Theory of Film Growth

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Theory of film growth

1)Processes in Adsorption:

Chemi- and Physi-sorption:A qualitative distinction can be made between Chemisorption and Physisorption in terms of

their 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, Ea, of the adatom is likely to be

comparable to or even more than the sublimation energy of the substrate (typically a few

eV/atom).

Physisorption is weaker, and is often being considered as having no chemical interaction

involved. The attractive interaction, in this case, is largely due to the van der Waals force. This

force is due to fluctuating dipole (and higher order) moments on the interacting adsorbate and

substrate, and is present between closed-shell systems. Physisorption energies are of order 100

meV/atom

Gas Impingement on a Surface

= Gas Impingement Flux. is the frequency with which molecules collide with a surface.

i.e.

=0

xxdnvn

At thermal equilibrium, the Maxwell Boltzmann velocity distribution is

( ) kTmv

x

xx

x

ekT

m

dv

dnvf

/2

121 2

2

==

So

21

0

/2

121

22

2

=

=

m

kTndvev

kT

mn x

kTmv

x

x

Using the gas law nkTp= , we will have

mkT

p

2

=

Substituting appropriate constants will giveMT

Px 221051.3=

where P is the pressure in Torr and M the atomic number of the gas molecules

How long does it take a surface to be covered by a monolayer of molecules ? This time c is

the inverse of the impingement flux .If a monolayer consists of 1015molecules. Then

P

MTx

P

MT

xc

8

22

15

1085.21051.3

10 ==

In air at atmospheric pressure sec1099.3 9= xc

In a vacuum of 10-10Torr hoursc 3.7=


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Condensation:

The adsorption of atoms/molecules is often preceded by condensation. The adsorbates need to

stay on the substrate surface for long enough time for the chemi-reaction to occur (chemi-

sorption) or simply stick in there (physi-sorption).

At certain temperature there is an equilibrium vapour pressure (sublimation pressure); no

deposit would occur at all unless one has supersaturation.

Supersaturation is normally achieved just above the substrate surface.

Under supersaturation, the density is so high, i.e. the atomic separation is so short, that

condensation occurs due to van der Waals force.

Substrate

Film

Atomic Separation

Potential

van der Waals potential

Increasing temperature


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2) Surface Diffusion and NucleationSurface Diffusion

Under supersatuaration the atoms/molecules are condensed onto the substrate surface. The

deposition rate or flux (of adatom) is related to the pressure as mkTpR 2=After the atoms being absorbed on the surface, they become adatoms with an (positive)

adsorption energy Ea, relative to zero in the vapour. (This sometimes called desorption energy)

The desorption rate of adatom is roughly given by( )kTE

aae

, where a is the

characteristic atomic vibration frequency and is expected to vary relatively slowly (not

exponentially) with T. Note that the desorption energy is assumed to come from lattice

vibration only.

The adatom can diffuse over the surface, with energy Ed (migration barrier energy) and the

corresponding frequency d (order of 1014s-1). Since ad EE


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Nucleation and growth of 2D islands

In the course of deposition, the atoms (monomer) arriving from the gas phase with a rate F

(units are atoms per surface unit cell and second, MLs-1). ML stands for monolayer. For

simplicity, we assume the surface temperature is low enough so that only monomers diffuse on

the surface and that dimers remain immobile. As deposition proceeds, the number of dimers

will increase roughly linearly until their concentration n2becomes comparable to the density of

monomers n1. From there on, the probabilities of a diffusing monomer to encounter one of its

own or a dimer become comparable and cluster growth competes with the creation of new

stable nuclei (dimers in our model). After the density of stable nuclei nx (x standing for any

size that is stable) has increased sufficiently, any further deposition will exclusively lead to

island growth. At this saturation island density, the mean free path of diffusing adatoms is

equal to the mean island separation and adatoms will attach themselves with much higher

probability to existing islands then to create new ones. Approaching coverage of about half a

monolayer, islands eventually coalesce which decreases their density.

If dimers are immobile and no re-evaporation occurs, then the rate equations for the density of

monomers and stable islands are:

( ) 11112111 22 nFntFFnnDnDFdt

dnxxx = ,

11

2

11 nFnDdt

dnx +=

The terms on the right-hand side of the first equation denote, respectively, the increase of

Low coverage High coverage


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monomer density due to deposition with flux F, its decrease due to the encounter of two

diffusing adatoms resulting in the creation of a dimer (associated with the disappearance of

two atoms), the decrease occurring when a monomer is captured by a stable island, and finally

two terms which denote the decrease caused by direct impingement onto stable island density,

nx, due to the creation of dimers, first when two monomers meet by diffusion, and second upon

direct deposition onto an adatom. In these equations coalescence is neglected; incorporation

would add a further term ( )dtdnFnx 12 to the second equation. In general the problem is

treated in the mean-field assumption, that is, outside the islands the monomer density

immediately takes on its average value. The time evolution of island and monomer densities

can be obtained from integration of these equations. Very often, one is interested in the

saturation island density, as this reflects the mean free path for monomer diffusion, The

temperature dependence of this quantity thus allows one to extract information on surface

diffusion. The power law expressed by these equations lead to an approximation

( )26

ln L

L

F

D

The characteristic length, L can either be identified with the mean island distance or with the

mean free path of diffusing adatoms before they create a new nucleus or are captured by

existing islands. The logarithmic correction term appearing in the denominator is small.

Omitting this term yields ( )

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FDL . So L depends only on the ratio D/F. This isdue to the fact that the flux is the only quantity introducing time, if there is no re-evaporation

and dimers are stable. Thus the ratio of deposition to diffusion rate determines the mean free

diffusion path and the mean island distance attained at saturation.

Variation of the saturation island density with temperature for deposition of 0.12 ML Ag

onto a Pt(111) surface at 80, 95, and 110K (F=1.1 x 10-3ML/s)


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If the deposition continues the nucleation centres become more numerous and coalescence of

2D islands occurs. As the size of the 2D island is increase and the coverage of the substrate

surface gets enlarged. Ultimately the concept of a film become imminent and we may now

consider film growth.

STM images showingthe transition from

the very early

nucleation phase to

island growth (2D)

for Ag deposited

(F=1.1 x 10-3 ML/s)

onto Pt(111) at 75 K.

The coverage and

mean island sizes are

indicated.


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3) Crystallization and film growthAmorphous no ordered structures

Polycrystalline randomly oriented grains, oriented grains, highly oriented grains,

epitaxy (homo- or hetero-)

Single crystal homoepitaxy, heteroepitaxy

Epitaxy

Epitaxy refers to the film growth phenomenon where a relation between the structure of the film

and the substrate esists. In particular it commonly denotes a single crystalline layer grown on a

single crystal surface. If the single crystalline film and the single crystal substrate are of the same

material, we call the growth homoepitaxy. If the film and the substrate are of different materials, we

call the growth heteroepitaxy. For a defined relation between nucleus and substrate orientation the

nucleus must consist of at least 3 atoms for a 3-fold symmetry and 4 atoms for a 4-fold symmetry.

Factors governing epitaxy

Substrate : Structural compatibility Lattice matching crystal structure and lattice constant.

Chemical compatibility chemical bonding, chemical diffusion

Temperature Above a more or less well defined elevated substrate temperature Te

good epitaxy is obtained. Te depends on the deposition rate, particle energy and the

surface contamination. The reason for the general need of higher temperature is

obviously the reduction surface contamination by desorption, the enhancement of

surface mobility of atoms to reach the favorable sites, and the enhancement of

diffusivity in the deposit thus favoring re-crystallization and defect annihilation.


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The classification of three growth modes was first introduced by Ernst Bauer in 1958. The Layer-

by-Layer, or Frank-van der Merwe, growth mode arises because the atoms of the deposit material

are more strongly attracted to the substrate than they are to themselves. In the opposite case, where

the deposit atoms are more strongly bound to each other than they are to the substrate, the Island

(3D), or Volmer-Weber mode results. An intermediate case, the Layer-plus-Island, or Stranski-

Krastanov growth mode is much more common than one might think. In this case, layers form first,but then for some reason or other the system gets tired of this, and switches to islands

Growth Modes

Frank-van der Merwe mode

(2 dimensional growth mode)

Volmer-Weber mode

(Island growth mode)

Stranski-Krastanov mode


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Lattice mismatch has a marked effect on film morphology. The strain resulting from lattice

mismatch contributes to the interface energy, a key parameter in determining the growth mode.

However, the surface free energies for the substrate and film materials also influence the mode of

growth. For heteroepitaxy in general, observed growth modes have been placed in three categories

depending on the resulting film morphology. These are: (1) Frank-van der Merwe (FM) or layer-by-

layer, (2) Volmer-Weber (VW) or 3D island, and (3) Stranski-Krastanow (SK) or 3D island-on-

wetting-layer growth. The interatomic interactions between substrate and film materials are stronger

and more attractive than those between the different atomic species within the film material in FM

growth, whereas just the opposite is true in VW growth. SK growth occurs for interaction strengths

somewhere in the middle. Bauer and van der Merwe have cast the energetics of film growth into a

particularly simple form under the assumption of equilibrium between the film components in the

gas phase and those in the film surface. In this formalism, layer-by-layer growth of A on B requires

that

0+= BiA (i)where A and B are the surface free energies of A and B, respectively, and i the interfacial free

energy. The latter quantity depends on the strain and the strength of chemical interactions between

A and B at the interface. Eq. (i) says in effect that the sum of the film surface energy and the

interface energy must be less than the surface energy of the substrate in order for wetting to occur.

Alternatively, it becomes easier for layer-by-layer growth to occur as the surface energy of the

substrate increases. Thus, FM growth is expected if Eq. (i) is obeyed. However, the strain energy,

which is a term in i, increases linearly with the number of strained layers. At some thickness, A+iexceeds B and the growth mode transforms from FM to SK resulting in 3D islands on the 2D

wetting layer. Alternatively, Amay be sufficiently in excess of Bthat Eq. (i) is never fulfilled even

for a strong attractive interaction between A and B and little strain (ifor B on A. However, the clever use of surfactants, which have the effect of lowering the surface

energy of the high-surface-energy component, can alleviate this problem. (Extract from Epitaxial

growth and properties of thin film oxides by Scott A. Chambers, Surface Science Report, 39, 105,

2000)


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Theory of Film Growth