Plasmonics DOI 10.1007/s11468-015-0110-4
Plasmonic Control of Spontaneous Emission of Quantum Dots in Sub-Wavelength Photonic Templates Chaitanya Indukuri 1 & Deepika Chaturvedi 1 & Jaydeep. K. Basu 1
Received: 12 July 2015 / Accepted: 7 October 2015 # Springer Science+Business Media New York 2015
Abstract Finite difference time domain (FDTD) simulations were performed on two different plasmonic sub-wavelength photonic templates embedded with CdSe quantum dots. Tunable loading of these templates with plasmonic nano antenna allowed control of the emission from the embedded quantum dots. We discuss how large loading of nano antenna can effectively control the optical density of states for the quantum dots leading to enhancement of their radiative decay rates as observed in experiments. On the other hand, at low level of loading, while FDTD fails to capture the observed enhancement of decay rates in experiment, an alternative mechanism is suggested to exist in such cases. Thus, subtle interplay of multiple mechanisms engineered by appropriate placement and loading of plasmonic nano antenna in such templates is demonstrated as an effective method to control optical density of states and hence spontaneous emission of embedded quantum dots. Keywords Plasmonics . Sub-wavelength structures . Nanostructures . Nanophotonics and photonic crystals . Cooperative emission
Introduction Several artificial templates have been developed to engineer the optical local density of states (LDOS) of quantum emitters [1–6]. Early research focused on photonic crystals and
* Jaydeep. K. Basu
[email protected] 1
Department of Physics, Indian Institute of Science, Bangalore, India
photonic templates to control light matter interaction typically at the length scale of the wavelength λ [7–11]. However, current research is driven, in part, by the need to integrate photonic in to further miniaturize integrated opto-electromechanical structures on the nanoscale. This, quite naturally, has led to increasing demands to develop and demonstrate control of light-matter interactions in general and LDOS in particular at the sub-wavelength scales [12–15]. Polymers, especially, block copolymers (BCPs), are often a unique platform to create sub-wavelength patterns which can not only act as photonic templates, but can also be easily integrated into hybrid opto-electronic structure [16–19]. This has led to increased research on the use of plasmonics especially plasmonic nano antenna assemblies and arrays to explore possibility of LDOS modification [20–22]. Recently, we have shown how such plasmonic nano structures can be combined with a block copolymer photonic template to control photoluminescence (PL) intensity and lifetime of embedded quantum dots (QDs) [23, 24]. Here, we show detailed comparison of the emission properties observed in such template embedded with Au nano particles (Au NPs) and try to model the observed experimental PL intensity variations in terms of the LDOS and PL decay rates. Both LDOS and emission decay rates are calculated using finite difference time domain (FDTD) simulations involving Au NPs and CdSe QDs, representing as dipole emitters, located in two types of complementary BCP templates. The main purpose of using the two templates is to explore nanoscale light-matter interaction through appropriate independent control of respective location, size, and density of the QDs and Au NPs. The two templates also allow two complementary aspects to be explored, namely low and high density of Au NPs. The role of the respective particle distribution can also be changed independently. Experiments reveal, apparently, similar PL enhancement for widely different loading of Au nano antenna, which originates from two different
Plasmonics
mechanisms. Studying the two complementary templates will allow us to explore the subtle aspects of the possible difference between LDOS and multi-emitter interactions in such templates at the nanoscale and its consequences on the emission properties of quantum emitters embedded in such templates. In one type of template, the simulations reveal a decay rate enhancement of the dipoles when separated at large distance from the Au NPs, which were present in low concentrations. Increase of concentration of Au NPs relative to the dipole as well as reducing their mutual separation leads to crossover from weak enhancement to sharp quenching of decay rates. The quenching behavior is similar to our experimental observation [23] and can be understood in terms of dominance of the non-radiative decay rate of the dipole, over the radiative rate, due to the increase in the number of Au NPs as well as the decrease in their separation from the dipoles. The, relatively small, enhancement observed for the low concentration of Au NPs is quantitatively quite far from the experimentally observed PL enhancement of CdSe QDs, and a completely different mechanism for such decay rate enhancement has been suggested in a separate work [23, 25], in terms of a collective emitter model. Varying the concentration of Au NPs in such sub-wavelength photonic templates, thus allows us to, engineer the emission properties of embedded quantum dots from an emitter-dominated regime to the plasmonic-dominated regime. On the other hand, for the complementary BCP template, consisting of an array of air cylinders in PS matrix, we obtain decay rate enhancement at all concentrations of Au NPs used, consistent with our earlier experimental observations [23]. The much larger loading of Au NPs possible in such templates compared to the previous template makes it a template with strong plasmonic-modulated LDOS. Interestingly, although PL of QDs are enhanced in both templates, under certain conditions, we discuss how the actual mechanism seems to be quite different in the two templates and points to the rich physics of nanoscale light-matter interactions at play in these versatile hybrid templates. In the following section, we discuss about FDTD simulations of these templates and also discuss similarities and differences in the extracted parameters like radiative Гr and non-radiative decay Гnr rates of the embedded emitters as well as the optical LDOS. The next section provides details of different methodologies used to perform FDTD simulations on the two different sub-wavelength photonic templates used, while the subsequent sections discuss the main results and detailed comparison with our experimental data on related systems. The final section concludes with the main observations from the simulations, comparison with experimental data, especially highlighting the reasons for failure of the simulation to capture the observed PL behavior of the QDs in one template while modeling the behavior of the PL of the same QDs in the other template, reasonably well.
FDTD Simulations Two types of templates, as used in our earlier experiments [23, 24], were used for the FDTD simulation (Lumerical, Inc, Canada) as indicated in Fig. 1. A Type Template In case of A type template CdSe QDs of diameter 4 and 6 nm are arranged in a hexagonally ordered cylindrical domains in uniform dielectric matrix containing Au NPs of diameter 3 and 6 nm. The dielectric constants of these two domains are almost equal (ϵmatrix =1.6, ϵcylinder =1.58). Hence, we have neglected the small difference in dielectric constants between the two domains of the BCP template in simulation and the template is considered as a medium with uniform dielectric ϵ=1.6 as shown in Table 1. In actual experiments, the cylinders exist as a real morphological feature and also allow preferential location of the QDs and Au NPs in respective mutually exclusive locations. This specificity of NP location is captured in simulations. Figure 1a shows distribution of Au NPs and CdSe QDs in the type A template. The distance between CdSe QDs and Au NPs in this template has been taken from the previous experimental observations and is further specified in Table 1 in terms of A series samples [26]. The total size of the simulation region was (1200 nm)×(1200 nm), corresponding to a unit cell of vertical aligned PVP cylinders in a matrix of PS, of height 160 nm, which corresponds to thickness of A type templates as indicated in Fig. 1a. This represents the average film thickness for these films in experiments [23]. For the same values of volume fraction of Au NPs (ϕAu), the number of Au NPs (NAu) and CdSe QDs (NCdSe) changes with particle size. ϕAu is average quantity obtained from the experiments, where as in simulation NAu and NCdSe are corresponding values in the unit cell. In experiment, it is easy to control and estimate volume fraction where as in simulations, one has control over both number and volume fraction. The NAu:NCdSe values for A series templates for large Au NPs varies from 1:25 to 13:25 for ASL1 to ALL2 samples, respectively. Similarly NAu:NCdSe values for A series templates for small Au NPs varies from 13:25 to 125:25 for ASS1 to ALS2 samples, respectively. Here, NAu, NCdSe are the number of Au NPs and CdSe QDs used in the simulation for one unit cell. NAu:NCdSe values of A series template with small Au NPs are high compared with the A type template with large Au NPs for same ϕAu. B Type Template The B type template has highly ordered hexagonal cylindrical porous lattice in uniform dielectric matrix filled with CdSe QDs of diameter 4.5 and 5.5 nm. These pores are filled with Au NPs of diameter 3 and 6 nm. Geometrically, these two
Plasmonics Fig. 1 Schematic diagram of photonic template used in the simulation. The radius of cylinder R, inter cylinder spacing d, and distance between center of cylinder to QDs r is shown. Figure showing the location of the Au NPs and the CdSe QD a A type template and b B type template. Thickness of the templates for A type is 160 nm and B type is 180 nm
templates are similar but considering the distribution of both Au NPs and QDs, they are complementary to each other. In case of B type template, NAu:NCdSe about 21:1 which is very high compared with the A type sample series. In case of B type template, there is a significant difference in dielectric constant between the domains which makes these templates structurally anisotropic. The type B templates are represented by the B series samples with two variants BN series corresponding to R=20 nm and BH corresponding to R=28 nm both having same value of inter cylinder spacing d=95 nm. The length of the cylinder was fixed at 180 nm corresponding to the typical film thickness used in experiments [24] as also indicated in Fig. 1b. In order to understand the optical properties of CdSe QDs in these plasmonic-photonic templates, especially their PL enhancement, we have performed FDTD simulations. To model the variation of PL of the CdSe QDs embedded in the polymeric templates, QDs are replaced with classical dipoles such that dipole emission matches with QDs PL. The FDTD simulations on the above templates were performed for various configurations, volume fraction of Au NPs (ϕAu) and dipoles to calculate various quantities like, LDOS, or Гr and Гnr decay rates of the embedded emitters using two different approaches, described below. For calculating decay rates, nonuniform meshing is used with mesh size 3 and 2 nm for large Au NP and small Au NP, respectively. In calculating LDOS, uniform meshing is used. All the simulation data presented in this work are normalized with respect to the respective templates without Au NPs (the base templates). For all the simulations, dielectric constants for Au NPs were taken from Johnson and Christy [27].
Method 1 Method 1 is based on calculation of emission decay rates of emitters in the presence or absence of Au NPs. The total or radiative power of the emitters are calculated and used to estimate change in the radiative and non-radiative decay rates. In order to calculate the normalized decay rates, we calculated the electromagnetic power radiated for dipole in homogeneous medium. The total decay rate can be written as [28, 29]. Γ total Ptotal ¼ o Γ rad Po
ð1Þ
where Γtotal, Γorad are total decay rate in photonic template and decay rate in homogeneous medium, respectively, while Ptotal and Pο are the total power emitted by dipole in photonic templates and power radiated in homogeneous medium, respectively. In case of a homogeneous medium, Γtotal =Γorad. To calculate these normalized decay rates, we used two monitors, in the form of boxes to calculate flux of Poynting vector S to find the Prad. One is a small box around the dipole to calculate Ptotal and the other is a box which covers both dipole and the photonic template. By calculating both powers, we have calculated normalized decay rates Γtotal, Γrad. Further, Γnonrad determined using Γrad and Γtotal. So, we calculated modified quantum yield of plasmonic template, Qplasmonic with respect to that of the base template Qbase using [30–32]. plasmonic Qplasmonic Γrplasmonic =Γtotal ¼ base Qbase Γrbase =Γtotal
ð2Þ
Plasmonics Table 1 The geometrical parameters of the two templates used in the simulations like, dielectric constants ϵmatrix and ϵcylinder, R (nm), and d (nm) are average cylinder radius and inter cylinder separation in respective templates. Distance between CdSe quantum dot and mean position of Au NP distribution r (nm). The sample nomenclature scheme is Bi L(S)L(S) α^, where Bi^ represents type of template A or B, first L(S) represents the size of QD, second L(S) represents the size of Au NP, and final term Bα^ represents the magnitude of ϕAu cylinder Sample
ϵmatrix
ϵcylinder
R
d
ϕAu/ϕCdSe
r
AS0
1.6
1.6
20
90
0
0
ASS1 ASS2
1.6 1.6
1.6 1.6
20 20
90 90
0.07 0.21
25 25
ASL1
1.6
1.6
20
90
0.07
35
ASL2 AL0
1.6 1.6
1.6 1.6
20 20
90 90
0.21 0
35 0
ALS1 ALS2
1.6 1.6
1.6 1.6
20 20
90 90
0.07 0.21
25 25
ALL1
1.6
1.6
20
90
0.07
35
ALL2 BNLL0 BNLL1
1.6 1.6 1.6
1.6 1.0 1.0
20 20 20
90 95 95
0.21 0 1.03
35 0 36
BNLL2 BNLL3
1.6 1.6
1.0 1.0
20 20
95 95
3.5 4.3
36 36
BNSL0
1.6
1.0
20
95
0
0
BNSL1 BNSL2 BNSL3
1.6 1.6 1.6
1.0 1.0 1.0
20 20 20
95 95 95
1.03 3.5 4.3
36 36 36
BHLL0 BHLL1 BHLL2 BHLL3
1.6 1.6 1.6 1.6
1.0 1.0 1.0 1.0
28 28 28 28
95 95 95 95
0 1.48 4.7 5.18
0 36 36 36
BHLS0 BHLS1 BHLS2 BHLS3
1.6 1.6 1.6 1.6
1.0 1.0 1.0 1.0
28 28 28 28
95 95 95 95
0 1.48 4.7 5.18
0 36 36 36
where Γplasmonic , Γplasmonic are radiative and total rates in plasr total base monic template and Γbase r , Γtotal radiative and total decay rates in base template, respectively. The Γrad and Γnonrad are calculated for different dipoles placed at various positions inside the cylinder and taking the average of these decay rates. Similarly, for the same positions of the dipoles, average decay rates are calculated for base samples. Method 2 Alternatively, we have calculated the LDOS inside the considered templates and estimated the rate of power radiated by the dipole, representing QDs, in the presence or absence of Au NPs. The radiative and non-radiative rates are estimated from the LDOS using formalism given by [33], to calculate emission properties of the quantum emitters in these templates. The
decay rates of quantum emitters in presence of arbitrary shaped Au NPs in the semi-classical theory described by [34, 35]. To calculate decay rates, we have to find electric field from the frequency ω, dependent Green’s dyadic function Gο(ro,ro,ω) [34–36]. These components of Green’s function are calculated by finding the self field of a dipole at its dipole position in space along the three orthogonal Cartesian coordinate systems. The partial LDOS along any orthogonal direction Bi^, ρi(ro,ω) can be obtained from imaginary part of the Green’s function [36]. ρi ð r o ; ω Þ ¼
6ω ℑGii ðro ; ro ; ωÞ π c2
ð3Þ
The total LDOS ρt(ro,ω) is then, ρ t ð r o ; ωÞ ¼
2ω ℑ½T r ðGo ðro ; ro ; ωÞ π c2
ð4Þ
where c is the electromagnetic wave velocity in free space. The diagonal terms represent independent emitter emission dominated by LDOS. The diagonal term in Go(ro,ro,ω) represents the inter emitter interaction among many emitters directly by virtue of Au NPs present in the plasmonic templates. The enhancement of decay rates of QDs in this template is calculated from ratio of LDOS with Au NP loading in the template with respect to the base template LDOS. Even though we used two different methods for calculation of optical properties of quantum emitters in these plasmonic templates, these two methods are related. In case of method 1, we have calculated radiative decay rates as, Γ Plasmonic PPlasmonic ¼ Γ Base PBase
ð5Þ
where PPlasmonic and PBase are power emitted by dipole in Au NPs loaded BCP template and base BCP template, respectively. ΓPlasmonic and ΓBase are decay rates inside template and base template, respectively. In case of method 2, we calculated radiative rate enhancement by Fermi golden rule through LDOS. LDOS is related to the imaginary part of the dyadic Green’s function as given in Eq. 3. From Poynting theorem, the rate of energy dissipation is related to Green’s function and dipole power P as [36], P¼
ω3 μ2 ℑGðr; r; ωÞ 2 c2 ε 0 ε r
ð6Þ
where μ is dipole moment, ε0 and εr are dielectric constants of air and a dielectric medium, respectively. In the case of a situation where P can be separated clearly in to a radiative and non-radiative component, the corresponding Γrad and Γnonrad can also be calculated. However, if this distinction does not exist then it is difficult to clearly separate Γrad and Γnonrad from ρ. However, the primary motivation of this
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study is to show how the emission properties of quantum emitters located in photonic templates can be modified by incorporating Au nano antenna embedded in discrete locations of the templates. In this context, using the LDOS perspective to calculate emission modification of quantum emitters appeared to be more insightful to us. Further, it turns out, it is more convenient to use LDOS formalism in such a highly heterogeneous photonic-plasmonic environment. But in case of A type template where LDOS variation is weaker, calculating radiative and non-radiative decay rates from LDOS is nontrivial. Hence, we calculated decay rates from the emitted power of dipoles placed in different environments and called it method 1. For consistency, we used this method for B type template also. The PL enhancement can be modeled by calculating ab 2 sorption enhancement F abs ¼ EEο where E and Eο are the electric fields at dipole position with Au NP and without Au NP, respectively, and emission enhancement Fem(ωem) in the presence of Au NPs, as F em ðωem Þ ¼
ρrPlasmonic QPlasmonic ρbase e Qbase
ð7Þ
where ρrPlasmonic and ρbase are LDOS in presence of Au NP and base template, respectively. F FDTD ¼ F abs ðωex Þ* F em ðωem Þ
ð8Þ
Here FFDTD is enhancement factor which quantifies the enhanced spontaneous emission of QDs in presence of Au NPs. This calculated enhancement from FDTD was compared Au where to the experimental enhancement defined as F exp ¼ IIbase IAu and Ibase are peak of PL spectra of QDs in presence of Au NP and base template, respectively. We have applied these two methodologies to calculate radiative and non-radiative decay rates of emitters to represent the observed PL enhancement or quenching of CdSe QDs due to presence of Au NPs in two broadly different templates used in our earlier reported experiments [23, 24].
Results and Discussions In Fig 2a, b, we show the variation of normalized Γr and Γnr, respectively, as a function of Br^ for larger Au NPs (ALL and ASL series samples). The radiative rates do not show much variation with r and ϕ within the parameter space explored. Similar is the case with non-radiative decay rates. The radiative and non-radiative decay rates for the smaller Au NPs (ASS and ALS series) are shown in Fig 2c, d, respectively. In this case as well, Γr ~1 and is almost independent of r. However, the Γnr shows strong increase with decrease in r especially for r
