Purcell effect in photonic crystal microcavities embedding InAs/InP quantum wires

Purcell effect in photonic crystal microcavities embedding InAs/InP quantum wires

Josep Canet-Ferrer,1,* Luis J. Martínez,2 Ivan Prieto,2 Benito Alén,2 Guillermo Muñoz-

Matutano,1 David Fuster,1 Yolanda González,2 María L. Dotor,2 Luisa González,2 Pablo

A. Postigo,2 and Juan P. Martínez-Pastor1 1UMDO (Unidad asociada al CSIC), P.O. Box 22085, E-46071 Valencia, Spain

2Instituto de Microelectronica de Madrid (CNM, CSIC), Isaac Newton 8, E-28760 Tres Cantos Madrid, Spain *[email protected]

Abstract: The spontaneous emission rate and Purcell factor of self-

assembled quantum wires embedded in photonic crystal micro-cavities are

measured at 80 K by using micro-photoluminescence, under transient and

steady state excitation conditions. The Purcell factors fall in the range 1.1 –

2 despite the theoretical prediction of ≈15.5 for the figure of merit. We

explain this difference by introducing a polarization dependence on the

cavity orientation, parallel or perpendicular with respect to the wire axis,

plus spectral and spatial detuning factors for the emitters and the cavity

modes, taking in account the finite size of the quantum wires.

©2012 Optical Society of America

OCIS codes: (230.0230) Optical devices; (250.0250) Optoelectronics; (020.5580) Quantum

electrodynamics; (050.5298) Photonic crystals; (230.5590) Quantum-well, -wire and -dot

devices.

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1. Introduction

Photonic crystal microcavities (PCM) combine high quality factors with small effective modal

volumes that lead to enhanced values of the spontaneous emission (SE) rate through the

Purcell effect [1–4]. In the case of a solid state emitter inside a PCM, the enhancement of the

SE rate is determined by the detuning of the emitter wavelength with respect to that of the

cavity mode (spectral detuning) [5], the position of the emitter in the cavity (spatial detuning)

[6], the cavity quality factor (Q), the quantum emitter linewidth [7] and the polarization

mismatch [8]. The optical properties of the emitter are crucial for its successful coupling to

the cavity modes. The quantum wires (QWRs) present optical properties in between of those

of quantum wells (QW) and quantum dots (QDs). Ideally, the density-of-states (DOS) in a

QD is described by a series of Dirac’s deltas while in the case of QWs and QWRs it is a

continuum of states above the fundamental transition, an important fact for technological


applications like nano-lasers. InAs/InP QWRs can be used for devices working at two

important optical telecom windows in 1.3 and 1.5 µm [9–11]. They also exhibit efficient

luminescence and lasing even at 1.6 µm [12]. In spite of their peculiar properties, the QWRs

are still less studied than other kind of nano-structures, and few works about the Purcell effect

on QWRs embedded in PCMs have been reported up to date. Among them, Atlasov et al.

demonstrated the integration of site-controlled InGaAs/GaAs V-groove QWRs into a PCM

where the coupling of the quantum emitter and the cavity modes exhibit a SE enhancement by

a factor 2.5 [13,14].

In this work we present a systematic study of linear PCMs with self-assembled InAs/InP

QWRs embedded together with a detailed model to simulate the Purcell effect in such system.

The fundamental optical modes of Ln defect PCMs (linear cavities formed by eliminating “n

holes” from the photonic lattice) are characterized by a large linear polarization anisotropy,

which is maximum along the direction perpendicular to the linear defect, as was

experimentally demonstrated for the L7-cavity [15]. QWRs present a linear polarization

anisotropy which is maximum along the QWR axis ([1–8,13,14] crystalline direction) [10].

To study the effect of polarization, two sets of L7-cavities have been fabricated and aligned

either parallel or perpendicular to the QWRs. We will show that the Purcell factor values

measured for both types of cavities are different. The average value and its dispersion will be

described considering the optical properties of a finite size of QWRs and the electromagnetic

field distribution of the optical modes. In particular, both the spatial and spectral detuning will

be estimated by creating a statistical distribution of QWRs within the area defined by the L7

cavity.

2. Experimental details of the active medium and Fabrication

2.1. QWR Epitaxy

The QWRs compose a single layer of self assembled nano-structures embedded in the middle

of a 237 nm thick slab of InP. Under the slab a 700 nm thick layer of In0.53Ga0.47As is

deposited on an InP(001) substrate. The growth is made by molecular beam epitaxy (MBE).

InAs QWRs are aligned along the [1–8,13,14] direction forming a quasi-periodic array. The

average height and width are 3.2 and 10.3 nm respectively, as determined from transmission

electron microscopy (TEM) studies carried out in samples with buried QWRs grown in the

same way than those studied in this work [16]. The photoluminescence (PL) emission band

measured at 80 K [Fig. 1(b)] can be deconvoluted into four Gaussian peaks, which correspond

to the QWR heights measured by AFM. The PL band of the QWR ensemble in this sample

exhibits around a 30% of linear polarization anisotropy along the QWR axis direction. This is

explained by the confinement potential anisotropy related with the QWRs geometry [10].

Indeed, the nanostructures form an almost periodic array of wires with a period close to 18

nm. The average length of the QWRs is estimated to be 200 nm from grazing incidence X-ray

diffraction measurements [17]. The exciton recombination dynamics as a function of the

excitation density and temperature in a QWR ensemble has been studied separately [11] but

some of the results have been used here for calculation of the experimental values of the

Purcell factor. The radiative recombination at very low temperatures is dominated by exciton

localization at the QWR fluctuations in width, but above 40-50 K the exciton dynamics is

dominated by free exciton recombination [11]. This is the reason why the study has been

carried out at 80 K.


Fig. 1. (a) Atomic force microscopy image of an uncapped sample of self-assembled InAs/InP

QWRs. (b) Photoluminescence spectra of the ensemble of QWRs for light linearly polarized

along the [110] and [1–8,13,14] directions measured at 80 K out of the photonic crystal

structure. (c) and (d) Scanning Electron Microscope (SEM) images of the microcavities

fabricated with the linear L7 defect parallel [type (-)] and perpendicular [type( + )] to the

longer axis of the QWRs [1–8,13,14].

2.2. Fabrication details of L7 microcavities

The L7 defect cavity is made by removing seven holes along the Γ-K direction (reciprocal

space) of a triangular photonic crystal lattice with a typical lattice parameter a = 410 nm. We

have fabricated two sets of L7 cavities, oriented parallel [type(-)] and perpendicular [type( +

)] to the QWR axis [1–8,13,14]. For each set we have fabricated structures with an evolution

in the hole radius (r) in order to tune the cavity modes over the PL band of the QWR

ensemble. The PCMs were fabricated by electron beam lithography on a

polymethylmetracrylate (PMMA-A4). The holes were opened by reactive ion beam etching

(RIBE) on a hard SiOx mask before being transferred to the active InP slab by reactive ion

etching (RIE). The remaining SiOx material was removed in a diluted HF solution. Finally we

have removed the InGaAs sacrificial layer underneath by a time controlled HF:H2O2:DI

solution. The resulting devices are shown in Figs. 1 (c)-(d). For more details about the process

see [18].

2.3. Set-up for optical micro-spectroscopy

The optical characterization of the QWR/L7-PCM structures was performed by micro-PL

(µPL) and time resolved µPL (µTRPL). The sample was held at 80 K by immersing a

confocal microscope in a liquid nitrogen bath. The µPL measurements were carried out by

using as excitation source a 980 nm pulsed laser diode (40 ps pulsewidth and 40 MHz of

repetition rate). The corresponding excitation energy (1.265 eV) is smaller than the InP

absorption band edge and hence carriers are photogenerated directly at the QWRs. The

excitation and emitted light were coupled to optical fibers and focused through the same

microscope objective (NA = 0.55), which determines a combined spatial resolution around

1.5 µm. The collected light was dispersed by a 0.5 m focal length monochromator and

detected with a cooled InGaAs photodiode array. The µTRPL measurements were performed

using the same optical set-up, except for the use of an InGaAs APD single photon detector

managed with electronics for time correlated single photon counting. In order to avoid


saturation effects in the PL intensity that might obscure the determination of the Purcell

factor, the excitation power was kept below 10µW.

3. Experimental determination of the Purcell Factor

The emission spectrum of a L7-cavity containing QWRs typically exhibits either three or four

emission resonances (Fig. 2) depending on the matching between its optical modes and the PL

band of the QWR ensemble. The optical modes can be labeled as O1, O2, O3 and E1

according to their different symmetry: the O-labeled modes are odd while the E-labeled ones

are even [15,19]. In this work we will focus our attention on the O1 and O2 modes since they

exhibit the highest intensity and the narrowest linewidths. Figure 2 shows the PL polarization

resolved spectra corresponding to a type(-) and a type( + ) cavity. The cavity modes are

linearly polarized in the direction [110] and [1–8,13,14] respectively, see Figs. 2(a) and 2(b).

Since the PL intensity of the QWRs is higher along the direction [1–8,13,14], we could expect

for a better polarization matching for the type( + ) than for the type(-) cavities. But notice that

in this kind of measurements it is not really possible to judge about how much light gets

coupled to the cavity modes if the QWRs are placed either parallel or perpendicular to the

cavity main axis. This is because QWR polarization anisotropy is never complete and the

emitted light will always couple to polarization of modes of both polarization types.

Moreover, polarization of the PCM modes throughout the cavity volume is not well defined.

On top of that every considered cavity can be slightly different due to fabrication, and the

self-assembled QWRs also differ from cavity to cavity. On the other hand, time-resolved

measurements will provide the solid basis for making conclusions.

Fig. 2. (a) Photoluminescence spectra of a type(-) cavity at the polarization directions [110]

(red) and [1–8,13,14] (blue), (b) Idem for a type( + ) cavity.

The steady state µPL (Fig. 3(a)) and µTRPL (Fig. 3(b)) spectra are registered

simultaneously. In this way, the wavelength of the cavity modes, their quality factor and their

decay time can be determined under the same excitation conditions. For comparison, the

TRPL transients for bare QWRs (emitting at the cavity mode wavelength) are measured in the

same epitaxy in a region of the sample without PCMs. An example is shown in Fig. 3(b) for

the O1 mode of a type(-) cavity emitting at 1509 nm. The decay time of QWRs decreases

smoothly from 2.6 to 2.3 ns between 1460 nm and 1510 nm, as shown in Figs. 3(c)-(d). The

decay times measured at the cavity modes of 12 different L7-cavities, although typically

smaller than that of the QWRs, exhibit a noticeable dispersion, as shown in the same figures.

We have summarized the obtained Purcell factors as τ0/τm, where τm is the cavity mode decay

time (m = 1 and 2 for modes O1 and O2) and τ0 is the decay time of the corresponding QWRs

at the mode wavelength (Tables 1 and 2). The 12 cavities studied in this work were selected to


have very similar Qs for comparison purposes. From these results, we point out three main

aspects:

i) For cavities of the same type [( + ) or (-)], the average Purcell factors for O1 and O2

modes are practically the same, even with their different narrowing, within the

dispersion error.

ii) For cavities of the same type [( + ) or (-)] the Purcell factors exhibit a great dispersion,

even in adjacent cavities with similar fabrication parameters.

iii) Although the µTRPL measurements were carried out for both type( + ) and type(-)

systems with similar Qs, the Purcell Factors are larger for the first ones.

Fig. 3. (a) µPL spectrum of a type(-) cavity. The decay times of the modes O1, O2 and O3

(blue crosses) and that of the QWRs emitting at the same wavelengths (red solid circles) are

also depicted. (b) µTRPL transients measured at the mode O1 of the a type(-) cavity and at

QWRs emitting at the same wavelength. (c) Decay time measured at the O1 mode of twelve

different cavities of both type(-) (x scatter) and type( + ) ( + scatter). The red solid circles stand

for the decay time of the QWRs emitting at the same wavelengths of the cavity modes. (d)

Idem at the O2 mode of the same cavities.

Table 1. SE enhancement of the type( + ) cavity modesa

τ0/τ1 Q1 τ0/τ2 Q2

1.78 7890 1.70 4720

1.35 7740 1.65 4550

1.41 7620 1.71 4830

1.72 7880 1.76 4330

2.03 6820 1.89 4540

1.32 7430 1.22 4390


Average 1.6 ± 0.3 7560 ± 170 1.65 ± 0.20 4560 ± 140

aObtained from the experimental data represented in Fig. 3 (c)-(d).

Table 2. SE enhancement of the type(-) cavity modesa

τ0/τ1 Q1 τ0/τ2 Q2

1.12 7500 1.07 4060

1.29 6880 1.34 3140

1.41 4910 1.28 2450

1.29 6540 1.18 4220

1.27 5740 1.33 3250

1.16 5420 1.22 3160

Average 1.26 ± 0.10 6200 ± 600 1.21 ± 0.12 3400 ± 500

aObtained from the experimental data represented in Fig. 3 (c)-(d).

4. Theory: Purcell Factor for a finite ensemble of extended emitters in a PCM

To explain our experimental results, we study first the case of a L7 cavity containing a single

QWR with a finite size. Using the results, we will be able to simulate by statistical approaches

the case of a QWR ensemble in the cavity. We start from the expression for the Purcell factor

of a point-like emitter exhibiting an emission linewidth similar to the cavity mode [20]:

3

2

0 2 2 2

3( / )

4 4( ) ( )

CAV

e c c e

ef e c e c

W n

W V

ω ω ω ωλξ

π ω ω ω ω∆ + ∆

=− + ∆ + ∆

(1)

where WCAV

and W0 are the SE rate of the emitter into the cavity and in the vacuum, Vef is the

effective modal volume, and λ/n is the cavity mode wavelength, respectively. The frequencies

of the cavity mode and the emitter are ωc and ωe, being ∆ωc and ∆ωe their linewidths defined

by their full widths at half maximum. The polarization mismatch between the optical mode

and the quantum emitter is represented by ξ2. This analytical expression was used to describe

the Purcell factor when the two linewidths are comparable and can be described by a

Lorentzian profile. If the effective quality factor is introduced in Eq. (1) [21]:

1 1 1

ef c eQ Q Q

= + (2)

we can factorize such expression to retrieve the well-known figure of merit for a perfectly

matched point-like emitter, FP:

0

CAV

p

WF

Wα= ×Π× (3)

3

2

3( / )

4

ef

p

ef

n QF

V

λ

π=with (4)

2ξΠ = (5)

2

2 2

1

4( ) ( )

e c c e

ef e c e cQ

ω ω ω ωα ξ

ω ω ω ω∆ + ∆

=− + ∆ + ∆

and (6)


Equation (6) accounts for the spectral detuning and finite linewidth of an emitter spatially

matched to the mode. If the emitter is not spatially matched, we can introduce a spatial

detuning factor, γ:

0

CAV

p

WF

Wα γ= ×Π× × (7)

For a point-like emitter (atom or QD) γ = γQD and is given by [5,22]:

2

2

MAX

( )

EQDγ =

E r (8)

Equation (8) stands for the ratio between the optical density of states available for the

emitter located at an arbitrary position (proportional to |E(r)|2) and the same emitter perfectly

placed at the point of maximum field amplitude (proportional to |EMAX|2). However, Eq. (8)

does not describe correctly situations where a large nano-structure has to be matched to a

particular electromagnetic mode. In that case, a possible approach would consist of

integrating Eq. (8) into the emitter volume [23]. Consequently, we propose a new definition of

γ to account for the spatial variation of the electric field of an optical mode:

2

2

( ) ( )d

( ) ( )d

V

V

H

Hγ

−=

−

∫∫

e

0

E r r r r

E r r r r (9)

where V stands for the volume of the optical mode, which can be substituted by the cavity

volume (VCAV) for tightly confined modes. We introduce the shape function H(r-re), to

account for the spatial extension of the emitter centered at re. Its value is H(r-re) = 1 into the

emitter volume and null in the rest of the space. In the spirit of Eq. (8), Eq. (9) gives the ratio

between the available optical density of states when the emitter is allocated at an arbitrary

position or at the antinodal position of the electric field, r0,where E(r0) = EMAX. For the QD

case after introducing HQD(x-xi,y-yi) = δ(x-xi)δ(y-yi):

2

2

0 0 MAX

( , )d d ( , )

( , )d d E

CAV

CAV

e eS e e

QD

S

H x x y y x y x y

H x x y y x yγ

− −= =

− −

∫∫

E (10)

If we apply this formalism for a QWR of length 2L and negligible width oriented along

the y direction, after introducing HQWR(x-xi,y-yi) = δ(x-xi) if |y-yi| ≤ L and HQWR(x-xi,y-yi) = 0,

γ can be determined as:

0

0

2 2

2 2

0 0 0

( , ) ( , )d d ( , ) d

( , ) ( , )d d ( , ) d

e

CAV e

CAV

y L

e e eS y L

QWR y L

S y L

x y H x x y y x y x x y y

x y H x x y y x y x x y yγ

+

−

+

−

− − == =

− − =

∫ ∫∫ ∫

E E

E E (11)

Lets discuss now the case of an ensemble of emitters. The cavity SE rate depends on the

homogeneous broadening of the emitter, because it determines the number of available states.

On the other side, there is an inhomogeneous broadening inherent to the emitter ensemble that

will affect the number of emitters coupled to the cavity mode. Therefore, in the same cavity

we can find perfectly tuned emitters together with off-resonance emitters whose emission is

inhibited [24]. Some authors have discussed about the Purcell Factor for an ensemble of QDs

by averaging the SE-rate [20,25–28] or by taken it smaller than in the case of isolated single

emitters coupled to the cavity [14,29–31]. Here we propose a simple method to estimate the

Purcell Factor of an ensemble of emitters by averaging the SE rate of the emitters embedded

into the cavity defect:


0

( )

0

0

i iCAV ii i iCAV i

p

CAV i ii i

PPW

FP PW

τα γτ τ

τ= = = ×Π×

∑ ∑∑ ∑

(12)

By using this expression we consider the optical intensity of the cavity mode as the

superposition of the emission of individual optical transitions. Fp and Π can be assumed to be

constant for all the emitters; hence they are out of the sum. The magnitudes αi and γi account

for the spectral and spatial detuning of each single emitter, labeled with the integer “i”. The

average is weighted considering that the emitters coupled to an optical mode contribute

differently to the PL depending on their coupling degree. This is accounted for by weighting

factor Pi which is proportional to the contribution of the emitter “i” to the PL intensity, I(i)

PL.

At the same time, this intensity is proportional to the SE rate of the emitter “i” inside the

cavity, WiCAV

.

( )i CAV

i PL iP I W∝ ∝ (13)

WiCAV

can be estimated by clearing Eq. (7) and assuming W0 as constant in the frequency

range around the cavity mode. Therefore the weighting factor becomes proportional to the

product of the spatial and the spectral detuning:

0CAV

i i p i i i iP W W F α γ α γ∝ = × ×Π× × ∝ (14)

As a result, the emitters with a higher coupling degree present a larger contribution to the

PL intensity and this leads to a higher influence in the SE of the optical mode. Finally Eq.

(12) can be rewritten as:

2

0( )

( )

i ii

p p

CAV i ii

F Fα γτ

α γτ α γ

= Π = Π ×∑∑

(15)

The theoretical approach developed here can be used for different kinds of emitters with

the proper definition of γ. Equation (7) can be used for the calculation of the Purcell factor in

the case of extended nanostructures, provided their size and shape. Equation (15) leads to an

appropriate estimation of the Purcell factor in cavities embedding more than an isolated

emitter. However, let notice that in order to apply Eq. (15) it is required the frequency

emission and position of each of the single emitters (single QWRs in our system) embedded

into the cavity, and these parameters are not usually known. Due to this fact, in the next

section the Purcell factor will be determined by simulating the SE rate of cavities with an

ensemble of QWRs. In such simulations the emission wavelength of the QWRs is randomly

distributed according to the PL spectrum shown in Fig. 1(b) while the position of the QWRs

into the cavity also varies accordingly to their geometry (size, shape and orientation with

respect to the cavity defect).

5. Discussion

5.1 Figure of Merit

In the analysis of the figure of merit in a L7 PCM, the key parameters are the linewidths of

both the cavity mode and the emitter. The modal volume of the three first optical modes are

approximately equal (V1ef = 1.09, V2ef = 1.17and V3ef = 1.14 in units of (λ/n)3) and the

theoretical Qs are Q1 = 82333, Q2 = 10567 and Q3 = 2141. The figure of merit becomes

directly proportional to the corresponding Qef for the three modes with negligible differences

attributable to the modal volume. For the estimation of Qef we have to calculate Qe which is

inversely proportional to the emitter PL linewidth. This magnitude can be taken from

measurements on single QWRs at 4 K [32]. At 80 K we should take into account the

temperature broadening of the optical transition by acoustic and optical phonon scattering:


0( )exp( / ) 1

LO

AC

LO

T TE KT

ΓΓ = Γ +Γ +

− (16)

From the single QWR characterization Γ0 is 0.5 meV, ΓAC is the scattering rate of excitons

by acoustic phonons and ΓLO the linewidth associated to the scattering of excitons by

longitudinal optical phonons of energy ELO. We take 0.035 meV/K and 25 meV as

approximate values for ΓAC and ΓLO (with ELO = 40 meV), as determined from PL

measurements in QWR ensembles [33]. In this way we can deduce Qe≈235 for QWRs

emitting at around 1550 nm (0.8 eV), one order of magnitude smaller than the cavity quality

factors. Therefore, Qef (Eq. (2) is mainly determined by the µPL linewidth of the single QWR

emitter. This is shown in Fig. 4. An upper limit for the figure of merit is given by the curve

Qef = Qc corresponding to a emitter with a narrow linewidth coupled to a wider optical mode:

it is the case, for instance, of QDs into a micropillar [22]. In contrast, the lower limit is given

by a curve with Qef = Qe = 40 corresponding to a broad band emitter coupled to a narrower

optical mode: this would be the case, for instance, of an InGaAsP QW emitting at 0.8 eV with

a 20 meV linewidth [34]. The figure of merit for the cavities in this work is Fp = 15.5 and it is

expected to be practically constant with Qc, because the emitters are wider than the cavity

modes. This result is consistent with observation i) in section 3, where minor differences in

the Purcell factor of O1 and O2 optical modes were noted for both kinds of cavities.

Fig. 4. Figure of merit of the Purcell factor as a funtion of Q for the three cases: narrow

emitters (doted black line), broad emitters (dashed black line) and self-assembled QWRs at the

lattice temperature of 80 K (red continuous line).

5.2 Polarization Mismatch

It is known than the non-polarized light coming from QDs reduces three times the Purcell

factor. Some reports introduce directly a factor 1/3 in Eq. (1) instead of the term Π [26,35]. In

the case of our QWRs, the value of the polarization factor is calculated as the projection of the

polarization vector of the emitter along the electromagnetic field direction:

2

Π = ×emitter modeP P (17)

In our cavities, the normalized polarization vectors of the modes for both types of cavities

( + and -) are:

2 2 2

0 01 1

1

0 0

Ex

Ey EyEyEx Ey Ez

Ez

= = =

+ +

(+)P (18)


2 2 2

11 1

0 0

0 0

Ex Ex

EyExEx Ey Ez

Ez

−

= = =

+ +

( )P (19)

If we analyze the emitted light of QWRs along the two directions defining each of the

three free faces of a sample (backscattering geometry) we can approximate the polarization

vector to:

1.471 1

2.454.92

1

PL

PL

PL PL PLz

PLz

I

II I I

I

+

−

+ −

= =

+ +

emitterP (20)

Here IPL + , IPL- and IPLz account respectively for the intensity of the polarized spectra of the

QWRs in the [110] [1–8,13,14], and [001] directions. The component IPLz can be compared

with IPL- and IPL+ by measuring the polarization resolved spectra in the direction [110] and [1–

8,13,14] respectively (not shown). Such component present a smaller contribution to the

QWR emission as has been also observed in InGaAsP/InP QWs [35]. With all of these, the

scalar product (Eq. (20) gives polarization factors Π(-) = 0.29 and Π( + ) = 0.49. This helps to

explain observation iii) in section 3, i.e. the smaller SE rates (and hence smaller Purcell

Factor) measured in type(-) L7-cavities respect to type( + ) ones.

5.3. Spectral and Spatial detuning factors

Finally we present a numerical simulation to obtain average values for the spectral and spatial

detuning factor, <α*γ>. The simulations consider a finite number of QWRs embedded in a

L7-cavity emitting with a wavelength randomly distributed but correlated to the PL emission

of the ensemble [Fig. 1(b)]. The L7-cavities can contain approximately from 260 to 620

QWRs depending of the QWR length and the type of cavity ( + or -). However, the

introduction of the weighting factor (Eqs. (15-18) will reduce the number of QWRs that

actually contribute: less than a 10% of the emitters will be coupled to the optical modes in the

most favorable case. We have developed an algorithm to evaluate <α*γ> in a system

composed by QWRs of 200 nm long embedded in type(-) [Fig. 5(a)] and type( + ) [Fig. 5(b)]

L7cavities. The extension of the emitter ensemble is determined by the cavity size, i.e. the

QWRs that overlap with holes forming the photonic crystal are removed from the calculation.

The emitter ensemble is constructed by considering each QWR located at a random position

into the cavity until the cavity is completely filled. Each of the QWR has an arbitrary

emission frequency selected by a random value function modulated by the DOS function of

the ensemble, which is proportional to the PL spectra shown in Fig. 1(b). The spatial

distribution of the electromagnetic field intensity for the mode O1 is illustrated in Fig. 5(c).

The cavity has a lattice parameter a = 410 nm and a filling factor r/a = 0.29.


Fig. 5. Distribution of QWRs in type(-) (a) and type( + ) (b) coupled to a L7-type PCM whose

electromagnetic field distribution for mode O1 is depicted in (c). (d) and (e) represent two

simulated distributions of QWRs leading to different values of <α*γ> (red and blue spectra

represent the emission of a single QWR and the optical mode O1, respectively). (f) One

hundred simulated values of <α*γ> for the optical mode O1.

Two simulated ensembles of QWRs are shown in Figs. 5(d) and (e). They give rise to

different values of <α*γ>. The two emission bands cannot be equal to the measured PL of an

ensemble due to the finite number of QWRs that can be embedded in the simulated cavity.

Therefore the peak energy of the simulated emission band is around 1470 nm [Fig. 5(d)] and

1410 nm [Fig. 5(e)] instead of the 1450 nm observed in the experimental PL [Fig. 1(b)].

Similarly, the number of QWRs emitting at 1522 nm [mode O1 in Figs. 5(d)-(e) that is

considered with Qc = 6500] is also different in the two simulations. This explains the different

values found for <α*γ> in Figs. 5(d)-(e). After 100 simulations (corresponding to 100


different QWR cavity ensembles) fluctuations of <α*γ> between 0.1 and 0.5 are obtained, as

shown in Fig. 5(f). The average value converges to <α*γ> ≈0.31 after the first 30 to 40

simulations.

Finally, we have performed the simulation of <α*γ> for 16 different type(-) and type( + )

L7-cavities and for the O1 and O2 modes. We have swept the most important region of the

QWR PL band and calculated the effective Purcell factor by taking into account the Purcell

figure of merit and the polarization factor discussed above. These values have been compared

to the experimental data (Fig. 6). From the comparison between simulation and measurement

we can conclude that the dispersion of the experimental data is produced by the finite content

and random distribution of QWRs. This reduces the SE rate by a factor 0.31 with respect to

the figure of merit, giving an average Purcell factor as low as 1.2-1.6 for type(-) L7-cavities

and for both O1 and O2 modes, in close agreement with the experiment. In the case of the

type( + ) L7-cavities the average Purcell Factor can increase up to 2.4, even when an

important reduction for optical modes above 1520 nm is predicted due to the low energy tail

of the PL band. The highest dispersion in the calculated Purcell factor is found for the mode

O1 of the type( + ) L7-cavity which is also in correspondence with the experimental

measurement (values in the range 1.3-2).

Fig. 6. (a)-(b) Calculated average values of the Purcell factor by using simulated values of

<α*γ>, as explained in the text, for the case of type(-) and type( + ) cavities for mode O1; (c)-

(d) idem for mode O2. The error bars stand for the dispersion in the simulated values of <α*γ>

as was illustrated in Fig. 5 (f).

6. Conclusions

In this work the spontaneous emission rate and Purcell factor of the L7-type PCM with

embedded self-assembled QWRs is studied, taking in account the effect of polarization, the

spectral and the spatial detuning of the QWRs inside the cavity. The measured Purcell factor

for cavities with QWRs parallel and perpendicularly oriented to the cavity modes are in

average 1.6 and 1.2 respectively, which differs considerably from the theoretical prediction


(15.5). The results have been discussed after grouping the parameters contained into the

Purcell factor expression into four factors: the figure of merit (Fp), the polarization mismatch

(Π), the spectral detuning factor (α) and the spatial detuning factor (γ). The two last factors

depend highly on the QWR content and position inside the cavity. Using simulations with

random contents of QWRs embedded into a L7-type(-) cavity we can conclude that the

moderate Purcell factor measured in our cavities is mainly attributed to the homogeneous

broadening of the QWRs, with an important reduction due to the polarization matching and

the spectral and spatial detuning factors. Finally, the important data dispersion in

experimental measurements is shown to be related to the finite number of QWRs embedded

into the cavity.

Acknowledgments

We want to acknowledge financial support from the Spanish MICINN through Grants: TEC

2005-05781-C03-01/03, TEC2008-06756-C03-01/03, S-0505-TIC-0191, Consolider-Ingenio

2010 QOIT (CSD2006-0019), and CAM (S2009ESP-1503). The main author, J. C.-F., thanks

also the Spanish MCI for his FPI grant BES-2006-12300.


Purcell effect in photonic crystal microcavities embedding InAs/InP quantum wires

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