Giant elastic tunability in strained BiFeO3 near an electrically induced phase transition

Elastic anomalies are signatures of phase transitions in condensed matters and have traditionally been studied using various techniques spanning from neutron scattering to static mechanical testing. Here, using band-excitation elastic/piezoresponse spectroscopy, we probed sub-MHz elastic dynamics of a tip bias-induced rhombohedral−tetragonal phase transition of strained (001)-BiFeO3 (rhombohedral) ferroelectric thin films from ∼103 nm3 sample volumes. Near this transition, we observed that the Young's modulus intrinsically softens by over 30% coinciding with two- to three-fold enhancement of local piezoresponse. Coupled with phase-field modelling, we also addressed the influence of polarization switching and mesoscopic structural heterogeneities (for example, domain walls) on the kinetics of this phase transition, thereby providing fresh insights into the morphotropic phase boundary in ferroelectrics. Furthermore, the giant electrically tunable elastic stiffness and corresponding electromechanical properties observed here suggest potential applications of BiFeO3 in next-generation frequency-agile electroacoustic devices, based on the utilization of the soft modes underlying successive ferroelectric phase transitions.

d-f, Piezoresponse loops (upper row) and associated resonance frequency loops (lower row) measured in bipolar (d) and unipolar negative/positive (e/f) waveforms. All of them are vertical loops expect those explicitly annotated in f. The loops (including the in-plane switching loops) in e,f were measured at the same location with one or two pre-poling cycles in between them.  Calculated Young's modulus.
Supplementary Table 2 | Phase-field modeling parameters and energy coefficients.

Supplementary Note 1. The extrinsic contribution to the measured on-field loops
Generally, the electrostatic effect contributes to measured first harmonic response, R(1ω), in PFM in proportion to the potential difference between the tip and sample surface via capacitive forces, according to the equation: where α is the sensitivity factor of the detection system, k C contact stiffness , z C capacitance gradient of the tip-sample system, V sp surface potential of the sample, and V ac /V dc applied a.c./d.c.
voltages on the tip. For typical ferroelectric samples, the electrostatic interaction in the tip-sample junction is conservative; i.e., the sample surface potential does not change significantly during BEPS measurements. Therefore, the electrostatic contribution to the on-field R(1ω) loop is a straight line with little hysteresis, as schematically shown in Supplementary Figure 1a. This is the reason why BEPS is predominantly performed in an off-field mode or only off-field loop data is paid attention to so as to minimize the electrostatic contribution. Supplementary Fig. 1b shows an example of bipolar switching loops measured on an epitaxial tetragonal-phase PZT thin film, which are in good agreement with the schematic. For our BFO thin film, the electromechanical response is found to be dominating compared to the electrostatic response even for on-field loops under our experiment conditions. Supplementary Fig. 1c,d shows two examples of on-field loops measured with the same parameters using a single tip, when it was relatively fresh (d) and significantly worn after long time usage (c). Note that electromechanical response in PFM is sensitive to the very tip apex; deterioration of the coatings therein reduces the effective local electrical fields and thus the measured response decreases (meanwhile the coercive biases increase and the loops appear to become broader). By contrast, the electrostatic forces mainly originate from the tip cone and cantilever beam and thus is far less sensitive to the wearing of tip apex coatings. The electrostatic contribution can be subtracted from the measured on-field loops based on the slopes shown on them, as illustrated in Suppl. Fig. 1c. In the cases of Suppl. Fig. 1d and most of the data presented in the Main Text, nevertheless, this subtraction can be of little significance for our analysis especially regarding the phase transition regions where the piezoresponse is markedly enhanced.
In BEPS, conduction current flowing through the tip-sample junction can cause Joule heating that induces thermal expansion of the junction thereby potentially contributing the measured signals. 4 The Joule heating induced strain is proportional to the power (P) consumed on the tip-sample junction and expansion coefficients (β) of the sample as well as the tip, x = β P. In a general sense, assuming an ohmic conduction behavior with constant resistance R, P = R I 2 . For on-field BEPS measurements, conduction current I = I dc + I ac sin(ωt) due to both d.c. and a.c.
applied voltages. This leads to a first harmonic response as a function of V dc : Analysis of this effect can be made from two aspects. First, let us consider the V dc dependence (i.e., loop shape) of this response based on the I-V curves simultaneously measured in BEPS. Since the conduction behavior here is no longer ohmic, we may consider the instantaneous power, i.e., dP/dV, which should be approximately correlated with thermal strain response at a.c. modulation frequencies. Supplementary Figure 2a,b shows the I-V curves and BEPS amplitude loops from the same data set of Figure 1d-g in the Main Text. Obviously, no correlation exists between the measured conduction current and piezoresponse. Second, we measured a p-type Boron-doped Si wafer with a conductivity of 5-10 S m -1 under similar conditions. The conduction current in this case is two orders of magnitude higher than the BFO case, and measurable first harmonic response does exist and is in strong correlation with the I-V curves. This response, however, is much lower than the true piezoresponse of our BFO sample (c.f. Suppl. Fig. 2b,c). The contact resonance frequency of Si shows minute (~100 Hz) softening at high (positive) biases, presumably as a result of Joule heating taking place at the tip-sample junction. Note that although BFO has slightly lower heat capacity and higher thermal expansion coefficients than Si amounting to somewhat stronger Joule heating expansion effect in principle, 5 our comparative results still provide good indication of the contribution from this effect to the measured signals in BEPS. In addition, we also carried out finite element modeling of the Pt-BFO junction using Comsol Multiphysics 4.4 software based on the experimental parameters and measured conduction current. The temperature rises at the junction were found to be less than a few Kelvin which can be neglected for the main issues addressed in this work.

Supplementary Note 2. Quantitative analysis of the local elasticity of BiFeO 3
We chose three high quality commercially available single crystals of Si(001) wafer, PPLN (periodically-poled LiNbO 3 , in z-cut) and SrTiO 3 (001) as reference samples. The elastic moduli of these materials are expected to bracket those of BiFeO 3 . We performed contact resonance atomic force microscopy (CR-AFM) measurements of these samples, using the photothermal excitation method as we recently reported. 6  Then we derived contact stiffness, k * , from the measured f C and free cantilever resonance frequency f 0 using cantilever dynamics models that we solved numerically based on the mathematic formalism of Rabe 7 and typical geometry factors of PPP-EFM cantilevers (see our previous results in Ref. [6]). With the contact stiffness values quantified, we were able to calculate elastic moduli of the samples based on the Hertzian contact mechanics model. The Hertzian model relates k C with contact radius C a and reduced Young's modulus  Y as: . Among them,  Y is defined as: where  S ,  T and Y S , Y T are the Poisson rations and Young's moduli of the sample and AFM tip (here, the values of the 25 nm conductive coating material, Pt, were used), respectively. The contact radius is related to the tip shape and applied force. Direct quantification of it from the Hertzian model could bring about large errors. Therefore, we followed a commonly used calibration approach, according to the relation: are the reduced Young's modulus and measured contact stiffness of the reference sample, respectively; m is tip geometry factor ranging from 1.5 for the case of hemispheric tip shape to 1 for a flat punch shape. The real tip geometry in practice is usually intermediate between these two ideal cases. We used m = 1.25 throughout our analysis as this value was found to yield the most self-consistent results for the three reference samples.
The elastic modulus of 50 nm BFO thin film in the pristine state (i.e., zero applied biases) was