Gupta D. (Ed.). Diffusion Processes in Advanced Technological Materials
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2DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS
great importance in their design, fabrication, and performance. Diffusion
research during the past hundred years is replete with numerous examples
from industry such as sintering, power generation, lighting, metal forming,
aviation, space, and information technology, to mention a few.
The objective of this chapter is to provide some basic and useful phe-
nomenological aspects of diffusion for engineering materials. It is illus-
trated by some model materials, in the context of enormously differing
time, temperature, and length scales used from industry to industry. A
comprehensive discussion of the driving forces for diffusion, both in lin-
ear and nonlinear regimes, is included, which should be relevant in the
context of the current technological trends of using thinner and multilevel
metallization schemes. Typical data on diffusion in bulk metals, semicon-
ductors, and thin metallic films, given in tabular form, with discussions on
the role of variable microstructure and chemistry, should be useful to the
materials community in general. Because the field of diffusion is vast, it
was impossible to provide a comprehensive survey of diffusion as a sci-
entific discipline. The recent literature
[611]
contains discussions of topics
that could not be covered here.
1.2 Diffusion in Single Crystals
1.2.1 Mathematical Basis
This section discusses the general aspects of diffusion without con-
sidering the role of the microstructure of the samples, which is discussed
in Sec. 1.3. The specimens only contain lattice point defects such as
vacancies, divacancies, interstitials, and impurity atoms in thermal equi-
librium. Textbooks by Shewmon,
[10]
Carslaw and Jaeger,
[12]
and Crank
[13]
present detailed mathematical solutions of specific diffusion experiments.
1.2.1.1 Fick’s First Law
Fick’s first law defines the diffusion coefficient (D
) of the component 1
and flux in an inhomogeneous single-phase binary alloy due to concen-
tration gradient as:
J
1
D
1
∂
∂
C
x
1
T
C
1
v, (1)
where J
1
is the flux at time t of atoms of the component 1, C
1
is its concen-
tration, n is the velocity of mass that moves due to application of forces such
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DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 3
as electromigration or a thermal or chemical potential gradient, and x is the
distance into the sample taken parallel to the concentration gradient. If the
concentration of the diffusant can be defined at the entrance and exit surfaces
of the sample separated by distance x in steady-state condition, the diffusiv-
ity can be measured easily by monitoring the flux in the absence of a driving
force. A number of such simple experiments can be found in the literature;
for example, diffusion of Ni in Si has been studied by Thompson et al.
[14]
using such a scheme. Because the measurement of flux contains no infor-
mation on the distribution of the diffusant inside the sample, an independent
study of the microstructure of the specimen and its chemistry is usually con-
ducted to describe the diffusion conditions and path in the physical sense.
It is possible to measure the diffusion coefficient in the absence of
external forces, including the chemical gradient, using isotope techniques.
The diffusant is the isotope of the host species; it may be a radioactive
tracer or a stable isotope. The latter is detected by mass spectroscopy tech-
niques such as the one used in secondary ion mass spectroscopy. Such
measurements are termed self-diffusion and are generally denoted by an
asterisk or by the isotope used.
Fick’s first law then reduces to:
J
∗
D
∗
∂
∂
C
x
∗
T
. (2)
For diffusion in a chemical gradient, the measured diffusion coeffi-
cient is affected by the motion of all the atomic species and is conse-
quently termed interdiffusion coefficient. As will be seen in Sec. 1.2.4, it
contains thermodynamical contributions due to the changing chemistry of
the alloy. The interdiffusion or chemical diffusion coefficient is generally
denoted by D
∼
; consequently, Fick’s first law is written as:
J D
∼
∂
∂
C
x
T
. (3)
In three-dimensional vector notations, the general statement of Fick’s first
law is:
J D
∼
C. (4)
1.2.1.2 Fick’s Second Law
Fick’s second law describes diffusion in a non-steady-state condition
when the concentration changes with time but particles are neither created
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nor destroyed. Thus, we have the continuity equation:
∂
∂
C
t
J. (5)
Fick’s second law can be written by combining Eqs. (2) and (5),
∂
∂
C
t
D
2
C, (6)
for linear flow and constant D (implying that it is not a function of posi-
tion). For variable D, Fick’s second law is written as:
∂
∂
C
t
∂
∂
x
D
∂
∂
C
x
. (7)
Fick’s second law is the basis of most diffusion measurements in solids in
general. It has been widely used in samples in rod, plate, and thin-layer
geometry. It has been used for measurements of diffusion in single-crystal
specimens, and along grain boundaries and dislocations. These individual
applications are discussed below.
1.2.1.3 Instantaneous Source Geometry
Imagine an infinitesimally thin layer of diffusant of strength M
deposited on the face of a cylindrical specimen so that its thickness, the
Dirac d, is a lot less than the diffusion distance 2(Dt)
1/2
, corresponding to
the initial condition. Assume that there is no flux on the surface or in
the imperfections, such as grain boundaries and dislocations. When the
boundary conditions:
C(x, 0) Md(x) (8)
and
∂
∂
C
x
(0, t) 0 (9)
are maintained, the Gaussian solution to Fick’s second law is given by:
C(x, t) (M2
pDt
)exp(x
2
4Dt), (10)
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DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 5
where t is time of diffusion and C is concentration at x and t.
This geometry has been widely used in the radioactive tracer work
where a Fermi-Dirac d of diffusant distribution can be initially obtained
and the mass and thickness involved are infinitesimally small. Much of
the basic diffusion work in solids has been carried out in this way.
1.2.1.4 Thick-Layer Geometry
Generally, it is not possible to obtain Dirac d function in nontracer dif-
fusion experiments. Consequently, a thick geometry constitutes a situation
where the initial thickness of the source of the diffusant (h) is of the order
of the diffusion distance (2
Dt
). The initial source condition for the
thick-layer geometry for t 0 is given by:
C(x, 0) C
0
for h x 0(11)
and
C(x, 0) 0 for x h (12)
For t 0, the boundary condition is:
∂
∂
C
x
(0, t) 0. (13)
The solution is then given by:
C(x, t)
C
2
0
erf
2
x
D
h
t
erf
2
x
D
h
t
. (14)
Thick-layer geometries commonly arise during diffusion in thin-film cou-
ples as well, which is described in Sec. 1.4.1.(The error function, erf, and
its complement, erfc, are defined by Shewmon
[10]
in the form:
erf (z) 1 erfc(z) 1
p
2
z
0
e
h
2
dh,
where h
2
x
Dt
.
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1.2.1.5 Infinite Couple with Constant Surface Composition
When a diffusion couple is formed from two samples that have uni-
form concentrations of C
0
and C
l
, the initial condition near the interface at
x 0 and t 0 is described as:
C(x, 0) C
l
for x 0, (15)
C(x, 0) C
0
for x 0, (16)
and the solution is given by:
C(
C
x,
l
t)
C
0
C
0
1
2
erfc
2
x
Dt
. (17)
1.2.1.6 Diffusion Under a Chemical Gradient:
Boltzmann-Matano Analysis
It has been known for a long time that diffusion under a chemical
gradient produces a net flow of matter relative to the initial interface. In
Fig. 1.1(a), such a diffusion couple consisting of an alloy AB and a pure
metal A is shown schematically from the article by Lazarus.
[15]
To keep
track of the transfer of matter, inert markers were placed at W, the initial
interface. The two species diffused at unequal rates, and the mass balance
was maintained by the compensating flow of vacancies. After diffusion,
the markers were found displaced relative to the edges of the couple, and
porosity was found at the original interface due to condensation of super-
saturated vacancies.
To calculate the diffusion coefficient, Eq. (7) was initially solved by
Boltzmann in 1894
[16]
and derived empirically by Matano
[17]
in 1933.
Their solution was based on a single D
∼
describing diffusion and bulk
motion as the same process. Thus, for the condition that the difference
between C
l
and C
0
is large and x 2Dt, as is the case in real experi-
ments, the diffusion coefficient will vary with composition and along the
distance x.
In Fig. 1.1(b), a Matano plot (concentration versus the distance x) is
shown schematically. The initial conditions are:
C C
0
for x 0, at t 0, (18)
C 0 for x 0, at t 0. (19)
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DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 7
Figure 1.1. (a) Diffusion in an inhomogeneous diffusion couple showing Kirkendall
effect: formation of porosity, motion of the tungsten wire markers (W).The Matano
interface position is shown at M.(After Lazarus
[15]
) (b) Procedure to compute inter-
diffusion coefficient D
∼
.Matano interface position is first obtained by equating areas
above and below the composition curve. D
∼
(C) is then computed from the slope of
the tangent at the chosen composition C and the hatched area under the curve,
according to Eq. (20).
(a)
(b)
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Following the determination of the concentration profile, the location of
the Matano interface along x is determined either graphically or by numer-
ical techniques such that the two areas under the curve are equal. The
value of chemical diffusivity D
∼
is then computed at the desired composi-
tion C according to the equation:
D
∼
(C)
2
1
t
d
d
x
c
C
C
0
xdc. (20)
The term (dxdc) is actually the slope of the curve at the concentration C.
The integrated hatched area under the curve between CC
0
0 and
CC
0
C is then computed. Reasonable values for D
∼
(C) can be com-
puted in the range 0.1 C0.9. Outside this range, the errors in com-
puting the slope and the area under the curve become rather large.
[10]
It is also possible to determine the diffusivities for the individual
components in an alloy using Matano-type analysis. Smigelskas and
Kirkendall
[18]
studied the motion of Mo markers in the CuZn/Cu diffusion
couple and measured unequal diffusion coefficients for the Zn and Cu
species. The marker velocity (v
m
) was related to the diffusivities as:
v
m
(D
Zn
D
Cu
)
∂C
∂
(
x
Zn)
. (21)
The interdiffusivity D
∼
itself was later related by Darken
[19]
to the intrinsic
diffusivities as:
D
∼
[D
Zn
C(Cu) D
Cu
C(Zn)]. (22)
Because Eqs. (21) and (22) involve two unknowns, D
Zn
and D
Cu
can be
determined individually. A complete description of the Darken’s analysis
for interdiffusion in non-ideal solid solutions is given in Sec. 1.2.4.
1.2.2 Atomistic Nature of Diffusion
1.2.2.1 Diffusion Mechanisms
A few possible mechanisms considered for atomic diffusion in single
crystals are shown in Fig. 1.2. The thermodynamics and computer model-
ing of these diffusion mechanisms are discussed in Chapters 2 and 3,
respectively. They are briefly discussed here.
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DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 9
Figure 1.2 Possible diffusion mechanisms in solids. (After Lazarus
[15]
)
(a) Interchange and ring mechanism. The diffusive motion in
this mechanism may take place by a correlated rotation of
two or more atoms about a common center without involv-
ing a defect. This mechanism has been found energetically
unfavorable in most solids.
(b) Interstitial mechanism. Small interstitial atoms can read-
ily diffuse by meandering in the interstices. Commonly,
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10 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS
gas atoms such as O, N, H, and C diffuse easily in the open
lattices of BCC metals, for example, Fe, Ta, W, and Mo.
There may be several variations of interstitial diffusion that
involve displacing a neighboring atom to an interstitial
position, termed interstitialcy or kickout mechanism, or a
correlated motion of a line of displaced atoms, the crow-
dion mechanism.
(c) Vacancy mechanism. Atomic diffusion into the missing
atomic sites (vacancies) has been found to be the most
favorable. Indeed, vacancies are present in pure metals
and alloys at all temperatures; their concentration at the
melting temperature (T
m
) is about 0.01%. This process
has been studied extensively in the past century by a vari-
ety of techniques, and the results support this mechanism
overwhelmingly on a wide basis in metals, alloys, and
oxides. The vacancies may also be in the form of dim-
mers (the divacancies), particularly near the melting tem-
perature (T
m
).
(d) Sub-boundary mechanism. The diffusing atom moves
along interconnecting dislocation pipes, which result from
naturally occurring low-angle boundaries. This mecha-
nism, discussed later, has been found to operate at low
temperatures, typically 0.5 T
m
, the absolute melting tem-
perature, in metals such as Au, Ag, and Cu.
(e) Relaxion mechanism. The diffusing atom moves more or
less freely within a disordered group of atoms within the
lattice. This mechanism has been ruled out in most crys-
talline solids but has been considered in recent years in the
context of radiation damage in amorphous metallic alloys
and some polymers.
This section focuses on the diffusion mechanism by vacancies and to
a lesser extent by interstitial atoms. Figure 1.3 shows the energy barrier as
a function of atomic position for a jumping atom under zero driving force,
as described by Manning.
[20]
Due to the absence of any driving force, the
probability of the forward and backward jumps is the same. Hence, the
velocity of the vacancy flow v in Eq. (1) is zero. A successful jump is
executed when an atom gathers the free energy G
m
, goes over the barrier,
and occupies an equivalent equilibrium position by an exchange with a
defect like a vacancy. The probability W of acquiring the energy G
m
is
given by the Boltzmann factor as:
W v
o
exp(G
m
kT), (23)
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where n
o
is the atomic vibration frequency. The success of the jump, how-
ever, depends on the availability of a defect in the adjoining position,
which is also given by a Boltzmann factor. Hence, the average probability
N
d
of finding the proper defect may be written as:
N
d
Z exp(G
f
kT), (24)
where G
f
is free energy necessary to form the defect and Z is a coordina-
tion factor that depends on the crystal type. As mentioned earlier, the
vacancy concentration in most metals and alloys is generally ~0.01% at
the melting temperature and even smaller at lower temperatures. The
frequency of jumping Γ is obtained by multiplying Eqs. (23) and (24):
Γ Zv
o
exp(G
m
G
f
). (25)
For the three-dimensional case, the diffusion coefficient is expressed as:
D Γ
a
6
2
f, (26)
where a is the nearest neighbor atomic distance. For tracer diffusion
measurements, which are discussed in Sec. 1.2.3, a correction term f 1
should be introduced for the non-random character of the jumps. The
factor f is known as a correlation factor. Its full implications are discussed
by Manning;
[20, 21]
it is briefly discussed in Sec. 1.2.3.
Finally, the diffusion coefficient can be written as:
D
1
6
a
2
Z f v
o
exp[(S
m
S
f
)k] exp[(H
f
H
m
)kT], (27)
DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 11
Figure 1.3 Schematic diagram for diffusion showing displacement of an atom in
the lattice when the driving force is zero. (After Manning
[21]
)
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