OPTIMIZATION OF SI/SIGE MICROREFRIGERATORS FOR HYBRID SOLID- STATE/LIQUID COOLING

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1 Proceedings of IPACK007 ASME InterPACK '07 July 8-1, 007, Vancouver, British Columbia, CANADA IPACK OPTIMIZATION OF SI/SIGE MICOEFIGEATOS FO HYBID SOLID- STATE/LIQUID COOLING Y. Ezzahri,. Singh, J. Christofferson, Z. Bian, and A. Shakouri Department of Electrical Engineering, University of California Santa Cruz, California USA ABSTACT Monolithic Solid-state microrefrigerators have attracted a lot of attention during last ten years since y have potential to solve some of problems related to localized heat generation and temperature stabilization in optoelectronic, microelectronics or even biological microchip applications. Combination of solid-state cooling with or conventional techniques like liquid cooling, offers an additional degree of freedom to control both overall temperature of chip and to remove hot spots. We present a new approach based on Thermal Quadrupoles Method to model behavior of a single stage Si/SiGe microrefrigerator in DC operating regime. The sensitivity and precision of this method come from its analytical expressions, which are based on solution of Fourier s heat diffusion equation in Laplace domain. The microrefrigerator top surface temperature is calculated by taking into account all possible mechanisms of heat generation and conduction within entire device. The 3D heat and current spreading in substrate is taken into account. The parasitic heat leakage to cold junction due to heat generation and heat conduction within metal lead is also considered. The oretical results are n compared with experimental ones for several microrefrigerator sizes. A good agreement is found between m. Based on simulations, structure of microrefrigerator is optimized for lowering overall chip temperature and removing high power density hot spots in several applications in conjunction with liquid cooling techniques. INTODUCTION Thermoelectric coolers are used for temperature stabilization and control of microelectronic and optoelectronic components. Due to an increasing demand for localized cooling and temperature control in last ten years, nanostructured cooling devices have attracted a lot attention for ir potential use in hot spot removal in microelectronic and optoelectronic devices. The nanostructured materials have manifested very interesting rmoelectric properties, enabling m to have a figure-of-merit ZT, exceeding 1 at room temperature. The ZT is σ S defined by ZT T, where σ, S, and β are respectively β electrical conductivity, Seebeck coefficient and rmal conductivity of rmoelectric material. SiGe is a well known bulk rmoelectric material for high temperature power generation applications. ecently, Si/SiGe superlattice structures have been investigated for room temperature applications [1]. Si-based microrefrigerators are attractive for ir potential monolithic integration with Si microelectronics. Anor advantage of se devices is ir ability to be combined with or conventional liquid cooling techniques [], which can offer an additional degree of freedom to both remove background heating and hot spots. In this paper, we use Thermal Quadrupoles Method (TQM) [3] to optimize microrefrigerator maximum cooling and maximum cooling power density as a function of several geometrical parameters and material properties; we will focus especially on properties of active Si/SiGe layer and substrate. In this single element devices, Joule heating and heat conduction in contact metallization to cold junction could reduce device performance significantly and geometry and thickness of this metallization should be optimized. The TQM is a general analytical model that can be used to calculate electrical and rmal responses in AC regime, thus making it possible to distinguish, in some cases, Peltier effect from Joule effect. In case of a pure sine wave electrical excitation, Peltier effect appears at same frequency as operating current, whereas Joule effect appears at double frequency. The precision of TQM allows its application in detailed characterization of rmoelectric 1 Copyright 007 by ASME

2 material properties [4]. This method has been used to model behavior of a conventional rmoelectric couple (Bi Te 3 ) [5], and recently behavior of Si/SiGe microrefrigerator [6]. The model presented here uses limit of TQM at long times (i.e. steady-state behavior). Thermophysical properties of microrefrigerator are assumed to be temperature independent. In this model, microrefrigerator top surface temperature variation is calculated by taking into account all possible mechanisms of heat generation and conduction within entire device. 3D Heat and current spreading in substrate is taken into account. Heat generation in metal leads connected to cold junction and also heat conduction through se leads are also calculated via TQM. All se features make this model more consistent, more complete and closer to real devices, than previous work of D. Vashaee et al [7, 8], and Y. Ezzahri et al [6]. NOMENCLATUE β i : rmal conductivity of medium i. σ i : electrical conductivity of medium i. Σ i : cross sectional area of medium i. θ in : temperature at input of medium. θ out : temperature at output of medium. φ in : heat flux at input of medium. φ out : heat flux at output of medium. h eff : effective heat transfer coefficient describing heat conduction within substrate. I e : amplitude of excitation current. l x l b l m : length of metal lead. J: Joule heat source. P: Peltier heat source. r: radius of contact disc at microrefrigerator/ substrate interface. : rmal resistance. ele : electrical resitance. S i : absolute Seebeck coefficient of medium i. T 0 : room temperature. w x w b w m : width of metal lead. ZT: figure of merit. SAMPLE DISCIPTION Figure 1 (a) shows a Scanning Electronic Microscope (SEM) picture of a set of Si/SiGe microrefrigerators with five different sizes. Figure 1 (b) illustrates schematic crosssectional view of a Si/SiGe microrefrigerator that we consider in our simulation. The detailed composition of device can be found in references [9, 10]. Figure 1: (a) SEM picture of a set of microrefrigerators with five different sizes, and (b) Schematic diagram of microrefrigerator cross sectional view. Arrows indicate direction of electron flux. X is direction of metal lead at top side, Z is cross plan direction, and Y is complementary direction. THEMOELECTIC AND THEMIONIC COOLING In a Si/SiGe microrefrigerator, Peltier cooling occurs at top metal layer/sige active layer junction and also at buffer layer/substrate junction when device is fed by a current. The density of heat exchanged with surrounding medium is characterized by effective Seebeck coefficient difference at se junctions, and it is proportional to both current intensity and junction temperature [1]. Because of difference in Seebeck coefficient values at various interfaces Peltier cooling or heating is created depending on direction of electrical current [1]. In case where active layer is a superlattice structure, in addition to rmoelectric cooling, re is a rmionic cooling as it was shown by A. Shakouri et al [11-1] and G. D. Mahan et al [13], which is an evaporative selective hot electrons filtering effect. Assuming small current densities, we can define an effective Seebeck coefficient for solid-state rmionic cooling analogous to linear rmoelectric effects. Heating or cooling density at interfaces can n be considered as a linear function of current [11-14]. In our model, we assume that effective Seebeck coefficient takes into account both rmoelectric and rmionic phenomena. Copyright 007 by ASME

3 THEOETICAL MODEL AND SIMULATION Our oretical model is based on limit of TQM at long times. Layer thicknesses are several orders of magnitude larger than mean free path of both electrons and phonons [15]. We can hence assume a diffusive transport regime, and Fourier Diffusion Heat Equation (FDHE) can n be used. When active layer is a superlattice, because individual layers within it are very thin, on order of nanometers, superlattice is considered as an effective medium. HEAT TANSPOT IN THE COSS-PLAN DIECTION OF THE MICOEFIGEATO Let us assume a one-dimensional heat transfer in Laplace domain for a passive, linear, homogeneous and isotropic medium. The solution of FDHE under adiabatic conditions gives a linear relationship between temperature-flux vectors at both ends of medium in form: θ in 1 θ i out () φin φout ti where: i, β i, t i, and Σ i are respectively rmal βiσ conductivity, thickness and cross sectional area of θin isormal surface in medium. θout and φ are in φout Laplace transforms of temperature and heat flux vectors at input and output of medium, respectively. The thickness of active SiGe layer is very small compared to that of substrate; moreover, all Peltier sources are uniform at all junction plans. We thus consider heat transfer across microrefrigerator to be one-dimensional in cross-plan direction of device. We neglect heat transfer at side surface area around mesa due to convection and radiation. We assume adiabatic conditions at se surfaces. This can be justified due to small dimensions of microrefrigerator and marginal cooling temperature reduction. Our structure is formed of four essential layers; transfer matrix of each layer can be written in form: θ in 1 θ i i out - J ( ) i φin φout The term J i indicates internal Joule heating source inside each layer, and is given by: ele J I ele i i i e () 3 and I e are electrical resistance of each layer, and amplitude of excitation current respectively. The subscript im, c, s, b stands for metallic layer, cap layer, active SiGe layer and buffer layer, respectively. HEAT TANSPOT WHITIN THE SUBSTATE egarding local character of microrefrigerator, silicon substrate underneath it is considered rmally thick and its effect will be contained in what is called esistance of constriction or spreading. Constriction and spreading resistances exist whenever heat flows from one region to anor of different cross sectional area. The term constriction is used to describe situation where heat flows out from a large cross sectional region into a narrower one, and term spreading is used to describe opposite case where heat flows out of a narrow region into a larger cross sectional area. Approximating both of microrefrigerator and substrate with a cylindrical geometry, rmal constriction/spreading resistance is given by [3]:,1 Sub 8 3π β r where r is radius of contact disc between two mediums, and β s is rmal conductivity of substrate. In fact, expression of this constriction/spreading resistance depends on form of temperature and heat flux distributions in [0, r] interval. Equation (4) is valid in case of uniform heat flux distribution in this interval which should match better physics. One should note that difference in expressions of constriction/spreading resistances for case of uniform heat flux distribution and case of uniform temperature distribution in [0, r] interval is only 8% [3]. In addition to rmal spreading inside substrate, re is an electrical current spreading. Joule heating is mainly localized at buffer-layer/substrate interface [16]. This electrical spreading is characterized by a spreading of electrical current density lines in substrate. The electrical constriction/spreading resistance is calculated in analogy with rmal resistance, and is given by equation: ele Sub s 8 3π σ r s where r is radius of contact disc between microrefrigerator and substrate, and σ s is electrical conductivity of substrate. HEAT TANSPOT WHITIN THE TOP SIDE CONTACT OF THE MICOEFIGEATO The top side metal lead is provided to carry current to cold junction, but it turns out that it is responsible for some part of Joule heating, which limits maximum cooling. The temperature change within this metal lead is dominant in longitudinal direction (X direction) and, refore, metal lead can be viewed as a rmal fin. To describe heat transfer within this metal lead, P. Wang et al [17] have solved directly heat diffusion equation in longitudinal direction with two appropriate boundary conditions, which are: (i) a constant temperature at interface metal lead/ microrefrigerator top surface, and (ii) a zero heat flux at or end of metal lead. In our analysis, we still use TQM in longitudinal direction of metal lead by assuming same boundary conditions as P. Wang et al [17]. As for microrefrigerator, top metal lead surface is considered adiabatic, but heat conduction from bottom surface of metal lead through SiN x layer and buffer layer into silicon substrate () 4 () 5 3 Copyright 007 by ASME

4 Figure : Thermal quadrupole network associated with heat transfer within whole SiGe microrefrigerator in a steady state regime (see figure 1 (b)), is described by an effective heat transfer coefficient given by: 1 heff () 6, ( SiN + Buffer + Sub ) lmw x m, where SiNx, Buffer and Sub are rmal resistances of SiN x layer, buffer layer and substrate, respectively. Heat conduction inside SiN x layer and buffer layer is assumed to be one dimensional and perpendicular to bottom surface of metal lead. The corresponding rmal resistances are thus given by: tx SiN x β xlxwx () 7 tb Buffer βblbwb where t x, t b, β x, β b, are thicknesses and rmal conductivities of SiN x layer and buffer layer, respectively. l x l b l m and wx wb wm Σ are length and width of metal lead. Σ is cross sectional area of microrefrigerator. As length of metal lead is larger than its width, heat conduction downwards in silicon substrate is assumed to take form of two dimensional spreading (in Y-Z plane), and average value of spreading rmal resistance can be calculated as [17, 18]: where, Sub δ δ β Sublm π ε n t Sub wsub, and ε 1 sin w m wsub ( nπε ) n 3 tanh ( nπδ ) ( 8). t Sub, W Sub, and β Sub are thickness, width and rmal conductivity of substrate, respectively. One can note here that substrate underneath metal lead is considered as a finite medium at opposite of microrefrigerator device. This is because of relatively larger size of side metal lead. Application of TQM to heat transfer along metal lead, in combination with second boundary condition (zero heat flux at beginning of metal lead) allows us to get relation between heat flux at interface metal lead/ microrefrigerator and temperature variation at top in surface of later. In fact, it is easy to see that φ ML 0 leads to: out ele β mt h m eff φml φc1 ML th lmi e lmh β eff mt m h eff β mtmwmheff th lmθ C1 () 9 β mt m ele lm where ML, t m and σ m are thickness and σ w t m m m electrical conductivity of metal lead, respectively. More detail on how we obtain equation (9) by application of TQM, will be presented in a future work [19]. θ C1 represents temperature variation at top surface of microrefrigerator. The first term of equation (9) describes Joule heating generation within metal lead whereas second term describes heat conduction due to temperature gradient within it. Both of those effects will reduce effective cooling rate of microrefrigerator and thus degrade cooling performance. We also demonstrated that ohmic contact Ohm resistance C between cap layer and metallic layer is anor important limiting factor on performance of microrefrigerator. THEMAL QUADUPOLE EPESENTATION Figure above, illustrates quadrupole system associated with whole microrefrigerator device. The application of Kirchhoff laws to this system, allows us to get a matrix relation, which represents heat transfer in whole θc1 structure between θc and φ, temperature-heat flux C1 φc vectors at top metallic layer and interface buffer layer/substrate, respectively: 4 Copyright 007 by ASME

5 θ 1 C m c s θ b C φc φc PBS J Sub b 1 + s m c s J m b c 0 1 J 0 1 Jb J s 1 c m m Jc Jm ( 10) 0 1 Jc + PMC + J MC J m where: P MC ( S M Ss ) I et0, PBS ( Ss SSub ) I et0 ele Ohm Ohm J Sub SubI e, J C C I e ( 11) θ C φc,1 Sub S M, S s, and S Sub are absolute Seebeck coefficients of metal layer, active SiGe layer, and substrate, respectively. The effective active layer Seebeck coefficients include both rmoelectric and rmionic contributions in case where active layer is a Si/SiGe superlattice, as we have mentioned above. T 0 is average temperature of junction that we take to be equal to room temperature 300K [6]. Combination of equations 9, 10, and 11 allows us to get expression of microrefrigerator top surface temperature variation θ C1 as a function of excitation current amplitude I e, as well as all physical and geometrical parameters of whole device. The final point we shall discuss in this section is related to cooling power density. This quantity is defined as heat load mass added to microrefrigerator top surface for which temperature at this surface becomes zero. ESULTS AND DISCUSSION Before we present result of optimization, we start our discussion by ensuring consistency of our model. For this we have used it to fit some of experimental data. Figure 3 shows results of this operation for two microrefrigerators with different device sizes, those microrefrigerators differ only by nature of active SiGe layer. In figure 3 (a) active layer is an alloy, whereas in figure 3 (b), active layer is a Si/SiGe superlattice. The thickness of active layer is same for both devices and equals 3 µm. The cooling of devices was measured using standard type E micrormocouple with a tip size of 50 µm [0]. As we can see in Figure 4 above, both microrefrigerators with bulk SiGe and Si/SiGe superlattice produce almost same cooling for each device size. Although separate 3ω measurements [1] show that bulk SiGe alloy has a rmal conductivity which is about 30% lower than that of Si/SiGe superlattice, later compensate this difference in power factor as we can see in table 1 below. This table recapitulates material parameters of bulk SiGe alloy and Si/SiGe superlattice measured and estimated at room temperature for a 60 60µm² s device size. This is because of this compensation effect why global cooling of microrefrigerators is same for same device size []. The power factor σs in case of Si/SiGe superlattice is almost 4% higher than case with bulk SiGe alloy. Figure 3: Measured cooling on thin film bulk SiGe microrefrigerator (a) and Si/SiGe superlattice microrefrigerator (b) for different device sizes, and ir corresponding fit based on Thermal Quadrupoles Method. Indeed, physics behind cooling effect is different for two microrefrigerators; for first one with SiGe alloy, cooling is purely rmoelectric, whereas for second one with Si/SiGe superlattice, in addition to rmoelectric cooling, re is a rmionic cooling, which consists on a selective evaporative emission of hot electrons from cathode (cap layer) through barrier (superlattice layer) to anode (buffer layer). The model fits very well big device sizes, but fails to fit small sizes (less than 50x50µm ). In fact, for small device sizes, because of rmal mass of micro-rmocouple, cooling is underestimated. 5 Copyright 007 by ASME

6 Table 1: Material parameters for bulk SiGe alloy and Si/SiGe superlattice. Maximum cooling temperature of microrefrigerator devices based on some of material is also given. Typical microrefrigerator device size is µm and thin film thickness is ~3 µm. ( ) refers to in-plane material properties and ( ) refers to cross-plane material properties. The estimated cross-plane power factors and ZTs for superlattices are based on measured maximum cooling of microrefrigerators and comparison with identical thin film devices based on alloy material. * refers to values used to get best fit for all microrefrigerator sizes. ^ refers to calculated power factors and ZTs based on best fit parameters. Material Si 0.8 Ge 0. alloys T max 4.0K Si/Si 0.75 Ge 0.5 (3nm/1nm) T max 4.K S (µv/k) 10 10* 00-0 ( ) 35* 180 ( ) σ (Ω/cm ) * 384 ( ) 384* β (W/m/K) 5.9 6* ( ) 8* S σ (10 3 W/K /m) 1.6* 1.7^. ( estimated) 1. ( ).1^ ZT ^ ( estimated) 0.080^ Now after model validation, let us see how we can use model to optimize cooling performance of microrefrigerator. To study how maximum cooling and maximum cooling power density are affected by geometrical and physical properties of microrefrigerator and to find methods of optimizing its performance, a series of simulation are performed and y are presented below. We start by discussing dependence of maximum cooling and maximum cooling power density on both cross sectional area of microrefrigerator and thickness of active layer. Figure 4 shows variation of maximum cooling (a) and maximum cooling power density (b) as a function of cross sectional area of microrefrigerator and thickness of active SiGe layer. As we can see in figure 4 (a), maximum cooling is increasing monotonically with respect to variables, i.e. optimum device size increases by increasing active layer thickness and vice versa. On or hand, maximum cooling power density (see figure 4 (b)) is decreasing monotonically with respect to variables as it is expected. The maximum thickness of Si/SiGe superlattice depends on growth time and for MBE growth we can take value of 10 µm. We take this thickness as fixed, and we look to behavior of maximum cooling and maximum cooling power density as a function of microrefrigerator cross sectional area for different values of Ohmic contact resistances at interface metal layer/cap layer. It turns out that this parameter is most important parasitic factor limiting cooling performance of microrefrigerator [0]. Figure 4: Calculated microrefrigerator maximum cooling (a) and maximum cooling power density (b) as a function of microrefrigerator cross sectional area and active SiGe layer thickness. In figure (a), black contours show maximum cooling value within range of used variables, and in figure (b), black contour show a particular value of 000 W/cm just as a reference. In figures 5 (a) and (b), we have plotted variation of maximum cooling and maximum cooling power density at microrefrigerator top surface, respectively, as a function of microrefrigerator cross sectional area for different values Ohm of C for realistic case with top side heat leakage within metal lead considered. In both figures, re is an optimum size of microrefrigerator. The results are very sensitive to ; by increasing contact resistance, Ohm C maximum cooling and maximum cooling power density decreases and optimum size increases. The existence of an optimum device size can be understood as a consequence of interplay between many competing effects. In fact main factors limiting maximum cooling are total electrical and rmal parasitic resistances at top side of cooler and also spreading 6 Copyright 007 by ASME

7 resistances in substrate. Small size devices have large electrical and rmal resistances, but y have a small optimum current for maximum cooling and a low rmal leakage. On or hand, for large devices, electrical and rmal resistances are small; optimum current for maximum cooling is large, and rmal leakage is more significant. Since heating terms (due to Joule effect) depend quadratically on current, while cooling terms (due to Peltier effect) depend linearly on current, a trade-off between se parameters gives an optimum device size. microrefrigerators, let us take a look at rmophysical properties. We will study effect of rmal and electrical conductivities of active SiGe layer and substrate. In figure 6, we have plotted variation of maximum cooling (a) and maximum cooling power density (b) of a µm² microrefrigerator, as a function of substrate rmal conductivity and active SiGe layer rmal conductivity. By increasing rmal conductivity of active SiGe layer, both maximum cooling and maximum cooling power density decrease. On or hand, by increasing rmal conductivity of substrate, maximum cooling power density increases, whereas maximum cooling decreases to rich a minimum and n starts to increase very slowly to saturate at about 1000 W/m/K. After this value, maximum cooling of microrefrigerator becomes almost insensitive to value of substrate rmal conductivity. These results show importance of using a high rmal conductivity substrate and a low rmal conductivity material as an active layer. Figure 5: Calculated maximum cooling (a) and and maximum cooling power density (b) as a function of device cross section area for different Ohmic contact resistances values. In order to optimize cooling performance of microrefrigerator device, we set value of Ohmic contact resistance at metal/cap layers interface equal to a small realistic value which is about 10-7 Ω.cm. For this value, optimum device size for maximum cooling is about µm as we can see in figure 5 (a). In next discussions, we will keep this device size fixed as well as value of. Ohm C Now that we have discussed, in some detail, how geometrical parameters affect cooling performance of Figure 6: Calculated maximum cooling (a) and maximum cooling power density (b) of a µm² device with 10 µm thick active SiGe layer as a function of active SiGe layer rmal conductivity and substrate rmal conductivity ( inset). In each case, solid line corresponds to current 7 Copyright 007 by ASME

8 realistic device with top side heat leakage, and dashed line corresponds to case without top side heat leakage. In figures 7 (a) and (b), we show effect of electrical conductivity of active SiGe layer and substrate, respectively, on maximum cooling of a 50x50µm device. The general trend is same, by increasing electrical conductivities, maximum cooling of microrefrigerator increase. The dependence on rmal and electrical conductivities of substrate and active SiGe layer can easily be explained. Increasing rmal conductivities increases heat conduction between top and bottom of microrefrigerator, while increasing electrical conductivities decreases Joule heating. The latter enhances cooling performance while former degrades it. Figure 7: Calculated maximum cooling of a µm² device with 10 µm thick SiGe layer as a function of active SiGe layer electrical conductivity (a) and substrate electrical conductivity (b). In each case, solid line corresponds to current realistic device with top side heat leakage, and dashed line corresponds to case without top side heat leakage. Figure 8: (a) Calculated maximum cooling as a function of active SiGe layer rmal conductivity and Seebeck coefficient. The black contour show value of 30 K. (b) and (c) correspond to calculated maximum cooling and maximum cooling power density, respectively, as a function of active SiGe layer ZT. The solid line corresponds to ZT change due to rmal conductivity variation, and dashed line corresponds to ZT change due to Seebeck coefficient variation, electrical conductivity was kept constant. 8 Copyright 007 by ASME

9 Finally, let s study maximum cooling and maximum cooling power density as a function of active layer factor of merit ZT including effect of rmal conductivity and Seebeck coefficient. In figure 8 (a), we have plotted calculated maximum cooling of microrefrigerator as a function of active layer rmal conductivity and Seebeck coefficient. The maximum cooling increases by increasing Seebeck coefficient and/or decreasing rmal conductivity as expected. The black line points value of 30 K (10% of room temperature). At cooling temperatures below ~30K, our assumption on temperature independence of physical properties of microrefrigerator are still valid as well as fact that we can still keep Peltier cooling power at interfaces independent of cooling temperature (model linearity). Figures 8 (b) and (c) show calculated microrefrigerator maximum cooling and maximum cooling power density, respectively, as a function of active layer ZT. We have plotted this dependence for two cases; first case corresponds to variation of ZT due to variation of rmal conductivity, and second case corresponds to variation of ZT due to variation of Seebeck coefficient. The electrical conductivity is assumed to be constant. To be able to compare two parameters, power factor σs is increased by same factor by which rmal conductivity β is decreased. As we can see in figure 8, both maximum cooling (b) and maximum cooling power density (c) are more sensitive to power factor variation than to rmal conductivity variation. This behavior proves advantage of using a superlattice structure as an active layer. As we have seen above, with se structures we can achieve higher power factors. In figure 9, cooling (a) and maximum cooling power density (b) of microrefrigerator are plotted as a function of excitation current for two different values of active SiGe layer figure of merit ZT. Figure 9: Calculated microrefrigerator cooling (a) and maximum cooling power density (b) as a function of applied current when active layer figure of merit ZT is increased by a factor of 5. The insets show performance curves for current devices. The insets, with dashed line, correspond to existing SiGe material with ZT0.1, we have a maximum cooling of about 7 K and a maximum cooling power density of 485 W/cm. Multiplying ZT by 5 allows a maximum cooling of about 4.75 K and a maximum cooling power density of 1188 W/cm, which are more than 00% higher for cooling and more than 150% higher for maximum cooling power density, than current values. We can see also that optimum current is almost doubled. Figure 10: (a) Schema of model configuration for combining microrefrigerator cooling with liquid cooling. (b) Variation of microrefrigerator cooling as a function of applied current for a heat load of 1000 W/ cm, in comparison with case of zero heat load ( inset). 9 Copyright 007 by ASME

10 In figure 10, we give an example to demonstrate microrefrigerator s effectiveness to remove hot spots in an integrated circuit, and furr advantage we can get by combining this technique with liquid cooling technique. We assume a µm hot spot with a heat load of 1000 W/cm located at center of a (1cm 1cm 500µm) microelectronic chip with an average heat load of 100 W/cm. In this simulation, we calculate temperature profile on top of chip with and without microrefrigerator integrated underneath hot spot. The heat sink is assumed to be operating at 70 C with a Th Chip convection coefficient of 1.3 W/cm /K []. Let s assume, A, and h are rmal resistance of chip, its cross sectional area, and convection coefficient at its back side, respectively. T ave, and ϕ ave are average temperature and heat flux at chip top side. T, and ϕ are room temperature and heat flux at chip back side (heat sink). We can write T ave as a function ofϕ ave : Th 1 Tave T Chip + ϕ ave ( 1) ha For a silicon chip with a rmal conductivity of 130 W/m/K, o chip heats up to T ave T 81 C. egarding local character of hot spot, temperature at its top side is calculated assuming chip as a semi-infinite medium, and n is expressed as [3]: Sub 8r 0 ϕ HS THS Tave π r0 ϕ HS Chip ( 13) 3πβChip where r A 0, β π Chip is rmal conductivity of silicon chip, andϕ HS is heat load density at top of hot spot. Using this expression, we find that hot spot heats o up to T HS T 83 C, C above average temperature at front side of microelectronic chip. The microrefrigerator integrated underneath hot spot is assumed to have a 10 µm SiGe layer with a factor of merit ZT0.5 with side metal contact geometry optimized as described earlier in paper. When we integrate this microrefrigerator at hot spot and apply an optimum current of 570 ma, hot spot cools down to ~79 C, which is C below chip average temperature. Figure 10 (b) recapitulates this behavior, where we can see that even with a heat load of 1000 W/cm, microrefrigerator allows us to have a maximum cooling of ~4 C, whereas for a zero heat load at top microrefrigerator surface, maximum cooling was ~5 C. Improving convection conditions by using for example direct liquid cooling at back side of chip makes whole chip temperature drops down even more. This simple example shows clearly effectiveness of microrefrigerators in removing hot spots from microelectronic chip. If we wanted to lower maximum temperature of chip (from 83 C to 81 C) by only increasing power of liquid cooling system, heat sink had to dissipate an additional.5w, while solution that includes monolithic microrefrigerator device require only a power of ~0.37W. CONCLUSION The Thermal quadrupoles method is used to calculate performance of Si/SiGe based microrefrigerators in steady state regime. The maximum cooling and maximum cooling power density of device were studied as a function of different geometrical and material properties. We emphasized effect of substrate and active SiGe layer properties. Simulation results match well existing experimental data and y show existence of an optimum device size and an optimum active layer thickness for maximum cooling performance. Existence of optimum size is closely related to top side heat leakage and electrical spreading resistance in substrate. Simulations also demonstrate effect of ohmic contact resistance at metal /cap layer interface. This parameter is mainly dependant on cleanliness of sample surface and metallization and annealing conditions. Minimizing this parameter by increasing doping in cap layer or decreasing potential barrier between metallic layer and cap layer is very important. The localized cooling performance of Si/SiGe based microrefrigerators makes m a potential candidate to be combined with liquid cooling techniques which offer an additional degree of freedom for rmal management in different length scales. ACKNOWLEDGMENTS The authors would like to thank IFC for supporting this work. EFEENCES [1]: D. M. owe, Handbook of Thermoelectrics, CC, []: A. B. Cohen, M. Arik, and M. Ohadi, Proceeding of IEEE, 94, (006). [3]: D. Maillet, S. André, J. C. Batsale, A. Degiovanni, and C. Moyne, THEMAL QUADUPOLES: Solving Heat Equation through Integral Transforms (John Wiley & Sons, 000). [4]: L. D. Patino-Lopez, S. Grauby, Y. Ezzahri, W. Cleays, and S. Dilhaire, Proceeding of 5 rd International Conference on Thermoelectrics, Vienna, Austria, August 6-10, (006). [5]: L. D. Patiño-Lopez, PhD Thesis, ON 79, University Bordeaux 1, (004). [6]: Y. Ezzahri, S. Dilhaire, S. Grauby, L. D. Patiño-Lopez, W. Claeys, Y. Zhang, Z. Bian, and A. 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