Electromagnetic Response and Optical Properties of Spherical CuSbS2 Nanoparticles

We study the electromagnetic response of individual spherical copper antimony disulfide $(CuSbS_{2})$ nanoparticles and layers embedded with them for solar applications and near infrared (NIR) sensors using computational methods. We first calculate the single particle scattering and absorption efficiencies using Lorenz-Mie theory. The absorption and the total scattering efficiencies broaden and shift to longer wavelengths with increasing particle radius from 1 to 100 nm. We further investigate the response of multiple nanoparticles embedded in a thin layer at a low volume fraction using a Monte Carlo method. Our results demonstrate that with increasing particle size and scattering NIR transmittance is strongly suppressed and absorption and reflectance enhanced. The high absorption coefficient and solar-compatible band gap of CuSbS2 make it a good candidate for nanocrystalline solar cell and other NIR device applications.


INTRODUCTION
Copper antimony disulfide (CuSbS 2 ) is emerging as an alternative solar absorber with earthabundant constituent elements [1]. Chalcostibite, CuSbS 2 , is a naturally occurring mineral orthorhombic system with space group Pnma [2,3]. The optical band gap ranges from 1.4 eV to 1.52 eV [4][5][6][7]. For CuSbS 2 the fundamental band gaps have been predicted to be indirect in nature; however, the difference between the lowest energy direct and indirect gaps is only of the order of 0.1 eV [8].
In recent years, CuSbS 2 thin films deposited by sputtering [9], thermal evaporation [10,11], or chemical bath deposition [12] have been characterized, but there are only a few studies on the synthesis of CuSbS 2 in powder form. In the synthesis of ternary materials, controlling the reaction conditions is essential. Solvothermal, hydrothermal and hot injection methods have already been reported for the formation of CuSbS 2 nanocrystals [4,6,[13][14][15]. Most recently, high-quality plateletlike CuSbS 2 nanocrystals with a well-defined shape and narrow size distribution were experimentally prepared using an improved hot injection method [7]. This opens up new possibilities to engineer nanoparticle shapes and sizes to best suited for a given application.
In this work, we focus on the fundamental size-dependent optical and NIR properties of CuSbS 2 nanoparticles as individual spheres and embedded in a layer at low volume fractions (0.01%). The material properties are described by the complex dielectric function previously calculated using Density Functional Theory (DFT) [16,17] and Many-Body Perturbation Theory [18].

Lorenz-Mie Scattering of Spherical Particles
The optical response of a small sphere in an incident electromagnetic field is calculated using Lorenz-Mie theory [19]. In general, the scattered field is a superposition of normal modes, each weighted by the electric, a n , and magnetic, b n , Mie coefficients. If the permeability of the particle and the surrounding medium are assumed to be equal, then the electric and magnetic Mie coefficients, a n and b n , for a sphere are given by where the prime indicates differentiation with respect to the argument of the corresponding function. The radial functions ψ n (x) = xj n (x), ξ n (x) = ψ n (x) + iχ n (x) and χ n (x) = −xy n (x) are Riccati-Bessel and Hankel functions where j n , y n are the spherical Bessel functions [19]. The size parameter, x = 2πn m R/λ, R is the radius of sphere, n m is the refractive index of the surrounding medium, and m 1 is the refractive index of the sphere relative to the surrounding medium (n m ). The order of the electric and magnetic modes are represented by n, where dipole corresponds to n = 1, quadrupole to n = 2, and so on. The total scattering (Q sca ) and absorption (Q abs ) efficiencies [19] can be computed from

Optical Properties of Layers Containing Particles
Monte Carlo simulations provide a method of determining the exact optical properties of thin layers containing large numbers of spheres which is useful for simulating systems with many coupled degrees of freedom. We apply a Monte Carlo method to investigate the optical behavior of the copper antimony disulfide nanoparticles in layers containing spherical particles. The transmittance, reflectance, and absorption of the layer were simulated using a modified Monte Carlo method [20] originally developed by Wang et al. [21]. We consider a layer with a thickness T = 200 µm embedded with nanoparticles at a volume fraction of f = 0.01%. The particles are embedded in a nonabsorbing medium with a refractive index of 1.0 and the layer is surrounded by air. The scattering and absorption coefficients, µ sca and µ abs per unit length of the film given by, The ensemble averaged particle asymmetry factor is where ρ(r i ) is the particle size distribution, and Q sca (r i ) is the scattering efficiency of a particle with radius r i , and i is the binning index [22]. The effective dielectric permittivity of the layer, eff , from Maxwell Garnett Effective Medium Theory is [23], where p and m are the dielectric permittivities of the particle and medium components. The Monte Carlo method used here records the path and termination result of 10 7 photons from an infinitesimally small beam normal to the composite surface. The grid resolution was dz = 2 µm and dr = 1 µm for the axial and radial directions, respectively.
The real and imaginary components of the permittivity of CuSbS 2 used herein were obtained from a previous study using Density Functional Theory [16] and have been averaged over the three orthogonal axes in order to estimate optical properties. Their values are provided in Figure 1.

RESULTS AND DISCUSSION
In this section, we discuss results of our calculations from the Lorenz-Mie theory and the Monte Carlo method. The maximum of the absolute value of the dipolar electric and magnetic Mie coefficients modes are in the ultraviolet (UV) to visible regime for particles with a radius less than 100 nm are shown in Figure 2. The electric and magnetic dipole modes broaden and shift to longer wavelength with increasing particle size. The magnitude of the electric modes considerably changes with size. Figure 2(b) depicts how the magnetic dipole modes redshift when the radius increases but the intensities of these modes for large particles, radii greater than 30 nm here, are more pronounced.   We have explicitly plotted the scattering and absorption efficiencies for a specific radius R = 100 nm in Figure 3(a). We have also plotted the absolute values of first two Mie coefficients, a 1 and b 1 , for particle with radius of 100 nm in Figure 3(a) to investigate the origin of the modes in the scattering efficiency curve. One can clearly see that the peak at the shorter wavelength 0.596 µm in the scattering efficiency is associated with the electric mode and the other one at 0.708 µm with the magnetic mode.
The amplitudes of the scattered electric field corresponding to the maxima of the Mie coefficients are shown in Figure 3. An electric field distribution inside and outside the particle at different wavelengths clearly shows the intensity variation. The high intensity of red color indicates the areas where the magnitude of Q sca is large while the blue color indicates small magnitudes. The electric nearfield in Figure 3(b) which spatially extends in the inner domain shows that the a 1 mode is characteristic to the dielectric electric dipolar mode [24].
The absorption, Q abs , and scattering, Q sca , efficiencies of a CuSbS 2 particle with radius up to 100 nm in the UV-Visible regime are presented in Figures 4(a)-(b), respectively. The intensity of the blue color indicates the magnitude of the efficiencies and the maximum values, corresponding to the electric and magnetic dipolar dielectric resonances, are in dark blue. The dark blue areas of Q abs and Q sca correspond to maxima in the Mie coefficients |a 1 | and |b 1 | where also the higher order modes are present. As expected from the behavior of the Mie coefficients, Figure 4(a) indicates that the absorption efficiency maxima broaden and shift to longer wavelengths with increasing particle size. Similarly, the total scattering efficiency broadens and shifts to longer wavelengths with increasing particle size, cf. Figure 4(b). The modes shift to longer wavelengths because m 1 , the relative refractive index of the particle and medium, is larger than one.

Optical Properties of Compacts Embedded with Spherical Particles
The spectra obtained from Monte Carlo simulations for a 200 µm thick layer embedded with nanoparticles of radii R = 10, 20, 40, 60, 80, and 100 nm at 0.01% volume fraction are shown in Figure 5. When comparing layers having different particle sizes, the data show that there are competing mechanisms between the reflectance and absorption when the transmittance is small. At short wavelengths λ < 0.6 µm, absorption by the CuSbS 2 particles becomes significant enough (> 90% even at low volume fraction) to be the main transport mechanism. Note that the minimum of absorption is due to the peaks in the reflectance. Regardless of the local variations in the absorption bands in the short-wavelength range, the decrease in the absorption bands occurs at longer wavelengths for a layer containing larger particles. Increasing the size of the particles embedded in the layers leads to transmittance beginning at longer wavelengths while the intensity of the transmittance decreases in the range considered here (0.1-1.1 µm). For example for layers containing R = 10 nm nanoparticles transmittance begins at λ = 430 nm while for R = 100 nm this happens beyond 855 nm. The intensity of the reflectance in the layer containing large particles is more pronounced than the intensity in the layer with small ones. In the reflectance curves multiple peaks appear and redshift when the particle size embedded in the layers increases. Subsequently, multiple peaks appear in the absorption curves for layers containing large particles of radii 60-100 nm.

SUMMARY AND CONCLUSIONS
In this work we have studied the scattering and absorption efficiencies of spherical CuSbS 2 nanoparticles with radii R up to 100 nm in the NIR regime using Lorenz-Mie theory. The electric and magnetic Mie coefficients of these nanoparticles strongly depend on the particle size. For the large particles the scattering modes redshift and their intensity significantly increases. Furthermore, we have characterized the properties of films of 200 µm in thickness containing spherical particles at a low volume using Monte Carlo methods. The results show how the optical and NIR properties in such layers can be tuned by changing the particle size. For the smallest 10 nm particles the decay of absorption and onset of transmittance rapidly occur around λ = 500 nm while reflectance remains negligible. With increasing particle size and scattering, transmittance is strongly suppressed and reflectance consequently increases. Such high tunability makes CuSbS 2 nanoparticles promising candidates for solar cells and sensors in the NIR regime.