Giant optical enhancement of strain gradient in ferroelectric BiFeO3 thin films and its physical origin

Through mapping of the spatiotemporal strain profile in ferroelectric BiFeO3 epitaxial thin films, we report an optically initiated dynamic enhancement of the strain gradient of 105–106 m−1 that lasts up to a few ns depending on the film thickness. Correlating with transient optical absorption measurements, the enhancement of the strain gradient is attributed to a piezoelectric effect driven by a transient screening field mediated by excitons. These findings not only demonstrate a new possible way of controlling the flexoelectric effect, but also reveal the important role of exciton dynamics in photostriction and photovoltaic effects in ferroelectrics.


Strain profile retrieval methods
The diffraction amplitude from a thin film with the contribution from the substrate is calculated using a kinetic diffraction model, Here is the layer-by-layer BFO phase shift of the unit cells, FBFO and FSTO are the structure factors, n=1,…, N and is the layer index and N is the number of epitaxial layers, and a is the fitting parameter for the relative intensity of the substrate and the film, respectively.
The fitting of the strain profile takes our knowledge of the film thickness and is accomplished by a spline algorithm 1 using 4 evenly distributed fitting points for adjusting the phase while using a spline interpolation to obtain the phase at each unit cell layer. Fitting with more points does not change the fitting qualitatively and only reduce the fitting error modestly by less than 10%. The fitting algorithm tries to minimize the following error while changing the phase: where |Amea(q)|=|I (q)| 1/2 is the measured diffraction amplitude along the truncation rod.
The measurement was repeated at different delays for samples with different laser fluences and film thicknesses. A complete set of data is depicted in Figs. S1-S3. The fitting matches the position and the relative amplitude of the fringes well except for the 20 nm film at time zero, which needs further study to determine the cause.

Figure S1
A complete data set of the mapping of the strain as a function of delay for a 35 nm film (88 monolayers) at 3.3 mJ/cm 2 . Left column: measured (red lines) and fitted (blue lines) diffraction amplitude |A| at different delays between the laser and the X-ray. Left column: corresponding fitted strain at different delays as a function of the film layer index n. (1) (red) and average strain change ∆ε(t) (blue). (1) (red) and average strain change ∆ε(t) (blue). The rhombohedral lattice parameters were then converted to pseudo-cubic. The out-ofplane strain was calculated according to Hooke's law, taking into account the coupling of epitaxial in-plane stress due to the cold substrate: where ν = 0.34 is the Poisson ratio 10 , and ε and are the out-of-plane and in-plane strain, respectively.
The resulting γ using our experimental data is shown in Table S1.

Disproving the charge separation by polarization field model
Bulk charge separation due to internal polarization or external field leads to strong distortion of the field in a pump probe experiment though it may have no effect in a CW experiment. We simulated the situation with a one dimensional particle-in-cell dynamic model for the 35 nm film where the motion of the carriers (Fig. S4(a)) and the field (Fig. S4(b)) are For carrier density higher than 3×10 18 cm -3 (corresponding roughly to an absorption fluence of 0.01 mJ/cm 2 ), 100% modulation of the field can be achieved, indicating the saturation of the applied field. For lower carrier densities, the maximum modulation is proportional to the carrier density. Fig. S3 shows a case with carrier density of 1.5×10 18 cm -3 (absorption fluence of 0.005 mJ/cm 2 ) As can be seen in Figure S3 (b), the charge separation induces a ±50% modulation of the field inside the film. When mapped into the piezoelectric response of the unit cells, the modulation will lead to a strain profile completely different from that arises from a uniform field expected from the exciton scenario where carriers only separate at the surface and the interface. The low carrier density needed for such modulation also demonstrates the sensitivity of the strain profile to such a bulk space charge effect. Simulation using strain profile following the field profiles with comparable strain range as measured from the experiment generates significantly asymmetric fringes around the central diffraction peak, which is not observed in the experimental data (Fig. S5), disproving the commonly accepted carrier separation by polarization field model.