First Principles Thermodynamics in Nanomaterials: Applications to Surfaces L. Liborio

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

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

3. First principle: W Q gas gas P0, T0, V0, U0 Pf, Tf, Vf, Uf Second principle: Thermodynamics Review 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. U is also known as a Characteristic Thermodynamical function.

4. 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 G can be used to study the stability of the nanosystem 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. DFT allow for the calculation of the total energy of a nanosystem First Principles Thermodynamics

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

6. Sr Ti O (1x1)-TiO2 terminated surface (1x1)-SrO terminated surface Strontium Titanate (001) • 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) 177-191. [2] M.Castell, Surf. Sci. 505 (2002) 1-13. [3] N. Erdman et al, J. Am. Chem. Soc. 125 (2003) 10050-10056.

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. (1x1)TiO2-terminated O= 0 Thermodynamics of Surface Reconstructions SrOTiO2 O2 O2 O2 Surface excesses: Components of the system: SrO, TiO2,O (2x1)Ti2O3-terminated O= -1/2

11. Gibbs free energy definition: Thermodynamics of Surface Reconstructions

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

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

14. ~1200K (1x1) Θ=1 (2x1) Θ=0.5 ~1500K c(4x2) Θ=0.25 Results: Kubo and Nozoye Coverage Θ As we increase the temperature,  tends to decrease (not monotonically) as the surface goes through a sequence of reconstructions. UHV=5x10-12 atm T. Kubo and H. Nozoye, Surface Science 542 (2003) 177-191

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-L230. 2005

16. ~1200K 1 2 4 3 ~1500K Results: Kubo and Nozoye Equilibrium with SrO 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.