1. Introduction

Semiconductor nanolasers, having low threshold, high energy-efficiency and small foot-print, have been under intensive study, with prospects for a wide range of applications from optical interconnection [1] to sensing [2,3]. Pursuit of one of the ultimate forms of such nanolasers comes down to the development of thresholdless nanolasers [4–8], in which almost all the spontaneous emission is funneled into the lasing cavity mode, thus leading to high energy efficiency [9] and other superior properties such as fast intensity modulation capability [10]. There are three key requirements for realizing thresholdless lasing [5,10]: (1) a near unity spontaneous emission coupling factor (β), which describes the fraction of spontaneous emission that couples into the laser cavity mode, (2) negligible non-radiative processes and (3) low gain-material absorption for intracavity photons. Some of the state-of-the-art nanocavities, including those based on photonic crystals [1,7,8] (PhCs) and plasmonic structures [6], are advantageous for increasing β by enhancing the spontaneous emission into the cavity mode and/or by suppressing unwanted coupling to other optical modes [6,11]. It is also known that careful choices of the gain material and its size lead to the suppression of the non-radiative recombination and the material photon absorption. Semiconductor quantum dots (QDs) are one of the best gain media for these purposes and, furthermore, can provide additional superior laser performances in terms of e.g. temperature stability [12] and narrower laser linewidth [13], thus offering a distinguished platform for exploring high-end thresholdless lasers.

Thresholdless lasers do not exhibit any apparent kinks in their light output curve, hindering the straightforward identification of the laser transition. This difficulty has resulted in long-lasting arguments [14–16] on how to recognize lasing in such devices. It is now widely accepted that an indispensable experiment is to measure the change in coherence properties of emitted light at least up to the second order [17–19]. The important point here is that the change should originate from the lasing itself, and not from other physical processes in the nanocavities and/or gain media [20]. Accordingly, a multifaceted assessment of various light output properties becomes essential [7,14]. Moreover, consistency between the experimental data and theoretical modeling should be carefully addressed. In this regard, to the best of our knowledge, a convincing experiment of the thresholdless lasing with a ‘coherent’ set of data, in particular for QD nanolasers, is still missing.

In this paper, we demonstrate thresholdless lasing in a PhC QD nanolaser driven under cavity resonant excitation [21,22], which enables focused carrier injection into QDs located in high cavity field regions. We observe a highly linear output curve of the nanolaser and carefully confirm its laser transition through combined experiments on the laser output power, linewidth, second order coherence and the relaxation oscillation. We also perform the same set of experiments on the same nanolaser under conventional above bandgap excitation, in which a threshold behavior in the output curve is detectable, thereby permitting a direct and systematic comparison between the threshold-less and thresholded laser transitions. We find that the two laser transitions can be consistently reproduced using a semiconductor laser model, shedding the light on the essential requirements for the realization of thresholdless QD lasers as energy efficient high-end nanolasers.

2. QD nanolaser design and basic characteristics

Figure 1(a) shows a schematic of our thresholdless laser operating under the cavity resonant excitation [21,22], through which we inject carriers only into QDs located in high cavity field regions. The injected carriers undergo efficient radiative coupling to the cavity mode with a high β, assisted by the Purcell effect and also by the photonic bandgap effect that suppresses the emission to non-lasing modes. The cavity is based on a L3 type photonic crystalnanocavity [23] made of GaAs and Figs. 1(b) and 1(c) show simulated electric field profiles for two cavity modes relevant to the experiment. The fundamental cavity mode resonating around ~1065 nm (Fig. 1(b)) sustains the lasing, while the 5th order mode at ~960 nm (Fig. 1(c)) is used for the cavity resonant excitation. The two modes have a large spatial overlap: roughly 80% of high field regions of the 5th mode are located within those of the fundamental mode. Here, the high field regions are defined as areas with an electric field intensity greater than 20% of the field maximum, in which strong QD-cavity interactions are supported. Besides, the overlap of the two modes (calculated for squared electric field) can be estimated to be ~55% (see Appendix for the details). Thus, the carriers injected through the 5th mode are captured into the InAs/GaAs QDs that strongly interact with the lasing cavity mode. In addition, the pump wavelength corresponds to the second excited state of the QD. Therefore, the three dimensional confinement effect for the generated carriers suppresses any carrier escape, further facilitating the focused carrier injection. Figure 1(d) shows a photoluminescence (PL) spectrum of the sample under weak above-bandgap excitation by an 808 nm laser source at 50 K. The broad peak centered at ~1070 nm originates from QD ground state emission, and the sharp peaks are due to several cavity modes. The fundamental cavity mode (~1065 nm) resonates with the QD ground state emission while the 5th mode (~960 nm) resonates with the 2nd excited state of the QDs.

 

Fig. 1 Structure of the PhC nanocavity QD lasers. (a) Schematic of the PhC nanocavity laser driven under the cavity resonant excitation. Carriers are injected only into the defect region of the nanocavity. (b,c) Electric field distributions of the fundamental (b) and the 5th order (c) cavity modes. The fundamental cavity mode is used for supporting the lasing oscillation while the 5th mode is for the cavity resonant excitation. It can be seen that the two mode has a large spatial overlap. The lattice constant of the PhC, a, is set to be 287 nm. (d) PL spectrum under above bandgap excitation at 50 K. The broad emission peak originated from QD ground state spontaneous emission and sharp peaks from cavity modes. Lasing and pumping modes are indicated by red arrows.

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3. Light output properties under different excitation conditions

Figure 2(a) shows experimental light-in-light-out (LL) curves plotted on a double logarithmic scale under above bandgap excitation (blue) and the cavity resonant excitation (red) taken at a lower temperature of 15 K. The detected light output powers, corresponding to peak integrated intensities of the cavity mode emission, were extracted by peak fitting. For the cavity resonant excitation, the pump laser wavelength is tuned to the resonance peak of the 5th cavity mode. The two LL curves are plotted so that they share the same lateral axis in a form of carrier injection rate, which is obtained from fitting the LL curves with a semiconductor laser model including two types of gain (as shown in Fig. 2(b) and also described in Appendix in detail). The two gains respectively express the QDs in the high field region (thus high β) and in the exterior of the defect cavity region (low β). For the above-bandgap excitation, the LL curve exhibits the well-known behavior of a high β nanolaser [10]: it starts from a low efficiency linear increase at low injection rates, then shows a mild intensity jump with kinks around the threshold injection rate of ~700 GHz (corresponding to a pump power of ~10 μW), and finally converges into the more-efficient linear increase well above the threshold. Meanwhile, the LL curve of the resonant excitation exhibits a featureless straight line without any noticeable kink. This thresholdless behavior can also be confirmed in linear scale plots of the LL curves in Fig. 2(c). Indeed, a very high β of ~0.97 is extracted for the LL curve under cavity resonant excitation (β ~0.22 for the above bandgap excitation) through the fitting by the model. In the logarithmic scale plot, it is apparent that the output powers of the laser cavity mode for the two excitation conditions become comparable each other in the high injection rate region (> 6000 GHz). This suggests that, at these high injection rates, the intracavity photon numbers sustained under the cavity resonant excitation nearly equals to those of the above bandgap excitation operated well above the threshold, since the two curves are measured for the same nanocavity having a single cavity leakage rate. Under these high injection conditions, the intracavity photon number in the simulation reaches over 100. Therefore, as long as the lasing oscillation occurs under the above-bandgap excitation, significant stimulated emission (and hence lasing oscillation) should also occur under the cavity resonant excitation. Moreover, the measured output powers reasonably agree with those calculated. The maximum measured output power for the above-bandgap excitation is 3.5 nW and the corresponding simulated output power is 3.9 μW. The three order difference is well explained by the detection efficiency of our PL setup on the order of 0.1%, which is composed of the coupling from the nanocavity to objective lens (30% or less) and transmission efficiency of the detection path (~1%). The agreement between the experiment and simulation further evidences the accumulation of the large photon number in the cavity mode.

 

Fig. 2 LL curves under the two excitation schemes at 15 K and their modeling. (a) Double logarithmic scale LL curves measured under the cavity resonant (red data points) and above bandgap (blue data points) excitation. Solid lines are obtained by fitting. (b) Laser model that contains two types of gain for describing our experiments. κ is the cavity leakage rate, β_i is the spontaneous emission coupling factor of the i-th gain medium, N₁^T is the transparence carrier number, γ_i is the total spontaneous emission rate, P_i is the pump rate. The attached table summarizes the parameters used for the fitting to the data. (c), Linear scale LL curves measured under the two excitation schemes, clearly showing the disappearance of the threshold behavior under the cavity resonant excitation.

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4. Linewidth behavior

Figures 3(a)-3(d) show emission spectra of the fundamental cavity mode under the above-bandgap and cavity resonant excitation. Under high pumping conditions, both the two lasingspectra exhibit similar line shape with significantly narrowed linewidths, as expected for the lasing oscillation. Under weak pumping conditions, the two spontaneous emission spectra exhibit a clear difference in the background emission. For the above-bandgap excitation, a significant background emission can be seen below the sharp emission peak. This background is not present under the cavity resonant excitation conditions. The background emission primarily originates from QD emission that did not couple to the cavity mode. Therefore, the disappearance of the background emission suggests much higher β for the cavity resonant excitation than that of the above-bandgap excitation. Figure 3(e) shows an evolution of emission linewidths when increasing the injection rate, exhibiting considerable linewidth narrowings of roughly one order of magnitude. At very low injection rates, the two linewidths measured under the cavity resonant and above-bandgap excitation converge into a single value. For low injection rates (< 1000 GHz), the linewidths for the cavity resonant excitation narrow faster than those of the above-bandgap excitation as increasing pumping, suggesting faster reduction of absorption loss due to more efficient carrier injection into high cavity field regions. These behaviors are well described by a linewidth model [10] (solid lines) consistently using the same gain model and parameters when analyzing the LL curves (see Appendix). When increasing injection rate, both the linewidth curves start to deviate from the simple linewidth model and exhibit well-documented plateaus, which are known as the result of the gain-refractive index coupling [24,25] and of the transition from the conventional Schawlow-Townes linewidth to its modified version. The points of the deviation in the linewidth curves are associated with the rise of gain in the system (see Appendix), which can be regarded as indicators of the onset of the laser transition [24]. Indeed, the points of the deviation correspond well to the points at which the positive gain appears in the simulation, which are indicated by vertical dash lines in Fig. 3(e). In the high injection regions, the observed saturation of the linewidth narrowing is likely due to a continuous increase the effect of the gain-refractive index coupling [26]. Note that the description of linewidth behaviors in this region will require more sophisticated linewidth models [27–29], the analysis of which lies beyond the scope of this paper.

 

Fig. 3 Comparison of emission spectra and linewidths under the two excitation schemes. (a,b), Emission spectra at 15 K at low injection rates about 17 GHz and (c,d) at high injection rates about 7700 GHz. Spectra under the above bandgap (a,c) are shown in blue while those under the cavity resonant excitation (b,d) are in red. (e) Linewidths plotted as a function of the injection rate. In particular for the low injection rates, linewidths for the cavity resonant excitation (red balls) exhibits faster narrowing than those for the above bandgap excitation (blue). Solid lines are of the calculation results using the simple linewidth model and the same parameter for simulating the LL curves.

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5. Evolution of the second order coherence function

Figure 4(a) and 4(b) show the evolutions of the second order coherence functions, g⁽²⁾(t), under different injection rates for both the above-bandgap and cavity resonant excitation, measured by a Hanbury Brown-Twiss (HBT) intensity auto-correlator equipped with ultrafast superconducting single photon counters [30]. For low pumping powers, both cases exhibit bunching at zero time delay with rapid monotonic decays as the delay time increases. In general, the decay slopes are determined by the coherence time of the emission [31], which is shorter than 10 ps in our nanolaser under low pumping conditions due to gain material absorption. Therefore, the measured decay slopes for low injection conditions are mainly determined by the time response of our HBT setup (~60 ps). Meanwhile, when increasing the injection rate, we observe relaxation oscillations, which in general can be regarded as a sign of the onset of the laser transition. An example oscillatory g⁽²⁾(t) obtained under cavity resonant excitation with an injection rate of 760 GHz is shown in Fig. 4(c). It is worth noting that the clear relaxation oscillation was observed only around the laser transition region, where the relatively strong intra-cavity intensity noise can trigger the oscillation [17,30,32]. For high injection rates, we observed flat g⁽²⁾(t) curves, as expected for the generation of the coherent state of light. Figure 4(d) shows a summary of zero-delay values of the second order coherence function, g⁽²⁾(0). For both the pumping conditions, transitions from thermal-like (g⁽²⁾(0) ~2) to coherent-like (g⁽²⁾(0) ~1) state are observed. The g⁽²⁾(0) under the cavity resonant excitation exhibits faster transition to the coherent state, reflecting the faster accumulation of intracavity photons and the faster development of strong stimulated emission.

 

Fig. 4 Second order coherence measurement results. (a,b) g⁽²⁾(t) curves measured under various pump powers for the cavity resonant (a) and above bandgap (b) excitation. For the above bandgap excitation, PL intensity when P = ~100 GHz was too low to measure the corresponding g⁽²⁾(t) curve. (c) g⁽²⁾(t) curve showing the relaxation oscillation measured under the cavity resonant excitation at P = 760 GHz. Gray arrows indicate the peak positions of the oscillation. For (a)-(c), solid lines correspond to fitting results. (d) Evolution of g⁽²⁾(0) values as a function of the injection rate. The plot for the cavity resonant excitation (red points) exhibits faster decrease toward unity than that for the above bandgap excitation (blue points).

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6. Temperature dependence of laser output curve

Next, we investigated changes in LL curves when increasing or decreasing the sample temperature. Figure 5(a) summarizes the evolution of LL curves taken under the cavity resonant excitation at 4.5, 15 and 40 K. For the cases of 4.5 and 40 K, we observe a change to thresholded lasing with mild kinks and reduced βs of 0.50 and 0.49, respectively. These β values are extracted from fitting to the LL curves with a simple single-gain laser model, in which we set N₁^T = 55 and β₂ = γ₂ = N₂^T = 0. Figure 5(b) summarizes the change in β as a function of temperature. β has a peak value at 15 K, and exhibits a gradual reduction for both increasing and decreasing the temperature. This behavior can be interpreted by considering the temperature dependence of the off-resonant emitter-cavity coupling [33,34], which is known to be prominent in QD-nanocavity systems and efficiently connects the emission from frequency-detuned QDs to the cavity mode. We primarily consider the contribution of the acoustic phonon mediated off-resonant coupling [35,36], in which the dominant acoustic phonons relevant to the process are assumed to have energies of a few meV. Therefore, the phonon population around the QDs should be highly temperature dependent in the measured temperature range (4K to 40K), which corresponds to an increase of the product of the Boltzmann constant and the temperature from 0.35 to 3.5 meV. Accordingly, at low temperatures such as 4.5 K, stimulated emission of phonons is too low to enable the efficient off-resonant coupling especially for higher energy detuned QDs, leading to the reduction of overall β. Meanwhile, at higher temperatures (40 K), the excess phonon-QD coupling and other sources of decoherence turned on by heating significantly degrade the efficiency of the QD-cavity coupling and hence the overall β. Another possible explanation for the reduced β at high temperatures could be the diffusion of carriers from QDs to low cavity field regions.

 

Fig. 5 Temperature dependence of the laser transition (a) LL curves measured under the cavity resonant excitation at 4.5 K (magenta), 15 K (red) and 40 K (dark yellow). Solid lines are of fitting using a laser mode with a single gain medium. Dashed lines indicates the β = 1 straight lines. The 4.5 K and 40 K curves are offset by multiplication factors of 10 and 0.1, respectively. (b) Evolution of the extracted β value when changing the temperature. Error bars are determined by laser model fitting to the observed LL curves.

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7. Discussion

Finally, we discuss the relevant number of QDs for the thresholdless lasing oscillation. The transparency carrier number of the main gain medium, N₁^T, is 47, which can be read roughly as the number of undoped QDs located in the high cavity field regions, supposing the contribution of various QD excitonic species such as biexcitons. This small number of emitters reduces the absorption-induced photon loss in the cavity and is essential for realizing the thresholdless operation [10] (see also Appendix). It is estimated that these 47 QDs have an average β₁ of 0.97, corresponding to an average cavity-enhanced spontaneous emission rate of 3.8 GHz. This somewhat-high average emission rate into the cavity mode for the inhomogeneously broadened QDs can be explained by taking into account the contribution of the phonon-mediated off-resonant QD-cavity coupling [35], which can efficiently mediate the radiation of QDs into a cavity mode detuned by a few meV. This phonon-assisted coupling is prominent for QDs placed in high Q/V cavities due to strong light matter interactions therein. This process does not presume dephasing of QDs’ zero-phonon linewidths as like in pure-dephasing induced off-resonant QD-cavity coupling [34]. Consideration of the phonon-mediated coupling is also important for explaining the estimated value of N₁^T. The total number of QDs that are located within the high field regions and are pumped by the cavity resonant excitation is estimated to be ~207. Here, the QDs in the high field regions, in which the electric field intensity is greater than 20% of the field maximum, are supposed to possess coupling constant to the cavity mode greater than 30 μeV, which is large enough to mediate the phonon-assisted processes [35]. Given the QD inhomogeneous broadening (42 meV), 46 out of those 207 QDs have the energy detunings less than ± 5 meV. These numbers reasonably agree with N₁^T ( = 47) and roughly with the range of the efficient acoustic phonon-mediated off-resonant coupling. The relatively large detuning range of ± 5 meV could be explained by the fact that single QD emission peaks have energy distributions over several meV due to the presence of various complex excitonic states, some of which would have small energy detunings with respect to the cavity mode and still efficiently couple to it. It is noteworthy that the experimental verification of the relatively fast decay rate into the cavity mode under cavity resonant excitation will be an important future work. Moreover, developing more detailed QD-cavity models for explaining their efficient radiative coupling will also be of future interest.

8. Summary

In summary, we have demonstrated thresholdless quantum dot nanolasing under cavity resonant excitation conditions. The lasing transition is verified through a multifaceted analysis, consistently explained by a laser model. Our study here points out the importance of the focused carrier injection, the reduction of the QD number and the phonon-mediated off-resonant QD-cavity coupling for the realization of the thresholdless operation. An immediate implication of this work in terms of the realization of truly energy efficient nanolasers is the use of buried-hetero gain structures [1,7] in order to perform the focused carrier injection by strictly limiting the gain region.

Appendix A Sample design and fabrication

We adapted a defect-type L3 photonic crystal nanocavity design [23], which is composed of three missing air holes in an air bridge 2D triangle PhC lattice with a lattice constant a of 287 nm. We tuned the sizes and positions of the air holes around the defect region so as to increase cavity Q factor. Calculated Q factor by the finite difference time domain method is 75,000 and the mode volume V is 0.83(λ/n)³. We also introduced a doubled periodic modulation of the air hole sizes around the defect region [37] for better photon extraction from the cavity mode. The resulting far-field coupling to an objective lens with a numerical aperture of 0.65 is calculated to be 30%. The PhC sample was patterned onto a QD wafer comprised of a 180nm thick GaAs slab on a 1-μm-thick AlGaAs sacrificial layer. The slab embeds a layer of nominally undoped InAs QDs (areal density ~ 5×10¹⁰cm⁻²) in the middle. The high QD density differs this work from previous efforts toward the realization of single or a few QD lasers [38–41]. The PhC patterning was done using a standard semiconductor nanofabrication processes with electron beam lithography and subsequent dry and wet etching.

As another quantity describing the overlap of the fundamental and the 5th order mode, we calculated their overlap integral as ${{\displaystyle \int }}^{\text{}}{\left|E_{1st}\left(r\right)\right|}^2{D}_{QD}\left(r\right){\left|E_{5th}\left(r\right)\right|}^{2}d{r}^2$, where E_1st(r) and E_5th(r) are respectively the field distribution of the fundamental and the 5th mode, and r is the spatial positions in the slab center, at which the QD layer is inserted. D_QD(r) describes the areal distribution of QDs and is unity (zero) at the positions with (without) QDs (assumed homogeneous, continuous distribution in the QD layer). D_QD(r) is set to be zero in the air holes. E_1st(r) and E_5th(r) are normalized such that the overlap integral between the individual modes becomes unity when being calculated with D_QD(r) =1 for any r. In short, we normalized the fields by the following equation ${{\displaystyle \int }}^{\text{}}{\left|E\left(r\right)\right|}^{4}d{r}^2=1$. The overlap integral is considered to correspond to the overlap of photo-excited QDs with the fundamental mode, since the both photon absorption and emission rates are proportional to square of the electric field intensity. The calculated value of 55% suggests the efficient injection of carriers into the high field regions of the fundamental mode via the cavity resonant excitation.

Appendix B PL measurement

Optical characterization was done by a micro-PL setup. The sample was kept in a liquid helium flow cryostat equipped with a temperature controller. An objective lens with a numerical aperture of 0.65 was used to address the sample. The primary pump source is from a fiber-coupled continuous-wave external cavity diode laser and is collimated by an achromatic reflective collimator. A fiber coupled continuous-wave diode laser source at 808 nm is also used for the above-bandgap excitation. The two laser source share an individual fiber attached to the collimator by using an optical fiber switch, keeping the spot positions of the two pump sources nearly identical. During all optical measurements, we carefully kept the relative position of the objective lens to the PhC nanolaser by routinely monitoring the PL counts under the cavity resonant excitation. The cavity resonant excitation requires a stricter alignment than the above bandgap excitation, since both the pumping and signal collection spots needed to be aligned. A half waveplate was used to tune the polarization of the pump laser source to that of the 5th cavity mode for the efficient cavity resonant excitation. A 50/50 beam splitter is used to pump the sample through the objective lens. For the cavity resonant excitation, we finely tuned the pump laser wavelength to the resonance of the 5th cavity mode before starting each measurement. Sample PL was analyzed by a spectrometer equipped with a cooled InGaAs camera. A spectral resolution of ~ 30 μeV was obtained when using a grating with a groove density of 1,350 mm⁻¹. In order to avoid overflow of the detector count, we inserted switchable neutral density filters with different optical densities. The power of the attenuation of each filter was measured by using actual sample signal every time we switched the filter. Detected laser output powers in Fig. 2 and Fig. 5 are simply deduced from integrated detector count per unit time for the cavity emission, extracted through peak fitting with a Voigt function. The second order coherence measurements were performed using a HBT setup equipped with a fiber beam splitter and ultrafast superconducting single photon counters. The overall timing resolution of the setup was estimated to be ~ 60ps.

Appendix C Laser model

We employed a semiconductor laser model comprising two types of gain media in order to describe our experiment in a simplified manner. The model is a straightforward extension of the model in the literature [10]. The laser rate equations are as follows:

(1)$$\begin{array}{l}\frac{dn}{dt}=-\kappa n+{\displaystyle \sum _{i=1,2}{\beta }_i{\gamma }_i\left(N_i-N_i^T\right)n}+{\displaystyle \sum _{i=1,2}{\beta }_i{\gamma }_i{N}_i},\\ \frac{d{N}_1}{dt}=P_1-{\gamma }_1{N}_1-{\gamma }_{nr}{N}_1-{\beta }_1{\gamma }_1(N_1-N_1^T)n\\ \frac{d{N}_2}{dt}=P_2-{\gamma }_2{N}_2-{\gamma }_{nr}{N}_2-{\beta }_2{\gamma }_2(N_2-N_2^T)n\end{array}$$

where n is the intracavity photon number for the lasing cavity mode, N_i is the carrier number for the ith gain medium (i = 1,2), β_i is the spontaneous emission coupling factor for emission from the emitter belonging to the ith gain medium. Similarly, γ_i is the total spontaneous emission rate, N_i^T is the transparent carrier number and P_i is the carrier injection rate. κ is the cavity decay rate at the transparency condition, which is chosen to be 122 GHz (80 μeV) as a typical value achieved in our PhC nanofabrication. γ_nr is the non-radiative recombination rate, which we neglect in the following analysis since our gain originates from defect-free self-assembled QDs. β describes the ratio of spontaneous emission coupled into the cavity mode to the total spontaneous emission. Thus, γ can be regarded as a function of β and expressed as ${\gamma }_i={\gamma }_{PBG}/\left(1-{\beta }_i\right)$, where γ_PBG describes the out-of-cavity spontaneous emission rate. In other words, we define the spontaneous emission coupling factor as ${\beta }_i=({\gamma }_i-{\gamma }_{PBG})/{\gamma }_i$. In our case, γ_PBG is significantly slowed down by the presence of the photonic bandgap effect and is chosen to be 0.1 GHz as a typical value for QDs in 2D PhCs [11], which is roughly ten times slower than that of QDs in unprocessed regions. We assume that QDs of the 1st gain medium are mainly located in high cavity field regions and have a high average β₁, while those of the second gain medium weakly couple to the cavity mode with low average β₂. We assume the carrier injection rates are expressed as $P_1=P{N}_1^T/\left(N_1^T+N_2^T\right)$ and $P_2=P{N}_2^T/\left(N_1^T+N_2^T\right)$ for the above bandgap excitation and $P_1=P$ and $P_2=0$ for the cavity resonant excitation, where P is the total carrier injection rate. Here, the above bandgap excitation is interpreted as the homogeneous carrier injection into both the gain media, while the cavity resonant excitation as the focused injection into the 1st gain media. We solved the equation in the steady state condition (d/dt = 0), as we employed the continuous wave pumping in the experiments. Here, we note that the use of more complex laser models might be useful to uncover detailed physics in thresholdless lasing [17,28].

The increase of quantum efficiency accompanied by the laser transition can be derived from Eq. (1). For both the strong and weak injection limit (P or P⁻¹ >> 1), the photon output rate takes a form of $\kappa n=AP$. The change in the coefficient, A, determines the amount of the intensity “jump” across the threshold, which equals to A of well above the threshold over that of well below the threshold. For the above bandgap excitation, the jump across the threshold J_above can be expressed as

(2)$$J_{above}={\overline{\beta }}^{-1}\left(1+\frac{{\gamma }_{nr}}{{\gamma }_i}+{\displaystyle \sum _{i=1,2}\left(1-{\beta }_i+\frac{{\gamma }_{nr}}{{\gamma }_i}\right)\frac{{\beta }_i{\gamma }_i{N}_i^T}{\kappa }}\right),$$

where $\overline{\beta }=\left({\beta }_1{N}_1^T+{\beta }_2{N}_2^T\right)/\left(N_1^T+N_2^T\right)$ is the average spontaneous emission coupling factor. For the cavity resonant excitation, the jump J_reso can be expressed as

(3)$$J_{reso}={\beta }_1^{-1}\left(1+\frac{{\gamma }_{nr}}{{\gamma }_i}+{\displaystyle \sum _{i=1,2}\left(1-{\beta }_i+\frac{{\gamma }_{nr}}{{\gamma }_i}\right)\frac{{\beta }_i{\gamma }_i{N}_i^T}{\kappa }}\right).$$

The ratio of the two jumps takes a simple form, $J_{reso}/J_{above}=\overline{\beta }/{\beta }_1$. Experimentally-obtained J_reso = 1.08 and J_above =4.34 are used as constraints for the fitting process. The expressions of J tell us three important conditions for realizing the thresholdless lasing (J~1) - high β, small γ_nr and small N^T. Note that βγN^T/κ expresses the intracaviy photon absorption before its leakage. Such absorbed photon eventually becomes loss with a probability of (1-β+γ_nr/γ), leading to the increase of J. In order to visually show the importance to reduce N^T for achieving the thresholdless operation, we calculated J_reso according to Eq. (2) with β₁ and N₁^T being the parameters and plot it in Fig. 6. It is clearly found that the reduction of the N₁^T roughly below a hundred is essential to achieve J_reso < 1.1 in our case.

 

Fig. 6 J_reso simulated for various β₁ and N₁^T. The rest of parameters are the same with those used for fitting the LL curves. White line expresses J_reso = 1.08. The red star corresponds to the parameter found in the fitting to the experimental data (β₁ = 0.973, N₁^T = 47).

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We also paid attention to the evolutions of the emission linewidths as a function of the pump power and try to fit them to a linewidth model. We employed an elementary linewidth model, which is based on a LC resonator connected with the gain and loss and is expected to well describe the linewidth behaviour below the threshold [10]. The emission linewidth, ∆ν, can be expressed as

(4)$$\Delta \nu =\kappa -{\displaystyle \sum _{i=1,2}{\beta }_i{\gamma }_i\left(N_i-N_i^T\right)}=\kappa \left(1+{\displaystyle \sum _{i=1,2}\frac{{\beta }_i{\gamma }_i{N}_i^T}{\kappa }\left(1-\frac{N_i}{N_i^T}\right)}\right).$$

This equation approaches asymptotically to the conventional Schawlow-Townes linewidth formula above the threshold. However, around the threshold and above, actual laser emission linewidths are influenced by the gain-refractive-index coupling, which significantly deviates the behaviour of the linewidth from the simple Eq. (4). Below the transparency condition (N_i<N_i^T), κ and intracavity photon absorption (βγN^T/κ) determines the emission linewidth. For the low pumping limit (P<<1), the linewidths under the two excitation schemes converge into the following expression:

(5)$$\Delta {\nu }_0=\kappa +{\displaystyle \sum _{i=1,2}{\beta }_i{\gamma }_i{N}_i^T}.$$

Experimentally, we observed a tendency of the convergence of the two linewidth curves to a single value as the injection rate reduces (Fig. 3(e)). The value of the zero injection linewidth was estimated to be ∆ν₀ = 195 μeV, which was used as a constraint in the fitting to the LL curves.

As mentioned above, mode linewidths of semiconductor lasers in general deviate from the Schawlow-Townes law due to the gain-refractive index coupling [42]. The deviation starts roughly when positive gain appears in the system. We calculated the gain term ${\displaystyle \sum {\beta }_i{\gamma }_i\left(N_i-N_i^T\right)}$ in the laser rate equation as a function of the injection rate and plotted it in Fig. 7. This gain term can be converted to the mode gain by dividing with the group velocity. The injection rate above which the positive gain is realized are indicated by the dashed lines. The same injection rates are also indicated in Fig. 3(e) in the same manner.

 

Fig. 7 Gain values simulated for the cavity resonant (red) and above bandgap (blue) excitation.

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In the fitting process, we have seven parameters in Eq. (1) (β_i, γ_i, N_i^T and κ) and six constrains (J_above, J_reso, ∆ν₀, γ_i=γ_PBG/(1-β_i) and κ). Therefore, we have eventually a single actual fitting parameter (N₁^T + N₂^T) for determining the shape of LL curves. In addition to this, we have three other fitting parameters: one is for converting the experimentally-measured output power to the simulation counterparts and the other two are for converting experimental pump powers to the injection rates. The extracted parameter values are exhibited in Fig. 2(b) in the main text. We found that reasonable fitting results exist only at the proximity of the deduced parameter set.

In order to discuss the relaxation oscillation in our nanolaser, we applied the small signal analysis on Eq. (1). We set steady state solutions for the photon and carrier number as n₀, N₁⁰ and N₂⁰ and the fluctuation of them as δn, δN₁ and δN₂. The evolutions of the fluctuations after the linearization are expressed by following rate equations after neglecting the non-radiative recombination (γ_nr=0):

(6)$$\frac{d}{dt}\left[\begin{array}{c}\delta n\\ \delta N_1\\ \delta N_2\end{array}\right]=\left[\begin{array}{ccc}-\kappa +{\displaystyle \sum _{i=1,2}{\beta }_i{\gamma }_i\left(N_i^0-N_i^T\right)}& {\beta }_1{\gamma }_1\left(1+n_0\right)& {\beta }_2{\gamma }_2\left(1+n_0\right)\\ -{\beta }_1{\gamma }_1\left(N_1^0-N_1^T\right)& -{\gamma }_1\left(1+{\beta }_1{n}_0\right)& 0\\ -{\beta }_2{\gamma }_2\left(N_2^0-N_2^T\right)& 0& -{\gamma }_2\left(1+{\beta }_2{n}_0\right)\end{array}\right]\left[\begin{array}{c}\delta n\\ \delta N_1\\ \delta N_2\end{array}\right]$$

By setting δN = δN₁ + δN₂ and introducing an approximation δN₁>>δN₂, Eq. (6) becomes

(7)$$\frac{d}{dt}\left[\begin{array}{c}\delta n\\ \delta N\end{array}\right]=\left[\begin{array}{cc}-\kappa +{\displaystyle \sum _{i=1,2}{\beta }_i{\gamma }_i\left(N_i^0-N_i^T\right)}& {\beta }_1{\gamma }_1\left(1+n_0\right)\\ -{\displaystyle \sum _{i=1,2}{\beta }_i{\gamma }_i\left(N_i^0-N_i^T\right)}& -{\gamma }_1\left(1+{\beta }_1{n}_0\right)\end{array}\right]\left[\begin{array}{c}\delta n\\ \delta N\end{array}\right].$$

This equation has the same form as in the small signal analysis for the conventional single gain medium model [30,32]. Then, the solution of δn can be expressed as

(8)$$\delta n(t)=\delta n(0)e^{-{\gamma }_{R}t}\cos ({\omega }_{R}t+\varphi )/\cos (\varphi ),$$

where φ is the initial phase of the oscillation and is set to zero in the following analysis. In the case that ω_R is imaginal, δn can be treated as a monotonic decay curve as follows:

(9)$$\delta n(t)=\delta n(0)e^{-{\gamma }_{R}t}.$$

These expressions of δn are connected to the second order coherence function g⁽²⁾(t) as follows:

(10)$$g^{(2)}(|t|)=1+\frac{〈\delta n(0)\delta n(|t|)〉}{n_0^2}.$$

In general, the decay slope of δn below lasing threshold is determined by the first order coherence time of the laser cavity. Indeed, for the low pumping limit (n₀, N₀ <<1), δn takes the form of the monotonic decay and its decay slope can be approximated as γ_R~∆ν₀. Therefore, it is in general required to use ultrafast techniques to measure the second order coherence function in low coherence lasers (such as nanolasers) below lasing threshold. We note that the approximation used here (δN₁>>δN₂) is very suitable for the cavity resonant excitation. We assume that the same forms of the solutions to δn can be used even for the case of the above bandgap excitation as often did in the literature [30,32]. Therefore, we fit the measured coincidence curves using either the relaxation oscillation or the monotonic decay curve, after convolved with a Gaussian function representing the measurement system time response of ~60ps.

Funding

Project for Developing Innovation Systems of MEXT, JSPS KAKENHI Grant-in-Aid for Specially promoted Research (15H05700); KAKENHI(16K06294) and NEDO project.

Acknowledgements

We thank D. Takamiya, K. Kamide and M. Holmes for their technical support and for fruitful discussions.

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