Dangling-bond spin relaxation and magnetic1∕fnoise from the amorphous-semiconductor/oxide interface: Theory

Abstract

We propose a model for magnetic noise based on spin flips (not electron trapping) of paramagnetic dangling bonds at the amorphous-semiconductor/oxide interface. A wide distribution of spin-flip times is derived from the single-phonon cross-relaxation mechanism for a dangling bond interacting with the tunneling two-level systems of the amorphous interface. The temperature and frequency dependence is sensitive to three energy scales: The dangling-bond spin Zeeman energy $(\ensuremath{\delta})$, as well as the minimum $({E}_{\mathrm{min}})$ and maximum $({E}_{\mathrm{max}})$ values for the energy splittings of the tunneling two-level systems. At the highest temperatures, ${k}_{B}T⪢\mathrm{max}$ $(\ensuremath{\delta},{E}_{\mathrm{max}})$, the noise spectral density is independent of temperature and has a $1∕f$ frequency dependence. At intermediate temperatures, ${k}_{B}T⪢\ensuremath{\delta}$ and ${E}_{\mathrm{min}}⪡{k}_{B}T⪡{E}_{\mathrm{max}}$, the noise is proportional to a power law in temperature and possesses a $1∕{f}^{p}$ spectral density, with $p=1.2--1.5$. At the lowest temperatures, ${k}_{B}T⪡\ensuremath{\delta}$, or ${k}_{B}T⪡{E}_{\mathrm{min}}$, the magnetic noise is exponentially suppressed. We compare and fit our model parameters to a recent experiment probing spin coherence of antimony donors implanted in nuclear-spin-free silicon [T. Schenkel et al., Appl. Phys. Lett. 88, 112101 (2006)], and conclude that a dangling-bond area density of the order of ${10}^{14}\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}2}$ is consistent with the data. This enables the prediction of single spin qubit coherence times as a function of the distance from the interface and the dangling-bond area density in a real device structure. We apply our theory to calculations of magnetic flux noise affecting superconducting quantum interference devices (SQUIDs) due to their $\mathrm{Si}∕\mathrm{Si}{\mathrm{O}}_{2}$ substrate. Our explicit estimates of flux noise in SQUIDs lead to a noise spectral density of the order of ${10}^{\ensuremath{-}12}\phantom{\rule{0.3em}{0ex}}{\ensuremath{\Phi}}_{0}^{2}{(\mathrm{Hz})}^{\ensuremath{-}1}$ at $f=1\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$. This value might explain the origin of flux noise in some SQUIDs. Finally, we consider the suppression of these effects using surface passivation with hydrogen, and the residual nuclear-spin noise resulting from a perfect silicon-hydride surface.