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 (V _m). In the weak-coupling regime, the artificial control of the radiation of photons, which is known as the Purcell effect, is determined by Q/V _m. In the strong-coupling regime, the light-matter coupling strength g is determined by V _m ^-1/2, which must be larger than γ _QD/4 and γ _cav/4, where γ _QD and γ _cav are the spectral linewidths of a QD and cavity mode, respectively. Therefore, a high-Q cavity with smaller V _m 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 V _m is essential to enhance the light-matter interaction in the C-QED systems. A small V _m 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 zero-cell 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 V _m of 0.26-0.29(λ ₀/n)³, where λ ₀ and n are the wavelength in vacuum and the refractive index of the material ( = 3.4), respectively. A square-lattice zero-cell PhC structure has been reported with a smaller V _m value of 0.21(λ ₀/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 V _m and a high Q.

In this paper, we report a high-Q design of a two-dimensional (2D) hexagonal-lattice zero-cell PhC nanocavity with a small V _m 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 V _m of 0.23(λ ₀/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 C-QED 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 energy-momentum 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-of-plane radiation loss is the fundamental guideline [30].

 

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 E _x, E _y and H _z components for fundamental mode.

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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 V _m than defect-type cavities produced via the removal of air holes. The cavity center is located at a C _{2 v},_σv symmetry point in the hexagonal lattice. The shifts of on-axis air holes S _ix (i = 1–3) in the x-direction (Γ-K) and of S _jy (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 E _x, E _y and H _z components are shown in Figs. 1(c)–1(e), respectively. The main electric field component is E _y, but E _x has comparable amplitude (~50%). The E _x component shows very strong localization in the cavity at two maxima. The calculated V _m slightly differs with a change in S _ix,y, but does not drastically change from a value of 0.24(λ ₀/n)³.

2.2 Optimization of Q by shifting multiple air holes

The FDTD simulations are performed by optimizing each shift in the order of S _1x, S _1y, S _2x, S _2y, and S _3x so that Q is a maximum. First, S _1x = 0.14a gives the largest Q of ~1.1 × 10⁵. S _1x is then fixed at 0.14a and the value of S _1y 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, S _1y = 0.04a gives the maximum value of 2.1 × 10⁵ and Q decreases at larger S _1y. The same series of calculations are performed for the tuning parameters S _2x, S _2y, and S _3x. The maximum value of Q of 2.8 × 10⁵ is found for S _1x = 0.14a, S _2x = 0, S _3x = 0.06a, S _1y = 0.04a, and S _2y = 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 V _m 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.

 

Fig. 2 (a) Calculated cavity Q of fundamental mode with different S _1y values with fixed S _1x 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 S _3x values for S _1x = 0.14a, S _2x = 0, S _1y = 0.04a, and S _2y = 0.02a. Q increases from 2.2 × 10⁵ to 2.8 × 10⁵. The mode volume V _m of Cavity C increased by only 6% as compared to that of Cavity A, whereas Q increased by more than twice.

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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 y-direction. 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.

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Figures 3(a) –3(c) show the calculated 2D spatial FT spectra of E _x for Cavities A, B, and C at the center of the slab. The FT spectra show E _x 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.

 

Fig. 3 (a–c) 2D spatial FT spectra of E _x 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.

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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 E _x for Cavities A (green), B (blue), and C (red) on the k _x and k _y 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 k _x = 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 k _y direction. The drastic reduction in the FT component at k _x = 0 is mainly due to the optimization of air-hole positions in the y-direction. 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 k _y axis [Fig. 4(b)] because there is no additional displacement of any air hole in the y-direction. However, the FT components k _y ~0 are reduced by the additional optimization of S _3x in the x-direction. This improvement is clearly shown in the FT spectrum along the k _x 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.

 

Fig. 4 FT components of E _x for Cavities A (green), B (blue), and C (red) on (a) k _x and (b) k _y axes. The gray region indicates the interior of the light cone. Smaller integrated components of FT(E _x) in the light cone result in less out-of-plane radiation loss. The optimization of S _ix and S _iy reasonably lead to smaller integration components of FT(E _x) in (a) and (b), respectively.

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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 (γ_QD + γ_cav)/2, where γ_QD and γ_cav are respectively the spectral linewidths of a QD and cavity mode. A state-of-the-art QD has γ_QD ~35 μeV in most C-QED experiments at cavity-mode photon energy of E _cav ~1.35 eV [6,17]. Therefore, γ_QD 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, γ_QD 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 γ_QD and strong light–matter coupling cannot be expected because of relatively large V _m > (λ ₀/n)³ in the C-QED experiments. It is thus concluded that a smaller cavity has better performance in C-QED experiments if Q is comparable to E _cav/γ_QD. 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(λ ₀/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 out-of-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 k _x and k _y axes, respectively. This optimized design for a zero-cell PhC nanocavity is applicable to Si- and GaAs-based semiconductor materials.