Structure functions – thermal CAT-scan?
I have always admired the medical diagnostic tools, especially imaging. X-rays have a long history, starting with Conrad Röntgenback in the 19th century. In fact in Hungary x-raying is called “röntgening” – yes, like this; the phrase became part of our language as an ordinary word; both as a verb (to take somebody’s x-ray image) or as an adjective to specifically name this very high frequency range of electro-magnetic radiation, for example. Computer tomography (CAT-scan as usually called in English speaking countries) is a real miracle of the late 20th century science and technology. The idea is that if x-ray images of the human body are taken from different angles around the axis of the body in small increments and the resulting series of the 1D projected x-ray images of a given cross section of the 3D body are post-processed, then it is possible to generate complete 2D images of cross-sectional cuts of the body. The image below was actually taken from my body 17 years ago. (At that time the images were photographed to an x-ray film, rather than saved on any long term computer storage media – remember, this was still the era of the 1.44M 3.5″ floppies; pen-drives, magnetic or optical storage with gigabytes of capacity were not yet available. So, I just re-digitized from the film one of the images my body’s cross sectional cuts and greatly reduced the pixel-count in order to be able to share this image here.)
Reconstruction of such images from the projections is possible with the help of some calculus using the Fourier-transform. (Now I do not want you to get bored – interested readers can find the principles of the calculations in lecture notes of graduate image processing courses, such as Vladimir Székely‘s KÉPFELDOLGOZÁS (Image processing) at the Budapest University of Technology and Economics, Faculty of Electrical Engineering and Informatics, but perhaps e.g. the book of Wim van Drongelen – Signal Processing for Neuroscientists - provides a real, comprehensive description of the topic.) Anyway, at the moment the great French scientist, Joseph Fourier is very important for us from the point of view of this story . Besides establishing the Fourier-series expansion for approximating functions in mathematics he also contributed to physics a lot, especially in the study of vibrations and in heat-transfer. The basic equation of heat-conduction is also known as the Fourier-equation.
Of course one may ask, how thermal analysis of semiconductor device packages – the main topic of this blog – and non-destructive analysis techniques like X-raying or CAT-scan are connected? In fact in thermal testing we also have a non-destructive analysis method. If one applies a sudden, stepwise change of the heating power and one measures the transient reponse of the junction temperature, one obtains every available information about the junction-to-ambient heat-flow path of a semiconductor package. (Here the word ‘ambient’ may represent any kind of standard test environment such as a JEDEC standard still-air chamber or a cold-plate; in this post I do not want to deal with details of test environment). So, back to the thermal transient testing: in a basic ‘signals and systems’ course for EE-s the mathematically pure representation of the sudden, step-wise change of the heating power as EXCITATION is called the Heaviside-function, and the RESPONSE of a linear system is called the unit-step response function.
In thermal transient testing the JEDEC JESD51-1 standard describes how such a sudden change of heating power of a pn-junction (a semiconductor diode) can be achieved and the same standard also defines two methods with which the junction temperature transient – as a response to the change of powering – can be measured. With Mentor Graphics’ T3Ster equipment we implemented the so called static test method – but again, this is not the main point of discussion in this blog post.
Until 1988 junction temperature transients were measured but there was no widely published and widely used method of extracting the information hidden in the measured junction temperature transients. The great breakthrough was Vladmir Székely‘s work with his PhD student, Tran Van Bien in 1986-1989. This was the era, when all the theory of the so called NID method (network identification by deconvolution) and the concept of structure functions was born. (Yet another scientist who’s name is connected to different fields of engineering – image processing on one hand and thermal problems of semiconductor devices on the other hand…). The first, original paper on the topic was published by prof. Székely and T. V. Bien in Solid-State Electronics in 1988 (“Fine structure of heat flow path in semiconductor devices: A measurement and identification method“, doi:10.1016/0038-1101(88)90099-8) – since then this paper is the basis of any study dealing with thermal transient measurement based structural analysis of semiconductor device packages and thermal management solutions.
Life is becoming more complicated, if there are multiple heat-flow paths from the chip’s junction towards the environment. Typical examples are the large area IC packages where heat leaves the package through the large top surface of the package as well as through the leads of the package or even through the bottom surface of the package when e.g. underfill is also applied during the assembly. In such situations what we measure during thermal transient testing is the net thermal impedance of all these parallel heat-flow paths. If this is the case, it is really hard to create compact thermal models for the packages. Therefore in the DELPHI project of the EU hard thermal boundary conditions were defined: the top/bottom surfaces of the package under test have to be directly attached to a cold-plate while the other surface is left ‘intact’. This way the majority of the heat-flow is directed through the actual package surface which is attached to the cold-plate. All together there are four different test cold-plate configurations defined in DELPHI. Since dual cold-plates were used in DELPHI, these conditions are abbreviated DCP1, DCP2, DCP3 and DCP4. In the DELPHI project only the steady-state characterization and compact modeling was the target. It was the PROFIT project of the EU in which the dynamic characteristics of semiconductor device packages were studied, using the DCP1, 2, 3 and 4 test conditions – and structure functions were used to represent the properties of the given heat-flow path by means of thermal resistance and capacitance values.
Now, one may ask, what are the structure functions? There could be many answers to this question. A very academic answer is that structure functions mean an alternate representation of the junction-to-ambient thermal impedanceof a packaged semiconductor device. Another, more practical answer is that structure functions provide thermal capacitance – thermal impedance maps of such heat-flow paths. In other words: with the help of structure functions we can identify which major element of the heat-flow path has how much thermal capacitance and thermal resistance. If the heat-flow follows an essentially one-dimensional path, there is a one-to-one correspondence between thestructure functions and the actual structural elements of the heat-flow path.
Essentially 1D covers a lot – this definition does not restrict the structure functions to really one-dimensional features like a thin, long rod. A heat-spreading showing a really three-dimensional pattern in the Euclidian space may also be considered as a 1D spreading, if there is a coordinate transformation with which the spreading can be mapped to 1D function. (Yet another connection between the real 3D world and its lower dimensional projections…). Radial spreading, cylindrical spreading or conical spreading are the most obvious cases of such essentially 1D heat-flow patterns. In most power semiconductor devices packages we find essentially 1D hear-flow from the chip towards the ‘ambient’.
The DCP1, 2, 3 conditions and application of the structure functions as non-destructive means of structural analysis of packages very much reminds the X-ray/CAT scan technique of medical diagnostics. We have just one projection of the complex 3D structure of an IC packages, but in the direction where the heat flows we can see all the details of the heat-flow path, as if we have “X-rayed the package”. We hardly can recognize the bonding wires – neither can doctors see the capillary veins of the body. But if there is a die-attach void for example, it can be well detected, just like a broken leg for example which is clearly visible even on conventional X-ray images. Measuring with the different DCP setups of DELPHI/PROFIT, one can have Rth-Cth distribution snapshots of the package body from different directions – just like in CAT-scan. Research is still needed to find ways of reconstructing package models from structure functions measured under different test conditions, but I sincerely believe, one day we’ll have the solution. When this day arrives, we can celebrate the birthday of something which resembles ‘thermal CAT-scan’.
At this point let me wish a merry Christmas now to all of my patient readers who continued digesting this post. Last Sunday I lit the 4th candel on the Advent-wreath (see my previous blog post) and in 2 days I will light my LED string on our Christmas tree – as in Hungary we celebrate Christmas in the evening of the 24th of December – the holy night. Therefore I say good bye to everybody – until the next year when I will continue this topic.
Best regards: Andras
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