A generalized Ginzburg-Landau approach to second harmonic generation

Abstract

We develop a generalized Ginzburg-Landau theory for second harmonic generation (SHG) in magnets by expanding the free energy in terms of the order parameter in the magnetic phase and the susceptibility tensor in the corresponding high-temperature phase. The non-zero components of the SHG susceptibility in the ordered phase are derived from the symmetries of the susceptibility tensor in the high-temperature phase and the symmetry of the order parameter. In this derivation, the dependence of the SHG susceptibility on the order parameter follows naturally, and therefore its nonreciprocal optical properties. We examine this phenomenology for the magnetoelectric compound Cr2O3 as well as for the ferroelectromagnet YMnO3. [3,4] in Cr 2 O 3 were obtained what confirms that these experiments can distinguish between the two magnetic states that are related to each other by the time reversal operation, and therefore its nonreciprocity. These observations can be explained by an interference effect between a time-symmetric magnetic dipole contribution and a time-antisymmetric electric dipole contribution [3]. Soon after these experiments were done, a microscopic theory was proposed [5] which could explain quantitatively the non-reciprocal effects observed in Cr 2 O 3 . It was shown that the electric dipole contributions were linearly proportional to the antiferromagnetic order parameter giving rise to the timeantisymmetric character to the electric dipole tensor. The study of this dependence is of crucial importance in order to understand the non-reciprocal character of the SHG tensors. The derivation of a micoscopic theory for a specific effect, in our case SHG, can be rather complicated depending on the properties of the material under study [5,6]. Concrete information of the SHG process in materials where a transition takes place, i.e. paramagnetantiferromagnet or paraelectric-ferroelectric, can be obtained at a simpler level, i.e. by considering only the symmetry arguments. In particular, we are interested in investigating the dependence of the SHG susceptibilities in the ordered phase on the order parameter.A powerful phenomenological theory suitable to describe phase transitions is the Ginzburg-Landau theory. The Ginzburg-Landau approach is based on the existence of an order parameter in the ordered phase and on symmetry considerations [7]. Pershan in 1963 [8] showed that the tensors for nonlinear electro-and magneto-optic effects could be derived from a phenomenological timeaveraged free energy. It is our purpose in this paper to extend the formulation of Pershan by including the order parameter explicitly in the expansion of the free energy for SHG. We shall show that the non-zero components of the SHG susceptibility tensor in the ordered phase are naturally obtained from the symmetry of the susceptibility tensor in the high-temperature phase and the symmetry of the order parameter. Once the dependence of the SHG susceptibility tensor on the order parameter is known, the non-reciprocal optical properties below the...

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