NUSOD Blog

Connecting Theory and Practice in Optoelectronics

A surprisingly unique concept for a virtual conference

For the third year in a row, we successfully completed our virtual NUSOD conference last week, which allowed for 24/7 access to more than 100 pre-recorded presentation videos on our customized YouTube-style platform. 250+ participants from 37 countries were able to watch and discuss these talks across many time zones whenever it was convenient for them. More than 500 written comments, questions, and answers were generated, with some detailed debates continuing for several days. In my view, this discussion was more widespread and extensive than at our pre-pandemic onsite meetings. The ability to rerun video segments also improved the educational value of the conference. I wonder why most other virtual conferences stick to a traditional real-time schedule.

However, nothing beats the personal contact at onsite meetings for forming or maintaining fruitful collaborations. The travel experience is certainly also an important driver of conference attendance. Therefore, our next NUSOD meeting will be organized in September 2023 by the Politecnico team onsite in Turin, Italy. We hope to see you there!

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Comparison of flux discretizations for varying band-edge energies

Random alloy fluctuations significantly affect the electronic, optical, and transport properties of (In,Ga)N-based optoelectronic devices. To bridge the gap between macroscale drift-diffusion simulations and atomistic band-edge fluctuations, recently a multiscale framework was developed to integrate the macroscopic and microscopic worlds [1]. In order to combine atomistic tight-binding theory and continuum-based drift–diffusion solvers, it is necessary to develop flux discretizations for variable band-edge energies (see Figure 1). While in [1] a first scheme was formulated, flux discretizations are conceivable which avoid the gradient of the fluctuating band-edge energies. The question of handling varying band edge energies is not limited to the framework of atomistic band-edge fluctuations: devices with heterojunctions are also subject to the same issues, since the band-edge energies can exhibit discontinuities at the junctions.

Figure 1: Profile of an in-plane valence band edge energy obtained with atomistic tight-binding model, and a typical (simplified) profile of random fluctuation in 1D.

Considering a drift-diffusion system with discontinuous band-edge energies leads to some mathematical issues, since the gradient of discontinuous functions will appear in the model. As a potential remedy, we introduce a new change of variable, based on a Slotboom trick, in order to reformulate the equations under an expression where the gradient of the discontinuous quantities no longer appears. From a numerical point of view, we use a finite volume method (see [2]), and we introduce schemes based on this change of variable. Such schemes are inspired by the work of [3], and are called “exponential-fitting” schemes. Then, we compare these numerical schemes with other “classical” schemes of [2], and exhibit the speed of convergence of the schemes (see Figure 2). We also look at the influence of the choice of the mesh and the fact that the discontinuities of the band-edge energies match the edges of the mesh.

Figure 2: errors graph for a 1D toy model with heterojunctions

Our results indicate that the different schemes have very different behaviors, and some of them are more suited to handle models with irregular band-edge energies.

Figure 3: Overview of the properties of some schemes for problems with varying band edge energies

More details about this work will be presented at NUSOD 2022 (paper MM05).

References

[1] M. O’Donovan, D. Chaudhuri, T. Streckenbach, P. Farrell, S. Schulz, T. Koprucki, “From atomistic tight binding theory to macroscale drift diffusion: multiscale modeling and numerical simulation of uni-polar
charge transport in (In,Ga)N devices with random fluctuations”, Journal of Applied Physics, vol. 130(6), pp. 065702, 2021.

[2] P. Farrell, T. Koprucki, J. Fuhrmann, Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics, Journal of Computational Physics, Volume 346, 2017, Pages 497-513Z.

[3] F. Brezzi, and L. D. Marini, and P. Pietra, Two-dimensional exponential fitting and applications to drift-diffusion models SIAM J. Numer. Anal., vol. 26, pp 1342–1355, 1989.


3-D multiscale simulations of uni- and bi-polar carrier transport in (In,Ga)N-based devices including alloy fluctuations

In recent years III-nitride materials have attracted interest for use in light-emitting diodes (LEDs) due to their potential for extensive bandgap engineering. The alloys (In,Ga)N and (Al,Ga)N span in principle the visible spectrum, and extend deep into ultraviolet wavelengths [1]. (In,Ga)N/GaN quantum well systems have proved very successful in the blue part of the visible spectrum, however their efficiency is significantly reduced when emission is pushed to longer wavelengths (i.e. higher indium content in the quantum well regions). In order to negate the efficiency reduction in devices it is important to be able to model their material properties accurately, and use these valid models to steer device design.

For guiding the design of III-N-based optoelectronic devices, one dimensional continuum-based transport models are widely employed in the literature. However, these approaches overlook completely alloy induced carrier localization effects, which dominate electronic and optical properties of (In,Ga)N/GaN quantum wells [2]. Only recently have alloy fluctuations been included in transport simulations of (In,Ga)N-based devices via modified continuum-based models. These calculations revealed that such effects are of central importance for an accurate description of the transport properties [3]. Still, an overarching question remains on how to connect atomistic models, which account for alloy fluctuations on a microscopic level [4], with continuum-based calculations, e.g. macroscale drift-diffusion simulations [5].

Figure 1: (a) Valence band edge and (b) conduction band edge energies of an In0.1Ga0.9N quantum well derived from an atomistic tight-binding model including alloy fluctuations and quantum corrections via localization landscape theory.

In this work we address this question by presenting a 3-D carrier transport solver that connects atomistic tight-binding electronic structure theory with the macroscale drift-diffusion solver ddfermi [6, 7]. To account for quantum corrections in our carrier transport calculations, we utilize localization landscape theory (LLT) [8]. Figure 1 gives an example of the energy landscape extracted from our atomitsic tight-binding model, and after LLT is applied, for a slice through the growth plane of an (In,Ga)N/GaN quantum well. The figure clearly reveals that alloy fluctuations are of secondary importance for the conduction band edge energy inside the well but lead to strong variations in the valence band edge energy. It is important to stress again that such fluctuations are not captured in 1D transport simulations available in many commercial software packages.

We have applied this framework to uni-polar electron (n-i-n) and hole (p-i-p) transport to understand the impact that alloy fluctuations and also quantum corrections have on transport properties of such devices [7, 9]. Furthermore, the framework is extended to bi-polar (LED) structures (n-i-p) to investigate the distribution of carriers across a multi-quantum well stack, and in particular study the impact of alloy fluctuations on recombination.

More details of this will be presented at NUSOD 2022 (paper MM02).

Acknowledgements
This work received funding from the Sustainable Energy Authority of Ireland, the Science Foundation Ireland (Nos. 17/CDA/4789,12/RC/2276 P2 and 21/FFP-A/9014) and the Deutsche Forschungsgemeinschaft (DFG) under Germanys Excellence Strategy EXC2046: MATH+, project AA2-15, as well as the Leibniz competition 2020.

References

  1. C. J. Humphreys, “Solid-State Lighting,” MRS Bulletin, vol. 33, p. 459, 2008.
  2. P. Dawson, S. Schulz, R. A. Oliver, M. J. Kappers, and C. J. Humphreys, “The nature of carrier localisation in polar and nonpolar InGaN/GaN quantum wells,” Journal of Applied Physics, vol. 119, no. 18, p. 181505, 2016.
  3. C.-K. Li, M. Piccardo, L.-S. Lu, S. Mayboroda, L. Martinelli, J. Peretti, J. S. Speck, C. Weisbuch, M. Filoche, and Y.-R. Wu, “Localization landscape theory of disorder in semiconductors. III. Application to carrier transport and recombination in light emitting diodes,” Phys. Rev. B, vol. 95, p. 144206, 2017.
  4. M. O’Donovan, M. Luisier, E. P. O’Reilly, and S. Schulz, “Impact of random alloy fluctuations on inter-well transport in InGaN/GaN multi-quantum well systems: an atomistic non-equilibrium Green’s function study,” J. Phys.: Condens. Matter, vol. 33, p. 045302, 2021.
  5. D. H. Doan, P. Farrell, J. Fuhrmann, M. Kantner, T. Koprucki, and N. Rotundo, “ddfermi – a drift-diffusion simulation tool,” ddfermi – a drift-diffusion simulation tool, Weierstrass Institute (WIAS), doi: http://doi.org/10.20347/WIAS.SOFTWARE.DDFERMI, 2020.
  6. D. Chaudhuri, M. O’Donovan, T. Streckenbach, O. Marquardt, P. Farrell, S. K. Patra, T. Koprucki, and S. Schulz, “Multiscale simulations of the electronic structure of III-nitride quantum wells with varied indium content: Connecting atomistic and continuum-based models,” J. Appl. Phys., vol. 129, p. 073104, 2021.
  7. M. O’Donovan, D. Chaudhuri, T. Streckenbach, P. Farrell, S. Schulz, and T. Koprucki, “From atomistic tight-binding theory to macroscale drift-diffusion: Multiscale modeling and numerical simulation of uni-polar charge transport in (In,Ga)N devices with random fluctuations,” Journal of Applied Physics, vol. 130, no. 6, p. 065702, 2021.
  8. M. Filoche, M. Piccardo, Y.-R. Wu, C.-K. Li, C. Weisbuch, and S. Mayboroda, “Localization landscape theory of disorder in semiconductors. I. Theory and modeling,” Phys. Rev. B, vol. 95, p. 144204, 2017.
  9. M. O’Donovan, P. Farrell, T. Streckenbach, T. Koprucki, and S. Schulz, “Multiscale simulations of unipolar hole transport in (In,Ga)N quantum well systems,” Optical and Quantum Electronics, vol. 54, p. 405, June 2022.

Impact of random alloy fluctuations on the electronic and optical properties of c-plane AlxGa1-xN/AlN quantum wells

The semiconductors gallium nitride (GaN), aluminium nitride (AlN), indium nitride (InN) and their connected alloys (Al,In,Ga)N have attracted considerable interest for optoelectronic device applications [1] since their direct band gaps span, in principle, from the infrared into the deep ultraviolet (UV). (In,Ga)N quantum wells (QWs) are at the heart of modern energy efficient light emitting diodes (LEDs) operating in the visible spectral range. However, (Al,Ga)N QW-based LEDs operating in the UV exhibit very low efficiencies, especially in the deep UV [2]. To improve the efficiencies of such devices, understanding the fundamental electronic and optical properties of (Al,Ga)N-based QWs is key.  

Research activities directed towards (In,Ga)N systems have already revealed that these alloys, and also respective heterostructures, exhibit vastly different properties when compared to other more conventional III-V material systems, e.g. InGaAs QWs. In particular, alloy fluctuation induced carrier localization effects can dominate the electronic, optical and transport properties in (In,Ga)N QWs [3-6].

Figure 1: Isosurface plots of the electron (red) and hole (blue) ground state charge densities for (a) side and (b) top view for a c-plane Al0.5Ga0.5N/AlN QW. The light and dark surfaces correspond to 10% and 50% of the maximum values, respectively. The dashed lines indicate, in the side view , (a), the top and bottom QW barrier interface, and in the top view, (b), the supercell boundaries.

Recently, experimental investigations targeting (Al,Ga)N QWs revealed features in their optical properties that are similar to (In,Ga)N systems and are indicative of carrier localization effects [7]. However, compared to (In,Ga)N structures, theoretical studies on alloy induced carrier localization effects in (Al,Ga)N QWs are sparse.

In the present study [8] we target this question by means of an atomistic tight-binding model. Our model accounts for random alloy fluctuations and connected fluctuations in strain and built-in field on a microscopic level. We focus here on the electronic and optical properties of (Al,Ga)N QW systems with aluminium compositions ranging from 10% to 75%. This span in aluminium content allows us to investigate the properties of (Al,Ga)N-based QWs relevant for devices operating in the UV-A to deep UV-C range.  

Our results reveal that already random alloy fluctuations are sufficient to lead to strong carrier localization effects in (Al,Ga)N QWs, independent of the aluminium content in the well. For electrons, our calculations indicate that the carriers may become localized by these alloy fluctuations at higher aluminium contents (>50%). Examples of such carrier localization effects are given in the figures below in terms of isosurface plots of electron and hole charge densities.

More details on this and how alloy fluctuations affect the degree of optical polarization of (Al,Ga)N-based QWs, a quantity of key importance for light extraction efficiencies in UV LEDs, will be presented at NUSOD 2022 (paper NM02). 

This work was financially supported by the Sustainable Energy Authority of Ireland and Science Foundation Ireland (Nos. 17/CDA/4789, 12/RC/2276 P2 and 21/FFP-A/9014)

[1] F. Chen, X. Ji, and S. P. Lau, “Recent progress in group III-nitride nanostructures: From materials to applications,” Materials Science and Engineering 142, 100578 (2020).

[2] M. Kneissl, T. Kolbe, C. Chua, et al., “Advances in group III-nitride-based deep UV light-emitting diode technology”, Semicond. Sci. Technol. 26, 014036 (2011).

[3] P. Dawson, S. Schulz, R. A. Oliver, M. J. Kappers, and C. J. Humphreys, “The nature of carrier localisation in polar and nonpolar InGaN/GaN quantum wells,” J. Appl. Phys. 119, 181505 (2016).

[4] C.-K. Li, M. Piccardo, L.-S. Lu, et al., “Localization landscape theory of disorder in semiconductors. III. Application to carrier transport and recombination in light emitting diodes”, Phys. Rev. B 95, 144205 (2017).

[5] M. O’Donovan, D. Chaudhuri, T. Streckenbach, et al., “From atomistic tight-binding theory to macroscale drift-diffusion: Multiscale modelling and numerical simulation of unipolar charge transport in (In,Ga)N devices with random fluctuations”, J. Appl. Phys. 130, 065702 (2021).

[6] M. O’Donovan, M. Luiser, E. O’Reilly, et al., “Impact of random alloy fluctuations on inter-well transport in InGaN/GaN multi-quantum well systems: an atomistic non-equilibrium Green’s function study”, J. Phys.: Condens. Matter 33, 045302 (2021).

[7] C. Frankerl, F. Nippert, A. Gomez-Iglesias, et al., “Origin of carrier localization in AlGaN-based quantum well structures and implications for efficiency droop”, Appl. Phys. Lett. 177, 102107, (2020)

[8] R. Finn, S. Schulz, Impact of random alloy fluctuations on the electronic and optical properties of (Al,Ga)N quantum wells: Insights frim tight-binding calculations, (2022), submitted, DOI:10.48550/arXiv.2208.05337

Modal properties of dielectric bowtie cavities with deep sub-wavelength confinement

It is a long-held belief in the nanophotonics community that the so-called diffraction limit bounds light confinement in dielectrics to volumes larger than the cubic half-wavelength.  However, recent dielectric cavity designs involving slits and bowties have proven this assumption to be incorrect, by demonstrating light confinement at a deep sub-wavelength scale. The emergence of this novel class of cavities, therefore, allows dielectrics to enter a long-thought inaccessible confinement regime and marks the initiation of a new field with important implications for classical and quantum nanoscale light-generation, detection, and manipulation.

The key figure of merit for many applications in the weak coupling regime is the Purcell factor, which quantifies the relative enhancement in the radiative decay rate of a dipole emitter in an electromagnetic environment.  This enhancement is related to both the temporal and spatial confinement of the field, and inverse design approaches, such as density-based topology optimization, have been used to optimize for this important quantity.

In this work, we take an inverse-design-optimized bowtie cavity and first show that the initial complex structure can be replaced by a much simpler design, which makes it much easier to study and manipulate.  Both designs are shown in Fig. 1a, along with a 3D render of the simplified design in Fig. 1b. Next, we use the framework of quasinormal modes to analyze the electromagnetic response of the cavity and show that a single mode strongly dominates the response, as depicted in Fig. 1c, greatly simplifying the study of the structure.  Finally, we use the established theory to assess the sensitivity of the cavity resonance and quality factor with respect to fabrication imperfections – a subject that severely impacts the manufacturing constraints. We found that both the resonance wavelength and the quality factor are extremely sensitive to even minor perturbations to the bowtie, presenting at the same time a significant challenge and opportunity for future applications.

Further details will be presented at the NUSOD-22 virtual conference.

Fig1. (a). The original inverse design structure and the extracted simplified design. (b). A 3D render of the simplified design showing the norm of the complex mode, scaled to 1. (c). The single mode approximation to the full response (top), showing an error of 0.3% near resonance (bottom). The middle figure shows the complex spectrum of quasinormal modes, with the highest Q mode corresponding to the one used in the single mode approximation.

Funding acknowledgment

This work was supported by the Danish National Research Foundation through NanoPhoton – Center for Nanophotonics, grant number DNRF147.  E.V.D. acknowledges support from Independent Research Fund Denmark through an International Postdoc Fellowship (Grant No. 0164-00014B).