Variable Range Hopping Conduction in BaTiO3 Ceramics ExhibitingColossal PermittivityHyukSu 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 exhibitroom temperature colossal permittivity (εr′ ≈ 105, tan δ ≈ 0.05 at 1 kHz) whensynthesized via the spark plasma sintering technique. Broadband dielectricspectroscopy analysis is here utilized to identify the active conduction mechanismsin the compound. Analysis of the temperature-dependent bulk dc conductivityreveals that the bulk conduction in fast-fired barium titanate is the result ofvariable-range hopping with relatively short hopping distance (∼0.5 nm)compared to other colossal permittivity materials. A common equivalent circuitthat can model the contributions of the different polarization mechanisms to boththe dc conductivity and colossal permittivity is established.

1. INTRODUCTION

Barium titanate (BaTiO3, BT) is the most widely usedferroelectric material in a variety of applications due to itsexcellent dielectric (ε′ = 3500) properties.1,2 It is well-knownthat for conventionally sintered BT the maximum relativepermittivity value at room temperature varies in the range of4000−6000 depending on grain size.3,4 However, the relativepermittivity of BT can be further enhanced via novel fast-firingsynthesis techniques such as spark plasma sintering (SPS) andmicrowave sintering (MWS).5−7 Recently, Han et al. revealedthat extremely high relative permittivity up to 104−105 withrelatively low dielectric loss (tan δ = 0.03−0.05) can beachieved in fast-fired BT.6,8 Such a high permittivity has beenreferred to as colossal permittivity (CP) following theterminology used for CaCu3Ti4O12 (CCTO).Several techniques have been investigated to discover novel

CP materials to be utilized in microelectronics, high energydensity storage applications, and high performance dielectricdevices.9−16 However, a strong temperature and frequencydependency is commonly observed in dielectric properties ofthe CP materials, which limits the use of CP materials inconventional capacitive applications.8 It is widely accepted thatin CP materials the temperature and frequency dependency ofdielectric properties is the result of extrinsic effects such asinterfacial polarization and hopping conduction contributing tothe bulk conductivity of the material.17−19

Thus, understanding the fundamental mechanisms of thebulk conduction in CP materials is essential to tailor thetemperature- and frequency-dependent dielectric response andthus fully realize their potential. Fundamental analyses of thiskind have been attempted in the past for some microcrystallineCP materials such as CCTO and Ni0.5Zn0.5Fe2O4 (NZFO)sintered by a conventional method,17,19 but those results cannotbe generalized to nanocrystalline fast-fired BT ceramics

exhibiting CP since the electrical properties of several oxides(including BT) differ significantly between nanocrystalline andmicrocrystalline structured ceramics. This is mainly due to thelower enthalpy for reduction/oxidation, lower vacancyformation energy at the grain boundaries, and the higherdensity of grain boundaries in nanocrystalline oxides whencompared to microcrystalline ones.20,21

In this article, a detailed analysis of the electrical conductivityof nanocrystalline BT ceramics exhibiting CP is presented, anda corresponding analytical model elucidating the conductionmechanisms is introduced. In addition, the relative contribu-tions of the respective polarization mechanisms to the dcconductivity and the relative permittivity are estimated.

2. EXPERIMENTAL PROCEDURE

An oxalate route was used for the synthesis of starting BTpowder which can be found in previous works.7,8 The SPStechnique was performed using a Dr. Sinter 2080 fromSumimoto Coal Mining (SPS Syntx Inc., Dr. Sinter 2080).Approximately 0.9 g of BT powder without binder was placedin an 8 mm diameter graphite die. A heating rate of 25 °C/minwas used to reach 1150 °C from 600 °C, and then the powderwas sintered at 1150 °C with a dwell time of 3 min. Thesintered body was then cooled down from 1150 to 600 °C at arate of 100 °C/min. A mechanical pressure of 50 MPa and anelectrical current up to 350 A were applied during the sinteringprocess. A thin carbon layer due to graphite contaminationduring sintering was observed at the surface of the as-sinteredceramic. This carbon layer was easily removed by polishing thesurface. For a postannealing treatment, the as-sintered ceramic

Received: March 6, 2014Revised: April 9, 2014Published: April 10, 2014

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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 airquenching was followed.The density of the as-sintered ceramics was determined by

using Archimedes’ method. Inductively coupled plasma−atomicemission spectroscopy (ICP-AES, Horiba Scientific., Thermo-Optec ARL 3580) was performed in order to investigate thechemical stoichiometry of the sintered ceramics. The micro-structure of the sintered ceramics was investigated by field-emission scanning electron microscopy (FE-SEM) using aJEOL 6335F FEG-SEM.An X-ray diffractometer equipped with a one-dimensional

detector (CPS120, Inel Inc.) was used to investigate the crystalstructure and phase of the sintered ceramics. The CPS120detector measures diffraction data over a wide 2θ rangesimultaneously. All the XRD data were measured over a 10 mintime period. For dielectric property measurements, the sinteredceramics were electroded with gold (Au) having a thickness ofapproximately 30 nm using a sputter coater (CressingtonScientific Inc., Cressington 108 Auto). Dielectric properties ofthe samples were measured as a function of temperature (300−20 K) at different frequencies (40 Hz−100 kHz) using anAgilient 4284 LCR meter and a closed cycle cryonicworkstation (CTI-Cryogenics, Model 22). In addition, broad-band dielectric spectroscopy (1 kHz−4 MHz) was performedon the samples at different temperatures (20−300 K) by usinga precision impedance analyzer (PIA, Agilent Ltd., Agilient4294A) coupled with a cryonic workstation.

3. RESULTS AND DISCUSSION

A Ti3+/Ti4+ ratio (=0.053) was determined for BT sintered bythe SPS technique (SPS BT) by using ICP-AES. The density ofthe sintered ceramic reached up to 98.00% of the theoreticaldensity of BT (6.02 g/cm3). A relatively high density wasachieved for SPS BT due to the applied mechanical stress (∼50MPa) during the SPS process. Figure 1a shows the micro-structure of the sample observed by FE-SEM. Grain size of 56± 6 nm was determined by using the mean linear interceptmethod of the ASTM E112 standard. Figure 1b shows thecollected XRD data of the sample. The XRD pattern verifiedthat the sintered ceramic was composed of mixture of a cubicand a tetragonal perovskite phase.6,8 Additionally, Ba4Ti12O27(∼7 vol %) was confirmed as a secondary phase existing in SPSBT. This phase is commonly found in fast-fired BT ceramicsdue to the reduced sintering atmosphere.6,8

Figure 2 shows dielectric properties and the real part of acconductivity (σ′) for SPS BT as a function of frequency (1kHz−4 MHz) at different temperature (110−220 K). In Figure2a, it is observed that CP up to 105 is maintained across afrequency range of 1 kHz−100 kHz at 220 K. As frequencyincreases, a Debye type dielectric relaxation occurs, which isnaturally accompanied by a dielectric loss peak at the relaxationfrequency as expected (Figure 2b). The relaxation frequencyshifts to lower frequencies as temperature decreases. It is worthmentioning that a relative permittivity of 650−1150 wasobserved at 10 MHz (see the inset of Figure 2a) in themeasured temperature range presented here (110−220 K).Also, the dielectric response at temperatures above 220 K toroom temperature did not vary substantially and was almostidentical to that at 220 K. By contrast, the relative permittivitywas significantly decreased to below 104 at temperatures lowerthan 110 K.

Toward a systematic analysis of this dielectric behavior, it isimportant to recall that according to Jonscher’s universaldielectric response (UDR) model, the conductivity of thematerial can be described as eq 122

σ σ σ′ = +f f( ) sdc 0 (1)

where σ0 and s are temperature-dependent constants, frepresents the experimental frequency ( f = ω/2π), and σdc isthe dc conductivity. Figure2c shows the experimentalconductivity data for SPS BT with fitted curves using eq 1. Itcan be observed that for a given temperature two different typesof response with characteristic frequency ranges occur. Onerange shows a strong decrease of the conductivity at lowfrequencies (below ∼104 Hz);17 however, in the high frequencyrange (above ∼104 Hz), the conductivity of SPS BT shows lowfrequency dependence that can be well fitted by using the UDRmodel (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 throughthe material,22 while an “s” value between 0.5 and 0.8 isobserved in materials with more localized charge carriersaffecting not only conductivity but also the dielectric polar-ization of the material.22,23 In addition, σdc can be extracted at agiven temperature from the fitted parameters using eq 1, whichgives valuable information to understand conduction mecha-nisms in the material.In trying to understand the origin of these localized charges,

it is worth remembering that materials sintered by fast-firingmethods, such as SPS and MWS, are significantly reducedduring the process.6,7,24 As such, a high concentration of oxygenvacancies, Ti3+, and free electrons can be induced in thesintered ceramic. These anticipated charge carriers can then

Figure 1. (a) FE-SEM image and (b) XRD pattern of SPS BT sample.

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contribute to the conductivity of the material via long-rangehopping conduction under applied electric field.In hopping conduction, the dc conductivity of the material

(σdc) can be well described by the Arrhenius equation

σ σ= −E k Texp( / )dc 1 1 B (2)

where σ1 is a constant and E1 is the activation energy forhopping conduction.25 In this model, hopping conductionalways occurs through the neighboring sites, and hence thehopping range and activation energy are independent oftemperature. This model is commonly referred to as thenearest-neighbor-hopping (NNH) conduction.25,26 In theNNH model, eq 2 can be applied to describe the temper-ature-dependent σdc where the concentration of charge carriersfor hopping conduction is temperature dependent. It is obviousfrom eq 2 that ln(σdc) should have a linear relationship with 1/T for materials with a conduction mechanism following theNNH model. To test this model, σdc values extracted from eachof the fitting curves in Figure 2c were plotted as a function ofinverse temperature (1/T) and are presented in Figure 3. It isclearly observed that σdc increases as temperature increases,indicating that thermally activated bulk conduction exists inSPS BT. An activation energy of 112.5 ± 12.9 meV is obtainedfrom the ln(σdc) vs 1/T plot.

However, contrary to the NNH model assumptions, Mott etal. have pointed out that hopping conduction in certainmaterials would not process through the nearest site at lowtemperatures. According to the theory proposed by Mott, thehopping range may vary as temperature decreases and becomelarger than the distance between neighboring sites due to thelower activation energy involved at lower temperatures. Thismodel is generally referred to as Mott’s variable-range-hopping(VRH) conduction. σdc that follows the Mott’s VRH model canbe represented as

σ σ= − T Texp[ ( / ) ]pdc 2 0 (3)

π ξ=T k N E24/[ ( ) ]0 B F3

(4)

where σ2 and T0 are constants and ξ and N(EF) are the decaylength of the localized wave function and the density oflocalized states at the Fermi level, respectively. The exponent pcan be 1/4, 1/3, and1/2 for different materials.26,28 Accordingto eq 3, the linear relationship between ln(σdc) and 1/Tp existsfor 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 isgiven by27

σ= −E kTd[ln( )]/d(1/ )A dc (5)

Most importantly, this equation also enables to investigate theactivation energy of the bulk conductivity (EA), regardless of aparticular conduction mechanism. The inset in Figure 3 depictsthe activation energies, calculated by using eq 5, as a function oftemperature. It is clearly shown that in Figure 3 EA istemperature dependent, with EA decreasing as temperaturedecreases. This result is in contradiction to the NNHconduction model, however consistent with the VRH theory,in which EA is supposed to be variable as a function oftemperature. Furthermore, it can be shown by comparing eq 5and eq 3 that 1 − p should be equal to the slope of the log EA vslog 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” inthis 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 thatthe linear fit of ln(σdc) against 1/T1/2 matches very well theexperimental data over a whole range of temperature from 110to 220 K. It is important to note that compared to the NNHmodel, the linear relationship of σdc and temperature following

Figure 2. (a) Relative permittivity, (b) dielectric loss, and (c)conductivity of SPS BT as a function of frequency (1 kHz−4 MHz) atdifferent temperatures (110−220 K). The inset of (a) shows themeasured permittivity up to 10 MHz.

Figure 3. Temperature dependence of dc conductivity (σdc) withrespect to 1/T and 1/T1/2. Black lines are the linear fits of the data.

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the VRH model can more satisfactorily describe theexperimental 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 SPSBT follows VRH model rather than NNH model with the valueof p as 1/2, which is similar to the case of well-known CPmaterials, such as CCTO.17

In the VRH model, the activation energy (W) can be writtenas the following equation by substituting eq 3 into eq 5.

=W k T T0.5 B 00.5 0.5

(6)

From the fitted value of T0 (4.48 × 104 K), W can be calculatedas a function of temperature, and the resulting values are shownin Figure 4. It is observed thatW varies from 95.7 to 135.3 meV

as temperature increases from 110 to 220 K, which is wellconsistent with the activation energies calculated by using thedefining equation (5) (EA = 98.0−125.7 meV at 110−220 K).This also apparently suggests that the conduction mechanismin 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 temperatureby using the equation17

ξ π=R N E k T/[8 ( ) ]1/4F B

1/4(7)

For SPS BT, ξ can be assumed as the distance between nearestTi ions, which in the case of BT is approximately 0.40 nm.Using these values for T0 and ξ, the density of localized states atthe Fermi level can be estimated as N(EF) ≅ 3.11 × 1022 (eV−1

cm−3), which is 3 orders of magnitude higher than that ofCCTO (N(EF) ≅ 3.25 × 1019 (eV−1 cm−3)).17 Then, thehopping range of R can be calculated for SPS BT as a functionof temperature, which is also presented in Figure 4. It is foundthat R increases from 0.41 to 0.48 nm as temperature decreases

from 220 to 110 K. The hopping distance at 220 K for SPS BTis almost identical with the distance between nearest Ti ions,and it varies approximately 0.07 nm as temperature changesfrom 220 to 110 K. It should be denoted that the variation ofhopping distance for SPS BT is much less than that of other CPmaterials such as CCTO, which also follows the VRHconduction mechanism. For example, Zhang et al. havedemonstrated that the hopping range of CCTO increasesfrom 2.32 to 2.76 nm as temperature decreases from 180 to 90K.17 The relatively high number of N(EF) and less variation ofhopping range for SPS BT, compared to other CP materials,might be due to the high concentration of Ti3+ and oxygenvacancies existing in SPS BT, since the sample was sinteredunder severe reduction conditions (i.e., vacuum and fast heatingrate).Moreover, the activation energy for dielectric polarization

(93 meV)8 of SPS BT calculated by using the Debye model isconsistent 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 correlationbetween bulk conduction and polarization exists in SPS BTceramics, which will be further demonstrated in the followingdiscussions.The dielectric response of CP materials can be well described

by the Koop’s equivalent circuit model.29 In the Koop’sequivalent circuit model, the grain and the grain boundary aresupposed to be in series, and each component is composed ofthree parallel elements: a capacitance, a conductance, and aconstant phase element (CPE). The analytical equation for thisequivalent circuit can be driven as18

εω σ ωε ε ω σ ωε ε η

ε ω ω σ η ω σ ω ε ε η* =

+ + + + +

+ + ⟨ + ⟩ − +

−_ ∞

−_ ∞

−_

−_ ∞

c i i c i

i c i c i

[ ( ) ][ ( ) i ](1 )

[ ( ) ( ) ] (1 )j j

j jBT

g1 Pg

g,dc BT, 0 gb1 Pgb

gb,dc BT, 0

0 gb1 Pgb

gb,dc g1 Pg

g,dc2

BT, 02

(8)

where σg(gb),dc_jj* represent frequency-independent part of theJonscher’s universal conductivity for grains (g) and grainboundaries (gb). ε0, i, η, and ω are the permittivity of the freespace, the imaginary unit, √−1, the ratio of thickness of grainboundary to thickness of grain, and angular frequency (=2πf),respectively. Also, 1 − pg(gb) is the exponent of angularfrequency for the grains and the grain boundaries, and cg(gb) isequal to σg(gb),0/[0.5π cos(1 − pg(gb))], where σg(gb),0 is a

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 capacitiveresponse 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 ineq 8. It is noticeable that the fitted curves are significantly in

Figure 4. Activation energies and hopping distance as a function oftemperature for VRH conduction in SPS BT.

Figure 5. Experimental data with fitting data of the real and imaginalypart of permittivity for SPS BT.

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good agreement with the measured experimental data over theentire frequency range. Furthermore, σg,dc is orders ofmagnitude higher than σgb,dc, which is in accordance with theinternal barrier layer capacitance (IBLC) model (Table 1). Asexpected for the IBLC model, the value of pg (≈ 0.91) is muchcloser to “1” compared to pgb (≈0.03), which demonstrates thatconductive grains and capacitive grain boundaries are present inSPS BT. In the IBLC model, σg,dc (1.39 × 10−3 S cm−1) of SPSBT is mainly ascribed by interfacial polarization, and thus thedc conductivity as a result of hooping polarization can beestimated by subtracting σg,dc (1.39 × 10−3 S cm−1) from thetotal bulk conductivity at given temperature (σtotal,dc = 1.14 ×10−2 S cm−1 from Figure 3). The net result indicates thathopping polarization of the material is attributed to about 88%of the dc conductivity (σhopping = 1.00 × 10−2 S cm−1) of SPSBT, and the remaining 12% of the dc conductivity is due tointerfacial 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 bothconductivity and relative permittivity in SPS BT ceramics. Inour previous work, excluding the electrode effect, it wasdemonstrated that ∼20% of relative permittivity of SPS BT isattributed to interfacial polarization while the rest of ∼80% isdue to hopping polarization.8 This indicates that the polar-ization in BT ceramics exhibiting CP is not only the result ofshort-range dipole motions but also the result of the bulkconductivity of the compound. Furthermore, hopping polar-ization plays much more significant role in the bulk conductionof SPS BT compared to interfacial polarization.Thus, in order to achieve BT ceramics exhibiting temper-

ature−frequency independent CP, hopping dipoles should behighly localized in the lattice such that prevent dipoles fromestablishing long-range ordering through the material. Recently,Hu et al. demonstrated that through novel defect engineeringstrategies, quasi-intrinsic CP (nearly temperature−frequencyindependent) can be achieved in (Nb + In) codoped TiO2 dueto highly localized defect-dipoles. Although detailed broadbanddielectric spectroscopy analyzes have not been performed onthat material, this approach suggested a promising strategy forthe development of new high-performance CP materials.30 Thehopping distance of (Nb + In) codoped TiO2 remains constantat 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 afascinating candidate as a novel CP material, since hoppingdistance of SPS BT is relatively short and the variation ofhopping range as a function of temperature is very smallcompared to other CP materials.

4. CONCLUSIONBroadband dielectric spectroscopy was performed on the sparkplasma sintered BT (SPS BT) ceramics exhibiting colossalpermittivity (CP). In-depth analysis for revealing theconduction mechanism was conducted for SPS BT ceramics.

The conduction mechanism in SPS BT ceramics followed thevariable-range-hopping (VRH) conduction mechanism ratherthan the nearest-neighboring-hopping (NNH) model. Inaddition, based on the Jonscher’s universial dielectric response(UDR) and Koop’s equivalent circuit models, the dcconductivity and the relative permittivity of SPS BT werededuced for each polarization mechanism.

■ AUTHOR INFORMATION

Corresponding Author*E-mail dducks82@ufl.edu; Tel +1 (352) 846 3793; Fax +1(352) 846 3355 (H.H.).

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

This work was supported partially by the U.S. National ScienceFoundation under Grant DMR 0449710 and the GraduateResearch Fellowship under Grant DGE-0802270.

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Table 1. Related Fitting Parameters for the SPS BT UsingKoop’s Equivalent Circuit Model

related parameters SPS related parameters SPS

cg 0.32 σg,dc (S cm−1) 1.39 × 10−3

cgb 1.27 × 10−8 σgb,dc (S cm−1) 4.54 × 10−10

pg 0.909 71 η (= tb/tg) 0.0100pgb 0.028 53 εBT,∞ 494.034 77

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