Simultaneous Existence of Confined and Delocalized Vibrational Modes in Colloidal Quantum Dots

TL;DR

This study reveals the coexistence of confined and delocalized vibrational modes in colloidal quantum dots, which strongly couple to excitons, using advanced spectroscopy and simulation to clarify phonon-exciton interactions.

Contribution

It demonstrates the simultaneous presence of confined and delocalized vibrational modes in colloidal quantum dots and their strong coupling to excitons, advancing understanding of phonon interactions.

Findings

Evidence of coexisting vibrational modes from lineshape analysis

Strong exciton-phonon coupling observed in experiments

Simulation supports the experimental interpretation

Abstract

Coupling to phonon modes is a primary mechanism of excitonic dephasing and energy loss in semiconductors. However, low-energy phonons in colloidal quantum dots and their coupling to excitons are poorly understood, since their experimental signatures are weak and usually obscured by unavoidable inhomogeneous broadening of colloidal dot ensembles. We use multi-dimensional coherent spectroscopy at cryogenic temperatures to extract the homogeneous nonlinear optical response of excitons in a CdSe/CdZnS core/shell colloidal quantum dot ensemble. Comparison to simulation provides evidence that the observed lineshapes arise from the co-existence of confined and delocalized vibrational modes, both of which couple strongly to excitons in CdSe/CdZnS colloidal quantum dots.

Equations

$$J_a(\omega )=A{\omega }^p{e}^{-{\omega }^2/{\omega }_c^2}$$

Full text

Semiconductor quantum dots, structures that confine electronic excitations in three dimensions, are a maturing technology that have potential applications in numerous areas of optoelectronics. These include areas such as single-photon emitters 1, 2, quantum computing 3, 4, 5, and photovoltaics 6, 7.

Though development of practical quantum dot devices is progressing rapidly, most areas of quantum dot optoelectronics suffer from the common issue of vibrational coupling to excitons. Interactions between electronic excitations and lattice vibrations facilitate energy loss through non-radiative decay processes and play a vital role in coherent control protocols 8, 9, 10. Acoustic vibrational modes are of particular importance, since their low energies mediate coupling between the fine-structure of an exciton manifold 11, 12, 13 and comprise the primary dephasing mechanism of interband coherences 14, 15, 16 in quantum dots. Though usually a continuum of modes, acoustic vibrations can assume discrete modes 17, 18 due to size-confinement in colloidal quantum dots 19, 20, 21, 22. However, the two pictures of acoustic vibrations as a bulk-like phonon continuum or discrete spherical harmonics are seldom considered simultaneously.

Vibrational coupling to excitons in quantum dots has been studied extensively, but most spectroscopic studies have utilized linear techniques such as absorption and photoluminescence. Linear spectrocopies encounter two main obstacles. First, ensembles of all types of quantum dots exhibit inhomogeneous broadening (due to dot size dispersion) of their absorption and emission profiles. Linear techniques only provide the inhomogeneous lineshape of a quantum dot ensemble that reflects its size distribution, and are largely insensitive to its microscopic dynamics. Second, single quantum dot spectroscopy studies that circumvent inhomogeneous broadening have inherent limitations such as time-resolution and dot-to-dot structural variations. These limitations have restricted the focus of most experimental studies to properties of discrete vibrational modes, since signatures of continuum mode coupling are generally weak. Studies that do observe continuum modes are heavily influenced by spectral diffusion at timescales shorter than the integration time 23, and vary greatly between dots 24.

A technique capable of extracting the homogeneous response of an inhomogeneously broadened ensemble is multi-dimensional coherent spectroscopy (MDCS) 25, which correlates absorption, intraband (Raman) coherence 26, and emission spectra. MDCS has recently been applied to a variety of quantum dot systems, including interfacial 27, 28, self-assembled 29, 30, and colloidal 31, 32, 33, 34, 35, 36 dots, to reveal physics normally obscured by inhomogeneous broadening and/or single-dot experiment limitations. MDCS provides material properties that are both ensemble-averaged and size-resolved for all resonance frequencies within the excitation bandwidth.

In this Letter, we use MDCS to study an ensemble of core/shell colloidal quantum dots (CQDs) at cryogenic temperatures. The third-order response that we measure via MDCS enhances vibrational lineshapes due to phonon coupling, allowing for a sensitive, direct characterization of acoustic phonon coupling in the material. We not only characterize coupling to discrete acoustic modes related to the dot geometry 18, but also observe strong features characteristic of coupling to a continuum harmonic bath. We attribute this bath to continuum acoustic modes of the quantum dot lattice 37, 38, and argue for the coexistence of both a bulk-like continuum and discrete torsional vibrations. By comparison to simulation, we characterize the continuum mode spectral density and coupling mechanism.

The sample is CQDs composed of a 2 nm mean radius CdSe core and 2.5 nm mean thickness CdZnS shell (shown in Fig. 1a), whose synthesis is detailed elsewhere 39. To study their properties at cryogenic temperatures the CQDs are dispersed in heptamethylnonane, which forms a transparent glass at temperatures below 100 K and is liquid up to room temperature. The colloidal suspension is diluted to an optical density of 0.3 at the room-temperature 1S exciton absorption peak.

The finite size of the CQD geometry introduces discrete vibrations 17, 18. These vibrations may be separated into the two classes of acoustic and torsional modes, which have longitudinal and transverse character respectively. However, CQDs embedded in a matrix may display an acoustic continuum characteristic of bulk materials 38 if the sound velocity mismatch between the CQD and surrounding matrix is small 37, since boundary conditions at the CQD surface may be modified such that vibrations of the combined matrix and embedded sphere must be jointly considered.

MDCS is ideal to investigate acoustic phonon coupling in CQDs for two main reasons. First, the ensemble-averaged homogeneous response may be retrieved in the presence of inhomogeneity as a function of resonance energy (corresponding to radius in CQDs). Second, vibrational lineshapes are enhanced by nonlinear spectroscopies such as MDCS.

A MDCS spectrum is generated by Fourier transforming a four-wave mixing signal along a combination of the two delays $\tau$ and $T$ between three excitation pulses and the evolution time $t$ after the last pulse. We acquire single-quantum spectra 40, 36 by Fourier transforming along $\tau$ and $t$ 41, which correlate absorption and emission spectra of the sample. Rephasing single-quantum spectra circumvent inhomogeneity by rephasing the evolution of excited coherences (due to a phase-conjugated first pulse 40), as reflected in the negative absorption energies $\hbar\omega_{\tau}$. Cross-diagonal slices (taken perpendicular to the diagonal line $|\hbar\omega_{\tau}|=|\hbar\omega_{t}|$) provide the homogeneous third-order response.

To perform MDCS, we use a Multi-Dimensional Optical Nonlinear Spectrometer (MONSTR) 41. 90 fs pulses at a 250 kHz repetition rate are split into four identical copies that are independently delayed in time and arranged in the box geometry. An excitation intensity of 4 W/cm2 generates a predominately third-order response as verified by the power-dependence of the heterodyned signal. All pulses are co-linearly polarized and centered at wavelength 605 nm.

Single-quantum spectra are shown in the top row of Fig. 2 and have three main features. First, a narrow zero-phonon line is present along the diagonal, which corresponds to absorption and emission at the same energy. Second, a prominent broad pedestal around the zero-phonon line 42, 43, which has the characteristic lineshape of localized excitons coupling to an acoustic phonon continuum bath 44, 45, grows with increasing temperature. At these low temperatures, the pedestal is asymmetric due to higher probability of emission than absorption of vibrational energy. Third, the acoustic phonon pedestal features two peaks next to the zero-phonon line (seen more clearly in Fig. 3). Although the zero-frequency cutoff of the spectral density may result in a sharp feature on the Stokes-side ($\hbar\omega_{t}-\hbar\omega_{\tau}<0$) of the zero-phonon line, we cannot explain the anti-Stokes ($\hbar\omega_{t}-\hbar\omega_{\tau}>0$) peak (marked by red arrows in Fig. 3) through solely an acoustic phonon continuum (see Supplemental Info).

We explain these results by proposing that discrete torsional modes related to the quantum dot geometry exist in conjunction with continuum longitudinal acoustic modes related to the crystal lattice. We were unable to find values for the low-temperature (glass phase) sound velocity for heptamethylnonane, though from its room temperature (liquid phase) speed $1.285\times 10^{5}$ cm/s 46 it is plausible that in solid form its longitudinal sound velocity may increase three- to four-fold and become comparable to the CQD longitudinal sound velocity (see Supplemental Info). In this case longitudinal vibrations may propagate into the surrounding glass matrix with minimal reflection at the dot boundary, giving rise to a continuum of longitudinal acoustic phonon modes. Simultaneously, transverse torsional modes involving no radial displacement into the surrounding matrix are supported by the dot geometry.

To investigate the validity of this interpretation, we simulate the acquired single-quantum spectra according to the following model. A discrete transverse torsional mode “dresses” the ground and excited electronic states to generate ladders of states separated by the torsional mode energy 47 (shown in Fig. 1b). By calculating allowed torsional mode energies (indexed by $\ell$ and $n$) according to our material parameters, we find that the $(\ell,n)=(2,0)$ mode matches the features of our experimental spectra (see Supplemental Info). We note that the higher-frequency $(\ell,n)=(1,0)$ mode, which involves oscillations of a core and shell layer in opposite directions, should couple weakly to the type-I CdSe/CdZnS quantum dots studied here that confine both carriers in the CdSe core 39. The transition strengths between these states are determined by the Huang-Rhys parameter $S$ 48. The energies of these states are then modulated by a continuum bath of longitudinal acoustic phonons via elastic interactions. Assuming the bath is harmonic, that is the coupling is taken to be linear coupling to a continuous distribution of harmonic oscillators, we can characterize the system-bath interaction by a spectral density function $J(\omega)$ (shown in Fig. 1c). The spectral density may be loosely interpreted as the frequency spectrum of the energy gap modulation by a coupled harmonic bath. For a spherical quantum dot with identical electron and hole localization radii, the spectral density of an acoustic phonon bath:

[TABLE]

may be derived analytically 49, where $A$ characterizes the coupling strength, $\omega_{c}$ is a cutoff frequency that determines the width of acoustic phonon spectral features, and $p$ is an integer that depends on the coupling mechanism ($p=(1)3$ for (deformation potential) piezoelectric coupling) 50. The experimental and simulated single-quantum spectra at temperatures 5, 10, and 16 Kelvin are shown in Fig. 2. Corresponding simulations without torsional mode coupling, which fail to reproduce the anti-Stokes peak, may be found in the Supplemental Info.

For the model described above, analytic expressions are not available to fit experimental lineshapes. We thus emphasize that the goal of our simulations is not to extract numerical values for specific quantities, but rather to elucidate the nature of the underlying microscopic dynamics. The torsional mode energy $E_{(2,0)}$ is calculated 17, while accounting for the core-shell structure of our dots (see Supplemental Info), to be 0.8 meV, and is the value used in our simulation. The torsional mode Huang-Rhys parameter used is $S=0.6$. We were unable to obtain good agreement between experiment and simulation for deformation potential coupling ($p=1$) with the continuum acoustic modes. Although excitons in free-standing few-nm radii CQDs are thought to couple to acoustic vibrations predominately via the deformation potential mechanism 17, our spectra suggest that the delocalized acoustic vibrations considered here couple through the long-range piezoelectric interaction ($p=3$) used in our simulation. Coherences between states $\Ket{g}$ and $\Ket{e}$ are simulated with a dephasing rate $\hbar\gamma$ = 0.4 meV to match the zero-phonon linewidth while coherences involving the vibrationally-excited states $\Ket{\tilde{g}}$ and $\Ket{\tilde{e}}$ dephase more quickly at $\hbar\gamma$ = 1.6 meV. The remaining parameters used for the spectral density are $A=0.47$ ps4 and $\hbar\omega_{c}=1.15$ meV. It should be noted that the simulated spectra in Fig. 2 are obtained after numerically including finite pulse bandwidth effects (see Supplemental Info) for a large inhomogeneously broadened ($\sigma=50$ meV) response function by using the respective experimental laser spectrum at each temperature.

In Fig. 3, we plot cross-diagonal slices centered at energy $|\hbar\omega_{\tau}|=|\hbar\omega_{t}|$ = 2060 meV. Comparison of the single-quantum spectrum slices with the simulated absorption and fluorescence lineshapes plotted inset contrasts the difference in strength (relative to the zero-phonon line) between linear and third-order vibrational lineshapes. The enhancement of vibrational lineshapes in the third-order response is due to additional terms involving the dephasing lineshape function which are absent in the linear response (see Supplemental Info). The nonlinear response of a system may therefore reveal vibrational couplings that are otherwise too weak to observe via linear spectroscopies.

In conclusion, we have observed vibrational lineshapes in the third-order nonlinear response of a core-shell CQD ensemble that indicate simultaneous existence of discrete and continuum acoustic vibrational modes. As a primary mechanism of energy loss and dephasing, understanding acoustic phonon coupling is crucial to the design and implementation of CQDs in optoelectronic devices. In particular, devices based on CQD-doped glasses 51 or superlattices 52 should be designed with an engineered spectral density to minimize energy loss and other dissipative processes due to interactions with acoustic vibrations.

{acknowledgement}

This work was supported by the Department of Energy grant number DE-SC0015782. D.B.A. acknowledges support by a fellowship from the Brazilian National Council for Scientific and Technological Development (CNPq). L.A.P. acknowledges support from FAPESP (Project numbers 2013/16911-2 and 2016/50011-7).