ISDRS 2011, December 7-9, 2011, College Park, MD, USA

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Student Paper

Overcoming Auger Recombination in Nanocrystal Quantum Dots Using Purcell

Enhancement

Shilpi Gupta1, Edo Waks1

1Department of Electrical and Computer Engineering, University of Maryland, USA, shilpi@umd.edu

Chemically synthesized semiconductor nanocrystal quantum dots (NQDs) are promising

candidates for gain media in micro/nano-lasers. The key advantage of using these NQDs is wide-

tunability of emission wavelength across the visible and near IR spectrum [1], which is achieved by

synthesizing NQDs of different sizes. One well-known decay process, which becomes significant with

size reduction in NQDs, is non-radiative Auger recombination, and it has been identified as the main

hindrance in achieving lasing with NQDs [2]. Lasing has been demonstrated in a variety of low-loss

microcavities with moderate spatial confinement [3][4], by employing close-packed films of NQDs – a

way to overcome Auger recombination process. Here we show that by using moderate-loss cavities that

provide strong spatial confinement, which results in Purcell enhancement [5], we can overcome Auger

recombination process, and achieve efficient lasing with only a few hundreds of NQDs.

The NQD-laser is composed of NQDs integrated with an optical cavity at room temperature (Fig.

1a). NQDs serve as the gain medium for the laser, and are excited by an above-band laser pump. The

optical cavity is chosen to have a resonant mode at the emission wavelength of NQDs. An example of

optical cavity is shown in Fig. 1(a). It is an L3 photonic crystal cavity formed by removing three holes

from a hexagonal lattice type 2D photonic crystal.

We model the NQD as an incoherently-pumped three-level system with two electron-hole pairs,

as described in Fig. 1(c). The single exciton state |X> consists of one electron-hole pair which, when

recombines, leaves the NQD in the ground state, |g>. The biexcitonic state |XX> is made up of two

electron-hole pairs. The rate γ12 describes spontaneous emission of photons in radiative leaky modes and

nonradiative decay of lasing transition |X> - |g>. Similarly, γ23 represents total decay rate for |XX> - |X>

transition, comprising of spontaneous emission of photons in radiative leaky modes, nonradiative decay

and Auger recombination rate. Incoherent pumping of the laser levels is described by rate .

The coupling strength of the transitions between lasing levels |X> - |g> and |XX> - |X>, and the

cavity mode is described by the coupling strength where √ is the

electric field amplitude, V is the quantization volume or mode-volume, n is the refractive index and is

the dipole moment [6]. Note that depends on the electric field amplitude at the position of the NQD.

Thus, if the electric field amplitude is not constant over the entire cavity region (Fig. 1b), NQDs in

different locations in the cavity couple with different strengths to the cavity mode. In order to properly

take this into account in the calculations, we divide the cavity region into several sub-regions, assuming

constant electric field amplitude across each sub-region. We also assume a uniform areal density of NQD,

for the entire cavity region.

To analyze the dynamics of the NQD-cavity system by properly taking into account cavity and

NQD damping, we use density matrix formalism governed by the master equation. The equations of

motion for the projections of density matrix on the NQD levels are obtained after tracing over photon

states, and applying the semiclassical approximation. Large dephasing rate of these NQDs allows us to

adiabatically eliminate the expectation value of the off-diagonal terms, yielding population rate equations

for the three-level system.

For calculations, we consider CdSe/ZnS core-shell NQDs embedded in the dielectric slab of

silicon nitride L3 photonic crystal cavity. Using the rate equations, we study the threshold requirement of

NQD density for lasing (Fig. 2). Our calculations show that for an L3 photonic crystal cavity with a Q of

3000 and maximal coupling strength of 80ns-1, the NQD density required for lasing is ~500 m-2. This

translates into ~200 NQDs in the cavity (cavity area ~0.38 m-2) which is much less than the number

required for a single close-packed layer (~5000). Another advantage of working in the Purcell-enhanced

regime is the increase in the fraction of the spontaneous emission into the lasing mode, Fig. 3), which

results in a lowering of the pump threshold requirement.

Figure 4 shows the typical laser input/output power curves (LL curves) for different coupling

strengths for a cavity Q of 3000 and total number of NQDs, N = 2000. With an increase in coupling

ISDRS 2011, December 7-9, 2011, College Park, MD, USA

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strength , we see an increase in the differential quantum efficiency (DQE) of the device, defined as the

slope of the LL curve above threshold (Fig. 5). The maximum DQE limit is set by the ratio of cavity

mode frequency to pump frequency, and our calculations show that in Purcell-enhanced regime the NQD-

laser works close to the maximum efficiency limit. Figure 6 shows the NQD-laser linewidth (calculated

using the Schawlow-Townes formalism) as a function of pump rate, for different coupling strengths .

At small pump rates, the linewidth is broader than the empty cavity linewidth, due to the presence of

absorbers, i.e NQDs. Thereafter, a decrease in laser linewidth (Fig. 6) and a linear input/output power

regime (Fig. 5) follow, both of which are clear signatures of lasing.

In conclusion, we show that by using high spatial confinement optical cavities, resulting in

Purcell enhancement of spontaneous emission, our scheme can overcome Auger recombination with only

a few hundred NQDs, compared to the close-packed film requirement for moderately confining cavities

lacking Purcell enhancement. Further, we show that the NQD-laser works close to the maximum

efficiency limit in the Purcell enhanced regime.

References

[1] H.-J. Eisler, et al. “Color-selective semiconductor nanocrystal laser,” Applied Physics Letters, vol. 80,

no. 24, pp. 4614, 2002.

[2] V. I. Klimov, et al. “Quantization of Multiparticle Auger Rates in Semiconductor Quantum Dots,”

Science, vol. 287, no. 5455, pp. 1011-1013, Feb 2000.

[3] P. T. Snee, Y. Chan, D. G. Nocera, and M. G. Bawendi, “Whispering-Gallery-Mode Lasing from a

Semiconductor Nanocrystal/Microsphere Resonator Composite,” Advanced Materials, vol. 17, no. 9, pp.

1131-1136, May 2005.

[4] B. Min, et al. “Ultralow threshold on-chip microcavity nanocrystal quantum dot lasers,” Applied

Physics Letters, vol. 89, no. 19, pp. 191124, 2006.

[5] E. M. Purcell, H. C. Torrey, and R. V. Pound, “Resonance Absorption by Nuclear Magnetic Moments

in a Solid”, Phys. Rev., vol. 69, pp. 681, 1946.

[6] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, England,

1997).

Fig. 5 DQE as a function of NQD-

cavity coupling strength 𝑔 for

different total number of NQD in

cavity, N = 500, 1000, 2000, 5000.

Fig. 1 Schematic of (a) a L3

photonic crystal cavity with

uniform density of NQDs, (b)

cavity mode profile, and (c) level

diagram of a three-level model

of a NQD.

Fig. 2 Variation of threshold

NQD density with coupling

strength 𝑔 for different cavity

quality factors Q = 1000, 3000,

10,000, 60,000.

Fig. 3 Dependence of the fraction

of spontaneous emission β coupled

to the cavity mode on NQD-cavity

coupling strength 𝑔 .

Fig. 4 Dependence of mean cavity

photon number on pump rate, for

different coupling strengths

𝑔 30, 40, 60, 150 ns-1.

Fig. 6 NQD-laser linewidth as

a function of pump rate, for

different coupling strengths

𝑔 = 20, 40, 60, 80, 100 ns-1.