Download First-Principles Studies of Surface Phonons and Electronic

thesis for the degree of doctor of philosophy
First-Principles Studies of Surface
Phonons and Electronic Properties
of Clean and Alkali Covered Metals
Vasse Chis
Department of Physics
University of Gothenburg
Gothenburg, Sweden 2008
First-Principles Studies of Surface Phonons and Electronic Properties of Clean
and Alkali Covered Metals.
Vasse Chis
ISBN: 978-91-628-7520-6
c
°Vasse
Chis, 2008
Department of Physics
University of Gothenburg
Gothenburg, Sweden 2008
Telephone +46 (0)31-7721338
(E-mail: [email protected])
Chalmers Reproservice
Göteborg, Sweden 2008
Cover: A compilation of results from this thesis.
Abstract
This thesis is focused on lattice dynamical properties of surfaces and electronic properties of clean and alkali covered metals. We give a detailed account
of the electronic states and vibrational modes using ab-initio calculation methods.
The atomic structure of a solid surface is of interest for the understanding
of surface processes from a fundamental and application point of view. We
describe how the surface localized electronic states influence the atomic positions at the surface with the example of Al(001). The relaxation of Al(001)
gives rise to surface and resonance phonon modes not previously reported,
such as two degenerate optical subsurface resonance modes known as Lucas
modes. Our calculations of Cu(111) surface lattice dynamics describe the longitudinal phonon resonance observed in Helium atom scattering and Electron
energy loss spectroscopy. It has been questioned whether this resonance is
an experimental artifact or due to a force constant anomaly in the surface.
The longitudinal phonon resonance is ascribed to a hybridization between a
first-layer longitudinal (L) mode and a strong second-layer shear-vertical (SV)
mode. The calculated projected density of states (PDOS) for the first-layer L
modes and the second-layer SV modes agree well with experimental data. Previous semi-empirical multipole expansion of the phonon-induced charge density
oscillations in the first layer shows a remarkable similarity to our PDOS for
the second layer SV displacements.
Adsorption of alkali layers on metal surfaces has been considered as perfect
systems for realizing nearly ideal quantum wells. When adsorption takes place,
for example K on Be(0001), charge is transfered to the interface inducing
a surface dipole moment which in turn yields a substantially lowered work
function. Furthermore, a variety of alkali induced electronic states emerge at
the surface/interface, such as quantum well states and surface resonances. At
saturation coverage of Cs on Cu(111) a not previously reported Cs induced
state appears. This state is peculiar in the sense that it is located above the
vacuum level, although strictly in to the Cs overlayer. Our analysis shows
that this behavior is due to the fact that a large fraction of the band energy
is attributed to kinetic energy parallel to the surface.
Keywords: Alkali metals, metallic quantum wells, surface phonons, electronic
structure, surface relaxation, surface states, phonon density of states, charge
density.
i
ii
List of Appended Papers
The thesis consists of an introductory text which provides a background for
the results presented in the following scientific papers:
I. Surface relaxation influenced by surface states
V. Chis and B. Hellsing,
Phys. Rev. Lett. 93, 226103 (2004).
II. Overlayer resonance and quantum well state of Cs/Cu(111)
studied with angle-resolved photoemission, LEED, and firstprinciples calculations
M. Breitholtz, V. Chis, B. Hellsing, S.-Å. Lindgren, and L. Walldén
Phys. Rev. B 75, 155403 (2007).
III. Two-dimensional localization of fast electrons in p(2×2)Cs/Cu(111)
V. Chis, S. Caravati, G. Butti, M. I. Trioni, P. Cabrera-Sanfelix, A.
Arnau, and B. Hellsing,
Phys. Rev. B 76, 153404 (2007).
IV. Sodium and potassium monolayers on Be(0001) investigated
by photoemission and electronic structure calculations
J. Algdal, T. Balasubramanian, M. Breitholz, V. Chis, B. Hellsing, S.Å. Lindgren, and L. Walldén
Submitted to Phys. Rev. B, (2008).
V. Evidence of longitudinal resonance and optical subsurface
phonons in Al(001)
V. Chis, B. Hellsing, G. Benedek, M. Bernasconi, and J. P. Toennies
J. Phys. Cond. Matt. 19, 305011 (2007).
VI. Optical phonon resonances at metal surfaces
V. Chis, B. Hellsing, G. Benedek, M. Bernasconi, and J. P. Toennies
To be submitted to Phys. Rev. Lett. (2008).
iii
iv
Acknowledgments
First of all I would like to thank Bo Hellsing, for all the support, guidance
and encouragement you have given me, from my diploma thesis work until
the present Licentiate thesis. I have enjoyed spending time in your presence,
whether is was private time or work related.
Special thanks to Giorgio Benedek and Marco Bernasconi for their hospitality and support during my visit as a Marie Curie Fellow at the University
of Milano-Bicocca, Italy. I am grateful to Prof. Benedek who has given me
invaluable information about surface phonons and physics in general.
The people who gets their hands dirty (the experimentalists) are acknowledged for their close collaborations with us: Lars Walldén, Stig-Åke Lindgren,
Markus Breitholtz and Jonathan Algdal.
A big thank you goes to my colleagues and the rest of the staff for their
help and for creating a pleasant and fun working environment.
I am thankful for the support, love and friendship of two special friends. My
partner, Pernilla Pihlström, and my friend Linda Landin. You are the best!
Finally, I would like to give a very special thought to my family and friends
for all their love and support. I wish you the best in life.
This work has been financially supported by the Swedish Research Council
(VR) and partially supported by an EU Marie Curie grant, No HPMT-CT2001-00242. Calculations in this work have been done using the QuantumESPRESSO computer program package [1]. Computer resources from the
Swedish National Infrastructure for Computing (SNIC) is acknowledged.
Vasse Chis
Göteborg
2008
v
vi
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Appended Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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v
1 Introduction
1
2 Quantum Well Systems: Alkali Overlayers
2.1 Quantum wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
3 Electronic Structure Calculations
3.1 Density functional theory . . . . . . . .
3.2 Exchange-correlation functionals . . . .
3.3 Plane waves and pseudo-potentials . . .
3.4 Relaxation . . . . . . . . . . . . . . . . .
3.5 Surface effects . . . . . . . . . . . . . . .
3.5.1 Bandstructure . . . . . . . . . .
3.5.2 Localized states . . . . . . . . . .
3.5.3 Charge density . . . . . . . . . .
3.5.4 Surface energy and work function
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25
4 Density Functional Perturbation Theory
4.1 Linear response . . . . . . . . . . . . . . .
4.2 Linearized Schrödinger equation . . . . . .
4.3 Force constants . . . . . . . . . . . . . . .
4.4 Phonons . . . . . . . . . . . . . . . . . . .
4.4.1 Phonon dispersion curves . . . . .
4.4.2 Phonon DOS . . . . . . . . . . . .
4.4.3 Induced charge density . . . . . . .
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vii
5
Summary of Appended Papers
5.1 Paper I: Surface Relaxation Influenced by Surface States . . . . . . .
5.2 Paper II: Overlayer Resonance and Quantum Well State of Cs/Cu(111)
Studied with Angle-Resolved Photoemission, LEED, and First-Principles
Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Paper III: Two-dimensional Localization of Fast Electrons in p(2×2)Cs/Cu(111) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Paper IV: Sodium and Potassium Monolayers on Be(0001) Investigated by Photoemission and Electronic Structure Calculations . . . .
5.5 Paper V: Evidence of Longitudinal Resonance and Optical Subsurface
Phonons in Al(001) . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Paper VI: Optical Phonon Resonances at Metal Surfaces . . . . . . .
Bibliography
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viii
Chapter
1
Introduction
In everyday life the first things we encounter when we wake up in the morning
and open our eyes are surfaces surrounding us with different structures and
colors.
In surface physics we study processes at surfaces down to the atomic level.
Understanding why and how different processes take place at the surface is the
aim of surface physics.
The surface acts as a boundary on the degree of freedom for electrons.
The electrons experience the interior of the material as a more attractive surrounding than the empty vacuum region outside the material. The surface
also introduces new possibilities for the electrons. Within the surface region
electrons can form additional states, known as surface states. Electron surface states confined between the bulk material and the vacuum region arise if
their energies is within the band gap of electron bulk states. Another type of
confinement of electrons is trapping in thin metal films, for example in alkali
metal films on a substrate with a band gap. The confinement in one dimension leads to quantum size effects - the energy spectrum depends on the film
thickness. These states are called quantum well states. Further understanding
of how to, in a controlled manner, create quantum well states may lead to new
applications like electronic devices.
The theoretical method that allows us to characterize the electronic structure of a fairly large collection of atoms like molecules or solids with high
accuracy is Density functional theory (DFT). DFT is a parameter free firstprinciples mean field theory which gives the ground state electron structure
given the atomic number. DFT can be used to study and predict properties of
new materials and surfaces, also for which experimental studies are inhibited.
1
1. Introduction
The relaxation of surfaces still poses challenging questions. Types of relaxations are (excluding surface reconstruction) inward [2], outward [3] and
oscillatory relaxation [4], respectively. In this thesis I have applied the DFT
scheme to analyze in details the mechanism responsible for the outward relaxation of the Al(001) surface. As a step towards a better understanding of the
outward relaxation we show how surface states contribute to this process.
Alkali covered metal surfaces are studied within DFT. Electrons confined
to the overlayer region form so-called quantum well states [5] by the vacuum
barrier on one side, and the substrate band gap on the other side. The aims of
these studies have been to characterize the alkali overlayer structure and the
alkali induced states.
Several complex processes take place at surfaces, for example adsorption,
desorption, growth, corrosion, desorption and catalysis. A deeper understanding of these process on a microscopic level demands a proper knowledge of
structural, electronic, and dynamical properties. At a surface, the atomic environment is different compared to the interior of the crystal. Due to the loss
of symmetry and periodicity we expect e. g that some vibrational modes are
present that differ from the modes of the perfect bulk material. These modes
are localized to and near the surface region, so-called surface modes.
An understanding of the collective vibrational modes, phonons, requires
lattice-dynamics calculations applying realistic models. A theoretical framework that allows a detailed determination of the phonon structure of materials is the Density functional perturbation theory (DFPT). DFPT is a firstprinciples theory based on DFT applying linear response.
The dynamical properties of the (111) noble metal surfaces, and in particular Cu(111) have been widely studied by means of simple models [6] and
more sophisticated ab-initio methods [7]. The interest in these surfaces arose
due to inconsistency between two experimental methods used for determining
the surface dynamical properties. One of the methods is based on electrons
probing the displacement of the atoms within the surface, while the second
method uses He atoms which are reflected by the phonon induced charge at
the surface.
Within this thesis the DFPT scheme is applied for calculating the dynamical properties of clean metal surfaces, Cu(111) and Al(001). The inconsistency
between the two experimental methods for Cu(111) is explained in terms of the
phonon induced charge density oscillations mediated by surface and subsurface
phonon modes.
This thesis is organized as follows. In Chapter 2 we give a brief presentation
2
to alkali metal overlayers acting as quantum well systems. In Chapter 3 we
present a short introduction to electronic structure calculations and a brief
description of DFT. In the subsections we discuss some surface effects which
also serve as an introduction to some of the results in Paper I, II, III and IV.
Chapter 4 presents the theoretical framework of DFPT for calculating lattice
dynamics from first-principles. At the end of this chapter we discuss some of
the results included in Paper V and VI. Finally, Chapter 5 contains a brief
summary of the appended papers.
3
Chapter
2
Quantum Well Systems:
Alkali Overlayers
Ever since the discovery by Langmuir [8] and Langmuir and Kindon [9], that
the current of electrons is greatly increased when a tungsten filament was exposed to various metal vapors, the properties of adsorbate-substrate systems
have been widely studied. Adsorption of alkali atoms on simple and noble
metal surfaces was long considered to be simple models of metal on metal
adsorption. During the past 20-30 years, experimental and theoretical studies have shown that the alkali-on-metal adsorption is far more complex then
thought [10, 11]. Certain interesting properties of the alkali-on-metal systems
derive from the fact that when the alkali atoms are adsorbed on the surface,
a significant fraction of its outer electron is displaced towards the metal surface inducing a surface dipole moment which reduces the work function of the
substrate [12].
The adsorption properties of a substrate is to a large extent determined
by the electron structure of its surface. Substrates exhibiting a local band
gap leads to quasi-localized electrons in the overlayer, confined between the
substrate and the vacuum barrier.
2.1
Quantum wells
In the introduction to quantum mechanics, a particle placed in a one-dimensional
square potential well is studied, the so called ”particle in a box”. Solving the
time-independent Schrödinger equation we obtain a discreet set of wave functions ψn (x) with energies ²n , where n = 1, 2, 3, .. (see Fig. 2.1a). The interpre5
2. Quantum Well Systems: Alkali Overlayers
tation of ψn (x) is that |ψn (x)|2 gives the probability distribution of the particle
in state n. The wave functions are strictly confined to the potential well. If
the potential is finite however, the wave functions decay exponentially outside
the barrier walls. The decay length is determined by the energy difference
between the state and the potential barrier hight.
Figure 2.1: Wave functions ψn of the four lowest particle states ²n in a onedimensional square well of width a and infinite (a) and finite (b) walls. Figure
from reference [13].
A atomic layers of alkali atoms adsorbed on substrates exhibiting a local
band gap constitute a system close to an ”ideal” quantum well system. A
potential with variations predominantly in the direction perpendicular to the
surface is formed and electrons are confined due to a barrier on the vacuum
side and the projected band gap on the substrate side. Adsorption of alkali
monolayers gives us some freedom to engineer our quantum well potentials.
For example, the depth of the potential depends on the choice of alkali atom
specie and the width of the overlayer potential depends on the number of alkali
layers.
The first discovery of an occupied quantum well state (QWS) was done
by Lindgren and Walldén [14] by means of angle resolved photoemission on
Na/Cu(111). An other popular experimental technique for probing QWS is
scanning tunneling spectroscopy [15, 16].
6
Chapter
3
Electronic Structure
Calculations
The quantum mechanical behavior of electrons in solids is in principle described
by the many-particle Schrödinger equation. The equation contains all the
physical information available, however in almost all cases this equation is far
too complex to solve.
Several methods to simplify the complexity of the many-particle equation
has been approached, such as the Free Electron, Nearly Free Electron and
the Tight-Binding model [17]. Non of the models treat the electron-electron
interaction explicitly. An early approach where the electrostatic interaction
between electrons is taken into account is the Hartree equations. The HartreeFock [17] methods extends the Hartree approximation to include also the effects
of exchange interaction. The Hartree-Fock method gives good results where
the effects of exchange are more important than the correlation effects. The
next step is to include correlation effects, which adds further complexity to
the problem since for realistic materials the correlation effect is impossible to
treat exactly.
A theory that, within a mean field approximation, includes both exchange
and correlation effects is the Density Functional theory (DFT).
3.1
Density functional theory
Density functional theory (DFT) has become the method of choice for calculating electron structure of most condensed-matter systems.
In 1964, Hohenberg and Kohn laid the ground stone for the DFT [18]. They
7
3. Electronic Structure Calculations
stated that if two electronic systems that have the same ground state electronic
density, then they must be equivalent in the sens that the ground state energy
and the external potential are the same. The result, which is the basis for
the DFT, implies that there exists a direct relation between the electronic
density and the ground state energy. If we could find an explicit formula for
the total energy, E[n(~r)], we could calculate the exact ground state density for
an arbitrary ionic configuration by minimizing the total energy. There exists
a functional, F [n(~r)], such that the total energy functional1 ,
Z
E[n(~r)] = F [n(~r)] +
n(~r)Vion (~r)d~r,
(3.1)
is minimized by the electron density of the ground state corresponding to the
external potential, Vion (~r). This states that all the physical properties of a
system of interacting electrons are uniquely determined by its ground state
electron density distribution.
Kohn and Sham [19] later showed that the total energy functional for the
interacting many-particle system can be transformed into one-particle equations. In order to determine the total energy, E[n(~r)], it is necessary to separate
the various known contributions to the total energy. These are, T0 [n(~r)], the
kinetic energy of a non-interacting electron gas, Eext [n(~r)] the coulomb energy
due to the ion potential, Vion (~r), ECoul [n(~r)] the classical energy due to the
mutual electron coulomb interaction and the many-body exchange-correlation
energy, Exc [n(~r)],
E[n(~r)] = T0 [n(~r)] + ECoul [n(~r)] + Exc [n(~r)] + Eext [n(~r)]
Z
≡ F [n(~r)] +
n(~r)Vion (~r)d~r.
(3.2)
When the strength of the electron-electron interaction vanishes, F [n(~r)]
defines the ground state kinetic energy of a system of non-interacting electrons
as a functional of its ground state electron density distribution [19]. Kohn and
Sham used this fact to map the problem of a system of interacting electrons
onto an equivalent non-interacting problem. The functional can then be cast
in the form:
F [n(~r)] = T0 [n(~r)] +
1
1 Z n(~r)n(~r0 )
d~rd~r0 + Exc [n(~r)].
0
2
|~r − ~r |
In the forthcoming equations we will use atomic units: me =h̄=e=1
8
(3.3)
3.1. Density functional theory
The last term contains the quantum-mechanical exchange and correlation energy and the difference between the true kinetic energy T [n(~r)] and T0 [n(~r)],
the kinetic energy of the non-interacting electrons.
Variations of the energy functional with respect to the electron density
formally leads to the same equation that would hold for a system of noninteracting electrons subject to an effective one-electron potential, known as
the self-consistent field (SCF) potential:
Z
VSCF (~r) = Vion (~r) +
n(~r0 )
d~r0 + vxc (~r),
0
|~r − ~r |
(3.4)
with
vxc (~r) ≡
δExc
δn(~r)
,
Vion (~r) = −Z
X
1
~
R
~
|~r − R|
,
where vxc is the functional derivative of the exchange-correlation energy, also
called the exchange-correlation potential and Vion the external ion potential,
respectively.
The problem of finding the ground state of an interacting many-particle
system is transformed into a mean-field problem where non-interacting electrons are moving in an effective potential. If one could determine the effective
potential VSCF (~r), the problem for non-interacting electrons could be solved
by applying the one-electron Schrödinger equation,
µ
¶
1
− ∇2 + VSCF (~r) ψn,~k (~r) = ²n,~k ψn,~k (~r).
2
(3.5)
The ground-state charge-density distribution and the non-interacting kineticenergy functional for a non-magnetic system are determined in terms of oneelectron wave functions (orbitals), ψn,~k (~r),
n(~r) = 2
X
|ψn,~k (~r)|2 ,
(3.6)
n,~k
and
T0 [n(~r)] = −
XZ
n,~k
∗
r)∇~2r ψn,~k (~r)d~r.
ψn,
~k (~
(3.7)
In periodic systems the sums in Eq. 3.6 and 3.7 runs over occupied states with
band index n and wave vector ~k belonging to the first Brillouin zone. Each of
9
3. Electronic Structure Calculations
the orbital states accommodates two electrons of opposite spin, thus the factor
2 in Eq. 3.6.
The equations, Eq. 3.4 - 3.6, are the famous Kohn-Sham equations, which
have to be solved self-consistently. This procedure yields the correct electron
density, which minimizes the total energy of the system. By assigning a trail
density, e.g. superposition of atomic densities, VSCF,in (~r) can be calculated according to (Eq. 3.4) and then inserted into the one-electron Schrödinger equation (Eq 3.5). Wave functions are then obtained by solving the Schrödinger
equation and a new density can be calculated (Eq. 3.6). The procedure is repeated until convergence is achieved for the density and total energy. The total
energy can also be expressed in terms of a sum over the occupied one-electron
energies ²n,~k
E[n(~r)] = 2
X
n,~k
²n,~k −
Z
1 Z n(~r)n(~r0 )
0
d~
r
d~
r
+
E
[n(~
r
)]
−
n(~r)Vxc (~r)d~r. (3.8)
xc
2
|~r − ~r0 |
DFT [18–21] yields an enormous simplification of the quantum-mechanical
problem of searching for the ground-state properties of a system of interacting
electrons. It replaces the description based on wave functions (which depends
on 3N independent variables, N being the number of electrons) with a description of the electron density which depends only on three variables x, y and
z.
3.2
Exchange-correlation functionals
So far, the self-consistent equations developed by Kohn and Sham are exact,
in the sense that there exists an exact formulation of the exchange and correlation energy. However, explicit formulas for the exchange-correlation energy
Exc are not known. Approximations for the exchange-correlation energy are
done within the so-called Local density approximation (LDA) and the Generalized gradient approximation (GGA), to name two of them. Within LDA, the
exchange-correlation energy of an electronic system is constructed by assuming
that the exchange-correlation energy per electron at a point ~r is equal to the
exchange-correlation energy per electron in a homogeneous electron gas with
an electron density as at the point ~r,
Z
LDA
Exc
[n(~r)] =
n(~r)²xc (n(~r))d~r.
10
(3.9)
3.3. Plane waves and pseudo-potentials
The connection between LDA and GGA is that LDA can be generalized to
include gradients of the electron density. The GGA [22] exchange-correlation
functional is a function of both the electron density and the gradients of the
electron density,
Z
GGA
[n(~r)] =
Exc
n(~r)²xc (n(~r), ∇n(~r))d~r.
(3.10)
In general, the GGA gives better results than LDA. LDA should work if
the electron density varies slowly. For surfaces however, the electron density
changes abruptly across the surface layers into the vacuum, this assumption is
harder to justify. In this case GGA is the obvious choice.
3.3
Plane waves and pseudo-potentials
In order to accurately describe the Kohn-Sham wave functions of a system, it is
necessary to choose a suitable set of basis functions in terms of which one can
write a series expansion of the electron wave functions. Several approaches
exists. One is to consider the most natural basis functions from real space
viewpoint, that is atomic-like basis functions.
Alternatively, one could employ a basis set more suitable for a momentum
space description of the material, that is plane waves. The momentum space
approach copes with extended systems such as surfaces in a much more natural
way because of its use of unit cells and periodic boundary conditions.
According to Bloch’s theorem [17], each electronic wave function can be
written as a product of a cell-periodic part and a wave-like part,
~
ψn,~k ∼ fn,~k (~r) · eik·~r .
(3.11)
Using a basis set consisting of a set of plane waves, we can expand the cellperiodic part of the wave function in terms of reciprocal lattice vectors,
fn,~k (~r) =
X
~
cn,~k+G~ eiG·~r .
(3.12)
~
G
Thus we have
ψn,~k (~r) =
X
~
~
cn,~k+G~ ei(k+G)·~r .
(3.13)
~
G
For practical reasons the plane wave basis set has to be truncated by choosing
a kinetic energy cutoff through the condition:
11
3. Electronic Structure Calculations
1~ ~ 2
|k + G| ≤ Ecut .
2
(3.14)
The electrons in a solid can be divided into two categories, core and valence electrons. The core electrons organize them self into closed shells which
screen the positively charged nucleus, while the valence electrons take part in
the bonding between atoms. The wave functions that describe the core electrons and the valence electrons oscillates rapidly in the region of the nuclei.
Using plane-waves as a basis set one would need a large number of expansion
coefficients to describe this region with a good accuracy. Fortunately, the core
electrons on different atoms are almost inert and only the valence electrons
participate in interactions between atoms. Hence, the core electrons may be
assumed to be fixed and a pseudo potential may be constructed which takes
into account the effects of the nucleus and core electrons.
The constructed pseudo potential should coincide with the true potential
at and beyond some given cut-off radius rc . At the cut-off radius and beyond,
the pseudo wave functions must match the corresponding true wave function,
while within the core region the pseudo wave functions are constructed to be
smoother than the true wave functions. Another desirable property is that
the pseudo potential is constructed to be norm conserving [23], which means
that the integral of the pseudo charge density will give the same total charge
inside the cut-off radius as the true charge. A second property, in order to
have optimum transferability, is that the logarithmic derivatives of the true
and pseudo wave functions agree at the cut-off radius. The smoothness of the
pseudo potential is essential in plane wave calculations to reduce the number
of expansion coefficients.
Further development of even softer pseudo potentials was proposed by Vanderbildt by introducing the concept of ultrasoft pseudo potentials [24]. As a
starting point the norm-conserving condition was dropped in such a way as
to optimize smoothness. The charge loss in the core region is compensated
by an augmentation charge, which is determined by the deviation between the
ultrasoft pseudo wave functions and the true wave functions.
3.4
Relaxation
A priori, the ground state atomic positions of extended systems of bulk terminated surfaces are usually unknown. In the course of DFT one need to find a
set of ionic coordinates that minimizes the total energy of the atomic system.
12
3.5. Surface effects
Several numerical minimization algorithms are used. One is the BroydenFletcher-Goldfarb-Shanno (BFGS) algorithm. The BFGS algorithm describes
the total energy of the system as a function of the atomic coordinates in the
unit cell, as Etot = Etot (R). Expanding the total energy, Etot to second order,
about the minimum energy of the collective configuration given by Rm yields,
Etot (R) = E(Rm ) + (R − Rm )
∂E 1
+ (R − Rm )i Hij (R − Rm )j ,
∂R 2
(3.15)
where Hij is the Hessian matrix, defined as
Hij =
∂ 2E
.
∂Ri ∂Rj
(3.16)
Taking the gradient of the total energy at the minimum energy configuration
results in that the first order term in Eq. 3.15 vanishes and the rest gives us
an expression for the forces as a function of the coordinates,
F(R) = −H (R − Rm ).
(3.17)
Inverting this expression gives us the equation for the minimum ionic position,
Rm = R + (H)−1 F. If the Hessian and the force are known the minimum
configuration could be determined directly. In general this is not the case and
the inverse Hessian matrix has to be calculated by iterative methods, taking
successive ionic steps and recalculating the Hessian in each step. This procedure is absolutely vital in order to obtain, in particular, the positions of the
surface atoms and hence a proper description of the ground state properties.
As an example of this procedure we show in Fig. 3.1 the iterative process
of a 15 layer Al(100) surface slab. The total energy is calculated for a supercell (described below) with one atom in each layer. Starting from a bulk-like
configuration, a total of 34 ionic steps are required until the minimum total
energy is achieved.
3.5
Surface effects
Electrons in a material experience the potential as a periodically repeated
entity. According to Bloch’s theorem an electron in a periodic potential can
be described by the product of a local, periodic function and a plane wave, Eq.
3.11. The power of the theorem is that all important properties of the crystal
can be studied within the unit cell.
13
3. Electronic Structure Calculations
-62.9438
Total Energy [Ryd]
-62.9439
-62.9440
-62.9441
-62.9442
-62.9443
-62.9444
0
10
20
# Iterations
30
Figure 3.1: Total energy versus number of ionic steps.
However, if the three dimensional translational symmetry is broken by a
surface, the theorem is not strictly applicable. One can avoid the problem of
lost periodicity normal to the surface by considering a large periodic cell, a so
called super-cell, where the periodicity is artificially restored and the surfaces
are far apart. A super-cell can be described within the slab model, consisting
of a finite number of atomic layers along the z-axis, normal to the surface, and
a region of reasonable amount of empty space separating the surfaces. In the
x,y-plane periodic boundary conditions are applied.
3.5.1
Bandstructure
Within DFT one calculates the ground state of a system. The Kohn-Sham
wave functions ψn,~k and eigenvalues ²n,~k extracted from DFT can be used to
determine the valence charge density and the bandstructure. Although the
parameters ²n,~k and ψn,~k that enters the Schrödinger-like equation (3.5) have
in principle no physical meaning, they are often interpreted successfully as
one-particle eigenfunctions and energies [25]. The physical entity in the DFT
scheme is the total charge density. However, the Kohn-Sham eigenvalues are
for metals usually in good correspondence with the physical eigenvalues, and
DFT-bandstructures compares well with photo-emission spectroscopy data.
A surface bandstructure calculation with DFT is done in a two steps process. At first the electron density is calculated according to the Kohn-Sham
14
3.5. Surface effects
self-consistent calculation scheme using a uniform sampling of the parallel
component of the ~k vector, ~kk , in the surface Brillouin zone (SBZ). In the
second step the eigenvalues at the desired ~kk -points along the high symmetry
directions are calculated non-self-consistently using the self-consistent electron
density from the first step.
An example of a bandstructure calculation is the surface projected bandstructure of Al(100) shown in Fig. 3.2. The super-cell is composed of a 23
layer slab with one atom per layer. A peculiarity of this surface is the slow
decay of a surface state into the bulk (shown in Fig. 3.5.b and further discussed in section 3.5.2). This slow decay leads to a splitting of the surface
state band (see Fig. 3.2 (blue bands)) due to the interaction between the slab
surfaces, Paper I. The average energy of the two splitted surface state bands
in the Γ̄-point is -3.03 eV, which is in reasonable agreement with the measured
value, -2.75 eV, from photo-emission experiment [26], and the value -2.98 eV
from calculated photo-emission spectra [27].
Figure 3.2: Calculated band structure of Al(100) surface, using a slab consisting of 23 atomic layers. The blue bands indicate the surface states.
Band structure calculations of clean surfaces are usually done for a (1×1)
surface unit cell, as in the example described above. When an atomic layer
is adsorbed onto a single crystal substrate a different periodicity parallel to
the surface is often introduced. The consequence is that the SBZ changes and
folding of bands occur.
15
3. Electronic Structure Calculations
Figure 3.3: Surface unit cell and surface Brillouin zone of the (1×1)-unit cell
and of the (2×2)-unit cell, respectively. The irreducible part of the (1×1) SBZ
is indicated by the dark gray area. Also shown are the symmetry points of the
respective SBZ
Adsorption of a saturated monolayer Cs on Cu(111) changes the surface
unit cell from (1×1) to (2×2). To understand the effect of band folding, it
is interesting to examine the band structure for the clean Cu(111) with both
a (1×1) and a (2×2) surface unit cell. Thus we introduce a so called empty
lattice with a (2×2) structure in the plane perpendicular to the surface normal.
The effect of expanding the area of the surface unit cell with a factor 4 yields
a SBZ area reduced by a factor 4. As an introduction to Brillouin zone band
folding we illustrate in Fig. 3.3 the two SBZ for the (1×1) (larger hexagon)
and for the (2×2) surface unit cell (smaller hexagon), respectively. The shaded
region indicates the irreducible Brillouin zone wedge of the (1×1) surface unit
cell. Due to the reduction of the SBZ we will obtain additional bands for the
(2×2) surface unit cell. In the Γ̄-M̄’ direction, each band along Γ̄-M̄ yields two
new band corresponding to M̄’-M̄ and M̄’-M̄”. Furthermore, in the direction
Γ̄-K̄’ each band along Γ̄-K̄ gives additional band corresponding to the direction
M̄’-K̄ and M̄’-K̄’. In the band structure of the (2×2)-unit cell, Fig. 3.4 (right
panel), we mark for illustrational purpose some of the folded bands. The solid
lines are arbitrary pair of Γ̄-M̄ and Γ̄-K̄ bands, the dashed dotted lines are the
corresponding folded bands M̄’-M̄ and M̄’-K̄ and the dashed lines correspond to
the bands in the M̄’-M̄” and M̄’-K̄ directions. Comparing the band structure
of the (1×1) and (2×2)-unit cells (Fig. 3.4) we realize that in the normal
direction (Γ̄-point) the lower band gap edge is moved up in energy, from 1.2
eV below EF to about 1.6 eV above EF .
Mapping out the band structure from experiments is done by Angle resolved
photo emission spectroscopy (ARPES) techniques [26], e.g. using synchrotron
radiation as a light source to excite electrons in a sample. The energy of the
16
3.5. Surface effects
Figure 3.4: Band structure of the (1×1)-Cu(111) (left panel) surface along
the K̄’-Γ̄-M̄’ symmetry lines of the SBZ and the band structure of the (2×2)Cu(111) (right panel) along the symmetry lines K̄-Γ̄-M̄ displayed in Fig. 3.3.
outgoing electron and its parallel wave vector is determined by varying the
angle of detection.
3.5.2
Localized states
Surface states
A surface state is defined as a bound state in the surface region [28]. In order for
a surface state to exist the surface must have a projected band gap, otherwise
the state will couple to bulk states. A usual property of surface states in metals
is that their energy bands disperse as free-electrons parallel the surface. The
wave function of a surface state decreases exponentially both into the vacuum
and into the bulk region. In general, the closer an occupied s, p surface band
appears relative to the bulk bands, the greater is the decay length of the state.
As exemplified in Fig. 3.5, the squared magnitude of the surface state wave
functions belonging to the Be(0001) and Al(001) have different decay length
into the bulk region. The surface band of Be(0001) is located ∼2 eV relative
the lower bulk band edge in the Γ̄ point and thus have a shorter decay length
than compared to the Al(001) surface band which is only ∼0.8 eV relative the
bulk band edge.
A true surface state band is not degenerate in energy with any bulk bands.
If that would be the case a surface resonance state, with an enhanced amplitude
at the surface, is formed.
17
3. Electronic Structure Calculations
Figure 3.5: Planar averaged squared magnitude of the surface state wave function at the Γ̄ point. Horizontal axis extends from the center of the slab (to
the left) towards the vacuum region (to the right) shown for a 17 atomic layer
Be(0001) slab (a) and a 23 atomic layer Al(001) slab (b). The vertical lines
indicates the topmost surface layer of the respective surface.
Surface resonance
Resonances are of particular interest in the study of surfaces since they allow
charge to be transferred dynamically between the surface and bulk regions.
As described in section 3.5.1, the Cu(111) surface exhibits a surface state
in a 5 eV wide projected band gap. When a saturated monolayer of Cs atoms
is adsorbed we change the periodicity of the surface from (1×1) to (2×2). This
leads to a reduction of the gap to 2 eV and the surface state is now located
within the folded bulk bands. This behavior is expected from the previous
analysis simply expanding the unit cell from (1×1) to (2×2) for clean Cu(111)
(Fig. 3.4). When Cu(111) is gradually exposed to Cs the energy of the surface
state shifts to lower energies with increasing Cs coverage, Paper II. According
to our analysis, the picture is that the Cu(111) surface state will hybridize
with the lowest band of the free standing (2×2)-Cs layer, forming a resonance
state, located about 1.2 eV below the Fermi level in the Γ̄ point.
The resonance character of this state is seen in the wave function plotted
in Fig. 3.6. In the Γ̄ point the state extends over all the Cu layers in the
slab and the Cs overlayer. As we examine the squared magnitude of the wave
function in several points along the Γ̄-M̄ direction one notes that the resonance
character develops when approaching the M̄-point with increasing weight in
the Cs layer.
18
3.5. Surface effects
2.0
B
0.04
1.5
0.02
0
E - EF [eV]
1.0
●
●
0.5
●
●
0.0
A
0.04
0.02
0
-0.5
Γ
0.04
-1.0 ●
0.02
●
-1.5
Γ
A
BM
0
●
●
●
●
●
●
Figure 3.6: Electron structure of the overlayer resonance state. To the left, the
band structure in the Γ̄-M̄ direction, and to the right, the even (red) and odd
(black) squared magnitude of the wave function, averaged in planes parallel to
the surface, at different locations along the overlayer resonance band.
Quantum well states
In the past and in the future, studies of quantum well states (QWS) have and
will attract interest, referring to further applications. QWS are suggested to
underlie phenomena such as the giant magnetoresistance [29] and mediators of
magnetic coupling through non-magnetic films [30, 31].
By intuition and curiosity the border between the school book example
”particle in a box” and reality is narrowed by alkali metal overlayers on substrates with a projected band gap. Several theoretical [32–34] and experimental [14,16] work has been devoted to QWS from alkali adsorbed metal surfaces.
The surface of Be(0001) has attracted a lot of interest due to several unusual properties, such as strong electron-phonon coupling for surface electrons.
Moreover, this surface exhibits a distinct difference between the bulk and surface electronic structure in that sense the substrate is semimetal-like in the
bulk and metal-like at the surface. It has a surface projected band gap which
is wide with respect to both energy and lateral wave vector. With respect to
these conditions, Be(0001) acting as a substrate for alkali metal adsorption is a
perfect candidate for realizing almost ideal simple metal quantum well states.
At saturated monolayer coverage of K on the Be(0001) surface the K atoms
form a (2×2) ordered pattern, Paper IV. The adsorption of a K overlayer gives a
19
3. Electronic Structure Calculations
Figure 3.7: Calculated QWS electron density at Γ̄ for a saturated K (2×2)
overlayer on Be(0001) in the [112̄0] plane of the substrate.
work function change due to a redistribution of electron charge. The calculated
work function for the (2×2)-K/Be(0001) is 2.3 eV, which is a reduction by 2.6
eV compared to the clean Be(0001), (4.9 eV). In addition, the electronic band
structure of this system exhibits a QWS band with a binding energy of 0.56 eV
in the Γ̄-point located within the Be(0001) band gap. Further characteristics
of the QWS is that, as shown in Fig. 3.7, the state has one node in the K layer
near the overlayer/substrate interface and with almost the entire charge in the
overlayer/vacuum interface. This degree of confinement is extreme compared
to similar cases.
In the case of a (2×2) Cs monolayer on Cu(111) the calculated QWS binding energy lies above the Fermi level, indicating an unoccupied state. Photoemission measurements of the QWS binding energy reveals a partially occupied
state with binding energy of 25 meV, Paper II. The discrepancy between the
experimental and the theoretical QWS binding energy can be understood in
terms of the number of sampling points in SBZ and the introduction of a Fermi
smearing. Following the results from table I in Paper II, we can see the sensitivity in the QWS energy with respect to the number of k-points and the
Fermi smearing width. In an ideal calculation one would like to have infinitely
many k-points and a smearing width approaching zero, though that would be
impossible due to computational cost. The Fermi smearing is introduced to
improve convergence and less number of k-points needs to be used.
For Cs alkali metal on Cu(111) the charge of the corresponding QWS, with
one node in the film, is shared more equally between the substrate and the
overlayer (Fig. 3.8, left panel) compared to the QWS of K/Be(0001). This
20
3.5. Surface effects
state is in principle a resonance state due to hybridization between the second
lowest band (Fig. 3.8, denoted p) of the free standing (2×2)-Cs layer with
folded copper bands. However, the hybridization effect turns out to be weak
illustrated by the almost exponential decay of the state into the bulk in Fig.
3.8 (left panel).
4
Intensity (arb. units)
d’
2
d
1
p
0
Energy - EF (eV)
3
-1
s
-2
Γ
Μ
Figure 3.8: (Left panel) Planar averaged squared magnitude of the quantum
well resonance wave function at the Γ̄ point of the (2×2)-Cs/Cu(111). Horizontal axis extends from the center of the slab (to the left) towards the vacuum region (to the right). The vertical lines indicates the Cs layer position
(solid) and the topmost Cu plane (dashed). (Right panel) Band structure of
Cs/Cu(111) and unsupported Cs layer. The unsupported Cs layer bands (s,
p, d, and d’ ) are denoted by solid lines.
Gap state
Adsorption of alkali overlayers on noble metals, such as Cs on Cu(111), introduces a significant change in the electronic structure of the surface and reveals
a manifold of surface localized electronic states. Referring to the electron density redistribution a lowering of the work function takes place due to charge
transfer from the alkali overlayer to the interface. For a saturated monolayer
of Cs on Cu(111) there appears a new, not previously discussed, one-electron
gap state (GS), Paper III. The GS is a result of folding the overlayer resonance
to the Γ̄ point in a bulk band gap at 2.7 eV above the Fermi level. The calculated and experimental work function of the system is 1.8 eV and hence the
GS is positioned 0.9 eV above the vacuum level. However, the planar averaged
21
3. Electronic Structure Calculations
squared wave function reveals a strong localization of the GS to the Cs layer.
The reason for this localization is that the main part of its band energy is attributed to kinetic energy parallel to the surface. The GS band is a Cs induced
band and the character of the GS band in the Γ̄ is analyzed further. From
the calculated self-consistent GS wave function, we can obtain the expectation
value of the parallel kinetic energy,
²GS
k = −
h̄2
~ 2 |ψGS i,
hψGS |∇
k
2m
(3.18)
~ 2 = ∂ 2 + ∂ 2 and ψGS is the first-principles GS wave function. At the
where ∇
k
∂x
∂y
~
Γ-point (kk = 0) the GS wave function is given by:
2
2
~ z) =
ψGS (R,
X
~
~ k ,g
G
~
CG~ k g eiGk ·R eigz ,
(3.19)
~ = (x, y)
where C are the Fourier components of the GS wave function, R
~ k and g are the parallel and perpendicular components of the reciprocal
and G
~ = (G
~ k , g). We then have
lattice vectors G
²GS
k =
h̄2 X ~ 2
|Gk | |CG~ k g |2 = 10.34 eV.
2m ~
(3.20)
Gk ,g
Subtraction of this energy from the band energy of the GS is illustrated in
Fig. 3.9. From the histogram, shown in the figure, we observe that about 60 %
~ k and the remaining
of the parallel kinetic energy originates from the shortest G
from higher order diffraction. The outstanding characteristics of the GS are:
• band energy above the vacuum level.
• spatially localized at the surface and the Cs overlayer.
• large in-surface-plane kinetic energy.
When this state is populated the electron will move fast along the Cs overlayer.
22
3.5. Surface effects
Figure 3.9: Planar averaged VSCF and the squared magnitude of the GS wave
function. The energy of the GS when subtracting the calculated ²GS
is indik
cated by the vertical arrow. The histogram shows the squared magnitude of
the GS wave function (red) and parallel kinetic energy (blue) resolved in terms
~k
of contribution from different parallel reciprocal lattice vectors G
3.5.3
Charge density
In a newly created surface the atoms located in the region of the topmost
layers will experience different forces compared to the bulk atoms. The rearrangement of the electronic charge and the new electronic distribution induces
a relaxation of the topmost atoms along the direction normal to the surface.
The relaxation is determined predominantly by forces on the ions set up by
the rapid redistribution of the valence electron density. Calculations of the
redistribution of charge can be used to understand the anomalous outward
relaxation of certain metal surfaces, such as e.g. the Al(100) surface.
The x,y-averaged valence electron density, ρ̄(z), has been calculated in
two different ways. One is by averaging ρ(~x, z) over the planar points, ~x =
(x, y), of the super-cell. The second method is based on the free-electron
approximation in the x,y plane with band masses, m∗v , determined by the selfconsistent bands, v, near the Γ̄ point, (|~kk | = 0). Integrating the occupied
bands over the surface Brillouin zone yields the average density. We have
applied the more simple latter method to evaluate the valence electron density
for the relaxed surface as well as for the unrelaxed surface. With this method
we can decompose the valence electron density in parts originating from surface
and bulk bands, respectively. This will give us the possibility to understand
23
3. Electronic Structure Calculations
Figure 3.10: Valence charge density calculated for the Al(100) surface. The
dashed line corresponds to a 13 atomic layer slab and the solid line corresponds
to a 23 atomic layer slab. The vertical lines represents the ion core positions
of the four uppermost layers.
how the redistribution of the valence charge density acts as we create a surface.
In Fig. 3.10 we exemplify the results from this analysis for two different slab
thickens. The contribution from bulk states is essentially the same for the two
slab thicknesses, while the difference in electron density at and outside the
uppermost surface plane is the charge carried by surface state bands.
Furthermore, of great importance in surface physics is to learn about the
induced charge density due to adsorption of different atom species. The information can be used to investigate electron density depletion and accumulation
at the surface or interface of the material and give an understanding of e. g.
bonding character and change of work function.
The induced charge density of the (2×2)-K/Be(0001) shown in Fig. 3.11 is
calculated by taking the difference between the charge density of the K/Be(0001)
slab and the sum of the charge density of the clean Be(0001) slab and the freestanding K overlayer corresponding to the adsorbate structure,
h
i
∆ρ(~x, z) = ρK/Be(0001) (~x, z) − ρBe(0001) (~x, z) + ρK
x, z) .
overlayer (~
(3.21)
When the K monolayer is adsorbed onto the Be(0001) surface a substantial
amount of charge is transfered from the K atoms to the interface leading to
an accumulation of charge between the K layer and the Be substrate and a
24
3.5. Surface effects
Figure 3.11: The adsorbate-induced charge density ∆ρ of the (2×2)K/Be(0001) structure plotted in the [112̄0] plane. The Be atoms are dark
gray and the K atoms are turquoise. The charge density range is shown by
the thermometer (a.u. units).
charge depletion in the plane of the K atoms. From Fig. 3.11 one notes that
the contribution to the charge accumulation at the interface does not entirely
originate from the K monolayer but even some minor contribution originates
from the first and second Be layer.
3.5.4
Surface energy and work function
The finite number of layers in a slab might lead to interaction between the
two surfaces within the super-cell. Another possibility for the surface-surface
interactions is through the finite size of the vacuum region separating the surfaces. The surface-surface interaction through the slab or through the vacuum
can however be controlled by increasing the number of atomic layers and increasing the size of the vacuum region. A bulk-like interior is crucial when
surface properties, such as e.g. surface energy and work function, are to be
determined.
The surface energy can be determined in two different approaches. Usually
the surface energy is determined from the basic relation [35],
Es (l) =
1
[El − lEb ] ,
2
25
(3.22)
3. Electronic Structure Calculations
where El is the energy of an l -layer super-cell slab and Eb is the bulk energy
corresponding to one layer calculated from an infinite solid. The factor 12 takes
account for the two surfaces of the slab. This method has the advantage that
only one calculation is needed in order to obtain the bulk energy, Eb , and
hence, the surface energy. The second method considers two different slab
calculations which only differs in the number of atomic layers. Then the bulk
energy is approximated by the energy difference between two adjacent slabs
with l and l -1 layers, ∆E(l) = El − El−1 . The surface energy is then:
1
Es (l) = [El − l∆E(l)].
2
(3.23)
The disadvantage of the latter method is the large computational time when
l increases. However, the latter method yields consistent results compared to
the previous method since the bulk energy is obtained from the reference slab.
Figure 3.12: (x, y)-average self-consistent one electron potential of a 17 atomic
layer Al(100) slab. The work function Φ is given by Ev − EF .
The work function Φ of a surface is the minimum energy required to remove
an electron from the Fermi level to a point far outside the surface. To determine
the work function within the slab method we need to have enough vacuum to
ensure that the self consistent one electron potential, VSCF , levels off with zero
slope at the mid point of the slab vacuum region. Subtracting the calculated
26
3.5. Surface effects
Fermi energy from the value of VSCF at this point yields the work function,
Fig. 3.12.
Alkali metal adsorption onto a metal surface lowers the work function drastically. At a saturated monolayer coverage of Cs on Cu(111) the work function
is lowered by about 3 eV compared to the calculated work function, 4.9 eV,
of the clean Cu(111) surface. The work function for the adsorption of a K
monolayer onto Be(0001) surface is reduced by 2.6 eV compared to 4.9 eV for
the clean Be surface. The responsible mechanism for the lowering of the work
function is the charge transfered from the alkali atoms to the interface (see
Fig. 3.11) creating a surface induced dipole moment.
27
Chapter
4
Density Functional
Perturbation Theory
Until now we have considered the DFT scheme for a rigid ion lattice. The
ground state electron structure of materials provides a lot of information about
their physical properties. However, many accessible features are related to
higher order derivatives of the ground state energy. For example vibrational
modes in a crystal are determined by the second derivative of the total energy
with respect to ionic displacements.
Density-functional perturbation theory [36–40] (DFPT) is a first-principles
scheme developed to handle higher order effects of materials, such as vibrational properties and electron-vibration coupling. Other approaches to
characterize the lattice dynamics, such as e.g., semi-empirical methods, firstprinciples frozen-phonon and molecular-dynamics simulations have been widely
used.
Dynamical matrices and phonon dispersion curves can be determined by
calculating the energy difference and forces acting on the atoms from selected
atomic displacements. Within the frozen-phonon method a suitable choice
of the atomic displacements is made in order to determine the inter-planar
force constants. The total energy and Hellmann-Feynman forces are calculated as a function of atomic displacements from the equilibrium positions. A
frozen-phonon calculation for lattice vibrations at a generic ~q vector requires a
super-cell having ~q as a reciprocal-lattice vector. This method has its limitations, since as for small ~q the super-cell becomes large, which in turn requires
extensively computational time.
In Molecular-dynamics (MD) simulations [41], the finite-temperature dy29
4. Density Functional Perturbation Theory
namics of atoms which vibrate about their equilibrium position are studied.
The harmonic approximation can be applied, for low enough temperatures, to
describe the atomic trajectories from the classical equations of motions. The
Hellmann-Feynman forces have to be essentially the exact derivatives of the total energy in order to obtain accurate trajectories and correct frequencies. As
in the frozen-phonon method MD requires large super-cells in order to describe
large wavelength phonons (small ~q).
Semi-empirical methods, such as the embedded-atom method, relies on
experimental values for e. g. the equilibrium lattice constants, bulk modulus,
etcetera, which are used as fitting parameters. While these values are usually
exactly reproduced, the force constants determining the lattice vibrations of
the crystal are harder to determine with good accuracy.
The approach of DFPT is based on the ground state results obtained from
DFT. The response to arbitrary (infinitesimal) displacements of the atoms
and to corresponding changes of the ionic effective one-electron potential is
calculated within linear response theory. DFPT allows an efficient treatment
of the response of an electron system to external perturbations and allows one
to calculate phonon frequencies at arbitrary ~q vector.
4.1
Linear response
Within DFT the electronic density n(~r) is determined by the self-consistent
potential, Eq. 3.4, which is the sum of the external potential acting on the
electrons and an effective one-electron potential which depends on the density itself. The total energy of the system is determined by solving Eq. 3.8
self-consistently. Within the Born-Oppenheimer adiabatic approximation it
has been shown that the force constants [42] can be extracted from the second order derivatives of the total energy of the system with respect to ionic
coordinates. Application of the Hellmann-Feynman [43] theorem shows that
the linear variation of the electron density upon application of an external
perturbation determines energy variations to second and higher order in the
perturbation [36]. The theorem states that the ”force” associated with the
variation of some external parameters, λ, is given by the ground-state expecλ
.
tation value of the derivative of the bare ion-electron potential, Vion
Assume that the external parameters, λ, represents ion displacements,
~ where R
~ indicates the position of the unit cell, i the ion position within
uαi (R),
the unit cell, and α = x, y or z. The force acting on the ions are then given
by the first derivative of the total energy with respect to ion displacements,
30
4.2. Linearized Schrödinger equation
Z
∂E
~
∂uαi (R)
=
n(~r)
∂Vion (~r)
d~r.
~
∂uαi (R)
(4.1)
Thus, determining the forces acting on the ions one needs the knowledge of
the ground-state electron density. To receive the force constants we need to
take the derivative of Eq. 4.1,
Z
∂ 2E
∂ 2 Vion (~r)
=
n(~r)
d~r
~ 0 )∂uαi (R)
~
~ 0 )∂uαi (R)
~
∂uβj (R
∂uβj (R
Z
∂n(~r) ∂Vion (~r)
+
d~r,
~ 0 ) ∂uαi (R)
~
∂uβj (R
(4.2)
~ 0 ) is the electron-density
where in the second term of Eq. 4.2, ∂n(~r)/∂uβj (R
response to displacements in the β-direction of the j th ion located in the unit
~ 0.
cell at R
Thus, the electronic response to ionic displacements gives access to interatomic force constants, which in turn can be used to compute the frequencies
of the collective vibrational modes, phonons.
4.2
Linearized Schrödinger equation
Once the ground-state problem has been solved applying the DFT scheme,
DFPT provides an efficient method to calculate the electronic linear response
to any external perturbation. Introducing a perturbation, ∆Vbare , on the external potential acting on the Kohn-Sham system, the self-consistent potential,
Eq. 3.4, is modified accordingly: VSCF −→ VSCF + ∆VSCF .
~ 0 ), appearing in Eq. 4.2 can
The electron-density response, ∂n(~r)/∂uβj (R
be evaluated by linearizing Eq. 3.6 with respect to the wave function using the
finite-difference operator ∆λ ,
∆λ F =
X ∂Fλ
i
∂λi
∆λi .
(4.3)
where F is a arbitrary physical quantity, and λ is a perturbing parameter. The
linear form of the electron density with respect to the wave function changes
as,
N/2
n(~r) = 2
X
|ψν (~r)|
2
N/2
∆ψ
−→ ∆n(~r) = 4Re
ν=1
X
ν=1
31
ψν∗ (~r)∆ψν (~r),
(4.4)
4. Density Functional Perturbation Theory
where ν = (n, ~k), n is the band index which runs over the occupied states and
~k is the wave vector belonging to the first Brillouin zone. The Schrödinger
equation, for the unperturbed system,
[HSCF − ²ν ]|ψν i = 0,
(4.5)
where HSCF = − 12 ∇2 + VSCF (~r), is expanded to first order giving,
(HSCF − ²ν )|∆ψν i = −(∆VSCF − ∆²ν )|ψν i.
(4.6)
The variation of the Kohn-Sham wave functions, ∆ψν (~r), are obtained by
solving the linear form of the Schrödinger equation, Eq. 4.6. The input of
∆VSCF in Eq. 4.6 is obtained by linearizing Eq. 3.4 with respect to the density,
Z
∆VSCF (~r) = ∆Vion (~r) +
¯
∆n(~r0 ) 0 dvxc (n) ¯¯
d~r +
∆n(~r),
|~r − ~r0 |
dn ¯n=n(~r)
(4.7)
which is the first-order correction to the self-consistent one-electron potential.
The first-order variation of the Kohn-Sham eigenvalue, ²ν , is given by ∆²ν =
hψν |∆VSCF |ψν i.
The equations , Eq. 4.4, 4.6 and 4.7, form a set of self-consistent equations
for the perturbed system completely analogous to the Kohn-Sham equations in
the unperturbed case. ∆VSCF (~r) is a linear functional of ∆n(~r), which in turn
depends linearly on ∆ψ(~r), the whole self-consistent calculation can be cast
in terms of generalized linear problem. Efficient iterative algorithms such as
conjugate gradient methods [44] can be used for solution of the linear system.
Staging the self-consistent scheme of DFPT is done by assigning a trial
solution for the first order change of the wave functions. From the first order
change in the wave functions the corresponding electron density is received
and the first order correction to the self-consistent potential, Eq. 4.7, and the
first order variation of the Kohn-Sham eigenvalue may be calculated. Inserting
these quantities into Eq. 4.6 a new set of first order change in wave functions
are received. The scheme is iterated until the first order electron density is
within some convergence criterion.
4.3
Force constants
The second order derivatives, Eq. 4.2, of the total energy with respect to
ion displacements gives access to the inter-atomic force constants [42]. The
32
4.4. Phonons
matrix of the inter-atomic force constants depends on i and j only through
~ ≡R
~i − R
~ j , as a result of the translational invariance [45],
the difference R
~ i, R
~ j) ≡
Cα,β (R
∂2E
~i − R
~ j ),
= Cα,β (R
~ i )∂uβ (R
~ j)
∂uα (R
(4.8)
where α and β indicate Cartesian components, x, y and z.
~ with respect to R,
~ the force constants
Taking the Fourier transform of Cα,β (R)
can be seen as the second derivative of the ground-state energy with respect
to the amplitude of a lattice distortion of definite wave vector ~q,
C̃α,β (~q) ≡
X
~
R
∂ 2E
~
~ = 1
e−i~q·R Cα,β (R)
.
Nc ∂ ũ∗α (~q)∂ ũβ (~q)
(4.9)
In the above equation, Nc defines the number of unit cells, and the vector
ũα (~q) is defined as,
~ =
uα (R)
X
~
ũα (~q)ei~q·R .
(4.10)
q~
Phonon frequencies are obtained by diagonalizing the dynamical matrix,
D̃αi,βj (~q), defined as
C̃α,β (~q)
D̃αi,βj (~q) = q
,
(4.11)
Mi Mj
where the M ’s are ionic masses.
The development so far allows for calculation of the vibrational frequencies
at any phonon wave vector ~q. Phonon frequencies are usually smooth functions
of the wave vector, so that suitable interpolation techniques can be used when
complete dispersion relations are needed. Once real-space inter-atomic force
constants have been obtained, dynamical matrices in reciprocal-space and,
hence, vibrational frequencies can be obtained at any wave vector by Fast
Fourier Transform techniques.
4.4
Phonons
Determining the phonon frequencies of a crystal surface is usually done within
the slab model. The slab method is concerned with the vibrational properties
of the entire system, both bulk and surface phonons, all of which, in principle,
play a role in surface scattering phenomena. Because of the limited number
of layers (N ) in a slab, there are only a correspondingly limited number of
33
4. Density Functional Perturbation Theory
vibrational frequencies for each two-dimensional wave vector (~qk ) [46], namely
3×N frequencies. Thick slabs will provide a better representation of the deeply
penetrating modes at long wavelengths. Beyond a certain N the surface modes
and band gaps do not change significantly [46, 47].
The in plane translational invariance of the force constants, Eq. 4.8, implies
that the normal-mode solutions of the equation of motion [48],
Mi
X
d2
~ i, R
~ j )uβ (R
~ j ),
~ i) = −
Cαβ (R
u
(
R
α
dt2
~
(4.12)
Rj β
have the form of two-dimensional Bloch functions,
~ i) =
uα (R
X
~
qk
1
~
√ A0 ξα (~qk )ei(q~k Ri −ωt) .
Mi
(4.13)
The amplitude of the vibration is given by A0 , ξα is the α-component of the polarization vector and ω is the vibrational frequency and ~qk is a two-dimensional
vector. Substituting the solution of Eq. 4.13 into Eq. 4.12 we obtain the dynamical matrix eigenvalue equation,
X
C̃α,β (~qk )ξβ (~qk ) = ω 2 ξβ (~qk ).
(4.14)
β
The phonon frequencies, ων (~
qk ), are solutions to the secular equation,
¯
¯
¯
¯
¯
¯
1
2¯
¯
q
det ¯
C̃αβ (~
qk ) − ω ¯ = 0.
¯ Mi Mj
¯
(4.15)
Experimental determination of surface phonon dispersion, ων (~qk ), can be
done by e.g. helium atom scattering (HAS). The momentum transfer is set by
fixing the incoming and outgoing beam direction with respect to the crystal
normal, while the kinetic energy of the scattered atom is measured in a time-offlight analysis. Measuring the time-of-flight spectra for a number of different
scattering angles (θi , θf ) gives information of intensities between diffraction
peaks resulting from elastic scattering and inelastic scattering. The frequency
and wave vector of the phonons involved can be determined from the energy
change and the scattering angles via the conservation equations [49]:
Ã
!
h̄2 ~ 2 ~ 2
(kf − ki ) = ±∆E = ±h̄ω(~qk )
2m
~kf sinθf − ~ki sinθi = ±∆K = G ± ~qk
34
4.4. Phonons
where ~ki and ~kf are the incident and final wave vector, respectively, and G is a
reciprocal lattice vector. The first successful measurements of surface phonon
dispersion curves out to the zone boundary were carried out by Toennies and
collaborators on LiF(001) in 1981 and on Ag(111) in 1983.
35
4. Density Functional Perturbation Theory
4.4.1
Phonon dispersion curves
In Fig. 4.1 we show the calculated phonon dispersion relation of Cu(111)
along the symmetry lines in the surface Brillouin zone, Paper VI. The phonon
dispersion curves in Fig. 4.1 corresponds to a 25 atomic layer slab.
30
Energy [meV]
25
20
15
10
5
0
Γ
M
X
Γ
Figure 4.1: Comparison between dispersion curves for Cu(111) obtained from
DFPT calculations, helium atom scattering (red circles) and electron energy
loss spectroscopy (blue circles).
The vibrational modes can be assigned to bulk modes, which when projected onto the surface Brillouin zone form a continuum for a very thick slab.
Additional contributions are from surface localized modes. The presence of vibrational modes localized at the surface was first pointed out by Lord Rayleigh
(1885). The motion of the atoms in a Rayleigh wave (RW) are mainly share
vertical (SV), defined by the direction normal to the surface. In Fig. 4.1 the
Rayleigh mode is the lowest dispersion branch below the bulk band edge. Surface localized modes are referred to as pure surface modes when they appear
in regions of the dispersion space not filled by bulk bands. Surface resonance
modes appear in regions occupied by bulk bands with which they may or may
not couple (hybridize), depending on the mode polarizations.
The displacement field of vibrational modes, in a crystal with a surface,
36
4.4. Phonons
can be classified in three categories [50]:
• Bulk modes: the displacement field is homogeneously distributed on the
bulk region of the slab.
• Surface modes: the field decreases in general exponentially into the bulk.
• Surface resonance modes: the field is distributed in the bulk with an
enhanced amplitude in the surface region.
Within this classification of vibrational crystal modes there are further
possible characteristics of the modes due to the polarization of the modes.
The polarizations are customarily referred to the sagittal plane. The sagittal
plane is defined by the propagation vector ~qk and the surface normal z. Surface
modes can be either transverse, longitudinal or shear-horizontal in polarization.
Shear-horizontal modes have a well-defined polarization which is normal to
the sagittal plane. Modes polarized within the sagittal plane are typically a
mixture of transverse and longitudinal polarization, referring to vibrational
motion perpendicular and parallel to the surface. If the modes are assigned
longitudinal or transverse depends on the relative amplitudes.
The eigenvector ξ~i (ν~qk ) in Eq. 4.14 gives the polarization of atom i. It
specifies the direction of motion of atom i when the crystal is in the mode
labeled by ν with the wave vector ~qk . Allen et. al. [48], derive that the motion
of an atom about its mean position traces out an ellipse with an orientation
~ which give the major and
which lies in the plane defined by Reξ~ and Imξ,
minor axis or vice versa.
37
4. Density Functional Perturbation Theory
4.4.2
Phonon DOS
Determining which vibrational modes are surface or resonance modes can be
a difficult task as can be seen from Fig. 4.1. If a mode lies below the manifold
of the bulk bands then it must be a surface mode. A mode that appears in
a gap of bulk modes, as in Fig. 4.1 at the M̄, must also be a surface mode.
While identifying surface modes is relatively easy, surface resonance modes
are harder to detect. Surface resonance modes can be found by calculating the
projected density of states (PDOS),
ρlα (~qk , E)
¯2 ¸
¯2 ¯
X ·¯¯
¯
¯
¯
¯Re ξα (l|ν~
qk )¯ + ¯Im ξα (l|ν~qk )¯ δ(h̄ω(ν~qk ) − E),
=
(4.16)
ν
where ξα (l|ν~qk ) is the α-component of the l:th layer polarization vector in
mode ν th . In order to properly describe the displacements of the atoms, both
the real and imaginary part of ξα must be taken into account.
Figure 4.2: Calculated PDOS for the shear-vertical polarization in the first and
second layer (SV1 and SV2) in the Γ̄-M̄ direction. Comparison is made with
HAS [51] (red circles) and EELS [52, 53] (yellow circles) experimental data.
The experimental data points are plotted on the energy- wave vector plane.
We can then study the projected phonon density of states along the symmetry lines of the surface Brillouin zone, as seen in Fig. 4.2 and Fig. 4.3. In
Fig. 4.2.a we show PDOS for the topmost layer (layer 1) of Cu(111) that are
shear-vertical in character (SV1). The intense dispersive ridge is attributed to
38
4.4. Phonons
Figure 4.3: Calculated phonon DOS projected onto the first and second layers
for longitudinal (L1 and L2) polarizations in the Γ̄-M̄ direction. Experimental
data points as in Fig. 4.2.
the RW mode. Fig. 4.2.b shows the second layer shear-vertical modes (SV2).
The longitudinal character of the vibrational modes projected on the first layer
(L1) and the second layer (L2) are presented in Fig. 4.3.a and 4.3.b.
A controversial and widely discussed mode is the so-called acoustic longitudinal resonance (LR) mode which show up below the lower edge of the
longitudinal polarized bulk bands. This mode, common to most metal surfaces, was first discovered by HAS in silver [54] and other noble metals [51].
The appearance of the LR was latter confirmed by EELS for example the
Cu(111) surface [52]. Although both experimental methods agree regarding
the location of the LR there is some disagreement regarding the scattering
intensities. The HAS intensity for LR being much larger than for the RW,
whereas the EELS intensity for LR is much smaller. The differences between
HAS and EELS in terms of detecting surface localized phonons is that the former is much more surface sensitive since He atoms are scattered by the charge
density profile at the surface, while in the latter method electrons penetrate
and are scattered by the ion displacements within 2-3 surface atomic layers.
The calculated longitudinal phonon resonance (denoted L in Fig. 4.3.a)
coincide well with the experimental points up to one half of the Γ̄-M̄ line.
While approaching the M̄-point the mode becomes a pure surface mode (S2 ) as
it appears in the bulk band gap. Close to the M̄-point in the direction Γ̄ to M̄,
the experimental points follow the modes that are shear-vertical in the second
39
4. Density Functional Perturbation Theory
layer. This is a resonance mode denoted S3 in Fig. 4.2.b. This resonance has
a weak L component in the first layer and a large SV component in the second
layer.
The detection of the longitudinal phonon resonance from experimental
point of view (HAS and EELS) is explained by the fact that the longitudinal
resonance strongly hybridize with an intense shear-vertical resonance localized
in the second atomic layer. The hybridization results from the interaction between modes belonging to the same symmetry, e. g. modes polarized within
the sagittal plane. This, in turn, will lead to a non-crossing behavior. Following the dispersion of the LR in Fig. 4.3.a we see a weak non-crossing behavior
between S3 and S2 . Resolving the phonon DOS in terms of both first layer and
second layer longitudinal and shear-vertical modes, respectively, in Fig. 4.4
this non-crossing becomes larger in the second layer between the high optical
S2 mode and the LR. As approaching the SBZ boundary (M̄-point) this hybridization ends up in two distinct modes, the gap mode S2 and a resonance
mode, S3 .
Figure 4.4: PDOS in the first and second layers for L and SV polarizations in
the Γ̄-M̄ direction.
The longitudinal resonance is not only observed for Cu(111) but also in
other metal surfaces e. g. Al(001), Paper V. The PDOS of the L modes in the
first layer (L1) shown in Fig. 4.5 along the Γ̄− X̄ direction, reveals a prominent
ridge starting in Γ̄ and terminating in the gap mode S2 at the X̄-point. This
resonance, as appears from the longitudinal amplitudes in the second layer
(L2), can be explained by a non-crossing phenomena between the LR and the
40
4.4. Phonons
Figure 4.5: Al(001) PDOS in the first and second surface layer for SV and L
polarizations along the Γ̄ − X̄ direction. The blue and black circles represents
HAS experimental data [55].
surface optical mode S4 . Surface phonons and phonon resonance of the Al(001)
surface is further discussed in Paper V.
4.4.3
Induced charge density
Using simple force-constant models, Hall and Mills, could explain the dispersion and scattering intensities of the longitudinal phonon resonance found from
EELS data for Cu(111) by a slight softening of the longitudinal force-constant
with 15% [52,53]. In order to reproduce the dispersion and the HAS intensities
of the LR, Bortolani et. al. had to soften the longitudinal force-constant by
67% and, in addition, a 60% stifening of the force-constant in the two first
layers [56]. This apparent contradiction between force-constants referring to
EELS and HAS data has been referred to as the ”Bortolani-Mills paradox” [57].
In HAS the He atoms are scattered by the phonon induced oscillations in
41
4. Density Functional Perturbation Theory
Figure 4.6: Induced charge density due to the RW mode, the gap mode S2 and
the S3 resonance mode in the M̄-point for Cu(111).
the surface electron density about 2-3 Å away from the first atomic layer. Due
to the differences in the displacement pattern the modes give rise to different
phonon induced charge density changes. For a qualitative prediction of whether
or not the S3 resonance mode could yield HAS intensities comparable with
those of the Rayleigh mode the induced charge density of the respective mode
should be comparable in the region where the He atoms are scattered. This is
a fair comparison, since the main ingredient in HAS cross section calculations
is the induced charge density [58]. In Fig. 4.6 we notice that the induced
density variations from the RW and S3 appear similar. A similar analysis based
on multipole expansion of the phonon induced charge density oscillations has
previously been preformed [58]. Our calculated PDOS, Fig. 4.3.a together
with Fig. 4.2.b, are almost identical with the charge density oscillations at the
surface in reference [58] Fig. 14 (b).
42
Chapter
5
Summary of Appended Papers
5.1
Paper I: Surface Relaxation Influenced by
Surface States
In this paper, the relaxation of the Al(100) surface is studied based on an analysis of the electron structure from first-principles applying density functional
theory. We model the Al(100) surface by slabs, ranging from 7 atomic layer up
to 23 atomic layers. We observed a slow convergence of the relaxation of the
atoms in the topmost layers with respect to the number of slab layers. Al(100)
exhibits a partially occupied surface band 3 eV below the Fermi level in the Γ̄
point and located just above the lower band gap edge. This explains the slow
convergence, as in general, the closer an occupied s, p surface band appears
relative to the bulk bands the greater is the decay length of the state.
The variation in the surface valence charge density distribution as a function of slab layers correlates with atomic forces and the interlayer relaxation
of the topmost layers. Decomposing the valence electron density in parts originating from surface and bulk states shows that the change in electron density
at and outside the uppermost lattice plane is within the charge carried by
the surface state band. In this region, the contribution from bulk states is
essentially the same for 13-23 layer slabs.
43
5. Summary of Appended Papers
5.2
Paper II: Overlayer Resonance and Quantum Well State of Cs/Cu(111) Studied
with Angle-Resolved Photoemission, LEED,
and First-Principles Calculations
This paper outline results of a joint experiment and theory investigation of the
surface electronic structure of a saturated p-(2×2) Cs monolayer on Cu(111).
Low energy electron diffraction was used to study the atomic structure and
angle-resolved photoemission spectroscopy to map out the band structure. The
Density functional theory based calculations yield band structure in agreement
with the experimental findings and give further information about the character of wave functions.
When the Cs monolayer is adsorbed on Cu(111) the 5 eV band gap for
clean Cu(111) is reduced to 2 eV due to folding of copper bands. Furthermore,
a quantum well state band and a surface resonance band is established. The
labeling - overlayer resonance - is due to our interpretation of this state as the
lowest band of the unsupported Cs layer in resonance with copper bands.
44
5.3. Paper III: Two-dimensional Localization of Fast Electrons in
p(2×2)-Cs/Cu(111)
5.3
Paper III: Two-dimensional Localization
of Fast Electrons in p(2×2)-Cs/Cu(111)
In this paper, we analyze the electronic band structure of p(2×2)-Cs/Cu(111),
based on density functional theory calculations.
The Cs/Cu(111) surface exhibits a variety of surface features such as, overlayer resonance, quantum well state, image states, and gap state (GS). Our
focus is mainly on the appearance of the gap state, which, to our knowledge,
has not been previously discussed.The main characteristics of the GS are; (i) a
band energy well above the vacuum level, (ii) appearance in a local bulk band
gap but within the continuum of vacuum states, (iii) strong localization to the
Cs layer, (iv) large in-surface-plane component of the kinetic energy.
From the self-consistent GS wave function we calculate the expectation
value of the parallel kinetic energy and resolve it in contributions from different
parallel reciprocal lattice vectors. 60 % originates from the shortest reciprocal
lattice vector and the remaining 40 % from higher order diffractions.
An electron populating this state will move fast parallel to the surface and is
expected to travel of the order 1000 Å prior to decay. A possible application as
a nano electron gun near step edges of adsorbed alkali overlayers is addressed.
45
5. Summary of Appended Papers
5.4
Paper IV: Sodium and Potassium Monolayers on Be(0001) Investigated by Photoemission and Electronic Structure Calculations
This paper is a result of an experimental investigation supported by electron
structure calculations. Monolayer coverage of K and Na on the Be(0001) surface is studied by means of angle-resolved photoemission spectroscopy, low
energy electron diffraction (LEED) and first-principles density functional theory based calculations.
The Be(0001) surface has some unusual property which has attracted much
interest over the years. When the surface is created there is a distinct difference
between the bulk and surface electronic structure, semi-metal in the bulk and
metal in the surface region.
This surface makes a perfect candidate for studying nearly ideal quantum
wells since the Be(0001) surface exhibits a band gap which is wide with respect
to energy and lateral wave vectors. The gap provides confinement of occupied
electronic states for an adsorbed alkali metal film.
At a full monolayer coverage LEED observations show that Na atoms form
a closed packed incommensurate overlayer with a surface mesh corresponding
to 1.65×1.65, while K form a 2×2 overlayer structure. In addition to the
experimental findings, we present atomic and electronic structure calculations
for the clean Be(0001) surface and the (2×2)-K/Be(0001) system. A detailed
study of the atomic arrangement resulted in an on-top K adsorption site with
a small induced rumpling of the substrate. We calculate the work function
change due to the adsorption of a K monolayer and present the redistribution
of the charge. The band structure and the wave functions for the states of
interest is presented in connection to the experimental results. The Be(0001)
Γ̄ surface state shifts to lower binding energy with increasing K coverage.
Compared to previous results for Cs/Cu(111) this state does not become a
resonance state, it is still confined to the surface.
46
5.5. Paper V: Evidence of Longitudinal Resonance and Optical
Subsurface Phonons in Al(001)
5.5
Paper V: Evidence of Longitudinal Resonance and Optical Subsurface Phonons
in Al(001)
In this paper, the phonon structure is calculated applying the density functional perturbation theory for a fully relaxed Al(001) surface using the slab
method. We present calculated phonon dispersion curves along the high symmetry directions in the surface Brillouin zone with experimental helium atom
scattering data. The agreement of the calculated Rayleigh wave dispersion
curve with experimental data along Γ̄-X̄ is perfect.
We analyze and characterize in detail the occurrence of surface states, surface resonances and sub-surface modes. In particular, phonon dispersion curves
and layer- and polarization-resolved phonon density of states reveals the intrinsic nature of the controversial longitudinal surface phonon resonance. The
nature of such a resonance is not trivial, as appears from the longitudinal amplitudes in the second layer there is an avoided crossing between a longitudinal
acoustic resonance and a surface optical branch running between Γ̄-X̄ leading
to a second layer shear vertical resonance in the X̄ point.
In addition, the phonon density of states also reveals the existence of new
subsurface optical modes, localized in the second layer. These modes are not
predicted by force constant models with no change in the interlayer force constants. Thus, they are likely induced by the change in the interlayer spacing
near the surface and may be regarded as typical effects of the extensive relaxation induced charge redistribution at s, p metal surfaces. These modes,
similar to the Lucas modes in ionic crystals, form a degenerate pair at the zone
center.
47
5. Summary of Appended Papers
5.6
Paper VI: Optical Phonon Resonances at
Metal Surfaces
In this paper, the lattice dynamics of the Cu(111) surface is studied applying
the first-principles density functional perturbation theory. A 25 layer slab
is constructed by means of slab-filling. The dynamical matrices calculated
for a fully relaxed 7 layer slab gives us the information of the inter-atomic
force constants in the surface region, whereas a separate bulk calculation is
used for filling the rest of the 25 layer slab. The calculated phonon dispersion
curves along Γ̄-M̄ in the surface Brillouin zone are presented with experimental
Helium atom scattering (HAS) and electron energy loss spectroscopy data. Our
calculations are in an excellent agreement regarding the Rayleigh wave and the
gap mode (S2 ) at the M̄ point.
Regarding Cu(111) and other noble metal surfaces, it has been a longlived debate whether the longitudinal phonon resonance, found in experimental studies, is an experimental artifact or due to an anomalous softening of
the surface atom force constants. In order to trace the longitudinal surface
resonance we analyze the DOS for different layers and polarizations. There is
a clear evidence of a longitudinal phonon resonance mode in the first layer.
The calculated DOS for longitudinal components agrees well with experimental
data from HAS and EELS to about half of the Brillouin zone, along the Γ̄-M̄
direction. Thereafter, the longitudinal character of the modes decrease in intensity, whereas the second layer phonon DOS shows an intense shear-vertical
character when approaching the M̄-point in agreement with the experimental
data.
We propose, based on our study, that the He atoms to some extent can
probe the second layer shear-vertical displacements through the associated
surface charge density oscillations. We conclude that the longitudinal resonance owes its complex nature to a strong hybridization with an intense shear
vertically polarized resonance localized in the second atomic layer.
48
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53
Paper I
PRL 93, 226103 (2004)
week ending
26 NOVEMBER 2004
PHYSICA L R EVIEW LET T ERS
Surface Relaxation Influenced by Surface States
Vasile Chis and Bo Hellsing
Solid State Physics, Experimental Physics, Göteborg University and Chalmers University of Technology,
S-412 96 Göteborg, Sweden
(Received 5 July 2004; published 23 November 2004)
A detailed theoretical investigation of the relaxation of the simple metal surface Al(100) is presented.
We show the influence of electronic surface states in this context. The sign and magnitude of the
relaxation of the topmost atomic layers is mainly determined by the rearrangements of the surface state
charge. The degree of surface relaxation convergence, with respect to the number of slab layers, is
determined by the location of the surface state band relative to bulk bands.
DOI: 10.1103/PhysRevLett.93.226103
PACS numbers: 68.35.Bs, 73.20.At, 79.60.Bm
0031-9007=04=93(22)=226103(4)$22.50
Based on simple arguments, one would expect that for
a nonmetallic surface, the relaxation would be inward
simply due to breaking of local bonds, while for a metallic surface the relaxation could be outward as the top
layer tries to adjust to the displaced valence electron
density. However, it has been shown [5] that for a simple
metal such as aluminum, both inward, Al(110), and outward, Al(100), relaxation take place.
Al(100) exhibits, as many other single crystal metal
surfaces [6], a surface localized electronic band. In the
case of Al(100) the binding energy of the surface state is
about 3 eV in the point and thus these band states will be
partly occupied. Our detailed investigations, resolving
the redistribution of valence electron charge in the relaxation process of the uppermost surface layers, show
that the surface state electrons play a dominant role. As a
consequence, we notice that in a slab calculation the
convergence of the surface layer relaxation, as function
of number of slab layers (N), is determined by the location
of the surface state band relative to the bulk bands. For
Al(100), the surface state band is close to the lower edge
2.0
∆d12 [%]
When a solid is cut, the atoms near the surface are
exposed to different forces from those in the bulk of the
material. The atoms near the newly created surface tend
to relax mostly perpendicular to the surface to minimize
the total energy. A proper description of the ionic forces is
crucial to obtain the relaxed atomic structure. The final
surface atomic positions will determine the surface electron structure properties such as, e.g., work function and
surface energy and also the surface lattice dynamics. We
show that electronic surface states could play a crucial
role. Although in general, only the uppermost two or
three atomic layers will show any appreciable relaxation,
an order of magnitude larger number of layers might be
required in a slab calculation due to deeply penetrating
surface states.
The relaxation and dynamical properties of the Al(100)
surface has been the objective of several investigations
based on different theoretical methods. Already more
than a decade ago, Bohnen et al. [1] showed that slight
variations of the force constants in the surface significantly influence surface phonon frequencies near
Brillouin zone boundaries. The Al(100) surface has an
occupied electronic surface state band and we have found
a clear correspondence between the relaxation and the
surface state.
Experimental investigations by Davis et al. [2] of the
relaxation of the interlayer spacing between the two top
atomic layers, d12 , of Al(100) yields an expansion of
1.8%. Semiempirical calculations yield an inward relaxation 2:7% (embedded atom method) [3] and 3:0%
(effective medium theory) [4], while first-principles studies by Bohnen et al. [1] and Zheng et al. [5] suggest an
outward relaxation (expansion) by 1.2% and 1.8%, respectively. The semiempirical methods obviously do not
describe the surface ionic forces correctly. Both of the
first-principles calculations yield correct outward relaxation. However, due to the deeply penetrating electronic
surface state, far too few slab layers have been used in
order to grasp the ground state relaxed atomic structure.
This is illustrated by our calculated d12 shown in Fig. 1
in comparison with the experimental result.
Exp
1.5
1.0
7
9
11
13
15
17
19
21
23
Number of layers
FIG. 1. Interlayer relaxation d12 . Filled circles are calculated results and the dashed line serves as a guide for the eye.
The horizontal solid line marks the experimental value
(Ref. [2]).
226103-1
 2004 The American Physical Society
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26 NOVEMBER 2004
PHYSICA L R EVIEW LET T ERS
of the projected bulk band gap and thus its wave functions
penetrate far into the bulk. Thus a large number of slab
layers are needed to avoid interference effects between
the two slab surfaces. To our knowledge, no previous
theoretical investigations have been presented where the
role of surface states have been pointed out as crucial for
the understanding of the surface relaxation.
The calculations were based on first-principles density
functional theory (DFT) with the local density approximation (LDA) for the exchange and correlation potential.
The electron wave functions were expanded in plane
waves with an energy cutoff of 12 Ry and the Al ion
pseudopotential described by norm-conserving pseudopotentials. Calculations are performed for slabs with N 7–23 and six layers of vacuum. The supercell
is defined
p
by Lx ; Ly ; Lz b; b; L, with b a= 2 5:3033 a:u,
where a is the conventional lattice parameter of Al and L
is determined by N. The Brillouin zone was sampled at k
points according to the Monkhorst-Pack 16 16 1
mesh. To improve the convergence, a finite temperature
smearing of kB T 0:7 eV was included. The relaxation
of the system is performed according to the BroydenFletcher-Goldfarb-Shanno (BFGS) algorithm. The
Hellman-Feynman forces acting on each atomic layer
perpendicular to the surface were calculated and the
structure was relaxed in the z direction until these forces
The surface projected band
were less than 2:6 meV=A.
structure of the relaxed surfaces was determined by calculating the Kohn-Sham eigenvalues in 81 k points along
M
in the surface Brillouin zone
the symmetry lines X(SBZ).
The calculation of the x; y-averaged valence electron
density, z,
has been done in two ways: one, by averag~ z over the planar points, x,
~ in the x; y plane of the
ing x;
supercell (Fig. 4). Second, by applying a well motivated
free-electron approximation in the x; y plane with band
masses determined by the self-consistent bands near the
point (Fig. 6). We sum the integrated parts of the
occupied bands over the SBZ to get the density. Thus
we can decompose the valence electron density in parts
originating from surface and bulk states.
We present part of the band structure for the 13 layer
and 23 layer slab in Fig. 2 in the reduced zone scheme.
The interaction of the two slab surfaces is reflected in the
splitting of the odd and even symmetry surface state band
(dashed lines in Fig. 2). An increase of N will reduce the
gap E between the surface states. E versus N is calculated for 7–23 layers slabs and is well approximated by
an exponential function, E E0 eN , where E0 3:235 eV, 0:132. The energy gap for the 13 and 39
layer slab is found to be 580 meVand 19 meV, respectively.
This compares well with calculations by Caruthers et al.
[7], 570 meV and 27 meV, respectively. According to this
analysis, N > 41 is required in order to obtain E <
10 meV for Al(100), which is optimal to obtain an accuracy compatible with the typical resolution of a DFT
0
−2
−4
−6
E−EF [eV]
PRL 93, 226103 (2004)
−8
−10
0
−2
−4
−6
−8
−10
X
Γ
M
FIG. 2. Energy bands of Al(100) for the slabs of 13-layer
(top) and 23-layer (bottom). The gray area represents bulk
bands areas and the dashed lines the surface states.
calculation and, e.g., high resolution photoemission spectroscopy data. The bulk band gap reduction with N shown
in Fig. 2 does not have a large effect on the surface
density variation with N (shown later) and thus not on
the surface relaxation. However, the reduced energy separation between the surface bands and the lower bulk band
edge with N will increase the decay length of the surface
state (approximately inversely proportional). Fortunately,
this increase is slower than the increase of the slab thickness with N. This phenomena is most likely partly responsible for the apparent convergence previously
reported [5] up to N 9.
The relaxation of the top layers of a slab is determined
by the forces on the ions set up by the redistribution of
valence electron charge density taking place as soon as
the surface is created. Within the Born-Oppenheimer
approximation, the electron density adjust instantaneously to a change in the ionic lattice structure, determined by the direction of the net ionic forces. As the
net ionic forces eventually vanish, the relaxed lattice
structure is obtained. This scenario can be mapped in
detail in the calculations.
The initial ionic forces perpendicular to the surface
prior to the actual ionic relaxation is determined by the
self-consistent electron density for the truncated bulk
lattice structure. In Fig. 3 we show these net forces acting
on the first and second layer as a function of N.
We note that independent of N, the initial force acting
on the first layer is outward, while slightly inward on the
second layer. Thus we expect that these two layers will
start to move apart as the ions start to respond to the
forces. Furthermore, the result for the N 13 indicates a
minimum of this tendency, which is consistent with the
dip in d12 shown in Fig. 1. The variation of the force
226103-2
∆σ[%] ∆φ[%] Force [eV/Angstrom]
0.1
0.0
1:st Layer
−0.1
0.1
0.0
2:nd Layer
−0.1
4
2
0
3
2
1
0
∆φ
∆σ
7
9
11
13
15
17
19
21
23
Number of layers
FIG. 3. Top: Initial forces (see text) acting on the first and
second layer. Positive and negative values indicate direction
towards vacuum and slab center, respectively. Middle: Work
function change; N N 23
=23. Bottom:
Surface energy change; N N=N 23=23
=
23=23.
with N in the region 7–23 layers indicates the slow
convergence. As the force is determined by the electron
density, the major changes in the electron density take
place near the topmost layer. To get further insight, we
have analyzed the electron density in more detail.
In Fig. 4 we compare the total valence electron density
for the unrelaxed and relaxed lattice structure. We note
the general trend that the relaxation drives the nearsurface electron density towards the vacuum region.
However, the magnitude of this tendency depends on N.
Comparing the 13 and 23 layer slabs, we observe a
consistent trend when comparing the force in Fig. 3,
valence electron density in Fig. 4, and finally d12 in
Fig. 1.
We now show that this layer dependence brings about
the evidence for the importance of the surface states in
the relaxation process.
To see this, we integrated the surface electron charge,
from the position of the unrelaxed second layer to a
position far into the vacuum, for both the unrelaxed
and the relaxed lattice. The percentage difference is
shown in Fig. 5 and shows a similar structure as the layer
dependence of d12 shown in Fig. 1.
To illustrate the connection between the relaxation and
the surface states, we resolve the relaxed electron density
of the 13 and 23 layer slabs in terms of contributions from
bulk and surface states, respectively (Fig. 6).
It is obvious from Fig. 6 that what makes the difference
in electron density at and outside the uppermost lattice
plane is the charge carried by surface state bands. In this
region, the contribution from bulk states is essentially the
same for the 13 and 23 layer slab (as well as for the rest of
the investigated slabs).
For very thick slabs (N ! 1) the odd and even surface
state wave functions, with respect to the slab center,
become equal in the surface region where they have any
appreciable amplitude and if x; y-averaged, their nodes
will coincide with the lattice planes. However, for a slab
thickness less than twice the surface state wave function
decay length, the odd and even wave functions will be
shifted relative to each other in the region of the surface
and their nodes will not coincide with the lattice planes;
thus the charge density in the surface region will vary
with N and accordingly the ionic forces and therefore the
lattice relaxation.
We thus have the following scenario. As the surface is
created, the valence electrons relax rapidly. The forces
built up drive the top ionic layer outwards. During this
process, the electron density will be pushed even further.
The further the electron density is pushed outwards the
further the top layer will be displaced. The electron
charge density from bulk band states will not change
13 Layers
Change in electrons [%]
Charge density [a.u.]
0.03
0.02
0.01
0.03
week ending
26 NOVEMBER 2004
PHYSICA L R EVIEW LET T ERS
PRL 93, 226103 (2004)
23 Layers
0.02
0.01
20
10
0
5
10
15
20
25
Number of layers
FIG. 4. Valence electron density, for a 13 and 23 layer slab.
The dashed and solid lines represent the unrelaxed and relaxed
geometry, respectively. The vertical lines represent the ion core
positions of the unrelaxed uppermost four layers.
FIG. 5. Percentage change of valence electron charge due to
the relaxation in the region from the unrelaxed second layer out
into the vacuum.
226103-3
Charge density [a.u.]
PRL 93, 226103 (2004)
0.03
TABLE I. Surface relaxation of Al(100) as function of number of slab layers (L). ( ) denotes results of this work.
Bulk
states
9L
15L
23L
Theory (9L)
Theory (15L)
Theory
Experiment
0.02
0.01
week ending
26 NOVEMBER 2004
PHYSICA L R EVIEW LET T ERS
Surface
states
0 00
FIG. 6. Valence electron density for a 13 (dashed line) and 23
layer slab (solid line). Contributions from bulk and surface
states indicated. The vertical lines represent the ion core
positions of the unrelaxed uppermost four layers.
appreciably with N in the focused region, while the
density from the surface states is responsible for most of
the electron density variation. The surface energy and
work function variations with respect to N are shown in
Fig. 3. The surface energies converge smoothly while the
work function shows a similar variation as the top layer
forces and relaxation. This is understandable as the surface energy is an integrated quantity based on total energies while the work function is really determined by the
dipole generated by the relaxed surface electron density.
For the surface Al(111), we expect the same mechanism
to be responsible for the observed outward relaxation, as
the observed surface band electron structure is similar
[3].
In Table I, some of our results for the relaxation are
shown together with results of previous calculations and
experimental data. Our calculated value for d12 , 1.93%
for the nine layer slab, agrees well with the result 1.89% in
a previous first-principles calculation by Zeng et al. [5].
By chance the nine layer results happen to agree well with
the experimental result (1.84%). However, our study
shows that d12 is far from convergence for a nine layer
slab. The 23 layer result for d12 , 1.50%, is more close to a
converged value. We find a value about 20% smaller than
experiment. This underestimate of d12 is not surprising
as LDA is well known to give an over-binding. The semiempirical results [3], d12 2:7%, which are based on
fitting to elastic bulk properties, indicate the importance
of including the surface electron structure.
In conclusion, we have shown that the relaxation of the
Al(100) is strongly influenced by the surface state bands.
The slow convergence of the relaxation, in particularly
for the uppermost layer, is a clear signal of a deeply
penetrating occupied surface state band. We have estimated that for Al(100), an optimal result for the relaxa-
d12
d23
d34
d45
Ref.
1.93
1.41
1.50
1.89
1.2
2:7
1.84
1.40
1.13
1.19
4.12
0.2
0:1
2.04
0.87
0.83
0.64
2.96
0:1
1.06
0.60
0.66
2.94
()
()
()
[5]
[1]
[3]
[8]
tion of a first-principles study with a typical accuracy of
10 meV, which also is about the present resolution in
HRPES experiments, requires at least a 41 layer slab. In
general, the closer an occupied s, p surface state band
appears relative to the bulk bands the greater the decay
length of the surface states and thus the greater the
number of slab layers is required to avoid interacting
surface states. A proper surface relaxation is important
for a realistic description of (i) the surface electron structure, e.g., the work function and surface energy, and
(ii) surface ionic forces that will determine the surface dynamics. Finally we propose more extensive slab
calculations of surfaces with an experimentally determined surface band structure similar to Al(100).
Photoemmision data indicate that, e.g., Ag(111) and
Mg(0001) have a partly occupied s, p surface state band
slightly above the lower bulk band edge, and thus they
serve as good candidates.
We thank Pieter Kuiper for stimulating discussions.
Calculations were done using the PWSCF package [9]
and were performed using the UNICC resources at
Chalmers, Göteborg, Sweden. Financial support is acknowledged from Swedish Science Council (VR).
[1] K.-P. Bohnen and K.-M. Ho, Surf. Sci. 207, 105 (1988).
[2] H. L. Davis, J. B. Hannon, K. B. Ray, and E.W. Plummer,
Phys. Rev. Lett. 68, 2632 (1992).
[3] E.V. Chulkov and I. Yu. Sklyadneva, Surf. Sci. 331, 1414
(1995).
[4] P. D. Levsen and J. K. Norskov, Surf. Sci. 254, 261 (1991).
[5] J.-C. Zheng, H.-Q. Wang, C. H. A. Huan, and A. T. S. Wee,
J. Electron Spectrosc. Relat. Phenom. 114, 501 (2001).
[6] E.V. Chulkov, V. M. Silkin, and P. M. Echenique, Surf.
Sci. 437, 330 (1999).
[7] E. Caruthers, L. Kleinman, and G. P. Alldredge, Phys.
Rev. B 8, 4570 (1973).
[8] W. Berndt, D. Weick, C. Stampfl, A. M. Bradshaw, and M.
Scheffler, Surf. Sci. 330, 182 (1995).
[9] S. Baroni, A. Dal Corso, S. de Gironcoli, and P.
Giannozzi, Plane-Wave Self-Consistent Field, http://
www.pwscf.org.
226103-4
Paper II
PHYSICAL REVIEW B 75, 155403 共2007兲
Overlayer resonance and quantum well state of Cs/ Cu„111… studied with angle-resolved
photoemission, LEED, and first-principles calculations
M. Breitholtz,1 V. Chis,2 B. Hellsing,2 S.-Å. Lindgren,1 and L. Walldén1
1Applied
Physics Department, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
2
Physics Department, Göteborg University, SE-412 96 Göteborg, Sweden
共Received 6 December 2006; published 4 April 2007兲
Angle-resolved photoemission spectroscopy and low-energy electron diffraction are used to study submonolayer coverages of Cs on Cu共111兲 at room temperature 共RT兲 and 170 K. At RT, the Cs saturation coverage is
approximately 90% of the full monolayer coverage. The full monolayer is characterized by a quantum well
state 共QWS兲 band having an energy of 25 meV below the Fermi level 共EF兲 in the ¯⌫ point and a resonance band
extending to energies below the Cu band gap. This is supported by our first-principles calculations. Lowenergy electron diffraction shows that the Cs overalyer forms a 共2 ⫻ 2兲 structure over a wide coverage range,
in which the QWS has energies from 50 meV above to 25 meV below EF. The continued energy shift of the
QWS after saturation of the diffraction angles is interpreted in terms of vacancies in the overlayer.
DOI: 10.1103/PhysRevB.75.155403
PACS number共s兲: 79.60.⫺i, 73.21.Fg, 73.20.At, 71.15.Mb
I. INTRODUCTION
Alkali-metal overlayers attract interest as prototype examples of simple metal quantum wells, in which valence
electrons are more or less confined depending on the choice
of substrate.1 Strong confinement is obtained when the substrate has a band gap, which allows a state in the overlayer to
have only an oscillating tail on the substrate side of the interface. If, as exemplified by Na/ Al共111兲, there is a potential
step at the interface to a more attractive potential in the substrate, the confinement is weaker but still sufficient for resonances to be observed by photoemission or inverse
photoemission.2,3 Alkali metals on Cu共111兲 give examples of
both strong and weak confinements. It is strong for energies
within the Cu band gap at the L point of the Brillouin zone.
Since this gap extends 0.75 eV below the Fermi energy4 and
the filled bandwidth of an alkali metal is appreciably larger
than this, most of the valence electrons in an adsorbed film
are found in resonances. The states within the gap give narrow lines in angle-resolved photoemission spectroscopy
共ARPES兲, as noted for Na 共Ref. 5兲 and Li,6 and narrow onsets in dI / dV curves recorded with scanning tunneling spectroscopy 共STS兲, as observed for Na.7 A detailed characterization of the electron structure has also been done for the
system Na/ Cu共111兲 applying first-principles methods.8 The
calculated properties are consistent with experiment.
In a report including dI / dV data, it was recently noted for
共2 ⫻ 2兲 ordered Na and Cs monolayers on Cu共111兲 that there
is actually no strict confinement of states within the gap.9
This follows from the different periodicities of the overlayer
and the substrate. In this case, a state, which extends across
the border of the two, adjusts to both periods. This is the
situation for many systems, in particular, low-dimensional
ones.10 In the case of a 共2 ⫻ 2兲 monolayer on Cu共111兲, the
adjustment to two periods makes the quantum well leaky due
to overlap with Cu bulk states when these are back folded
into the smaller Brillouin zone defined by the overlayer. A
quantum well state 共QWS兲 at the center of the surface Brillouin zone with an energy in the lower half of the gap will
overlap with Cu s , p-band bulk states along the L-X direc1098-0121/2007/75共15兲/155403共9兲
tion. This gives a small resonant width, which was estimated
theoretically to be somewhat less than 10 meV. If this is
correct, it explains why the dI / dV onset, which was recorded
at 5 K, is broader than the calculated lifetime width of a
strictly confined and, thus, discrete state in a 共2 ⫻ 2兲 monolayer.
Characteristic of alkali-metal monolayers on Cu共111兲 is
that energies of surface located states gradually shift with
coverage. This applies to the surface state of Cu共111兲 0.4 eV
below EF,11,12 as well as to the strongly confined quantum
well state that becomes populated at high monolayer
coverage.14,13 The ability to tune the electronic structure and
properties has been manifested and exploited in a number of
different experiments, where various aspects of the interaction between radiation and matter are studied.15–17
In the present work, we use ARPES at low photon energy
to monitor the states characteristic of Cs monolayers at different coverages and of a full Cs monolayer exposed to
oxygen.25 The effect of exposing the Cs overlayer to oxygen
has previously been studied. These studies were, however,
limited to very small Cs coverages using ARPES and
electron-energy-loss spectroscopy combined with workfunction measurements for Cs coverages in the monolayer
coverage range. It was suggested that oxygen acts as a diluter
of the electron gas in the Cs layer, a picture which is supported by the present work.
Regarding the Cu共111兲 surface state, we note that it shifts
upon Cs deposition to energies below the edge of the L gap,
suggesting that the surface state at high coverage ends up as
a monolayer resonance. This interpretation is supported by
our first-principles calculations. Qualitatively, the same electronic structure is obtained, however, not discussed, in a recent calculation for Na/ Cu共111兲.7
A recent scanning tunneling microscopy 共STM兲
investigation,24 with the sample at room temperature 共RT兲
during the Cs deposition but at 9 K during imaging, revealed
a commensurate low coverage phase with a very large distance 共11 Å兲 between Cs atoms and well described as a
共冑19⫻ 冑19兲 R23.4° structure. At intermediate coverages, the
observed incommensurate hexagonal structures are some-
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PHYSICAL REVIEW B 75, 155403 共2007兲
BREITHOLTZ et al.
what rotated with respect to the substrate surface lattice. The
reciprocal lattice vectors of these structures will mediate a
coupling to Cu bulk states, as do the scattering vectors of the
particular overlayer discussed by Corriol et al.,9 and this will
give the downshifted Cu surface state a resonant width as Cs
is adsorbed.
The present experiments were made with the sample at
295 K 共RT兲 and around 170 K. We find that the linewidth of
the QWS does not depend significantly on the coverage and,
thus, not on whether or not the overlayer has an ideal 共2
⫻ 2兲 structure. The state is robust even with respect to oxygen exposure and shifts to higher energies gradually with
increasing oxygen exposure.
The paper is organized as follows. First, in Sec. II, we
describe the experimental and theoretical methods used. In
Sec. III, we outline briefly our results. The results will be
discussed in more detail in Sec. IV and, finally, we make
some concluding remarks in Sec. V.
II. EXPERIMENTAL AND CALCULATIONAL METHODS
Cs was evaporated onto the Cu共111兲 crystal from a heated
Cs glass ampoule broken in situ and kept at constant temperature during an experimental run such that a full monolayer is obtained after 100– 350 s evaporation time. All of
the photoemission spectra presented below were recorded
with a spectrometer 共Leybold兲 designed for high-resolution
electron-energy-loss spectroscopy and a 1 mW He-Cd laser
共h␯ = 3.82 eV兲 as light source. The light was made incident at
an angle of 80°. Only p-polarized light gave off an emission
from the states studied so this is the polarization used to
record the presented spectra. The sample was cooled via its
holder, which was attached to a LN2 container. The sample
temperature was obtained by fitting the high-energy cutoff to
the Fermi-Dirac distribution in spectra where no QWS is
close to the Fermi level. For a range of coverages, the QWS
of interest lies above the Fermi level. The QWS is then observed as a shoulder in the raw spectra, but a peak may be
recovered by multiplying the raw spectra by the inverse of
the Fermi-Dirac 共FD兲 distribution function as exemplified by
the spectra in Fig. 1. When this procedure has been used, the
spectra presented are labeled FD normalized.
The electronic structure calculations were performed with
the total energy, plane-wave code PWSCF,18 which is based on
the density-functional theory. We adopt ultrasoft19 pseudopotentials for Cu and Cs and the general gradient
approximation20 GGA-PBE for the exchange and correlation
energy functional. A kinetic-energy cutoff of 30 Ry was used
for the plane waves and a charge density cutoff energy of
480 Ry. In general, the charge-density cutoff is about four
times the kinetic-energy cutoff. However, ultrasoft pseudopotentials require a substantially higher value.21 The
Brillouin-zone sampling was done over several uniform
Monkhorst-Pack grids, and a finite temperature smearing of
first-order Methfessel-Paxton22 type was used with a Fermi
distribution broadening ranging from ␴ = 0.136 to ␴
= 0.680 eV. The atomic arrangement of the 共2
⫻ 2兲-Cs/ Cu共111兲 was obtained by minimizing the total energy with respect to the positions of the Cs atoms in the
FIG. 1. The spectrum labeled PE Spectrum is recorded for a Cs
coverage such that a QWS lies 85 meV above the Fermi level. The
QWS is seen as a weak shoulder around 70 meV above EF. By
dividing the spectrum with the Fermi-Dirac function, given by the
line labeled FD, the line FD norm is obtained. This line has a
distinct peak above EF, which can be fitted with a Lorentzian line,
also shown in the diagram.
overlayer and the Cu atoms in a slab. The supercell was built
by including ten Cu layers with four Cu atoms in each layer.
The Cs atoms were placed on both sides of the copper slab.
In order to exclude any surface to surface interaction through
the vacuum region, the surfaces are separated by 25.3 Å of
vacuum. During the relaxation process, the Cu atoms where
free to move in all directions, whereas the Cs atoms where
restricted to move along the surface normal. The electronic
band structure was calculated along the high-symmetry di¯ -M̄. Sampling was performed over 130 k points
rections K̄-⌫
in a non-self-consistent manner using the self-consistent
electron density obtained by optimizing the atomic arrangement.
III. RESULTS
A. Coverage and temperature dependence of QWS
A QWS monitored along the surface normal shifts to
lower energy as the Cs coverage is increased and saturates at
around 25 meV below the Fermi energy 共Fig. 2兲. The QWS
energy changes almost linearly with the evaporation time
共Fig. 3兲. Initially, upon continued deposition, the energy remains nearly constant while the emission line broadens. Assuming that the energy of the QWS saturates at full monolayer coverage, one thus gets a clear demarcation of this
coverage. We present this observation first since the lowenergy electron diffraction 共LEED兲 pattern and the surfacestate data are less distinct in this respect. At coverages for
which the state is above EF, the emission line is modified by
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FIG. 2. Fermi-Dirac normalized photoemission spectra recorded
along the surface normal for various Cs evaporation times. The
evaporation time is given by each spectrum, and a complete Cs
monolayer is obtained after 220 s of evaporation.
the FD cutoff function, but a peak is recovered when the raw
spectra are multiplied by the inverse of the FD distribution
共Fig. 1兲.
Figure 2 shows a plot of FD normalized spectra recorded
at different Cs coverages as given by the evaporation time.
The substrate temperature is 170 K. If the substrate is kept at
RT, the QWS saturation energy is around 10 meV above the
Fermi energy. If the sample is cooled to 170 K after satura-
FIG. 4. Photoemission spectra recorded for the RT Cs saturation
coverage. The lower spectrum is recorded with the sample at RT,
and the upper spectrum after cooling the sample to 170 K. The
widths given by the spectra are obtained from Lorentzians fitted to
the Fermi-Dirac normalized spectra.
tion has been reached, the emission line narrows from a full
width at half maximum 共FWHM兲 of 57 to 40 meV, but the
energy remains nearly the same 共Fig. 4兲. Upon a small additional Cs deposited onto the cooled sample, a further downshift by around 20 meV is observed, indicating that the saturation coverage is higher at 170 K than at RT.
B. Oxygen exposure
Upon exposure to oxygen, the QWS shifts to higher energy. This effect was studied for a saturated Cs monolayer
共Fig. 5兲. The pressure gauge was not positioned such that
correct exposure values were obtained, but it could be used
to increase the exposure in approximately equally large
doses. As a reference regarding exposure, one may measure
the work function, which passes a minimum value around
0.5 eV lower than for the Cs saturated monolayer. In previous work, the minimum was obtained after 0.4 L exposure.25
The work-function values obtained at the different exposures
are plotted in Fig. 6.
C. LEED observations
FIG. 3. Energy of the QWS plotted against the Cs evaporation
time. For a full Cs monolayer, obtained after 220 s of evaporation,
the QWS is 25 meV below the Fermi level.
As observed for several alkali-metal adsorption systems,
Cs deposition is noted as a ring pattern, which, on continued
evaporation, transforms into a spot pattern.26 The LEED pattern was registered at an electron energy 共73 eV兲 for which
the low-index Cu spots as well as Cs deposition induced
changes are observed. Figure 7 shows a plot of 共d / D兲2 versus evaporation time, where d is the diameter of the Cs induced diffraction rings observed at low coverage and, at
higher coverage, the distance between opposite half-order
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BREITHOLTZ et al.
FIG. 5. Fermi-Dirac normalized photoemission spectra showing
the evolution of the QWS formed in a complete monolayer Cs on
Cu共111兲 upon O2 exposure. The oxygen-induced work-function
change is given by each spectrum. The temperature is 170 K.
spots and D is the distance between opposite substrate spots.
Depending on the Cs induced pattern on the LEED display,
different symbols are used in the diagram. Circles correspond to a ring pattern, circles with an inside dot to a ring
pattern with azimuthal intensity variations, and filled circles
to a spot pattern on the display. As can be seen in Fig. 7,
共d / D兲2 increases linearly with deposition time until satura-
FIG. 7. The squared ratio of the distance between Cs induced
features 共d兲 and the Cu spots 共D兲 on the LEED screen plotted
against evaporation time. Different symbols are used depending on
the Cs induced diffraction pattern. Circles correspond to a ring,
circles with an inside dot to a ring with intensity variations, and
dots to a distinct spot pattern. The inset shows Fermi-Dirac normalized photoemission spectra for coverages where the LEED pattern
has saturated.
tion of the monolayer Cs coverage at a value around 0.25,
expected for a 共2 ⫻ 2兲 order. The inset shows FD normalized
photoemission spectra recorded in the same experimental
run, which was made with the sample at 170 K. The observation of main interest is that the QWS continues to shift
with increased deposition even after saturation of the diffraction angles.
D. Surface-state shift
FIG. 6. The change in work function for an oxygen-exposed full
monolayer Cs on Cu共111兲 plotted against O2 exposure. The inset
shows the shift of the QWS versus the work-function change.
It was noted in a previous study that the Cu共111兲 surface
state shifts to lower energy with increasing Cs coverage, and
that the emission intensity decreases such that the state is no
longer observed when it has reached the lower edge of the L
gap.14 According to the present results, however, a weak
peak is resolved below the edge of the gap as the deposition
is continued, indicating that the binding energy actually
shifts into the resonant range 共Fig. 8兲. The strong peak at
around 2 eV binding energy is due to emission from the
uppermost Cu 3d band. This emission line is, using 3.82 eV
photons, revealed only at coverages near the work-function
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FIG. 9. 共Color online兲 Relaxed lattice structure of 共2
⫻ 2兲-Cs/ Cu共111兲. The distance 共A兲 between the Cs atom and the
Cu atom positioned vertically below is 3.16 Å, and the vertical
distance between Cs and the surrounding Cu atoms 共B兲 is 3.03 Å.
FIG. 8. Photoemission spectra showing the shift of the surface
state upon Cs deposition. The evaporation time is given by the
spectra. A full Cs monolayer is obtained after 225 s of evaporation.
The high intensity around 2 eV binding energy in some of the spectra is due to the emission of Cu 3d electrons. Note how the Cs
induced change of work function is reflected in spectra by a change
of the low-energy cutoff. To observe the emission from states at EF,
at least a 30 s Cs deposition, corresponding to around 15% of a full
monolayer, is necessary.
minimum, which is obtained after approximately half the
deposition time required for obtaining a full monolayer.
E. Lattice and electron structures
Two different sites for the Cs atoms in the 共2 ⫻ 2兲 monolayer on Cu共111兲 have been investigated, the on-top and hcp
hollow sites. The lowest total energy was obtained with the
Cs atoms in the on-top position with a Cs-Cu distance of
3.06 Å. This is in good agreement with previous LEED data,
3.01± 0.05 Å, by Lindgren et al.23 The Cs atoms induce a
small rumpling of the Cu surface atoms, see Fig. 9. For the
relaxed structure, the Cu atom positioned under the Cs atom
is pushed down by 0.13 Å.
The electronic structure and, in particular, the QWS energy relative to the Fermi level were tested by varying the
smearing width ␴ and the number of k points used in the
Brillouin-zone integration. The results are summarized in
Table I. The conclusion from the table is that the QWS energy converges relatively slowly with respect to the number
of k points. For the ␴ = 0.136 eV case, we fitted the results to
a damped oscillation-type function giving us an asymptotic
value for the QWS energy of 0.035 eV above the Fermi
level.
The band structure of the 共2 ⫻ 2兲 Cs overlayer, presented
in Fig. 10, along the high-symmetry directions ¯⌫-M̄ and ¯⌫-K̄
of the surface Brillouin zone 共SBZ兲 is based on calculation
for ␴ = 0.136 eV and a 共10⫻ 10兲 Monkhorst-Pack 共MP兲
mesh. Due to the finite thickness of the slab, each overlayer
induced state is split into a doublet, represented by odd and
even wave functions with respect to the center of the slab. In
Fig. 10, the dark shaded area represents the clean 共1 ⫻ 1兲 Cu
surface projected band structure, while the light shaded area
is the additional projected bulk band structure due to band
folding when the 共2 ⫻ 2兲 Cs layer is adsorbed. The band
structure for the clean Cu共111兲 surface is based on calculations of an 18 atomic layer Cu slab. The mean values of the
doublets of the three surface localized bands are shown by
lines. The three bands are 共i兲 the clean Cu共111兲 surface state,
共ii兲 the QWS state, and 共iii兲 the overlayer resonance state.
The energy, E-EF, of these bands in the ¯⌫ point is −0.39 eV
关clean Cu共111兲 surface state兴, 0.033 eV 共QWS兲, and
−1.14 eV 共overlayer resonance兲, respectively. The QWS
band has a free-electron-like dispersion with an effective
electron mass of 0.86me. When the Cs overlayer is introduced, there is a substantial reduction of the band gap, which
is illustrated by the light area in the figure.
IV. DISCUSSION
In this section, we first present details of the overlayer
resonance state, referring to our photoemission spectroscopy
共PES兲 data and calculated electron structure. Then an analysis of the monolayer lattice versus Cs exposure is presented.
The coverage dependence of the QWS energy and the temperature dependence of its width with reference to the
electron-phonon coupling will be discussed.
A. Overlayer resonance state
When the clean Cu共111兲 surface is exposed to Cs, the PES
data reveal an energy downshift of the surface state 共Fig. 5兲.
After 120 s 共⬃0.5 ML兲 of Cs evaporation, the state has
shifted to energies below the Cu bulk band gap. The crossing
of the gap edge is accompanied by a broadening of the line
since the state couples to Cu bulk states. Continued Cs
evaporation leads to an additional shift and broadening of the
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BREITHOLTZ et al.
TABLE I. Convergence test of the QWS energy, E-EF 共eV兲, in
the ¯⌫ point with respect to selected k points used in the Brillouinzone integration and with respect to the Methfessel-Paxton smearing width, ␴ 共eV兲. The k points are given in terms of MonkhorstPack 共MP兲 mesh with the corresponding number of k points in the
irreducible Brillouin zone.
␴
0.680
0.408
0.340
0.272
0.136
MP-mesh 共6 ⫻ 6兲 共8 ⫻ 8兲 共10⫻ 10兲 共12⫻ 12兲 共14⫻ 14兲
No. of
k points
13
21
31
43
57
0.047
0.062
0.082
0.058
0.050
0.040
0.013
0.048
0.033
0.052
0.054
0.031
line. The state is now better described as an overlayer resonance characteristic of the monolayer rather than as a downshifted surface state. The resonance has its origin in the freeelectron-like energy-band characteristic of a freestanding,
close-packed Cs monolayer, which, according to the calculations of Wimmer, extends 1.28eV below EF.27 The integrity
of this band is thus maintained upon adsorption of the monolayer though the states are broadened due to hybridization
with copper states.
The calculated overlayer resonance band along the ¯⌫-M̄
direction is shown in Fig. 11 together with the squared magnitude of the wave function at specific points. In the ¯⌫ point,
the state is spread over the Cu layers and the Cs overlayer,
with not much of an overlayer resonant character. However,
as one moves out toward the M̄ point, the resonance character develops. In point A, about two-thirds of the distance to
M̄, a substantial part of the state is localized to the Cs layer.
The increasing weight in the Cs overlayer as the SBZ edge is
approached is seen at point B. Further characteristics of the
band which support the labeling—overlayer resonance—is
the absence of a node of the wave function in the Cs layer,
which is also the case for the lowest band of the unsupported
共2 ⫻ 2兲 Cs monolayer.27
The resonance character of an alkali-metal overlayer state
is clearly revealed by applying an embedding method with a
FIG. 10. Band structure of 共2 ⫻ 2兲-Cs/ Cu共111兲 along the highsymmetry directions ¯⌫-M̄ and ¯⌫-K̄ in the surface Brillouin zone.
The dark shaded area represents the clean Cu共111兲 surface projected
bulk band structure and the light shaded area the additional projected bulk band structure due to band folding when the 共2 ⫻ 2兲 Cs
layer is adsorbed. The solid bright line represents the surface-state
band, of the clean Cu共111兲 surface, the dashed-dotted line the QWS
band, and the dashed line the overlayer resonance band.
semi-infinite substrate. Such a calculation was recently reported for a full monolayer of Na on Cu共111兲.7 This monolayer has a structure which may be approximated with a 共3
⫻ 3兲 lattice and four Na atoms in the cell. The discussion in
that paper is focused on the energies and dispersion of the
strongly confined states and the image potential states with
energies high in the Cu L gap. Here we note that the Na
projected density of states for different k points along symmetry lines in the Brillouin zone of the adlayer, in addition,
shows resonances 共Fig. 2 in Ref. 7兲. By reading off the energies of the resonances, one finds E共k兲 branches with a
nearly free-electron dispersion. This is shown in Fig. 12,
where the energy is plotted versus the square of the wave
vector for the resonances. The resonance band extends
1.5 eV below EF and the band mass is 0.93me. This may be
compared with the filled bandwidth of 2.12 eV calculated for
a freestanding monolayer of Na.27
FIG. 11. 共Color online兲 Electron structure of
the overlayer resonance state. To the left, the
band structure in the ¯⌫-M̄ direction, and to the
right, the even 共red兲 and odd 共black兲 squared
magnitude of the wave function, averaged in
planes parallel to the surface, at different locations along the overlayer resonance band.
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FIG. 12. Energy versus squared parallel wave vector for the
resonance state formed in 1 ML Na on Cu共111兲. The data points are
obtained by reading off the peak positions in the density of state
curves presented by Butti et al. 共Ref. 7兲. The straight line is a fit to
the data points.
To conclude, our picture, based on PES data and electron
structure calculations, is that the surface state shifts in energy
upon alkali-metal adsorption and eventually turns into a
resonance characteristic of the overlayer at high monolayer
coverage. This view is supported by the calculation for Na
on Cu共111兲 by Butti et al.,7 according to our analysis of their
data.
B. QWS-shift and LEED observations
The coverage dependence of the QWS energy is quite
similar to the observations for Na and Li monolayers.5,28 The
saturation energy in the monolayer range is below the Fermi
level, but closer to it for Cs 共EF − 25 meV兲 than for Li 共EF
− 135 meV兲 共Ref. 6兲 or Na 共EF − 97 meV兲.14 The energy obtained for Na with STS is 共EF − 127 meV兲.29 There are two
general reasons why different energies may be obtained with
the two methods. Using STS, a much smaller selected area is
probed and the perturbation of the sample is different. If a
wide terrace is chosen, the lateral confinement will be less
significant and will give a lower energy than when a large
area is probed. This could explain the difference between
STS and photoemission energies for the Na case. For Cs this
explanation is not applicable since the STS gives a higher
energy 共EF + 40 meV兲.9 This energy was measured at 5 K for
a sample cooled after deposition at RT of an amount that
gave a 共2 ⫻ 2兲 LEED pattern at RT. This energy may be
compared with the saturation ⬃10 meV above EF in the
present experiment for a sample prepared at RT and measured at RT or 170 K. Assuming an insignificant energy shift
between 170 and 5 K and constant sticking probability, the
deposition time dependence observed here for the QWS energy would suggest that the tunneling results are representative of around 93% of the saturation coverage at RT. According to the present observations, a 共2 ⫻ 2兲 order is observed
for a range of coverages and, for the QWS energies observed
with STS, the coverage is within this range. Our LEED observations made with the sample at 170 K suggest an inhomogeneous Cs surface density at high monolayer coverages.
The lattice parameter saturates at a value which deviates insignificantly from that of the 共2 ⫻ 2兲 order, but the QWS
energy continues to change with evaporation even after saturation of the lattice parameter. A possible explanation for this
may be that completion of the layer occurs via occupation of
vacant sites in the layer. A STM image for the 共2 ⫻ 2兲 layer
indicates that there are rather large holes in the Cs film 共Fig.
5 in Ref. 24兲, but this observation was made at 5 K. One may
note that the fraction of the surface area covered by Cs islands is not far from the above coverage estimate based on
the QWS energies.
The details discussed above are of interest for the resonance broadening of the QWS, indicated by the theoretical
evaluation of the STS data, which was based on the assumption of p共2 ⫻ 2兲 order for the Cs atoms. The present results
indicate that this assumption is valid. The structure is very
close to this order even though the RT saturation coverage is
lower than a full monolayer. For the 共2 ⫻ 2兲 overlayer considered by the authors, the QWS couples to Cu bulk states
along the L-X direction, but this mechanism should apply
also for other overlayer coverages and orders. If, for example, the overlayer is sparser and rotated compared to the
共2 ⫻ 2兲 overlayer, the only difference would be that the overlayer states couple to states some distance away from the
L-X line. The point of this comment is that the resonance
mechanism is expected to be rather robust with respect to
deviations from the ideal order for the overlayer. We note
that the mechanism should be applicable also at low Cs coverage and then to the Cu共111兲 surface state. This is clearly
the case when the adsorbate forms ordered structure found
with STM at low Cs coverage and temperature.9 At higher
temperature, the diffraction rings observed with LEED define
the scattering vectors which couple the surface state at ¯⌫ to
bulk states on a cylinder on the L-X direction.
An outstanding question concerns to what extent the inhomogeneity of the Cs monolayer affects not only the QWS
energy but also adds one more contribution to the measured
linewidth.
C. Width of QWS
Although an inhomogeneous density and elastic
scattering9 may contribute, we expect that the phonon processes give important contributions to the linewidth of the
QWS peak. A sign of this is the narrowing of the peak between 295 and 170 K, from 57 to 40 meV 共FWHM兲. At the
saturated Cs coverage, the QWS binding energy is stable
close to the Fermi level for these temperatures 共see Fig. 4兲.
The phonon-induced linewidth is then given by30
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⌫ep共T兲 = 2␲ប
冕
␻m
␣2F共␻兲关1 + 2n共␻ ;T兲 + f共− ⑀b + ␻ ;T兲 − f共
0
− ⑀b − ␻ ;T兲兴d␻ ⯝ 4␲ប
冕
␻m
0
␣2F共␻兲
d␻ ,
ប␻
sinh
k BT
冉 冊
共1兲
where ␣2F共␻兲 is the electron-phonon coupling Eliashberg
function associated with the QWS, f and n the electron and
phonon distribution functions, respectively, ␻m the maximum
phonon frequency of the system, and ⑀b the QWS binding
energy. As long as the ប␻m 艋 kBT, the width is approximately
linear in temperature
⌫ep共T兲 = 4␲ប
冕
␻m
0
␣2F共␻兲
d␻kBT = 2␲␭0kBT,
ប␻
共2兲
where ␭0 is the zero-temperature electron-phonon coupling
parameter. At this point, we actually do not know ␣2F共␻兲 for
the QWS and, thus, we have no information about the maximum frequency of the phonons which are assisting in the
filling of the QWS photohole in the ¯⌫ point. If we take the
bulk copper Debye energy 共ប␻D = 30 meV兲 as an upper limit
for the maximum phonon energy, Eq. 共1兲 gives a slight reduction of the slope of the width versus temperature at the
lower temperature T = 170 K 共kBT = 15 meV兲 in comparison
with the slope at 295 K. Equation 共2兲 is then adjusted by a
factor ␰
⌫ep共T兲 = 2␲␰␭0kBT,
共3兲
where ␰ = 共1 / 2兲共ប␻ / kBT兲2关cosh共ប␻ / kBT兲 − 1兴−1, with the
property ␰ → 1 as T → ⬁. From the fitting to our two measured widths, assuming constant additional non-phononinduced broadening, we obtain the low-temperature lambda
value ␭0 = 0.18. Equation 共3兲 then yields ⌫ep共170 K兲
= 12 meV and ⌫共295 K兲 = 29 meV. Comparing these values
with the measured widths gives an estimate of the additional
contributions from elastic broadening, inelastic electronelectron scattering, and inhomogeneous broadening of
28 meV.
2
Applying the Debye model, ␣2F共␻兲 = ␭0共 ␻␻D 兲 , and experimental QWS binding energy ⑀b = 25 meV, the zerotemperature 共T = 0兲 phonon-induced width is determined by
Eq. 共1兲. The upper integration limit is now given by ⑀b / ប as
no phonons with higher frequencies can be emitted at T=0.
We then obtain
⌫ep共0兲 = 2␲ប
冕
⑀b/ប
0
width estimated here and in the work by Corriol et al.9 is not
conclusive, as they both are partly based on an oversimplified Debye model for the Eliashberg function. Preliminary
phonon calculations of the system indicate low-frequency
surface localized modes which might be of crucial importance for an appropriate Eliashberg function of the QWS.31
D. Oxygen exposure
First to note is that the integrity of the QWS is maintained
upon oxygen exposure. The gradual energy shift is in accordance with a model presented in a previous work25 based on
a gradual downshift observed for the plasmon loss energy,
namely, that oxygen acts as a diluter of the electron gas in
the Cs layer. The Cs atoms collectively transfer electrons to
the oxygen atoms such that a rather homogeneous electron
gas with reduced density is obtained. The upward shift of the
QWS means that part of the transferred charge is donated via
the depopulation of the QWS band. In concert with the energy shift, the line broadens from 44 meV FWHM at zero
exposure to 105 meV when the QWS energy is 45 meV
above EF. This is in contrast to a pure Cs overlayer where the
line is only marginally broader when found at 45 meV above
EF than when it is observed for full monolayer coverage at
25 meV below EF. While the QWS is clearly seen in spectra
at 100 meV above EF for the pure Cs overlayer, the broader
line for the oxygen exposed Cs together with a lower intensity makes it difficult to observe for state energies higher
than 75 meV above EF at 170 K.
The upshift of the monolayer QWS upon oxygen exposure provides part of the explanation for the decrease of the
work function. The work function reflects the electron density of a free-electron metal.32 For a film, the work function
is also sensitive to the balance between populated states with
long and short tails into vacuum.33 Oxygen exposure changes
both, such that the work function is reduced. The plasmon
energy of the overlayer is reduced gradually with increasing
exposure.25 This signals that the electron gas in the film becomes thinner, which results in a lower work function. The
reduction is particularly large if the electrons removed from
the film extend far into vacuum.33 This applies to the QWS,
which has a node in the film. This is seen in Fig. 13, where
the square of the magnitude of the wave function of the
QWS is shown in the ¯⌫ point. As the QWS band is depopulated on oxygen exposure, this will thus give a stronger contribution to the work-function change than the removal of
electrons from the resonant states below the L gap.
⑀3
2
␣2F共␻兲d␻ = ␲␭0 2 b 2 ⯝ 7 meV.
3
ប ␻D
V. CONCLUSIONS
共4兲
This value is approximately the same as estimated in the
previous study of this system,9 7.5± 3 meV. The fact that the
data of a local probe measurement, STS, are consistent with
our PES data regarding the phonon-induced linewidth suggests that the present data are not much affected by an inhomogeneous adatom density across the much larger area
probed. Still, we want to emphasize that the phonon-induced
The electronic structure of Cs/ Cu共111兲 at 170 K and full
monolayer coverage is characterized by a QWS band extending 25 meV below EF and a resonance band extending to
energies below the edge of the Cu L gap. At RT the QWS
energy is around 35 meV higher than at 170 K, which is
ascribed to a smaller saturation coverage at RT. A LEED
pattern characteristic of 共2 ⫻ 2兲 order is observed in a range
of monolayer coverages, giving the QWS energies in the
range from EF − 25 meV to EF + 50 meV. This is in accor-
155403-8
PHYSICAL REVIEW B 75, 155403 共2007兲
OVERLAYER RESONANCE AND QUANTUM WELL STATE OF…
FIG. 13. The average of the squared magnitude of the even and
odd wave functions of surface localized states in the ¯⌫ point. From
the top: surface state 共SS兲 of clean Cu共111兲 and overlayer resonance
共OR兲 state and quantum well state 共QWS兲 of 共2 ⫻ 2兲-Cs/ Cu共111兲.
The horizontal axis 共left to right兲 ranges from the slab center toward
the vacuum region. The dashed line represents the mean position of
the first Cu layer and the dotted line the position of the Cs atom.
dance with the QWS energy EF + 40 meV obtained with STS
by Corriol et al.9 and the energy obtained from the present
and previously reported9 first-principles calculations. Our
1 S.-Å.
Lindgren and L. Walldén, in Electronic structure, edited by
S. Holloway, N. V. Richardson, K. Horn, and M. Scheffler,
Handbook of Surface Science Vol. 2 共Elsevier Science, Amsterdam, 2000兲.
2 S. R. Barman, P. Häberle, K. Horn, J. A. Maytorena, and A.
Liebsch, Phys. Rev. Lett. 86, 5108 共2001兲.
3 D. Heskett, K.-H. Frank, E. E. Koch, and H. J. Freund, Phys. Rev.
B 36, 1276 共1987兲.
4 I. Lindau and L. Walldén, Phys. Scr. 3, 77 共1971兲.
5 A. Carlsson, S.-Å Lindgren C. Svensson, and L. Walldén, Phys.
Rev. B 50, 8926 共1994兲.
6 D. Claesson, S.-Å Lindgren, and L. Walldén, Phys. Rev. B 60,
5217 共1999兲.
7 G. Butti, S. Caravati, G. P. Brivio, M. I. Trioni, and H. Ishida,
Phys. Rev. B 72, 125402 共2005兲.
8 J. M. Carlsson and B. Hellsing, Phys. Rev. B 61, 13973 共2000兲.
9
C. Corriol, V. M. Silkin, D. Sánchez-Portal, A. Arnau, E. V.
Chulkov, P. M. Echenique, T. von Hofe, J. Kliewer, J. Kröger,
and R. Berndt, Phys. Rev. Lett. 95, 176802 共2005兲.
10 for a review, see M. Grioni, Ch. R. Ast, D. Pacilé, M. Papagna, H.
Berger, and L. Perfetti, New J. Phys. 7, 106 共2005兲.
11 S.-Å Lindgren and L. Walldén, Surf. Sci. 89, 319 共1979兲.
12
S.-Å Lindgren and L. Walldén, Phys. Rev. B 38, 3060 共1988兲.
13 S.-Å Lindgren and L. Walldén, Solid State Commun. 28, 283
共1978兲.
14
A. Carlsson, B. Hellsing, S.-Å Lindgren, and L. Walldén, Phys.
Rev. B 56, 1593 共1997兲.
15
S. Ogawa, H. Nagano, and H. Petek, Phys. Rev. Lett. 82, 1931
共1999兲.
data give no clear demarcation of the optimum coverage for
the Cs p共2 ⫻ 2兲 structure. A likely interpretation, which
would reconcile the continued shift of the QWS energy with
the constant LEED angles, is that this structure is only complete near full monolayer coverage and that the incomplete
monolayer has vacancies.
The gradual downshift of the Cu共111兲 surface state to energies below the Cu L gap suggests that the surface state
gradually transforms with increasing coverage to an overlayer resonance. Such a resonance is predicted by the present
calculations for Cs/ Cu共111兲 and recently also for
Na/ Cu共111兲.7
Regarding the resonance broadening proposed for QWS,9
characteristic of ordered Na and Cs monolayers on Cu共111兲,
we point out that the mechanism should be relevant also for
less ordered overlayers as exemplified by alkali-metal overlayers at low and intermediate coverages. This means that a
coverage-dependent resonance broadening is expected for
the Cu共111兲 surface state as this is downshifted upon alkalimetal adsorption. For the monolayer QWS, the width is predicted to depend mildly on deviations from the ideal ordered
structure considered in the previous work.9 An indication of
the rather modest broadening induced by disorder is that the
integrity of the monolayer QWS is maintained upon oxygen
exposures sufficient to give appreciable energy shifts.
16 P.
Johansson, G. Hoffmann, and R. Berndt, Phys. Rev. B 66,
245415 共2002兲.
17 G. Hoffmann, R. Berndt, and P. Johansson, Phys. Rev. Lett. 90,
046803 共2003兲.
18 S. Baroni, A. Dal Corso, S. de Gironcoli, P. Giannozzi, C. Cavazzoni, G. Ballabio, S. Scandolo, G. Chiarotti, P. Focher, A. Pasquarello, K. Laasonen, A. Trave, R. Car, N. Marzari, and A.
Kokalj, http://www.pwscf.org/
19 D. Vanderbilt, Phys. Rev. B 41, 7892 共1990兲.
20 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,
3865 共1996兲.
21 P. Giannozzi, F. De Angelis, and R. Car, J. Chem. Phys. 120,
5903 共2004兲.
22 M. Methfessel and A. T. Paxton, Phys. Rev. B 40, 3616 共1989兲.
23
S.-Å Lindgren, L. Walldén, J. Rundgren, P. Westrin, and J. Neve,
Phys. Rev. B 28, 6707 共1983兲.
24
Th. von Hofe, J. Kröger, and R. Berndt, Phys. Rev. B 73, 245434
共2006兲.
25
S.-Å Lindgren and L. Walldén, Phys. Rev. B 22, 5967 共1980兲.
26 R. D. Diehl and R. McGrath, Surf. Sci. Rep. 23, 46 共1996兲.
27
E. Wimmer, J. Phys. F: Met. Phys. 13, 2313 共1983兲.
28 A. Carlsson, D. Claesson, G. Katrich, S.-Å Lindgren, and L.
Walldén, Phys. Rev. B 57, 13192 共1998兲.
29 J. Kliewer and R. Berndt, Phys. Rev. B 65, 035412 共2001兲.
30
B. Hellsing, A. Eiguren, and E. V. Chulkov, J. Phys.: Condens.
Matter 14, 5959 共2002兲.
31 A. Nojima and B. Hellsing 共unpunblished兲.
32
N. D. Lang and W. Kohn, Phys. Rev. B 3, 1215 共1971兲.
33
J. K. Schulte, Surf. Sci. 55, 427 共1976兲.
155403-9
Paper III
PHYSICAL REVIEW B 76, 153404 共2007兲
Two-dimensional localization of fast electrons in p„2 Ã 2…-Cs/ Cu„111…
V. Chis,1 S. Caravati,2 G. Butti,2 M. I. Trioni,2 P. Cabrera-Sanfelix,3 A. Arnau,4 and B. Hellsing1
1Department
of Physics, Göteborg University, S-412 96 Göteborg, Sweden
and Dipartimento di Scienza dei Materiali, Università di Milano-Bicocca, Via R. Cozzi 53, I-20125 Milano, Italy
3
Donostia International Physics Center (DIPC), P. Manuel de Lardizabal 4, San Sebastian 20018, Spain
4Departamento de Fisica de Materiales UPV/EHU and Unidad de Fisica de Materiales Centro Mixto CSIS-UPV/EHU, Facultad de
Quimica, Apartado 1073, San Sebastian 20080, Spain
共Received 4 July 2007; revised manuscript received 29 August 2007; published 10 October 2007兲
2CNISM
The system p共2 ⫻ 2兲-Cs/ Cu共111兲 reveals a manifold of surface localized electronic states, such as image
states, quantum well states, and surface resonances. We find an electronic state with a band energy above the
vacuum level while strictly localized at the Cs monolayer. An electron populating this state will have a large
in-surface-plane kinetic energy and, presumably, a very long lifetime. The state could be an interesting tool for
control of electron induced surface reactions.
DOI: 10.1103/PhysRevB.76.153404
PACS number共s兲: 73.20.⫺r, 71.15.Mb, 71.20.⫺b, 73.21.Fg
Adsorption of alkali atoms on simple and noble metal
surfaces has for a long time been considered as bench mark
models for metal adsorption. During the recent past two decades, experimental1,2 and theoretical3–6 studies have shown
that the alkali-on-metal adsorption system is far more complex and rich with electronic properties than expected.
It is well known from experiments and theory that the
共111兲 surface of noble metals exhibits a local band gap perpendicular to the surface hosting surface localized electronic
states. The state with the lowest energy is referred to as the
surface state, and the sequence of bands with less binding
energy is the image state series. Electrons occupying surface
states close to the Fermi level extend far toward vacuum and,
therefore, play an important role in the adsorption of weakly
physisorbed species.7,8 Unoccupied image states may be important for hot electron transport at surfaces. For surface
states and image states appearing in the surface projected
directional gap, the lifetime broadening, determined by inelastic processes due to electron-electron and electronphonon scatterings, is found to be in the range of
1 – 50 meV.9 This corresponds to lifetimes in the range of
0.01– 1 ps.
Adsorption of alkali overlayers introduces a significant
change in the electronic structure of the surface. Charge
transfer from the alkali layer to the metal reduces appreciably the work function and new states appear at the interface,
such as quantum well states 共QWSs兲 and overlayer resonances 共ORs兲.10 The lifetime of these states will depend on
the amount of both elastic 共coupling to bulk bands兲 and inelastic scattering processes.
In this Brief Report, we demonstrate the existence of a,
not previously discussed, one-electron gap state 共GS兲 found
for the system of a full monolayer of Cs on Cu共111兲. Outstanding characteristics of the GS are 共i兲 a band energy well
above the vacuum level, 共ii兲 appearance in a local bulk band
gap but within the continuum of vacuum states, 共iii兲 strong
localization to the Cs layer, and 共iv兲 large in-surface-plane
component of the kinetic energy. An electron excited to this
state will travel fast parallel to the surface being strictly confined within the Cs layer. Control of electron induced reactions of atomic species decorating surface terrace step edges
is a possible application—a nanoscale electron gun.
1098-0121/2007/76共15兲/153404共4兲
The electronic structure calculations are based on the density functional theory. We have applied two different 共ultrasoft pseudopotentials兲 plane-wave codes11 and a Green’s
function based embedding scheme12 within a full potential
linearized augmented plane-wave approach. All of them
make use of the generalized gradient approximation for the
exchange and correlation energy functional.13
To obtain the ground state atomic structure of
p共2 ⫻ 2兲-Cs/ Cu共111兲, we used the PWSCF code14 and a slab
of ten layers of Cu atoms with double-sided adsorption of the
Cs atoms and 24 Å of vacuum. The Cu atoms were allowed
to relax in any direction and the Cs atom, in the on-top15
position, in the direction perpendicular to the Cu planes. Our
determination of the relaxed geometry agrees well with
experiment.16
For a detailed investigation of the surface electronic
states, we take advantage of the embedding scheme.6 In this
scheme, the Cs overlayer and the two topmost copper layers
are considered in the calculation. The correct matching with
the vacuum and bulk solutions is guaranteed by a nonlocal
energy-dependent potential acting on the boundaries of the
embedded region. This extended substrate approach allows
one to properly take into account the continuum of bulk
states and consequently the presence of the resonances and
their elastic width, if any.
The electronic structure of p共2 ⫻ 2兲-Cs/ Cu共111兲 reveals a
variety of surface features: Overlayer resonance, quantum
well state, image states 共ISs兲, and gap state. In Fig. 1, we
show the calculated band structure, applying the embedding
scheme, along the ⌫M path. In such a method, the semiinfinite character of the substrate 共i兲 prevents the appearance
of surface localized band doublets, as in the case of finite
slab calculations, and secondly, 共ii兲 allows the determination
of the inherent elastic width of states.
The p共2 ⫻ 2兲 periodic perturbation introduced by the Cs
layer yields folded copper bulk bands which reduce appreciably the projected band gap of Cu共111兲.4 As a result of this
folding, hybridization of surface localized states takes place,
which gives elastic widths to the bands. We find that the
elastic width of the QWS 共shown in Fig. 1兲 increases from
2.4 meV in the ¯⌫ point to 35.4 meV halfway out to the M̄
153404-1
©2007 The American Physical Society
PHYSICAL REVIEW B 76, 153404 共2007兲
BRIEF REPORTS
QWS
GS
IS
(a)
0.04
0.02
OR
OR
0
Γ
(b)
0.04
0.02
|Ψ|
2
QWS
0
(c)
0.15
0.10
0.05
-2
-1
0
2
3
GS
0
0.05
d’
(d)
0.04
0.03
Energy -EF (eV)
s
0.02
FIG. 1. 共Color online兲 Cs/ Cu共111兲 k储-resolved local density of
states along the ⌫M path in the SBZ. The local density of states has
been integrated over the vacuum volume.
point. The minimum of the parabolic band centered at ¯⌫ is
slightly above the Fermi level 共⬃40 meV兲, which is in
agreement with a recent theoretical and experimental scanning tunneling spectroscopy study.4 As described below,
other states such as image states and the overlayer resonance,
shown in Fig. 1, with energies below and above the QWS are
found to have elastic widths at ¯⌫ significantly larger or comparable to the QWS, respectively. At ¯⌫, an exceptionally
sharp feature is found above the vacuum level in the projected bulk band gap. This gap state, analyzed in detail below, is the focus of this Brief Report.
A new method has been proposed6 to merge the KohnSham potential asymptotically far from the surface into the
correct image form −1 / 4z. Implementation of this method in
the embedding scheme has made it possible to determine the
共ISs兲. The manifold of the parabolic IS bands are shown in
Fig. 1, with an energy at the ¯⌫ point of 1.04, 1.46, and
1.60 eV above the Fermi level for the lowest three states.
The OR band is located about 1.2 eV below the Fermi
level at the ¯⌫ point which is in agreement with recent reported photoemission data.10 The broad feature of this band
reflects its resonance character, as clearly seen in Fig. 1. The
wave function at the surface Brillouin zone 共SBZ兲 center,
presented in Fig. 2共a兲, also shows the resonance behavior
with a persisting oscillation 共nonexponential decay兲 into the
slab.
In addition to these bands, we observe an interesting, and
to our knowledge not previously analyzed, Cs induced band
共GS兲 which meets the ¯⌫ point in a bulk band gap at about
2.7 eV above the Fermi level. As the calculated and experimental work function of the system is 1.8 eV, the GS is
located 0.9 eV above the vacuum level in the SBZ center.
Still, the planar averaged wave function squared at the SBZ
center is surprisingly localized to the Cs layer 关Fig. 2共c兲兴.
This localized character is consistent with zero width calculated applying the embedding scheme. As it is shown below,
the reason for this localization is that the main part of its
one-electron energy is attributed to kinetic energy parallel to
d
p
0.01
0
10
5
20
15
Z (Å)
FIG. 2. 共Color online兲 Planar averaged squared magnitude of
wave functions at the ¯⌫ point. 共a兲 Even 共solid line兲 and odd 共dashed
line兲 of the OR, 共b兲 even 共solid line兲 and odd 共dashed line兲 of the
QWS, 共c兲 GS, and 共d兲 s, p, d, and d⬘ bands of the unsupported Cs
layer. The vertical lines correspond to the outermost Cu atom layer
共dashed兲 and the Cs layer 共dotted兲. Zero on the horizontal axis is the
middle of the slab.
the surface. As the GS band has zero elastic width despite its
energy position in the continuum of vacuum states, the inherent width should thus be due to the inelastic processes,
such as electron-electron and electron-phonon scatterings.
The origin of the GS band, as well as the OR and QWS
bands, can be understood from the band structure of the freestanding Cs monolayer. Referring to Figs. 2 and 3, we can
associate the lowest band s with the OR band, the p band
with the QWS band, and the folded s band, labeled d⬘, with
the GS band. In Fig. 1, we see the gap opening at the M̄ point
4
3
2
d’
d
1
p
0
Energy - EF (eV)
Μ
1
4
-1
s
-2
⎯Γ
⎯Μ
FIG. 3. Band structure of Cs/ Cu共111兲 and unsupported Cs layer.
The unsupported Cs layer bands 共s, p, d, and d⬘兲 are denoted by
solid lines. The dotted lines show the bulk band edges due to
folding.
153404-2
PHYSICAL REVIEW B 76, 153404 共2007兲
BRIEF REPORTS
共as for the unsupported Cs layer between the s and the d⬘
band, as seen in Fig. 3兲. This band gap opening is not present
at the K̄ point both for the unsupported Cs layer17 nor for the
adsorbed Cs layer 共not shown兲. The reason is that along the
⌫M direction, the Cs nearest neighbor 共nn兲 distance is about
twice the nn distance in the ⌫K direction and, thus, the oneelectron potential is more corrugated in the ⌫M direction.
The band corresponding to the d band for
p共2 ⫻ 2兲-Cs/ Cu共111兲 is not seen in Fig. 1 in the region from
the center to about halfway out to the SBZ boundary. This is
due to the fact that the potential is too shallow to form this
bound state. This is indicated by the lower work function for
Cs/ Cu共111兲 than for the freestanding Cs layer, 1.8 eV in
comparison with 2.21 eV, respectively.
Furthermore, if we compare the wave functions of the
freestanding Cs monolayer with the ones for Cs/ Cu共111兲 at
the ¯⌫ point 共Fig. 2兲, we have additional support for this picture. In the SBZ center, the wave functions of s, d⬘, OR, and
GS have no nodes in the Cs layer, while the p and QWS
band has one node. The pronounced confinement of the GS
to the Cs layer is also seen for the corresponding d⬘ state
wave function for the unsupported Cs layer. The reason for
the even stronger localization compared to the s band state is
due to the strong hybridization with the d band which, in
turn, is responsible for the avoided crossing of the bands
forming lower bonding and upper antibonding bands, d and
d⬘ 共see Fig. 3兲, respectively.
Our calculated band structure of the unsupported Cs
monolayer is similar to the one obtained by Wimmer,17
who symmetrically analyzed the bands. He concludes
that the symmetry of the d and d⬘ bands is the same, which
is consistent with the observed avoided crossing phenomenon. The avoided crossing phenomenon persists for
p共2 ⫻ 2兲-Cs/ Cu共111兲 halfway between ¯⌫ and the SBZ
boundary, as shown in Fig. 1.
It is interesting to note that both the PWSCF and the embedding scheme yield the same GS energy, 2.7 eV above the
Fermi level, and the same Cs layer localized wave function,
as pictured in Fig. 2共c兲. The embedding scheme calculation
yields the correct asymptotic image form of the selfconsistent one-electron potential, while PWSCF does not.
However, this is not an important issue for the GS which is
well localized to the Cs layer plane. However, what is the
origin of the localization? We conclude that the GS band is a
Cs induced band and analyze now further the character of the
GS band in the ¯⌫ point.
Due to the fact that the GS appears as a folded OR state,
we expect that a substantial part of the GS band energy in the
¯⌫ point is attributed to parallel kinetic energy. Applying the
free electron empty lattice model, the parallel contribution to
2 ជ 2
ជ
the total GS energy is ⑀GS
储 = ប 兩G 1兩 / 2m = 7.41 eV, where G 1
is the shortest reciprocal lattice vector parallel to the surface.
However, the gap opening at the M̄ point signals a periodic
potential in the Cs layer, and thus we cannot trust this estimate. If the potential is not strictly sinusoidal, we would also
have contributions from larger reciprocal lattice vectors to
the GS wave function. From the calculated self-consistent
GS wave function, we can obtain the expectation value of the
parallel kinetic energy,
FIG. 4. 共Color online兲 Planar averaged vSCF 共solid line兲 and the
squared magnitude of the GS wave function 关green 共gray兲兴. The
energy of the GS when subtracting the calculated ⑀GS
is indicated
储
by the vertical arrow. The histogram shows the squared magnitude
of the GS wave function 关red 共gray兲兴 and parallel kinetic energy
关blue 共dark gray兲兴 resolved in terms of contribution from different
ជ 储. The horizontal axis shows 兩G
ជ 储兩
parallel reciprocal lattice vectors G
in units of 2␲ / a, where a = 5.2 Å is the nearest Cs-Cs distance in
the Cs layer.
⑀GS
=−
储
ប2
ជ 2储 兩␺ 典,
具␺GS兩ⵜ
GS
2m
共1兲
ជ 2储 = ⳵ 2 + ⳵ 2 and ␺ is the first principles gap state
where ⵜ
GS
⳵x
⳵y
wave function. At the ¯⌫ point 共kជ 储 = 0兲, the GS wave function
can be written as
2
2
␺GS共Rជ ,z兲 =
ជ ជ
CGជ geiG ·Reigz ,
兺
ជ
储
储
共2兲
G储,g
ជ = 共x , y兲 and G
ជ 储 and g are the parallel and perpendicuwhere R
ជ = 共G
ជ 储 , g兲.
lar components of the reciprocal lattice vectors G
We then have
⑀GS
=
储
ប2
兺 兩Gជ 储兩2兩CGជ 储g兩2 = 10.34 eV.
2m ជ
共3兲
G储,g
Subtraction of this energy from the band energy of the GS
is illustrated in Fig. 4. The free electron estimate was
7.41 eV, which is about 72% of the 10.34 eV. This is approximately in agreement with our findings when resolving
⑀kin
in terms of its contributions from different parallel recip储
rocal lattice vectors. From the histogram in Fig. 4, we oboriginates from the shortest
serve that only about 63% of ⑀kin
储
ជ
G储 vector and thus 37% from higher order diffraction.
ជ 储 can be understood
The contributions from the different G
from the x , y plot of the GS wave function, as shown in Fig.
5. The Cs next-nearest-neighbor direction corresponds to the
ជ 兩, the
magnitude of the shortest reciprocal lattice vector 兩G
1
ជ
nearest-neighbor distance to the next smallest 兩G2兩, and the
153404-3
PHYSICAL REVIEW B 76, 153404 共2007兲
BRIEF REPORTS
plane kinetic energy, and 共iv兲 is expected to have an exceptionally large lifetime in the range of 0.1– 1 ps due to inelastic scattering by phonons or electrons. The estimated lifetime
yields an electron traveling-distance parallel to the surface of
500– 5000 Å. We propose that when the GS is populated by
means of photon absorption or electron injection in a STM or
inverse photoemission setup, an electron strictly confined to
the Cs overlayer moving fast parallel to the surface is
present. A possible application is control of electron stimulated reactivity of reactants decorating edges of single crystal
terraces of the alkali overlayer.
FIG. 5. 共Color online兲 Electron density map of the gap state in
the plane parallel to the surface intersecting the Cs layer.
ជ 兩 = 2兩G
ជ 兩 related to the shorter wave length corrugathird 兩G
3
1
tion seen in the next-nearest-neighbor direction.
In conclusion, we find a different kind of electronic state,
denoted as gap state, located in the projected bulk band gap
which 共i兲 have a band energy above vacuum level, 共ii兲 is
spatially localized at the surface, 共iii兲 have a large in-surface-
R. D. Diehl and R. McGrath, Surf. Sci. Rep. 23, 43 共1996兲.
Lindgren and L. Walldén, in Handbook of Surface Science,
Electronic Structure, edited by S. Holloway, N. V. Richardson,
K. Horn, and M. Scheffler 共Elsevier Science, Amsterdam, 2000兲,
Vol. 2.
3 J. M. Carlsson and B. Hellsing, Phys. Rev. B 61, 13973 共2000兲.
4
C. Corriol, V. M. Silkin, D. Sánchez-Portal, A. Arnau, E. V.
Chulkov, P. M. Echenique, T. von Hofe, J. Kliewer, J. Kröger,
and R. Berndt, Phys. Rev. Lett. 95, 176802 共2005兲.
5 D. Sánchez-Portal, Prog. Surf. Sci. 82, 313 共2007兲.
6 G. Butti, S. Caravati, G. P. Brivio, M. I. Trioni, and H. Ishida,
Phys. Rev. B 72, 125402 共2005兲.
7 F. Silly, M. Pivetta, M. Ternes, F. Patthey, J. P. Pelz, and W.-D.
Schneider, Phys. Rev. Lett. 92, 016101 共2004兲.
8 W. Xiao, P. Ruffieux, K. Aït-Mansour, O. Gröning, K. Palotas, W.
A. Hofer, P. Gröning, and R. J. Fasel, J. Phys. Chem. B 110,
21394 共2006兲.
9
B. Hellsing, A. Eiguren, and E. V. Chulkov, J. Phys.: Condens.
Matter 14, 5959 共2002兲.
10 M. Breitholtz, V. Chis, B. Hellsing, S.-Å. Lindgren, and L. Walldén, Phys. Rev. B 75, 155403 共2007兲.
11
VASP 共Ref. 18兲 using GGA-PW91 共Ref. 19兲, and PWSCF 共Ref. 20兲
with the GGA-PBE 共Ref. 13兲.
V.C. and B.H. thank Lars Walldén and Stig Andersson
for stimulating discussions. Computer resources for the
project have been provided by the Swedish National
Infrastructure for Computing 共SNIC兲. This work was
共partially兲 supported by the EU Network of Excellence
Nanoquanta 共Grant No. NMP4-CT-2004-500198兲. P.C.-S.
and A.A. thank the Basque Departemento of Industria
共ETORTEK programme兲, UPV/EHU, and MEC for financial
support.
H. Ishida, Phys. Rev. B 63, 165409 共2001兲.
P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,
3865 共1996兲.
14
A kinetic energy cutoff of 30 Ry was used for the plane waves
and 480 Ry for the charge density. A Monkhorst-Pack sampling
grid of 10⫻ 10 was used corresponding to 31 k points in the
irreducible SBZ. A finite temperature first-order spreading of
Methfessel-Paxton 共Ref. 21兲 type of value 0.01 Ry was used.
15 S.-Å. Lindgren and L. Walldén, J. Rundgren, P. Westrin, and J.
Neve, Phys. Rev. B 28, 6707 共1983兲.
16 In the LEED study of Walldén et al. 共Ref. 15兲, the distance between the Cs atom and the Cu surface layer is 3.01± 0.05 Å
which compares well with the calculated averaged value to
3.06 Å 共Ref. 10兲.
17 E. Wimmer, J. Phys. F: Met. Phys. 13, 2313 共1983兲.
18 G. Kresse and J. Hafner, Phys. Rev. B 49, 14251 共1994兲.
19 J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R.
Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671
共1992兲.
20
S. Baroni, A. Dal Corso, S. de Gironcoli, and P. Giannozzi 共http://
www.pwscf. org兲.
21
M. Methfessel and A. T. Paxton, Phys. Rev. B 40, 3616 共1989兲.
1
12
2 S.-Å.
13 J.
153404-4
Paper IV
Sodium and potassium monolayers on Be(0001) investigated by photoemission and
electronic structure calculations.
J. Algdal,1 T. Balasubramanian,2 M. Breitholtz,1 V. Chis,3 B. Hellsing,3 S.-Å. Lindgren,1 and L. Walldén1
1
Applied Physics Department, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
2
MAX-lab, Lund University, SE-221 00 Lund, Sweden
3
Physics Department, Göteborg University, SE-412 96 Göteborg, Sweden
Photoemission spectra show that Be(0001) surface states shift to lower energy with increasing Na
or K coverage in the monolayer range. At an intermediate monolayer coverage a quantum well state
appears near the Fermi edge and shifts to lower energy reaching saturation energy at full monolayer
coverage. At full monolayer coverage LEED shows 2×2 order for K while Na forms an incommensurate close packed structure aligned with the substrate. As a result of the different structures, the
photoemission spectra show qualitative differences that are explained by diffraction. First-principles
calculations for Be(0001) and p(2×2) K/Be(0001) reproduce reasonably the measured energy shifts
and dispersions. Spectra recorded for the shallow alkali metal core levels show that the adlayer is
inhomogeneous in an intermediate coverage range. While this is not noted when the valence state
energies are measured a line width change observed for one of the surface states is ascribed to this
inhomogeneity. It is suggested that the onset of inhomogeneity is associated with the occupation
of states in the quantum well band. Occupation of these states, which are highly localized to the
adlayer, gives metal character to the layer over an increasing area as the coverage is increased making the film homogeneous at high monolayer coverage. An anomalous emission line is observed for
both Na and K as a low energy companion to the quantum well state line becoming increasingly
separated from this as the coverage increases. We suggest that the satellite is due to an energy loss
associated with collective oscillations in the overlayer.
PACS numbers: 79.60.Dp, 73.20.At, 73.21.-b, 71.15.Mb, 71.20.-b
I.
INTRODUCTION
We have used angle resolved photoemission to study
the Na/Be(0001) and K/Be(0001) systems in mainly the
monolayer coverage range and also observed low energy
electron diffraction (LEED) patterns at full monolayer
coverage. After noting that a full monolayer of K has
2×2 order a first principles calculation of atomic and electronic structure was made for this system.
Be(0001) has been extensively studied by photoemission. Aside from providing information about bulk and
surface states with respect to dispersion, symmetry and
charge distribution1–5 the extreme properties of Be, high
vibration frequencies and a strong electron-phonon coupling for surface state electrons, the results have been important for advancing the understanding of how phonons
affect photoemission line shapes and widths for valence
as well as core electrons6–11 . Another unusual property
of Be(0001) of interest when a metal layer is adsorbed is
the distinct difference between the bulk and surface electronic structure, semimetal in the bulk and metal near
the surface.
Characteristic of Be(0001) is the existence of a band
gap, which is wide with respect to energy and lateral wave
vector. The gap provides confinement over the entire
energy-wave vector range of occupied electronic states in
an adsorbed alkali metal film. Due to the confinement
all states characteristic of the overlayer systems are expected to be discrete as for a thin film in vacuum with
the difference that the tail on the substrate side of the
adsorbed film is oscillatory. The boundary conditions are
thus favorable for realizing simple metal quantum wells
with all states confined and discrete. Only few substrates
meet this requirement considering also that the sample
must be conducting for photoemission to be applicable.
Graphite is an alternative but when an alkali metal is adsorbed a low temperature is needed to avoid intercalation
or, for Na, 3D growth12–14 .
A motivation for finding systems that realize near ideal
metal quantum wells is that these provide unique opportunities to study many aspects of solid state properties as
witnessed by the wide range of experiments performed on
the metal quantum well systems already uncovered15–22 .
In the present work the emphasis is on the characterization of Na/Be(0001) and K/Be(0001) in the monolayer
range but some measurements made at higher coverage
indicate that both systems are benign in the sense that
additional layers can be grown with atomic layer defined
thickness.
In part the results are as expected and foreshadowed by
those reported for the Li/Be(0001) system23,24 . Surface
states shift gradually with increasing monolayer coverage and the same holds for a quantum well state (QWS),
which becomes populated above an intermediate coverage. The gradual energy shifts could suggest that the
adlayer is homogeneous with uniform adatom density
and electronic structure. However, the spectra obtained
for the shallow alkali metal core levels reveal an inhomogeneity in an intermediate coverage range in which
the overlayer obtains metal character. Furthermore the
combination of photoemission and inverse photoemission
data obtained for Li indicated that the population of the
2
QWS does not occur via a gradual downshift of the energy across the Fermi level in the manner observed for Na
or Cs covered Cu(111)25–27 . Instead the QWS appears
separated from the Fermi edge at all coverages for which
it was observed.
At full monolayer coverage the photoemission spectra
of K/Be(0001) and Na/Be(0001) show qualitative differences although the electronic structures are quite similar.
As discussed below these differences may be ascribed to
diffraction, the 2×2 ordered K layer producing emission
lines not noted for the close packed but incommensurate
Na layer.
Perhaps most interesting among the observations is an
emission line noted for both Na and K and previously
for Li23 . The line appears in concert with the quantum
well state line becoming increasingly separated from this
as the alkali metal coverage is increased. The energy
band calculation predicts no state that may account for
the line and as far as we can understand there are no
single particle excitations that could explain the line in
terms of an energy loss. We suggest that the line is due
to an energy loss associated with a collective oscillation
involving the quantum well state electrons.
II.
EXPERIMENTAL
The angle resolved photoemission spectra are recorded
in the MAX synchrotron radiation laboratory at BL33,
which covers the photon energy range 15-150 eV . The
Be(0001) surface was prepared by cycles of Ar-ion sputtering followed by heating repeated in cycles until the
zone center surface state appeared as distinct as in a
recent report9 . The alkali metal was evaporated from
heated glass ampoules broken is situ and held at constant
temperature during an experimental run. For deposition
and measurement the sample holder had to be moved
between manipulators in the spectrometer and preparation chambers. In the spectrometer chamber and in the
chamber used for K deposition the sample holder was in
contact with a LN2 filled tube that gives a sample temperature of around 100 K
Spectra recorded before and after the sample had been
exposed to the LEED beam for a few minutes showed
that this degraded the sample with respect to measured
peak intensities. Apart from this contamination effects
seem modest. After leaving a sample with a full alkali
metal monolayer in vacuum for 12 hours without cooling
contamination gives an increased intensity at around 8
eV binding energy. This is typical of oxygen which is the
main contaminant in Be. In spite of this the emission
lines of present interest are still observed with reduced
intensity but with little change of the binding energy.
III.
CALCULATIONS
The electronic structure calculations were performed
with the total energy, plane wave code PWSCF28 , which
is based on the density functional theory. We adopt
norm-conserving pseudopotentials for K and Be and the
local density approximation for the exchange and correlation energy. A kinetic cut-off of 300 eV was used for the
plane wave basis and the finite temperature smearing of
first-order Methfessel-Paxton type29 was set to 0.68eV.
The atomic arrangement of (2×2)-K/Be(0001) was obtained by minimizing the total energy with respect to the
positions of the K and Be atoms using a 17 atomic layers
thick Be slab as substrate. The supercell contains four
Be atoms in each layer and one K atom on the vacuum
sides of the cell. In order to marginalize surface to surface
interaction through the vacuum region the surfaces were
separated by 22.5 Å (Be) and 26.5 Å (K/Be) of vacuum.
The Be(0001) slab was relaxed until the resulting force
acting on an atom was was less than 25 meV/Å. Our calculation gives an expansion of the first and second interlayer distances by 3.2 % and 0.93 % respectively while
the relaxation of deeper layers is negligible. A sizable
first interlayer expansion is in agreement with previous
calculations, 3.8 %30 and LEED results, 5.8 %31 . As for
the first interlayer distance our second interlayer separation is somewhat smaller than found in the previous
calculation, 2.2 %30 .
Based on the optimized structure for the Be(0001) slab
three different K adsorption sites were investigated. The
on-top and the two different hollow sites were compared
with respect to total energy after having fully relaxed all
atoms in a two step procedure. In the first step the adsorbed K atom was allowed to relax freely but the structure optimized for the Be substrate was kept rigid. This
resulted in the following order of preference: 1. fcc hollow, 2. hcp hollow, 3. on-top, with 12 meV lower energy
for the fcc site. In the second step all atoms are allowed
to relax. The result is that the on-top site is favored by
15 meV per K atom. With the atom in this site there
is, in the second step, an inward relaxation for both the
K atom and the uppermost Be layer such that the K-Be
distance is reduced by 2.6 %. Furthermore, as shown in
Fig. 1, the adatom induces a rumpling of the Be surface
layer. The on-top K atom resides 2.9 Å above the Be
atom and this distance is 0.13 Å larger than the distance
to the plane of the surrounding Be atoms in the top layer.
The adsorption of a K monolayer gives a work function change and a redistribution of the charge compared
to that of the constituents. The calculated work function of Be(0001) is 4.9 eV and is reduced by 2.6 eV when
the K layer has been adsorbed. The charge density redistribution shown in Fig. 1 was obtained by summation
of the charge density distributions of a free standing K
monolayer and the Be(0001) slab and subtraction of the
sum from the charge density distribution for the overlayer
system. One notes that most of the charge accumulation
is found in the interface region. This increase is mainly
3
1 ML K
hv=33eV
S
100K
Intensity (Arb Units)
M
A
QWS
QWS
1 ML Na
S
A
S
Be(0001)
FIG. 1: a) Optimized structure of a K (2×2) monolayer on
Be(0001). b) Calculated electron density redistribution in the
(1120) plane due to adsorption of a K (2×2) monolayer onto
Be(0001).
due to transfer from the K layer but there is also a small
contribution from the first and second Be layers.
The energy band structure was calculated along high
symmetry directions of the respective Brillouin zones of
Be(0001) and (2×2)-K/Be(0001). The bands and the
wave functions for the states of interest will be presented
below in connection to the experimental results.
IV.
A.
RESULTS
Band structure and diffraction effects
The differences and similarities between the photoemission spectra of the two systems are illustrated in Fig.
2, which shows energy distributions recorded along the
surface normal for Be(0001) and this substrate covered
with a full monolayer of Na or K. Like most of the spectra
recorded in the present work the results in Fig. 2 were
obtained at photon energies in the 30- 35 eV range since
all the states of interest then appear with reasonable intensity. The spectrum recorded for the clean substrate
is dominated by the surface state peak at 2.8 eV binding energy. As alkali metal is deposited this state shifts
to lower energy gradually with increasing surface coverage approaching a saturation binding energy of 3.7 eV
for Na and 3.4 eV for K (Fig. 3). The peaks labeled
QWS in Fig. 2 are due to quantum well like states in
the adsorbed monolayer. A corresponding state was observed for 1 ML of Li23 . The QWS lines appear near
4
3
2
1
0
Binding Energy (eV)
FIG. 2: Photoemission spectra recorded at 33 eV photon energy along the surface normal for Be(0001) (lower spectrum),
and this substrate covered with 1 ML of Na or 1 ML of K.
The adsorbates shift the Γ̄ surface state (labeled S in the
diagram) to higher binding energy and quantum well states
(QWS in the diagram) are observed at around 0.5 eV binding
energy. The peak labeled M, observed for K but not for Na,
is explained by the different structures of the two overlayers.
Similar peaks, labeled A, are observed for both Na and K and
these are not well understood. The light is incident at angle
of 45◦ and polarized in the incidence plane.
the Fermi edge at a certain coverage and shift to higher
binding energy with saturation at 0.55 eV and 0.40 eV
binding energy for Na and K respectively (Fig. 3). Peak
A is poorly understood and will be discussed last in this
section. The shown spectra were recorded for a cooled
sample. Spectra recorded with the sample at RT show
the same features although somewhat less distinct. At
the lower temperature the QWS binding energy at saturation is slightly higher, by around 0.15 eV for Na and
0.1 eV for K. A possible reason is that more atoms may
be accommodated in the monolayer at the lower temperature.
Using saturation of the binding energy shifts as demarcation of full monolayer coverage the LEED patterns
were recorded at this stage (Fig. 4). These show that K
atoms form a 2×2 overlayer as might be expected from
the relative sizes of K and Be atoms. The Na atoms are
smaller and form a close packed incommensurate overlayer with a surface mesh corresponding to 1.65x1.65 and
4
S
hv=31eV K on Be(0001)
100K
A QWS
S
M
hv=33eV
100K
240s
170s
130s
210s
110s
150s
90s
80s
90s
S
70s
S
45
36
0
0
0s
2
1
0
Binding Energy (eV)
o
o
o
0s
4
3
hv=33 eV
100 K
o
30s
40s
4
hv=33 eV 1ML K on Be(0001)
100 K
QWS
210s
Intensity (Arb. Units)
1 ML Na on Be(0001)
A
Intensity (Arb. Units)
Na on Be(0001)
4
3
2
1
0
3
2
1
Binding Energy (eV)
0
4
3
2
1
Binding Energy (eV)
0
Binding Energy (eV)
FIG. 3: Photoemission spectra recorded along the surface normal of Be(0001) for different Na (left panel) and K coverages
(right panel). The coverages are given via the evaporation
times in seconds. Full monolayer coverages are obtained after around 210 s for Na and 240 s for K. For labeling and
explanation of peaks we refer to Fig. 2 and to the text.
FIG. 4: LEED patterns obtained at full monolayer alkali
metal coverage on Be(0001). For Na (left panel), the overlayer forms an incommensurate structure, surface mesh approximately 1.65×1.65, aligned with the substrate although
with some azimuthal disorder. For K, the pattern corresponds
to 2×2 order (right panel). Both patterns are recorded at 100
K with an electron energy of 112 eV. The appearance of the
zero order spot just below the center shadow in the lefthand
panel is explained by a somewhat larger incidence angle in
that case.
with the symmetry directions along those of the substrate
although with a small azimuthal elongation of the spots.
The peak labeled M in Fig. 2 appears only for potassium and becomes prominent only at high monolayer coverage. The origin of this peak becomes evident from the
LEED pattern and the emission angle dependence of the
photoemission spectra (Fig. 5). Fig. 6 shows the dispersion of states along the Γ̄-M̄ direction obtained from the
FIG. 5: Photoemission spectra recorded with 33 eV photon
energy at different polar angles probing downshifted Be surface states and quantum well states along Γ̄-M̄ for Na (left
panel) and K (right panel) at 1 ML coverage. The p-polarized
light beam is in the plane of measurement and incident at angle of 45◦ .
angle dependence of the spectra shown in Fig. 5. For
comparison the calculated bands for slabs of Be(0001)
and p(2 × 2)K/Be(0001) are shown in Fig. 7. For
Be(0001) the calculation shows surface states at Γ̄ and M̄
with binding energies (2.7 eV, and 1.8 eV respectively)
near the experimental values (2.8 eV and 1.8 eV). The
main qualitative difference between the two calculated
band structures is that a QWS band appears for the overlayer case extending 0.56 eV below the Fermi energy at
Γ̄. Aside from this there is an increased number of bands
for the p(2×2)K/Be(0001) structure due to the folding
of bands into the smaller Brillouin zone defined by the
overlayer. In this zone the surface state at the M̄ point
of the larger Be(0001) zone is found at Γ̄.
From the dispersion it is clear that peak M in Fig. 2 is
due to the upper surface state at the M̄-point of the surface Brillouin of the substrate. Upon adsorption the state
has become downshifted by 0.4 eV for Na and by 0.3 eV
for K from the energy for clean Be(0001). The shift calculated for K is 0.2 eV. Also the larger shift of the Γ̄ surface
state, 0.7 eV for K, is well reproduced by the calculation
(0.70 eV). Although the shift of the Γ̄ state is large there
is only a modest increase of the number of electrons in
the band since the dispersion is changed. The experiment
(calculation) gives an effective band mass m∗ /m of 1.25
(1.25) for Be(0001) and 1.14 (1.15) when the substrate is
covered with 1ML K. Assuming the length of Fermi wave
vector in the Γ̄-M̄ direction to be representative the population of the Γ̄ surface band increases by around 5 %, from
0.7 electrons per Be atom for the clean substrate. A more
significant change for the charge balance at the interface
5
FIG. 6: Binding energies, read off from the spectra in Fig.
5, plotted versus parallel wave vector for 1 ML of Na (black
dots) and 1 ML of K (red dots). The arrows at ±1.59 Å−1
on the horizontal axes indicate opposite M̄-points of the Be
(0001) surface Brillouin zone.
is that the state has shifted to energies closer to the lower
edge of the bulk band gap, at around 4.8 eV (4.3 eV) below EF according to experiment (calculation)2 . More of
the charge is therefore deposited further inside the substrate (Fig. 8a and 9a). In addition the tail outside the
substrate becomes longer than for the clean surface due
to the attractive potential in the overlayer. Also the upper M̄ state shifts towards the edge of the bulk gap but
in this case, at least in the (1120) plane, the charge is
redistributed mainly in the lateral direction (Fig. 8b and
9b).
For the QWS bands the dispersion is repeated with a
lateral period that agrees well with the LEED patterns.
In the case of Na one might expect the substrate to introduce another repetition period for the QWS band but
this is not observed. Regarding periodicity it is as if
there had been no substrate. The dispersion of the QWS
bands corresponds to a band mass m∗ /m of 1.65 for Na
and 1.55 for K and according to the band calculation
it is 1.05 for K. As noted in Table I, where experimental
and calculated energies and effective masses may be compared, the only significant difference concerns the mass
of the QWS band. The discrepancy may reflect the poor
energy resolution in the calculations close to the Fermi
level. This is probably due to the introduced smearing
of the Fermi level to improve convergence. In these calculations the smearing parameter is 0.68 eV, which even
exceeds the binding energy 0.56 eV of the QWS at Γ̄.
This underestimate of the effective mass is also obtained
in a previous DFT calculation for the QWS band of the
FIG. 7: Calculated band structure for 17 atomic layers
Be(0001) (a), and for this film covered with 2×2 ordered K
atoms occupying on top sites (b). For clean Be(0001) there
are surface bands with binding energies of 2.7 eV at Γ̄ and
1.8 eV and 2.9 eV at M̄. For the adsorbate covered substrate
the corresponding energies are 3.4 eV, 2.0 eV, and 2.9 eV respectively. In addition there is a quantum well state band
extending 0.56 eV below EF .
FIG. 8: Calculated electron density for Be(0001) in the (1120)
plane for the Γ̄ surface state (a), the upper M̄ surface state
(b) and the lower M̄ surface state (c).
saturated Na ML on Cu(111)32 , when applying a smearing parameter 0.1 eV, which similarly slightly exceeded
the binding energy at Γ̄ of 0.06 eV. The measured dispersion gives occupancies of 0.54 and 0.50 electrons per
alkali metal atom for Na and K respectively. For K also
the downshifted Γ surface band appears displaced by a
reciprocal lattice vector defined by the 2×2 overlayer,
while no corresponding observation is made for Na (Fig.
6).
In addition to the states discussed above Be(0001) has
a second surface state or resonance at the lower edge of
the bulk band gap at the M̄ point. The state is observed
only in a small range of lateral wave vectors near the
6
1 ML K on Be(0001)
TABLE I: Experimental and calculated surface and quantum well state binding energies and effective band masses for
Be(0001) and alkali metal covered Be(0001) at 1 ML coverage.
Li/Be(0001) from Watson et al23 .
Γ̄
Be
Li/Be
Na/Be
K/Be
EB (eV)
Exp Cal
2.8 2.7
4.0
3.7
3.5 3.4
m∗ /m
Exp Cal
1.25 1.25
1.03
1.19
1.14 1.15
M̄upper
EB (eV)
Exp Cal
1.8 1.8
2.3
2.2
2.1 2.0
QWS
EB (eV)
m∗ /m
Exp Cal Exp Cal
0.47
1.8
0.55
1.65
0.4 0.56 1.55 1.05
M̄-point. For the K covered surface this surface state is
difficult to resolve from the downshifted and k// - shifted
Γ̄ surface state but overlap between these emission lines
may give the distorted looking dispersion near the band
minimum in Fig. 6. As noted in a previous calculation37
the lower M̄ state deposits its charge below the surface
(Fig. 8c) and is therefore not much affected by an adsorbate.
Regarding peak A in Fig. 2, a similar feature was
observed for Li covered Be(0001)23 . It was mentioned
that it might be a loss component to the QWS line but
it was believed to be due to a state translated to Γ̄ by a
reciprocal lattice vector for the Li overlayer. Support for
this came from the observation of a band maximum with
the same energy as the peak a reciprocal lattice vector
away from Γ̄. For Na or K there is no similar coincidence
with respect to energy making the interpretation in terms
of Umklapp unlikely. Although we will not be able to
give an interpretation we report some observations. As
shown in Fig. 3 peak A appears in concert with the QWS
line and becomes increasingly separated from this line as
the coverage increases. As for Li the peak is observed
only at angles near the surface normal (Fig. 6) but for
K it is observed also near Γ̄ of the second zone of the
overlayer. If the photon energy is changed the intensity
of peak A varies approximately as the QWS peak (Fig.
39
Intensity (Arb.)
FIG. 9: Calculated electron density at Γ̄ for a K 2×2 monolayer on Be(0001)in the (1120) plane of the substrate for the
downshifted Be (0001) Γ̄ surface state (a), the downshifted
upper M̄ state and the quantum well state (c) .
100 K
hv
(eV)
33
S
M
QWS
A
26
15
4
3
2
1
0
Binding Energy (eV)
FIG. 10: Photoelectron energy spectra recorded at different photon energies along the surface normal for 1 ML K
on Be(0001) showing a strong photon energy dependence for
cross sections of the downshifted Be surface states, S and M,
and the QWS. Note that the intensity of peak A varies in
concert with the QWS intensity.
10). As noted previously34,35 the cross section for a QWS
can vary rapidly with photon energy (Fig. 10).If the
coverage is increased beyond the monolayer range peak
A decays in intensity. For multilayers no peak similar
to A is observed in the energy range between the QWS
lines and the line due to the downshifted Γ̄ surface state
(Fig. 11). The latter appears at approximately the same
energy as for the monolayer but with reduced intensity.
Only a marginal energy shift is expected since the state
already at 1 ML coverage extends only with a tail in
the overlayer. No satellite with an intensity comparable
to peak A is observed adjacent to the Γ̄ surface state
line. Although this line is broader than the QWS line
one would still expect to resolve such a satellite.
The QWS energies for multilayers (Fig. 12) change
with overlayer thickness in the manner observed for many
metal overlayer systems as may be explained in terms of
simple model potentials33 . At the photon energy and
detection angle used to record the spectra in Fig. 10
only QWS with low binding energy give a high intensity.
B.
Shallow core levels
The gradual energy shifts observed for the valence
states might suggest that the overlayer is homogeneous
with respect to adatom density and electronic structure.
The shallow core level spectra shown in Fig. 12 for
Intensity (Arb. Units)
7
K on Be(0001)
hv=27eV
4ML
Intensity (Arb. Units)
100 K
3ML
2ML
M
3 ML
2 ML
1 ML
32.0
QWS
A
hv = 70 eV
100 K
Na on Be(0001)
31.5
31.0
30.5
30.0
Binding Energy (eV)
1ML
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Binding Energy (eV)
FIG. 11: Quantum well state emission lines for 1, 2, 3 and 4
ML of K on Be(0001) recorded along the surface normal at 27
eV photon energy. For overlayers thicker than 1 ML no peaks
similar to A are observed.
Na on Be(0001)
hv =150eV
100 K
Intensity (Arb. Units)
210s
170s
130s
110s
90s
80s
FIG. 13: Na 2p core level spectra recorded at 70 eV photon
energy for Be(0001) covered with 1, 2 and 3 ML Na showing
atomic layer dependent binding energies.
constant binding energy and out of this grows a second
line, which shifts to lower binding energy with increasing
coverage and dominates the spectrum at high coverage.
At high coverage this line is accompanied by a satellite
on the low kinetic energy side. The peak separation is
around 0.7 eV at full monolayer coverage. This separation is thus similar to that between the QWS line and the
anomalous peak A. While a second atomic layer might
start to form before the first layer is complete this does
not explain the satellite. As shown in Fig. 13 the satellite observed for 1 ML Na falls outside the energy range
of the layer dependent binding energies of thicker films.
In the case of K there are bigger differences between the
3p binding energies for overlayers with different thickness
(Fig. 14). Also for K a weak low energy component is
noted for 1 ML but this falls at approximately the same
binding energy as the K 3p line assigned to the uppermost atomic layer of a two atomic layers thick overlayer.
70s
40s
34
33
32
31
C.
Coverage dependent line width for the Γ̄ surface
state
30
Binding Energy (eV)
FIG. 12: Na 2p core level spectra recorded at 150 eV photon
energy for Na/Be(0001) at different coverages given by the
deposition times. Full monolayer coverage is obtained after
210 s deposition time.
Na/Be(0001) demonstrate that this is not the case, at
least not in a range of intermediate monolayer coverages
where two emission lines are observed. To save recording time these spectra were obtained with less resolution
than required to resolve the Na 2p spin-orbit doublet.
The line characteristic of low coverage remains at nearly
After noting in several experimental runs that the
linewidth of the Γ̄ surface state increases markedly in an
intermediate coverage range this was studied in detail for
Na together with the Na 2p spectrum described above.
This increase is observed at approximately the same coverage for which the QWS starts to appear. When energy shifts were monitored the coverage was typically increased in doses. Due to the time required it was suspected that the spectra recorded at high coverages might
get an increased line width due to contamination. In
the line width measurement the sample was therefore
cleaned between each dose such that each spectrum could
be recorded within minutes after the deposition. Initially
upon Na deposition the width increases modestly. In an
intermediate coverage range it increases more rapidly by
8
0.0
hv=38 eV
100 K
3 ML
2 ML
1 ML
19.5
19.0
18.5
18.0
17.5
17.0
QWS
Na on Be(0001)
100 K
0.5
Binding Energy (eV)
Intensity (Arb. Units)
K on Be(0001)
1.0
A
a
S
3.0
3.5
b
31
Na 2p
32
Binding Energy (eV)
FWHM (eV)
FIG. 14: K 3p core level spectra recorded at 38 eV photon
energy for Be(0001) covered with 1, 2 and 3 ML K showing
atomic layer dependent binding energies with the spin orbit
splitting resolved.
c
33
S
0.5
d
0.4
0
around 100 meV reaching an almost constant value at
high monolayer coverage. The coverage dependence of
the line width is shown in Fig. 15 together with the
binding energies for the states of interest.
V.
A.
DISCUSSION
Charge distributions, diffraction effects and
energy shifts
The states characteristic of an overlayer system are hybrids with distinctly different character in the overlayer
and the substrate. The electrons experience a potential
with different spatial variations in the substrate and the
adsorbed layer. The mere existence of the QWS derives
from the potential variations in the direction normal to
the surface and laterally there are different periods in the
substrate and the overlayer. That there are two lateral
periods is not expected to be important for the states observed here since each state is well anchored in either the
substrate or the overlayer. As shown for K in Fig. 9c the
QWS at Γ̄ has one node in the K layer near the overlayersubstrate interface and deposits almost the entire charge
in the overlayer, with maximum density outside a plane
through the cores of the K atoms. This degree of confinement is extreme compared to similar cases. For alkali
metals on Cu(111) the charge of the corresponding QWS,
with one node in the film, is shared more equally between
the substrate and the overlayer26,36 .
In contrast to the QWS the downshifted Γ̄ surface state
has nearly all its charge within the substrate (Fig. 9a).
The energy is lower than the valence band bottom of K
metal and the state therefore extends with a tail outside the substrate although the tail is longer than for the
50
100
150
200
Deposition time (s)
FIG. 15: Na coverage dependence for the binding energies of
the quantum well state and peak A (a), of the Γ surface state,
S (b), the Na 2p electrons (c), and the width (FWHM) of the
Γ̄ surface state emission line (d). Full monolayer coverage is
obtained after 210 s evaporation time.
clean Be(0001) surface. The bottom state in a square
potential well is nodeless in the well and the downshifted
surface state is the corresponding state for a K monolayer
on Be(0001). For K the energy bands of the downshifted
Γ̄ and upper M̄ states appear translated by a reciprocal
lattice vector of the overlayer (Fig. 6). This is explained
by diffraction by the (2×2)K layer of the photoelectrons
excited in the substrate or at the interface. No similar
observation was made for Na. In that case the reciprocal lattice vector is larger. The M̄ states are translated
to different k-points in the overlayer zone making the
diffracted intensity weaker than for the K layer when all
the M̄ states are backfolded to the Γ̄ point. A possible reason for the difference between Na and K is that
the heavier alkali metal atom scatters low energy electrons more strongly. It should also be pointed out that
we did not investigate the photon energy dependence of
the diffracted intensity. As in LEED the diffracted intensity will vary with electron energy with different energy
dependence for different adsorbates.
B.
Line width of the Γ̄ surface state
The Γ̄ surface state has served as a test object for
theoretical calculations of hole lifetimes9,38 . From the
9
temperature dependence of the line width the electronphonon contribution to the width could be extracted9 .
When the line width due to the electron-phonon interaction and the experimental resolution is accounted for
there is a good agreement with the contribution to the
width from electron-electron scattering. The theoretical
estimate is 265 meV, most of this, 225 meV, coming from
scattering within the surface state band, with smaller
contributions from bulk states, 35 meV, and from the
upper M̄ surface state, 5 meV9 .
As noted in Fig. 15 the line width of the Γ̄ surface
state increases significantly in the coverage range where
the QWS appears near the Fermi edge. A possible reason for the onset of width increase is that the population of the QWS band opens a new channel of decay for
the photohole in the Γ̄ surface state. This is certain to
contribute to the line width but, considering the small
overlap between the QWS and the surface state (Fig. 9a
and 9c), it would be surprising to find the effect to be as
large as measured. We believe a more likely reason for
the increased width is the inhomogeneity of the overlayer
demonstrated by the core level spectra (Fig. 12). The
surface state energy is no local probe and the inhomogeneity affects the binding energy weakly such that the
spread in binding energy is reflected by the line width
increase.
At high monolayer coverage the overlayer is homogeneous and there is still a larger line width than at low
coverage. This we ascribe to the difference in electronic
structure between the clean substrate surface and this
surface covered with 1 ML of the alkali metal. In particular one would expect the contribution to the line width
from bulk states to increase. One reason is the larger
overlap with bulk states as the surface state extends
deeper into the bulk. Furthermore, as shown for K in
Fig. 7b, the downshifted surface states is found among
backfolded bulk states and may couple to these states.
As a result the surface state will no longer have the discrete character it has for the clean substrate. This type
of resonance broadening was recently noted to be small,
< 10 meV half width, but still significant for QWS in
1 ML of Na or Cs on Cu(111)20 .
C.
Nonmetal-metal transition
While the binding energies of the valence electron
states shift gradually with coverage this is not observed
for the Na 2p level. Initially upon deposition one emission
line is observed but this vanishes upon continued deposition and becomes replaced by a somewhat narrower line
with a gradually shifting binding energy (Fig. 12). In an
intermediate coverage range both emission lines are observed (80 s- 130 s evaporation time). At high monolayer
coverage the line obtains a satellite on the low kinetic
energy side.
The coverage dependence for the Na 2p emission is
quite similar to that noted for the Li 1s line of the
Li/Be(0001) system24 . The Li data included observations with inverse photoemission of the coverage dependence of the energy for an empty state near above the
Fermi level. A plot versus coverage of this energy and
the QWS energy measured at k// =0 indicated that that
there is a gap between the lowest energy observed for the
empty state and the highest QWS energy. As for Na and
K the highest QWS energy for Li is close to the Fermi
level. Based on this, the Li 1s data and the observed
LEED patterns it was suggested that there is a discontinuous nonmetal to metal transition in the overlayer at
an intermediate monolayer coverage.
The same interpretation may be made for the present
systems. The metal character of the full monolayer is
assured by the QWS band, which at full monolayer coverage contains around 0.54 and 0.50 electrons per alkali
metal atom for Na and K respectively. When the QWS
first appears near the Fermi edge this means that the
density of states at the Fermi energy obtains a value typical of a metal. At lower coverage the states in the Γ̄
surface band extends across the Fermi level but little of
this charge falls in the overlayer. Assuming the QWS to
have a distinct energy when it appears at a certain coverage this would give a stepwise increase of the density
of states at the Fermi level at that coverage. In practice
the adatom density will not be homogeneous making the
QWS energy less distinct and the onset of metal character
less abrupt. Such observations are made for Na or Cs covered Cu(111) where the corresponding QWS can be observed by photoemission as it shifts with increasing coverage from above to below the Fermi energy26,39 . With the
present set up the intensity was too low to monitor states
above the Fermi energy. We thus do not know whether
there actually exists a QWS above the Fermi level having an energy, which may be downshifted in a gradual
manner into the populated range by increasing the coverage. According to the inverse photoemission and photoemission results this is not the case for Li/Be(0001)24
but further information on this for the present systems
would be of interest.
With the above considered it seems possible to explain
the present data in terms of an almost stepwise transition to metal character in patches that become larger
as the coverage is increased making the transition gradual when the whole film is considered. In the first stage
the QWS is only patchwise occupied. Thus, if this electron gas is spatially inhomogeneous only a fraction of
the alkali metal atoms reside in the electron gas. This
would explain the existence of two different core lines in
an intermediate coverage range. Beyond this range the
electron gas hangs together and only one Na 2p emission
line is observed. As with increasing coverage the QWS
band shifts to lower energy the electron density in the
overlayer increases and the Na 2p emission line shifts to
lower binding energy. In concert with the occupation of
the QWS, maybe driven by this, there could be some
change of the order in the overlayer. The results however indicate that this change is modest with regards to
10
the adatom density. This is demonstrated by the gradual
energy shift of the surface state and by the observation
that the Na 2p line of the high coverage phase starts out
at an energy near that observed at low coverage.
Finally we note that the present systems seem to have
properties intermediate between those, which may be
obtained for alkali metals deposited on metal and on
graphite. Although it is not typical one may for metal
substrates find a gradual change of alkali metal core level
binding energies or excitation thresholds throughout the
monolayer coverage range with a higher binding energy
at low coverage when the adatoms are more ionic than at
high coverage when the layer is almost neutral40,41 . This
is as observed for the high coverage phase on Be(0001).
On graphite by contrast there is a distinct change of
atomic order and electronic structure in the monolayer
coverage range. Beyond a coverage threshold the alkali
metal atoms form a condensed phase coexisting with a
dispersed phase found at low coverage42 . For K and Rb
the high coverage phase consists of monolayer thick islands with close packed atoms in 2×2 order13,43 . The
effect of increasing the adatom coverage is just a larger
island area but no significant compression of the atoms.
This is reflected by coverage independent binding energies for core and valence states for the high coverage
phase. As demonstrated by a LEED and core level study
of K/Ag(100) one system can show both of the two types
of behavior referred to above44 . At 90 K core level binding energies change in a gradual manner as the coverage
is increased but at 220 K this is only observed in a low
coverage range. Beyond approximately 0.2 ML coverage an island phase is observed which is reflected in the
spectra by a nearly stepwise change in binding energy,
this remaining almost constant when the coverage is increased.
D.
The anomalous peak
Peak A in Fig. 2 could be due to an energy loss with
the QWS line as the primary. Since there is a similar
separation in energy between the Na 2p components observed at high monolayer coverage (Fig. 12) the low energy component may be a loss satellite due to the same
excitation. We find no single particle excitations that
could account for a loss peak and speculate that the loss
is due to collective oscillations in the overlayer. For the
clean Be(0001) a loss due acoustic plasmons in the surface layer was recently observed and the dispersion was
1
2
U.O. Karlsson, S. A. Flodström, R. Engelhardt, W.
Gadeke, and E.E. Koch, Solid State Commun. 49, 711
(1984)
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VI.
SUMMARY
The Γ̄ surface state of Be(0001) shifts gradually to
lower energy with increasing Na or K coverage in the
monolayer range. At least for Na/Be(0001) which was
studied in detail the gradual shift masks the inhomogeneity evident from the shallow core level spectra in an
intermediate coverage range. In this range a quantum
well state characteristic of 1ML coverage becomes occupied marking the onset of metal character locally as well
as the onset of inhomogeneity. According to our interpretation the inhomogeneity is noted as a line width change
for the Γ̄ surface state. Although the population of the
quantum well state opens a new channel of decay for the
photohole in the Γ̄ surface state we believe this to give
a minor increase of the line width. At full monolayer
coverage the spectra for K and Na show qualitative differences, which reflect the different structures obtained
with prominent diffraction effects noted for the commensurate K monolayer. The band structure obtained from
a structure optimized DFT calculation made for 2×2 ordered K on Be(0001) agrees well with the experiment.
An unexplained emission line is observed for both 1ML
Na and K. The data indicate that this is a loss companion to the quantum well state emission peak. We suggest
that some collective mode is excited involving the QWS
electrons, which form a thin sheet of electron gas in the
adsorbed monolayers.
Acknowledgments
Financial support from the Swedish Research Council
is gratefully acknowledged.
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Paper V
IOP PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 19 (2007) 305011 (10pp)
doi:10.1088/0953-8984/19/30/305011
Evidence of longitudinal resonance and optical
subsurface phonons in Al(001)
Vasile Chis1 , Bo Hellsing1 , Giorgio Benedek2 , Marco Bernasconi2 and
J P Toennies3
1
Physics Department, Göteborgs University, Fysikgränd 3, S-412 96 Göteborg, Sweden
Dipartimento di Scienza dei Materiali, Università di Milano-Bicocca, Via R. Cozzi 53,
20125 Milano, Italy
3 Max-Planck-Institut für Dynamik und Selbstorganization, Bunsenstrasse 10, 37072 Göttingen,
Germany
2
E-mail: [email protected]
Received 5 February 2007, in final form 6 February 2007
Published 13 July 2007
Online at stacks.iop.org/JPhysCM/19/305011
Abstract
A calculation of the surface phonon dispersion curves of Al(001) based on
density functional perturbation theory confirms the intrinsic nature of the
controversial longitudinal surface phonon resonance, which has been reported
to occur in most metal surfaces. The results support previous density–
response pseudopotential and semi-empirical Born–von-Kàrmàn calculations
by Franchini, Bortolani, et al, proving the extended nature of the surface
perturbation. The latter implies the existence of a Lucas pair of optical surface
modes localized in the second layer.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Many important properties of metal surfaces, such as the relaxation of the surface atomic
layers and the surface-dependent work function, are determined by the redistribution of the
surface electronic charge with respect to the charge density in the bulk [1, 2]. Another
feature, apparently common to most metal surfaces, is the longitudinal (L) surface phonon
resonance, which was first discovered by means of inelastic helium atom spectroscopy (HAS)
in silver [3] and other noble metals [4], and subsequently in practically all investigated simple
and transition metals [5]. Most of these observations have been confirmed by electron energyloss spectroscopy (EELS) [5]. This ubiquitous acoustic resonance invariably occurs well below
the edge of the longitudinally polarized bulk band, even for the densely packed (111) surface
of fcc metals, where no surface longitudinal mode is expected for an ideal surface with simple
nearest-neighbour force constants [6]. For this reason this new surface mode was frequently
0953-8984/07/305011+10$30.00 © 2007 IOP Publishing Ltd
Printed in the UK
1
J. Phys.: Condens. Matter 19 (2007) 305011
V Chis et al
referred to as anomalous longitudinal resonance. The above observations led to considerable
theoretical activity in search of a simple explanation. A thorough discussion of the ab initio
studies devoted to the surface dynamics of metals, with a comparison to former semi-empirical
approaches, can be found in a review by Heid and Bohnen [7].
Whereas HAS and EELS are in agreement as regards the location of the L resonance,
there is however a substantial difference between the respective scattering amplitudes, the
HAS intensity for the L resonance being often much larger than that for the Rayleigh wave
(RW) [3–5], whereas the EELS amplitude for the L mode is normally much smaller [7, 8].
This was regarded as a paradox since electrons in the inelastic impact regime are essentially
scattered by the oscillation of the atom cores of the first two or three surface layers, whereas
He atoms are scattered by the oscillations of the surface electron density about 2–3 Å away
from the first atomic layer and should be rather insensitive to the atomic displacements parallel
to the surface. For these facts Heid and Bohnen [7] conjectured that the L resonance may be
attributed to some special property, if not an artefact, of the He–surface interaction and not to a
real spectral feature of metal surface dynamics.
This rather unsatisfactory explanation for an apparently universal property of metal
surfaces motivated the present study of the surface phonon dispersion curves of Al(001) based
on density functional perturbation theory (DFPT) [8]. Since the displacement field of a surface
resonance penetrates substantially into the bulk, the calculation has been performed for a slab
of 17 atomic layers, rather than for a few surface layers artificially matched to an inner set of
bulk layers. The latter procedure has been largely used in previous ab initio calculations [7] in
order to have a sufficient statistics in the surface-projected densities of phonon states. However,
such a matching procedure requires the rotational invariance (RI) of the dynamical matrix to
be restored through the addition of artificial non-diagonal force constants [9]. The violation of
the RI condition at layers where the L-resonance amplitude in not negligible may pose some
problem. This difficulty is avoided in the present pure slab calculation and the results are
quite satisfactory. This calculation confirms the intrinsic nature of the surface L resonance
and the previous analysis based on density–response pseudopotential [10–12] and semiempirical Born–von-Kàrmàn [10, 11] calculations for the low-index surfaces of aluminium.
The calculation also reveals the existence of new subsurface optical modes, localized in the
second layer. Since these modes are not predicted by force constant models with no change in
the interlayer force constants [13], they are likely to be induced by the change in the interlayer
spacings near the surface. These modes, similarly to the Lucas modes in ionic crystals, form a
degenerate pair at the zone centre.
2. DFPT calculations and results for Al(001)
The dynamical properties of the Al(001) surface were calculated by applying first-principles
density functional perturbation theory (DFPT) [8, 14] with ultrasoft pseudopotential [15] and
general gradient approximation (GGA-PBE) [16] for the exchange and correlation energy
functional. The electron wave functions were expanded in plane waves with an energy cutoff
of 25 Ryd and a cutoff of 525 Ryd for the charge density. The irreducible Brillouin zone
was sampled over a Monkhorst–Pack grid of 10 × 10, resulting in 21 k-points. Brillouin
zone integration was performed with the smearing technique [17], with a smearing width of
σ = 50 mRyd. The theoretical bulk value for the in-plane lattice constant of a0 = 7.62 au
was used. Starting from an ideal surface the relaxation was performed by minimizing the total
energy with respect to the atomic positions in the slab. At equilibrium the forces acting on the
−1
atoms are less than 0.26 meV Å .
2
J. Phys.: Condens. Matter 19 (2007) 305011
V Chis et al
Table 1. The elastic relaxation of the first four surface interlayer spacings calculated in this work for
Al(001) slabs of four different thicknesses (expressed in the number of layers (ML)). Comparison is
made with a previous DFT calculation for seven layers [19], a first-principles calculation based on
the full-potential linearized augmented plane wave (FLAPW) method [20] and LEED experimental
data [21, 22].
Al(001)
d12 (%)
d23 (%)
d34 (%)
d45 (%)
Reference
7 ML
9 ML
15 ML
17 ML
23 ML
2.11
1.93
1.41
1.65
1.65
1.14
1.40
1.13
2.30
2.30
1.01
0.87
0.83
0.86
0.86
—
1.06
0.60
0.40
0.40
[18, 19]
[19]
[19]
This work
This work
FLAPW (9 ML)
FLAPW (15 ML)
FLAPW (17 ML)
1.54
1.60
1.60
0.43
0.55
0.44
−0.02
0.02
−0.02
−0.89
−0.43
−0.68
[20]
[20]
[20]
LEED
LEED
1.84
2.0 ± 0.8
2.04
1.2 ± 0.7
—
—
—
—
[21]
[22]
The relaxation of the interlayer spacing for the first four surface layers calculated in this
work for two different slab thicknesses are given in table 1 and compared with a previous
DFT calculation for 7 [18, 19], 9 and 15 layers [19], with recent calculations based on the
full-potential linearized augmented plane wave (FLAPW) method [20] and with the available
low-energy electron diffraction (LEED) experimental data [21, 22]. The dependence on the
slab thickness shows in all cases an oscillatory convergence to the thick-slab limit. The first
interlayer spacing d12 calculated in this work agrees with the FLAPW result, whereas the
relaxations between inner planes are found to be substantially larger than the FLAPW ones,
and are all positive. It is noted however that the present results are in a good agreement with the
LEED data, though the latter are affected by a large experimental error and are only available
for the first two spacings.
The phonons were calculated for a fully relaxed Al(001) surface using 17 atomic layers
in the slab and correspondingly 8 atomic layers of vacuum. The dynamical matrices of the
17-layer slab were evaluated on a 4 × 4 mesh of q-points in the surface Brillouin zone. The
Fourier transforms of these matrices provide the real-space interatomic force constants of the
slab layers and five neighbouring atoms within the layers. The phonon dispersion curves of
Al(001) calculated with DFPT for a 17-layer slab are shown in figure 1(a) together with the
HAS experimental data reported by Gaspar et al [23]. A few more data points obtained with
EELS by Mohamed and Kesmodel [24] along the Rayleigh wave dispersion curve (S1 ) also in
the ¯ –M̄ direction are perfectly superimposed to the HAS data and are not shown, for clarity4 .
The agreement of the calculated S1 dispersion curve with the HAS data is perfect. In addition
to the S1 mode, HAS data show a set of points at larger energies inside the bulk continuum
which have been assigned, on basis of an ab initio density response calculation [23], to two
distinct branches: the upper one (labelled by S2 ) is attributed to a longitudinal mode, whereas
the one at lower energy (S3 ) has prevalent shear-vertical character and a weaker intensity. Both
surface modes fall in the continuum of the bulk phonon density and are therefore to be regarded
as resonances. In order to show the evolution of the slab modes into genuine surface modes the
calculation for 17-layer slab is compared to a previous DFPT calculation [18] for a 7-layer slab
4 Surface modes are labelled, as they should always be, according to their polarization. Thus S labels the Rayleigh
1
mode also in the ¯ –X̄ direction. In this direction (and for this particular surface) the lowest mode frequency corresponds
to a shear-horizontal mode and is therefore labelled S7 .
3
J. Phys.: Condens. Matter 19 (2007) 305011
V Chis et al
Figure 1. (a) DFPT calculated surface phonon dispersion curves for a 17-layer slab of Al(001). The
experimental points corresponding to the Rayleigh wave (S1 ) and to two upper branches (S2 and S3 )
have been measured by Gaspar et al with HAS spectroscopy [23]; (b) a comparison is made with a
previous DFPT calculation for a 7-layer slab [19].
(figure 1(b)). It is important to stress that for the 17-layer slab the full dynamical matrix has
also been generated without insertion of bulk layers into a thinner slab with a matching of the
force constants. Rotational invariance conditions are preserved, which allows us to reproduce
with good accuracy the experimental velocity of the Rayleigh wave.
In order to investigate the nature of the two resonances, notably of the controversial
longitudinal surface resonance, the spectral intensities of the surface modes, projected onto
both the first and second surface layer, have been calculated for shear-vertical (SV1 and SV2 ,
respectively), longitudinal (L1 , L2 ) and shear-horizontal (SH1 , SH2 ) components. Their contour
plots are mapped in a colour scale, increasing from grey (light grey), then red (dark grey)
4
J. Phys.: Condens. Matter 19 (2007) 305011
V Chis et al
Figure 2. Contour plots of the spectral intensities of the shear-vertical components of the surface
modes and resonances of Al(001) projected onto the first (SV1 ) and second (SV2 ) surface layer. The
DFPT calculation has been performed for a 17-layer slab. The width of the S2 resonance (present
in the lower figure to the left) is larger than the separation of the bulk dispersion curves and appears
to be spread over several bulk lines.
to violet (black), on top of the dispersion curves (figures 2–4). Figure 2 shows that the SV
components contribute the most, as expected, to the Rayleigh wave (S1 ). The decrease in the
second-layer contribution (SV2 ) towards the zone boundaries reflects the decreasing penetration
for increasing wavevectors. This is compensated by the second SV mode S3 (actually a
resonance), which has a large amplitude in the second layer and near the zone boundaries,
whereas it only shows a weak intensity in the first layer at intermediate wavevectors, as actually
observed in HAS experiments.
5
J. Phys.: Condens. Matter 19 (2007) 305011
V Chis et al
Figure 3. Same as figure 2 for the longitudinal components projected onto the first (L1 ) and second
(L2 ) surface layer.
The longitudinal components (figure 3) clearly show a large intensity just below the LA
(longitudinal acoustic) bulk edge in the first layer (L1 ), terminating, along the ¯ –X̄ direction,
in the gap mode S2 at X̄. In the ¯ –M̄ direction Al(001) exhibits a complex situation which deserves further experimental investigation. The S2 resonance undergoes a considerable softening
with respect to the LA edge, ending at about 26 meV at the M̄ point, just above the SV resonance
S3 in the second layer (23 meV). In the ¯ –M̄ direction towards ¯ the latter resonance converts
into an enhancement of both the longitudinal components L1 and L2 near the TA (transverse
acoustic) band edge. On the other hand the gap mode S2 , which is longitudinal in the first layer,
acquires an SV character in the second layer, whereas most of the zone-boundary longitudinal
amplitude in the second layer is concentrated in the high-energy mode S2 slightly above S2 .
6
J. Phys.: Condens. Matter 19 (2007) 305011
V Chis et al
Figure 4. Same as figure 2 for the shear-horizontal components projected onto the first (SH1 ) and
second (SH2 ) surface layer.
The S2 mode (actually a resonance except in the middle of the zone boundary X̄–M̄)
is an example of a surface phonon whose amplitude is mostly concentrated in the second
layer. Its longitudinal character at the zone boundary gradually turns into a very broad, albeit
intense, resonance towards the zone centre. This is due to a hybridization with a secondlayer L resonance (S4 ), which also has a purely optical character since its energy remains
finite (20 meV) at the zone centre. Since empirical models with no change of interlayer
force constants do not predict subsurface modes [13], these second-layer resonances may be
attributed to the changes in the second and third interlayer spacing (see table 1) and may be
regarded as typical effects of the extensive charge redistribution at s, p metal surfaces.
7
J. Phys.: Condens. Matter 19 (2007) 305011
V Chis et al
Figure 5. Comparison of HAS experimental data (black and blue circles) [23] with calculated
phonon density of states projected onto the first and second surface layer for shear-vertical (SV)
and longitudinal (L) polarizations along the ¯ –X̄ direction. The main features of the first-layer
densities of states are the Rayleigh wave (S1 ) and the acoustic L resonance, whereas the secondlayer densities (SV2 , L2 ) show two surface optical resonances of SV and L polarization and a
complex hybridization (avoided-crossing) pattern with the surface acoustic modes. The data points
are plotted onto the energy–wavevector plane. Note that the same colour scale (topological intensity
scale) is used as in previous figures.
Also the SH surface modes (figure 4) present an interesting structure due to the complex
hybridization scheme between the acoustic SH surface mode S7 and the second-layer optical
resonance S5 , further complicated by the avoided crossing with the S1 mode along the zone
boundary X̄–M̄ where the sagittal plane mirror symmetry is broken. In the (001) surface of
cubic crystals the acoustic SH mode S7 is strongly localized only along the ¯ –X̄ direction, and
indeed its second-layer intensity vanishes at X̄, whereas along ¯ –M̄ it degenerates into the bulk
band-edge SH mode. There is also a weak SH resonance occurring at long waves below the LA
edge in the ¯ –M̄ direction which is not found in any empirical model calculations [13]. Even
more interesting is the optical second-layer SH resonance S5 which at the zone centre becomes
degenerate with the optical second-layer L resonance S4 . This is required by symmetry and
is reminiscent of the well-known pair of optical surface modes S4 and S5 , discovered by
Lucas [26] in cubic ionic crystals and also shown to have L and SH polarizations, respectively,
and to be degenerate at ¯ [27, 9].
The hybridization and the resulting avoided crossing of the surface modes observed with
HAS spectroscopy can be better appreciated from the densities of phonon states (DOSs)
projected onto the sagittal (SV and L) components of the first and second surface layer along the
¯ –X̄ direction (figure 5). Also the experimental points (red circles) attributed to the quasi-SV
8
J. Phys.: Condens. Matter 19 (2007) 305011
V Chis et al
and quasi-L polarizations are plotted on the respective SV1 and L1 density of states. As already
noted in figure 1, the S1 amplitude is the main feature in the SV1 and it shows the typical sinelike increase towards the X̄-point. Moreover the points at higher energy in SV1 correspond to a
much weaker ridge arising from the small contribution of S3 in the first layer (see figure 1). A
similar result was found by Franchini et al in their density–response calculations [10].
As appears in figure 5 (L1 ), the longitudinal component of S1 is very small, indicating a
small ellipticity. The main feature in L1 is the ridge starting from the origin at ¯ and ending
into a gap mode S2 at the X̄-point, also corresponding very well to the experimental points: this
is clear confirmation that the longitudinal acoustic resonance occurs also for s, p-metal surfaces
and is not just an artefact of helium atom scattering, as speculated by Heid et al [7]. The nature
of such resonance, however, is not trivial: as appears from the longitudinal amplitudes in the
second layer (L2 ), there is an avoided crossing between the longitudinal acoustic resonance and
the surface optical branch S4 running between 20 meV at ¯ and 17 meV at X̄. This longitudinal
optical resonance as well as the SV optical resonance occurring in the SV2 around 34 meV have
the largest amplitude on the second layer, but almost no signature on the first layer. As remarked
above, this is an indication that the surface perturbation involves several layers, as correctly
argued in the original analysis by Gaspar et al [23] and Franchini, Bortolani, et al [10, 11]. The
nature of such an extended perturbation is likely to depend on the oscillating, slowly decaying
relaxation field, which, according to the results listed in table 1, affects a considerable number
of surface layers.
A careful theoretical analysis of the dynamics of aluminium low-index surfaces was carried
out by Irvine and Modena groups in which the results of semi-empirical Born–von-Kàrmàn
calculations were compared to those of perturbative and non-perturbative pseudopotential
density–response theories [10, 11]. An important conclusion was that the longitudinal
resonance in aluminium is not related to a softening of a single force constant, as originally
argued for the close-packed noble metal surfaces [25, 28–30], but is rather due to the cumulative
effect of changes in the entire force constant field extending over several layers in the vicinity
of the surface. This can be interpreted as indicating that the major changes occur in the global
coupling mediated by conduction electrons.
The present DFPT calculation strongly confirms that analysis and reveals another
important consequence: the existence of surface optical modes localized in the second layer of
metals. They show a surprising, albeit qualitative, similarity to the microscopic surface optical
modes of alkali halides, the S4 and S5 resonances corresponding to the Lucas mode pair arising
from the TO bulk bands and the S2 resonance being the analogue of the Wallis mode associated
with the LO (longitudinal optical) bulk band. The typical oscillatory charge redistribution of
metals may provide some hint for this analogy. It is a fact that such a dynamical structure with
optical surface modes on the second layer is also found in a DFPT calculation for Cu(111), as
will be shown in a forthcoming paper [31].
Acknowledgments
Two of us (GB and JPT) acknowledge a quarter of a century of collaboration and intense
discussion with Professor Virginio (Bibi) Bortolani and his Modena group, which have
triggered so much progress in the theory of surface dynamics and of the surface phonon
spectroscopy. VC acknowledges an EU Marie Curie grant, No HPMT-CT-2001-00242.
Computer resources for the project have been provided by the Swedish National Infrastructure
for Computing (SNIC). GB and JPT acknowledge the Alexander von Humboldt foundation for
support in the framework of the AvHre-invitation program.
9
J. Phys.: Condens. Matter 19 (2007) 305011
V Chis et al
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[26]
[27]
[28]
[29]
[30]
[31]
10
Zangwill A 1988 Physics at Surfaces (Cambridge: Cambridge University Press)
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Springer) chapter 3, at press
Franchini A, Bortolani V, Santoro G, Celli V, Eguiluz A G, Gaspar J A, Gester M, Lock A and Toennies J P 1993
Phys. Rev. B 47 4691
Wallis R F, Maradudin A A, Bortolani V, Eguiluz A G, Quong A A, Franchini A and Santoro G 1993 Phys. Rev.
B 48 6043
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(Amsterdam: North-Holland)
Black J E 1990 Dynamical Properties of Solids vol 6, ed G K Horton and A A Maradudin (Amsterdam: Elsevier)
p 179
http://www.quantum-espresso.org/
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Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
Methfessel M and Paxton A T 1989 Phys. Rev. B 40 3616
Chis V 2003 First principles calculations of phonons at a metal surface Växjö University Report Nr. 5
Chis V and Hellsing B 2004 Phys. Rev. Lett. 93 226103
da Silva J L F 2005 Phys. Rev. B 71 195416
Berndt W, Weick D, Stampfl C, Bradshaw A M and Scheffler M 1995 Surf. Sci. 330 182
Petersen J H, Mikkelsen A, Nielsen M M and Adams D L 1999 Phys. Rev. B 60 5963
Gaspar J A, Eguiluz A G, Gester M, Lock A and Toennies J P 1991 Phys. Rev. Lett. 66 337
Mohamed M H and Kesmodel L L 1988 Phys. Rev. B 37 6519
Bortolani V, Santoro G, Harten U and Toennies J P 1984 Surf. Sci. 148 82
Lucas A A 1968 J. Chem. Phys. 48 3156
Benedek G and Miglio L 1991 Surface Phonons ed F W de Wette and W Kress (Heidelberg: Springer) p 37
Santoro G, Franchini A, Bortolani V, Harten U, Toennies J P and Wöll Ch 1987 Surf. Sci. 183 180
Bortolani V, Franchini A, Nizzoli F and Santoro G 1984 Phys. Rev. Lett. 52 429
Bortolani V, Franchini A and Santoro G 1985 Electronic, Dynamic Quantum Structural Properties of Condensed
Matter ed J T Devreese and P E van Camp (New York: Plenum) p 401
Chis V, Hellsing B, Benedek G, Bernasconi M and Toennies J P 2007 to be published
Paper VI
Optical Phonon Resonances at Metal Surfaces
V. Chis,1, 2 B. Hellsing,1 G. Benedek, M. Bernasconi,2 and J. P. Toennies3
1
Department of Physics, Göteborgs University, Fysikgården 6B, S-412 96 Göteborg, Sweden
2
CNISM, Dipartimento di Scienza dei Materiali,
Universitá di Milano-Bicocca, Via Cozzi 53, 20125 Milano, Italy
3
Max-Planck-Institut für Dynamik und Selbstorganization, Bunsenstrasse 19, 37073 Göttingen, Germany
(Dated: April 24, 2008)
A density functional perturbation theory investigation of Cu(111) surface dynamics shows that
the surface inward relaxation causes the appearance of strong surface optical resonances of both
longitudinal and shear-vertical polarizations and with the largest amplitude in the second layer.
Moreover it is proved that the ubiquitous longitudinal acoustic resonance is not an artifact of
inelastic He atom scattering spectroscopy but a genuine dynamical feature of metal surfaces and
suggests the ability of He atoms to probe the atomic displacements in the second layer via the
associated surface charge modulation.
PACS numbers: 62.20.-e, 63.20.Pw
There are common features characterizing all metal
surfaces which originate from the redistribution of the
surface electronic charge with respect to the charge density in the bulk. Textbook examples are the relaxation
of the surface atomic layers, causing a change of the first
interlayer distances, and a surface-dependent work function [1, 2]. Another apparently common feature is the
longitudinal (L) surface phonon resonance, which was
first discovered by means of inelastic Helium atom spectroscopy (HAS) in silver [3] and other noble metals [4],
and subsequently in practically all investigated simple
and transition metals [5].
Most of these observations have been confirmed by
electron energy loss spectroscopy (EELS) [5]. This ubiquitous acoustic resonance was invariably found well below
the edge of the longitudinally polarized bulk band, with
a frequency significantly smaller than anticipated for the
longitudinal mode of an ideal surface and was therefore
referred to as an anomalous longitudinal resonance.
The above observations led to considerable theoretical
activity in search of a simple explanation. A thorough
account and discussion of the ab-initio studies devoted
to the surface dynamics of metals, including noble metals, has been given by Heid and Bohnen[6] in a recent
review with a comparison to former semi-empirical approaches. Hall, Mills et. al. [7, 8] were able to detect
the L resonance in Cu(111) by means of electron-energy
loss spectroscopy (EELS) and to explain the data with
just a simple force constant model and a moderate 15 %
reduction of the in-plane longitudinal force constant βk .
There is however a substantial difference between the
observed HAS and EELS scattering amplitudes from the
L resonance, the former being often much larger than
the amplitude of the Rayleigh waves (RW) [3–5], whereas
the EELS amplitude for the L modes are normally much
smaller [7, 8]. This appeared soon as a paradox since
electrons in the inelastic impact regime are essentially
scattered by the oscillation of the atom cores of the first
two or three surface layers, whereas He atoms are scattered by the oscillations of the surface electron density
about 2-3 Å away from the first atomic layer and should
be rather insensitive to the atomic displacements parallel
to the surface. Thus the occurrence of the anomalous L
resonance was soon interpreted as pointing to the crucial role of surface electrons and led to the application of
the multipole expansion (ME) method, originally devised
by Phil Allen for the lattice dynamics of transition metals with Kohn anomalies [9], to the problem of surface
phonons in noble metals [10].
The ME method, by providing the phonon-induced
charge density oscillations on a simple, albeit phenomenological local multipole basis, gave for copper surfaces an excellent fit of the RW and L dispersion curves
as well as of their HAS amplitudes [11, 12].
Although an arsenal of other semi-empirical and abinitio methods have been successfully deployed for the
calculation of the phonon structure of copper surfaces,
no calculation purely based on the atom-core displacement amplitudes and the corresponding density of states
(DOS) could provide a satisfactory account of the measured HAS reflection coefficient [13–20]. This difficulty
led Heid and Bohnen to the conclusion that the L resonance may be attributed to some special property, if not
an artifact, of the He-surface interaction and not to a real
spectral feature of metal surface dynamics[6].
In view of the basic relevance of the L phonon resonance in metal surfaces it is important to try to solve
the still open question about the very nature of this resonance by means of density-functional perturbation theory (DFPT)[21]. In this Letter DFPT calculations of the
surface phonon DOS of Cu(111) are presented and discussed for different layers and polarizations. Moreover
the surface charge density (SCD) oscillations associated
with surface phonons have been calculated in order to
account for the observed inelastic HAS intensities. It is
established that the L-resonance owes its complex na-
2
ture to a strong hybridization with an intense sagittallypolarized resonance localized on the second atomic layer,
as argued by former calculations with the embedded
atom [13], effective-medium [14, 15] and first-principle
frozen-phonon [16, 17] methods. The dispersion of such
a second-layer resonance has an optical character. The
existence of surface optical phonon branches localized on
the second layer is another general property of metal surfaces induced by the relaxation field of the first atomic
layers. The analysis of the electron-density oscillations
induced by the surface phonon displacements reveals that
a sizeable part of the SCD oscillations is produced by
the shear-vertical (SV) motion of the underlying atoms
in the second layer. This provides a definitive proof that
the L phonon resonance at metal surfaces is not an artifact of the inelastic He atom scattering mechanism but
a genuine and general dynamical feature. The He atoms
are able to probe the atomic displacements in the second layer through the surface charge modulation they
produce even in a close-packed surface.
are calculated for bulk Cu and for a seven layer Cu(111)
slab. The latter provides the interatomic interactions
within the surface layers whereas the former describes
the force constants in the bulk when the 25 layer slab
is constructed[23]. The dynamical matrices of the seven
layer slab are evaluated on a 6 × 6 mesh of q points
in the surface Brillouin zone. A Fourier transform of
these matrices provides the real-space interatomic force
constants of the three surface layers and six neighboring
atoms within the layers. The in-plane longitudinal force
constant, βk , of the bulk atoms are found to be 11 %
stiffer compared with the ones in the surface plane.
In Fig. 1 we present the dispersion curves for the
phonons of the 25 layer slab along the Γ̄ − M̄ direction of
the SBZ. The calculated phonon dispersion curves are
compared with experimental HAS[4] (red circles) and
EELS[7, 8] (blue circles) data. Our results are in excellent agreement with experiment referring to the RW
mode and the gap mode S2 in the M̄ point.
Cu(111)
30
7L
18L [24] [25] [26] [27] [28]
∆d12 (%) -1.19 -1.18 -1.0 -0.7 -1.58 -0.9 -1.05
25
∆d23 (%) -0.49 -0.82 -0.2 — -0.73 -0.6 -0.07
Energy [meV]
20
TABLE I: DFPT calculated interlayer distance of a 7-layer
and 18-layer Cu(111) slab compared with medium energyion scattering (MEIS)[24] at room temperature, low-energy
electron diffraction (LEED)[25] data, full-potential linear
augmented plane waves (FP-LAPW)[26] calculation, density
functional theory[27] and embedded atom method[28].
15
10
5
0
Γ
M
FIG. 1: Comparison between the calculated dispersion curves
and the helium atom scattering [4] (red circles) and EELS
[7, 8] (blue circles) data for the Cu(111) surface.
The vibrational properties of the Cu(111) surface
are calculated applying DFPT[21] with ultrasoft pseudopotential and GGA exchange and correlation energy
functional according to Perdew, Burke, and Ernzerhof
(PBE)[22]. Plane wave basis with a 30 Ry energy cutoff
for the wave functions and 480 Ry for the charge density
are used.
A 25 layer slab is used with 13 Å of vacuum space. The
irreducible Brillouin zone was sampled over a MonkhorstPack grid of 12 × 12, resulting in 19 k-points. The interlayer spacing is relaxed requiring that the forces are less
than 0.35 meV/Å. The relaxation is confined to the first
few layers and compared in Table I, with experimental
and theoretical results. The interatomic force constants
Fig. 2 and Fig. 3 displays the DFPT phonon DOS
projected onto the shear-vertical (SV1, SV2) and longitudinal (L1, L2) components of the atomic displacements
in the first and second layers, respectively, as functions
of energy and wave vector for the Γ̄ − M̄ symmetry direction. The HAS and EELS data are represented by red
and yellow full circles, respectively.
The calculated DOS show first of all the intense ridge
marked RW in both SV1 and SV2 and a much weaker
intensity in the longitudinal components L1 and L2, as
expected from the quasi-SV character of the RW. Moreover there is a clear evidence of the L resonance, appearing in Fig. 2(b) (L1) as a ridge ending at M̄ as a gap
mode (S2 ). It appears however that both the HAS and
EELS data points follow the L branch up to one half of
the zone, then deviate towards the S3 resonance, which
only has a weak L component in the first and second layer
but a SV component in the second layer which becomes
very large at the M̄ point. The origin of such intensity
transfer from the L to the S3 resonance, interpreted as a
dramatic softening of the L resonance, is well understood
for EELS data since electrons penetrates sufficiently to
give a sizeable scattering intensity also from the SV dis-
3
FIG. 2: Calculated phonon DOS projected on the first layer
shear-vertical (SV1) and longitudinal (L1) polarizations in the
Γ̄-M̄ direction. Comparison is made with HAS[4](red circles)
and EELS[7, 8] (yellow circles) experimental data. The data
points are plotted in the energy- wavevector plane.
FIG. 3: Calculated phonon DOS projected on the second layer
shear-vertical (SV2) and longitudinal (L2) polarizations in the
Γ̄-M̄ direction. Experimental data points as in Fig. 2.
placements of the 2nd layer, whereas for He atoms this
would pose a problem since they do not penetrate at all.
This long-standing puzzle in now elucidated by the
present calculation of the phonon-induced SCD oscillations (Fig. 4). Results for RW, S3 and S2 frozen-phonon
displacements for wavevectors Q at the M point and at
the mid-point (M/2) of the Γ̄-M̄ direction. Only contour lines for SCD oscillations ∆n between +10−4 and
−10−4 a.u. are displayed, together with the contour line
for 10−4 a.u. of the rigid-lattice (RL) SCD (Fig. 4,
heavy lines). This approximately correspond to the locus of turning points for incident He atoms from a roomtemperature source. The SCD oscillations contour lines
for ∆n = ±0.0143 × 10−4 a.u. have been added in order
to show that the amplitudes of SCD oscillations at the
turning point are about this size for all modes. Since the
squared ratio of the SCD oscillation to the RL SCD at
the turning point is a good estimation of the inelasticto-elastic (specular) HAS intensity ratio, the present calculation predicts intensity ratios of the order of 10−4 ,
in good agreement with experiment[11]. It is interesting
to note that similar HAS amplitudes are predicted for
the RW and L-resonance S3 despite the fact that the S3
atomic displacements are much larger in the second layer
and weaker firs-layer displacements are essentially longitudinal. It is the SCD modulation which carries the interaction of the He atoms with the ”buried” S3 phonons.
Incidentally it appears that the DOS for the SV atomic
motion in the 2nd layer (Fig. 3 (a)) is almost identical to the DOS of the multipolar oscillations in the first
layer reported in Ref. [11], thus indicating that the surface charge density deformations closely reflect the SV
motion of the underlying atoms.
In Fig. 4 we show the S3 mode at the M̄ point projected
in the sagittal plane. It is clear that a substantial weight
of this mode is of SV character in the second layer. Thus
it is likely that He atoms can probe to some extent also
2:nd-layer atomic motion. This effect explains the very
nature of the L resonance, which remained a puzzle over
two decades because of the assumption that He atoms
can only probe the displacements of surface atoms.
A closer inspection of Fig. 2 (b) shows a weak noncrossing between S3 and S2 in the first layer, which, however, becomes rather large in the second layer for the SV
components (SV2, Fig. 3 (a)). On the other hand the
L mode in the second layer (L2, Fig. 3 (b)) has a noncrossing with another optical resonance, not detected by
0
any probe, which ends at S2 .
In summary, this complex scenario is fully consistent
with the effective-medium picture [14, 15], ascribing the
L resonance to a second-layer z-polarized mode, but also
with the interpretation based on the embedded-atom theory [13] which ascribes the L resonance to a non-crossing
effect between two longitudinal branches. The present
analysis also vindicates the substantial correctness of
the multipole expansion paradigm [10] and the related
pseudo-charge model [11] which provided a unifying picture in the discussion of electron-phonon effects at metal
surfaces. Here it has been shown that the inward surface
relaxation, common to most metal surfaces, induces two
branches of optical surface phonon resonances of approximate SV and L polarizations, having the largest am-
4
FIG. 4: Surface charge density modulation (in unit of 10−4 a.u.) induced by frozen-phonon displacements (arrows) for the
Rayleigh wave, the S3 and S2 surface modes at the M̄ point (RW(M), S3 (M), S2 (M), respectively) and at the midpoint of Γ̄-M̄
(RW(M/2), S3 (M/2), S2 (M/2), respectively). The atomic displacement length scale is provided by the 2nd layer displacement
of S3 (M) equal to 0.006 Å. 15 contour lines for equal SCD change ∆n (red for ∆n ≥ 0, blue for ∆n ≤ 0) are shown for
equally-spaced values between +10−4 and −10−4 a.u. (spacing = 0.143 × 10−4 a.u.); the contour lines for ∆n = ±0.0143 × 10−4
a.u. are added in order to show the density modulation near the turning point locus (heavy lines, for a rigid-lattice (RL) SCD
of 10−4 a.u.) for a typical HAS experiment.
plitudes on the second layer (buried resonances) (Fig.
4). The hybridization (avoided crossing) of the optical
L-polarized resonance with the acoustic longitudinal surface mode is responsible for the anomalous L-resonance.
Finally the SV components of buried resonances cause
oscillations of the SCD comparable to those of genuine
surface modes like RWs, and can be detected by HAS
even on close-packed surfaces. It is suggested that these
effects are common to most metal surfaces.
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