Article pubs.acs.org/JPCC
Variable Range Hopping Conduction in BaTiO3 Ceramics Exhibiting Colossal Permittivity HyukSu Han,* Calvin Davis, III, and Juan C. Nino Department of Materials Science and Engineering, University of Florida, Gainesville, Florida 32611, United States ABSTRACT: Fast-fired barium titanate ceramics have been shown to exhibit room temperature colossal permittivity (ε′r ≈ 105, tan δ ≈ 0.05 at 1 kHz) when synthesized via the spark plasma sintering technique. Broadband dielectric spectroscopy analysis is here utilized to identify the active conduction mechanisms in the compound. Analysis of the temperature-dependent bulk dc conductivity reveals that the bulk conduction in fast-fired barium titanate is the result of variable-range hopping with relatively short hopping distance (∼0.5 nm) compared to other colossal permittivity materials. A common equivalent circuit that can model the contributions of the different polarization mechanisms to both the dc conductivity and colossal permittivity is established.
1. INTRODUCTION Barium titanate (BaTiO3, BT) is the most widely used ferroelectric material in a variety of applications due to its excellent dielectric (ε′ = 3500) properties.1,2 It is well-known that for conventionally sintered BT the maximum relative permittivity value at room temperature varies in the range of 4000−6000 depending on grain size.3,4 However, the relative permittivity of BT can be further enhanced via novel fast-firing synthesis techniques such as spark plasma sintering (SPS) and microwave sintering (MWS).5−7 Recently, Han et al. revealed that extremely high relative permittivity up to 104−105 with relatively low dielectric loss (tan δ = 0.03−0.05) can be achieved in fast-fired BT.6,8 Such a high permittivity has been referred to as colossal permittivity (CP) following the terminology used for CaCu3Ti4O12 (CCTO). Several techniques have been investigated to discover novel CP materials to be utilized in microelectronics, high energy density storage applications, and high performance dielectric devices.9−16 However, a strong temperature and frequency dependency is commonly observed in dielectric properties of the CP materials, which limits the use of CP materials in conventional capacitive applications.8 It is widely accepted that in CP materials the temperature and frequency dependency of dielectric properties is the result of extrinsic effects such as interfacial polarization and hopping conduction contributing to the bulk conductivity of the material.17−19 Thus, understanding the fundamental mechanisms of the bulk conduction in CP materials is essential to tailor the temperature- and frequency-dependent dielectric response and thus fully realize their potential. Fundamental analyses of this kind have been attempted in the past for some microcrystalline CP materials such as CCTO and Ni0.5Zn0.5Fe2O4 (NZFO) sintered by a conventional method,17,19 but those results cannot be generalized to nanocrystalline fast-fired BT ceramics © 2014 American Chemical Society
exhibiting CP since the electrical properties of several oxides (including BT) differ significantly between nanocrystalline and microcrystalline structured ceramics. This is mainly due to the lower enthalpy for reduction/oxidation, lower vacancy formation energy at the grain boundaries, and the higher density of grain boundaries in nanocrystalline oxides when compared to microcrystalline ones.20,21 In this article, a detailed analysis of the electrical conductivity of nanocrystalline BT ceramics exhibiting CP is presented, and a corresponding analytical model elucidating the conduction mechanisms is introduced. In addition, the relative contributions of the respective polarization mechanisms to the dc conductivity and the relative permittivity are estimated.
2. EXPERIMENTAL PROCEDURE An oxalate route was used for the synthesis of starting BT powder which can be found in previous works.7,8 The SPS technique was performed using a Dr. Sinter 2080 from Sumimoto Coal Mining (SPS Syntx Inc., Dr. Sinter 2080). Approximately 0.9 g of BT powder without binder was placed in an 8 mm diameter graphite die. A heating rate of 25 °C/min was used to reach 1150 °C from 600 °C, and then the powder was sintered at 1150 °C with a dwell time of 3 min. The sintered body was then cooled down from 1150 to 600 °C at a rate of 100 °C/min. A mechanical pressure of 50 MPa and an electrical current up to 350 A were applied during the sintering process. A thin carbon layer due to graphite contamination during sintering was observed at the surface of the as-sintered ceramic. This carbon layer was easily removed by polishing the surface. For a postannealing treatment, the as-sintered ceramic Received: March 6, 2014 Revised: April 9, 2014 Published: April 10, 2014 9137
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was placed into the furnace which was preheated to 850 °C. Then, the sample was left to dwell for 15 min, and air quenching was followed. The density of the as-sintered ceramics was determined by using Archimedes’ method. Inductively coupled plasma−atomic emission spectroscopy (ICP-AES, Horiba Scientific., ThermoOptec ARL 3580) was performed in order to investigate the chemical stoichiometry of the sintered ceramics. The microstructure of the sintered ceramics was investigated by fieldemission scanning electron microscopy (FE-SEM) using a JEOL 6335F FEG-SEM. An X-ray diffractometer equipped with a one-dimensional detector (CPS120, Inel Inc.) was used to investigate the crystal structure and phase of the sintered ceramics. The CPS120 detector measures diffraction data over a wide 2θ range simultaneously. All the XRD data were measured over a 10 min time period. For dielectric property measurements, the sintered ceramics were electroded with gold (Au) having a thickness of approximately 30 nm using a sputter coater (Cressington Scientific Inc., Cressington 108 Auto). Dielectric properties of the samples were measured as a function of temperature (300− 20 K) at different frequencies (40 Hz−100 kHz) using an Agilient 4284 LCR meter and a closed cycle cryonic workstation (CTI-Cryogenics, Model 22). In addition, broadband dielectric spectroscopy (1 kHz−4 MHz) was performed on the samples at different temperatures (20−300 K) by using a precision impedance analyzer (PIA, Agilent Ltd., Agilient 4294A) coupled with a cryonic workstation.
Figure 1. (a) FE-SEM image and (b) XRD pattern of SPS BT sample.
3. RESULTS AND DISCUSSION A Ti3+/Ti4+ ratio (=0.053) was determined for BT sintered by the SPS technique (SPS BT) by using ICP-AES. The density of the sintered ceramic reached up to 98.00% of the theoretical density of BT (6.02 g/cm3). A relatively high density was achieved for SPS BT due to the applied mechanical stress (∼50 MPa) during the SPS process. Figure 1a shows the microstructure of the sample observed by FE-SEM. Grain size of 56 ± 6 nm was determined by using the mean linear intercept method of the ASTM E112 standard. Figure 1b shows the collected XRD data of the sample. The XRD pattern verified that the sintered ceramic was composed of mixture of a cubic and a tetragonal perovskite phase.6,8 Additionally, Ba4Ti12O27 (∼7 vol %) was confirmed as a secondary phase existing in SPS BT. This phase is commonly found in fast-fired BT ceramics due to the reduced sintering atmosphere.6,8 Figure 2 shows dielectric properties and the real part of ac conductivity (σ′) for SPS BT as a function of frequency (1 kHz−4 MHz) at different temperature (110−220 K). In Figure 2a, it is observed that CP up to 105 is maintained across a frequency range of 1 kHz−100 kHz at 220 K. As frequency increases, a Debye type dielectric relaxation occurs, which is naturally accompanied by a dielectric loss peak at the relaxation frequency as expected (Figure 2b). The relaxation frequency shifts to lower frequencies as temperature decreases. It is worth mentioning that a relative permittivity of 650−1150 was observed at 10 MHz (see the inset of Figure 2a) in the measured temperature range presented here (110−220 K). Also, the dielectric response at temperatures above 220 K to room temperature did not vary substantially and was almost identical to that at 220 K. By contrast, the relative permittivity was significantly decreased to below 104 at temperatures lower than 110 K.
Toward a systematic analysis of this dielectric behavior, it is important to recall that according to Jonscher’s universal dielectric response (UDR) model, the conductivity of the material can be described as eq 122 σ ′(f ) = σdc + σ0f s
(1)
where σ0 and s are temperature-dependent constants, f represents the experimental frequency (f = ω/2π), and σdc is the dc conductivity. Figure2c shows the experimental conductivity data for SPS BT with fitted curves using eq 1. It can be observed that for a given temperature two different types of response with characteristic frequency ranges occur. One range shows a strong decrease of the conductivity at low frequencies (below ∼104 Hz);17 however, in the high frequency range (above ∼104 Hz), the conductivity of SPS BT shows low frequency dependence that can be well fitted by using the UDR model (eq 1). Moreover, it should be noted that the exponent “s” obtained from the fitted curves is around 0.50−0.64. Typically, it is understood that an “s” value closer to “0” indicates that charge carriers are more free to conduct through the material,22 while an “s” value between 0.5 and 0.8 is observed in materials with more localized charge carriers affecting not only conductivity but also the dielectric polarization of the material.22,23 In addition, σdc can be extracted at a given temperature from the fitted parameters using eq 1, which gives valuable information to understand conduction mechanisms in the material. In trying to understand the origin of these localized charges, it is worth remembering that materials sintered by fast-firing methods, such as SPS and MWS, are significantly reduced during the process.6,7,24 As such, a high concentration of oxygen vacancies, Ti3+, and free electrons can be induced in the sintered ceramic. These anticipated charge carriers can then 9138
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Figure 3. Temperature dependence of dc conductivity (σdc) with respect to 1/T and 1/T1/2. Black lines are the linear fits of the data.
However, contrary to the NNH model assumptions, Mott et al. have pointed out that hopping conduction in certain materials would not process through the nearest site at low temperatures. According to the theory proposed by Mott, the hopping range may vary as temperature decreases and become larger than the distance between neighboring sites due to the lower activation energy involved at lower temperatures. This model is generally referred to as Mott’s variable-range-hopping (VRH) conduction. σdc that follows the Mott’s VRH model can be represented as (3)
T0 = 24/[πkBN (E F)ξ 3]
(4)
where σ2 and T0 are constants and ξ and N(EF) are the decay length of the localized wave function and the density of localized states at the Fermi level, respectively. The exponent p can be 1/4, 1/3, and1/2 for different materials.26,28 According to eq 3, the linear relationship between ln(σdc) and 1/Tp exists for the material following the VRH conduction mechanism. In order to calculate the value of exponent “p” in eq 3, one should invoke the activation energy defining equation which is given by27
Figure 2. (a) Relative permittivity, (b) dielectric loss, and (c) conductivity of SPS BT as a function of frequency (1 kHz−4 MHz) at different temperatures (110−220 K). The inset of (a) shows the measured permittivity up to 10 MHz.
contribute to the conductivity of the material via long-range hopping conduction under applied electric field. In hopping conduction, the dc conductivity of the material (σdc) can be well described by the Arrhenius equation σdc = σ1 exp( −E1/kBT )
σdc = σ2 exp[− (T0/T ) p ]
EA = −d[ln(σdc)]/d(1/kT )
(5)
Most importantly, this equation also enables to investigate the activation energy of the bulk conductivity (EA), regardless of a particular conduction mechanism. The inset in Figure 3 depicts the activation energies, calculated by using eq 5, as a function of temperature. It is clearly shown that in Figure 3 EA is temperature dependent, with EA decreasing as temperature decreases. This result is in contradiction to the NNH conduction model, however consistent with the VRH theory, in which EA is supposed to be variable as a function of temperature. Furthermore, it can be shown by comparing eq 5 and eq 3 that 1 − p should be equal to the slope of the log EA vs log T plot. As can be seen in the inset of Figure 3, the slope of ∼0.5 is obtained from the linear fit of the log EA vs log T plot. Thus, it can be inferred that the most probable value of “p” in this case is 1/2. The temperature dependence of ln(σdc) with respect to 1/ T1/2 for SPS BT is represented in Figure 3. It is observed that the linear fit of ln(σdc) against 1/T1/2 matches very well the experimental data over a whole range of temperature from 110 to 220 K. It is important to note that compared to the NNH model, the linear relationship of σdc and temperature following
(2)
where σ1 is a constant and E1 is the activation energy for hopping conduction.25 In this model, hopping conduction always occurs through the neighboring sites, and hence the hopping range and activation energy are independent of temperature. This model is commonly referred to as the nearest-neighbor-hopping (NNH) conduction.25,26 In the NNH model, eq 2 can be applied to describe the temperature-dependent σdc where the concentration of charge carriers for hopping conduction is temperature dependent. It is obvious from eq 2 that ln(σdc) should have a linear relationship with 1/ T for materials with a conduction mechanism following the NNH model. To test this model, σdc values extracted from each of the fitting curves in Figure 2c were plotted as a function of inverse temperature (1/T) and are presented in Figure 3. It is clearly observed that σdc increases as temperature increases, indicating that thermally activated bulk conduction exists in SPS BT. An activation energy of 112.5 ± 12.9 meV is obtained from the ln(σdc) vs 1/T plot. 9139
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the VRH model can more satisfactorily describe the experimental results in the whole range of temperatures, especially for the experimental data at low temperature regions. All these results support that the conduction mechanism in SPS BT follows VRH model rather than NNH model with the value of p as 1/2, which is similar to the case of well-known CP materials, such as CCTO.17 In the VRH model, the activation energy (W) can be written as the following equation by substituting eq 3 into eq 5. W = 0.5kBT0 0.5T 0.5
(6)
From the fitted value of T0 (4.48 × 10 K), W can be calculated as a function of temperature, and the resulting values are shown in Figure 4. It is observed that W varies from 95.7 to 135.3 meV 4
Figure 5. Experimental data with fitting data of the real and imaginaly part of permittivity for SPS BT.
from 220 to 110 K. The hopping distance at 220 K for SPS BT is almost identical with the distance between nearest Ti ions, and it varies approximately 0.07 nm as temperature changes from 220 to 110 K. It should be denoted that the variation of hopping distance for SPS BT is much less than that of other CP materials such as CCTO, which also follows the VRH conduction mechanism. For example, Zhang et al. have demonstrated that the hopping range of CCTO increases from 2.32 to 2.76 nm as temperature decreases from 180 to 90 K.17 The relatively high number of N(EF) and less variation of hopping range for SPS BT, compared to other CP materials, might be due to the high concentration of Ti3+ and oxygen vacancies existing in SPS BT, since the sample was sintered under severe reduction conditions (i.e., vacuum and fast heating rate). Moreover, the activation energy for dielectric polarization (93 meV)8 of SPS BT calculated by using the Debye model is consistent with the activation energy of hopping conduction (W ≅ 90−120 meV) for the temperature range of 200−100 K. This point to the possibility that an inherent correlation between bulk conduction and polarization exists in SPS BT ceramics, which will be further demonstrated in the following discussions. The dielectric response of CP materials can be well described by the Koop’s equivalent circuit model.29 In the Koop’s equivalent circuit model, the grain and the grain boundary are supposed to be in series, and each component is composed of three parallel elements: a capacitance, a conductance, and a constant phase element (CPE). The analytical equation for this equivalent circuit can be driven as18
Figure 4. Activation energies and hopping distance as a function of temperature for VRH conduction in SPS BT.
as temperature increases from 110 to 220 K, which is well consistent with the activation energies calculated by using the defining equation (5) (EA = 98.0−125.7 meV at 110−220 K). This also apparently suggests that the conduction mechanism in SPS BT follows the VRH model. Furthermore, in the VRH model, it is possible to investigate the most probable hopping range (R) at a certain temperature by using the equation17 R = ξ1/4 /[8πN (E F)kBT ]1/4
(7)
For SPS BT, ξ can be assumed as the distance between nearest Ti ions, which in the case of BT is approximately 0.40 nm. Using these values for T0 and ξ, the density of localized states at the Fermi level can be estimated as N(EF) ≅ 3.11 × 1022 (eV−1 cm−3), which is 3 orders of magnitude higher than that of CCTO (N(EF) ≅ 3.25 × 1019 (eV−1 cm−3)).17 Then, the hopping range of R can be calculated for SPS BT as a function of temperature, which is also presented in Figure 4. It is found that R increases from 0.41 to 0.48 nm as temperature decreases * = εBT
[cg(iω)1 − Pg + σg,dc_j + iωεBT, ∞ε0][cgb(iω)1 − Pgb + σgb,dc_j + iωεBT, ∞ε0](1 + η) iε0ω[cgb(iω)1 − Pgb + σgb,dc_j + η⟨cg(iω)1 − Pg + σg,dc_j⟩] − ω 2εBT, ∞ε0 2(1 + η)
where σg(gb),dc_jj * represent frequency-independent part of the Jonscher’s universal conductivity for grains (g) and grain boundaries (gb). ε0, i, η, and ω are the permittivity of the free space, the imaginary unit, √−1, the ratio of thickness of grain boundary to thickness of grain, and angular frequency (=2πf), respectively. Also, 1 − pg(gb) is the exponent of angular frequency for the grains and the grain boundaries, and cg(gb) is equal to σg(gb),0/[0.5π cos(1 − pg(gb))], where σg(gb),0 is a
(8)
prefactor. It is worth noting that in the model the value of pg(gb) should be in between 0 and 1, and pg(gb) becomes closer to “1” for a more conductive response while a more capacitive response results in the pg(gb) value closer to “0”. The experimental data with theoretical fitted curves using eq 8 for the relative permittivity of SPS BT is shown in Figure 5. Additionally, Table 1 summarizes the fitting parameters used in eq 8. It is noticeable that the fitted curves are significantly in 9140
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The conduction mechanism in SPS BT ceramics followed the variable-range-hopping (VRH) conduction mechanism rather than the nearest-neighboring-hopping (NNH) model. In addition, based on the Jonscher’s universial dielectric response (UDR) and Koop’s equivalent circuit models, the dc conductivity and the relative permittivity of SPS BT were deduced for each polarization mechanism.
Table 1. Related Fitting Parameters for the SPS BT Using Koop’s Equivalent Circuit Model related parameters
SPS
related parameters
SPS
cg cgb pg pgb
0.32 1.27 × 10−8 0.909 71 0.028 53
σg,dc (S cm−1) σgb,dc (S cm−1) η (= tb/tg) εBT,∞
1.39 × 10−3 4.54 × 10−10 0.0100 494.034 77
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good agreement with the measured experimental data over the entire frequency range. Furthermore, σg,dc is orders of magnitude higher than σgb,dc, which is in accordance with the internal barrier layer capacitance (IBLC) model (Table 1). As expected for the IBLC model, the value of pg (≈ 0.91) is much closer to “1” compared to pgb (≈0.03), which demonstrates that conductive grains and capacitive grain boundaries are present in SPS BT. In the IBLC model, σg,dc (1.39 × 10−3 S cm−1) of SPS BT is mainly ascribed by interfacial polarization, and thus the dc conductivity as a result of hooping polarization can be estimated by subtracting σg,dc (1.39 × 10−3 S cm−1) from the total bulk conductivity at given temperature (σtotal,dc = 1.14 × 10−2 S cm−1 from Figure 3). The net result indicates that hopping polarization of the material is attributed to about 88% of the dc conductivity (σhopping = 1.00 × 10−2 S cm−1) of SPS BT, and the remaining 12% of the dc conductivity is due to interfacial polarization (σinterfacial = 1.39 × 10−3 S cm−1). It is important to note that the contributions for the different polarization mechanisms have similar values for both conductivity and relative permittivity in SPS BT ceramics. In our previous work, excluding the electrode effect, it was demonstrated that ∼20% of relative permittivity of SPS BT is attributed to interfacial polarization while the rest of ∼80% is due to hopping polarization.8 This indicates that the polarization in BT ceramics exhibiting CP is not only the result of short-range dipole motions but also the result of the bulk conductivity of the compound. Furthermore, hopping polarization plays much more significant role in the bulk conduction of SPS BT compared to interfacial polarization. Thus, in order to achieve BT ceramics exhibiting temperature−frequency independent CP, hopping dipoles should be highly localized in the lattice such that prevent dipoles from establishing long-range ordering through the material. Recently, Hu et al. demonstrated that through novel defect engineering strategies, quasi-intrinsic CP (nearly temperature−frequency independent) can be achieved in (Nb + In) codoped TiO2 due to highly localized defect-dipoles. Although detailed broadband dielectric spectroscopy analyzes have not been performed on that material, this approach suggested a promising strategy for the development of new high-performance CP materials.30 The hopping distance of (Nb + In) codoped TiO2 remains constant at different temperatures due to highly localized defect dipoles, which consequently gives rise to the excellent temperature− frequency-stable CP. From this point of view, SPS BT is a fascinating candidate as a novel CP material, since hopping distance of SPS BT is relatively short and the variation of hopping range as a function of temperature is very small compared to other CP materials.
AUTHOR INFORMATION
Corresponding Author
*E-mail dducks82@ufl.edu; Tel +1 (352) 846 3793; Fax +1 (352) 846 3355 (H.H.). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported partially by the U.S. National Science Foundation under Grant DMR 0449710 and the Graduate Research Fellowship under Grant DGE-0802270.
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