High-order Ho multipoles in HoB2C2 observed with soft resonant x-ray diffraction

TL;DR

This study uses soft resonant x-ray diffraction to analyze high-order multipole moments in HoB$_2$C$_2$, revealing dominant hexadecapole order and distinct magnetic and quadrupolar ordering behaviors.

Contribution

The paper provides a detailed energy and azimuthal dependence analysis to determine high-rank multipole moments in HoB$_2$C$_2$, highlighting the dominance of hexadecapole order and the interplay between magnetic and quadrupolar phases.

Findings

Hexadecapole (rank 4) is the dominant multipole moment.

Magnetic and quadrupolar order parameters are distinct.

Quadrupolar order depends on magnetic order inducing a quasi doublet ground state.

Abstract

Soft resonant x-ray Bragg diffraction (SRXD) at the Ho M$_{4,5}$ edges has been used to study Ho $4f$ multipoles in the combined magnetic and orbitally ordered phase of HoB$_2$C$_2$. A full description of the energy dependence for both $\sigma$ and $\pi$ incident x-rays at two different azimuthal angles, as well as the ratio $I_\sigma/I_\pi$ as a function of azimuthal angle for a selection of energies, allows a determination of the higher order multipole moments of rank 1 (dipole) to 6 (hexacontatetrapole). The Ho 4f multipole moments have been estimated, indicating a dominant hexadecapole (rank 4) order with an almost negligible influence from either the dipole or the octupole magnetic terms. The analysis incorporates both the intra-atomic magnetic and quadrupolar interactions between the 3d core and 4f valence shells as well as the interference of contributions to the scattering that behave differently under time reversal. Comparison of SRXD, neutron diffraction and non resonant x-ray diffraction shows that the magnetic and quadrupolar order parameter are distinct. The $(0 0 1/2)$ component of the magnetic order exhibits a Brillouin type increase below the orbital ordering temperature T$_Q$, while the quadrupolar order increases more sharply. We conclude the quadrupolar interaction is strong, but quadrupolar order only occurs when the magnetic order gives rise to a quasi doublet ground state, which results in a lock-in of the orbitals at T$_Q$.