ELECTRONIC STRUCTURE OF SURFACES : Ramin Lotfollahi : Dr. Peter Johansson

ELECTRONIC STRUCTURE OF SURFACES : Ramin Lotfollahi : Dr. Peter Johansson
ELECTRONIC STRUCTURE OF
SURFACES
By: Ramin Lotfollahi
Supervisor: Dr. Peter Johansson
Examiner: Dr. Andreas Oberstedt
Contents
Chapter 1
Introduction
1.1 Nearly free electron model
1.2 Surface states
1.3 Lattice and image techniques
Chapter 2
Crystal structure
2.1 Crystal structure
2.2 Defects and dislocations
2.3 Crystallography
2.4 Surface crystallography
Chapter 3
One-dimensional band theory and surface states
3.1 The jellium model and surface electronic structure
3.2 One dimensional band theory
3.2.1 Solution of the Schrödinger equation in the bulk
3.2.2 Boundary conditions at the surface
3.2.3 Image potential
3.3 Numerical calculations and results
Chapter 4
Two-dimensional lattice electronic structure
4.1 Theoretical descriptions
4.2 Steps
4.3 Calculation method
1
1
2
3
5
5
6
8
9
12
13
13
14
16
18
19
23
23
24
26
4.3.1 The density of states
4.3.2 The Green’s functions
4.3.3 Equation of motion
4.4 Green’s function for two-dimensional terrace
4.5 Results and discussions
26
27
28
30
33
Summary
35
Appendix A: Introduction to Green’s Functions
36
Appendix B: Listing of programs used for the numerical
calculations
38
References
47
Preface
The electronic structure of surfaces has become a very active area of research in the
past few decades. Recent advances in experimental techniques have produced an
abundant supply of reliable data on clean, well-ordered single crystal surfaces. This has
urged theoretical research in the development and application of models and calculational
techniques. The extent of this essay is limited to theoretical studies of the electronic
structure and an simple calculation of electronic states of solid surfaces in the nearly-free
and free-electron approximation. For investigation of the electronic properties of clean
solid surfaces, it is helpful for the development of intuition to start with a base of detailed
knowledge about simple definitions. I have therefore introduced a number of useful
concepts in the first chapter. In chapter two I give a brief general view of the crystal
forming and a series of technologies to identify the atomic and electronic structure of the
crystals. The theory of surface states and the electronic properties at the metal surfaces
have been investigated in third and fourth chapter. Finally, short discussions on the
results, my working method and also some mathematical material have been placed in
appendix at the end of this paper.
Chapter 1
Introduction
Electronic properties associated with condensed matter have been studied in many
years and with many different aims. The effect of technological evolutions however
accelerated the theoretical and experimental developments in this area until the solid state
physics and particularly electronic structure of surfaces has become one of the major
research areas. As a consequence of the technological needs and the developments of new
techniques in studying the crystal structure, the scientists were able to detect the
electronic properties of the crystalline metal surfaces. In this context and for a better
understanding of the electronic properties of solid surfaces we will investigate solid
surfaces theoretically to explain a more extensive perspective of the electronic structure
of surfaces.
1.1 Nearly free electron model
The free electron model of metals gives us better insight into the electronic
properties of solid bulk of metals. In this model the allowed energy values vary
continuously from zero to infinity with wave functions of the form,
2 2
k
ψ k (r ) = exp ik ⋅ r , with E =
,
(1-1)
2m
They are propagating waves in all directions. There is no explanation of the band
structure of a crystal in this model. The band structure, however, can often be explained
by the nearly free electron model that considers the effect of a weak periodic potential
(Kronig-Penny model) of the ion cores. This model explains why electrons are arranged
in energy bands and the behaviour of electrons in metals. The wave functions in this
(
)
model are a product of a function ϕ k (r ) with the same periodicity as the potential and a
(
plane wave, exp ik ⋅ r
)
(
ψ k (r ) = ϕ k (r ) exp ik ⋅ r
)
(1-2)
A wave function of this form is called Bloch function and describes electrons that
propagate through the potential field of the ion cores. The wave equation can be treated in
detail for a general potential, at general values of k which lies within the Brillouin zone
boundary. As a consequence of interaction between the conduction electrons waves and
the ion cores of the crystal, there are forbidden regions in energy space which are called
energy gaps or band gaps. It means there are no electron states in these regions [Ref.2].
We will talk more detailed about this later.
1.2 Surface states
To discuss the electronic properties at the surface of metals we must have an
understanding of the surface. In general the surface is the last few atomic layers of the
solid at the nearby vacuum. In this region, the atomic geometry may depart from that of
the bulk, and the effective potential seen by the electrons changes from that characteristic
of the bulk to the constant vacuum level. In the direction parallel to the surface, however,
we have a periodic potential similar to the bulk. In dealing with wave functions in solids,
the problem is simplified by considering an infinite crystal or else a part of an infinite
crystal. This is done so as to deal with wave functions which satisfy simple boundary
conditions. For an idealized one-dimensional crystal in which the atomic fields are
represented by square potential (The Kronig-Penny model) this simplification introduces
certain features of the surface of the crystal. With this consideration Tamm found that
one surface state was possible for each energy gap between the ordinary allowed bands of
energies. In other words there were energy levels whose wave functions were localized at
the surface of the crystal. These have been found and probed by different experimental
techniques. The two most used probes of the surface spectrum are ultraviolet
photoemission spectroscopy and inelastic low-energy electron scattering. Both techniques
derive their surface sensitivity from the fact that the mean free path of the electrons
involved in each is quite short, about 5-10 Å [Ref.1,7].
The solution of Schrödinger equation for energies in the energy gap gives an infinite
number of solutions with complex κ n . Waves associated with these complex wave
vectors can exist just at the surface region. They are normally neglected in dealing with a
bulk solid since they diverge in amplitude as z goes to either + ∞ or − ∞ . This means
that outside the surface region, in the vacuum, the electron wave functions decay
exponentially and in the bulk because of the energy gap, these waves do not exist (There
is no electron states in the energy gap region). With the matching wave function model
one can solve the Schrödinger equation at the surfaces, which means that the wave
functions and their derivatives must match to each other. With this method and depending
on the vacuum potential model, one or more surface electron states at the surface can be
calculated. [Ref. 1,2,5,6,7].
1.3 Lattice and image techniques
With the discovery of X-rays, it was soon possible to image crystals, which are built
of a periodic array of atoms. Studying crystal structure through the diffraction of X-rays,
electron and neutron beams or other techniques shows that the crystals are made of a
periodic lattice planes. The Bragg law is one of the consequences of this periodicity of
the lattice.
2d ⋅ sin θ = nλ
(1-4)
Where θ is measured from the plane, λ wavelength and d spacing of parallel atomic
planes [Ref.2]. A crystal is often not well-ordered and defects, dislocations and disorders
in the crystal structure can affect its electronic and mechanical properties. The surface of
solids exhibits interatomic distances and physical properties different from the bulk.
There exist also specific properties associated to steps at a surface. Surfaces with steps
have a practical interest too. It is generally easier to study the specific properties of steps
on a surface that presents a regular arrangement of steps, i.e. on a vicinal surface. In
general, a crystal surface cut at a small angle to a symmetry direction is called a vicinal
surface. Such a miscut surface consists of terraces of the symmetry plane, separated by
monatomic steps running across the sample in a preferred direction dictated by the cut.
(Fig. 1.1) Structure of the Cu (211) vicinal surface.
The surface consists of three-atom (111) terraces separated by single-atom (100) steps.
Due to the importance of these structures, the dynamics of stepped surfaces have
been the subject of several experimental investigations using modern surface-science
techniques within the last several years.
Scanning tunneling microscopy is a technique developed in the eighties and allows
imaging solid surfaces with extraordinary resolution. The operation of a scanning
tunneling microscope (STM) is based on the so-called tunneling current, which starts to
flow when a sharp tip approaches a conducting surface at a distance of approximately one
nanometer. The tip is mounted on a piezoelectric tube, which allows tiny movements by
applying a voltage at its electrodes. Thereby, the electronics of the STM system control
the tip position in such a way that the tunneling current is kept constant while it is
scanning a small area of the sample. This movement is recorded and can be displayed as
an image of the surface topography. It should be noted, however, that STM images not
only display the geometric structure of the surface, but also depend on the electronic
density of states of the sample. Under ideal circumstances, the individual atoms of a
surface can also be resolved and displayed [Ref.22-26].
(Fig.1-1) Here is a STM image showing iron atoms adsorb on a copper (111) surface forming a so called
"quantum corral” in a very low temperature (4K). Actually, the image shows the form of the local density
of electron states in a box of the iron atoms. The rings in the quantum corral show the corrugations in the
density of states.
Although the STM itself does not need vacuum to operate (it works in air as well as
in liquids), ultrahigh vacuum is required to avoid contamination of the samples from the
surrounding medium.The scanning tunneling microscope (STM) is widely used in both
industrial and fundamental research to obtain atomic-scale images of metal surfaces. It
provides a three-dimensional profile of the surface which is very useful for characterizing
surface roughness, observing surface defects, and determining the size and conformation
of molecules and aggregates on the surface.
In following chapters we will discuss more details about crystals, image techniques,
the local density of states, electronic properties of surfaces and so on, but the
investigation of electronic properties at the surfaces will be the main object in this essay.
Chapter 2
Crystal structure
The beauty and symmetry of crystals have fascinated people for thousands of years.
Somehow, crystals seem different from more ordinary matter. However, most solid
objects are made of many tiny interlocking crystals. It is only occasionally that the
individual crystals are large enough to catch our attention. As we know the crystal form
develops as identical building blocks are added continuously. The building blocks here
are atoms or groups of atoms. However, the atoms in a crystal form a regular repeating
pattern called the crystalline lattice. In this chapter we will study the crystal structure and
the techniques to image and discover crystal structure in three dimensions.
2.1 Crystal structure
We begin by considering a crystal and develop our knowledge about the solid
crystal structure and the electronic properties. The systematic study of solid state physics
began with the discovery of X-ray diffraction by crystals and the publication of a series
of simple calculations of the properties of crystals and of electrons in crystals. The
diffraction work proved with certainty that crystals are built up as a periodic array of
atoms or groups of atoms. An ideal crystal results from the infinite repetition of identical
groups of atoms. Each of these groups are called “basis”, and the set of mathematical
points that the basis is attached to is called “lattice”.
There are 14 distinct types of lattices in three dimensions (3D) which are grouped in
seven different systems, triclinic, monoclinic, orthorhombic, tetragonal, cubic, trigonal
and hexagonal. There are three lattices in the cubic system: the simple cubic (sc) lattice,
the body-centered cubic (bcc) lattice and the face-centered cubic (fcc) lattice (cubic
closed-packed). The conventional cells of these cubic space lattices can contain one or
more lattice points, for example a sc lattice cell contains only one lattice point, but the
bcc lattice cell has two lattice points in the conventional cell and four points for a
conventional cell of fcc lattice [Ref.2]. Fig.3-1 shows the conventional cells of these
cubic lattices.
simple cubic (sc)
body-centered cubic (bcc)
face-centered cubic (fcc, ccp)
(Fig.2-1)
2.2 Defects and dislocations
It is useful to think about solids in terms of a regular repeating pattern of planes of
particles. But it is important to recognize that solids are seldom perfectly ordered, to be
exact; defects and dislocations are some of the usual causes of disorder in crystals. Some
basic mechanisms for introducing a point defect into the structure of a solid are:
•
Vacancies are sites that are usually occupied by an atom but now are unoccupied.
If a neighbouring atom moves to occupy the vacant site, the vacancy moves in the
opposite direction, to the site which used to be occupied by the moving atom. The
stability of the surrounding crystal structure guarantees that the neighbouring
atoms will not simply collapse around the vacancy. (In some materials,
neighbouring atoms actually move away from a vacancy, because they can better
form bonds with atoms in the other directions.)
•
Interstitials are atoms which occupy a site in the crystal structure at which there is
usually not an atom. They are generally high-energy configurations. Small atoms
can in some crystals occupy interstices without high energy penalty, such as
hydrogen in palladium.
(a)
(b)
(Fig.2-2) schematically shows (a) a vacancy and (b) an interstitial defect
•
Impurities occur because a material is never 100% pure. In the case of an
impurity, the atom is often incorporated at a regular atomic site in the crystal
structure. This is neither a vacant site nor is the atom on an interstitial site and it is
called a substitution defect. The atom is not supposed to be anywhere in the
crystal, and is thus an impurity.
•
Anti-site defects occur in an ordered alloy. For example, some alloys have a
regular structure in which every other atom is a different species. For example
assume that type A atoms sit on the cube corners of a cubic lattice, and type B
atoms sit in the centre of the cubes. If one cube has an A atom at its centre, the
atom is on a site usually occupied by an atom, but it is not the correct type. This is
neither a vacancy nor an interstitial, nor an impurity, but anti-site defect.
•
Complexes can form between different kinds of point defects. For example, if a
vacancy encounters an impurity, the two may bind together if the impurity is too
large for the lattice. Interstitials can form 'split interstitial' or 'dumbbell' structures
where two atoms effectively share an atomic site, resulting in neither atom
occupying the actual site.
•
Dislocations are line defects around which some of the atoms of the crystal lattice
are misaligned. There are two basic types of dislocations, the EDGE dislocation
and the SCREW dislocation. ("MIXED" dislocations combining aspects of both
types are also common).
Edge dislocations are caused by the termination of a plane of atoms in the middle of
a crystal. In such a case, Fig.3-2, the adjacent planes are not straight, but instead bend
around the edge of the terminating plane so that the crystal structure is perfectly ordered
on either side. The screw dislocation is more difficult to visualise, but basically
comprises a structure in which a helical path is traced around the linear defect
(dislocation line) by the atomic planes of atoms in the crystal lattice. It is similar to a
spiral stair case.
(a)
(b)
(Fig2-3) schematic shows structure of a screw dislocation in (a) and an edge dislocation in (b).
Dislocations can move if the atoms from one of the surrounding planes break their
bonds and rebinds with the atoms at the terminating edge. It is the presence of
dislocations and their ability to readily move (and interact) under the influence of stresses
induced by external loads that leads to the characteristic malleability of metallic
materials. Dislocations can be observed using transmission electron microscopy, field ion
microscopy and atom probe techniques. Disinclinations are line defects corresponding to
"adding" or "subtracting" an angle around a line. Basically, this means that if you track
the crystal orientation around the line defect, you get a rotation [Ref.2].
2.3 Crystallography
In the last decade the ability of materials scientists to synthesize materials has
enabled the development of new technologies with applications that range from the
spectacular to the ordinary. For example, X-ray mirrors composed of alternating, thin
(less than 20-nanometer) films of molybdenum and silicon constitute the optics that are
used to produce high-resolution pictures of the sun. Optoelectronic components
composed of alternating atomic layers of different elements are the devices that enable us
to extract information from video compact disks and to generate and detect transoceanic
telephone signals by fiberoptic cables. The alternating, ultra thin layers of cobalt and iron
in new high-density magnetic storage heads, and increasingly miniature microelectronics,
are fundamental constituents of powerful desktop computers, portable laptops, and
pocket-size wireless telephones. The smaller these devices become, the more their
performance depends on the atomic ordering of their component materials. Such details
include the arrangement of atoms in crystal structures and the presence, size, and density
of grain boundaries, impurities, dislocations, or other imperfections.
To identify the effect of atomic arrangement on material performance, materials
scientists use a variety of techniques. X-ray diffraction, transmission electron microscopy
and scanning tunnelling microscopy are some of these techniques. Diffraction occurs as
waves interact with a regular structure whose periodicity is about the same as the
wavelength. The phenomenon is common in physics, and occurs on all length scales. For
example, light can be diffracted by a grating having scribed lines spaced on the order of a
few thousand angstroms, about the wavelength of light. X-rays happen to have
wavelengths on the order of a few angstroms, the same as typical interatomic distances in
crystalline solids. That means X-rays can be diffracted from minerals which, by
definition, are crystalline and have regularly repeating atomic structures. When certain
geometric requirements are met, X-rays scattered from a crystalline solid can
constructively interfere, producing a diffracted beam.
The Transmission Electron Microscope (TEM) allows the user to determine the
internal structure of materials, either of biological or non-biological origin. The energy of
the electrons in the TEM determines the relative degree of penetration of electrons in a
specific sample, or alternatively, influence the thickness of material from which useful
information may be obtained. Because of the high spatial resolution obtained, TEMs are
often employed to determine the detailed crystallography of fine-grained, or rare,
materials. Thus, for the physical and biological sciences, TEM is a complementary tool to
conventional crystallographic methods such as X-ray diffraction.
The scanning tunnelling microscope (STM) provides a picture of the atomic
arrangement of a surface by sensing corrugations in the electron density of the surface
that arise from the positions of surface atoms. A finely sharpened tip usually made of
tungsten wire is first positioned within 2 nanometers of the specimen by a piezoelectric
transducer, a ceramic positioning device that expands or contracts in response to a change
in applied voltage. This arrangement allows us to control the motion of the tip with
subnanometer precision. At this small separation, as explained by the principles of
quantum mechanics, electrons "tunnel" through the gap, the region of vacuum between
the tip and the sample. If a small voltage (bias) is applied between the tip and the sample,
a net current of electrons (the "tunnelling current") flows through the vacuum gap in the
direction of the bias. For a suitably sharpened tip (one that terminates ideally in a single
atom) the tunnelling current is confined to a channel with a width of a few Angstroms.
The remarkable spatial resolution of the STM is due to this lateral confinement of the
current [Ref.22-25].
(Fig.2-4) an image of the STM technique. The probe tip as held by tripod, which consists of three
piezoelectric cylinders that expand or contract in the directions (x, y, z) shown to displace the tip.
In this context, the STM offers a new opportunity for direct diagnosis of how the
processing conditions affect the atomic details of surfaces. This combination of surface
diagnostics is used to study the structural development of metal surfaces and electron
density for different appearances which is the central topic in this essay.
2.4 Surface crystallography
The surface of a crystalline solid in vacuum is generally defined as the few
outermost atomic layers of the solid that differ significantly from the bulk. The surface
may be entirely clean or it may have foreign atoms deposited on it or incorporated in it.
However, in this essay we consider that the surface of solid, particularly metals, is clean.
For a better understanding of a solid surface we need knowledge about atoms and their
place (which atoms and where). Actually it is the geometrical arrangement of the surface
atoms that determines the electronic and magnetic properties at the surface. The bulk
structure problem is resolved by x-ray diffraction but unfortunately the extremely large
penetration depth limits their routine use for surface crystallography. A number of
techniques have been developed for surface-specific structural examination. Some of
these common techniques will be discussed below.
The surface structure is in general periodic in two dimensions (but needless to say
there is no periodicity in the third direction). At a surface, the electrons are free to
rearrange their distribution in space to lower their energy. The net force on the ions points
primarily into the crystal and a contraction relaxation of the surface plane occurs until
equilibrium is re-established. In this point an STM image can thus give a complete
picture of the local density of states. The local density of states (LDOS) represents the
amount of electrons that exist at specific values of energy. Keeping the gap distance
constant, measuring the current change with respect to the bias voltage can probe the
LDOS of the sample. Moreover, changing the polarity of bias voltage can get local
occupied and unoccupied states. In Fig.2-5, when the tip is negatively biased, electrons
tunnel from the occupied states of the tip to the unoccupied states of the sample. If the tip
is positively biased, electrons tunnel from the occupied states of sample to the
unoccupied states of the tip.
(Fig.2-5) a demonstration of STM technique (a) with positive biased tip that the electrons tunnel from the
sample to the unoccupied states of the tip and (b) negative biased tip with tunnelled electrons from the tip
to the unoccupied states of the sample
Surfaces of planes nominally of high indices may be built up of low index planes
separated by steps one or two atoms in height (Fig.1-1). Such terrace-step arrangements
are important in evaporation and desorption because the attachment energy of atoms is
often low at the steps and at kinks in the steps. The chemical activity of such sites may be
high. Double and triple beams of diffraction in LEED experiments may detect the
presence of periodic arrays of steps. Long experience with diffraction methods in the
bulk suggests a search for a similar methodology at the surface. As always, a diffraction
experiment designed for crystal structure analysis requires a probe with de Broglie
wavelength less than typical interatomic spacing, for example about 1Å, which gives a
2
h
kinetic energy about E = λ
≅ 150eV . Electrons with energies in the range 20-500
2m
eV that is elastically backscattered from the crystal surface will form a diffraction pattern
that is the Fourier transform of the surface atom arrangement. This forms the basis for
“low energy electron diffraction” (LEED) from solid surfaces.
( )
(Fig.2-6)An image of LEED techniques
An image of the low energy electron diffraction, LEED in Fig.3-5 shows
schematically how this technique works. Electrons from an electron gun diffract off the
sample and head back towards the electron gun and the grids surrounding it. The two
middle grids are set at energy slightly less than the electron beam to prevent any
inelastically scattered electrons from reaching the screen. The screen is biased positive
between 0 and 7 keV accelerating the electrons towards it. The screen is coated in
phosphor, which glows green when electrons hit it. The resulting LEED pattern is viewed
from the rear end of the apparatus.
Chapter 3
One-dimensional band theory
and surface states
The theory of surface states has been studied for many years. Tamm was the first to
consider the wave function for a semi-infinite crystal with a simplified piecewise constant
periodic potential (Kronig-Penney). He found that it was possible to have energy levels
whose wave functions were localized at the surface of the crystal and one surface level
was possible for each energy gap between the ordinary allowed bands of energies.
Shockley followed Tamm’s consideration and found that there are more electronic states
localized at the surface region as a consequence of the vacuum potential model [Ref.5].
Both of them used the method of matching the wave function and its derivative at crystal
surface.
We will make considerable use of simple methods to exhibit the most characteristic
features of surface electronic structure. We know that there are different interactions
between atoms (ions and electrons). The Hamiltonian that describes the surface electronic
structure contains three terms; the kinetic energy of electrons, the ion-electron attraction
and the electron-electron repulsion, ignoring ion motion.
= Kinetic energy + ion-electron attraction + electron-electron repulsion
(3-1)
Here we are going to solve this problem within a simple one-dimensional nearly-free
electron model which neglects the electron-electron interaction. The effective potential
includes only the ion cores and an image surface barrier [Ref.1].
3.1 The jellium model and surface electronic structure
In this chapter I will give a basic solution of Schrödinger equation in one
dimension. While we must assume that we have at least a passing familiarity with
common concepts and calculation techniques in the bulk solids, a few general concepts
related to surface electronic structure will be introduced below.
A basic concept that we already mentioned is the surface region, which refers to the
volume of space containing the last few atomic layers of the solid. In this region the
atomic geometry and the effective potential may not be the same as in the bulk. If we
know the position of all atoms (almost) we get the Hamiltonian that describes the surface
electronic structure as a function of the kinetic energy, ion-electron attraction and
electron- electron repulsion. For present purpose, we rephrase the exact solution to the
electronic structure problem in term of the “self-consistent “calculation. Self-consistency
involves redistribution of charge throughout the cluster until a minimum energy is
reached, thereby hopefully producing an exact reproduction of the charge distribution in
an equivalent real system. In this model, which can be used for the case of simple metals,
we use an approximation that replaces the ions by a uniform semi-infinite positive charge
background of density. This method is called the jellium model and was used for the first
time by Bennett, Duke and Smith [Ref.1&6].
The positive background charge density, ñ,
often is expressed in term of an inverse sphere
volume
(4 ⋅ π 3 )⋅ r
3
s
= 1 . Typical values of rs
ñ
range from about two to five. (N.D.Lang &
W.Kohn)
(Fig.3-1) self-consistent charge density near metal surface for
(Uniform positive background model)
rs = 2 and rs = 5
The jellium description of a metal surface is a one- dimensional model that
neglects the details of the electron-ion interaction and emphasizes the nature of the
smooth surface barrier.
3.2. One dimensional band theory
The one-dimensional band structure approach to surface electronic structure emphasizes
the lattice aspects of the problem and simplifies the form of the surface barrier. The basic
argument of the band structure models is the influence of a boundary condition for the
Schrödinger equation that reflects the presence of a free surface. These boundary
conditions lead to the existence of surface states for the free-electron model.
3.2.1 Solution of the Schrödinger equation in the bulk
Now it is time to find a basic solution of Schrödinger equation that leads to an
understanding of band structure in solids. The one-dimensional nearly free-electron
model gives:
−
d2
+ V ( z ) ψ ( z ) = Eψ ( z ) ,
dz 2
(3-2)
Here V ( z ) is a weak periodic pseudopotential,
V ( z ) = −V0 + 2V g cos gz ,
(3-3)
and g = 2π
is the shortest reciprocal lattice vector of the chain. We know the solution
a
of these problems especially in the bulk which we have already seen. In the free electron
model ( V g = 0 ) the allowed energy values are distributed continuously from zero to
infinity and in the case of nearly-free electron model, as we have seen (C.Kittel) the band
structure of a crystal can be explained because the band electrons are treated as perturbed
only weakly by the periodic potential of the ion cores.
From before we know that the wave function of an electron in a periodic potential
may be expressed as a Fourier series summed over all values of the wave vectors
permitted by boundary conditions. For our purpose, however, we consider a superposition
of two plane waves, which is the most important part of the Fourier series at the first
Brillouin zone boundary [Ref.1&2]:
ψ (z ) = α ⋅ e ikz + β ⋅ e i⋅(k − g ) z ,
[
(3-4)
]
Substituting this function into our Schrödinger equation leads to;
2
−d
− V0 + 2V g cos gz ⋅ (α ⋅ e ikz + β ⋅ e i (k − g )z ) = E ⋅ (α ⋅ e ikz + β ⋅ e i (k − g )z ) ,
(3-5)
dz 2
e igz + e −igz
If we use, cos gz =
and shift the zero of the energy scale so that it coincides
2
with − V0 , then we have;
[(k
2
)
]
− E ⋅ α + β ⋅ V g e ikz +
{[(k − g )
2
]
}
− E ⋅ β + α ⋅ V g e i (k − g ) z
(3-6)
+ α ⋅ Vg ei (k + g )z + β ⋅ Vg ei ( k − 2 g )z = 0 ,
The two last terms in (3-6), are placed in the second Brillouin zone and are insignificant
for this solution. Therefore we can write our equation in its secular shape;
k2 −E
α
0
=
0
k −g −E β
Vg
(
Vg
)
(Central equation),
2
(3-7)
This equation has none-trivial solutions only when the determinant of the matrix is zero.
If the wave vector k should be written in terms of its deviation from the Brillouin zone
boundary, k = g + κ then we have the solution for the energy eigenvalues;
2
(k
2
) (
−E ⋅ k −g
g
2
2
+κ
2
E= g
)
2
− E − Vg = 0
2
g
−E ⋅
2
(
+ κ 2 ± g 2 ⋅ κ 2 + Vg
2
g
2
)
1
2
(3-8a)
+ E ± g 2 ⋅ E + Vg ,
(3-8b)
2
2
− E − Vg = 0
,
2
And for the wave vectors,
κ2 =
2
+κ − g
( )
For the one-dimensional semi-infinite nearly-free electron model E κ 2 shows the
( ) , is a continuous
familiar energy gap at the Brillouin zone boundary. However, E κ
function of κ 2 if one allows κ to take imaginary values.
(FIG.3-2) E
2
(κ ) for the one-dimensional semi-infinite nearly free electron model [Ref.1]
2
Now when we have the energy eigenvalues, it is obvious that the wave function would be
found by solving the “Central equation”:
(k
2
)
2
− E ⋅ α + Vg ⋅ β = 0
(
Vg ⋅ α + k − g
)
2
(3-9a)
− E ⋅ β = 0,
(3-9b)
In these equations α , β must be chosen so that the both (3-9a) and (3-9b) are valid. The
result is simplified if we represent α = A′ ⋅ e − iδ and β = A′ ⋅ e iδ , there A′ is a constant and
δ is ”phase shift” between the forward-moving waves and backward-moving waves.
Actually δ varies smoothly in energy band gap between π
β
α
=
E−k2
Vg
exp(2iδ ) =
2
and zero.
E−k2
,
Vg
ψ κ (z ) = A′ ⋅ e −iδ ⋅ exp i ⋅ g 2 + κ ⋅ z + A′ ⋅ e iδ ⋅ exp i ⋅ κ − g 2 ⋅ z
, with A = A′
2
we
have;
ψ κ (z ) = A ⋅ e iκz ⋅ cos
1
⋅ g ⋅ z −δ ,
2
(3-10)
This wave function is a product of an exponentially decaying function for
imaginary κ and a periodic function and increases for positive z and decreases for
negative values of z. This is acceptable since it will be matched (at z = a ) onto a
2
function that describes decay of the wave function in the vacuum. In this way we get a
wave function describing a surface state.
3.2.2 Boundary conditions at the surface
We have already seen that the wave function for the bulk increases when z goes to
positive values, but it must also match the vacuum function, which is a decreasing
exponential wave function. This means that these two functions and their derivatives
must be continuous at the surface. For the last layer of the crystal, this point counts to be
at the half of the crystal lattice constant a ;
ψ (z ) = Ae i⋅κ ⋅z cos
ψ (z ) = Ce − q⋅z
1
⋅ g ⋅ z −δ
2
a
2
a
z
2
(3-11a)
z
, where q 2 = V0 − E ,
(3-11b)
If the logarithmic derivative of ψ κ ( z ) can be made continuous at z = a , an electronic
2
state exists that is localized near the surface of the lattice chain. The energy of this
surface state lies in the bulk energy gap.
(Fig.3-3) One-dimensional semi-infinite lattice model potential
and an associated surface state
dψ in
ψ in
( 2) =
dz a
dψ out
ψ out
(ψ )′ (a ) = (ψ )′ (a )
2
2
( 2)
in
dz a
out
ψ in
ψ out
(3-12)
1
⋅ g ⋅a −δ
4
i⋅κ ⋅a
1
g
1
⋅ cos ⋅ g ⋅ a − δ − ⋅ e 2 ⋅ sin ⋅ g ⋅ a − δ
4
2
4
e
i ⋅κ ⋅ e
i ⋅κ ⋅a
2
i ⋅κ ⋅a
2
⋅ cos
−e
− q ⋅a
2
q⋅e
− q⋅a
2
D1 D 2
A
0
,
⋅
=
D3 D 4 C
0
⋅
A
0
=
C
0
(3-13)
This secular equation has solution if and only if, det
D1 D 2
=0
D3 D 4
D1 ⋅ D 4 − D 2 ⋅ D3 = 0
e
−
q ⋅a
2
⋅e
i ⋅κ ⋅a
2
1
⋅ q ⋅ cos ⋅ g ⋅ a − δ + i ⋅ κ ⋅ e
4
i⋅κ ⋅a
2
⋅ cos
1
g
1
⋅ g ⋅ a − δ − ⋅ sin ⋅ g ⋅ a − δ
4
2
4
=0
(3-14)
This equation determines E. Note that κ , q and δ are functions of the energy. For
each κ and energy E , the electron states will be given by the solution of Schrödinger
equation for the bulk potential (periodic in direction normal to the surface). The electron
states exist if the wave functionsψ κ ( z ) and their derivatives are continuous at the surface.
We should not forget that there are an infinite numbers of solutions for complex κ but
just those that satisfy the continuous boundary conditions at the surface are important for
our case.
.
(Fig.3-4) the wave function associated to surface, z=0 corresponds to solid atomic layer position, a=2.08Å
for Cu (111)
3.2.3 Image potential
If an electron has enough energy to exist in the neighbourhood of a metal surface,
the presence of the electron affects the charge distribution on the metal surface. A
positive charge will be distributed along the metal surface with the same magnitude. To
calculate the electrostatic interaction force produced by the electron and the positive
charge distribution we use one positive charge with the same magnitude as the positive
charge distribution and with the same distance from the surface, and calculate the
electrostatic force between these two point charges. It means that the electron generates a
hole in the crystal at the same distance from the surface in opposite direction. This
positive point charge is called the image charge. If the electron is at a distance z from the
surface, the distance between the electron and its image will be 2 z . At this case the
electrostatic force of attraction between the two charge particles e + and e − will be;
1 e− ⋅ e+
F=
,
(3-15)
4πε 0 d 2
Here d = 2 z is the distance between the electron and its image. Then,
1
e2
1 e2
F=
,
(3-16)
F=
4πε 0 (2 z )2
16πε 0 z 2
The image potential energy is therefore the integral of the electrostatic force from infinity
to the distance z from the surface, i.e.
∞
∞
e2
1
V ( z ) = F ( z )dz
V (z ) =
dz
16πε 0 z z 2
z
V (z ) = −
e2
,
16πε 0 z
1
(3-17)
(Fig3-5) Field lines for an electron and its image in the crystal
Then the electrons outside the surface would be affected by a constant vacuum
potential plus the image potential (3-17). This is a potential that changes with the distance
from the surface and for large distances approaches to a constant vacuum level (zero).
3.3 Numerical calculations and results
To solve the Schrödinger equation (3-2) one needs to know the
potential, V ( z ) = −V0 + 2V g cos gz because a detailed calculation of this equation would
require the knowledge of the bulk structure. In this purpose I used copper (Cu (111)) as
an example. Cu (111) is one of the most studied metal surfaces. This surface represents a
simple kind of potential and on this surface the energy gap is located below the vacuum
level and the first state (actually there is just one state if the vacuum potential barrier is
high enough) is at about the middle of this gap. V0 = 11,562eV , V g = 2,5eV and
a = 2.08 nm are relevant parameter values for the copper (111).
In the vacuum, however, the potential is constant and the wave function is an
exponentially decaying function. All details about this wave function can be calculate by
(3-11b) and (3-12) i.e. matching the wave functions and their derivatives at the surface
( z = a ).
2
level
Potential (eV)
Gap (eV)
Energy ( eV )
1
8,003
V0 = 11,562eV
E midgap = 8,797eV
2V g = 5eV
E gb = 6,300eV
E gt = 11,293eV
Tab.3-1 the only Tamm state for the square potential model.
the gap and
E midgap is the energy of its middel
E gb and E gt are the lowerand upper edge of
(Fig.3-6) square potential and the associated wave function
Fig 3-6 shows the wave function associated with the square potential model. This
model potential was considered by Tamm and therefore these surface states are called
“Tamm states”. The charge density localization shows that these states are localized
mainly at the surface atomic layer. For a square model potential, irrespective of vacuum
level, I always got just one surface state which agrees with the Tamm state. In this case
there is one electron state whose energy lies at about the middle of the band gap. This is
shown in Tab.3-1. The higher the vacuum level, the closer the energy level is to the
middle of the band gap.
With a image potential that is the result an electron located in the neighbourhood of
a surface, the wave function changes significantly compared to our first case. In this case
the potential varies with distance and therefore to calculate the wave function we use
numerical solution [Ref.21]. This numerical solution gives us the shape of the wave
function in the vacuum and as I previously mentioned these two wave functions (twoplane wave function and function in the vacuum) must be continuous at z = a (i.e.
2
logarithmic derivative). Tab.3-2 shows the values of the image states for the vacuum
model potential.
level
1
Energy ( eV )
7,407
Potential
V0 = 11,562eV
Gap
E midgap = 8,797eV
2
11,161
2V g = 5eV
E gb = 6,300eV
E gt = 11,293eV
Tab. 3-2 the Tamm state (level 1) and the image state (level 2) of the image potential model
(Fig.3-7) image potential and the associated wave functions
Fig.3-7 and 3-8 shows the wave functions and the probability amplitude of the n = 1
surface state and n = 2 image state obtained with the use of the image potential. In these
figures we see a corresponding potential energy diagram for an electron in front of a Cu
(111) surface. For this surface the energy gap lies below the vacuum level and the surface
state is located almost in the middle of the band gap but the image state (the only image
state) is at the top of this gap just below the upper edge of the gap. The maximum of the
probability density for the image potential state shown in Fig.3-8 is several Angstrom
away from the surface. If the vacuum level is high and the energy gap lies below the
vacuum level, we have one electron state that is similar the to the one of the square
potential model but this state has lower energy, 7,523 (eV) compare to the surface state of
the squre potential model, 8,003 (eV) (Tab.3-3).
level
1
Energy ( eV )
7,523
Potential (eV)
V0 = 12,00eV
Gap (eV)
E midgap = 8,797eV
2V g = 5eV
E gb = 6,300eV
E gt = 11,293eV
Tab.3-3 The Tamm state for high image potential case
For the case that the upper edge of the energy gap lies above the vacuum level, there are
many electron states with energies just below the vacuum level. In this case, which is
shown in Tab.3-4, the image states are localized far outside the surface in the vacuum.
A surface state near the centre of the gap is highly localized in the surface region,
these states as we already seen are Tamm states. However one with its energy near the
edge of the gap will have a lot of its weight in a slowly decaying tail in the vacuum. Thus
a state merging into the edge of the gap might gradually fade away in terms of its
amplitude at the surface, rather than unexpectedly vanish [Ref.7-9].
(Fig.3-8) the probability amplitude of the n=1 and 2 image states obtained
from the image potential calculation, z=0 corresponds to the
surface atomic layer position.
level
1
Energy ( eV )
7,318
Potential
V0 = 11,25eV
Gap
E midgap = 8,797eV
2
10,845
2V g = 5eV
E gb = 6,300eV
3
11,096
4
5
6
7
8
9
10
11,164
11,192
11,206
11,214
11,219
11,222
11,225
E gt = 11,293eV
Tab. 3-4 the Tamm state (level 1) and the nine first image states of the image potential model, because the
upper edge of the energy gap lies above the vacuum level there are more surface states.
Chapter 4
Two-dimensional lattice
electronic structure
In the previous chapter we obtained a type of wave function that could exist for
particular energies in the gaps for a given imaginary κ . These wave functions consist
purely of decaying waves in the bulk, and thus, are confined to the surface. The existence
of one or more surface states in an energy gap depends on the detailed nature of the
potential in the surface region. In this chapter we investigate the electronic structure in
direction parallel to the surface. We will investigate the electronic structure of wellordered, clean and stepped Cu (111) surface as a function of wavevector by means of the
scanning tunnelling microscopy.
4.1 Theoretical descriptions
The behaviour of a two-dimensional (2D) electron gas is receiving general attention
for both fundamental and technological reasons. A 2D electron gas can be found at metalsemiconductor interfaces, in artificially layered semiconductors interfaces, or in some
organic charge-transfer salts. The electrons occupying the sp-like surface state band in
close-packed noble metal surfaces (Cu, Ag, Au) are confined normal to the surface
between the vacuum barrier and the crystal band gap. They effectively behave as free
electrons in two dimensions, providing a useful ground for testing the properties of the
2D electron gas. In fact, electron scattering processes at point defects, steps, and isolated
atoms result in the formation of standing waves that can be visualized by means of the
scanning tunnelling microscope (STM) [Ref10-11].
As a consequence of the 3D periodicity of a bulk solid, electrons eigenstates can be
classified by a Bloch wave vector k . The presence of a surface destroys periodicity in the
direction normal to the surface, but periodicity parallel to the surface remains. The
electron states in the presence of a surface can thus be characterized by a 2D Bloch wave
vector k p . To calculate the electron movement in direction parallel to the surface we will
follow our solution of the Schrödinger equation with the wave function:
(
)
ψ (r ) = C exp ik p ⋅ rp ⋅ψ n (z ) , where
Etot = E n + E p
k p = k x ⋅ xˆ + k y ⋅ yˆ , and
Etot = E n +
2
2m
(k
2
x
+ ky
2
)
rp = x ⋅ xˆ + y ⋅ yˆ
(4-1)
In this wave function,ψ n ( z ) is defined as (2-10) and describes motion along z axis. In our
()
2D electron gas model, not only the E k relation but also the density of states is of
particular interest.
4.2 Steps
We have already seen that the atomic arrangement can be seen by the scanning
tunnelling microscopy, STM, (Fig.2-3). By sensing the corrugations in the electron
density of the surface, Shockley type surface states have been probed by this method that
shows the quantum mechanical probability density distribution of electrons of metals
(Fig.3-5). These corrugations in the electron density introduce a well-defined theoretical
concept,” the Local Density of States”. In this context the present work studies the
influence of steps on the electron states density at the surfaces. In order to study a sample
with well-defined steps I have investigated Cu (111) surface. Copper is an ideal system
that has been studied a lot and its vicinal surface distributes steps regularly. At a vicinal
surface the average surface normal is tipped away from a high symmetry low-index
direction, and the microscopic structure consists of an ordered array of atomic steps
which separate low-index terraces.
(Fig.4-1) Schematic diagram of ideally terminated Cu (410) surface
in plan and perspective views. The Cu atom row at the down-step
edge of each (100) terrace is shown shaded darker for clarity.
[Ref.20]
The microscopic atomic structure near a step edge is very different from that found
in the bulk or at a low-index surface, and as a consequence one expects a substantial
redistribution of the valence electronic charge density near the step edges. This can play
an important role in modifying the local dynamical properties and the chemical reactivity.
Therefore, stepped surfaces are important systems to study experimentally and
theoretically. In fact, the influence of steps on the surface chemistry can be so large that
the macroscopic reactivity of many real surfaces is governed by dynamical phenomena at
steps. Understanding these processes at the microscopic level can lead to control of
surface dynamical phenomena through control and design of the surface structure.
Stepped surfaces can be produced by purposely miscutting a crystal forming a
vicinal surface. Usually reconstruction will create a surface characterized by wide
terraces separated by monatomic steps. In some cases the steps may be two atomic layers
high. In extreme cases, height of these steps may become of comparable size to the
terraces and the surface forms relatively large facets (areas of low index planes).
(Fig.4-2)
In the case of the FCC surface shown above (Fig.4-2), the surface is composed of
terraces 7 rows wide (including the atoms forming the step edges) which are parallel to
the simple low index (100) surface. The steps are 1 layer high and have a (110) structure.
Therefore the surface is designated as FCC (S)-[7(100)*(110)] often shortened to
7(100)*(110). If the terrace were 10 atoms wide and the step 2 layers high the designation
would be (S)-[10(100)*2(110)]. Remember that this is only a model: the real surface will,
on average, show this type of structure but locally there will be regions with a range of
terrace widths.
4.3 Calculation method
In this essay I have investigated two important questions. First, is there any surface
state contained in the band gap and if so, at what energy does it occur? Second, how is
the charge distributed? For nearly free electrons (which we consider in this essay) the
first question was answered by Shockley for the gaps at the zone boundary (In chapter
three, we found out that there is at least one electron state which lies in the middle of the
gap). The second question has also been investigated here and we will compare our
results to the other theoretical and experimental studies.
4.3.1 The density of states
We already know that in the ground state of a system of free electrons, occupied
orbitals of the system at 0 K fill the Fermi sphere. The number of orbitals per unit energy
is called the density of states. This is actually the density of one-particle states or density
of orbitals. Density of single-particle states as a function of energy, for free electron gas
is shown schematically in Fig.4-3. In three-dimensional free-electron gas the density of
1
2
states is a function of ε , but it is constant for the two-dimensional case. Now if a threedimensional free-electron gas confined in one dimension and free in the other two, the
density of states will be constant between energies that the electrons satisfy the particle in
a box conditions in the third dimension. If a two-dimensional electron gas is confined in
one dimension, the density of states is on average constant. The peaks appear when the
electrons have enough energy to satisfy the conditions of a particle in a box as in first
case. Between the peaks the density of states decreases like in one dimensional case
i.e. ε
−
1
2
.
(Fig.4-3) Density of states for free electron gas, for 3D, 2D and 1D
The general definition of density of states (DOS) is:
D( E ) =
δ (E − E n ) ,
(4-2)
n
What the above equation tells us is the average number of states over the entire solid.
Notice that the density of states changes on an atomic scale. Distribution of eigenfunction
2
amplitudes, ϕ n ( x ) characterizes the spatial fluctuations of well defined energy levels. It
is simply the total density of states for the system weighted by the squared magnitude of
the wave functions in an energy interval at the spatial point in question. The definition of
the local density of states (LDOS) in the case of one dimensional free-electron gas is:
D( E , x ) =
ϕ n ( x ) δ (E − E n )
2
(4-3)
n
4.3.2 The Green’s functions
In basic mathematical physics Green’s functions play an important role for the
solution of linear ordinary and partial differential equation, and are a key component to
the development of boundary integral methods [Appendix & Ref.3]. The Green’s
function for electrons is defined as:
ϕ n (x )ϕ ∗ n (x ′)
g ( E , x, x ′ ) =
,
(4-4)
E − En
n
This is a function of energy and distance and is a sum of all the eigenstates. E is the
enery variable and E n is the energy of the state n. Actually g (E , x, x ′) is the amplitude for
an electron with energy E , released at point x ′ , to reach point x . g (E , x, x ′) is in general
complex. Now if we let E have a tiny imaginary part, ∆ , self-energy, we can write:
ϕ n (x )ϕ n ∗ (x ′)
= x′
Im[g (E + i∆, x, x ′)] = Im
x

→
n E + i∆ − E n
=
ϕ n (x )
Im
∆ →0
n
2
E + i∆ − E n
ϕ n (x ) δ (E − E n ) (4-5)
= −π
2
n
Note that the Green’s functions in our solutions are calculated at x = x ′ .
(Fig.4-4) a delta function with imaginary value
In (4-5), ϕ n ( x ) is real and we can show that;
2
Im
E − E n − i∆
1
= Im
=
E − E n + i∆
(E − E n )2 + ∆2
(4-6)
−∆
= −πδ (E − E n )
(E − E n )2 + ∆2 ∆→0
To prove the last step in (4-6), we look at the integral over energy:
∞
−∆
dE
there, (E − E n ) = x
dE = dx then;
2
2
− ∞ (E − E n ) + ∆
∞
∞
−∆
1
dx
dx = −
2
2
∆ −∞
x
−∞ x + ∆
1+
∆
−
∞
x
∆
dx = ∆dt
t=
2
∞
1 ∆dt
dt
=−
= − I −∞∞ arctan t = −π
2
2
∆ −∞1 + t
1
+
t
−∞
(4-7)
We already know that LDOS is the squared magnitude of the wave functions in a small
energy range at the spatial point, then (4-5) becomes;
(4-8)
Im[g (E , x, x ′)] = −π ⋅ D(E , x )
′
Then the local density of states in each point ( x = x ) is:
1
(4-9)
D(E , x ) = − ⋅ Im g (E , x, x ′)
π
4.3.3 Equation of motion
We can deduce an equation of motion for g by letting the (Schrödinger-like)
operator H − E operate on g:
(
)
ϕ n (x )ϕ n ∗ (x ′)
[H − E ]g (E, x, x′) = [H − E ]
n
E − En
(E n − E )ϕ n (x )ϕ n ∗ (x′)
=
=−
ϕ n (x )ϕ n ∗ ( x ′)
(4-10)
E − En
n
The right hand side in (4-10) is a Dirac delta function. To show that, there is a general
theorem in quantum mechanic that states: an arbitrary function ϕ n ( x ) can be expanded in
a complete set of eigenfunctions of Hamiltonian, that is [Ref.4],
(4-11)
f (x ) =
f nϕ n ( x ) ,
n
n
If we choose the eigenfunctions of Hamiltonian to be normalized, and if we take into
account that the eigenfunctions are orthogonal, the expansion coefficients can be
determined by:
+∞
f n = ϕ n ( x ) f ( x )dx ,
∗
(4-12)
−∞
We take:
f ( x ) = δ ( x − x ′) ,
(4-13)
Then the expansion coefficients,
∞
f n = ϕ n ( x )δ ( x − x ′)dx = ϕ n ( x ′) ,
∗
∗
(4-14)
−∞
Then (4-11) gives:
δ ( x − x ′) =
ϕ n ∗ (x ′)ϕ n (x )
(4-15)
n
Now with comparing (4-9) and (4-14) and for any point but x ′ , ( x ≠ x ′ )
H − E g ( E , x, x ′ ) = 0
(4-16)
Then, the most general solution is:
g ± (E , x, x ′) = C ± e ± ikx
(4-17)
These correspond to outgoing waves in both directions from x ′ . At this point, x ′ , g must
be continuous but its derivative is not because of a cusp at x = x ′ that make matching
process difficult. At this point and for the incoming waves,
1. g + (E , x, x ′) x = x′ = g − (E , x, x ′) x = x′
(4-18)
(
x′ +δ
2.
x′ −δ
)
(H − E )g (E, x, x′)dx =
x′ +δ
− δ ( x − x ′)dx = −1
(4-19)
x′ −δ
This means:
x′ +δ
2a.
x ′ −δ
x ′ +δ
2
d2
−
+ V ( x ) g (E , x, x ′) − Eg (E , x, x ′) = −1
2m dx 2
x′ −δ
x′ +δ
2
dg (E , x, x ′)
dg (E , x, x ′)
dg (E , x, x ′)
−
=−
(
x = x′ + ) − (
x = x ′ − ) = −1
2m
dx
2m
dx
dx
x′ −δ
2
2
dg
dg
dg
dg
2m
(4-20)
+ −
− =
+ −
− =1
2
dx
dx
2m dx
dx
With these two conditions, (4-17) and (4-18), we can find the eigenfunctions for our onedimensional problem;
− im ikx′
− im −ikx′
&
C+ =
e
C− =
e
2
2
kn
kn
− im
(4-21)
g ( E , x, x ′ ) =
exp(ik x − x ′ )
k 2
Using (4-19) in (4-8), shows that the density of states is proportional to the inverse wave
1
vector and thus, proportional to
, which agrees with our expectation of one
E
dimensional electron gas from earlier [Ref.2];
D(E , x ) = −
1
π
⋅ Im g (E , x, x ′)
− im
exp(ik x − x ′ )
π
k 2
1
− im
D(E , x ) = − ⋅ Im
π
k 2
D (E , x ) = −
1
⋅ Im
= x′
x

→
(4-22)
4.4 Green’s functions for two-dimensional terrace
The eigenfunctions problem for the electrons between edges at a terrace, that we
already considered as hard walls will be similar to the problem for a particle in a box.
This implies that the wave functions must vanish for y ≤ 0 and y ≥ a with a being the
terrace width. The solutions of the Schrödinger equation in this case are sin kx
and cos kx . The boundary conditions imply that:
n = 1, 2, 3…
(4-23)
kx = nπ
The normalized sine solution is:
ϕn (y) =
2
nπy
sin
a
a
(Fig.4-5) Eigensolutions for particle in a box. (n=1 and n=2)
(4-24)
It turns out that the Green’s function for the two-dimensional terrace can be written as:
nπy
nπy ′
,
(4-25)
G ( x − x ′, y, y ′, E ) =
g n ( x − x ′, E ) ⋅ sin
⋅ sin
a
a
n
g n ( x − x ′, E ) = g n x − x ′, E −
n 2π 2 2
2ma 2
(4-26)
n 2π 2 2
is the energy left for the motion along the x direction.Then the
2ma 2
Schrödinger equation with this function that contains all variables can be written:
Here E −
[H (x, y ) − E ]G(x − x′, y, y′, E ) = −δ (x − x′) ⋅ δ ( y − y′) ,
(4-27)
Where:
H =−
2
∂2
+∂
2
(4-28)
+V ,
∂x
∂y 2
2m
In this case the Green’s function, G ( x − x ′, y, y ′, E ) is a function of both x and y. To get
the density of states from the Green’s function we set x = x ′ and y = y ′ as a
consequence, the local density of states becomes a function of its position at the terrace,y
but it is independent of x (a function of E and y). The local density of states turns out to
be a function of the position at the terrace in a small range at the spatial point;
= x′& y = y ′
(4-29)
→ D( y, E ) ,
G ( x − x ′, y, y ′, E ) x
2
In this context, there is another factor that affects the shape of the local density of
states drastically, namely τ , the lifetime. Lifetime is defined as the time between two
collisions that an electron collides with other electrons and loses its energy until it falls
down to the bulk states. It is the similar for an excited electron when a photon is
absorbed. The electron goes up to an excited state but it goes down eventually after
releasing a photon. The time that electron spends at this excited state is also called
lifetime. These two definitions are actually the same because both are about the time an
electron can be in an excited state. The potential energy that we consider is an imaginary
potential, can be defined as;
V = −i∆ , then;
(4-30)
E − V = E + i∆ ,
Then the equation for G ( x − x ′, y, y ′, E ) with ∆ absorbed in E ( E → E + i∆ ) becomes:
−
n
∂ 2 g n ( x − x ′, E )
−
2m
∂x 2
2
⋅
n 2π 2
nπy
nπy ′
+ E g n ( x − x ′, E ) sin
sin
=0
2
a
a
2ma
2
∂2
2mE n 2π 2
′
(
)
g
x
−
x
,
E
+
− 2 g n ( x − x ′, E ) = 0
n
2
∂x 2
a
The solution, as we have seen before is:
(4-31)
g n ( x − x ′, E ) = C n exp[ik n x − x ′ ]
(
Where, k n = k1 − k 2
2
2
(4-32)
) = 2mE − naπ
2
2
2
2
kn =
2
2mE
2
−
n 2π 2
a2
(4-33)
nπ
2ma 2
2
1. E
2
2
k n is
real
and
g
is
a
propagating
plane
wave
along the step.
nπ
k n is imaginary and g decrease exponentially along the step.
2ma 2
The wave function is continuous at x = x ′ , but not its derivative. This is the same solution
we provided for the one dimensional case. Then the solution for the 2D electron gas is:
− im ikn x − x′
nπy
nπy ′
⋅ sin
⋅ sin
(4-34)
Gn ( x − x ′, y, y ′, E ) =
e
2
a
a
kn
n
Now we can develop our expression for the local density of states (4-9), for the 2D
electron gas:
1
D(E , x, y ) = − Im Gn ( x − x ′, y, y ′, E )
2
2
2
2. E
π
D ( E , x, y ) = −
1
π
Im
n
− im ikn x − x′
nπy
nπy ′
e
⋅ sin
⋅ sin
2
a
a
kn
= x′& y = y ′
x
→
− im
nπy
⋅ sin 2
(4-35)
2
π
a
n kn
In our solution we considered a terrace i.e. we contained the electron in a box with
length a. The system is two dimensional but in this case we have confined the electron in
one direction, it is the reason why we do not get a constant density of states. In following
section we will discuss the origin of these peaks. Fig (4-6) shows a plot of the local
density of states, (4-35) at the stepped Cu (111) surface with terrace width 50Å.
D ( E , x, y ) = −
1
Im
(Fig.4-6) Plot of the local density of states (eqv. (4-35)).
4.5 Results and discussions
In our investigation of electronic structure of stepped surfaces, we began with a
series of approximations and put some conditions to simplify our problem in a sensible
way. Nearly-free and free electron approximation resulted in shape of the band theory
and the free electron gas model for the electrons at the surface. With these
approximations and conditions like clean and well-ordered surfaces, the solution of
Schrödinger-like Green’s function equation, (4-27) gave a two-dimensional wave
function (4-33). Then we were able to find the density of states for this two-dimensional
case.
For a three dimensional free-electron gas that is confined in two dimensions the
density of states is a step function with steps occurring at the energy of each quantized
level for the third dimension (Fig.4-3). In our case, on the other hand, the system (each
step at the surface) represents a two dimensional free-electron gas that is confined in one
dimension therefore the density of states must be an the average constant with peaks at
the energy of each quantized level for the other dimension (Fig.4-6).
Equation (4-34) gives the local density of states at a terrace of a Cu (111) surface. It
means the local density of states (LDOS) for each vicinal surface with terraces at this
direction. Figure (4-7) shows a plot of LDOS for a terrace with width 50Å in direction
(111) but different lifetimes. It does not look like the plot of density of states for the 2D
electron gas, but it is approximately the same. We have already seen that the local density
of states for our case, terrace with hard walls, is an average constant function of energy
and the peaks appear when the electrons satisfy the boundary conditions for a particle in a
box, i.e. energies associated with the quantized level. From (4-35) one can see that the
peaks appear for odd values of n ( n 2 = 1, 9, 25, 49).
(a)
(b)
(Fig.4-7) density of states associated with stepped Cu (111) surfaces at the middle of the terrace with
lifetime, (a); τ = 100 fs and (b); τ = 10 fs
The electrons with energies of the peaks are able to satisfy the conditions for particle
in a box in direction perpendicular to the steps and also the conditions for onedimensional electron gas at the other direction. For energies between the peaks, the local
density of states decreases just like the one-dimensional case. Fig.4-6 shows the lifetimedependence of the local density of states from Cu (111). The longer the lifetime, the
higher the peaks in LDOS diagram.
Photo-electron spectroscopy is a very powerful tool for investigation of the
three dimensional bulk as well as the two dimensional surface bands. Not only the E k
relation but also the lifetimes of the excited electronic states are of particular interest. The
lifetime is of interest as it determines the mean free path of the surface state electrons,
and hence the effective range of surface state mediated interactions. This can be
determined by line width analysis in photoemission spectra. These studies of line widths
and line shapes contain important information of lifetime broadening of the final hole and
electron states, on electron-hole pair excitations and conduction-electron screening in the
bulk and at the surface. The effects of the lifetime on Shockley surface state electron can
also be measured by the scanning tunnelling microscopy, STM. Using low-temperature
STM spectroscopy one can calculate the relation between the geometrical width of the
onset and the imaginary self-energy ( Σ =
= ∆ ). In general, the natural line width is
2τ
2
the inverse lifetime of the photohole. A number of investigations with photoemission
spectra have provided an important picture of the relations between the local density of
states, lifetime and the vicinal faces width. All these investigations lead to that the peaks
width for the vicinal surfaces increases compared to the peak width of the unstepped
surfaces and this increasing is actually of the same order as the lifetime width [Ref.1215]. This is illustrated in Fig.4-7.
()
Summary
Summarizing I can state that the calculations in this paper yield an understanding of
the surface electronic states and the electronic structure of vicinal faces of Cu (111) by
the means of the scanning tunneling microscopy (STM).
For an idealized one-dimensional crystal it is possible to have energy levels whose
wave functions are localized at the surface. These states are called surface states. There is
one surface state for each energy gap between the ordinary allowed bands of energies.
These electron states are called Tamm states. This Tamm state has an energy that lies
almost at the middle of the energy gap and is mainly localized at the surface atomic layer.
The image potential states are generated by a potential well formed by the Coulomb-like
image potential barrier. These image states that are also called Shockley states are
localized in a slowly decaying tail in the vacuum.
I also studied the lateral (in-plane) motion of electrons confined to terraces between
steps on a vicinal Cu (111) surface. The local density of states showed a number of peaks
at energies where electrons can occupy new quantum-well states on a step. I also tested
the influence of the electron lifetime on the local density of states.
Acknowledgements: I am grateful to my teacher and supervisor Dr. Peter Johansson for
many discussions on the subject of solid state physics and critical comments on various
aspects of this work. I would also like to thank the examiner, Dr. Andreas Oberstedt.
Appendix A: Introduction to Green's Functions
Green’s function is a basic solution to a linear differential equation, a building block
that can be used to construct many useful solutions. For heat conduction, for example, the
Green’s Function is proportional to the temperature caused by a concentrated energy
source. The exact form of the Green’s Function depends on the differential equation, the
body shape, and the type of boundary conditions present. Green’s functions are named in
honour of English mathematician and physicist George Green (1793-1841). Green’s
functions play an important role in the solution of linear ordinary and partial differential
equations, and are a key component to the development of boundary integral equation
methods.
Consider a linear differential equation written in the general form
L( x )u ( x ) = f ( x )
(1)
where L(x) is a linear, self-adjoint differential operator, u(x) is the unknown function, and
f(x) is a known non-homogeneous term. Operationally, we can write a solution to
equation (1) as
u ( x ) = L−1 ( x ) f ( x )
(2)
where L-1 is the inverse of the differential operator L. Since L is a differential operator, it
is reasonable to expect its inverse to be an integral operator. We expect the usual
properties of inverses to hold,
LL−1 = L−1 L = I
(3)
where I is the identity operator. More specifically, we define the inverse operator as
L−1 f = G ( x, x ′) f ( x ′)dx ′
(4)
where the kernel G ( x, x ′) is the Green's Function associated with the differential operator
L. Note that G ( x, x ′) is a two-point function which depends on both x and x'. To complete
the idea of the inverse operator L, we introduce the Dirac delta function as the identity
operator I. Recall the properties of the Dirac delta function δ ( x ) are
∞
δ (x − x ′) f (x ′)dx ′ = f (x )
−∞
∞
(5)
δ (x ′)dx ′ = 1
−∞
The Green’s function G ( x, x ′) then satisfies
L( x )G ( x, x ′) = δ ( x − x ′)
(6)
The solution to equation (1) can then be written directly in terms of the Green’s function
as
∞
u ( x ) = G ( x, x ′) f ( x ′)dx ′
(7)
−∞
To prove that equation (7) is indeed a solution to equation (1), simply substitute as
follows:
∞
Lu ( x ) = L G ( x, x ′) f ( x ′)dx ′
−∞
=
∞
LG ( x, x ′) f ( x ′)dx ′
−∞
∞
= δ ( x − x ′) f ( x ′)dx ′
−∞
= f (x )
(8)
Note that we have used the linearity of the differential and inverse operators in addition
to equations (4), (5), and (6) to arrive at the final answer.
Appendix B: Listing of programs used for the numerical
calculations
Calculations of the surface states in one-dimensional band theory
The main program, “test”, is actually the program that uses all the calculated values
in subroutines, functions and modules to identify the boundary conditions, energy levels,
shape of the wave functions in different positions and derive the wave functions
associated with these energies. To derive the wave functions we need the wave vector,
k and the phase shift, δ . These are both variables of energy and are calculated in the
module “kappa_mod”. It is important to know that this program picks up just the
imaginary values of the wave vector which we are most interested. Module “det_mod”
calls these calculated values and uses them to solve the “central equation”, (3-7) for both
the bulk and the vacuum as well. For the vacuum part, we have two different form of
potential. For the image potential model which must be solved numerically, there are
three programs to do that. In modules “ystat_mod” and “yimage_mod” a series of
vectors are made with the initial values of the wave function, its derivative, energy and
normalized value of the wave function. These vectors are known after the numerical
solution in module “odeint_mod” and finally in module “imderiv_mod”. Now we have
calculated all values we need to derive the wave functions back in the program “test”.
program test
USE param_mod, only: pi, hbar, imun, el, epsi0, ett, noll
USE zriddr_mod, only: zriddr
USE kappa_mod, only: kappa
USE det_mod, only: detfunction
USE pot_mod, only: V0, Vg, l, mass
USE odeint_mod, only: odeint
USE yimage_mod, only: yimage
USE imderiv_mod, only: imagederiv
USE wave_mod, only: vagfunction
USE ystart_mod, only: yborja
implicit none
real(8) V0, Vg, g, q, xskarv, delta, res
real(8) E, l, V0_rec, E_rec, Vg_rec, Emidgap, Em_rec
real(8) imkappa, kappa2, A1, A2, A3, A4
real(8) Emin,Emax, P, term, V, Eprev, Enow, smallest_step
real(8) DET, DET_prev
real(8) A, x1, x2, eps, h1, hmin, I2, N, C, W1,W2,W, NORM,Itot,xstart
integer, PARAMETER :: maxnivaer=10, steg=1000
real(8), dimension(maxnivaer) :: Evec
integer i, antal, punkter, j, steg
real(8), dimension(steg) :: zvec, psivec
complex(8), dimension(4) :: ystart
complex(8) I1,D,y1,y2
! read indata
open(1,file='test.inda')
read(1,*) V0, Vg, l, mass
V0 = V0*el
Vg = Vg*el
write(6,*) "V0 after conversion", V0
write(6,*) "Vg after conversion", Vg
close(1)
xskarv=l/2
g=(2*pi)/l
Emidgap=((hbar**2)*(g/2)**2)/(2*mass)
Em_rec=(2*mass/(hbar**2))*Emidgap
Emax = Emidgap+Vg
Emin = Emidgap-Vg
if (Emax > V0) then
Emax = V0
end if
smallest_step = abs(1/((maxnivaer+1.d0)**2) - 1/((maxnivaer+0.d0)**2))
smallest_step = (13.6*el/16.d0) * smallest_step
punkter = 2*int((Emax-Emin)/smallest_step)
write(6,*) Emax/el, Emin/el, smallest_step
write(6,*) "punkter", punkter
write(6,*)"Emidgap=", Emidgap/el,"Em_rec=", Em_rec/el
Write(6,*)"gap-bottom", (Emidgap-Vg)/el
Write(6,*)"gap-top", (Emidgap+Vg)/el
i=1
E = Emin + i * (Emax-Emin) / punkter
call kappa(E,imkappa,kappa2,delta)
res = detfunction(E)
DET_prev = res
write(6,*)"E(1)=", E
write(6,*)"DET_prev=",DET_prev
write(6,*)"delta i test innan loppet=", delta
open(11, file='vagfunc.dat')
write(11,*) "#E, z, psi"
write(11,*) " "
write(11,*) "V0=",V0
write(11,*) " "
antal = 0
LOOP: do i=2,punkter-1
E = Emin + i * (Emax-Emin) / punkter
res=detfunction(E)
write(6,*) i,E/el,res
if ((DET_prev*res) .le. 0.d0) then
antal = antal + 1
write(6,*)"egentillstand"
Eprev = Emin + (i-1) * (Emax-Emin) / punkter
Enow = Emin + i * (Emax-Emin) / punkter
E=zriddr(detfunction,0.d0,Eprev,Enow,1.d-24)
Evec(antal) = E
write(6,*) "antal=", antal
write(6,*) "E=", E
call kappa(E,imkappa,kappa2,delta)
write(6,*) " in i test ar logderiv=",imkappa-(g/2)*tan((g*xstart/2)-delta)
! normering av vagfunktionen
call yborja(ystart(1),ystart(2),xstart,1.d0,E)
ystart(3) = E * ett
ystart(4) = noll
x1 = xstart
x2 = xskarv
h1 = abs(x2-x1)
hmin = 1.d-12
eps = 1.d-6
call odeint(imagederiv,ystart,x1,x2,eps,h1,hmin)
W = real(ystart(1))/((exp(imkappa*xskarv))*cos((g*xskarv/2)-delta))
I1 = 0.5d0*(xskarv-(imun/g)*exp((2*imun)*((g*xskarv/2)-delta)))
I1 = (abs(W))**2 * real(I1)
call yimage(A2,A4,NORM,E)
I2 = NORM
N = I1+I2
C = 1.d0/sqrt(N)
A = W/sqrt(N)
g=(2*pi)/l
write(6,*) "W=",W,"I1=",I1,"I2=",I2
write(6,*) "N=",N,"C=",C,"A=",A
call vagfunction(zvec,psivec,E,l,A,C)
do j=1, steg
write(11,*) E, zvec(j), psivec(j)
end do
write(11,*) " "
write(11,*) " "
else
write(6,*)"inget tillstand"
end if
DET_prev = res
write(6,*)"DET_prev=",DET_prev
if (antal == maxnivaer) exit LOOP
end do LOOP
write(6, *) " "
write(6, *) "TABELL OVER ENERGINIVAER"
if (antal > 0) then
do i=1, antal
write(6, *) "Niva", i, Evec(i)/el
end do
end if
write(6,*)" "
close(11)
end program test
module kappa_mod
USE pot_mod, only: V0, Vg, l, mass
contains
!
for att berakna k-varden for gransen (Imaginar delen)
!
subroutine kappa(E,imkappa,kappa2,delta)
USE param_mod, only: pi, hbar, imun
implicit none
real(8), intent(in) :: E
real(8), intent(out) :: imkappa, kappa2, delta
real(8) B, B1, B2
real(8) q, g, P, term, z
real(8) E_rec, V0_rec, Vg_rec
real(8) A1, A2, A3, A4
real(8) x0
complex kvot
E_rec=(2*mass/(hbar**2))*E
V0_rec=(2*mass/(hbar**2))*V0
Vg_rec=(2*mass/(hbar**2))*Vg
q=sqrt(V0_rec-E_rec)
g=(2*pi)/l
P=((g/2)**2)+E_rec
term=sqrt((g**2)*E_rec+(Vg_rec)**2)
kappa2=P-term
imkappa = sqrt(-kappa2) ! Detta ar egentligen minus Im(kappa).
! slut pa att rakna imkappa, borjar med determinanten.
kvot = sqrt((E_rec-kappa2-((g/2)**2)+(imun*imkappa*g))/(Vg_rec))
write(6,*)"kvot=", kvot
delta = atan2(aimag(kvot),real(kvot))
write(6,*)"delta i kappa_mod=", E/1.60219d-19, delta
end subroutine kappa
! SLUT PA Det har har att gora med determinanten...
end module kappa_mod
module det_mod
USE pot_mod, only: V0, Vg, l, mass
contains
function detfunction(E) result(res)
USE param_mod, only: pi, hbar,imun
USE kappa_mod, only: kappa, V0, Vg, l, mass
USE yimage_mod, only: yimage
USE imderiv_mod, only: imagederiv
USE odeint_mod, only: odeint
implicit none
real(8), intent(in) :: E
real(8) res, NORM
real(8) imkappa, kappa2, delta, A1, A2, A3, A4
real(8) g, Vg_rec, V0_rec, E_rec, x0, q
real(8) x,x1,x2,h1,hmin,eps
complex(8), dimension(4) :: ystart
complex(8) kvot
g=(2*pi)/l
x0 = l/2.d0
E_rec=(2*mass/(hbar**2))*E
V0_rec=(2*mass/(hbar**2))*V0
Vg_rec=(2*mass/(hbar**2))*Vg
call kappa(E,imkappa,kappa2,delta)
A1=(exp(imkappa*x0))*cos(((g/2)*x0)-delta)
A3=imkappa*exp(imkappa*x0)*cos(((g/2)*x0)-delta)
A3= A3-(g/2.d0)*exp(imkappa*x0)*sin(((g/2)*x0)-delta)
! This is for a square barrier, for now.
!
q=sqrt(V0_rec-E_rec)
!
A2=-exp(-q*x0)
!
A4=q*exp(-q*x0)
! This is for an imaginary barrier.
call yimage(A2, A4, NORM, E)
res=(A1*A4)-(A2*A3)
write(6,*)"A1=",A1,"A2=",A2
write(6,*)"A3=",A3,"A4=",A4
end function detfunction
end module det_mod
module yimage_mod
contains
subroutine yimage(A2, A4, NORM, E)
USE param_mod, only: pi, el, epsi0, imun, noll, ett, hbar
USE pot_mod, only: V0,Vg,l, mass, zimage
USE imderiv_mod, only: imagederiv
USE ystart_mod, only: yborja
USE odeint_mod, only: odeint
implicit none
real(8), intent(out) :: A2, A4, NORM
real(8), intent(in) :: E
real(8) h1, hmin, x1, x2
real(8) eps
complex(8) y1, y2
complex(8), dimension(4):: ystart
x2 = l/2
call yborja(y1, y2, x1, 1.d0, E)
ystart(1) = y1
ystart(2) = y2
ystart(3) = E * ett
ystart(4) = noll
eps = 1.d-6
h1 = abs(x1-x2)
hmin = 1.d-12
call odeint(imagederiv,ystart,x1,x2,eps,h1,hmin)
A2 = real(ystart(1))
A4 = real(ystart(2))
NORM = real(ystart(4))
write(6,*) " in i yimage ar logderv=", A4/A2, real(ystart(2))/real(ystart(1))
end subroutine yimage
end module yimage_mod
module ystart_mod
contains
subroutine yborja(y1,y2, xstart,C,E)
USE param_mod, only: pi, el, epsi0, imun, noll, ett, hbar
USE pot_mod, only: V0,Vg,l, mass, zimage
USE imderiv_mod, only: imagederiv
USE odeint_mod, only: odeint
implicit none
complex(8), intent(out) :: y1, y2
real(8), intent(out) :: xstart
real(8), intent(in) :: E,C
real(8) zstart, h1, hmin, x1, x2, x0
real(8) V_rec, E_rec, V, q, eps
real(8) keff, zeff
keff = sqrt((2*mass/(hbar**2))*abs(V0-E))
zeff = 1/keff
xstart = zimage+(el**2)/(16*pi*epsi0*(V0-E)) + 10*zeff
write(6,*) "keff=", keff
write(6,*) "V0-E", (V0-E)/el
write(6,*) "xstart=", xstart
x0 = l/2.d0
x1 = xstart
x2 = x0
eps = 1.d-6
h1 = abs(xstart-x0)
hmin = 1.d-12
V= V0 - (el**2)/(16*pi*epsi0*(xstart-zimage))
V_rec = (2*mass/(hbar**2))*V
E_rec = (2*mass/(hbar**2))*E
q = sqrt(V_rec-E_rec)
y1 = C*ett * exp(-q*xstart)
y2 = -q*C*exp(-q*xstart) * ett
end subroutine yborja
end module ystart_mod
module odeint_mod
contains
-----\\----------\\----------\\----------\\-----subroutine odeint(dersub,ystart,x1,x2,eps,h1,hmin)
implicit none
complex(8), dimension(:), intent(inout) :: ystart
real(8), intent(in) :: x1, x2, eps, h1, hmin
integer nok, nbad
integer, PARAMETER :: maxstp=1000
integer nstp,i
real(8) h,x
real(8) hdid,hnext
real(8), dimension(size(ystart)) :: yscal
complex(8), dimension(size(ystart)) :: y,dydx
real(8), PARAMETER :: zero=0.d0, tiny=1.d-30
interface
subroutine dersub(x,y,dydx)
real(8), intent(in) :: x
complex(8), dimension(:), intent(in) :: y
complex(8), dimension(:), intent(out) :: dydx
end subroutine dersub
end interface
x=x1
h=sign(h1,x2-x1)
nok=0
nbad=0
y(:)=ystart(:)
do nstp=1,maxstp
call dersub(x,y,dydx)
yscal(:)=abs(y(:)) + abs(h*dydx(:)) + tiny
!
write(6,*) "yscal", yscal(:)
!** avoid taking too long a step.
if ((x+h-x2)*(x+h-x1).gt.zero) h=x2-x
call bsstep(dersub,y,dydx,x,h,eps,yscal,hdid,hnext)
if (hdid.eq.h) then
nok=nok+1
else
nbad=nbad+1
end if
if ((x-x2)*(x2-x1) >= zero) then
!** we are done
ystart(:)=y(:)
RETURN
!** this is the normal exit.
endif
if (dabs(hnext) < hmin) then
write(6,*)'too small step in odeint', x1,x2,hnext, hmin
stop
end if
h=hnext
end do
write(6,*)'too many steps in odeint'
stop
end subroutine odeint
end module odeint_mod
module imderiv_mod
USE param_mod, only: pi, el, epsi0, imun, noll, hbar
USE pot_mod, only: V0, mass, zimage
contains
subroutine imagederiv(x, y, dydx)
implicit none
real(8), intent(in) :: x
real(8) V
complex(8), dimension(:), intent(in) :: y
complex(8), dimension(:), intent(out) :: dydx
V= V0 - (el**2)/(16*pi*epsi0*(x-zimage))
dydx(1) = y(2)
dydx(2) = ((2*mass/(hbar**2))*(V-y(3)))*y(1)
dydx(3) = noll
dydx(4) = -(abs(y(1)))**2
end subroutine imagederiv
end module imderiv_mod
Calculations of the Local Density of States (LDOS)
To calculate the Local Density of States, we must first calculate the wave vectors
and the Green’s functions (4-34). Module “green_mod” calculate the wave vectors,
energies and finally the Green’s functions. These calculated values then can be used in
the main program, “el_density” to investigate and derive the Local Density of States,
LDOS.
program el_density
USE param_mod, only: pi, hbar, imun, mass, el
USE green_mod, only: green
implicit none
real(8) E, Emin, Emax
real(8) y, ymin, ymax
real(8) a, t, delta
complex(8) Kp, g
real(8) D
integer i,j, steps, points
open(1,file="surf.inda")
read(1,*) t, a
read(1,*) Emin, Emax, steps
read(1,*) ymin, ymax, points
close (1)
delta = hbar/t
write(6,*) "delta=====", delta,a
write(6,*) "Energies=======", Emin, Emax
write(6,*) "avstand=======", ymin, ymax
open (11,file='density.dat')
write(11,*) "#E(eV), y(nm), D(1/(nm2*eV))"
do i=1,steps
E = Emin + i*(Emax-Emin)/steps
do j=1,points
y = ymin+j*(ymax-ymin)/points
call green(g,E,y,a,delta)
D = (-1/pi)*aimag(g)
write(11,*) E/el,y,(el*1.d-18)*D
end do
end do
write(6,*) " YYYYYYYYYYY==",y
write(6,*) "EEEEEEEEEEEEE=",E
write(6,*) "DDDDDDDDDDDDDDD=",D
end do
write(11,*) " "
write(11,*) " "
close (11)
end program el_density
module green_mod
contains
!
To calculate the wave vector and Grean's function
subroutine green(g,E,y,a,delta)
USE param_mod, only: pi, hbar, imun, mass, noll
implicit none
real(8), intent(in) :: E, y, a, delta
complex(8), intent(out) :: g
real(8), PARAMETER :: factor=10.d3
integer, PARAMETER :: nmax=100000
integer, PARAMETER :: nmax=1000
complex(8) Kp, kx, ky
complex(8) term, term1, E1
integer n
g = noll
do n=1,nmax
E1 = E+imun*delta
kx =((2*mass)/(hbar**2))*E1
ky = (n**2)*(pi**2)/(a**2)
Kp = sqrt(kx-ky)
term=(-imun*mass)/(Kp*(hbar**2))
term1 = term*(sin(n*pi*y/a))**2
g = g + term1
write(6,*) " in i green ar E1",E1
write(6,*) " in i green ar kx", kx
write(6,*) " in i green ar ky", ky
write(6,*) " in i green ar kp", kp
write(6,*) " in i green ar term", term
write(6,*) " in i green ar term1", term1
write(6,*) " in i green ar g ", g
write(6,*) " n#########", n
write(6,*) " nmax¤¤¤¤¤¤", nmax
write(6,*) "aimag(g)= ", aimag(g)
write(6,*) "aimag(term1)*factor= ", aimag(term1)*factor
if (abs(aimag(g)) >= abs(aimag(term)*factor)) exit
end do
end subroutine green
end module green_mod
References
1. Zangwill, Andrew, Physics at surfaces. (Press Syndicate of the University of
Cambridge1988), Chap.4.
2. Kittel, Charles, Introduction to Solid State Physics (8th ed.)
3. George B. Arfken & Hans J. Weber. Mathematical Methods for Physicists.
(Academic Press Limited, 4th ed.)
4. Gasiorowicz, Stephan. Quantum Physics. (John Wiley & Sons, Inc. 1974)
5. William Shockley, Phys. Rev. 56, 317–323 (1939)
6. N. D. Lang and W. Kohn. Phys. Rev. B 1, 4555–4568 (1970)
7. Joel A. Appelbaum and D. R. Hamann, Rev. Mod. Phys. 48, 479–496 (1976)
8. E. V. Chulkov, V. M. Silkin and P. M. Echenique, Surface science 437(1999)
330-352
9. P. M. Echenique, R. Berndt, E. V. Chulkov, Th. Fauster, A. Goldmann and U.
Höfer, Surface science reports 52 (2004) 219-317
10. J. M. García, O. Sánchez, P. Segovia, J. E. Ortega, J. Alvarez, A. L. Vázquez de,
Parga and R. Miranda, Applied Physics A 61, 609 (1995)
11. O. Sanchez, J. M. Garcia, P. Segovia, J. Alvarez, A. L. Vázquez de Parga, J.E.
Ortega, M. Prietsch and R. Miranda, Physical Review B 52, 7894 (1995)
12. A. Beckmann, Ch. Ammer, K. Meinel, H. Neddermeyer, Surface science 432
(1999) L589-L593
13. F. Theilmann, R. Matzdorf, A. Goldmann, Surface science 387 127-135 (1997)
14. J. Li, W. D. Schneider, R. Berndt, O. R. Bryant, S. Crampin, Phys. Rev. Lett. 81,
4464-4467 (1998)
15. R. Matzdorf, A. Goldmann, Surface science 400 (1998) 329-334
16. P. Heimann, J. Hermanson, and H. Miosga, Phys. Rev. B 20, 3059–3066 (1979)
17. F. J. Himpsel and J. E. Ortega, Phys. Rev. B 46, 9719–9723 (1992)
18. S. J. Gurman, Phys. Rev. Lett. 31, 637–639 (1973)
19. J. G. Gay, J. R. Smith, and F. J. Arlinghaus, Phys. Rev. Lett. 42, 332–335 (1979)
20. L. Bürgi, O. Jeandupeux, A. Hirstein, H. Brune, and K. Kern, Phys. Rev. Lett. 81,
5370–5373 (1998)
21. S. M. Driver, D. P. Woodruff, Surface science 560 (2004) 35-44
22. William H. Press, Saul A. Tenkolsky, William T.Vetterling, Brian P. Flannery,
Numerical Recipes in Fortran 77. (The Art of Scientific Computing 2nd edition)
23. Y. C. Cheng, & K. C. Lin, Chinese Journal Physics 26, nr. 4 october1988
24. M. D. Stiles (1997). Generalized Slater-Koster method for fitting band structure
25. http://physweb.spec.warwick.ac.uk/~spsd/, The Warwick STM Gallery
26. http://www.boulder.nist.gov/div853/greenfn/tutorial.html?.html, National Institute
of Standard and Technology, NIST
27. Science & Technology review
28. CEM 924 "Special Topics in Analytical Chemistry"
29. http://www.iap.tuwien.ac.at/www/surface/STM_Gallery/dislocations.html
30. http://en.wikipedia.org/wiki/Crystallographic_defect
31. http://www.physik.uni-marburg.de/of/dynamics/wp.html
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