Temporal and spatial tracking of ultrafast light-induced strain and polarization modulation in a ferroelectric thin film

Ultrashort light pulses induce rapid deformations of crystalline lattices. In ferroelectrics, lattice deformations couple directly to the polarization, which opens the perspective to modulate the electric polarization on an ultrafast time scale. Here, we report on the temporal and spatial tracking of strain and polar modulation in a single-domain BiFeO3 thin film by ultrashort light pulses. To map the light-induced deformation of the BiFeO3 unit cell, we perform time-resolved optical reflectivity and time-resolved x-ray diffraction. We show that an optical femtosecond laser pulse generates not only longitudinal but also shear strains. The longitudinal strain peaks at a large amplitude of 0.6%. The access of both the longitudinal and shear strains enables to quantitatively reconstruct the ultrafast deformation of the unit cell and to infer the corresponding reorientation of the ferroelectric polarization direction in space and time. Our findings open new perspectives for ultrafast manipulation of strain-coupled ferroic orders.


II. SUPPLEMENTARY NOTE 2. BRILLOUIN SIGNAL ANALYSIS
In this note we discuss in more details the values of the Brillouin frequencies, which we deduced from our Wavelet analysis shown in Fig. 3B of the main manuscript, and the corresponding sound velocities.In Fig. S4 we show typical acoustic phonon spectra extracted at different time delays (100, 115 and 175 ps).As mentioned in the main text, the Brillouin frequency of the LA phonon mode in BiFeO 3 is f LA, BiFeO 3 ≈ 43 GHz, in agreement with reported values in bulk single crystals and ceramic samples of rhombohedral BiFeO 3 (17,18,34).With a refractive index n BiFeO 3 ≈ 2.9 at λ probe = 587 nm (51)(52)(53), the Brillouin frequency f LA, BiFeO 3 corresponds to a LA sound velocity of v L, BiFeO 3 = f LA, BiFeO 3 λ probe •(2n BiFeO 3 ) −1 ≈ 4500 m.s −1 in agreement with literature (15,17,(56)(57)(58).The shear wave component in BiFeO 3 is observed at the Brillouin frequency f TA ≈ 31 GHz, which leads to a sound velocity v T, BiFeO 3 = f TA, BiFeO 3 λ probe • (2n BiFeO 3 ) −1 ≈ 3100 m.s −1 in good agreement with previous measurements on rhombohedral BiFeO 3 (17,18,57,58).Finally, the LA wave in SrTiO 3 is detected with a Brillouin frequency of f SrTiO 3 ≈ 66 GHz.With the SrTiO 3 refractive index of n SrTiO 3 = 2.4, we obtain the sound velocity v L,SrTiO 3 = 8100 m.s −1 , in agreement with literature (59).In our work all sound velocities are given for a propagation direction of (100) m .This corresponds in the pseudo-cubic to the (110) pc direction (see Figs.There is around 13% of variation.This variation is of the same order than 5 those reported in the literature for bulk BFO where discrepancies between works approach sometimes 20 % (see Ref. (11,18,57,58)).When a material is subjected to a photoinduced stress, a strain pulse is generated in the material (semi-infinite geometry).Considering the simplest linear elastic response, we can assume that the photoinduced stress has the same form as the initial distribution of 6 light absorption with σ = σ 0 e − z ξ where ξ stands for the light beam penetration depth.In the case of presence of light-induced longitudinal and shear stresses we can consider two kind of stress : σ L = σ 0,z e − z ξ and σ T = σ 0,y e − z ξ .z stands as the direction of propagation which is perpendicular to the thin.In the case of BiFeO 3 , the pump penetration depth is ξ ∼ 60 nm for a pump wavelength of 400 nm and using the optical data known for the rhombohedral structure (31,(51)(52)(53).This penetration depth is smaller than the total layer thickness H=180 nm.The equation of motion, in the case of a longitudinal strain, is given by (32): where u z (t,z), v L , and ρ are the out-of-plane atomic displacement, the longitudinal sound velocity, and the mass density, respectively.In our experiment, it is indeed a one-dimensional problem since the lateral extension of the pump beam (spot size) is much larger (tens of micrometers) than the distance of propagation of acoustic waves in our study.For a time-delay of 200 ps, acoustic waves propagate typically over a distance of 1 micrometer only.As a consequence, the excitation has a piston-shape and the motion equation is 1D.
Solving this equation for a semi-infinite system (with a mechanically-free surface at z=0 which corresponds to the BiFeO 3 surface) leads to the following general expression of the longitudinal strain (η L = ∂uz ∂z ) (32): The formulation shown above for a longitudinal strain can be straightforwardly rewritten for a shear acoustic pulse with η T (z, t) = ∂uy ∂z , and only the sound velocity has to be replaced by v T .u y (z, t) is here the in-plane atomic motion propagating along z.In this model we assume, for simplification, that the longitudinal (LA) and shear (TA) waves are 7 pure LA and TA waves and we do not consider the possible quasi-longitudinal (QLA) and quasi-transverse (QTA) nature of these waves (11).Since the motion equation (Eq.S1) is a linear partial differential equation, the general solution can be a linear combination of a longitudinal and shear strains.
Once the strain pulse has left the region near the photoexcited surface, it propagates within the film and arrives to the interface BiFeO 3 -SrTiO 3 where the propagating acoustic strain pulses undergo some reflection and transmission.To take into account these multiple reflections, we have simulated the evolution of a longitudinal and shear strains with the software udkm1Dsim developed by Schick et al., available for free in the literature (40).In this calculation, the strain is modeled following the equation presented above (Eq.S1) with all the necessary optical parameters of the pump beam.The elastic properties of BiFeO 3 , SrTiO 3 and those of the buffer layer SrRuO 3 are taken into account.The thermal diffusion in the BiFeO 3 layer, which can lead to an in-depth extension of the induced stress, is also taken into account but has negligible effect as already discussed previously so that the emitted strain pulse is very well described by Eq.S2 (10,18).The calculation has been performed with the following acoustic velocities, as deduced from our Brillouin analysis (see Sup-  S5.In this calculation we can observe that there is a nearly perfect acoustic matching between the acoustic impedance of the LA waves and that of the substrate (see 8 also subsection B of Supplementary Note III for the discussion of the acoustic reflection coefficients).As a consequence, once the LA acoustic wave has travelled across the film, it 110 is transmitted into the substrate and only the thermal component of the LA strain remains.
Said differently there is no acoustic reflection of the LA waves on the substrate.For this reason, the compressive front (green area depicted in Figs.6A and 6C) can be detected and analysed only during around 30 ps.We can see in Fig. S5 that the propagating front incident on the substrate and seen for a time delay of 40 ps, does not give rise to a reflected front at the time 40 ps since the elastic energy is indeed transferred into the substrate.By contrast, our calculations show that the shear (TA) waves do exhibit an acoustic mismatch with the substrate.As a consequence, these waves are reflected and are then confined in the film.
Moreover, since the acoustic reflection coefficient for the TA waves is negative, there is a change of the sign of the strain perfectly in agreement with our observations (see Fig. 6B).
The simulation of the strain field will be used in the part C of Supplementary Note III for the simulation of the transient optical reflectivity signal discussed in Fig. 3 in the main text.In this simulation we have taken into account the propagation of the longitudinal and shear acoustic phonons in BiFeO 3 and their reflections at the BiFeO 3 -SrTiO 3 interface.
We also simulate the transmitted longitudinal strain in SrTiO 3 .As mentioned in the main text, we cannot detect for symmetry reasons the shear acoustic phonon in SrTiO 3 (39).
As a consequence, we did not include it in the calculation.Finally we did not include the possible mode conversion (L to T and T to L) that could arise at the interface between an anisotropic (BiFeO 3 ) and isotropic (SrTiO 3 ) material when the incident longitudinal wave in BiFeO 3 impinges on the interface.Even if we cannot exclude a possible contribution of this mode-conversion, at this stage of the analysis it does not appear necessary to have a good agreement between the experimental time-resolved Brillouin signal and the simulated one as shown in Fig. 3.

B. Acoustic reflection
The acoustic reflection coefficient is given by where Z denotes the acoustic impedance with Z=ρ • v, where ρ and v are the mass density and the sound velocity (either LA or TA), respectively.The subscript 1 defines the medium where the wave originates, while the subscript 2 denotes the medium from which the wave is reflected.
Considering the sound velocities determined from our experiment (Brillouin frequencies, see Supplementary Note II) and those from the literature for SrRuO 3 , all summarized in Table S1, we have then estimated the respective acoustic reflection coefficients for the L and T waves.These values are shown in Table S2.S2, the reflection of the is notably larger than that of the longitudinal wave (R L ).The small R L coefficient (around 2 %) is consistent with our observation of the absence of acoustic echo in the temporal evolution of the longitudinal strain presented in Fig. 6A (see main text).Our calculations show that R T is negative independently on the nature of the interface (SrRuO 3 , SrTiO 3 ), which is consistent with the change of sign of the shear strain signal in Fig. 6B.In summary, these estimates predict a reversal of the shear strain upon reflection from the substrate/film interface, in agreement with our X-ray measurements (Figs.6A and B) and in agreement with the simulation of the Brillouin signal discussed in Fig. 3.The simulation of the Billouin signal is detailed in the next subsection.

C. Simulation of the time-resolved Brillouin signal
The time and space dependence of the strain presented previously (Fig. S5) is used to simulate the transient optical reflectivity signal (∆R/R) shown in Fig. 3.In an optical pumpprobe experiment, the strain modulates the refractive index (photoelastic effect or so-called Brillouin process) and also causes a motion of interfaces (interferometric effect).To simulate the contribution of the strain to our ∆R/R signal, we have applied the standard approach 12 already described in many previous works (35)(36)(37)(38).In the particular case of the system studied in this work, since the buried SrRuO 3 layer is thin compared to both the BiFeO 3 layer and the probe wavelength (λ=587 nm), we assume that from the optical point of view the system can be considered as the assembly of a transparent layer (BiFeO 3 ) deposited onto a semi-infinite transparent substrate (SrTiO 3 ).Note that the probe wavelength λ=587 nm corresponds to a quanta of energy smaller than both the band gaps of SrTiO 3 and BiFeO 3 justifying the transparency of this assembly.With that assumption, the transient optical reflectivity magnitude becomes ∆R/R (35)(36)(37)(38): is the acoustic strain, H is the thickness of the BiFeO 3 layer.The notation r BiFeO 3 −SrTiO 3 is used for the optical reflection coefficient at the interface between the media BiFeO 3 and SrTiO 3 , for a normal incident probe beam: We use a similar expression to calculate r air−BiFeO 3 .The first integral term in Eq.S4 is the photoelastic contribution due to the scattering of the probe beam electric field by the acoustic strain η(z, t) within the BiFeO 3 transparent layer, i.e. the so-called Brillouin scattering.
The last integral term accounts for the photoelastic contribution of the substrate.Like the first integral term, it describes how the electric field of the light is scattered in the substrate by the acoustic strain.For a probe wavelength below the band gap of both BiFeO 3 and SrTiO 3 , a narrowband acousto-optic detection (detection of Brillouin oscillations) is involved in our experiment (35), i.e. we detect only the Brillouin components as mentioned 192 in the manuscript Methods.Note that in Eq.S4, the strain field η(z, t) can be either the 193 longitudinal and shear strain components.Cosnequently, the photoelastic coupling term ∂k BiFeO 3 ∂η has to be adapted for the longitudinal (η L ) and shear (η T ) strains. ∂η The simulation uses as adjustable parameters only the photoelastic coefficients ∂k BiFeO 3 ∂η which all are real in our case since (for both the longitudinal and shear strain) and ∂k SrTiO 3 we have a probe photon energy smaller than the BiFeO 3 and SrTiO 3 band gaps.The computed signal is shown in Figs.3A-B in the main manuscript.We are able to reproduce the experimental signal with a good agreement.

IV. SUPPLEMENTARY NOTE 4. SELECTED BRAGG PLANES IN TIME-RESOLVED X-RAY DIFFRACTION MEASUREMENTS AND THEIR RELATION WITH THE STRAIN A. Selected Bragg planes in time-resolved X-ray diffraction measurements
For the time-resolved X-ray diffraction experiment we selected two Bragg plane families being either sensitive (403) m or not (530) m to the shear strain.As shown in Fig. S6, when 14 the pump laser impinges on the sample (i.e., on the (b m ,c m ) plane in the monoclinic frame),

¯3
for symmetry reasons, the light-induced strain is composed of longitudinal (LA) and shear (TA) strain components.Since the plane (a m ,c m ) is a mirror plane, the shear motion cannot be along b m direction and is restricted to the c m direction, i.e., this shear strain induces a sliding of the   the light penetrates over a characteristic skin depth of 60 nm [20,21].The time resolution of the Cristal beamline is 10 ps with the low-↵ mode of the SOLEIL synchrotron facility.In this configuration the X-ray beam penetrates over several micrometers and the recorded Bragg peaks is the result of the di↵raction of the entire atomic planes of the thin film.
In order to measure the strain along each of the three lattice parameters and to be able to reconstruct the temporal evolution of the unit cell symmetry after the light pulse excitation, we have selected 4 Bragg planes families (two sets of antisymmetric Bragg planes).The (1 10) c plane (equivalent to (010) m ) and containing the ferroelectric polarization (pointing out along the along the [111] c direction as shown as a blue arrow in Fig. 2(a) ) is a symmetry plane.Consequently, when the light impinges on the BiFeO 3 layer with a surface illumination much larger than the thickness of the system (i.e.1D geometry [11]), the in-plane atomic motion can only occur in that (1 10) c plane and along the [001] c direction (or equivalently along the [001] m direction in the monoclinic frame).This in-plane motion (shear) is depicted by the red arrow (TA) in Fig. 2(a).In-plane motion perpendicular to the plane containing the ferroelectric polarization is forbidden for symmetry reasons.As a matter of fact, the interplanar distance of (223) c ((403) m ) and ( 223)c ((40 3) m ) depicted in Fig. 2(a) are coupled to the shear and longitudinal strains, i.e. the interplanar distance is a function of both the out-of (LA) and in-plane (TA) transient lattice parameters as we will quantitatively detail later on in Eq. 1. Importantly, if an in-plane symmetry breaking associated to a shear motion is induced by the light excitation, a di↵erent transient dynamic of the interplanar distances d (223) and d (22 3) is expected.On the opposite, for symmetry reasons, the interplanar distances of the families (140) c ((530) m ) and (140) c ((5 30) m ) are not a↵ected by the shear motion which induces only a sliding of the planes since the in-plane atoms displacement is parallel to this later family planes [11].Using this set of Bragg planes o↵ers then a perfect local probe to follow the light-induced unit cell distorsion.
As detailed in the Supplementary File, the analysis of the Bragg peak in the 2D UFXC images indicates that the Bragg peak displacement upon illumination occurs only along the so-called Q z direction, i.e. the di↵raction takes place within the incident plane of the the X-ray beam.Typical time dependent k! scans, after a 2D integration of the intensity recorded on the UFXC detector, are shown in Fig. 2(c) (Bragg ( 223)).The Bragg peak depicted in red is registered before the light excitation.The blue curves are the Bragg peaks recorded at di↵erent time delays between the optical excitation and the probing by the pulsed X-ray beam.The insets represent the di↵erence between the blue and the red curves.We can see how the Bragg peak evolves as a function of time over the time range 0-200 ps.  the light penetrates over a characteristic skin depth of 60 nm [20,21].The time resolution of the Cristal beamline i 10 ps with the low-↵ mode of the SOLEIL synchrotron facility.In this configuration the X-ray beam penetrates ove several micrometers and the recorded Bragg peaks is the result of the di↵raction of the entire atomic planes of the thin film.
In order to measure the strain along each of the three lattice parameters and to be able to reconstruct the tempora evolution of the unit cell symmetry after the light pulse excitation, we have selected 4 Bragg planes families (two set of antisymmetric Bragg planes).The (1 10) c plane (equivalent to (010) m ) and containing the ferroelectric polarization (pointing out along the along the [111] c direction as shown as a blue arrow in Fig. 2(a) ) is a symmetry plane Consequently, when the light impinges on the BiFeO 3 layer with a surface illumination much larger than the thicknes of the system (i.e.1D geometry [11]), the in-plane atomic motion can only occur in that (1 10) c plane and along the [001] c direction (or equivalently along the [001] m direction in the monoclinic frame).This in-plane motion (shear) i depicted by the red arrow (TA) in Fig. 2(a).In-plane motion perpendicular to the plane containing the ferroelectric polarization is forbidden for symmetry reasons.As a matter of fact, the interplanar distance of (223) c ((403) m ) and ( 223)c ((40 3) m ) depicted in Fig. 2 In the figure S7, we present the geometry for the X-ray diffraction experiment based on 220 a rocking curve (ω) scans.In the case of asymmetric Bragg planes (i.e.not parallel to the 221 surface of the sample), when the light induces some longitudinal and shear strains (denoted η L and η T , respectively), there is not only a change of the interplanar distance (∆d/d), but also a change of the Bragg plane orientation with respect to the direction of the incident X-ray beam.This tilt angle is depicted as ∆α in the Figure S7 and the latter one also depends on the strain components.As a consequence, the variation of the ω angle we have is not only governed by the Bragg angle change (∆θ) but also by this tilt angle ∆α = α ′ − α (α ′ is the angle when the unit cell is submitted to the strain).Both contributions depend on the strain components following: where ω ′ and ω 0 denote the time-dependent diffraction angle, and the diffraction angle at equilibrium, respectively.
We consider first the case of (h0l) planes.Using the reasonable orthorhombic approximation (monoclinic angle β taken as 90 • instead of 89.5 • ), we can show first that the interplanar distance ∆d(η L ,η T ) d can be written as function of the variation of the α angle with: We have also the two following expressions for α ′ = α + ∆α and α: With the trigonometric relation sin(α + ∆α) = sin(α) cos(∆α) + cos(α) sin(∆α) and con-238 sidering small variation of the α angle (∆α ≪ 1), and the corresponding Taylor developments cos(∆α) ∼ 1 and sin(∆α) ∼ ∆α, we arrive to: with further Taylor developments, we arrive to: With Eq.S7 and Eq.S6, Eq.S11 is rewritten as: A similar calculation can be realized for the (hk0) planes.We have first: and We finally arrive to: For numerical applications, we have considered for the pseudo-cubic approximation that α=30   hence plane (221) c could not be selected.
From the photoinduced strain process point of view, it is important to have in mind that these Bragg planes have a different orientation regarding to the normal We note that the effect of the pump light polarization is negligible.Considering the values of the refractive index available in the literature only for the rhombohedral BiFeO 3 material, we have calculated that pumping with an electric field along either the ordinary or extraordinary axe leads to a respective optical absorption of 69% and 71%.Similar conclusions were already discussed in the case of an excitation of a BiFeO 3 single crystal having a 3m point group symmetry (11).
FIG. S2.Estimation of the fraction of ferroelastic variants in the BiFeO 3 thin film on 1 and S6), so equivalent to the direction defined by the vector (⃗ a− ⃗ b) where ⃗ a and ⃗ b are the two first unit cell vectors in the rhombohedral frame.Recently, along this direction (58) a longitudinal (LA) and shear (fast TA mode is the one we detect) velocities of V LA = 5100 m.s −1 and V T A = 2700 m.s −1 (see angle ϕ=0 • in Figure 2 of Ref. (58)) have been measured.These values compare with our experimentally determined values given above (V LA = 4500 m.s −1 , V T A = 3100 m.s −1 ).
FIG. S4.Acoustic phonon spectrum at different time-delays.A Time-frequency domain

III. SUPPLEMENTARY NOTE 3 :
GENERATION, PROPAGATION, REFLEC-TION OF THE STRAIN WAVES IN THE BIFEO 3 /SRRUO 3 /SRTIO 3 FILM AND SIMULATION OF THE BRILLOUIN SIGNAL A. Model of the photoinduced strain FIG. S5.Numerical simulation of the time dependence of the light-induced strain in

(
530) m and (530) m planes.Consequently, there is no contribution of this shear component on the light-induced change of the interplanar distance of the (530) m planes.On the opposite, this shear strain (atomic displacement along c m direction) induces a modification of the interplanar distance that is different for (403) m and (40 ) m .In the next subsection B of this Supplementary Note IV, the relation between diffraction angle ω and strain component values are derived for each of the above-mentioned lattice planes.

Figure 2 :
Figure 2: (color online) (a) Selected Bragg planes families (223)c and (140)c during the time resolved X-ray di↵raction experiment.The illuminated surface of BFO is the free surface corresponding to the (110)c.(b) Sketch of the time-resolved X-ray di↵raction setup.A ✓/2✓ configuration was chosen while k! scan (rocking curve) was realized to record the Bragg peaks.(c) Typical evolution of the Bragg peak (in blue) after an ultrashort laser pulse excitation.The Bragg peak depicted by the red curve is the one corresponding to the material before the optical excitation.

Figure 2 :
Figure 2: (color online) (a) Selected Bragg planes families (223)c and (140)c during the time resolved X-ray di↵raction exper iment.The illuminated surface of BFO is the free surface corresponding to the (110)c.(b) Sketch of the time-resolved X-ray di↵raction setup.A ✓/2✓ configuration was chosen while k! scan (rocking curve) was realized to record the Bragg peaks.(c Typical evolution of the Bragg peak (in blue) after an ultrashort laser pulse excitation.The Bragg peak depicted by the red curve is the one corresponding to the material before the optical excitation.

FigureChapter 3
Figure S8 shows images of the 403 m Bragg peak as detected before (left) and at a time of 200 ps after laser excitation (right).The analysis of the 2D images obtained by ω scans reveals that the diffraction takes place within the incident plane of the X-ray beam, indicating that the light-induced atomic motion takes place in the (a m , c m ) plane consistently with the symmetry of the BiFeO 3 layer and the model.For each ω angle, we integrate the 2D 261

Fig 3 .
Fig 3.11: (a) Relation between the variations of the ✓ angle and the kw angle.Note that here only the Bragg plane are shown and not the free surface of a thin film that can be not parallel to (hkl) in case of asymmetric Bragg planes.(b, c) X-ray diffraction image recorded by the 2D detector at Bragg condition for "unpumped" and "pumped" signals.The displacement of center of mass (black dot) is calculated by N ⇥ D, with N the number of pixel and D the pixel size of UFXC3.2kdetector.

TABLE S1 .
Elastic parameters and density