Cao Z. (Ed.) Thin Film Growth: Physics, materials science and applications

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26 Thin ﬁlm growth

of momentum – but the third component k

along the surface normal is not

directly determined. Applying the technique to three-dimensional systems

then requires some input assumptions or special methods. Nonetheless, it

has been used with great utility for measuring the band structures of many

materials. ARPES gives the complete E(k) relation for two-dimensional

systems such as surface states, and can be applied readily also to systems that

are quasi-two-dimensional, such as supported graphene (Ohta, Bostwick et al.

2006, 2007), certain crystal planes in high-T

materials (Kondo, Khasanov

et al. 2007), or layered materials such as charge-density wave compounds

(Wilson and Yoffe 1969, Kidd, Miller et al. 2002), etc. The ‘k

problem’

can also be bypassed in the application of ARPES to thin lm systems that

support quantum-well states (Loly and Pendry 1983).

An example of ARPES data from thin lms is shown in Fig. 2.2 (Luh,

Miller et al. 2002). The system is Ag on Fe(100), and the lms were grown to

promote smooth layer-by-layer growth by depositing at low temperature and

subsequently annealing (Paggel, Miller et al. 2000). The Ag lms are oriented

so the surface is (100). Each curve is a photoemission energy distribution

curve (EDC), created by scanning the electron analyser energy setting while

keeping the emission angle and the exciting photon energy constant. The

energy scale has been shifted so that zero is at the Fermi level. The different

curves are from lms of different thicknesses in monolayers (ML) as shown,

Energy (eV)

–2

–4

–6

–8

G D X S G L L

Crystal momentum

[100] [110] [111]

2.1 Energy band structure of silver along three high-symmetry

directions. The dashed circle marks the area of the sp bands that

contribute to the quantum-well states in Fig. 2.2.

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27Quantum electronic stability of atomically uniform ﬁlms

and are displaced vertically for clarity. The emission direction was normal

to the sample surface. Each spectrum shows a set of discrete peaks which

crowd together and become more numerous as the lm thickness is increased.

The spectra can be understood as follows: because the spectra were taken

in normal emission, the parallel components of momenta are zero and the

relevant part of the band structure is along the [100] direction. In the bulk,

we see a continuous band of states in the energy range represented by these

spectra (see Fig. 2.1). That band is derived from the sp states of Ag; it is

highly dispersive and can be regarded as nearly-free-electron-like. However,

in a thin lm, the surface and substrate interface of the Ag lm behave

as reecting barriers to electron motion, and as in the simple ‘particle-in-

a-box’ problem of elementary quantum mechanics, the states which are

continuous in the bulk become quantized. The allowed states are given by

the Bohr–Sommerfeld condition:

Ag/Fe(100)

2 1 0

Binding energy (eV)

Photoemission intensity (arb. units)

2.2 Normal emission photoemission spectra from different

thicknesses of Ag on Fe(100). The thickness for each spectrum

is shown as the number of monolayers N. The energy scale is

referenced to the Fermi level. Peaks corresponding to four different

quantum numbers are marked with dashed lines.

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28 Thin ﬁlm growth

2np = 2k(E)d + f(E) [2.1]

where k(E) is the electron wavevector component perpendicular to the

surface, d is the thickness of the lm, f(E) is the total phase shift of the

wavefunction on reection from both the substrate interface and the surface,

and n, an integer, is the quantum number. The term k(E) is the inverse of

the dispersion relation such as those represented in Fig. 2.1. With reference

to the band structure, roughly (ignoring the phase shift), the allowed states

can be obtained by dividing up the interval in k space between G and x by

the number of layers in the lm and reading off the corresponding energies.

As in the spectra, the allowed states become more numerous and approach

a continuum as the lm thickness increases.

2.4 Atomically uniform ﬁlms

The curves in Fig. 2.2 are labeled by integer thicknesses. Generally, lms

deposited using a quartz microbalance to gauge lm thickness will not be

controlled to such precision. However, using the two-step deposition and

annealing process, these lms are of atomically uniform thickness, which

makes absolute thickness measurement by photoemission possible. This is

illustrated by the sequence of deposition and measurement represented in

Fig. 2.3 (Paggel, Miller et al. 1998). Data from three sequential depositions

are shown, going up in the gure. The bottom and top curves are from

atomically-uniform lms 38 and 39 ML thick, respectively. They both show a

set of individual peaks roughly evenly spaced in energy. The middle curve is

from a lm of intermediate thickness and, in contrast, shows pairs of peaks.

By inspection, each peak in a pair corresponds to a peak in either the top or

bottom spectrum. We can conclude that the middle spectrum is composed

of emission from distinct areas of the lm that differ in thickness by 1 ML,

whereas the other two show evidence of only one thickness each.

This ability to grow atomically uniform lms is extremely important for

fundamental studies. Generally, a lm will have a distribution of thicknesses,

and photoemission spectra will be composed of many, possibly unresolved,

peaks. This makes assignment of energies uncertain. Further, the peaks may

appear to smoothly evolve with deposition, making an absolute determination

of thickness impossible. In contrast, for the growth represented in Fig.

2.3, one knows that exactly one monolayer has been added in going from

the bottom to the top spectrum, so that the microbalance can be precisely

calibrated and atomically uniform lms of any desired thickness can be grown.

The individual sharp peaks afforded by uniformity allow photoemission

measurements of quantum behavior even from rather thick lms. Figure

2.4 shows spectra from lms up to 119 ML (Paggel, Miller et al. 1999b,

2000). The presence of quantum-well peaks to this thickness demonstrates

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29Quantum electronic stability of atomically uniform ﬁlms

a long coherence length for the electronic states in the lm. Once a set of

spectra like those in Figs 2.3 and 2.4 have been collected, photoemission

becomes a tool for measuring the physical structure of the lm on an atomic

scale. Atomically uniform lms have been obtained in a variety of systems,

including Ag on semiconductor substrates Si(111) and Ge(111), and Pb on

Si. Using these uniform lms, a variety of measurements have been made

of quantum behavior in thin lms, including band structure (Paggel, Miller

et al. 1999b), electron–phonon coupling (Paggel, Miller et al. 1999c, Paggel,

Luh et al. 2004, Luh, Miller et al. 2002), variations in work function with

thickness (Paggel, Wei et al. 2002), and thermal stability (Luh, Miller et al.

2001, Upton, Wei et al. 2004).

2.5 Quantum thermal stability of thin ﬁlms

The ability of photoemission to measure the thickness of an atomically uniform

lm makes it an effective tool to investigate thermal stability. An example

is shown in Fig. 2.5 (Luh, Miller et al. 2001). The two panels show the

Photoemission intensity (arb. units)

Ag/Fe(100)

hn = 13 eV

39 ML

38.5 ML

38 ML

2 1 0

Binding energy (eV)

2.3 Normal emission photoemission spectra from Ag ﬁlms on

Fe(100) with thicknesses as shown. The top and bottom spectra

are from atomically uniform ﬁlms, whereas the middle spectrum is

from a ﬁlm of intermediate thickness. Arrows mark the positions of

corresponding peaks.

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30 Thin ﬁlm growth

evolution of spectra as two Ag lms of different thicknesses (6 and 3 ML),

each initially of uniform thickness, change under annealing to successively

higher temperatures. The uniform lms were prepared on a Fe(100) substrate

by depositing at low temperature (~110K) and then annealing to produce

atomically uniform lms, then the sample was cooled back down before

starting the measurements. The top curve in each panel was measured at

this low temperature as indicated and shows a single quantum-well peak

indicative of the single thickness of the lm. Subsequent spectra, going down

in the gure, were taken at successively higher temperatures as indicated.

As the temperature is increased, some broadening of this initial peak can be

seen. This is expected due to the increasing thermal agitation of the system.

At some point, the spectra change qualitatively. Two other peaks on either

side of the original one appear, and the original one eventually fades out.

The bottom curves were taken last in the sequence, after cooling the sample

back down to the base temperature. Cooling reduces the thermal broadening,

sharpening the peaks, and now it can be clearly seen that the original peak

has been replaced by the two new ones. By comparison to spectra taken from

Photoemission intensity (arb. units)

Ag/Fe(100)

hn = 16 eV

Thickness

(ML)

119

1 0

Binding energy (eV)

2.4 Normal emission photoemission spectra from atomically uniform

Ag ﬁlms of different thicknesses as shown.

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31Quantum electronic stability of atomically uniform ﬁlms

samples of different uniform lm thicknesses, the new peaks can be identied

as shown in the gure: they belong to lms with thicknesses differing by

one monolayer from the original lm. In other words, heating these lms has

caused them to bifurcate from the initial uniform N monolayers to a system

with N + 1 and N – 1 layers.

Figure 2.6 shows the intensity of the original peak as a function of

annealing temperature for a few lms of different starting thicknesses (Luh,

Miller et al. 2001). Except for the plot for the 5 ML starting thickness, each

curve exhibits a well-dened breakpoint which can be assigned to a ‘stability

temperature’ for that thickness. The 5 ML lm survived the highest annealing

temperature achievable in the experiment (900K), so only a lower limit

could be established for its stability temperature. Stability temperatures for

a range of coverages are shown in Fig. 2.7(a) (Luh, Miller et al. 2001). The

data show dramatic differences in thermal stability between lms that differ

in thickness by even just one atomic layer. As mentioned above, 5 ML is

Photoemission intensity (arb. units)

6 ML

3 ML

2 ML

5 ML

4 ML

7 ML

3 ML

T (K)

111

159

199

240

279

319

361

398

453

494

115

118

162

202

246

279

320

350

374

414

451

150

2 1 0

Binding energy (eV)

2 1 0

Binding energy (eV)

2.5 Evolution of spectra from initially uniform ﬁlms under annealing.

The top spectrum in each stack is an atomically uniform ﬁlm of 6

ML (left) and 3 ML (right) thickness. For both cases, each spectrum

below the top one was taken at successively higher temperatures as

indicated, except the bottom one which was taken after the samples

were allowed to cool back down to near the starting temperature.

Each stack shows a single peak indicative of a uniform ﬁlm which

splits into two peaks as annealing proceeds. The end result is a

bifurcated ﬁlm.

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32 Thin ﬁ lm growth

particularly stable, while 6 ML requires care to prepare. Most  lms broke down

at temperatures not far above RT (<400K or so), but  lms of thickness 1, 2,

and 5 ML all survived to at least 800K. In addition, the variation in stability

is not monotonic. These features point to quantum effects as the source of

the variability. A clue to the exceptional stability of a 5 ML  lm can be seen

in the spectra of quantum-well state energies for different thicknesses (Fig.

2.2). It can be seen that the highest energy state in the 5 ML spectrum lies

below that of both the 4 ML and 6 ML spectra. This suggests that, in order

for a 5 ML  lm to bifurcate, energy would have to be added to the system,

making that thickness energetically stable.

A measure of stability against bifurcation can be written as:

()

[(

+ 1)

(

– 1

)]

– A

()

()NA()

=NA =

[(NA[(

+ 1)NA + 1)

+NA +

(NA (

(NN (

– 1NN – 1

)]NN)]

– ANN – A

()NN()

where A(N) is the total electronic energy of the system. D(N) is the discrete

Initial uniform

Ag thickness

(ML)

0 100 200 300 400 500

Temperature (°K)

Normalized quantum-well peak intensity (arb. units)

2.6 Breakdown of initially uniform ﬁ lms under annealing. Each

curve shows the intensity of a photoemission quantum-well peak

characteristic of the thickness of an initially atomically uniform ﬁ lm

as the ﬁ lm was heated to successively higher temperatures. Except

for the 5 ML ﬁ lm, all showed a distinct temperature where this peak

intensity started to decrease. Dots are data points and the lines are

guides to the eye.

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33Quantum electronic stability of atomically uniform ﬁlms

second derivative of the total electronic energy, and represents the energy

needed to transform the system from an original at lm of thickness N to

a bifurcated one consisting of equal areas of N + 1 and N – 1 thicknesses.

So a large positive D (N) would characterize a stable lm thickness. Here we

are considering only the energy of the ‘bulk’ states of the lm; in general

other contributions, such as edge terms, are possible. To calculate D (N),

we start with Eq. 2.1 which gives the discrete allowed values of k

for

each thickness provided the phase shift function is known. The phase shift

can be experimentally determined by an analysis of the quantum well as a

Fabry–Pérot interferometer with the input of quantum-well peak positions from

a large number of thicknesses, as in Fig. 2.4 (Paggel, Miller et al. 1999b).

This analysis is involved but the basic idea can be illustrated by considering

Lower limit

(a)

(b)

(c)

0 2 4 6 8 10 12 14

Thickness N(ML)

1000

800

600

400

0.04

0.00

–0.04

0.2

0.0

–0.2

Temperature (K)

D(N) (eV/site)D(N)

= k

= 0 (eV)

2.7 (a) Experimentally measured bifurcation temperatures for ﬁlms

of different starting thicknesses. The point for 5 ML is a lower limit;

this was the highest temperature achieved in the experiment but the

ﬁlm did not break down. (b) Calculated change in energy per surface

site due to bifurcation. (c) Same as (b) except only considering states

visible in normal emission.

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34 Thin ﬁlm growth

a hypothetical pair of spectra from lms of two different thicknesses that

happen to show a quantum-well peak at the same binding energy. In this

case, k

(E) and the phase shift are the same for the peaks in both spectra,

but they will have different quantum numbers n. The thickness of the lm

is also quantized to an integer number of layers N. By writing Eq. 2.1 once

for each of the two lms, the system can be solved for f(E) and k

(E) for the

energy of the peak. In fact, this is a way of determining the band structure as

well as the phase shift as a function of energy (Mueller, Miller et al. 1990,

Mueller, Samsavar et al. 1989). A(N) involves all of the lled states, not

only those as seen in normal emission, but also those with non-zero parallel

components of momenta as well. That is, there is a set of discrete subbands

each with continuous k

and k

. generally, a consideration of the entire three-

dimensional band structure is needed. Figure 2.7(b) shows calculated results

for D (N), using a realistic band structure (Smith and Mattheiss 1974). Only

the nearly-free-electron-like sp bands are included – Ag also has a set of

more deeply bound d states as part of its valence band structure, but these

do not cross the Fermi level and are completely lled. The results can be

compared to the measured stability temperatures. Two features stand out:

the calculation, like the experimental results, shows signicant and non-

monotonic variations as a function of lm thickness, and also reproduces

the relatively high stability of lms with N = 2 and N = 5. experimentally,

a single monolayer is also very stable. The calculation does not extend to

this thickness, however, because A(0) is undened. Physically, for a single

monolayer to bifurcate it must leave half of the substrate area bare. Bonding

of Ag to the Fe surface might be expected to be stronger than Ag to itself,

based on the fact that it does tend to wet the Fe surface. This would tend to

stabilize the single monolayer without respect to quantum-well effects.

These results clearly show an important role for the valence electrons

in the stability of lms, even for metallic bonding where the states are

distributed throughout. The extreme stability of particular thicknesses (such

as the 5 ML one here) suggests that, for applications, these effects need to

be considered and may be useful. Atomically uniform lms were created

in these studies by a fairly cumbersome procedure; however, it was noted

in experiments that a 5 ML lm could readily be created by desorbing a

thicker lm simply by heating. For these fundamental studies, the uniform

lms allow a simple theoretical treatment. The experimental fact that the

lms break down rst by bifurcation also helps – the nal system can be

regarded as essentially two samples with different thicknesses but still each

with atomic uniformity. In general, life may be more complicated, with

the formation of many different thicknesses, or clusters where lateral QSE

and surface energies would have to be considered along with the detailed

geometries of the constituent parts.

The sp band of Ag is generally nearly-free-electron-like near the Fermi

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35Quantum electronic stability of atomically uniform ﬁlms

level, but the Fermi surface becomes non-spherical as one goes out along the

(111) directions. In fact the band does not cross E

along that direction, so

the Fermi surface develops ‘necks’ at the zone boundaries. This potentially

complicates the effect of quantization on the thermal stability, as this area

would be included in the sum of energies over all states. To gain physical

insight, a simpler ‘stability function’ can be calculated without summing over

and k

, using only states with zero parallel momenta. The result is shown

in Fig. 2.7(c). Comparing the two calculations we see that the simpler one

actually preserves the essential features of the system’s behavior. In other

words, a reasonable picture of the electronic stability of these lms comes

from looking only at the states at the zone center, namely, the peak positions

that can be seen in Figs 2.2 and 2.4. As the lm thickness is increased one

layer at a time, the electronic states shift towards higher binding energies,

increasing the stability of the system. However, at intervals, a new state

appears near the Fermi level, which increases the electronic energy in a

discontinuous fashion. The physics is reminiscent of the shell model of the

stability of atoms as one runs through the elements of the periodic table.

With this picture in mind, the importance of the phase shift can be readily

appreciated. The entire spectrum of allowed energies of a lm would shift

with changing this quantity. This directly affects the total electronic energy,

and also changes the points at which a new state would appear from the top

of the spectrum as the thickness is increased. It is then reasonable to expect

the energy landscape and the electronic stability to depend sensitively on

this phase shift. In fact that is the case, as will be discussed below.

2.6 General principles of ﬁlm stability and

nanostructure development

Atomically uniform lms of Ag on Fe(100) can be prepared over a very

large range of thicknesses and their photoemission spectra display a number

of peaks over a relatively wide spectrum of binding energies. These features

make this system ideal for illustrating the basics of quantum electronic

stability. The model presented above gives an intuitive picture, but it relies

on a band structure as input (the bulk band structure was assumed) and all

of the details of the surface and interface are rolled up into the phase shift

parameter. In this section, the system Pb on Si(111) will be examined to

extend the theoretical picture and explore some general principles of quantum

electronic stability.

Atomically uniform lms of Pb can be grown by depositing with the

substrate at liquid nitrogen temperatures. The substrate presents a barrier

to electron transmission for energy levels within the Si band gap. Outside

the gap, partial transmission occurs and the quantum-well peaks become

broader resonances that are more difcult to see in photoemission spectra.

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