We study the indirect exchange interaction between two localized magnetic moments, known as Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, in a one-dimensional spin-polarized electron gas. We find explicit expressions for each term of this interaction, study their oscillatory behaviors as a function of the distance between two magnetic moments, R, and compare them with the known results for RKKY interaction in the case of one-dimensional standard electron gas. We show this interaction can be written in an anisotropic Heisenberg form, E( R) = λ 2 χxx(S1xS2x + S1yS2y) + λ 2 χzz S1zS2z, coming from broken time-reversal symmetry of the host material.

We extend the original work of Ruderman, Kittel, Kasuya, and Yosida (RKKY) on the interaction between two magnetic moments embedded in an electron gas to the case where the electron gas is spin polarized. The broken symmetry of a host material introduces the Dzyaloshinsky-Moriya (DM) vector and tensor interaction terms, in addition to the standard RKKY term, so that the net interaction energy has the form:We find that for the spin-polarized electron gas, a non-zero tensor interaction ↔ Γ is present in addition to the scalar RKKY interaction J, while D is zero due to the presence of inversion symmetry. Explicit expressions for these are derived for the electron gas both in 2D and 3D. The RKKY interaction exhibits a beating pattern, caused by the presence of the two Fermi momenta k F ↑ and k F ↓ , while the R −3 distance dependence of the original RKKY result for the 3D electron gas is retained. This model serves as a simple example of the magnetic interaction in systems with broken symmetry, which goes beyond the RKKY interaction.

The exchange interaction between two magnetic moments embedded in a host metal is fundamental to the description of the magnetic behavior of solids. In the standard spin‐degenerate electron gas, it leads to the well known Ruderman–Kittel–Kasuya–Yoshida (RKKY) interaction, which is of the Heisenberg form JboldS1·boldS2. Here, we study the more general case of the spin‐polarized electron gas both in two and three dimensions, by evaluating the interaction strength as an integration over the product of the host Green's functions. We find that an additional Ising‐like term appears in the magnetic interaction, so that the net interaction for the spin‐polarized gas is of the form J1boldS1·boldS2+J2S1zS2z. The anisotropic interaction is due to the broken rotational symmetry because of the spin polarization in the truezˆ direction. Furthermore, the interaction shows a beating pattern as a function of distance, caused by the two different Fermi momenta for the two spins.

Chiral order in magnetic structures is currently an area of considerable interest and leads to the skyrmion structures and domain walls with certain chirality. The chiral structure originates from the Dzyaloshinskii-Moriya interaction caused by broken inversion symmetry and the spin-orbit interaction. In addition to the Rashba or Dresselhaus interactions, there may also exist substantial spin polarization in magnetic thin films. Here, we study the exchange interaction between two localized magnetic moments in the spin-polarized electron gas with both Rashba and Dresselhaus spin-orbit interaction present. Analytical expressions are found in certain limits in addition to what is known in the literature. The stability of the Bloch and Néel domain walls in magnetic thin films is discussed in light of our results.

The exchange interaction between two localized magnetic moments embedded in a host material is fundamental to the description of the magnetic behavior of solids and chiral order in magnetic structures. This interaction was originally found by Ruderman, Kittel, Kasuya and Yosida (RKKY). It is currently an area of considerable interest. The potential spintronics applications of chiral magnetic structures, originating from the competition between RKKY and Dzyaloshinsky-Moriya (DM) interactions caused by broken symmetries and the presence of the spin-orbit interaction, have stimulated lots of recent theoretical studies. In the standard spin-degenerate electron gas, it leads to the well-known RKKY interaction. In this dissertation, we study RKKY and DM interactions in the spinpolarized electron gas in one, two and three dimensions. In addition to the Heisenberg form of interaction, we found an Ising-like term appears in the magnetic interaction for the spin-polarized gas. The anisotropic interaction is due to the broken time-reversal symmetry, based on the fact that spin polarization is in the z direction. Furthermore, we study RKKY and DM interactions in a spin-polarized two-dimensional electron gas in the presence of both Rashba and Dresselhaus spin-orbit coupling (SOC). We find that in addition to the scalar RKKY interaction, there are also vector and tensor interactions between the two localized moments. Analytical expressions are found for each term of the total magnetic interaction, and their oscillatory behaviors are studied and compared to the numerical results. We also use our model to find energy of Neel and Bloch magnetic walls. Using our theoretical results we are able to predict which wall is preferred. In the last chapter, we study impurity states in the kagome lattice. The kagome lattice is a 2D lattice that has been of recent interest owing to its graphene-like band structure, the existence of flat band states and exotic quasiparticle excitations. In this chapter, we study the electronic states introduced by impurities in the system by applying the Green's function approach within a tight-binding model Hamiltonian. The impurities introduce localized states close to the Dirac point, in many ways similar to graphene, which will be discussed.