1. Introduction

Nanolasers [1–13] make it possible to reduce actual device size to a wavelength scale where even a dense laser array with a pitch of several μm can be utilized [13]. An laser diode (LD) essentially exhibits its highest efficiency just above the threshold current [14] and is usually designed to operate at a current (I) of ~10 times the threshold current (I_th) or less. For a nanolaser operated around 10 μA an output power of ~μW is expected. Such a low power is suitable for both on-chip optical interconnect and optical processor applications where low power consumption is mandatory [15]. Here our goal is a nanolaser diode capable of highly dense integration smaller than a pitch of ~10 μm or less [13,16] for both parallel single wavelength and multiwavelengths on-chip laser sources [17]. Few-cell point-defect PhC nanocavities (such as L3 and H1 type cavities [2–9]), have several unique characteristics including an ultra-small mode volume (V_m), a small device footprint advantageous for dense integration, and a large mode spacing [18] advantageous as regards a high spontaneous-emission coupling coefficient (β) [2,4–10,19–21], which are promising for energy-efficient densely integrable on-chip laser light sources enhanced by the cavity QED effect [22–25]. And importantly, the cavity effect may enhance the optical confinement factor (Γ) [1,14] as discussed later. To achieve this goal with current injection CW lasing at room temperature (RT), a high quality factor (Q) is essential, but conventional few-cell nanocavities [2,4–10,13] do not have a sufficiently high Q. We have realized CW PhC nanolaser diodes based on line defects with a BH MQW active region operated at less than 10 μA at RT with a laser lightemission of up to ~μW into an on-chip waveguide [11,26], but as yet there have been very few reports of nanolaser diodes/nano-LEDs with few-cell nanocavities [3,4,10,27,31] and RT CW lasing by current injection has yet to be demonstrated. Line-defect-based nanolaser diodes confine light with a modegap formed in a PhC line defect by employing the index contrast between the BH and the surrounding InP PhC and utilize an ultrahigh Q. Meanwhile, few-cell nanocavities, which do not have a long line defect, are more suitable for highly dense integration. In addition, we found that L_x cavities can exhibit a much larger β than line-defect cavities [21]. Since, unlike the optical pumping [6,9], a few QDs cannot support the threshold gain [5,28], for current injection CW laser applications, many QDs or MQWs have been employed for the active region [3,4,10–12,26–30]. However, in PhC nanolasers (H2 [3] and larger whispering-gallery-mode-based cavities are excluded), only pulse lasing has been utilized at RT [4,12,23] except with modegap nanocavities [11,26]. (Note: the cavity of a non-membrane-type nanobeam-like nanolaser diode [30] is much longer than the wavelength scale.) When we compare the BH nanolaser diodes [11,26] and the previous H1 [4,27,29] and L3 [10,31] nanolaser diodes/nano-LEDs, the obvious advantages of the former are a cold Q factor (Q_th) (corresponding to the linewidth at the lasing threshold) exceeding 10⁴ and good heat management. The implementation of a BH active region with L3 nanocavities [21] utilized the latter, but previous nanocavity design employed a Q comparable to that of conventional H1 and L3 nanolasers. To clarify the importance of the Q value as regards current injection lasing, here we design L_x (x = 0,1,2,3: the number of cells) nanocavities [32] with an ultrahigh Q and apply it to nanolasers with a BH active region [18,33]. Since a smaller footprint is better in terms of making the LD array pitch smaller and because the horizontal footprint of the smallest nano-pillar lasers [28,34] is smaller than that of the L3 nanocavity, we mainly explore an ultrahigh-Q design with x smaller than 3.

2. Ultrahigh-Q few cell nanocavities

2.1 Multi-hole-tuned L_x nanocavities

We focus on the fundamental cavity mode (0th mode). To significantly enhance the Q of such L_x nanocavities, we refer to the topologically optimized zero-cell (H0) and three-cell (L3) nanocavities reported by Minkov and Savona [35], which suggest that the evolution of the multi-hole shifting of axial holes [36,37] greatly enhances Q. Following Minkov and Savona, we first analyzed the Si/air air-bridge nanocavity with similar PhC parameters (lattice constant a ~400 nm, slab thickness d ~220 nm, air-hole radius r~0.25a, refractive index n of Si: 3.46). We reveal that axial multi-hole-tuning [35–37], which systematically shifts the axial holes labeled 1 to 5 (shown in red; corresponding outward shift: s₁-s₅), works efficiently in other few-cell L_x nanocavities including single-cell (L1) and two-cell (L2) nanocavities as summarized in Fig. 1 We specify the almost fixed empirical relations between s₁-s₅ as shown in Figs. 1(c) and 5 (See Appendix 1 for details). We abbreviated the zero-cell nanocavity to L0 (we don’t use H0 [8,37] since it apparently loses hexagonal symmetry and the cavity modes are similar to those of L1-L3 nanocavities as shown in Fig. 1(b)). There is an similar empirical shifting relationship between L0A, L1A, L2A and L3A, whereas L2B that greatly shifts s₄ and s₅ is similar to L3B revealed by Minkov and Savona. (The difference betweenrules A and B is clarified in the Appendix 2). By using axial tuning alone, a theoretical Q value exceeding 10⁶ was obtained for all L_x designs in the order of L0<<L1<<L2< = L3 when d was 100-300 nm, and at d>300 nm, the Q values of L1 and L3 were saturated but L2 exceeded L3 (6 × 10⁶ at d = 400 nm). Our numerical study revealed that these L_x nanocavities exhibited ultrahigh-Q at d/a>> 0.5, which is different from Minkov and Savona (Note: we optimized s₁-s₅ at every d as shown in Fig. 5(a)) and the trend in literature where d/a was kept to < = 0.5 [38]. We studied additional side hole tuning (S-U in Fig. 1(a)) and revealed that Q was additionally enhanced nearly 10 times (L0A) and 2-3 times (L2A/B), respectively, by the very small horizontal shifting of S (~0.025a) in L0A and S/T (~0.005a) in L2A/B) with the axial hole positions almost fixed. The relationships between the shift (s_S, s_T) and Q in L0A and L2B are shown in Figs. 5(c) and 5(d), respectively. In contrast, Q was not enhanced by side hole tuning (S-U) in L1 and L3. The highest Q values were 1.8 × 10⁷ in L0 and 1.6 × 10⁷ in L2A/B, as high as the double-heterostructure nanocavity [39], were theoretically demonstrated. This tuning keeps V_m at ~1 (λ/n)³ or smaller as shown in Fig. 5(b), so Q/V_m is theoretically ultrahigh (~10⁷). Both the axial and side hole tunings are highly sensitive to small deviations in the shift (actually fabrication error) as shown in Fig. 5(c), suggesting that in an experiment Q probably becomes much lower than the theoretical value and fluctuates due to fabrication error [40,41]. Although the side hole tuning was highly sensitive to the fine shifting of S and T (~10 nm in L0 and ~2 nm in L2), we obtained very good agreement between a finite-difference time-domain (FDTD) simulation mainly used in a theoretical study and finite element method simulations (COMSOL) reported in Fig. 5(e), suggesting that the side-hole-tunings were theoretically valid. We concluded that the use of Q values exceeding 10⁷ L0 and L2 are good solutions but L1 and L3 are poor solutions.

 

Fig. 1 FDTD simulation results. (a) L-type nanocavities studied here. a. Multi-hole-tunings studied here. Annotated holes are shifted for tuning. 1-5 are axial holes and S, T, and U are side holes. Colored holes are shifted along the arrowheads and black holes are not shifted. (b) (Left panel) E_x mode profile of the 0th mode of multi-hole-tuned L0-L3 nanocavities obtained by FDTD simulation. PhC parameters are presented in (c). (Right panel) the color bar of the left panel and the relation between the color bar scale (arbitrary unit) and the |E_x|. (c) Table of the empirical multi-hole-tuning designs (L01-L3B) (L3B is according to Minkov and Savona.) Theoretical Q, λ_c, and V are those of the 0th mode. (d) Multi-hole-tuning results (left) and corresponding theoretical Q, of the 0th mode of L0-L3 nanocavities as a function of d at fixed PhC parameters (a = 370 nm, r = 0.25a). λ_c and V are shown in Fig. 5(b). s1-s5 were tuned to each d value as shown in Fig. 5(a). In side-hole-tuning, s_x = s_s in L0A and s_X = s_S = s_T in L2 (A/B), respectively.

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2.2 Microscopic model of high-Q nanocavity design in 2D PhCs

To explore the internal mechanism that determines Q, in right panel of Fig. 1(b) we significantly enhance the bottom of the x component of the electronic field E_x, which largely determines the cavity loss [42,43], as it is seen. It reveals the existence of two types of boundaries separating the localized mode from the outer (evanescent) modes, on which the wave front and phase are shifted. The axial boundary (blue broken lines) terminates the cavity mode in the axial (z) direction and the side boundary (yellow broken lines) separates the axial antinodes from the other antinodes. From a careful comparison of the various nanocavity mode patterns shown in Figs. 1 and 6, we can obtain a microscopic model that explains how Q is determined in these L_x nanocavities. (See Appendix 1 and 2 for details.) In these nanocavities, the mode pattern vicinity of the cavity center is very similar to that of the even gap-guided mode in a line-defect waveguide (W1) [44], which is the origin of the cavity mode. In multi-hole-tuned L_x as shown in Figs. 1(b) and 6(e), the side (yellow) boundary unambiguously appears and stretches as they nearly reaches the axial (blue) boundary. The outer axial hole positions (3-5) in the multi-hole-tuning were strongly correlated with the location of the axial (blue) boundary. In contrast, in partially-optimized L_x nanocavities as shown in Figs. 6(a)-6(d), the side boundary is significantly shortened and/or ambiguous, and the localized modes are strongly compressed by the axial and side holes (the stress is shown by green arrowheads) and thereby seriously deviated from the ideal W1 mode. The poor Q (10⁵ or lower) in the partially optimized nanocavities is apparently due to a mismatch between the cavity mode and the unoptimized holes. It is suggested that how well the side and axial boundaries are formed (W1 mode is ideally lossless) is a good benchmark of whether Q is low (10³) or high (10⁷). This idea is supported by the fact that in a line-defect-based nanocavity [45] in which all axial holes are eliminated Q exceeds 10⁸ where the axial boundary is not formed in the vicinity of the cavity center and almost straight side boundary appears with very long stretching as shown in Fig. 6(f). (Actually, such line-defect-based designs demonstrated highest theoretical [43] and experimental [46] Q factors). To include the axial holes in the W1 mode, the waveguide must terminate the localized mode, where multi-hole-tuning is a compromise between the localized mode and the conflicting holes as Q becomes 10⁶ to 10⁷ in L_x nanocavities. Hence, such a tuning mechanism is microscopic rather than macroscopic [42,43,47]. This is more apparent with the side hole tuning in L0 and L2 where the change of the mode pattern in Fig. 1(b) caused by the small shift (0.025a and 0.005a) is not visible despite the large change in Q. Since the side holes are on the border of the side boundary where the electromagnetic field changes abruptly, even a small mismatch between the localized mode and the hole (position) might cause serious cavity loss. To tune the side hole is to add substantial disorder to the W1 mode and therefore likely to result in a poor Q. We think that L0 and L2 are rare lucky cases where accommodating the mismatch caused by side hole tuning overcomes the loss caused by the addition of disorder and makes Q >10⁷, which is difficult for L1 and L3.

2.3 Experimental demonstration of ultrahigh Q factor

To validate the ultrahigh-Q multi-hole-tuned L_x nanocavities experimentally, we fabricated Si PhCs. (See Appendix 4 for details.) First, we fabricated L0A, L1A, L2B, and L3B with the same PhC parameters (Fig. 2(a), d = 230 nm, a = 400 nm, r = 95 nm). As in previous work [32,45] we made two on-chip waveguides coupled with a nanocavity as shown in Figs. 8(a)-8(b) and evaluated the Q factor from the cavity mode linewidth in thetransmission spectrum. At the same time, we collected the light radiated from the nanocavity using an objective lens with a numerical aperture (NA) of 0.40 placed right above the nanocavity. As compared in Fig. 2(b), the light intensity and spectrum waveform collected by the top lens (top) are very similar to those of the transmitted light collected from the on-chip waveguide (WG) coupled by a lensed fiber. As a result, the Q values measured with the output WG and with the top lens agreed well as shown in Fig. 2(c). All multi-hole-tuned (L0A, L1A, L2B, L3B) nanocavities exhibited an experimental (loaded) Q higher than 1.0 × 10⁶ (Fig. 2(d)). As mentioned above, the multi-hole-tuned Lx is essentially sensitive to fabrication error and actually the experimental Q was dispersed. The effect of side-hole-tuning is significant in L0A (s_x = 11nm) but ambiguous in L2B (s_x = 2nm). Since our simulation as shown in Fig. 1(d) anticipated an ultrahigh Q in a very thick slab (d/a>>0.5), we fabricated PhCs with d of 300 nm (d/a ~0.8) and realized an ultrahigh Q (1.2x10⁶ in L2B.) As previously reported [48] there is no substantial limitation to Q in a d/a range of 0.5-1.0 although in past studies d/a was kept at ~0.5 or smaller. Finally, we evaluated the coupling efficiency (η) of the ultrahigh-Q mode to a NA = 0.40 lens. (See Appendix 3 for details.) Despite the ultrahigh Q factor (~10⁶), the experimentally evaluated η in these nanocavities was ~4% (L2) or higher, which agreed with theoretical values obtained from FDTD (See Fig. 8(e)). These results are unlike those of a previous report [49], which predicted an η of only ~1% in L3, L5, and L7 (Q<<10⁶) without far-field optimization [49,50]. Here, we tuned the Lx nanocavities solely considering Q and disregarding far-field optimization. Thanks to the ultrasmall cavity size, which was smaller than the spot size of the objective lens, ultrahigh-Q L0-L2 nanocavities are inherently compatible with lenses with a normal NA lens, a high f number, and a long working distance without far-field optimization. Far-field optimization results in a significantly sacrifice as regards Q but useful in an application requiring η >10%.

 

Fig. 2 Experimental results in Si L-type nanocavities. (a) Top view SEM images of L0-L3 nanocavities. (d, r = 200, 95 nm) The length of white arrow lines shows 1 μm. (b) Cavity mode spectra detected at the same time from an on-chip output waveguide (WG) and the top-side objective lens (top) in L0A, L1A, and L2B at d = 230 nm. (c) Dispersion of experimental Q among several nanocavity samples and evaluated from the light transmitted to an on-chip output waveguide (WG) and light dropped to the top-side objective lens (top). In L2B (s_x = 2nm, top), two plots are missing due to the poor output power. (d) Highest Q obtained in the L-type nanocavities at different d values. (As for L3B we fabricated a very limited number of samples). Experimental total (loaded) Q factors reported here are evaluated from the cavity mode linewidth [44]. Detailed parameters of the nanocavities are given in Appendix 4.

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3. Few cell BH nanolaser diodes

3.1 RT CW operation of L1 and L2 nanolaser diodes

Now the multi-hole-tuned L_x design is combined with a BH nanolaser diode. The size of the BH is limited as it is included in the center cells (point-defects), which makes the active volume (V_gain), namely the volume of the gain material (here the well part of the QW), smaller as x becomes smaller. A smaller active volume is better as regards reducing the threshold current (I_th) [1,51], whereas it also makes the optical confinement factor (Γ) poor [1,14] and makes lasing difficult. And such a small BH active region can interact with only few electric field (E) antinodes of the nanocavity mode, where the intensity of the light-matter interaction between them should be considered. We excluded L0 because Γ was too poor and the center of the cavity where the BH was placed was the E node. We designed BH-L1 and BH-L2 nanolaser diodes as shown in Figs. 3(a) and 3(e) with the parameters summarized in Table 1. The multi-hole tuning gives the 0th modes of both L1 and L2 an unprecedentedly high theoretical Q value that is much higher than 10⁵ with V_m < = (λ/n)³. A higher Q factor contrast and a wider mode spacing against the 1st mode compared with L3 (See Table 1) are advantageous for lasing. Here we fabricated L1 and L2 nanolaser diodes as well as previous nanolaser diodes [11,26] (see Appendix 5 for details) employing the parameter shown in Table 1. To address various fabrication error, we prepared nanolaser diodes with different a.

 

Fig. 3 Experimental electrical CW lasing characteristics of BH-L1 and BH-L2 nanolaser diodes measured at 23°C. a-d: Schematics of nanocavity design and multi-hole-tuning (a), E field profiles of 0th and 1st cavity modes (b), laser microscope image (c) and (d) SEM images of 6QW L1 nanolaser diode (a = 445 nm, d = 250 nm, r = 97 nm.). e-g: nanocavity design (e), E field profiles of 0th and 1st cavity modes (f), and a SEM image (g) of a 6QW L2 nanolaser diode (a = 440 nm, d = 250 nm, r = 97 nm.). (h)-(i): I-L curves around the threshold of L1 (h) and L2 (i) nanolaser diodes with the application of a BPF. (j)-(m): Enlarged current dependent lasing mode spectra around the lasing threshold (j:L1, l:L2) and wide band spectrum above the threshold (k: L1, m: L2.) In the latter, the 0th, 1st, and 5th cavity modes are even modes and the 2nd-4th cavity modes are odd modes [52].

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Table 1. Design of L_x nanolaser diodes and its theoretical characteristics evaluated by FDTD. 0th and 1st correspond to fundamental and 1st high order cavity modes. L3 and L5 (*) were designed according to Ref. 33.

View Table

Finally, we found that L1 (a = 445 nm) and L2 (a = 440 nm) nanolaser diodes operated in the optical fiber communication (telecom) band by current injection as shown in Figs. 3(c)-3(d) (L1) and 3g(L2). The nanolasers did not have a on-chip waveguide and we employed the same measurement system as shown in Fig. 2 that we used for ultrahigh-Q Si L_x nanocavities and collected the light emitted to the top-side via the same NA = 0.40 objective lens. We applied a DC bias voltage (V_b) and obtained current-voltage (I-V) and current-light-output (I-L) curves as shown in Figs. 3(h)-3(i) at RT (23°C). The InGaAs/AlGaInP MQW [18] exhibited gain over a wide wavelength range (1,400 nm to 1,600 nm) and thereby excited many high-order (1st-5th) nanocavity modes [52] in addition to the 0th mode as shown in Figs. 3(k) and 3(m). Electroluminescence (EL) in the high-order modes was highly intense compared with that in the 0th mode around the lasing threshold. We employed a tunable band-pass filter (BPF), which cut off the high-order modes and revealed a clear kink in the I-L curves, and that exhibited CW lasing at a threshold current I_th of 5.6 μA (V_b = 0.88 V) at L1 and 4.1 μA (V_b = 0.85 V) at L2. With these nanolaser diodes, we confirmed the lasing transition by evaluating the second order photon correlation function g²(t) of the LD emission [21] just above the threshold as shown in Figs. 9(a) and 9(b) in a Hanbury Brown and Twiss quantum interferometer setup [53], where both the L1 and L2 nanolaser diodes exhibited g²(0) >>1 around the I_th but as I was increased g²(0) finally became 1.0, exhibiting a transition from bunching light (spontaneous emission) to coherent light (stimulated emission). Another feature of the lasing transition is a change in the cavity spectrum as shown in Figs. 3(j) and 3(l). Below the threshold, the cavity wavelength λ_c was blue-shifted as I was increased due to the carrier plasma effect [11,26] and at the threshold, the linewidth became very narrow, corresponding to Q_th values of 59,000 (L1) and 65,000 (L2), respectively. This high Q_th for RT current injection CW lasing, which was difficult to achieve with conventional designs [3,4,10,27,30] was thanks to the multi-hole-tuning employed here. We note that these nanolaser diodes have a sub-micron BH (0.2 × 0.65 × 0.15 μm³ in L1 and 0.2 × 0.99 × 0.15 μm³ in L2) and include six 5-nm-thick QWs corresponding to V_gain values of 4 × 10⁻¹⁵ cm³ and 6 × 10⁻¹⁵ cm³ respectively, which are unprecedentedly small among RT CW LDs [3,11,16,26,30,54,55] and comparable with that of nanowire lasers [56–58]. We also utilized the CW lasing of three QW L1 nanolaser diodes as shown in Figs. 9(c)-9(e) where V_gain was only 2 × 10⁻¹⁵ cm³. Although I_th was comparable for the L1 and L2 nanolaser diodes, the slope efficiency just above the threshold was 4.2 mW/A (external differential quantum efficiency: 0.59%) in the former and 0.42 mW/A in the latter by assuming η values of 11% and 4%, respectively. The surprisingly large slope efficiency difference was not simply explained by the overlapping between the cavity mode and the BH active region. |E|/|E_max| (|E_max|: maximum value of |E|) [59] averaged over the BH was ~0.6 for 0th mode of L1 and L2 and ~0.5 for 1st mode of L1 and L2. A possible hypothesis is that BH active region may not contribute to lasing uniformly but its center may play a major role. As shown in Figs. 3(b) and 3(f), the centers of 0th L1 cavity mode and 1st L2 cavity mode are E antinode and those of 1st L1 cavity mode and 0th L2 cavity mode are E node, where the former (the latter) should efficiently (inefficiently) excite the emission. In fact, in L2 the EL of the 1st mode was more intense than that of the 0th mode even after lasing of the latter at I<10I_th. Despite the high EL intensity, the 1st mode in L2 never lased since the Q value was 5,000 or less even with enhancement by high current injection. The L1 and L2 nanolaser diodes both always exhibited single-mode lasing at the 0th mode under CW operation.

3.2 RT CW operation of L3 and L5 nanolaser diodes

We also fabricated 6QW BH-L3 and BH-L5 nanolaser diodes, having larger Γ and V_gain than BH-L1 and BH-L2, on the same wafers. L3 and L5 employed different multi-hole-tuning [32,33] that previously realized monolithically-integrated multi-bit optical RAMs [18]. As shown in Figs. 4 and 10, Both L3 and L5 nanolaser diodes demonstrated RT CW lasing at an I_th value much smaller than 10 μA and V_b lower than 1V. The Q_th values of 56,000 (L3), and 29,000 (L5) were also ultrahigh. As summarized in Table 1, L5 exhibited the highest lasing efficiency thanks to the highest Γ and the largest V_gain, once the output power had been corrected according to η. But here surprisingly, if η is neglected, L1 outperformed L5 in terms of net output power coupled to the NA = 0.40 lens as shown in Fig. 4(d) despite the fact that both V_gain and Γ were ~1/3 in the former. We show that L1 is very efficient when it is vertically out-coupled in the same way as VCSELs. Figure 4(e) shows the relation between Γ and V_gain among the L_x nanolaser diodes studied here. Reducing the number of cells (x) makes Γ small, but it also shrinks V_gain significantly. As a result, Γ/V_gain is almost constant in L_x nanolaser diodes and similar to that in larger nanolaser diodes [11]. This is important for dynamic LD performance since the relaxation oscillation frequency f_r, which determines the upper limit of LD direct modulation speed, is simplified as f_r = 1/2π × (cg’/n_g)^1/2 × (p₀/τ_p)^1/2 Hz [14] (c: speed of light in a vacuum, n_g = 3.5: group index, g’: differential gain in active region, p₀: instantaneous photon density) and p₀ is proportional to Γ/V_gain × τ_p (I-I_th) since current I (pumping power P) is an integrated value over unit time and τ_p is thereby multiplied when I (or P) is converted into p₀. Hence, in a high Q regime (>10⁴), f_r is proportional to Γ/V_gain. According to Fig. 4(e), we can expect fairly high direct modulation rate (~10 Gbit/s) for theseL_x nanolaser diodes. Among these BH nanolaser diodes Q_th is always high (> = 10,000). Such a high Q_th was observed in many other nanolasers exhibiting a fairly low threshold [5–9,28],which suggests that the lack of gain resulting in the small V_gain is probably recovered by enhancing Q [5,50], thus a high Q is important for our work. Note that the threshold current density J_th for the L1 nanolaser diode in Fig. 3(h) (I_th = 5.6 μA) was 5.5 kA/cm³ and the injection of too large a current into a nanolaser likely causes serious gain saturation, free carrier absorption, and device heating and thus making lasing difficult. An issue with the nanolaser diodes described here is their low external differential efficiency of smaller than 1%. One of the main reasons for this is the wideband gain of the MQW, which causes detuning from the lasing mode [21] and the non-negligible EL emission from the higher cavity modes. Leakage current which was dominant in I_th in the previous work [11,26] was still main limiting factor of I_th as well here and we believe that there is room for improving the LD characteristics by optimizing the details of the p-i-n diode design since it has yet to be optimized in the current L_x nanolaser diodes.

 

Fig. 4 Experimental electrical CW lasing characteristics of BH-L3 and BH-L5 nanolaser diodes measured at 23°C. (a). Schematics of BH nanocavities and multi-hole tunings. (b). I-L curves with BPF (left: L3, right: L5). (c). Enlarged cavity mode spectra around the threshold (left: L3, right: L5). d. Comparison of output power without (top) and with (bottom) a BPF. In the latter, the coupling efficiency η and insertion loss of the BPF (5 dB) were not corrected. (e) Relation between Γ and V_gain in various BH nanolaser diodes.

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3.3 Discussion

Finally, we compare this work with other small lasers that have been reported and discuss the impact and importance of the nanolaser diodes presented here. A single cell nanolaser had been one of the ultimate goals of the PhC community [2,4], but current injection CW lasing proved difficult. An approach for utilizing current injection pulse lasing at RT involves putting a post beneath the cavity center and setting the cavity center E node [4,12,27]. However, this approach is incompatible with a high Q factor and the laser becomes inefficient if the active region is placed at the center. In last decade, L3 [5–7,9,10,21,31] has been regarded as the best way of providing an appropriate Q/V_m for lasing and space for putting the active region [6,9,21]. but this work has shown that L1, whose footprint is significantly smaller than that of L3, exhibits CW lasing at least without sacrificing LD performance. The L2 nanolaser diode is also highlighted by its lowest threshold (~4 μA). The poor output power efficiency in the latter is a serious issue and we are addressing this issue by employing a different approach. A subwavelength BH active region can be embedded in the small center cell(s) of L1 and L2 nanolaser diodes and can utilize current injection RT CW lasing. The footprint of L1 nanolaser is essentially close to that of the L0 nanocavity [13] and about ten times of a (~440 nm in present work). Although various realization of nanolasers with footprint smaller than present work was demonstrated by optical pumping and/or low temperature [1], current injection CW nanolaser with foot-print actually smaller than 5 μm × 5 μm operated at RT was only demonstrated by metallic-cavity semiconductor lasers with larger V_gain and I_th higher than 1 mA [34,55]. Actual active region size of VCSELs with oxide aperture smaller than 3 μm [54] were significantly enhanced by lateral carrier diffusion and no current injection RT CW operation of all-dielectric VCSEL with a total device footprint smaller than 5 μm × 5 μm was presented. L_x nanolasers whose does not need thick (>> 1 μm) vertical distributed Bragg reflection mirrors unlike VCSELs and the micropillar lasers [28] had a vertical size of only 0.25 μm in present work. An L1 nanolaser diode can realize a dense LD array at least with a pitch smaller than 10 μm.

4. Summary

We have demonstrated ultrahigh-Q few cell nanocavities and their application to nanolasers. In experiments, multi-hole-tuned L0-L2 nanocavities exhibited Q values exceeding 10⁶, a theoretical V_m of ~1 (λ/n)³, and a fairly high coupling efficiency (several percent for NA = 0.4 objective lens) with a smaller footprint than an L3 nanocavity, which are highly useful characteristics for various fundamental studies. Newly designed L1 and L2 nanolaser diodes with active volumes smaller than 10⁻¹⁴ cm³ exhibited RT CW lasing with a DC current injection of only several μA. The smallest RT CW semiconductor LDs were operated at a bias voltage as low as 1 V. We have revealed that we should design a nanolaser with a Q of ~10⁴ or higher and enhance Γ/V_gain by employing a ultrasmall BH active region in order to obtain good LD performance at RT and overcome the detuning between the lasing mode and active region (here the MQW). The high-Q L_x nanolasers will exhibit a high cavity-QED effect and/or a high β with a tuned narrowband active region. The ultra-compact BH-L_x nanolaser diodes have high potential as on-chip laser sources. And these BH-L_x nanolasers diodes with a low operating power may allow us to realize a highly dense LD array.

5 Appendix 1: microscopic model for high-Q 2D PhC nanocavities based on W1 waveguide mode (Figs. 1 and 6.)

We have demonstrated ultrahigh-Q few cell nanocavities and their application to nanolasers. In experiments, multi-hole-tuned L0-L2 nanocavities exhibited Q values exceeding 10⁶, a We assume a 2D PhC slab and a triangular lattice of triangular holes. The W1-based defect mode is formed by zero (L0/H0) [8,37,60], one (L1), or several (Lx) missing holes on the z-axis combined with shifting axial holes on the z-axis. The orientation of the x-axis is +-90° in the slab. The y-axis is vertical to the slab. Axial holes are holes placed in the W1 line defect. Side holes are the innermost holes of the W1 line defect. According to studies by a team at Kyoto University [42,43], we can focus on E_x and neglect the impact of E_z. By analyzing various point-defect and line-defect nanocavities, we find the following rules empirically, which satisfy the Maxwell equations.

• I. Out of the axial holes (on the z-axis), electric field antinodes form standing waves as they are parallel to the hexagonal lattice ( ± 60°, corresponding to Bragg reflection by the lattice). If its origin is W1-like mode (anti-crossing [44]), no standing waves are formed along 0°.
• II. Side field discontinuities (A: yellow broken lines) are formed on both sides of the localized mode (W1 mode). At the center, the phase shift is 90°. When a nanocavity is formed, the phase shift becomes smaller than 90° as it moves further from the center. When there are only a few point defects (L0, L1,…), A become ambiguous if substantial axial hole tuning is not performed because of the conflict between the axial and side holes as a result of rule IV.
• III. In a cavity whose center is exactly at the mid-point between two lattice points (L0,L2,L4,…) the electric field has a nodal plane on the xy-plane because of the odd symmetry.
• IV. A magnetic field antinode is located inside a hole. An electric mode antinode is located outside or on the edge of the holes but never inside the holes. (III may contradict IV). According to this rule (IV) when there is a significant mismatch between the axial/side hole position and the cavity mode, the holes push the latter (green arrows in Figs. 6(a)-6(d) which is severely distorted and it significantly disturbs and/or almost eliminates A.

• V. In L-type nanocavities field discontinuities (B: blue broken lines) appeared from the m-th axial holes (m = 1: innermost hole; usually m< = 4) and extend along Γ-M orientation ( ± 30°). Out of B standing wave is like a plane wave and the wave front is nominal to the z-axis (0°) suggesting a slab mode rather than a W1-mode. The location of B depends on the hole tuning and B terminates the side field discontinuities (A) in II.
• VI. In addition to the radiation loss originating from the k-component inside the light cone, additional out-of-plane loss is caused by the conflict (or mismatch) between the nanocavity mode and the holes. A nearly ideal design and a good nanocavity mode pattern is a line-defect-based mode-gap cavity with Q ~10⁸ [45,46]. In untuned L-type cavities Q is seriously restricted by the field compression [induced?] by both the axial and the side holes, and the result is radiation loss due to the mismatch between the cavity mode and the hole. The relatively large outward shift of the axial holes (1-5) makes the mismatch small and the side discontinuities (A) smooth as it is close to that of the W1 mode, and also results in a Q value of 10⁶-10⁷. But because of the excess loss caused by putting the axial holes in the line defect, the Q of the L-type nanocavities never reaches 10⁸. Note that the axial hole tuning dominantly defines the cavity mode whereas the side hole tuning only accommodates the mismatch between the side discontinuity (A) and the holes, although the latter has a significant impact on Q. In partially-tuned L-type nanocavities as shown in Figs. 6(a)-6(d) the axial and the side holes compress the cavity mode significantly and shrink the mode volume V_m (which is smallest in L0 where V_m is 0.25 (λ/n)³ or smaller [8,60].)
• VII. To reduce the disorder-induced loss of the W1 mode, the side holes (A is formed on the inner border of the side holes) should be placed exactly on the lattice point. But this contradicts the side hole tuning used to minimize the mismatch between the cavity mode and the hole position. In very small nanocavities such as L0 and L2, the small lateral shift of the side holes (S and T) may be greatly enhanced Q to ~10⁷ because the yield obtained by minimizing the conflict between the cavity mode and the holes (S and/or T) may be much greater than the cost of the disorder induced by tuning them . In contrast, with L1 and L3, tuning S, T, and U does not improve the mismatch effectively and the tuning was invalid. An analysis of position dependent loss [41,43] suggests that leakage mainly occurs in the vicinity of the cavity center, suggesting that Q is limited by the mismatch induced by the axial holes (1-5) and the side holes (S-U). The availability of side hole tuning separates the designs (L0 and L2) where Q can exceed 10⁷ or the designs (L1 and L3) where Q remains below 10⁷.

 

Fig. 5 Theoretical simulation results. (a). Hole shifting (s₁-s₄) of multi-hole-tuned L-type nanocavities (L0A-L3B) at a = 370 nm, r = 0.25a. s₅ was 0 in L0A, L1A, L2A, and L3A. S5 was s1/2 in L2B and L3B. (b). λ_c and V_m at the settings in A as a function of d. (c). Change in Q caused by side hole shifting (s_x = s_S) in L0A (a = 375 nm, d = 300 nm, r = 0.25a, s1 = 0.390a, s2 = 0.351a, s3 = 0.312a, s4 = 0.230a). (d). Change in Q caused by deviation of s₁-s₄ and s_x from the optimized value. Nanocavity is identical to that in c. (e). 2D mapping of Q by side-hole-shifting (s_S and s_T) in L2B (d = 300 nm in Fig. 1B). (f). FEM simulation results (COMSOL). The nanocavities were identical to B/C at L0A and D at L2B. a-e were obtained by FDTD simulations. Fig. 5

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Fig. 6 E_x mode profiles of the 0th nanocavity mode of various nanocavities obtained by FDTD. (a)-(f) (left panel) schematic of the nanocavity and (right panel) its E_x profile. As regards the PhC parameters, a = 400 nm, d = 220 nm, r = 100 nm in (a) (L0), a = 408 nm, d = 210 nm, r = 100 nm in (b)-(e) (L1-L3), and a = 420 nm, d = 204 nm, and r = 108 nm in (f) (width modulated modegap nanocavity [5]). The color bar in the left panel is the same as that in Fig. 1(b).

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The optimized geometrical designs can be clarified in accordance with rules I-VII. Macroscopic models [42,43,47] need a good Gaussian-like mode distribution for realizing an (ultra) high-Q nanocavity mode, but this microscopic model emphasizes the formation of a W1-like localized mode and its good boundary (A/B). They are realized using axial-hole tuning (1-5). Tuning other holes including the side holes usually induces disorder for the W1-based mode and is likely to degrade Q. But in some cases, shifting the side holes a little {such as shifting S (and T) in L0 (L2)} greatly improves the serious mismatch between the cavity mode and the hole position and thereby improves Q. To tune three (1-3) or more consecutive axial holes appears to be ineffective for L4 and larger L-type cavities since the location of B on the z-axis where the W1 mode is terminated is inside 4a from the cavity center with or without tuning. However, another multi-hole-tuning rule [9], which can shift the location of the termination according to the line-defect length, works effectively for L4 and larger nanocavities. The optimized position of 4-5 in L1A and 3-5 in L2B/L3B appears somewhat strange, but Fig. 1(c) and Fig. 6 reveal that the tuning is strongly related to the termination of the W1 mode (location of boundary B).

6 Appendix 2: classification in multi-hole tunings (A/B)

Type-A multi-hole tunings (L1A/L2A/L3A) refer to the simplified multi-hole tuning in L0A. Step-by-step tuning can optimize the Q value [36,41] but Type-A and another type of multi-hole tuning [32] can optimize several holes by combining their tuning. With Type-A, tuning can be started from an initial setting in which all the shift parameters (s₁, s₂,…) are zero. A parameter space including (s₁=s₂=…=0) may have a local maximum whereas another parameter space may have a higher local maximum (Q). In fact, the tunings reported here set s₁ much higher than 0.3a, which optimized Q to the local maxima. This is different from the case where the tuning set s₁ at 0.2a [36,41,42]. In Type-B multi-hole tunings (L2B/L3B) s₄ is larger than 0.9a and s₅ is exactly 0.5a, which is as good as in another tuning [9] (Fig. 6(e)) s_C is larger than 0.03a. If we start the tuning from the initial parameter set (s₁=s₂=…=0) it is difficult to find the tuning rules described above. The genetic algorithm employed by Minkov and Savona [35], which can disperse the parameters greatly, is a powerful tool for tuning where the optimized parameter is far from 0.

7 Appendix 3: coupled model theory for comparing transmitted light to the output on-chip waveguide and with dropped light radiated vertically [36,61]

As shown in Fig. 7, we assume a PhC nanocavity coupled with two on-chip PhC waveguides. Port 1 and port 2 are coupled with input/output optical fibers. We assume that the length (L) and propagation constant (β), and the cavity-waveguide coupling (Q_C) of the two PhC waveguides are the same. We assume no radiation to the PhC slab and a total vertical radiation rate of 1/γ_V (Q factor to cavity-vacuum (air) coupling Q_V.) We neglect the absorption loss in the nanocavity and then the horizontal Q factor Q_H is Q_C/2 and Q_V is equal to theoretical Q. The unloaded Q factor, which is the Q of an isolated nanocavity, is Q_V. And the loaded Q factor, which is directly evaluated in an experiment, is total Q (Q_T). At a cavity mode frequency of ω₀, we obtain the relations described in Fig. 7.

 

Fig. 7 Analyzed model and derivations of relations by coupled mode theory for evaluating coupling efficiency η.

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Here we assume that the vertical radiation to the bottom is completely dissipated and only the radiation to the top can be collected. We can specify T, Q_V, and η×s_rad,up (η: coupling efficiency to objective lens) experimentally and then evaluated η as shown in Fig. 2. We highlight that even if Q_V is 10⁶ or 10⁷, we can set s_rad,up at ~0.5 (or T~1) by appropriately setting the coupling Q factors. It is evident that an ultrahigh intrinsic Q does not exclude the high coupling efficiency of a nanocavity.

8 Appendix 4: Nanofabrication of Si PhC samples

In accordance with previous studies [18,32,45], we fabricated the device using electron beam lithography. We drew the PhC patterns with d=230 nm at an acceleration voltage of 125 kV and other PhC patterns were drawn at an acceleration voltage 100 kV. Si PhCs were patterned by dry etching with the direct use of a positive tone resist (ZEON ZEP-520A) as a mask. For L_x nanolaser diodes a SiN mask was used to etch an InP-based PhC. The Si membrane PhC pattern was released from underneath the oxide layer by wet etching using buffered HF. In Fig. 2, the PhC parameters are as follows. (d=200 nm:) a=415 nm, r=100 nm in L0A, a=410 nm, r=100 nm in L1A and L2B. (d=230 nm:) a=400 nm, r=95 nm in L0A, L1A, L2B, and L3B. (d=300 nm:) a=375 nm, r=90 nm in L0A, a=370 nm, r=90 nm in L2B and L3B.

9 Appendix 5: nanofabrication of nanolaser diode samples.

The nanolaser diodes reported here are technically similar to previously reported nanolaser diodes with an InGaAs/InAlAsP MQW active region [26]. The etching mask (SiN) length of the BH was designed to be 1.4a, 2.25a, 3.1a, and 5.1a for L1, L2, L3, and L5, respectively. The mask width was 0.29 μm or narrower in all L_x nanolaser diodes. In the wet-etching process the BH width was reduced by 0.1 μm, and so the final BH width was ~0.2 μm. The BH length was not changed from the mask length by the etching. The device with the exception of the BH and PhC was patterned by a stepper. A p-doped region was formed by Zn diffusion and an n-doped region was formed by Si ion implantation. The shape of p/n-doped regions was trapezoidal, and the width under the Au electrodes was 10 μm. The width of the BH side of the p/n-doped regions was x×a for Lx nanocavities. The gap between the BH and the doped region was less than 1 μm. Every nanolaser diode had 50×50 μm ² p and n electrode pads to allow access to a pair of probes for electronic operation.

10. Appendix 6: detail of measurement of ultrahigh-Q few cell Si nanocavities and evaluation of coupling efficiency to lens by FDTD simulation

 

Fig. 8 Measurement setups and samples. (a). Schematic of transmission measurement and drop measurement from the top side. The right image is from the NA = 0.40 objective lens actually used in the experiments. (b). Bird’s-eye views of the thick (d = 300 nm, d/a = 0.80) Si L0A nanocavity obtained by SEM. (c). Ultrahigh Q cavity mode spectra obtained at d = 200 nm (left: L0) and d = 300 nm (right: L2). (d). Coupling efficiency with NA = 0.4 objective lens evaluated by comparing the transmitted light and the dropped light. (See Appendix 3 for details.) *η_av was the average value. Data points where η was >0.4 were excluded. The abnormally high η was probably due to additional fabrication-related loss in the output waveguide. (e). Expanded far field patterns inside the light cone (white dashed circles) and the cone corresponding to an NA = 0.40 objective lens. Coupling efficiency η was obtained by integration over the two cones. All theoretical data were obtained by FDTD simulations. There was no substantial difference in η among the L0-L2 nanocavities. The large difference in the laser output powers (or efficiencies) of the Lx nanolaser diodes in Fig. 4(e) are considered to be caused by differences in internal quantum efficiency and/or differences in mode competition. Fig. 8

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11. Appendix 7: measurement of nanolaser diode samples

 

Fig. 9 (a)-(b). Second order photon correlation function g²(t) measured at 1. 3I_th, 2. 1I_th, and 3. 0I_th In L1 nanolaser diode (a) and I_th and 3.2I_th In L2 nanolaser diode (b). (c–e). Measurement results 3QW L1 nanolaser diode (a = 435 nm, BH size: 0.2x0.61x0.15 μm³, V_gain: 2x10⁻¹⁵ cm³) (c). Expanded I-L curve. (d-e) Expanded (d) and wideband (e) EL spectra. Q_th was 69,000. Fig. 9

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We used an optical spectrum analyzer (wavelength accuracy: ±0.04 nm, minimum resolution: 0.015 nm) for our spectrum measurements. We did not use an optical amplifier such as an Er-doped fiber amplifier. To obtain I-L curves a tunable BPF was used to extract the lasing mode only. The passband was set at 10 nm (insertion loss: 5 dB) and the center wavelength wastuned to that of the lasing mode. For photon correlation measurements, we used a superconducting nanowire single photon detector (SNSPD) with ultrahigh sensitivity cooled by liquid helium and a time-correlated single photon counting (TSCPC) module.

 

Fig. 10 (a)-(b). Blue laser microscope images (a) and wideband EL spectra including the lasing mode above the threshold in L3 and L5 nanolaser diodes (b). Fig. 10

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Funding

CREST, Japan Science and Technology Agency (JPMJCR15N4).

Acknowledgments

We thank H. Onji, T. Tamamura, J. Asaoka, D. Takagi, O. Moriwaki, K. Ishibashi, and Y. Shouji for sample preparation, H. Onji for performing measurements, Prof. T. Aoki of the Tokyo Institute of Technology for providing GPGPU-based FDTD simulation codes, S. Kita for helping with COMSOL simulations, K. Takata for designing the BH PhC nanolaser diodes, and T. Gotoh and T. Sogawa for constant support.

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