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

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36 Thin ﬁlm growth

This limits the number of peaks that can be seen in any given spectrum.

Figure 2.8 shows normal emission photoemission intensity as a function of

binding energy and lm thickness for a series of depositions ranging from

1.0

0.5

0.0

Binding energy (eV)

Thickness N (ML)

Binding energy (eV)

Fit

Expt

n = 0

0 2 4 6 8 10 12

Thickness (ML)

2.8 Top: 3-D mesh plot of normal emission photoemission intensity

as a function of binding energy and ﬁlm thickness for Pb ﬁlms on

Si(111). Bottom: modeled (circles) and measured (crosses) quantum-

well peak binding energies for Pb on Si(111). The model is ﬁtted to

the measured peak positions and includes the band structure of Pb

and the phase shift. Quantum-well peak positions corresponding to

particular quantum numbers n are joined by lines. The dashed line

represents the position of the Si band edge: quantum conﬁnement

in the ﬁlm occurs for energies between the band edge and the Fermi

level at zero energy.

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37Quantum electronic stability of atomically uniform ﬁ lms

about 5 ML to 9 ML (Upton, Wei et al. 2004). Pairs of peaks like those in

Fig. 2.3 are not seen. However, for 6 ML no peak at all appears. The pattern

can be understood by reference to the plot of the binding energies of the

allowed states as a function of thickness shown in the bottom part of the

 gure (Luh, Miller et al. 2002, Smith 1985, Upton, Wei et al. 2004, Wei

and Chou 2002). Over the thickness range from 5 to 9 ML there will be an

allowed quantum-well state between the Fermi level and the Si band edge

only for the odd thicknesses of 5, 7, and 9 ML. Starting at 5 ML and going

up in thickness a monolayer at a time, we see the state in the ‘con nement

gap’ for 5 ML being pulled down below the Si band edge at 6 ML, while

another allowed state is still above the Fermi level and so is unoccupied,

leaving no states to be seen with photoemission. Continuing up in thickness,

at 7 ML this unoccupied state becomes occupied and falls within the gap

region, and appears in the ARPES data. At 8 ML this state is just below the

gap. It actually does appear in the photoemission data as a weak, broad feature

– it is a quantum-well resonance that is not quite bound in the well. This

pattern prevents the observation of pairs of peaks characteristic of adjacent

monolayer thicknesses. On the other hand, the complete disappearance of

peaks when no states are within the con nement gap indicates that those

 lms are in fact atomically uniform.

The alternating-layer pattern is re ected in the growth of Pb on Si and

diverse physical properties of  lms, including their thermal stability and

even superconducting transition temperatures (Guo, Zhang et al. 2004). As

previously mentioned, preference for bilayer height steps in the population

of islands grown at different temperatures has been observed in STM

measurements (Hupalo, Kremmer et al. 2001, Hupalo and Tringides 2002).

The thermal stability of uniform Pb  lms has been measured by photoemission;

results are shown in Fig. 2.9(a), where oscillations with a bilayer period are

seen (Upton, Wei et al. 2004). The physical basis for the bilayer pattern is

in the valence band structure. Along the (111) growth direction, the Pb sp

band crosses the Fermi level about halfway through the Brillouin zone. As

discussed above, sharp variations in physical properties are expected to be

related to quantum-well states crossing the Fermi level as the  lm grows.

As a simple illustration, consider a case where the relevant band crosses

the Fermi level exactly in the middle of the zone. Ignoring the phase shift,

the allowed quantum-well states are determined by dividing up the zone by

the number of layers. It can easily be seen that as each layer is deposited,

the spectrum of allowed energies alternates between having a state right

at the Fermi level, or not. In fact, based on this picture, one can de ne a

periodicity for quantum property variations derived from the Fermi wave

vector k

along the growth direction:

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38 Thin ﬁlm growth

where Dh is the periodicity in thickness at which Fermi level crossings

occur and is equal to half of the Fermi wavelength. Here, h is dened as a

continuous variable, but a uniform lm is constrained to quantized values

that differ by integer numbers of monolayers. This discrete sampling is

important because it can lead to aliasing which produces very different

observed periods of oscillation in properties (Miller, Chou et al. 2009). A

related complication is the denition of the Fermi wave vector. For example,

using a free-electron value of k

= 1.58 Å

–1

for Pb lms gives Dh = 0.7 ML,

which is obviously not observed. However, taking into account the nite

sampling by monolayer increments, the aliased period is 2.2 Ml, a reasonable

value in line with observation. Accounting for the sampling in this case is

(a)

(b)

(c)

Expt

Pb ﬁlms supported on Si

Theory

Envelope function

Theory

Envelope function

Freestanding Pb ﬁlms

2 4 6 8 10 12 14

Thickness (ML)

250

200

150

0.1

0.0

–0.1

0.1

0.0

–0.1

Temperature (K)S (eV)S (eV)

2.9 (a) Measured stability temperatures for Pb ﬁlms on Si(111) as a

function of thickness. Multiple measurements were made for most

thicknesses: the lines are guides to the eye connecting average

temperatures measured at each thickness. (b) Density-functional

calculation of the surface energy per site S for Pb ﬁlms supported on

a Si substrate. Lines connect values calculated at integer monolayer

thicknesses. The dotted curves show an envelope function based

on a Friedel model; this illustrates the beating effect of sampling at

discrete thicknesses (Upton, Wei et al. 2004). (c) Similar to (b) except

for unsupported Pb ﬁlms.

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39Quantum electronic stability of atomically uniform ﬁlms

equivalent to measuring the Fermi wave vector from the Brillouin zone

boundary instead of from the zone center (Wei and Chou 2002).

While 2.2 ML is not exactly an integer number of layers either, it is close

enough so that over a limited range of thicknesses, bilayer oscillations in

properties appear. However, over a great enough range the remainder becomes

signicant, leading to a beating effect and a phase reversal. Figure 2.9(b) is a

plot of the second derivative of the calculated surface energies per site from

rst-principles slab calculations of the electronic structure (Wei and Chou

2002). Analogous with the analysis of the previous section, this quantity

is a measure of the stability of a given thickness against bifurcation. The

calculated results reect the bilayer oscillations in the measured stability

temperatures. The theoretical values extend to greater thicknesses than the

experiment, and the beating effect leads to a diminishing of the quantum

oscillations and an even–odd phase reversal around 12 ML.

Previously, the importance of the phase shift was mentioned. The rst-

principles calculation presented in Fig. 2.9(b) includes implicitly the nature

of the interface between the lm and the substrate as well as the barrier

at the surface of the lm. This is built-in by having substrate layers and

vacuum layers as well as the lm material as part of the slab model used.

We note here that inclusion of the substrate is problematic – the interface is

mismatched and may involve reconstructions or special adlayers which are

only approximately handled in the calculation – and this could be the source

of some of the discrepancies in detail with the experimental result. In any

case, here the effects of the phase shift are implicitly included, and in fact

an effective phase shift can be extracted from the calculation if desired. One

can explore the effect of substrate interactions, for example, by removing

the substrate layers from the slab model. Results for such a calculation are

shown in Fig 2.9(c), which represents the stability function for a freestanding

Pb lm (Wei and Chou 2002). The important point to note here is that the

calculated result is very sensitive to changing the substrate interface. We

see cases of reversal in stability on a monolayer-by-monolayer basis, as

well as a large shift in the beating pattern. Again, one can look at this as

an indication of the sensitivity of quantum physical properties on the phase

shift.

The foregoing examples establish atomically uniform lms as an instructive

‘laboratory’ for the study of quantum-electronic stability. Spectra measured

with ARPES from such samples give both morphological and electronic

information at the same time. With the Pb on Si(111), a variety of growth

modes are possible, depending on the deposition temperature and annealing

recipes. How far can one go in applying the concepts learned from simple at

lms to more complicated morphologies? For the determination of morphology,

X-ray diffraction can be used and in some ways is more convenient than

photoemission. Thickness is determined by the appearance of multilayer

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40 Thin ﬁlm growth

interference peaks between the main Bragg peaks in the diffraction pattern.

Like photoemission, the measurement typically is macroscopic, the probe

beam typically being of millimeter to sub-millimeter size, so measurements

taken from a non-uniform sample will necessarily give superpositions of

patterns from different areas of the sample (Hong, Wu et al. 2000). However,

photoemission peak intensities can be difcult to interpret. For example, in a

spectrum from a sample composed of equal areas of 5 and 6 ML thicknesses,

the peak intensities associated with the two areas will in general not be

equal. Even worse, the relative intensities can vary markedly with changing

the exciting photon energy or the quantum numbers n of the states involved

(Paggel, Miller et al. 1999a). This is because photoexcitation involves a

matrix element that is sensitive to the relative phase of the initial and nal

state wavefunctions. In contrast, the analysis of X-ray diffraction data is a

rened art that is readily applied to thin lms.

Figure 2.10 shows X-ray diffraction data from Pb deposited on Si(111)

at 180K (Hong, Wei et al. 2003). The stacked curves show the intensity

along the Pb(10L) truncation rod at different times during a continuous

deposition at a constant rate. The two main peaks are bulk Bragg peaks

as noted. Between these appear smaller satellite peaks from multilayer

diffraction – these peaks carry information about the thicknesses present on

the sample. The behavior of the diffraction can be summarized as follows:

rst, there is a period of time where Pb diffraction peaks do not appear. At

the same time, Si diffraction peaks (not shown) increase in intensity. This

is consistent with growth limited to an adlayer, commensurate with the

Si(111) surface. Then there is a period corresponding to about 1.5 ML of

total deposition where nothing really happens to the intensities, suggesting

Pb is present as a disordered gas on top of the adlayer. Beyond this point

Pb features abruptly appear. These features include both the bulk diffraction

peaks and satellite peaks, all of which increase in intensity as deposition

proceeds. The line shapes, however, remain the same during this time. The

pattern is characteristic of 5 ML high structures above the wetting layer,

or 6 ML altogether. Deposition beyond 6 ML causes the pattern to change

apparently in accord with layer-by-layer growth. The growth is summarized

schematically in Fig. 2.11 (Chiang 1997). First, Pb wets the surface. Subsequent

deposition leads to island growth, where all the islands are 6 ML high (from

the starting surface). Eventually a completely lled 6 ML lm is obtained,

followed by layer-by-layer growth.

This behavior can be understood by looking at a plot of surface energies vs.

lm thickness (Fig. 2.12) (Boettger, Smith et al. 1998, Chiang 2000, hong,

Wei et al. 2003, Kresse and Furthmüller 1996, Materzanini, Saalfrank et al.

2001, Wei and Chou 2002). This plot differs from the earlier one showing

the second derivative of the total energies because here we are not dealing

with bifurcation of a at layer, but rather a more general behavior, so there is

ThinFilm-Zexian-02.indd 40 7/1/11 9:39:36 AM

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41Quantum electronic stability of atomically uniform ﬁlms

need to examine the global energy landscape. The surface energy per surface

atom shows layer-by-layer variations but the important features are the deep

minima at 1 and 6 ML thicknesses. The energy of any conguration of only

these two thicknesses, considered as non-interacting regions, would lie on

the straight line segment shown between 1 and 6 ML. It is clear since this

line lies below all of the points for intermediate thicknesses that the system

will prefer to phase-separate into the wetting layer plus 6 ML high islands

for all intermediate amounts deposited.

So the physics of the at lm in this case predicts the behavior of a

slightly more complicated system of a wetting layer plus preferred-height

island growth. This study does not give information about the lateral size

distribution, but apparently this is not essential to the understanding of the

basics of growth. The growth temperature is important. For example, in Fig.

Pb (101)

Pb (102)

Intensity (arb. units)

Perpendicular crystal momentum L (arb. units)

2.10 X-ray diffraction intensities along the Pb(10L) rod during

deposition of Pb on Si(111). Time and the deposition amount

increase uniformly for each successive scan going up in the

ﬁgure; scans are displaced vertically for clarity. Two large peaks

characteristic of a 6 ML thick ﬁlm are marked, smaller peaks between

these are multilayer diffraction for larger thicknesses. The dotted

curves indicate scans at 1.7 ML (lower dotted curve) and 5.7 ML

(upper dotted curve) thicknesses.

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42 Thin ﬁlm growth

2.12 we see also a signicant minimum in the energy at 4 ML. A system

prepared at low temperature could stay at that thickness rather than phase

separate. As a general yardstick for thermal energies, k

T, where k

Boltzmann’s constant and T is the temperature, can be used, and for 180K

Layer-by-layer growth

Islands merge into uniform ﬁlm

Pb adlayer + islands of uniform height

Pb adlayer

Bare Si(111) substrate

2.11 From bottom to top, schematic of the growth of Pb on Si(111).

After completion of an adlayer, growth favors the development of

multilayer islands of uniform height, followed by layer-by-layer

growth.

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43Quantum electronic stability of atomically uniform ﬁlms

is shown by a bar in Fig. 2.12. Evidently this is large enough compared to

the layer-by-layer variations between 1 and 6 ML so the system does not get

‘stuck’ at 4 ML at this temperature. It is reasonable that if the temperature

is chosen suitably the bilayer oscillations in favored thicknesses would be

seen instead of this phase separation.

Another X-ray diffraction study was reported on this system that explicitly

explored the role of the temperature on lm morphology. Figure 2.13

(Czoschke, Hong et al. 2005) shows the fraction of a surface covered by

Pb islands of different heights, starting with an 11 ML thick nearly uniform

lm and subsequently annealing to the temperatures indicated. 11 ML is

an unstable thickness, and with heating it starts to bifurcate and appears

to be fully bifurcated into regions of 10 and 12 ML at 200K. Annealing

further, we see the development of a single peak at 12 ML. From the surface

energy, 10 ML should be more stable than 12 ML; however, the system

started with 11 ML and the total amount of material must be conserved. To

accommodate this, the system can have 12 Ml islands separated by some

amount of the very stable wetting layer. At higher temperatures, the lm

becomes disordered, and at 280K it has reached a state of local thermodynamic

equilibrium with many thicknesses present. Inspection of the top curve shows

that the distribution of island heights is not random, but follows what is by

now a familiar pattern – a jagged series of bilayer oscillations but with the

appearance of occasional phase reversals (beating effect) indicated by the

Pb/Si ﬁlms

0 2 4 6 8 10 12 14 16

Film thickness (ML)

Relative surface energy (eV)

–0.6

–0.7

–0.8

2.12 Relative surface energy derived from a density-functional

calculation for different thicknesses of Pb on Si(111). Dots are

calculated values and lines are guides to the eye. The dashed line

joining the points at 1 and 6 ML thicknesses represents the energy

of a system consisting of the adlayer plus 5 ML high islands with

average thickness as plotted. The ‘error bar’ shows the scale of

thermal energy at the growth temperature.

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44 Thin ﬁlm growth

arrows in Fig. 2.13. Indeed, with suitable thermodynamic analysis, data

such as these can be related to the calculated surface energies with good

agreement. It is perhaps remarkable that even in this ‘more realistic’ system,

which should involve many complicating factors such as lateral connement,

shape effects, registry with the substrate, etc., the simple picture afforded

by atomically uniform lms still gives essential insight as to its quantum

electronic stability.

The importance of the phase shift has been demonstrated and emphasized in

the foregoing. A natural question is, can one inuence growth by adjustment

of the phase shift by modication of the surface or substrate interface? The

encouragement of layer-by-layer growth modes through the use of surfactants

has been studied and reported on in the literature. As the name implies, these

are expected to operate by reducing the surface energy. Typically, a foreign

species is introduced at the start of growth that tends to ‘oat’ on the surface,

stabilizing it. In some sense this is a ‘catalytic’ process in that the surfactant

is not consumed but acts continuously as growth proceeds (Rioux and Höchst

1992). However, typically these are viewed as acting locally, for example

by saturating the surface dangling bonds, and do not involve states in the

bulk of the lm. For the purpose of exploiting quantum electronic effects to

inuence stability, intuitively it would seem to be much easier to modify the

Local equilibrium

253 K

Preferred height islands

110 K

Initial 11 ML deposit

200 K

Bifurcation

280 K

Pb on Si(111)

Annealing

0 5 10 15 20 25 30 35

Film thickness (ML)

% Surface area covered

2.13 Distribution of thicknesses of Pb on Si(111) as annealling

proceeded following an initial 11 ML deposition, derived from

X-ray diffraction measurements. The arrows mark points where

the alternating layer pattern of stability switches from even to odd

numbers of monolayers and vice versa.

ThinFilm-Zexian-02.indd 44 7/1/11 9:39:36 AM

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45Quantum electronic stability of atomically uniform ﬁ lms

phase shift at the substrate/ lm interface than at the surface. However, for

fundamental studies it is important that the interface remain well ordered. For

Pb on Si(111), one can make several well-ordered reconstructed surfaces using

‘foreign’ atoms and still be able to grow Pb  lms on top. It turns out that by

doing so, it is indeed possible to tailor the landscape of thermal stability in

a dramatic fashion. Figure 2.14 shows results of experiments using Au and

In as ‘interfactants’, compared to a straight Pb system (Ricci, Miller et al.

2005). These systems were prepared by  rst depositing the interfactant atoms

and annealing to produce the following ordered surfaces: In

)

, Au

(6 ¥ 6), and Pb

)

. The amounts of the interfactant materials to make

these reconstructions are 0.333, 0.96, and 0.33 ML, respectively. Here a ML

is de ned to be with respect to the Si(111) surface layer. Then, Pb layers

were grown on top at 60K. The thermal stability was measured in the same

way that has been outlined earlier, using photoemission to monitor the  lm

structure. The interfactant atoms at the substrate surface change the interface

2.14 Stability temperatures for Pb ﬁ lms on Si(111) as a function of

ﬁ lm thickness using three different interfactants, (a) In, (b) Au, and

points, the crosses joined by dashed lines are a ﬁ t to a model from

which the phase shift can be extracted (Ricci, Miller et al. 2005,

Czoschke, Hong, et al. 2005). The same scales are used in all three

panels so that the effect of the interfactant on thermal stability is

directly visible.

Expt

Fit

Pb/In/Si(111)

Pb/Au/Si(111)

Pb/Pb/Si(111)

2 3 4 5 6 7 8 9 10

Thickness (ML)

Stability temperature (K)

300

200

100

300

200

100

300

200

100

(a)

(b)

(c)

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