## 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 OSA

## 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 In_{0.53}Ga_{0.47}As 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.

#### 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 SiO_{x} mask before being transferred to the active InP slab by reactive ion etching (RIE). The remaining SiO_{x} material was removed in a diluted HF solution. Finally we have removed the InGaAs sacrificial layer underneath by a time controlled HF:H_{2}O_{2}: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.

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 τ_{0}/τ_{m}, where τ_{m} is the cavity mode decay time (m = 1 and 2 for modes O1 and O2) and τ_{0} 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.

## 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]:

*W*and

^{CAV}*W*are the SE rate of the emitter into the cavity and in the vacuum, V

^{0}_{ef}is the effective modal volume, and λ/n is the cavity mode wavelength, respectively. The frequencies of the cavity mode and the emitter are

*ω*and

_{c}*ω*, being

_{e}*Δω*and

_{c}*Δω*their linewidths defined by their full widths at half maximum. The polarization mismatch between the optical mode and the quantum emitter is represented by

_{e}*ξ*. 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]:we can factorize such expression to retrieve the well-known figure of merit for a perfectly matched point-like emitter,

^{2}*F*:

_{P}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:

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

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 |E_{MAX}|^{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:

_{CAV}) for tightly confined modes. We introduce the shape function H(

**r-r**

_{e}), to account for the spatial extension of the emitter centered at

**r**

_{e}. Its value is H(

**r-r**

_{e}) = 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,

**r**

_{0},where

**E**(

**r**

_{0}) = E

_{MAX}. For the QD case after introducing H

_{QD}(x-x

_{i},y-y

_{i}) = δ(x-x

_{i})δ(y-y

_{i}):

If we apply this formalism for a QWR of length 2L and negligible width oriented along the y direction, after introducing H_{QWR}(x-x_{i},y-y_{i}) = δ(x-x_{i}) if |y-y_{i}| ≤ L and H_{QWR}(x-x_{i},y-y_{i}) = 0, γ can be determined as:

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:

By using this expression we consider the optical intensity of the cavity mode as the superposition of the emission of individual optical transitions. *F _{p}* and Π can be assumed to be constant for all the emitters; hence they are out of the sum. The magnitudes α

_{i}and γ

_{i}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 P

_{i}which is proportional to the contribution of the emitter “i” to the PL intensity,

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

^{(i)}_{PL}*W*.

_{i}^{CAV}*W _{i}^{CAV}* can be estimated by clearing Eq. (7) and assuming W

^{0}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:

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:

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 (V1_{ef} = 1.09, V2_{ef} = 1.17and V3_{ef} = 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 Q_{ef} for the three modes with negligible differences attributable to the modal volume. For the estimation of Q_{ef} we have to calculate Q_{e} 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:

From the single QWR characterization Γ_{0} is 0.5 meV, Γ_{AC} is the scattering rate of excitons by acoustic phonons and Γ_{LO} the linewidth associated to the scattering of excitons by longitudinal optical phonons of energy E_{LO}. We take 0.035 meV/K and 25 meV as approximate values for Γ_{AC} and Γ_{LO} (with E_{LO} = 40 meV), as determined from PL measurements in QWR ensembles [33]. In this way we can deduce Q_{e}≈235 for QWRs emitting at around 1550 nm (0.8 eV), one order of magnitude smaller than the cavity quality factors. Therefore, Q_{ef} (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 Q_{ef} = Q_{c} 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 Q_{ef} = Q_{e} = 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 F_{p} = 15.5 and it is expected to be practically constant with Q_{c}, 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.

#### 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:

In our cavities, the normalized polarization vectors of the modes for both types of cavities ( + and -) are:

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:

Here I_{PL + ,} I_{PL-} and I_{PLz} account respectively for the intensity of the polarized spectra of the QWRs in the [110] [1–8,13,14], and [001] directions. The component I_{PLz} can be compared with I_{PL-} and I_{PL+} 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.

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 *Q _{c}* = 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).

## 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 (F_{p}), 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.

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