Temperature effects in metal-clad semiconductor nanolasers

2.1 Thermal dependence of material gain

To understand experimental nanolaser performance at various temperatures, a temperature dependent model for the material gain and EM cavity modes is essential. For the lasers discussed in this review, the active materials are multi-quantum wells (MQW) of 1.6Q/1.3Q InGaAsP (In_0.564Ga_0.436As_0.933P_0.067/In_0.737Ga_0.263As_0.569P_0.431) for the optically pumped devices, and bulk In_0.53Ga_0.47As for the electrically pumped devices. The gain spectra of these materials, calculated according to the semiclassical model of [44], are shown in Figure 1, for various temperatures, at carrier densities of 2.0×10¹⁸ cm^-3 and 7.07×10¹⁸ cm^-3, respectively. The reason for plotting the material gain with these parameters will become clear in the following sections. For now, we focus on the main trends. Generally, for a fixed carrier density, the material gain decreases with rising temperature. This may be explained if we consider the total carrier density N to be the sum of thermally excited carriers N_therm and externally injected carriers (either through optical or electronic injection) N_ext. As the temperature rises, N_therm increases, which for a fixed N means that N_ext decreases, and hence the inversion factor, f_inv, decreases. The material gain is directly proportional to f_inv, resulting in a lower peak gain at higher temperatures. Additionally, for a fixed N, the gain spectra for both material systems red shift as the temperature rises. This is primarily due to the fact that the bandgap energy decreases with increasing temperature [45]. Finally, for a fixed T, as the carrier density increases the spectra broaden and the peak gain blue-shifts [44].

Figure 1

Material gain spectra with temperature as a parameter for (A) 10 nm 1.6Q/1.3Q InGaAsP QW near the transparent carrier density and (B) bulk In_0.53Ga_0.47As near the threshold carrier density.

2.2 Thermal dependence of cavity resonance

The resonant wavelengths of the cavity modes of metal-clad nanolasers also depend upon temperature, although to a much lesser extent than the gain spectrum. The thermo-optic coefficient (TOC), dn_r/dT, in InGaAs and InGaAsP is on the order of 10^-4 RIU/K [46], where RIU refers to refractive index units. This means that a change in temperature by 100 K only changes the refractive index by a factor of 0.01. In Figure 2, we plot both the real and imaginary part of silver’s permittivity at 1550 nm via a temperature dependent Drude model, scaled to match the empirical data of Johnson and Christy [47] at room temperature [35]. Similarly to the semiconductor behavior, the real part of the permittivity of the metal cladding varies little with respect to temperature. On the other hand, the imaginary part of the permittivity of the metal cladding may increase by an order of magnitude with a 100 K increase in temperature; the most dramatic part of this increase occurs from around 100 K–130 K. As it turns out, even this large change in imaginary permittivity has little effect on the resonant wavelength, since the effective index of a waveguide mode is essentially invariant with respect to the cladding permittivity [40]. Temperature, therefore, should have the greatest effect on the metal absorption loss experienced by the cavity modes, and a modest effect on resonant wavelength.

Figure 2

Real and imaginary parts of the permittivity of silver as a function of temperature at λ=1.55 μm. (Reprinted from [35]).

A rigorous calculation using finite-element modeling, empirical room-temperature optical properties, and the temperature-dependent Drude model verifies that the variation of cavity resonant wavelength is small over the temperature range 75 K–400 K [35]. As an example, we simulate a nanolaser geometry similar to that shown in Figure 3A, but with a gain radius of 250 nm and optimal shield thickness of 100 nm, using the commercial software COMSOL Multiphysics’ EM module. In Figure 3D, the resonant wavelengths of the first six cavity modes are plotted as a function of temperature, for both a positive [46] and effectively negative TOC [48, 49] of the gain medium. The latter corresponds to the empirical observations of Massum et al. [48] and Ramoo et al. [49]. In these works on InGaAsP-based cavities, the resonant wavelength was observed to increase with the temperature. However, the analysis did not consider the temperature dependence of the individual materials; hence our use of the term “effectively” negative TOC. The lasing mode of the 250 nm cavity is the TE₀₁₁ mode; for the case of positive (effectively negative) TOC, its resonant wavelength ranges from 1.465 μm (1.568 μm) at 77 K–1.514 μm (1.468 μm) at 400 K.

Figure 3

(A) 3D optically pumped metallo-dielectric nanolaser schematic. The air gap at the bottom of the laser is formed after selective etch of the InP substrate In the designed cavity the values for h₁, h₂, and h₃ are 200 nm, 550 nm and 250 nm, respectively. (B, left) Metal-coated gain disk with vertical confinement provided by low index waveguide cutoff sections. (A, right) The structure shown in (B, left) but now including a dielectric shield layer with thickness Δ. (C) Variation of gain threshold of the composite the dielectric shield thickness Δ is changed, assuming a fixed total radius R_tot=460 nm. (D) Cavity resonances as a function of temperature for a cavity similar to that of (A) except R_tot=350 nm and the optimal SiO₂ shield layer is 100 nm. Hence the lower order modes cover the near-infrared wavelengths. (Reprinted from [20] and [35]).

2.3 Thermal effects on the optimum shield thickness

One aspect of metallo-dielectric nanolaser design is the choice of dielectric shield thickness. The optimum shield thickness for a given total device diameter can be calculated using the numerical technique outlined in [39], or by the analytical approximations outlined in [40]. Both techniques involve first finding the optimal shield thickness in a composite gain 2D waveguide, which shares the same optimal shield thickness with its 3D counterpart. Such a waveguide is described in the conceptual diagram in Figure 3B, where the upper and lower waveguide sections extend to infinity. The mode is only supported in the central region where the high-index gain material resides. Figure 3C shows the numerically determined optimal SiO₂ shield thickness and its corresponding threshold gain for this infinite waveguide, for a total device radius of 460 nm [39]. This optimal shield thickness depends on the real part of the material permittivities, which depend weakly on temperature, as discussed in Section 2.2. However, it would intuitively seem that the large increase in metal loss with increasing temperature would necessitate a large increase in optimum shield thickness, to isolate the mode from the increasingly lossy metal; this proves not to be the case. In fact, the optimum shield thickness is virtually temperature-independent [40]. However, this optimum shield thickness only takes electromagnetic considerations into account. The choice of shield thickness and material will impact the laser’s ability to dissipate heat, as we show in Section 4, and through its effect on the laser’s internal operating temperature, will dynamically affect the material gain spectrum, as well as metal loss.

3.1 Purcell effect in semiconductor nanolasers

Apart from understanding the temperature dependence of material gain and electromagnetic cavities, the understanding of the temperature dependence of the Purcell factor F_p and the spontaneous emission factor β is equally important, largely motivated by the quest for energy-efficient operations in nano-scale devices. While the concept of thresholdless operation continues to be a subject of debate [25, 50], the modulation and efficiency improvements enabled by wavelength-scale cavities, which are directly related to the modification of the spontaneous emission (Purcell effect) in these cavities, is fairly well understood [30–32]. For example, with proper design, the cavity of a sub-wavelength laser may be designed such that most of the spontaneous emission is channeled into the lasing mode. In so doing, unwanted emission into non-lasing modes is mitigated, and the below-threshold efficiency is limited only by non-radiative recombination. In nano-scale lasers, enhanced emission together with a reduced number of cavity modes relative to large lasers can have significant effects, especially on sub-threshold behavior. These effects are generally desirable, as they tend to increase the utilization of spontaneous emission into the lasing mode, thus lowering the lasing threshold and increasing β. If the desired cavity mode has the highest Purcell factor amongst all cavity modes, a high β laser can be realized even in a multi-mode cavity. With this design goal in mind, it is important to accurately evaluate the F_p of all cavity modes, taking into account the emitter environment and the semiconductor gain material properties. While the original Purcell effect evaluation was for radio frequency micro-cavities, the formal treatment of Purcell effect specific to nano-scale devices wasn’t presented until the work of [26, 32, 51]. Tailored for nanolasers, these studies provide insight into the fundamental physics in the nano-cavities, although the temperature dependence of F_p was not included.

For a cavity of quality factor Q, active region volume V_a and refractive index n, emitting at the free-space wavelength λ, the Purcell factor F_p describes the rate of spontaneous emission relative to the emission in bulk. In the simplified expression of a two-level system and a single cavity mode that are on resonance and spatially overlapping, F_p usually takes the form

This simplified form is the one most commonly seen and used in the literature [52] to quantify the cavity-enhanced or -inhibited rate of spontaneous emission, relative to the emission in freespace. However, using the emitter-field-reservoir model in the quantum theory of damping, effects ignored in the commonly used Purcell factor expression can be captured. The result revealed that, this expression is far from accurate in semiconductor nanolasers with bulk or MQW gain media.

We apply the results from the non-relativistic QED treatment of 2-level systems to a 3-level laser, in which emitters are pumped from the ground state ∣1> to an excited state ∣3> and quickly decay from state |3> to a lower state |2>; the lasing transition is between states |2> and |1> [26]. Semiconductor lasers in particular are frequently modeled in this manner, even though their underlying physics differs: state |2> describes the condition where a conduction band state is occupied and the valence band state of the same crystal momentum is vacant, while state |1> describes the condition when the conduction band state is vacant and the valence band state is occupied [53]. To describe such a system, we construct a basic model similar to that in ([54]. §9) and [55]. We suppose each emitter to interact with all modes of the cavity, but ignore direct interaction among emitters. The cavity modes, on their part, undergo damping as a result of loss at the cavity boundaries, and we model the loss as a thermal reservoir.

Loss at the cavity boundary, such as loss in a metallic mirror, or loss of energy through the mirror and its eventual conversion to heat at some remote point in space, generally satisfies the assumptions of a reservoir model: it is weak interaction with a large stochastic system that is disordered and does not retain memory of past interactions. Further, this reservoir is passive, as it does not return energy to the mode. Rather, it drains the mode energy over time and is commonly known as the zero temperature condition. The Hamiltonian describing each single emitter in this system can be expressed as

where H^A, H^F and H^R are the emitter, field and reservoir Hamiltonian, respectively. H^AF denotes interaction between the emitter and the field modes, while H^FR denotes interaction between the field modes and the reservoir.

We note that even if, by assumption, a given emitter does not directly interact with other emitters, the field modes still interact with all emitters present, rather than only with a single emitter. This interaction is not included in the Hamiltonian in Eq. (2), either explicitly or as part of the reservoir. In a solid-state system where an emitter interacts with the field, and the field interacts with a thermal reservoir, the cavity Purcell factor F_p is defined as the ratio of spontaneous emission in a cavity to that in bulk material of effective index n_r, with no cavity [56, 57]. The spontaneous emission probability in the bulk material, P2→1,|0…0〉material, takes the same form as in free space, except that ε₀ is replaced by the permittivity of the medium εr=nr2ε0 and that c is scaled down by the refractive index n_r. It is expressed as

where ω₂₁ is the mode resonant frequency, ℘12(ω21) is the dipole matrix element, and D(ω₂₁) characterizes the inhomogeneity of the system. The intraband collision time, τ_coll, is the average time between carrier-carrier and carrier-phonon collisions, and decreases with increasing temperature [58]. In the second line of Eq. (3), we evaluate ω213 and ℘12(ω21) at the center frequency ω¯21 of the inhomogeneous broadening spectrum D(ω₂₁) and pull them out of the integration, because these quantities vary relatively little over the homogenous broadening range.

In a damped cavity, the mode interacts with the reservoir. Provided that equilibrium between the mode and the reservoir is reached, we obtain the photonemission probability in steady-state,

where R(ω-ω₂₁, τ_coll) is the homogeneous broadening function and depends on τ_coll. Viewed as a function of ω, R(ω) peaks at ω₂₁, has a width on the order of 1/τ_coll, and satisfies ∫R(ω-ω21,τcoll)dω=2π⋅τcoll [59]. The Lorentzian L_k(ω-ω_k) in Eq. (4) is expressed as

and the quality factor is defined as Q≡ω_k/Δω_k. From the discussion in Section 2, the quality factor is a strong function of temperature due to the dependence of the full-width-at-half-maximum (FWHM), Δω_k, on the imaginary part of the metal-cladding permittivity. This is especially true for well-confined modes where the Q is dominated by absorptive, rather than radiation, losses. The convolution in Eq. (4) determines the emission probability in a cavity for an inhomogeneously broadened ensemble of emitters, when the mode-reservoir equilibrium has been reached. The effect of the reservoir on the emission probability is described by L_k(ω-ω_k), whose spectral property is described by Eq. (5).

The Purcell factor F_p is then

The emission probability in Eq. (4), and hence the Purcell factor in Eq. (6), depends on the location r_e of the emitter. More precisely, it depends on the normalized mode field at the location of the emitter e_k(r_e), as well as on the orientation of the emitter’s dipole moment matrix element ℘12(ω¯21) relative to the field. If the emitters are randomly oriented and uniformly distributed over an active region of volume V_a, the quantity |℘12(ω¯21)⋅ek(re)|2 is replaced by its average over all locations and orientations.

where the coefficient 1/3 accounts for the random emitter orientation.

In certain situations, the carrier distribution over V_a may become non-uniform. For example, in MQW structures, the carrier distributions in the well and barrier regions differ significantly. Even in bulk semiconductors, the recombination of carriers may vary spatially, with the highest rates occurring at field antinodes. This is the case if the recombination at field antinodes is so rapid that diffusion of carriers from other parts of the active volume is not fast enough to avoid depletion. Carrier depletion at field antinodes and subsequent diffusion from the nodes toward the antinodes leads to the spatial inhomogeneity of the recombination. At room temperature, the diffusion length of carriers in InGaAsP (i.e., average distance traveled before recombination) is on the order of 1–2 μm [60]. The distance between the field node and antinode in visible and near infra-red sub-wavelength semiconductor cavities, on the other hand, is usually <0.5 μm [19, 20]. Thus, the depletion regions would remain relatively depleted due to the finite diffusion time. Under these circumstances, Eq. (7) should then be replaced by an appropriately weighted average.

where Γ_k is the energy confinement factor of mode k. Equation (8) permits several observations. Firstly, the double integral in Eq. (8) is the convolution of inhomogeneous broadening D(ω₂₁), cavity Lorentzian L_k(ω-ω_k), and homogeneous broadening R(ω-ω₂₁, τ_coll). It should be noted that although the homogenous broadening function R(ω) and the inhomogeneous broadening function D(ω) appear symmetrically in Eq. (8), they may in principle exhibit different dynamics. In particular, rapid recombination of carriers near the mode frequency ω_k may deplete the carrier population at that frequency faster than it is replenished by intraband scattering (this phenomenon is known as “spectral hole burning”). In such cases, it could be meaningful to disaggregate the integral in dω₂₁ in Eq. (8) and define separate Purcell factors for carriers at different frequencies ω₂₁ [61]. More typically, however, especially at room temperatures, the intraband relaxation time τ_coll ∼0.3 ps of InGaAsP is much shorter than photonemission time, and the distribution of carriers D(ω₂₁) is at all times the equilibrium distribution ([44]. Appendix 6). This equilibrium distribution closely resembles the photoluminescence spectrum [62]. In semiconductor lasers utilizing bulk or MQW gain material, it is the broadest of the three convolution factors in Eq. (8) and therefore dominates the convolution. For InGaAsP at room temperature, the FWHM of D(ω₂₁) and R(ω-ω₂₁, τ_coll) are approximately 7×10¹³ rad/s and 6.7×10¹² rad/s, respectively. D(ω₂₁) dominates the convolution in Eq. (8) as long as the cavity Q factor is above 19, which corresponds to a FWHM of 7×10¹³ rad/s. For practical cavities, the Q factor will be significantly larger; thus diminishing the contribution of L_k(ω-ω_k) to the resulting Purcell factor. In fact, R(ω-ω₂₁, τ_coll), alone, dominates L_k(ω-ω_k) if the Q factor is greater than 200 [58, 63]. Consequently, in typical III-V semiconductor lasers with MQW or bulk gain material, the cavity Q factor plays a negligible role in determining the spontaneous emission rate and F_p. Further, while the cavity lineshape broadens with temperature for well-confined cavity modes, the homogeneous lineshape broadens as well. Secondly, F_p may be large in small laser cavities due to its inverse proportionality to the active region volume V_a. However, F_p is actually inversely proportional to the effective size of the mode, V_a/Γ_k, where the mode-gain overlap factor Γ_k is defined in Eq. (8) and describes the spatial overlap between the mode and the active region. Thus, if the mode is poorly confined, Γ_k <<1, F_p will remain small, despite a small active region.

3.2 Purcell factor of metal-clad nanolasers operating at room-temperature

Using the optically pumped room temperature metallo-dielectric nanolaser reported in Nezhad et al. [20], whose geometry is similar to that in Figure 3, Figure 4 shows its lasing mode’s electric field profile and the three spectra in the evaluation of the Purcell factor: cavity lineshape L_k(ω-ω_k) (Figure 4B), homogeneous broadening R(ω-ω₂₁, τ_coll) (Figure 4C), and inhomogeneous broadening D(ω₂₁) approximated by the photoluminescence (PL) spectrum (Figure 4D). It is then evident that the role of L_k(ω-ω_k) and R(ω-ω₂₁, τ_coll) are diminished in the determination of Purcell factor for the cavity mode. We evaluate the F_p of the lasing mode to be 0.215 from Eq. (8), whereas F_p takes the value of 8.79 if evaluated using Eq. (1). In fact, Eq. (8) simplifies to Eq. (1) if the cavity lineshape is much wider than the gain medium inhomogeneous broadening lineshape, which is not the case in moderate- to high-Q semiconductor lasers. Finally, note that the Purcell factor F_p is the sum of contributions Fp(k) from each cavity mode present, as is the emission probability in Eq. (4).

Figure 4

(A) The lasing mode’s electric field profile and the three spectra in the evaluation of the Purcell factor, (B) cavity lineshape, (C) homogeneous broadening lineshape and (D) PL spectrum (Reprinted from [26]).

3.3 Temperature dependence of the spontaneous emission factor

In Section 2 we discussed the temperature dependence of the material gain spectra and cavity resonances. The PL spectra used in Section 3.1 and 3.2 is closely related to the gain spectra [44]; they both, for example, depend on material properties such as bandgap energy. Because of the temperature dependence of the PL spectra, as well as that of homogeneous and cavity lineshapes, the Purcell factor varies with temperature. This can perhaps best be seen by rewriting Eq. (8) with temperature made as an explicit variable.

In Eq. (9), Z_SP(ω₂₁, T) is the rate of spontaneous emission in the absence of the cavity, describing inhomogeneous broadening similar to D(ω₂₁) in Eq. (8), but calculated according to [44]. The temperature dependence of the spontaneous emission factor, β, follows directly from Eq. (9). We can isolate the effect of the temperature on the competition between modes by defining β_max as the ratio of the Purcell factor of the lasing mode to the sum of Purcell factors of all cavity modes,

To illustrate the evaluation of Eqs. (9) and (10) for a specific cavity, we identify the transparent carrier densities corresponding to the cavity modes of the resonator presented in [35], which has a total radius of 350 nm and an optimized SiO₂ shield of 100 nm. If a large spontaneous emission factor is desired, this resonator is more favorable compared to the resonator of [20] because its smaller size admits fewer modes and a larger free-spectral range. The TE₀₁₁ mode has the lowest threshold gain of the 350 nm resonator and, depending upon the temperature, its resonant wavelength ranges from ∼1465 nm to ∼1515 nm for positive TOC and ∼1570 nm to ∼1470 nm for effectively negative TOC (see Figure 3D). The transparent carrier densities are found by computing the material gain and noting the carrier density at which the gain changes sign for a particular wavelength. Over the temperature range 77 K–400 K, the transparent carrier density of the 10 nm 1.6Q/1.3Q InGaAsP QW ranges from ∼2e17 cm^-3 to ∼3e18 cm^-3, as shown in Figure 5A. The spontaneous emission factor is plotted as a function of temperature, in Figure 5B, for both positive and effectively negative TOCs.

Figure 5

(A) Transparent wavelength versus carrier density and (B) spontaneous emission factor versus temperature for MQW InGaAsP metal-clad nanolasers. The cases of positive and effectively negative TOCs are denoted by dn_r/dT>0 and dn_r/dT<0, respectively. (Reprinted from [35]).

At and above room-temperature β_max is over 0.5 because the TE₀₁₁ mode lies close to the peak of the spontaneous emission spectrum. While the TE₀₁₁ mode has a much higher Q than neighboring modes, the neighboring modes still overlap with the emission spectrum, preventing β_max to ever reach unity. For the case of positive TOC, the maximum of β_max occurs at 200 K when the TE₀₁₁ mode most nearly coincides with the peak spontaneous emission. Below 200 K, however, β_max drops significantly as the emission spectrum blue-shifts and the TE₀₁₁ mode wavelength remains nearly constant. The sharp roll-off in the MQW gain spectrum (see Figure 1A), which comes from the step-like density of states function, is responsible for the sharp decrease in β_max. At low temperatures λ_TE011 falls to the red side of the spontaneous emission spectrum and neighboring modes receive an increasing ratio of the spontaneous emission. The resonant wavelength of the HE₂₁₁ mode, especially, always resides near the peak of the emission for low temperatures. Nonetheless, for the positive TOC case, the TE₀₁₁ mode maintains sufficient overlap with the gain to remain the lasing mode for all T.

For the case of effectively negative TOC, below 150 K the TE₀₁₁ mode has negligible overlap with the gain spectrum, while the HE₁₂₁ and HE₂₁₁ modes have a strong overlap. In this case, it is possible that the hybrid modes may lase, depending on their threshold gains. The HE₂₁₁ mode, in particular, has a threshold gain of g_th<2000 cm^-1 for T<150 K. Based on Figure 1A, large, but not unattainable, carrier densities would be required for the HE₁₂₁ mode to lase. For T=77 K and effectively negative TOC, we compute β_max=0.39 for the HE₁₂₁ mode. Due to the two-fold degeneracy of this mode, β_max cannot exceed 0.5.

Qualitatively, we see similar behavior for the cases of positive and effectively negative TOCs. The major quantitative difference results from the fact that, below 300 K, λ_TE011 of the latter case exceeds that of the former. For example, at 200 K λ_TE011(200 K)≈1.483 for positive TOC, whereas λ_TE011(200 K)≈1.529 for negative TOC. As indicated by the material gain of Figure 1A, the spontaneous emission spectrum blue-shifts significantly with decreasing temperature, so the observation that the shorter wavelength has a larger β_max for a given temperature below 300 K agrees with our expectation.

At 300 K, λ_transp for effectively negative TOC is equal to λ_transp for positive TOC, leading to identical β_max’s. Above 300 K, λ_transp for negative TOC is less than λ_transp for positive TOC but the difference is not as significant as for low temperatures. While the peak spontaneous emission red-shifts with increasing T, we observe that β_max for effective negative TOC is almost identical to β_max for positive TOC above 300 K. This result is most likely due to the broadening of the emission with T. The broader emission enables more cavity modes to participate in the coupling process, leading to a less dramatic difference between the two TOC cases for β_max.

The major effect of the temperature in the calculation of β_max(T) is to shift the gain and spontaneous emission spectra significantly. The effect of the temperature on the real parts of the cladding and semiconductor permittivities is not as dramatic. The rapid rise in β_max with temperature and the subsequent broad maximum was similarly observed for VCSELs and microcavities [48, 49]. However, the critical temperature at which β decreases sharply appears to be lower for a subwavelength cavity. The transition temperatures of [48] and [49] were approximately 240 K and 290 K, respectively. In this sense, our analysis demonstrates the rather non-intuitive result that β of a subwavelength cavity may be more robust to temperature variation than that of a larger laser. This appears to be a direct consequence of the sparse mode density in the cavity that we have considered. Furthermore, as a consequence of the small number of total modes, β_max is much higher in the metallo-dielectric lasers that we analyze, compared to those analyzed in [48] and [49].

Although we have analyzed a metallo-dielectric laser cavity supporting a photonic-type mode, the method may also be applied to lasers supporting plasmonic modes. For example, the coaxial plasmonic laser of [24], with an inner conductor radius of 100 nm and semiconductor annulus of width 100 nm (Structure A), exhibited near-unity-β at T=4.5 K. Using the results of our analysis we may make several hypotheses concerning its performance at room-temperature. Firstly, the TEM-like mode supported by this cavity was simulated to have a resonant wavelength of λ_sim,A≈1.43 μm, but lasing was observed at λ_obs,A≈1.38 μm. The material system employed in [24] was 10 nm 1.6Q/1.3Q InGaAsP MQW, so, using Figure 1A and the fact that the nearest neighbor modes are each about 200 nm from the lasing mode, we would predict the TEM-like mode to have β≈1 for T<100 K. The red-shift of the bandgap energy with T, so far discussed theoretically, is corroborated by the ∼160 nm shift in wavelength of the lasing mode of Structure B of [24], from λ_obs,B(4.5 K)≈1.32 μm to λ_obs,B(300 K)≈1.48 μm. The bandgap energy for 1.6Q/1.3Q InGaAsP red-shifts ∼140 nm, from 0 K to 300 K. Given that the TEM-like mode has the lowest threshold gain of the two modes in Structure B, and the experimentally observed far-field radiation patterns, we can attribute the majority of the experimentally observed red-shift to the temperature dependence of the gain medium. While the analysis may explain the observations of Structure B, perhaps more significantly, it suggests that the room-temperature β of Structure A would decrease due to the higher order mode at the simulated resonance of λ_sim,A≈1.61 μm. Furthermore if the Q of the higher order mode is large with respect to that of the TEM-like mode, then the latter may not lase at all.

Admittedly, the applicability of this analysis becomes questionable at very low temperatures, when the assumption of continuous gain and spontaneous emission spectra in the absence of a cavity may become invalid. At low enough temperatures, the intraband lifetime may become comparable to the spontaneous emission lifetime. Hence, the assumption that the carriers interact with one another on a time scale much shorter than the time between photo-emission events may no longer be valid. This means that the use of the quasi-Fermi levels becomes invalid, and we must treat the classes that contribute to the continuous spectra at high temperatures, on a class-by-class basis. It is not immediately clear how this would affect the resulting Purcell and spontaneous emission factors. For the temperatures that we have considered in this report, this is not a concern for semiconductors, but for experiments at T≈4.5 K, more sophisticated emission spectra models may be required.

4.1 Calculation of self-heating

In addition to the insight into the effects of temperature on nanolaser behavior described in Sections 2 and 3, the design of thermally-robust nanolasers also requires the ability to simulate the temperature performance of nanolaser designs. The internal operating temperature of a nanolaser depends not only on the ambient temperature, but also on the amount of self-heating the nanolaser experiences. For an optically-pumped nanolaser, the self-heating will depend on the power absorbed from the optical pump; most of this absorbed power is converted to heat, and only a small portion is utilized in the generation of emitted light. For an electrically-pumped nanolaser, self-heating can be calculated using the effective heat source model used in VCSELs ([29]. §5.3), with modification to include the heat generated from non-radiative recombination in the active region, which is insignificant in micro- or large-scale lasers, but can play an important role in the self-heating of nanolasers.

In the modified effective heat source model, the cavity self-heating in an electrically-pumped nanolaser can be categorized into three mechanisms: 1) junction and heterojunction heating, 2) Joule heating, and 3) non-radiative recombination heating.

The first type of heating is generated at the interface of the differently doped semiconductor layers. It consists of junction heating, a term that is designated to describe the heat generated at the interfaces between the doped semiconductors and the un-doped gain region; it also consists of heterojunction heating, which accounts for the heat generated at all doped semiconductor layer interfaces. Both terms are expressed as I·V_Jn, where I denotes current and V_Jn is the potential difference at the nth junction. These two terms take the same form below threshold, where I is the injection current I_inj. Above threshold, I for junction heating is clamped at the threshold current I_th, while the heterojunction heating continues to use I_inj.

The second type of heating is Joule heating due to the series electrical resistance in all doped semiconductor layers, and takes the form (I_inj)²·R_i, where the resistance R_i of the ith layer is calculated using the layer’s thickness t_i, cross-sectional area A_i, doping concentration n_i, and carrier mobility μ_i [64], using the formula

where q is the carrier charge. Each doped semiconductor layer, therefore, becomes a distributed source of Joule heating.

The third type of heating is generated by non-radiative recombination inside the gain region. In nanolasers, the non-radiative recombination heating is generated by Auger recombination and surface recombination. Auger recombination is significant at high temperatures and/or high carrier densities, and surface recombination, is significant at high temperature and/or large surface-to-volume ratios, the latter being especially significant relevant to nanolasers. Therefore, the gain region becomes a distributed heat source whose power is given by the non-radiative recombination, assuming that all non-radiative energy is converted to lattice vibrations through the creation of phonons.

The above calculation of junction and heterojunction heating requires knowledge of the potential differences needed to forward bias each junction. These can be obtained using software such as SILVACO’s ATLAS, a two-dimensional electronic device simulator. Given the operating temperature and properties of the device’s constituent materials, ATLAS self-consistently solves the Poisson equation, the Schrodinger equation, and the carrier transport equations, considering Fermi-Dirac statistics, and obtains at each injection level the carrier density, the electron and hole quasi-Fermi levels, and the potential difference necessary to forward bias the junctions.

4.2 Simulation of nanolaser heat dissipation

Once the amount of nanolaser self-heating is known, the internal operating temperature of the nanolaser can be calculated using finite element software such as COMSOL’s heat transfer module. Each layer and junction can be treated as a heat source according to the effective heat source model described in Section 4.1 (or by the amount of pump absorption, for optically pumped lasers), and the transient or steady-state temperature in the laser can be obtained.

Accurate thermal analysis of a nanolaser design requires knowledge of the thermal conductivity, heat capacity, and density of each material used in the nanolaser. Since these parameters are themselves temperature-dependent, the thermal analysis should ideally include thermal feedback mechanisms to update the material parameters as the device’s temperature rises. However, experimental or experimentally-validated thermal parameters are lacking for most commonly-used nanolaser materials; temperature-dependent study of these material properties would be valuable to future nanolaser research.

Also valuable would be investigation into microscale heat transfer, as applied to nanolasers. The heat conduction models in most commercial finite element software, such as COMSOL, use macroscopic heat transfer equations, which may break down on the micro-/nano-scale. When the device dimension becomes comparable to or smaller than the mean free path of constituent materials’ heat carriers, we enter the microscale heat transfer regime [65]. Microscale conductive and radiative heat transfer in VCSELs and convective heat transfer in carbon nanotubes have been studied [65], but this has not yet been a subject of attention in the field of nanolasers.

The above analysis also does not include the effects of non-ideal Ohmic contacts, defects at material interfaces, or the effects of surface passivation on surface recombination [66]. Further refinements of nanolaser models to include these and other parameters will serve to increase the accuracy and value of nanolaser thermal simulations, as well as to suggest avenues for design improvement.