Magnetic exchange interaction in the spin‐polarized electron gas

Abstract

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.

Anisotropic Heisenberg form of RKKY interaction in the one-dimensional spin-polarized electron gas

“…Similar difficulties appear for the case of the one-dimensional spin polarized electron gas. Effect of the broken time-reversal symmetry on the RKKY interaction has been studied in two-and three-dimensions 17,18 . In this brief manuscript, we study this effect for the case of one-dimensional electron gas, which provides a useful addition to the literature.…”

Magnetic interactions in the spin-orbit coupled electron gas

“…Although Eq. (2.19) is a known formula and has been widely used in the literature [5,6,7,29], working through its derivation still holds value. A pedagogical derivation of this equation, which is given in the Appendix, provides some important information about limitation of using this equation to get the indirect exchange interaction between two localized magnetic moments, and can be helpful for both experts and beginners in this research field.…”

“…This a standard integration, which can be evaluated by using the Jacobi-Anger expansion of the exponential term in terms of the Bessel's functions and by performing the angular integration. [25,26] The result is [28,29]…”

“…RKKY interaction yields a ferromagnetic or antiferromagnetic coupling of localized moments, while vector DM interaction would like to produce a nonlinear coupling of them. It has been shown [24,28,29] that in the presence of inversion symmetry vector DM interaction vanishes, the tensor DM can be potentially present.…”

“…Another limit in which one can obtain the long-distance behavior for the magnetic interactions is the limit where there is spin polarization, but both the Rashba and the Dresselhaus SOC terms are absent, i. e., α = β = 0, but ∆ = 0. This case was treated in our earlier work, [28,29] and we quote the results here for completeness.…”