## Modulation response of nanoLEDs and nanolasers exploiting Purcell enhanced spontaneous emission

Optics Express, Vol. 18, Issue 11, pp. 11230-11241 (2010)

http://dx.doi.org/10.1364/OE.18.011230

Acrobat PDF (1579 KB)

### Abstract

The modulation bandwidth of quantum well nanoLED and nanolaser devices is calculated from the laser rate equations using a detailed model for the Purcell enhanced spontaneous emission. It is found that the Purcell enhancement saturates when the cavity quality-factor is increased, which limits the maximum achievable spontaneous recombination rate. The modulation bandwidth is thereby limited to a few tens of GHz for realistic devices.

© 2010 OSA

## Introduction

1. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Letters to Nature **425**(6961), 944–947 (2003). [CrossRef]

3. Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A **44**(1), 657–668 (1991). [CrossRef] [PubMed]

6. E. K. Lau, A. Lakhani, R. S. Tucker, and M. C. Wu, “Enhanced modulation bandwidth of nanocavity light emitting devices,” Opt. Express **17**(10), 7790–7799 (2009). [CrossRef] [PubMed]

*et al.*[5

5. H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. **2**(7), 484–488 (2006). [CrossRef]

*et al.*[6

6. E. K. Lau, A. Lakhani, R. S. Tucker, and M. C. Wu, “Enhanced modulation bandwidth of nanocavity light emitting devices,” Opt. Express **17**(10), 7790–7799 (2009). [CrossRef] [PubMed]

*et al.*were the result of the measurement method and are in reality only achievable in non-lasing devices with ultralow mode volume. Because of the potentially very high Purcell factor, the dynamics of the device is very dependent on the details of the spontaneous emission and a rigorous treatment of spontaneous emission is needed. In this paper we therefore calculate the spontaneous emission from fundamental principles and study the effect on the dynamic properties of nanoLEDs and nanolasers. We find that the high-speed properties predicted in ref [6

6. E. K. Lau, A. Lakhani, R. S. Tucker, and M. C. Wu, “Enhanced modulation bandwidth of nanocavity light emitting devices,” Opt. Express **17**(10), 7790–7799 (2009). [CrossRef] [PubMed]

## General rate equations and modulation response

*N*) and photon (

*S*) densities in the laser rate equations [6

**17**(10), 7790–7799 (2009). [CrossRef] [PubMed]

*J*is the carrier injection into the active volume,

*Γ*is the confinement factor and

*τ*is the lifetime of the photon in the cavity given by the quality (Q) factor and the cavity resonance frequency [8] (see Table 1 for parameter values used in this paper). The total carrier recombination rate has been separated into contributions from stimulated emission (

_{p}= Q/ω_{0}*R*), spontaneous emission into the cavity (

_{st}*R*), spontaneous emission into all other modes (

_{c}*R*) and non-radiative losses (

_{b}*R*). For the latter we approximate

_{nr}*R*with

_{nr}= N/τ_{nr}*τ*being the non-radiative life time. For the stimulated emission we usewhich is suitable for the quantum well type devices considered [7]. Here

_{nr}*v*is the group velocity,

_{g}= c/n*G*is the material gain,

_{0}*N*is the transparency carrier density and

_{tr}*N*is a gain parameter [7]. The gain,

_{s}*G*, in Eq. (3) is inversely proportional to 1

*+ εS*[9

9. A. Mecozzi and J. Mørk, “Saturation induced by picosecond pulses in semiconductor optical amplifiers,” J. Opt. Soc. Am. B **14**(4), 761–770 (1997). [CrossRef]

*R*and

_{c}*R*are spontaneous emission processes that depend only on the carrier density and not on the photon density, therefore, with Eq. (1) and (2),

_{b}*ω*and

_{R}*γ*are given byandwhere

_{R}*N*and

_{0}*S*are steady-state carrier and photon densities and

_{0}*a = dG/dN*,

*a*,

_{p}= -dG/dS*R*and

_{c,N}= dR_{c}/dN*R*. In the following we specify

_{b,N}= dR_{b}/dN*R*and

_{c}*R*and use Eq. (5) - (7) to calculate the modulation response.

_{b}## Purcell enhanced spontaneous emission

*Q*and low mode volume (

*V)*the Purcell effect should be taken into account when calculating the spontaneous emission [2,5

5. H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. **2**(7), 484–488 (2006). [CrossRef]

**17**(10), 7790–7799 (2009). [CrossRef] [PubMed]

*V*is the mode volume in half wavelengths. There exist a number of different definitions of the Purcell effect, but here we take

_{n}= V/(λ/(2n))^{3}*F*to be the spontaneous emission rate into the cavity mode relative to the spontaneous emission rate in a bulk (homogeneous) medium at the same spectral position and use Eq. (8) to describe its magnitude. We consider two models for spontaneous emission, which include the Purcell enhancement.

### Linear Model

*R*, where

_{sp}= N/τ_{sp}*τ*is the spontaneous lifetime in bulk. In the linear model the spontaneous emission is often split so that

_{sp}*R*and

_{c}= βR_{sp}*R*, where

_{b}= (1-β)R_{sp}*β = R*is the spontaneous emission factor.

_{c}/R_{sp}*F*onto

*R*[5

_{c}5. H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. **2**(7), 484–488 (2006). [CrossRef]

**17**(10), 7790–7799 (2009). [CrossRef] [PubMed]

12. H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. **66**(10), 4801 (1989). [CrossRef]

13. T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Top. Quantum Electron. **3**(3), 808 (1997). [CrossRef]

**17**(10), 7790–7799 (2009). [CrossRef] [PubMed]

**17**(10), 7790–7799 (2009). [CrossRef] [PubMed]

*β*~1 this model assumes that the Purcell effect acts on all the carriers in the system as it does not take into account the details in the optical density-of-states (DOS), which are important when discussing devices with large Purcell factors. Furthermore, the linear model is only valid close to its Taylor expansion point.

### Full Model

*dN*, as

_{2}*dR*, where the Einstein coefficient (

_{sp}= A_{21}dN_{2}*A*) is proportional to the optical DOS (

_{21}*ρ*) and the

_{op}*B*coefficient, i.e.

_{21}*A*.

_{21}= hνρ_{op}B_{21}*dN*can be written in terms of the electron (

_{2}*f*) and hole (

_{2}*f*) Fermi functions and the reduced electronic DOS (

_{1}*ρ*) as

_{el}*dN*and homogeneous broadening is taken into account by convoluting with the lineshape function

_{2}= ρ_{el}f_{2}(1-f_{1})dE_{21}*L(E - hν)*, which is usually taken to be a Lorentzian. Integrating over energy we arrive at an expression for spontaneous emission rate (ref [7]. p. 459-472)

14. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. **95**(1), 013904 (2005). [CrossRef] [PubMed]

*H*is the Heaviside function and the cavity is centered at

*hν*, with

_{0}= (hν_{U}- hν_{L})/2*hν*(

_{L}*hν*) being the lower (upper) edge of the photonic band gap and

_{U}*B*coefficient is found by setting

_{21}*A*, where the differential recombination time (

_{21}(hν_{0}) = 1/τ_{21}= hν_{0}B_{21}ρ_{op}(hν_{0})*τ*) is chosen so that the bulk spontaneous emission time (

_{21}*τ*) is recovered in the absence of the cavity, i.e. so that

_{sp}*R*(

_{sp}*N*) =

_{tr}*N*/

_{tr}*τ*in bulk. We then haveThis model for the optical DOS thus describes the Purcell enhancement as the redistribution of the modes, which are suppressed by the photonic bandgap, into the cavity. The increase in the DOS near the edges of the photonic bandgap, which is usually observed in photonic crystals [17

_{sp}17. M. P. Marder, *Condensed Matter Physics* (John Wiley & Sons, inc., New York, 2000). [PubMed]

15. S. M. Barnett and R. Loudon, “Sum rule for modified spontaneous emission rates,” Phys. Rev. Lett. **77**(12), 2444–2446 (1996). [CrossRef] [PubMed]

*m*is the reduced effective mass,

_{r}= (m_{e}*m_{h}*)/(m_{e}* + m_{h}*)*m*(

_{e}**m*) is the effective electron (hole) mass,

_{h}**W*is the well width,

*E*is the bandgap energy and

_{g}*E*is the energy of the first energy level in a quantum well with infinitely high barriers. The use of a more accurate description of the quantum well does not change the conclusions of the paper.

_{1}= h^{2}/(8m_{r}W^{2})*R*is the integral over the bulk part of Eq. (13) while the Lorentzian part gives

_{b}*R*. Although the full model specifically describes quantum well devices it is also suitable for bulk devices if an appropriate electronic DOS is chosen. Quantum dot devices, however, are governed by a different set of laser rate equations than Eq. (1) and (2) and should account for the dynamics in both the wetting layer and quantum dot levels. Furthermore, the electronic DOS for quantum dots is markedly different from that of bulk and quantum wells and including quantum dots in the treatment would make the discussion less clear. Although the use of quantum dot structures may be very promising, we therefore only treat quantum well (and bulk) devices in this paper.

_{c}### Saturation of Purcell enhancement

*N*. To see clearly the contribution from the electronic part of Eq. (11) we also plot (for a

*Q*of 100)As can be seen from Fig. 2a,

*R*approaches the slope of the linear model for high carrier density, i.e. for the quasi-Fermi level separation high above the electronic band edge, where the integral in Eq. (15) is almost equal to the carrier density. For low carrier density

_{el}*R*follows

_{el}*R*, although

_{c}*R*is somewhat smaller due to the limiting effect of the cavity DOS. The saturation of

_{c}*R*for

_{c}*N*>

*N*is not seen in

_{tr}*R*and originates from the finite quantum well DOS and associated band filling effects at the cavity resonance

_{el}*E*, i.e. the quasi-Fermi level separation becomes much larger than

_{0}*E*so that

_{0}*f*(1 –

_{2}*f*) ~1 near

_{1}*E*.

_{0}*R*does not increase with the Q-factor in the full model. This can be explained by evaluating the integral over frequency in Eq. (11). This is usually done by assuming the homogeneous broadening term,

_{c}*L (E - hν)*, to be sharply peaked at

*hν*=

*E*compared to the other terms in Eq. (11) so that it can be replaced by a Dirac delta-function. However, for high-Q cavities the cavity linewidth can easily become smaller than the homogeneous broadening and in this case the integral over frequency in Eq. (11) should be evaluated exactly. The results in the present model, however, are not significantly changed when evaluating the frequency integral exactly using a Lorentzian for the homogeneous broadening as the electronic DOS is much wider than the homogeneous broadening. Therefore, we will use

*L(E – hν) = δ(E - hν)*from here on.

*Q*larger than a few hundreds. The reduction thus originates from the mismatch between cavity bandwidth and the effective inhomogeneous broadening expressed by the electronic DOS. This agrees with the qualitative discussion given in ref [16

16. T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. **85**(18), 3989–3991 (2004). [CrossRef]

*V*. Note also that we here neglect features in the electronic DOS such as the exciton peak near the band edge, which may challenge the requirement that the electronic DOS be slowly varying. Deviations of this type from the assumed smooth electronic DOS do not affect the conclusions, but rather increase the Q-factor at which the Purcell enhancement saturates. However, a rigorous treatment of the electronic DOS is beyond the scope of this work.

_{n}*R*) shown in Fig. 2a is negligible for

_{b}*N*< 10

*N*. Reducing the size of the photonic bandgap increases

_{tr}*R*, but the change is insignificant compared to

_{b}*R*.

_{c}### Effective Purcell factor

*R*is the total spontaneous emission in the absence of a cavity, i.e.The effective Purcell factor encompasses a number of the interesting effects described above, namely: the reduction of the Purcell enhancement due to the limited cavity bandwidth as compared to the inhomogeneous broadening, the saturation effect that occurs when the quasi-Fermi level separation is much larger than the cavity resonance frequency and the increase in the β-factor, which follows from the Purcell enhanced emission into the cavity.

_{bulk}*R*in Eq. (9) the effective Purcell factor reduces to

_{c}*F*=

_{eff}*βF*in the linear model. In the full model the effective Purcell factor is a function of the optical and electronic DOS and thus depends on

*Q*,

*V*and

_{n}*N*and is therefore markedly different from the

*F*in the linear model.

_{eff}*N*<

*N*we have

_{tr}*F*in the full model almost constant, but lower than

_{eff}*F*in the linear model due to the reduction originating from the limited cavity bandwidth. The decrease for

_{eff}*N*>

*N*in the full model is a combination of the bandfilling effect and

_{tr}*R*growing large.

_{b}## Results

^{4}and a mode volume of 10

*V*and device B has a Q-factor of 10

_{n}^{2}and a mode volume of 0.1

*V*. The results for device A are plotted in Fig. 3a-c , and for device B in Fig. 3d-f.

_{n}*J*=

_{0}*N*). For

_{tr}/τ_{sp}*J*<

*J*the carrier density in the linear model is much lower than in the full model. This is an effect of the reduction of the effective Purcell factor for high Q-factors in the full model, which limits

_{0}*F*to ~1 for low carrier density, whereas

_{eff}*F*~610 in the linear model. The higher spontaneous emission rate in the linear model explains the lower carrier density.

_{eff}*J*~

*J*, when lasing sets in, and remains constant until

_{0}*J*~100

*J*, where it again begins to increase. This increase is due to the gain suppression that becomes a significant process for

_{0}*S*0.05

_{0}~1/ε ~*N*and must be compensated by an increase of the linear gain. The β-factor is close to unity for the entire pumping range, which explains why the photon densities for the two models are equal for

_{tr}*J*> 10

*J*even though the linear model is dominated by spontaneous emission, while the full model is dominated by stimulated emission.

_{0}*J*~100

*J*, where the carrier density begins to increase again due to the gain suppression effect discussed above. In the linear model the carrier density is much lower than in the full model, giving a lower stimulated emission. This pushes the threshold pump up to around

_{0}*J*~1000

*J*.

_{0}*J*~

*J*, after which stimulated emission dominates until gain suppression becomes important around

_{0}*J*~100

*J*. Over the entire pumping range the 3dB-bandwidth does not exceed 20 GHz. In the linear model the high 3dB-bandwidth for

_{0}*J*< 100

*J*is due to the high spontaneous emission and the drop-off for

_{0}*J*>100

*J*is due to the damping rate (

_{0}*γ*) increasing more rapidly than the resonance frequency (

_{R}*ω*) as explained in ref [2].

_{R}*V*, are 100 times lower than in device A, so that

_{n}*F*remains ~610 in the linear model, but changes to ~70 in the full model (for low carrier density). For this device the photon loss is 100 times larger than for device A and therefore the photon densities for the two models in Fig. 3d are 100 times lower. In the linear model, the relatively lower photon density is reflected in the carrier density, which is also lowered to balance the photon loss.

_{eff}*J*<

*J*. This is because the spontaneous emission rate in the full model follows another dependence on

_{0}*N*(approximately

*N*) than in the linear model and therefore a smaller adjustment of

^{2}*N*is necessary to compensate the lower photon density.

*F*, but is also modified by the lower carrier density. In neither models the gain becomes large enough to initiate lasing and this is reflected in the carrier densities, which do not clamp in this device.

_{eff}*J*< 50

*J*the effective Purcell factor is almost constant and

_{0}*R*≈0, so that the first term in Eq. (23) determines

_{b,N}*τ*. For 50

_{eff}*J*<

_{0}*J*< 800

*J*the term

_{0}*R*is still low, while

_{b,N}*F*starts to decrease so that the first term of Eq. (23) becomes smaller and the second term becomes negative, leading to a decrease in the 3dB-bandwidth. For

_{eff}*J*> 800

*J*the background emission increases sharply, making

_{0}*τ*decrease and leading to the final increase in the 3dB-bandwidth. Thus the 3dB-bandwidth in the full model is roughly an order of magnitude lower than in the linear model and this clearly underlines the necessity for a detailed description of the spontaneous emission.

_{eff}*J*< 2

*J*, as spontaneous emission is also dominant in this pumping range. Here the effective Purcell factor in the full model is 100 times lower than for device B, making 1/

_{0}*τ*the dominating term in Eq. (23) and giving the low 3dB-bandwidth compared to device B. Another way of expressing the behavior is that the Purcell enhancement only affects the carriers associated with transitions within the bandwidth of the cavity resonance. When the cavity resonance is much narrower than the electronic bandwidth the influence from the enhanced spontaneous emission only has a small effect on the total carrier density life time and thus also the speed.

_{nr}*J*=

*J*and

_{0}*J*= 100

*J*. The white line in the figure shows, for the given pump current, the device parameters for which stimulated and spontaneous emission are equal in magnitude. Devices to the right of this boundary are dominated by stimulated emission. The black line shows the limiting value of this boundary for large pump current, i.e. devices to the left of this line will always be dominated by spontaneous emission, independently of the strength of the pump, and are thus always in the LED regime.

_{0}*Q*is due to the photon life time that becomes smaller at low

*Q*and thereby increases the 3dB-bandwidth in Eq. (21).

*Q*is seen in Fig. 4b, where the 3dB-bandwidth exceeds 200 GHz in the lower left corner, i.e. in the LED regime. In the top right area, which corresponds to conventional laser structures, the effective Purcell factor saturates at a

*Q*of a few hundreds and the stimulated emission therefore becomes the dominant recombination process so that the term

*aS*in Eq. (6) becomes large, giving the large modulation speed in this area. In the lower right corner the photon loss is too large to meet the lasing condition and the effective Purcell factor is low, giving the lower 3dB-bandwidth. In general the ultrahigh modulation speeds previously reported [6

_{0}/τ_{p}**17**(10), 7790–7799 (2009). [CrossRef] [PubMed]

## Discussion and conclusions

**17**(10), 7790–7799 (2009). [CrossRef] [PubMed]

18. R. S. Tucker, “High-speed modulation of semiconductor lasers,” J. Lightwave Technol. **3**(6), 1180–1192 (1985). [CrossRef]

**17**(10), 7790–7799 (2009). [CrossRef] [PubMed]

## Acknowledgements

## References and links

1. | Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Letters to Nature |

2. | E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. |

3. | Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A |

4. | O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim I, “Two-dimensional photonic band-Gap defect mode laser,” Science |

5. | H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. |

6. | E. K. Lau, A. Lakhani, R. S. Tucker, and M. C. Wu, “Enhanced modulation bandwidth of nanocavity light emitting devices,” Opt. Express |

7. | L. A. Coldren, and S. W. Corzine, |

8. | J. D. Jackson, |

9. | A. Mecozzi and J. Mørk, “Saturation induced by picosecond pulses in semiconductor optical amplifiers,” J. Opt. Soc. Am. B |

10. | J. D. Joannopoulos, S. G. Johnson, and J. N. Winn, |

11. | J.-M. Gerard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots”, in |

12. | H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. |

13. | T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Top. Quantum Electron. |

14. | D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. |

15. | S. M. Barnett and R. Loudon, “Sum rule for modified spontaneous emission rates,” Phys. Rev. Lett. |

16. | T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. |

17. | M. P. Marder, |

18. | R. S. Tucker, “High-speed modulation of semiconductor lasers,” J. Lightwave Technol. |

**OCIS Codes**

(060.4080) Fiber optics and optical communications : Modulation

(140.5960) Lasers and laser optics : Semiconductor lasers

(230.3670) Optical devices : Light-emitting diodes

(320.7090) Ultrafast optics : Ultrafast lasers

(140.3948) Lasers and laser optics : Microcavity devices

**ToC Category:**

Optical Devices

**History**

Original Manuscript: March 31, 2010

Revised Manuscript: May 6, 2010

Manuscript Accepted: May 9, 2010

Published: May 12, 2010

**Citation**

T. Suhr, N. Gregersen, K. Yvind, and J. Mørk, "Modulation response of nanoLEDs and nanolasers exploiting Purcell enhanced spontaneous emission," Opt. Express **18**, 11230-11241 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-11230

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### References

- Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Letters to Nature 425(6961), 944–947 (2003). [CrossRef]
- E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
- Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44(1), 657–668 (1991). [CrossRef] [PubMed]
- O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999). [CrossRef] [PubMed]
- H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. 2(7), 484–488 (2006). [CrossRef]
- E. K. Lau, A. Lakhani, R. S. Tucker, and M. C. Wu, “Enhanced modulation bandwidth of nanocavity light emitting devices,” Opt. Express 17(10), 7790–7799 (2009). [CrossRef] [PubMed]
- L. A. Coldren, and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (John Wiley & Sons, inc., New York, 1995).
- J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, inc., New York, 1998)
- A. Mecozzi and J. Mørk, “Saturation induced by picosecond pulses in semiconductor optical amplifiers,” J. Opt. Soc. Am. B 14(4), 761–770 (1997). [CrossRef]
- J. D. Joannopoulos, S. G. Johnson, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, New Jersey, 2008).
- J.-M. Gerard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots”, in Single Quantum Dots, Fundamentals, Applications and New Concepts, P. Michler (Springer, Berlin, 2003), pp. 269–314.
- H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66(10), 4801 (1989). [CrossRef]
- T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Top. Quantum Electron. 3(3), 808 (1997). [CrossRef]
- D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005). [CrossRef] [PubMed]
- S. M. Barnett and R. Loudon, “Sum rule for modified spontaneous emission rates,” Phys. Rev. Lett. 77(12), 2444–2446 (1996). [CrossRef] [PubMed]
- T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. 85(18), 3989–3991 (2004). [CrossRef]
- M. P. Marder, Condensed Matter Physics (John Wiley & Sons, inc., New York, 2000). [PubMed]
- R. S. Tucker, “High-speed modulation of semiconductor lasers,” J. Lightwave Technol. 3(6), 1180–1192 (1985). [CrossRef]

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