In this chapter, we perform an analysis of the fraction of the secondary thermal power sources in the total power dissipation for MP LED packages. First, we analytically estimate the power losses related to the trapped light ${P}_{hC}$. We demonstrate a method to estimate ${P}_{hC}$ based on the values of IQE, EQE, and LEE parameters and on the geometries of the LED chip and the package. Next, we estimate the thermal losses caused by phosphor light conversion ${P}_{hD}$. Finally, we propose an experimental method that enables separation of the secondary heat sources power ${P}_{h}^{\prime}$ from the total thermal power ${P}_{h}$. The method is based on a revised analytical solution of an initial thermal transient response. We analyze the applicability of the proposed method for LEDs with transparent domes and for LEDs with phosphor light conversion.

#### 2.5.1. Secondary Heat Sources

In this subchapter, we aim to estimate the fraction that secondary heat sources contributing to the total heat dissipation. First, we analyze LEDs without phosphor light conversion. In this case the secondary heat sources are originated from the light trapped in the encapsulating dome layer due to total inner reflections (TIRs), (see

Figure 5). This light is partially absorbed by the reflector cup walls, the reflective metal contact pads, and the pn junction when light re-enters the LED crystal [

47].

In general, evaluation of the trapped light losses requires sophisticated ray tracing modeling and extensive LED characterization. Yet, it is possible to make a coarse analytical estimation employing approximations of the

LEE and

IQE coefficients and other LED parameters, such as cup reflectivity, dome curvature, die and package dimensions, etc. First, we define the total thermal power

${P}_{h}$ dependence on the applied electrical power. The total

LEE of an LED package is a product of the chip-to-dome light extraction coefficient

$LE{E}_{chip}$ and the dome-to-air light extraction coefficient

$LE{E}_{dome}$, thusly:

$EQE$ is a product of

$IQE$ and

$LEE$, that is:

Unlike

$LE{E}_{chip}$ and

$LE{E}_{dome}$,

$IQE$ is dependent on the forward current

$I$. The total thermal power dissipated in an LED is:

The fraction of the light initially left the die and trapped on the reflection cup walls is:

where coefficient

$\mathsf{\Lambda}$ provides a correction for the crystal light re-absorption.

Appendix A analyzes

$\mathsf{\Lambda}$ dependence on the LED’s package geometry and other parameters.

The

$LE{E}_{dome}$ coefficient is difficult to measure directly without manufacturing packages of custom calibration LEDs and without advanced measurement setups. Therefore, we use the results of ray tracing simulation presented by Tran et al. [

42]. The results enable estimation of the

$LE{E}_{dome}$ for MP LED packages. The authors define the

$LE{E}_{dome}$ of an LED, dependence on the dome curvature, and angle of the reflector cup. The simulation results evidence that the

$LE{E}_{dome}$ coefficient for conventional dome designs varies from 0.65–0.92. The

$LE{E}_{dome}$ coefficient for conventional multiple-chip LEDs with flat light-emitting surface (LES) and the absence of special light extraction enhancement structures is around 60% [

48,

49,

50]. Optimization of the LEDs packages by using a gradient refractive index encapsulant, roughened or patterned lead-frame substrates, and the scattering effect of phosphor particles can increase the

$LE{E}_{dome}$ up to 85%.

Now we estimate the fraction of

${P}_{hC}$ in the total power dissipated by an LED:

We then estimate the

${P}_{hC}/{P}_{h}$ ratio for blue GaN LEDs. Modern state-of-the-art high brightness blue LEDs can have a

$LE{E}_{chip}$ coefficient of 85% for double-side textured-crystals [

51]. The reflectance coefficient of the cup is set to 93%. The ratio between the areas of the cup and the crystal

${S}_{cup}/{S}_{cry}$ plays an important role. It defines the probability of crystals re-entry by the trapped light. We consider two

${S}_{cup}/{S}_{cry}$ ratios: 7 and 2. The first corresponds to the MP LED architecture presented in

Table 1. The second represents a smaller single die or a multiple die LED package. Indeed, multiple-die packages with low

${S}_{cup}/{S}_{cry}$ ratios have increased shielding of the light by the neighboring dies (denser chip placement leads to an increased light re-absorption by the LED package chips) [

52]. The results are presented in

Figure 6.

Figure 6 plots evidence that LED chips with high

IQE in low

$LE{E}_{dome}$ packages have the highest relative

${P}_{hC}$ thermal losses. The fraction of

${P}_{hC}$ rapidly increases with increase of

IQE. A decrease of the

${S}_{cup}/{S}_{cry}$ value leads to a decrease of the

${P}_{hC}$ fraction due to the shielding effect.

The IQE of blue GaN LEDs approaches its theoretical limit of 95%. Nevertheless, in the majority of high-power applications, blue LEDs are driven in the droop regime when the IQE is approximately 70% (marked at the plot). These parameters indicate that approximately up to 25% of the total thermal power ${P}_{h}$ can be dissipated on the LED cup walls due to TIR.

LEDs with a phosphor light conversion layer always have extra heat losses due to Stokes effect. This emphasizes the fact that the secondary heat sources are significant. Next, we propose a methodology of their experimental estimation.

#### 2.5.2. Estimation of the Secondary Heat Sources

Modern LED transient testing methods determine the total thermal power

${P}_{h}$ as a difference between the applied electrical power

${P}_{el}$ and the emitted radiant flux

${P}_{opt}$ [

32,

33,

34,

35]. It is impossible to separate the secondary heat sources

${P}_{h}^{\prime}$ with this approach. In this section, we propose a method of experimentally estimate

${P}_{h}^{\prime}$. The method is based on a revised solution of the initial

${T}_{j}$ transient response.

In practice, it is challenging to measure the initial

${T}_{j}$ response during the first tens of milliseconds due to the electrical transient processes in the pn junction, the connecting wires and the transient measurement equipment. Therefore, correction methods were developed to restore this data. One of these methods is a square root correction. The method is based on the analytical solution of heat propagation into a semi-infinite material from a homogeneous surface heat source: the pn junction substrate can be often approximated as a surface heat source and a semi-infinite body in the beginning of thermal transient. The initial

${T}_{j}$ transient response can be approximated with the following equation [

53,

54]:

The form of the equation evidences that the initial

${T}_{j}$ transience response plotted versus a square root of time is linear. The coefficient

K bounds the slope of this plot with the junction dissipated thermal power

${P}_{hJ}$ and the chip surface

${S}_{pn}$. An example of measured LED data and of the applied square root initial transient correction based on 50–400 μsec interval is shown in

Figure 7.

The correction method assumes a unilateral one-dimensional heat propagation in the beginning of the transient. While LEDs chips are typically encapsulated in a dome to enhance light extraction and shape the light beam. Thus, the heat dissipated by the pn junction propagates bilaterally, both to the substrate and the dome. We solve a problem of the bilateral heat transfer into two semi-infinite bodies representing the sapphire crystal substrate and the encapsulating dome. We solve a problem of the bilateral heat transfer into two semi-infinite bodies representing the sapphire crystal substrate and the dome. The solution is presented in

Appendix B. The resulting equation retains the square root time-dependence. After substituting the power density

$q$ with

${P}_{hJ}/{S}_{pn}$ in Equation (A17) we obtain the coefficient

${K}_{bi\_lat}$ characterizing

${T}_{j}$ initial response for the case of bilateral heat propagation. The original coefficient

${K}_{uni\_lat}$ and the derived

${K}_{bi\_lat}$ are presented below:

The similar form of the bilateral heat propagation solution justifies the application of the square root initial correction for LEDs. Nevertheless, the bilateral solution bounds quite accurately the initial ${T}_{j}$ response to the thermal power dissipated by the junction ${P}_{hJ}$.

We determine the values of the heat flows toward the dome and the substrate during the initial thermal transient by substituting the correspondent thermal properties in

Appendix B Equations (A15) and (A16). We derive that approximately 5% of the total heat is dissipated by the junction propagates to the dome. This corresponds to approximately 5% systematic error for

${P}_{hJ}$ power evaluation with the classic method when no heat propagation into the dome is considered.

The ${T}_{j}$ response follows the square root time dependency only for a finite amount of time while the assumption of one-dimensional heat propagation is valid on both sides of the active region. On the one hand, it is limited by the characteristic time constant of the substrate ${\tau}_{s}$. On the other hand, by the requirement of sufficiently one-dimensional heat propagation into the dome.

The substrate time constant

${\tau}_{s}$ can be found as a product of the partial thermal resistance and thermal capacitance of the die crystal, which can be expressed with sapphire thermal properties and the crystal height

${H}_{sap}$:

The heat propagation into the dome can be considered sufficiently one-dimensional if the characteristic width of the pn junction

${l}_{pn}^{\prime}$ is much larger than the characteristic depth of heat propagation into the dome

${x}_{d}^{\prime}$:

We estimate

${l}_{pn}^{\prime}$ as a one half of the minimal dimension of the top of the sapphire crystal. Carslaw and Jaeger [

55] have derived a closed-form solution of the time-dependent temperature profile for heat propagation into a semi-infinite media:

The form of the exponential term of the Equation (17) yields the

${x}_{d}^{\prime}$ dependence on time:

Thus, the characters time

${\tau}_{d}$ at which

${x}_{d}^{\prime}\approx {l}_{pn}^{\prime}$ is:

Therefore, the initial heat propagation is sufficiently one-dimensional on both sides of the active region if:

${\tau}_{s}$ and

${\tau}_{d}$ are estimated as

$3.8\mathrm{msec}$ and

$250\mathrm{msec},$ respectively, considering the data of

Table 1. Estimated value of

${\tau}_{s}$ is significantly smaller than

${\tau}_{d}$. Thus, we conclude that the heat propagation into the silicone dome is always sufficiently one dimensional until the heat flux has not reached the DAL via sapphire substrate. Therefore,

${P}_{hJ}$ can be reliably extracted by Equations (12) and (14). Then,

${P}_{h}^{\prime}$ can be found as a difference between

${P}_{h}$ and

${P}_{hJ}$.

#### 2.5.3. Applicability of the Approach for LEDs with Phosphor Light Conversion

In this section, we investigate the applicability of the proposed method of ${P}_{hJ}$ estimation for LEDs with silicone/phosphor composite domes. The heat generation by phosphor particles may disturb the initial thermal transient which can affect the accuracy. We investigate the impact of the presence of phosphor on the accuracy of the proposed ${P}_{hJ}$ extraction method.

Firstly, we perform estimation of the thermal properties of the phosphor/silicone composite material. We use the phosphor filler volume fraction

$f$ parameter defined with the volumes of the silicone

${V}_{sil}$ and the phosphor

${V}_{pho}$ fractions as:

Numerous models have been proposed to model the effective thermal conductivity of this type of composites [

56,

57,

58]. These models are typically derived for a certain range of the phosphor volume fraction. We use a high volume fraction limit model proposed by Every [

59]. The effective thermal conductivity

${k}_{d}$ of the silicone/phosphor composite material dome is expressed as:

Here, ${k}_{sil}$ is the thermal conductivity of the silicone and $\mathsf{\alpha}$ is a nondimensional parameter bounding the particle size and particle-composite matrix interface effect.

Zhang et al. [

60] have fitted the model of the equation to experimental measurements of the typical silicone/phosphor composite used in LEDs They used and Ce

^{3+} doped YAG (

${Y}_{3}A{l}_{5}{O}_{12}$) phosphor particles of

$13.0\pm 2.0\mathsf{\mu}\mathrm{m}$ diameter encapsulated in high optical transparency silicone. They achieved an excellent agreement with the experimental results for high volume concentrations

$f$ from 20–40%. It was found that

$\alpha $ is 0.004. This fitting result slightly overestimate the thermal conductivity for the composites with low phosphor volume fractions.

The YAG phosphor density

${\rho}_{pho}$ and the specific heat

${C}_{pho}$ are

$4.56\text{}\mathrm{g}/{\mathrm{cm}}^{3}$ and

$0.6J/\left(gK\right)$, respectively [

61]. The density

${\rho}_{d}$ and specific heat

${C}_{d}$ of the dome composite are estimated based on the volume fraction of the phosphor particles

$f$ as:

The phosphor dome light conversion efficiency was shown to be dependent on the phosphor particles type, concentration, temperature, experienced thermal stress [

13,

44,

62,

63]. Yet, we assume the silicone/phosphor composite properties to be constant during the fast initial transient processes due to small temperature variations. Consequently, the heat transfer problem remains linear. Therefore, we address the multiple heat source initial thermal transient analysis using the principle of superposition. We analyze heating of the LED package with

${P}_{hD}$ and

${P}_{hJ}$ heat sources separately and compare the heating rates.

We start with the analysis of the silicone/phosphor composite dome. The data presented by Chung [

64] shows the evidence that the heating time constant of the remote phosphor layers is significantly slower than the pn junction heating time constant (

$1\mathrm{min}$ vs.

$0.02\mathrm{sec}$). However, the phosphor layers deposited over the pn junction may have significant higher heating rates due to the higher optical power density. Lou et al. [

20,

65] determined the phosphor energy conversion efficiency both experimentally and numerically. For warm white LEDs with high phosphor volume fractions, up to 45% of blue light optical power can be dissipated as heat during light conversion [

66]. If we assume that

$WP{E}_{chip}=70\%$ for the LED chip (a typical value for modern blue LEDs under typical operational conditions), then it will mean that up to half of the total thermal power

${P}_{h}$ for white LED package can be related to the phosphor thermal losses

${P}_{hD}$. Similar power ratio results were previously determined in the literature [

19,

22].

The rate at which the temperature of the silicone/phosphor composite dome increases during the first hundreds of milliseconds after turning an LED on is linear. It is determined mainly by the capacitive thermal effects due to the low thermal conductivity of the silicone/phosphor composite. Thus, we estimate the composite dome temperature increase

$\Delta {T}_{d}$ at times

$\tau \ll {\tau}_{d}$ as:

Here ${V}_{d}$ is the characteristic dome volume per die. Multiple die LEDs have higher dome optical power density than the one-die LEDs. This leads to a faster rate of silicone/phosphor composite heating. To consider a worst-case scenario we perform an estimation for a case of two-die MP 3030 white LED package. Thus, we chose ${V}_{d}$ as a half of the correspondent LED dome volume.

We compare the characteristic dome temperature increase with the junction temperature increase $\Delta {T}_{j}\left(\tau \right)$ estimated by Equation (12) and ${K}_{bi\_lat}$ coefficient. The comparison yields evidence that for $\tau $ below $400\text{}\mathsf{\mu}\mathrm{sec}$ (which is a typical upper time limit used for transient correction), $\Delta {T}_{d}$ is less than $0.03\xb7\Delta {T}_{j}$. Thus, the initial heating rate of the silicone/phosphor composite is significantly slower than one of the pn junction. Moreover, the thermal conductivity of the silicone/phosphor composite is considerably lower than one of the sapphire crystal. All these factors indicate that the phosphor-related thermal losses have insignificant impact on the initial transients, as will be confirmed in the next chapter.

The unilateral coefficient

${K}_{uni\_lat}$ and the derived dependence of the bilateral coefficient

${K}_{bi\_lat}$ as a function of the phosphor fraction

$f$ are shown in

Figure 8. The data confirms 5–15%

${K}_{uni\_lat}$ relative error if compared with more precise

${K}_{bi\_lat}$ values. The same error will have

${P}_{hJ}$ when estimated by Equation (12) under the classical unilateral heat propagation assumption.