A model for positron annihilation in multi-layer systems by solving the diffusion equation using different positron affinities

TL;DR

Interpreting depth-resolved DBS signals in layered materials requires linking implantation profiles to diffusion- and annihilation-driven S(E) curves. LIMPID provides a two-stage diffusion solver that combines implantation modeling with a Markov-chain transport between layer boundaries and includes positron affinity via a Boltzmann factor and optional epithermal corrections. The framework delivers per-energy annihilation fractions and depth-resolved S parameters, enabling robust extraction of diffusion lengths and layer thicknesses while improving cross-group comparability. Demonstrated on a Cu/Si system, LIMPID achieves excellent agreement with data and highlights the importance of affinities and epithermal corrections for accurate parameter inference. As an open-source tool, LIMPID offers a transparent platform for standardized analysis of positron defect measurements across research communities.

Abstract

We present a method for solving the positron diffusion equation in multi-layer systems. Our approach incorporates material-specific implantation profiles, diffusion parameters, and positron affinities. It utilizes a Markov chain approach to model annihilation probabilities and provides fitting capabilities for experimental S (lineshape) parameter data. We have implemented this algorithm in Python and made it available for free under the name LIMPID. To demonstrate its performance, we analyze depth-resolved Doppler-Broadening Spectroscopy measurements of a Cu layer on a Si substrate, achieving excellent agreement with the experimental profiles. The LIMPID tool enhances the reproducibility and comparability of positron defect characterization measurements across different research groups.

Figures (6)

• Figure 1: Schematic drawing of a generic sample explaining the nomenclature used by limpid. The sample consists of $N$ layers, whereby the $N^{\textrm{th}}$ layer has an infinite thickness.
• Figure 2: Schematic drawing of the first diffusion step inside the $i^{th}$ layer. $P(z)$ represents the positron implantation profile within a material layer of thickness $d$. After diffusion, the positron distribution at the layer boundaries is determined by integrating the product of $P(z)$ and the probability of a positron reaching the left [right] boundary $c_{\textrm{left}}(z)$ [$c_{\textrm{right}}(z)$]. The corresponding mathematical formulation is provided in Equation \ref{['eq:n_left']}. $c_{\textrm{ann}}$ is the probability of a positron to annihilate before reaching a boundary. All three probabilities sum to 1.
• Figure 3: Schematic drawing of the fluxes in the second diffusion step. After reaching a layer boundary, positrons either jump between boundaries or annihilate within the layer. The transition probabilities calculated from the fluxes form a Markov chain, which describes how positrons move through the system until they are annihilated.
• Figure 4: Depth-resolved dbs data for a Cu layer on a Si substrate, fitted using the limpid algorithm. The experimental data (symbols) and limpid models (solid lines) demonstrate almost perfect agreement for the $S(E)$ profile. The orange line represents the best fit achieved (and is identical for the fit without positron affinities) using the correct positron affinities and an epithermal correction. The green line shows a fit with the epithermal correction disabled, resulting in significant deviation for energies $\leq3$ keV. The red line shows an affinity-less model of the best-fit result. All fit parameters are listed in Table \ref{['tab:fit-params']}.
• Figure 5: Calculated positron implantation profiles in the Cu/Si sample for two representative positron implantation energies (12 keV and 27 keV). We used the thicknesses resulting from the best fit in Figure \ref{['fig:cusi']} and Table \ref{['tab:fit-params']}, i.e., 448 nm Cu on top of the Si substrate. The profiles show how the implantation depth and spread depend on energy and (mainly) how density shapes multi-layer implantation.