# Effect of Carrier Localization on Recombination Processes and Efficiency of InGaN-Based LEDs Operating in the “Green Gap”

## Abstract

**:**

## 1. Introduction

_{0.2}Ga

_{0.8}N alloys. No electron localization was identified in the study. Additional simulations of spherical In

_{0.76}Ga

_{0.24}N/In

_{0.33}Ga

_{0.67}N quantum dots 2.8 nm in diameter have shown that hole localization is insensitive to the quantum-confining potential formed in the quantum dot, but occurs at a dot corner due to a local strain [35].

_{0.25}Ga

_{0.75}N and a GaN/AlGaN QW grown by metalorganic vapor phase epitaxy were restored by atomic-probe tomography by making necessary statistical corrections. Then, the obtained composition profiles were used in quantum-mechanical simulations of the conduction and valence band profiles perturbed by fluctuations and, finally, the emission spectra of the Al

_{0.25}Ga

_{0.75}N alloy and GaN/AlGaN QW were calculated, providing good agreement with the measured spectra. Based on the above data, the localization radius of holes of about 0.9–1.5 nm and the respective localization energies of about 40–60 meV were estimated. Though these results are relevant to AlGaN alloys, they clearly indicate that the hole localization by composition fluctuations occurs on an atomic scale and provides rather high localization energies.

## 2. Model

**ρ**is the in-plane radius-vector, z is the coordinate normal to QW interfaces and directed along the (0001) axis of the crystal in the case of a polar LED, u

_{hh}is the heavy-hole Bloch function, a is the localization radius assumed for simplicity to be the same for lateral and normal directions, and z

_{0}(0 ≤ z

_{0}≤ d) is the hole localization point inside the QW (due to translational symmetry, the lateral position of the localized hole can be chosen at

**ρ**= 0).

_{e}and ψ(z) are the Bloch and the envelope wave functions of the electron, respectively, having the normalization width ${w}_{e}={{\displaystyle \int}}_{-\infty}^{\infty}dz\xb7{\psi}^{2}\left(z\right)$; S is the plane-wave normalization area; and

**k**is the in-plane wave vector of the electron. The assumption that the localized hole does not “feel” the confinement potential of the InGaN QW requires a to be much less than d. In this case, the electron envelope function ψ(z) varies slightly inside the region of hole localization.

#### 2.1. Radiative Recombination Coefficient

_{eh}, the envelope function ψ(z) is approximated by the first two terms of its Taylor series in the vicinity of z

_{0}. Due to the even symmetry of the hole wave function, the second term does not contribute to the matrix element, giving:

_{t}is the Kane’s matrix element coupled S-like electron Bloch function u

_{e}with X(Y)-like hole Bloch function u

_{hh}(see, e.g., [37]). Expression (3) is valid, if ${\left(a/d\right)}^{2}\ll 1$. The accuracy of the above expression may be further improved by accounting for an additional (third) term in the Taylor series expansion of the electron envelope function. This results in substituting ψ(z

_{0}) with ψ(z

_{0}) + a

^{2}ψ″(z

_{0}) in Equation (3), where ψ″ is the second derivative of the electron envelope wave function. In this case, the model applicability becomes determined by a less strict inequality: ${\left(a/d\right)}^{4}\ll 1$.

**k**-selection rule [38,39]. First, using the Fermi’s Golden Rule and accounting for the equilibrium photon population, the recombination rate of a localized hole with electrons in all the allowed quantum states is found. Then, the recombination rate is averaged over all possible positions z

_{0}of the localized hole inside the QW. Finally, dividing the above recombination rate by the electron concentration provides the RRC. Essential simplification can be achieved by substituting in the calculations the momentum-dependent electron kinetic energy ${\hslash}^{2}{k}^{2}/2{m}_{e}$ (ħ is the Planck constant and m

_{e}is the electron effective mass) with a mean electron energy E

_{e}. This energy is equal to kT (k is the Boltzmann constant and T is temperature) in the case of non-degenerate electrons and nearly equal to ½F

_{n}(F

_{n}is the quasi-Fermi level position counted from the conduction band edge), if electrons are degenerate. In this case, the optical matrix element M

_{eh}defined by Equation (3) becomes momentum-independent and summation over all possible states of electrons in the QW can be reduced to their sheet concentration. As a result, the following analytical expression can be obtained for the sheet RRC:

_{r}is the group refractive index, E

_{g}is the InGaN energy gap, ${E}_{P}=2{m}_{0}{P}_{t}^{2}$ is the energy associated with the Kane’s matrix element P

_{t}, m

_{0}is the electron mass, c is the light velocity, and $\langle {\psi}^{2}\rangle $ is the square of the electron envelope wave function averaged over all possible positions z

_{0}of the localized hole. Using the material-dependent frequency ν

_{B}provides compact expressions for the RRCs not only for the case considered above, but also for radiative processes with

**k**-selection rules involving free electrons and holes in bulk materials and QWs [31].

#### 2.2. Auger Recombination Coefficient

_{sh}is the sheet electron density in InGaN QW, and I is the parameter proportional to the square of the overlap integral between the electron and hole wave functions. For two-dimensional electrons: $\sigma =\pi {a}^{2}/v{\tau}_{\u03f5}$, where ${\tau}_{\u03f5}=\frac{1}{2}{m}_{e}^{2}{v}^{2}/{\pi}^{2}\hslash {E}_{D}{n}_{sh}$ is the energy relaxation time [41], and E

_{D}is the binding energy of a shallow donor in InGaN calculated within effective mass approximation. Estimating the overlap integral in a manner suggested in [40], $\cong 2a{\xi}^{3}\left({E}_{e}/{E}_{g}\right)\langle {\psi}^{2}\rangle /{w}_{e}$, and accounting for the fact that the electron kinetic energy, $\frac{1}{2}{m}_{e}^{2}{v}^{2}$, should be equal to the energy transferred from one electron to another, i.e., ${E}_{g}-{E}_{L}$, one can derive the following expression for the sheet ARC:

#### 2.3. Localization Energy

_{x}Ga

_{x}

_{−1}N, implies selecting an averaging volume V

_{a}centered at the point with the radius-vector

**r**, counting the number of atoms of a certain kind, e.g., those of In, contained in the volume, and then determining a local alloy composition x(

**r**) as the ratio of the number of In atoms to the total number of all cations in the volume V

_{a}. At the next step, spatial variations of the conduction and valence band edges, E

_{C}(

**r**) and E

_{V}(

**r**), are determined using the obtained x(

**r**) and known relationships between the alloy composition, its bandgap, and band offsets. Finally, Schrödinger equations are solved with E

_{C}(

**r**) and E

_{V}(

**r**) serving as the potential energy profiles for electrons and holes and confined states of the carriers are found, providing their localization energy. Such an approach has been used, in particular, in [36], where the composition fluctuations were evaluated experimentally using atomic probe tomography.

_{a}results in lower relative composition fluctuations and, eventually, in a weaker carrier localization. In order to avoid the uncertainty in the choice of V

_{a}, the averaging volume should be found self-consistently with the solution of the Schrödinger equations in such a way as to nearly correspond to the volume occupied by a carrier wave function in the localized state. The above self-consistence exists ab initio in the method of optimal fluctuation (see [43] for a review of the method and relevant results), which predicts that the density of localized states g(E) vary with energy E counted from the edge of either the conduction or valence band towards the mid of the bandgap as $g\left(E\right)\propto {E}^{3/2}exp\left(-\sqrt{E/{E}_{L}}\right)$ with the localization energy:

_{eff}is the carrier effective mass, and n

_{cs}is the concentration of cation sites in the crystal lattice. Expression (6) for the energy E

_{t}is valid if: (i) composition fluctuations have a thermodynamic character; (ii) they occur in an ideal alloy, i.e., that having negligible enthalpy of mixing; and (iii) the number of In atoms in the optimal fluctuation $N=x{n}_{cs}{V}_{a}\gg {N}^{1/2}$. Estimations show that the latter two assumptions are not satisfied in real InGaN compounds. First, the InGaN alloys are far from ideal, tending to phase separation due to a large lattice constant mismatch between InN and GaN. Second, the typical number of In atoms in the optimal fluctuation is about six to eight because of the atomistic character of hole localization. This makes the condition $N\gg {N}^{1/2}$ invalid. Therefore, Equation (6) is used in this study only as a general relationship between the localization energy E

_{L}and alloy composition x, whereas E

_{t}is regarded as a fitting parameter. A similar approach has been previously applied to the bulk InGaN alloys [31].

## 3. Results and Discussion

#### 3.1. Recombination Coefficients

_{e}varied from 26 meV (thermal energy at room temperature) to 80 meV. One can see that the experimental points corresponding to LED structures grown on sapphire substrates agree well with the theory, whereas the data for the structures grown on silicon substrates lie slightly beyond the theoretical predictions.

_{t}was changed from 1.85 eV recommended in [31] to 2.0 eV, corresponding to the localization energy variation from 24 meV at 420 nm to 96 meV at 550 nm. These values are comparable with localization energies reported in [33] and associated with two- and three-atomic In clusters. The wavelength dependence of the respective localization radius is shown in Figure 2a. Here, the condition ${\left(a/d\right)}^{2}\ll 1$, at which Equations (3) and (4) are valid, is met at the wavelengths longer than ~435 nm. At shorter wavelengths, down to ~410 nm, the approximations provide an inaccuracy of less than ~20%, which is within the scatter of the measured recombination coefficients (see Figure 3). At E

_{t}= 2.0 eV, the localization radius a varies from 0.92 nm at 420 nm to 0.47 nm at 550 nm, as is shown in Figure 2. The sub-nanometer scale of the localization radius variation agrees well with the mean dimensions (nearly equal to 2a) of the hole wave functions obtained in [35,46,47] by ab initio calculations.

_{P}, etc. The comparison demonstrates good quantitative agreement between the theory and experiment achieved simultaneously for both RCs. RRC is found to depend weakly on E

_{e}, whereas ARC is proportional to the mean electron energy. RCs calculated for nonpolar LED structure (grey solid lines in Figure 2), appear to be much higher than those corresponding to the polar structure, which is the consequence of the difference in the $\langle {\psi}^{2}\rangle /{w}_{e}$ seen in Figure 2a. Since the theoretical B/C ratio does not depend on $\langle {\psi}^{2}\rangle /{w}_{e}$, according to Equations (3) and (4), the ratios plotted in Figure 2b by lines actually correspond to both polar and nonpolar LED structures.

_{max}would grow with the wavelength, which contradicts the observed efficiency decline of the nonpolar LEDs in the “green gap” [15].

_{max}via Q-factor shows the emission efficiency to decline when the hole localization becomes stronger, i.e., when the localization radius decreases. The prediction of the model is in line with the data reported in [49], where the efficiency droop in nonpolar InGaN QWs was observed at lower optical excitation powers, and if the InGaN material exhibited a stronger disorder, this resulted in a deeper carrier localization.

#### 3.2. Efficiency of Polar and Nonpolar LEDs in the “Green Gap”

^{5}s

^{−1}being nearly constant in the spectral range of 440–520 nm (see Figure 1), the factor $Q=B/{\left(AC\right)}^{1/2}$ and then IQE

_{max}were calculated for the polar LED structure versus emission wavelength (Figure 4). The obtained curve fits the experimental points obtained from independent IQE evaluations very well (see [10] for summary of the data) at wavelengths shorter than ~540 nm. However, at longer wavelengths, the experimental IQE

_{max}declines much faster than it is predicted by the theory (dotted line in Figure 3 shows this trend). It looks like intensive defect formation starts to occur at about 540 nm, resulting in additional IQE reduction.

^{6}s

^{−1}obtained in this manner was expanded over the whole spectral range, similar to the case of polar LEDs. Finally, Q-factor and IQE

_{max}were calculated for nonpolar LEDs, using the estimated SRH coefficient and wavelength-dependent RRC and ARC from Figure 3.

_{max}of nonpolar LEDs is found to decrease substantially with the emission wavelength, similar to the case of polar LEDs (Figure 4). Being in line with the experimental behavior reviewed in [29], this result points out the importance of carrier localization for the efficiency of nonpolar LEDs. Breakdown of theoretical IQE losses in the “green gap” shown in Figure 4 demonstrates that about 53% of the efficiency decline can be attributed to hole localization, whereas only 29% is related to QCSE; the remaining 18% of the IQE reduction corresponds to variation of the native InGaN parameters, E

_{g}, E

_{P}, and carrier effective masses, with wavelength. Since composition fluctuations are unavoidable in InGaN-based LEDs of any crystal orientation, the carrier localization seems to be the major mechanism limiting the device efficiency in the “green gap”.

#### 3.3. Model Limitations and Outlook for Future Studies

_{t}is properly fitted. The model has, however, some intrinsic limitations originating from the fact that the most important parameter of the model, localization energy or localization radius, is fitted rather than derived theoretically and calculated. Since the composition fluctuations in InGaN alloys produce a variety of localized holes distributed inside the energy gap with a certain density of states (DOS), g(E), the localization radius in Equations (4) and (5) should be regarded as a parameter averaged over the ensemble of localized holes [22]. If g(E) is known, the averaging accounted for the Fermi distribution function of holes would enable a prediction of the dependences of the RCs on temperature and carrier concentration. However, it is impossible with the current model based on fitting to the data obtained at room temperature.

_{sh}/B

_{sh}ratio does not include the localization radius a and is proportional to the mean electron energy E

_{e}nearly equal to thermal energy kT in the case of non-degenerate electrons. Figure 5 shows the experimental C

_{sh}/B

_{sh}ratios versus temperature estimated from the data of [22,52]. The bars at the data points corresponding to green LED indicate the errors originating from an evaluation of the Q-factor from the current dependence of LED efficiency; the errors are especially pronounced at low temperatures, 50–100 K, where the efficiency is hardly approximated by the ABC-model (see [22] for more details on the experimental method). Similar bars for blue LEDs do not exceed the size of symbols. One can see from Figure 5 that the C

_{sh}/B

_{sh}ratios depend nearly linearly on temperature, maybe except for the points corresponding to T = 350 K. Moreover, the slopes of the linear dependences are scaled with the LED emission wavelengths as ${\left({E}_{g}^{3}{E}_{P}\right)}^{-1}$, in accordance with the suggested model. Remarkable deviation of the experimental points from the above trends observed at T = 350 K may be evidence for the transition from the localization-mediated to conventional free carrier-mediated recombination processes. The latter issue, however, requires a special examination.

**k**-selection rule. Prediction of the transition point requires knowing the DOS of localized holes, which is a task for future studies. Experimentally, such a transition has not been identified to date.

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Experimental Shockley-Read-Hall recombination coefficients as a function of emission wavelength of polar LEDs grown on different substrates.

**Figure 2.**Averaged square of electron envelope function computed for [0001]-polar and nonpolar SQW LED structures and localization radius (

**a**) and theoretical and experimental B/C ratio of polar SQW LEDs (

**b**) as a function of wavelength. Symbols are data from [20,21]; dashed, dash-dotted, and solid lines in (

**b**) correspond to the electron energy E

_{e}of 26, 80, and 40 meV, respectively. In the plot (

**a**), there is an arrow pointing out to the right vertical axis with the corresponding title. In the plot (

**b**), red, orange, and purple lines are mentioned as “solid”, “dashed”, and “dash-dotted” lines, respectively.

**Figure 3.**RRC (

**a**) and ARC (

**b**) of [0001]-polar and nonpolar SQW LED structures as a function of emission wavelength. Symbols are data from [20,21]; dashed, dash-dotted, and solid lines correspond to calculations with mean electron energies of 26, 80, and 40 meV, respectively. Dotted lines show the recombination coefficients reported in [48]. The red, orange, and purple lines are mentioned in the capture as “solid”, “dashed”, and “dash-dotted” lines, respectively.

**Figure 4.**Maximum IQE versus emission wavelength. Balls are experimental points summarized in [10]. Solid lines are theoretical estimates for [0001]-polar and nonpolar SQW LED structures, whereas the dotted line indicates the experimental trend in IQE variation at wavelengths longer than 540 nm. Dash-dotted line shows the IQE decrease caused by variation of the intrinsic material properties with the wavelength.

**Figure 5.**Experimental C

_{sh}/B

_{sh}ratios for blue (445 nm) and green (530 nm) LEDs obtained from the data of [22,52] as a function of temperature. Circles and squares correspond to blue and green LEDs, respectively. Bars indicate the errors originating from uncertainty in the Q-factor evaluation. Lines drawn for eyes indicate the experimental trends.

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Karpov, S.Y.
Effect of Carrier Localization on Recombination Processes and Efficiency of InGaN-Based LEDs Operating in the “Green Gap”. *Appl. Sci.* **2018**, *8*, 818.
https://doi.org/10.3390/app8050818

**AMA Style**

Karpov SY.
Effect of Carrier Localization on Recombination Processes and Efficiency of InGaN-Based LEDs Operating in the “Green Gap”. *Applied Sciences*. 2018; 8(5):818.
https://doi.org/10.3390/app8050818

**Chicago/Turabian Style**

Karpov, Sergey Yu.
2018. "Effect of Carrier Localization on Recombination Processes and Efficiency of InGaN-Based LEDs Operating in the “Green Gap”" *Applied Sciences* 8, no. 5: 818.
https://doi.org/10.3390/app8050818