Coupling of Excitons and Discrete Acoustic Phonons in Vibrationally

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Coupling of Excitons and Discrete Acoustic Phonons in Vibrationally Isolated Quantum Emitters Florian Werschler,† Christopher Hinz,† Florian Froning,† Pascal Gumbsheimer,† Johannes Haase,† Carla Negele,‡ Tjaard de Roo,‡ Stefan Mecking,‡ Alfred Leitenstorfer,† and Denis V. Seletskiy*,† †

Department of Physics and Center for Applied Photonics, University of Konstanz, P.O. Box 696, D-78457 Konstanz, Germany Department of Chemistry, University of Konstanz, P.O. Box 737, D-78457 Konstanz, Germany

‡

S Supporting Information *

ABSTRACT: The photoluminescence emission by mesoscopic condensed matter is ultimately dictated by the finestructure splitting of the fundamental exciton into optically allowed and dipole-forbidden states. In epitaxially grown semiconductor quantum dots, nonradiative equilibration between the fine-structure levels is mediated by bulk acoustic phonons, resulting in asymmetric spectral broadening of the excitonic luminescence. In isolated colloidal quantum dots, spatial confinement of the vibrational motion is expected to give rise to an interplay between the quantized electronic and phononic degrees of freedom. In most cases, however, zerodimensional colloidal nanocrystals are strongly coupled to the substrate such that the charge relaxation processes are still effectively governed by the bulk properties. Here we show that encapsulation of single colloidal CdSe/CdS nanocrystals into individual organic polymer shells allows for systematic vibrational decoupling of the semiconductor nanospheres from the surroundings. In contrast to epitaxially grown quantum dots, simultaneous quantization of both electronic and vibrational degrees of freedom results in a series of strong and narrow acoustic phonon sidebands observed in the photoluminescence. Furthermore, an individual analysis of more than 200 compound particles reveals that enhancement or suppression of the radiative properties of the fundamental exciton is controlled by the interaction between fine-structure states via the discrete vibrational modes. For the first time, pronounced resonances in the scattering rate between the fine-structure states are directly observed, in good agreement with a quantum mechanical model. The unambiguous assignment of mediating acoustic modes to the observed scattering resonances complements the experimental findings. Thus, our results form an attractive basis for future studies on subterahertz quantum opto-mechanics and efficient laser cooling at the nanoscale. KEYWORDS: Exciton−phonon coupling, discrete acoustic phonons, scattering resonances, colloidal quantum dots, semiconductor−polymer compound, nanoscale opto-mechanics

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the particle−matrix boundary. Indeed, the quantized character of acoustic phonons has been directly or indirectly confirmed via either photoluminescence signals from single20−22 or ensembles14,23,24 of nanoemitters or via ultrafast transient absorption measurements.25,19 In this work we synthesize compound particles consisting of semiconductor cores, which are individually coated by organic polymer shells.26,27 In this way we achieve a model system for a vibrationally isolated individual quantum nanosphere with fully discrete electron− phonon interaction. By carefully analyzing spectral and temporal emission characteristics of multiple nanocrystals, we systematically explore the coupling between two fine-structure exciton states, which is resonantly mediated by the discrete acoustic phonon modes. This ability to select nanoparticles

he interaction between charge carriers and lattice vibrations represents the most elementary coupling mechanism in condensed matter. Attractive properties such as superconductivity and thermoelectricity are just two of the illustrative examples where this phenomenon even dominates the macroscopic physics. Additional richness of electron− phonon interactions arises in mesoscopic structures where either one or both of these subsystems display a discrete spectrum of eigenenergies.1−16 Fully quantized electronic states occur in colloidal semiconductor quantum dots with nanometer dimensions matching the de Broglie wavelength, resulting in a large variety of optoelectronic and photonic functionalities.3,5,7 On a substrate, the discrete low-energy vibrational motion in such nanocrystals is strongly dampened due to the environmental influence in the form of near-field-enhanced radiative heat exchange17,18 or direct mechanical coupling.19 The vibrational decoupling can be achieved by placing the colloidal nanoparticles in a polymer matrix thereby resulting in confinement of the phonons due to an acoustic mismatch at © 2016 American Chemical Society

Received: June 28, 2016 Revised: August 18, 2016 Published: August 23, 2016 5861

DOI: 10.1021/acs.nanolett.6b02667 Nano Lett. 2016, 16, 5861−5865

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Nano Letters

Figure 1. Emission properties of a single CdSe/CdS/PMMA nanoparticle. (a) High-resolution transmission electron micrographs of a CdSe/CdS quantum dot with radius RCdS of 4.5 nm (dashed circle, top) and CdSe/CdS/PMMA hybrid particle (bottom). (b) PL spectrum of a hybrid particle at a temperature of 5 K, resonantly excited with an average intensity of 18 kW/cm2 at a wavelength of 570 nm. The zero-phonon line (ZPL) structure results from an exciton doublet consisting of an allowed (A) and a nominally dipole-forbidden (F) emission peaks at wavelengths of 609.2 and 609.6 nm, respectively. Multiple acoustic phonon sidebands are resolved on the Stokes side of the ZPLs, in agreement with the calculated vibrational eigenenergies of the inorganic center, labeled El,n (vertical dashed lines). (c) Time-resolved (black) second-order intensity correlation function g(2)(τ) with strong antibunching signature (g(2)(0) = 0.05 ± 0.01) proves the singular nature of a quantum emitter (least-squares fit shown in red). (d) A sketch of an inorganic core/shell quantum dot of radius (RCdS) where blue and red lines represent radial distributions of the squared deformation amplitude |ul,n|2 for the three low-energy vibrational modes (0,1), (2,1), and (2,2) of a uniform sphere with unclamped boundary conditions. The diagrams on the right show cross-sectional representations of the spatial symmetries of those eigenmodes, color-coded to match the schematic on the left. Solid black lines: position of the stationary sphere. Solid gray line: stationary position corresponding to an additional radial node of the (2,2) mode. Dashed black and gray lines: amplitude extrema of the deformation. Vectors depict velocities associated with the positions marked with solid lines; green and red arrows mark velocities shifted by a half-period of the vibration.

interferometer. A strong reduction of the signal at zero time delay is characteristic of antibunched photon statistics. This fact proves emission from a single quantum system and therefore a microscopic nature of the spectral substructure. The behavior of the Zeeman splitting under high magnetic field (Figure S3 in Supporting Information) assigns two of the observed lines with the X0 doublet of total angular momentum F = 2. These transitions correspond to the recombination of a “bright” exciton state |A⟩ and a “dark” state |F⟩ with total angular momentum projections mF = ±1 and mF = ±2, respectively.1,2,4,6 To address the origin of the additional PL maxima, the QD is modeled as a homogeneous elastic sphere suspended in PMMA with unclamped boundary conditions. The vibrational eigenenergies El,n are obtained with Lamb’s theory10−14 where non-negative integers l and n represent orbital and radial quantum numbers associated with a particular mechanical mode (l,n) of the deformation amplitude ul,n (see Figure 1d). In Figure 1b, we plot the energy positions for Stokes-shifted sidebands of the bare transitions A and F as expected from the calculated acoustic resonances El,n (vertical dashed lines). Apart from the known mechanical constants of the inorganic sphere,34 only the QD radius of RCdS = 4.2 nm enters those calculations. This value is consistent with the average size of the ensemble of our nanoparticles (compare Figure 1a). The resulting frequencies are in excellent agreement with the additional PL maxima we observe experimentally. Consequently, we argue that the sidebands originate from the interaction of fully quantized electronic and vibrational degrees of freedom.

with desired exciton−phonon coupling strength motivates an extreme limit of quantum opto-mechanics28,29 operating completely at nanometer length and on subterahertz frequency scales as well as laser cooling29−31 at the ultimate singleelectron limit. The center of the hybrid particles26,27 studied in this work consists of a CdSe core with a mean radius RCdSe of 2 ± 0.4 nm and a CdS shell with an outer mean radius RCdS of 4.5 ± 0.6 nm, together forming the crystalline quantum dot (QD, see top of Figure 1a). A protective layer of poly(methyl methacrylate) (PMMA) with a radius varying from 15 to 30 nm encapsulates the QD (bottom of Figure 1a and Figure S1 of the Supporting Information). The large discrepancy in sound velocities between the inorganic center and the PMMA shell sustains high-quality mechanical vibrations in the QD. Additionally, the organic cover32 improves the photochemical stability of colloidal emitters, providing high and uninterrupted photoluminescence (PL) yield even under strong optical pumping.26 Electron−hole pairs are resonantly excited in the inorganic centers by tunable picosecond pulses33 (see Figure S2 in Supporting Information). At low temperatures, the emission spectra of typical CdSe/CdS QDs exhibit two transitions associated with the fine-structure doublet of the uncharged exciton (X0), which is split1 by a characteristic energy ΔEX. In contrast, a typical PL spectrum of our CdSe/CdS/PMMA particles reveals up to six additional narrow lines spanning the relative energy interval from 0.5 to 4 meV (Figure 1b). For exclusion of multiparticle emission, we analyze second-order intensity correlations with a Hanbury Brown and Twiss 5862

DOI: 10.1021/acs.nanolett.6b02667 Nano Lett. 2016, 16, 5861−5865

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we now map out the energy spectrum of the coupling rate γ0 as a function of the fine-structure splitting ΔEX while keeping the vibrational structure fixed? To this end, we evaluate γ0 in multiple emitters that possess varying values of ΔEX while at the same time having a nearly fixed spectrum El,n of the vibrational excitations. First, we preselect hybrid particles with X0 emission wavelengths of (615 ± 6) nm while rigorously ensuring that the energetic positions of the prominent vibrational sidebands E2,2 remain within a 10% interval around the average value of 1.1 meV (see Figure S5 in Supporting Information). In doing so, we choose quantum dots with almost identical core radii RCdSe and outer shell radii RCdS. Within this subset, the fine-structure splitting energy ΔEX still varies significantly as a result of its nonlinear dependence on minor deviations from spherical symmetry1 and the approximate inverse-cubic scaling with the core radius1,5 (see Supporting Information). Out of more than 200 compound particles, which were optically characterized, approximately 35 fit this protocol. The resulting variation of the exciton−phonon coupling rate as a function of the finestructure splitting ΔEX is shown in Figure 3a. A drastic modulation of γ0 is observed with pronounced maxima arising at fine-structure splitting energies of ΔEX = 0.6, 1.15, 1.6, 2.1, and 2.7 meV.

Realizing the identical energy scales for the exciton finestructure splitting ΔEX and the vibrational spectrum El,n, one might wonder how the interaction between these subsystems would manifest in the coupling between the |A⟩ and |F⟩ states. For the exploration of this physics we perform a detailed analysis of the time-resolved PL emission. Figure 2a depicts the

Figure 2. Temperature dependence of PL from CdSe/CdS/PMMA nanoparticles. (a) Spectrally integrated PL emission of a single hybrid particle versus time after photoexcitation at various temperatures, as recorded via time-correlated single-photon counting (see Supporting Information). At low temperatures the signal exhibits a double exponential decay with clearly distinguishable short τS and long τL time constants (corresponding exponential functions marked by dashed lines). (b) Measured temperature dependence of the relaxation rate ΓL = τL−1 (gray-filled circles) together with a fit (red solid line) based on a three-level rate-equation model (eq 1). The inset depicts a schematic of the energy level structure and the relevant rates (wavy lines: radiative rates).

spectrally integrated intensity of a single quantum emitter with ΔEX of 1.7 meV, measured at temperatures T ranging from 4.2 to 50 K. For T < 30 K, a short exponential decay time τS and a long one τL are clearly distinguished. Furthermore, τL shows a pronounced decrease with increasing T. To analyze this dependence, we use a three-level model sketched in the inset of Figure 2b: the relaxation of the X0 doublet to the ground state |GS⟩ occurs via two distinct radiative pathways with intrinsic (bare) transition rates ΓA and ΓF. In addition, the nonradiative coupling between the |A⟩ and |F⟩ states is mediated via the zero-temperature phonon emission rate γ0. Following an analytic solution,21,35 the measured rates ΓL,S = τL,S−1 are given by ΓL , S =

Figure 3. Exciton−phonon coupling mediated by discrete acoustic modes. (a) Phonon emission rate γ0 as measured for each preselected nanoparticle (see text for details) depicted versus fine-structure energy splitting ΔEX (data shown with circles, solid line is a guide to the eye). The spectrum γ0(ΔEX) reveals distinct resonances at the characteristic eigenenergies of the dominant acoustic phonon modes (l = 2,n). (b) Calculated energy spectrum of the transition rate between exciton finestructure states |A⟩ and |F⟩, proportional to γ0 (eq 2). A phenomenological broadening of 0.1 meV is assumed. Insets depict measured and calculated γ0(ΔEX) on extended logarithmic scales with assignment of the resonant vibrational modes, highlighting the agreement between experiment (a) and simulation (b).

⎛ ⎛ ΔE ⎞ 1⎜ γ0 coth⎜ X ⎟ + ΓA + ΓF ⎜ 2⎝ ⎝ 2kBT ⎠ ∓

⎛ ΔE ⎞ ⎞ (ΓA − ΓF + γ0)2 + γ0 2 sinh−2⎜ X ⎟ ⎟⎟ ⎝ 2kBT ⎠ ⎠

Here kB is the Stefan−Boltzmann constant. For T ≫

To gain quantitative insight into this microscopic interaction, we model the exciton−phonon coupling via the deformation potential.16 Within the effective mass approximation, the exciton state is given as a product of electron and hole wave functions taking into account the crystal field splitting and the electron−hole exchange interaction, respectively.1,16 The total wave function ΨF,mF is predominantly s-type (l = 0) with an admixture of d-wave (l = 2) symmetry reflecting the anisotropy of the valence band. By applying an independent boson model,16,36 we analyze the energy spectrum of the coupling rate γ0, which is proportional to the squares of the transition matrix elements of the operator ∇·ul,n, summed over the l and n vibrational quantum numbers:

(1) ΔEX , kB

which is above 40 K for our compound particles (see Figure S4 in Supporting Information), the two rates ΓL,S become virtually indistinguishable owing to nearly equal thermal occupation of the two excited states (eq 1). At very low temperatures ΔE T ≪ k X , the two decay rates ΓL and ΓS approach ΓF and ΓA + B

γ0, respectively. The temperature dependence of ΓL for this specific nanoparticle is summarized in Figure 2b (gray-filled circles). Together with the asymptotic tendency of ΓL at low temperatures, a least-squares fit based on eq 1 (red line in Figure 2b) uniquely determines the coupling rate to be γ0 = (0.084 ± 0.008) ns−1 (see also Supporting Information). Can

γ0(E) ∝

∑ δ(E − El ,n)|⟨Ψ2,±1|∇·ul ,n|Ψ2,±2⟩|2 l ,n

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(2) DOI: 10.1021/acs.nanolett.6b02667 Nano Lett. 2016, 16, 5861−5865

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Nano Letters δ(E) is a Dirac delta function. The radial dimensions of the QD are taken to be RCdSe = 2 nm and RCdS = 5 nm, matching the quantum dot subset used for the measurements presented in Figure 3a. The band offset in the conduction band is 300 meV with type-I band alignment at low temperatures.37−40 Figure 3b shows the computed transition rate (∝ γ0) between |F⟩ and |A⟩ as a function of the relative energy ΔEX, where a numerical estimate of eq 2 includes a phenomenological broadening. The calculation reveals resonant energies at 0.6, 1.1, 1.8, 2.6, and 2.9 meV, arising from the dominant contributions of (2,1), (2,2), (2,3), (2,4), and (2,5) spheroidal vibrational modes (see Supporting Information). These resonances are in excellent agreement with the energies of the measured maxima in Figure 3a. The acoustic modes that mediate the exciton−phonon coupling share the same value of l = 2 with a small contribution of l = 4, as imposed by the selection rules of the studied transition (eq 2). As observed in the experiment (Figure 3a) and verified in the calculation (Figure 3b), the interaction becomes stronger with increasing n due to an increasing overlap between the exciton and vibrational wave functions. To our knowledge, these results constitute the first systematic mapping of excitonic coupling rates mediated by quantized acoustic phonon modes in mesoscopic systems. The microscopic treatment confirms the experiment and reveals the inherent symmetries of exciton−phonon interaction in a fully discrete limit. In summary, we showed that by embedding colloidal quantum dots into individual PMMA shells we achieve a systematic vibrational decoupling of the CdSe/CdS center from the environment. A large discontinuity in sound velocities at the semiconductor−polymer interface renders the hybridparticles high-quality acoustic resonators capable of sustaining multiple discrete vibrational modes. In this way, we can study the dynamics of excitons in a completely zero-dimensional system where both electronic states and vibrational motion are fully quantized. Specifically, we demonstrate that the coupling between the bright and the dark exciton states is strongly enhanced when the eigenenergy of a discrete phonon mode of proper symmetry resonantly matches the fine-structure splitting. In contrast, thermalization of the electronic subsystem might even become slower than interband recombination in a limit where the vibrational spectrum is antiresonant with the energy difference between excitonic states. As a result, we conclude that the optical emission properties of colloidal quantum dots can be tailored by precise control over the radii of the core/shell structures. Vibrationally isolated nanoemitters are also attractive for possibilities of coupling to single plasmonic resonators, motivating novel ultrafast quantum photonic functionalities.41 The nanoscale opto-mechanical system implemented here will enable future studies on advanced topics ranging from nonclassical phononic states to laser cooling of nanostructures in the single-electron regime.

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lines via measurements in external magnetic fields and temperature-dependent photoluminescence measurements, experimental analysis of influence of quantum dot size and shape on the exciton fine-structure, additional details on the calculations using the threelevel rate-equation model mentioned in the main text, and differential equations for the calculation of acoustic phonon eigenenergies for an elastic sphere (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: +49-(0)753188-3072. Present Addresses

(F.F.) Department of Physics, University of Basel, Klingelbergstr. 82, CH-4056 Basel, Switzerland. (J.H.) Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland. (C.N.) Henkel AG & Co. KGaA, Henkelstr. 67, D-40589 Düsseldorf, Germany. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors wish to thank A. S. Moskalenko for helpful discussions. This work was funded by the Deutsche Forschungsgemeinschaft (DFG) through Collaborative Research Center SFB 767. D.V.S. acknowledges support from Marie Curie Zukunftskolleg at the University of Konstanz, Baden-Württemberg Stiftung, and DFG Grant #SE 2443/2-1.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.6b02667. Description of CdSe/CdS/PMMA hybrid particle synthesis, optical characterization of nanoparticles including the specification of the experimental setup, assignment of fine-structure transitions to photoluminescence emission 5864

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