Quantitative theory of magnetic properties of elemental praseodymium

TL;DR

Elemental Pr in the dhcp structure remains nonmagnetic due to a localized $f^2$ configuration that forms a singlet crystal-field ground state. The authors derive an ab initio effective Hamiltonian $\hat{H}_{\mathrm{eff}}=\sum_i \hat{H}_i^{\mathrm{CF}}-\sum_{ij} I_{ij}\mathbf{J}_i\cdot\mathbf{J}_j$ within the $^3H_4$ multiplet using DFT+DMFT in Hubbard-I to compute crystal-field parameters and intersite exchange, finding singlet ground states at both sites and exchange insufficient to close the CF gap. At the (0001) surface, CF splittings shrink (down to ~1 meV) while the singlet ground state persists; the reduced gaps raise the possibility of exotic two-dimensional multipolar orders, such as octupolar arrangements under strain. Overall, the study demonstrates that ab initio CF and exchange calculations can capture the nonmagnetic behavior of bulk Pr and point to surface-driven multipolar phenomena in rare-earth metals, with potential experimental routes via strain to detect such orders.

Abstract

Elemental Pr metal crystallizes in the double hexagonal close packed (dhcp) structure and is unique among rare-earth elements in featuring a localized partially filled 4f shell without ordered magnetism. Experimental evidence attributes this absence of magnetism to a singlet crystal-field (CF) ground state of the Pr 4f$^2$ configuration, which is energetically well isolated from excited magnetic doublets. Here, we construct a realistic effective magnetic Hamiltonian for dhcp Pr, by combining density-functional theory with dynamical mean-field theory, in the quasiatomic Hubbard-I approximation. Our calculations fully determine the CF potential and predict singlet CF ground states at both inequivalent sites of the dhcp lattice. The intersite exchange interactions, obtained from the magnetic force theorem, are found to be insufficient to close the CF gap to the magnetic doublets. Hence, ab-initio theory is demonstrated to explain the unusual, non-magnetic state of elemental Pr. Extending this analysis to the (0001) surface of Pr, we find that the singlet ground state remains robust preventing conventional magnetic orders. Nevertheless, the gap between the ground state and the lowest excited singlet is significantly reduced at the surface, opening the possibility for exotic two-dimensional multipolar orders to emerge within this two-singlet manifold.

Figures (5)

• Figure 1: Calculated crystal-field splitting of the Pr $^3H_4$ configuration for the cubic (a) and hexagonal (b) site in bulk dhcp Pr. The CF wavefunctions are written in the $|M\rangle\equiv |J=4;M_J\rangle$ basis and are defined in the same coordination frame as the CFPs in Table \ref{['tab:CF_pars']}. In panel (c), we reproduce the experimentally inferred CF level scheme of Ref. Houmann1979 for the hexagonal site.
• Figure 2: Exchange interactions $\tilde{I}_{ij}$ for the dhcp crystal structures of Pr (current work) and Nd Kamber2020, as well as for the hcp Pr (current work)
• Figure 3: Evolution of the $B_2^0$ CFP on hexagonal sites (top panel) and the $B_4^3$ CFP on cubic sites (bottom panel) vs the layer depth with respect to the surface. The surface layer is the first one; the corresponding values for bulk are shown on the right-hand side. Since the sign of $B_4^3$ in the global coordination frame is flipped between the two dhcp cubic sites related by the inversion symmetry, we display the value $B_4^3$ in its local frame (i. e., we flip the sign of $B_4^3$ for every second cubic layer starting from the bulk one).
• Figure 4: Calculated crystal-field level splitting of the Pr $^3H_4$ multiplet at the (0001) dhcp surfaces with hexagonal (a) and cubic (b) termination. The CF wavefunction representation and coordination frame are the same as in Fig. \ref{['fig:CF_lev']}. For both cases, we show the levels for the surface and subsurface site. In the subsurface layer the site symmetries are reversed with respect to the surface one, becoming cubic in (a) and hexagonal in (b), respectively.
• Figure 5: Magnetic moment of the hexagonal site in the bulk and in the surface layer vs. applied field.