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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