Anisotropic Materials

About the Anisotropic Materials

Anisotropy is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties (absorbance, refractive index, conductivity, tensile strength, etc.). Currently, anisotropic materials are widely used in catalytic thin films, thermoelectric devices.... Studying the optical properties of anisotropic materials is essential for applications.

Figure 1(a) is the 3D spatial structure of SnS material, a typical material with anisotropic properties. SnS has a similar structure to black phosphorus. The difference in structure is clearly shown in Fig. 1(b), Brillouin structure, all three axes a, b, and c are different resulting in different dielectric functions shown in Figs. 1(c) and 1(d).

Figure 1. (a) Atomic structure of SnS, (b) Brillouin zone of the SnS structure, (c) Change in dielectric function from the zigzag-direction to the armchair-direction, at room temperature, (d) Mapping of the full azimuthal rotation angle of the dielectric function on the cleavage plane of single-crystal SnS [1].

Figure 2. Real and imaginary parts of the ielectric tensor of SnS in the cleavage plane at [(a) and (b)] 27 K and [(c) and (d)] 300 K. Dashed, solid, and dasheddotted curves show the dielectric tensor at azimuthal angles of 0^o (a-axis), 45^o, and 90^o (b-axis), respectively.

Figure 3. Temperature dependences of the CP energies (open symbols) of SnS and the best fits (solid lines) for CPs of (a) a, (b) b, and (c) c axes [2].

Figure 3 shows Critical Points (CPs) analysis results of the dielectric function of SnS along the a, b, and c axes with the temperature range from 27 đến 350 K. From this analysis, we can see the similar value of CPs in different directions. This can be explained by band structure and Partial orbitals.

Table 1. CP energies at 27 and 300 K [2] compared to data at room temperature in Ref. [3]

The CP energies that are the best fits at 27 and 300 K are listed in Table 1 in comparison to room temperature report, Ref. 3. At 27 K, we could observe more CPs, and therefore obtain closer access to the intrinsic properties of the material.

Figure 4. Band structure of SnS with partial orbitals of atoms presented for each energy band. Blue, magenta, green, and red represent s, p_x , p_y , and p_z, respectively.

Figure 4 shows the energy band structure of single-crystal SnS. This is plotted with partial orbitals, where blue, magenta, green, and red represent the orbitals s, p_x , p_y , and p_z, respectively. The band-to-band transitions corresponding to the fundamental bandgap E₀, E₁, and exciton are denoted by arrows. The fundamental direct bandgap (E₀) occurs between the first VB and the first CB in the Γ–X line, This region is constructed mainly by s and p_x orbitals. The calculated result indicates that the energy of this CP is about 1.33 eV. This is in good agreement with the data, which give about 1.39 eV at 27 K. The E₁ CP that occurs at 1.66 eV along the armchair-direction is predicted to be the saddle point M₁ near the Y-point.

Most of the CPs have the same peak positions [1-3], even though the intensities and lineshapes of the CPs may be much different along the different principal axes. This phenomenon can be interpreted by the natural anisotropy of the p orbitals, which leads to the anisotropic localization of wave functions. Figure 4(a) shows the partial charge density calculated at the top of the VB in the Γ–X region where the E₀ peak arises. It is obvious that the charge density is extended along the zigzag-direction, which results in the dominance of the E₀ peak in the zigzag-direction, as shown in Fig. 4(b). Similarly, we also observe the preferred orientation of charge density along the armchair-axis at the saddle point in the Γ–Y region where the E₁ peak and exciton are etected, as shown in Figs. 4(c) and 4(d). This explains the dominance of the exciton along the armchair-axis at low temperatures.

Figure 5. Partial orbitals (partial charge density) [(a) and (b)] at the top of the valance band in the Γ–X line and [(c) and (d)] at the saddle point near the top of the valance band in the Γ–Y line. The Sn atoms are dark gray, while the S atoms are bright gray [1].

[1] Le. L. V et al. AIP Advances 10, 105003 (2020).

[2] Nguyen, H. T. et al. Scientific Reports 10, 18396 (2020).

[3] Nguyen, X. A. et al. Crystals 11, 548 (2021).