Quantitative probe for in-plane piezoelectric coupling in 2D materials

Piezoelectric response in two-dimensional (2D) materials has evoked immense interest in using them for various applications involving electromechanical coupling. In most of the 2D materials, piezoelectricity is coupled along the in-plane direction. Here, we propose a technique to probe the in-plane piezoelectric coupling strength in layered nanomaterials quantitively. The method involves a novel approach for in-plane field excitation in lateral Piezoresponse force microscopy (PFM) for 2D materials. Operating near contact resonance has enabled the measurement of the piezoelectric coupling coefficients in the sub pm/V range. Detailed methodology for the signal calibration and the background subtraction when PFM is operated near the contact resonance of the cantilever is also provided. The technique is verified by estimating the in-plane piezoelectric coupling coefficients (d11) for freely suspended MoS2 of one to five atomic layers. For 2D-MoS2 with the odd number of atomic layers, which are non-centrosymmetric, finite d11 is measured. The measurements also indicate that the coupling strength decreases with an increase in the number of layers. The techniques presented would be an effective tool to study the in-plane piezoelectricity quantitatively in various materials along with emerging 2D-materials.


Device Fabrication
b) Initially, metal contacts are patterned on the substrate. To achieve this, S1805 optical resist is spin-coated on the substrate, and the photolithography process is carried out with the direct writing technique (μpg 501 Heidelberg tool). Then, metallization of Platinum (Pt) contacts with Titanium (Ti) as adhesion metal layers with the thickness of 15/50 nm is carried out using Techport e-beam evaporator. This is followed by a lift-off process in acetone. c) To make a circular trench in between the electrodes, PMMA resists (495 A4 and 950 A2) are spin-coated and a circular opening is formed between the contacts using the e-beam lithography technique (Raith eline tool). SiO2 is completely etched away in the circular drum portion by using Reactive ion etching (RIE from Oxford instruments) followed by buffered hydrofluoric acid etch.  MoS2 is suspended in that region. The depth measured across the trench region is the addition of indentation depth because of the applied normal force while scanning and the initial sag of the MoS2 layer. Fig. S2(c) shows the image of the same monolayer device after it has collapsed (after multiple measurements and SHG). When the MoS2 layer is collapsed on the trench, the line profile across the trench is shown in fig. S2(d) and the depth measured at the circular trench region is close to 285nm, which is the actual etch depth.

Second harmonic generation microscopy (SHG)
For d11 measurement, the direction of the electric field must be along the armchair edge of the MoS2 flake. The direction of the metal pads with respect to the flakes is found based on the polarised SHG studies. SHG experiments are carried out to find the edge chirality of the flakes. A linearly polarised femtosecond pulsed laser operating at 1040 nm is used for the measurements, and the corresponding second harmonic component is detected at 520 nm. An optical image collected at 1040 nm (incident wavelength) is used to locate the region of interest from the sample (fig S4(a)). Fig. S4(b) shows the SHG mapping of the monolayer MoS2 flake. It is observed that SHG intensity is relatively high on the circular drum region when MoS2 is suspended in that region. It is also observed that if MoS2 is collapsed in the circular drum region, SHG intensity vanishes in some samples. Fig. S4(c) shows the plot with SHG intensity vs. incident power. SHG power shows quadratic dependence with the input power.
The second harmonic field amplitude of the parallel component is given by 2

=
(2) cos(3 + 0 ) (S1) Where is the incident laser frequency, 2 is the second harmonic frequency, C is the proportionality constant which is a function of electric field component at ( ) and the dielectric environment. The direction 'x' corresponds to the armchair direction of the 2D-MoS2 flake, is the sample rotation angle w.r.t the laboratory x-coordinate of the experimental system and 0 is the initial crystallographic orientation of MoS2 flake. Fig S4(d) shows the polar plot of the polarised SHG data collected at different rotation angles, here laser is polarized along the y-axis. By fitting the intensity(I) plot to IαE 2, we get

Vertical Deflection sensitivity calibration
To calculate the vertical deflection sensitivity of the AFM tip, force-distance curves are obtained on the sapphire substrate. Fig. S5(a) shows the deflection error plot when the tip is approaching and is retracting from the sample. The slope of the linear region in the approach/retract curve is defined as inverse vertical deflection sensitivity of the tip. From this plot, we calculate the vertical deflection sensitivity (nd) of one of the SCM-PIC tips to be 207.5 nm/V. Vertical deflection sensitivity of the SCM-PIC tip is calculated using the force-distance method, and it is found to be 200+/-10 nm/V (nd) for various tips used for the measurement. Fig S5(b) shows the thermal tuning data of the AFM cantilever (cantilever is not in contact with the sample, and it is free to vibrate in the air). The normal spring constant (kN) of the cantilever is estimated from thermal tuning data 3 . The normal spring constant calculated from this is 0.17 N/m for one of the tips.
This information is needed to estimate the amount of normal force (FN=kN*nd*setpoint voltage) applied on the sample corresponding to the deflection voltage set point of the system. This force is maintained constant across all the piezo and pseudo measurements for a given sample.
where, kt -Torsional spring constant, μcoefficient of friction, H=h+t/2, h-is the height of the tip, half the tip side angle and Ns-Normal force The angle conversion factor relates the lateral deflection voltages and the twist angles of the cantilever.
The torsional spring constant is given by Where G is the shear modulus of the cantilever, W, L, t are the width, length, and thickness of the cantilever, respectively.
Solving the equations S2 to S5, we get equation S6, which relates the applied normal forces with twist To determine the lateral deflection sensitivity for the AFM probes, the tip is scanned laterally across the standard silicon step gratings (step height s=180 nm) as shown in the schematic below ( fig. S6(a,c)).
Friction images are captured in LFM mode (Lateral force microscopy) by keeping applied normal force constant ( fig. S6(d)). When the tip climbs up along the step grating, it experiences a maximum twist angle at the apex and fig. S6(b) shows the lateral deflection voltage signals. The peak voltage read while the tip is climbing is termed V1, while V0 is the average deflection signal when the tip is moving on the flat surface. This procedure is repeated at various normal forces ranging from 20nN to 60nN of force.  This lateral spring constant is not to be confused with the true lateral spring constant of the tip, which is related to the stiffness of the cantilever and corresponding true lateral deflections 6 . When the tip edge is interacting with the sample in contact mode of AFM, the PSPD relates the lateral deflections of the cantilever related to kl given by eq. S8.
Lateral deflection sensitivity factor ld can be calculated from the angle conversion factor as follows Where h is the height of the AFM tip, t is the thickness of the cantilever, and s is the height of the silicon grating step. These are measured from the scanning electron micrographs of the tip taken by placing it on a stub (holder) used for cross-section imaging ( fig. S7(a) & S7(b)). SEM images are taken before and after the measurements to check if the tip is intact post measurements. Shear modulus(G) for the calculations is taken as 60Gpa for silicon cantilevers. Geometrical parameters measured from the SEM images and estimated parameters for some of the tips used for the measurements are tabulated in Table   S1.    When the voltage is applied between the source and the drain electrodes ( fig. S8(a)), because of the inplane piezoelectric effect in MoS2, the net lateral displacement of the AFM tip is zero when the tip is placed at the center. Even if the tip is not placed precisely at the center, we do not get optimal results.
Whereas in the current scheme ( fig. S8(b)), the tip can detect net lateral deflection in one direction so that the effective displacement can be measured. It can be observed from fig. S8(c) that the signal strength is much higher when voltage is applied between the electrode and the tip in contact with the sample when compared to the signal obtained when voltage is applied between the two electrodes. The observed signal in that case ( fig. S8(b)) was much smaller, and it is comparable to the pseudo piezoresponse (Fig 4(c) of the manuscript). This is considered as instrument noise floor for the measurement of lateral deflection. Contact resonance can enhance the signal levels from pico meter to nano meter scale (refer Fig. S9) and thus enables the measurement of low piezoelectric coefficients.

Selection of the torsional resonance frequency range for monolayer MoS2 Device
In the manuscript, a specific drive frequency range (110 kHz to 220 kHz) is selected to discuss operation of the lateral PFM near the torsional resonance. To identify the peak frequency, we fit a Lorentzian equation to the drive frequency response of the lateral piezoresponse curve. Then, the identified local maxima position of the data is taken as peak frequency datapoint. A constant baseline is chosen from the local minima which is based on the statistical median of the amplitude data near the torsional resonance peak. The x-coordinates (frequencies) corresponding to the intersection of the baseline with the experimental data are chosen as base frequencies. Identified peak and base frequency positions are indicated in Fig. S10 (a). Fig S10(b) shows the Lorentzian peak fit of the torsional resonance peak. For this particular device, the resonance peak is near 190 kHz (vertical line marked in red), and base frequency positions are near 100 and 220 kHz frequencies (vertical lines marked in blue). From this analysis, the base frequency (ω0) is selected at 100 kHz for the measurements. Hence, the operating frequency range is selected above the baseline frequencies, i.e. from 110 kHz to 220 kHz, to study lateral PFM near torsional resonance.

Contribution from the Pseudo piezo response
In the current measurement scheme, d11_pseudo contributes to the pseudo piezo response to the measured d11 coefficient. Fig. S11 shows the d11pseudo measured near the torsional resonance frequency range. Here, the pseudo response to the d11 is compared with the measured and effective d11 coefficients. d11pseudo measured near contact torsional resonance peak (190 kHz) is ~0.8pm/V for this set of measurements. From this, we emphasize that pseudo response contribution becomes significant for estimating the coefficients in the sub pm/V range. Note: The amplitude mapping images presented in Fig. S12(a & c) are for the qualitative illustration.
The amplitude mapping images are acquired using the line scan, and relatively less contact force is applied to avoid damage to the tip. Furthermore, to acquire the amplitude mapping images shown in fig.   12(a & c), we use relatively less contact force. The smaller force ensures that the tip does not get damaged while scanning and avoids a short circuit when the tip is in contact with the drain electrode.
In Fig. S1(e), the data is obtained from a single point at constant applied force. Quantitative information is extracted from the point measurements.

d11 coefficient for different layers of MoS2
Fig. S13 shows the measured and effective d11 coefficients for MoS2 of 2 to 5 layers calculated using the methods discussed in the manuscript. It can be observed that the frequency bandwidth of the resonance is wider for the mono and 3-Layers when compared to even number of layers. This indicates that lateral piezoelectric response in odd no. of layers of MoS2 enhances the torsional resonance.

LPFM on quartz
To test if these methods can be applied to the bulk samples, LPFM is carried out on a quartz substrate to find the lateral piezo coefficient. Fig. S14(a) shows the XRD plot of the AT-cut quartz crystal, which confirms the crystalline behaviour of the quartz. Metal electrodes (Cr/Au 10/100 nm) are patterned on the surface quartz substrate as shown in Fig. S14(b). The in-plane piezo coefficient for AT-cut quartz crystal was found to be 1.6 -2.1 pm/V, which is close to other experimentally verified values, i.e., 1.9 pm/V 7 (Note: Pseudo-piezoresponse is not separated here as the metal electrode is also on quartz). Fig.   S14(c) shows the lateral piezoresponse obtained on the quartz sample at its base frequency of resonance.