Exploring negative refraction conditions for quantum cascade semiconductor metamaterials in the terahertz spectral range

In order to avoid losses in metamaterial unit cells at frequencies of interest, caused by metallic inclusions, an active medium design has been proposed. As candidate structures for this active medium, we have chosen quantum cascade lasers because of their high output gain. Here we analyze and compare two quantum cascade structures that emit at 4.6 THz and 3.9 THz, respectively, placed under the influence of a strong magnetic field. We first solve the full system of rate equations for all relevant Landau levels, and obtain the necessary information about carrier distribution among the levels, after which we are able to evaluate the permittivity component along the growth direction of the structure. With these data one can determine the conditions under which negative refraction occurs, and calculate the values of the refractive index of the structure, as well as the range of frequencies at which the structure exhibits negative refraction for a predefined total electron sheet density