Open Access
Add to BibSonomyAbstract
We present simulation results for optical modes in micro-pillar cavities that were computed with the finite element method and that show good agreement with experimental data. By means of this viable tool various influences on the quality factor of the fundamental mode were calculated: Firstly, the light confinement depends strongly on the absorption of the semiconductor cavity material. Here we were able to determine absolute maximum quality factors achievable in a GaAs/AlAs Bragg micro-pillar cavity. Furthermore, small pillar diameters as well as the inclination of pillar sidewalls show critical features with respect to light confinement. Additional effects of the top and bottom Bragg stacks in the pillar were calculated as well.
©2009 Optical Society of America
1. Introduction
Both the stimulated as well as the spontaneous emission rate of an optically active material can be influenced by a modified light-matter interaction inside a cavity. While the former effect is applied in laser optics the latter is exploited for example in the field of cavity quantum electrodynamics. Micro-cavity modes with high quality (Q) factors are required in order to study cavity quantum electrodynamics phenomena like the Purcell effect [1, 2] or strong coupling [3, 4]. Moreover, recent research activities in applied semiconductor optics focus on single photon sources [5] and devices for quantum information processing where quantum bits stored in quantum dots shall be coupled via cavity modes [6, 7, 8, 9]. Besides micro-cavities created by defects in photonic crystal slabs [4], micro-disks [10] and micro-pyramids [11, 12], micro-pillars exhibit high-Q factors [3, 13] and small mode volumes especially for high index-contrast distributed Bragg reflectors (DBRs) [14]. In order to achieve high-Q cavities one requires a thorough understanding and concideration of all major impacts on light confinement. Therefore it is essential to compute the characteristics of micro-cavity modes. One of the most common methods for these electromagnetical calculations is based on the finite-difference time-domain approach [15, 16]. However, so-called eigenmode expansion [17] or Fourier expansion techniques [18, 19] have been used to model micro-pillars as well.
In this paper we present a new simulation approach which is based on a finite element method. The main advantages of finite element methods for this application are the availability of adaptive grid refinement based on an a posteriori error estimator, and the ability to resolve geometric properties of the pillar such as the tilted sidewalls in the simplicial mesh without the negative side-effects of stair-casing. Edge elements provide “built-in” dielectric boundary conditions that are crucial for a high precision simulation of step index profiles such as the mirror pairs attached to the cavity and due to the mathematical structure of the algebraic problem, very efficient numerical solvers can be used. Relying on this sophisticated model we compute the high impact of optical absorption and manufacturing parameters concerning the Bragg mirror which limit the maximum achievable Q factors for micro-pillars.
After an introduction to this simulation method and a comparison with experimental data we study various influences on the Q factor of the fundamental mode in a Bragg micro-pillar in detail. The simulated micro-pillars consist of top and bottom GaAs/AlAs DBRs and a wavelength-thick GaAs cavity in between. First the absorption of the GaAs material, i.e., the imaginary part of the refractive index is varied resulting in an accurate theoretical upper bound for the maximum achievable Q factor, which depends obviously on the pillar diameter and the number of DBR pairs as well. With respect to the pillar diameter Q factors have been calculated in the range from 1 to 4μm revealing fluctuations in the small diameter range. Additionally, an inclination (0° to 3°) of the pillar sidewalls as it occurs due to fabrication imperfections was simulated for different numbers of DBR pairs. It turns out that the Q factor is reduced seriously for tilted sidewalls but may increase for larger angles again. Furthermore, a strong dependence of light confinement on the number of top and bottom DBR pairs is of course found, demonstrating that Q factors up to about 250 000 for at least 40 DBR pairs are possible in theory.
2. Simulation method
For the simulation described in this paper we used JCMwave’s JCMsuite, a finite element package for solving Maxwell’s equations. We utilize high order finite elements with adaptive mesh refinement. The advantages of applying finite element methods for the problem class investigated here are:
- Resolving the main geometrical features of the structure including the tilted side-walls with the triangular mesh is readily done without the negative side effects such as stair-casing.
- The scattering solver allows for quick and exact solution of the underlying equation, resulting in high-quality results.
- With the eigenmode solver of the software package one can find many different modes in one calculation.
- Adaptive mesh refinement allows for smaller computation times for high precision results, especially when the field is strongly localized.
The geometry of our pillars and the relevant parameters are depicted in Fig. 1. For both the scattering problem as well as the eigenvalue problem, the basic equations are Maxwell’s equations which can be formulated as second order curl curl equation for the electric field
2.1. Scattering Problem
For our numerical simulation we have considered several problem classes: to find the resonance frequency of the cavity and to estimate its response, we excited the structure with plane waves and computed the resulting scattered field. This requires solving a rotationally symmetric 3D scattering problem in an unbounded domain for a given frequency ω [20]. The formulation of a scattering problem relies on an incident field E in exciting a scattered field E sc. The incident field E in has to satisfy Eq. (1) in the computational domain Ω and the scattered field E sc is required to satisfy Maxwell’s Eq. (1) on the exterior ℝn\Ω.
2.2. Eigenvalue Problem
For a closer investigation of the resonances we found in these calculations, we solved an eigenvalue problem on the unbounded rotationally symmetric domain: a time-harmonic ansatz for the vectorial magnetic field E leads to the eigenvalue problem
where Ω is the computational domain, n is the normal vector of the boundary of the computational domain and ω is an eigenvalue corresponding to a resonance frequency. The boundary conditions Eq. (4) and Eq. (5) for this formulation of the problem are decisive for the quality of the simulation. Transparent boundary conditions are required for realistic modelling. Our simulation uses the adaptive perfectly matched layers method (PML) as described in [21, 22] for the realization of the transparent boundary conditions.
Fig. 1. Schematic of a pillar with all relevant parameters. Parameters taken into account here are the sidewall inclination φ, the cavity radius rcavity, the number of mirror pairs above and below the cavity, n bilayers,bottom and n bilayers,top, as well as the etching depth detch.
2.3. Convergence
To examine the convergence of our simulation we chose a pillar with a diameter of 3.3μm. We solved the scattering problem for exciting the structure at the frequency of its fundamental resonance.
Figure 2 shows a plot of the number of unknowns versus the relative error of the field energy within the cavity. This simulation was done using finite elements of degree 2 and adaptive mesh refinement. As can be seen by comparing with the dashed reference line, the convergence rate is close to the order of 3 that we expect for a finite elements degree of 2. In fact, fitting the relative error to a linear polynomial in the least square sense, reveals a convergence rate of 2.815. The convergence of the methods used here has been shown in [23].
Fig. 2. Convergence of the finite element method for a resonance frequency. The solid line represents the relative error and the dashed line represents a reference line with a gradient of -3.
2.4. Simulation Approach
In order to reproduce the experimental setup with a fast and reliable method we found that a scattering setup is best suited for the tasks at hand. We used an axis-symmetric setup with an incoming plane wave from above the structure with different angles of incidence, φ. In order to detect the modes, we determined the energy of the computed time-harmonic electromagnetic field E within the cavity. The field energy of the electric field W el,j on the jth subdomain Dj of the computational domain is defined as
where the energy density wel of the electric field can be obtained from
Since E is computed as complex field, the real part of this dot product is
With a harmonic time dependency most of the parts of the dot product vanish when averaging over one period, yielding the electric field strength within the cavity as:
In Fig. 3 this quantity is plotted as a function of wavelength of the incoming plane wave for two different angles of incidence, φ= 0° and φ= 0.5°. For better visibility the response for φ= 0.5° is enlarged by a factor of 40.
Fig. 3. Simulation (top) and experimental data (bottom) of a micro-pillar with a diameter of 4.8μm and 25 DBR pairs. The inset shows the measured pillar fabricated by focused ion beam milling [9].
Fabrication errors regarding the thickness by molecular-beam epitaxy are in the order of 3%. Within this range the fundamental mode (HE11) at 931.7nm was fitted to the experimental data for perpendicular excitation parallel to the rotational axes (φ = 0°) by adapting the thicknesses of the layer structure, see Fig.3. Another mode for the 0° excitation appeared at 927.75nm and was confirmed by the excitation under 0.5° where modes at 928.4nm and 930.3nm additionally were found. More precisely, the calculated mode at 930.3nm consists of two peaks which agrees with theoretical expectation for the second mode [26]. Comparing the simulation to the experimental data, four different modes are observable in both cases with equal relative spectral distances, e.g., the third and fourth mode are spectrally relatively close together in both cases. The Q factor is discussed below, while the non-overall spectral agreement should be caused by two issues: the actual diameter in the experiment may differ slightly and, with more impact, the measurements were performed at low temperature with lower refractive indices in the semiconductor material while the simulation was based on data at room temperature. Particularly the last issue can also explain why only relative and no absolute spectral distances between simulation and experimental data agree: Absolute spectral positions are very sensitive to the pillar diameter and the refractive index. (With a slightly stronger lateral confinement due to a higher refractive index in the simulation an effectively smaller diameter with a wider spectral spacing of the modes is expected.)
The fast and mathematically proven convergence of this finite element method together with the advantage of an exact resolution of the studied geometries and the good agreement to experimental data, including the splitting of the second mode, reflects the high capability of the described simulation. In the following we will concentrate on the fundamental mode of the micro-pillar and various influences on its Q factor. If not stated otherwise the calculations rely on results of the scattering problem. Q factors are computed by fitting a Lorentzian peak to the calculated energy spectrum as shown in Fig. 3. This method shows very good agreement with the eigenvalue problem.
3. Variation of absorption in GaAs
We found a significant impact of the optical absorption in the GaAs material on the Q factor in our simulation. Generally, absorption in the bulk material at energies below the band-gap is originated by the Urbach tail and defects in the grown material. We studied the dependency of the Q factor of a micro-pillar with 3.3μm diameter and 30 DBR pairs, see Fig. 4. The complex refractive index ñ can be written in general as ñ = n+ik, where the imaginary part k = αλ/(4π) with the vacuum wavelength λ of light and the absorption coefficient α. For the permittivity ε it holds ε= ε 1 +iε 2 = ñ 2 and thus, the relationship between the imaginary part of the permittivity ε 2 and the absorption coefficient α is given by: ε 2 = 2nαλ/(4π).
Since the absorption in AlAs is known to be orders of magnitude smaller than the absorption in GaAs in the relevant wavelength range we only considered a non-zero imaginary part of the permittivity in GaAs, ε 2,GaAs. The material properties of GaAs in the λ-cavity and the DBR layers were chosen identically. For an increased absorption in GaAs, Fig. 4 clearly shows that the upper limit for the Q factor decreases from 417,000 without any absorption down to 68,000 for ε 2,GaAs = 0.00025 (equivalent to α GaAs = 4.659/cm). Hence, the grown material quality is decisive. The absorption in GaAs depends on temperature [27, 28] and may be critical to the exact growth conditions as well. Supposing a realistic absorption in the order of α= 1/cm at a wavelength of 950nm [29] the Q factor for such a 3.3μm pillar would be limited to 200,000. Notice, that Q factors have already been measured reaching values up to 165,000 [13]. This is already very close to the theoretical limit.
For further calculations we used the data for the dielectric function of GaAs and AlAs from a software for ellipsometry [30] (ε 1,GaAs = 12.6, ε 2,GaAs = 0.000075, ε 1,AlAs = 8.8, and ε 2,AlAs =0 for λ= 950nm). Since absorption also depends on the wavelength of light it could be of advantage to design the cavity for longer wavelengths in order to reduce the material absorption. Further (re-)absorption processes in experiment are caused by processing-induced defect states at the surface (see below) as well as the optically active material, usually quantum wells or dots, embedded at the antinodes of the field [31, 32]. The latter reabsorption effect has already been considered and optimized for high-Q pillar cavities by reducing the number of quantum dots in the spectral vicinity of the cavity mode [13, 33].
As we will see in the next section, diameter-dependent calculations showed that at least for up to 30 DBR pairs the Q factor for thicker pillars did not exceed the value for a 3.3μm pillar and thus could be set to the maximum Q factor of the planar structure. And since other Q-reducing effects like surface scattering and surface absorption or sidewall inclinations were neglected so far the values given in Fig. 4 are the maximum achievable Q values for GaAs/AlAs based Bragg pillar cavities with 30 DBR pairs.
4. Q factor dependency on diameter
Four main channels for optical losses can be constituted: (i) the material absorption in the cavity and the DBRs themselves, as discussed in the last section, (ii) absorption due to defect states at the surface, (iii) radiation losses into vertical directions as well as (iv) radiation losses into radial directions including any scattering due to surface roughness. For decreasing diameters the losses into radial directions are supposed to become more and more important since the Q factor decreases and the DBRs are not the limiting parameter.
Fig. 5. Q factors for micro-pillars of different diameters (top). Top and bottom DBRs are equal, i.e., 20 or 30 pairs each. The thicknesses of the layer structure are either matched to the effective refractive index n eff of the fundamental mode or not. The lines are guides to the eye. For small diameters the resonance wavelength of the fundamental mode shifts to higher energies (bottom).
Figure 5 shows the calculated Q factors for the fundamental mode in micro-pillars with diameters in the range from 1 to 4μm. For 20 DBR pairs the Q factor reaches a level of 11,500 for a diameter of 1.5μm and for diameters greater than 2.5μm the Q value shows some kind of saturation at 13,000. This indicates that radial losses for 20 DBR pairs are of minor importance for diameters greater than about 1.25μm. In this case the light confinement is limited by the DBRs in vertical directions.
For 30 DBR pairs a saturation seems to be reached for diameters greater or equal to 3μm, Q = 165,000 for an absorption equivalent to ε 2,GaAs = 0.000075 and Q around 400,000 without any material absorption. For pillar diameters below 1.5μm the Q factor for 30 DBRs (calculated with absorption in GaAs) shows an irregular behavior. Q values fluctuate between some maxima at 30,000 (1.0μm), 52,000 (1.1μm) and 90,000 (1.3μm) while dropping down to 6,500 (1.05μm), 33,000 (1.15μm) and 48,500 (1.33μm) in-between. This Q fluctuation has already been reported for submicron-diameter pillars [18, 19] and could be found in this range as well. It was explained by a coupling of the fundamental mode in the cavity with propagative Bloch modes in the DBRs.
Fig. 6. Color coded plot of normalized electrical field intensity of the fundamental mode in a cross section of the micro-pillar: (a) Micro-pillar with 30 top and bottom DBR pairs and a diameter of 1.33μm. (b-c) Micro-pillars with 25 top and bottom DBR pairs, a diameter of 3.3μm in the center of the cavity and a sidewall inclination of 0° and 2.2°, respectively; in addition to the contour the initial grid (before any refinements) is shown on the right halves.
Figure 6(a) shows the electrical field intensity in a cross section of a relatively low-Q micro-pillar with diameter 1.33μm. Intensity plots in this diameter range (not shown here except for 1.33μm) hardly differ from each other. Nevertheless, substantial energy densities outside the λ-cavity, i.e. in the DBR pairs, are found in the cases of low-Q diameters according to the simulation. Hence, we would agree that this Q factor fluctuation—especially in the limit of small diameters—has its origin in the coupling to lossy (propagating) modes in the DBRs.
For small pillar diameters the resonance of the fundamental mode shifts to higher energies when using the same layer thicknesses, see Fig. 5 (bottom). Extrapolating the trend in Fig. 5 the resonance wavelength of the planar structure is close to the resonance at a diameter of 4μm, i.e., the resonance shift for a 1μm pillar in comparison to the planar structure can be estimated to 20nm. On the other hand, the stop band of a GaAs/AlAs DBR has a width in the order of 100nm. Therefore, some groups designed their layer thicknesses of the cavity length and the DBRs with respect to the effective refractive indices of the fundamental mode in dependence on the diameter [17, 18]. We calculated the effective indices n eff in GaAs and AlAs for these diameters and resonance wavelengths as well and matched the thicknesses to λ/(4n eff) in the DBRs and λ/n eff for the cavity (design resonance wavelength λ in vacuum) but no Q factor improvement could be found in this diameter range, see Fig. 5 (data points are overlapping for 30 DBR pairs).
As already stated, experimental Q values were reported with values up to 165,000 [13] for a diameter of 4μm with a similar number of DBR pairs (32 top DBR pairs). In our simulation we determined Q ≈ 165,000 as well. But for smaller diameters (3μm / 2μm /1.5μm) the Q factor decreased faster in the experiment (137,000 / 62,000 / 25,000 [13]) than in the simulation (165,000 / 110,000 / 76,000, see Fig. 5). To explain this faster decrease some effects which dominate for small diameters have to be taken into account, e.g., surface scattering at the sidewalls due to roughness. Our utilized simulation technique is capable to model surface roughness as well to study such scattering effects. But calculations of this type require a great calculational effort since some statistics have to be made (and hence are beyond the scope of this paper). However, first results suggest that surface scattering rather plays a minor role. Therefore, it seems that absorption in the surface region of the pillar might be much more important for the deterioration of the Q factor in thin pillars than theoretically predicted (Fig. 5). Such an absorption nappe due to defects is, e.g., implemented during the ion bombardment of the fabrication process (focused ion beam or reactive ion etching).
5. Q factor dependency on sidewall inclination
In this section we take advantage of one great convenience of our calculational approach. Owing to an implementation based on finite elements an inclination of the pillar side-walls can be realized easily and more accurately than with a stair-case approximation on the Yee lattice used for finite-difference time-domain simulations [17]. Dependencies of the Q factor on the pillar sidewall inclination have already been reported [17, 18]. Here, we extend these calculations on a wider inclination range since for focused ion beam milled pillars we measured inclinations of up to three degrees. Additionally, we study the impact of the amount of DBR pairs and suggest an explanation based on the electric field distribution.
For our chosen thicknesses of 66.9nm (80.2 nm) for GaAs (AlAs) in the DBR and 269.6nm in the cavity the Q factors of the fundamental mode show their maximum values for the case of non-tilted sidewalls (0° inclination), see Fig. 7 (top). For any angle of sidewall inclination greater than zero, i.e., the diameter at the top end of the pillar is smaller than the diameter at the bottom, the Q value declines. This configuration of inclination was chosen since it develops for pillars fabricated by focused ion beam milling, see inset in Fig. 3 (whereas a fabrication by reactive ion etching can induce a tapered pillar towards the bottom). The decrease of the Q factor turned out to depend critically on the amount of DBR pairs, see Fig. 7 (top): In the range from zero to three degrees of sidewall inclination the Q value of the fundamental mode for 20 DBR pairs drops down slightly by seven percent at 2.2° of inclination relatively to its maximum at 0° while for 25 DBR pairs the Q factor declines tremendously from 57,250 (0°) down to 4,900 at 2.2° (note the breaks in the scale).
Fig. 7. Q factors for micro-pillars with tilted sidewalls for 20, 25, and 30 DBR pairs (top). The pillar diameter is fixed to 3.3μm at the center of the cavity. At the bottom the ratio of electric field energy in the DBR pairs normalized to the energy in the cavity at resonance (949.66 nm) is shown for 25 and 30 DBR pairs. Note the breaks in the scale.
The plotted field intensity depicts the situation. Figure 6(b) shows the 3.3μm pillar with 25 DBR pairs and perpendicular sidewalls in cross section, i.e., the case of highest light confinement. However, a sidewall inclination of 2.2° (Fig. 6(c)) gives an intensity distribution that reveals unambiguously a mode which is located in the top DBR and competes with the considered fundamental mode. This loss channel causes and explains a dramatic decline of the Q value below 5,000 for this geometry. For this case the competing mode can clearly be resolved in the top DBR, here the Q deterioration sums up to over 90%.
Whenever the fundamental mode interacts with higher modes in the DBR stacks the ratio of energy stored in the DBRs and energy in the λ-cavity should change significantly. Therefore, we studied this ratio in detail, see bottom of Fig. 7. It is remarkable that peaks in this ratio appear exactly at the inclination angles where there are dips in the Q plot above, namely at 2.2° for 25 DBR pairs and 0.85° for 30 DBR pairs. In the vicinity of these inclination angles higher modes in the DBRs gradually extract the energy of the fundamental HE11 mode and thus cause optical losses.
The calculations verify that in both cases, small diameters and tilted sidewalls, the same effect should be responsible for Q value fluctuations overlaid by any trend.
6. Q factor dependency on number of DBR pairs
This section is dedicated to the influence of the top and bottom DBRs. Figure 8(a) shows on a logarithmic scale how the Q factor improves with more DBR pairs. (Since the substrate consisted of GaAs like the lower layer of one DBR pair itself there were half a pair less in the bottom DBR than in the top DBR.) The Q factor of the simulated 3.3μm pillar with a present absorption in GaAs as mentioned above increases strongly up to 170,000 at 30 DBR pairs. From there on absorption losses are dominating and the Q value saturates at 250,000. In the absence of any absorption the Q factor is almost the same for up to 25 DBR pairs but then increases further exceeding values of 1,000,000, see Fig. 8(a). These calculations demonstrate that the optical absorption of GaAs around 950 nm is of little importance for micro-pillar cavities of less than 25 DBR pairs (Q ≈ 60,000).
Fig. 8. Q factors for micro-pillars with a diameter of 3.3μm. (a) Variation of top and bottom DBR pairs simultaneously with or without absorption in the cavity material (GaAs). (b) Variation of top DBR pairs with fixed number of bottom DBR pairs.
Figure 8(b) shows results for different top and bottom DBRs. The calculations were done with a fixed number of bottom DBR pairs (29.5 pairs) while the number of top DBR pairs was varied around that value. The Q factor increases continuously for up to 31 top DBR pairs where it saturates at around 168,000. As a consequence it can be concluded that the Q factor crucially depends on the minimum number of top or bottom DBR pairs since this is the main channel for optical losses. Hence, growing some more DBR pairs in the bottom part of the micro-pillar than in the top DBR as often done may improve the light outcoupling towards the measurement set-up but in general more DBR pairs at one end of the pillar cavity do hardly effect the optical confinement.
7. Q factor dependency on etching depth
Finally we study a last geometrical parameter. Due to the fact that the pillars are milled or etched out of a planar layer structure, the top DBR pairs have a diameter equal to the diameter of the λ cavity unless there is a sidewall inclination, see above. But the bottom DBR could remain planar or only partly etched, see for example [34]. The latter group reports about a relatively low Q factor.
Here we investigate the impact of the etching depth into the bottom DBR pairs in detail. The Q factors of 2.0μm diameter micro-pillars without sidewall inclination are calculated with a varied etching depth (measured in units of DBR pairs) into the bottom DBR. Apart from previous calculations now an eigenvalue solver for better time efficiency is used performing with a comparable accuracy as the solver for the scattering problem before. A strong decrease in the Q value is found for few etched bottom DBR pairs, see Fig. 9. For an etching depth of less than five DBR pairs the Q factor drops down to less than 10,000 independent of the total amount of DBR pairs. In all the calculated cases (20, 25 and 30 DBR pairs in total) the Q factor remains almost unaffected if less than ten bottom DBR pairs are left unetched. For the pillar fabrication this parameter can be controlled easily, thus it is advisable to etch away this minimum number.
Fig. 9. Q factors obtained from eigenvalue calculations for micro-pillars with a diameter of 2.0μm. Top and bottom DBR pairs are fixed to 20, 25 and 30, respectively while the etching depth (measured in units of DBR pairs) into the bottom DBR is varied from 0 up to the total number of bottom DBR pairs.
8. Mode volume
Beside the Q values we computed the mode volume of some micro-pillars. This was done by integrating the field intensity within the whole computational domain and normalizing to the maximum field intensity [35]. For 30 DBR pairs we get the following values in units of cubic resonance wavelength in the effective medium: 6.5 (λ/n eff)3 (1μm diameter), 8.7 (λ/n eff)3 (1.3μm diameter), and 20.4 (λ/n eff)3 (2μm diameter). These values are in good agreement with results obtained by other groups [14].
9. Summary and conclusion
We presented a versatile simulation method based on finite elements which we used to calculate resonance modes in micro-pillars. Its reliability was proven by a comparison with experimental data of a 4.8μm pillar and convergence analysis. In particular, various influences on the Q factor of the fundamental HE11 mode were studied:
For an ideal 3.3μm pillar with 30 top and bottom DBR pairs the maximum achievable Q factor was calculated to be 475,000 in the absence of any absorption. Taking a realistic optical absorption of GaAs into account we found Q ≤ 170,000. Furthermore, this value could be found as an upper limit for 30 DBR pairs, since Q factors for thicker pillars did not exceed the Q factor of a 3.3μm pillar. To overcome the natural absorption of GaAs cavities could be designed for resonances shifted further to the infrared.
For a further improvement of the Q factor the number of DBR pairs had to be increased. An upper limit at Q ≈ 250,000 is reached for more than 40 DBR pairs where saturation took place. This limit can be regarded as a maximum Q value for Bragg cavities with a realistic absorption and resonances around 950 nm. However, in real structures, lower values are found, in particular for small pillar diameters. Preliminary calculations seem to indicate that Q is limited by absorption at surface defects rather than roughness scattering.
Furthermore, extensive simulations revealed a decrease in the Q factor for thin pillars and geometries with tilted sidewalls. While pillars with 20 DBR pairs already showed Q saturation at a diameter of 2μm and Q ≈ 13,000 pillars with 30 DBR pairs reached Q ≈ 165,000 for diameters larger or equal to 3μm, both for the case of a realistic optical absorption in the GaAs material. Neglecting absorption, pillar cavities with 30 DBR pairs exceeded Q = 400,000 for diameters greater or equal to 3μm. Beyond the main Q-reducing tendency for small diameters and tilted sidewalls an irregular behavior was found having its origin in an interaction of the fundamental HE11 mode of the λ-cavity with propagating higher modes in the DBR stacks which causes additional optical losses. Since this coupling tremendously depends on the exact diameter or sidewall inclination, fabrication tolerances in the geometry are crucial to succeed with high-Q micro-pillars in the small diameter range. Nevertheless, a simple design rule for the etching depth into the bottom DBR could be obtained.
A multitude of imperfections like sidewall inclination, optical absorption and even surface roughness are crucially affecting the performance of a micro-pillar cavity and have to be understood to optimize the figure of merits like Q factor and mode volume. We presented a versatile simulation based on finite elements to implement and study all the mentioned influences on micro-pillar cavities allowing to calculate upper limits for the Q factors under different conditions.
Acknowledgments
We would like to thank P. Brenner for technical assistance at the FIB and S. Li for growing the layer structure by MBE. This work has been performed within project A2 of the DFG Research Center for Functional Nanostructures (CFN). It has been further supported by a grant from the Ministry of Science, Research and the Arts of Baden-Württemberg (Az: 7713.14-300).
We thank A. Schädle for fruitful discussion, and we acknowledge support by the initiative DFG Research Center Matheon of the Deutsche Forschungsgemeinschaft, DFG.
References and links
1. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
2. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]
3. J. P. Reithmaier, G. Sȩk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature (London) 432, 197–200 (2004). [CrossRef]
4. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature (London) 432, 200–203 (2004). [CrossRef]
5. P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoğlu, “A Quantum Dot Single-Photon Turnstile Device,” Science 290, 2282–2285 (2000). [CrossRef] [PubMed]
6. A. Imamoğlu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum Information Processing Using Quantum Dot Spins and Cavity QED,” Phys. Rev. Lett. 83, 4204–4207 (1999). [CrossRef]
7. G. Ortner, M. Bayer, Y. Lyanda-Geller, T. L. Reinecke, A. Kress, J. P. Reithmaier, and A. Forchel, “Control of Vertically Coupled InGaAs/GaAs Quantum Dots with Electric Fields,” Phys. Rev. Lett. 94, 157401 (2005). [CrossRef] [PubMed]
8. M. Benyoucef, S. Kiravittaya, Y. F. Mei, A. Rastelli, and O. G. Schmidt, “Strongly coupled semiconductor microcavities: A route to couple artificial atoms over micrometric distances,” Phys. Rev. B 77, 035108 (2008). [CrossRef]
9. M. Karl, S. Li, T. Passow, W. Löffler, H. Kalt, and M. Hetterich, “Localized and delocalized modes in coupled optical micropillar cavities,” Opt. Express 15, 8191–8196 (2007). [CrossRef] [PubMed]
10. K. Srinivasan, M. Borselli, T. J. Johnson, P. E. Barclay, O. Painter, A. Stintz, and S. Krishna, “Optical loss and lasing characteristics of high-quality-factor AlGaAs microdisk resonators with embedded quantum dots,” Appl. Phys. Lett. 86, 151106 (2005). [CrossRef]
11. F. M. Weber, M. Karl, J. Lupaca-Schomber, W. Löffler, S. Li, T. Passow, J. Hawecker, D. Gerthsen, H. Kalt, and M. Hetterich, “Optical modes in pyramidal GaAs microcavities,” Appl. Phys. Lett. 90, 161104 (2007). [CrossRef]
12. M. Karl, T. Beck, S. Li, H. Kalt, and M. Hetterich, “Q-factor and density of optical modes in pyramidal and cone-shaped GaAs microcavities,” Appl. Phys. Lett. 92, 231105 (2008). [CrossRef]
13. S. Reitzenstein, C. Hofmann, A. Gorbunov, M. Strauss, S. H. Kwon, C. Schneider, A. Löffler, S. Höfling, M. Kamp, and A. Forchel, “AlAs/GaAs micropillar cavities with quality factors exceeding 150.000,” Appl. Phys. Lett. 90, 251109 (2007). [CrossRef]
14. A. J. Bennett, D. J. P. Ellis, A. J. Shields, P. Atkinson, I. Farrer, and D. A. Ritchie, “Observation of the Purcell effect in high-index-contrast micropillars,” Appl. Phys. Lett. 90, 191911 (2007). [CrossRef]
15. J. Vučkovií, M. Pelton, A. Scherer, and Y. Yamamoto “Optimization of three-dimensional micro-post microcavities for cavity quantum electrodynamics,” Phys. Rev. A 66, 023808 (2002). [CrossRef]
16. Y.-L. D. Ho, T. Cao, P. S. Ivanov, M. J. Cryan, I. J. Craddock, C. J. Railton, and J. G. Rarity “Three-Dimensional FDTD Simulation of Micro-Pillar Microcavity Geometries Suitable for Efficient Single-Photon Sources,” IEEE J. Quantum Electron. 43, 462–472 (2007). [CrossRef]
17. N. Gregersen, T. R. Nielsen, B. Tromborg, and J. Mørk, “Quality factors of nonideal micro pillars,” Appl. Phys. Lett. 91, 011116 (2007). [CrossRef]
18. Ph. Lalanne, J. P. Hugonin, and J. M Gérard, “Electromagnetic study of the quality factor of pillar microcavities in the small diameter limit,” Appl. Phys. Lett. 84, 4726–4728 (2004). [CrossRef]
19. G. Lecamp, J. P. Hugonin, P. Lalanne, R. Braive, S. Varoutsis, S. Laurent, A. Lemaître, I. Sagnes, G. Patriarche, I. Robert-Philip, and I. Abram, “Submicron-diameter semiconductor pillar micro-cavities with very high quality factors,” Appl. Phys. Lett. 90, 091120 (2007). [CrossRef]
20. J. Pomplun, S. Burger, L. Zschiedrich, and F. Schmidt, “Adaptive Finite Element Method for Simulation of Optical Nano Structures,” Phys. Status Solidi B , 244, 3419–3434 (2007). [CrossRef]
21. L. Zschiedrich, R. Klose, A. Schädle, and F. Schmidt “A new finite element realization of the Perfectly Matched Layer Method for Helmholtz scattering problems on polygonal domains in 2D,” J. Comput Appl. Math. , 188, 12–32, (2006). [CrossRef]
22. L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, “Advanced Finite Element Method for Nano Resonators,” in Physics and Simulation of Optoelectronic Devices XIVM. Osinski, F. Henneberger, and Y. Arakawa, eds. SPIE 6115, 164–174 (2006).
23. S. Burger, R. Köhle, L. Zschiedrich, W. Gao, F. Schmidt, R. März, and C. Nölscher “Benchmark of FEM, Waveguide and FDTD Algorithms for Rigorous Mask Simulation,” Proc. SPIE 5992, 378–289 (2005).
24. J. L. Jewell, S. L. McCall, A. Scherer, H. H. Houh, N. A. Whitaker, A. C. Gossard, and J. H. English, “Transverse modes, waveguide dispersion, and 30 ps recovery in submicron GaAs/AlAs microresonators,” Appl. Phys. Lett. 55, 22–24 (1989). [CrossRef]
25. J. M. Gérard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. Thierry-Mieg, and T. Rivera, “Quantum boxes as active probes for photonic microstructures: The pillar microcavity case,” Appl. Phys. Lett. 69, 449–451 (1996). [CrossRef]
26. M. Benyoucef, S. M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, “Correlated photon pairs from single (In,Ga)As/GaAs quantum dots in pillar microcavities,” J. Appl. Phys. 97, 023101 (2005). [CrossRef]
27. B. Monemar, K. K. Shih, and G. D. Pettit, “Some optical properties of the AlxGa1-xAs alloys system,” J. Appl. Phys. 47, 2604–2613 (1976). [CrossRef]
28. D. D. Sell and H. C. Casey Jr., “Optical absorption and photoluminescence studies of thin GaAs layers in GaAs-AlxGa1-xAs double heterostructures,” J. Appl. Phys. 45, 800–807 (1974). [CrossRef]
29. M. D. Sturge, “Optical Absorption of Gallium Arsenide between 0.6 and 2.75 eV,” Phys. Rev. 127, 768–773 (1962). [CrossRef]
30. WVase32 J.A. WoollamCo. Inc., Lincoln, USA.
31. G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, “Vacuum Rabi splitting in semiconductors,” Nature Physics 2, 81–90 (2006). [CrossRef]
32. J. Hendrickson, B. C. Richards, J. Sweet, S. Mosor, C. Christenson, D. Lam, G. Khitrova, H. M. Gibbs, T. Yoshie, A. Scherer, O. B. Shchekin, and D. G. Deppe, “Quantum dot photonic-crystal-slab nanocavities: Quality factors and lasing,” Phys. Rev. B 72, 193303 (2005). [CrossRef]
33. T. Tawara, H. Kamada, Y.-H. Zhang, T. Tanabe, N. I. Cade, H. Gotoh, D. Ding, S. R. Johnson, E. Kuramochi, M. Notomi, and H. Nakano, “Role of Re-absorption Effect to Quality Factor in Quantum-Dot Photonic-Crystal Nanocavities,” in Proceedings of IEEE International Conference on Indium Phosphide and Related Materials (Institute of Electrical and Electronics Engineers, New York, 2007), pp. 490–492. [CrossRef]
34. M. Benyoucef, S.M. Ulrich, P. Michler, J. Wiersig, F. Jahnke, and A. Forchel, “Enhanced correlated photon pair emission from a pillar microcavity,” New J. Phys. 6, 91 (2004). [CrossRef]
35. J. M. Gérard, B. Sermarge, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced Spontaneous Emission by Quantum Boxes in a Monolithic Optical Microcavity,” Phys. Rev. Lett. 81, 1110–1113 (1998). [CrossRef]
