Download 4 Surface Electronic Structure 2010 1

Surface Electronic Structure
Surface periodicity and the two dimensional Bloch property
An electron in the surface region moves in a potential field V(r) originating from its
interaction with the positive nuclei and with the static charge density due to the
other electrons (Hartree potential). In addition the other electrons tend to get out of
the way of our electron lowering its energy (exchange and correlation term)
This potential has a periodic modulation of the potential along coordinates x and y
defining the surface plane, while the periodicity is lost along the vertical direction z
r
r r
V ( R, z ) = V ( R + RI , z )
z
Asymptotically the potential behaves like
the classical image potential, which for a
metal surface (perfect screening) goes as:
V ( z) ≈ −
1
4 | z − z0 |
RI
With z0 a reference plane measured with
respect to the geometrical surface position
x,y
selvedge
The potential goes smoothly from the flat region in the vacuum to
the bulk potential determined by exchange and correlation and by
the surface electric dipole.
The difference between the internal potential and the Fermi
energy EF, is the work necessary to extract an electron from the
surface, the so called work function, Φ
The electronic wavefunctions have to satisfy the the Schroedinger equation
Following Bloch theorem the system goes into itself when it is displaced by one
lattice vector RI
the wavevector K is parallel to the surface and
is defined only within a reciprocal lattice vector
G given by G⋅⋅RI=2πn, i.e. within the Brillouin
zone., implying:
Let’s consider a beam of electrons incident on the surface and scattered by it:
The total wavefunction is given by
The amplitudes AG contain information
about the location of the atoms with respect
to the surface mesh.
If the beam impinges from the outside of the
crystal we have the phenomenon of
(low energy) electron diffraction (LEED)
If it impinges from the inside it gives rise
to the band structure
Bulk States: one dimensional case :
Inside the crystal the total wavefunction is
given by the sum of the amplitudes of
incoming and reflected waves, while
outside it is described by an evanescent
wave
The coefficients r and t can be readily obtained
by matching amplitude and derivative of the wavefunction in the two domains.
While φ- and φ+ are propagating waves, ψ is a standing wave.
3D: analogous to 1D but more complicated situation since φ+ consists of all
diffracted waves and in addition evanescent waves are now possible also
inside the crystal . Inside the crystal we have:
The latter waves are evanescent. They are not allowed for an extended system,
but may exist in presence of a surface since it prevents them to grow indefinitely
For the total wavefunction we get inside the crystal:
ψ =φ− +
∑
rG φ K+ , G ; E
G
While the transmitted waves outside of the crystal
for a step like barrier (see figure) are :
ψ = Σt G e
)
i ( K + G )⋅ R ) γ G z
e
Matching amplitude and derivative at z=0 we obtain 2N equations for NrG and NtG
The square of the wavefunction
gives the density of states
Interference is negative at the
extremes of the 3D band at the
surface leading to vanishing
densities
band narrowing.
and to extra features in the gap:
the surface states SS
------- bulk density
_____ surface density
σ surf ∝ | E − E0 |
SS
Relation between 3D and
2D Brillouin zones
Photoemission spectroscopy
The energy of bulk states depends on photon
energy, hν, because they disperse with kz.
The surface state has on the contrary an
energy independent of kz.
Evac
EF
hν
SS
Band narrowing in the tight binding picture of atomic orbitals
Tight binding scheme, φα,I: localized atomic orbital α on atom I
ψ = aα , I φα , I + aβ , J φβ , J + aγ , K φγ , K + ...
The coefficients aα,I , aβ,J , … are complex number representing the contribution to
the state ψ of the atomic wavefunctions φ with orbitals α, β…localized on the
atoms I, J, … They form a vector which satisfies the matrix form of the
Schroedinger equation
With Hi,j the matrix element between orbital i and j , corresponding to the
hopping probability of an electron if i≠j (in Dirac’s notation <αI|H|βJ>) and the
energy if i=j. The Schroedinger equation is still a differential equation which
may be solved using the Green function method
The shape of σ(E) can be described in terms of its moments with respect to E
(series expansion)
The integral can be calculated with the method of the residues (extension of the
integral into the imaginary plane - contour integration or method of Cauchy)
Im
residue
which can be rewritten inserting the expansion
over the basis functions
yielding:
•
Re
The width of a distribution is given by its
second momentum
surface
bulk
Experimental verification of band narrowing by photoemission
Given λ(E), mean free path of the electrons
(see figure) in a solid at the kinetic energy E
(universal curve)
ln λ
10nm
1nm
d-band width
50 eV 500 eV ln E
Photoemission spectroscopy
Modification of the density of states at EF
W(100)
The increased density of states at EF
implies a charge unbalance which
generates an electric field which
causes a rigid shift of the electronic
structure of the surface atoms
Surface core level shifts in photoemission
Ta less than 5 d-electron 4f pulled down by 0.4 eV
Ir more than 5 d-electrons 4f pushed up by 0.7 eV
Surface Core level
shift and chemical
core level shifts
Shockley Surface States for the free electron model
Let’s assume a fairly weak pseudopotential V(z)=2Vgcos gzz
2π
π gz
g
=
k
=
±
=
with z
and a periodicity in the z direction which opens a gap at z
a
a 2
giving bulk solutions for the Schroedinger equation of the form:
φ = ae ik z + be i ( k
z
z
−gz )z
The eigenvalue equation Hφ=Eφ in matrix form contains off diagonal elements
arising from the mixing of the two parts of the wavefunction:
 k z2

 2

 Vg

 a 
a
 
Vg
 

 = E 
2 
( k z − g z )  
b
 b 
 
2

or
 k z2

−E
 2

 Vg

written in atomic units for which
ћ=me=e=1 and the energy is
measured in Hartrees
(1 H=2Rydberg ~27,2 eV, twice the
ionization energy of the H atom)
 a 
 
Vg
  = 0
2
(k z − g z )

− E   b 
2

The energy E is calculated from
the determinant of the matrix
which gives:
 k z2

−E
 2

 Vg




2
(k z − g z )

−E
2

Vg
kz
(k z − g z ) 2
( − E )(
− E ) − Vg2 = 0
2
2
k z2 (k z − g z ) 2 k z2 (k z − g z ) 2
−V +
−( +
)E + E 2 = 0
2
2
2
2
2
g
2
2
2
2
2
2
1
k
(
k
−
g
)
k
(
k
−
g
)
k
(
k
−
g
)
z
z
z
E± = { z + z
± ( z + z
− Vg2 ) }
) 2 − 4( z z
2
2 2
2
2
2
2
Since the gap is at the zone boundary we can put kz = gz /2 and obtain
2
g
E ± = z ± | Vg |
8
i.e. the band gap is of 2|Vg| and is at the average energy of gz2/8
In order to find the wavefunctions we have
to substitute E± into the Schroedinger
equation and look for propagating waves
2
g
E ± = z ± | Vg |
8
 k z2

− E±
 2

 Vg


  a 
  = 0
2
(k z − g z )

− E ±   b 
2

Vg
and multiply the rows of the matrix with the columns of the vector,
obtaining:

from upper row


k z2
±
(
−
E
)a + V gb = 0

2

2
(
k
−
g
)
±
z
z
V g a + [
−
E
]b = 0

2

a

yielding:
= 
b
E


from lower row

Vg
k z2
E −
2
(k z − g z )2
−
2
Vg
±
±
g z2
± | Vg |
At the zone boundary kz=gz/2 and remembering that E =
8
Vg

 g2
2
g
 z ± |Vg | − z
±
a

8
ik z z
i(kz − g z ) z
we obtain:
=  82
φ
=
ae
+
be
2
b
 g z ± |Vg | − g z
8
 8

Vg

±
±
gz
φ =e
i
gzz
2
±
| Vg |
Vg
if Vg>0 it reads at
z=a/2
e
−i
gz z
2
wavefunction of the bulk states at the
upper and lower border of the gap
gz z
φ = 2 cos
2
g z
φ g− = 2i sin z
2
+
g
‘p’-like wave, node at the
position of the nuclei
‘s’-like wave, belly at the
position of the nuclei
normal gap: ‘s’ states are lower in energy than ‘p’ states
However, if is Vg is negative the characters of the bulk
wavefunctions at the upper and at the lower side of the gap are
inverted:
the ‘p’ like wave has now a lower energy than the ‘s’ like wave.
This situation corresponds to an avoided crossing of the electron
states and is called inverted Shockley gap,
(present typically at the L point of the 3D BZ)
The density of states which cannot be in the gap moves to the
upper or to the lower branch
E
-2Vg
kz
kz=π/a
Surface States
Solutions corresponding to surface states have an imaginary wavevector since
they decay towards the bulk
kz=κ+iγ
Substituting the new kz in the eigenvalue equation we get for the energy:
which for κ=gz/2 becomes:
E g±z
2
+ iγ
2 2
g z2 γ 2
g
2
zγ
=
− ± Vg −
8
2
4
Since the energy is real by definition we must have Vg2>gz2 γ2/4
i.e. γ≤2Vg/gz or
Since at the borders of the gap γ=0, the imaginary part is largest close to the
center of the gap.
In general two solutions are expected for E± giving rise to two Shockley States
Let’s analyse the wavefunction of the surface states.
φ = ae
i ( κ + iγ ) z
+ be
i ( κ + iγ − g z ) z
which becomes at gz/2
Inserting E± and
substituting kz with κ+iγ
to find the a and b
coefficients of the
wavefunction.
For κ=gz/2 it becomes:
φ =e
− γz
(e
i
gzz
2
 k z2

− E±
 2

 Vg

 gz
2
 ( + iγ )
 2
− E±
2



Vg


b
+ e
a
−i
gzz
2
)
 a 
 
  = 0
2
(k z − g z )

− E ±   b 
2

Vg


 a
Vg
  = 0
b
gz

2
 
(−
+ iγ )

2
− E± 
2

 g z2 γ 2
ig z γ
±
) a + Vg b = 0
( − − E +
8
2
2

2
2
g
γ
Vg a + ( z − − E ± − ig z γ )b = 0

8
2
2
±
We then obtain either:
a
=
b
2
+ iγ
− Vg
g z2γ 2 ig zγ
m V −
+
4
2
2
g
g z2γ 2 ig zγ
m V −
+
±
a
4
2
=
b
− Vg
2
g
or:
Multiplying the two solutions we get
φ g = e − γz ( e
z
E g±z
2 2
g z2 γ 2
g
2
zγ
=
− ± Vg −
8
2
4
i
gzz
2
+ e
−i
gzz
2
a 2
| | =1
b
i.e.
a
= e 2 iχ
b
e − 2 iχ ) = e − γ z e − iχ ( e
i
gzz
2
is a phase factor
e iχ + e
−i
gzz
2
2
this is the wavefunction of the Surface State inside the crystal
e − iχ )
For the total wavefunction we have to match
amplitude and derivative (or the logaritmic
derivative) of the wavefucntion inside with
the wavefunction outside of the crystal at
z=0:
outside
Vsb
inside
φ' )
= =γ
φ
gz
)
− γ − tan χ = γ
2
matching is possible only for negative tan χ
values, i.e for -½π<χ<0 or ½π<χ< π
Let’s analyse χ
At the border of the gap γ=0 and (a/b) ±=±Vg/|Vg|;
E- =gz2/8-|Vg| at the lower border and
E+=gz2/8+|Vg| at the upper border
if Vg<0
a/b=-1 at the lower border 2χ=π
a/b=+1 at the upper border 2χ=2π
at half gap γ ≈
| Vg |
π /a
π
a
=
b
− Vg
g z2γ 2 ig z γ
m V −
+
4
2
2
g
Vg
− Vg
a
1
a
=
=
= = −i
ig
γ
2
π
i
b
z
| Vg | i
2
2a
if Vg>0
at the upper border a/b=1 χ=0
at the lower border a/b=-1 χ=π
while at the center a/b=-Vg|(i|Vg|)=+i
so that
π < 2 χ < 2π
or
π
2
< χ <π
wave matching is possible
0 < 2χ < π 0 < χ <
π
2
no wave matching is possible
Conclusion:
For the existence of the Shockley Surface State Vg has to be
negative, i.e. the potential has to be less attractive on the nuclei
and the charge density accumulates outside of the outermost
atomic plane.
This condition is realized
for gaps originating from
the projection of the L
point of the 3D fcc
Brillouin zone which
presents an inverted
Shockley gap
Surface charge density is
largest outside of the
outermost atomic plane
Surface band structure of Au(111) spin resolved
photoemission spectra
Spin resolved photoemission spectra for Au(111)
Scheme of the surface band
structure of Au(111)
M. Hoesch et al., Phys. Rev. B, 69, R241401 (2004)
F. Reinert et al., Phys. Rev. B, 63, 115415 (2001)
Au(788) quasi-one dimensional Surface States
If the SS are above EF they are empty and
may be observed by inverse photoemission
Inverted
Shockley
gap at L
S3
S1 is an Image State, S2 and S2 Shockley States
Shockley Surface states on Si(111) 2x1
one dimensional π bonded chains
along (0-11) Γ-J
Interband transitions induced
by infrared photons
(Attenuated total reflection)
anisotropic signal
- transitions possible only when
electric field is along the π
bonded chains
Notice: gap 0.6 eV instead of
1.2 eV as for the bulk
Si(111) 7x7 quasi - elastic peak width
in electron scattering (HREELS) due to
the Shockley Surface States. Excitation
of a low energy surface plasmon on
the metallic surface
Peak shape vs
crystal
temperature:
The peak
broadens
with T
Localized states: Surface Tamm states
The free electron model cannot be applied to localised states as the d-states. The
tight binding model, describing the electron wavefunctions as the superposition
of atomic states is then more appropriate. Let’s take a one-dimensional chain:
ψ = a0φ0 + a1φ1 + a2φ2 + ...
φ0 orbital on the surface atom (0), φ1 orbital on the subsurface atom (1), ...
The coefficients a may have real and imaginary parts and must satisfy the
Schroedinger equation:
where the atomic energy levels, corresponding to the diagonal elements, are set to
zero in the bulk, while the off diagonal elements denote the hopping probability
between nearest neighbours. v is the energy shift of the surface atom caused by
the broken bonds. h implies a dispersion of the bulk band which extends from
-2h<E<2h
Localized states: Surface Tamm states
We seek for solutions in which each coefficient is related to
the next by a factor α
Substituting we obtain:
From the first row :
and from the second and following rows:
This system can be solved graphically for the
unknowns E and α.
The bulk band extends over
Surface solutions must have |α|<1, so that the wavefunction decays away from the
surface and E>2h or E<2h so that it lies outside of the bulk band (BB).
This is possible for:
for v/h>1 at positive α and for E>2h
for v/h<-1 , negative α and E<-2h
Surface Tamm state pulled out above BB
Surface Tamm state pulled out below BB
Localized states: Surface Tamm states
Either v is large or h is small.
Case of Ag(100). The potential v at the surface
atoms is only slightly less attractive than in the
bulk (positive v). Tamm Surface State at M-bar:
Matrix element in photoemission <f|A·p|i>
with A magnetic vector potential lying in the same
direction as the electric field of the photon and p the
momentum operator. Matrix element has to be even.
Γ-M corresponds to a mirror plane of the surface, dxy
(i.e. |i> ) is odd with respect to it, while |f> is even.
Image potential states
Electrons may be trapped inside the image potential if their energy is in the band
gap and they cannot propagate into the bulk. This gives rise to an approximately
triangular well.
Reminding that the image potential has the asymptotic form: V ( z ) ≈ −
1
4 | z − z0 |
Neglecting z0 and looking for solutions like:
Inserting into the Schroedinger eq we obtain
This eq is identical to the one of the hydrogenic atom with nuclear charge Z
Substituting back
and Z=1/4 we get:
Hence is therefore an infinite number of image states and they are dense
at the vacuum level
Image potential states
Image states are close to the
vacuum energy and are
therefore empty:
Inverse photoemission
experiments
If the SS are above EF they are empty and
may be observed by inverse photoemission
Inverted
Shockley
gap at L
S3
S1 is an Image State, S2 and S2 Shockley States
Image potential state measurement
by selective adsorption and desorption
of electron in HREELS
2PPE
In conclusion
the existence of the surface states depends on the
atomic structure.
Shockley states depend on the scattering properties of
the atoms,
Tamm states originate directly from the superposition of
atomic orbitals,
and the image states depend on having a band gap
(originating from the bulk band structure)
The Jellium Model
This model considers the positive charge of the
nuclei smeared out over the unit cell into a
positive uniform background, a valid first
approximation for free electron metals.
It gives a reasonably accurate picture of charge
density , inner potential and work function
The electrons feel a positive potential V+ , the negative electrostatic potential due to
all the other electrons VH (Hartree potential)
where
is the
charge density.
and the exchange correlation potential Vxc .
The latter is a functional of the electron density and depends on the variation of the
electron density with position. Since the functional is unknown it is usually set to
the value of a uniform electron gas with a density equal to the local density
Vxc(r)=Vxc(ρ(r))
The homogeneous electron gas and density functional theory
The Jellium Model
The exchange potential Vxc(r) is
determined by consistency, iterating the
calculation until input and output values
are the same
The density is thereby calculated from the
standing waves generated from the fact
that the electron waves are reflected from
the surface
Tricks to achieve self-consistency include to add only a small fraction
of the calculated additional density to the next iteration
The Jellium Model
The work function doesn’t vary much
for the different elements since the
higher electron density is compensated
by the increased surface dipole
The actual work function value
depends on the surface density which
depends on the crystallographic face
Slab calculations
The difficulty :dealing with the absence of
periodicity in the vertical direction.
Trick compute the wavefunctions for a 5 to 21
layer thick slab.
periodic wavefunctions in the x-y plane and
standing waves in the z direction (expanded in sin
and cos terms with coefficients which satisfy the
Schroedinger equation)
The wavefunction basis has to be augmented with
atomic like functions to describe the rapid
oscillations close to the nuclei
Ni(100)
Slab calculations