Download G - Dipartimento di Fisica

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

 
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, 
The electronic wavefunctions have to satisfy 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=2n, 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 - and + are propagating waves,  is a standing wave.
3D: analogous to 1D but more complicated situation since + 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:
      rGK ,G ; E
G
While the transmitted waves outside of the crystal
for a step like barrier (see figure) are :
  tG e

i ( K G )R )  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
 surf  | E  E0 |
SS
Relation between 3D and
2D Brillouin zones
Photoemission spectroscopy
The energy of bulk states depends on photon
energy, h, because they disperse with kz.
The surface state has on the contrary an
energy independent of kz.
Evac
EF
h
SS
Band narrowing in the tight binding picture of atomic orbitals
Tight binding scheme, α,I: localized atomic orbital α on atom I
  a , I  , I  a , J  , J  a , K , K  ...
The coefficients aα,I , aβ,J , … are complex number representing the contribution
to the state  of the atomic wavefunctions  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
bulk
Van Hove
singularities
surface
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
2
 gz
with g z 
and a periodicity in the z direction which opens a gap at k z   
a
a 2
giving bulk solutions for the Schroedinger equation of the form:
  aeik z  bei ( k
z
z
gz ) z
The eigenvalue equation H=E 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 
 
   0
2
(k z  g z )

 E  b 
2

Vg
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
) 2  4( z z
 Vg2 )}
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  Vg b  0


2

2
(
k

g
)

z
Vg a  [ z

E
]b  0

2

Vg
2
k

E  z
a 
2
yielding:

2
(
k

g
)
b E  z
z
2


Vg
from lower row


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


ae

be
2
g
g
b
z
z

 | Vg | 
8
 8

Vg


 e

gz
i
gz z
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
g z
  2 cos z
2
g z
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=π/a
kz
Surface States
Solutions corresponding to surface states have an imaginary wavevector since
they decay towards the bulk
kz=κ+i
Substituting the new kz in the eigenvalue equation we get for the energy:
which for κ=gz/2 becomes:
E gz
2
 i
2 2
g z2  2
g
2
z

  Vg 
8
2
4
Since the energy is real by definition we must have Vg2>gz2 2/4
i.e. 2Vg/gz or
Since at the borders of the gap =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.
  ae
i ( i ) z
 be
i ( i  g z ) z
which becomes at gz/2
Inserting E± and
substituting kz with κ+i
to find the a and b
coefficients of the
wavefunction.
For κ=gz/2 it becomes:
  e z (e
i
gz z
2
 k z2

 E
 2

 Vg

 gz
2
 (  i )
 2
 E
2



Vg


b i
 e
a
gz z
2
)
 a 
 
Vg
   0
2
(k z  g z )

 E   b 
2



 a
Vg
   0
b
gz

2
 
(
 i )

2
 E 
2

 g z2  2
ig z

(


E

) a  Vg b  0

8
2
2

2
2
g

Vg a  ( z   E   ig z )b  0

8
2
2

We then obtain either:
a

b
2
 i
 Vg
g z2 2 ig z
 V 

4
2
2
g
g z2 2 ig z
 V 


a
4
2

b
 Vg
2
g
or:
Multiplying the two solutions we get
 g  e z (e
z
E gz
2 2
g z2  2
g
2
z

  Vg 
8
2
4
i
gz z
2
e
i
gz z
2
a 2
| | 1
b
i.e.
a
 e 2 i
b
e  2i )  e z e i (e
i
gz z
2
is a phase factor
ei  e
i
gz z
2
2
this is the wavefunction of the Surface State inside the crystal
e i )
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
' 
 

gz

   tan   
2
matching is possible only for negative tan 
values, i.e for -½<<0 or ½<< 
Let’s analyse 
At the border of the gap =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  2=π
a/b=+1 at the upper border  2=2π
at half gap  
| Vg |
 /a

a

b
 Vg
g z2 2 ig z
 V 

4
2
2
g
Vg
a  Vg
1

 a
  i
2i
b ig z 
| Vg | i
2
2a
if Vg>0
at the upper border a/b=1  =0
at the lower border a/b=-1  =π
while at the center a/b=-Vg|(i|Vg|)=+i
so that
  2  2
or

2
  
wave matching is possible
0  2   0   

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
Semiconduttori: ricostruzione superficiale
• L’assenza di atomi limitrofi da un lato del cristallo altera le forze interatomiche
nei piani più vicini alla superficie
• Le condizioni di equilibrio per la superficie sono modificate rispetto al volume
• Le posizioni atomiche e la struttura atomica di superficie possono essere
differenti da quella attesa dalla terminazione del volume
• Differenza tra metalli e semiconduttori:
Metalli:
Gas di elettroni fortemente delocalizzato
Legami chimici essenzialmente non direzionali
Semiconduttori: Legami tetraedrici (Si, Ge, GaAs, InP, InSb…)
Legami fortemente direzionali
La rottura dei legami modifica in modo considerevole la configurazione
atomica di superficie
Semiconduttori III-V: GaAs
• Superficie di clivaggio: (110)
• Il legame sp3 del Gallio si deibridizza
per formare un legame di tipo quasi sp2
(planare).
• Ricostruzione superficiale
Ga
As
• Piccola variazione nella lunghezza dei
legami
• Inclinazione dei legami covalenti (27°)
Vista tridimensionale laterale
..
Silicio
Si(111)
•Un “dangling bond” per atomo di superficie
•Ricostruzioni principali: (2x1), (7x7)
•Sono possibili anche (5x5), (9x9)…
Configurazione più stabile
Transizioni di fase:
(2x1)
(7x7) a ~ 800 K irreversibile
(7x7)
(1x1) a ~ 1100 K reversibile
Si(001)
• 2 legami rotti per atomo di superficie
• Ricostruisce (2x1), (2x2), c(4x2)
Transizione di fase
(2x1)
c(4x2) a T ~ 200 K reversibile
stabile a temperatura ambiente
STM
Fornisce un’immagine nello spazio
reale della topologia di superficie su
scala
atomica.
Permette lo studio della struttura
elettronica di superficie fornendo
informazioni sulla presenza di stati
occupati o vuoti.
Immagini STM del Si(111) 2x1
Shift tra gli stati vuoti e pieni
“Buckling” Trasferimento di carica
tra atomi della superficie
Stati occupati
Stati vuoti
E’ necessario determinare un modello
strutturale che spieghi la
localizzazione spaziale degli stati di
superficie.
Modelli alternativi per spiegare la ricostruzione 2x1
La ricostruzione coinvolge gli atomi
del 2° strato, rompendo alcuni
legami e formandone di nuovi.
Si forma una catena di legami  a
“zig-zag”
Stati occupati
Stati vuoti
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 0.5 eV instead of
1.2 eV as for the bulk
Stati elettronici di superficie:
Il metodo della rifessione totale attenuata mostra una forte dipendenza
nell’assorbimento nell’infrarosso, alla frequenza corrispondente alla transizione
interbanda tra gli stati superficiali, dalla direzione cristallografica lungo la
quale viene allineato il campo elettrico dei fotoni.
Tale asimmetria corrisponde ad una asimmetria nella dispersione degli stati
elettronici superficiali, predetta correttamente dal modello di Pandey ma non
riprodotta da modelli in cui la sovrastruttura sia determinata solo dallo
spostamento verticale degli atomi alla superficie.
Spettroscopia locale: STS
Misure di conducibilità (dI/dV) fatte
con l’STM confermano la presenza di
densità elettronica superficiale sia per
stati pieni che per stati vuoti che
riproduce quella calcolata per il modello
di Pandey. Tale modello può pertanto
dirsi pienamente confermato.
Densità degli stati
di superficie
Densità degli
stati di bulk
Si(111) 7x7
Ricostruzione complessa
Modello, composto da due layers ricostruiti + 12 adatomi, caratterizzato da:
 12 adatomi che compensano alcuni dangling bonds dello strato sottostante
 6 atomi con un dangling bond per atomo
 9 dimeri sul bordo delle sottocelle triangolari
 vacanze profonde ad ogni apice della cella unitaria
 posizionamento errato (stacking fault) degli atomi nella sottocella di sinistra
Immagini STM del Si(111) 7x7
 Si possono notare le profonde vacanze agli apici
della cella.
 Le due metà della cella unitaria non sono equivalenti.
La differenza nella corrente di tunnelling non è
attribuita ad una reale variazione in altezza degli
atomi, ma è dovuta ad una differente densità degli
stati.
Immagini STM del Si(111) 7x7
Sequenze di immagini al variare della tensione
Immagine topografica della superfici (+2V)
Stato posto 0.35 eV al di sotto di EF
Stato localizzato sui 12 adatomi.
Si può notare l’asimetria della cella. La corrente
è maggiore nella metà con l’impilamento sbagliato.
A piccole tensioni positive si osservano immagini
del tutto simili
stati metallici di superficie
Stato posto 0.8 eV al di sotto di EF
Stato generato dai dangling bonds dei 6 atomi
del secondo strato che non sono legati
direttamente agli adatomi.
Si osservano i dangling bonds sul fondo delle
vacanze agli angoli della cella.
Stato posto 1.8 eV al di sotto di EF
Stato “back-bond” dovuto agli orbitali 3px e 3py
degli adatomi legati agli orbitali 3pz degli atomi
sottostanti
Si(001) 2x1
Le immagini LEED mostrano una ricostruzione (2x1). Diversi modelli furono proposti:
Si(100) non ricostruito
Missing Row
(scartato dai dati di fotoemissione)
Dimeri simmetrici
Dimeri asimmetrici
Come si formano i dimeri?
I due dangling bond sp3 per atomo di superficie si
deibridizzano formando orbitali quasi spz, px, py.
I primi danno luogo a legami , creando dimeri
nella direzione [110] con stati doppiamente
occupati.
Gli altri formano stati parzialmente occupati nel
piano della superficie di carattere px e py.
Dimeri
asimmetrici
Comportamento non-metallico
Dimeri
simmetrici
Comportamento metallico
Sperimentalmente non si osservano stati di superficie in prossimità nel livello
di Fermi.
La superficie è quindi semiconduttrice, in accordo con il modello asimmetrico.
Immagini STM del Si(001) 2x1
In questa struttura i dimeri danno origine a
dipoli, a causa del trasferimento di carica tra gli
atomi.
L’interazione tra i dimeri di righe vicine è
piccola e le strutture (2x2) e c(4x2) hanno circa
la stessa energia. Bastano, quindi, deboli effetti
termici per avere una transizione dalla fase
(2x1) a queste due strutture.
Si(111) 7x7 (n doped) 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. The pinned surface Fermi
level causes a band bending and a surface
depletion layer (d~1000 Angstrom) forms.
Peak shape vs
crystal
temperature:
The peak
broadens
with T
Fermi level pinning:
the work function of Si(111) (i.e. The position of the Fermi level)
is independent of the doping level of the bulk
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:
  a00  a11  a22  ...
0 orbital on the surface atom (0), 1 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.
Tamm surface state on Cu(111)
(Yang, PRB 54, 5092 (1996))
The energy of A,B e C depends on k 
 bulk states
The energy of S does NOT depend on k 
Surface State
GaAs(110):
Shockley or Tamm surface states?
Shockley or Tamm states
depending on the more
or less covalent nature of
the Ga-As bonds
Tamm state since this
band comes from a non
bonding orbital on the
As atoms
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 there 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
Image potential states extend far into the vacuum
Two Photon Photoemission : 2PPE
n=1
n=2
Note scale change!
Inserting a “buffer layer” the image potential states are stabilized
Dispersion of the image states parallel to the surface plane
In conclusion
the existence of the surface states depends on the
atomic structure.
Shockley states depend on the scattering properties of
the atoms and originate from propagating states
Tamm states are described by the superposition of
atomic orbitals and originate from localized bulk states
Image states depend on the presence of a band gap
(originating from the bulk band structure). Their energy
is determined by the detailed form of the surface image
potential
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 the positive potential, V+ , due to the ion cores, the negative
electrostatic potential due to all the other electrons, VH (Hartree potential),
where
is the
charge density
and the exchange correlation contribution to the 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
reflection of the electron waves from the
surfaces
Tricks to achieve self-consistency include to add only a small fraction
of the calculated additional density to the next iteration step
The Jellium Model: contributions to the work function
+
The work function doesn’t vary much
for the different elements since the
higher electron density is compensated
by an increased surface dipole
The actual work function value
depends on the surface electron
density which depends on the
crystallographic face
Surface Band Structure: Slab calculations and spaghetti diagrams
The difficulty: dealing with the absence of periodicity
in the vertical direction.
The 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 functions 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