Textbook on Surface Science
Klaus Wandelt
WILEY-VCH Verlag Berlin GmbH
February 8, 2011
Contents
1
Electron dynamics at surfaces
(Th. Fauster, P. M. Echenique, and E. V. Chulkov)
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Theoretical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Electron-electron interaction . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Electron-phonon interaction . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Electron-defect interaction . . . . . . . . . . . . . . . . . . . . . . .
1.3 Time-resolved measurements . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Shockley surface states . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Image-potential states . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Adsorbate states . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.4 Semiconductor surface states . . . . . . . . . . . . . . . . . . . . . .
1.3.5 Lifetimes of hot electrons . . . . . . . . . . . . . . . . . . . . . . .
1.4 Energy-resolved measurements . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Photoemission lineshape analysis . . . . . . . . . . . . . . . . . . .
1.4.2 Scanning tunneling spectroscopy . . . . . . . . . . . . . . . . . . . .
1.5 Spatially-resolved measurements . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Scattering patterns at steps . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Scattering patterns in adatom arrays . . . . . . . . . . . . . . . . . .
1.6 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Electron-electron interaction . . . . . . . . . . . . . . . . . . . . . .
1.6.2 Electron-phonon interaction . . . . . . . . . . . . . . . . . . . . . .
1.6.3 Electron-defect interaction . . . . . . . . . . . . . . . . . . . . . . .
1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography
5
5
9
9
17
21
22
22
23
27
28
29
30
30
34
35
35
37
37
37
38
39
40
41
1 Electron dynamics at surfaces
Thomas Fauster1 , Pedro M. Echenique2 , and E. V. Chulkov2
1
Lehrstuhl für Festkörperphysik, Universität Erlangen-Nürnberg,
Staudtstr. 7, D-91058 Erlangen, Germany
2
Departamento de Fı́sica de Materiales, Centro de Fı́sica de Materiales CFM-MPC, and Centro Mixto CSIC-UPV/EHU,
Facultad de Ciencias Quı́micas, UPV/EHU,
Apdo. 1072, 20080 San Sebastián/Donostia, Basque Country, Spain and
Donostia International Physics Center (DIPC), Paseo de Manuel Lardizabal 4,
20018 San Sebastián/Donostia, Basque Country, Spain
1.1
Overview
Electron energy
In many surface processes energy is deposited and the surface is brought far off from thermal
equilibrium. In particular when photons or electrons hit the surface, the energy is transferred
initially to electrons in the surface region. By electron-electron scattering the excess energy is
distributed between many electrons on a subpicosecond time scale. These hot electrons transfer their energy to the lattice vibrations (phonons) on a significantly longer time scale, because
the electron-phonon coupling is much weaker than the electron-electron interaction. At this
time the deposited energy is still in the surface region and only on time scales of microseconds
Initial
excitation
s
n
tro
ec
-el
Hot
electrons
ron
ct
ele
ng
eri
tt
ca
s
n
no
o
-ph
Hot
surface
n
o
ctr
ng
eri
tt
ca
ele
on
on
no
o
ph
h
n-p
g
rin
tte
a
sc
Warm
bulk
Time
Figure 1.1: Pathways of energy relaxation from an initial excitation at the surface to thermal
equilibrium of the bulk-surface system.
6
1
Electron dynamics at surfaces
or more surface and bulk reach thermal equilibrium via phonon-phonon scattering, i. e. thermal conduction. These pathways of energy relaxation are sketched schematically in Fig. 1.1
illustrating electron energy or temperature as a function of time. If the scattering processes of
electrons with electrons or phonons occur near the surface, the related electron dynamics can
be studied with surface sensitive techniques. More conveniently, signatures of it can be found
by careful analysis of most obtained signals.
Before we turn to the details, it should be pointed out, that this energy relaxation scheme
is quite general and common. It happens when sunlight hits our skin leading to a warm,
comfortable feeling. More useful are schemes where not all the energy is converted to heat,
e. g. in the conversion of light to chemical energy in photosynthesis or to electrical energy in
a solar cell. These processes rely on a sufficiently long lifetime of the electronic excitation,
so charge can be transferred and used for the intended purpose. An understanding of electron
dynamics is therefore important in many areas of research and applications.
We now consider an excited electron at the surface, e. g. in a surface or adsorbate state. It
is going to be scattered and the possible processes can be divided into the following categories:
• Electron-electron scattering:
This most important process for energy relaxation leads to decay into bulk or surface
states at lower energy with the simultaneous creation of an electron-hole pair. The coupling to the electronic system can also include other quasiparticles such as excitons,
plasmons, and magnons which are not covered in this chapter.
• Electron-phonon scattering:
This scattering process changes mainly the direction of the electron motion and embraces
scattering to bulk bands as well as scattering within the surface state band. For molecular
vibrations the electron might also loose a significant amount of energy.
• Electron-defect scattering:
Real surfaces contain always a non-negligible amount of defects, such as steps or impurity atoms. The associated electron-defect scattering changes mainly the electron momentum. If we regard a phonon as a distortion of the ideal lattice it is in many aspects
similar to electron-phonon scattering.
• Electron transfer:
This mechanism is not related to energy relaxation processes, but it describes the contribution from the energy-conserving resonant electron transfer from an excited electron
state to bulk and/or surface states. The electron-transfer mechanism is important for resonance surface and image-potential states on clean or adlayer-covered surfaces. Defects
may also lead to a charge transfer between bulk and surface elecronic states.
Various scattering processes involving several surface states on the clean Cu(111) and
Cu(100) surfaces are sketched in Fig. 1.2. On the way to thermal equilibrium many of these
scattering processes occur and in particular electron-electron scattering leads to a cascade of
electrons and holes. All scattering processes apply as well to holes in the sense that an electron is scattered into the hole leaving another hole behind. The recombination of a hole with
an excited electron is unlikely due to the high density of valence electrons. Only electronelectron scattering is connected with a significant loss of energy of the primary electron.
7
Overview
Cu(111)
Cu(100)
6
n=1
Φ n=1n=2
4
ep,def
n=0
2
0
n=0
-0.5
ep,def
ee
Energy relative to EF (eV)
1.1
ee
EF
0.0
0.5 -0.5
0.0
Parallel momentum (Å-1)
0.5
Figure 1.2: Projected bulk-band structure (shaded areas) for the Cu(111) and Cu(100) surfaces
with intrinsic Shockley surface-state bands (n = 0) and image-potential bands (n ≥ 1). Arrows indicate possible electron-electron (ee), electron-phonon (ep) and defect (def ) scattering
processes.
Electron-phonon or -defect scattering involves coupling to atoms and changes mainly momentum. These observations are in agreement with the expectation for the scattering between two
particles from classical mechanics. Energy and momentum conservation hold also in quantum
mechanics, but one should be aware of momentum provided by reciprocal lattice vectors or
collective excitations in condensed matter physics. From an experimental point of view the
effect of electron-phonon and -defect scattering can be identified by a systematic variation of
the number of phonons through temperature or the concentration of defects through sample
preparation, respectively. The remaining many-particle effect of electron-electron scattering
may then be estimated by extrapolation to zero temperature and to zero defect concentration.
A successful study and understanding of the decay of electronic excitations in surface
states requires appropriate theoretical and experimental methods. The theoretical calculations
use different models depending on the interactions relevant for the scattering processes [1].
The decay rate due to inelastic electron-electron scattering, Γe−e , is normally calculated using the GW or GW + T-matrix approximation for the self energy [2, 3, 4, 5, 6, 7, 8, 9]. The
contribution to the decay rate from the energy-conserving resonant electron-transfer mech1e
anism, Γe−e
, can be evaluated by using a wave propagation method [1, 10] or/and Green
function method [1]. For the electron-phonon contribution, Γe−ph , the Eliashberg function
is used [11, 12, 13]. The contribution due to scattering by defects, Γe−def , has been treated
for the elastic case by a wave-packet propagation method [14, 15, 16, 17]. Three different
experimental techniques can be employed to study the decay of electronic excitations at surfaces. Photoelectron spectroscopy (PES) as described in detail in chapter . . . (see also Refs.
[18, 19, 20]) accesses the decay rate through the spectral lineshape and linewidth [21]. It is
limited to occupied states below the Fermi energy. The complementary method of inverse
8
1
Electron dynamics at surfaces
Table 1.1: Methods to study the various scattering processes, quasiparticles and measurement
domains.
Scattering
processes
Quasiparticles
Measurement
domain
electron-electron
electron-phonon
electron-defect
electrons
holes
time
energy
space
Theory
X
X
X
X
X
2PPE
X
X
X
X
X
X
X
—
PES
X
X
X
—
X
—
X
—
STM
X
X
X
X
X
—
X
X
photoemission permits the spectroscopy of unoccupied surface states [22, 23]. However, its
limited energy resolution does not allow to extract useful information on linewidths except
in favorable cases [24]. Scanning tunneling microscopy (STM) and in particular scanning
tunneling spectroscopy (STS)as described in detail in chapter . . . (see also Refs. [25, 26, 27])
obtains detailed information on the decay properties of surface states. The topographical
images monitor simultaneously the quality and detailed structure of the surface area under
investigation. Two-photon photoemission (2PPE) as described in detail in chapter . . . and in
Refs. [28, 29] in the time-resolved mode is the only technique which is able to study the decay
in the time domain [30, 31]. By combining this information with spectroscopic measurements
a very detailed picture of the electron dynamics emerges [1, 32].
Table 1.1 lists the different scattering processes and quasiparticles which can be studied by
the various techniques. Each of the techniques has its weaknesses and strengths in addressing
specific questions about the electron dynamics at surfaces. Photoelectron spectroscopy can
access only occupied states and is therefore restricted to the hole decay. Two-photon photoemission is most successful for measuring image-potential states (see chapter . . . ) but the
application to other states has a lot of potential. The scanning tunneling methods work particularly well for Shockley-type surface states. As illustrated in Table 1.1 the various methods
can tackle almost all the involved problems.
The theoretical description can specifically address the various scattering processes by
the development of appropriate models which are presented in Section 1.2. In experiment
the combined effect of all scattering processes is usually observed and only by extensive and
careful analysis the individual contributions can be disentangled. Therefore, the presentation
of the experimental methods is organized according to the physical observables as listed in
Table 1.1. The electron dynamics in the time domain is accessible only in time-resolved twophoton photoemission and is discussed in Section 1.3. Photoelectron spectroscopies as well
as scanning tunneling spectroscopy access the electron dynamics via the spectral line shape.
These aspects are presented in Section 1.4. Real space imaging of electron waves is possible
only in scanning tunneling microscopy which is the topic of Section 1.5. The final Section
1.6 compares the information obtained by the various experimental methods with theoretical
calculations and presents the current understanding of electron dynamics at surfaces.
1.2
Theoretical description
1.2
9
Theoretical description
As discussed in the overview, the decay rate of an excited electron or hole can be presented as
a sum of four contributions:
Γtot = Γe−e + Γ1e
e−e + Γe−ph + Γe−def .
(1.1)
The first term, Γe−e , describes the contribution from inelastic electron-electron (e − e) scattering. The second term, Γ1e
e−e , describes the contribution from the energy-conserving resonant
electron-transfer mechanism. The third contribution, Γe−ph , represents the electron-phonon
(e − ph) channel for relaxation of excited electron (hole) via e − ph scattering. This mechanism is the only one which carries the temperature dependence of the electron (hole) decay in
paramagnetic metals. The fourth term, Γe−def , describes elastic scattering of excited electron
by defects. Below we give a short description of the calculational methods. For more details
we refer the reader to respective original publications. Electron transfer in particular has been
reviewed recently in Ref. [1].
In this section atomic units (~ = e2 = m = 1) are used, unless stated otherwise.
1.2.1
Electron-electron interaction
The physics of electron dynamics in image-potential and surface states (as well as in bulk
states) can be qualitatively understood in terms of a few key ingredients such as initial and final
states, density of final states as well as screened electron-electron interaction and electronphonon coupling. In particular, the linewidth gets wider with (i) a larger overlap between
initial and final states, (ii) more states at the energy of the final states (Density of states (DOS)
factor), (iii) a larger imaginary part of the screened interaction and the associated weaker
screening and (iv) an increased electron-phonon interaction.
Quantitatively, in the framework of many-body theory the damping rate Γe−e , i. e. lifetime
broadening contribution for an excited electron with energy i > EF can be obtained on
the energy-shell approximation in terms of the imaginary part of the complex non-local selfenergy operator σ [2, 5] as
Z
Z
Γe−e = −2 d~r d~r0 φ∗i (~r)ImΣ(~r, ~r0 ; i )φi (~r0 ),
(1.2)
where φi and i are the eigenfunctions and eigenvalues of the one-electron Hamiltonian. For
a quantum state characterized by one-electron energy i and spin σ the self-energy can be
represented as [6, 7, 8, 9]
Σσ (~r, ~r0 , i , σ) = ΣGW
(~r, ~r0 , i , σ) + ΣTσ (~r, ~r0 , i , σ),
σ
(1.3)
where ΣGW
is the self-energy derived within the GW approximation [2]. In this approximaσ
tion for the self-energy (G means a single particle Green function) only the first term in the
series expansion in terms of the screened Coulomb interaction W is retained [2, 4]. The ΣTσ
is the self-energy contribution obtained from the T-matrix approximation which accounts for
multiple electron-hole, electron-electron, and hole-hole scattering [33, 6, 7, 8, 9]. The GW
10
1
Electron dynamics at surfaces
approximation is normally used to describe electronic structure and excited electron dynamics in paramegnetic systems while the GW+T method represents a generalization of the GW
approximation by including the higher-order self-energy terms that allow for the calculation
of the quasiparticle decay in ferromagnetic systems on the same footing as in paramagnets..
In the following subsections we discuss mostly the GW approximation. A very brief
description of the GW + T method we give in subsection 1.2.1.2.
1.2.1.1
GW Method
In this subsection we consider paramagnetic systems only and thus omit spin index. The imaginary part of the self energy is evaluated in the GW approximation, in terms of the screened
interaction and the allowed final states for the decay process:
X
0 ∗ 0
ImΣ(~r, ~r0 , i > EF ) =
φf (~r )ImW (~r, ~r0 ; ω)φf (~r).
(1.4)
f
Here ω = i − f and the prime in the summation indicates that the final-state energies lie
between the initial state and the Fermi level. For the holes these energies are below the Fermi
level.
The final expression for the inverse lifetime then becomes
Z
X Z
0
Γe−e = 2
d~r d~r0 φ∗i (~r) φ∗f (~r0 ) Im [−W (~r, ~r0 ; ω)] φi (~r0 ) φf (~r).
(1.5)
f
The screened interaction W is given by
W (~r, ~r0 ; ω)
=
v(~r − ~r0 ) +
(1.6)
Z
Z
d~r1 d~r2 [v(~r − ~r1 ) + K xc (~r, ~r1 )]χ(~r1 , ~r2 ; ω)v(~r2 − ~r0 ),
or in short notation
W = v + (v + K xc )χ v.
(1.7)
Here v is the bare Coulomb interaction and χ the linear density-density response function,
which is given by the following integral equation
χ = χ0 + χ0 (v + K xc )χ.
(1.8)
χ0 is the density-density response function of the non-interacting electron system:
χ0 (~r, ~r0 ; ω) = 2
X θ(EF − i ) − θ(EF − j )
φi (~r)φ∗j (~r)φj (~r0 )φ∗i (~r0 ).
i − j + (ω + iη)
(1.9)
i,j
In this equation η is an infinitesimally small positive constant. The kernel K xc entering
Eqs. (1.7) and (1.8) accounts for the reduction of the electron-electron interaction due to the
existence of short-range exchange and correlation effects associated with the probe electron
1.2
11
Theoretical description
(Eq. (1.7)) and with screening electrons (Eq. (1.8)). Most calculations that have been performed to date of the lifetimes of electrons and holes for surface and image-potential states
use the so-called Random Phase Approximation (RPA). In this approximation the exchange
and correlation kernel K xc is omitted from both Eqs. (1.7) and (1.8). Inclusion of exchange
and correlation effects in the screened interaction (Eq. (1.7)) and in the screening (Eq. (1.8))
act in opposite directions as the evaluation of the lifetimes is concerned [5].
1.2.1.2
GW + T Method
In the GW+T extension of the GW method the central quantity is the T-matrix operator which
is defined as a solution of the Bethe-Salpeter (BS) equation [34]
Tσ1 ,σ2 (1, 2|3, 4)
=
+
W (1, 2)Zδ(1 − 3) δ(2 − 4)
W (1, 2) d10 d20 Kσ1 ,σ2 (1, 2 | 10 , 20 )Tσ1 ,σ2 (10 , 20 | 3, 4) (1.10)
Here we use short-hand notation 1 ≡ (~r1 , t1 ) etc.. W is a screened potential and the kernel
Kσ1 ,σ2 is a two-particle propagator. In the case of multiple electron-hole scattering the kernel
(electron-hole propagator) is a product of electron and hole time-ordered Green functions
Kσe−h
(1, 2 | 10 , 20 ) = iGσ1 (1, 10 )Gσ2 (20 , 2)
1 ,σ2
(1.11)
For electron-electron scattering it is a product of two electron Green functions
Kσe−e
(1, 2 | 10 , 20 ) = iGσ1 (10 , 1)Gσ2 (20 , 2),
1 ,σ2
(1.12)
and for hole-hole scattering it is a product of two-hole Green functions
Kσh−h
(1, 2 | 10 , 20 ) = iGσ1 (1, 10 )Gσ2 (2, 20 ).
1 ,σ2
(1.13)
Diagrams used for the GW and T-matrix self-energy are shown in Fig. 1.3. With these
diagrams and kernel of Eq. (1.11) the self-energy term describing multiple electron-hole scattering can be expressed as
XZ
Σe−h
(4,
2)
=
−i
d1d3Gσ1 (3, 1)Tσ2 ,σ1 (1, 2|3, 4)
(1.14)
σ2
σ1
Similar equations can be obtained for multiple e − e and h − h scattering [8]. In general,
the screened potential W is energy dependent. However, to make computations feasible the
local and static approximation, W (1, 2) = W (~r1 , ~r2 )δ(~r1 − ~r2 )δ(t1 − t2 ), is frequently used.
Most calculations for excited electrons lifetimes have been done within this approximation.
For details, we refer the reader to Refs. [6, 8].
1.2.1.3
Screened interaction
The Fourier transforms of the screened interaction are very useful entities to understand the
physics of electron (hole) dynamics at surfaces. The screened interaction given by Eq. (1.7)
12
1
Electron dynamics at surfaces
Figure 1.3: Feynman diagrams for GW and T-matrix self-energy of an excited electron. A: GWterm; B: T-matrix direct terms with multiple electron-electron scattering; C: T-matrix direct
terms with electron-hole scattering; D: T-matrix exchange terms. The vertical wiggly lines
represent static screened potential, and the lines with arrows are Green functions. The time
direction is right. By changing time direction, one obtains analogous diagrams for the selfenergy of an excited hole. From Ref. [1]
can be written for the homogeneous electron gas using the Fourier transform parallel to the
surface:
Z
0
1
W (z, z 0 , ~qk , ω)ei~qk (~rk −~rk ) d~qk ,
(1.15)
W (~r, ~r0 , ω) =
2
(2π)
where
1
W (z, z , ~qk , ω) =
(2π)
0
Z
0
W (q, ω)eiqz (z−z ) dqz .
(1.16)
Here q 2 = qk2 + qz2 . In reciprocal space the screened interaction, Eq. (1.7), can be presented
by using RPA in terms of dielectric function (q, ω) = 1 (q, ω) + i2 (q, ω) as
W (q, ω) = −1 (q, ω)v(q)
(1.17)
In the static limit and for the case of a Thomas-Fermi dielectric function [35], (q) = 1 +
qT2 F /q 2 , we find
2π
W (z, z 0 , qk ) = q
qk2 + qT2 F
−
e
q
0
2
qk2 +qT
F |z−z |
,
(1.18)
1.2
13
Theoretical description
70
||
Im[-W(z=z’,q’=q||/qTF)]/ω rs
2
10
60
50
8
Total
40
6
Without surface
state contribution
Jellium
30
4
20
10
2
0
0
0
0.1
0.2
0.3
0.4
0.5
-30
-20
-10
0
10
20
q’
Figure 1.4: Left: Im −W (z = z 0 , qk /qT F , ω) /(ω rs2 ) for the case of a homogeneous electron gas as a function of the reduced q 0 = qk /qT F is a universal function independent of energy
ω and density parameter rs . Right: Diagonal part of the imaginary part of the screened interaction for the case of a Cu(111) surface, described by the model potential. The results obtained
omitting the surface state in the calculation of the response function and for a jellium model
with density parameter rs = 2.67 are also shown.
where the 3D Thomas-Fermi momentum qT F is related to the 3D density parameter rs of the
electron gas of density n0 = 3/4πrs3 (rs is given in units of the Bohr radii, a. u.).
qT2 F =
4 9π 1/3 −1
( ) rs .
π 4
The imaginary part of the screened interaction is given by
Z
0
4π
2
1
0
eiqz (z−z ) dqz .
Im −W (z, z , qk , ω) =
2π
(qk2 + qz2 ) 21 + 22
(1.19)
(1.20)
A simpler approximation appropriate for the case of low excitation energies, can be obtained
from the low-ω expansion of the imaginary part of the bulk RPA response function [36].
The case z = z 0 can be of guidance as the bulk limit of more complicated surface calculations. The diagonal part of the imaginary part of the screened interaction is then given
by
Z ∞
dqz
q
.
(1.21)
Im −W (z = z 0 , qk , ω) = 8ω
0
qk2 + qz2 (qk2 + qz2 + qT2 F )2
As a function of the variable qk0 = qk /qT F , ImW given by Eq. (1.21) scales as rs2 . In the
bulk the maximum of Im −W (z, z 0 , qk , ω) occurs at z = z 0 and is independent of the actual
value of z. The linear dependence on ω holds for almost the whole range of energy transfers
that are relevant to the dynamics of surface states.
14
1
Electron dynamics at surfaces
0.15
0.10
crystal edge
||
1
Im [-Σ(z,z';q =0,E )] (meV/bohr)
0.05
0.00
-0.05
300
250
200
solid
vacuum
150
100
50
0
-50
60
40
20
0
-20
-20
-15
-10
-5
0
5
10
15
20
z (a.u.)
Figure 1.5: Left: Imaginary part of the self-energy of the n = 0 surface-state hole,
Im −Σ(z, z 0 ; qk = 0, E0 ) , versus z, in the vicinity of the (111) surface of Cu, with the use of
the model potential. z 0 is fixed at −11.3 (bottom), 0 (middle) and 10 a. u. (top). Dotted lines
represent the contribution from transitions to the n = 0 surface state itself. Right: Same for the
n = 1 image-potential state. From Ref. [40].
The left part of Fig. 1.4 presents the diagonal part of the scaled imaginary part of the
screened interaction as a function of the reduced momentum qk /qT F . When scaled by the
energy ω and the square of the density parameter rs2 the diagonal part of the imaginary part of
the screened interaction is a universal function of the reduced variable qk /qT F .
Figure 1.4 shows at the right the diagonal part of the screened interaction for Cu(111)
calculated using the one-electron model potential described above. The result of neglecting the surface state in the calculation of the response function is also shown, together with
what would be obtained describing the system with a jellium model of density equivalent to
that of the 4s valence of Cu (rs = 2.67). The surface state makes a strong contribution to
Im [−W ] /ω at the surface. This contribution increases with ω and decreases with qk [37].
The enhancement of Im [−W ] /ω at the surface depends on the extent of the wave function
into the vacuum and is increased for lower work function [37, 38] or metal overlayers [39].
1.2.1.4
Electron self-energy
The imaginary part of the self-energy involves a sum over final states of the screened interaction weighted by the final-states wave functions (Eq. (1.4)).
Figure 1.5 shows the
results
of calculations of the imaginary part of the self-energy Im −Σ(z, z 0 ; kk = 0, En ) for the
n = 0 surface-state hole and the n = 1 image-potential state at the (111) surface of Cu. The
imaginary part of the self-energy is represented in these figures as a function of z and for a
fixed value of z 0 . In the bottom panels, z 0 is fixed at a few atomic layers within the bulk,
1.2
Theoretical description
15
showing that Im(−Σ) has a maximum at z = z 0 , as in the case of a homogeneous electron
gas. When z 0 is fixed at the crystal edge (z 0 ∼ 0), as shown in the middle panels of Fig. 1.5,
we find that Im(−Σ) is still maximum at z = z 0 , but the magnitude of this maximum is now
enhanced with respect to the bulk value. The top panels of Fig. 1.5 correspond to z 0 being
fixed far from the surface into the vacuum. In this case, the maximum magnitude of Im(−Σ)
occurs at z 0 ∼ 0 rather than for z = z 0 , as occurs in the case of the imaginary part of the
screened interaction. As the phase space available for real transitions from the n = 1 imagepotential state is larger than that from the excited hole at the edge of the n = 0 surface state
[E1 − EF > EF − E0 ], the imaginary part of the self-energy of the n = 1 image-potential
state is larger than in the case of the n = 0 surface-state hole. Dotted lines in Fig. 1.5 represent the contribution to the imaginary part of the n = 0 surface-state-hole self-energy from
transitions to the n = 0 surface state itself, the so-called intraband transitions. One sees that
at the vacuum side of the surface, intraband transitions dominate, while within the bulk they
represent a minor contribution.
1.2.1.5
Decay rates
Homogeneous electron gas For energies very close to the Fermi level and in the high density limit, substitution of the RPA response function of the homogeneous electron gas [36]
gives the well known Quinn and Ferrell [41] formula for the excited electron lifetime in a
3-dimensional (3D) system:
Γ3D = 2.5rs5/2 (E − EF )2
(1.22)
Here rs is given in units of the Bohr radii (a. u.). The energy difference from the Fermi energy
is given in eV and the inverse lifetime (lifetime broadening) is obtained in meV.
For a 2D electron gas the lifetime broadening is given by [42, 43]
Γ2D = −
E − EF
2q 2D
(E − EF )2
[ln(
) − ln( T F ) − 0.5].
4πEF
EF
kF
(1.23)
In the last equation the Thomas-Fermi screening vector for 2D space is qT2D
F = 2. The lifetime
τ can be easily evaluated from Γ τ = 658 meV fs.
Surface states The evaluation of the decay rate involves a double integral of the self energy bracketed with the initial-state wave functions (see Eq. 1.2). For an efficient evaluation
a free-electron approximation parallel to the surface simplifies the calculations considerably.
Different effective masses may be used for bulk and surface states. The results of such calculations for the occupied Shockley surface state on noble-metal (111) surfaces is shown in
Table 1.2.
Separate contributions from intraband (within the surface state itself) and interband (between bulk states and the surface state) transitions to the decay of Shockley surface-state holes
at the Γ point of the projected bulk band gap of the (111) surfaces of Cu, Ag and Au are displayed in Table 1.2. We also show the decay rate evaluated within a 3D electron gas model
(EGM) for a hole with energy at the bottom of the surface state band. The comparison of the
calculated decay rates shows that the EGM gives decay rates values which are significantly
16
1
Electron dynamics at surfaces
Table 1.2: Decay rates in meV of the Shockley surface-state hole at the Γ point of the noblemetal (111) surfaces. The decay rate Γe−e is decomposed into interband (Γinter ) and intraband
(Γintra ) contributions. Decay rates in a 3D electron gas model (ΓEGM , see Eq. (1.22)) of holes
with the energy of the Shockley surface-state at Γ are also displayed. From Ref. [37].
Surface
Energy(eV)
ΓEGM
Γinter
Γintra
Γe−e
Cu(111)
Ag(111)
Au(111)
−0.445
−0.067
−0.505
5.9
0.18
10
6
0.3
8
19
2.7
21
25
3
29
smaller than those obtained using the model potential. The EGM takes into account only 3D
transitions and neglects both band structure and surface effects. The results of the calculations
displayed in Table 1.2 show that these effects are crucial for the hole lifetimes in surface states.
In particular, intraband transitions within the surface state band itself (2D → 2D transitions)
contribute ∼ 80% of the total electron-electron decay rate. These transitions are more efficient
in filling the hole than those arising from bulk states 3D → 2D (interband) transitions because
of the greater overlap of the initial- and final-state wave functions which exists in the region
where the imaginary part of the screened interaction is larger than in the bulk (see Fig. 1.4
right).
Differences between the full interband calculations using the model potential and those
obtained from the electron gas model arise from (i) the enhancement of Im[−W ] at the surface, which increases the decay rate, (ii) localization of the surface-state wave function in the
direction perpendicular to the surface and (iii) the restriction that only bulk states with energy
lying outside the projected band gap are allowed. Both localization of the surface-state wave
function and the presence of the band gap reduce the decay rate, and therefore they tend to
compensate the enhancement of Im[−W ] at the surface. In the case of Cu(111) this compensation is almost complete, thereby yielding an interband decay rate that nearly coincides with
the 3D decay of free holes. However, as can be seen from Table 1.2, this is not necessarily
the case for other materials such as Ag or Au and depends in particular on the surface band
structure. In the case of the hole at the Γ point at Be(0001) the model potential interband value
is 40 meV while the 3D electron gas result is 90 meV [44].
Bulk states Calculations of Γe−e for bulk metals have been done by using various ab initio
methods for simple [46, 47, 48, 49, 50], noble [46, 47, 48, 51], and transition [6, 45, 49, 52, 53,
54] metals. These calculations showed that, in general, the excited electron decay rate Γe−e
(lifetime τe−e ) is strongly anisotropic, i. e. strongly momentum dependent [52, 53]. Only in
few simple and noble metals it is practically momentum independent in an energy interval
0–5 eV [46, 48, 51]. At the same time, momentum average lifetime τe−e shows a smooth
behavior as a function of energy. As an example we show in Fig. 1.6 experimental data and
theoretical results for τe−e in bulk Mo [45]. The measurements have been done by using thin
polycrystalline films. One can see that for energies 1–3 eV a good agreement is observed
between the experiment and calculations if the T-matrix contribution to the self-energy is
included. This contribution also leads to better agreement for other metals [45, 54]. However,
1.2
17
Theoretical description
80
50
2
τ(E-EF) (fs eV )
70
40
30
2
60
τ (fs)
50
40
20
10
0
0.0
30
0.5
1.0
0
0.0
2.0
2.5
3.0
3.5
E-EF (eV)
20
10
1.5
GW
GW+T
sheet
thin film
0.5
1.0
1.5
2.0
2.5
3.0
3.5
E-EF (eV)
Figure 1.6: Calculated (solid and dashed lines) and experimental momentum-averaged excited
electron lifetimes τ in Mo [45]. Open circles are lifetimes obtained from a polycrystalline
Mo film, filled circles are lifetimes measured on a polycrystalline Mo sheet. Inset, scaled
momentum-averaged electron lifetimes, τ (E − EF )2 .
the T-matrix effects are qualitatively important for the description of a spin-dependent lifetime
in ferromagnetic metals [6]. The inclusion of the T-matrix in the calculation results in a strong
spin-dependent Γe−e and good agreement with the measured data for spin-minority bulk states
in Fe and Ni [6].
1.2.2
Electron-phonon interaction
The phonon-induced linewidth broadening Γe−ph of surface states of energy E and momentum parallel ~kki that takes into account both the phonon absorption and emission processes
can be written as [55]
Z ωm
~
Γe−ph (E, kki ) = 2π
α2 F~kki (ω)[1 + 2n(ω) + f (E + ω) − f (E − ω)]dω, (1.24)
0
where f and n are the electron and phonon distribution functions, respectively, and ωm is the
maximum phonon frequency. The Eliashberg function α2 F (ω) which is the phonon density
of states weighted by the electron-phonon-coupling function g, can be written in a quasielastic
approximation as
X
ν
α2 F~kki (ω) =
|gi,f
(1.25)
(~qk )|2 δ(ω − ων (~qk )) δ(f − ~kki ),
ν,~
qk ,f
where ων (~qk ) is the phonon frequency, ν is the phonon index and the last δ function indicates
that we consider the quasielastic approximation [13], neglecting the change of the energy
of the scattered electron due to absorption or emission of a phonon. The electron-phonon
coupling function includes the matrix element between the initial (i) and final (f ) electron
band states. For a translationally invariant system the matrix element involves a z-integration
18
1
Electron dynamics at surfaces
only
X
1
ν
~ µ) · ∇
~ ~ Ṽ µ |ii.
gi,f
hf |
~q~k ν (R
(~qk ) = p
~k
Rµ q
2M N ων (~qk )
µ
(1.26)
In this expression i and f refer to the z-dependent wave functions φ(z). The static screening
of the electron-ion potential is used and thus one neglects the frequency dependence of the
ν
coupling function gi,f
(~qk ). The coupling function in Eq. (1.26) is the result of the standard
first-order expansion of the screened electron-ion potential Ṽq~µk with respect to the vibrational
~ µ . N is the number of ions in each atomic layer, M is the ion mass, µ is the layer
coordinate R
~ µ ) are the phonon polarization vectors. From Eq. (1.24) we easily obtain the
index and ~q~k ν (R
T = 0 result (n() = 0) for Γe−ph
Γe−ph (E, ~kki ) = 2π
Z
|E|
0
α2 F~kki (ω)dω.
(1.27)
For binding energies |E| > ωm the upper limit for the integration should be the maximum
phonon frequency ωm .
From the knowledge of the Eliashberg function [11, 12], the electron-phonon coupling
parameter λ can be calculated as the first reciprocal moment of α2 F
λ(~kki ) = 2
Z
0
ωm
α2 F~kki (ω)
ω
dω.
(1.28)
If the high temperature limit (kB T ωm , here kB is the Boltzmann constant) of Eq. (1.24) is
considered, Grimvall [13] has pointed out a very useful result which enables an experimental
determination of the mass enhancement parameter
Γe−ph (E, ~kki ) = 2π λ(~kki ) kB T.
(1.29)
The above equations show that the Eliashberg function α2 F is a basic function to calculate.
Given this function most of the interesting quantities can be calculated, such as the temperature and also binding energy dependence of the linewidth broadening and the electron-phonon
coupling parameter. However, this is not a simple task, as all the physics connected to the
electron-phonon interaction is buried in α2 F , the phonon dispersion relation, phonon polarization vectors, one-electron wave functions and the gradient of the screened electron-ion potential – the deformation potential. All these quantities, and finally α2 F for bulk metal electron
states, can be obtained from first-principles calculations [56, 57], whereas for electron states
on metal surfaces these evaluations are very time consuming [58]. Some approximations are
needed to make these computations feasible for surfaces.
In particular, the phonon dispersion relations and polarization vectors can be calculated
with reasonable accuracy using force constant models [59] or the embedded atom method
[60, 61, 62]. In recent calculations of Γe−ph and λ for surface states, wave functions obtained
from the one-electron model potential [63, 64] have been used. For the description of the
deformation potential the screened electron-ion potential as Rdetermined by the static dielecµ
(z 0 ; qk ),
tric function and the bare pseudopotential is used, Ṽqµk (z) = dz 0 ˜−1 (z, z 0 ; qk )Ṽbare
1.2
19
Theoretical description
Table 1.3: Eliashberg function and phonon contribution to decay for Einstein and Debye models.
α2 F (ω)
Einstein
(Ref. [13])
Debye 3D
(Ref. [13])
1
2 λωE δ(ω
Γe−ph
− ωE )
2
λω 2 /ωD
0
ω < ωD
ω > ωD
2
Debye 2D (λ/π)ω/(ωD
− ω 2 )1/2 ω < ωD
(Ref. [66])
ω > ωD
0
πλωE
0
E > ωE
E < ωE
2πλωD /3
2πλωD (ω/ωD )3 /3
E > ωD
E < ωD
2λω
E > ωD
qD
E 2
2λωD (1 − 1 − ( ωD ) ) E < ωD
30
ω(meV)
20
10
b)
a)
0
0.02
Γ
Μ
0.06
0.10
2
0.14
0.18
α F(ω)
Figure 1.7: (a) The phonon-dispersion from a 31 layer slab calculation in the ΓM direction
of the Surface Brillouin Zone. (b) The Eliashberg function of the hole state in the Γ point for
Cu (111) (solid line) and the contribution from the Rayleigh mode to the Eliashberg function
(dashed line). From Ref. [67].
where qk is the modulus of the phonon momentum wave vector parallel to the surface and
µ
the 2D-Fourier transform parallel to the surface of the bare electron-ion pseudopotential
Ṽbare
[65].
ν
A drastic simplification is obtained by taking gi,f
(qk ) to be constant. In this case the
Eliashberg function is proportional to the product of the phonon and electron densities of
states. Assuming a free-electron-gas model for the one-electron states one can obtain simple analytical expressions for the Eliashberg function and for the phonon contribution to the
linewidth broadening Γe−ph within the Debye and Einstein models (see chapter . . . ) for vibrational spectra in terms of the energy-dependent electron-phonon coupling parameter and
the characteristic Debye (Einstein) energy ωD (ωE ). They are listed in Table 1.3.
20
1
Electron dynamics at surfaces
Energy
n=3
n=2
n=1
Parallel momentum
Figure 1.8: Adsorbate-induced scattering of the Cu(100) image-potential state electrons. The
figure shows the energy of the image-potential states (dashed lines) as a function of the electron
momentum parallel to the surface, kk . The shaded area represents the 3D bulk states. The
intraband and interband scattering processes, that lead to dephasing and population decay of the
image-potential states and that are allowed by energy conservation, are indicated schematically
by horizontal arrows. Gray arrows indicate the intraband scattering and black arrows indicate
the interband scattering. From Ref. [1]
A proper theoretical analysis of the electron-phonon contribution to the linewidth broadening of surface electron states requires to take into account all electron and phonon states
involved in the electron-phonon scattering process. The theoretical analysis of Eiguren et al.
[67, 68], based on a calculation of the full Eliashberg spectral function, is a step forward in
this direction. In this approach the contributions from different phonon modes, in particular
the Rayleigh surface mode as well as bulk phonons and the general temperature dependence
are taken into account. They also obtain the high temperature behavior represented by λ.
Their approach is based on (i) Thomas-Fermi screened Ashcroft electron-ion potentials, (ii)
one-electron states obtained from the model potential (iii) a simple force constant phonon
model calculation that gives results for the phonon spectrum in good agreement with recently
published experimental data [69, 70, 71, 72].
In Fig. 1.7 the calculated phonon dispersion and the Eliashberg function at the Γ point
calculated by Eiguren et al. [67, 68] is presented for the Cu(111) surface. The Rayleigh
surface mode is split off from the bulk phonon band, which gives a lower energy peak in
the Eliashberg function of the hole state at the Γ point of about ∼ 13 meV in Cu(111). The
oscillations in the Eliashberg function reflect the finite number of layers of the model potential
calculation (31 layers) of electron wave functions and thus have no physical significance [67].
The use of the 3D and 2D Debye models in the evaluations of Γe−ph for holes in surface
states on the (111) surface of noble metals [69] and Al(100) normally leads to a fairly good
agreement with the results of more sophisticated calculations [55, 67, 68].
1.2
Theoretical description
1.2.3
21
Electron-defect interaction
The presence of defects (adatoms, vacancies, steps, . . . ) on surfaces results in the loss of
2D-translational symmetry and, in turn, to additional broadening of the image-potential and
surface states. Adatoms, in particular, can be very efficient in perturbing the dynamics of
excited electrons at surfaces. A single adatom on a surface acts as a scatterer for the excited
electrons traveling on the surface in e. g. image-potential or surface states thus perturbing their
dynamics. The adatom can also support transient excited states localized at this adatom. When
an excited electron moves in an image-potential state two different scattering processes are
possible, as shown in Fig. 1.8. In one process, the electron is scattered from an image-potential
state into a substrate bulk state or into another state of the image-potential state continuum.
This interband scattering process results in a decrease of the excited state population, i. e.
to population decay. In another energy-conserving process, the electron remains in the same
electronic band: it changes only its momentum. This intraband scattering process leads to the
decay of the coherence of the electron, without population decay (process also called ’pure
dephasing’). In the optical Bloch equations formalism which is very often used to analyze the
coherence dynamics of a system, these two processes correspond to the population lifetime T1
and to the pure dephasing time T2∗ .
For a low density of adsorbates randomly distributed on the surface one can assume that
the different scattering centers are independent and incoherent. In this case the corresponding
population decay rate ΓDecay and ’pure dephasing’ rate ΓDeph can be presented as [15, 16, 17]
ΓDecay
ΓDeph
=
=
kk n0 σinter ,
kk n0 σintra ,
(1.30)
where σinter and σintra are the scattering cross sections for interband and intraband scattering,
respectively. n0 is the adsorbate surface density and kk is the electron traveling momentum in
the initial state. Both population decay and ’pure dephasing’ contribute to the broadening of
the level, and the total broadening rate of the excited state is given by:
Γtotal = kk n0 (σinter + σintra ) = kk n0 σtotal
(1.31)
where σtotal is the total scattering cross section, sum of the interband and intraband scattering
cross sections.
As an example of scattering by adsorbates of an electron in image-potential states, Fig. 1.9
shows the adsorbate-induced decay rate of the n = 1 and n = 2 image-potential states on a
Cu(100) surface with Cs adsorbates [15]. The decay rate is given for a Cs coverage of the
surface equal to one Cs adsorbate per 1000 Cu surface atoms.
One can see that the adsorbate-induced decay rate exhibits a few ((1)-(5)) sharp structures.
The structure (1) is associated to the opening of the n = 2 image-potential state continuum as a
decay channel for scattering of an n = 1 electron. Structures labeled (2) and (5) are associated
to the thresholds of the n = 3 and n = 4 states. Other structures, like structures (3) and
(4), located slightly below the image-potential state continua thresholds are due to adsorbateinduced resonances, associated to the localization of the image-potential state continua [15].
The decay induced by Cs adsorbates on Cu(100) is quite efficient: for energies close to Γ, the
n = 2 decay rate induced by Cs adsorbates amounts to 1.5 meV for a 0.001 ML coverage.
This is comparable to the decay rate of the n = 2 image-potential state on Cu(100) which
22
1
3
4
b)
3
(2)
(4)
(1)
2
(3)
1
Decay rate (meV)
Decay rate (meV)
a)
0
Electron dynamics at surfaces
(4)
(5)
2
(2)
1
0
-0.5
-0.4
-0.3
-0.2
-0.1
Energy (eV)
-0.16
-0.12
-0.08
-0.04
Energy (eV)
Figure 1.9: (a) Total and partial decay rates (in meV) of the n = 1 image-potential state on a Cu(100)
surface induced by electron scattering from Cs adsorbates. The theoretical results [15] are presented as
functions of the electron total energy measured with respect to the vacuum level. The abscissa-axis starts
at the bottom of the n = 1 image-potential state continuum (-0.573 eV). The Cs adsorbate coverage
corresponds to 1 Cs adsorbate per 1000 Cu surface atoms. Assignment of the structures is explained
in the text. Solid line: total decay rate; dashed line: partial decay rate corresponding to the interband
transition into the n = 2 image-potential state continuum. (b) The same as above for the n = 2 imagepotential state. The abscissa-axis starts at the bottom of the n = 2 image-potential state continuum
(−0.177 eV). Solid line: total decay rate; dashed line: partial decay rate corresponding to the interband
transition into the n = 1 image-potential state continuum. From Ref. [15].
is in the 4–5 meV range [73, 74]. Thus, for Cs coverages in the few 10−3 ML range, the
lifetime of the n = 2 image-potential state is dominated by adsorbate scattering. So, even
trace concentrations of alkali adsorbates are able to significantly affect the dynamics of the
image-potential states at surfaces.
1.3
Time-resolved measurements
The most detailed and direct information on the electron dynamics at surfaces stems from
time-resolved two-photon photoemission experiments. The method is explained in Section
. . . and this section presents important examples for the electron dynamics at surfaces.
1.3.1
Shockley surface states
The Shockley surface states commonly found on many fcc(111) surfaces are occupied for
kk = 0 (see also chapter . . . ). The study of electron dynamics by time-resolved two-photon
photoemission, however, is restricted to unoccupied states, because the laser intensities required to induce a detectable change in the population of occupied states are close to the
damage threshold of the sample. The Pd(111) surface is one notable exception where the
Shockley surface state is found 1.35 eV above the Fermi energy EF [75]. With time-resolved
two-photon photoemission the lifetime of this state was measured to 13 fs as shown in Fig. 4
of Section . . . .
1.3
23
Time-resolved measurements
105
104
Intensity
Cu(001)
n=1
41.3 fs
n=2
150 fs
103
n=3 (4)
406 fs
102
101
0
500
1000
Delay (fs)
1500
2000
Figure 1.10: Two-photon photoemission signal for the lowest image-potential states on Cu(001)
as a function of pump-probe delay. From Ref. [32].
1.3.2
Image-potential states
The attractive image potential experienced by an electron in front of a metal surface leads
to a special class of surface states (see chapter . . . ). These image-potential states [76] exist
on many metal surfaces. The binding energies relative to the vacuum level are EB (n) ≤
0.85/n2 eV. The low values indicate a weak coupling to the surface. Correspondingly imagepotential states can have relatively long lifetimes (> 10 fs) and can serve as almost ideal
spectator states to monitor subtle changes of the surface [28, 77].
Time-resolved two-photon photoemission measurements for the lowest image-potential
states on the Cu(001) surface are shown in Fig. 1.10. Due the logarithmic ordinate axis the
exponential decay of the population appears as a linear decrease at large pump-probe delays.
The lifetimes for the n = 1 and n = 2 states are τ1 = 40 and τ2 = 150 fs, respectively.
Note that the curves cross at large delay times indicating that the n = 2 population exceeds
eventually the n = 1 population. The trace for the n = 3 state shows regular oscillations on
top of a linear slope corresponding to a lifetime τ3 = 400 fs. These quantum beats arise from
a coherent excitation of the n = 3 and n = 4 image-potential states by the short laser pulse
with a spectral band width comparable to the energy separation of the states [78]. From the
oscillation period of T = 117 fs the energy difference can be determined very accurately to
|EB (3) − EB (4)| = h/T = 35 meV (see Eq. 3 of Section . . . ).
The lifetimes τn shown in Fig. 1.10 depend strongly on the quantum number n and consequently binding energy EB . Data for the decay rates Γn = ~/τn are plotted in Fig. 1.11 as
3/2
a function of binding energy EB for several copper surfaces. For n ≥ 2 an EB dependence
indicated by the dashed lines is observed. It corresponds to the classical expectation of the
round trip oscillation of the electron in the potential well formed by the image potential and
the solid represented by a hard wall [79]. This picture assumes that decay processes occur predominantly when the electron hits the surface. Since the binding energy EB is proportional
to n−2 , it follows for the lifetimes τn ∝ n3 . The same result is obtained when the probability
24
1
Electron dynamics at surfaces
100
10
n=1
Cu(117)
10
100
n=2
Cu(001)
n=3
Lifetime (fs)
Decay rate (meV)
Cu(111)
1
n=4
1000
n=5
0.1
Binding energy (eV)
1.0
Figure 1.11: Decay rates for the image-potential states as a function of binding energy for
Cu(111) (squares), Cu(117) (circles) and Cu(001) (diamonds). The dashed lines indicate an
3/2
EB dependence.
to find the electron in the bulk is evaluated from the wave function of the image-potential
state. The overlap with bulk bands explains also the increase of the decay rate from Cu(001)
to Cu(117) in Fig. 1.11. The energies of the image-potential states get closer to the effective
band gap and therefore the penetration of the wave function into the bulk becomes larger. The
limiting case is reached for Cu(111) where the states n ≥ 2 are degenerate with bulk bands.
The energy of the n = 1 state is still in the band gap and has a decay rate smaller than the one
of the n = 2 state. This state shows a dependence on temperature [80] which is negligible
on most surfaces where the image-potential states have only a small overlap with bulk states
[81].
The overlap of image-potential states with bulk bands can be reduced by insulating overlayers. In particular for noble gas overlayers the lifetime can be significantly prolonged into
the picosecond range [82].
1.3.2.1
Momentum dependence of lifetimes
The overlap with bulk states is an important factor determining the lifetimes of surface or
image-potential states. For dispersing bands the decay depends also on the parallel momentum. An example is shown in Fig. 1.12 for the n = 1 image-potential band on Cu(001) [83]
which follows the expected dispersion for a free electron parallel to the surface. The left panel
shows that the lifetime decreases with increasing parallel momentum. Two effects contribute
to the increase of the decay rate: (i) The energy increases with parallel momentum and the
available phase space for inelastic decay increases. (ii) Inelastic scattering processes within
the image-potential band become possible for electrons with energies above the band bottom
located at kk = 0. These inelastic intraband scattering processes gain importance with energy.
Both processes show a linear dependence on energy as shown in the right panel of Fig. 1.12.
25
Time-resolved measurements
Evac
101
k|| (Å–1)
n=2
0 0.1
Cu(001)
n=1
0.3
n=1
n=3
0
40
10
0.0
n=2
15
n=1
10–1
0
k||
0.00 Å–1
0.14 Å–1
0.24 Å–1
10–2
0.6
30
20
–1
0.0 k|| (Å ) 0.3
n=1
30
20
10
n=2
10–3
Lifetime (fs)
EF
Decay rate (meV)
2PPE–Signal (arb. units)
0.2
EB (eV)
1.3
40
experiment
full linewidth
60
interband contribution
100
0
–200
0
200
400
Time Delay (fs)
600
0.0
0.2
E|| (eV)
0.4
0.6
Figure 1.12: Left: Time-resolved 2PPE signal of the n = 1 state of Cu(001) for three different
values of the parallel momentum kk . Interband decay to the bulk and intraband decay within
the n = 1 band are the two basic kk -dependent decay processes mediated by bulk electrons
as sketched in the inset. Right: Experimental (dots) and theoretical (solid lines) decay rates of
the first two image-potential states n = 1, 2 as a function of excitation energy E(kk ) above the
respective band bottom. Dashed lines show the calculated contribution of interband relaxation.
Inset: Measured dispersion of n = 1, 2 parallel to the surface. Adapted from Ref. [83].
For the n = 1 band these inelastic scattering processes are calculated to contribute about the
same amount to the increase of the decay rate. For the n = 2 band interband scattering to
the n = 1 band is an important scattering channel and will be discussed in more detail in the
following section.
1.3.2.2
Elastic interband scattering
As indicated at the top right of the inset of Fig. 1.12 by a dashed arrow elastic scattering
between image-potential states is another important scattering channel. Because the lifetime
increases rapidly with quantum number n, electrons can still be scattered from the n = 2 state
to the n = 1 state when the initial population of the n = 1 band has already decayed (see
Fig. 1.10). This leads to a biexponential decay as seen in the time-resolved measurements of
Fig. 1.12. The elastic interband scattering involves a negligible change of energy, but a large
change of momentum. At the same time the electron moves closer to the surface. The elastic
character of the scattering can be proven by the fact that the decay rate of the second (slower)
decay measured on the n = 1 band is identical to the one of the n = 2 state at the same
energy [85] (see also top of right panel in Fig. 1.13). Detailed studies have shown that strong
elastic interband scattering is caused mainly by adatoms or steps [84, 85]. The scattering
of electrons by these heavy obstacles changes the momentum leaving the energy unchanged.
Electron-electron scattering on the other hand involves two particles of equal mass and leads
26
1
Electron dynamics at surfaces
8
Cu(119)
EB
2PPE intensity (arb. units)
II I
k ||
-1
61 meV, 0.38 Å
II
I
4
III
-1
81 meV, 0.36 Å
I
10
-1
121 meV, 0.34 Å
-1
256 meV, 0.29 Å
0
10
0
200
Delay (fs)
400
Percentage (%) Decay rate (meV)
10
40
ΓΙΙ
Γ2
10
Γ3
ΓΙΙΙ
Γ4
60
II
40
III
20
0
300
200
100
0
Binding energy (meV)
Figure 1.13: a) Time-resolved 2PPE measurements of the n = 1 image-potential state on
Cu(119) as a function of binding energy plotted on a semilogarithmic scale. Spectra are recorded
either for kk running upstairs (solid circles) and downstairs (open circles). Components I, II and
III are indicated by solid lines. b) Decay rates and percentage of components II and III. Adapted
from Ref. [84]
to an efficient energy transfer. The inelastic interband scattering from the n = 2 to the n = 1
band only amounts to a contribution of a few percent [85].
Elastic interband scattering is particularly effective at steps. An example is shown in
Fig. 1.13 for the Cu(119) surface [84]. The left panel presents the time-resolved data for
different parallel momenta and associated binding energies on the n = 1 band. At lower
binding energies the oscillations indicate that also scattering from the n = 3 (or higher)
bands with their characteristic quantum beats occurs. The intensity of the scattered component
shown in the bottom right panel of Fig. 1.13 has a maximum corresponding to almost 60% at
an energy ≈ 100 meV above the minimum of the n = 2 band. At this point the backfolding of
the bands by the periodic step array leads to a crossing of the n = 1 and n = 2 bands and very
efficient interband scattering [84, 86]. The elastic interband scattering shows a pronounced
asymmetry with respect to the electrons moving in the ’upstairs’ or ’downstairs’ direction of
the stepped surface. As seen in the time-resolved measurements of Fig. 1.13 the interband
scattering components is almost negligible for electrons in the downstairs direction (open
circles). A similar asymmetry is seen in the decay rates [84]. These asymmetries indicate that
scattering at steps involves also a significant probability for scattering to bulk states, but also
an asymmetry of the elastic intraband scattering might be relevant [86].
1.3.2.3
Elastic intraband scattering
The preceding section showed that elastic interband scattering can be an important process
induced by defects or steps. In elastic interband scattering the electron jumps between two
1.3
27
Time-resolved measurements
Photoemission intensity
Cu(001)
n=3,4
+ 0.004 ML Cu
+ 0.04 ML CO
Cu(001)
0
500
1000
Delay (fs)
Figure 1.14: Influence of adsorbates on the quantum beats on Cu(001). From Ref. [87].
bands of different quantum number which have a small overlap. Even more efficient is the
elastic intraband scattering, because it involves the spatial overlap between two states with
different parallel momenta in the same band. Elastic intraband scattering can not be detected
in the usual way by time-resolved two-photon photoemission, because it does not change
the population in the band. The scattering however leads to a change of the phase of the
wave function which contributes to the spectral linewidth. The details have been discussed in
Section . . . . The importance of the elastic intraband scattering for the n = 1 image-potential
band has been studied for Cu adatoms on Cu(001) [88, 89]. For higher image-potential states
the loss of phase coherence can be directly seen in the disappearance of the quantum beats.
An example is shown in Fig. 1.14 where small amounts of CO or Cu destroy the quantumbeat oscillations quite effectively. For Cu this is associated with the observed decrease of
the lifetime. For CO the lifetime remains almost unchanged indicating that CO molecules
are strong elastic scatterers and induce only little scattering to bulk bands or other inelastic
channels [87].
1.3.3
Adsorbate states
So far we have discussed the electron dynamics mainly for image-potential states. Due to their
relatively long lifetimes they serve as a model system to study electron scattering processes at
surfaces by time-resolved two-photon photoemission. Studies on other systems are relatively
scarce, because the overlap of adsorbate or surface states with bulk bands is usually relatively
large and the lifetime is below the attainable time resolution. Figure 1.15 shows time-resolved
data for the lowest unoccupied molecular orbital (LUMO) of C6 F6 on Cu(111) [90]. The
population dynamics is rather complex and involves a slow filling of the state as evident from
the delayed onset of the intensity.
28
1
C6F6/Cu(111)
state A
40
(b)
τdecay(A)
30
5 ML
τrise(A)
4 ML
20
3 ML
Time (fs)
2PPE intensity (norm.)
(a)
Electron dynamics at surfaces
τdecay(B)
2 ML
10
1 ML
SS
-100
0
100 200
Pump-probe delay (fs)
0
0 1 2 3 4 5
Coverage (ML)
Figure 1.15: Time-resolved 2PPE spectroscopy of the lowest unoccupied molecular orbital
(LUMO) of C6 F6 on Cu(111). From Ref. [90].
Other adsorbate systems with measurable lifetimes are C60 on Cu(111) [91] or alkali metals on copper surfaces [92, 93]. For recent reviews see [94, 95]. Resonant Auger spectroscopy
is an alternate approach to measure charge-transfer times for adsorbate systems [96].
1.3.4
Semiconductor surface states
At metals electron-electron scattering can proceed at arbitrarily small energies, because of
the continuum of states at the Fermi level. For semiconductors the band gap energy is the
minimum amount of energy needed to excite an electron from the valence to the conduction
band. Accordingly the decay of electrons close to the conduction band minimum can proceed
only via recombination which is a rather slow process. The same argument applies to electrons
at semiconducting surfaces with the associated surface band gap. Several examples have been
studied by Haight [98] and the field is expanding [99, 100, 101]. One particularly well-studied
surface is the Si(100) c(4 × 2) surface [97, 102, 103, 104]. The dimerization of the dangling
bonds leads to occupied and unoccupied surface bands. The unoccupied Ddown state shows
a complex dynamics as shown in the left panel of Fig. 1.16. The state is populated on the
time scale of 1.5 ps and shows a biexponential decay with time constants of 5 and 220 ps.
The excitation and decay scheme is indicated in the band structure diagram together with
calculated bands in the right panel of Fig. 1.16. The infrared laser pulse excites an electron
0
from the occupied Dup
band to the unoccupied Ddown band. Within 1.5 ps the electron relaxes
to the band minimum by phonon scattering. The formation of a bound exciton state occurs
within 5 ps and this state has a lifetime of almost 100 ns as can be estimated from the top
curve in the left panel of Fig. 1.16. The long time constant of 220 ps is attributed to the filling
1.4
29
Energy-resolved measurements
[011]
[011]
2.0
’
Ddown
X
220 ps
crosscorrelation
0
2 4
Delay (ps)
2
1
1.0
1.5 ps
Ddown
hν=1.71
Ddown
3
Energy E - EVBM (eV)
2PPE - intensity (arb. units)
4
X
5 ps
C
EF
0.0
Dup
Ddown
Si(100) c(4x2), 90 K
0
0
100
’
Dup
-1.0
-0.2
Delay (ps)
-0.1
0.0
0.1
0.2
Parallel momentum k|| (Å-1)
Figure 1.16: Left: Time-resolved 2PPE measurements from Si(100) c(4×2) in normal emission
recorded with the analyzer tuned to the peak maxima of X (top) and the Ddown -state (bottom).
The inset depicts the dynamics of Ddown population on an enlarged time scale; Right: Measured
(symbols) and calculated (solid lines, shaded areas) surface band structure of Si(100) c(4 × 2)
at 90 K. Adapted from Ref. [97].
of the surface band from the bulk conduction band minimum [103].
1.3.5
Lifetimes of hot electrons
At the end of the section on time-resolved measurements we want to mention the work on
lifetimes of hot electrons. These data are collected at energies where no specific surface state
exists and are, therefore, attributed to the lifetimes of hot electrons in the continuum of bulk
states. Because two-photon photoelectron spectroscopy is intrinsically surface sensitive some
contribution from the surface is inherent to such data. An example is shown in Fig. 1.17
for Ag(100) films of various thicknesses on a MgO(100) substrate [105]. The left panel of
Fig. 1.17 shows two-photon photoemission spectra and the right panel presents the measured
lifetimes. The lifetimes decrease approximately with (E − EF )−2 as expected for a free
electron gas [41]. Further information about this topic can be found in Refs. [5, 105, 106, 107].
30
1
70
60
2PPE spectra
15 nm
20 nm
30 nm
50
silver
15 nm
20 nm
30 nm
bulk
60
50
40
40
30
Lifetime (fs)
Counts (arb. units)
Electron dynamics at surfaces
20
30
20
10
10
0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
1.0
Intermediate state energy (eV)
1.5
2.0
Intermediate state energy (eV)
Figure 1.17: Left: 2PPE spectra for three film thicknesses of Ag on MgO plotted on an energy
scale for the intermediate state above the Fermi level. Right: Measured relaxation times of
excited electrons For comparison data for bulk Ag are shown. Adapted from Ref. [105].
1.4 Energy-resolved measurements
1.4.1
Photoemission lineshape analysis
The complementary method to time-resolved measurements is the spectroscopy in the energy domain. Energy-resolved experiments determine the Fourier transform of the exponential decay in the time domain which yields a Lorentzian for the intrinsic spectral lineshape.
As mentioned in Section . . . , photoemission does not permit to separate inelastic and elastic decay processes, which both contribute to the linewidth. Because the energy resolution
of inverse photoemission is usually not sufficient for a lineshape analysis, this section is devoted to photoemission of occupied states providing an additional complementary aspect to
the time-resolved measurements of transiently populated unoccupied states by two-photon
photoemission. Occupied (initial) states can also be observed by two-photon photoemission
spectroscopy. Their linewidth is similar to the values obtained in regular photoemission [108].
In measured data the intrinsic Lorentzian lineshape is broadened by finite resolution of
the experiment. In photoemission this comprises the limited energy and angular resolution
as well as the spectral linewidth of the photon source. These factors can be accounted for
by a convolution of the Lorentzian lineshape with a Gaussian of an appropriate width. The
demands on the energy and angular resolution can be fairly high for the strongly dispersing
Shockley surface states as illustrated in Fig. 1.18 for the case of Cu(111) [21]. At the bottom
left a series of spectra is shown for different emission angles at a constant angular resolution
of ∆ϑ = ±0.4◦ . The bottom right panel illustrates the influence of the angular resolution on
the spectra at the band bottom. The influence on the observed linewidth is illustrated in the top
panel of Fig. 1.18 in context with the measured dispersion relation. Neglecting the influence of
a finite energy resolution it is obvious that for an angular resolution of ∆ϑ = ±3◦ the spectra
1.4
31
Energy-resolved measurements
EF=0
98meV
a)
Initial state energy (meV)
Cu(111) Γ
–100
ϑ
Contribution
to linewidth
T = 115 K
HeI
–200
∆ϑ = ±3°
59meV
110meV
–300
±1°
12meV
2meV
±0.4°
–400
∆ϑ
–0.2
–0.1
0.0
0.1
0.2
Parallel momentum k|| (Å-1)
ϑ=0°
b)
1.5
T = 115 K
c)
Cu(111) Γ
ϑ=3°
HeI
ϑ=5°
1.0
0.5
Normalized intensity
Intensity (103 counts/sec)
Cu(111)
T = 300 K
HeI
∆ϑ=±3°
∆ϑ=±1°
∆ϑ=±0.4°
–0.6
–0.4
–0.2
Initial state energy (eV)
EF=0
–0.6
–0.4
–0.2
EF=0
Initial state energy (eV)
Figure 1.18: (a) Parabolic dispersion Ei (kk ) of the Cu(111) surface state around Γ, the center
of the surface Brillouin zone. Vertical solid lines indicate the contribution of different angular
resolution ∆ϑ on the observed linewidth of spectra taken at ϑ = 0 (kk = 0). Dotted lines
show the increase of broadening with increasing kk due to the dispersion, at constant ∆ϑ. (b)
Spectra taken at different ϑ with fixed ∆ϑ. Solid lines are results of a k-space integration, with
the experimental dispersion from (a) and a fixed ∆ϑ = ±0.4◦ . (c) Spectra taken at normal
emission with different ∆ϑ as indicated. All data collected at ~ω = 21.2 eV. Adapted from
Ref. [21].
become asymmetric and the linewidth exceeds 100 meV. A comparable value is obtained for
the best value of ∆ϑ = ±0.4◦ at an emission angle ϑ = 3◦ .
Using a state-of-the art two-dimensional analyzer with a resolution of 3.5 meV on the
energy axis and 0.15◦ in the angular direction the Shockley surface states for the (111) surfaces
on Cu, Ag, and Au have been measured with high accuracy [72]. The agreement between
various techniques for the intrinsic linewidth is excellent and is discussed in Section 1.6.1.
32
1
Electron dynamics at surfaces
-1.90
T=10 K
Γ=7 meV
-1.80
Energy (eV)
UPS-FWHM [meV]
Intensity (arb. units)
100
Cu(100)
SS at M
HeI
s-pol
-1.70
75
Cu(100)
SS at M
HeI
50
25
0
0.0
T=300 K
E kin =127 eV
2.5 5.0 7.5 10.0
LEED-FWHM [%SBZ]
Figure 1.19: Left: Tamm state at M measured at 10 K with an energy resolution of 5 meV. The
fit to the data gives a Lorentzian peak of 7 meV FWHM. From Ref. [109]. Right: Photoemission
peak width of the Tamm-type surface state observed at M on Cu(100) in its dependence on
surface disorder as monitored by the width of the corresponding LEED spots. From Ref. [71].
Such high-quality results require a careful sample preparation and are very sensitive to sample
contamination.
The more localized Tamm surface states show less dispersion and the angular resolution
is less important. As an example a spectrum and the corresponding fit for the M Tamm state
on Cu(100) is shown in the left panel of Fig. 1.19 [109]. The state shows a narrow intrinsic
linewidth of Γ = 7 meV even at an energy of −1.8 eV below the Fermi energy. This is
attributed to the localized character of the d-states and the small overlap with the sp-bands
which provide the main decay channel for inelastic decay.
1.4.1.1
Influence of defects
While striving to reach the best surface quality systematic, quantitative studies of the influence of defects on photoemission spectra have been somewhat neglected. As an example we
show the sensitivity of the M Tamm state of Cu(100) to the surface quality in the right panel
of Fig. 1.19 [71]. The surface quality was varied by argon-ion sputtering and subsequent insufficient annealing and monitored by the width of low-energy electron diffraction spots. The
observed linear dependence permits an extrapolation to zero defect density which yields a total
linewidth around 30 meV including the experimental resolution. The main difference between
the two experiments shown in Fig. 1.19 is the sample temperature. Extrapolating the data of
Ref. [71] to low temperature [32] yields an upper limit of Γ ≤ 13 ± 4 meV in reasonable
agreement with the value from Ref. [109].
1.4
33
Energy-resolved measurements
k-kF (Å-1)
.094
.063
EF
ωph
Be(0001)
Energy (eV)
Normalized intensities (arb. units)
-.1
.031
-.2
.016
-.3
-0.08
-0.06
-0.04
-0.02
0.00
0.02
k|| - kF (Å-1)
0
-.017
-.033
-.049
-.065
-.098
-1.0
-0.8
-0.6
-0.4
-0.2
EF
Energy (eV)
Figure 1.20: Left: Experimental surface-state linewidth W at Γ on Be(0001). The solid line
is a fit using the Debye model. The inset shows experimental spectra at Γ for various sample
temperatures (dots) and the corresponding fits using Lorentzians (solid line) plus a linear background. From Ref. [110]. Right: Photoelectron spectra from Be(0001) close to EF at 12 K. The
inset shows the measured and calculated quasiparticle dispersion. Adapted from Ref. [111].
1.4.1.2
Electron-phonon coupling
The most obvious influence of electron-phonon coupling on photoelectron spectra is the increase of the linewidth with temperature [58]. An example is shown in the inset of the left
panel of Fig. 1.20 for the case of a surface state on Be(0001) [110]. When plotted against temperature the linewidth increases linearly for sufficiently high temperatures. From this slope
the mass enhancement parameter λ can be determined using Eq. (1.29). The electron-phonon
coupling in Be is strongly enhanced at the surface [32, 110].
Strong electron-phonon coupling has also consequences on the photoelectron spectra. The
lineshape changes and split-off peaks appear [111, 112, 113]. The right panel of Fig. 1.20
shows the spectra of the surface state on Be(0001) near the Fermi energy measured at a sample
temperature of 12 K [111]. A sharp peak emerges at EF and the data show in addition to
linearly dispersing peak an additional feature saturating near the maximum phonon energy.
The dispersion E(kk ) is plotted in the inset and agrees well with the calculated quasiparticle
dispersion [112].
34
dI/dV (arb. units)
1
Ag(111)
Au(111)
Electron dynamics at surfaces
Cu(111)
30mV
23mV
8mV
–90
–70
–50
–700 –500 –300
–500 –400 –300
Sample Voltage (mV)
Figure 1.21: dI/dV spectra for the surface states on Ag(111), Au(111) and Cu(111). Adapted
from Ref. [69].
1.4.2
Scanning tunneling spectroscopy
Scanning tunneling microscopy uses the exponential dependence of the tunneling current on
distance and voltage to image the surface with resolution on the atomic scale (see chapter . . . ).
In the spectroscopic mode the voltage is varied at fixed distance and the current is sensitive
to the electronic structure of the surface (and tip). The high spatial resolution implies an
integration over a range of parallel momenta kk . Therefore it seems impossible to determine
energetic linewidths in scanning tunneling spectroscopy. However, the features in any energy
spectrum have some intrinsic width and in suitable experiments the measured information may
be related to the linewidth as measured with other techniques.
1.4.2.1
Spectroscopy of flat surfaces
The first method simply records the onset of the tunneling into the Shockley surface states.
The corresponding spectra are shown in Fig. 1.21 for the (111) surfaces of Ag, Au, and Cu
[69]. All spectra were taken at least 200 Å away from impurities and are averages of different
single spectra from varying sample locations and tips. In particular the Au(111) spectrum
constitutes an average over 17 single spectra taken across various positions across the surface
reconstruction (see chapter . . . ). A step-like onset is observed with a material-dependent width
∆. This width is in good approximation proportional to the intrinsic linewidth Γ measured in
photoelectron spectroscopy [114]: ∆ = π2 Γ.
1.4.2.2
Spectroscopy of confined electrons
The previous method is limited to the onset, i. e. to the bottom of the surface state band. In
order to select states at higher energies one can resort to select specific kk values by confining
electrons by suitable barriers. Figure 1.22 presents in the left panel a topographic image of
such a confining structure consisting of a ring of 35 Mn atoms on a Ag(111) surface. The
right panel shows the spectroscopy results for the Shockley surface state on Ag(111) obtained
at the center of the ring [115]. Pronounced peaks are observed which become broader with
increasing energy away from the Fermi energy. These peaks correspond to the energy levels
1.5
35
Spatially-resolved measurements
a
b
35 ATOM RING
dI/dV (arb.units)
Experiment
–100
Theory
0
100
200
Sample Voltage (mV)
Figure 1.22: (a) Circular array of 35 Mn atoms on Ag(111). Diameter ∼ 22 nm. (b) dI/dV
spectra from the center of the array as measured (top) and calculated (bottom). Adapted from
Ref. [115].
for a particle in a two-dimensional box with non-zero probability amplitude in the center. The
calculated spectrum shows a very good agreement with the experimental data and permits a
quantitative determination of the linewidth as a function of energy. This method inherently
includes some disturbance by the confining barriers which can be minimized for sufficiently
large structures. The spectroscopy of the onset on the other hand can be performed on large
defect-free terraces.
1.5
Spatially-resolved measurements
The spectroscopy of confined electrons presented in the previous section is closely related
to the topic to be discussed now. In the topographic image at the left panel of Fig. 1.22 a
circular fine structure is observed inside the ring of Mn atoms. This structure depends strongly
on tunneling voltage and correspondingly the spectroscopic results vary with the location
inside the ring where the measurements are performed. This leads to the topic of spatiallyresolved measurements which are somewhat easier to perform, because data are collected at
fixed voltage. In addition no confining structure is needed (see also chapter . . . ).
1.5.1
Scattering patterns at steps
The most simple structures are steps which are ubiquitous on real surfaces. The left panel of
Figure 1.23 shows spectra at a descending step located at x = 0 on a Cu(111) surface. The
current or more pronounced the differential conductivity dI/dV show an oscillatory pattern
as a function of the distance x from the step. This can be explained by the interference of the
electron wave reflected by the step with the original wave at the tip position [116]. The data
36
1
(a)
3
Triangle of 51 Ag
atoms on Ag(111)
E-EF = 1 eV
dI/dV (arb.units)
Electron dynamics at surfaces
Cu(111)
Topography
L φ = 89 Å
at V = 0.14 V
2
50 nm x 50 nm
(b)
Measured dI / dV
1
E-EF = 2 eV
of marked area
L φ = 31 Å
at V = 1.7 V
Lφ = ∞
0
(c)
Calculation
0
50
100
x (Å)
150
Figure 1.23: Left: dI/dV data perpendicular to a descending Cu(111) step obtained by averaging over several line scans of a dI/dV image. Solid lines indicate fits. The significance of the
deduced L is demonstrated by the dotted line: neglecting inelastic processes by setting L → ∞
leads to a much slower decay rate than observed. Adapted from Ref. [116]. Right: (a) Topographic image of a triangle constructed of 51 Ag atoms on a Ag(111) surface. (b) dI/dV data
taken at V = 1.7 V in the square marked in (a). (c) Calculated image for optimized parameters.
Adapted from Ref. [117].
can be fitted very well by the following function [118]:
x
dI
∝ 1 − |r| exp(− )J0 (2kk x)
dV
Lφ
(1.32)
The Bessel function J0 arises from the summation of all possible scattering paths and the
exponential term includes additional damping described by the dephasing length Lφ . The
arbitrary scaling and offset of the experimental data makes a fit of the reflectivity |r| difficult.
The dephasing length Lφ can be fitted reliably and shows a strong energy dependence as
illustrated in Fig. 1.23 (left panel). The analysis of the data yields also the dispersion relation
E(kk ) for the Shockley surface state in good agreement with results from photoemission or
inverse photoemission. Scanning tunneling microscopy or spectroscopy can be performed at
positive and negative voltages which makes occupied as well as unoccupied states accessible
(see also Fig. 1.22). The obtained dephasing lengths can be converted to lifetimes τ = Lφ /vg
using the group velocity vg = ∂E/∂~kk = ~kk /m∗ for a parabolic band with effective mass
m∗ .
1.6
37
Synopsis
1.5.2
Scattering patterns in adatom arrays
The scattering at steps permits an accurate determination of dephasing lengths, but is limited
by the necessity to find large enough defect-free terraces in order to exclude interference by
scattered waves from other structures. The damping of the oscillations by the finite dephasing
length is effective on any length scale and the disturbances by unwanted and unidentified
defects can be eliminated by confining the scattering to a well-defined artificial geometry.
Figure 1.23 shows in the right panel at the top a topographic image of a triangle constructed
by 51 Ag atoms on a Ag(111) surface [117]. The center image shows the dI/dV data taken
in the square marked in the topographic image at a higher tunneling voltage. The calculated
image using an optimized parameter set for the dephasing length and scattering properties of
the adatom array is presented at the bottom of the right panel of Fig. 1.23.
1.6
Synopsis
The previous sections presented the available methods to obtain information about the electron
dynamics at surfaces. This section compares the results obtained by different techniques according to the various scattering processes. More extensive tabulations of results and detailed
discussions may be found in recent reviews [1, 32].
1.6.1
Electron-electron interaction
1.6.1.1
Shockley surface states
The Shockley surface states on Cu(111) and Ag(111) are the best studied examples for the
electron dynamics at surfaces. They have been measured at low temperatures with photoelectron spectroscopy and scanning tunneling spectroscopy. The experimental results for the
bottom of the band at kk = 0 are listed in Table 1.4 and compared to calculations. The agreement is perfect and it is worth noting that even at temperatures < 30 K (see also Fig. 1.25) the
electron-phonon contribution is significant.
With increasing energy away from the Fermi energy the lifetimes decrease rapidly. Data
obtained for the Shockley surface state on Ag(111) by scanning tunneling spectroscopy at
steps [118, 119, 120] and in circles formed by Mn atoms [115] are shown in Fig. 1.24. For
comparison the results of a full calculation (circles [119]) and the free-electron gas behavior
(dashed line) matched to high energy asymptotic are shown. The decrease of the lifetimes for
Table 1.4: Calculated and measured decay rates Γ at the band minimum, in meV, for the Shockley surface states of Cu(111) and Ag(111). Γe−e is the calculated lifetime width due to electronelectron scattering [69], Γe−ph from electron-phonon scattering [68], ΓST M from STM measurements at T = 4.6 K [69], ΓARU P S from ARUPS measurements [72].
Cu(111)
Ag(111)
Γe−ph
7.3
3.7
Γe−e
14.0
2.0
Γe−e + Γe−ph
21
5.7
ΓST M
24
6
ΓARU P S
23 ± 1.0
6 ± 0.5
38
1
Ag(111)
100
Lifetime (fs)
Electron dynamics at surfaces
10
0.1
1.0
Energy (eV)
Figure 1.24: Lifetime for the Shockley surface state on Ag(111) for energies above the Fermi
energy measured by scanning tunneling spectroscopy at steps (filled squares [119]) and in confining structures (open diamonds [115]). For comparison the results of a calculation (open
circles [119]) and the free-electron gas behavior (dashed line) are shown.
Table 1.5: Inelastic and elastic scattering rates for image-potential states on on clean and Cuadatom covered Cu(001) [85, 121] compared with theoretical calculations [73, 83, 122]. The
inelastic scattering rates Γ are given in meV and the elastic scattering rates γ are given in
meV/%ML. The experimental error limits are typically 10%.
Inelastic
Experiment
Calculation
Elastic
Experiment
Calculation
Γ1b
15.9
17.5
γ1b
2.5
3
Γ2b
4.3
3.9
γ2b
< 0.2
1
Γ21
0.36
0.4
γ21
1.1
4
dΓ1 /dE
0.035
0.034
γ2b + γ21
0.9
5
dΓ2 /dE
0.010
0.006
γ11
8.0
17
Γ3
2.0
1.6
γ3
2.5
1.8
γ33
1.4
0.7
energies above 0.7 eV is attributed to the crossing of the surface band with the bulk band edge
and enhanced coupling to bulk states.
1.6.1.2
Image-potential states
The image-potential states on (001) surface of copper have been studied by several groups in
considerable detail. The inelastic decay rates for the image-potential states on Cu(001) are
given in the first two rows of Table 1.5. The agreement between experiment [85, 121] and
calculations [73, 83, 122]. is almost perfect. This holds for the various scattering rates Γ at
kk = 0 as well as for the increase with energy denoted by the slope dΓ/dE.
1.6.2
Electron-phonon interaction
The effect of the electron-phonon interaction can be observed most conveniently by an increase of the linewidth with temperature. An example has been presented in Fig. 1.20. For
energies close to EF the contribution to the decay from electron-electron scattering becomes
1.6
39
Synopsis
Linewidth (meV)
14
Cu(111)
12
10
8
6
4
2
0
20
40
60
80
Binding Energy (meV)
Decay rate (meV)
Figure 1.25: Left: Linewidth broadening of the Cu(111) surface hole state as a function of
binding energy, Γe−e + Γe−ph (solid line), Γe−ph (dotted line), the Rayleigh mode contribution
to Γe−ph (dashed line), and photoemission data (diamonds). Adapted from Ref. [67].
25
Cu(100)
20
n=1
n=2
15
10
n=3
5
0
0
0.01
0.02
0.03
Coverage (ML)
0.04
Figure 1.26: Decay rate the n = 1, 2, and 3 image-potential states on Cu(100) as a function of
Cu adatom coverage. (Adapted from [122].)
negligible which opens the opportunity to observe the effects of electron-phonon scattering
in more detail. As an example the linewidth broadening for the Shockley surface state on
Cu(111) is shown as a function of binding energy in the left panel of Fig. 1.25. The experimental data from high-resolution photoemission experiments are compared to calculations
[67]. The decrease towards low binding energies is explained by the decrease of phase space
for energy loss to phonons. The calculation reveals the coupling to the Rayleigh phonon modes
as a significant decay channel and the negligible contribution by electron-electron scattering.
1.6.3
Electron-defect interaction
The effect of electron-defect scattering has been studied in photoemission by Kevan [71, 123].
For the Shockley surface state an increase of the linewidth with increasing adsorbate coverage was observed. More detailed studies have been done for image-potential states. For Cu
and Co adatoms on Cu(001) it was found that adatoms cause mainly elastic scattering within
40
1
Electron dynamics at surfaces
or between image-potential bands as well as scattering to bulk bands [85, 88, 121, 89]. Figure 1.26 shows that the decay rate of the image-potential states increases linearly with Cu
adatom coverage for coverages up to a few percent of a monolayer [122]. The corresponding elastic scattering rates are compared quantitatively in the bottom rows of Table 1.5 with
calculations using a wave-packet propagation method [122].
Defects might also be created by thermal excitations through anharmonic effects. In such
cases special care has to be taken to separate the contributions from electron-phonon and
electron-defect scattering [124].
1.7
Conclusions
This chapter attempted to give an overview on the dynamics of excited electrons at surfaces
and the various methods to study it. Most of the techniques are still under active development
and the frontiers of their applicability are still under exploration. Considering the tremendous
progress in experimental techniques and theoretical methods during the last decade it is to be
expected that the field is moving rapidly from exploratory model systems to real life questions
in surface physics.
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