Venables J. Introduction to Surface and Thin Film Processes

Подождите немного. Документ загружается.

ing. Assuming that the energy channels A, B and C are equally spaced, with A over the

peak, B just above the peak and C an equal distance to higher energy:

(a) Show that (A22B1C)/(2B2C) is the simplest linear measure of the P/B ratio, and

that this reduces to (A2B)/B if the background spectrum has zero slope.

(b) Assuming that the measured counts are limited by electron shot noise, ﬁnd the

SNR of the peak height (A 22B1 C) and of the peak to background ratio, explain-

ing your reasons carefully.

linear measure, and convince yourself that it is typically higher for comparable

values of the numbers of counts.

Student talks related to chapter 3 have included the following

In each case a page of suggestions for narrowing the topic, and suggested references

have been given out. The aim is to give the main features of the techniques clearly, with

adequate visual aids, in about 20 minutes, taking questions from the class.

1. A comparison of XPS (X-ray photoelectron spectroscopy) and AES.

2. Angular resolved AES and/or X-ray Photoelectron diﬀraction (XPD).

3. ARUPS and inverse photoemission.

4. Electron stimulated desorption (ESD) and the angular distribution of ions

(ESDIAD).

5. High energy ion, or Rutherford back scattering (HEIS-RBS) or medium energy

(MEIS).

6. Low energy ion scattering and ICISS.

7. Scanning tuneling spectroscopy (STS) and microscopy (STM) in UHV.

8. STM in solution.

9. Field ion microscopy (FIM) studies of atomic mobility on surfaces.

10. SIMS and SNMS (sputtered neutral mass spectroscopy).

11. Secondary electron spectroscopy and microscopy in UHV.

12. Low energy and photo-electron microscopy (LEEM/PEEM).

13. RHEED and REM.

14. Optical techniques for monitoring semiconductor crystal growth.

15. Surface magneto-optic Kerr eﬀect (SMOKE).

16. SEM with polarization analysis (SEMPA).

17. Film thickness measurements.

18. Nanoindentation.

19. Reactive ion etching.

Problems, talks and projects for chapter 3 107

4 Surface processes in adsorption

4.1 Chemi- and physisorption

A qualitative distinction is usually made between chemisorption and physisorption, in

terms of the relative binding strengths and mechanisms. In chemisorption, a strong

‘chemical bond’ is formed between the adsorbate atom or molecule and the substrate.

In this case, the adsorption energy, E

, of the adatom is likely to be a good fraction of

the sublimation energy of the substrate, and it could be more. For example, in chapter

1, problem 1.2(a), we found that in a nearest neighbor pair bond model, E

52 eV for

an adatom on an f.c.c. (100) surface when the sublimation energy L

53 eV. In that case

the atoms of the substrate and the ‘adsorbate’ were the same, but the calculation of the

adsorption stay time,

, would have been valid if they had been diﬀerent. Energies of

1–10 eV/atom are typical of chemisorption.

Physisorption is weaker, and no chemical interaction in the usual sense is present.

But if there were no attractive interaction, then the atom would not stay on the surface

for any measurable time – it would simply bounce back into the vapor. In physisorp-

tion, the energy of interaction is largely due to the (physical) van der Waals force. This

force arises from ﬂuctuating dipole (and higher order) moments on the interacting

adsorbate and substrate, and is present between closed-shell systems. Typical systems

are rare gases or small molecules on layer compounds or metals, with experiments per-

formed below room temperature. Physisorption energies are ⬃50–500 meV/atom; as

they are small, they can be expressed in kelvin per atom, via 1 eV⬅11604 K, omitting

Boltzmann’s constant in the corresponding equations. These energies are comparable

to the sublimation energies of rare gas solids, as given in section 1.3, table 1.1.

Adsorption of reactive molecules may proceed in two stages, acting either in series or

as alternatives. A ﬁrst, precursor, stage has all the characteristics of physisorption, but

the resulting state is metastable. In this state the molecule may reevaporate, or it may

stay on the surface long enough to transform irreversibly into a chemisorbed state. This

second stage is rather dramatic, usually resulting in splitting the molecule and adsorb-

ing the individual atoms: dissociative chemisorption. The adsorption energies for the

precursor phase are similar to physisorption of rare gases, but may contain additional

contributions from the dipole, quadrupole, and higher moments, and from the aniso-

tropic shape and polarizability of the molecules. The dissociation stage can be explosive

– literally. The heat of adsorption is given up suddenly, and can be imparted to the

resulting adatoms. Examples are O

/Al(111) and O

/Pt(111), which will be discussed

brieﬂy in section 4.5. O

and N

can be condensed at low temperatures as (long-lived)

108

physisorbed molecules on many substrates. Bulk solid F

is, however, quite dangerous,

and has an alarming tendency to blow up by reacting dissociatively with its container.

This chapter starts by considering adsorption at low coverage, where the statistical

mechanics of adsorption can be worked out precisely in terms of simple models; two of

these limiting models are considered in some detail in section 4.2. The next section 4.3

discusses the application of thermodynamic reasoning to the adsorbed state of matter,

including how to describe phases and phase transitions. The ﬁnal two sections 4.4 and

4.5 discuss the application of thermodynamic and statistical models to ﬁrst physisorp-

tion and then chemisorption, with experimental examples and literature references.

4.2 Statistical physics of adsorption at low coverage

4.2.1 General points

We have already discussed, in section 1.3.1, the sublimation of a pure solid at equilib-

rium, given by the condition

, with

1kT ln (p), and the standard free energy

52kT ln (kT/

). (4.1)

Now we wish to consider adsorbed layers in more detail, with a corresponding chem-

ical potential

. Thus we have two possible conditions:

for equilibrium with

the vapor, and

for equilibrium with the solid. The ﬁrst case is discussed in the

following sections 4.2.2 and 4.2.3. The second case was the subject of problem 1.2(b)

in chapter 1.

4.2.2 Localized adsorption: the Langmuir adsorption isotherm

In the Langmuir picture, each adatom is adsorbed at a well-deﬁned adsorption site on

the surface. The canonical partition function for the adsorbed atoms is Z

5兺

exp

(2E

/kT), and in general the Helmholtz free energy F52kTln(Z), where E

represents

the energies of all the quantized states of the system. For N

adsorbed atoms distrib-

uted over N

sites, each of which have the same adsorption energy E

, Z

Q(N

)exp(N

/kT), where Q represents the conﬁgurational (and vibrational)

degeneracy. The new element is the conﬁgurational entropy, since there are many ways,

at low coverage, to arrange the adatoms on the available adsorption sites. This Q is

given by (e.g. Hill 1960, chapter 7.1) as

Q5N

!/(N

!)((N

)!), (4.2)

multiplied by a factor q

if vibrational eﬀects are included, as discussed below. The

expression for ln(Q) is evaluated using Stirling’s approximation for ln(N!)5N

ln(N)2 N, valid for large N, to give

5F/N

52(kT/N

) ln (Z

)5kT ln (

/(12

))2 E

2kT ln(q), (4.3)

where

4.2 Statistical physics of adsorption at low coverage 109

The ﬁrst term is the conﬁgurational contribution in terms of the adatom coverage

the second the adsorption energy (measured positive with the vacuum level zero),

and the last term is the (optional) vibrational contribution. We can now see that if

, the density of adatoms in ML units is determined, in the high temperature

Einstein model, by

53kT ln(h

/kT)2 L

. (4.4a)

Using the form of

in (4.3), and rearranging to ﬁnd

gives, at low coverage,

5C exp{(2L

)/kT}, (4.4b)

where the pre-exponential function C depends on vibrations in both the solid and the

adsorbed layer, and the important exponential term depends on the diﬀerence between

the sublimation and the adsorption energy.

The Langmuir adsorption isotherm results from putting

, using this to calcu-

late the vapor pressure p in equilibrium with the adsorbed layer. We now have

p5C

/(12

) exp (2E

/kT), or (4.5a)

(T)p/(11

(T)p), with

(T)5C

exp(E

/kT); (4.5b)

the constant C

can be shown by direct substitution to be kT/q

. The form of this iso-

therm is shown in ﬁgure 4.1(a), using parameters appropriate to xenon adsorbed on

graphite. The coverage starts out linearly proportional to p, but goes to 1 as p → `.

The internal partition function q is the product of vibrational functions for the three

dimensions, i.e. q5q

. If the Einstein model is chosen, then we can think of the z-

direction, perpendicular to the surface, having a vibrational frequency

; this is the fre-

quency appropriate to desorption, and in the high temperature limit q

5(kT/h

). The

other two (x,y) frequencies, in the plane of the surface, will be the same on the square

(or triangular, hexagonal) lattice, and correspond to diﬀusion frequencies

. Thus q is

inversely proportional to an ‘eﬀective’ value

, for the adsorbed state, namely

.As

we will see in section 4.4.4, this model is very good for the z-vibrations, but is certainly

not exact for vibrations in the surface plane.

The pre-exponential constant in (4.5) is

5kT/q

5(2pm

)

3/2

(kT)

21/2

. (4.6)

It is instructive to note that this is in exactly the same form as that for the vapor pres-

sure in section 1.3.1. Moreover, the value of E

includes the zero-point motion, analo-

gously to the sublimation energy L

. Inserting reasonably realistic values for the

vibration frequencies in (4.6) gives the full curve of ﬁgure 4.1(a), to be compared with

the dashed curve in which vibrational eﬀects are neglected.

4.2.3 The two-dimensional adsorbed gas: Henry law adsorption

If the entropy due to vibrations in the adsorbed layer becomes even more important, the

adsorbate can eventually translate freely in two dimensions. This case is appropriate to

a very smooth substrate, with shallow potential wells, and/or at high temperatures. Thus,

110 4 Surface processes in adsorption

4.2 Statistical physics of adsorption at low coverage 111

Figure 4.1. Vapor pressure isotherms of a monolayer using parameters approximating to

xenon on graphite, with E

51925 K/atom (166 meV/atom), but ignoring lateral interactions.

(a) Langmuir isotherms for T560 K: full line, Einstein model for vibration frequencies

51,

50.2 THz; dashed line, without vibrational eﬀects so that q51. (b) Comparison of

Langmuir with 2D gas isotherm at T580 K: full and dashed lines as (a), dot-dash line, 2D gas

with average adsorption energy E

51889 K/atom. Note the lower coverage scale (4100 with

respect to (a)) and the extra factor of 10 for the dashed curve without vibrational eﬀects.

0.0

2.0x10

–7

4.0x10

–7

6.0x10

–7

8.0x10

–7

1.0x10

–6

0.0

0.2

0.4

0.6

0.8

1.0

(b)

–

= 36 K

=80 K, Einstein vibrations

No vibrations (x10)

2D gas

Coverage (

) x10

–

Pressure (Torr)

2x10

–7

4x10

–7

6x10

–7

8x10

–7

1x10

–6

0.0

0.2

0.4

0.6

0.8

1.0

(a)

= 1925 K

= 1,

= 0.2 THz

T=60 K, Einstein vibrations

No vibrations

Coverage (

)

in contrast to the previous section, the other limit of isolated adatom behavior is the 2D

gas. The mobile adatoms see the average adsorption energy E

, rather than the maximum

energy E

at the bottom of the potential wells. In compensation, they gain additional

entropy from the gaseous motion. The chemical potential is now

52E

1kT ln(N

/Aq

), (4.7)

where this expression is valid at suﬃciently low density for the distinction between clas-

sical, Bose–Einstein and Fermi–Dirac statistics not to be important. The derivation

involves evaluating the partition function by summing over 2D momenta, analogously

to a 3D gas, while retaining the z-motion partition function q

. The diﬀerence between

2D and 3D accounts for

rather than

, and the N

/A, the number of adsorbed atoms

per unit area, is the 2D version of the 3D density N/V, as in pV5NkT.

In fact, there is a 2D version of the perfect gas law of the form

A5N

kT, where

is known as the spreading pressure. This means that we could write

52E

1kT ln(

/kTq

), (4.8a)

52E

1kT ln(

), (4.8b)

where

52kT ln (kTq

) is the standard free energy of a 2D gas. This makes the

correspondence between 3D gases and 2D adsorption clear: p↔

↔

, and the

energy is lower in the 2D case by E

. Note that it is easy to forget the q

term, as is often

done, since the various qs are dimensionless: this doesn’t make them unimportant

numerically.

By equating

we get Henry’s law for 2D gas adsorption:

p5C

/A)exp(2E

/kT), or (4.9a)

/A)5

9(T)p, (4.9b)

with

9(T)5C

exp(E

/kT), and C

5kT/q

You may feel that detailed discussion of these constants is rather laboring the point,

but it is instructive if we stick with it for a while. Note that the 2D gas form has (N

/A)

proportional to p, whereas the localized form has the coverage

proportional

to p. These can be reconciled if we write (N

/A)5

/A). Here we have deﬁned the

monolayer coverage (N

/A), and then deﬁned (N

/A), the areal density of adsorbed

atoms in terms of this, rather artiﬁcial, constant. (Both N

and N

are numbers here,

not areal densities, though we can think of them as densities by choosing A51). This

is the identical problem we discussed in section 2.1.4, emphasizing the need for consis-

tency in the deﬁnition of the ML unit. If, however, we do make this deﬁnition, we can

recast the 2D gas equation as

p5(kTN

/Aq

)

exp(2E

/kT), (4.10)

which can be compared directly with the corresponding equation for localized adsorp-

tion.

This comparison shows that there is a transition from localized to 2D gas-like behav-

ior as the temperature is raised, because E

, whereas the pre-exponential (entropic)

term is larger for the 2D gas. The ratio of coverages at a given p for the two states is

112 4 Surface processes in adsorption

(

gas

loc

)5(2pmkT/h

) (A/N

) exp{(E

)/kT} (4.11a)

5(2pma

/kT) exp{(E

)/kT}, (4.11b)

where the length a is an atomic dimension (a

5A/N

). The comparison of Langmuir

and 2D gas isotherms is illustrated in ﬁgure 4.1(b) for Xe/graphite parameters at

T580K, using the reasonably realistic value of the well depth (E

)536 K

(Kariotis et al. 1988). Note that in both models the coverage varies linearly with pres-

sure at low coverage. However, as shown here, the 2D gas model is most appropriate,

but if the well depth were much larger, the localized model with vibrations would be a

better description. The model without vibrations is numerically quite poor in all such

situations.

The second equality (4.11b) is only true for the Einstein model at high temperature.

In this limit, where equipartition of energy holds (no term in h), the following argu-

ment can be made. Localized atoms vibrate with amplitude x, and 4p

is the

energy associated with this 2D oscillation, which is equal to 2kT at high temperature,

if a harmonic approximation is good enough (a big if ). Thus, the pre-exponential is

just a ratio of free areas (a

/px

), the numerator associated with the 2D gas, and the

denominator with the potential well in which the adatom vibrates. Clearly the vibra-

tional model starts to fail as x increases towards px

4.2.4 Interactions and vibrations in higher density adsorbates

To consider the statistical mechanics of higher density adsorbates, we need both the

interaction potentials and suitable models of the atomic vibrations. In analogy with the

3D case, moderate densities in a 2D ﬂuid phase can be described by virial expansions

(Hill 1960, chapter 15, Bruch et al. 1997, section 4.2.2). The spreading pressure is given

/kT5(N

/A)1B

(T)(N

/A)

(T)(N

/A)

1. . ., (4.12a)

in which the ﬁrst term in an expansion in powers of the 2D density (N

/A) is the second

virial coeﬃcient, B

(T), given by

(T)521/2

兰

[exp (2U(r)/kT) – 1]dr, (4.12b)

where the interaction potential U(r) is between two atoms; the 2D integral is performed

over the substrate area A, where for cylindrical symmetry dr ⬅2prdr. In a relatively

low-density gas at high T, this integral is small due to the fact that the atoms spend

most of their time outside the range of inﬂuence of U(r).

There is a continuous line of reasoning between the argument leading to (4.11),

(4.12) and the cell model of lattice vibrations. This model was originally introduced by

Lennard-Jones and Devonshire (1937, 1938) as an approximation of the 3D liquid

state, and described e.g. by Hill (1960, chapter 16) and Bruch et al. (1997, chapter 5).

The free area, A

5px

in the discussion following (4.11), is deﬁned by integrating the

Boltzmann factor over the ‘cell’ in which the atom vibrates, namely

兰

exp (2U(r)/kT)dr; (4.13)

4.2 Statistical physics of adsorption at low coverage 113

the corresponding quantity in 3D is a free volume, V

. In a high-density adsorbate with

large U(r) at moderate T, the integral is negligible except at positions r close to the equi-

librium spacing. This approximate, classical theory, is very eﬀective in computations

for solids at high temperature, since it includes thermal expansion – the response to the

spreading pressure exerted by the anharmonic vibrations – which many, apparently

more sophisticated models, ignore. For these reasons at least, it deserves to be better

known and more widely used as a teaching aid.

One anharmonic model which aims to have to have the correct low temperature

limit, and to be useful at higher T, is the quantum (or quantum-corrected) cell model

(Holian 1980, Barker et al. 1981, Bruch et al. 1997, chapter 5). A comparison of lattice

dynamical models for a 2D solid monolayer interacting via the approximate Lennard-

Jones potential, with parameters appropriate to xenon, is shown in ﬁgure 4.2.

4.3 Phase diagrams and phase transitions

One of the intriguing aspects of both physi- and chemisorption is the large number of

phases that can exist at the surface, and the transitions that occur between these phases.

114 4 Surface processes in adsorption

Figure 4.2. Thermal expansion of a 2D Lennard-Jones triangular monolayer solid on a

smooth substrate, computed for Xe interaction potential parameters (Bruch et al. 1997, after

Phillips et al. 1981, replotted with permission). Results for the quasi-harmonic theory (QHT)

are good at low temperature, with the (quantum-corrected) cell model agreeing closely with

classical Monte Carlo calculations at high T. The triple point of 2D Xe (on graphite) is 99 K.

There is a comparable richness of structure to that displayed in high pressure physics,

where there is both a density

, and a corresponding structure, at a given p and T. The

relation

5f(p,T ) is called the Equation of State (EOS) in the (3D) physics of bulk

matter, or the (p,V,T ) relation.

4.3.1 Adsorption in equilibrium with the gas phase

The corresponding equilibrium equation for (2D) adsorbed layers is

5f(p,T ), and

since we have already used

1kT ln (p), and

, we can think of

5f(

,T)

equivalently. For

, read N

if we do not deﬁne the ML unit in the standard way. So, as

we compress a 2D gas or localized adlayer by increasing the (gas) pressure p, the

adatoms will come within range of their mutual attractive or repulsive forces, and

phase transitions may result, ﬁrst within the ML, and subsequently from ML to multi-

layer.

If the substrate and adsorbate are well ordered, the condensation may proceed in

well deﬁned steps, as shown in ﬁgures 4.3 and 4.4 for physisorbed Xe and Kr/graphite

respectively at ML and sub-ML coverages. As studied by several French groups espe-

cially (Thomy et al. 1981, Thomy & Duval 1994, Suzanne & Gay 1996), these volumet-

ric studies, using high quality exfoliated graphite, established the existence of 2D solid,

liquid and gaseous layers. The (p,T) positions of the phase transitions (including multi-

layer transitions) and ﬁxed points such as triple points and critical points in these layers

were also accurately measured.

More recent experimental thermodynamic work has automated the measurement

process, and has achieved very high accuracy for quantities which depend on the slope

4.3 Phase diagrams and phase transitions 115

Figure 4.3. Sub-ML isotherms for Xe/graphite between 97 and 117 K. The isotherms, from left

to right, are at 97.4, 100.1, 102.4, 105.4, 108.3, 112.6 and 117.0 K. Between 110.1 and 117 K,

the layer undergoes two ﬁrst order phase transitions, showing 2D gas (G), liquid (L) and solid

(S) phases, whereas at 97.4 K only the G to L transition occurs; the 2D triple point is at 99 K

(after Thomy et al. 1981, reproduced with permission).

of the isotherm, such as the isothermal compressibility. Examples for Kr in the multi-

layer region are given by Gangwar & Suter (1990) and for Xe near ML melting by

Gangwar et al. (1989) and Jin et al. (1989). An example showing the much improved pre-

cision of these data is given in ﬁgure 4.5 which can be directly compared with ﬁgure 4.3.

Before we examine these results, we can note the diﬀerent forms of graphs and phase

diagrams that can be plotted. The problem arises because we have three variables, T,

ln(p)or

, and

, but we typically want to output onto paper, so that one of these three

is not plotted; the corresponding (third piece of) information may either not be known,

or may be discarded, or it may be given as a parameter.

An isotherm is a graph of

against ln(p) or

, with T as the parameter, as used in

the previous section. A phase diagram using log(p) and 1/T as axes is very convenient

for (physisorption) experimentalists, because the pressure can be varied over many

orders of magnitude, and this plot results in straight lines for phase transitions (e.g.

gas–solid or monolayer–bilayer transitions) which show an Arrhenius behavior – the

slope of these lines give the corresponding energies. But typically the coverage infor-

mation is lost. Theorists are fond of phase diagrams as a function of T and

: this gives

them the chance to investigate the adsorbed phase, and ignore the 3D gas phase, which

provides the value of

, typically D

with respect to the bulk 3D phase, when the com-

parison with experiment is made later.

116 4 Surface processes in adsorption

Figure 4.4. Sub-ML isotherms for Kr/graphite between 77 and 110 K. The isotherms are at

77.3, 79.8, 82.3, 84.8, 86.0, 88.0, 91.8, 96.6, 102.6 and 109.5 K. These show 2D gas, liquid

(maybe) and two solid phases, with a presumed solid-solid phase transition at point A

0, which

is not ﬁrst order (after Thomy et al. 1981, reproduced with permission).