Journal of Nanomaterials

Volume 2016 (2016), Article ID 7659074, 9 pages

http://dx.doi.org/10.1155/2016/7659074

## Induced Magnetic Anisotropy in Liquid Crystals Doped with Resonant Semiconductor Nanoparticles

Displays and Photonic Applications Group, Electronic Technology Department, Carlos III University of Madrid, Leganés, 28911 Madrid, Spain

Received 10 April 2016; Accepted 28 June 2016

Academic Editor: Nathan C. Lindquist

Copyright © 2016 Vicente Marzal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Currently, there are many efforts to improve the electrooptical properties of liquid crystals by means of doping them with different types of nanoparticles. In addition, liquid crystals may be used as active media to dynamically control other interesting phenomena, such as light scattering resonances. In this sense, mixtures of resonant nanoparticles hosted in a liquid crystal could be a potential metamaterial with interesting properties. In this work, the artificial magnetism induced in a mixture of semiconductor nanoparticles surrounded by a liquid crystal is analyzed. Effective magnetic permeability of mixtures has been obtained using the Maxwell-Garnett effective medium theory. Furthermore, permeability variations with nanoparticles size and their concentration in the liquid crystal, as well as the magnetic anisotropy, have been studied.

#### 1. Introduction

Plasmonics is a consolidated research field, due to its potential applications in many nanotechnology domains [1–3]. Plasmon resonances occur when the photon frequency of an incident radiation is resonant with the collective oscillation of the conduction electrons in a metal. This process involves the concentration of light in subwavelength dimensions which allows the manipulation of light at nanoscale. This phenomenon is the basis of a myriad of applications, ranging from new all-optical devices [4, 5] to the design of metatoms to perform metamaterials with tunable optical properties [6–8].

Although huge efforts have been made on plasmon nanoparticles and plasmonic applications [9], metallic samples have the inherent problem of absorption in the visible range. This absorption produces thermal effects leading to material degradation. Semiconductor materials with high refractive index can be the alternative to plasmonic materials (e.g., gold and silver) in this kind of applications. The magnetic and electric Mie resonances in light scattering of semiconductor materials [10, 11] are analogous to the localized surface plasmon resonances in metallic nanoparticles. These semiconductor nanoparticles can act as photonic nanoresonators due to the high-refractive-index contrast between them and the surrounding medium and the ratio between their size and the incident radiation wavelength. These properties lead to the appearance of both magnetic and electric resonances in the visible range of the electromagnetic spectrum [12]. The excitation of these resonances is being deeply studied nowadays. For instance, astonishing phenomena, like scattering directionality, have been reported [13, 14] in these systems. In addition, the low optical absorption of semiconductors has overcome in part thermal issues [15].

The tunability of resonances of these nanostructures is a key aspect for several applications. For instance, the spectral coincidence of these resonances with the absorption peak of different molecules is the basis of chemical sensors [16]. As in plasmonics, Mie resonances of dielectric nanoparticles can be tuned changing their size, shape, composition, or the optical properties of the surrounding medium. The variation of the properties of the nanoparticles is a static method: it involves the fabrication of new ones in order to change the output response, limiting its application. Nevertheless, the change of the surrounding medium properties may be dynamic. By using an active material as surrounding medium, its optical properties can be changed with real time control and reversibility capacity, changing the resonant wavelength in the same way. This dynamic response is very interesting for applications such as active polarizers [17] or new tools for active photonic circuits [18].

Among all the active media used in plasmonic devices, liquid crystals are a good choice: they have capacity to easily modify their refractive index by electrical [19], optical [20], or thermal techniques [21] because of their high birefringence and small elastic constants. This control feature besides their high compatibility with optoelectronic materials makes them the perfect option as active dielectric media. Liquid crystal properties have also led to the development of a large variety of photonic devices (modulators, filters, optical switches, and so on) [22–24].

While liquid crystals have a remarkable dielectric anisotropy, their magnetic properties in the visible range do not have a noticeable behavior. In this sense, this work is devoted to analyze the magnetic properties of mixtures composed of a liquid crystal doped with semiconductor nanoparticles with magnetic resonances. The magnetic permeability of the nanoparticles and the effective magnetic permeability of the mixture have been studied in a set of simulations. The Maxwell-Garnett effective medium approach has been considered in the calculus of the permeability [25]. Searching a noticeable magnetic anisotropy, several liquid crystals and semiconductor nanoparticles have been considered.

#### 2. Theory

The effective permeability, , of mixtures composed of a liquid crystal (LC) doped with resonant nanoparticles (NPs) can be obtained through several effective medium approximations. In particular, a generalization of the Maxwell-Garnett Theory (MG) [25] has been considered in this work. This theory establishes that the effective dielectric function, , of a matrix with homogeneous spherical inclusions can be expressed in terms of the optical properties of each component and the volume fraction of the inclusions () [26] as follows:where and are the dielectric permittivity of the inclusions and the matrix material, respectively.

Due to the symmetric nature of the electromagnetic field, an analogue expression can be derived for the effective magnetic permeability, :where and are the magnetic permeability of the host medium and the inclusion, respectively.

In this work, a LC host medium with high-refractive-index semiconductor nanoparticles as inclusions is considered. Selected NPs have both electric and magnetic resonances in the visible range, although they are composed of nonmagnetic materials (e.g., silicon and germanium) [11]. Thus, effective optical constants can also be derived for the individual nanoparticles. Focusing attention on the magnetic properties, the effective magnetic permeability of these nanoparticles has been obtained through the Mie theory [26].

The scattered electromagnetic field () of a spherical particle (of radius ) made of a certain material, embedded in a homogeneous and isotropic media, can be calculated through the Mie theory [26]. This theory considers an expansion of the scattered fields in vector spherical harmonics (), when the sphere is radiated with an incident electromagnetic plane wave with wavelength *λ* and amplitude where is the wave vector and is the frequency of the incident field, is the magnetic permeability of the surrounding medium, and , , are the so-called Mie coefficients for the scattered field. Applying the boundary conditions between the sphere and the surrounding medium, analytical expressions for these coefficients can be deduced [26]: and are the Riccati-Bessel functions and is the relative refractive index between the sphere () and the surrounding medium (). The factor is given by , where is the magnetic permeability of the sphere material. The size parameter is , which is given by the relation , where is the radius of the sphere and the incident wavelength, as was commented. While coefficients are related to the electric response of the system, are usually related to the magnetic character. In addition, and correspond to the dipolar behavior, and with the quadrupolar one, and so on.

These complex expressions can be simplified under certain assumptions, for instance, applying the Rayleigh approximation, which is satisfied whenIn this case, only the dipolar components remain, and the expressions of Mie coefficients can be simplified as follows [26]:This approximation is only satisfied by dipole-like spheres. Although the magnetic response of semiconductor nanoparticles has been observed in large nanoparticles (diameter > 50 nm), a simple analysis of the Mie coefficients shows that the dipolar character dominates in these nanoparticles in the visible wavelength range. Figure 1 shows the first four Mie coefficients of an AlAs nanoparticle with a radius of * *nm as a function of the incident wavelength, considering the ordinary refractive index (a) and the extraordinary refractive index (b) for the surrounding UCF-N3b LC host [27]. Although liquid crystals are not isotropic media, they can be approximated as isotropic considering either their ordinary or extraordinary refractive index. For the wavelength range where the magnetic resonance arises ( nm–450 nm), it can be seen that the dipolar contributions ( and ) are dominant, allowing the assumption of a dipole-like sphere.