Why thermodynamics for materials?

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2 Why thermodynamics for materials?

3 Example p 2mkT T For = 300 K, = 1 atm ~ 10 8 site -1 s -1 p p Requires atm to keep a clean surface clean; surface can also lose atoms

4 Example

5 Thermodynamic potentials (,, { }),, = +,, =,, = + = + =, =, =, =,

6 Reaching the equilibrium

7 Statistical thermodynamics = log - number of microstates for a given macrostate This is only a postulate - but it works! Why it should work: (i) in equilibrium max, so that max; (ii) is additive, but is multiplicative

8 Statistical thermodynamics = /, = /, = 1

9 Statistical thermodynamics =!!!! =!!!!! = = /, = ln = ln! ln! - canonical partition function ln! ln

10 Statistical thermodynamics = ln = ln + / = = = = / = = = ln = ln + = ln + ln = = ln = + = ln + (, ) =, = ln +,

11 Statistical thermodynamics ( T, O p O 2 2 ) equilibrium Δ = =

12 Statistical thermodynamics Δ,, = 1,,,, min = Δ + Δ Δ Δ + Δ

13 Statistical thermodynamics =! translational states are invariant with respect to any permutations of molecules (indistinguishable molecules)

14 Statistical thermodynamics (, ) = ( ln + ln! ln ln ln ln + ),, = ln ln ln ln ln + = ħ k = ħ = ħ required input - molecule s mass = (2 + 1) ln(2 + 1) required input -,

15 Statistical thermodynamics = (2 + 1) ( ) = = ħ, =, is the bond length ln ħ, required input - rotational constant (calculated or from microwave spectroscopy)

16 Statistical thermodynamics = Π ħ = Π ħ ħ = = Π ħ ħ (used the fact that sum over is a geometric series) required input - vibrational frequency = ħ + ln 1 ħ /

17 Ab initio atomistic thermodynamics It is convenient to define a reference for : ( T, p) ( T, p) E0 ( T, p) Alternatively: ( T, p) ( T, p ) kbt ln( p / p ) and ( T, p 1 atm) from thermochemical tables (e.g., JANAF)

18 Ab initio atomistic thermodynamics 2 ( T, O p O 2 ) Δ,, = 1 Δ + Δ Δ Δ + Δ Δ electronic structure calculations Δ, =,, ( ) - phonon density of states,, = ħ + ln 1 ħ /

19 Ab initio atomistic thermodynamics surface bulk Pd metal Δ (, ) = 0 Δ (, )

20 Δ (, ) = Δ Δ (, ) p(2x2) O/Pd(100) ( 5x 5)R27 o PdO(101)/Pd(100) M. Todorova et al., Surf. Sci. 541, 101 (2003); K. Reuter and M. Scheffler, Appl. Phys. A 78, 793 (2004)

21 First-principles atomistic thermodynamics: constrained equilibria 2 ( T, O p O 2 ) X CO ( T, pco) constrained equilibrium Δ, = 1, Δ, Δ, 1 2 C.M. Weinert and M. Scheffler, Mater. Sci. Forum 10-12, 25 (1986); E. Kaxiras et al., Phys. Rev. B 35, 9625 (1987); K. Reuter and M. Scheffler, Phys. Rev. B 65, (2001); Phys. Rev. B (2003)

22 Surface phase diagrams p O (atm) K 600 K CO oxidation on RuO 2 (110) p CO (atm) K. Reuter and M. Scheffler, Phys. Rev. Lett. 90, (2003) O (ev)

23 When vibrations do matter µ O (ev) ZnO (0001) surface phase diagram in H 2 O-O 2 atmosphere no vibrations µ H (ev) No structure with (2x2) periodicity as seen at the ZnO(0001) surface annealed in a dry oxygen atmosphere (containing at maximum 2 ppm water) M. Valtiner, M. Todorova, G. Grundmeier, and J. Neugebauer, PRL 103, (2009)

24 When vibrations do matter A (2x2)-O adlayer structure is stabilized by vibrational entropy effects (bar) Observed at humid conditions (bar) M. Valtiner, M. Todorova, G. Grundmeier, and J. Neugebauer, PRL 103, (2009)

25 Very small concentrations of defects can significantly alter materials properties 50 ppm Fe

26 My precious! : Perfect defected gems Cr:Al 2 O 3 V:Al 2 O 3 Fe:Al 2 O 3 Fe:Al 2 O 3 Impurities are responsible for the color of sapphire and many other precious stones Typical concentrations: ppm Fe,Ti:Al 2 O 3

27 Entropy G U pv TS S k lnw W number of microstates 1) Solid: vibrational entropy (phonons) 2) Solid: electronic entropy 3) Gas: vibrational, rotational, translational, etc. (part of ) 4) Solid: defect disorder i

28 G Configurational entropy ~ [ U pv T ( S S )] TS G TS N config config equivalent defect sites in the sold n defects If defects do not interact: S config config N! k ln n!( N n)! Stirling s formula: ln( n!) n(ln n 1 ), n 1, ~ ln(2n) 2n S config k N ln N n ln n ( N n)ln( N Good approximation only on a macroscopic scale n)

29 Defect concentration Minimize the free energy of the system with respect to the number of defects ~ G( n) G ng TS ( n) 0 f config If defects do not interact: n N 1 expg kt 1 f n N exp G f ( T, p) n N 1 exp G kt 1 kt textbook formula f

30 Charged defects and charge compensation n N 1 expg kt 1 f for non-interacting defects But can charged defects be considered as non-interacting?! Q 1 0 Q 2 0 V interact Q Q 1 2 r1 r2 Coulomb interaction long-range!

31 Charged defects and charge compensation + Q 1 0 Q V For a system of charges: interact V Q Q 1 2 r1 r2 interact 1 2 i j Q Q r i i r j j + + In the thermodynamic limit (N ) the electrostatic energy of charges with any finite concentration diverges Charged defects must be compensated in realistic materials

32 Charged defects and charge compensation For a system of charges: + + V interact 1 2 i j Q Q r i i r j j + + In the thermodynamic limit (N ) the electrostatic energy of charges with any finite concentration diverges Typical dependence of the defect formation energy as a function of unit cell size

33 Charged defects and charge compensation Typical dependence of the defect formation energy as a function of unit cell size In standard periodic calculations the charge per unit cell is compensated by a uniform background charge (occurs naturally as a regularization of the Ewald summation) The compensated defects interact much weaker with each other But they do interact strongly with the compensating charge (~1/L)

34 Local and global effects of doping In realistic semiconductors, charged defects can be compensated by the depletion of charge carriers (electrons or holes) interaction electrons occupying hole states (localized or not) Local effect of doping (chemical bond formation) Global effect of doping (interaction with the compensating charge) Formation energy and concentration of charged defects depend strongly on the distribution of the defects and the compensating charge

35 Defect-defect interactions G( T,{ p}, e,{ n}) n i 0 n i ~ n sites exp( G f kt ) Local interactions: Local relaxation Chemical bonding Long-range (global) interactions: Charging Fermi level shifting Charged defects at any finite concentration cannot be considered non-interacting

36 Charged defects in a doped material G( n) ng f ( n 0) 0 ( r) E d r TSconfig( n) 2 formation energy in the dilute limit S config k ln m g m electrostatic energy at finite n ( n)exp( E m ( n) / kt ) The charged defects are screened by the compensating charge: S config k ln N! n!( N n)!

37 Space charge formation and band bending Space charge region = / causes band bending and electric field -- dopant concentration -- surface charge due to charged vacancies

38 Using Gauss theorem: Band bending = - electric field For surface density of charged defects with charge per defect the surface charge density is The thickness of the compensating charge layer: =, where is a dopant concentration (assuming uniform distr.) Electrostatic energy per area : = = =

39 Band bending For surface density of charged defects with charge per defect the surface charge density is The thickness of the compensating charge layer: =, where is a dopant concentration (assuming uniform distr.) Electrostatic energy per unit area: = The surface Gibbs free energy per unit area: = Δ 0 +

40 F 2+ concentration at p-mgo(001)

41 Coarse-graining potential-energy surface (PES) ( )

42 Coarse-graining potential-energy surface (PES)

43 Ab initio atomistic thermodynamics approach allows to model materials in thermodynamic equilibrium at realistic temperatures and pressures from first principles Surface phase diagrams and defect concentrations as a function of temperature and pressure are two prominant exapmples Doping should be considered as a thermodynamic variable, along with temperature and pressure

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