Emission properties of photonic crystal nanolasers

6 Emission properties of photonic crystal nanolasers S. STRAUF, Stevens Institute of Technology, USA

Abstract: Nanolasers built on a photonic crystal platform have generated tremendous interest in the research community driven by their device application potential for on-chip optical interconnects, beam steering, biochemical sensing and quantum information processing. This chapter reviews the optical emission properties of photonic crystal nanolasers with a focus on devices operating with a few or even an individual semiconductor quantum dot as a gain medium. Concepts underlying the design and operation of these devices, as well as recent experimental results and lasing signatures are discussed. A particular focus will be the critical discussion of the ‘self-tuned gain’ mechanism, which gives rise to quantum-dot mode coupling in the off-resonant case. Key words: nanolaser, photonic crystals, quantum dots, cavity-QED, photon statistics, coherence.

6.1

Introduction

The continuous miniaturization of solid-state lasers has led to optical microcavities with dimensions down to the diffraction limit of the laser light (Vahala, 2003, 2005). When light is confined down to submicron dimensions the underlying light-matter interaction processes become increasingly dominated by cavity-quantum electrodynamic (cavity-QED) effects, ultimately resulting in faster and more energy efficient nanolasers. A large variety of micro- and nanocavities have been developed in the past two decades including microdisks (McCall et al., 1992), microspheres (Sandoghdar et al., 1996), micropillars (Gutbrod et al., 1998; Vahala, 2003), as well as two-dimensional photonic crystal (PC) nanocavities (Painter et al., 1999a). Lasing action has been demonstrated in all of these geometries for both optical pumping and electrical carrier injection. This chapter will focus on the emission properties of two-dimensional PC nanolasers. For a review of lasing in micropillars see Chapter 4 by Reitzenstein and Forchel and for a review of lasing in intentionally deformed microdisks with highly directional output see Chapter 7 by Shim et al. Photonic band gap materials in the visible range were first introduced by Krauss et al. (1996) based on two-dimensional arrays of air holes etched 186 © Woodhead Publishing Limited, 2012

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into a thin slab semiconductor waveguide. To achieve lasing action both quantum wells (QWs) (Painter et al., 1999a; Noda et al., 2001) and quantum dots (QDs) (Yoshie et al., 2002; Hendrickson et al., 2005; Strauf et al., 2006a; Nomura et al., 2010) have been incorporated as an active gain medium in the center of the thin slab waveguide geometry. In order to provide enough gain to overcome the losses, the traditional approach utilized either several layers of QWs or a high density of QDs as an active gain medium. With the advent of ultra-small mode volume PC nanocavities, the development of cavity-QED lasers with few or even a single-QD emitter (Nomura et al., 2010) became possible, which has generated tremendous interest in the research community (Noda, 2006). Device applications of PC nanolasers are driven by the prospects for on-chip optical interconnects and beam steering, biochemical sensing, and on-chip quantum information processing. PC nanolasers have many advantages compared to other small scale lasers such as wavelength-tunable coherent light emission, small footprints and ultra-low lasing thresholds of a few nW optical pump power (Strauf et al., 2006a; Nomura et al., 2010) or injection currents as low as 180 nA (Elis et al., 2011), fast signal modulation in excess of 100 GHz (Altug et al., 2006; Englund et al., 2008), and they can be directly integrated with other optical elements such as waveguides, beam splitters, modulators and detectors, to create planar photonic circuits on a chip. Recently, three-dimensional woodpile PC nanocavities containing self-assembled QDs as active medium have been fabricated using micromanipulation techniques (Aoki et al., 2008; Tandaechanurat et al., 2009) and signatures of lasing have been observed (Strauf, 2011; Tandaechanurat et al., 2011). The omnidirectional reflectivity property of the complete photonic band gap (Yablonovitch, 1987) in these woodpile lasers might even allow creating three-dimensional integrated photonic circuits. Another interesting development is the controlled utilization of disorder in PC cavities and waveguides, which can lead to Anderson localization of light via multiple scattering (Sapienza et al., 2010). Here disorder is understood as a resource rather than a limitation of cavity-QED devices, as further discussed in Chapter 12 by Lodahl. PC nanocavities provide furthermore an ideal test bed to study light-matter interaction and cavity-QED effects of individual QDs in the weak and strong coupling regime, as well as few-body Coulomb interaction effects in these artificial few electron systems. Traditionally the discrete level structure observed in optical experiments on individual QDs has led to the development of QD laser models based on the analogy to atomic laser models with discrete energy levels. Recent experiments, however, have raised the question of whether atomistic models are adequate for QDs since their energy spectra are more complex due to the possibility of multiple excited carriers, that is, exciton manifolds, and their interactions. The QD carriers can

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furthermore interact with the environment in the form of phonon mediated dephasing or with the spatial and energetic close lying continuum of delocalized wetting-layer states, from which the QDs are formed during the growth process. A particular consequence of the various carrier interactions is that nanolasers display a ‘self-tuned gain’ effect, also known as a ‘cavity feeding’ effect, that is, QD-mode coupling even under pronounced offresonance energy conditions between exciton transitions and cavity mode, which was first discovered by Strauf et al. (2006a) and recently reviewed by Strauf and Jahnke (2011). Recent progress in microscopic theories for single-QD nanolasers that go beyond the artificial atom model and tackle the various carrier interaction mechanisms within the QD and with the environment are reviewed by Gies et al. in Chapter 3. The goal of this chapter is to review the nanocavity design and characterization of the emission properties of PC nanolasers with few or individual QDs as active gain medium from an experimental point of view. A particular focus will be the critical discussion of the ‘self-tuned gain’ mechanism that gives rise to QD-mode coupling in the off-resonant case for both QD nanolasers and single-QD cavity-QED systems. A detailed review of recent progress in modeling the cavity feeding effect based on Monte Carlo simulations and a microscopic configuration interaction approach for the excitation manifolds can be found in Chapter 10 by Tarel et al. This chapter is organized as follows. Section 6.2 introduces the design aspect of high-Q and low mode volume PC nanocavities displaying desired mode profiles and emission wavelengths. Section 6.3 describes optical emission properties of individual QDs in PC nanocavities and experimental techniques to establish resonance conditions between emitter and cavity mode. Section 6.4 discusses experimental signatures of lasing for nanolasers with small mode volume and high spontaneous emission coupling efficiencies. Section 6.5 focuses on QD-mode detuning experiments for randomly positioned as well as actively positioned QDs and discusses the self-tuned gain mechanism for small and large detuning regimes.

6.2

Design of photonic crystal (PC) nanocavities

Light-matter interaction strength in nanocavities is characterized by two key figures, the optical mode volume V and the Q-factor, which is defined as Q = ω0U/P, where U is the time averaged energy stored in the cavity, P is the energy loss per cycle, that is, the far-field radiation intensity and ω0 is the frequency of the confined mode. Q is a measure of how long the cavity stores light, that is, the photon hold time τcav = Q/ω0. Since the square of the electrical field strength E is inversely proportional to the mode volume, high-Q factors and ultra-small mode volumes are required to maximize the

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interaction strength between the optical field and, for example, the transition dipole moment of excitons inside a QD. To this end, PC nanocavities display the smallest mode volumes among the various micro and nanocavities created with modern fabrication technologies, with typical values of about V ~ 0.5(λ/n)3. PCs are dielectric materials engineered with periodicity on a length scale comparable to the wavelength of light (John, 1987; Yablonovitch, 1987), which are often achieved by etching a regular array of air holes into a planar slab of dielectric materials. Nanocavities are formed by breaking that periodicity in a controlled way, for example by leaving out one or several air holes. In these 2D slab cavities light is confined in the in-plane direction by distributed Bragg reflection and out of plane by total internal reflection (TIR), thereby creating a quasi-3D optical confinement. This is in contrast to pillar vertical-cavity surfaceemitting laser (VCSEL) resonators, in which Bragg reflection is used in one dimension and TIR in the two other directions, or the whispering gallery modes in microdisks with light confinement in all three dimensions by TIR. Designing an ideal PC nanocavity for a nanolaser supporting only a single mode with high Q and low V at a particular mode wavelength λm with a desired mode profile is nontrivial (Painter et al., 1999b; Srinivasan and Painter, 2002, 2003; Andreani et al., 2005; Benisty, 2005; Englund et al., 2005a; Lalanne et al., 2008). A widespread technique is to start with an initial geometry and calculate the parameters and optical properties using finitedifference time domain (FDTD) simulations, a technique that timesteps Maxwell’s equation over a spatially discretized structure. Further optimization can be achieved by varying the geometry in a parametric search. Using FDTD one can also predict dynamical effects such as the Purcell effect (Xu et al., 2000; Ryu et al., 2003b), coupling to input or output channels (Kim et al., 2004; Yao et al., 2010), or the β-factor of a cavity mode (Vuckovic et al., 1999). The inverse approach starts by choosing a desired mode defined in a Bloch-wave distribution and seeks the matching cavity geometry supporting that mode (Englund et al., 2005a). For a recent review on other techniques to model PC nanocavities, such as the mesoscopic confinement description, which is based on approximating the distributed Bragg reflection field profile with a Fabry-Perot model, see Lalanne et al. (2008).

6.2.1 Comparison of S1- and L3-type cavities This section focuses on optical properties of two-dimensional PC nanocavities made from GaAs. For a review of recent developments of one-dimensional silicon PC nanobeam cavities and their advantages for dense optical integration or applicability for cavity optomechanics see Chapter 13 by Hendrickson et al. A first requirement is that the resulting cavity mode

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(b)

500 nm (c)

500 nm (d)

0

Max

6.1 Nanocavity design. SEM images of a GaAs based square lattice S1-type cavity (a) and a hexagonal lattice L3-type cavity (b). The corresponding FDTD calculations of the electrical field intensity profiles for the S1 and L3 geometry are shown in (c) and in (d), respectively. (Source: Figures are taken from Hennessy et al. (2005).)

wavelength λm must be close to the QD emission wavelength. For a given semiconducting material with refractive index n and slab thickness d one can choose a lattice type, for example a square lattice with a lattice constant a and air hole radius r and only one missing air hole in the center. This geometry is known as the S1-type cavity and a corresponding scanning electron microscope (SEM) image is shown in Fig. 6.1a, which was fabricated by electron beam lithography (EBL) and inductively coupled plasma etching (Hennessy et al., 2005; Strauf et al., 2006a, 2006b). The resulting λm can be calculated in FDTD simulations by placing broadband magnetic dipole sources selectively exciting transversal electromagnetic (TE) modes of the slab. Comparisons to experiments are achieved by mapping the resulting mode wavelength using micro photoluminescence (µ-PL) spectroscopy (Hennessy et al., 2003). Typical parameters for an S1 cavity fabricated into a GaAs slab designed to match λm with InGaAs QDs emitting around 930 nm are n = 3.4, d = 180 nm, a = 300 nm and r/a = 0.38. With an effective mode volume of only 0.1(λ/n)3 the S1 cavity has one of the smallest mode volumes and is on a first glance attractive for making a nanolaser (Han-Youl et al., 2003). The corresponding 3D mode profile can be calculated with FDTD simulations by exciting a single mode at its

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location of maximum field intensity with a spectrally narrow source. As can be seen in Fig. 6.1c, the highest field intensity of the quadrupole mode in the S1 cavity is unfortunately predominantly localized in the air holes, where the QDs cannot be positioned, or close to the air/semiconductor interface. Since QDs in close proximity (∼ 40 nm or smaller) to these interfaces have detrimental optical properties (Wang et al., 2004), the field profile of a regular S1 cavity is thus not very well suited for creating a single-QD nanolaser, in particular since QDs near the interface can introduce absorption losses rather than gain. However, lasing with InGaAsP QWs operating at 1.5 µm can be achieved with S1 cavities, for both the dipole mode and the quadrupole mode, due to the low nonradiative surface recombination rate in this material (Ryu et al., 2002b, 2003a). The first optically pumped PC laser was demonstrated by Painter et al. (1999a) utilizing a hexagonal lattice with one air hole missing (H1 or L1 cavity) and QWs as the active material. An electrically pumped PC laser based on a similar H1 cavity was demonstrated by Park et al. (2004). In contrast to the S1 cavity, the H1 cavity localizes the light better within the dielectric material but the Q-factor in these lasers was limited to Q = 250 (Q = 2500) for the optically (electrically) pumped H1 laser. Akahane et al. (2003a) realized that the abrupt interface between the defect region and the DBR mirror region causes large losses reducing the Q-factor, which becomes more serious with decreasing cavity size. In order to reach higher Q/V values they suggested that one has to confine the light in a more gentle way. One way is to create a nanocavity by taking out 3 holes in a row, a geometry known as L3-type cavity (Akahane et al., 2003a). As an example, Fig. 6.1b shows an SEM image of an L3-type cavity and Fig. 6.1d the corresponding mode profile. In contrast to the S1 cavity the mode profile of the L3 cavity avoids the air/semiconductor interface and is thus better suited to realize a single-QD nanolaser as it provides higher Q/V values and avoids coupling to lossy QDs near the interface. Optimizing nanocavities to achieve high Q-factors can be done by modeling  the cavity field by a set of plane waves with corresponding wave vectors k using Fourier analysis of the mode profile (Srinivasan and Painter, 2002, 2003). Momentum components of the mode profile with k values smaller than kTIR = 2π/λm are responsible for losses since they don’t satisfy the TIR conditions in the vertical direction and escape the GaAs slab as radiation into the far field. These losses can dramatically reduce the Q-factor of the cavity and special care must be taken to eliminate those k components. To this end one can shift and shrink adjacent holes. Theoretical Q-factors approaching 100 000 can be achieved by shifting holes for about 15–18% of a and shrinking about 15–25% of r (Akahane et al., 2003a; Andreani et al., 2005). Improved out-coupling into the vertical direction while maintaining a high Q can be achieved by superposing a grating-like structure into the

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arrangement of several hole rows adjacent to the cavity area (Toishi et al., 2009; Tran et al., 2009; Portalupi et al., 2010). This skilful fine-tuning of the resonator geometry to maximize Q is somewhat analogous to the fine-tuning of a musical instrument utilizing standing sound waves, such as a violin. Experimentally Q-factors of 45 000 (62 000) have been observed for Si-based L3-type (L7-type) cavities probed around 1.5 µm, which are close to the theoretical values (Akahane et al., 2003b; Portalupi et al., 2010). In contrast, GaAs based L3-type cavities are limited to experimental Q values around 20 000 in the 930 nm wavelength region where the InAs QDs emit (Strauf et al., 2006a; Nomura et al., 2010), while the designed Q is about 100 000 (Andreani et al., 2005). The discrepancy between the fabricated device and the intended design can be caused by fabrication errors such as hole size or lattice constant variations and rough side walls (Yoshie et al., 2001; Borselli et al., 2005), but also by absorption losses at etched interfaces or by intrinsic absorption effects in the GaAs slab. In particular, it was found that GaAs has at 980 nm about six-fold higher absorption losses for TE modes as compared to 1460 nm, which was assigned predominantly to bulk (defect related) absorption and partly to the Urbach tail and surface state absorption (Michael et al., 2007). This effect generally limits the achievable Q/V values for the case of single-QD nanolasers made from self-assembled InAs QDs.

6.2.2 Ultra-high Q cavities Another principle to design nanocavities is based on a photonic double heterostructure confining light within a 1D line defect acting as a waveguide and resulting in experimental Q-factors up to 6 × 105 measured in Si at a mode volume of 1.2(λ/n)3 (Song et al., 2005) and 8 × 105 for slightly larger V and local width modulation (Kuramochi et al., 2006). Similar high Q-factors up to 7 × 105 have also been achieved in the GaAs material system (Weidner et al., 2006; Combrie et al., 2008). Here light is confined due to the mode-gap effect along the waveguide rather than the photonic band gap effect. The mode gap is created at the interface where two PC lattices with different lattice constant are combined. This is somewhat analogous to the lattice mismatch in heteroepitaxy when two crystalline materials with different lattice constants grow on top of each other, which can result in electronic confinement (Kroemer, 2001). Further improvement can be achieved by tapering the hole radius slowly over the length of the waveguide, which yields theoretical Q values up to 8 × 107 (Kwon et al., 2008). These heterostructure nanocavities allow the trapping and delaying of photons in Si for up to 1 ns (Tanabe et al., 2007a) and can perform all-optical switching (Tanabe et al., 2005, 2007b; Nozaki et al., 2010), but they have not yet been applied to the GaAs material system for the purpose of creating a single-QD nanolaser.

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Quality factors in excess of 108 can be implemented on a chip by using silica microtoroids that display relatively large mode volumes (Armani et al., 2003). At these extremely high Q-factors light is trapped for more than 40 ns, which enables the new field of cavity optomechanics, reporting exciting new phenomena such as on-chip frequency comb generation (Del’Haye et al., 2007) and cooling of mechanical resonator modes by dynamic backaction (Kippenberg and Vahala, 2008; Schliesser and Kippenberg, 2010). Although generally very promising, integration with epitaxially grown QDs has not yet been demonstrated.

6.2.3 Lithographic tuning of PC nanocavity modes In order to study cavity-QED effects between a single QD and a nanocavity mode or to realize lasing one must tune the cavity mode energy into resonance with the emitter spectrum. This section discusses lithographic tuning, that is, tuning by changing the design of the PC pattern in the fabrication process. Detailed discussions about how one can fine-tune a QD and mode emission into resonance after a device was fabricated will be discussed in Section 6.3. The resonant mode wavelength λm of the PC nanocavity can be tuned over a wavelength range of several hundred nanometers by scaling the lattice constant a and filling fraction r/a accordingly. In the experiment the cavity mode spectrum can be mapped out after fabrication by placing an internal broadband emitter into the thin slab PC. To this end, one can integrate a layer with high-density QD material in the molecular beam epitaxy growth process in such a way that the layer is located at the center of the thin slab GaAs membrane, that is, at the field anti-node of the cavity mode in the growth direction. Optical excitation of the QDs with a µ-PL setup reveals the mode spectrum, which is decorated by the broadband emission of the QD material under moderate to high pump power settings. Typical mode spectra of L3, L7 and L11 cavities are shown in Fig. 6.2. As can be seen, the L3-type cavity has only one cavity mode in the energy range while the larger L7-type cavity shows two modes and the L11 three modes. Three different tuning scales are observed in the experiments. Large shifts in λm, on the order of 25–40 nm, can be achieved by changing the lattice constant by Δa = 10 nm. Moderate tuning of λm by about 10 nm can be achieved by altering the filling fraction of an initial pattern by about 6%, and final, subtle shifts of about Δλm 1–2 nm can be produced by varying the electron dose in the electron beam lithography, which in effect causes an increase in filling fraction. The lithographic tuning of a nanocavity mode is summarized in Fig. 6.3a, which plots λm for an S1 type cavity mode as a function of filling

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μ-PL intensity (a.u.)

Ey –

e1

0

+

L3

e1

e2

L7

L11 1.285

e1

e2 1.290

e3

1.295 Energy (eV)

1.300

1.305

6.2 Nanocavity mode spectra for triangular-lattice PC structures with 3, 7 and 11 missing holes in the Γ-K direction, denoted as L3, L7, L11 cavities, supporting one, two and three non-degenerate modes polarized along the y direction. The insets show the corresponding Ey field profiles calculated by FDTD simulations. (Source: Figure is taken from Choi et al. (2007).)

a = 290 nm a = 300 nm a = 310 nm

Wavelength (nm)

960 940 920 900 880 0.36

0.38

0.4 r/a

0.42

(b)

T = 4.5 K P = 5 μW

Intensity (arb. units)

(a)

900

920 940 960 Wavelength (nm)

980

6.3 (a) Tuning map for an S1 nanocavity showing mode wavelength λm as a function of filling fraction. (Source: Figure is taken from Hennessy et al. (2003).) (b) Normalized μ-PL spectra for a series of nine L3-type nanolasers with varying lattice constant and filling fraction effectively covering the p-shell and s-shell emission from the embedded InAs QDs. (Source: Figure is taken from Strauf et al. (2006a).)

fraction for three different lattice constants. Each point in this resonant mode map represents the average wavelength of several devices defined with the same pattern with an error bar of 1 nm. Figure 6.3b demonstrates lithographic tuning of an L3 laser mode over the entire emission range of the embedded QD material.

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6.3

195

Optical emission properties of quantum dots (QDs) in PC nanocavities

6.3.1 Controlling spontaneous emission The controlled engineering of spontaneous emission (SE) rates and the corresponding far-field emission profiles of optical cavities can lead to photonic devices with largely improved performance. For a laser, SE from the active material that does not couple to the lasing mode constitutes an unwanted loss of excitation in the gain medium that raises the lasing threshold. As a result, there is a strong motivation to achieve control over SE and inhibit it when it is not desired or, alternatively, concentrate it into useful forms. According to Fermi’s golden rule, the SE rate of a semiconductor QD depends on the transition matrix element between initial and final states for electron-hole pair recombination as well as the available density of states of the electromagnetic field at the position of the emitter. While the transition matrix element is largely determined by the particular geometry and choice of the QD material, the density of states can be modified by placing the QD in an optical cavity (Yokoyama et al., 1992). In the weak coupling regime between emitter and mode the SE rate is determined by the Purcell effect (Purcell et al., 1946; Haroche and Kleppner, 1989). By placing the emitter at a cavity field anti-node (node) or by tuning the cavity into (out of) resonance with the emitter frequency, the SE rate can be either enhanced or inhibited. The altered SE due to modifications of the photonic density of states and the electric-field strength at the position c of the emitter can be quantified by the Purcell factor FP = τ 0sp / τ sp , which describes the SE lifetime changes of a two-level emitter in the presence of c 0 the cavity ( sp ) in comparison to the vacuum SE lifetime ( sp ). For a singlefrequency emitter tuned in resonance with the cavity mode and positioned at the cavity-field anti-node, one finds FP =

3 λm Q 4π 2 n3 V

[6.1]

where n is the refractive index of the cavity material. Note that for a QD embedded in a host material with refractive index n, its radiative rate is n times what it would be in vacuum (Thraenhardt et al., 2002). Furthermore note that this relationship is valid for dielectric cavities while plasmonic nanocavities require a modified treatment (Koenderink, 2010). If the interaction strength of the coupled cavity and emitter system overcomes the dissipative losses, then it is in the so-called strong coupling regime where the light-matter interaction becomes reversible (non-perturbative) and the strongly coupled emitter modifies the cavity spectrum itself. If the

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discrete transition of a single emitter is coupled to a single mode of the electromagnetic field (selected by the cavity), then the Jaynes–Cummings model describes the appearance of new dressed states belonging to the coupled system. The periodic energy exchange between an excited emitter and the empty cavity is known as vacuum-field Rabi oscillations and can be observed as vacuum-field Rabi splitting in the anti-crossing of the cavity and emitter resonances. Much of the initial work in this field was accomplished in the 1980s with Rydberg atoms displaying SE enhancement (Goy et al., 1983) and inhibition (Hulet et al., 1985). Strong coupling has been demonstrated via normal-mode splitting with many atoms (Meschede et al., 1985) and, after systematically increasing the cavity quality and lowering the mode volume, vacuum-field Rabi splitting (Thompson et al., 1992) and vacuum-field Rabi oscillations (Haroche and Kleppner, 1989) have been achieved with a single atom. For semiconductor systems in the perturbative cavity-QED regime, enhanced SE from GaAs QWs (Yokoyama et al., 1990) and InGaAs QDs (Graham et al., 1999) in monolithic VCSEL microcavities has been demonstrated. In the non-perturbative cavity-QED regime, normal-mode coupling of QW excitons in planar VCSEL microcavities has been realized (Weisbuch et al., 1992) and the transition from strong to weak coupling in connection with excitonic nonlinearities has been studied (Khitrova et al., 1999). Vacuumfield Rabi splitting of a single-QD emitter in a three-dimensional nanocavity recently became possible with pillar VCSELs (Reithmaier et al., 2004) and PC resonators (Yoshie et al., 2004). In principle, semiconductor QDs are governed by the same cavity-QED physics (described by the Jaynes–Cummings Hamiltonian) as atomic systems. However, the technical implementations are quite different, which has major consequences. i. QDs are permanently embedded in a solid state and are a priori scalable. However, two self-assembled QDs are not alike, since they nucleate at random positions over the wafer and they vary in their transition frequencies due to size, strain, and composition fluctuations (Petroff et al., 2001). This constitutes severe technological challenges for the realization of single-QD nanolaser and other deterministic and scalable cavityQED devices. ii. QDs interact with their environment via carrier–phonon interaction and Coulomb interaction. This leads to particular dephasing properties, like the pure dephasing due to LA phonons (Borri et al., 2001) with a characteristic signature in the lineshape of optical QD transitions and a specific temperature dependence (Besombes et al., 2001; Muljarov and Zimmermann, 2004). Furthermore, the QD dephasing is excitation

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dependent and linked to energy renormalizations that additionally influence the optical properties (Lorke et al., 2006). iii. While control of the mode wavelength λm in the relatively large cavities used for experiments with individual atoms can be carried out, for example by actuating one of the mirrors, tuning the mode frequency of PC nanocavities is not so straightforward. Thus research in QD cavityQED has to deal with techniques to control the spectral resonance condition between λm and the exciton emission wavelength λe as well as the spatial position of the emitter with respect to the electric field maximum at the anti-node E max. Deviations from the ideal resonance condition and emitter position lead to modifications of Equation [6.1], which can be described as 2 3λ 3m Q λm FP = 4 π 2 n3 V λ 2m + 4Q2 ( e −

m

)2

 2 E (r )  2 Emax

[6.2]

  where E(r ) is the electric field at the QD location (Gayral and Gerard, 2003).

6.3.2 Frequency tuning of randomly positioned QDs A first generation of QD cavity-QED experiments was carried out with selfassembled QDs that nucleate randomly out of their wetting layer (Petroff et al., 2001). After fabrication of a cavity the relative position between field maxima and QDs is fixed and not necessarily optimized. Weak coupling signatures such as inhibition and enhancement of SE from a single QD have been observed by achieving the spectral resonance condition via temperature tuning (Michler et al., 2000b; Santori et al., 2001; Pelton et al., 2002; Englund et al., 2005b; Gevaux et al., 2006; Michler, 2009; Strauf et al., 2006a). While λm is relatively unaffected by temperature, λe is pinned to the semiconductor band gap, which becomes smaller with increasing temperature, such that relative energy shifts of about 3–5 meV can be realized. To achieve strong coupling the QD must be located at the anti-node of the cavity field with a precision usually better than 50 nm. Thus the corresponding generation of strong coupling experiments relied on random chance and, depending on QD density, often required the measurement of hundreds of devices before finding signatures of strong coupling in temperature tuning experiments (Reithmaier et al., 2004; Yoshie et al., 2004; Peter et al., 2005; Khitrova et al., 2006; Michler, 2009). A drawback of the temperature tuning technique in QD cavity-QED is that the exciton dephasing strongly increases with temperature. For an InGaAs QD, a reduction of the dephasing time from 630 ps at 7K down to 11 ps at 75K has been demonstrated in the weak excitation regime (Borri

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et al., 2001). The correspondingly reduced coherence length in the photon emission is a severe problem for quantum information processing schemes relying on efficient two-photon interference at a beam splitter (Hong et al., 1987; Knill et al., 2001; Santori et al., 2002). It was found that λm of PC nanocavities and microdisks is strongly affected by residual gas deposition (adsorption) at cryogenic temperatures (Strauf et al., 2006b). This effect can be utilized to achieve the QD-mode energy resonance condition at 4K, thereby avoiding strong QD dephasing at higher temperatures. Figure 6.4 shows a tuning experiment carried out at 4K using residual gas deposition. On resonance the data exhibit a clear intensity enhancement but anti-crossing between QD and mode was not observed, which is indicative of a QD in the weak coupling regime. Similar experiments with controlled Xe gas deposition have been performed in L3 cavities that reached into the strong coupling regime (Mosor et al., 2005). In order to confirm that light originates from a single quantum emitter one typically records the photon antibunching signature, that is, the secondorder or intensity autocorrelation function that depends on the average number of photon pairs, consisting of photons appearing at the times t and t + τ according to,

)=

1.3179

Energy (eV)

1.3176

1.3173

b† ( t ) b† ( t +

) b (t + ) b (t ) b† ( t ) b ( t ) b† ( t + ) b ( t + )

[6.3]

(a)

(b)

Mode QD

150 nW, 4.2 K

QD

1.3170

1.0 Coincidences (arb. units)

2 g( ) (t

0.5 0.35

Mode –600 –400 –200 0 200 400 Detuning energy (μeV)

0.0

–25 0 25 Delay time (ns)

6.4 (a) Energy tuning experiment based on residual gas deposition to achieve the spectral resonance condition. The insets show the corresponding μ-PL spectra. (b) Photon antibunching signature of a single QD in resonance with an L3-type PC mode. (Source: Figure is taken from Strauf and Jahnke (2011).)

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where b† and b are photon creation and annihilation operators for the cavity mode. For the special case of zero delay time τ = 0, the simultaneous appearance of two photons is analyzed. In a stationary situation, g(2) does not depend on t and the τ = 0 result reflects characteristic properties of the photon statistics. For thermal light with photon bunching in the count statistics, the normalized stationary autocorrelation function leads to g(2)(τ = 0) = 2 while for coherent radiation obeying Poisson statistics one finds g(2)(τ = 0) = 1. For light emission from an individual two-level system g(2)(τ = 0) drops down to zero, indicating that simultaneous emission of two photons does not occur. Antibunching was first observed by Kimble et al. (1977) for single atoms. Later antibunching was demonstrated for other kinds of quantum emitters, such as individual molecules (Basche et al., 1992), semiconductor QDs (Michler et al., 2000a), nitrogen vacancy centers (Kurtsiefer et al., 2000), acceptor-bound excitons (Strauf et al., 2002) and single-walled carbon nanotubes (Högele et al., 2008). Figure 6.4b shows the corresponding antibunching signature for a QD-exciton tuned into resonance with an L3-type cavity mode under 82 MHz pulsed laser excitation. The normalized zero delay time peak area of 0.35 is indicative of single-photon emission from a single QD in the PC nanolaser. Figure 6.5a compares the pump power dependence of exciton emission from a single QD located in an unprocessed area of the GaAs slab (bulk) with the intensity of a single QD near the cavity center. For the S1 or L3-type PC nanocavities, which are not optimized for far-field emission (Toishi et al., 2009; Tran et al., 2009; Portalupi et al., 2010), one typically observes a 10–12-fold intensity enhancement for single QDs tuned into spectral resonance, as demonstrated in Fig. 6.5a. The QD on spectral resonance with the cavity appears also to saturate at a higher pump power (Happ et al., 2002). Interestingly, the enhanced emission on resonance is not directly connected to the Purcell effect, that is, a faster lifetime of the exciton emission. The corresponding measurements of the single-QD lifetime are shown in Fig. 6.5b. This lack of QD-cavity mode coupling is caused by the spatial misalignment between the QD and the areas of high field intensity of the mode, which is particularly pronounced at the very low InGaAs QD areal densities of about 1–5 × 108 cm−2 typically used in these devices (Strauf et al., 2006a, 2006b). The measurements in Fig. 6.5b demonstrate that the photonic band gap effect can lead to three-fold inhibition of the exciton transition rate in an S1 cavity and up to one order of magnitude inhibition in an L3-type cavity, while the emission efficiency into the far-field can be simultaneously enhanced by an order of magnitude. Concurrent SE inhibition and enhanced light extraction was also demonstrated in PC

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6.5 (a) Single QD intensity versus pump power for a single QD located in an unprocessed area of the GaAs slab (i) and a QD near the cavity center in an L3-type device (ii). (b) Lifetime measurements for single QDs in three different dielectric environments recorded by time-correlated photon counting with an avalanche photodiode. The system resolution was optimized to 200 ps. Data are recorded at 4K. (Source: Figure is taken from Strauf and Jahnke (2011).)

nanocavities with QWs as the active medium (Fujita et al., 2005; Noda et al., 2007). Thus one cannot directly interpret the measured single-QD intensity enhancement as being caused by SE enhancement and one has to carefully distinguish contributions from SE enhancement and light extraction when analyzing time integrated data (Badolato et al., 2005; Ota et al., 2008). In contrast, the SE lifetime is a direct measure for the Purcell effect, and SE enhancement has often been observed in the time domain in nanocavities with higher QD densities between 109 and 1010 cm−2, where the chances of spatial alignment are significantly increased (Englund et al., 2005b; Kuroda et al., 2008; Laucht et al., 2009). For a recent review on SE control using PC nanocavities see Noda et al. (2007). Another ex situ tuning technique is digital etching, which is based on gradually reducing the surface oxide layer of the GaAs slab in citric acid, resulting in removal of less than a nm of material per etch step (Hennessy et al., 2005). With this technique λm can be tuned up to 80 nm in steps of 2 nm. Additionally, removal of the surface oxide increases the Q-factor by about 30%. Regrowth of the surface oxide can be avoided by removing the PC chip out of the liquid, drying with nitrogen gas and immediate capping with a glass slide and loading into the cryostat.

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6.3.3 Actively positioned QDs To drastically improve the chances of finding pronounced light-matter interaction, mode maxima and QD position must be actively controlled. Recently, a sophisticated controlled coupling method was introduced based on strain mediated stacking (Xie et al., 1995) of several QDs combined with atomic force microscopy tracing to locate the strain sites of a buried seed QD, followed by subsequent fabrication of PC cavities with respect to these markers (Badolato et al., 2005; Hennessy et al., 2007). When QD and mode are tuned into spectral resonance a characteristic anti-crossing signature is observed in these actively positioned samples, confirming that emitter and mode are in the strong coupling regime. While signatures of weak and strong coupling can now be observed in a deterministic way there are also drawbacks of this particular positioning technique. The tracer QDs contribute partly to the optical emission and the quality factor of the L3 mode does significantly degrade since the topmost tracer QD creates a hill at the center of the L3 mode with lateral dimension of 100–300 nm. One way to overcome these limitations is based on locating the position of the QD emission at 4K with a confocal microscope and tracing this position with respect to markers on the surface, thereby avoiding complications caused by a tracer stack of QDs. At first glance this approach seems limited as a typical laser spot size is about 1 micron and carrier diffusion can further blur the QD location. However, the position of an object can be determined with much higher precision (experimentally 10 nm accuracy was demonstrated (Thon et al., 2009)) than the lateral separation of two nearby objects, which is diffraction limited. Using this far-field optical lithography technique, signatures of single-QD strong coupling have been demonstrated for micropillars (Dousse et al., 2008; Suffczynski et al., 2009) and for PC nanocavities (Thon et al., 2009). In all of these approaches QDs still nucleate at random positions in the epitaxial growth, such that no lateral order exists between QDs. Therefore, spatial arrangement of several QDs with respect to a cavity mode is still an open issue and formation of quantum networks (Imamoglu et al., 1999) seems quite challenging. To overcome these limitations direct control of the QD nucleation sites was carried out with patterned templates in the epitaxial growth (Gerardot et al., 2002; Mano et al., 2002; Mehta et al., 2010) and first signatures of single-QD weak coupling have been recently reported for micropillars (Schneider et al., 2009). For a recent review see Dalacu et al. (2010). While these demonstrations of deterministic spatial QD-mode coupling are very promising, a single-QD nanolaser based on active positioning was not yet demonstrated and more efforts are necessary to create robust, deterministic and scalable QD cavity-QED systems with several actively positioned QDs.

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6.4

Signatures of lasing in PC nanolasers

Conventional lasers operate in the weak coupling regime. In lasers with cavity dimensions strongly exceeding the light wavelength, such as edge-emitting laser diodes, fiber lasers, gas lasers, etc., the free spectral range of cavity modes is small and a large number of modes interact with the active material. As a result, a large fraction of the SE from the active material is coupled into nonlasing modes. This is quantified with the β factor, which is defined as the ratio of the SE rate into the laser mode over the total SE rate. In conventional lasers, this ratio is typically of the order of 106. The input/output curves of conventional lasers reveal a sharp onset of stimulated emission at the laser threshold. According to a rate equation analysis (Yokoyama and Brorson, 1989; Bjoerk and Yamamoto, 1991), the intensity jump at the threshold scales with 1/β. In the limit of β = 1 the intensity increases linearly, which has led to the term of a ‘thresholdless laser’. This limit can be realized in a microcavity with high cavity Q and low mode volume by utilizing the Purcell effect: the SE into the laser mode is enhanced while SE into nonlasing modes is inhibited. As a result, the nonlinear kink input/output curve becomes less pronounced, which raises the question about the verification of the onset of lasing in high-β nanolasers. The transition from SE into the lasing regime for conventional low-β lasers can be experimentally characterized by several key signatures, which are, however, to some extent interrelated and not altered independently: • • • • •

Observation of a sharp onset in the input/output curve. Schawlow–Townes linewidth narrowing of the cavity mode. Coherence time/length build up. Modification of the probability of two-photon coincidences described by a transition of g(2)(τ = 0) from 2 (SE) to 1 (coherent state). Mode competition.

To illustrate the behavior of conventional low-β lasers three typical signatures of lasing are shown in Fig. 6.6, which have been recorded for a commercial VCSEL diode operating at room temperature based on a multilayer QW gain medium. First, a sharp onset in the input/output curve in Fig. 6.6a occurs at a threshold current of 36 mA. Second, many transverse modes compete for the gain since the β factor of planar VCSELs is about 0.001, such that the output spectrum exhibits a transition from multimode to singlemode emission above threshold, until ultimately the mode with the lowest loss dominates the spectrum, as shown in the insets of Fig. 6.6b. And third, Fig. 6.6b demonstrates the transition in the intensity autocorrelation function g(2)(τ = 0), as defined in Equation [6.3], from a value of 2 (representing thermal light) to 1 (characteristic for coherent light) at the laser threshold.

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6.6 Three signatures of the phase transition into the lasing regime for a VCSEL diode (β = 0.001) recorded at room temperature. (a) Sharp onset in the input/output curve. (b) g(2)(τ = 0) measured as the bunching height, spectrally filtered at the lasing mode and plotted as a function of pump current normalized to the threshold current Ith = 36 mA. The inset shows the corresponding mode spectra taken at a drive current of 32.5 mA (left), 34.8 mA (middle) and at Ith (right), with the lasing mode highlighted by the shading. (Source: Figure is taken from Choi et al. (2007).)

In theory g(2)(τ = 0) should display a value of two for all pump powers below threshold. However, in the experiment one typically observes a deviation below threshold from the expected value of thermal light, which is a measurement artifact in these Hanbury–Brown and Twiss type experiments carried out with single-photon counting avalanche photodiodes (APDs). Due to the typical APD timing jitter of about 600 ps (optimized 150 ps) g(2)(τ) measurements of short coherence times (τcoh < 100 ps) emitted by a laser driven at or below threshold are hampered. The origin is that due to the finite time resolution also pair events for τ > 0 are detected and that g(2)(τ) decays to unity on a scale of the coherence time. More detailed explanations are found in Choi et al. (2007) and Ulrich et al. (2007). To overcome these measurement limitations, an alternative method based on a streak camera in single-photon counting mode has recently been introduced (Wiersig et al., 2009) that provides a track record of the individual photon-emission events as well as a strongly increased time resolution of 2 ps. In these experiments, the decay of g(2) from 2 to 1 for smaller β, the broadening of the transition region for microlasers with larger β (Wiersig et al., 2009), as well as the time evolution of the g(2)(t, τ) function was directly measured (Assmann et al., 2010). For a detailed review of these findings see Chapter 5 by Assmann and Bayer.

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6.7 (a) Linewidth narrowing and (b) input/output curve for an L3-type PC nanolaser with a very low QD density of 8 QDs per μm2. The solid lines are fits to the rate equation based on atomic models. (c) and (d) Same for a non-lasing device located on the same wafer showing monotonic behavior without a kink. (e) g(2)(τ = 0) traces similar to Fig. 6.6b but recorded for PC nanocavities with 11, 7 and 3 missing air holes. The solid lines are calculated using the standard photon number probability distribution. The lasing transition region is shown in a light gray shade and broadens as the corresponding β factor becomes higher. (Source: Figure 6.7a and 6.7b is taken from Strauf et al. (2006a) and Fig. 6.7e from Choi et al. (2007).)

In contrast to the results shown in Fig. 6.6, for high-β nanocavity lasers, which are strongly influenced by cavity-QED effects, the signatures of lasing are less pronounced, making their detection in the experiment more challenging. The laser transition in a system with a small number of optical modes available for emission processes is analogous to a phase transition in the limit of a small system size (Rice and Carmichael, 1994), that is, a sharp and clearly defined transition does not exist anymore. For example, an L3-type PC cavity with a β parameter approaching unity confines only one (polarization non-degenerate) mode in the entire QD emission spectrum. Thus the typical signature of mode selection/competition shown in Fig. 6.6b for the microcavity laser does not apply anymore to a singlemode nanolaser. Figure 6.7 shows three signatures of lasing for an L3-type nanocavity with a very low density of QDs (8 QDs per µm2) as active medium. The s-shaped input/output curve in Fig. 6.7b displays a rather soft turn-on behavior and when fitted to the atomistic rate equation model (solid lines) yields β = 0.85, which is near unity where a vanishing lasing threshold behavior is expected (Yokoyama et al., 1992). For comparison, FDTD simulations for a nanocavity with a mode profile similar to the L3-type cavity yields β = 0.87, assuming a SE bandwidth of 25 nm, which is reduced from unity due to the presence of leaky modes in the thin slab PC design (Vuckovic et al., 1999).

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A soft threshold behavior has been also seen, for example, in Hendrickson et al. (2005), Ulrich et al. (2007), and Xie et al. (2007). Care must be taken when using the kink in the input/output curves as the only signature for QD lasing, since several QD configurations (exciton, biexciton, etc.) can contribute to the total photon production rate. For example, a kink can also occur in the input/output trace when the biexciton state contribution takes over at higher pump powers and the exciton contribution quenches (Ritter et al., 2010). Thus in order to demonstrate lasing for high-β nanolasers it is essential to consider the interplay of several characteristic signatures for lasing. Another signature is the linewidth of the lasing mode,which varies for a conventional laser inversely with the output power according to the Schawlow–Townes limit, but remains well above this limit for small mode volume semiconductor lasers. In particular, it was predicted that VCSEL with QWs as a gain medium display a pronounced plateau in the linewidth data around the threshold regime (Bjoerk et al., 1992). The underlying cause of the plateau formation is the coupling between intensity and phase noise (gain refractive-index coupling), which is of particular importance in semiconductor lasers with QD gain.To quantify this linewidth broadening Henry (1982) introduced the linewidth enhancement factor α, which increases the linewidth at and above threshold according to 1 + α2. The experimental data for an L3-type QD nanolaser display such a plateau in the linewidth narrowing trace, although at a reduced magnitude, as shown in Fig. 6.7a. Note that to compare the linewidth narrowing with the intensity data in Fig. 6.7b the linewidth is plotted here as a function of pump power (instead of output power). A similar plateau was recently observed in three-dimensional PC QD nanolaser (Tandaechanurat et al., 2011). In contrast, the L3-type cavity shown in Figs. 6.7c and 6.7d display only a monotonous linewidth narrowing and a linear input/output curve until saturation, indicating that this device does not reach into the lasing regime. Such behavior corresponds to operation in the light-emitting-diode regime where there is a constant cavity loss but an increase in gain (Coldren and Corzine, 1995). To some extent it is surprising that QD lasers show such a pronounced linewidth re-broadening around threshold since simple models predict a small if not zero α factor due to their discrete density of states. However, several experiments have shown that α factors of QD lasers can have a comparable magnitude to those of QDs with a continuous density of states (Melnik et al., 2006). One explanation put forward is that the finite carrier capture time and the plasma effect in QDs might be responsible for this behavior (Melnik et al., 2006). Another explanation is that the presence of excited QD states results in an α factor above one (Vazques et al., 2006). Recent calculations of the α factor on the basis of a microscopic semiconductor theory (Lorke et al., 2007) also predicted large values for a homogeneously broadened QD system, which are strongly reduced by inhomogeneous broadening. While the physical origin seems still under debate, it appears that a signature for

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lasing in PC QD nanolaser is not just monotonous linewidth narrowing but rather the observation of a characteristic plateau in the linewidth narrowing trace, as shown in Fig. 6.7a, which is caused by the nonlinear interplay between carrier density and photon density. A direct signature for lasing action in high-β nanolasers is the transition from SE into the coherent light state, which is shown in Fig. 6.7e for three different devices with 11, 7 and 3 missing holes in the Γ-K direction forming L11, L7 and L3-type nanocavities, respectively. The larger the line defect the more modes can fit into the nanocavity. In particular, three modes into the L11, two modes into the L7 and one mode into the L3 device, corresponding to ideal β factors of 0.33, 0.5 and 1 from a simple mode counting argument when leaky modes are not taken into consideration. The solid line fits are calculated using the standard photon number probability distribution leading also to the stated β factors (Choi et al., 2007). With this set of devices the phase transition into the lasing regime was approached systematically. As the nominal number of modes in the laser decreases the transition to the coherent state broadens and occurs over a much wider range of pump powers (shaded region in Fig. 6.7e). Similarly, the maximum value and the slopes of the g(2)(t, τ) data decrease suggesting reduced photon number fluctuations in high-β nanolasers (Ulrich et al., 2007). In contrast, the lasing transition of the VCSEL diode shown in Fig. 6.6 occurs in a normalized pump region that is orders of magnitude narrower than those of the QD PC nanolasers, making the transition easily detectable in the experiment. The lasing threshold values extracted from the high-β device shown in Fig. 6.7 are of the order of 100 nW and, when corrected for the actual absorbed pump power, are as low as 4 nW (49 mW/cm2) (Strauf et al., 2006a). These values are remarkably low, in fact orders of magnitude lower than observed in any macroscopic laser, VCSEL, microdisk laser (Michler et al., 2000c; Srinivasan et al., 2006; Xie et al., 2007), micropillar laser (Ulrich et al., 2007; Bajoni et al., 2008; Reitzenstein et al., 2008; Wiersig et al., 2009) or PC nanocavities (Painter et al., 1999a; Loncar et al., 2002; Ryu et al., 2002a; Yoshie et al., 2002; Ryu et al., 2004; Hendrickson et al., 2005; Altug et al., 2006; Nomura et al., 2006; Englund et al., 2007; Martinez et al., 2009), and comparable to recent work on PC nanolasers (Nomura et al., 2009).

6.5

Detuning experiments: the quest for the gain mechanism

6.5.1 First generation experiments utilizing randomly positioned QDs Assuming ideal two-level-like QDs that exhibit spectrally sharp exciton resonances and a statistical distribution of the QD emission over about 50 nm

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due to unavoidable QD size, strain, and composition fluctuations, it is rather surprising that simultaneous spectral and spatial coupling as described in Equation [6.2] seems to occur in the L3-type nanocavities of Fig. 6.7 with only about 8 ± 2 QDs per µm2 (Strauf et al., 2006a). As the chances that even one of these QDs is both spectrally and spatially coupled to the lasing mode are below 1%, the question arises what does actually lase in these QD nanolasers? To address this important question several related experiments are discussed in this section. The results will also challenge the analogy between the single atom laser and the single-QD nanolaser. Unlike atoms, the energy spectra of QDs are more complex. A striking difference between cavity-QED experiments with atoms and with QDs is that in the latter case efficient light-matter coupling is also observed if the sharp exciton transitions of the s-shell are spectrally detuned from the cavity mode wavelength. This was first recognized in experiments on L3-type nanocavities with an ultra-low areal QD density (Strauf et al., 2006a), for which the lasing signatures have been discussed in Section 6.4. Figure 6.8 shows the corresponding mode spectra, where a few sharp exciton recombination lines are visible at pump powers well below the lasing threshold (1 nW). When spectrally filtered, these sharp lines displayed pronounced photon antibunching signatures, both for on- and off-resonance conditions, confirming that they originate from a single quantum emitter (see Fig. 6.4b). Temperature tuning experiments revealed that typically none of these sharp QD transitions was by chance in spectral resonance with the cavity mode at 4K. In addition, it was often observed (>100 devices) that s-shell excitons display pronounced inhibition of SE up to an order of magnitude (Strauf et al., 2006a) as shown in Figs. 6.5b and 6.8c, indicating that these QDs are spatially off resonance, although spectrally in the vicinity of the mode emission. In contrast, the broad background visible on the log scale in Fig. 6.8c was found to be strongly Purcell enhanced, if spectrally on resonance with the cavity mode, with lifetimes down to 145 ps. The vanishing probability for simultaneous spatial and spectral coupling of the sharp s-shell exciton lines and the cavity mode is also supported by FDTD calculations as shown in Fig. 6.1d. For the L3 mode, areas with a field intensity I equal or larger than one tenth of its maximum Imax cover about 0.1 µm2. Hence less than one QD within the mode volume has efficient coupling due to a favorable spatial position. Even for the cavity field intensity reduced to one hundredth of its maximum value (areal coverage of 0.9 µm2) only 6–8 QDs are expected within this region. These QDs are not necessarily spectrally near the cavity mode emission. What can be concluded from these experiments is that lasing in PC cavities with ultra-low QD densities occurs regardless of the spectral resonance condition between the sharp s-shell exciton emission lines and the cavity mode

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6.8 Mode spectra of an L3-type nanolaser recorded at pump powers below the lasing threshold (a) and around the lasing threshold (b). Several individual QD s-shell exciton recombination lines are visible in the vicinity of the cavity mode that saturate close to the lasing threshold (∼100 nW). Data are recorded at 4K. (Source: Figure is taken from Strauf and Jahnke (2011).) (c) Spectra of randomly positioned QDs in another L3-type cavity shown on a log scale (right y-axis) with the corresponding lifetimes (left y-axis) recorded at several spectral positions. While individual QDs show pronounced inhibition of SE, the region of the broad background that is spectrally on resonance with the cavity mode is strongly Purcell enhanced (145 ps), under various spectral detuning conditions. (Source: Figure is taken from Strauf et al. (2006a).)

spectrum, while individual QD lines show pronounced photon antibunching, and the broad background is Purcell enhanced and thus clearly coupled, which is in strong contrast to atomic systems (Strauf et al., 2006a; Strauf, 2010a). On the other hand, the localized QD states are required to excite the cavity mode emission since devices fabricated in a region without QDs (but with the InAs wetting layer) show no cavity mode emission at all, that is, lasing is not sustained by the energetic tail from the wetting layer alone. It was furthermore found that lasers operating at shorter wavelengths near the QD p-shell centered at 920 nm, despite having similar Q-factors, exhibit substantially higher lasing threshold powers, as shown in Fig. 6.9. To tune the cavity mode over the entire emission range of the QD gain medium 12 PC nanolasers with different lattice constants were fabricated. The threshold power under optical pumping, as estimated from the kink in

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the input/output curve (red squares) or the onset of linewidth narrowing (blue circles), increased by over an order of magnitude when the mode is sustained by emission from the QD p-shell. This behavior can be understood by the fact that the QD p-shell becomes only significantly occupied after the s-shell states have been saturated. Xie et al. (2007) used a microdisk with a larger mode volume and a QD density of 50 QDs µm2, where the coupling probability is significantly increased. A pronounced reduction in lasing thresholds by a factor of three demonstrates that the s-shell exciton of a single QD, when tuned into resonance with the cavity mode, can indeed influence the lasing characteristics. However, in that work it was also found that lasing signatures of the cavity mode are present under off-resonant conditions similar to earlier findings (Strauf et al., 2006a). Almost identical results, that is, a three-fold reduced lasing threshold and sustained lasing under off-resonance conditions was also reported by other groups using micropillars with about 200 QDs per cavity (Reitzenstein et al., 2008) and L3-type PC cavities with an ultra-low density of less than one randomly positioned QD per cavity (Nomura et al., 2009). In addition it was shown that the sharp s-shell emission displayed pronounced photon antibunching, under off-resonance (Strauf et al., 2006a; Xie et al., 2007) and when tuned into spectral resonance with the mode (Reitzenstein et al., 2008; Nomura et al., 2009). While observation of antibunching from a sharp s-shell emission is an unambiguous proof that the light emission stems from an individual quantum

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emitter with an anharmonic energy spectrum, it is not a priori clear that lasing itself is sustained by that s-shell exciton. Since lasing occurs already under off-resonance conditions, tuning of the s-shell exciton into resonance with the cavity mode creates a higher photon number inside the mode, thereby reducing the lasing threshold. Furthermore, the cavity mode was shown to display photon antibunching even when the s-shell exciton was spectrally detuned from the cavity mode (Press et al., 2007; Englund et al., 2010). Thus in order to reveal the gain mechanism in present single-QD nanolaser experiments, it is essential to understand the cavity mode feeding under off-resonance conditions.

6.5.2 Second generation experiments utilizing actively positioned QDs A second generation of experiments pioneered by Imamoglu et al. (1999) utilized actively positioned QDs (as described in Section 6.3.3) to achieve stronger cavity-QED effects as only one QD is present inside the device, and that QD is near or at the field maximum of the cavity mode (Badolato et al., 2005; Hennessy et al., 2007; Dousse et al., 2008; Thon et al., 2009). See also Chapter 10 by Tarel et al. Applying the gas-deposition technique described in Section 6.3.2 it was found that the cavity mode is visible for spectral detuning from the sharp s-shell exciton resonance in the energy range from −20 to +4 nm, a remarkably broad energy range (Hennessy et al., 2007). Furthermore, it was demonstrated with second-order photon cross-correlation measurements that the photon emission from the cavity mode and the s-shell excitons is anti-correlated at the level of single quanta, proving that the mode is driven solely by the QD, despite the large energy mismatch up to 20 nm (Hennessy et al., 2007). However, the mode emission itself did not display photon antibunching under these very large detunings. Similar signatures in photon cross-correlation experiments between s-shell excitons and the cavity mode were also found in micropillars with a larger density of randomly positioned QDs. In this case, resonant excitation of the p-shell of a particular QD was used to suppress the contribution from other QD emitters. As a result the cavity mode revealed photon antibunching when detuned by up to 0.7 meV from the s-shell exciton emission (Press et al., 2007). Recently Englund et al. (2010) demonstrated similar signatures for a single QD that was 1.2 nm detuned from an L3-type PC cavity mode, and also showed Rabi splitting when tuned into resonance. This cavity emission for arbitrary detuning clearly challenges the standard picture that QDs may be fully described as artificial atoms with discrete energy levels. On the flip side it also offers new possibilities for resonant single-QD spectroscopy and background-free photon statistics as recently

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demonstrated by several groups (Ates et al., 2009; Englund et al., 2010; Ulhaq et al., 2010). It was also proposed as a resource for advanced solid-state single-photon sources (Strauf et al., 2007; Strauf, 2010b) since the common spectral diffusion of a quantum emitter (Frantsuzov et al., 2008) is effectively canceled out by the ‘self-tuned’ coupling effect (Auffeves et al., 2009).

6.5.3 Microscopic mechanism for non-resonant QD-mode coupling The question remains what microscopic mechanism could be responsible for the non-resonant QD-cavity coupling. Most likely more than one mechanism contributes, depending on the magnitude of the detuning. Strauf et al. (2006a) originally proposed both a contribution from phonon mediated coupling, which is visible in the broadened spectral lineshape of s-shell excitons, and a contribution from a spectrally broad and Purcell enhanced quasi-continuum created by multiexcitons inside the s-shell and p-shell that are coupled to extended wetting-layer states, which was called for short a ‘self-tuned gain mechanism’, however, without further quantifying it. Calculations of the carrier–phonon interaction show that the exciton lineshape broadens in the base up to 2 meV at 7K due to quadratic coupling of acoustic phonons (Muljarov and Zimmermann, 2004), in agreement with experiments (Besombes et al., 2001). With respect to QDs inside cavities, it was predicted that dephasing for nonzero detuning leads to a qualitative change in the cavity spectrum as dephasing shifts the emission intensity towards the cavity frequency. This intensity shifting effect was predicted for detuning up to 1 meV (Naesby et al., 2008; Auffeves et al., 2009). Further theoretical work on phonon-assisted cavity feeding predicted QD-mode coupling up to 2 meV at liquid helium temperatures, based on phonon scattering rates calculated by the independent Boson Hamiltonian (Hohenester et al., 2009) or the Schrieffer–Wolff perturbation approach (Hohenester, 2010). In time resolved experiments on deterministically positioned QDs it was found for detuning energies up to three mode linewidth (about 2–3 meV) that the emission within the cavity mode arises from the spectrally closest s-shell exciton state, since the mode follows the exact SE lifetime of the detuned exciton state, where either the neutral, charged or the biexciton states were probed, strongly suggesting phonon induced dephasing as the coupling mechanism (Suffczynski et al., 2009). The phonon mediated dephasing process was recently further confirmed in temperature dependent experiments under resonant s-shell excitation in micropillar cavities (Ates et al., 2009) and in L3-type PC cavities (Englund et al., 2010). In light of the theoretical predictions and experimental observations it seems thus

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2X

X–

M Etch 0

Intensity (a.u.)

Etch 3

Etch 5

Etch 7 935

940 945 950 Wavelength (nm)

955

6.10 Spectra of an actively positioned QD with respect to an L3-type cavity mode labeled M (shaded gray) for various detuning energies achieved by digital etching of the surface oxide in citric acid. The number of etch steps are indicated. (Source: Figure is taken from Hennessy et al. (2005).)

plausible that strong dephasing causes exciton-mode coupling for detuning energies up to about 3 meV. On the other hand, spectral detuning and cross-correlation experiments demonstrate that single-photon feeding into the cavity mode occurs for detuning energies up to 20 nm (27 meV), which are about an order of magnitude larger (Hennessy et al., 2007), as illustrated in the experiment in Fig. 6.10. These observations can thus not be explained by pure dephasing or by the phonon-assisted mechanisms alone. Kaniber et al. (2008) carried out similar experiments on L3-type cavities with randomly positioned QDs and found cross-correlation signatures up to detuning of 19 meV. It was furthermore shown that shifting the mode away from the s-shell exciton transition enhances the purity of the photon antibunching signature, while the broad background coupled to the cavity mode does not display photon antibunching, that is, there is multi-photon emission. As a possible explanation it has been suggested that charged excitons decay into a continuum of final states mediated by the photo-induced hybridization effect with the wetting layer, that is, by photon-induced shake-up processes that are enhanced by the dielectric environment of the cavity, although no further evidence is provided (Kaniber et al., 2008). Recently Winger et al. (2009) revealed a more detailed picture of the microscopic origin of the broad single-QD background emission and regarded it

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as an intrinsic feature in QD cavity-QED systems with multiexciton states (see also Chapter 10 by Tarel et al.). Excitation (or capture) of carriers into the p- and d-shell leads to a series of manifolds. Using a configuration interaction approach, multiexciton eigenstates up to four electron-hole pairs were calculated. Since the higher orbital angular momentum states are subject to strong hybridization with the wetting layer, as was found earlier by Karrai et al. (2004), these multiexciton transitions merge into an excited-state quasi-continuum distributed over an about 15 meV broad energy range. Possible optical transitions between the various broadband exciton manifolds, separated roughly by the band gap energy, leads to the omnipresent single-QD background emission. As this background also appears in the energy range where the sharp s-shell excitons emit, the model provides a microscopic picture of the above-discussed ‘self-tuned gain’ mechanism. In support of this non-resonant feeding mechanism, Laucht et al. (2010) have recently demonstrated that temporal correlation between the multiexciton background and the cavity mode exists, whereas emission from the sharp s-shell exciton occurred a few ns delayed and uncorrelated from the mode emission. This is also consistent with earlier second-order photon cross-correlation signatures which show that mode and s-shell exciton emissions do not occur simultaneously (Hennessy et al., 2007; Winger et al., 2009). Interestingly, the intensity of the cavity mode itself is slightly nonlinear in the input/output curve of Winger et al. (2009), resembling the essential features of high-β lasing as presented for example in Fig. 6.7. However, the authors interpret the behavior of the cavity to first follow the s-shell exciton at low pump powers, which is linear, then the biexciton at intermediate pump powers, which is superlinear, and finally the mode follows the p-shell, which is linear up to saturation. Moreover, the work by Winger et al. (2009) challenges the view that the transition of the cavity mode emission from pronounced photon bunching at lower pump powers to Poissonian emission at higher pump powers is solely caused by QD lasing. Using a Monte Carlo random walk of excitation and photon-emission events within the multiexciton configurations of a single QD, they calculate that the cavity mode displays pronounced photon bunching at lower pump powers and a Poissonian emission at higher pump powers, which matches the observation in their experiments. See also Chapter 10 by Tarel et al. Ritter et al. (2010) have recently introduced a microscopic description of a single-QD nanolaser going beyond the atomistic models, which also takes into account multiexciton configurations together with excitation-induced dephasing and energy renormalization (see also Gies et al. in Chapter 3). This model also reveals a nonlinear input/output curve and attributes the change in slope to the photon contribution from s-shell exciton, biexciton and p-shell. In contrast to the random walk model, the theory by Ritter

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et al. (2010) traces the experimentally observed transition from photon antibunching to bunching with increasing pumping back to the appearance of additional emission channels while the occurrence of Poissonian emission for large pump rates is traced back to stimulated photon emission. The conclusion from recent experiments (Laucht et al., 2010) is that a sharp s-shell exciton line and a broad emission background feeding a detuned cavity resonance do not coexist. For strong excitation, a large number of various multiexciton states, possibly hybridizing also with wettinglayer states (Karrai et al.,2004), provide a broad emission background that can efficiently feed the cavity mode, while in the same single-QD nanolaser system for weak excitation, discrete s- and p-shell exciton resonances appear. Note that in the strong excitation regime the s-states likely remain occupied with excitons. However, they recombine at different energies due to their Coulomb interaction with other excited carriers. Since the experimental spectra collect data from many repeated excitation cycles, the results represent statistical averages over many realizations of various multiexciton states. These results challenge the interpretation that the s-shell exciton alone lases and provides the gain in present high-β nanolasers. On the other hand, it can be argued that if there is spatially only one QD within the cavity mode, then 100% of the gain can be provided by that single QD through the ‘selftuned gain mechanism’ (Strauf et al., 2006a; Winger et al., 2009), involving both the contributions from the single-QD excitations and their coupling to the delocalized wetting-layer states. As the single-QD background effect is not solely from the QD alone, one should understand the single-QD gain medium, that is, the single-QD nanolasers, as a hybrid system between QD and QW, in stark contrast to atomic systems.

6.6

Conclusions

Recent progress in design and fabrication of ultra-small mode volume PC nanocavities has led to the development of cavity-QED lasers operating with a few or even a single QD as the active gain medium. These nanolasers feature ultra-low lasing thresholds and ultra-fast device operation due to a strongly enhanced SE coupling into the lasing mode. Typical lasing signatures such as the nonlinear kink in the light input/output curve are less pronounced in PC nanolasers, making it essential to consider other lasing signatures as well, such as the transition from SE into the coherent light state visible in the second-order coherence function and the characteristic plateau in the linewidth narrowing trace implying a large α factor for QDs. Recent experiments also indicate that the idealistic picture of QDs as artificial atoms and the often utilized atomistic laser models are insufficient to describe optical emission properties of nanolasers. A prominent example is the ‘self-tuned

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gain’ effect, also known as the ‘cavity feeding’ effect, which was first observed in 2006. Recent results show that strong dephasing can cause exciton-mode coupling for small detuning energies up to 3 meV, while coupling at larger detunings up to 27 meV is attributed to the presence of various multiexciton charge configuration states of the exciton manifold as well as their coupling to the surrounding delocalized wetting-layer states. While the self-tuned gain effect can in some cases have detrimental consequences for cavity-QED experiments with individual QDs, it can also be understood as a resource for making better single-photon sources with reduced spectral diffusion or for building more efficient nanolasers under relaxed fabrication constraints. Future research in this field is expected to aim at achieving more atomiclike systems in order to reduce the cavity feeding effect, either by using QDs with stronger confinement or by finding ways to grow QDs without wetting layers, such as GaN QDs. Another way could be to utilize top-down fabricated vertical QDs, which are known to emit brightly (Kalliakos et al., 2007), and also allow positioning of several QDs deterministically with respect to a waveguide or cavity mode. Progress is also expected on the theoretical side to improve microscopic theories for single-QD nanolasers that tackle the various carrier interaction mechanisms within the QD and with the environment, ultimately replacing atomistic laser models with static configurations, which are of limited applicability for QD nanolasers. PC nanolaser research remains both challenging and exciting and future progress will reveal deeper insights into the light-matter interaction mechanisms of multiexciton gain media, while real word applications such as on-chip optical interconnects with ultra-low electrical power consumption come within reach.

6.7

Acknowledgments

The author would like to acknowledge the partial financial support provided through the National Science Foundation (NSF-ECCS CAREER Award #1053537). I would like to thank my collaborators Dirk Bouwmeester, Pierre Petroff, Matthew Rakher, Evelyn Hu, Larry Coldren, Kevin Hennessy, Yong-Seok Choi, Antonio Badolato, and Lucio Claudio Andreani, for the joint work on nanolasers. I would also like to thank Frank Jahnke, Peter Michler, and Glenn Solomon for stimulating discussions.

6.8

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