Kramers degeneracy and relaxation in vanadium, niobium and tantalum clusters

A Diaz-Bachs,M I Katsnelson,A Kirilyuk

doi:10.1088/1367-2630/aab5ca

Abstract

In this work we use magnetic deflection of V, Nb, and Ta atomic clusters to measure their magnetic moments. While only a few of the clusters show weak magnetism, all odd-numbered clusters deflect due to the presence of a single unpaired electron. Surprisingly, for the majority of V and Nb clusters an atomic-like behavior is found, which is a direct indication of the absence of spin–lattice interaction. This is in agreement with Kramers degeneracy theorem for systems with a half-integer spin. This purely quantum phenomenon is surprisingly observed for large systems of more than 20 atoms, and also indicates various quantum relaxation processes, via Raman two-phonon and Orbach high-spin mechanisms. In heavier, Ta clusters, the relaxation is always present, probably due to larger masses and thus lower phonon energies, as well as increased spin–orbit coupling.

Highlights

Kramers degeneracy theorem[1] states that every energy eigenstate of a time-reversal symmetric system with noninteger total spin is at least doubly degenerated
The same work estimates that a chain between 2 and 7 vanadium atoms would achive a magnetic moment around 4 μB per atom[13]
DorantesDávila et al studied the magnetic moments as function of the parameter J/W, where J is the exchange integral and W is the bulk bandwidth; they found a total magnetic moment between of 0-4 μB for ferromagnetic V9 cluster[14], while the value decreased to 0-3 μB for an antiferromagnetic order

Summary

Introduction

Kramers degeneracy theorem[1] states that every energy eigenstate of a time-reversal symmetric system with noninteger total spin is at least doubly degenerated. The basis states of the system are Kramers-conjugate, i.e. they are related to each other by the time-reversal operator. The immediate consequence of this is that for such a system, spin-lattice coupling is prohibited, because any spin-phonon operator is invariant under time reversal, and has zero matrix elements for the transitions between such states. This selection rule, known as the Van Vleck cancellation[2], implies that the lattice excitations cannot be responsible for the relaxation between two Kramers conjugated states. There are other, more materialdependent mechanisms of the relaxation, such as for example the electronuclear spin entanglement[4]

Methods

Results

Conclusion