The Influence of Chemical Strains on the Electrocaloric Response, Polarization Morphology, Tetragonality and Negative Capacitance Effect of Ferroelectric Core-Shell Nanorods and Nanowires

Using Landau-Ginzburg-Devonshire (LGD) approach we proposed the analytical description of the chemical strains influence on the spontaneous polarization and electrocaloric response in ferroelectric core-shell nanorods. We postulate that the nanorod core presents a defect-free single-crystalline ferroelectric material, and the elastic defects are accumulated in the ultra-thin shell, where they can induce tensile or compressive chemical strains. The finite element modeling (FEM) based on the LGD approach reveals transitions of domain structure morphology induced by the chemical strains in the BaTiO3 nanorods. Namely, tensile chemical strains induce and support the single-domain state in the central part of the nanorod, while the curled domain structures appear near the unscreened or partially screened ends of the rod. The vortex-like domains propagate toward the central part of the rod and fill it entirely, when the rod is covered by a shell with compressive chemical strains above some critical value. The critical value depends on the nanorod sizes, aspect ratio, and screening conditions at its ends. Both analytical theory and FEM predict that the tensile chemical strains in the shell increase the nanorod polarization, lattice tetragonality, and electrocaloric response well-above the values corresponding to the bulk material. The physical reason of the increase is the strong electrostriction coupling between the mismatch-type elastic strains induced in the core by the chemical strains in the shell. Comparison with the earlier XRD data confirmed an increase of tetragonality ratio in tensiled BaTiO3 nanorods compared to the bulk material.

nanorods and nanowires, can be useful for strain engineering of advanced ferroelectric nanomaterials for energy storage, harvesting, electrocaloric applications and negative capacitance elements.

I. INTRODUCTION
The influence of shape and size effects, defects, and elastic strains on the phase state, polar and structural properties, and related working performances of various nanosized ferroelectrics is still poorly explored.In particular, the physical explanation and theoretical description of the strongly enhanced spontaneous polarization and lattice tetragonality observed in BaTiO3 core-shell ferroelectric nanoparticles [1,2,3,4] have been absent for a long time [5].Recent X-ray spectroscopic measurements [2] revealed a large Ti-cation off-centering in 10-nm quasi-spherical BaTiO3 core-shell nanoparticles near 300 K confirmed by the tetragonality ratio   ≈ 1.0108, which is higher than the bulk value,   ≈ 1.010, and significantly higher in comparison with   ≈ 1.0075 for 50 nm nanoparticles.The off-centering of Ti-cations is a key factor in producing the enhanced spontaneous polarization (up to 130 C/cm 2 at room temperature) in the core-shell nanoparticles, and the barium oleate component in the core-shell matrix (resulting from mechanochemical synthesis during the ballmilling process [6]) stabilizes the enhanced polar structural phase of the BaTiO3 core.Only recently the theoretical models [7,8,9], which postulate the appearance of elastic strains caused by elastic defects accumulated in the shell, have been proposed, and numerical and analytical solutions for the strain-induced polarization changes in spherical core-shell nanoparticles have been derived.
Depending on the nature of elastic defects (e.g., dilatation centers, such as oxygen or cation vacancies, divacancies, OH-complexes, or isovalent impurity atoms), the defects can create compressive or tensile elastic strains in the oxide ferroelectrics, which are usually called chemical (or compositional) strains [10,11].Instead of "chemical" or "compositional" strains we use more narrow terminology in Refs.[7 -9], as well as in some places in this work, namely "Vegard" strains [12,13].
Furthermore, we consider that the strain is linearly proportional to the concentration of elastic defects (the Vegard law for chemical strains), and the proportionality coefficient is named the Vegard tensor.
Assuming that the formation energy of elastic defects is much smaller near the surface than in the bulk of the ferroelectric [14], elastic defects (and corresponding strains) are accumulated in a thin layer under the surface.It was shown that the Vegard strains are responsible for the strong increase of the Curie temperature (above 440 K) and tetragonality (up to 1.032) near the surface of a BaTiO3 film with injected oxygen vacancies [15].
To the best of our knowledge, analytical solutions for the strain-induced polarization changes for other shapes of core-shell ferroelectric nanoparticles are absent.However, an enhanced polarization, electrocaloric response, and high lattice tetragonality can be observed in non-spherical nanoparticles (e.g., / ≈ 1.013 is observed for BaTiO3 nanorods and nanowires [16]), where the core-shell structure can be formed spontaneously, because various defects are accumulated at the surface and under the surface due to the strong (e.g., exponential) lowering of the defect formation energy when approaching the surface [14].Thus, analytical solutions are important for fundamental physics and can help to achieve significant progress in the energy storage [17,18,19,20], harvesting [21] and electrocaloric applications [22] of the non-spherical ferroelectric core-shell nanoparticles.
Since the shape variation is one of the most effective means of controlling depolarization factors in ferroelectric nanoparticles, very long nanorods and nanowires with the spontaneous polarization directed along their axis have negligibly small depolarization fields, which cannot decrease the polarization.Because of this, several theoretical papers [23,24,25] predict the increase of a reversible spontaneous polarization in homogeneous (without the core-shell structure) ferroelectric nanorods and nanowires, when the spontaneous polarization is directed along their axis.The increase of the spontaneous polarization can appear due to the positive surface tension coefficient  and negative electrostriction coupling coefficients  12 of ABO3-type perovskites, because the dependence of the Curie temperature   on the particle radius R is proportional to the positive value − 4   12 in the nanowire (see e.g., Table 1 in Ref. [26]).The increase of   becomes significant for  ≤ 5 nm and requires very high  > (5 − 10) N/m [26].The flexo-chemical effect [27], being the joint action of the chemical strains and flexoelectric effect, can increase   , spontaneous polarization, and   in ultrasmall (5 nm or less) spherical or cylindrical BaTiO3 nanoparticles, although the effect rapidly disappears with a radius increase (as 1  2 ) and requires very high values of the flexoelectric coupling coefficients and strain gradients.
The chemical strains in the shell influence the core polarization almost independently on its size (until the strain relaxation via e.g., mismatch dislocations, appear).For this reason the chemical strains can significantly increase the Curie temperature, spontaneous polarization, lattice tetragonality, pyroelectric effect and electrocaloric response of the (5 -50) nm spherical core-shell nanoparticles [7,8,9].Furthermore, the higher-order electrostriction coupling [28], which needs to be considered for chemical strains higher than 1%, can be very important for a correct description of the core-shell nanoparticle polar properties [29,30].
Using these ideas, this work analyzes polar properties of cylindrical core-shell BaTiO3 nanorods and nanowires in the framework of Landau-Ginzburg-Devonshire (LGD) free energy functional, which includes the 8-th power of polarization, and thus allows high chemical strains in the shell and the nonlinear electrostriction coupling in the core to be considered.The analytical description of polar and electrocaloric properties of single-domain core-shell ferroelectric nanowires and long nanorods are presented in Section II.The finite element modeling (FEM) results, which show the ranges of analytical solutions applicability and demonstrate the strain-induced domain morphology in core-shell nanorods, are presented and analyzed in Section III.Comparison with available experiments, discussion of the controllable negative capacitance effect, and conclusion are in Section IV. [31] contain calculation details.

A. The problem formulation for a single-domain ferroelectric nanorod
Let us consider a core-shell nanorod, whose core of radius   and length 2  is a single-domain ferroelectric with a spontaneous polarization  ⃗⃗  directed along the polar axis  3 .The nanorod geometry is shown in Fig. 1.The core permittivity is ̂, which contains a background contribution,   [32], and the ferroelectric contribution,   .The defect-free core is crystalline, has a tetragonal symmetry, and is insulating, since a bulk BaTiO3 has a wide band gap (around 3.4 eV).The core is covered with a crystalline shell of cubic symmetry, which has the average thickness Δ =   −   .We postulate that the shell is semiconducting due to the high concentration of free charges.The charge screening is formed spontaneously due to the multiple mechanisms of a spontaneous polarization screening by internal and external charges in nanoscale ferroelectrics (e.g., Ref. [33] and refs.therein).The free charges provide effective screening of the core spontaneous polarization and prevent domain formation.The effective screening length in the shell, λ, is relatively small (less than 1 nm), and its relative dielectric permittivity tensor,    , is isotropic,    =     , and can be large enough (e.g., several hundred or higher) which correspond to the paraelectric phase.
We postulate the presence of elastic defects (e.g., oxygen or cation vacancies, divacancies, OHcomplexes, or isovalent impurity atoms) in the shell and assume that they can induce strong tensile or compressive chemical strains [10,11].We also assume the validity of the Vegard law [12,13] for chemical strains: the strain is linearly proportional to the concentration  of elastic defects, and the proportionality coefficient is the Vegard tensor, denoted as   , ( ).The Vegard tensor, whose components can be calculated from the first principles for certain cases [10], is assumed to have a cubic symmetry in average,   , =    , , where   is the Kronecker-delta symbol,   and   are the averaged tensor magnitudes in the core (denoted by the superscript "c") and shell (denoted by the superscript "s"), respectively.According to the Vegard law, the chemical strains   , are equal to   , .Because we assumed that the Vegard tensor has a cubic symmetry, corresponding chemical strains have the same symmetry, namely   , =    , .As a rule, the difference   −   can reach (0.5 -3)%, and it is unlikely that it can exceed (5 -7)% because the concentration of elastic defects cannot exceed (5 -10) % near the surface in accordance with many experiments [14,15].

B. Analytical solution for elastic strains, quasi-static polarization and electrocaloric properties of single-domain ferroelectric nanowires and nanorods
The LGD free energy density includes the Landau-Devonshire expansion in even powers of the polarization  3 (up to the 8-th power); the Ginzburg polarization gradient energy; and the elastic, electrostriction, and flexoelectric energies, which are listed in Table AI in Appendix A [31].Material parameters corresponding to the bulk BaTiO3 are taken from Refs.[34,35].Components of the polarization gradient tensor are taken from Ref. [36].
In the case of natural boundary conditions for polarization vector at the ends and side surface of the nanorod (which are used hereinafter) and polarization gradient coefficients higher than 10 −11 C - 2 m 3 J (which are used hereinafter), the polarization gradient effects can be neglected inside the nanowires (i.e., for   → ∞) and long nanorods with a small effective screening length (i.e., for     ≪ 0.1 ⁄ and  ≪ 0.1 nm).In this case a single-domain state is revealed to have minimal energy compared to polydomain states.The field dependence of a quasi-static single-domain polarization can be found from the following equation [8]: The parameters  * ,  * , , and  are the 2-nd, 4-th, 6-th, and 8-th order expansion coefficients in the  3 -powers of the Landau free energy. 3  is the static external field inside the core.The depolarization field and elastic stresses contribute to the "renormalization" of the first Landau expansion coefficient  1 () =   ( −   ), which becomes the temperature-, radius-, stress-, and screening length-dependent function  * [30]: ( The second term in Eq.( 2) is the depolarization field contribution, which is derived in Refs.[23,29,30].Here ε 0 = 8.85 pF/m is a universal dielectric constant,   is the dielectric permittivity of ferroelectric background [37]; 2 is the "effective" depolarization factor of the nanorod in the direction of the spontaneous polarization  3 [38].The third term originates from the strainelectrostriction coupling.Here  3 are the components of the second-order electrostriction tensor components and   are elastic stresses in the core, written in the Voigt notations.
Due to the nonlinear electrostriction coupling, the coefficient  * is "renormalized" by elastic stresses as The values  33 are the components of the higher-order electrostriction strain tensor in the Voigt notation [28].The values The temperature-dependent values  1 () and  11 () and the constants  111 and  1111 are listed in

Table AI
in Appendix A [31].
Elastic stresses and strains can be calculated analytically in a cylindrical core-shell nanorod, as derived in Appendix B [31].For a very long nanorod or nanowire, the nonzero components of the core strains,    , written in the Voigt notations, are: ].
Here the relative shell volume () and "effective" mismatch strain () are introduced as: In Eq.(4c) we define the shell volume as   =   (  2 −   2 ) and the nanorod volume as  =     2 .
In a general case, the effective mismatch strain  is created not only by the difference between the core and the shell chemical strains (  and   ), as assumed in Eq.(4c), but also be the lattice constants mismatch and/or different thermal expansion coefficients in the core and the shell.However, here we consider the simplest case when the elastic defects are postulated to be present in the shell only, and other contributions to  are absent, i.e.,  ≡   and   = 0 in Eqs.(4).Hence, below we can consider that  is the effective chemical strain.
Consideration of the surface tension leads to the appearance of terms proportional to −2s in the expressions (4a) and (4b), respectively (see Appendix B [31] for details).
Due to the reasons discussed in the Introduction, the terms appear negligibly small (less than 0.01 %) for the considered radii of nanorods and nanowires (  ≥ 10 nm) and realistic surface tension The tetragonality ratio of the lattice constants  and , is given by expression: From Eq.( 4), the tetragonality ratio is equal to: The first two terms in Eq.(5b) coincide with the expression for a bulk ferroelectric with the spontaneous polarization  3 in the tetragonal ferroelectric phase, which has a cubic parent phase.The next two terms, proportional to the relative shell volume , are caused by the elastic anisotropy between the tetragonal core and cubic shell, as well as by the effective chemical strain, .From Eq.( 5), the nonzero tetragonality can exist in the paraelectric core-shell nanorods and is equal to After substitution of the elastic strains from Eq.(4) into Eq.( 1) we obtain the equation of state for the electric polarization  3 : The renormalized coefficients in Eq.( 6) are given by expressions: For a very thick shell (i.e., for  → 1 at   ≫   ) expressions (7) transform into the well-known expressions [39] for the renormalized Landau coefficients in the ferroelectric thin film with in-plane spontaneous polarization, clamped to the infinitely thick substrate.In the case the "effective" strain, , is determined by the different chemical strains, and/or lattice constants, and/or thermal expansion coefficients in the film and its substrate.In the opposite case of a very thin shell (i.e., for  → 0 at   →   ) expressions (7) transform to the coefficients of a bulk ferroelectric.
The field dependence of a static single-domain pyroelectric coefficient Π 3 and the electrocaloric (EC) temperature change Δ  in an external field  3  are given by the following expressions [40]: where , and   =   .For the case when  3  is equal to the coercive field   , Eq.( 9) contains several contributions to the EC effect, which are proportional to the even powers of the spontaneous polarization   and the factor Let us underline that Eqs.(10) are valid only for a single-domain quasi-homogeneous distribution of  3 , because the derivation of the right-hand side in Eq.( 9), given in Ref. [40], accounts for neither the domain structure appearance nor the possible polarization rotation in the core-shell BaTiO3 nanorods.Also, it is necessary to consider the heat dissipation and temperature gradient in the case of significant  -dependence of  3 ( ,  3  , ) for realistic thermal boundary conditions, as well as consider that all experimental measurements are performed at the finite rate of the temperature change (e.g., in adiabatic conditions).To describe the real experimental measurements of the EC effect, it is necessary to solve the thermal problem taking into account the finite rate of the heat transfer and the non-uniform temperature distribution in a multilayer and/or multidomain system (see, e.g., Ref. [41]).

C. Quasi-static polarization, tetragonality, and electrocaloric properties of singledomain ferroelectric nanowires
Analytical results, calculated using Eqs.( 1)- (10)  The dependence of the core EC temperature change Δ  on the relative shell volume  and chemical strain  calculated at room temperature is shown in Fig. 4(a).It is seen that compressive strains decrease the EC cooling effect, which corresponds to Δ  < 0; and tensile strains significantly increase the magnitude of Δ  < 0. The increase of  leads to the increase of negative Δ  for  > 0, and supports the paraelectric state with Δ  = 0 for  ≤ 0. The temperature dependence of maximum monotonically decreases with a  increase for  ≤ 0. It is seen from Fig. 4(e), where  = 0.3%, that the EC cooling decreases with a  increase for  < 400 K, and then increases with a  increase for  > 400 K.For  ≫   , the EC cooling effect monotonically increases with a  increase and exists up to 900 K, which is the ferroelectric-paraelectric transition temperature for Analytical results, shown in Figs.2-4, demonstrate that the synergy of electrostriction coupling and tensile chemical strains can significantly increase the ferroelectric-paraelectric temperature (up to 900 K in comparison with 400 K for bulk BaTiO3), tetragonality (up to +1.015 in comparison with 1.011 for bulk BaTiO3), and EC cooling (up to -6 K in comparison with -3 K for bulk BaTiO3) of the core-shell BaTiO3 nanowires with co-axial polarization.We would like to underline that the results should also be valid for the very long single-domain nanorods whose ends are well-screened.However, the single-domain state should become unstable, as well as the polarization vector should rotate in the unscreened and/or not very long core-shell nanorods.Thus, the analytical results, derived in this section, require numerical verification (especially in the case  ≥ 0.05 nm and     > 0.1 ⁄ ).
Corresponding FEM results are presented in the next section.

III. FINITE ELEMENT MODELING
The FEM is performed in COMSOL@MultiPhysics software.The COMSOL@MultiPhysics model uses the electrostatics module for the solution of the Poisson equation, solid mechanics, and general math (PDE toolbox) modules for the self-consistent solution of time-dependent LGD equations listed in Table AI in Appendix A [31].FEM is performed for different discretization densities of the self-adaptive tetragonal mesh, and randomly small initial polarization distributions.The size of the computational region is not less than 4040160 nm 3 .Material parameters of BaTiO3 are listed in Table AI in Appendix A [31].The minimal size of a tetrahedral element in a mesh with fine discretization is equal to the unit cell size, 0.4 nm, the maximal size is (0.8 -1.2) nm, and 4 nm in the dielectric medium outside the nanorod.The dependence on the mesh size is verified by increasing the minimal size to 0.8 nm.We verified that this results in minor changes in the electric polarization, electric field, and elastic stress and strain, such that the spatial distribution of each of these quantities becomes less smooth (i.e., they contain numerical errors in the form of a small random noise).
FEM are performed for cylindrical core-shell nanorods of different sizes (5 nm <   < 25 nm, 20 nm <   < 100 nm) and aspect ratios (    ≥ 0. The characteristic features of polarization vector morphology in the middle and near the ends of the core-shell nanorod are shown in Fig. 6 in the form of arrow fields in the lateral { 1 ,  2 } crosssections.Figure 7 shows corresponding distributions of the radial polarization component,   .It is seen from Fig. 6, that the polarization vector becomes curled and forms the vertex-like or chiral meronlike structures near the rod ends, or vortex-like structure in the rod volume, in dependence on the chemical strain magnitude in the shell.Analytical calculations and FEM results, performed in Ref. [43] for strain-free unscreened BaTiO3 nanorods (i.e., for  → ∞ and  = 0), reveal the similar chiral meron-like structures near the rod ends, which axial polarization has the flexoelectric nature.They termed them "flexon" because a change of the flexoelectric coefficient sign leads to a reorientation of their axial polarization.FEM performed in this work for tensiled screened BaTiO3 nanorods (i.e., for 0.01 nm ≤  ≤ 1 nm and 0.3% ≤  ≤ 3%) proved that the flexoelectric coupling determines the meron-like structures chirality and related domain morphology.
The curled structures in the system tend to minimize the free energy consisting of the negative Landau energy, and from the positive polarization gradient energy, elastic and depolarization field energies (see Appendix A [31] for details and Refs.[43,44]).The negative Landau energy is maximal and the positive polarization gradient energy is minimal in the single-domain state of the nanorod.The elastic energy can significantly increase or decrease (which is dependent on the sign and value of the chemical strain ) the Landau energy due to the electrostriction coupling.For instance, see Eqs. (7) for qualitative understanding of the Landau energy coefficients renormalization by the strains.The domain formation, which leads to the decrease of the depolarization field divergency, simultaneously decreases the positive depolarization field energy.The polarization screening is incomplete near the rod ends (even for relatively small  = 0.1 nm), and the depolarization field is maximal in the spatial regions.The curled domain structures, which emerge near the ends of the rod for all considered , minimize the positive energy of the depolarization electric field.Since the length of the rod is 3 times bigger than its width, the vortices vanish approaching the central part of defect-free and tensiled rods, where the negative Landau energy dominates for  > 0. At the same time, the vortices fill the core of the compressed rods, where the negative Landau energy is much smaller for  < 0.
Hence, the FEM reveals that the chemical strain in the shell can induce vertex-like (see Fig. FEM results shows that the vortex intergrowth occurs for chemical compressive strains above some critical value,    , which depends on the temperature, nanorod sizes, aspect ratio and screening conditions at the nanorod ends.The value    can be estimated as: From Eq.( 11), larger  (i.e., thicker shells) decreases    .Since we consider the case  <   , the first term in Eq.( 11) is negative and the second term is always positive.Thus, the condition    = 0 becomes valid for a definite nanorod aspect ratio and length (assuming the fixed effective screening length and temperature).Hence, the shape and strain changes allow the control of the domain morphology in the core-shell nanorods of multiaxial ferroelectrics.Botton end To resume, both analytical LGD-based theory and FEM predict that the chemical strains in the shell can increase the nanorod core polarization, lattice tetragonality, and EC cooling effect well-above the values corresponding to the bulk material, as well as the strain control of the domain morphology is possible.The physical reason of the effects is the strong electrostriction coupling between the mismatch-type elastic strains induced in the core by the chemical strains in the shell.

A. Evidence of tetragonality increase obtained from XRD results
The BaTiO3 nanorods were obtained using a single-step hydrothermal technique and studied by Kovalenko et al. [16].In the work [16], the phase and structure of the as-prepared BaTiO3 nanopowder were determined using an X-ray diffractometer (XRD) with Cu-Kα radiation.The crystallite size was evaluated based on the size of the coherent scattering region calculated using the Scherrer equation from the full width at half-maximum of the ( 100) and ( 001 In this work we refined the XRD data found in Ref. [16], and the lattice constants ratio, /, appears as high as 1.013 for two powder samples consisting with nanorods, marked as NR1 and NR2, respectively (compare Table I in this work with Table I in Ref. [16]).The nanorods average aspect ratio,     ⁄ , is 0.17 and 0.12, their average diameter is 70 nm and 90 nm, and their average length is 410 nm and 770 nm for the samples NR1 and NR2, respectively.The / ratio of the samples is higher than the known value for the bulk BaTiO3 single crystal,   = 1.010 [45].Moreover, the high / corresponds to different average aspect ratios and radii of the rods, which indicates a weak relation between / and the depolarization field effects.The shifting of the diffraction lines (002) and ( 200) towards lower angles compared to a bulk BaTiO3 was observed for the samples and indicated the lattice expansion due to the presence of OH groups in the crystalline nanorods, apparently into trans-position [46], which leads to the high tetragonality equal to 1.013.The lattice strain in the (001) direction is 2 -2.5 times greater compared to those in the (100) direction, being unrelated with the nanorod aspect ratio.However, the difference in lattice strains does not affect the degree of anisotropy and tetragonality of the crystalline nanorods, which is consistent with the statement about the effect of OH groups on tetragonality [46].Furthermore, comparison with the XRD data [16] confirmed the increase of tetragonality ratio in tensiled BaTiO3 nanorods compared to the bulk material.

B. The negative capacitance effect
It was experimentally demonstrated that in a double-layer capacitor made of paraelectric strontium titanate (SrTiO3) and ferroelectric lead zirconate-titanate (PbxZr1-xTiO3), the total capacitance is greater than it would be for the SrTiO3 layer of the same thickness as used in the doublelayer capacitor [47].This proves the stabilization of PbxZr1-xTiO3 in the state of negative differential capacitance (NC) [48].The NC effect is very important for advanced applications in nanoelectronics [IRDS™ 2021: Beyond CMOS].Replacing the standard insulator in the gate stack of a field-effect transistor (FET) with a ferroelectric NC insulator of the appropriate thickness has several advantages.
The main advantage is that it is a relatively simple replacement for conventional FETs, which significantly reduces heat dissipation of nano-chips with a high density of critical electronic elements.
However, it is very difficult to find the analytical conditions of the NC effect appearance and stability (materials pairs, geometry, temperature and thicknesses ranges) in a general case.Many empirical demonstrations of the NC effect in ferroelectric double-layer capacitors are available [49,50,51,52], and only several works, which contain semi-analytical expressions for the conditions of the NC effect appearance and consider the inevitable appearance of the domain structure in the ferroelectric layer, exist (see e.g., Refs.[53,54,55]).
Our analytical calculations and FEM show that ferroelectric BaTiO3 nanorods, which ends are covered by the thin layer (thickness ℎ  ≤ 10 nm) of paraelectric SrTiO3, can be suitable candidates for the controllable reduction of the SrTiO3 layer capacitance due to the NC effect emerging in the BaTiO3.
In this case the SrTiO3 layers act as a cover for the BaTiO3 core.The physical origin of the NC effect is the specific energy-degenerated metastable states of the spontaneous polarization in BaTiO3 In Appendix C [31] we derived that the NC effect exists in the range of thicknesses ℎ  and ℎ  , chemical strains , shell relative volume , and temperatures , which satisfy the conditions: In Eq.( 12) we regard that ℎ  ≪ 2  .The term is the decrease of   originated from the depolarization field of a single-domain BaTiO3 core.Hence, the conditions ( 12) are valid for the BaTiO3 core in the region of size-induced paraelectric (PE) phase coexisting with the "shallow" ferroelectric (FE) phase.Notably that the energy-degenerated metastable domain states occur exactly in the region of PE and FE phases coexistence in the ferroelectrics with the first order FE-PE phase transition.The difference of the three-layer capacitance and reference capacitance is given by the expression: The dependence of the dimensionless ratio,

∆𝐶 𝐶 𝑟
, on the relative strain  and thickness ratio  It is seen from Eqs.( 12)-( 13) and Fig. 8(d) that the magnitude of the NC effect is controlled by the chemical strain and relative shell volume (namely, by the product ), as well as by the thickness ratio ℎ  ℎ . Since it is relatively easy to change the sizes and geometry of core-shell nanoparticles (i.e., parameters , ℎ  and ℎ  ), they are suitable objects for the NC effect control.

C. Conclusions
Using the LGD approach, we derive analytical expressions for the spontaneous polarization, tetragonality, and electrocaloric response in core-shell nanorods.We postulate that the nanorod core presents a defect-free single-crystalline ferroelectric material, and the elastic defects are accumulated in an ultra-thin shell, where they can induce tensile or compressive chemical strains.
The FEM reveals the strain-induced transitions of domain structure morphology in the nanorods.Namely, tensile chemical strains induce and support the single-domain state in the central part of the nanorod, while the curled domain structures appear near the unscreened or partially screened ends of the rod.The vortex-like domains propagate towards the central part of the rod and fill it entirely when the rod is covered with the shell compressed by elastic defects.The vortex intergrowth occurs for compressive strains above some critical value, which depends on the nanorod sizes, aspect ratio, and screening conditions at the nanorod ends.
Both analytical theory and FEM predict that the tensile chemical strains in the shell increase of the nanorod polarization, lattice tetragonality, and electrocaloric cooling effect well-above the values corresponding to the bulk material.The physical reason of the increase is the strong electrostriction coupling between the mismatch-type elastic strains induced in the core by the chemical strains in the shell.Comparison with the XRD data published earlier confirmed the increase of tetragonality ratio in tensiled BaTiO3 nanorods compared to the bulk material.
Analytical calculations and FEM show that BaTiO3 nanopellets, which ends are covered by SrTiO3 layers, can be suitable candidates for the controllable the NC effect.Obtained analytical expressions, which are suitable for the description of strain-induced changes in a wide class of multiaxial ferroelectric core-shell nanorods, nanowires and nanopellets, can be useful for prediction and strain engineering of advanced ferroelectric nanomaterials.Similar equations are valid in the diffuse shell except for the absence of the electrostriction term.From general symmetry consideration, we can suggest that the displacement vector has the radial and axial components,   (, ) and   (, ), which depend on the polar radius  and axial coordinate z.A general homogeneous solution for the mechanical displacement of a radially symmetric wire is [7]: Here the constants  0 , , , and  should be determined from the boundary and/or interfacial conditions.In this case, the components of strain tensor in cylindrical coordinate system are: Let consider a bilayer nanorod which has a ferroelectric core and a paraelectric shell.For the sake of simplicity, we suppose that the core and the shell have the same isotropic elastic compliances tensor.Note that the radially symmetric solution is impossible even for the cubic anisotropy of elastic properties.From Eq.(B.5) the solution for the core can be written as

APPENDIX С. Analytical calculations of the negative capacitance effect
In the case ℎ  ≪ 2  the electric potential  of the three-layer capacitor, shown in Fig. 8 The NC effect (∆ > 0) corresponds to the condition

FIGURE 1 .
FIGURE 1. (a)the radial cross-section of the core-shell ferroelectric nanorod, (b)the side-view of the core-shell nanorod.
mismatch strain u = us -uc
Δ  calculated for different values of , negative, zero, and positive chemical strains  are shown in Fig. 4(b)-4(f).Purple curves, which correspond to  = 0, are the Δ  values of an unstrained bulk BaTiO3, which reach the maximal value -(3.4 -3.8)K in the temperature range (280 -380) K. Red curves, which correspond to  = 1 (no core), show the maximal strain-induced changes of Δ  , which can reach -5.8 K for the tensile strain +1.5%.It is seen from Figs. 4(b)-4(d) that |Δ  |

FIGURE 4 .
FIGURE 4. (a) The dependence of the BaTiO3 nanowire EC temperature change   on the relative shell volume  and chemical strain .Color scale is the temperature change   in Kelvin.(b-e) The dependence of   on temperature  calculated for different values of  varying from 0 (purple curves) to 1 (red curves) with a step of 0.05; and chemical strains  =-1.5% (b), -0.3 % (c), 0 (d), 0.3 % (e), and 1.5% (f).Other parameters are the same as in Fig. 2.

6 (a) and 7 (
a)), meron-like (see Fig. 6(b) and 7(b)), or vortex-like (see Fig. 6(c) and 7(c)) transitions of domain structure morphology in the nanorod core.In particular, tensile chemical strains induce and support the single-domain state in the central part of the nanorod core, meanwhile the curled domain structures appear near the unscreened or partially screened ends of the rod.The vortex-like domains propagate towards the central part of the rod and fill it entirely, when the rod is covered with the compressed shell.

µC/cm 2 (FIGURE 5 .FIGURE 6 .where 5 .FIGURE 7 .
FIGURE 5.The distribution of spontaneous polarization components, strains, and tetragonality c/a in the defectfree (a), tensiled (b), and compressed (c) core-shell BaTiO3 nanorods.Color scales are the polarization components in μC/cm 2 , strain components in %, and tetragonality in dimensionless units.Chemical strains are absent for the top row (a), where  = 0. Chemical strains are localized under the side surface of the rod in the 2 nm thick shell, being equal to  = +1% for the middle row (b), and  = −1% for the bottom row (c).The rod radius is 10 nm, the length is 60 nm, the screening length is 0.1 nm, and the temperature  =298 K.The "rotated" coordinate  1 ′ = ( 1 −  2 ) √2 ⁄ .
) diffraction peaks.The tetragonality (c/a) was determined by the splitting of (200) peak into (200) and (002) reflections, which are characteristic of the tetragonal structure of BaTiO3.The broadening of the low-angle diffraction lines was used to estimate the sizes of coherent scattering regions, strains, and anisotropy using Williamson-Hall technique.The sizes of the BaTiO3 nanorods were analyzed using a field-emission scanning electron microscope employing a voltage of 3 kV, and the size distribution was obtained from the SEM images.
nanocylinders, some examples of which are schematically shown in Fig. 8(b).The free energy potential of these states has relatively flat negative wells, which couple to the positive parabolic potential of the SrTiO3 layers (see read and blue curves in Fig. 8(c)).In result, the total potential relief of the BaTiO3 becomes significantly flatter that the SrTiO3 potential, and the charge  stored at the electrodes covering the three-layer SrTiO3-BaTiO3-SrTiO3 capacitor of the thickness 2ℎ  + ℎ  can become bigger than the charge   at the electrodes covering the SrTiO3 layer of the thickness 2ℎ  .The effective differential capacitance of any electroded system,   , is equal to the first derivative of the  over applied voltage U,   =   .If the voltage dependence () is steeper than   (), the differential capacitance of the SrTiO3-BaTiO3-SrTiO3 capacitor (thickness 2ℎ  + ℎ  ) can be greater than the capacitance   =  0   2ℎ  of the reference SrTiO3 capacitor (thickness 2ℎ  ).

1 𝜀 12 −≥ 5 ,
temperature is shown in Fig. 8(d).The ratio ∆   is negative (which corresponds to   <   ) in the lower rectangular region Fig. 8(d).The region corresponds to the PE phase of a bulk BaTiO3.The ratio ∆   is zero along the black horizontal line  =   + corresponds to the NC effect) between the black horizontal line and the black hyperbolae,   ( −   ).The hyperbolae is the boundary between the sizeinduced PE phase and the single-domain FE phase, and thus   sharply increases approaching the PE-FE boundary and diverges (  → ∞) at it.Note that the red color in Fig. 8(d) corresponds to ∆   and white color corresponds to the region of the "deep" single-domain FE phase, where −∞ approaching the FE-PE boundary.

FIGURE 8 .
FIGURE 8. (a) Three-layer capacitor consisting of the BaTiO3 nanocylinder, which ends are covered by the paraelectric SrTiO3 layers.(b) Typical metastable states of the spontaneous polarization in the SrTiO3-BaTiO3-SrTiO3 nanocapacitor.(c) Schematic illustration of the free energy dependence on the polarization for the singledomain bulk BaTiO3 (the green curve), paraelectric SrTiO3 shell (the blue curve) and the BaTiO3 core with the metastable polarization states (the red curve).(d) The dependence of the dimensionless ratio, ∆   , on the relative Authors contribution.A.N.M. generated the research idea, formulated the problem, performed most of analytical calculations and wrote the paper draft.E.A.E.wrote the FEM codes and prepare figures.The treatment of XRD data is performed by O.A.K. D.R.E.worked on the results interpretation, discussion, and paper improvement.

B. 1 .
= 0,   = 0. (B.4b)It is seen from Eqs.(B.2) that the solution (B.3), obtained for the single-domain rod with a homogeneous polarization, is valid in a general case too, and thus the strain field (B.4) corresponds to the following stress tensor:   = ,   =  −   2 ,   =  +   2 ,   =   =   = 0. (B.5)Here the constants , , and  should be determined from the boundary and/or interfacial conditions.They are related with the constants  0 , , , and  by Eqs.(B.2).Below we apply the general solution (B.4)-(B.5) to the considered physical problem.The core-shell model for long nanorods and nanowires

Table AI .
LGD coefficients and other material parameters of a BaTiO3 core in Voigt notations.

Analytical solution of elastic problem for a ferroelectric nanowire
The free energy expansion on polarization  3 and stress   powers has the following form:Here   are electrostriction coefficients,  3 is the electric field component along the wire axis, and   is the chemical and/or thermal expansion strain.Hereinafter we use the Voigt notations for   or matrix notation for   ( → 1,  → 2,  → 3,  → 4,  → 5 and  → 6) when necessary.