Quantitative measurements of shear displacement using atomic force microscopy Free

Ferroelectric materials, which possess switchable spontaneous polarization, have wide application in nonvolatile random-access memory,¹ nanoscale sensors,² photonic crystals,³ and biologically inspired nanostructures.⁴ In order to fabricate these nanoscale devices, it is imperative to develop microscopy techniques to visualize domain patterns with high sensitivity and high resolution. Piezoresponse force microscopy (PFM), which detects a local converse piezoelectric effect, has been widely used in domain characterization of ferroelectrics and multiferroics^5–7 because of its non-destructive nature and nanoscale spatial resolution.

In PFM, an AC voltage is applied between a conductive atomic force microscopy (AFM) tip and the bottom electrode of a ferroelectric sample. The resultant mechanical deformation of the sample, due to piezoelectric effect, causes deflections of the AFM cantilever which is in contact with the sample surface. For a material with unknown orientation, there could be both out-of-plane and in-plane piezoelectric responses. The out-of-plane deformation leads to vertical deflection of the cantilever, while the in-plane deformation leads to torsional and buckling deformation of cantilever.^8,9 The torsional mode of the AFM cantilever results in a “lateral” signal of the AFM position sensor, while deflection and buckling result in a “deflection” signal which complicates the analysis of ferroelectric domains in PFM images. In this letter, we only consider the simple case of in-plane polarization in PFM measurements, i.e., without the complication of out-of-plane PFM signal. Although the domain analysis can be simplified for ferroelectrics with only in-plane polarization, it is not straightforward to determine the local polarization orientation from one PFM image. This issue can be addressed by angle-resolved piezoresponse force microscopy (AR-PFM),^10–13 where a series of lateral PFM images are taken at different cantilever-sample orientations. Ferroelectric domains in a material with multiple in-plane polarization directions can be determined using the angular dependence. However, it is tedious and time-consuming to perform AR-PFM measurements. The major bottleneck is to find the same scan area after sample rotation.

Our approach to overcome this difficulty is to collect both deflection and lateral signals at the same location simultaneously. Buckling mode measures the shear displacement parallel to the cantilever, while torsional mode measures the shear displacement perpendicular to the cantilever, as shown in Figures 1(a) and 1(b). The total in-plane displacement can be obtained by combining both Cartesian components, which require quantitative calibrations of the position sensors of the AFM system. Different methods have been reported to calibrate lateral force by using contact-mode AFM cantilever, e.g., wedge method,^14,15 by attaching a lever and tungsten to the cantilever,¹⁶ or using a piezoresistive sensor.¹⁷ However, studies on lateral displacement calibration with AFM cantilevers are still lacking. Choi et al. proposed a simple method to calibrate lateral displacement by monitoring the lateral signals of different scan sizes.¹⁸ However, the resolution of this method is relatively poor because of the non-linear piezoelectric response of the scanner tube with typical scan sizes (⁠$≥1 nm$⁠).

Inspired by the aforementioned method, we propose an improved method with much better sensitivity to calibrate both buckling and torsional modes using a calibrated AFM scanner tube. By applying a small AC voltage to the X (or Y) electrodes of the AFM scanner, the sample attached to the scanner undergoes an in-plane oscillating motion with a known amplitude. The tip in contact with the sample surface experiences buckling (or torsional) deformation, giving rise to the deflection (or lateral) voltage signals from AFM. Thus, the coefficient of the proportionality between these two quantities allows the conversion of PFM voltage signal to physical displacement.¹⁸ With this calibration method, the shear displacement of the electrically actuated y-cut α-quartz was quantitatively measured at various cantilever-sample orientations. The direction of the shear displacement is verified by recording the rotation angles. The piezoelectric coefficient obtained agrees well with the value reported in literature. This method of quantitative measurement of shear displacement without rotating the sample would facilitate domain imaging and analysis of ferroelectrics with complex in-plane polarizations.

This shear displacement calibration is conducted on a multi-mode AFM system (Bruker) together with a lock-in amplifier (SR830), as shown in Fig. 1. A homemade breakout box was used to control individual in-and-out signal lines, e.g., the voltage line to X or Y electrodes of the AFM scanner head. Conducting diamond coated AFM tips (DD-ACTA, AppNano) were used in our calibration process for PFM measurements. The buckling (or torsional) motion of the AFM tip can be detected by position sensors of the AFM system as deflection (or lateral) signals which are demodulated by the lock-in amplifier. The calibration coefficient of the buckling (or torsional) mode can be obtained by taking the ratio of the deflection (or lateral) signal to the in-plane displacement, which can be accurately estimated by taking the product of the applied voltage and calibrated piezoelectric coefficient of the scanner tube (⁠$71.4 nm/V$ in our AFM system). A single-point PFM measurement on y-cut α-quartz with various orientations was performed by using this calibration method.

A frequency dependent deflection (or lateral) signal with an amplitude of 1.4 mV_rms (0.1 nm) applied to the X (or Y) scanner is shown in Fig. 2(a). A peak was observed at approximately 1 kHz, which is the resonant frequency of the AFM scanner tube. Therefore, these curves are fitted by a driven harmonic oscillator model

where F represents the driving force exerted on the scanner tube, ω₀ is the resonance frequency of the scanner tube, and γ represents the damping coefficient. The fitting resonance frequencies of the scanner tube are 996 ± 8 Hz and 1017 ± 5 Hz for voltages applied on X and Y electrodes, respectively. In this work, typical displacement calibration was conducted at $100 Hz$⁠. The static (DC) limit of the calibration coefficient is extrapolated according to Eq. (1). The calibration does not depend on the modulation frequency as long as it is much lower than ω₀. Typical calibration curves at $100 Hz$ for both buckling and torsional modes are shown in Fig. 2(b). After taking into account the frequency factor, the calibration coefficients at DC limits are $26.77±0.03 nm/V$ and $14.34±0.02 nm/V$⁠, respectively. The voltage-displacement curve deviates from linear behavior at high driving amplitudes (⁠$≥1 nm$⁠). In addition to the non-linear piezoelectric response of the scanner tube at large drive amplitudes, it is possible that the displacement is too large for the AFM tip to maintain solid contact with the sample surface without sliding.¹⁸ Our experience indicates that the calibration (slope of the voltage-displacement curve) is sensitive to tip-surface contact conditions. Therefore, it is necessary to perform such a calibration before starting each PFM measurement.

A single-point PFM measurement on y-cut α-quartz was performed after both linear and buckling modes were carefully calibrated. A layer of 50 nm thick Au film is deposited on α-quartz to ensure a uniform out-of-plane electric field. During the PFM measurement, the AFM tip was grounded. The modulated voltage (⁠$5 Vrms$⁠) was applied to the back electrode to eliminate the system-inherent background,¹⁹ as shown in Figure 3(a). For an arbitrary sample orientation, we recorded the frequency dependence of both deflection (buckling) and lateral (torsional) signals. The deflection signal shows a weak frequency dependence, while the lateral signal shows low-pass filter behavior, as shown in Fig. 3(b). The DC values were extrapolated to be $1.15±0.02 mV$ and $0.479±0.005 mV$ after curve fitting. With the calibration factors, the two Cartesian components of in-plane shear displacement are $2.52±0.03 nm$ and $3.29±0.06 nm$⁠, respectively. Therefore, the direction and magnitude of the in-plane displacement can be calculated. In this case, the angle between the cantilever and the shear displacement was estimated to be $52.5±0.6°$⁠, which is consistent with the visually measured value (⁠$∼52°$⁠). In addition, we obtain the quantitative value of the effective piezoelectric coefficient d₂₆, approximately $4.14±0.05 pm/V$⁠. The excellent agreement substantiates our calibration method.

To further establish the validity of our method, we performed similar calibrations and PFM measurements at various sample orientations with respect to the AFM cantilever. The results of these measurements are summarized in Table I. The visually measured rotation angles agree well with the angles calculated from PFM measurements. Therefore, our single-point PFM measurement is effective in determining the direction of the local in-plane polarization direction. The magnitude of the piezoelectric response d₂₆ at various angles exhibits small variations, as shown in Fig. 4(b). Piezoelectric coefficient $d11≈2.3$ pm/V of α-quartz was experimentally established in early measurements.^20,21 The crystallographic symmetry of α-quartz (point group 32) requires $d26=2d11$⁠, thus $d26≈4.6$ pm/V. The averaged value of measured d₂₆ (4.33 ± 0.03 pm/V) is only $∼6%$ off of the literature value. The slight difference might come from imperfection of the scanner tube calibration.

In conclusion, we established an effective method to quantitatively measure the local shear deformation of piezoelectric materials with high sensitivity. We devised a recipe for calibration of the buckling and torsional motion of the AFM cantilever using the scanner tube, which allows quantitative measurements of local shear displacement of piezoelectric materials. This method can be potentially implemented in lateral PFM measurements for imaging in-plane domains of ferroelectric materials. Because both vertical deflection and buckling of the cantilever contribute to the deflection signal of position sensors, our method is not applicable to more complicate situations when both in-plane and out-of-plane polarizations are present. Separation of these two signals still requires a careful alignment of in-plane polarization with respect to cantilever orientation.

This work at Rutgers is supported by the Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, U.S. Department of Energy under Award No. DE-SC0008147.