Chemical capacitance proposed for manganite-based ceramics

The measured value of effective electric permittivity \varepsilon_{eff} of several compounds, e.g., (BiNa)(MnNb)O_{3}, (BiPb)(MnNb)O_{3}, and BiMnO_{3} increases from a value \approx 10-100 at the low temperature range (100-300 K) up to the high value reaching the value 10^5 at high temperature range, e.g., 500-800 K. Such features suggest the manifestation of thermally activated space charge carriers, which effect the measured capacitance. The measured high-value effective permittivity of several manganite compounds can be ascribed to the chemical capacitance C_{\mu}=e^2\partial N_{i}/\partial \mu_{i} expressed in terms of the chemical potential \mu. The chemical capacitance C_{\mu}^{(cb)} = e^2 n_{C}/k_{B}T depends on temperature when the conduction electrons with density n_{C} = N_{C} \exp(\mu_{n}- E_{C})/k_{B}T are considered. The experimental results obtained for the manganite compounds, at high temperature range, are discussed in the framework of the chemical capacitance model. However, the measured capacitance dependence on geometrical factors is analysed for BiMnO_{3} indicating that the non-homogeneous electrostatic capacitor model is valid in 300-500 K range.


Introduction
Recently, scientists have turned their attention to the development of high power electrical energy storage devices. Such a system includes electrochemical and electrostatic capacitors. The promising materials are oxides and composites of oxides, e.g., MnO 2 , RuO 2 , Fe 2 O 3 [1]. The occurrence of defects produces an effect on the electric properties of a sample. Primarily, they affect the electronic structure and thus the conductivity. However, due to the relation ε * = iσ * , the effective value of electric permittivity is usually measured in case of the materials with perovskite structure that contain defects. In such a case, contribution of the lattice and the space charge or charge carrier responses ought to be considered [2] ε * (ω) = ε * (ω) lattice + ε * (ω) carries . (1.1) The concept of a capacitance is usually related to an electrostatic geometric capacitor determined by the electric field E between two metal electrodes storing opposite charges. When a dielectric fills the capacitor, the electric field is modified due to the appearance of induced and ordered dipoles. Hence, the capacitance of such a system can be modified in accord with the dielectric material properties C electrostatic = ε r ε 0 A d .

A. Molak
The perovskite stoichiometric ABO 3 compounds, e.g., niobates, tantalates and titanates, show a large gap E g ∼ 3 eV in the electronic structure. Hence, their dielectric properties are described using the electrostatic geometric capacitance [equation (1.2)]. However, several perovskites that contain the 3d transition metal ions in the B sublattice, e.g., BiMnO 3 , (BiNa)(MnNb)O 3 [4,5], (BiPb)(MnNb)O 3 [6], have a narrow gap E g < 1 eV. Therefore, they show a marked conductivity at temperature ranges above the room temperature. It was reported that these perovskite materials (see also references in [4][5][6] for other materials) exhibit the effective permittivity which reaches high values ε eff ∼ 10 5 ÷ 10 6 at high temperature when measured at radio-frequencies. This effect can be ascribed to the occurrence of defects, not only to electric current charges and to oxygen vacancies but also to chemical and structural non-homogeneity. Such features correspond to the semiconductor behaviour of the electric conduction. The reported activation energy values varied within 0.2 ÷ 1.0 eV range [4][5][6].
It seems worthwhile to discuss whether the electric properties of the non-homogeneous perovskite compounds can be considered in terms of the chemical capacitance. Such an approach would be limited, e.g., to the manganite-based compounds, their solid solutions, and to compounds containing a high amount of oxygen vacancies.
The electrochemical capacitance includes electrical and chemical contributions where Q denotes the charge, φ -the electric potential at the electrode, µ * = (1/ez)µ denotes the normalized chemical potential, e -the elementary charge, and z -charge number (the notation which discriminates the type of the charge carriers and non-homogeneity of the material is omitted here for simplicity) [7]. The general electrochemical capacitance has been defined for mesoscopic systems [8]. The capacitance relates to the density of states C DOS = e 2 dN dE , (1.4) where N denotes the density of charges. The other definition of the chemical capacitance is related to the carriers density c, since µ = µ * + k B T ln(c/N ), (1.5) Therefore, the chemical capacitance is proportional to the volume, V = Ad , which contains the charge carriers. Correspondingly, the electrochemical resistor can be defined and the electric current density was obtained [7] I (r, t ) = ze J = −σ(r )∇ µ * + φ . (1.6) This idea of the chemical capacitance was proposed e.g., for the solar cells based on TiO 2 nanocomposites working on the redox processes [8,9] and (La,Sr)(Co,Fe)O 3−δ electrodes applied for solid oxide fuel cells [10]. In such cases, the modification of the electrochemical potential dV produces a change in the chemical potential dµ n of electrons, associated both with the variation of free electron density dn C and the variation of a localized electron density in band gap states dn L . The chemical capacitance defined locally for a small volume element, reflects the capability of a system to accept or release additional charge carriers with the density N i due to a change in their chemical potential µ i = µ 0 i + k B T . The chemical capacitance per unit volume is formulated as follows: Hence, when both free and localized electrons are considered, one obtains the chemical capacitance dependent on temperature (1.9) The density of localized states in the band gap, at energy E and for exponential distribution is [8,11,12] g (1.10) where N L is the total density below the conductivity band and T 0 is a parameter with temperature units that determines the depth of distribution. Similarly, for electrons placed in a conduction band (1.11) where in accord with the Boltzmann distribution function (1.12) Therefore, the occurrence of the free charge carriers offers a thermally activated contribution to the chemical capacitance.
The geometrical dependence of the capacitance was measured for BiMnO 3 and NaNbO 3 . Several ceramics BiMnO 3 samples with different thickness d and surface area A were prepared. Moreover, one chosen sample of BiMnO 3 ceramics was polished step by step to obtain a series with d varying from 3.7 mm to 0.833 mm while the surface A = 0.72 mm was constant. The capacitance C and conductance G of the BiMnO 3 samples were measured using a HP 4263B LCR meter in a parallel mode. The results obtained for the NaNbO 3 and NaNbO 3−x crystals were analyzed for comparison. The crystals have been obtained from the solution of melted salts [13]. The amount of the oxygen vacancies was increased by the electrochemical procedure conducted at high temperature (950 K). The samples were reduced at a chamber in the air at a pressure lowered to 0.1 Pa. The concentration of the space charge was estimated from depolarization currents. The electric measurements of sodium niobate crystals were conducted using a Tesla BM 595 capacitance meter at f = 1 MHz.

Capacitance dependence on temperature
The former studies conducted for the (BiNa)(MnNb)O 3 and (BiPb)(MnNb)O 3 compounds showed their chemical and structural disorder. The XRD test showed the coexistence of long range electric order and short-range disorder related to defects [14]. The microanalysis carried out with the SEM tests showed a chemical non-homogeneity in the micro-scale [4][5][6]15]. The analysis of the electric conductivity and the effective permittivity was consistent with these features of the materials. The electric conductivity temperature dependences were described within the small polaron model with the activation energy varying in the 0.2÷0.5 eV range. The model of a small variable hopping polaron, which has been proposed 31801-3 for these compounds, indicated that a distribution of traps with different energies takes place in the (BiNa)(MnNb)O 3 and (BiPb)(MnNb)O 3 ceramics.
The effective permittivity ε eff ( f , T ) estimated from the geometrical formula (1.2) showed a steep increase in the high temperature range, above 500 K. Moreover, a marked dispersion in ε eff took place. The permittivity estimated at the measuring frequency f = 100 kHz, showed moderate values between ε eff 100÷1000. On the other hand, the permittivity at the frequency of 100 Hz reached high values about ε eff 10 5 ÷ 10 6 in the high temperature range 600 ÷ 800 K. This effect was ascribed to the mutual effect of the conductivity and the capacitance in accord with the general dependence ε * = iσ * .
However, another approach, based on the chemical capacitance concept is considered herein. This model includes the participation of defects and electric current carriers [7][8][9][10][11][12]. The occurrence of the point defects, which form the traps, consecutively enables the participation of thermally activated charge carriers in the measured capacitance of the samples.
The Arrhenius-type dependences of the measured capacitance C on temperature T were plotted to check the contribution of the defect subsystem ( figure 1 and 2). It turned out that the C (T, f ) dependences

The effect of the concentration of oxygen vacancies on capacitance
The effect of the concentration of oxygen vacancies on the capacitance is shown for the case of sodium niobate crystals. The estimated concentration of the space charge was equal to N Q ∈ (1·10 17 ÷6·10 17 ) cm −3 in case of the as-grown crystals. The estimated concentration of the charge carriers was one order higher, and it can be ascribed to the appearance of the oxygen vacancies N Q N (VO) ∈ (2 · 10 17 ÷ 1 · 10 18 ) cm −3 .

31801-4
Chemical capacitance for manganite based ceramics  Moderate values measured at f = 100 kHz were obtained for the electric permittivity i.e., ε eff (T, 100 kHz) < 2000 within the range 300 ÷ 900 K, even for the samples with high Bi-Mn content. The Bi and Mn co-doping caused a steep increase of the ε eff (T, 100 kHz) value at a high temperature range. This effect corresponded to the increase in the loss factor tan δ with temperature and it was ascribed to a thermally activated effect. The electric permittivity ε eff (T ) of the lightly doped ceramics (x = 0.015, y = 0.01), measured at f = 100 Hz, exhibited the values close to the values measured for the crystal ε eff (T, 100 Hz) < 10 3 .  Hence, the addition of Bi and Mn ions to the sodium niobate host induced the high value effective permittivity when measured at low frequency and at high temperature.   When a non-homogeneous multi-layer model of a condenser is assumed [3,18], the thickness L of subelectrode layers can be estimated from the formula  where d is the thickness of the sample, L is the thickness of the sub-electrode layer [ figure 5 (c)]. The thickness L = (1/2)ε L tan α of the sub-electrode layer in NaNbO 3 crystal was estimated as L = 40 ± 10 µm (numerical fit for plots in figure 5 (b), correlation coefficient 0.964 ÷ 0.990). The ε L contribution varied from 300 to 1000. Therefore, the crystal lattice contribution dominated in case of NaNbO 3 crystals, when measured at a frequency f = 1 MHz.

31801-6
The dependence of capacitance on the geometrical factors, obtained for the BiMnO 3 ceramics, is shown in figure 6. It turned out that the capacitance measured at f = 100 kHz increased with the A/d ratio [ figure 6 (a)]. The C dependence on the volume of the samples was not clear. However, a decreasing tendency can be deduced from the data obtained for the sample whose thickness was decreased by  In case of bismuth manganite, the thickness L of sub-electrode layers was estimated using the data presented in figure 7 fitted with the equation (3.2). Assuming the value ε L ∼ 200, extrapolated for thin samples, and the slope tan α = 4.5 · 10 −4 mm, the sub-electrode thickness L = (1/2)ε L tan α = 50 ± 30 µm was estimated.

Discussion
The studied ceramics BiMnO 3 , (BiNa)(MnNb)O 3 , and (BiPb)(MnNb)O 3 show thermally activated capacitance in a high temperature range. Such a behaviour corresponds to the semiconductor-type features of the electric conductivity. When the contribution of thermally activated charges to the measured capacitance is assumed, the chemical capacitance model can be applied. In accord with this model, the exponential dependence on temperature C (cb) describes the capacitance temperature dependence. One can notice that the estimated values of parameter ∆E ∈ 0.06 ÷ 2.52 eV correspond to the activation energy E a obtained from the electric conductivity temperature dependence since the E a value varies within 0.2 ÷ 1.1 eV range. Moreover, in case of the BiMnO 3 , the values of the ∆E can be compared to the energy gap E g obtained from optical absorption measurements 1.1 eV and 1.6 eV [19] as well as to the result obtained from ab initio calculations of the electronic structure where E g = 0.33 eV [20].
In case of the perovskite structure, the crystal lattice component is usually on the ε lattice ∼ 100 and higher values 10 3 ÷ 10 4 occur in the vicinity of structural phase transitions. From this point of view, the common origin for higher values of conduction and the capacitance temperature dependence, apart from the narrow energy gap, would be the occurrence of defects. Such a deduction is consistent with the results of former studies which proved the occurrence of a local structural disorder and chemical nonhomogeneity in these materials by XRD and EPMA tests. The estimated density of states in the vicinity of the Fermi energy was ∼ 10 18 ÷ 10 20 eV −1 cm −3 [4-6, 14, 15]. Hence, the high value of the measured capacitance and thus the estimated effective electric permittivity are consistent with the high concentration of defects in the samples studied, since C chem = [(ez) 2 On the other hand, the chemical capacitance and the electrostatic capacitance show different dependences on the geometrical factors. Therefore, the test was conducted for the end members of the studied series of compounds, i.e., for NaNbO 3 and BiMnO 3 .
In case of the stoichiometric sodium niobate crystals, the chemical capacitance contribution was not expected. The dielectric permittivity did not increase with temperature increase. However, a low concentration of the space charge was determined ∼ 10 17 cm −3 . Hence, the model of electrostatic, nonhomogeneous, multi-layer condenser was used to estimate the participation of the space charge in the effective permittivity. The thickness of sub-electrode layers was of the order of several tenth of µm.
In case of sodium niobate and bismuth manganite, one can expect a correlation of the capacitance with the volume of the samples, in accord with the chemical capacitance model. However, it turned out that the capacitance was not proportional to the volume of the samples when they were measured and analysed in the temperature range 300 ÷ 500 K. Instead, it was proportional to the A/d ratio which indicated that the electrostatic, non-homogeneous model is appropriate for bismuth manganite. The estimated thickness of sub-electrode layers was also of the order of several tenth of µm.
Hence, in case of BiMnO 3 , contradictory results were obtained from the analysis of the geometrical factor dependence and from the high temperature dependence of the capacitance. One can notice that the geometrical factors analysis was carried out in the 300 ÷ 500 K range where the capacitance dependence on temperature was weak.
Nevertheless, the chemical capacitance model should be used for high temperature ranges for (Bi 0.5 Na 0.5 )(Mn 0.5 Nb 0.5 )O 3 , (Na 0.5 Pb 0.5 )(Mn 0.5 Nb 0.5 )O 3 , BiMnO 3 compounds (see figure 1 and 2) where a steep increase in C (T ) took place. The thermal generation of the charge carriers, which contribute to the capacitance and induce its high value, is likely to occur only in a part of the volume of the samples.
Therefore, it seems that further studies conducted at T > 600 K would be quite useful in order to discern the ambiguities.