BCS Theory on Lasing ZnO NanowiresA.J. van Lange1, M.A.M. Versteegh1, J.I. Dijkhuis1, H.T.C. Stoof2

1 Debye Institute for Nanomaterials Science, Utrecht2 Institute for Theoretical Physics, Utrecht

Cooper pairs of an electron and a hole. Required attractive interaction is the screened Coulomb interaction. When we approach Tc 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.

3.10 3.15 3.20 3.25 3.30 3.350

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Photon energy (eV)

1224 J/m2

1385 J/m2

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) ne = nh (photocreation of e-h pairs) me* = mh* = me

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 ne= 1025 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 μm. 3-photon absorption. Fluence above (red) and below the laser threshold.

In the normal state. T = 30 K , me* = 0.28 me, mh*= 0.59 me

Photon energy measured from top of the band gap. Large difference between free and interacting carriers.

On a wood of nanowires of ~20 μm length. Calculated gain of ~105 m-1. Carrier recombination on ps timescale.

Pump-probe measurement

AcknowledgementsWe 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.