## Abstract

A Strong temperature dependence of microdisk lasers and photonic crystal nanocavity lasers with InAs quantum dot active regions is reported. These lasers operate at 1.3 μm at room temperature under optical pumping conditions. T_{0, microdisk} = 31 K. T_{0, photonic crystal nanocavity} = 14 K. The lasing threshold dependence on the lasing wavelength is also reported. We observe a minimum absorbed threshold pump power of 9 μW. This temperature and wavelength dependent lasing behavior is explained qualitatively by a simple model which attributes the experimental observations predominantly to surface recombination at threshold and the high quality factors of these cavities.

© 2007 Optical Society of America

## 1. Introduction

Microcavity lasers with InAs quantum dot (QD) active regions combine the high quality factor (Q) small volume (V) optical modes of the cavities with the discrete and narrow gain profile of the quantum dots. They are promising candidates for high speed, low power devices in high density photonic integrated circuits [1,2]. In this paper, we report the characterization and modeling of the room temperature lasing behavior of InAs QD microdisk lasers and photonic crystal nanocavity lasers as a function of temperature and wavelength. A strong temperature dependence is reported and attributed predominantly to surface recombination at threshold. The minimum absorbed threshold pump power of 9 μW for photonic crystal nanocavities is reached at a wavelength longer than that corresponding to the maximum QD density, which is possible because of the high Q values. This is the lowest room temperature threshold value reported to date for photonic crystal lasers. Our model shows that the underlying mechanisms for the temperature dependence and the wavelength dependence of these devices are closely connected.

## 2. Experiment

The devices were fabricated in a 180 nm thick GaAs membrane clad by 20 nm thick Al_{0.25}Ga_{0.75}As layers. The membrane contains five layers of self-assembled InAs QDs. The QD density per layer is 2-3 10^{10}/cm^{2}. Each layer of QDs is inside a 5 nm thick In_{0.15}Ga_{0.85}As quantum well (QW). The QWs are separated by 30 nm thick GaAs barriers. Figure 1 shows the scanning electron micrographs of a microdisk cavity and a photonic crystal nanocavity. The photonic crystal cavities are formed by removing 3 holes from a triangular lattice. The two holes on both ends of each photonic crystal cavity are shifted outwards to increase the Q [3].

The devices were optically pumped at normal incidence with an 850 nm diode laser under pulsed conditions. The pump beam was focused to a 1-1.5 μm diameter spot. Emission from the devices was collected into an optical spectrum analyzer. Experimental details on device fabrication and characterization can be found in [4].

## 3. Temperature dependence of lasing behavior

To study the temperature dependence of the lasing behavior of these devices, they were mounted on a copper bar whose temperature was controlled by a thermo-electric cooler.

Figure 2(a) shows the lasing spectrum of a 3.1 μm diameter microdisk at room temperature [5]. The four resonances at the wavelengths of 1254nm, 1266nm, 1302nm, and 1319 nm can be well approximated by whispering gallery modes (WGM) of TE_{i+1,1}, TE_{i-3,2}, TE_{i,1}, and TE_{i-4,2}, i = 17 ± 1 [6]. Figure 2(b) shows the output lasing power versus incident pump power characteristic of a lasing mode at 1.30 μm in a 3.2 μm diameter microdisk at varied substrate temperatures. Pumping conditions for Fig. 2(b) are 24 ns pulse widths and a 0.9% duty cycle. A characteristic temperature for the lasing threshold, T_{0}, of 32 K and a characteristic temperature for the lasing slope efficiency, T_{η}, of 65 K were extracted from these data.

Figure 3(a) shows the lasing spectrum of a photonic crystal nanocavity [4]. The data points in Fig. 3(b) represent the output lasing power versus incident pump power characteristic of a lasing mode at 1.33 μm in a photonic crystal nanocavity at varied substrate temperatures [4]. Pumping conditions are 16ns pulse widths and a 1% duty cycle. The photonic crystal cavities are in suspended membranes and therefore have slower heat dissipation than the microdisks supported on posts. Heating of the photonic crystal cavities by the pump light is estimated to occur at a rate of 0.19 K/ pJ during the pump pulses. This temperature increase rate was estimated by: (1) experimental observation of lasing wavelength and linewidth change with pump power and pulse width; (2) a finite difference time domain (FDTD) calculation of lasing wavelength shift with cavity temperature. This results in a 3K temperature increase within a single pump pulse of 1 mW and 16 ns. This temperature increase is comparable to the increments in substrate temperatures in Fig. 3(b) and should be considered in order to obtain characteristic temperatures. The curves in Fig. 3(b) are fitting results, where the experimental data points are fitted to the exponential temperature dependence of lasing threshold and slope efficiency. The pump heating effect is included in this fitting, and T_{0} = 14 K and T_{η} = 18 K were obtained.

The temperature dependences of both the microdisk lasers and the photonic crystal nanocavity lasers are much stronger than those reported for conventional InAs QD lasers. A T_{0} value of 126 K was reported for Febry-Perot lasers under continuous wave (CW) operation which have similar InAs QDs as in our samples and which were grown by the same group [7]. The strong temperature dependence is attributed to carrier recombination at threshold being dominated by surface recombination for these microcavity devices. Surface recombination has a much stronger temperature dependence than the carrier recombination inside the QDs. A detailed explanation and numerical modeling will be addressed in section 5.

## 4. Lasing threshold versus lasing wavelength

In this section, we report experimental results on the lasing threshold versus lasing wavelength behavior for the InAs QD microcavity lasers at a constant room temperature. Only photonic crystal nanocavities are studied due to their single mode property [4]. Microdisks have multiple high Q modes.

Ninety photonic crystal nanocavities were fabricated in an array and characterized. The lattice constant, a, varied from 337 nm to 358 nm; the hole radii, r, had values 0.3 a, 0.31 a, and 0.32 a; and the shift distances of the two holes on the two ends of the cavity were 0.12 a and 0.15 a. Lasing threshold values were recorded under 8ns pulse width, 0.5% duty cycle pumping conditions in order to reduce the pump heating effects. Because of variations in cavity structure, sample growth, and fabrication, lasers with similar lasing wavelengths do not all have identical lasing thresholds. The threshold values that are the lowest among the devices with similar lasing wavelengths are plotted in Fig. 4. Also plotted in Fig. 4 is the photoluminescence (PL) spectrum from an unpatterned region on the same sample. This PL spectrum was taken with a high enough peak pump power but a small enough pump heating caused red-shift in order to approximately represent the QD density distribution versus wavelength. The PL spectra were quite uniform across the whole sample, with a 1-1.5 μm diameter pump spot.

As shown in Fig. 4, near the wavelength corresponding to the maximum QD density, the threshold pump power decreases with increasing wavelength. This occurs even when the QD density is decreasing. This indicates that the threshold gain required by the Q of these cavities is much smaller than the maximum gain available from the QDs. This will be further explained in section 5. The 0.2 mW threshold incident pump power around 1330 nm to 1340 nm corresponds to an estimated absorbed threshold pump power of 9 μW. This is the lowest threshold value for room temperature photonic crystal lasers reported to date.

As an aside, at the wavelength corresponding to the maximum QD density, approximately 24 QDs on average are estimated to contribute to the lasing. For the device with the longest lasing wavelength in Fig. 4, 4 to 5 QDs on average are estimated to contribute to the lasing. In this estimation, a 45 nm inhomogeneous broadening and a 5 nm homogeneous broadening of the InAs QD ensemble at room temperature are considered [8].

## 5. Modeling of temperature and wavelength dependence

To understand the mechanism behind the lasing performance reported in section 3 and section 4, we will describe a simple model in this section which gives qualitative predictions of the lasing behavior of the QD microcavity devices as a function of temperature and wavelength.

#### 5.1 The model

Because of the high surface to volume ratios of these microcavity devices, and the large surface recombination velocities of GaAs and In_{0.15}Ga_{0.85}As near room temperature [9], carrier consumption at lasing threshold is dominated by surface recombination. This can be confirmed as follows: The spontaneous emission power from the ground states of the InAs QDs has an upper limit in which case all the QD ground states are excited. This upper limit is approximately 0.2 μW for 5 layers of QDs within a 1 μm diameter pump spot with a QD density per layer of 2 × 10^{10}/cm^{2}, a luminescence wavelength of 1300 nm, and a spontaneous emission lifetime of 1 ns [8]. Therefore, at the absorbed threshold pump powers above 9 μW at 850 nm, spontaneous emission from the QDs only consume a tiny fraction of all the excited electron-hole pairs. Using a simple model for estimating the radiative and nonradiative recombination rates, as in [10], it can be concluded that at threshold most of the pumped carriers recombine via surface recombination.

In this model, we will just calculate the surface recombination rate at threshold and take it as the lasing threshold. To simplify the calculation three assumptions are made without losing the essence of the physical mechanisms.

### Assumption 1: surface recombination rate ∝ N,

where N is the density of carriers outside the QDs in the optical mode volume.

The carrier capture time from the wetting layers of the InAs QDs has been reported to be on the order of 1 ps near room temperature [11]. Therefore the carrier distribution is at quasi-equilibrium at and below the lasing threshold, and the density of carriers outside the QDs is obtained by non-degenerate Fermi-Dirac distribution [12]:

where N_{bulk} is the carrier density in bulk material, NQW is the 3-D carrier density in a confined QW state, m^{*} is the effective mass of carriers, k_{B} is the Boltzmann constant, T is the temperature, E_{F} is the quasi Fermi level, and E_{0} is the band edge of the carrier energy bands. The 2-D carrier density in a QW state is divided by the thickness of the GaAs membrane, 180 nm, to be converted to an equivalent 3-D carrier density. The band diagram and the effective mass values that will be used in the model are shown in Fig. 5.

### Assumption 2: N = (N_{e} N_{h})^{1/2},

where N is the nominal carrier density outside of the QDs that we will use for surface recombination rate calculation. N_{e} and N_{h} are the electron density and the hole density outside the QDs respectively.

According to Fig. 5 and Eq. (1), we have:

$$=({N}_{\mathit{e}\mathit{,}\mathit{bulk}}+{N}_{\mathit{e}\mathit{,}\mathit{QWc}})\left({N}_{\mathit{hh}\mathit{,}\mathit{bulk}}+{N}_{\mathit{lh}\mathit{,}\mathit{bulk}}+{N}_{\mathit{hh}\mathit{,}\mathit{QWv}\mathit{1}}+{N}_{\mathit{hh}\mathit{,}\mathit{QWv}\mathit{2}}\right)$$

$$=2{\left[\frac{{m}_{e}{k}_{B}T}{{2\pi \mathit{\u0127}}^{2}}\right]}^{\frac{3}{2}}2{\left[\frac{{\left({m}_{\mathit{hh}}^{\frac{3}{2}}+{m}_{\mathit{lh}}^{\frac{3}{2}}\right)}^{\frac{2}{3}}{k}_{B}T}{{2\pi \mathit{\u0127}}^{2}}\right]}^{\frac{3}{2}}{e}^{\frac{({E}_{\mathit{Fe}}-{E}_{\mathit{Fh}})-({E}_{\mathit{c}\mathit{,}\mathit{GaAs}}-{E}_{\mathit{v}\mathit{,}\mathit{GaAs}})}{{k}_{B}T}}$$

$$+2{\left[\frac{{m}_{e}{k}_{B}T}{{2\pi \mathit{\u0127}}^{2}}\right]}^{\frac{3}{2}}\left[\frac{{m}_{\mathit{hh\rho}}{k}_{B}T}{{\pi \mathit{\u0127}}^{2}\times 180\phantom{\rule{.2em}{0ex}}\mathrm{nm}}\right]{e}^{\frac{({E}_{\mathit{Fe}}-{E}_{\mathit{Fh}})}{{k}_{B}T}}\left({e}^{\frac{{E}_{\mathit{c}\mathit{,}\mathit{GaAs}}-{E}_{\mathit{QWv}\mathit{1}}}{{k}_{B}T}}+{e}^{\frac{{E}_{\mathit{c}\mathit{,}\mathit{GaAs}}-{E}_{\mathit{QWv}\mathit{2}}}{{k}_{B}T}}\right)$$

$$+\left[\frac{{m}_{\mathit{e\rho}}{k}_{B}T}{{\pi \mathit{\u0127}}^{2}\times 180\phantom{\rule{.2em}{0ex}}\mathrm{nm}}\right]2{\left[\frac{{\left({m}_{\mathit{hh}}^{\frac{3}{2}}+{m}_{\mathit{lh}}^{\frac{3}{2}}\right)}^{\frac{2}{3}}{k}_{B}T}{{2\pi \mathit{\u0127}}^{2}}\right]}^{\frac{3}{2}}{e}^{\frac{({E}_{\mathit{Fe}}-{E}_{\mathit{Fh}})-({E}_{\mathit{QWc}}-{E}_{\mathit{v}\mathit{,}\mathit{GaAs}})}{{k}_{B}T}}$$

$$+\left[\frac{{m}_{\mathit{e\rho}}{k}_{B}T}{{\pi \mathit{\u0127}}^{2}\times 180\phantom{\rule{.2em}{0ex}}\mathrm{nm}}\right]\left[\frac{{m}_{\mathit{hh\rho}}{k}_{B}T}{{\pi \mathit{\u0127}}^{2}\times 180\phantom{\rule{.2em}{0ex}}\mathrm{nm}}\right]{e}^{\frac{({E}_{\mathit{Fe}}-{E}_{\mathit{Fh}})}{{k}_{B}T}}\left({e}^{\frac{{E}_{\mathit{QWc}}-{E}_{\mathit{QWv}\mathit{1}}}{{k}_{B}T}}+{e}^{\frac{{E}_{\mathit{QWc}}-{E}_{\mathit{QWv}\mathit{2}}}{{k}_{B}T}}\right)$$

We can see that now N is only a function of T and (E_{Fe} - E_{Fh}).

### Assumption 3: Occupancies of a QD’s conduction band ground state and valence band ground state are equal.

This assumption is mathematically expressed as, according to Fermi-Dirac distribution:

At threshold, the population inversion of the electron and hole ground states of each QD participating in the lasing is proportional to the threshold gain divided by the number of QDs participating in the lasing. This can be mathematically expressed as:

where ρ_{QD} is the QD density per unit wavelength, λ is the lasing wavelength, λ_{0} is the wavelength that corresponds to the maximum QD density, G_{th} is the threshold gain which is determined by the cavity Q value and will be assumed as a constant from now on, and G_{0} is the gain value at λ_{0} when the all the QD ground states are inverted. Inside the square brackets on the left side of Eq. (4) is the population inversion of one pair of QD electron and hole ground states at the lasing wavelength λ. G_{th}/G_{0} is the threshold population inversion of one such pair when the lasing wavelength is λ_{0}.

Substituting Eq. (3) into Eq. (4), and noticing that for the QDs participating in the lasing we have E_{Qc} - E_{Qv} = hc/λ, it is easy to obtain:

Substituting Eq. (5) into Eq. (2), we can see that N can be written as a function of T, λ, and ρ_{QD}, with one fitting parameter G_{th}/G_{0}. ρ_{QD} is obtained from the PL spectrum from an unpatterned area on the sample. Now we can qualitatively predict the lasing threshold’s dependence on temperature and wavelength.

### 5.2 Modeling results

Let’s look at the modeling results on temperature dependence first. This model predicts a T_{0} = 36.0 K at T = 40 °C and a lasing wavelength of 1.30 μm, and a T_{0} = 30.9 K at T = 20 °C and a lasing wavelength of 1.33 μm. The sample material’s energy band shift and the lasing wavelength shift under different temperatures and pump powers have been determined to be insignificant and ignored in the calculation. In section 2 we have reported for the microdisk laser T_{0} = 31 K at T = 40 °C and a lasing wavelength of 1.30 μm; for the photonic crystal nanocavity laser T_{0} = 14 K at T = 20 °C and a lasing wavelength of 1.33 μm. The calculation and experimental results for the microdisk are close. The experimental T_{0} value of the photonic crystal nanocavity is quite smaller than that of the microdisk. The reason is not clear but it may be attributed to the different carrier diffusion behaviors, which are also temperature dependent. Carries in the microdisks are confined within the disk while there are no such confinement boundaries in photonic crystal.

The mechanism behind the large difference between the temperature dependence of the InAs QD microcavity lasers and that of the conventional InAs QD lasers can be seen by simply looking at the Fermi-Dirac occupancy of a carrier state:

For the conventional InAs QD lasers, E in Eq. (6) predominantly represents the energy levels of the carriers in the QDs. For the microcavity lasers, E predominantly represents the energy levels of the carriers in the bulk and the QWs which contribute to the surface recombination. Therefore (E-E_{F})/k_{B}T is a much larger number for the microcavity lasers than for the conventional lasers. By simple mathematical derivation, it can be seen that the large difference in the values of (E-E_{F})/k_{B}T leads to a large difference in the temperature dependence of the value of Eq. (6).

Assuming the QDs have a carrier capture time of 1ps [11], we estimate that the carrier density outside the QDs is not clamped above threshold, and that surface recombination is still a large portion of the total recombination rate above threshold. This explains the rapid decrease of the slope efficiency with increasing temperature and the rollover of the output power versus pump power characteristic in Fig. 2(b) and Fig. 3(b). These phenomena are not observed with the Febry-Perot InAs QD laser [7].

This study indicates that reduction of the surface recombination in such InAs QD microcavity lasers is important in order to obtain low threshold, high slope efficiency, high thermal stability, and CW operation. Surface passivation, p-type modulation doping [17], and incorporating Al-bearing materials around the QDs are three possible routes.

Now let’s look at the modeling results on lasing wavelength dependence. To explain the data points in Fig. 4 which show that the lasing wavelength corresponding to the lowest threshold is not aligned with the wavelength corresponding to the maximum QD density, we define G(λ) = [ρ_{QD}(λ)/ρ_{QD}(λ_{0})]G_{0}. G(λ) represents the maximum available gain for lasers at λ. When G_{th} ≪ G(λ), as λ increases the value of Eq. (5) will predominantly follow the change of the term hc/λ and decrease. Therefore, N and the surface recombination rate at threshold will decrease. Consequently, in Fig. 4, the threshold decreases with increasing λ up to around 1340 nm. After 1340 nm, G_{th} ≪ G(λ) is no longer true, and the decrease in the number of QDs participating in the lasing becomes important, which is represented by the first term on the right side of Eq. (5).

The dashed curves in Fig. 4 are the modeling results with one single fitting parameter G_{th}/G_{0}. Threshold values in this model are up to a constant multiplication factor. Again, the sample material’s energy band shift and the lasing wavelength shift under different pump powers have been determined to be insignificant and ignored in the calculation. Modeling results with G_{th}/G_{0} = 1/7 agree well with the experimental results. Modeling results with G_{th}/G_{0} = 1/100 shows how thresholds decrease with increasing lasing wavelength when G_{th} ≪ G(λ).

This study of lasing wavelength dependence indicates how to engineer the lasing threshold values. A constant threshold value in a broad lasing wavelength range is favorable for photonic integrated circuits.

## 6. Conclusion

We have reported InAs QD microdisk lasers, photonic crystal nanocavity lasers, and their lasing behaviors as a function of temperature and lasing wavelength. A simple model attributes the strong temperature dependence of these devices predominantly to surface recombination near room temperature. High Q values of the photonic crystal nanocavity lasers result in threshold decreases with increasing lasing wavelength. These studies are important for understanding, improving, and engineering the QD microcavity devices in order to be applied in high density photonic integrated circuits.

## Acknowledgments

This study is based on research supported by the National Science Foundation (NSF) and the Defense Advanced Research Projects Agency (DARPA).

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