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Ab Initio Thermodynamics

Leandro Liborio

Computational Materials Science Group

MSSC2008

Ab Initio Modelling in Solid State Chemistry

(001)

Sr

Ti

O

Double layer model

Castell’s

model

Sr-adatom

model

c(4x2) surface reconstruction

Experimental Motivation

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)

The systems’ total energy can be linked to the Gibbs free energy, from Thermodynamics

G can be used to study the properties of the nanosystem

The nanosystems are assumed to be in equilibrium.

Ab Initio Thermodynamics

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

T

T const.

PO2

reference chemical potential

Variation with temperature

Variation with pressure

Gibbs Free Energy: Gas Phase

T > 298 K and PO2< 2 atm

Gibbs Free Energy: Gas Phase

NIST-JANAF Thermochemical Tables, Fourth edition Journal of Physical and Chemical Reference Data, Monograph 9 (1998)

P.J.Linstrom. http://webbook.nist.gov/chemistry/guide

Gibbs Free Energy: Gas Phase

Gibbs Free Energy: Gas Phase

- Experimental errors
- Neglect of the thermal contributions to the Gibbs free energies of solids.
- DFT exchange and correlation approximations
- Presence of pseudopotentials (depends on the code)

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)

T

T const.

PO2

Gibbs Free Energy: Gas Phase

CASTEP, SIESTA: GGA and LDA functionals.

Method 2: Calculating the oxygen molecule’s properties from ab initio:

(4) W. Li et al., PRB, Vol. 65, pp. 075407-075419, 2002.

Equate the derivatives from the analytical and polynomial expressions

Get

Gibbs Free Energy: Gas Phase

Ab initio calculations of bulk metals and oxides

Polynomial fitting of experimental data

Equating the analytical results with the polynomial fitting

Ab initio calculations of the oxygen molecule

Gibbs Free Energy: Gas Phase

Method 1: Using experimental Gibbs formation energies

Method 2: Calculating the oxygen molecule’s properties from ab initio:

Gibbs Free Energy: Solid Phase

Helmholtz vibrational energy

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.

(001)

Sr

Ti

O

Double layer model

Castell’s

model

Sr-adatom

model

c(4x2) surface reconstruction

M. Castell, Surface Science, 1-13505 (2002)

Gibbs Free Energy: Solid Phase

- 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 .

with

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

Gibbs Free Energy: Solid Phase

(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

(1) K. Refson et al, Phys. Rev. B, 73, 155114, (2006).

C(4x2) reconstruction

Gibbs Free Energy: Solid Phase

T=1000 K

c(4x2) reconstructions

Gibbs Free Energy: Solid Phase

Vibrational Helmholtz free energies contribution

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).

Ab initio calculation

Calculated, approximated, considered negligible

Ab initio calculations of bulk metals and oxides

Polynomial fitting of experimental data

Equating the analytical results with the polynomial fitting

Ab initio calculations of the oxygen molecule

Gibbs Free Energy: Summary

Solid phases

Gas phases

(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

Figure 1b

Figure 1a

TnO2n-1 composition, .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.

View along the a lattice parameter

View of Hexagonal oxygen arrangement

Rutile unit cell

View of Hexagonal oxygen network

Magneli Phases: T4O7 crystalline structure

Figure 3c

Figure 3b

Figure 3a

Figure 3e

Figure 3d

Metal nets in antiphase. (121)r Cristallographic shear plane.

Technical details of the calculations

CASTEP

CRYSTAL

Local density functional: LDA

Ultrasoft pseudopotentials replacing core electrons

Plane waves code

Supercell approach

Hybrid density functional: B3LYP,

GGA Exchange

GGA Correlation

20% Exact Exchange

All electron code. No pseudopotentials

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.

Final state

Initial state

T, pO2

T, pO2

TiO2-x or TinO2n-1

TiO2 bulk

+

nDef oxygen atoms

Phonon contribution

pV contribution

Defect Formation Energies

Figure 5a

Limits for the oxygen chemical potential:

Hard limit

Soft limit

Formation Energies: Oxygen chemical potential

CASTEP

CRYSTAL

Relationship between pO2 and T in the phase equilibrium.

Results for the Magneli phases

Equilibrium point Ti4O7-TiO9:

Equilibrium point Ti3O5-Ti4O7:

Experiment Figure 10c

CRYSTAL Figure 10b

CASTEP Figure 10a

Results for the Magneli phases

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.

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