quantum size effect in low energy electron diffraction of thin films
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Quantum size effect in low energy electron diffraction of thin ®lms
M.S. Altmana,*, W.F. Chunga, Z.Q. Hea, H.C. Poonb, S.Y. Tongb
aDepartment of Physics, Hong Kong University of Science and Technology, Kowloon, Hong Kong, PR ChinabDepartment of Physics, The University of Hong Kong, Hong Kong, PR China
Received 19 July 1999; accepted 5 October 1999
Abstract
Low energy electron microscopy (LEEM) is used to study the quantum size effect (QSE) in electron re¯ectivity from thin
®lms. Strong QSE interference peaks are seen below 20 eV for Cu and Ag ®lms on the W(1 1 0) surface and Sb ®lms on the
Mo(0 0 1) surface. Simple inspection of QSE interference peaks reveals that all three metals grow atomic layer-by-atomic
layer. Layer-speci®c I(V) spectra obtained with LEEM permit structural analysis by full dynamical multiple scattering LEED
calculations for a layer-by-layer view of thin ®lm structure. # 2001 Elsevier Science B.V. All rights reserved.
Keywords: Low energy electron diffraction; Low energy electron microscopy; Thin ®lm structure
1. Introduction
It is well-known that the size of an object can have
an important effect on its electronic properties. In
particular, a `̀ particle-in-a-box'' quantisation of elec-
tronic states is expected to occur when electrons are
con®ned in a very small space. Thin ®lms are prime
examples of systems that exhibit quantum size elec-
tronic effects. In the thin ®lm geometry, con®nement
imposed by the vacuum on one side and the supporting
substrate on the other effectively creates a one-dimen-
sional quantum well in the direction perpendicular to
the ®lm. Thin ®lm quantum size effects (QSE) invol-
ving electrons near the Fermi level have been studied
extensively in photoemission [1±10], inverse photo-
emission [10], electrical resistivity [9,11], Hall effect
[12], and scanning tunneling microscopy [13,14] in
the past. A QSE model has also been invoked to
explain and predict thin ®lm growth behaviour [15].
Observations of a QSE in electron re¯ectivity invol-
ving electrons above the vacuum level has also been
reported [16±22].
The QSE in electron re¯ectivity occurs when the
electron wavelength and penetration depth become
comparable to ®lm thickness at very low energy. The
QSE has been commonly understood to be an inter-
ference phenomenon between the electron waves
which are re¯ected from the surface of a thin ®lm
and from the interface between ®lm and substrate. The
phase shift between these two waves which deter-
mines the nature of their interference is related to ®lm
thickness according to f � k0d � k0(2t), where k0 is
the electron wave vector in the thin ®lm, d � 2t is the
path length difference, and t is the ®lm thickness.
Taking the inner potential of the thin ®lm, V0, into
account and adopting the free electron dispersion in
the thin ®lm that has been often assumed in the past,
the phase shift may be written as
f � 2t
�h
� � ������������������������2m�E � V0�
p(1)
Applied Surface Science 169±170 (2001) 82±87
* Corresponding author. Tel.: �852-2358-7478;
fax: �852-2358-1652.
E-mail address: phaltman@ust.hk (M.S. Altman).
0169-4332/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 9 - 4 3 3 2 ( 0 0 ) 0 0 6 4 4 - 9
where E is the incident beam energy. Although this
kinematic model is simplistic, it serves well to demon-
strate the key feature of the QSE, i.e., the QSE
interference periodically modulates the re¯ected
intensity as a function of both the energy and the ®lm
thickness.
For a ®lm consisting of N atomic layers, there are
exactly (N ÿ 1) QSE interference peaks between con-
secutive Bragg peaks. Because of this relationship, the
prominent QSE interference peaks that arise at very
low energy are sensitive indicators of total ®lm thick-
ness. This sensitivity immediately suggests the utility
of the QSE in the study of thin ®lm structure. Com-
bined with surface structure determination by low
energy electron diffraction (LEED) at conventional
energies, the QSE at very low energies can provide
unprecedented insight into the structure of buried
interfaces. However, the QSE was not exploited in
such a manner, nor even studied in much depth, until
recently because of the challenges posed by a number
of experimental and theoretical dif®culties. First,
electron beams are increasingly dif®cult to handle
at lower energies due to their sensitivity to stray
magnetic ®elds. A major experimental challenge to
study the QSE stems also from the dif®culty to prepare
®lms with uniform thickness, or equivalently to mea-
sure re¯ected intensity from regions of uniform thick-
ness. For these reasons, it has not been worth putting
much emphasis on the quantitative comparison of
experimental data with model predictions.
The experimental dif®culties to study the QSE have
recently been solved by the use of the low energy
electron microscope (LEEM) as an `̀ electron inter-
ferometer'' [21]. The novel use of the LEEM has
several advantages over earlier work on the QSE.
First, LEEM's extensive magnetic shielding and
immersion objective lens are ideally suited for precise
control of very low energy electron beams. The second
advantage is that LEEM allows the re¯ected intensity
from regions of different ®lm thickness to be distin-
guished with atomic precision and therefore to obtain
layer-speci®c I(V) curves. This has not been possible
with any of the laterally-averaging techniques which
have been used in the past to study the QSE in its
various forms. The LEEM approach, therefore, sig-
ni®cantly simpli®es the comparison of experimental
data to model predictions because no separate knowl-
edge of the ®lm thickness variation is required. This
simpli®cation has encouraged more sophisticated
modelling of the QSE, including a quantum mechan-
ical Kronig±Penney (KP) model [21], relativistic
Green functions calculations of thin ®lm local density
of states [20] and dynamical multiple scattering cal-
culations [22].
2. Experimental
We report here on measurements of Cu and Ag ®lms
on the W(1 1 0) surface and Sb ®lms on the Mo(0 0 1)
surface. There have been no prior studies of Sb ®lm
growth on the Mo(0 0 1) surface. Sb in bulk is known
to have a rhombohedral structure which is quite close
to simple cubic with edge length of 3.18 AÊ . Since this
is very close to the Mo bcc lattice constant of 3.15 AÊ , it
is therefore expected that sightly strained layers of Sb
can be grown on the Mo(0 0 1) surface. In the present
work, growth of four well-ordered layers was discern-
ible with LEEM at 400 K. The ®rst layer grew pseu-
domorphically and to completion before nucleation of
the second layer. Similarly, the second layer grew with
(11� 2) periodicity and was completed before
nucleation of subsequent layers. On the contrary,
fourth layer islands appeared already prior to comple-
tion of the third layer. Deposition of additional mate-
rial on top of the four layer ®lms caused the sudden
generation of a high density of dislocations. We report
here only on the ®rst four well-ordered layers.
The growth and structure of Cu on the W(1 1 0)
surface have been discussed previously [23,24]. For
this work, a (15� 1) Cu double-layer was grown
initially at 960 K, which resulted in ®lms of the
highest quality. Kinetically limited layer-by-layer
growth of Cu ®lms beyond two layers was carried
out at 420 K on top of the wetting Cu double-layer. In
this case, the (15� 1) is replaced by an approximate
(1� 8) periodicity by the completion of the third
layer. The (1� 8) structure corresponds to a nearly
undistorted Cu(1 1 1) lattice.
Ag ®lms on W(1 1 0) are known from earlier stu-
dies [25±27] to pass through a sequence of ordered
structures with increasing coverage. An initially [100]
row-matched structure with higher order periodicity in
the [110] direction transforms between 1 and 2 ML
coverage to a distorted Ag(1 1 1) bilayer structure
above 800 K. Thicker metastable ®lms of the same
M.S. Altman et al. / Applied Surface Science 169±170 (2001) 82±87 83
structure can be prepared by deposition at somewhat
lower temperature. In our work, two and three layer
Ag ®lms were grown at 795 K. Thicker ®lms were
grown at 500 K on top of three ML ®lms that were
prepared at higher temperature.
Examples of LEEM images of Sb ®lms on the
Mo(0 0 1) surface obtained at two energies are shown
in Fig. 1. Contrast in these images stems from the
dependence of the QSE interference condition
(Eq. (1)) upon ®lm thickness, and is aptly named
quantum size contrast. In other words, the various
intensities in each image identify regions of different
®lm thickness. Laterally resolved measurements of the
QSE were made with LEEM in the bright-®eld ima-
ging mode. Re¯ected intensities were integrated over
small areas of uniform thickness in the image (Fig. 1).
This amounts to measuring the layer-speci®c (0,0)
beam intensity. The QSE interference peaks for Sb/
Mo(0 0 1), Cu/W(1 1 0), and Ag/W(1 1 0) are shown
in Figs. 2±4, respectively. Inspection of these results
reveals that the number of QSE interference peaks
increases by one for each additional layer. In the case
of Cu and Ag ®lms, this veri®es the atomic layer-by-
atomic layer growth that has been observed previously
by other methods [23±27]. For thicker Ag and Cu
Fig. 1. LEEM images of two and three MLSb ®lms on the Mo(0 0 1) surface (a) 4.5 eV; (b) 5.5 eV. The dark lines are monoatomic steps
generated at the Sb/Mo interface.
Fig. 2. Experimental QSE interference peaks for N atomic layer
thick Sb ®lms on Mo(0 0 1) and for clean Mo(0 0 1). A Bragg peak
associated with the Sb ®lm appears at approximately 10 eV.
84 M.S. Altman et al. / Applied Surface Science 169±170 (2001) 82±87
®lms and for the Sb ®lms, the QSE data give a clear
indication that all three metals grow in a simple atomic
layer-by-atomic layer manner, as opposed to multi-
layer growth.
3. Theoretical modelling
The layer-speci®c I(V) spectra that are obtained
with LEEM permit a straightforward comparison with
model predictions. In this section two options for
theoretical modelling of the QSE are outlined. A
rather simple approach that has been taken is to model
the QSE with a quantum mechanical KP model [21].
This model is the simplest which predicts elementary
features of electron band structure [28]. The KP model
as implemented in the thin ®lm geometry consists of a
re¯ective potential step from the vacuum level to a
periodic KP potential that mimics the periodicity of
the ®lm. Then there is another re¯ective potential step
down to the substrate potential. As seen in Fig. 3, the
KP model is able to reproduce the positions of the
nearly equally spaced QSE interference peaks as well
as the peak shapes quite well for Cu/W(1 1 0). This is
clearly an improvement over the kinematical model
discussed above (Eq. (1)) which predicts a non-linear
relationship between QSE peak position and energy,
even for realistic values of the inner potential [21].
The QSE peak amplitudes are not reproduced as well
because of the neglect of absorption in the KP model.
Despite the apparent success of the KP model, it is
Fig. 3. QSE interference peaks for N atomic layer thick Cu ®lms
on W(1 1 0): experimental (thick line) and Kronig±Penney model
predictions (thin line) [21]. The large feature above 20 eV in the
predicted curve is a Bragg peak associated with the Cu ®lm.
Fig. 4. Experimental QSE interference peaks for N atomic layer
thick Ag ®lms on W(1 1 0) and for clean W(1 1 0). A Bragg peak
associated with the Ag ®lm appears at approximately 15 eV.
M.S. Altman et al. / Applied Surface Science 169±170 (2001) 82±87 85
dif®cult to assign any physical signi®cance to its
phenomenological ®t parameters.
Now that the demand for high quality data can be
met by using the LEEM approach, it is worth con-
sidering the use of full dynamical multiple scattering
calculations to evaluate the QSE. However, details of
dynamical inputs which are usually neglected in
LEED calculations above 50 eV may become impor-
tant below 30 eV, where the QSE interference peaks
are the strongest. Factors such as the surface barrier
and the energy dependence of the self-energy (inner
potential, electron damping) may have a large in¯u-
ence on the diffraction spectra at very low energies. In
LEED at conventional energies, the inner potential
and electron damping are taken to be constant, and the
surface potential is normally treated as a non-re¯ect-
ing barrier which is only responsible for electron
refraction. It is also common to match the inner
potential of the thin ®lm and substrate.
LEED dynamical analysis has so far been applied to
Ag ®lms on W(1 1 0) [22]. Energy dependent electron
damping in Ag and W was determined directly from
the measured optical dielectric functions [29,30]. To
account for electron re¯ection at the surface barrier,
the image potential model of Jones, Jennings, and
Jepsen (JJJ) was used [31]. This potential is charac-
terised by three independent parameters, the barrier
height, U0, width, l, and position of the image plane,
z0 Electron damping and the re¯ective surface barrier
have been found to affect mainly the relative inten-
sities of the diffraction peaks. More important for
reproducing the positions of the QSE features are
the inner potential and its energy dependence. We
have assumed a linear form below 25 eV, V�E� �aE � b.
Details of the dynamical inputs (surface barrier and
inner potential parameters) are determined by com-
paring theory with the experimental data for W(1 1 0)
and for thick Ag(1 1 1)-like ®lms on W(1 1 0). In
order to determine the dynamical parameters, top
layer surface relaxations obtained previously for
Ag(1 1 1) and W(1 1 0) were used [32,33]. There
are four free parameters in the dynamical model,
namely, l, z0, a, and b, since the quantity U0 is a
function of a and b. The optimised Ag surface barrier
parameters U0 � 0:89 Ry, l � 0:75 a.u. and z0 �ÿ1:6 a.u. and inner potential parameters (a, b) of
(ÿ0.1, 12.2 eV) and (ÿ0.04, 18.5 eV) for Ag and
W, respectively, were determined. The surface barrier
was not used for W(1 1 0) ®rst of all because it only
affects the peak intensities and secondly because the
vacuum-W(1 1 0) barrier is irrelevant in the Ag ®lm
system. With the exception of the Ag inner potential,
these parameters were used in subsequent modelling
of the QSE interference peaks for thin Ag ®lms on
W(1 1 0). As a consistency check, the Ag inner
potential was also treated as a quantity to be opti-
mised, along with the layer spacing at the buried
interface in the modelling of the QSE interference
peaks. The results of optimisation show consistency of
the inner potential determined by the two methods.
From the analysis of the QSE peak positions by
dynamical theory, it was found that the Ag/
W(1 1 0) interface layer spacing decreases as the ®lm
thickness is increased [22]. The comparison of calcu-
lated with experimental QSE interference peaks will
be discussed in detail elsewhere [34].
Whether the QSE in electron re¯ectivity can be
generally of use for studying ®lm structure depends
crucially upon the existence of QSE interference
peaks. Based upon the models discussed above,
QSE interference peaks can be expected if there is
a suf®ciently large discontinuity at the interface, for
example, arising from the difference of the inner
potential of the ®lm and substrate. In comparison to
Ag/W(1 1 0) and Cu/W(1 1 0), the somewhat weaker
QSE interference that is seen for Sb/Mo(0 0 1) may be
related to the different substrate scattering for
Mo(0 0 1) and W(1 1 0) that are shown in Figs. 2
and 4, respectively. In addition to the three systems
discussed in this paper, QSE interference has been
observed for Cu/Co(1 0 0) [19] and Co/W(1 1 0) [20].
However, we have found no interference peaks for Ag/
Cu(1 1 1) and Co/W(1 1 1). The absence of QSE
interference in these systems may be an indication
of interface mixing that effectively diminishes the
interface discontinuity.
4. Conclusion
The QSE in electron re¯ectivity gives rise to pro-
minent interference peaks at very low energy which
are sensitive indicators of ®lm thickness. We have
described a novel experimental approach to study the
QSE based upon low energy electron microscopy
86 M.S. Altman et al. / Applied Surface Science 169±170 (2001) 82±87
(LEEM). Examples of QSE interference peaks for the
systems Ag/W(1 1 0), Cu/W(1 1 0) and Sb/Mo(0 0 1)
are presented. These data reveal that all three metals
grow atomic layer-by-atomic layer. In agreement with
expectations, Sb is found to grow epitaxially on the
Mo(0 0 1) surface. The layer-speci®c I(V) data that are
obtained with LEEM permit a straightforward com-
parison with model predictions. Approaches to study
®lm structure based upon a simple quantum mechan-
ical Kronig±Penney model and more sophisticated full
dynamical multiple scattering LEED calculations
were outlined. In addition to gaining a layer-by-layer
view of thin ®lm structure, these investigations help to
clarify several issues which are of concern in the use of
very low energy electron diffraction.
Acknowledgements
This work is suported by HK RGC grants HKU 260/
95P and HKUST 6129/98P, U.S. DOE Grant No. DE-
FG02-84ER45076 and NSF Grant No. DMR-
9214054.
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