Kubo spins in nanoscale aluminum grains: A muon spin relaxation study

We report muon spin relaxation rate measurements on films composed of aluminum grains having a size of a few nm, with a large energy level splitting of the order of 100 K. The films range from weakly metallic to insulating. In the insulating case, the low-temperature relaxation rate is consistent with the presence of single electron spins in grains having an odd number of electrons, known as Kubo spins. The relaxation rate temperature dependence follows an activation law having an energy scale in agreement with the average level splitting. In weakly metallic films, inter-grain junction spins may contribute substantially to the smaller relaxation rate.

Figure 1

Time evolution of the polarization asymmetry (symbols) at selected temperatures for insulating granular Al sample S100K with room-temperature resistivity of $100000\mu \mathrm{\Omega }\mathrm{cm}$ with the corresponding fit (line) according to Eq. (1). Here $A$ is a constant determined by the muon decay properties and the geometry of the $\mu \text{SR}$ spectrometer.

Figure 2

Simulation of the normalized asymmetry plot using the function given in Eq. (1) of the main text. The curves were generated for the ratio of the electronic relaxation rate $\lambda$ and the nuclear relaxation rate $\sigma$ from 0 to 2 in steps of 0.2.

Figure 3

Temperature dependence of the muon spin relaxation rate of electronic origin $\lambda$ for sample S100K. The red dashed line follows Eq. (2) with $E_a\approx 125\text{K}$ and $\lambda \left(0\right)\approx 0.19\mu {\mathrm{s}}^{-1}$.

Figure 4

Summary of asymmetry curves at selected temperatures for samples measured in this work (see Table 1): (a) $380\mu \mathrm{\Omega }\mathrm{cm}$, (b) $1222\mu \mathrm{\Omega }\mathrm{cm}$, (c) $1816\mu \mathrm{\Omega }\mathrm{cm}$, (d) $9440\mu \mathrm{\Omega }\mathrm{cm}$, and (e) $100000\mu \mathrm{\Omega }\mathrm{cm}$. Open symbols represent the experimental data and full lines represent fits to Eq. (1). The insulating sample data are presented now at the same temperatures as those of the metallic/superconducting samples.

Figure 5

Temperature dependence of the muon spin relaxation rate of electronic origin, $\lambda$, for all samples in this work. The inset shows a comparison of $\lambda$ between samples S380 and S10K.

Figure 6

Temperature dependence of the muon spin relaxation rates $\lambda$ and $\sigma$ for samples S10K and S100K.

Figure 7

(a) Trim.SP Monte Carlo simulations [16] of the muon stopping distribution in a 100 nm thick granular Al sample where the implantation energy is 8 keV. The distribution has a mean depth value of 55 nm with straggling of 13 nm. (b) The mean depth and straggling values of the muon stopping distribution as a function of the implantation energy.