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Vacancy defect positron lifetimes in strontium titanate

The results of positron annihilation lifetime spectroscopy measurements on undoped, electron irradiated, and Nb doped SrTiO3 single crystals are reported. Perfect lattice and vacancy defect positron lifetimes were calculated using two different first-principles schemes. The Sr vacancy defect related positron lifetime was obtained from measurements on Nb doped, electron irradiated, and vacuum annealed samples. Undoped crystals showed a defect lifetime component dominated by trapping to Ti vacancy related defects.


I. INTRODUCTION
Vacancies are normally assumed to be the dominant point defects in perovskite oxide (ABO 3 ) titanates, and often strongly influence the material properties such as electrical conductivity and domain stability and dynamics. 1,2 Strontium titanate is one of the most widely studied oxide materials, having a model cubic perovskite oxide structure at room temperature and exhibits an antiferrodistortive phase transition at 105 K. The material is a wide band gap semiconductor (with 0 g E = 3.3 eV, 3 ) but can be made a good conductor by doping, for example, with Nb. 4 Single crystal SrTiO 3 substrates are readily available, and high quality epitaxial layers can be grown. [5][6][7] The recent observation of two-dimensional electron gas formation at the interface between SrTiO 3 and LaAlO 3 , 8

residing in the
SrTiO 3 , has further demonstrated the importance of the material for future multifunctional oxide electronic device structures. 9 The electronic quality of SrTiO 3 is ultimately limited by the presence of electrically active point defects.
The presence and importance of vacancy-type defects in SrTiO 3 has been inferred from electrical conductivity measurements as a function of thermal treatments, and the Sr/Ti ratio, analyzed using defect chemistry models. [10][11][12][13][14] Evidence has also been presented that vacancy defects can also control the grain boundary structure, and hence grain growth. 15 The SrTiO 3 structure is shown in Fig. 1; the Ti 4+ ion is at the perovskite B-site and has octahedral coordination, and the Sr 2+ is the A-site ion with twelve O nearest neighbors. A good theoretical understanding of point-defect formation energies in SrTiO 3 has been developed using static lattice calculations with empirical and semiempirical interatomic could have an even smaller formation energy. 16 A detailed first principles study as a function of Fermi level, at a number of points in the phase diagram, has confirmed the potential importance of the Sr partial Schottky defect compared to the full, and Ti partial, Schottky defects, and the authors have also calculated formation energies for the isolated vacancy defects as a function of the charge state varying from the ionic value to the neutral state. 18 In addition, Tanaka et al. 18 report the atomic and electronic structures of the isolated vacancy defects and have provided detailed information on structural relaxations.
Evidence for the presence of vacancy-type defects in SrTiO 3 has also been provided by tracer diffusion measurements using appropriate isotopes of Sr, Ti, or O. 19 The model is readily extended to include more defect states, and can also allow for detrapping from shallow states. 25 As the concentration of defects increases, the intensity of the longer lifetime, I 2 , increases towards unity. Eventually, all positrons will annihilate from the defect. This is known as saturation trapping and sensitivity to defect concentration is restricted or lost completely; the concentation at which this occurs depends on the specific values of τ B and µ d . Assuming a plausible specific trapping coefficient ~ 2 × 10 15 s -1 at. for a negatively charged monovacancy, 25  However, their interpretation was hampered by the lack of available theoretical positron lifetimes. A nanosecond orthopositronium pick-off lifetime was also detected from La doped ceramics, providing evidence for the presence of nanovoids. 27,28 More recently the atomic superposition method has been used to calculate the lifetimes of both delocalized and trapped positron states in perovskite oxide materials. 30

II. POSITRON LIFETIME CALCULATIONS
Calculations of the positron lifetime were performed using two different density functional theory (DFT) methods: one using the MIKA/Doppler package, 35 and the other with the linear muffin-tin orbital (LMTO) method. 36 In the MIKA/Doppler package, the electron density of the solid is approximated by the non-self-consistent superposition of free atom electron densities in the absence of the positron (the so-called 'conventional scheme'). 37 This approximation to the complete two-component density functional theory (TCDFT) has been found to give positron lifetimes close to TCDFT as well as experimental values. In our calculations of the positron lifetimes, two schemes for the electron-positron enhancement factor (that accounts for the enhancement of the electron density at the positron) have been used, that due to Barbiellini and co-workers, 38, 39 as a parameterization of the data of Arponen and Pajanne, 40 (referred to as AP) and that due to Boronski and Nieminen (BN), 41 both of which have been described within the generalised gradient approximation (GGA) of the enhancement factor. Calculations were performed with the room temperature Pm3m structure of SrTiO 3 , 42, 43 using a 6×6×6 supercell.
Calculations that included the relaxation of neighbor atoms surrounding the isolated vacancies were also made.
Additionally, self-consistent calculations of the electron density (where the charge density is iteratively computed until self-consistency is achieved) have been performed using the LMTO method within the atomic sphere approximation, and including combined correction terms. 36

III. EXPERIMENT
Undoped SrTiO 3 single crystals were obtained from a number of different suppliers, were performed using a movable tube furnace connected with a base pressure better than 2x10 -6 Pa.
The positron lifetime experiments were performed using two conventional fast-fast spectrometers. One is optimized for room temperature measurements and with a time resolution function of 205 ps, the second for variable temperature studies in the range 12-300 K with a resolution function of 235 ps. 44 A detailed description of the resolution function required for spectrum analysis used three Gaussian functions with intensities 80%, 10%, 10%, and appropriate relative shifts. All spectra contained at least 5 × 10 6 counts. Positron sources were made from aqueous 22  ps for Al (~5 %). Care is required to correctly account for source annihilations, measurements were made with at least two types of foil source to establish source independent fittings. 44 This procedure involved a systematic variation of the source correction terms and a minimization of the fit chi-squared and allowed both material and source correction terms to be determined and checked.
Studies were also performed using simulated spectra. These were generated from the relevant simple trapping model using the calculated positron lifetimes, and assumed values for positron trapping coefficients and the defect concentrations. The resulting sum of exponential decays was then convolved with the experimental spectrometer timing resolution function, noise added and the area normalized to simulate a 5 × 10 6 count spectrum. These spectra were then analyzed using POSITRONFIT.

IV. RESULTS AND DISCUSSION
The calculated positron lifetime values for the bulk and for the relevant vacancy defects in SrTiO 3 are given in Table I, along with previously reported values. 31 The LMTO calculation results shown were made using 3×3×3 supercells.
Comparing the results from the present atomic superposition method using the Boronski and Nieminen (BN) positron enhancement factor with the previously reported values, 31 given in Table I 31 The inward relaxation for V Ti causes a reduction in the lifetime, but there is essentially no effect for the more open V Sr .
The formation of complexes between cation vacancies and nearest neighbor charge  (Table I).
The experimental room temperature positron lifetimes for a series of commercially available, nominally undoped, single crystal SrTiO 3 samples are shown in Fig. 2, and those for Nb-doped, electron irradiated, and vacuum annealed, single crystal SrTiO 3 and ceramic SrTiO 3 are given in Fig. 3. Differences were observed between different batches from the same supplier (Fig. 2). Apart from Nb-doped samples from one supplier, all analyzed lifetime spectra showed significantly smaller variances (< 1.10) for two lifetimes-component fits, and relevant results are summarized in Table II Table II Table I).
The results from the undoped SrTiO 3 single crystals are given in Fig. 2 and Table II (Table I).  (Table II).
To further investigate this model simulations were performed. The best fit to the 293 K spectrum required a two-defect trapping model simulation, and the resulting POSITRONFIT contained two terms and is shown in Fig. 5  The second resolved lifetime for the 3×10 18 e -/cm 2 irradiated sample was 261(5) ps, and is comparable to that observed for the vacuum-annealed samples, 262(6) ps (Fig. 3).
These are slightly shorter than the above, and may indicate that the component results from more than one vacancy defect type. The as-received MaTecK GmbH substrates used for electron irradiation showed lifetime spectra similar to that given in Table II. Both electron irradiation and vacuum annealing were observed to have a marked effect on the first lifetime component (see Table II (Table I)