Mean-Field Approach to Dielectric Relaxation in Giant Dielectric Constant Perovskite Ceramics

The dielectric properties of CaCu3Ti4O3 (CCTO) and A2FeNbO6 (AFN, A = Ba, Sr, and Ca) giant dielectric constant ceramics were investigated in the frequency range from 1 Hz to 10 MHz. The relaxation properties can be perfectly described by a polaron model, indicating that the dielectric relaxation is intimately related to the hopping motion caused by localized charge carriers.


Introduction
As driven by the impetus of smaller and smaller feature size of devices in microelectronics, researchers are looking for the so-called high- materials.Perovskites such as lead zirconate titanate (PZT) usually possess high dielectric constant of a few hundreds at room temperature [1,2] and are widely used as capacitive components.The perovskite-based oxide CaCu 3 Ti 4 O 12 (CCTO, space group Im3) was reported to have a colossal dielectric constant (CDC) in the order of 10 4 which is nearly constant from 400 K down to 100 K but drops rapidly to less than 100 below 100 K [3].A huge amount of work [3][4][5][6][7] has thereafter been carried out in attempts to understand the origin of the remarkable dielectric properties.It has been found that the temperature at which the steplike decrease in dielectric constant takes place strongly depends on the measuring frequency and roughly follows an Arrhenius behavior.Similar phenomenon has also been reported in a number of materials, such as A 2 FeBO 6 (A = Ba, Sr, and Ca; B = Nb, Ta, etc.) [8], La 1− Ca  MnO 3 (LCMO) [9], Pr 0.7 Ca 0.3 MnO 3 (PCMO) [10], LaCuLiO 4 [11], LaSrNiO 4 [12], TbMnO 3 [13], and Li/Ti doped NiO [14].The remarkable low temperature dielectric relaxation in many manganites, cuprates, and nickelates [9][10][11][12][15][16][17] has been attributed to localized hopping of polarons between lattice sites with a characteristic timescale.Furthermore, this relaxation behavior clearly displays a freezing temperature following a glass-like process, suggesting that an electronic glass state is realized.It seems plausible to think of the influence of the electric state on the dielectric response of a solid.
Polaronic relaxation usually involves a variable range hopping (VRH) or a nearest-neighbor hopping conduction process.Polarons as the main sources contributing to the conduction of CCTO ceramics at the low temperature relaxation region has been confirmed [18].Tselév et al. [19] found a power-law frequency dependent dielectric response in CCTO epitaxial thin films suggesting that the dielectric contribution came from hopping carriers.Based on these results, the low temperature dielectric response in CCTO has been attributed to polaron relaxation related localized charge carriers [20].It should be mentioned that Maxwell-Wagner (MW) origin, such as the internal barrier layer capacitance (IBLC) should not be excluded in the explanation of the colossal dielectric constants, especially at high temperatures [4,6,7,21].Although, the MW type mechanisms which were resulted from the surface layer [21], IBLC [4], and/or contact electrode [6] do have an influence on the dielectric constant values of CDC materials, they nearly have no influence on the relaxation process at low temperature.
In this paper, we present a systematic study on the low temperature dielectric relaxation process of CCTO and A 2 FeNbO 6 (AFN, A = Ba, Sr, and Ca) systems.A polaron relaxation model was proposed as a possible explanation to the low temperature dielectric relaxation in CCTO and AFN ceramics.It should be mentioned that MW model and polaron model are both related to the moving of space charge carriers.The difference is that MW model emphasizes the interface, but polaron model considers the space electrons/ions exist in the whole bulk materials.Although the polaron concept has been used to explain the dipolar effects induced by charge carrier hopping motions inside the CCTO grains [20,22], a clear theory formula to describe polaron relaxation is absence.Based on polaron theory, Jonscher's law (universal dielectric response (UDR)) could be used to fit the frequency dependence of the dielectric permittivity.By using a mean-field approach, we will present a simplicity expression on the temperature dependence of the dielectric permittivity driven from polaron relaxation.

Experimental
Single phase CCTO and AFN (A = Ba, Sr, and Ca) ceramics were prepared through a conventional mixed oxide route, and the detailed processing parameters can be found elsewhere [23,24].The Wolframite method, using FeNbO 4 as the site precursor, has been used to synthesize AFN with no secondary iron-oxide phases.
X-ray diffraction (XRD) was conducted on sintered ceramics samples of CCTO and AFN.Data were collected on an automated diffractometer (X'Pert PRO MPD, Philips) with Cu  1 radiation.The XRD results confirm that the sintered ceramics are single phase.The fracture surfaces of the ceramic pellets were examined by scanning electron microscopy (SEM, JSM-6335F, and JEOL).Micro-Raman spectral measurements were performed on a JY HR800 Raman spectrometer under backscattering geometry.An Argon ion laser was used as the excitation source with an output power of 15 mw at 488 nm.For dielectric measurement, top and bottom electrodes were made by coating silver paint on both sides of the sintered disks and followed by a firing at 650 ∘ C for 20 minutes.The temperature and the frequency dependences of the dielectric permittivity of the samples were measured by a frequency response analyzer (Novocontrol Alpha-analyzer) over a broad frequency range (1 Hz-10 MHz).

Results and Discussion
3.1.Crystal Structure and Microstructure.Figure 1 shows the room temperature XRD pattern of sintered CCTO ceramics.All the diffraction peaks can be indexed according to a cubic cell of space group Im3 [3].Standard group theory analysis predicts that the Raman active modes are distributed among the irreducible representations as 2  + 2  + 4  [25].Inset of Figure 1 shows the Raman spectrum at room temperature.The Raman lines at 448 and 512 cm −1 have exact   symmetry, and a line of   symmetry is clearly pronounced at 575 cm −1 , which is assigned to the Ti-O-Ti antistretching mode of the oxygen octahedra [25].The Raman spectrum also confirms a single cubic phase of our CCTO sample.
The XRD patterns of the Ba 2 FeNbO 6 (BFN), Sr 2 FeNbO 6 (SFN), and Ca 2 FeNbO 6 (CFN) are given in Figure 2. The diffraction peaks of BFN samples can be indexed according to a monoclinic structure [26], while SFN and CFN samples show an orthorhombic structure with the space group Pnma(62) [27,28].Figure 3 displays typical SEM photographs of the fracture surface of CCTO and SFN ceramics.The average grain size of CCTO is found to be in the range of 2∼8 m, while SFN exhibits a morphology with the bimodal distribution of grain size.One has the grain size of several micrometers, and the other has the smaller grain size in the order of several hundreds of nanometers.It should be noted that BFN and Ca 2 FeNbO 6 (CFN) display a similar morphology to that of SFN, while BFN has a slightly larger average grain size than SFN and CFN.

Dielectric Behavior.
Figure 4 shows the frequency dependences of the real [  ()] and imaginary [  ()] parts of the dielectric constant of CCTO, BFN, CFN, and SFN under different temperatures.The temperatures were chosen because they reveal the typical relaxation features of the mentioned ceramics.The high frequency region of CCTO and BFN shows the presence of a Debye-like dipolar relaxation process, and SFN and CFN show a generalized dipolar loss peak possessing a more serious departure from Debye, as summarized by Jonscher [29].The peak frequency of   () decreases with decreasing temperatures for all the samples, indicating the freezing in dipolar moment.The width of the   () peak of AFN (A = Ba, Sr, Ca) is much larger than that of CCTO (as shown in Figure 4), which implies a broader distribution of relaxation time.Above the   () peak frequencies, the high-frequency branches of CCTO and AFN follow the fractional power law [29]: where  is the angular frequency (2) and  is the exponential constant.The calculated exponent  from Figure 4 is 0.11, 0.12, 0.25, and 0.38 for CCTO, BFN, CFN, and SFN, respectively.It should be noted that  is nearly independent of temperature for all these CCTO and AFN samples.Larger  means a much lower loss (  ()/  () = cot(/2)) and flatter frequency dependence than that in a Debye relaxation.The peak frequency of the above relaxation process obeys the following Arrhenius law: where  0 is a prefactor and   is the activation energy.The Arrhenius fits for CCTO, BFN, SFN, and CFN ceramics are summarized in Figure 5, where   = 0.1, 0.18, 0.23, and 0.34 eV for CCTO, BFN, SFN, and CFN, respectively.The obtained values are very close to the previous reported ones for the same materials [4][5][6][7]30].
Although, the Maxwell-Wagner type mechanism is always responsible for the dielectric permittivity of ceramics as an extrinsic source due to the existence of grain boundaries.This model cannot explain why the one with very similar microstructure and electrical resistivity (BFN, CFN, and SFN) shows so different relaxation regions and activation energies.In addition, in terms of the Maxwell-Wagner model, it is very difficult to explain the similar dielectric relaxation behavior appearing in the same temperature region for single crystals, polycrystalline ceramics, and epitaxial thin films whose microstructure is certainly different.In Figure 6, the imaginary part of dielectric permittivity   of CCTO and BFN is shown as a function of the real part   , namely, the Cole-Cole plot.From Figure 6, all the data at different temperatures collapse on an arc described by the Cole-Cole equation, showing universal scaling behavior in the dielectric response.We note that in the lower frequency region there is a relaxation relating to the extrinsic effect.This relaxation is probably due to a Schottky-type barrier existing in this ceramic [31].

Mean-Field Approach.
In a previous study, It has been confirmed that localized charge carriers are the main sources contributing to the conduction of CCTO [18] and AFN [32] ceramics at low temperature relaxation region.Under an external electric field, the hopping motion of localized carriers like polarons gives rise to dipolar effect and sizable polarization contribution to the dielectric permittivity.Therefore a quantitative description of the polaron contribution to the dielectric response is highly demanding.
Ramirez et al. [33] have proposed a defect model as a possible explanation of the low temperature dielectric response of CCTO.In this model, isolated defects such as Cu vacancies produce a local disruption of the ideal cubic structure and then the defective regions relax between alternate equivalent configurations.However, high resolution experiments [33] have revealed that Cu vacancies most probably exist in the grain boundary regions, and thus they cannot be shown in the bulk response.Therefore, polarons are mainly responsible for the bulk dielectric relaxation.Essentially, it is reasonable to presume that polarons are isolated; then the cooperative effects are absent at low densities of polarons.The relaxation of polarons occurs at the rate  determined by the energy  barrier between alternate equivalent configurations and the temperature: where  0 depends on the effective mass of the defect and is in the order of 10 10 -10 12 Hz.The local polarizability of a polaron is where   is the static polarizability of a polaron.It should be noted that (3) and ( 4) could be obtained directly from [33].As suggested by Ramirez et al. [33], a mean-field approximation can be adopted and the dielectric response can be calculated by the Clausius-Mossotti equation: where  is the density of the polarons, and the details can be found in [34].It is worth noting that the Clausius-Mossotti equation works best for gases and is only approximately true for liquids or solids, particularly for polar materials, or the dielectric constant is large.In addition, the limitations of the approximation in (5), which does not consider the variations around each polaron properly, should also be kept in mind.Let  = (4/3) 0   ; then we get The real and imaginary parts of the dielectric permittivity are The measured temperature dependent   () and   () (see Figures 7, 8, and 9) for CCTO, BFN, and CFN are similar to those reported in previous work [3][4][5][6][7]30].The real part dielectric permittivity   reveals a pronounced steplike increase accompanied by a loss peak.The dielectric permittivity of CCTO, BFN, and CFN is fitted according to (7) by a least-square method.For CCTO measured at 100 kHz, the best fitted result was obtained with  = 0.971,  0 = 3.4 × 10 9 , and Δ = 924 K.It is very interesting to note that the obtained barrier energy  (obtained from Δ = /  ) is 0.08 eV, a value quite close to the characteristic activation energy obtained from (2) (Section 3.2,   = 0.1 eV for CCTO).The same fitting parameters were used to calculate the temperature dependence of   and   at different frequencies.The solid lines in Figure 7 are calculated from the previous model, which reproduce the experimental data quite well.We have also performed the least square fitting process on BFN and CFN.The best result was obtained with  = 0.987,  0 = 9.710, and Δ = 1951 K for BFN, and  = 0.987,  0 = 1.98E9, and Δ = 2695 K for CFN, respectively.The calculated barrier energy  is 0.17 eV for BFN, which is also quite close to the result obtained from (2) (see Figure 5).However, the calculated barrier energy for CFN is only 0.24 eV, which is much smaller than the one obtained from (2) (0.34 eV).Presumably, this may be attributed to an overlap between the polaron relaxation and a high temperature relaxation process as shown in Figure 9.

Conclusions
In summary, we have prepared single phase CCTO and AFN (BFN, CFN, and SFN) ceramics by conventional solid state reaction.Based on the dielectric study, it can be concluded  that the low temperature step-like dielectric relaxation in CCTO and AFN can be well explained by a polaron model involving a dipolar-type relaxation.By using a mean-field approach, we get a dielectric function which describes the dielectric relaxation in CCTO and BFN quite well.

Figure 3 :
Figure 3: SEM images of the fracture surface of (a) CCTO and (b) SFN ceramics.The arrows denote nanograins in SFN.

Figure 4 :
Figure 4: Frequency dependence of the real and imaginary parts of dielectric permittivity for (a) CCTO, (b) BFN, (c) CFN, and (d) SFN.The straight gray line in (a) is a guide to the eye.

− 1 )Figure 5 :
Figure 5: Arrhenius fit of the frequency dependence of the dynamic transition temperatures   for the dielectric relaxation in CCTO, BFN, SFN, and CFN.

Figure 6 :
Figure 6: The imaginary part of dielectric permittivity   of CCTO and BFN is plotted against the real part   .

Figure 7 :
Figure 7: The temperature dependence of dielectric constant of CCTO.The solid lines are the best fits from polaron model (see text).

Figure 8 :
Figure 8: The dielectric constant of BFN is shown against temperature.The solid lines are the beat fits of polaron model.

Figure 9 :
Figure 9: The dielectric constant of CFN is shown as a function of temperature.The solid lines are the beat fits of polaron model.