Relaxors are complex materials with unusual properties that have been puzzling the scientific community since their discovery. The main characteristic of relaxors, that is, their dielectric relaxation, remains unclear and is still under debate. The difficulty to conduct measurements at frequencies ranging from ≃1 GHz to ≃1 THz and the challenge of developing models to capture their complex dynamical responses are among the reasons for such a situation. Here, we report first-principles-based molecular dynamic simulations of lead-free Ba(Zr0.5Ti0.5)O3, which allows us to obtain its subterahertz dynamics. This approach reproduces the striking characteristics of relaxors including the dielectric relaxation, the constant-loss behaviour, the diffuse maximum in the temperature dependence of susceptibility, the substantial widening of dielectric spectrum on cooling and the resulting Vogel–Fulcher law. The simulations further relate such features to the decomposed dielectric responses, each associated with its own polarization mechanism, therefore, enhancing the current understanding of relaxor behaviour.
Relaxor ferroelectrics are fascinating materials whose major unusual characteristics are related to dipolar relaxation, hence the name relaxor given to them. Such relaxation manifests itself by the fact that the real part of their dielectric response versus temperature exhibits a broad peak, at T=Tm, which is strongly frequency dependent1,2. The origin of this dipolar relaxation is still open for discussion. A commonly proposed explanation is that it arises from the motion of polar nanoregions (PNRs), which appear on cooling at the so-called Burns temperature, TB (refs 3, 4, 5). However, such a widely accepted mechanism has been recently challenged in (lead-based) relaxors on the ground of analysis of measured structural data6,7,8 (note that the controversy also extends to atomistic simulations, since some computational works did report the existence of PNRs in relaxor ferroelectrics4,9,10 while others did not11,12,13). This lack of consensus is largely due to the experimental challenges in observing PNRs directly and measuring some properties they are expected to exhibit, such as the dielectric spectra for frequencies ranging between 109 and 1012 Hz. As a matter of fact, while low frequencies (typically between 10−3 and 109 Hz) can be accessed by dielectric spectroscopy and high frequencies (typically phonons with frequencies above 1012 Hz) by the infrared reflectivity technique2,14,15,16,17, very little data at limited temperatures are currently available, primarily from time-domain terahertz transmission spectroscopy15,16, in the intermediate frequency range 109–1012 Hz. As a result, details of the relaxation spectrum (and, consequently, the details of the dipolar dynamics) remain unknown at T≃TB, where relaxation exists at subterahertz frequencies only. At lower temperatures, the part of the spectrum where relaxation and phonon dynamics merge, is thus also poorly understood.
This article reports first-principles-based simulations on the lead-free Ba(Zr0.5Ti0.5)O3 (BZT) relaxor ferroelectrics in the subterahertz frequency range that resolve this outstanding problem. Atomistic features of our numerical tool further reveal the role of the Ti- and Zr-centred dipoles, and their cross-correlations in determining some relaxor characteristics such as the Vogel–Fulcher law (VF) obeyed by Tm. Note that BZT is chosen here because BaTiO3-based relaxors proved both experimentally5,17 and theoretically4,18,19,20 to exhibit PNRs and because some experimental data on BZT (to test the predictions against) are available at some terahertz and subterahertz frequencies.
As indicated in the Methods section below, molecular dynamics simulations using the effective Hamiltonian of ref. 4 are conducted to predict the complex dielectric response at given temperatures. Practically, a series of molecular dynamics simulations is performed at 50 different temperatures between 5 and 1,000 K. After computing the dielectric response via equation (2) indicated in the Methods section, we analyse the obtained spectrum at each temperature. To this end, we apply an approach that is typically used in experimental dielectric spectroscopy, that is, to fit the data with an analytical expression written as a sum of standard empirical functions, each of which is the contribution of a particular polarization mechanism. We use the following expression (for the average diagonal element of the dielectric tensor):
The first term of this sum is the Havriliak–Negami (HN) function, which represents a relaxation mechanism. In this function, α and γ are parameters characterizing the shape of the relaxation spectrum, and χRO and νHN are the dielectric strength and the characteristic frequency of this relaxation mechanism, respectively. The second term of equation (1) describes a coupled oscillator-relaxator (COR) mechanism, with SCO, νCO, and γCO being the dielectric strength, resonant frequency and damping constant of the coupled oscillator mode, respectively, while νRO is the bare relaxation frequency of the coupled relaxation mode and δ is the coupling constant characterizing the strength of interaction between the oscillator and the relaxator. The last two terms represent two damped harmonic oscillators (DHO) of dielectric strength Sk, resonant frequency νk and damping constant γk, with k=1, 2. Note that, during our nonlinear least square fitting procedure, the real and imaginary parts are fitted simultaneously in the entire frequency range varying between 109 and 2 × 1013 Hz (0.033–667 cm−1), where meaningful data are available.
The simulated spectra and the results of their fitting to equation (1) are shown for selected temperatures in Fig. 1 (fitting results at some other temperatures can be found in the Supplementary Movie 1). One can see that, in all cases, the isothermal frequency dependence of the complex total susceptibility can be well described by equation (1), which justifies the choice of such an equation. Figure 1 further provides the decomposition of χ into four separate terms of equation (1), and thereby demonstrates that χR is the dominant effect for the lowest frequencies, while χCOR and the DHOs contribute to the dielectric response for the intermediate and highest frequencies, respectively. Note that, at microscopic level, our simulations reveal that the intermediate DHO1 mode is mostly related to the phonon vibration of the Zr ions, while the DHO2 mode is associated with the high-frequency phonon vibrations of the Ti ions, which is consistent with ref. 20. The COR mechanism corresponds to a mixed relaxation-oscillation mode within the Ti subsystem, that is a single Debye relaxation mode coupled to an optical phonon mode, while the HN function corresponds to a more complex relaxation mode arising from the influence of different chemical environments on Ti-centred dipoles. Moreover, Fig. 1c,d also display the experimental data at 150 K of ref. 16 by means of triangles. The agreement between the simulations and the measured data is quite remarkable for both the real and imaginary parts of the dielectric response, especially given that Fig. 1 shows that these two parts are very sensitive to temperature. Such an agreement asserts the validity and accuracy of the present simulations.
Having demonstrated the accuracy of our simulation, let us now use the best-fit relaxation parameters of equation (1) to calculate χ(T, ν) at selected temperatures and probing frequencies. The resulting real and imaginary parts of this total dielectric response as a function of temperature are reported in Fig. 2a,b, respectively, for different frequencies ranging from 1 GHz to 1 THz. One can see that both the real and imaginary parts of the predicted dielectric response exhibit a broad peak over temperature, which shifts in position when the probing frequency changes. In other words, our present first-principles-derived simulations reproduce the main characteristics (dielectric dispersion) of relaxor ferroelectrics. Moreover, Fig. 2b also reveals that the simulated imaginary part of the dielectric response is practically insensitive to frequency for low temperatures (below ∼150 K), which is experimentally known to occur in relaxor ferroelectrics21,22, and is often referred to as the constant-loss behaviour.
Figure 2a also displays the dielectric response obtained from the extrapolation of our numerical molecular dynamics data down to zero frequency, that is the dielectric response given by χRO+χCOR(ν=0)+S1+S2, along with the dielectric response obtained from our previous effective Hamiltonian simulations using the Monte–Carlo method4. Interestingly, as evidenced in Fig. 2a, such two static responses agree very well with each other, which further supports the scheme described by equation (1) to fit the molecular dynamics outputs. In particular, these two responses are found to possess a maximum at some finite temperature (of the order of 100 K), similar to the peak experimentally found for the static susceptibility in some lead-based and lead-free relaxors2,22,23.
To gain a deeper insight into the results depicted in Fig. 2a,b, we take a close look at the temperature dependences of some parameters appearing in equation (1). Such dependences are shown in Fig. 3. More precisely, Fig. 3a reports the total static susceptibility, χ(ν=0), associated with equation (1), as well as its separate contributions from the mechanisms involving relaxation, that is, χRO+χCOR(ν=0). Figure 3b displays the α parameter associated with the HN function (equation (1)), while Fig. 3c shows the frequency, νm, at which ImχR(ν) peaks for any given temperature (νm is known as the most probable relaxation frequency and may differ from νHN). The Supplementary Fig. 1 and Supplementary Note 1 provide information about other parameters not shown in Fig. 3. It is clear from Fig. 3a that, for temperatures above the well-known Burns temperature TB≃450 K of this system4,17, the total static dielectric susceptibility is dominated by the relaxation mechanisms, or, equivalently, that the DHOs contribute only very slightly to it (the contribution from DHOs is the difference between the black and green curves of Fig. 3a, which is found to be ∼20 for any investigated temperature). Such features contrast with the case of lead-based perovskite relaxors, where the response is purely of phonon (DHO) nature for T>TB and relaxation appears only at T<TB (refs 24, 25). The large and small contributions of the relaxation mechanisms and the DHOs, respectively, result in the Curie–Weiss law behaviour for T>TB with a negative critical temperature, TCW, in agreement with the Monte–Carlo method simulations4 and experimental results17. Moreover, χRO+χCOR(ν=0) exhibits a peak at T≃100 K, which is found to be fully responsible for the maximum in the temperature dependence of the total static susceptibility reported in Fig. 2a (note also that the broadness aspect of the total susceptibility is found to originate from the temperature behaviour of the static contributions of the relaxation mechanisms). Supplementary Fig. 1 further shows the separate contribution of χCOR(ν=0) and χRO as a function of temperature. In particular, χRO is found to practically vanish for temperatures above TB (implying that the COR mechanism is the most predominant player there), while χRO is about twice as big as χCOR(ν=0) for temperatures close to 100 K. However, we numerically found that the HN and COR contributions significantly overlap in frequency, especially at comparatively high temperatures. As a result, some of the parameters of these two contributions can only be estimated rather than reliably determined. One should therefore consider the temperature dependence of the sum of χRO and χCOR rather than look at them separately. This explains why this sum is shown in Fig. 3a.
It is also important to realize that Fig. 3b reveals that the α parameter departs from zero at the temperature of ≃240 K, that is, at the so-called T* characteristic temperature of BZT, below which Ti-rich PNRs were found to begin interacting with each other in an antiferroelectric-like fashion in ref. 4. This departure from zero may indicate the existence of a distribution of Debye modes with different relaxation times. α then increases up to ≃0.5 when cooling the system from T* to ≃130 K. It remains quite large when further decreasing the temperature down to 5–10 K, which results in frequency broadening of the ImχR(ν) peak. In fact, the peak becomes so wide at temperatures below ≃100 K (Fig. 1f) that the loss is practically constant with frequency (cf. Fig. 2b). Note that the broadening of relaxation spectrum on cooling and the constant-loss behaviour are two characteristic features of relaxor ferroelectrics, in contrast to the behaviour of most other dielectrics where the shape of the spectrum is temperature independent26. Figure 3b,c also indicate that the extracted α and νm parameters exhibit some scattering for temperatures ranging between ≃15 and 80 K. This is because the characteristic relaxation frequency, νm, reaches the low-frequency boundary of the available molecular dynamics data in that temperature range (as shown in Fig. 3c) and, as a result, the HN parameters become uncertain. On the other hand, at the very low temperatures of 5–10 K, the relaxation response moves back to the simulation frequency window, as evidenced in Fig. 1g,h, which allows us to precisely extract again the values of α. Interestingly, this parameter is now very close to zero as revealed by Fig. 3b, which explains why the loss peak is now narrower (cf. Fig. 1h) and which is characteristic of a simple Debye (exponential in time domain) relaxation. Note that the remaining relaxation at this temperature is numerically found to be caused by dipoles associated with Ti ions surrounded mostly by Zr ions (not shown here).
Arrhenius and VF laws
Let us now discuss the temperature dependence of νm depicted in Fig. 3c. Fitting this relation for temperatures above 70 K with the VF law, νm=νa exp[−Ea/kB(T−T0)] shows that T0=(−2±9) K, that is, T0 can be safely assumed to vanish. In other words, νm rather follows an Arrhenius law νm=νa exp(−Ea/kBT) above ≃70 K, with the best-fit parameters νa=1.6 × 1012 Hz (53 cm−1) and Ea=0.043 eV. Interestingly, an Arrhenius law having a similar νa, but four times larger Ea, has been experimentally found in BZT (refs 15, 16) for a different, lower-in-frequency relaxation mode. The fact that both the present simulations and the experiments indicate that T0 vanishes (that is, it is not finite) in BZT can be understood in terms of the semi-phenomenological model proposed by Pirc and Blinc27. This model suggests that T0 corresponds to the temperature at which the PNRs grow into an infinite cluster as temperature decreases, which explains why some relaxor systems exhibit a VF law behaviour. Such a scenario is relevant to lead-based relaxors according to a recent study28. However, PNRs in BZT are confined to Ti-rich regions and cannot grow into an infinite cluster with decreasing temperature (cf. ref. 4) because these PNRs are immersed into a Zr-rich matrix that is only weakly polarizable. Consistent with the fact that Pirc and Blinc have shown that the VF law would reduce to the Arrhenius law if PNRs did not grow, we can conclude that the Arrhenius law, rather than the VF law, applies to BZT. Note that Fig. 3c further shows that the Arrhenius law is not obeyed at lower temperatures, which is consistent with ref. 20.
Furthermore, the data displayed in Fig. 2a also allow us to obtain the temperature, Tm, at which the real part of the dielectric response peaks, for any probing frequency, ν (the Tm versus ν for the imaginary part of the susceptibility is shown in Supplementary Fig. 2 and discussed in Supplementary Note 2). The dependence of Tm on ν is reported in Fig. 2d, and it neither follows Arrhenius behaviour nor a single VF relation. In fact, Fig. 2d shows that Tm obeys two different VF laws of the form ν=ν0 exp[−U/(Tm−TVF)]: one for temperatures below ≃280 K (with the best-fit parameters ν0=322 cm−1, U=637 K and TVF=87 K) and one for temperatures above it (ν0=1,950 cm−1, U=1,005 K and TVF=88 K). These two VF laws have significantly different U and ν0, but similar TVF≃90 K. Interestingly, this temperature is consistent with the finding in ref. 4 that the thermally activated reorientations of the (Ti-based) dipoles inside the existing PNRs are numerically found to be frozen in BZT below ≃90 K. This TVF is also close to the temperature of the static susceptibility maximum shown in Figs 2a and 3a, as predicted by the phenomenological theory of VF relationship for relaxors proposed by Tagantsev29. In particular, it is interesting to realize that refs 22, 29 indicate that Tm can follow a VF law even if the characteristic relaxation frequency obeys the Arrhenius behaviour provided that the static dielectric response possesses a maximum at a finite temperature, which is consistent with the present data shown in Fig. 2d (for Tm), Fig. 3c (for νm) and Fig. 2a (for the static dielectric response). Note that Tagantsev’s model predicts the fulfillment of a single VF law for the Tm(ν) values belonging to the temperature interval above Tm(ν=0), but neither the size of this interval nor the Tm(ν) behaviour at higher temperatures are specified (they are determined by several factors including the shape of temperature dependence of static susceptibility, the temperature dependence of characteristic times of the contributing relaxation modes, and the shape of relaxation spectrum22). In fact, two different VF laws have been observed experimentally in some relaxors30, with the second VF law occurring in a higher temperature range that cannot be described by Tagantsev’s model.
It is also worth emphasizing that, in Tagantsev’s model for Tm, U is not an activation energy and TVF is not the temperature where the relaxation frequency vanishes, that is, TVF could be different from T0 (the critical temperature involved in the temperature behaviour of νm), while in canonical lead-based relaxors these two temperatures are experimentally found to be close to each other. Therefore, like in other relaxors, the dielectric response in BZT differs fundamentally from that in ideal paraelectrics, where the static susceptibility and the characteristic relaxation time diverge simultaneously (in proportion to 1/T) as the temperature approaches zero (ref. 26). However, in canonical lead-containing relaxors the relaxation time diverges at the characteristic VF temperature T0>0 (ref. 21), while, in BZT, the relaxation time remains finite at T>0, but the dielectric strength (static susceptibility) of relaxation mode peaks at another characteristic temperature, Tm(v=0)≃TVF, and tends towards zero on further cooling.
Note also that a detailed analysis (Supplementary Fig. 3 and Supplementary Note 3) shows that the two relaxation-related modes (HN and COR), we have predicted in our simulations, are also seen in experiments (they are traditionally called central modes). However, their behaviour cannot be accurately verified because of insufficiency of experimental data. Moreover, experiments also report an additional relaxation mode in BZT which dominates at very low frequency (ν<1 GHz). This latter mode cannot be mimicked by our present simulations and was suggested to originate from the collective dynamics inside PNRs or the thermally activated reorientations of the PNRs (ref. 20).
To elucidate further the origin of the dielectric response in BZT and to better understand the existence of two VF laws depicted in Fig. 2d, we take advantage of another strength provided by our atomistic simulations, that is the possibility to separate the total dielectric response into its constituent parts: the χTi,Ti contribution from the 5-atom cells centred on Ti sites, the χZr,Zr contribution from the 5-atom cells centred on Zr sites and the χTi,Zr correlation between these two different types of 5-atom cells. This decomposition of the total dielectric response is (unfortunately) impossible to obtain in experiments but is made feasible here, thanks to the previously developed technique utilizing, in equation (2), the fact that d=dTi+dZr, where d is the total electric dipole moment of the whole system, while dTi and dZr are the electric dipole moments of the 5-atom cells centred on the Ti and Zr ions, respectively20 [note that the effective Hamiltonian used here4 possesses the so-called local modes as degrees of freedom. These local modes are technically centred on the B-sites (Ti or Zr), represent the collective displacements of the Ba, B and oxygen ions within any 5-atom unit cell, and are directly proportional to the electric dipole moments of each 5-atom cell]. Figure 2c shows the resulting real parts of χTi,Ti, χTi,Zr and χZr,Zr as a function of temperature, for different probing frequencies. The real part of χTi,Ti resembles the total susceptibility of Fig. 2a in every aspect except that its maximum value is ∼20% smaller. In contrast, the real parts of χTi,Zr and χZr,Zr are much smaller in both magnitude (approximately one tenth of χTi,Ti) and dispersion. In particular, the susceptibility of χZr,Zr is essentially a constant over temperatures and frequencies. These results therefore indicate that the total dielectric response of disordered BZT and its accompanying dispersion are mainly dominated by the response arising from the Ti-centred 5-atom cells, which is consistent with our aforementioned finding that the HN and COR mechanisms (which concern the Ti subsystem) are mostly responsible for the unusual dielectric features of BZT below TB. Interestingly, one can also extract from Fig. 2c the temperature Tm,Ti at which the real part of χTi,Ti peaks for any probing frequency. The resulting Tm,Ti-versus-ν curve is also reported in Fig. 2d, and can be well fitted at any temperature by a single VF law, with the best-fit parameters ν0=361 cm−1, U=745 K and TVF=69 K. It is striking to realize that the Tm,Ti(ν) function is virtually identical to that of the total response for any temperature ≲240 K (which is, incidentally, T*), but differs for the temperatures above. For these higher temperatures, one can see in Fig. 2c that the real part of χTi,Zr disperses much less than that of χTi,Ti, which leads to a reduction of Tm with respect to Tm,Ti above 280 K.
In summary, we report first-principles-derived simulations in BZT that reproduce the main characteristics of relaxor ferroelectrics, that is the frequency dependence of the real and imaginary parts of the dielectric response versus temperature. Analysis of these simulated results reveals that the relaxation-type polarization processes dominate in the dielectric response of disordered BZT for all temperatures, which contrasts with lead-based perovskite relaxors in which the response is purely of phonon nature for T>TB and the relaxation only appears below TB (refs 24, 25). Such a dramatic difference may originate from the fact that BZT has been reported to exhibit PNRs below TB (refs 4, 5, 18, 19, 20), while the PNRs as compact regions of ferroelectric order might not exist inside a nonpolar matrix in the classical lead magnesium niobate relaxor according to recent diffuse neutron and x-ray scattering6,7 and x-ray fluorescence holography experiments8. Furthermore, the significance of the characteristic relaxor temperature T* of BZT in terms of dipolar relaxation is also clearly determined by the present study: above this temperature, the relaxation consists of a single Debye mode (corresponding to α=0) coupled to an optical phonon mode within the Ti subsystem, while below T*, the Ti-centred dipoles subject to different chemical environments result in a relaxation spectrum that significantly broadens (α>0), signifying the distribution of Debye relaxation times. The present computational results also predict that the frequency dependence of the Tm temperature at which the real part of the total dielectric response peaks is well described (below ≃280 K) by a single VF law. The second VF law having a different parameter U but identical TVF is further found to describe the behaviour of Tm versus frequency for the temperatures above ≃280 K. We therefore believe that the present study deepens our knowledge on relaxor ferroelectrics, in general, and their characteristic dipolar relaxations, in particular, which are responsible for their unusual properties.
We employ the first-principles-based effective Hamiltonian that has been developed in ref. 4 and successfully used to model and study different static and dynamical properties of the BZT systems18,19,20. This Hamiltonian is presently implemented within the molecular dynamics technique described in refs 20, 31, 32, 33, 34, 35, and is applied to a 12 × 12 × 12 supercell made from disordered Ba(Zr0.5Ti0.5)O3 solid solution. The complex dielectric susceptibility, χlm(ν), can be obtained from the molecular dynamics simulations via refs 20, 31, 33, 36, 37:
where ν is the frequency, while l and m define Cartesian components, V is the volume of the chosen supercell, ɛ0 is the vacuum permittivity and d(t) is the electric dipole moment of the system at time t, and 〈...〉 represents thermal averages. At any given time t, d(t) is obtained by summing over every dipole of the system, and therefore contains contributions from different compositional regions that have different dipole dynamics, including relaxational dynamics. Note the important technical differences with respect to previous studies20,32,33,34,35,38,39: the total molecular dynamics simulation time is extended to 16 ns, which is about eight times longer than the simulation time used in refs 20, 32, 33, 34, 35, 38, 39, to accurately mimic the low-frequency part of the dielectric response; and the real part of the dielectric response is presently investigated in detail, in addition to the imaginary part.
How to cite this article: Wang, D. et al. Subterahertz dielectric relaxation in lead-free Ba(Zr,Ti)O3 relaxor ferroelectrics. Nat. Commun. 7:11014 doi: 10.1038/ncomms11014 (2016).
D.W. and L.B. acknowledge the support of the Air Force Office of Scientific Research under Grant FA9550-16-1-0065 and ONR Grant N00014-12-1-1034. A.A.B. and Z.G.Y. thank the ONR (Grant No. N00014-12-1-1045) and the NSERC (Grant No. 203773) for support. J.H. thanks the Czech Science Foundation (Project 15-04121S). D.W. also acknowledges the support from National Natural Science Foundation of China (Grant Nos. 51390472 and 11574246). Note that Z.G.Y.'s primary affiliation is Simon Fraser University.
This video shows the fitting of the simulated dielectric response using Eq. (1) at various temperatures.