Electron spin exchange of shallow donor muonium states

Abstract

Shallow donor muonium states with small hyperfine frequencies, recently observed in II–VI semiconductor compounds, have a number of unique features that present both opportunities and challenges in understanding muon spin dynamics in the presence of Heisenberg spin exchange. First, the shallow muonium state in CdSe with hyperfine frequency ω0/2π ≈ 0.1 MHz is already in the high field regime even in the earth's magnetic field, where only two precession frequencies are observable by the muon spin rotation (μSR) technique. Second, unlike in the case of more conventional muonium species with a larger hyperfine frequency, the μSR signal of shallow muonium states can be observed even in the transition region, between the slow spin–flip regime and the fast spin–flip regime, where the spin–flip rate and the hyperfine frequency are comparable. The muon spin dynamics in the transition region has not been theoretically explored previously, mainly because normal muonium in vacuum gives no observable signal in this region. Third, in the case of shallow muonium states, the incoherent process defined to be those spin–flip collisions that cause changes in muon spin precession frequencies, becomes crucially important in the transition region, where the incoherent process is entirely negligible in more conventional muonium species. By taking incoherent multiple collisions into account, an analytical expression for the time evolution of the muon spin polarization in Mu is derived, where Mu undergoes repeated spin–flip collisions. Comparisons with Monte Carlo calculations show that the analytical expression obtained in this work can reliably be used to analyse experimental data for shallow donor states not only in the slow spin–flip regime, but also in the transition region up to the onset of the fast regime. The present work confirms a recent experimental finding that, in the transition region, the initial phases of the two precession components of shallow donor states exhibit a strong and unexpected dependence on the spin–flip rate. Such a rate-dependent initial phase is expected to be widely observed in Mu species with hyperfine frequencies less than several tens of megahertz. In order to understand spin dynamics in the transition region in a wider and more general context, Monte Carlo calculations are also carried out for more conventional muonium species with large hyperfine frequencies in which no observable signal can be seen in a transverse field in the transition region and in a large part of the fast regime.