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ISDRS 2011, December 7-9, 2011, College Park, MD, USA
ISDRS 2011 – http://www.ece.umd.edu/ISDRS2011
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, [email protected]
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
ISDRS 2011 – http://www.ece.umd.edu/ISDRS2011
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.