Single frequency vertical piezoresponse force microscopy

Piezoresponse Force Microscopy (PFM) is a mode of atomic force microscopy (AFM), enabling users to map and manipulate the electric dipole moment in piezoelectric and ferroelectric materials.^1–5 PFM, since its inception in 1992,⁶ has attracted attention from the ferroelectric community. It has been an important part of major ferroelectric related conferences such as the International Meeting on Ferroelectricity (IMF), International Symposium on Integrated Functionalities (ISIF), IEEE International Symposium on Applications of Ferroelectrics (IEEE-ISAF), and PFM workshops. Recent advances in PFM have expanded the list of its applications^7–14 and inspired the birth of electrochemical strain microscopy (ESM)^15–18 that can map and manipulate ionic species in ion conductors and electrode materials.

PFM is actively used to prove ferroelectricity in nanoscale objects and leaky samples as PFM is relatively insensitive to leakage current and can detect very small polarization at the nanoscale.^19–24 However, PFM suffers from its own artifacts and has been used cautiously when interpreting images. PFM contrast can be obscured by electrostatic (capacitive coupling and charge injection/trapping), electrochemical (ionic motion-induced strain), thermal (thermal expansion due to Joule heating through the sample), electrostrictive, flexoelectric, and capacitive buckling of the cantilever, to name a few.^25–34 

However, for most researchers, PFM is not an easy technique to understand. The images obtained by PFM sometimes pose a challenge due to the various artifacts mentioned above. Although there are more advanced PFM techniques such as multifrequency PFM [e.g., band excitation PFM³⁵ and dual ac resonance tracking (DART) PFM³⁶], I will focus on the simplest and oldest PFM, which is the single frequency vertical PFM (SFV-PFM). This tutorial provides an overall and detailed step by step instruction for graduate students and interested researchers to conduct PFM using atomic force microscopy and a lock-in amplifier and help them learn how to interpret the PFM images and the piezoresponse hysteresis loops correctly.

Figure 1 shows the schematic diagram of a tip–cantilever–holder system. It is important to know the shape and size of each component. Depending on the purpose of your measurement, you may use tips with different radii of curvature, height, shape (bird beak, pyramid, and cone), and material (silicon, silicon nitride, and platinum) with various coating materials (platinum, gold, titanium, iridium, tungsten carbide, diamond, etc.). You may also choose cantilever with different shapes (“A” shape or diving board), sizes (length, width, and thickness), and material (silicon, silicon nitride, and platinum) with various coatings (single sided or double sided, e.g., platinum or aluminum coating).

For single frequency vertical PFM (SFV-PFM), a platinum coated silicon tip with a silicon cantilever having a double sided metallic coating is frequently used. Furthermore, a high aspect ratio (tip height higher than 10 μm and tip radius less than 50 nm) tip with high stiffness (spring constant higher than 2 N/m) is usually adopted in SFV-PFM to reduce any artifacts emanating from the electrostatic effect, which will be further discussed in Sec. VIII.

Figure 2 depicts the core components we need to know in understanding how PFM works. However, it should be noted that there are AFMs with slightly different configurations in the market (e.g., piezoelectric tube scanners with X, Y, Z actuators integrated in one body). The X–Y stage [Fig. 2(a)] uses piezoelectric ceramics as linear actuators and mechanical levers to amplify the stroke of an actuator. Usually, the range covered by X and Y actuators is from 30 μm to 100 μm. For a larger range, one has to use a linear motor stage underneath the X–Y stage. In order to guarantee the precision and accuracy of the stage, the stage is run in a close loop where the distance moved is measured by the position sensors (capacitive, laser Doppler effect, etc.), which guarantees a position accuracy of less than 1 nm. The Z stage [Fig. 2(b)] consists of a piezoelectric actuator, which can usually expand up to 12 μm—some extended actuators can expand up to 40 μm—and a Z sensor that monitors the movement of an Z actuator. A quadrant photodiode, named position sensitive photodiode (PSPD), is used to measure the vertical displacement and lateral torsion of a cantilever [see Fig. 2(c)]. For a laser diode, the most popular one is the red laser with a wavelength of 633 nm (class 2), but there are vendors using an IR laser (class 1) or blue laser as well [see Fig. 2(d)]. A cantilever holder comes in many shapes, including alumina and steel-based holders. In order to apply a voltage to a tip, the electrical connection to the holder as well as to the cantilever and the tip is very important for conducting SFV-PFM experiments [see Fig. 2(e)].

Once we assemble all components, we have a complete AFM ready for topography acquisition, but not one yet for PFM experiments (see Fig. 3).³⁷ 

We will first understand how the vertical displacement of a tip (or the vertical deflection of a cantilever) is measured. The so-called beam bouncing method uses a laser beam, which is emitted from a laser diode and reflected on a cantilever surface to hit a quadrant photodiode. It should be noted that in the beam bouncing method, the cantilever bending angle is measured and correlated with the tip displacement through the equations [e.g., Eqs. (16) and (17)] discussed in Sec. VIII.

Because of the geometry, the displacement on the cantilever can be amplified by a factor of a few hundred when the laser beam arrives at the diode. For example, 1 nm of vertical displacement of the tip translates into at least 100 nm vertical motion of laser beam at the diode. In this way, AFM can measure tens of picometer displacements over a wide frequency range.

How do we measure the tip position using the beam bouncing method? Figure 4 provides us with the answer. First of all, we define deflection signal as the position of the laser beam at the quadrant photodiode. Numerically, D is determined by the difference between the laser intensity in upper segment A and lower segment B. The current signal from the photodiode is proportional to the intensity of the laser beam, which is proportional to the beam area impinging upon the area of interest in the photodiode. As the laser beam diameter is around 40 μm, the vertical motion of the beam in the range of 100 nm will translate into a linear increase in the beam area as a function of tip displacement.

As shown in Fig. 4(a), we position the laser beam on the upper part of the cantilever (we can position the laser with x and y motor control of the laser diode) and move the photodiode to align the center of the laser beam with the center of the photodiode. This reference deflection will be set to zero. Depending on the loading force we want to apply to the sample, we will set the deflection to, e.g., 100 nm. However, for simplicity, we will assume that the reference state is when D = 0.

If the tip moves up [see Fig. 4(b)], the laser beam will move up (some vendors design the geometry in the way that the laser will move to the left instead of in the upward direction). As such, D will become positive. If the tip moves down [see Fig. 4(c)], the laser beam will also move down, resulting in a negative D.

As we now understand how AFM measures the vertical displacement of a tip, we can start approaching our tip to the sample and see what happens. Here, we can think of the cantilever–tip system [Fig. 5(a)] as a spring–mass system [Fig. 5(b)]. Before the tip meets the sample, the only forces acting on the tip will be gravitational force and restoring force (we will ignore the thermal expansion of the cantilever due to the heating induced by the laser beam bouncing off the cantilever). As the gravitational force is also negligible, we will ignore it and also its corresponding restoring force. As such, there will be no force acting on the cantilever (F = 0).

Here, we will define a few parameters shown in Fig. 5. Z_actuator will represent the end position of the Z actuator. Z_tip will be the position of the tip from the end position of the Z actuator. Gap is the distance between the outermost tip surface and the outermost sample surface. If we differentiate the Lennard–Jones potential with respect to the gap, then we will obtain the force acting on the tip.³⁸ Assuming that we can change the gap linearly toward the sample surface, we will get the force curve as a function of gap. In Fig. 5(c), r₀ represents the equilibrium distance between two atoms where one atom is from the outermost surface and the other from the outermost tip. It is clear that down to r₀, the dominant force is the attractive van der Waals force, while the repulsive atomic force emanating from the Pauli exclusion principle kicks as the gap becomes smaller than r₀. Therefore, once we are in the repulsive force region, reasonably large fluctuations in loading force will not change the gap between the outermost tip atom and the outermost surface atom, which makes the contact mode AFM imaging insensitive to the loading force.

Now, we will cover the force–distance curve, which is often used to calibrate the photodiode signal. Here, the force refers to the applied force by the tip on the sample, which is mostly the elastic restoring force of the cantilever. As such, the force will be proportional to the spring constant k of the cantilever (k_lever) and the relative displacement of the tip from the equilibrium position (Δz_tip). The distance refers to the distance traveled by the actuator (Δz_actuator).

As shown in Fig. 6, the force–distance curve acquired by AFM looks different from the ideal force–gap curve illustrated in Fig. 5(c). As we cannot directly control the gap between the tip and the sample surface, we control the position of the Z actuator. Let us assume that we have an ample gap between the tip and the sample surface, so the gap is a large positive number. In such a situation, the force acting on the tip will be zero [see the green dot in Fig. 6(a)]. As we increase z_actuator, the gap will decrease accordingly, whereas z_tip will remain constant. As such, until the tip is in contact with the sample surface, this motion of the Z actuator is equivalent to reducing the gap.

However, once the gap is less than 1 nm, the attractive van der Waals force becomes significant, so the cantilever will bend toward the surface until it is countered by the elastic restoring force of the cantilever. Moving the Z actuator closer to the surface will straighten the bent cantilever and enter into the boundary between the attractive and repulsive force regime in Fig 5(c). Therefore, the net force will again be zero as depicted by a yellow dot in Fig. 6(a). From here, the gap will remain almost constant because the force applied by the cantilever is countered by the steeply increasing repulsive force with a very small change in the gap. As such, z_tip will decrease linearly, as shown in Fig. 6 (red dot).

Let us look at this in more detail. We will first cover what happens between the green and yellow dots in Fig. 6. We will compare the force–gap curve and the force–distance curve side by side and try to understand what happens when we approach the tip by controlling z_actuator. Figure 7 shows the stepwise details of the force between the outermost atoms of the tip and the sample surface, and the change in z_tip as z_actuator increases (the blue arrows indicate the increasing direction for both z_tip and z_actuator). Up until the second stage [Figs. 7(a), 7(b), 7(e)–7(f), 7(i), and 7(j)], the van der Waals attractive force is increasing but still negligible so that z_tip remains constant. However, as the gap is reduced to the third stage [Figs. 7(c), 7(g), and 7(k)], the attractive force becomes large enough to bend the cantilever toward the sample surface, which results in the increase in z_tip by Δz_tip. As we further approach the tip to the sample surface, the attractive force will diminish and z_tip will be restored to its original value [Figs. 7(d), 7(h) and 7(l)]. Here, we can see that Δz_actuator and Δz_tip will be equal to each other. As such, we can use this relationship to calibrate the photodiode signal that measures Δz_tip in terms of deflection signal in voltage to “nm” by acquiring Δz_actuator from the Z sensor (we often use the term “inverse optical lever sensitivity, which is Δz_actuator/ΔDeflection).

From Fig. 7(d) where the atomic force changes its polarity from attractive to repulsive, we are entering into a contact mode regime. Figure 8 depicts the contact mode regime in more detail. As the tip moves toward the sample surface, the surface will be deformed due to the increasing repulsive force [ΔF_repulsive in Fig. 8(b)] by the tip loading force [ΔF_lever in Fig. 8(c)]. The equilibrium distance, r₀, between the tip and the sample surface will decrease, and this will lead to a local indentation by Δz_sample, as shown in Fig. 8(d). Figures 8(e) and 8(f) show how r₀ and Δz_sample are determined from the force–gap curve and their corresponding meanings in the locally deformed surface by the tip.

We are now ready to learn two main contact mode AFM operations (see Fig. 9): constant deflection [Fig. 9(a)] and constant height [Fig. 9(b)] modes.^39,40 In the constant deflection mode, we use a feedback loop to maintain a constant bending of the cantilever. This is achieved by setting a target deflection (e.g., setpoint = 1 V or 100 nm in case if the inverse optical lever sensitivity is 100 nm/V) and an appropriate gain for the feedback loop. The Z actuator (z_actuator) will move up and down to keep z_tip constant at the setpoint based on the error signal, which is the difference between the target setpoint and the actual deflection signal. As the tip loading force applied to the sample surface will also be constant, the gap will also remain constant, as depicted in Fig. 8(a). In this mode, the movement of the Z actuator (z_actuator) will represent the topography of the sample surface. The deviation of the deflection signal from the target setpoint (Δz_tip) represents the error signal.

In the constant height mode, the Z feedback loop is disabled. As such, z_actuator will stay constant during the scan. The tip will move up and down along the surface topography to find the dynamic equilibrium between the tip loading force and the atomic repulsive force. As the loading force will vary as z_tip changes, the gap will also change, as shown in Fig. 9(b). Here, the change in gap will constitute the error signal, which we cannot directly measure using AFM. In this mode, the deflection of the cantilever, which will result in a change of z_tip, constitutes the surface topography of the sample.

You may ask: How should I decide the mode when I want to acquire the surface topography using AFM? It depends on the purpose of your experiment. For example, if you want to image a surface reconstruction of your sample at an atomic scale, then your best mode will be the constant height mode.⁴¹ However, if you want to image composite electrode materials with roughness up to a few μm over a 100 μm scan range, your best mode will be the constant deflection mode.¹⁸ 

The constant height mode has a limited z_tip range (usually less than 1 μm), because the variation in loading force will be too large to ignore the error due to the variation in gap (wearing of both the tip and the sample can occur at a very high loading force of, e.g., 1 μN) or so large that your cantilever will be broken. The photodiode will usually saturate at 10 V, which corresponds to z_tip of 1 μm in case if the the inverse optical lever sensitivity is 100 nm/V. However, since it does not use the Z feedback loop and Z actuator, one can use the maximum scan rate to minimize any drift along the x and y directions and obtain a high spatial frequency of the z_tip component, which will enhance the spatial resolution of the AFM topography image. That is why the constant height mode is more suitable for topography images with a small scan range but very high spatial resolution.⁴² 

The constant deflection mode has a limited bandwidth of z_actuator because the resonance frequency of the Z actuator (1–10 kHz) is lower than the resonance frequency of the cantilever (10 kHz–1 MHz). In addition, the Z feedback loop imposes another constraint in the speed of topography acquisition. However, as the Z actuator can move up to usually 10 μm, it is more suited to visualize rough composite samples over a very large scan range (e.g., 100 μm). The other advantage is that one can keep the loading force constant during the scan, as z_tip will stay constant. A nanoscale feature such as 0.1 nm height change over 1 nm lateral periodicity can still be visualized by the constant deflection mode, because the Z sensor offers a sufficient spatial resolution. For example, if we image the 0.1 nm height change over 1 nm lateral periodicity with a scan rate of 1 Hz and a scan range of 0.5 μm, z_actuator will vibrate at 1 kHz with an amplitude of 1 nm, of which frequency is usually lower than the resonance frequency of the Z actuator.

We will do some thought experiments to understand the potential artifacts in AFM topography images. Let us assume that we have a perfectly flat composite sample that consists of a hard and a soft phase with equal volume in an alternating fashion, forming a lamellar structure, as demonstrated in Fig. 10. Even when the equilibrium gap r₀ is the same for both phases, the gap will decrease more in the soft phase under the same loading force than in the hard phase, as schematically depicted in Fig. 10(b). Therefore, the flat surface (ground truth) will appear like a mesa structure shown in Fig. 10(a). The gap will also be different for both soft and hard phases, as shown in Figs. 10(b) and 10(c).

What happens when we apply voltage to an AFM tip? We will start from a very simple situation where the sample of interest is a conducting material such as Pt. Figure 11 shows the schematic diagram where the AFM tip is in contact with a conductor, and a voltage supplier is applying a voltage to the tip while the conductor is connected to the ground.

When we measure the current vs voltage curve to measure the spreading resistance of the sample, we usually use a triangle shaped voltage pulse, as depicted in Fig. 12(a). Here, we denote the time to reach the first peak voltage from zero as t₀ so that the full period of the pulse becomes 4t₀. From the relationship between capacitance (C), voltage (V), and charge (Q), we can derive the equation of I_C as follows [see Eq. (1)]:

Therefore, the slope of the curve in Fig. 12(a) will give us the current through the capacitor in Fig. 11(b). As the slope is positive and constant up until t = t₀, a straight line corresponding to CV₀/t₀ will be drawn in the first part of the graph in Fig. 12(b). After the voltage peak of V₀ at t = t₀, the voltage decreases linearly down to –V₀. Here, the slope reverses its sign to minus but maintains its absolute magnitude, so another straight line from t₀ to 3t₀ will be drawn at −CV₀/t₀. Finally, from t = 3t₀ to 4t₀, the voltage increases again to zero where the slope is identical to the first part of the graph. This will yield a straight line in Fig. 12(b) at CV₀/t₀.

Figure 12(c) shows the current through the resistance (I_R), which has the same shape as that shown in Fig. 12(a). This is easy to understand if we remember that V = IR according to Ohm's law, where R is the resistance of the resistor, as shown in Fig. 11(b). As such, the first peak current of I_R corresponds to V₀/R.

Figure 13 shows the I–V curves obtained from the conducting AFM tip in contact with a conducting sample like the one in Fig. 11(a). Triangular voltage pulse [Fig. 12(a)] is used, and the total current (I_C+I_R) is collected through the current amplifier. In order to protect the circuit, a compliance current of 20 nA is set, so the resulting I–V curve from an equivalent circuit shown in Fig. 11(b) will look like that shown in Fig. 13(a). Real experimental datasets are shown in Figs. 13(b) and 13(c), in which the sample is highly ordered pyrolytic graphite (HOPG) and the AFM tip is either a Pt wire tip [Fig. 13(b)] or a conducting diamond coated tip [Fig. 13(c)].⁴³ The voltage offset seen in Figs. 13(b) and 13(c) is due to the offset inherent to the current amplifier. The slope of the curve represents the spreading resistance between the AFM tip and the sample.

Another way to measure the I–V curve is to use an external multimeter and connect the circuit outside the AFM. Figure 14 shows an example of such an experiment with a standard resistor added to the sample.⁴⁴ By first measuring the resistance of the standard resistor, as shown in Fig. 14(b), one can obtain the resistance between the AFM tip and the sample by subtracting the resistance obtained from Fig. 14(c) by that from Fig. 14(d).

Now let us add one more complexity to Fig. 11. What happens if we add a dielectric layer on top of the conducting substrate, make our AFM tip to come into contact with the surface of the dielectric material, and apply a voltage to the tip? We will discuss a few interesting phenomena that can happen in such a situation, which is depicted in Fig. 15.

First, electron or hole injection can occur from the tip via tunneling (e.g., Fowler–Nordheim tunneling or thermionic emission).^45–47 In fact, this is how a standard FLASH memory works in, e.g., a Si/SiO₂/Si₃N₄/SiO₂/Si (SONOS) structure.⁴⁸ Park et al. observed such an injection through a carbon nanotube into an ONO structure using Kelvin probe force microscopy (KPFM).⁴⁹ The injected electrons or holes can be trapped in surface states or inner trap sites (such as vacancies and interstitials). They can react with surrounding oxygen, hydroxyl, or hydrogen to create electrochemical reactions either on the tip or on the sample surface.⁵⁰ 

Second, defects such as oxygen vacancies or metal interstitials can be extracted by the high electric field near the AFM tip.⁵¹ These defects can be ordered to make a connecting path to the bottom layer, which is one of the mechanisms used for resistance change memory devices (e.g., ReRAM, RRAM, and OxRAM).^52–56 Those connected paths could be the source of Joule heating, which can induce local phase transformation (e.g., PRAM).^57,58

Figure 16 illustrates an example of the second case mentioned above. Kim et al.⁵¹ found that the region, where they applied positive voltage to the tip during AFM scan, expanded in the thickness direction with a decreasing partial pressure of oxygen surrounding the single crystal TiO₂. This is evidenced by a bright contrast in the topography images in Fig. 16(a). This positive voltage will attract an oxygen anion to the surface. The tip will take the electron from the oxygen anion and oxygen gas will come out of the surface, leaving a positively charged oxygen vacancy on the surface [see Fig. 16(b)]. The related chemical reaction can be described as follows:

The amount of oxygen vacancies can be controlled by changing the oxygen partial pressure surrounding the sample surface. For example, one can change the oxygen partial pressure by flowing an Ar diluted H₂ gas (low oxygen partial pressure), Ar (medium oxygen partial pressure), and O₂ (high oxygen partial pressure), as shown in Fig. 16(a).⁵¹ 

Why do we see a distinct change of topography up to 10 nm in TiO₂? Is this a real change in surface morphology? In order to answer this question, we need to think what force is in play when we create charged defects near the sample surface. One hypothesis we can think of is the repulsive force between the oxygen vacancy in the sample and positive charges in the AFM tip. A Pt coated AFM tip often creates a thin layer of PtO_x on top of the tip and can be charged by the applied voltage to the tip. As such, a positive voltage to the tip can leave positive charges being trapped in PtO_x, making the tip slightly positive. In such a case, the positively charged oxygen vacancies on the sample can repel the tip to create a fictitious topography change, as seen in Fig. 16(a).⁵¹ The bump of about 10 nm above the surface is physically unfeasible unless there is additional deposition of new materials on top of the surface or there is a big crack inside due to enormous compressive stress built up near the surface. Either scenario can be quickly checked by ascertaining the reversibility of the feature with different environments. As we change the environment from oxygen poor (Ar + 4% H₂) to oxygen rich (100% O₂), the topography becomes flat, which is indicative of the important role of oxygen vacancies in the topography image.⁵¹ 

Figure 17 illustrates the reason why we observe a topography artifact on a charged surface. Figure 17(a) shows a schematic diagram of the case described in Fig. 16. Oxygen vacancies created by the positively charged AFM tip create an electric field that exerts a strong repulsive force larger than the van der Waals attractive force on the tip. As such, the force–gap curve of a neutral surface will be modified by a positive offset, as shown in Fig. 17(b). For the same repulsive force, z_actuator increases to keep z_tip constant. So, the topography image obtained by using the constant deflection mode will show a bright contrast on the square region where the oxygen vacancies were formed by the AFM tip. This is an artifact, as shown in Figs. 17(c) and 17(d).

Now, we are ready to tackle the basic operation principle of piezoresponse force microscopy (PFM).^59,60 PFM is designed to visualize the electric polarization of ferroelectric materials at a nanoscale. PFM uses the piezoelectric strain as the marker for the polarization through the following equations:^59–61 

where d_zz is the piezoelectric coefficient obtained from the strain along the z direction when an electric field is applied along the same z direction, ɛ₀and ɛ_r are the vacuum permittivity and relative permittivity of the sample, respectively, Q is the electrostrictive coefficient, P_s is the spontaneous polarization of the sample, ɛ_z is the strain along the z direction when stress is applied along this direction, and E_z is the electric field applied along the z direction. Here, it should be noted that the E field induced by a PFM tip usually has significant in-plane components that can couple with multiple coefficients (e.g., d₃₁, d₁₅, and d₃₃ for a transversally isotropic piezoelectric solid in an isotropic elastic approximation and d₃₁, d₃₂, d₃₃, d₂₄, and d₁₅ for an anisotropic piezoelectric solid in the limit of dielectric and elastic isotropies) in the piezoelectric tensor and contribute to the measured longitudinal strain.⁶² In addition, if the sample is polycrystalline in nature, the effective d_zz will be more complicated and contributed to by other components of the piezoelectric coefficient tensor.⁶² However, we will ignore these complexities in the following analysis and assume that d_zz is the only one contributing to the vertical piezoelectric strain.

Therefore, PFM uses a biased tip to create an electric field along the z direction (E_z) to induce a piezoelectric strain along the same direction (ɛ_z), as shown in Fig. 18. There are two ways to induce this piezoelectric strain. One is to use a dc voltage to induce a static strain, while the other is to use an ac voltage to induce a dynamic strain in the ferroelectric sample.

Let us consider the static strain first. If the topography is flat like in the case of a single crystal, one can simply apply a dc voltage to the tip that is smaller than the coercive voltage but large enough to make a sufficient change in topography to visualize the up and down domains.⁶³ 

Figure 19 shows the operation principle of the static PFM imaging on a flat single crystal with up and down polarization domains. If we apply a positive voltage to the AFM tip that is well below the coercive voltage of the ferroelectric sample, then the up domain will shrink as the electric field is acting against the polarization and the down domain will expand because the electric field is aligned with the polarization.

If the piezoelectric strain vs electric field shows a symmetric butterfly shape,⁶¹ then the piezoelectric strain will have the same magnitude with opposite polarity, as depicted in Fig. 19(b). As such, the Z actuator will move down and up as the tip scans across the up and down domains. Even though the ground truth will be a flat surface topography, the topography image will show a distinct contrast, as shown in Fig. 19(c) due to the motion of z_actuator [see Fig. 19(d)].

However, the static PFM suffers from a low signal to noise ratio, as the topography is not always as flat as a single crystal and the piezoelectric strain is usually much smaller than the topographic variation in the sample. Background thermal noise and mechanical vibration can also pose a big challenge to using the static PFM to visualize polarization domains in ferroelectric samples.

In order to circumvent the problems of interference by topography variation and various background noises, single frequency vertical PFM (SFV-PFM) was invented in 1992⁶ and used by many research groups since then.^64–70 In order to understand the operating principle of SFV-PFM, one has to understand the “lock-in technique.”^71–73 

Lock-in amplifiers are used to detect and measure very small AC signals—all the way down to a few nanovolts. If you think about the deflection signal of 1 V from the photodiode in AFM and the typical inverse optical lever sensitivity of 100 nm/V, it means that lock-in amplifiers can potentially detect a sub-femtometer deflection. Accurate measurements may be made even when the small signal is obscured by noise sources many thousands of times larger. Lock-in amplifiers use a technique known as phase-sensitive detection (PSD) to single out the components of signals at specific reference frequencies and phases. Noise signals, at frequencies other than reference frequencies, are rejected and do not affect measurements. The reason why lock-in amplifiers are better than low-noise amplifiers or bandpass filters is well explained in Ref. 71 

Figure 20 illustrates the basic principle of a lock-in amplifier using a phase sensitive detector (PSD). A lock-in reference signal can either be fed into an amplifier by an external source or generated by an internal source. Let us assume that the reference signal is a sinusoidal one at a frequency of ω_r and a phase of zero. Then, an input source signal of a sine wave at a frequency of ω_s and a phase of θ, V_s sin(ω_st + θ), is multiplied with a reference signal, as shown in Figs. 20(a) and 20(b), in which ω_s is equal to ω_r [Fig. 20(a)] and ω_s is not equal to ω_r [Fig. 20(b)], respectively. The resulting PSD signal, V_psd, can be expressed as Eq. (5) [see Fig. 20(c)],

The PSD output, V_psd, is two AC signals, one at the difference frequency (ω_s−ω_r) and the other at the sum frequency (ω_s + ω_r). However, if ω_s = ω_r, the output is one DC signal and one AC signal. Therefore, if the PSD output is passed through a low-pass filter, the AC signals are removed and only the DC signal is left. As such, the dc component of V_psd will contain information only about the source signal having the same frequency (ω_s = ω_r) as that of the reference signal [see Eq. (6) and Figs. 20(d) and 20(e)],⁷¹ 

This is how a single-phase lock-in amplifier works. Early PFM works used this single-phase lock-in amplifier and mapped the piezoresponse, which is the convolution of the PFM amplitude (A, V_s can be converted to A if multiplied by the inverse optical lever sensitivity) and cosine value of the PFM phase (cos θ).^74,75

In order to deconvolute the piezoresponse into PFM amplitude (A) and PFM phase (θ), we need to use a dual phase lock-in amplifier where we have another lock-in reference signal shifted by 90°. In other words, the second PSD signal, V_psd2, is generated by multiplying the second lock-in reference [V_r sin(ω_st+90)] by the same source signal as expressed by

Therefore, the dc component of V_psd2 can be expressed by

Then, we can rewrite Eq. (6) in terms of V_psd1 as follows:

With Eqs. (8) and (9), we can calculate V_s and θ from the known value of V_r and measured values of V_psd1,dc and V_psd2,dc from the lock-in amplifier as follows:

How do we connect the PFM amplitude and phase with V_s and θ measured from the lock-in amplifier? We apply an ac voltage to the tip using the reference signal of the lock-in amplifier. Therefore, the sample will be exposed to an oscillating electric field from the AFM tip induced by V_r sin ωt. This will induce a sinusoidal piezoelectric strain under the tip, which will change z_tip accordingly. The change in z_tip will be detected by the photodiode in the form of V_s sin(ωt + θ) that is being fed into the lock-in amplifier as the source signal. One can define PFM amplitude as the piezoelectric displacement, in which case one should multiply the photodiode voltage signal (V_s) with the inverse optical lever sensitivity (1/S_lever). The PFM phase will be the same as θ in Eq. (11).

A typical setup of SFV-PFM is shown in Fig. 21 and can also be found in the literature.^64,76 The deflection signal from the photodiode is connected to the input signal of the lock-in amplifier. The internal reference signal in the lock-in amplifier is being fed into the AFM tip, while the sample is either grounded or dc biased. The topography image is obtained from the Z actuator signal (z_actuator), while the PFM amplitude and phase images are acquired from the amplitude and phase signals of the lock-in amplifier. The X–Y stage is being controlled by the scanner controller where x and y sensors are located to control the x and y positions of the stage in a closed loop.

How do we separate the topography signal and the PFM signal? In order to answer this question, let us focus on how we acquire the topography using the constant deflection mode (see Fig. 22). We already know from Fig. 9 that the deflection signal represents the topography signal in the constant deflection mode. In this mode, as the tip scans the surface from left to right [see Fig. 22(a)], the deflection signal detected by the photodiode can be plotted as a function of time, which is depicted in Fig. 22(b).

The height as a function of x can be described by f(x), where x is the scan distance. As the scan speed is v = dx/dt, we can get the height as a function of time, g(t), by dividing x by v [i.e., g(t) = f(x/v)], as shown in Fig. 22(b). If we use Fourier transform, we can decompose any arbitrary signal into the summation of sine waves with different frequencies and phases. One simple example of decomposing a simple topography composed of sin(x) and sin(10x) is depicted in Fig. 22(c). The ratio of amplitude is 5:1 for sin(x) and sin(10x). Converting this topography into the time domain by using the scan speed v will enable us to plot the amplitude of each component in the frequency domain (Hz), as shown in Fig. 22(e).

For a rough composite sample, the topography will be more complex than just a superposition of two sine waves with different frequencies. It will be a superposition of numerous, if not infinite, numbers of constituent sine waves with harmonics where the amplitude of each wave usually decreases as the frequency increases. Why is this the case? Usually in a composite material, a large component has a large peak height with a wide grain size, which will contribute to a low frequency component wave (large periodicity) with a high amplitude (large peak height), whereas a small component has a small peak height with a small grain size, which will contribute to a high frequency component wave (small periodicity) with a low amplitude (small peak height).

We can calculate the maximum frequency of the deflection signal for a wide range of samples with the boundary condition set by the scan range and the number of pixels we use to obtain the topography image. Considering that the smallest scan range is usually 500 nm and the resolution of the X–Y scanner is 1 nm, the maximum number of pixels we should use is 512. To make the calculation simple, let us assume that the number of pixels is 500 instead of 512. Then, our image will have a 1 nm/pixel resolution. Therefore, the smallest periodicity of the topographic feature will be 1 nm. For the scan rate, let us assume that we use 1 Hz, which will give us a scan speed of 1 μm/s. If we assume a sine wave approximation for this feature, the frequency of the deflection caused by the smallest topographic feature will be 1 kHz.

Therefore, placing a low-pass filter of, e.g., 2 kHz–10 kHz in Fig. 21 before the Z regulator/Z sensor unit will not affect the quality of the topography signal obtained by using the constant deflection mode.

Now, we can answer the question of “How do we separate the topography signal and PFM signal?” We split the deflection signal into low and high frequency regimes and use the constant deflection mode to compensate the tip movement induced by the topography and use the constant height mode and a lock-in amplifier to pinpoint the piezoelectric displacement at the frequency at which we excited the sample with the reference signal from the lock-in amplifier. This will enable us to image both the AFM topography and the PFM image at the same time in the same region.

From the perspective of an AFM tip, the topography looks flat because z_actuator is compensating the change in height from the close loop operation. It is like an active damping system of a luxurious car where the passenger feels like moving on a flat ground when, in reality, the road is very bumpy but the actuators are moving up and down to compensate the bumpy road. In this situation, any change in the topography contributing to the deflection signal will be the high frequency PFM signal. As represented by a line in Fig. 23, we can plot it as a sine wave in the time domain.

Before moving on, we will think about what can happen when we apply ac voltage to the tip with a drive frequency (e.g., 17 kHz) above the low-filter cutoff frequency. We will ignore the effects of charge injection, flexoelectricity, electrostriction, thermal expansion due to Joule heating, additional defect formation (e.g., charged oxygen vacancy and conducting filament), and electrochemical reaction as well as ionic motion across the lattice or within the lattice (i.e., electrochemical strain).

Even then there can be several factors affecting the deflection at high frequencies, as described in Table I.¹ We can divide the interacting components into two categories: tip–sample and cantilever–sample interactions.

For the tip–sample interaction, there are two driving forces: piezoelectric force and electrostatic force. The piezoelectric force is the repulsive atomic force applied to the tip by a piezoelectric strain. As such, the gap will be less than 1 nm. As the piezoelectric strain follows Eq. (4), it will be a sine wave with the same frequency as that of the sine wave of ac drive voltage applied to the tip. The electrostatic force can be either attractive or repulsive depending on the nature of the sample surface. If electroneutrality is conserved without any excess charge near a surface, the electrostatic force will always be attractive, no matter what polarity the voltage applied to the AFM tip is. If one has a parallel plate capacitor and connects it through an ac voltage supplier, the plates will always be attracted to each other, no matter what voltage is applied between them. However, if a surface is charged by, e.g., oxygen vacancies or trapped electrons, then the excess charge can exert repulsive force on the AFM tip. In this case, the tip will tap on the surface due to the repulsive electric force.

For the cantilever–sample interaction, the driving force is the electrostatic force between the cantilever and the sample surface. If the tip is away from the surface, then this force will move the cantilever in the same way it moves the AFM tip. The cantilever will bend toward the surface, making the gap between the tip and the sample smaller. However, if the tip is in contact with the surface, it will buckle the cantilever toward the sample, making the cantilever deformed in a “U” shape and reducing the gap between the cantilever and the sample without causing any significant change in the gap between the tip and the sample surface.³¹ 

Now, let us assume that we designed the PFM experiment in such a way that all other driving forces are suppressed and only the piezoelectric force is acting on the AFM tip. In such a simple case, the ac drive voltage fed into the AFM tip will either expand or contract the surface depending on the relationship between the polarization and the electric field. Figure 24 shows a schematic diagram of the SFV-PFM operation principle.

For an up domain, the ac voltage to the tip, which is smaller than the coercive voltage, is being applied while the bottom electrode is grounded [see Fig. 24(a)(i)]. When the tip is positively charged, the electric field will direct from the tip to the sample. As the electric field is against the polarization, the polarization will shrink (in case if the piezoelectric coefficient, d₃₃, is positive). Therefore, the sample surface will undergo a contraction that will move the tip downward. When the tip is negatively charged, the opposite phenomenon will occur, leading to an expansion of the sample surface along the direction of the thickness. This will make the tip move upward. If we compare the lock-in reference signal fed into the tip in the form of a sine wave and the resulting deflection signal coming from the photodiode, we find a 180° phase difference between them. Therefore, the PFM phase signal will display 180°, which is colored white in Fig. 24(b)(iii).

For a down domain, a similar process can be applied. The positive voltage on the tip will make the surface expand and the negative voltage will make the surface contract. If we compare the lock-in reference signal with the deflection signal, the phase difference will be 0°, which will result in a PFM phase image of 0° [colored black in Fig. 24(b)(iv)]. If we have symmetric polarizations for the up and down domains in terms of their polarization magnitude, then the PFM amplitude image will be identical to each other, as shown in Figs. 24(b)(iii) and 24(b)(iv).

Figure 25 shows an example of a PFM image with a mixture of the up and down domains in a 150 nm thick ⟨111⟩ preferentially oriented Pb(Zr_0.40Ti_0.60)O₃ film on a Pt/Ti/SiO₂/Si substrate prepared by chemical solution deposition.⁷⁷ A negative dc bias voltage of −10 V, the magnitude of which is larger than the coercive voltage, was applied to the bottom electrode, while a grounded AFM tip was scanned over an area of 10 × 10 μm² to create a square area with the down polarization domain. Then, a positive voltage pulse of +10 V with a duration of 5 ms was applied to the bottom electrode at a rate of 24 Hz while the grounded AFM tip was scanning over the same area. This procedure created a fully penetrating up domain in a down domain matrix, as schematically depicted in Fig. 25(d).⁷⁸ An area of 5 × 5 μm² was scanned where the topography, PFM amplitude, and PFM phase images were obtained [see Figs. 25(a)–25(c)]. Using the PFM amplitude and phase images shown in Fig. 24(b), one can easily draw the expected PFM amplitude and phase images [see Figs. 25(e) and 25(f)] for an up-domain bit written in a down-domain matrix shown in Fig. 25(d).

In Sec. VII, we assumed that all other factors listed in Table I were suppressed, except the piezoelectric force acting on the tip. Here, we will look into the artifacts listed in Table I one by one. The first artifact will be the electrostatic force acting on the tip due to the capacitive force. This is an artifact in PFM but can be the main contrast mechanism of contact mode electrostatic force microscopy (EFM).^79,80

As shown in Fig. 26, when there is an excess positive charge on the surface (an unscreened positive polarization or injected/trapped positive charge), the ac voltage applied to the tip will cause a mechanical vibration of the tip by the electrodynamic force between the tip and the sample surface. The positive charge on the tip will repel the tip from the positively charged surface, resulting in an effect similar to the expansion of the surface, whereas the negative charge on the tip will attract the tip toward the same surface, resulting in an effect similar to the contraction of the surface. Therefore, the EFM phase will be 0°, and the EFM amplitude will be proportional to the total amount of excess positive charges on the surface [see Figs. 26(a)(i) and 26(b)(iii)]. The opposite case will be easily understood, which is schematically depicted in Figs. 26(a)(ii) and 26(b)(iv).

Figure 27 shows the EFM images along with the topography image at the same place, which were operated in a similar manner to PFM imaging.⁵¹ Here, instead of using the contact mode AFM, the researchers used AC mode AFM (tapping mode AFM) with a drive frequency of 145 kHz (slightly lower than the resonance frequency of the cantilever), and EFM images of the surface were simultaneously acquired by applying an AC modulation voltage of amplitude (V_ac) of 3 V and a frequency (ω_r) of 20 kHz to the tip. Therefore, the average gap between the AFM tip and the sample surface is expected to be much larger (∼40 nm) than that (∼0.3 nm) of the contact mode AFM. One can clearly see that the EFM phase on the positively charged region with oxygen vacancies is 0° [compare Fig. 26(b)(iii) with Fig. 27(c)]. The EFM amplitude shows a relatively strong contrast when compared with the background, which is indicative of highly concentrated oxygen vacancies in the square region. Based on this example, one can understand that if the electrostatic force dominates in the PFM experiment, the phase will be reversed by 180° for the same polarization domain. However, it should be noted that the surface must not be screened by any screening charges, which is very unlikely in an ambient condition.

Figure 28 shows the operation principle of EFM, when either the ferroelectric surface or the charged dielectric surface is fully screened. The same argument can be applied to this case, as in Fig. 26, but with one important difference. The surface with positive polarization or injected/trapped positive charge will be screened by the negative screening charges that are closer to the AFM tip. Therefore, if the tip is positively charged, it will be attracted to, rather than be repelled from, the surface. Accordingly, the EFM phase will be 180° instead of 0°, as shown in Fig. 28(b)(iii). Furthermore, the EFM amplitude will be much smaller as the net charge is zero in the vicinity of the AFM tip.

Sometimes, it is hard to remember all of these contrast mechanisms contributing to the PFM images. Therefore, a step by step discussion is needed many times. So, let us compare Figs. 26 and 28 again. There are two distinct features that need our attention. The polarity of the closest charges to the AFM tip is opposite to each other. Next, the overall net charge is not zero for Fig. 26 but zero for Fig. 28. As such, the EFM phase is opposite and the amplitude is strong for the unscreened case but weak for the screened case.

This could be understood in terms of the surface potential and electric field distribution, as schematically depicted in Fig. 29.^81,82 For the unscreened up polarization domain, the surface potential is positive and the electric field is directing upward [denoted as negative in Fig. 29(e)] as E = σ/(2ɛ₀) = P•n/(2ɛ₀). For a fully screened up polarization, the surface potential is negative and the electric field is pointing downward [denoted as positive in Fig. 29(f)]. From the magnitude of both the surface potential and the electric field, the screened case is always smaller than the unscreened one, which can explain the difference in the EFM amplitude images between these cases [compare Fig. 26(b) with Fig. 28(b)].

Is there a way to distinguish between PFM and EFM? Yes. One can conduct both PFM and Kelvin probe force microscopy (KPFM) imaging on the same place. However, even if we can determine which contrast mechanism plays the major role, it does not mean that the minor factors are absent. They are there, but can be ignored in many cases due to their insignificant contribution.

Kim et al. used both PFM and KPFM modes to visualize the switched ferroelectric domains and injected charges through the AFM tip.⁸³ As shown in Figs. 30(a) and 30(b), the dc bias voltage applied between the AFM tip and the bottom electrode was varied between −8 and 8 V with a step of 2 V over a rectangular area of 4.5 × 3.6 μm². From the KPFM and PFM images, one can determine the amount of surface charges and polarization. The relative potential ΔV can be expressed as follows:

where E is the surface electric field, gap is the distance between the AFM tip and the surface charge (see Figs. 5 and 8), ɛ₀ is the permittivity of vacuum, P is the polarization of the ferroelectric domain, and n is the surface normal vector.

From Figs. 26 and 28, one sees that the EFM phase will be determined by the degree of screening on the ferroelectric surface. In other words, the sign of ΔV in Eq. (12) will determine whether the EFM phase is 0° or 180°. One can clearly see that if the EFM is the main contributing contrast mechanism of PFM in Fig. 30, then the PFM phase should change from 180° to 0° when ΔV changes its polarity from positive to negative [see the red line in Figs. 30(a) and 30(b)]. However, the PFM phase changed from 180° to 0° at the dc voltage of −4 V, indicative of the other factors determining the phase rather than the electrostatic force between the tip and the sample surface. In this case, the most probable factor is the piezoresponse. This could also be confirmed by the shape of the piezoresponse hysteresis loop obtained separately and shown in the inset of Fig. 30(c).

One may argue that the charge injection may also have a threshold voltage for both polarities. Therefore, showing the prototypical ferroelectric like hysteresis loop may not be sufficient. In such a case, there is another method to discriminate the piezoresponse from the electrostatic effect from the surface charges. By changing the amplitude and duration of voltage pulse applied to the tip or the bottom electrode, we can form a matrix of dot array in both the PFM phase and the resistive probe images.^84,85

Figure 31 shows an example of such dot arrays. Voltage pulses with different amplitudes and durations were applied to the bottom electrode. The diameter of each dot is a function of the amplitude and duration, a quantitative relationship of which can be found in Ref. 78. The white triangle region is where either the amplitude or duration is not sufficient to switch the polarization all the way through the film thickness. Still, one can see the bright contrast in most of the dots within the same triangle in Fig. 31(b). This indicates that the charge injection onto the ferroelectric thin film surface is occurring for most of the amplitudes and durations of the voltage pulses, whereas there is a certain range of the amplitude and duration for which the polarization domain can switch through the film thickness. If the PFM phase were dominated by the EFM effect, then we would see no difference between Figs. 31(a) and 31(b).

So, up to this point, we learned how to separate out each factor listed in the first interacting component, the AFM tip, in Table I. We learned to use PFM¹ and KPFM,⁸³ or EFM⁷⁹ or scanning resistive probe microscopy (SRPM)⁸⁴ to discriminate the piezoresponse from the electrostatic effect. Excess charge can create an additional electrostatic effect, which can be visualized by the contact mode AFM topography under different environments.

Next, we will focus on the second interacting component, the cantilever, in Table I. As depicted in Fig. 11, the cantilever and bottom electrode can be viewed as a parallel plate capacitor where the voltage applied to the tip will also be applied to the cantilever.⁶⁰ As such, there will be an attractive electrostatic force acting on the cantilever, no matter what polarity of voltage we apply to the tip. However, because of the contact potential difference between the cantilever and the bottom electrode as well as the additional excess charges on the surface or in the sample or at the interface between the sample and the bottom electrode, the attractive electrostatic force may be asymmetric depending on the polarity of the applied voltage even with the same magnitude. This will lead to a first harmonic signal to the cantilever vibration, contributing to the deflection signal at the reference frequency of the lock-in amplifier.

Figure 32 shows the force–distance curve obtained on a TGS single crystal surface.⁷⁹ Overlaying the graph is the PFM/EFM amplitude as a function of z_actuator. When the Z actuator is at z = 0.4 μm in Fig. 32, the deflection signal is zero and the EFM amplitude is at around 6.24 a.u. This situation is similar to the green dot in Fig. 6(b). The gap is at least 150 nm judging from the point of contact in Fig. 32 [see the yellow dot in Fig. 6(b)].

Hong et al. estimated the sustained vibration shown in Fig. 32 to be ca. 1 nm and assumed that this vibration is occurring above the surface.⁷⁹ One can see that the PFM/EFM amplitude after the contact point between the AFM tip and the sample surface is almost constant (1.38 a.u.) up until z_actuator = 0.126 μm from z_actuator = 0.282 μm. How is this possible? I will come up with an explanation in detail, but the short answer to this question is “via the cantilever buckling induced by the capacitive force.”

We know for sure that the vibration amplitude at the reference frequency of ω_r solely emanates from the electrostatic force between the tip–cantilever and the sample surface when the AFM tip is off contact in the z_actuator range between 0.236 μm and 0.4 μm. Also, from the calculation reported elsewhere,^86,87 there is a competition between the two different capacitive forces: the force between the tip and the sample surface vs the force between the cantilever and the sample surface. As the gap decreases, the former force will prevail over the latter force, and often the crossing point is somewhere between 2 nm and 10 nm, but it depends on the tip radius, tip height, and the cantilever area.⁸⁶ 

Once the tip lands on the sample surface, what happens? We know at least four factors play their roles in the PFM signal. Because of the contact electrification, triboelectric effect or charge injection, we can have both indentation and tapping of the AFM tip by the electric force.¹ Another important factor is the cantilever buckling due to the capacitive force.³¹ All three cases can lead to the sustained vibration of the tip with the amplitude of 1 nm. However, the piezoelectric strain cannot be so large. The piezoelectric coefficient, d₂₂, of the TGS single crystal is 20 pm/V, and the resulting piezoresponse will be about 0.01 nm, which is much smaller than the vibration amplitude of 1 nm measured in Fig. 32. As such, the piezoresponse seems to play a minor role in the PFM image acquired in Ref. 79. This is why the mode is called dynamic contact mode EFM.⁷⁹ 

However, it is not easy to understand how the AFM tip can jump above the surface without any excess charge on the surface. So, I will revisit their model with the assumption that the surface has excess charge due to charge injection/trapping or the triboelectric effect or flexoelectric effect. Then, I will also include all possibilities and analyze each possibility in depth.

As shown in Fig. 33, if there is no voltage applied to the tip, the forces that are being balanced are the atomic repulsive force (ΔF_repulsive) and the elastic restoring force (ΔF_lever) of the cantilever when in contact. Figure 33(a) schematically depicts this situation, where ΔF_repulsive = ΔF_lever. Δz_sample is the equilibrium indentation depth and r₀ is the equilibrium distance between the outermost atom of the tip and that of the sample surface.

Figure 33(b) illustrates the force balance from the perspective of the force between the outermost atom of the tip and that of the sample surface and from the viewpoint of the force acting on the cantilever due to its bending motion. The gray dot in Fig. 33(b) corresponds to the point where there is no restoring elastic force by the cantilever and the net force between the outermost atom of the tip and that of the sample surface is zero. As we move the tip closer to the outermost surface atom, both the repulsive atomic force and the elastic restoring force increase to the red point. Then, Δz'_tip will cause the cantilever to bend upward and exert the elastic restoring force, ΔF_lever, to the sample surface. This force will cause the tip to move toward the sample surface by Δz_sample, which, in turn, will increase the repulsive atomic force, ΔF_repulsive, to match the restoring elastic force of the cantilever, ΔF_lever. The equilibrium distance, r₀, will slightly decrease, as shown in Fig. 33(b).

Let us imagine that we apply ac voltage to the tip while the tip is in contact with the sample surface, as depicted in Fig. 33(a), which is the case of Fig. 32. Furthermore, we assume that there is no piezoelectric displacement and only an electrostatic force induced vibration of the tip and the cantilever. Then, there can be three different scenarios to account for the vibration amplitude measured in Fig. 32.

Figures 33(c)–33(e) depict such scenarios. Figure 33(c) shows the case where the attractive capacitive force induces indentation of the tip. Using the Hertzian model, the radius of the tip, and the elastic moduli of both the tip and the sample, one can calculate the amplitude of indentation when the ac electric force is applied to the tip. The tip radius is 40 nm. Assuming that SiO₂ is formed at the tip, Young's modulus, E_SiO2, is 74 GPa⁸⁸ and that of the TGS single crystal is 21 GPa.⁸⁹ 

How do we calculate the electric force acting on the tip due to the surface charge? We will use some basic knowledge of electrodynamics to solve this problem. One thing we can use is the maximum field from the fully screened ferroelectric surface shown in Fig. 29(f), where the maximum field is about 0.6 MV/m. This field scales with the polarization charge, which is 25 μC/cm² in Fig. 29 but is 2.7 μC/cm² for the TGS single crystal. Therefore, the maximum electric field from the TGS surface will be 64.8 kV/m. As we know the electric field acting on the tip, we can find the maximum effective charge accumulated on the tip due to the applied ac voltage of 3 V. We can use the method reported by Woo et al.⁷⁸ and calculate the charge using the following equation:

where V is the voltage applied to the tip (we will use the maximum of ac voltage applied to the tip, i.e., 3 V), q is the charge trapped in a conducting sphere (we assume that the AFM tip is a conducting sphere), ɛ_e is the effective permittivity of the medium (we will use the permittivity of the TGS single crystal, 100ɛ₀)⁹⁰ and r is the radius of the sphere (we will use the tip radius, 40 nm, here). You can calculate Q using Eq. (13), which is 1.33 fC.

If we use the equation of F = qE, then the force acting on the tip will be 1.33 fC × 64.8 kV/m = 0.08 nN. This force is orders of magnitude smaller than the loading force we use for the contact mode AFM, which is between 10 nN and 100 nN. Using the Hertzian contact model⁹¹ with

where F_electric is the electric force acting on the tip, E_SiO2 and E_TGS are the elastic moduli of SiO₂ and the TGS single crystal, respectively, ν is Poisson's ratio (we used 0.25), Δz_sample is the indentation depth, and r is the tip radius. Then, the calculated Δz_sample becomes 7 pm. This value is far less than 1 nm, so we conclude that it is not possible to have 1 nm of indentation with the electric force acting on the tip with 3 V of applied voltage if the surface is fully screened.

What if the surface is not fully screened? Let us assume the extreme case of an unscreened surface. In such a case, we need to add more charges to the tip (the image charge of surface polarization) and use the surface electric field of an unscreened polarization, which is shown in Fig. 29(e). In such a case, the electric force, F_electric, increases by two orders of magnitude and becomes 2 nN. So does the indentation depth, Δz_sample, which becomes 57 pm. Even when the surface is unscreened, the indentation induced by the electric force is far less than the vibration amplitude of 1 nm. Hence, we can conclude that this is not what was measured in Fig. 32 and discard the scenario of Fig. 33(c).

What about the scenario of Fig. 33(d)? We can use the same electric force we calculated for the scenario of Fig. 33(c). However, instead of using the Hertzian contact mechanics model, we will use Hooke's law because the cantilever will move upward, in the direction away from the sample rather than toward the sample. Using the formula of F = kΔz_sample, we can calculate Δz_sample using the fact that k is equal to 1.9 N/m. For the unscreened surface where F_electric is 2 nN, Δz_sample = 1.06 nm. This is exactly the same value as reported by Hong et al.⁷⁹ For the fully screened surface where F_electric is 0.08 nN, Δz_sample = 46 pm that is far less than 1 nm. Hence, Δz_sample can vary from 46 pm to 1.06 nm depending on the degree of surface screening. But remember that usually under ambient conditions, the surface is fully screened as reported by many groups.⁵⁰ 

Finally, what about the scenario of Fig. 33(e)? We need to first calculate the capacitive force acting on the cantilever. Let us first assume that the cantilever is a diving board type. Then, we use the equations for a parallel plate capacitor. The tip used in Fig. 32 is 3 μm high, and the sample is 400 μm thick. Therefore, we can think of parallel plate capacitors connected in series with the first layer being 3 μm thick and the second layer being 400 μm thick. Furthermore, the permittivity of the first layer is ɛ₀ while that of the second layer is 100ɛ₀. Using the principle of a virtual work, we can calculate the force on the cantilever using the following equation:

where C₁ is the capacitance of the air layer between the cantilever and the sample surface, d₁ is the distance between the cantilever and the sample surface, d₂ is the thickness of the TGS single crystal, ɛ₁ (=ɛ₀) is the permittivity of air, ɛ₂ (=100ɛ₀) is the permittivity of the TGS single crystal, and V is the amplitude of ac voltage applied to the tip (3 V).

In case the ferroelectric surface is unscreened, which is highly unlikely, we need to add additional electric force in addition to that of Eq. (15). Using the same approach as the one for the electric force acting on the tip, we will use the electric field from the unscreened surface, E_unscreened. This will be 1.08 MV/m. The charge quantity on the cantilever is obtained from Q = CV and is 27.3 fC. Therefore, the additional force, F_unscreened, is 29.49 nN. The total force acting on the cantilever will be 35.34 nN. Hence, depending on the degree of screening on the surface, the electric force acting on the cantilever can vary from 5.85 nN to 35.34 nN.

To understand how the cantilever will buckle when the tip is in contact, we will use the Euler–Bernoulli beam theory.⁹² For the cantilever far away from the surface, we use the equation of the cantilever beam with a uniformly distributed load, and for the cantilever where the tip is in contact with the surface, we use the equation of the cantilever beam with the uniformly distributed load and with the free end supported on a roller.

The measured displacement of the tip, Δz_tip, is proportional to the slope of the cantilever, dw/dx, where w is the displacement of the cantilever in the z direction and x is the axis along the undeformed cantilever long axis. Therefore, we need to understand that the slope measured when the tip is in contact will be different from that measured when the tip is off contact. One can intuitively see that the sign of the slope will change from negative to positive or vice versa and the magnitude will decrease. Therefore, the photodiode will interpret the bending motion of the cantilever toward the surface by the attractive electric field as the tip moves upward away from the surface. In addition, the same force acting on the cantilever will result in a smaller slope when the tip is in contact with the surface than when the tip is far away from the surface. Therefore, the absolute value of Δz_tip will be much smaller. We can interpret this as the effective stiffness of the cantilever being increased.⁴⁴ 

Let us assume that the cantilever length is L, and the laser beam is located on x = 0.75L, where x = 0 means the pivot of the cantilever.⁹³ Then, the displacement of the cantilever as a function of x, w(x), can be described by using Eq. (16),⁹² 

where E is the elastic modulus and I is the second moment of area of the beam's cross section, and q is the distributed load in terms of force per length. The slope of the beam is dw/dx, which can be described as follows:⁹² 

One can clearly see that both the displacement, w, and the slope of the beam, dw/dx, linearly scale with the distributed load, q. As such, the laser beam movement on the photodiode shown in Fig. 4, which is proportional to dw/dx, is linearly proportional to w. Therefore, once we measure the inverse optical lever sensitivity, we can convert the deflection signal into Δz_tip.

What happens when the tip is in contact and the electric force on the cantilever bends it toward the sample surface? The displacement of the cantilever as a function of x, w(x) in such a case can be described by using Eq. (18),⁹² 

The slope of the beam, dw/dx, can be described as follows:

If we compare the ratio between Eqs. (17) and (19) when x = 0.75L, it will indicate how much we should multiply the nominal stiffness to obtain the effective stiffness when the tip is in contact. The ratio becomes −14, indicating that the stiffness increases by 14 times and the displacement, Δz_tip, reverses its sign. Therefore, the effective stiffness, k_eff = 14 × 1.9 = 26.6 N/m. Using Hooke's law again, Δz_sample = Δz_tip = F/k_eff = 5.85 nN/26.6 N m⁻¹ = 0.2 nm for fully a screened surface and Δz_sample = 1.3 nm. The range of possible displacement from 0.2 nm to 1.3 nm is in good agreement with the measured vibration of 1 nm in Fig. 32. However, it is highly unlikely that the surface is unscreened, so the vibration by the electric force on the cantilever will probably be closer to 0.2 nm.

Figure 34 shows the model experiment that proved that the inset of Fig. 32 is dominated by the cantilever bending by the electric force and not by the sustained vibration of the tip induced by the electric force. The sample was a single crystal n-type Si wafer covered with native oxide. As such, there will be no detection of piezoelectric displacement. The experiment was conducted under two conditions: off contact, where the gap is 0.3 μm, and contact, where the gap is ∼0.3 nm. The scan size was set to 0 nm, so the image represents the time evolution of the AFM tip deflection and vibration signals at the same spot. An AC voltage of 1 V_pp at 17 kHz was applied to the tip, while sinusoidal shaped pulses with an amplitude of 20 V_pp and a period of 0.05 s (20 Hz) were applied to the sample at intervals of 0.2 s. The pulse was synchronized with each scan line. The static deflection and the first harmonic (ω_r) signal of the induced tip vibration were recorded simultaneously.

In the off-contact mode, the static deflection shows that the tip is moving toward the sample surface regardless of the polarity of the voltage applied to the bottom electrode. The asymmetry in the static deflection can be explained by the contact potential difference between the p-type Si tip and the n-type Si wafer.⁴⁴ The first harmonic amplitude and phase signals are the EFM amplitude and phase, respectively. We can use Fig. 26 to understand the results in Fig. 34(a). The EFM amplitude linearly scales with the electric charge on the sample, which is proportional to the voltage applied to the sample. As such, the EFM amplitude will follow the absolute value of the sinusoidal wave, which is clearly the case in the line profile of the EFM amplitude image. The asymmetry in the peak heights could be explained by the contact potential difference as well. The EFM phase is determined by the polarity of the surface charge on the sample. As such, the positive voltage that induces the positive charge on the sample will cause the EFM phase to be 0°, whereas the negative voltage will make the EFM phase 180°.

In the contact mode, the static deflection shows that the tip is moving away from the surface, which, at first glance, seems to prove that the tip is being repelled by the surface charge. In addition, the EFM amplitude is linearly proportional to the absolute value of the voltage applied to the sample. However, the EFM phase shows an inversed result when compared with the EFM phase of the off-contact mode.

To understand the results of Fig. 34, I prepared Fig. 35. Figures 35(a)–35(c) illustrate the three separate scenarios that can contribute to the static deflection and the EFM amplitude and phase signals. Figure 35(a) corresponds to the tip indentation scenario, where the capacitive force between the tip and the sample causes the tip to further indent the sample by the ac voltage. In such a case, the static deflection, EFM amplitude, and EFM phase will be similar to the off-contact case shown in Fig. 34(a). This is schematically depicted in Fig. 35(d). However, when we compare the experimental results shown in Fig. 35(g), it is clear that the static deflection and the EFM phase are different from the expected results from the tip indentation scenario.

Figure 35(b) shows the case where the tip is jumping above the surface to have a sustained vibration as explained by Hong et al.,⁷⁹ which corresponds to the tip tapping scenario. Figure 35(e) illustrates the expected results, which show good agreement with the static deflection and the EFM amplitude, but fail to match the EFM phase. When the positive charge comes to the sample surface, the positive charge supplied to the tip will feel a repulsive force and increase the gap between the tip and the sample. On the other hand, the negative charge supplied to the tip will feel an attractive force and reduce the gap between the tip and the sample. The average effect will result in half the gap of the positive charge on the tip case, so the average static deflection will be higher than the neutral case.

The EFM phase will be determined by the sequence of repulsive and attractive forces between the tip and the sample, as depicted in Fig. 26. As such, the EFM phase will be the same as that of Fig. 34(a). When the bottom voltage is positive, the phase will be 0°, and when it is negative, the phase will be 180°. However, the experimental result shows the opposite. When the bottom voltage is positive, the phase is 180°, and when it is negative, the phase is 0°.

This leaves us with the last scenario, which is depicted in Fig. 35(c). As discussed in Fig. 33(e), the cantilever bending induced by the capacitive force will result in the inversion of the slope of the beam and the increase of effective stiffness of the cantilever. The inversion of the slope will reverse the sign of Δz_sample, which means that the cantilever bending toward the sample will be interpreted as a protrusion of the sample. This will let the Z actuator move upward to compensate the bending, which will result in the bright contrast in the static deflection [see Fig. 35(f)]. Furthermore, as dictated by the same mechanism of the tip indentation but with a deflection of a reversed sign, the EFM phase will be inversed as well. The expected results from the cantilever buckling scenario match with all of the experimental results, as shown in Figs. 35(f) and 35(g).

Figure 36 shows the schematic diagram of the tip–cantilever system for both off-contact and contact conditions. The position sensitive photodiode (PSPD) is split into two segments and the laser beam bouncing scheme in this case is slightly different from that in Fig. 4. However, the basic operation principle is the same, where the slope of the cantilever determines the displacement of the laser beam hitting the PSPD. The PSPD signal generates the deflection signal, which is linearly proportional to the slope of the cantilever, dw/dx, which, in turn, is linearly proportional to the displacement, w, itself.

The inverse optical lever sensitivity in terms of the displacement per PSPD voltage (nm/V) is obtained based on this principle. In Fig. 36, the laser beam will move to the left side, which reduces the deflection signal, if the tip moves downward and the cantilever rotates around the pivot B in a counterclockwise direction. This is because the angle, θ_d, between the surface normal vectors of the sample (N_sample) and the cantilever (N_lever) increases when the tip moves downward. The opposite case can also be explained in a similar way when the tip moves upward. θ_d decreases and the laser beam moves to the right side, which increases the deflection signal.

For the off-contact case, the electric force between the tip–cantilever system and the sample surface makes the tip to move downward and the cantilever to rotate in the counterclockwise direction around the pivot B. As such, the static deflection will always be negative, indicating that the tip is moving toward the sample. However, for the contact case, due to the pivot C created between the tip and the sample surface, the electric force mainly bends the cantilever in a way that can be described by the cantilever beam with a uniformly distributed load and with the free end supported on a roller. We already learned from Eq. (19) that the slope of the cantilever at x = 0.75L reverses its sign. In other words, the attractive electric force acting on the cantilever will be recorded as a repulsive force acting on the tip, because the laser beam will move to the right side, which increases the deflection signal. This is exactly what happened in Fig. 34(c).

Now we are ready to quantitatively analyze the results of Fig. 34. Let us define some important parameters. The ac voltage applied to the tip at 17 kHz with an amplitude of 1 V_pp will be denoted as Eq. (20),

where V₁ = 0.5 V and ω₁ = ω_r = 2πf = 2π × 17 kHz. Here, ω₁ will be the reference frequency of the lock-in amplifier, which is ω_r. We will simplify the sinusoidal pulse as a constant dc voltage, V_dc.

From the discussion related to Fig. 35, we know that the most relevant scenario for the results of Fig. 34 is the cantilever buckling scenario. Therefore, we will ignore the other scenarios for the following analysis.

The capacitive force, F_c,lever, between the cantilever and the sample can be described as follows:⁴⁴ 

where C is the capacitance between the cantilever and the sample, V_c is the contact potential difference between the cantilever and the sample, and Γ is ½(∂C/∂z). If we rearrange Eq. (21) into a dc term, first harmonic term (ω₁), and second harmonic term (2ω₁), then we get Eq. (22),

Since the tip displacement, Δz_tip, is linearly proportional to F_c,lever, we can describe the static and the first harmonic components as follows:

where Δz_tip|_static and Δz_tip|_ω1 are the static tip displacement and the tip displacement at the frequency of ω₁. In the case of Fig. 34, the tip was a p-type Si (heavily doped, 10¹⁷–10¹⁹/cm³) and the wafer was n-type Si (medium doped, 10¹⁵–10¹⁶/cm³). As such, V_c is between –0.7 and –0.8 V.⁴⁴ 

We already know that the deflection signal measured by the photodiode, D, is linearly proportional to both the slope of the cantilever and the displacement of the cantilever. However, the specific dependence will change when the tip is in contact with the sample in case where only the electric force is being applied to the tip and the cantilever.

How can we calculate D for the off-contact case? We will simplify the situation and assume that the trajectory of the laser beam on the photodiode follows the arc of a circle with a radius of R, which is the distance between the cantilever and the photodiode. The conversion ratio between the beam displacement on the photodiode and the deflection signal D will be α,

where x is the beam position on the cantilever (usually, x = 0.75L, where L is the length of the cantilever) and Δθ_D is the angular change of the surface normal vector of the cantilever in Fig. 36. Therefore, if we combine Eqs. (25) and (26), then we can express D in terms of Δz_tip,

where S_lever is the optical lever sensitivity (OLS). It usually varies from 5 to 20 mV/nm. From Eq. (27), we can see why the laser beam bouncing scheme is so effective in amplifying the tip displacement, Δz_tip. For example, if R = 20 mm and x = 0.75 × 180 μm = 135 μm, then R/x = 148. This means that the displacement will be 148 times amplified when the laser beam reaches the photodiode. Therefore, if the position sensitive photodiode has a spatial resolution of 1 nm, then the z resolution of our AFM will be 7 pm. This is how AFM can visualize the atomic structure as well as the piezoelectric displacement and electric field induced displacement on the sample surface.

By combining Eqs. (23), (24), and (27), we obtain the following equations:

According to Eq. (28), D_static will always be negative since Γ is negative and all the other terms are positive, no matter what values V₁, V_dc, and V_c have. This is exactly what we see in Fig. 34(a) for the off-contact case.

Usually, researchers use Eq. (24) as the EFM signal. Therefore, the EFM amplitude and phase can be derived from Eq. (24) rather than from Eq. (29). However, it should be noted that D_ω1 is the signal that is being measured during the experiment,

We see that the line profile of the EFM amplitude in Fig. 34(a) has a shape similar to |sin(ω₂t)| where ω₂ is the angular frequency (2π × 20 Hz) of the sinusoidal voltage pulse applied to the sample. The asymmetry comes from the fact that V_c is non-zero in Eq. (30). Replacing V_dc in Eq. (30) by V_dcsin(ω₂t) will result in the curve shown in Fig. 34(a). What about the EFM phase? As can be seen in Fig. 34(a), the phase is 0° when V_dc > V_c (= −0.7 V) and 0° when V_dc < V_c. This is in good agreement with Eq. (31).

Then, how can we calculate D for the contact case? We will simplify the situation again and assume that the trajectory of the laser beam on the photodiode follows the arc of a circle with a radius of R, which is the distance between the cantilever and the photodiode. The conversion ratio between the beam displacement on the photodiode and the deflection signal D will be α. Therefore, we can use the same equation here as Eq. (25). However, we need to modify Eq. (26) because the way the cantilever rotates about the pivot changes when the tip is in contact. It rotates around pivot C rather than pivot B in Fig. 36. If we assume that the tip height (3 μm) is much smaller than the cantilever length (180 μm), then we can modify Eq. (26) as follows:

Combining Eqs. (25) and (32) yields Eq. (33),

If we use x = 0.75L, then D = 3S_leverΔz_tip. For the same Δz_tip, the deflection signal will be three times larger than that of the off-contact case. Here, we need to understand that the effective tip displacement, Δz_tip,eff, is simply the deflection signal divided by S_lever. If we want to know the real displacement of the cantilever when the buckling occurs, then we will have to divide Δz_tip,eff by [x/(L−x)], which is 3 in the case of x = 0.75L.

When x = 0.75L, we obtained the ratio, r, of the slope of the beam under the same distributed load between the contact and the off-contact cases, which is −14. This ratio can be used to calculate the effective tip displacement, Δz_tip,eff, under the same capacitive force acting on the cantilever. By doing so, we can understand how the cantilever buckling translates into the equivalent tip movement in the deflection signal. The ratio, r, in general can be expressed by way of Eq. (34),

As discussed above, r = −14 when x = 0.75L. Combining Eqs. (27), (33), and (34), we obtain the following equation when x = 0.75L:

Therefore, for the same capacitive force, the magnitude of the deflection signal will be reduced by 14 times and the sign of the deflection signal will be reversed. Then, the static deflection can be described by using Eq. (36),

The static deflection shown in Fig. 34(c) shows an opposite trend (repulsive vs attractive) to that of Fig. 34(a) and a 15 times smaller magnitude (0.8 nm vs 12 nm). This result can be explained by using Eq. (36). However, the capacitive force acting on the cantilever will be 4 times stronger for the contact case than that for the off-contact case. As such, considering this factor [in other words, Γ in Eq. (36) is 4 times larger than Γ in Eq. (28)], the expected deflection signal in contact mode is 3.5 times smaller than that in the off-contact case.

The EFM amplitude and phase for the contact case can be derived in a similar way as follows:

From Eqs. (37) and (38), one can see that the EFM amplitude will follow the same trend as in Eq. (30) but with a much smaller magnitude, and the EFM phase will be reversed from Eq. (31). This is exactly the case with Fig. 34(c).

Figure 37 shows the EFM amplitude and phase as a function of V_dc for the off-contact and contact cases. As can be seen from Figs. 37(a) and 37(b), the amplitude shows a “V” shape curve, which could be predicted from Eqs. (30) and (37). By locating the minimum point in Figs. 37(a) and 37(b), one can find the contact potential difference between the AFM tip and the sample. The measured V_c's from the plot show good agreement with the expected values of −0.7 V. The discrepancy of 0.2 V in the contact case may arise from the excess surface charge contribution to the electric force acting on the cantilever, which becomes significant as the gap decreases.⁶⁰ 

The notable difference between the off-contact and the contact cases can be found from the plots of the EFM phase as a function of V_dc, as shown in Figs. 37(c) and 37(d). As we increase V_dc, the EFM phase changes from 180° to 0° for the off-contact case, while it changes from 0° to 180° for the contact case. This can be well explained by using Eqs. (31) and (38), which attests to the fact that the cantilever buckling induced by the capacitive force plays the main role in the EFM amplitude and phase signals in Fig. 37.⁴⁴ 

With the presence of the cantilever buckling, the PFM amplitude and phase signals will be modified as follows when x = 0.75L,

If we replace −Γ/(7k_lever) with β, then Eq. (39) becomes Eq. (40). For k_lever = 0.26 N/m and cantilever length and width of 180 μm and 50 μm, respectively, β becomes 2.43 × 10⁻⁹ m/V²,

Figure 38 shows the result of the EFM/PFM amplitude and phase signals as a function of V_dc applied to the Pt electrode. The sample was a 250 nm thick Pb(Zr_0.4TiO_0.6)O₃ ferroelectric thin film deposited on a Pt/SiO₂/Si substrate.⁴⁴ 

For the off-contact case [see Figs. 38(a) and 38(d)], the EFM amplitude shows a “V” shape and the EFM phase changes from 180° to 360° (= 0°) when V_dc > V_c, which is expected from Eq. (31). From the work function of the Pt bottom electrode (5.3–5.6 eV) and p-type Si tip (4.9–5.0 eV), we found that V_c = 0.3–0.7 V. The measured V_c of 0.45 V shows a fair agreement with the expected V_c.⁴⁴ 

Hong et al.⁴⁴ measured the EFM/PFM amplitude and phase signals as a function of V_dc on a negatively polarized domain [Figs. 38(b) and 38(e)] and a positively polarized domain [Figs. 38(c) and 38(f)]. As the tip contacts the sample surface, V_c increases from 0.45 V to 0.61∼0.66 V. This increase upon contact is similar to the increase of V_c from −0.7 V to −0.5 V upon contact, as observed in Fig. 37. This increase should be related to the electric force acting on the tip, the details of which are discussed in Ref. 60.

An interesting comparison is between the up (positive) and the down (negative) domains. From Eq. (40), we can predict that the apparent V_c of a positive domain will be smaller than that of a negative domain by 2d₃₃/β. As β = 2.43 × 10⁻⁹ m/V² in Fig. 38 and ΔV_c = 0.05 V (between down and up domains), d₃₃ is calculated to be 60.78 pm/V. This value is similar to 58 pm/V measured by PFM on a sandwich structure of Pt/Pb(Zr_0.45Ti_0.55)O₃/Pt.⁹⁴ 

Based on the knowledge obtained from Fig. 38, we can predict the PFM domain image of the up and down domains when the cantilever buckling by the capacitive force is playing an important role, as indicated in Eq. (40). Figure 39 shows the schematic diagram of the graphs shown in Figs. 38(b), 38(c), 38(e), and 38(f) along with the PFM amplitude and phase images of down polarized circular domain arrays in a negatively polarized matrix.

As the PFM image is collecting the PFM amplitude and phase at V_dc = 0, we need to focus on the y intercepts of the PFM amplitude and phase of the up and down domains. As shown in Fig. 39(a), the down domain has a slightly higher amplitude than the up domain. Therefore, the down polarized circular domains in Fig. 39(a) show a brighter contrast than the surrounding up domain matrix.

The PFM phase for both up and down domains is 0°, so the phase of both the down polarized circular domains and the surrounding up domain matrix will be 0°. This is exactly the case in Fig. 39(d). What happens if we only have the piezoelectric displacement and no cantilever buckling induced by the capacitive force? Then, we will see the PFM amplitude and phase images like in Fig. 25.

Even with the presence of the cantilever buckling induced by the capacitive force, there is a way to extract the piezoelectric coefficient as well as the magnitude and direction of polarization at the nanoscale using Eqs. (3), (4), and (40). However, it would be more convenient if we can remove such an effect from the beginning of the measurement or imaging experiment.

Hong et al.^44,95 suggested three ways to achieve the goal of reducing the cantilever buckling induced by the capacitive force and maximizing the piezoelectric strain effect on the PFM experiment, as shown in Fig. 40.

Figure 40(a) shows the high aspect ratio tip with the high cantilever stiffness method, which will reduce Γ and increase k_lever in Eq. (39). As the coefficient of Γ/k_lever approaches zero, Eq. (39) can be rewritten as Eq. (41),

Figure 40(b) illustrates the large top electrode method, where the AFM tip is in contact with the top conducting electrode layer. In such a case, the AFM tip and the top electrode are short-circuited so that the electric potential difference becomes zero (in some cases where the tip radius too small, one needs to connect the cantilever and the top electrode through external wiring or additional micro-needle contact). Therefore, the cantilever will be electrically shielded and have no potential difference, V in Eq. (21). This will lead to zero F_c,lever. Therefore, the piezoresponse signal defined as $Aω1cosΦω1$ becomes the same as that in Eq. (41).

Figure 40(c) shows the measurement at the edge of the sample method, where Γ approaches zero. Therefore, the piezoresponse signal approaches Eq. (41).

Figure 41 shows the PFM images obtained on Pb(Zr,Ti)O₃ thin films using a low (tip height: 3 μm, UL20B Boron doped tip, k_lever = 0.24 N/m, tip radius = 20 nm) and a high aspect ratio (tip height: 18 μm, US90B Pt coated tip, k_lever = 6 N/m, tip radius = 35 nm) tips.

Figures 41(a) and 41(c) are the same images as Figs. 39(c) and 39(d). The expected line profile across the circularly polarized down domain in the up-domain matrix is shown in Fig. 41(e), which can be easily understood with the help of Figs. 39(a) and 39(b).

Figures 41(b) and 41(d) are similar to Figs. 25(b) and 25(c). With only the piezoelectric strain presence in the PFM image, the expected line profile across the circularly polarized down domain in the up-domain matrix can be understood as well as shown in Fig. 41(e), which is exactly the same as in Figs. 25(e) and 25(f).

Figure 42 shows the PFM images obtained on the sandwich structure of Pt/Pb(Zr,Ti)O₃/Pt where the Pt top electrode was short-circuited with the AFM tip and connected to the ground.⁹⁴ As such, the ac voltage was applied to the bottom electrode. This is why the PFM phase in Fig. 42 should be interpreted with care when compared with the other PFM images. The down domain shows 180° or −180° instead of 0°, as shown in Fig. 42(a), whereas the up domain shows 0° instead of 180° or −180°.

It is clearly seen in Figs. 42(a) and 42(b) that the PFM amplitude is almost the same for the down and up domains, which is expected from Eq. (41). In addition, the PFM phase shows a clear change from −180° to 0° when the polarization is switched from the down to up direction.

Figure 42(c) shows the evolution of polarization domains as a function of dc voltage pulse applied to the bottom electrode. As the magnitude of the pulse increases, more domains switched from the down to up domains as evidenced by the conversion of black regions to white regions in Fig. 42(c). The domain boundary between the up and down domains shows the minimum in the PFM amplitude as evidenced by the dark contrast in the PFM amplitude images in Fig. 42(c), 2), and 3).

Figure 43 shows the PFM images obtained from the center and the edge of Pb(Zr,Ti)O₃ thin films using a low aspect ratio tip with low cantilever stiffness (UL06B).⁹⁵ Figures 43(a) and 43(c) show similar features to Figs. 41(a) and 41(c). However, by just moving the tip near the edge of the sample, the PFM amplitude and phase images become similar to those obtained by a high aspect ratio tip with a high stiffness cantilever [see Figs. 43(b) and 43(d) and compare them with Figs. 41(b) and 41(d)].

This method would be suitable for conducting PFM experiments on soft materials, which may undergo local phase transition or suffer from the mechanical annealing effect if one uses cantilevers with high stiffnesses.^96,97

PFM images are used to study the spatial distribution of up and down polarization domains. However, PFM images themselves may not be sufficient to prove ferroelectricity in the sample of interest. In order to show the switchability of electric polarization by an external bias field, researchers conduct piezoresponse hysteresis loop measurements, which are analogous to polarization-electric field hysteresis loop measurements. There are two main methods to measure the loop. One is the continuous dc mode (on mode), and the other is the pulse dc mode (off mode).⁴⁴ In the continuous dc mode, the piezoresponse is being measured, while a stepwise increasing V_dc is applied to the bottom electrode. In the pulse dc mode, the piezoresponse is being measured right after a stepwise increasing dc voltage pulse is applied to the bottom electrode. Therefore, the piezoresponse in the pulse dc mode is always being measured while V_dc is equal to zero.

However, due to the presence of artifacts such as the electric field induced tip vibration and cantilever buckling, and the charging effect, one need to identify these artifacts from the measurement result.¹ 

Let us start with the presence of cantilever buckling induced by the capacitive force. In such a case, Eq. (39) will dictate the piezoresponse. As such, the expected piezoresponse hysteresis loops in the continuous dc and pulse dc modes will be different, as shown in Figs. 44(a) and 44(b). In the continuous dc mode, the cantilever component in the piezoresponse can be described as Eq. (42). In Fig. 44, V_dc was applied to the AFM tip instead of the bottom electrode, so the sign of V_dc in Eq. (39) should be reversed. In such a case, Eq. (42) will change to Eq. (43),

Since β is positive, Eq. (43) represents a line with a positive slope, as shown by a green line in Fig. 44(a). The x intercept corresponds to −V_c, while the difference in the y intercept corresponds to 2d₃₃V₁.

In the pulse dc mode, Eq. (43) changes to Eq. (44) because V_dc is always equal to zero when the piezoresponse is being measured,

As such, the contribution of the cantilever buckling will be constant for all V_dc. Since β, V_c, and V₁ are all positive, the contribution will be a positive vertical offset of the piezoresponse hysteresis loop, as shown in Fig. 44(b). Again, the difference in the y intercept corresponds to 2d₃₃V₁.

Figures 44(c) and 44(d) show the experimental piezoresponse hysteresis loops measured while both ac and dc voltages were applied to the AFM tip in the continuous dc and pulse dc modes, respectively. The overall shape of the loops is very similar to the expected one, which attests to the validity of Eqs. (43) and (44).

Using a high aspect ratio tip with a high stiffness cantilever, one can obtain a well-defined piezoresponse hysteresis loop.^98,99 Similar loops can be obtained by placing the AFM tip near the edge of the sample.^60,95 Some of the loops acquired using such tips or placing the tip near the edge of the sample are shown in Fig. 45.

Figure 45(a) shows the piezoresponse hysteresis loop measured on a 436 nm thick Pb(Zr_0.4Ti_0.6)O₃ thin film composed of ⟨111⟩ and ⟨100⟩ oriented crystals. The AFM tip was positioned near the edge of the sample. Neither vertical offset as in Fig. 44(d) nor slanting of the curve as in Fig. 44(c) is observed in the loop, which is indicative of the absence of the cantilever buckling effect due to the capacitive force.

Figure 45(b) shows the piezoresponse hysteresis loops measured on 20 nm thick PbTiO₃ thin films grown on a Pt electrode by single-step and multi-step growth methods. A high aspect ratio Pt coated tip with high cantilever stiffness was used for the measurement. Figure 45(c) shows the piezoresponse hysteresis loops measured on square and circle shaped BiFeO₃ islands. Well-defined loops without either vertical offset or significant slanting are obtained. Figure 45(d) illustrates the piezoresponse hysteresis loops measured on a PbTiO₃ hollow nanotube grown on TiN nanopillars. Again, by using a high aspect ratio tip with high cantilever stiffness, the authors could acquire a well-defined piezoresponse hysteresis loop without an artifact.

By placing a large area top electrode between the AFM tip and the sample surface, one can obtain a well-defined piezoresponse hysteresis loop.^66,76 Figure 46 shows the example of measuring the piezoresponse hysteresis loop on the Pt top electrode using PFM and a double-beam interferometer.⁶⁰ As can be seen by comparing Figs. 46(a) and 46(b), the coercive voltages measured by both methods show a very good agreement. Furthermore, the piezoresponse can be calibrated by the value obtained from the interferometer. The sample used in Fig. 46 is the same as that used in Fig. 38. d₃₃ obtained from the difference of V_c was 60.78 pm/V. This value is very close to that measured by the interferometer, which is 70 pm/V.⁶⁰ 

What are the questions we can answer using PFM images and piezoresponse hysteresis loops? The answers to these questions will determine their applications.

One of the questions that have been answered by PFM and the piezoresponse hysteresis loop is whether the material of interest is ferroelectric.^103–106 By showing the polarization domain in the PFM image and measuring the piezoresponse hysteresis loop, one can prove ferroelectricity, which is defined by the presence of spontaneous polarization and its switchability from one state to the other by an external electric field. As PFM and the piezoresponse hysteresis loop are relatively insensitive to the leakage current component in the film, many researchers choose PFM and the piezoresponse hysteresis loop as the main tools to validate ferroelectricity. However, there are other artifacts (e.g., electrochemical strain) than the electrical force-induced ones in PFM, so care should be taken when interpreting PFM and the piezoresponse hysteresis loop as a way to prove ferroelectricity.

Another popular question is, “What is the minimum domain size?”^107–114 Ferroelectricity is a collective phenomenon of soft phonon vibrations and depends on an internal electric field set up by neighboring unit cells.¹¹⁵ As such, there exists a critical thickness under which ferroelectricity vanishes and the equilibrium distance between the up and down polarization domains that keep them stable and sustainable.^{112,116–118} Researchers in the field of information storage devices are interested in using the ferroelectric domain as a unit for information.^{2,20,119–121} As such, the minimum sustainable domain size will determine the physical limit of information storage devices using the ferroelectric domain as a bit.¹⁰⁷ Therefore, the search for smaller domain sizes in new materials is a quest worth the investment, which makes PFM and the piezoresponse hysteresis loop measurement important tools for them.

The nucleation and the growth theory have been very popular in the field of phase transformation.¹²² The commonly accepted approach to modeling the polarization switching kinetics in ferroelectrics uses the Kolmogorov–Avrami–Ishibashi model developed by the group of Ishibashi¹²³ based on the classical nucleation and growth theory developed by Kolmogorov¹²⁴ and Avrami.¹²⁵ Polarization switching can be conducted by chemical pressure such as oxygen partial pressure, in which case the polarization can be switched without nucleation of the opposite domain nuclei.⁵⁰ 

In the case of nucleation and growth mediated polarization switching, the rate-determining step should be well identified if one is interested in designing the shape of ferroelectric capacitors.^76,94 The rate-determining step will govern the overall switching speed, which is an important performance specification for memory devices. For example, if the forward domain growth along the thickness direction is the rate-determining step, then ferroelectric capacitors with small thicknesses and large areas will guarantee both a high signal-to-noise ratio and a high switching speed. If the sideways domain growth in the direction perpendicular to the thickness is the rate-determining step, then ferroelectric capacitors in the form of nanowires or nanotubes would be the preferred choice for high-speed switching devices.¹ 

The reduction of switchable polarization and back-switching of polarization is important for achieving a reliable performance of ferroelectric capacitors.^{60,66,74,75,126–129} As such, identifying the underlying mechanism can result in a breakthrough of nonvolatile memories based on ferroelectric capacitors. PFM and the piezoresponse hysteresis loop, in conjunction with the polarization-electric field hysteresis loop, play an important role in understanding the pinning of domains at various interfaces such as the grain boundary, electrode/ferroelectric interface, and domain wall.^130,131

One of the interesting questions in materials science and engineering is, “How does the property scale with the grain size?” The Hall–Petch equation predicts the inverse square root dependence of the yield strength on the grain size.¹³² As such, materials with finer grain sizes tend to have a higher yield strength. Then, one may ask, “How does the grain size affect the polarization magnitude?”¹³³ This could also extend to the question of “How does the size of isolated ferroelectric islands affect polarization?”^134–136 

The more specific question of “How does the grain size affect the size distribution of circularly polarized domains formed by the voltage pulse applied to the AFM tip?” is important for those designing ferroelectric hard disk drives because the size distribution of written bits will determine the error rates of information storage devices.^110,133

Enhanced ferroelectricity at the domain boundary¹³⁷ or grain boundary¹³⁸ could be an interesting topic as well. Hafnia-based ferroelectricity depends on the local strain state of a film.^104,139 The grain boundary between two different grains might induce enhanced ferroelectricity due to stress concentration along the grain boundary. A comparison between the topography and the PFM image can answer the question of enhanced polarization at the grain boundary.

Nanopillars, nanotubes, and nanowires are difficult to characterize using conventional methods such as polarization–electric field (P–E) hysteresis loop measurement.^140–142 For such complex material structures, PFM and piezoresponse hysteresis loop measurement are the tools of choice for characterizing the ferroelectric and piezoelectric properties.^143,144

A leaky ferroelectric sample will add an ellipsoid type of component in the P-E hysteresis loop measurement.⁴⁵ As the current flow through the AFM tip is limited by the contact resistance, and the Joule-heating by the current flow results in a negligible thermal expansion, PFM and piezoresponse hysteresis loops are relatively insensitive to leakage current. As such, PFM and piezoresponse hysteresis loop measurements are often used to check the ferroelectricity of such samples and characterize the ferroelectric properties.

Single-frequency vertical PFM is the oldest method among many PFM methods and is still being used among researchers. There are more advanced PFM modes, such as band excitation (BE),³⁵ dual ac resonance tracking (DART),³⁶ vector PFM,¹⁴⁵ angle-resolved PFM,¹⁴⁶ switching spectroscopy (SS) PFM,¹⁴⁷ tomographic PFM,^148,149 etc. More interested readers are encouraged to read the related references.

This work was supported by the KAIST funded Global Singularity Research Program for 2020, the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIT) (No. 2020R1A2C201207811), the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (No. NRF-2019S1A5C2A03081332), the Global Frontier Hybrid Interface Materials (GFHIM) of the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2013M3A6B1078872), Samsung Electronics Semiconductor Research Project funded by Samsung Electronics, the Samsung Electronics' University R&D program (Developing Three-dimensional Measurement Technology based on Analysis of Semiconductor Material Properties), and the Technology Innovation Program (No. 20012389, Developing Supercritical Haptic Materials with High Piezoelectric Coefficient and Elastic Strain using Artificial Intelligence) funded by the Ministry of Trade, Industry & Energy (MOTIE, Korea).

The data that support the findings of this study are available from the corresponding author upon reasonable request.