Coldren L. A.

Gerard J.

J. Mørk

Joannopoulos J. D.

M. Lorke

N. Gregersen

T. Suhr

The modulation bandwidth for a quantum dot light-emitting device is calculated using a detailed model for the spontaneous emission including the optical and electronic density-of-states. We show that the Purcell enhancement of the spontaneous emission rate depends critically on the degree of inhomogeneous broadening relative to the cavity linewidth and can improve the modulation speed only within certain parameter regimes

Skovgård, Troels Suhr

Gregersen, Niels

Lorke, Michael

Mørk, Jesper

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 18, 2017Modulation response of quantum dot nanolight-emitting-diodes exploiting purcell-enhanced spontaneous emissionSkovgård, Troels Suhr; Gregersen, Niels; Lorke, Michael; Mørk, JesperPublished in:Applied Physics LettersLink to article, DOI:10.1063/1.3592571Publication date:2011Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Skovgård, T. S., Gregersen, N., Lorke, M., & Mørk, J. (2011). Modulation response of quantum dot nanolight-emitting-diodes exploiting purcell-enhanced spontaneous emission. Applied Physics Letters, 98(21), 211109.DOI: 10.1063/1.3592571Modulation response of quantum dot nanolight-emitting-diodes exploitingpurcell-enhanced spontaneous emissionT. Suhr,a N. Gregersen, M. Lorke, and J. MørkDepartment of Photonics Engineering, DTU Fotonik, Technical University of Denmark, Ørsteds Plads 343,DK2800 Kgs. Lyngby, DenmarkReceived 17 January 2011; accepted 28 April 2011; published online 24 May 2011The modulation bandwidth for a quantum dot light-emitting device is calculated using a detailedmodel for the spontaneous emission including the optical and electronic density-of-states. We showthat the Purcell enhancement of the spontaneous emission rate depends critically on the degree ofinhomogeneous broadening relative to the cavity linewidth and can improve the modulation speedonly within certain parameter regimes. © 2011 American Institute of Physics.doi:10.1063/1.3592571The continued increase in data processing speed is lim-ited by the power consumption and heat generation of elec-trical wiring. It has been suggested instead to exploit opticalbusses, which in turn require ultrasmall and low-power opti-cal sources that can be modulated at very high speeds.1Nanolight-emitting-diodes nanoLEDs are nanostructuredlight emitting devices and appear as a promising technologythat also has the potential of cheap mass fabrication. Inthese structures, a photonic crystal defect nanocavity is em-ployed to strongly confine the optical mode, thereby enhanc-ing the carrier recombination rate via the Purcell effect.2However, a better understanding of the limitations of thesedevices is needed. It was recently shown3 that in quantumwell QW nanoLEDs and nanolasers the Purcell enhancedspontaneous emission is severely limited compared to earlierpredictions.4,5 The limitation originates from the broad-bandnature of QW transitions as compared to the narrow opticaldensity of states implied by a high-quality nanocavity. Quan-tum dots QDs, on the other hand, have discrete energystates, but are subject to inhomogeneous broadening due tosize fluctuations stemming from the fabrication techniques.6In this letter, we report a detailed investigation of the modu-lation speed of QD based nanoLEDs and establish necessaryconditions to achieve high modulation speed. We limit thescope to below threshold operation, where spontaneousemission dominates the recombination processes.We consider an ensemble of QDs embedded in a high-Qcavity and model the dynamics using the rate equationsN˙ = J − Rc − Rb, 1S˙ = Rc −Sp, 2where N is the total QD ground level and wetting layercarrier density, S is the photon density, J is the injectioncurrent density, Rc Rb is the spontaneous emission coupledinto the cavity mode to all other modes,  is the confine-ment factor and p=Q /c is the photon life time given by thecavity quality factor and resonance frequency. An importantdifference from standard rate equations7 is that the spontane-ous emission is Purcell enhanced and is calculated using adetailed model of the interaction between a confined opticalmode and the possible electronic transitions3,7Rc + Rb =  ef21 − f1A21LdhdE. 3For the electronic density-of-state DOS e, a wettinglayer and a Gaussian distribution of QD energies with stan-dard deviation  representing inhomogeneous broadening, istaken into account. The factor A21 contains the optical DOSop and is stated in full in Ref. 3. It is also given by A21=ophB21, where B21 is the Einstein B coefficient. op ismodeled as a Lorentzian lineshape times the Purcell en-hanced two-state recombination rate in bulk7 within a pho-tonic band gap and with a bulk background. The homoge-neous broadening L is modeled as a Lorentzian with full-width at half-maximum  and the hole and electron quasi-Fermi functions f1 and f2 are determined from the totalcarrier density. The Purcell enhancement is given by8 F= 6Q / 	2Vn, where Vn is the mode volume measured inhalf wavelengths cubed  / 2n3. An example of the elec-tronic and optical DOS is shown in Fig. 1. Notice that themodel assumes local equilibrium between QD and wettinglayer states,9 which is consistent if the capture time, usuallyon the order of a few picoseconds or less,10 is smaller thanthe inverse of the modulation frequency predicted by themodel. Furthermore, scattering rates need to be low enough,aElectronic mail: suhr@fotonik.dtu.dk.FIG. 1. Color online Illustration of the inhomogeneously broadened elec-tronic DOS and the optical DOS. The latter consists of a Lorentzian cavityresonance inside a background DOS with a photonic band gap. The bandgap of the wetting layer is 1 eV.APPLIED PHYSICS LETTERS 98, 211109 20110003-6951/2011/9821/211109/3/$30.00 © 2011 American Institute of Physics98, 211109-1Downloaded 01 Jun 2011 to 192.38.67.112. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsthat the modulation response is not reduced by the scatteringitself.11From this model, we calculate the 3 dB-bandwidth be-low threshold using a small-signal analysis. We set f2=1− f1= f and use the transparency condition f =1 /2 as alower bound for the threshold and plot the maximum modu-lation speed below this value. The result is displayed in Fig.2 for a range of Q-factors and inhomogeneous widths usingthe parameters in Table I. We here, choose a value for ho-mogeneous broadening appropriate for low temperature andlow density in order to be able to see the effect of inhomo-geneous broadening. Two interesting trends are seen: 1when increasing the Q-factor, the 3 dB-frequency goesthrough a maximum at Q500. The initial rise is due to thePurcell enhancement increase with Q. The decrease at largeQ-factors is an intrinsic feature of all resonators and is due tothe photon life time becoming large, prohibiting fast opera-tion. 2 When increasing the inhomogeneous broadeningabove 0.5 meV, the 3 dB-frequency decreases, whereas itis almost constant below 0.5 meV. Both features can beunderstood from a small-signal expression for the 3 dB-bandwidth derived from Eqs. 1 and 2f3dB =12	p2 + r2−1/2,1r= NRc + NRb. 4Below transparency, r is well described by NRb=0 andNRc=2BN0. Here, N0 is the steady state carrier density andB can be approximated byB =F21QDpp +  + ,  = 2 ln2 , 5where 21 is the electron-hole recombination time for twostates in bulk,7 QD is the three-dimensional 3D density ofdots and p=2 /p is the cavity linewidth. At low Q-factorsf3 dB is governed by 1 /r, which increases with Q, but athigh Q-factors the bandwidth is limited by the term p=Q /0. The Q-factor giving the maximum bandwidth isfound for Qf3 dB=0 which for +p equals for N0=QD /2Qopt =	26 021Vn. 6In Fig. 2, Qopt is indicated by the blue line and is seen todeviate from the value in Eq. 6 when + becomes com-parable to p. For increasing inhomogeneous broadening,Eqs. 4 and 5 reproduce the decrease observed in Fig. 2 forlarge .Figure 3 plots f3 dB and the recombination rates againstthe injection current density J for the two devices markedin Fig. 2, one with =100 eV device A and one with =10 meV device B, both with Q=2000. The arrows in Fig.3 indicate lower bounds on the threshold, namely where f=1 /2, indicating the validity range of the present model. The3 dB-frequency for device A is seen to increase with J untilit becomes limited by the photon life time and the spontane-ous recombination is dominated by emission into the cavity.For device B, Rc increases with J until the quasi-Fermi levelseparation becomes larger than the cavity resonance energy.After this point Rc becomes constant and Rb rises sharply.This is reflected in f3 dB which goes through a maximum andthen decreases as NRc→0 see Eq. 4. The final increase isdue to Rb that becomes large when the wetting layer statesbecome important.FIG. 2. Color online The maximum 3 dB-bandwidth as function ofQ-factor and inhomogeneous broadening. The contour lines are drawn forconstant modulation speed in gigahertz. The devices A and B, treated indetail in Fig. 3, and the optimum Q are indicated. The parameter values usedare listed in Table I.TABLE I. Parameters used in the calculations. Notice that the differentialbulk life time 21 is the transition time between two states Ref. 7 and ischosen so the bulk carrier life time is 1 ns.Parameter ValueCavity resonance 0 0.8 eVRefractive index n see Ref. 7 3.4Mode volume Vn see Ref. 12 1Homogeneous broadening  see Ref. 13 100 eVTemperature T see Ref. 13 100 K3D density of dots QD see Ref. 14 2104 m−3Confinement factor  0.01Photonic band gap see Ref. 15 0.2 eVDifferential bulk life time 21 125 psFIG. 3. Color online The 3 dB-bandwidth solid, Rc dashed and Rbdotted at Q=2000 for an inhomogeneous broadening of 100 eV deviceA, top and 10 meV device B, bottom. The dot-dashed indicates the photonlife time limit and the arrows indicate the transparency current for A and B.For device A, Rb100 m−3 ps−1 and is, therefore, not shown.211109-2 Suhr et al. Appl. Phys. Lett. 98, 211109 2011Downloaded 01 Jun 2011 to 192.38.67.112. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsIn Fig. 4, we plot f3 dB for the same range of Q-factorsand  as in Fig. 2, but for a homogeneous broadening, , of10 meV corresponding to room temperature operation.13,16The same general dependence on Q and  is seen, but themaximum 3 dB-bandwidth is significantly lower. The modu-lation bandwidth is seen to become independent of the inho-mogeneous broadening, , for . Note that f3 dB contin-ues to increase even for  as the width of the distributionis inversely proportional to + see Eq. 5.In conclusion, we have calculated the 3 dB-bandwidthfor QD nanoLEDs using a detailed model for the spontane-ous recombination. It was found that for inhomogeneousbroadening smaller than the homogeneous broadening, themodulation bandwidth initially increases with the Purcell en-hancement and is limited by the photon escape rate for highQ-factors, giving an optimal Q-factor. At large broadenings,the 3 dB bandwidth is reduced due to the electronic DOSbecoming much broader than the optical DOS. Figures 2 and4 show that with today’s technology nanoLEDs are not com-petitive with conventional devices and improvements of thetechniques to produce QD ensembles with low broadeningare therefore required in order to unlock the full potential ofnanoLEDs.The authors would like to thank Villum Fonden for fi-nancial support via the NATEC Nanophotonics for TErabitCommunications center.1D. A. B. Miller, Proc. IEEE 97, 1166 2009.2E. M. Purcell, H. C. Torrey, and R. V. Pound, Phys. Rev. 69, 681 1946.3T. Suhr, K. Yvind, N. Gregersen, and J. Mørk, Opt. Express 18, 112302010.4H. Altug, D. Englund, and J. Vučković, Nat. Phys. 2, 484 2006.5E. K. Lau, A. Lakhani, R. S. Tucker, and M. C. Wu, Opt. Express 17,7790 2009.6D. Leonard, K. Pond, and P. M. Petroff, Phys. Rev. B 50, 11687 1994.7L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic IntegratedCircuits Wiley, New York, 1995.8J. Gerard, Single Quantum Dots: Fundamentals, Applications and NewConcepts, edited by P. Michler Springer, New York, 2004, Chap. 7.9T. W. Berg and J. Mørk, IEEE J. Quantum Electron. 40, 1527 2004.10T. Markussen, P. Kristensen, B. Tromborg, T. W. Berg, and J. Mørk, Phys.Rev. B 74, 195342 2006.11M. Lorke, T. R. Nielsen, and J. Mørk, Appl. Phys. Lett. 97, 2111062010.12M. Nomura, Y. Ota, N. Kumagai, S. Iwamoto, and Y. Arakawa, Appl.Phys. Lett. 97, 191108 2010.13M. Bayer and A. Forchel, Phys. Rev. B 65, 041308 2002.14K. Jacobi, Prog. Surf. Sci. 71, 185 2003.15J. D. Joannopoulos, S. G. Johnson, and J. N. Winn, Photonic Crystals:Molding the Flow of Light Princeton University Press, New Jersey, 2008.16M. Lorke, T. R. Nielsen, J. Seebeck, P. Gartner, and F. Jahnke, Phys. Rev.B 73, 085324 2006.FIG. 4. Color online The maximum 3 dB-bandwidth as function ofQ-factor and inhomogeneous broadening. The contour lines are drawn forconstant modulation speed in gigahertz. The parameters are listed in TableI except that the homogeneous broadening is 10 meV and room temperatureis considered.211109-3 Suhr et al. Appl. Phys. Lett. 98, 211109 2011Downloaded 01 Jun 2011 to 192.38.67.112. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions

Modulation response of quantum dot nanolight-emitting-diodes exploiting purcell-enhanced spontaneous emission

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Modulation response of quantum dot nanolight-emitting-diodes exploiting purcell-enhanced spontaneous emission

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