Many electron theory of $1∕f$ noise in hopping conductivity

We show that $1∕f$ noise in the variable-range hopping regime is related to transitions of many-electrons clusters (fluctuators) between two almost-degenerate states. Giant fluctuation times necessary for $1∕f$ noise are provided by a slow rate of simultaneous tunneling of many localized electrons and by large activation barriers for their consecutive rearrangements. The Hooge constant steeply grows with decreasing temperature because it is easier to find a slow fluctuator at lower temperatures. Our conclusions qualitatively agree with the low-temperature observations of $1∕f$ noise in $p$-type silicon and GaAs.

Figure 1

A chessboard cluster of donors, responsible for a $1∕f$ noise. The dark circles indicate occupied (neutral) donors, while the open circles indicate empty donors. All donors are in the energy band of the width $2e^2∕\kappa R$ around the chemical potential $(E=0)$. Arrows show the direction of the electron transition between two energy minima corresponding to two possible ways to occupy cluster sublattices. The length $R$ stands for the size of the cluster cell.

Figure 2

(Color online) Thermally activated transition of the “chessboard” cluster between two energy minima $A$ and $C$ along the route $A\to B\to C$. The activation energy is proportional to the number of electrons $N^{2∕3}$ in the domain wall shown in panel (B) by dashed lines.

Figure 3

Comparison of the theory [solid line, Eq. (30)] and the experiment [solid line with circles, Eq. (34)] for the case $\omega ∕2\pi =1\mathrm{Hz}$ and $T_{\mathrm{ES}}=11\mathrm{K}$. We have used $\lambda =2.2$ and $n_d{a}^3=0.0084$ to make the optimum data fit in the range of temperatures $(0.07–0.3\mathrm{K})$ studied in the experiment.

Figure 4

Theoretical predictions for the frequency dependence of the Hooge parameter [Eq. (30)] at different temperatures for samples with $T_{\mathit{ES}}=11\mathrm{K}$. All parameters are as in Fig. 3.