Reentrant phase in nanoferroics induced by the flexoelectric and Vegard effects

Reentrant phase in nanoferroics induced y the flexoelectric and Vegard effects Anna N. Morozovska 1 and Maya D. Glinchuk, 1 Institute of Physics, National Academy of Sciences of Ukraine, 46, pr. Nauky, 008 Kyiv, Ukraine Institute of Prolems for Material Sciences, National Academy of Sciences of Ukraine,, Krjanovskogo, 0068 Kyiv, Ukraine Astract We explore the impact of the flexoelectric effect and Vegard effect (chemical pressure) on the phase diagrams, long-range polar order and related physical properties of the spherical ferroelectric nanoparticles using Landau-Ginzurg-Devonshire phenomenological approach. The synergy of these effects can lead to the remarkale changes of the nanoparticles' phase diagrams. In particular, a commonly expected transition from ferroelectric to paraelectric phase at some small itical size is asent; so that the itical size loses its sense. Contrary, the stailization of the ferroelectric phase manifests itself y the enhancement of the transition temperature and polarization with the particle size deease. Ferroelectric phase reentrant phenomenon was oserved earlier in the tetragonal BaTiO nanospheres of radii 5 50 nm [Zhu et al., JAP 11, 064110 (01)] and stayed unexplained up to now. Our calculations have shown the physical mechanism of the exciting phenomenon is the flexo-chemo-effect. Since the spontaneous flexoelectric coupling, as well as ion vacancies, should exist in any nanostructured ferroelectrics, otained analytical results can e valid for many nanoferroelectrics, where reentrant phases appearance can e forecasted. Corresponding author: glim1@voliacale.com 1

I. Introduction Unique physical properties of nanosized multiferroics attract the permanent attention of researchers [1]. Many experimental and theoretical studies of nanograined ceramics or nanopowders conclude that the transition from the ferroelectric, (ferromagnetic) phase to the paraelectric (paramagnetic) phase occurs at some itical size with the size deease. Using the conception of the itical size within the theory of size effects, an excellent aility to manage the temperature of the ferroelectric phase transition, the magnitude and position of the maximum of the dielectric susceptiilty and other properties for many different forms of nanoparticles [1]. In particular, it has een demonstrated for spherical nanoparticles and nanograined ceramics of BaTiO and PTiO [,, 4]. Disappearance of ferroelectric phase for the sizes smaller than itical one seems to e in agreement with fundamental physics statement aout necessity to have large amount of interacting ions for eation of mean field as the characteristic feature of a long-range order (so called correlation effect) [5]. Theoretical consideration of size effects allows one to estimate the itical size, estalish the physical origin of transition temperature shift relatively to the ulk value, as well to calculate the changes of phase diagrams appeared under deeasing of nanoparticles sizes and shape. In particular, using the continual phenomenological approach Niepce [6], Huang et al [7, 8], Ma [9] and Morozovska et al [10, 11, 1, 1, 14, 15] have shown, that the itical sizes existence, size-driven changes of the transition temperatures leading to the enhancement or weakening of polar properties are conditioned y the different shape, surface and strain effects in nanoparticles. In particular, there are several intriguing cases. The conservation of polar properties was discovered y Yadlovker and Berger [16, 17, 18] in cylindrical nanoparticles of Rochelle salt with radius less than 15 nm at temperatures higher than the ulk Curie temperature up to the decomposition temperature. The ehaviour.was theoretically explained in Ref.[10, 1, 14] on the ase of the ordering role of the iaxial intrinsic surface stress (surface tension) [19, 0] in nanocylinders, while the hydrostatic pressure induced y the surface tension in nanospheres [1, 15] as well as the ond contraction [7, 8] can only inease the itical size. The correlation and depolarization effects can only inease the itical size [10-15] and so they deease the ferroelectric phase region. Zhu et al [1] oserved the appearance of reentrant tetragonal phase in BaTiO nanospheres at room temperature along with the enhancement of polarization and transition temperature for the particle sizes less than 0 nm and narrow distriution of the particle sizes. Therefore the transition to cuic phase is asent and the itical size does not exist for the case. As it was shown earlier [], the size-induced phase transition from a ferroelectric to paraelectric

phase disappears in ferroelectric nanopills and nanowires due to the spontaneous flexoelectric effect. However, the physical nature of the reentrant phase stays unclear up to now, although its understanding can e very important oth for fundamental science and for practical applications. This prolem solution will e useful not only for BaTiO nanoparticles, ut practically for other nano-ferroelectrics. Therefore the elaoration of the theoretical explanation that can pour light on physical mechanisms of reentrant phase appearance in nano-ferroelectric materials is the main goal of the work. We explore the impact of the flexoelectric effect and Vegard effect (chemical pressure), intrinsic surface stresses, correlation and depolarization effects on the phase diagrams, longrange polar order and related physical properties of the spherical ferroelectric nanoparticles using Landau-Ginzurg-Devonshire phenomenological approach. Below we will show oth effects can e a decisive physical mechanism ruling the oserved phenomena. Flexoelectric effect, firstly predicted theoretically y Mashkevich and Tolpygo [] in 1957, exists in any material, making the effect universal [4, 5, 6, 7]. The flexoelectricity impact is of great importance in nanoscale ojects, for which the strong strain gradients are inevitale present near the surfaces, in thin films [8, 9, 0], at the domain wall or ferroelectric interfaces [9, 1,, ], around point and topological defects [5, 6]. Owing to the surface or interface effects, flexoeffect appears spontaneously near the surface of any nanoparticle [], everywhere where the spontaneous polarization distriution ecomes inhomogeneous. Notaly, that the influence of flexoelectricity is important not only in thin films and nanoparticles, ut also in mio- and nanograined ceramics [4, 5]. Despite the great importance of the flexoelectricity, its tensorial components strength and manifestations, as estimated y Kogan [6] in early 196, remained poorly known for most of the ferroic materials, except for the experimental measurements of some components for ferroelectric perovskites [7, 8, 9] and a initio calculations [40, 41]. However all of these experimental and theoretical results do not allow otaining the complete information aout the full tensor and are rather contradictory [4], indicating on a limited understanding of the effect nature. Overall, much more is predicted theoretically than found experimentally. The nature of the chemical pressure effect [4], that other names are compositionally induced Vegard strains [44, 45] or elastic dipole [45], is the following. Any point defect (interstitial atoms, impurities and vacancies) leads to the local deformation of the ystal lattice, and the action of many defects causes the strain proportional to their concentration [45, 46, 47]. The influence of Vegard strain, coming from the diffusion and the accumulation of defects near the interfaces, results in the pronounced change of their polar properties [48, 49]. Unfortunately

exact values of the Vegard strain tensor components are purely known, ut its order of magnitude is availale for perovskites [45]. II. Analytical theory LGD-functional ulk () and surface (s) energy densities for a nanoparticle with uniaxial r ferroelectric polarization P = ( 0,0, P ) acquires a relatively simple form []: Φ = V 4 6 P g σ F P P P P k α ( T ) + β + γ + P 4 6 xi x j x k d r d E s kl P + E Qσ P σσ kl WσδN σ P x k (1a) The coefficient α (T ) S S α β 4 S Φ S = d r P + P + µ αβuαβ +... (1) 4 S typically depends on the temperature T. Here we assume the linear dependence, α ( T ) = αt ( T TC ), where T C is a Curie temperature. Coefficients β and γ are regarded temperature independent, d γ > 0. P stands for the ferroelectric polarization, is E depolarization field, Q are the ulk electrostriction tensor coefficients, g is the gradient kl coefficients tensor, F is the flexoelectric stress tensor, σ is the stress tensor, W is the elastic kl dipole (or Vegard strain) tensor, () r δn = N ( r ) N e is the difference etween the concentration of defects N r in the point r and their equilirium (average) concentration N. Hereinafter the Vegard tensor is regarded diagonal, i.e. W Wδ ( δ is delta Kroneker symol). = S S S Coefficients α and β are supposed to e positive and temperature independent, is kl e µ αβ the surface stress tensor [0, ], u is the stress tensor. As one can see from Eq.(1a), the coefficient α (T ) ecomes renormalized y elastic stresses via electrostriction. More rigorously, direct variational method shows that renormalized coefficient α depends on temperature, particle shape and sizes, polarization orientation, correlation and depolarization effects [10-15]. Below we will calculate the renormalized coefficient using definite models for elastic stresses and depolarization field acting on the inner part of a spherical particle of radius R covered with an ultra-thin defect layer of thickness R 0 ( R << R ). Most of defects are located in the ultra-thin layer and their concentration deeases 0 exponentially towards the particle ulk [51]. The "core" of radius R R 0 R is free from defects, i.e. δn = 0 in the core and δ N = N in the shell. [Figure 1] 4

R S R 0 ε shell ε core P R Figure 1. Schematics of the spherical particle with radius R covered y the shell of thickness R 0 with accumulated defects. Separation etween the seening charge and the spontaneous polarization arupt at the core-shell interface is R S. In particular we assume that the flexoelectric effect, intrinsic surface stress, ond contraction and Vegard chemical pressure contriute into the elastic stress and strain field under the curved surface of the particle. Acting elastic fields distriution strongly depends on the mechanical oundary conditions, which corresponds to the fixed normal stresses at the free surfaces and displacements at the clamped surfaces of the particle. Additional terms in the oundary conditions can originate from the flexocoupling energy [, 50]. A. Flexoelectric stresses. General expressions for the flexoelectic stresses have the form: P k δσ = Fkl. () xl The strain induced y the spontaneous polarization gradient we are interested in, are noticale in near the surface, where the gradient is essential. Since we consider a sphere within core-andshell model [6, 10] and regard that the shell thickness R 0 is much thinner than the core radius R R 0, we can approximate the flexoelectric strain value in every small part of the shell y the ones of the thin film or pill derived y Eliseev et al []. Thus we can use the expressions [] for the flexoelectric strains, ( f11 c11 ) P r and δu rθ = ( f 44 c 44 ) P r δu rr = ( P = P cos θ ), as estimation in the local reference frame. Here c are elastic stiffness, f are S the flexoelectric strain coefficients. 5

B. Vegard-type chemical stresses. Defects accumulation under a curved surface produces effective stresses of the inner part of the particle due to the lattice expansion or contraction [51,15]. The characteristic thickness of the layer enriched y defects is determined y the seening length, and their maximal concentration is limited y steric effect [48, 49]. Hence the hydrostatic pressure excess in the inner part of the particle can originate from the surface tension, ond contraction and Vegard mechanisms. For a spherical particle of radius R the pressure excess acquires the form [1, 15]: µ η R0 δσ rr ( R) =. () R s s R 11 + Here susipt "rr" denotes that the pressure is radial. R is the particle radius, µ is the surface 1 tension coefficient, are elastic compliances modulus of the material, the characteristic size s 0 R is the particle surface layer thickness, where accumulated defects eate elementary volume changes. Parameter η WδN is a "compositional" Vegard strain. For perovskites ABO the asolute values of W related with vacancies can e estimated as W (10 10 ) Å [45] and so corresponding Vegard strains δu = W δn can reach percents for relatively high variation of defect concentration 7 δn ~ 10 m -. C. Depolarization field model. Let us estimate the depolarization field existing inside a r P = 0,0, P, singe-domain spherical nanoparticle core. The core ferroelectric polarization is ( ) relative dielectric permittivity is core ε and free carriers are regarded asent. The core is covered y a non-ferroelectric paraelectric shell with relative dielectric permittivity shell ε, at that the strong inequality ε >> shell core ε is likely when the shell is regarded to e in a paraelectric phase close to the ferroelectric transition. The seening charge is located either immediately outside the shell or near its outer surface, so the "effective" separation etween the seening charge and the spontaneous polarization arupt at the core-shell interface is and R S R (see Figure 1). The electrostatic prolem can e solved exactly for a spherical particle: E d 0 RS 0 shell ( R ( R RS ) )( P P ) =. (4) ε shell core shell ( R R ) ( ε ε ) + R ( ε + ε ) core Under the typical condition R S << R the expression can e approximated as E d shell ( P P ) RS. If the shell is semiconducting the value R shell S acquires the sense of the ε ε R 0 Tomas-Fermi seening radius and its value can e much smaller than a lattice constant [ 5]. S 6

D. Size effect of the transition temperature, spontaneous polarization and tetragonality. Using expressions for elastic stresses ()-() and depolarization field (4) in a spherical particle, the renormalization of the coefficient α for the first order phase transitions, Curie (T C ) and FE transition (T FE ) temperatures acquire the form: A B ( T T ) + + α = αt C (5) R R 1 A B 1 A B TC ( R) = TC +, TFE ( R) = T θ + (6) α R R α R R T T Where T C β + Tθ 16γαT T C β + 4γα T due to the temperature hysteresis [5]. The parameters: A RS g + 4g1 + + ( 4Q1 + Q ) µ, (7a) shell ε ε = 11 0 ( 4Q + Q ) 11 ( R + λ ) z 1 ηr 1 f R B = (7) 11 0 z ( s + s ) ( R + λ ) 11 The renormalized gradient coefficients g11 = g11 f11 c11, g1 = g1 f 44 c44, extrapolation length λ 1 g 11 f 11 = 1 and correlation length R S z = g11 ε0ε are introduced in Eqs.(7) α c11g11 [, 54]. Flexo-parameter f is proportional to the components of the flexoelectric tensor f and elastic stiffness c. Expression for f is listed the Appendix A in the Suppl.Mat. [55]. As one can see from Eqs.(5)-(7) the size-dependent shift of Curie and ferroelectric phase transition temperatures is determined y the contriutions of depolarization field (the first term in the parameter A); correlation effect (the second term in A); surface tension (the third term in A); Vegard strain (the first term in the parameter B) and the flexoelectric effect (the second term in the parameter B). Equations (5)-(7) show that Curie and transition temperatures R- dependences are governed y the signs and values of parameters A and B. That constant A is always positive for all known ferroics with perovskite structure, ecause z Q1 + Q11 > for the materials and µ should e positive for the surface staility, depolarization and correlation terms are also positive. The parameter B can e positive, zero or negative, ecause the sign of the Vegard strain η is not fixed. In particular it is positive for high enough tensile Vegard strains η>0 and relatively small flexo-parameter f, ecause its contriution to Eq.(7) is always negative. The condition B<0 is delierately true for zero or compressive strains η 0. The type of the phase diagram depends essentially on the itical radius (none, the single one min max R, or two ones and R ) that can e otained from the solution of equation T ( ) = 0 R 0 FE R (see details in the Appendix B in [55]). For the case A>0 and B>0 the only itical radius exists, 7

R ( D + A) T αt = θ, parameter D = A + 4BTθ αt. The size-induced transition to the paraelectric (PE) phase occurs at R<R. Under the conditions A>0, B<0 and A 4BT α > 0, there exist two itical radii, R ( A D) T αt min = θ + θ T max and R = ( D + A) Tθ αt Correspondingly FE to PE phase transition occurs at R max and PE to FE transition at R min. Under the conditions B<0 and A + 4BTθ αt < 0 the itical radius does not exist and the nanoparticle maintains its ferroelectric state up to the ultra-small sizes. Exactly the reentrant ferroelectric phase appear in the case and enhances at small R. Since oth conditions B<0 and A + 4BTθ αt < 0 can e valid for relatively high flexo-parameter f and/or compressive Vegard strains η<0, only these two mechanisms can enhance the ferroelectric properties of nanospheres (in contrast to the nanowires, where the surface tension can maintain and improve the properties due to the condition Q < 0 1 [10]). The spontaneous polarization is ( R T ), = β( T ) 4α( R, T ) γ( T ) β( T ) γ( T ) the lattice constants c/a reflecting the tetragonality is ( ) P S 1 Q11 Q1 k PS.. Ratio of +, where the coefficient k = cosθ reflects the random disordering of the ystallographic axes in the ensemle of nanoparticles or ceramic grains orientation in the ordered FE phase, For the case of tetragonal or rhomohedral symmetry of the grains k=0.81 or k=0.866 correspondingly [56, 57]. III. Reentrant phase in the spherical BaTiO nanoparticles: theory and experiment Since the parameters µ, η and f are included in the expressions for A and B in an additive way, only their cominations can e relialy defined from the fitting to experimental data. Below we demonstrate the possiility for the spherical BaTiO particles. The known parameters α, β, γ k, Q, s, c, g, f kl, F kl and R z of the ulk BaTiO along with the reasonale ranges of the surface influence related parameters R 0, R S, f, λ, µ and η are listed in the Tale SI in Appendix C [55]. Note that we chose very small "seeding" values of λ in order to show clearly the influence of the flexocoupling mechanism. The chosen value R S = 0.1 nm corresponds to a rather small, ut reasonale depolarization effect. R 0 varies in the range (0.8 4) nm depending on the defect concentration gradient. Dependence of the transition temperature T FE on the particle radius was calculated at different flexo-parameter f and Vegard strain η. Results are shown in the Figures a-. Different curves 1-4 in the Figure a correspond to zero Vegard strain (η=0) and different effective flexo-parameter f, f 1 <f <f <f 4. The ferroelectric transition temperature monotonically ineases with particle radius inease for the smallest value f 1 =.01 10-6 V/Pa (curve 1 in the 8

Figure a). This type of ehaviour used to e the conventional one [1]. The value of f 1 was calculated using the reliale values of the flexoelectric coefficients f and taulated elastic modulus c. and η=0. The itical radius R (that corresponds to T=0) is aout 6 nm for oth cases f=0 and f=f1. The further inease of f value leads to the dramatic change of phase diagram form and transition temperature ehaviour (see curves -4 in the Figure a). Note, that the parameter f can e several times higher than the calculated value.01 10-6 V/Pa due to the symmetry change at the surface, so the highest value f 4 =.1 10-6 V/Pa is reasonale. The min max curve has two itical radiuses, and, the minimal is aout 1 nm and the maximal is aout 5 nm. It is not clear whether the phenomenological theory is applicale for reliale determination of the minimal itical radii. The phase sequence corresponding to the curve is the ferroelectric (FE) phase at R R min and again FE phase at R R ( T ) R max min max > R ( T ), paraelectric (PE) phase at R ( T ) R < R ( T ) <, <. Here we are faced with the new type of tetragonal ferroelectric phase reentrance with R deease. Classical reentrant phase without PE phase and R occurs when T FE firstly deeases with R deease and then inease at R<R r (curves and 4 in the Figure a). R is an analog of a "turning radius" and T is the "turning temperature", at ( ) r r that T FE Rr = T. The temperature Tr and radius Rr strongly ineases with f inease (compare the curves and 4 in the Figure a). The transition temperature is insensitive to the sign of the parameter f, ecause its contriution to the constant B is quadratic. Different curves 1-4 in the Figure correspond to zero flexo-parameter (f=0) and compressive Vegard strains (η<0) with the asolute value η 1 <η <η <η 4. The general form of the phase diagram in Figs.a and is similar. Actually, the ferroelectric transition temperature monotonically ineases with particle radius inease for η=0 and f=0 (curve 1 in the Figure ). However, quantitatively, the inease of T FE with the particle radius deease appears under the moderate inease of the compressive strain η from 0.4% to 1.5% (see curves -4 in the Figure min max ). The curve has two itical radiuses, R 1.5 nm and 5. nm. The "turning point" r R temperature T r strongly ineases with η inease (compare the curves and 4 in the plots). Tensile strains η>0 favour the temperature deease with the particle radius deease. Since oth chemical pressure and flexoelectric contriutions are additive to the constant B given y Eq.(7), the curves in the Figures a and demonstrate the same trends with either f inease or η inease. Room temperature polarization vs. the particle radius was calculated at different effective flexo-constant f and Vegard strain η. It is shown in the Figures c-d. Different curves 1-4 in the Figure c correspond to zero Vegard strain (η=0) and different effective flexo-parameter 9

f 1 <f <f <f 4 exactly the same as in the Figure a. Different curves 1-4 in the Figure d correspond to zero flexo-parameter (f=0) and different compressive Vegard strains η 1 <η <η <η 4 exactly the same as in the Figure. Reentrant FE phase is clearly seen for the curves 4. Curves and have two regions of FE phase, separated y PE phase in the radii range R min max ( T ) R < R ( T ) < at T=9 K. The itical radii can e defined graphically as the intersection points of the horizontal line T=9 K with the curves or in the Figures a-. Note that this statement is value for curves at all temperatures and for the curves for T > T r only. At T < T r one can expect the polarization ehavior similar to the curves 4. Curves 1 has only one PE region at R < R ( T ) and FE region at R > R ( T ) (corresponding curves 1 in the Figures a- have only one intersection point wit the horizontal line T=9 K. Temperature TFE (K) 400 00 PE 4 00 FE T r 100 η = 0 f 1 <f <f <f 4 PE R 0 0 R r 5 10 15 0 (a) Particle radius R (nm) 1 Temperature TFE (K) () 400 00 T r 00 100 PE PE 4 R f = 0 η 1 <η <η <η 4 0 0 5 R r 10 15 0 Particle radius R (nm) 1 Polarization P (C/m ) 0.5 0.4 0. 0. 0.1 4 T = 9 K η = 0 f 1 <f <f <f 4 1 Polarization P (C/m ) 0.5 0.4 0. 0. 0.1 T = 9 K f = 0 η 1 <η <η <η 4 4 1 (c) 0 5 10 15 0 Particle radius R (nm) (d) 0 5 10 15 0 Particle radius R (nm) Figure. (a) Transition temperature T FE vs. the particle radius calculated for zero Vegard strain η=0 and different effective flexo-parameter f: f 1 =.01 10-6 V/Pa (dashed curve 1), f =5f 1 (solid 10

curve ), f =7.5f 1 (dotted curve ), f 4 =11f 1 (solid curve 4). () Transition temperature T FE vs. the spherical particle radius calculated for zero effective flexo-constant f=0 and different Vegard strain η: η 1 =0 (dashed curve 1), η = 0.004 (solid curve ) η = 0.01 (dotted curve ), η 4 = 0.0145 (solid curve 4). (c)-(d) Spontaneous polarization vs. the spherical particle radius calculated for different flexo-parameter and Vegard strain values exactly the same as in the plots (a) and (). Temperature T=9 K. Other parameters of BaTiO are listed in the Tale SI. Tetragonality vs. the particle radius is shown in the Figures a-. Different curves 1-4 in the Figure a correspond to zero Vegard strain (η=0) and different flexo-parameter f 1 <f <f <f 4. Different curves 1-4 in the Figure correspond to zero flexo-parameter (f=0) and different compressive Vegard strains η 1 <η <η <η 4. Ratio c/a (a) 1.006 1.005 1.004 1.00 1.00 4 T = 9 K η = 0 f 1 <f <f <f 4 1 Ratio c/a 0 5 10 15 0 5 Particle radius R (nm) () 1.006 1.005 1.004 1.00 1.00 4 T = 9 K f = 0 η 1 <η <η <η 4 0 5 10 15 0 5 Particle radius R (nm) Figure. (a) Tetragonality vs. the particle radius calculated for zero Vegard strain η=0 and different effective flexo-parameter f: f 1 =.01 10-6 V/Pa (dashed curve 1), f =10f 1 (dotted curve ), f =11f 1 (solid curve ), f 4 =1f 1 (solid curve 4). () Tetragonality vs. the spherical particle radius calculated for zero flexo-parameter f=0 and different Vegard strain η: η 1 =0.001 (dashed curve 1), η = 0.010 (dotted curve ) η = 0.015 (solid curve ), η 4 = 0.016 (solid curve 4). Temperature T=9 K. Other parameters of BaTiO are listed in the Tale SI. 1 The tetragonality ratio c/a monotonically ineases with particle radius inease for the curves 1 corresponding to zero or smallest f and η. The inease of c/a ratio with the particle radius deease appears under the inease of either flexo-parameter f or compressive strains η min max (see curves -4). The curve has two itical radiuses, R (.5 5) nm and (11 R 11

T r 1) nm. The "turning point" temperature strongly ineases with f or η inease (compare the curves and 4). Reentrant FE phase is clearly seen for the curves and 4. Figure 4a shows the est fitting of our theory (solid curve) to experimental data [1] (symols with error ars) achieved for the definite values of the parameters A = 1.6079 K m J/C and B = 6.81869 10 9 K m J/C in Eq.(6). We did not aim to fit well the tetragonality for the smallest particle of radius.5 nm, ecause here the phenomenological continuous approach may e invalid. The fitting procedure does not allow us to extract the values η and f separately, ut only their comination in the constant B. Figures 4 and 4c illustrate the dependences of the spontaneous polarization and transition temperature on the spherical particle radius calculated for the same parameters A and B than in the Figure 4a. So that we "reconstruct" the polarization and transition temperature from to the est fitting of the tetragonality measured experimentally [1]. The reconstructed dependences clearly demonstrate the strong (more than times) enhancement of polarization and transition temperature for the particle radius less than 10 nm. The reentrant phase region appears for radii R<10 nm. One can see that theoretical Figure desie pretty good the spontaneous polarization and transition temperature reconstructed from experimental c/a value. 1

Ratio c/a (a) 1.006 1.005 1.004 1.00 1.00 T = 9 K Experiment Theory 0 10 0 0 40 50 Particle radius R (nm) Polarization P (C/m ) () 0.5 0.4 0. 0. 0.1 T = 9 K 0 0 10 0 0 40 50 Particle radius R (nm) Temperature TFE (K) (c) 400 80 60 40 0 00 80 0 10 0 0 40 50 Particle radius R (nm) Figure 4. Room temperature tetragonality ratio c/a (a), spontaneous polarization (), and transition temperature (c) vs. the particle radius. Plot (a) shows the est fitting of our theory (solid curve) to experimental data [1] (symols with error ars) achieved for the values of the constants A = 1.6079 K m J/C and B = 6.81869 10 9 K m J/C in Eq.(6). T=9 K. Other parameters of BaTiO are listed in the Tale SI. IV. Discussion It follows from the Figures -4 that oth considered mechanisms, namely the spontaneous flexoeffect and Vegard mechanism reflecting the influence of elastic dipoles originated from defects (in the oxides the defects are mainly the oxygen vacancies), desied experimental points good enough, although the est fitting was otained y the mixture of the mechanisms (Fig.4). However up to now the value of the coefficients f and η for nanostructured BaTiO were neither measured nor calculated on the asis of the first principles approach. 1

Because of this it seems reasonale to propose a method that can deease one of these two mechanisms contriution. While the spontaneous flexoelectric effect exists in any nanostructured ferroelectrics, the contriution of oxygen vacancies can e deeased y annealing in oxygen atmosphere similarly to the procedure used to find out mechanism of d 0 magnetization in nanooxides (see e.g. [58, 59, 60, 61, 6]). It can e expected that the annealing should deease essentially Vegard mechanism contriution and to otain the spontaneous flexoeffect coefficient experimentally for the first time. It is worth to underline that it could e essential size dependence of flexocoefficient, ecause general properties of flexoelectricity and piezoelectricity are similar and their estimated value is e/a, where a is characteristic atomic separation [4, 6]. Strong size dependence of piezoelectric coefficient made it possile to explain large value of magnetoelectric effect in some nanostructured multiferroics [6]. Moreover it was shown [1] that piezoelectric tensor components related to unit cell deformation is proportional to flexoeffect coefficient. Possile size dependence of the flexoelectric coefficient will open the way to inease the effective flexoelectric coefficient and so its contriution to the appearance of reentrant phases in any nanoferroelectric. On the other hand investigation of the phase diagrams in other nanostructured perovskites similar to BaTiO, e.g. ferroelectric KTa 1-x N x O (x>0.05) with different coefficients f and η in comparison in BaTiO, and so with different numer of R and therefore the type of the phase diagrams, namely FE-PE at some R, reentrant FE phase without PE and any R, FE to PE phase transition at R max and PE to FE transition at R min. The theoretical forecast is waiting for experimental verification. V. Conclusion Using Landau-Ginzurg-Devonshire phenomenological approach we estalished the impact of the flexoelectric and Vegard effect on the phase diagrams, long-range polar order and related physical properties of the spherical ferroelectric nanoparticles. The synergy of these effects can lead to the remarkale changes of the nanoparticles' phase diagrams. In particular, a commonly expected transition from ferroelectric to paraelectric phase at some small itical size is asent; so that the itical size loses its sense. The stailization of the ferroelectric phase manifests itself y the enhancement of the transition temperature and polarization with the particle size deease (reentrant ferroelectric phase). Appeared that oth flexoelectric and Vegard effects, areviated as flexo-chemo, can e a decisive physical mechanism ruling the oserved phenomena. Since the spontaneous flexoelectric effect, as well as ion vacancies, should exist in any nanostructured ferroelectrics, otained analytical results can e valid for many nanoferroelectrics, where reentrant phases appearance can e forecasted. 14

Acknowledgements A.N.M. acknowledges National Academy of Sciences of Ukraine (grant 5-0-15 and joint Ukraine-Belarus grant 07-06-15). Authors acknowledge useful remarks and multiple discussions with E.A. Eliseev (NASU). Authors' contriution M.D.G. generated the idea that the spontaneous flexoelectric effect can induce the reentrant phase in nanoferroics; she is the main contriutor to the introductive and discussion part of the manusipt. A.N.M. performed the analytical calculations of the flexo-chemical mechanism impact, generated figures and compared theory with experiment; she wrote the original part of the manusipt. References 1 M.D. Glinchuk, A. V. Ragulya, V.A. Stephanovich, Nanoferroics, Springer (01), p.78. M. H. Frey and D. A. Payne. "Grain-size effect on structure and phase transformations for arium titanate." Phys. Rev. B 54, 158-168 (1996).. Z. Zhao, V. Buscaglia, M. Viviani, M.T. Buscaglia, L. Mitoseriu, A. Testino, M. Nygren, M. Johnsson, and P. Nanni. "Grain-size effects on the ferroelectric ehavior of dense nanoystalline BaTiO ceramics." Phys. Rev. B 70, 04107-1-8 (004). 4. E. Erdem, H.-Ch. Semmelhack, R. Bottcher, H. Rumpf, J. Banys, A.Matthes, H.-J. Glasel, D. Hirsch and E. Hartmann. "Study of the tetragonal-to-cuic phase transition in PTiO nanopowders." J. Phys.: Condens. Matter 18 861 874 (006). 5 M. E. Lines and A. M. Glass, Principles and Application of Ferroelectrics and Related Materials (Clarendon Press, Oxford, 1977). 6 P. Perriat, J. C. Niepce, and G. Caoche. "Thermodynamic considerations of the grain size dependency of material properties." Journal of Thermal Analysis and Calorimetry 41, 65-649 (1994). 7 Haitao Huang, Chang Q Sun and Peter Hing, "Surface ond contraction and its effect on the nanometric sized lead zirconate titanate." J. Phys.: Condens. Matter 1, L17 L1 (000). 8 Huang, Haitao, Chang Q. Sun, Zhang Tianshu, and Peter Hing. "Grain-size effect on ferroelectric P (Z r 1 x Ti x) O solid solutions induced y surface ond contraction." Phys. Rev. B 6, 18411 (001). 9 Wenhui Ma. "Surface tension and Curie temperature in ferroelectric nanowires and nanodots." Appl. Phys. A 96, 915 90 (009). 10 M.D. Glinchuk, A.N. Morozovskaya. "Effect of Surface Tension and Depolarization Field on Ferroelectric Nanomaterials Properties." Phys. Stat. Sol. () Vol. 8, 1, 81 91 (00). 11. A. N. Morozovska, E. A. Eliseev, and M.D. Glinchuk, " Ferroelectricity enhancement in confined nanorods : Direct variational method " Phys. Rev. B 7, 14106 (006). 15

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Supplementary Materials to "Reentrant phase in nanoferroics induced y the flexoelectric and Vegard effects" Anna N. Morozovska 1 and Maya D. Glinchuk, 1 Institute of Physics, National Academy of Sciences of Ukraine, 46, pr. Nauky, 008 Kyiv, Ukraine Institute of Prolems for Material Sciences, National Academy of Sciences of Ukraine,, Krjanovskogo, 0068 Kyiv, Ukraine Appendix A The equations of state δφ δp = 0 and δ Φ δσ = u (δ is the symol of variation derivative) are otained y variation of the ulk energy density functional (1) in the particle core: σ ( k xi x, (A.1a) j xk 5 P d ( T Q ) P P P g ) σ + β + γ + = F + E E α + u P = sklσkl + Q P Fl + WδN. (A.1) x Here σ is the stress tensor, u is the strain tensor. The equations should e solved along with jk the equations of mechanical equilirium jk σ l ( x ) x = 0 and compatiility relation equivalent to the mechanical displacement vector u i continuity [1]. Variation of the surface and ulk free energy (1) on Pi yields to Euler-Lagrange equations with the oundary conditions [ ]: i Here P F m S jkim g + + σ = 0 kjimnk a Pj jk nm. (A.) x j S n r r = ( e,0, 0 ) r is the external normal to the particle spherical surface, polarization in θ spherical coordinates P = cos, P = sin θ, P = 0. The most evident consequence of r P S θ P S the flexo-coupling is the inhomogeneous oundary conditions. We consider mechanically free nanoparticles without misfit dislocations. The flexoeffect leads to the renormalization of the gradient coefficients g11 = g11 f11 c11, ϕ g 1 g1 f 44 c44 = and extrapolation length f = λ 11 1 c11g λ 11, where S λ = g11 α. Corresponding author: glim1@voliacale.com 1

Hereinafter c are elastic stiffness, f are the flexoelectric strain coefficients. The renormalized gradient coefficients should e positive for the system staility, and the staility conditions g 0 and g 0 determine the upper limits for the values of flexoelectric coefficients 11 > 1 > (compare with Ref.[ ]). The characteristic length, The parameter ( f c c ) ( c + c c ) 11 11 11 1 c1 R z = g11 ε0ε. g 1 11 f11 is proportional to the flexocoupling constants f c f 1 elastic stiffness c and geometrical dimensionless factor g. For estimations the expression for f for a sphere is regarded two times different from the one derived for the pills in Ref.[]. It is worth to underline that f values can e different from the ulk ones due to the symmetry change at the surface [ 4]., Appendix B The monotonic deease of the transition temperature with R deease appears under the condition A>0 and B>0, ecause the size-dependent contriution A R B + R is positive and monotonically ineases with R deease, so that TFE(R) deeases. Corresponding itical radius defined from the condition T ( ) = 0 is equal to B>0). FE R R = A + 4BT T θ θ α T α T + A (here A>0 and The inease of the transition temperature for some R-range can appear in the case B<0 under A B the condition + < 0. Since the first term R R A R is always positive, it favors the nanosphere transition into the non-polar paraelectric cuic phase, correspondingly deeases the spontaneous polarization and tetragonality c/a with R-deease. When the second term B R is negative and thus ineases it the transition temperature, enhances polar properties and tetragonality with R deease, since 1/R is stronger that 1/R at small R. The competition of these two terms results into the appearance of two itical radiuses, R min = A A T + 4BT θ α T θ α T and R max = A + 4BT θ T θ αt α T + A, which exist under the conditions A>0, B<0 and A 4BT α > 0. + θ T At that min R tends to zero at BT αt. Under the condition + 4BTθ αt < 0 A >> θ A the itical radiuses do not exist and the nanoparticle maintains its ferroelectric state up to the ultrasmall sizes. Exactly the reentrant ferro electric phase can appear in the case and enhances at small

+ θ T R. Since oth conditions B<0 and A 4BT α < 0 can e valid for relatively high flexoparameter f and/or compressive Vegard strains η<0, only these two mechanisms can enhance the ferroelectric properties of nanospheres (in contrast to the nanowires, where the surface tension can maintain and improve the properties due to the condition Q < 0 1 [Ошибка! Закладка не определена.]). Appendix C Tale SI. Material parameters used in the simulations coefficient ferroelectric BaTiO (collected and recalculated mainly from Ref. [a, ]) Symmetry at room T tetragonal ε (the same is shell) 7 (Ref. []) α (C - mj) 6.68(T 81) 10 5 β (C -4 m 5 J) β 11 = 18.76(T 9) 10 8.08 10 8, β1= 1.9 10 γ k (C m J) γ 111 =.1(T 9) 10 +16.56 10 9, γ 11 =6.8 10 9, γ 1 =9.46 10 9-4 Q (C m ) Q 11 =0.11, Q 1 = 0.04, Q 44 =0.059 s ( 10-1 Pa -1 ) s 11 =8., s 1 =.7, s 44 =9.4 c ( 10 11 Pa) c 11 =1.76, c 1 = 8.46, c 44 =1.08 g ( 10-10 C - m J) g 11 =5.1, g 1 = 0., g 44 = 0. [c], g 11 =.6 f kl (V) ~100 (estimated from measurements of Ref. [d]) f 11 = 5.1, f 1 =., f 44 0.045 ± 0.015 [e] F iklj ( 10-11 C -1 m ) ~100 (estimated from measurements of Ref. [d]) F 11 = +.46, F 1 =0.48, F 44 =0.05 (recalculated from [e] using F αγ =f αβ s βγ ) R z (nm) R z = 0.15 R 0 (nm) R 0 = 0.8-4 (reasonale interval is 0.4 5) R S (nm) R S = 0.1 (reasonale interval is 0.05 ) -6 f ( 10 V/ Pa).01 0.01 (reasonale interval for the ulk material is 1 8) λ (nm) <0.1 (reasonale interval is 0 4) µ(n/m) 1.5 (reasonale interval is 1 ) η 0.001-0.015 (reasonale interval of dimensionless strain 0.001 0.0) [a] A.J. Bell. J. Appl. Phys. 89, 907 (001). [] J. Hlinka and P. Márton, Phys. Rev. B 74, 104104 (006). [c] P. Marton, I. Rychetsky, and J. Hlinka. Phys. Rev. B 81, 14415 (010). [d] W. Ma and L. E. Cross, Appl. Phys. Lett., 88, 90 (006). [e] I. Ponomareva, A. K. Tagantsev, L. Bellaiche. Phys.Rev B 85, 104101 (01). References 1 S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, McGraw-Hill, N. Y., 1970. E.A. Eliseev, A.N. Morozovska, M.D. Glinchuk, and R. Blinc. Spontaneous flexoelectric/flexomagnetic effect in nanoferroics. Phys. Rev. B. 79, 16, 1654-1-10, (009).

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