[Paper Review] Fluorescence Intermittency of A Single Quantum System and Anderson Localization

Xiang Xia, R. Silbey|arXiv (Cornell University)|Jan 5, 2006

Semiconductor Quantum Structures and Devices2 references3 citations

TL;DR

This paper proposes that fluorescence intermittency in single quantum dots and molecules arises from Anderson localization, where random environmental interactions lead to power-law on-time distributions. Using a tight-binding Hamiltonian, it shows universal on-time statistics with a robust exponent of $ m_{\text{on}} = -2 $, explaining temperature-independent blinking via quantum delocalization and tunneling dynamics.

ABSTRACT

The nature of fluorescence intermittency for semiconductor quantum dots (QD) and single molecules (SM) is proposed as a manifestation of Anderson localization. The power law like distribution for the \emph{on} time is explained as due to the interaction between QD/SM with a random environment. In particular, we find that the \emph{on}-time probability distribution behaves differently in localized and delocalized regimes. They, when properly scaled, are \emph{universal} for different QD/SM systems. The \emph{on}-time probability distribution function in the delocalized QD/SM regime can be approximated by power laws with exponents covering $-2\le m <0$. QD/SM switches to a dark (\emph{off}) state when a charge of QD/SM hops into the trap states, which becomes localized after stabilization by the surrounding matrix.

Motivation & Objective

• To explain the origin of fluorescence intermittency (FI) in single quantum dots and molecules as a quantum phenomenon rooted in Anderson localization.
• To resolve the puzzle of temperature-independent power-law on-time distributions with exponent $ m_{\text{on}} \approx -2 $, which contradicts classical diffusion models.
• To develop a microscopic quantum-mechanical Hamiltonian model that does not rely on anomalous diffusion or quadratic potential assumptions.
• To establish a universal on-time probability distribution function across different systems by scaling with characteristic lifetime $ t_0 = 1/\Gamma_0 $.
• To differentiate the on-time and off-time statistics by modeling recovery via quantum tunneling from localized trap states.

Proposed method

• Formulates a tight-binding Hamiltonian with random on-site energies $ \varepsilon_j $ and coupling terms $ v_j $, modeling the QD/SM and its disordered environment.
• Uses zeroth-order Green's functions $ G_d^{(0)}(\omega) = 1/(\omega - \omega_d + i\eta) $ and self-energy $ \Sigma_s(\omega) $ to compute renormalized energy and decay rate $ \Gamma $.
• Derives the on-time distribution $ P_{\text{on}}(t) \propto t^{-1/2} \exp(-\sqrt{4\pi\gamma_0 t}) $, showing power-law behavior at long times with exponent $ -1/2 $, approaching $ -2 $ in the universal limit.
• Models off-time recovery via quantum tunneling, with $ P_{\text{off}}(t) \propto t^{-m_{\text{off}}} $, where $ m_{\text{off}} = 1 + \sqrt{m_d^* E_{\text{ion}} / (m_t^* E_t)} $, dependent on effective masses and dielectric constants.
• Applies scaling theory of localization to show universality of on-time statistics across different QD/SM systems, independent of microscopic details.
• Uses effective mass theory to model wave functions and decay rates, with $ \gamma_{\text{off}}(r) \propto \exp(-2r/b) $, leading to power-law off-time distributions.

Experimental results

Research questions

• RQ1 Why do on-time distributions in single emitters follow power laws with exponents near $ -2 $, independent of temperature?
• RQ2 How can fluorescence intermittency be explained without relying on classical diffusion or anomalous diffusion in energy space?
• RQ3 What is the quantum mechanical origin of the observed universality in on-time statistics across different quantum dot and single molecule systems?
• RQ4 Why do on-time and off-time exponents differ in experiments, contrary to predictions from diffusion-based models?
• RQ5 How does Anderson localization of electronic states in a disordered environment lead to intermittent emission dynamics?

Key findings

• The on-time probability distribution function $ P_{\text{on}}(t) $ exhibits a power-law regime with exponent $ -1/2 $ at intermediate times, approaching $ -2 $ in the long-time universal limit.
• The on-time distribution is universal when scaled by the characteristic lifetime $ t_0 = 1/\Gamma_0 $, independent of system-specific parameters.
• The off-time distribution $ P_{\text{off}}(t) \propto t^{-m_{\text{off}}} $ has an exponent $ m_{\text{off}} = 1 + \sqrt{m_d^* E_{\text{ion}} / (m_t^* E_t)} $, which depends on effective masses and ionization energies.
• The model explains the robustness of $ m_{\text{on}} = -2 $ as a universal feature arising from quantum delocalization and disorder, not from anomalous diffusion.
• The theory predicts that $ m_{\text{on}} $ and $ m_{\text{off}} $ can differ, consistent with experimental observations where $ m_{\text{on}} \approx -2 $ and $ m_{\text{off}} \approx -1.6 $ to $ -2.0 $.
• The stretched-exponential tail in $ P_{\text{on}}(t) $ is a purely quantum effect dependent on the disorder strength $ W/\langle v(r) \rangle $, not classical relaxation.