Complex Refractive Index Extraction for Spintronic Terahertz Emitter Analysis

TL;DR

This work tackles the challenge of extracting dielectric properties for thin multilayer spintronic terahertz emitters (STEs) from THz-TDS data, where conventional transfer-function analyses struggle due to substrate effects and multiple reflections. It introduces a practical Transfer Matrix Method framework with an analytic substrate-removal modification and uses a dry-air THz reference to enable direct fitting of layer refractive indices and extinction coefficients from time-domain data. The authors demonstrate significant improvements over literature dielectric constants in quartz and Pt-on-sapphire samples, including thickness-dependent measurements that yield parameters consistent with STE behavior and enable more accurate THz simulations. Thanks to a lightweight, open-source implementation, the method offers a readily adoptable tool for STE design and THz-TDS analysis, improving material characterization and device optimization.

Abstract

Spintronic terahertz emitters (STEs) generate broadband terahertz (THz) radiation, which is essential for spectroscopy, imaging, and communication. The performances and the essential physical parameters of STE devices are linked to the dielectric properties of the constituent materials. Terahertz time-domain spectroscopy (THz-TDS) is an effective tool to measure these properties, but conventional analysis struggles with thin or complex multilayered systems due to simplifying approximations or complex transfer functions. In this work, we present a practical method to extract dielectric properties of STE multilayers using the Transfer Matrix Method (TMM). By comparing the THz pulse calculated using the Transfer Matrix Method (TMM) with the experimentally measured pulse transmitted through the sample, we can extract the dielectric properties of STEs, enhancing THz-TDS analysis and facilitating STE design and optimization. This method avoids constructing complex transfer functions, accommodates diverse sample geometries, and is designed to be accessible, with a publicly available codebase, making it a useful tool for STE research.

Figures (5)

• Figure 1: The description of the variation problem when directly using the database for calculation and the recipe of the TMM-based dielectric property extraction.
• Figure 2: Block diagram illustrating the key stages of the TMM-based dielectric property extraction process.
• Figure 3: Surface plot of the error function used in TMM fitting for dielectric constant extraction. The figure shows the error landscape as a function of refractive index ($n$) and extinction coefficient ($k$) for 1mm thick quartz. Color indicates the magnitude of the error in arbitrary units. The error minima identify the optimal $(n, k)$ pairs for initializing frequency-dependent fitting and for estimating material parameters.
• Figure 4: Terahertz (THz) propagation analysis for two samples: quartz (1 mm) and Pt (6 nm)/sapphire (500 $\mu$m). Panels (a–b) show results for the quartz sample, while panels (e–f) correspond to the Pt/sapphire heterostructure. Time-domain electric field waveforms are shown in panels (a) and (e), and their respective frequency-domain spectra are shown in panels (b) and (f). Experimental data are represented by cyan dots ($E_{\text{exp}}^{\text{sample}}$), theoretical calculations using literature dielectric constants from Ref. franta2016optical (quartz) and Ref. ordal1985optical (Pt) are shown as purple curves ($E_{\text{theo}}^{\text{sample}}$), and the fitted results obtained via the TMM are shown as red curves ($E_{\text{theo,fit}}^{\text{sample}}$). Panels (c–d) present the literature input (purple) and TMM-extracted (red) optical constants $n$ and $k$ for quartz. Panels (g–h) show the literature input and TMM-fitted real and imaginary parts of the permittivity $\varepsilon$ for the Pt layer (calculated using Drude model). These results demonstrate the refinement of optical parameters through TMM fitting to accurately reproduce experimental THz transmission.
• Figure 5: THz transmission through NiFe(3nm)/Pt(x-nm)/Quartz(1mm) as a function of Pt thickness. Main panel: normalized peak-to-peak amplitude of the transmitted THz field versus $d_{\mathrm{Pt}}$ (0–10 nm). Circles are measurements; the dashed line is a guide to the eye. Solid curves are transfer-matrix simulations using (i) Pt dielectric parameters from the literatures ordal1985opticalrakic1998opticalseifert2018terahertz (blue lines), and (ii) Pt Drude parameters obtained from our fit to the transmission data (red). For comparison, a simulation using Drude parameters extracted independently from spintronic-emitter (STE) samples (“reference-3-STE”, light blue) is also shown. Inset: representative time-domain THz waveforms for several $d_{\mathrm{Pt}}$; traces are horizontally offset for clarity and illustrate the progressive attenuation with increasing Pt thickness.