defect structure in zirconia-based materials

Seminar, Department of Condensed Matter TheorySeminar, Department of Condensed Matter Theory,, Institute of PhysicsInstitute of Physics, CSAS,, CSAS, March 23,March 23, 20 201010

Defect structure in zirconia-based materials

JanJan KuriplachKuriplach

Department of Low Temperature PhysicsFaculty of Mathematics and Physics

Charles University, Prague, Czech Republic

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CooperationCooperation

J. Čížek, O. Melikhova, I. Procházka

W. Anwand, G. Brauer (Dresden)

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OutlineOutline

Introduction to positron annihilation

Positron calculations

Zirconia (ZrO2)

Conclusions

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Positron annihilationPositron 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.

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Positrons are self-seeking probes of various defects:

e+

Positrons can get trapped in this well.

e+e-

PA properties are defect specific.

e+

Many defects represent a potential well for positrons.

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Ni3Al bulk = 104 ps

Al vacancy = 174 ps

Al+Ni vacancy = 193 ps

GB (210) = 126 ps

GB+Al vacancy = 191 ps

Perfect vs imperfect Ni3Al lattice

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Positrons sensitive to: open volume defects

• vacancies and their agglomerates (+loops)

• dislocations (+loops)• grain boundaries• stacking faults (+tetrahedra)

some precipitatesneutral and negatively charged defects

Positrons insensitive to:isolated impuritiesinterstitialssome precipitatespositively charged defects

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How to mesure positron lifetime?

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How to measure momentum distribution of e-p pairs?

coincidence

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Comparison with other techniques:

STM=scanning tunneling microscopy, AFM=atomic force microscopy, NS=neutron scattering, OM=optical microscopy, TEM=transmission electron microscopy

After Howell et al., Appl. Surf. Sci. 116 (1997) 7

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Positron calculationsPositron 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))

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Mathematical formulation:

Vext is the (electrostatic) potential of nuclei,T is the kinetic energy functional,Exc is the exchange-correlation functional.

][|'|

)'()('

2

1

][][

)()(][][

nEnn

dd

nTnF

nVdnFnE

xc

ext

rr

rrrr

rrr

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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.

)()()()()(

][)(

2

1rrrr

rr

ext

xc Vn

nE

F

nn

d

2|)(|2)(and|'-|

)'(')( rr

rr

rrr

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Practical calculations:

1. Start from a reasonable electronic density (e.g. the superposition of atomic densities).

2. Generate the Coulomb potential () and exchange-correlation potential (Vxc) using this density.

3. Solve the Kohn-Sham equations.

4. Find the Fermi level and determine the new electron density.

5. Mix the new and old electron densities; go to point 2 until self-consistency is reached.

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Positrons are included from now on two component DFT (B. Chakraborty, Phys. Rev. B 24, 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).

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Mathematical formulation:

Ece-p is the electron-positron correlation

functional.

],[|'-|

)'()('

2

1

)]()()[(

][][],[

nnE

nndd

nnVd

nFnFnnE

pec

ext

rr

rrrr

rrrr

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One-particle equations:

where

)()()(

],[)()(

)(

][)(

2

1rr

rrr

rr

n

nnEV

n

nE pec

extxc

)()()(

],[)()(

)(

][)(

2

1rr

rrr

rr

n

nnEV

n

nE pec

extxc

|'-|

)'()'(')(

rr

rrrr

nn

d

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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.

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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.

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Electronic structure is calculated first not considering positrons.The Schrödinger equation for positrons is then simplified as follows:

with

.0for)(

],[))((

nn

nnEnV

pec

corr rr

)()())(()()()(2

1rrrrrr nVV corrext

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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).

))(()()( rrr nVVV corrCoul

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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.

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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.

VnIN

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Cross section for 2 annihilation (QED)

where (Lorentz factor)

and (electron radius)

1

31ln

1

14

1 2

22

220

2g

ggg

g

gg

g

r

22 /1/1 cg v

m103 150

r

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Non-relativistic approximation

Furthermore

Annihilation rate (number of annihilation events per unit time when only one positron is in the sample) is as follows:

1/ cv

vc

r 202

IPMIPM

IPM ncrVnnVnN

/1

)1( 202

v

v nI

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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– .

)(/1 20 nncr

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When the system is non-homogeneous

supposing

Assumption: is the same as for homogeneous electron gas.

rrrr dnnncr )())(()(/1 20

.1)( rr dn

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Practical expressions: Enhancement factor:

The Boroński-Nieminen parametrization (of Lantto’s results) is widely used at present.

)6/3286.026.1

8295.023.11(32/52

2/320

sss

ssBN

rrr

rrncr

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Practical expressions:

The annihilation rate (lifetime) approaches 2 ns-1 (500 ps) as electron density decreases [or rs increases].

PsoPsp 4

3

4

1

13

4 3 nrs

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Positron binding energy to a defect:

Positron affinity:

)defect()bulk( EEEb

.)(

ΦΦ

EEA F

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Momentum distribution of e-p pairs = MD of annihilation -quanta:

where

is the state dependent enhancement factor and

2

,2 )()().exp()(

Fj

di jj

rrrrpp

jIPM

jENHj

,

,

.)())((|)(| 2,

20, rrrr dnncr XjjX

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ZrOZrO22

ZrO2 (zirconia):high melting point (2700 C)low thermal conductivitygood oxygen-ion conductivity (higher temperatures)high strengthenhanced fracture toughness

For applications stabilization of the tetragonal or cubic phase is necessary.Zirconia is often stabilized by yttria (Y2O3) yttria stabilized zirconia (YSZ)

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ZrOZrO22

cubic structure (CaF2)above ~1380 C

tetragonal structureabove ~1200 C

monoclinic structure

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ZrOZrO22

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

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ZrOZrO22

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’s along [111] direction)

positron trapping at VZr only (but too long lifetime)

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ZrOZrO22

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.

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ZrOZrO22

ZrO2 (CaF2) structure:

Zr, fcc (8 fold)

O, sc (4 fold)

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ZrOZrO22

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) !!

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ZrOZrO22

Positron induced forces:can be treated within the scheme developed by Makkonen et al., PRB 73 (2006) 035103 conventional scheme for positron calculations consideredHellman-Feynman theorem used to calculate forces

V+ is constructed using atomic (Coulomb) potentials and densities and force calculation is thus very fastsuch 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

|}){,(| ijj V RrF

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ZrOZrO22

Configurations examined:

Interstitial hydrogen

Hydrogen in VZr

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ZrOZrO22

Interstitial hydrogen:

z

y

x

-0.03[1,-1,1]

-0.03[1,1,1]

-0.03[1,-1,-1]

0.01[1,0,0]

0.01[0,0,1]

0.01[-1,0,0]

0.01[0,0,-1]

0.01[0,1,0]

-0.03[-1,-1,-1]

-0.03[-1,1,-1]

-0.03[-1,1,1]

-0.03[1,-1,1]

0.01[0,-1,0]

z

y

x

-0.23[1,-1,1]

0.22[-1,1,1]

-0.05[-1,2,2]

-0.05[2,2,-1]

-0.14[-1,-1,2]

-0.05[2,-1,2]

-0.11[-1,-1,-1]

0.22[1,-1,1]

-0.23[-1,1,1]

0.22[-1,1,-1]

-0.14[-1,2,-1]

-0.53[1,1,1]

0.68[-1,-1,-1]

-0.14[2,-1,-1]

O

Zr

H-0.05[-1,1,1]

-0.05[-1,1,-1]

-0.05[-1,-1,-1]

-0.05[1,-1,-1]

-0.05[1,1,-1]

-0.05[1,-1,1]

-0.05[1,1,1]

-0.05[-1,-1,1]

0.06[0,-1,-1]

0.06[-1,0,-1]

0.06[-1,-1,0]

z

y

x

cubic tetragonal

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ZrOZrO22

Hydrogen in VZr:

cubic, EB = 2.3 eV tetragonal, EB = 5.8 eV

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ResultsResults

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 !!

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ConclusionsConclusions

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.

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OutlookOutlook

More H-related configurations needed.

The question of the cubic YSZ structure needs to be solved.

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TT hh aa nn k k yy oo uu ! !

defect structure in zirconia-based materials

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