Chap ter
5
Diffusion
T
he photograph at the top of this page is of a steel gear that has
been case hardened. Its outer surface layer was selectively hardened
by a high-temperature heat treatment during which carbon from the
surrounding atmosphere diffused into the surface. The “case” appears
as the dark outer rim of that segment of the gear that has been sectioned. This increase in the carbon content raises the surface hardness
(as explained in Section 10.7), which in turn leads to an improvement
of wear resistance of the gear. In addition, residual compressive
stresses are introduced within the case region; these give rise to an
enhancement of the gear’s resistance to failure by fatigue while in
service (Chapter 8).
Case-hardened steel gears are used in automobile transmissions,
similar to the one shown in the photograph directly below the gear.
(Top: courtesy of Surface Division Midland-Ross; center: courtesy
Ford Motor Company; bottom right: © BRIAN
KERSEY/UPI/Landov LLC; bottom inset: © iStockphoto.)
WHY StUdy Diffusion?
Materials of all types are often heat-treated to improve their properties. The phenomena that occur
during a heat treatment almost always involve atomic
diffusion. Often an enhancement of diffusion rate is
desired; on occasion measures are taken to reduce it.
Heat-treating temperatures and times and/or cooling
rates are often predictable using the mathematics of
diffusion and appropriate diffusion constants. The steel
gear shown on page 122 (top) has been case hardened
(Section 8.10); that is, its hardness and resistance to
failure by fatigue have been enhanced by diffusing excess
carbon or nitrogen into the outer surface layer.
In the processing/structure/properties/performance
scheme, reasons for studying diffusion are as follows:
• Diffusional processes (conducted at elevated temperatures) are often utilized to introduce impurity
atoms into silicon semiconductors. Thus, a knowledge of the time and temperature dependences of
appropriate diffusion parameters is necessary.
• Likewise, the heat treatment of steels (and the
development of their mechanical properties)
involves diffusion.
Learning Objectives
After studying this chapter you should be able to do the following:
1. Name and describe the two atomic mechanisms
concentration of diffusing species at the
of diffusion.
surface is held constant. Define all parameters
2. Distinguish between steady-state and nonsteadyin this equation.
state diffusion.
4. Calculate the diffusion coefficient for some
3. (a) Write Fick’s first and second laws in equamaterial at a specified temperature, given the
tion form and define all parameters.
appropriate diffusion constants.
(b) Note the kind of diffusion for which each
of these equations is normally applied.
4. Write the solution to Fick’s second law for
diffusion into a semi-infinite solid when the
5.1 INTRODUCTION
diffusion
Many reactions and processes that are important in the treatment of materials rely
on the transfer of mass either within a specific solid (ordinarily on a microscopic
level) or from a liquid, a gas, or another solid phase. This is necessarily accomplished
by diffusion, the phenomenon of material transport by atomic motion. This chapter discusses the atomic mechanisms by which diffusion occurs, the mathematics of
diffusion, and the influence of temperature and diffusing species on the rate of
diffusion.
The phenomenon of diffusion may be demonstrated with the use of a diffusion couple, which is formed by joining bars of two different metals together so
that there is intimate contact between the two faces; this is illustrated for copper
and nickel in Figure 5.1, which includes schematic representations of atom positions and composition across the interface. This couple is heated for an extended
period at an elevated temperature (but below the melting temperature of both
metals) and cooled to room temperature. Chemical analysis will reveal a condition
similar to that represented in Figure 5.2—namely, pure copper and nickel at the
two extremities of the couple, separated by an alloyed region. Concentrations of
• 123
124 • Chapter 5 / DiffUsion
Figure 5.1 (a) A copper–nickel diffusion couple
before a high-temperature heat treatment. (b)
Schematic representations of Cu (red circles) and
Ni (blue circles) atom locations within the diffusion
couple. (c) Concentrations of copper and nickel as a
function of position across the couple.
Ni
Cu
(a)
Concentration of Ni, Cu
(b)
100
Cu
Ni
0
Position
(c)
interdiffusion
impurity diffusion
both metals vary with position as shown in Figure 5.2c. This result indicates that
copper atoms have migrated or diffused into the nickel, and that nickel has diffused into copper. This process, whereby atoms of one metal diffuse into another,
is termed interdiffusion, or impurity diffusion.
Diffusion of Cu atoms
Cu
Cu–Ni alloy
Diffusion of Ni atoms
(a)
Concentration of Ni, Cu
(b)
(c)
100
Cu
0
Ni
Position
Ni
Figure 5.2 (a) A copper–nickel diffusion couple
after a high-temperature heat treatment, showing the
alloyed diffusion zone. (b) Schematic representations
of Cu (red circles) and Ni (blue circles) atom
locations within the couple. (c) Concentrations of
copper and nickel as a function of position across
the couple.
5.2 DiffUsion Mechanisms • 125
self-diffusion
Interdiffusion may be discerned from a macroscopic perspective by changes in
concentration that occur over time, as in the example for the Cu–Ni diffusion couple.There is a net drift or transport of atoms from high- to low-concentration regions.
Diffusion also occurs for pure metals, but all atoms exchanging positions are of the
same type; this is termed self-diffusion. Of course, self-diffusion is not normally subject to observation by noting compositional changes.
5.2 DIFFUSION MECHANISMS
From an atomic perspective, diffusion is just the stepwise migration of atoms from
lattice site to lattice site. In fact, the atoms in solid materials are in constant motion,
rapidly changing positions. For an atom to make such a move, two conditions must
be met: (1) there must be an empty adjacent site, and (2) the atom must have sufficient energy to break bonds with its neighbor atoms and then cause some lattice
distortion during the displacement. This energy is vibrational in nature (Section 4.8).
At a specific temperature some small fraction of the total number of atoms is
capable of diffusive motion, by virtue of the magnitudes of their vibrational energies. This fraction increases with rising temperature.
Several different models for this atomic motion have been proposed; of these
possibilities, two dominate for metallic diffusion.
Vacancy DiffUsion
vacancy diffusion
Figure 5.3
Schematic
representations of
(a) vacancy diffusion
and (b) interstitial
diffusion.
One mechanism involves the interchange of an atom from a normal lattice position
to an adjacent vacant lattice site or vacancy, as represented schematically in Figure
5.3 a. This mechanism is aptly termed vacancy diffusion. Of course, this process necessitates the presence of vacancies, and the extent to which vacancy diffusion can
occur is a function of the number of these defects that are present; significant
Motion of a host or
substitutional atom
Vacancy
Vacancy
(a)
Position of interstitial
atom before diffusion
(b)
Position of interstitial
atom after diffusion
126 • Chapter 5 / DiffUsion
concentrations of vacancies may exist in metals at elevated temperatures (Section
4.2). Because diffusing atoms and vacancies exchange positions, the diffusion of
atoms in one direction corresponds to the motion of vacancies in the opposite direction. Both self-diffusion and interdiffusion occur by this mechanism; for the latter, the impurity atoms must substitute for host atoms.
Interstitial DiffUsion
interstitial diffusion
The second type of diffusion involves atoms that migrate from an interstitial position to a neighboring one that is empty. This mechanism is found for interdiffusion of impurities such as hydrogen, carbon, nitrogen, and oxygen, which have
atoms that are small enough to fit into the interstitial positions. Host or substitutional impurity atoms rarely form interstitials and do not normally diffuse via
this mechanism. This phenomenon is appropriately termed interstitial diffusion
(Figure 5.3b).
In most metal alloys, interstitial diffusion occurs much more rapidly than diffusion by the vacancy mode, because the interstitial atoms are smaller and thus
more mobile. Furthermore, there are more empty interstitial positions than vacancies; hence, the probability of interstitial atomic movement is greater than for
vacancy diffusion.
5.3 STEADY-STATE DIFFUSION
diffusion flux
Definition of
diffusion flux
Diffusion is a time-dependent process—that is, in a macroscopic sense, the quantity of an element that is transported within another is a function of time. Often it
is necessary to know how fast diffusion occurs, or the rate of mass transfer. This
rate is frequently expressed as a diffusion flux (J ), defined as the mass (or, equivalently, the number of atoms) M diffusing through and perpendicular to a unit crosssectional area of solid per unit of time. In mathematical form, this may be
represented as
J=
M
(5.1a)
At
where A denotes the area across which diffusion is occurring and t is the elapsed
diffusion time. In differential form, this expression becomes
1 dM
J=
A dt
steady-state diffusion
concentration profile
concentration
gradient
(5.1b)
The units for J are kilograms or atoms per meter squared per second (kg/m 2. s or
atoms/m2. s).
If the diffusion flux does not change with time, a steady-state condition exists.
One common example of steady-state diffusion is the diffusion of atoms of a gas
through a plate of metal for which the concentrations (or pressures) of the diffusing species on both surfaces of the plate are held constant. This is represented
schematically in Figure 5.4a.
When concentration C is plotted versus position (or distance) within the solid x,
the resulting curve is termed the concentration profile; the slope at a particular
point on this curve is the concentration gradient:
concentration gradient =
dC
dx
(5.2a)
5.3 Steady-State DiffUsion • 127
PA > P B
and constant
Thin metal plate
Gas at
pressure PB
Gas at
pressure PA
Direction of
diffusion of
gaseous species
Concentration of diffusing species, C
Figure 5.4
(a) Steady-state
diffusion across a
thin plate. (b) A
linear concentration
profile for the
diffusion situation
in (a).
CA
CB
xA
xB
Position, x
(b)
Area, A
(a)
In the present treatment, the concentration profile is assumed to be linear, as depicted in Figure 5.4b, and
¢C CA — CB
concentration gradient =
=
(5.2b)
¢x
xA — xB
For diffusion problems, it is sometimes convenient to express concentration in terms
of mass of diffusing species per unit volume of solid (kg/m 3 or g/cm3).1
The mathematics of steady-state diffusion in a single (x) direction is relatively
simple, in that the flux is proportional to the concentration gradient through the
expression
Fick’s first law—
diffusion flux for
steady-state diffusion
(in one direction)
diffusion coefficient
Fick’s first law
driving force
J = —D
dC
dx
(5.3)
The constant of proportionality D is called the diffusion coefficient, which is expressed in square meters per second. The negative sign in this expression indicates
that the direction of diffusion is down the concentration gradient, from a high to a
low concentration. Equation 5.3 is sometimes called Fick’s first law.
Sometimes the term driving force is used in the context of what compels a reaction to occur. For diffusion reactions, several such forces are possible; but when
diffusion is according to Equation 5.3, the concentration gradient is the driving force.
One practical example of steady-state diffusion is found in the purification of
hydrogen gas. One side of a thin sheet of palladium metal is exposed to the impure
gas composed of hydrogen and other gaseous species such as nitrogen, oxygen, and
water vapor. The hydrogen selectively diffuses through the sheet to the opposite
side, which is maintained at a constant and lower hydrogen pressure.
1
Conversion of concentration from weight percent to mass per unit volume (in kg/m 3) is
possible using Equation 4.9.
128 • Chapter 5 / DiffUsion
EXAMPLE PROBLEM 5.1
DiffUsion FlUx CompUtation
A plate of iron is exposed to a carburizing (carbon-rich) atmosphere on one
side and a decarburizing (carbon-deficient) atmosphere on the other side at
700°C (1300°F). If a condition of steady state is achieved, calculate the diffusion flux of carbon through the plate if the concentrations of carbon at
positions of 5 and 10 mm (5 × 10—3 and 10—2 m) beneath the carburizing surface are 1.2 and 0.8 kg/m3, respectively. Assume a diffusion coefficient of 3 ×
10—11 m2/s at this temperature.
Solution
Fick’s first law, Equation 5.3, is utilized to determine the diffusion flux. Substitution of the values above into this expression yields
J = —D
11.2 — 0.82 kg/m3
CA — CB
= —13 × 10—11 m2/s 2
xA — xB
15 × 10—3 — 10—2 2 m
= 2.4 × 10—9 kg/m2. s
5.4 NONSTEADY-STATE DIFFUSION
Most practical diffusion situations are nonsteady-state ones. That is, the diffusion
flux and the concentration gradient at some particular point in a solid vary with
time, with a net accumulation or depletion of the diffusing species resulting. This is
illustrated in Figure 5.5, which shows concentration profiles at three different diffusion times. Under conditions of nonsteady state, use of Equation 5.3 is no longer
convenient; instead, the partial differential equation
0C 0
0C
= aD
b
0t
x
0x
Fick’s second law—
diffusion equation
for nonsteady-state
diffusion (in one
direction)
known as Fick’s second law, is used. If the diffusion coefficient is independent of
composition (which should be verified for each particular diffusion situation), Equation 5.4a simplifies to
0C
02 C
(5.4b)
=D 2
0t
0x
Solutions to this expression (concentration in terms of both position and time) are
possible when physically meaningful boundary conditions are specified.
Concentration of diffusing species
Fick’s second law
(5.4a)
t3 > t2 > t1
t3
t2
t1
Distance
Figure 5.5 Concentration profiles for
nonsteady-state diffusion taken at three
different times, t1, t2, and t3.
5.4 Nonsteady-State DiffUsion • 129
Comprehensive collections of these are given by Crank, and Carslaw and Jaeger
(see References).
One practically important solution is for a semi-infinite solid2 in which the surface concentration is held constant. Frequently, the source of the diffusing species
is a gas phase, the partial pressure of which is maintained at a constant value. Furthermore, the following assumptions are made:
1. Before diffusion, any of the diffusing solute atoms in the solid are uniformly
distributed with concentration of C0.
2. The value of x at the surface is zero and increases with distance into the solid.
3. The time is taken to be zero the instant before the diffusion process begins.
These boundary conditions are simply stated as
Solution to Fick’s
second law for the
condition of constant
surface concentration (for a semiinfinite solid)
For t = 0, C = C0 at 0 ≤ x ≤ ∞
For t > 0, C = Cs (the constant surface concentration) at x = 0
C = C0 at x =∞
Application of these boundary conditions to Equation 5.4b yields the solution
Cx — C0
x
= 1 — erf a
b
C s— C 0
21 Dt
(5.5)
where Cx represents the concentration at depth x after time t. The expression
erf 1x/21Dt 2 is the Gaussian error function,3 values of which are given in mathematical tables for various x/21Dt values; a partial listing is given in Table 5.1. The
Table 5.1 TabUlation of Error FUnction ValUes
z
erf(z)
z
erf(z)
z
erf(z)
0
0.025
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0
0.0282
0.0564
0.1125
0.1680
0.2227
0.2763
0.3286
0.3794
0.4284
0.4755
0.5205
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.0
1.1
1.2
0.5633
0.6039
0.6420
0.6778
0.7112
0.7421
0.7707
0.7970
0.8209
0.8427
0.8802
0.9103
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.2
2.4
2.6
2.8
0.9340
0.9523
0.9661
0.9763
0.9838
0.9891
0.9928
0.9953
0.9981
0.9993
0.9998
0.9999
2
A bar of solid is considered to be semi-infinite if none of the diffusing atoms reaches the
bar end during the time over which diffusion takes place. A bar of length l is considered to
be semi-infinite when l 7 102Dt.
3
This Gaussian error function is defined by
erf 1z 2 =
2
z
∫e
1p
0
where x/21Dt has been replaced by the variable z.
— y2
dy
130 • Chapter 5 / DiffUsion
Figure 5.6 Concentration profile for
nonsteady-state diffusion; concentration
parameters relate to Equation 5.5.
Concentration, C
Cs
Cs – C0
Cx
Cx – C 0
C0
Distance from interface, x
concentration parameters that appear in Equation 5.5 are noted in Figure 5.6, a
concentration profile taken at a specific time. Equation 5.5 thus demonstrates the
relationship between concentration, position, and time—namely, that Cx, being a
function of the dimensionless parameter x/ 1Dt, may be determined at any time
and position if the parameters C0, Cs, and D are known.
Suppose that it is desired to achieve some specific concentration of solute, C1,
in an alloy; the left-hand side of Equation 5.5 now becomes
C1 — C0
= constant
Cs — C0
This being the case, the right-hand side of this same expression is also a constant,
and subsequently
x
= constant
(5.6a)
22Dt
or
x2
Dt
= constant
(5.6b)
Some diffusion computations are thus facilitated on the basis of this relationship, as demonstrated in Example Problem 5.3.
EXAMPLE PROBLEM 5.2
Nonsteady-State Diff U sion Time Comp U tation I
carburizing
For some applications, it is necessary to harden the surface of a steel (or
iron–carbon alloy) above that of its interior. One way this may be accomplished is by increasing the surface concentration of carbon in a process termed
carburizing; the steel piece is exposed, at an elevated temperature, to an atmosphere rich in a hydrocarbon gas, such as methane (CH4).
Consider one such alloy that initially has a uniform carbon concentration
of 0.25 wt% and is to be treated at 950°C (1750°F). If the concentration of
carbon at the surface is suddenly brought to and maintained at 1.20 wt%, how
long will it take to achieve a carbon content of 0.80 wt% at a position 0.5 mm
below the surface? The diffusion coefficient for carbon in iron at this temperature is 1.6 × 10—11 m2/s; assume that the steel piece is semi-infinite.
5.4 Nonsteady-State DiffUsion • 131
Solution
Because this is a nonsteady-state diffusion problem in which the surface composition is held constant, Equation 5.5 is used. Values for all the parameters
in this expression except time t are specified in the problem as follows:
C0 = 0.25 wt% C
Cs = 1.20 wt% C
Cx = 0.80 wt% C
x = 0.50 mm = 5 × 10—4 m
D = 1.6 × 10—11 m2/s
Thus,
Cx — C0
15 × 10 —4 m 2
= 0.80 — 0.25 = 1 — erf c
d
1.20 — 0.25
Cs — C0
2211.6 × 10 —11 m2/s 2 1t 2
0.4210 = erf a
62.5 s1/2
b
1t
We must now determine from Table 5.1 the value of z for which the error
function is 0.4210. An interpolation is necessary, as
z
erf(z)
0.35
z
0.40
0.3794
0.4210
0.4284
z — 0.35
0.4210 — 0.3794
=
0.40 — 0.35 0.4284 — 0.3794
or
z = 0.392
Therefore,
62.5 s1/2
= 0.392
2t
and solving for t,
t= a
62.5 s1/2 2
b = 25,400 s = 7.1 h
0.392
EXAMPLE PROBLEM 5.3
Nonsteady-State DiffUsion Time CompUtation II
The diffusion coefficients for copper in aluminum at 500 and 600°C are 4.8 ×
10—14 and 5.3 × 10—13 m2/s, respectively. Determine the approximate time at
500°C that will produce the same diffusion result (in terms of concentration
of Cu at some specific point in Al) as a 10-h heat treatment at 600°C.
Solution
This is a diffusion problem in which Equation 5.6b may be employed. The
composition in both diffusion situations will be equal at the same position (i.e.,
x is also a constant), thus
132 • Chapter 5 / DiffUsion
Dt = constant
(5.7)
at both temperatures. That is,
D500 t500 = D600 t600
or
t500 =
D600 t600
D500
=
15.3 × 10 —13 m2/s 2 110 h 2
4.8 × 10—14 m2/s
= 110.4 h
5.5 FACTORS THAT INFLUENCE DIFFUSION
DiffUsing Species
The magnitude of the diffusion coefficient D is indicative of the rate at which atoms
diffuse. Coefficients, both self- and interdiffusion, for several metallic systems are
listed in Table 5.2. The diffusing species as well as the host material influence the
diffusion coefficient. For example, there is a significant difference in magnitude between
self-diffusion and carbon interdiffusion in α iron at 500°C, the D value being greater
for the carbon interdiffusion (3.0 × 10—21 vs. 2.4 × 10—12 m2/s). This comparison also
provides a contrast between rates of diffusion via vacancy and interstitial modes as
discussed earlier. Self-diffusion occurs by a vacancy mechanism, whereas carbon diffusion in iron is interstitial.
TemperatUre
Temperature has a most profound influence on the coefficients and diffusion rates.
For example, for the self-diffusion of Fe in α-Fe, the diffusion coefficient increases
approximately six orders of magnitude (from 3.0 × 10—21 to 1.8 × 10—15 m2/s) in rising temperature from 500°C to 900°C (Table 5.2). The temperature dependence of
Table 5.2 A TabUlation of DiffUsion Data
Diffusing
Species
Activation Energy Q d
2
Calculated Value
Host Metal
D0(m /s)
kJ/mol
eV/atom
T(°C)
Fe
α-Fe
(BCC)
2.8 × 10—4
251
2.60
500
900
Fe
ç-Fe
(FCC)
5.0 × 10—5
284
2.94
900
1100
C
α-Fe
6.2 × 10—7
80
0.83
500
900
C
μ-Fe
2.3 × 10—5
148
1.53
900
1100
Cu
Zn
Al
Cu
Mg
Cu
Cu
Cu
Al
Al
Al
Ni
10—5
10—5
10—4
10—5
10—4
10—5
211
189
144
136
131
256
2.19
1.96
1.49
1.41
1.35
2.65
500
500
500
500
500
500
7.8 ×
2.4 ×
2.3 ×
6.5 ×
1.2 ×
2.7 ×
D(m 2/s)
3.0 ×
1.8 ×
1.1 ×
7.8 ×
2.4 ×
1.7 ×
5.9 ×
5.3 ×
4.2 ×
4.0 ×
4.2 ×
4.1 ×
1.9 ×
1.3 ×
10—21
10—15
10—17
10—16
10—12
10—10
10—12
10—11
10—19
10—18
10—14
10—14
10—13
10—22
Source: E. A. Brandes and G. B. Brook (Editors), Smithells Metals Reference Book, 7th edition, ButterworthHeinemann, Oxford, 1992.
5.5 Factors That InflUence DiffUsion • 133
the diffusion coefficients is
Dependence of the
diffusion coefficient
on temperature
D = D 0 exp a —
Qd
RT
b
(5.8)
where
D0 = a temperature-independent preexponential (m 2/s)
activation energy
Qd = the activation energy for diffusion (J/mol or eV/atom)
R = the gas constant, 8.31 J/mol·K or 8.62 × 10—5 eV/atom·K
T = absolute temperature (K)
The activation energy may be thought of as that energy required to produce
the diffusive motion of one mole of atoms. A large activation energy results in a
relatively small diffusion coefficient. Table 5.2 also contains a listing of D0 and Qd
values for several diffusion systems.
Taking natural logarithms of Equation 5.8 yields
Q
ln D = ln D — d 1
(5.9a)
a b
0
R T
or, in terms of logarithms to the base 10,
log D = log D —
0
Qd 1
a b
2.3R T
(5.9b)
Because D0, Qd, and R are all constants, Equation 5.9b takes on the form of an
equation of a straight line:
y = b + mx
where y and x are analogous, respectively, to the variables log D and 1/T. Thus, if
log D is plotted versus the reciprocal of the absolute temperature, a straight line
should result, having slope and intercept of —Qd/2.3R and log D0, respectively. This
is, in fact, the manner in which the values of Qd and D0 are determined experimentally. From such a plot for several alloy systems (Figure 5.7), it may be noted
that linear relationships exist for all cases shown.
Concept Check 5.1
Rank the magnitudes of the diffusion coefficients from greatest to least for the following systems:
N in Fe at 700°C
Cr in Fe at 700°C
N in Fe at 900°C
Cr in Fe at 900°C
Now justify this ranking. (Note: Both Fe and Cr have the BCC crystal structure, and
the atomic radii for Fe, Cr, and N are 0.124, 0.125, and 0.065 nm, respectively. You
may also want to refer to Section 4.3.)
[The answer may be found at www.wiley.com/college/callister (Student Companion Site).]
134 • Chapter 5 / DiffUsion
Temperature (C)
1500 1200 1000 800
10 –8
600
500
400
300
10–10
C in α -Fe
Diffusion coefficient (m2/s)
C in μ -Fe
10–12
Zn in Cu
10–14
Fe in μ -Fe
10–16
Al in Al
Fe in α -Fe
Cu in Cu
10–18
10–20
0.5
1.0
1.5
2.0
Reciprocal temperature (1000/K)
Figure 5.7 Plot of the logarithm of the diffusion coefficient versus the reciprocal of absolute
temperature for several metals. [Data taken from E. A. Brandes and G. B. Brook (Editors),
Smithells Metals Reference Book, 7th edition, Butterworth-Heinemann, Oxford, 1992.]
Concept Check 5.2
Consider the self-diffusion of two hypothetical metals A and B. On a schematic
graph of ln D versus 1/T, plot (and label) lines for both metals given that D0(A) >
D0(B) and also that Qd(A) > Qd(B).
[The answer may be found at www.wiley.com/college/callister (Student Companion Site).]
EXAMPLE PROBLEM 5.4
DiffUsion Coefficient Determination
Using the data in Table 5.2, compute the diffusion coefficient for magnesium
in aluminum at 550°C.
Solution
This diffusion coefficient may be determined by applying Equation 5.8; the
values of D0 and Qd from Table 5.2 are 1.2 × 10—4 m2/s and 131 kJ/mol, respectively. Thus,
D = 11.2 × 10 —4 m2/s 2 exp c—
= 5.8 × 10—13 m2/s
1131,000 J/mol 2
1 8.31 J/mol · K 2 1550 + 273 K 2
d
5.5 Factors That InflUence DiffUsion • 135
EXAMPLE PROBLEM 5.5
DiffUsion Coefficient Activation Energy and
Preexponential CalcUlations
In Figure 5.8 is shown a plot of the logarithm (to the base 10) of the diffusion
coefficient versus reciprocal of absolute temperature, for the diffusion of copper
in gold. Determine values for the activation energy and the preexponential.
VMSE
¢ 1log D2
§
1
¢a b
T
Qd = —2.3R 1slope 2 = —2.3R £
log D1 — log D2
= —2.3R £
1
T1
—
1
§
T2
where D1 and D2 are the diffusion coefficient values at 1/T1 and 1/T2, respectively. Let us arbitrarily take 1/T1 = 0.8 × 10—3 (K)—1 and 1/T2 = 1.1 × 10—3
(K)—1. We may now read the corresponding log D1 and log D2 values from the
line segment in Figure 5.8.
[Before this is done, however, a parenthetic note of caution is offered. The
vertical axis in Figure 5.8 is scaled logarithmically (to the base 10); however,
the actual diffusion coefficient values are noted on this axis. For example, for
D = 10—14 m2/s, the logarithm of D is —14.0, not 10—14. Furthermore, this logarithmic scaling affects the readings between decade values; for example, at a
location midway between 10—14 and 10—15, the value is not 5 × 10—15 but, rather,
10—14.5 = 3.2 × 10—15.]
Figure 5.8 Plot of the
logarithm of the diffusion
coefficient versus the
reciprocal of absolute
temperature for the
diffusion of copper in gold.
10–12
Diffusion coefficient (m2/s)
Do and Qd from
Experimental Data
Solution
From Equation 5.9b the slope of the line segment in Figure 5.8 is equal to
—Qd/2.3R, and the intercept at 1/T = 0 gives the value of log D0. Thus, the
activation energy may be determined as
10–13
10–14
10–15
10–16
10–17
0.7
0.8
0.9
1.0
1.1
Reciprocal temperature (1000/K)
1.2
136 • Chapter 5 / DiffUsion
Thus, from Figure 5.8, at 1/T1 = 0.8 × 10—3 (K)—1, log D1 = —12.40, while
for 1/T2 = 1.1 × 10—3 (K)—1, log D2 = —15.45, and the activation energy, as
determined from the slope of the line segment in Figure 5.8, is
Q d= — 2.3R £
log D 1— log D2
§
1
1
—
T1 T 2
= —2.318.31 J/mol # K 2 c
—12.40 — 1—15.452
0.8 × 10
—3
1K 2 — 1 — 1.1 × 10 —3 1K 2 —1
d
= 194,000 J/mol = 194 kJ/mol
Now, rather than trying to make a graphical extrapolation to determine
D0, a more accurate value is obtained analytically using Equation 5.9b, and a
specific value of D (or log D) and its corresponding T (or 1/T) from Figure
5.8. Because we know that log D =—15.45 at 1/T = 1.1 × 10—3 (K)—1 , then
log D0= log D +
Qd
1
a b
2.3R T
= —15.45 +
1194,000 J/mol 2 11.1 × 10 —3 [ K] —1 2
12.32 18.31 J/mol # K 2
= —4.28
—4.28
Thus, D0 = 10
m2/s = 5.2 × 10—5 m2/s.
DESIGN EXAMPLE 5.1
DiffUsion TemperatUre–Time Heat Treatment Specification
The wear resistance of a steel gear is to be improved by hardening its surface.
This is to be accomplished by increasing the carbon content within an outer
surface layer as a result of carbon diffusion into the steel; the carbon is to be
supplied from an external carbon-rich gaseous atmosphere at an elevated and
constant temperature. The initial carbon content of the steel is 0.20 wt%,
whereas the surface concentration is to be maintained at 1.00 wt%. For this
treatment to be effective, a carbon content of 0.60 wt% must be established at
a position 0.75 mm below the surface. Specify an appropriate heat treatment
in terms of temperature and time for temperatures between 900 and 1050°C.
Use data in Table 5.2 for the diffusion of carbon in μ-iron.
Solution
Because this is a nonsteady-state diffusion situation, let us first of all employ
Equation 5.5, utilizing the following values for the concentration parameters:
C0 = 0.20 wt% C
Cs = 1.00 wt% C
Cx = 0.60 wt% C
5.6 DiffUsion in SemicondUcting Materials • 137
Therefore
0.60 — 0.20
x
Cx — C0
a
b
Cs — C0 = 1.00 — 0.20 = 1 — erf
22Dt
and thus
x
0.5 = erfa
b
22Dt
Using an interpolation technique as demonstrated in Example Problem 5.2 and
the data presented in Table 5.1,
x
= 0.4747
(5.10)
22Dt
The problem stipulates that x = 0.75 mm = 7.5 × 10—4 m. Therefore
7.5 × 10 —4 m
= 0.4747
22Dt
This leads to
Dt = 6.24 × 10—7 m2
Furthermore, the diffusion coefficient depends on temperature according to
Equation 5.8; and, from Table 5.2 for the diffusion of carbon in μ-iron, D0 =
2.3 × 10—5 m2/s and Qd = 148,000 J/mol. Hence
Qd
Dt = D0 exp a—
b 1t 2 = 6.24 × 10 —m7 2
RT
12.3 × 10 —5 m2/s 2 exp c —
and, solving for the time t,
1
148,000 J/mol
—7
2
#8.31 J/mol d 1t 2 = 6.24 × 10 m
K2 1T 2
t 1in s 2 =
0.0271
17,810
b
T
Thus, the required diffusion time may be computed for some specified temperature (in K). The following table gives t values for four different temperatures
that lie within the range stipulated in the problem.
exp a —
Temperature
(°C )
900
950
1000
1050
Time
s
106,400
57,200
32,300
19,000
h
29.6
15.9
9.0
5.3
5.6 DIFFUSION IN SEMICONDUCTING MATERIALS
One technology that applies solid-state diffusion is the fabrication of semiconductor
integrated circuits (ICs) (Section 18.15). Each integrated circuit chip is a thin square
wafer having dimensions on the order of 6 mm by 6 mm by 0.4 mm; furthermore,
138 • Chapter 5 / DiffUsion
Concentration of diffusing species
millions of interconnected electronic devices and circuits are embedded in one of the
chip faces. Single-crystal silicon is the base material for most ICs. In order for these IC
devices to function satisfactorily, very precise concentrations of an impurity (or impurities) must be incorporated into minute spatial regions in a very intricate and detailed
pattern on the silicon chip; one way this is accomplished is by atomic diffusion.
Normally two heat treatments are used in this process. In the first, or predeposition step, impurity atoms are diffused into the silicon, often from a gas phase, the
partial pressure of which is maintained constant. Thus, the surface composition of
the impurity also remains constant over time, such that impurity concentration
within the silicon is a function of position and time according to Equation 5.5. Predeposition treatments are normally carried out within the temperature range of 900
and 1000°C and for times typically less than one hour.
The second treatment, sometimes called drive-in diffusion, is used to transport
impurity atoms farther into the silicon in order to provide a more suitable concentration distribution without increasing the overall impurity content. This treatment
is carried out at a higher temperature than the predeposition one (up to about
1200°C), and also in an oxidizing atmosphere so as to form an oxide layer on the
surface. Diffusion rates through this SiO2 layer are relatively slow, such that very
few impurity atoms diffuse out of and escape from the silicon. Schematic concentration profiles taken at three different times for this diffusion situation are shown
in Figure 5.9; these profiles may be compared and contrasted to those in Figure 5.5
for the case wherein the surface concentration of diffusing species is held constant.
In addition, Figure 5.10 compares (schematically) concentration profiles for predeposition and drive-in treatments.
If we assume that the impurity atoms introduced during the predeposition treatment are confined to a very thin layer at the surface of the silicon (which, of course,
Figure 5.9 Schematic concentration profiles
for drive-in diffusion of semiconductors at
three different times, t1, t2, and t3.
t3 > t2 > t1
t1
t2
t3
Concentration of diffusing species (C)
Distance
Figure 5.10 Schematic concentration
profiles taken after (1) predeposition and (2)
drive-in diffusion treatments for
semiconductors. Also shown is the junction
depth, xj.
Cs
After predeposition
After drive-in
CB
xj
Distance into silicon (x)
5.6 DiffUsion in SemicondUcting Materials • 139
is only an approximation), then the solution to Fick’s second law (Equation 5.4b)
takes the form
Q0
x2
C1x, t 2 =
exp a —
b
(5.11)
4Dt
2pDt
Here Q0 represents the total amount of impurities in the solid that were introduced
during the predeposition treatment (in number of impurity atoms per unit area);
all other parameters in this equation have the same meanings as previously. Furthermore, it can be shown that
Q0 = 2Cs
Dptp
(5.12)
B p
where Cs is the surface concentration for the predeposition step (Figure 5.10), which
was held constant, Dp is the diffusion coefficient, and tp is the predeposition treatment time.
Another important diffusion parameter is junction depth, xj. It represents the
depth (i.e., value of x) at which the diffusing impurity concentration is just equal to
the background concentration of that impurity in the silicon (CB) (Figure 5.10). For
drive-in diffusion xj may be computed using the following expression:
xj = c 14Dd td 2 ln a
1/2
Q0
bd
CB 2pD dtd
(5.13)
Here Dd and td represent, respectively, the diffusion coefficient and time for the
drive-in treatment.
EXAMPLE PROBLEM 5.6
DiffUsion of Boron into Silicon
Boron atoms are to be diffused into a silicon wafer using both predeposition
and drive-in heat treatments; the background concentration of B in this silicon
material is known to be 1 × 1020 atoms/m3. The predeposition treatment is to
be conducted at 900°C for 30 minutes; the surface concentration of B is to be
maintained at a constant level of 3 × 1026 atoms/m3. Drive-in diffusion will be
carried out at 1100°C for a period of 2 h. For the diffusion coefficient of B in
Si, values of Qd and D0 are 3.87 eV/atom and 2.4 × 10—3 m2/s, respectively.
(a) Calculate the value of Q0.
(b) Determine the value of xj for the drive-in diffusion treatment.
(c) Also for the drive-in treatment, compute the concentration of B atoms at
a position 1 µm below the surface of the silicon wafer.
Solution
(a) The value of Q0 is calculated using Equation 5.12. However, before this
is possible, it is first necessary to determine the value of D for the predeposition treatment [Dp at T = Tp = 900°C (1173 K)] using Equation 5.8.
(Note: For the gas constant R in Equation 5.8, we use Boltzmann’s constant
k, which has a value of 8.62 × 10—5 eV/atom # K). Thus
D p = D 0exp a —
Qd
kTp
b
140 • Chapter 5 / DiffUsion
= 12.4 × 10 — 3 m2/s 2 exp c—
3.87 eV/atom
18.62 × 10 — 5 eV/atom # K 2 11173 K 2
d
= 5.73 × 10—20 m2/s
The value of Q0 may be determined as follows:
Dptp
Q0 = 2Cs
26
= 122 13 × 10 atoms/m
2
3
B p
15.73 × 10 — 20 m 2/s 2 130 min 2 160 s/min 2
p
B
= 3.44 × 1018 atoms/m2
(b) Computation of the junction depth requires that we use Equation 5.13.
However, before this is possible it is necessary to calculate D at the temperature of the drive-in treatment [Dd at 1100°C (1373 K)]. Thus,
Dd = 12.4 × 10 — 3 m2/s 2 exp c—
3.87 eV/atom
d
18.62 × 10 — 5 eV/atom # K 2 11373 K 2
= 1.51 × 10—17 m2/s
Now, from Equation 5.13,
xj = £ 14Dd td 2 ln a
1/2
Q0
CB 2pDd td
bd
= • 142 11.51 × 10 — 17 m2/s 2 17200 s 2 ×
ln £
3.44 × 1018 atoms/m2
11 × 1020 atoms/m 32 21p2 1 1.51 × 10 —17 m 2/s 217200s2
1/2
§¶
= 2.19 × 10—6 m = 2.16 µm
(c) At x = 1 µm for the drive-in treatment, we compute the concentration of
B atoms using Equation 5.11 and values for Q0 and Dd determined previously as follows:
C1x,t 2 =
Q0
2pDd t
=
exp a —
x2
b
4D dt
11 × 10 — 6 m 2 2
3.44 × 1018 atoms/m2
exp c —
d
142 11.51 × 10 — 17 m2/s 2 17200 s 2
21p2 11.51 × 10 — 17 m2/s 2 17200 s 2
= 5.90 × 1023 atoms/m3
MATERIALS OF IMPORTANCE
Aluminum for Integrated Circuit Interconnects
S
ubsequent to the predeposition and drive-in
heat treatments just described, another important step in the IC fabrication process is the
deposition of very thin and narrow conducting
circuit paths to facilitate the passage of current
from one device to another; these paths are called
5.6 DiffUsion in SemicondUcting Materials • 141
interconnects, and several are shown in Figure 5.11,
a scanning electron micrograph of an IC chip. Of
course the material to be used for interconnects
must have a high electrical conductivity—a metal,
because, of all materials, metals have the highest
conductivities. Table 5.3 cites values for silver,
copper, gold, and aluminum, the most conductive
metals. On the basis of these conductivities, and
discounting material cost, Ag is the metal of
choice, followed by Cu, Au, and Al.
Once these interconnects have been deposited, it is still necessary to subject the IC chip
to other heat treatments, which may run as high as
500°C. If, during these treatments, there is significant diffusion of the interconnect metal into the
silicon, the electrical functionality of the IC will be
destroyed. Thus, because the extent of diffusion is
dependent on the magnitude of the diffusion coefficient, it is necessary to select an interconnect
metal that has a small value of D in silicon. Figure 5.12 plots the logarithm of D versus 1/T for the
diffusion, into silicon, of copper, gold, silver, and
aluminum. Also, a dashed vertical line has been
constructed at 500°C, from which values of D for
the four metals are noted at this temperature. Here
it may be seen that the diffusion coefficient for aluminum in silicon (2.5 × 10—21 m2/s) is at least four
orders of magnitude (i.e., a factor of 104) lower than
the values for the other three metals.
Table 5.3 Room-TemperatUre Electrical CondUctivity ValUes for Silver, Copper,
Gold, and AlUminUm (the FoUr
Most CondUctive Metals)
Electrical Conductivity
[(ohm-meters)—1]
Metal
6.8 × 107
6.0 × 107
4.3 × 107
3.8 × 107
Silver
Copper
Gold
Aluminum
Aluminum is indeed used for interconnects
in some integrated circuits; even though its electrical conductivity is slightly lower than the values for silver, copper, and gold, its extremely low
diffusion coefficient makes it the material of
choice for this application. An aluminum–copper–
silicon alloy (94.5 wt% Al-4 wt% Cu-1.5 wt% Si)
is sometimes also used for interconnects; it not
only bonds easily to the surface of the chip, but
is also more corrosion resistant than pure aluminum.
More recently, copper interconnects have also
been used. However, it is first necessary to deposit
a very thin layer of tantalum or tantalum nitride
beneath the copper, which acts as a barrier to deter diffusion of Cu into the silicon.
Interconnects
Temperature (C)
1200 1000 900 800 700 600 500 400
4 µm
10–12
Cu in Si
Diffusion coefficient (m2/s)
Au in Si
10—14
4  10—13
2.5  10—15
Ag in Si
4.2  10—17
10—16
Al in Si
10—18
2.5  10—21
10—20
10—22
0.6
Figure 5.11 Scanning electron micrograph of an
integrated circuit chip, on which is noted aluminum
interconnect regions. Approximately 2000×.
(Photograph courtesy of National Semiconductor
Corporation.)
0.8
1.0
1.2
Reciprocal temperature (1000/K)
1.4
Figure 5.12 Logarithm of D-versus-1/T (K) curves
(lines) for the diffusion of copper, gold, silver, and
aluminum in silicon. Also noted are D values at
500°C.
142 • Chapter 5 / DiffUsion
5.7 OTHER DIFFUSION PATHS
Atomic migration may also occur along dislocations, grain boundaries, and external surfaces. These are sometimes called “short-circuit” diffusion paths inasmuch as
rates are much faster than for bulk diffusion. However, in most situations short-circuit
contributions to the overall diffusion flux are insignificant because the crosssectional areas of these paths are extremely small.
SUMMARY
Introduction
• Solid-state diffusion is a means of mass transport within solid materials by stepwise atomic motion.
• The term interdiffusion refers to the migration of impurity atoms; for host atoms,
the term self-diffusion is used.
Diffusion Mechanisms
• Two mechanisms for diffusion are possible: vacancy and interstitial.
Vacancy diffusion occurs via the exchange of an atom residing on a normal
lattice site with an adjacent vacancy.
For interstitial diffusion, an atom migrates from one interstitial position to
an empty adjacent one.
• For a given host metal, interstitial atomic species generally diffuse more rapidly.
Steady-State Diffusion
• Diffusion flux is defined in terms of mass of diffusing species, cross-sectional area,
and time according to Equation 5.1a.
• Concentration profile is represented as a plot of concentration versus distance
into the solid material.
• Concentration gradient is the slope of the concentration profile curve at some
specific point.
• The diffusion condition for which the flux is independent of time is known as
steady state.
• For steady-state diffusion, diffusion flux is proportional to the negative of the
concentration gradient according to Fick’s first law, Equation 5.3.
• The driving force for steady-state diffusion is the concentration gradient (dC/dx).
Nonsteady-State Diffusion
• For nonsteady-state diffusion, there is a net accumulation or depletion of diffusing
species and the flux is dependent on time.
• The mathematics for nonsteady state in a single (x) direction (and when the
diffusion coefficient is independent of concentration) are described by Fick’s
second law, Equation 5.4b.
• For a constant surface composition boundary condition, the solution to Fick’s
second law (Equation 5.4b) is Equation 5.5, which involves the Gaussian error
function (erf).
Factors That Influence Diffusion
• The magnitude of the diffusion coefficient is indicative of the rate of atomic motion and depends on both host and diffusing species as well as on temperature.
• The diffusion coefficient is a function of temperature according to Equation 5.8.
SUmmary • 143
Diffusion in Semiconducting Materials
• The two heat treatments that are used to diffuse impurities into silicon during
integrated circuit fabrication are predeposition and drive-in.
During predeposition, impurity atoms are diffused into the silicon, often from
a gas phase, the partial pressure of which is maintained constant.
For the drive-in step, impurity atoms are transported deeper into the silicon
so as to provide a more suitable concentration distribution without increasing the overall impurity content.
• Integrated circuit interconnects are normally made of aluminum—instead of metals such as copper, silver, and gold that have higher electrical conductivities—on
the basis of diffusion considerations. During high-temperature heat treatments,
interconnect metal atoms diffuse into the silicon; appreciable concentrations will
compromise the chip’s functionality.
EqUation SUmmary
Equation
Number
Equation
M
At
5.1a
J=
5.3
J = —D
5.4b
0C
02 C
=D 2
0t
0x
5.5
5.8
Cx — C0
Cs — C0
Page
Number
Solving for
dC
dx
= 1—erf a
x
Diffusion flux
126
Fick’s first law—diffusion flux for steady-state diffusion
127
Fick’s second law—for nonsteady-state diffusion
128
b Solution to Fick’s second law—for constant surface composition
129
2 2Dt
D = D exp a—
0
Qd
b
RT
Temperature dependence of diffusion coefficient
133
List of Symbols
Symbol
A
C
C0
Cs
Cx
D
D0
M
Qd
R
t
x
Meaning
Cross-sectional area perpendicular to direction of diffusion
Concentration of diffusing species
Initial concentration of diffusing species prior to the onset of the diffusion process
Surface concentration of diffusing species
Concentration at position x after diffusion time t
Diffusion coefficient
Temperature-independent constant
Mass of material diffusing
Activation energy for diffusion
Gas constant (8.31 J/mol # K)
Elapsed diffusion time
Position coordinate (or distance) measured in the direction of diffusion,
normally from a solid surface
144 • Chapter 5 / DiffUsion
Processing/StructUre/Properties/Performance SUmmary
Diffusion in semiconducting materials was discussed in Section 5.6. For both predeposition and drive-in treatments, diffusion is nonsteady-state—solutions to
Fick’s second law were provided for both. Nonsteady-state diffusion and these
treatments are two of the processing components for silicon, as noted in the
following diagram:
Silicon
(Processing)
Nonsteady-state diffusion
(Chapter 5)
Diffusion in semiconductors
(Chapter 5)
In the design of heat treatments to be used for introducing impurities into
semiconductors (i.e., doping, Chapter 18), and, in addition, in the production of
steel alloys (Chapter 10), an understanding of the temperature dependence of the
diffusion coefficient (i.e., Equation 5.8) is essential. The following diagrams illustrate the preceding relationships for these two materials.
Silicon
(Processing)
Iron–Carbon
Alloys (Steels)
(Processing)
Temperature dependence
of diffusion coefficient
(Chapter 5)
Temperature dependence
of diffusion
(Chapter 5)
Diffusion in semiconductors
(impurity doping)
(Chapters 5 and 18)
Isothermal transformation
diagrams
(Chapter 10)
Tempering (tempered
martensite)
(Chapter 10)
Important Terms and Concepts
activation energy
carburizing
concentration gradient
concentration profile
diffusion
diffusion coefficient
diffusion flux
driving force
Fick’s first and second laws
interdiffusion (impurity diffusion)
interstitial diffusion
nonsteady-state diffusion
self-diffusion
steady-state diffusion
vacancy diffusion
REFERENCES
Carslaw, H. S., and J. C. Jaeger, Conduction of Heat
in Solids, 2nd edition, Oxford University Press,
Oxford, 1986.
Crank, J., The Mathematics of Diffusion, Oxford
University Press, Oxford, 1980.
Gale, W. F., and T. C. Totemeier (Editors), Smithells
Metals Reference Book, 8th edition, ButterworthHeinemann, Woburn, UK, 2003.
Glicksman, M., Diffusion in Solids, WileyInterscience, New York, 2000.
Shewmon, P. G., Diffusion in Solids, 2nd edition,
The Minerals, Metals and Materials Society,
Warrendale,
PA,