materials

Article The Effect of Acceptor and Donor Doping on Oxygen Vacancy Concentrations in Lead Zirconate Titanate (PZT)

Christoph Slouka 1, Theresa Kainz 2, Edvinas Navickas 1, Gregor Walch 1, Herbert Hutter 1, Klaus Reichmann 2 and Jürgen Fleig 1,*

1 Institute of Chemical Technologies and Analytics, Vienna University of Technology, Getreidemarkt 9/164EC, 1060 Vienna, Austria; [email protected] (C.S.); [email protected] (E.N.); [email protected] (G.W.); [email protected] (H.H.) 2 Institute for Chemistry and Technology of Materials, Graz University of Technology, Stremayrgasse 9, 8010 Graz, Austria; [email protected] (T.K.); [email protected] (K.R.) * Correspondence: j.fl[email protected]; Tel.: +43-1-588-0115800

Academic Editor: Sergey Kustov Received: 7 July 2016; Accepted: 14 November 2016; Published: 22 November 2016

Abstract: The different properties of acceptor-doped (hard) and donor-doped (soft) lead zirconate titanate (PZT) ceramics are often attributed to different amounts of oxygen vacancies introduced by the dopant. Acceptor doping is believed to cause high oxygen vacancy concentrations, while donors are expected to strongly suppress their amount. In this study, La3+ donor-doped, Fe3+ acceptor-doped and La3+/Fe3+-co-doped PZT samples were investigated by oxygen tracer exchange and electrochemical impedance spectroscopy in order to analyse the effect of doping on oxygen vacancy concentrations. Relative changes in the tracer diffusion coefficients for different doping and quantitative relations between defect concentrations allowed estimates of oxygen vacancy concentrations. Donor doping does not completely suppress the formation of oxygen vacancies; rather, it concentrates them in the grain boundary region. Acceptor doping enhances the amount of oxygen vacancies but estimates suggest that bulk concentrations are still in the ppm range, even for 1% acceptor doping. Trapped holes might thus considerably contribute to the charge balancing of the acceptor dopants. This could also be of relevance in understanding the properties of hard and soft PZT.

Keywords: defect chemistry; oxygen vacancies; doping; diffusion; lead zirconate titanate

1. Introduction The ferroelectric and piezoelectric properties of perovskite-type titanates are employed in many technological devices such as positive temperature coefficient (PTC) resistors and multilayer capacitors [1], generators, motors, ultrasonic transductors, actuators, capacitors, or non-volatile memories [2,3]. Knowledge of the ionic and electronic conductivity in these oxides is highly relevant for establishing, optimizing, and retaining their functionality. For barium titanate and strontium titanate, many studies are available dealing with defect concentrations and transport properties [4–14]. In lead zirconate titanate (PZT), defect-related properties such as the mixed ionic and electronic conductivity are much less understood. This lack of knowledge is partly caused by the limited control of cation stoichiometry due to PbO losses during sintering. Still, doping is often used to tailor properties of PZT. So-called hard PZT ceramics mainly result from acceptor doping, e.g., by Fe3+, and these ceramics exhibit less sharp hysteresis loops and lower dielectric constants. Their behaviour, such as difficult poling but also high mechanical quality factor, is often attributed to the supposedly high concentration of oxygen vacancies and their interaction with

Materials 2016, 9, 945; doi:10.3390/ma9110945 www.mdpi.com/journal/materials Materials 2016, 9, 945 2 of 22

domain walls. Early electron paramagnetic resonance (EPR) spectra of Fe in PbTiO3 gave hints for two different iron centres, one of them being interpreted as a defect associate of Fe3+ with its compensating oxygen vacancy [15]. This was later confirmed [16,17], again with EPR spectra. Recently, further details on point defect associates and their interaction with domain walls were revealed by density functional theory calculations [18,19] and analysis of hysteresis loops [20]. Also trapping of oxygen vacancies in crystallographic shear planes was reported [21]. Soft PZT ceramics result, for example, by doping with Nb5+ and exhibit properties such as square hysteresis loops, low coercive fields, high remnant polarization, high dielectric constants, maximum coupling factors, higher dielectric loss, high mechanical compliance, and reduced aging. Essential in this context is the fact that mainly immobile cation vacancies result from donor doping. For details on the current status of the interpretation of hardening and softening of PZT, see [18,22,23]. The effects of donor doping were already summarised in 1971 in [24]. A more comprehensive study on rare earth doping can be found in [25]. It was also shown that donor doping does not lead to complete annihilation of the p-type conduction behaviour in PZT and furthermore does not entirely suppress the formation of oxygen vacancies [26–31]. Oxide ion motion was analysed for undoped and donor doped PZT as well as co-doped PZT with donor excess [32–36]. Among others, an enhanced oxide ion conductivity along grain boundaries was found in nominally donor-doped PZT [35,37,38]. Still, quantitative data on the defect chemical properties of doped and co-doped PZT are scarce. Additional studies on the charge balancing defects in doped PZT are therefore highly desirable in order to further elucidate the reasons behind hard and soft piezoelectric behaviour. In this contribution, we discuss the transport properties of donor-doped PZT, acceptor-doped PZT, and combined acceptor–donor-doped PZT with acceptor excess. Defect chemical information is gained from a combined study using tracer diffusion, analysed by subsequent time-of-flight secondary ion mass spectrometry (ToF-SIMS) and electrochemical impedance spectroscopy (EIS). Oxygen tracer diffusion coefficients scale with the oxygen vacancy concentration and thus comparison between samples reveals relative changes of the vacancy concentration caused by doping. EIS and the partial pressure dependence of the conductivity allow the analysis of the conducting species and relative concentration changes of the electronic charge carriers with doping. Applying this combination of techniques to sample series of La3+ donor-doped, Fe3+ acceptor-doped, and Fe/La co-doped PZT allowed us to draw conclusions on oxygen vacancy concentrations in differently doped PZT.

2. Experimental

2.1. Sample Preparation and Characterisation

As base composition PbZr0.6Ti0.4O3 was chosen, a rhombohedral ferroelectric perovskite far off the morphotropic phase boundary in the PZT system. For donor-doped samples divalent lead was substituted by trivalent lanthanum with the assumption of ionic compensation by lead vacancies, i.e., two La3+ replacing three Pb2+. For acceptor-doped samples, tetravalent titanium was substituted by trivalent iron. Ionic compensation would thus lead to oxygen vacancies. In the case of Fe/La co-doping a self-compensation may take place and only the excess of acceptor should be charge-compensated by oxygen vacancies. In all samples the starting material contained 1 mol % of PbO excess to compensate for losses during calcination and sintering. The doping concentrations and compositions of all samples are summarised in Table1. All compositions were prepared by conventional solid state synthesis from commercially available powders of Pb3O4 (99.99%, Penox GmbH, Köln, Germany), Fe2O3 (99.99% purity, Merck, Darmstadt, Germany), La2O3 (min. 99%, Treibacher Industrie AG, Treibach, Austria), ZrO2 (Grade 15, MEL Chemicals, Manchester, UK), and TiO2 (99.8% purity, Tronox Pigments, Krefeld, Germany). All raw materials were dried at 220 ◦C and cooled down in a desiccator to avoid moisture effects on weighing. For homogenization the powder mixtures were ball milled in ethanol with tungsten carbide-milling balls 5 mm in diameter in stainless steel beakers lined with a tungsten carbide-inlay using a planetary Materials 2016, 9, 945 3 of 22 mill (Pulverisette 7, Fritsch, Idar-Oberstein, Germany). The ball-milling was carried out for 30 min at 300 rpm. After milling, ethanol was removed from the suspensions by keeping them in a drying oven (LUT 6050, Heraeus, Hanau, Germany) at 80 ◦C over night. The dry mixtures were then sieved through a 500 µm test sieve, transferred into alumina crucibles, covered with lids, and allowed to undergo the solid state reaction at 850 ◦C for 3 h with a heating rate of 5 K/min in a box furnace (N7/H with C290 controller, Nabertherm, Lilienthal, Germany).

Table 1. Summary of lead zirconate titanate (PZT) samples investigated in this study. The amounts of dopant are given in mol % according to the perovskite formula. In the formula of the composition ionic compensation of donors is taken into account by cation vacancies.

Dopants Net Acceptor Doping (Singly Charged) Composition

Undoped 0% PbZr0.60Ti0.40O3 0.5% La - Pb0.9925La0.005Zr0.6Ti0.4O3 1.5% La - Pb0.9775La0.015Zr0.6Ti0.4O3 0.5% Fe 0.5% PbZr0.60Ti0.395Fe0.005O3 1% Fe 1% PbZr0.60Ti0.39Fe0.01O3 2% Fe, 1.5% La 0.5% Pb0.985La0.015Zr0.60Ti0.38Fe0.02O3 6.5% Fe, 6% La 0.5% Pb0.94La0.06Zr0.60Ti0.335Fe0.065O3 7% Fe, 6% La 1% Pb0.94La0.06Zr0.60Ti0.33Fe0.07O3

After the solid state reaction the compounds were again milled in ethanol under the same conditions as before, dried at 80 ◦C, and sieved to reduce the agglomerate size to less than 180 µm. Before pressing the powder into disc-shaped samples, 5 wt % polyethylene-glycol PEG 20000 (per analysis, Merck, Kenilworth, NJ, USA) was added as a binding agent to ensure sufficient mechanical strength for handling. The samples were pressed with 150 MPa for 5 min into discs of 13 mm diameter and about 1 mm height. To remove the binder before sintering, the samples were ◦ heated up to 500 C in open alumina crucibles to promote the decomposition of PEG to H2O and CO2. Pellets were arranged in a coin roll separated by ZrO2 powder. Additionally, atmospheric powder, a 1:1 mixture of Pb3O4 and ZrO2, was put close to the setup to prevent loss of PbO. All was covered with two alumina crucibles. Sintering was performed at 1250 ◦C for 3 h with a heating rate of 5 K/min in the box furnace (Nabertherm N7/H with C290 controller). After solid state reaction and sintering, the samples were checked by X-ray diffraction (XRD) (Bruker AXS D5005, Cu Kα emitter, graphite secondary monochromator, Bruker, Karlsruhe, Germany). The microstructure of the samples was analysed by a scanning electron microscope (Zeiss Ultra 55, FEG, Carl Zeiss AG, Oberkochen, Germany) in backscattered mode with a working distance of 3–8 mm. This working distance was used in order to perform orientation contrast (OC) imaging. Samples were immersed in epoxy resin, ground with silicon carbide paper, and subsequently polished using 0.25 µm diamond paste. Fine mechanical polishing was performed with 0.04 µm SiO2 emulsion for 2 h to obtain a smooth surface and remove any stresses induced during grinding, which would disturb OC imaging. To prevent charging, samples were sputtered with carbon.

2.2. 18O Diffusion Experiments and ToF-SIMS Analysis Information on the oxygen diffusivity of the differently doped PZT samples was obtained by tracer oxygen exchange experiments and subsequent time-of-flight secondary ion mass spectrometry 18 (ToF-SIMS) analysis. The O2 employed for tracer diffusion was purchased from Campro Scientific 18 18 GmbH (97.1 atm % O). To avoid significant changes in the O2 partial pressure during the experiment, the tracer amount in the gas phase was very large compared to the oxygen needed for tracer exchange [39]. Prior to evacuation and tracer filling, the samples were pre-annealed for 4 h in ambient air. Tracer diffusion at either 560 ◦C or 715 ◦C lasted 30 min. These temperatures turned out to be appropriate in terms of materials stability and depth of the tracer profiles. Since evacuation Materials 2016, 9, 945 4 of 22 was performed at annealing temperatures, chemical oxygen diffusion processes could not be avoided, but all measured tracer concentrations were so high that significant effects caused by chemical 18O diffusion to equilibrate the oxygen stoichiometry in the sample can safely be neglected. After tracer 18 exchange, the samples were quickly cooled in O2 atmosphere. Any modification of the established tracer profiles by additional tracer diffusion in ferroelectric PZT close to the Curie temperature Tc (during quenching) or at room temperature (before SIMS analysis) is believed to be restricted to distances much smaller than the depths of measured profiles. This is due to low diffusion coefficients at room temperatures and small times with exposure to temperatures close to Tc. To obtain 18O depth profiles, a TOF.SIMS 5 instrument (ION-TOF, Münster, Germany) was used with a pulsed bismuth primary ion gun (25 keV, Bi1+) working in the “Collimated Burst Alignment” (CBA) mode [40,41]. Applying the CBA mode enables an accurate determination of 18 oxygen isotopic fractions and an improved lateral resolution. The O fraction f18O is calculated 16 − 18 − from the experimentally obtained O and O secondary ion intensities (I16O , I18O ) according   to f18O = I18O / I18O + I16O . During analysis the primary ion gun scanned a “field-of-view” of 1024 × 1024 pixel, typically in the range of 50 µm × 50 µm. For depth profiling (up to 1.5 µm depth) a 2 keV Cs+ sputter gun rastered an area of 400 µm × 400 µm. The depth of the tracer profiles was calculated by normalizing the sputter times to the sputter crater depths, measured by means of a ZEISS Axio CSM-700 microscope (Carl Zeiss AG, Oberkochen, Germany). In some cases, additional cross-sectional profiles were measured to gain tracer information on a larger length scale. Since PZT is poorly conductive, a 20 nm gold layer was sputtered onto the PZT surface (BAL-TEC MED 020 Coating System, BAL-TEC, Balzers, Liechtenstein) and charge compensation was achieved by an electron flood gun (20 eV).

2.3. Impedance Measurements For electrical characterization, a PZT sample with sputter-deposited Pt electrodes (200 nm) was heated in ambient air to 560 ◦C, controlled by a thermocouple positioned next to the sample. After temperature stabilization to ±0.5 ◦C impedance measurements were performed in the range of 10−1–106 Hz using an N4L PSM 1735 (Newton4th, Leicester, UK) in combination with a current amplifier DHPCA-100 (Femto, Berlin, Germany) and 100 mV rms. Most measurements were performed in ambient air. Additionally, different gas compositions were established by using pure O2, 1% O2 in N2 and pure N2. Before performing impedance measurements, all virgin samples underwent an equilibration in ambient air until a constant conductivity was reached [27]; this equilibration took place on a time scale of several hours to days and thus probably included also some cation motion. Mechanistic details on the exact processes during this equilibration are not available yet. Conductivity values of differently doped PZT samples are compared after this equilibration.

2.4. P–E Measurements Hysteresis curves were investigated in polarization (P)–electrical field (E) measurements. Samples were contacted with silver paste (Leitsilber 200, Ögussa, Austria) and P–E curves were recorded at room temperature with an aixACCT aixPES system (AixACCT Systems, Aachen, Germany). The electric field was applied with a frequency of 0.1 Hz (triangular shape).

3. Results and Discussion

3.1. Oxygen Tracer Diffusion in Differently Doped PZT Samples Samples with five different compositions, varying from 1.5% La donor to 1% Fe acceptor doping, were exposed to tracer oxygen for 30 min at 560 ◦C and 715 ◦C. The resulting depth profiles were analysed by ToF-SIMS and are shown in Figure1a–d. For 715 ◦C cross-sectional profiles, revealing further information on depths > ca. 1 µm, were also measured and are shown in Figure1e,f. Moreover, Materials 2016, 9, 945 5 of 22 in-plane tracer distribution images of the near-surface regions are given in Figure2 for 715 ◦C. TheMaterials depth 2016 profiles, 9, 945 reveal substantial differences between the samples. 5 of 22

FigureFigure 1.1. ((aa)) DepthDepth profilesprofiles ofof fivefive differentlydifferently dopeddoped leadlead zirconatezirconate titanatetitanate (PZT)(PZT) samplessamples inin linearlinear plotsplots afterafter tracertracer diffusiondiffusion (30(30 min)min) atat 560560 ◦°C;C; ((bb)) logarithmiclogarithmic plotsplots ofof thethe samesame profiles;profiles; ((cc,,dd)) depthdepth profilesprofiles ofof tracertracer exchangeexchange experimentsexperiments atat 715715 ◦°CC (30 min) in linear plots (c) and logarithmic plots (d); ((ee)) cross-sectionalcross-sectional measurements after after the the tracer tracer ex exchangechange experiments experiments at at715 715 °C◦ Cin inlinear linear plots; plots; (f) (cross-sectionalf) cross-sectional profiles profiles for for 715 715 °C◦ Cin in logarithmic logarithmic plots plots with with fit fit curves curves (Equation (Equation (2)) (2)) for for 0.5% and 1% Fe-dopedFe-doped samples.samples.

The highest tracer concentration in several hundred nm depth is found for the high Fe The highest tracer concentration in several hundred nm depth is found for the high Fe concentration (1%), while the lowest tracer concentration in that depth accounts for the high La concentration (1%), while the lowest tracer concentration in that depth accounts for the high concentration (1.5%). This is in accordance with the expectation that donors supress and acceptors La concentration (1.5%). This is in accordance with the expectation that donors supress and enhance the oxygen vacancy concentration and thus the tracer diffusion coefficient. However, acceptors enhance the oxygen vacancy concentration and thus the tracer diffusion coefficient. quantification of the measured data in terms of diffusion coefficients turns out to be non-trivial, since most depth profiles cannot be described by the simple solution of ∂ ∂ =∗ (1) ∂ ∂

Materials 2016, 9, 945 6 of 22

However, quantification of the measured data in terms of diffusion coefficients turns out to be non-trivial, since most depth profiles cannot be described by the simple solution of

2 ∂ f18O ∂ f18O Materials 2016, 9, 945 = D∗ 6 of(1) 22 ∂t b ∂x2 forfor diffusiondiffusion intointo aa homogeneoushomogeneous halfhalf infiniteinfinite samplesample fromfrom aa constantconstant sourcesource withwith limitedlimited oxygenoxygen exchangeexchange (quantified(quantified byby thethe oxygenoxygen exchangeexchange factorfactork k*),*), i.e.,i.e., byby [[39]39]:: ∗ ∗ ∗ ∗ f18 − f18 !! ∗ ∗2 " ∗ s !# O O =1erf k x+ k t 1 erf + ∗ (2) bg x ∗ D∗ D∗ x∗ ∗ t = − 2 − b b − 2 + 1 erf p ∗ e 1 erf p ∗ k ∗ (2) f18 − f18 2 D t 2 D t D gasO bgO b b b In Equation (2), denotes the fraction of tracer oxygen in the gas phase and the In Equation (2), f18 denotes the fraction of tracer oxygen in the gas phase∗ and f18 the natural natural abundance ofgas tracerO oxygen, x = depth, t = diffusion time, and = bulk tracerbgO diffusion ∗ abundancecoefficient. of tracer oxygen, x = depth, t = diffusion time, and Db = bulk tracer diffusion coefficient.

FigureFigure 2.2. Tracer distributiondistribution imagesimages ofof differentlydifferently dopeddoped PZTPZT afterafter diffusiondiffusion atat 715715 ◦°CC withwith clearclear indicationindication of of fast fast grain grain boundary boundary diffusion diffusion in in the the donor donor doped doped samples; samples;18 18OO tracertracer fractions fractions are are shown. shown.

The discrepancies between experimental data and Equation (2) are exemplified in Figure 3. For The discrepancies between experimental data and Equation (2) are exemplified in Figure3. La-doped samples, the first part of the profile fits acceptably well to Equation (2) but particularly at For La-doped samples, the first part of the profile fits acceptably well to Equation (2) but particularly 715 °C deviations are very pronounced in some depth. The reason becomes obvious from the tracer at 715 ◦C deviations are very pronounced in some depth. The reason becomes obvious from the tracer distribution images in Figure 2 (715 °C): In La-doped PZT high tracer concentrations are visible at distribution images in Figure2 (715 ◦C): In La-doped PZT high tracer concentrations are visible at grain grain boundaries, indicating fast grain boundary diffusion. This is also in accordance with recent boundaries, indicating fast grain boundary diffusion. This is also in accordance with recent publications publications showing fast grain boundary diffusion in Nd3+-doped PZT [42] and Sr/Nb-doped PZT showing fast grain boundary diffusion in Nd3+-doped PZT [42] and Sr/Nb-doped PZT [29,37]. [29,37]. Hence, the near-surface part of the profiles of La-doped samples was attributed to bulk diffusion Hence, the near-surface part of the profiles of La-doped samples was attributed to bulk diffusion and for both temperatures an approximate bulk diffusion coefficient was calculated from a fit of this and for both temperatures an approximate bulk diffusion coefficient was calculated from a fit of this profile part to Equation (2), see Figure3a,c. All resulting values are summarized in Figure4. We may profile part to Equation (2), see Figure 3a,c. All resulting values are summarized in Figure 4. We may compare the bulk diffusion coefficient of 1.5% La-doped PZT at 560 ◦C (1.3 × 10−14 cm2/s) with the compare the bulk diffusion coefficient of 1.5% La-doped PZT at 560 °C (1.3 × 10−14 cm2/s) with the value for Nd3+-doped PZT in [42]: There, a bulk diffusion coefficient of ca. 5 × 10−14 cm2/s is found value for Nd3+-doped PZT in [42]: There, a bulk diffusion coefficient of ca. 5 × 10−14 cm2/s is found at at 550 ◦C, which is in reasonable agreement with our results. Also a tracer diffusion study with higher 550 °C, which is in reasonable agreement with our results. Also a tracer diffusion study with higher La concentrations (ca. 8%) leads to very similar diffusion coefficients in the 10−14 cm2/s range at 560 °C [36].

Materials 2016, 9, 945 7 of 22

La concentrations (ca. 8%) leads to very similar diffusion coefficients in the 10−14 cm2/s range at ◦ 560 C[Materials36]. 2016, 9, 945 7 of 22

1 1 a) b) a) 1.5 % La b) 1.0 % Fe 715°C Fit to Eq. 2 Fit to Eq. 2 near-surface part Fit to Eq. 2 beyond SCL

0.1 0.1 O-fraction O-fraction O-fraction O-fraction 18 18 18 18

0.01 0.01 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 depth [m] depth [m]

1 c) d) 1 0.5 % La 1.0 % Fe 560°C Fit to Eq. 2 Fit to Eq. 2 near-surface part Fit to Eq. 2 beyond CL

0.1 0.1 O-fraction O-fraction O-fraction O-fraction 18 18 18 0.01 18 0.01

1E-3 1E-3 0.00 0.15 0.30 0.45 0.60 0.75 0.00 0.25 0.50 0.75 1.00 1.25 depth [m] depth [m] Figure 3. (a) Tracer depth profile of 1.5% La-doped PZT at 715 °C with fit according to Equation (2). Figure 3. (a) Tracer depth profile of 1.5% La-doped PZT at 715 ◦C with fit according to Equation (2). A clear grain boundary contribution is indicated by the diffusion tail. (b) Tracer depth profile of 1% A clear grain boundary contribution is indicated by the diffusion tail. (b) Tracer depth profile of 1% Fe-doped PZT at 715◦ °C with fits according to Equation (2) in the near surface regime and in the region Fe-dopedin some PZT depth at 715 (beyondC with SCL fits = accordingspace charge to layer); Equation (c) tracer (2) in depth the near profile surface of 0.5% regime La-doped and in PZT the at region in some560 depth°C with (beyond fit according SCL to = Equation space charge (2); (d) layer); tracer depth (c) tracer profile depth of 1% profile Fe-doped of 0.5%PZT at La-doped 560 °C with PZT at ◦ ◦ 560 fitsC with according fit according to Equation to Equation (2) in the (2);near (d surfac) tracere regime depth and profile in the of 1%region Fe-doped in some PZTdepth at (beyond 560 C with fits accordingSCL). to Equation (2) in the near surface regime and in the region in some depth (beyond SCL).

-11 D* from depth profiles at 560 °C 10 b * Db from depth profiles at 715 °C * Db from cross-section at 715 °C

10-12 /s] /s] 2 2 [cm [cm * b * b D D 10-13

10-14 -2 -1 0 1 dopant content [%] Figure 4. Bulk diffusion coefficients of the five analysed sample compositions. Negative dopant Figure 4. Bulk diffusion coefficients of the five analysed sample compositions. Negative dopant concentrations refer to donor dopant (La), positive to acceptor dopant (Fe). concentrations refer to donor dopant (La), positive to acceptor dopant (Fe).

Materials 2016, 9, 945 8 of 22

Materials 2016, 9, 945 8 of 22 Provided that all requirements of Harrison type B diffusion are fulfilled [43], e.g., diffusion length in grainsProvided much smaller that all than requirements the grain size, of aHarrison grain boundary type B diffusiondiffusion coefficient are fulfillDed∗ [43],can bee.g., extracted diffusion gb ∗ fromlength the tailin grains part of much the profile smaller by than using the the grain equation size, a of grain Whipple boundary [44] and diffusion Le Claire coefficient [45]: can be extracted from the tail part of the profile by using the equation of Whipple [44] and Le Claire [45]: ∗ 1/2 !−5/3  D  ∗ ⁄ ∂ln f18 ⁄ ∗ 1.322 b ∂O D = ∗ 1.322 − (3) gb δ= 6/5 (3) δ t ∂x∂⁄ withwithδ being δ being the grainthe grain boundary boundary width. width. For 715 For◦C, for715 example, °C, for aexample, grain boundary a grain diffusion boundary coefficient diffusion ∗ − 9 ∗ 2 −9 2 Dgbcoefficient= 8 × 10 cm = /s8 × is 10 found cm for/s is 0.5% found La assumingfor 0.5% Laδ = assuming 2 nm. Hence, δ = 2 grain nm. boundaryHence, grain diffusion boundary is ordersdiffusion of magnitude is orders fasterof magnitude than bulk faster diffusion than bulk in La-doped diffusion PZT. in La-doped However, PZT. this However, is not in the this focus is not of in thisthe paper; focus a of detailed this paper; discussion a detailed of grain discussion boundary of grain diffusion boundary in PZT diffusion is given in in PZT [27 ,is29 given,37]. in [27,29,37]. AlsoAlso the the profiles profiles in Fe-dopedin Fe-doped PZT PZT could could not not be be well well described described by by Equation Equation (2); (2); severe severe differences differences betweenbetween the the shapes shapes expected expected fromfrom Equation (2) (2) and and ex experimentalperimental curves curves are areclearly clearly visible visible in Figure in Figure3b,d3 forb,d 1% for Fe. 1% However, Fe. However, tracer tracerdistribution distribution images images (Figure (Figure 2) do not2) give do not any give indication any indication of fast grain ofboundary fast grain diffusion boundary and diffusion SIMS measurements and SIMS measurements with a region with of interest a region (ROI) of interestlimited (ROI)to the interior limited of toa the single interior grain of showed a single the grain same showed profile theas larger same sample profile asregions larger (see sample Figure regions 5). Moreover, (see Figure the 5near-). Moreover,surface part the near-surfaceoften showed part clear often deviations showed from clear Equa deviationstion (2). from The Equation much sharper (2). The drop much of the sharper oxygen droptracer ofthe fraction oxygen in the tracer near-surface fraction in region the near-surface is in agreement region with is inresults agreement found for with other results acceptor-doped found for otherperovskite-type acceptor-doped oxides perovskite-type (Fe-doped oxidesstrontium (Fe-doped titanate strontium single titanatecrystals singleand undoped crystals and PZT undoped films in PZT[38,46,47]). films in [ 38There,,46,47 this]). There, sharp thisdrop sharp was interpreted drop was interpreted in terms of in a terms near-surface of a near-surface space charge space layer charge (SCL) layerwith (SCL) vacancy with depletion vacancy depletion and thus and slow thus (depth slow depen (depthdent) dependent) tracer diffusion. tracer diffusion. We assume We assume the same the for sameour for acceptor-doped our acceptor-doped samples samples and thus and conclude thus conclude that in that Fe-doped in Fe-doped samples samples the deeper the deeper parts parts of the of theprofiles profiles represent represent bulk bulk diffusion. diffusion. Such Such a space a space charge charge with withoxygen oxygen vacancy vacancy depletion depletion can also can explain also explainwhy the why near-surface the near-surface tracer tracer concentration concentration of Fe-doped of Fe-doped samples samples is lower is lower than than for forthe theLa-doped La-doped ones ones(see (see Figure Figure 1a,b),1a,b), despite despite supposedly supposedly more more oxygen oxygen vacancies vacancies in in the the bulk. bulk.

FigureFigure 5. Tracer5. Tracer depth depth profiles profiles in 1%in 1% Fe-doped Fe-doped PZT PZT measured measured after after 30 30 min min tracer tracer diffusion diffusion at 560at 560◦C. °C. TheThe circles circles represent represent the the data data after after integration integration over over the the entire entire region region shown shown in thein the tracer tracer distribution distribution imageimage (see (see insets, insets,f (18 fO)).(18O)). Profiles Profiles obtained obtained for for small small regions regions of of interest interest (ROIs) (ROIs) within within single single grains grains (indicated(indicated in thein the SE SE inset inset image) image) lead lead to theto the same same profiles. profiles.

InIn accordance accordance with with the discussionthe discussion in [29 ,in48 ],[29,48], we included we included the space the charge space region charge into region an effective into an surfaceeffective exchange surface coefficient exchange (cf.coefficient [27]) and (cf. only [27]) analysed and only theanalysed bulk-related the bulk-related profile part profile in some part depth in some bydepth Equation by Equation (2) (see Figure (2) (see3b,d). Figure This 3b,d). method This of method analysis of may analysis cause may some cause inaccuracy, some inaccuracy, since setting since thesetting limits ofthe the limits (rather of short)the (rather bulk typeshort) profile bulk parttype in profile some depthpart in suffers some fromdepth a certainsuffers arbitrariness.from a certain arbitrariness. However, cross-section profiles confirmed the validity of the analysis for Fe-doped samples (see below). All values deduced in this manner are plotted in Figure 4. For example, in 0.5% Fe samples a bulk diffusion coefficient of 4.6 × 10−13 cm2/s was found at 560 °C.

Materials 2016, 9, 945 9 of 22

However, cross-section profiles confirmed the validity of the analysis for Fe-doped samples (see below). All values deduced in this manner are plotted in Figure4. For example, in 0.5% Fe samples a bulk diffusion coefficient of 4.6 × 10−13 cm2/s was found at 560 ◦C. Undoped PZT did not show clear indication of fast grain boundary diffusion in tracer distribution images but still a kind of tail in the depth profiles at 560 ◦C that resembles the shape of profiles for donor doping. The first part of the curve fits acceptably well to Equation (2) and hence this part of the profile was attributed to bulk diffusion. At 715 ◦C, however, such a grain boundary tail cannot be distinguished and the entire profile is used to determine a bulk diffusion coefficient, despite only ∗ moderate fit quality. The Db values for both temperatures are also given in Figure4. The measured cross-sectional profiles (Figure1e,f) are difficult to quantify, owing to their limited depth and low spatial resolution. For acceptor-doped samples, an analysis of profiles beyond the first µm is possible. In Figure1f such fits are shown for 0.5% and 1% Fe. The resulting bulk diffusion coefficients are in reasonable agreement with those obtained from the depth profiles (see Figure4) and this supports the validity of the analysis described above. For donor-doped samples, the near surface bulk part (visible in depth profiles) is not accessible in cross-sectional profiles. Therefore, one might only analyse the grain boundary part of the profile, which was not done in this study. Still, a qualitative comparison of the donor- and acceptor-doped samples is interesting: All effective tracer diffusion lengths visible in Figure1e are rather similar, even though the mechanisms of tracer transport (bulk diffusion in Fe-doped or grain boundary diffusion in La-doped) are very different. Figure4 summarises all bulk diffusion coefficients determined in the abovementioned ways for 560 and 715 ◦C. The negative values on the abscissa refer to La donor doping, the zero-point to nominally undoped PZT, and positive values to the Fe-doped samples. These tracer diffusion coefficients are related to the normalized concentration of mobile oxygen vacancies XV by [49]:

∗ Db = fcXVDV (4) with f c = correlation factor (=0.69 in the perovskite lattice), DV = vacancy diffusion coefficient. Trends in vacancy concentrations appear to be reasonable: the much higher tracer diffusion coefficients for ∗ acceptor doping are due to much higher oxygen vacancy concentrations. The highest and lowest Db values are found for the highest acceptor and highest donor concentration, respectively. A quantitative discussion in terms of defect concentrations is given in Section 3.5.

3.2. Electrical Conductivity of Differently Doped PZT Samples Electrical conductivity measurements were performed to get insight into the electrically conducting species (electrons or electron holes) and their doping dependence. In Figure6, the impedance responses of donor-doped (1.5% La), undoped, and acceptor-doped (1% Fe) samples at 560 ◦C are presented in Nyquist plots. The La-doped samples show an almost ideal semicircle in the complex plane. Fitting of such spectra was done using a parallel connection of a resistor R and a constant phase element (CPE) with impedance:

1 Z = (5) CPE (iω)n Q

Fit parameters n (~0.98) and Q can be used to determine a capacitance [50] and the values fit to the bulk of PZT (relative permittivity of 1576 in the specific case of 1.5% La at 560 ◦C). This capacitance, together with the absence of any second arc, suggests that the measured conductivity is indeed the bulk conductivity of La-doped PZT, even though a contribution of highly conducting grain boundaries cannot be excluded based on these data [51]. A comparison of the absolute resistance values in Figure6 shows that Fe-doped samples are much less resistive than the others. Moreover, the impedance response of Fe-doped PZT significantly differs from one semicircle (Figure6c). Rather, two overlapping arcs are visible and fitting was Materials 2016, 9, 945 10 of 22 done to a combination of two serial R-CPE elements. The capacitances of those two semicircles (calculated from n, Q and R [50]) differ by about a factor of five. Similar spectra with two overlapping arcs are found e.g., in [52] for La doped PZT ceramics and there the second arc is attributed to resistive grain boundaries according to the brick layer model [53]. The same interpretation is applied in our caseMaterials and 2016 the, 9, only 945 moderate difference of the two capacitances is in qualitative agreement with10 of the 22 assumption that the grain boundary resistance is caused by space charge layers: Due to low bulk carrier concentrations and high permittivities, rath ratherer thick space charge layers and thus low grain boundary capacitances may result.

50000 2000 400 1.5% La a) b) undoped PZT c) 1% Fe 40000 1600 300

] 30000 1200 ]  ]  [ [ 

[ 200 img img -Z img 20000 800 -Z -Z

100 10000 400

0 0 0 0 10000 20000 30000 40000 50000 0 400 800 1200 1600 2000 0 100 200 300 400 Zre [] Z [] Zre [] re Figure 6. (a) Impedance spectrum of the 1.5%1.5% La-dopedLa-doped PZTPZT atat 560560 ◦°C—anC—an almost ideal semicircle results; ( b)) impedance impedance spectrum spectrum of of the the undoped undoped sample sample at at 560 560 °C;◦ C;the the semicircle semicircle is more is more distorted; distorted; (c) (impedancec) impedance spectrum spectrum of the of the1% 1%Fe-doped Fe-doped PZT PZT at 560 at 560°C. A◦C. splitting A splitting into intotwo twosemicircles semicircles is visible. is visible. The Theabsolute absolute resistance resistance values values of the of three the three spectra spectra differ differ by two by twoorders orders of magnitude. of magnitude.

Space charge zones with thicknesses in the few 100 nm range are also in accordance with the Space charge zones with thicknesses in the few 100 nm range are also in accordance with the tracer exchange results. There, a space charge depletion layer of similar size was concluded near to tracer exchange results. There, a space charge depletion layer of similar size was concluded near to the the surface, with depletion of oxygen vacancies (see Section 3.1). Undoped PZT shows spectra similar surface, with depletion of oxygen vacancies (see Section 3.1). Undoped PZT shows spectra similar to to those of La-doped PZT, even though some asymmetry of the arc is visible in Figure 6b. Still, only those of La-doped PZT, even though some asymmetry of the arc is visible in Figure6b. Still, only a a single R-CPE element was used to fit the data and the extracted conductivity is assumed to be that single R-CPE element was used to fit the data and the extracted conductivity is assumed to be that of of the bulk. the bulk. The partial pressure dependence of the conductivity often gives important information on the The partial pressure dependence of the conductivity often gives important information on the type of charge carrier responsible for the electrical current. The oxygen partial pressure dependence type of charge carrier responsible for the electrical current. The oxygen partial pressure dependence of impedance spectra of two different samples is shown in Figure 7. The Fe-doped samples (Figure of impedance spectra of two different samples is shown in Figure7. The Fe-doped samples 7a) show a drastic increase in total resistance when switching from ambient air to nitrogen (typically (Figure7a) show a drastic increase in total resistance when switching from ambient air to nitrogen containing a few ppm O2). This increase refers to both the bulk and grain boundary resistance. The (typically containing a few ppm O2). This increase refers to both the bulk and grain boundary resistance. occurrence of an additional arc at low frequencies (probably an electrode effect) is not further The occurrence of an additional arc at low frequencies (probably an electrode effect) is not further considered. A detailed defect chemical analysis of the partial pressure dependent defect chemistry of considered. A detailed defect chemical analysis of the partial pressure dependent defect chemistry the samples is beyond the scope of this paper, but the increase in bulk resistance with decreasing of the samples is beyond the scope of this paper, but the increase in bulk resistance with decreasing oxygen partial pressure is a strong indication of predominant hole conduction in these Fe-doped oxygen partial pressure is a strong indication of predominant hole conduction in these Fe-doped samples. The corresponding defect chemical reaction is samples. The corresponding defect chemical reaction is O+2hx  12O+V (6) xO2O• •• OO + 2h ⇔ 1/2O2+VO (6) This hole conduction in acceptor-doped PZT is not surprising and also in accordance with [26,27,31].This hole conduction in acceptor-doped PZT is not surprising and also in accordance withAfter [26,27 thermal,31]. equilibration in ambient air, La-doped PZT was also exposed to a lower oxygen partialAfter pressure thermal (0.01 equilibration bar O2) and in showed ambient an air, increase La-doped in resistance PZT was also(Figure exposed 7b). The to a corresponding lower oxygen partialdecrease pressure of conductivity (0.01 bar when O2) and lowering showed the an oxygen increase partial in resistance pressure (Figure was reproducibly7b). The corresponding measured in decreaseall La-doped of conductivity samples. This when is again lowering a strong the indication oxygen partial of hole pressure conduction was reproduciblyin the material, measured cf. oxygen in allequilibration La-doped samples.reaction in This Equation is again (6). a strong On the indication one hand, of occurrence hole conduction of hole in theconduction material, in cf. a oxygendonor- equilibrationdoped perovskite-type reaction in Equationoxide might (6). be On surprising. the one hand, On occurrence the other ofhand, hole this conduction finding inis ain donor-doped accordance with several other studies [26,27,31]. Very similar results are reported in a detailed analysis of Nd3+ donor-doped PZT, which revealed hole conduction in a wide partial pressure range [27]. The reason is the high volatility of PbO, which during sintering most probably leads to further cation (lead) vacancies. The surplus of cation vacancies compared to the donor causes a small effective (electron- related) acceptor doping and thus electron holes.

Materials 2016, 9, 945 11 of 22 perovskite-type oxide might be surprising. On the other hand, this finding is in accordance with several other studies [26,27,31]. Very similar results are reported in a detailed analysis of Nd3+ donor-doped PZT, which revealed hole conduction in a wide partial pressure range [27]. The reason is the high volatility of PbO, which during sintering most probably leads to further cation (lead) vacancies. The surplus of cation vacancies compared to the donor causes a small effective (electron-related) acceptorMaterials doping 2016, 9, and 945 thus electron holes. 11 of 22

Materials 2016, 50009, 945 60000 11 of 22 1% Fe - ambient air 1.5% La - ambient air a) 1% Fe - N b) 1.5% La - 1% O 5000 2 60000 2 4000 50000 1% Fe - ambient air 1.5% La - ambient air a) 1% Fe - N b) 2 1.5% La - 1% O2 4000 4000050000

] 3000 ]   [ [ 3000040000 img img

] 3000 -Z ] 2000 -Z   [ [ 2000030000 img img -Z 2000 -Z 1000 1000020000

1000 0 100000 0 1000 2000 3000 4000 5000 0 10000 20000 30000 40000 50000 60000 Z [] re Z [] 0 0 re 0 1000 2000 3000 4000 5000 0 10000 20000 30000 40000 50000 60000 Z [] Figure 7. Partial pressure dependencere of impedance spectra of 1% Fe-dopedZ [] (a) and 1.5% La-doped Figure 7. Partial pressure dependence of impedance spectra of 1% Fe-dopedre (a) and 1.5% La-doped (b) (b) samples at 560 °C. Both show characteristics of hole conducting material, having a higher samples at 560 ◦C. Both show characteristics of hole conducting material, having a higher resistance for resistanceFigure 7. Partialfor lower pressure oxygen dependence partial pressure. of impedance spectra of 1% Fe-doped (a) and 1.5% La-doped lower(b oxygen) samples partial at 560 pressure. °C. Both show characteristics of hole conducting material, having a higher Forresistance the following for lower comparison,oxygen partial conductivity pressure. values were taken from measurements in ambient Forair after the followinga rather constant comparison, conductivity conductivity value is reached values (donor-doped were taken fromsamples measurements still showed a in small ambient For the following comparison, conductivity values were taken from measurements in ambient air aftervariation a rather in time). constant In Figure conductivity 8, the resulting value bulk is reached (hole) conductivity (donor-doped values samples of all samples still showed at 560 °C a small air after a rather constant conductivity value is reached (donor-doped samples still showed a small variationare shown. in time). The In plot Figure resembles8, the that resulting found for bulk tracer (hole) diffusion: conductivity The hole values conductivity of all samples is lowest at for 560 1.5%◦C are Lavariation and highest in time). for 1%In Figure Fe. 8, the resulting bulk (hole) conductivity values of all samples at 560 °C shown.are Theshown. plot The resembles plot resembles that found that found for tracer for tracer diffusion: diffusion: The The hole hole conductivity conductivity is lowest lowest for for 1.5% 1.5% La and highestLa and highest for 1% for Fe. 1% Fe. 10-2

10-2 10-3

10-3 10-4

10-4 10-5

-5

conductivity [S/cm] 10 10-6 donor acceptor conductivity [S/cm] 10-6 10-7 -2 -1donor 0acceptor 1 10-7 dopant content [%] -2 -1 0 1 Figure 8. Bulk conductivity values of differentlydopant doped content PZT [%] at 560 °C in ambient air; negative dopant concentrations refer to donor dopant (La), positive to acceptor dopant (Fe). FigureFigure 8. Bulk 8. Bulk conductivity conductivity values values of of differently differently dopeddoped PZT at at 560 560 °C◦ Cin inambient ambient air; air;negative negative dopant dopant concentrations refer to donor dopant (La), positive to acceptor dopant (Fe). concentrations3.3. Co-Doping of refer Iron to and donor Lanthanum dopant in (La), Net positive Acceptor to Doped acceptor PZT dopant (Fe). 3.3. Co-DopingIn a second of sample Iron and series, Lanthanum La and in Fe Net dopants Acceptor were Doped added PZT such that a net acceptor doping of 0.5% 3.3. Co-Dopingor 1% resulted of Iron in the and ceramic. Lanthanum A net in acceptor Net Acceptor dopant Doped level of PZT 0.5% was realized by the two pairs (2% Fe/1.5%In a La), second (6.5% sample Fe/6% series, La), while La and an Fe effective dopants acce wereptor added dopant such concentration that a net acceptor of 1% dopingis expected of 0.5% for In(7%or a1% Fe/6% second resulted La). sample in Figure the ceramic. series, 9 illustrat LaA net andes acceptorthe Fe microstructure dopants dopant were level of of added the0.5% co-doped was such realized that samples. a by net the acceptorThe two grain pairs doping size(2% of 0.5%decreasesFe/1.5% or 1% resulted La), with (6.5% increasing in Fe/6% the ceramic. La),dopant while Aconcentration. netan effective acceptor acceOnly dopantptor a slight dopant level diffe ofconcentrationrence 0.5% is was observed realized of 1% for is byexpectedthe thesamples two for pairs (2% Fe/1.5%with(7% Fe/6%(2% LaFe/1.5% ),La). (6.5% Figure La) Fe/6% and 9 (6.5%illustrat La), Fe/6% whilees the La), an microstructure effectivewhere the acceptoreffective of the dopantacceptor co-doped concentrationdopant samples. level equalsThe of 1%grain to is 0.5%. expectedsize for (7%Adecreases marked Fe/6% withdecrease La). increasing Figure of grain9 illustratesdopant size is concentration.found the in microstructure the sa Onlymple awith slight of(7% diffe the Fe/6% co-dopedrence La) is with observed samples. an effective for the The acceptorsamples grain size decreasesdopantwith (2% with level Fe/1.5% increasing of 1%. La) and dopant (6.5% concentration.Fe/6% La), where Only the effective a slight acceptor difference dopant is observed level equals for to the 0.5%. samples A marked decrease of grain size is found in the sample with (7% Fe/6% La) with an effective acceptor with (2% Fe/1.5% La) and (6.5% Fe/6% La), where the effective acceptor dopant level equals to 0.5%. dopant level of 1%.

Materials 2016, 9, 945 12 of 22

A marked decrease of grain size is found in the sample with (7% Fe/6% La) with an effective acceptor dopantMaterials level 2016 of,1%. 9, 945 12 of 22

Materialsa) 2016, 9, 945 b) c) 12 of 22 a) b) c)

Figure 9. SEM images of co-doped samples, (a) 2% Fe/1.5% La; (b) 6.5% Fe/6% La; (c) 7% Fe/6% La. Figure 9. SEM images of co-doped samples, (a) 2% Fe/1.5% La; (b) 6.5% Fe/6% La; (c) 7% Fe/6% La. Figure 9. 10a SEM shows images the of co-dopedtracer depth samples, profiles (a) 2% after Fe/1.5% oxygen La; (b tracer) 6.5% Fe/6%exchange La; (c at) 7% 560 Fe/6% °C. La.The shapes ◦ Figureresemble 10 thosea shows of only the Fe-doped tracer depth samples profiles (also shown after oxygenin Figure tracer 10): a steeper exchange near-surface at 560 C.decay The that shapes resemblecannotFigure those be described of10a only shows Fe-dopedby the Equation tracer samples depth(2); and profiles (alsoa second after shown part, oxygen in which Figure tracer can 10exchange be): fitted a steeper atto 560Equation near-surface°C. The (2). shapes Since decay that cannotresemblesome of bethe those describedgrains of only were Fe-doped by rather Equation large, samples profiles (2); (also and within shown a second a insingle Figure part, grain 10): which can a steeper again can benear-surface be compared fitted to decayto Equation profiles that (2). cannot be described by Equation (2); and a second part, which can be fitted to Equation (2). Since Sinceaveraged some of theover grains larger wereareas rather(see Figure large, 10b). profiles Both within profiles a singleare almost grain identical, can again supporting be compared the to some of the grains were rather large, profiles within a single grain can again be compared to profiles profilesinterpretation averaged overthat fast larger grain areas boundary (see Figure diffusion 10b). do Bothes not profiles play any are role almost and that identical, the deeper supporting part of the averagedthe profile over again larger corresponds areas (see to bulk Figure diffusion, 10b). Bothwhile profiles near-surface are almost regions identical, are determined supporting by space the interpretation that fast grain boundary diffusion does not play any role and that the deeper part of the interpretationcharge depletion that layers. fast grain Hence, boundary data analysis diffusion was do donees not as describedplay any roleabove and for that Fe-doped the deeper PZT. part of profilethe again profile corresponds again corresponds to bulk to diffusion, bulk diffusion, while while near-surface near-surface regions regions are determinedare determined by by space space charge depletioncharge layers. depletion Hence, layers. data Hence, analysis data wasanalysis done was as done described as described above above for Fe-doped for Fe-doped PZT. PZT.

Figure 10. (a) Tracer depth profiles after diffusion at 560 °C for all Fe- and Fe/La-doped samples; (b) tracer depth profiles of 6.5% Fe/6% La with a region of interest (ROI) integrated over a larger area and FigureROIs situated 10. (a) Tracerwithin depth single profiles grains (seeafter SE diffusion inset), respectively.at 560 °C for Theall Fe- second and Fe/La-doped inset displays samples; the lateral (b) Figure 10. (a) Tracer depth profiles after diffusion at 560 ◦C for all Fe- and Fe/La-doped samples; tracer depthdistribution. profiles of 6.5% Fe/6% La with a region of interest (ROI) integrated over a larger area and (b) tracerROIs depthsituated profiles within ofsingle 6.5% grains Fe/6% (see La SE with inset) a region, respectively. of interest The (ROI)second integrated inset displays over the a largerlateral area and ROIstracerFigure situated distribution. 11a shows within all single bulk diffusion grains (see coefficients SE inset), respectively.deduced from The this second analysis inset for displays both net the acceptor lateral tracerconcentrations distribution. (0.5% and 1%) as a function of the La content. Despite nominally identical acceptor levels,Figure the diffusion 11a shows coefficient all bulk diffusionof the 0.5% coefficients series strongly deduced increases from this when analysis adding for La, both while net theacceptor exact Figureconcentrationsamount 11 ofa La shows (1.5% (0.5% all or and bulk6.5%) 1%) diffusion does as a notfunction play coefficients such of the a role.La deduced content. Also, for fromDespite nominally this nominally analysis 1% acceptor identical for both doping, acceptor net acceptor co- levels, the diffusion coefficient of the 0.5% series strongly increases when adding La, while the exact∗ concentrationsdoping by La (0.5% causes and an 1%) increase as a functionin the tracer of thediffusion La content. coefficient. Despite In both nominally cases the increase identical ofacceptor amountby co-doping of La is(1.5% on the or order6.5%) ofdoes a factor not play of 20. such We a may role. assume Also, for that nominally the oxygen 1% vacancy acceptor mobility doping, and co- levels, the diffusion coefficient of the 0.5% series strongly increases when adding La, while the∗ exact dopingthus the by vacancy La causes diffusion an increase coefficient in the D Vtracer is the disameffusion in all coefficient. our PZT samples. In both Thencases thethe tracerincrease diffusion of amount of La (1.5% or 6.5%) does not play such a role. Also, for nominally 1% acceptor doping, bycoefficient co-doping is directly is on the related order toof thea factor concentration of 20. We ofmay mobile assume oxygen that vacanciesthe oxygen by vacancy Equation mobility (4). Hence, and co-dopingthusthe concentration the by vacancy La causes diffusion of (mobile) an increase coefficient oxygen in Dvacancies theV is tracerthe same in diffusion all in these all our samples coefficient. PZT samples. seems In to Then bothdiffer the cases much tracer themore diffusion increase than of ∗ Db bycoefficientone co-doping would is expect directly is on for the related samples order to ofwiththe a concentration factorthe entire of 20.net of Weacceptor mobile may dopingoxygen assume beingvacancies that compensated the by oxygen Equation vacancyfor (4).by oxygenHence, mobility and thusthevacancies. concentration the vacancy This is discussedof diffusion (mobile) in oxygen coefficientmore detail vacancies DinV Sectionis in the all 3.5.these same samples in all our seems PZT to samples.differ much Then more the than tracer diffusionone would coefficient expect is for directly samples related with the to theentire concentration net acceptor ofdoping mobile being oxygen compensated vacancies for by by Equation oxygen (4). Hence,vacancies. the concentration This is discussed of (mobile) in more oxygen detail in vacancies Section 3.5. in all these samples seems to differ much more than one would expect for samples with the entire net acceptor doping being compensated for by oxygen vacancies. This is discussed in more detail in Section 3.5.

Materials 2016, 9, 945 13 of 22 Materials 2016, 9, 945 13 of 22

10-10 10-2 a) b) 10-11 10-3 10-11 co-doped 10-12 10-4 /s] 2 /s] 2 [cm

*

b -13 -5 [cm D * * Materials 2016, 9, 945 b 13 of 22

D 10 10 b

D conductivity 10-12 10-10 10-2 0.5 % effective acceptor 10-14 10-6 conductivity [S/cm] a) 1 % effective acceptor b) 10-11 donor acceptor 10-3 10-11 10-15 10-7 -101234567 -2 -1 0 1 2co-doped 6 7 10-12 10-4 /s] 2 /s] 2 La - content [%] dopant content [%] [cm

*

b -13 -5 [cm D * * b

D 10 10 b

FigureD 11. (a) Bulk diffusion coefficients of PZT at 560 °C with effective acceptor concentrations of 0.5% Figure 11. (a) Bulk diffusion coefficients of PZT at 560 ◦C with effective conductivity acceptor concentrations of and 1%,10 -12drawn as function of the La content; (b) summary of all bulk diffusion coefficients and hole 0.5% and 1%, drawn as function 0.5 % of effective theLa acceptor content; (10b-14) summary of all bulk diffusion coefficients10-6 conductivity [S/cm] and conductivities in ambient air, 1 % measuredeffective acceptor at 560 °C. For co-doped and acceptor doped samples the ◦ donor acceptor hole conductivities in ambient air, measured at 560 C.-15 For co-doped and acceptor doped samples-7 the abscissa indicates the Fe content. 10 10 abscissa indicates-101234567 the Fe content. -2 -1 0 1 2 6 7 La - content [%] dopant content [%] Also impedance measurements were performed on co-doped samples at 560 °C (Figure 12) and the spectraFigure show 11. two(a) Bulk semicircles, diffusion coefficients as for purelyof PZT at Fe-doped560 °C with effectivesamples. acceptor However, concentrations for high◦ of 0.5% co-doping Also impedanceand 1%, drawn measurements as function of the were La content; performed (b) summary on co-dopedof all bulk diffusion samples coefficients at 560 andC hole (Figure 12) and levels, resistances are so small that a reasonable separation of the arcs is no longer possible due to the the spectra showconductivities two semicircles, in ambient air, as measured for purely at 560 °C. Fe-doped For co-doped samples. and acceptor However, doped samples for the high co-doping limited frequency range. For 2% Fe/1.5% La, determination of the bulk conductivity is still possible, levels, resistancesabscissa are indicates so small the Fe that content. a reasonable separation of the arcs is no longer possible due to the while for the higher co-doping only total conductivities including grain boundary resistance could limitedbe evaluated frequencyAlso from impedance range. the Forlow measurements 2%frequency Fe/1.5% intercept.were La, performed determination Possibly on co-doped the grain of thesamples boundary bulk atconductivity 560 resistance°C (Figure 12)was is and still absent possible, whileanyway for thethe in spectra higher these showsamples, co-doping two semicircles,cf. Figure only total 12. as Thefor conductivities purely results Fe-doped are summarized including samples. However, in grain Figure boundary for 11b, high together co-doping resistance with all could levels, resistances are so small that a reasonable separation of the arcs is no longer possible due to the be evaluatedother conductivities from the lowand frequencybulk diffusion intercept. coefficients Possibly determined the grain in this boundary study. The resistance trend is obvious: was absent limited frequency range. For 2% Fe/1.5% La, determination of the bulk conductivity is still possible, anywaythe inconductivity these samples, of co-doped cf. Figure PZT 12 increases. The results for higher are summarized Fe/La concentrat in Figureions, despite11b, together nominally with all while for the higher co-doping only total conductivities including grain boundary resistance could otheridentical conductivitiesbe evaluated net acceptor andfrom bulklevels.the low diffusion Itfrequency is also coefficientsworthintercept. mentioning Possibly determined the that grain for boundary in all this PZT study. resistance samples The was(donor-doped, trend absent is obvious: the conductivityacceptor-doped,anyway in of these and co-doped samples,co-doped) PZTcf. Figurethe increases relative 12. The changes results for higher are of summarizedhole Fe/La conductivity concentrations,in Figure and 11b, oxygen together despitebulk with diffusion all nominally identicalare very netother similar. acceptor conductivities The levels. corresponding and bulk It is diffus also curvesion worth coefficients are almo mentioningst determined in parallel that in (Figure th foris study. all 11b) PZT The and samplestrend the differenceis obvious: (donor-doped, from the conductivity of co-doped PZT increases for higher Fe/La concentrations, despite nominally acceptor-doped,the lowest to and the co-doped)highest value the is relative almost changesidentical of(a holefactor conductivity of 2000 and and2300 oxygen for conductivity bulk diffusion and are diffusionidentical coefficient, net acceptor respectively). levels. It is Thisalso worthis even mentioning more interesting that for all sincePZT samplesthe two (donor-doped, properties were very similar. The corresponding curves are almost in parallel (Figure 11b) and the difference from the obtainedacceptor-doped, by completely and different co-doped) experimental the relative changes methods. of hole conductivity and oxygen bulk diffusion lowest to theare very highest similar. value The iscorresponding almost identical curves are (a almo factorst in of parallel 2000 and(Figure 2300 11b) for and conductivity the difference from and diffusion the lowest to the highest value is almost identical (a factor of 2000 and 2300 for conductivity and coefficient, respectively). This is160 even more interesting since the two properties were obtained by diffusion coefficient, respectively). This is even more2 % Fe interesting/ 1.5 % La since the two properties were completely different experimental methods. obtained by completely different experimental methods. 6.5 % Fe / 6 % La 120 160 2 % Fe / 1.5 % La 6.5 % Fe / 6 % La

] 80  120 [ img

-Z 40 ] 80  [ img

-Z 0 40

00 40 80 120 160 200 Z [] re 0 40 80 120 160 200 Figure 12. Impedance spectra of PZT with 2% Fe/1.5%Z [ ]La and 6.5% Fe/6% La, measured in ambient re air at 560 °C. Figure 12. Impedance spectra of PZT with 2% Fe/1.5% La and 6.5% Fe/6% La, measured in ambient Figure 12. airImpedance at 560 °C. spectra of PZT with 2% Fe/1.5% La and 6.5% Fe/6% La, measured in ambient air3.4. at P–E 560 Curves◦C. P–E3.4. P–Ecurves Curves of differently doped PZT samples are shown in Figure 13a. La-doped samples 3.4. P–Edevelop Curves slimP–E P–Ecurves curves of differently with increased doped PZTsaturation samples polari are shownzation incompared Figure 13a. to undopedLa-doped samplesor Fe-doped samples.develop Fe-doped slim P–E samples, curves with on increasedthe other saturation hand, polaridevelopzation an compared inclined to P–E undoped curve or withFe-doped reduced P–E curvessamples. ofFe-doped differently samples, doped on the PZT other samples hand, develop are shown an inclined in Figure P–E curve 13a. with La-doped reduced samples develop slim P–E curves with increased saturation polarization compared to undoped or Fe-doped samples. Fe-doped samples, on the other hand, develop an inclined P–E curve with reduced saturation Materials 2016, 9, 945 14 of 22

Materials 2016, 9, 945 14 of 22 polarizationsaturation compared polarization to undoped compared PZT. to un Figuredoped PZT.13b comparesFigure 13b compares the P–E curvesthe P–E ofcurves samples of samples doped with 0.5% Fedoped developing with 0.5% at differentFe developing levels at different of maximum levels of electric maximum field. electric The pinchingfield. The pinching of the P–E of thecurves P–E at low field amplitudecurves at indicateslow field amplitude domain wall indicates pinning, domain which wall ispinning, typical which for acceptor-doped is typical for acceptor-doped samples. The same feature issamples. observed The same for samples feature is that observed are co-doped for samples with that 6.5%are co-doped Fe and with 6% 6.5% La (Figure Fe and 6%13c). La This(Figure pinching again indicates13c). This a pinching net acceptor again dopingindicates of a thenet co-dopedacceptor doping samples. of the These co-doped measurements samples. These thus reflect measurements thus reflect typical hard and soft PZT behaviour for Fe-doped (or co-doped) and La- typical hard and soft PZT behaviour for Fe-doped (or co-doped) and La-doped samples, respectively. doped samples, respectively.

40 a) 30 ] 2 20

C/cm 10  [ E [kV/mm] 0 -4 -2 0 2 4 -10 undoped -20 0.5% Fe Polarization 0.5% La -30 1.0 % Fe 1.5% La -40

40 b) 40 c) 30 30 ] ] 2 20 2 20 C/cm C/cm 10

 10  [ E [kV/mm] [ E [kV/mm] 0 0 -4 -2 0 2 4 -4 -2 0 2 4 -10 -10 0.5% Fe - 2 kV -20 -20 Polarization

Polarization 0.5% Fe - 3 kV -30 0.5% Fe - 4.5 kV -30 6.5% Fe / 6% La - 2 kV 6.5% Fe / 6% La - 3 kV -40 -40 6.5% Fe / 6% La - 4.5 kV

Figure 13. (a) P–E curves of different PZT samples (see legend). La-doped samples develop slim P–E Figure 13. (a) P–E curves of different PZT samples (see legend). La-doped samples develop slim P–E curves with increased polarization. Fe-doped samples develop inclined P–E curves with reduced curves with increased polarization. Fe-doped samples develop inclined P–E curves with reduced maximum polarization;. (b) P–E curves of a sample doped with 0.5 mol % Fe measured for different maximummaximum polarization; electric fields (b) P–E (seecurves legend). of Pinching a sample of the doped P–E withcurve 0.5at low mol field % Fe amplitude measured indicates for different maximumdomain electric wall fields pinning, (see which legend). is typical Pinching for acceptor-doped of the P–E curve samples; at low (c) P–E field curves amplitude of a sample indicates dopeddomain wall pinning,with 6 mol which % La isand typical 6.5 mol for % Fe acceptor-doped measured for different samples; maximum (c) P–E electriccurves fields of (see a legend). sample Again doped with 6 mol %the La pinchingand 6.5 of mol the %P–E Fe curves measured at low field for different amplitude maximum indicates net electric acceptor fields doping (see of the legend). sample.Again the pinching of the P–E curves at low field amplitude indicates net acceptor doping of the sample. 3.5. Problems of Interpretation in Terms of a Simple Defect Chemical Model

3.5. ProblemsQualitatively, of Interpretation the measurements in Terms of a of Simple the first Defect sample Chemical series (only Model one dopant) are in agreement with simple defect chemical models regarding the effect of dopants on oxygen vacancy and hole ∗ Qualitatively,concentrations: the the measurements more acceptors the of higher the first sample as well as series hole conductivity, (only one dopant)and the more are donors in agreement with simplethe lower defect both chemical values are. models Howeve regardingr, a quantitative the effect analysis of dopantsreveals that on simple oxygen defect vacancy chemical and hole models cannot explain the results. ∗ concentrations: the more acceptors the higher Db as well as hole conductivity, and∗ the more donors The first remarkable fact deals with the relative difference between measured values at the lower both values∗ are. However, a quantitative analysis reveals that simple defect chemical models 560 °C. The ratio of 1.5% La and 1% Fe is only about 371; for 715 °C it is even less (54). This cannot explaindifference the is small results. compared to what one might expect for donor- or acceptor-doped perovskites: If 3+ ∗ ◦ Theall first the 1% remarkable Fe dopants fact were deals in the with Fe thestate relative and charge difference balanced between by oxygen measured vacancies, Dweb wouldvalues get at 560 C. ∗ a normalized oxygen vacancy concentration XV of 1667 ppm with◦ respect to oxygen sites (i.e., 5000 The Db ratio of 1.5% La and 1% Fe is only about 371; for 715 C it is even less (54). This difference is small comparedppm when tonormalizing what one to might the formula expect unit). for From donor- this orvalue acceptor-doped and the measured perovskites: ratio of bulk diffusion If all the 1% Fe coefficients, we then have to conclude that XV = ca. 4 ppm in the 1.5% La-doped sample at 560 °C. dopants were in the Fe3+ state and charge balanced by oxygen vacancies, we would get a normalized oxygen vacancy concentration XV of 1667 ppm with respect to oxygen sites (i.e., 5000 ppm when normalizing to the formula unit). From this value and the measured ratio of bulk diffusion coefficients, ◦ we then have to conclude that XV = ca. 4 ppm in the 1.5% La-doped sample at 560 C. This value is rather high for donor-doped perovskites though not impossible, cf. the discussion above on PbO volatility. If, on the other hand, the vacancy concentration in the donor doped samples is much lower, Materials 2016, 9, 945 15 of 22 say in the 0.01 ppm range, as suggested in [42] for similar Nd-doped PZT, then Fe doping would lead to many fewer oxygen vacancies than expected (only in the 1 to 10 ppm range, e.g., XV = 4 ppm for 1% Fe at 560 ◦C). As a second consideration, let us assume similar oxygen vacancy diffusion coefficients of the −6 2 ◦ two perovskite type materials PZT and SrTiO3 (STO). We may take DV = 1.1 × 10 cm /s at 560 C for STO from [10] (activation energy = 0.86 eV) and the measured bulk diffusion coefficient in PZT (Figure4) to calculate vacancy concentrations by Equation (4). This leads to XV = 0.005 ppm for 1.5% La and 2 ppm oxygen vacancies for 1% Fe at 560 ◦C, i.e., much less than required for complete compensation of the acceptor by oxygen vacancies. Complete acceptor compensation by mobile oxygen vacancies would mean a three orders of magnitude lower vacancy diffusion coefficient in PZT compared to STO. This difference would require an activation energy for oxygen vacancy motion in PZT that is 0.5 eV higher than in STO, provided the pre-exponential factors are the same. However, DFT calculations indicate very similar activation energies of oxygen vacancy motion in most perovskite-type oxides [54,55]. Also, a direct comparison of the tracer diffusion coefficients of Fe-doped STO and ∗ Fe-doped PZT for similar conditions reveals much larger Db in STO, despite the existence of significant amounts of Fe4+ in STO: 0.3% Fe-doped STO at 700 ◦C exhibits 3.6 × 10−10 cm2/s [48,56,57] compared to 3.9 × 10−12 cm2/s at 715 ◦C for our 0.5% Fe-doped PZT. (Please note that the ferroelectricity of PZT should not play a role here, due to the high temperatures.) The third unusual effect is the non-linear increase of the tracer diffusion coefficient at 560 ◦C by one order of magnitude when increasing the Fe concentration from 0.5% to 1% (see Figure4). This again contradicts the assumption that the Fe-dopant is largely counter-balanced by oxygen vacancies, since in this case only a factor of two should be found. In line with this result is also the strong dependence of the tracer diffusion coefficient on co-doping: despite identical net acceptor concentration, the tracer diffusion coefficient and thus most probably also the oxygen vacancy concentration varies by a factor of about 20 (at 560 ◦C). This again supports the interpretation that there are generally many fewer oxygen vacancies in PZT than calculated from the upper limit given by complete vacancy compensation of the acceptor dopants in Fe-doped PZT. Fourth, the conductivity values also reveal discrepancies with a simple defect chemical model: One may assume that mobile holes instead of oxygen vacancies counter-balance the acceptor doping. Then we would have a hole concentration of 0.01 per formula unit or 1.56 × 1020 cm−3 (with a cubic lattice constant of 0.4 nm) for 1% Fe-doped PZT. However, if such high concentrations of mobile holes were indeed present, the conductivity should be much larger. Assuming the hole mobility of STO in [10], we would expect more than 1 S/cm, which is several orders of magnitude higher than the measured value. From the mobility of electron holes in Fe-doped STO [10] and the measured conductivity, we might rather estimate the mobile hole concentration in the 1% Fe-doped PZT sample to about 7 ppm per formula unit at 560 ◦C. Also, the trends in hole conductivities (e.g., an increase by a factor of 6.7 from 0.5% to 1% Fe) contradict the assumption of mobile holes as majority charge carriers that balance acceptors. Moreover, the effect of co-doping on the conductivity cannot be explained on the basis of a simple defect chemical model: the conductivity changes despite identical net acceptor concentration. Finally, the conductivity of hole conducting PZT was often reported to be strongly temperature-dependent, with an activation energy in the 1.1–1.4 eV range [7,58–60]. This indicates that a large number of holes is trapped and thus the concentration of mobile holes is strongly temperature-dependent.

3.6. Suggestion of a Modified Defect Chemical Model Based on all these arguments, we have come up with the hypothesis that neither (mobile) oxygen vacancies nor (mobile) holes are the majority charge carriers that counterbalance the acceptor dopant in PZT. Before detailing this hypothesis, we consider the possibility of explaining the results by vacancy-dopant associates and thus immobile oxygen vacancies. Associations of oxygen vacancies and Fe3+ were indeed found in EPR studies [16,17,61] and their properties were modelled in DFT Materials 2016, 9, 945 16 of 22 calculations [18,19]. However, the concentration of associated oxygen vacancies should increase with increasing Fe-doping and thus for identical net acceptor doping the fraction of free and thus highly mobile oxygen vacancies should decrease for increasing Fe content in co-doped samples, in contrast ∗ to the experimental results for oxygen tracer diffusion. Also, the non-linear increase of Db with increasing Fe content in Fe-doped PZT disagrees with a simple vacancy association model. Finally, in such a model the concentration of “free” mobile oxygen vacancies is strongly temperature-dependent and thus rather large activation energies of the tracer diffusion coefficient should result. We do not have sufficient data for a reliable analysis of the activation energy, but based on the data given in Figure4, we expect an activation energy in the 1 eV range or even lower. This is close to the activation energy of oxygen vacancies in perovskite-type lattices [42] and thus does not indicate strong additional temperature-dependent vacancy concentrations. Accordingly, defect associates may be present but they do not solve the problem regarding the charge balancing of acceptors. Therefore, we suggest trapped holes as the positive majority defect in acceptor-doped PZT. There are several candidates for hole traps in PZT: (i) Cation vacancies may act as hole traps but those have negative relative charges (also with a trapped hole) and thus cannot compensate the charge of any negative acceptor dopant. Rather, they need further positive charge carriers for charge compensation. Hence, cation vacancies, possibly formed during preparation by PbO evaporation, cannot solve our problem regarding the charge balancing. Still, some differences in the PbO loss of our samples during sintering may cause unknown further differences in the net acceptor concentrations and could affect ∗ 3+ 4+ 4+ the absolute values of Db and σh; (ii) Fe can trap holes in STO, thus forming Fe [10]. Significant Fe concentrations may also exist in Fe-doped PZT and we consider such an iron trap as possibly relevant in our PZT samples. However, for several reasons we assume that it is not the only kind of relevant ∗ hole trap and possibly also not the most important one: Explaining the 10-fold increase of Db between 0.5% and 1% Fe by iron traps would require a significant decrease of the trap energy with increasing Fe content. Calculations based on the STO defect chemical data set of [10] also reveal that an increase ∗ in the Fe content without changing the net acceptor doping should strongly lower Db, in complete contrast to the experimental data of co-doped samples (see Figure 11b). Fe3+ + h+ => Fe4+ as the only hole-trapping reaction cannot explain the strongly temperature-dependent hole conductivity found in PZT without Fe dopants [42]. Moreover, preliminary measurements with Cu2+ doping (0.25%) ∗ revealed even lower Db values than found for analogous iron doping (0.5%), despite the absence of the iron-based hole trap; (iii) We therefore suggest the existence of additional intrinsic neutral traps Tx in PZT which become positively charged by holes according to

Tx + h• ⇔ Th• or (7)

Tx + 2h• ⇔ T2h•• (8)

The existence of Pb4+ on A-sites of perovskites is discussed in several publications [62–65] and such Pb4+ ions can be considered as Pb2+ with two localized (immobile) trapped holes. Energies of these states are not available so far, but we consider Pb4+ as a realistic doubly filled hole trap in PZT. De-trapping in accordance with Equation (8) would then lead to mobile holes. A further possible intrinsic trap is an oxide ion that might form O− or superoxide ions. Such a hole trap was suggested 2+ 3+ for TiO2 [66] but might be rather shallow. Pb /Pb may also act as a neutral hole trap and the existence of Pb3+ is indeed discussed in the literature, but this trap is assumed to be shallow [67–69]. In the following we discuss (semi-)quantitatively whether the existence of an intrinsic deep trap could explain some of our observations. More specifically, we assume Pb4+ as the positive majority counter-defect of the acceptor dopants. (The same type of calculation, with similar results, can be performed for a singly charged trap.) As a first approximation we quantify the trapping reaction in Equation (8) by its mass action law: C = T2h KT 2 (9) CT · Ch Materials 2016, 9, 945 17 of 22

Symbol C denotes concentrations; Ch is thus the concentration of mobile holes. The total trap concentration CT,tot is assumed to be limited and given by

CT,tot = CT + CT2h (10) Materials 2016, 9, 945 17 of 22 Introducing this limitation of Pb2+/Pb4+ trap states is considering that Pb4+–Pb4+interaction Symbol C denotes concentrations; Ch is thus the concentration of mobile holes. The total trap 4+ may changeconcentration the corresponding CT,tot is assumed energy to be limited levels and and given make by hole trapping less favourable for high Pb concentrations. In a more accurate model, we would have to assume a C -dependence of KT in CCCT,tot T T2h T2h (10) Equation (9). Introducing this limitation of Pb2+/Pb4+ trap states is considering that Pb4+–Pb4+interaction may The charge neutrality equation for mobile holes (h), oxygen vacancies (V), trapped holes (T2h), change the corresponding energy levels and make hole trapping less favourable for high Pb4+ and acceptorsconcentrations. (A) then In reads: a more accurate model, we would have to assume a CT2h-dependence of KT in Equation (9). CA = 2CT2h + 2CV + Ch (11) The charge neutrality equation for mobile holes (h), oxygen vacancies (V), trapped holes (T2h), In equilibriumand acceptors with (A) then the reads: gas phase, holes and oxygen vacancies are coupled by Equation (6) and with mass action constant Kδ and oxygen partial pressure pO this leads to CCAT2hVh22 CC 2 (11)

In equilibrium with the gas phase, holesKδ and oxygenC vacancies are coupled by Equation (6) and = K0 = V (12) with mass action constant Kδ and oxygenp partial pressureδ pO2 2 this leads to O2 Ch KC  V K 2 (12) Combining Equations (9)–(12) results inpCO2h Combining Equations (9)–(12) results in q CA 1 CV − CV − 0 2C 1 2C Kδ K = A V (13) T  CV q  CV 22CA K 1 CV 0 CT,tot − + CV+ 0 KTK 2 2 K (13) δ δ CCVVCA 1 CCT,tot V KK22 0 which can be used to calculate the oxygen vacancy concentration for given KT, Kδ, CT,tot, and CA.

Figure 14 exemplarily displays the concentrations normalized toT a formulaT,tot unitA for a certain which can be used to calculate the oxygen vacancy concentration for given K , K , C , and C . parameter set.Figure A 14 nonlinear exemplarily increase displays the of CconcentrationsV is found normalized for CT,tot tobeing a formula not unit much for a largercertain than CA; parameter set. A nonlinear increase of CV is found for ◦CT,tot being not much larger than CA; the 2+ 4+ the measured increase of CV by a factor of 10 at 560 C can be reproduced in a total Pb /Pb 2+ 4+ trap concentrationmeasured increase of 0.56%. of C ForV by 1% a factor Fe we of then10 at find560 °C ca. can 10 be ppm reproduced oxygen in vacancies a total Pb and/Pb 10trap ppm holes, concentration of 0.56%. For 1% Fe we then find ca. 10 ppm oxygen vacancies and 10 ppm holes, both both withwith respect respect to to a formulaa formula unit unit (3.3 (3.3 ppm ppm with with respect respect to oxygen to oxygensites). sites).

Figure 14. Concentrations of oxygen vacancies and holes for different acceptor concentrations. All

Figure 14.concentrationsConcentrations are normalized of oxygen to a formula vacancies unit (concentration and holes for of ABO different3 cell = C acceptor0) and calculated concentrations. from All 2 concentrationsEquation are (13) normalized with CC/0.5610,0.810,110 to a formula unit2115 K (concentration  CK of ABO 3 cell = C C0). and calculated from T,tot 0−2 T11 02 0 0 5 Equation (13) with CT,tot/C0 = 0.56 × 10 , KT = 0.8 × 10 × (C0) , Kδ = 1 × 10 × C0.

Accordingly, this model explains the comparatively low oxygen vacancy concentrations in Fe-doped PZT as well as the strong change of oxygen tracer diffusivity when increasing the Fe 0 content from 0.5% to 1%. Owing to the temperature dependence of Kδ and KT, the CA dependence Materials 2016, 9, 945 18 of 22

of CV should also be affected by temperature and may become smaller at higher temperatures, in accordance with the experiments. Moreover, the severe effect of co-dopants on defect concentrations, despite identical net acceptor level, is no longer surprising, since the large amount of doping ions might easily change the electronic energy level. For example, if co-doping reduces the Pb2+/Pb4+ 0 trap energy of one hole (ET < 0) by only 80 meV (in KT = KTexp (2 |ET| /kT), the oxygen vacancy concentration increases by an order of magnitude for the dataset in Figure 14. Hence, the assumption of deep hole traps can explain a number of experimental facts that are in contradiction with the simple oxygen vacancy compensation model. Finally, we may also consider the strong correlation found between the hole conductivity and the tracer diffusion coefficient (Figure 11b). Qualitatively, the oxygen exchange equilibrium of Equation (6) explains why hole conductivity and tracer diffusion coefficient, i.e., hole concentration and oxygen vacancy concentration, change simultaneously. Equation (12) relates the two concentrations, and 0 for a dopant independent Kδ the vacancy concentration is expected to change even more than the 2 hole concentration (CV ∝ Ch). Table2 summarizes all relative changes of hole conductivity σh and ∗ 2 ∗
oxygen bulk diffusion coefficient Db for the first measurement series. Ideally, (σh ratio) / Db ratio (last column in Table2) should be unity. However, in all cases the hole conductivity changes more than expected (or the tracer diffusion coefficient less than expected) and hence an additional deviation 0 from a simple defect chemical situation seems to be present in PZT, e.g., Kδ may depend on the dopant concentration due to defect interactions.

Table 2. Ratios of conductivities and tracer bulk diffusion coefficients at 560 ◦C.

2 * (σh Ratio) Samples σh Ratio D Ratio * b (Db Ratio) 0.5% La/1.5% La 3.3 3.0 3.63 undoped/0.5% La 6.3 2.4 16.5 0.5% Fe/undoped 4.0 5.1 3.2 1% Fe/0.5% Fe 6.7 10.2 4.4

Based on the reasonable agreement of many experimental facts with the model of deep hole traps in PZT, we consider it a realistic hypothesis that acceptor-doped PZT has many fewer oxygen vacancies than one might expect from the dopant level. In our case we suggest only a few ppm oxygen vacancies for 1% Fe. Possibly, acceptors are largely charge-balanced by trapped holes. Presumably the trapped holes are not only present as Fe4+ but also populate an intrinsic hole trap such as Pb2+/Pb4+. Donor-doped PZT, on the other hand, has a measureable bulk vacancy concentration, probably in the sub-ppm range (0.01–0.1 ppm in our case), and a substantial oxygen vacancy concentration in grain boundaries. The latter is discussed in more detail in [32,42]. These results might also affect the discussion of PZT hardening by acceptor doping, since in the current models of hardening oxygen vacancies play a key role (see e.g., [18–20,22,23]). It is an open question whether oxygen vacancies have a large effect on domain motion even if their concentration in acceptor-doped PZT were low (in the several ppm range). Moreover, the interaction of domain walls with the supposed trapped holes might affect the ferroelectric properties of PZT. Future experimental and theoretical work may prove or disprove our hypothesis of deep hole traps. Experimentally, for example, additional oxygen partial pressure dependent measurements of conductivity and tracer diffusion may help elucidate the defect chemistry of PZT.

4. Conclusions Oxygen tracer diffusion in donor (La3+)- and acceptor (Fe3+)-doped PZT revealed different predominant diffusion mechanisms. In donor-doped PZT grain boundary diffusion prevails, with little though still measureable oxygen bulk diffusion. Acceptor-doped PZT shows higher bulk diffusion coefficients without any indication of fast grain boundary diffusion. The different bulk diffusion Materials 2016, 9, 945 19 of 22 coefficients of donor- and acceptor-doped PZT reflect the varying oxygen vacancy concentrations in the two materials; differences are up to about a factor of 400 for the 1.5% La and 1% Fe concentrations used here. However, variations are less pronounced than one might expect for this severe dopant change, and in particular differences are less than one could expect from an almost complete oxygen vacancy compensation of the acceptor dopants. Oxygen vacancy concentrations of only several ppm are estimated for our Fe-doped PZT. Conductivity measurements revealed hole conduction in both doping cases, with higher conductivity values in acceptor-doped PZT. However, the conductivity is much lower than supposed for a substantial compensation of acceptor doping by mobile holes. Co-doping by acceptors and donors changes the diffusion coefficients and conductivities even when keeping the net acceptor dopant concentration constant. The combination of all these observations cannot be simply explained by common defect chemical models, e.g., by the assumption of a predominant oxygen vacancy compensation of the net acceptor doping. Instead, we suggest that neither oxygen vacancies nor mobile holes, but trapped holes are the majority defect counterbalancing the acceptor doping in PZT. Besides Fe4+, intrinsic hole traps such as Pb4+ ions on A sites may play an important role. This hypothesis of acceptor compensation by trapped holes might also affect the further discussion of processes leading to the hardening of PZT.

Acknowledgments: SEM images were taken at the Austrian Centre for Electron Microscopy and Nanoanalysis, FELMI-ZFE in Graz, by Angelika Reichmann. Juergen Fleig, Edvinas Navickas, and Gregor Walch gratefully acknowledge financial support by the Austrian Science Fund (FWF), project 4509-N16. Author Contributions: Christoph Slouka performed the conductivity measurements, did the analysis of the diffusion profiles, and wrote parts of the paper; Theresa Kainz prepared the samples and did the polarization experiments; Edvinas Navickas performed the tracer exchange experiments and their SIMS analysis; Gregor Walch performed parts of the defect chemical modelling; Herbert Hutter supported the SIMS measurements; Klaus Reichmann supervised sample preparation and polarization experiments, and was strongly involved at all stages of the study, from planning to analysis to mechanistic discussion; and Jürgen Fleig supervised all conductivity and tracer measurements, was strongly involved in the planning of the study and the interpretation of the results, and wrote parts of the paper. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Moos, R.; Hardtl, K.H. Defect chemistry of donor-doped and undoped strontium titanate ceramics between 1000 ◦C and 1400 ◦C. J. Am. Ceram. Soc. 1997, 80, 2549–2562. [CrossRef] 2. Heywang, W.; Lubitz, K.; Wersing, W. Piezoelectricity, Evolution and Future of a Technology; Springer Science & Business Media: Berlin, Germany, 2008; Volume 114. 3. Haertling, G.H. Ferroelectric ceramics: History and technology. J. Am. Ceram. Soc. 1999, 82, 797–818. [CrossRef] 4. Raymond, M.V.; Smyth, D.M. Defects and charge transport in perovskite ferroelectrics. J. Phys. Chem. Solids 1996, 57, 1507–1511. [CrossRef] 5. Rodewald, S.; Fleig, J.; Maier, J. Resistance degradation of iron-doped strontium titanate investigated by spatially resolved conductivity measurements. J. Am. Ceram. Soc. 2000, 83, 1969–1976. [CrossRef] 6. Yoo, H.-I.; Chang, M.-W.; Oh, T.-S.; Lee, C.-E.; Becker, K. Electrocoloration and oxygen vacancy mobility of Bariumtitanate. J. Appl. Phys. 2007, 102, 093701. [CrossRef] 7. Smyth, D.M. Defect structure in perovskite titanates. Curr. Opin. Solid State Mater. Sci. 1996, 1, 692–697. [CrossRef] 8. Maier, J. On the conductivity of polycrystalline materials. Ber. Bunsenges. Phys. Chem. 1986, 90, 26–33. [CrossRef] 9. Maier, J. Complex oxides: High temperature defect chemistry vs. low temperature defect chemistry. Phys. Chem. Chem. Phys. 2003, 5, 2164–2173. [CrossRef]

10. Denk, I.; Münch, W.; Maier, J. Partial conductivities in SrTiO3: Bulk polarization experiments, oxygen concentration cell measurements, and defect-chemical modeling. J. Am. Ceram. Soc. 1995, 78, 3265–3272. [CrossRef] Materials 2016, 9, 945 20 of 22

11. Daniels, J.; Härdtl, K.H. Part I: Electrical conductivity at high temperatures of donor-doped barium titanate ceramics. Philips Res. Rep. 1976, 31, 489–504. 12. Daniels, J.; Härdtl, K.H.; Wernicke, R. Part II. Defect equilibria in acceptor doped barium titanate. Philips Technol. Rev. 1976, 38, 73–82.

13. De Souza, R.A.; Metlenko, V.; Park, D.; Weirich, T.E. Behavior of oxygen vacancies in single-crystal SrTiO3: Equilibrium distribution and diffusion kinetics. Phys. Rev. B 2012, 85, 174109. [CrossRef] 14. Waser, R.; Baiatu, T.; Härdtl, K.H. dc Electrical Degradation of Perovskite-Type Titanates Part 1–3. J. Am. Soc. 1990, 73, 1654–1662. 3+ 15. Gainon, D.J. Electron paramagnetic resonance of Fe in the strong axial field of PbTiO3 host. Phys. Rev. 1964, 134, A1300. [CrossRef] 16. Meštri´c,H.; Eichel, R.-A.; Dinse, K.-P.; Ozarowski, A.; van Tol, J.; Brunel, L.C. High-frequency electron 3+ paramagnetic resonance investigation of the Fe impurity center in polycrystalline PbTiO3 in its ferroelectric phase. J. Appl. Phys. 2004, 96, 7440–7444. [CrossRef] 17. Meštri´c,H.; Eichel, R.-A.; Kloss, T.; Dinse, K.-P.; Laubach, S.; Schmidt, P.; Schönau, K.; Knapp, M.;

Ehrenberg, H. Iron-oxygen vacancy defect centers in PbTiO3: Newman superposition model analysis and density functional calculations. Phys. Rev. B 2005, 71, 134109. [CrossRef] 18. Chandrasekaran, A.; Damjanovic, D.; Setter, N.; Marzari, N. Defect ordering and defect–domain-wall interactions in PbTiO 3: A first-principles study. Phys. Rev. B 2013, 88, 214116. [CrossRef] 19. Chandrasekaran, A.; Wei, X.-K.; Feigl, L.; Damjanovic, D.; Setter, N.; Marzari, N. Asymmetric structure of ◦ 90 domain walls and interactions with defects in PbTiO3. Phys. Rev. B 2016, 93, 144102. [CrossRef] 20. Rojac, T.; Drnovsek, S.; Bencan, A.; Malic, B.; Damjanovic, D. Role of charged defects on the electrical and

electromechanical properties of rhombohedral Pb(Zr,Ti)O3 with oxygen octahedra tilts. Phys. Rev. B 2016, 93, 014102. [CrossRef] 21. Batuk, D.; Batuk, M.; Tsirlin, A.A.; Hadermann, J.; Abakumov, A.M. Trapping of oxygen vacancies at crystallographic shear planes in acceptor-doped Pb-Based ferroelectrics. Angew. Chem. Int. Ed. 2015, 54, 14787–14790. [CrossRef][PubMed] 22. Martin, A.; Kakimoto, K.-I.; Hatano, K.; Doshida, Y. Temperature dependence of mechanical degradation in lead-free alkali niobate ceramics under unipolar loading. Mater. Lett. 2016, 175, 300–304. [CrossRef] 23. Morozov, M.I.; Einarsrud, M.-A.; Tolchard, J.R.; Geiger, P.T.; Webber, K.G.; Damjanovic, D.; Grande, T. In-situ structural investigations of ferroelasticity in soft and hard rhombohedral and tetragonal PZT. J. Appl. Phys. 2015, 118, 164104. [CrossRef] 24. Jaffe, B.; Cook, W.R.; Jaffe, H. Piezoelectric Ceramics; Academic Press: London, UK; New York, NY, USA, 1971. 25. Wu, L.; Wei, C.-C.; Wu, T.-S.; Liu, H.-C. Piezoelectric properties of modified PZT ceramics. J. Phys. C Solid State Phys. 1983, 16, 2813. [CrossRef] 26. Boukamp, B.A.; Pham, M.T.N.; Blank, D.H.A.; Bouwmeester, H.J.M. Ionic and electronic conductivity in lead–zirconate–titanate (PZT). Solid State Ion. 2004, 170, 239–254. [CrossRef]

27. Slouka, C.; Andrejs, L.; Fleig, J. Defect chemistry and transport properties of Nd-doped Pb(ZrxTi1−x)O3. J. Electroceramics 2014, 33, 221–229. [CrossRef] 28. Donnelly, N.J.; Randall, C.A. Mixed conduction and chemical diffusion in a PZT buried capacitor structure. Appl. Phys. Lett. 2010, 96, 052906. [CrossRef]

29. Frömling, T.; Hutter, H.; Fleig, J. Oxide ion transport in donor-doped Pb(ZrxTi1−x)O3: Near-surface diffusion properties. J. Am. Ceram. Soc. 2012, 95, 1692–1700. [CrossRef] 30. Donnelly, N.J.; Randall, C.A. Impedance Spectroscopy of PZT Ceramics—Measuring Diffusion Coefficients, Mixed Conduction and Pb Loss. In Proceedings of the Applications of Ferroelectrics (ISAF/PFM), 2011 International Symposium on and 2011 International Symposium on Piezoresponse Force Microscopy and Nanoscale Phenomena in Polar Materials, Vancouver, BC, Canada, 24–27 July 2011; pp. 1–5.

31. Donnelly, N.J.; Randall, C.A. Pb loss in Pb(Zr,Ti)O3 ceramics observed by in situ ionic conductivity measurements. J. Appl. Phys. 2011, 109, 104107. [CrossRef] 32. Izaki, T.; Haneda, H.; Watanabe, A.; Tanaka, J.; Shirasaki, S.-I.; Tsuji, K. Self diffusion of oxygen in PLZT ceramics: Optical materials and their applications. Nippon Seramikkusu Kyokai Gakujutsu Ronbunshi 1993, 101, 133–138. [CrossRef] 33. Schloss, L.F.; McIntyre, P.C.; Hendrix, B.C.; Bilodeau, S.M.; Roeder, J.F.; Gilbert, S.R. Oxygen tracer studies of

ferroelectric fatigue in Pb (Zr,Ti)O3 thin films. Appl. Phys. Lett. 2002, 81, 3218–3220. [CrossRef] Materials 2016, 9, 945 21 of 22

34. Cross, J.; Kurihara, K.; Kamehara, N.; Haneda, H.; Sakaguchi, I. Oxygen tracer diffusion in Pb(Zr,Ti)O3 thin film enhanced by catalytic platinum. Appl. Phys. Lett. 2005, 86, 141909. [CrossRef]

35. Gottschalk, S.; Hahn, H.; Flege, S.; Balogh, A. Oxygen vacancy kinetics in ferroelectric PbZr0.4Ti0.6O3. J. Appl. Phys. 2008, 104, 114106. [CrossRef] 36. Hummelt, S.; Flege, S.; Ensinger, W. 18Oxygen diffusion in polycrystalline lead lanthanum zirconate titanate. Phys. Status Solidi (RRL)-Rapid Res. Lett. 2010, 4, 203–205. [CrossRef]

37. Frömling, T.; Schintlmeister, A.; Hutter, H.; Fleig, J. Oxide ion transport in donor-doped Pb(ZrxTi1−x)O3: The role of grain boundaries. J. Am. Ceram. Soc. 2011, 94, 1173–1181. [CrossRef] 18 38. Wang, R.-V.; McIntyre, P.C. O tracer diffusion in Pb(Zr,Ti)O3 thin films: A probe of local oxygen vacancy concentration. J. Appl. Phys. 2005, 97, 023508. [CrossRef] 39. De Souza, R.A.; Chater, R.J. Oxygen exchange and diffusion measurements: The importance of extracting the correct initial and boundary conditions. Solid State Ion. 2005, 176, 1915–1920. [CrossRef] 40. Holzlechner, G.; Kubicek, M.; Hutter, H.; Fleig, J. A novel ToF-SIMS operation mode for improved accuracy and lateral resolution of oxygen isotope measurements on oxides. J. Anal. At. Spectrom. 2013, 28, 1080. [CrossRef] 41. Kubicek, M.; Holzlechner, G.; Opitz, A.; Larisegger, S.; Hutter, H.; Fleig, J. A novel ToF-SIMS operation mode for sub 100 nm lateral resolution: Application and performance. Appl. Surf. Sci. 2014, 289, 407–416. [CrossRef][PubMed] 42. Slouka, C.; Holzlechner, G.; Andrejs, L.; Navickas, E.; Hutter, H.; Fleig, J. Oxygen ion conduction in bulk and grain boundaries of nominally donor-doped lead zirconate titanate (PZT): A combined impedance and tracer diffusion study. J. Am. Ceram. Soc. 2015, 98, 3259–3269. [CrossRef] 43. Harrison, L. Influence of dislocations on diffusion kinetics in solids with particular reference to the alkali halides. Trans. Faraday Soc. 1961, 57, 1191–1199. [CrossRef] 44. Whipple, R. CXXXVIII. Concentration contours in grain boundary diffusion. Philos. Mag. 1954, 45, 1225–1236. [CrossRef] 45. Le Claire, A. The analysis of grain boundary diffusion measurements. Br. J. Appl. Phys. 1963, 14, 351. [CrossRef] 46. De Souza, R.A.; Zehnpfenning, J.; Martin, M.; Maier, J. Determining oxygen isotope profiles in oxides with Time-of-Flight SIMS. Solid State Ion. 2005, 176, 1465–1471. [CrossRef] 47. Donnelly, N.J.; Shrout, T.R.; Randall, C.A.; Reaney, I.M. Thermochemical reactions between PZT and Ag/Pd powders: Relevance to cofiring of multilayer actuators. J. Am. Ceram. Soc. 2008, 91, 1013–1018. [CrossRef] 48. De Souza, R.A.; Martin, M. Using 18O/16O exchange to probe an equilibrium space-charge layer at the surface of a crystalline oxide: Method and application. Phys. Chem. Chem. Phys. 2008, 10, 2356–2367. [CrossRef][PubMed] 49. Kessel, M.; De Souza, R.A.; Martin, M. Oxygen diffusion in single crystal barium titanate. Phys. Chem. Chem. Phys. 2015, 17, 12587–12597. [CrossRef][PubMed] 50. Fleig, J. The grain boundary impedance of random microstructures: Numerical simulations and implications for the analysis of experimental data. Solid State Ion. 2002, 150, 181–193. [CrossRef] 51. Fleig, J.; Maier, J. Microcontact impedance measurements of individual highly conductive grain boundaries: General aspects and application to AgCl. Phys. Chem. Chem. Phys. 1999, 1, 3315–3320. [CrossRef] 52. Cui, Z.H.; Gregori, G.; Ding, A.L.; Guo, X.X.; Maier, J. Electrical transport properties of transparent PLZT ceramics: Bulk and grain boundaries. Solid State Ion. 2012, 208, 4–7. [CrossRef] 53. Fleig, J.; Maier, J. The impedance of ceramics with highly resistive grain boundaries: Validity and limits of the brick layer model. J. Eur. Ceram. Soc. 1999, 19, 693–696. [CrossRef]

54. De Souza, R. Oxygen diffusion in SrTiO3 and related perovskite oxides. Adv. Funct. Mater. 2015, 25, 6326–6342. [CrossRef] 55. Mastrikov, Y.A.; Merkle, R.; Kotomin, E.A.; Kuklja, M.M.; Maier, J. Formation and migration of oxygen

vacancies in La1−xSrxCo1−yFeyO3−δ perovskites: Insight from ab initio calculations and comparison with Ba1−x SrxCo1−y FeyO3−δ. Phys. Chem. Chem. Phys. 2013, 15, 911–918. [CrossRef][PubMed] 56. De Souza, R.A.; Fleig, J.; Merkle, R.; Maier, J. SrTiO3: A model electroceramic: Dedicated to Professor Dr. Dr. hc Manfred Rühle on the Occasion of his 65th Birthday. Z. Metall. 2003, 94, 218–225. [CrossRef] 57. Claus, J.; Leonhardt, M.; Maier, J. Tracer diffusion and chemical diffusion of oxygen in acceptor doped SrTiO 3. J. Phys. Chem. Solids 2000, 61, 1199–1207. [CrossRef] Materials 2016, 9, 945 22 of 22

58. Andrejs, L.; Fleig, J. Resistance degradation in donor-doped PZT ceramic stacks with Ag/Pd electrodes: I. Phenomenology of processes. J. Eur. Ceram. Soc. 2013, 33, 779–794. [CrossRef]

59. Raymond, M.V.; Smyth, D.M. Defects and transport in Pb(Zr 1 Ti 1 )O3. Ferroelectrics 1993, 144, 129–135. 2 2 [CrossRef] 60. Dih, J.; Fulrath, R. Electrical conductivity in lead zirconate-titanate ceramics. J. Am. Ceram. Soc. 1978, 61, 448–451. [CrossRef] 61. Eichel, R.-A. Structural and dynamic properties of oxygen vacancies in perovskite oxides—Analysis of defect chemistry by modern multi-frequency and pulsed EPR techniques. Phys. Chem. Chem. Phys. 2011, 13, 368–384. [CrossRef][PubMed] 62. Eyraud, L.; Gonnard, P.; Claudel, B. Causes of instability and aging of piezoelectric power ceramics. J. Am. Ceram. Soc. 1990, 73, 1854–1856. [CrossRef] 63. Inaguma, Y.; Tanaka, K.; Tsuchiya, T.; Mori, D.; Katsumata, T.; Ohba, T.; Hiraki, K.-I.; Takahashi, T.; Saitoh, H. Synthesis, structural transformation, thermal stability, valence state, and magnetic and electronic properties

of PbNiO3 with perovskite-and LiNbO3-type structures. J. Am. Chem. Soc. 2011, 133, 16920–16929. [CrossRef] [PubMed] 64. Tsuchiya, T.; Saito, H.; Yoshida, M.; Katsumata, T.; Ohba, T.; Inaguma, Y.; Tsurui, T.; Shikano, M. High-Pressure

Synthesis of a Novel PbFeO3; MRS Proceedings; Cambridge University Press: Cambridge, UK, 2006. 65. Goodenough, J.B.; Zhou, J. Varied roles of Pb in transition-metal PbMO3 perovskites (M = Ti, V, Cr, Mn, Fe, Ni, Ru). Sci. Technol. Adv. Mater. 2015, 16. Available online: http://china.tandfonline.com/doi/full/10. 1088/1468-6996/16/3/036003 (accessed on 17 November 2016). 66. Yamamoto, T.; Ohno, T. A hybrid density functional study on the electron and hole trap states in anatase titanium dioxide. Phys. Chem. Chem. Phys. 2012, 14, 589–598. [CrossRef][PubMed] 67. Warren, W.; Robertson, J.; Dimos, D.; Tuttle, B.; Smyth, D. Transient hole traps in PZT. Ferroelectrics 1994, 153, 303–308. [CrossRef]

68. Frantti, J.; Lantto, V.; Kakihana, M. Raman scattering studies of Pb(ZrxTi1−x)O3 and Pb1–3y/2Ndy (ZrxTi1−x)O3 ceramics with uv and visible laser lights. Jpn. J. Appl. Phys. 1998, 37, 5406. 69. Robertson, J.; Warren, W.; Tuttle, B. Band states and shallow hole traps in Pb(Zr,Ti)O3 ferroelectrics. J. Appl. Phys. 1995, 77, 3975–3980. [CrossRef]

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).