Page 187 - Materials Science and Engineering An Introduction

Summary • 159

Fick’s First Law • Diffusion flux is defined in terms of mass of diffusing species, cross-sectional area, and
time according to Equation 5.1.
• Diffusion flux is proportional to the negative of the concentration gradient according
to Fick’s first law, Equation 5.2.
• 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.
• The driving force for steady-state diffusion is the concentration gradient (dC/dx).

Fick’s Second Law— • For nonsteady-state diffusion, there is a net accumulation or depletion of diffusing
Nonsteady-State species, and the flux is dependent on time.
Diffusion • 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 second
law (Equation 5.4b) is Equation 5.5, which involves the Gaussian error function (erf).

Factors That • The magnitude of the diffusion coefficient is indicative of the rate of atomic motion
Influence Diffusion 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.

Diffusion in • The two heat treatments that are used to diffuse impurities into silicon during inte-
Semiconducting grated circuit fabrication are predeposition and drive-in.
Materials 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 diffu-
sion considerations. During high-temperature heat treatments, interconnect metal atoms
diffuse into the silicon; appreciable concentrations will compromise the chip’s functionality.

Equation Summary

Equation Page
Number Equation Solving For Number
M
5.1 J = Diffusion flux 143
At
dC
5.2 J = -D Fick’s first law 143
dx
2
0C 0 C
5.4b = D Fick’s second law 145
0t 0x 2
5.5 C x - C 0 = 1 - erfa x b Solution to Fick’s second law—for constant surface composition 146
C s - C 0 21Dt
Q d
5.8 D = D 0 expa - b Temperature dependence of diffusion coefficient 149
RT