Optimisation of GaN LEDs and the reduction of efficiency droop using active machine learning

A fundamental challenge in the design of LEDs is to maximise electro-luminescence efficiency at high current densities. We simulate GaN-based LED structures that delay the onset of efficiency droop by spreading carrier concentrations evenly across the active region. Statistical analysis and machine learning effectively guide the selection of the next LED structure to be examined based upon its expected efficiency as well as model uncertainty. This active learning strategy rapidly constructs a model that predicts Poisson-Schrödinger simulations of devices, and that simultaneously produces structures with higher simulated efficiencies.

. Schematic of the LED that the learning algorithm was given the task to optimise. Left: reference LED structure. Right: conduction band of the active region of the reference LED structure, at high current density (75 A/cm 2 ).  ciencies, one uses Bayes rule to obtain a 'posterior' normal distribution for LED efficiency 21 . Specifically, for each as-yet unobs er ved LED str uc ture x , the GP mo del pro duces a p osterior distribution , with μ the expected efficiency and σ its standard deviation. In this study, we use the gaussian_process module of the scikit-learn Python package 22,23 . We choose a squared-exponential auto-correlation function and hyper-parameters determined by the maximum likelihood principle.
of LED structures and their measured efficiencies, the GP model produces a normal posterior distribution  P y x ( , ) n with mean efficiency μ and standard deviation σ for the structure x. The remaining question concerns experimental design: what is the best strategy to select new LED structures ...
such that we most quickly find LEDs with very high efficiencies? The EGO method provides a simple approximate answer: at each step in the design loop, the next LED structure to sample should be selected to optimise the expected improvement in efficiency, after accounting for model uncertainty 12 . Concretely, let y max denote the efficiency of the best LED device currently in our dataset n  . The expected efficiency improvement may be expressed as: where erfc (·) denotes the complementary error function and α µ σ = − y ( ) / 2 max is the scaled difference between the expected efficiency of x and the best LED in our dataset.
We select the next sample point to optimise this objective function, n n x 1 The new LED structure x n+1 becomes an input to a Poisson-Schrödinger code, described below, which calculates the simulated efficiency y n+1 . Next, we extend our dataset,  1 and update the GP posterior, , from which we can select another sample point via Eq. (2). This iterative process is repeated until a satisfactory LED structure is found. Simultaneously, we obtain a predictive ML model of LED efficiency over a broad range of inputs.
To better understand the objective function in Eq. (1), we evaluate it in two asymptotic limits. In the limit of vanishing uncertainty, σ → 0 (equivalently α → ± ∞), we observe . That is, when the model is very certain, the objective function seeks primarily to select points x with expected efficiency μ better than the best known, y max . Conversely, imagine that the model uncertainties σ are relatively large compared to μ − y max . In this α → 0 limit we observe → . Thus, when there is no obvious opportunity to improve on the best known LED, the learning strategy becomes primarily exploratory and favors points x with the largest model uncertainty. For intermediate α ~ 1 the strategy of Eq. (2) balances exploitation (maximizing μ) and exploration (maximizing σ). In this way, we avoid getting stuck in local maxima: once a region of very efficient LEDs has been well explored, the algorithm samples from a region of larger uncertainty, even if the predicted efficiency is not great.

Automated LED Design
In this work, we take the point x to represent the structure of the 5-well active region in a GaN-based LED (see Fig. 1 for a schematic). Each input point x has 6 parameters: the indium composition of each quantum well and the collective indium composition of the quantum barriers. The quantum well width varies with the indium composition of both well and barrier to keep the wavelength approximately constant. To determine the simulated efficiency of each structure, we use the APSYS software package with materials parameters taken from 24 and current density 75 A/cm 2 . The band structure was calculated using the 6 × 6 k.p method 25 in a finite volume approximation. The carrier transport equations were self-consistently computed and coupled with Schrödinger's equation to determine the confined states in the QWs. Schrödinger and Poisson equations are solved iteratively to account for the band structure deformation with carrier redistribution. The carrier transport consists of drift-diffusion of electrons and holes, Fermi statistics, and thermionic emission at hetero-interfaces, as well as band-to-band tunneling.
We use the machine learning algorithm in Eqs. (1) and (2) to optimise the internal quantum efficiency within the 6-dimensional space (the In content of each the 5 wells and the average In content of the barriers) of our LED structures. As can be seen in Fig. 2a, the procedure converges rapidly, finding a nearly optimal simulated LED efficiency in about 75 iterations. Subsequent iterations make little improvement upon optimal LED efficiency (Fig. 2b), and instead focus on decreasing model uncertainty. Between learning steps 150 through 1000 (Fig. 2c,d), this procedure constructs a very robust model over the global space of LED structures. At iteration 1000 the algorithm is fully converged, and the coefficient of determination is R 2 > 0.99, as determined by cross-validation.
The very high accuracy model provides also some physical insight into the Poisson-Schrodinger simulations. While the drift-diffusion model predicts that most of the light emission of a standard LED structure comes from the 2 top wells, in agreement with electro-luminescence experiments 26 , it also informs us that allowing the indium content of the individual wells to vary across the active region increases the carrier and light emission spreading, in agreement with recent electro-luminescence experiments 27 . As can be seen in Fig. 3, our active learning algorithm finds several optima, which have in common a diminishing indium content in the quantum wells from the n-side to the p-side and the use of InGaN barriers rather than GaN barriers. The diminishing indium content reduces the confinement in the p-side wells 28 , which otherwise concentrate most carriers. The diminishing indium content and the use of InGaN quantum barriers increases the thermionic emission and tunnelling through the hetero-interfaces 29 , allowing the carriers to spread more easily across the active region. The decreasing indium content with increasing well number is associated with increasing well widths for a constant peak emission wavelength. At high current, Auger recombination grows more rapidly with carrier concentration than radiative recombination, and wider wells that compensate for the low indium content become beneficial 30 , as the carrier spreading within each well is increased. Figure 4 draws a comparison between the simulation of a standard LED structure that has GaN barriers and identical wells with the simulation of a machine learning optimised LED structure. The optimised structure achieves an increased spreading of the radiative recombination events: within the wells due to wider wells, which should be beneficial at high currents 31 , and between the wells due to a high barrier indium content and a decreasing indium content towards the p-side of the active region.
To summarize, our active learning strategy rapidly finds LED structures with nearly optimal quantum efficiency while simultaneously building a GP regression model that is predictive for a wide range of LEDs. We used the objective function in (1) for experimental design, which balances the trade-off between exploitation (high predicted efficiency) and exploration (high model uncertainty). At each iteration in our algorithm, the objective function guides the selection of a new LED structure which we simulate, and then use to expand our GP model.
Interestingly, this automated approach finds LEDs that a human expert would strive to design: a structure that spreads evenly the carrier recombination events through the active region of the LED, maximising the radiative recombination events. Leaving the algorithm to optimise the indium content of the active region, we find much higher simulated efficiencies than in standard LEDs. This structure employs a high barrier indium content and a decreasing well indium content towards the p-side of the active region to prevent the accumulation of carriers on the p-side and improve the spreading of the carriers and the radiative recombination events. It also employs wider wells to compensate for wavelength changes with indium content, and to achieve a carrier spreading within the quantum wells that is desirable at high currents.
Our modelling of gallium nitride devices with Poisson-Schrödinger solvers provides qualitative information rather than quantitative predictions. Nevertheless, the algorithm we present demonstrates the power of machine learning for device design. Our method also applies to the optimisation of different LED structures than those presented here. When used in conjunction with actual materials fabrication, our method readily extends to the design of experimental devices. This work is currently ongoing.

Conclusions
Materials informatics is an emerging field 32-34 with great promise for functional materials design [35][36][37][38][39] . This approach has not yet been adopted by the LED community, despite great potential for improving physical understanding and for accelerating structural design of devices. In this work, we demonstrate that active learning based global optimisation can rapidly and automatically explore Poisson-Schrödinger simulations of gallium nitride devices, and can accelerate the discovery of efficient LEDs. We are currently using this machine-learning approach to guide the growth of experimental structures.