High-Q design of semiconductor-based ultrasmall ... - OSA Publishing

High-Q design of semiconductor-based ultrasmall photonic crystal nanocavity Masahiro Nomura1*, Katsuaki Tanabe1, Satoshi Iwamoto1,2, and Yasuhiko Arakawa1,2 1

Institute for Nano Quantum Information Electronics, The University of Tokyo, Tokyo 153-8505, Japan 2 Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan *[email protected]

Abstract: We report a high-Q design for a semiconductor-based twodimensional zero-cell photonic crystal (PhC) nanocavity with a small mode volume. The optimization of displacements of hexagonal-lattice air holes in the Γ-M direction, in addition to the Γ-K direction, resulted in a cavity quality factor Q of 2.8 × 105 sustaining the small modal volume of 0.23(λ0/n)3. The momentum space consideration of out-of-plane radiation loss showed that the optimization of air hole displacements in both the inplane x and y directions reduced FT components in the leaky region along the kx and ky axes, respectively. This high-Q cavity design is applicable to Si and GaAs semiconductor materials. © 2010 Optical Society of America OCIS codes: (050.5298) Photonic crystals; (270.5580) Quantum electrodynamics; (130.3990) Micro-optical devices.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

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Received 1 Mar 2010; revised 26 Mar 2010; accepted 26 Mar 2010; published 1 Apr 2010

12 April 2010 / Vol. 18, No. 8 / OPTICS EXPRESS 8144

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1. Introduction Strong light-matter interaction in solid state materials have been investigated because of their unique physical properties based on cavity quantum electrodynamics (C-QED) and their potential applications such as quantum information processing. One of the most promising light-matter coupling systems is a coupled system of a single semiconductor quantum dot (QD) [1] and a semiconductor nanocavity [2]. Thus far, semiconductor microcavities with a coupled single QD have been fabricated using microdisk, micropillar, and photonic crystal (PhC) structures. In these C-QED systems, vacuum Rabi splitting in the strong-coupling regime [3–6] and highly efficient lasing in the weak-coupling regime [7–15] have been observed. The key physical parameters of such systems are the cavity quality factor (Q) and the modal volume (Vm). In the weak-coupling regime, the artificial control of the radiation of photons, which is known as the Purcell effect, is determined by Q/Vm. In the strong-coupling regime, the light-matter coupling strength g is determined by Vm-1/2, which must be larger than γQD/4 and γcav/4, where γQD and γcav are the spectral linewidths of a QD and cavity mode, respectively. Therefore, a high-Q cavity with smaller Vm is a better C-QED system. Recently, it has been theoretically predicted [16] and experimentally shown [17] that laser oscillations can occur even in the strong-coupling regime in excellent C-QED systems. In such solid state C-QED systems, fragile quantum physical interactions between a single QD and cavity field is hindered by several factors including the phonon-induced emission process, pure dephasing, and manifolds of the single QD near the cavity resonance [18–20]. Therefore, in C-QED experiments, a cavity design with both a high Q and a small Vm is essential to enhance the light-matter interaction in the C-QED systems. A small Vm is the most important parameter in practical C-QED experiments, where the advantage of a high-Q cavity is lost owing to the broad spectral linewidths of QDs. Many types of PhC cavity designs have been reported [21–28]. A hexagonal-lattice zerocell PhC cavity, referred to as an H0-type or point-shift cavity, with air hole displacement in only the Γ-K direction has been proposed [21,29]. These designs have theoretical cavity Q values that exceed 1 × 105 with a small Vm of 0.26-0.29(λ0/n)3, where λ0 and n are the wavelength in vacuum and the refractive index of the material ( = 3.4), respectively. A #124861 - $15.00 USD

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square-lattice zero-cell PhC structure has been reported with a smaller Vm value of 0.21(λ0/n)3, but a moderate theoretical cavity Q value of 4200 [27]. The main focus of this study is to develop a cavity design that has both a small Vm and a high Q. In this paper, we report a high-Q design of a two-dimensional (2D) hexagonal-lattice zerocell PhC nanocavity with a small Vm for semiconductor materials, such as Si and GaAs, with a refractive index of 3.4. Careful tuning of several air hole positions in both the Γ-K and Γ-M high-symmetry directions reduces the coupling of the cavity mode’s dominant Fourier components with radiating components and reduces the out-of-plane radiation loss. The tuning of the air hole positions increases Q to a maximum value of 2.8 × 105 in zero-cell PhC nanocavities with a small Vm of 0.23(λ0/n)3. This result indicates the importance of the optimization of the air hole positions in the Γ-M direction that has not been adopted in previous works. These high-Q nanocavities can be used to produce excellent solid-state CQED systems with strong light–matter coupling that will enable further C-QED experiments to reach the multiquantum regime with high pumping. 2. Design of high-Q H0-type PhC nanocavity We investigate an air-clad 2D PhC nanocavity with a hexagonal air hole lattice [Fig. 1(a)]. In a 2D PhC slab, the distributed Bragg reflection due to the surrounding PhC structure results in the in-plane confinement of photons. On the other hand, standard wave guiding through total internal reflection determines the radiation loss into the out-of-plane direction. The energymomentum dispersion relationship for this structure is k// = (ω/c)2 defines a light cone [light blue region in Fig. 1(b)], where k// is the in-plane momentum component, ω is the angular frequency, and c is the speed of light in air. Modes that lie within the light cone of air have small |k//| and radiate vertically as leaky modes. Thus, designing cavities to reduce the out-ofplane radiation loss is the fundamental guideline [30]. 2.1 2D PhC H0-type nanocavity The studied optical cavity, which we refer to as the H0-type cavity in this work, comprises a defect created by shifting several air holes in a 2D PhC slab structure without removing any air holes as shown in Fig. 1(a). This cavity has much smaller Vm than defect-type cavities produced via the removal of air holes. The cavity center is located at a C2v,σv symmetry point in the hexagonal lattice. The shifts of on-axis air holes Six (i = 1–3) in the x-direction (Γ-K) and of Sjy (j = 1, 2) in the y-direction (Γ-M) are optimized in this study and are defined in Fig. 1(a). The band structure is calculated using the three-dimensional (3D) plane-wave expansion method. In the simulation, the thickness and n of the slab are 0.6a (where a is the period of the lattice) and 3.4, respectively. The radius of the air hole r is 0.26a. The results indicate that the photonic band gap (PBG) region, colored light green in Fig. 1(b), ranges from 0.252 to 0.302 in units of normalized frequency (a/λ). The normalized frequencies of the fundamental modes of the cavities investigated in this study are in the range from 0.285 to 0.29, which lies within the PBG as shown by the orange line in Fig. 1(b). 3D finite-difference time domain (FDTD) simulations are performed to obtain the frequency and profile of the fundamental mode of the example of an H0-type PhC nanocavity. The spatial distributions of Ex, Ey and Hz components are shown in Figs. 1(c)– 1(e), respectively. The main electric field component is Ey, but Ex has comparable amplitude (~50%). The Ex component shows very strong localization in the cavity at two maxima. The calculated Vm slightly differs with a change in Six,y, but does not drastically change from a value of 0.24(λ0/n)3.

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Fig. 1. (a) Schematic diagram of hexagonal-PhC H0-type nanocavity (top view). Band structure of fundamental TE-like mode calculated using 3D plane-wave expansion method. The air light cone (blue), PBG (light green), and typical cavity mode (orange) are shown. (c-e) 3D FDTD simulated profiles of Ex, Ey and Hz components for fundamental mode.

2.2 Optimization of Q by shifting multiple air holes The FDTD simulations are performed by optimizing each shift in the order of S1x, S1y, S2x, S2y, and S3x so that Q is a maximum. First, S1x = 0.14a gives the largest Q of ~1.1 × 105. S1x is then fixed at 0.14a and the value of S1y is changed between 0 and 0.1a to search for the local maximal value of Q as shown in Fig. 2(a). The cavity Q is sensitive to the air-hole displacement in the y-direction. In the series of calculations, S1y = 0.04a gives the maximum value of 2.1 × 105 and Q decreases at larger S1y. The same series of calculations are performed for the tuning parameters S2x, S2y, and S3x. The maximum value of Q of 2.8 × 105 is found for S1x = 0.14a, S2x = 0, S3x = 0.06a, S1y = 0.04a, and S2y = 0.02a (Cavity C). This value is more than twice the previously reported value for this type of cavity [11,21]. It is significant that Vm increased by only 6% as compared with the initial design (Cavity A), whereas Q is increased by more than twice. This is the most important point of cavity design in C-QED physics. In this study, the displacements of only on-axis air holes are optimized, and further improvement of Q may be possible with the additional optimization for off-axis air holes.

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Fig. 2. (a) Calculated cavity Q of fundamental mode with different S1y values with fixed S1x of 0.14a. The value of Q increases from 1.2 × 105 to 2.1 × 105. (b) Calculated cavity Q of fundamental mode with different S3x values for S1x = 0.14a, S2x = 0, S1y = 0.04a, and S2y = 0.02a. Q increases from 2.2 × 105 to 2.8 × 105. The mode volume Vm of Cavity C increased by only 6% as compared to that of Cavity A, whereas Q increased by more than twice.

3. Momentum space consideration of out-of-plane radiation loss In this section, we discuss the change in the out-of-plane radiation loss in the optimization of the positions of the surrounding air holes. The light confined in the very small cavity consists of a significant number of plane wave components with various k values. When |k//| lies within the range of 0–2π/λ, the plane wave can escape to the air cladding because the total internal reflection condition is not fulfilled. On the other hand, the plane wave with |k//| larger than 2π/λ is strongly confined to the cavity. Therefore, we can investigate the out-of-plane radiation loss of the cavity mode by calculating the 2D spatial Fourier transformation (FT) of the in-plane electric field of the mode in the slab. Here, three cavities, referred to as Cavities A, B, and C and indicated in Figs. 2(a) and 2(b), are compared. The design of each cavity is presented in Table 1. Cavity A is the initial design with a simple shift of the two closest air holes to create the cavity. Cavity B has an additional optimized shift of the air holes in the ydirection. The design of Cavity C is optimized for both the x- and y-directions within i = 1–3 and j = 1–2, respectively. Table 1. Cavity Q and shift of air holes for each type of cavity. Cavity A B C

Q ( × 105) 1.2 2.2 2.8

S1x 0.14a 0.14a 0.14a

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Figures 3(a)–3(c) show the calculated 2D spatial FT spectra of Ex for Cavities A, B, and C at the center of the slab. The FT spectra show Ex to be primarily composed of momentum components located around four M points. The intensity increases, indicating a stronger confinement of light, as the cavities are more optimized. Figures 3(e), 3(f) show magnified FT spectra of Figs. 3(a)–3(c). The white circles indicate the cross section of the surface of the light cone for the cavity mode’s value of ω. The electric field components inside the circle have small |k//| that result in out-of-plane radiation loss. The ratio of integrated FT components inside the air light cone to the total of FT components at the middle of the slab for each cavity is 0.165%, 0.148%, and 0.146%. A better cavity has a reasonably smaller fraction of leaky modes.

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2

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Fig. 3. (a–c) 2D spatial FT spectra of Ex for Cavities A, B, and C at center of slab. (d–f) Magnified FT spectra of (a–c), respectively. The white circles indicate the cross section of the surface of the light cone for the cavity mode’s ω. The electric field components inside the circle are leaky components that result in out-of-plane radiation.

Here, we individually discuss the changes in the out-of-plane radiation loss with the optimization of the displacements of air holes along the x and y axes. Figures 4(a) and 4(b) show the FT components of Ex for Cavities A (green), B (blue), and C (red) on the kx and ky axes, respectively. The gray region indicates the interior of the light cone, which corresponds to the leaky region. Cavity A has a large FT component at kx = 0 and large integrated FT components in the leaky region. Cavity B has additional air-hole displacement along the y axis and obviously smaller FT components along the ky direction. The drastic reduction in the FT component at kx = 0 is mainly due to the optimization of air-hole positions in the ydirection. Therefore, the optimization of the air-hole displacement in the y-direction is important in obtaining a high-Q H0-type PhC nanocavity. The FT spectra of Cavities B and C are almost the same along the ky axis [Fig. 4(b)] because there is no additional displacement of any air hole in the y-direction. However, the FT components ky ~0 are reduced by the additional optimization of S3x in the x-direction. This improvement is clearly shown in the FT spectrum along the kx direction in Fig. 4(a). These comparisons of FT components in the leaky region among the three cavities are consistent with the improvements in cavity Q.

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Fig. 4. FT components of Ex for Cavities A (green), B (blue), and C (red) on (a) kx and (b) ky axes. The gray region indicates the interior of the light cone. Smaller integrated components of FT(Ex) in the light cone result in less out-of-plane radiation loss. The optimization of Six and Siy reasonably lead to smaller integration components of FT(Ex) in (a) and (b), respectively.

4. Aptitude of a zero-cell PhC nanocavity for C-QED experiments Finally, we discuss the importance of a smaller cavity in C-QED experiments. In the strong coupling regime, the spectral linewidth, which influences the clarity of photoluminescence spectra, of polariton doublets is given by (γQD + γcav)/2, where γQD and γcav are respectively the spectral linewidths of a QD and cavity mode. A state-of-the-art QD has γQD ~35 µeV in most C-QED experiments at cavity-mode photon energy of Ecav ~1.35 eV [6,17]. Therefore, γQD limits the spectral linewidth of each peak of the polariton doublets for a high-Q cavity with Q ≥ 4 × 104 (~1.35 eV/35 µeV). However, γQD can potentially be small at about 5 µeV, which corresponds to Q ~2.7 × 105 for the cavity mode, under an ideal pumping condition [31]. The proposed design can provide an H0-type PhC nanocavity that does not degrade the spectral clarity in C-QED experiments even under this ideal pumping case. There are cavity designs with extremely high Q values on the order of 108, but this advantage is restricted by γQD and strong light–matter coupling cannot be expected because of relatively large Vm > (λ0/n)3 in the C-QED experiments. It is thus concluded that a smaller cavity has better performance in CQED experiments if Q is comparable to Ecav/γQD. The proposed H0-type PhC nanocavity satisfies these essential requirements of an excellent cavity in C-QED experiments. 5. Summary A high-Q design of a two-dimensional hexagonal-lattice zero-cell (H0-type) PhC nanocavity was proposed. The optimization of the air-hole position in both Γ-K and Γ-M directions increased cavity Q to a maximum value of 2.8 × 105 in zero-cell PhC nanocavities while maintaining a small mode volume of ~0.23(λ0/n)3. This result indicates the importance of the optimization of the air hole positions in the Γ-M direction, in addition to the Γ-K direction, that has not been adopted in previous works. The momentum space consideration of the outof-plane radiation loss showed that the optimization of displacements of air holes around the cavity in the x- and y-directions reduced FT components in the leaky region along the kx and ky axes, respectively. This optimized design for a zero-cell PhC nanocavity is applicable to Siand GaAs-based semiconductor materials. Acknowledgments We thank A. Tandaechanurat for the fruitful discussions. This research was supported by the Special Coordination Funds for Promoting Science and Technology and Kakenhi 20760030, from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

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