Solid State Many Body Physics course

Newns-Anderson model

Bo Hellsing

January 16 2008

Newns-Anderson model for chemisorption

⌉ a >

⌉ k >

V ka = hopping matrix element

V ka

>>

a| |k

:stateatom }{ :states metalatom

surface

V ε

Fε

levelenergy occupiedhighest energy Fermi ==Fε

Model Hamiltonian

>

++

++=

−+−

+++

++

∑

∑∑

|a-σσU

cccUcccVccV

ccccH

aaaaakkakk

aak

aaakk

kk

orbitalin and spin between n interactio coulomb theis where

)( σσσσσσσσ

σ

σσ

σσσ

σ εε

σσσσ

σσσ

σ εε −+ ++++= ∑∑∑ aak

kaakaa

kkk nUnCHccVnnH .).(

operatorsnumber with

We now apply the Hartree-Fock approximation, neglecting correlations. The matrix elements of the retarded Greens' function GR , assuming that ( |a>, { |k> } ) form a complete basis set:

)commutator-(antiBA ABB}{A,average state ground theis 0|.......|0...... where

(2) )}0(),({)()(

(1) )}0(),({)()(

+=><=><

><−=

><−=+

+

σσσ

σσσ

θθ

akka

aaaa

ctctitGctctitG

(3) ] )(Im1[lim)(0

δεπ

ερ σ

δσ iGaaa +−=

→

Determining GR will give us the adsorbate projected density of states

Time derivative of Eq. (1)

)0( where

(4) },)({)( }),({)()(

++

++

=

>∂

∂<−><−=∂

∂

σσ

σσ

σσ

σ

θδ

aa

aa

aaaa

cc

ct

tctictctit

tG

Eq. (4) then takes the form

(5) ][)( (t),Hcit

tcaσ

a −=∂

∂ σ

The derivative of the destruction operator in Eq. (3) is evaluated with the Heisenberg relation for the time derivative of an operator

(6) },][{)( )()( ><−=∂

∂ +−σ

σ

θδ aiHt

aσiHtaa ce,Hcetit

ttGi

Home problem 1

Show in details that the Newns-Anderson model hamiltonian gives

][ ∑ −++=k

σaaσkσakaσaaσ ncUcVcεH,c

(7) }),()()({)(

}),({)(

}),({)( )()(

><−

><−

><−=∂

∂

+−

+−

+

+

∑σσσσ

σσ

σσ

σ

θ

θ

θδ

aaaa

kakak

aaaa

ctctctcUti

ctcVti

ctctitt

tGi

We then have

Comparing Eq. (7) with Eq. (1) and (2), we realize that the fourth term in Eq. (7) is a higher order Greens' function, thus we have an infinite hierarchy of equations. One way to break this is to apply the Hartree-Fock approximation. This means that we neglect the correlation between the “up” and “down” electron in the adsorbate orbital

number simple a is where

)()(

><

><≈=

−

−−−+−

σ

σσσσ

a

aaaa

n

n(t)ntctc

The approximation is ok if :

1) we consider the “up” spin and the fluctuations of the “down” spin is small

2) if U is small compared to the band width W (the energy range of the | k > states)

><≈><+><−= −−−−− σσσσσ aaaaa nnn(t)n(t)n ) (

(8) )( )()( ~ )()( tGtV tGtt

tGi kak

akaaaaa σσ

σ

σ

εδ ∑++=∂

∂

We then have

(9) ~ ><+= −σσ εε aaa nU

where

Now proceed in the same way, taking the time derivative of Eq. (2). We obtain

(10) )()( )( )( t GtVtGt

tGi aaakkakka σσσ

ε +=∂

∂

Home problem 2

Evaluate the energy dependent Green's functon and show that you obtain the result

whith the self-energy of state ”a”

)(~

1εεε σ

σ

Σ−−=

aaaG

||)(2

∑ +−=Σ

k k

ak

iV

δεεε

is states ofdensity projected adsorbate the(3) Eq. toAckording

|V| function shift )(

)(|V| functionion chemisorpt )( where

(11) )()()(

k

2ak

2ak

∑

∑

−==Λ

−==∆

∆−Λ=Σ

k

kk

P

i

εεε

εεδπε

εεε

The self-energy

)12()()](~[

)(1

)()(~1Im1 ] )(Im1)(

22 εεεεε

π

εεεεπε

περ

σ

σ

σσ

∆+Λ−−∆=

∆+Λ−−−=−=

a

aaaa i

G

Simple 1 D model for calculation of the self-energy

)13()1

1sin(1

2|

"amplitudes site" with |||

1constant latticeSet .energy eigen with Eigenstate

, 0|| , 1||

hoppingelectron neighbour nearest Only ""index site Atom end. at the adsorbatean with atoms metal 1 ofchain Linear

0i

k

++

+=><

>><=>

=>

=><=><−=>±<

+

∑=

Nik

Nki

kiik

a|k

δi|jiHiiHi

.iN

N

ij

π

ε

β

Home problem 3

Show that the energy spectrum is given by

)14(,...,3,2,1 where, )1

cos(2 NkNk

k =+

−= πβε

|00||||| ||

bygiven is "0" denoted ,atom" surface" theand adsorbate ebetween thelement matrix hopping The .0 denoted atom

end the toatom adsorbateour chemisorb now We

. width band large thusand large large, hopping ,(metallic) states ddelocalizeFor

4W

0∑

=

>><<≈>><<=>=<

=

N

iak kVakiiVakVaV

""

Wβ

β

The band width

)()1

(sin1

2||)(||)(

bygiven then isfunction ion chemisorpt The

0with ,)1

sin(1

2

have weEq.(13) From

k

220

2

00

∑ ∑ −++

=−=∆

>=<++

=

ak

kak

ak

Nk

NVV

a|V| VNk

NVV

εεδππεεδπε

π

2

20

1

1

22

0220

0

2

)2(1||4)(

)2

(1||)2(x-1||2)(

sin;coswith

)cos2(sin||2)(

1,{........}1{.......}

integralan k toover sum thechange

)1

cos(2

Now

WWV

VdxxV

ddxx

d

NkdN

Nk

k

k

εε

βε

ββεδε

ϕϕϕ

ϕϕβεϕδβε

πϕϕπ

πβε

π

−=∆

−=+=∆

−==

+′=∆

+=+→

+−=

∫

∫

∫∑

−

1D Linear chain model

a23 01

0V

0

W/2

In this model we have obtained a semi-eliptical chemisorption function

-W/2

)(ε∆

ε>k|

22

0 )2(1||4)(WW

V εε −=∆

And the shift function is linear in energy within the band

0-W/2 W/2

1|2

| when 0

and

1|2

|when ,||8)( 2

20

>→Λ

<=Λ

W

WWV

ε

εεπε

)(εΛ)(ε∆

0)(~ when enhanced iswhich )()](~[

)(1)(

have we(13) Eq. toAckording

22

=Λ−−∆+Λ−−

∆=

εεεεεεε

επ

ερ

σ

σσ

a

aa

ε

0

Weak coupling

)(εΛ

σεε a~-

ε

)(ε∆

)(ερa

ε

ε ′

)(ε ′∆

)(ε ′Λ

Localized states outside the band !

Strong coupling

)(ε∆ )(εΛ

σεε a~-

ε

ε

)(ερa

W

Weak coupling Strong coupling

Hydrogen atom chemisorption

e-

occupied

un-occupied

METAL H-ATOM

ε

)(ερa

ε

Fε

)(ερa

∫

∫

∞−↓↓

↑↓↓

↓

∞−↑↑

↓↑↑

↑

↓

>=<

><+=∆+Λ−−

∆=

>=<

><+=∆+Λ−−

∆=

><

F

F

dn

nU

dn

nU

n

aa

aaaa

a

aa

aaaa

a

a

ε

ε

εερ

εεεεεε

επ

ερ

εερ

εεεεεε

επ

ερ

)(

~with )()](~[

)(1)(

)(

~ where)()](~[

)(1)(

loop consistent-self

22

22

><+><><−><

=↓↑

↓↑

aa

aa

nnnn

ζ

Spin polarization

}2

)~

{arctan(1)(

)! Lorentzian(]~[

1)(

0

0

20

20

0

πεεπ

εερ

εεπερ

ε

−∆

Λ−−=>=<

∆+Λ−−∆=⇒

↑

∞−↑↑

↑↑

∫ aaa

aa

F

dn

0

0

)()(

Λ≡Λ∆≡∆

εε

simplifications

(a)

(b)

states ofdenstity substrate)(,)(||

)(||)(||)(

2

22

==

−≈−=∆ ∑∑ερερπ

εεδπεεδπε

sso

kkok

kak

V

VV

(c)

||2

)(|| re whe,)(

:scattering surfaceatomin modeling

2

20

20

><+=

≈∆∆=∆

−

↓↑

−↑

↑

aa

Fsoz

nUm

Vez

εγ

ερπσγ

REFERENCES

D. M. Newns, Phys. Rev. 178(1969)1123

P.W. Anderson, Phys. Rev. 124(1961)41

book:

S. Doniach and E.H. Sondheimer, Green's functions for solid state physics,

Benjamin/ Cumings publ. Comp. Mass. 1974

Application in atom-surface scattering:

“Effect in intra-atomic Coulomb repulsion on

charge transfer in atom scattering on metal surfaces”

B. Hellsing and V.P. Zhdanov, Surf. Sci. 274(1992)411