## Abstract

An analytical model is developed to study the thermal properties of microscale, inorganic light-emitting diodes (μ-ILEDs) with ultra-thin geometries and layouts. The predicted surface and μ-ILED temperatures agree well with experiments and finite-element simulations. A simple scaling law is obtained for the normalized μ-ILED temperature versus the normalized μ-ILED size. This study provides a theory to guide the design of layouts that minimize adverse thermal effects on the performance of μ-ILEDs not only for solid-state lighting but also for applications integrating μ-ILED devices on complex/soft substrate as are currently of interest in optogenetics and other emerging areas in biology.

## 1. Introduction

Indium gallium nitride-based (InGaN) light-emitting diodes (LEDs) offer combined characteristics in cost, efficiency and lifetime, which make them attractive for broad classes of application in solid-state lighting (Schubert & Kim 2005; Tsao *et al.* 2010). Conventional routes to LEDs involve epitaxial growth of active materials followed by wafer dicing and pick-and-up robotic manipulation into individually packaged components that are then interconnected by bulk wire bonding and mounted on millimetre-scale heat sinks for thermal management. Although suitable for many uses, such designs do not allow practical realization of potentially valuable engineering options, such as those that involve large collections of small devices or unusual substrates. Kim *et al.* (2011) recently reported schemes that avoid these limitations by use of advanced methods in epitaxial lift-off and deterministic assembly. The result is a realistic route to classes of LEDs that can be much thinner and smaller than those accessible using conventional procedures, in which interconnection is accomplished with thin-film metallization and photolithography. The process begins with the growth of InGaN epitaxial material on silicon wafers with (111) orientation followed by the definition of arrays of individual LEDs with microscale dimensions (down to approx. 25×25 μm), referred to as microscale inorganic LEDs (μ-ILEDs). Anisotropic wet chemical etching of the underlying silicon releases devices for transfer printing onto foreign substrates (e.g. glass or plastic). A photodefinable layer of benzocyclobutene (BCB; figure 1*a*) provides a top layer encapsulant that leaves the p- and n-type ohmic contacts (figure 1*a*) of each device exposed. Depositing and patterning of a metal layer (figure 1*a*) yields electrical interconnects that serve simultaneously as heat spreaders. Figure 1*a* shows the layouts of a single μ-ILED on a glass substrate where the small gap within the metal layer isolates p- and n-contacts to avoid short circuit. This gap also serves as a good approximation of the μ-ILED temperature because the BCB layer on the μ-ILED is very thin (approx. 1 μm). The cross section is shown in figure 1*b*. Related procedures have also been reported for AlInGaP μ-ILEDs (Kim *et al.* 2010).

The thermal properties of devices with this type of layout are critically important because excessive heating can limit the stability, reliability and efficiency. Changes in temperature can be even more important for applications of μ-ILEDs in biology, where even modest levels of heating can induce adverse responses. Our objective was to study the thermal properties of μ-ILEDs using analytical and rigorous finite-element methods and provide simple expressions for calculating the device temperatures in terms of the material and geometry parameters. Unlike the accurate analysis (Kim *et al.* 2011) in which a very complex expression for μ-ILED temperature is given, a simple scaling law for μ-ILED temperature is derived in this paper for convenient designing of μ-ILED layout to minimize the adverse thermal effects.

## 2. Heat transfer model for the μ-ILED system

The thickness of a typical μ-ILED (5 μm or less) is smaller than its in-plane size *L* (100×100 μm) such that heat transfer occurs mainly through the top and bottom surfaces. As shown in figure 1*c*, the μ-ILED is modelled as a planar heat source with the input power *Q* at the BCB–glass interface. The metal interconnect of thickness *H*_{m} is on top of the BCB. The thicknesses of the BCB layer and glass substrate are *H*_{B} and *H*_{g}, respectively. Their corresponding thermal conductivities are *k*_{m},*k*_{B} and *k*_{g}. The ambient temperature is *T*_{0}.

For simplicity, an axisymmetric model is adopted to obtain the analytical solution. The heat source is modelled as a circular disc of radius such that its area is the same as that (*L*^{2}) of the μ-ILED. Such an assumption, as validated by a three-dimensional finite-element analysis (FEA) and experiments for rectangular μ-ILEDs, significantly simplifies the analysis and enables analytical solutions. The steady-state heat conduction equation is
2.1
where Δ*T*=*T*−*T*_{0} is the difference between temperature *T* and the ambient temperature *T*_{0}, and (*r*,*z*) are the cylindrical coordinates with the origin at the centre of the heat source (figure 1*c*). Because the in-plane dimensions (approx. 50×50 mm) of the glass substrate, BCB and metal are much larger than the μ-ILED size (100×100 μm), the axisymmetric model is taken as infinity in the *r* direction. The natural convection condition at the top surface of the metal is
2.2
where *h* is the coefficient of heat convection. The ambient temperature at the bottom surface of the glass substrate as well as the remote boundary (of metal, BCB and glass) gives
2.3
and
2.4
Continuity of temperature and heat flux across the metal–BCB interface (*z*=*H*_{B}) requires
2.5
Across the BCB–glass interface (*z*=0), the temperature is continuous
2.6a
The heat flux outside the μ-ILED (*r*>*r*_{0}, *z*=0) is also continuous, which, together with the heat source condition inside the μ-ILED (*r*≤*r*_{0}, *z*=0), gives
2.6b
The Hankel transform (Carslaw & Jaeger 1959) of the steady-state heat conduction equation (2.1) gives the following ordinary differential equation:
2.7
where *J*_{0} is the 0th-order Bessel function of the first kind. This equation has the solution
2.8
where the functions *A*(*ξ*) and *B*(*ξ*) are *A*_{m} and *B*_{m} for metal, *A*_{B} and *B*_{B} for BCB, and *A*_{g} and *B*_{g} for the glass substrate, respectively. The Hankel transform of the boundary and continuity conditions (2.2), (2.3), (2.5) and (2.6) gives
2.9
2.10
2.11
and
2.12
where *J*_{1} is the first-order Bessel function of the first kind.

Substitution of (2.8) into (2.9)–(2.12) gives
2.13
for the glass substrate,
2.14
for the BCB layer and
2.15
for the metal layer, where
The inverse Hankel transform then gives the temperature distributions Δ*T*_{g}(*r*,*z*), Δ*T*_{B}(*r*,*z*) and Δ*T*_{m}(*r*,*z*) in the glass substrate, BCB and metal layers. The temperature distribution in the μ-ILED is obtained from Δ*T*_{g}(*r*,0), as . The temperature distribution at the metal surface is , which can be rewritten to obtain the surface temperature
2.16
where
A three-dimensional FEA is used to study the temperature distributions in the metal, BCB, μ-ILED and glass. The μ-ILED (100×100×50 μm) is enveloped by a layer of BCB, with a total thickness of 6 μm, and is modelled as a three-dimensional volume heat source with the total input power *Q*. The thicknesses of the metal and glass are the same as those used in the analytical model. The in-plane dimensions of the metal, BCB and glass are taken from the experiments (50×50 mm) (Kim *et al*. 2011). The metal layer is subjected to natural convection with the ambient temperature *T*_{0} at the top surface, while the lateral surfaces of the metal, BCB and glass, together with the bottom surface of the glass, are at a constant temperature *T*_{0}. The continuum element DC3D8 in the ABAQUS package (see ABAQUS 2009) is used in the FEA.

Figure 2 compares the surface temperature distribution in equation (2.16) with the experiments and three-dimensional FEA, which accurately accounts for the square-shaped μ-ILED with finite thickness, for the input power *Q*=12 mW (figure 2*a*) and *Q*=37.6 mW (figure 2*b*) used in the experiments, the μ-ILED size *L*=100 μm (*r*_{0}=56.4 μm) and ambient temperature *T*_{0}=50^{°}*C*. The thermal conductivity and thickness are (Lee & Cahill 1997) and *H*_{g}=800 μm for glass, (Christiaens *et al.* 2005) and *H*_{B}=1 μm for BCB, and *k*_{m}=70 and 160 W m^{−1} *K*^{−1} for the 300 and 1000 nm thick aluminium interconnects, respectively, because of their thickness-dependent thermal conductivity (La Spina *et al.* 2006; Stojanovic *et al.* 2007; Bourgoin *et al.* 2010; Schmid *et al.* 2010). The coefficient of heat convection is *h*=25 W m^{−2} *K*^{−1}. The analytical model agrees well with the FEA, and both agree well with the experiment, except along the gap (shown in the left two experimental images in figure 2, and also in figure 1*a*). This is because the surface temperature at the gap in the experiments is the temperature at the next (BCB) layer, which is very close to the μ-ILED temperature because the BCB layer directly above the μ-ILED is very thin (figure 1*a*). The μ-ILED temperature is approximately the same as the surface temperature at the gap.

## 3. Simple scaling law for thermal management

The FEA suggests that the temperature in the μ-ILED is very uniform except near its small side surfaces. This is because the in-plane size of the μ-ILED (*L*) is much larger than its thickness such that heat transfer from the μ-ILED occurs mainly through its top and bottom surfaces. The temperature increase in the μ-ILED can then be obtained analytically by averaging the temperature at the BCB–glass interface over the radius *r*_{0} as
3.1
Both the analytical model and FEA suggest that the natural convection at the top surface of the metal interconnect has a negligible effect on the temperature in the μ-ILED. The thick glass substrate (*H*_{g}≫*H*_{m}, *H*_{B}, *r*_{0}) can be modelled as a semi-infinite solid. These, together with the low thermal conductivity of BCB, *k*_{B}≪*k*_{m}, simplify equation (3.1) to
3.2
For a μ-ILED with size much larger than the metal and BCB layer thickness (*r*_{0}≫*H*_{m}, *H*_{B}), and (and in the integrand decreases rapidly with *x*), expression (3.2) can be further simplified to
3.3
It gives a simple scaling law such that the normalized temperature increase *k*_{m}*H*_{m}Δ*T*_{μ−ILED}/*Q* in the μ-ILED depends on only two combinations of material and geometry parameters: the normalized μ-ILED size *k*_{g}*r*_{0}/(*k*_{m}*H*_{m}) and the BCB layer thickness *k*_{g}*H*_{B}/(*k*_{B}*r*_{0}). The latter is very small (*k*_{g}*H*_{B}/(*k*_{B}*r*_{0})≪1) because *r*_{0}≫*H*_{B} and the thermal conductivities of the BCB and glass are of the same order of magnitude. Numerical integrations suggest that [*k*_{g}*H*_{B}/(*k*_{B}*r*_{0})]*x*^{2} can be neglected in equation (3.3) (also because decreases rapidly with *x*). This leads to a simple, analytical expression for the temperature in the μ-ILED,
3.4
The normalized temperature increase *k*_{m}*H*_{m}Δ*T*_{μ-ILED}/*Q* in the μ-ILED depends only on the normalized radius *k*_{g}*r*_{0}/(*k*_{m}*H*_{m}). It is independent of the BCB property and thickness because the BCB layer is very thin and has low thermal conductivity. This simple expression agrees very well with the accurate solution in equation (3.1) and the FEA, within a few per cent of error. Numerical integration in equation (3.4) gives an approximate but simple and explicit expression of temperature in the μ-ILED as
3.5
where the μ-ILED size *L* has substituted , and the first two terms 0.451*Q*(*k*_{g}*L*)^{−1}{1−0.842[*k*_{g}*L*/(*k*_{m}*H*_{m})]^{−1}} on the right-hand side result from the asymptotic analysis of equation (3.4) for relatively large *k*_{g}*L*/(*k*_{m}*H*_{m}). The temperature increase (*T*_{μ-ILED}−*T*_{0}) obtained from equation (3.5) is only a few per cent different from that given by equation (3.4). Figure 3 shows the normalized temperature increase in the μ-ILED versus the normalized μ-ILED size. The solid line is for the approximate solution in equation (3.5), while the accurate solution in equation (3.1), three-dimensional FEA and experiments are also shown. Their excellent agreement validates the analytical expression for the temperature in the μ-ILED in equation (3.5). This equation can serve as a guideline for the thermal management design of μ-ILEDs by selecting metal interconnects to minimize the adverse thermal effects. It suggests that thick metal interconnects or their large thermal conductivity will be effective in dissipating heat in μ-ILEDs. The scaling law in equation (3.5) (also shown in figure 3) indicates that the increase in μ-ILED size or the substrate thermal conductivity, for a given input power, also decreases the temperature in μ-ILEDs.

## 4. Conclusion

In conclusion, we have developed an analytical model to study the thermal properties of μ-ILEDs. The agreement between the predicted surface and μ-ILED temperatures with experiments and finite-element simulations validates the analytical model. We further approximate the μ-ILED temperature in terms of one combination of material and geometry parameters. Although the heat transfer in the current μ-ILED system should be in a three-dimensional manner, for which a solution for three-dimensional heat transfer in Cartesian coordinates is also obtainable, it is almost impossible to derive a simple scaling law such as equation (3.5). Such a simple analytical expression is very useful in the design of the μ-ILED layout to reduce the thermal effects on the performance of μ-ILEDs. The model in this paper can be easily extended to study other μ-ILED systems with other layouts and materials, for different types of applications.

## Acknowledgements

C.L. acknowledges support from the NSFC (grant no. 11172263) and the New Star Program from Zhejiang University. C.L. and Y.L. acknowledge support from the CSC. J.Z., Y.H. and J.A.R. acknowledge support from the NSF (OISE-1043161, 1043143 and 1043135). Y.H. also acknowledges support from the NSFC.

- Received April 13, 2012.
- Accepted May 17, 2012.

- This journal is © 2012 The Royal Society