Atomic force microscopy (AFM) is an enabling technique for the nanoscale mapping of topography and surface properties of interfaces in a wide range of environments [1]. Kelvin probe force microscopy (KPFM) utilizes the application of a bias and a conductive probe to map the local electrical properties of an interface at the nanoscale [2], allowing for the determination of the local contact potential difference (CPD) between the probe and the sample. This, in turn, allows the work function of the sample to be measured if the work function of the probe is known and vice versa. The mapping of local electrical properties of the interface is essential to further our understanding of corrosion, sensing, solar cells, energy storage devices, and bioelectric interfaces [3-8]. Since its first application in 1991 [2], there have been significant developments in the field of KPFM [6,9,10] with significant advances in both temporal [11-14] and spatial resolution [13,15-19]. These advances have enabled investigations mapping light-induced surface potential dynamics [20], ferroelectric domains [19], individual quantum dots [21,22], and even submolecular charge distributions [23-27]. These applications demonstrate that KPFM is capable of atomic-scale spatial resolution and nanosecond time resolution under specific conditions.

KPFM-based techniques can largely be classified as either “open loop” (OL) or “closed loop” (CL). CL techniques employ a feedback loop to apply a bias to compensate for the electrostatic force (or force gradient) between the tip and sample. CL techniques are more common than OL techniques due to the ease of implementation, wide-scale availability, and direct measurement of the apparent CPD. OL techniques, by contrast, are feedback-free and can be used to determine the CPD without the need to apply a DC bias [28,29]. OL techniques are increasingly being adopted to enable the mapping of voltage-sensitive materials [30-32], to enable investigations of fast electrodynamic processes [11-14] and to enable measurements in liquid environments (where bias application could lead to stray currents and unwanted electrochemical reactions) [9,33-35]. OL techniques avoid the limitations and artefacts that can arise when using a feedback loop, for example, bandwidth limitations due to the time constant of the feedback loop [29], increased noise [36,37], and electrical crosstalk [38,39]. Whilst the application of DC bias is not required for OL operation it can still be utilized to allow CPD to be determined via bias sweeps [28,40] or to investigate gate-dependent potential profiles of interfaces [22,41,42]. There are a wide range of OL KPFM techniques beyond those examined in this paper, including pump-probe KPFM [13,20,43], time-resolved KPFM [11,12,44-47], fast free force recovery KPFM (G-Mode) [14,48-50], intermodulation electrostatic force microscopy (EFM) [42,51], and PeakForce tapping KPFM [52].

The fundamental detection sensitivity to electrostatic forces in KPFM is generally expressed as the minimum detectable CPD [53], and is directly limited by the geometry of the interaction, thermal noise of the cantilever, and the detection noise limits of the AFM [36,54]. Cantilevers have a number of eigenmodes, ωn, where there is a mechanical enhancement in the response of the lever proportional to the quality factor of that mode, Qn, where n is the mode number [36,55]. KPFM techniques can be applied off resonance (ω ≠ ωn), where ∝ 1/kn, where kn is the spring constant of the n-th eigenmode. More generally, KPFM techniques are applied at, or close to ωn where ∝ Qn/kn and there is a significant enhancement in the oscillation amplitude of the cantilever in response to the electrostatic force, thereby increasing the signal-to-noise ratio (SNR) [10]. In this paper we define the SNR as the ratio of the measured signal (oscillation amplitude of the cantilever) at a given frequency to the noise at that frequency. Furthermore, we use the conventional definition of whereby SNR = 1 [2,53,56-59].

The desire to take advantage of the SNR enhancement on eigenmodes of the cantilever have led to the adoption of a number of imaging strategies [10]. The regulation of tip–sample distance in KPFM imaging is generally performed by employing a feedback loop that maintains the mechanical oscillation amplitude of the cantilever at the fundamental eigenmode, ω1, at a fixed value. This precludes simultaneous measurement of electrostatic forces on this eigenmode (the SNR is highest on ω1). As such, strategies for achieving high SNRs generally focus on three areas: (1) Lift mode – here the surface under investigation is mapped in two passes, the first pass with only a mechanical excitation applied at ω1 and the second pass with only the electrical excitation applied at ω1 as it traces the topography measured in the first pass at a specific lift height above the surface. Lift height can be set such that the electrostatic forces are isolated from stronger short range forces at the expense of spatial resolution [10,58]. By setting the lift height to match the mean tip–sample distance of the lever during the mechanical imaging pass, topography and potential can be correlated. (2) Sideband modes – here the electrical signal, ωe, is applied as a low frequency (ωe ≪ ω1) such that the electrical and mechanical drive, ωm, form mixing products ωm ± ωe. These mixing products have enhanced sensitivity to electrostatic forces at the expense of localization to small tip–sample separations. By choosing ωe such that the mixing products fall on the sidebands of ω1, the SNR is improved whilst enabling single-pass scanning. There are trade-offs here in that ωe should be higher than the topography feedback bandwidth to prevent crosstalk yet low enough that the mixing products are not too far from ωm to take advantage of gains in the SNR. This limits the accessible bandwidth and, therefore, the scanning speed [10]. (3) Higher eigenmodes – by applying the electrical signal to a higher eigenmode, the electrostatic response can be measured simultaneously with topography in a single pass [17,60-62]. Higher eigenmodes typically have poorer SNRs than the fundamental eigenmode since kn increases more rapidly than Qn [60,61]. However, these modes still offer significant SNR enhancements over off-resonance techniques, higher spatial resolution due to reduced influence of the cantilever to the electrostatic forces [17,60,63], and higher bandwidth than side-band techniques [10,58]. All modes of KPFM can in principle be applied in any of these scenarios.

In addition, there are more exotic approaches to mapping using KPFM, for example, force volume [64], PeakForce tapping [19,52], and, although not yet reported, KPFM could be combined with the recently introduced photothermal off-resonance tapping (PORT) mode [65]. KPFM can also be combined with other techniques to yield multidimensional data sets and aid in isolation of the influence of electrostatic potential, for example, PeakForce infrared-KPFM (PFIR-KPFM) [66], nanomechanical mapping + KPFM [67,68], magnetic force microscopy (MFM) + KPFM [69], piezoresponse force microscopy (PFM) + KPFM [70], and G-mode [14,48-50].

The most common application of KPFM in AFM is CL AM-KPFM on the fundamental eigenmode where a bias feedback loop is employed to cancel the electrostatic force and to extract VCPD[10,60,61]. This single-frequency technique can also be used under OL conditions without a feedback loop using phase-based detection [71], frequency sweeps [40,64,72], or bias modulation [10,52,73]. The advantages of CL AM-KPFM are that it is easy to implement, is standard on most commercial AFMs, and has high bias sensitivity [74]. The disadvantages of this technique are that it is limited by the properties of the feedback loop (and its associated artefacts) and is rarely fully quantitative due to the large influence of the cantilever beam on the electrostatic response [75-77]. This limits the spatial resolution. However, some authors address this by deconvolving the probe geometry from measurements in order to access a true surface potential map [78,79].

A natural extension of AM-KPFM is dual-harmonic KPFM (DH-KPFM), which is an OL technique that utilizes the measurement of both the first and second harmonic of the electrostatic response (ωe and 2ωe). By combining these two components, VCPD can be obtained directly without the need to employ a feedback loop, knowledge of the tip–sample capacitance gradient, or application of a DC bias. Initially implemented in ultrahigh vacuum by Takeuchi et al. [30], the method was extended to liquids by Kobayashi et al. [80] and to ambient environments by Collins et al. [81]. This technique exhibits a similar cantilever capacitive contribution to VCPD as AM-KPFM and is only quantitative if the relative gain of the two measured frequencies is known either through an additional measurement or through modelling [29]. Since the electrostatic response occurs at ωe and 2ωe, it is not possible to place both on eigenmodes in single-pass scanning, which adversely affects the SNR. To overcome this limitation, two passes could be made using excitation at ωe and ωe/2 such that the required harmonics are measured at the same frequency. This approach is known as half-harmonic KPFM [82]. Alternatively, two electrical drives, ωe1 and ωe2, can be applied such that the required harmonics occur on eigenmodes. This allows for a direct OL access to VCPD in a single pass with enhanced SNR. In addition, the mixing product, ωmix = ωe1 ± ωe2 occurs and can be placed on an eigenmode in order to measure VCPD with enhanced SNR since In this paper we refer to the use of two electrical drives and their products as electrodyne-KPFM (ED-KPFM).

In order to access higher spatial resolution, bandwidth, and SNR, heterodyne-KPFM (Het-KPFM) was developed [57] whereby the mechanical oscillation, ωm, at one eigenmode, used to track the topography, is mixed with ωe such that ωm ± ωe occurs on another eigenmode [57]. Typically, the probe would be driven mechanically at ω1 and ωe = ω2 − ω1 such that the first harmonic of the electrostatic response occurs at ω2. The positioning of the mechanical and electrical drives can also be applied such that the topography is measured on ω2 and the electrostatic response is on ω1[58]. This allows for single-pass scanning with enhanced SNR with greater bandwidth than other KPFM techniques [10,58]. This technique combines the enhanced sensitivity from operating on eigenmodes with the enhanced spatial resolution due to the electrostatic response being proportional to the second derivative of the capacitance gradient, C″ [57,76,83,84]. This enhanced sensitivity to short range forces (up to three times more sensitive than frequency modulation KPFM [53]) removes the influence of the cantilever and delivers enhanced bandwidth due to the high frequency of ωe[58]. Axt et al. found that Het-KPFM was the most accurate of all modes tested and was able to measure 99% of an applied potential difference even in the presence of strong stray fields [10]. Het-KPFM is generally operated in CL configurations [57,58] but could also be operated OL [58,85,86], either through bias sweeping techniques or through the simultaneous measurement of ωm ± ωe and ωm ± 2ωe similar to DH-KPFM. Het-KPFM has been demonstrated to achieve atomic resolution of the surface potential [53,76,86] and has enhanced our understanding of perovskite solar cells [10,87,88] and patch potentials in the Casimir force [89,90]. Implementations of Het-KPFM to date have primarily focused on the measurement of the first harmonic of the electrostatic force [57,58].

Furthermore, Het-KPFM can be extended using two electrical drive signals combined with a mechanical drive signal to aid in the positioning of the required harmonics on eigenmodes, enhancing both SNR and spatial resolution [91]. Examples include intermodulation AFM, which applies two electrical signals, ωe1 and ωe2, off resonance that mix with ωm to generate sideband signals around ωm[42,51], and harmonic mixing KPFM (HM-KPFM), which allows for the application of a combination of mechanical and electrical drives such that the first and second harmonic of the electrostatic response occurring at ωm ± ωe1 and ωm ± 2ωe2 both fall on an eigenmode, resulting in enhanced SNR [91]. Additionally, a ωm ± ωmix term occurs [83], which could also be placed on an eigenmode for enhanced SNR. The division of these electrostatic response components thereby enables the measurement of VCPD in OL. Here, the signals may be coupled with the mechanical drive, which may be applied either at a higher eigenmode or off resonance, for example, in the PORT mode [65]. In this paper we refer to any application of two electrical signals with a mechanical drive as HM-KPFM.

For OL multifrequency KPFM techniques (DH, Het, ED, HM), there is a need to be able to determine the sign of VCPD measured since we are dividing amplitude responses that are always positive. These techniques rely upon the sign of the cosine of the phase of the first harmonic electrostatic response [80]. However, for small VCPD values, the phase is strongly affected by noise [42]. An alternative approach is to measure the phase difference between the electrostatic harmonics to enable the determination of the sign of the measured VCPD[42]. In addition, the relationship between the response at different frequencies is strongly influenced by the transfer function of the cantilever. This frequency-dependent gain, XGain, represents the sensitivity ratio of the cantilever at the two frequencies of interest. XGain is relatively stable for a given tip–sample distance in a given environment and, as such, can be approximated mathematically [29]. However, changes in environment, tip–sample distance, and the influence of piezo-based mechanical activation significantly complicates these relationships and, as such, many techniques require the explicit measurement of XGain in order to be quantitative [29]. Spectral KPFM techniques (e.g., band excitation KPFM (BE-KPFM) [40,64,72], half-harmonic band excitation (HHBE-KPFM) [28,82], and G-mode [14,48-50]) that can measure the amplitude response of the cantilever as a function of frequency and DC bias, can access XGain directly as part of the measurements. Lastly, KPFM-based techniques can also utilize changes in XGain due to changes in the conservative [92,93] and dissipative [93-95] forces in order to access VCPD.

In order to assess the performance of OL and CL techniques and to establish the best route to obtain VCPD in any environment with the smallest required bias, we directly compare AM, DH, Het, ED, and HM KPFM techniques in terms of the minimum detectable CPD, the minimum AC bias required for operation, and the SNR. We compare and contrast the performance for three specific scenarios. The first scenario is “off resonance”, where the first harmonic of the electrostatic responses occurs on eigenmode ω1, and where the first harmonic of the electrostatic responses occurs on eigenmode ω2. We also compare the performance in air vs liquid (e.g., water), where both the transfer function of the cantilever changes (reducing Q enhancement at the eigenmodes) and the relative permittivity increases such that the electrostatic response is greatly enhanced. Other more complex effects in liquid environments are excluded from our analysis, for example, effects of the double layer, electrodynamics, or changes in permittivity with salt concentration. For a review of the impact of these effects on KPFM operation in liquid please see Collins et al. [9]. Our goal in this paper is to identify the OL techniques that provide the greatest performance with the smallest required VAC for operation in liquid environments, where bias application could lead to stray currents and unwanted electrochemical reactions. Here, we restrict our analysis to the first two eigenmodes of a cantilever, where the SNR is highest, but these calculations could be extended to higher eigenmodes if desired [96].