November 15, 2004 / Vol. 29, No. 22 / OPTICS LETTERS 2659

Enhanced single-photon emission from quantum dots inphotonic crystal waveguides and nanocavities

S. Hughes

NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan

Received June 4, 2004

A theoretical formalism is presented to investigate enhanced radiative decay of excited dipoles in photoniccrystal waveguides and nanocavities with a view to achieving efficient single-photon emission from embeddedquantum dots. Surprisingly, large enhancement effects are achievable in both waveguides and nanocavities,and enhanced emission in the waveguide is shown to scale proportionally (inversely) with the photon groupindex (velocity). Further, a way to include radiative coupling of the quantum dot is shown, and the impor-tance of its inclusion is subsequently demonstrated. © 2004 Optical Society of America

OCIS codes: 130.0130, 270.0270.

One of the major goals in semiconductor nanoscaleoptics research is to achieve lower dimensionality ofcharge-carrier and photonic wave packets, ultimatelydriving toward spatial conf inement in all dimensions.In this regard, single quantum dots (QDs) are currentlyreceiving an enormous amount of interest motivatedby their potential for use in scalable single-photonapplications.1 However, semiconductor microcavitiescapable of exploiting the optical properties of singleQDs have been somewhat hampered by the lack ofhigh quality factors �Q� and small volume cavities,although modest enhanced spontaneous-emissionfactors ��5� have been demonstrated in semiconductornanostructures such as micropost cavities.2

If future systems are to practically exploit thesingle-photon emission potential of QDs, improve-ments in the microcavity excitation, enhanced cavitycoupling, and efficient extraction of the emitted pho-tons are all required. Theoretical analysis of singleQD enhanced emission rates within pillar semicon-ductor microcavities3 show that intermediate-to-strongcoupling4 is not achievable since the Q�Vm ratiosare typically too small, where Vm is the effectivemode volume. It has been proposed that planarphotonic crystal (PPC) nanocavities may offer severaladvantages over the pillar microcavity system,5 and,recently, improved enhanced emission factors (EEFs)have been experimentally demonstrated in PPC sys-tems,6 with .17 obtained even at room temperature.7

In this Letter a real-space Green function tensor(GFT) technique8,9 is applied that allows one to derivesemiquantitative analytical expressions for enhancedemission rates in various PPC structures. Subse-quent calculations yield impressive enhancements,and these are compared and contrasted with enhancedemission in other cavity systems, such as the micro-cavity posts. The theoretical formalism is appliedto both side-coupled nanocavities and simple defectwaveguides and lends itself naturally to includingboth radiative coupling from the QD and randommanufacturing imperfections10 in a realistic way (bothof which are usually ignored altogether). Figure 1shows plan-view schematics of interest for this in-vestigation: (a) a defect waveguide with a QD at anantinode position of the propagating Bloch mode and

0146-9592/04/222659-03$15.00/0

(b) a coupled QD–nanocavity and PPC waveguidesystem with a QD at the antimode of the nanocavitymode. Both these configurations can be excited viathe defect photonic crystal waveguide.

The spontaneous-emission rate of a QD at r � rdcoupled to a single-cavity mode a can be expressed asGSE�rd� � Fa�rd�G0�rd�, where GSE is the free decay ratein the medium and Fa is the EEF. The EEF can beevaluated by employing a microscopic model for the di-electric medium11 or from the classical GFT

$G�r,r0;v�

through Fai�rd, v� � ��

$Gii

a�rd, rd;v���G0�v� whereG0 � v3ped��6pc3� is the homogeneous solution withed � e�rd� as the relative electric permittivity at theQD location and i is the polarization direction of theoscillating dipole that is assumed here to be polarizedin the y direction (this is not a restriction). Sponta-neous emission, which occurs from the interaction ofthe excited QD and the ground state of the quantizedelectric field, can be highly position and frequencydependent and essentially depends on the local

Fig. 1. (a) Schematic diagram of the propagating mode�ek� inside a PPC. The QD is placed near the field antin-ode location. The modes are shown by the dark shadedconcentric circles, and the f illed circles represent the air-holes in a semiconductor. (b) Schematic of the proposedPPC waveguide–nanocavity coupling scheme. The QD islocated within the nanocavity, which has an exciton reso-nance close to the cavity resonance. The propagating fieldexcites the defect cavity ec, which couples to the propagat-ing Bloch mode ek and to the embedded QD.

© 2004 Optical Society of America

2660 OPTICS LETTERS / Vol. 29, No. 22 / November 15, 2004

photonic density of states; this is completely containedwithin the photonic GFT.

We first turn attention to the PPC defect wave-guide depicted in Fig. 1(a). To help define a GFTwe assume that any change in the perfectly periodicwaveguide is incorporated into a change in the relativeelectric permittivity De�r�. This change can comefrom imperfections, a cavity, a QD, optical non-linearities, or a combination of all of these. We canthen write a general solution for the electric f ield asE�r,v� � Ew �r,v� 1

Rdr0$Gw�r,r0;v� ? De�r0�E�r0�,

or more simply, in operator form E � Ew 1 Gw ? DeE,where Ew is the unperturbed electric field vector forthe propagating waveguide Bloch mode and

$Gw is the

ideal GFT of the defect PPC waveguide.It is assumed that the PPC waveguide presented

here is single mode and is periodic along the directionof propagation, with period a. Applying Bloch’stheorem to the unperturbed bound field solutions, weobtain the bound contribution to the Green function an-alytically as

$G h̄

w�r, r;v� � i0.5aveh�r� ≠ eh��r��vg,

where eh is the homogeneous forward-propagatingBloch mode for wave number k � h, vg is the associ-ated group velocity, and ≠ is the dyadic multiplier.The waveguide modes are normalized throughRcell jeh�r�j2e�r� dr � 1, carried out over one unit cell

of the periodic structure, with permittivity e�r�. Itis assumed that the bound mode contribution to GFTdominates the radiation modes, which is certainly thecase for the studies addressed below. Subsequently,we derive the EEF within the PPC waveguide as

Fw�rd,v� �3pc2ang�v�

ped v2 jeh�rd� ? n̂y j

2, (1)

with ng � c�vg as the group index and n̂y as the unitvector parallel to the polarization direction of the ex-cited dipole. Immediately we recognize that the emis-sion rate is proportional to the group index and thuscan be substantially increased as one tends toward thephotonic band edge.

Here a prototype waveguide in which numeri-cal calculations are performed is considered, anda semiconductor PPC membrane slab of thickness210 nm is employed, with a triangular lattice ofholes of pitch a � 420 nm, radius r � 0.275a, anded � 12. We carry out quantitative finite-differencetime-domain simulations12 for this structure, andexperimental results in the literature show that thiswaveguide yields extremely large group indices.13 Asillustrated in Figs. 1(a) and 2(a), the waveguide isformed by a missing line of holes. First we locatethe fundamental lossless propagating defect Blochmode with wave vector k � 0.4 �2p�a� and ng � 11.The y component of jekj

2 is shown in Fig. 2, alongwith the dielectric interface regions. Assuming ay-polarized dipole, then at the f ield antinode positionwe find jek�rd�j2 � 3.5 3 1018 m23, and thus Fw � 3.This first example, which is completely unoptimized,already suggests enhancements that produce resultsquite comparable with those for state-of-the-art micro-cavities such as the microcavity posts. Additionally,

we also perform similar calculations with a wave vec-tor k � 0.45 �2p�a� and obtain Fw � 10 (with ng � 42).This value obviously becomes even greater as one getscloser and closer to the photonic band edge, which isassociated with the slowing down of the propagatinglight. The EEF can also be further increased byreducing the width of the PPC waveguide; e.g., ithas been numerically conf irmed that a 30% widthreduction results in approximately a threefold en-hancement of the EEF. Such large enhancements areremarkable on several fronts: (i) there is no nanocav-ity, (ii) PPC waveguides are routinely available, and(iii) any photon emission will directly lead to anefficient waveguide mode output channel.

Next a side-coupled nanocavity is introduced, asshown schematically in Fig. 1(b). Similar to before,we can write the solution for the propagating electricfield as E � Ew 1 G ? DecEw, where G is now thetotal GFT of the PPC, including both the defectcavity and the waveguide, and Dec is the changein permittivity obtained by adding the nanocavityto the PPC waveguide. To derive the appropriateGFT for the coupled cavity–waveguide system wewrite the general solution by expanding into the PPCmodes:

$G�r, r0;v� �

Pa,b Aa,b�v�ea �r� ≠ eb

��r0�,where Aa,b are the expansion coefficients that weseek to obtain and ea/b �r� represent the modes ofthe PPC system at frequency v that may belong toeither the cavity or waveguide; we also assume thatboth these sets of modes are weakly coupled to eachother. We define the fundamental cavity mode nor-malized from the E field of the cavity as ec � fc�

pVm,

with jfcj2 � jEcj

2�max�ecjEcj2�, where the effective

mode volume Vm �Rall space drec�r��jfc �r�j2. The

relevant GFT expansion coeff icients thus take theform Aa,b��c,c� � v2��vc

2 2 v2 2 ivGc�,8,9 wherevc is the cavity resonance frequency and Gc �v2a��2vg� j

Rall space drDec�r�eh�r�exp�ihx�ec

��r�j2 isthe cavity linewidth rate. The GFT at the QD location$G�rd,rd;v� � v2ec�rd� ≠ ec

��rd���vc2 2 v2 2 ivGc�,

which yields an EEF for the QD. Assuming that theQD is positioned near the f ield antinode position inthe nanocavity, we have

Fc�rd,vc� �6pc3

VmGced3�2vc

2. (2)

Fig. 2. (a) Top and (b) side views of the slab defectphotonic crystal waveguide. Example contour plots of thecalculated y-polarized Bloch vector jekj are also shown inboth views. The position of the QD is indicated schemat-ically by D. White circles and rectangles highlight theairhole boundaries within the high-index (semiconductor)slab. Material parameters are given in the text.

November 15, 2004 / Vol. 29, No. 22 / OPTICS LETTERS 2661

Employing the representative experimental pa-rameters of Q � 4000�� vc�Gc�, Vm � 0.05 mm3, andGc�vc� � 0.2 meV, we obtain Fc � 494. Although ourQ and Vm are for good-quality samples, we highlightthat PPC Q of up to 45,000 have recently been ob-tained experimentally,14 suggesting the possibility ofenormous Fc values (around 5000).

How do these results compare with the typical(ongoing improvements are expected) microcavityposts? Following Ref. 3, we use theoretically opti-mized parameters of radius 0.5 mm, Q � 2000, andVm � 0.13 mm3, which yields an EEF of Fp � 27.As noted in the introduction to this Letter, typicalexperimental values are nearer to 5, likely due topoor sample quality with the post structures. Thus,besides the advantage of lending itself naturally toefficient excitation and detection, the PPC systemyields larger EEFs, as was confirmed by recentexperiments.6,7

To include radiative coupling from the QD one canexpand the GFT in a Dyson equation to include theQD nonperturbatively through Gd � G 1 G ? Ded ? Gd,where Gd now includes the QD response and Ded is theQD permittivity. We assume a single QD with a vol-ume Vd and a diameter much smaller than the wave-length of light; thus the local permittivity Ded�rd� �jdj2��2Vde0h̄�vd 2 v 2 iGd��, where Gd is the non-radiative decay time; vd is the resonance frequency;d � pcv

Rdr�Flsjce

y�r�fhy�r�j0� is the exciton dipole

moment with pcv as the interband matrix element,fe/h

y as the creation operators of the correspondingelectron–hole, and Fls as the 1s exciton wave function.We assume fundamental 1s exciton resonance only. Itfollows that, with radiative coupling of the QD, thedenominator of Eq. (2) has the term Gc replaced withGc 2 i

Pd�v�, where the self-energy

Pd�v� � V2��vd 2

v 2 iGd� with V2 � vjdj2��2e0edh̄Vm�. Thus, assum-ing v � vc � vd, the EEF for the QD within thenanocavity can be rewritten as

Fc�rd,vc� �6pc3

Vm�Gc 1 V2�Gd�ed3�2vc

2. (3)

Consequently the EEF will decrease at the onsetof strong coupling, which is expected when V . Gd,Gc, and depends on the optical properties of the QDand the cavity. At this point Fermi’s golden rulebreaks down completely, and one must solve the self-consistent equation given above to obtain the correctradiative decay rate. When a representative QDdipole moment of d � 60 Debye15 and Gd � 0.2 meVare employed, V � 0.14 meV, thereby yielding asubstantially smaller Fc � 84 in the nanocavity. Twoimportant points are highlighted: first, radiativecoupling is rarely included when one is investigatingenhanced emission in photonic crystal nanocavities,16

and, second, coupling back into a waveguide Blochmode is conveniently expected to be the dominantscattering channel, although there will be some finitebut small overlap with radiation modes above the lightline.

To conclude, a timely Green function formalismfor calculating enhanced emission factors of single

quantum dots in planar photonic crystal wave-guides and nanocavities has been presented.Spontaneous-emission rates in the waveguide areshown to scale proportionally (inversely) with thephoton group index (velocity) and thus substantialincreases in the decay rates are expected for defectmodes close to the photonic bandedge. A methodto include radiative coupling of the QD, evidentlyan important effect even with reasonable cavityQ values of 4000, has also been described. Theseresults have important applications for achievingefficient single-photon emitters from embedded QDsin photonic crystal nanostructures.

The author (hughes@will.brl.ntt.co.jp) is grateful toL. Ramunno, J. F. Young, J. E. Sipe, H. Gotoh, andH. Kamada for useful discussions.

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