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 two-dimensional van der Waals 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. Here, we study the polar, piezoelectric, electrocaloric, and pyroelectric properties of thin strained films of a ferrielectric CuInP2S6 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 and its magnitude determine the complicated behavior of piezoelectric, electrocaloric, and pyroelectric responses. 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 responses increase strongly for tensile strains and decrease or vanish for compressive strains.

The piezoelectric, pyroelectric, and electrocaloric effects are inherent to ferroelectrics and are 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, and electromechanical and electrocaloric energy converters.1,2

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 features that determines the strain-polarization coupling in ferrielectric CuInP2(S,Se)66,7 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, whose 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 strain-polarization coupling in the vicinity of the states.10–12 

The spontaneous polarization of crystalline CuInP2S6 (CIPS) is directed normally to its structural layers, being a result of antiparallel shifts of the Cu+ and In3+ 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, we study the size- and strain-induced changes in the spontaneous polarization, piezoelectric, pyroelectric, and electrocaloric properties of thin strained CIPS films covered by semiconducting electrodes. The original part of this work contains a physical description of the problem (Sec. II) and an analysis of the strain-induced transitions of the piezoelectric, electrocaloric, and pyroelectric properties of the CIPS films (Sec. III). Section IV summarizes the obtained results. The supplementary material elaborates on a mathematical formulation of the problem and the table of material parameters.

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 P3, directed along the X3-axis. The perfect electric contact between the film and the electrodes provides effective screening of the out-of-plane polarization by the electrodes and precludes domain formation for small enough λ. An electric voltage is applied between the electrodes. The misfit strain um originates from the film-substrate lattice constants mismatch and exists throughout the film depth15–17 because the film thickness h is regarded as smaller than the critical thickness hd of misfit dislocations.

FIG. 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. The arrow shows the direction of the single-domain spontaneous polarization. (b) Schematics of the possible strain-induced changes in polarization reversal hysteresis loops.

FIG. 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. The arrow shows the direction of the single-domain spontaneous polarization. (b) Schematics of the possible strain-induced changes in polarization reversal hysteresis loops.

Close modal

Within the LGD approach, the value and orientation of the spontaneous polarization Pi in thin ferroelectric films are controlled by the temperature T and mismatch strain um. For the validity of the continuum media approximation, the film thickness is regarded as much bigger than the lattice constant c. As a rule, the condition ch < hd is valid for the film thickness range (5–50) nm.

It has been shown in Refs. 9–12 that the LGD free energy density of CIPS, gLGD, has four potential wells at E=0. The density gLGD includes the Landau–Devonshire expansion in even powers of the polarization P3 (up to the eighth power), the Ginzburg gradient energy, and the elastic and electrostriction energies. The behavior of polarization P3, piezocoefficient d33, and pyroelectric coefficient Π3 in the electric field E3 follows from the time-dependent Euler–Lagrange equations, which have the form9,10
ΓP3t+α2σiQi3+Wij3σjP3+β4Zi33σiP33+γP35+δP37g33kl2P3xkxl=E3,
(1a)
Γd33t+α2σiQi3+Wij3σj+3β4Zi33σiP32+5γP34+7δP36d33g33kl2d33xkxl=2Q33+2Wi33σiP3+4Z333P33,
(1b)
ΓΠ3t+α2σiQi3+Wij3σj+3β4Zi33σiP32+5γP34+7δP36Π3g33kl2Π33xkxl=αTP3+βTP33+γTP35.
(1c)
Here, Γ is the Khalatnikov kinetic coefficient.18 The coefficient α depends linearly on the temperature T, namely αT=αTTTC, where TC is the Curie temperature of a bulk ferrielectric. The coefficients β, γ, and δ are temperature independent. The values σi denote diagonal components of a stress tensor in the Voigt notation, and the subscripts i and j vary from 1 to 6. The values Qi3, Zi33, and Wij3 denote the components of second order and higher order electrostriction strain tensors in the Voigt notation, respectively.19,20 The values g33kl are polarization gradient coefficients in the matrix notation and the subscripts k, l = 1–3. The boundary condition for P3 at the film surface S is regarded as “natural,” i.e., g33klnkP3xlS=0, where n is the outer normal to the surface.

The values of TC, αT, β, γ, δ, Qi3, Wij3, and Zi33 have been derived in Ref. 21 from the fitting of temperature-dependent experimental data for the dielectric permittivity,22,23 spontaneous polarization,24 and lattice constants25 as a function of hydrostatic pressure. Elastic compliances, sij, were calculated from ultrasound velocity measurements.26,27 The CIPS parameters are listed in Table SI in Appendix A (see supplementary material).

Modified Hooke’s law, relating elastic strains ui and stresses σj, is obtained from the relation ui = −∂gLGD/∂σi,
ui=sijσj+Qi3P32+Zi33P34+Wij3σjP32.
(2)

For the considered geometry of a CIPS film, the following relations are valid for homogeneous stress and strain components: σ3 = σ4 = σ5 = σ6 = 0, u1 = u2 = um, and u4 = u5 = u6 = 0.

The value E3 in Eq. (1a) is an electric field component co-directed with the polarization P3. E3 is a superposition of external (E0) and depolarization (Ed) 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 P3, are energetically favorable, and domain formation is absent because the corresponding depolarization field, Ed=P3ε0εb+h/λ, is very small for h/λ ≫ 1. To analyze a quasi-static polarization reversal, we assume that the period, 2π/ω, of the sinusoidal external field E0 is very small in comparison with the Landau–Khalatnikov relaxation time, τ=Γ/α.

The electrocaloric (EC) temperature change, ΔTEC, can be calculated from the expression28,
ΔTEC=TE1E21ρPCPPTEdETE1E21ρPCPΠ3dE,
(3)
where ρP is the volume density, T is the ambient temperature, and CP is the CIPS specific heat. For ferroics, the specific heat depends on polarization (and so on the external field) and can be modeled as follows:
CP=CP0T2gT2,
(4)
where CP0 is the polarization-independent part of specific heat and g is the density of the LGD free energy. According to the experiment, the specific heat usually has a maximum at the first order ferroelectric phase transition point, whose height is about (10–30)% of the CP value near TC (see, e.g., Ref. 29). The mass density and the polarization-independent part of the CIPS specific heat are ρP = 3.415 × 103 kg/m3 and CP0=3.40×102 J/(kg K),30,31 respectively.

The out-of-plane spontaneous polarization, Ps, piezoelectric coefficient, d33, pyroelectric coefficient, Πs, and electrocaloric temperature change, ΔTEC, as a function of the film thickness h and misfit strain um, are shown in Figs. 24 for the temperatures T = 250, 293, and 330 K, respectively. The abbreviations “PE,” “FE1,” and “FE2” mean the paraelectric phase, high-polarization, and low-polarization ferrielectric states, respectively.

FIG. 2.

The spontaneous polarization Ps (a), piezoelectric coefficient d33 (b), pyroelectric coefficient Πs (c), and the electrocaloric temperature change ΔTEC (d) of a single-domain CIPS film calculated as a function of the film thickness h and misfit strain um for λ = 0.1 nm and temperature T = 250 K. Color scales show the values of Ps, d33, Πs, and ΔTEC in μC/cm2, pm/V, mC/(K m2), and K, respectively. White areas in the plots of d33 and Πs correspond to the regions where these values diverge. The abbreviations “PE,” “FE1,” and “FE2” mean the paraelectric phase and high and low polarization ferrielectric sates, respectively.

FIG. 2.

The spontaneous polarization Ps (a), piezoelectric coefficient d33 (b), pyroelectric coefficient Πs (c), and the electrocaloric temperature change ΔTEC (d) of a single-domain CIPS film calculated as a function of the film thickness h and misfit strain um for λ = 0.1 nm and temperature T = 250 K. Color scales show the values of Ps, d33, Πs, and ΔTEC in μC/cm2, pm/V, mC/(K m2), and K, respectively. White areas in the plots of d33 and Πs correspond to the regions where these values diverge. The abbreviations “PE,” “FE1,” and “FE2” mean the paraelectric phase and high and low polarization ferrielectric sates, respectively.

Close modal
FIG. 3.

The spontaneous polarization Ps (a), piezoelectric coefficient d33 (b), pyroelectric coefficient Πs (c), and the electrocaloric temperature change ΔTEC (d) of a CIPS film calculated as a function of the film thickness h and misfit strain um for the room temperature T = 293 K. Other parameters and designations are the same as in Fig. 2.

FIG. 3.

The spontaneous polarization Ps (a), piezoelectric coefficient d33 (b), pyroelectric coefficient Πs (c), and the electrocaloric temperature change ΔTEC (d) of a CIPS film calculated as a function of the film thickness h and misfit strain um for the room temperature T = 293 K. Other parameters and designations are the same as in Fig. 2.

Close modal
FIG. 4.

The spontaneous polarization Ps (a), piezoelectric coefficient d33 (b), pyroelectric coefficient Πs (c), and the electrocaloric temperature change ΔTEC (d) of a CIPS film calculated as a function of the film thickness h and misfit strain um for the temperature T = 330 K. Other parameters and designations are the same as in Fig. 2.

FIG. 4.

The spontaneous polarization Ps (a), piezoelectric coefficient d33 (b), pyroelectric coefficient Πs (c), and the electrocaloric temperature change ΔTEC (d) of a CIPS film calculated as a function of the film thickness h and misfit strain um for the temperature T = 330 K. Other parameters and designations are the same as in Fig. 2.

Close modal

The color maps of the spontaneous polarization, shown in Figs. 2(a)4(a), are calculated by a conventional numerical minimization of the free energy (S.1) listed in Appendix B (see supplementary material). The color scale in the maps shows the absolute value of Ps in μC/cm2, calculated in the deepest potential well of the LGD free energy. A sharp wedge-like region of the PE phase, which is stable at small thickness h < 10 nm and small strains um<1%, separates two ferrielectric states, FI1 and FI2, which correspond to big and small amplitudes of the out-of-plane spontaneous polarizations P3±, respectively. The area and height of the PE phase region significantly increase with the temperature increase, being the smallest at 250 K and the biggest at 330 K. The increase occurs because higher compressive or tensile strains are required to support the FI1 or FI2 polar states, respectively, when the growing temperature increases the coefficient αT in Eq. (1). The unusual features of the color maps are the high-polarization FI1 state existing at compressive strains um < 0, and the low-polarization state FI2 existing for tensile strains um > 0. The FI1 and FI2 states either transform into one another for very small um or undergo the first or second order phase transitions to the PE phase. The situation, shown in Fig. 2 for um > 0, is anomalous for the most uniaxial and multiaxial ferroelectric films, where the out-of-plane polarization is absent or very small at um > 0, and the region of the FE c-phase vanishes or significantly constricts for um > 0.15 

The color maps of the piezoelectric coefficient, d33, shown in Figs. 2(b)(b), are calculated using Eqs. (1a) and (1b) for E0 → 0. The color scale in the maps shows the absolute value of d33 in pm/V. Thin white curves in the plots correspond to the regions where d33 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 it is absent inside the wedge-like region of the PE phase separating the FI states. Relatively small values of d33 in the high-polarization FI1 state are explained by the weak field dependence of the saturated out-of-plane spontaneous polarization. Note that d33 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, Πs, shown in Figs. 2(c)4(c), are calculated using Eqs. (1a) and (1c) for E0 → 0. The color scale in the maps shows the absolute value of Πs in mC/(K m2). Thin white curves in the plots correspond to the regions where Πs 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 it is absent in the PE phase. Higher values of Πs in the low-polarization FI2 state are explained by the stronger field dependence of the unsaturated out-of-plane low-polarization. Despite the fact that Πs does not exceed 1 mC/(K m2) far from the paraelectric–ferrielectric boundary, the field behavior of Πs determines the features of electrocaloric properties in accordance with Eq. (3).

The color maps of the electrocaloric temperature change, ΔTEC, shown in Figs. 2(d)4(d), are calculated using Eqs. (1a), (1c), (3), and (4) for E0Ec, where Ec is the coercive field of the film. The color scale in the maps shows the value of ΔTEC in K. The electrocaloric response is smaller in the FI2 state and significantly higher in the FI1 state, and it is absent in the PE phase. Namely, ΔTEC reaches minimum ∼−(2–2.5) K near the PE-FI1 boundary for −1% < um < − 0.5% and 40 nm < h < 10 nm. Higher negative values of ΔTEC in the high-polarization FI1 state are related to the increase of spontaneous polarization and the features of the electrostriction coupling in CIPS, where the second order and higher order coefficients, Qi33 and Zi33, linearly depend on temperature (see Table SI in Appendix A from the supplementary material). Note that ΔTEC cannot exceed −2.5 K for a bulk BaTiO3,28 whose spontaneous polarization (about 25 μC/cm2 at room temperature) is much higher than the CIPS polarization (about 5 μC/cm2 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 ultra-thin nano-coolers.

  • We consider an epitaxial thin CIPS film sandwiched between semiconducting electrodes and clamped on a thick rigid substrate, which creates the mismatch strain in the film. Using the 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 increase strongly for tensile strains and decrease or vanish for compressive strains. The negative sign of the electrocaloric effect and its maximum predicted in compressed CIPS films can be useful for the strain engineering of ultra-thin nano-coolers. In addition, we studied the conditions under which piezoelectric, electrocaloric, and pyroelectric responses can reach maximal values for small electric fields.

The supplementary material contains Appendix A with LGD parameters for bulk ferroelectric CuInP2S6 and Appendix B with the renormalized free energy caused by misfit stress.

A.N.M. and A.L.K. acknowledge support from the Horizon Europe Framework Program (HORIZON-TMA-MSCA-SE), Project No. 101131229, Piezoelectricity in 2D-materials: materials, modeling, and applications (PIEZO 2D). A.N.M. also acknowledges funding from the National Academy of Sciences of Ukraine (Grant N 4.8/23-Π “Innovative materials and systems with magnetic and/or electrodipole ordering for the needs of using spintronics and nanoelectronics in strategically important issues of new technology”). A.N.M., E.A.E., and L.P.Y. acknowledge support from the National Academy of Sciences of Ukraine. The research was partly supported by the Czech Science Foundation under Project No. 23-05578S (V.L.). S.V.K. was supported by the Center for 3D Ferroelectric Microelectronics (3DFeM), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, under Award No. DE-SC0021118. This work (A.L.K.) was developed within the scope of project CICECO-Aveiro Institute of Materials (Nos. UIDB/50011/2020 and UIDP/50011/2020), financed by national funds through the FCT-Foundation for Science and Technology (Portugal).

The authors have no conflicts to disclose.

Anna N. Morozovska: Conceptualization (lead); Formal analysis (equal); Writing – original draft (lead); Writing – review & editing (equal). Eugene A. Eliseev: Formal analysis (equal); Software (lead); Visualization (equal). Lesya P. Yurchenko: Software (supporting); Visualization (equal). Valentyn V. Laguta: Writing – review & editing (equal). Yongtao Liu: Writing – review & editing (equal). Sergei V. Kalinin: Conceptualization (supporting); Writing – review & editing (equal). Andrei L Kholkin: Conceptualization (supporting). Yulian M. Vysochanskii: Conceptualization (supporting); Supervision (lead); Writing – review & editing (equal).

The numerical results presented in the work are obtained and visualized using the specialized software Mathematica 13.2.32 The Mathematica notebook, which contains the codes, is available per reasonable request.

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Supplementary Material