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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 3 — Feb. 10, 2014
  • pp: 3013–3027
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Overcoming Auger recombination in nanocrystal quantum dot laser using spontaneous emission enhancement

Shilpi Gupta and Edo Waks  »View Author Affiliations


Optics Express, Vol. 22, Issue 3, pp. 3013-3027 (2014)
http://dx.doi.org/10.1364/OE.22.003013


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Abstract

We propose a method to overcome Auger recombination in nanocrystal quantum dot lasers using cavity-enhanced spontaneous emission. We derive a numerical model for a laser composed of nanocrystal quantum dots coupled to optical nanocavities with small mode-volume. Using this model, we demonstrate that spontaneous emission enhancement of the biexciton transition lowers the lasing threshold by reducing the effect of Auger recombination. We analyze a photonic crystal nanobeam cavity laser as a realistic device structure that implements the proposed approach.

© 2014 Optical Society of America

1. Introduction

Room-temperature nanolasers have applications in fields ranging from optical communications and information processing [1

1. M. T. Hill, “Nanophotonics: lasers go beyond diffraction limit,” Nat. Nanotechnol. 4, 706–707 (2009). [CrossRef] [PubMed]

] to biological sensing [2

2. S. Kita, S. Hachuda, S. Otsuka, T. Endo, Y. Imai, Y. Nishijima, H. Misawa, and T. Baba, “Super-sensitivity in label-free protein sensing using a nanoslot nanolaser,” Opt. Express 19, 17683–17690 (2011). [CrossRef] [PubMed]

] and medical diagnostics [3

3. P. L. Gourley, J. K. Hendricks, A. E. McDonald, R. G. Copeland, K. E. Barrett, C. R. Gourley, and R. K. Naviaux, “Ultrafast nanolaser flow device for detecting cancer in single cells,” Biomed. Microdevices 7, 331–339 (2005). [CrossRef]

]. Colloidally synthesized nanocrystal quantum dots are a promising gain material for nanolasers. These quantum dots are efficient emitters at room temperature [4

4. W. J. Parak, D. Gerion, T. Pellegrino, D. Zanchet, C. Micheel, S. C. Williams, R. Boudreau, M. A. Le Gros, C. A. Larabell, and A. P. Alivisatos, “Biological applications of colloidal nanocrystals,” Nanotechnology 14, R15–R27 (2003). [CrossRef]

, 5

5. L. Qu and X. Peng, “Control of photoluminescence properties of CdSe nanocrystals in growth,” J. Am. Chem. Soc. 124, 2049–2055 (2002). [CrossRef] [PubMed]

], have broadly tunable emission frequencies [6

6. A. P. Alivisatos, “Semiconductor clusters, nanocrystals, and quantum dots,” Science 271, 933 (1996). [CrossRef]

, 7

7. V. I. Klimov, Semiconductor and Metal Nanocrystals: Synthesis and Electronic and Optical Properties (Marcel Dekker, 2003). [CrossRef]

] and are easy to integrate with photonic structures [8

8. H.-J. Eisler, V. C. Sundar, M. G. Bawendi, M. Walsh, H. I. Smith, and V. Klimov, “Color-selective semiconductor nanocrystal laser,” Appl. Phys. Lett. 80, 4614 (2002). [CrossRef]

10

10. B. Min, S. Kim, K. Okamoto, L. Yang, A. Scherer, H. Atwater, and K. Vahala, “Ultralow threshold on-chip microcavity nanocrystal quantum dot lasers,” Appl. Phys. Lett. 89, 191124 (2006). [CrossRef]

].

Nanocrystal quantum dot lasers have been demonstrated using resonant structures such as distributed feedback gratings [8

8. H.-J. Eisler, V. C. Sundar, M. G. Bawendi, M. Walsh, H. I. Smith, and V. Klimov, “Color-selective semiconductor nanocrystal laser,” Appl. Phys. Lett. 80, 4614 (2002). [CrossRef]

], microspheres [9

9. P. T. Snee, Y. Chan, D. G. Nocera, and M. G. Bawendi, “Whispering-gallery-mode lasing from a semiconductor nanocrystal/microsphere resonator composite,” Adv. Mater. 17, 1131–1136 (2005). [CrossRef]

], and micro-toroids [10

10. B. Min, S. Kim, K. Okamoto, L. Yang, A. Scherer, H. Atwater, and K. Vahala, “Ultralow threshold on-chip microcavity nanocrystal quantum dot lasers,” Appl. Phys. Lett. 89, 191124 (2006). [CrossRef]

]. However, these devices have exhibited high lasing thresholds due to fast non-radiative decay caused by Auger recombination [11

11. V. I. Klimov, A. A. Mikhailovsky, S. Xu, A. Malko, J. A. Hollingsworth, C. A. Leatherdale, H.-J. Eisler, and M. G. Bawendi, “Optical gain and stimulated emission in nanocrystal quantum dots,” Science 290, 314–317 (2000). [CrossRef] [PubMed]

, 12

12. V. I. Klimov, A. A. Mikhailovsky, D. W. McBranch, C. A. Leatherdale, and M. G. Bawendi, “Quantization of multiparticle Auger Rates in semiconductor quantum dots,” Science 287, 1011–1013 (2000). [CrossRef] [PubMed]

]. Nanocrystal quantum dots have a fast Auger recombination rate owing to the tight spatial confinement of carriers [12

12. V. I. Klimov, A. A. Mikhailovsky, D. W. McBranch, C. A. Leatherdale, and M. G. Bawendi, “Quantization of multiparticle Auger Rates in semiconductor quantum dots,” Science 287, 1011–1013 (2000). [CrossRef] [PubMed]

]. One approach to reduce Auger recombination is by engineering quantum dots with decreased spatial confinement. For example, elongated nanocrystals (quantum rods) can reduce Auger recombination [13

13. H. Htoon, J. Hollingsworth, R. Dickerson, and V. Klimov, “Effect of zero- to one-dimensional transformation on multiparticle Auger recombination in semiconductor quantum rods,” Phys. Rev. Lett. 91, 1–4 (2003). [CrossRef]

, 14

14. H. Htoon, J. A. Hollingworth, A. V. Malko, R. Dickerson, and V. I. Klimov, “Light amplification in semiconductor nanocrystals: quantum rods versus quantum dots,” Appl. Phys. Lett. 82, 4776 (2003). [CrossRef]

] to achieve lower threshold lasing [15

15. M. Kazes, D. Lewis, Y. Ebenstein, T. Mokari, and U. Banin, “Lasing from semiconductor quantum rods in a cylindrical microcavity,” Adv. Mater. 14, 317 (2002). [CrossRef]

]. Core/shell heteronanocrystals may also reduce the carrier spatial confinement [16

16. S. A. Ivanov, J. Nanda, A. Piryatinski, M. Achermann, L. P. Balet, I. V. Bezel, P. O. Anikeeva, S. Tretiak, and V. I. Klimov, “Light amplification using inverted core/shell nanocrystals: towards lasing in the single-exciton regime,” J. Phys. Chem. B 108, 10625–10630 (2004). [CrossRef]

, 17

17. J. Nanda, S. A. Ivanov, H. Htoon, I. Bezel, A. Piryatinski, S. Tretiak, and V. I. Klimov, “Absorption cross sections and Auger recombination lifetimes in inverted core-shell nanocrystals: Implications for lasing performance,” J. Appl. Phys. 99, 034309 (2006). [CrossRef]

], but have yet to be successfully integrated into a laser structure.

Here we show that spontaneous emission rate enhancement in a small mode volume cavity [18

18. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

] can overcome Auger recombination and enable low threshold lasing. We derive a model for a nanocrystal quantum dot laser using a master equation formalism that accounts for both Auger recombination and spontaneous emission enhancement. Using this model we show that spontaneous emission enhancement reduces the effect of Auger recombination, resulting in up to a factor of 17 reduction in the lasing threshold. We analyze a nanobeam photonic crystal cavity as a promising device implementation to achieve low threshold lasing in the presence of Auger recombination.

In section 2 we derive the theoretical formalism for a nanocrystal quantum dot laser. Section 3 presents numerical calculations for a general cavity structure under the uniform-field approximation. In section 4 we propose and analyze a nanobeam photonic crystal cavity design as a potential device implementation of a nanocrystal quantum dot laser.

2. Derivation of numerical model

Figure 1(a) illustrates the general model for a nanocrystal quantum dot laser. The laser is composed of an ensemble of quantum dots coupled to a single cavity mode. The level structure of the quantum dots, shown in Fig. 1(b), consists of four states: a ground state |1〉 which contains no carriers, the single exciton states |2〉 and |3〉 which contain a single electron-hole pair, and the biexciton state |4〉 which contains two electron-hole pairs. In the single exciton states, the quantum dot absorbs and emits a photon with nearly equal probability. Thus, only the biexciton state can provide optical gain [19

19. V. I. Klimov, “From fundamental photophysics to multicolor lasing,” Los Alamos Science 28, 214–220 (2003).

]. However, this state suffers from Auger recombination where an electron-hole pair recombines and transfers energy non-radiatively to a third carrier [12

12. V. I. Klimov, A. A. Mikhailovsky, D. W. McBranch, C. A. Leatherdale, and M. G. Bawendi, “Quantization of multiparticle Auger Rates in semiconductor quantum dots,” Science 287, 1011–1013 (2000). [CrossRef] [PubMed]

]. The strong carrier confinement in the quantum dots leads to fast Auger recombination, resulting in a low biexciton radiative efficiency.

Fig. 1 (a) Schematic of a laser composed of nanocrystal quantum dots coupled to an optical cavity. (b) Level diagram for a four-level model of a nanocrystal quantum dot.

Figure 1(b) also shows the relevant decay rates for our quantum dot model. The biexciton state decays to each exciton state with the rate γ2 = γ0 + γa/2, where γ0 is the spontaneous emission rate and γa is the total Auger recombination rate of the biexciton state. We assume the single exciton states decay predominantly by spontaneous emission. We also assume equal spontaneous emission rates for all four allowed transitions, and ignore long-lived trap states that are responsible for blinking behavior [20

20. M. Nirmal, B. O. Dabbousi, M. G. Bawendi, J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Fluorescence intermittency in single cadmium selenide nanocrystals,” Nature 383, 802–804 (1996). [CrossRef]

, 21

21. M. Jones, J. Nedeljkovic, R. J. Ellingson, A. J. Nozik, and G. Rumbles, “Photoenhancement of luminescence in colloidal CdSe quantum dot solutions,” J. Phys. Chem. B 107, 11346–11352 (2003). [CrossRef]

]. These states can be incorporated as additional energy levels in the model. The quantum dot is incoherently pumped with an external source characterized by the excitation rate R.

In bare nanocrystal quantum dots Auger recombination is an order of magnitude faster than spontaneous emission [12

12. V. I. Klimov, A. A. Mikhailovsky, D. W. McBranch, C. A. Leatherdale, and M. G. Bawendi, “Quantization of multiparticle Auger Rates in semiconductor quantum dots,” Science 287, 1011–1013 (2000). [CrossRef] [PubMed]

]. It therefore dominates the decay of the biexciton state and quenches the optical gain. However, when the quantum dot spectrally couples to an optical cavity, its spontaneous emission rate increases by the factor [22

22. J. Gerard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots,” Top. of Appl. Phys. 90, 283–327 (2003).

].
F(r0)=1+2g2(r0)γ0KXX
(1)
where g(r0) is the cavity-quantum dot coupling strength given by
g(r0)=μe^h¯h¯ωc2ε0Vm|E(r0)||E(r)|max
(2)
Here, E(r0) is the electric field amplitude, ê is the polarization direction of the cavity mode at the quantum dot position r0, ωc is the cavity mode resonant frequency, μ is the quantum dot dipole moment, Vm = ∫d3rε(r)|E(r)|2/[|E(r)|2]max is the cavity mode-volume [23

23. Y. Zhang, I. Bulu, W.-M. Tam, B. Levitt, J. Shah, T. Botto, and M. Loncar, “High-Q/V air-mode photonic crystal cavities at microwave frequencies,” Opt. Express 19, 9371–9377 (2011). [CrossRef] [PubMed]

], ε0 is the permittivity of free space and ε(r) is the relative dielectric permittivity. Since ε(r) is dimensionless, Vm has dimensions of volume. The rate KXX = (γ0 + 2γ2 + γd)/2 represents the total linewidth of the biexciton state, which is dominated by the dephasing rate γd at room-temperature [12

12. V. I. Klimov, A. A. Mikhailovsky, D. W. McBranch, C. A. Leatherdale, and M. G. Bawendi, “Quantization of multiparticle Auger Rates in semiconductor quantum dots,” Science 287, 1011–1013 (2000). [CrossRef] [PubMed]

, 24

24. W. G. J. H. M. van Sark, P. L. T. M. Frederix, D. J. Van den Heuvel, H. C. Gerritsen, A. A. Bol, J. N. J. van Lingen, C. de Mello Donegá, and A. Meijerink, “Photooxidation and photobleaching of single CdSe/ZnS quantum dots probed by room-temperature time-resolved spectroscopy,” J. Phys. Chem. B 105, 8281–8284 (2001). [CrossRef]

, 25

25. B. Lounis, H. A. Bechtel, D. Gerion, P. Alivisatos, and W. E. Moerner, “Photon antibunching in single CdSe/ZnS quantum dot fluorescence,” Chem. Phys. Lett. 329, 399–404 (2000). [CrossRef]

]. We note that Eq. (1) is different from the more common expression for F that depends on the ratio of the cavity quality factor Q and the cavity mode-volume Vm [18

18. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

,22

22. J. Gerard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots,” Top. of Appl. Phys. 90, 283–327 (2003).

]. This difference occurs because at room temperature the dephasing rate of nanocrystal quantum dots is much larger than the cavity linewidth. The device therefore operates in the bad emitter regime, where F becomes independent of the cavity Q. By engineering cavities with small mode-volumes, we can achieve large F and enhance the spontaneous emission rate, thereby increasing the radiative efficiency of the quantum dot in the presence of Auger recombination.

To analyze the nanocrystal quantum dot laser in the presence of Auger recombination and spontaneous emission enhancement, we begin with the master equation
ρt=ih¯[ρ,H]+Lρ
(3)
where ρ is the density matrix of the combined cavity-quantum dot system, H is the Hamiltonian, and L is the Liouvillian superoperator that accounts for incoherent damping and excitation processes. The Hamiltonian of the system is given by Hcavity + HNQD + HJC, where
Hcavity=h¯ωca^a
(4)
HNQD=m=1Nh¯ωmX(σ22,m+σ33,m)+h¯ωmXXσ44,m
(5)
HJC=i=mNh¯gmX(rm)(σ21,ma+σ12,ma+σ31,ma+σ13,ma)+h¯gmXX(rm)(σ42,ma+σ24,ma+σ43,ma+σ34,ma)
(6)
In the above equations a and a are the bosonic annihilation and creation operators of the cavity mode. The summation is carried out over all quantum dots in the cavity, where we denote the total number of quantum dots by N. For the mth quantum dot, σjk,m = |j〉 〈k| represents the atomic dipole operator when jk and the atomic population operator when j = k, for the single exciton states (j = 2, 3) and the biexciton state (j = 4). We set the energy of the quantum dot ground state to zero. We define ωmX and ωmXX as the resonant frequencies of the single-exciton and biexciton transitions, respectively. Similarly, the cavity-quantum dot coupling strengths for the exciton and biexciton transitions are gmX(rm) and gmXX(rm) for the mth quantum dot at position rm. At room temperature, the homogenous linewidth of these quantum dots is much larger than the biexcitonic shift [26

26. J.-M. Caruge, Y. Chan, V. Sundar, H. Eisler, and M. Bawendi, “Transient photoluminescence and simultaneous amplified spontaneous emission from multiexciton states in CdSe quantum dots,” Phys. Rev. B 70, 1–7 (2004). [CrossRef]

29

29. D. Norris and M. Bawendi, “Measurement and assignment of the size-dependent optical spectrum in CdSe quantum dots,” Phys. Rev. B, 53, 16338–16346 (1996). [CrossRef]

]. We therefore assume all four transitions of each quantum dot are resonantly coupled to the cavity mode ( ωc=ωmX=ωmXX/2). The Liouvillian superoperator L is fully defined in Appendix A.

The master equation is difficult to solve both analytically and numerically when the number of quantum dots becomes large. However, we can simplify the calculations by applying the semi-classical approximation in which the coherence between the atoms and the field is neglected [30

30. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).

, 31

31. O. Benson and Y. Yamamoto, “Master-equation model of a single-quantum-dot microsphere laser,” Phys. Rev. A 59, 4756–4763 (1999). [CrossRef]

] and the density matrix can be factorized into a product of the state of the field and atoms (see Appendix B). Under this approximation, the system is described by the average cavity photon number, p, and the quantum dot population density, nj(r)=limΔV0mσjjm/ΔV, where the sum is carried out over all quantum dots contained in a small volume ΔV at location r. We note that nj(r) is a function of the position r inside the cavity because of the non-uniform cavity field distribution. We derive the equations of motion of nj(r) from the master equation (see Appendix B) as
n1(r)t=ΓX(r)[(p+1)(n2(r)+n3(r))2pn1(r)]+γ0[n2(r)+n3(r)]2Rn1(r)
(7)
n2(r)t=ΓX(r)[(p+1)n2(r)pn1(r)]+ΓXX(r)[(p+1)n4(r)pn2(r)]γ0n2(r)+γ2n4(r)+R[n1(r)n2(r)]
(8)
n3(r)t=ΓX(r)[(p+1)n3(r)pn1(r)]+ΓXX(r)[(p+1)n4(r)pn3(r)]γ0n3(r)+γ2n4(r)+R[n1(r)n3(r)]
(9)
n4(r)t=ΓXX(r)[2(p+1)n4(r)p(n2(r)+n3(r))]2γ2n4(r)+R[n2(r)+n3(r)]
(10)

In the above equations, ΓX (r) = 2g2(r)/KX and ΓXX (r) = 2g2(r)/KXX are the modified spontaneous emission rates of the single-exciton and biexciton transitions, where KX = (γ0 + γd + 3R)/2 and KXX = (γ0 + 2γ2 + γd + R)/2. Here, we assume equal coupling strength for the single-exciton and biexciton transitions. We also treat the quantum dots in a small volume ΔV of the cavity to be identical, and therefore drop the subscript m from the coupling strength ( g(r)=gmX(rm)=gmXX(rm)).

The average cavity photon number satisfies a rate equation given by (see Appendix C for derivation)
pt=pκ+pG(p)+α(p)
(11)
where κ = ωc/Q is the cavity energy decay rate. The above equation is coupled to the quantum dot population density rate equations through the cavity gain coefficient
G(p)=d3r{ΓX(r)[n2(r)+n3(r)2n1(r)]+ΓXX(r)[2n4(r)n2(r)n3(r)]}
(12)
and the spontaneous emission rate into the lasing mode
α(p)=d3r{ΓX(r)[n2(r)+n3(r)]+2ΓXX(r)n4(r)}
(13)
where the integral is over all space. We use the notation G(p) and α(p) to highlight the fact that the above coefficients have a p dependence because the atomic densities nj(r) depend on the cavity photon number. The absorbed pump power of the nanocrystal quantum dot laser is given by
Pabs=h¯ωpRVpd3r[2n1(r)+n2(r)+n3(r)]
(14)
where ωp is the pump frequency and Vp is the optically pumped volume. The output power of the laser is given by
Pout=h¯ωcpκ
(15)

An important figure of merit for small mode-volume cavities is the spontaneous emission coupling efficiency, denoted by β. This parameter quantifies the fraction of photons spontaneously emitted to the cavity mode. A β approaching unity achieves thresholdless lasing [32

32. G. Bjork and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27, 2386–2396 (1991). [CrossRef]

]. In the quantum dot model, the single exciton and biexciton transitions have different coupling efficiencies given by
βX(r)=ΓX(r)ΓX(r)+γ0
(16)
βXX(r)=ΓXX(r)ΓXX(r)+γ2
(17)
The above coupling efficiencies depend on the position r due to the spatially varying cavity field intensity. The rate equations Eqs. (7)(11) describe the dynamics of a general nanocrystal quantum dot laser. We will use these equations in the remaining sections.

3. Lasing analysis under uniform-field approximation

Under the uniform field approximation the atomic population densities nj(r) are no longer spatially varying. We can therefore express the equations of motion in terms of the total number of quantum dots in state j given by Nj = Vmnj where Vm is the cavity mode volume. These quantum dot populations must satisfy the constraint that ∑jNj = N, where N is the total number of quantum dots contained in the cavity. With these definitions, the equations of motion become the standard cavity-atom rate equations, given by
N1t=Γ¯X[(p+1)(N2+N3)2pN1]+γ0(N2+N3)2RN1
(20)
N2t=Γ¯X[(p+1)N2pN1]+Γ¯XX[(p+1)N4pN2]γ0N2+γ2N4+R(N1N2)
(21)
N3t=Γ¯X[(p+1)N3pN1]+Γ¯XX[(p+1)N4pN3]γ0N3+γ2N4+R(N1N3)
(22)
N4t=Γ¯XX[2(p+1)N4p(N2+N3)]2γ2N4+R(N2+N3)
(23)
pt=pκ+pG¯(p)+α¯(p)
(24)
where
G¯(p)=Γ¯X(N2+N32N1)+Γ¯XX(2N4N2N3)
(25)
and
α¯(p)=Γ¯X(N2+N3)+2Γ¯XXN4
(26)
are the gain coefficient and spontaneous emission rate into the lasing mode. The absorbed power is given by
P¯abs=h¯ωpR(2N1+N2+N3)
(27)
The output power of the laser is still given by Eq. (15).

We first determine the minimum number of quantum dots required to achieve lasing. We define Nth as the total number of quantum dots in the cavity required to achieve a small signal gain equal to the cavity loss (limp→0 (p) = κ), and calculate it by using the analytical steady-state solutions to Eqs. (20)(23) along with the condition ∑jNj = N (see Appendices D, E). To perform calculations, we consider the specific example of colloidal CdSe/ZnS core-shell quantum dots that emit in a wavelength range of 500–700 nm. We perform simulations using a dephasing rate of γd = 4.39×104 ns−1 [24

24. W. G. J. H. M. van Sark, P. L. T. M. Frederix, D. J. Van den Heuvel, H. C. Gerritsen, A. A. Bol, J. N. J. van Lingen, C. de Mello Donegá, and A. Meijerink, “Photooxidation and photobleaching of single CdSe/ZnS quantum dots probed by room-temperature time-resolved spectroscopy,” J. Phys. Chem. B 105, 8281–8284 (2001). [CrossRef]

], a spontaneous emission rate of γ0 = 1/18 ns−1 [25

25. B. Lounis, H. A. Bechtel, D. Gerion, P. Alivisatos, and W. E. Moerner, “Photon antibunching in single CdSe/ZnS quantum dot fluorescence,” Chem. Phys. Lett. 329, 399–404 (2000). [CrossRef]

], and an Auger recombination rate of γa = 1/300 ps−1 [12

12. V. I. Klimov, A. A. Mikhailovsky, D. W. McBranch, C. A. Leatherdale, and M. G. Bawendi, “Quantization of multiparticle Auger Rates in semiconductor quantum dots,” Science 287, 1011–1013 (2000). [CrossRef] [PubMed]

, 34

34. M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alivisatos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998). [CrossRef] [PubMed]

]. Nanocrystal quantum dots can be incorporated into photonic devices in a variety of ways such as spin-casting [10

10. B. Min, S. Kim, K. Okamoto, L. Yang, A. Scherer, H. Atwater, and K. Vahala, “Ultralow threshold on-chip microcavity nanocrystal quantum dot lasers,” Appl. Phys. Lett. 89, 191124 (2006). [CrossRef]

,35

35. R. Bose, X. Yang, R. Chatterjee, J. Gao, and C. W. Wong, “Weak coupling interactions of colloidal lead sulphide nanocrystals with silicon photonic crystal nanocavities near 1.55μm at room temperature,” Appl. Phys. Lett. 90, 111117 (2007). [CrossRef]

38

38. M. T. Rakher, R. Bose, C. W. Wong, and K. Srinivasan, “Spectroscopy of 1.55μm PbS quantum dots on Si photonic crystal cavities with a fiber taper waveguide,” Appl. Phys. Lett. 96, 161108 (2010). [CrossRef]

] and immersion in liquid suspension [9

9. P. T. Snee, Y. Chan, D. G. Nocera, and M. G. Bawendi, “Whispering-gallery-mode lasing from a semiconductor nanocrystal/microsphere resonator composite,” Adv. Mater. 17, 1131–1136 (2005). [CrossRef]

, 39

39. Z. Wu, Z. Mi, P. Bhattacharya, T. Zhu, and J. Xu, “Enhanced spontaneous emission at 1.55μm from colloidal PbSe quantum dots in a Si photonic crystal microcavity,” Appl. Phys. Lett. 90, 171105 (2007). [CrossRef]

]. In these cases, the quantum dots reside on the surfaces of the devices, so we set εeff = 1.

Figure 2(a) plots Nth as a function of pump rate R for Vm = 0.01μm3, 1μm3 and 100μm3 and γa = 1/300 ps−1. Each mode-volume exhibits an optimum pump rate where the threshold quantum dot number is minimum. We denote this minimum threshold quantum dot number by Nopt. Figure 2(b) plots Nopt as a function of Vm. The figure shows that Nopt scales linearly with mode-volume.

Fig. 2 (a) Nth as a function of pump rate for Vm = 0.01μm3, 1 μm3 and 100μm3, γa = 1/300 ps−1. (b) Nopt for different mode-volumes for γa = 1/300 ps−1.

Next, we investigate the laser input-output power characteristics. We calculate the laser output power (using Eq. (15)) and the absorbed pump power (using Eq. (27)) using the numerical steady-state solutions to Eqs. (20)(24). Figure 3(a) plots Pout as a function of abs (also known as the light-in light-out curve), under the uniform-field approximation, for two different mode-volumes of Vm = 0.01 μm3 and 100 μm3, as well as two different Auger recombination rates of γa = 1/300 ps−1 and 0. We set Q = 20000 and N = 2Nopt (Fig. 2(b)) for each respective mode-volume. We calculate the curves in Fig. 3(a) using the same range of R values for both the mode-volumes. We note that the curves for the small mode volume cavity terminate earlier than those of the large mode volume cavity because the number of quantum dots contained inside the cavity mode-volume is much lower, which reduces the maximum output power.

Fig. 3 (a) Laser output power as a function of the absorbed pump power for Vm = 0.01μm3 and 100μm3. (b) Threshold absorbed pump power as a function of mode-volume. (c) η as a function of mode-volume.

The cavities with Vm = 100μm3, indicated by the dashed curves in Fig. 3(a), exhibit a pronounced threshold. Near threshold, the light-in light-out curve takes on the well-known S-curve behavior as it transitions from the below-threshold to above-threshold regime. Auger recombination increases the threshold by quenching the gain, which causes the S-curve region to occur at higher absorbed powers. Similar to Nth, we define the threshold power as the absorbed power where the small signal gain equals the cavity loss. We calculate this value numerically using the steady state solutions to Eqs. (20)(23), along with Eq. (27). The threshold power for Vm = 100 μm3 is 122.7 μW when γa = 1/300 ps−1, and 5.9 μW when γa = 0. Auger recombination therefore increases the lasing threshold by a factor of 21. When the mode volume is Vm = 0.01 μm3 the light-in light-out curve exhibits a thresholdless lasing behavior. The output power is nearly a linear function of the input power. Using the same definition of threshold, we determine the threshold powers with and without Auger recombination to be 97 nW and 84 nW respectively, corresponding to an increase of only 1.2. Thus, not only does the small mode volume cavity exhibit a much lower overall lasing threshold, but the lasing threshold is also largely unaffected by Auger recombination.

Figure 3(b) plots the absorbed pump power at threshold as a function of Vm for γa = 1/300 ps−1 and 0. We set the total quantum dot number in the cavities to N = 2Nopt for each value of Vm (Fig. 2(b)). Figure 3(c) plots the ratio of absorbed pump powers at threshold for γa = 1/300 ps−1 and 0, η, as a function of Vm. From this curve, we observe that below a mode-volume of 0.1 μm3 the lasing threshold is largely unaffected by Auger recombination. Above this mode volume, η rapidly increases and eventually reaches a saturated value. At large mode-volumes, η becomes independent of the mode volume itself and achieves an asymptotic limit. From the upper and the lower limits of η (21 and 1.2, respectively), we determine that spontaneous emission enhancement can reduce the lasing threshold up to a factor of 17.

To verify that the improvement in lasing threshold is due to spontaneous emission enhancement, we calculate the spontaneous emission coupling efficiency for the exciton and biexciton transition as a function of Vm. Using the uniform field approximation, we replace Γi(r) (i = X, XX) in Eqs. (16)(17) with its spatially averaged value Γ̄i which removes the spatial dependence and results in the simplified expressions for the coupling efficiencies given by
β¯X=Γ¯XΓX+γ0
(28)
β¯XX=Γ¯XXΓXX+γ2
(29)

Figure 4 plots spontaneous emission coupling efficiencies for the single-exciton transition β̄X and the biexciton transition β̄XX as a function of Vm using γa = 1/300 ps−1. At Vm = 100 μm3, β̄XX is more than an order of magnitude smaller than β̄X. As the mode volume decreases the two efficiencies approach unity. The coupling efficiency of the biexciton transition begins to increase sharply and approach unity around the same mode-volume where η (Fig. 3(b)) begins to saturate to unity. Thus, at small mode-volumes β̄XX is insensitive to Auger recombination, and therefore the threshold pump power does not significantly change as indicated in Fig. 3(b).

Fig. 4 Spontaneous emission coupling efficiency for single-exciton transition β̄X and biexciton transition β̄XX as a function of Vm for γa = 1/300 ps−1

4. Cavity device structure for low-threshold laser

Figure 5 shows the nanobeam photonic crystal cavity design that we consider for low threshold lasing. Nanocrystal quantum dots are typically spin cast onto the device and therefore reside outside the dielectric. We therefore design the cavity mode to be localized in the air holes rather than the dielectric material. This design choice maximizes the field overlap with the quantum dots.

Fig. 5 The electric field intensity (|E|2) of the resonant cavity mode of a nanobeam photonic crystal cavity. The seven holes in the center form the cavity defect.

The structure is composed of a silicon nitride beam with a one-dimensional periodic array of air holes (radius r = 0.24a, where a is the lattice constant). The cavity is composed of a defect in the structure created by gradually reducing the radius of the three holes on either side of the hole labelled C to a minimum of r0 = 0.2a. The adiabatic reduction of hole radius creates a smooth confinement for the photon and minimizes scattering due to edge states [52

52. Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96, 203102 (2010). [CrossRef]

]. The cavity is designed with beam thickness d = 0.727a and beam width b = 1.163a. The index of refraction of silicon nitride is set to 2.01 [53

53. M. Barth, J. Kouba, J. Stingl, B. Löchel, and O. Benson, “Modification of visible spontaneous emission with silicon nitride photonic crystal nanocavities,” Opt. Express 15, 17231–17240 (2007). [CrossRef] [PubMed]

]. We calculate the mode of the cavity using three dimensional finite-difference time-domain simulation (Lumerical Solutions, Inc.). Figure 5 shows the calculated electric field intensity overlaid on the structure. The computed mode-volume is Vm = 0.38λ3 (= 0.11 μm3) and the quality factor is Q = 64, 000.

Nanobeam photonic crystal cavities achieve mode-volumes that are on the order of a cubic wavelength. When the confinement volume of the cavity approaches the spatial variation of the field distribution, the uniform-field approximation can break down. We therefore analyze the nanobeam laser both with and without this approximation. We calculate εeff = 1.9 for the cavity by numerically integrating Eq. (19) using the computed electric field intensity profile of the simulated cavity structure (Fig. 5). Calculations under the uniform-field approximation follow the same approach as in the section 3.

Figure 6(a) plots Pout as a function of Pabs for the nanobeam photonic crystal cavity with simulated Q = 64, 000 using γa = 1/300 ps−1 and 0, both with and without the uniform-field approximation. The calculations show good agreement between the predicted input-output characteristics of the laser with and without the uniform-field approximation. Without the uniform-field approximation, the absorbed pump power at threshold for the nanobeam laser is 109.8 nW for γa = 1/300 ps−1 and 29.9 nW for γa = 0, resulting in η = 3.7. With the uniform-field approximation, the absorbed pump power at threshold for the nanobeam laser is 112.6 nW for γa = 1/300 ps−1 and 30 nW for γa = 0, resulting in η = 3.8.

Fig. 6 (a) Output power as a function of the absorbed pump power for nanocrystal quantum dot laser comprised of nanobeam photonic crystal cavity, using γa = 1/300 ps−1 and 0, both with and without uniform-field approximation (abbreviated as UFA in the legend). (b) η as a function of mode-volume under the uniform-field approximation for εeff = 1.9 and Q = 64,000.

5. Conclusion

In conclusion, we have theoretically shown that cavity-enhanced spontaneous emission of the biexciton reduces the effect of Auger recombination, leading to a lower lasing threshold. We developed a numerical model for a laser composed of an ensemble of nanocrystal quantum dots coupled to an optical cavity. The model can be expanded to incorporate more complex behavior of nanocrystal quantum dots, such as blinking, by introducing additional trap states into the quantum dot level structure [54

54. J. Zhao, G. Nair, B. R. Fisher, and M. G. Bawendi, “Challenge to the charging model of semiconductor-nanocrystal fluorescence intermittency from off-state quantum yields and multiexciton blinking,” Phys. Rev. Lett. 104, 1–4 (2010). [CrossRef]

, 55

55. M. Kuno, D. Fromm, S. Johnson, A. Gallagher, and D. Nesbitt, “Modeling distributed kinetics in isolated semiconductor quantum dots,” Phys. Rev. B 67, 1–15 (2003). [CrossRef]

]. This model can also be used to study lasing with other room-temperature emitters such as quantum rods [13

13. H. Htoon, J. Hollingsworth, R. Dickerson, and V. Klimov, “Effect of zero- to one-dimensional transformation on multiparticle Auger recombination in semiconductor quantum rods,” Phys. Rev. Lett. 91, 1–4 (2003). [CrossRef]

, 15

15. M. Kazes, D. Lewis, Y. Ebenstein, T. Mokari, and U. Banin, “Lasing from semiconductor quantum rods in a cylindrical microcavity,” Adv. Mater. 14, 317 (2002). [CrossRef]

], and other types of cavities such as plasmonic apertures [56

56. J. T. Choy, B. J. M. Hausmann, T. M. Babinec, I. Bulu, M. Khan, P. Maletinsky, A. Yacoby, and M. Lončar, “Enhanced single-photon emission from a diamondsilver aperture,” Nat. Photonics 5, 738–743 (2011). [CrossRef]

]. Our results provide a direction for development of low-threshold and highly tunable nanolasers that use nanocrystal quantum dot as gain material at room temperature.

Appendices

A. Liouvillian superoperator L

The Liouvillian superoperator L can be expressed as L = LNQD + Lpump + Lcavity, where LNQD accounts for the spontaneous relaxation of the quantum dot level structure, Lpump accounts for the incoherent pumping of the quantum dot population, and Lcavity accounts for the cavity decay. These operators are
LNQDρ=m=1Nγ0,m2(2σ12,mρσ21,mσ21,mσ12,mρρσ21,mσ12,m+2σ13,mρσ31,mσ31,mσ13,mρρσ31,mσ13,m)+γ2,m2(2σ24,mρσ42,mσ42,mσ24,mρρσ42,mσ24,m+2σ34,mρσ43,mσ43,mσ34,mρρσ43,mσ34,m)
(30)
Lpumpρ=m=1NR2(2σ21,mρσ12,mσ12,mσ21,mρρσ12,mσ21,m+2σ31,mρσ13,mσ13,mσ31,mρρσ13,mσ31,m+2σ42,mρσ24,mσ24,mσ42,mρρσ24,mσ42,m+2σ43,mρσ34,mσ34,mσ43,mρρσ34,mσ43,m)
(31)
Lcavityρ=κ2(2aρaaaρρaa)
(32)
The cavity energy decay rate is κ = ωc/Q.

B. Equations of motion: projected on quantum dot levels

The equations of motion for the projections of ρ on the levels (ij) of the mth quantum dot and photon states (pp’) ρip,jpm=mi,p|ρ|j,pm (i,j = 1, 2, 3, 4) and (p, p′= 0 to ∞) are obtained using Eq. (3):
ρ1p,1pmt=igmp(ρ1p,2p1mρ2p1,1pm+ρ1p,3p1mρ3p1,1pm)2Rρ1p,1pm+γ0(ρ2p,2pm+ρ3p,3pm)+κ((p+1)ρ1p+1,1p+1mpρ1p,1pm)
(33)
ρ2p,2pmt=igm(p+1(ρ2p,1p+1mρ1p+1,2pm)+p(ρ2p,4p1mρ4p1,2pm))(γ0+R)ρ2p,2pm+Rρ1p,1pm+γ2ρ4p,4pm+κ((p+1)ρ2p+1,2p+1mpρ2p,2pm)
(34)
ρ3p,3pmt=igm(p+1(ρ3p,1p+1mρ1p+1,3pm)+p(ρ3p,4p1mρ4p1,3pm))(γ0+R)ρ3p,3pm+Rρ1p,1pm+γ2ρ4p,4pm+κ((p+1)ρ3p+1,3p+1mpρ3p,3pm)
(35)
ρ4p,4pmt=igmp+1(ρ4p,2p+1mρ2p+1,4pm+ρ4p,3p+1mρ3p+1,4pm)2γ2ρ3p,3pm+R(ρ2p,2pm+ρ3p,3pm)+κ((p+1)ρ3p+1,3p+1mpρ3p,3pm)
(36)
ρ1p,2p1mt=igmp(ρ1p,1pmρ2p1,2p1m)KXρ1p,2p1m
(37)
ρ2p,4p1mt=igmp(ρ2p,2pmρ4p1,4p1m)KXXρ2p,4p1m
(38)
ρ1p,3p1mt=igmp(ρ1p,1pmρ3p1,3p1m)KXρ1p,3p1m
(39)
ρ3p,4p1mt=igmp(ρ3p,3pmρ4p1,4p1m)KXXρ3p,4p1m
(40)
Here, KX = (γ0 + γd + 3R)/2 and KXX = (γ0 + 2γ2 + γd + R)/2 are the total relaxation rates of the diagonal terms, and γd is the dephasing rate of the quantum dot (added phenomenologically). We set dephasing rate to be much greater than the cavity decay rate γdκ, allowing us to drop the cavity decay contributions from the equations of motion of off-diagonal terms (Eqs. (37)(40)). Large dephasing rate also allows us to adiabatically eliminate the expectation value 〈ρip,jp′〉 of the off-diagonal terms (ij)from Eqs. (37)(40), and reduces Eqs. (33)(36) to
ρ1p,1pmt=2gm2KX(ρ2p1,2p1m+ρ3p1,3p1m2ρ1p,1pm)p2Rρ1p,1pm+γ0(ρ2p,2pm+ρ3p,3pm)+κ((p+1)ρ1p+1,1p+1mpρ1p,1pm)
(41)
ρ2p,2pmt=2gm2KX(ρ2p,2pmρ1p+1,1p+1m)(p+1)+2gm2KXX(ρ4p1,4p1mρ2p,2pm)p(γ0+R)ρ2p,2pm+Rρ1p,1pm+γ2ρ4p,4pm+κ((p+1)ρ2p+1,2p+1mpρ2p,2pm)
(42)
ρ3p,3pmt=2gm2KX(ρ3p,3pmρ1p+1,1p+1m)(p+1)+2gm2KXX(ρ4p1,4p1mρ3p,3pm)p(γ0+R)ρ3p,3pm+Rρ1p,1pm+γ2ρ4p,4pm+κ((p+1)ρ3p+1,3p+1mpρ3p,3pm)
(43)
ρ4p,4pmt=2gm2KXX(2ρ4p,4pmρ2p+1,2p+1mρ3p+1,3p+1m)(p+1)2γ2ρ4p,4pm+R(ρ2p,2pm+ρ3p,3pm)+κ((p+1)ρ4p+1,4p+1mpρ4p,4pm)
(44)
Now, tracing over all the photon states in Eq. (41)(44), and applying semi-classical approximation to factorize full density matrix element into quantum dot and field parts such that ρip,ip = ρiiρpp, we get
ρ11mt=2gm2KX(ρ22m+ρ33m2ρ11m)p+2g2KX(ρ22m+ρ33m)2Rρ11m+γ0(ρ22m+ρ33m)
(45)
ρ22mt=2gm2KX(ρ22mρ11m)p+2gm2KXX(ρ44mρ22m)p2gm2KXρ22m+2gm2KXXρ44m(γ0+R)ρ22m+Rρ11m+γ2ρ44m
(46)
ρ33mt=2gm2KX(ρ33mρ11m)p+2gm2KXX(ρ44mρ33m)p2gm2KXρ33m+2gm2KXXρ44m(γ0+R)ρ33m+Rρ11m+γ2ρ44m
(47)
ρ44mt=2gm2KXX(2ρ44mρ22mρ33m)p4gm2KXXρ44m2γ2ρ44m+R(ρ22m+ρ33m)
(48)
where 〈p〉 = ∑p pp is the mean photon number. We define nj(r)=limΔV0mσjjm/ΔV as the quantum dot population density of the jth lasing level where the sum is carried out over all quantum dots contained in small volume ΔV at location r and get Eqs. (7)(10).

C. Rate equation for mean cavity photon number

The rate equation for the mean cavity photon number is given by
p˙=ppρ˙pp
(49)
Using Eqs. (41)(44)
p˙=mpp{2gm2KX(ρ2p1,2p1mρ1p,1pm)p2gm2KX(ρ2p,2pmρ1p+1,1p+1m)(p+1)+2gm2KX(ρ3p1,3p1mρ1p,1pm)p2gm2KX(ρ3p,3pmρ1p+1,1p+1m)(p+1)+2gm2KXX(ρ4p1,4p1mρ2p,2pm)p2gm2KXX(ρ4p,4pmρ2p+1,2p+1m)(p+1)+2gm2KXX(ρ4p1,4p1mρ3p,3pm)p2gm2KXX(ρ4p,4pmρ3p+1,3p+1m)(p+1)κ(pρpp(p+1)ρp+1p+1)}
(50)
Applying semi-classical approximation to factorize full density matrix element into quantum dot and field parts ρip,ip = ρiiρpp, and identifying p=0pρp,p=p gives
p˙=κp+m{2gm2KX(ρ22m+ρ33m2ρ11m)p+2gm2KXX(2ρ44mρ22mρ33m)p+2gm2KX(ρ22m+ρ33m)+4gm2KXXρ44m}
(51)
Eq. (51) leads us to Eq. (11).

D. Expression for Nj under the uniform-field approximation

Assuming total number of quantum dots in the cavity, N, such that ∑i Ni = N, Eqs. (20)(23) can be solved in the steady-state as
N1=((p+1)Γ¯X+γ0pΓX+R)N2ζ
(52)
N2=N3=N2ζ
(53)
N4=(pΓ¯XX+R(p+1)Γ¯XX+γ2)N2ζ
(54)
ζ=(p+1)Γ¯X+γ02(pΓ¯X+R)+1+pΓ¯XX+R2((p+1)Γ¯XX+γ2)
(55)
where ζ is the ratio of the total quantum dot population to the total single-exciton quantum dot population.

E. Quantum dot number required for achieving lasing threshold

Under uniform-field approximation
Nth=ωcQ(Γ¯X+γ02R+1+R2Γ¯XX+2γ2Γ¯X(1Γ¯X+γ0R)+Γ¯XX(RΓ¯XX+γ21))
(56)

Acknowledgments

We acknowledge funding support from the Physics Frontier Center at the Joint Quantum Institute (grant number PHY-0822671).

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44.

M. Khan, T. Babinec, M. W. McCutcheon, P. Deotare, and M. Loncar, “Fabrication and characterization of high-quality-factor silicon nitride nanobeam cavities,” Opt. Lett. 36, 421–423 (2011). [CrossRef] [PubMed]

45.

S. Gupta and E. Waks, “Spontaneous emission enhancement and saturable absorption of colloidal quantum dots coupled to photonic crystal cavity,” Opt. Express 21, 29612–29619 (2013). [CrossRef]

46.

P. Velha, E. Picard, T. Charvolin, E. Hadji, J. C. Rodier, P. Lalanne, and D. Peyrade, “Ultra-High Q/V Fabry-Perot microcavity on SOI substrate,” Opt. Express 15, 16090–16096 (2007). [CrossRef] [PubMed]

47.

A. R. Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express 16, 12084–12089 (2008). [CrossRef] [PubMed]

48.

Y. Gong and J. Vuckovic, “Photonic crystal cavities in silicon dioxide,” Appl. Phys. Lett. 96, 031107 (2010). [CrossRef]

49.

A. Rundquist, A. Majumdar, and J. Vuckovic, “Off-resonant coupling between a single quantum dot and a nanobeam photonic crystal cavity,” Appl. Phys. Lett. 99, 251907 (2011). [CrossRef]

50.

K. Rivoire, S. Buckley, and J. Vuckovic, “Multiply resonant high quality photonic crystal nanocavities,” Appl. Phys. Lett. 99, 013114 (2011). [CrossRef]

51.

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a zipper photonic crystal optomechanical cavity” Opt. Express 17, 3802–3817 (2009). [CrossRef] [PubMed]

52.

Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96, 203102 (2010). [CrossRef]

53.

M. Barth, J. Kouba, J. Stingl, B. Löchel, and O. Benson, “Modification of visible spontaneous emission with silicon nitride photonic crystal nanocavities,” Opt. Express 15, 17231–17240 (2007). [CrossRef] [PubMed]

54.

J. Zhao, G. Nair, B. R. Fisher, and M. G. Bawendi, “Challenge to the charging model of semiconductor-nanocrystal fluorescence intermittency from off-state quantum yields and multiexciton blinking,” Phys. Rev. Lett. 104, 1–4 (2010). [CrossRef]

55.

M. Kuno, D. Fromm, S. Johnson, A. Gallagher, and D. Nesbitt, “Modeling distributed kinetics in isolated semiconductor quantum dots,” Phys. Rev. B 67, 1–15 (2003). [CrossRef]

56.

J. T. Choy, B. J. M. Hausmann, T. M. Babinec, I. Bulu, M. Khan, P. Maletinsky, A. Yacoby, and M. Lončar, “Enhanced single-photon emission from a diamondsilver aperture,” Nat. Photonics 5, 738–743 (2011). [CrossRef]

OCIS Codes
(230.5590) Optical devices : Quantum-well, -wire and -dot devices
(270.5580) Quantum optics : Quantum electrodynamics
(230.5298) Optical devices : Photonic crystals

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: October 21, 2013
Revised Manuscript: January 24, 2014
Manuscript Accepted: January 24, 2014
Published: February 3, 2014

Citation
Shilpi Gupta and Edo Waks, "Overcoming Auger recombination in nanocrystal quantum dot laser using spontaneous emission enhancement," Opt. Express 22, 3013-3027 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3013


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