1. Introduction

Recent progress in nanotechnology, nanophotonics and plasmonics gave the possibility to bring optical signals to the nanoscale and develop a novel class of metal-semiconductor lasers that feature both ultra-low mode volume and physical footprint [1–15]. Metal is the essential part of these devices since its negative dielectric function allows to confine optical fields to a subwavelength scale, which reduces the crosstalk in high-dense on-chip photonic circuits [7] and increases the sensitivity of nanolaser-based sensors [16,17]. At the same time, the use of metal gives the possibility to increase the overlap of the laser mode with the gain medium, so that the coupling of spontaneously emitted photons to the laser mode (β-factor) can approach unity even in the case of the broadband gain media, such as bulk semiconductors or quantum wells [1,8,15]. In macroscopic lasers, the β-factor is typically very small, which results in a well pronounced amplified spontaneous (ASE) kink in the input-output characteristic of nanolaser, which is the dependence of the laser output power on the pump current or optical pump power [18]. However, in most metal-semiconductor nanolasers, the input-output characteristic is very smooth due to the high β-factor. At a low non-radiative recombination rate and β ≈1, the ASE kink completely disappears, which is known as “thresholdless lasing” [8,19]. The input-output characteristic is just a straight line in the log-log scale and does not show where the transition to lasing occurs. Nevertheless, it is evident that at low pump levels, such a thresholdless nanolaser acts as an LED, while at very high pump levels, it emits photons in a coherent state [20]. The small spectral linewidth of the light source, or equally the first-order coherence measured by the visibility of the interference pattern in Michelson, Mach-Zehnder, or Sagnac interferometers, is not enough to prove the laser action since it can be achieved simply by narrow-bandpass filtering of thermal radiation [21–23]. The laser action can be proven by proving the Poisson statistics of emitted photons [21,24,25], which can be done by measuring the second-order autocorrelation function g⁽²⁾(τ) = <I(t)I(t + τ)>/<I(t)>² [21]. In lasers based on bulk semiconductors, heterostructures, or quantum wells, g⁽²⁾(0) gradually decreases from 2 (thermal state) to 1 (coherent state) as the pump rate increases [20]. Hence, the decrease of g⁽²⁾(0) below ~1.5 can be considered as a clear indicator of the transition to lasing. However, direct measurements of the g⁽²⁾-function for nanolasers are complicated since its characteristic time is in the picosecond range, i.e., well below the time resolution of most photodetectors. This leads to large errors due to the greatly underestimated values of g⁽²⁾(0) in a standard Hanbury Brown and Twiss experiment, while more complicated measurements are not always possible [26,27]. Therefore, a deep understanding of coherence properties of metal-semiconductor photonic and plasmonic nanolasers is required for estimation whether the studied nanolaser achieves the threshold using the available experimental and numerical data.

In this work, we theoretically study the second-order coherence properties of metal-semiconductor photonic and plasmonic nanolasers based on a bulk semiconductor, heterostructure, or quantum wells. Taking into account both radiative and nonradiative recombination, we find the position of the lasing threshold defined as the degree of coherence of the emission of the nanolaser. We also derive a simple analytical expression for the threshold current, which can be applied to most metal-semiconductor photonic and plasmonic nanolasers, including thresholdless nanolasers. Finally, we analyze the impact of the β-factor, nonradiative recombination, and temperature on the position of the lasing threshold.

2. Results and discussion

A typical metal-semiconductor nanolaser is shown in Fig. 1. Many characteristics of such a nanolaser can be found using rate equations, which couple the density of non-equilibrium electrons and holes with the number of photons in the laser mode. Here, we assume the nanolaser to be electrically pumped due to the high importance of electrical pumping for practical applications. However, our analysis can be easily extended to optically pumped devices by replacing the pump current by the pump power multiplied by the appropriate coefficient since both electrical and optical pumping just provide non-equilibrium electrons and holes for the nanolaser. Two balance equations can fully describe the nanolaser in the steady state:

 

Fig. 1 Schematic illustration of a typical electrically pumped metal-semiconductor nanolaser. It consists of the semiconductor layer stack etched into a pillar and covered by a thin insulator layer. The semiconductor-insulator structure is encapsulated in silver, gold or copper to form a metallic cavity, which usually has a quality factor in the range from 100 to 300 [1,3,6–9,15,28,32].

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So-called thresholdless lasing is achieved when each act of carrier recombination produces a spontaneously emitted photon which goes into the laser mode, i.e., β = 1 and U_nr = 0. In this case, the output power is directly proportional to the pump current as follows from Eq. (1), and the light-current (L-I) curve does not show a characteristic kink at the lasing threshold. A typical thresholdless L-I curve is shown in Fig. 2(a). At low pump currents, the thresholdless nanolasers act as an LED, namely the output power is produced due to spontaneous emission and stimulated emission rate is negative (photons are absorbed in the active region rather than generated). At the same time, at very high currents, the rate of stimulated emission greatly exceeds that of spontaneous one, i.e., the laser emits light in an almost coherent state. However, there is no qualitative difference between these two regimes in the L-I curve as can be seen in Fig. 2(a). There are many indicators of the transition to the lasing regime [24,26], but only the second-order autocorrelation function g⁽²⁾(τ) provides direct evidence of coherence of the output power in a quantitative manner [24].

 

Fig. 2 (a) L-I curve of the metal-semiconductor nanolaser based on the In_0.53Ga_0.47As bulk gain medium at T = 20 K. The operating wavelength is 1.55 μm, the quality factor of the nanolaser cavity is Q = 100, the confinement factor of the laser mode to the active region is Γ = 0.5 and the volume of the active region is V = 0.3 μm³, the β-factor is equal to 1. These parameters are similar to those of the experimentally measured metal-clad and plasmonic nanolasers [1,6,8,9,15,28]. The material gain is calculated in the parabolic band approximation, the material parameters of InGaAs are adopted from [29]. (b-d) g⁽²⁾-function at pump currents of 1 μA (panel (b)), 80 μA (panel (c)) and 1000 μA (panel (d)). These currents are shown in panel (a) by points A, B, and C, respectively. Note that the time scales are different for different panels. (e) g⁽²⁾(τ = 0) as a function of U_stim/U_spont. The red line is the result of direct calculations, while the blue dashed line is obtained using Eq. (11).

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The measured second-order autocorrelation function characterizes fluctuations of the laser output power, which are equivalent to the fluctuations of the photon number in the laser mode:

The autocorrelation function <δN(t)δN(t + τ)> can be found by diagonalizing the system of Eqs. (4) and solving it using the method of variation of parameters. Figure 2(b)-2(d) shows the calculated g⁽²⁾-functions at three different pump currents. As the pump current increases, the behavior of the g⁽²⁾-function changes from ultrafast decay to damped oscillations. At low pump currents, the characteristic decay time is determined by the photon number relaxation time τ_p = 1/(1/τ_cav−G) since the dynamics of the photon number and carrier density are independent. As the pump current increases, the photon number and carriers density relaxation times τ_p and τ_c approach each other. At J_pump = 1 μA, τ_p = 0.1 ps and τ_c = 1.5 ns (Fig. 2(b)), while at J_pump = 80 μA, τ_p = 7 ps and τ_c = 125 ps (Fig. 2(c)). At high currents, the increased feedback from the gain medium makes the dynamics of the photon number and carrier density in the nanolaser inseparable, which gives rise to the relaxation oscillations in the g⁽²⁾-function (Fig. 2(d)). In macroscopic lasers, above the threshold, the frequency of relaxation oscillations is in the MHz range, whereas the considered nanolaser features ultrafast relaxation oscillations at a frequency of about 80 GHz. Such an ultrafast dynamics enables high-speed optical communication by direct modulation of the nanolaser. We should also note that at any pump current, the slowest characteristic time remains lower than 20 ps, which makes direct measurements of the g⁽²⁾-function in the Hanbury Brown and Twiss experiment very challenging [21,26,27,32]. Although it is possible to measure such a ‘fast’ g⁽²⁾-function [26,27], most measurements of the second-order coherence of nanolasers are unreliable due to the limited time resolution of the photodetectors.

At any pump level, the g⁽²⁾-function reaches its maximum at τ = 0. This maximum value steadily decreases as the pump current increases approaching g⁽²⁾(0) = 1 at very high currents, which corresponds to a coherent state of light. At the same time, the rate of stimulated emission U_stim rapidly increases with the pump current, while the rate of spontaneous emission U_spont is almost unchanged above the lasing threshold. In other words, the fraction of photons coherently emitted into the laser mode increases leading to the Poisson statistics well above the threshold. Therefore, it is reasonable to consider g⁽²⁾(0) as a function of U_stim/U_spont since U_stim/U_spont can be relatively easy obtained in direct numerical simulations of the metal-semiconductor nanolaser, as well as estimated in the experiment (see Fig. 2(e)). g⁽²⁾(0) = 2 at low pump levels. At U_stim/U_spont ≈50 (J_pump≈0.1 mA, N ≈50), g⁽²⁾(0) starts to decrease with J_pump and reaches g⁽²⁾(0) = 1.5 at U_stim/U_spont = 500 (J_pump = 1 mA, N = 500). This point can be treated as a threshold of the thresholdless nanolaser, since it is in the middle of the transition from the Bose-Einstein to Poisson statistics of photons in the laser mode. At very high pump levels, g⁽²⁾(0)-1 shows a power-law dependence on U_stim/U_spont. To understand the origin of this power-law dependence, we obtain an analytical solution of Eq. (4) for g⁽²⁾(0):

Here, we draw attention to the fact that Eq. (9), which is obtained using the classical description, is exactly the same as the expression for g⁽²⁾(0) obtained using the quantum Langevin equations [33] except the absence of the additional term (−1/N), which vanishes at large N. At the same time, we note that the quantum Langevin equations predict that g⁽²⁾(0) →−∞ as N approaches zero, which is not feasible. Thus, the standard quantum Langevin approach for lasers is not valid at small N, where it fails to account for large quantum fluctuations. It is not surprising since the quantum Langevin equations for lasers are initially derived under the assumption of large N [34], despite that the quantum Langevin approach itself is rigorous at any photon number [35]. The presented classical Langevin approach correctly predicts that g⁽²⁾(0) = 2 at low pump levels in spite of the fact the approach is rigorous only at significantly high N. To analyze the laser action at small N, methods based on the cluster-expansion technique [36–39] and birth-death equations [40] were proposed. They are developed for the gain media based on a finite number of independent emitters and are difficult to be used for metal-semiconductor nanolasers considered in this work. However, we note that, in the most important region, i.e., near the lasing threshold, the average number of photons in the laser mode is significantly larger than 10 for the majority of metal-semiconductor photonic and plasmonic nanolasers. Hence, the presented classical Langevin approach can be successfully used.

The analytically derived Eq. (9) can be greatly simplified in the case of thresholdless nanolasers with relatively low (≲ 1000) quality factors, which is typical for metal-semiconductor nanolasers:

The term AG/2 in the numerator is responsible for the amplitude squeezing [41,42] at very strong pumping. This term is negligibly small at N lower than 10⁶, which is valid for most nanoscale devices. Therefore, we can simplify Eq. (10) to

Equation (11) is a good approximation of the g⁽²⁾-function at τ = 0 as evident from Fig. 2e. At high pump levels, (g⁽²⁾(0) − 1) ∝ 1/(U_stim/U_spont)², which accurately describes the observed power-law dependence in Fig. 2(c). The remarkable implication of Eq. (11) is that it clearly shows that g⁽²⁾(0) is equal to 1.5 at J_pump = J_th, which justifies the chosen definition of the lasing threshold as

The developed theoretical approach can also be applied to non-thresholdless lasers. Figure 3 shows the L-I curves for the nanolasers with different β-factors obtained at T = 20 K. At such a low temperature, the Auger recombination rate is negligibly low. Therefore, the transition to lasing is determined only by the interplay between the stimulated and spontaneous emission. We neglect other nonradiative recombination mechanisms to demonstrate how the β-factor affects the lasing threshold. As the β-factor decreases from one to zero, a pronounced kink appears in the L-I curve (Fig. 3(a)). Figure 3(b) shows the derivative of the L-I curve. The maximum of the derivative roughly corresponds to the center of the kink in the corresponding L-I curve. However, direct calculations of the g⁽²⁾-function reveal that the transition to lasing occurs at the current higher than that corresponds to the ASE kink (Fig. 3(c) and 3(d)). This is especially pronounced at high β-factors. For example, at β = 0.1, the actual threshold current is two orders of magnitude higher than that given by the kink in the L-I curve. This feature contrasts with that of conventional macroscopic lasers, where the transition to a coherent state occurs exactly at the kink in the L-I curve. It is interesting that although Eqs. (11) and (12) are derived for the thresholdless laser, they can be applied to non-thresholdless metal-semiconductor lasers if the β-factor exceeds 0.01 and U_nr is very low, which can be seen in Fig. 3(c) and 3(d). The threshold current almost does not depend on β. For β > 0.01, the accuracy of the derived equation is better than 1%. Such a high accuracy can be explained if we note that parameters A and C can be written as the sum of two terms. The first term is responsible for the stimulated emission (∂U_stim/∂N_e) and the second term represents the spontaneous emission and non-radiative recombination (∂U_spont/∂N_e for A and ∂(U_spont/β + U_nr)/∂N_e for C). The first term is directly proportional to the photon number N, while the second term is independent of N. At high N near or above the lasing threshold, the first term dominates over the second one and A/C ≈1, which was used to derive Eqs. (10)–(13). Therefore, Eqs. (12) and (13) can also be applied to many non-thresholdless metal-semiconductor nanolasers. Equation (12) shows that the threshold current is proportional to τ_cav^-3/2 ∝Q^-3/2. However, one should also take into account that J_th depends on $dg/d{n}_{\text{e}}{|}_{g=g_{\text{th}}}$, which is a function of threshold gain determined by the quality factor. Hence, the dependence of J_th on Q is slightly more complicated than a simple power dependence (see Fig. 3(e)).

 

Fig. 3 (a) L-I curve of the metal-semiconductor nanolaser for different β-factors. Other parameters are the same as in Fig. 2. (b) First-order derivatives of the L-I curves shown in panel (a). (c-d) g⁽²⁾(0) as a function of pump current (panel (c)) and U_stim/U_spont (panel (d)) calculated using Eq. (9). The color code in panel (d) is the same as in panel (c). The blue area indicates the region where g⁽²⁾(0) < 1.5. (e) Dependence of the threshold current defined by Eq. (12) on the quality factor of the laser cavity. Open dots in panel (a) indicate the lasing threshold obtained using Eqs. (9) and (14).

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We also evaluate how the nonradiative recombination affects the lasing threshold. For simplicity, we limit our attention to the Auger recombination, but one can easily implement other recombination mechanisms into Eqs. (1)–(12). Assuming that all spontaneously emitted photons go into the laser mode, we did calculations for the nanolaser at room temperature varying the Auger recombination constant C_Aug from 0 to 5 × 10⁻²⁸ cm⁶/s. The presence of the nonradiative recombination changes the L-I curve (Fig. 4). Despite that β = 1, one can see a pronounced ASE kink. However, Fig. 4(d) shows that the transition to lasing occurs above the kink if the Auger recombination constant is lower than C_Aug³⁰⁰ = 1.25 × 10⁻²⁸ cm⁶/s, which corresponds to In_0.53Ga_0.47As at room temperature [43]. As C_Aug increases, the position of the kink approaches the threshold point (see Fig. 4). At C_Aug > 2 × 10⁻²⁸ cm⁶/s, the transition to lasing always occurs at the kink, and all definitions of the lasing threshold become equivalent to each other. At C_Aug < 2 × 10⁻²⁹ cm⁶/s, g⁽²⁾(0) is determined only by the ratio U_stim/U_spont, and one can again use Eq. (12).

 

Fig. 4 (a) L-I curve of the metal-semiconductor nanolaser for different Auger recombination constants, T = 300 K, other parameters are the same as in Fig. 2. (b) First-order derivative of the L-I curves shown in panel (a). (c-d) g⁽²⁾(0) as a function of pump current (panel (c)) and U_stim/U_spont (panel (d)) calculated using Eq. (9). The color code in panel (d) is the same as in panel (c). The blue area indicates the region where g⁽²⁾(0) < 1.5. Open dots in panel (a) indicate the lasing threshold obtained using Eqs. (9) and (14).

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Finally, we evaluate the impact of temperature on the threshold current of the metal-semiconductor nanolaser. We assume the β-factor to be equal to 0.5 at any temperature. We also ignore the temperature dependence of the quality factor of the cavity, since it depends on both the radiation losses and absorption in the metal, while the latter depends on the polycrystalline structure of the metal [44,45] and can vary from the strong decrease in the case of the single-crystalline gold to the temperature independent behavior in the case of the polycrystalline copper. At the same time, we consider the temperature dependence of the material gain and the Auger recombination constant C_Aug [43]. Other parameters of the considered nanolaser are the same as in Fig. 2. The results of the calculations are shown in Fig. 5. One can see a pronounced kink in the L-I curve at room temperature, which vanishes at low temperatures. The “threshold current” calculated as a maximum of the first-order derivative of the L-I curve in the log-log scale shows the exponential dependence on temperature (see Fig. 5(b)). However, the actual threshold defined based on the second-order coherence does not show a rapid reduction with the temperature decrease. At liquid nitrogen temperature, the threshold current is only twice lower than at room temperature, while the position of the kink in the L-I curve shifts by nearly two orders of magnitude. Figure 5(b) shows that at temperatures below 200 K, Eq. (12) gives a very good approximation of the exact dependence of the threshold current. At the same time, at temperatures above 350 K, one can simply use the kink in the L-I curve as a measure of the lasing threshold. Figure 5(b) shows that the threshold current is roughly equal to the maximum of the currents given by approximate Eq. (12) and by the position of the ASE kink.

 

Fig. 5 (a) L-I curve of the metal-semiconductor nanolaser at different temperatures. Open dots indicate the lasing threshold (g⁽²⁾(0) = 1.5), which is obtained using Eq. (9). (b) Temperature dependence of the threshold current obtained using three different definitions: green line – using the first-order derivative of the L-I curve in the log-log scale, red line – using the second-order coherence based on Eqs. (9) and (14), blue line – using the second-order coherence based on approximate Eq. (12).

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3. Summary

We present a systematic theoretical study of the second-order coherence properties of metal-semiconductor plasmonic and photonic nanolasers based on a gain medium which can be a bulk semiconductor, heterostructures, or quantum wells. The derived equations give the possibility to find the lasing threshold of any electrically or optically pumped metal-semiconductor nanolaser. Our numerical calculations demonstrate that even if the light-current (L-I) or light-light (L-L) curve is a straight line and does not show an ASE kink, the coherence properties of the emitted photons at low and high pump levels are remarkably different. Therefore, one can easily find the lasing threshold of such a thresholdless nanolaser. We derive a simple closed-form analytical expression for the threshold current of high-β metal-semiconductor nanolasers that are characterized by a low non-radiative recombination rate and find that this expression can be successfully applied to most metal-semiconductor nanolasers with β ≳ 0.01 and moderate non-radiative recombination rates. Here, we should note that this approximate equation is valid only for metal-semiconductor nanolasers, which feature relatively low quality factors of the laser cavity (≲ 500) due to absorption in the metal, and cannot be applied to lasers based on high-quality-factor all-dielectric resonators. The approximate equation shows that the threshold current depends on the small perturbation response dg/dn_e of the gain medium, which shows the key role of coupling between the photon number and carrier density fluctuations in the formation of the coherent state. We also present a study of coherence properties of non-thresholdless metal-semiconductor nanolasers and demonstrate that very often, a clearly observed ASE kink in the L-I or L-L curve cannot be used as a measure of the coherence of the nanolaser emission. The actual threshold current may be more than three orders of magnitude higher than that given by the ASE kink. At the same time, at high temperatures, very high non-radiative recombination rates or very low β-factors, one can still use the kink as an indicator of the lasing threshold. The presented theoretical findings give clear and easy instructions for the estimation of the lasing threshold knowing the basic parameters of the laser cavity and the gain medium. These results can be used for better understanding of lasing at the nanoscale and for the development of novel efficient light sources for different practical applications.