Combination of Optical Microspectroscopy with a 4x4 Transfer Matrix Model for Thickness Determination of 2D Materials

Session

Poster Session

Authors

Julian Schwarz (3), Michael Niebauer (3), Maria Koleśnik-Gray (2), Maximilian Szabo (1), Leander Baier (1), Phanish Chava (4), Artur Erbe (4), Vojislav Krstić (2), Mathias Rommel (1), Andreas Hutzler (3, 5)

Affiliations

1. Fraunhofer Institute for Integrated Systems and Device Technology IISB, Schottkystraße 10, Erlangen, 91058
2. Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Applied Physics, Staudtstraße 7, Erlangen, 91058
3. Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Electron Devices, Cauerstraße 6, Erlangen, 91058
4. Helmholtz-Zentrum Dresden-Rossendorf (HZDR), Institute of Ion Beam Physics and Materials Research, Bautzner Landstraße 400, Dresden, 01328
5. Forschungszentrum Jülich GmbH, Helmholtz Institute Erlangen-Nürnberg for Renewable Energy (IEK-11), Cauerstraße 1, Erlangen, 91058

Keywords

microspectroscopy, anisotropy, birefringence, thickness determination, transfer matrix method, 2D materials

Abstract text

In our work, optical microspectroscopy (i.e., combining a light microscope with a spectrometer) is used to measure reflectance spectra of 2D material flakes. These measured spectra are then compared with the expected reflectance spectra modeled with a 4x4 transfer matrix formalism. A least square fit is performed to determine the thickness of the investigated material flakes. 

2D materials are a promising material class for future (opto-)electronic devices. To fully exploit their advantages, the materials’ thickness must be precisely known to optimize the performance of a specific device as many of their features are thickness dependent. Even though there exist a variety of different techniques for thickness determination of 2D materials – for example atomic force microscopy (AFM) or Raman spectroscopy – they all have different drawbacks, such as being destructive or not capable of measuring encapsulated 2D material flakes sufficiently precise. Therefore, we present a nondestructive and technologically easily implementable approach for fast and accurate thickness determination, which is suitable for small flakes or structures (due to a measurement spot size of approx. 3 µm), as they are often found when working with 2D materials [1].

The procedure for thickness determination is exemplarily demonstrated for different 2D materials thicknesses ranging from atomic layers up to more than 100 nm. These are black phosphorus (BP), hexagonal boron nitride (hBN), molybdenum disulfide (MoS₂), tungsten diselenide (WSe₂), and highly oriented pyrolytic graphite (HOPG). The 2D flakes are placed on a substrate (typically an oxidized silicon wafer) and – in the case of BP, which degrades at ambient atmosphere – covered with a protective layer consisting of Poly(methyl methacrylate) (PMMA). The reflectance data is obtained by a spectrometer attached to a standard light microscope and is then combined with a modular model containing a 4x4 transfer matrix method and the optical components relevant to light microspectroscopy, especially objective lenses with high numerical aperture [1]. By least square fitting of the optical model to measured spectra, the thickness of the investigated 2D flakes is obtained. The basic principle of our approach is shown in Fig. 1.

Figure 1: Basic principle for thickness determination via combination of measured and modeled reflectance spectra [1]. Reproduced under terms of the CC-BY license [2]. Copyright 2023, The Authors, published by Wiley. https://doi.org/10.1002/smtd.202300618

For several samples of each of the mentioned materials, the thickness is determined with this procedure and afterwards the step heights of the 2D materials are measured with AFM as reference. In Fig. 2, the modeled thickness is compared to the corresponding AFM thickness for all samples.

Figure 2: Comparison between the thicknesses of various 2D material samples, determined via 4x4 transfer matrix modeling of microspectroscopic measurements and via AFM, respectively [1]. Reproduced under terms of the CC-BY license [2]. Copyright 2023, The Authors, published by Wiley. https://doi.org/10.1002/smtd.202300618

As can be observed, the thicknesses obtained by our model is highly comparable to the thicknesses obtained by AFM for all samples. Immediately before the AFM measurements, the PMMA capping layer of the BP samples was removed in acetone, since otherwise the step height and therefore thickness cannot be measured properly. For the application in device fabrication, this automatically would lead to the destruction of a device since the BP is no longer protected against atmosphere and therefore prone to degradation. At this point, the thickness determination via microspectroscopic reflectance measurements is clearly at an advantage.

In summary, the approach presented in our work is reliable and precise for thickness determination of various, even anisotropic materials like black phosphorus. It is also suitable for small flakes and structures and the determined thicknesses are highly comparable to the results of other measurement techniques established for 2D materials like AFM. However, an advantage in contrast to AFM is the possibility to measure even encapsulated layers without destroying them. Due to the modularity of the approach, it is applicable to different measurement systems with only small adaptions. Besides, an application for hyperspectral imaging as well as the measurement of refractive indices is possible, and the approach can be extended to transmission spectroscopy and the usage of polarized light.

References

[1] Schwarz, J., Niebauer, M., Koleśnik-Gray, M., Szabo, M., Baier, L., Chava, P., Erbe, A., Krstić, V., Rommel, M., Hutzler, A., Correlating Optical Microspectroscopy with 4×4 Transfer Matrix Modeling for Characterizing Birefringent Van der Waals Materials. Small Methods 2023, 2300618. https://doi.org/10.1002/smtd.202300618

[2] https://creativecommons.org/licenses/by/4.0/