Photoluminescence Line Shapes of Nanocrystals: Contributions from First- and Second-Order Vibronic Couplings

Abstract

We present a microscopic, parameter-free approach for computing the photoluminescence spectra of a single semiconductor nanocrystal. The method derives exciton-phonon coupling directly from the semi-empirical pseudopotential framework and systematically incorporates both diagonal and off-diagonal interactions, expanded to second-order in the phonon modes. The dipole-dipole correlation function was calculated using a Dyson expansion within the Kubo-Toyozawa formalism, enabling a consistent description of the role of pure dephasing and population-transfer on the photoluminescence spectral features. Applied to CdSe/CdS core-shell nanocrystals, the approach quantitatively reproduces experimental photoluminescence spectra over a wide temperature range, revealing that quadratic phonon couplings account for nearly half of the homogeneous linewidth above 100-150 K, while off-diagonal couplings leading to exciton thermalization play only a minor role and only as T approaches 300K.

Figures (5)

• Figure 1: (a) Schematic of the potential energy surfaces (PES) for the ground state (black line) and excited states (blue lines) of an NC system. The origin of the photoluminescence signal is indicated by the orange-glowing black arrow. (b) Spectral densities for linear diagonal (black solid line), linear off-diagonal (blue solid line), and quadratic diagonal (red solid line) phonon couplings of a $3$ nm diameter CdSe/3 ML CdS core-shell nanocrystal. Note that the linear couplings are plotted against the left y-axis, while the quadratic couplings are shown on the right y-axis. An inset zooms into the phonon energy range of $30-35$ meV to highlight the role of the linear off-diagonal spectral features. The spectral densities are broadened by Gaussian functions with widths of $0.05$ meV.
• Figure 2: Experimental lin2023theory and simulated photoluminescence spectra of a $3$ nm diameter CdSe/$3$ ML CdS core-shell nanocrystal measured at temperatures from $4$K to $290$K. Experimental spectra are shown as green solid lines. Simulated spectra obtained with (i) linear diagonal phonon couplings only (black solid lines, labeled as "1d"), (ii) both linear diagonal and linear off-diagonal phonon couplings (blue dashed lines, labeled as "1d + 1od"), and (iii) both linear and quadratic phonon couplings (red solid lines, labeled as "1d + 1od + 2d") are presented for comparison.
• Figure 3: (a) Full width at half maximum (FWHM) of the homogeneous PL spectrum as a function of temperature for the results shown in Figure \ref{['fig:fig2']}. (b) Simulated dephasing rates as a function of temperature under various exciton-phonon coupling terms.
• Figure 4: Calculated dephasing function $\left\langle F_{n}\left(t\right)\right\rangle$ for the lowest bright excitonic state of a $3$ nm diameter CdSe/$3$ ML CdS core-shell nanocrystal, computed for the different exciton-phonon coupling orders and for three temperatures. At temperatures below $150$ K, the dephasing functions obtained with only diagonal linear exciton-phonon couplings (black solid lines) closely overlaps with the dephasing function obtained using both diagonal and off-diagonal linear phonon couplings (blue solid lines).
• Figure 5: Homogeneous PL spectra at (a) $60$K, (b) $150$K, (c) $290$K obtained using the Kubo-Toyozawa method (solid lines, labeled as "KT") and the conventional second-order cumulant expansion method (dashed lines, labeled as "2nd") for different orders of phonon couplings of a $3$ nm diameter CdSe/$3$ ML CdS core-shell nanocrystal.