1. Introduction

Solid-state optical cavities with high quality factors and small photonic mode effective volumes are relevant for several applications, both technological and for fundamental physics studies [1]. These applications take advantage of the enhancement of the spontaneous rate of emission, the so-called Purcell effect [2], and/or the strong photon-exciton coupling enhanced by the cavity confinement. For these cavities, an embedded quantum dot constitutes a nearly ideal source of photons, due to its discrete energy levels and high emission efficiency. In this context, the coupling of the quantum dot emission to the photonic modes of an optical cavity is a very important feature. The requirements to obtain a large Purcell effect in such structures are well known. One needs not only quantum dots located in a region of high electromagnetic mode intensity but, to achieve a good coupling, the dot dipoles must also be aligned with the photonic mode local electric field. These requisites have been discussed extensively [3, 4] theoretically. However, experimentally, a direct correlation between dot dipole position and orientation with photonic mode excitation efficiency is difficult to probe [4], as one would need to measure dot position and polarization independently.

On another side, there has been an extensive effort for the investigation of the lowest energy mode of optical cavities and its coupling to the excitonic dot emission but relatively little attention has been given to the higher energy modes [5, 6]. These can have a somewhat surprising behavior, such as the predicted [7] increase in the quality factor as the confinement increases. For applications, the requirement is not just to maximize the quality factor Q but also to minimize the modal volume V since the enhancement of spontaneous emission is proportional to Q/V while for strong coupling the parameter that has to be maximized [8, 9] is Q/√V. Higher energy photonic modes may provide better figures of merit than the fundamental one, since some of them have smaller effective volumes and/or may present higher Q values [7].

In the present work, we investigate experimentally the effect of the quantum dot dipole orientation on the intensity of the photonic modes of microcavity pillars, with special attention to the higher energy modes. Microphotoluminescence measurements on 33 nominally identical pillars show statistically significant differences in the relative intensity of the photonic modes. We demonstrate that these differences are linked to the orientation of the quantum dot dipoles that act as the photon suppliers for the cavity allowed modes. We calculate numerically the emission energies and relative intensities of the photonic modes for several different positions and orientations of the quantum dot dipole. Since the higher energy modes are particularly sensitive to the polarization of the dot emission, we can use the intensity of these modes as a probe to obtain information on the polarization state of the quantum dots. We find that a surprisingly high proportion of the dots present polarized emission, including a detectable amount that has a [001], growth direction, polarization component.

2. Samples and experimental setup

The samples were grown by molecular beam epitaxy. They have a one λ -thick GaAs microcavity with lower and upper Bragg mirrors composed of, respectively, 27 and 20 periods, each one consisting of a 69.3 nm thick GaAs layer and a 78.0 nm thick Al _0.8Ga_0.2As layer. A layer of self-assembled InAs quantum dots with a density less than 10¹⁰ cm^-2 is grown at the center of the cavity. Circular pillars of different diameters were fabricated by electron beam lithography and inductively coupled plasma reactive ion etching [10].

The photonic mode structure of each individual micropillar was investigated with a standard microphotoluminescence setup. Excitation was provided by a titanium-sapphire laser tuned at 740 nm. A microscope objective with numerical aperture of 0.4 is used to focus the excitation beam to a spot with a diameter ≤ 2 µm. The emission from the pillar was collected by the same objective, focused on a 0.75 m monochromator, and detected by a nitrogen cooled charge-coupled device detector. The spectral resolution is 0.05 nm. All measurements were done at 4 K with a helium flow cryostat.

3. Results and discussion

We investigated 33 nominally identical pillars with circular cross sections of 1.5 µm diameter. The fundamental mode in these samples shows an experimental full width at half maximum of (0.22 ± 0.05) nm, which corresponds to a Q around 4300. The 33 pillars investigated were selected from a set of 62 as those in which neither the fundamental nor the first three excited modes showed a splitting, which would be an indicative of deviations from circular symmetry. We also chose only pillars which had the fundamental mode emission wavelength inside a 0.22 nm range. We therefore can be reasonably sure of the circularity and regularity of our pillars, an assumption that was subsequently confirmed by scanning electron microscope images of some of the samples.

Figure 1 shows representative spectra for the four lowest energy photonic modes of the 1.5 µm pillars. The modes are identified by the standard name of the guided modes of a circular dielectric waveguide [4, 5]. For all pillars, the fundamental HE₁₁ mode is always the most intense but there is a clear distinction in the spectra concerning the relative intensities of the three higher energy modes. Of the 33 pillars investigated, 70% showed spectra similar to the one shown in Fig. 1(a), with the TE₀₁ mode more intense than the HE₂₁ while the TM₀₁ is weaker. The TM₀₁ mode is seen with relatively high intensity in only one sample (3%), shown in Fig. 1(b). The remaining 27% of the pillars showed spectra such as the one displayed in Fig. 1(c), with the HE₂₁ as the most intense mode of the group of three excited modes.

The experimental spectra can be reproduced theoretically using the CAMFR code [11], which has been widely used to calculate the mode spectra of vertical cavity surface emitting lasers (VCSEL) and similar structures. CAMFR is based on an eigenmode expansion method where the basis states are the eigenmodes of a circular infinite waveguide with homogeneous refractive index. It uses the perfect matched layer method [12] to absorb unbounded modes at the simulation window boundary in order to eliminate parasitic reflections. To obtain the modes in a multi-layer structure such as ours, the boundary conditions are the usual ones: at the interfaces the tangential components of the electric field are continuous and the discontinuity of the normal components is determined by the change in the refractive index. The refractive indices we used are n=3.510 for GaAs and n=2.975 for the Al _0.8Ga_0.2As layers. In this way the scattering matrices [13] are obtained. The scattering matrix for the lateral wall of the pillar allows us to compute the leaky modes, which are used to calculate the quality factors of the photonic modes. The mode spectra are obtained from a calculation of the Poynting vector at the pillar top. Convergence is established by increasing the dimension of the basis until a change in the position of the modes of less than the experimental spectral resolution is obtained. We find that a basis of 120 modes is usually accurate enough for our structures. The linewidth of each mode is obtained from a Lorentzian fit.

 

Fig. 1. (Color online) Representative experimental spectra for the set of 1.5 µm diameter micropillars studied. The spectra are normalized at the fundamental mode. The spectrum shown in (a) is characteristic of the major (70%) part of the samples, with the TE₀₁ mode as the most intense of the group of the three first higher energy modes. For the sample shown in (b), although the TE₀₁ mode is still of higher intensity, the TM₀₁ mode is significantly excited, being more intense than the HE₂₁ mode. Only one pillar (3%) is in this category. The type of spectrum illustrated in (c), with the HE21 mode as the most intense of the high energy group, is displayed by the remaining 27% of the samples.

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The quantum dot is modeled as an electric dipole with six degrees of freedom that determine its position and orientation. These can be reduced to four since, by sample construction, the quantum dots are located in the center plane of the cavity. About the polarization of the emission, if one assumes rotational symmetry around the [001] growth axis the fundamental excitonic transition is constituted of two degenerate circularly polarized transitions. Nevertheless, the In(Ga)As/GaAs quantum dot is, in general, asymmetric in shape and composition and is subjected to strain. Due to the electron-hole exchange interaction, the observed transitions corresponding to the excitonic ground states are two orthogonally linearly polarized emissions which are separated by 10’s, or even 100’s, of µeV. Most theoretical calculations predict equal oscillator strengths for these two transitions and experimental observations generally agree with this prediction. However, there have been several reports of excitonic emission from self-assembled quantum dots with a dominant linear polarization. Favero et al. [14] observed in-plane linearly polarized emission from low density InAs quantum dots, with degrees of polarization above 80%. Krizhanovskii et al. [15] reported high degrees of in-plane linear polarization for neutral and charged excitons in InGaAs quantum dots and attributed this result to light-heavy hole mixing due to the quantum dot anisotropy. Polarization along the growth axis has also been seen, in the cleaved edge geometry, for InAs dots capped with an InGaAs layer [16] and for InGaAs/AlGaAs dots grown in inverted tetrahedral pyramids [17].

 

Fig. 2. (Color online) Calculated photoluminescence spectra for the 1.5 µm diameter micropillars. The top shows cross-sections of the electric field profiles for the lowest energy photonic modes. The double arrows labeled 1 and 2 represent X-polarized quantum dot dipoles at two different positions in the cavity. The PL spectrum in (a) is obtained with a dipole located in the horizontal axis, displaced 0.225 µm (0.3 times the pillar radius) from the center, with polarization in the plane with components X and Y of equal intensities. To obtain spectrum (b), the dipole is located 20° from the horizontal axis, 0.225 µm from the center. Its polarization has Y and Z components of equal magnitude and a X component which is 60% of the other two. For spectrum (c), the dipole is located in the horizontal axis, displaced 0.225 µm from the center, with polarization in the plane with components X and Y, with the Y component half the magnitude of the X component. The insets in (b) show the calculated electric field intensity for each mode.

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The top part of Fig. 2 shows the calculated electric field profiles of the four lowest energy photonic modes of the 1.5 µm diameter micropillars. The double arrows labeled 1 and 2 represent quantum dot dipoles at two different positions inside the pillar. It is clear that if the dots are, for instance, X-polarized as represented, a dot in position 1 will be able to excite relatively strongly the TE₀₁ mode and will not excite TM₀₁ since a X-polarized dipole at this position is parallel to the TE₀₁ electric field but perpendicular to the electric field of the TM₀₁ mode. If the same X-polarized dot is located at position 2 the reverse happens. The efficiency of excitation of the HE₁₁ and HE₂₁ modes by this X-polarized quantum dot at position 1 will depend on the exact location of position 1 along the Y-axis, increasing towards the center of the pillar. What is important to note here is that the coupling strengths of a polarized quantum dot with each photonic mode will vary strongly with dot position. It is clear also that if the dot has equal probability of emitting in the X and Y polarizations, i.e., if it is unpolarized, then all four modes will be excited, independently of the dot position. For our 1.5 µm diameter pillars, we estimate a number between 20 and 100 dots randomly located in the central microcavity plane. The relative intensity with which the photonic modes are observed for each pillar will depend on the position and polarization of the dots in this pillar.

The relative mode intensities displayed by the majority of pillars, illustrated in Fig. 1(a), can be reproduced considering equal probability of excitonic dot emission in the X and Y directions, which can be identified with the [110] and [110] crystalline directions. The theoretical spectrum shown in Fig. 2(a) is obtained in this way. Note that the same spectrum can be obtained assuming linearly polarized (along [110] for instance) dots located at different positions inside the microcavity (for example, two dots at positions 1 and 2 shown in the top of Fig. 2). However, for the spectrum shown in Fig. 1(b), the only way to reproduce theoretically the observed relative intensity of the excited photonic modes is to assume a significant degree of [001], Z-polarization for at least some of the dots. It is not possible to obtain a theoretical spectrum with the TM₀₁ more intense than the HE₂₁ mode, as observed experimentally, without assuming a considerable degree of Z-polarization. This is a rare case, as it should be expected. Finally, the spectrum exemplified in Fig. 1(c) can only be obtained theoretically if one assumes a degree of linear polarization along the X ([110]) or the Y ([110̄]) direction, in addition to a certain amount of non-uniformity in the spatial distribution of the dots. For example, the spectrum shown in Fig. 2(c) was obtained assuming a dot located on the X axis (position 2), with X-polarized emission two times stronger than emission along Y. In this way, the coupling with the TM₀₁ and HE₂₁ modes is enhanced relative to the coupling with the TE₀₁ mode, as can be qualitatively understood by inspecting the diagrams at the top of Fig. 2. Naturally, exactly the same theoretical spectrum can be obtained assuming a predominantly Y-polarized dot located at the position labeled as 1 in these diagrams.

The results discussed above imply that the emission of at least some quantum dots inside the pillars is polarized. This can be verified directly looking at the photoluminescence at low excitation power. Under relatively high power excitation such as that used in Fig. 1, 300 µW, the emission lines from single dots within the pillars are saturated and only the photonic modes of the pillars are seen. With low power excitation, ~1 µW, luminescence from single quantum dots can be resolved [5, 18, 19]. Looking at the microphotoluminescence at low excitation power, we do find that pillars that show a spectrum of the type demonstrated in Fig. 1(c) present several linearly polarized dots which are in resonance or close to resonance to the photonic modes. Not all dots are polarized and the number of polarized dots varies from pillar to pillar. The polarization is usually found to be along the [110] crystalline direction, although we also find a few dots with polarization along [110]. The degree of polarization can be as high as 90%. For the spectrum shown in Fig. 1(b), it is much more difficult do verify directly the polarization of the dots. As discussed above, the relative intensities of the excited modes seen for this pillar imply a significant degree of [001], Z-polarization. Lateral detection of the microphotoluminescence of a particular microcavity pillar would be required to directly measure the degree of this [001] polarization. Nevertheless, the relative intensity profile depicted in Fig. 1(b) and reproduced theoretically in Fig. 2(b) is an unmistakable evidence of [001] polarization. Therefore, the relative intensity of the electromagnetic modes of photonic cavities can be used to infer information about the polarization of emission of quantum dots. On the other hand, a control of the polarization of the dot emission could be used to maximize coupling to selective photonic modes.

4. Conclusion

In conclusion, we verify experimentally that the efficiency of excitation of the various photonic modes of microcavity pillars depends significantly on the position and orientation of the quantum dot dipoles that act as the photon suppliers for the cavity allowed modes. We show that the relative intensity of the electromagnetic modes in photonic cavities provides information about the polarization of quantum dot emission. The results show that a percentage of the dots in our pillars has a significant degree of linear polarization, an assumption that has been experimentally confirmed. We demonstrate that the analysis of the relative intensity of the photonic modes allows one to estimate the overall degree of in-plane polarization of the quantum dot ensemble and also to give information on polarization along the growth axis, a quantity which is usually difficult to measure.