Tennant’s Law, Part 2

In the first part of this article, I talked about the empirically determined Tennant’s Law:  the areal throughput (A_t) of a direct-write lithography system is proportional to the resolution (R) to the fifth power.  In mathematical terms,

A_t = k_T R⁵

where k_T is Tennant’s constant, and was equal to about 4.3 nm^-3 s^-1 in 1995 according to the data Don Tennant collected [1].  The power of 5 comes from two sources:  (1) areal throughput is equal to the pixel throughput times R², and (2) pixel throughput is proportional to the volume of a voxel (a three-dimensional pixel), R³.  The first part is a simple geometrical consideration:  for a given time required to write one pixel, doubling the number of pixels doubles the write time.  It’s the second part that fascinates me:  the time to write a voxel is inversely proportional to the volume of the voxel.  It takes care to write something small, and it’s hard to be careful and fast at the same time.

The implication for direct write is clear:  the economics of writing small features is very bad.  Granted, Tennant’s constant increases over time as our technology improves, but it has never increased nearly fast enough for high resolution direct-write lithography to make up for the R⁵ deficit.

But does Tennant’s Law apply to optical lithography?  Yes, and no.  Unlike direct-write lithography, in optical lithography we write a massive number of pixels at once:  parallel processing versus serial processing.  That makes Tennant’s constant very large (and that’s a good thing), but is the scaling any different?

For a given level of technology, the number of pixels that can fit into an optical field of a projection lens is roughly constant.  Thus, a lower-resolution lens with a large field size can be just as difficult to make as a higher-resolution lens with a small field size if the number of pixels in the lens field is the same.  That would give an R² dependence to the areal throughput, just like for direct write (though, again, Tennant’s constant will be much larger for projection printing).

But is there a further R³ dependence to printing small pixels for optical projection printing, just as in electron-beam and other direct-write technologies?  Historically, the answer has been no.  Thanks to significant effort by lithography tool companies (and considerable financial incentives as well), the highest resolution tools also tend to have the fastest stages, wafer handling, and alignment systems. And historically, light sources have been bright, so that resist sensitivity (especially for chemically amplified resists) has not been a fundamental limit to throughput.

But all that is changing as the lithography community looks to Extreme UV (EUV) lithography.  EUV lithography has the highest resolution, but it is slow.  Painfully slow, in fact.  Our EUV sources are not bright, so resist sensitivity limits throughput.  And resist sensitivity for EUV, as it turns out, is a function of resolution.

For some time now, researchers in the world of EUV lithography have been talking about the “RLS trade-off”:  the unfortunate constraint that it is hard to get simultaneously high resolution (R), low line-edge roughness (L), and good resist sensitivity (S, the dose required to properly expose the resist).  Based on scaling arguments and empirical evidence, Tom Wallow and coworkers have found that, for a given level of resist technology [2],

R³ L² S = constant

Since throughput is limited by the available intensity from the EUV source, we find that, for a fixed amount of LER,

Throughput ~ 1/S ~ R³

Finally, since the number of pixels in a lens field is fixed, the areal throughput will be

A_t ~ (lens field size) R³ ~ R⁵

Thus, like direct-write lithographies, EUV lithography obeys Tennant’s law.  This is bad news.  This means that EUV will suffer from same disastrous economics as direct-write lithography:  shrinking the feature size by a factor of 2 produces a factor of 32 lower throughput.

Ah, but the comparison is not quite fair.  For projection lithography, Tennant’s constant is not only large, it increases rapidly over time.  Tim Brunner first noted this in what I call Brunner’s Corollary [3]:  over time, optical lithography tends to increase Tennant’s constant at a rate that more than makes up for the R² dependence of the lens field size.  As a result, optical lithography actually increases areal throughput while simultaneously improving resolution for each new generation of technology.  Roughly, it seems that Tennant’s constant has been inversely proportional to about R^2.5 as R shrank with each technology node.

But that was before EUV, and before the R³ dependence of the RLS trade-off kicked in.  At best, we might hope for an effective Tennant’s law over time that sees throughput go as R².  This is still very bad.  This means that for every technology node (when feature sizes shrink by 70%) we’ll need our source power to double just to keep the throughput constant.  The only way out of this dilemma is to break the RLS “triangle of death” so that resolution can improve without more dose and worse LER.

Is the RLS trade-off breakable?  Can LER be lowered without using more dose?  This is a topic receiving considerable attention and research effort today.  We’ll have to stay tuned over the next few years to find out.  But for all the risks involved with EUV lithography for semiconductor manufacturing , we can add one more:  Tennant’s law.

[1]  Donald M. Tennant, Chapter 4, “Limits of Conventional Lithography”, in Nanotechnology, Gregory Timp Ed., Springer (1999) p. 164.

[2]  Thomas Wallow, et al., “Evaluation of EUV resist materials for use at the 32 nm half-pitch node”, Proc. SPIE 6921, 69211F (2008).

[3]  T. A. Brunner, “Why optical lithography will live forever”, JVST B 21(6), p. 2632 (2003).