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One of the term which often resonates inside a semiconductor company is “split lots”. Even though I vaguely knew what it referred to, the haziness around was deeper than the meat of the term. Borrowing Mark Twain, ” So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover”. If not fully,  I should know what I am completely dumb about. In the end it is a fairly simple terminology. Here is what I understood. Most of what I gathered are a result of an information gathering attempt, after repeated bugging on many of my VLSI colleagues coupled with my dump fight with Mr. Google.

Firstly, the hero term, “corner lot”. Corner lot is a term referring to the semiconductor fabrication process corners. Basically, the fabrication parameters are deliberately (and carefully) changed to create extreme corners of a circuit etched in semiconductor wafer. These corner chips are expected to run slower or faster than the nominal (average behaviour of) chip produced in large volume. These corner lot chips also function at lower or higher temperature and voltages than a nominal chip. In short they differ from the typical parts, in terms of performance.

Why do this? Well, the simple answer is: To study the robustness of semiconductor chips mass manufactured out of a complicated fabrication process. When millions of chips are produced at the assembly, statistics come into play (think of the law of large numbers and central limit theorems). In order to ensure that, the manufactured chips function within a certain confidence interval  (within certain variance from the typical parts), it is a practice to fabricate corner lots.

When the volume of samples is large, the manufactured VLSI chips are likely to have performance variation, which admit a Gaussian statistical distribution. The mean of the distribution is the nominal performance of the chips. The three sigma or six sigma performance from the nominal one is that entails upon the corner lot chips. The process parameters are carefully adjusted to the three sigma (or six sigma depending on the need) from the nominal doping concentration in transistors on a silicon wafer. This way, one can deliberately mimick the corner chips which comes out in the volume production. In the manufacturing routine, the variation in performance may occur for many reasons, due to minor changes in temperature or humidity present in the clean room. The variation can also happen with variation in the die position relative to the center of the wafer.

In essence, the corner lots are groups of wafers whose process parameters are carefully adjusted according to chosen extremes. How are these corners produced? In other words, what exactly is changed to carefully achieve these extreme wafers? The exact details are not presented here. From this forum, I infer that the main parameters are 1) the doping concentration. 2) The process variation. 3) The resistance of the actives 4) the properties and thickness of oxides 5) The effective width, length of the stray capacitances etc.

What do we do with these corner lot chips? These extreme corners are characterized at various conditions such as temperature, voltages etc. Once the characterization across these corners is proved to be within accepted limits, then the mass manufactured volume of semiconductor chips falls within the required confidence interval. If all the corner lot chips meet he performance, it is safe to assume that, the huge volume of chips mass manufactured also will fall within the performance limits. That way, humongous saving of time and effort from testing of each chips is achieved.

What are the different process corner types? There is no end to the terminologies here. The nomenclature of the corners are based on two letters (we limit attention to CMOS semiconductors alone). The first letter attributed to the NMOS corner and the second letter for the PMOS. Three types exist in each, namely typical (T), fast (F) and slow (S).Here, slow and fast refers to the speed (mobility) of electrons and holes. In all ${3 \choose 2 }$ (i.e., $6$) corners and they are FF,FS,FT,SS,ST,TT. Among these, FF,SS and  TT are even corners since both PMOS and NMOS are affected equally. The corners FS and SF are skewed. The even corners are expected to function less adversely compared to the skewed corners, for valid reasons.

By the way, the obvious impact of the “corner” feature is the difference in device switching speed, which is related to the mobility of electrons and holes. The mobility of electron is related to the doping concentration (among others). A rough empirical (as shown) model shows the following relationship. The mobility $\mu$ depends also on the impurity and doping concentration $\rho$. The parameters vary, depending on the type of impurity. The three common impurity elements are arsenic, phosphorus, boron (See this link for further details)

$\mu=\mu_{0}+\frac{\mu_{1}-\mu_{0}}{1+\left(\frac{\rho}{\rho_{0}}\right)^{\alpha}}$

where, $\mu_{0}$ and $\mu_{1}$ are the minimum and maximum limits of the mobility. $\alpha$ is a fitting parameter.

In the figure (below), the mobility of electrons due to different impurities and doping concentration are shown. The three commonly used dopants are arsenic, boron and phosphorus.

It is an irony that, the dopant which is deliberately added  to increase conductivity of semiconductor itself slows down the mobility of electrons and holes due to collision (of electrons or holes) with the dopants.

A touch of note on the term mobility too. Broadly speaking, the device switching speed is directly related to the mobility of charged particles (electrons and holes). So, higher mobility somewhat implies that better and faster switching logic etc. When an electric field $E$ is applied to a semiconductor, the electrostatic force will drive the carriers to a constant average velocity $v$, when the carriers scatter though the impurities and lattice vibrations. The ratio of the velocity to the applied electric field is called the mobility $\mu$. That is., $\mu=v/E$. Initially, the velocity increases with increase in electric field and finally reach a saturation velocity at high electric field. When the carriers flow at the surface of semiconductor, additional scattering may occur and that will pull down the mobility.

The kind of interactions which happens at atomic and sub atomic levels within and outside a transistor are way too complex to comprehend (in a blog for sure!). Besides, the millions (and these days billions) of these transistors all must work in tandem to make the system function as desired; And that too, with not a single one failing! To make business, certain economic sense should prevail as well. It goes without saying that, time is extremely critical to get the product into action. The split lot is one heck of a way not to stretch that time window.

(Photo courtesy)

I didn’t see this book before. While trying to dig a little deeper into the sigma delta modulation theory, I bumped across this book by Robert Gray. The book, first published in 1990 hasn’t really become a mainstream reference on source coding, but the book is awesome. I didn’t read the whole book, but the chapter on uniform quantization noise is simply a treat for someone who loves the theory. Among other things, it discusses the Benetts’s conditions for the exactness of the uniform quantization error. I am now going through the noise analysis of the delta modulation and sigma delta modulation schemes.

Initially, I just managed to read a near full chapter content at books.google.com, but later I was convinced myself to get the book at Amazon. After losing the 7.5 USD used book offer, I finally had to content myself from ordering the next cheapest option of 40USD. I am expecting this to arrive in a few days. A brand new book would cost 130 bucks anyway!

I am at Dublin attending the IEEE Information Theory Workshop ITW 2010. The city is small, but beautiful. I will write a bit about my Dublin experience later. Apart from the Whiskey (which I don’t drink anyway, but I am going to go for the whiskey tasting tour organized as part of the ITW program, on thursday evening) there is rich literary legacy. The Bernard Shaw and James Joycee belongs to this famous city. And oh dear, Yeats, my favourite poet in the adolescent years. My wife (then my fiancee) still says that she was truly impressed when I’d recited those Yeats poems, many years ago!  But then, I was so fond of poetry back then. Now, with (lack of) time and tide, things just got locked in nostalgia, alone. I am definitely going to spend sometime in the Dublin literary museum, this weekend, before flying to Geneva.

My talk, (on the joint work with Emmanuel) is on Tuesday afternoon. Need to work a bit on the slides too! Feeling a touch tired, after the sleep being deprived for the last two days! I will also try to pen down a bit on some of the interesting talks I am going to attend.

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