It is well known that the static permittivity of ferroelectric materials is related to frequencies of all polar phonon modes through the Lyddane–Sachs–Teller formula.^[Reference Lyddane, Sachs and Teller^1] Near the phase transition, however, an additional low-frequency mode has to be often taken into account—the so-called central mode (CM).^[Reference Petzelt, Kozlov and Volkov^2–Reference Onodera^5] A generic reason for this additional Debye-type excitation seems to be large-amplitude fluctuations between quasi-stable off-center ionic positions. Existence of such intrinsic CM could be very clearly demonstrated, e.g., below the cubic-tetragonal phase transition T _C of BaTiO₃.^[Reference Hlinka, Ostapchuk, Nuzhnyy, Petzelt, Kuzel, Kadlec, Vanek, Ponomareva and Bellaiche^6]

Similar CM is also known to exist in the paraelectric phase. Phenomenological theories of the paraelectric CM have been developed by several authors.^[Reference Petzelt, Kozlov and Volkov^2–Reference Onodera^5, Reference Pirc and Blinc^7] All these approaches led to a coupled relaxator–oscillator dielectric response. However, an important question has been left open so far: whether the CM persists up to the highest temperatures, or rather it progressively disappears, or whether it disappears at some well-defined temperature T _CM (>T _C).

Unfortunately, it is much more difficult to obtain a clear-cut experimental evidence for the dielectric CM in the cubic perovskite phase.^[Reference Petzelt, Kozlov and Volkov^2, Reference Girshberg and Yacobi^4, Reference Vogt, Sanjurjo and Rossbroich^8, Reference Ponomareva, Bellaiche, Ostapchuk, Hlinka and Petzelt^9] The characteristic frequencies of the soft phonon-oscillator and CM in KNbO₃ and BaTiO₃ are so broad and close together that they can hardly be disentangled. Here, we describe a combined experimental and theoretical study of the technologically relevant mixed Ba_xSr_(1–x)TiO₃ system (BST). Established characteristic temperature trends in the model parameters of the relaxator–oscillator dielectric response allow one to understand the existence of the temperature T _CM, at which the CM in the dielectric spectra is appearing or disappearing.

Basic dielectric properties of BST solid solutions were discussed, e.g., in Refs. Reference Shirokov, Torgashev, Bakirov and Lemanov10 and Reference Walizer, Lisenkov and Bellaiche11 and works cited therein. Present experiments were carried out with a set of high-density BST ceramics with Ba:Sr ratio ranging from x = 0 to x = 1, prepared by methods described elsewhere (see Refs. Reference Ostapchuk, Petzelt, Kuzel, Savinov, Hlinka, Tkach, Vilarinho, Lisenkov, Ponomareva and Bellaiche12 and Reference Ostapchuk, Petzelt, Hlinka, Bovtun, Kuzel, Ponomareva, Lisenkov, Bellaiche, Tkach and Vilarinho13). Low-frequency permittivity obtained from standard dielectric measurements (10 kHz) in the paraelectric phase was fitted to a Curie–Weiss law:

(1)$$\varepsilon \left(0 \right)^{ - 1} = \left({T - T_0 } \right)/C.$$

As expected,^[Reference Petzelt, Kozlov and Volkov^2, Reference Shirokov, Torgashev, Bakirov and Lemanov^10] Curie constants were of the order of 10⁵ K for all concentrations, while the extrapolation temperature T ₀(x) turned out to be strongly dependent on the concentration [see the inset of Fig. 1(a)].

Figure 1.

(Color online) Dielectric loss spectra of BST ceramics determined (a) by combined IR reflectivity and time-domain THz transmission spectroscopic techniques and (b) by MD simulations using the effective Hamiltonian of Ref. Reference Walizer, Lisenkov and Bellaiche11. The spectra are obtained at 100 K above the extrapolation temperature T ₀ of the Curie–Weiss law, indicated in the insets by full line. Point symbols in insets indicate temperature–concentration coordinates associated with the spectra shown in the main panel.

These findings may suggest that at a fixed temperature interval above the extrapolation temperature T ₀, the static permittivity is roughly independent of x. However, this is not the case for the THz-range dielectric spectra of BST ceramics. In [Fig. 1(a)], we compare dielectric loss spectra obtained similarly as in Refs. Reference Ostapchuk, Petzelt, Kuzel, Savinov, Hlinka, Tkach, Vilarinho, Lisenkov, Ponomareva and Bellaiche12 and Reference Ostapchuk, Petzelt, Hlinka, Bovtun, Kuzel, Ponomareva, Lisenkov, Bellaiche, Tkach and Vilarinho13 from simultaneous fits to time-domain THz transmission and far-IR reflectivity spectra. Although all these spectra are obtained at 100 K above the corresponding temperature T ₀, we can see a clear crossover between two rather distinct regimes: a broad-band regime for BaTiO₃-rich compounds, and a narrow-band regime for SrTiO₃-rich compounds. The narrow response band can be easily adjusted with a single damped harmonic oscillator (DHO) model, while the broad band suggests a more complex spectral shape, such as that of the coupled relaxator–oscillator model.

To prove the intrinsic origin of this spectral broadening phenomenon, we have also calculated^[Reference Zhong, Vanderbilt and Rabe^14–Reference Ramer and Rappe^16] dielectric spectra of disordered BST solutions by conducting molecular dynamics (MD) simulations using effective Hamiltonians.^[Reference Hlinka, Ostapchuk, Nuzhnyy, Petzelt, Kuzel, Kadlec, Vanek, Ponomareva and Bellaiche^6, Reference Ponomareva, Bellaiche, Ostapchuk, Hlinka and Petzelt^9, Reference Walizer, Lisenkov and Bellaiche^11] Chemical disorder in the Ba and Sr ion sublattice has been included directly by specifying randomly chosen Ba and Sr ion positions within a simulation supercell representing 12 × 12 × 12 perovskite formula units of BST, assuming periodic boundary conditions. To verify that the results are independent of the particular configuration, simulations were also carried out for several different configurations.

The resulting spectra [see Fig. 1(b)] are quite monotonously varying with the average Ba concentration in the simulated supercell, even though the Ba ion distribution is obviously different for each concentration. This also confirms that the observed trends are driven by the overall concentration, rather than by the degree of the occupational disorder. As consistent with the experiments, the predicted static dielectric permittivity above T _C obeyed the Curie–Weiss law [Eq. (1)]. Moreover, simulated loss spectra for temperatures about 100 K above T ₀ indeed confirmed a similar broadening for BaTiO₃-rich compounds as in the measurements [see Fig. 1(b)].

Despite this considerable broadening, the spectra of Fig. 1 do not show a very clear 2-maximum dielectric loss response, such as the one observed in the A₁ spectra of the tetragonal phase of BaTiO₃ single crystal.^[Reference Hlinka, Ostapchuk, Nuzhnyy, Petzelt, Kuzel, Kadlec, Vanek, Ponomareva and Bellaiche^6] We expect that CM and soft mode (SM) components are much better separated in the spectra within the first few tens of K above T _C. However, in this temperature range the THz measurements were not feasible even with very thin ceramic samples because they become too opaque for THz radiation.

Motivated by this difficulty, we have calculated dielectric spectra of Ba_0.5Sr_0.5TiO₃ (BST50) in a broad range of frequencies and temperatures down to the vicinity of the ferroelectric phase transition using the effective-Hamiltonian approach of Ref. Reference Walizer, Lisenkov and Bellaiche11. To obtain well-behaved smooth dielectric spectra especially closer to T ₀ in the low THz regime, an MD time-step was chosen to be 0.5 fs and the simulations were conducted for a length of 13.2 ns to obtain 10,000 samples for averaging the autocorrelation functions.^[Reference Ponomareva, Bellaiche, Ostapchuk, Hlinka and Petzelt^9]Figure 2 displays the frequency dependence of the imaginary part of dielectric permittivity derived from MD simulations at 270 K (C = 0.6 × 10⁵ K and T ₀ = 245 K for simulated BST50). The presence of a lower frequency Debye-like CM in addition to the higher frequency SM is clearly seen there.

Figure 2.

(Color online) Dielectric loss spectra of Ba_0.5Sr_0.5TiO₃ determined by MD simulations corresponding to T = 270 K (at 25 K above the theoretical extrapolation temperature T ₀), T = 340 K, and T = 700 K. Smooth continuous lines are fit to the coupled oscillator-relaxator model of Eq. (2) (fitting parameters are shown in Fig. 3).

To account for the simultaneously appearing SM and CM, we have used the coupled oscillator–relaxator model (following the notation of Ref. Reference Hlinka, Ostapchuk, Nuzhnyy, Petzelt, Kuzel, Kadlec, Vanek, Ponomareva and Bellaiche6):

(2)$$\varepsilon \left(\omega \right)= \varepsilon _\infty + \displaystyle{{\Omega _{\rm S}^2 } \over {\omega _{\rm S}^2 - \omega ^2 - i\omega \Gamma _{\rm S} - \lpar \gamma _{\rm D} \delta ^2 /\lpar \gamma _{\rm D} - i\omega \rpar \rpar }}.$$

Here, Ω_S, ω _S, Γ_S are the plasma frequency, oscillator frequency, and damping constant of the SM, respectively; γ _D is the bare relaxation frequency of the CM, δ is the coupling coefficient, and ɛ _∞ = 1 is the background permittivity. Model parameters of the unconstrained fitting of Eq. (2) to the MD simulation data at various temperatures are shown as filled symbols in Fig. 3. For temperatures above ~400 K, the CM is not visible in the simulated spectrum and the simple DHO formula [Eq. (2) with δ = 0] provides a satisfactory fit (see Fig. 2).

Figure 3.

(Color online) Temperature dependence of the parameters of Eq. (2) obtained from MD dielectric spectra of BST50. Filled symbols correspond to the unconstrained fit by Eq. (2) for temperatures below 400 K, the full line in panel (b) represents adjustment of the Arrhenius law [Eq. (9)] to γ _D data below 400 K. Open symbols were obtained for temperatures above T _CM from a constrained fit by Eq. (2) with δ = 57 cm⁻¹ and γ _D prescribed by Eq. (9) with U = 84 meV and γ _∞ = 1160 cm⁻¹.

What can we learn from the temperature trends revealed in Fig. 3? First, let us note that Eq. (2) can be considered as an asymptotic case of the dielectric response in the model of two coupled DHOs, one representing the normal phonon SM and the other representing the extraneous excitation D^[Reference Petzelt, Kozlov and Volkov^2]:

(3)$$\varepsilon \left(\omega \right)= \varepsilon _\infty + \left[{\Omega _{\rm D} \; \; \Omega _{\rm S} } \right]\; \hat {\rm G}\left(\omega \right)\left[{\matrix{ {\Omega _{\rm D} } \cr {\Omega _{\rm S} } \cr } } \right]\comma \;$$

with

(4)$$\hat {\rm G}^{ - 1} \left(\omega \right)= \left({\matrix{ {\omega _{\rm D}^2 - \omega ^2 - i\omega \Gamma _{\rm D} } & \Delta \cr {\Delta ^{ \ast} } & {\omega _{\rm S}^2 - \omega ^2 - i\omega \Gamma _{\rm S} } \cr } } \right)\comma \;$$

where the quantities Ω_D, ω _D, Γ_D related to the CM have analogous meaning as Ω_S, ω _S, Γ_S. The coupling term Δ can be chosen positive. [Note that for ω = 0, the matrix in Eq. (4) reduces to a dynamic matrix corresponding to the two Brillouin zone center polar phonon modes under consideration. Therefore, it can be taken as real, and the frequency-independent parameter Δ can be taken as real. Moreover, the sign of the Ω_D, Ω_S, and Δ parameters enter in the expression of Eq. (3) only through the product Re(Ω_D·Ω_S·Δ). Since only the product of signs of these three parameters matters, any two of them can be taken positive. Such a preliminary choice thus does not limit the family of possible shapes of the dielectric function defined by Eq. (3).] Indeed, in the Debye-relaxator limit (ω≪Γ_D and ω≪ω _D) the ω ² term can be neglected against $\omega _D^2 - i\omega \Gamma _D $ and the above matrix reduces to that of the coupled oscillator–relaxator model.^[Reference Petzelt, Kozlov and Volkov^2] In this case, four parameters Ω_D, ω _D, Γ_D, and Δ can be replaced by only three parameters—the relaxational frequency γ _D, the coupling frequency δ, and an auxiliary frequency ρ:

(5)$$\gamma _{\rm D} = \omega _{\rm D}^2 /\Gamma _{\rm D} \comma \; \quad \delta = \Delta /\omega _{\rm D} \comma \; \quad \rho = \omega _{\rm D} \Omega _{\rm S} /\Omega _{\rm D} .$$

With these substitutions [Eq. (5)] and for ω≪Γ_D, ω≪ω _D, and Ω_S≫Ω_D or ρ≫δ, the expression of Eq. (3) reduces to the formula of Eq. (2).

Previously, several authors^[Reference Petzelt, Kozlov and Volkov^2, Reference Girshberg and Yacobi^4, Reference Pirc and Blinc^7] treated the problem of paraelectric CM in perovskites and derived a frequency dependence of dielectric permittivity in the form equivalent to Eqs. (2) or (3), (4). Common assumption in all these approaches^[Reference Petzelt, Kozlov and Volkov^2, Reference Girshberg and Yacobi^4, Reference Pirc and Blinc^7] is that the Curie–Weiss behavior of static permittivity results from the Cochran-like temperature dependence of the SM frequency ω_S² = A(T − T _C′) with a “Cochran” temperature T _C′ < T ₀ ~ T _C. This is indeed well obeyed here (with A ≈ 50 cm⁻² K⁻¹ and T ′_C = 165 K for BST50, see Fig. 3). However, the implications for the temperature dependence of δ and γ _D are quite different among these models. In the spirit of Ref. Reference Petzelt, Kozlov and Volkov2, one could assume that δ and γ _D either do not change with temperature since the bare D-mode is not contributing to the ordering mechanism, or perhaps, that (model A)

(6)$$\delta ^2 = {\rm const}.$$

and

(7)$$\gamma _{\rm D} \sim 1/T\comma \;$$

when assuming in Eq. (5) that the D-mode damping increases with the temperature like Γ_D ~ T. Bare pseudospins in the model of Ref. Reference Girshberg and Yacobi4 are considered as freely fluctuating bistable dipoles forming an ideal paraelectric gas with Arrhenius relaxation law, which leads to (model B)

(8)$$\delta ^2 = \alpha _0 /T$$

and

(9)$$\gamma _{\rm D} = \gamma _\infty \exp \left[{ - \displaystyle{U \over {kT}}} \right]\comma \;$$

where U is a potential barrier and γ _∞ is an attempt frequency. In a similar model applied to BaTiO₃ in Ref. Reference Pirc and Blinc7, one assumes that the bare pseudospin subsystem would order at a finite temperature T _L < T _C, with T _L ≈ T _C and the resulting formulas for δ and γ _D read (model C)

(10)$$\delta ^2 = \alpha _0 /\left({T - T_{\rm L} } \right)$$

and

(11)$$\gamma _{\rm D} = \gamma _\infty \exp \left[{ - \displaystyle{U \over {kT}}} \right]\; \displaystyle{{T - T_{\rm L} } \over T}.$$

Comparison of the anticipated temperature trends with the outcome of our MD simulations shown in Fig. 3 indicates that none of the above three models are fully satisfactory. As a matter of fact, the MD results suggest that the coupling term δ is constant (or it perhaps slightly increases with the increasing temperature), which is compatible with the model A only, while the increasing γ _D rather supports the thermally activated flipping, assumed in models B or C. Tentatively, we have fitted the temperature dependence of γ _D to the Arrhenius law of Eq. (9) in the 260–400 K temperature range. It gives reasonable values of U = 84 meV and γ _∞ = 1160 cm⁻¹.

In fact, it turns out that the BST50 dielectric spectra can be well fitted at all temperatures (i.e., even above T _CM) when keeping δ = 57 cm⁻¹ and γ _D described by the aforementioned Arrhenius law (fitted parameters are shown by open symbols in Fig. 3). In other words, the coupled relaxator–oscillator model Eq. (2) can be used in the whole temperature range above T _C, without any particular discontinuity in the temperature course of the model parameters near temperature T _CM.

So why does the CM seem to “disappear” from the BST50 spectrum at T _CM? To understand this behavior, it is convenient to consider the formula of Eq. (2) as the response of a DHO with a frequency-dependent self-energy term

(12)$$\hat \Pi \left(\omega \right)= - \displaystyle{{\delta ^2 \gamma _{\rm D}^2 } \over {\gamma _{\rm D}^2 + \omega ^2 }} - i\omega \left[{\Gamma _{\rm S} + \displaystyle{{\delta ^2 \gamma _{\rm D} } \over {\gamma _{\rm D}^2 + \omega ^2 }}} \right].$$

From this expression, it is obvious that the simple DHO formula is recovered when γ_D ≫ δ ²/Γ_S, since in this case the imaginary part of the self-energy becomes simply ${\mathop{\rm Im}\nolimits} \left[{\hat \Pi } \right]\approx - \omega \Gamma _S $. Therefore, it is reasonable to define the temperature T _CM as the temperature at which δ ² ≈ γ_DΓ_S. Since the temperature dependence of γ _D is steep, the temperature-driven crossover between γ _D ≪ δ ²/Γ_S and γ _D ≫ δ ²/Γ_S is quite sharp. With the data of Fig. 3, this crossover temperature is about T _CM ≈ 360–370 K. Moreover, in the case of the Arrhenius law for γ _D, one obtains

(13)$$T_{\rm CM} \approx \displaystyle{U \over {k\ln \left({\gamma _\infty \Gamma _{\rm S} /\delta ^2 } \right)}}\comma \;$$

showing that T _CM scales with the activation energy U. Perhaps this kind of crossover temperature T _CM bears some resemblance with the so-called Burns temperature or T* temperature, which have been observed in some complex perovskite materials.^[Reference Dul'kin, Petzelt, Kamba, Mojaev and Roth^17–Reference Dkhil, Gemeiner, Al-Barakaty, Bellaiche, Dul'kin, Mojaev and Roth^20]

In summary, we provide interpretation of experiments and MD simulations suggesting that permittivity paraelectric perovskites show a crossover between a simple SM high-temperature spectrum and a lower-temperature spectrum with a more complex shape involving an additional CM. Analysis of MD simulations allowed us to select a convenient formula for temperature evolution of the model parameters. As a matter of fact, these MD results indicate a thermally activated dynamics in a postulated pseudospin subsystem and an almost temperature-independent pseudospin–phonon coupling coefficient. Furthermore, the dielectric spectra of BST50 from MD simulations could also be analyzed using this original coupled CM–SM model at any temperature above T _C (i.e., even in the high-temperature regime). As a result, the appearance of the CM in the dielectric spectra is understood as a crossover between fast and slow pseudospin dynamics. Arrhenius-type temperature dependence of bare pseudospin dynamics allows one to understand the abrupt change of the spectrum at this crossover, which thus appears almost as a phase transition.