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# Tuning photostriction in PbTiO3/SrTiO3 superlattices: An ab initio study

Carmel Dansou, Charles Paillard, Laurent Bellaiche

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Carmel Dansou, Charles Paillard, Laurent Bellaiche. Tuning photostriction in PbTiO3/SrTiO3 superlattices: An ab initio study. Physical Review B, 2024, 110 (22), pp.224303. 10.1103/Phys-RevB.110.224303. hal-04855864

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# Tuning photostriction in PbTiO₃/SrTiO₃ superlattices: An ab-initio study

Carmel Dansou¹, Charles Paillard¹,², Laurent Bellaiche¹,³

¹Smart Ferroic Materials Center, Institute for Nanoscience & Engineering and Department of Physics,
University of Arkansas, Fayetteville, Arkansas 72701, USA

²Université Paris-Saclay, CentraleSupélec, CNRS,
Laboratoire SPMS, 91190 Gif-sur-Yvette, France and

³Department of Materials Science and Engineering,
Tel Aviv University, Ramat Aviv, Tel Aviv 6997801, Israel.

(Dated: February 18, 2025)

Light-induced mechanical deformations in single-domain PbTiO₃/SrTiO₃ superlattices were simulated using first-principle calculations. By varying thickness or the chemical fraction of PbTiO₃, we discover that these heterostructures' photostrictive behavior can be tuned quantitatively and qualitatively. Additionally, we present simple analytical models to explain the calculated deformations and predict a critical PbTiO₃ fraction with no photostriction. Our results present two pathways for tuning the photostrictive behavior of PbTiO₃/SrTiO₃ superlattices, which could be utilized for innovative optomechanical applications.

## I. INTRODUCTION

In the last decade, ferroelectric/dielectric superlattices (FE/DE SLs) have gained significant interest because of their potential in next-generation multifunctional devices [1]. Among these FE/DE SLs, $(PbTiO_3)_n/(SrTiO_3)_m$ have been widely studied as a model system [2–12]. Many peculiar properties have been evidenced in such nanosystems, such as polar vortices and skyrmions or negative capacitance effects [3–12]. Recently, significant photostriction - the light-induced mechanical deformation in a material - has been revealed in these superlattices [13–15], adding yet another dimension to the rich number of their functionalities. Yet, there is little microscopic understanding of the origin of this effect in FE/DE SLs. Moreover, existing experimental reports are limited to as-grown SLs, with no systematic exploration of the parameters involved in their photo-induced deformations. For instance, only light-induced lattice expansion has been reported in the prototype PbTiO₃/SrTiO₃ (PTO/STO) SLs [13, 14]. There exists thus an urgent need to develop models and understanding of the photo-induced deformation in FE/DE SLs.

In this report, we used first-principles calculations to explore the space of parameters susceptible of tuning the photostrictive behavior of the SLs while saving time and avoiding material waste. Firstly, we show how the electrostatic coupling strength between the layers of a PTO/STO SL with fixed chemical composition can strongly change the photostrictive response from a compression to an extension by varying the overall SLs period from 2 to 10 perovskite unit cells. Secondly, we demonstrate how the chemical composition, at fixed SLs period, can also strongly impact the amplitude and sign of the photostriction.

Specifically, first, we consider $(n|n)$ SLs with polarization along the [001] direction, generally referred to as *c* phase in literature [16]. We vary the periodicity *n* from 1 to 5 (and thus the chemical period from 2 to 10 perovskite cells). We thus sample SLs with electrostatic coupling

between the PTO and STO layers varying from strong ($n \le 3$) to weak ($n \ge 3$) as defined in Ref. [12]. Second, we consider $(PbTiO_3)_n/(SrTiO_3)_m$ SLs ($m + n = 10$) where we generalize and extend the discussion of the results on ($n|n$) SLs, changing the chemical period of the SLs. Our aim is to provide a general theoretical understanding into the mechanisms and ways to tune photostriction in these SLs with any chemical composition. In the present report, we aim at uncovering the dependence of photostriction in PTO/STO SLs on its thickness and the PTO volume fraction. Our report is organized as follows. Construction of the simulated PTO/STO SLs and the computational methods are detailed in Sec. II A and Sec. II B. In Sec. III A, we present and discuss the photostrictive behavior of the ($n|n$) SLs. In Sec. III B we explore a phenomenological model to quantify light-induced strains in these SLs. In Sec. III C, we present and analyze the microscopic origin of the photostrictive behavior of the ($n|m$) SLs as function of PTO ratio. In Sec. III D, we first extend the phenomenological model presented in Sec. III B to the ($n|m$) SLs. Additionally, we present another model that highlight the converse piezoelectric response in each layer of the SLs. And we close this report by a summary of our findings and future perspectives. We thus hope that the photostrictive tunability of PTO/STO SLs explored in this article may be of significant interest to the application of FE/DE SLs for optomechanics application and may open new avenues for fundamental explorations.

## II. SYSTEMS AND METHODS

### A. Computational details

Calculations in this report are based on density functional theory (DFT) using the projector augmented wave method [17] as implemented in the Abinit package [18–20]. The Perdew-Burke-Ernzerhof generalized gradient approximation revised for solids is used for the exchange-
correlation functional [21]. The plane-wave energy cutoff is set to 20 Hartree. Integration over the Brillouin zone is done using a Monkhorst-Pack [22] scheme equivalent to 6 × 6 × 6 k-point mesh in a five-atom perovskite cell. To simulate the c phase, the Ti atoms are displaced up along the z axis from their centrosymmetric positions, and the out-of-plane cell dimension, atomic positions and angles are relaxed until forces on the supercell are less than $10^{-8}$ Hartree Bohr$^{-1}$ and the force on ions less than $2 \times 10^{-6}$ Hartree Bohr$^{-1}$. At the end of the relaxation, the obtained structures have the P4mm space group symmetry with polarization along the +$\hat{z}$ direction. Starting from the optimized structure in the dark, photoexcitation is simulated using the two quasi-Fermi levels method to mimic thermalized electrons and holes recently implemented in the Abinit [23] code by constraining a fixed number $n_{ph}$ of electrons/holes to the conduction/valence bands. Under the constraint of $n_{ph}$ electrons/holes in the conduction/valence bands, the structures were relaxed again with the same convergence criteria as in the dark ($n_{ph} = 0$). It should be noted that this constrained occupation scheme does not explicitly include the effect of the polarization of light on the number of photo-excited carriers. However, different absorption coefficients along the principal optical axes would result in different photo-excited carrier concentrations for different polarization directions of the incident light. As such, the effect of the polarization of light on photostriction in the considered systems can be deduced from the knowledge of the full absorption tensor and the photostrictive behavior for varying numbers of photo-excited carriers, which is discussed in the present study.

## B. Systems

To simulate the PTO/STO SLs, we adopt a supercell approach used in many previous studies on FE/DE SLs [4–6, 24, 25]. Stoichiometric SLs are built by stacking alternatively *n* unit cells of PTO on top of *m* unit cells of STO along the [001] direction. To construct the SLs, we first optimized the lattice parameter of both PTO and STO (we have *n* = *m* for the (*n*|*n*) SLs). The optimized values are summarized in the table I below, showing excellent agreement with previous experimental [26–32] as well as theoretical values [33–36].

Before structural relaxation, atoms are placed in their ideal cubic positions, and the out-of-plane lattice constant of the supercell is set to ($ma_{STO} + na_{PTO}$), with $a_{STO}$ and $a_{PTO}$ being the theoretical optimized values of cubic PTO and STO, respectively. A schematic of the relaxed (5|5) SLs is shown in Fig. 1. Only SLs with polarization along the [001] direction commonly denoted the c phase in literature are considered in this study [16]. Structural, electronic and physical properties of the c phase of PTO/STO SLs are well documented [4, 7, 8, 10, 12, 25, 37, 38]. Additionally, we build reference paraelectric phases for the calculation of the layer-

<table><caption>TABLE I. Optimized lattice parameters of PTO and STO</caption><thead><tr><th colspan="4">Pm<span style="text-decoration: overline">3</span>m</th></tr><tr><th></th><th>This work</th><th>Other</th><th>Expt</th></tr></thead><tbody><tr><th scope="row">a<sub>STO</sub> (Å)</th><td>3.897</td><td>3.904 [33]</td><td>3.905 [26–28]</td></tr><tr><th scope="row">a<sub>PTO</sub> (Å)</th><td>3.928</td><td>3.969 [35]</td><td>3.926 [29]</td></tr></tbody><tbody><tr><th colspan="4">P4mm (PbTiO<sub>3</sub>)</th></tr><tr><th scope="row">a (Å)</th><td>3.885</td><td>3.888 [34, 35]</td><td>3.905 [30–32]</td></tr><tr><th scope="row">c/a</th><td>1.071</td><td>1.074</td><td>1.064</td></tr></tbody></table>

by-layer polarization. We aim at simulating PTO/STO SLs grown on an STO substrate as commonly done in experiment. We therefore fix the in-plane lattice constant of the SLs to the theoretical computed value for STO. We did not consider oxygen octahedral tilts modes in our simulation as they would require considering ($\sqrt{2} \times \sqrt{2} \times 1$) supercell and thus doubling the volume of the unit cell making our simulations computationally involved especially during photoexcitation. Also only single domain SLs are considered in this study. Due to the complexity of simulating polydomain SLs structures within DFT, the single domain approximation is usually used even at the thickness considered in the presence study [4, 6, 25, 39]. Monodomain structures still capture the main qualitative feature of the polarization and the coupling between layers of the superlattices (as we will show, the coupling between layer is one of the essential factors driving the qualitative photostrictive behavior of the superlattice)

## III. RESULTS AND DISCUSSION

### A. Photostriction as function of PTO/STO Superlattices thickness

We start by discussing the photostrictive behavior of the ($n|n$) SLs. We vary the periodicity *n* from 1 to 5 (and thus the chemical period from 2 to 10 perovskite cells). We thus sample SLs with electrostatic coupling between the PTO and STO layers varying from strong ($n \le 3$) to weak ($n \ge 3$) as defined in Ref. [12]. Let us first look at the photo-induced strain $\eta_{33} = (c(n_{ph}) - c(0))/c(0)$ (where $c(n_{ph})$ is the out-of-plane lattice constant at the concentration of photo-excited carriers $n_{ph}$) as a function of *n* in the ($n|n$) SLs. As shown in Fig. 2, one can distinguish two behaviors when focusing on the sign of the light-induced strain. For the ultrathin chemical period ($n = 1, 2$) SLs i.e., in the strong electrostatic coupling regime of the PTO and STO layers, light induces a compression, i.e. negative photostriction (see Fig 2(a) for
<figure><img src="image_3.png" /><figcaption>FIG. 1. Vesta schematic of the relaxed P4mm (PbTiO$_{3}$)$_{5}$/(SrTiO$_{3}$)$_{5}$ superlattices. The direction of the polarization is shown by the black arrow.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 2. a) Lattice deformation in the (n|n) SLs as function of $n_{ph}$. b) Average c/a of the PTO layer as function of $n_{ph}$, c) Average c/a of the STO layer as function of $n_{ph}$.</figcaption></figure>

different chosen values of $n_{ph}$). In contrast, photostric- tion is positive (i.e., the SLs expands in the out-of-plane direction) for thicker SLs $n \ge 3$ (i.e., in the weak electro- static coupling regime of the PTO and STO layers). To investigate in more details these two regimes, we exam- ine the light-induced tetragonal distortion in the different layers. Consequently, in Fig 2(b) & c) is shown the av- erage tetragonal distortion in the PTO and STO layers of the SLs as a function of the concentration of photoex- cited carriers for all values of n. STO is an incipient ferroelectric material [40]. However, the proximity with PTO induces, in DFT calculations, a polarization in the STO layer (via an internal polarizing field) and thus a significant tetragonal distortion. For utrathin SLs with $n = 1, 2$, both the distortion in the PTO and STO layers (black and red curves in Figs 2 b&c) decrease under opti- cal excitation leading to the overall observed compression in the out-of-plane lattice constant; a feature reminiscent of the behavior of bulk ferroelectric titanates [23, 41]. On the other hand, in the SLs $n \ge 3$, upon optical excita- tion, the distortion in the STO layer decreases whereas the distortion in the PTO layer increases (see blue, green and purple curves in Fig. 2 b&c). The positive distor-

tion in the PTO layers dominates and thus the overall expansion is observed. The data depicted in Fig. 2 thus suggest that the photostrictive response (negative or pos- itive) of the PTO layer and the SL as a whole is strongly dependent on the strong or weak electrostatic coupling between the layers.

We now examine the evolution of the electrical polar- ization of the SLs as a function of the concentration of photo-excited carriers. The polarization in the SLs is computed in a layer-by-layer fashion using the following equation:

$$P_i = \frac{e}{\Omega_i} \sum_{j \in i} w_j \delta_j Z_j^* \quad (1)$$

where $\Omega_i$, $e$, and $\delta_j$ are the volume of the local unit cell $i$ (local unit cell is used here to refer to an effective five atoms unit cell centered on a Ti atom), the elementary charge and the atomic displacement from a built refer- ence nonpolar SLs (we built this reference structure for each of the SLs) respectively. The weight factor $w_j$ tells how many of such local cells share the $j$ ions. Its values are 1/2, 1, 1/8 for the O, Ti and (Pb,Sr) atoms respec- tively. $Z_j^*$ is the z component of the Born effective charge
of atom j, with j running through all the atoms in the local cell i. The Born effective charges $Z_j^*$ are those of the atoms in dark conditions and are taken from Ref. [42]. We do not expect a significant alteration of $Z_j^*$ under the concentration of photo-excited carriers investigated here, and thus do not employ the more refined approach devised in Ref. [43]. The total polarization in the SLs is the average of the polarization in PTO and STO layers. As shown on Fig 3 for selected SLs (See Supplementary Material [44] for the other SLs), optical excitation reduces the total polarization in ultra-thin ($n|n$) SLs ($n = 1$ and $n = 2$). Additionally, polarization in both the STO and PTO layers decreases upon optical excitation. This is consistent with the decrease of the lattice distortion shown on Fig 2 (b& c) for the (1|1) and (2|2) SLs. When the SLs layers are thick enough, that is for $n \ge 3$, the overall polarization is enhanced by optical excitation as shown by the black curves on Figs 3 (b&c). Additionally, the polarization in the PTO layer increases (see red curves on Fig 3 b & c) whereas the induced polarization in the STO layer is decreased (see blue curves on Fig 3 b & c) – implying that STO becomes less polar under light. **Notice that the ratio c/a in each layer of the SLs is a direct probe of polarization in the layer. As such, polarization in the layer and the c/a ratio is correlated. We, however, show them separately for clearer understanding and to show consistency across these quantities under varying photoexcited carrier concentration, which may not be trivial to capture by a single plot.**

To gain further insight on the role of the free charges in the above observed changes in the lattice constant and polarization inside the SLs, we computed the macro-average [45] excited charges density and the electrostatic potential inside the SLs under illumination for the considered ($n|n$) SLs. As shown in Fig 4 (a)-(b), in the ultrathin ($n|n$) SLs ($n = 1$ or $n = 2$), the photo-excited carriers are more delocalized through the SLs than for $n \ge 3$. When the thickness of the films is increased ($n$ is equal or greater than 3), the free charges become strongly localized on TiO$_2$ planes forming the interface, as shown in Fig. 4 (a) to (c) for the (1|1), (3|3) and (5|5) SLs respectively (see the Supplemental Materials [44] for (2|2) and (4|4) SLs). The spatial distribution of the free charges in the ($n|n$) SLs provide us with the following pictures: the ultrathin SLs, where there is a strong electrostatic coupling between PTO and STO layers, behave like bulk system [23, 41, 46] where photo-excited carriers are spread out within the SL and lead to an overall decrease of polarization and thus out-of-plane lattice compression in both PTO and STO layers. For thick enough SLs, the free charges strongly localize at the interfaces. Electrons and holes populate given interfaces and produce fields that oppose the existing field in the STO and PTO layers within the SLs. The light-induced field in STO thus opposes the polarizing field and leads to the decrease in the polarization in STO and to the observed compression of this specific layer. On the other hand,

the light-induced field in PTO opposes the depolarizing field there too, which then results in an increase of the electrical polarization, therefore generating the observed expansion of the PTO layer (see the Supplemental Materials [44] for a schematic). We further elucidate this by mapping out the electrostatic potential in the SLs before and after illumination (see Fig. 4d for the (4|4) SLs). In dark, the electrostatic potential has a non-zero slope in both the STO and PTO layers (see blue curve in Figure 4d). This suggests the presence of a finite field as postulated above. Upon illumination, the electrostatic potential flattens leading to vanishingly small electric fields within layers of the SLs (see orange curve in Fig 4d in the case of the (4|4) SLs). Essentially, in thick SLs, photo-excitation provides enough free carriers to screen the polarization bound charges and quench the resulting local electrostatic fields in each layer.

## B. Analytical models of the photostriction in the ($n|n$) SLs

Now, we aim at finding a simple analytical model to match the light-induced strain from first principle calculations. Previous studies based on Landau-Guinzburg-Devonshire (LGD) theory have shown that, when the direction of the polarization is locked along the z axis (the argument is also true for the other axes), the tetragonality of the SLs can serve as a probe of polarization of the SLs [7, 8]. It was established that the polarization is proportional to an exponent of the tetragonality $\eta_T = (c - c_p)/a$ with c and $c_p$ being the out-of-plane lattice constant of the polar SLs and the corresponding paraelectric SLs, and a the in-plane lattice parameter of the SLs. Earlier works on single domain PTO/STO SLs in dark have shown that $P_z \propto \eta_T^{1/2}$ [8], however, in a recent work, it was found that $\alpha$ differs from 1/2 when free charges screening effect is considered. The study found that the values of $\alpha$ depends on the SLs, its structure and on the pump used for the excitation. Typically, in presence of free charge, it was found that $P_z \propto \eta_T^{-1}$ [14] for a polydomain (PbTiO$_3$)$_8$/(SrTiO$_3$)$_3$ SLs. In view of these previous works, we plot $\ln(P_z(n_{ph}))$ as function $\ln(\eta_T(n_{ph}))$ for the considered SLs. As shown on the top panel of Fig 5 (shown for (1|1), (3|3) and (5|5) SLs), there is a linear relation between $\ln(P_z(n_{ph}))$ and $\ln(\eta_T(n_{ph}))$ – implying that $P_z \propto \eta_T^\alpha = [(c - c_p)/a]^\alpha$. We obtained $\alpha$ (0.65, 0.56, 0.40, 0.62 and 0.49 for the (1|1), (2|2), (3|3), (4|4) and (5|5) respectively) by taking the slope of a linear fit into $\ln(P_z(n_{ph}))$ vs $\ln(\eta_T(n_{ph}))$ for each SLs.

$$P_z \propto \eta_T^\alpha = [(c - c_p)/a]^\alpha. \quad (2)$$

Taking the logarithmic derivative of Eq 2, gives the following:

$$\frac{dP_z}{P_z} = \alpha \frac{dc}{(c - c_p)}. \quad (3)$$
<figure><img src="image_5.png" /><figcaption>FIG. 3. Top panel: polarization as a function of $n_{ph}$ in the $(n|n)$ SLs. a)-c) (1|1), (3|3), and (5|5) SLs respectively. Bottom panel: projection to five-atom unit cell of the polarization in the d)-f) (1|1), (3|3), and (5|5) SLs respectively. The vertical dashed lines in e) and f) are to separate local cells corresponding to PTO and STO layer in each SLs.</figcaption></figure>

From Eq 3, it can be readily established that the strain $dc/c$ in the out-of-plane lattice constant is given by

$$dc/c = \eta_{33} = \frac{\eta_d}{\alpha} \frac{dP_z}{P_z}. \quad (4)$$

where we define $\eta_d = (c - c_p)/c$. The scaling relation in Eq. 4 provides us with a simple analytical expression of the light induced strain $\eta_{33}$ as function of the light-induced relative change in the polarization. The estimated strains using Eq. 4, along with the DFT data, are shown on the bottom panel in Fig 5 for each of the SLs. It is in excellent agreement with the DFT data. Also we plot the estimated strain using reported values of $\alpha$ from Refs. [8, 14] for comparison.

Although not the focus of this study, we would like to comment on the obtained values of $\alpha$. As found in Ref. [14], it would be expected that $\alpha$ depends on the SLs. Also one would expect the values of $\alpha$ to deviate from 1/2 as we find in our study. However, with the low excited carriers concentration considered in this study and the simple structure of the SLs, we do not expect a drastic deviation from 1/2, consistent with the range of values 0.4 – 0.65 found for the considered $(n|n)$ corresponding to a maximum deviation of 0.15 from 0.5. As discussed previously in Ref. [14], the physical significance of $\alpha$ is unclear to us. It would interesting to investigate in more details the effect of free charge on $\alpha$ and its behavior as function of the SLs.

We finish the discussion on the photostriction in the $(n|n)$ SLs by commenting on their quantitative photo-

strictive advantage over bulk FE materials. For the considered $(n|n)$ SLs and the examined range of excited carrier concentrations, the lowest light-induced strain is observed in the (2|2) SL, which is –0.012% at $n_{ph}$ = 0.05e/f.u. This value is 1.2 times higher than the theoretical values reported for bulk ferroelectric materials such as BiFeO$_3$, PbTiO$_3$, and BaTiO$_3$ [36, 46], as well as experimental findings for BiFeO$_3$, which range between $10^{-2} – 10^{-3}\%$. The most considerable deformation (0.23%) occurs in the (5|5) SL at $n_{ph}$ = 0.20 e/f.u, which is 20 – 200 times higher than the values reported for bulk FE mentioned above and about twice the reported value for 2D ferroelectrics (0.1%) [47]. However, the largest photostriction in the (5|5) SL is smaller than the recent predicted attainable 0.9% in BiFeO$_3$ film [48], and previously reported 1.1% in SrCoO$_3$ [49], 1.12% in SrRuO$_3$ [50], 1% in SrIrO$_3$ [51] as well as predicted photostriction (as large as 9.2%) in halide perovskites [52, 53]

### C. Photostriction as function of PTO volume fraction

In this section, we look at the photostrictive behavior of the $(n|m)$ SLs $(m + n = 10)$ as a function of the PTO fraction. As shown in Fig. 6, the photostrictive behavior of the $(n|m)$ SLs is dependent on the PTO fraction. Like for the $(n|n)$ SLs, two behaviors can be distinguished when we look at the sign of the light-induced deformation in the out-of plane lattice of the $(n|m)$ SLs. As shown
<figure><img src="image_6.png" /><figcaption>FIG. 4. Excited carriers density for a) (1|1) SLs, b) (3|3) SLs, c) (5|5) SLs, computed at $n_{ph} = 0.4$ e/f.u. (d) Electrostatic potential in dark (blue) and under illumination (orange) for the (4|4) SLs. Plots in a)- c) are the difference between the macro-average density under illumination and in dark.</figcaption></figure>

in Fig. 6, for PTO fractions less than a critical fraction $x_c \approx 0.34$ (estimated from the graphs on Fig. 6; this value is also independent of $n_{ph}$), the SLs shows negative photostriction ($\eta_{33} < 0$) while for fraction between $x_c$ and 0.9, the SLs shows positive photostriction ($\eta_{33} > 0$). Note also that beyond $x_c$, the light-induced deformation in the SLs, for each considered $n_{ph}$, as function of PTO fraction, reaches a maximal value at $x = 0.6$ and subsequently decreases back to negative values in pure bulk PTO, consistent with previous calculations [36, 46]. The layer-by-layer decomposition of the photo-induced average tetragonality ($c/a$) shows that, for PTO fraction $x \le 0.2$, optical excitation penalizes distortions in both PTO and STO layers (see Fig 7a where we show data for the (2|8) SLs), similar to (1|1) and (2|2) SLs. On the other hand, for PTO fraction $x \ge 0.3$, the free carriers decrease distortion in STO layer while they enhance distortion in the PTO layer (see Figs 7b, c& d for selected SLs, see [44] for the other SLs), similar to our observation in the $(n|n)$ SLs with $n \ge 3$. On Fig 8, we show the estimated total polarization as well as polarization in each layer for selected SLs as function of $n_{ph}$ (See [44] for the rest of the SLs). The decomposed polarization per local five atoms unit cell of the SLs are also shown in Fig 9 (we only show for selected SLs, see the rest in

the supplemental materials). For PTO fraction $x \le 0.2$ optical excitation reduces polarization in both PTO and STO layer and thus the reducing of the total polarization in the SLs, see Figs 8a and Figs 9a. When the fraction of PTO is increased above 0.2 ($x \ge 0.3$), polarization in STO is decreased by optical excitation whereas polarization in PTO is enhanced, see Figs 8 and Figs 9 (b,c,d). The total polarization of the SLs decreases for $x \le 0.3$ and increases for $x \ge 0.4$ PTO fraction. These light-induced change in the polarization in the SLs are in line with the observations from the lattice distortion in the SLs discussed above. The overall light-induced strain in these SLs is thus the resultant of the light-induced strain in PTO layer and STO layer as shown in the piezoelectric model in section IIID).

We also compute the macro-average excited charges density and the electrostatic potential inside the SLs under illumination for the different $(n|m)$ combinations. The results are shown in the supplemental materials [44] as they are similar to the case of the $(n|n)$ SLs. In the considered SLs, the free charges are strongly localized at the interface between the two layers on the $TiO_2$ group. In addition, from the plot of the electrostatic potential, it can be noticed that there is a finite field in each layer of the SLs in the dark. Under illumination, these lo-
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>FIG. 5. Top panel a) to c) plot of ln(*P*) vs ln(*η*<sub>T</sub>) for the (1|1), (3|3) and (5|5) SLs respectively. Bottom panel d) to f) plot of the DFT data and fit using eq 3 for different values of α for the (1|1), (3|3) and (5|5) SLs respectively.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>FIG. 6. Light-induced strains as function of PTO fraction in the (*n*|*m*) SLs.</figcaption></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>FIG. 7. a) Average c/a for PTO and STO layer in the (2|8) SLs, b) Average c/a for PTO and STO layer in the (3|7) SLs, c) Average c/a for PTO and STO layer in the (7|3) SLs, d) Average c/a for PTO and STO layer in the (8|2) SLs as function of $n_{ph} = 0.0 - 0.2$ e/f.u.</figcaption></figure>

cal fields are quenched. Therefore, the action of light in the (*n*|*m*) SLs is intrinsically the same as in the (*n*|*n*) SLs discussed above. The decrease of the field in the STO layer translates into the observed compression in the layer. In the PTO layer, it could be expected that the screening of the depolarizing field will induce an expansion of the layer for all the (*n*|*m*) SLs. It can be noted, however, that for PTO fraction less than 0.3, the free charges significantly tail in the PTO layer, making the

layer strongly photo-doped, which leads to the decrease in its polarization that translates into the observed contraction. For $x \ge 0.3$, the free charges become more and more strongly localized at the interface as the number of PTO cells increases, and the depolarizing field screening overcomes the dipole screening due to photo-doping, thus causing the layer to expand. Similarly to the (*n*|*n*)
<figure><img src="image_18.png" /><figcaption>FIG. 8. Polarizations as function of number of excited carriers for the a) (2|8) SLs, b) (3|7) SLs, c) (7|3) SLs, d) (8|2) SLs.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>FIG. 9. Layer-by-layer decomposition of the polarizations as function of number of excited carriers for the a) (2|8) SLs, b) (3|7) SLs, c) (7|3) SLs, d) (8|2) SLs. are to separate local cells corresponding to PTO and STO layer in each SLs.</figcaption></figure>

SLs, the overall observed strain in the SLs results from these light-induced microscopic actions in each layer of the SLs. Note that the reports in Refs [13, 14] explained the observed lattice expansion in term of depolarizing field screening in PTO/STO SLs without much emphasis on of the light induced effect at the layer level. We believe our proposed explanation is more nuanced and thus provide a general microscopic picture of the effect of light on the lattice PTO/STO SLs. This seems to be a general feature of FE/DE SLs, since a recent experiment on (2|4) BaTiO₃/CaTiO₃ SLs showed that optical excitation induced compression in CaTiO₃ and oppositely an expansion in BaTiO₃ [54].

While the lone pair is known to participate in the

stability of polar distortion's stability in bulk PTO, it plays little role in the observed light induced strain observed in PTO/STO superlattices. As shown on the projected density of state of the (5|5) SLs (see [44]), the Pb 6s and 6p from which the lone pair originate are far from the top/bottom of the valence and conduction band of the SLs where free carriers are excited from. As explained above, the driving force behind the behavior in each layer is the electrostatics induced by light that leads to strain via the converse piezoelectric effect, which is well-captured in the second model proposed in the manuscript. This picture is much consistent with existing experimental reports

### D. Analytical models of the photostriction in the ($n|m$) SLs

$P_z \propto \eta_T^\alpha$ model: To describe the light-induced mechanical response within a phenomenological method, we applied the $P_z \propto \eta_T^\alpha$ model, described in Sec. IIIB, to each of the ($n|m$) SLs. Recall that in this phenomenological model, the light-induced strain is related to the relative change in the polarization via Eq. 4. As described above (see Sec. IIIB) for the ($n|m$), for each of the ($n|m$) SLs, we obtained the value of $\alpha$ by a linear fit of $\ln(P_z(n_{ph})) \propto \ln(\eta_T(n_{ph}))$ (see the supplemental materials for the values of $\alpha$). On figure 10, we show the results (for selected SLs, the results for other SLs can be found in the Supplemental Material [44]) of our fit along with the estimation of the strain using $\alpha = 0.5$ and $\alpha = 1.1$ for comparison. The model with estimated

<figure><img src="image_20.png" /><figcaption>FIG. 10. Different fits of the DFT data ($\eta_{33}$ vs $n_{ph}$) using Eq 1 for a) (2|8) SLs, b) (3|7) SLs, c) (7|3) SLs.</figcaption></figure>

$\alpha$ from the scaling relation fits better the DFT data for the corresponding SLs. As discussed **for the ($n|m$) SLs** and by Lee *et al.* [14], we are unable to attach a physical meaning to $\alpha$ and its changes as function of photoexcited charges and the type of SLs. We leave this as an
interesting future exploration.

**Converse piezoelectric model:** Here, we aim at finding another simple model to describe and reproduce our DFT data **on the** ($n|m$) **SLs.** The total light-induced strain can be approximated as the weighted average of the deformation in STO and PTO layers in the SLs (see Supplemental Material [44] for details and discussion of the derivation):

$$ \eta_{33} \approx x\delta\eta_{33}^p + (1-x)\delta\eta_{33}^s \quad (5) $$

where $x$ is the volume fraction of PTO in the SLs. $\delta\eta_{33}^p$ is the light-induced strain in the PTO layer and $\delta\eta_{33}^s$ the light-induced strain in the STO layer. Note that the volume fraction $x$ changes as function of $n_{ph}$. Introducing the changes in the polarization ($\delta P_1 = P_1(n_{ph} \neq 0) - P_1(n_{ph} = 0)$) for PTO and ($\delta P_2 = P_2(n_{ph} \neq 0) - P_2(n_{ph} = 0)$) for STO), and writing the following converse piezoelectric relation between strain and change in polarization [36]:

$$ \delta\eta_{33}^p = \epsilon_0^{-1} \frac{d_{33}^1}{\chi_{33}^1} \delta P_1; \quad \delta\eta_{33}^s = \epsilon_0^{-1} \frac{d_{33}^2}{\chi_{33}^2} \delta P_2, \quad (6) $$

we have:

$$ \eta_{33} \approx \epsilon_0^{-1} \left[ x \left( \frac{d_{33}^1}{\chi_{33}^1} \delta P_1 - \frac{d_{33}^2}{\chi_{33}^2} \delta P_2 \right) + \frac{d_{33}^2}{\chi_{33}^2} \delta P_2 \right]; \quad (7) $$

where ($d_{33}^1$, $\chi_{33}^1$) and ($d_{33}^2$, $\chi_{33}^2$) are the coupled piezoelectric constants, dielectric susceptibility of the PTO layer and the STO layer respectively. Note that although STO is cubic in its bulk structure, it becomes tetragonal, with a finite internal field, in the SLs due to its proximity with the PTO layer and thus has a nonvanishing piezoelectric constant $d_{33}$. The quantities $d_{33}^i$, and $\chi_{33}^i$ change with illumination and their values can be computed, in principle, within density functional perturbation theory (DFPT). However, the current implementation of DFPT in Abinit does not account for the presence of free charge in the conduction band. As such, we define the ratio $r^i = d_{33}^i/\chi_{33}^i$ and estimate $r^i$ from the DFT data. The estimation of $r^i$ are detailed in the Supplemental Materials [44]. Eq. 7 can therefore be rewritten as:

$$ \eta_{33} \approx \epsilon_0^{-1} \left[ x (r^1 \delta P_1 - r^2 \delta P_2) + \delta P_2 r^2 \right] \quad (8) $$

As observed in the DFT data, this model predicts a critical PTO fraction for a critical SLs that is at the boundary of negative and positive photostrictive SLs:

$$ x_c = \frac{1}{1 - \frac{\delta P_1 r^1}{\delta P_2 r^2}} \quad (9) $$

Interestingly, the DFT data suggests that this critical volume fraction is independent of $n_{ph}$. On Fig 11, we show the results from this model and the DFT data for $n_{ph} = 0.00-0.20$ e/f.u for few selected SLs (the rest of the

SLs is shown in the Supplemental Material [44]). For such a simple model, it agrees rather well and captures the underlying physics. The model, however, fails to quantitatively capture the light induced strains at higher $n_{ph}$ for some of the SLs. This is because at high $n_{ph}$, it can be noticed that the photostrictive behavior of some of the SLs becomes very nonlinear and a rigorous calculation of the $r^i$ ratio in Eq. 4 using DFPT might improve the model at large $n_{ph}$. Also, note that this model intrinsically depends on $c/a$ in each layer of the SLs. It serves primarily as a qualitative tool for understanding the relationship between light-induced strain and change in polarization in the SLs under illumination. It can be easily adapted to a new system. However, it should be used cautiously and the different ratios ($r^i$) involved in Eq. 8 should be computed for the new system. We also

<figure><img src="image_21.png" /><figcaption>FIG. 11. Different fits of the DFT data ($\eta_{33}$ vs $n_{ph}$) using Eq 3 for a) (2|8) SLs, b) (3|7) SLs, c) (7|3) SLs, d) (8|2) SLs.</figcaption></figure>

would like to relate this study to the two only existing experimental studies we are aware of on light-induced strain in PTO/STO SLs. In Refs. [13, 14], a film of 8 units of PTO and 3 units of STO was used. This corresponds to roughly a 0.73 PTO fraction. They all reported an overall positive photostrictive behavior, qualitatively consistent with the observation in our study. Interestingly, changing the PTO chemical fraction shows that maximum light-induced strain is achieved at a 60% PTO composition and a strain as high as 0.48% can be reached.

## IV. CONCLUSION AND PERSPECTIVES

In summary, using DFT within a constrained occupation scheme, we investigate photostriction in single domain PTO/STO SLs with varying thickness and chemical composition. We considered ($n|n$) PTO/STO SLs ($n=1-5$), and we show that, for very thin SLs ($n \le 2$), free carriers are delocalized and screen existing dipoles, which induces contractions in the out-of-plane lattice of
the SLs. On the other hand, when the layers are thick enough ($n \ge 3$), the free charges are localized in the interfaces regions. They produce a field that opposes existing depolarizing fields in the SLs and thereby induces a contraction in the STO layers and oppositely an expansion in the PTO layers and leads to an overall expansion of the SLs. These show that the photostrictive behavior of PTO/STO SLs can be tuned through its thickness. Subsequently, we investigate the photostrictive behavior as function of the PTO fraction in the SLs. For that, we consider ($n|m$) PTO/STO SLs ($n + m = 10$). We find that, for PTO fraction $x \le 0.2$, both layers of the SLs contract upon optical excitation, leading an overall contraction of the SLs. On the other hand, for $x \ge 0.3$, optical excitation induces a contraction in the STO layers and an expansion in the PTO layers. Depending on the fraction of PTO in the SLs, an overall contraction or expansion is observed. Our study shows that the photostrictive behavior of these SLs can be tuned not only through the SLs composition but also its thickness. The possibility of tuning the light-induced strain by simply changing the SLs composition or thickness is appealing as it presents potential for novel applications using light as modulator.

In the present study, we restrict ourselves, because of the computational cost of simulating optical excitation in large system, to single domain PTO/STO SLs with no octahedral tilts (oxygen octahedral tiltings only appear in STO below 105K [55] and does not exist in pure PTO. Additionally existing experimental studies[13–15] were done at much higher temperature and did not consider any octahedral tiltings, thus neglecting their effect is a realistic consideration). Also our supercells consist at most of a total of 10 five-atom perovskite cells. Consequently, this choice limits the predictive power of our studies and a quantitative comparison with reported experimental values in Refs.[13, 14] can be misleading. However, note that our results are qualitatively in line with these reports. Additionally our study suggests that the PTO ratio used in existing experiment does not lead

to optimal photostrictive response. Larger light-induced strain can be obtained by changing the chemical composition of PTO/STO SLs. As such, we believe that, through this investigation, we have provided first-principle insights into the photostrictive behavior of PTO/STO SLs and our studies can be used by experimentalists as a starting point to optimize the photostrictive response of PTO/STO SLs. For a full fundamental exploration of the photostrictive behavior in PTO/STO SLs, various extension direction can be under taken. A straightforward future development is to extend these studies to thicker SLs (i.e., SLs with more than 10 five-atoms perovskite cells) and investigate how the dependence on PTO chemical fraction of the photostrictive behavior (shown on Fig 6) scales with the SLs thickness. Another approach would be to extend this study to SLs with domain structures, octahedral tilts or super-orders [56, 57]. Expanding on the ideas presented in our reports in these directions offers an interesting avenue for future explorations. However, pursuing these ideas may require treating the “constrained occupation scheme” in an effective way through large scale methods such as effective Hamiltonians or second-principles approaches, as simulating photoexcitation in these large systems using DFT becomes computationally cumbersome.

## ACKNOWLEDGMENTS

The work is supported by the ARO Grants No. W911NF-21-2-0162 (ETHOS) and W911NF-21-1-0113, and the Vannevar Bush Faculty Fellowship (VBFF) grant no. N00014-20-1-2834 from the Department of Defense. C.P. acknowledges the support from a public grant overseen by the French Agence Nationale de la Recherche under grant agreement no. ANR-21-CE24-0032 (SUPERSPIN). We also acknowledge the computational support from the Arkansas High Performance Computing Center (AHPCC) for computational resources.

[1] S. Das, Z. Hong, M. McCarter, P. Shafer, Y. T. Shao, D. Muller, L. Martin, and R. Ramesh, A new era in ferroelectrics, APL Mater. 8, 120902 (2020).

[2] J. Jiang, X. Pan, W. Tian, C. Theis, and D. Schlom, Abrupt PbTiO₃/SrTiO₃ superlattices grown by reactive molecular beam epitaxy, Appl. Phys. Lett. 74, 2851 (1999).

[3] A. Y. Abid, Y. Sun, X. Hou, C. Tan, X. Zhong, R. Zhu, H. Chen, K. Qu, Y. Li, M. Wu, J. Zhang, J. Wang, K. Liu, X. Bai, D. Yu, X. Ouyang, J. Wang, J. Li, and P. Gao, Creating polar antivortex in PbTiO₃/SrTiO₃ superlattice, Nature Commun. 12, 1 (2021).

[4] P. Aguado-Puente, P. Garcia-Fernandez, and J. Junquera, Interplay of couplings between antiferrodistortive, ferroelectric, and strain degrees of freedom in monodomain pbtio₃/srtio₃ superlattices, Phys. Rev. Lett.

107, 217601 (2011).

[5] P. Aguado-Puente and J. Junquera, Structural and energetic properties of domains in PbTiO₃/SrTiO₃ superlattices from first principles, Phys. Rev. B 85, 184105 (2012).

[6] E. Bousquet, M. Dawber, N. Stucki, C. Lichtensteiger, P. Hermet, J. M. T. S. Gariglio, and P. Ghosez, Improper ferroelectricity in perovskite oxide artificial superlattices, Nature 452, 732 (2008).

[7] M. Dawber, C. Lichtensteiger, M. Cantoni, M. Veithen, P. Ghosez, K. R. K. Johnston, and J. M. Triscone, Unusual behavior of the ferroelectric polarization in PbTiO₃/SrTiO₃ superlattices, Phys. Rev. Lett. 95, 177601 (2005).

[8] M. Dawber, N. Stucki, C. Lichtensteiger, S. Gariglio, P. Ghosez, and J. M. Triscone, Tailoring the properties of
artificially layered ferroelectric superlattices, Adv. Mater. **19**, 4153 (2007).

[9] S. Das, Y. Tang, Z. Hong, M. Gonçalves, M. Mc-Carter, C. Klewe, K. Nguyen, F. Gómez-Ortiz, P. Shafer, E. Arenholz, V. A. Stoica, S. L. Hsu, B. Wang, C. Ophus, J. F. Liu, C. T. Nelson, S. Saremi, B. Prasad, A. B. Mei, D. G. Schlom, J. Íñiguez, P. García-Fernández, D. A. Muller, L. Q. Chen, J. Junquera, L. W. Martin, and R. Ramesh, Observation of room-temperature polar skyrmions, Nature **568**, 368 (2019).

[10] P. Zubko, N. Jecklin, N. Stucki, C. Lichtensteiger, G. Rispens, and J. M. Triscone, Ferroelectric domains in PbTiO₃/SrTiO₃ superlattices, Ferroelectrics **433**, 127 (2012).

[11] P. Zubko, J. C. Wojdel, M. Hadjimichael, S. Fernandez-Pena, I. L. A. Sené, J. M. Triscone, and J. Íñiguez, Negative capacitance in multidomain ferroelectric superlattices, Nature **534**, 524 (2016).

[12] P. Zubko, N. Jecklin, A. Torres-Pardo, P. Aguado-Puente, A. Gloter, C. Lichtensteiger, J. Junquera, O. Stéphan, and J.-M. Triscone, Electrostatic coupling and local structural distortions at interfaces in ferroelectric/paraelectric superlattices, Nano Lett. **12**, 2846 (2012).

[13] Y. Ahn, J. Park, A. Pateras, M. B. Rich, Q. Zhang, P. Chen, M. H. Yusuf, H. Wen, M. Dawber, and P. G. Evans, Photoinduced domain pattern transformation in ferroelectric-dielectric superlattices, Phys. Rev. Lett. **119**, 057601 (2017).

[14] H. J. Lee, Y. Ahn, S. D. Marks, E. C. Landahl, S. Zhuang, M. H. Yusuf, M. Dawber, J. Y. Lee, T. Y. Kim, S. Unithrattil, S. H. Chun, S. Kim, I. Eom, S. Y. Park, K. S. Kim, S. Lee, J. Y. Jo, J. Hu, and P. G. Evans, Structural evidence for ultrafast polarization rotation in ferroelectric/dielectric superlattice nanodomains, Phys. Rev. X **11**, 031031 (2021).

[15] D. Daranciang, M. J. Highland, H. Wen, S. M. Young, N. C. Brandt, H. Y. Hwang, M. Vattilana, M. Nicoul, F. Quirin, J. Goodfellow, et al., Ultrafast photovoltaic response in ferroelectric nanolayers, Phys. Rev. Lett. **108**, 087601 (2012).

[16] N. Pertsev, A. Zembilgotov, and A. Tagantsev, Equilibrium states and phase transitions in epitaxial ferroelectric thin films, Ferroelectrics **223**, 79 (1999).

[17] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B **50**, 17953 (1994).

[18] X. Gonze, B. Amadon, G. Antonius, F. Arnardi, L. Baguet, J.-M. Beuken, J. Bieder, F. Bottin, J. Bouchet, E. Bousquet, et al., The abinit project: Impact, environment and recent developments, Comput. Phys. Commun. **248**, 107042 (2020).

[19] F. Bottin, S. Leroux, A. Knyazev, and G. Zérah, Large-scale ab initio calculations based on three levels of parallelization, Comput. Mater. Sci. **42**, 329 (2008).

[20] A. H. Romero, D. C. Allan, B. Amadon, G. Antonius, T. Applencourt, J. B. L. Baguet, F. Bottin, J. Bouchet, E. Bousquet, F. Bruneval, G. Brunin, M. C. D. Caliste, J. Denier, C. Dreyer, P. Ghosez, M. Giantomassi, Y. Gillet, O. Gingras, D. R. Hamann, G. Hautier, F. Jollet, G. Jonard, A. Martin, H. P. C. Miranda, F. Naccarato, G. Petretto, N. A. Pike, V. Planes, S. Prokhorenko, T. Rangel, F. Ricci, G. M. Rignanese, M. Royo, M. Stengel, M. Torrent, M. J. van Setten, B. V.

Troeye, M. J. Verstraete, J. Wiktor, J. W. Zwanziger, and X. Gonze, Abinit: Overview and focus on selected capabilities, J. Chem. Phys. **152**, 124102 (2020).

[21] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Restoring the density-gradient expansion for exchange in solids and surfaces, Phys. Rev. Lett. **100**, 136406 (2008).

[22] H. J. Monkhorst and J. D. Pack, Special points for brillouin-zone integrations, Phys. Rev. B **13**, 5188 (1976).

[23] C. Paillard, E. Torun, L. Wirtz, J. Íñiguez, and L. Belaiche, Photoinduced phase transitions in ferroelectrics, Phys. Rev. Lett. **123**, 087601 (2019).

[24] N. Bristowe, E. Artacho, and P. Littlewood, Oxide superlattices with alternating p and n interfaces, Phys. Rev. B **80**, 045425 (2009).

[25] M. Gu, J. Wang, Q. Xie, and X. Wu, Structural and electronic properties of PbTiO₃/SrTiO₃ superlattices from first principles, Phys. Rev. B **82**, 134102 (2010).

[26] V. M. Longo, M. d. G. S. Costa, A. Z. Simoes, I. L. V. Rosa, C. O. P. Santos, J. Andres, E. Longo, and J. A. Varela, On the photoluminescence behavior of samarium-doped strontium titanate nanostructures under uv light. a structural and electronic understanding, Phys. Chem. Chem. Phys. **12**, 7566 (2010).

[27] N. F. Muhamad, R. A. M. Osman, M. S. Idris, and M. N. M. Yasin, Physical and electrical properties of SrTiO₃ and SrZrO₃, in EPJ Web of Conferences, Vol. 162 (EDP Sciences, 2017) p. 01052.

[28] O. Nakagawara, M. Kobayashi, Y. Yoshino, Y. Katayama, H. Tabata, and T. Kawai, Effects of buffer layers in epitaxial growth of SrTiO₃ thin film on si (100), J. Appl. Phys. **78**, 7226 (1995).

[29] G. Shirane and S. Hoshino, On the phase transition in lead titanate, Journal of the Physical Society of Japan **6**, 265 (1951).

[30] R. Nelmes and W. Kuhs, The crystal structure of tetragonal PbTiO₃ at room temperature and at 700 k, Solid State Commun. **54**, 721 (1985).

[31] G. Shirane, R. Pepinsky, and B. Frazer, X-ray and neutron diffraction study of ferroelectric PbTiO₃, Phys. Rev. **97**, 1179 (1955).

[32] A. Glazer and S. Mabud, Powder profile refinement of lead zirconate titanate at several temperatures. ii. pure PbTiO₃, Acta Crystallographica Section B: Structural Crystallography and Crystal Chemistry **34**, 1065 (1978).

[33] R. Eglitis and D. Vanderbilt, First-principles calculations of atomic and electronic structure of SrTiO₃(001) and (011) surfaces, Phys. Rev. B **77**, 195408 (2008).

[34] R. A. Ribeiro and S. R. d. Lázaro, Theoretical investigations of the bulk modulus in the tetra-cubic transition of PbTiO₃ material, Química Nova **37**, 1165 (2014).

[35] Y. Umeno, B. Meyer, C. Elsässer, and P. Gumbsch, Ab initio study of the critical thickness for ferroelectricity in ultrathin Pt/PbTiO₃/Pt films, Phys. Rev. B **74**, 060101 (2006).

[36] C. Paillard, S. Prosandeev, and L. Bellaiche, Ab initio approach to photostriction in classical ferroelectric materials, Phys. Rev. B **96**, 045205 (2017).

[37] Z. Zhen-Ye, W. Biao, W. Hai, Z. Yue, and L. Qing-Kun, The first-principles study of ferroelectric behaviours of PbTiO₃/SrTiO₃ and BaTiO₃/SrTiO₃ superlattices, Chinese Physics **16**, 1780 (2007).

[38] R. Eglitis, S. Piskunov, and Y. F. Zhukovskii, Ab initio
calculations of PbTiO₃/SrTiO₃ (001) heterostructures, Physica Status Solidi (c) **13**, 913 (2016).

[39] V. R. Cooper, K. Johnston, and K. M. Rabe, Polarization enhancement in short period superlattices via interfacial intermixing, Phys. Rev. B—Cond. Matt. and Mat. Phys. **76**, 020103 (2007).

[40] K. A. Müller and H. Burkard, SrTiO₃: An intrinsic quantum paraelectric below 4 k, Phys. Rev. B **19**, 3593 (1979).

[41] C. Paillard, S. Prosandeev, and L. Bellaiche, Ab initio approach to photostriction in classical ferroelectric materials, Phys. Rev. B **96**, 045205 (2017).

[42] W. Zhong, R. King-Smith, and D. Vanderbilt, Giant lo-to splittings in perovskite ferroelectrics, Phys. Rev. Lett. **72**, 3618 (1994).

[43] C. E. Dreyer, S. Coh, and M. Stengel, Nonadiabatic born effective charges in metals and the drude weight, Phys. Rev. Lett. **128**, 095901 (2022).

[44] See supplemetal material at [url] for data on the other superlattices, derivations and discussion,.

[45] J. Junquera, M. H. Cohen, and K. M. Rabe, Nanoscale smoothing and the analysis of interfacial charge and dipolar densities, J. Phys: Cond. Mat. **19**, 213203 (2007).

[46] C. Paillard, B. Xu, B. Dkhil, G. Geneste, and L. Bellaiche, Photostriction in ferroelectrics from density functional theory, Phys. Rev. Lett. **116**, 247401 (2016).

[47] R. Haleoot, C. Paillard, T. P. Kaloni, M. Mehboudi, B. Xu, L. Bellaiche, and S. Barraza-Lopez, Photostrictive two-dimensional materials in the monochalcogenide family, Phys. Rev. Lett. **118**, 227401 (2017).

[48] Y. Yang, C. Dansou, C. Paillard, J. He, H. Xiang, and L. Bellaiche, Large photostriction near the phase boundary in bife o 3 under varying epitaxial strain, Phys. Rev. B **109**, 184111 (2024).

[49] Giant photoinduced lattice distortion in oxygen vacancy ordered SrCoO₂.₅ thin films, author=Zhang, Bingbing and He, Xu and Zhao, Jiali and Yu, Can and

Wen, Haidan and Meng, Sheng and Bousquet, Eric and Li, Yuelin and Ge, Chen and Jin, Kuijuan and others, journal=Phys. Rev. B, volume=100, number=14, pages=144201, year=2019, publisher=APS, .

[50] T.-C. Wei, H.-P. Wang, H.-J. Liu, D.-S. Tsai, J.-J. Ke, C.-L. Wu, Y.-P. Yin, Q. Zhan, G.-R. Lin, Y.-H. Chu, et al., Photostriction of strontium ruthenate, Nature Commun **8**, 15018 (2017).

[51] J.-C. Yang, Y.-D. Liou, W.-Y. Tzeng, H.-J. Liu, Y.-W. Chang, P.-H. Xiang, Z. Zhang, C.-G. Duan, C.-W. Luo, Y.-C. Chen, et al., Ultrafast giant photostriction of epitaxial strontium iridate film with superior endurance, Nano Lett. **18**, 7742 (2018).

[52] B. Peng, D. Bennett, I. Bravić, and B. Monserrat, Tunable photostriction of halide perovskites through energy dependent photoexcitation, Phys. Rev. Mat. **6**, L082401 (2022).

[53] C. Paillard and L. Bellaiche, Light: A new handle to control the structure of cesium lead iodide, Phys. Rev. B **107**, 054107 (2023).

[54] D. Sri Gyan, H. J. Lee, Y. Ahn, R. B. Carson, J. Carnis, T. Y. Kim, S. Unithrattil, J. Y. Lee, S. H. Chun, S. Kim, et al., Optically induced picosecond lattice compression in the dielectric component of a strongly coupled ferroelectric/dielectric superlattice, Adv. Electron. Mater. **8**, 2101051 (2022).

[55] J. Scott, Soft-mode spectroscopy: Experimental studies of structural phase transitions, Rev. Mod. Phys. **46**, 83 (1974).

[56] R. Xu, F. Delodovici, B. Dkhil, and C. Paillard, Oxygen tilt driven polar superorders in BiFeO₃-based superlattices, Phys. Rev. B **109**, L220101 (2024).

[57] R. Gu, R. Xu, F. Delodovici, B. Carcan, M. Khiari, G. Vaudel, V. Juvé, M. Weber, A. Poirier, P. Nandi, et al., Superorders and acoustic modes folding in BiFeO₃/LaFeO₃ superlattices, arXiv preprint arXiv:2401.06647 (2024).