Ab Initio Atomistic Thermodynamics

Transcription

1 Ab Initio Atomistic Thermodynamics (Elizabeth C. Beret) Luca M. Ghiringhelli Fritz-Haber-Institut Berlin

2 Extending the scale Thermodynamics: p, T, V, N Length (m) continuum 0 e or m -3 Potential Energy Surface: {Ri} 0-6 (3N+)-dimensional 0 E -9 average over all processes many atoms Mesoscopic regime few atoms many processes Microscopic regime few processes 0-5 {Ri} Macroscopic regime ils a t de e or m pr es s es c o Time (s) Essentials of computational chemistry: theories and models. nd edition. C. J. Cramer, JohnWiley and Sons Ltd (West Sussex, 004). 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., (Ed.) S. Yip, Springer (Berlin, 005).

3 Equilibrium concentration of point defects On entropic grounds there will always be a finite concentration of defects at finite temperature, even though the creation of a defect costs energy (Edef > 0). S conf =k B ln Z n =k B ln N!/ [ N N def! N def! ] (assumes no interaction between defects) In equilibrium: def dg d perf def conf = G N ΔG TS pv ] =0 [ def dn def dn def N def N def exp E /k B T def exp ΔG /k B T N sites N def defects ( << N )

4 Outline Ab initio atomistic thermodynamics: General concepts Oxide formation on a metal surface Adsorption of O and CO on a metal oxide Formation of defects in semiconductors

5 Outline Ab initio atomistic thermodynamics: General concepts Oxide formation on a metal surface Adsorption of O and CO on a metal oxide Formation of defects in semiconductors

6 Ab initio atomistic thermodynamics Consider several structures for our system PES Microscopic Which ones are preferred as a function of (T,p)? Extend the length scale Free energy Macroscopic A surface cannot be separated from a gas (or liquid) above it p ν= πmkt For T = 300 K, p = atm => ν ~ 08 site- s- Requires p 0- atm to keep a clean surface clean; surface can also lose atoms

7 (DFT) internal versus free energy At constant T a system minimizes its free energy, not internal energy U If also volume V is constant, the energy minimized is Helmholtz free energy F F = U TS If (T,p) are constant, the energy minimized is Gibbs free energy G G = U + pv TS = µi N i i Chemical potential µ i of the i-th atom type is the change in free energy as the number of atoms of that type in the system increases by one In thermodynamic equilibrium, µ i is the same in the whole system (surface, bulk, gas) Statistics plays a crucial role due to a macroscopically large number of particles in the system

8 Computation of free energies: ideal gas Q= N q N! pv = NkT q=qtrans q rot q vib qel q conf qnucl m k T trans Translational: q = h 3 V Need: particle mass m Use the ideal gas law to relate V and p rot q = Rotational: 8 I A k T h rot kt q =8 h Rigid rotor Linear molecules 3 I A I BIC Non-linear molecules Need: moments of inertia IA, IB, IC

9 Computation of free energies: ideal gas Q= N q N! pv = NkT q=qtrans q rot q vib qel q conf qnucl [ vib ] Vibrational: q = exp h i h i exp kt kt Need: vibrational modes Harmonic oscillator Electronic: q el = v i e Ei kt v0 e v i=si Assuming that the first excited state is energetically unaccessible Conformational : q conf Diatomic molecules: σ= (heteroatomic) = E0 kt Need: total (DFT) energy of the ground state and its degeneration Need: symmetry number (N. equivalent orientations) Polyatomic molecules: σ = N. symmetry operations according to the symmetry point group

10 Computation of free energies: ideal gas pv = NkT From the partition function Q: Helmholtz free energy: F= kt ln Q Canonical ensemble (NVT) Gibbs free energy: G= kt ln Q pv Isothermal-isobaric ensemble (NpT) Chemical potential: F = N G = N V,T p, T =kt ln N kt ln q (NVT) =kt [ ln N ln q ] (NpT) Statistical Mechanics, D. A. McQuarrie, University Science Books, 000 Statistical Thermodynamics, R. Fowler and E. A. Guggenheim, Cambridge University Press, 949

11 Computation of free energies: solids G T, p =E tot F trans F rot F vib F conf pv E tot Total electronic energy F trans Translational free energy F rot Rotational free energy F vib Vibrational free energy DFT Discarded (consider fixed solids) F vib = d F vib T, F vib T, [ h h kt ln exp kt F conf Conformational free energy Needs more advanced methods (Ex: cluster expansion) pv Expansion term Usually small, often neglected G T, p E tot F vib ]

12 Outline Ab initio atomistic thermodynamics: General concepts Oxide formation on a metal surface Adsorption of O and CO on a metal oxide Formation of defects in semiconductors

13 Oxide formation on Pd(00) Heterogeneous catalysis: O, R O, P T, p p(o) Which oxide forms at given experimental conditions (T,p)? Reuter and Scheffler, Appl. Phys. A, 78, , 004 J. Rogal and K. Reuter, Ab initio atomistic thermodynamics for surfaces: A primer. In: Experiment, Modeling and Simulation of Gas-Surface Interactions for Reactive Flows in Hypersonic Flights. Educational Notes RTO-EN-AVT-4, Neuilly-sur-Seine (007)

14 Oxide formation on Pd(00) Pd(00) + x O Pd (O)x O gas G = G Pd O G Pd 00 x O A ad x G T, p E total F vib E total F vib ±5 mev/å up to T = 600 K surface O T, p = O T, p Pd O T, p =EODFT EOZPE O T, p DFT ZPE O T, p = EO EO O T, p G ad [ E Pd O E Pd 00 x EODFT EOZPE O T, p A x ]

15 Oxide formation on Pd(00) G ad x Clean Pd(00) surface O/Pd(00) adlayers: p(x): 0.5 ML c(x): 0.50 ML PdO(0)/Pd(00) surface oxide: ( 5x 5)R7 : 0.80 ML Bulk PdO -00 G ad [ E Pd O E Pd 00 x EODFT EOZPE O T, p A ]

16 Relation between chemical potential and (T, p) DFT ZPE O T, p =EO EO O T, p From the O partition function: T, p = kt ln EZPE [ m h 3 kt h kt ln e 5 ] 8 I A kt kt ln p kt ln h h kt E DFT kt ln v 0 Δ μ (T, p) From thermochemical tables: T, p = T, p o T o, po kt ln p/ p o Tables Example: JANAF Thermochemical tables, D.R. Stull, H. Prophet. US National Bureau of Standards, Washington DC, 97 Sº (T), Hº (T)

17 Relation between chemical potential and (T, p) DFT ZPE O T, p =EO EO O T, p From thermochemical tables: T, p = T, p o T o, po kt ln p/ p o Tables Example: JANAF Thermochemical tables, D.R. Stull, H. Prophet. US National Bureau of Standards, Washington DC, 97 Sº (T), Hº (T)

18 Oxide formation on Pd(00) [ E Pd O E Pd 00 x EODFT EOZPE O T, p A x Clean Pd(00) surface O/Pd(00) adlayers: p(x): 0.5 ML c(x): 0.50 ML PdO(0)/Pd(00) surface oxide: ( 5x 5)R7 : 0.80 ML Bulk PdO 600 K 300 K G ad G ad ]

19 Limitations 600 K 300 K Number of screened structures: structures not considered cannot be predicted Neglecting Fvib: (slight) horizontal shift of lines Neglecting Fconf: smearing of phase transitions Systems in equilibrium: possible kinetic effects in experiment G ad

20 Comparing with experiment: kinetic effect in-situ SXRD Theory metal E. Lundgren et al., Phys. Rev. Lett. 9, 0460 (004) ) ) p( p( 5 )R 7 ( 5 5)R7 ( 5 bu lk ox id e bulk oxide metal

21 Outline Ab initio atomistic thermodynamics: General concepts Oxide formation on a metal surface Adsorption of O and CO on a metal oxide Formation of defects in semiconductors

22 Adsorption of O and CO on RuO(0) CO oxidation over a Ru catalyst: O CO CO RuO What are the preferred structures for RuO (O)x (CO)y? This reaction does not take place in the gas phase: Singlet + Triplet Singlet E= -3.7 ev (DFT-PBE+vdW) Spin forbidden! K. Reuter and M. Scheffler, Phys. Rev. Lett. 90, (003); Phys. Rev. B 68, (003)

23 Constrained equilibrium X μo (T, p) μco(t, p) G(T, p) Etot RuO(0) + x O + y CO RuO (O)x (CO)y G = [G RuO O CO G RuO 0 x O y CO] A ad x G [ E RuO O A y ad x DFT ZPE E O E O O CO E RuO 0 x y DFT ZPE y ECO ECO CO ]

24 Obr / Gad E RuO O A [ x CO E RuO 0 x y DFT ZPE E O E O O DFT ZPE y ECO ECO CO Obr / COcus CObr / COcus Gad (mev/å) Obr / Ocus ]

25 Obr / - Obr / COcus ΔμCO (ev) Obr / Ocus CObr / COcus ΔμO (ev)

26 Adsorption of O and CO on gold clusters

27 Outline Ab initio atomistic thermodynamics: General concepts Oxide formation on a metal surface Adsorption of O and CO on a metal oxide Formation of defects in semiconductors

28 Electronic structure of solids Metal Insulator Semiconductor E Conduction Band EG (Band gap) EF (Fermi level) Conduction Band EF Valence Band EF EG Valence Band The electrical conductivity of semiconductors can be enhanced by the presence of defects (vacancies, interstitials, impurities...) which can contribute: electrons to the CB: donors, n-type conductivity holes to the VB: acceptors, p-type conductivity The presence of defects can affect the position of the Fermi level

29 Defects and material properties Defects can transform insulator into a semiconductor or a metal (doping) Defects can determine optical properties (color) At surfaces, defects have unique chemical properties Defects can have different number of electrons associated with them (charge), and each charge state can have very different chemical properties Measuring concentration of defects, especially at temperatures and pressures relevant for practical applications, is very difficult Mg O

30 Defect thermodynamics in GaAs Ga: sp As: sp3 Perfect crystal Zincblende structure: Each Ga binds 4 As Each As binds 4 Ga in a tetrahedral environment a) Ga vacancy Ga: As: b) Ga interstitial Ga me- 5e- remain in the vacancy Acceptor: m 0 m= m= m= m= V0 VVV3- Ga Ga can donate up to 3eDonor: n 0 ne- n= n= n= n= I0 I+ I+ I+3

31 Defect thermodynamics in GaAs a) Ga vacancy b) Ga interstitial m 0 Ga V Ga m e n 0 Ga I Ga n e G= Ga G V m G0 G=G I n G 0 Ga Need to know: G0, G V, G I,, Ga G 0 E 0 GV E V G I E I =E F E VB Ga From DFT calculations (use the same cell size) Discard vibrational contributions (very small here) Energy of Fermi level, wrt the maximum of the valence band Varies from EVB to ECB Depends on the environmental conditions: define upper and lower limiting values

32 Limiting values for μga μga and μas are related: Ga l As g GaAs s Ga As = GaAs = 6.8 ev Ga-rich (As-poor) conditions: formation of Ga droplets Ga g Ga l Ga.8 ev G =.8 ev As 4.0 ev As-rich (Ga-poor) conditions: formation of As gas As.98 ev Ga GaAs As g Ga 4.8 ev G =.98 ev Energies taken from: R. Hultgren et al, Selected values of the Thermodynamic Properties of the Elements, American Society for Metals, Ohio (973). R. C. Weast, Handbook of Chemistry and Physics. 60th edition, CRC Press (980).

33 Defect thermodynamics in GaAs a) Ga vacancy b) Ga interstitial m 0 Ga V Ga m e n 0 Ga I Ga n e G= Ga G V m G0 G=G I n G 0 Ga Need to know: G0, G V, G I,, Ga G 0 E 0 GV E V G I E I =E F E VB Ga From DFT calculations (use the same cell size) Discard vibrational contributions (very small here) Energy of Fermi level, wrt the maximum of the valence band Varies from EVB to ECB Depends on the environmental conditions: 4.8 ev Ga.80 ev define upper and lower limiting values

34 Defect thermodynamics in GaAs a) Ga vacancy b) Ga interstitial G= Ga G V m G0 G=G I n G 0 Ga As-rich limit Ga = 4.8 ev Ga-rich limit Ga =.80 ev ΔG (ev) n 0 Ga I Ga n e ΔG (ev) m 0 Ga V Ga m e 0 V-Band μ (ev) High formation energy Reasonable good acceptor.5 C-Band 0 V-Band μ (ev) Energetically favored Good donor.5 C-Band

35 Defect thermodynamics in GaAs (m+n)e- ΔG (ev) c) (Independent) Ga vacancy and Ga interstitial m 0 V Ga I nga m n e ΔG (ev) VGa and IGa have opposite charges: Partial compensation of effects Possibly annihilate ΔG (ev) G=GV G I m n G0 0 V-Band μ (ev).5 C-Band

36 Summary Ab initio atomistic thermodynamics provides a connection between the microscopic and macroscopic regimes With ab initio atomistic thermodynamics one can predict the preferred structure of a material as a function of environmental conditions the regions of enhanced catalytic activity Limitations: Only actually sampled structures enter the free energy contest: Need for an exhaustive structural sampling Equilibrium assumption: possible kinetic hindrance in experiment Acknowledgements Sergey Levchenko

37 Appendix: configurational entropy and phase transitions γ (mev/å ) clean surface p(x ) No lateral interactions: Fconf = kbt ln (N+n)! / (N!n!) 0.5 < Θ (O) > T = 300 K T = 600 K Langmuir adsorption-isotherm <θ (Ocus)> = µ O (ev) exp((ebind - µ O)/kBT) Configurational entropy smears out phase transitions

38 Appendix: estimation of ΔFvib [ vib x vib vib F = F Pd O F Pd 00 O A vib ] vib F d F T, Pd O A vib d F T, Pd 00 vib ΔFvib (mev/å) h O x h O kt ln exp kt νo = 96 mev phase 0 mev 80 mev 40 mev gas Pd O T (K) J. Rogal and K. Reuter, Ab initio atomistic thermodynamics for surfaces: A primer. In: Experiment, Modeling and Simulation of Gas-Surface Interactions for Reactive Flows in Hypersonic Flights. Educational Notes RTO-EN-AVT-4, Neuilly-sur-Seine (007)

39 Appendix: chemical potential from tables [ ] m T, p = kt ln h 3 kt 5 h 8 I A kt kt ln p kt ln h DFT h kt ln e k T E kt ln v 0 T, p =kt ln p E DFT E ZPE f T T, po =E DFT E ZPE f T 0, p o = E DFT EZPE T, p = T, p 0, p o =kt ln p f T T, po = T, po 0, p o =f T T, p = T, p o 0, p o kt ln p Tables