Advanced Kelvin Probe Force Microscopy (KPFM)

Introduction

Kelvin probe force microscopy (KPFM) is one of the essential electrical modes in scanning probe microscopy. It is measuring a fundamental physical property of materials – a surface potential. Unlike many other modes, where the extraction of quantitative data is subject to various, often complex calibration routines, quantification of a KPFM measurement is rather straightforward – difference between the surface potentials of a scanning probe and the sample is measured in volts. This voltage is often called a contact potential difference (CPD), and in case of metals, it is equal to the difference in work functions between the tip and the sample materials.

The work function is of great importance in studies of semiconductor materials, photovoltaics of solar cells, chemistry of fuel cells, lithium battery research, properties of alloys, conducting polymers, material differentiation in composite compounds, 2D materials, and many other areas1. A wider adoption of traditional KPFM, was sluggish in the beginning, due to the lack of spatial resolution and topography artifacts. However, new advanced methods of KPFM have pushed these limits and have shown significant improvement in resolving smaller features, this, in turn has made KPFM one of the leading electrical modes. This application note is describing the physics behind the KPFM technique, the difference between various modes, and gives measurement examples of a few typical samples. The measurements in this note were made in single pass, and not in a lift mode.

Figure 1: Principles of Kelvin probe force microscopy. A cantilever with a resonant frequency f<sub>0</sub>, spring constant k and quality factor Q is mechanically excited at its resonant frequency by an oscillator. The AC signal on the photodetector on that frequency is detected by a lock-in amplifier and is used for the topography measurement and the control of the cantilever position above the sample. This is a so-called mechanical loop. Another oscillator is set to modulate the sample potential on a modulation frequency fm. The electrical AC excitation is added to a DC bias voltage, and this V<sub>AC</sub> + V<sub>DC</sub> potential is applied to the sample. The AC electrostatic interaction between the sample and the cantilever depends on the value of the V<sub>DC</sub>.
A PID controller is set to zero this interaction (the value of the Demodulator X) by adjusting the value of the V<sub>DC</sub>. When the electrostatic interaction between the tip and the sample is zero, both are at the same potential, and the value of V<sub>DC</sub> would be the contact potential difference or (CPD).
Figure 1: Principles of Kelvin probe force microscopy. A cantilever with a resonant frequency f0, spring constant k and quality factor Q is mechanically excited at its resonant frequency by an oscillator. The AC signal on the photodetector on that frequency is detected by a lock-in amplifier and is used for the topography measurement and the control of the cantilever position above the sample. This is a so-called mechanical loop. Another oscillator is set to modulate the sample potential on a modulation frequency fm. The electrical AC excitation is added to a DC bias voltage, and this VAC + VDC potential is applied to the sample. The AC electrostatic interaction between the sample and the cantilever depends on the value of the VDC.
A PID controller is set to zero this interaction (the value of the Demodulator X) by adjusting the value of the VDC. When the electrostatic interaction between the tip and the sample is zero, both are at the same potential, and the value of VDC would be the contact potential difference or (CPD).

Background

The origins of the method lie in the experiment, measuring electrostatic forces, between the plates of a parallel plate capacitor, made of different metals, conducted by Lord Kelvin2, hence the name of the technique. A more or less modern interpretation of the CPD measurement was realized by Walter Zisman3. First KPFM experiments were conducted shortly after the invention of the atomic force microscope (AFM) at Watson Research Center (IBM) by Weaver4 and Nonnenmacher5 working at separate labs. In a KPFM measurement the AFM probe is placed to a short distance above the sample surface, via the mechanical feedback loop (Fig. 1). The KPFM controller is applying an AC voltage between the tip and the sample, creating an electrostatic AC force, and is automatically minimizing this force by applying the bucking DC voltage. Both AC and DC voltages can be applied to the sample or to the tip, while the other one (tip or sample) is on the ground. The variation of the bucking voltage while scanning the tip above the sample is recorded as an XY map of CPD. The simplest KPFM controller has at least one lock-in amplifier for the AC excitation and detection and a PID controller to maintain the minimal response of the system. In practice advanced KPFM controllers would have several lock-ins and an additional phase locked loop (PLL).

AM and FM KPFM

To understand the underlying physics of the KPFM we have to consider the force F coming from the electrostatic interaction, when the AC voltage is applied between the tip and the sample (Eq. 1):

F= -(∆V2)/2 ∂C/∂z (1)

here δV is the voltage between the tip and the sample, C is the tip-sample capacitance, and z is the distance between them. The expression for the δV can be written as follows (Eq. 2):

∆V=VDC-VCPD+VAC sin⁡ωt (2)

Where the VCPD is the contact potential difference, and the VDC and the VAC are the bucking voltage and the AC voltage, applied by the KPFM controller. When we replace the δV in the Eq. 1 with the expression in the Eq. 2, we would have the more extended expression for the force (Eq. 3):

F=∂C/∂z [(VAC2)/2+(VDC-VCPD )2+(VDC-VCPD ) VAC sin⁡ωt+(VAC2)/4 cos⁡2ωt ] (3)

The first two terms do not change with time and in that sense are DC terms. The third term is where the principle of the amplitude modulated (AM) KPFM lies. The lock-in amplifier is measuring the amplitude and phase of the VACsinωt signal, and the PID controller is adjusting the VDC to nullify this signal. There are two slightly different ways to choose the working frequency for the electrostatic excitation: the first one is to apply the AC voltage at a frequencyfmwell below the mechanical resonance of the tip f0, and a second one – to apply this excitation at the frequency of the second mechanical eigenmode of the cantilever f1 (Fig. 2). In the latter case one is having a better signal-to-noise ratio (SNR) due to the extra amplification of the electrostatic signal by the mechanical resonator. In both cases of the AM-KPFM, the KPFM is measuring the electrostatic force, and the electrical excitation and detection are done on the same frequency.

The frequency modulated (FM) KPFM, relies on the ∂C⁄∂z (or C’) part of the Eq. 3. When performing a polynomial expansion of C’ and looking at the first two members of the expansion, it can be shown6 that:

C'=a0+C’’ (4)

where a0 is a constant. Which essentially means, that the C' depends on the force gradient, rather than on the force amplitude6. This is the main difference between AM and FM KPFM modes. There are many implementations of the FM KPFM, and the most notable is heterodyne FM KPFM. Heterodyne FM KPFM is using frequency mixing between the mechanical and the electrical excitations, and the excitation frequencyfmis chosen to be at the frequency difference between the second and the first mechanical eigenmodes of the cantilever: (f1 – f0). In this case the mixed product will be positioned directly at f1 allowing the use of the mechanical amplification, same as in the case of the AM KPFM on the 2nd eigenmode (Fig. 2). Due to the higher excitation and detection frequencies, the heterodyne FM KPFM has higher bandwidth, than the sideband AM KPFM.

Figure 2: Different KPFM implementations in a frequency domain. Black curves show mechanical motion of the cantilever, and black dotted lines show its mechanical excitation. Red dotted lines show electrical excitation and detection. (a) Off resonance AM KPFM. The electrical excitationf<sub>m</sub>is done below the mechanical resonance of the cantilever f<sub>0</sub>. Both excitation and detection are done on the same frequency. (b) AM KPFM at 2nd eigenmode. The electrical excitation fe is done above the mechanical resonance of the cantilever f<sub>0</sub> and at it’s 2nd mechanical eigenmode f<sub>1</sub>. Both excitation and detection are done on the same frequency. The advantage of this mode is the amplification of the electrostatic signal by a mechanical oscillator. (c)  Heterodyne FM KPFM. The excitation frequencyf<sub>m</sub>= (f<sub>1</sub> - f<sub>0</sub>) is chosen such, that the mixed product would fall directly on f<sub>1</sub>. Because of higher excitation frequency, the heterodyne FM KPFM has much higher bandwidth than the AM KPFM. The second harmonics of the excitation and the mixed product carry information about the ∂C⁄∂z and ∂<sup>2</sup>C/∂<sup>2</sup>z.<sup>7</sup>
Figure 2: Different KPFM implementations in a frequency domain. Black curves show mechanical motion of the cantilever, and black dotted lines show its mechanical excitation. Red dotted lines show electrical excitation and detection. (a) Off resonance AM KPFM. The electrical excitationfmis done below the mechanical resonance of the cantilever f0. Both excitation and detection are done on the same frequency. (b) AM KPFM at 2nd eigenmode. The electrical excitation fe is done above the mechanical resonance of the cantilever f0 and at it’s 2nd mechanical eigenmode f1. Both excitation and detection are done on the same frequency. The advantage of this mode is the amplification of the electrostatic signal by a mechanical oscillator. (c) Heterodyne FM KPFM. The excitation frequencyfm= (f1 - f0) is chosen such, that the mixed product would fall directly on f1. Because of higher excitation frequency, the heterodyne FM KPFM has much higher bandwidth than the AM KPFM. The second harmonics of the excitation and the mixed product carry information about the ∂C⁄∂z and ∂2C/∂2z.7

Interaction volume in AM and FM KPFM

In AM KPFM the signal is proportional to the electrostatic force, and in FM KPFM to the force gradient. The consequence of this is that the contributions to the KPFM signal in AM and FM modes are dominated by different parts of the AFM probe. To summarize it shortly, in AM mode at tip-sample distances bigger than 1 nm the signal is dominated by the cantilever part, only close to 1 Å the signal from the tip apex approaches to the one from the cantilever. This limits the spatial resolution of the method, and the AM KPFM is well known for its poor performance, when it comes to measuring small samples. In the FM mode already at about 10 nm above the surface, the most dominant contributions are the ones from the tip apex and the tip cone, and at 1 Å the signal is fully dominated by the one from the tip apex8 (Fig. 3). Because of a larger interaction volume, the AM KPFM usually has better signal-to-noise ratio, but suffers from the lateral resolution. The FM KPFM on the other hand, has lower SNR and better ability in resolving small features.

Figure 3: Comparison of contributions of different parts of the cantilever to the KPFM signal in AM and FM modes. In the AM KPFM mode, the signal is dominated by the interaction between the cantilever part of the AFM probe, and only at distances around 1 Å, the tip apex contribution becomes noticeable, but never exceeds the one from the cantilever. In the FM KPFM mode even at relatively large distances the main contribution comes from the tip cone and the tip apex, and at shorter distances, the signal if fully dominated by the tip apex contribution8. As a result, the AM KPFM has better signal-to-noise ratio, but the FM KPFM has better XY resolution and provides more credible measurement of V<sub>CPD</sub>. Image is taken from Ref 8.
Figure 3: Comparison of contributions of different parts of the cantilever to the KPFM signal in AM and FM modes. In the AM KPFM mode, the signal is dominated by the interaction between the cantilever part of the AFM probe, and only at distances around 1 Å, the tip apex contribution becomes noticeable, but never exceeds the one from the cantilever. In the FM KPFM mode even at relatively large distances the main contribution comes from the tip cone and the tip apex, and at shorter distances, the signal if fully dominated by the tip apex contribution8. As a result, the AM KPFM has better signal-to-noise ratio, but the FM KPFM has better XY resolution and provides more credible measurement of VCPD. Image is taken from Ref 8.

Cantilevers

Already in the first publications on the KPFM, the researchers have experienced the limit to sensing the CPD voltage, which can be described as5 (Eq. 5):

VCPD, min=1/(ϵ0 VAC ) d/R √((2kB TkB)/(π3 Qf0 )) (5)

where the R – tip radius, d – tip-sample distance T – temperature, k – cantilever spring constant, Q – quality factor, B – bandwidth, f0 – resonance frequency of the cantilever, kB – Boltzmann constant, VAC – amplitude of the AC voltage between the tip and the sample, and the ϵ0 is the dielectric constant in VACuum. As one can see from this formula, for the smallest detectable signal one needs a large tip radius, short distance to the sample, large AC voltage and small sprint constant. On practice choosing softer and shorter cantilevers is beneficial for the KPFM sensitivity, as well as using larger AC amplitudes, however the latter one comes at a cost of reduced lateral resolution.

Experimental comparison between AM and FM KPFM

In order to demonstrate the differences between different KPFM modes, we have measured some typical KPFM samples and compared the CPD signal and resolution in the obtained images. First sample of interest consists of the aluminum dots on gold. Fig. 4 is showing the topography and the KPFM measurements with three different KPFM modes: the off-resonance AM KPFM, the AM KPFM on the 2nd mechanical eigenmode and the heterodyne FM KPFM. From the data we can see that indeed the 2nd eigenmode improves the signal-to-noise ratio of the AM KPFM, due to the amplification of the signal by the mechanical resonance. Both AM modes, however, demonstrate noticeable topography cross-talk – i.e. grains of metal are visible in the surface potential. Heterodyne FM KPFM is much less influenced by this cross-talk. The second sample is the SRAM sample with the areas of p- and n-type doped silicon (Fig. 5). Those usually are showing good contrast in the KPFM. Again, the AM KPFM on the 2nd mechanical resonance is showing improved SNR, but the FM KPFM is demonstrating bigger signal span on the signal cross-section and the overall good image quality.

Figure 4: Comparison between the AM and FM KPFM modes on aluminium dots on gold sample. The image demonstrates the amplification of the KPFM signal between the AM KPFM on 2nd eigenmode and the off-resonance AM KPFM. One can also see the topography signal mixing with the KPFM signal for both AM modes. The heterodyne KPFM is showing cleaner signal without topography cross-talk.
(a) Image size: 20 x 20 μm<sup>2</sup>, height range: 250 nm,
(b) Image size: 7 x 7 μm<sup>2</sup>, CPD range: 0.4 V,
(c) Image size: 7 x 7 μm<sup>2</sup>, CPD range: 0.45 V,
(d) Image size: 7 x 7 μm<sup>2</sup>, CPD range: 0.47 V.
Figure 4: Comparison between the AM and FM KPFM modes on aluminium dots on gold sample. The image demonstrates the amplification of the KPFM signal between the AM KPFM on 2nd eigenmode and the off-resonance AM KPFM. One can also see the topography signal mixing with the KPFM signal for both AM modes. The heterodyne KPFM is showing cleaner signal without topography cross-talk.
(a) Image size: 20 x 20 μm2, height range: 250 nm,
(b) Image size: 7 x 7 μm2, CPD range: 0.4 V,
(c) Image size: 7 x 7 μm2, CPD range: 0.45 V,
(d) Image size: 7 x 7 μm2, CPD range: 0.47 V.
Figure 5:
Comparison between the AM and FM KPFM modes on SRAM sample. The cross sections of the CPD scan demonstrate steady increase of difference between the maxium and minimum in CPD starting from the off-resonance AM KPFM, followed by the AM KPFM on 2nd eigenmode and heterodyne FM KPFM.
Image size: 40 x 40 μm<sup>2</sup>.
(a) height range: 600 nm, (b) CPD range: 0.55 V,
(c) CPD range: 0.61 V, (d) CPD range: 0.72 V.
(c), (e), (g) demonstate cross sections of (b), (d), (f), made along red lines
Figure 5: Comparison between the AM and FM KPFM modes on SRAM sample. The cross sections of the CPD scan demonstrate steady increase of difference between the maxium and minimum in CPD starting from the off-resonance AM KPFM, followed by the AM KPFM on 2nd eigenmode and heterodyne FM KPFM.
Image size: 40 x 40 μm2.
(a) height range: 600 nm, (b) CPD range: 0.55 V,
(c) CPD range: 0.61 V, (d) CPD range: 0.72 V.
(c), (e), (g) demonstate cross sections of (b), (d), (f), made along red lines

The third sample we have measured is the semi fluorinated alkane (F(CF2)14(CH2)20H or f14H20) sample. The semi fluorinated alkanes consist of two parts – the soft hydrocarbon part, and the more rigid fluorinated part. The softer part usually attaches to the surface and these molecules form different self-organized structures on the substrate9. Semi fluorinated alkanes typically show good KPFM contrast and often used as a standard sample to demonstrate the spatial resolution of the technique. Fig. 6 shows such a sample imaged with the same three modes, and it is very evident, that the FM KPFM shows the best lateral resolution of all three techniques. This confirms the theoretical work about the size of the interaction volume in the FM KPFM mode.

Figure 6:  Comparison between the AM and FM KPFM modes on fluorinated alkane sample f<sub>1</sub>4H20.
Image size: 2 x 2 μm<sup>2</sup>. FM KPFM shows best contrast.
(a) height range: 4 nm, (b) CPD range: 0.9 V,
(c) CPD range: 1.1 V, (d) CPD range: 2 V.
Figure 6: Comparison between the AM and FM KPFM modes on fluorinated alkane sample f14H20.
Image size: 2 x 2 μm2. FM KPFM shows best contrast.
(a) height range: 4 nm, (b) CPD range: 0.9 V,
(c) CPD range: 1.1 V, (d) CPD range: 2 V.

Surface charge and 2D materials

While the phenomenological explanation of the Kelvin probe describes the measurement of work-functions, the KPFM technique, in fact, responds to any electrostatic interaction and surface potentials. As such, it can also image the distribution of surface charges, that may appear after a local gate voltage is applied10. Another interesting area of application is the study of charge-polarized superlattices formed by layers of twisted hexagonal boron nitride11 (hBN). Fig. 7 is showing such a sample imaged with AM KPFM at 2nd resonance and with the heterodyne FM KPFM. The charge superlattice is clearly observed with both modes, and the image is a bit sharper with the FM mode.

The 2D materials in general can also be well imaged with the KPFM, while extracting the information about their surface potentials, which is of extreme importance for the materials with thicknesses as small as one atom. Fig. 8 is showing KPFM images of CVD graphene, and here, due to the relatively large scan size, and smaller KPFM signal, the image quality is better with the AM KPFM at 2nd resonance.

Figure 7:  Comparison between the AM and FM KPFM modes on fluorinated alkane sample f<sub>1</sub>4H20.
Image size: 2 x 2 μm<sup>2</sup>. FM KPFM shows best contrast.
(a) height range: 4 nm, (b) CPD range: 0.9 V,
(c) CPD range: 1.1 V, (d) CPD range: 2 V.
Figure 7: KPFM measurements of charge-polarized superlattices in twisted hexagonal boron nitride made with AM and FM KPFM. The FM KPFM is showing sharper image with finer details than the AM KPFM.
Image size: 5.8 x 5.8 μm2, (a) height range: 6.8 nm,
(b) CPD range: 260 mV, (c) CPD range: 340 mV.
Figure 8: Comparison between the AM and FM KPFM modes on CVD graphene sample.
Image size: 4.5 x 4.5 μm<sup>2</sup>.
(a) height range: 20 nm, (b) CPD range: 325 mV,
(c) CPD range: 470 mV, (d) CPD range: 540 mV.
Figure 8: Comparison between the AM and FM KPFM modes on CVD graphene sample.
Image size: 4.5 x 4.5 μm2.
(a) height range: 20 nm, (b) CPD range: 325 mV,
(c) CPD range: 470 mV, (d) CPD range: 540 mV.

Calibration of the tip potential

Since KPFM is measuring the CPD voltage, which is defined as the difference of the potentials, for the precise determination of the work function of the sample, one needs to know the work function of the tip. Over the years the researchers have come up with several calibration procedures. One of them is measuring the VCPD on highly oriented pyrolytic graphite (HOPG). HOPG is well studied and often is the preferred calibration sample due to its low cost, flatness and ease of manipulation. The reported values for the work function lie in the range 4.5-5 eV. The dispersion of the work function is mainly related to the surface contamination or inhomogeneities12. Another way to calibrate the probe is to measure the VCPD of a gold film, which was shortly prior characterized with ultraviolet photoelectron spectroscopy12,13. The choice of gold is due to its stability in air, non-oxidizing properties, and being less prone to formation of monolayer of water in ambient conditions13. It has to be noted that the calibrated KPFM measurements in ambient conditions often suffer from water, oxidation and absorption of other contaminants on the surface. While the measurements in ultra-high VACuum conditions are relatively expensive and cumbersome, a compromise medium such as a nitrogen or argon-filled glovebox could be an acceptable and affordable solution.

Conclusion

To summarize, modern KPFM is a powerful electrical mode of AFM. It is measuring a surface potential, which is essential in multiple domains of material research. The heterodyne FM KPFM is a relatively new KPFM mode, which is providing the most adequate measurement of the CPD voltage and together with that – the best spatial resolution amongst other KPFM modes – a conclusion, which is supported by the research community14. Choice of shorter and softer cantilevers helps to achieve better sensitivity to the surface potentials. For the absolute measurement of the CPD, the calibration of the tip potential and ambient control are necessary.

References

1. Melitz, W., et al., Surface Science Reports 66 (2011) 1-27
2. Kelvin, Lord (1898). "V. Contact electricity of metals". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 46 (278) (1898) 82–120
3. Zisman, W. A., Rev. Sci. Inst. 3 (7) (1932) 367–370
4. J.M.R. Weaver and D.W. Abraham, J. VAC. Sci. Technol. B 9 (3) (1991) 1559-61
5. M. Nonnenmacher, M.P. O’Boyle, and H.K. Wickramasinghe, Appl. Phys. Lett. 58 (1991) 2921-23
6. R. Borgani et al., Appl. Phys. Lett. 105 (2014) 143113
7. S. Magonov and J. Alexander, Beilstein J. Nanotechnol. 2 (2011) 15–27
8. T. Wagner et al., Beilstein J. Nanotechnol. 6 (2015) 2193–2206
9. A. Mourran et al., Langmuir 21 (2005) 2308-2316
10. S. Ye et al., Nanotechnology 32 (2021) 325206
11. C. Woods et al., Nature Communications 12 (2021) 347
12. E. Castanon et al., J. Phys. Commun. 4 (2020) 095025
13. V. Panchal et al., Scientific Reports 3 (2013) 2597
14. A Axt et al., Beilstein J. Nanotechnol. 9 (2018) 1809–1819

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