Reflectivity-based refractive index measurement of van der Waals materials

TL;DR

The paper presents a reflectivity-based method to measure the in-plane refractive index of transparent van der Waals flakes as small as $3 \times 3\,\mu\mathrm{m}^2$, circumventing the large-area and spectral-model requirements of traditional ellipsometry. By modeling a three-layer air–vdW film–non-transmissive substrate system and analyzing the spectral positions of reflectivity minima $\lambda_q$ at normal incidence, the in-plane index $n_2(\lambda)$ is extracted using thickness $d$ and known substrate index, without assuming a specific spectral form. The method is validated on hafnium disulfide (HfS$_2$), revealing $n_2(\lambda)$ from $460$ to $850$ nm across 45 data points that align with both density functional theory and imaging ellipsometry data. This approach expands accessible refractive-index characterization for vdW materials, enabling measurements on micron-sized flakes and offering a robust cross-check for existing optical characterization techniques.

Abstract

We present a reflectivity-based method for measuring the in-plane refractive index of transparent van der Waals (vdW) materials. The approach enables the characterization of as small as $3 \times 3$~{\textmu}m$^2$ exfoliated flakes on a non-transmissive substrate without assuming any specific spectral shape of the refractive index. Exfoliated flakes are most commonly obtained through mechanical exfoliation, which generally produces vdW flakes with tens-of-micron lateral dimensions. As a result, conventional ellipsometry - which depends on large, uniform areas and specific spectral models - becomes challenging to apply. Our method determines the refractive index directly from the spectral position of reflectivity minima, provided the flake thickness and the substrate complex refractive index are known. We demonstrate the technique on hafnium disulfide (HfS$_2$), a vdW semiconductor with high refractive index and low absorption, retrieving its in-plane refractive index across the visible range. The results both validate previous ellipsometry measurements and establish this method as an accessible and spectral-model-free alternative for refractive index characterization of vdW materials.

Figures (4)

• Figure 1: Schematic of a three-layered system, composed of (1) a semi-infinite layer of air, (2) a film of a transparent vdW material of thickness $d$, and (3) a semi-infinite non-transmissive substrate. We denote $r_{kl}$ as the reflection coefficients from medium $k$ to $l$ and $\tilde{n_l} = n_l - i k_l$ as the complex refractive of medium $l$.
• Figure 2: Normalized reflectivity spectra of HfS$_2$ flakes of different thicknesses $d$ on top of a $100$ nm gold layer. For clarity, the curves are offset with $+0.2,+0.4,+0.7,+1.05$ a.u., respectively. The thinnest flake ($d=72.6$ nm) yields only a $q=1$ reflectance dip. The $q=1$ reflectivity minima is observable in our spectral range up to the flake thickness of $d=178.9$ nm. The thickest flake ($d=491.0$ nm) displays five different minima of orders $q=4,..,8$. All curves are normalized to the reflection of the gold substrate. An optical image of each flake is shown to the right of each reflectivity curve.
• Figure 3: (a) Diagram showing the method used to extract $\lambda_{q,d}$ and the refractive index $n_2(\lambda_{q,d})$ from reflectivity measurements such as the ones shown in Fig. \ref{['fig:reflectivity']}. (b) Reflectivity of a HfS$_2$ flake with a thickness of $d=178.9$ nm along with a 3-Lorentzian fit (see Eq. (\ref{['eq:Lorentzians']})). (c) Zoom-in reflectivity plot of the $q=1$ minimum shown in (b), as well as two different fits using a single and a 3-Lorentzian model.
• Figure 4: In-plane refractive index of HfS$_2$ measured with the reflectivity-based method described in this work (red dots) compared to density functional theory (DFT) calculations (black line) and imaging ellipsometry measurements (black dashed lines). The spectral range covers wavelengths from 460 nm to 850 nm. The DFT and imaging ellipsometry data are taken from Ref. Zambrana2025.