## Abstract

A theoretical model of heat conduction is developed for wirelessly powered microscale inorganic light-emitting diodes (μ-ILEDs). Analytical solutions are obtained for the three-dimensional temperature distribution of each component in the system, which agree reasonably well with the finite-element analyses and experiment results. A simplified scaling law is presented between the non-dimensional temperature of the μ-ILEDs, and the combined geometrical parameters and thermal conductivities of the inductive receiver coil and the substrate. These results provide useful design guidelines for avoiding adverse heating of wireless μ-ILEDs systems, of critical importance for bio-implanted applications.

## 1. Introduction

Indium gallium nitride-based (InGaN) blue-light-emitting diodes (LEDs), owing to their superior advantages in efficiency and lifetime, are attractive for broad classes of applications in solid-state lighting (Schubert & Kim 2005; Tsao *et al*. 2010). Newer potential uses are in biotechnology, where InGaN LEDs can be used to accelerate wound healing (Corazza *et al*. 2007; Liang *et al*. 2008; Bastos *et al.* 2009), to activate photosensitive drugs (Bai *et al*. 2010; Timko *et al*. 2010; Xu *et al*. 2010) or to perform imaging and spectroscopic characterization of internal tissues (Waxman 2008; Balas 2009). In these clinical applications, the use of wireless power, instead of batteries, could bring additional advantages and alternative modes of use, avoiding needs in battery replacement and battery biocompatibility. Recently, Kim *et al*. (2012) reported routes to integrate flexible and stretchable collections of microscale inorganic LEDs (μ-ILEDs) systems with wireless powering schemes, and demonstrated their potential applications in implantable devices through *in vivo* experiments on animal models. These devices incorporate rectangular spiral inductor coils and straight interconnect lines terminating at the p and n-type ohmic contacts of μ-ILEDs, which are coated with epoxy (SU8–2, Microchem., USA; relative dielectric const.=3.2 at 10 MHz) and delivered to the PET (polyethylene terephthalate) substrate via transfer printing (Meitl *et al*. 2006; Kim *et al*. 2008). The electrical energy transfers through radio frequency coupling into the inductor coil and then to the μ-ILEDs, without any physical contact, on the basis of the mechanism of resonant magnetic induction.

The temperature increase in the μ-ILEDs must be as small as possible to ensure efficient and reliable operation, particularly when used in bio-integrated applications. The thermal behaviour of such systems under different heating conditions, and the relationships between the device temperature and the various geometrical and material parameters are of crucial importance. Kim *et al*. (2011) and Lu *et al*. (2012) investigated the heat conduction of μ-ILEDs systems with traditional power supplies both experimentally and theoretically, the latter through an axisymmetric model capable of quantitatively capturing the temperature distribution in the substrate and μ-ILEDs. Unlike the layout of Kim *et al*. (2011), which incorporated metal interconnect structures designed in large, film geometries to facilitate thermal transport away from the active devices, the wireless μ-ILEDs system, as reported by Kim *et al*. (2012), incorporates only two straight line connections between the μ-ILEDs and the inductive coil, as shown in figure 1*a*,*b*. The thermal analyses in such structures involve coupled one-dimensional heat conduction in the straight inductive coil, and three-dimensional heat conduction in the substrate, which makes the solution rather complicated. Kim *et al*. (2012) measured the temperature increases in the μ-ILEDs under different input powers, and only a simplified model was introduced to account for the basic behaviours, where the dimension of μ-ILEDs (also the heat generation) was not taken into account. The objective of this study is to develop a rigorous analytical model and finite-element analyses (FEA) of heat conduction for the wireless μ-ILEDs system (figure 1*a*,*b*), and provide simple solutions of device temperatures for design optimization.

## 2. Heat conduction model for the wireless μ-ILEDs system

In the device of Kim *et al*. (2012), the μ-ILED (length *a*_{LED}=100 μm) is much shorter than the inductive coil (length *L*_{IC}=30 mm), but both have similar thermal conductivities. For simplicity, the μ-ILED and coil are modelled as a single structure with thermal conductivity *k*_{IC}, thickness *H*_{IC} and width *W*_{IC}. The μ-ILED and coil are encapsulated in a thin epoxy layer (thickness *H*_{epoxy}=10.3 μm), such that these components are very close to the top surface, with a distance of approximately 3.76 μm, when compared with the substrate thickness of *H*_{PET}=50 μm. Thus, it is reasonable to simplify the analyses by assuming that the μ-ILED and coil are placed on the top surface of the whole structure. Furthermore, the thermal conductivity (*k*_{epoxy}=0.2 W m^{−1} K^{−1}) of epoxy is close to that of PET (*k*_{PET}=0.24 W m^{−1} K^{−1}) such that the two layers can be modelled as a single PET layer for simplicity. Figure 1*c* provides a schematic of this simplified model, where the PET substrate, with the μ-ILED and coil on its surface, is placed on a Petri dish with a thickness of *H*_{PD} and thermal conductivity of *k*_{PD}. A Cartesian coordinate system is set such that the origin is located at the centre of the bottom surface of Petri dish. The heat generation, with an input power of *Q*, is modelled as body heat flux in the area occupied by the μ-ILED (i.e. |*x*|≤*a*_{LED}/2).

The one- and three-dimensional heat conduction models are adopted for the coil and bi-layer structure (i.e. substrate and Petri dish), respectively, and their coupling is taken into account through the continuity condition across the inductor coil/PET interface. For the coil, the one-dimensional steady-state heat conduction equation can be derived from the energy balance of an infinitesimal segment as
2.1where *T* is the temperature, and *q*_{generation} is the body heat flux due to heating of μ-ILED, which is given by
2.2*q*_{z} is the heat flux from the inductor coil into the PET substrate, and is to be determined by the continuity condition across the inductor coil/PET interface. For the bi-layer structure of the PET substrate and Petri dish, the three-dimensional steady-state heat conduction equation is
2.3The followings are the boundary and continuity conditions of inductor coil and bi-layer structure;

—

*z*=0 (constant temperature at the bottom surface of Petri dish): 2.4awhere*T*_{0}is the ambient temperature;—

*z*=*H*_{PD}=*h*_{1}(continuity of temperature and heat flux across the PET/Petri dish interface): 2.4b— |

*y*|≤*W*_{IC}/2,*z*=*H*_{PD}+*H*_{PET}=*h*_{2}(continuity of temperature and heat flux across the inductor coil/PET interface): 2.4c— |

*y*|>*W*_{IC}/2,*z*=*H*_{PD}+*H*_{PET}=*h*_{2}(adiabatic condition at the top surface of PET substrate which does not contact the inductor coil): 2.4d

Employing the Fourier transform [and its inverse transform ], equations (2.1) and (2.3) become
2.5and
2.6A double Fourier transform of equation (2.6) along the *y* direction yields
2.7where and The temperature solution of equation (2.7) and the corresponding heat flux are given by
2.8where *D*_{1} and *D*_{2}, together with and in equation (2.5), are determined by the Fourier transforms of boundary and continuity conditions equation () as
2.9
2.10
and
2.11where
2.12
2.13
and
2.14The temperatures of all components in the wireless μ-ILED system, including the μ-ILED, the inductive coil, the PET substrate and the Petri dish, are obtained by performing inverse Fourier transforms of equation (2.8) and in equation (2.9), i.e.
2.15
2.16
and
2.17where *α* denotes ‘PET’ or ‘PD’, corresponding to PET substrate or Petri dish; and are given in equations (2.10) and (2.11); and the μ-ILED temperature in equation (2.15) is obtained by its average over the entire active region (|*x*|≤*a*_{LED}/2).

We then compare the analytical solutions of temperatures of μ-ILED and inductor coil, with the experiment results (Kim *et al*. 2012) and FEA. The μ-ILED size is 100 μm×100 μm, and the ambient temperature is *T*_{0}=21.9^{°}C. The thermal conductivity and thickness are *W*_{IC}=100 μm, *H*_{IC}=1 μm and for inductor coil (Kim *et al*. 2012); *H*_{PET}=50 μm and *k*_{PET}=0.24 W m^{−1} K^{−1} for PET substrate (Wang *et al*. 2010; Guo *et al*. 2011); *H*_{PD}=1200 μm and for Petri dish (Pasquino & Pilswort 1964; Algaer *et al*. 2009). On the basis of equation (2.15), the temperature increase of μ-ILED due to heating of unit input power (*Q*) is 6.34 K m^{−1} W^{−1}, which agrees reasonably well with the experiment (in the range from 5.12 to 6.38 K m^{−1} W^{−1} for four different input powers), and the accurate three-dimensional FEA (5.92 K m^{−1} W^{−1}) for the original structure layout (figure 1*a*,*b*) as used in experiments. Figure 2 demonstrates the temperature distribution within the core elements (μ-ILED and inductor coil) of the wireless μ-ILED system for input power of *Q*=2.6 mW, as predicted by the analytical solution in equation (2.16) and accurate three-dimensional FEA, which shows both qualitative and quantitative accordance.

## 3. A scaling law between the device temperature and the geometrical and material parameters

To understand clearly the influences of various geometrical and material parameters on the μ-ILED temperature, we normalize all thicknesses and widths by the length (*a*_{LED}) of μ-ILED, i.e.
3.1The normalized temperature is . For μ-ILED, the normalized temperature can then be written as
3.2where
3.3

Equations (3.2) and (3.3) indicate that the non-dimensional temperature increase of μ-ILED depends on five non-dimensional parameters, *k*_{PD}/*k*_{PET}, , , and . For the Petri dish thickness (1200 μm, Kim *et al*. (2012)) much larger than the thickness of PET substrate (50 μm) and inductor coil (1 μm), can be further simplified as
3.4Figure 3 shows the non-dimensional temperature increase of device versus the combined parameter of inductor coil for *k*_{PD}/*k*_{PET}=0.65, and , obtained from the accurate and approximate analytical models, three-dimensional FEA and experiments. The analytical models and FEA show a rapid decrease of device temperature with increase of before the latter reaches approximately 15. This suggests that the device temperature can be well reduced by adopting a thicker coil with a larger thermal conductivity. Figure 4 shows versus the normalized substrate thickness for several normalized thermal conductivities of the substrate *k*_{PD}/*k*_{PET}, with and fixed at 13.25 and 1, respectively. The analytical results are in good agreement with the three-dimensional FEA, both indicating that the device temperature is more sensitive to the thermal conductivity of substrate than its thickness. For a substrate having the same thermal conductivity as the Petri dish (i.e. *k*_{PD}/*k*_{PET}=1), the device temperature is independent of the substrate thickness because the Petri dish is very thick. For a substrate having a larger thermal conductivity than the Petri dish (i.e. *k*_{PD}/*k*_{PET}>1), the device temperature increases with the substrate thickness. This trend is reversed for *k*_{PD}/*k*_{PET}<1.

## 4. Conclusion

In summary, a three-dimensional heat conduction model of an emerging class of thin, wireless μ-ILED system is developed, and analytical solutions of temperature distribution are obtained via the approach of Fourier transformation, which are validated by experiment and FEA results. A scaling law for the device temperature is established, which shows its dependence on various geometrical and material parameters. For the microscale optoelectronic device system of current interest, which generally consists of key devices, metal interconnect and substrate, it is important to design the interconnect with a large cross-sectional area and high thermal conductivity, for efficient thermal management. Compared with the geometric parameters of interconnect (e.g. thickness), the device temperature is usually not so sensitive to those of substrate. This theoretical study provides routes to structural design of wireless μ-ILED system with regard to the thermal performance, which could be beneficial for emerging biomedicine-related applications.

## Acknowledgements

Y.H. and J.A.R. acknowledge the support from NSF (ECCS-0824129 and CMMI-0749028). Y.H. also acknowledges the support from NSFC.

- Received July 12, 2012.
- Accepted September 7, 2012.

- This journal is © 2012 The Royal Society