NUSOD Blog

Connecting Theory and Practice in Optoelectronics

COVID-19 Forecasting Challenges

Our 21st NUSOD conference is scheduled for September 2021 in Turin, Italy. However, the ongoing COVID-19 pandemic makes the future somewhat uncertain. Many local epidemic forecast models have been developed, some of which became a basis for lockdown policies. Mathematical modeling is an important tool for studying real-world processes; however, achieving realistic predictions is always challenging. Even the retroactive modeling of the first COVID-19 wave produced mixed results. For example, a much cited paper from Germany revealed that the lockdown ordered by the government beginning March 23, 2020, accomplished only a modest reduction of the infection reproduction number R from 1.25 to 0.75 (R is the average number of people infected by a sick person). In contrast, a leading British group compared lockdown effects among European countries and found a major reduction of R from 3.9 to 0.8 in Germany on that same day (see picture). In the following, I’ll briefly discuss some features and challenges of epidemic models, most of which employ mathematical methods similar to what we use in our NUSOD community. [1,2]

Traditional epidemic models are based on rate equations that describe transitions between population sub-groups, e.g., between Susceptible, Infected, and Recovered people in the so-called SIR model. These transitions are mediated by parameters such as infection rate and recovery rate, averaged over the entire population. Besides this deterministic class of epidemic models, there are also various stochastic models utilizing, e.g., Monte-Carlo methods. They may include infections down to the individual level and offer a more detailed representation of an epidemic. Countless variations and combinations of both approaches can be found in the literature. [1,2]

However, all models simplify reality and are limited by underlying assumptions which may become invalid in a real-world situation. For instance, the population is often treated as homogeneous and closed system in which each person may have contact with any other person. However, human interaction is hard to predict and quite heterogeneous even within national borders, so that the often envisioned exponential growth may not happen in reality. Using an average infection rate across such different settings as homes, schools, work places, public transportation, shops, and elderly homes seems inappropriate. Many models don’t distinguish between such groups of people, so that a general lockdown may seem to be the only way out. More complex models are accompanied by more unknown parameters. Stochastic models need to make various assumptions at the micro level that are also hard to validate. Environmental factors such as air circulation or temperature modify the coronavirus spread but are typically neglected.

As we know from our own work, a key requirement of predictive modeling is the parameter calibration based on real-world data. Without sufficient and reliable data, epidemic simulations can easily lead to wrong conclusions. Many modeling groups adjust their parameters to the widely reported number of newly infected people, typically based on a positive PCR test. However, this test is not sufficient to detect a coronavirus infection, as recently emphasized by the WHO. In fact, some empirical studies observe no evidence of coronavirus transmission from asymptomatic PCR-positive cases to traced close contacts. Models often don’t distinguish between asymptomatic and pre-symptomatic cases, which depend on virus load and natural immune response. These and other issues seriously undermine the reliability of COVID-19 forecast models, I think. Further challenges of epidemic modeling are discussed here, here, and here.

Considering all these forecasting problems as well as vaccination uncertainties, I think we should pursue a dual-track approach to our NUSOD-21 conference and decide between online and on-site options based on the actual situation in late spring.

[1] W. Duan et al.: Mathematical and computational approaches to epidemic modeling: a comprehensive review. Front. Comput. Sci. 9, 806–826 (2015).
[2] L. Tan et al.: A Review of Multi‐Compartment Infectious Disease Models. Int. Statist. Rev. 88, 462–513 (2020).

Challenges of GaN-LED Modeling and Simulation

Light emitting diodes (LEDs) based on Gallium Nitride (GaN) have been revolutionizing various applications in lighting, displays, biotechnology, and other fields. However, their energy efficiency is still below expectations in many cases. An unprecedented variety of modeling and simulation papers has been published, mainly focusing on efficiency analysis and GaN-LED design optimization. In this open access review paper, I recently tried to provide an overview of the GaN-LED modeling landscape with special emphasis on the influence of III-Nitride material properties [1]. Even after 20+ years of intense worldwide research activities, I still see some key challenges as briefly listed below.

(1) The employment of realistic material parameters remains a fundamental issue for GaN-LED simulations. For example, some simulations of experimental characteristics validate competing efficiency droop models by simple variation of uncertain parameters (see figure) [2].

(2) GaN-LEDs are three-dimensional (3D) objects but most LED simulations are performed in 1D or 2D. Even with uniform material properties in each semiconductor layer, the current flow is often non-uniform in real devices, leading to local self-heating, non-uniform carrier density in each quantum well (QW), and non-uniform light emission. While 1D and 2D simulations are valuable in studying specific mechanisms, they are unable to fully reflect the internal physics and the measured performance of real LEDs.

(3) Another major challenge arises from the non-uniform nature of InGaN quantum wells. QWs with low Indium content may exhibit an average Indium atom distance that is larger than the QW thickness. QWs with larger Indium concentration allow for Indium accumulation regions with lower bandgap, larger free carrier concentration, and stronger Auger recombination.  Thus, the typical assumption of uniform QW properties is often invalid, giving rise to various atomistic LED modeling approaches.

(4) Artificial intelligence (AI) methods also represent a serious challenge. Several simulation-trained AI methods have been utilized for GaN-LED design optimization. However, they produce unreliable results due to various simulation uncertainties. The strength of AI actually lies in the analysis of experimental data. The combination of reality-trained AI methods with numerical simulations could lead to the creation of realistic digital twins that support the LED design and production process.

More details are given in [1] and references therein.

[1] Efficiency Models for GaN-based Light-Emitting Diodes: Status and Challenges, MDPI Materials 13, 5174 (2020)

[2] How to decide between competing efficiency droop models for GaN-based light-emitting diodes. Appl. Phys. Lett. 107, 031101 (2015)




Pros and Cons of Online Conferences

Thanks to the new Corona virus, I have been attending many online conferences this year, most of them for free. Such virtual conferences reach a much larger worldwide audience than onsite meetings. However, I always wonder why organizers stick to the rigid concept of live conferences. Online attendance is spread over many time zones and people are still busy with their daily routines. In fact, I usually managed to view only 2-3 presentations. What is more, a brief live Q&A period after each talk is clearly too short for useful discussions.

Therefore, we developed a more flexible concept for our 2020 NUSOD online conference, which ended last week. With great success, it seems, as 324 participants are triple our usual number. It was also our most international audience with attendees from 43 countries. 70 pre-recorded presentation videos were available 24/7, attracting 59 views on average, five talks even more than 150 views, according to website statistics. Questions and answers could be posted continuously in the comment section of each video. A few in-depth discussions went on for several days.

However, more than half of our presentations did not receive any feedback at all, despite our conference extension by one week. Google Analytics reveals that almost all site visits lasted less than 30 minutes. In most cases, videos were not watched in full. In other words, online attendees are too distracted by their daily life and pay far less attention than at onsite conferences.

Virtual conferences are convenient, inexpensive, and accessible, but by no means generate the full benefits of traditional conferences. Onsite meetings are crucial to inspire new ideas and to establish personal connections. We hope to accomplish this at our 2021 NUSOD conference in Turin, Italy. See you there!

Electronic structure of lonsdaleite SiGe alloys

The indirect band gaps of the elemental group-IV semiconductors silicon (Si) and germanium (Ge) make these materials intrinsically inefficient light emitters. This limits their applications in active photonic devices such as light-emitting diodes and lasers, which in turn limits the development of Si photonics due to the unavailability of direct-gap semiconductors compatible with established complementary metal-oxide semiconductor (CMOS) fabrication processes. The so-called “holy grail” of Si photonics is to realise CMOS-compatible, group-IV materials possessing direct band gaps, to enable the development of LEDs and lasers which can be integrated monolithically on a Si platform [1]. Such devices have huge potential for practical applications, enabling new functionalities while simultaneously delivering improved system-level energy efficiency by allowing for removal of power-hungry electrical interconnects on-chip and in data centres.

Ge possesses a “weak” indirect band gap, with the L-point conduction band (CB) minimum lying only 145 meV lower in energy than the Γ-point CB edge. Efforts to obtain a direct-gap group-IV semiconductor have therefore largely centred on engineering the band structure of Ge, in an attempt to reverse the ordering of the L- and Γ-point CB edge states. Proposed approaches to achieve this include application of tensile strain to Ge [2], or alloying Ge with small concentrations of carbon (C) [3,4], tin (Sn) [5] or lead (Pb) [6]. Impressive initial experimental demonstrations related to each of these concepts have driven significant activity related to materials growth and device fabrication in recent years. Theory and simulation underlying several of these advancements have been described at NUSOD, including (i) analysis of the electronic structure of Ge1-xSnx and Ge1-xPbx alloys [7,8], and (ii) simulation of prototypical Ge1-xSnx-based photonic [9,10] and electronic [11] devices. Of these approaches, the current leading contender is the Ge1-xSnx alloy, with optically and electrically pumped lasing having been demonstrated by several groups [12,13]. However, high quality Ge1-xSnx alloys are extremely challenging to achieve via epitaxial growth, and their reduced temperature stability compared to Ge presents further challenges for device fabrication and reliability. This mandates continued efforts to develop alternative routes to achieve enabling material platforms for CMOS-compatible active photonic devices.

Si and Ge conventionally crystallise in the cubic diamond phase. However, recent advancements in non-equilibrium growth of semiconductor nanowires have made it possible to grow these materials in the metastable hexagonal lonsdaleite (or “hexagonal diamond”) phase [14]. Just as alloying or application of strain can strongly modify the band structure of a semiconductor material, the emerging ability to grow conventional materials in non-conventional crystal structures provides a new approach to engineer the band structure for practical applications. The transition from the diamond to lonsdaleite phase in Ge modifies the electronic structure such that the lonsdaleite phase possesses a so-called “pseudo-direct” band gap – i.e. the CB minimum lies at Γ, but this direct gap possesses weak oscillator strength [15]. This band structure modification is illustrated in Fig. 1, which shows the band structure and density of states (DOS) of diamond- and lonsdaleite-structured Si and Ge, calculated from first principles. Diamond-structured (a) Ge and (c) Si are indirect-gap semiconductors. The cubic to hexagonal phase transition gives rise to a pseudo-direct band gap in lonsdaleite-structured (c) Ge, while lonsdaleite-structured (d) Si remains indirect-gap. These band structure calculations indicate that Ge-rich lonsdaleite SiGe alloys should possess a direct band gap, suggesting a radical new approach to obtain a tunable direct band gap group-IV material.

Fig. 1. First principles calculated band structures of (a) diamond-structured (cubic) Ge, (b) lonsdaleite-structured (hexagonal) Ge, (c) diamond-structured Si, and (d) lonsdaleite-structured Si. Conventional cubic Ge and Si are indirect-gap semiconductors, as is hexagonal Si. Hexagonal Ge admits a “pseudo-direct” band gap, making it a potential candidate material for Si photonics applications.

Indeed, recent experimental measurements have revealed room temperature light emission from lonsdaleite SiGe nanowires [16], constituting a highly promising demonstration of a novel approach to obtain light emission from a conventional group-IV semiconductor alloy, with extremely strong potential for Si photonics applications. Given the pseudo-direct nature of the lonsdaleite SiGe band gap, the measured optical properties – in particular the high radiative recombination rate – of SiGe are surprising. To begin to understand this unusual behaviour, we have undertaken first principles calculations of the electronic structure evolution of lonsdaleite SiGe alloys. By combining alloy supercell electronic structure calculations with zone unfolding methods, we elucidate the nature and evolution of the alloy band structure. We comment on the consequences of the calculated electronic properties for optical properties, and evaluate our findings and their implications for optical properties in the context of emerging experimental data.

These results will be presented in talk NM02, “Electronic structure of lonsdaleite SixGe1-x alloys“, at the free online NUSOD 2020 conference.

Acknowledgements

This work was supported by the National University of Ireland Post-Doctoral Fellowship in the Sciences, and by Science Foundation Ireland (SFI; project no. 15/IA/3082).

References

[1] R. Geiger, T. Zabel, and H. Sigg, “Group IV direct gap photonics: methods, challenges, and opportunities”, Front. Mater. 2, 52 (2015)

[2] X. Sun, J. Liu, L. C. Kimerling, and J. Michel, “Direct gap photoluminescence of n-type tensile-strained Ge-on-Si”, Appl. Phys. Lett. 95, 011911 (2009)

[3] C. A. Stephenson, W. A. O’Brien, M. W. Penninger, W. F. Schneider, M. Gillett-Kunnath, J. Zajicek, K. M. Yu, R. Kudraweic, R. A. Stillwell, and M. A. Wistey, “Band structure of germanium carbides for direct bandgap silicon photonics”, J. Appl. Phys. 120, 053102 (2016)

[4] C. A. Broderick, M. D. Dunne, D. S. P. Tanner, and E. P. O’Reilly, “Electronic structure evolution in dilute carbide Ge1-xCx alloys and implications for device applications”, J. Appl. Phys. 126, 195702 (2019)

[5] J. Doherty, S. Biswas, E. Galluccio, C. A. Broderick, A. Garcia-Gil, R. Duffy, E. P. O’Reilly, and J. D. Holmes, “Progress on germanium-tin nanoscale alloys”, Chem. Mater. 32, 4383 (2020)

[6] C. A. Broderick, E. J. O’Halloran, and E. P. O’Reilly, “First principles analysis of electronic structure evolution and the indirect- to direct-gap transition in group-IV Ge1-xPbx alloys”, arXiv:1911.05679 (2019)

[7] S. Schulz, C. A. Broderick, E. J. O’Halloran, and E. P. O’Reilly, “The nature of the band gap of Ge1-xSnx alloys”, Proc. NUSOD (2018)

[8] C. A. Broderick, E. J. O’Halloran, and E. P. O’Reilly, “Comparative analysis of electronic structure evolution in Ge1-xSnx and Ge1-xPbx alloys”, Proc. NUSOD (2019)

[9] H. S. Maczko, R. Kudrawiec, and M. Gladysiewicz, “Designing and analysis of SiGeSn-based quantum wells integrated with Si platform for laser applications”, Proc. NUSOD (2017)

[10] H. Kumar and R. Basu, “Comprehensive study and noise analysis of GeSn-based p-n-p heterojunction phototransistors for efficient detection”, Proc. NUSOD (2018)

[11] M. D. Dunne, C. A. Broderick, M. Luisier, and E. P. O’Reilly, “Atomistic analysis of band-to-band tunnelling in direct-gap Ge1-xSnx group-IV alloys”, Proc. NUSOD (2020)

[12] S. Wirths, R. Geiger, N. von den Driesch, G. Mussler, T. Stoica, S. Mantl, Z. Ikonic, M. Luysberg, S. Chiussi, J. M. Hartmann, H. Sigg, J. Faist, D. Buca, and D. Grützmacher, “Lasing in direct-bandgap GeSn alloy grown on Si”, Nature Photonics 9, 88 (2015)

[13] Y. Zhou, Y. Miao, S. Ojo, H. Tran, G. Abernathy, J. M. Grant, S. Amoah, G. Salamo, W. Du, J. Liu, J. Margetis, J. Tolle, Y.-H. Zhang, G. Sun, R. A. Soref, B. Li, and S.-Q. Yu, “Electrically injected GeSn lasers on Si operating up to 100 K”, arXiv:2004.09402 (2020)

[14] H. I. T. Hague, S. Conesa-Boj, M. A. Verheijen, S. Koelling, and E. P. A. M. Bakkers, “Single-crystalline hexagonal silicon-germanium”, Nano. Lett. 17, 85 (2016)

[15] C. Rödl, J. Furthmüller, J. R. Suckert, V. Armuzza, F. Bechstedt, and S. Botti, “Accurate electronic and optical properties of hexagonal germanium for optoelectronic applications“, Phys. Rev. Materials 3, 034602 (2019)

[16] E. M. T. Fadaly, A. Dijkstra, J. R. Suckert, D. Ziss, M. A. J. van Tilburg, C. Mao, Y. Ren, V. T. van Lange, K. Korzun, S. Kölling, M. A. Verheijen, D. Busse, C. Rödl, J. Furthmüller, F. Bechstedt, J. Stangl, J. J. Finley, S. Botti, J. E. M. Haverkort, and E. P. A. M. Bakkers, “Direct-bandgap emission from hexagonal Ge and SiGe alloys”, Nature 580, 205 (2020)

The dilemma of simulation-based machine learning

Machine learning applications typically perform statistical analyses and predictions based on real-world data collection (Fig. 1). This can be very valuable when the amount of data is large and difficult to digest. Deep learning is currently one of the most popular artificial intelligence methods, utilizing multi-layered artificial neural networks (ANNs). ANNs connect a set of input numbers to a set of output numbers by mathematical operations. However, these operations are not derived from the meaning behind these numbers. Instead, thousands of training data sets are required to teach an ANN. But sufficient training data are sometimes hard to obtain. Therefore, such data are often generated by numerical simulations, especially in the fields of materials science and photonics. This simulation-based approach has the advantage of creating consistent data sets in agreement with existing theoretical models.

Fig. 1: Computational analysis methods.

However, models always simplify reality, and simulation results often disagree with measurements. There are many possible reasons for such disagreements, some of which are listed here. Strictly speaking, numerical simulations produce a virtual reality in which various artificial effects may happen (Fig. 1). Machine learning from flawed simulations ignores such flaws and renders them untraceable. Resulting design optimizations of optoelectronic devices are often unreliable. [1]

Even a perfect simulation model reflects at best our present understanding of reality. This would be quite appropriate for design optimization projects. But we cannot expect artificial intelligence to discover new laws of physics this way. Scientific discoveries are typically triggered by a conflict between existing models and real-world observations. For that, we still need human intelligence, I think.

Further details and possible remedies will be presented at the NUSOD-20 conference.

[1] J. Piprek, Opt Quant Electron 53, 175 (2021)