The strain-induced transitions of the piezoelectric, pyroelectric and electrocaloric properties of the CuInP 2 S 6 films

The low-dimensional ferroelectrics, ferrielectrics and antiferroelectrics are of urgent scientific interest due to their unusual polar, piezoelectric, electrocaloric and pyroelectric properties. The strain engineering and strain control of the ferroelectric properties of layered 2D Van der Waals materials, such as CuInP 2 (S,Se) 6 monolayers, thin films and nanoflakes, are of fundamental interest and especially promising for their advanced applications in nanoscale nonvolatile memories, energy conversion and storage, nano-coolers and sensors. Here, we study the polar, piezoelectric, electrocaloric and pyroelectric properties of thin strained films of a ferrielectric CuInP 2 S 6 covered by semiconducting electrodes and reveal an unusually strong effect of a mismatch strain on these properties. In particular, the sign of the mismatch strain


I. INTRODUCTION
The piezoelectric, pyroelectric and electrocaloric effects are inherent to ferroelectrics, being a consequence of their spontaneous polarization dependence on strain and temperature, which becomes especially strong in the vicinity of the paraelectric -ferroelectric phase transition.
These properties determine the indispensable value of ferroelectrics for modern actuators, pyroelectric sensors, electromechanical and electrocaloric energy converters [1,2].
The low-dimensional ferroelectrics, ferrielectrics and antiferroelectrics are of urgent scientific interest due to their unusual polar, piezoelectric, electrocaloric and pyroelectric properties [3,4].The strain engineering and strain control of the ferroelectric properties of layered two-dimensional Van der Waals (V-d-W) materials, such as CuInP2(S,Se)6 monolayers, thin films and nanoflakes, are of fundamental interest and especially promising for their advanced applications in nanoscale nonvolatile memories, energy conversion and storage, nano-coolers and sensors [5].
One of the most important feature, which determine the strain-polarization coupling in ferrielectric CuInP2(S,Se)6 [6,7], is the existence of more than two potential wells [8], which are responsible for strain-tunable multiple polar states [9].Due to the multiple potential wells, which height and position are temperature-and strain-dependent, the energy profiles of a uniaxial ferrielectric CuInP2(S,Se)6 can be flat in the vicinity of the nonzero polarization states.The flat energy profiles give rise to the unusual polar and dielectric properties associated with the strainpolarization coupling in the vicinity of the states [10,11,12].
The spontaneous polarization of crystalline CuInP2S6 (CIPS) is directed normally to its structural layers being a result of antiparallel shifts of the Cu + and In 3+ cations from the middle of the layers [13,14].The strain effect on the polarization reversal in CIPS is opposite, i.e., "anomalous", in comparison with many other ferroelectric films, for which the out-of-plane remanent polarization and coercive field increase strongly for tensile strains, and decrease or vanish for compressive strains [9 -12].
Using the Landau-Ginzburg-Devonshire (LGD) approach here we study the size-and strain-induced changes of the spontaneous polarization, piezoelectric, pyroelectric and electrocaloric properties of thin strained CIPS films covered by semiconducting electrodes.The original part of this work contains the physical description of the problem (Section II), analysis the strain-induced transitions of the piezoelectric, electrocaloric and pyroelectric properties of the CIPS films (Section III).Section IV summarizes the obtained results.Supplementary Materials elaborate on a mathematical formulation of the problem and the table of material parameters.

II. PROBLEM FORMULATION
Let us consider an epitaxial thin CIPS film sandwiched between the semiconducting electrodes with a screening length λ, which is clamped on a thick rigid substrate [see Fig. 1(a)].
Arrows show the out-of-plane ferroelectric polarization  3 , directed along the X3-axis.The perfect electric contact between the film and the electrodes provides the effective screening of the out-ofplane polarization by the electrodes and precludes the domain formation for small enough λ.An electric voltage is applied between the electrodes.The misfit strain   originates from the filmsubstrate lattice constants mismatch and exists entire the film depth [15,16,17], because the film thickness ℎ is regarded smaller than the critical thickness ℎ  of misfit dislocations appearance.Within the LGD approach, the value and orientation of the spontaneous polarization   in thin ferroelectric films are controlled by the temperature  and mismatch strain   .For the validity of the continuum media approximation, the film thickness is regarded to be much bigger than the lattice constant .As a rule, the condition  ≪ ℎ < ℎ  is valid for the film thickness range (5 -50) nm.

CIPS film
Semiconducting electrode with a screening length 

P3
It has been shown in Refs.[9 -12] Here,  is the Khalatnikov kinetic coefficient [18].The coefficient  depends linearly on the temperature , namely () =   ( −   ), where   is the Curie temperature of a bulk ferrielectric.The coefficients , , and  are temperature independent.The values   denote diagonal components of a stress tensor in the Voigt notation, and the subscripts  and  vary from 1 to 6.The values  3 ,  33 , and  3 denote the components of a second order and higher order electrostriction strain tensors in the Voigt notation, respectively [19,20].The values  33 are polarization gradient coefficients in the matrix notation and the subscripts ,  = 1 − 3. The boundary condition for  3 at the film surfaces S is regarded "natural", i.e.,  33 where  ⃗ is the outer normal to the surface.
The value  3 in Eq.( 1a) is an electric field component co-directed with the polarization  3 .
3 is a superposition of external ( 0 ) and depolarization (  ) fields.In the considered case of a very high screening degree by the semiconducting electrodes with a small screening length λ ≤ 0.1 nm, the solutions, corresponding to the almost constant  3 , are energetically favorable and the domain formation is absent, because the corresponding depolarization field, , is very small for ℎ λ ⁄ ≫ 1.To analyze a quasi-static polarization reversal, we assume that the period, 2  ⁄ , of the sinusoidal external field  0 is very small in comparison with the Landau-Khalatnikov relaxation time,  =  || ⁄ .
The electrocaloric (EC) temperature change Δ  , can be calculated from the expression [28]: where   is the volume density,  is the ambient temperature, and   is the CIPS specific heat.
For ferroics the specific heat depends on polarization (and so on external field) and can be modeled as following: where   0 is the polarization-independent part of specific heat and g is the density of the LGD free energy.According to experiment, the specific heat usually has a maximum in the first order ferroelectric phase transition point, which height is about (10 -30) % of the   value near TC (see e.g.[29]).The mass density and the polarization-independent part of the CIPS specific heat are   = 3.415 • 10 3 kg/m 3 and   0 = 3.40 • 10 2 J/(kg K) [30,31], respectively.

III RESULTS AND DISCUSSION
The out-of-plane spontaneous polarization,   , piezoelectric coefficient,   3)-( 4) for  0 →   , where   is the coercive field of the film.The color scale in the maps shows the value of Δ  in K.The electrocaloric response is smaller in the FI2 state and significantly higher in the FI1 state; and is absent in the PE phase.
Namely, Δ  reaches minimum ~-(2 -2.5)K near the PE-FI1 boundary for −1% <   < −0.5% and 40 nm< ℎ <10 nm.Higher negative values of Δ  in the high-polarization FI1 state are related with the increase of the spontaneous polarization and with the features of the electrostriction coupling in CIPS, where the second order and higher order coefficients,  33 and  33 , linearly depend on temperature (see Table SI in Appendix A from Supplementary materials).Note that Δ  cannot exceed -2.5 K for a bulk BaTiO3 [28], which spontaneous polarization (about 25 µC/cm 2 at room temperature) is much higher than the CIPS polarization (about 5 µC/cm 2 at room temperature).The negative sign of the electrocaloric effect and its maximum predicted in compressed CIPS films, can be useful for the strain engineering of ultrathin nano-coolers.where these values diverge.The abbreviations "PE", "FE1" and "FE2" mean the paraelectric phase, high and low polarization ferrielectric sates, respectively.

IV. SUMMARY
• We consider an epitaxial thin CIPS film sandwiched between semiconducting electrodes, and clamped on a thick rigid substrate, which create the mismatch strain in the film.Using LGD phenomenological approach, we study the piezoelectric, electrocaloric and pyroelectric properties of the strained film and reveal an unusually strong effect of a mismatch strain on these properties.
• We revealed that the sign of the mismatch strain and its magnitude determine the behavior of piezoelectric, electrocaloric and pyroelectric responses.In particular, the strain effect on these properties is opposite, i.e., "anomalous", in comparison with many other ferroelectric films, for which the out-of-plane remanent polarization, piezoelectric, electrocaloric and pyroelectric LGD parameters for a bulk ferroelectric CuInP2S6 Table SI.

APPENDIX B. Free energy with renormalized coefficients
Using results [41] and making the Legendre transformation of Eq.( 1) to the strain-polarization representation,  ̃=  + , the renormalized free energy density is [34]:

FIGURE 1 .
FIGURE 1.(a) Schematics of a thin epitaxial CIPS film sandwiched between semiconducting electrodes with a small screening length λ, and clamped on a rigid substrate.Arrow shows the direction of the single-domain spontaneous polarization.(b) Schematics of the possible straininduced changes of polarization reversal hysteresis loops.

Fig. 2 for
Fig.2for   > 0, is anomalous for the most uniaxial and multiaxial ferroelectric films, where the out-of-plane polarization is absent or very small at   > 0, and the region of the FE c-phase vanishes or significantly constricts for   > 0[15].The color maps of the piezoelectric coefficient,  33 , shown in Figs.2(b), 3(b) and 4(b), are calculated using Eqs.(1a) and (1b) for  0 → 0. The color scale in the maps shows the absolute value of  33 in pm/V.Thin white curves in the plots correspond to the regions where  33 diverges at the boundary of the paraelectric-ferrielectric phase transition.The piezoelectric response is smaller in the FI1 state and higher in the FI2 state; and is absent inside the wedge-like region of the PE phase separating the FI states.Relatively small values of  33 in the high-polarization FI1 state are explained by the weak field dependence of the saturated out-of-plane spontaneous polarization.Note that  33 reaches (60 -200) pm/V near the PE-FI2 boundary, being the smallest for 330 K and the biggest for 250 K.The color maps of the pyroelectric coefficient, Π  , shown in Figs.2(c), 3(c) and 4(c), are calculated using Eqs.(1a) and (1c) for  0 → 0. The color scale in the maps shows the absolute value of Π  in mC/(K m 2 ).Thin white curves in the plots correspond to the regions where Π  diverges at the boundary of the paraelectric-ferrielectric phase transition.The pyroelectric response is smaller in the FI1 state and significantly higher in the FI2 state; and is absent in the PE phase.Higher values of Π  in the low-polarization FI2 state are explained by the stronger field dependence of the unsaturated out-of-plane low-polarization.Despite the Π  does not exceed 1 mC/(K m 2 ) far from the paraelectric-ferrielectric boundary, the field behavior of Π  determines the features of electrocaloric properties in accordance with Eq.(3).The color maps of the electrocaloric temperature change, Δ  , shown in Figs.2(d), 3(d) and 4(d), are calculated using Eqs.(1a), (1c) and (3)-(4) for  0 →   , where   is the coercive field

FIGURE 2 .
FIGURE 2. The spontaneous polarization   (a), piezoelectric coefficient  33 (b), pyroelectric coefficient Π  (c), and the electrocaloric temperature change Δ  (d) of a single-domain CIPS film calculated as a function of the film thickness ℎ and misfit strain   for  = 0.1 nm and

FIGURE 3 .FIGURE 4 .
FIGURE 3. The spontaneous polarization   (a), piezoelectric coefficient  33 (b), pyroelectric coefficient Π  (c), and the electrocaloric temperature change Δ  (d) of a CIPS film calculated as a function of the film thickness ℎ and misfit strain   for the room temperature  =293 K. Other parameters and designations the same as in Fig. 2.

FilmFE1
Data availability statement.Numerical results presented in the work are obtained and visualized using a specialized software,Mathematica 13.2 [32].The Mathematica notebook, which contain the codes, is available per reasonable request.