Light absorption in hybrid silicon-on-insulator/quantum dot waveguides

: We analyze the absorption coefﬁcient of planarized silicon-on-insulator waveguides coated by close packed mono- and multilayers of colloidal PbS/CdS quantum dots (QDs). Experimental data clearly show the inﬂuence of the QDs on the waveguide absorbance around 1500 nm, where we ﬁnd that QDs absorb stronger in thicker layers. To simulate the absorption coefﬁcient of QD functionalized waveguides, the QD layer is replaced by an effective medium with a dielectric function determined by dipolar coupling between neighbouring QDs. Using the host dielectric constant ε h as an adjustable parameter, excellent agreement with the experimental results is obtained. In this way, the increase in absorption cross section with layer thickness can be traced back to an increasing ε h . We argue that this reﬂects the decreasing inﬂuence of the surroundings on the ε h , which therefore evolves from an extrinsic property for monolayers to a more intrinsic ﬁlm property for multilayers.


Introduction
The optical properties of semiconductor nanocrystals show important deviations from their bulk counterparts. Quantum confinement makes the band gap depend on the nanocrystal dimensions, it reduces the quasi continuous energy bands to a set of discrete energy levels [1] -especially at the band edges -and it leads to a relaxation of selection rules, thus allowing for example light absorption by transitions within an energy band [2,3]. As a result, the application of semiconductor nanocrystals in optical or photonic devices is widely investigated, using either self-assembled or colloidal quantum dots (QDs) made by epitaxial growth techniques or wet chemical methods, respectively. Compared to self-assembled QDs, colloidal QDs stand out since they can be synthesized in bulk quantities with low size dispersion and may be combined with various materials technology platforms by straightforward solution-based processing methods. Over the last 5 years, their use has been demonstrated in, e.g., light emitting devices, [4], amplifiers [5], lasers [6], displays and photodetectors [7] and they have been combined with integrated photonics platforms based on silicon-on-insulator (SOI) or silicon nitride [8][9][10][11][12].
In device applications, colloidal QDs are mostly deposited as close packed nanocrystal thin films, either within a layered stack [13] or as a surface coating [8] where they interact with the internal or evanescent optical field, respectively. A typical example here involves QDs embedded in a resonator, where the coupling of the QD light emission to the resonator modes eventually leads to lasing [6]. Essential to the development and optimization of such QD-based devices is a proper understanding of the optical field in materials with embedded or surfacecoated QD films, where the influence of the QD film on the optical field is properly taken into account. In the literature, it is well known that a dilute dispersion of QDs, either in a liquid or a solid host, can be described as an effective optical medium according to the Maxwell-Garnett effective medium theory in the local field approximation [14]. On the other hand, far fewer studies have addressed the effective medium description of close packed QD films, let alone the experimental investigation and theoretical simulation of the optical properties of hybrid materials containing close packed QD films.
In this letter, we analyze the absorption coefficient of SOI planarized waveguides (PWGs) coated with close-packed mono-and multilayers -generally denoted as i-layer with i = 1, 2, ... -of PbS/CdS QDs. We retrieve the fingerprint of the QDs in the waveguide absorbance and find that the absorbance per QD increases with the number of QD layers. The experimental data are compared with simulation results, where the QD i-layers are described as an effective medium in which the optical properties depend on dipolar coupling between neighboring QDs [15]. Close agreement between the experimental values and the simulation results is obtained using the dielectric constant ε h of the QD host material as the only adjustable parameter. We find that the increased absorbance in thicker layers makes that a higher value of ε h is needed to match the simulated and the experimental data. We interpret this as a transition from a regime where the field lines coupling the QDs mainly pass through the surroundings (monolayer case) to a situation where these field lines are mainly confined within the QD stack (thicker multilayers).

Experimental
The PbS/CdS core/shell QDs used in this work were synthesized using an established cationic exchange procedure [16,17]. Core diameter and shell thickness were adjusted to have a band gap absorption at ≈ 1450 nm (see Fig. 1(a)). In this way, the long wavelength side of the absorption peak covers the 1500 − 1550 nm bandpass window of the SOI grating couplers, which are used to couple light in and out of the waveguides. This will make the QD absorption well discernible in the measurements. Attempts of measuring the QD absorption of only core PbS QDs on the waveguides, showed a blue shift of the QD wavelength spectrum. This has been observed previously for PbS QDs deposited on glass substrates [21]. These optically unstable films will hamper a quantitative study of the QD absorption. The growth of the CdS shell allowed for a better passiviation of the PbS core QDs yielding optically stable PbS/CdS particles, which will enable a quantitative comparison between the experimental and theoretical QD absorption.
Using the absorption spectrum of the original PbS QDs and the resulting PbS/CdS QDs, core diameter and shell thickness were estimated to be 5.4 and 0.4 nm, respectively [18,19]. The QDs were locally deposited on the PWGs by combining optical lithography and Langmuir-Blodgett deposition, [20,21] forming strips of 200, 500, 1000, 1500 μm on otherwise identical waveguides (see Fig. 1(b)). The scanning electron microscopy image shown in Fig. 1(c) is indicative of the close packing and locally hexagonal ordering of the QDs on top of the PWG, while atomic force microscopy imaging reveals that the PWGs are depressed by 5 − 10 nm relative to the surrounding silica, with the QD layer conformally following this geometry (see Fig. 1(d)). The use of QD strips with different interaction length on identical waveguides enables us to quantify the light absorption in the QD functionalized sections of the PWGs, regardless of coupling losses (see Fig. 1(b)). Indeed, denoting the absorption coefficient of a bare and QD coated PWG α 0 and α, respectively, the net absorption coefficient α QD = α − α 0 of a QD coated PWG can be derived from the transmitted power P t through PWGs with different QD strip lengths L. More specifically, using the length of and the power transmitted through one of the QD coated waveguides as a reference, the net waveguide absorbance A reads: In the determination of A we neglect any contribution of the QD emission to the measured P t [16].

Results and discussion
The net waveguide absorbance is related to the net waveguide loss (dB) as (10A log e). According to Eq. (1), α QD can be obtained from a plot of A versus the strip length difference ΔL = L − L re f . This is exemplified by Fig. 2(a), which shows A as obtained from measurements on a PWG coated by a PbS/CdS QD 2-layer using the 200 μm strip as a reference. Clearly, A is proportional to ΔL and α QD can thus be obtained as the slope of the best fitting line passing through the origin. The difference between the absorption coefficient thus obtained -6.6(3) cm −1 -and the determined 1.8dB/cm (i.e., 0.41 cm −1 ) loss of an uncoated PWG provides a first indication that the QD coating has a strong influence on the waveguide absorbance. This conclusion is further supported by the wavelength dependence of α QD . As shown in Fig. 2(b), the α QD spectrum for a mono-and a 2-layer coating strongly resembles the absorption spectrum of the PbS/CdS QDs used in a dilute tetrachloroethylene dispersion. Similar results are obtained using films consisting of up to 7-layers (see Supplemental Material [16]). Remarkably, we find a higher absorption coefficient per layer for thicker layers (see Fig. 2(b)), meaning that the absorption cross section of a QD in, e.g., a 7-layer is larger than in a monolayer. To compare the experimental α QD with model predictions, we use an approach where the real QD i-layer covering the PWG is replaced by an effective medium with a dielectric function ε e f f (see Fig. 3(a)). Using the real geometry of the PWG -including the slightly submerged waveguide top surface, coated by a 2 nm thin native silica layer -this enables us to extract a theoretical absorption coefficient α QD,th from the simulated effective refractive indexñ e f f = n e f f + ıκ e f f of the propagating quasi-TE mode: This approach however requires that the dielectric function of each material or medium involved is known. For silicon and silica, we use typical values at 1520 nm of 3.45 and 1.45, respectively.
For ε e f f , we build on the recent finding that the absorption cross section of QDs in close-packed monolayers similar to the ones used here can be well described by taking dipolar coupling between neighboring QDs explicitly into account [15]. As shown in the Supplemental Material [16], this coupled dipole model (CDM) also applies to all PbS/CdS QD i-layers used in this study. Therefore, the CDM applies for core/shell particles as in [15] only the absorption cross section of core PbS and CdSe core-QD monolayers deposited on glass was studied. Moreover, the study of imaginary part of ε e f f through the absorption cross section of QDs in [15] can be extended to yield a generalised expression for ε e f f [16]: Here, N s is the QD surface density in the layer, L t is the thickness of the effective layer, a QD is the polarizability of a single PbS/CdS QD and S is the so-called dipole sum, which sums up the influence of the dipolar field of neighboring QDs on an individual QD in the layer. In general, S is different for fields parallel (S ) or perpendicular (S ⊥ ) to the QD film. However, since the main field component of the quasi-TE modes in the PWG used here lies parallel to the QD film, only S -which was analysed experimentally in [15] -is of relevance here. Opposite from S, which only depends on the position of the particles relative to each other, a QD is a function of ε h and the dielectric function ε QD = ε QD,R + ıε QD,I of the QDs. While we consider ε h as an adjustable parameter in this study, we use calculated values for ε QD,R and ε QD,I [16], taking care that they yield the experimental absorption coefficient spectrum of the QDs in a dilute dispersion while obeying the Kramers-Kronig transformation [22]. Importantly, in this analysis, we assume that the absorption coefficient of the PbS/CdS core/shell QDs at wavelengths shorter than 400 nm can be derived from the bulk dielectric function of PbS and CdS, respectively -as was demonstrated for PbSe/CdSe QDs [19] -and we neglect possible quantization effects in the CdS shell.
Combining the geometry of the PWG cross section and the expression for ε e f f -based on the coupled dipole model and the self-consistently determined ε QD -the electric field of the guided optical mode in the PWG can be calculated, resulting in theoretical values forñ e f f and α QD,th . As an example, Fig. 3(b) represents the electric field at a wavelength of 1520 nm for a PWG covered by a QD monolayer as obtained using Fimmwave 3.4 complex mode solver. The figure clearly shows the overlap between the QD film and the evanescent field, which makes that light absorption by the QDs affects κ e f f and leads to a non-zero α QD,th . The determination of α QD,th would not be possible through a transmission measurement of monolayer coated glass in a spectrophotometer as the absorbance would fall within the noise range. As shown in Fig. 3(c), a close match can be obtained between the simulated and experimental α QD spectrum for a QD monolayer-coated PWG by adjusting ε h . It should be noted that the ε h value needed to match the experimental and simulated α QD somewhat depends on the geometry of the PWG. Looking at the AFM cross section of the PWG (see Fig. 1(d)), an exact value of the height difference l between the top surface of the PWG and its silica cladding is hard to determine. Varying l between 6 and 10 nm as extreme cases, we obtain agreement between experiment and simulation for ε h = 1.0 (l = 6nm) to ε h = 1.16 (l = 10 nm). For QDs capped by oleic acid (ε=2.1 at 2000 nm [23]), both figures are relatively low yet they agree with the ε h = 1.0 found for PbS and CdSe QD monolayers deposited on glass [15].
The approach as outlined above can be readily extended to simulate α QD for PWG covered by QD i-layers. Using once more ε h as an adjustable parameter, correspondence between experimental and simulated values can be obtained as shown by the example of a 7-layer in Fig. 4. Importantly, the increasing QD absorption cross section with thicker layers makes that a larger ε h is needed to fit the simulations to the experimental data when the number of layers increases.
As shown in Fig. 4(b), an ε h in the range of 1.00-1.16 is obtained in the case of a monolayer, whereas values between 1.47 and 1.50 are found for the simulation of the 7-layer using l = 6 and l = 10 nm, respectively [16]. This demonstrates that in the case of a QD i-layer with i close to one, ε h is not an intrinsic property of the QD film. Since the field lines that couple neighboring QDs in a monolayer mainly pass through the surroundings, the combined effect of the layers surrounding the QDs -air and native silica on silicon -and the organic ligands separating the QDs will determine ε h in this case. For thicker layers however, the larger part of these coupling fields remains within the QD film. In that respect, the trend shown in Fig. 4(b) can be interpreted as the progressive evolution of ε h from an extrinsic value, determined by the layer and its surroundings, in the monolayer case to a value that is an intrinsic property of a QD multilayer.
This results are not only essential for a proper understanding of the optical field in QD mono- layers and the of optical development of QD-based devices, but could have an implication for electronic devices. Thicker QD layers, as outlined above, will allow for more absorption per particle. On the other hand thicker QD layers will hamper charge extraction for photovoltaic or detector devices. In addition, the electric field within a QD layer due to an applied voltage accross the layer will be limited by increasing the layer thickness. This means that the opposing effects of increased absorption and limited electric field will lead to a optimum QD thickness for electronic devices.

Conclusion
In conclusion, we have studied light absorption in planarized SOI waveguides functionalized with a top coating of PbS/CdS QD mono-to multilayers. The experimental absorption coefficients can be simulated using an approach where the QD layer is replaced by an effective medium with an effective dielectric function determined by dipolar coupling between neighboring QDs. This approach leaves the host dielectric constant ε h as the only adjustable parameter and provides a generic scheme to model optical properties of composite materials containing close packed QD films. Using ε h to match experimental and simulated absorption coefficients, we find that ε h systematically increases for thicker films. We interpret this as an evolution of ε h from an extrinsic property, both determined by the QD films and the surrounding layers, to a more intrinsic property of the QD layer. To the best of our knowledge, this is the first demonstration of how ε h -a typical parameter introduced in effective medium theories -depends on the dimensions of the layer modeled. Our results are essential for the optimization and development of QD-based devices.

PbS/CdS synthesis
The PbS QDs were prepared using a procedure as described by [24], where the synthesis conditions were chosen such that oleate capped PbS QDs with a first exciton absorption at 1590 nm -corresponding to a diameter of 6.2 nm -were formed (see blue line in Fig. 5). For the CdS shell growth, a cationic exchange procedure was used, starting from a 5.7μM QD dispersion in toluene. The dispersion was heated to 125 • C in a reaction flask placed in an nitrogen atmo-sphere. Cadmium oleate was added in a 20:1 Cd to Pb ratio. This starts a cationic exchange process in which the outer Pb 2+ cations are replaced by Cd 2+ cations, leading to a heterostructure with a PbS core and a CdS shell. The reaction was stopped by quenching with a double amount of ethanol as compared to the reaction volume. After centrifugation and decantation the PbS/CdS Qdots were suspended in toluene. The reduction of the PbS core size by cationic exchange leads to a blueshift of the first exciton absorption. For the PbS/CdS used here, the exciton absorption is shifted to 1450 nm (red line in Fig. 5) leading to particles particles with a total size of 6.2 nm, a shell thickness of 0.4 nm and a size dispersion of 8%.

Absorption coefficient spectra for one to seven layer coated PWGs
We used the method as outlined in the paper to determine the spectrum of the absorption coefficient of PWGs coated with a QD i-layer with i = 1, 2, 3, 4, 5 and 7. All results are grouped in Fig. 6.

Dielectric constant and polarizability of the PbS/CdS core shell Qdots
The dielectric response of a core/shell QD ε QD depends on the dielectric function of the core (ε c ) and the shell (ε sh ). To determine ε c and ε sh , we use the iterative matrix inversion method (IMI) as described in [22] for core particles, but we adapt it to the use of a core/shell system. In this method, a self-consistent ε QD is obtained by minimizing the error between the experimentally measured QD intrinsic absorption coefficient (μ i,exp ) and the intrinsic absorption coefficient μ i,th as calculated according to the Mawell-Garnett mixing rule in the local field approximation, while requiring that the real and imaginary part of ε QD are related by the Kramer-Kronig transformation. μ i,exp can be directly determined from the absorbance A susp of a diluted dispersion of PbS/CdS QDs, where we use tetrachloroethylene (TCE) as the solvent: Here, f susp denotes the volume fraction of the QDs in the suspension, i.e., the ratio between the volume of all suspended QDs and the total volume of the suspension, while L cuv is the cuvette length (1 cm). In the case of PbSe/CdSe core/shell QDs, the correspondence between μ i,exp and μ i,th has been confirmed by [19], using the expression of μ i,th according to [25]: Here, ε s , n s denote the dielectric constant and refractive index of the solvent, respectively, while q is ratio of the shell volume to the total core/shell volume. When implementing the procedure outlined above, we have excluded possible quantization effects in the QD shell and used the dielectric constant of bulk CdS [26] for ε sh . The obtained spectral dependence of ε c,R , ε c,I is shown in Fig. 7(a).

Coupled Dipole Model: Derivation of ε e f f
In the coupled dipole model, we consider a collection of polarizable point particles embedded with a volume density N in a host with permittivity ε h [15]. We can define an effective dielectric constant ε e f f for this composite medium by relating the average dielectric displacement to the average electric field: In the latter sum, the polarization is split into the contribution of the host and the additional contribution of the dipoles p 0 induced on the point particles. The latter term can be rewritten in terms of the polarizability (a 0 ) of the point particles and the local electric field E L that polarizes them: Using the expression derived by [27] for spherically symmetrical core/shell particles, a 0 can be expressed as: Here, p denotes the ratio of the volume of the core and the volume of the whole core/shell particle.
In the case of a close packed layer, the local field that drives an individual dipole i is the sum of the external field and the field of the neighboring dipoles j: A crucial element in writing down equation 9 is the assumption that because of symmetry reasons, E L is the same for all particles: Here, S is the dipole sum which contains all contributions from the neighboring particles on the local field driving a central particle. For an applied field parallel to the QD film (S ), the dipole sum is given by: To obtain a consistent expression for N in the case of QD i-layers, we determine it as the ratio between the QD surface density in the layer N s -which is the total number of QDs in the i-layer per unit of surface area -and the thickness L t attributed to the i-layer. In this study, we take L t as the product of the number of layers i and the spacing between close-packed planes in an fcc stacking of QDs. With d the QD diameter (6.2 nm) and l lig the thickness of the ligand shell (taken as 1.8 nm), we thus have: Using this definition of N and L t , ε e f f is obtained as: In Fig. 7(b), we plot the spectrum of the real and imaginary part of the refractive index as derived from ε e f f for a PbS/CdS QD monolayer with a surface density N s =1.2 10 12 1 cm 2 and taking ε h =1 and L t =8 nm.

Coupled Dipole Model: Multi-layers
The use of the coupled dipole model to analyze the effective permittivity of nanocrystal composites was elaborated by [15] for the case of a single component monolayer. Their analysis showed a remarkable absorption enhancement effect absent in other effective medium models for dilute systems, such as the Maxwell-Garnett mixing rule. However, the experimental and theoretical analysis was limited to monolayers of QDs. As shown in Fig. 8(a), the ratio between the absorbance of a QD i-layer and the number of layers is in a good approximation constant, which means that the same absorption enhancement as for monolayers holds for i-layers. We can thus extend the expression of ε e f f that follows from the coupled dipole model to the analysis of QD i-layers.

Simulated α QD spectrum of the PbS/CdS core shell Qdots
To simulate the absorption coefficient of the QD functionalized waveguides, the QD layer is replaced by an effective medium with a dielectric function determined by dipolar coupling between neighbouring QDs as outlined above. Using the host dielectric constant ε h as an adjustable parameter we match the simulated absorption coefficient to the experimental spectrum. As shown in Fig. 9, a good match between both is obtained for all i-layers, provided that a larger ε h is used when the number of layers increases.  4  7  1  -l  a  y  e  r  ,  2  -l  a  y  e  r  ,  3  -l  a  y  e  r  4  -l  a  y  e  r  ,  5  -l  a  y  e  r  ,  7 l a y e r Fig. 9. The full lines represent the absorption spectrum of the simulated α QD spectrum with the needed ε h values for l = 6nm to match the experimental and simulated absorption coefficient. Similar curves where obtained for l = 10nm.

Contribution of the QD PbS/CdS core shell emission to the measured A and P t
In dispersion, the PbS/CdS QDs used here have a photoluminescence peaking at 1520 nm (see Fig. 10) and a quantum yield of 37%, as determined using an integrating sphere [28]. When deposited on waveguides, part of this luminescence may be coupled to the waveguides, thus raising the transmitted power and lowering the resulting absorption coefficient. In general, the fraction of the absorbed light entering the waveguide in the direction of the detector is determined by the product QY × η coupling of the photoluminescence QY and a coupling factor η coupling , which is the ratio between the power emitted in the waveguide in one of the two directions to the total power emitted by the QDs. By comparing the transmitted power through waveguides with and without QDs, the net QD absorption coefficient α − α 0 is obtained. Clearly, when each QD covering the WGs would re-emit each absorbed photon in the waveguide in the direction of the detector (QY = 1, η coupling = 1), then the QD-emission will fully compensate the QD-absorption and α − α 0 would amount to zero. Otherwise, if QY × η coupling << 1, the QDemission contribution to the measured absorbance will be very low and the measured additional waveguide loss can be directly linked to light absorption by the QDs. As mentioned above, the QY of the PbS/CdS QDs in tetrachloroethylene dispersions used here was determined at 0.37. In general, this QY goes down after Langmuir-Blodgett deposition on silicon [29,30], such that the value of 0.37 should be seen as an upper limit. For determining η coupling , we use an FDTD numerical solver to determine the fraction of the power emitted by a dipole oscillator that is coupled to the waveguide modes. The oscillator is located at a distance of 20 nm from the surface, which should be regarded as an average position of the QD-monolayers relative to the waveguide surface. Along the light propagation direction, the oscillator is put at the center of the waveguide and we find that η coupling amounts to 0.15. Here, we accounted for the fact that only half of the emitted light will be collected at one of the waveguide ends and we averaged over the three spatial directions in which the dipole can oscillate. This shows that the emitted light will amounts to, at its best, 5.6% of the absorbed light. This number falls within the ≈10% error of our α QD measurements, and the contribution is therefore discarded.