Magnetism of dimer ensemble with random exchange energy

Abstract

We present the study of magnetism of dimer ensemble deposited on a non-magnetic conducting substrate. The dimers consist of ions with spin 1/2 and have a random size. The magnetic ions are coupled by RKKY-interaction with random energy value. We consider two interaction models. The first assumes that dimer coupling energy has a Gaussian distribution, and the second model is based on a log-normal distribution of dimer size. The magnetization and susceptibility are evaluated from the exact solution for dimer energy and eigenstates. We show that the average exchange value and its dispersion have a strong effect on the system properties. The magnetic saturation exists under strong magnetic fields. High-temperature leads to the Curie’s law for susceptibility. The saturation magnetization of system with ferromagnetic coupling lowers, while the antiferromagnetic dimer ensemble can have non-zero magnetization under the strong magnetic field. It is determined by the dispersion of exchange energy.

GraphicAbstract

Appendix A: Laplace’s formula

The Laplace’s approximation used to estimate the integrals of the form

$$\begin{aligned} {\mathcal {I}} = \int \limits _a^b f(x) \exp (\lambda S(x)) dx, \end{aligned}$$

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where $\lambda$ is a large number and S(x) has a local maximum $x_0 \in [a, b]$. Using the Taylor series expansion near $x_0$, one obtains

$$\begin{aligned} {\mathcal {I}} \approx \sqrt{\frac{-2\pi }{ \lambda S''(x_0) }} \exp (\lambda S(x_0)) \left( f(x_0) + O(\lambda ^{-1}) \right) . \end{aligned}$$

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In the current paper, $\lambda$ is a $1 / \varDelta ^2$, S(x) is the

$$\begin{aligned} (2 \beta J_0 \varDelta ^2 - (J - J_0)^2) / 2 \end{aligned}$$

for the Gaussian function and

$$\begin{aligned} \beta J(r) \varDelta _r^2 - \ln ^2(r / r_0) / 2 \end{aligned}$$

for the log-normal distribution.