Interlaboratory study on Sb2S3 interplay between structure, dielectric function, and amorphous-to-crystalline phase change for photonics

Summary Antimony sulfide, Sb2S3, is interesting as the phase-change material for applications requiring high transmission from the visible to telecom wavelengths, with its band gap tunable from 2.2 to 1.6 eV, depending on the amorphous and crystalline phase. Here we present results from an interlaboratory study on the interplay between the structural change and resulting optical contrast during the amorphous-to-crystalline transformation triggered both thermally and optically. By statistical analysis of Raman and ellipsometric spectroscopic data, we have identified two regimes of crystallization, namely 250°C ≤ T < 300°C, resulting in Type-I spherulitic crystallization yielding an optical contrast Δn ∼ 0.4, and 300 ≤ T < 350°C, yielding Type-II crystallization bended spherulitic structure with different dielectric function and optical contrast Δn ∼ 0.2 below 1.5 eV. Based on our findings, applications of on-chip reconfigurable nanophotonic phase modulators and of a reconfigurable high-refractive-index core/phase-change shell nanoantenna are designed and proposed.


INTRODUCTION
Phase-change materials (PCMs) that can be switched between amorphous and crystalline phases by thermal annealing or laser irradiation are being exploited in a wide range of reconfigurable photonic platforms that expand from programmable photonics (Stegmaier et al., 2017;Delaney et al., 2021), neuromorphic computing (Feldmann et al., 2019), non-volatile and rewritable data storage (Ríos et al., 2015;Wuttig and Yamada, 2007), to tunable metasurfaces and flat optics with amplitude/phase control (Abdollahramezani et al., 2020), cloaking (Lepeshov et al., 2019), and reflective displays (Carrillo et al., 2019). Most PCMs contain chalcogenides elements such as tellurium (Te) and/or pnictogens such as antimony (Sb), like the prototypical Ge 2 Sb 2 Te 5 (GST) and its alloys (Wuttig and Yamada, 2007). GST is established as a mature technology because of its large optical contrast (Dn = 2.17 and Dk = 0.785 at l = 1,550 nm) (Pernice and Bhaskaran, 2012) and resistance difference (Dr > 1,000 U,m at l = 1,550 nm) between the amorphous and crystalline phases, fast phase switching (nanosecond scale), large number of cycles of reversible transitions (billions of reproducible cycles), and relatively low transition temperature (crystallization temperature T c z 170 C and melting temperature of T m z 600 C) (Wuttig and Yamada, 2007). Nevertheless, GST has a non-null extinction coefficient of the crystalline (k = 1.49) and amorphous phases (k = 0.12) at telecommunication wavelengths, introducing absorption losses that impose limitations to applications in phase modulation schemes where phase control independent of amplitude changes of the propagating signal is required.
In general, as the chalcogenide element increases in atomic number, i.e., sulfur / selenium / tellurium, the bandgap of the PCM tends to decrease. Therefore, Te-based PCMs have rather small gaps, typically below 1 eV, making difficult to realize reconfigurable photonic components at visible frequencies.
Thus, in the last decade, numerous attempts have focused on identifying PCMs with large band gap and a pronounced optical contrast at photon energies higher than 1 eV to enable optical amplitude and phase modulations schemes in the visible range (Mü ller et al., 2021).
In the quest for low-loss and high-bandgap PCMs, the sesqui-chalcogenide antimony sulfide, Sb 2 S 3 , with octahedral-like atomic arrangement and a ratio of glassy and melting temperatures T g (220 C)/T m (540 C) z 0.5 (Wuttig and Yamada, 2007) is gaining increasing interest (Dong et al., 2019;Delaney et al., 2020) also because it has been shown to exhibit topological state transition under pressure (Sorb et al., 2016). Sb 2 S 3 has been reported to have an optical band gap of 2.05 and 1.72 eV for the amorphous and crystalline phase, respectively, and a refractive index contrast Dn z 0.6 with negligible losses as the extinction coefficient, k, is less than 10 À5 in both amorphous and crystalline phases at 1,550 nm (Delaney et al., 2020;Dong et al., 2019). Interestingly, the phase transition temperature of Sb 2 S 3 is accessible by diode laser irradiation as Sb 2 S 3 has a crystallization activation energy of 2.0 eV , which is similar to that of Ge 2 Sb 2 Te 5 (2.3 eV) (Guo et al., 2019). Therefore, crystallization is achieved by thermal annealing Sb 2 S 3 to temperatures around 250 C, whereas amorphization involves heating above its T m of 540 C and quenching to ''freeze-in'' the disordered state. Sb 2 S 3 synthesis over large area has been reported using a wide range of methods including chemical bath deposition (Kondrotas et al., 2018;Cobianu et al., 2021), electrophoretic deposition (Hassam et al., 2021), atomic layer deposition (Kim et al., 2014) and RF sputtering (Gao et al., 2019). In the 1990s, Sb 2 S 3 was widely studied for write-once-read-many times (WORM) optical storage applications, as problems related to sulfur loss upon phase change leading to film degradation were reported Vedeshwar, 1996, 1997). Very recently, Dong et al. (2019) reported the laser amorphous-crystalline-amorphous switching and questioned the WORM classification, although they did not measur the cycle endurance of Sb 2 S 3 . Delaney et al. (2020) showed a decay in reflection change from 5% to 1% over 2,000 cycles of phase-change. On the other hand, very recently, Gao et al. (2021) demonstrated that multi-pulse laser irradiation with low pulse energy can improve cycling durability with respect to single-pulse, high-energy irradiation. Despite further studies on the cycles endurance and reversibility of Sb 2 S 3 are needed, integration of Sb 2 S 3 in photonic circuits is already being exploited in the design of Mach-Zehnder interferometers (MZIs) operating in the C and O communication bands (Faneca et al., 2021).
In order to exploit the potential of Sb 2 S 3 in those applications, it is highly desirable to understand better the changes of structural property and to control the optical contrast that can be realized.
Considering the technological relevance that Sb 2 S 3 is gaining, this work presents results of an interlaboratory study aimed at understanding how the structure and bonding nature of Sb 2 S 3 determines its electronic and optical properties and how those properties change upon the amorphous-to-crystalline transformation triggered by thermal annealing and laser irradiation. In the Round Robin test, Raman spectroscopy was considered for establishing statistically structural fingerprints, as it is a sensitive probe for isostructural transitions (Efthimiopoulos et al., 2016), whereas spectroscopic ellipsometry was exploited to obtain the spectral dependence of the refractive index, n, and extinction coefficient, k, in the broad range of 1,700-190 nm (0.75-6.50 eV), in order to evaluate the spectral applicability of Sb 2 S 3 in reconfigurable devices. Those analyses were further corroborated by extensive chemical (energy dispersive X-ray spectroscopy -EDX; X-ray photoelectron spectroscopy -XPS), morphological (atomic force microcopy -AFM; scanning electron microcopy -SEM) and optical (polarimetry) characterization. Therefore, first, the crystalline c-Sb 2 S 3 and amorphous, a-Sb 2 S 3 , properties are statistically analyzed in order to control and validate the interplay between phase transformation and optical contrast during thermal and laser crystallization. Specifically, independently of the annealing atmosphere and of film thickness, and depending on temperature we have identified two regimes of annealing, namely 250 % T < 300 C resulting in Type-I crystallized spherulitic structural network with an optical contrast Dn z 0.4, and 300 % T < 350 C, resulting in Type-II crystallized bended structural network with different dielectric function and optical contrast Dn z 0.2 at telecom wavelengths. Then, based on the established optical properties, applications of an on-chip reconfigurable nanophotonic phase modulator and of a reconfigurable high-refractive-index (HRI) core/PCM-shell (Si/Sb 2 S 3 ) nanoantenna are designed and proposed.

RESULTS
Single crystal Sb 2 S 3 (c Sb 2 S 3 ): definition of band structure, dielectric function, and Raman spectrum This paragraph aims at defining the reference single crystal, c-Sb 2 S 3 , structure, band structure, and corresponding anisotropic dielectric function. iScience Article As shown in Figure 1A, c-Sb 2 S 3 is a biaxial anisotropic material that crystallizes in an orthorhombic structure of space group symmetry Pnma (no. 62) with lattice parameters a = 1.13107 nm, b = 0.3863 nm, and c = 1.12285 nm (Bayliss and Nowacki, 1972). This Pnma phase resembles a layered structure consisting of (Sb 4 S 6 ) n ribbon-like chains along the short b-axis held together by longer weaker bonds with significant (but not entirely) van der Waals character, as they involve the electrons lone-pair in the 5s 2 orbital of Sbatoms. The coordination number of both Sb-and S-atoms is not straightforward, as can also be inferred from Figures 1A and 1B, as three different S-sites and two different Sb-sites can be identified, with the Sb(1) coordinated to three S-atoms in SbS 3 (E), and the Sb(2) coordinated to five S-atoms in SbS 5 (E) units, being (E) the electrons lone-pair of Sb. Furthermore, those SbS 3 (E) and SbS 5 (E) units also have different kinds of bonds, namely short and strong intra-ribbon bonds and long and weaker inter-ribbon bonds; specifically, Sb(2) in SbS 5 (E) forms intra-ribbon bonds with the five S-atoms and two more inter-ribbon bonds with most distant S-atoms (McKenna, 2021). Those weak bonds as well as the Sb electrons lone-pair introduce Peierls-like distortion (Gaspard, 2016), giving flexibility in reconfiguring the Sb 2 S 3 structure as it will be discussed below.
Because of those various types of Sb-and S-sites and bonds, the Raman spectrum of c-Sb 2 S 3 is very complex, and various spectra are reported in literature, also depending on the excitation laser source. Figure 1C shows the statistical distribution of the relative amplitude of the unpolarized Raman modes of c-Sb 2 S 3 resulting from interlaboratory measurements. According to the group theory, 30 Raman-active modes are expected (Liu et al., 2014), i.e., G = 10 A g + 5B 1g + 10B 2g + 5B 3g . As shown in Figure 1D, in the range 50-350 cm À1 , we can resolve 10 Raman bands assigned according to (Liu et al., 2014), i.e., the modes B 1g /B 3g at 70.4 cm À1 , A g at 89.7 cm À1 and B 2g at 125.9 cm À1 (Ibá ñ ez et al., 2016), A g at 154.5 cm À1 and the modes A g at 191.1 cm À1 , B 1g at 206.6 cm À1 , B 1g at 237.7 cm À1 , A g at 281.4 cm À1 , A g at 303.3 cm À1 , Figure 1. Interplay between structure and Raman spectrum of crystalline Sb 2 S 3 (A) Sb 2 S 3 crystalline structure. Arrangement of the nearest sulfur S(1), S(2), and S(3) atoms for Sb in the two different sites of Sb(1) and Sb(2), forming (B) SbS 3 (E) and SbS 5 (E) units; (E) indicates the Sb electrons lone-pairs. (C) Statistical analysis of the Raman spectra of single crystal Sb 2 S 3 ; the shadow indicates the statistical variation of the relative amplitude of the modes. The red line spectrum represents the mean value.
(D) Raman spectra of single crystal Sb 2 S 3 indicating fitting of the Raman modes with Gaussian line shape function.
(E) Picture of the bulk crystal and of $150-nm-thick Sb 2 S 3 samples as deposited (amorphous) and after annealing at different temperatures (crystallized) for 5 min.

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iScience 25, 104377, June 17, 2022 3 iScience Article and B 2g at 311.2 cm À1 . The 191 and 238 cm À1 bands represent the anti-symmetric and symmetric S-Sb-S bending modes, respectively. Although the 281.4 and 303.3 cm À1 anti-symmetric and symmetric S-Sb-S stretching modes are usually ascribed to the trigonal pyramidal vibration modes of the SbS 3 (E) units (Benjamin et al., 2015), their link with neighboring SbS 3 (E) and SbS 5 (E) units makes the unique assignment of pure symmetric and anti-symmetric stretching modes of SbS 3 and SbS 5 units difficult (Ibá ñ ez et al., 2016). Noteworthy, the unit cell in Figure 1B shows that c-Sb 2 S 3 has 50% of SbS 3 (E) units and 50% of SbS 5 (E) units. From the mean value spectrum resulting from the statistical analysis, we identify the following modes ratio as representative of c-Sb 2 S 3 , namely the A g (303cm À1 )/A g (281cm À1 ) z 1.5 and the A g (303cm À1 )/ A g (191cm À1 ) z 1.5. Considering that the A g (303cm À1 ) clearly has more contributions, we make the hypothesis that the A g (303cm À1 ) is more representative of SbS 5 (E) units, whereas the A g (281cm À1 ) is more representative of the SbS 3 (E) units; this hypothesis will be further supported during the discussion in the next paragraphs on structural modifications during the amorphous-to-crystalline transformation.
Similarly, scattered data about the crystalline Sb 2 S 3 dielectric function are present in literature, as often referred to a crystalline state without clarifying if referring to a single crystal or a polycrystalline material. Interestingly, optical constants were measured at specific wavelengths many years ago on natural cleavage surfaces by Tyndall and Drude (Drude, 1888;Tyndall, 1923). More recently, Shubert et al. (Schubert et al., 2004) reported the complex dielectric function tensor of Sb 2 S 3 obtained by cutting and measuring by generalized spectroscopic ellipsometry the Sb 2 S 3 crystal along the (100) (a-plane), (010) (b-plane), and (001) (c-plane). Six different interband transitions associated with critical points (CPs) in the dielectric function were identified, in good agreement with those measured by Shutov et al. (1969) by polarized reflectivity. Nevertheless, no information about the bands involved in the interband transitions was provided. In order to establish a correlation between the dielectric function and interband transitions based on the band structure and the dipolar transition matrix elements between occupied and unoccupied single-electron eigenstates, the complex dielectric function has been calculated for c-Sb 2 S 3 using first-order time-dependent perturbation theory as implemented in SIESTA . Figures 2A and 2B show the real,˛1, and imaginary,˛2, part of the complex dielectric function,˛=˛1 + i˛2, calculated for c-Sb 2 S 3 for polarization parallel to the crystallographic axes (E||a, E||b, E||c). Noteworthy, the calculated dielectric function is in good agreement with those reported by Schubert et al. (2004). A critical points analysis by using the minimum of the second derivative spectra of the dielectric function, d 2 1 /dE 2 and d 2˛2 /dE 2 (not shown here) calculated numerically reveals CPs in agreement with the experimental measurements by Schubert (Schubert et al., 2004), as shown in the table at the bottom of Figure 2. For the assignment of the CPs, Figure 2C shows the orbital-projected bands (fatbands) and the projected density of states (PDOS) of bulk Sb 2 S 3 representing the contribution of the S 3p, Sb 5s, and Sb 5p orbitals to each band. We notice that the highest valence bands are comprised primarily of S orbitals with 3p character and a minor contribution of a mixing between Sb 5s and Sb 5p levels and the featured structure of the PDOS at around À1 eV has being suggested to indicate the formation of electron lone-pairs on the Sb-ions by Carey et al. (2014). The valence band maximum (VBM) is slightly displaced from the G-point in the high symmetry line GZ (this direction corresponds to the c-axis of the (Sb 4 S 6 ) n chains perpendicular to the zigzag arrangement). Likely, the conduction band minimum (CBM) is located at the same high symmetry line but closer to the Z-point. This results in an indirect fundamental band gap of 1.30 eV and a higher direct gap of 1.33eV. Our results are in agreement with previously reported first principles calculations using PBE functional (Sun et al., 2008;Carey et al., 2014), although the location of the VBM and CBM in the Brillouin zone does not have total consensus throughout the related bibliography. The shape of the band structure resembles the anisotropy of the crystal structure of Sb 2 S 3 , having flatter bands in directions along a-and c-axes and larger band widths perpendicular to both.
Considering the different contributions of the Sb and S atomic orbitals as in Figures 2D-2F), the following interband transitions can be assigned to the CPs, as in the table. The peak E 1 involves direct transitions near the G-point (in the GY direction) for all polarization directions between the highest valence bands and lower conduction bands comprising mostly occupied S3p and Sb5s orbitals and empty Sb5p. The peak labeled E 2 presents contributions spread over the momentum space, weaker for E||a case with respect to perpendicular polarizations. Direct transitions occur between valence bands at À0.9 eV and higher conduction bands around 1.5 eV and also between the lower valence bands at À1.5 eV and conduction bands in the minimum valleys. The peak E 3 is also composed of transitions delocalized in k-space. The contributions of E||b and E|| c are larger with respect to E||a polarization; for the first case, the stronger transition probabilities arise ll OPEN ACCESS close to G, Z and X-points. The peak E 4 comprise vertical excitations from low lying valence bands (<À1 eV) and unoccupied conduction bands above 1.5eV mostly, also showing a wide distribution over k-space. Peaks E 5 and E 6 present relative lower transition probabilities with respect to the formers;, it is noticeable that intense contributions to E 5 are found for E||a polarization in GY and XS directions.

Amorphous Sb 2 S 3 films (a-Sb 2 S 3 ): interlaboratory study of the structural (Raman) and optical (ellipsometry) properties
This paragraph aims at defining and validating the structural and optical fingerprints of a-Sb 2 S 3, namely the dielectric function and the corresponding Raman spectrum. The XPS analysis revealed that the analyzed a-Sb 2 S 3 films were almost stoichiometric with an S/Sb atomic ratio of 1.45 G 0.05 estimated by both XPS and EDX analyses. Figure 3A shows the statistical analysis of Raman spectra of a-Sb 2 S 3 films from two different batches (Batch #1 (B-1) and Batch #2 (B-2)) produced 6 months apart on various substrates of glass, sapphire, and Si(100) measured in different laboratories with different instruments. The Raman spectra of a-Sb 2 S 3 show good reproducibility and is characterized by a broad and intense mode at z 285 cm À1 , which could originate from the mixture of stretching and bending modes of distorted SbS 3 (E) and SbS 5 (E) units and, hence, from the envelope of broadened A g at 281.4 cm À1 , B 1g at 303.3 cm À1 , and A g at 311.2 cm À1 vibrational modes. Indeed, a-Sb 2 S 3 has larger Peierls-like distortion (Gaspard, 2016) than c-Sb 2 S 3 , with bonding angles deviated slightly from 90 . Specifically, it has been reported that chalcogen-centered S-bonds slightly shift to larger angles (94.11 ), whereas the corresponding Sb-centered (C) Orbital-projected band diagram and density of states of Sb 2 S 3 on the S 3p, Sb 5p, and Sb 5s shells. The size of the marker is proportional to the strength of the contribution. Band diagram of Sb 2 S 3 representing the bands contributing to the interband transitions responsible of the CPs in the dielectric function for polarizations along the crystallographic (D) a-axis, (E) b-axis, and (F) c-axis (the size of the symbol is a measure of the probability amplitude). The table at the bottom summarizes the energy of the main critical points also compared with literature data from Schubert et al. (2004) and Shutov et al. (1969)).

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iScience 25, 104377, June 17, 2022 5 iScience Article angles decrease to 88.81 (Xu et al., 2021). Interestingly, such structural distortions are responsible for the wider band gap of a-Sb 2 S 3 than c-Sb 2 S 3 and for the different electronic and optical properties, as measured by ellipsometry. Figure 3C shows the statistical variation of the energy-dependent behavior of the real, ε 1 , and imaginary, ε 2 , parts of the isotropic a-Sb 2 S 3 dielectric function, resulting in a bandgap of 2.2 G 0.1 eV; all analyzed a-Sb 2 S 3 samples behave in a similar way, regardless the thickness; the shadowed area indicates the effect of the local Peierls-like distortions on the dielectric function. The isotropy of the a-Sb 2 S 3 dielectric function was proven by imaging polarimetry, as the Mueller Matrix measured on different a-Sb 2 S 3 films revealed the identity matrix, typical of isotropic materials, as shown in Figure S1 (Gutierrez et al., 2022).
The a-Sb 2 S 3 index of refraction (n) and extinction coefficient (k) are compared with literature data in Figure 3D.
The different amplitude of the curves can be explained by the different contribution of voids resulting from a large number of Sb-atoms electrons lone-pair (Xu et al., 2021), depending on the stoichiometry of the deposited amorphous samples. Furthermore, more voids mean larger Peierls-like distortions, therefore, emphasizing the role of those distortions on the a-Sb 2 S 3 dielectric function. Specifically, the strength of an optical transition and, hence, the maximum of ε 2 , decreases with increasing Peierls distortion, i.e., shortening Sb-S bonds. Therefore, tailoring Peierls-like distortion in the a-Sb 2 S 3 could be a way to increase the optical contrast between the amorphous and crystalline phases. (B) Topographical images recorded by SEM (gray) and AFM (colored) in various laboratories for a-Sb 2 S 3 films. (C) Statistical variation of the energy-dependent behavior of the real, ε 1 , and imaginary, ε 2 , parts of the a-Sb 2 S 3 dielectric function.
The surface topographical images recorded by SEM and AFM in various laboratories for a-Sb 2 S 3 films with thicknesses of z150 nm, shown in Figure 3B, confirms the presence of voids, with films characterized by a grainy-like morphology. The measured grain size and roughness (root-mean-square roughness, RMS) are statistically as 30 G 5 and À3.5 G 0.5 nm, respectively. The irregular shapes of the grains stem from the fact that at room temperature during deposition, the kinetic energy is not enough to induce the coalescence of the grains (Tigau, 2007).
Sb 2 S 3 amorphous-to-crystalline phase change by thermal annealing and laser crystallization Here we address the interplay between structure and optical properties induced by both thermal and laser crystallization. Figure 4 shows the representative optical, SEM, and AFM images of crystallized Sb 2 S 3 upon thermal annealing. The annealing was run under various conditions in different laboratories (see STAR Methods). Noteworthy, data from the various laboratories could be grouped in two different kinds of structures/morphologies independently of the annealing in vacuum, argon, and air, revealing the role mainly of the annealing temperature (T = 250-300 C and 300-350 C) on the evolution of structure upon thermal crystallization. Specifically, Type-I crystallized c-Sb 2 S 3 , with an almost unchanged S/Sb ratio of 1.40 G 0.05 as from XPS analysis, consists of spherulitic crystals as in Figures 4C and 4D (Shtukenberg et al., 2012;Crist and Schultz, 2016) surrounded by randomly oriented smaller crystallites as in Figure 4E. The spherulites consist of dense fibrils radially grown from a single nucleation point, as can also be seen in the AFM image of Figure 4D. This spherulitic thermal crystallization of a-Sb 2 S 3 films is consistent with previous work (Bolotov et al., 1970;Sokol et al., 1986;Gutierrez et al., 2022). Those spherulitic faceted crystallites obtained in the range 250-300 C increase in the size and uniformity of distribution as well as compactness with the increase in annealing temperature, as shown in Figures 4C (T = 300 C) and 4E (T = 250 C). The grain boundaries and surface features are clearly seen in the AFM and SEM images in Figures 4B-4F, with large grains of the order of tens of micrometers. Interestingly, Type-I c-Sb 2 S 3 is characterized by a very low surface roughness (RMS = 1.9 G 0.3 nm).
The Raman spectrum of Type-I crystallized Sb 2 S 3 was statistically studied as shown in Figure 4A. Large amplitude variations characterize the modes at 191, 281, and 303 cm À1 . Interestingly, the statistically analysis of the intensity ratio between modes at 303 and 281 cm À1 has an average value A g (303cm À1 )/ A g (281cm À1 ) z 1. The role of the crystallites orientations on the Raman amplitude is further emphasized in Figure 4G, where the Raman maps of the modes at 191, 281, and 303 cm À1 taken in a region of randomly oriented crystallites exposing different faces show a different amplitude of the modes. The large spherulitic crystals as those considered in the map of Figures 5G-5I are characterized by the A g (303cm À1 )/ A g (281cm À1 ) z 1.5 similarly to the c-Sb 2 S 3.
Type-II crystallized Sb 2 S 3, which resulted in a changed S/Sb ratio of 0.95 G 0.05 from both XPS and EDX analyses, shows banded patterns spherulites (see Figure S4), characterized by periodic rings with a width of approximately 10 mm as shown in Figures 5G-5I. Careful SEM and optical inspections as in Figures 5D-5G show hexagonal and octahedral Sb and Sb 2 O 3 (see Raman analysis Figure 5A) crystals localized on the white bright bands. Statistical Raman analysis on the different bright and dark bands shows significant differences in the intensity of the Raman modes at 191, 281, and 303 cm À1 in those bands. Specifically, Raman spectra at the dark bands are characterized by an intensity ratio A g (303cm À1 )/A g (281cm À1 ) >1. Conversely, Raman spectra at the bright bands are characterized by A g (303cm À1 )/A g (281cm À1 ) < 1 with an increased intensity of the mode at 191 cm À1 . This is further shown in Figure 5I by the Raman maps of the modes at 191, 281, and 303 cm À1 taken in an area comprising both dark and bright bands. These maps clearly show a higher intensity of the modes at 191 and 281 cm À1 and lower intensity of the 303 cm À1 mode at the bright bands. This behavior is inverted for the dark bands. The AFM of Type-II crystallized Sb 2 S 3 also shows different morphologies and roughness of the bright (RMS = 5.1 nm) and dark bands (RMS = 4.0 nm), which is higher than the RMS of Type-I. Thus, from this contrast of bands, we infer the mobility of Sb atoms along different crystallographic directions. On this, more detailed study is in progress to identify them.
A further insight into the structural and optical properties of the Type-I and Type-II c-Sb 2 S 3 was obtained by imaging polarimetry of the thermally crystallized Sb 2 S 3 films. A differential analysis of the Mueller matrix (MM) was performed using the Mueller Matrix Differential Decomposition (Arteaga and Kahr, 2013), from which the linear and circular birefringence and dichroism of the crystallized film can be obtained. Specifically, the dichroic properties are owing to the anisotropy of extinction coefficient Dk p,q for two ll OPEN ACCESS iScience 25, 104377, June 17, 2022 7 iScience Article orthogonal polarization states (p,q). The birefringent properties stem from the anisotropy in the real refractive index Dn p,q , entailing different phase shifts for two orthogonal polarization states (Gil Pé rez and Ossikovski, 2017). Therefore, expressions for linear dichroism in the x-y axes, linear dichroism in the 45-135 and circular dichroism (LD xy , LD 45 and CD), and linear birefringence in the x-y and 45-135 axes and circular birefringence (LB xy , LB 45 and CB) can be written as  Figure S1, for Type I and Type II c-Sb 2 S 3 , strong signals are observed in the non-diagonal elements of the MM, especially on the elements m 24 , m 42 , m 34 and m 43 as seen in Figures S2 and S3. As shown in Figure 6, through the MMDD, the LD xy , LD 45 , CD, LB xy , LB 45 , and CB parameters were extracted from the MMs.
In the case of Type I in Figure 6A, these parameters show a strong birefringent behavior (LB s 0) with a lower LD contribution arising from the different refractive index and extinction coefficients associated iScience Article with the dielectric tensor of Sb 2 S 3 (Schubert et al., 2004). Both CD and CB are negligible. Cross-polarization micrographs reveal big crystalline domains showing a slightly distorted maltase cross-pattern (i.e., the four dark perpendicular cones diverging from the center of the crystal) typical of spherulites (as those in Figure 4C) (Bolotov et al., 1970;Gutierrez et al., 2022;Sokol et al., 1986). Therefore, Type-I c-Sb 2 S 3 has a polycrystalline texture consisting of randomly oriented micron-size crystallites surrounding larger spherulitic crystalline domains.
For Type II, polarimetric measurements in Figure 6B support a crystallization process forming banded spherulitic crystals, as confirmed by the banded Maltese cross pattern seen in the cross-polarization. The MMDD for Type-II shows strong signals of linear birefringent and dichroism taking alternating signs between consecutives bands. Similar behavior, although with lower signals, can be seen in both CD and Both Type-I and Type-II c-Sb 2 S 3 show local birefringent and dichroic behavior within the microscopic crystalline domains. Nevertheless, at the macroscopic level, considering the radial symmetry of spherulites as well as the randomly oriented microcrystallites, they result in an overall isotropic response for the crystallized film, supporting the isotropic dielectric function determined by ellipsometry.
Those two types of crystallization and structures can induce a different optical contrast during the amorphous-to-crystalline transformation. This is confirmed by the different optical properties reported in Figure 7. Figure 7A shows the real, ε 1 , and imaginary, ε 2 , part of the dielectric function of Type-I and Type-II c-Sb 2 S 3 as determined from the ellipsometric analysis. Interestingly, ε 2 of spherulitic Type-I c-Sb 2 S 3 shows broadened CPs and a shape of ε 2 similar to that along the a-axis (see Figure 2), leading to an optical contrast of Dn $ 0.4. Consistent with the different structure highlighted in Figures 5 and 6, a different dielectric function spectrum characterizes Type-II c-Sb 2 S 3 . By comparing those spectra with c-Sb 2 S 3 in Figure 2, it can be inferred that Type-II crystallized samples have a dielectric function resembling the average of the three components along the three axis of the dielectric tensor in Figure 2 (consistently with the bending scheme in Figure S4), leading to an optical contrast during the amorphous-to-crystalline transition of Dn $ 0.2 in the spectral region <1.5 eV. The spectra of the refractive index, n, and extinction coefficient, k, are compared with data from literature in Figures 7C and 7D, whereas the values of absorption edge are compared with literature data in Table 1. iScience Article Laser crystallization has also been investigated using green CW lasers (514 and 532 nm) with wavelengths above the absorption edge of both c-Sb 2 S 3 and a-Sb 2 S 3 , operated at different power. Raman spectra taken after laser irradiation at increasing power are shown in Figure 8. Comparing the Raman spectrum for the lower power of 2.7 mW with those of Type-I and Type-II thermally crystallized samples, it can be noticed that, statistically, the A g (303cm À1 )/A g (281cm À1 ) < 1 and the modes below 150 cm À1 are predominant than the 191 cm À1 mode, similarly to the Raman spectra of Type-II bright region, as also confirmed by the AFM morphology shown in Figures 8C-8E, which is similar to the morphology reported in the inset of Figure 4C. Therefore, we can infer that laser crystallization results in Type-II-like bright regions, consistent with the fact that the segregation of Sb clearly appears by increasing the laser power, similarly to the increase of temperature above 300-350 C. Optical and AFM images in Figures 8B-8D also show that for laser powers higher than 7.4 mW, ablation of Sb 2 S 3 at the center of the laser's Gaussian intensity profile occurs, indicating that the local temperature may exceed 450 C. At T > 450 C, we have also observed thermally the loss of samples'mass, with the complete loss of the Sb 2 S 3 at T $ 550 C (Makreski et al., 2013). This underlines the difficulty to reach the melting temperature of Sb 2 S 3 of 540 C without the loss of material, which could be a challenge for the reproducibility over a large number of cycles of Sb 2 S 3 phase change.

DISCUSSION
Sb 2 S 3 crystallization by thermal annealing and green laser irradiation has highlighted two main regimes and structures. A deeper insight into those regimes can be further gained by the XPS analysis reported in Figure 9. Specifically, the fitting of the Sb 3d3/2 photoelectron core level indicates three main states of Sb, namely the Sb +5 at binding energy (BE) of 540.3 eV owing to the SbS 5 (E) units, the Sb +3 at BE = 539.9 eV owing to the SbS 3 (E) units, and the Sb-electrons lone-pair at the low BE = 538.9 eV (Stamenkovi c et al., 2021). The a-Sb 2 S 3 has a certain relative volume fraction of those units that of course depends on the disorder induced by the deposition method and conditions, with a predominance of SbS 3 (E) units in our case as inferred by the higher Sb +3 fitting component (the perfect c-Sb 2 S 3 has 50% of SbS 3 (E) and SbS 5 (E) units -see Figure 1). Interestingly, Type-I crystallized samples show a ratio of the SbS 5 (E)-(40%) and SbS 3 (E)-(46%) units approaching the ideal 1:1 ratio of c-Sb 2 S 3 , consistent with the Raman A g (303cm À1 )/A g (281cm À1 ) $ 1 (see Figure 4). Conversely, Figure 9 indicates that Type-II crystallized samples show a ratio of the SbS 5 (E)-(22%) and SbS 3 (E)-(51%) units lower than 1, consistent with the Raman A g (303cm À1 )/A g (281cm À1 ) < 1 (see Figure 5). Therefore, we can infer that thermal annealing at 250 C < T < 300 C yields a crystallized structure with SbS 5 (E) and SbS 3 (E) units close to the single crystal structure with large spherulitic crystals as in Figures 4C and 4D, resulting in the absence of the oxidation or degradation of this kind of crystallized structure during the phase transformation. This also supports the use of Type-I Sb 2 S 3 without any capping layer, as in the present study.
Conversely, Type-II crystallization by thermal annealing at T > 300 C or laser irradiation, because of some loss of sulfur, results in a reduced presence of SbS 5 (E) units and the appearance of Sb-segregation that further oxidizes.
However, the possibility to switch between contrasts, i.e., Type-I and Type-II in controlled manner gives extra freedom in terms of photonic memory and the devices.
Those two identified regimes, and the negligible oxidation at T < 300 C, are consistent with a previous study on the kinetics of the oxidation of Sb 2 S 3 at temperatures from 300 C up to 500 C, i.e., below its iScience Article melting temperature (Padilla et al., 2014). Specifically, oxidation of Sb 2 S 3 starts at 350 C and its rate increases by more than two orders of magnitude to 500 C, staying negligible in an N 2 /O 2 atmosphere at T < 250 C. As supported by the change of the S/Sb ratio by XPS ad EDX, the oxidation at T > 350 C is due mainly to the loss of sulfur, and mainly occurs at the surface of Sb 2 S 3 , whereas Type-II crystallization homogenously involves the rest of the film depth. In this direction, crystallization could be improved even at T > 350 C by a capping layer, although this study is still ongoing and requires optimization. Conversely, for GST, the oxidation is more problematic and related to the different reactivities of Ge, Sb, and Te, i.e., the oxygen exposure leads to a depletion of Te from GST surface and heterogeneous formation of the Sb-and Ge-oxides, with Sb-oxides more in depth and Ge-oxides more toward the surface (Golovchak et al., 2015).

Reconfigurable nanophotonic phase modulator on-chip
High-performance, low-loss, compact reconfigurable photonic devices on-chip are essential building blocks of modern adaptive optical network (Cheng et al., 2018;Feldmann et al., 2019Feldmann et al., , 2021. Compact and energy efficient phase modulator represent the first evolutionary step from reconfigurable devices into programmable circuits. Thus, our goal is the realization of low-loss nanophotonic phase modulator on-chip based on pioneerinvestigated PCM cell transition a-Sb 2 S 3 4 c-Sb 2 S 3 . Controllable optical modulator is based on Si 3 N 4 MZI on-chip, where the input optical signal splits into two arms by multimode interference (MMI) junction, iScience Article after a certain distance both modes recombine at the output MMI combiner in a single arm producing the interference of both beams as in Figure 10B. Output intensity of the signal depends on the phase difference between optical modes in two branches. The establishment of reconfigurable PCM cell atop one of the arms of MZI results in the relative phase shift of the carried optical signal.
The design of the established nanophotonic device is based on the numerical investigation of the evanescent interaction between the transverse electric (TE 0 ) mode in the waveguide and integrated PCM cell atop it. The reversible switching of PCM phase state generally can be triggered by appropriate heat stimulation via evanescent part of intense guided mode (Feldmann et al., 2019(Feldmann et al., , 2021Ríos et al., 2015;Stegmaier et al., 2017) or by help of extra low-loss microheaters deposited in the vicinity of PCM cell (Wu et al., 2019;Zheng et al., 2020bZheng et al., , 2020a. Joule heating elevates the temperature of PCM cell above the melting point following quenching results in an amorphous phase-state, whereas an increase in the temperature above the glass transition results in a crystalline phase-state. The modeled evanescent interaction between the nanoguide mode and atop Sb 2 S 3 cell in the amorphous and crystalline states are shown in Figure 10A. We benefit from utilization Sb 2 S 3 in the visible and infrared (IR) region owing to negligible optical loss of both states and good optical contrast ( Figure 7B). Therefore, we consider reconfigurable MZI phase modulators equipped with PCM cell for wide application range modulation of the signal in the visible and IR region, in particular at telecommunication band wavelength l = 1,310 nm where standard optical fiber exhibits zero chromatic dispersion and at telecommunication wavelength l = 1,550 nm obtaining minimum optical propagation loss. Upon phase transitions a-Sb 2 S 3 4 c-Sb 2 S 3 , the guided mode is lifted up owing to the refractive index change of PCM resulting in the modification of the mode phase.
The essential parameter to be determined is required PCM length L p needed to attain p-phase difference between optical modes in both branches as a result of the switching between cell states thus ensuring amplitude key-switch (switching on/off) of the output signal with corresponding extinction ratio at necessary wavelength. Simulation results are shown in Figure 10C, where owing to higher contrast of the refractive index of amorphous and crystalline states of Sb 2 S 3 at visible wavelength range required thickness of PCM cell is lower to obtain p-shift in comparison with IR guided modes. iScience Article cell in another arm -in the crystalline state and vice versa), which can be obtained by utilization controllable individual microheaters (Ovvyan et al., 2016).
Another improvement strategy of the switching the states of PCM cell is patterning it into subwavelength-nanostructures atop the center of the waveguide above the maximum electric field of the nanoguide mode ( Figure 10B) following in highly uniform heating of each micro-cell, resulting in an increase of the switching contrast and a decrease of required PCM area in comparison with the unpatterned cell (Wu et al., 2019).
Further potential application of explored PCM material is the introduction of a phase-gradient Sb 2 S 3 metasurface atop photonic device, resulting in programmable broadband mode converter for realization unique scalable optical functionalities (Karvounis et al., 2016;Li et al., 2017;Wu et al., 2021).
The cycling endurance of Sb 2 S 3 is a matter of ongoing investigation and optimization worldwide. Presently, from literature, the cycling record reported for Sb 2 S 3 varies from 30 cycles for a phase change degree (PCD) of 90%-7,000 cycles for PCD of 20% (Gao et al., 2021), clearly indicating that work is needed  (Kuzum et al., 2012). Noteworthy, the number of cycles is not the only relevant parameter to consider.
Indeed, it should be noted that the absorption losses of both amorphous and crystalline phases of GST PCM cell leads to fundamental limitation phase modulation circuits, namely it prevents optical phase control independently of changes in the amplitude of propagated light through the photonic devices in the telecommunication band. Therefore, one of the ways to build efficient lossless phase modulator based on integrated PCM cell on one of the arms of the MZI interferometer is Sb 2 S 3 , which shows extremely low insertion loss (k, is less than 10 À5 ), whereas remaining non-volatile at operating temperatures, where crystallization and amorphization temperature are comparable with GST.
Furthermore, another potential benefit of Sb 2 S 3 is the existence of two crystalline states of Sb 2 S 3 (Type-I and Type-II) explored in this work, which enables the multilevel operation of the phase modulator as well as provides extra degree of freedom of non-volatile memory enabling applications of hybrid photonic devices with integrated PCM cells.

Design and modeling of an Sb 2 S 3 as high refractive index dielectric reconfigurable antenna
Dielectric nanoparticles, with a high refractive index (HRI), are able to locate and intensify electromagnetic fields with Joule losses almost null. For characteristic incident wavelengths, confined displacement currents lead to resonances known as whispering gallery modes. These can be understood as rays enclosed and traveling in a spherical cavity supported by multiple total internal reflections. According to the Mie theory, these resonances can be either of electric or of magnetic character and can be independently excited by changing the incident wavelength. For some situations, they can overlap in certain spectral regions, leading to coherent effects between the scattered electromagnetic fields. These effects produce characteristic angular distributions of the scattered electromagnetic energy; hence, HRI nanoparticles can be used to control the direction of the scattered radiation. When those coherent effects are predominantly between electric and magnetic dipole resonances, interesting directionality phenomena can be found, like a minimum scattering in the forward direction or a null backscattered intensity phenomenon.
In the literature, these are known as the Kerker conditions (Kerker et al., 1983), light is scattered predominantly either in the forward direction, being null in the backward direction (first Kerker condition or zero backward (ZB)) or mostly in the backscattered direction, being minimum in forward (second Kerker condition or minimum forward (MF)). In general, there are other coherent effects owing to interference with higher order resonances that lead to a wider gamut of directionality effects (Tribelsky et al., 2015). Consequently, HRI dielectric nanoparticles can be considered as nanoantennas whose radiation directionality can be controlled depending on the incident wavelength, the external medium, and the geometry of iScience Article the nanoparticle (Geffrin et al., 2012). Here, a core-shell geometry schematized in Figure 12 is analyzed, with the core made of silicon (a conventional HRI material in the NIR; García-Etxarri et al., 2011) and the shell made of Sb 2 S 3 as PCM, whose optical properties depend on its phase (amorphous or crystalline) as described in Figure 7. Depending on the phase of Sb 2 S 3 , the direction of the scattered light can be either in the forward or in a mixture of both directions (see Figure 12A). A core-shell geometry is selected to have enough degrees of freedom to obtain the suitable conditions to generate the desired directionality effects, as the optical properties of the phases of Sb 2 S 3 do not permit by themselves to get these conditions from a single nanoantenna made only of this material.
The internal and external radii have been optimized to get directional switching of the scattered radiation.
Here we consider the external radius of 90 nm and the internal radius of 61 nm. The first and second Kerker conditions for both phases (either amorphous or crystalline) are plotted as a function of the nanoantenna internal radius and the incident wavelength in Figure 12B. For an internal radius of 61 nm and an incident wavelength of 590 nm, both MF and ZB conditions coincide for both phases. Therefore, by fixing these parameters, a reconfigurable nanoantenna could be designed with the scattering diagram for an incident wavelength of 590 nm shown in Figure 12C. For the crystalline phase, radiation is scattered in both forward and backward directions, whereas for the amorphous phase, light is scattered predominantly in the forward semi space direction, being almost null in the backscattered direction. It is important to point out that Kerker conditions are only valid when coherent effects between the electrical and magnetically dipole resonances are predominant. In this case, higher order resonances have to be considered (in this case, the quadrupolar contributions is not negligible. See Mie coefficients a 2 and b 2 (Bohren and Huffman, 1998) in Figure 12D. Nevertheless, a very efficient switching effect in the backward direction can be achieved for these optimized parameters by controlling the crystallinity of Sb 2 S 3 . Figures 12D and  12E show the extinction efficiency (Q ext ) together with the electric and magnetic dipole modes contribution (a i and b i Mie coefficients; Bohren and Huffman, 1998) for the amorphous and crystalline phases, respectively.
The angular scattering diagrams of the proposed core-shell nanoantenna are plotted for the two phases of Sb 2 S 3 in Figure 13. In this case, the intensity of the scattered radiation has been integrated for every scattering angle to calculate the percentage of radiation scattered in either the forward semi space or the backward semi space, as indicated in Figure 13 by arrows. A switching effect by changing the Sb 2 S 3 phase is clearly obtained in the forward and backward directions. For the former, we get a contrast ratio Am/Cr of 2 (3 dB), and for the latter, we get a contrast ratio of 0.17 (8 dB). Furthermore, it is important to point out that for the crystalline phase, the nanoantenna acts as a ''beam splitter'' by dividing the scattered power between the forward and backward semi-spaces. The amorphous/crystalline cycle endurance of Sb 2 S 3 has not been measured here. This study does not make claims about the reversibility over a high number of cycles of phase changes. Based on our methodology, we analyze statistically the structural and corresponding optical changes only over one cycle, as after the reported crystallization, we went back to the same Rama spectra of the amorphous phase.

STAR+METHODS
Detailed methods are provided in the online version of this paper and include the following:

DECLARATION OF INTERESTS
The authors declare no competing interests.

Chemical and morphological characterizations
Scanning electron microscopy (SEM) was carried out for the morphological characterization of the samples with a Zeiss Supra 219 40 FEG SEM equipped with a Gemini field emission gun. Energy-dispersive-X-ray analysis (EDX) was used to check elemental composition of the films. Analyses were carried out at an extraction voltage of 3 kV and 221 a 30-mm aperture.
The elemental composition was cross-checked by x-ray photoelectron spectroscopy (XPS). XPS measurements were carried out by a Scanning XPS Microprobe (PHI 5000 Versa Probe II, Physical Electronics) equipped with a monochromatic Al Ka x-ray source (1,486.6 eV), with a spot size of 200 mm. Survey (0-1,200 eV) and high-resolution spectra (C1s, S2p, S2s, Sb3d) were recorded in FAT mode at a pass energy of 117.40 and 29.35 eV, respectively. Spectra were acquired at a take-off angle of 45 with respect to the sample surface. Surface charging was compensated using a dual beam charge neutralization system, and the hydrocarbon component of C1s spectrum was used as internal standard for charging correction, and it was fixed at 285 eV.
Morphology was investigated by atomic force microcopy (AFM) in the interlaboratory study. The topography images were acquired in three different laboratories using two instruments of AFM NX10 from Park Systems in non-contact mode using PPP-NCHR tip probe with radius of curvature less 10 nm, and Autoprobe CP (Thermomicroscope) used also in non-contact single-pass mode using a gold-coated Si tips (their frequency is $80 Hz).

Raman spectroscopy
In the interlaboratory study, five different Raman setups in confocal configuration were used at different laboratories for unpolarized Raman spectra acquisition. (i) A LabRam Horiba instrument operating with a 532 nm wavelength laser and a 3100 microscope objective (NA = 0.9). (ii) A Invia Reflex Renishaw Raman instrument was used operating with a 532 nm wavelength laser and a 3100 microscope objective (NA = 0.9). (iii) A Invia Reflex Renishaw Raman instrument was used operating with a 514 nm wavelength laser and a 3100 microscope objective (NA = 0.9); (iv) A Xploraplus Horiba Raman operating with 532 nm laser and a 3100 microscope objective (NA = 0.9); (v) A Horiba LabRam Aramis VIS operating at 532 nm with a3100 microscope objective (NA = 0.9) was used. The size of the laser spot on the sample was approximately 1 mm.
For the statistical analysis of the interlaboratory measurements ten random Raman spectra for each sample were taken in each laboratory. The laser power was 0.2 -2 mW for the characterization f materials without inducing any phase change.

Spectroscopic ellipsometry
In the interlaboratory study, two different spectroscopic ellipsometers were used for the optical characterization at the various laboratories. (i) A phase-modulated spectroscopic ellipsometer (UVISEL Horiba) operating in the 0.75 -6.0 eV energy range with a resolution of 0.05 eV and with the angle of incidence varying in the range 55 -70 . (ii) A J.A. Woollam VASE M-2000 spectrocopic ellipsometer operating in the spectral range 193 nm to 1690 nm at variable angles from 55 to 75 . All measurements were performed at ambient conditions and at room temperature.
Ellipsometric spectra of the pseudodielectric function were analyzed by a simple isotropic model substrate/Sb 2 S 3 /surface roughness. The substrate was experimentally measured before the deposition; the surface roughness was modelled by a Bruggeman effective medium approximation (BEMA) of 50% Sb 2 S 3 +50%voids with the thickness set at the RMS value from AFM.
The a-Sb 2 S 3 was parameterized with a Tauc-Lorentz dispersion equation (Jellison and Modine, 1996); the Tauc-Lorentz parameters and the thickness of the film were the fit parameters.
The crystallized Sb 2 S 3 films were parameterized with an ensemble of Tauc-Lorentz oscillators for the different electronic transitions of Sb 2 S 3 as in Figure 2. The best fit was obtained by three oscillators,

Polarimetry
Polarimetric measurements were performed with an adapted microscope set-up (Nikon Eclipse LV-N). The microscope base structure includes a monochromatic unpolarized light source (CoolLED pE-100, 633 nm), a polarizer (P1) before de sample holder, objectives (5x, 20x and 50x with NA of 0.15, 0.4 and 0.6 respectively) and an analyzer (P2) before the CCD camera (The Imaging Source DMK 33UX174). To achieve the measurement of the Mueller Matrix (MM) with the microscope two l/4 waveplate retarders (R1 and R2) with controllable angular orientation of the fast axes were included. One after P1 and another before P2. Both retarders are identical (Thorlabs WPH05M-633) and are mounted onto motorized rotation stations (Thorlabs PRM1/MZ8) controlled by an ordinary laptop. We worked out a MATLAB routine that controls the angular position of R1 and R2 fast axis synchronously with the acquisition process of the camera. The routine consisted in measuring N images, with angular increments in the fast axis orientation of 180 /N for R1 and 900 /N for R2 (5:1 speed ratio) between each image acquisition. As a result, in each pixel of the camera we measured a full Fourier measurement cycle that was processed to obtain the corresponding MM of a given part of the sample. As a result, a polarimetric image is obtained, i.e., an image that describes with spatial resolution what changes are produced in the light polarization by a sample. The full process (data acquisition plus post processing) took less than 20 minutes.
A differential analysis of the Mueller matrices using the Mueller Matrix Differential Decomposition (MMDD) was performed (Arteaga and Kahr, 2013). The differential matrix m relates the Mueller matrix M as dM dz = mM (Equation 2) If the differential matrix, m does not depend on z, i.e., distance travelled along the direction of propagation, the solution of this differential equation is found by taking the logarithm of M, thus, m = lnðMÞ (Equation 3) The most general form of the differential Mueller matrix for a non-depolarizing medium is given as follows (Azzam, 1978;Ossikovski, 2011;Arteaga and Kahr, 2013) m = 0 B B @ a À LD À LD 0 CD À LD a CB LB 0 À LD 0 À CB a À LB CD À LB 0 LB a 1 C C A (Equation 4) where LD, LD 0 and CD are linear dichroism in the x-y axes, linear dichroism in the 45-135 and circular dichroism. LB, LB 0 and CB stand for linear birefringence in the x-y and 45-135 axes and circular birefringence, respectively. The value a is the absorption coefficient.