1. Introduction

Recently, many groups have considered the use of metallic cladding as a possible solution for reducing the optical resonator size below a free space resonance wavelength (λ₀) in all dimensions [1–4]. Metallic materials would also satisfy the electrode requirement for current injection and electric field application in electrically-driven nano- and micro-cavity photonic devices such as current-injection nano-lasers. However, subwavelength (sub-λ) size metallic resonators suffer from low Q-factors in the order of a few hundreds since conventional metallic materials such as gold and silver are extremely lossy in the visible and near-infrared wavelengths. In order to improve the Q-factors of metal-incorporating resonators, a common design approach has been the reduction of the overlap of metallic material and the optical mode at the expense of complicated device structure and deteriorated device performance [5]. Current-injection nano-lasers based on these resonators would suffer from (1) small thermal conductance, (2) large electric resistance due to large current path, and (3) large laser threshold current due to significant leak current induced by indirect current injection. As a different approach, compact optical resonators clad by low-loss transparent conductive oxide (TCO) electrodes are promising for room temperature (RT), continuous-wave (CW), current injection low-threshold micro-lasers due to high Q-factors and proximate placement of electrodes to a laser mode [6].

In this letter, we show that both the radiation and absorption losses are significantly reduced in simple nano-scale disk optical resonators incorporating planar indium tin oxide (ITO) electrodes by optimizing the cavity geometry. Calculated Q-factors exceed 10⁴ for optical cavities of the subwavelength (sub-λ) size in all dimensions.

2. Resonator structure

Analyzed ITO-containing disk resonators are composed of a high refractive index (n_s = 3.4) disk of thickness h and radius r, a top ITO electrode of thickness h_e and a bottom AlO_x layer of thickness h_o; see Fig. 1 . The bottom AlAs cylinder with radius r_a provides a current-injection path. The refractive indices of the AlO_x layer and the AlAs cylinder used in the analysis are n_o = 1.7 and n_a = 2.89. We consider ITO as the electrode material as well as a cladding medium, since the optical and electrical properties are suitable for electrode-containing sub-λ scale high-Q optical resonators needed for current-injection nano-lasers. The refractive index of ITO is appropriate for electrically-pumped high-Q nanolasers, in that the real part of the refractive index is smaller than that of semiconductors (typically ~3.4) and the absorption is negligible in visible wavelength spectra. Even at infrared wavelengths, optical absorption is still small compared to gold and silver [7]. In this study, 670 nm is chosen as the resonance wavelength and we measure the corresponding refractive index of RF-sputtered ITO as n_e = 1.9 + 0.01i which is used in the modeling, by spectroscopic ellipsometry. The measured refractive indices of ITO deposited under different conditions vary between 1.852 + 0.009i and 2.012 + 0.078i [8]. Thus, the refractive index value that we use for our modeling can be considered as a reasonable assumption but not the best case apparently. Moreover, ITO can form ohmic contacts on both n- and p-GaAs—required for smooth current injection and a negligible voltage drop at the semiconductor-metal contact [9].

 

Fig. 1 (Color online) Sub-λ-scale, ITO-clad disk resonator. Schematic representation of the disk optical cavity with ITO top disk electrode, an AlO_x bottom cladding layer and an AlAs current path. A substrate can be GaAs.

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3. Optical modes and mode properties

Disk optical resonators support TE_mpq-like and TM_mpq-like WGMs originated from slab modes for large aspect ratio (r/h) whereas HE_mpq-like and EH_mpq-like hybrid cylinder modes (m≠0)—arisen from optical fiber modes—are formed as the ratio is small where indices m, p and q correspond to quantization in azimuth, radial and vertical directions. As the aspect ratio decreases from 2 to 0.5, the optical modes are evolved from slab-like modes to optical fiber-like modes, and interact with each other for the similar mode frequencies. For TE-like (TM-like) modes, E_z-field (H_z-field) component is negligible. Hybrid modes have m > 0, and contain both the E_z-field and H_z-field components; designation of these modes is based on relative contribution of these field components to a transverse field component (e.g. E_r or E_ϕ). When the mode is TM-dominant (E_z-field makes larger contribution), the mode is named as HE_mpq-like. The mode is named as EH_mpq-like when the mode is TE-dominant (H_z-field component makes larger contribution) [10]. For small azimuthal mode numbers m, the optical modes suffer from the radiation loss; the Q-factors are below 50 for m < 3. Although metal enclosures of the disk reduce the radiation loss by photon recycling, optical absorption in the metal limits maximum Q-factors to a few hundred [2, 4]. Thus, we consider the modes HE_m11-like, TE_m11-like, EH_m21-like, and TE_m12-like (the indices are chosen based on the quantization of major field component of the corresponding mode) with m = 6 and 7 that can maintain large Q-factors and, at the same time, sub-λ-scale cavity dimensions. Figure 2 shows the relative intensity distributions of the E_z-field and H_z-field components of the corresponding optical modes with m = 7 for various aspect ratios. As the aspect ratio decreases, the modes are stretched vertically. For the mode labeled as TE_m11-like, E_z field component becomes significant as the aspect ratio decreases below 0.9 making the mode a hybrid one. On the other hand, E_z field component of EH_m11-like mode becomes negligible and radial mode number decreases to 1 about r/h = 0.7. However, we will use the same notation for consistency.

 

Fig. 2 Intensity z-r plane (r > 0) distributions of E_z and H_z field components of HE₇₁₁, TE₇₁₁, EH₇₂₁ and TE₇₁₂ modes respectively for various aspect ratios (r/h). Outline of the structure profile is overlaid onto image where the top rectangle is ITO cladding. The intensity is normalized by the maximum field component. Modes are stretched vertically as the aspect ratio decreases.

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We analyzed cavity properties—mode frequencies, Q-factors, effective mode volumes (V_eff), resonator sizes and confinement factors (Γ) with the disk aspect ratio (r/h) as a geometric parameter for optimization—by two-dimensional (r-z plane) finite-difference time-domain (FDTD) method, considering the azimuthal symmetry of the disk geometry. The disk height, h is chosen as the unit length and resolved by 20 mesh points. For most of the calculations, the ITO cladding thickness is equal to the disk height and resolution is 20. For specific cases, where ITO thickness is assumed as 100 nm, the resolution is 9 nm. The normalized mode frequencies with respect to h and V_eff as a function of aspect ratio are presented in Figs. 3(a) and 3(b), respectively. In our analysis, we intentionally exclude the bottom AlAs current injection path, but the high-Q disk modes are typically maintained for r/r_a ratios smaller than 0.82. Figure 3(c) shows that the Q-factor increases as the aspect ratio decreases from 2.0 to 0.5 where the Q-factor is the maximum for HE_m11-like and EH_m21-like modes. However, the Q-factors of the TE_m11-like and TE_m12-like modes keep increasing with the further reduction of the aspect ratio.

 

Fig. 3 Mode characteristics (a) Normalized mode frequency, (b) effective mode volume (V_eff), (b) Q-factor, (d) disk diameter, (e) Purcell factor (F_p), and (f) threshold material gain (g_th) as a function of disk aspect ratio (r/h). Some zigzag lines are caused by mode coupling.

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We obtain the Q-factor as high as 5.3x10⁴ for EH₇₂₁-like mode for an optical cavity with 0.99 λ₀² footprint size, corresponding to 0.44μm² at λ₀ = 670 nm. Assuming a 100 nm thick ITO electrode and a 100 nm AlO_x bottom layer, we obtain the Q-factor of HE₇₁₁-like mode as high as 3x10⁴ for the cavity with all sub-λ-size dimensions. This is much higher than the Q-factors of any other past sub-λ size, current injection, and optical cavity designs [2–4]. The corresponding aspect ratio is 0.8 and the disk diameter (d = 2r) and the total cavity height (h_e + h + h_o) is 0.944λ₀ and 0.973 λ₀, respectively. Figure 3(d) shows the normalized disk diameter with respect to the aspect ratio for the analyzed modes. For r/h < 1.2, the cavity diameter is below λ₀ for the HE₆₁₁-like, TE₆₁₁-like, EH₆₂₁-like and HE₇₁₁-like modes. Our results show that absorption is the main loss mechanism for large aspect ratio disks. The fraction of the energy density stored in ITO cladding (Γ₂) decreases for all the modes as the aspect ratio decreases, resulting in improvement of Q-factors. Γ₂ values are given in Table1 for r/h = 2 & r/h = 0.6. For a detailed discussion about the effect of Γ₂ on Q-factors of low-absorption electrode media clad microcavities, see [6]. In addition to reduction of absorption loss, scaling down the aspect ratio also reduces the radiation loss. For HE_m11-like and EH_m21-like modes, reduction of absorption loss is more drastic than the reduction of the radiation loss. Hence, Q-factors of these modes for small aspect ratio disks become eventually radiation loss limited. On the contrary, for TE_m11-like and TE_m12-like modes radiation loss reduction is prominent, resulting in absorption loss limited Q-factor. As a result, the increase in absorption loss would significantly degrade the Q-factors of the TE_m11-like and TE_m12-like modes, but would have a negligible effect on the Q-factors of HE_m11-like and EH_m21-like modes. The zigzag behavior of the HE₆₂₁-like and TE₆₁₂-like data about r/h = 1.6 is attributed to interaction of these two modes.

Table 1. Fractions of the energy density stored in ITO (Γ₂) for ITO-clad disk modes

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The Purcell factor (F_p) is an indicator of the spontaneous emission enhancement that can be prominent for sub-λ-scale lasers. We calculate F_p using $F_p=3Q{({\lambda }_0/n)}^3/4{\pi }^2{V}_{eff}$. Among all the modes, HE₇₁₁-like modes exhibit the highest F_p in the analyzed parameter space, i.e., $r/h \in (0.5,2.0)$. We obtain 456 as the highest F_p for the cavity design with all sub-λ dimensions; see Fig. 3(e). This is an order of magnitude higher than those of reported sub-λ-size metallic optical resonators [4].

In order to examine the possibility of current-injection lasing of the presented ITO-clad disk resonators, we calculate the threshold material gain (g_th) as $g_{th}={\omega }_0/[\text{Γ}Q\left(c/n_{eff}\right)]$ where ω₀ is the angular resonance frequency, c is the speed of light, n_eff is the effective mode index, and Γ is the energy confinement factor. Assuming a bulk gain medium, calculated g_th values are below 10 cm⁻¹ for modes with m = 7. Figure 3(f) shows the threshold gain values with respect to the aspect ratio for all the modes. HE₇₁₁-like modes require the smallest bulk threshold gain of 11 cm⁻¹ at r/h = 0.8 among the analyzed disk resonators with sub-λ size in all dimensions. This threshold gain is significantly small compared to those of sub-λ-size metallic resonators incorporating bulk gain medium [3, 11], showing an advantage of using our designs. We also calculate the threshold gain values per well for a single quantum well (SQW) and three quantum wells (3QWs) for 7nm/10nm thick well/barrier layers placed in the middle of the disk layer. For HE₇₁₁-like mode, g_th per well is calculated as 364 cm⁻¹ for SQW and 41 cm⁻¹ for 3QWs. Even for SQW layer, g_th per well is still smaller than those of metallic optical cavities with seven or nine QWs [4]. According to our gain analysis results, the reduction of threshold current is expected. The calculated threshold gain is achievable in compressively strained InGaP/AlGaInP QW material system at RT [12].

4. Conclusion

In conclusion, we have optimized the disk optical resonator geometries incorporating ITO electrodes for enhanced cavity characteristics of ITO-clad, sub-λ-size disk resonators. The optimized cavities exhibit Q-factors larger than 10⁴ and Purcell factors as high as 500. Both the demonstrated Q factors and Purcell factors are significantly higher than those of the optical metallic cavities and the designed structures still have sub-λ sizes in all dimensions. Among the analyzed modes, the HE₇₁₁-like modes exhibit the highest Q-factor and Purcell factor among resonators that are sub-λ size in all dimensions. Owing to high Q-factors, small threshold gain values and the direct electrode placement above the lasing mode, the optimized optical cavity designs can be utilized to build current injection low-threshold nanolasers at room temperature. The proposed cavities would also be useful for many other electrically-driven optoelectronic devices such as modulators, switches, logic gates, photodiodes, and single photon sources.