This document discusses the application of BCS theory to the lasing behavior of ZnO nanowires, focusing on the electron-hole plasma created during intense optical excitation. It describes the conditions under which lasing occurs, particularly above the Mott density, and explores the theoretical development of cooper pairs in high-density states. The authors aim to compare their theoretical results with experimental data to enhance understanding of the optical responses in these nanowires, which present an advantage over bulk materials due to their low defect structure.

1. BCS Theory on Lasing ZnO Nanowires A.J. van Lange 1 , M.A.M. Versteegh 1 , J.I. Dijkhuis 1 , H.T.C. Stoof 2 1 Debye Institute for Nanomaterials Science, Utrecht 2 Institute for Theoretical Physics, Utrecht Cooper pairs of an electron and a hole. Required attractive interaction is the screened Coulomb interaction. When we approach T c from above the BCS gap equation is Luminescence Spectra Gain calculations CCD images of a single lasing nanowire with increasing distance from the microscope objective. ZnO nanowires are among the smallest known lasers. In an effort to explain their UV laser action, we have developed a many body theory of the electron hole (e-h) plasma in ZnO. In experiment ZnO nanowires are excited by intense 800-nm 120-fs pulses to create an e-h plasma with high carrier densities. For low carrier densities laser action in ZnO is described by exciton polaritons. In our experiment lasing occurrs above the Mott density, where excitons no longer exist. We study the possibility of a new bound state at high densities: the Cooper pair of an electron and a hole. Our goal is to extract the optical response from the theory and to compare those results to our experiments on ZnO nanowires. Although the theory does not incorporate any finite size effects, nanowires are preferable over bulk in experiment, because their crystalline structure shows very little defects. Introduction BCS theory in an e-h plasma Solving the gap equation requires a number of assumptions, of which the most important are stated below. The first two are reasonable for our experiments, the last is used to shorten expressions. Only s-wave interactions contribute (low temperatures) n e = n h (photocreation of e-h pairs) m e * = m h * = m e Angle Averaged Gap Equation This equation can be discretized to , with Assumptions Gallery Introducing mass imbalance. Establishing the critical temperature. Calculating gain spectra in normal state at room temperature. Calculating gain spectra in BCS-state. Luminescence measurements at low temperature. Future work Screened Coulomb Interaction Figure 1: Angle averaged screened potential at n e = 10 25 m -3 . Value of V0 ≈ 10 -46 Jm -3 . The gap equation is integrated over the angles to yield where is the angle averaged Coulomb interaction . As we lower the temperature, the eigenvalue of U becomes one at the critical temperature. In statistical field theory screening is described by the Bethe-Salpeter equation: Because we are looking at with long wavelength behaviour, we set ω = 0. To find the angle averaged potential , we write and perform the integrals over the angles. In the evaluation of the bubble diagram Fermi-Dirac distributions are approximated by stepfunctions. Here, is the bare Coulomb interaction and the amplitude of the bubble diagram is called . The effective screened interaction is: Pump: 800-nm 120-fs. Single nanowire of length 17 μ m. 3-photon absorption. Fluence above (red) and below the laser threshold. In the normal state. T = 30 K , m e * = 0.28 m e , m h *= 0.59 m e Photon energy measured from top of the band gap. Large difference between free and interacting carriers. On a wood of nanowires of ~20 μ m length. Calculated gain of ~10 5 m -1 . Carrier recombination on ps timescale. Pump-probe measurement Acknowledgements We thank P. Jurrius and C.R. de Kok for technical support and D.H. van Dorp and D.A.M. Vanmaekelbergh for the nanowire samples.