1. Computational Materials Science Group
First Principles Thermodynamics in Nanomaterials: Applications to Surfaces
L. Liborio
Computational Materials Science Group

2. DFT Review ETot=Ts+Eee+Ene +Enn+Tn
Write the electronic density in terms of a set of non-interacting orbitals:
kinetic energy
nuclei potential
electrostatic interaction.
exchange and correlation
If Exc[] were known, the exact ground state could be found.

3. Thermodynamics Review
Pf, Tf, Vf, Uf
gas Q W
First principle:
P0, T0, V0, U0
gas
Examples of processes:
a) dU=0 (Complete cycle)
b) dU=0 (W=-Q, steady state)
c) dU=W (Q=0, thermal insulation)
Natural and Reversible processes
Reversible process, closed phase, no chemical reactions, absorbs Q and performs W.
Second principle:
U is also known as a Characteristic Thermodynamical function.

4. Thermodynamics Review
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.
This energy can be linked to the internal energy, U, from Thermodynamics
U can be used to define the Gibbs free energy, G, of the nanosystem
DFT allow for the calculation of the total energy of a nanosystem
G can be used to study the stability of the nanosystem
First Principles Thermodynamics

5. Nanosystems Unit cell Lattice param. Defective bulk Surface Metals
Crystalline structures: atoms are arranged in a periodic spatial arrangement
Metals
Ceramics
Oxides
Unit cell
Lattice param.
Defective bulk
Surface
Atomic Scale surface reconstructions in a Ceramic: Strontium Titanate (SrTiO3).
Neutral oxygen defects in an Oxide: Titanium Dioxide (TiO2) in the rutile structure.

6. (1x1)-TiO2 terminated surface (1x1)-SrO terminated surface
Strontium Titanate
(1x1)-TiO2 terminated surface
(001) Ti O Sr
(1x1)-SrO terminated surface
Substrate for superconducting thin films.
Buffer material for the growth of Ga As on Si.

7. Overview of the problem M. Castell in Surface Science 505 (2002) 1-13
Double layer model
Castell’s
model
Sr-adatom
model
c(4x2) surface reconstruction

8. Overview of the problem
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, among which are the ones presented in the previous slide.
Under which circumstances are any of these models representing the observed surface reconstructions? Are any of these in equilibrium?
[1] T.Kubo and H.Nozoye, Surf. Sci. 542 (2003)
[2] M.Castell, Surf. Sci. 505 (2002) 1-13.
[3] N. Erdman et al, J. Am. Chem. Soc. 125 (2003)

9. Calculation Technique
Simulations within DFT theory using LDA approximation (T=0K)
Core electrons replaced by Troullier-Martin pseudopotentials
Calculations were carried out using the SIESTA program
Static calculations to predict equilibrium states (minimun energy)
Geometry:
Reconstructions using SrTiO3 bulk lattice constant
7-layer slabs separated by 3 layers of vacuum
3 outermost layers fully relaxed

10. Thermodynamics of Surface Reconstructions
SrO TiO2 O2 O2 O2
Surface excesses:
(1x1)TiO2-terminated O= 0
Components of the system:
SrO, TiO2,O
(2x1)Ti2O3-terminated
O= -1/2

11. Thermodynamics of Surface Reconstructions
Gibbs free energy definition:

12. Thermodynamics of Surface Reconstructions
Oxygen Gibbs free energy
We used 12 oxides: SrO, TiO2, MgO, SiO2, Al2O3, CaO, PbO2, CdO, SnO2, Cu2O, Ag2O, ZnO
Experimental Value

13. Thermodynamics of Surface Reconstructions
Calculated from first principles
First principles + analytical expression
The dependence of the surface energy with p and T comes through the gas phase.

14. Results: Kubo and Nozoye
Coverage Θ
(1x1) Θ=1
As we increase the temperature,  tends to decrease (not monotonically) as the surface goes through a sequence of reconstructions.
(2x1) Θ=0.5
UHV=5x10-12 atm
c(4x2) Θ=0.25
T. Kubo and H. Nozoye, Surface Science 542 (2003)

15. Results: Kubo and Nozoye
0: TiO2-terminated (11) =0,
1: (1313) =0.0769,
2: c(44) =0.125,
3: (55) =0.20,
4: (22) =0.25 .
L. Liborio, et al. J. Phys.: Condensed Matter 17. L223-L

16. Results: Kubo and Nozoye
Equilibrium with SrO
~1200K
~1500K 4 2 3 1
0: TiO2-terminated (11) =0,
1: (1313) =0.0769, 2: c(44) =0.125,
3: (55) =0.20, 4: (22) =0.25 .

17. Conclusions
We have calculated the surface energy of the Sr adatom structures. These structures were proposed by Kubo and Nozoye to explain a set of structural phase transitions on the SrTiO3 (001) surface. The different surface structures were observed using an STM.
Only the surface with coverage =0.20 is stable for the ranges of temperature and pressure reported by Kubo and Nozoye. Our calculations show that the lower Sr coverages implied in the Sr adatom model can only be explained if the surface is far from equilibrium, in a transient state as it loses Sr to the enviroment.