Internal Quantum Efficiency in Light-Emitting Diodes

Abstract

In this chapter, we present different techniques used to assess the internal quantum efficiency (IQE) in light-emitting diodes (LEDs). The commonly used technique based on temperature-dependent photoluminescence relies in strong assumptions which are discussed in this chapter.

Appendices

Appendix A: Theoretical Model of Light Emission in LEDs: QW Emission Described by Classical Dipoles

QWs in LEDs are usually more than one, distributed over a finite thickness within the device, as each is a few nanometers wide and spaced apart by barrier layers of tens of nanometers thick. One might assume that each well is equally excited and thus emits an equal number of photons in a spherically symmetric pattern. However, it is just as plausible to assume that under EL excitation, either the first well that carriers encounter, the last well (perhaps immediately preceding an electron barrier layer), or one of the center wells (being most equally accessible to both electrons and holes) would have a disproportionately larger share of the emission. David et al. [12] examined this question with regard to GaN/InGaN LEDs, and reported that nearly all of the emission from an LED comes from the QW nearest to the p-doped side of the device. The predominant emission from the top QW was attributed to the poor hole transfer between QWs and occurs regardless of the number of quantum wells, often requiring some special design such as a double heterostructure to modify the carrier distribution [12]. That determination was made possible due to the fact that the various QWs have different emission patterns due to their different distances to the strongly reflecting Ag/GaN interface.

To address the question of calculating light emission per QW, we refer to Benisty et al. [26], who used dipole emitters as a photon source term in LED emission models. Benisty et al. model the emission from the dipoles as a discontinuity in the scattering matrix propagation technique [27], as we describe below. The use of dipole emitters is justified by the similarity between the normalized expressions of the rate of spontaneous emission in QWs and the power emitted by classic electric dipoles. The rate of spontaneous transitions of e–h pairs between the conduction and valence bands in a QW, given by Fermi’s golden rule, is proportional to $\mathcal {M}_{c-v} = \vert \langle \psi _{c} \mid \hat{e} \hat{p}\mid \psi _{v} \rangle \vert ^2$, where $\hat{e}$ is the light polarization, $\hat{p}$ is the momentum operator, and $\psi _{c}$ and $\psi _{v}$ are the conduction and valence band wavefunctions, respectively [28, 29]. The angular dependence of $\mathcal {M}_{c-v}$ is well described by combinations of horizontal and vertical dipole-like terms [30]. The normalized radiation patterns, given by the power per unit of solid angle, of vertical (v) and horizontal (h) dipoles, for both TE and TM polarizations are given by [26]

$$\begin{aligned} \frac{dP_{\text {source}}^{(v,TM)}}{d\Omega }(\theta _L)&=\frac{3}{8\pi }\text {sin}^2\theta _L \end{aligned}$$

(6.10a)

$$\begin{aligned} \frac{dP_{\text {source}}^{(h,TE)}}{d\Omega }(\theta _L)&=\frac{3}{16\pi }\end{aligned}$$

(6.10b)

$$\begin{aligned} \frac{dP_{\text {source}}^{(h,TM)}}{d\Omega }(\theta _L)&=\frac{3}{16\pi }\text {cos}^2\theta _L , \end{aligned}$$

(6.10c)

where $\theta _L$ is the angle with respect to the vertical direction.

The replacement of the electron–hole excitons by a uniform distribution of dipoles, whose emission is propagated through the multilayered structure by transfer matrix formalism, allows the calculation of the electric field in any layer of the structure by propagating the electric field from the dipole layer, which are determined from (6.10) as

$$\begin{aligned} E_{\text {source}}^{(\gamma ,\rho )}(\theta _L)=\sqrt{\frac{dP_{\text {source}}^{(\gamma ,\rho )}}{d\Omega }(\theta _L)} \end{aligned}$$

(6.11)

for a $\gamma$ dipole orientation and a $\rho$ polarization.

6.1.1 Analytical Model for Light Extraction Efficiency

The theoretical assessment of the light extraction efficiency $\eta _{{\text {extr}}}$ in an LED structure is based on the fraction of the integrated emitted power that exits the LED structure and propagates to air. This is determined from the calculation of the electric field radiated from the QWs, based on the propagation of the dipole electric field throughout the structure. The external power per solid angle $dP_{\text {out}}^{(\gamma ,\rho )} / d\Omega dS(\theta )$, in the external direction $\theta$, is given by the flux of the Poynting vector emitted from the dipoles, transmitted through the structure and corrected by the change in solid angle when the medium is changed [26]:

$$\begin{aligned} \frac{dP_{\text {out}}^{(\gamma ,\rho )}}{d\Omega dS}(\theta )=\underbrace{\left( \vert E_{\text {out}}^{(\gamma ,\rho )}(\theta ) \vert ^2 \frac{n_{\text {out}}\text {cos}(\theta )}{n_{L}\text {cos}(\theta _{L})}\right) }_{\text {transmitted power from dipoles}}\underbrace{\left( \frac{n_{\text {out}}^2\text {cos}(\theta )}{n_{L}^2\text {cos}(\theta _{L})}\right) ,}_{\text {change in solid angle}} \end{aligned}$$

(6.12)

where $E_{\text {out}}^{(\gamma ,\rho )}$ is the external electric field for a $\gamma$ dipole orientation and a $\rho$ polarization, and $n_{\text {out}}$ and $n_{L}$ are the refractive indexes of the external and LED media, respectively. $\theta$ and $\theta _{L}$ are respectively the external and internal angles with respect to the vertical direction. The angles in each medium $\theta _i$, between the vertical and the propagation direction of emitted light, are determined from Snell’s law $n_i\text {cos}(\theta _i)=n_j\text {cos}(\theta _j)$. The first term in (6.12) corresponds to the transmitted power from the dipole source to the external medium, given by the Fresnel transmission coefficient and the second term corresponds to the changes in solid angle from different media, obtained from the derivative of Snell’s law.

The light extraction efficiency of the LED through a single facet is given by the ratio of the total output power, calculated from the integration of (6.12) for $\theta$ from 0 to $\pi /2$, to the total emitted power. The total emitted power by the normalized dipole source is unity for dipoles in bulk material. However, the total radiation by the dipole source inside an LED heterostructure is modified by the Purcell factor, which corresponds to the relative change in spontaneous emission from a source within an optical cavity compared to the same source in a bulk material [31]. The Purcell factor depends largely on the position of the QWs relative to the interface between the metallic contact and GaN, as well as on the choice of metal (or more specifically, on the metal reflectivity), which is treated in more details in Appendix B. Figure 6.13 in Appendix B shows the deviation from unity of the total emitted power by the dipoles for different metals (Al, Au, and Ag) versus the distance of the QW to their interface with GaN. A judicious choice of contact metal, such as Au, can largely reduce this dependence and for most of the practical cases as well as for the simple geometry structure treated here, the Purcell factor is close to one and the total emitted power by the dipoles can be approximated to unity (deviation from unity of 2.8% at most). Figure 6.13 of Appendix B also offers a correction factor in case a different metal is used as contacts.

Therefore, the light extraction efficiency of the LED through its top facet is given by

$$\begin{aligned} \eta _{{\text {extr}}}={\int }^{\frac{\pi }{2}}_0 2\pi \dfrac{dP_{\text {out}}^{(\gamma ,\rho )}(\theta )}{d\Omega dS}\text {sin}(\theta )\, d\theta \end{aligned}$$

(6.13)

summed for $\rho =\text {TE}\, \text {and}\, \text {TM}$, and $\gamma =v\, \text {and}\,h$. The e-HH recombinations in the QWs, in both TE and TM polarizations, can be well described by horizontal electric dipoles. The e-LH recombinations can only be partially described by the combination of horizontal and vertical dipoles and will be neglected at low current injections due to the nondegeneracy of the light hole band at the minimum of energy and to the lower density of states of this energy band [30]. Thus $\gamma =h$. Therefore, the only requirement to determine $\eta _{{\text {extr}}}$ is to calculate the external electric field $E_{\text {out}}^{(\gamma ,\rho )}$, which is treated in the following section.

6.1.2 Exact Calculation of the Electric Field in a Multilayer Structure

The calculation of the electric field in a multilayer structure is based on the transfer matrix formalism, where a wave $E_{\uparrow }e^{+ik_{z}z}+E_{\downarrow }e^{-ik_{z}z}$ is represented by $\left( \begin{array}{c} E_{\uparrow }\\ E_{\downarrow }\end{array}\right)$, which is the electric field of the upward and downward plane waves, respectively, and $k^i_{z}=n_ik_{0}cos(\theta _i)$ in the medium i.

In a multilayer structure, the source electric field is propagated through the structure by multiplying it to propagation $M_{\text {prop}}$ matrices in the homogeneous layers and interface $M_{\text {interf}}$ matrices in the interface between two different media. A convenient property of this method is that an equivalent matrix $\mathcal {M}$ for the entire structure can be simply obtained by multiplying all the interface and propagation matrices corresponding to the structure (Fig. 6.10), as

$$\begin{aligned} \mathcal {M}=...\,M^{i+1,i}_{\text {interf}}M^{i}_{\text {prop}}M^{i,i-1}_{\text {interf}}\,... \end{aligned}$$

(6.14)

The propagation matrix from $z_{1}$ to $z_{2}$ in a homogeneous medium is

$$\begin{aligned} M_{\text {prop}}= \left[ \begin{array}{cc} e^{i(k_{z}(z_{2}-z_{1}))} &{} 0 \\ 0 &{} e^{-i(k_{z}(z_{2}-z_{1}))} \\ \end{array} \right] . \end{aligned}$$

(6.15)

The interface matrices for TE and TM polarizations are given by

$$\begin{aligned} M^{\text {TE}}_{\text {interf}}= \left[ \begin{array}{cc} \dfrac{k_{z}^{(2)}+k_{z}^{(1)}}{2k_{z}^{(2)}} &{} \dfrac{k_{z}^{(2)}-k_{z}^{(1)}}{2k_{z}^{(2)}} \\ \dfrac{k_{z}^{(2)}-k_{z}^{(1)}}{2k_{z}^{(2)}} &{} \dfrac{k_{z}^{(2)}+k_{z}^{(1)}}{2k_{z}^{(2)}} \\ \end{array}\right] \;,\;M^{\text {TM}}_{\text {interf}}= \left[ \begin{array}{cc} \dfrac{n_{2}^{2}k_{z}^{(1)}+n_{1}^{2}k_{z}^{(2)}}{2n_{1}n_{2}k_{z}^{(2)}} &{} \dfrac{-n_{2}^{2}k_{z}^{(1)}+n_{1}^{2}k_{z}^{(2)}}{2n_{1}n_{2}k_{z}^{(2)}} \\ \dfrac{-n_{2}^{2}k_{z}^{(1)}+n_{1}^{2}k_{z}^{(2)}}{2n_{1}n_{2}k_{z}^{(2)}} &{} \dfrac{n_{2}^{2}k_{z}^{(1)}+n_{1}^{2}k_{z}^{(2)}}{2n_{1}n_{2}k_{z}^{(2)}} \end{array}\right] . \end{aligned}$$

(6.16)

To obtain analytical expressions for the electric field in the outer media, we impose, as boundary condition of this problem, that the electric field of the incoming waves in the outer media is zero (Fig. 6.10). The external electric fields, from the top and bottom media, are then propagated to the source, where the discontinuity from the source is applied as

Fig. 6.10

Schematic of a propagation of the electric fields in a multilayer structure. The external electric fields $E_u$ and $E_d$, from the top and bottom media, are propagated to the source, where the discontinuity from the source electric field is applied. The matrices $M_a$ and $M_b$ correspond to the transfer matrix of the bottom (a) and top (b) halves of the LED structure

$$\begin{aligned} \mathcal {M}_b \left( \begin{array}{c} E^{(\gamma ,\rho )}_{u}\\ 0\end{array}\right) - \mathcal {M}_a \left( \begin{array}{c} 0\\ E^{(\gamma ,\rho )}_{d}\end{array}\right) = \left( \begin{array}{c} E^{(\gamma ,\rho )}_{\text {source}}(\theta )\\ E^{(\gamma ,\rho )}_{\text {source}}(\theta )\end{array}\right) \end{aligned}$$

(6.17)

where

$$\begin{aligned} \mathcal {M}_a= \left[ \begin{array}{cc} a_{11} &{} a_{12} \\ a_{21} &{} a_{22} \\ \end{array} \right] \; \text {and} \; \mathcal {M}_b= \left[ \begin{array}{cc} b_{11} &{} b_{12} \\ b_{21} &{} b_{22} \\ \end{array} \right] \end{aligned}$$

(6.18)

are calculated from (6.14) for the bottom (a) and top (b) halves of the LED structure (Fig. 6.10). The electric fields in the top (u) and bottom (d) outer media are

$$\begin{aligned} E_u^{(\gamma ,\rho )}(\theta )=E_{\text {source}}^{(\gamma ,\rho )}\dfrac{\frac{1}{b_{11}}(1+\frac{a_{12}}{a_{22}})}{1-\frac{b_{21}}{b_{11}}\frac{a_{12}}{a_{22}}} \text { and } E_d^{(\gamma ,\rho )}(\theta )=E_{\text {source}}^{(\gamma ,\rho )}\dfrac{\frac{1}{a_{22}}(1+\frac{b_{21}}{b_{11}})}{1-\frac{b_{21}}{b_{11}}\frac{a_{12}}{a_{22}}}. \end{aligned}$$

(6.19)

In the case of the simple geometry considered in this chapter, the matrix terms are

$$\begin{aligned} \frac{1}{b_{11}}=\dfrac{2k_z^b}{k_z^b+k_z^u}e^{ik_z^bL_b}\; \text {and} \; \frac{1}{a_{22}}=\dfrac{2k_z^a}{k_z^a+k_z^d}e^{ik_z^aL_a}, \end{aligned}$$

(6.20)

which correspond to transmission coefficients and

$$\begin{aligned} \frac{b_{21}}{b_{11}}=\dfrac{k_z^b-k_z^u}{k_z^b+k_z^u}e^{2ik_z^bL_b} \; \text {and} \; \frac{a_{12}}{a_{22}}=\dfrac{k_z^a-k_z^d}{k_z^a+k_z^d}e^{2ik_z^aL_a}, \end{aligned}$$

(6.21)

corresponding to reflection coefficients.

The electric field after the metal layer can be written as

$$\begin{aligned} E_{\text {out}}^{(\gamma ,\rho )} =\dfrac{E_{\text {source}}^{(\gamma ,\rho )}t_{L,m,a}^{\rho }e^{i\phi '}(1+r_{L,s}^{\rho }e^{i2(\phi -\phi ')})}{1-r_{m,L,a}^{\rho }r_{L,s}^{\rho }e^{i2\phi }} = \dfrac{E_{\text {source}}^{(\gamma ,\rho )}t_{L,a}^{\rho }e^{i\phi '}(1+r_{L,s}^{\rho }e^{i2(\phi -\phi ')})}{1-r_{L,a}^{\rho }r_{L,s}^{\rho }e^{i2\phi }} \dfrac{1-r_{L,a}^{\rho }r_{L,s}^{\rho }e^{i2\phi }t_{L,m,a}^{\rho }}{1-r_{m,L,a}^{\rho }r_{L,s}^{\rho }e^{i2\phi }t_{L,a}^{\rho }}. \end{aligned}$$

(6.22)

The first term is the external electric field without metal contacts and the second is the metal transmission function. Using the general property: $x,y \in \mathbb {C}$, $\vert xy \vert ^2=\vert x \vert ^2\vert y \vert ^2$, we obtain [13]

$$\begin{aligned} \vert E_{\text {out}}^{(\gamma ,\rho )} \vert ^2 = \mathcal {T}^\rho _m \vert E_0^{(\gamma ,\rho )} \vert ^2. \end{aligned}$$

(6.23)

Let us first calculate the external electric field $E_0^{(\gamma ,\rho )}$ in a structure without metal contact. Let the total LED thickness be L and the position of the dipole source within the LED be z. The corresponding phase shifts in the LED are $\phi (\theta _L)=n_{L}k_{0}L\cos (\theta _{L})$ and $\phi '(\theta _L)=n_{L}k_{0}z\cos (\theta _{L})$, where $k_{0}=2\pi /\lambda$, $\lambda$ is the wavelength of the emitted light, $n_{L}$ and $\theta _{L}$ are the refractive index and angle in the LED layer, respectively. The transmission and reflection coefficients for a polarization $\rho$, at each interface from a medium i to j, are given respectively by $t_{i,j}^{\rho }$ and $r_{i,j}^{\rho }$.

The external electric field can be determined from the propagation of the dipole electric field $E_{\text {source}}^{(\gamma ,\rho )}$ [18]:

$$E_0^{(\gamma ,\rho )}(\theta )=E_{\text {source}}^{(\gamma ,\rho )}e^{i\phi '}t_{L,a}^{\rho }(1+r_{L,a}^{\rho }r_{L,s}^{\rho }e^{2i\phi }+\cdots )+ E_{\text {source}}^{(\gamma ,\rho )}e^{i(2\phi -\phi ')}r_{L,s}^{\rho }t_{L,a}^{\rho }(1+r_{L,a}^{\rho }r_{L,s}^{\rho }e^{2i\phi }+ \cdots )$$

$$\begin{aligned} E_0^{(\gamma ,\rho )}(\theta )=\frac{E_{\text {source}}^{(\gamma ,\rho )}t_{L,a}^{\rho }e^{i\phi '}(1+r_{L,s}^{\rho }e^{i2(\phi -\phi ')})}{1-r_{L,a}^{\rho }r_{L,s}^{\rho }e^{i2\phi }}. \end{aligned}$$

(6.24)

We recall the transmission and reflection expressions for both TE and TM polarizations for a plane wave going from a medium i to j:

$$\begin{aligned} r_{i,j}^{\text {TE}}=\dfrac{n_i\text {cos}(\theta _i)-n_j\text {cos}(\theta _j)}{n_i\text {cos}(\theta _i)+n_j\text {cos}(\theta _j)} \text {,}\, r_{i,j}^{\text {TM}}=\dfrac{n_i\text {cos}(\theta _j)-n_j\text {cos}(\theta _i)}{n_i\text {cos}(\theta _j)+n_j\text {cos}(\theta _i)}\nonumber \\ t_{i,j}^{\text {TE}}=\dfrac{2n_i\text {cos}(\theta _i)}{n_i\text {cos}(\theta _i)+n_j\text {cos}(\theta _j)} \text {,}\, t_{i,j}^{\text {TM}}=\dfrac{2n_i\text {cos}(\theta _i)}{n_j\text {cos}(\theta _i)+n_i\text {cos}(\theta _j)}.\nonumber \end{aligned}$$

The transmission function $\mathcal {T}^\rho _m$ for the metal contact is a simple replacement of the transmission and reflection coefficients of a simple LED/air interface (L, a) with those of an LED/metal/air interface (L, m, a). It contains all the metal parameters separated from the more general expression of $E_0^{(\gamma ,\rho )}(\theta )$ and it is written as

$$\begin{aligned} \mathcal {T}^\rho _m={{{{\vert }}}}\dfrac{1-r_{L,a}^{\rho }r_{L,s}^{\rho }e^{i2\phi }}{1-r_{m,L,a}^{\rho }r_{L,s}^{\rho }e^{i2\phi }}\;\dfrac{t_{L,m,a}^{\rho }}{t_{L,a}^{\rho }}{{{{\vert }}}} ^2, \end{aligned}$$

(6.25)

where now $r_{L,m,a}^{\rho }$ and $t_{L,m,a}^{\rho }$ accounts for the complex refractive index $\tilde{n}_m=n_m+ik_m$ and thickness $t_m$ of the metal layer as in a homogeneous film [18]:

$$\begin{aligned} r_{L,m,a}^{\rho }=\dfrac{r_{L,m}^{\rho }+r_{m,a}^{\rho }e^{i\beta }}{1+r_{L,m}^{\rho }r_{m,a}^{\rho }e^{i2\beta }}\text {,}\;\;\;t_{L,m,a}^{\rho }=\dfrac{t_{L,m}^{\rho }t_{m,a}^{\rho }e^{i\beta }}{1+r_{L,m}^{\rho }r_{m,a}^{\rho }e^{i2\beta }} \end{aligned}$$

(6.26)

where $\beta =k_0\tilde{n}_mt_m\text {cos}(\theta _m)$. Therefore, the external field after the metal contact is calculated using (6.24) and (6.25) in (6.23).

The angular dependence of the external power per solid angle can be directly calculated as

$$\begin{aligned} \frac{dP_{\text {out}}^{(\gamma ,\rho )}}{d\Omega dS}(\theta )= \mathcal {T}^\rho _m(\theta ) \vert E_0^{(\gamma ,\rho )}(\theta ) \vert ^2 \frac{n_{\text {out}}^3\text {cos}(\theta )^2}{n_{L}^3\text {cos}(\theta _{L})^2}. \end{aligned}$$

(6.27)

This equation predicts the angular emission of the LED which is used to corroborate the theoretical and experimental results, as shown in later sections. The light extraction efficiency from the top surface in such structure is then calculated using (6.13), which can be easily done numerically.

6.1.3 Model for Light Extraction in a Simple LED Geometry

The analytical model presented in the previous section considered a monochromatic emission from the dipoles, but the QW emission in practical LEDs has a broad spectrum (typically 5–10% of the central wavelength). In this section, the spectral broadening of the source is considered and, while the results are not much different from the monochromatic model, it allows us to simplify the analytical model for $\eta _{{\text {extr}}}$.

The effect of the QW lineshape is included in the previous model by averaging the extraction efficiency at different wavelengths with the normalized spectral emission $s(\lambda )$ from the QWs as $\eta ^{poly}_{{\text {extr}}}=\int _0^{\infty } \! s(\lambda )\eta _{{\text {extr}}}(\lambda ) \, d\lambda$.

The QW spectral emission can be approximated by a Gaussian function $s(\lambda )=\frac{1}{\sqrt{2\pi \sigma ^2}}e^{-\frac{(\lambda -\lambda _c)^2}{2\sigma ^2}}$ and the asymmetry of its lineshape can be taken into account by combining two Gaussian functions, with different variances $\sigma ^2$, at their central wavelength. The polychromatic extraction efficiency is thus explicitly written as

$$\begin{aligned} \eta ^{poly}_{{\text {extr}}}={\sum }_{\rho =\text {TE},\text {TM}}{\int }_0^{\infty } \! s(\lambda ){\int }_0^{\frac{\pi }{2}} \! 2\pi \mathcal {T}^\rho _m(\theta ,\lambda ) \dfrac{dP_{\text {0}}^{(\gamma ,\rho )}(\theta ,\lambda )}{d\Omega dS}sin(\theta )\, d\theta \, d\lambda , \end{aligned}$$

(6.28)

where $dP_{\text {0}}^{(\gamma ,\rho )}(\theta ,\lambda ) / d\Omega dS$ is the external power per unit of solid angle of a structure without metal contacts and its dependence with $\lambda$ is due to Fabry–Perot interferences on the interfaces of the LED layer, which is generally not very pronounced in most of LED structures (thick LED structures), allowing us to make the following approximation [13]:

$$\begin{aligned} \eta ^{poly}_{{\text {extr}}} \simeq {\sum }_{\rho =\text {TE},\text {TM}}{\int }_0^{\frac{\pi }{2}} \! \underbrace{\left[ {\int }_0^{\infty } \! s(\lambda )\mathcal {T}^\rho _m(\theta ,\lambda )d\lambda \right] }_{\left\langle \mathcal {T}^\rho _m(\theta ) \right\rangle _\lambda } 2\pi \dfrac{dP_{\text {0}}^{(\gamma ,\rho )}(\theta ,\overline{\lambda })}{d\Omega dS}sin(\theta )\, d\theta \end{aligned}$$

(6.29)

for a fixed $\overline{\lambda }$ and only the metal transmission function is averaged with respect to $\lambda$. The averaged function $\left\langle \mathcal {T}^\rho _m(\theta ) \right\rangle _\lambda$ varies very slowly with $\theta$ for both TE and TM polarizations (an example is shown in Fig. 6.11). This is because the reflection and transmission coefficients do not vary much for $\theta _{L}$ inside the air cone, which is small in the case of high refractive index semiconductors, such as GaN and GaAs. Thus we can approximate (6.29) by a Variable-Incidence (VI) approximation:

$$\begin{aligned} \eta _{{\text {extr}}} \simeq \left\langle \mathcal {T}^{\text {TE}}_m \right\rangle _{(\lambda ,\theta )}\eta ^{0,\text {TE}}_{{\text {extr}}}+\left\langle \mathcal {T}^{\text {TM}}_m \right\rangle _{(\lambda ,\theta )}\eta ^{0,\text {TM}}_{{\text {extr}}}, \end{aligned}$$

(6.30)

where $\left\langle \mathcal {T}^{\rho }_m\right\rangle _{(\lambda ,\theta )}$ is the metal transmission function averaged in both $\lambda$ and $\theta$.

A simpler expression can be obtained by noticing that the averaged function $\left\langle \mathcal {T}^\rho _m(\theta ) \right\rangle _\lambda$ is nearly constant inside the air cone for both TE and TM polarizations (Fig. 6.11). Moreover, its value at $\theta = 0^{\circ }$ is the same for both polarizations $\left\langle \mathcal {T}^{\text {TE}}_m(0^{\circ }) \right\rangle _\lambda$ = $\left\langle \mathcal {T}^{\text {TM}}_m(0^{\circ }) \right\rangle _\lambda$ = $\left\langle \mathcal {T}_m(0^{\circ }) \right\rangle _\lambda$ (reflection and transmission coefficients are equal for both polarizations at normal incidence), thus we obtain the Normal-Incidence (NI) approximation:

$$\begin{aligned} \eta _{{\text {extr}}} \simeq \left\langle \mathcal {T}_m(0^{\circ }) \right\rangle _\lambda \eta ^{0}_{{\text {extr}}}. \end{aligned}$$

(6.31)

Notice that we started calculating the polychromatic extraction efficiency but ended with expressions for the monochromatic extraction efficiency. The spectrally averaged transmission function $\left\langle \mathcal {T}_m(0^{\circ }) \right\rangle _\lambda$ is numerically calculated as

$$\begin{aligned} \left\langle \mathcal {T}_m(0^{\circ }) \right\rangle _\lambda = \sum _{\lambda =\lambda _1}^{\lambda _2} \mathcal {T}_m(0^{\circ },\lambda ) \frac{1}{\sqrt{2\pi \sigma ^2}}e^{-\frac{(\lambda -\lambda _c)^2}{2\sigma ^2}}\Delta \lambda , \end{aligned}$$

(6.32)

where $\Delta \lambda = \frac{(\lambda _2-\lambda _1)}{n-1}$, n is the number of discrete values of $\lambda$, and the interval $[\lambda _1,\lambda _2]$ needs to be at least a few times larger than $\sigma ^2$.

Fig. 6.11

Plot of $\mathcal {T}^\rho _m(\theta )$ for $\theta$ within the air cone ($\theta < \theta _c$) and $\lambda '$ = 445 nm for TE (dashdotted) and TM (dotted), along with the plot of the averaged $\left\langle \mathcal {T}^\rho _m(\theta ) \right\rangle _\lambda$ for TE (squares) and TM (circles) polarizations. The structure is composed of a 4869-nm-thick GaN-based LED over a sapphire substrate, and the QWs are 140 nm below the top surface. The 15.4-nm-thick metal contact is Ni–Au alloy with experimentally determined optical properties [14]

The determination of the extraction efficiency of any LED structure is very simple using one of the two approximative models above. The monochromatic extraction efficiency for the LED structure with metal contacts for any wavelength is simply obtained from the multiplication of the averaged transmission function of the metal, averaged also in the air cone or at $\theta =0^{\circ }$, and the monochromatic extraction efficiency of the LED without metal contact ($\eta ^{0}_{{\text {extr}}}$).

$\eta ^{0}_{{\text {extr}}}$ is not very sensitive to the LED parameters in this simple LED geometry (treated in Appendix B), which can be generally calculated for any LED structure for a given material. The most sensitive parameters in these models are those from the metal, which are contained solely in the metal transmission function $\mathcal {T}^{\rho }_{m}$ and can be separately calculated using (6.25). In the following section, we apply these models to the case of a GaN-based LED.

Determination of the extraction efficiency: evaluation of $\eta _{{\text {extr}}}$, $\eta ^{0}_{{\text {extr}}}$, $\mathcal {T}^\rho _m(\theta ,\lambda )$ and $\left\langle \mathcal {T}_m(0^{\circ }) \right\rangle _\lambda$

Let us evaluate the terms $\mathcal {T}^\rho _m(\theta ,\lambda )$, $\left\langle \mathcal {T}_m(0^{\circ }) \right\rangle _\lambda$ and $\eta ^{0}_{{\text {extr}}}$ and compare the results of the extraction efficiency $\eta _{{\text {extr}}}$ from the approximative models (6.30) and (6.31) with the analytical model (6.27) to validate our approximations.

Fig. 6.12

a Extraction efficiency of the GaN LED structure versus wavelength using the analytical (solid-red), variable-incidence (VI) (cross-brown) and normal-incidence (NI) (dashdotted-green) approximations along with the extraction efficiency to air (dotted-blue) and its linear fit (dashed-black). The structure is composed of a 4869-nm-thick GaN-based LED over a sapphire substrate, and the QWs are 140 nm below the top surface. The 15.4-nm-thick metal contact is Ni–Au alloy with optical properties determined experimentally. Four different emission wavelengths were considered, $\lambda '_1$ = 405 nm, $\lambda '_2$ = 445 nm, $\lambda '_3$ = 485 nm, and $\lambda '_4$ = 525 nm in the function $s(\lambda )$ to evaluate the approximative models. b Extraction efficiency of the GaN LED structure versus metal thickness using the analytical with a monochromatic (solid) and polychromatic (squares) emission, VI (circle), and NI (dashdotted) approximations

The transmission function $\mathcal {T}_m(\theta ,\lambda )$ was averaged using (6.32) to obtain the approximative models (6.30) and (6.31), for which the main assumption was that $\left\langle \mathcal {T}^\rho _m(\theta ) \right\rangle _\lambda$ varied slowly with $\theta$ for both TE and TM polarizations. The evaluation of such function for the case of the GaN LED, for both polarizations, is presented in Fig. 6.11, which shows the plot of $\mathcal {T}^\rho _m(\theta )$ for a fixed $\lambda ' = 445$ nm for TE (dashdotted) and TM (dotted), as well as the averaged $\left\langle \mathcal {T}^\rho _m(\theta ) \right\rangle _\lambda$ for TE (squares) and TM (circles) polarizations, for $\theta$ within the air cone. The oscillations with respect to $\lambda$ observed in $\mathcal {T}^\rho _m(\theta )$ are completely eliminated when this function is averaged with the lineshape of the QWs. As a matter of fact, the averaged function is nearly constant for both polarizations, within the air cone, which supports both approximations made to obtain (6.30) and (6.31). This is due to the weak variation of the reflection and transmission coefficients inside the air cone which, in its turn, is small in the case of high refractive index semiconductors, such as GaN and GaAs. Let us now calculate the extraction efficiency to air $\eta _{{\text {extr}}}^0$ (without metal contact) for the GaN LED treated here, which is shown in Fig. 6.12a (dashed line). The small oscillations in extraction efficiency to air with $\lambda$ for a monochromatic emission average out in real devices with polychromatic (poly) emission—note that the oscillation period in this example is around 7 nm, which is substantially less than the typical 25 nm EL linewidth for blue GaN-based LEDs. As shown later, the extraction efficiency to air, through the top surface of the LED, depends weakly on the LED ‘fine’ structure (for a simple LED geometry), and therefore we can determine a general extraction efficiency to air for GaN LEDs grown in sapphire substrates by a linear fit of $\eta _{{\text {extr}}}^0$, which is represented by the dashed line in Fig. 6.12a:

$$\begin{aligned} \eta _{{\text {extr}}}^0(\lambda )=5.31\times 10^{4}\lambda + 2.43\times 10^{-2}. \end{aligned}$$

(6.33)

To validate the models, we calculate the extraction efficiency of the GaN LED structure using the analytical model (6.27) and compare the results to the ones from the approximative models.

The extraction efficiency of the GaN LED structure after the metal contact ($\eta _{{\text {extr}}}$) versus wavelength calculated using the analytical model is shown in the solid-red curve in Fig. 6.12a, along with the results of the VI (cross-brown) and the NI (dashdotted-green) approximations. Four different emission wavelengths were considered in this plot, $\lambda '$ = 405, 445, 485, and 525 nm, which were used in the function $s(\lambda )$ to evaluate the approximative models.

The application of both models for same structure when the metal thickness is varied is shown in Fig. 6.12b, where the circles correspond to the VI and the dashdotted line corresponds to the NI approximations. Again, there is a very good agreement with the analytical results (solid line).

These results were compared to the polychromatic extraction efficiency ($\eta _{{\text {extr}}}^{\text {poly}}$) taking into account the lineshape of the QWs, which was calculated analytically from (6.28) and represented in Fig. 6.12b by the full squares. The polychromatic $\eta _{{\text {extr}}}^{\text {poly}}$ is very similar to the monochromatic results which is due to the peaked QW emission at the center wavelength (variance $\sigma ^2$ is small compared to the center wavelength). The polychromatic extraction efficiency to air $\eta _{{\text {extr}}}^{\text {poly,0}}$ (corresponding to 0 nm of metal contact) agrees well with the monochromatic $\eta _{{\text {extr}}}^{\text {0}}$ calculated at the center wavelength of the QW linewidth.

Appendix B: Sensitivity of Model to LED Parameters

We present in this section, the investigation of the sensitivity of the calculated extraction efficiency to LED parameters, revealing that for large range of LED thicknesses and QW positions within the LED, the extraction efficiency through the top facet does not change significantly. Indeed the estimation of the extraction efficiency is mostly, if not solely, affected by the metal contact properties. Moreover, the sensitivity of our model to different metal contacts is presented, which shows that the appropriate choice of metal contacts reduces significantly the dependence of the modeling results on the LED parameters.

Fig. 6.13

a Extraction efficiency as a function of LED thickness for a QW positioned at a fixed 140 nm from the top surface for four cases of metal contacts: no metal, Au, Ag, and Al and all metals are 15.4-nm-thick. b Deviation from unity of total radiated power by the source within the LED as a function of the QW position for the same metals. c Total radiated power by the dipole source inside the LED as a function of LED thickness and QW position

Let us first present the effect of the LED thickness on the calculated extraction efficiency $\eta _{{\text {extr}}}$. Figure 6.13a shows the $\eta _{{\text {extr}}}$ of a GaN LED on sapphire substrate for $\lambda$ = 445 nm, where the QW is positioned at a fixed 140 nm distance from the top surface. Four cases of metal contacts are considered: no metal, Au, Ag, and Al, where the metal thickness is 15.4 nm. In all cases, the extraction efficiency oscillates significantly for an LED structure thinner than 1 $\upmu$m and tends to a fixed value for thicker structures. In practical LEDs, the fast oscillating behavior shown in Fig. 6.13a is averaged due to thickness fluctuations and polychromatic emission from the source which yields a much smoother function almost independent on LED thickness. Suffice to notice that the half period of such oscillations is $\sim$50 nm for a fixed $\lambda$ = 445 nm, which is much shorter than the thickness fluctuations present in real devices. A more general map of the extraction efficiency is shown in Fig. 6.4b, where an LED structure without metal contacts was considered, showing that the extraction efficiency $\eta ^0_{{\text {extr}}}$ is fairly constant for thick LED structures.

Let us investigate the effect of the LED parameters on the total power emitted from the source inside the LED cavity. In the models presented in this chapter, the dipole terms were normalized to yield a unity emitted power in a homogeneous medium. When these dipoles are placed inside the LED heterogeneous medium, or in an optical cavity, the total power emitted from the dipole source may vary because its amplitude is kept constant but the optical medium is modified. This effect is related to a change in radiative emission rate from a source within an optical cavity, or Purcell effect [31]. One assumption made in our models was that the total emitted power from the dipole sources inside the LED would be considered unity due to a negligible Purcell effect in common LED structures.

Here, we test this assumption by calculating the deviation from unity of the total emitted power by the dipole sources inside the LED cavity. Let us consider the effect of the most sensitive LED parameter which is the distance of the QWs to the top surface. The deviation of the total emitted power from unity by the dipole sources inside the LED structure calculated as a function of the QW distance to the top surface z for a total LED thickness of 4729 nm is shown in Fig. 6.13b. The same four cases of metal contacts are considered: no metal, Au, Ag, and Al, where the metal thickness is 15.4 nm and it is interesting to notice that the metal contact plays a significant role in this case.

The deviation from unity in total emitted power due to a cavity effect can be significantly reduced by judiciously choosing the metal contacts, for example, in the case of Au, it is at most 2% for realistic LED structures (z > 150 nm). In case other metals such as Al or Ag are used as contacts, the deviation given in this plot can be used as a corrective factor for the theoretical extraction efficiency. The total emitted power tends to unity as the QW is placed farther from the LED top surface. While the oscillations observed in Fig. 6.13b are only slightly averaged to the polychromatic emission from the source, the use of several QWs or thick active regions, as well as thickness fluctuations of the LED structure, averages these oscillations resulting in a smaller deviation of the emitted power from unity.

A more general map of the deviation from unity of the power emitted from the dipole sources inside the LED is shown in Fig. 6.13c, where an LED structure without metal contacts was considered. As can be seen, the total emission from the dipole source is fairly constant for a thick LED structure (($L/\lambda$) > 5).

Therefore, under appropriate choice of metal contacts, the technique presented in this chapter is robust to LED parameters and the results presented in Sect. 6.3.1, or more specifically shown in Fig. 6.4 can be generally applied to a large range of LED configurations.

Appendix C: Modeling the Angle-Resolved Emission from LEDs: Accounting for Surface Roughness

The model presented in this chapter (details in Appendix A) can be used to predict the angular emission of an LED structure, as shown in Fig. 6.6. In particular, it is useful to check whether light emission occurs according to theoretical predictions. The power per solid angle of an LED can be experimentally measured using an angle-resolved setup [32] and compared to the theoretical angle-resolved emission, as demonstrated in [14]. The light radiated to air from the LED structure corresponds to $dP_{\text {out}}^{(h,\text {total})} / d\Omega dS(\theta ) = dP_{\text {out}}^{(h,\text {TE})} / d\Omega dS + dP_{\text {out}}^{(h,\text {TM})} / d\Omega dS$. To compare this angular power flow to the corresponding angle-resolved measurement, a correction term cos($\theta$) needs to be used to take into account the projection of the power flow (perpendicular to the LED top surface) into the plane perpendicular to the rotating detector.

Fig. 6.14

a Theoretical angle-resolved radiation ($dP_{\text {out}}^{(h,\text {total})}/d\Omega dS$) versus $\theta$ (for $\theta$ from 0$^\circ$ to 90$^\circ$) and $\lambda$ showing the Fabry–Perot interferences at the interfaces of the LED structure. The LED structure consisted of a 4.87-$\upmu$m-thick GaN with QW embedded at 140 nm below the top surface. b Effect of surface roughness on the angular power flow, normalized by the total power emitted to air, for a smooth LED with $F_{\text {rough}}=0$ (dashed line), for $F_{\text {rough}}=30\%$ (solid line) and for $F_{\text {rough}}=100\%$ (dotted line))

Figure 6.14a shows the radiation from the LED (total power per solid angle) versus $\theta$ and $\lambda$, where the oscillations observed correspond to the Fabry–Perot interferences on the interfaces of the LED structure. While experimental results for LEDs with flat surface match such angular emission (Fig. 6.6a), LEDs with rough surface or pits present diminished oscillations in their angular diagram. The effect of surface roughness can be considered as a damping on the Fabry–Perot oscillations through the assumption that the roughness randomizes the angular distribution of the power flow. Therefore, a fraction $F_{\text {rough}}$ of the original power flow is now randomly emitted in all angles, which can be approximated by an isotropic emission normalized by the total power emitted to air $\eta _{{\text {extr}}}$, as $R=\eta _{{\text {extr}}}/2\pi$. A more rigorous investigation of the effect of rough surfaces is presented in [33, 34]. The normalized angular output power flow due to the rough surface is approximated by

$$\begin{aligned} \dfrac{F_{\text {rough}}R+(1-F_{\text {rough}})dP_{\text {out}}^{(h,\text {total})} / d\Omega dS}{\eta _{{\text {extr}}}}. \end{aligned}$$

(6.34)

Figure 6.14b illustrates this effect, where the dashed line is the corrected angular power flow, normalized by the total power emitted to air, for a smooth LED with $F_{\text {rough}}=0$. The solid line corresponds to $F_{\text {rough}}=30\%$ and the dotted line is for $F_{\text {rough}}=100\%$. The angle-resolved measurement is a useful technique to assess the contribution of light randomly scattered at rough surfaces to the measured EQE, which ultimately can be used to validate the model requirements of flat interfaces. The pronounced Fabry–Perot oscillations in this measurement fade with increase of $F_{\text {rough}}$ which largely reduces the accuracy of this model.