Showing 1–49 of 49 results for author: Jahnke, F

Wannier-Function-Based Approach to Coupled Exciton-Phonon-Photon Dynamics in Two-Dimensional Semiconductors

Authors: Alexander Steinhoff, Frank Jahnke, Matthias Florian

Abstract: Marrying the predictive power of ab initio calculations with many-body effects remains a challenging task in two-dimensional (2d) materials, where efficient carrier-carrier interaction challenges established approximation schemes. In particular, understanding exciton-phonon interaction from first principles is a field of growing interest. Here, we present a many-body theory for coupled free-carrie… ▽ More Marrying the predictive power of ab initio calculations with many-body effects remains a challenging task in two-dimensional (2d) materials, where efficient carrier-carrier interaction challenges established approximation schemes. In particular, understanding exciton-phonon interaction from first principles is a field of growing interest. Here, we present a many-body theory for coupled free-carrier, exciton, phonon and photon dynamics based on carrier-carrier and carrier-phonon interaction matrix elements obtained via projection on Wannier orbitals. The framework is applied to study the impact of carrier-phonon correlations on optical spectra and coupled nonequilibrium carrier-phonon kinetics in monolayer MoSe$_2$. We find that non-Markovian effects and dynamical buildup of quasi-particles are only properly described if correlations at least on the two-phonon level are included. Our studies open a perspective to advance the material-realistic description of nonequilibrium physics in 2d nanostructures to new many-body realms. △ Less

Submitted 17 March, 2025; originally announced March 2025.

Material-realistic modelling of quantum many-body effects in a monolayer TMDC nanolaser device

Authors: Joel Buchgeister, Alexander Steinhoff, Daniel Erben, Michael Lorke, Frank Jahnke

Abstract: The efficient light-matter interaction in combination with the small volume occupied by monolayer transition-metal dichalcogenides (TMDCs) makes this material class a notable option as gain layer in future opto-electronic devices. Many-body effects of excited carriers influence the emission dynamics due to the introduction of optical non-linearities following excitation, but the exact mechanisms r… ▽ More The efficient light-matter interaction in combination with the small volume occupied by monolayer transition-metal dichalcogenides (TMDCs) makes this material class a notable option as gain layer in future opto-electronic devices. Many-body effects of excited carriers influence the emission dynamics due to the introduction of optical non-linearities following excitation, but the exact mechanisms remain unexamined from a theoretical point of view. In this paper, we present a material-realistic microscopic theory of a device based on an MoS2-monolayer, which demonstrates stimulated emission activity at room temperature. The modelling procedure combines Coulomb and light-matter interaction matrix elements with doublet-level Quantum Laser Equations (QLEs). These give access to the dynamics of the photon-assisted polarisation, populations, and photon number while allowing the solution of a multi-scale problem ranging from femto- to nanoseconds. The input-output curve, hole burning and spectral clamping obtained from this theory present strong indications of electron-hole-plasma-based lasing occurring at densities above $5 \cdot 10^{13}\,\text{cm}^{-2}$ in this device. △ Less

Submitted 10 November, 2024; v1 submitted 26 September, 2024; originally announced September 2024.

Comments: 8 pages, 6 figures

Strategies for the alignment of electronic states in quantum-dot tunnel-injection lasers and their influence on the emission dynamics

Authors: Michael Lorke, Igor Khanonkin, Stephan Michael, Johann Peter Reithmaier, Gadi Eisenstein, Frank Jahnke

Abstract: In quantum-dot tunnel-injection lasers, the excited charge carriers are efficiently captured from the bulk states via an injector quantum well and then transferred into the quantum dots via a tunnel barrier. The alignment of the electronic levels is crucial for the high efficiency of these processes and especially for the fast modulation dynamics of these lasers. In particular, the quantum mechani… ▽ More In quantum-dot tunnel-injection lasers, the excited charge carriers are efficiently captured from the bulk states via an injector quantum well and then transferred into the quantum dots via a tunnel barrier. The alignment of the electronic levels is crucial for the high efficiency of these processes and especially for the fast modulation dynamics of these lasers. In particular, the quantum mechanical nature of the tunneling process must be taken into account in the transition from two-dimensional quantum well states to zero-dimensional quantum dot states. This results in hybrid states, from which the scattering into the quantum-dot ground states takes place. We combine electronic state calculations of the tunnel-injection structures with many-body calculations of the scattering processes and insert this into a complete laser simulator. This allows us to study the influence of the structural design and the resulting electronic states as well as limitations due to inhomogeneous quantum-dot distributions. We find that the optimal electronic state alignment deviates from a simple picture in which the of the quantum-dot ground state energies are one LO-phonon energy below the injector quantum well ground state. △ Less

Submitted 19 November, 2024; v1 submitted 28 February, 2024; originally announced February 2024.

Comments: 5 pages, 5 figures

Exciton-exciton interactions in van der Waals heterobilayers

Authors: Alexander Steinhoff, Edith Wietek, Matthias Florian, Tommy Schulz, Takashi Taniguchi, Kenji Watanabe, Shen Zhao, Alexander Högele, Frank Jahnke, Alexey Chernikov

Abstract: Exciton-exciton interactions are key to understanding non-linear optical and transport phenomena in van der Waals heterobilayers, which emerged as versatile platforms to study correlated electronic states. We present a combined theory-experiment study of excitonic many-body effects based on first-principle band structures and Coulomb interaction matrix elements. Key to our approach is the explicit… ▽ More Exciton-exciton interactions are key to understanding non-linear optical and transport phenomena in van der Waals heterobilayers, which emerged as versatile platforms to study correlated electronic states. We present a combined theory-experiment study of excitonic many-body effects based on first-principle band structures and Coulomb interaction matrix elements. Key to our approach is the explicit treatment of the fermionic substructure of excitons and dynamical screening effects for density-induced energy renormalization and dissipation. We demonstrate that dipolar blue shifts are almost perfectly compensated by many-body effects, mainly by screening-induced self-energy corrections. Moreover, we identify a crossover between attractive and repulsive behavior at elevated exciton densities. Theoretical findings are supported by experimental studies of spectrally-narrow interlayer excitons in atomically-reconstructed, hBN-encapsulated MoSe$_2$/WSe$_2$ heterobilayers. Both theory and experiment show energy renormalization on a scale of a few meV even for high injection densities in the vicinity of the Mott transition. Our results revise the established picture of dipolar repulsion dominating exciton-exciton interactions in van der Waals heterostructures and open up opportunities for their external design. △ Less

Submitted 19 October, 2023; originally announced October 2023.

Journal ref: Phys. Rev. X 14, 031025 (2024)

Carrier dynamics in quantum-dot tunnel-injection structures: microscopic theory and experiment

Authors: Michael Lorke, Igor Khanonkin, Stephan Michael, Johann Peter Reithmaier, Gadi Eisenstein, Frank Jahnke

Abstract: Tunneling-injection structures are incorporated in semiconductor lasers in order to overcome the fundamental dynamical limitation due to hot carrier injection by providing a carrier transport path from a cold carrier reservoir. The tunneling process itself depends on band alignment between quantum-dot levels and the injector quantum well, especially as in these devices LO-phonon scattering is domi… ▽ More Tunneling-injection structures are incorporated in semiconductor lasers in order to overcome the fundamental dynamical limitation due to hot carrier injection by providing a carrier transport path from a cold carrier reservoir. The tunneling process itself depends on band alignment between quantum-dot levels and the injector quantum well, especially as in these devices LO-phonon scattering is dominant. Quantum dots with their first excited state near the quantum well bottom profit most from tunnel coupling. As inhomogeneous broadening is omnipresent in quantum dot structures, this implies that individual members of the ensemble couple differently to the injector quantum well. Quantum dots with higher energy profit less, as the phonon couples to higher, less occupied states. Likewise, if the energy difference between ground state and quantum well exceeds the LO phonon energy, scattering becomes increasingly inefficient. Therefore, within 20-30meV we find Quantum Dots that benefit substantially different from the tunnel coupling. Furthermore, in quantum dots with increasing confinement depth, excited states become sucessively confined. Here, scattering gets more efficient again, as subsequent excited states reach the phonon resonance with the quantum well bottom. Our results provide guidelines for the optimization of tunnel-injection lasers. Theoretical results for electronic state caluluations in connection with carrier-phonon and carrier-carrier scattering are compared to experimental results of the temporal gain recovery after a short pulse perturbation. △ Less

Submitted 31 May, 2022; originally announced May 2022.

Quantum fluctuations and lineshape anomaly in a high-$β$ silver-coated InP-based metallic nanolaser

Authors: A. Koulas-Simos, J. Buchgeister, M. Drechsler, T. Zhang, K. Laiho, G. Sinatkas, J. Xu, F. Lohof, Q. Kan, R. K. Zhang, F. Jahnke, C. Gies, W. W. Chow, C. Z. Ning, S. Reitzenstein

Abstract: Metallic nanocavity lasers provide important technological advancement towards even smaller integrable light sources. They give access to widely unexplored lasing physics in which the distinction between different operational regimes, like those of thermal or a coherent light emission, becomes increasingly challenging upon approaching a device with a near-perfect spontaneous-emission coupling fact… ▽ More Metallic nanocavity lasers provide important technological advancement towards even smaller integrable light sources. They give access to widely unexplored lasing physics in which the distinction between different operational regimes, like those of thermal or a coherent light emission, becomes increasingly challenging upon approaching a device with a near-perfect spontaneous-emission coupling factor $β$. In fact, quantum-optical studies have to be employed to reveal a transition to coherent emission in the intensity fluctuation behavior of nanolasers when the input-output characteristic appears thresholdless for $β= 1$ nanolasers. Here, we identify a new indicator for lasing operation in high-$β$ lasers by showing that stimulated emission can give rise to a lineshape anomaly manifesting as a transition from a Lorentzian to a Gaussian component in the emission linewidth that dominates the spectrum above the lasing threshold. △ Less

Submitted 14 January, 2022; originally announced January 2022.

Electron dynamics in a 2D nanobubble: A two-level system based on spatial density

Authors: Roberto Rosati, Frank Lengers, Christian Carmesin, Matthias Florian, Tilmann Kuhn, Frank Jahnke, Michael Lorke, Doris E. Reiter

Abstract: Nanobubbles formed in monolayers of transition metal dichalcogenides (TMDCs) on top of a substrate feature localized potentials, in which electrons can be captured. We show that the captured electronic density can exhibit a non-trivial spatiotemporal dynamics, whose movements can be mapped to states in a two-level system illustrated as points of an electronic Poincaré sphere. These states can be f… ▽ More Nanobubbles formed in monolayers of transition metal dichalcogenides (TMDCs) on top of a substrate feature localized potentials, in which electrons can be captured. We show that the captured electronic density can exhibit a non-trivial spatiotemporal dynamics, whose movements can be mapped to states in a two-level system illustrated as points of an electronic Poincaré sphere. These states can be fully controlled, i.e, initialized and switched, by multiple electronic wave packets. Our results could be the foundation for novel implementations of quantum circuits. △ Less

Submitted 17 November, 2021; originally announced November 2021.

Comments: This document is the Author's version of a Work that was accepted for publication in Nano Letters, copyright American Chemical Society after peer review

Trions in MoS$_2$ are quantum superpositions of intra- and intervalley spin states

Authors: Julian Klein, Matthias Florian, Alexander Hötger, Alexander Steinhoff, Alex Delhomme, Takashi Taniguchi, Kenji Watanabe, Frank Jahnke, Alexander W. Holleitner, Marek Potemski, Clément Faugeras, Andreas V. Stier, Jonathan J. Finley

Abstract: We report magneto-photoluminescence spectroscopy of gated MoS$_2$ monolayers in high magnetic fields to 28 T. At B = 0T and electron density $n_s\sim 10^{12}cm^-2$, we observe three trion resonances that cannot be explained within a single-particle picture. Employing ab initio calculations that take into account three-particle correlation effects as well as local and non-local electron-hole exchan… ▽ More We report magneto-photoluminescence spectroscopy of gated MoS$_2$ monolayers in high magnetic fields to 28 T. At B = 0T and electron density $n_s\sim 10^{12}cm^-2$, we observe three trion resonances that cannot be explained within a single-particle picture. Employing ab initio calculations that take into account three-particle correlation effects as well as local and non-local electron-hole exchange interaction, we identify those features as quantum superpositions of inter- and intravalley spin states. We experimentally investigate the mixed character of the trion wave function via the filling factor dependent valley Zeeman shift in positive and negative magnetic fields. Our results highlight the importance of exchange interactions for exciton physics in monolayer MoS$_2$ and provide new insights into the microscopic understanding of trion physics in 2D multi-valley semiconductors for low excess carrier densities. △ Less

Submitted 13 September, 2021; originally announced September 2021.

Comments: Main manuscript: 7 pages, 3 figures ; Supplemental material: 16 pages, 4 figures

Journal ref: Phys. Rev. B 105, L041302 (2022)

Atomistic spin textures on-demand in the van der Waals layered magnet CrSBr

Authors: Julian Klein, Thang Pham, Joachim Dahl Thomsen, Jonathan B. Curtis, Michael Lorke, Matthias Florian, Alexander Steinhoff, Ren A. Wiscons, Jan Luxa, Zdenek Sofer, Frank Jahnke, Prineha Narang, Frances M. Ross

Abstract: Controlling magnetism in low dimensional materials is essential for designing devices that have feature sizes comparable to several critical length scales that exploit functional spin textures, allowing the realization of low-power spintronic and magneto-electric hardware. [1] Unlike conventional covalently-bonded bulk materials, van der Waals (vdW)-bonded layered magnets [2-4] offer exceptional d… ▽ More Controlling magnetism in low dimensional materials is essential for designing devices that have feature sizes comparable to several critical length scales that exploit functional spin textures, allowing the realization of low-power spintronic and magneto-electric hardware. [1] Unlike conventional covalently-bonded bulk materials, van der Waals (vdW)-bonded layered magnets [2-4] offer exceptional degrees of freedom for engineering spin textures. [5] However, their structural instability has hindered microscopic studies and manipulations. Here, we demonstrate nanoscale structural control in the layered magnet CrSBr creating novel spin textures down to the atomic scale. We show that it is possible to drive a local structural phase transformation using an electron beam that locally exchanges the bondings in different directions, effectively creating regions that have vertical vdW layers embedded within the horizontally vdW bonded exfoliated flakes. We calculate that the newly formed 2D structure is ferromagnetically ordered in-plane with an energy gap in the visible spectrum, and weak antiferromagnetism between the planes. Our study lays the groundwork for designing and studying novel spin textures and related quantum magnetic phases down to single-atom sensitivity, potentially to create on-demand spin Hamiltonians probing fundamental concepts in physics, [6-10] and for realizing high-performance spintronic, magneto-electric and topological devices with nanometer feature sizes. [11,12] △ Less

Submitted 21 July, 2021; v1 submitted 30 June, 2021; originally announced July 2021.

Comments: Main manuscript: 11 pages, 4 figures ; Extended data: 22 pages, 19 figures

Journal ref: Nature Communications 13, 5420 (2022)

Microscopic Theory of Exciton-Exciton Annihilation in Two-Dimensional Semiconductors

Authors: Alexander Steinhoff, Matthias Florian, Frank Jahnke

Abstract: Auger-like exciton-exciton annihilation (EEA) is considered the key fundamental limitation to quantum yield in devices based on excitons in two-dimensional (2d) materials. Since it is challenging to experimentally disentangle EEA from competing processes, guidance of a quantitative theory is highly desirable. The very nature of EEA requires a material-realistic description that is not available to… ▽ More Auger-like exciton-exciton annihilation (EEA) is considered the key fundamental limitation to quantum yield in devices based on excitons in two-dimensional (2d) materials. Since it is challenging to experimentally disentangle EEA from competing processes, guidance of a quantitative theory is highly desirable. The very nature of EEA requires a material-realistic description that is not available to date. We present a many-body theory of EEA based on first-principle band structures and Coulomb interaction matrix elements that goes beyond an effective bosonic picture. Applying our theory to monolayer MoS$_2$ encapsulated in hexagonal BN, we obtain an EEA coefficient in the order of $10^{-3}$ cm$^{2}$s$^{-1}$ at room temperature, suggesting that carrier losses are often dominated by other processes, such as defect-assisted scattering. Our studies open a perspective to quantify the efficiency of intrinsic EEA processes in various 2d materials in the focus of modern materials research. △ Less

Submitted 24 September, 2021; v1 submitted 30 June, 2021; originally announced June 2021.

Bright electrically controllable quantum-dot-molecule devices fabricated by in-situ electron-beam lithography

Authors: Johannes Schall, Marielle Deconinck, Nikolai Bart, Matthias Florian, Martin von Helversen, Christian Dangel, Ronny Schmidt, Lucas Bremer, Frederik Bopp, Isabell Hüllen, Christopher Gies, Dirk Reuter, Andreas D. Wieck, Sven Rodt, Jonathan J. Finley, Frank Jahnke, Arne Ludwig, Stephan Reitzenstein

Abstract: Self-organized semiconductor quantum dots represent almost ideal two-level systems, which have strong potential to applications in photonic quantum technologies. For instance, they can act as emitters in close-to-ideal quantum light sources. Coupled quantum dot systems with significantly increased functionality are potentially of even stronger interest since they can be used to host ultra-stable s… ▽ More Self-organized semiconductor quantum dots represent almost ideal two-level systems, which have strong potential to applications in photonic quantum technologies. For instance, they can act as emitters in close-to-ideal quantum light sources. Coupled quantum dot systems with significantly increased functionality are potentially of even stronger interest since they can be used to host ultra-stable singlet-triplet spin qubits for efficient spin-photon interfaces and for a deterministic photonic 2D cluster-state generation. We realize an advanced quantum dot molecule (QDM) device and demonstrate excellent optical properties. The device includes electrically controllable QDMs based on stacked quantum dots in a pin-diode structure. The QDMs are deterministically integrated into a photonic structure with a circular Bragg grating using in-situ electron beam lithography. We measure a photon extraction efficiency of up to (24$\pm$4)% in good agreement with numerical simulations. The coupling character of the QDMs is clearly demonstrated by bias voltage dependent spectroscopy that also controls the orbital couplings of the QDMs and their charge state in quantitative agreement with theory. The QDM devices show excellent single-photon emission properties with a multi-photon suppression of $g^{(2)}(0) = (3.9 \pm 0.5) \cdot 10^{-3}$. These metrics make the developed QDM devices attractive building blocks for use in future photonic quantum networks using advanced nanophotonic hardware. △ Less

Submitted 8 April, 2021; v1 submitted 10 January, 2021; originally announced January 2021.

Optical nonlinearities in the excited carrier density of atomically thin transition metal dichalcogenides

Authors: Daniel Erben, Alexander Steinhoff, Michael Lorke, Frank Jahnke

Abstract: In atomically thin semiconductors based on transition metal dichalcogenides, photoexcitation can be used to generate high densities of electron-hole pairs. Due to optical nonlinearities, which originate from Pauli blocking and many-body effects of the excited carriers, the generated carrier density will deviate from a linear increase in pump fluence. In this paper, we use a theoretical approach th… ▽ More In atomically thin semiconductors based on transition metal dichalcogenides, photoexcitation can be used to generate high densities of electron-hole pairs. Due to optical nonlinearities, which originate from Pauli blocking and many-body effects of the excited carriers, the generated carrier density will deviate from a linear increase in pump fluence. In this paper, we use a theoretical approach that combines results from ab-initio electronic-state calculations with a many-body treatment of optical excitation to describe nonlinear absorption properties and the resulting excited carrier dynamics. We determine the validity range of a linear approximation for the excited carrier density vs. pump power and identify the role and magnitude of optical nonlinearities at elevated excitation carrier densities for MoS2, MoSe2, WS2, and WSe2 considering various excitation conditions. We find that for above-band-gap photoexcitation, the use of a linear absorption coefficient of the unexcited system can strongly underestimate the achievable carrier density for a wide range of pump fluences due to many-body renormalizations of the two-particle density-of-states. △ Less

Submitted 14 December, 2020; originally announced December 2020.

Twist Angle Dependent Interlayer Exciton Lifetimes in van der Waals Heterostructures

Authors: Junho Choi, Matthias Florian, Alexander Steinhoff, Daniel Erben, Kha Tran, Dong Seob Kim, Liuyang Sun, Jiamin Quan, Robert Claassen, Somak Majumder, Jennifer A. Hollingsworth, Takashi Taniguchi, Kenji Watanabe, Keiji Ueno, Akshay Singh, Galan Moody, Frank Jahnke, Xiaoqin Li

Abstract: In van der Waals (vdW) heterostructures formed by stacking two monolayers of transition metal dichalcogenides, multiple exciton resonances with highly tunable properties are formed and subject to both vertical and lateral confinement. We investigate how a unique control knob, the twist angle between the two monolayers, can be used to control the exciton dynamics. We observe that the interlayer exc… ▽ More In van der Waals (vdW) heterostructures formed by stacking two monolayers of transition metal dichalcogenides, multiple exciton resonances with highly tunable properties are formed and subject to both vertical and lateral confinement. We investigate how a unique control knob, the twist angle between the two monolayers, can be used to control the exciton dynamics. We observe that the interlayer exciton lifetimes in $\text{MoSe}_{\text{2}}$/$\text{WSe}_{\text{2}}$ twisted bilayers (TBLs) change by one order of magnitude when the twist angle is varied from 1$^\circ$ to 3.5$^\circ$. Using a low-energy continuum model, we theoretically separate two leading mechanisms that influence interlayer exciton radiative lifetimes. The shift to indirect transitions in the momentum space with an increasing twist angle and the energy modulation from the moiré potential both have a significant impact on interlayer exciton lifetimes. We further predict distinct temperature dependence of interlayer exciton lifetimes in TBLs with different twist angles, which is partially validated by experiments. While many recent studies have highlighted how the twist angle in a vdW TBL can be used to engineer the ground states and quantum phases due to many-body interaction, our studies explore its role in controlling the dynamics of optically excited states, thus, expanding the conceptual applications of "twistronics". △ Less

Submitted 26 January, 2021; v1 submitted 29 July, 2020; originally announced July 2020.

Journal ref: Phys. Rev. Lett. 126, 047401 (2021)

Controlling exciton many-body states by the electric-field effect in monolayer MoS$_2$

Authors: Julian Klein, Alexander Hötger, Matthias Florian, Alexander Steinhoff, Alex Delhomme, Takashi Taniguchi, Kenji Watanabe, Frank Jahnke, Alexander W. Holleitner, Marek Potemski, Clément Faugeras, Jonathan J. Finley, Andreas V. Stier

Abstract: We report magneto-optical spectroscopy of gated monolayer MoS$_2$ in high magnetic fields up to 28T and obtain new insights on the many-body interaction of neutral and charged excitons with the resident charges of distinct spin and valley texture. For neutral excitons at low electron doping, we observe a nonlinear valley Zeeman shift due to dipolar spin-interactions that depends sensitively on the… ▽ More We report magneto-optical spectroscopy of gated monolayer MoS$_2$ in high magnetic fields up to 28T and obtain new insights on the many-body interaction of neutral and charged excitons with the resident charges of distinct spin and valley texture. For neutral excitons at low electron doping, we observe a nonlinear valley Zeeman shift due to dipolar spin-interactions that depends sensitively on the local carrier concentration. As the Fermi energy increases to dominate over the other relevant energy scales in the system, the magneto-optical response depends on the occupation of the fully spin-polarized Landau levels in both $K/K^{\prime}$ valleys. This manifests itself in a many-body state. Our experiments demonstrate that the exciton in monolayer semiconductors is only a single particle boson close to charge neutrality. We find that away from charge neutrality it smoothly transitions into polaronic states with a distinct spin-valley flavour that is defined by the Landau level quantized spin and valley texture. △ Less

Submitted 13 September, 2021; v1 submitted 28 July, 2020; originally announced July 2020.

Comments: Main manuscript: 7 pages, 4 figures ; Supplemental material: 20 pages, 8 figures

Journal ref: Phys. Rev. Research 3, 022009 (2021)

Imaging strain-localized exciton states in nanoscale bubbles in monolayer WSe2 at room temperature

Authors: Thomas P. Darlington, Christian Carmesin, Matthias Florian, Emanuil Yanev, Obafunso Ajayi, Jenny Ardelean, Daniel A. Rhodes, Augusto Ghiotto, Andrey Krayev, K. Watanabe, T. Taniguchi, Jeffrey W. Kysar, Abhay N. Pasupathy, James C. Hone, Frank Jahnke, Nicholas J. Borys, P. James Schuck

Abstract: In monolayer transition metal dichalcogenides, quantum emitters are associated with localized strain that can be deterministically applied to create designer nano-arrays of single photon sources. Despite an overwhelming empirical correlation with local strain, the nanoscale interplay between strain, excitons, defects and local crystalline structure that gives rise to these quantum emitters is poor… ▽ More In monolayer transition metal dichalcogenides, quantum emitters are associated with localized strain that can be deterministically applied to create designer nano-arrays of single photon sources. Despite an overwhelming empirical correlation with local strain, the nanoscale interplay between strain, excitons, defects and local crystalline structure that gives rise to these quantum emitters is poorly understood. Here, we combine room-temperature nano-optical imaging and spectroscopy of excitons in nanobubbles of localized strain in monolayer WSe2 with atomistic structural models to elucidate how strain induces nanoscale confinement potentials that give rise to highly localized exciton states in 2D semiconductors. Nano-optical imaging of nanobubbles in low-defect monolayers reveal localized excitons on length scales of approximately 10 nm at multiple sites along the periphery of individual nanobubbles, which is in stark contrast to predictions of continuum models of strain. These results agree with theoretical confinement potentials that are atomistically derived from measured topographies of existing nanobubbles. Our results provide one-of-a-kind experimental and theoretical insight of how strain-induced confinement - without crystalline defects - can efficiently localize excitons on length scales commensurate with exciton size, providing key nanoscale structure-property information for quantum emitter phenomena in monolayer WSe2. △ Less

Submitted 3 March, 2020; originally announced March 2020.

Comments: 18 pages, 4 figures

Scalable single-photon sources in atomically thin MoS2

Authors: Julian Klein, Lukas Sigl, Samuel Gyger, Katja Barthelmi, Matthias Florian, Sergio Rey, Takashi Taniguchi, Kenji Watanabe, Frank Jahnke, Christoph Kastl, Val Zwiller, Klaus D. Jöns, Kai Müller, Ursula Wurstbauer, Jonathan J. Finley, Alexander W. Holleitner

Abstract: Real-world quantum applications, eg. on-chip quantum networks and quantum cryptography, necessitate large scale integrated single-photon sources with nanoscale footprint for modern information technology. While on-demand and high fidelity implantation of atomic scale single-photon sources in conventional 3D materials suffer from uncertainties due to the crystals dimensionality, layered 2D material… ▽ More Real-world quantum applications, eg. on-chip quantum networks and quantum cryptography, necessitate large scale integrated single-photon sources with nanoscale footprint for modern information technology. While on-demand and high fidelity implantation of atomic scale single-photon sources in conventional 3D materials suffer from uncertainties due to the crystals dimensionality, layered 2D materials can host point-like centers with inherent confinement to a sub-nm plane. However, previous attempts to truly deterministically control spatial position and spectral homogeneity while maintaining the 2D character have not been realized. Here, we demonstrate the on-demand creation and precise positioning of single-photon sources in atomically thin MoS2 with very narrow ensemble broadening and near-unity fabrication yield. Focused ion beam irradiation creates 100s to 1000s of mono-typical atomistic defects with anti-bunched emission lines with sub-10 nm lateral and 0.7 nm axial positioning accuracy. Our results firmly establish 2D materials as a scalable platform for single-photon emitters with unprecedented control of position as well as photophysical properties owing to the all-interfacial nature. △ Less

Submitted 20 February, 2020; originally announced February 2020.

Comments: 13 pages, 4 figures

Journal ref: ACS Photonics 8, 669-677 (2021)

Dynamical screening effects of substrate phonons on two-dimensional excitons

Authors: Alexander Steinhoff, Matthias Florian, Frank Jahnke

Abstract: Atomically thin materials are exceedingly susceptible to their dielectric environment. For transition metal dichalcogenides, sample placement on a substrate or encapsulation in hexagonal boron nitride (hBN) are frequently used. In this paper we show that the dielectric response due to optical phonons of adjacent materials influences excitons in 2d crystals. We provide an analytic model for the cou… ▽ More Atomically thin materials are exceedingly susceptible to their dielectric environment. For transition metal dichalcogenides, sample placement on a substrate or encapsulation in hexagonal boron nitride (hBN) are frequently used. In this paper we show that the dielectric response due to optical phonons of adjacent materials influences excitons in 2d crystals. We provide an analytic model for the coupling of 2d charge carriers to optical substrate phonons, which causes polaron effects similar to that of intrinsic 2d phonons. We apply the model to hBN-encapsulated WSe2, finding a significant reduction of the exciton binding energies due to dynamical screening effects. △ Less

Submitted 8 November, 2019; originally announced November 2019.

Journal ref: Phys. Rev. B 101, 045411 (2020)

Quantum-dot-like states in molybdenum disulfide nanostructures due to the interplay of local surface wrinkling, strain, and dielectric confinement

Authors: Christian Carmesin, Michael Lorke, Matthias Florian, Daniel Erben, Alexander Schulz, Tim O. Wehling, Frank Jahnke

Abstract: The observation of quantum light emission from atomically thin transition metal dichalcogenides has opened a new field of applications for these material systems. The corresponding excited charge-carrier localization has been linked to defects and strain, while open questions remain regarding the microscopic origin. We demonstrate that the bending rigidity of these materials leads to wrinkling of… ▽ More The observation of quantum light emission from atomically thin transition metal dichalcogenides has opened a new field of applications for these material systems. The corresponding excited charge-carrier localization has been linked to defects and strain, while open questions remain regarding the microscopic origin. We demonstrate that the bending rigidity of these materials leads to wrinkling of the two-dimensional layer. The resulting strain field facilitates strong carrier localization due to its pronounced influence on the band gap. Additionally, we consider charge carrier confinement due to local changes of the dielectric environment and show that both effects contribute to modified electronic states and optical properties. The interplay of surface wrinkling, strain-induced confinement, and local changes of the dielectric environment is demonstrated for the example of nanobubbles that form when monolayers are deposited on substrates or other two-dimensional materials. △ Less

Submitted 13 February, 2019; originally announced February 2019.

Prospects and limitations of transition-metal dichalcogenide laser gain materials

Authors: Frederik Lohof, Alexander Steinhoff, Michael Lorke, Matthias Florian, Daniel Erben, Frank Jahnke, Christopher Gies

Abstract: Nanolasers operate with a minimal amount of active material and low losses. In this regime, single layers of transition-metal dichalcogenides (TMDs) are being investigated as next generation gain materials due to their high quantum efficiency. We provide results from microscopic gain calculations of highly excited TMD monolayers and specify requirements to achieve lasing with four commonly used TM… ▽ More Nanolasers operate with a minimal amount of active material and low losses. In this regime, single layers of transition-metal dichalcogenides (TMDs) are being investigated as next generation gain materials due to their high quantum efficiency. We provide results from microscopic gain calculations of highly excited TMD monolayers and specify requirements to achieve lasing with four commonly used TMD semiconductors. Our approach includes band-structure renormalizations due to excited carriers that trigger a direct-to-indirect band-gap transition. As a consequence, we predict a rollover for the gain that limits the excitation regime where laser operation is possible. A parametrization of the peak gain is provided that is used in combination with a rate-equation theory to discuss consequences for experimentally accessible laser characteristics. △ Less

Submitted 24 September, 2018; originally announced September 2018.

Carrier Dynamics in a Tunneling Injection Quantum Dot Semiconductor Optical Amplifier

Authors: I. Khanonkin, M. Lorke, S. Michael, A. K. Mishra, J. P. Reithmaier, F. Jahnke, G. Eisenstein

Abstract: The process of tunneling injection is known to improve the dynamical characteristics of quantum well and quantum dot lasers; in the latter, it also improves the temperature performance. The advantage of the tunneling injection process stems from the fact that it avoids hot carrier injection, which is a key performance-limiting factor in all semiconductor lasers. The tunneling injection process is… ▽ More The process of tunneling injection is known to improve the dynamical characteristics of quantum well and quantum dot lasers; in the latter, it also improves the temperature performance. The advantage of the tunneling injection process stems from the fact that it avoids hot carrier injection, which is a key performance-limiting factor in all semiconductor lasers. The tunneling injection process is not fully understood microscopically and therefore it is difficult to optimize those laser structures. We present here a numerical study of the broad band carrier dynamics in a tunneling injection quantum dot gain medium in the form of an optical amplifier operating at 1.55 um. Charge carrier tunneling occurs in a hybrid state that joins the quantum dot first excited state and the confined quantum well - injection well states. The hybrid state, which is placed energetically roughly one LO phonon above the ground state and has a spectral extent of about 5 meV , dominates the carrier injection to the ground state. We calculate the dynamical response of the inversion across the entire gain spectrum following a short pulse perturbation at various wavelengths and for two bias currents. At a high bias of 200 mA, the entire spectrum exhibits gain; at 30 mA, the system exhibits a mixed gain - absorption spectrum. The carrier dynamics in the injection well is calculated simultaneously. We discuss the role of the pulse excitation wavelengths relative to the gain spectrum peak and demonstrate that the injection well responds to all perturbation wavelengths, even those which are far from the region where the tunneling injection process dominates. △ Less

Submitted 22 July, 2018; originally announced July 2018.

Journal ref: Phys. Rev. B 98, 125307 (2018)

Performance of quantum-dot-based tunnel-injection lasers: A theoretical analysis

Authors: Michael Lorke, Stephan Michael, Marian Cepok, Frank Jahnke

Abstract: Tunnel-injection lasers promise advantages in modulation bandwidth and temperature stability in comparison to conventional laser designs. In this paper, we present results of a microscopic theory for laser properties of tunnel-injection devices and a comparison to a conventional quantum-dot laser structure. In general, the modulation bandwidth of semiconductor lasers is affected by the steady-stat… ▽ More Tunnel-injection lasers promise advantages in modulation bandwidth and temperature stability in comparison to conventional laser designs. In this paper, we present results of a microscopic theory for laser properties of tunnel-injection devices and a comparison to a conventional quantum-dot laser structure. In general, the modulation bandwidth of semiconductor lasers is affected by the steady-state occupations of electrons and holes via the presence of spectral hole burning. For tunnel-injection lasers with InGaAs quantum dot emitting at the telecom wavelength of 1,55$μ$m, we demonstrate that the absence of spectral hole burning favors this concept over conventional quantum-dot based lasers. △ Less

Submitted 22 June, 2018; originally announced June 2018.

Comments: 5 pages, 4 figures

Observation of exciton redshift-blueshift crossover in monolayer WS2

Authors: Edbert J. Sie, Alexander Steinhoff, Christopher Gies, Chun Hung Lui, Qiong Ma, Malte Rosner, Gunnar Schonhoff, Frank Jahnke, Tim O. Wehling, Yi-Hsien Lee, Jing Kong, Pablo Jarillo-Herrero, Nuh Gedik

Abstract: We report a rare atom-like interaction between excitons in monolayer WS2, measured using ultrafast absorption spectroscopy. At increasing excitation density, the exciton resonance energy exhibits a pronounced redshift followed by an anomalous blueshift. Using both material-realistic computation and phenomenological modeling, we attribute this observation to plasma effects and an attraction-repulsi… ▽ More We report a rare atom-like interaction between excitons in monolayer WS2, measured using ultrafast absorption spectroscopy. At increasing excitation density, the exciton resonance energy exhibits a pronounced redshift followed by an anomalous blueshift. Using both material-realistic computation and phenomenological modeling, we attribute this observation to plasma effects and an attraction-repulsion crossover of the exciton-exciton interaction that mimics the Lennard-Jones potential between atoms. Our experiment demonstrates a strong analogy between excitons and atoms with respect to inter-particle interaction, which holds promise to pursue the predicted liquid and crystalline phases of excitons in two-dimensional materials. △ Less

Submitted 27 April, 2018; originally announced April 2018.

Journal ref: Nano Letters 17, 4210 (2017)

Excitation-induced transition to indirect band gaps in atomically thin transition metal dichalcogenide semiconductors

Authors: D. Erben, A. Steinhoff, G. Schönhoff, T. O. Wehling, C. Gies, F. Jahnke

Abstract: Monolayers of transition metal dichalcogenides (TMDCs) exhibit an exceptionally strong Coulomb interaction between charge carriers due to the two-dimensional carrier confinement in connection with weak dielectric screening. High densities of excited charge carriers in the various band-structure valleys cause strong many-body renormalizations that influence both the electronic properties and the op… ▽ More Monolayers of transition metal dichalcogenides (TMDCs) exhibit an exceptionally strong Coulomb interaction between charge carriers due to the two-dimensional carrier confinement in connection with weak dielectric screening. High densities of excited charge carriers in the various band-structure valleys cause strong many-body renormalizations that influence both the electronic properties and the optical response of the material. We investigate electronic and optical properties of the typical monolayer TMDCs MoS$_2$, MoSe$_2$, WS$_2$ and WSe$_2$ in the presence of excited carriers by solving semiconductor Bloch equations on the full Brillouin zone. With increasing carrier density, we systematically find a reduction of the exciton binding energies due to Coulomb screening and Pauli blocking. Together with excitation-induced band-gap shrinkage this leads to redshifts of excitonic resonances up to the dissociation of excitons. As a central result, we predict for all investigated monolayer TMDCs that the $Σ$-valley shifts stronger than the K-valley. Two of the materials undergo a transition from direct to indirect band gaps under carrier excitation similar to well-known strain-induced effects. Our findings have strong implications for the filling of conduction-band valleys with excited carriers and are relevant to transport and optical applications as well as the emergence of phonon-driven superconductivity. △ Less

Submitted 23 April, 2018; originally announced April 2018.

Journal ref: Phys. Rev. B 98, 035434 (2018)

Interplay of structural design and interaction processes in tunnel-injection semiconductor lasers

Authors: Stephan Michael, Michael Lorke, Marian Cepok, Christian Carmesin, Frank Jahnke

Abstract: Tunnel-injection lasers promise various advantages in comparison to conventional laser designs. In this paper, we present a theoretical analysis for the physics of the tunnel-injection process in quantum-dot based laser devices. We describe the carrier dynamics in terms of scattering between states of the coupled system consisting of injector quantum-well, tunnel-barrier, and quantum-dots. Our ana… ▽ More Tunnel-injection lasers promise various advantages in comparison to conventional laser designs. In this paper, we present a theoretical analysis for the physics of the tunnel-injection process in quantum-dot based laser devices. We describe the carrier dynamics in terms of scattering between states of the coupled system consisting of injector quantum-well, tunnel-barrier, and quantum-dots. Our analysis demonstrates how current quantum-dot based lasers can benefit from the tunnel-injection design. We find that the often assumed LO-phonon resonance condition for the level alignment only weakly influences the injection rate of carriers into the quantum-dot states. On the other hand, our investigations show that the energetic alignment of quantum-dot and quantum-well states modifies the injection efficiency, as it controls the hybridization strength. Our description of tunneling includes the phonon-mediated and the Coulomb scattering contributions and is based on material realistic electronic structure calculations. △ Less

Submitted 30 October, 2018; v1 submitted 10 March, 2018; originally announced March 2018.

Journal ref: Phys. Rev. B 98, 165431 (2018)

Biexciton fine structure in monolayer transition metal dichalcogenides

Authors: Alexander Steinhoff, Matthias Florian, Akshay Singh, Kha Tran, Mirco Kolarczik, Sophia Helmrich, Alexander W. Achtstein, Ulrike Woggon, Nina Owschimikow, Frank Jahnke, Xiaoqin Li

Abstract: The optical properties of atomically thin transition metal dichalcogenide (TMDC) semiconductors are shaped by the emergence of correlated many-body complexes due to strong Coulomb interaction. Exceptional electron-hole exchange predestines TMDCs to study fundamental and applied properties of Coulomb complexes such as valley depolarization of excitons and fine-structure splitting of trions. Biexcit… ▽ More The optical properties of atomically thin transition metal dichalcogenide (TMDC) semiconductors are shaped by the emergence of correlated many-body complexes due to strong Coulomb interaction. Exceptional electron-hole exchange predestines TMDCs to study fundamental and applied properties of Coulomb complexes such as valley depolarization of excitons and fine-structure splitting of trions. Biexcitons in these materials are less understood and it has been established only recently that they are spectrally located between exciton and trion. Here we show that biexcitons in monolayer TMDCs exhibit a distinct fine structure on the order of meV due to electron-hole exchange. Ultrafast pump-probe experiments on monolayer WSe$_2$ reveal decisive biexciton signatures and a fine structure in excellent agreement with a microscopic theory. We provide a pathway to access biexciton spectra with unprecedented accuracy, which is valuable beyond the class of TMDCs, and to understand even higher Coulomb complexes under the influence of electron-hole exchange. △ Less

Submitted 12 January, 2018; originally announced January 2018.

Journal ref: Nature Physics 14, 1199 (2018)

Excitons versus electron-hole plasma in monolayer transition metal dichalcogenide semiconductors

Authors: Alexander Steinhoff, Matthias Florian, Malte Rösner, Gunnar Schönhoff, Tim Oliver Wehling, Frank Jahnke

Abstract: When electron-hole pairs are excited in a semiconductor, it is a priori not clear if they form a fermionic plasma of unbound particles or a bosonic exciton gas. Usually, the exciton phase is associated with low temperatures. In atomically thin transition metal dichalcogenide semiconductors, excitons are particularly important even at room temperature due to strong Coulomb interaction and a large e… ▽ More When electron-hole pairs are excited in a semiconductor, it is a priori not clear if they form a fermionic plasma of unbound particles or a bosonic exciton gas. Usually, the exciton phase is associated with low temperatures. In atomically thin transition metal dichalcogenide semiconductors, excitons are particularly important even at room temperature due to strong Coulomb interaction and a large exciton density of states. Using state-of-the-art many-body theory including dynamical screening, we show that the exciton-to-plasma ratio can be efficiently tuned by dielectric substrate screening as well as charge carrier doping. Moreover, we predict a Mott transition from the exciton-dominated regime to a fully ionized electron-hole plasma at excitation densities between $3\times10^{12}$ cm$^{-2}$ and $1\times10^{13}$ cm$^{-2}$ depending on temperature, carrier doping and dielectric environment. We propose the observation of these effects by studying excitonic satellites in photoemission spectroscopy and scanning tunneling microscopy. △ Less

Submitted 18 May, 2017; v1 submitted 15 May, 2017; originally announced May 2017.

Journal ref: Nature Communications 8, 1166 (2017)

Non-invasive control of excitons in two-dimensional materials

Authors: Christina Steinke, Daniel Mourad, Malte Rösner, Michael Lorke, Christopher Gies, Frank Jahnke, Gerd Czycholl, Tim O. Wehling

Abstract: We investigate how external screening shapes excitons in two-dimensional (2d) semiconductors embedded in laterally structured dielectric environments. An atomic scale view of these elementary excitations is developed using models which apply to a variety of materials including transition metal dichalcogenides (TMDCs). We find that structured dielectrics imprint a peculiar potential energy landscap… ▽ More We investigate how external screening shapes excitons in two-dimensional (2d) semiconductors embedded in laterally structured dielectric environments. An atomic scale view of these elementary excitations is developed using models which apply to a variety of materials including transition metal dichalcogenides (TMDCs). We find that structured dielectrics imprint a peculiar potential energy landscape on excitons in these systems: While the ground-state exciton is least influenced, higher excitations are attracted towards regions with high dielectric constant of the environment. This landscape is "inverted" in the sense that low energy excitons are less strongly affected than their higher energy counterparts. Corresponding energy variations emerge on length scales of the order of a few unit cells. This opens the prospect of trapping and guiding of higher excitons by means of tailor-made dielectric substrates on ultimately small spatial scales. △ Less

Submitted 20 April, 2017; originally announced April 2017.

Journal ref: Phys. Rev. B 96, 045431 (2017)

Long-lived direct and indirect interlayer excitons in van der Waals heterostructures

Authors: Bastian Miller, Alexander Steinhoff, Borja Pano, Frank Jahnke, Alexander Holleitner, Ursula Wurstbauer

Abstract: We investigate the photoluminescence of interlayer excitons in heterostructures consisting of monolayer MoSe2 and WSe2 at low temperatures. Surprisingly, we find a doublet structure for such interlayer excitons. Both peaks exhibit long photoluminescence lifetimes of several ten nanoseconds up to 100 ns at low temperatures, which verifies the interlayer nature of both. The peak energy and linewidth… ▽ More We investigate the photoluminescence of interlayer excitons in heterostructures consisting of monolayer MoSe2 and WSe2 at low temperatures. Surprisingly, we find a doublet structure for such interlayer excitons. Both peaks exhibit long photoluminescence lifetimes of several ten nanoseconds up to 100 ns at low temperatures, which verifies the interlayer nature of both. The peak energy and linewidth of both show unusual temperature and power dependences. In particular, we observe a blue-shift of their emission energy for increasing excitation powers. At a low excitation power and low temperatures, the energetically higher peak shows several spikes. We explain the findings by two sorts of interlayer excitons; one that is indirect in real space but direct in reciprocal space, and the other one being indirect in both spaces. Our results provide fundamental insights into long-lived interlayer states in van der Waals heterostructures with possible bosonic many-body interactions △ Less

Submitted 28 March, 2017; originally announced March 2017.

Comments: 18 pages, 5 figures

Electric-field switchable second-harmonic generation in bilayer MoS$_{2}$ by inversion symmetry breaking

Authors: Julian Klein, Jakob Wierzbowski, Alexander Steinhoff, Matthias Florian, Malte Rösner, Florian Heimbach, Kai Müller, Frank Jahnke, Tim O. Wehling, Jonathan J. Finley, Michael Kaniber

Abstract: We demonstrate pronounced electric-field-induced second-harmonic generation in naturally inversion symmetric 2H stacked bilayer MoS$_{2}$ embedded into microcapacitor devices. By applying strong external electric field perturbations ($|F| = \pm 2.6 MVcm^{-1}$) perpendicular to the basal plane of the crystal we control the inversion symmetry breaking and, hereby, tune the nonlinear conversion effic… ▽ More We demonstrate pronounced electric-field-induced second-harmonic generation in naturally inversion symmetric 2H stacked bilayer MoS$_{2}$ embedded into microcapacitor devices. By applying strong external electric field perturbations ($|F| = \pm 2.6 MVcm^{-1}$) perpendicular to the basal plane of the crystal we control the inversion symmetry breaking and, hereby, tune the nonlinear conversion efficiency. Strong tunability of the nonlinear response is observed throughout the energy range ($E_ω \sim 1.25 eV - 1.47 eV$) probed by measuring the second-harmonic response at $E_{2ω}$, spectrally detuned from both the A- and B-exciton resonances. A 60-fold enhancement of the second-order nonlinear signal is obtained for emission at $E_{2ω} = 2.49 eV$, energetically detuned by $ΔE = E_{2ω} - E_C = -0.26 eV$ from the C-resonance ($E_{C} = 2.75 eV$). The pronounced spectral dependence of the electric-field-induced second-harmonic generation signal reflects the bandstructure and wave function admixture and exhibits particularly strong tunability below the C-resonance, in good agreement with Density Functional Theory calculations. Moreover, we show that the field-induced second-harmonic generation relies on the interlayer coupling in the bilayer. Our findings strongly suggest that the strong tunability of the electric-field-induced second-harmonic generation signal in bilayer transition metal dichalcogenides may find applications in miniaturized electrically switchable nonlinear devices. △ Less

Submitted 16 March, 2017; originally announced March 2017.

Comments: 8 pages, 5 figures

Journal ref: Nano Lett. 17, 392 (2017)

Emission from quantum-dot high- microcavities: transition from spontaneous emission to lasing and the effects of superradiant emitter coupling

Authors: S. Kreinberg, W. W. Chow, J. Wolters, C. Schneider, C. Gies, F. Jahnke, S. Höfling, M. Kamp, S. Reitzenstein

Abstract: Measured and calculated results are presented on the emission properties of a new class of emitters operating in the cavity quantum electrodynamics regime. The structures are based on high-finesse GaAs/AlAs micropillar cavities, each with an active medium consisting of a layer of InGaAs quantum dots and distinguishing feature of having substantial fraction of spontaneous emission channeled into on… ▽ More Measured and calculated results are presented on the emission properties of a new class of emitters operating in the cavity quantum electrodynamics regime. The structures are based on high-finesse GaAs/AlAs micropillar cavities, each with an active medium consisting of a layer of InGaAs quantum dots and distinguishing feature of having substantial fraction of spontaneous emission channeled into one cavity mode (high-beta factor). This paper shows that the usual criterion for lasing with a conventional (low-beta factor) cavity, a sharp nonlinearity in an input-output curve accompanied by noticeable linewidth narrowing, has to be reinforced by the equal-time second-order photon autocorrelation function for confirming lasing. It will also show that the equal-time second-order photon autocorrelation function is useful for recognizing superradiance, a manifestation of the correlations possible in high- microcavities operating with quantum dots. In terms of consolidating the collected data and identifying the physics underlying laser action, both theory and experiment suggest a sole dependence on intracavity photon number. Evidence for this comes from all our measured and calculated data on emission coherence and fluctuation, for devices ranging from LEDs and cavity-enhanced LEDs to lasers, lying on the same two curves: one for linewidth narrowing versus intracavity photon number and the other for g(2)(0) versus intracavity photon number. △ Less

Submitted 13 October, 2016; originally announced October 2016.

Journal ref: Light: Science & Applications 6, e17030 (2017)

Strong light-matter coupling in the presence of lasing

Authors: Christopher Gies, Fabian Gericke, Paul Gartner, Steffen Holzinger, Caspar Hopfmann, Tobias Heindel, Janik Wolters, Christian Schneider, Matthias Florian, Frank Jahnke, Sven. Höfling, Martin Kamp, Stephan Reitzenstein

Abstract: The regime of strong light-matter coupling is typically associated with weak excitation. With current realizations of cavity-QED systems, strong coupling may persevere even at elevated excitation levels sufficient to cross the threshold to lasing. In the presence of stimulated emission, the vacuum-Rabi doublet in the emission spectrum is modified and the established criterion for strong coupling n… ▽ More The regime of strong light-matter coupling is typically associated with weak excitation. With current realizations of cavity-QED systems, strong coupling may persevere even at elevated excitation levels sufficient to cross the threshold to lasing. In the presence of stimulated emission, the vacuum-Rabi doublet in the emission spectrum is modified and the established criterion for strong coupling no longer applies. We provide a generalized criterion for strong coupling and the corresponding emission spectrum, which includes the influence of higher Jaynes-Cummings states. The applicability is demonstrated in a theory-experiment comparison of a few-emitter quantum-dot--micropillar laser as a particular realization of the driven dissipative Jaynes-Cummings model. Furthermore, we address the question if and for which parameters true single-emitter lasing can be achieved, and provide evidence for the coexistence of strong coupling and lasing in our system in the presence of background emitter contributions. △ Less

Submitted 23 May, 2017; v1 submitted 17 June, 2016; originally announced June 2016.

Comments: 12 pages, 9 figures

Journal ref: Phys. Rev. A 96, 023806 (2017)

On thresholdless lasing features in high-$β$ nitride nanobeam cavities: a quantum optical study

Authors: Stefan T. Jagsch, Noelia Vico Triviño, Frederik Lohof, Gordon Callsen, Stefan Kalinowski, Ian M. Rousseau, Roy Barzel, Jean-François Carlin, Frank Jahnke, Raphaël Butté, Christopher Gies, Axel Hoffmann, Nicolas Grandjean, Stephan Reitzenstein

Abstract: Exploring the limits of spontaneous emission coupling is not only one of the central goals in the development of nanolasers, it is also highly relevant regarding future large-scale photonic integration requiring energy-efficient coherent light sources with a small footprint. These studies are accompanied by a vivid debate on how to prove and interpret lasing in the high-$β$ regime. We investigate… ▽ More Exploring the limits of spontaneous emission coupling is not only one of the central goals in the development of nanolasers, it is also highly relevant regarding future large-scale photonic integration requiring energy-efficient coherent light sources with a small footprint. These studies are accompanied by a vivid debate on how to prove and interpret lasing in the high-$β$ regime. We investigate close-to-ideal spontaneous emission coupling in GaN nanobeam lasers grown on silicon. Due to their high optical quality, such nanobeam cavities allow for efficient funneling of spontaneous emission from the quantum well gain material into the laser mode. By performing a comprehensive optical and quantum-optical characterization, supported by microscopic modeling of the nanolasers, we identify high-$β$ lasing at room temperature and show a lasing transition in the absence of a threshold nonlinearity at 156 K. This peculiar characteristic is explained in terms of a temperature and excitation power dependent interplay between 0D and 2D gain contributions. △ Less

Submitted 16 August, 2017; v1 submitted 21 March, 2016; originally announced March 2016.

Journal ref: Nature Communications 9, 564 (2018)

Nonequilibrium Carrier Dynamics in Transition Metal Dichalcogenide Semiconductors

Authors: Alexander Steinhoff, Matthias Florian, Malte Rösner, Michael Lorke, Tim O. Wehling, Christopher Gies, Frank Jahnke

Abstract: When exploring new materials for their potential in (opto)electronic device applications, it is important to understand the role of various carrier interaction and scattering processes. Research on transition metal dichalcogenide (TMD) semiconductors has recently progressed towards the realisation of working devices, which involve light-emitting diodes, nanocavity lasers, and single-photon emitter… ▽ More When exploring new materials for their potential in (opto)electronic device applications, it is important to understand the role of various carrier interaction and scattering processes. Research on transition metal dichalcogenide (TMD) semiconductors has recently progressed towards the realisation of working devices, which involve light-emitting diodes, nanocavity lasers, and single-photon emitters. In these two-dimensional atomically thin semiconductors, the Coulomb interaction is known to be much stronger than in quantum wells of conventional semiconductors like GaAs, as witnessed by the 50 times larger exciton binding energy. The question arises, whether this directly translates into equivalently faster carrier-carrier Coulomb scattering of excited carriers. Here we show that a combination of ab-initio band-structure and many-body theory predicts carrier relaxation on a 50-fs time scale, which is less than an order of magnitude faster than in quantum wells. These scattering times compete with the recently reported sub-ps exciton recombination times, thus making it harder to achieve population inversion and lasing. △ Less

Submitted 11 March, 2016; originally announced March 2016.

Journal ref: 2D Mater. 3 031006 (2016)

A few-emitter solid-state multi-exciton laser

Authors: S. Lichtmannecker, M. Florian, T. Reichert, M. Blauth, M. Bichler, F. Jahnke, J. J. Finley, C. Gies, M. Kaniber

Abstract: We report a combined experimental and theoretical study of non-conventional lasing from higher multi-exciton states of a few quantum dot-photonic crystal nanocavity. We show that the photon output is fed from saturable quantum emitters rather than a non-saturable background despite being rather insensitive to the spectral position of the mode. Although the exciton transitions of each quantum dot a… ▽ More We report a combined experimental and theoretical study of non-conventional lasing from higher multi-exciton states of a few quantum dot-photonic crystal nanocavity. We show that the photon output is fed from saturable quantum emitters rather than a non-saturable background despite being rather insensitive to the spectral position of the mode. Although the exciton transitions of each quantum dot are detuned by up to $160$ cavity linewidths, we observe that strong excitation populates a multitude of closely spaced multi-exciton states, which partly overlap spectrally with the mode. The limited number of emitters is confirmed by a complete saturation of the mode intensity at strong pumping, providing sufficient gain to reach stimulated emission, whilst being accompanied by a distinct lasing threshold. Detailed second-order photon-correlation measurements unambiguously identify the transition to lasing for strong pumping and, most remarkably, reveal super-thermal photon bunching with $g^{(2)}(0)>2$ below lasing threshold. Based on our microscopic theory, a pump-rate dependent $β$-factor $β(P)$ is needed to describe the nanolaser and account for the interplay of multi-exciton transitions in the few-emitter gain medium. Moreover, we theoretically predict that the super-thermal bunching is related to dipole-anticorrelated multi-exciton recombination channels via sub- and super-radiant coupling below and above lasing threshold, respectively. Our results provide new insights into the microscopic light-matter-coupling of spatially separated emitters coupled to a common cavity mode and, thus, provides a complete understanding of stimulated emission in nanolasers with discrete emitters. △ Less

Submitted 3 March, 2016; v1 submitted 12 February, 2016; originally announced February 2016.

Coulomb-assisted cavity feeding in the non-resonant optical emission from a quantum dot

Authors: Matthias Florian, Paul Gartner, Alexander Steinhoff, Christopher Gies, Frank Jahnke

Abstract: Recent experiments have demonstrated that for a quantum dot in an optical resonator off-resonant cavity mode emission can occur even for detunings of the order of 10 meV. We show that Coulomb mediated Auger processes based on additional carriers in delocalized states can facilitate this far off-resonant emission. Using a novel theoretical approach for a non-perturbative treatment of the Auger-assi… ▽ More Recent experiments have demonstrated that for a quantum dot in an optical resonator off-resonant cavity mode emission can occur even for detunings of the order of 10 meV. We show that Coulomb mediated Auger processes based on additional carriers in delocalized states can facilitate this far off-resonant emission. Using a novel theoretical approach for a non-perturbative treatment of the Auger-assisted quantum-dot carrier recombination, we present numerical calculations of the far off-resonant cavity feeding rate and cavity mean photon number confirming efficient coupling at higher densities of carriers in the delocalized states. In comparison to fast Auger-like intraband scattering processes, we find a reduced overall efficiency of Coulomb-mediated interband transitions due the required electron-hole correlations for the recombination processes. △ Less

Submitted 3 October, 2013; v1 submitted 9 August, 2013; originally announced August 2013.

Journal ref: Phys. Rev. B 89, 161302(R) (2014)

Quantum-Confined Stark Effect in polar and nonpolar Wurtzite InN/GaN Heterostructures: Influence on Electronic Structure and Compensation by Coulomb Attraction

Authors: Stefan Barthel, Kolja Schuh, Oliver Marquardt, Tilmann Hickel, Jörg Neugebauer, Frank Jahnke, Gerd Czycholl

Abstract: In this paper we systematically analyze the electronic structures of polar and nonpolar wurtzite-InN/GaN quantum dots and their modification due to the quantum-confined Stark effect caused by intrinsic fields. This is achieved by combining continuum elasticity theory with an empirical tight binding model to describe the elastic and single-particle electronic properties in these nitride systems. Ba… ▽ More In this paper we systematically analyze the electronic structures of polar and nonpolar wurtzite-InN/GaN quantum dots and their modification due to the quantum-confined Stark effect caused by intrinsic fields. This is achieved by combining continuum elasticity theory with an empirical tight binding model to describe the elastic and single-particle electronic properties in these nitride systems. Based on these results, a many-body treatment is used to determine optical absorption spectra. The efficiency of optical transitions depends on the interplay between the Coulomb interaction and the quantum-confined Stark effect. We introduce an effective confinement potential which represents the electronic structure under the influence of the intrinsic polarization fields and calculate the needed strength of Coulomb interaction to diminish the separation of electrons and holes. △ Less

Submitted 11 January, 2013; originally announced January 2013.

Comments: 10 pages, 10 figures, submitted to Phys. Rev. B (2012)

Journal ref: Eur. Phys. J. B (2013) 86: 449

Pronounced Purcell enhancement of spontaneous emission in CdTe/ZnTe quantum dots embedded in micropillar cavities

Authors: Tomasz Jakubczyk, Wojciech Pacuski, Tomasz Smoleński, Andrzej Golnik, Matthias Florian, Frank Jahnke, Carsten Kruse, Detlef Hommel, Piotr Kossacki

Abstract: The coupling of CdTe/ZnTe quantum dot (QD) emission to micropillar cavity eigenmodes in the weak coupling regime is demonstrated. We analyze photoluminescence spectra of QDs embedded in monolithic micropillar cavities based on Bragg mirrors which contain MgSe/ZnTe/MgTe superlattices as low-index material. The pillar emission shows pronounced cavity eigenmodes and their spectral shape is in good ag… ▽ More The coupling of CdTe/ZnTe quantum dot (QD) emission to micropillar cavity eigenmodes in the weak coupling regime is demonstrated. We analyze photoluminescence spectra of QDs embedded in monolithic micropillar cavities based on Bragg mirrors which contain MgSe/ZnTe/MgTe superlattices as low-index material. The pillar emission shows pronounced cavity eigenmodes and their spectral shape is in good agreement with simulations. QD emission in resonance with the cavity mode is shown to be efficiently guided toward the detector and an experimental Purcell enhancement by a factor of 5.7 is determined, confirming theoretical expectations. △ Less

Submitted 29 July, 2012; originally announced July 2012.

Comments: 3 figures

Journal ref: Appl. Phys. Lett. 101, 132105 (2012)

Systematic study of carrier correlations in the electron-hole recombination dynamics of quantum dots

Authors: T. Berstermann, T. Auer, H. Kurtze, M. Schwab, D. R. Yakovlev, M. Bayer, J. Wiersig, C. Gies, F. Jahnke, D. Reuter, A. D. Wieck

Abstract: The ground state carrier dynamics in self-assembled (In,Ga)As/GaAs quantum dots has been studied using time-resolved photoluminescence and transmission. By varying the dot design with respect to confinement and doping, the dynamics is shown to follow in general a non-exponential decay. Only for specific conditions in regard to optical excitation and carrier population, for example, the decay can… ▽ More The ground state carrier dynamics in self-assembled (In,Ga)As/GaAs quantum dots has been studied using time-resolved photoluminescence and transmission. By varying the dot design with respect to confinement and doping, the dynamics is shown to follow in general a non-exponential decay. Only for specific conditions in regard to optical excitation and carrier population, for example, the decay can be well described by a mono-exponential form. For resonant excitation of the ground state transition a strong shortening of the luminescence decay time is observed as compared to the non-resonant case. The results are consistent with a microscopic theory that accounts for deviations from a simple two-level picture. △ Less

Submitted 25 June, 2007; originally announced June 2007.

Comments: 8 pages, 7 figures

Influence of symmetry and Coulomb-correlation effects on the optical properties of nitride quantum dots

Authors: N. Baer, S. Schulz, P. Gartner, S. Schumacher, G. Czycholl, F. Jahnke

Abstract: The electronic and optical properties of self-assembled InN/GaN quantum dots (QDs) are investigated by means of a tight-binding model combined with configuration interaction calculations. Tight-binding single particle wave functions are used as a basis for computing Coulomb and dipole matrix elements. Within this framework, we analyze multi-exciton emission spectra for two different sizes of a l… ▽ More The electronic and optical properties of self-assembled InN/GaN quantum dots (QDs) are investigated by means of a tight-binding model combined with configuration interaction calculations. Tight-binding single particle wave functions are used as a basis for computing Coulomb and dipole matrix elements. Within this framework, we analyze multi-exciton emission spectra for two different sizes of a lens-shaped InN/GaN QD with wurtzite crystal structure. The impact of the symmetry of the involved electron and hole one-particle states on the optical spectra is discussed in detail. Furthermore we show how the characteristic features of the spectra can be interpreted using a simplified Hamiltonian which provides analytical results for the interacting multi-exciton complexes. We predict a vanishing exciton and biexciton ground state emission for small lens-shaped InN/GaN QDs. For larger systems we report a bright ground state emission but with drastically reduced oscillator strengths caused by the quantum confined Stark effect. △ Less

Submitted 14 December, 2006; originally announced December 2006.

Comments: 15 pages, 17 figures

Radiative emission dynamics of quantum dots in a single cavity micropillar

Authors: M. Schwab, H. Kurtze, T. Auer, T. Berstermann, M. Bayer, J. Wiersig, N. Baer, C. Gies, F. Jahnke, J. P. Reithmaier, A. Forchel, M. Benyoucef, P. Michler

Abstract: The light emission of self-assembled (In,Ga)As/GaAs quantum dots embedded in single GaAs-based micropillars has been studied by time-resolved photoluminescence spectroscopy. The altered spontaneous emission is found to be accompanied by a non-exponential decay of the photoluminescence where the decay rate strongly depends on the excitation intensity. A microscopic theory of the quantum dot photo… ▽ More The light emission of self-assembled (In,Ga)As/GaAs quantum dots embedded in single GaAs-based micropillars has been studied by time-resolved photoluminescence spectroscopy. The altered spontaneous emission is found to be accompanied by a non-exponential decay of the photoluminescence where the decay rate strongly depends on the excitation intensity. A microscopic theory of the quantum dot photon emission is used to explain both, the non-exponential decay and its intensity dependence. Also the transition from spontaneous to stimulated emission is studied. △ Less

Submitted 22 June, 2006; v1 submitted 30 May, 2006; originally announced May 2006.

Comments: 9 pages, 6 figures, figure captions corrected

Relaxation properties of the quantum kinetics of carrier-LO-phonon interaction in quantum wells and quantum dots

Authors: Paul Gartner, Jan Seebeck, Frank Jahnke

Abstract: The time evolution of optically excited carriers in semiconductor quantum wells and quantum dots is analyzed for their interaction with LO-phonons. Both the full two-time Green's function formalism and the one-time approximation provided by the generalized Kadanoff-Baym ansatz are considered, in order to compare their description of relaxation processes. It is shown that the two-time quantum kin… ▽ More The time evolution of optically excited carriers in semiconductor quantum wells and quantum dots is analyzed for their interaction with LO-phonons. Both the full two-time Green's function formalism and the one-time approximation provided by the generalized Kadanoff-Baym ansatz are considered, in order to compare their description of relaxation processes. It is shown that the two-time quantum kinetics leads to thermalization in all the examined cases, which is not the case for the one-time approach in the intermediate-coupling regime, even though it provides convergence to a steady state. The thermalization criterion used is the Kubo-Martin-Schwinger condition. △ Less

Submitted 17 February, 2006; v1 submitted 20 October, 2005; originally announced October 2005.

Comments: 7 pages, 8 figures, accepted for publication in Phys. Rev. B

Quantum kinetic theory of phonon-assisted carrier transitions in nitride-based quantum-dot systems

Authors: J. Seebeck, T. R. Nielsen, P. Gartner, F. Jahnke

Abstract: A microscopic theory for the interaction of carriers with LO phonons is used to study the ultrafast carrier dynamics in nitride-based semiconductor quantum dots. It is shown that the efficiency of scattering processes is directly linked to quasi-particle renormalizations. The electronic states of the interacting system are strongly modified by the combined influence of quantum confinement and po… ▽ More A microscopic theory for the interaction of carriers with LO phonons is used to study the ultrafast carrier dynamics in nitride-based semiconductor quantum dots. It is shown that the efficiency of scattering processes is directly linked to quasi-particle renormalizations. The electronic states of the interacting system are strongly modified by the combined influence of quantum confinement and polar coupling. Inherent electrostatic fields, typical for InGaN/GaN quantum dots, do not limit the fast scattering channels. △ Less

Submitted 13 January, 2006; v1 submitted 27 September, 2005; originally announced September 2005.

Comments: 4 pages, 3 figures, accepted for publication in EPJ B

Optical properties of self-organized wurtzite InN/GaN quantum dots: A combined atomistic tight-binding and full configuration interaction calculation

Authors: N. Baer, S. Schulz, S. Schumacher, P. Gartner, G. Czycholl, F. Jahnke

Abstract: In this work we investigate the electronic and optical properties of self-assembled InN/GaN quantum dots. The one-particle states of the low-dimensional heterostructures are provided by a tight-binding model that fully includes the wurtzite crystal structure on an atomistic level. Optical dipole and Coulomb matrix elements are calculated from these one-particle wave functions and serve as an inp… ▽ More In this work we investigate the electronic and optical properties of self-assembled InN/GaN quantum dots. The one-particle states of the low-dimensional heterostructures are provided by a tight-binding model that fully includes the wurtzite crystal structure on an atomistic level. Optical dipole and Coulomb matrix elements are calculated from these one-particle wave functions and serve as an input for full configuration interaction calculations. We present multi-exciton emission spectra and discuss in detail how Coulomb correlations and oscillator strengths are changed by the piezoelectric fields present in the structure. Vanishing exciton and biexciton ground state emission for small lens-shaped dots is predicted. △ Less

Submitted 21 September, 2005; originally announced September 2005.

Comments: 3 pages, 2 figures

Influence of carrier-carrier and carrier-phonon correlations on optical absorption and gain in quantum-dot systems

Authors: M. Lorke, T. R. Nielsen, J. Seebeck, P. Gartner, F. Jahnke

Abstract: A microscopic theory is used to study the optical properties of semiconductor quantum dots. The dephasing of a coherent excitation and line-shifts of the interband transitions due to carrier-carrier Coulomb interaction and carrier-phonon interaction are determined from a quantum kinetic treatment of correlation processes. We investigate the density dependence of both mechanisms and clarify the i… ▽ More A microscopic theory is used to study the optical properties of semiconductor quantum dots. The dephasing of a coherent excitation and line-shifts of the interband transitions due to carrier-carrier Coulomb interaction and carrier-phonon interaction are determined from a quantum kinetic treatment of correlation processes. We investigate the density dependence of both mechanisms and clarify the importance of various dephasing channels involving the localized and delocalized states of the system. △ Less

Submitted 6 February, 2006; v1 submitted 21 September, 2005; originally announced September 2005.

Comments: 12 pages, 10 figures

Coulomb scattering in nitride based self-assembled quantum-dot systems

Authors: Torben R. Nielsen, Paul Gartner, Michael Lorke, Jan Seebeck, Frank Jahnke

Abstract: We study the carrier capture and relaxation due to Coulomb scattering in nitride-based quantum dots on the basis of population kinetics. For the states involved in the scattering processes the combined influence of the quantum-confined Stark effect and many-body renormalizations is taken into account. The charge separation induced by the built-in field has important consequences on the capture a… ▽ More We study the carrier capture and relaxation due to Coulomb scattering in nitride-based quantum dots on the basis of population kinetics. For the states involved in the scattering processes the combined influence of the quantum-confined Stark effect and many-body renormalizations is taken into account. The charge separation induced by the built-in field has important consequences on the capture and relaxation rates. It is shown that its main effect comes through the renormalization of the energies of the states involved in the collisions, and leads to an increase in the scattering efficency. △ Less

Submitted 20 October, 2005; v1 submitted 4 July, 2005; originally announced July 2005.

Comments: 11 pages, 7 figures

Coherent Propagation of Polaritons in Semiconductor Heterostructures: Nonlinear Pulse Transmission in Theory and Experiment

Authors: S. Schumacher, G. Czycholl, F. Jahnke, I. Kudyk, L. Wischmeier, I. Rückmann, T. Voss, J. Gutowski, A. Gust, D. Hommel

Abstract: The influence of coherent optical nonlinearities on polariton propagation effects is studied within a theory-experiment comparison. A novel approach that combines a microscopic treatment of the boundary problem in a sample of finite thickness with excitonic and biexcitonic nonlinearities is introduced. Light-polarization dependent spectral changes are analyzed for single-pulse transmission and p… ▽ More The influence of coherent optical nonlinearities on polariton propagation effects is studied within a theory-experiment comparison. A novel approach that combines a microscopic treatment of the boundary problem in a sample of finite thickness with excitonic and biexcitonic nonlinearities is introduced. Light-polarization dependent spectral changes are analyzed for single-pulse transmission and pump-probe excitation. △ Less

Submitted 5 April, 2005; originally announced April 2005.

Correlated Photon-Pair Emission from a Charged Single Quantum Dot

Authors: S. M. Ulrich, M. Benyoucef, P. Michler, N. Baer, P. Gartner, F. Jahnke, M. Schwab, H. Kurtze, M. Bayer, S. Fafard, Z. Wasilewski

Abstract: The optical creation and recombination of charged biexciton and trion complexes in an (In,Ga)As/GaAs quantum dot is investigated by micro-photoluminescence spectroscopy. Photon cross-correlation measurements demonstrate the temporally correlated decay of charged biexciton and trion states. Our calculations provide strong evidence for radiative decay from the excited trion state which allows for… ▽ More The optical creation and recombination of charged biexciton and trion complexes in an (In,Ga)As/GaAs quantum dot is investigated by micro-photoluminescence spectroscopy. Photon cross-correlation measurements demonstrate the temporally correlated decay of charged biexciton and trion states. Our calculations provide strong evidence for radiative decay from the excited trion state which allows for a deeper insight into the spin configurations and their dynamics in these systems. △ Less

Submitted 16 July, 2004; originally announced July 2004.

Comments: 5 pages, 3 figures, submitted for publication

Polarons in semiconductor quantum-dots and their role in the quantum kinetics of carrier relaxation

Authors: J. Seebeck, T. R. Nielsen, P. Gartner, F. Jahnke

Abstract: While time-dependent perturbation theory shows inefficient carrier-phonon scattering in semiconductor quantum dots, we demonstrate that a quantum kinetic description of carrier-phonon interaction predicts fast carrier capture and relaxation. The considered processes do not fulfill energy conservation in terms of free-carrier energies because polar coupling of localized quantum-dot states strongl… ▽ More While time-dependent perturbation theory shows inefficient carrier-phonon scattering in semiconductor quantum dots, we demonstrate that a quantum kinetic description of carrier-phonon interaction predicts fast carrier capture and relaxation. The considered processes do not fulfill energy conservation in terms of free-carrier energies because polar coupling of localized quantum-dot states strongly modifies this picture. △ Less

Submitted 20 January, 2005; v1 submitted 23 June, 2004; originally announced June 2004.

Comments: 6 pages, 6 figures, accepted for publication in Phys.Rev.B

Coulomb effects in semiconductor quantum dots

Authors: Norman Baer, Paul Gartner, Frank Jahnke

Abstract: Coulomb correlations in the optical spectra of semiconductor quantum dots are investigated using a full-diagonalization approach. The resulting multi-exciton spectra are discussed in terms of the symmetry of the involved states. Characteristic features of the spectra like the nearly equidistantly spaced s-shell emission lines and the approximately constant p-shell transition energies are explain… ▽ More Coulomb correlations in the optical spectra of semiconductor quantum dots are investigated using a full-diagonalization approach. The resulting multi-exciton spectra are discussed in terms of the symmetry of the involved states. Characteristic features of the spectra like the nearly equidistantly spaced s-shell emission lines and the approximately constant p-shell transition energies are explained using simplified Hamiltonians that are derived taking into account the relative importance of various interaction contributions. Comparisons with previous results in the literature and their interpretation are made. △ Less

Submitted 11 June, 2004; originally announced June 2004.

Comments: 7 pages, 2 figures