Implantation studies of low-energy positive muons in niobium thin films

TL;DR

This study uses low-energy μSR to quantify the range of μ^+ implanted in Nb-based thin films, correlating the diamagnetic μ^+ fraction with implantation energy. By modeling μ^+ stopping with TRIM.SP and updated electronic stopping cross sections (notably for Nb) and incorporating a muonium formation model in SiO2 via a transmitted energy $E^{*}$, the authors demonstrate excellent agreement between experiment and simulation in Nb, while older tabulations underestimate the μ^+ range. The work highlights the critical role of accurate stopping data for depth-resolved μSR and suggests that previous Nb-derived length scales, such as the magnetic penetration depth, may be biased if older stopping data are used. It also discusses energy straggling, sample-dependent Mu formation, and the broader applicability of updated stopping data to other elements with sparse low-energy information.

Abstract

Here we study the range of keV positive muons $μ^+$ implanted in Nb$_2$O$_5$($x$ nm)/Nb($y$ nm)/SiO$_2$(300 nm)/Si [$x$ = 3.6 nm, 3.3 nm; $y$ = 42.0 nm, 60.1 nm] thin films using low-energy muon spin spectroscopy (LE-$μ$SR). At implantation energies 1.3 keV $\leq E \leq$ 23.3 keV, we compare the measured diamagnetic $μ^+$ signal fraction $f_{\mathrm{dia.}}$ against predictions derived from implantation profile simulations using the TRIM.SP Monte Carlo code. Treating the implanted $μ^+$ as light protons, we find that simulations making use of updated stopping cross section data are in good agreement with the LE-$μ$SR measurements, in contrast to parameterizations found in earlier tabulations. Implications for other studies relying on accurate $μ^+$ stopping information are discussed.

Figures (11)

• Figure 1: Sketch of the $\mu^{+}$ implantation experiment using . Muons with their spin direction $\mathbf{S}_{\mu}$ perpendicular to their momentum $\mathbf{p}_{\mu}$ are implanted in Nb/SiO2 films at energies $E \in [0.5, 30]$, such that their stopping profile $\rho(z,E)$ overlaps with the two material layers. An external magnetic field $\mathbf{B}_{\mathrm{applied}} \approx \qty{10}{\milli\tesla} \parallel \hat{\mathbf{z}}$ is applied perpendicular to $\mathbf{S}_{\mu}$, such that only the fraction of $\mu^{+}$ stopped in a diamagnetic environment $f_{\mathrm{dia.}}$ is observable. As $f_{\mathrm{dia.}}$ deviates significantly between the two layers (see the sketch's inset), the signal amplitude is expected to decrease as more $\mu^{+}$ stop in the SiO2 layer.
• Figure 2: Typical stopping profiles $\rho(z, E)$ for $\mu^{+}$ implanted in Nb2O5($x$)/Nb($y$)/SiO2 [$x = \qty{3.6}{\nano\meter}, \qty{3.3}{\nano\meter}$; $y = \qty{42.0}{\nano\meter}, \qty{60.1}{\nano\meter}$] targets at different energies $E \leq \qty{30}{\kilo\electronvolt}$. The profiles were simulated using the Monte Carlo code TRIM.SP1984-Biersack-APA-34-731991-Eckstein-SSMS-101994-Eckstein-REDS-1-239 for e5 projectiles, with the results represented as histograms with 2 bins. Note that these simulations make use of our revised Varelas-Biersack 1970-Varelas-NIM-79-213 parameterization of each target atom's electronic stopping cross section (see \ref{['sec:cross-sections']}). The evolution of the $\mu^{+}$ stopping fraction $f$ with $E$ is shown in each plot's inset. Further simulation details are described in \ref{['sec:experiment:trimsp']} and Refs. 2002-Morenzoni-NIMB-192-2452023-McFadden-PRA-19-044018.
• Figure 3: Typical time-differential data in the Nb2O5($x$)/Nb($y$)/SiO2 [$x = \qty{3.6}{\nano\meter}, \qty{3.3}{\nano\meter}$; $y = \qty{42.0}{\nano\meter}, \qty{60.1}{\nano\meter}$] films measured at $T = \qty{200}{\kelvin}$ in a $B_{\mathrm{applied}} = \qty{10}{\milli\tesla}$ transverse-field at different implantation energies $E$ and a $\mu^{+}$ transport bias $\mathrm{Tr} = \qty{15}{\kilo\electronvolt}$. Note that only the signal in one of four detectors (binned by a factor of 500.0) is shown for clarity. The solid colored lines denote fits to \ref{['eq:counts', 'eq:asymmetry', 'eq:polarization']}, in excellent agreement with the data. The amplitude $\mathcal{A}(t=\qty{0}{\micro\second}) \equiv \mathcal{A}_{0}$ of the observable single is proportional to the population of implanted $\mu^{+}$ in a diamagnetic state, corresponding (predominantly) to $\mu^{+}$ stopped in the Nb layer, which decreases with increasing $E$.
• Figure 4: Implantation energy $E$ dependence of the main fit parameters derived from the analysis of the data in the Nb2O5($x$)/Nb($y$)/SiO2 [$x = \qty{3.6}{\nano\meter}, \qty{3.3}{\nano\meter}$; $y = \qty{42.0}{\nano\meter}, \qty{60.1}{\nano\meter}$] films at $T = \qty{200}{\kelvin}$ in a $B_{\mathrm{applied}} = \qty{10}{\milli\tesla}$ transverse-field using \ref{['eq:counts', 'eq:asymmetry', 'eq:polarization']}. Here, $\mathcal{A}_{0}$ is the initial asymmetry, proportional to the population of $\mu^{+}$ stopped in a diamagnetic environment, $\lambda$ is the exponential damping rate, and $\langle B \rangle$ is the mean local field experienced by the spin-probes. While systematic differences are evident for the $\mathcal{A}_{0}$s determined at different transport biases $\mathrm{Tr}$ (e.g., due to slightly different initial spin states), their $E$-dependence is identical up to a normalization factor. The sigmoidal $E$-dependence of $\lambda$ reflects the different damping rates for the Nb and SiO2 layers, whose asymptotic values ($\lambda_{\ch{Nb}}$ and $\lambda_{\ch{SiO2}}$) are given in the plot inset. Values for the $E$-independent local field values $\bar{\langle B \rangle}$ are also indicated.
• Figure 5: Measured diamagnetic fraction $f_{\mathrm{dia.}}$ as a function of $\mu^{+}$ implantation energy $E$ in the Nb2O5($x$)/Nb($y$)/SiO2 [$x = \qty{3.6}{\nano\meter}, \qty{3.3}{\nano\meter}$; $y = \qty{42.0}{\nano\meter}, \qty{60.1}{\nano\meter}$] thin films at $T = \qty{200}{\kelvin}$ in a transverse applied field $B_{\mathrm{applied}} \approx \qty{10}{\milli\tesla}$. Data points correspond to measurements by using two different $\mu^{+}$ beam extraction biases $\mathrm{Tr}$. The lines indicate predictions by the Monte Carlo code TRIM.SP1984-Biersack-APA-34-731991-Eckstein-SSMS-101994-Eckstein-REDS-1-239 using different muonium formation probabilities $\tilde{p}_{\ch{Mu}}(E^{*})$ and electronic stopping cross section parameterizations. Simulations using revised inputs derived from the ['s] stopping database 2017-Montanari-NIMB-408-502024-Montanari-NIMB-551-165336 (extending the work in Ref. 2023-McFadden-PRA-19-044018) are in excellent agreement with the data, in contrast to predictions based on older cross section tabulations 1977-Anderson-SRIM-31993-ICRU-49, which systematically underestimate the range of $\mu^{+}$ (i.e., predict larger $f_{\mathrm{dia.}}$s). Note that the $f_{\mathrm{dia.}}$s simulated using the older tabulations 1977-Anderson-SRIM-31993-ICRU-49 are nearly indistinguishable, owing to their similar stopping coefficients $A_{i}$ for O, Si, and Nb (see \ref{['tab:coeff']}).