Chemisorption -- basic concepts and models

Transcription

1 Chemisorption -- basic concepts and models Bedřich Velický VII. NEVF 514 Surface Physics Winter Term Troja, nd December 011

2 This class is an introduction into the microscopic physics of adsorption two model treatments of chemisorption of atoms a sequence of simple models of electronic states culminating with the Anderson-Grimley-Newns m. semi-infinite jellium with adsorbates as a model treated in LDA an ab initio approach

3 Adsorption of atoms Adsorption Physisorption Chemisorption weak long range forces (v.d.w.) true chemical bond Description Dynamical Adiabatic full process in time static nuclei, no time UNEVEN PARTNERS BULK mechanical thermodynamic framework energy bands electron pool E F surface molecule ATOM isolated discrete levels fixed el. number 3

4 Questions the adiabatic theory may answer Trustworthy answers are given by a fully ab initio theory at the cost of an extensive computational effort; models provide illustrative partial results of a limited relability achieved in a cheap manner INPUT 4 STANDARD AB INITIO LOOP GEOMETRY TOTAL ENERGY sc potential inner loop 1-el. orbitals QUESTIONS AT TWO LEVELS adsorption energy E E E E ADS = ( clean)+ ( atom) 0 adsorption site adsorption geometry vibrational properties TOT chemical bond + electron structure electronic properties

5 Model one-electron Hamiltonians to study in a non- selfconsistent way the orbital structure of electrons near a surface with adsorbates (instead of the inner green loop)

6 One-electron potential at a clean surface atomic around each site periodic in the bulk the zero fixed by the barrier with respect to infinity in the vacuum the potential inside may be smoothed using the pseudopotential concept PSEUDOPOTENTIAL BARRIER VACUUM MODELS I. AND II. of Lecture 3.

7 MODELS: A. Sommerfeld B. tight binding (LCAO) C. AGN One-electron potential with an adatom atomic around each site periodic in the bulk the zero fixed by the barrier with respect to infinity in the vacuum the potential inside may be smoothed using the pseudopotential concept the adatom creates a local disturbance just above the surface VACUUM PSEUDOPOTENTIAL ADATOM BARRIER

8 What we are going to do ( I. ) ( II.) 8 solve the one-electron Schrödinger ( ( )) ( ) ( m + V ψ E e α = αψα r r r) equation Many-electron state sequence of occup. numbers { nασ } add spin σ =, Pauli principle 0 nασ 1 Aufbau principle fill from the bottom up ( e) ( e) ( e) (III.) Charge balance nασ = N + N = N = Z defines Fermi energy... HOMO highest occupied n = ϑ( EF E ) ασ α... LUMO lowest unoccupied (IV.) Observables local particle density n ( r) = n ( r) + n ( r) local spin densit y m( r) = n ( r) n ( r) n σ ( r) = nˆ σ ( r) = nασ ψασ ( r) orbital interpretation double average J

9 Three of the ways to solve the Schrödinger eq. one-electron Schrödinger equation ( ( )) ( ) ( ) m + V r ψ E e α r = αψα r 9 in real space on-shell at E α matching of the wave function (boundary conditions) textbook approach MODEL A. choose simple V(r) tight binding on-shell at E α basis of atomic-like orbitals sparse matrix techniques today a recognized approach MODEL B. fitted matrix elem. orbital representation Schrödinger eq. matrix problem ψ = c ϕ ( λ Hˆ µ E λ µ ) cµ = 0 µ H S λµ λ orbital representation Schrödinger eq. matrix eq. basis of fragment eigenstates Green s function technique elementary use of universal method MODEL C. AGN λ λ λµ

10 Three of the ways to solve the Schrödinger eq. one-electron Schrödinger equation ( ( )) ( ) ( ) m + V r ψ E e α r = αψα r in real space on-shell at E α matching of the wave function (boundary conditions) textbook approach MODEL A. choose simple V(r)

11 On the way to the Sommerfeld type model A. CLEAN SURFACE The smooth pseudopotential bottom is modeled by a completely flat Sommerfeld plateau The barrier is modeled by a step-like abrupt rise from the plateau to the vacuum zero PSEUDOPOTENTIAL BARRIER VACUUM MODELS I. Lecture 3

12 On the way to the Sommerfeld type model A. SURFACE DECORATED BY AN ADATOM The smooth pseudopotential bottom is modeled by a completely flat Sommerfeld plateau The barrier is modeled by a step-like abrupt rise from the plateau to the vacuum zero The atomic potential is modeled by an attractive δ - well Unrealistic model soluble by hand and giving very good insight VACUUM PSEUDOPOTENTIAL ADATOM BARRIER MODEL A.

13 MODEL A. 1D Sommerfeld, step barrier, δ-atom 1 E < 0 V x E F Fermi sea d Aδ ( x d) region I. region II. region III. Three parameters V, Ad, Three matching regions

14 MODEL A. 1D Sommerfeld, step barrier, δ-atom E < 0 E F SUBSTRATE ALONE we know that as model I. of Lecture 3 Fermi sea V x RESULT standing wave in region I. exponential leaking into region II. energy dependent phase shift needed for smooth matching region I. region II. Solution of the Schrödinger equation by matching A. SEMI-INFINITE SAMPLE ikx ikx I. ψ = α + β = + bound states e e k E V k V k V < E < 0 κ x I. I ψ = γ e κ = E i α = α e tan = κ k ψ ( x) = Acos( kx+ ) x< 0 i β = α e sin = κ k κ x = Acos e 0 x > 0 γ = α cos cos = k k 0 m + κ = m 0 m

15 MODEL A. 1D Sommerfeld, step barrier, δ-atom 3 THE δ -ATOM ALONE reminescent of deuter ium x region II. d Aδ ( x d) E b region III. E κ = b b = m A A m 4 select A so that ψ = εe κb x d V < E < b 0 ψ Aδ( x d ) ψ = Eψ dx m m ( ψ ( d + ω) ψ ( d ω) ) d + ω d ω Aψ( d) = O( ω)

16 MODEL A. 1D Sommerfeld, step barrier, δ-atom 4 SUBSTRATE + THE E < 0 i κ = = region I. α = α e, β = α e κ tan = k y k κb κ b 1 e 1+ e E V E full model V i κd κd δ -ATOM d region II. ε α Aδ ( x d) region III. x ikx I. ψ = αe + β e II. κx ψ = γ e + δ e III. κ ( x d) ψ = εe ( ) κd e κb = ikx κx ( κd 1 y ) ( κd κ 1+ y ) (1 + y) 1 e + e b resonance shifted exponential broadening For the adsorbate, the atomic bound state dissolves into a resonance in the bulk band The coupling and the reonance width depends on e κd

17 MODEL A. 1D Sommerfeld, step barrier, δ-atom 5 SUBSTRATE + THE E < 0 E F Fermi sea region I. full model V δ -ATOM d region II. Aδ ( x d) region III. x GLOBAL AND LOCAL EFFECTS locally, the band states are strongly affected global effect of a single atom is small (change in the phase shift) π 1. quantization k = L n+. normalization dx ψ = α L + o( L) unchanged by the adsorbate m 3. density of states ge ( ) = L 1 h V E L

18 Refresher Band filling without spin N k ( e) N( E ) = 1 = ϑ( E E ) F F k E E k E = k k = π n+ n = m L L k F { p p + 1 general definition 1 π n( E) me L for L a suitable for smoothing N 1 ( e) in particul 1 π N π ( E) = me L, ar = me L F DOS density of states hustota stavů dn N ( E) dηg( η) ge ( ) = DOS de ϑ( E) = δ( E) ge ( ) = δ( E E) IDOS d de E ge ( ) k m 1 h E k = L general definition basic form our model

19 Refresher LDOS local density of states E LDOS nx ( ) = ψ ( x) dηδ( η E) dη gx (, η) k gxe (, ) = ψ ( x) δ ( E E) k dx g( x, E) = dx ψ k ( x) δ ( E Ek ) = g( E) k 1 LDOS f or our model k k F gxe (, ) = ψ k( E) ( x) δ ( E Ek ) = ψ k( E) ( x) ge ( ) gde (, ) = ε ge ( ) k k k E F SUM RULE I. SUM RULE II.

20 Three of the ways to solve the Schrödinger eq. one-electron Schrödinger equation ( ( )) ( ) ( ) m + V r ψ E e α r = αψα r tight binding orbital representation Schrödinger eq. matrix problem ψ = c ϕ ( λ Hˆ µ E λ µ ) cµ = 0 µ H S λµ λ λ λ λµ on-shell at E α basis of atomic-like orbitals sparse matrix techniques today a recognized approach MODEL B. fitted matrix elem. 0

21 Tight binding method ε write ϕ ε t t ϕ 1 T T 0 A ε A ϕ A 1D s-band with nearest neighbor hopping orbital representation Schrödinger eq. matrix problem ψ = c ϕ ( λ Hˆ µ E λ µ ) cµ = 0 µ H S Schrödinger equation becomes a difference equation for the coefficients c ( ε E) a + Ta = 0 termination of the chain of equations A ψ = c ϕ + cϕ + c ϕ + cϕ + Ta + ( ε E) a + ta = ta + ( ε E) a + ta = ta + ( E) a + ta = 0 n ε n+ 1 n+ The states form a band, ε t E ε + t In a chain of N substrate atoms, there are N+1 states ( N of the substrate+one dissolved adatom state) (to infinity) 1 λµ λ n λ λ λµ

22 Projected density of states (PDOS) Again, distinguish the global and Orbital occupancy the local properties 3 * α Probability amplitude = d rψ ( r) ϕ( r) = α ϕ ϕ = α α ϕ = FD( ) δ ( α) α ϕ α α n n de f E E E PDOS g ( E) SUM RULE de g ( E) = 1 ϕ In words, the orbital is spread over all eigenenergies with the total weight equal to 1. PDOS for the adatom in our model ϕ α α A δ α 0 α α g ( E) = δ( E E ) = ( E E ) c ( E ) A 0 δ α 0 α g ( E) = c ( E) ( E E ) = c ( E) g( E) A α resonant behavior just like in the previous model width T

23 Three of the ways to solve the Schrödinger eq. one-electron Schrödinger equation ( ( )) ( ) ( ) m + V r ψ E e α r = αψα r orbital representation Schrödinger eq. matrix problem ψ = c ϕ ( λ Hˆ µ E λ µ ) cµ = 0 µ H S λµ λ λ λ λµ orbital representation Schrödinger eq. matrix eq. basis of fragment eigenstates Green s function technique elementary use of universal method MODEL C. AGN 3

24 Anderson - Grimley - Newns model 1 Widely used Conceptually crucial also today System = substrate + adsorbate + coupling Hˆ = Hˆ + Hˆ + Vˆ While we may visualize these parts in space, their description is abstract Hˆ = S 0 A b Eb b + AE A A + bv ba A + h.. c b b unperturbed Hˆ perturbation Vˆ Somewhat like "tunneling Hamiltonians" P.W. Anderson thought about a d-level impurity in bulk Not 1D, it may cover any 3D situation, if the bands and the couplings are properly chosen

25 band Anderson - Grimley - Newns model E A E A resonance bound state Schrödinger eq. ( E Hˆ ) ψ = 0 ψ = α A + β b { ( E EA) α + VAbβb = 0 ( E E ) β + V α = 0 E = E + A b b ba V ba secular equation E E b for the bound state energy provided it does not fall into the band Inside the band, the adsorbate level becomes unstable Golden rule decay rate (for weak coupling) b this gives the resonant energy its imaginary part π w= VbA δ ( E E b ) Self-consistent exact equation for the complex energy of the resonant state E = E + A ba V E+ i0 E b

26 Anderson - Grimley - Newns model 3 Dirac identity 1 E Eb + i0 = 1 E E iπδ ( E E ) b b Making sense of the equation for the complex energy of the resonance E = E + A adsorption function (self- energy) First iteration of (*) yields E = E + Λ( E ) i ( E A A A V ba E+ i0 E b V ba Σ E Eb (*) ( E+ i0) = iπ V δ ( E E ) Σ ( E + i0) = Λ( E) i ( E) ) agrees with Golden Rule EXAMPLE ba Na on (synopsis of ab initio calculations) the level goes down broadens the Fermi energy fixed partial occupancy results Al is more electronegative than Na back effect of occupancy on the level Al b

27 Anderson - Grimley - Newns model 4 A gentle introduction into the kingdom of Green's functions PDOS ga( E) = δ( E Eζ ) A ζ 1 1 transformation ga( E) = π Im A ζ ζ A E+ i0 Eζ ˆ 1 Green's function GE ( + i0) = ζ ζ E+ i0 Eζ Schrödinger eq. ( z HGz ˆ ) ˆ( ) = 1ˆ= ζ ζ 1 projected GF ˆ VbA AGz ( ) A = ; Σ ( z) = z E z E ( ) b A Σ z 1 PDOS again g( ) Im ˆ A E = π AGE ( + i0) A 1 π ( E) = ( E E Λ( E)) + ( E) A nearly Lorentzian, except for the energy dependence of Λ and

28 Anderson - Grimley - Newns model 5 Zeros of eq. E E Λ( E) = 0 Two basic situations w A 0 Grimley semi-elliptic model of the chemisorption function 0 ½ ε (E)= (1 ( ) ) Λ( E) by Kramers-Kronig coupling strength, adsorbate induced < 1 weak coupling > 1 strong coupling w { w E w bound states resonances half-band width

29 Anderson - Grimley - Newns model 6

30 Semi-infinite jellium in DFT-LDA fully self-coonsistent ab initio study of the simplest model of a solid incorporating the exchange and correlations

31 Recapitulation of bulk jellium 1 Definition neutral crystal ionic charge evenly spread + J = Jδ J = Ω Ω n Z ( r R ) n d rn ( r) single parameter n = n determines all equilibrium properties WIGNER RADIUS 1 = 4π n 3 Bohr radius n 30 3 = r 3 s m simple metals r =... 6 s + volume per electron ( ar) 0 s 3 dimensionless parameter

32 Recapitulation of bulk jellium Jellium is TOTAL ENERGY/ELECTRON atomic units εtot = εh + εx + εc WIGNER FORMULA = r r r s s s stabilized by exchange and further by the correlations XC hole gr ( ) n n ( r) 3

33 Recapitulation of bulk jellium 3 0 εc Ry r 0 s Correlation energy in jellium Wigner... historical, approx. compared with QMC numerical, heavy computation in practice -- Perdew-Zunger fit by analytic expressions range of previous slide

34 Recapitulation of DFT-LDA 1 Write Euler-Lagrange equations e = 0 non-interacting el 's e [ ] = s[ ] + υ0( ) ( ) [ ] = s[ ] δts δ n( r) 34 0 ( e) ( e) Kohn-Sham E n T n dr r n r E n T n G[ n] dr υ( r) n( r) δ E µδ N = 0 δ E µδ N = 0 µ + υ ( r) = µ Here, the solution is known δt ( ) υ ( r) s δn( r) + δn( r) + υ r = µ ( m 0( r)) E ( eff ( )) e m r e eff n r ψα r n( r) = ψ ( r) ( ) = ( ) = α = s + r υ0 = s + r 3 E = Eα dr theory Lagrange multiplier Use the eff. potential as a real one + υ ψ = ψ + υ ψ = E ψ α α α α α α α E E T d n E T G+ d υn µ = δe δ N ( e) δg ( υ υ) n+ G eff This is the complete set of equations of the Kohn-Sham theory

35 Recapitulation of DFT-LDA Exact decomposition H XC rn r d r e 1 e n d n d r r r r KS G[ n] = d ( ) ( ) + ( ) hxc (, r r r r r r ) U + E defines KS XC energy KS XC hole different from the true one, but similar properties sum rule d (, r r ) = 1 3 KS r hxc XC energy per particle Effective potential ε KS 3 1 KS XC n = d e hxc (, r ) r ( r, r ) r r 35 eff δg δexc δn( r) ) H ( ) δn( r) υ ( r) = υ( r) + = υ( r + υ r + Final expression for the total energy 3 H r υxc XC E = E U d n++ E α

36 Recapitulation of DFT-LDA 3 For jellium, KS theory is exact ε KS XC LDA Ansatz ε υ KS XC xc δe δ n( r) J XC (, r n) = ε () n J XC ( r,[ n]) = ε ( n( r)) ( r) XC = n LDA KS equations ( nε ) XC ( m ( ) H ( ) xc( ) e defines the LDA XC energy + υ r + υ r + υ r ) ψ = E ψ (1) As easy to solve as Hartree or Slater equations () Of course, by iteration in a self-consistent cycle J α α α

37 Clean semi-infinite jellium semi-infinite jellium far more realistic than truncated Sommerfeld + simplest theory of surface ever -- a single parameter n or r s treated in DFT-LDA means fully ab initio, no additional fudging with parameters etc. the calculation permitted to find also total energies, in paticular the surface energy here, we inspect only a few figures of surface charge distribution and related quantities (N.D. Lang and W. Kohn, PRB 1(1970), 4555)

38 Semi-infinite jellium as a substrate 1 semi-infinite jellium (parameter rs ) is approached by a point charge Z treated in DFT-LDA means fully ab initio the calculation permitted to find also total energies and the equilibrium distances of the adsorbed atoms here, we inspect only a few figures ( N.D. Lang and A.R. Williams, PRB 18(1978), 616)

39 Semi-infinite jellium as a substrate

40 Semi-infinite jellium as a substrate 3

41 The end