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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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 = ?
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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)
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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)
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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)
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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)
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
Electronic structure of CoO defected by cationic vacancies
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
Electronic structure of CoO defected by cationic vacancies
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
Electronic structure of CoO defected by cationic vacancies
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
Electronic structure of CoO defected by cationic vacancies
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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.
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
Lattice dynamics of cobalt-deficient CoO
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
Lattice dynamics of cobalt-deficient CoO
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%
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
Lattice dynamics of cobalt-deficient CoO
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
Lattice dynamics of cobalt-deficient CoO
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
Lattice dynamics of cobalt-deficient CoO
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%
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
Electronic structure of 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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
Electronic structure of Co(Fe)O and Co(Fe,V)O
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.
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
Electronic structure of Co(Fe)O and Co(Fe,V)O
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
Introduction Method Results for undefected CoO Phonons. Harmonic approximation. Direct method Cobalt-deficient CoO Fe-doped
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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