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The role of interstitial Ti atoms and O vacancy
sites in the Ti(3d) defect states
in the band gap of Titania
Gilberto Teobaldi
Department of Physics and Astronomy
University College London
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Rutile titania faces are among the key models for metal oxide surfaces:
- “easy” to prepare
- thermodynamically stable
- extremely reactive in the presence of point-defects
- high-k
Applications in the fields of:
- biocompatible materials
- catalysis
- photocatalysis
- gas sensing
- MOSFET (gate)
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Grand canonical (µVT) formation energies[1]
(from DFT calculations)
defect
CLEAN
nO = -1
nH = -2
CLEAN
H2O
X
( )= E (q)− E
Ef X
q
X
q: charge state of the defect X
clean
− ∑ ni µi − q∆µe + ∆Fvib
CLEAN
i
ni: number of atoms (of chemical potential µi) to be added (ni<0) or removed (ni>0) to create the defect X
µi: chemical potential of atoms i.
Energy of a given atom when it is exchanged with the corresponding reservoir in order to create or remove X
∆µe: change in electron potential
∆Fvib: change in vibrational free energy due to the presence of X on the clean surface
[1] J. Appl. Phys. 95, 3851 (2004)
THERMODYNAMIC EQUILIBRIUM !!!
∑n µ
i
i
i
=0
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(Kohn-Sham) Density Functional Theory
1)
E [ρ (r )] = T [ρ (r )] + VNe [ρ (r )] + VH [ρ (r )] + E XC [ρ (r )] ground state energy E as a functional of the electronic density ρ(r)
2)
ρ (r ) = 2∑ ϕi (r ) =2∑ ρ i (r )
n
2
i
3)
4)
T [ρ (r )] = −
n
Minimize E[ρ(r)] (eq. 1) with respect to occupied orbitals
i
Solve (self-consistently) a set of
one-particle (mean-field, coupled) equations
1 n
∑ dr ϕi* (r )∇ 2ϕi (r )
2 i ∫
⎡ 1 2
⎤
⎢⎣− 2 ∇ + Vion (r ) + VH (r ) + VXC (r )⎥⎦ϕi (r ) = ε iϕi (r )
VNe [ρ (r )] = ∫ dr Vion (r ) ρ (r )
ρ
1
ρ (r )ρ (r')
dr dr'
∫∫
2
r − r'
5)
VH [ρ (r )] = J [ρ (r )] =
6)
LDA
[ρ (r )] = ∫ dr VXC (ρ ) ρ (r )
E XC
7)
GGA
E XC
[ρ (r )] = ∫ dr VXC (ρ , ∇ρ ) ρ (r )
ρ
HEG
LDA
VXC(ρ)
GGA
exact EXC[q] are not known except for the free electron gas
[1] R. G. Parr, W. Yang, Density Functional Theory of Atoms and Molecules
(Oxford University Press, New York, 1989)
[2] J. P. Perdew, A. Zunger, Phys. Rev. B 23, 5048 (1981)
ρ i (r ) = ϕi (r )
2
one electron (no Vee ) : VH [ρ i ] + VXC [ρ i ] = 0
SIE = VH [ρ i ] + VXC [ρ i ] ≠ 0
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GGA on OxHy/TiOz
• slightly (~5%) overestimation of lattice parameters
• (10-20%) underestimation of elastic constants
• ~25% underestimation of band gaps
• erroneous energy localisation in the band gap of defect states associated with point defects
• erroneous real space localisation of defect states associated with point defects
• erroneous description of charged point defects
• erroneous description of atomization energies of molecules (O2, H2, H2O)
(mind atomic chemical potential within µVT framework)
EF
[1] Surf. Sci. Rep. 48 (2003) 53-229
[2] Phys. Rev. Lett. 97, 166803 (2006)
[3] Phys. Rev. B 46, 6671-6687 (1992)
[4] J. Chem. Phys. 118, 8207-8215 (2003)
[5] J. Chem. Phys. 98, 5648 (1993)
Experiment
VB
GGA
VB
CB
CB
EF
E
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On site Coulomb interaction: L(S)DA+U[1]
The L(S)DA often fails to describe systems with localized (strongly correlated) d and f electrons
(unrealistic one-electron energies).
On site replacement of the L(S)DA by introduction of strong intra-atomic interaction (J,U)
in a (screened) HF-like manner
U: on site effective Coulomb e-e interaction parameter
J: on site effective exchange interaction parameter
n: on site (real) occupancy matrix of d (f) electrons
σ: spin component (1,2)
L(S)DA/GGA calculation[2]
CAS-(CI,S,SD→SDact)
embedded cluster calculations[3]
‘tuning’ of (U-J) on experimental data[4]
(band gap, energy position of BGS)
Energy penalization to push nσ towards idempotency: nσ = nσnσ
(eigenvalues of nσ are to be either 1 or 0)
[1] Phys. Rev. B 57, 1505 (1998)
[2] Phys. Rev. B 52, R5467 (1995)
[3] Phys. Rev. B 77, 045118 (2008)
[4] Phys. Rev. B 75, 195212 (2007)
Either fully occupied or unoccupied one-electron levels
Improved one-electron energies
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Hybrid DFT
Exc: Exchange-correlation energies. Exc = Ex + Ec
Etot = Ekin + Eps + EH + Exc
Standard DFT: LDA, GGA (local)
Non-local Fock exchange energy Ex (real space):
Ex (hybrid) = linear combination of Ex(HF) with some other Ex functional(s)
Ec(hybrid) = combination of other Ec functional(s)
a0=0.20; ax=0.72; ac=0.81
µ=0
(B3LYP) J. Chem. Phys. 98, 5648 (1993).
(PB0) J. Chem. Phys. 122, 234102 (2005)
(HSE) J. Chem. Phys. 124, 219906 (2006)
the slowly decaying long-ranged part of the Fock exchange
is replaced by the corresponding PBE counterpart
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3-fold coordinated O (as in bulk)
Rutile TiO2(110)
+ H2O(g)
2.97 Å
6.5 Å
[001]
bridging OH (OHbr)
_
[110]
O vacancy (Ovac)
2-fold coordinated bridging O (Obr)
5-fold coordinated Ti (Ti5c)
+ O2(g)
O2(g)
H2O(g) which desorbs from the surface
(300-320 K)1
O adatom (Oad)
[1] J. Phys. Chem. B 107, 534-545 (2003)
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BGS EELS-peak
O vacancy (Ovac)
Band Gap States
bridging OH (OHbr)
Band Gap States
O adatom (Oad)
NO Band Gap States
(EELS) J. Phys. Chem. B 107, 534-545 (2003)
(STS) Phys. Rev. B 66, 235401 (2002)
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• Besides O vacancies (Ovac), also interstitial Ti atoms (Tiint)
are present in rutile TiO2 samples (up to 2x1019 cm-3)1
• Negligible Tiint mobility for T< 400 K1
• The larger its formal charge (≥4), the larger the stability for Tiint2
• Very low (LSDA+U) polaron hopping barriers for rutile TiO2 (<0.3 eV)3
• High e-polaron affinity reported for O vacancy SITES in other high-k systems (HfO2)4
• (de)-localization of Tiint-donated electrons (at RT)?!?
[1] Surf. Sci. 419, 174-187 (1999)
[2] Phys. Rev. B 75, 073203 (2007)
[3] Phys. Rev. B 75, 195212 (2007)
[4] Phys. Rev. B 75, 205336 (2007)
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OHbr
Ovac
rPBE (GGA)
hybrid DFT (B3LYP)
Phys. Rev. Lett. 97, 166803 (2006)
Science 320, 1755 (2008)
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1.9 e donated from Tiint
HSE03-3x2-4L (2-2)
(ih)-Tiint
as from HVEM and HAADF-STEM
images of reduced TiO2(110)[1]
QBader(Oad_1) = -0.9
QBader(Oad_2) = -0.8
QBader(H2O2) = +0.03 QBader(Oad) = -0.8
1. Both Ob-vac and Tiint induce BGS
2. In the presence of Tiint, Oad favoured over H2O2
[1] N. Shibata et al., Science 322, 570 (2008)
3. Tiint-donated e (rather than Tiint) induce BGS
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+1.5 V, 0.2 nA
+1.5 V, 0.3 nA
A
F
E
B
C
Experimentally, NO STM footprint of Tiint for positive biases (at room temperature)
Chem. Phys. Lett. 437, 73 (2007)
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Obr
TH: +1.5V, 10-7 eÅ-3
Ti5c
x
Å
Å
clean
Ovac/Tiint(1st/2nd)
Å
Å
Oad/Tiint(2nd/3rd)
Tiint(2nd/3rd)
x
Å
Ovac/Tiint(2nd/3rd)
Å
2Oad/”Tiint(2nd/3rd)”
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Rlxd
(216)
Tiint(4-5), Ueff: 5.5 eV1 (8 tri-layers)
~13 Å
(ih)
fixed
(72)
TH: +1.5V, 10-7 eÅ-3
ρBGS: 10-5 e Å-3
Tiint(4/5) not detectable by STM for positive biases
_
[110]
[001]
[1] Phys. Rev. B 77, 045118 (2008)
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Ob-vac
Ob-vac/Tiint
Tiint
Ob-vac/Tiint (-1.85 eV)
OHb/Tiint
OHb/Tiint (-1.36 eV)
OHb
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STM: +2V
Figure S2. Modelled STM and CITS appearance for pristine surface
defects. Simulated STM images (left: +2V, 10-7 e Å-3) and LDOS
maps (right: -2 V, same height above the surface as from the
corresponding left-side topography) are shown for Ob-vac in (a) and
(b), for Ob-vac/Tiint in (c) and (d), for OHb in (e) and (f), and for Obvac/Tiint in (g) and (h). Heights are reported in Å, LDOS in 10-9 e Å-3.
Ti5c rows are indicated by black lines. (X) marks the Ob-vac site, (■)
marks the OHb site and (+) marks the lateral position of Tiint on the
scan areas.
CITS: -2V
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d 2V (z )
= −4πρ (z )
2
dz
µσ =
z0 − c / 2
∫
(
)()
dz z − z0 ρ z
z0 − c / 2
∆V = 4πµσ
∆Vvac
µ
δ+
(110)
(011)-2x1
-0.3 eV
+2.0 eV
*e-
h+
(110)
δJ. Phys.: Cond. Mat. 19, 213203 (2007).
δ(011)-2x1
h+
*e-
δ+
µ
z
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STM: +2V
(HSE03) Ob-vac/OHb e-charging
Figure 3 Calculated electronic structure of Ob-vac and
OHb. (a) Right: The integrated number of electrons in the
BGS calculated as ∆n=n(E) – n(EVBmax) where EVBmax < E <
EFermi and n(EVBmax) is the number of electrons below the
valence band maximum. Left: the total density of states for
the optimized layers.
Figure 4 STM and STS of the Ob-vac along with theoretical
simulations. (a) 10 Å × 13 Å topographic STM image at ~4.5 K (2 V,
0.03 nA) recorded simultaneously with the STS data. (b) Current
map (-2 V) recorded at the same tip-surface distance as in (a).
Simulated STM images (+2V, 10-7 e Å-3) and current maps (-2V)
are shown for Ob-vac0 in (c) and (d), for Ob-vac1- in (e) and (f), and
for Ob-vac2- in (g) and (h). Ti5c rows are indicated by black lines and
an ‘X’ marks the Ob-vac.
CITS: -2V
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µσ =
z0 − c / 2
∫
(
)()
dz z − z0 ρ z
z0 − c / 2
( )
()
E f X q = E X q − Eclean − ∑ ni µi − q∆µe + ∆Fvib
i
Fermi energy for polarized Xq system
∆µ e = E F − EVBM + ∆V
Change in electron potential
∆V = 4πµσ
Shift in electrostatic potential due to surface polarization (Xq)
Fermi energy for unpolarized bulk system (VB maximum)
[1] a) J. Appl. Phys. 95, 3851 (2004); b) Phys. Rev. B 46, 16067 (1992); c) Phys. Rev. B 51, 4014 (1995);
d) Phys. Rev. B 59, 12301-12304 (1999); e) J. Phys.: Condens. Mat. 19, 213203 (2007)
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Conclusions
• Both Tiint and Ovac induce Band Gap States (BGS)
• Both Tiint and Ovac induced BGS are scavenged by exposure to O2(g)
• Tiint-donated electron surplus is necessary to account for TiO2(110) reactivity towards O2(g)
(and other oxidants...)
• Questionable presence of Tiint in the immediate subsurface region (1st-3rd tri-layer)
• TiO2(110) reactivity governed by accumulation of Tiint-donated electron surplus
at surface traps (Ovac/OHbr)
• Suggested mechanism fully compatible with the absence of BGS for TiO2 thin-films
(high-temperature annealing of pre-adsorbed Ti layers in the presence of O2(g))
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Acknowledgments
(involved .AND. bothered)
• Werner A. Hofer
• Nikolaos S. Beglitis, Andrew J. Fisher
• Anthoula C. Papageorgiou, Chi L. Pang, Gregory Cabailh, Qiao Chen, Geoff Thornton
(bothered)
• Alex Shluger, David Muñoz Ramo
• Daniel Sanchez Portal
• Georg Kresse
EPSRC-UK (EP/C541898/1)
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Outlook
Experimental maximum Tiint concentration:
1 every 800 cells1 (4800 atoms)
finite size effects
vs
accurate description of 3d states
HSE06: 3x2-4L slab (2-2)
24 cells (144 atoms)
LSDA+U: 3x2-8L slab (2-6)
48 cells (288 atoms)
Surf. Sci. 419, 174-187 (1999)
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Linear scaling O(N) methods
NO CUTOFF ↔ NO SPARSE MATRIXES
↨
cpu: O(N3), mem: O(N2)
Phys. Stat. Sol. (b), 243, 989 (2006)
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xσ e −2 xσ / 3 2 2 / 3 ρσ5 / 3
= π
3
xσ − 2
Qσ
(
)
1
Qσ = ∇ 2 ρσ − 2γDσ , γ = 0.8
6
(
1 ∇ρσ
Dσ = tσ −
4 ρσ
2 fitted parameters: α, β
)
2
F. Tran et al., Phys. Rev. Lett. 102, 226401 (2009)
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• electronic affinity at (3d) site (?!?)
• energy localization of BGS vs CB and vacuum level (?!?)
• real space localization of BGS (?!?)
• polaron hopping barries (?!?)
F. Tran et al., Phys. Rev. Lett. 102, 226401 (2009)
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N. Marzari, D. Vanderbilt, Phys. Rev. B 56, 12847 (1997)
• MLWF exponentially localized
• ground-state energy invariant
with respect to Umn(k)
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FFT: NFFTln(NFFT)
ρij(r) i,j=1,Ne
PW implementation: Ne2 x NFFTln(NFFT)
X. Wu et al., Phys. Rev. B 79, 085102 (2009)
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Bloch orbitals -> MLWF
ϕ j (r ) < ρ cut ⎯
⎯→υij (r )ϕ j (r ) = 0
2
• reduced number of non-zero pair exchanges (vij)
• calculated on reduced (real space) grid
• independent of system size (ρcut)
Discretization of eq. 9)
finite-difference linear matrix equations: Ax=b
CD-optimization of f(x)=½xTAx - bTx + c
X. Wu et al., Phys. Rev. B 79, 085102 (2009)
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• ρcut vs electronic affinity at (3d) site (?!?)
• ρcut vs energy localization of BGS vs CB and vacuum level (?!?)
• ρcut vs real space localization of BGS (?!?)
• ρcut vs polaron hopping barries (?!?)
X. Wu et al., Phys. Rev. B 79, 085102 (2009)
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THANK YOU