Fast time-resolved electrostatic force microscopy: Achieving sub-cycle time resolution

Electrostatic force microscopy (EFM) is a non-contact atomic force microscopy (AFM) technique, commonly utilized to study electronic properties of materials, with nanoscale spatial resolution. Researchers have used EFM methods to investigate different phenomena such as characterizing local defects,^1–4 measuring local dielectric constants,^5–7 and probing ionic conduction.⁸ Despite these many applications, conventional EFM methods do not capture dynamic information at time scales relevant to many physical properties of interest. Time-resolved EFM (trEFM) has been used to measure events on time scales ranging from the order of seconds^1,2,8,9 to milliseconds.³ A variety of methods have also been employed to measure local dynamics in a sub-diffraction limited volume with scanning probes; these include methods such as near-field scanning optical microscopy^10,11 and scanning tunneling microscopy with pulsed laser optics.^12,13 However, these methods typically require complex optics and expensive hardware beyond the scanning probe microscope.

Motivated by the need for new methods to understand sub-diffraction-limited processes in emerging thin-film materials,^14–17 our research group reported a trEFM method to capture information on time scales as fast as 100 μs using frequency-shift feedback.¹⁸ With a focus on probing optoelectronic processes in nanostructured materials, we showed that trEFM measurements can be used to make local photovoltaic quantum efficiency maps as a function of many parameters, such as materials composition and processing,^18–20 photodegradation,^21,22 and excitation wavelength.^20,22 We recently used trEFM to demonstrate non-contact imaging of local currents with the equivalent of attoampere sensitivity.²⁰ 

However, many interesting processes occur on time scales faster than it can be resolved with feedback-mode trEFM. For example, monomolecular carrier lifetimes in hybrid perovskites^23,24 can be in the order of 10 ns–1 μs in thin films used for solar cell applications, while carrier lifetimes in state-of-the-art polymer bulk heterojunctions are typically on the order of 1 μs or less under AM1.5G illumination.^25,26 Some biological processes, such as ion channel gating,²⁷ synaptic transmission,²⁸ and electron transport in photosynthesis,²⁹ happen at time scales in the order of 1 ns–1 μs.

In our initial implementation of trEFM,¹⁸ the time resolution was primarily limited by the bandwidth of the frequency-shift feedback loop. By combining a feedback-free approach with signal averaging and analyzing the cantilever motion, we later demonstrated sub-microsecond time resolution in trEFM.¹⁹ Here, we describe in detail a robust implementation of feedback-free trEFM. We place particular emphasis on the importance of phase-locking the excitation signal, the opportunities for reducing system noise via methods such as photothermal drive,^30–34 and the selection of cantilever parameters. Finally, we show that with improved software implementation, it is possible to process the gigabyte-scale datasets required to render trEFM images concurrently with data acquisition, realizing approximately 10-fold improvement in image display speed.

The experimental apparatus consists of a commercially available AFM (Asylum Research Cypher-ES), an off-the-shelf digitizer (Gage CSE1622), trigger circuitry (built in-house, see supporting information for details),³⁵ and a pulsed excitation source: either light or voltage (here a function generator, Agilent 33500B). Figure 1(a) depicts the experimental scheme. The goal of the trEFM method is to measure how quickly the electrostatic interaction between the cantilever and the sample is changing in response to the excitation. We use a scan-lift-rescan approach^36–38 to track the topography while minimizing short-range interactions. The instrument first acquires a topography scan in AC mode. The cantilever is then raised to a user-defined height from the surface (typically 10-100 nm) and then performs the trEFM pass during a second scan with the cantilever retracing the recorded topography line at a fixed height chosen to minimize short range interactions, such as van der Waals forces, while remaining sensitive to longer range electrostatic forces.

During the trEFM pass, we first apply a voltage bias between the cantilever and the sample (typical V_bias ∼ 5-10 V). After the cantilever reaches equilibrium (2-4 ms), we apply a user-defined perturbation, such as a light pulse or voltage between the cantilever and the sample, at a user chosen point in the cantilever oscillation cycle. This perturbation induces transient deviations from the sinusoidal motion of the cantilever, which we record by digitizing the cantilever displacement signal with a fast acquisition card. We record the cantilever motion at a sampling rate of 10 MHz with 16-bit precision over a user-defined window (typically 0.8-3.2 ms) before and after the sample perturbation. The entire process is typically 16 ms per data point. In a line scan, we bin collected signals to pixels and average them before processing; in a point scan, we collect signals at a single spot and average them before processing. A line scan is typically 1920 signals over 64 pixels (30 averages per pixel) or 128 pixels (15 averages per pixel), though higher averages-per-pixel can be acquired at the expense of time-per-scan. We then process the data to extract the frequency information as described below.

We introduced the data analysis process in previous work.^19,39 In brief, we use the Hilbert transform to convert the cantilever motion into time-dependent frequency information.^40,41 Importantly, we first average the deflection vs. time traces for a number of trigger signals acquired at the same spot and at the same phase to improve our signal-to-noise ratio. We then multiply the average signal by a windowing function to reduce edge effects and spectral leakage, which is the temporal effect of a finite sampling window becoming apparent in the Fourier transform. We chose a Blackman window because it is designed to minimize spectral leakage and optimize its peak’s concentration around the center frequency.^42,43 We filter the signal with a finite-impulse-response (FIR) filter, since FIR filters can be built to have linear phase response and are stable.⁴³ The FIR filter we use is a causal filter to preserve the time-ordering of the data series. However, as all filters, the FIR filter delays the signal by (N − 1)/(2f_s), where N is the number of taps and f_s is the sampling frequency. As a result, we correct the delay in the location of trigger. We then apply a Hilbert transform on the windowed and filtered signal to obtain the analytical signal from which we extract the instantaneous phase of the cantilever.^19,40 We take the derivative of the instantaneous phase with respect to time to yield the instantaneous frequency of the AFM cantilever. From the instantaneous frequency curve, we find when the minimum frequency occurred and the duration between the event and the minimum, which we use as a metric “time to first peak” (t_FP) as shown in Figure 1(b) for two different excitations of varying time constants. The procedure for finding the minimum of the curve is described in the Supplementary material.³⁵ 

trEFM measures a time-dependent evolution in the surface potential and/or tip-sample capacitive force gradient. trEFM measurements are therefore limited to dynamic processes that have an effect on the resonance frequency of the cantilever and cannot be employed on measurements of arbitrary transient events, such as uncharged motion in the substrate. Here, we focus on characterizing the system’s response to single-exponential signals, because many systems of interest exhibit either first-order, or pseudo-first-order dynamics,^18,44,45 and indeed we have observed such behavior in trEFM data on real systems.^18,20,21 To create the perturbation, we apply an exponentially rising voltage to a gold electrode of the form,

where V_pp is peak-to-peak voltage, t is time, and τ is the characteristic time, ranging from 10 ns to 1 ms. For AFM cantilevers, we use commercially available Bruker DDESP-V2, BudgetSensors ElectriTap300-G, ElectriTap190-G, and ElectriMulti75-G with manufacturer-specified resonance frequencies of 450, 300, 190, and 75 kHz and spring constants of 80, 40, 48, and 3 N/m. All BudgetSensors tips are expected to have less than a 25 nm radius and Bruker DDESP-V2 tips to have less than a 150 nm radius. In addition to using artificial signals to characterize the system, in Sec. V we also demonstrate our fast time-resolved EFM method on real-world samples that we investigate.

To thoroughly understand the underlying physics of the fast-time dynamics of the cantilever, we compare our experimental data with numerical simulations of the cantilever motion. We assume that the cantilever behaves as a damped driven harmonic oscillator (DDHO),

where z is displacement of the tip, ω₀ is the resonance frequency of the cantilever, Q is the quality factor, F₀ is the driving force, and m is the effective cantilever mass.

The voltage pulse applied to the cantilever will affect its motion in two primary ways.⁴⁶ First, the time-varying voltage will induce a time varying electrostatic force (F_e(t)) acting on the cantilever. Second, the change in dF_e(t)/dz, due to the height-dependence of the tip-sample capacitance, affects the cantilever motion.⁴⁷ In our experiments with time-dependent tip-sample potentials, we model this term as a time-dependent resonant frequency, ω₀(t), that changes to a value of ω₀(t) + Δ ω₀ with the same time constant, τ, as the electrostatic perturbation,

where Δ ω₀ is the maximum frequency shift due to the electrostatic force gradient and k is the spring constant of the cantilever.

The electrostatic force F_e and the electrostatic force gradient dF_e/dz are both proportional to square of the total potential difference between the tip and sample,

We can write the voltage pulse as a time-dependent force and time-dependent force gradient modeling the system as a capacitor (C) of a conducting sphere floating above a conducting plane,⁴⁷ 

where we obtain the parameters R (radius of the tip) and $z̄$ (the average distance between the tip and the sample) from experiment. Although Eqs. (6) and (7) are an approximation to the tip-sample capacitance, we emphasize that here we are interested primarily in the functional relationship, such as F_e ∼ Δ V² and dF_e/dz ∼ Δ V² rather than the exact prefactors. Combining Eqs. (2), (3) and (6), we can rewrite Eq. (2) as follows:¹⁹ 

In all simulations below, we have solved Eq. (8) using an ordinary differential equation solver with input parameters such as cantilever parameters (ω₀, k, m, Q) and force parameters (F_e, dF_e/dz) determined from experiment. We note that it is also possible for Q to exhibit a time dependence,^48,49 an effect we ignore here as it is not necessary to model our voltage pulse data.^19,46

In our experiments, we used a gold electrode deposited on a silicon wafer in order to characterize the response of our system to known transient signals. We utilized a waveform generator as our excitation source to create exponential voltage transients (Eq. (1)) with programmed rise times in order to simulate transient charge buildups or decays that might occur in systems of interest such as an optoelectronic device under optical excitation.

Using this model system to compare simulations and experimental data, we now explore the parameter space that defines the operation of trEFM in detail. Figure 2 shows typical experimental displacement (z), phase (θ), and frequency shift (Δ f) vs. time following voltage pulses with 10 ns and 10 μs rise times applied to the cantilever with a resonance frequency of 503.327 kHz (and thus cantilever period of 1.98 678 μs). We obtain phase and frequency shift via Hilbert transform of the deflection data as explained in Section II. Figure 2(c) shows that there is no change in the frequency of the cantilever until excitation time (t = 0), and then the instantaneous frequency starts decreasing due to the transient motion of the cantilever being induced by the time-dependent electrostatic force and force gradient (Figure 2(c) also shows the effect of the causal filtering causing a lag in the frequency shift). After reaching a minimum in the frequency shift (t = t_FP), the transient change in the cantilever frequency diminishes as the cantilever returns to a sinusoidal oscillation at the drive frequency with a new amplitude and relaxation time constant given by τ_R = 2Q/ω₀. The exact value of t_FP depends on time constants of the excitation and the relaxation, as well as the processing parameters, such as the filter bandwidth and the number of FIR filter coefficients (throughout this paper, we keep the filter parameters the same for all experiments and simulations). Since the relaxation time constant is dominated by Q of the cantilever (rather than the electrostatic change in the sample), the transient behavior prior to the relaxation is key to recovering information about fast perturbations, which forms the center of our attention here. We focus on the time for the cantilever to reach the maximum instantaneous frequency deviation from the drive frequency, t_FP, as a simplifying metric to parameterize the data.¹⁹ We are able to extract information about the fast perturbations that are much shorter than a single oscillation period because cantilever oscillations start developing a phase lag after the excitation that propagates over time scales longer than the excitation’s characteristic time constant. For example, in Figure 2(c), t_FP values of 136.41 μs and 153.7 μs are longer than the excitation with exponential time constants of 10 ns and 10 μs, respectively.

Figure 2(d) shows a plot of measured t_FP as a function of the input signal rise time, τ, which we call a calibration curve. The calibration curve serves as an experimental map to understand how t_FP is related to the excitation time and characteristics of the instrument, because there is one-to-one correspondence between t_FP and τ, down to a minimum distinguishable rise time. We consider “distinguishable rise time” to be where the change in t_FP’s for two consecutive τ’s is greater than the sum of standard deviations of t_FP’s for those τ values,

In the inset of Figure 2(d), we show that the minimum distinguishable rise time is 10 ns with our current implementation, representing a 10-fold improvement over our previous efforts.¹⁹ We next examine the factors required to obtain this monotonic calibration curve and reach a time-resolution of 10 ns, such as the dependence on the cantilever phase³⁹ during excitation, as well as other factors.

As discussed in Section III and our earlier work,¹⁹ a time-varying tip-sample voltage will affect the instantaneous cantilever motion through two terms: the electrostatic force and the force gradient. In previous studies, the figure of merit for a frequency-modulation (FM) system has been the minimum detectable force gradient.^46,50,51 The minimum detectable force gradient for an FM system in the absence of the electrostatic force term is^46,50,51

where $zosc2$ is mean square displacement, ω₀ is the resonance frequency of the cantilever, Q is the quality factor, k_bT is the thermal energy at ambient temperature, k is the spring constant, and B is the measurement bandwidth.

We model our experiments in two ways. First, the experiment is modeled using only the force gradient, then it is modeled using both the electrostatic force and the force gradient, unlike in previous studies. Figure 3(a) demonstrates that simulations including both the force and its gradient capture the experimental trend for faster rise times, whereas the calibration curve of simulations including only the force gradient but not the force is flat for time scales faster than 100 ns. For rise times slower than 1 μs, simulations with and without the force term show the same trend and follow the experimental curve (for simulations and experiment, excitation is applied at the cantilever phase of 180°). For the discrepancy between simulation and experimental results, we speculate that there are two main reasons: (i) the invalidity of electrostatic force assumption as the transient signals get faster since the fast motion of electrons requires an electrodynamic model and (ii) simulations’ inability to capture 3-dimensional interaction of the cantilever with the sample. Both of these behaviors are not captured by 1D DDHO simulations, where we assume slower or at a similar time scale as the cantilever’s oscillation. Based on these models, we propose that the electrostatic force is an important component of trEFM method in the sub-cycle regime. This assumption is in contrast to previous literature that only included the force gradient instead of both the electrostatic force and the force gradient.⁴⁶ As discussed below, we conclude that, for rise times shorter than a cantilever period, the trEFM method is more sensitive to the force than the force gradient, and the electrostatic force is a significant component of the detection.

As discussed in Sec. IV A, the electrostatic force and force gradient components affect the minimum distinguishable rise time and the curvature of the calibration curve. We also expect the direction of the electrostatic force relative to the cantilever motion to have an effect on our measurements. In Figure 4, we illustrate the cantilever motion at different points in time and show velocity and acceleration vectors at four key phases. We hypothesized that the relative alignment of the electrostatic force vector with the damping force (⁠$Fdamping=−2βżZˆ$⁠, where β = ω₀/2Q) should affect the results. Following this qualitative analysis, we simulated the cantilever motion when it is perturbed by a time-varying electrical potential beginning during different relative phases of the oscillation cycle (Figure 5(a)).

In accordance with the qualitative discussion above, our numerical simulations show that for rise times longer than 1 μs, calibration curves behave monotonically at all phases (inset of Figure 5(a)). However, for shorter rise times, we can see that only calibration curves where the excitation was turned on at 180° and 270° behave monotonically, whereas other phases behave non-monotonically. This is a key observation; since we desire a one-to-one mapping between t_FP and τ values, we need t_FP to shift monotonically with τ.

To investigate further, we built a computer-controlled trigger circuit that takes the deflection signal, trigger request, and desired phase as inputs and then triggers the digitizer and the excitation source at the given phase as described in Section II and the Supplementary material.³⁵ In our experiments, we varied the trigger phase from 0° to 360° and measured t_FP for different rise times. Figure 5(b) shows measured t_FP’s of a cantilever with f₀ = 272.218 kHz, k = 26.2 N/m, and Q = 432 at V_bias = 5 V, for voltage pulses with V_pp = 5 V at different values of τ at different phases. The data in Figure 5(b) support our argument that the phase of the cantilever oscillation at the excitation time has a significant effect on the results. Indeed, the chosen phase-delay can change the minimum detectable rise time by more than 10 folds. The ability to discriminate the rise time of a transient event is determined by both the slope of the calibration curve and the repeatability of a measurement (being able to distinguish two measurements in time) as described in Eq. (9). Both experiment and simulation suggest that a phase of 180° is the best phase for maximum slope in calibration curves, which corresponds to the ability to differentiate rise times with minimum error.

To understand the cause of the phase effect, we simulated our experiment with and without the electrostatic force term because we hypothesize that the t_FP encodes time-dependent information from both F_e and dF_e/dz.¹⁹ In Figure 6, we show the results of these simulations. We simulated the same cantilever described above, with a voltage pulse that has a rise time of 100 ns, over all phases for two different cases: using only the force gradient and using both the electrostatic force and the force gradient. One can easily see that there is significant variation in t_FP and Δ f when both terms are present (solid blue lines). However, when we remove the electrostatic force term and run the simulations again with only the force gradient term present, the effect of excitation phase on t_FP and Δ f vanishes (dashed red lines in Figure 6). This result implies that the electrostatic force is responsible for the phase effects.

The t_FP is at its minimum and Δ f is maximum at 180° with respect to cantilever deflection only when the electrostatic force is included in our simulations. We speculate that this result is because, at a cantilever phase of 180°, the electrostatic force is in the same direction as the cantilever velocity (towards the surface) and opposite to the damping force. At 90° and 270°, the cantilever is at the maximum of its motion range and has no velocity. At 0°, the cantilever is moving away from the surface and at the center of its motion. Since t_FP is at its minimum and the frequency shift is maximum at 180°, it is also easier to detect the minimum of the instantaneous frequency because the convexity of the instantaneous frequency curve increases. Experimental and simulated instantaneous frequency curves can be found in Figures S3 and S4.³⁵ 

In an FM detection system, there are several noise sources such as detector noise, shot noise, 1/f noise, and thermal noise. In our system, acoustic noise and mechanical noise due to coupling of cantilever to the piezoacoustic drive element are also seen in the frequency spectra (see Figure S5).³⁵ 

While piezoacoustic excitation is the most commonly available drive mechanism, it relies on the physical connection between the drive piezo and the cantilever. The mechanical coupling of the cantilever to a larger mechanical system, such as the piezoacoustic element itself, can give rise to a complex resonance spectrum. In an FM detection scheme, this complex resonance spectrum can be problematic, since the resulting drive response is not flat with frequency,³⁴ and the region of interest is generally a small bandwidth around the resonance frequency. However, the cantilever now has a response function that is dependent on the insufficiently flat drive spectrum, in addition to sample properties. This effect results in frequency tracking instabilities.⁵² In order to access a cleaner cantilever response, one can use an actuation method that does not mechanically couple to the cantilever, such as photothermal excitation.

Photothermal excitation is a new method for driving cantilevers, most often used in liquid environments for AC atomic force microscopy.^30–34 In photothermal excitation mode, a power-modulated laser is focused on the cantilever base to drive the cantilever oscillation, achieving a cleaner response function that is largely uncontaminated by spurious resonances. Furthermore, photothermal excitation can be used with frequency modulated AFM, where it has proven to be more accurate and stable.^33,34,53 The recovery of the cantilever’s frequency shift and damping is easier due to the stability, accuracy, and linearity of the transfer functions of the excitation and detection systems.³³ We thus anticipated that using photothermal excitation instead of piezoacoustic excitation should increase the stability of cantilever oscillation by providing a drive response that is flat with frequency.

In order to test the noise-reducing effect of photothermal excitation, we employed a commercially available system (Asylum Research’s BlueDrive) in our experiments.³⁴ Figure 7 shows experimental Δ f vs. τ and t_FP vs. τ (calibration curve) data taken with BlueDrive and conventional piezoacoustic drive for comparison. In Figure 7(a), we see that frequency shifts for both drive types follow the same trend and lie in close proximity. The calibration curves are also quite similar in shape for both drive modes, as shown in Figures 7(b) and 7(c) (Figure 7(c) shows a zoomed in view at faster rise times). As we have predicted, the error bars, which are caused by the instability of the drive response, for the BlueDrive data are ∼5× smaller at the fastest rise times. Having smaller error bars means that we get closer to having one-to-one mapping between t_FP and τ. Practically, this result means that for a given number of averages, when the cantilever is driven by piezoacoustic excitation, rise times faster than 100 ns are not distinguishable from each other due to the frequency noise. With BlueDrive, the lower frequency noise floor permits us to distinguish rise times down to 25 ns with a cantilever of f₀ = 255.920 kHz, k = 17.9 N/m, and Q = 428, and we can distinguish rise times down to 10 ns with a cantilever of f₀ = 503.327 kHz, k = 72.7 N/m, and Q = 499. We discuss the noise around the cantilever resonance frequency in more detail in Figure S5.³⁵ 

In any system operating at ambient temperatures, thermal noise (Brownian motion of the cantilever) is always the fundamental limiting factor of the achievable noise floor, while other noise resources can be mitigated by changes to hardware. Due to its Brownian nature, thermal noise can be modeled as additive white Gaussian noise (AWGN) and one of the easiest and well-known methods to remove it (at the expense of acquisition time) is to signal average. In the present case, we average the raw cantilever deflection signals for multiple excitation pulses before analysis. In Figure 8, we show different numbers of averages (offset upwards by 5 μs for clarity), where deflection signals were acquired at the same point and at cantilever phase of 180° driven by BlueDrive. As expected, increasing the number of averages reduces the uncertainty in t_FP without altering the trend of the calibration curves. When driven by BlueDrive and with 2000 averages per point, we are able to distinguish rise times down to 10 ns, which can be seen in the inset of Figure 2(d). In this case, 10 ns corresponds to 0.5% of a cycle.

In Figure 9, we present standard deviations for experimental t_FP’s from 30 different runs of the same cantilever for BlueDrive and piezoacoustic excitation at 100 ns rise time, as well as simulated thermal noise, where dashed lines denote $1/n$ law for averaging. The standard deviations for experiments acquired with piezoacoustic excitation and BlueDrive start decreasing at first with averaging; however, the gains begin to level off rapidly. We take this deviation from the expected $1/n$ trend as indicating other noise sources related to the instrument, since the variation due to thermal noise is expected to follow $1/n$⁠.⁴⁶ The BlueDrive data benefit more from signal averaging, only to a certain extent. We conclude that with increasing number of averages, both BlueDrive and piezoacoustic excitation modes become limited by non-thermal noise sources such as detector noise. On the other hand, our simulated results, where we added AWGN calculated by the fluctuation-dissipation theorem to the simulated cantilever deflection, clearly follow the expected $1/n$ behavior and show a lower overall noise floor. For any given number of averages, photothermal excitation has approximately 5 times less variation in t_FP compared to piezoacoustic excitation, and simulated thermal noise has approximately 10 times less variation in t_FP compared to photothermal excitation. This result indicates that, while the work presented here represents a greater than five-fold improvement over previous results, there is still significant room to improve the performance of the trEFM method in the future.

We investigated how different approaches work for removing different types of noise, namely, frequency and thermal noise, and what we can achieve in terms of time resolution using these methods. Additionally, selection of other experimental parameters, such as the cantilever, affects how the system performs. Unfortunately, it is impossible to scan such an immense parameter space experimentally. Therefore, we combine experimental data with modeled data to understand better how the selection of cantilever parameters affects our results. Here, we consider the effects of resonance frequency (f₀), spring constant (k), and the quality factor (Q). The resonance frequency and the spring constant of a cantilever depend on the cantilever geometry and materials that are used. The quality factor of the cantilever is determined in part by the environment; for example, moving from a liquid environment to ultra-high vacuum would increase the quality factor by 4-5 orders of magnitude.^33,46,54 Here, we look into effects of these parameters on time resolution in fast trEFM.

In order to test the frequency dependence of calibration curves, we performed experiments and simulations with four different types of commercial cantilevers, with resonance frequencies 60.219 kHz, 154.806 kHz, 255.920 kHz, and 503.327 kHz. In Figure 10, we can clearly see that the slowest cantilever (60 kHz) has the shallowest slope, where the calibration curve is almost completely flat for time scales shorter than 1 μs. Cantilevers with resonance frequencies 154 kHz and 255 kHz have similar behavior; however, the calibration curve for 154 kHz flattens after 100 ns. The fastest cantilever (503 kHz) has the minimum distinguishable rise time of 10 ns, whereas the cantilever with resonance frequency of 255 kHz can resolve down to 25 ns. Although the calibration curve of the 255 kHz cantilever looks similar to the fastest cantilevers, the standard deviations for the 255 kHz cantilever are larger than the 503 kHz cantilevers (see Figure S6).³⁵ Thereby, the 255 kHz cantilever has a higher minimum distinguishable rise time. Cantilevers with resonance frequencies of 60 kHz and 155 kHz have minimum distinguishable rise times of 1 μs and 100 ns, respectively. These results and Eq. (10) imply that, all things being equal, higher frequency cantilevers are more sensitive to the force and the force gradient.

We also consider the effects of Q, as we expect Q to have an effect on calibration curves: looking at Eq. (10) (which considers only the effects of a force gradient), one would expect the minimum detectable force gradient to decrease (increasing sensitivity) with increasing Q.⁴⁶ We simulate this effect in Figure 11, where one can clearly see the effect of Q on the calibration curve with relaxation times (τ_R) ranging from 20 μs to 0.2 s. Having a high Q cantilever increases the slope of the calibration curve for rise times longer than 10 μs, where it becomes comparable to τ_R; however, it does not affect the calibration curve for shorter rise times as much, where it is much less than τ_R. This result is consistent with Eq. (10) at long times but not at short times. We propose that this discrepancy at short times between the prediction of Eq. (10), and our experiments and simulations, which we have confirmed by both simulation and experiment, arises from the fact that the electrostatic force and its gradient both contribute to the t_FP signal at shorter rise times. In summary, having higher Q has a favorable effect on the detection of slower events (τ > cantilever period), while the effect of Q is more complicated and less important at faster times.

We next examine the effect of spring constant. Based on the force gradient alone, one would expect that increasing the spring constant should increase the minimum detectable force gradient (Eq. (10)).⁴⁶ On the other hand, we find this expectation only holds for a model where there are no external forces other than the driving force or where the effects of external forces are small compared to the frequency shift due to the force gradient. Thus, the effect of spring constant on the detection of rise times is not as clear as the effect of Q. When we include the electrostatic force term in our simulations, we do not observe a substantial change in the slope of the calibration curve for slower rise times while the spring constant is changing as seen in inset of Figure 12. However, contrary to the previous literature, Figure 12 shows that as the cantilever gets stiffer, the curvature of the calibration curve slightly increases for faster rise times (<1 μs) with the inclusion of the force term, and higher curvature makes distinguishing adjacent τ’s easier to some extent. Furthermore, stiffer cantilevers have lower levels of thermal noise.⁵⁵ As a result, we expect stiffer cantilevers to perform slightly better in detection of rise times that are shorter than a cantilever period and to have negligible effects on detection of longer rise times.

In summary, Figures 10-12 show that a combination of faster cantilever resonance frequency, larger spring constant, and higher Q will tend to improve the time resolution in fast trEFM. There is a large range of spring constants that is commercially available, and Q is contingent upon the environment. In addition, we tested the effects of other experimental parameters, such as lift height and voltage bias between the tip and the sample, and found that the frequency shift follows Eq. (4) as expected (Figures S7 and S8³⁵), while there is no substantial change in calibration curves. In other words, cantilever parameters are the primary experimental parameters, other than the number of averages and the cantilever excitation method, that affect the time-resolution in fast trEFM. Exciting the cantilever at a phase of 180° using BlueDrive with more than 250 averages per point where the cantilever has a high resonance frequency, larger spring constant, and high Q, the data in Figure 2(d) show that it is possible to obtain robust calibration curves.

We have implemented trEFM as an imaging technique that can acquire and analyze data concurrently. The trEFM software suite consists of two parts: measurement and analysis. The measurement part is coded in Igor Pro (Wavemetrics, Inc.) using Asylum Research’s software and uses the GaGe Software Development Kit to interface with the digitizer hardware in C+ +; the analysis is coded in Python using NumPy/SciPy.^56,57 Using application-programming interfaces, we are able to transfer the digitized deflection signal to the analysis program and send the results back to be interpreted by the user. During imaging, the digitized deflection signals for each line of pixels are processed upon acquisition. Signals for each line are passed along to the analysis software with the number of pixels contained in a line, where the instantaneous frequency is extracted and returned to the user. The duration of this operation is dependent upon the number of averages and the number of pixels specified for binning, as well as the duration of each signal acquisition. To expedite acquisition at the expense of concurrent imaging, it is possible to acquire image data and post-process that into an image on a separate machine. We have made the software suite freely available online.⁵⁸ 

Currently, we are able to take a 64 × 32 image in approximately an hour with 60 averages per pixel. Figure 13 shows a fast trEFM image of a model organic photovoltaic blend (poly[2-methoxy-5-(3′,7′-dimethyloctyloxy)-1,4-phenylenevinylene]:^6,6-phenyl-C61-butyric acid methyl ester) (MDMO-PPV:PCBM). Here, we used a modulated laser (488 nm) to photoexcite the material, similarly to what we have done in previous reports where we used modulated light-emitting diodes (LEDs).^18–22 Instead of a shaped voltage pulse as in the gold film experiments earlier, here the film exhibits a physical response due to absorption of the incident laser resulting in a build-up of photogenerated charge beneath the tip. In this case, the trEFM response reflects the time constant for the photogenerated charges to fill the tip-sample capacitor. This blend’s film morphology can be easily adjusted by changing the solvent,⁵⁹ and it has large domains that can be easily distinguished and thus makes it a useful test bed for trEFM. The structure of MDMO-PPV:PCBM is well known and the data here qualitatively agree with previous trEFM work.²⁰ Areas of low (high) 1/t_FP values correspond to low (high) external quantum efficiency regions in the film.¹⁹ Importantly, fast trEFM as implemented here allows for illumination intensities many orders of magnitude higher than previous trEFM work on these materials²⁰ and are closer to realistic device conditions; we show additional light intensity-dependent data in Figure S9.³⁵ This image clearly shows that variation in t_FP on the nanoscale can be directly measured using the trEFM technique.

trEFM offers high time resolution with high spatial resolution as a supplement to bulk measurements of electronic and photonic properties of materials. In our experiments and simulations, we investigated the effect of the electrostatic force in detection of fast time transients. We found that the electrostatic force and the phase of cantilever motion during which the force is applied both have an effect on the ability to resolve rise times shorter than a period of the cantilever. Importantly, by controlling the cantilever phase at the perturbation time in order to align the electrostatic force with the cantilever while it is at the midpoint of its motion moving downward, we can increase the time resolution of fast trEFM as well as making measurements more reliable.

Using photothermal excitation instead of piezoacoustic excitation, which reduces the measurement noise by increasing the stability of the drive, and increasing the number of averaged signals to decrease the thermal noise, we are able to differentiate rise times down to 10 ns. We anticipate that these results will lead to continued improvements in the future by using stiffer and faster cantilevers with high quality factors. We expect that trEFM will find applications beyond the development and characterization of organic photovoltaics with its improved time resolution, such as perovskite solar cell development, battery electrode characterization, and biological measurements.

The paper is based upon work supported by the National Science Foundation (NSF MRI Grant No. DMR-1337173). The authors wish to thank the staff of University of Washington—Chemistry Electronics Shop, in particular Jim Gladden and Bill Beaty for the design and construction of the trigger circuitry, and Phil Cox for the help with the fabrication of MDMO-PPV:PCBM thin films.