Magnetic Impurity Corrections to Specific Heat of Alumina (and Alumina Substrates)

Scottky-type anomalies have been observed in a variety of RuO_2 thick film resistors on alumina substrates (Volokitin 1994) and alumina ceramics with magnetic impurities (Tarasenko 2010).

In the presence of a magnetic field, the energy levels of a nucleus with spin \(I\) split into \(2I+1\) levels. This effect — known as the nuclear Zeeman effect — leads to hyperfine splitting of energy levels when the nuclei are in a local magnetic field and a field and temperature dependent contribution to the specific heat.

In closed form (Phillips 1971, Leyarovski 1988),

\begin{equation} \label{eq:ClosedFormSolution}c_{N}=R\left(\frac{x}{2I}\right)^{2}\left[{\textrm{Csch}}^{2}\left(\frac{x}{2I}\right)-(2I+1)^{2}{\textrm{Csch}}^{2}\left[(2I+1)\left(\frac{x}{2I}\right)\right]\right]\\ \end{equation}where

\begin{equation} \label{eq:ClosedFormVariable}x=\frac{g_{N}{\mu}_{N}IH}{k_{B}T}\\ \end{equation}The Zeeman splitting of the nuclear energy levels leads to a broad peak in the specific heat when \(k_{b}T\) is on the order of the energy splitting \(\mu H\), but because the energy splittings are small^{1}^{1}the nuclear magneton is a factor of 1836.1 smaller than the Bohr magneton \(\mu_{B}\), only the high temperature \((\mu H/k_{B}T)^{2}\) tail of specific heat peak is usually seen. In that limit, and assuming that the zero field field splitting (ZFS) is negligible in comparison to that from the applied field \(H\), then (Lounasmaa 1974, Leyarovski 1988),

where \({\mu}_{0}=4\pi\cdot 10^{-7}\textrm{ H/m}\) and the nuclear Curie constant \(\lambda_{N}\) is given by

\begin{equation} \label{eq:CurieConstant}{\frac{\lambda_{N}}{\mu_{0}}=N_{A}I(I+1)\frac{\left({\mu}_{N}g_{N}\right)^{2}}{3k_{B}}\left[\frac{\textrm{ J K}}{\textrm{mol T}^{2}}\right]}.\\ \end{equation}where here \(g_{N}=\mu/I\) and \(\mu_{N}\) is the nuclear magneton \(\mu_{N}=\frac{he}{4\pi M_{p}c}=5.05110^{-27}\textrm{ J/T}\).

It is often convenient to express this Zeeman splitting of the nuclear energy levels in temperature units. In this case

\begin{equation} \label{eq:NuclearEnergySplittingInKelvin}\Delta=g\mu_{n}H/k_{B}\textrm{ [K]}.\\ \end{equation}\(\kappa\)-(BEDT-TTF)\({}_{2}\)Cu(NCS)\({}_{2}\) — with chemical formula C\({}_{22}\)H\({}_{16}\)N\({}_{2}\)S\({}_{18}\)Cu\({}_{1}\) (Urayama 1988) — has a molar mass \(

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