Temperature Dependence of Local Magnetic Moments in Paramagnetic Metals: Hubbard's Alloy Analogy

Abstract

The temperature $T$ dependence of the local magnetic moments and the static uniform spin susceptibility $\ensuremath{\chi}(T)$ in paramagnetic metals is studied using the Hubbard Hamiltonian for non-degenerate bands. Hubbard's alloy analogy, which correctly describes both strongly interacting and weakly interacting systems of electrons, is applied to determine the average local-moment magnitude ${m}_{l}(T)\ensuremath{\equiv}{N}^{\ensuremath{-}1}\ensuremath{\Sigma}{i}^{}|{n}_{i\ensuremath{\uparrow}}\ensuremath{-}{n}_{i\ensuremath{\downarrow}}|$ and $\ensuremath{\chi}(T)$. It is found that the magnetic-moment magnitude decreases gradually with temperature and does not undergo a phase transition to a zero value. This is in contrast to the results of previous theories which predict a sharp phase transition associated with a disappearance at high temperatures of the moments. The moment magnitude which is a function of the ratio of the Coulomb repulsion energy $U$ to the half-bandwidth $W$ and of the number of electrons per site $2n$ approaches the finite value ${m}_{l}(T\ensuremath{\rightarrow}\ensuremath{\infty})=2n(1\ensuremath{-}n)$ at temperatures high compared to $U$ and $W$, which is just the probability that a noninteracting electron and hole of opposite spin are at the same site. At these temperatures $\ensuremath{\chi}(T\ensuremath{\rightarrow}\ensuremath{\infty})$ obeys a Curie law with Curie constant given by ${m}_{l}(T\ensuremath{\rightarrow}\ensuremath{\infty})$. For $U\ensuremath{\ll}W$, ${m}_{l}(T)$ is nearly temperature independent and approximately equal to $2n(1\ensuremath{-}n)$. For $U\ensuremath{\gg}W$ and $\mathrm{kT}\ensuremath{\ll}U$ the local moment has its maximum possible magnitude, which is either $2n$ or $2(1\ensuremath{-}n)$, corresponding to electrons ($n<\frac{1}{2}$) or holes ($n>\frac{1}{2}$), respectively. In addition, the behavior of the paramagnetic susceptibility $\ensuremath{\chi}(T)$ which is found to be smoothly varying with temperature does not suggest the existence of a phase transition associated with a disappearance of local magnetic moments.