4.3 Atomic force microscopy
4.3.3 Piezoresponse force microscopy
The first experiments using atomic force microscopy in order to image ferroelectric domains were performed in dynamic mode with the tip oscillating 1-100 nm away from the surface [119–121] and where the oscillating tip experiences a force gradient. This gradient affects the effective spring constant of the cantilever. The change in sign of the charges at the surface of the ferroelectric material as the tip scans across a domain wall leads to changes in the force gradient, which in turn affects the cantilever resonance frequency.
These changes can be used to map the position of the underlying ferroelectric domains. Measurements were also performed in contact mode with electrically insulating Si3N4 tips, where a contrast in the lateral deflection was detected and later attributed to the tip acquiring an electric polarisation from the ferroelectric material, which, it was assumed, relaxes slower than the scanning time [120–122]. This tip polarisation was though to cause torques on the cantilever due to the electric field created by either the bound charge at the surface or the charge of the layer screening this bound charge.
Other works  used metallic tips scanning in contact mode and a DC bias applied to the tip to cause surface deformation of the ferroelectric material due to the converse piezoelectric effect. The material undergoes a mechanical deformation in response to the application of an electric field through the relationSi=djiEj, whereS,dandE are the strain and piezo-electric tensors and the piezo-electric field respectively. These deformations were visible in the topography signal and subsequent scans using insulating tips were used to separate topographical from ferroelectric domain contributions.
At around the same time, other studies [124–126] were performed using what is now called single-frequency piezoresponse force microscopy (PFM).
In this technique, an oscillating electric field at frequencyω is applied to the material by a metallic tip in contact with the sample. The converse piezoelectric effect causes a periodic deformation of the material under the tip,
which in turns leads to a periodic bending of the cantilever, also at frequency ω. A lock-in amplifier is used to demodulate the vertical deflection amplitude and phase at ω, where the amplitude signal is proportional to the local polarisation magnitude. Polarisation vectors oriented anti-parallel to each other yield a 180◦phase shift with respect to each other upon application of the AC bias, as illustrated in figure4.10, showing the polarisation-dependent deformation of a ferroelectric material upon application of a down-oriented electric field. Therefore, the phase signal from the lock-in amplifier gives information on the polarisation orientation of the material under the tip.
Figure 4.10:Snapshot of the expected behaviour of a piezoelectric or ferroelectric material under a positive tip bias (downward-oriented electric field). Upon application of an AC field, up and down polarised materials show a 180◦shift in the phase of the cantilever oscillation with respect to each other.
When scanning across a ferroelectric domain wall in PFM, a 180◦contrast is therefore expected in the PFM phase. A drop in the PFM amplitude is expected at the domain wall position. This is due to the cancelling contributions to the PFM amplitude at the domain wall itself, the typically 10-15 nm resolution of PFM being larger than the domain wall width, and to the null vertical polarisation at the domain wall.
Typically, the cantilever is driven at frequencies of∼1−100 kHz and the corresponding oscillation amplitudes are close to the AFM noise levels.
To increase the signal to noise ratio, the drive frequency can be set close to the contact resonance frequency∼300−500 kHz of the tip-sample system.
The drawback of this technique is that the contact resonance frequency can vary across a sample, for example when scanning over topographical features, which can potentially lead to significant cross-talk between the topography and PFM phase and amplitude signals. One way to solve this problem is to track the resonance peak, as is done in dual frequency resonance tracking (DFRT) PFM.
Dual frequency resonance tracking
In this technique, the resonance peak is tracked by applying a bias through the tip-sample system at two frequencies simultaneously, typically 5-15 kHz
4.3 Atomic force microscopy
Figure 4.11: Left: Expected PFM phase and amplitude for a line scan crossing through an-up oriented domain in a down-oriented area. Right: PFM phase and amplitude image of a stripe shaped ferroelectric domain.
apart and centred around the contact resonance frequency. Two lock-in amplifiers track the cantilever oscillation amplitude and phase at each of the two drive frequencies and a feedback loop dynamically controls the central frequency by using the difference in amplitudesA2−A1 as an error signal.
Figure 4.12:Operating principle of DFRT PFM. (a) Two carrier frequencies are used to drive the tip-sample system and two lock-in amplifiers track the amplitudes of the cantilever oscillations at the two frequencies. (b) A feedback loop shifts the central frequency in order to keep the oscillation amplitude difference at 0. Image from 
Driving the sample at resonance frequencies results in a significant im-provement of the signal to noise ratio as can be seen in figure 4.13. This allows detection and imaging of finer features without introducing crosstalk with the topography. DFRT has become a standard for PFM imaging and several manufacturers now offer built-in DFRT implementations in their systems.
Further improvements have been made, extending the idea of increasing the number of simultaneously applied frequencies to applying a whole fre-quency range. In band-excitation  for example, the tip bias is applied as a band of frequencies around the resonance peak, allowing full tracking of the resonance, determination of the Q-factor, giving information on mechanical properties of the sample.
Figure 4.13:Line scans of a domain in single frequency (left) and DFRT (right).
The PFM phases and amplitudes have a much higher signal to noise ration in DFRT.
Switching spectroscopy PFM
Measurements of ferroelectric hysteresis loops can be brought down to the nanoscale with switching spectroscopy PFM . In this technique, the AFM tip is held stationary on the sample surface, while a bias waveform shown in figure 4.14is applied.
The waveform consists of a superposition of a DC component used to locally affect the polarisation with an AC component used simultaneously to measure the PFM phase and amplitude. The DC waveform has a triangular envelope but is split in HIGH and LOW sections illustrated in figure4.14in order to separate electrostatic and ferroelectric contributions to the acquired signals. The PFM phase in the LOW sections is used to reconstruct the ferroelectric hysteresis, while the amplitude shows a characteristic butterfly shape where the minima positions correspond to the coercive bias as shown in figure4.15. These measurements are usually performed on a grid in order to extract ferroelectric hysteretic behaviour with spatial resolution [113,130], down to single defects .
4.3 Atomic force microscopy
Figure 4.14: Illustration of the bias waveform applied in SSPFM. A "reading" AC bias used to acquire PFM phase and amplitude signals is superimposed with a DC bias with a triangular envelope. The PFM phase in the LOW steps allows the ferroelectric hysteresis loop to be constructed, while the amplitude typically shows a characteristic butterfly shape. An example of acquired signals is shown in figure4.15. From .
Figure 4.15:Example of an SSPFM measurement performed on one point, showing two hysteresis loops in the phase signal on top and butterfly loops in the amplitude at the bottom.
Ferroelectric domain lithography
The ability to use the AFM tip to locally switch the polarisation allows lithography of ferroelectric domains at the nanoscale by applying electric fields higher than the coercive field of the studied material. During this process, the tip can be moved in any desired pattern, allowing domains of various sizes and shapes to be written. This aspect has been used extensively to study the switching dynamics of ferroelectric domains [132,133], measure ferroelectric hysteresis loops at the nanoscale , as well as study funda-mental aspects of disordered elastic systems and ferroelectric domains [25,
Figure 4.16:DFRT PFM phase and amplitude of an artificially written ferroelectric domain of complex shape1.
It is worthy to note that the sharpness of AFM tips lead to locally high electric fields which can often lead to injection or redistribution of defects and damage to the film [134,135] as illustrated in figure4.17. While this is often an unwanted effect, it can in some cases be used to locally alter the defect landscape in a material.
Figure 4.17:Effects induced by voltages applied by AFM tips can range from simple polarisation switching in ferroelectrics to charging, defect ordering and reinjection to film damage. These effects are usually not well separated in tip bias and often a mixture of multiple reversible and/or irreversible effects can occur simultaneously.
Lateral and Vector PFM
Imaging of ferroelectric materials using PFM is not limited to domains where the polarisation points out of the sample plane. Because ferroelectric
1Simon’s cat: https://shop.simonscat.com/products/simons-cat-feed-me-car-sticker
4.3 Atomic force microscopy
materials can have non-zero shear piezoelectric tensor components, if domains in the sample plane are present, the tip field can also cause torsion of the cantilever through the shear motion of the material shown in figure4.18. The periodic torsion of the cantilever due to the AC tip field can be picked up by demodulating the amplitude and phase of the corresponding oscillation of the horizontal deflectionDH. The measurement setup is the same as for vertical PFM, with the only difference being that the lateral deflection signal corresponding to torques of the cantilever are demodulated instead of the vertical deflection.
Figure 4.18:Non-zero d15components of the piezoelectric tensor cause a torsion of the cantilever which depends on the polarisation orientation and for an AC tip bias, can be picked-up as a non-zero oscillation amplitude in the horizontal deflection.
In vector PFM , the vertical and lateral oscillation amplitudes are acquired simultaneously in order to image the vertical and lateral components of the sample polarisation vector. In general, the corresponding resonance modes would have very different frequencies and the tip is simultaneously biased at two different frequencies chosen to excite both modes.
Interpretation of lateral PFM images is more difficult than in vertical PFM. First, the stiffness of the cantilever is much higher for rotations around the long axis than bending, which typically leads to a much lower signal in lateral PFM than in vertical PFM. Second, any single lateral PFM image only acquires information on the in-plane polarisation component perpendicular to the long axis of the cantilever. This means that at least two images need to be obtained at different cantilever orientations and the polarisation directions need to be reconstructed from the set of images obtained. An example of this is displayed in figure4.19. Third, in vertical PFM, the absolute orientation of the polarisation can easily be determined by using electric field pulses of known polarity to locally switch the polarisation, then comparing the PFM phase of the written domain of known polarisation orientation with
that of the as-grown polarisation state. This is not easily done in lateral PFM. Though domains can in principle be written using the trailing lateral field of the tip , this is requires a more complex writing process .
Furthermore, lateral PFM signals are subject to artefacts linked to broken symmetries in the investigated materials. These artefacts are discussed in further detail in chapter7.2.4.
Figure 4.19:Example of an in-plane polarisation pattern and the corresponding expected lateral PFM phase and amplitude contrasts for different orientations of the cantilever with respect to the polarisation pattern. After .