Quantitative estimation of inter-dipole interaction energy in giant-permittivity CaCu 3 Ti 4 O 12 solid bulks

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AIP Advances ^ARTICLE scitation.org/journal/adv

Quantitative estimation of inter-dipole interaction energy in giant-permittivity CaCu ₃ Ti ₄ O ₁₂ solid bulks

Cite as: AIP Advances9, 115108 (2019);doi: 10.1063/1.5126442 Submitted: 4 September 2019•Accepted: 23 October 2019• Published Online: 14 November 2019

Ran Ma,^1,a) Yongqiang Li,^2,a) Yicheng Huang,^1,a) Shicheng Dong,¹ Qirui Yang,¹ Yazhou Jin,¹ Zhengjun Zhang,¹ Meng Yan,¹ Kai Chen,^1,b) Junming Liu,² and Jinsong Zhu²

AFFILIATIONS

1School of Science, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China

2National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, People’s Republic of China

a)Contributions:R. Ma, Y. Li, and Y. Huang contributed equally to this work.

b)Author to whom correspondence should be addressed:[email protected]

ABSTRACT

In the CaCu3TiO4solid bulks with the giant permittivity of∼10⁴, the 100-fold permittivity-drop below 200 K has been observed without any long-range structural transition. In the frame of the comprehensive model, the Ising-model analysis of dielectric polarization is applied and the energy of the nearest interaction among the dipoles is deduced from the experiments by utilizing the particle swarm optimization.

The interaction energy of∼56 meV is reasonably smaller than the experimental value of the thermal activation energy of∼88 meV. The interaction energy is independent of the temperatures, the dipole-chain lengths, and the frequencies of the applied AC electric field. Therefore, such appropriate, stable, and intrinsic interaction energy makes the many-body dielectric relaxation rather than the structural transition responsible for the permittivity drop.

© 2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).https://doi.org/10.1063/1.5126442., s

Giant-permittivity solid insulators are potential devices for nanoscale miniaturization of microelectronics and capacitive power storage of renewable energies. Double-perovskite-structure CaCu3TiO4(CCTO) solid bulks with space group Im¯3 exhibit a giant permittivity of∼10⁴from 200 K to 400 K, which is almost independent of low frequencies.^1–3 In particular, the permittivity drops 100-fold below 200 K, which has been suggested as a signature of the Maxwell-Wagner relaxation in the electrical inhomogeneity.⁴ However, such interpretation is recently put in doubt by the neg- ative magnetodielectric effect, which suggests that the drop arises from the “frozen” state of one-dimensional, antiparallel, and independent finite-dipole-chains in a comprehensive model.⁵ In com- parison with the ambiguous relaxation units in the Maxwell-Wagner relaxation, the dipole chains in the comprehensive model are clearly suggested to originate from the finite correlated off-center displacements of B-site titanium (Ti) ions in the individual⟨001⟩columns, the length of which is within the 5–10 crystallographic unit cells

or the 10–20 TiO6octahedra (right panel ofFig. 1).⁵The independent⟨001⟩columns forbid the correlation in the transverse direction and then a long-range ordered ferroelectric state. Dynamically, an applied AC electric field triggers a parallel-dipole-orientation branch (left panel ofFig. 1), and the inversion symmetry restriction of the ionic dielectric polarization is externally broken. These 1-D finite dipole chains are independent of each other, and the thermal activation process of the independent dipole chains is approximated to that of one dipole chain (the single-body approximation), and the Arrhenius law is applied well. The relaxation process of the 10–20 dipoles in the finite dipole chain is well described by the Hamilto- nian in the Ising model analysis of dielectric polarization,⁵ and the dielectric relaxation is the many-body process rather than the Debye behavior.^6,7

Genuine Debye behavior is usual in most liquids but seldom observed in solids. In liquids, the van der Waals force among molecules is weak and the dielectric relaxation of the molecules

AIP Advances9, 115108 (2019); doi: 10.1063/1.5126442 9, 115108-1

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FIG. 1. Schematic of the TiO6octahedra along the [001] crystallographic direction (left panel) approximated to a parallel-dipole-orientation branch (right panel).

can be approximated by the single-body model. However, the interactions in solids are strong, and the empirical expressions for the dielectric relaxation in solids include the Cole-Cole, Cole- Davidson, and Havliak-Nigami forms,⁸ in which the relevant fac- tors are induced for the correlation strength among many-body relaxation units, regardless of the materials, the physical structures, the types of bonding, the chemical types, the polarizing species, and the geometrical configurations. These phenomenological forms put us in an awkward situation of why the CCTO solid bulks with the inversion symmetry exhibit the giant permittivity or even the unusual large dielectric polarization while assuming that the dielectric polarization is proportional to the permittivity. In the CCTO solid bulks, either single crystals or polycrystalline ceram- ics, the type and density of the defects are very different; however, all these solids exhibit similar dielectric relaxation,⁹ the origin of which is hardly related to the defects. Hitherto, the dipole chains in the comprehensive model are the most probable origin for the dielectric relaxation below 200 K, which is implicit in the electrical, antiferromagnetic, optical, and photo-related phenomena. The off-center displacement of the single Ti ion is ∼0.04 Å, and the displacement sum in one dipole chain is∼0.40 to 0.8.^5,10Intrigu- ingly, in the cubic-to-tetragonal structural phase transition of the classic ferroelectric BaTiO3bulks, the off-centered displacement of the Ti ion in the single TiO6octahedra is as large as∼0.10 Å, and thus, the ferroelectric solid bulks exhibit the permittivity of∼10³.

A question arises in the CCTO solid bulks that why such collec- tive and larger off-center displacements are accompanied with the dielectric relaxation rather than the structural phase transition. To the best of our knowledge, none of the research work has been hitherto reported to answer the question. The interaction energy among the dipoles may be a clue to elucidate the question although the concrete example is scarce and, further, give us an insight into the fundamental nature of the correlated dipoles in the solid bulks.

Because of the screening effect in the solids, it is rather appropriate to simplify the interactions among the dipoles to only consider the nearest neighbor interaction. For the one-dimensional dipole chain with the nearest neighbor interaction, the Hamiltonian in the Ising model analysis of dielectric polarization is given by

H=W−Jsj(sj−1+sj+1),

whereWis the energy barrier between two localized sites, apart from the nearest neighbor interaction energy of dipoles,J.^11,12sjis the unit vector of thejth dipole located on sitej, which is a stochastic function of time and makes random transitions between values±1, due to interactions with a phonon bath. The experimental value ofW is about 88 meV. The Hamiltonian can be applied to the parallel- dipole-orientation branch in the CCTO solid bulks. The resultant complex dielectric function is

ε(w) =1−2(fγ)⁻¹iw

[η⁻¹D_N/2+D_N/2−1]^N/2−1∑

r=0

η^{∣r−N/2∣}Dr

(2Λ−C)DN−1−pDN−2

,

where the parameters are cited from Refs.11–13. Based on our published experimental data that the relaxation temperatures and the relaxation frequencies are measured in the dielectric relaxation of the CCTO solid bulks (the inset ofFig. 2),^5,7,13the particle swarm optimization is programmed in the MATLAB (2015b) to deduce the value of the interaction energy in the frame of the Ising model analysis. The convergence accuracy is better than 1%.

Figure 2 shows the relation between the relative calculation error and the interaction energy in different dipole chains. All of the relative calculation errors are better than 3%, and the best one in the dipole chain of 20 dipoles is about 0.7%. Interestingly, the different-length dipole chains show the same change between the relative calculation error and the interaction energy. The relative calculation error decreases when the value of the interaction energy increases, and finally, the relative calculation error reaches the minimum value when the interaction energy is∼56 meV. When the interaction energy is larger than 56 meV, the relative calculation errors increase. Although the relaxation temperatures and the relaxation frequencies are different in the same dipole chain, all of the interaction energies converge to about 56 meV when the relative calculated errors are the minimum value. Therefore, the interaction energy of about 56 meV is independent of the relaxation temperatures and the relaxation frequencies.

Figure 3clearly shows the intrinsic nature of the interaction energy. The calculated results shown inFig. 2are summarized in Fig. 3. The interaction energy is the dependent variable, when the relaxation temperature is the independent variable, “Temperature,”

AIP Advances9, 115108 (2019); doi: 10.1063/1.5126442 9, 115108-2

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FIG. 2. Relation between the relative calculation error and the interaction energy in the different-length dipole chains at different relaxation temperatures and relaxation frequencies.

the dipole-chain length is transformed into “Dipole Number in One Dipole Chain,” and the relaxation frequency is also the “Frequency of Applied AC Electric Field.” The interaction energy is independent of the temperatures, which indicates that the interaction energy

keeps against the thermal destruction below 200 K. The interaction energy is in accordance with the structural transition temperature of∼224 meV (645 K)/crystalline unit cell, and thus, the thermal energy below 200 K could not break up the dipole correlations.

AIP Advances9, 115108 (2019); doi: 10.1063/1.5126442 9, 115108-3

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FIG. 3. Relation between the interaction energy and the temperature, the dipole number in one dipole chain, or the frequency of the applied AC electric field.

The rather strong interaction could fasten these local lattice distor- tions, keep the lattice firm, and stop the avalanche effect. When the applied AC electric field disturbs the lattice and the temperature is decreased, the thermal energy only freezes the dipole chains, and the permittivity drop is accompanied with the dielectric relaxation rather than the long-range structural transition. The independence of the interaction energy on the dipole numbers and the external frequencies further reflects the reliability of the calculated interaction energy.

In conclusion, the particle swarm optimization is utilized to deduce the interdipole interaction energy from the experimental data, in the comprehensive model with the Ising model analysis of dielectric polarization. The interaction energy is so strong that it keeps the local lattice distortion from thermal-energy destruction below 200 K, which theoretically proves the self-consistency of the comprehensive model to elucidate the dielectric relaxation in the CaCu3Ti4O12solid bulks.

The work was supported by the National Natural Science Foun- dation of China (Grant No. 11004106) and by the National 973 Program of China (Grant No. 2015CB946502).

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