# Rate Equations for a Three-Level Model in Low-Dimensional Exciton Systems

Rate Equations for a Three-Level Model in Low-Dimensional Exciton Systems

This Demonstrations shows a three-level scheme useful for describing the luminescence decays of low-dimensional quantum systems, such as single-walled carbon nanotubes and quantum dots. The excited-state photophysics of these systems can be modeled with an optically active exciton state, namely bright exciton , with an optically forbidden level below, known as dark exciton , separated by a splitting energy (of the order of a few meV). Recombination from both levels to the electronic ground state is recovered by the related rates (mainly nonradiative) and (purely nonradiative), respectively. The inverse quantities give the lifetimes of these excitons.

B〉

|D〉

Δ

B-D

GS〉

Γ

B

Γ

D

Thermalization between dark and bright states can occur through coupling to acoustic phonon modes, involving one phonon absorption and emission process, with upward =n and downward =(n+1) scattering rates, where is the phonon occupation number given by the Bose–Einstein distribution function and is the temperature-independent scattering rate (set to a typical value of 0.05 for simplicity).

γ

↑

γ

0

γ

↓

γ

0

n=1/[exp(/T)-1]

Δ

B-D

k

B

γ

0

-1

ns

The kinetic equations for the bright and dark state populations (t) and (t) as a function of time are given by the following system of coupled differential equations:

P

B

P

D

′ P B Γ B γ ↓ P B γ ↑ P D |

′ P D γ ↓ P B Γ D γ ↑ P D |

This Demonstration implements the numerical solution of this system of equations and lets you explore dynamically the solutions at a given temperature as a function of scattering rates, bright-dark energy splitting, and the initial population of bright state (t). The initial population of the dark state is set to (0)=1-(0).

P

B

P

D

P

B