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# Thermal Metamaterials for Heat Flow Control in ElectronicsPUBLIC ACCESS

[+] Author and Article Information
Ercan M. Dede

Electronics Research Department,
Toyota Research Institute of North America,
1555 Woodridge Avenue,
Ann Arbor, MI 48105
e-mail: eric.dede@toyota.com

Feng Zhou, Paul Schmalenberg, Tsuyoshi Nomura

Electronics Research Department,
Toyota Research Institute of North America,
1555 Woodridge Avenue,
Ann Arbor, MI 48105

Contributed by the Electronic and Photonic Packaging Division of ASME for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received September 22, 2017; final manuscript received December 19, 2017; published online March 2, 2018. Assoc. Editor: Kaushik Mysore.

J. Electron. Packag 140(1), 010904 (Mar 02, 2018) (10 pages) Paper No: EP-17-1093; doi: 10.1115/1.4039020 History: Received September 22, 2017; Revised December 19, 2017

## Abstract

Rapid advancement of modern electronics has pushed the limits of traditional thermal management techniques. Novel approaches to the manipulation of the flow of heat in electronic systems have potential to open new design spaces. Here, the field of thermal metamaterials as it applies to electronics is briefly reviewed. Recent research and development of thermal metamaterial systems with anisotropic thermal conductivity for the manipulation of heat flow in ultra-thin composites is explained. An explanation of fundamental experimental studies on heat flow control using standard printed circuit board (PCB) technology follows. From this, basic building blocks for heat flux cloaking, focusing, and reversal are reviewed, and their extension to a variety of electronics applications is emphasized. While device temperature control, thermal energy harvesting, and electrothermal circuit design are the primary focus, some discussion on the extension of thermal guiding (TG) structures to device-scale applications is provided. In total, a holistic view is offered of the myriad of possible applications of thermal metamaterials to heat flow control in future electronics.

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## Introduction

The revolutionary field of electromagnetic metamaterials [13] served as a foundation for the field of thermal metamaterials, which has grown significantly within the last decade. Specifically, based on the established coordinate transformation approach [48], the idea of heat flux manipulation for novel thermal function has been expanded greatly. A highly influential experimental demonstration [9] used engineered materials for heat flux manipulation including cloaking, focusing, and reversal. This study concentrated on conduction heat transfer effects in relatively thick, ∼5 cm depth, multilayer composites. The sole heat flux cloak concept was then extended to a two-dimensional (2D) planar structure in Ref. [10], again under assumed pure heat conduction. In parallel, ultra-thin, ∼500 μm thick, composites in commercially fabricated printed circuit board (PCB) form were exploited in Ref. [11] to realize the three heat flux cloaking, focusing, and reversal basic complimentary functions, while considering both conduction and convection effects, with an aim toward eventual electronics applications. Building from early computational heat flow control anisotropic composite design optimization studies [12], and in contrast with either a coordinate transformation [7] or a scattering cancellation [13] approach, additional work on the development of a highly versatile gradient-based numerical optimization method for the design of thermal-composite metamaterials in arbitrary geometries was described in Ref. [14]. The central idea behind anisotropic composite (or thermal metamaterial) engineering for heat flow control is to design thermally conductive paths in a heterogeneous material by orienting high thermal conductivity fibers or asymmetric particles in a preferred direction within a low thermal conductivity matrix. As a result, heat tends to flow in a direction generally parallel to the fiber or particle long axis, while heat flow normal to the fiber direction is typically substantially reduced. More recently, the literature has become rich with follow on experimental demonstrations of thermal metamaterials in both 2D [1517] and three dimensions (3D) [18], as well as multiphysics (i.e., thermal–electrical) studies [19,20], thermal camouflage concepts [21], and experiments on active heat flux alteration [22,23], to name a few. However, while these investigations show great potential, the majority of studies are fundamental in nature and provide limited concrete examples of how to extend basic thermal metamaterial functions to practical electronics applications.

Regarding thermal management for PCB-based electronics, the existing macroscale approaches to controlling heat transport include the use of through-hole thermal vias in communication with large-area thick metal planes or cores [24], creating a cavity in the PCB to further reduce through-plane thermal resistance to a heat sink [24], using high thermal conductivity metal/graphite inlays [25], or novel PCB-based heat pipes [26]. Basic printed circuit board copper (Cu) plane layout for anisotropic thermal conductivity has also been exploited as a design strategy. An existing example, Ref. [27], is to enhance PCB thermal uniformity and reduce luminous flux output difference between light-emitting diode devices via Cu trace patterning. While anisotropic material design has been considered in the current state-of-the-art, detailed surface patterning of Cu traces through standard mass manufacturing methods for complex thermal metamaterial function (i.e., anisotropy) is a comparatively new concept and a logical first step before moving to the micro-to-nano device scale.

In this paper, the anisotropic thermal-composite design optimization method from Refs. [12] and [14] is briefly reviewed, and an experimental demonstration, Ref. [11], of the three thermal metamaterial basic building block functions (i.e., heat flux cloaking, focusing, and reversal) using commercial PCB technology is summarized. Three experimental studies of circuit board prototypes that exploit basic thermal metamaterial heat flow control building blocks are then explored to convey the importance of thermal metamaterial design for enhanced heat flow control in electronics. The first case study pertains to the temperature reduction of a heat-sensitive device located in close proximity to a higher heat generating device through the informed design of a conventional multilayer PCB [28]. A second study on the collection of low-grade waste heat for enhanced thermal energy harvesting using a PCB thermal-composite substrate is then outlined [29]. Finally, a recent third investigation into the electrothermal circuit design of a synchronous buck converter using PCB heat flow control technology is summarized. In the latter case study, ramifications of the thermal-composite design on electrical circuit response are emphasized. Some discussion on the extension of thermal guiding (TG) structures to device-scale advanced computing (e.g., Boolean-logic), as suggested in the literature [30], is additionally provided as a possible new research direction.

## Thermal Metamaterial Design

Aforementioned, a coordinate transformation approach is a popular method for the design of anisotropic thermal-composite metamaterials. Typically, the steady-state diffusion equation within a predefined region in original Cartesian, (x, y, z), coordinates is modified by stretching into a transformed (e.g., polar, spherical, or ellipsoidal) space. In this way, closed-form solutions of transformed material properties in the distorted space and coordinate system are obtained to achieve, for instance, heat flux cloaking. The reader is referred to the literature for numerous examples [48,17].

Here, instead of a coordinate transformation approach, thermal metamaterial design optimization based on composite theory is briefly reviewed. For a general anisotropic composite material subject to steady-state heat conduction, heat transport is governed by Fourier's law [31] and is written in a Cartesian coordinate system using Einstein summation convention asDisplay Formula

(1)$Q=−∂∂xi(kij∂T∂xj)$

where Q is the volumetric heat generation in the solid, T is the temperature state variable, and kij is the thermal conductivity tensor.

In 2D, the symmetric thermal conductivity tensor involves four conductivity coefficients, as follows:Display Formula

(2)$kij=[k11k12k21k22], where$
Display Formula
(3)$kij=kji i,j=1,2$

For a planar composite involving a high thermal conductivity fiber/inclusion, e.g., Cu, embedded in a low thermal conductivity matrix, e.g., FR-4 glass epoxy or similar, the thermal conductivity coefficients, kij, are expressed as a function of the fiber orientation angle, θ, asDisplay Formula

(4)$k11=k1 cos2θ+k2 sin2θ$
Display Formula
(5)$k22=k1 sin2θ+k2 cos2θ$
Display Formula
(6)$k12=k21=(k1−k2)sin θ cos θ$

where k1 and k2 are principal thermal conductivities parallel and transverse to the fiber direction, respectively, based on a preferred composite theory; refer to Refs. [3234] for options.

As thoroughly explained in Ref. [14], the optimization problem for the design of a thermal-composite metamaterial is then expressed as

$find γminimize fosubject to Eqs.(1)–(6)–θ=μπ−R2∇2μ+μ=γ−1≤γ≤1$

where fo is the optimization objective function, μ is a smooth scalar function that is transformed into θ to cover fiber angles spanning $−180 deg$ to $+180 deg$, γ is the material design variable constrained on the interval $[−1,1]$, and R is a filter radius for Helmholtz filtering, per [35], to avoid oscillations during the optimization procedure. The reader is referred to Refs. [12] and [14] for extensive details on implementation using commercial software [36] with an established optimizer [37].

The optimization objective, fo, depends on the selected thermal-composite metamaterial function that the designer ultimately seeks. An extensive treatment of this specific topic is provided in Ref. [14] with detailed objective function formulas for a heat flux cloak (i.e., shield), concentrator (i.e., focusing device), and rotator (i.e., reversal device). Simply stated, to design a heat flux cloak, the thermal compliance, and hence the temperature gradient, in a target region should be minimized, whereas for a focusing device it should be maximized. In the case of heat flux reversal, the direction of heat flow through a given spatial position is controlled by defining a multi-objective function that re-aligns the heat flux vector with a target unit vector, per [14].

Representative designs obtained via numerical optimization for each of these cases are shown in Figs. 1(a), 1(b), and 1(c) for an annular anisotropic thermal conductivity metamaterial object. High thermal conductivity fibers are designed within a lower thermal conductivity matrix inside this annulus. The thermal response of the annulus when placed in a background medium with isotropic thermal conductivity and subjected to a left-to-right decreasing temperature differential is shown in Figs. 1(d), 1(e), and 1(f). Observe that cloak fibers generally run parallel to each other and initially normal to the heat flux. Fiber orientation for rotators is more closely aligned with the heat flux guiding it to a new angle, while concentrator fibers fall into each other causing abrupt ends to some thermal traces as spacing becomes insufficient. A description of a fundamental experimental demonstration in commercial PCB form for each of these basic thermal metamaterial devices follows.

###### Fundamental Demonstration in Ultra-Thin Printed Circuit Board.

The optimization method is a highly flexible approach to the design of anisotropic thermal-composite metamaterials. This heat flux manipulation is readily achievable using standard commercial PCB technology. In Ref. [11], a fundamental demonstration is provided using 508 μm thick RO4350B substrate material as the “matrix,” $km=0.69$ W/(m-K), with 1 oz (i.e., 35 μm) thick front and back side etched Cu traces as the “fibers,” kf = 400 W/(m-K). Figures 2(a)2(d) show a zoomed view image of the center region of a baseline, heat flux cloaking, heat flux concentrator, and heat flux rotator device, respectively.

To achieve heat flux cloak, concentrator, and rotator functions with minimal disturbance to thermal response outside of the annulus, the effective thermal conductivity of the surrounding medium, ko, is designed to be close to the reduced average of the materials comprising the ring, i.e., $ko2∼kikm$, per [9]. This function is achieved with a 200 μm thick Cu trace mesh at a 2.5 mm unit cell spacing for $ko=4.58$ W/(m-K); refer to Fig. 2(a). The effective thermal conductivity of the inclusion region where the Cu traces exist is found to be ki = 49 W/(m-K) using a basic composite slab model [34].

The steady-state thermal response of each PCB-based device, measured using a calibrated infrared (IR) camera, is shown in Fig. 2 with corresponding simulation results; see Ref. [11] for full details. Note that all Cu trace patterns are explicitly modeled in the numerical studies. Logically, these devices represent thermal-composite metamaterial building blocks that enable sophisticated heat transfer function.

## Heat Flow Control in Multilayer Printed Circuit Board

In this section, the heat flux cloak, concentrator, and rotator devices are exploited as key ingredients in three applications. First, a heat shielding application is summarized. Second, heat collection or thermal energy harvesting of low-grade waste heat is described. Third, the electrothermal design of a functional PCB-based power electronics synchronous buck converter circuit with optimized heat flow control is introduced.

###### Heat Shielding.

While an extensive number of fundamental studies in the literature uses some variant of a basic thermal cloak device, a majority of the work focuses on a closed cloak (e.g., ring) structure and the creation of an isothermal region in which temperature gradients are nearly eliminated. While novel, a more relevant objective for electronics might be to design an open cloak structure [38], or heat shield/thermostat, that allows for temperature reduction in a desired region of a circuit.

This core idea is explored in Ref. [28] and shown in Fig. 3, where the optimal design of anisotropic thermal conductivity in multilayer PCBs is emphasized. In this figure, a PCB with a custom-designed composite microstructure thermally shields a heat-sensitive device located next to a heat generating device.

Figure 4 shows the optimized multi-layer PCB design from Ref. [28] obtained using the numerical algorithm described previously with a simple average temperature minimization objective, i.e., $fo=Tavg$, for the heat-sensitive device attachment pad area. Here, electrothermal traces are adopted to conceptualize the manner in which electrical and thermal function might be combined. Additionally, the zoomed views of the optimized electrothermal traces highlight the three basic thermal metamaterial building blocks. Between the heat generating device (i.e., hot region) and the temperature-sensitive device, a heat flux cloak is utilized to prevent direct thermal communication. Heat flux rotator-like trace features are then built to transport heat around the temperature-sensitive device toward the cooler end of the PCB. At the cold end of the board, a heat flux concentrator allows enhanced thermal communication between the coldest portions of the PCB and the temperature-sensitive device.

A comparison is provided in Fig. 5 of the measured thermal performance of a baseline multilayer PCB design with a standard electrical trace connection plus full Cu ground plane versus a second modified design with a closed square-shaped thermal cloak and the multilayer PCB with an optimized heat shield. All three board designs have approximately the same Cu volume fraction. Each final PCB is composed of three FR-4 layers interspersed between four Cu layers as shown in the cross section detail in Fig. 5. Each board was tested in free convection, and it was found that the optimized design provides the greatest temperature reduction of the heat-sensitive device (i.e., $ΔTd=−10.5 °C$) with the smallest temperature rise penalty for the heat source (i.e., $ΔTh_max=+2.3 °C$) [28]. Thus, the thermal metamaterial building blocks revealed in Fig. 4 are critical to optimal performance.

###### Thermal Energy Harvesting.

Heat shielding is logical for thermal management purposes. However, the three basic thermal metamaterial devices enable alternative functions such as thermal energy harvesting of low-grade waste heat. In lieu of optimizing heat flow between source and sink, a thermal composite may be designed to mold the flow of heat and collect it in a central location for energy recovery using a thermoelectric generator (TEG). Following Ref. [29], a baseline PCB structure is illustrated in Fig. 6, where the heat from an array of four heat sources is transferred via a large Cu plane on the PCB to a center location where a heat sink plus TEG assembly is mounted for thermal energy recovery.

A second PCB thermal-composite substrate was designed using the optimization methodology described previously. This PCB has the same Cu volume fraction as the baseline design for straightforward comparison. In contrast with heat shielding, the optimization objective function is to maximize the average temperature, i.e., $fo=−Tavg$, of the center region of the PCB, where the TEG is attached. Again, elements of the three basic thermal metamaterial devices shown in Figs. 1 and 2 are evident in the top view of this PCB design in Fig. 7.

Under free and forced convection, the optimized thermal metamaterial composite more effectively guides and collects the waste heat into the center TEG attachment region leading to a larger $ΔT$ across the TEG. Figure 8 shows the steady-state baseline and optimized PCB temperature contours obtained using IR imaging below the TEG-to-PCB attachment region in free convection. Electrical power harvested by each TEG from the low-grade waste heat was measured [29] and is provided in Fig. 9. Since TEG output power, Pout, is proportional to the square of the temperature differential across the device [39], i.e., $Pout∝ΔT2$, a significant increase in power output may be realized for a relatively small, $+5.1°C$, change in the TEG hot side temperature [29]. Observe in Fig. 9 that for both composites, the power generated by the TEG under forced convection increases relative to the natural convection situation. This is an artifact of the increased TEG hot-to-cold side temperature difference. This generally larger TEG temperature difference reduces the percent increase in power due to the optimized composite. As explained in Ref. [29], a smaller percent increase in power is linked to a smaller ratio of temperature differences between the optimized and baseline cases when compared with the natural convection test. Nonetheless, when coupled with efficient power management strategies, such technology may be valuable for ultra-low power remote wireless sensing systems that are autonomously powered by scavenging low-grade waste heat.

###### Electrothermal Circuit Design.

Distinct from the prior two application studies, the electrothermal co-design of a functional buck converter circuit with an optimized thermal-composite PCB for enhanced heat flow control [40] is briefly summarized in this section.

Here, a standard commercial reference design [41] is modified based on the framework shown in Fig. 10. For step 1, the circuit diagram represents a single-phase synchronous buck converter rated for a 0.9 V output at 45 A from a 12 V input. In step 2, the baseline PCB layout comprises four 2 oz Cu layers. The second and third, ground and power planes, respectively, were then modified in step 3 based on an optimization study following the approach discussed herein. The optimization objective function was to minimize the thermal compliance of the PCB heat flow path from source (i.e., the two power metal-oxide-semiconductor field-effect transistor devices and the inductor) to sink (i.e., the four corner standoff and bus bar connections). The first and fourth Cu, power and signal layers, respectively, were left unmodified. The electrical and thermal co-design process was realized using the saved file format of an open source electronics design automation software [42] to splice in optimized thermal trace spline data. Both baseline and optimized PCBs were fabricated and tested, step 4, to confirm design intent.

Figure 11 shows the second Cu layer with the featured regions once more displaying the basic thermal metamaterial functions. Observe, as an example, that heat flux cloak-like thermal traces are constructed by the optimizer between the metal-oxide-semiconductor field-effect transistor devices (i.e., hot regions) such that heat from one device does not spread readily toward the other device. Heat flux rotator and concentrator features are then built in other areas to route heat toward the cooler portions of the circuit board.

The electrical and thermal performance of a baseline PCB and the modified PCB (each 65 mm × 65 mm in footprint area size) were experimentally evaluated both under free convection and with active cooling. In free convection, the PCBs were simply mounted to nylon standoffs and tested in an open lab environment, while for the active cooling experiments the PCBs were in thermal communication with a thermoelectric cooler via aluminum standoffs [40]. IR temperature measurement results for the baseline and modified designs at 1, 4, and 20 A output current, Iout, levels under natural convection are shown in Fig. 12. Observe that the measured IR temperature contour results for the modified PCB are noticeably different from the baseline PCB at the low output current levels of 1 A and 4 A; the switch and inductor temperatures are generally lower with more heat transferred via the thermal traces to the heat sinking locations. However, at the higher output current level of 20 A, the IR temperature measurement results indicate nearly identical switch and inductor temperatures.

At higher output current levels, the high thermal resistance of the nylon standoffs becomes the dominant thermal resistance bottleneck in the free convection experiments and limits heat transfer by conduction to thermal ground. The addition of active cooling allows for recovery of a $3−5°C$ temperature reduction effect due to the optimized thermal composite, as shown in Fig. 13. In this figure, a zoomed IR image of the central portion of each PCB is provided when the standoff temperatures are fixed to $10°C$ by way of the thermoelectric cooler.

Table 1 provides a summary of the experimentally measured change in circuit efficiency, $Δη$, the average change in the maximum switch temperature, $ΔTs_avg$, and the change in the inductor maximum temperature, $ΔTi$, for the modified PCB at various output current levels [40]. From a thermal performance perspective, the modified design provides a modest benefit at low output current levels, and these benefits only improve with active cooling. From an electrical response perspective, the modified design also interestingly exhibits improvements such as higher efficiency, up to $∼7.4%$, at low output current levels, <5 A, and reduced input current ripple, as illustrated in Fig. 14. This reduced input current ripple is related to a filter effect that is a likely artifact of the modification of the second and third Cu layers of the PCB design. Specifically, following Ref. [40], the electrothermal traces of the modified PCB act as a distributed element low pass filter that both increases the high frequency impedance and reduces the low frequency impedance with a shifted resonant frequency.

Accurate simulation and assessment of the full impact of the electrothermal traces on circuit performance is still challenging with today's state-of-the-art numerical tools. Regardless, the results in this section indicate that the thermal-composite metamaterial building blocks may enhance heat transfer performance for functional circuit designs and provide additional opportunities for electrical performance improvement, as well. Thus, although complex, the co-design process holds further potential for improving both thermal and electrical response and is an active line of investigation for power conversion applications [43].

## New Research Directions

The applications involving thermal-composite metamaterials for heat flow control presented in this paper are centered at the macroscale electronics circuit board level. Additional macroscale applications of the method are emerging including the design optimization of multifunctional structural composites [44] and cases for electronics [45,46], where the control of the flow of heat between source and sink is important; refer to Fig. 15 for a representative image of an electronics enclosure.

A range of studies on the manipulation of the flow of heat at the micro-to-nanoscale also exists including, for example, the informed structuring of nanoscale features in silicon (Si) to control phonon conduction [47,48], the design of silicon substrates for optical devices for thermo-optic tuning of performance [49], and nanoscale material structuring to introduce thermal flow asymmetry [50,51]. The reader is referred to Ref. [52] for a good review of progress in understanding and manipulation of energy dissipation and transport in nanoscale solid-state structures.

Specific to thermal metamaterials involving heat flux cloaking, concentration, and rotation functions, the demand for faster CMOS electronics, optoelectronics, and photonic devices has motivated research, e.g., Ref. [30], on thermal guiding structures built off these basic functions. Such anisotropic TG structures may facilitate thermal management for next-generation high power density devices [53], act as dynamic reconfigurable heat flow paths in microelectromechanical systems (MEMS), enable advanced brain-like (neuromorphic) computing [30], or serve as Boolean-logic gates for thermal computing [30,54].

Figure 16 provides interesting example results adapted from a numerical study by Loke et al. [30], of Boolean-logic carried out thermally using nanoscale four-way graphene-on-silica (GOS) anisotropic thermal-diffusion guides in a Si base substrate. The structure was sized at 140 × 140 nm2, and the widths of the guides were fixed at 4 nm. The 0 and 1 input boundary conditions were set to 300 K and 400 K, respectively. The no-bias and bias settings were set to 300 K and 400 K, respectively, as well. The right flux guide boundary was fixed at 300 K, and the reference temperature, used to distinguish logical outputs, was set at 330 K, per [30]. Simulation times were kept constant at 4 and 5 ps. Using such TG structures, Loke et al. posited an application of anisotropic thermal-composite metamaterial GOS structures for thermally based Boolean-logic computations that could be fabricated using established CMOS-based Si trench etching and patterning technology. The authors of Ref. [30] further proposed exploiting the dependence of thermal-diffusion properties on the delay of excitation timings for neuromorphic computing; the reader is referred to the literature.

## Conclusions

This paper presented a brief review of the field of thermal metamaterials, which has undergone a rapid expansion over the last ten years. The core thermal metamaterial building block structures for heat flux cloaking, focusing, and reversal were highlighted. An application-focused perspective on the use of anisotropic thermal-composite metamaterials was provided in relation to heat shielding, thermal energy harvesting, and electrothermal power conversion circuit design. These three applications were focused at the macro-scale printed circuit board level. Additional thoughts were provided on opportunities found in the literature for applying thermal guiding structures at the device micro-to-nanoscale. Such thermal guiding structures are inspired by the thermal metamaterial concept, and it is expected that these heat flow control principles will become widespread in advanced electronics, optoelectronics, and photonic devices, where the tight integration of electrical, optical, and thermal function will be central to next-generation power converters, computers, and sensors.

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Kawamoto, A. , Matsumori, T. , Yamasaki, S. , Nomura, T. , Kondoh, T. , and Nishiwaki, S. , 2011, “Heaviside Projection Based Topology Optimization by a PDE-Filtered Scalar Function,” Struct. Multidiscip. Optim., 44(1), pp. 19–24.
COMSOL AB, 2008, “COMSOL Multiphysics Ver. 3.5a,” Stockholm, Sweden.
Svanberg, K. , 1987, “The Method of Moving Asymptotes—A New Method for Structural Optimization,” Int. J. Numer. Methods Eng., 24(2), pp. 359–373.
He, X. , and Wu, L. , 2013, “Design of Two-Dimensional Open Cloaks With Finite Material Parameters for Thermodynamics,” Appl. Phys. Lett., 102(21), p. 211912.
Min, G. , and Rowe, D. M. , 1992, “Optimisation of Thermoelectric Module Geometry for ‘Waste Heat’ Electric Power Generation,” J. Power Sources, 38(3), pp. 253–259.
Dede, E. M. , Wang, C.-M. , Liu, Y. , Schmalenberg, P. , Zhou, F. , Shin, J.-W. , and Ishigaki, M. , 2017, “Electrothermal Circuit Design With Heat Flow Control—Synchronous Buck Converter Case Study,” IEEE Trans. Compon. Pack. Manuf. Technol. (in press).
TI, 2016, “High Power Density Voltage Regulator Module for CPU Core Power in Enterprise Switching,” Texas Instruments Inc., Dallas, TX, accessed Jan. 23, 2018,
Lohan, D. J. , Allison, J. T. , Dede, E. M. , and Ishigaki, M. , 2016, “Combined Lumped and Continuum Parameter Design Optimization of Electro-Thermal Systems,” ASME Paper No. DETC2016-60218.
Petrovic, M. , Nomura, T. , Yamada, T. , Izui, K. , and Nishiwaki, S. , “Orthotropic Material Orientation Optimization Method in Composite Laminates,” Struct. Multidiscip. Optim. (epub).
Nomura, T. , 2016, “Structural Thermodynamics With Topology Optimization,” accessed Jan. 23, 2018,
Nishiwaki, S. , 2017, “Development of Conceptual Design Methods Based on Topology Optimization,” JSDE Des. Eng., 52(7), pp. 439–444.
Yu, J.-K. , Mitrovic, S. , Tham, D. , Varghese, J. , and Heath, J. R. , 2010, “Reduction of Thermal Conductivity in Phononic Nanomesh Structures,” Nat. Nanotechnol., 5, pp. 718–721. [PubMed]
Park, W. , Romano, G. , Ahn, E. C. , Kodama, T. , Park, J. , Barako, M. T. , Sohn, J. , Kim, S. J. , Cho, J. , Marconnet, A. M. , Asheghi, M. , Kolpak, A. M. , and Goodson, K. E. , 2017, “Phonon Conduction in Silicon Nanobeam Labyrinths,” Sci. Rep., 7, p. 6233. [PubMed]
Lei, S. , Alexandre, S. , and Enright, E. , 2017, “Thermo-Optic Tuning Efficiency of Micro Ring Resonators on Low Thermal Resistance Silicon Photonics Substrates,” ASME Paper No. IPACK2017-74181.
Markus, S. , Maier, J. , Scheer, E. , and Leiderer, P. , 2011, “A Thermal Diode Using Phonon Rectification,” New J. Phys., 13(11), p. 113027.
Yang, N. , Zhang, G. , and Li, B. , 2008, “Carbon Nanocone: A Promising Thermal Rectifier,” Appl. Phys. Lett., 93(24), p. 243111.
Pop, E. , 2010, “Energy Dissipation and Transport in Nanoscale Devices,” Nano Res., 3(3), pp. 147–169.
Han, H. , Zhang, Y. , Wang, N. , Samani, M. K. , Ni, Y. , Mijbil, Z. Y. , Edwards, M. , Xiong, S. , Saaskilahti, K. , Murugesan, M. , Fu, Y. , Ye, L. , Sadeghi, H. , Bailey, S. , Kosevich, Y. A. , Lambert, C. J. , Liu, J. , and Volz, S. , 2016, “Functionalization Mediates Heat Transport in Graphene Nanoflakes,” Nat. Commun., 7, p. 11281. [PubMed]
Wang, L. , and Li, B. , 2007, “Thermal Logic Gates: Computation With Phonons,” Phys. Rev. Lett., 99, p. 177208. [PubMed]
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Hull, D. , and Clyne, T. W. , 1996, An Introduction to Composite Materials, 2nd ed., Cambridge University Press, Cambridge, UK.
Kawamoto, A. , Matsumori, T. , Yamasaki, S. , Nomura, T. , Kondoh, T. , and Nishiwaki, S. , 2011, “Heaviside Projection Based Topology Optimization by a PDE-Filtered Scalar Function,” Struct. Multidiscip. Optim., 44(1), pp. 19–24.
COMSOL AB, 2008, “COMSOL Multiphysics Ver. 3.5a,” Stockholm, Sweden.
Svanberg, K. , 1987, “The Method of Moving Asymptotes—A New Method for Structural Optimization,” Int. J. Numer. Methods Eng., 24(2), pp. 359–373.
He, X. , and Wu, L. , 2013, “Design of Two-Dimensional Open Cloaks With Finite Material Parameters for Thermodynamics,” Appl. Phys. Lett., 102(21), p. 211912.
Min, G. , and Rowe, D. M. , 1992, “Optimisation of Thermoelectric Module Geometry for ‘Waste Heat’ Electric Power Generation,” J. Power Sources, 38(3), pp. 253–259.
Dede, E. M. , Wang, C.-M. , Liu, Y. , Schmalenberg, P. , Zhou, F. , Shin, J.-W. , and Ishigaki, M. , 2017, “Electrothermal Circuit Design With Heat Flow Control—Synchronous Buck Converter Case Study,” IEEE Trans. Compon. Pack. Manuf. Technol. (in press).
TI, 2016, “High Power Density Voltage Regulator Module for CPU Core Power in Enterprise Switching,” Texas Instruments Inc., Dallas, TX, accessed Jan. 23, 2018,
Lohan, D. J. , Allison, J. T. , Dede, E. M. , and Ishigaki, M. , 2016, “Combined Lumped and Continuum Parameter Design Optimization of Electro-Thermal Systems,” ASME Paper No. DETC2016-60218.
Petrovic, M. , Nomura, T. , Yamada, T. , Izui, K. , and Nishiwaki, S. , “Orthotropic Material Orientation Optimization Method in Composite Laminates,” Struct. Multidiscip. Optim. (epub).
Nomura, T. , 2016, “Structural Thermodynamics With Topology Optimization,” accessed Jan. 23, 2018,
Nishiwaki, S. , 2017, “Development of Conceptual Design Methods Based on Topology Optimization,” JSDE Des. Eng., 52(7), pp. 439–444.
Yu, J.-K. , Mitrovic, S. , Tham, D. , Varghese, J. , and Heath, J. R. , 2010, “Reduction of Thermal Conductivity in Phononic Nanomesh Structures,” Nat. Nanotechnol., 5, pp. 718–721. [PubMed]
Park, W. , Romano, G. , Ahn, E. C. , Kodama, T. , Park, J. , Barako, M. T. , Sohn, J. , Kim, S. J. , Cho, J. , Marconnet, A. M. , Asheghi, M. , Kolpak, A. M. , and Goodson, K. E. , 2017, “Phonon Conduction in Silicon Nanobeam Labyrinths,” Sci. Rep., 7, p. 6233. [PubMed]
Lei, S. , Alexandre, S. , and Enright, E. , 2017, “Thermo-Optic Tuning Efficiency of Micro Ring Resonators on Low Thermal Resistance Silicon Photonics Substrates,” ASME Paper No. IPACK2017-74181.
Markus, S. , Maier, J. , Scheer, E. , and Leiderer, P. , 2011, “A Thermal Diode Using Phonon Rectification,” New J. Phys., 13(11), p. 113027.
Yang, N. , Zhang, G. , and Li, B. , 2008, “Carbon Nanocone: A Promising Thermal Rectifier,” Appl. Phys. Lett., 93(24), p. 243111.
Pop, E. , 2010, “Energy Dissipation and Transport in Nanoscale Devices,” Nano Res., 3(3), pp. 147–169.
Han, H. , Zhang, Y. , Wang, N. , Samani, M. K. , Ni, Y. , Mijbil, Z. Y. , Edwards, M. , Xiong, S. , Saaskilahti, K. , Murugesan, M. , Fu, Y. , Ye, L. , Sadeghi, H. , Bailey, S. , Kosevich, Y. A. , Lambert, C. J. , Liu, J. , and Volz, S. , 2016, “Functionalization Mediates Heat Transport in Graphene Nanoflakes,” Nat. Commun., 7, p. 11281. [PubMed]
Wang, L. , and Li, B. , 2007, “Thermal Logic Gates: Computation With Phonons,” Phys. Rev. Lett., 99, p. 177208. [PubMed]

## Figures

Fig. 8

Experimental temperature profile in free convection for each PCB below TEG attachment region: (left) baseline and (right) optimized

Fig. 9

A comparison of the experimentally measured power generated using each thermal-composite substrate. Reprinted with permission from Dede et al. [29] used in accordance with the Creative Commons Attribution (CC BY) license.

Fig. 7

Top view of optimized PCB thermal-composite from Ref. [29] designed for waste heat recovery. Featured regions display basic thermal metamaterial functions including: (1) heat flux cloaking; (2) heat flux concentration; and (3) heat flux rotation.

Fig. 6

Baseline PCB assembly with array of four heat sources ( red-dashed boxes) and square Cu plane for heat transfer to a centrally positioned heat sink/TEG assembly for energy recovery [29]

Fig. 5

Multilayer PCB designs (top row) described in Ref. [28]. Middle row shows measured IR temperature contours for baseline configuration (left), square-shaped thermal cloak (center), and optimized thermal metamaterial cloak with electrothermal traces (right). The zoomed view for each case (lower row) uses a reduced temperature scale for clarity with the temperature-sensitive device region indicated by the red-dashed box.

Fig. 4

Optimized multilayer PCB with electrothermal traces from Ref. [28]. Featured regions display basic thermal-composite metamaterial functions including (1) heat flux cloaking; (2) heat flux concentration; and (3) heat flux rotation.

Fig. 3

Conceptual illustration of heat flow control on a multilayer PCB using a designed anisotropic thermal-composite microstructure. Reprinted with permission from Dede et al. [28]. Copyright 2015 by IEEE.

Fig. 2

Zoomed view of center region of each PCB: (a) baseline, (b) heat flux cloaking device, (c) heat flux focusing device, and (d) heat flux reversal device. Numerical thermal contours of each device center region: (e) baseline, (f) heat flux cloaking, (g) heat flux focusing, and (h) heat flux reversal. Experimental thermal contours: (i) baseline, (j) heat flux cloaking, (k) heat flux focusing, and (l) heat flux reversal. Points p1 and p2 indicate locations for temperature gradient measurement [11].

Fig. 1

Numerically optimized thermal-composite metamaterial designs. (a)–(c) show the high thermal conductivity fiber layout in a low thermal conductivity matrix for, respectively, heat flux cloaking, focusing, and reversal [14]. (d)–(f) show the temperature contours with proportionally scaled heat flux vectors across each heat flux cloaking, focusing, and reversal device, respectively, as heat flows from left to right.

Fig. 10

Framework for electrothermal co-design integrating thermal metamaterial functions into a PCB for heat flow control [40]

Fig. 11

Top view of second Cu layer for optimized synchronous buck converter. Featured regions display basic thermal metamaterial functions including: (1) heat flux cloaking; (2) heat flux concentration; and (3) heat flux rotation.

Fig. 12

Temperature contour results for baseline (upper row) and modified (lower row) PCBs under natural convection. The left, middle, and right columns correspond to converter output current levels of 1 A, 4, A, and 20 A, respectively, [40]. The white-dashed boxes in the upper left image denote the switch locations (small boxes) and inductor location (large box). Units: °C.

Fig. 13

Zoomed view of temperature contour results for baseline (left) and modified (right) PCBs at 20 A output current with active cooling. Units: °C.

Fig. 14

Input current waveforms for baseline and modified synchronous buck converter PCB designs at 1 A and 16 A output current levels. Reduced input current ripple for the modified design is an artifact of a distributed element low pass filter effect due to the electrothermal traces [40]. Note: waveforms are plotted using 5 μs x-axis divisions.

Fig. 15

Representative electronics enclosure based on thermal metamaterial design [45,46]. The case is an injection molded polymer with an electroplated Cu trace pattern for heat flow control.

Fig. 16

GOS thermal-diffusion-guide structural models for thermal-switching Boolean-logic operations. (left) Temperature profiles of Si structures containing crossed four-way GOS thermal-diffusion guides with varied thermal inputs at the left and bottom flux-guide boundaries, with and without thermal biases at the top flux-guide boundary, used for Boolean-logic operations. (right) Dependence of the temperature values at the intersection (marked X) of the models shown on the left used for AND and OR Boolean-logic computations. (Reprinted (adapted) with permission from Loke et al. [30]. Copyright 2016 by American Chemical Society).

## Tables

Table 1 Measured change in electrical circuit efficiency and device temperatures for modified PCB relative to baseline PCB at same Iout value
Note: FC: free convection and AC: active cooling.

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