De Mey, Gilbert

Vermeersch, Bjorn

English

Ghent University Academic Bibliography

Microelectronics Reliability 46 (2006) 174–177Research NoteThermal impedance plots of micro-scaled devicesB. Vermeersch, G. De Mey *Department of Electronics and Information Systems (ELIS), University of Ghent, Sint Pietersnieuwstraat 41, 9000 Ghent, BelgiumReceived 21 February 2005; received in revised form 2 May 2005Available online 5 July 2005www.elsevier.com/locate/microrelAbstractThe complex thermal impedance Zth of a microelectronic heat source on the surface of a silicon wafer has been cal-culated semi-analytically as a function of the frequency. By representing the results in a Nyquist plot, almost perfectcircular curves are obtained. This result is analogous to complex loci of the dielectric constant obtained for somematerials. 2005 Elsevier Ltd. All rights reserved.1. IntroductionIn present day electronics and microelectronics, ther-mal problems are found to become more dominant dueto the high power densities. The latter being propor-tional to the clock frequency (for a digital circuit), it isexpected that thermal problems will need more andmore attention in the future.Usually the high operating temperature of an inte-grated circuit is considered as the only thermal problem.However, even if the total dissipated power is quitemoderate, the temperature distribution can still give riseto malfunctioning circuits. The reason is quite simple: alltransistors have temperature dependent characteristics.Even a temperature ﬂuctuation of a few degrees mayhave an inﬂuence on the electric behaviour of the circuit.It must be emphasised here that this problem is not lim-ited to uniform heating of a chip but is much moreimportant for non-uniform and time dependent heating.0026-2714/$ - see front matter  2005 Elsevier Ltd. All rights reservdoi:10.1016/j.microrel.2005.05.014* Corresponding author. Tel.: +32 9 264 33 86; fax: +32 9 26489 61.E-mail addresses: bjorn.vermeersch@ugent.be (B. Ver-meersch), demey@elis.ugent.be (G. De Mey).Unwanted couplings between diﬀerent parts of a circuitcan be induced thermally.Therefore, the need arises not to limit one self to thestudy of the thermal resistance Rth but also to the ther-mal impedance Zth of a transistor in order to deal withtime or frequency dependent problems.In this paper we limit ourselves to a linear thermalanalysis. Hence the temperature dependence of the ther-mal conductivity k or other thermal parameters has notbeen taken into account. Otherwise, the thermal conduc-tion problem becomes non-linear and strictly speaking athermal impedance cannot be deﬁned. Experimentally,thermal impedance measurements are always performedat a maximum junction temperature (above ambient) ofless than 20 C [1].The originality of the present research is proved bythe fact that only a few papers on related topics couldbe found in the literature [2]. The most interesting resultis that the impedance plot turned out to be circulargraphs. As a consequence, the AC thermal behaviouris characterised by a limited number of parameters (cen-tre and radius of the circle). For the point of view of reli-ability it might be useful to measure these parameters asa function of time in order to detect the reliability in theclose neighbourhood of the heat sources.ed.B. Vermeersch, G. De Mey / Microelectronics Reliability 46 (2006) 174–177 175Although a very limited number of papers also dis-played circular Nyquist plots, no one pointed out theperfect analogy with the theory of dielectrics publishedby Fuoss and Kirkwood long time ago [3].2. Calculation of the thermal impedance ZthDue to the small dimensions of the heat sources in-volved in microelectronics one can consider the semicon-ductor substrate as a half inﬁnite volume (z < 0) with aheat source on the top surface z = 0 (Fig. 1).In phasor notation the temperature distributionT(x,y,z) in the substrate satisﬁes the equation [4]:kr2T ðx; y; zÞ  jxCvT ðx; y; zÞ ¼ 0; ð1Þwhere k is the thermal conductivity (in Wm1K1), Cvthe thermal capacity per unit volume (in Jm3K1),j ¼ ﬃﬃﬃﬃﬃﬃ1p the imaginary unit and x = 2pf the angularfrequency (in Hz).For a point shaped heat source dissipating a power P(in W) and located on the surface z = 0 in the point(x 0,y 0, 0), the temperature distribution is given by:T ðx; y; zÞ ¼ P2pkrexp ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃjxCvkrr !; ð2Þwhere r is the distance between the ﬁeld point (x,y,z)and the source point (x 0,y 0, 0). The square root of j mustbe chosen asﬃjp ¼ ﬃﬃ2p2ð1þ jÞ in order to ensure conver-gence for r!1.If the power P is omitted and the factor 2 in thedenominator of (2) is replaced by 4, one will recognisethe Greens function or the fundamental solution ofEq. (1) in the three-dimensional space. The factor 2 isdue to the fact that the semiconductor material occupiesonly the half space z < 0.On the surface z = 0 the adiabatic boundary condi-tion has been assumed. The thermal conductivityk = 160 Wm1K1 of silicon is relatively high with re-spect to the value k = 0.025 Wm1K1 of the dry air ﬁll-Fig. 1. Rectangular heat source on top of hing (z > 0) in an electronic package. Taking this intoaccount, the adiabatic condition is suﬃciently accurate.Although this boundary condition may be questionablefor the DC analysis of the electronic package, for timedependent or AC problems it is suﬃcient to consideronly the temperature distribution inside the silicon. Wehave veriﬁed from our calculations that the AC temper-ature ﬁeld is completely decayed (i.e., jTj < 103jTjmax)at a distance of 100 lm from the heat source for frequen-cies f > 100 kHz.For a rectangular heat source with dimensions a andb (Fig. 1), and a power dissipation P or a power densityp ¼ Pab (in W/m2), the temperature distribution can befound by superposition:T ðx; y; zÞ ¼ p2pkZ a=2a=2dx0Z b=2b=2dy0exp ﬃﬃﬃﬃﬃﬃﬃjxCvkq ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðx x0Þ2 þ ðy  y0Þ2 þ z2q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðx x0Þ2 þ ðy  y0Þ2 þ z2q .ð3ÞThe integral (3) has been evaluated numerically by divid-ing the heat source area into N small sub areas. Thenumerical convergence was checked by using diﬀerentvalues of N. The singularity in the integrand of (3),which occurs for x = x 0, y = y 0 and z = 0, has been calcu-lated separately by an exact analytical integration [5].Strictly speaking, the thermal impedance is deﬁned asthe temperature diﬀerence between a heat source (as-sumed at a uniform temperature) and the temperatureof a heat sink (here T = 0 at inﬁnity) divided by the heatﬂow. In our situation the temperature of the heat sourceis not uniform at all. Hence the impedance has beenevaluated by taking (i) the average temperature and(ii) the maximum temperature (i.e. the maximum abso-lute value) of the heat source evaluated by (3).A Nyquist plot of both kinds of impedances is shownin Fig. 2.alf inﬁnite thermal conductor (z < 0).Fig. 2. Nyquist plot of thermal impedance Zth.176 B. Vermeersch, G. De Mey / Microelectronics Reliability 46 (2006) 174–177These results have been obtained for the thermalparameters of Si (k = 160 Wm1K1 and Cv = 1.784 ·106 Jm3K1) and a square shaped heat source of200 lm · 200 lm. However, by rearranging (3) in adimensionless form, it can be easily proved that thoseparameters have no inﬂuence on the shape of the Ny-quist plot. Just the frequency scale has to be adjustedif smaller heat sources or another semiconductor mate-rial is investigated.The same kind of Nyquist plots have been obtainedfor rectangular heat sources with aspect ratios a/b upto 10. The central frequency fc, where the imaginary partof Zth reaches its minimum value, has also been deter-mined. From all these data the following relationshipwas found by curve ﬁtting:fc ¼ 1.20 1abkCv. ð4ÞIt has been veriﬁed that (4) can even be applied with anaccuracy of 10% for aspect ratios a/b up to 48.A more careful inspection of both curves in Fig. 2 re-veals that a curve ﬁtting with a circle intersecting thehorizontal axis at 45, is almost perfect. Although itcan be proved mathematically that (3) does not give riseto an exact circular plot. The particular form of the Ny-quist curve corresponds with the experimental measure-ments performed on electronic packages. The highfrequency part of these results were found to be a circlewith the same intersection angles [1,6].Experimentally, the temperature of a microcircuit atchip level is usually measured by using sensor diodesor any other junction. These devices deliver a signal pro-portional to the average device temperature. Hence, theuse of the impedance using the average temperature ismore sound (Fig. 2).3. DiscussionThe most remarkable fact of the Nyquist plot ofFig. 2 is the almost perfect circular shape. Similar resultshave been reported in the literature many years agoabout the complex dielectric constant of some materials[7,8]. A Nyquist plot of this quantity is called the Cole–Cole plot.The explanation for this phenomenon was the exis-tence of a Debye relaxation mechanism in the dielectriclayer. If one single time constant was taken into account,the Nyquist plot of the complex dielectric constantturned out to be a circle with its centre point locatedon the real axis. Experimentally the Nyquist plots wereindeed circles, but with their central point above the realaxis just like the plot of Fig. 2. This aspect has been ex-plained successfully by introducing a probability distri-bution for the relaxation time constants [3].For the thermal impedance plot on the other hand,similar plots have been obtained although no probabil-ity distribution has been used. All parameters like k,Cv were kept constant. It proves that a homogeneousthermal conductor involves a spectrum of thermal timeconstants rather than a single value. This conclusionalso agrees with the experimental results published bySze´kely et al. obtained from transient temperature mea-surements on electronic packages [9–11].4. ConclusionThe Nyquist plot of the thermal impedance of a rect-angular heat source on a half inﬁnite thermal conductorhas been evaluated numerically. All the Nyquist plotsshow a circular curve just like the Cole–Cole plots ob-served for dielectric layers involving Debye relaxationmechanism. Hence, the theories developed for dielectriclayers can be directly applied to the AC thermal imped-ance analysis, for example to determine the time con-stant distribution.References[1] Kawka P. Thermal impedance measurements and dynamicmodelling of electronic packages. PhD thesis, University ofGhent, 2005.[2] Sze´kely V. Identiﬁcation of RC networks by deconvolu-tion: chances and limits. IEEE Trans Circuits Syst—I:Fund Theor Appl 1998;45(3):244–58.[3] Fuoss RM, Kirkwood JG. Electrical properties of solids:VIII dipole moments in polyvinyl chloride-diphenyl sys-tems. J Am Chem Soc 1941;63:385–94.[4] De Mey G. Various applications of the boundary elementmethod. Oradea: Ed. University of Oradea, Romania;2002.[5] Dean DJ. Thermal design of electronic circuit boards andpackages. Ayr, Scotland: Electrochemical publications;1985.[6] Kawka P, De Mey G, Napieralski A. Electronic packagethermal description by dielectric polarization theory. In:Proceedings of the 11th MIXDES conference 2004, June24–26, Szczecin, Poland. p. 287–90.B. Vermeersch, G. De Mey / Microelectronics Reliability 46 (2006) 174–177 177[7] Cole KS, Cole RH. Dispersion and absorption in dielec-trics: I––alternating current characteristics. J Chem Phys1949;9:341–51.[8] Jonscher AK. Dielectric relaxation in solids. J Phys D:Appl Phys 1999;32:R57–70.[9] Sze´kely V, Van Bien T. Fine structure of heat ﬂow path insemiconductor devices: a measurement and identiﬁcationmethod. Solid-State Electron 1988;31:1363–8.[10] Sze´kely V. Thermal monitoring of microelectronic struc-tures. Microelectron J 1994;25(3):157–70.[11] Sze´kely V, Rencz M. Thermal investigations of microsys-tems. In: Proceedings of NATO advanced research work-shop on multichip models with integrated sensors, May18–20, 1995, Budapest, Hungary. p. 1–10.

Thermal impedance plots of micro-scaled devices

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Thermal impedance plots of micro-scaled devices

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