Bedřich Velický - Fyzikální Ústav Akademie Věd ... · Chemisorption -- basic concepts and models NEVF 514 Surface Physics ... Questions the adiabatic theory may answer 4 Trustworthy

Chemisorption -- basic concepts and models

NEVF 514 Surface Physics

Winter Term 2011 - 2012

Troja, 2nd December 2011

Bedřich Velický

VII.

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

Adsorption of atoms

3

weak long range forces (v.d.W.) true chemical bond

AdsorptionPhysisorption Chemisorptio

Dynamical

n

full process in time static nuclei, no time

Description Ad a i batic

BULKmechanical

thermodynamicframework

energy bandselectron pool EF

ATOMisolated

discrete levelsfixed el. number

“surface molecule”

U N E V E N P A R T N E R S

STANDARD AB INITIO LOOP

Questions the adiabatic theory may answer

4

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

TOTAL ENERGY

GEOMETRY

inner loop

1-el. orbitals

sc potential

INPUT

ADS TOT

clean atom

adsorption energy

adsorption site

ad

vibrational pro

QUESTIONS AT TWO LEVEL

sorption geo

S

chemical bond + electron struc

( )+

perties

metr

( )

y

ture

0= − ≥E E E E

electronic properties

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)

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

BARRIER

VACUUM

MODELS I. AND II. of Lecture 3.

PSEUDOPOTENTIAL

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

ADATOM

VACUUM

MODELS: A. SommerfeldB. tight binding (LCAO)C. AGN

PSEUDOPOTENTIAL

BARRIER

What we are going to do

8

( )2

2

solve the one-electron Schrödinger

( ( )) ( ) (

equation

)

I.

em V Eα α αψ ψ− ∆ + =r r r

Many-electron state sequence of occup. numbers ,

add spinPauli principle Auf

(

b

II.)

fill from the bottomau principle up 0 1

n

n

ασ

ασ

σ ↓↑

≡• =• ≤ ≤•

( ) ( ) ( )

... highest occupied

... lowest

HOMO

unoccupiL dO

eUM( )

C(III.) harge balance

e e eJ

F E

n N N N Z

n Eασ α

ασ

ϑ

↑ ↓

=

∑ = + = = ∑

−defines Fermi energy

( )(IV.)

( ) ( )

Observables local particle density local spin densit ( ) ( )y ) (

n n nm n n

↑ ↓

↑ ↓

= += −

r r rr r r

2ˆ ( ) ( )( ) orbital interpretationnn nσ ασσ ασ ψ= = ∑r rrdouble average

Schrödinger eq. → matrix problem

Schrödinger eq. → matrix eq.basis of fragment eigenstatesGreen’s function technique

elementary use of universal methodMODEL C. AGN

Three of the ways to solve the Schrödinger eq.

9

2

2

one-electron Schrödinger equati

( ( )) ( ) (

)

on


orbital representation

ˆ 0( )HH c

S

c

Eλµ

λ λλ

µ

µµ

λ

ψ ϕ

µ µλ λ

= ∑

∑ − =

orbital representationon-shell at Eα

matching of the wave function(boundary conditions)

textbook approachMODEL A. choose simple V(r)

on-shell at Eαbasis of atomic-like orbitals

sparse matrix techniquestoday a recognized approach

MODEL B. fitted matrix elem.

in real space

tight binding


2

2


( ( )) ( ) (

)

on


on-shell at Eαmatching of the wave function

(boundary conditions)textbook approach

MODEL A. choose simple V(r)

in real space

On the way to the Sommerfeld type model A.

BARRIER

VACUUM

MODELS I. Lecture 3

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

On the way to the Sommerfeld type model A.

ADATOM

VACUUM

MODEL A.

PSEUDOPOTENTIAL

BARRIER

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

MODEL A. 1D Sommerfeld, step barrier, δ-atom 1

x→

V

FE

0E <

Fermi sea d ( )A x dδ− −

region III.region I. region II.

T

T

h

hr

re

ee p

e ma

arameter

tching r

,

e

,

g

s

ions

V A d


x→

V

FE

0E <

Fermi sea

region I. region II.

RESULT• standing wave in region I.• exponential leaking into

region II.• energy dependent phase shift

needed for smooth matching

Solution of the Schrödinger equation by matching A. SEMI-INFINITE SAMPLE2

22

22 2 2

022 2

2

Ibound sta e e 0 e

.

I .te

Is ikx ikx

xm m

m

k E Vk V kV E Eκ

ψ α βκ

ψ γ κ

−

−

= + = + + = ≡− < < = = −

( ) 2 cos( ) 0

2 cos e 0x

A k x xx

A xκ

ψ ∆

∆ −

= + <

= ⋅ >0

0

| | e tan

| | e sin

2 | | cos cos

i

i

k

k

k

k

∆

∆

κ

κα α ∆

β α ∆

γ α ∆ ∆

−

= =

= =

= =

we know that as model I

SUBST

. of

RATE ALO

Lect

NE

ure 3


x→

d

( )A x dδ− −

region III.region II.

reminesc

THE -A

ent of

TOM

deut

ALO

er

NE

ium

δ

( )

2

2

2

2

| |

( )

( ) ( )( ) ( )

ed

dm

m

b x d

A x d E dx

A d Od d

ω

ω

κ

ψ δ ψ ψ

ψ ωψ ω ψ ω

ψ ε+

−

− −

′′− − − =

′ ′− − =+ − −

=

∫

bE2

2 2

22

2 4

select a

0

so th t

b

b

b

m A

Am

A

V E

E

κ =

= −

− < <

⋅

⋅


x→

V0E <

d ( )A x dδ− −


( )

I. e eII. e eIII. e

ikx ikx

x x

x d

κ κ

κ

ψ α βψ γ δψ ε

−

−

− −

= += +=

S

full model

UBSTRATE + THE -ATOM δ

2

2

2

2

| | e , | | e

1 etan

1 e

| || |

b

b

i i

d

dk

EyV Ek

∆ ∆

κ

κ

κκκκ

α α β α

κ∆

κ

−

−

−

= =

− −= ⋅

− +

= =−

( )( ) ( )

222

2 2 22 211

2e

| | (1 ) 1 e 2e

b

b

d

d dyyy

κ

κ κ

κκ

κκ

εα

−

− −−+

= + − − +

resonance shifted exponential broadening

For the adsorbate, the atomic bound state dissolvesinto a resonance in the bulk bandThe coupling and the reonance width depends on 2e dκ−


x→

V

FE

0E <

Fermi sea d ( )A x dδ− −


S

full model

UBSTRATE + THE -ATOM δ

GLOBAL AND LOCAL EFFECTS

• locally, the band states are strongly affected• global effect of a single atom is small (change in the phase shift)

2 2

2

2

1

1. quantization2. normalization | | 2 | | ( ) unchanged by the adsorbate3. density of states ( ) m

V E

L L

Lh

k ndx L o L

g E

π ∆

ψ α

−

= ⋅ +∫ = + =

Refresher

2

( )

2

( )

1

1 1

22

general definition

for suitable for smoothing

in particul

Band filling( ) 1 ( )

22 1

( ) 2

( ) 2

without spin

, ar 2

k F

eF F k

E E k

k

eF

m L L

E E E

pE k k n n p

n E mE L L a

E mE L mE L

π

π π

π ∆

ϑ≤

≡ = ∑ = ∑ −

= = ⋅ + = +

≈ ⋅

= ⋅ = ⋅

N N

N N

IDOS DOS

2 1

general definiti

DOS density of states

( ) ( ) ( )

( ) ( ) ( ) ( )

( )

on

basic fo

our

r

mo l

m

de

E

kk

ddE

mEh

dE d g g EdE

E E g E E E

g E

hustota stav

L

ů

η η

ϑ δ δ−∞

≡ =

= = ∑ −

= ⋅ ⋅

∫N

N

2

2

LDO

2

2( )

SSUM RULE I.

SUM RULE II.

LDOS local densit

( )( ) ( ) ( , )

( )( , ) ( )

( )( , ) ( ) ( )

LDOS f or our model( )

y of states

( )

1

, (

F FE E

k kk

k kk

k kk

k Ek

xn x d E d g x

xg x E E E

xdx g x E dx E E g E

xg x E E E

ψ ηδ η η η

ψ δ

ψ δ

ψ δ

−∞ −∞

= ∑ − ≡

= ∑ −

∫ = ∑ ∫ − =

= ∑ −

∫ ∫

2( )

2

( )) ( )

( , ) | | ( )

k Ek x g E

g d E g E

ψ

ε

=

=

Refresher



20

2

2


( ( )) ( ) (

)

on


ˆ 0( )HH c

S

c

Eλµ

λ λλ

µ

µµ

λ

ψ ϕ

µ µλ λ

= ∑

∑ − =


on-shell at Eαbasis of atomic-like orbitals

sparse matrix techniquestoday a recognized approach

MODEL B. fitted matrix elem.

tight binding


Tight binding method

21

ˆ 0( )HH c

S

c

Eλµ

λ λλ

µ

µµ

λ

ψ ϕ

µ µλ λ

= ∑

∑ − =

orbital representationAε

Aϕ

εt

2ϕt

ε T

1ϕT

0 1 1 2 2 3 3

0 1

write

Schrödinger equation becomes a difference equation for the coefficient

termination of the chain of

( equa

1D -band with nearest neighbor h

) 0 ti

oppi

ns

n

o

g

sA

n

A

c c c c

c

s

E a Ta

T

ψ ϕ ϕ ϕ ϕ

ε

= + + + +

− + =

0 1 2

1 2 3

1 2

( ) 0

( ) 0

( ) 0

(to infinity)n n n

a E a ta

ta E a ta

ta E a ta

ε

ε

ε + +

+ − + =

+ − + =

+ − + =

The states form a band,

In a chain of substrate atoms, there are states

( of the substrate+one d

2 2

issolved adatom state)

+1

t E t

N N

N

ε ε− ≤ ≤ +

Projected density of states (PDOS)

22

3 *

2 2FD

Probability amplitude = d (

Orbital occupancy

In word

Again, distinguish the global and

PDOS SUM RU

the local prop

) ( )

| | ( ) ( ) | |

( ) ( )

ert

,E

i

sL

s

1

e

n n dE f E E E

g EdE g E

α

ϕ α αα α

ϕ

ϕ

ψ ϕ ϕα

δϕ ϕα α

∫ =

= ∑ = ∫ ∑ −

∫ =

r r r

2 20

2 20 0

the orbital is spread over all eigenenergieswith the total weight equal to 1.PDOS for the adatom in

( ) ( ) | | ( ) | ( ) |

(

our model

) | ( ) | ( ) | (

resona t b

) (

n

|

)

A

A

g E E E E E c EA

g E c E E E c E g E

α α αα α

αα

δ δα

δ

= ∑ − = ∑ −

=

•

∑ − = ⋅

2 just liehavior ke i width n the prev| | ious modelT• ∝


Schrödinger eq. → matrix eq.basis of fragment eigenstatesGreen’s function technique

elementary use of universal methodMODEL C. AGN


23

2

2


( ( )) ( ) (

)

on



ˆ 0( )HH c

S

c

Eλµ

λ λλ

µ

µµ

λ

ψ ϕ

µ µλ λ

= ∑

∑ − =


Anderson - Grimley - Newns model 1W i d e l y u s e dC o n c e p t u a l l y c r u c i a l a l s o t o d a y

System = substrate + adsorbate + couplingˆ ˆ ˆ ˆ

While we may visualize these parts in space,

their description is abstractˆ

S AH H H V

H

•

= + +

•

=

0ˆperturbation

Somewhat like "tunneling Hamiltonians"

P.W. Anderson thought about a d-level impurit

Not

| |

y in bulk

| |

ˆunperturbed

1D, it

|

| . .

may

b A bAb b

b E b A E A b V A h

H

c

V

•

∑ ⟩ ⟨ + ⟩ ⟨ + ∑ ⟩ ⟨

•

•

+

cover any 3D situation, if the bands and

the couplings are properly chosen

Anderson - Grimley - Newns model 2

band

resonance AE

bound state AE

2| | secular equation

for the bo

Schrödinger eq.ˆ( ) | 0 | | |

( ) 0( ) 0

Inside the band, the adsorbate

und state energy

provided it does not fall into the band

bAb

b

A Ab b

b b bA

AVE E

E H A b

E E VE E V

E E

ψ ψ α β

α ββ α

−

− ⟩ = ⟩ = ⟩ + ∑ ⟩

− + ∑ =− + ∑ =

= + ∑

2

22

| |0

Self-consistent e

level becomes unstable

xact equation for

the c

Golden rule decay rate (for weak cou

omplex energy of the resonant sta

pling)

t

| )

e

| (

bAb

bA b

AV

E i E

w V E E

E E

π δ

+ −

= ∑ −

= + ∑

this gives the resonant energy its imaginary part


2

0| |

Making sense of the equation for

the complex energy of the

resonance

(*)AbA

b

VE i EE E + −= +∑

22| |

agrees with Golden Rule

( 0) ( ) ( )

First it

( 0) |

eratio

| ( )

n o

adsorption function

(self-

(*

energy)

f yield)

( ) (

s

)

bAb bA b

A A A

VE EE i i V E E

E E E

i

E

EE i

i

E

Σ

Σ ∆

δ

Λ

Λ

π

∆

−+ = ∑ − ∑ −

−+

= + −

=

10

1 ( )

Dirac identity

b

b

b

E E i

E E i E Eπδ

− +

− − −

=

the level goes downbroadensthe Ferm

i energy fixedpartial occupancy resu

EXAMPLE Na on

ltsAl is mo

Al

(synopsis of ab initio calculations)

re electronegative t back e

hanffect of o

Nac

•••••• cupancy on the level


2

1

PDOS ( ) ( ) | | |1transformation ( ) Im | |0

1ˆGreen's function ( 0) | |0

ˆ ˆˆSchrödinger eq. ( )

A gentle introduction into the kingdom of Green's func

(

tions

)

A

A

g E E E A

g E A AE i E

G E iE i E

z H G z =1

ζ

ζ

ζ

δ ζ

π ζ ζ

ζ ζ

−

= ∑ − ⟨ ⟩

= − ∑⟨ ⟩ ⟨ ⟩+ −

+ = ⟩ ⟨+ −

− =

∑

2

1

1

2 2

| |1ˆprojected GF | ( ) | ; ( )( )

ˆPDOS again ( ) Im | ( 0) |

| |

( )( ( )) ( )

nearly Lorentzian, except for the energy dependence of and

bAb

A

A

A

Vz EA G z A z

z E zg E A G E i A

EE E E E

ΣΣ

ππ ∆Λ ∆

ζ ζ

Λ ∆

−

−

−⟨ ⟩ = = ∑− −

= − ⟨ + ⟩

=

∑ ⟩⟨

− − +


2 ½0

0

Grimley semi-elliptic modelof the chemisorption function

(E)= (1 ( ) )

( ) by Kramers-Kronig

coupling strength, half-band width

Ew

E

w

ε∆ ∆

Λ

∆

−−

0 0

bound statesadsorbate induced resonances

Two basic situations

Zeros of eq. ( ) 0

1 we1 2ak coupling 1 strong coupling

A

w w

E E E

∆ ∆

Λ− − =

< >


Semi-infinite jellium in DFT-LDA

fully self-coonsistent ab initio study of the simplest model of a solid incorporating

the exchange and correlations

Recapitulation of bulk jellium 1

J 31

neutral crystal ionic charge evenl

( ) ( )

y spread

Definit

n

i

o

J Jn Z n d nΩ

Ωδ+ +

→

= ∑ − → = ∫r R r r

( )Bohr radius dimensionless paramet

0er

3

3

volume per electron

41

single parameter determines all equilibrium properties

WIGNER RADIUS

3

1.612 10

m

s

n n

a rn

n

π

+=

=

= ×

30 3

simple metals 2 ... 6

s

s

r

r

−

=


32

2

WIGNER FORMULTOT H X AC

8.72.21 0.916 0.88

TOTAL ENERGY/ELECTRON atomic units

Jellium is stabilized by exchangeand further by the correlations

XC

hol )e (

s s sr r r

g r

ε ε ε ε

+

= + +

= − −

22

)

(

n r

n≡


0 10 20sr →

0

0.06−

0.12−

CRyε Wigner ... historical, approx.

compared with QMC numerical, heavy computation

in practice -- Perdew-Zunger fit by a

Correlation energ

nalytic expressio

y in jelliu

ns

m

range of previous slide

Recapitulation of DFT-LDA 1

34This is the complete set of equations

of the Kohn-Sham theory

2 2

3 30 0

( ) ( )0

0( )

Lagrange multiplier

non-interacting el 's Kohn-Sham0 0

[ ] [ ] ( ) ( ) [

Write Euler-Lagrange eq

] [ ]

uati

[ ] ( ) ( )

0 0

(

on theory

)

s

s s

s se e

n

e e

n n d n n n n d n

δ δδ

υ υ

δ µδ δ µδ µ

υ µ

′ ′= ≠

= + ∫ = + + ∫

− = − =

+ =r

r r r r r r

rT T

E T E T G

E N E N

2 2

eff

0 eff

2 2

3 30 0

( ) ( )

2 2

Here, the solution is known Use the eff. potential as a real one

( )

( )

( ( )) ( ( ))

( ) ( )( ) ( )e e

s s

n n

m mE E

n n

E d n d n

E

α α α α α α

α α

α

α

δδ δ υ µ

υ

υ ψ ψ υ ψ ψ

ψ ψ

υ υ

+ + =

− ∆ + = − ∆ + =

= ∑ = ∑

= ∑ = + ∫ = + ∫

= ∑

r r r

r

r r

r rr r

r r

G

E T E T G +

E

( )3eff

( )

e

d nδ

δ

υ υ

µ

−− ∫ +

=

rE

N

G


35

2 23 3 3 3

H C

K

X

S1 12 2| | | |

defines KS XC energy

Exact decompositi

KS XC hole different from the true one, but similar properties

sum

[ ] ( )

on

( ,

rule

( ) ( )

)xce en d hn d n d n d′ ′− −′ ′′ ′ ′= ∫ ∫ + ∫ ∫

≡

′

+

r r r r r rr r r r r r rG

U E

XC

2

3 KS

KS 3 KSXC

eff H

3H xc XC

12

( ) ( )

| |

( , ) 1

( , ) (

Effective pote

XC energy p

ntial

Final expression for the

, )

( ) ( ) (

er particle

total ener

)

y

)

g

(

xc

xc

n n

e

d h

n d h

E d nα

δ δδ δ

ε

υ υ υ υ

υ

′− ′

′ ′∫ = −

′ ′= ∫

= + = + +

= ∑ − − ∫ + +

r r

r r

r r r

r r r r

r r r r

r

G E

E U E


( )

2

XC

KS JXC XC

KS JXC XC

XCxc

JH xc

( )

2

defines the LD

For jellium, KS theory is exact

LDA Ansatz

LDA KS equation

( , ) ( )

( ,[ ]) ( ( ))

( )

( ( ) ( ) ( )

A XC energy

(1) As eas

)

y

s

e

n n

m

n n

n n

n

Eα α α

δδ

ε ε

ε ε

ευ

υ υ υ ψ ψ

∂∂

=

=

= →

− ∆ + + + =

r

r

r r

r

r r r

E

to solve as Hartree or Slater equations

(2) Of course, by iteration in a self-consistent cycle

Clean semi-infinite jellium semi-infinite jellium far more realistic than truncated Sommerfeld simplest theory of surface ever -- a single parameter or treated in DFT-LDA means fully ab initio,

no additional fud

ging wi

t

sn r+

h 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)

Semi-infinite jellium as a substrate 1

semi-infinite jellium (parameter ) is approached by a point charge treated in DFT-LDA means fully ab initio the calculation permitted to find also total energies and the equilibrium

dis

tances

sr Z

of the adsorbed atoms here, we inspect only a few figN.D. Lang and A.R. Williams, PRB 18(1978)

ur,

es ( )616



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Bedřich Velický - Fyzikální Ústav Akademie Věd ... · Chemisorption -- basic concepts and models NEVF 514 Surface Physics ... Questions the adiabatic theory may answer 4 Trustworthy