WHY STUDY Imperfections in Solids?
3. The time is taken to be zero the instant before the diffusion process begins
5.7 OTHER DIFFUSION PATHS
Summary • 139
D for the four metals are noted at this temperature.
Here it may be seen that the diffusion coefficient for aluminum in silicon (3.6 × 10−26 m2/s) is at least eight orders of magnitude (i.e., a factor of 108) lower than the values for the other three metals.
Aluminum is indeed used for interconnects in some integrated circuits; even though its electrical conductiv- ity 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 be- neath the copper, which acts as a barrier to deter dif- fusion of copper into the silicon.
Table 5.3 Room-Temperature Electrical Conductivity Values for Silver, Copper, Gold, and Aluminum (the Four Most Conductive Metals)
Electrical Conductivity Metal [(ohm-m)−1]
Silver 6.8 × 107
Copper 6.0 × 107
Gold 4.3 × 107
Aluminum 3.8 × 107
Figure 5.11 Logarithm of D-versus-1/T (K) curves (lines) for the diffusion of copper, gold, silver, and alu- minum in silicon. Also noted are D values at 500°C.
1200 1000 800 700 Temperature (°C)
Cu in Si Au in Si
Ag in Si
Al in Si
600 500 400
0.6
× 10–3 0.8
× 10–3 1.0
× 10–3
Reciprocal temperature (1/K) Diffusion coefficient (m2/s)
1.2
× 10–3 1.4
× 10–3 10–28
10–24 10–20 10–16
7.1 × 10–10
2.8 × 10–14
6.9 × 10–18
3.6 × 10–26 10–12
10–8
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 contribu- tions to the overall diffusion flux are insignificant because the cross-sectional areas of these paths are extremely small.
• Diffusion flux is proportional to the negative of the concentration gradient according to Fick’s first law, Equation 5.2.
• The diffusion condition for which the flux is independent of time is known as steady state.
• The driving force for steady-state diffusion is the concentration gradient (dC/dx).
• 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) may be described by Fick’s second law, Equation 5.4b.
• For a constant surface composition boundary condition, the solution to Fick’s sec- ond law (Equation 5.4b) is Equation 5.5, which involves the Gaussian error func- tion (erf).
• 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.
• The two heat treatments that are used to diffuse impurities into silicon during inte- grated 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.
Fick’s First Law
Fick’s Second Law—
Nonsteady-State Diffusion
Factors That Influence Diffusion
Diffusion in Semiconducting Materials
Equation Summary
Equation
Number Equation Solving For
5.1 J= M
At Diffusion flux
5.2 J= −D dC
dx Fick’s first law
5.4b ∂C
∂t =D ∂2C
∂x2 Fick’s second law 5.5 Cx−C0
Cs−C0
=1−erf( x
2√Dt) Solution to Fick’s second law—for constant surface composition 5.8 D=D0exp(−Qd
RT) Temperature dependence of diffusion coefficient
References • 141
List of Symbols
Symbol Meaning
A Cross-sectional area perpendicular to direction of diffusion C Concentration of diffusing species
C0 Initial concentration of diffusing species prior to the onset of the diffusion process
Cs Surface concentration of diffusing species Cx Concentration at position x after diffusion time t D Diffusion coefficient
D0 Temperature-independent constant M Mass of material diffusing
Qd Activation energy for diffusion R Gas constant (8.31 J/mol·K)
t Elapsed diffusion time
x Position coordinate (or distance) measured in the direction of diffusion, normally from a solid surface
Important Terms and Concepts
activation energy carburizing
concentration gradient concentration profile diffusion
diffusion coefficient
diffusion flux driving force Fick’s first law Fick’s second law
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, Elsevier Butterworth- Heinemann, Oxford, 2004.
Glicksman, M., Diffusion in Solids, Wiley-Interscience, New York, 2000.
Shewmon, P. G., Diffusion in Solids, 2nd edition, The Minerals, Metals and Materials Society, Warrendale, PA, 1989.
F
igure (a) shows an apparatus that measures the mechanical properties of metals using applied tensile forces (Sections 6.3, 6.5, and 6.6). Figure (b) was generated from a tensile test performed by an apparatus such as this on a steel specimen. Data plotted are stress (vertical axis—a measure of applied force) versus strain (horizontal axis—related to the degree of specimen elongation).The mechanical properties of modulus of elasticity (stiffness, E), yield strength (𝜎y), and tensile strength (TS) are determined as noted on these graphs.
Figure (c) shows a suspension bridge. The weight of the bridge deck and automobiles imposes tensile forces on the vertical suspender cables. These forces are transferred to the main suspension cable, which sags in a more-or-less parabolic shape. The metal alloy(s) from which these cables are constructed must meet certain stiffness and strength criteria. Stiffness and strength of the alloy(s) may be assessed from tests performed using a tensile-testing apparatus (and the resulting stress–strain plots) similar to those shown.
C h a p t e r 6 Mechanical Properties of Metals
0 0 1000
Stress (MPa)
2000
0
0.000 0.040
0.010
Strain
Strain
0.080
Stress (MPa)
2000
1000 TS
E
𝜎y
© Mr. Focus/iStockphoto
(a)
(b)
(c)
Model H300KU Universal Testing Machine by Tinius Olsen
142 •
It is incumbent on engineers to understand how the vari- ous mechanical properties are measured and what these properties represent; they may be called upon to design structures/components using predetermined materials such that unacceptable levels of deformation and/or failure
will not occur. In Design Examples 6.1 and 6.2 we pre- sent two typical types of design protocols; these examples demonstrate, respectively, a procedure used to design a tensile-testing apparatus and how material requirements may be determined for a pressurized cylindrical tube.