Relaxation and condensation kinetics of trapped excitons at ultra-low temperatures: numerical simulation
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ORIGINAL PAPER
Relaxation and condensation kinetics of trapped excitons at ultra-low temperatures: numerical simulation S Som* Department of Physics, Nehru Gram Bharati University, Allahabad, Uttar Pradesh 221505, India Received: 07 February 2019 / Accepted: 09 July 2019
Abstract: This work is a theoretical study of the excitons relaxation dynamics in cuprous oxide at ultra-low temperatures and within a potential trap. Exciton–phonon collisions and exciton–exciton collisions have been included as relaxation processes. Special attention is given to the thermal distribution, cooling process and the condensation of the exciton gas with high number of excitons. It has been done by solving the Boltzmann equation numerically, using MATLAB. In this work, the relaxation behaviour has been analysed for the temperatures between 0.1 and 3 K. It has been found that for the temperatures above 0.1 K the local effective temperatures are coming down to the bath temperatures, i.e. the excitons reach the local equilibrium with the lattice, but for 0.1 K it does not happen. For 3 K, the global distributions do not change at all with time, but for 0.1 K this change is significant. Maybe this thermal loss of excitons for 0.1 K is related to the Bose– Einstein condensation of excitons. Interestingly, for the temperature of 0.1 K the condensate has been formed with sufficiently high number of excitons, for the times larger than 300 ns. Also, the results have been compared with the other theoretical results. Keywords: Relaxation kinetics of trapped excitons; Bose–Einstein condensation of trapped excitons; Numerical simulations of Boltzmann equation; Excitons in cuprous oxide; Exciton–phonon scattering; Exciton–exciton scattering PACS No.: 71.35.Lk; 63.20.kr; 67.85.-d
1. Introduction The exciton, a bound electron–hole pair, is a natural candidate for observing the Bose–Einstein condensation (BEC) due to its boson nature and its light effective mass [1–3]. Cu2O is a well-known semiconductor for realizing the excitonic BEC [4, 5] because it has large binding energy, long lifetime and no electron–hole droplet formed. Cu2O is a direct-gap semiconductor with a band gap of 2.173 eV [6]. The exciton consists of a hole in the highest valence band and an electron in the lowest conduction band. Both have even parity at the zone centre, which results a long excitonic lifetime against the radiative recombination. The excitonic binding energy of 150 meV ˚ . The spin-decorresponds to a small Bohr radius of 7A generate 1 s state is split by the exchange interaction into a J = 1 orthoexciton and a J = 0 paraexciton with a splitting
*Corresponding author, E-mail: [email protected]
energy of D ¼ 12 meV. J is the eigenvalue of the angular momentum. Though phonon-assisted recombination is weakly allowed, the paraexcitons transition is optically forbidden, and the orthoexcitons transition is weakly optically allowed. The motivation of this work is to see the relaxation kinetics of trapped excitons, at ultra-low temperatures and at longer times, and the
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