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Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped

Struktura elektronowa i dynamika sieciczystego i zdefektowanego CoO

Urszula D. Wdowik, Krzysztof Parlinski

Wydzial Matematyczno-Fizyczno-Techniczny,Uniwersytet Pedagogiczny, Krakow

Seminarium WFiIS 3/04/2009


Outline

1 Introduction

structure, electronic and magnetic properties of CoO2 Methodology

3 Results for defect-free CoO

4 Phonons (direct method)

5 Cobalt-deficient CoO

electronic structurelattice dynamics

6 Fe-doped CoO

lattice dynamicscharge and spin states of Fe in CoO and vacancy-defectedCoOelectronic structure


Structure, electronic and magnetic properties of CoO

CoO structure

High temperature structure

Paramagnetic (Fm3m)

aexp = 4.26 Å

Low temperature structure

distorted NaCl structure(R3m) below TN = 293 K

rhombohedral distortionalong <111>, 0.2 − 0.3

2nd kind ofantiferromagnetic ordering,AFII structure


Structure, electronic and magnetic properties of CoO

Electronic and magnetic properties of CoO

Experiment

Insulator, Eg = 2.5 − 2.8 eV (experiment)

Mtotal = 3.8 − 3.98µB

Theory (DFT)

Charge-transfer insulator

Strong on-site Coulomb interactions in 3d shell cannot beneglected (strongly correlated electron system)

DFT+U formalism required

Interactions between correlated states - Hubbard potentialU and the exchange interaction J


Calculations with +U formalism

Pseudopotential plane-wave method - VASP code

spin-polarized DFT for AFII structure (64-atom supercell)

Projector-augmented wave pseudopotentials (PAWs)

Exchange-interaction: GGA+U

Full relaxations of AFII structure

U = ?, J = ?


Energy gap, spin magnetic moment, lattice constant vs U

4.20

4.22

4.24

4.26

4.28

4.30

a (1

0-10 m

)a(exp.) = 4.26x10-10 m

2.30

2.50

2.70

2.90

M (

µ B)

M(exp.) = 3.98 µB

0.000.501.001.502.002.503.00

1 2 3 4 5 6 7 8 9

Eg

(eV

)

U (eV)

Eg(exp.) = 2.5-2.8 eV

Resultsa - weak dependence upon U

MS changes by ca. 17%

Underestimation of Eg for U < 5 eV

Eg vanishes for U = 1 eV (GGA limit)


Defect-free CoO. Results for U = 7.1 eV, J = 1 eV

-40

-20

0

20

40

DO

S (

stat

es/e

V)

Co t2gCo eg

-20 0

20

-8 -6 -4 -2 0 2 4 6

Energy (eV)

O 2p

U = 7.1 eV, J = 1 eV

a = 4.27ÅRhombohedral distortionalong <111> (0.3o)

space group D53d (R3m)

Eg = 2.77 eV

Ms = 2.74µB


Method to calculate phonons

Harmonic approximationDirect method - PHONON software by K. Parlinski

displace crystallographically nonequivalent atoms from theirequilibrium positionscalculate Hellmann-Feynman forces F(n, µ) = −δE

δR(n,µ)

Fi(n, µ) = −∑

m,ν,j Φij(n, µ, m, ν)Uj(m, ν)solve dynamical matrix D(k)ω

2(k, j)e(k, j) = D(k)e(k, j)⇓

ω(k, j) - phonon frequenciese(k, j) - polarization vectors of phonons


Phonon dispersion relations for U = 0 eV and U = 3 eV

Ueff = 0 eV

-2

0

2

4

6

8

10

12

14

Fre

quen

cy (

TH

z)

Γ X K Γ L W X

[ξ00] [ξξ0] [ξξξ ]

0 0.1 0.2

-2

0

2

4

6

8

10

12

14

DOS (1/THz)

Ueff = 2 eV

-4

-2

0

2

4

6

8

10

12

14

16

Fre

quen

cy (

TH

z)

Γ X K Γ L W X

[ξ00] [ξξ0] [ξξξ ]

0 0.2 0.4-4

-2

0

2

4

6

8

10

12

14

16

DOS (1/THz)

imaginary frequencies of acoustic modes ⇒ instability of such CoOstructure

significantly underestimated HF forces ⇒ artificial mode softening

small Ueff ⇒ too low repulsion in 3d shell


Dispersion relations and phonon DOS for U = 7 eV

U = 7.1 eV, J = 1 eV

0

2

4

6

8

10

12

14

16

18

Fre

quen

cy (

TH

z)

Γ X K Γ L W X

[ξ00] [ξξ0] [ξξξ ]

0 0.15 0.30

2

4

6

8

10

12

14

16

18

DOS (1/THz)

Data for LO-TO splittingD(k) =D0(k) + DN(ǫ∞, Z∗

, . . . )

high-frequency dielectricconstantǫ∞ = 5.3[1]

Born effective charges

|Z ∗| = 2.06

Experimental data from neutron scattering (T = 110 K)

[1] J. Sakurai, W.J.L. Buyers, R.A. Cowley, and G. Dolling, Phys. Rev. 167, 510 (1968)


Comparison with experiment

Experiment vs theory

Γ-point frequencies Neutron Infrared Theory(THz) scattering[1] spectroscopy[2]

ωTO 10.50 10.40-10.50 10.25ωLO 15.75 16.30-16.40 15.73

Experimental data

[1] J. Sakurai, W.J.L. Buyers, R.A. Cowley, and G. Dolling, Phys. Rev. 167, 510 (1968)[2] J.P. Gielisse, J.N. Plendl, L.C. Mansur, R. Marshall, S.S. Mitra, R. Mikolajewicz, andA. Smakula, J. Appl. Phys. 36 2426 (1965)


Heat capacity and Debye-Waller factors

Heat capacity

0

10

20

30

40

50

60

70

0 100 200 300 400 500 600 700 800 900 1000

Hea

t cap

acity

(J/

mol

K)

Temperature (K)

Lattice contribution

magnetic contribution below TN

Dulong-Petit law @ HT

Debye-Waller factors

0.000

0.003

0.006

0.009

0.012

0.015

0.018

0 100 200 300 400 500 600 700 800 900 1000

Mea

n-sq

uare

d di

spla

cem

ents

(10

-20 m

2 )

Temperature (K)

Cobalt (γ-ray)Cobalt (x-ray)

Oxygen (γ-ray)Oxygen (x-ray)

57Fe in CoO

calculated ΘD = 500 K

Mossbauer ⇒ ΘD = 440 K

[3] E.G. King nad A.U. Christensen, U.S. Bur.Mines. Tech. Paper 80, 1800 (1956); E.G. King, ibid. 80, 2399 (1956)[4] W. Jauch and M. Reehuis, PRB 65, 125111 (2002)

[5] S. Sasaki F. Fujino, and K. Takeuchi, Proc. Jpn. Acad. B 55, 43 (1979)[6] K.Ruebenbauer and U.D. Wdowik, J. Phys. Chem. Solids 65, 1785 (2004)


Electronic structure of CoO defected by cationic vacancies

Is CoO a stoichiometric material?

Native defects in CoO - cationic vacanciesCo deficiency in ’almost stoichiometric’ samples: 0.1-3%

uncharged, singly and doubly charged vacancies

nonstoichiometry depends on temperature and oxygenpartial pressure

How to simulate point defects (vacancies, impurities)

supercell approach

remove atoms ⇒ vacancies

replace host atoms by different kind of atoms ⇒ impurities



Modeling of non-stoichiometric CoO

Co0.97O

Co @ (12 ,

12 ,

12) removed

from 64-atom SC⇓CoO with 3% vacancies

Co0.94O

Co @ (12 ,

12 ,

12) and (1

2 ,12 , 0)

removed from 64-atom SC⇓CoO with 6% vacancies

Atomic relaxation in defected systems

oxygens surrounding vacancies displace outward vacancysite by 0.12 Å

initial Co-O distance (2.15 Å) changes by ca. 0.01 Å



Trivalent cations in CoO

Pure CoOMS = 2.74µB

valence charges qCo = +1.33e, qO = −1.33e

Co2+-O 2.15 Å

Co0.97O

MS = 3.15µB

qCo = +1.65e

Co3+ − O 2.05 Å

2 Co3+ in SC

Co0.94O

MS = 3.16µB

qCo = +1.68e

Co3+ − O 2.05 Å

4 Co3+ in SC



Location of trivalent cations

Co0.97O

-2.0

-1.0

0.0

1.0

2.0

2 3 4 5 6 7 8

Val

ence

cha

rge

(e)

(a)

← Co3+ Co3+ →

-1.0

0.0

1.0

2.0

3.0

4.0

2 3 4 5 6 7 8

Mag

netic

mom

ent (

µ B)

R (Angstrom)

(b)

← Co3+ Co3+ →

Co3+ distance from vacancy

2.96 Å 7.41 Å

Co0.94O

-2.0

-1.0

0.0

1.0

2.0

2 3 4 5 6 7 8

Val

ence

cha

rge

(e)

(a)

Co3+ → ← Co3+

-1.0

0.0

1.0

2.0

3.0

4.0

2 3 4 5 6 7 8

Mag

netic

mom

ent (

µ B)

R (Angstrom)

(b)

Co3+ → ← Co3+

Co3+ distance from vacancy

4.29 Å 6.04 Å



Electronic structure of vacancy-defected CoO

Co0.97O

-40

-20

0

20

40

DO

S (

stat

es/e

V)

Co t2gCo eg

-20

0

20

-8 -6 -4 -2 0 2 4 6

Energy (eV)

O 2p

Co0.94O

-40

-20

0

20

40

DO

S (

stat

es/e

V)

Co t2gCo eg

-20

0

20

-10 -8 -6 -4 -2 0 2 4 6

Energy (eV)

O 2p

charge transfer from Co to O (via s,p states)

Co2+ ⇒ Co3+ + e−

2 holes associated with vacancy are compensated by 2 Co3+

acceptor states in Co0.97O

1 eV above VBM

acceptor states in Co0.94O

1.5 eV above VBM


Lattice dynamics of cobalt-deficient CoO

Description of phonons within SC approach

CoO

primitive UCN = 2 ⇒ 6 branches

CoO as 64 atom SC

NS = 64 ⇒ 192 branches ?

Co0.97O

SC - already primitive UCNS = 63 ⇒ 189 branches ?

dimensions of D(k) increases to 3NS

number of phonon dispersion curves increases to 3NS

size of BZ conjugated with SC shrinks

selection of a different kind of unit cell⇓number of phonon dispersion curves is blown up.



Phonon form factor and filter

Phonon form factor

F (p)(k, j) = 1k2

∣

∣

∣

∣

∑

µ

k·e(k,j ;µ)√Mµ

∣

∣

∣

∣

2

∫

Ω dΩ F (p)(k, j) = 13 F (s)(k, j)

Filter

F (s)(k, j) =

∣

∣

∣

∣

∑

µ

e(k,j ;µ)√Mµ

∣

∣

∣

∣

2

Fake phonon modes

F (p)(k, j) = 0



Application of filter to stoichiometric CoO

Intensities of modes for CoO

0

2

4

6

8

10

12

14

16

18

Fre

quen

cy (

TH

z)

X K Γ L W X Γ

Intensity (%)

72.5 - 100.0

50.0 - 72.5

27.5 - 50.0

20.0 - 27.5

12.5 - 20.0

5.0 - 12.5

experimental dataclose to branches withintensities > 30%



Application of filter to nonstoichiometric CoO

Co0.97O

0

2

4

6

8

10

12

14

16

18

Fre

quen

cy (

TH

z)

X K Γ L W X Γ 0 0.1 0.20

2

4

6

8

10

12

14

16

18

DOS (1/THz)

Co0.94O

0

2

4

6

8

10

12

14

16

18

Fre

quen

cy (

TH

z)

X K Γ L W X Γ 0 0.1 0.20

2

4

6

8

10

12

14

16

18

DOS (1/THz)

perturbed phonons with wavelength ∼ size of region disturbed by vacancy

missing Co ⇒ additional O vibrations (new modes)

acoustic branches at small k not affected by vacancies⇓

long wavelength phonons insensitive to point defects



Oxygens surrounding vacancies

Partial DOS

0.000

0.005

0.010

0.015

0.020

0.025

0.030

2 4 6 8 10 12 14 16 18

pDO

S (

1/T

Hz)

Frequency (THz)

CoO

CoO with 3% vacancies

CoO with 6% vacancies

vacancies affect thehighest frequency LOmodes

no change inlow-frequency acousticregion

Γ-point frequencies

frequency CoO Co0.97O Co0.94O(THz)ωTO 10.25 9.81 9.61ωLO 15.73 15.87 16.08



Mean-squared displacements vs T

MSD for Co and O

0.000

0.003

0.006

0.009

0.012

0.015

0.018

Mea

n-sq

uare

d di

spla

cem

ents

(10

-10 m

)

CationsCoO

Co0.97O

Co0.94O

γ-ray (Co)

x-ray (Co)

0.000

0.003

0.006

0.009

0.012

0.015

0.018

0 200 400 600 800 1000

Temperature (K)

AnionsCoO

Co0.97O

Co0.94O

γ-ray (O)

x-ray (O)

Co and O at low T5% larger MSD than instoichiometric CoO

negligible differencebetween Co0.97O andCo0.94O

Co and O above RT

Co OCo0.97O 13% 19%Co0.94O 22% 23%


Lattice dynamics of Fe-doped CoO

Dispersion curves.UFe = UCo = 7.1 eV

0

2

4

6

8

10

12

14

16

18

Fre

quen

cy (

TH

z)

X K Γ L W X Γ 0 0.1 0.20

2

4

6

8

10

12

14

16

18

DOS (1/THz)

localized modes

mass defect negligible

change in on-site forceconstant at Fe site

at Co site Φ = 10.013 (eV/Å2)const. vs UFe

Force constant at Fe site

UFe (eV) 5.1 6.1 7.1Φ (eV/Å2) 12.264 12.787 13.313


Dynamics of Fe impurity

Partial phonon DOSUFe = UCo = 7.1 eV

0.000

0.002

0.004

0.006

DO

S (

1/T

Hz)

Fe

Co

0.000

0.010

0.020

0.030

0 2 4 6 8 10 12 14 16 18

Frequency (THz)

Oxygens surrounding Fe

Oxygens surrounding Co

splitting of ωTO

modes corresponding to:oxygens vibrating aroundCo

ωTO = 10.51 THzconst. vs UFe

oxygens vibrating around Fe

oxygens neighboring Fe

UFe (eV) 5.1 6.1 7.1

ωTO (THz) 9.67 9.68 9.72


Debye-Waller factors in Fe-doped CoO

Vibrational amplitudes

0.000

0.003

0.006

0.009

0.012

0.015

0.018

Mea

n-sq

uare

d di

spla

cem

ents

(10

-20 m

2 ) Cations FeCo in CoO

Co in (Fe)CoO57Fe

γ-ray (Co)x-ray (Co)

0.000

0.003

0.006

0.009

0.012

0.015

0.018

0 200 400 600 800 1000

Temperature (K)

Anions O in CoOO in (Fe)CoO

γ-ray (O)x-ray (O)

MSDFe < MSDCo < MSDO

5% increase in MSDFe withdecreasing UFe

Low T (10 K)

MSDCo > MSDFe by 7%mass + force constant defect

MSDCo , MSDO in Co(Fe)O are4% higher than respective MSDsin CoO(const. vs UFe)

Above RT ( UFe = 7.1 eV)

MSDFe slopecalc. = exp. Co(57Fe)O

Co(Fe)O Co(57Fe)O CoO

ΘD (K) 440 440 500


Electronic structure of Co(Fe)O and Co(Fe,V)O

Valence and spin states of Fe in CoO and CoO0.97O

Experiment - M ossbauer spectroscopy

2 singlets @ RTferrous line Fe2+

ferric line Fe3+

vicinity of TN = 293 Kmagnetically split component due to Fe2+

T = 78 K magnetically split Fe2+ and Fe3+

G. K. Wertheim, Phys. Rev. 124, 764 (1961)

MethodFP-(L)APW+lo (WIEN2k code)Full-potential(linearized) augmented plane-wave plus local orbitals⇓

Isomer shift and EFG on Fe in Co(Fe)O and Co(Fe,V)O



Charge and spin states

different Fe-V complexes in CoO considered

negligible changes vs UFe

α = −0.291a.u.3mms−1

Q = +0.17 bU.D. Wdowik and K. Ruebenbauer, PRB 76, 155118 (2007)

Results

Co(Fe)O Co0.97(Fe)O Exp.[6]MCo (µB) 2.74 2.74 Co2+

3.13 Co3+

MFe (µB) 3.67 4.27IS (mm/s) -1.031 -0.371 -1.1245(8) Fe2+

-0.37(5) Fe3+

EFG (mm/s) 0.27 0.28 00.3-0.4 mm/s linewidth

η 0.03 0.07 0



Site preferences

Fe-V complex in CoO

-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5

∆E (

meV

/ato

m)

d (10-10 m)

Trivalent and divalent impurities invacancy-defected CoO

-55.0

-45.0

-35.0

-25.0

-15.0

-5.0

5.0

15.0

25.0

35.0

45.0

55.0

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

∆E (

meV

/ato

m)

Cation radius (10-10 m)

Al3+ Ga3+ Fe3+ Mg2+ In3+

First neighbor site

Second neighbor site

magnetic impurities

Fe3+ in 1st n.s.

Co3+ in 2nd n.s.exception: Fe3+ in 2nd n.s.

non-magnetic impurities

’small’ trivalent cations in 2nd n.s.

’large’ trivalent cations in 1st n.s.

divalent cations in 2nd n.s.



Density od states for Co(Fe)O and Co0.97(Fe)O

DOS of Fe-doped CoO

-60

-40

-20

0

20

40

60

-6 -4 -2 0 2 4 6

DO

S (

stat

es/e

V)

Energy (eV)

CoFeO

DOS of Fe-doped Co0.97O

-120

-80

-40

0

40

80

-6 -4 -2 0 2 4 6

DO

S (

stat

es/e

V)

Energy (eV)

CoFeO

Co3+ and Fe3+ states inside gap


Summary and conclusions

1 DFT+U can predict correct ground states of CoO2 Trivalent Co are found in vacancy-defected CoO3 Filter allows to present phonon-dispersion curves to be more close to

the dispersion relations of a real defected sample4 Point defects influence mainly optical phonon region. Long-wavelength

acoustic phonons are practically not affected by defects, i.e.,smallconcentration of defects does not disturb those crystal properties whichare due to the low-frequency acoustic phonons

5 Average MSDs of ions increase with increasing vacancy concentration(decreased intensity of scattered radiation)

6 Differences in the vibrational dynamics of a dopant and host atomsarise from the dirrefence in their force constants

7 Divalent Fe is found in the host CoO lattice undisturbed by cobaltvacancies

8 Trivalent Fe is found in the host CoO lattice decorated with cobaltvacancies

9 Details can be found in PRB 75, 104306 (2007); PRB 77, 115110 (2008); PRB 78, 224114;

J.Phys.:Condens. Matter 21, 125601 (2009)

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