4.3.1
β factors in PhCW
The most common metric for assessing the coupling efficiency is theβ factor [104]:
β = Γw g
Γw g+ Γuc
=Γc− Γuc
Γc
(4.11)
whereΓc= Γw g+ Γucis the emission rate of an exciton coupled to the waveguide
band,Γuc= Γr ad+Γnr is the emission rate of an exciton not coupled to the waveguide
band,Γw g is the exciton emission rate into the waveguide mode, Γnr is the non-
radiative exciton recombination rate andΓr ad is the exciton emission rate into the
radiation modes (i.e., non-waveguided modes) as depicted on Fig. 4.11(a). Theβ factor is usually obtained from the lifetimes of QDs in the photonic bandgap and that of QDs emitting into the PhCW band. However, lifetimes of QD ensembles in the bandgap suffer from statistical variations which translate into uncertainties of
the measuredΓuc. A first source of uncertainty lies in the position of QDs in the
PhCW. Indeed, the coupling to radiation modes is position and and dipole-orientation dependent, which can cause into variations ofΓuc by one order of magnitude [98].
Besides, the top collection schemes used in the past [107, 105] for measuringΓuc
do not guarantee that the QD is placed inside the PhCW in which the coupling to radiation modes is different [104], which may lead to overestimations of theβ factor. Thyrrestrup et al. demonstrated a beta factor value of 0.85 by temperature tuning a QD line across the photonic bandedge [106]. However, the result was made possible by the absence of Anderson localized modes caused by a significant disorder. The losses induced by such a disorder may cancel the advantage of enhanced collection.
β factors as high as 0.98 were reported by Arcari et al. [104] by measured Γucfrom
one single QD weakly coupled to the waveguide. This technique has the advantage of better controlling the QD position which is necessarily close to the waveguide channel. However, although variations ofΓucandβ are more controlled, this method does not
account for other sources of uncertainties: first, all QDs may not have the same dipole moment, which translates into uncertainties ofΓr ad. Secondly, QDs may exhibit
statistical fluctuations of their non-radiative recombination ratesΓnr as emphasized
by Wang et al. [98]. In our design, the site-controlled QDs were all positioned at the same place with respect to the waveguide Bloch cell. However, statistical variations of Γr adorΓnr are still possible. A statistical measurement ofΓucis, therefore, necessary
to infer theβ factor with a high certainty.
4.3.2
β factors measurements with site-controlled QDs
4.3.2.1 β factor measurement robust to QD inhomogeneities
In this thesis, the average rateΓucwas estimated from time-resolved PL measurement
of 18 QDs placed in W1 waveguides with r=80nm, i.e., spectrally positioned in the photonic bandgap. These 18 QDs were selected from the 25 QDs placed in 5 nominally identical PhC waveguides: only the QDs in which the trion could be unambiguously identified were analyzed. Notice that this measurement is based on the assumption that the averageΓucis weakly varying when changing the PhC hole size. OnceΓucis
known, theβ factor of 21 trions on resonance with the waveguide band were deduced from their time-resolved PL (TRPL) decay. These decay traces were selected out of those acquired on 47 QDs. Only structures where the trion could be unambiguously identified were measured. The lower ratio of selected QDs is caused by the more difficult identification of QD lines in the presence of FP modes. The spectrum of QDs in the bandgap were acquired by collecting the emission directly from above the QD, and that of QDs coupled to the PhCW band were collected from the coupler.
тwg тuc |e> |g> TLS тnr тrad тuc=тrad+тnr TLS coupled to a waveguide (a) (c) :DYHOHQJWKѤ>QP@ (d) /LIHWLPHќ X >QV@ In bandgap 5HVRQDQW &RXQWV>DX@ Fit /DVHUSXPS 5HVRQDQW with the wg band
In the bandgap 0 2 4 6 8 0 7LPHѬ>QV@ (b) 0 2 3 954 950 946 943 939 &RXSOLQJHIILFLHQF\ћ 3KRWRQHQHUJ\>H9@ 0 :DYHOHQJWKѤ>QP@ 954 950 946 943 939 3KRWRQHQHUJ\>H9@
Figure 4.11 – (a) sketch of a two level system (TLS) representing the QD (left) and a TLS coupled to a semi-waveguide (right); (b) Time resolved PL of trion lines of QD5 in the bandgap (green) and on resonance with the waveguide band (blue); (c) Measured lifetimes of QDs in the bandgap (green) and on resonance with the waveguide band (blue); (d) Extractedβ factors of QDs on resonance with the waveguide band (P = 2μW , T=10K)
Fig. 4.11(b) illustrates the time resolved PL trace of a trion in the bandgap (green curve) and in resonance with the PhCW band (in blue). The TRPL of the QD on resonance shows a biexponential decay. Biexponential decay is typically assumed to come from the interplay between dark and bright states in excitons. However, trions do not have dark states. Rather, the biexponential decay was interpreted as arising from the feeding of the trion states by two channels: the usual fast, phonon-assisted decay and another slow decay channel caused by the trion charge feeding by a nearby impurity [173, 112]. This model is encompassed by the following rate equations:
∂tpph= −γphpph (4.12)
∂tpsl= −γslpsl (4.13)
∂tpX= −γXpX+ γphpph+ γslpsl (4.14)
where pphis the population of excited states feeding the trion via a fast phonon
assisted decay scheme with a decay rateγph, pslandγslthe corresponding parameters
for slow decay channel, and pX is the trion population decaying at a rateγX, which
resonance with the waveguide band. The solution for pX is: pX = pph(0)γph γX− γph (e−γXt− e−γpht)+pC(0)γsl γX− γsl (e−γXt− e−γslt) (4.15)
This result yields a biexponential decay if the rise time, determined byγph, is
neglected. Exponential or biexponential decays were observed both for QDs in the bandgap and on resonance with the waveguide band. Each trion decay curve mea- sured was fitted with either a single exponential or a biexponential. The single ex- ponential fit for the non-resonant trace (green curve on Fig. 4.11(b)) yields the trion decay rateγX= 0.36 ± 0.01ns−1, while the biexponential fit for the resonant QD (the
blue curve on Fig. 4.11(b)) yields the trion decay rateγX = 1.6 ± 0.2ns−1 and the
slow trion feedingγsl = 0.23 ± 0.02ns−1. The measured decay values of each of the
18 QDs in the bandgap are given on Fig. 4.11(c) as green dots and range from 1.9 to 3.2ns. Their measured average yields the average decay rate of trions in the bandgap:
Γuc= 0.43ns−1, with a standard deviation of±0.06ns−1(orτuc= 2.3 ± 0.3ns). The fast
decay ratesγX of each of the 21 trions are shown on Fig. 4.11(d) as blue dots. Due to
the Purcell effect, the average lifetime of QDs in resonance with the waveguide band is reduced. However, some trions still exhibit lifetimes as high as the ones of uncoupled trions, confirming that some QDs have a negligible spatial or spectral overlap with the FP modes. Theβfactors for each QD estimated from these measurements are given on Fig. 4.11(d). The error-bars were estimated from the measured standard deviation ofΓuc; We obtain values up to 0.88±0.02 for r = 54.5nm. Although it is lower than the
best reported value ofβ = 0.98 [104], our measurements take into account statistical variations of the QD decay rate in the bandgap, which, if not properly treated, may lead to overestimations of theβ factors. The lower values of β measured here arises mainly from our rather large measured value ofΓuc= 0.43ns−1compared to that obtained in
[104]:Γuc= 0.098ns−1. Note however that our measurement is comparable to that
obtained in other published results: Chang et al. [93] obtained:Ãuc= 0.40ns−1, Balet
et al. [95] obtained:Ãuc= 0.28ns−1, Luxmoore et al. [174] obtainedÃuc= 0.40ns−1.
The difference between our measure value ofΓuc, and that obtained by Arcari et al.
[104] can arise from a higher non-radiative emission rateΓnr and also possibly from
the dipole moment of trions in our QDs which may not be in plane and would thus benefit less from the bandgap induced reduction of the emission rate.
4.3.3 Optimal coupling of a QD to a semi-waveguide
Using the knowledge acquired in the previous parts, an optimal design for coupling inhomogenously broadened single QDs to a semi PhC waveguide with a perfect reflectivity on one side, connected to a coupler with a reflectivity r can be inferred. First, as noted in other publications [175] the QD should be placed at the maxima of
the photonic Bloch mode.
Second, although the FP resonances may help increasing the spontaneous emis- sion rate into the target waveguide mode, the discretization of the LDOS caused by the coupler reflection is detrimental to achieving a deterministic emission of QDs into the PhCW modes. Retrieving the broadband spectrum of a PhC waveguide requires reducing the coupler reflection r. The FP modes modulation of the waveguide-related Purcell enhancement is given by the FP modes visibility: 2r /(1+ r2). The visibility in blue and finesse in green of a waveguide and a semi-waveguide are plotted on Fig. 4.12(a). Obtaining a visibility of 0.5 requires a reflection coefficient R= r2≈ 0.25 in a waveguide with couplers at each ends, but only R≈ 0.07 in a semi-waveguide. How- ever, in a waveguide, the QD emission is divided into the two propagation directions, which is a disadvantage for realizing on chip photon sources. Besides, the reflection from the closed edge can enhance theβ factor through a positive interference ef- fect [176]. A design for a broadband low reflection coupler restoring a broadband extraction will be exposed in Chapter 5.
A third element of optimization is the group index. In the presence of FP modes, the propagating waveguide modes are recycled several times which enhances the QD emission rate. In this context, coupling the QD to the highest group index is not necessarily a good idea, not only because more light will be lost via propagation losses after the QD emission (see Chapter 3), but also because propagation losses reduce the FP modes Q-factors. The highest achievable Purcell factor is not necessarily reached at the highest group index. This point is especially clear from the measured Q-factors of Fig. 4.6(b) where similar values of the Q-factor are obtained for both high and low values of the group index. On the contrary, if the coupler reflectivity is so low that the waveguide mode recycling is negligible, QDs are coupled only to truly propagating modes and high group indexes will enhance the Purcell factor. However, even in this case, losses after the emission may compromise the advantage of higher extraction rates.
A fourth factor that requires a precise optimization is the spatial and spectral envelope modulation of the FP modes. Notice, that this effect is completely general to semi-waveguides where it cannot be avoided even when r=1. The Purcell enhance- ment of a QD placed at the maximum of Bloch mode, in a W1 PhC semi-waveguide with no reflection from the coupler is given by:
Pw g= 3 4π ng n λ2 A si n(( π a− k)rd) 2 (4.16)
where A is the W1 mode area, n is the GaAs refractive index, ng is the waveguide
mode group index and k is its wavevector. The Purcell factor as a function of energy and QD position along the PhCW is shown on Fig. 4.12(b). It is oscillating in energy with a periodΔω = vgπ/rd and in space with a period: P= a/(1 − ka/π). Obtaining a
broadband and efficient coupling is obtained via a tradeoff between two opposing effects:Δω is larger close to the semi-waveguide edge, but too close to the edge, the Purcell enhancement is zero because there is a destructive interference between the two counter-propagating Bloch modes. Therefore, an optimal position is between 2a and 3a, where the envelope modulation period is larger than 60meV ensuring a Purcell factor around 1 in this energy range. Notice that a higher Purcell effect is obtained near the band edge, although this is achieved within a lower energy range.
Finesse 1 10 100 Visibility 0 0.5 1 R 0 0.5 1 Finesse wg Finesse semi-wg Visibility wg Visibility semi-wg (a) (b) QD position, r d [a] 0.1 1 10 Purcell factor ng=30
Photon energy [eV] 0
5 10 15
1.28 1.3 1.32 1.34
Figure 4.12 – (a) Finesse and reflectivity in a waveguide closed by couplers with intensity reflection coefficients R at each sides and a semi-waveguide closed by a perfect reflector on one side and a coupler with reflection coefficient R on the other side; (b) Purcell enhancement in a semi-waveguide (The red line indicates the disorder induced limit of light propagation: ng = 30, the waveguide mode area used for this
picture: A= 0.3(n/λ)2, with n=3.46.