Defect structure in zirconia-based materials

1. Defect structure in zirconia-based materials JanKuriplach Department of Low Temperature Physics Faculty of Mathematics and Physics Charles University, Prague, Czech Republic

2. Cooperation • J. Čížek, O. Melikhova, I. Procházka • W. Anwand, G. Brauer (Dresden)

3. Outline • Introduction to positron annihilation • Positron calculations • Zirconia (ZrO2) • Conclusions

4. Positron annihilation • A positron is the antiparticle of an electron: i.e. mass = me , charge = +e , spin = ½ , rest energy (mec2) = 511 keV. • In solids positrons reside in the interstitial region due to their repulsion from ionic cores. • When a positron and an electron meet, they can annihilate producing two -quanta. • Positrons can be effectively used for defect studies.

5. Many defects represent a potential well for positrons. • Positrons can get trapped in this well. e+ • PA properties are defect specific.  e- e+ e+  Positron annihilation Positrons are self-seeking probes of various defects:

6. Positron annihilation • Perfect vs imperfect Ni3Al lattice Ni3Al bulk  = 104 ps Al vacancy  = 174 ps Al+Ni vacancy  = 193 ps GB (210)  = 126 ps GB+Al vacancy  = 191 ps

7. Positron annihilation • Positrons sensitive to: • open volume defects • vacancies and their agglomerates (+loops) • dislocations (+loops) • grain boundaries • stacking faults (+tetrahedra) • some precipitates • neutral and negatively charged defects • Positrons insensitive to: • isolated impurities • interstitials • some precipitates • positively charged defects

8. Positron annihilation • How to mesure positron lifetime?

9. coincidence Positron annihilation • How to measure momentum distribution of e-p pairs?

10. Positron annihilation • Comparison with other techniques: After Howell et al., Appl. Surf. Sci. 116 (1997) 7 STM=scanning tunneling microscopy, AFM=atomic force microscopy, NS=neutron scattering, OM=optical microscopy, TEM=transmission electron microscopy

11. Positron calculations • Density functional theory (DFT) = theoretical background for electronic structure calculations. • A many-electron system is considered as a collection of many one-electron subsystems. • Basic DFT statement: the ground-state energy of a system composed of electrons (and nuclei) is the functional of the electron density (and nuclear positions). (Hohenberg and Kohn, Phys. Rev. (1964))

12. Positron calculations • Mathematical formulation: Vext is the (electrostatic) potential of nuclei, T is the kinetic energy functional, Exc is the exchange-correlation functional.

13. Positron calculations • Kohn-Sham equations (Kohn and Sham, Phys. Rev. (1965)) for one-electron ‘wave functions’ and ‘energies’: where are the electrostatic potential generated by electrons and the electron density, respectively.

14. Positron calculations • Practical calculations: • Start from a reasonable electronic density (e.g. the superposition of atomic densities). • Generate the Coulomb potential () and exchange-correlation potential (Vxc) using this density. • Solve the Kohn-Sham equations. • Find the Fermi level and determine the new electron density. • Mix the new and old electron densities; go to point 2 until self-consistency is reached.

15. Positron calculations • Positrons are included from now on  two component DFT (B. Chakraborty, Phys. Rev. B24, 7423 (1981)). • Basic TCDFT statement: the ground-state energy of a system composed of electrons and positrons (and nuclei) is the functional of the electron and positron density (and nuclear positions).

16. Positron calculations • Mathematical formulation: Ece-p is the electron-positron correlation functional.

17. Positron calculations • One-particle equations: where

18. Positron calculations • One has to solve a system of coupled equations for electrons and positrons. • This requires knowledge of the Ece-p[n-,n+] functional (in addition to the Exc one). • The knowledge of the Ece-p functional is far from being complete.

19. Positron calculations • The TCDFT is quite time and computationally demanding. • When n+ 0 (delocalized positron state), TCDFT can be simplified considerably. • Then we obtain the zero positron density limit or the so-called ‘conventional scheme.’ • In this case, electronic structure (n-) is not influenced by the presence of positrons.

20. Positron calculations • Electronic structure is calculated first not considering positrons. • The Schrödinger equation for positrons is then simplified as follows: with

21. Positron calculations • The positron potential can be written as where VCoul is the Coulomb potential for electrons (obtained from electronic structure calculations). • Correlation energy and enhancement: Boroński and Nieminen, PRB 34 (1986) 3820; correction for incomplete positron screening: Puska et al., PRB 39 (1989) 7666. • Vxc is effectively cancelled by the positron part of  (due to self-interaction).

22. Positron calculations • Summary: • The electronic structure of a given system is determined. • The positron potential is calculated using the electron Coulomb potential and density. • The Schrödinger equation for positrons is solved (+, +). • Further positron quantities of interest are calculated.

23. Positron calculations • Positrons with intensity I+ [m-2s-1] are coming into the sample with volume V [m3]. The number N of annihilation events per unit time is • where n–is the positron density and •  is the cross section for positron annihilation.

24. Positron calculations • Cross section for 2 annihilation (QED) where (Lorentz factor) and (electron radius)

25. Positron calculations • Non-relativistic approximation • Furthermore • Annihilation rate (number of annihilation events per unit time when only one positron is in the sample) is as follows:

26. Positron calculations • So far the independent particle model (IPM) has been considered: e– and e+ interact at the moment of the annihilation only. • However, in reality positrons attract electrons: where  is the so-called enhancement factor. • Assumption:  depends on n– .

27. Positron calculations • When the system is non-homogeneous supposing • Assumption:  is the same as for homogeneous electron gas.

28. Positron calculations • Practical expressions: • Enhancement factor: • The Boroński-Nieminen parametrization (of Lantto’s results) is widely used at present.

29. Positron calculations • Practical expressions: • The annihilation rate (lifetime) approaches 2 ns-1 (500 ps) as electron density decreases [or rs increases].

30. Positron calculations • Positron binding energy to a defect: • Positron affinity:

31. Positron calculations • Momentum distribution of e-p pairs = MD of annihilation -quanta: where is the state dependent enhancement factor and

32. ZrO2 • ZrO2 (zirconia): • high melting point (2700 C) • low thermal conductivity • good oxygen-ion conductivity (higher temperatures) • high strength • enhanced fracture toughness • For applications stabilization of the tetragonal or cubic phase is necessary. • Zirconia is often stabilized by yttria (Y2O3)  yttria stabilized zirconia (YSZ)

33. ZrO2 cubic structure (CaF2) above ~1380 C tetragonal structure above ~1200 C monoclinic structure

34. ZrO2 • Zr has +4 charge state (Zr’’’’) and Y is +3 only (Y’’’), i.e. -1 with respect to the lattice •  compensation by O vacancies having +2 charge state (VO) • there are two Y’’’ per one VO • stability ranges: • > 8 mol% of Y2O3  cubic phase • > 3 mol% of Y2O3  tetragonal phase •  large amount of vacancies in YSZ materials

35. ZrO2 • Identification of defects in ZrO2 is of large importance. • Positrons may help to identify open volume defects. • Cubic YSZ single crystal (8 mol% of Y2O3)  175 ps positron lifetime. • Before we studied the following vacancy-like defects: • VZr, VO, VO-2Y, 2VO-4Y (VO’salong [111] direction) • positron trapping at VZr only (but too long lifetime)

36. ZrO2 • H is everywhere … • First NRA results indicate that we have appreciable amount of H in our sample. • Now we look at positions of H in ZrO2 lattice.

37. ZrO2 • ZrO2 (CaF2) structure: Zr, fcc (8 fold) O, sc (4 fold)

38. ZrO2 • Defect configurations: • obtained using VASP-PAW, LDA & GGA • 96 atom based supercells • total energy of studied defect configurations relaxed with respect to atomic positions • cubic structure unstable (without Y) !!

39. ZrO2 • Positron induced forces: • can be treated within the scheme developed by Makkonen et al., PRB 73 (2006) 035103 • conventional scheme for positron calculations considered • Hellman-Feynman theorem used to calculate forces • V+is constructed using atomic (Coulomb) potentials and densities and force calculation is thus very fast • such forces added to ionic forces calculated by VASP and atomic positions where total forces vanish are found • though the method is not fully self-consistent and does not use TCDFT, it is sufficient to get reliable results

40. ZrO2 • Configurations examined: • Interstitial hydrogen • Hydrogen in VZr

41. ZrO2 • Interstitial hydrogen: cubic tetragonal

42. ZrO2 • Hydrogen in VZr: tetragonal, EB = 5.8 eV cubic, EB = 2.3 eV

43. Results • Positron lifetimes and binding energies (LDA/GGA):  (ps) Eb (eV) • ZrO2 bulk 133/151 -- • VZr 220/287 2.8/2.3 • VZr+1H-c 167/195 0.9/1.1 • VZr+1H-t 171/175 1.4/1.8 H in VZr can in principle explain lifetime data !!

44. Conclusions • We have studied H positions in the zirconia lattice. • In the case of the interstitial H the lowest energy configuration is for H bound to an O atom near the center of the interstitial space (tetragonal, O-H bond formed). • As for the VZr , H prefers position close to a neighboring O atom (tetragonal, O-H bond formed). • The latter defect traps positrons and could be responsible for the observed positron lifetime.

45. Outlook • More H-related configurations needed. • The question of the cubic YSZ structure needs to be solved.

46. Thank you !