Kinetics of Vacancy Doping in SrTiO 3 Studied by in situ Electrical Resistivity

The kinetics of annealing to transform stoichiometric SrTiO3 insulator into a vacancy-doped semiconductor-superconductor was revisited by in situ electrical resistivity measurements. SrTiO3 single crystals were grown by Floating Zone Method. Using a homemade apparatus several electrical resistivity as a function of time were measured at different temperatures, which allows one to study the creation of vacancies during annealing under vacuum. The activation energy for the oxygen vacancies formation/charge doping in SrTiO3 was estimated as 1.4±0.3 eV using solid-state kinetics approach.


Introduction
Strontium titanate (SrTiO 3 ) is one of the most studied cubic perovskites due to the fascinating physical properties [1][2][3][4] .This compound exhibits structural cubic to tetragonal transition at 105 K with a ferroelectric order simultaneous to antiferrodistorsive transition order 5 .The interplay between these distortions has been focus of many studies [6][7][8] .
During last five decades, much attention has been given to the electronic transport mechanisms in doped SrTiO 3 .This is special due to the anomalous superconductivity within a limited doping window [9][10][11] in the vicinity of a ferroelectric transition 12 .
Stoichiometric SrTiO 3 is an insulator with 3.2 eV band gap, but one can introduce electron donor states by replacing Ti 4+ by Nb +513 or Sr 2+ by La 3+ like a n-type silicon based semiconductor 14 either oxygen can be removed with heat treatment in vacuum at high temperatures in order to reach SrTiO 3-δ vacancy self-doped semiconductors 15 .It is known that oxygen vacancies remain double ionized in quenched samples after annealing under vacuum, therefore, two free-electrons became available for each oxygen removed from the crystal lattice after the reduction process 16 .
Several authors have studied the vacancy doping of SrTiO 3 in form of single crystals, thin films or ceramics [17][18][19][20] .Some of these authors have shown chemical diffusion coefficients for the SrTiO 3 crystal lattice using the Neumann solution of the Fick's second law carried out by in situ transient electrical resistivity measurements [18][19][20] .Additionally, in this paper we report that the solid-state kinetic models reviewed by Khawam and Flanagan 21 can describe the in situ electrical resistivity measurements as a function of time, which allows us to determine the energy involved in oxygen removal process from the SrTiO 3 .
Thus, this paper deals with the SrTiO 3 single crystals grown by Floating Zone Method 22 .The reported data show that the self-doping in SrTiO 3-δ is well described by a first order mechanism with an activation energy of 1.4±0.3eV to produce oxygen vacancies, which is comparable to the recent data available on literature 23 .

Crystal growth and samples quality
Polycrystalline SrTiO 3 phase was synthetized from mixed SrCO 3 and TiO 2 powders with 99.9% purity (Sigma-Aldrich reagents).Precursor powders were mixed in liquid acetone under magnetic stirring.The resultant mixture was dried and then calcined in air at 1200ºC for 24 hours.X-ray powder diffractometry technique with CuKα radiation and Ni filter using PanAnalytical Empirian diffractometer was employed to verify the phases present in the samples.Due to the heavier mass, the impurities have lower diffusion compared to oxygen atoms.Therefore, in this work we suppose that the sample impurities contents do not chance during the heat treatment and are not responsible for the electrical resistance change.Materials Research In order to grow single crystals of SrTiO 3 using the Floating Zone Method 22 , polycrystalline powder was sealed in cylindrical rubbers and hydrostatically compressed into rods with 5 to 6 mm in diameter and 70 to 130 mm in length.The polycrystalline feed rods were sintered in air at 1300ºC for 12 hours in which 80 to 90% of theoretical density was obtained.
Floating Zone Method consists in directional solidification of a polycrystalline feed rod casted by concentrated radiation emitted from bulb lamps at the focus of concave ellipsoid mirrors.The desired single crystal grows by controlled solidifying of molten above the single crystal used as a seed while both are pulling downward from heating zone 24 .Our single crystals were grown at normal atmosphere using 8 mm/h growth rate, 35 rpm in clockwise, and 5 rpm in counterclockwise rotations for feed rod and seed.The Quantum Design Infrared Image Furnace used here has two bulb lamps with 650 W maximum power in each concave mirrors, one facing another.At least, 95% of the power lamp was required to achieve 2040ºC in order to melt the SrTiO 3 rods 25 .
Obtained SrTiO 3 single crystals are transparent with light yellow color or brown color.It is well known that oxygen content and stoichiometry deviations can affect the color of single crystals 26 .In in this sense, eventual evaporations during crystal growth should explain the differences observed in the coloration of samples produced here.
After the growing, circular wafers were cut from single crystal bars with an IsoMet cutting machine.Rectangular shapes with 1 mm in thickness were also cut for in situ electrical resistivity measurements from the same single crystal rod.The X-ray diffraction on single crystals surface was measured in texture analysis Philips X'Pert with CuKα radiation and Ni filter to determine the preferential orientation of the obtained samples.Figure 1 shows some pictures of sample preparation procedures.

In situ electrical resistivity measurements
Single crystals in rectangular blocks with the longest side parallel to the growth axis were used to study the kinetics of oxygen removal process by in situ electrical resistivity measurements.
For the electrical resistivity measurements, a sample holder was constructed with an alumina tube.Four platinum wires were connected to the sample with special epoxy, and the sample holder has a thermocouple with its tip installed near the specimen.The sample holder part is involved by a quartz tube and introduced inside of a heated stainless-steel enclosure continuously evacuated at 10 -5 Torr by a turbomechanic vacuum pump.Software developed in Matlab programing language acquires the electrical resistance of the sample by standard four-probe method using Keithley

Results and Discussion
Figure 2 shows the X-ray diffraction (XRD) pattern for the SrTiO 3 polycrystalline powder and the structural refinement by GSAS program 27 .One can observe good agreement between experimental and refined patterns 28 , which confirms SrTiO 3 single phase with lattice parameter a = 3.905 Å and Pm-3m space group 2,25 .The goodness refinement indices are: R exp = 4.39%; R p = 6.42%;R wp = 7.76% and χ 2 = 3.12.The results of preferential orientation in single crystals are shown in Figure 3. Figure 3a shows that the surface on the disk shape samples has only diffraction patterns of the {100} family planes.This is an indicative of preferential growth axis parallel with [100] crystallographic orientation.On the other hand, one of the single crystals samples grew with [110] direction (see Figure 3b).
Figure 4 shows the evolution in electrical resistivity of a sample during thermal annealing at 650ºC under a vacuum of 10 -5 Torr.One can observe four regimes in Figure 4: i) a fast warming up to the temperature of the measurement, in which there is a decreasing of the electrical resistivity due to insulator behavior of the stoichiometric SrTiO 3 ; ii) a plateau up to 200 to 300 min depending on the annealing temperature; iii) a regime of stable temperature longer than 1200 minutes in which the oxygen removal occurs; and iv) a fast cooling down to room temperature.The behavior in the cooling regime indicates that under these experimental conditions used here, the amount of oxygen atoms removed from the samples was not high enough to produce metallic samples, such as reported before by other authors 29 .
The equilibrium degree α during oxygen removal process at constant temperature regime can be written in terms of electrical conductivity by 18,20,21 : ( where σ is electrical conductivity (1/ρ) at elapsed time t, σ 0 is the electrical conductivity at initial condition (α ≈ 0) and σ ∞ is the electrical conductivity when a new equilibrium is achieved (α ≈ 1).
In solid-state kinetic formalism, the evolution in the equilibrium degree can be expressed by a linear form allowed by some g(α) model-function 21 : ( where the reaction rate constant k, is temperature dependent and given by: where A represents the frequency reaction factor, E is the activation energy of the process, k B is the Boltzmann constant, and T is the absolute temperature.The first-order reaction model reviewed in reference 21 fits better our experimental data.It is implies: (4)   providing: (5) The results for two equilibrium temperatures are shown in Figure 5.The solid lines show the best fit for the obtained data in this work.
From the fit of the function expressed by equation 5 or from the slope of lines shown in insets of the Figure 5 (a) and (b), one can obtain the Arrhenius kinetic factor (k) at the analyzed temperature.The values are (3.7 ± 0.7) 10 -3 min -1 and (8.6 ± 0.1) 10 -2 min -1 for 650°C and 850°C, respectively.In such way, the activation energy value for the charge doping through oxygen removal process in SrTiO 3 can be estimated as 1.4±0.3eV using equation 3.This activation energy is based upon the evolution of the equilibrium degree observed in the electrical resistivity during the annealing in vacuum at constant temperature, which can be related to possible three mechanisms: i) the formation of vacancies; ii) the oxygen self-diffusion; and iii) the oxygen desorption from the sample surface.A recently mechanism has been predicted theoretically in which easy diffusion path such as oxygen dislocations in SrTiO 3 has impact on resistive switch applications 30 .
Within the standard deviation, our data is in good agreement with the recent value reported by Zhu et al. (1.0 eV), where a first-order kinetic model also fits their experimental data for in situ optical properties measurements 23 .
Other similar energy values can be found on literature, however they are based upon oxygen diffusion models 18 .Some of these results are based upon similar electrical measurements reported here, but the authors attribute the influence on the electrical conductivity to the oxygen diffusion 18,20 or possible easy diffusion path such as oxygen dislocations 31 instead charge doping reaction as stressed out in this report.

Conclusions
In situ electrical resistivity measurements during reduction annealing under vacuum in SrTiO 3 single crystals grown by Floating Zone Method were well described by the general solid-state kinetics approach as a first-order mechanism providing information on the oxygen vacancies/charge doping formation.Additionally, this allowed us to determine the activation energy for the formation of oxygen vacancies as 1.4±0.3eV.This result is in good agreement with the data reported on literature within the experimental deviation.

Figure 1 .
Figure 1.a) SrTiO 3 powder inside rubber mold (up) and SrTiO 3 feed rod after compressed and sintered (down).b) Single crystal as seed is connected to the Floating Zone furnace holder (up) and SrTiO 3 feed rood prepared for crystal growth process (down).In the pictures (c) to (f) one can see: c) feed rod (up) and seed (down) before feed rod melting; d) feed rod tip melted at hot zone; e) tip melted joined on the single-crystalline seed; and f) both feed rod and seed are pulling down with crystal growing below the hot zone.In (g) in shown a typical SrTiO 3 single crystal rod after the crystal growth.2410 source meter and Keithley Dmm-2000 multimeter controlled by NI-GPIB interface.Samples were excited with direct current of 10 µA in which characteristic I-V curves are in ohmic regime.

Figure 3 .
Figure 3. XRD of the single crystal surface with: a) [100] preferential orientation and b) [110] preferential orientation.The insets show the analyzed samples.

Figure 4 .
Figure 4. a) Evolution of the electrical resistivity of a SrTiO 3 single crystal as a function of time during annealing under vacuum of 10 -5 Torr.In (b) is shown the temperature during the annealing.The Inset displays the acquired data at 850 °C setting the initial time just after the plateau mentioned in the text.

Figure 5 .
Figure 5. Equilibrium degree evolution calculated by equation (1) at isothermal temperatures of 650ºC (a) and 850ºC (b) under vacuum of 10 -5 Torr.The initial time was set up at the starting point where the annealing temperature becomes stable, such as observed in Figure 4 (a).The insets show the linear model − ln (1 − α) = kt due to the first-order kinetics for the oxygen removal process.The lines are the fitted models.