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KINETICS
MACROSCOPIC TRANSPORT
Whenever a material system is not in thermodynamic equilibrium, driving forces
arise naturally to push it toward equilibrium. The resulting atomic concentration
gradients trigger time-dependent mass transport effects that reduce free-energy
variations in the system. In solids, mass transport is accomplished by diffusion.
Fickestablished the phenomenological connection between concentration
gradients and the resultant diffusional transport through the equation
WhereD is Diffusion constant and increases in exponential fashion with
temperature according to a Maxwell-Boltzmann relation
Where Do is a constant and EDis activation energy of Diffusion
The time-varying accumulations or depletions
of atomic species is governed by equation:
Where C(x,t) is Concentration as a function of
space and time.
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Consider an initially pure thick film into which some solute diffuses from the surface. If the
film thickness is very large or effectively infinite compared to the extent of diffusion, the
situation can be physically modelled as following initial and boundary conditions
A second boundary condition that must be specified concerns the nature of the
diffusant distribution maintained at the film surface x = 0.
Substrate
Diffusantx=0Two simple cases are possible.
i) ii)
In the first, a thick layer of diffusant provides an essentially limitless external supply of atoms
maintaining a constant surface concentrationCo for all time.
In the second case, a very thin layer of diffusant provides an instantaneous source Soof
surface atoms per unit area. Here the surface concentration diminishes with time as atoms
diffuse into the underlying
substrate.
MACROSCOPIC TRANSPORT Cont...
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The two solutions are
i)
ii)
MACROSCOPIC TRANSPORT Cont...
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The fraction of vacant lattice sites was
previously given by
Where Ef is surface energy generally 1eV.
exp( )f
B
E
K T
The number of times per second that an atomsuccessfully reaches the activated state is :
Where Each atom vibrates about its equilibrium
position with a characteristic lattice frequencyv
and Emis the vacancy jump or migration energy
per atom.
exp( )m
B
EvK T
Atom fluxes that pass from plane 1 to 2 and from plane 2 to 1 are respectively given by
1 2 0
1exp( ) exp( )
6
fm
B B
EEJ v Ca
K T K T
0
2 1 0
1exp( ) exp( )( )
6
fm
B B
EE dCaJ v C a
K T K T dx
where we have substituted Ca0 for n(x) and used the factor of 1/6 to account for
bidirectional jumping in each of the three coordinate directions.
The net flux JNis the difference or 20
1exp( ) exp( )( )
6
fmN
B B
EE dCJ a v
K T K T dx
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By association with Pick's law and the expression for D
where
We get [ ]D f m
E E E 20 0
1
6D a v
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Atom Transport in a Force Field
Consider neighbouring atomic positions in a
crystalline solid where no fields are applied. The
free energy of the system has the periodicity of
the lattice and varies schematically as shown inFig a.
Imposition of an external field now biases or tilts
the system such that the free energy is lower in
site 2 relative to 1 by an amount 2G as shown in
Fig b.
The rate at which atoms move from 1 to 2 or 2 to
1 is given by
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and the net rate rNis given by the difference
By multiplying both sides of Eq a0, we obtain the atomic velocity v:
The term in brackets is essentially the diffusivity D, where GDis a diffusional activation free
energy that for all practical purposes we may equate with ED
The term 2G/a0is a measure of the molar free-energy gradient or applied force F
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When chemical equilibrium prevails, thecompeting rates are equal and rN= 0. Therefore
where G* is the molar free energy of activation
If CRis the concentration of reactants at coordinate position 1 and
CPthe concentration of products at 2, then the net rate of reaction
is proportional to
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Substrate Surfaces and Thin-Film Nucleation
Many observations of subsequent film formation have pointed to three basic growth modes:
(1) island(or Volmer-Weber), (2)layer (or Frank Van der Merwe), and (3) SK(Stranski-
Krastanov)
Island growth occurs when the smallest stable clustersnucleate on the substrate and grow in three
dimensions to form islands. This happens when atoms
or molecules in the deposit are more strongly bound
to each other than to the substrate.
the extension of the smallest stable nucleus occursoverwhelmingly in two dimensions, resulting in the
formation of planar sheets. In this growth mode the
atoms are more stronglybound to the substrate than
to each other. The first complete monolayer is then
covered with a somewhat less tightly bound second
layer. Providing the decrease in bonding energy is
continuous toward the bulk-crystal value,
the layer growth mode is sustained.
The layer plus island or Stranski-Krastanov (S-K) growth mechanism is an intermediate
combination of the preceding two modes. In this case after forming one or more
monolayers, subsequent layer growth becomes unfavorable and islands form.
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AN ATOMIC VIEW OF SUBSTRATE SURFACES
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THERMODYNAMIC ASPECTS OF NUCLEATION
SURFACE ENERGIES
Atoms at free surfaces are more energetic than
atoms within the underlying bulk because they makefewer bonds with surrounding atoms and are thus
less constrained. The difference in inter atomic
energy of atoms at these two locations is the origin
of surface energy. Alternatively, there is a
thermodynamic driving force to reduce the number
of necessarily cut, dangling bonds at the surfacethrough rebonding between atoms. We may then
equivalently view the surface energy (J/m2) in
terms of the energy reduction per unit area.
surface energies roughly span the range 0.2 to 3
J/m2 with 1 J/m2 ( = 1000 erg/cm2) being typical.
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CAPILLARITY THEORY OF HETEROGENEOUS NUCLEATION
We start by assuming that film forming
atoms or molecules in the vapor phaseimpinge on the substrate creating nuclei of
mean dimension r. The free-energy change
accompanying the formation of such an
aggregate is given by
For the spherical cap-shaped solid nucleus shown in Fig. the curved surface area (a1r2), the
projected circular area on the substrate (a2r2), and the volume (a3r3) are involved
0d G
dr
At equilibrium And the size is called critical radius r*
Correspondingly
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An aggregate smaller in size than r* disappears by shrinking, lowering G in the
process. Critical nuclei grow to supercritical dimensions by further addition of
atoms, a process which lowers G still more. In heterogeneous nucleation the
accommodating substrate catalyzes vapour condensation by lowering the energybarrier G* through a reduction of the contact angle. After substitution of the
geometric constants, it is easily shown that
If, for example, the film nucleus is elastically strained throughout because of the
bonding mismatch between film and substrate, then a term a3r3Gs, where Gsis the
strain free-energy change per unit volume, would be appropriate. In the calculation for
AG*, the denominator of Eq. 7-9 would then be altered to 27a32/ (Gv + Gs)2,
Because the sign of Gv is negative while Gs is positive, the overall energy barrier to
nucleation increases in such a case. If, however, deposition occurred on an initially
strained substrate, i.e., one with emergent cleavage steps or screw dislocations, then
stress relieve during nucleation would be manifested by a reduction of G*. Substrate
charge and impurities would similarly influence G* by affecting terms related to either
surface and volume electrostatic, chemical, etc., energies.
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The most important thing to remember about G* is its strong influence on the
density (N*) of stable nuclei that can be expected to survive.
where nsis the total nucleation site density
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