RKKY interaction with diffused contact potential

The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction is considered when the contact potential is replaced by an arbitrary distribution instead of the conventional Dirac’s $\delta$ function. The appropriate formulas for the RKKY exchange integrals, in the case of one-dimensional, two-dimensional, and three-dimensional systems, are derived. In order to exemplify the modification, the three distributions are used for numerical calculations of the interaction vs spin-spin distance, namely: Gaussian, uniform, and exponential. One of the results shows that “diffusion” of the contact potential removes an unphysical divergency of the RKKY integral at zero distance and the finite value obtained depends strongly on the distribution width.

Figure 1

The dependence of RKKY exchange integral on dimensionless distance $2k_{F}r$ in one dimension for Gaussian (a), uniform (b), and exponential (c) probability distributions. Solid line is for standard deviation $2k_F\sigma =0$, dashed for $2k_F\sigma =1$, and dotted for $2k_F\sigma =2$.

Figure 2

The dependence of RKKY exchange integral on dimensionless distance $2k_{F}r$ in two dimensions for Gaussian (a), uniform (b), and exponential (c) probability distributions. Solid line is for standard deviation $2k_F\sigma =0$, dashed for $2k_F\sigma =1$, and dotted for $2k_F\sigma =2$.

Figure 3

The dependence of RKKY exchange integral on dimensionless distance $2k_{F}r$ in three dimensions for Gaussian (a), uniform (b), and exponential (c) probability distributions. Solid line is for standard deviation $2k_F\sigma =0$, dashed for $2k_F\sigma =1$, and dotted for $2k_F\sigma =2$.

Figure 4

The value of RKKY exchange integral at the point $r=0$ as a function of standard deviation $2k_F\sigma$ for one (a), two (b), and three dimensions (c). The solid line is plotted for the Gaussian and uniform probability distribution; the dotted one for exponential distribution.

Figure 5

The damping factor for RKKY exchange integral at large distances as a function of standard deviation $2k_F\sigma$ in one (a), two (b), and three dimensions (c). The solid line is plotted for the Gaussian probability distribution, the dashed line for uniform one, and the dotted line for exponential one.

Figure 6

The position of the first zero of the RKKY exchange integral as a function of standard deviation $2k_F\sigma$ in one (a), two (b), and three dimensions (c). The solid line corresponds to the Gaussian probability distribution, the dashed line to the uniform one, and the dotted line to the exponential one.

Figure 7

The MFA critical temperature for model Ising spin-$S$ system on fcc lattice with RKKY interactions vs “diffusion” of the contact potential. The solid curve corresponds to the charge carrier concentration $N_e/N=0.05$, the dashed one to $N_e/N=0.2$, and the dashed-dotted one to $N_e/N=0.5$.