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DUANE BONING: OK, so today
is a little bit different.

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00:00:24,240 --> 00:00:26,220
Today we're going to talk
about yield modeling.

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And this is
unabashedly connected

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to semiconductor manufacturing--

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00:00:31,932 --> 00:00:33,390
although I think
many of the things

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I talk about here are more
widely applicable, especially

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to anything that's large
area manufacturing.

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So for example, a few years ago,
I did a roll coding process,

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rolling out these giant
plastic sheets of film material

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at Kodak.

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And defect modeling,
and the yield of--

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on a per area basis was
a big deal there as well.

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00:00:59,530 --> 00:01:01,750
Similarly, many of
the MEMS processes,

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thin film processes, yield
modeling associated especially

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00:01:06,120 --> 00:01:08,310
with defects is very important.

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Many of the other ideas
I'll be talking about here

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00:01:10,470 --> 00:01:14,160
in this lecture
are also connected

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to the idea of
assemblies or systems

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that consist of many,
many, many, many parts,

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00:01:21,150 --> 00:01:26,190
whether it may be a failure
of probability or deviations

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00:01:26,190 --> 00:01:27,270
in individual parts.

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And part of the question
is how those aggregate

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across the whole system.

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So all of the examples here
will be pretty much drawn

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00:01:35,045 --> 00:01:36,420
from semiconductor
manufacturing,

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00:01:36,420 --> 00:01:39,450
but I think they are
more broadly applicable.

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And perhaps many of the
tools actually developed here

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00:01:42,510 --> 00:01:47,730
in semiconductor manufacturing
can be used and propagate

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00:01:47,730 --> 00:01:50,290
to other processes.

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00:01:50,290 --> 00:01:55,170
So the material will mostly
be drawn from chapter 5,

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00:01:55,170 --> 00:01:58,900
so on-- need to add a note
as a reading assignment.

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00:01:58,900 --> 00:02:02,700
This is drawn from
[INAUDIBLE] chapter 5.

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00:02:02,700 --> 00:02:04,380
But I'm also showing
some examples

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00:02:04,380 --> 00:02:08,340
from a couple of other papers,
and I'll put those papers

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on the website as well.

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I realized this morning
they're not up yet.

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One is a paper--

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00:02:15,420 --> 00:02:17,850
sort of a classic
paper by Stapper

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00:02:17,850 --> 00:02:22,320
on integrated circuit yield
management, yield analysis--

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00:02:22,320 --> 00:02:24,570
and then a more recent one--

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00:02:24,570 --> 00:02:27,210
I guess somewhat more recent--

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00:02:27,210 --> 00:02:30,420
2000-- on predictive
yield modeling.

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00:02:30,420 --> 00:02:32,730
So both of those will be
available on the website.

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00:02:35,540 --> 00:02:37,060
So we've already
talked a little bit

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about some of the kinds of
variations that lead to--

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ultimately can lead to failures,
or failures in specification.

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When a deviation in some
continuous parameter

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exceeds some spec limit
for normal operation,

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00:02:53,380 --> 00:02:56,300
we can think of those
as parametric failures.

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00:02:56,300 --> 00:02:58,240
So we've talked about
things like line width

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or so on that might lead
to either direct functional

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failure--

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meaning it really won't work
because the parameter's too far

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away--

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or a more fuzzy failure--

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this continuous quality
loss in performance,

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meaning that I've
gotten such a deviation,

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the thing might still
turn on or might operate,

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but it does so with decayed
or degraded performance.

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In addition to
that, we also want

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to talk about random failures.

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And these are generally
thought of as more

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uncoordinated-- or uncorrelated
random failure in some element.

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They aren't necessarily due
to some continuous parameter,

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but maybe more of
a lumped failure.

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We'll talk in
particular about area

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dependent kinds of failures.

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In semiconductor manufacturing,
the main source of these

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are point defects associated
with very small particles,

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dust, or debris that
interfere with the operation

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00:04:07,690 --> 00:04:11,440
of an electrical
element, generally.

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And we previewed some of those
in one of the first lectures.

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So we're going to talk about
these kinds of defects.

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So the key idea in these is many
of the area-dependent failures

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are ones where, depending
on the total square area

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of your circuit, you have
more or less opportunity

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for those kinds of failures.

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So it becomes an
interesting problem

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in terms of analyzing the
probabilities associated

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with failure for
different sized circuits.

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And we'll talk about those.

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So here's an example I pulled
out of the [INAUDIBLE] paper

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00:04:53,560 --> 00:04:56,560
on an integrated
circuit yield tree--

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so looking at, say, 100 ASIC
chips that are manufactured,

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00:05:02,080 --> 00:05:05,500
and what the breakdown
of those might

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00:05:05,500 --> 00:05:08,620
be in terms of
their ultimate fate.

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00:05:08,620 --> 00:05:13,930
And in this particular
process, out of the 100 chips,

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00:05:13,930 --> 00:05:16,820
about 30 are ultimately
shipped to the customer.

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So in some sense, you've
got a yield of 30%--

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00:05:20,170 --> 00:05:26,640
not great, but may not
be entirely unrealistic.

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And then, within that 30
chips shipped to the customer,

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there's already some
parametric variation

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00:05:35,430 --> 00:05:41,130
going on that is
essentially a reflection

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of that kind of quality
loss degradation

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that we talked about before.

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Typically, chips are tested,
and we'll do a breakdown,

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00:05:56,340 --> 00:05:58,960
give you a feel for the
kinds of tests that are done.

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00:05:58,960 --> 00:06:01,740
But at the end,
they're often tested

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for a few key
performance parameters--

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in particular, speed.

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And then a thing called
speed binning is done,

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and you can see here, binning
down into three different speed

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00:06:16,710 --> 00:06:20,490
categories, where you've
got a few that are operating

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at 400 megahertz that--
presumably, you can sell

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00:06:22,950 --> 00:06:25,290
those chips a little bit more.

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00:06:25,290 --> 00:06:28,230
350 megahertz you might not
have quite the same price

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00:06:28,230 --> 00:06:32,580
premium on, and maybe you
have to sell with almost no--

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00:06:32,580 --> 00:06:35,780
or very limited profit
the 300 megahertz chips,

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00:06:35,780 --> 00:06:37,600
or something like that.

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So there's already still the
driver and process control

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to get as tight
control as you can

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and push the speed limit
as much as you can.

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But what we want to talk
about especially today

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are some of the sources for the
chips and sources of variation

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and kinds of failures
that are affecting

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the chips that are rejected.

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And we can break those
down here on this chart

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00:07:01,950 --> 00:07:04,560
into these other categories.

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We've got growth functional
fail, unrepairable cache, speed

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less than 300 megahertz,
and all others.

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So out of these, first off, the
speed less than 300 megahertz--

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00:07:15,090 --> 00:07:17,280
that's just our
cut-off on the speed,

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and that's probably more
of a parametric variation.

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We can break that down and
start to look at what devices,

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or what components, or what's
responsible for the slowness

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perhaps to improve either
design or manufacturing control.

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And here, for example, the clock
speed perhaps is a little bit

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too slow, either because
of the interconnect--

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the interconnect delay might
be a little bit too long--

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or because the transistors,
the active devices

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are not quite strong enough.

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Knowing those two
could tell you a lot

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about the sources of variation.

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For example, if
it's interconnect,

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it's probably something
with your back-end process.

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Some of your resistances
in the interconnect wires

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might be a little too high, or
some of those capacitances--

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whereas, if you've got an active
device strength failures--

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the devices are too slow--

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that's likely something to
do with, say, channel length,

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or perhaps gate oxide
thicknesses a little too thick.

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So it can start to lead to good
knowledge that can give rise

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to some improvement efforts.

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Now, a little bit more
interesting for today

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are these two other categories--
this gross functional

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fail and unrepairable cache.

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The unrepairable cache--
this might be an ASIC chip,

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and it might have different
components or different regions

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on it.

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Some of it may be
random logic performing

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particular combinatorial or
combination logic functions.

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But another component is
likely to be embedded memory.

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And in fact, as we get to
larger and larger chips

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and integration scale, most
of that additional area

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these days is going to memory.

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So I don't know what percentage
of the new 2 billion transistor

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Intel chip is cache, but it's--

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AUDIENCE: 90%--

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DUANE BONING: 90% is cache--

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something you can
do with the area

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that helps with performance.

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But very interesting
issue here is

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these are among the most
dense, most tightly packed

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and smallest scaled structures--

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highly repeated transistor SRAM
cells, little memory cells.

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But there you're
already expecting--

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and we'll talk about the
sources of some of these--

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you're already expecting some
number of those memory cells

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to fail, perhaps because of
particle-oriented defects.

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And so one builds in a
certain amount of redundancy

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into the cache so
that one can detect--

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or into the memory so one can
detect particular failed cells

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and program in or fold
in additional redundant

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capability.

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So that's the repair, the
direct repair of the cache.

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But at some point, depending
on where the failure is,

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you may not-- you may have
too many of those failures,

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or you may fail perhaps even in
some of the redundant switching

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in the circuitry.

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So you get to a
point where you can't

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repair all of those caches.

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And so you might start to
look then inside and start

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trying to say, OK, what
are the sources there?

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Some of those you
might not know--

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00:10:55,950 --> 00:10:58,080
these unobservable root causes.

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Others-- a big chunk maybe due
to these points defects landing

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00:11:02,130 --> 00:11:05,760
in particularly
damaging locations.

195
00:11:05,760 --> 00:11:08,740
And similarly, if you then
look at growth functional fail,

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you just try to-- you
can't even test the memory.

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The chip simply won't
power up or won't operate.

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00:11:15,330 --> 00:11:19,230
One might then also do different
kinds of inspection approaches

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00:11:19,230 --> 00:11:22,690
to try to detect what the
source of those errors are.

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And again, random
defects are typically

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a really large part of that.

202
00:11:29,160 --> 00:11:31,380
Systematic defect we'll talk
about a little bit later

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00:11:31,380 --> 00:11:32,160
as well.

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00:11:32,160 --> 00:11:35,730
Those might be not quite
random point defects,

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00:11:35,730 --> 00:11:41,890
but some other failure that's
not quite a parametric failure,

206
00:11:41,890 --> 00:11:45,630
but it's something that's
affecting an awful lot

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00:11:45,630 --> 00:11:47,190
of the components all together.

208
00:11:47,190 --> 00:11:49,350
And a typical
example here might be

209
00:11:49,350 --> 00:11:52,830
things like overlay
between different layers

210
00:11:52,830 --> 00:11:54,310
of the process.

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00:11:54,310 --> 00:11:56,310
So everything's just
a little bit off

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00:11:56,310 --> 00:12:00,450
set in terms of the alignment
from one layer to the next,

213
00:12:00,450 --> 00:12:08,790
causing a substantial yield loss
in lots and lots of components

214
00:12:08,790 --> 00:12:11,060
together.

215
00:12:11,060 --> 00:12:16,300
So what we've bolded here
in that big black rectangle

216
00:12:16,300 --> 00:12:17,860
are these random defects.

217
00:12:17,860 --> 00:12:20,350
And that's one of
the key elements

218
00:12:20,350 --> 00:12:23,530
I want to talk about here are
these point defects associated

219
00:12:23,530 --> 00:12:27,430
with particle-oriented
problems, and how

220
00:12:27,430 --> 00:12:32,360
to model the impact
of these basically

221
00:12:32,360 --> 00:12:34,280
from a statistical
point of view.

222
00:12:34,280 --> 00:12:38,330
But getting back to the
sources or the mental picture

223
00:12:38,330 --> 00:12:42,710
for these kinds of
defects, there's a--

224
00:12:42,710 --> 00:12:45,470
you can be a little bit more
careful with the terminology

225
00:12:45,470 --> 00:12:48,710
here and talk or differentiate--

226
00:12:48,710 --> 00:12:51,770
talk about the
difference between, say,

227
00:12:51,770 --> 00:12:54,720
particles and defects.

228
00:12:54,720 --> 00:12:57,950
So a particle-- you can think
of any kind of foreign matter

229
00:12:57,950 --> 00:12:59,840
that might be sitting
on the surface

230
00:12:59,840 --> 00:13:04,700
or might be embedded
in a layer on the chip.

231
00:13:04,700 --> 00:13:07,980
Now, some of these
might be benign,

232
00:13:07,980 --> 00:13:12,330
so a particle is not
necessarily a defect.

233
00:13:15,240 --> 00:13:17,250
A defect would be
when it affects

234
00:13:17,250 --> 00:13:20,520
some functionality of the chip.

235
00:13:20,520 --> 00:13:26,400
So a picture here, qualitatively
giving you this sense--

236
00:13:26,400 --> 00:13:29,040
we've got three
different dust particles

237
00:13:29,040 --> 00:13:33,870
on a particular feature on--

238
00:13:33,870 --> 00:13:35,880
lined up with particular
features on the mask.

239
00:13:35,880 --> 00:13:43,270
And one can easily imagine
that perhaps this particle,

240
00:13:43,270 --> 00:13:48,640
if it's conductive and those
crosshatch areas are also

241
00:13:48,640 --> 00:13:50,890
conductive-- or
say, metal lines--

242
00:13:50,890 --> 00:13:55,400
that's going to be a
problem, potentially.

243
00:13:55,400 --> 00:14:00,050
Actually, if someplace
else, these two things

244
00:14:00,050 --> 00:14:04,470
are the same wire, maybe
it's not a problem.

245
00:14:04,470 --> 00:14:06,920
So it's actually
kind of interesting.

246
00:14:06,920 --> 00:14:11,480
It depends on both
the particular layout

247
00:14:11,480 --> 00:14:15,650
and on the location
of a particle,

248
00:14:15,650 --> 00:14:19,280
whether it will lead to
degradation or functional

249
00:14:19,280 --> 00:14:20,840
failure.

250
00:14:20,840 --> 00:14:25,170
You can also imagine
perhaps this structure,

251
00:14:25,170 --> 00:14:30,770
this particle might be a defect.

252
00:14:30,770 --> 00:14:32,585
In what case would
this be a defect?

253
00:14:36,430 --> 00:14:38,260
Your initial
inclination might be,

254
00:14:38,260 --> 00:14:40,390
if I'm just looking
at this layer,

255
00:14:40,390 --> 00:14:43,910
it's not necessarily bridging
from this wire to that wire.

256
00:14:43,910 --> 00:14:46,482
So why would that be a problem?

257
00:14:46,482 --> 00:14:49,397
AUDIENCE: [INAUDIBLE]
capacitor then [INAUDIBLE]

258
00:14:49,397 --> 00:14:50,230
DUANE BONING: Right.

259
00:14:50,230 --> 00:14:52,960
So if it's conductive and
these two are conductive,

260
00:14:52,960 --> 00:14:56,230
I might have some additional
capacity of linking even

261
00:14:56,230 --> 00:14:58,720
within that one layer.

262
00:14:58,720 --> 00:15:02,450
But I'm trying to give
hints, saying one layer here.

263
00:15:02,450 --> 00:15:02,950
Yes?

264
00:15:02,950 --> 00:15:03,992
AUDIENCE: This direction.

265
00:15:03,992 --> 00:15:06,530
DUANE BONING: Yes,
in the z-direction--

266
00:15:06,530 --> 00:15:10,460
remember also, we've got
build up of many, many layers.

267
00:15:10,460 --> 00:15:12,670
Remember that stacking of
the interconnect layer.

268
00:15:12,670 --> 00:15:15,250
And even within the device
layer, there's many--

269
00:15:15,250 --> 00:15:19,570
so it can propagate
potentially bridge or short

270
00:15:19,570 --> 00:15:25,220
or cause deviations in the next
layer of processing as well.

271
00:15:25,220 --> 00:15:27,610
How about this last one here?

272
00:15:27,610 --> 00:15:28,390
Is that a problem?

273
00:15:32,120 --> 00:15:37,420
I'm seeing maybe,
yes, yes, probably.

274
00:15:37,420 --> 00:15:39,151
Why do you think
that's a problem?

275
00:15:39,151 --> 00:15:42,360
AUDIENCE: [INAUDIBLE] the
resistance on the [INAUDIBLE]??

276
00:15:42,360 --> 00:15:44,110
DUANE BONING: Well,
if it were conductive,

277
00:15:44,110 --> 00:15:47,590
it might not reduce
the resistance--

278
00:15:47,590 --> 00:15:50,070
or increase the resistance.

279
00:15:50,070 --> 00:15:57,120
AUDIENCE: [INAUDIBLE]

280
00:15:57,120 --> 00:15:58,330
DUANE BONING: Right.

281
00:15:58,330 --> 00:15:58,830
Yeah.

282
00:15:58,830 --> 00:16:01,050
So a couple of examples
there is certainly,

283
00:16:01,050 --> 00:16:03,540
if it's non-conductive,
what you've got now

284
00:16:03,540 --> 00:16:07,620
is increased resistance in
that segment right there,

285
00:16:07,620 --> 00:16:11,940
which can also cascade
to reliability problems.

286
00:16:11,940 --> 00:16:14,640
A well-known problem
is migration,

287
00:16:14,640 --> 00:16:19,710
where basically, the electron
flux of high current flowing

288
00:16:19,710 --> 00:16:22,920
very, very high current
density up in the 10

289
00:16:22,920 --> 00:16:26,160
to the fifth to 10 to the
sixth per centimeter squared

290
00:16:26,160 --> 00:16:29,550
kind of amps centimeter
squared can actually

291
00:16:29,550 --> 00:16:36,130
cause the metal atoms to move.

292
00:16:36,130 --> 00:16:38,910
So they will migrate
with the current flow--

293
00:16:38,910 --> 00:16:41,070
or with the electronic
flow, and you'll

294
00:16:41,070 --> 00:16:43,530
get-- you can very
often get voiding,

295
00:16:43,530 --> 00:16:46,530
especially in these
particular locations,

296
00:16:46,530 --> 00:16:49,020
where the wire gets
thinner and thinner,

297
00:16:49,020 --> 00:16:52,110
and ultimately, in
fact, may be an open.

298
00:16:54,750 --> 00:16:57,000
But again, maybe it's OK.

299
00:16:57,000 --> 00:16:59,550
Maybe you've got enough
latitude and you're not really

300
00:16:59,550 --> 00:17:01,990
pushing current through there.

301
00:17:01,990 --> 00:17:06,390
So one side message here is
you never want particles.

302
00:17:06,390 --> 00:17:08,250
You always want to
minimize the number

303
00:17:08,250 --> 00:17:12,480
of particles and the
opportunity for failure.

304
00:17:12,480 --> 00:17:13,960
But the other
message, of course,

305
00:17:13,960 --> 00:17:17,790
is how bad they
are kind of depend

306
00:17:17,790 --> 00:17:21,641
on particulars of your circuit
and your specifications.

307
00:17:21,641 --> 00:17:23,099
So we'll actually
talk a little bit

308
00:17:23,099 --> 00:17:24,690
about some of the
tools that have

309
00:17:24,690 --> 00:17:30,120
evolved to be able to analyze
some of those sorts of things.

310
00:17:30,120 --> 00:17:35,180
So I've talked a lot about opens
and-- or I guess short circuits

311
00:17:35,180 --> 00:17:35,870
here.

312
00:17:35,870 --> 00:17:38,990
You might lead to
an open circuit,

313
00:17:38,990 --> 00:17:42,300
in terms of losing
a conductive path,

314
00:17:42,300 --> 00:17:46,580
but there's also a lot of
other kinds of failures

315
00:17:46,580 --> 00:17:49,550
that-- where we're dust
particles or other defects

316
00:17:49,550 --> 00:17:52,670
might impact the
device operation so

317
00:17:52,670 --> 00:17:55,920
that you get failure.

318
00:17:55,920 --> 00:17:59,600
It could be, for example in
an active device, where you've

319
00:17:59,600 --> 00:18:03,530
got perturbation of
transistor parameters,

320
00:18:03,530 --> 00:18:05,840
not necessarily just
a short or an open.

321
00:18:08,910 --> 00:18:10,940
So what is yield?

322
00:18:13,830 --> 00:18:16,440
Very qualitatively,
yield is the percentage

323
00:18:16,440 --> 00:18:19,850
of parts meeting
some specification--

324
00:18:19,850 --> 00:18:21,560
set of specification.

325
00:18:21,560 --> 00:18:25,340
But what we often do
in IC yield terminology

326
00:18:25,340 --> 00:18:28,940
is look at different
points in the process--

327
00:18:28,940 --> 00:18:31,460
flow or different
points of testing--

328
00:18:31,460 --> 00:18:35,210
and we'll differentiate the
yield in some cases that way.

329
00:18:35,210 --> 00:18:39,020
And the other is we'll
sometimes break down

330
00:18:39,020 --> 00:18:45,950
what kind of yield losses
or what size of a bucket

331
00:18:45,950 --> 00:18:50,090
we're talking about for
thinking about yield.

332
00:18:50,090 --> 00:18:53,660
By size of a bucket, what
we've got in semiconductor

333
00:18:53,660 --> 00:18:56,600
manufacturing is a-- and many
other manufacturing process is

334
00:18:56,600 --> 00:18:58,970
a very hierarchical--

335
00:18:58,970 --> 00:19:02,660
spatially as well as
temperately-- structure.

336
00:19:02,660 --> 00:19:08,180
What I mean by that is we've got
within the fab many different

337
00:19:08,180 --> 00:19:09,740
lots of wafers.

338
00:19:09,740 --> 00:19:15,080
Each lot maybe is 25 wafers
being processed together.

339
00:19:15,080 --> 00:19:17,510
Within each lot, I've got--

340
00:19:17,510 --> 00:19:20,570
of those 25 wafers, I pull
out one wafer and I've got,

341
00:19:20,570 --> 00:19:24,480
what, 50 to thousands
of chips on it.

342
00:19:24,480 --> 00:19:27,845
So I can talk about
yield in terms of,

343
00:19:27,845 --> 00:19:32,190
well, what fraction of lots
make it through the line?

344
00:19:32,190 --> 00:19:36,120
It's possible I might scrap
a whole lot of wafers.

345
00:19:36,120 --> 00:19:40,560
Or what fraction of the wafers
within the lot make it through?

346
00:19:40,560 --> 00:19:44,460
I might have a
dropped wafer, and you

347
00:19:44,460 --> 00:19:49,560
might break it, or other kinds
of large-scale mechanical

348
00:19:49,560 --> 00:19:51,430
failure along the line.

349
00:19:51,430 --> 00:19:53,430
You're not going to
scrap the whole lot

350
00:19:53,430 --> 00:19:57,270
if one wafer breaks, but--

351
00:19:57,270 --> 00:20:01,320
so the actual percentage
of wafers that

352
00:20:01,320 --> 00:20:02,820
make it to the end of the line.

353
00:20:02,820 --> 00:20:05,520
And that's just,
mechanically, have

354
00:20:05,520 --> 00:20:07,410
the wafers made it to the end?

355
00:20:07,410 --> 00:20:09,130
Then you can start
looking and saying,

356
00:20:09,130 --> 00:20:10,920
OK, what fraction
of those wafers

357
00:20:10,920 --> 00:20:15,690
appear to be coarsely
are grossly within spec?

358
00:20:15,690 --> 00:20:17,400
And then similarly,
you start looking

359
00:20:17,400 --> 00:20:19,950
at the die and say,
which of the die

360
00:20:19,950 --> 00:20:26,290
are likely to be
able to function?

361
00:20:26,290 --> 00:20:29,650
What percentage of
die yield do I have?

362
00:20:29,650 --> 00:20:33,460
Before I go and I invest the
two hours of intense burn in

363
00:20:33,460 --> 00:20:35,590
and testing for
each of those chips,

364
00:20:35,590 --> 00:20:37,630
and packaging of
each of those chips,

365
00:20:37,630 --> 00:20:42,280
I might then also want to
do some on-chip electrical

366
00:20:42,280 --> 00:20:44,158
testing.

367
00:20:44,158 --> 00:20:45,700
And then, once it's
packaged, I might

368
00:20:45,700 --> 00:20:48,190
do a full functional
testing at speed

369
00:20:48,190 --> 00:20:52,130
to try to determine which
fraction of those are working.

370
00:20:52,130 --> 00:20:54,910
So we've got some of these
different terminologies here.

371
00:20:54,910 --> 00:20:57,670
So wafer yield is just--

372
00:20:57,670 --> 00:21:00,910
actually, let me go
to the next picture.

373
00:21:00,910 --> 00:21:06,460
This graphically defines some
of these different yields.

374
00:21:06,460 --> 00:21:10,510
We've drawn the
wafer fab itself,

375
00:21:10,510 --> 00:21:17,940
the sequence of the processing
steps shown up above.

376
00:21:17,940 --> 00:21:24,100
And very often, inline tests are
being performed all the time--

377
00:21:24,100 --> 00:21:28,740
not necessarily on every
piece of equipment or test

378
00:21:28,740 --> 00:21:32,550
of the wafer after
every individual fab

379
00:21:32,550 --> 00:21:36,090
step, but one will be making
measurements at various points

380
00:21:36,090 --> 00:21:40,140
along the flow to check on
the status of the wafer,

381
00:21:40,140 --> 00:21:42,720
as well as check on the
status of the equipment.

382
00:21:42,720 --> 00:21:46,080
One might also have some amount
of real-time measurements

383
00:21:46,080 --> 00:21:48,030
actually on the
equipment itself.

384
00:21:48,030 --> 00:21:50,760
And a whole
additional problem is

385
00:21:50,760 --> 00:21:52,650
correlating equipment
measurements

386
00:21:52,650 --> 00:21:55,020
to what's happening
on the wafer--

387
00:21:55,020 --> 00:21:57,420
that's especially
useful for debug,

388
00:21:57,420 --> 00:21:59,220
but it tends not
to be used directly

389
00:21:59,220 --> 00:22:01,450
for yield calculations.

390
00:22:01,450 --> 00:22:03,720
So it is possible that, as
you're coming along here,

391
00:22:03,720 --> 00:22:11,550
you may scrap out wafers based
on some of the inline tests.

392
00:22:11,550 --> 00:22:14,490
Then what's often referred
to as the back end--

393
00:22:14,490 --> 00:22:16,945
although it's kind of
confusing, because front end

394
00:22:16,945 --> 00:22:18,570
and back end can mean
different things,

395
00:22:18,570 --> 00:22:21,330
depending on who you're
talking to in an IC fab.

396
00:22:23,840 --> 00:22:26,570
If you're within the
IC fab, often they'll

397
00:22:26,570 --> 00:22:29,930
talk front end processing
as the transistor formation

398
00:22:29,930 --> 00:22:32,870
and back end as the
interconnect formation.

399
00:22:32,870 --> 00:22:39,080
But once you emerge out of the
fab itself into the testing,

400
00:22:39,080 --> 00:22:42,170
that's also referred
to as the back end.

401
00:22:42,170 --> 00:22:46,730
Once the wafer has
finished its fabrication,

402
00:22:46,730 --> 00:22:48,470
and is now going
both in the testing

403
00:22:48,470 --> 00:22:54,660
and then dicing up into
individual chips for packaging,

404
00:22:54,660 --> 00:22:57,200
that's also referred
to as the back end.

405
00:22:57,200 --> 00:22:58,670
So we've got some wafer fab.

406
00:22:58,670 --> 00:23:01,820
We've got wafer yield that
maybe those wafers that

407
00:23:01,820 --> 00:23:06,080
make it through the very
coarse electrical test.

408
00:23:06,080 --> 00:23:09,560
Then you do a more detailed
functional test on each die,

409
00:23:09,560 --> 00:23:12,770
and as pictured here, the
die yield or functional yield

410
00:23:12,770 --> 00:23:16,880
would be those fraction
that are making it through.

411
00:23:16,880 --> 00:23:20,600
Those set of a little bit more
elaborate functional tests--

412
00:23:20,600 --> 00:23:23,300
now you go ahead and
package up those--

413
00:23:23,300 --> 00:23:26,960
just those chips
that are successfully

414
00:23:26,960 --> 00:23:28,940
passing those
tests, and then you

415
00:23:28,940 --> 00:23:34,530
might do a binning or parametric
test on all of the chips.

416
00:23:34,530 --> 00:23:39,120
There is another
additional failure point

417
00:23:39,120 --> 00:23:40,840
that's really important.

418
00:23:40,840 --> 00:23:44,340
And this distinction between
yield and reliability

419
00:23:44,340 --> 00:23:46,910
is a little bit fuzzy.

420
00:23:46,910 --> 00:23:49,310
We've got die yield as being--

421
00:23:49,310 --> 00:23:53,990
the full parametric die
yield as those chips that

422
00:23:53,990 --> 00:23:55,430
meet all the specifications.

423
00:23:55,430 --> 00:23:56,900
You ship them to the customer.

424
00:23:56,900 --> 00:23:58,340
They go into parts.

425
00:23:58,340 --> 00:24:03,260
And three months later,
they fail in the field.

426
00:24:03,260 --> 00:24:06,710
That's also a yield
loss, in some sense.

427
00:24:06,710 --> 00:24:10,310
It's more described
as a reliability loss,

428
00:24:10,310 --> 00:24:15,840
but that's that field
loss can be really bad.

429
00:24:15,840 --> 00:24:19,920
You really want to avoid that,
because the costs associated

430
00:24:19,920 --> 00:24:25,230
typically with dealing with
in-field failures is very high.

431
00:24:25,230 --> 00:24:28,378
What's interesting
is, very often--

432
00:24:28,378 --> 00:24:30,420
and I'm not going to talk
too much about it here,

433
00:24:30,420 --> 00:24:32,850
but what's very
often the case is

434
00:24:32,850 --> 00:24:37,770
there is a relationship
between reliability failures

435
00:24:37,770 --> 00:24:42,900
and yield loss sources
back in the fab.

436
00:24:42,900 --> 00:24:45,570
The intuition is not
that hard to imagine.

437
00:24:45,570 --> 00:24:49,560
We even talked about it
back on in this picture.

438
00:24:49,560 --> 00:24:55,770
If I have a low yielding process
because, on a critical metal

439
00:24:55,770 --> 00:24:58,290
layer, I've got lots of
point defects leading

440
00:24:58,290 --> 00:25:03,840
to these kinds of problems, it
might survive through the test,

441
00:25:03,840 --> 00:25:06,090
but that problem
of, say, migration

442
00:25:06,090 --> 00:25:10,190
might be more prone
to occur in the field.

443
00:25:10,190 --> 00:25:14,530
So in general, actually,
yield problems in the fab

444
00:25:14,530 --> 00:25:17,530
can actually be a
great warning signal

445
00:25:17,530 --> 00:25:23,100
that you may have ultimate
reliability failures.

446
00:25:23,100 --> 00:25:25,550
So what we want to do
is try to get a handle

447
00:25:25,550 --> 00:25:33,240
on ways to model and
understand the yield loss,

448
00:25:33,240 --> 00:25:35,430
and be able to make
some predictions

449
00:25:35,430 --> 00:25:39,000
not just for-- based on
historical data for product A,

450
00:25:39,000 --> 00:25:41,340
but also make some
predictions for product B

451
00:25:41,340 --> 00:25:46,170
on your line-- what you might
expect the yield loss to be.

452
00:25:46,170 --> 00:25:49,890
What's interesting here is we've
done so much with the Gaussian

453
00:25:49,890 --> 00:25:52,230
distribution--

454
00:25:52,230 --> 00:25:56,070
this is a great case where
the normal distribution

455
00:25:56,070 --> 00:26:00,330
is generally not the
operative distribution--

456
00:26:00,330 --> 00:26:05,340
that the probability functions
of the binomial and Poisson

457
00:26:05,340 --> 00:26:08,700
statistics are
typically more at work

458
00:26:08,700 --> 00:26:11,320
with random kinds
of point failures.

459
00:26:11,320 --> 00:26:15,210
So we'll review that
just real briefly.

460
00:26:15,210 --> 00:26:17,460
And then what I think
is really interesting

461
00:26:17,460 --> 00:26:20,130
is the spatial nature
of some of these kinds

462
00:26:20,130 --> 00:26:23,090
of defect processes,
this area dependence.

463
00:26:23,090 --> 00:26:25,998
So we want to talk about how--

464
00:26:25,998 --> 00:26:28,020
what some of the basic
modeling approaches

465
00:26:28,020 --> 00:26:31,927
are for area-dependent failures.

466
00:26:31,927 --> 00:26:33,510
So earlier in the
semester, we already

467
00:26:33,510 --> 00:26:35,640
talked about the
binomial distribution.

468
00:26:35,640 --> 00:26:38,190
I think this may be the
same slide, or almost

469
00:26:38,190 --> 00:26:39,270
the same slide.

470
00:26:39,270 --> 00:26:42,540
Remember, the
binomial distribution

471
00:26:42,540 --> 00:26:45,750
is kind of a nice one
dealing with just this notion

472
00:26:45,750 --> 00:26:47,950
of success or failure.

473
00:26:47,950 --> 00:26:51,480
So if I have a point defect
that leads to success or failure

474
00:26:51,480 --> 00:26:54,300
with some probability
p, and I've

475
00:26:54,300 --> 00:26:57,510
got now lots of opportunities
for that failure--

476
00:26:57,510 --> 00:27:02,280
n trials for that particular
failure or success--

477
00:27:02,280 --> 00:27:06,990
then we can count up or
associate the probability

478
00:27:06,990 --> 00:27:14,940
of some number of successes x
using a binomial distribution.

479
00:27:14,940 --> 00:27:19,860
This very often is, in fact,
probably the most important

480
00:27:19,860 --> 00:27:22,170
underlying function--

481
00:27:22,170 --> 00:27:24,450
it and its approximation
on the next slide--

482
00:27:24,450 --> 00:27:27,600
for thinking about
ways to aggregate

483
00:27:27,600 --> 00:27:30,510
when I've got
multiple opportunities

484
00:27:30,510 --> 00:27:32,310
or multiple
structures, and I want

485
00:27:32,310 --> 00:27:36,270
to estimate, given
my probability,

486
00:27:36,270 --> 00:27:38,790
that any one-- say,
any one chip on a wafer

487
00:27:38,790 --> 00:27:41,560
is bad, assuming they
were uncorrelated,

488
00:27:41,560 --> 00:27:45,000
and just due to
random defects, what

489
00:27:45,000 --> 00:27:48,510
is the probability that, in
a wafer with 100 chips on it,

490
00:27:48,510 --> 00:27:55,380
I would have 95, 96, 9-- or
better number of chips actually

491
00:27:55,380 --> 00:27:56,340
coming out?

492
00:27:56,340 --> 00:28:00,930
What's my probability associated
with the yield of at least 95%

493
00:28:00,930 --> 00:28:02,010
on that?

494
00:28:02,010 --> 00:28:09,270
And that falls out directly
from a binomial distribution.

495
00:28:09,270 --> 00:28:13,050
You could add up then
the probabilities of F,

496
00:28:13,050 --> 00:28:21,520
with x being 95, 96, 97, 98,
99, 100, and there you go.

497
00:28:21,520 --> 00:28:24,070
Now, the Poisson distribution
we also talked about,

498
00:28:24,070 --> 00:28:29,320
and this is a good
one when we have

499
00:28:29,320 --> 00:28:35,260
in particular very large numbers
of opportunities for failure,

500
00:28:35,260 --> 00:28:40,090
but the failure probability
for any one of those occurring

501
00:28:40,090 --> 00:28:42,932
is exceptionally small.

502
00:28:42,932 --> 00:28:44,890
So we'll talk about this,
especially when we're

503
00:28:44,890 --> 00:28:48,580
talking about, say, the tens
of thousands or millions

504
00:28:48,580 --> 00:28:53,140
of devices or structures
within an individual chip.

505
00:28:53,140 --> 00:28:55,360
Typically, binomial,
the probabilities

506
00:28:55,360 --> 00:28:58,450
of failure for any one
chip are fairly large,

507
00:28:58,450 --> 00:29:00,760
and the numbers of
chips on the wafer

508
00:29:00,760 --> 00:29:05,800
are fairly moderate, so you
can directly use the binomial.

509
00:29:05,800 --> 00:29:10,810
But when we start to talk about
very, very small probabilities

510
00:29:10,810 --> 00:29:14,440
of failure, the Poisson ends up
being very, very interesting.

511
00:29:14,440 --> 00:29:19,360
And this is the case also where
you start to not just think

512
00:29:19,360 --> 00:29:22,780
about discrete
failure opportunities,

513
00:29:22,780 --> 00:29:29,140
but it's more useful to think
about a failure rate lambda.

514
00:29:29,140 --> 00:29:33,250
So for example, what-- if
you have a particular failure

515
00:29:33,250 --> 00:29:37,780
rate per unit area, just
as with queuing systems,

516
00:29:37,780 --> 00:29:43,170
you have an opportunity
for arrival per unit time.

517
00:29:43,170 --> 00:29:46,480
Now think, I've got the
opportunity for an arrival

518
00:29:46,480 --> 00:29:49,310
of a defect per unit area.

519
00:29:49,310 --> 00:29:53,450
How does the probability
then, for different areas,

520
00:29:53,450 --> 00:29:57,320
give rise to the
different probabilities

521
00:29:57,320 --> 00:30:01,200
of certain numbers of
successes or failures?

522
00:30:01,200 --> 00:30:03,470
So this ends up
being very useful

523
00:30:03,470 --> 00:30:08,780
for defect-oriented,
area-dependent oriented

524
00:30:08,780 --> 00:30:11,060
modeling, very often.

525
00:30:11,060 --> 00:30:12,830
It's also great
for any other case

526
00:30:12,830 --> 00:30:16,130
where you've got just a large
number of discrete failures.

527
00:30:16,130 --> 00:30:19,640
And a typical one
that we'll talk about

528
00:30:19,640 --> 00:30:23,360
are things like via yield
failures or contact yield

529
00:30:23,360 --> 00:30:24,330
failures.

530
00:30:24,330 --> 00:30:26,330
These are the little
electrical connections

531
00:30:26,330 --> 00:30:28,820
from one layer to the
next in the wiring,

532
00:30:28,820 --> 00:30:31,700
or from the wiring down
to the transistor level.

533
00:30:31,700 --> 00:30:36,650
You can imagine any one
metal layer may have millions

534
00:30:36,650 --> 00:30:41,720
to perhaps, in some
layers, billions of these.

535
00:30:41,720 --> 00:30:44,080
The opportunity for
failure of any one of those

536
00:30:44,080 --> 00:30:46,160
is exceptionally
small, but you've

537
00:30:46,160 --> 00:30:48,133
got a heck of a lot of them.

538
00:30:48,133 --> 00:30:49,550
And so then you
might want to ask,

539
00:30:49,550 --> 00:30:55,800
what's the probability of having
perfect operation within that?

540
00:30:55,800 --> 00:30:57,128
So I saw a hand somewhere.

541
00:30:57,128 --> 00:30:57,920
Was there question?

542
00:30:57,920 --> 00:30:59,503
AUDIENCE: No, you
answered [INAUDIBLE]

543
00:30:59,503 --> 00:31:01,740
DUANE BONING: Great, great--

544
00:31:01,740 --> 00:31:05,730
OK, so we're going
to be using those.

545
00:31:05,730 --> 00:31:07,245
Here is the via example.

546
00:31:10,480 --> 00:31:13,350
We could use the
binomial distribution.

547
00:31:13,350 --> 00:31:16,470
Well, first, let me give you
a couple of definitions here.

548
00:31:16,470 --> 00:31:20,100
So we're looking, say, at
one particular metal layer,

549
00:31:20,100 --> 00:31:24,720
and the probability of
failure for any one via

550
00:31:24,720 --> 00:31:26,100
is exceptionally small.

551
00:31:26,100 --> 00:31:29,940
We'll call that p sub v,
probability of failure

552
00:31:29,940 --> 00:31:31,380
for that.

553
00:31:31,380 --> 00:31:37,890
However, again, we have
n opportunities or n vias

554
00:31:37,890 --> 00:31:39,747
in each layer of the chip.

555
00:31:39,747 --> 00:31:41,580
So you might want to
ask the question, well,

556
00:31:41,580 --> 00:31:44,430
what's the probability that
I have one via failure,

557
00:31:44,430 --> 00:31:45,600
then I have 10--

558
00:31:45,600 --> 00:31:47,910
I have failures in some range?

559
00:31:47,910 --> 00:31:49,530
Or ultimately, you
want to really ask

560
00:31:49,530 --> 00:31:53,670
the question, what's the
probability I have zero

561
00:31:53,670 --> 00:31:56,610
via failures, so I
don't have any wiring

562
00:31:56,610 --> 00:31:59,750
problems on that chip?

563
00:31:59,750 --> 00:32:05,250
One could go ahead and directly
use the binomial distribution.

564
00:32:05,250 --> 00:32:08,550
Alternatively, what we
can start to think of as,

565
00:32:08,550 --> 00:32:11,730
what is a failure rate
or the average number

566
00:32:11,730 --> 00:32:19,230
of total via failures, lambda
v, for those vias on a layer?

567
00:32:19,230 --> 00:32:21,502
So here, we're essentially--

568
00:32:21,502 --> 00:32:22,460
what the heck happened?

569
00:32:22,460 --> 00:32:23,793
That's supposed to be an equals.

570
00:32:26,940 --> 00:32:30,180
So this failure rate
is simply the product

571
00:32:30,180 --> 00:32:33,060
of the opportunities in the
individual failure, which

572
00:32:33,060 --> 00:32:36,660
gives you per chip
now this failure rate.

573
00:32:36,660 --> 00:32:42,210
What is the average number
of failures-- via failures

574
00:32:42,210 --> 00:32:44,460
per chip for that layer?

575
00:32:44,460 --> 00:32:47,490
And now you can use the
Poisson distribution, again,

576
00:32:47,490 --> 00:32:53,142
because the conditions of
very small P, very large n.

577
00:32:53,142 --> 00:32:59,070
And just plugging in, we've
got this expression here.

578
00:32:59,070 --> 00:33:02,220
And again, what I said is what
we're really interested in

579
00:33:02,220 --> 00:33:05,130
is the probability that
the whole chip is good,

580
00:33:05,130 --> 00:33:08,940
that none of these via
failures are catastrophic.

581
00:33:08,940 --> 00:33:11,350
None of them occur.

582
00:33:11,350 --> 00:33:13,740
And so I'm really looking
for the probability

583
00:33:13,740 --> 00:33:15,270
that x equals 0--

584
00:33:15,270 --> 00:33:16,890
I have zero via failures.

585
00:33:16,890 --> 00:33:21,690
With an average number of
failures of three per chip,

586
00:33:21,690 --> 00:33:25,440
I'd like to know, well,
what's the likelihood then--

587
00:33:25,440 --> 00:33:30,940
what's my probability that
the whole chip is good?

588
00:33:30,940 --> 00:33:32,620
It's not zero.

589
00:33:32,620 --> 00:33:35,140
It's not 100%, because
on average, I've

590
00:33:35,140 --> 00:33:38,410
got three defects, or three
via failures per chip.

591
00:33:38,410 --> 00:33:42,370
But I've got some out
there in a tail that

592
00:33:42,370 --> 00:33:44,270
are still going to be good.

593
00:33:44,270 --> 00:33:49,945
And so even with
non-zero failure rates--

594
00:33:49,945 --> 00:33:53,740
fact, it's hard to imagine that
you have a full zero failure

595
00:33:53,740 --> 00:33:54,730
rate--

596
00:33:54,730 --> 00:33:56,145
you can still have good chips.

597
00:33:56,145 --> 00:33:57,520
Now, of course,
you'd like lambda

598
00:33:57,520 --> 00:34:01,780
to be perhaps less
than 1 on average.

599
00:34:01,780 --> 00:34:06,370
But now we can basically use
Poisson statistics to aggregate

600
00:34:06,370 --> 00:34:10,780
and calculate, given individual
failure likelihoods or failure

601
00:34:10,780 --> 00:34:12,580
rates, what the
probabilities are

602
00:34:12,580 --> 00:34:16,100
that the whole assembly works.

603
00:34:16,100 --> 00:34:17,889
Question here--

604
00:34:17,889 --> 00:34:25,010
AUDIENCE: [INAUDIBLE]

605
00:34:25,010 --> 00:34:25,900
DUANE BONING: Right.

606
00:34:25,900 --> 00:34:28,370
Right.

607
00:34:28,370 --> 00:34:32,179
In fact, what this has already
done is multiplied by the area,

608
00:34:32,179 --> 00:34:35,420
and so the area multiplication
here was per chip.

609
00:34:35,420 --> 00:34:41,460
So you ultimately get to
some per unit unitless.

610
00:34:41,460 --> 00:34:46,239
So lambda is unitless
in this case.

611
00:34:46,239 --> 00:34:48,780
We'll see other examples
when we do some other area

612
00:34:48,780 --> 00:34:53,520
dependencies, where you might
be looking within the chip,

613
00:34:53,520 --> 00:34:59,190
and actually explicitly adding
in or calculating some area,

614
00:34:59,190 --> 00:35:03,390
and then multiplying the failure
per unit area times the area

615
00:35:03,390 --> 00:35:09,460
that you're sensitive to to
get to a lambda-like parameter.

616
00:35:09,460 --> 00:35:11,960
OK?

617
00:35:11,960 --> 00:35:13,610
This is just a little example--

618
00:35:13,610 --> 00:35:15,440
I'm actually not going
to go through it--

619
00:35:15,440 --> 00:35:18,810
that's just working through
the two cases for the binomial

620
00:35:18,810 --> 00:35:20,630
and Poisson distributions--

621
00:35:20,630 --> 00:35:26,280
particularly in the case when
n is large and pv as small.

622
00:35:26,280 --> 00:35:29,030
So this is just looking
at the particular binomial

623
00:35:29,030 --> 00:35:33,170
distribution when x is 0, or
the Poisson distribution for x

624
00:35:33,170 --> 00:35:35,930
equals 0 for no failures
in the two cases,

625
00:35:35,930 --> 00:35:41,540
and just showing that, for
small lambda or for small pv,

626
00:35:41,540 --> 00:35:51,160
they both go to the
same approximate result.

627
00:35:51,160 --> 00:35:53,410
So we have our
simplest yield model.

628
00:35:53,410 --> 00:35:55,913
We have the binomial
distribution,

629
00:35:55,913 --> 00:35:57,330
which you might
use, for example--

630
00:35:57,330 --> 00:36:02,500
aggregating chip yield.

631
00:36:02,500 --> 00:36:07,990
We've got via kinds of
individual failure models--

632
00:36:07,990 --> 00:36:10,270
so per component
or a failure rate,

633
00:36:10,270 --> 00:36:14,500
and how to aggregate those on
a per unit or per area basis.

634
00:36:14,500 --> 00:36:18,460
But I do want to get, actually,
to exactly the question you

635
00:36:18,460 --> 00:36:21,550
just asked.

636
00:36:21,550 --> 00:36:24,280
How do we get our minds
around the situation

637
00:36:24,280 --> 00:36:28,570
when I've got those little dust
particles falling on some area,

638
00:36:28,570 --> 00:36:34,290
and I'm trying to understand the
area dependence of the circuit?

639
00:36:34,290 --> 00:36:36,220
And so what we're
going to do is actually

640
00:36:36,220 --> 00:36:39,130
want to build a yield model
that's a little bit more

641
00:36:39,130 --> 00:36:44,710
broken out, that explicitly
allows us to make predictions

642
00:36:44,710 --> 00:36:46,750
based on the area
of the circuit,

643
00:36:46,750 --> 00:36:50,530
the area of opportunity
for these failures,

644
00:36:50,530 --> 00:36:56,080
and a defect density or
knowledge about the number

645
00:36:56,080 --> 00:36:59,080
of defects on
average per unit area

646
00:36:59,080 --> 00:37:01,450
that we are likely to have.

647
00:37:01,450 --> 00:37:05,270
And the reason is, if
you think about it,

648
00:37:05,270 --> 00:37:06,910
the chip gets bigger and bigger.

649
00:37:06,910 --> 00:37:08,590
It's got larger area.

650
00:37:08,590 --> 00:37:14,200
And if it's-- only takes
one defect to fail,

651
00:37:14,200 --> 00:37:18,820
the larger it becomes, the more
likely that chip is to fail.

652
00:37:18,820 --> 00:37:22,630
So one key driver in this
that interacts a little bit

653
00:37:22,630 --> 00:37:27,160
with design is, how
big can I make the chip

654
00:37:27,160 --> 00:37:29,500
without incurring
undue yield loss,

655
00:37:29,500 --> 00:37:33,940
just because I'm going to have
some likelihood of defects

656
00:37:33,940 --> 00:37:36,050
per unit area?

657
00:37:36,050 --> 00:37:41,020
So we want to understand
that interplay.

658
00:37:41,020 --> 00:37:43,330
We'll start with
just overall area,

659
00:37:43,330 --> 00:37:46,540
but quickly get to this
notion of a critical area

660
00:37:46,540 --> 00:37:49,180
on the chip, which
is really just

661
00:37:49,180 --> 00:37:53,410
that area where the defect
has to fall or a particle

662
00:37:53,410 --> 00:37:56,410
has to fall in order for
it to actually be a defect,

663
00:37:56,410 --> 00:37:58,630
and cause an electrical
open, or a short,

664
00:37:58,630 --> 00:38:05,300
or some other fault, some
other failure in the circuit.

665
00:38:05,300 --> 00:38:08,240
So how might we go
about modeling these?

666
00:38:08,240 --> 00:38:11,500
Well, first, to help
with the mental model

667
00:38:11,500 --> 00:38:14,920
here, with spatial defects,
we're going to make,

668
00:38:14,920 --> 00:38:18,280
in the simplest yield
model, a few assumptions.

669
00:38:18,280 --> 00:38:20,890
And then I'll show you,
over the course of time,

670
00:38:20,890 --> 00:38:24,940
some of the improved versions
of these defect-oriented models

671
00:38:24,940 --> 00:38:28,870
that have arrived that account
for a little bit more--

672
00:38:28,870 --> 00:38:33,230
or additional effects or relax
a few of these assumptions.

673
00:38:33,230 --> 00:38:34,635
So what I've pictured here--

674
00:38:34,635 --> 00:38:36,040
[INAUDIBLE] it does show up--

675
00:38:36,040 --> 00:38:40,390
is a wafer with some
number of chips on it--

676
00:38:40,390 --> 00:38:43,360
I don't know-- 100,
150 different chips,

677
00:38:43,360 --> 00:38:49,480
and a splattering of a
few little red particles.

678
00:38:49,480 --> 00:38:51,020
These actually are defects.

679
00:38:51,020 --> 00:38:54,370
Each one of these
red particles falls

680
00:38:54,370 --> 00:38:57,040
into place that causes a
failure, some kind of a short.

681
00:38:57,040 --> 00:39:00,170
They're big enough that they
actually sort things out.

682
00:39:00,170 --> 00:39:06,740
And you can start to see,
essentially, an assumption here

683
00:39:06,740 --> 00:39:08,510
is that each one
of these defects

684
00:39:08,510 --> 00:39:16,850
corresponds to killing one
chip in this simple model.

685
00:39:16,850 --> 00:39:19,850
Some other assumptions
are they are, in fact,

686
00:39:19,850 --> 00:39:26,270
randomly distributed by
Poisson kinds of statistics.

687
00:39:26,270 --> 00:39:31,090
They're also randomly
spatially distributed.

688
00:39:31,090 --> 00:39:33,610
Knowing where one
defect is tells you

689
00:39:33,610 --> 00:39:37,000
nothing about where
another defect is.

690
00:39:37,000 --> 00:39:39,920
So they are spatially
uncorrelated.

691
00:39:39,920 --> 00:39:42,580
So those are some of
the initial assumptions,

692
00:39:42,580 --> 00:39:45,430
and under those
assumptions, what

693
00:39:45,430 --> 00:39:50,560
has been observed is a
very interesting or natural

694
00:39:50,560 --> 00:39:55,870
relationship between
the density, d0--

695
00:39:55,870 --> 00:40:01,120
the average number per
unit area of defects--

696
00:40:01,120 --> 00:40:04,000
in this case, I've
got, what, 1, 2, 3, 4--

697
00:40:04,000 --> 00:40:09,670
eight defects here per the
total unit area of the wafer,

698
00:40:09,670 --> 00:40:18,260
and the number of or percentage
of chips that fail, depending

699
00:40:18,260 --> 00:40:20,450
on the area of each chip.

700
00:40:24,670 --> 00:40:28,480
And it's pretty obvious,
especially if I go to extremes.

701
00:40:28,480 --> 00:40:33,250
What if the area of my chip were
the area of the entire wafer?

702
00:40:33,250 --> 00:40:37,110
I had one chip per wafer.

703
00:40:37,110 --> 00:40:40,350
If I had eight
defects on average

704
00:40:40,350 --> 00:40:45,240
per wafer, that means
pretty much every wafer,

705
00:40:45,240 --> 00:40:50,100
every time, I'm going to for
sure have most likely at least

706
00:40:50,100 --> 00:40:54,700
one defect, and my yield's
going to be extremely low.

707
00:40:54,700 --> 00:40:56,980
At some point,
though, my chip size

708
00:40:56,980 --> 00:41:00,430
gets small enough
that this assumption

709
00:41:00,430 --> 00:41:05,620
of every defect killing only
one chip is a very good one.

710
00:41:05,620 --> 00:41:10,270
And then I saturate out to
basically a relationship

711
00:41:10,270 --> 00:41:14,500
that's very close to
just being determined

712
00:41:14,500 --> 00:41:19,660
by counting the number of
defects I have per unit area.

713
00:41:19,660 --> 00:41:21,230
And what was done--

714
00:41:21,230 --> 00:41:23,350
and this is either
in the Stapper paper

715
00:41:23,350 --> 00:41:26,830
or referred to another
paper from Stapper--

716
00:41:26,830 --> 00:41:31,510
is very early on,
this dependence

717
00:41:31,510 --> 00:41:36,970
on the chip area and that
percentage functioning

718
00:41:36,970 --> 00:41:41,580
was observed, and it was
observed to be exponential.

719
00:41:41,580 --> 00:41:44,300
So this is empirical
observation that

720
00:41:44,300 --> 00:41:48,090
gives credence to this
notion of the Poisson

721
00:41:48,090 --> 00:41:49,800
statistics are really
what's at work,

722
00:41:49,800 --> 00:41:54,000
that exponential
dependence on area.

723
00:41:54,000 --> 00:41:55,980
And what he basically
found is that,

724
00:41:55,980 --> 00:42:00,450
as the chip area in square
millimeters went up,

725
00:42:00,450 --> 00:42:02,100
the yield went down.

726
00:42:02,100 --> 00:42:04,830
So when the chip area
was small enough--

727
00:42:04,830 --> 00:42:07,230
very close to 100% yield.

728
00:42:07,230 --> 00:42:10,160
And then-- this is
on a log scale--

729
00:42:10,160 --> 00:42:12,000
notice, this is on a long scale.

730
00:42:12,000 --> 00:42:15,726
So there appeared
to be roughly a--

731
00:42:15,726 --> 00:42:18,840
on the log scale,
a linear decrease

732
00:42:18,840 --> 00:42:21,270
in yield as the wafer area--

733
00:42:21,270 --> 00:42:25,470
or excuse me-- as the
chip area increased--

734
00:42:25,470 --> 00:42:28,510
so that kind of an
exponential dependence.

735
00:42:28,510 --> 00:42:32,280
And so very early on, the first
model that was really used

736
00:42:32,280 --> 00:42:36,120
was a Poisson defect model that
basically treated each defect

737
00:42:36,120 --> 00:42:37,140
as a point--

738
00:42:37,140 --> 00:42:38,760
said, again, these
same assumptions.

739
00:42:38,760 --> 00:42:41,040
Each defect results in a
fault, and these things

740
00:42:41,040 --> 00:42:43,210
are spatially uncorrelated.

741
00:42:43,210 --> 00:42:44,610
So then what you
can really do is

742
00:42:44,610 --> 00:42:49,080
start to say, for any
circuit, any chip,

743
00:42:49,080 --> 00:42:51,630
with some critical
area a sub c--

744
00:42:51,630 --> 00:42:53,220
so that's the area
within the chip

745
00:42:53,220 --> 00:42:56,490
that's sensitive to the
falling of these particles--

746
00:42:56,490 --> 00:43:00,060
maybe ac is equal to the
whole chip area, maybe not--

747
00:43:00,060 --> 00:43:02,940
and some defect
density, then the yield

748
00:43:02,940 --> 00:43:06,670
is simply exponential--

749
00:43:06,670 --> 00:43:09,250
e to the minus ac times d0.

750
00:43:09,250 --> 00:43:13,830
And recognize,
that ac times d0--

751
00:43:13,830 --> 00:43:18,150
that gives rise to something
like a lambda parameter,

752
00:43:18,150 --> 00:43:23,730
a failure rate
kind of parameter.

753
00:43:23,730 --> 00:43:29,490
Now, he did a little
bit of further breakdown

754
00:43:29,490 --> 00:43:32,070
here, which I'm not going
to go too much into.

755
00:43:32,070 --> 00:43:35,490
This actually distinguishes
between a given circuit

756
00:43:35,490 --> 00:43:39,290
and then the whole chip, which
might have n circuits on it.

757
00:43:39,290 --> 00:43:43,250
Looking individually at each
critical circuit on the chip,

758
00:43:43,250 --> 00:43:45,860
you could say for that
particular circuit--

759
00:43:45,860 --> 00:43:48,890
the wiring pattern, say,
of that particular circuit

760
00:43:48,890 --> 00:43:50,210
for a metal layer--

761
00:43:50,210 --> 00:43:53,970
what is the critical area
a sub c for that layer?

762
00:43:53,970 --> 00:43:57,980
And you can get the yield
statistics for metal layer 3

763
00:43:57,980 --> 00:43:59,990
for circuit--

764
00:43:59,990 --> 00:44:04,010
maybe it's the adder circuit
in the upper left corner

765
00:44:04,010 --> 00:44:05,780
of the device--

766
00:44:05,780 --> 00:44:07,760
or the chip.

767
00:44:07,760 --> 00:44:12,450
And then, if you
have n circuits, each

768
00:44:12,450 --> 00:44:17,670
with a critical area A sub
C, for all of them to work,

769
00:44:17,670 --> 00:44:20,250
you've just got a
multiplicative probability

770
00:44:20,250 --> 00:44:24,030
so that your yield is
a multiplicative yield

771
00:44:24,030 --> 00:44:29,160
factor for all of those
individual circuits.

772
00:44:29,160 --> 00:44:31,950
So you can read
this as your yield

773
00:44:31,950 --> 00:44:37,930
for an individual circuit,
just to the n-th power.

774
00:44:42,280 --> 00:44:46,120
And so what they're doing here
is just simply aggregating

775
00:44:46,120 --> 00:44:48,430
and saying the
total critical area

776
00:44:48,430 --> 00:44:51,190
might be across all
of your circuits.

777
00:44:51,190 --> 00:44:54,130
If each one of them
had equal area a sub c,

778
00:44:54,130 --> 00:44:56,950
the total area would be
just the product n times ac.

779
00:44:56,950 --> 00:44:59,410
Or you might do a
summation, might simply

780
00:44:59,410 --> 00:45:01,480
add up all of the
different critical areas.

781
00:45:05,280 --> 00:45:09,630
Now, an expansion on this
starts to pull in a little bit

782
00:45:09,630 --> 00:45:11,730
more statistics.

783
00:45:11,730 --> 00:45:14,760
And in particular, one of the
really interesting statistics

784
00:45:14,760 --> 00:45:21,480
is an observation that not every
wafer observes the same defect

785
00:45:21,480 --> 00:45:22,350
density--

786
00:45:22,350 --> 00:45:24,390
that there, in fact, is
a probability density

787
00:45:24,390 --> 00:45:28,380
function associated
with the defect density.

788
00:45:28,380 --> 00:45:31,320
Some wafers might
see larger numbers

789
00:45:31,320 --> 00:45:33,420
of defect per unit area.

790
00:45:33,420 --> 00:45:35,940
Other wafers may see fewer.

791
00:45:35,940 --> 00:45:37,680
In the natural
operation-- you've

792
00:45:37,680 --> 00:45:40,000
got the fab as clean
as you can make it--

793
00:45:40,000 --> 00:45:42,990
there's still a range of
different defect densities

794
00:45:42,990 --> 00:45:45,840
you expect on any one wafer.

795
00:45:45,840 --> 00:45:51,070
So the first extension
is to characterize

796
00:45:51,070 --> 00:45:54,090
the probability density
function associated

797
00:45:54,090 --> 00:45:56,730
with defect density--

798
00:45:56,730 --> 00:45:59,790
just number of
defects per unit area.

799
00:45:59,790 --> 00:46:06,545
And now you can integrate up
what your expected yield is,

800
00:46:06,545 --> 00:46:08,210
accounting for
the fact that I've

801
00:46:08,210 --> 00:46:11,810
got a whole range of, or a
whole PDF for different defect

802
00:46:11,810 --> 00:46:12,830
densities.

803
00:46:12,830 --> 00:46:17,990
And so the first extension
here is referred to

804
00:46:17,990 --> 00:46:21,080
as the Murphy yield
model, discussed, again,

805
00:46:21,080 --> 00:46:22,310
in [INAUDIBLE].

806
00:46:22,310 --> 00:46:26,420
And all we do is we
have, for any given d,

807
00:46:26,420 --> 00:46:30,080
we have our Poisson
yield model, and then

808
00:46:30,080 --> 00:46:35,990
I'm simply averaging
that over my PDF.

809
00:46:35,990 --> 00:46:40,770
So I'm integrating that over
all possible defect densities.

810
00:46:40,770 --> 00:46:43,020
Now, we can get back to
the Poisson yield model,

811
00:46:43,020 --> 00:46:44,880
and now we actually
recognize that that's

812
00:46:44,880 --> 00:46:48,210
the special case when we
assume there's only one defect

813
00:46:48,210 --> 00:46:50,400
density, and it
applies to every wafer.

814
00:46:50,400 --> 00:46:55,500
That is, our PDF, our f of
d, is just a delta function.

815
00:46:55,500 --> 00:46:58,950
All of the defects are at d0.

816
00:46:58,950 --> 00:47:03,270
So we can recover and get back
to our Poisson yield model.

817
00:47:03,270 --> 00:47:05,910
But what's interesting
now is, depending

818
00:47:05,910 --> 00:47:08,970
on the statistics associated
with defect densities,

819
00:47:08,970 --> 00:47:13,920
I might end up with different
final yield formulas.

820
00:47:13,920 --> 00:47:15,810
And so a number of
different-- whoops--

821
00:47:15,810 --> 00:47:20,940
yield distributions, PDF
associated with defect density

822
00:47:20,940 --> 00:47:24,180
have been explored, and
then some empirical fits

823
00:47:24,180 --> 00:47:27,090
done to data--

824
00:47:27,090 --> 00:47:31,410
yield data to try to see which
matched a little bit better.

825
00:47:31,410 --> 00:47:34,060
And what's nice is there
are at least a few PDFs

826
00:47:34,060 --> 00:47:35,970
that, if you plug them
into that integral,

827
00:47:35,970 --> 00:47:38,590
you're going to have a
closed form solution.

828
00:47:38,590 --> 00:47:43,410
So for example, if you have
a uniform probability density

829
00:47:43,410 --> 00:47:48,390
function associated
with defect density,

830
00:47:48,390 --> 00:47:52,990
that yields or gives rise to
this uniform yield formula,

831
00:47:52,990 --> 00:47:54,900
which is no longer exponential--

832
00:47:54,900 --> 00:47:56,460
or just an exponential.

833
00:47:56,460 --> 00:48:00,060
It's also got a 1 minus the--
this exponential in a scaling

834
00:48:00,060 --> 00:48:01,320
factor--

835
00:48:01,320 --> 00:48:04,470
can also do it for a
triangular distribution.

836
00:48:04,470 --> 00:48:09,700
You get a squared version.

837
00:48:09,700 --> 00:48:11,620
If I plug in a
Gaussian, we already

838
00:48:11,620 --> 00:48:15,490
know that an integral over a
Gaussian is kind of nasty--

839
00:48:15,490 --> 00:48:18,680
doesn't have a
closed form solution.

840
00:48:18,680 --> 00:48:20,830
So it's not directly integrable.

841
00:48:20,830 --> 00:48:22,900
One can certainly
do it numerically,

842
00:48:22,900 --> 00:48:27,920
and things like the
phi function does that.

843
00:48:27,920 --> 00:48:31,220
The Murphy yield
model was done back

844
00:48:31,220 --> 00:48:36,015
when people really wanted
closed form kinds of solutions.

845
00:48:36,015 --> 00:48:38,420
Oh, I thought I had a picture.

846
00:48:38,420 --> 00:48:40,080
Here we go.

847
00:48:40,080 --> 00:48:46,280
So here's a comparison of some
of these different PDFs that

848
00:48:46,280 --> 00:48:48,260
have been examined.

849
00:48:48,260 --> 00:48:51,380
Again, Poisson assumes
everything is at a d0--

850
00:48:51,380 --> 00:48:53,720
should be a d0 there--

851
00:48:53,720 --> 00:48:55,940
a uniform distribution.

852
00:48:55,940 --> 00:48:59,270
I might have defect densities
across that whole range,

853
00:48:59,270 --> 00:49:01,340
or some triangular
distribution where

854
00:49:01,340 --> 00:49:07,250
you might, in fact, restrict it
in some additional form related

855
00:49:07,250 --> 00:49:08,570
to some d0--

856
00:49:08,570 --> 00:49:16,070
or an exponentially decaying
defect density function.

857
00:49:16,070 --> 00:49:23,660
What was interesting is, if
you go back to the literature,

858
00:49:23,660 --> 00:49:27,320
Seeds proposed that--

859
00:49:27,320 --> 00:49:32,270
based on some experimental
data, that an exponential defect

860
00:49:32,270 --> 00:49:35,570
density distribution
appeared to make sense.

861
00:49:35,570 --> 00:49:37,850
First off, the
qualitative reason

862
00:49:37,850 --> 00:49:41,600
was its vanishingly
small likelihood

863
00:49:41,600 --> 00:49:46,880
that you've got lots and lots
of defects, because if you do,

864
00:49:46,880 --> 00:49:50,180
your overall process yield
is not going to be very good,

865
00:49:50,180 --> 00:49:54,230
and so you would have done
process correction or process

866
00:49:54,230 --> 00:49:56,390
development to remove that.

867
00:49:56,390 --> 00:50:00,380
But as the defect density
gets smaller and smaller,

868
00:50:00,380 --> 00:50:02,870
a good manufacturing process--

869
00:50:02,870 --> 00:50:04,040
that's where you want to be.

870
00:50:04,040 --> 00:50:05,915
You want to have much,
much higher likelihood

871
00:50:05,915 --> 00:50:11,270
of small numbers of defects
per unit area than high ones.

872
00:50:11,270 --> 00:50:15,290
So this is not really a
statement about physics.

873
00:50:15,290 --> 00:50:19,010
It's a statement about
manufacturing operation

874
00:50:19,010 --> 00:50:22,940
that drives a particular kind
of shape of defect density

875
00:50:22,940 --> 00:50:24,140
distributions.

876
00:50:24,140 --> 00:50:26,450
It says all of the--

877
00:50:26,450 --> 00:50:28,670
or a huge amount
of energy is put

878
00:50:28,670 --> 00:50:32,990
in to driving down the
defect density distribution.

879
00:50:32,990 --> 00:50:35,300
And what that should
lead to is something

880
00:50:35,300 --> 00:50:41,600
like an exponential
falloff, as pictured here.

881
00:50:41,600 --> 00:50:44,690
You would expect and hope that
your manufacturing process

882
00:50:44,690 --> 00:50:47,420
would have a much higher
likelihood of a small number

883
00:50:47,420 --> 00:50:49,340
of defects per unit area.

884
00:50:49,340 --> 00:50:52,340
And what's nice is, when you
plug that in, you can get

885
00:50:52,340 --> 00:50:54,920
a closed form
expression that's--

886
00:50:54,920 --> 00:50:58,250
that, for the exponential
defect density,

887
00:50:58,250 --> 00:51:04,500
is very nice and simple.

888
00:51:04,500 --> 00:51:05,010
Question--

889
00:51:05,010 --> 00:51:07,302
AUDIENCE: Does that really
make sense, though, compared

890
00:51:07,302 --> 00:51:11,250
to, say, using half a
Gaussian instead with, say,

891
00:51:11,250 --> 00:51:13,770
v centered at 0 and just
cropping half of it.

892
00:51:13,770 --> 00:51:16,290
It kind of seems
as you approach 0,

893
00:51:16,290 --> 00:51:18,790
it actually becomes
more difficult to remove

894
00:51:18,790 --> 00:51:22,530
those last couple of defects,
rather than going exponentially

895
00:51:22,530 --> 00:51:24,498
up that curve, that it
kind of flattens off.

896
00:51:24,498 --> 00:51:25,290
DUANE BONING: Yeah.

897
00:51:25,290 --> 00:51:28,950
So the question is, what
really is the defect density

898
00:51:28,950 --> 00:51:29,670
distribution?

899
00:51:29,670 --> 00:51:30,880
And does this make sense?

900
00:51:30,880 --> 00:51:35,580
Especially with the singularity,
as the defect density

901
00:51:35,580 --> 00:51:40,140
goes to 0, what's the
relative probability?

902
00:51:40,140 --> 00:51:45,630
Might you model this with a
Gaussian or a half Gaussian?

903
00:51:45,630 --> 00:51:49,020
There's all kinds of arguments.

904
00:51:49,020 --> 00:51:54,690
And it's actually difficult to
get enough data to really nail

905
00:51:54,690 --> 00:51:58,470
down the distribution.

906
00:51:58,470 --> 00:52:01,060
Think of how many
wafers, if you will--

907
00:52:01,060 --> 00:52:03,360
to get a very
careful description

908
00:52:03,360 --> 00:52:06,690
of defect density
per unit area--

909
00:52:06,690 --> 00:52:09,150
on average you might need.

910
00:52:09,150 --> 00:52:13,600
It's hard to fully get the
amount of data that you need.

911
00:52:13,600 --> 00:52:17,550
So you're really getting
a few data points in here

912
00:52:17,550 --> 00:52:20,460
that you're trying to get at
least the right trend with.

913
00:52:20,460 --> 00:52:24,360
And so it actually doesn't
matter too critically, as long

914
00:52:24,360 --> 00:52:27,330
as you've got the basic
essence of the shape.

915
00:52:29,910 --> 00:52:32,160
And I will show you
at the end a few

916
00:52:32,160 --> 00:52:33,660
of the kinds of
test structures that

917
00:52:33,660 --> 00:52:37,500
are used to try to approximate
or get at these defect density

918
00:52:37,500 --> 00:52:38,700
distributions.

919
00:52:38,700 --> 00:52:42,840
And in fact, what is very
often done, just to give you

920
00:52:42,840 --> 00:52:45,030
a little bit of a peek--

921
00:52:45,030 --> 00:52:51,330
people might use the
exponential with a fit

922
00:52:51,330 --> 00:52:55,380
to just a couple of points
or a couple of parameters.

923
00:53:02,520 --> 00:53:07,770
OK, so that's basically
the Seeds model,

924
00:53:07,770 --> 00:53:10,330
and that often is used.

925
00:53:10,330 --> 00:53:13,740
But I want to return to a
couple of other further extended

926
00:53:13,740 --> 00:53:16,500
models and give you a little
bit of a feel for them,

927
00:53:16,500 --> 00:53:22,450
because the arguments about
defect density distributions

928
00:53:22,450 --> 00:53:23,710
continue.

929
00:53:23,710 --> 00:53:26,650
But also, reassessing
or looking back

930
00:53:26,650 --> 00:53:28,960
at some of the other
assumptions that I've mentioned,

931
00:53:28,960 --> 00:53:31,490
arguments about
those also exist.

932
00:53:31,490 --> 00:53:33,580
And one of the
most important ones

933
00:53:33,580 --> 00:53:40,630
is this notion of no spatial
correlation in your defect

934
00:53:40,630 --> 00:53:42,260
locations.

935
00:53:42,260 --> 00:53:44,800
Rather than show this,
let me show this first.

936
00:53:44,800 --> 00:53:48,040
So we assumed the
picture over on the left,

937
00:53:48,040 --> 00:53:50,500
that all of your
defects were randomly

938
00:53:50,500 --> 00:53:52,030
distributed across the wafer.

939
00:53:52,030 --> 00:53:54,790
What's very often
observed in practice

940
00:53:54,790 --> 00:53:59,080
is that these defects tend
to cluster near each other.

941
00:53:59,080 --> 00:54:03,670
And maybe there's some process
going on in your chamber that

942
00:54:03,670 --> 00:54:07,720
occasionally splattering
particles, accelerating

943
00:54:07,720 --> 00:54:09,290
particles in some direction.

944
00:54:09,290 --> 00:54:12,850
And so those may naturally
send multiple particles

945
00:54:12,850 --> 00:54:23,760
all together, and they may very
often tend to cluster together.

946
00:54:23,760 --> 00:54:26,290
So that is very interesting.

947
00:54:26,290 --> 00:54:29,460
If, instead of each
and every particle

948
00:54:29,460 --> 00:54:34,410
being spatially distributed
and causing a fault

949
00:54:34,410 --> 00:54:38,730
on an individual
chip, now, well, I've

950
00:54:38,730 --> 00:54:42,510
got multiple particles
all falling and perhaps

951
00:54:42,510 --> 00:54:45,600
causing defects on
these two chips,

952
00:54:45,600 --> 00:54:48,780
but now that assumption
that every single defect is

953
00:54:48,780 --> 00:54:56,700
causing its own unique kill
event is no longer really true.

954
00:54:56,700 --> 00:54:59,730
You really can't keep
killing the same chip

955
00:54:59,730 --> 00:55:01,770
and causing
additional yield loss.

956
00:55:01,770 --> 00:55:04,150
So if you've got clustering
of your defects, in fact,

957
00:55:04,150 --> 00:55:06,090
you may be in better
shape than you

958
00:55:06,090 --> 00:55:10,950
would have assumed per the count
of defects over on the left.

959
00:55:10,950 --> 00:55:15,450
And a distribution that has an
additional parameter in it--

960
00:55:15,450 --> 00:55:17,340
this alpha parameter--

961
00:55:17,340 --> 00:55:24,410
that gives a defect
density distribution

962
00:55:24,410 --> 00:55:26,420
with an extra degree
of freedom that you

963
00:55:26,420 --> 00:55:28,010
can play with this shape--

964
00:55:28,010 --> 00:55:30,350
not necessarily even Gaussian.

965
00:55:30,350 --> 00:55:33,140
But some other
amounts of skewness

966
00:55:33,140 --> 00:55:41,480
away from that exponential is
a negative binomial or gamma

967
00:55:41,480 --> 00:55:44,150
probability distribution
that gives rise

968
00:55:44,150 --> 00:55:47,300
to this negative binomial model.

969
00:55:47,300 --> 00:55:50,810
And so empirically, there's
this additional alpha parameter

970
00:55:50,810 --> 00:55:55,130
that lets one tweak, or fit
your data to tweak the defect

971
00:55:55,130 --> 00:55:57,090
density distribution.

972
00:55:57,090 --> 00:55:59,540
So if you wanted something
that was a little bit more

973
00:55:59,540 --> 00:56:02,750
like a Gaussian or
a half Gaussian,

974
00:56:02,750 --> 00:56:05,420
but maybe behaved a
little bit differently

975
00:56:05,420 --> 00:56:08,300
right near your
low defect density,

976
00:56:08,300 --> 00:56:10,550
you've got that opportunity.

977
00:56:10,550 --> 00:56:14,340
And what's nice about
it is it actually

978
00:56:14,340 --> 00:56:18,150
correlates or relates to this
notion of spatial clustering

979
00:56:18,150 --> 00:56:19,120
of your defects.

980
00:56:19,120 --> 00:56:22,080
So there's a reasonable
physical explanation

981
00:56:22,080 --> 00:56:26,400
for these situations.

982
00:56:26,400 --> 00:56:27,570
Yes, question--

983
00:56:27,570 --> 00:56:32,870
AUDIENCE: [INAUDIBLE]

984
00:56:32,870 --> 00:56:33,870
DUANE BONING: I'm sorry.

985
00:56:33,870 --> 00:56:34,560
Say that again.

986
00:56:34,560 --> 00:56:39,330
AUDIENCE: [INAUDIBLE]

987
00:56:39,330 --> 00:56:41,610
DUANE BONING: Oh, well, if
they're doing the d0 Murphy

988
00:56:41,610 --> 00:56:43,290
model, it's probably--

989
00:56:48,140 --> 00:56:53,240
they might be just using
the delta function--

990
00:56:53,240 --> 00:56:56,030
simple kind of a model.

991
00:56:56,030 --> 00:56:58,190
But you'd actually have
to probe a little bit,

992
00:56:58,190 --> 00:57:01,130
because they might also
have a clustering parameter,

993
00:57:01,130 --> 00:57:04,470
and really, then what's going
on is something like this.

994
00:57:04,470 --> 00:57:07,670
So where your d0
is in here, there

995
00:57:07,670 --> 00:57:11,900
is still a scaling factor
to this distribution.

996
00:57:11,900 --> 00:57:14,030
You can see the d0 in here.

997
00:57:14,030 --> 00:57:19,700
So they might be using, in
fact, a negative binomial yield

998
00:57:19,700 --> 00:57:20,430
model.

999
00:57:20,430 --> 00:57:22,610
In fact, I think
that, right now, this

1000
00:57:22,610 --> 00:57:24,560
is a dominant
model that is used,

1001
00:57:24,560 --> 00:57:27,860
with clustering accounted for.

1002
00:57:27,860 --> 00:57:30,800
And so the d0 is
still your average--

1003
00:57:30,800 --> 00:57:34,850
it's your central scaling or
average on this distribution.

1004
00:57:34,850 --> 00:57:40,480
AUDIENCE: [INAUDIBLE] 10
years or 5 years, then

1005
00:57:40,480 --> 00:57:45,780
that [INAUDIBLE]
changing [INAUDIBLE]

1006
00:57:45,780 --> 00:57:48,030
DUANE BONING: So the question
is, for a long lifetime,

1007
00:57:48,030 --> 00:57:49,740
how do these parameters change?

1008
00:57:49,740 --> 00:57:54,360
And generally, they do
change on your fab--

1009
00:57:54,360 --> 00:57:57,660
not necessarily the
lifetime of your product

1010
00:57:57,660 --> 00:58:00,240
so much, because I think of--

1011
00:58:00,240 --> 00:58:04,020
your d0 tends to be more a
characteristic of your unit

1012
00:58:04,020 --> 00:58:06,150
process or your
integrated process.

1013
00:58:06,150 --> 00:58:10,080
But as you learn more, you
have this yield learning,

1014
00:58:10,080 --> 00:58:15,390
where you hope you drive your
defect density down with time--

1015
00:58:15,390 --> 00:58:17,310
drive your particle size down.

1016
00:58:17,310 --> 00:58:21,810
So you do continue to
improve the process.

1017
00:58:21,810 --> 00:58:25,320
In fact, in some of the
yield projections for product

1018
00:58:25,320 --> 00:58:27,810
you might run in
your fab in a year,

1019
00:58:27,810 --> 00:58:30,030
you might also include
some projections

1020
00:58:30,030 --> 00:58:32,790
on what you think
d0 will improve to,

1021
00:58:32,790 --> 00:58:35,550
based on historical
trends over time.

1022
00:58:35,550 --> 00:58:42,010
AUDIENCE: [INAUDIBLE]

1023
00:58:42,010 --> 00:58:44,650
DUANE BONING: Yes, yes--

1024
00:58:44,650 --> 00:58:49,000
typically, perhaps
more with the d0--

1025
00:58:49,000 --> 00:58:51,400
probably less
projections on alpha.

1026
00:58:51,400 --> 00:58:54,130
Alpha tends to be
this clustering, which

1027
00:58:54,130 --> 00:58:56,560
has two limit that
I'll talk about--

1028
00:58:56,560 --> 00:59:01,780
low clustering and very
highly, tightly clustered.

1029
00:59:01,780 --> 00:59:07,030
I don't think that's assumed
to change that much with time.

1030
00:59:07,030 --> 00:59:10,060
But the d0 is the main
thing that goes down

1031
00:59:10,060 --> 00:59:13,675
with improved processing.

1032
00:59:13,675 --> 00:59:20,200
AUDIENCE: [INAUDIBLE]

1033
00:59:20,200 --> 00:59:22,630
DUANE BONING: They
should, whether they're

1034
00:59:22,630 --> 00:59:25,690
drawn to actually
integrate out to 1 or not.

1035
00:59:25,690 --> 00:59:28,660
But they are all still defect
[INAUDIBLE] probability density

1036
00:59:28,660 --> 00:59:31,600
functions.

1037
00:59:31,600 --> 00:59:34,600
So we do have this alpha
clustering parameter.

1038
00:59:34,600 --> 00:59:41,650
What's nice is, amazingly, you
plug that PDF into the integral

1039
00:59:41,650 --> 00:59:52,960
with an exponential Poisson
kernel in e to the minus acd,

1040
00:59:52,960 --> 00:59:55,750
you get a closed form
yield formula here,

1041
00:59:55,750 --> 01:00:00,030
as shown at the bottom,
which has the alpha

1042
01:00:00,030 --> 01:00:01,890
clustering parameter in it.

1043
01:00:01,890 --> 01:00:04,290
And we can take two limits.

1044
01:00:04,290 --> 01:00:06,990
One limit is the large
alpha limit, which

1045
01:00:06,990 --> 01:00:09,970
is very little clustering.

1046
01:00:09,970 --> 01:00:13,270
So think of maybe
alpha as the distance

1047
01:00:13,270 --> 01:00:17,200
between individual defects,
and as that gets large,

1048
01:00:17,200 --> 01:00:19,120
you don't have any clustering.

1049
01:00:19,120 --> 01:00:22,660
And that limit converges
to the Poisson model.

1050
01:00:22,660 --> 01:00:24,340
And in the very
small alpha limit,

1051
01:00:24,340 --> 01:00:27,700
with very, very strong
clustering, that actually

1052
01:00:27,700 --> 01:00:31,690
converges in the limit for alpha
going to 0 to the Seeds model

1053
01:00:31,690 --> 01:00:36,650
that we saw earlier, which
was the pure exponential.

1054
01:00:36,650 --> 01:00:41,660
So you see, as alpha
gets smaller and smaller,

1055
01:00:41,660 --> 01:00:47,300
this approach is more and more
the exponential defect density

1056
01:00:47,300 --> 01:00:49,710
model.

1057
01:00:49,710 --> 01:00:53,070
Turns out that,
generally, people

1058
01:00:53,070 --> 01:00:56,130
are fitting based on
experimental data, their D0.

1059
01:00:56,130 --> 01:00:59,610
And they're also fitting,
empirically, alpha.

1060
01:00:59,610 --> 01:01:04,290
And alpha tends to be related
both to the clustering,

1061
01:01:04,290 --> 01:01:07,020
but also a little bit to
the sensitivity of your type

1062
01:01:07,020 --> 01:01:10,990
of circuit to clustering.

1063
01:01:10,990 --> 01:01:13,450
So it's not purely--

1064
01:01:13,450 --> 01:01:15,550
if I did this just
on blanket wafers

1065
01:01:15,550 --> 01:01:18,040
and looked at the clustering,
that may actually not

1066
01:01:18,040 --> 01:01:21,610
tell me what is going to happen
for different kinds of product.

1067
01:01:21,610 --> 01:01:27,550
So you actually might end up
with different components of

1068
01:01:27,550 --> 01:01:29,110
or different
products, whether it

1069
01:01:29,110 --> 01:01:32,830
be a memory product or a
microprocessor product,

1070
01:01:32,830 --> 01:01:35,650
or different components
on a big multi-product

1071
01:01:35,650 --> 01:01:37,600
that has a lot of
memory cache on it,

1072
01:01:37,600 --> 01:01:41,380
and also has the
combinational logic on it.

1073
01:01:41,380 --> 01:01:43,690
You might have slightly
different yield model

1074
01:01:43,690 --> 01:01:46,690
components or slightly
different alphas for those two

1075
01:01:46,690 --> 01:01:49,944
different cases, and
you would fit those.

1076
01:01:49,944 --> 01:01:55,480
AUDIENCE: [INAUDIBLE] is
actually designed using

1077
01:01:55,480 --> 01:01:56,795
different alpha parameters--

1078
01:01:56,795 --> 01:01:57,920
DUANE BONING: Interesting--

1079
01:01:57,920 --> 01:02:07,880
AUDIENCE: [INAUDIBLE]

1080
01:02:07,880 --> 01:02:08,985
DUANE BONING: Yeah.

1081
01:02:08,985 --> 01:02:11,360
So the observation, if you
didn't hear that in Singapore,

1082
01:02:11,360 --> 01:02:15,170
was that, in practice, with
those memory redundancy

1083
01:02:15,170 --> 01:02:18,470
schemes, those
also affect alpha,

1084
01:02:18,470 --> 01:02:21,770
and so it's an empirical fitting
process with different kinds

1085
01:02:21,770 --> 01:02:24,440
of redundancy to see
how that affects alpha--

1086
01:02:24,440 --> 01:02:28,910
and what your ultimate yield
would be based on that.

1087
01:02:28,910 --> 01:02:33,880
OK, so so far, we've talked
about a probability density

1088
01:02:33,880 --> 01:02:38,750
associated with the number
of defects per unit area.

1089
01:02:38,750 --> 01:02:43,180
We can also think about
another probability function,

1090
01:02:43,180 --> 01:02:45,670
another statistical
relationship.

1091
01:02:45,670 --> 01:02:48,550
So far, we've talked about
every defect abstractly

1092
01:02:48,550 --> 01:02:53,120
as being infinitesimally small.

1093
01:02:53,120 --> 01:02:55,970
So one defect doesn't cover
20 different chips, right?

1094
01:02:55,970 --> 01:02:58,280
It's just infinitesimally small.

1095
01:02:58,280 --> 01:03:02,480
But what if there is an area
dependence to the size--

1096
01:03:02,480 --> 01:03:08,150
or a probability associated
with the size of those defects?

1097
01:03:08,150 --> 01:03:10,430
That could interact
very importantly

1098
01:03:10,430 --> 01:03:15,060
with some of those original
shorting and open physics

1099
01:03:15,060 --> 01:03:17,110
that we talked about earlier.

1100
01:03:17,110 --> 01:03:20,910
So for example, if I have
wiring lines like this,

1101
01:03:20,910 --> 01:03:23,490
and I'm really worried
about either open or short,

1102
01:03:23,490 --> 01:03:29,070
and my defect is substantially
smaller than either the spacing

1103
01:03:29,070 --> 01:03:33,330
or the width of the line,
it can fall almost anywhere

1104
01:03:33,330 --> 01:03:36,630
and not cause at least
an immediate failure--

1105
01:03:36,630 --> 01:03:40,140
might still be a reliability
or a parametric resistance

1106
01:03:40,140 --> 01:03:42,540
change that I'd
be worried about--

1107
01:03:42,540 --> 01:03:45,660
whereas, if the defect
were much larger,

1108
01:03:45,660 --> 01:03:48,480
it can fall almost
anywhere on my circuit

1109
01:03:48,480 --> 01:03:51,030
and cause either
an open or a short.

1110
01:03:51,030 --> 01:03:54,390
So the effect of particles
of different sizes

1111
01:03:54,390 --> 01:03:57,420
can be very different,
and so I might also

1112
01:03:57,420 --> 01:04:03,520
want to characterize the size
distribution of particles.

1113
01:04:03,520 --> 01:04:07,120
And the interaction of
those science distributions

1114
01:04:07,120 --> 01:04:09,940
with the particular
feature sizes on my circuit

1115
01:04:09,940 --> 01:04:13,580
is going to be very important.

1116
01:04:13,580 --> 01:04:19,990
So it interacts with this
notion then of a critical area.

1117
01:04:19,990 --> 01:04:21,850
Let's see if I've
got a better picture.

1118
01:04:21,850 --> 01:04:24,900
Nope, this is pretty much it.

1119
01:04:24,900 --> 01:04:27,240
There is a formal notion
of critical error--

1120
01:04:27,240 --> 01:04:32,490
area for any particular defect
size that you can actually

1121
01:04:32,490 --> 01:04:38,610
analyze for your particular
layout and say, which area--

1122
01:04:38,610 --> 01:04:43,120
which fraction of the
area on that layer

1123
01:04:43,120 --> 01:04:47,260
does the center of the particle
have to fall in order for it

1124
01:04:47,260 --> 01:04:49,030
to cause either an
open or a short?

1125
01:04:51,827 --> 01:04:53,410
Let me try to erase
this a little bit.

1126
01:04:57,360 --> 01:04:59,540
So for example,
the critical area

1127
01:04:59,540 --> 01:05:05,630
perhaps associated with the
smallest particle may be 0.

1128
01:05:05,630 --> 01:05:09,050
It can fall anywhere
and not cause a problem.

1129
01:05:09,050 --> 01:05:15,330
This particle is perhaps
a more interesting one,

1130
01:05:15,330 --> 01:05:19,185
in that maybe it's exactly
equal to the size of--

1131
01:05:21,770 --> 01:05:29,790
actually, let's do an
example [INAUDIBLE]

1132
01:05:29,790 --> 01:05:36,760
Let's do an example
where I've got something

1133
01:05:36,760 --> 01:05:40,950
that's, say, equal to--

1134
01:05:40,950 --> 01:05:45,840
or just slightly larger
than the spacing size.

1135
01:05:45,840 --> 01:05:49,680
But in some places,
I've got wires where--

1136
01:05:49,680 --> 01:05:53,640
and spaces that are
smaller than that,

1137
01:05:53,640 --> 01:05:57,705
such as pictured here, but
I've also got other places--

1138
01:05:57,705 --> 01:05:59,790
let's say, I have
now another wire

1139
01:05:59,790 --> 01:06:06,300
up here, where the spacing is
larger than the particle size.

1140
01:06:06,300 --> 01:06:09,600
Now this same particle
can fall right there

1141
01:06:09,600 --> 01:06:12,340
and not cause a short.

1142
01:06:12,340 --> 01:06:18,040
So you can calculate across your
entire particular layout what

1143
01:06:18,040 --> 01:06:21,160
is the band where the
center of the particle

1144
01:06:21,160 --> 01:06:25,480
has to fall to cause
either an open or a short

1145
01:06:25,480 --> 01:06:29,800
and some up that
area of sensitivity

1146
01:06:29,800 --> 01:06:34,060
for failure for
each of the layers.

1147
01:06:34,060 --> 01:06:38,650
So there's this interaction
between a critical area

1148
01:06:38,650 --> 01:06:41,410
per particle size, and
then a distribution

1149
01:06:41,410 --> 01:06:43,840
associated with the
particle sizes that are very

1150
01:06:43,840 --> 01:06:46,980
important to also characterize.

1151
01:06:46,980 --> 01:06:48,960
And here are some examples--

1152
01:06:48,960 --> 01:06:51,240
again, going way back--

1153
01:06:51,240 --> 01:06:55,310
for defect size distributions.

1154
01:06:55,310 --> 01:06:58,100
These are back
characterized in mils.

1155
01:06:58,100 --> 01:07:01,590
Anybody know what a mil is?

1156
01:07:01,590 --> 01:07:06,720
Thousandth of an inch,
or about 25 microns--

1157
01:07:06,720 --> 01:07:09,870
so these are giant,
giant particles.

1158
01:07:09,870 --> 01:07:12,990
We have driven down
defect sizes a bit.

1159
01:07:12,990 --> 01:07:15,870
But what is very interesting
is the same trend

1160
01:07:15,870 --> 01:07:17,880
has continued to be observed--

1161
01:07:17,880 --> 01:07:20,550
that there is generally
believed to be something

1162
01:07:20,550 --> 01:07:25,980
close to an exponential
dependence in defect size,

1163
01:07:25,980 --> 01:07:29,800
not just in the number
of defects as well.

1164
01:07:29,800 --> 01:07:34,960
And that exponential-- or
power law kind of dependence

1165
01:07:34,960 --> 01:07:40,480
is very often used in
modeling the size distribution

1166
01:07:40,480 --> 01:07:41,455
for defects.

1167
01:07:45,040 --> 01:07:47,260
Now, there's a couple
of parameters-- this n

1168
01:07:47,260 --> 01:07:49,240
and this p, which,
again, end up being

1169
01:07:49,240 --> 01:07:51,610
technology dependent
and generally fitting

1170
01:07:51,610 --> 01:07:54,680
parameters to the data.

1171
01:07:54,680 --> 01:08:00,220
So now, if we take in that
notion of a distribution

1172
01:08:00,220 --> 01:08:03,970
in defect sizes and the
probability associated

1173
01:08:03,970 --> 01:08:07,630
with that, one can
form an aggregate sort

1174
01:08:07,630 --> 01:08:09,530
of approximate parameter.

1175
01:08:09,530 --> 01:08:12,880
And so sometimes that's
also the d0 that is quoted.

1176
01:08:12,880 --> 01:08:18,490
So it's not only averaged in
terms of numbers per area,

1177
01:08:18,490 --> 01:08:22,840
but it's also kind of a boiled
down approximate parameter

1178
01:08:22,840 --> 01:08:27,609
giving you a sense of the basic
central moment of the size

1179
01:08:27,609 --> 01:08:30,550
distribution as well.

1180
01:08:30,550 --> 01:08:34,200
But if you really wanted to
do careful defect modeling,

1181
01:08:34,200 --> 01:08:37,200
you actually need to know
the p parameters and the n

1182
01:08:37,200 --> 01:08:37,890
parameters.

1183
01:08:37,890 --> 01:08:40,170
You would like to have
that full defect size

1184
01:08:40,170 --> 01:08:41,880
distribution at hand.

1185
01:08:44,399 --> 01:08:45,819
Here are some examples.

1186
01:08:45,819 --> 01:08:48,370
The typical ranges
of that exponent--

1187
01:08:48,370 --> 01:08:51,180
I guess it's a p exponent
in the previous slide--

1188
01:08:51,180 --> 01:08:54,300
is two, three, four.

1189
01:08:54,300 --> 01:08:58,680
And it may depend on
the defect failure mode

1190
01:08:58,680 --> 01:09:00,689
that you're looking
at-- so for example,

1191
01:09:00,689 --> 01:09:04,740
extra metal or short versus
missing metal and opened.

1192
01:09:04,740 --> 01:09:10,840
They may have slightly different
sensitivity to that defect size

1193
01:09:10,840 --> 01:09:11,340
as well.

1194
01:09:11,340 --> 01:09:17,302
AUDIENCE: [INAUDIBLE] more
distributions [INAUDIBLE]

1195
01:09:17,302 --> 01:09:19,649
DUANE BONING: This is
basically this power law.

1196
01:09:19,649 --> 01:09:23,500
1/x to the p is the assumed--

1197
01:09:23,500 --> 01:09:24,870
AUDIENCE: That's [INAUDIBLE]

1198
01:09:24,870 --> 01:09:25,890
DUANE BONING: Yes.

1199
01:09:25,890 --> 01:09:28,400
AUDIENCE: But for [INAUDIBLE]

1200
01:09:28,400 --> 01:09:30,500
DUANE BONING: For the d0,
the d0 is actually still

1201
01:09:30,500 --> 01:09:34,189
not counting-- adding
in the defect density.

1202
01:09:34,189 --> 01:09:36,710
That will come in in
the other distribution.

1203
01:09:39,660 --> 01:09:42,260
Oh, OK, I did have a slide.

1204
01:09:42,260 --> 01:09:43,970
I think I've already
explained this,

1205
01:09:43,970 --> 01:09:46,430
but this is talking again
about the critical area

1206
01:09:46,430 --> 01:09:49,430
for different size dependencies.

1207
01:09:49,430 --> 01:09:51,350
And what's interesting
is then you can also

1208
01:09:51,350 --> 01:09:56,580
start to produce a plot of this
critical area versus defect

1209
01:09:56,580 --> 01:09:57,080
size.

1210
01:10:00,170 --> 01:10:04,580
And what's really important
here is very intuitive trends

1211
01:10:04,580 --> 01:10:08,930
that, for larger defects,
I've got larger critical area.

1212
01:10:08,930 --> 01:10:11,810
More part of my chip
is sensitive to it.

1213
01:10:11,810 --> 01:10:16,400
But once it gets smaller than
a certain dimension, when

1214
01:10:16,400 --> 01:10:20,690
my defects tend to be smaller
than my minimum feature size

1215
01:10:20,690 --> 01:10:23,060
on the device, I
start to be less

1216
01:10:23,060 --> 01:10:25,260
sensitive to immediate failures.

1217
01:10:25,260 --> 01:10:27,350
So if they're a fraction--

1218
01:10:27,350 --> 01:10:31,330
your particles are a fraction
of your minimum dimension.

1219
01:10:31,330 --> 01:10:33,080
And that's good, because
you probably have

1220
01:10:33,080 --> 01:10:35,210
lots of defects of those sizes.

1221
01:10:35,210 --> 01:10:38,590
It's very hard to
get rid of those.

1222
01:10:38,590 --> 01:10:40,800
Now, what you can do is
start to put these together.

1223
01:10:40,800 --> 01:10:42,240
You aggregate these.

1224
01:10:42,240 --> 01:10:45,630
If you have a defect
size distribution that

1225
01:10:45,630 --> 01:10:50,380
goes as one of these
1/x to the p's, you

1226
01:10:50,380 --> 01:10:55,360
can start to try to empirically
fit that to your data.

1227
01:10:55,360 --> 01:10:58,850
One thing that happens with
these distributions, of course,

1228
01:10:58,850 --> 01:11:00,520
is the same thing
you were worried

1229
01:11:00,520 --> 01:11:01,930
about with the exponential.

1230
01:11:01,930 --> 01:11:07,470
When x gets really small,
that goes sky high--

1231
01:11:07,470 --> 01:11:10,030
1 over a very small number.

1232
01:11:10,030 --> 01:11:13,560
And so what people
basically do is

1233
01:11:13,560 --> 01:11:18,310
they're mostly worried about
the defect sizes larger

1234
01:11:18,310 --> 01:11:21,280
than their minimum feature size.

1235
01:11:21,280 --> 01:11:24,520
And then they'll basically
truncate it either

1236
01:11:24,520 --> 01:11:28,090
as a constant or, in fact,
as a linear drop-off,

1237
01:11:28,090 --> 01:11:31,900
once you get down below
some minimum size.

1238
01:11:31,900 --> 01:11:38,230
I should have drawn
that near x0 instead.

1239
01:11:38,230 --> 01:11:42,280
So there is a size at which it's
hard to either detect these,

1240
01:11:42,280 --> 01:11:44,890
and you don't care
about them, and so

1241
01:11:44,890 --> 01:11:46,510
you're not really
trying to model

1242
01:11:46,510 --> 01:11:48,220
that part of this distribution.

1243
01:11:51,613 --> 01:11:53,280
In one of the first
lectures, I gave you

1244
01:11:53,280 --> 01:11:56,010
a little bit of a preview
and an example here

1245
01:11:56,010 --> 01:11:59,520
of how you might measure these
defect size distributions.

1246
01:11:59,520 --> 01:12:06,120
Characterization test
vehicles, especially early

1247
01:12:06,120 --> 01:12:08,280
in process development,
might be used

1248
01:12:08,280 --> 01:12:11,160
to try to characterize the
capability of the process

1249
01:12:11,160 --> 01:12:12,390
in terms of these--

1250
01:12:12,390 --> 01:12:18,300
both your defect density,
but also your defect sizes.

1251
01:12:18,300 --> 01:12:20,340
And imagine now that
I've got a whole array

1252
01:12:20,340 --> 01:12:25,500
of these nested
metal lines that I

1253
01:12:25,500 --> 01:12:28,440
can make electrical
measurements or connections to.

1254
01:12:28,440 --> 01:12:31,770
And if I have now
a defect that's

1255
01:12:31,770 --> 01:12:34,440
kind of small in general--

1256
01:12:34,440 --> 01:12:39,630
or usually-- it might be even so
small that I rarely get short,

1257
01:12:39,630 --> 01:12:42,330
but I might get some amount
of resistance change in these

1258
01:12:42,330 --> 01:12:43,380
lines--

1259
01:12:43,380 --> 01:12:47,820
versus other defects that are so
big that they start to bridge,

1260
01:12:47,820 --> 01:12:51,780
on average, two or three lines.

1261
01:12:51,780 --> 01:12:55,050
You can start to build up
some electrical measure

1262
01:12:55,050 --> 01:12:59,010
of the likelihood of having--

1263
01:12:59,010 --> 01:13:03,490
or the relative counts of
defects of different sizes.

1264
01:13:03,490 --> 01:13:05,250
So here's an empirical--

1265
01:13:05,250 --> 01:13:09,860
this is, again,
from a 2003 paper.

1266
01:13:09,860 --> 01:13:16,090
You can start to see that 1/x
to the p empirical relationship,

1267
01:13:16,090 --> 01:13:22,570
and it gives you a sense for how
you can actually measure those.

1268
01:13:22,570 --> 01:13:26,920
So now you can have a
more careful definition

1269
01:13:26,920 --> 01:13:31,210
of the critical area, where
you have your defect size

1270
01:13:31,210 --> 01:13:32,080
distribution.

1271
01:13:32,080 --> 01:13:36,930
I might use dsd
to indicate that.

1272
01:13:36,930 --> 01:13:39,520
So there's our defect
size distribution,

1273
01:13:39,520 --> 01:13:41,860
that 1/x to the p--

1274
01:13:41,860 --> 01:13:46,450
again, really only worrying
about it above some x0.

1275
01:13:46,450 --> 01:13:48,580
And then you also
have this probability

1276
01:13:48,580 --> 01:13:53,880
of failure, which folds in
this notion of critical area.

1277
01:13:53,880 --> 01:13:55,550
You can actually
look at your layout

1278
01:13:55,550 --> 01:13:58,190
and say, if I've got
a defect of this size,

1279
01:13:58,190 --> 01:14:01,370
this is my probability
of failure integrated

1280
01:14:01,370 --> 01:14:03,590
across my whole chip.

1281
01:14:03,590 --> 01:14:06,260
And now you can aggregate
the two of those

1282
01:14:06,260 --> 01:14:09,320
into a net total critical area.

1283
01:14:09,320 --> 01:14:12,730
So all I'm doing
is saying there is

1284
01:14:12,730 --> 01:14:16,480
a probability of failure
associated with defects

1285
01:14:16,480 --> 01:14:17,870
of different sizes.

1286
01:14:17,870 --> 01:14:19,210
The defect is really small.

1287
01:14:19,210 --> 01:14:25,760
Again, I have that critical area
plot, where I'm down in here,

1288
01:14:25,760 --> 01:14:27,500
and it's very small.

1289
01:14:27,500 --> 01:14:29,930
But empirically, for
my particular circuit,

1290
01:14:29,930 --> 01:14:36,750
I may have different critical
area dependence curves, where

1291
01:14:36,750 --> 01:14:39,870
I aggregate the total pof.

1292
01:14:39,870 --> 01:14:42,330
And then I look at
the product of those

1293
01:14:42,330 --> 01:14:44,940
two, integrate it
up, and that gives me

1294
01:14:44,940 --> 01:14:46,890
a good aggregate sense--

1295
01:14:46,890 --> 01:14:52,320
which really says where I'm
mostly worried is in this range

1296
01:14:52,320 --> 01:14:54,510
here, where I've
got defects that

1297
01:14:54,510 --> 01:14:58,350
are close to my feature size.

1298
01:14:58,350 --> 01:15:01,640
The small ones aren't
going to kill me.

1299
01:15:01,640 --> 01:15:03,658
The bigger ones are
going to kill me.

1300
01:15:03,658 --> 01:15:06,200
Really big ones are not going
to kill me, because there's not

1301
01:15:06,200 --> 01:15:08,420
that many big ones.

1302
01:15:08,420 --> 01:15:14,510
So really, it tells you
the area to worry about.

1303
01:15:14,510 --> 01:15:18,830
OK, I'm going to skip
over most of this.

1304
01:15:18,830 --> 01:15:21,950
I just want you to get
a feel for this notion

1305
01:15:21,950 --> 01:15:23,460
of critical area.

1306
01:15:23,460 --> 01:15:24,860
And this is an
area where there's

1307
01:15:24,860 --> 01:15:28,340
a lot of design automation
tools, where it can actually

1308
01:15:28,340 --> 01:15:31,910
look at your particular
layout and start

1309
01:15:31,910 --> 01:15:34,280
to do those kinds of
drawings that I showed you

1310
01:15:34,280 --> 01:15:37,490
for critical area, if you
will, and shade in-- maybe

1311
01:15:37,490 --> 01:15:40,400
a little hard to see, but
you can shade in and say,

1312
01:15:40,400 --> 01:15:45,000
what is the gray critical area
for that particular circuit?

1313
01:15:45,000 --> 01:15:47,060
Where am I sensitive
to the likelihood

1314
01:15:47,060 --> 01:15:50,710
of a short for defects
of a particular size?

1315
01:15:50,710 --> 01:15:53,120
Where am I sensitive?

1316
01:15:53,120 --> 01:15:55,910
Where would a-- the
center of a particle

1317
01:15:55,910 --> 01:16:02,080
have to fall in order
to cause an open?

1318
01:16:02,080 --> 01:16:05,410
Or where might I have
to have a particle

1319
01:16:05,410 --> 01:16:08,840
fall that would cause a short
between two different layers?

1320
01:16:08,840 --> 01:16:11,980
And you can actually do these
calculations for your layout,

1321
01:16:11,980 --> 01:16:14,560
and then do design
modifications that

1322
01:16:14,560 --> 01:16:17,950
would come back in
and say, oh, if that's

1323
01:16:17,950 --> 01:16:22,010
my critical area where I might
be susceptible to a break,

1324
01:16:22,010 --> 01:16:24,880
let's make those lines
a little bit wider.

1325
01:16:24,880 --> 01:16:26,770
If I make those
lines wider, I've

1326
01:16:26,770 --> 01:16:29,830
improved my yield, because
now, if the particle falls

1327
01:16:29,830 --> 01:16:33,460
in those lines, I'm not
as likely to actually have

1328
01:16:33,460 --> 01:16:35,180
an open failure.

1329
01:16:35,180 --> 01:16:37,750
So there is yield
improvement strategies

1330
01:16:37,750 --> 01:16:41,830
that go with these
notions of critical area,

1331
01:16:41,830 --> 01:16:46,480
and the probabilities
associated with them.

1332
01:16:46,480 --> 01:16:48,190
The last notion
is simply you can

1333
01:16:48,190 --> 01:16:50,740
integrate these up and
get aggregate notions

1334
01:16:50,740 --> 01:16:54,040
of overall yield.

1335
01:16:54,040 --> 01:16:57,070
This is also described in
[INAUDIBLE] Just reminding

1336
01:16:57,070 --> 01:16:59,860
you-- there's also other
kinds of yield detractors,

1337
01:16:59,860 --> 01:17:02,800
where you might have
gross yield losses,

1338
01:17:02,800 --> 01:17:05,950
and so you might have a
global factor y0 that's

1339
01:17:05,950 --> 01:17:09,190
associated with alignment
errors or other kinds

1340
01:17:09,190 --> 01:17:12,070
of gross factors.

1341
01:17:12,070 --> 01:17:13,810
And then this last
is a little example

1342
01:17:13,810 --> 01:17:16,660
that you can read about in
[INAUDIBLE],, which is basically

1343
01:17:16,660 --> 01:17:20,650
simply saying, in
practice, your chip yield

1344
01:17:20,650 --> 01:17:23,110
is all so aggregated
that it doesn't really

1345
01:17:23,110 --> 01:17:25,750
tell you what's gone wrong.

1346
01:17:25,750 --> 01:17:28,630
So very often, you
want to slice yield

1347
01:17:28,630 --> 01:17:31,120
into your different layers--

1348
01:17:31,120 --> 01:17:33,850
your different process
layers, or slice them

1349
01:17:33,850 --> 01:17:35,770
into different
functional blocks--

1350
01:17:35,770 --> 01:17:40,540
maybe the memory or cache
block, a logic block--

1351
01:17:40,540 --> 01:17:43,600
and basically look
at where you are

1352
01:17:43,600 --> 01:17:45,700
most sensitive to yield loss.

1353
01:17:45,700 --> 01:17:52,570
For example, that 95% yield
loss factor or via 2 inside

1354
01:17:52,570 --> 01:17:54,730
of your SRAM might
be where you're

1355
01:17:54,730 --> 01:17:56,220
losing most of your yield.

1356
01:17:59,420 --> 01:18:04,150
There's quite a bit of
development of these test chips

1357
01:18:04,150 --> 01:18:08,170
that have those things like
those nested via or nested

1358
01:18:08,170 --> 01:18:13,900
snake structures in them so
that one can characterize

1359
01:18:13,900 --> 01:18:17,950
defectivity distributions,
as well as sensitivity

1360
01:18:17,950 --> 01:18:21,410
of different kinds of
circuits to those failures.

1361
01:18:21,410 --> 01:18:25,840
So that's a whirlwind tour
there of semiconductor yield,

1362
01:18:25,840 --> 01:18:28,090
these notions of not
just functional yield

1363
01:18:28,090 --> 01:18:31,810
that we saw last
time, but also this--

1364
01:18:31,810 --> 01:18:34,540
or parametric yield, but
defect yield as well.

1365
01:18:34,540 --> 01:18:39,040
And you'll have a little
bit of fun playing around

1366
01:18:39,040 --> 01:18:42,160
with some of these notions
of area-dependent yield,

1367
01:18:42,160 --> 01:18:45,100
which I think is really
kind of the cool idea,

1368
01:18:45,100 --> 01:18:49,480
the important idea in yield
modeling for semiconductors.

1369
01:18:49,480 --> 01:18:51,760
OK, so we'll see you
again on Thursday.

1370
01:18:51,760 --> 01:18:54,760
Thursday is the quiz.

1371
01:18:54,760 --> 01:18:57,600
I think here, you had also
posted for office hours.

1372
01:18:57,600 --> 01:18:58,600
I think you have those--

1373
01:18:58,600 --> 01:18:58,900
AUDIENCE: Yes.

1374
01:18:58,900 --> 01:19:00,010
DUANE BONING:
--tomorrow as well.

1375
01:19:00,010 --> 01:19:01,000
AUDIENCE: [INAUDIBLE] to 6:00.

1376
01:19:01,000 --> 01:19:03,125
DUANE BONING: Tomorrow,
5:00 to 6:00, if you want--

1377
01:19:03,125 --> 01:19:05,590
any last questions
before the quiz--

1378
01:19:05,590 --> 01:19:07,160
Hayden's available
for those as well.

1379
01:19:07,160 --> 01:19:08,080
So thanks.

1380
01:19:08,080 --> 01:19:09,744
We'll see you on Thursday.