Chapter 12-Newns-Anderson model

Introduction to the Newns-
Anderson model
The NA model has two equally valid aspects
a) the dynamic aspect, given by the Hamiltonian H
∂ϕ(
t)
ih
= Hϕ(
t)
∂t
H determines the evolution of a general state in time.
b) the aspects of eigenfunctions.
H determines the eigen functions of constant energy
E
= hω
ϕ ( t)
= ϕ
eigen 0
e
−iωt

creation and annihilation operators automatically take Pauli’s
principle into account
state
1 filled with one electron,
empty state
creation(
appearance)
operator
annihilation ( disappearance)
a
∗
for
one can fill state a only with one electron:
state
a
0
operator a for state a
a
∗
∗
0 = 1 a 1 =
0
one can take only one electron out of a state a : a 1 = 0 a 0 = 0
With these definition it is easy to derive:
∗
∗
a a 1 = 1 and a a 0 = 0 = 0×
a
Therefore, the operator a*a gives the number of electrons in state
a
namely 0 or 1.

Newns-Anderson-model,
dynamics, with condition
V ak < filled metal bandwidth
single electron energy ε
adsorbate is „reduced“ to the
lowest unoccupied orbital
(LUMO) |a>
Fermi
energy
E CT
k final
k initial
metal with
electron states k
„dynamical charge transfer“
Note, that Heisenberg‘s
uncertainty principle allows
residence in |a> for times
shorter than ħ/E CT .
V ak
grows, the nearer the adsorbate
comes to the surface
z
H = ε a † a + ∑ε
a † a + ∑( V a † a + V * a † a)
NA a k k k ak k ak k
k
k
energy of
LUMO
number of
electrons in
|a>
energy of
state k
number
of
electrons
in k
Without energy for the excitation of
an electron-hole pair: k final= k initial

Newns-Anderson model, adsorbate induced density of states
H = ε a † a + ∑ε
a † a + ∑( V a † a + V * a † a)
NA a k k k ak k ak k
k
k
For weak coupling (V ak < metallic bandwidth) H NA yields eigenfunctions | i >of the total system
adsorbate and metal with a continuum of energies ε i .
„adsorbate induced density of states“ ρ a
(ε)dε=Σ i
|| 2 δ(ε-ε i
)dε
is the projection of the LUMO of the combined system.
Result:
ε i
ρ a
(ε) = (1/π) (∆(ε)/(ε-ε a
-Λ(ε)) 2 +∆(ε) 2
This is a Lorentzian with center ε a
+Λ(ε),
2∆
ε a
+Λ(ε)
with shift Λ(ε)=(1/π) P∫(∆(ε‘)dε‘)/(ε-ε‘)
Fermi energy ε F
and halfwidth ∆(ε) = π∫⏐V ak
⏐ 2 δ(ε-ε k
)dk
The usual approximation is
ρ
a
( ε ) =
1
π
∆
( ε ε )
a
2
− +
∆
2
0
ρ a
(ε)
time average of electrons in |a>

Why is it possible that k final ≠ k initial (under condition of energy input)?
|a>
Fermi
energy
E CT
k final
k initial
electron after
}
hole dynamic CT
† † † * †
NA a k k k ak k ak k
k
k
number of
energy of electrons in energy of
LUMO
state k
|a>
H = ε a a + ∑ ε a a + ∑ ( V a a + V a a )
transformation without loss of generality
z
k
∑
initial
∑
V a a + V a a
† * †
ak k ak k
k
initial initial final final
final

Self consistent calculations by N.D. Lang, A.R .Williams, Phys. Rev.B 18 (1978) 616,
Cl, Si and Li atom adsorbed on jellium (r s =2), similarity to NA-resonances
above: total electron density plots
edge of the positive
background
Li Si Cl
below: total electron density-minus superposition
of atomic and bare metal electron density.
Is it a Li ion in front of a sharp boundary or is it a polarised Li atom? This distinction is not unambiguously possible!
∞
However the surface averaged dipole moment d is still a good definition:
d < 0 for electronegative adsorbates (e.g. Cl), d >0 for electropositive adsorbates e.g. Li
d =−∫ ezδ
n( r)
d r
One might call the SERS model of temporary charge transfer correctly the model of temporary
change of the surface dipole moment, but I will not do that!
−∞