Felix
Gunkel
*a,
Sebastian
Wicklein
a,
Susanne
Hoffmann-Eifert
a,
Paul
Meuffels
a,
Peter
Brinks
b,
Mark
Huijben
b,
Guus
Rijnders
b,
Rainer
Waser
a and
Regina
Dittmann
a
aPeter Grünberg Institut and Jülich Aachen Research Alliance – Fundamentals of Future Information Technology (JARA-FIT), Forschungszentrum Jülich GmbH, Jülich, Germany. E-mail: f.gunkel@fz-juelich.de; Fax: +49 2461 612550; Tel: +49 2461 615339
bMESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500AE, Enschede, The Netherlands
First published on 21st November 2014
The electrical properties of the metallic interface in LaAlO3/SrTiO3 (LAO/STO) bilayers are investigated with focus on the role of cationic defects in thin film STO. Systematic growth-control of the STO thin film cation stoichiometry (defect-engineering) yields a relation between cationic defects in the STO layer and electronic properties of the bilayer-interface. Hall measurements reveal a stoichiometry-effect primarily on the electron mobility. The results indicate an enhancement of scattering processes in as-grown non-stoichiometric samples indicating an increased density of defects. Furthermore, we discuss the thermodynamic processes and defect-exchange reactions at the LAO/STO-bilayer interface determined in high temperature equilibrium. By quenching defined defect states from high temperature equilibrium, we finally connect equilibrium thermodynamics with room temperature transport. The results are consistent with the defect-chemistry model suggested for LAO/STO interfaces. Moreover, they reveal an additional healing process of extended defects in thin film STO.
A general understanding of the conduction mechanism at oxide interfaces is an important prerequisite for the development of real electronic devices. Furthermore, it is desirable to obtain a transfer from the scientifically relevant STO-single-crystal-based LAO/STO interface to the technologically more relevant STO-thin-film-based interface system and superlattices. However, given the crucial impact of crystal defects on the electronic properties of LAO/STO interfaces,5–15 the use of STO thin films adds new challenges to the field, namely understanding and controlling the defect structure of thin film STO. This is particularly important keeping in mind that electron transport along the LAO/STO interface actually takes place on the STO side of the interface.
Yet, conducting interfaces between thin film STO and LAO have been obtained for STO grown on a Si template16 as well as for hetero- and homoepitaxial STO grown on perovskite templates.6,17–19 However, their performance in terms of low temperature electron mobility is much lower than that of LAO/STO interfaces to single crystals.16,17 This particularly hints at the presence of growth-induced defects in these samples. For thin film STO, cation non-stoichiometry and the associated appearance of cationic defects have been identified as crucial issues.20–25 To date, however, cationic defects in STO have barely been considered as relevant defect species in the discussion of origin and appearance of the conducting interface between LAO and STO. Being acceptor-type defects, Sr-vacancies have been proposed to be responsible for the reduced electron density observed for LAO/STO interfaces after annealing in oxygen-rich atmosphere.6 Moreover, the effect of extrinsic acceptor doping has been addressed by the insertion of single monolayers of chemically acceptor-doped STO close to the LAO/STO interface.26,27 However, the influence of inherent intrinsic acceptor-type defects in thin film STO has not been considered so far.
Here, we follow the potential approach to alter the intrinsic cation stoichiometry in STO by growth-control. We investigated the electrical properties of LAO/STO bilayers comprising STO thin films grown under varied growth conditions. In particular, we varied the laser fluence, F, during the pulsed laser deposition (PLD) of the STO thin film. This procedure enables to continuously tune the composition of the STO thin film from being Sr-rich to stoichiometric to Ti-rich. The role of cationic defects for the electronic properties of LAO/STO interfaces is discussed from two perspectives, (1) thermodynamics and defect chemistry and (2) thin film growth. A combination of defect-engineering techniques for PLD-grown oxide thin films and defect chemistry models of perovsikte oxide interfaces is then applied to understand and to control the electronic properties of LAO/STO bilayer interfaces.
In STO, one has to consider acceptor-type defects, too. Besides extrinsic acceptor-doping, e.g. by doping Fe on Ti-sites (Fe′Ti), STO can generate intrinsic acceptors – namely cation vacancies – in analogy to intrinsic donors.‡ Specifically for STO, one has to consider doubly charged strontium vacancies (V′′Sr) and four times negatively charged titanium vacancies (V′′′′Ti), while V′′′′Ti are energetically less favorable than V′′Sr.29
Cationic point defects form in thermodynamical (entropy-driven) processes as will be discussed in the following section. In the case of STO thin films, however, the role of cationic defects is further pronounced as they might moreover be induced and controlled in excess of their equilibrium concentrations during the growth process.
Energetically, acceptor-type defect states are usually located deep inside the band gap of STO. Therefore, p-type conduction is hardly observable in STO at room temperature as electron holes freeze out already above 300 K.
As we discuss below, the defect chemistry model established in ref. 5,6 can be extended to a more general case accounting for the compensation of an internal electrical dipole. Charge transfer into the interface can (in the simplest case, i.e. assuming homogeneous charge injection) be formally modeled by a modification of the charge neutrality condition. Assuming that after charge transfer the STO close to the interface is no longer charge neutral, but carries an overall negative net charge −QP (associated with charge density −ρP), the charge balance of the system is given by
p + 2[V˙˙O] − n − 2[V′′Sr] − 4[V′′′′Ti] = −ρP, |
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Fig. 1 Schematic illustration of equilibrium defect concentrations at the LAO/STO interface (in analogy to donor-doped STO) derived from defect chemistry calculations for (a) inactive and (b) active Schottky-equilibrium. Electrons (n), holes (p), oxygen vacancies (V˙˙O), and strontium vacancies (V′′Sr) compensate the interfacial charge associated with ρ. The characteristic decrease of n at high pO2 appearing in (b) is a fingerprint for ionic charge compensation. For more details, please refer to ref. 6, 30. |
As a result of non-stoichiometric growth, STO thin films incorporate defects, in particular vacancies on the sub-stoichiometric site. Both types of vacancies, V′′Sr and V′′′′Ti, cause an expansion of the STO unit cell volume due to the enhanced Coulomb repulsion between the unscreened oxygen anions surrounding the vacancy site.39 For the case of Sr-rich films, non-stoichiometry is accommodated also by the insertion of Ruddlesden–Popper-like phases and Sr-rich antiphase boundaries rather than by titanium vacancies alone.24,40 Such extended defect structures similarly result in a net expansion of the lattice. Therefore, the c-lattice constant of STO films or the actual elongation with respect to the bulk reference value, Δc = (c − c0), represents a qualitative measure of (non-) stoichiometry of a STO film. For a stoichiometric growth, a minimum Δc is expected.41 In order to separate the lattice expansion caused by non-stoichiometry from general strain effects,35 homoepitaxial growth is studied. Having calibrated the growth-control of the thin film cation stoichiometry, the results can then be transferred to the heteroepitaxial case as the stoichiometry-determining processes are mainly decoupled from the choice of the substrate material.
The c-lattice parameter of the thick homoepitaxial films was determined by means of X-ray diffraction experiments around the (200) STO substrate peak (Fig. 2(a)). For the lowest and the largest laser fluences a clear shoulder appears in the XRD-spectra indicating an elongated c-lattice parameter. Fig. 2(b) depicts the extracted lattice expansion, Δc, as a function of applied laser fluence. A minimum lattice expansion is found at a laser fluence of FSTO = 1.9 J cm−2 indicating stoichiometric conditions, whereas the non-zero lattice expansion at lower and larger laser fluences indicates non-stochiometric growth. Comparison to various publications allows to conclude that the STO thin films become Sr-rich for FSTO < 1.9 J cm−2, while they become Ti-rich for FSTO > 1.9 J cm−2. At the chosen growth conditions, one can thus adjust the stoichiometry of the STO thin film from being Sr-rich to stoichiometric to Ti-rich by the variation of the laser energy. (However, Δc does not provide any quantitative information.)
Subsequently, heteroepitaxial STO thin films were grown on (La,Sr)(Al,Ta)O3 (LSAT) substrates using identical growth conditions providing the systematic variation of the STO thin films’ cation stoichiometry. The growth was monitored using reflection high energy electron diffraction (RHEED). In order to accurately control the thickness of the STO layer, the laser repetition rate was reduced to f = 1 s−1. The thickness of the heteroepitaxial layers was kept to 10 unit cells. In order to establish LAO/STO bilayers, the STO growth was directly followed by the growth of 10 unit cells LAO at Tdep = 1070 K, pLAOdep = 4 × 10−5 mbar, f = 1 s−1, and FLAO = 1.4 J cm−2. All samples were cooled down to room temperature with a constant rate of 10 K min−1 in deposition pressure.
Using nanometer-thin heteroepitaxial STO films on LSAT substrates, we confine the electron transport at the LAO/STO interface to the stoichiometry-controlled STO and, at the same time, preserve layer-by-layer growth mode under any growth condition as depicted in Fig. 3(a). Clear RHEED intensity oscillations are observed during the growth of both the STO-layer and the LAO-layer (Fig. 3(b)) indicating layer-by-layer growth mode for the entire bilayer stack. In accordance with this, the step-terrace-structure of the substrate surface is conserved for all bilayer samples as verified by atomic force microscopy (Fig. 3(c–e)). This ensures an atomically defined LAO/STO interface in all bilayer samples. Using the insulating LSAT6 as substrate and keeping the STO layer thickness small we furthermore avoid any major contribution of reduced STO interfering with interface conduction in our electrical measurements (even when working with extremely reducing conditions).6,17
As the LAO-growth parameters were kept constant the LAO-stoichiometry is identical in all samples, so that any effect of LAO-stoichiometry is expected to be negligible.37,42–44 The only parameter varied is the stoichiometry and thus the defect structure of the STO layer.
Electrical characterization in high-temperature equilibrium was conducted using a custom in-house system based on a yttria-stabilized ZrO2-oxygen pump. The setup allows four-point measurements of the high temperature equilibrium conductance (HTEC) at temperatures between 900 K < T < 1100 K in thermodynamic equilibrium with the surrounding atmosphere. The surrounding oxygen partial pressure, pO2, can be controlled within the wide range of 10−23 bar < pO2 < 1 bar. At each individual equilibration temperature and pO2, the resistance of the LAO/STO bilayer samples was monitored until no further change was observed. In order to assure thermodynamic equilibrium, the samples were then kept in constant conditions for additional 1–2 hours. The HTEC was finally obtained from the slope of the linear I–V-curve recorded at each individual equilibration condition. Further details can be found in ref. 5, 45.
Gas mixture | pO2/bar |
---|---|
Dry 4%H2/Ar | 2 × 10−23 |
Wet 4%H2/Ar | 2 × 10−19 |
SGM5 (Ar) | 1 × 10−13–1 × 10−12 |
Ar | 1 × 10−5 |
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Fig. 4 Temperature dependence of the sheet resistance of LAO/STO bilayer interfaces comprising stoichiometric (red circles) and Sr-rich (black squares) STO thin films. |
For the stoichiometric growth of STO, the LAO/STO bilayers typically exhibit a sheet resistance, RS, in the range of 10 kΩ–25 kΩ at room temperature. This is comparable to the values found for standard LAO/STO heterostructures comprising single crystal STO. The sheet resistance of the stoichiometric bilayer decreases with decreasing temperature generally reflecting a metallic temperature dependence. An resistance upturn is observed around 150 K, before RS saturates at a value of several kilo ohms at temperatures below 50 K. This low temperature limiting value is about one to two orders of magnitude higher than the typical residual resistance observed for standard LAO/STO interfaces to single crystals (typically a few hundred ohms) hinting at an increased defect density in the STO thin film as compared to a single crystal even under stoichiometric growth conditions. Such an increased density of defects in stoichiometric STO films was observed also by positron annihilation spectroscopy.20
In general, non-stoichiometric samples deliver higher sheet resistance values than stoichiometric samples as displayed in Fig. 4 for an LAO/STO interface to slightly SrO-rich STO (FSTO = 1.67 J cm−2). Starting at 300 K, the non-stoichiometric sample exhibits a metallic temperature dependence. Below 120 K, the resistance shows an upturn and increases constantly indicating an even more pronounced effect of defect scattering or carrier freeze-out than for the stoichiometric case.
Fig. 5 depicts the room temperature sheet resistance, RS, of the as-grown LAO/STO bilayers as a function of the laser fluence applied during growth of the STO layer. For comparison, the c-lattice expansion of the homoepitaxial reference samples – used as a measure of the cation non-stoichiometry of the STO films – is added to the plot. Obviously, RS exhibits a minimum for the heterostructures with almost stoichiometric STO layers indicated by the vanishing c-lattice expansion of the homoepitaxial films at FSTO = 1.9 J cm−2. For Sr-rich as well as Ti-rich STO layers, the sheet resistance of the LAO/STO bilayer increases with increasing non-stoichiometry, finally reaching several hundreds of kilo-ohms at FSTO = 1.1 J cm−2 and at FSTO = 3.05 J cm−2, respectively. Thus, the interface conduction is affected for both types of non-stoichiometry, Ti-rich composition and Sr-rich composition. In fact, the sheet resistance of the LAO/STO interface can be tailored and controlled by the (non-) stoichiometry of the involved STO layer.
As discussed earlier, one way to accommodate non-stoichiometry is the incorporation of cation vacancies on the under-stoichiometric site. Being acceptor-type, these cation vacancies are expected to (partially) suppress the donor-type interface conduction, provided that their concentration is sufficiently high. In a simple picture, one could expect an effective charge density
ρeff = ρ − 4[V′′Sr]inherent − 2[V′′′′Ti]inherent |
To clarify this point, several samples were investigated by Hall measurements at 300 K. The results are presented in Fig. 6. The non-stoichiometry of the STO layer primarily influences the electron mobility which exhibits an opposite dependence on laser fluence as RS. For a stoichiometric growth of the STO layer, μn reaches its maximum value of 1.6 cm2 V−1 s−1. For Sr-rich and Ti-rich growth of the STO layer, the mobility is reduced by up to one order of magnitude. An increased amount of defects in the non-stoichiometric STO layers causes additional scattering centers for the electrons at the LAO/STO interface. In contrast to that, nS is for all samples of the same order of magnitude (≈ 1014 cm−2), while it shows a slight tendency to increase with increasing laser fluence (within a factor of 3). The lowest sheet carrier density is found for the interface to the Sr-rich STO layer and the highest sheet carrier density is found for the interface to the Ti-rich STO layer.
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Fig. 6 Room temperature Hall data obtained for as-grown LAO/STO bilayers with varied STO stoichiometry. The dashed lines are guides to the eye. |
In bulk STO (or standard LAO/STO heterostructures), μn at room temperature is typically limited by phonon scattering yielding μph ≈ 5–10 cm2 V−1 s−1.32 Hence, the low mobility values (μn ≲ 1–2 cm2 V−1 s−1) obtained for LAO/STO bilayers in this study as well as in literature16,17 indicate that defect scattering affects and ultimately limits μn in the thin film LAO/STO interface samples even at 300 K. This is the case also for stoichiometric STO thin film composition. Such an effect of μn being limited by cationic defects has been reported for donor-doped STO thin films, too.46 The further decrease in mobility when tuning the STO stoichiometry indicates that non-stoichiometry of the STO layer causes a further increase in the concentration of scatter centers, i.e. defects, in the vicinity of the LAO/STO interface.
Generally, data taken for differently grown samples refer to non-equilibrium states. Therefore, the trend found for nS may be due to the fact that the oxidation state of the deposited thin films depends on the actual laser fluence applied during STO growth,22,37i.e. oxygen loads and associated defect states may slightly differ from sample to sample. It may also be related to a partial change of the surface termination during STO growth as recently reported by Reinle-Schmitt et al.47 This effect should be most pronounced for SrO-rich growth. However, a termination effect alone cannot account for the observed maximum of the mobility for stoichiometric growth conditions. For a mere termination effect, one would rather expect a monotonous trend when tuning from Sr-rich to Ti-rich STO composition.
Fig. 7 presents the HTEC characteristics of LAO/STO bilayers on LSAT for (a) a stoichiometric STO layer, (b) a Ti-rich STO layer, and (c) a Sr-rich STO layer. For clarity, Fig. 7(d) compares the HTEC characteristics of the three samples at an equilibration temperature of 950 K. All samples show a similar behavior. Two characteristic features can be unambiguously assigned to interface conduction:6 (1) a conductance plateau, this is the HTEC's tendency to saturate in reducing atmosphere, and (2) a temperature-activated decrease in conductance at intermediate oxygen partial pressures. In constrast, the increase in conductance (3) observed at pO2 ≳ 10−8 bar is due to conduction through the LSAT substrate as discussed in ref. 6. As argued therein and in section 2, feature (1) can be attributed to electronic compensation, ρeff = n, while feature (2) results from ionic charge compensation due to the insertion of Sr-vacancies via the Schottky-equilibrium, ρeff = 2[V′′Sr].
In essence, a comparable donor-type conduction mechanism as well as a comparable ionic charge compensation mechanism can be assumed for all samples. The conductance plateau is less pronounced for the non-stoichiometric samples than for the stoichiometric sample. In fact, it is slightly shifted downwards. This might indicate a slightly lower ρeff in the non-stoichiometric samples, provided that the electron mobility in this high temperature regime is identical for all samples. Applying a literature value for μn obtained for bulk STO at high temperatures,32 one receives an equilibrium sheet carrier density, nS = G/eμn, of about 1 × 1014 cm−2 in the electronic compensation regime (reducing atmosphere).
In the intermediate pressure range, the characteristics of the non-stoichiometric samples are shifted towards lower oxygen partial pressures and lower conductance values, too (as indicated by the arrows in Fig. 7(b), (c)). These shifts become more obvious in direct comparison (Fig. 7(d)). As this intermediate pO2 region was identified with a Schottky-like compensation mechanism,6 the shift of the conductance characteristics indicates a dependence of the associated thermodynamics – in particular the one of the Schottky equilibrium – on the actual inherent stoichiometry and defect structure of the STO layer.
(One may further expect that an increased concentration of cation vacancies in (non-stoichiometric) thin film STO comes along with facilitated cation diffusion. However, this refers to dynamic processes not accessible in HTEC measurements.)
In order to estimate ρeff in the LAO/STO bilayers with different STO stoichiometry, the samples were equilibrated in reducing atmospheres where charge compensation is mostly electronic. Fig. 8 shows the sheet resistance, electron mobility, and sheet electron density in the quenched state after 6 hours of equilibration at 1070 K in reducing 4% H2/Ar atmosphere. The clear dependence of the sheet resistance (top) and the electron mobility (middle) on the inherent cation stoichiometry of the as-grown samples tends to even out after the equilibration in the plateau-region. In particular, the electron mobility is increased in the non-stoichiometric samples and almost reaches a common level of about 1 cm2 V−1 s−1 independent of the particular inherent defect configuration. This indicates, that the presumed additional defects at the LAO–STO interface to non-stoichiometric STO layers can be healed – at least partially – by the high temperature treatment. Thus, the enhanced scattering processes in the as-grown non-stoichiometric samples may be caused not only by inherent cationic point defects, but also by associated extended structural defects incorporated during non-stoichiometric growth.24,25,40 This healing effect is most pronounced for bilayers comprising Sr-rich STO which is known to incorporate Ruddlesden–Popper phases to accommodate non-stoichiometry. These extended defects seem to partially reorganize during the thermal treatment causing the recovery of μn.
The sheet carrier density at the LAO–STO interface (bottom panel) evens out after the equilibration, too. Hence, there is only a minor effect of inherent cation vacancies on ρeff, i.e. the background acceptor doping by cation vacancies is small compared to ρ. This sets an upper limit for the growth-induced inherent cation vacancy concentrations of the non-stoichiometric STO layers
ρeff ≈ ρ ≫ 4[V′′Sr]inherent + 2[V′′′′Ti]inherent |
In order to further probe the influence of the Schottky equilibrium on the electrical properties of the LAO/STO interface, 4 samples cut from the same LAO/STO/LSAT heterostructure were equilibrated for 6 hours at 1070 K in various atmospheres before quenching the achieved defect state down to room temperature. Initially, the LAO/STO/LSAT sample had a sheet resistance of about 100 kΩ corresponding to a sheet carrier density of 2 × 1014 cm−2 and an electron mobility of 0.35 cm2 V−1 s−1. The low initial mobility indicated a slight non-stoichiometry of the subjacent STO layer, as discussed above. The results of the quenching experiments are displayed in Fig. 9. After equilibration, the carrier density of the samples which had been annealed in wet and dry 4%H2–Ar gas mixture did not differ significantly from the initial value. By contrast, the samples which had been equilibrated in higher oxygen partial pressure showed a considerable decrease in the carrier density. For an oxygen partial pressure of about 10−12 bar, the sheet carrier density decreased by one order of magnitude to 2 × 1013 cm−2. The sample annealed in Ar gas even exhibited insulating behavior after the equilibration. The mobility of all samples – showing a measurable conductance – increased after the heat treatment. Thus, the presumed healing effect takes place independent of the actual pO2 during heat treatment.
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Fig. 9 Room temperature Hall data obtained for a quenched LAO/STO bilayer after equilibration in various pO2 (cf. Table 1 for details on the used gas mixtures). After equilibration at pO2 = 1 × 10−5 bar, the sample turned insulating and no data could be recorded (indicated by the black crosses). The dashed lines indicate as grown values. |
The dependence of the sheet carrier density on the equilibration atmosphere corresponds well with the predicted equilibrium values displayed in Fig. 7 (nS = G/eμn). For reducing conditions, a constant plateau-like behavior was expected due to the electronic compensation of the interface charge. For intermediate and oxidizing conditions, a decrease of nS was expected due to the ionic compensation of ρeff by V′′Sr induced via the Schottky-equilibrium (see Fig. 1(b)). After annealing in Ar atmosphere (pO2 ≈ 10−5 bar) ionic charge compensation through strontium vacancies is as pronounced as the electron density at the LAO/STO bilayer interface drops below the room temperature conduction threshold resulting in insulating behavior.
Interestingly, the sheet carrier density obtained for the as-grown LAO/STO bilayers corresponds with the carrier densities obtained for samples equilibrated in very reducing atmosphere (pO2 ≲ 10−19 bar). Considering that the nominal oxygen background pressure in the PLD chamber was pLAOdep = 4 × 10−8 bar in the final stage of the fabrication process, this emphasizes that the PLD growth process and the resulting defect state is far from thermodynamic equilibrium. During growth, the sample is driven into a non-equilibrium defect state which in equilibrium corresponds to much more reducing atmosphere than expected from the nominal growth pressure. A similar effect can be concluded from the growth-induced reduction of STO substrates reported in ref. 49.
As shown, as-grown LAO/STO heterostructure devices obtained by PLD exhibit a drastic non-equilibrium defect structure involving both oxygen and cation species. Understanding and controlling this defect structure is still a key issue in the field of oxide heterostructures and interfaces.
In this context, we discussed two possible ways of manipulating and controlling the cation stoichiometry of the STO layer in the vicinity of the LAO/STO interface. Both strategies, (1) growth-induced variation of the cation stoichiometry (a defect engineering process) and (2) thermal insertion of strontium vacancies via the Schottky reaction (a thermodynamical process), have been shown to modify the transport properties of the LAO/STO interface.
A variation of the STO growth parameters primarily tailors the electron mobility at the LAO–STO interface due to the incorporation of point defects as well as extended defects in non-stoichiometric STO thin films. However, the amount of acceptor-type point defects induced during growth seems insufficient to alter the effective charge density ρeff and the resulting electron concentration at the LAO/STO interface. By contrast, thermal annealing in various oxygen partial pressures varies the electron density at the LAO/STO interface. In this case, charge compensation shifts from being mostly electronic in reducing atmosphere to being mostly ionic in oxidizing atmosphere. In addition, high temperature annealing leads to an increase in mobility. This indicates that extended structural defects such Ruddlesden–Popper-like defects induced during non-stoichiometric growth contribute to electron scattering and that these can rearrange and heal during a thermal treatment.
The results underline the direct interrelation between interfacial conductivity and cationic defects in the STO layer adjacent to the LAO/STO interface. Such defects evidently exist even in nominally stoichiometric STO and limit the electron mobility which at room temperature did not exceed 1–2 cm2 V−1 s−1 for any bilayer sample. In order to achieve sufficient performance of thin film LAO/STO devices or superlattices with multiple conducting interfaces, it is thus necessary to further decrease the defect concentrations in thin film STO beyond the stoichiometry control presented in this study. In order to overcome this challenging task we suggest to use of extreme growth temperatures46,50 or to utilize alternative growth techniques.51
Footnotes |
† In Kröger-Vink notation28 the main index corresponds to the prevailing element sitting on a lattice site of the crystal. Vacancies are indicated by V. The subscript refers to lattice site of the original, perfect lattice. The superscript indicates the relative charge of a defect, with ˙ indicating a positive relative charge, and ′ indicating a negative relative charge. |
‡ Obviously, a missing cation cannot donate its electrons to the lattice (when forming a cation from the neutral atom). As a result, the total number of electrons (electron holes) in the system is reduced (increased) in presence of cation vacancies legitimating their classification as acceptor-type defects. Relative to the original lattice site, cation vacancies thus carry a negative net charge. |
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