supporting Information File 1).
The Fourier approach presented above also accurately models the behaviour of the sideband amplitude and phase for modulation frequencies approaching the cantilever bandwidth ω_{c}. In Figure 2a, we show the expected and experimentally measured sideband amplitudes and phases. The excellent agreement with the above model proves the validity of our derivation. Each sideband is phaseshifted by ±φ_{k} + argG(ω_{d} ± ω_{m}).
For narrowband frequency modulation, we can define a complex modulation index by the sideband and carrier amplitudes as . With Equation 8, the dc response (ω → 0) of thus is
Figure 2b shows the amplitude and phase of for pure narrowband FM and the harmonic oscillator as a function of ω_{m}/ω_{c}. The amplitude and phase only agree with the result for pure narrowband frequency modulation when the cantilever bandwidth is negligible compared to the modulation frequency. For low modulation frequencies approaches instead (Equation 9).
To further demonstrate the validity of the sideband transfer function, we show in Figure 3 the response to a step in from both the approximation in Equation 8 and from a numerical simulation of the perturbed harmonic oscillator, Equation 1, including lockin amplifiers at ω_{d} ± ω_{m}. Each change in the force gradient modulation also excites a transient oscillation at the resonant frequency of the cantilever, which appears in the sideband signal and decays exponentially with 1/ω_{c}. Therefore, the filter settings of the lockin amplifier should be set accordingly to provide sufficient rejection near ω_{m}.
We conclude that sidebands evolve as soon as k_{ts} gets modulated and it is not important whether the resonant frequency is actually tracked or not. The main benefit of tracking the resonant frequency (e.g., with a phaselocked loop) is merely to keep the carrier phase constant, which would otherwise affect the sideband phases.
Electrostatic force and force gradient
The electrostatic force between the AFM tip and sample is
where is the effective capacitance gradient, U_{ts} is the tip–sample voltage, and U_{lcpd} is the local contact potential difference.
For Kelvin probe force microscopy, U_{ts} is modulated around a dc voltage: U_{ts} = U_{dc} + U_{ac} cos(ω_{m}t). Therefore, the electrostatic force and likewise its gradient, , are modulated at ω_{m} and 2ω_{m},
where
and
These modulations of the force gradient cause sidebands of the cantilever deflection at ω_{d} ± ω_{m} and ω_{d} ± 2ω_{m}, which can be detected directly with lockin amplifiers at the respective frequencies. The lockin amplifiers return, relative to the reference oscillator, amplitude and phase of each sideband as well as their cartesian projection: the inphase component X and the quadrature component Y. In the narrowband approximation for ω_{m} >> ω_{c}, the inphase components of the modulation at ω_{m} and the amplitudes at 2ω_{m} are
and
The reference phase offsets of the lockin amplifier for the first set of sidebands at ±ω_{m} are chosen to maximise their respective inphase components, taking into account the 180° phase shift of the lower sideband. Then, X_{ω} = X(ω_{d} + ω_{m}) − X(ω_{d} − ω_{m}) = is the total inphase component, which depends linearly on the applied dc bias. Furthermore, when U_{dc} matches U_{lcpd}, X_{ω} is nullified and the ±ω_{m} sidebands disappear.
The total amplitude of the second set of sidebands, A_{2ω} = A(ω_{d} + 2ω_{m}) + A(ω_{d} − 2ω_{m}) = , only depends on the ac modulation amplitude and the second order capacitance gradient, . This signal thus provides a handle for imaging variations in the tip–surface capacitance, surface dielectric properties [25], or lateral dopant profiling [26].
In Figure 4, we show experimental data of modulation indices and , calculated from the ω_{m} and 2ω_{m} sidebands, respectively, as a function of U_{dc} for different electrostatic modulation amplitudes, U_{ac}. During this experiment, the tip was positioned above a nickel electrode with amplitude modulated topography feedback enabled in netattractive mode.
As expected from Equation 13, the ω_{m} sideband amplitudes (Figure 4a) vanish when U_{dc} = U_{lcpd}. While they change linearly with U_{dc} close to this point, there are nonlinear deviations at larger voltage offsets, which are caused by changes in C''. This is also evident from the 2ω_{m} sideband amplitudes (Figure 4b), showing the decrease of C'' with increasing voltage offsets. Since these sweeps are acquired with topography feedback enabled, the observed variations in C'' are most likely due to changes in the tip–surface separation: The AM topography feedback is sensitive to the static force gradient, which contains electrostatic interactions, Equation 10, that increase as the dc bias does not match the surface potential; consequently, the topography feedback retracts the tip, reducing C''.
In Figure 4c, we plot the ratio of the ω_{m} and 2ω_{m} sideband amplitudes, normalised to U_{ac}. As apparent from Equation 13 and Equation 14, this process cancels out the nonlinearities and collapses the sweeps at different U_{ac} to a single curve.
Additionally, we show the similarly normalised amplitudes due to the electrostatic force at ω_{m} and 2ω_{m} in the deflection signal, which we acquired simultaneously with the sidebands at ω_{d} ± ω_{m}. They show the same vshaped relationship, with their minimum being slightly shifted with respect to the FM case. This shift is due to the different weights of contributions in the AM signal (cf. Figure 1). Setting the dc bias to the minimum obtained by AMKFM does not guarantee to compensate the electrostatic force gradient and can cause height errors in topography. At the minimum determined from the sidebands, in Figure 4b reaches its maximum value, corresponding to the closest approach.
There are two major methods to find the local contact potential difference at every point during the scan. Openloop KFM exploits the fact that the 2ω_{m} amplitudes do depend on C'' but not on U_{dc} − U_{lcpd}. As demonstrated in Figure 4, the ratio of the ω_{m} and 2ω_{m} sidebands is independent of changes in C'' and only depends on the chosen modulation amplitude and dc bias,
hence U_{lcpd} can easily be determined. Note, however, that the above definition of the prefactor K^{′} is only valid for modulation frequencies well beyond the cantilever bandwidth. In the general case, K^{′} = (4/U_{ac}) G(ω_{d} ± ω_{m})/G(ω_{d} ± 2ω_{m}), that is, it also depends on the resonant frequency and the quality factor, which may change while scanning. Furthermore, there may be differences in the sideband phase shift when Q or ω_{0} are not constant (cf. Figure 2). Together, such inaccuracies in the model easily translate into uncertainties of U_{lcpd} in an openloop method. A PLL can reduce these effects, but then its transfer function needs to be considered as well [27], and the bandwidth must be larger than 2ω_{m}.
In closedloop KFM, the local contact potential difference is found by nullifying the inphase components of the ω_{m} sidebands (Equation 13) with a feedback loop adjusting the applied dc voltage [13,16]. Thus, the 2ω_{m} sidebands are not necessary to determine the CPD, and the effect of model deviations and nonlinearities is cancelled by the feedback. Furthermore, the nulling process also minimises the dc electrostatic force and force gradient (Equation 10), reducing electrostatically induced height errors [14,28,29].
However, a few critical issues remain with simple Kelvin feedback loops. For example, when the sidebands are not completely nullified by the feedback, leaving a small error δ, it follows from Equation 13 that [30]. C'' depends strongly on the electrostatic interactions between tip and surface and may change significantly on structured surfaces even for a welltuned topography controller. During a scan, imperfect Kelvin feedback therefore leads to errors in the measured CPD, constituting a source of topographyinduced crosstalk.
If additional apparent forces (or force gradients) are detected at the frequencies used for KFM, the Kelvin feedback does not compensate the CPD, but rather nullifies the inphase component affected by offsets [31]. Such crosstalk is due to parasitic capacitive coupling and observed mainly in AMKFM, where the electrostatic modulation is at high frequencies [32]. When coupling to the shaker piezo [33], cantilever resonances can amplify this effect.
Another source of crosstalk can appear when ω_{m} is set too low and the Kelvin lockins capture the modulation of k_{ts} induced by topography. This can happen on highly structured surfaces when the bandwidth of the topography feedback is insufficient for the scan speed. By monitoring the deflection power spectral density near the driving frequency, an upper frequency bound of the remaining k_{ts} modulations can be determined. In order to avoid crosstalk, ω_{m} should be chosen above this bound, considering both the bandwidth and filter steepness of the Kelvin lockins.
As already mentioned above, the tuning of the Kelvin feedback loop itself can be a challenge because its sensitivity depends on C''. This becomes even more acute for small tip–sample distances and singlescan techniques on structured surfaces, where the tip–surface interaction is not limited to the apex.
In order to address the topography crosstalk due to C'', Lee et al. [34] suggested to use a feedback signal normalised to the 2ω_{m} sideband, thus rendering the CPD tracking error independent of C'' (cf. Equation 15). However, as shown in Figure 4c, the normalisation procedure may introduce additional noise when dividing by small signals, e.g., for low U_{ac}.
In the following section we introduce a novel Kelvin feedback scheme that resolves these subtleties.
Results and Discussion
Optimal CPD estimation and Kelvin control
Most instruments provide a generic PID controller for Kelvin control, which compares the signal (X_{ω}) to a setpoint (0), yielding the error signal e. The sum of e, ∫dte, and de/dt, scaled by respective proportional (P), integral (I), and derivative (D) gains, is fed back into the system. In case of KFM, the resulting dc voltage compensates the electrostatic interactions. This standard PID feedback loop is illustrated in Figure 5a. Knowing the system dynamics, a multitude of tuning rules can be applied [35]. In practice, however, the feedback gains are often tuned by trial and error, and the derivative part is omitted altogether [36]. In many cases, only the integral part is necessary for good tracking and to eliminate steadystate errors. Integralonly controllers are therefore prevalent for topography or Kelvin feedback.
Controllers basing their actions on an error signal only are unaware of the systems they control. Thus, they need to be retuned as soon as the system bandwidths or gains change considerably, either due to different operator settings or, more importantly during KFM scans, due to local variations of electronic properties and topography of the sample. To maintain best feedback settings at every location during a scan, we introduce a novel controller for FMKFM based on stochastic optimal control [37]. Optimal control and modelbased controllers have been successfully used before in AFM, e.g., for active damping of cantilevers [38] or fast scanning [39]. According to the separation principle [37], the optimal controller that minimises the expected error can be constructed by finding an optimal ‘observer’ and an optimal ‘regulator’. As an observer, we use a Kalman filter [40], which continuously blends the sideband measurements at ±ω_{m} into an estimate of the contact potential difference, , based on a simplified model of the FMKFM detection system. The Kalman filter is the stochastically optimal observer that minimises the state error covariance [37], taking into account both measurement noise and the uncertainties in the knowledge of its state. Adapted for KFM control, the Kalman filter minimises the estimation error variance of the surface potential, . Since the aim of the regulator in KFM is to minimise electrostatic interactions, the optimal regulator is found by matching the dc bias to , thereby closing the feedback loop (Figure 5b).
Our Kalman filter design based on a model of the KFM detection system includes the sideband dynamics, Equation 8, the electrostatic force gradients acting on the cantilever, Equation 11 and Equation 12, and the transfer function of the demodulating lockin amplifier. Since the lockin bandwidths must be kept well below ω_{m} to avoid carrier and topography crosstalk, the sideband transfer functions reduce to an effective gain and phase, Equation 9. The lockin transfer function can either be measured or is known from its filter properties. We focus on a particularly simple case, the nth order critically damped lowpass filter, which is formed by n consecutive first order stages with a time constant τ. With these considerations, the transfer function for the inphase lockin components is G(s = iω) = K(1 + τ s)^{−n}, where, following Equation 13, we find the static gain and the system output G(s)(U_{dc} − U_{lcpd}).
Based on the transfer function, we find a statespace model of the system, in which we incorporate U_{lcpd} as a hidden state, and U_{dc} is the control signal. We further model the uncertainties of state transitions () and our measurements as uncorrelated, zeromean white noise with power spectral densities V and W, respectively. Hence, U_{lcpd} follows a Wiener process or Brownian motion [37]. For a derivation of the continuoustime Kalman(Bucy) filter [41], see Supporting Information File 1. In discrete time, the Kalman filter is similarly found from a discretetime statespace model [40]. In this formulation the state estimate and covariances are refined recursively as new measurements are incorporated:
At the time t, an a priori state and covariance estimate is found using the state and covariances at the time t − Δt, based on the system model. Then, the Kalman gain L is computed from the covariance matrices of the a priori estimated state and the system model. L controls the innovations process, in which the measurements at the time t are incorporated to the a posteriori estimate of state and covariances [42].
This recursive predictor–corrector structure allows for updates of the system parameters, such as the static gain K, at each instant of the state update. With Equation 15, the 2ω sideband amplitudes can thus be exploited to continuously update K = K'A_{2ω}. Consequently, the observer model will follow changes in the Kelvin signal strength due to variations of C''. This strategy avoids normalisation by potentially noisy C'' signals [34], yet changes in C'' do not affect closedloop performance. We demonstrate this in Figure 6, where we compare step responses of the closedloop Kalman observer and PI controller. As soon as the gain K drops, the noise level increases with a PI controller, whereas the Kalman estimate remains clean.
To further elucidate the performance of the controller, we plot in Figure 7 its −3 dB closedloop bandwidth, normalised to the −3 dB filter bandwidth, as a function of the normalised noise power spectral densities and of state transitions and observations, respectively. As the noise at the output, , increases for a fixed , the bandwidth is reduced (Figure 7a). The ratio resembles a signaltonoise ratio (SNR), which increases for large K and small filter bandwidths BW. The closedloop bandwidth is a function of this SNR. Therefore, in addition to avoiding divisions by small signals, the Kalman filter improves noise performance by bandwidth adjustments. For normalised closedloop bandwidths ≤ 1, the bandwidth is adjusted following (Figure 7b). Larger bandwidths are not desired, since they would counteract the lockin lowpass action.
Our setup is shown in Figure 8. We implemented the Kalmanfiltering Kelvin controller as a realtime program on the digital signal processor (DSP) of a digital lockin amplifier and PLL (HF2, Zurich Instruments), which demodulates the sidebands at ω_{d} ± ω_{m} and ω_{d} ± 2ω_{m} as well as the carrier signal at ω_{d}. Since our implementation of the Kalman filter is integrated into the lockin, all signals are available without additional digital/analog/digital conversions. Additional offsets that might affect the feedback accuracy are avoided. We have implemented the Kalman filter as a reusable component in C++ using the Eigen template library for linear algebra [43], allowing us to perform offline tests with the same code that is compiled for the DSP. In its current state, our custom FMKFM controller can work at sampling rates of up to 7200 Sa/s.
Since the sideband signals are detected individually, we do not depend on the Δf signal as in a typical FMKFM setup. Therefore, the Kelvin feedback remains the same for AM and FM topography feedback schemes. For example, on samples with coarse topography one may use AM topography feedback to avoid instabilities typically experienced with FM operation. In vacuum, this may require additional application of active Q control [44,45] to lower the Qfactor of the cantilever.
Performance on a nanowire device
To demonstrate the performance of our KalmanKFM controller, we examine an active nanowire device as depicted in Figure 8 and Figure 9. Such devices exhibit some of the most typical and relevant issues hindering reliable KFM measurements in the past: a combination of large topography with a multitude of different materials including oxides prone to charging. In Figure 9, we show a scan of a 70 nm diameter indium arsenide (InAs) nanowire with nickel (Ni) contacts (height ≈ 120 nm), obtained at a bias current of 1.4 μA under ambient conditions using a commercial AFM (Cypher, Asylum Research). The steep edges at the electrodes necessitated AM topography feedback. Oftentimes the contact resistances between nanowire and metal contacts are uneven and large, obscuring the electrostatics of devices. Traditional fourpoint measurements are limited at submicron length scales because the contact length can become comparable to the channel dimensions. For such samples, KFM appears to be an ideal tool to characterise the electrostatics in order to optimise device performance, for example as field effect transistors.
Figure 9b displays the simultaneously acquired C'', calculated from the 2ω_{m} sideband amplitudes, Equation 14. To ensure highest lateral potential resolution, we used highly doped silicon AFM tips (Olympus AC160TSR3) without a metal coating. These tips are sharp and not symmetrical at the apex (schematically depicted in the inset), explaining the increased C'' on the edge of the left electrode.
Figure 9c shows as estimated by the KalmanKFM controller. Since its gain is continuously updated using the 2ω_{m} sidebands, crosstalk due to changes of C'' is absent from the scan. Near the left electrode edge the measured U_{lcpd} displays less spatial variation because also the sides of the tip are in close proximity to the electrode edge, increasing their contribution to the tip–sample capacitance and widening the KFM point spread function. Even though the KalmanKFM controller remains stable and works unaffected by the increased C'', reaching up to seven times the mean value of the scan, the geometry of both tip and sample fundamentally limit the attainable resolution. The disturbances remaining on the edge of the right contact are due to imperfect topography feedback and accidental switches from netattractive to netrepulsive tip–sample interactions. Most importantly, edge effects are absent at the boundaries of the nanowire. Longrange potential averaging due to the cantilever beam is absent due to the gradientsensitive FM detection.
For an extraction of contact resistances, the voltage profile due to current flow needs to be separated from additional offsets in U_{lcpd}, such as spatial variations in work function. These are easily obtained from a scan at zero bias. Knowing the potential drop at the contacts, the bias current, and assuming uniform material or transport properties, for the nanowire device in Figure 9 we hereby obtain a contact resistance of 40 kΩ at the left electrode, 150 kΩ at the right electrode, and a channel resistance of 50 kΩ.
In Figure 10, we compare the performance of our KalmanKFM controller with a standard integral controller. Both controllers are tuned for optimised closedloop performance on the nanowire. The integral controller exhibits ringing artefacts at electrode edge, indicated by an arrow in Figure 10a, since the gain margin of the controller is exceeded due to the increased C''. Such feedback oscillations should be avoided particularly in singlescan techniques, because they may perturb the topography controller. Next to the nanowire, where C'' is slightly reduced because of the gate oxide, the bandwidth of the feedback loop drops due to a lower gain.
As shown in Figure 10b, with the KalmanKFM controller the feedback performance and image quality remain consistent during the scan. The error signal (X_{ω}) is almost featureless and its standard deviation reduces by about 30% on average (excluding edge effects). Better tracking is also apparent from the power spectral densities of the error signals, depicted in Figure 11. For given lockin filter bandwidths, the KalmanKFM controller can nullify the ±ω_{m} sidebands faster and better than the integral controller, without adding to the noise level or introducing feedback artefacts.
Conclusion
We have demonstrated a novel feedback controller for high resolution, frequency modulated Kelvin probe force microscopy. Based on Kalman filtering and stochastic optimal control, our feedback employs a modeldriven estimation process, which allows one to integrate sensitivity information from the 2ω_{m} sidebands. In comparison to normalisation approaches [34], this strategy does not increase the noise level.
We have tested performance on an InAs nanowire device with rough surface and abrupt height variations, which pose severe challenges to both traditional singlescan and liftmode FMKFM setups. Since direct sideband demodulation allowed us to perform FMKFM irrespective of the topography feedback, we could perform these scans with amplitude modulation in air. Similarly, Magonov and Alexander [46] demonstrated a setup in which the modulated force gradients are detected from the phase output of the carrier oscillation lockin, requiring ω_{m} to be within its bandwidth. With direct sideband detection, the detour via a phase modulation is avoided, and ω_{m} can be chosen independently of the lockin bandwidth to achieve best separation from topography.
We have provided a detailed quantitative description of the evolution of sidebands in dynamic AFM modes. Precise knowledge of their frequency dependence in low and high Q environments is not only neccessary for accurate openloop KFM techniques, but also offers a direct approach to noise performance and optimisation of frequency modulated KFM [19]. For example, ω_{m} should ideally be chosen below the thermal noise limited bandwidth of the cantilever [24], but the modulation induced by rough surfaces as well as the desired scan bandwidth establish lower limits. Furthermore, the sideband transfer function explains the higher resolution obtained by heterodyne amplitudemodulated KFM [47]. In this technique, the cantilever is driven mechanically at ω_{0} and electrically at ω_{m} = ω_{1} − ω_{0}, where ω_{0} and ω_{1} are the lowest two eigenfrequencies of the cantilever. Accordingly, the sideband at ω_{0} + ω_{m} coincides with the second eigenmode of the cantilever, resulting in an amplified signal proportional to the electrostatic force gradient instead of the electrostatic force.
Although we have found our feedback loop superior to existing controllers, we see several aspects for improvement in the future. For example, the dynamics of U_{lcpd} are currently modelled as white noise. Since successive lines in AFM scans only change slightly, information from the previous line could be incorporated, similar to a feedforward controller [29]. Other state estimators could also be integrated, including H∞ filters for minimising the worstcase error [42].
Finally, we would like to point out that using our KalmanKFM controller is not complicated. Since it automatically incorporates the lockin filter settings and the system sensitivity, the only parameters left to tune are the noise power spectral density of the U_{lcpd} transitions and sideband observations. Because the latter is easily determined from a power spectrum near the sidebands, the controller performance can be tuned in practice using the transition noise only. Independent of the chosen parameters, the feedback loop will be stable thanks to the Kalman filter structure.
Supporting Information
Supporting Information File 1: Detailed derivations of the effective forces and the statespace KFM controller.  
Format: PDF  Size: 156.8 KB  Download 
Acknowledgements
The authors thank Daniel Wright (Zurich Instruments) for his assistance with the realtime programming interface, and Romain Stomp (Zurich Instruments) for many stimulating discussions as well as joint first experiments on the frequency dependence of the sidebands. InAs nanowire fabrication was facilitated by Siegfried Karg and Ute Drechsler (IBM). We gratefully acknowledge financial support from the Swiss National Science Foundation (Project No. 134777).
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27.  Kou, L.; Ma, Z.; Li, Y. J.; Naitoh, Y.; Komiyama, M.; Sugawara, Y. Nanotechnology 2015, 26, 195701. doi:10.1088/09574484/26/19/195701 
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25.  Martin, Y.; Abraham, D. W.; Wickramasinghe, H. K. Appl. Phys. Lett. 1988, 52, 1103. doi:10.1063/1.99224 
15.  Albrecht, T. R.; Grütter, P.; Horne, D.; Rugar, D. J. Appl. Phys. 1991, 69, 668. doi:10.1063/1.347347 
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