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MICROLITHOGRAPHY

9.24 WAFER PROCESSING

the high clockspeed bins on the final test of the device, can provide significant revenue improve-
ments for the fab.
The preceding discussion describes why CD control is important, using the gate layer of a CMOS
process as an example. But how can CD control be improved? What are the root causes of CD errors?
What is the best approach to attacking the problem? Fundamentally, errors in the final dimension of
a feature are the result of errors in the tools, processes, and materials that affect the final CD. An
error in a process variable (the temperature of a hotplate, for example) propagates through to become
an error in the final CD based on the various physical mechanisms by which the variable influences
the lithographic result. In such a situation, a propagation of errors analysis can be used to help under-
stand the effects. Suppose the influence of each input variable on the final CD were expressed in a
mathematical form, such as
CD = f(v , v , v , . . .) (9.23)
1 2 3
where v are the input (process) variables.
i
Given an error in each process variable ∆v , the resulting CD error can be computed from a Taylor
i
expansion of the function in Eq. (9.23)
∞  ∂ ∂  n
∆CD = ∑  ∆v 1 + ∆v 2 + . . .  fv v , . . .) (9.24)
(,
1
2
n =1  ∂v 1 ∂v 2 
This imposing looking summation of powers of derivatives can be simplified if the function is
reasonably well behaved (and of course we hope that our critical features will be so) and the errors
in the process variables are small (we hope this is true as well). In such a case, it may be possible to
ignore the higher-order terms (n > 1), as well as the cross terms of Eq. (9.24), to leave a simple, lin-
ear error equation
∂ CD ∂ CD
∆CD = ∆v + ∆v + . . . (9.25)
1
∂v 1 2 ∂v 2
Each ∆v represents the magnitude of a process error. Each partial derivative ∂CD/∂v represents
i i
the process response, the response of CD to an incremental change in the variable. This process
response can be expressed in many forms; for example, the inverse of the process response is called
process latitude.
The linear error equation (Eq. (9.25)) can be modified to account for the nature of the errors at
hand. In general, CD errors are specified as a percentage of the nominal CD. For such a case, the goal
is usually to minimize the relative CD error ∆CD/CD. Equation (9.25) can be put in this form as

∆CD ∂ln CD ∂ln CD
= ∆v + ∆v + . . . (9.26)
CD 1 ∂v 2 ∂v
1 2
Also, many sources of process errors result in errors that are a fraction of the nominal value of
that variable (e.g., illumination nonuniformity in a stepper produces a dose error that is a fixed per-
centage of the nominal dose). For such error types, it is best to modify Eq. (9.26) to use a relative
process error ∆v /v
i i
∆CD ∆v ∂ln CD ∆v ∂ln CD
= 1 + 2 + . . . (9.27)
CD v ∂ln v v ∂ln v
1 1 2 2
Although Eqs. (25) to (27) may seem obvious, even trivial, in their form, they reveal a very
important truth about error propagation and the control of CD. There are two distinct ways to reduce

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