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Characterization of black phosphorus as Microstructured Anisotropic 2D Material by Imaging Mueller Matrix Ellipsometry


Thin-film layers of black phosphorus - the most stable allocate of the element phosphorus - is a promising material for future semiconductor electronics based on novel 2D-materials. Ultra-thin films or even monolayers of black phosphorus (b-P) may be fabricated by mechanical exfoliation from bulk material (similar to graphene). 2D-b-P is of particular interest for semiconductor electronics as it features a widely tunable bandgap via the layer thickness (from ~2 eV for monolayers down to 0.3 eV for bulk material). Thus, it closes the energy gap between the other known 2D- materials graphene (zero bandgap) and transition-metal dichalcogenides (bandgap typically 1-2 eV) such as MoS 2.[1]
Regarding its optical properties, b-P stands out from other 2D-materials as well because its crystal structure causes an in-plane optical anisotropy in the visible range of the electromagnetic spectrum. However, the ellipsometric characterization of exfoliated flakes of b-P is extremely challenging as flake sizes are typically on the order of some 10μm, and the thickness often varies within a single flake.
  In this work, for the first time we report the ellipsometric characterization of micro-structured b-P-flakes by means of spectroscopic Imaging Mueller Matrix Ellipsometry (IMME). It features much higher x-y-resolution compared to conventional ellipsometers since for IMME this resolution is not limited by the size of the illuminating spot. By using IMME with a microscope objective lens, the measurement of anisotropic refractive indices and the orientation of the optical axes comes feasible even on microstructured thin-film samples such as b-P-flakes.

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Imaging Mueller Matrix Ellipsometry θ-scans

- non-zero off-diagonal blocks of MM-micrographs reveal the b-P-flake’s anisotropy (wavelength of illumination λ=550 nm)
- θ-scan: rotate the flake around its center point & record MM-micrographs at each position (rotation from 0° to 340° in 20°-steps)
- evaluating central region of interest (ROI, c. fig. right) on each MM-micrograph for all θ-positions yields Mueller-Matrix θ- spectrum
- observed periodicity of 180° complies with an orthorhombic, biaxial model for b-P (one axis orthogonal to the sample surface)
- using the flake thickness obtained from AFM-measurements (43 nm) for optical modeling yields the in-plane refractive indices and the orientation of the in-plane principal axes of the index ellipsoid.

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MM-micromaps of black phosphorus (MM-element m 31, λ=550 nm) at different rotational angles θ: MM- value changes due to in-plane anisotropy


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Micromaps of normalized MM (m11=1) at θ=0° and λ=550 nm. Mind different scale bars for block diagonal (blue frames) and block off-diagonal (green frames) matrix elements. Arrows in m34-graph indicate the orientation of in- plane optical axes obtained from ellipsometric modelling of θ-scan.


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θ-spectrum of the normalized 3x4 MM of black phosphorus at different angles of incidence obtained from ROI-evaluation of MM-micromaps. Indices of fitted complex refractive indices and refer to principal in-plane axes of index ellipsoid (c. figure left, inset in m 34-micrograph). Mind different y-scales for block diagonal and block off-diagonal MM-element.


AFM measurements

- atomic force microscopy (AFM) topography measurements yield layer thicknesses d of different parts of the flake.
- statistical analysis of central region (circular mask, c. image right): d = 43 ± 3 nm.
- line profiles yield approximate thicknesses of different plateaus
AFM measurements were carried out at University of Goettingen by Ann

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Topography image obtained from AFM measurements (left) and horizontal line profiles with derived layer thicknesses (right)


A microstructured flake of exfoliated black phosphorus was characterized by means of Imaging Mueller Matrix Ellipsometry (IMME). The flake-size was below the x-y-resolution limits of conventional ellipsometers. Thus, IMME is a technique to extend Mueller-matrix analysis of anisotropic 2D-materials to microstrucured samples.

- non-zero off-diagonal MM-blocks of b-P-flake (MM-micromaps) reveal the anisotropy of thin-layer flake
- recorded θ-scans were successfully modeled by assuming a simple orthorhombic crystal structure with biaxial anisotropy for b-P-layer with one axis orthogonal to the sample’s surface
- in-plane complex refractive indices and orientations of in-plane optical axes were obtained for a single wavelength (λ=550 nm) from fitting the model to IMME-θ-scans for three different angles of incidence using predetermined layer thickness (AFM) and real refractive index in z-direction
- in-plane fit results are almost independent of refractive index in z-direction

pectroscopic IMME:
- once having found an ellipsometric model and in-plane orientations of optical axes obtained from IMME-θ-scans, a spectroscopic IMME measurement yielded the optical dispersions of in-plane components of b-P for the wavelength-range of 480-690 nm (assuming constant real refractive index for z-direction)
- MM raw data and fitted dispersions both revealed significant reduction of in- plane anisotropy for spectral range of 600-700nm
- Obtained dispersions might still depend on assumed value for refractive index in z-direction. Additional data at different AOI or sample rotations θ might decouple in-plane and out-of-plane refractive indices upon numerical analysis.


[1] A. Castellanos-Gomez, J. Phys. Chem. Lett., vol.6, no.21 (Nov. 2015)
[2] A. Castellanos-Gomez et al., 2D Materials, vol.1, no.2 (June 25, 2014)