Zn Vacancy Formation Energy and Diffusion Coefficient of CVT ZnO Crystals in the Sub-Surface Micron Region

By using positron annihilation spectroscopy methods, we have experimentally demonstrated the creation of isolated zinc vacancy concentrations >1020 cm−3 in chemical vapor transport (CVT)-grown ZnO bulk single crystals. X-ray diffraction ω-rocking curve (XRC) shows the good quality of ZnO single crystal with (110) orientation. The depth analysis of Auger electron spectroscopy indicates the atomic concentrations of Zn and O are almost stoichiometric and constant throughout the measurement. Boltzmann statistics are applied to calculate the zinc vacancy formation energies (Ef) of ~1.3–1.52 eV in the sub-surface micron region. We have also applied Fick’s 2nd law to calculate the zinc diffusion coefficient to be ~1.07 × 10−14 cm2/s at 1100 °C. The zinc vacancies began annealing out at 300 °C and, by heating in the air, were completely annealed out at 700 °C.

a study of the formation energy and diffusion coefficient that is more reliable relative to an approach where zinc vacancies are created by a non-thermodynamic-equilibrium process such as, an electron irradiation or bombardment using a laser exposure.

Experimental Methods
Chemical vapor-transport-grown ZnO crystals were placed in Heraeus high-purity quartz ampoules that were evacuated using roughing and turbo molecular pumps and that were then baked at ~150 °C overnight to remove residual water vapor. The vacuum pressure was ~10 −7 Torr prior to back filling with ultra-high-purity oxygen to ~300 Torr. The ampoules were then sealed using an oxy-hydrogen torch and placed in a tube furnace. The annealing process was carried out at 1100 °C or 1200 °C for 24 hours, and the ampoule remained in the furnace during cooling.
Quenching experiments were also carried out by dipping the hot ampoule (1200 °C) in water at room temperature (RT). This quenching process reduced the ZnO temperature from 1200 °C to RT in a few seconds. Annealing (i.e., annihilation) out of the zinc vacancies was performed by heating the crystals in air for one hour at either 300 °C, 500 °C or 700 °C.
Positron annihilation spectroscopy (PAS) 12 is a well-known tool to characterize negatively charged defects, such as zinc vacancies (V Zn ) in ZnO crystals 13 . Positrons are positively charged and become trapped in negatively charged native defects, which reduces their Doppler momentum. The trapped positrons eventually annihilate the surrounding electrons, emitting two photons of 511 keV energy 14 . Emitted photons with Doppler broadening are the signature of the annihilation site. Though, PAS is not very useful for investigating positively charged defects, it has been quite helpful in investigating neutral defects 15 . The signal-to-noise ratio can be increased by a significant amount by performing two-detector coincidence measurements 16 . However, the coincidence-mode process takes a much longer time to collect data. In this letter, depth-resolved positron annihilation spectroscopy (PAS) Doppler broadening measurements were performed at the Washington State University (WSU). The 511 keV annihilation peak was recorded using a liquid-nitrogen-cooled HPGe detector. The S parameter is sensitive to the annihilation fraction with low-momentum valence electrons and is proportional to the concentration of trapping centers 17 . The W parameter comprises the wings of the peak where higher momentum Doppler shifts dominate, and it relates chemical species to the annihilation site. Together, the S and W parameters were used to characterize positron-trapping centers in the ZnO crystals. Further positron experimental details and analysis are discussed elsewhere 6,18 .
X-ray diffraction (XRD) measurement was carried out using a high-resolution ATX-G, Rigaku, triple-axis diffractometer system, using Cu Kα radiation, with a scintillation counter (0-D detector).
The elemental composition along the depth direction was measured by Auger electron spectroscopy (AES). The crystal surfaces were measured with an AES, PHI 700 (ULVAC-PHI, INC) system, and the accelerating voltage of the first exciting electron beam was 5 kV.
Data Analysis X-ray diffraction (XRD). Figure 1 shows the X-ray diffraction ω-rocking curve (XRC) of the as-grown CVT ZnO single crystal. An omega scan was performed for the reflection from the (110) crystal surface. The full-width at half-maximum (FWHM) was measured to be 0.018°, indicating the good crystallinity of the CVT-grown ZnO single crystal. Electron back scattering diffraction (EBSD) measurement, also suggested a good quality of ZnO crystal, with a (110) plane orientation (not shown).

Auger electron spectroscopy (AES). AES depth analysis was done by etching a ZnO crystal with an
Ar-ion gun, and the variation of O(KLL) and Zn (LMM) intensity along the depth direction of the crystals was measured. In all the depth profiles, the main elements present in ZnO, i.e. zinc (Zn) and oxygen (O) were detected. As shown in Fig. 2 the atomic concentrations of Zn and O were found to be almost stoichiometric Defect concentration. The concentration of a point defect depends on its formation energy. In thermodynamic equilibrium and in the dilute regime (i.e., neglecting defect-defect interactions), the concentration of a point defect is given by where, E f is the formation energy, N sites is the number of sites the defect can be incorporated on, k B the Boltzmann constant, and T is the temperature. Equation (1) shows that defects with high formation energies will occur in low concentrations. Annealing CVT-grown ZnO crystals at 1200 °C created ~5 × 10 20 cm −3 defects in the top (100-150 nm) and ~1 × 10 18 cm −3 in the mid (200-700 nm) crystal region. The bulk crystal region (>3 µm) remains with the minimal zinc vacancy concentration (~10 15 cm −3 ),thereby matching the pristine state (i.e., as-grown) of the ZnO crystals 6 [Fig. 3]. Details of the calculation of the zinc vacancy (V Zn ) concentration are described elsewhere 6 .
Zinc vacancy formation energy. The zinc vacancy formation energies were calculated using the Boltzmann Equation -Eq. 1. A CVT sample annealed at 1200 °C yields E f ~ 0.64 eV in the top region and ~1.39 in the mid region. The discrepency in these formation energies can be understood since Eq. 1 is only valid in the dilute limit. Positron measurements showed that, in the top region, the SW data deviated from a straight line -suggesting a saturation of positrons with excess (V zn ) > 10 20 cm −3 for the 1200 °C oxygen-annealed sample. The zinc vacancy formation energy (E f ) is 1.38 eV (top) and 1.52 eV (mid) region for the 1100 °C annealed CVT sample. The deviation of ~0.14 eV for this sample in two regions is quite reasonable, and Eq. 1 can be assumed to be  Diffusion coefficients. The zinc diffusion coefficient at 1100 °C was calculated by using Fick's 2 nd law: where, C is the zinc vacancy concentration [ Fig. 4], erf(z) is the Gaussian error function, x is the diffusion distance, z is the approximated value of the Gaussian error function, D is the diffusion coefficient or diffusivity, and t is the diffusion time. The diffusion coefficients are summarized in the Table 2.
Quenching. Positron depth-resolved data were fit (from 4 keV onwards) using the VEPFIT 19 computer code.
The S-parameter is ~3% higher with respect to the bulk in the top layer [ Fig. 5(a)    The zinc vacancy concentration (V zn ) (atom −1 ) was calculated by using the equation: where, μ is the specific trapping rate, and [λ τ ] is the positron lifetime in the bulk semiconductor. The subscripts refer to the measured τ or S in the case of average (ave), the bulk value for (b) and the defect specific value in the case of defect. The ZnO atomic density was assumed (ρ ~ 8.3 × 10 22 cm −3 ).
In a quenched sample, S ave increased by only ~3%, i.e., a much lower value than for the slowly cooled samples -leading to a value of only (V zn ) ~ 1.15 × 10 17 cm −3 [ Table 3]. Generally, it is believed that the actual concentration of vacancies will be higher than the equilibrium value if the crystal is annealed at an elevated temperature and then cooled suddenly, thereby freezing in the vacancies. Our observation, however, was exactly the opposite since the zinc vacancy formation energy in the quenched condition [ Table 4] is somewhat higher than that compared to the slow cooling case. A precise comparison is not justified, however, since the quenching process does not follow thermodynamic equilibrium conditions. Annealing out zinc vacancies. A CVT crystal, oxygen annealed at 1100 °C, [(V zn ) ~ 1 × 10 18 cm −3 (top layer)], was air annealed at various temperature steps (300 °C, 500 °C and 700 °C) for 1 hour each and this procedure was followed by performing the positron measurements [ Fig. 6] after every air-annealing step. The positron data suggest that the (V zn ) completely anneals out (i.e., to the as-grown (virgin) level) after a 700 °C air anneal for    1 hour. The S-W data followed a straight-line trend suggesting that no zinc vacancy-related clusters were formed during the annihilation process.
A comparative study. Janoti et al. 20 , have reported a zinc vacancy formation energy of ~3.7 eV using first-principles calculations under p-type conditions (E F at the VBM), in an oxygen-rich atmosphere. Kohan et al. 21 Tables 5 and 6. Also, Lany et al. 22 , have performed first principle calculations and showed that the intrinsic defects (V O and Zn i ) can't lead to shallow donors due to a large formation energy, when E F is close to the CBM. Tomlins et al. 11 , performed zinc self-diffusion experiment in single crystal ZnO using nonradioactive 70 Zn as the tracer isotope and reported zinc vacancies formation energy >3.51 eV. Azarov et al. 23 , did zinc-diffusion measurements using isotopically modulated ZnO crystals and showed zinc vacancies formation energy varies as, is the difference in the Fermi level and conduction band energy) and depends on the position of the Fermi level. It can be seen that there exist a large discrepancies in zinc vacancies formation energy values calculated theoretically or experimentally.
Based on the positron analysis for the zinc vacancy concentration, the ZnO crystal region was divided into 3 parts: (1) top region (100-150 nm), (2) mid-region (200-700 nm) and, (3) bulk region (>3 µm). Our calculated values are based on the near-surface (~100-700 nm) experimental data (V zn ), that requires less formation energy as compared to the bulk region. The calculated (V zn ) formation energy is much lower than any previously reported values. This was observed experimentally, since there was no increase in the V zn concentration in the bulk region, which is consistent with the high formation energy V zn in the bulk region.

Conclusions
The good quality of ZnO crystal with (110) orientation was used in this study as suggested by X-ray diffraction ω-rocking curve (XRC). The depth analysis of AES indicates the atomic concentrations of Zn and O are almost stoichiometric and uniform throughout the measurement. In general, the diffusivity is greater through the less restrictive structural regions, such as the near surface (micron region), as compared to the bulk. Based on the zinc vacancy concentration and using Fick's 2 nd law, we have calculated zinc diffusion coefficient of ~1.07 × 10 −14 cm 2 /s at 1100 °C in the sub-micron region. The zinc vacancy formation energy (E f ) is calculated to be 1.38 eV (100-150 nm) and 1.52 eV (200-700 nm) region for 1100 °C oxygen anneal samples. These values are significantly lower than the reported values as obtained by first-principles calculations based on the bulk region of the ZnO crystal and few other reported experimental results.