Polarization rotation associated critical phenomena in epitaxial PbTiO₃ thin films near room temperature

Properties of a material usually show anomaly at the phase transition point, and critical phenomena for a material system operated near a first-order phase transition can be utilized to further optimize the physical properties. Experimental studies revealed that ferroelectric relaxors electrically driven near the critical end points exhibit giant electromechanical coupling due to critical phenomena.^1,2 It is widely accepted that electric-field-induced polarization rotation and polarization extension may have a significant impact on the piezoelectric response in some relaxor ferroelectric single crystals of complex perovskite-oxide solid solutions such as Pb(Zn_1/3Nb_2/3)O₃ − PbTiO₃ (PZN-PT) and Pb(Zr_1−xTi_x)O₃ (PZT),^3–8 in which the intermediate monoclinic structures play a crucial role. It was suggested that critical phenomenon leads to reduction of free energy barrier, and therefore permitting ease of polarization rotation and large enhancement of electromechanical response.¹ 

Critical phenomena in ferroelectrics are mostly studied at the ferroelectric tetragonal to paraelectric cubic phase transition in bulk crystals,^9,10 with electric field applied parallel to the four-fold polar axis (i.e., polarization extension rather than polarization rotation is involved). Landau theory based phenomenological analyses in perovskite-type ferroelectrics suggested that dielectric and piezoelectric properties are indeed enormously enhanced near the critical end point for ferroelectric to paraelectric phase transition.^2,9 When electric field is applied along axes other than the polar axes (e.g. in [110] and [111] crystallographic directions), critical end points that involve ferroelectric tetragonal, orthorhombic, and rhombohedral phases have been phenomenologically analyzed.¹¹ However, monoclinic structures that are often linked to ultrahigh piezoelectric response driven by the mechanism of polarization rotation, were not considered.¹¹ So far, polarization rotation associated critical phenomena in epitaxial systems remain to be explored.

PbTiO₃ is a typical displacive-type ferroelectric substance of perovskite oxide family that exhibits only a single ferroelectric state of tetragonal symmetry under stress- or strain-free conditions. Monoclinic phases were experimentally observed in 5 nm-thick strained epitaxial PbTiO₃ thin film grown on DyScO₃.¹² Recent experimental studies¹³ showed that local polarization rotation can occur in 30 nm-thick polydomain epitaxial PbTiO₃ grown on SrRuO₃/DyScO₃ that shows tetragonal symmetry macroscopically, which was attributed to strain gradient associated flexoelectric field.^14,15 Recently an engineered rotation of polarization direction was observed in artificially layered epitaxial PbTiO₃/CaTiO₃ superlattices through multiple x-ray diffraction measurements.¹⁶ In PbTiO₃ crystal, first-principles¹⁷ and phenomenological calculations¹⁸ showed that monoclinic structures can be stabilized under electric field. In PbTiO₃ epitaxial thin films the electric-field-induced polarization rotation path and electromechanical coupling were shown to be sensitive to anisotropy of in-plane strain.¹⁹ Epitaxial thin films can be grown on various substrates where a variety of isotropic or anisotropic strained states are possible through selection of appropriate substrates, and properly strain-engineered ferroelectric thin films may promise optimized functional response.

The paraelectric-ferroelectric phase transition in bulk PbTiO₃ occurs at ∼492 ^oC, which is less favorable for room-temperature microelectronic device applications. In the present paper it is shown that in epitaxial thin films there exist electric field dependent strain-driven and temperature-driven first-order ferroelectric-ferroelectric phase transitions that involve polarization rotation and monoclinic states. The present study indicates that for epitaxial PbTiO₃ subject to highly anisotropic in-plane strain, critical phenomena at the first-order monoclinic-orthorhombic transition can be tuned to room temperature. Due to critical behavior rather large dielectric and piezoelectric response may occur in strained epitaxial PbTiO₃ near room temperature because easy polarization rotation and polarization extension are allowed in the vicinity of a critical point.

Over the past decade electrocaloric effect in ferroelectric thin films has attracted great attention due to their potential applications in miniaturized electric cooling devices.^20–22 In the present paper I will also present the impact of polarization rotation associated critical phenomena on room-temperature electrocaloric response in strained epitaxial PbTiO₃ thin films.

(001)-oriented ferroelectric thin films epitaxially grown on dissimilar substrates are considered here. Electric field (E₃) is applied in a direction perpendicular to the film surface. The present theoretical calculations are based on a free energy function shown below,

where T is temperature, u_j and P_i are components of misfit strain and polarization, respectively (i, j = 1, 2, 3). F(T, u_j, P_i) is the appropriate free energy potential for epitaxial ferroelectric thin films.^23–25 The equilibrium polarization states are determined by minimizing F.

Anisotropy of in-plane strain strongly affects the field-induced polarization path.¹⁹ Here strain anisotropy is defined by a factor $ϕ= ( u 1 − u 2 ) / u 1$⁠, which ranges from 0 to 2: ϕ = 0 for isotropic strain (u₁ = u₂), and ϕ = 2 for highly anisotropic strain (i.e., equal but opposite in-plane strain, u₁ = -u₂). Due to anisotropic in-plane strain, pseudo-cubic perovskite unit cell turns into orthorhombic one (a₁≠a₂≠c, see Fig. 1). Based on previous studies,¹⁹ three ferroelectric phases may exist for ϕ = 2: (1) Orthorhombic O(a₁), $P 1 2 ≠0$⁠, $P 2 2 = P 3 2 =0$⁠; (2) Orthorhombic O(c), $P 3 2 ≠0$⁠, $P 1 2 = P 2 2 =0$⁠; (3) Monoclinic M_C(a₁c), $P 1 2 ≠ P 3 2 ≠0$⁠, $P 2 2 =0$⁠. Both $M C I$ and $M C II$ in Fig. 1 refer to c-type monoclinic M_C(a₁c) structure²⁶ for which the polarization vector is confined to (010) plane, the difference lies in the relative magnitude of P₁ and P₃.

For a first-order transition, thermodynamic state functions (e.g. polarization and enthalpy) change discontinuously and there is a latent heat at the transition point where two phases coexist. For a phase transition under external pressure²⁷ or electric field,¹ above a critical point the changes in thermodynamic state functions become continuous and there is no phase transition any longer. Critical point of a material system can be determined by calculating the latent heat of a first-order phase transition and monitoring its disappearance. In the case of epitaxial PbTiO₃, entropy of the orthorhombic and monoclinic states is written as S = − ∂F/∂T = − 3.765 × 10⁵P², where $P= P 1 2 + P 3 2$ is magnitude of the polarization vector. Latent heat L for the ferroelectric-ferroelectric transition in PbTiO₃ at temperature T^* is thus given by,

where ΔS is entropy change at the transition point.

Relative dielectric susceptibility η_ij for monoclinic M_C and orthorhombic O(a₁) and O(c) states are calculated from dielectric stiffness χ_ij as follows,

where ε₀ is the vacuum permittivity. The non-zero dielectric stiffness in Eq. (3) are listed below,

where $α i *$ and $α i j *$ are renormalized coefficients²³ obtained from α_i and α_ij; α_i, α_ij, and α_ijk are dielectric stiffness and higher order dielectric stiffness coefficients at constant stress. According to Eqs. (3) and (4), non-zero components of dielectric tensors for O(c) and O(a₁) phases are η₁₁≠η₂₂≠η₃₃, consistent with the orthorhombic symmetry.

Longitudinal piezoelectric coefficient d₃₃ can be derived from field-induced longitudinal strain S₃.²⁸ For M_C or orthorhombic phases d₃₃ is given by

where s_ij are the elastic compliance at constant polarization, and Q_ij the electrostrictive constants.

When electric field is applied to a ferroelectric, there is usually entropy change associated with field induced polarization rotation and/or polarization extension. However, if the polarization changes adiabatically, the entropy would remain unchanged, which is only possible if the temperature changes simultaneously. At a given temperature T, adiabatic temperature change in PbTiO₃ due to electrocaloric effect²⁹ is given by

where C_p is the specific heat capacity. For PbTiO₃, near room temperature C_p is taken as 2.8x10⁶ J/m³K.³⁰ 

The free energy coefficients and material parameters (in SI units, T in ^oC) taken from the literature^23,31,32 are used in the present calculations: a₁ = 3.765(T-478.8) × 10⁵, a₁₁ = − 7.252 × 10⁷, a₁₂ = 7.5 × 10⁸, a₁₁₁ = 2.606 × 10⁸, a₁₁₂ = 6.1 × 10⁸, a₁₂₃ = − 3.66 × 10⁹, Q₁₁ = 0.089, Q₁₂ = − 0.026, s₁₁ = 8.0 × 10⁻¹², s₁₂ = − 2.5 × 10⁻¹².

For epitaxial PbTiO₃ subject to highly anisotropic in-plane strain, triclinic phase is absent and only monoclinic M_C phase occurs during the entire electric-field-induced polarization rotation process.¹⁹ This condition of high strain anisotropy may bring about considerable enhancement of piezoelectricity at medium-level electric field.¹⁹ In the present work, we focus on the effect of electric field induced critical phenomena on the monoclinic structure associated phase transition and the electromechanical coupling of epitaxial ferroelectrics.

Polarization components versus anisotropic in-plane strain (u₁ = -u₂) for epitaxial PbTiO₃ thin film at various electric fields and 25 ^oC are shown in Fig. 2. At zero E₃, O(c) and O(a₁) exist at small and large strain level, respectively, while at the intermediate strain level M_C(a₁c) phase is stabilized. With increasing strain, P₁ increases while P₃ decreases, and there is a strain-driven first-order monoclinic-orthorhombic phase transition close to a critical strain of u_C = 0.01152. As can be seen in Fig. 2, there are polarization jumps for both P₁ and P₃ at the phase transition point (u₁ = -u₂ = 0.01152) where $M C I$ transforms into O(a₁). Under E₃ field, P₁ decreases while P₃ increases so that the polarization vector rotates from in-plane to out-of-plane position and monoclinic M_C(a₁c) phase region is accordingly expanded. In the meantime, the critical strain at which polarization jumps occur is shifted to higher values, and polarization jumps decrease, suggesting that the first-order phase transition is gradually weakened. Above a critical field of ∼2.7 MV/m, polarization discontinuity disappears.

Fig. 3(a) presents magnitude of the polarization vector as a function of in-plane strain (u₁ = -u₂) at 25 ^oC under electric field, from which the E₃-field dependent strain-driven $M C I$ to O(a₁) ferroelectric-ferroelectric phase transition is clearly seen. Latent heat shown in Fig. 3(b) was calculated from the polarization jump shown in Fig. 3(a) using Eq. (2). At zero field the latent heat exceeds 9000 kJ/m³. Under E₃-field, polarization vector in O(a₁) phase rotates away from in-plane position so that the strain-driven ferroelectric-ferroelectric phase transition now happens between $M C I$ and $M C II$⁠. The polarization discontinuity is reduced under E-field so that the latent heat drops with increasing E₃. The critical field E_CP was obtained by data fitting and extrapolation as shown in Fig. 3(b). From Fig. 3 it is clear that E_CP is about 2.72 MV/m at which latent heat L approaches zero.

Fig. 4 shows strain vs. electric field (u₁-E) phase diagram indicative of an E-field-induced critical phenomenon in epitaxial PbTiO₃. The solid line (phase coexistence line) represents locus of the first-order $M C I - M C II$ transition which vanishes at the critical end point indicated by the arrow in Fig. 4. The critical point (u_CP, E_CP) is determined to be about (0.01215, 2.72 MV/m). Below the critical point, there are polarization jumps and non-zero latent heat. Energy barrier for a transition between two phases is associated with latent heat at the transition point. Appling E-field effectively weakens the first-order transition and lowers the free energy barrier. Near the critical point the $M C I − M C II$ transition barrier dramatically diminishes, so that polarization rotation is greatly eased. Above the critical point changes in polarization state take place continuously, i.e., there is no phase transition any longer. However, there are points at which slopes of the P-u₁ curves or ∂P/∂u₁ (derivative of polarization with respect to epitaxial strain) reaches maxima (see Fig. 3(a)). These inflection points form supercritical line or the so-called Widom line.²⁷ 

In the above discussion, critical phenomenon observed at the field-biased strain-driven monoclinic-orthorhombic phase transition has been presented. In the following I will present a similar critical phenomenon observed at the temperature-driven monoclinic-orthorhombic phase transition. For this purpose let us choose an epitaxial system subject to a highly anisotropic in-plane strain of u₁ = -u₂ = 0.012 that is very close to the zero-field critical strain for M_C(a₁c)-O(c) phase transition (see Fig. 2). It is easily understood that the critical electric field should drop significantly with in-plane strain approaching the critical value for zero-field strain-driven monoclinic-orthorhombic phase transition.

Fig. 5 shows polarization components as a function of temperature for strained epitaxial PbTiO₃ (u₁ = -u₂ = 0.012) at various E₃ fields. At zero field, M_C(a₁c) phase exists over a broad range of temperature below about -1 ^oC, at which there is a first-order phase transition from monoclinic M_C(a₁c) to orthorhombic O(a₁). Above -1 ^oC, O(a₁) is stabilized. Under electric field, the polarization vector in O(a₁) phase rotates from in-plane to out-of-plane position, accompanied by polarization extension, and the M_C(a₁c) to O(a₁) transition temperature is shifted up with increasing electric field strength.

Similar to the case of strain-driven M_C-O transition, for the temperature-driven M_C-O transition both polarization discontinuity and latent heat L at the transition decrease with electric field and disappear at a critical field E_CP. Fig. 6 presents temperature dependence of P and L for strained epitaxial PbTiO₃ thin film (u₁ = -u₂ = 0.012) at various electric fields. Under electric field the first-order transition becomes drastically weakened, and the energy barrier for polarization rotation is significantly decreased as well. The E_CP is estimated to be about 2.73 MV/m.

Fig. 7 shows temperature vs. electric field (T-E) phase diagram indicative of an E-field-induced critical phenomenon in epitaxial PbTiO₃. The solid line is phase boundary between $M C I$ and $M C II$ states, which ends at a critical point. The critical point (T_CP, E_CP) is evaluated to be about (34 ^oC, 2.73 MV/m). At high field above the critical value, the two monoclinic states merge and the epitaxial system is at a supercritical regime where the phase boundary disappears and instead there exists a supercritical line that can be determined from calculations of the maxima of ∂P/∂T (derivative of polarization with respect to temperature) above the critical end point. It is clear that near the critical point energy barrier for the temperature-driven phase transition is remarkably lowered and polarization rotation is greatly eased.

From Figs. 4 and 7 it can be inferred that there should exist a line of critical end points in a T-E-u₁ three-dimensional phase diagram. More detailed studies are needed to clarify this issue. E_CP should increase drastically with the in-plane strain deviating from the critical strain for M-O phase transition (u_C = 0.01152). Previous studies have indeed shown that for u₁ = -u₂ = 0.015, the critical electric field necessary for polarization rotation from in-plane to out-of-plane is greatly enhanced.¹⁹ 

Fig. 8 displays η₃₃ as a function of temperature and in-plane strain (u₁ = -u₂) for epitaxial PbTiO₃ under various electric fields. Figs. 8(a) and 8(b) show that one can adjust magnitude of the electric field to obtain giant η₃₃ at room temperature. The giant dielectric susceptibility and its great tunability under applied biasing electric field suggest that large electromechanical coupling can be made possible. As shown in Fig. 9, the present phenomenological calculations indeed predict that piezoelectric properties of epitaxial PbTiO₃ thin film at room temperature can be tuned by the applied field. For O(c) (distorted tetragonal structure) in strained PbTiO₃ d₃₃ values are very close to that for the ordinary tetragonal structure in stress-free PbTiO₃ bulk crystal. However, for the low-symmetry intermediate monoclinic state on the polarization path d₃₃ may have huge enhancement, depending on the anisotropic in-plane strain and the applied electric field. At zero field d₃₃ of the M_C(a₁c) phase reaches maximum at a critical strain where M_C(a₁c)-O(a₁) transition takes place, i.e. strain-induced M_C state already possesses fairly large piezoelectric coefficient d₃₃. Under electric field, lowering of energy barrier and ease of polarization rotation may bring about even higher electromechanical response. As shown in Fig. 9, at an E₃-field of just 1 MV/m, d₃₃ now peaks at a greater value, much higher than that for the zero-field strain-induced state.

A general feature of various critical phenomena is that thermodynamic response functions become divergent as a critical point is approached.^1,27 The prominent feature of strain dependent and temperature dependent dielectric and piezoelectric spectra as shown in Fig. 8 and Fig. 9 is manifest of polarization rotation associated critical phenomena in epitaxial PbTiO₃. From Figs. 8 and 9 it is noted that for E < < E_CP the dielectric and piezoelectric peaks are characteristic of first-order phase transitions. Near E_CP, however, those peaks tend to become divergent. Slightly above the critical point, the dielectric and piezoelectric maxima remain prominent. For E > > E_CP, those dielectric and piezoelectric peaks are greatly lowered and smeared, which are characteristic of a supercritical regime.

Fig. 10(a) shows electrocaloric effect induced adiabatic temperature change in strained epitaxial PbTiO₃ (u₁ = -u₂ = 0.012) under various electric fields as a function of temperature. The room-temperature electrocaloric response as a function of in-plane strain is displayed in Fig. 10(b). Under an electric field of 1 MV/m, near room temperature ΔT can well exceed 3 K. Fig. 10 indicates that the electrocaloric activity below the critical end point is enhanced due to the latent heat for ferroelectric-ferroelectric phase transition. Similar enhancement was observed at the paraelectric-ferroelectric transition in BaTiO₃ single crystal¹⁰ below the critical end point, for which only polarization extension is involved. The calculated ΔT is comparable to the room-temperature multicaloric effect³³ calculated for bulk PbTiO₃ subjected to much large fields (ΔE ∼ 40 MV/m, stress Δσ ∼ 1 GPa), but much greater than the room-temperature electrocaloric effect calculated for (001)-textured polycrystalline PbTiO₃ thin films.³⁴ 

Fig. 11 presents E-field dependence of polarization, dielectric constant η₃₃, piezoelectric coefficient d₃₃, and electrocaloric properties of the induced M_C structure in epitaxial PbTiO₃ at room temperature. The initial state at zero field is O(a₁) phase, and M_C state is formed by field-induced polarization rotation. If the epitaxial strain is below the critical point u_CP = 0.01215 (see Fig. 4), there is polarization jump in the P vs. E curve as shown in Fig. 11(a). Above u_CP, polarization increases continuously with the applied field. Below u_CP, dielectric constant η₃₃ and piezoelectric coefficient d₃₃ peak at the E-field where polarization jump occurs. Above u_CP, η₃₃ and d₃₃ peak at the inflection point of the P vs. E curve. Clearly the η₃₃ and d₃₃ peaks below u_CP are considerably higher and sharper than those above u_CP. As shown in Fig. 11(c), below u_CP there is discontinuous change in entropy that occurs in parallel to the polarization jump in Fig. 11(a); electrocaloric responsivity ΔT/E exhibits a similar jump at the same E-field but quickly peaks afterwards, and further increasing field leads to gradual decrease of electrocaloric responsivity. Above u_CP, however, the entropy changes continuously with E-field and the ΔT/E peak is considerably lowered and broadened.

In the present paper, I use a modified Landau-Ginzburg-Devonshire phenomenological model to compute properties of PbTiO₃ under strain and applied electric field. Under zero E-field and zero strain, as discussed by Vanderbilt and Cohen,²⁶ the classical sixth-order Devonshire theory does not support a monoclinic phase in which the polarization is confined to a symmetry plane, rather than to a symmetry axis, while an eighth-order theory allows for three kinds of monoclinic structures. Under E-field, however, previous studies in BaTiO₃ showed that both sixth-order³⁵ and eighth-order free energy model³⁶ can be used to describe the E-field-temperature phase diagrams, and the calculated phase boundary agrees well with the experiments.³⁷ Experimentally the monoclinic M_C phase has been observed in (001)-field cooled BaTiO₃ single crystals.³⁸ In strained PbTiO₃^12,16 and BaTiO₃³⁹ epitaxial thin films, monoclinic phases have been experimentally confirmed. Such strain-induced low-symmetry structures can be explained using the misfit strain-temperature phase diagrams²³ derived from a modified sixth-order model with renormalized free energy coefficients. Theoretical studies showed that polarization rotation and the induced monoclinic states in epitaxial thin films are dependent upon both electric and elastic boundary conditions.^40,41 Based on a modified sixth-order model, detailed phenomenological analyses considering electrostatic boundary conditions as well as strain and domain formation showed that occurrence of the monoclinic structures in PbTiO₃ thin films depends on both epitaxial strain and film thickness.⁴¹ So far, there has been no direct experimental evidence on the E-field induced monoclinic structures in PbTiO₃ single crystals. However, the reported flexoelectric rotation of polarization due to local strain gradient in PbTiO₃ thin films¹³ may be regarded as indirect evidence since the effect of mechanical strain gradient and the associated flexoelectric field^14,15 is in some way analogous to that of E-field. Very recently, long-range ordered vortex-antivortex arrays that exhibit nearly continuous polarization rotation were experimentally observed in PbTiO₃/SrTiO₃ superlattices grown on (001)-DyScO₃, which resulted from the balance of epitaxial strain associated elastic energy, electrostatic energy at the interface, and gradient energy in the well strain-engineered heteroepitaxial systems.⁴² 

In summary, polarization rotation associated critical phenomena have been investigated in epitaxial PbTiO₃ thin films. The critical point can be shifted to room temperature by manipulating in-plane strain. Near the critical point the energy barrier for polarization rotation and polarization extension diminishes, concomitantly dielectric and piezoelectric coefficients become divergent. Large electromechanical and electrocaloric coupling near room temperature are highly desirable for technological applications of epitaxial ferroelectrics. Phenomenological analyses indicate that epitaxial strain can tune the electric-field-induced critical phenomena to room temperature, resulting in large enhancement of dielectric, piezoelectric, and electrocaloric response.

The financial support from the National Natural Science Foundation of China (NSFC Grant No. 10874109) and the Program for New Century Excellent Talents in University (Grant No. NCET-08-0648) is gratefully acknowledged.