Written by Anne Helena Arkenbout
Abstract
When the dimensions of a material are in the range of the de Broglie wavelength, a change in physical properties can be observed with respect to the bulk behaviour. Small islands and thin films of lead are synthesised, which indeed showed these quantum size effects. This paper gives a review on the theory, the synthesis techniques and the experiments that are relevant for these confined lead structures. Although many experiments are known, the details of these lead samples remain unclear. For example, it is not yet known whether a wettinglayer is present between the lead and the substrate. Another example is the recent discussion on the stability of these confined lead structures. This paper will discuss and compare the experiments that contributed to these controversies. Finally, suggestions are presented for future research that might unravel the disagreements.
Contends
1. Introduction:
2. Theory:
3. Film growth:
3.1. Techniques to monitor the growth
2.1. Particle in a box and quantum well states
2.2. Friedel oscillations
2.3. Stability
2.4. Influence on the physical properties
3.2 The growth of confined Pb structures
4. Observation of QSE
4.1. Determining the electronic structure
4.1.1. Electron tunnelling
4.1.2. Scanning Tunnelling Microscopy
4.1.3. Photoemission
4.2. Step height oscillations
4.3. Stability
4.3.1. Magical height
4.3.2. Stability versus electronic structure
4.4. Physical properties
5. Summary and conclusions
6. Acknowledgements
7. References
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1. Introduction
Last few decades several new technologies were developed which allow for the synthesis of very small structures.
Nowadays, films with atomic layer thicknesses and islands which have the size of nanometres can be produced. One example of these new techniques is molecular beam epitaxy. This technology gives the possibility of synthesising film that consists of only a few atomic layers. Threedimensional island as well as layer-by–layer growth is possible. The theory allows us to predict whether it is possible to create a certain structure or not. This creates the opportunity to synthesize the smallest structure with a great precision.
Interestingly enough, these small structures have rather different physical properties than the bulk materials. Properties as the conductivity and the critical temperature for superconductivity can differ more than ten orders of magnitudes from the bulk. The deviation of these small structures from bulk behaviour is referred to as Quantum size effects. Quantum size effects begin to appear as the size of a small object becomes comparable to the de Broglie wavelength of the electrons confined in it. Confined structures not only have different values for certain physical properties, they also show quantisation of these. This is because the electrons have only a confined area to go to and so the separation between the energy levels becomes much bigger. The energy levels in the bulk, which can be considered to be continuous, split up into electronic levels that can no longer be approached as a continuum.
In 1971, quantum size effects were observed for the first time by Jaklevic and co-workers
[1] . They measured the tunnelling current of several materials and observed that the thin
Pb films showed an oscillation in the dI/dV and d 2 I/dV 2 curves. The oscillations are due to the presence of electron sub bands, which exist because the film is very thin.
These quantum size effects are not only very interesting in the perspective of research but they also attract the attention of the industry.
The tendency in computer industry is to make the components smaller to increase the working speed. The component, however might loose their function when the size is reduced beyond the limit where quantum size effects will start to occur.
Of course the quantum size effects are not really a bore to the industry. The quantized energy levels in these small structures open the door to totally new devices, because the distance between energy levels can be easily tuned by changing the dimensions of the system. In the future quantum confined structures might find use in lasing devices, sensors, switches, etc.
The research on quantum confined structures has enhanced now the techniques of synthesis and detection are more advanced.
Mostly, the development of the scanning techniques as Scanning Tunnelling
Microscopy (STM) contributes to the study of quantum size effects. This technique made it easier to determine the electronic and topological characteristics of a system.
The probing area of the STM is of atomic scale, which makes it possible to measure the electronic structure of a single island.
This is a great advantage because then it is possible to observe these interesting properties without the tedious synthesis of thin films.
Actually, the amount of research on quantum size effects is so much that it is impossible to review them all in one article.
Therefore this paper mainly focuses on quantum confined lead (Pb) structures deposited on insulating substrates. Pb is often studied because the valence electrons almost behave as a free electron gas, which allows the experimental observations to be compared to the theory.
This paper will give an overview of the synthesis techniques, theory and electronic properties of confined lead structures. It will try to explain the existence of the of stepheight variations. And in particular, this paper will focus on the recent controversy about the correlation between the stability and the total amount of monolayers in lead films and islands. In this paper I will try to point out the difficulties and inconsistencies of the experiments, and their connection to the theory. Finally, I will try to unravel this disagreement about the stability of the confined lead structures.
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2. Theory
Normal bulk materials are characterized by an electronic structure, in which the levels are so close to each other that it, more or less, consists of bands. When a material is confined in one dimension, like in thin films, the electrons with the wave vector perpendicular to the surface will no longer form bands. The electron band of the bulk will split up in a number of sub bands, which are also referred to as quantum well states. The electrons with a wave vector parallel with the film surface will not change their electronic structure. The energy levels that are present in the confined direction can be described by a particle in a box, which, in the basis, is familiar to everyone who ever opened a quantum mechanics book. This simple theory is not sufficient to describe all phenomena that are observed in experiment.
Therefore also more sophisticated theories will be presented here.
2.1. Particle in a box and quantum well states [2]
Electrons in a one-dimensional square well with infinite barriers on each side of the well are described by the following wave function:
1 / 2
 
2 sin
L
(1) k
 n

L
Where L is the length of the box and n is a positive integer, which can be referred to as the quantum number. The wave functions are plotted in figure 1.
Figure 1: wave functions of a particle in a box for different quantum numbers (n).
The energy (E) of these electrons can be found by solving the Schrödinger equation:
H
 
E

H
 
 2 d
2 (2)
2 m dx
2
In which m is the mass of the electron. By filling in the wave equation (1) into the
Schrödinger equation (2) one finds that the energy is given by: n
2 h
2
E

(3)
8 mL
2
So, the energy of the electron in a box is quantized, it can only get discreet values.
These discreet energy levels are often referred to as quantum well states. As the length of the box increases, the separation between the energy becomes smaller, and finally, at large thicknesses, all sub bands will coincide, and form one big electron band. The distance between the energy levels near the Fermi level can be approximated by:
 

 v f
L
(4)
Where v f
is the electron velocity at the
Fermi level. In the bulk, where L is very big the sub bands will no longer be observed.
Moreover, there are three dimensions in the bulk instead of one in this model, which will also change the electronic structure.
Only when the length is of the order of the de Broglie wavelength (

) sub bands can be observed. The de Broglie wavelength is defined as:
  h p
(5)
In which p is the momentum of the electron.
In a metal the electrons near the Fermi level have a very large momentum. That is why quantum size effects can only be observed for very thin films of only a few atomic layers. For semiconductors and isolators the momentum of the electron in the highest occupied states is much smaller. For lead,
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quantum size effects are observed up to thicknesses of 900 Å [1] , while for semiconductors and insulators the quantum states are detectable already for much thicker films.
However, not all quantum well states are really present in the material. The states that can be obtained are called commensurate states. Commensurate states are quantum well states that have an electron wavelength, which is commensurate with the crystal.
These states occur because the box can only take thicknesses that are a multiple of the lattice constant. So, not all quantum-well states that are predicted by the theory can be realised in a real material. Only the commensurate states can be observed for which the wave vector is given by: n
 k

(6)
Nd
Which is the same as formula (1), but now the length of the box is given by an integer
(N) times the lattice spacing (d).
In a thin film the assumption that the barriers are infinitely high is not always valid. When the barrier has a finite height and a finite thickness electrons have the possibility of tunnelling through the barrier.
This of course changes the energy levels of the system, and it can eventually lead to the total disappearance of the discreet energy levels. However, according to Schulte et al.
[3] , insulator and semiconductor substrates will not allow the wave functions at the
Fermi level to penetrate, and so infinite barrier heights can be assumed. For metallic substrates this assumption is not valid.
2.2. Friedel oscillations
In a thin film, electrons as well as nuclei are present. To describe the electronically structure of the thin film properly, one needs to take both into account. Schulte was the first to describe quantum size effects in thin films theoretically [3] . He described how the film thickness can induce changes in the electron densities, work functions and potentials. For his calculations, he used the planar uniform background model to describe the properties of very thin films as a function of the mean electron density r s
.
First a free standing slab of a constant positive background was assumed, which is described by: n


 n

 0
0 x x

1 / 2 D

1 / 2 D
(7)
In which D is the thickness of the film. For this Jellium slab they calculated the ground state electron density, n(r) by solving the
Schrödinger equation.


1 / 2
  v eff

 i

E i
 i n

N  i

1
 i
2
(8)
Because the Schrödinger equation can not be solved exactly, density functional theory was used, as described by Hohenberg and
Kohn and by Kohn and Sham. However, the actual calculation is beyond the scope of this paper and therefore only the result will be presented here: n

1

 n


E f
( E f
 n
 
 v
0

1 / 2

 2
2 n 2
D

2


3
8

 
2

3
E f
 v
0

1 / 2 k f
2
  n
)
 n
2
2 / 3 r s
(9)
Wherein D is the layer thickness and v
0
is the positive background potential in the slab.
They found a behaviour of the eigenfunctions
 n
as is shown in figure 2. In this figure the wavefunctions are plotted as a function of the distance. The surface of the positively charged slab is located at x=10 a.u.. The oscillations of the wave function induce oscillations in the electron density, as they are related by equation (8).
These electron density oscillations are called
Friedel oscillations. Friedel oscillations are frequently observed in a free electron gas with a delta function charge. Friedel oscillations are a feature of defects in metals.
The surface of a material is an example of such a defect.
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Figure 2: Electron wavefunctions in the model of
Schulte [3] .
The electrons are so effective in screening
(Lindhard screening) this defect that there are ripples in the response. One of the most famous Friedel oscillations are the ones in a quantum corral as was made by IBM, see picture 3. The blue peaks in this STM picture are atoms that are arranged in a circle on a flat substrate. The atoms behave as defects and induce Friedel oscillations.
The oscillations of all the atoms interfere, which results in the spherical waves in the electron density, shown in red.
Figure 3: Friedel oscillations in a quantum corral, taken from [4].
Figure 4: Friedel oscillations in the electron density as a function of the wavevector , taken from [5].
According to Zhang et al.
[5] these Friedel oscillations can be described with the following formula: n
  n
0

1


3  cos
 sin u
2

 / u
2
(10)
Where u=2k f z and z is the distance from the surface and k f
is the Fermi wave vector. The
Friedel oscillations for are shown in yellow in figure 4.
The thicknesses for which a minimum in electron density is observed are denoted with D n which is given by:
D n d

 
2 d
3

8  

2

3
1
2
 f n
3 / 2 r s
(11)
Calculations of Wei and Chou [6] in 2002 for
Pb confirm that this formula is correct for describing s-p metals. They studied freestanding Pb films with a density functional theory model that makes use of pseudo potentials and plane waves. In this model they included the d-orbitals of the Pb as valence states. They calculated that the period between two minima in the electron density will be 2.2. However, this value strongly depends on how k f
is chosen. They concluded that the free electron model could not be used and that one should use the real electron density at the Fermi level, which is located in the p-band of Pb.
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According to Schulte, the predicted Friedel oscillations can induce charge oscillation at the interfaces. This is because at every Dn a new function
 n
starts to contribute to the electron density. This new function has a larger probability outside the layer (see figure 2), which increases the electron density outside the slab. In order for the whole film to be neutral this charge should be compensated for by a charge inside the film. So, at the interface of the thin film a dipole layer is formed, of which the strength oscillates as a function of the thickness.
Friedel oscillations are present for every surface. However in the bulk, these Friedel oscillations have no appreciable influence on the overall properties because of the small surface to volume ratio.
2.3. Stability
Studies on Pb islands and layers have shown that sometimes islands are created which are very flat and have a strong preference for certain thicknesses. The preferred heights are called magical heights. In a metal film on a semiconductor substrate, the conduction electrons are confined by the vacuum on the one side and the metalsemiconductor interface on the other. In
1988, Zhang et al.
[5] made a simple model, the electronic growth model, to describe such systems. The model takes into account three ingredients; the quantum confinement, the charge spilling and the interface induced
Friedel oscillations.
They state that the compressibility should be positive to obtain stable islands. This implies that:
 2
E

L 2

0 (12)
Where E(L) is the total energy of the system.
The barrier height at the interfaces of the slab is not infinitely large. This allows electrons to have a probability outside the slab, which is called charge spilling. This spilling induces a charge at the metalsemiconductor interface, which can reduce the total energy with E c
.
Thus the total energy of the system becomes:
E t

E
0

E c
(13)
Where E
0
is the energy of the free electron gas in a well with infinitely high barriers.
And E c
is given by:
E c

0 .
5 CV
2
(14)
Where V is the potential and C the capacitance of the interface.
However, also Friedel oscillations should be taken into account when describing the energy. Every surface induces an oscillation in the electron density. When, in a thin film, the other surface coincides with a minimum of these Friedel oscillations caused by the other surface, there is an additional energy gain. In this situation fewer electrons need to be pushed up in energy by the confinement of the outer surface, and so the energy is reduced. If these more stable states can be formed depends on the energy of the electrons in the highest occupied states and of the lattice parameter. For Pb, these parameters are such that the position of the first minimum coincides with the thickness of 1ML. For many other metals, however, the Friedel oscillations mostly do not coincide with the lattice parameter, and so the quantum size effects are not observed.
The relative stability of islands can also be predicted from the position of the electron sub bands relative to the position of the
Fermi level. The bigger the separation between the highest occupied energy band
(HOB) and the Fermi level the more stable the Pb island. This subject will be discussed in more detail in chapter 4.
Another measure for the stability can be the surface roughness, as is described by
Floreano et al.
[7] . They state that the more rough the surface, the higher the energy of the thin film. An atom favours to have as much neighbours as possible. At a surface however, the number of neighbours is incomplete. The most stable configuration for the surface will be the configuration in which the amount of neighbours for the atom is the highest. For a smooth film the amount of neighbours is higher than for a rough film, and therefore the smooth film is the most stable configuration.
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2.4. Influence on the physical properties
All discussed theories above predict oscillations in the energy, the electron density and surface charge as a function of the thickness in confined structures. These oscillations also can influence the other physical properties of the material. Schulte et al described the change in the work function and the potential.
The electron confinement can induce a change in the conductivity, as is described by the theoretical article of Meyerovich and
Ponomarev [8] . They studied the conductivity multilayer metal films. The conductivity depends on the scattering of the electrons.
Therefore the change in the electronic structure due to the confinement, does not directly influence the conductivity. However the change in the occupation of sub bands, as the film grows thicker, can be accompanied by a change in the number of scattering channels. This can lead to a saw like dependence of the conductivity on the film thickness as is shown in picture 5 for a single layer of metal.
The positions of the singularities correspond to the values of the thickness at which a new energy sub band becomes accessible. However, the exact shape of the conductivity plot strongly depends on the surface roughness.
The charge oscillations at the surface induce oscillations as well in the work function as in the potential. Wei et al.
[6] found that the work function and the surface energy were described by the following formula:
E

A sin
 k f dN
 
0


B

C (15)
N

Where A, B, C, α and 
0
are N independent constants. The behaviour of the workfunction and the surface charge is shown in figure 6. Also Schulte [3] predicted to see oscillations in the work function.
These oscillations showed minima at film thicknesses of Dn, just as the oscillations in the electron density.
3. Film growth
To show quantum size effects, films must be as thin as the de Broglie wavelength. For metals this length is just a few atomic layers.
Films of this thickness are very difficult to prepare.
The most widely used technique for making quantum confined structures is molecular beam epitaxy. Molecular beam epitaxy uses an ultra high vacuum chamber, in which the pressure can be lowered to 1x10 -11 Torr. In the first place this low pressure is needed because the system needs to be very clean.
The amount of atoms that are present in a vacuum system is very small, and so very little pollution is present. Due to the low concentration of atoms the evaporated particles can travel for very long distances
(50 km) before they collide with another atom. The second reason for this low pressure is that it takes quite a while before a single layer of gas is formed on the substrate.
This low growth rate is essential to tune the amount of monolayers that will be deposited.
Figure 5: The saw like dependence of the conductivity on the thickness, according to [8].
Figure 6: Oscillations in the work function and the surface energy as a function of the thickness.
The figure is taken from [6].
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In the UHV chamber the material is evaporated in an effusion cell, from which it diffuses to the substrate and condenses to form a film.
Depending on the deposition conditions and the materials that are used, one can distinguish three different types of film growth. The first type of growth is Stranski-
Krastanov growth in which first a wetting layer is formed followed by the growth of three dimensional clusters. The second mode is the Franck-van der Merwe growth in which the film grows monolayer by monolayer. The third mode is called the
Volmer-Weber mode. In this mode three dimensional islands are formed immediately, without the creation of a wetting layer. The sort of mode can be tuned by changing the temperature and deposition rate and also by choosing an appropriate substrate.
As already mentioned in the theory, the substrate has a great influence on the appearance of quantum size effects. To be sure that the electrons are confined to the grown films or islands, only isolating substrates can be used. This paper reviews articles which use mostly Ge (001) and
Si(111) as a substrate, but the first tunnelling diodes were grown on aluminium oxide and magnesium oxide. Before the film or islands can be grown, the substrate need to be cleaned. Even the tiniest pollution can change the growth completely and ruin your system. Cleaning methods that are often used are combinations of chemical etching,
Ar+ bombardment and annealing. Although cleaning is very important, most used techniques are sufficient. Therefore it will be assumed the cleaning process has not influenced the growth in the synthesis processes.
3.1. Techniques to monitor the growth [21]
As the quantum size effects strongly depend on the size of the system, accurate determination of the height is necessary. To monitor the growth and obtain the height several techniques are available. The most researchers use a combination of the following techniques; RHEED, SPA-LEED, the quartz balance, STM, Has and x-ray diffraction. The working of the STM will be discussed in chapter 4. The other techniques will be shortly discussed here.
The quartz balance is an instrument that can measure the amount of material deposited.
The instrument consists of a crystal, which is placed near the substrate. As the film is growing, material will be deposited on the quartz. The increase in mass will change the eigen frequency of the crystal, and so by measuring this frequency, the thickness of the crystal can be calculated. From this thickness the amount of material deposited can be derived.
Reflection High Energy Electron Diffraction
(RHEED) makes use of an electron beam, which is almost parallel to the substrate. The electrons are scattered from the substrate and detected by a fluorescent screen. From the diffraction pattern one can obtain the surface morphology of the crystal. One can determine whether the surface is rough or flat and whether islands or layers are formed.
In the Franck-van der Merwe mode, the
RHEED diffraction pattern shows oscillations which are a measure for the growth rate. RHEED is often used to monitor the film growth. However, RHEED can not always be used to monitor the real time growth, because it needs the sample to be fixed. For thick films, it is necessary to rotate the sample to observe a good uniform film. For thin films rotating the substrate is not necessary, and thus RHEED can be used.
Spot Profile Analysis- Low Energy Electron
Diffraction (SPA-LEED) is a good technique to monitor the real time growth, but it has the same problem as RHEED, that it does not work on rotating samples. In this technique the electrons approach the surface perpendicularly. Then the electrons are diffracted by the surface and detected by a fluorescent screen. Because the energy of the electrons is so low, they will interact with the periodicity of the lattice. Moreover, they can only penetrate the film for a few atomic layers. Therefore LEED is a good technique to obtain the lattice structure from the surface (in contrast to normal X-ray that probes the bulk.). Spot Profile Analysis can give even more information about the surface. The spot profile is a measure for the surface structure, one can see whether the layer consists of steps or it is smooth.
Helium atom scattering (HAS) is a technique in which a surface is bombarded with He atoms. He atoms do not penetrate the film but are reflected immediately, after
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which they are detected and the momentum transfer is determined. The technique can be used to determine the surface symmetry and ordering.
In X-ray diffraction high frequency light is send to a material. Due to the periodicity of the lattice the light is scattered, and an interference pattern is created. From the interference pattern the lattice symmetry can be deduced. When the incoming beam makes only a small angel with the film, mostly the surface is probed. This technique can be used to determine the thickness and the lattice structure of the film.
3.2. The growth of confined Pb structures
When the lead is deposited, it will form a film or islands on the substrate. The lead structures will have different lattice parameters and surface tension than the substrate, and so a mismatch can occur at the substrate. To overcome this mismatch, the deposited can first form a wettinglayer before the film or islands are grown. The wettinglayer does not have a very clear lattice structure and as well atoms from the substrate as from the deposited material can be present in there. For lead structures indeed sometimes a wettinglayer is observed, for example by Hupalo et al.
[18] . They grew lead islands on Si(111) at several temperatures starting from 130 K. Their
STM and SPA-LEED experiments showed that a wetting layer was formed before the island began to grow. They summarised their different measurements in a phase diagram shown in figure 7. In this figure is shown that, depending on the temperature, a wettinglayer is formed of thicknesses varying between 1 to 3 monolayers, before the islands were observed. However, Feng et al.
[20] grew the same kind of lead islands on
Si and their x-ray diffraction experiments showed that no wetting layer was present between the islands and the substrate as is shown in figure 8. The islands are deposited directly on top of the substrates, while between the islands the substrate is covered by a wettinglayer. Experiments, in which the lead structures were grown on Ge(001), report the formation of a pseudomorphic layer before the lead structures are formed.
This pseudomorphic layer is formed to reduce the lattice mismatch between the substrate and the deposited material, just as the wettinglayer. But in contrast to the wettinglayer, this pseudomorphic layer does have a crystalline structure. This pseudomorphic layer has been observed for instance by Zhao et al.
[22] They grew Pb films on Ge (001) and observed that first a pseudomorphic layer was formed before pure Pb films were observed. The pseudomorphic layer consisted of a crystalline phase containing both Pb and Ge atoms.
Depending on the conditions both lead films and lead islands can be formed. Film growth has been observed when the lead was deposited at temperatures below 130K. The lead film has a (111) surface as is reported by Jaklevic et al.
[1] . These films can be grown on different surfaces. The synthesis on bare Si is tedious and can only be prepared when the first few layers of Pb are annealed as is described by Upton et al.
[13]
More often, however, the Si (111) is covered by a single monolayer of gold, on top of which layer-by layer growth is observed immediately, without annealation.
Figure 7: The phase diagram of the Pb island growth according to Hupalo et al.
[18] .
Figure 8: The structure of the Pb islands according to Feng et al.
[20]
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This is reported, for example, by
Jalochowski et al.
[10] [11] [12] . The first quantum size effects, however, were observed by Jaklevic et al.
[1] in Pb films on aluminium and magnesium oxide. They made tunnelling diodes consisting of a metal
(Mg or Al) an oxide layer and on top of it the lead layer. The substrate was cooled to about 100K and with x-ray, they observed that the growth was of the Franck-van der
Merwe type.
Generally, island growth is observed, when growing Pb at higher temperatures than
130K. The islands grow very easily on bare
Si; no annealing or so is needed. For example, Altfeder et al.
[15] report the synthesis of Pb islands on stepped Si (111) in 1997. They deposited Pb at a temperature of 273 K and with STM, they observed the formation of flat islands with steep edges.
The observed Pb islands have an interesting shape and size; They have a flat top and steep edges, while normally more pyramidal structures are observed. Moreover the islands have a narrow height distribution.
These features are discussed for instance by
Hupalo et al.
[18]
Summarizing, films of Pb can be grown on several isolating substrates when the lead is deposited at a temperature between 100 and
130K. When Pb is deposited at higher temperatures islands are formed. The islands are flat on the top and they have steep edges.
One part of the researchers state that before the growth of islands or thin films a wettinglayer is formed. However, no consensus is reached yet on that issue.
4. Observation of QSE
As discussed in chapter 2, the theory predicts that quantum size effects can be observed for a lot of physical properties like work function and conductivity. However, all these different physical properties are, in the basis, caused by a change in electron distribution. In this chapter we will first discuss several techniques that are being used to determine the electronic structure of confined systems. Moreover a literature review will be given on the methods, the results and the conclusions of the most important experimentalists in this field.
Finally, experiments on step height oscillations, stability and other
Figure 9: As a voltage is applied, the Fermi levels shift with respect to each other and a current can be observed experimentally observed quantum size effects will be discussed.
4.1. Determining the electronic structure
The electronic sub band structure is in the basis of all quantum size effects. To find out what the electronic structure of a certain material looks like, one should extract or inject electrons, with a known energy, from or to the system. In this paragraph roughly two different techniques will be discussed.
The first is tunnelling, in which the electron current is determined trough a certain barrier.
The second method is photoemission, in which electrons are removed from a material with light and subsequently their kinetic energy is determined. In the following text an overview will be given on the scientific reports that determined the electronic structure of confined lead systems, making use of tunnelling or photoemission.
4.1.1. Electron tunnelling
By far the most used technique to observe quantum effects in confined Pb structures is electron tunnelling. In 1971 Jaklevic et al.
[1] observed electron standing wave states
(quantum well states) in thin Pb films by electron tunnelling. Tunnelling is the movement of electrons trough a barrier of finite thickness. Imagine two pieces of metal
A and metal B, separated by an insulating material. Classically, electrons from metal A can not reach metal B. However, when the electrons are described by wave functions the electrons from metal A still have a probability to be in metal B, but the thicker the barrier the lower the probability. The tunnelling probability, P, is given by:
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P
 
0
2 e

2 kd
2 m (
 
E )
(16) k


Where

0
is the wave function of the electron, d is the thickness of the barrier.
 is the work function of the material (the height of the barrier) and E is the energy of the electron. When no voltage is applied over the insulating material, the tunnelling current in both directions is the same, and so no net current will be observed. As a voltage is applied over the insulating slab, the Fermi levels of material A and B will shift with respect to each other, as is shown in figure 9.
In this energy diagram the Fermi levels are marked by the black continuous lines.
Tunnelling from one side to the other is only possible between a filled level on the one side and an empty level on the other side. If the Fermi level of material A is not aligned with some empty energy level of material B, no tunnelling occurs. Hence, by changing the applied voltage, the electronic structure can be determined.
Jaklevic et al. used tunnelling diodes of aluminium and magnesium on which an oxide layer was formed by exposure to a gas discharge. After that the Pb was deposited to form a thin layer of approximately 250Å.
They put contacts on both sides of the oxide layer and obtained tunnelling resistances in the range 50-1000

for areas of several square millimetres. They measured the current versus the voltage at 4 K, and observed oscillations in the dI/dV and the d 2 I/dV 2 curves, which are plotted in figure
10. The spectra look symmetric with respect to the prominent peak which is located at 0.8 volt. This peak stays at 0.8 volt when the thickness was changed, while the others move. This state at 0.8 volt is the commensurate state with k=(1/2

)/d. In thick films, it sometimes is energetically more favourable to form grains. These grains are small confined crystals. The sizes of these grains are not all equal, but are given by a certain distribution. The decay of the intensity of the oscillations is due to this grain size distribution.
The equally spaced oscillations where observed with the Pb positive with respect to the Aluminium, so the graph involves the
Figure 10: oscillations in the tunnelling current, observed by Jaklevic et al.
[1] . energy levels of the Pb, which are located above the Fermi level, which are empty.
Thus the electrons tunnel from the aluminium to the Pb. The spacing between the oscillations varied with changing the thickness of the Pb layer in the range of 250-
900 Å and they disappeared for thicknesses larger than 1000 Å. The spacing between the oscillations decreased linearly when the thickness was increased.
Four years later, the same researchers used this technique to observe also two commensurate states beneath the Fermi level with wave vectors of respectively (2/5

)/d and (1/3

)/d. Moreover they observed that as they increased the temperature, from 4 to
200 K, the energy of the commensurate states shifted due to the thermal expansion of the lead lattice.
They also determined the boundary effects by depositing 25 Å of Ag on top of the Pb layer. The Ag overlayer reduced the amplitude of the oscillations dramatically.
By deposition of lead oxide on the lead layer, however, the oscillations hardly changed.
4.1.2. Scanning Tunnelling Microscopy
In the experiment discussed above the tunnelling current is measured, averaged over a quite big area. However, the tunnelling can also be determined locally by making use of a Scanning Tunnelling
12

Microscope (STM). STM measures locally the tunnelling from or to the material by putting a conducting tip on the surface. The tip does not touch the surface and the air between the tip and the sample is the tunnelling barrier. 90% of this tunnelling current goes through the last atom on the tip, which is the nearest to the sample. Therefore the tunnelling is measured at a very small area and atomic resolution can be reached.
Due to this small probing area fluctuations in the surface roughness have less influence than in the normal tunnelling experiment.
Hence, STM results are expected to be more accurate than the tunnelling experiments discussed above. Another advantage is that thin layers do not necessarily need to be formed, because, with STM one can also measure the properties of one single island.
The STM can be used in two modes, the constant current mode and the constant height mode. In the constant current mode the current is kept constant while the tip scans along the surface. When the electronic properties or the distance to the surface is changed, the tip changes the distance to the surface to keep the current constant. This method is good for measuring the topology through the density of states (DOS) of a surface. To determine the occupied and unoccupied states of a material, however, the constant height mode is used. This means that the tip is kept in a fixed position, laterally and vertically, and that the voltage is smoothly changed. When the tip voltage is changed from negative to positive values, the electrons will first tunnel out of the occupied states to the tip and then from the tip to the unoccupied states. The changes in the current are due to changes in the DOS.
In 1997 Altfeder et al.
[15] synthesised a wedge of Pb on a stepped Si surface which is shown in figure 11.
Although the substrate was stepped, STM measurements showed that the Pb layer was smooth, so at every step in the Si substrate the thickness of the
Pb surface changed. The thickness of the Pb film varied between 20 and 100 Å. The wedge was examined by scanning tunnelling microscopy at both 4.8 and 77 K with tip voltages varying between -1 and +8 volt.
The experiments showed the presence of interference fringes, which are schematically shown in figure 11. These fringes are due to a change in the electronic structure as the thickness changes. For each of these fringes the I-V characteristics were determined by varying the tip voltage between -2 and +2
Volt. The results of this experiment are shown in figure 12. The numbers on the right side of the figure refers to the number of Pb layers with respect to the top of the wetting layer and the energy levels are marked by vertical lines. The 0 voltage point can be viewed as the Fermi level of the film, and the cusps in the current indicate the energy levels. One can clearly see that the distance between the energy levels (

) decreases linearly as the thickness of the Pb
Figure 11: a quantum wedge, taken from
Altfeder et al.
[15] .
Figure 12: The electronic structure of the quantum wedge according Altfeder et al.
[15] .
13

Figure 13: The electronic structure of lead islands according to Su et al.
[17] . to layer increases as was predicted by the theory of equation (4) in chapter 2.1.
Moreover, the height of the wettinglayer could be determined by plotting 1/

against the height of the Pb layer. The intersection of this straight line with the axis gives a value for the thickness of the wetting layer, which was found to be 2 ML.
As Altfeder et al. showed that the observation of QSE with STM can be very successful, many researchers followed their example. STM was used to detect the electronic structure of thin films as well as islands of Pb. In 2001 Su et al.
[17] used STM to show the correlation between electronic properties and thickness for individual islands. Their results are shown in figure 13.
Be aware of the fact that in figure 12 the tip voltage is on the x-axis and in figure 13 it is the sample voltage. So -2 Volt in figure 12 corresponds to 2 Volt in figure 13. The downward pointing arrows are indicating the highest occupied sub band (HOS) and the upward pointing arrow indicates the lowest unoccupied sub band. (N) is the number of
Pb layers, calculated from the wetting layer.
Clear differences in the electronic structure are observed for odd and even thickness of the islands. This trend can also be observed in the results of Altfeder et al. in figure 12.
With the same technique as Altfeder et al. they determined that the graph of 1/

versus thickness intersects the 1/

axis at -3.
According to Su et al. this was due to the fact that the wetting layer was 2 ML thick
Figure 14: the photoemission spectrum for fixed layer height according to Jalochowski et al.
[11] and that the electrons could penetrate in the substrate for approximately 1 ML.
In 2002, Hupalo and Tringides [19] preformed the same experiment and confirmed the results found by Su et al.
4.1.3. Photoemission
Measuring the electron tunnelling one can determine the electronic structure of a material. However, tunnelling is a technique that mostly probes the interfaces of the material. The electronic structure of the surface is not always the same as that of the ”bulk” because of the presence of surface states and surface roughness.
Photoemission is a technique that can determine the electronic structure of the first few monolayers of the material. Hence when one does photoemission on thin films, approximately the whole system can be probed.
Photoemission is a technique where intense monochromatic light is shone on a material.
If the energy of the light is high enough electrons in the material are kicked out.
These electrons are being detected and their
14

kinetic energy is determined. Because the energy of the incoming light is know, the kinetic energy of the electrons allows one to calculate the binding energy of the electron in the material with respect to the vacuum.
There are many different techniques that make use of photoemission. The two that are mostly used will be shortly introduced here.
UPS, ultraviolet photoemission spectroscopy is a method that probes the valence electrons of a material. The energy of the UV light is relatively low and so it can only be used to remove electrons near the Fermi level. XPS,
X-ray photo emission spectroscopy can be used to obtain the core electrons of the material. The energy of the x-ray radiation is so high that these core electrons are easily be removed form the material.
For the observation of quantum size effects the valence electrons are most important and therefore UPS is the best technique to observe them. Photoemission probes a large area (mm), and so reliable measurements on quantum size effects can only be found for very smooth films.
In 1992 Jalochowski et al.
[11] did photon emission experiments on Pb film (on a Si with one monolayer of Au) during deposition as well as with fixed coverage.
The RHEED spectra showed that the films were smooth The intensity of the reflected electrons for different energies is measured for different fixed thicknesses of the Pb film and also during the film growth, as is shown in figure 14. The peak at 4.4 eV which appears at a film thickness of 0-4 monolayers is due to the Au. The other peak that appears is due to Pb. Although the presence of this narrow peak (instead of an electron band) might already indicate the presence of quantum size effects, it is hard to see the presence of oscillations in the actual energy levels in this figure, as they are predicted by the theory and observed by
STM. However, the authors of this paper state that the quantum size effects are present.
In 1995, Jalochwski [12] did photoemission experiments on the same Pb layers as described above. They only measured photoemission experiments during the film growth and measured exactly the same as in
1992. The result were compared with calculations of a simple finite-potential quantum well, they are shown in figure 15 and 16. In both graphs some oscillations are present, however as the experiment is done for a growing film it is hard to say if the oscillations are due to the quantum size effects or due to a change in surface
Figure 15: The electronic structure of Pb films according to Jalochowski et al.
[11] .
Figure 16: Theoretical calculations of the electronic structure of Pb films.
15

roughness. Moreover, it is hard to determine the exact periodicity of the oscillations, from this data.
In 2004, Upton et al.
[13] determined, with photoemission, the electronic structure of uniform Pb films on Si. They observed quantum well states only for thicknesses that consist of an odd number of monolayers.
The results are shown in figure 17. The dots refer to the energy levels of the quantum well states from the theoretical calculations.
The crosses are the measured values for the quantum well states. The graph shows also some crosses for even thicknesses, like for 8,
10 and 12. The thicknesses are defined with respect to the wetting layer. The discontinuous line at -0.5 volt give the limit, from witch the electrons in the Pb are confined by the band gap of the silicium; the electrons that have energies outside the 0 to
-0.5 e V binding energy region are not confined by the silicium band gap. So, according to Upton et al., the observations of states outside this energy range are not due to quantum well states but due to partial reflection at the Si- Pb. boundary. However, many papers, which are discussed above, reported to have observed these quantum well states as well for odd as well as even amounts of monolayers.
4.2. Step height oscillations
Studies on the growth of Pb layers with
HAS and STM have shown quite a surprising behaviour in the thickness of the monolayers. In contrast to what is assumed in the theory (in chapter 2), the thickness of one monolayer in a thin film does not have the same value as for bulk Pb. Depending on the total thickness of the film the thickness of the outermost monolayer can oscillate.
In 1997, Crottini et al.
[14] reported step height oscillations for Pb on Ge(001), with variations as large as 15%. They studied the thin film of Pb with HAS during the film growth. As already discussed in chapter 3, first pseudomorphic growth was observed followed by layer by layer growth. From the intensity and the peak width of the He spectrum they calculated the step height
(=thickness of the monolayer) at different stages in the film growth. Their results are shown in figure 18. The white squares mark the observed thicknesses and the black discontinuous line indicates the lattice parameter that is observed for the bulk.
The oscillations can be explained by the fact that the thicknesses for which quantum well states occur are multiples of the Fermi electron half- wavelength,
 f
/2=1.83Å. To minimize the mismatch between the growing film and the multiple of
 f
/2 the thickness of the outermost layer is changed, while no lateral variations in the lattice can be observed. For bulk Pb the lattice spacing d=2.85Å and so 2d~3  f
/2. So for every two new monolayers three new states can be formed. In layer-by-layer growth, this will result in a sequence of one followed by two new states per monolayer. In one way or another, the film thickness relaxes according to the amount of quantum well states present inducing an oscillation in the thickness.
In 2002, Su et al.
[17] observed oscillations in the film thickness for Pb islands on Si. They measured the apparent height of the islands at different biases, to exclude electronic changes from being the cause of the oscillations. Their observations are plotted in figure 19. The black squares mark the measured heights for complete islands. The
 t=0 point denotes the monolayer height
Figure 17: Binding energy versus thickness according to Upton et al.
[13]
Figure 18: Step height oscillations according to
Crottini et al.
[14]
16

observed for bulk Pb. Although, the same 2
ML periodicity is observed as by Crottini et al. the minima in the film thickness are now for odd layers, while according to Crottini et al. the minima are observed for the even layers. The difference in substrate can have caused a change in wetting layer thickness, which could result in another way of numbering the monolayers. But this is not very clear from the text. In any case, Su et al. attribute the oscillations to the same minimization of the mismatch between lattice and electron wavelength as Crottini et al..
In 2003 Floreano et al.
[7] reported HAS experiments on Pb films on Ge (001). They observed an oscillation in the HAS reflectivity with a period of 2 ML. From the specular peak width they calculated the differences in step height which are shown in figure 20. The black dots give the measured heights of the Pb islands. The value of 1.0 refers to the height observed for bulk Pb. The minima in the step heights are observed for the even amount of monolayers.
Their Pb island heights were measured relatively to the substrate. The results are in agreement with the results reported by
Crottini et al..
4.3. Stability
In the discussed papers the difference often appeared between the films that consist of odd or even amount of monolayers. In this paragraph the stability of both types of films and islands will be discussed. The observation of strongly preferred heights
(magical heights) and relaxation processes will be discussed. Moreover, the relation of the energy sub band structure and the stability will be described.
However, the researchers have not found any agreement yet on which off the two types of layers is the most stable.
4.3.1 Magical height
In 2000, Budde et al.
[16] observed the formation of uniform seven layer high, steep- edged, flat top Pb islands, with SPA-
LEED. During the growth, they first observed the formation of a two monolayer thick wetting layer. Then the 7 ML high islands where formed. The number of islands increased until a flat film of 7 ML was formed. Because only seven ML high islands where observed, it is likely that this is a very stable thickness for the Pb. The quantum size effects make it energetically favorable to form islands that consist of seven monolayers of Pb, measured from the substrate or the wettnglayer, because the actual mismatch between the length of the potential well and a multiple of the half-
Fermi wavelength is very small. This can be calculated with the following formula:
2 nd
 w
 f
(17)
Where n and w are integers of respectively 7 and 11, d =2.86Å and  f
=3.66Å. This small mismatch implies a relatively low energy, and therefore the seven step height islands are very stable.
Figure 19: The step height oscillations according to Su et al.
[17] .
Figure 20: Step height oscillations for Pb on Ge
(001), taken from Floreano et al.
[7] .
17

Hupalo et al.
[18] did both STM and SPA-
LEED experiments on Pb islands that were grown under different conditions. They observed that, after the deposition of 4 ML, islands were formed that mainly have heights of seven monolayers, but also other heights were observed, as is shown in figure
21. The peak at zero height is due to the substrate. The peaks at finite height are due to the Pb islands. The picture shows the biggest peak for 6.7 monolayers height, so islands with this height are the most abundant. But also islands of 3.8 and 5.7 monolayers high are present.
When more than 8 ML are deposited two by two layer growth has been observed. The heights are measured with respect to the wetting layer, of which the height can change with temperature as is shown in figure 7.
4.3.2. Stability versus electronic structure
From the observation of islands with magical heights it seems that a thickness of seven monolayers is very stable. In this part the possible correlation between the electronic structure and the stability of a confined structure will be discussed.
Hupalo et al.
[19] determined the electronic structure for Pb islands with STM. Their results are shown in figure 22.
The plotted voltage is the substrate bias, so at negative voltages one tunnels from the sample to the tip. For an even amount of monolayers, the HOS was much nearer to the Fermi level than for the odd layers. They state that for this even layers the spilling and the charge density is higher, because the
HOS is nearer to the Fermi level. Therefore the energy of this even layer islands is higher than for the odd. According to the authors, also the level of the tunneling current is a measure for the stability. The I/V curves for the stable islands lay lower than for the unstable islands. During the synthesis also even-layer islands were observed, however, after some time they relaxed into odd numbered islands.
As already discussed in chapter 4.1.2 Su et al.
[17] determined the electronic structure of
Pb islands with STM. Their results were shown in figure 13. For an even amount of monolayers, the HOS was much nearer to the Fermi level than for the odd layers. The middle positions between the HOS and the
LUS were marked with the short dotted lines.
The deviation, of this middle point, from the
Fermi level, is given in figure 23.
For the odd thickness islands the middle point is much nearer to the Fermi level than for the even ones. Surprisingly enough, the authors state that the surface charge density of the odd thickness islands is higher, due to this large separation between the HOS and
Figure 21: Observed island heigths according to
Hupalo et al.
[18]
Figure 22: electronic structure of Pb island of different heights according to Hupalo et al.
[19]
18

Figure 23: The difference between the middle point of the LUS and the HOS and the Fermi level, taken from Su et al.
[17] the Fermi level. According to the authors this high surface charge density is an indication that the odd-layer islands are most stable. And so, although their argumentation is completely opposite to that of Hupalo et al., Su et al. come to the same conclusion that the odd layer island are most stable.
They observed indeed that the six-layer islands were unstable at 200 K and decayed into 7-layer islands.
In 2003, Floreano et al.
[7] examined thin film of Pb on Ge. Although they did not determine the electronic structure of the material, they made an important statement on the stability. And therefore their results are mentioned in this chapter. In their HAS spectra they observed minima in the roughness of the film with a 2 ML periodicity. They observed that the minima occur for the even-layers deposition. And so the even layer films where more smooth than the odd layer films. This implies that the even number layers are energetically more stable. The authors noticed the disagreement of this statement with the result of Budde et al.. They addressed this difference to the fact that Hupalo et al. measured the layer thickness with respect to the wetting layer, while Floreano et al. measured the height with respect to the substrate.
In 2004 Upton et al.
[13] determined the electronic structure of Pb layers with photo emission. They observed quantum well states only for odd numbered layers, but according to the authors the even numbered layers are the most stable ones. They did annealing experiments on films of different thicknesses. And they observed that the even-layered film decayed at higher temperature than the odd-layered films. The film thicknesses were defined with respect to the wetting layer. They observed that the odd-layer films have their HOS nearer to the
Fermi level than the even-layer films, as is shown in figure 17.
Summarizing, all publications agree on the fact that situations with the largest distance between the HOS and the Fermi level are the most stable. However, no agreement has been found yet on which system, the oddlayer or the even-layer, fulfills this condition.
This might be due to the different techniques that are used to observe the energetic structure. Another possibility is that islands give rise to other energy levels than thin films, but this is not supported by the theory.
Also the influence of the different substrates might make the difference. The most likely cause, however, is that the researchers did define the island/film thickness differently.
The cause of this different definition may have its origin in the discussion whether or not a wettinglayer is present, as is already discussed in chapter 3.
4.4. Physical properties
The split up of the electronic structure in sub bands due to the confinement can be expressed in a change in the behaviour of physical properties. For example, the theory has predicted that quantum sizes effects can cause oscillations in the resistivity. However, also other features can cause resistivity oscillations. And therefore not all observed oscillations need to be caused by quantum size effects.
In 1988 Jalochowski and Bauer [10] studied
Pb films with resistivity measurements.
They grew two different samples; one with
Pb deposited on bare silicium and one which had a one monolayer of Au between the Pb and the Si. During the deposition of Pb the resistivity was measured for both samples.
They put a voltage on the silicon and measured the current. The resistance of the
Si was known to be 1 k

and so from the current measurement the resistivity for the
Pb layer could be calculated. In the plots of the resistivity against the thickness of the Pb layer oscillations are observed for both films, as is shown in figure 24. It is clear that the behaviour deviates from linear; however, the oscillations are hard to see in this figure.
According to the authors, the period of the oscillations in the type 2 material (Si covered with a monolayer of Au) is 2 ML and in type 1(bare Si) it is 1 ML.
19

Figure 24: The resistivity as a function of the layer thickness, according to Jalochowski and
Bauer [10]
The oscillations of the type 1 material are not quantum size effects. The reason for this conclusion is that the Fermi length of Pb is not commensurate with a single monolayer, and therefore oscillations with a period of 1
ML can not be caused by quantum size effects. The observed oscillations are the result of the scattering of the electrons at the growing film surface. The roughness of the surface varies periodically with 1 ML as the film grows in the layer by layer mode.
Because steps are created when the film starts growing the roughness is at minimum when the film is complete and ad maximum when half a monolayer has been deposited.
The electrons are scattered differently as the roughness is changed, and so oscillations with a 1ML period are observed in the resistivity.
The oscillations in the type 2 material are caused by quantum size effects, as the Fermi wavelength of lead is commensurate with the period of 2 ML. Moreover these oscillations can by no means be induced by the roughness, because than the period would have been 1 ML. Thus it can be concluded that the quantum size effects can induce oscillations in the resistivity.
Also, superconducting features as the critical field and the transition temperature can strongly be influence by quantum size effects. It is predicted that these size effects become important when the material has dimensions in the order of the coherence length. The coherence length of Pb is 87 nm.
Li et al.
[23] produced Pb particles with an average diameter of 8 nm. The critical field for the 9 nm particles was 50 times higher than that of bulk Pb. Also the critical
Figure 25: critical field and temperature in different Pb particles according to Li et al.
[23] temperature strongly depends on the size of the Pb particles. The size dependence of the critical field and the critical temperature are shown in figure 25. At zero field the 4.5 nm particle shows a critical temperature of 4.5
K, while all other particles have a Tc around
7 K. Moreover, the field dependence of the two smallest particles of 4.5 and 6 nm is little, while the critical temperature for the larger particles goes to zero relatively fast for an increasing magnetic field.
5. Summary and conclusions
As well the simple particle in a box model as the other theoretical models, discussed in this paper, predicted the formation of electronic sub bands in confined lead structures. As a consequence of this, the theoreticians predicted that oscillations of could be observed in the resistivity, the work function and the surface charge. In the case of lead the oscillations would occur with a periodicity of 2 ML. The stability of the confined lead structures would be high when the surface of the material coincides with a minimum in the Friedel oscillations.
The predicted sub bands were indeed observed by STM and photoemission. Also the 2 ML periodicity in the resistivity was observed in experiment. However, the theory lacks to give good descriptions of the stability of layers of certain thickness. To give accurate predictions, I think it is necessary to take thickness oscillations and the wetting layer into account.
Many experiments showed the presence of oscillations in the monolayer thickness as a function of the thickness of the complete layer. The discussed theory, however,
20

assumes that this lattice parameter is a constant. This assumption can lead to wrong conclusions as has been proved that the quantum size effects strongly depend on the total width of the potential well.
The described theoretical models do not describe the wetting layer. The presence of this layer can influence the Friedel oscillations as well as the thickness of the material has therefore an important influence on the quantum size effects.
The review on the film growth showed that films of Pb can be grown on several isolating substrates when the lead is deposited at a temperature between 100 and
130K. When Pb is deposited at higher temperatures islands are formed. The islands are flat on the top and they have steep edges.
Part of the researchers state that before the growth of islands or thin films a wetting layer is formed. However, no consensus is reached yet on that issue.
The experimental review showed that the behaviour of the confined systems indeed deviates from the bulk material. STM and
Photoemission experiments show that electronic sub bands are formed and also oscillations in the resistivity where observed.
However, no agreement is found yet on the fact whether this quantum well states should be present in films of all thicknesses or only the even ones. The differences between the experiments might be due to the different techniques that were used to obtain the electronic structure. So, my suggestion is to determine the electronic structures of the same materials with different techniques to see if the methods can indeed be the cause of the observed differences.
Oscillations were observed in the thickness of the monolayers. They are probably due to minimization of the mismatch between the growing film and the quantum well states.
Uniform steep- edged, flat top Pb islands were observed in many experiments.
However the height of the magical islands still is an object for debate. I think the observed differences have their origin in the fact that it is not sure whether our not a wettinglayer is present.
All reported publications agree on the fact that situations with the largest distance between the HOS and the Fermi level are the most stable. Thus the disagreement, on which islands are the most stable, is probably due to the lack of convention in numbering the monolayers.
Concluding, more research needs to be done on the wettinglayer and the influence of the used technique on the observed electronic structure. New theories should be developed that include thickness oscillations and the wettinglayer. The disagreement on the stability of the confined lead structures is most likely due to deviating definitions of the thickness of the confined structure. This can be easily solved, by always measuring the height with respect to the substrate and not the wettinglayer.
6. Acknowledgements
I would like to thank Petra Rudolf for supervising me during this project.
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The picture on the cover was taken from
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22