Venables J. Introduction to Surface and Thin Film Processes

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nucleation experiments described in section 5.4.1: E

52.260.1 and E

50.1560.1eV;

the values of (E

– E

) are rather similar! Use of the more accurate value for E

50.10

60.01eV from STM experiments (Brune et al. 1995, Brune 1998) makes the agreement

even closer.

This type of annealing is a particular case of Ostwald ripening, which is a term

usually applied to situations where large clusters grow or coarsen, and small clusters

shrink or disappear. In the case of ripening controlled via adatom surface diﬀusion, the

eﬀective diﬀusion coeﬃcient is given by (5.24) with activation energy Q5(L2 E

It is noticeable that the high temperature limit of the nucleation model in complete con-

densation gives this same energy, since L5E

13E

for large 2D (hexagonal) clusters.

The nucleation density expression in table 5.1 has E

53E

for large critical nucleus sizes

i, so the corresponding activation energy is again (E

13E

); this is thus an internal

check on the validity of the nucleation model. Ostwald ripening is generally important

in materials science (Martin & Doherty 1976, Voorhees 1985); 2D ripening on surfaces

is described more fully by Zinke-Allmang et al. (1992) and Zinke-Allmang (1999).

STM experiments have been instrumental in pinning down some of these energies,

and the ES barrier has been investigated speciﬁcally for Ag(111), where authors have

5.5 Steps, ripening and interdiffusion 177

Figure 5.19. Distribution of ML Au nuclei in relation to steps on a Ag(111) surface at the

temperatures indicated. The sense of the step train corresponds to the TEM image of the Au

islands. Note that the up-steps are decorated with a continuous thin strip of Au, and what is

plotted is the island position histogram on the terraces, demonstrating an active ES barrier at

the lower temperatures (after Klaua 1987, reproduced with permission).

(b)

found values of E

50.1260.02 eV (Bromann et al. 1995) and 0.1360.04 eV

(Morgenstern et al. 1998). This latter experiment is particularly elegant, in that the

authors were able to create ML height pits on the surface by ion bombardment, and

then place either islands (by deposition) or smaller ML deep pits (by further bombard-

ment) close to the centers of the larger pits. The small pits were found to ﬁll in more

slowly than the islands evaporated, in annealing experiments lasting several hours

around and below room temperature. This diﬀerence in rate is directly due to the pres-

ence of the ES barrier, whereas the average rate is governed by the same Q as above in

(5.24), with Q50.7160.03 eV from the comparison of the model with experiment.

Comparison with the SEM results on Ag on 1 ML Ag/Fe(110) which gave 0.8660.05

eV is interesting, in that the SEM experiments (Noro et al. 1996) involve diﬀusion over

a large number of steps; in this case the maximum activation energy would correspond

to (Q1E

), i.e. (0.7110.13)50.84 eV, good agreement! But such arguments are at the

limit of current experimental accuracy, and there are uncertainties in frequency factors,

and other important eﬀects to consider. One which has been demonstrated is the eﬀect

of strain, where Ag adatoms diﬀusing on the compressed ML phase of Ag/Pt(111) have

a lower value of E

than on Ag(111) (Ruggerone et al. 1997, Brune & Kern 1997, Brune

1998), as indicated in table 5.4. These calculations also give a lower value of E

on com-

178 5 Surface processes in epitaxial growth

Figure 5.20. (a) Expansion of patches of Ag/Fe(110), initially 20 mm wide and 3ML

deposited, during annealing at a typical temperature T5323613 °C, the arrow indicating the

time at which islands on the second ML disappear; (b) Eﬀective diﬀusion coeﬃcient D

function of 1000/T, with activation energies Q indicated for an assumed K54 (after Noro et

al. 1996, reproduced with permission). See text for discussion.

0246810

islands

disappear

(a)

2nd ML

1st ML

= 323 +/–13

Patch width (

1/2

(min

1/2

)

1.5 1.6 1.7 1.8

1E-10

1E-9

(b)

0.88 eV

0.81 eV

(cm

/s)

1000/ (K

–1

)

pressed Ag(111). Other Ostwald ripening experiments of a similar nature have yielded

Q5 0.766 0.04 eV and E

50.22 60.01 eV for adatoms on Cu(111) (Giesen & Ibach

1999). These authors and their co-workers have also observed step ﬂuctuations, and

rapid decay processes when multilayer islands coalesce, and have recently invoked elec-

tronic mechanisms in explanation. Thus, as we have seen in the previous two sections,

research on atomistic processes at metal surfaces has encountered the need for elec-

tronic structure calculations in the search for complete explanation. The background

needed for this understanding is given in section 6.1.

5.5.3 Interdiffusion in magnetic multilayers

Magnetic multilayers are typically formed by interspersing a magnetic metal (Fe, Ni,

Co, Cr, etc.) with a non-magnetic spacer, often a noble metal (Cu, Ag, Au, Pt etc.). The

sequence Co/Cu/Co . . . for example has a giant magnetoresistance whose properties

are controlled by the various layer thicknesses and perfection; there are many such

systems, whose properties have been extensively reviewed in the last few years, as dis-

cussed later in sections 6.3 and 8.3. In this section we concentrate on the growth mode,

taking Fe/Ag/Fe(110) as the example.

As described in section 5.4.1, Ag/Fe(110) is a typical SK growth system, with two

layers before islands form, the ﬁrst of which has the c531 structure, which has a

nominal coverage of 0.8ML, and the second is close to a compact Ag(111) layer. Auger

amplitudes from this structure have been measured (Noro et al. 1995, Venables et al.

1996, Venables & Persaud 1997); there is nothing unusual about the Ag/Fe(110) inter-

face. However, deposition of Fe on thin ﬁlms of Ag/Fe(110) results in some inter-

diﬀusion, the extent of which depends on the Ag ﬁlm thickness, deposition and

annealing conditions. An example is shown in ﬁgure 5.21, where the ratio of Ag/Fe

AES intensities is plotted against Fe coverage, and is compared with a layer growth cal-

culation (Persaud et al. 1998).

The lower curves are calculated assuming no surface segregation, the two curves

reﬂecting some uncertainty in the correct inelastic mean free path for the Auger elec-

trons. For deposits of under 1 ML at room temperature, the data follow this layer

growth curve, more or less. But between 1 and 2 ML, there is clearly some segregation,

where the calculation assumes that all of the ﬁrst 0.8 ML Ag has moved to the surface;

this is clearly not a bad approximation. But annealing to around 250°C results in more

segregation, and deposition at 250°C results in almost complete segregation. Results

for other Ag layer thicknesses show a similar trend: interdiﬀusion at the ML level pro-

ceeds even at room temperature, and there is long range interdiﬀusion already at a few

hundred degrees Celsius.

From the arguments given in section 5.1.1, we can see that metal deposition systems

should follow the island growth mode, if the surface energy of the deposit (Fe ⬃2.9

J/m

) is greater than that of the substrate (Ag ⬃1.2 J/m

); surface energy values are

discussed and tabulated in section 6.1.4. Thus islands of the strongly bound material,

Fe, once formed, could lower their energy by allowing themselves to be coated with

a thin skin of Ag substrate material! This corresponds to a curious form of

5.5 Steps, ripening and interdiffusion 179

interdiﬀusion, in which islands or layers, rather than single atoms, bury themselves in

(i.e. burrow into) the substrate. At low temperatures this will not happen, because the

substrate atoms will not diﬀuse. However, STM studies of surface steps on noble

metals have shown that steps can move quite rapidly, even at room temperature

(Poensgen et al. 1992) and burying of deposited clusters has been observed by in situ

TEM at elevated temperatures (Zimmerman et al. 1999). It is now clear that the

diﬃculties various groups have experienced in producing well-deﬁned thin ﬁlms of

magnetic metals on noble metal surfaces are related to eﬀects of this nature. Such

magnetic metals generally have higher surface energies than the substrates; they also

undergo structural phase changes with increasing thickness; the magnetic features are

discussed in sections 6.3 and 8.3.

One case which has been studied by STM is Ni/Ag(111) (Meyer & Behm 1995). Here

Ni can both diﬀuse by hopping over the surface, or, at higher temperature, can

exchange with a silver atom. This immobile Ni atom now acts as a nucleus for further

growth of Ni clusters. In a ﬁxed temperature deposition, this corresponds to creation

of nuclei at a rate proportional to the adatom concentration; if the Ni–Ni bond is

strong enough, then i50. Similar cases are Fe, Co and Ni/Au(111), with a complex

180 5 Surface processes in epitaxial growth

Figure 5.21. AES deposition curves for Fe/1.8 ML Ag/Fe(110), showing (a) rearrangement of

the Ag layer between 1 and 2ML, and segregation on annealing, or during deposition at

elevated temperature. The Fe/5 ML Ag/Fe(110) curves (b) show rather less complete

segregation. The parameter p is the amount interchanged with the surface in the model

(Persaud et al. 1998, reproduced with permission).

0123456789

0.0

0.2

0.4

0.6

(a)

Calculation

(parameter

= 0.8 ML)

Calculation

(No segregation)

SUBSTRATE: 1.8ML Ag/Fe(110)

RT deposition

250

C deposition

Annealed to 250

Annealed to 300

Ratio of Ag to Fe Intensities

Fe coverage (ML)

0123456789

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(b)

Calculation

(No segregation)

Calculation

(parameter

= 0.8 ML)

SUBSTRATE: 5ML Ag/Fe(110)

RT deposition

250

C deposition

RT dep, anneal 250

" 300

250 dep, ann 250

" 300

Fe coverage (ML)

reconstruction which orders the Fe, Co or Ni islands (Voigtländer et al. 1991,

Chambliss et al. 1991, Stroscio et al. 1992, Meyer et al. 1995, Tölkes et al. 1997), and

Fe or Co/Cu(100), where subsurface and surface ML islands can co-exist (Kief &

Egelhoﬀ 1993, Chambliss & Johnson 1994, Healy et al. 1994). The size distributions of

clusters nucleated on point defects have been studied by KMC; a broad distribution is

found, with a high proportion of small islands, as shown earlier in ﬁgure 5.17(b) (Amar

& Family 1995, Zangwill & Kaxiras 1995, Bales 1996, Brune 1998).

All these cases take us back to ﬁgure 5.3, and the diﬃculty of making high quality

multilayers from A/B/A systems: if one interface is ‘good’, typically an example of SK

growth as described in this section, then the other interface is ‘bad’. These systems may

formally be an example of island growth, but active participation of the substrate

makes this classiﬁcation too naive, in some cases even at room temperature. Nuclei

form by exchanging deposit and substrate atoms; clusters of deposited atoms start to

form, and then tend to get covered by a substrate ‘skin’. Once one realizes what is hap-

pening on a microscopic scale, the evidence is already there in the classical surface

science results, e.g. from AES as shown in ﬁgure 5.21. These data show that segrega-

tion of Ag to the surface already happens at the ML level at room temperature; at 250

°C there is widespread interdiﬀusion.

Further reading for chapter 5

King, D.A. & D.P. Woodruﬀ (Eds.) (1997) Growth and Properties of Ultrathin Epitaxial

Layers (The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis,

Elsevier), 8.

Liu, W.K. & M.B. Santos (Eds.) (1999) Thin Films: Heteroepitaxial Systems (World

Scientiﬁc).

Matthews, J.W. (Ed.) (1975) Epitaxial Growth, part B (Academic).

Tringides, M.C. (Ed.) (1997) Surface Diﬀusion: Atomistic and Collective Processes

(Plenum NATO ASI) B360.

Problems and projects for chapter 5

Problem 5.1. Growth laws and the condensation coefﬁcient

The rate equation (5.10) is a good approximation when the coverage of the substrate

by islands, Z ,,1. When Z is not so small, one might like to correct (5.13a) for direct

impingement, by writing R(12Z) in place of R. The condensation coeﬃcient is the

ratio of the amount of material in the ﬁlm to the amount in the depositing ﬂux, and

comes in two forms

(t) and

(t).

(a) Identify the terms in the modiﬁed (5.10) which lead to the increase in size of clus-

ters, and write down an expression for the cluster growth rate, in atoms per unit

area per second.

Problems and projects for chapter 5 181

(b) Assuming 2D islands, express the instantaneous condensation coeﬃcient

(t) in

terms of the rate of atom arrival and departure (per unit area per second) at

time t.

coeﬃcient

(t), assuming deposition was started at t50.

(d) Using (5.11) in addition, we can now compute n

(t),

(t) and Z(t). However, as

explained in the text, it is preferable to use Z as the independent variable. Compute

(Z),

(Z) and t(Z ) for parameter values which illustrate the initially incomplete

condensation regime for a critical nucleus size i51, and 10

,Z,0.5, and iden-

tify on the surface processes which dominate at diﬀerent values of Z.

Problem 5.2. Capture numbers and Bessel functions

The rate equation treatment of capture numbers needs an ancilliary diﬀusion equation

with cylindrical symmetry. Consider the formulation of such an equation in order to

understand how the solutions (5.12) arise for incomplete condensation (

), and

how they can be generalized to the more general case when growth also occurs (

(a) Express the adatom diﬀusion equation



t5D

in cylindrical polar coordi-

nates, and determine the simpliﬁed equation which results when the solution does

not depend on the angular variable

(b) Now consider a particular cluster with radius r

, centered at r50, and add the

source and sink terms from the rate equation (5.13) for the adatom concentration.

Explore the relation between the resulting steady state equation for n

(r) outside

the cluster and Bessel’s equation.

concentration n

(r)5R

(12 K

)/K

(X)), where the arguments X and X

of the

Bessel function K

are as deﬁned in the text following (5.12).

(d) Given that the derivative of K

(X)52K

(X), derive (5.12) for the capture

numbers

and

by considering the diﬀusion ﬂux J(r)52D

(r) at the cluster

boundary. Show this result is exact for small clusters when

, and that it is a

good approximation when

. For complete condensation, show that

the result for

only depends on the island coverage Z.

Project 5.3. Step capture and diffusion barriers between layers

Adatoms being captured by steps can be formulated as in problem 1.3 by considering

an individual terrace and the boundary conditions at the steps at either end. Consider

some further 1D step capture problems along the following lines.

(a) The presence of an Ehrlich–Schwoebel barrier at a down step means that an

adatom has a temperature-dependent probability to be reﬂected there. Formulate

this problem and apply your equations to the data shown in ﬁgure 5.19. What

surface processes determine the nucleation density N(x,T) shown, on the assump-

182 5 Surface processes in epitaxial growth

tion of no intermixing between Au and Ag, and what value for the corresponding

energies might you deduce from the comparison with experiment.

(b) Problems on a mesoscopic scale require a suitable mixture of atomistic and con-

tinuum modeling. One such problem is how to model the eﬀects of step capture on

a scale large compared to the distance between the steps. Show that in this limit,

step capture contributes and extra characteristic inverse time

to (5.13). Evaluate

in terms of the step spacing d and the adatom diﬀusion coeﬃcient D to prove the

limit given by (5.23), and see if it is valid in general.

ation and growth of islands on a vicinal surface. Using such a formulation, discuss

the occurrence of denuded zones parallel to the steps, and the reduction in nucle-

ation density on vicinal surfaces. What are the eﬀects of step movement and island

incorporation into steps at later stages?

Project 5.4. Clustering and intermixing during diffusion

The formulation of adatom diﬀusion in terms of hopping via (1.16) provides the sim-

plest description of intrinsic diﬀusion, valid at low coverages. Consider some of the

forms of the diﬀusion coeﬃcent which are appropriate to long range diﬀusion, valid at

higher coverage and/or temperatures.

(a) The mass transport or chemical diﬀusion coeﬃcient D* is expressed here as n

D in

(5.24). Consider the surface processes involved in Ostwald ripening of clusters, and

show that this expression is reasonable at low concentrations when only adatoms

are mobile.

(b) At higher adatom concentrations some of the adatoms will spend part of the time

in small clusters, size j, which have typically smaller (maybe zero) intrinsic diﬀusion

coeﬃcients, D

. Show that in this case, the chemical diﬀusion coeﬃcient is concen-

tration dependent, and is given by D*5兺

/{ 兺

}. Hence show, using the

Walton relation (5.9) in its simplest form, that at non-zero concentrations D*

depends exponentially on E

/kT as well as on E

/kT, and may also depend on other

energies due to diﬀusion of small clusters.

at even higher temperatures the surface may become rough over several layers. In

alloys we can have exchange diﬀusion with unlike species involved. Consider how

these possibilities aﬀect the interpretation of D* in terms of individual surface

(and near surface) processes.

Problems and projects for chapter 5 183

6 Electronic structure and emission

processes at metallic surfaces

This chapter gives, in section 6.1, some generally accessible models of metallic behav-

ior, and tabulates the values of work function and surface energies of selected metals.

In section 6.2 we discuss electron emission properties of metals, concentrating on the

role of low work function, high surface energy materials as electron sources; we also

show that electron emission and secondary electron microscopy can be used to study

diﬀusion of adsorbates. An introduction to magnetism in the context of surfaces and

thin ﬁlms is given in section 6.3.

6.1 The electron gas: work function, surface structure and energy

6.1.1 Free electron models and density functionals

Free electron models of metals have a long history, going back to the Drude model of

conductivity which dates from 1900 (Ashcroft & Mermin 1976). The partly true,

partly false predictions of this classical model were important precursors to quantum

mechanical models based on the Fermi–Dirac energy distribution. If words in the fol-

lowing description don’t make sense, now is the time to take a second look at section

1.5. Modern calculations start from a description of the electron density,

(r) (

(z)

in 1D) in the presence of a uniform density

(r or z) of metal ions. This is the jellium

model, where the positive charge is smeared out uniformly. At a later stage we can add

the eﬀects of the ion cores D

(r) by pseudopotentials or other approximations. This

division into a uniform

, with a step function to zero at the surface, allows us to con-

sider the electron density

as the response to this discontinuity. Clearly, a long way

inside jellium,

, and there is overall charge neutrality. But at the surface there

is a charge imbalance, and the electrostatic potential V varies as a function of z.

To see this response, we draw an energy diagram as in ﬁgure 6.1, with V(2`)

,V(1`), with the Fermi energy E

¯, the chemical potential for the electrons, and

note that

2V(2`)5

¯, (6.1)

the Fermi level with respect to the bottom of the conduction band, and that the work

function,

5V(1`)2V(2`)2

¯, or equivalently

184

¯5V(1`)2V(2`)⬅ DV. (6.2)

From this simple manipulation we can understand the following points: (1)

¯ is a bulk

property, determined by the kinetic energy and exchange-correlation energy of the

electron gas; (2) the fact that the work function

depends on the surface face {hkl}

means that DV has to be a surface property also. This has various consequences, which

are spelled out in the next sections; but ﬁrst, we need a bit of background theory. The

details can be quite complicated, especially considering that there are (at least) two

length scales in the problem, one connected with the electron gas, and another con-

nected with the lattice of ions.

It is a good idea to understand the elements of density functional theory (DFT),

even if only in outline, in the form that Lang and Kohn used in the early 1970s to derive

values for the work function and surface energies of monovalent metals (Lang & Kohn

1970, 1971; Lang 1973). In order not to lose the thrust of the argument, this material

is relegated to Appendix J. These calculations characterize free electron metals in

general in terms of the radius (r

) which contains one electron; in particular, their cal-

culations spanned the range 2,r

,6 (in units of the Bohr radius a

) which includes

the alkali metals Li to Cs. Figure 6.1 is drawn to scale for r

54, which is close to the

value needed to describe sodium.

6.1 The electron gas 185

Figure 6.1. Energy diagram deﬁning the terms

, DV,

¯ and the eﬀective potential V

eﬀ

(z) in

relation to the Fermi level and the bottom of the conduction band of a metal. This diagram is

drawn to scale from the data in table 1 of Lang & Kohn (1970) for r

54. See text for

discussion.

–1.5 –1.0 –0.5 0.0 0.5 1.0

–4

–3

–2

–1

∆

–

eff

(

)

(

)

Fermi level,

Energy (eV)

Distance

(Fermi wavelengths)

The main aspect of this model is the replacement of the (insoluble) many electron

N-body problem by N one-electron problems with an eﬀective potential, V

eﬀ

in ﬁgure

6.1, which is a functional of the electron density. This potential contains the original

electron–nuclei and electron–electron terms, and also has a term to describe exchange

and correlation between electrons. These terms have been worked out precisely for a

uniform electron gas, corresponding to the interior of jellium, so that explicit, numer-

ical values can be given to these energies as a function of electron density. The trick

now is to apply these same numerical recipes to non-uniform densities, whence the term

local density approximation (LDA). There are many further methods which try to

correct LDA for non-local eﬀects and density gradients, such as the generalized gradi-

ent approximation (GGA), but it is not clear that they always produce a better result.

In any case, we are now getting into the realm of arguments between specialists.

Some results of Lang & Kohn’s work on jellium are indicated on ﬁgures 6.1 to 6.3.

The electron density (ﬁgure 6.2(a)), electrostatic potential and eﬀective potential (ﬁgure

6.1) have oscillations normal to the surface in the self-consistent solution obtained;

there are substantial cancellations between the various terms. The work function of

these model alkali metals (ﬁgure 6.3) varies weakly from Li (r

about 3.3) to Cs (r

about

5.6), whereas the individual components of the work function vary quite a lot. This

model was the ﬁrst to get the order of magnitude, and the trends with r

correct: a big

achievement. Note that the position of the ions do not enter this model at all: every-

thing is due to the electron gas, and the importance of the exchange-correlation term

, and the variation of the electrostatic contribution, are evident in table 6.1.

In the quarter century since Lang & Kohn’s initial work, there have been major

developments within the jellium model. As computers have improved, this method has

also been applied to clusters, especially of alkali metals, of increasing size. Figure

6.2(b) shows the comparison of the electron density in a spherical sodium atom cluster

of more than 2500 atoms, modeled as jellium, compared with the free planar jellium

surface on the same scale (Brack 1993). The only diﬀerence of note between the two

curves is that the oscillations in the cluster produce a standing wave pattern at the

center of the cluster, whereas they die away from the planar surface. This central peak

186 6 Electronic structure and emission processes

Table 6.1. The work function of jellium and its

components. Columns 2 and 3 represent the

kinetic, and exchange-correlation energy

respectively (after Lang & Kohn, 1971)

¯ DV

(eV) (eV) (eV) (eV) (eV)

2 12.52 29.61 22.91 6.80 3.89

3 5.57 26.75 21.18 2.32 3.50

4 3.13 25.28 22.15 0.91 3.06

5 2.00 24.38 22.38 0.35 2.73

6 1.39 23.76 22.37 0.04 2.41