Transcript Slide 1
Ab Initio Thermodynamics
Leandro Liborio
Computational Materials Science Group
MSSC2008
Ab Initio Modelling in Solid State
Chemistry
Experimental Motivation
(001)
Double layer
model
Sr
Ti
Castell’s
O
model
c(4x2) surface reconstruction
Sr-adatom
model
A great variety of surface reconstructions have been observed, namely: (2x1),
c(4x2) [1][2][3], (2x2), c(4x4), (4x4) [1][2], c(2x2), (√5x√5),(√13x√13) [1].
And several structural models have been proposed.
Under which circumstances are any of these models representing the observed
surface reconstructions?
[1] T.Kubo and H.Nozoye, Surf. Sci. 542 (2003) 177-191.
[2] M.Castell, Surf. Sci. 505 (2002) 1-13.
[3] N. Erdman et al, J. Am. Chem. Soc. 125 (2003) 10050-10056.
General Idea and Considerations
• DFT provides the 0K Total energy: E({RI}).
• Classical thermodynamics studies real systems.
• The systems are assumed to be in equilibrium. For the nanosystems
considered here –surfaces and defective systems- this approximation is
good enough.
• We want to calculate appropriate thermodynamic potentials: F, G, U, etc.
• Ab initio thermodynamics might have a different “flavour” depending on
the first principles code we are using: CRYSTAL, CASTEP, SIESTA,
VASP, etc.
Ab initio atomistic thermodynamics and statistical mechanics of surface properties and functions. K. Reuter, C.
Stampfl and M. Scheffler, in: Handbook of Materials Modeling Vol. 1, (Ed.) S. Yip, Springer (Berlin, 2005).
( http://www.fhi-berlin.mpg.de/th/paper.html)
General Idea and Considerations
Helmholtz free energy: F=U-TS, independent variables (T,V)
Enthalpy: H=U+PV, independent variables (S,P)
Gibbs Free Energy: G=U-TS+PV, independent variables (T,P)
If, for a given P and T, G(T,P) is a minimum, then the system is
said to be in a stable equilibrium.
DFT allow for the
calculation of the
total energy of a
nanosystem
The nanosystems
are assumed to
be in equilibrium.
The systems’
total energy can
be linked to the
Gibbs free
energy, from
Thermodynamics
Ab Initio Thermodynamics
G can be used
to study the
properties of the
nanosystem
Gibbs Free Energy: Gas Phase
O ( p , T )
0
O2
T
T > 298 K and PO2< 2 atm
O ( pO , T )
2
O2
p
O0 (T )
T const.
pO2
O ( pO2 , T ) O ( p , T ) kT ln 0
pO
2
0
O2
O0 O ( pO0 , T 0 )
2
kT
p
T
2
PO2
pO2
O ( pO2 , T ) (T ) kT ln 0
pO
2
0
O
reference chemical potential
0
O
Variation with temperature
Variation with pressure
Gibbs Free Energy: Gas Phase
Ti(s) O2 ( g ) TiO2 (s)
0
TiO2
(T ) (T ) 2 (T ) G
0
0
Ti
0
0
O
0
0
TiO2
0
(T )
1 0
0
0
(T ) TiO2 (T 0 ) Ti0 (T 0) GTiO
(
T
)
2
2
0
O
0
0
0
GTiO
(
T
) Experimental (JANAF)
2
0
0
0
0
0
0
Ab initio: TiO
(
T
)
E
(
0
K
)
(
T
)
E
TiO2
Ti
Ti (0K )
2
pO2
1 0
0
0
0
0
O ( pO2 , T ) ETiO2 (0 K ) ETi (0 K ) GTiO2 (T ) O (T ) kT ln 0
pO
2
2
NIST-JANAF Thermochemical Tables, Fourth edition Journal of Physical and Chemical Reference Data,
Monograph 9 (1998)
Gibbs Free Energy: Gas Phase
pO2
1 0
0
0
0
0
O ( pO2 , T ) ETiO2 (0 K ) ETi (0 K ) GTiO2 (T ) O (T ) kT ln 0
pO
2
2
1
1
1
E
O0 (T ) A(T T ln(T )) BT 2 CT 3 DT 4
F GT
2
2
2
2T
Parameter
Value
A
29,659x10-3 KJ/(mol.K)
B
6,1373x10-6 KJ/(mol.K2)
C
-1,1865x10-9 KJ/(mol.K3)
D
0,09578x10-12 KJ/(mol.K4)
E
0,2197x103 KJ.K/mol
F
-9,8614 KJ/mol
G
237,948x10-3 KJ/(mol.K)
P.J.Linstrom. http://webbook.nist.gov/chemistry/guide
Gibbs Free Energy: Gas Phase
pO2
1 0
0
0
0
0
O ( pO2 , T ) EM xOy (0 K ) xEM (0 K ) GM xOy (T ) O (T ) kT ln 0
pO
y
2
O0 (T 0 )
• Experimental errors
• Neglect of the thermal contributions to the Gibbs free
0
0
energies of solids. TiO
(T 0 ) ETiO
(0K ) Ti0 (T 0 ) ETi0 (0K )
2
2
•DFT exchange and correlation approximations
•Presence of pseudopotentials (depends on the code)
O0 (T 0 ) O0 Err
pO2
O ( pO2 , T ) ( Err) (T ) kT ln 0
pO
2
0
O
0
O
Ab initio atomistic thermodynamics of the (001) surface of SrTiO3. L. Liborio, PhD Thesis.
(http://www.ch.ic.ac.uk/harrison/Group/Liborio/Docs/liborio-phdthesis.pdf)
Gibbs Free Energy: Gas Phase
0
0
TiO
(T 0 ) ETiO
(0K ) Ti0 (T 0 ) ETi0 (0K )
2
2
CASTEP, SIESTA: GGA
and LDA functionals.
Method 2: Calculating the oxygen molecule’s properties from ab initio:
O ( pO0 , T )
2
T
O ( pO , T )
2
Exp.
PW-GGA
(4)
CRYSTAL
(B3LYP)
Binding energy [eV]
2.56
3.6
2.53
Bond length [ang]
1.21
1.22
1.23
O2
p
O0 (T )
kT
p
T
T const.
O0
pO2
O ( pO2 , T ) O ( p , T ) kT ln 0
pO
2
0
O2
PO2
(4) W. Li et al., PRB, Vol. 65, pp. 075407-075419, 2002.
Gibbs Free Energy: Gas Phase
pO2
O ( pO2 , T ) (T ) kT ln 0
pO
2
0
O
O2
T
T
p
0
O
H
with H E0 CpT
2
T
PO2
T 5k
T
O2 ( pO2 , T ) E 0 ( E0 ) 0 T ln 0 kT ln 0
PO
T
2
T
2
0
O2
O2 ( pO2 , T )
T
T T 0
p pO2
Equate the derivatives from the
analytical and polynomial expressions
Get
O0
2
Gibbs Free Energy: Gas Phase
Method 1: Using experimental Gibbs formation energies
pO2
O ( pO2 , T ) ( Err) (T ) kT ln 0
pO
2
0
O
Ab initio calculations of bulk metals and oxides
0
O
Polynomial fitting of experimental data
Method 2: Calculating the oxygen molecule’s properties from ab initio:
PO2
T 5k
T
O2 ( pO2 , T ) E 0 ( E0 ) 0 T ln 0 kT ln 0
PO
T
2
T
2
0
O2
Ab initio calculations of the oxygen molecule
Equating the analytical results with the polynomial fitting
Gibbs Free Energy: Solid Phase
G E TS pV
G( p, T ) E(0K ) E vib (T ) T (S vib (T ) S config ) pV
F vib (T ) E vib (T ) TS vib (T )
Helmholtz vibrational energy
G(T , p) E(0K ) TS config F vib (T ) pV
E(0K): Total ab initio energy.
Sconfig: Configurational entropy.
pV: Related with the systems’ volume,
(0.005 J/m2 in the SrTiO3 surfaces.)
Fvib(T): Helmholtz vibrational energy.
The quantities of interest to us, namely surface energies and defect formation
energies, depend on differences of Gibbs free energies.
Gibbs Free Energy: Solid Phase
G(T , p) E(0K ) TS config F vib (T )
•E(0K): total energy of the system calculated ab initio. This is the dominant term and the difficulty
in calculating it depends essentially on the type of system and the chosen ab initio code.
•Sconfig=0 The system configuration is known .
(001)
Double layer
model
Sr
Ti
Castell’s
O
model
c(4x2) surface reconstruction
Sr-adatom
model
M. Castell, Surface Science, 1-13 505 (2002)
Gibbs Free Energy: Solid Phase
G(T , p) E(0K ) F vib (T )
F (T ) F (T , ) g ()d
vib
vib
with
F (T , )
k BT ln1 exp
2
k BT
vib
Lattice Dynamics of Bulk
Rutile
Calculated using the
implementation of Density
Functional Perturbation
Theory in the CASTEP code
(1). The agreement with
experimental results is
excellent (2).
Rutile unit cell
(1) K. Refson et al, Phys. Rev. B, 73, 155114, (2006).
(2) J. G. Taylor et al, Phys. Rev. B, 3, 3457, (1971).
(3) N. Ashcroft and D. Mermin, Solid State Physics, (1976).
Gibbs Free Energy: Solid Phase
G(T , p) E(0K ) F vib (T )
F vib(T ) F vib (T , ) g ()d
(1) K. Refson et al, Phys. Rev. B, 73, 155114, (2006).
Gibbs Free Energy: Solid Phase
G(T , p) E(0K ) F vib (T )
C(4x2) reconstruction
Fx
x
Fy
K
x
Fz
x
Fx
y
Fy
y
Fz
y
Fx
z
Fy
Fi (u ) Fi (u h) Fi (u h)
with
z
u
2h
Fz
z
j
O
j
j
i
1 exp
FOvib
(
T
,
)
k
T
ln
O
B
i
i
k BT
2
Oj i
F (T , )
vib
j
Oi
N
3
i 1
j 1
Oj i
k BT ln1 exp
k BT
2
Oj i
Oj as eigenvalues.
i
Gibbs Free Energy: Solid Phase
G(T , p) E(0K ) F vib (T )
T=1000 K
Vibrational Helmholtz free energies contribution
Compound
Vib. Cont. J/m2
PdO (1)
0.2
RuO2 (2)
0.15
αAl2O3 (3)
0.18
NiO[100] (4)
0.2
SrTiO3[001]
0.2
c(4x2) reconstructions
The quantities of interest to us, namely surface energies and defect formation
energies, depend on differences of Gibbs free energies.
(1) J. Rogal et al, PRB, 69, 075421, (2004).
(2) K. Reuter et al, PRB, 68, 045407, (2003).
(3) A. Marmier et al, J. Eur. Cer. Soc, 23, 2729, (2003).
(4) M.B. Taylor et al, PRB, 59, 6742, (1999).
Gibbs Free Energy: Summary
Solid phases
G(T , p) E(0K ) F vib (T )
Ab initio calculation
Calculated, approximated, considered negligible
Gas phases
pO2
O ( pO2 , T ) ( Err) (T ) kT ln 0
pO
2
0
O
Ab initio calculations of bulk metals and oxides
0
O
Polynomial fitting of experimental data
PO2
T 5k
T
O2 ( pO2 , T ) E 0 ( E0 ) 0 T ln 0 kT ln 0
PO
T
2
T
2
0
O2
Ab initio calculations of the oxygen molecule
Equating the analytical results with the polynomial fitting
Magneli Phases
Figure 1a
Figure 1b
TnO2n-1 composition, 4 n 9 .Oxygen defects in {121} planes.
Ti4O7 at T<154K insulator with 0.29eV band gap(1).
T4O7 Metal-insulator transition at 154K, with sharp decrease of
the magnetic susceptibility.
(1) K. Kobayashi et al., Europhysics Lett., Vol. 59, pp. 868-874, 2002.
( 2) W. Masayuki et al., J. of Luminiscence, Vol. 122-123, pp. 393-395, 2007.
(3) P. Waldner and G. Eriksson, Calphad Vol. 23, No. 2, pp. 189-218, 1999.
Magneli Phases: T4O7 crystalline structure
Figure 3a
Rutile unit cell
Figure 3d
View of Hexagonal
oxygen network
Figure 3b
Figure 3c
View along the a
lattice parameter
View of Hexagonal
oxygen arrangement
Figure 3e
Metal nets in antiphase. (121)r
Cristallographic shear plane.
Technical details of the calculations
CASTEP
Local density functional: LDA
CRYSTAL
Ultrasoft pseudopotentials replacing core
electrons
Hybrid density functional: B3LYP,
GGA Exchange
GGA Correlation
20% Exact Exchange
Plane waves code
All electron code. No pseudopotentials
Supercell approach
Local basis functions: atom centred Gaussian
type functions.
Ti: 27 atomic orbitals, O: 18 atomic orbitals
Supercell approach
•SCARF cluster. Facility provided by STFC’s e-Science facility.
•HPCx, UK’s national high-performance computing service.
Defect Formation Energies
Figure 5a
G Def
(T , pO2 )
f
Initial state
T, pO2
TiO2 bulk
nTiO2
1
nTiO2
G Def
(T , pO2 )
f
Phonon contribution
pV contribution
Final state
T, pO2
TiO2-x or TinO2n-
nDef oxygen atoms
sup cell
n
bulk
TiO
(T , pO )
TiO2
1
nTiO2
1
nTiO2
1
nTiO2
(T , pO2 ) nODef Oref (T , pO2 )
2
E
F
sup cell
2
sup cell
Vib
G Def
(T , pO2 )
f
1
nTiO2
(1)
bulk
(0 K ) nTiO2 ETiO
(0 K )
2
bulk
(T ) nTiO2 FVib
TiO2 (T )
Supcell
bulk
pO2 VVib
nTiO2VTiO
2
nODef ref
O (T , pO2 )
nTiO2 2
1
+
G
1
E
sup cell
(2)
bulk
(0 K ) nTiO2 ETiO
(0 K )
2
nODef ref
O (T , pO2 )
nTiO2
(3)
Formation Energies: Oxygen chemical potential
G
Def
f
(T , pO2 )
1
nTiO2
E
sup cell
(0 K ) nTiO2 E
bulk
TiO2
nODef
(0 K )
O (T , pO2 )
nTiO2
CASTEP
pO2
O ( pO2 , T ) ( Err) (T ) kT ln 0
pO
2
0
O
0
O
CRYSTAL
PO2
T 5k
T
O2 ( pO2 , T ) E 0 ( E0 ) 0 T ln 0 kT ln 0
PO
T
2
T
2
0
O2
Limits for the oxygen chemical potential:
E0
G Rutile
(0 K )
f
2
Hard limit
O ( pO2 , T ) E0 (4)
Soft limit
(3)
Results for the Magneli phases
Figure 8a
Isolated defects
G
Def
f isolat .
nODef
nTiO
2
nODef sup cell
bulk
(T , pO2 )
E
(0 K ) nTiO2 ETiO
(0 K ) Oref (T , pO2 )
2
nTiO2
1
T4O7 4
GDef
Ti
like Ti 4O7
O
(T , pO2 )
1 Supcell
bulk
ref
E
(0K ) 27ETiO
(
0
K
)
(T , pO2 )
O
2
4
Magneli phases
Figure 8b
G
Def
f
nODef
nTiO
2
(T , pO2 )
1
nTiO2
E
sup cell
(0 K ) nTiO2 E
bulk
TiO2
nODef ref
(0 K )
O (T , pO2 )
nTiO2
1
T4O7 4
GTiDef4O7 (T , pO2 )
1 Ti 4O7
1 ref
bulk
E
(0 K ) 8ETiO
(
0
K
)
O (T , pO2 )
2
16
4
Results for the Magneli phases
Equilibrium point Ti4O7-TiO9:
GTiDef4O7 (O ) GTiDef5O9 (O ) Tieq4O7 Ti5O9
Equilibrium point Ti3O5-Ti4O7:
GTiDef4O7 (O ) GTiDef3O5 (O ) Tieq4O7 Ti3O5
T
0
4 7
5 9
2
T
PO2
5k
T
T ln 0 kT ln 0
PO
2
T
2
Tieq O Ti O E 0 ( O0 E0 )
GTiDef3O5 ( O ) KTi 3O5
GTiDef4O7 ( O ) KTi 4O7
GTiDef5O9 ( O ) KTi 5O9
O
3
O
4
O
5
Relationship between pO2
and T in the phase
equilibrium.
Results for the Magneli phases
Experiment
Figure 10c
CRYSTAL
Figure 10b
CASTEP
Figure 10a
Log(pO2)
T(K)
Exper.
-15
1538.5
CASTEP
-15
1515.2
CRYSTAL
-15
1379.3
P. Waldner and G. Eriksson, Calphad Vol. 23, No. 2, pp. 189-218, 1999.
Conclusions
• Ab initio thermodynamics uses DFT to estimate Gibbs free energies.
• Ab inito thermodynamics allows general thermodynamic reasoning with
nanosystems and it can be implemented using different ab initio codes.
• It can be used to simulate systems under real environmental conditions.
• For the Magneli phases, ab initio thermodynamics reproduce the experimental
observations reasonably well.
• The equilibrium experimental (P,T) diagrams were reproduced from first
principles.
•At a high concentration of oxygen defects and low oxygen chemical potential,
oxygen defects prefer to form Magneli phases.
•But, at low concentration of oxygen defects and low oxygen chemical potential,
titanium interstitials proved to be the stable point defects.
Acknowledgements
Prof. Nic Harrison
Dr. Giuseppe Mallia
Dr. Barbara Montanari
Dr. Keith Refson