Download Photoemission studies of quantum well states in thin films

Photoemission studies of
quantum well states in thin ®lms
T.-C. Chianga,b
a
Department of Physics, University of Illinois, 1110 West Green Street,
Urbana, IL 61801-3080, USA
b
Frederick Seitz Materials Research Laboratory, University of Illinois,
104 South Goodwin Avenue, Urbana, IL 61801-2902, USA
Amsterdam±Lausanne±New York±Oxford±Shannon±Tokyo
182
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
Contents
1. Introduction
2. Basic properties of thin ®lm quantum wells: Ag on Au(1 1 1)
2.1. Con®nement by a relative gap
2.2. Model calculations
2.3. Wave functions and quantum numbers
3. Resonances, coupled quantum wells, and superlattices
3.1. Incomplete con®nement and resonance states: Ag(1 1 1) + Au + Ag
3.2. Coupling between quantum wells: Au(1 1 1) + Ag + Au + Ag
3.3. Superlattices
4. Mismatched interfaces
4.1. Ag on Cu(1 1 1)
4.2. Ratcheting quantum well peaks Ð Ag on Ni(1 1 1)
5. Atomically uniform ®lms as quantum wells and electron interferometers
5.1. Discrete layer thicknesses
5.2. Preparing atomically uniform ®lms
5.3. Intensity modulations
5.4. The Bohr±Sommerfeld quantization rule and band structure determination
5.5. Quantum wells as electron interferometers
5.6. Temperature dependence of the band structure
5.7. Quasiparticle lifetime and scattering by defects, electrons, and phonons
5.8. Interfacial re¯ectivity and phase shift
6. Magnetic effects and spin polarization
7. Summary and conclusions
Acknowledgements
References
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Surface Science Reports 39 (2000) 181±235
Photoemission studies of quantum well states in thin ®lms
T.-C. Chianga,b,*
a
Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801-3080, USA
b
Frederick Seitz Materials Research Laboratory, University of Illinois, 104 South Goodwin Avenue,
Urbana, IL 61801-2902, USA
Manuscript received in final form 9 May 2000
Abstract
Con®nement of electrons in small structures such as a thin ®lm results in discrete quantum well states. Such
states can be probed by angle-resolved photoemission, as ®rst predicted by theory in 1983 and later observed
experimentally in 1986. Since then, numerous advances have been made in this ®eld, and the purpose of this
report is to review the basic physics and applications of quantum well spectroscopy. The energies and lifetime
widths of quantum well states in a ®lm depend on the ®lm thickness, the dynamics of electron motion in the ®lm,
and the con®nement potential. A detailed study allows a determination of the bulk band structure of the ®lm
material, the lifetime broadening of the quasiparticle, and the interfacial re¯ectivity and phase shift, as will be
demonstrated with simple examples. Quanti®cation of the photoemission results can be achieved via a simple
phase analysis based on the Bohr±Sommerfeld quantization rule. Explicit forms of wave functions can also be
constructed for additional information regarding the spatial distribution of the electronic states. From such
studies, a detailed understanding of the behavior of simple quantum wells including the effects of lattice
mismatch can be developed, which provides a useful basis for investigating the properties of multilayers.
Examples of multilayer systems including coupled quantum wells and superlattices will be presented and
discussed. An important recent development in this ®eld is the preparation of atomically uniform ®lms, as
demonstrated for Ag(1 0 0) grown on Fe(1 0 0) whisker substrates. The elimination of atomic layer ¯uctuation
allows a precise measurement of the intrinsic line widths of quantum well states, which are related to the
quasiparticle lifetime and the interfacial re¯ectivity. The underlying physics can be described in terms of electron
interferometry, and a Fabry±PeÂrot analysis yields the quasiparticle lifetime. It consists of three major
contributions including electron±electron scattering, electron±phonon scattering, and defect scattering. The
experimental results are compared with theoretical predications based on the Fermi-liquid theory and a
perturbative treatment of the electron±phonon coupling. Since the absolute ®lm thickness in terms of monolayers
is known from layer counting, the same interferometric analysis yields an electronic band structure of Ag with
unprecedented accuracy (< 30 meV). The resulting Fermi wave vector challenges the de Haas±van Alphen value.
Film structures containing magnetic materials can exhibit spin-split quantum well states, and their magnetic
*
Corresponding address: Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801-3080,
USA. Tel.: ‡1-217-333-2593; fax: ‡1-217-244-2278.
E-mail address: [email protected] (T.-C. Chiang).
0167-5729/00/$ ± see front matter # 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 5 7 2 9 ( 0 0 ) 0 0 0 0 6 - 6
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T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
sensitivity makes them suitable candidates for device applications. Quantum well spectroscopy is useful for
clarifying the relationship between quantum con®nement and translayer magnetic coupling effects. # 2000 Elsevier Science B.V. All rights reserved.
1. Introduction
Discrete states form when electrons are con®ned in space by a potential well. An elementary example
is the hydrogen atom problem described in standard quantum mechanics textbooks. The electron in a
hydrogen atom is bounded by a three-dimensional Coulomb well, and the resulting states form a set of
energy levels that are well described by the Bohr theory, which is a major cornerstone for modern
quantum mechanics. Likewise, electronic states are quantized in molecules, which represent more
complex con®nement potentials.
Quantum well states, the subject of this report, are usually associated with discrete quantized
electronic states in small arti®cial structures with adjustable physical dimensions. Although the basic
physics is similar, energy levels in atoms or molecules are generally not referred to as quantum well
states. Arti®cial structures allow tailoring of properties, and metastable and nonequilibrium structures
that are not found in nature due to thermodynamic requirements may offer unique opportunities for
applications. Many electronic devices are made of thin ®lms. When ®lms become thin enough, quantum
mechanical effects become important. Because of a widespread interest in electronic device
applications and a long history of research and development in this area, thin ®lms have been a
major playground and test ground for quantum well effects. However, in recent years, interest in
quantum structures of other forms has been growing. These include quantum dots, corrals, wires,
stripes, etc. Research in such low-dimensional systems has been greatly aided by advances in
nanofabrication techniques including self-assembly, template growth, and direct atomic manipulation.
As the system dimensions reduce, there is an increased localization and overlap of electronic wave
functions, and consequently, an increase in electron correlation. Thus, low-dimensional systems offer a
convenient platform for experimenting with many-body effects.
This paper will focus on quantum well effects in thin ®lms. A simple beginner's model for electronic
motion perpendicular to the ®lm surface is that of a free electron con®ned in a one-dimensional box.
Although this is a very crude model, it serves to illustrate the basic ideas and provides a good starting
point. The allowed wave vectors k for stationary states, or quantum well states, are determined by the
requirement that standing wave patterns ®t into the geometry
np
(1)
kˆ ;
d
where n is an integer quantum number and d is the ®lm thickness or box dimension. The energy levels
are given by
h2 k 2
h2 np2
;
(2)
Eˆ
ˆ
2m
2m d
where m is the free electron mass, and the wave functions are given by
npz
c…z† / sin
:
d
(3)
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
Fig. 1.
size d.
185
Probability densities for the n ˆ 1, 2, and 3 quantum well states of a particle con®ned in a one-dimensional box of
Fig. 1 shows the probability density for the ®rst three quantum well states n ˆ 1, 2, and 3. The
quantum number n is just the number of antinodes (or maxima). These results can be found in standard
textbooks of quantum mechanics. Although this is effectively a one-dimensional problem, the
electronic wave functions are extended within the ®lm along the x and y directions. The con®nement
does not generally lead to enhanced electron correlation effects compared to the bulk case (except
possibly when the ®lm thickness is reduced to atomic dimensions).
For a solid ®lm, the E(k) band dispersion relation is generally different from the free electron
dispersion given in Eq. (2). This has a direct bearing on the measured energies of the quantum well
states, En. Thus, a measurement of En can provide useful information about E(k). Determination of E(k)
has been a major issue in solid state physics [1±6]. It will be shown in the ensuing discussion that
quantum well spectroscopy is a powerful method for bulk band structure determination. Note that the
quantization condition given in Eq. (1) is valid only for an abrupt in®nite barrier. For solid±solid and
solid±vacuum interfaces, the con®nement potential is generally ®nite and rounded. Thus, the
quantization condition must be modi®ed by a phase shift. This phase shift information can be deduced
from quantum well spectroscopy. It is an important quantity characterizing the boundary potential.
Quantum well lasers and quantum devices are familiar to all, and many would associate quantum
well effects with semiconductor systems. The fundamental gap in semiconductors plays a critical role
in many device applications. Electrons with energies in a gap cannot propagate, and therefore, it is quite
common to employ the fundamental gap as a means for carrier con®nement. The situation with a metal
is less obvious. There are no absolute gaps at the Fermi level in metals, and electrical conduction
through a metal-to-metal junction is an everyday experience. There are no such things as threshold
voltages or rectifying junctions. For these reasons, one might have the impression that con®nement in a
metallic system would be impossible or rare. It is not. For epitaxial ®lms, all that is required is a
``relative gap'' in the substrate. Namely, if there is a gap for a particular direction (usually
perpendicular to the ®lm), electrons propagating along that particular direction can be con®ned. If one
performs direction-speci®c spectroscopic measurements, such as angle-resolved photoemission, such
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T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
con®nement effects can be easily observed. Relative gaps are actually fairly common in metals [7,8].
Electrical conduction through a metal-to-metal junction is not a direction-speci®c probe of the junction
properties. It involves a wide range in k space, and therefore the gap effect is generally not apparent.
In addition to the relative gap, con®nement can be accomplished by the so-called symmetry gap or
hybridization gap in a metallic system. An example to be presented below is Ag on Fe. The d bands in
Fe are near the Fermi level, where the sp band exhibits an effective gap due to hybridization with the d
bands. This is not a true gap, but the re¯ectivity for an sp electron incident on a Fe substrate can be
high, and spectroscopic studies can yield results very similar to the truly con®ned case. These partially
con®ned states are often referred to as quantum well resonances.
A gap is effective for electron con®nement if the overlayer and substrate are lattice matched or have a
commensurate epitaxial relationship. However, more often than not, the interface is incommensurate or
has a rather complex atomic arrangement. This can cause non-specular or umklapp re¯ections, resulting
in damping of the wave function. How this affects the photoemission spectra is an interesting issue and
will be addressed.
Most of the examples below are drawn from metallic systems. In many of them, a noble metal such as
Ag is chosen as the overlayer material. The reason for this choice is simplicity. The band structure of a
noble metal near the Fermi level is nearly free-electron-like, and there is only one band crossing the
Fermi level. One does not need to worry about issues such as different band masses for the heavy and
light holes as in typical semiconductor structures [9]. The spectra are thus much simpler, and
interpretation of experimental data is straightforward. Likewise, the alkali metals are excellent
candidates for exploring the basic behaviors of quantum well states.
Here, we will make some historical remarks in regard to the development of quantum well
spectroscopy for ®lm studies. In 1975, Jaklevic and Lambe reported the observation of oscillatory I±V
curves in a transport measurement of a ®lm structure [10]. They recognized this as a manifestation of
quantum size effects, and pointed out the connection among the oscillations, ®lm thickness, and band
structure. Later, a group led by Park observed oscillatory I±V transmission curves for a ®lm using a low
energy electron diffraction setup [11±14]. This was attributed to quantum interference effects. These
early measurements clearly established the importance of quantum size effects in ®lms. At about the
same time, semiconductor devices based on quantum con®nement, such as the quantum well laser, were
being developed [15]. An important signature was the blue shift of lasing frequencies as the active
region became thin.
Angle-resolved photoemission is an extremely powerful technique for measuring the electronic
properties of solids [1±6]. Unlike photoluminescence, which is widely employed for studies of
semiconductor structures and mostly yields the energy difference between the highest occupied state
and the lowest unoccupied state, photoemission yields information about the occupied states directly. In
the early 1980s, a large number of solid ®lms were examined with this technique. The ®rst paper
recognizing the importance of quantum size effects was a theoretical paper by Loly and Pendry [16].
They pointed out that photoemission from thin ®lms should reveal a set of sharp peaks, and a
measurement of the peak positions and widths should permit a precise determination of the band
structure and photohole lifetime. However, their paper was largely ignored by the community at that
time because nobody had reported such observations despite the many photoemission experiments
performed on ®lms. In fact, it was often argued, even demonstrated experimentally [17], that
photoemission was such a surface sensitive probe that a ®lm as thin as a few monolayers should exhibit
spectral lineshapes indistinguishable from the bulk. In retrospect, many of the failures were likely
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
187
caused by ®lm roughness. The ®lms were probably so rough that signatures of quantum size effects
were smeared out beyond recognition.
The ®rst photoemission observation of quantum size effects was reported in 1986 [18]. The evidence
was clear but the quantum well peaks were very broad, again due to ®lm roughness. Later work,
however, clearly established the importance of quantum size effects in ®lms [19±25]. The argument that
photoemission senses only the top few atomic layers and is therefore insensitive to the ®lm thickness is
¯awed. This will become clear from an interferometric formulation of the problem to be presented
below. Other related methods for experimental observation of quantum size effects in ®lms include
re¯ection high energy electron diffraction [26], inverse photoemission [27±29], low energy electron
microscopy [30], scanning tunneling microscopy [31,32], low energy electron re¯ection [33], etc. Any
experiment involving either an electron beam incident on a ®lm or electron emission from a ®lm is
likely to reveal quantum interference effects under appropriate conditions.
This report is organized as follows. In Section 2 the results from a photoemission study of the Ag±Au
system will be reviewed. This is a very simple model system, and provides a nice illustration of the
basic features including the thickness dependence and the effect of the con®nement gap. The analysis
will be based on a two-band model, which yields realistic wave functions that can be used for analyzing
more complex structures. The topics for discussion in Section 3 include partial con®nement, quantum
well resonances, coupled quantum wells, and superlattices. Section 4 focuses on issues related to lattice
mismatch. Section 5 presents results obtained from Ag on Fe(1 0 0), which can be prepared in the form
of atomically uniform ®lms. An interferometric formulation will be developed and used to analyze the
lineshape of the quantum well peaks, from which the quasiparticle lifetime widths can be deduced and
decomposed into various contributions in terms of the interactions of elementary excitations. Section 6
contains a discussion of the spin polarization of quantum well states in systems containing magnetic
materials. The relationship between quantum well effects and magnetic behavior will be addressed.
2. Basic properties of thin ®lm quantum wells: Ag on Au(1 1 1)
The lattices of Ag and Au are almost perfectly matched. Ag grows epitaxially on Au(1 1 1) to form a
(1 1 1) ®lm. The interface is abrupt for growth at room temperature or below. For such a commensurate
interface, the momentum component parallel to the surface and the interface, kk , is a conserved
quantity. Most of the experimental work to be discussed below employs a normal-emission geometry.
The photoemitted electron has kk ˆ 0, and so does the initial state. The only momentum component of
interest would be that perpendicular to the surface, k?, along the [1 1 1] direction. For simplicity, we
will use k to denote this perpendicular component, as in the discussion above for the one-dimensional box.
2.1. Con®nement by a relative gap
Before proceeding, it is useful to review the band structures of Ag and Au. Fig. 2 shows the band
structure of Ag along major symmetry directions [7,8,34]. The complicated manifold at energies 4±
8 eV below the Fermi level consists of the Ag 4d bands. Between these bands and the Fermi level, there
is only one band, the free-electron-like sp band, in each direction. It crosses the Fermi level along
[1 0 0] and [1 1 0], but does not quite reach the Fermi level along the [1 1 1] direction, the direction
probed by the normal-emission geometry. Thus, the Fermi level lies in a relative gap along the [1 1 1]
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T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
Fig. 2. Band structure of Ag. The dashed circle indicates the region of interest for normal-emission from Ag(1 1 1).
direction. The dashed circle in Fig. 2 highlights the region of interest. The band structure of Au is
similar, with the d bands being closer to the Fermi level.
Fig. 3 shows a magni®ed view within the region of interest for Ag and Au. Here, we have changed
the vertical axis from ``energy'' as used in Fig. 2 to ``binding energy''. The two scales are related by a
sign reversal. For photoemission, it is customary to work with the binding energy scale, since the main
interest is in the occupied states. The top of the Au sp band is at 1.1 eV binding energy (marked by a
horizontal dashed line), compared to 0.3 eV for Ag. Along this particular direction, each crystal looks
like a semiconductor with a gap. This relative gap supports an occupied Shockley surface state [35±38].
Fig. 3. Band structure of Ag and Au near the Fermi level along the [1 1 1] direction. The dashed line indicates the valence
band maximum of Au, which is also the threshold of con®nement for the quantum well states in Ag.
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
189
Fig. 4. Normal-emission spectra taken with hn ˆ 10 eV for Au(1 1 1), Ag(1 1 1) ‡ 20 ML Au, Ag(1 1 1), and
Au(1 1 1) ‡ 20 ML Ag.
Imagine sending an electron with energy within the gap from vacuum towards the crystal surface. The
electron cannot propagate inside the crystal and must be re¯ected backwards. If the energy is also
below the vacuum level (work function), the electron cannot escape and is therefore con®ned. This
results in a surface quantum well, which can support a set of quantized states, or surface states. For
Ag(1 1 1) and Au(1 1 1), there is one such surface state lying between the top of the sp band and the
Fermi level [39±41].
Fig. 4 shows normal-emission spectra taken from Au(1 1 1), Ag(1 1 1) ‡ 20 ML Au, Ag(1 1 1), and
Au(1 1 1) ‡ 20 ML Ag [20], where ML denotes a monolayer. Here, the Au(1 1 1) substrate is really a
thick Au ®lm grown on a bulk crystal of Ag(1 1 1). The Ag(1 1 1) spectrum shows a peak just below the
Fermi level followed by a smooth background-like continuum emission. The peak represents emission
from the Shockley surface state just mentioned. The continuum emission is mostly derived from surface
photoemission from the sp band, and the contribution from inelastic scattering events is relatively
minor [42±46]. The rise at high binding energy is caused by a tail from a direct-transition peak. Surface
photoemission is nonselective in k?, and thus the entire sp band contributes, giving rise to a continuum
emission. The spectrum from Au(1 1 1) is very similar. The surface state is at a larger binding energy,
and the rise at the high binding energy end is due to d band emission.
The spectrum for Ag(1 1 1) ‡ 20 ML Au looks very similar to that from bulk Au(1 1 1). The same
surface state is observed. The wave function of the surface state does not penetrate very far into the
®lm, and is therefore insensitive to the ®lm±substrate interface. For this reason, the binding energy
should be essentially the same, as the experiment shows. Likewise, the surface state peak for
Au(1 1 1) ‡ 20 ML Ag looks the same as that for Ag(1 1 1). However, the part of the spectrum below
the surface state looks different, with the ®lm sample showing three peaks. These are quantum well
peaks corresponding to electrons in the Ag ®lm con®ned by the relative gap in Au. These peaks are
labeled n ˆ 1, 2, and 3, where the state label n denotes the nth peak below the surface state. The n used
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T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
Fig. 5. Normal-emission spectra taken with hn ˆ 10 eV for Ag overlayers with indicated thicknesses on Au(1 1 1). The
dashed curves indicate the evolution of the n ˆ 1 ± 3 quantum well peaks.
here is related to, but not the same as, the quantum number n used in Eqs. (1)±(3), and its physical
meaning will be clari®ed below.
A simple test of the quantum size effect is to change the ®lm thickness. Eq. (1) shows that as d
increases, the difference between neighboring allowed values of k decreases, and so does the energy
difference. In the limit of a very large d, the allowed states should form a continuum, and one should
recover the band structure of the bulk solid. Fig. 5 shows a set of normal-emission spectra for Ag ®lms
of various thicknesses. Clearly, the peaks become more crowded as d increases, in qualitative
agreement with the expectation. As the ®lm thickness increases, the observed quantum well peaks
evolve and converge toward the top of the sp valence band. In the limit of an in®nitely thick ®lm, one
can imagine that the peaks merge to form a continuum.
The vertical dashed line in Fig. 5 indicates the threshold for con®nement. Intense quantum well peaks
are observed only to the right of this line. This line, at a binding energy of 1.1 eV, corresponds to the top
of the Au sp band (dashed horizontal line in Fig. 3). Above this energy, the electrons in the Ag ®lm are
within the relative gap of Au and therefore con®ned by the Au potential, forming quantum well states.
Below this, the electrons can couple to the bulk states in the Au substrate and are therefore uncon®ned.
Some weak and broad features are present in the spectra, which can be attributed to quantum well
resonances due to partial re¯ection by the boundary potential. The same reasoning explains why the
spectrum for Ag(1 1 1) ‡ 20 ML Au in Fig. 4 shows no quantum well peaks. The Au sp band does not
overlap the gap in Ag. None of the electrons in the Au ®lm is con®ned, and therefore there should not
be any quantum well states. These results illustrate the concept of con®nement by a relative gap. It is
important to note that con®nement by a relative gap is generally dependent on the probing direction.
Here, if the detection direction is moved off normal, there might not be a gap for a suf®ciently large
emission angle [47]. Again, if one measures the electrical resistance across a Ag±Au junction, one
would not detect a gap, because such a measurement is not k selective.
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
191
2.2. Model calculations
The sp band structures of Ag and Au are nearly free-electron-like, and can be well described by the
usual two-band model [35,38,48] with a minimal number of parameters. Despite the simplicity, this
model is extremely powerful for simulating experimental results, and can be easily extended to
calculations involving multilayer ®lm structures. In the two-band model, two plane wave components
and a single pseudopotential form factor are employed. The wave vector k as a function of binding
energy E (referred to the Fermi level) is given by
2 2
2 2 1=2 )1=2
( 2 2
2mi;f 1=2 hp
h
p
h p
ÿ V ‡ Evbm ÿ E ÿ V 2 ‡ 4
ÿ V ‡ Evbm ÿ E
;
ki;f ˆ p ÿ
2
mi;f
2mi;f
2mi;f
h
(4)
where p B p/t, t is the monolayer thickness and p is the wave vector at the Brillouin zone boundary (the
L point for the [1 1 1] direction). The subscripts ``i'' and ``f'' refer to the lower and upper sp bands,
respectively, separated by the relative gap. For photoemission from a bulk single crystal using low
photon energies for excitations across the gap, bands i and f are just the initial and ®nal bands [42,43].
The quantities mi,f are the effective masses associated with the two bands. The quantity V is the absolute
value of the pseudopotential form factor; it equals one-half of the gap at the zone boundary. The
quantity Evbm is the binding energy of the valence band maximum (the top of the lower sp band). It is
positive for the [1 1 1] direction, and negative for the [1 0 0] direction (see Fig. 2).
For each crystallographic direction, this model contains just four adjustable parameters mi,f, V, and
Evbm. The parameters V and Evbm determine the positions of the band edges, and they are the only
parameters used in the standard nearly free-electron model. The free electron mass is replaced by the
parameters mi,f here in order to set the curvatures of the lower and upper bands correctly. They
represent higher order corrections from multi-band effects. Note that mi,f do not equal the inverse
curvatures of the bands. There are other ways to parameterize the band structure [38], but four
parameters represent the minimum requirement. Eq. (4) can be inverted to express the binding energy E
(increasing downward) in terms of k. The formula is
!1=2
h2 …p ÿ k†2
h2 p2 h2 …p ÿ k†2
2
4
‡V
;
(5)
E ˆ Evbm ÿ V ÿ
2mi;f
2mi;f 2mi;f
where the ‡ sign and mi should be used for the lower band and the ÿ sign and mf should be used for the
upper band.
The wave functions are given by
2
…
h2 p2 =2mi;f † ÿ V ‡ Evbm ÿ E ÿ …h2 ki;f
=2mi;f †
exp…i…k ÿ 2p†z†;
(6)
ci;f …z† / exp…ikz† ÿ
V
where the two plane wave components are explicitly shown. This wave function can be either a
propagating Bloch state or an exponentially damped (or exponentially growing) oscillatory state
depending on the energy relative to the gap. If the energy is within the gap, k is complex, giving rise to
an exponential factor. Only the exponentially damped term should be included in the solution for a
semi-in®nite substrate (corresponding to total re¯ection). For a ®lm, however, both the exponentially
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damped and growing terms must be included, and the coef®cients are determined by the boundary
conditions.
Since the two-band model is based on the nearly free-electron approximation (or the lowest order
pseudopotential model), it provides a good representation of the wave function only in the interstitial
region. Orthogonalization to the core wave functions (including the d wave functions) is not taken into
account explicitly, and the real wave functions could look much more complicated near the cores. The
size of the core region is a fraction of the inter-atomic distance. It has little effect on the matching of
wave functions across a boundary, which is in the interstitial region. Photoemission intensity is
governed by the square of the transition matrix element, and the contribution from the core region,
proportional to the core volume, is generally negligible.
Another simple model that is often used for quantum well calculations is the tight-binding model
[49,50]. A set of overlap integrals is used as parameters in a model Hamiltonian in the form of a matrix.
These parameters are adjusted to reproduce the bulk band structure. Diagonalizing the Hamiltonian yields
the wave functions and eigenvalues of the system. Additional parameters may be used for coupling terms
across an interface. This model is better suited for bands with small dispersions, such as the d states.
2.3. Wave functions and quantum numbers
The following discussion will be based on the two-band model. To construct the wave functions for
quantum wells such as Ag(1 1 1) on Au(1 1 1), one begins with appropriate choices of the four
parameters each for Ag(1 1 1) and Au(1 1 1) to reproduce the bulk band structure. First, let us consider
the symmetric case where a Ag ®lm is sandwiched between two semi-in®nite Au(1 1 1) crystals. The
wave function in the Ag ®lm is a linear combination of ci(z) and ci(ÿ z). This wave function must be
joined to an exponentially damped wave function in the Au crystal to the right. The usual boundary
conditions apply; namely, the wave function and its ®rst derivative must be continuous (or the
logarithmic derivative must be continuous if the normalization of the wave function is of no concern).
This ®xes the mixing ratio between ci(z) and ci(ÿ z) in Ag. The same boundary condition must also
apply to the interface at the left. In general, this cannot be satis®ed except at certain energies. These
special energies de®ne the allowed energies of quantum well states.
The probability densities for several quantum well states obtained from such a calculation for a
24 ML Ag ®lm sandwiched between two semi-in®nite Au crystals are shown in the left-hand side panel
of Fig. 6. Here the state label n is the same as that used in Fig. 5. In other words, the quantum number n
is used to label the nth state below the valence band maximum, and is not the same as that used in
Eqs. (1)±(3). Going back to the particle-in-a-box model, the quantum number n is the number of
antinodes in the probability density. Counting the number of antinodes between the two vertical lines
representing the two interface boundaries, there are 23, 22, and 21 of them for the bottom, middle, and
top states depicted in the left-hand side panel of Fig. 6. Thus, the quantum numbers would be 23, 22,
and 21, respectively, based on that scheme. If we call the quantum number used in Fig. 6, n, and the
quantum number based on the number of antinodes in the probability density, n0 , the relationship is
n ˆ N ÿ n0 , where N ˆ 24 is the number of monolayers in the ®lm. The quantum number n0 is large in
this case, because we are probing states near the zone boundary, where the wave vector k p/t is large.
Substituting k p/t in Eq. (1) yields n0 N.
The rapid oscillations of the wave functions in Fig. 6 are associated with the zone boundary wave
vector. These oscillations are modulated by a slowly varying envelope function [9] due to beating
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193
Fig. 6. Left-hand side panel: theoretical probability densities for the n ˆ 1 ± 3 quantum well states in a 24 ML Ag(1 1 1) ®lm
sandwiched in-between two semi-in®nite Au(1 1 1) crystals. Right-hand side panel: the same except that the Au(1 1 1) crystal
on the right is replaced by vacuum. The vertical lines indicate the Ag±Au and Ag±vacuum boundaries.
between the plane wave components. Focusing on the envelope function, one sees that the three states,
from bottom to top, have 1±3 antinodes, respectively. Thus, the quantum number n shown in Fig. 6 can
be identi®ed as the number of antinodes in the envelope function. Clearly, as the quantum number n
increases, the envelope function will vary more rapidly, and eventually will lose its meaning. For this
reason, envelope functions are useful only for quantum well states near the zone boundary.
Referring back to Fig. 3, the band structure near the valence band maximum is roughly an inverted
parabola (hole-like). One can imagine reformulating the problem by making a local expansion about
the valence band maximum relative to k measured from the zone boundary. This is basically a Wannier
transform to remove the rapid oscillatory part of the wave function, and the resulting wave function will
resemble the envelope function. The quantum number n is then again compatible with the particle-in-abox model provided that the energy scale is inverted with the zero at the valence band maximum. This
approach is often adopted in semiconductor physics.
There is a good reason that we will want to use the quantum number n as shown in Figs. 5 and 6
rather than the other. In Fig. 5, the state n ˆ 1 is always the one closest to the valence band maximum.
As the ®lm thickness increases, the ®rst peak evolves smoothly, and always has the same quantum
number. If we use the other scheme, the quantum number would be changing from 9, to 11, to 14, etc.
Changing the state label for a seemingly continuously evolving peak is a little awkward. It is also more
natural to start counting the quantum numbers from unity. Of course, this is largely a matter of personal
taste.
The calculation for a Ag ®lm on a Au substrate is similar. The boundary condition at the Ag±vacuum
interface can be modeled in many ways. The simplest approximation is an abrupt interface [38]. The
wave function in vacuum is an exponential function damped towards the vacuum, and the damping is
determined by the difference between the energy of the electron and the vacuum level. There is,
however, a question about the location of the boundary. Physically, the electrons in the Ag spill out
a little into vacuum, and the effective boundary should be slightly outside the boundary de®ned by the
so-called positive background. The positive background refers to the region in space covered by
modeling each atomic layer as a uniform slab of thickness t centered about the atomic plane. Thus, the
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edge of the positive background is just 12 t beyond the last atomic plane. For the two-band calculation,
this vacuum boundary position is adjusted to reproduce the measured binding energy of the Shockley
surface state.
The calculation for the Ag surface state is straightforward. An exponentially damped gap state in Ag
is matched to an exponentially damped plane wave state in vacuum. Solving the resulting eigenvalue
equation gives the surface state energy. Matching the surface state energy to the experimental value
guarantees that the phase shift is exactly correct at the energy of the surface state. Since we are
concerned mostly with energies near the Fermi level (also near the Shockley surface state), the error
introduced by using this model should be minimal. Other boundary potentials such as the image
potential, a linear potential, etc., can be employed, but the differences are minor and do not warrant the
extra effort at this level of approximation [51±53].
The wave functions for the n ˆ 1 ± 3 quantum well states for a 24 ML Ag ®lm on a Au substrate are
shown in the right-hand side panel of Fig. 6. The wave functions look similar to the ones in the lefthand side panel, but are no longer symmetric. They are damped very rapidly on the vacuum side,
because the energy involved is far below the vacuum level. It is interesting to note that the n ˆ 3 state
has a long tail going into the Au substrate. Referring to Fig. 5, this state at N ˆ 24 is just slightly above
the threshold of con®nement. In other words, this state is just barely con®ned, and there should be a
long exponential tail going into the substrate. At a slightly smaller ®lm thickness, this state becomes
Fig. 7. (a) Left-hand side panel: sample con®guration involving a Ni monolayer sandwiched in-between two Cu wedges.
Right-hand side panel: the Ni layer position within the Cu quantum well across the diagonal BD. (b) Photoemission intensity
map at the Fermi level across the sample surface (®gure taken from [57]).
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195
uncon®ned, and the wave function extends to z ˆ ÿ 1. For the same reason, the wave function for
n ˆ 2 penetrates a little deeper into the Au substrate than n ˆ 1. Similar calculations for this system can
be found in [54,55]. The evolution of the quantum well states from a ®lm exposed to vacuum to a ®lm
capped by the substrate material has been examined experimentally in a related system [56].
The surface state in Fig. 5 is labeled as n ˆ 0. This state can be regarded as one of the quantum well
states. The main difference from the other states is that its binding energy and wave function become
independent of the ®lm thickness for ®lms thicker than just a few monolayers.
An interesting way to probe the envelope function experimentally is illustrated in Fig. 7(a) [57]. The
sample con®guration consists of two mutually orthogonal Cu(1 0 0) wedges separated by a 1 ML Ni
layer grown on a Co(1 0 0) substrate. The Ni layer here is employed as a perturbing layer in a Cu
quantum well made of the two wedges. Along the diagonal AC, the Ni layer is always at the center of
the Cu quantum well as depicted, and the well itself has a continuously varying thickness. Along the
other diagonal BD, the total Cu thickness is constant, while the Ni layer is swept continuously from one
side of the Cu quantum well to the other. The grey scale image in Fig. 7(b) is a measure of the normal
photoemission intensity at the Fermi level across the sample surface. Two types of oscillations are
detected. The oscillation along AC is due to different quantum well peaks moving through the Fermi
level for increasing quantum well width (thin at A, thicker at C). The oscillation along BD can be
related to the envelope function of the quantum well state. The interpretation is that, as the Ni layer
position in the quantum well varies continuously, it perturbs the wave function in different ways. When
its position coincides with a node in the envelope function, the perturbation is a minimum, and the
photoemission intensity remains a maximum at the Fermi level crossing. When it coincides with an
antinode, the perturbation is a maximum, causing the emission intensity to decrease signi®cantly. This
example demonstrates that wedge samples can greatly facilitate experiments designed to probe the
quantum well behavior as a function of layer con®guration.
3. Resonances, coupled quantum wells, and superlattices
The above example illustrates the basic physics of thin ®lm quantum wells. One can go beyond and
explore the physics of multilayers and superlattices. Layering of different materials offers opportunities
for tailoring properties and for exploring interesting physical phenomena. A few examples based on the
Ag±Au system will be given to cover some major topics of interest, including leaky quantum wells,
coupling between quantum wells, and superlattice effects [58±64]. Again, some of the advantages of
Ag±Au are a close lattice match, a simple nearly free-electron-like band structure, and the availability
of high quality ®lms. For Ag±Au multilayer systems, the two-band model described above can be
extended to simulate the results and to provide a framework for making predictions.
3.1. Incomplete con®nement and resonance states: Ag(1 1 1) ‡ Au ‡ Ag
The system under consideration is Ag(1 1 1) ‡ x ML Au ‡ y ML Ag [58]. When x is large, the
situation reduces to the previous case of a simple quantum well as far as the electrons in the Ag
overlayer are concerned. What happens when x is small? The quantum well states in the Ag overlayer
do penetrate somewhat into the Au. If the Au layer thickness becomes comparable to the penetration
depth, the wave function in the Ag overlayer can couple to the continuum states in the Ag substrate.
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Fig. 8. Normal-emission spectra taken with hn ˆ 10 eV for the leaky quantum well system Ag(1 1 1) ‡ x ML Au ‡ 15 ML
Ag. The three spectra are for x ˆ 0, 3, and 2 as indicated. Also shown are theoretical spectra for x ˆ 3 and 2.
Effectively, one will have a leaky quantum well, and the quantum well states will become resonance
states as the con®nement becomes incomplete. The Au layer is employed here as a barrier, and varying
its thickness allows a continuous change in the degree of con®nement. The decay of the Ag quantum
well states is exponential in the Au, and therefore even a very thin Au layer can be an effective barrier.
The degree of con®nement also is a function of binding energy. A state near the Au band edge (1.1 eV)
would be much less con®ned than a state near the Fermi level (see Fig. 6).
Fig. 8 shows representative spectra. The bottom spectrum was taken with high resolution from a bulk
single crystal Ag(1 1 1). In addition to the surface state, the spectrum shows a direct-transition peak,
which corresponds to a vertical optical transition from band i to f as discussed above in connection with
the two-band model. The continuum part of the spectrum, between the surface state peak and the directtransition peak, is dominated by surface photoemission as discussed above [42±46]. A threshold for this
emission is seen just below the surface state peak, and corresponds to the valence band maximum. This
was not seen in Fig. 4, because those spectra were taken with a lower resolution. The other two spectra
in Fig. 8 are for Ag(1 1 1) ‡ x ML Ag ‡ 15 ML Ag, with x ˆ 2 and 3, respectively. Other than the
quantum well peaks labeled n ˆ 1 ± 3, these spectra resemble the single crystal spectrum. The surface
state is minimally affected by the Au barrier because the overlayer thickness of 15 ML is much larger
than the penetration depth of the surface state. The direct-transition peak involves the sp wave function
over many atomic planes (limited by the ®nal state mean free path), which should be fairly insensitive
to the 2 or 3 ML of Au inserted into the lattice.
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197
Note that the Au band edge is at 1.1 eV (Fig. 3). Only the n ˆ 1 peak has an energy within the Au
band gap. Nevertheless, the Au layer is so thin that none of the three peaks n ˆ 1 ± 3 are truly con®ned.
These are all resonance states. Measurements carried out at other photon energies indicate that these
resonance peaks have ®xed binding energies. Also shown in the ®gure are calculated spectra based on
the two-band model. The ®nal state for the photoemission process in the calculation is taken to be the
time-reversed low-energy-electron-diffraction state in accordance with the one-step model [3,6,65]. The
dipole matrix element for optical transition is evaluated using the two-band wave functions, and the
spectral lineshape is calculated using Fermi's golden rule. The quantum well peak positions are well
reproduced by the calculation. However, the observed quantum well peak widths are much broader than
predicted. This broadening is most severe for larger quantum numbers at higher binding energies. The
n ˆ 3 peak is broadened almost beyond recognition, an effect that can be attributed to sample
imperfection. Despite this extrinsic broadening, it is clear that the peaks for x ˆ 3 ML are sharper and
more intense than x ˆ 2 ML, both experimentally and theoretically. This is because a larger x provides
a higher degree of con®nement; a less con®ned state should have a larger width.
The n ˆ 1 and 2 peaks are at binding energies of 0.60 and 1.23 eV, respectively. Fig. 9 shows some
calculated initial state wave functions at these and other energies for x ˆ 2 ML. None of the states with
energies below the Ag valence band maximum are truly con®ned, and the allowed energies form a
continuous band. In this ®gure, the vertical dashed lines indicate the boundaries between Ag, Au, and
vacuum. The wave functions again appear as short-period oscillations superimposed on an envelope
function because the wave vector is near the zone boundary. Focusing on the envelope function, it is
clear that the n ˆ 1 resonance state is characterized by a good ®t of the ®rst antinode of the envelope
function into the Ag slab. When this happens, the electron becomes partially trapped in the slab, as
Fig. 9. Probability densities at binding energies 0.40, 0.60, 0.90, and 1.23 eV for the leaky quantum well system
Ag(1 1 1) ‡ 2 ML Au ‡ 15 ML Ag. The n ˆ 1 and 2 resonances are indicated.
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indicated by a higher probability density. Likewise, the n ˆ 2 resonance state is characterized by a good
®t of the ®rst two antinodes into the Ag slab. Between the two resonances, the wave function has a
reduced probability density within the Ag slab, as evidenced by the curve for 0.9 eV binding energy.
This example illustrates that quantum well resonances are associated with enhanced probability
densities within the Ag slab when the wave functions can ®t into the slab geometry forming a pattern
resembling a standing wave. Because of the ®nite probing depth governed by the mean free path in the
®nal state, an enhanced probability density within the ®lm should lead to a higher photoemission
intensity and thus a peak in the spectrum. In a sense, quantum well resonances are just broadened
quantum well states superimposed on a background of continuum.
When n becomes large, more antinodes in the envelope function are cramped into the Ag slab. The
Ag slab is likely to have imperfections such as atomic steps or layer thickness ¯uctuations. The
scattering probability caused by the imperfections is proportional to the thickness scale of the
imperfections divided by the wave length of the envelope function. Thus, imperfections in the Ag slab
structure at either the surface or the interface will cause a stronger damping for quantum well
resonances with larger n's. In the limiting case where the wavelength of the envelope function becomes
the same as the layer thickness ¯uctuation, the n and (n ‡ 1) resonances will become indistinguishable,
and the peaks will simply merge. Although a quantitative measure of the ®lm roughness is not available
in this case, it is conceivable that atomic steps and incomplete layering during ®lm growth can easily
lead to an effective layer thickness ¯uctuation of 1 ML or more. For the n ˆ 3 state, this would give
2
n ˆ 40%, and the binding energy of the resonance peak can
rise to a perturbation on the order of 15
become broadened by the same order, or 1 eV. A 1 eV broadening would make the n ˆ 3 peak nearly
unrecognizable, as seen in our data.
3.2. Coupling between quantum wells: Au(1 1 1) ‡ Ag ‡ Au ‡ Ag
Here, we will consider the coupling between two Ag quantum wells [59±61]. The sample is made by
starting with a Au(1 1 1) substrate and sequentially depositing 8 ML of Ag, 3 ML of Au, and x ML of
Ag. The two Ag slabs form quantum wells, and the 3 ML Au is a barrier layer. The Au substrate and the
vacuum provide con®nement for all occupied states up to the Au band edged at 1.1 eV binding energy.
The quantum well states in the two Ag wells are decoupled in the limit of an in®nitely thick Au barrier.
As the thickness of the outer well is varied, the binding energies of the outer well states evolve and
sweep across the energies of the inner well states. Such level crossings must be avoided if there is any
coupling between the two wells. By using a thin Au barrier to facilitate the coupling, the anticrossing
behavior can be probed directly with photoemission, and the dispersion is a direct measure of the
coupling strength. This coupling strength is an essential parameter for modeling more complex layer
structures.
The same two-band model can be employed to make predictions. The left-hand side panel of Fig. 10
summarizes the behavior of single quantum wells. The dashed curves labeled On, for n ˆ 1 ± 4, indicate
the binding energies of Ag single quantum well states for various thicknesses x. They represent the
outer well states of a double quantum well in the limit of an in®nitely thick Au barrier. They all
converge to the Ag valence band maximum as x becomes large. For an 8 ML Ag quantum well bounded
on both sides by semi-in®nite Au, only one inner well state is allowed (labeled I). In this limit, it is
decoupled from the outer well, and its binding energy is independent of x, as indicated by the horizontal
dashed line at E ˆ 0.67 eV in the ®gure. Note that state I intersects O1 and O2 at x ˆ 14 and 28 ML,
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
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Fig. 10. Left-hand side panel: theoretical binding energies (circles) for the Au(1 1 1) ‡ 8 ML Ag ‡ 3 ML Au ‡ x ML Ag
double quantum well system as a function of the outer well thickness x. The area of each circle is proportional to the
photoemission intensity. The dashed curves show binding energies for isolated inner (I) and outer (On, n ˆ 1 ± 4) well states.
The energy positions of the Au and Ag valence band maxima (VBM) and the Shockley surface state (S) are also indicated.
Right-hand side panel: the experimental results compared to the theoretical dispersions (dashed curves).
respectively, so the states involved are degenerated for decoupled quantum wells. In these places we
expect to observe avoided crossings if the Au barrier thickness is ®nite (3 ML in the experiment).
In the two-band calculation, the coupling is mediated by the wave functions in the Au barrier layer,
which are oscillatory wave functions modulated by exponential functions. Solving the boundary
conditions at all four boundaries in the system yields an eigenvalue equation. The solutions are plotted
in the left-hand side panel of Fig. 10 using circles at integer values of x. These represent the energies of
the coupled quantum well states, and are labeled A±E. The avoided crossings are apparent. For
example, state A starts out as state I for small x, becomes a mixture of I and O1 at x ˆ 14, and ®nally
becomes O1 for large x. Likewise, state B evolves from O1, to a mixture of I and O1 at x ˆ 14, to mostly
I at x ˆ 20, to a mixture of I and O2 at x ˆ 28, and ®nally to O2. The two gaps of the avoided crossings
at x ˆ 14 and 28 ML are 0.17 and 0.12 eV, respectively.
The measured dispersions are shown as circles in the right-hand side panel of Fig. 10. The solid
curves connect the data points, and there is a good overall agreement with the two-band prediction
as indicated by the dashed curves. While the experimental curves are about 40 meV higher in energy
than the predictions, the sizes of the gap and the rates of dispersion are all well reproduced. The reason
for the slight discrepancy is likely that the modeling of the 3 ML Au barrier layer is not entirely
accurate. In particular, the Ag±Au boundary potential must be somewhat rounded due to a ®nite
metallic screening length, but in the model, this is assumed to be abrupt. The sizes of the circles in the
®gure are a rough indication of the predicted (left-hand side panel) and measured (right-hand side
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Fig. 11. Probability densities of states A and B (see Fig. 10) for a few representative outer well thicknesses x in the double
quantum well system Au(1 1 1) ‡ 8 ML Ag ‡ 3 ML Au ‡ x ML Ag. All states are normalized. The vertical dashed lines
indicate the various Ag±Au and Ag±vacuum boundaries in the system.
panel) photoemission intensities. Again, the trends of the intensity variation are well reproduced by the
model.
Additional insight can be gained from an examination of the wave functions. The calculated
probability densities for a few representative cases are shown in Fig. 11. For x ˆ 10 ML, state A
roughly corresponds to state I, and state B to state O1 (see Fig. 10). Thus, the wave functions are mostly
concentrated in the inner and outer wells, respectively. At x ˆ 14 ML, states I and O1 are fully mixed.
The higher energy state A has an antinode in its envelope function at the barrier, while the lower energy
state B has a node. They resemble the n ˆ 1 and 2 states in the combined well with the Au replaced by
Ag. At x ˆ 20 ML and beyond, state A becomes essentially the n ˆ 1 state in the outer well, O1. State
B, at x ˆ 20 ML, becomes essentially the inner well state I, and upon further increasing of x, evolves
into O2.
The above illustrates how coupling between quantum wells can lead to signi®cant modi®cations of
the bound state energies. In the two-band model, the coupling is mediated by the wave functions in the
barrier layer, and the coupling strength is determined by wave function matching at the boundaries.
There are no additional parameters speci®c to the interface. It is remarkable that the anticrossing gaps
are well reproduced by this model where the only inputs are the bulk band structures of the constituent
materials. This is unlike the tight-binding method where the Ag±Au overlap integrals must be
introduced as additional parameters for the interface.
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The coupling between the two quantum wells decays rather rapidly as the Au barrier thickness is
increased, because the interaction is via evanescent (exponentially decaying) waves within the Au
barrier. Longer-range interaction with oscillatory structures can be expected if the coupling is via
propagating waves, i.e., quantum well states or resonances. Later in Section 6, this point will be
addressed again in connection with the magnetic behavior of multilayer systems.
3.3. Superlattices
Superlattices are often employed in device structures. The main effects of superlattice modulation
include band folding and mini-gap formation. There are two relevant parameters for a binary
superlattice, namely, the thicknesses of the two repeating constituent layers. Varying these parameters
allows detailed tuning of the band structure, providing opportunities for property enhancement and
avenues for device engineering. Here, we consider a superlattice made of repeating layers of Ag and Au
[62±64]. It is expected that the sp band dispersion should lie somewhere between those of Ag and Au,
with superlattice gaps at the mini-zone boundaries.
The same two-band model can be easily applied to the superlattice. Assume that each superlattice
period is made of N1 monolayers of Ag and N2 monolayers of Au. The wave functions in the Ag and Au
are linear combinations of ci(z) and ci(ÿz), and there are four coef®cients in each period. The
boundary condition at the Ag±Au interface requires that the wave function and its derivative be
continuous. Further, the Bloch theorem requires that the wave function and its derivative advance by a
phase factor of exp[ik(N1 ‡ N2)t] for a displacement by a period. Thus, the four coef®cients must
satisfy a set of four homogeneous algebraic equations. For a solution, the determinant must vanish, and
the resulting eigenvalue equation generally has only one solution at each k in the extended zone. This
eigenvalue equation can be cast into a simple transcendental form, and the reader is referred to the
original publication for details [64].
Fig. 12 shows a set of normal-emission spectra for a superlattice with N1 ˆ 8 ML (Ag) and
N2 ˆ 4 ML (Au). These spectra were taken with various photon energies in the usual range for ``band
mapping''. This particular superlattice is terminated by a Ag slab, and the surface state looks like that
of pure Ag(1 1 1). The dispersive peak labeled N (for normal) is very similar to the direct-transition
peak seen in pure Ag (see the bottom spectrum in Fig. 8). This superlattice is made of more Ag than
Au, and therefore it is not surprising that the peak position is closer to that of Ag than Au. As the
photon energy is varied, the direct transition moves to different positions in k (see Fig. 2), and the peak
position moves in accordance with the dispersions of both the initial and ®nal bands. The weaker
dispersive peaks labeled U (for umklapp) are derived from the folded bands of the superlattice, and
have no counterparts in pure Ag or Au. The two U branches are separated by a mini-gap, in which there
is a non-dispersive peak (S). This peak, observed over a wide photon energy range, is a surface state
peak speci®c to the superlattice geometry.
Fig. 13 displays the measured positions of the N and U peaks as a function of k in the extended zone,
assuming that the ®nal band is the same as that for pure Ag. The N and U peak positions agree where
they overlap. The vertical dashed lines indicate the expected superlattice zone boundaries, where minigaps should form. The data indeed show gaps at these positions. The Ag and Au band dispersions are
included for comparison. The superlattice band dispersion falls in-between, and is closer to the Ag
dispersion as expected. Also shown in Fig. 13 are solid curves that represent the calculated superlattice
band structure based on the two-band model as discussed above. The overall behavior of the data is well
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Fig. 12. Normal-emission spectra for an 8 ML Ag ‡ 4 ML Au superlattice taken with various photon energies as indicated.
The dashed curves are guides to the eye for the various peaks in the spectra.
reproduced by the calculation. Most encouraging is that the measured gaps are in excellent agreement
with the calculation, which predicts that the upper gap is 0.30 eV and the lower gap is 0.12 eV. The
lower gap can be detected if the data points are viewed at a glancing angle relative to the page; it can be
quanti®ed by curve ®tting of the two adjacent branches. The measured dispersion curves are slightly
higher than predicted. Discrepancy of this magnitude should not be surprising, considering the extreme
simplicity of the model (abrupt, ideal interfaces).
Extensive data show that the energy position of peak S (see Fig. 12) is independent of photon energy
and falls within the superlattice gap. This establishes that peak S is derived from a surface state rather
than a bulk state. Fig. 14 focuses on this state and shows a set of spectra taken at hn ˆ 11.6 eV so that
the N and U peaks are well separated from peak S. The bottom spectrum is taken from the Agterminated sample, as before. The other spectra, from bottom to top, are taken from the same sample
after adding one atomic layer at a time in going through one superlattice period. While the position of
peak S depends on the surface termination, it always remains within the superlattice gap, as indicated
by the vertical dashed lines. As the surface termination is changed, peak S moves across the superlattice
band gap over one half of the superlattice period, disappears for the other half of the period, and
reappears as the surface termination is restored after one complete period (the top spectrum in the
®gure). The triangles indicated the predicted positions of the surface state based on a two-band
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
203
Fig. 13. Experimental band dispersion (diamonds) derived from the normal and umklapp peaks in Fig. 12. The dashed curves
are the bands for pure Ag and Au. The solid curves are derived from a two-band calculation.
calculation and a phase shift analysis. Again, there are no adjustable parameters beyond the bulk band
structures of Ag and Au, and yet the agreement is excellent. The success of such a simple model should
not be surprising, because the behavior follows some very general principles and should be independent
of the numerical details [63].
4. Mismatched interfaces
The overwhelming majority of elemental pairs in the periodic table are lattice mismatched, and this
mismatch affects overlayer growth. Some exhibit strained growth to force a match. Defects form in
others to accommodate the mismatch. Interfacial reactions, intermixing, and segregation represent
complicating factors. For metal epitaxial systems, it is common to have unstrained overlayer growth.
For example, Ag grows unstrained along [1 1 1], the low energy direction, on Cu(1 1 1), Ni(1 1 1), and
Co(1 1 1). The mismatch is too large for a strained growth. Unlike semiconductors where the bonding is
covalent and highly directional, metals are held together by metallic bonding involving a sea of
electrons. These electrons are fairly diffuse and can easily accommodate the bonding requirements at
an interface, even for the highly directional dangling bonds on a semiconductor substrate. In all of the
cases mentioned above, there is an orientational relationship between the overlayer and the substrate.
The major crystallographic axes are parallel, and in some cases, there is signi®cant twin formation
suggesting that second nearest neighbor interaction is a weak effect.
What are the rami®cations of a mismatched interface for quantum well states? The mismatch may
give rise to an incommensurate interface or a reconstructed interface involving a large unit cell. The
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Fig. 14. Normal-emission spectra for various surface terminations of the Ag±Au superlattice taken with a photon energy of
11.6 eV. Starting from the bottom spectrum, which is for the Ag-terminated con®guration, each successive spectrum
corresponds to the addition of one atomic layer in going through one superlattice period. The top and bottom spectra
correspond to the same sample con®guration. The superlattice gap is indicated by two vertical dashed lines. The triangles are
the energy positions of the surface state from a calculation.
interface potential is characterized by a linear combination of the two sets of interface reciprocal lattice
vectors from the two materials, and can cause an electron incident normally on the interface to scatter
off to many different directions. A quantum well state is essentially a standing wave involving multiple
re¯ections of the electron between the two ®lm boundaries. Interface scattering to other directions
causes the standing wave to decay over time, and the result is similar to partial con®nement. Another
issue of importance is the probability for electron transmission through an interface. In the Ag±Au case
discussed above, the gap in the substrate allows no transmission. For a lattice mismatched case,
however, the matching of the wave functions at the interface becomes more complicated. The kk ˆ 0
wave functions on the two sides of the interface are no longer phase matched at every lattice site. A full
calculation would consider coupling of wave functions involving many different kk . An enhanced
re¯ection probability can result if the wave functions do not match well. Specular re¯ectivity at the
interface is unity reduced by the transmissivity and the non-specular re¯ectivity. If this reduction is not
too large, the system should support well-de®ned quantum well resonances.
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Fig. 15. Normal-emission spectra taken with hn ˆ 10 eV for various Ag coverages on Cu(1 1 1).
4.1. Ag on Cu(1 1 1)
Ag on Cu(1 1 1) represents a case that supports well-de®ned quantum well resonances [24]. Fig. 15
shows a set of normal-emission spectra for Ag ®lms of various thicknesses on Cu(1 1 1) taken with a
photon energy of 10 eV. The main features include a Shockley surface state S and quantum well peaks
that evolve in an expected manner. Note that the valence band maximum of Cu(1 1 1) is at about
0.85 eV, above which there is a relative gap. Quantum well peaks below this can still be seen, although
broader. Data taken at different photon energies indicate that the peak positions are essentially
independent of the photon energy. This is consistent with earlier ®ndings for the leaky quantum well
system of Ag(1 1 1) ‡ x ML Au ‡ 15 ML Ag. In both cases, the resonance peak positions are governed
by the standing wave patterns in the initial state, and are independent of the photon energy.
Fig. 16 shows the dependence of these quantum well peaks on the polar emission angle. As the
analyzer is moved off normal to either side, the peaks move toward the Fermi surface following a
roughly parabolic behavior, and so is the surface state S. One by one, these peaks cross the Fermi level
and become invisible. The measured peak dispersions can be used to deduce the Ag sp band dispersions
along the direction of kk . This is suf®cient to provide a complete speci®cation for the surface state
because it has no k? dependence. Likewise, the quantum well states are completely speci®ed by n and
kk . In other words, the quantum number k? for a bulk single crystal is replaced by n in the case of a
®lm.
It is clear from Fig. 16 that the quantum well peaks become weaker as the polar emission angle
becomes larger. This can be explained as follows. Each quantum well peak corresponds to a standing
wave pattern involving multiple re¯ections of the electron between the two ®lm boundaries. With an
off-normal geometry, each round trip of the electron within the ®lm is associated with a sideways
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Fig. 16. Photoemission spectra for 30 ML of Ag on Cu(1 1 1) taken with hn ˆ 11 eV. The polar emission angle y and the
direction of azimuth are indicated. The dashed curve is a guide to the eye, and shows the approximately parabolic dispersion of
a quantum well peak.
displacement. This displacement accumulates until it becomes larger than the lateral coherence length
of the wave function. At that point, multiple beam interference terminates. The effective number of
beams contributing to the interference determines the intensity and sharpness of the quantum well peak.
For a large polar angle, this number is small, and therefore the peak becomes weaker and broader. The
argument here is similar to that for a leaky quantum well.
4.2. Ratcheting quantum well peaks Ð Ag on Ni(1 1 1)
One would expect Ag on Ni(1 1 1) to be very similar to Ag on Cu(1 1 1) because both systems have a
large lattice mismatch. A detailed examination of the spectra shows interesting differences [66]. The
left-hand side panel of Fig. 17 shows photoemission spectra from a 14 ML Ag ®lm on Ni(1 1 1) taken
with photon energies ranging from 5.5 to 13.75 eV in 0.25 eV steps. In each spectrum, the sharp peak
just below the Fermi level is emission from the Ag(1 1 1) surface state. In addition, four quantum well
peaks, n ˆ 1 ± 4, are observed. There is also an intense, broad feature that moves toward higher binding
energies as the photon energy is increased. This is the counterpart of the sp direct-transition peak seen
in Ag(1 1 1) (see Fig. 12 for a similar peak for the Ag±Au superlattice discussed above). At a high
photon energy, where the direct-transition feature is away from the region of interest, one can clearly
see that the quantum well peaks evolve toward the Ag valence band maximum for increasing ®lm
thicknesses, as in Ag on Cu(1 1 1).
One important difference for Ag on Ni is the cyclic or ``ratcheting'' behavior of the quantum well
peaks as a function of photon energy. This is indicated by the dashed curves in Fig. 17. As the photon
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207
Fig. 17. Left-hand side panel: normal-emission spectra for 14 ML Ag on Ni(1 1 1). Scans are shown for each 14 eV photon
energy from 5.5 to 13.75 eV. The dashed lines are guides to the eye showing the peak motions. Right-hand side panel:
calculated spectra presented in the same fashion as the left-hand side panel. The surface state is not included in the calculation.
energy is varied, each peak emerges from one end of the range of movement, moves to the other end,
pops back, and repeats its motion. The range of peak movement is a fraction of an electron volt for each
peak, and is different for different peaks. Its intensity diminishes near the two ends of the movement,
and reaches a maximum at the midpoint. Such a ratcheting behavior is not evident in the case of Ag on
Cu(1 1 1).
The right-hand side panel of Fig. 17 shows results from a model calculation based on the
methodology discussed above. The initial state is again a linear combination of ci(z) and ci(ÿz), with
the coef®cients determined by the vacuum boundary condition and by appropriate normalization. The
®nal state is the time-reversed low-energy-electron-diffraction state made of a two-band Bloch state in
the crystal and plane wave states in vacuum normalized to a unity outgoing wave. The Ag±Ni interface,
being incommensurate, is assumed to be a perfectly diffuse boundary, so that an electron incident on the
boundary is completely lost by incoherent scattering. Thus, the matrix element integral for optical
transition between the initial and ®nal states is truncated at the Ag±Ni interface. This is a physically
reasonable model and conveniently bypasses the problem of setting up the wave functions in the Ni
substrate. The calculated matrix element, together with the density of states, yields the photoemission
spectra shown at the right of Fig. 17. The results reproduce all of the essential features of the data,
including the ratcheting behavior of the peak positions, the cyclic evolution of the intensities, and the
movement of the direct-transition-like peak. Even the amplitudes of the ratcheting for different peaks
and the relative widths of the various peaks are fairly well reproduced. The good overall agreement is
gratifying, considering that the only parameters entering the calculation are the bulk band structure of
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Ag and the work function. This provides strong support for the assumed diffuse scattering boundary
condition.
The difference between Ag on Ni and Ag on Cu is interesting. Ag on Cu behaves like a leaky
quantum well, where the quantum well peaks are characterized by standing wave patterns that are
broadened by coupling effects. The Ag on Ni case does not support such standing wave patterns.
Instead, the quantum-well-like features are a consequence of the truncation of the matrix element for
the optical transition, as would be the case for a negligible specular re¯ectivity at the Ag±Ni interface.
This truncation creates an interference effect, and yields the unusual ratcheting peak positions and
intensities. While both the Ag±Cu and Ag±Ni interfaces are mismatched, the results suggest that the
specular re¯ectivity for Ag±Cu is signi®cantly higher than that for Ag±Ni. This difference may be
related to the presence of a relative gap in Cu, but not in Ni. The gap in Cu suppresses the interfacial
transmissivity, and thus the specular re¯ectivity is higher.
5. Atomically uniform ®lms as quantum wells and electron interferometers
The quantum well peaks seen so far are broad, with widths that are generally much larger than
expected based on the photohole lifetime. This broadening can be attributed to ®lm roughness. The
same broadening explains why the quantum well peaks appear to evolve continuously as a function of
coverage. The ®lm thickness should be discrete, and the quantum well peak positions should be discrete
as well. Variations in the ®lm thickness on a real structure smear out such discreteness, and the
positions of the broad quantum well peaks simply re¯ect the average ®lm thickness. Until recently, it
has been generally thought that roughness on the atomic scale would be inevitable over a macroscopic
lateral distance. If this roughness can be eliminated somehow, highly accurate determination of the
electronic properties will become feasible. These include the band structure, photohole (or
quasiparticle) lifetime, interfacial re¯ectivity, and phase shift. These properties are of basic importance
to solid state physics and interface science. This section will focus on the making of atomically uniform
®lms and using these ®lms for precision spectroscopic studies [67±72]. An analysis in terms of electron
interferometry will be presented, which provides a useful framework for understanding the quantum
well peak positions and lineshapes.
5.1. Discrete layer thicknesses
Countless investigations by scanning tunneling microscopy have shown that surfaces contain steps
and defects. Film growth involves a stochastic deposition process and the resulting ®lms are also
affected by steps, defects, and impurities. Thermodynamic ¯uctuations and growth kinetics tend to
create roughness, and the roughness tends to increase as a function of thickness, often according to
certain scaling laws [73±75]. A ®lm with a nominal thickness of N atomic layers typically consists of
multiple thicknesses including N, N 1, N 2, . . . over different domains. If the domain sizes are
large, photoemission spectra should exhibit a linear combination of quantum well peaks derived from
these different ®lm thicknesses (as well as effects unique to the steps).
An example is shown in Fig. 18 [76]. The spectrum in the lower panel is taken from a Ag ®lm of
8 ML nominal thickness grown on a cleaved graphite substrate. The peak just below the Fermi level is
the Shockley surface state. The broad hump centered about 1.3 eV is the usual n ˆ 1 quantum well peak.
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209
Fig. 18. Measured and calculated photoemission spectra for Ag on graphite. The quantum number n and the layer thickness
N are indicated. In the lower panel the dispersion is shown to illustrate that the different sub-peaks are derived from different
discrete layer thicknesses. In the upper panel, the ®lm is too thick for the sub-peaks to be resolved (®gure taken from [76]).
The little bumps superimposed on the broad peak correspond to individual quantum well peaks derived
from N ˆ 6; 7; . . . ; 10. The lower part of the panel shows the Ag band structure and the predicted peak
positions for different layer thicknesses. These individual peaks are barely resolved in this case. As the
®lm gets thicker, the peak spacing becomes smaller, and eventually the individual peaks can no longer
be resolved. This is illustrated by the results shown in the upper panel of the ®gure for a ®lm of 28 ML
nominal thickness. A survey of the photoemission literature shows that such atomic layer resolved
quantum well peaks are the exception rather than the norm. It is likely that most ®lms have such a high
step and/or defect density that random lateral con®nement and scattering cause signi®cant smearing of
the quantum well peaks, rendering peaks from different thicknesses unresolvable.
Another example of such atomic layer resolved quantum well peaks is shown in Fig. 19. The spectra
are for a Ag(1 0 0) ®lm of 12 ML nominal thickness deposited on a Fe(1 0 0) whisker substrate at room
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Fig. 19. Normal-emission spectra taken from 12 ML Ag deposited on Fe(1 0 0) at room temperature using various photon
energies as indicated. The vertical dashed lines indicate the positions of quantum well peaks. Several thicknesses are present in
this ®lm.
temperature. Within the binding energy ranging 0±2 eV, there should be just two quantum well peaks
for a thickness of 12 ML, and yet many more peaks are observed due to the presence of several
thicknesses around 12 ML. Note that in this case, the ®lm orientation is (1 0 0). Referring back to the
band structure of Ag shown in Fig. 2, the Ag sp band crosses the Fermi level along the [1 0 0] direction.
The gap is above the Fermi level, and there is no surface state peak in the spectrum as in the case of a
(1 1 1) ®lm.
The reason for using a Fe whisker for Ag growth is that such whisker crystals represent perhaps the
best possible substrates available. Each whisker contains just one bulk defect by nature. The
observation of atomic layer resolved quantum well peaks suggests that the domain sizes in the Ag ®lm
are unusually large, and the system looks promising. The growth of Ag on Fe(1 0 0) has been
investigated extensively. The interest stems from a good lattice match (within 0.8%) and a nearly ideal
epitaxial relationship. There is no intermixing or reaction between Ag and Fe even at elevated
temperatures.
5.2. Preparing atomically uniform ®lms
The technique of molecular beam epitaxy has been widely employed for growth of semiconductor
device con®gurations. The basic strategy is to grow the ®lm at an elevated temperature to ensure
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211
adequate adatom mobility, thus promoting layer-by-layer growth and minimizing the chance of defect
formation caused by statistical ¯uctuation of the deposition process. The growth temperature is usually
set as high as possible to optimize adatom diffusion, but within the limits determined by re-evaporation
and intermixing. This strategy, however, does not work for Ag on Fe(1 0 0). Growth at room
temperature or higher always results in ®lms with multiple thicknesses, as illustrated by the spectra
shown in Fig. 19. It was noticed in several recent experiments that a highly nonequilibrium pathway
often led to smoother ®lms in related systems [77±79]. This involves the deposition of the overlayer at
very low temperatures followed by annealing. The deposition temperature is so low that the resulting
®lm has a ®ne-grained texture, and epitaxial crystalline order is restored by post-deposition annealing.
This technique has been demonstrated in the growth of Ag on GaAs and Si. In those cases, the ®lms are
much better, but not atomically uniform.
Employing this low temperature growth technique for Ag on Fe(1 0 0) resulted in dramatic
improvement in ®lm structure relative to growth at room temperature. In the experiment, the base
temperature of the substrate was 100 K, and the annealing temperature employed was 600 K. Fig. 20
shows normal-emission spectra taken at various photon energies for a 6.6 ML Ag ®lm deposited on
Fe(1 0 0) [67]. There are two main peaks, and the peak intensities vary signi®cantly as a function of
photon energy due to cross section variations. The two peaks arise from quantum well states
corresponding to N ˆ 6 and 7, as indicated in the ®gure. Hence, the ®lm exhibits only two thicknesses.
This strongly suggests that if the amount of deposition is just right (an integer of monolayers), the
resulting ®lm will become atomically uniform, as has been veri®ed by experiment. However, it is never
easy to lay down the exact amount of coverage in one step, and a correction is often required.
Fig. 20. Normal-emission spectra taken from 6.6 ML Ag on Fe(1 0 0) prepared by low temperature deposition followed by
annealing. The two peaks can be assigned to ®lm thicknesses of 6 and 7 ML, respectively. The peak heights oscillate as a
function of photon energy.
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Fig. 21. Normal-emission spectra for a 38 ML (bottom), 38.5 ML (center), and 39 ML (top) Ag ®lm on Fe(1 0 0) taken
at a photon energy of 13 eV. The quantum well peak positions are indicated by vertical dashed lines. The spectrum for the
38.5 ML ®lm shows two sets of quantum well peaks indicating the simultaneous presence of areas covered by 38 and 39 ML
of Ag.
Fortunately, the same procedure of low temperature deposition followed by annealing can be repeated
to build up a ®lm gradually without any adverse effect on ®lm roughness. Atomic layer ¯uctuation
remains suppressed even after many cycles of deposition and annealing, and the ®lm thickness is
always either an integer, uniform across the surface, or a linear combination of two neighboring integer
thicknesses.
Fig. 21 demonstrates the atomic uniformity of ®lms of integer monolayer coverages [68]. The bottom
spectrum is for a ®lm made by adding incremental amounts to a thinner ®lm to reach a coverage of
38 ML. Five very sharp quantum well peaks are observed. Adding 0.5 ML to this ®lm yields the middle
spectrum, which exhibits two sets of quantum well peaks. One set is at the same positions as the 38 ML
case, and the other corresponds to a thickness of 39 ML. This is veri®ed by adding another 0.5 ML
because the peaks corresponding to 38 ML are suppressed, and only the 39 ML peaks remain. The same
discrete layer behavior has been seen for many different starting thicknesses. Since the area probed in
the photoemission experiment is about 1 mm in size, the above result establishes that the ®lm is
uniform on an atomic scale over a macroscopic distance ( 1 mm). If the overlayer or substrate is not
optimally prepared (contamination of the substrate surface, too high a substrate temperature, etc.), there
will be a multiple set of thicknesses. Once this happens, no amount of annealing or deposition can
restore the atomic-level uniformity, and the sample must be stripped clean for a new attempt. For more
severely contaminated substrates (by deliberate gas dosing), the quantum well peaks can become very
broad.
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
213
Fig. 22. Normal-emission spectra for atomically uniform ®lms of 12 and 38 ML of Ag on Fe(1 0 0). The curves are results
from a ®t using Voigt lineshapes. The Lorentzian full width is indicated for each ®tted peak, and the Gaussian width is the
instrumental resolution obtained from a ®t to the Fermi edge.
Fig. 22 shows the results from a least-squares ®t, using Voigt line shapes, to the quantum well peaks
for ®lms of thicknesses 12 and 38 ML [68]. The Gaussian width is the instrumental resolution
determined from a ®t to the Fermi edge line shape. The Lorentzian width is a ®tting parameter, and is
given in the ®gure for each ®tted peak. These peaks are much sharper than the usual direct-transition
peaks seen in bulk single crystals. The trend is that the peak width is smaller at lower binding energies
and for larger ®lm thicknesses. A quantitative discussion of the peak width in terms of the quasiparticle
lifetime will be given below.
Clearly, growth at low temperatures is key to the making of such uniform ®lms. It is likely that
thermal atomic diffusion is suppressed at low temperatures, allowing the ®lm to build up ``uniformly''
before annealing to restore the atomic order. In contrast, growth at higher temperatures involves a large
diffusion length, and kinetic effects related to the interaction of diffusing atoms and atomic steps can
lead to the formation of surface roughness [73±75]. For example, an adatom approaching an atomic
step can sense a potential barrier (Ehrlich±Schwoebel barrier) [80,81]. Such a barrier can cause island
formation on an incomplete layer, and continued growth can lead to the formation of multi-level
terraces.
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Fig. 23. Quantum well peak intensities in normal-emission for ®lms of thicknesses 6, 7, 8, and 9 ML of Ag on Fe(1 0 0). The
horizontal axis is the ®nal state energy.
5.3. Intensity modulations
Fig. 20 shows that the quantum well peak intensities can depend strongly on the photon energy and
the ®lm thickness. Fig. 23 presents the measured intensities for ®lm thicknesses 6, 7, 8, and 9 ML as a
function of the ®nal state energy [67]. The intensity modulation is almost 100%. In other words, at
certain photon energies, the emission intensity is essentially zero. For example, the 7 ML peak vanishes
at hn ˆ 16.5 eV, and the 6 ML peak vanishes at hn ˆ 18 eV. An important conclusion is that the
observed photoemission intensities are not necessarily a good measure of the area covered by a
particular layer thickness. For example, based on the spectrum taken with hn ˆ 16.5 eV alone (see
Fig. 20), one might erroneously think that the 6.6 ML ®lm is a pure 6 ML ®lm.
The results in Fig. 23 demonstrate that the cross section depends on the ®nal state energy. This
behavior is somewhat similar to the case of Ag on Ni(1 1 1) discussed earlier, where the photoemission
intensity of each quantum well peak is cyclic. For Ag on Fe(1 0 0), the ®nal state of photoemission
involves an interference pattern related to the two boundaries of the ®lm, just as the initial state. This
interference pattern evolves as a function of energy, and the dipole matrix element between the initial
and ®nal states oscillates depending on the relative phases. This gives rise to an intensity modulation.
Similar intensity modulations have been reported for Ag on V [82], Cu on Co [83], and Na and Cs on
Cu [84,85].
Because the cross section modulations are different for different thicknesses, a rough ®lm consisting
of multiple thicknesses will exhibit a reduced intensity modulation representing the average behavior.
The modulation also depends on the photon energy and ®lm thickness. When the photon energy is high,
the ®nal state mean free path becomes short. This effectively introduces a broadening in k, and the
intensity modulations are reduced. Likewise, if the ®lm is thick, the interference pattern in the ®nal
state becomes less pronounced, leading to a reduced intensity modulation.
5.4. The Bohr±Sommerfeld quantization rule and band structure determination
An important application of angle-resolved photoemission is band structure determination [1±6].
Photoemission involves an optical transition from an occupied (initial) state to an unoccupied (®nal)
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
215
state, and the resulting spectra generally depend on both the initial and ®nal band properties. The
information is thus convoluted, and much of the historical development of the angle-resolved
photoemission technique has focussed on methods to untangle this information in order to extract the
initial state properties. While this is straightforward for two-dimensional systems such as layer
compounds and surface states, it is a major problem for three-dimensional systems. Because the surface
of a crystal breaks the translational symmetry, momentum conservation does not hold along the surface
normal direction. The momentum component perpendicular to the surface, k?, of the photoelectron
outside the crystal can be measured accurately, but this information is generally insuf®cient for a
determination of k? for the initial state inside the crystal. In contrast, the parallel component of the
momentum, kk , is conserved, and this is the only component of interest for two-dimensional systems.
This ``k? problem'' for three-dimensional systems has been the subject of much research, and many
methods have been devised to overcome this problem with varying degrees of success and utility.
Approximations, interpolations, and/or theoretical calculations are often invoked in these methods,
resulting in an uncertainty of Dk? typically on the order of one-tenth of the Brillouin zone size at an
arbitrary point in k space. Another related problem is that the measured photoemission lineshape is
often quite broad because it is dominated by a very large ®nal state lifetime width [86±88].
Furthermore, the lineshape can be distorted by interference from surface photoemission [42±46]. As a
result of these complications, an energy uncertainty of DE ˆ 0.1±0.2 eV is typical. These energy and
momentum uncertainties are much too large for modern research, and essentially all recent highresolution studies have been limited to two-dimensional systems.
Quantum well spectroscopy provides a solution to this k? problem. The basic idea is that k? is
quantized in a ®lm as a result of electron con®nement. The allowed k? values are determined by the
®lm thickness and boundary conditions. Measurements of quantum well peak positions for many
different ®lm thicknesses should permit a unique solution of E(k?). Although this method was proposed
and implemented rather early on [16,23,24], the uncertainty Dk? remained quite large in those early
studies due to ®lm thickness ¯uctuation and uncertainty. Recent success in preparing atomically
uniform ®lms has ®nally made it possible to carry out accurate band structure determination using
quantum well spectroscopy. This procedure will be presented below for Ag on Fe(1 0 0). It is important
that the absolute ®lm thickness N be known. In the actual experiment, a series of absolute ®lm thickness
calibrations was made layer by layer at N ˆ 1; 2; 3; . . . ; 20 as the ®lm was built up gradually by
submonolayer depositions. This discrete layer counting provided a data set large enough to establish
a unique functional relationship between the absolute ®lm thickness N and quantum well peak
positions, which could be used for higher thicknesses by extrapolation [69,70]. Additional layers at
N > 20 were prepared and the peak positions were veri®ed to be consistent with this relationship. Fig. 24
presents one subset of such data taken at normal-emission with a relatively high photon energy of
55 eV.
While it is easy to carry out accurate calculations for Ag±Au as discussed above, Ag±Fe presents a
much more dif®cult case. The d wave functions in Fe near the Fermi level are much more complicated.
Furthermore, several d bands can be simultaneously involved in the interfacial coupling. Fortunately,
inasmuch as the peak positions and band structure are concerned, there is no need to construct the wave
functions explicitly. It suf®ces to carry out a simple analysis based on the Bohr±Sommerfeld
quantization rule
2k…E†Nt ‡ F…E† ˆ 2np;
(7)
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Fig. 24. Normal-emission spectra for atomically uniform ®lms of Ag on Fe(1 0 0) taken with a photon energy of hn ˆ 55 eV.
The ®lm thicknesses are indicated.
where F, a function of the binding energy E, is the total phase shift of re¯ection at the surface and the
Ag±Fe interface. This equation states that the total phase shift in a round trip perpendicular to the
surface is equal to an integer (quantum number n) times 2p. This is the same as requiring the de Broglie
wave associated with the electron to form a standing wave. Eq. (7) is sometimes referred to as the phase
accumulation rule in the literature [38], and its origin dates back to the beginning of quantum
mechanics. Eq. (1) is a special case of Eq. (7) with F ˆ 0 (or 2p).
In Eq. (7), both k and F are dependent on E, and it is impossible to solve this equation uniquely based
on a single measurement. However, there is a multiplicative factor N associated with k only, and thus
measurements at different values of N should lead to additional independent equations that can be used
to extract a unique solution. If quantum well state n for thickness N happens to be at the same energy E
as that of quantum well state n0 for thickness N0 , we have the additional phase relation
2kN 0 t ‡ F ˆ 2n0 p;
(8)
where k and F are the same as before in Eq. (7) because E is the same. Eqs. (7) and (8) can be solved to
yield k and F at E in terms of the known quantities N, N0 , n, n0 , and t. Of course, it is rather seldom that
quantum well peaks from different thicknesses happen to have the same energy, but mathematical
interpolation can be employed because k and F are continuous functions of E. In practice, a number of
thicknesses are used in the experiment to determine k(E) and F(E) via a least-squares ®tting procedure.
The band dispersion E(k) is obtained by inverting k(E).
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
217
Fig. 25. Normal-emission spectra for a bulk single crystal Ag(1 0 0). The photon energies are indicated.
Additional information is available from normal-emission data taken from bulk single crystal
Ag(1 0 0). The direct-transition peaks, shown in Fig. 25, are determined by
ÿEi …k† ‡ hn ˆ ÿEf …k†;
(9)
where hv is the photon energy, and Ei,f are the initial and ®nal binding energies, respectively (Ei is
positive and Ef is negative). A unique solution of the band structure is impossible with Eq. (9) alone.
One could rescale k in Eq. (9) by an arbitrary factor, and the resulting equation would still be consistent
with the normal-emission data. The lack of k constraints is the origin of the k? problem mentioned
earlier. If the initial band structure is known, as would be the case with the quantum well data analyzed
in the manner outlined above, the ®nal band structure is then uniquely determined by Eq. (9). Note that
the peaks in Fig. 25 are broad and asymmetric. The large width is caused by a large ®nal state lifetime
broadening, and the asymmetry is due to surface photoemission. A careful analysis of the lineshape is
needed to yield an accurate measure of Ei [42±46].
In the actual data analysis [70], analytic expressions of Ei,f(k) from the two-band model are used. The
phase shift F(E) is modeled by a third order polynomial. Eqs. (4), (7) and (9) are used to ®t the quantum
well data and the normal-emission data from bulk Ag(1 0 0) simultaneously (with N ˆ 1 ± 3 quantum
well data excluded; see below). In all, a total of 46 quantum well peak positions and nine normalemission peak positions are used in this simultaneous ®t. The resulting four band structure parameters
are presented in Table 1. As mentioned above, the quantum well data alone are suf®cient to determine
the initial band dispersion. The addition of the normal-emission data from the bulk crystal to the ®tting
procedure allows the ®nal band dispersion to be determined.
The circles in Fig. 26 are the measured quantum well peak positions used as input to the ®t, plotted as
a function of N, and the solid curves are results derived from the ®t. N, an integer by de®nition, is taken
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T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
Table 1
Parameter values for the Ag(1 0 0) band structure (me denotes the free electron mass)
Parameter
Value
V
Evbm
mi
mf
3.033 eV
ÿ1.721 eV
0.759me
0.890me
to be a continuous variable in this calculation. These continuous curves illustrate the evolution of the
peak position for each quantum number n, which is given in the ®gure and corresponds to the scheme
that k is measured from the zone boundary. The differences between the curves and the data points are
very small. The bottom panel in the same ®gure shows the differences using an ampli®ed vertical scale.
There are no systematic deviations except for N ˆ 1, 2, and possibly 3. Ignoring these data points, the
average deviation is 20 meV. The larger and systematic deviation for N ˆ 1 and 2 can be attributed to
overlap of the surface and interface potentials. The screening length in a metal is short. Nevertheless,
there can be a signi®cant overlap at very small ®lm thicknesses, and Eq. (7), based on the assumption
that the phase shifts at the two boundaries are independent and additive, is no longer valid. For this
reason, the quantum well data for N ˆ 1 ± 3 are excluded from the ®t as mentioned above. The same
reason explains why this model fails at N ˆ 0 (the quantum well states reduce to a surface state) [49].
Fig. 26. The top panel is a structure plot showing the quantum well peak binding energies as a function of ®lm thickness of
Ag on Fe(1 0 0). The circles are data points and the curves are computed using parameters from a best ®t. The quantum
number n for each curve is shown. The bottom panel shows the difference between the best ®t and experiment using an
ampli®ed vertical scale.
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
219
Fig. 27. Band dispersions of Ag from a best ®t to the photoemission data (solid curves) and from calculations by Eckardt
et al. [90] (dash-dotted curves) and Fuster et al. [91] (dashed curves). The circles indicate the ®nal states based on the normalemission spectra from bulk Ag(1 0 0). The shaded region indicates data from both quantum wells and bulk single crystals are
available. Outside this range, the dispersion curves are simply an extrapolation based on the best-®t two-band model.
The quantum well data can also be modeled by the tight-binding method [49]. While the general
behavior is correctly reproduced, the agreement is not quite as good. A recent ab initio layer Korringa±
Kohn±Rostoker calculation, on the other hand, has yielded quantum well peak positions in reasonably
good agreement with the experiment [89].
The initial and ®nal band dispersion relations of Ag(1 0 0) deduced from the ®t are plotted as solid
curves in Fig. 27. The shaded region indicates the k range spanned by the quantum well data. The
curves outside this region represent an extrapolation. The circles indicate the ®nal states reached by
direct transitions from the initial band based on the normal-emission data from bulk Ag(1 0 0).
Additional data analyses show that the band structure within the shaded region is accurate to within
30 meV, which sets a new standard in bulk band structure determination. The dashed and dash-dotted
curves indicate the results of two band structure calculations [90,91], which are chosen as
representative for the large number of available results. The differences between these two calculations
are much larger than our experimental accuracy, and illustrate the level of accuracy of modern band
structure calculations.
5.5. Quantum wells as electron interferometers
The above analysis is simple and yet powerful. However, it leaves out the lineshape, or the width of
the quantum well peak, which is related to the quasiparticle lifetime. A more detailed analysis will
require an examination of the wave function. The photoemission signal from a quantum well state is
typically dominated by surface photoemission. Consider the time-reversed process. The photoelectron
is sent back from the detector to the ®lm, and makes an optical transition to the initial state at the
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T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
Fig. 28. A schematic diagram illustrating the time-reversed photoemission process for a Ag ®lm on Fe. The time-reversed
photoelectron travels from the detector to the surface, where it makes an optical transition to the initial state. It then travels to
z ˆ ÿ 1, and during this process, it undergoes multiple re¯ections.
surface. The electron then travels to z ˆ ÿ 1 to complete the electrical circuit. In the case of a
quantum well, the electron must undergo multiple re¯ections between the two boundaries, as illustrated
by the diagram in Fig. 28. The initial state wave function is thus modulated by an interference factor
1
;
1 ÿ R exp‰i…2kNt ‡ F†Š exp…ÿNt=l†
(10)
where R B r1r2 is the product of the re¯ectivities at the surface and the interface, and l is the
quasiparticle mean free path. The mean free path gives rise to damping of the wave function, and is
related to the quasiparticle inverse lifetime G via the group velocity v by G ˆ v/l.
The photoemission intensity is modulated by the absolute square of the factor in Eq. (10), and the
spectrum for a quantum well becomes
I/
1
1‡
…4f 2 =p2 † sin2 …kNt
‡ …F=2††
A…E† ‡ B…E†;
(11)
where A is a smooth function of E, its prefactor comes from the absolute square of the interference
factor, and B is a smooth background function due to inelastic scattering and incoherent emission. The
quantity f is the Fabry±PeÂrot ®nesse (ratio of peak separation to peak width) given by
p
p R exp…ÿNt=2l†
:
(12)
f ˆ
1 ÿ R exp… ÿ Nt=l†
A short mean free path corresponds to a low ®nesse, so does a low re¯ectivity. Eqs. (11) and (12) are
the same as the usual Fabry±PeÂrot formula for an optical interferometer ®lled with an absorptive
medium [92,93].
Eq. (11) yields a set of peaks at positions where the sine function in the denominator equals zero. The
resulting condition is just the Bohr±Sommerfeld quantization rule given in Eq. (7). The peak width dE
depends on N, R, and l
1 ÿ R exp… ÿ 1=Z†
dE ˆ GZ p
;
R exp…ÿ1=2Z†
(13)
where Z ˆ l/Nt. Generally, dE > G. The only exception is dE ˆ G, when R ˆ 1 and l @ Nt. In other
words, the measured width is just the quasiparticle inverse lifetime for an ideal quantum well (R ˆ 1)
with a long mean free path. The Ag±Fe quantum well has R 0.8. The peaks in Fig. 22 are thus
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
221
Fig. 29. Normal-emission spectra for Ag on Fe(1 0 0) at various coverages as indicated. The photon energy is hn ˆ 16 eV,
and the spectra are normalized to the incident ¯ux. The data points are shown as dots, while the ®t to the data and the
background function are shown as curves.
broadened by this effect. This broadening is relatively minor for the thicker ®lms, but can be quite
signi®cant for the thinner ®lms (due to loss caused by more frequent re¯ections). As mentioned earlier,
the peaks become narrower near the Fermi level. This is mostly due to a smaller G at lower binding
energies.
The quantities k and G (related to the real and imaginary parts of the band dispersion relations) and R
and F (related to the boundary con®nement potentials) are of basic interest, and completely specify the
interferometer properties. While k and F are related to the peak positions through Eq. (7), G and R are
related to the peak width through Eq. (13). They all depend on E, but not on N (except possibly for
N ˆ 1 ± 3). Thus, a set of data spanning a wide range in N (> 3) allows a unique determination, and
crosschecks, of these four quantities at a given E. A ®tting procedure is adopted, and some of the
spectra employed in this ®t together with the ®tting results (solid curves through the data points) are
shown in Fig. 29 [71]. In this ®gure, all of the spectra with integer monolayer coverages correspond to
atomically uniform ®lms. The 27.5 and 42.5 ML spectra show a mixture of peaks derived from
neighboring integer thicknesses and are not included in the ®t. The band structure k(E) is parameterized
using the two-band model, G and R are modeled by quadratic functions of E, and F is modeled by a
cubic function of E. In addition, A in Eq. (11) is taken to be a polynomial of E (up to fourth order and
no higher than the number of peaks in each spectrum), and B is also a polynomial (fourth order for the
119-ML spectrum, and third order for the rest). The resulting energy dependence of k (band structure),
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T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
Fig. 30. Results deduced from Ag(1 0 0) quantum well data: (a) band dispersion relation for the Ag sp band; (b) quasiparticle
inverse lifetime as a function of binding energy; (c) re¯ectivity as a function of binding energy; (d) phase shift in units of p as
a function of binding energy (solid curve). Phase shift deduced from a semi-empirical formula, with an arbitrary vertical
offset, is shown as a dashed curve.
G (quasiparticle lifetime), F (phase shift), and R (re¯ectivity) is shown in Fig. 30 [71]. The band
dispersion is essentially indistinguishable from that shown in Fig. 27.
Fig. 31 illustrates the effects of surface roughness on the quantum well peak width [71]. The bottom
spectrum is for a 29 ML atomically uniform ®lm, where the circles are data points and the solid curves
indicate the ®t and the background function. After adding 0.05 ML Ag to the sample at the base
temperature without annealing, the quantum well peaks become much weaker and broader, as shown by
the top spectrum. This is caused by a mere 5% defect on the surface, and the large effect
demonstrates the high sensitivity of the quantum well peaks to sample imperfection. The increased
width can be modeled by a decrease in specular re¯ectivity at the surface [71].
5.6. Temperature dependence of the band structure
Fig. 32 shows the effect of raising the sample temperature [72]. The quantum well peaks for a 19 ML
®lm, compared to the vertical reference lines, are seen to shift slightly to lower binding energies for
increasing temperatures. More noticeable are the reduction in peak height and increase in peak width.
The shifts can be attributed to three possible causes: (1) thermal expansion of the Ag lattice resulting in
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
223
Fig. 31. Normal-emission spectra taken at a photon energy of 16 eV from a 29 ML atomically uniform ®lm of Ag on
Fe(1 0 0) (bottom) and the same ®lm after the addition of 0.05 ML deposited at 100 K (top). The solid curves represent a ®t to
the data and the background function for each case.
a change in the interferometer path length, (2) temperature dependence of the band structure, and (3) a
change in phase shift at the Ag±Fe boundary. These factors could contribute to the shift according to the
Bohr±Sommerfeld quantization rule, Eq. (7). A detailed analysis shows that the thermal expansion
accounts only for part of the observed shifts. The remaining two contributions can be distinguished by
the fact that a change in k is ampli®ed by a factor of N in the phase shift, while there is no such
ampli®cation for F. An analysis of data taken at different ®lm thicknesses shows that F does not change
as a function of T, and the remaining shift must be caused by a band structure change.
The same interferometric analysis, with the thermal expansion taken into account, yields the
temperature dependence of the band structure. The results are summarized in Fig. 33. The top panel
shows the dispersion of Ag along [1 0 0] at two temperatures as a function of the reduced wave vector
(k normalized to the Brillouin zone size kGX). The dispersion is slightly reduced at higher temperatures,
because the interatomic overlap integral decreases as the lattice expands. Due to charge conservation,
the low and high temperature bands must coincide near the Fermi level, leading to a very small
temperature dependence of the reduced Fermi wave vector kF/kGX. This temperature dependence is
shown in the lower panel. The extrapolated value at absolute zero temperature is kF/kGX ˆ
0.829 0.001. The error represents the systematic error deduced from an analysis of Ag layers of
various thicknesses. This result challenges the value kF/kGX ˆ 0.819 deduced from a de Haas±van
Alphen measurement [94]. The de Haas±van Alphen method is the standard method for Fermi surface
measurements, and its development is widely regarded as a milestone in solid state physics. This
method, however, gives only the circumference of the Fermi surface, and the determination of the
Fermi wave vector depends on the precision of the parameterization of the Fermi surface. It has been
shown that in a study of the radio-frequency size effect, the Fermi surface of Au as determined by the
de Haas±van Alphen method is in fact not entirely accurate. Namely, the Fermi wave vector in the
[1 0 0] direction is too small by 1%. This is very similar to the 1% discrepancy, with the same sign,
found for Ag in this study [95].
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T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
Fig. 32. Normal-emission spectra (dots) from a 19 ML thick Ag ®lm on Fe(1 0 0) taken at various temperatures. The curves
represent a ®t using a generalized Fabry±PeÂrot formula. The vertical reference lines highlight peak shifts.
An issue of concern is the possibility of a lattice distortion in the ®lm. Ag(1 0 0) and Fe(1 0 0) have a
small lattice mismatch of 0.8%. For semiconductor epitaxial systems with such a small mismatch,
strained growth to a critical thickness followed by unstrained growth and defect formation is typical.
For metal epitaxial systems, the situation is less clear. Since metallic bonding is not directional as the
covalent bonds in semiconductors, lattice relaxation is likely to occur already at the early stages of ®lm
growth. The post-deposition anneal to 3008C is likely to further suppress lattice strain. Assuming the
worst case that the ®lms are fully strained by 0.8% in the interface plane, this translates into a 0.6%
expansion along the surface normal direction with a Poisson ratio of 0.37. Incorporating this change in
lattice constant in the analysis leaves the band edge parameters in the ®t unchanged. However, it
changes the effective masses by about 1.5%, and the effect is essentially the same as a rescaling in k.
The band structure, when plotted against the normalized wave vector k/kGX, changes by less than 1 meV
in the energy range of interest, and the change in the normalized Fermi wave vector is negligible. This
analysis suggests that lattice strain is very unlikely an explanation for the discrepancy between the de
Haas±van Alphen and quantum well results.
5.7. Quasiparticle lifetime and scattering by defects, electrons, and phonons
Eq. (13) shows that the quantum well peak width is determined by the quasiparticle inverse lifetime
and the re¯ectivity at the interface. Since the re¯ectivity is tied to the band structure mismatch between
Ag and Fe, this is not expected to change signi®cantly as a function of temperature. The thermally
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
225
Fig. 33. (a) Ag sp band dispersion at 100 and 400 K in the GX-direction plotted as a function of the reduced wave vector
k/kGX. These curves are deduced from an interferometric analysis of the quantum well data. (b) The reduced Fermi wave vector
k/kGX as a function of T. The straight line represents a linear regression of the data.
induced peak broadening seen in Fig. 32 simply re¯ects an increase in lifetime broadening due to
phonon scattering. As the temperature rises, the phonon population increases, leading to an enhanced
scattering rate.
The quasiparticle inverse lifetime is approximately given by
G…E; T† ˆ G0 ‡ G1 …E; T† ‡ 2bE2 ;
(14)
where E is the binding energy [96±99]. The ®rst term, being independent of T and E, represents the
contribution from defect scattering. The second term represents phonon scattering, and a perturbation
calculation yields
Z ED 0 2
E
‰1 ÿ f …E ÿ E0 † ‡ 2b…E0 † ‡ f …E ‡ E0 †Š dE0 ;
(15)
G1 …E; T† ˆ 2pl
E
D
0
where the electron±phonon mass enhancement parameter l enters as a proportionality constant, ED is
the Debye energy (0.0194 eV for Ag), and f and b are the Fermi±Dirac and Bose±Einstein distribution
functions, respectively. For E > 50 meV, G1(E, T)  G1(T); in other words, the phonon contribution
becomes asymptotically independent of E as long as E is much larger than ED. The third term in Eq. (4)
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T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
Fig. 34. The circles represent experimental results for the sum of phonon and defect scattering contributions to the
quasiparticle inverse lifetime. The solid curve is a ®t, and the dashed curves indicate the separate phonon and defect
contributions. Note that the phonon contribution does not vanish at T ˆ 0 because phonon emission is still possible.
represents the contribution of electron±electron scattering. The E2 dependence is often regarded as a
test of the Fermi-liquid theory [96]. An experiment carried out at the base temperature of 100 K yields
2b ˆ 25.6 meV/eV2 (see Fig. 30(b)). Similar experiments carried out at elevated temperatures allow a
deduction of G0 ‡ G1(T), provided E > 50 meV. This last condition is easily satis®ed if quantum well
peaks very close to the Fermi level are ignored.
Fig. 34 shows such an analysis [72]. The circles represent G ÿ 2bE2, where G is determined from a
Fabry±PeÂrot ®t to the data. The solid curve is a ®t using G0 ‡ G1(T), and the ®tting parameters are
l ˆ 0.29 0.05 and G0 ˆ 8 meV. The latter corresponds to a quasiparticle coherence length of about
Ê in the absence of phonon scattering and electron±electron scattering. The parameter l is a
1000 A
measure of the strength of electron±phonon coupling, and is closely related to the transition
temperatures of traditional BCS superconductors. Theoretical and experimental determination of l is
dif®cult, and often only rough estimates are available. The above result appears to be the only
experimental determination of l involving a bulk-derived state for Ag. It is larger than the available
theoretical estimates by about a factor two [97,98], and puts Ag very close to the borderline of
superconductivity. This study demonstrates the utility of quantum well spectroscopy for detailed studies
of the various scattering contributions to the quasiparticle lifetime. This information is critical for
understanding the interactions among elementary excitations in solids.
5.8. Interfacial re¯ectivity and phase shift
The phase shift function F(E) for Ag on Fe(1 0 0) is shown in Fig. 30(d) as a solid curve. Much of
this variation is associated with a hybridization gap in Fe. This gap is de®ned by the effective width of
the d bands in Fe. The phase shift variation at the interface is expected to change by p across the
gap. The phase shift variation at the Ag surface over the same energy range is much smaller. A detailed
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
227
calculation of the phase shift will require matching wave functions in Ag and Fe, which is dif®cult.
Smith et al. [49] has derived a semi-empirical formula for F(E) with the upper and lower edges of the
Fe hybridization gap, Eu and El, respectively, as the only two adjustable parameters. If Eu ˆ 0 eV and
El ˆ 2 eV, are assumed, the result of the semi-empirical formula, with the addition of an arbitrary
vertical offset, yields the dashed curve in Fig. 30(d). It agrees very well with the experiment, thus
suggesting that the hybridization gap in Fe covers the range from 0 to 2 eV below the Fermi level. The
lower edge position is con®rmed approximately by the observation that quantum well peaks become
signi®cantly weaker and broader at binding energies greater than 2 eV, where the electrons in Ag
become uncon®ned.
The hybridization gap in Fe(1 0 0) has been previously identi®ed as the range between the G12
critical point and the highest point of the lowest lying D1 band [49,100]. Based on available band
structure calculations, this estimate would yield Eu ˆ E(G12) ˆ ÿ 1.3 eV and El ˆ E…D1 max † ˆ 2:7 eV,
or a gap about 4 eV. This is much larger than the value of 2 eV noted above. The factor of two
discrepancy is too large to be accounted for by inaccuracies in band structure calculations. A likely
explanation is that the hybridization between the sp and d states involves a gradual shift in orbital
character, and estimating the gap boundary by visual inspection of the band dispersions is not accurate.
The re¯ectivity shown in Fig. 30(c) is another important interfacial property. It is close to, but less
than, unity, suggesting that the interface potential at the Ag±Fe interface is not fully con®ning. This is
not surprising, since a hybridization gap, rather than an absolute gap, provides the con®nement
potential. Also, the lattice mismatch between Ag and Fe, though small, could lead to non-specular
re¯ection, resulting in a reduced specular re¯ectivity. The ®nesse of the electron interferometer with
R 0.8 is about 20; this is almost comparable to that of a simple optical etalon.
6. Magnetic effects and spin polarization
The discussion so far has ignored the spin polarization, and yet spin effects can be very important for
layer systems containing magnetic materials. Much of the interest in this area is driven by applications
in ``spintronics'' (electronics based on the spin rather than the charge of carriers) or ``magnetoelectronics'' [101±105]. A well-known example is the use of the giant magnetoresistance effect for device
applications. This effect relates to the phenomenon that two ferromagnetic layers separated by a
nonmagnetic layer can exhibit oscillatory magnetic coupling depending on the thickness of the
interlayer. If the magnetic alignment is ferromagnetic, carriers can travel across the interfaces without a
spin ¯ip transition, and the electrical resistance is thus signi®cantly lower than the case of an
antiferromagnetic alignment. A carefully engineered multilayer structure that exhibits a large resistance
change upon the application of a magnetic ®eld can be used as a reader for magnetic recording media
[101±109].
As discussed earlier in connection with multilayer systems, translayer coupling can be mediated by
states that are either propagating or nonpropagating. The propagating states can be quantum-well-like
or continuum-like. In most magnetic systems of technological interest, it seems that translayer coupling
via quantum-well-like states is the most common. The coupling is generally believed to be of the
RKKY type, and the oscillation periods are thus determined by the nesting Fermi wave vectors of the
interlayer material. An attempt to describe such magnetic coupling effects in terms of electronic states
(quantum well states) has led to much interest in the spin properties of simple quantum wells made of
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T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
nonmagnetic ®lms on magnetic substrates. Translayer magnetic coupling is a rapidly growing area of
research. A detailed review of the applications and technologies is beyond the scope of this paper. The
focus here will be on the basic physics of spin-polarized quantum wells.
As seen earlier, quantum well spectroscopy is an excellent tool for measuring k(E), including the
Fermi wave vector kF ˆ k(0). At the Fermi level, the Bohr±Sommerfeld quantization rule becomes
2kF Nt ‡ F…0† ˆ 2np:
(16)
As N (the average thickness of a ®lm) increases, quantum well peaks move through the Fermi level oneby-one. Each passage is marked by a maximum in photoemission intensity at the Fermi level. Two
neighboring maxima (Dn ˆ 1) are separated by
p
:
(17)
DN ˆ
tkF
A measurement of DN gives kF. The appearance of intensity maxima at the Fermi level is thus a fairly
direct measure of the oscillation periods [28,29]. This argument can be generalized to emission
directions other than the surface normal.
Fig. 35. Spin-resolved normal-emission spectra from Cu ®lms of various thicknesses taken with hn ˆ 13 eV. The minority
quantum well states are marked by shaded areas (®gure taken from [118]).
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
229
Cu on Co [28,29,100,110±127] and Ag on Fe [28,29,67±72,89,100,128,129] represent two magnetic
quantum well systems that have attracted much interest. In both cases, the lattices are nearly matched,
and high quality ®lms can be prepared by standard growth techniques (although only Ag on Fe(1 0 0)
has been made with atomic uniformity). Fe and Co are ferromagnetic, while Ag and Cu are not. The
magnetic substrate materials are characterized by an exchange splitting of the bands. As a result, both
the hybridization gap and the window of con®nement depend on the spin. The phase shifts can be
different for the two spin orientations as well. Thus, photoemission should reveal two sets of quantum
well states, one spin up and the other spin down, generally at different binding energies. Spin-polarized
photoemission should yield unambiguous signatures for these splittings [100].
Fig. 35 shows spin-resolved photoemission spectra from Cu ®lms of different thicknesses on
Co(1 0 0) [118]. The majority spin spectrum shows a broad hump, with no clear evidence for quantum
well peaks, while the minority spin spectrum shows well-de®ned quantum well peaks. This behavior
can be explained in terms of the Co band structure. The Cu sp band near the Fermi level overlaps with
the minority hybridization gap in Co, and thus the minority electrons in Cu are well con®ned. The
corresponding Co majority gap is lower and overlaps the Cu d band region, and thus no sp quantum
well peaks near the Fermi level are expected.
For the Ag on Fe(1 0 0) system, the situation is similar. A recent study has shown that the observed
quantum well peaks are of the minority spin character at low coverages. Fig. 36 shows spin-resolved
photoemission spectra taken from a 1 ML Ag ®lm on Fe(1 0 0) [100,129]. The quantum well peak is
clearly of the minority spin character. As mentioned above, the Fe hybridization gap covers a range of
2 eV below the Fermi level. This is actually the minority gap, and so it is not surprising that minority
quantum well states are observed in this energy range. Available band structure calculations show that
Fig. 36. Normal-emission spectra for one monolayer of Ag on Fe(1 0 0). The spin-integrated spectrum is indicated by the
curve, and the majority and minority spin components are indicated by full and open triangles, respectively (®gure taken from
[129]).
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T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
Fig. 37. (a) Sample con®guration. (b) Photoemission intensity at the belly of the Fermi surface oscillates with 5.6 ML
periodicity of the Cu thickness. (c) Photoemission intensity at the neck of the Fermi surface oscillates with 2.7 ML periodicity
of the Cu thickness. (d) Interlayer coupling from a magnetic X-ray linear dichroism measurement. The bright and dark regions
correspond to antiferromagnetic and ferromagnetic couplings, respectively. (e) Calculated interlayer coupling based on a
phased sum of the two oscillations (®gure taken from [127]).
T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
231
G12 and D1 max are at about 0.8 and 3.2 eV binding energies, respectively, for the majority spin states in
Fe. Assuming that the same factor of two correction is needed as discussed above for the minority gap,
the majority gap should be about 1.2 eV wide. Applying the same linear mapping needed to go from the
calculated G12 and D1 max to Eu and El for the minority states, we can estimate that Eu ˆ 1.6 eV and
El ˆ 2.8 eV for the majority states. This range, though far below the Fermi level and relatively small,
should support some majority quantum well states. However, experimental search in the form of extra
peaks not explainable by the minority spin states has not turned up anything that can be identi®ed as
majority spin states. Perhaps they are just too weak or broad (the Bohr±Sommerfeld quantization rule
says nothing about the peak intensity). However, evidence for majority unoccupied states has been
detected by spin-polarized inverse photoemission [89,128].
We close this section by presenting an example in which quantum well measurements provide an
elegant explanation of the magnetic coupling effect in a Co±Cu±Co(1 0 0) sandwich structure.
Translayer coupling of the RKKY type can involve multiple nesting Fermi wave vectors. For Cu(1 0 0),
there are two relevant Fermi wave vectors, one associated with the ``belly'', and the other associated
with the ``neck''. This can be probed by photoemission from quantum wells as discussed before. Data
taken from a linear wedge sample with a con®guration depicted in Fig. 37(a) are shown in Fig. 37(b)
and (c) for the belly and neck oscillations, respectively [127]. One can use a simple semi-empirical
formula to estimate the relative phases between the two oscillations. A phased linear superposition of
the two oscillations, shown in Fig. 37(e), should reproduce the observed magnetic coupling as a
function of interlayer thickness. This is indeed the case, as revealed by a comparison with Fig. 37(d),
which is obtained by a magnetic X-ray linear dichroism measurement.
7. Summary and conclusions
This paper reviews the physics of quantum wells made of thin ®lms. The basic manifestation of
quantum con®nement is the formation of discrete states as observed by angle-resolved photoemission.
Concepts including con®nement by gaps, interfacial re¯ection, phase shift, barrier penetration, electron
propagation and scattering, layer±layer coupling are addressed in terms of the band structure and wave
functions. For commensurate overlayers, all that is required is a relative gap for con®nement. Partial
con®nement by hybridization gaps and thin barriers give rise to quantum well resonances. A signi®cant
lattice mismatch can cause a substantial reduction in specular re¯ectivity, and the resulting spectral
response can be qualitatively different. The foundation laid by studies of simple quantum wells
provides the knowledge base needed for understanding the properties of multilayer systems. The
concepts of interlayer coupling, translayer coupling, and band folding are clari®ed and illustrated by
examples involving multilayer and superlattice con®gurations.
An important application of quantum well spectroscopy is to determine the band structure,
quasiparticle lifetime, interface re¯ectivity, and phase shift, which are fundamental to solid state
physics and interface science. These quantities completely specify the electron dynamics within a ®lm
and at the ®lm interfaces, but are dif®cult to determine using any other means. The accuracy of
quantum well measurements is greatly improved by the use of atomically uniform ®lms. An
interferometric analysis of the photoemission data yields a band structure that is suf®ciently accurate to
challenge the de Haas±van Alphen method for Fermi surface determination. The fundamental reason
for this improvement is that k?, being a continuous variable in a bulk crystal and highly uncertain in
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T.-C. Chiang / Surface Science Reports 39 (2000) 181±235
photoemission, becomes quantized and precisely determined by the geometry of a ®lm. Temperaturedependent line width determination yields accurate measures of the quasiparticle lifetime, allowing a
detailed analysis of the electron±electron, electron±phonon, and electron±defect interactions. The phase
shift and re¯ectivity deduced from quantum well spectroscopy can be used to extract useful information
about the interface potential and the band structure of the substrate material.
Much of the current interest in quantum wells is driven by technological issues related to device
con®gurations made of magnetic layer structures. In such systems, quantum well states can be spin
polarized, giving rise to a magnetically sensitive response. The giant magnetoresistance effect is a
prominent case, and many technological issues can be addressed by photoemission measurements using
model structures. This provides an interesting example for a fairly direct connection between basic
research and device design and industrial applications.
Atomically uniform ®lms could become a dominant theme in this area of research if such ®lms can
be routinely prepared for materials other than just Ag on Fe whiskers. Beyond issues of academic
interest, atomically uniform ®lms represent the ultimate limit for device con®gurations made of ®lms.
The response function can be much sharper, and modeling can be much simpler. The current push for
nanotechnology will undoubtedly continue, and soon it would be up against the atomic limit. Atomic
level manipulation and control will remain a subject of intense research in the foreseeable future. Of
great interest would be the fabrication of other forms of quantum structures such as dots, wires, stripes,
etc., with atomic uniformity. Controlled dimensionality allows ®ne tuning of properties including
electron correlation. Studies of the interplay between quantum con®nement and magnetism in ®lms
have already resulted in useful applications. Similar studies involving other physical phenomena such
as superconductivity, charge ordering, and colossal magnetoresistance may be equally fruitful for both
basic science and technology.
Acknowledgements
The author wishes to thank present and former group members T. Miller, J.J. Paggel, A.L. Wachs,
W.E. McMahon, E.D. Hansen, M.A. Mueller, A. Samsavar, A.P. Shapiro, T.C. Hsieh, G.E. Franklin,
and D.-A. Luh for their contributions to the work on quantum wells. He also wishes to thank J. Weaver
for reading the manuscript and providing feedback. Much of the material presented here is based upon
work supported by the US National Science Foundation, under grant Nos. DMR-99-75470 and DMR99-75182. An acknowledgment is made to the Donors of the Petroleum Research Fund, administered
by the American Chemical Society, and to the US Department of Energy, Division of Materials
Sciences, (grant No. DEFG02-91ER45439) for partial support of the synchrotron beamline operation
and for support of the central facilities of the Frederick Seitz Materials Research Laboratory. The
Synchrotron Radiation Center of the University of Wisconsin is supported by the National Science
Foundation under grant No. DMR-95-31009.
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