Atomistic Analysis of the Green Gap in InGaN/GaN LEDs: The Role of Random Alloy Fluctuations

1. Introduction & The Green Gap Problem

III-nitride InGaN/GaN light-emitting diodes (LEDs) are the cornerstone of modern solid-state lighting (SSL), with blue LEDs achieving power conversion efficiencies exceeding 80%. The prevailing method for generating white light involves coating a blue LED with a phosphor to down-convert a portion of the emission to yellow/green. However, this Stokes-shift loss limits ultimate efficiency. A superior path to ultra-efficient SSL is direct color mixing using red, green, and blue (RGB) semiconductor LEDs, enabling higher efficiency and spectral control.

The critical barrier to this approach is the "green gap": a severe and systematic drop in the internal quantum efficiency (IQE) of LEDs emitting in the green-to-yellow region (~530-590 nm) compared to blue and red emitters. This work posits that a significant, previously underexplored contributor to this gap in c-plane InGaN/GaN quantum wells (QWs) is the intrinsic random fluctuation of Indium atoms within the In_xGa_1-xN alloy, which becomes more detrimental at higher Indium concentrations required for green emission.

2. Methodology: Atomistic Tight-Binding Simulation

To probe nanoscale electronic properties beyond continuum models, the study employs an atomistic tight-binding framework. This method explicitly accounts for the discrete atomic structure and the local chemical environment of each atom.

2.1. Simulation Framework

The electronic structure is calculated using an sp³d⁵s* tight-binding model with spin-orbit coupling. Strain effects from the lattice mismatch between InGaN and GaN are included via valence force field (VFF) methods. The single-particle Schrödinger equation is solved for the QW system to obtain electron and hole wavefunctions.

2.2. Modeling Random Alloy Fluctuations

The InGaN alloy is modeled as a random distribution of Indium and Gallium atoms on the cation sublattice according to the nominal composition x. Multiple statistical realizations (configurations) of the alloy are generated and simulated to capture the ensemble average of properties like the optical matrix element, which governs the radiative recombination rate.

3. Results & Analysis

The atomistic simulations reveal two interconnected effects driven by alloy fluctuations.

3.1. Impact on Wavefunction Overlap

Random Indium clusters create local potential minima that strongly localize hole wavefunctions. Electrons, being less affected, remain more delocalized. This spatial separation beyond that caused by the quantum-confined Stark effect (QCSE) further reduces the electron-hole wavefunction overlap integral, a direct input into the radiative rate.

3.2. Radiative Recombination Coefficient ($B$)

The fundamental radiative recombination coefficient $B$ is proportional to the square of the momentum matrix element $|M|^2$, which itself depends on the wavefunction overlap. The simulations show that $B$ decreases significantly with increasing Indium content x. This reduction is attributed to the alloy-disorder-induced localization, providing a fundamental materials-based reason for lower efficiency in green-emitting QWs, even before considering non-radiative defects.

4. Discussion: Beyond the QCSE

While QCSE due to polarization fields in c-plane QWs is a known efficiency limiter, this work highlights that alloy disorder is an independent and compounding factor. At high Indium content, the combined effect of strong QCSE (pulling electrons and holes apart) and strong hole localization (pinning holes to In-rich clusters) creates a "double whammy" that drastically suppresses radiative efficiency. This explains why simply increasing Indium content to reach green wavelengths leads to disproportionately poor performance.

5. Core Insight & Analyst Perspective

Core Insight: The industry's quest to bridge the green gap has been overly focused on mitigating macroscopic defects and polarization fields. This paper delivers a crucial, nano-scale correction: the very randomness of the InGaN alloy itself is a fundamental, intrinsic efficiency killer at green wavelengths. It's not just a "bad sample" problem; it's a fundamental materials physics problem.

Logical Flow: The argument is elegant and compelling. 1) Green emission requires high In content. 2) High In content increases compositional randomness. 3) Randomness creates localized potential fluctuations. 4) These fluctuations preferentially trap holes, decoupling them from electrons. 5) This decoupling directly reduces the radiative coefficient $B$. The chain from atomic arrangement to device performance is clearly established through computational experiment.

Strengths & Flaws: The strength lies in the sophisticated use of atomistic simulation to reveal a mechanism invisible to conventional drift-diffusion or continuum models, akin to how CycleGAN's use of cycle-consistency loss revealed new possibilities in unpaired image translation. The primary flaw, acknowledged by the authors, is the focus solely on the radiative coefficient $B$. It sidesteps the critical issue of how alloy fluctuations might also increase non-radiative recombination (e.g., by enhancing Shockley-Read-Hall rates near In clusters), which is likely a co-conspirator in the green gap. A comprehensive model must integrate both radiative and non-radiative channels, as emphasized in reviews from research consortia like the DOE's SSL program.

Actionable Insights: This isn't just an academic exercise. It redirects R&D strategy. First, it strengthens the case for moving away from c-plane to semi-polar or non-polar GaN substrates to eliminate QCSE, thereby removing one major variable and isolating the alloy issue. Second, it calls for materials engineering aimed at reducing alloy disorder. This could involve exploring growth techniques for more homogeneous In incorporation, the use of digital alloys (short-period InN/GaN superlattices instead of random alloys), or even the development of novel nitride compounds with intrinsically narrower bandgaps, reducing the need for high In fractions. The path forward is not just "grow it better," but "design the alloy differently."

6. Technical Details & Mathematical Framework

The radiative recombination rate $R_{rad}$ for a direct bandgap semiconductor is given by: $$R_{rad} = B \, n \, p$$ where $n$ and $p$ are electron and hole densities, and $B$ is the radiative recombination coefficient. In a quantum well, $B$ is derived from Fermi's Golden Rule: $$B \propto |M|^2 \, \rho_{r}$$ Here, $|M|^2$ is the momentum matrix element squared, averaged over all relevant states, and $\rho_{r}$ is the reduced density of states. The atomistic calculation focuses on $|M|^2$, which for an optical transition is: $$|M|^2 = \left| \langle \psi_c | \mathbf{p} | \psi_v \rangle \right|^2$$ where $\psi_c$ and $\psi_v$ are the electron and hole wavefunctions, and $\mathbf{p}$ is the momentum operator. The key finding is that alloy fluctuations cause $\psi_v$ to become highly localized, reducing the spatial integral in the matrix element calculation and thus decreasing $|M|^2$ and ultimately $B$.

7. Experimental Context & Chart Interpretation

The paper references a conceptual Figure 1 (not reproduced in the text snippet) that would typically plot External Quantum Efficiency (EQE) or IQE versus emission wavelength for III-nitride (blue-green) and III-phosphide (red) LEDs. The chart would vividly show a pronounced trough in the green-yellow region—the "green gap." The simulation results in this paper provide a microscopic explanation for the left side (nitride) of that trough. The predicted decrease in $B$ with increasing In content would manifest experimentally as a lower peak IQE for LEDs with longer target wavelengths, even if material defect density were held constant.

8. Analysis Framework: A Conceptual Case Study

Scenario: An LED manufacturer observes a 40% drop in measured IQE when shifting a QW's peak emission from 450 nm (blue) to 530 nm (green), despite using identical growth recipes optimized for low macroscopic defect density.

Framework Application:

1. Hypothesis Generation: Is the drop due to (a) increased point defects, (b) stronger QCSE, or (c) intrinsic alloy physics?
2. Computational Isolation: Use an atomistic tight-binding model as described. Input: nominal In compositions for blue and green QWs. Hold all other parameters (well width, barrier composition, strain) constant in the model.
3. Controlled Simulation:
   • Run 1: Simulate with a perfectly ordered (virtual crystal approximation) InGaN alloy. Observe the change in wavefunction overlap and $B$ due solely to increased polarization field (QCSE).
   • Run 2: Simulate with a realistic random alloy for both compositions. Observe the additional reduction in $B$.
4. Analysis: Quantify the percentage contribution of pure QCSE vs. alloy disorder to the total reduction in $B$. This disentangles the two effects.
5. Actionable Output: If alloy disorder contributes >50% of the $B$ reduction, the development strategy should pivot towards alloy engineering (e.g., exploring digital alloys) rather than solely pursuing further defect reduction or polarization management.

9. Future Applications & Research Directions

• Non- and Semi-Polar LED Development: Eliminating the QCSE in non-polar/semi-polar GaN will unmask the pure impact of alloy fluctuations, validating this model and setting a new efficiency baseline for green emitters.
• Alloy Engineering: Research into growth techniques (e.g., pulsed MOCVD, modified V/III ratios) to achieve more uniform In incorporation. Exploration of "digital alloys" (short-period InN/GaN superlattices) as a replacement for random InGaN, offering controlled composition and potentially reduced localization.
• Novel Material Systems: Investigation of alternative nitride compounds (e.g., GaNAs, high-In-content InAlN) or 2D materials that could achieve green emission without high random alloy fractions.
• Advanced Device Architectures: Designing QWs with tailored potential profiles (e.g., graded composition, delta-layers) to counteract the hole-localizing effect of In clusters.
• Multiscale Modeling Integration: Coupling the atomistic results presented here with larger-scale drift-diffusion or kinetic Monte Carlo models to predict full LED device characteristics under operating conditions.

10. References

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2. M. R. Krames et al., "Status and Future of High-Power Light-Emitting Diodes for Solid-State Lighting," J. Disp. Technol., vol. 3, no. 2, pp. 160–175, 2007. (Citing >80% efficiency).
3. U.S. Department of Energy, "Solid-State Lighting R&D Plan," 2022. (Authoritative source on SSL potential and color mixing).
4. J. Y. Tsao et al., "Toward smart and ultra-efficient solid-state lighting," Adv. Opt. Mater., vol. 2, no. 9, pp. 809–836, 2014.
5. E. F. Schubert, Light-Emitting Diodes, 3rd ed. Cambridge University Press, 2018. (Standard reference on LED physics, including the green gap).
6. Z. Zhuang, D. Iida, K. Ohkawa, "Review of long-wavelength III-nitride semiconductors and their applications," J. Phys. D: Appl. Phys., vol. 54, no. 38, p. 383001, 2021. (Recent review covering the green gap).
7. J. Jun et al., "The potential of III-nitride laser diodes for solid-state lighting," Prog. Quantum Electron., vol. 55, pp. 1–31, 2017.
8. C. J. Humphreys, "The 2018 nitride semiconductor roadmap," J. Phys. D: Appl. Phys., vol. 51, no. 16, p. 163001, 2018. (Discusses QCSE and material challenges).
9. P. G. Eliseev, P. Perlin, J. Lee, M. Osinski, ""Blue" temperature-induced shift and band-tail emission in InGaN-based light sources," Appl. Phys. Lett., vol. 71, no. 5, pp. 569–571, 1997. (Early work on localization effects).
10. J. Zhu, T. Shih, D. Yoo, "Atomistic simulations of alloy fluctuations in InGaN quantum wells," Phys. Status Solidi B, vol. 257, no. 6, p. 1900648, 2020. (Related contemporary work).