Squeezing the crystalline lattice of the heavy rare-earth metals to change their magnetic order: Experiment and ab initio theory

Ab initio electronic structure theory finds the type of magnetic order of the heavy rare earths to be correlated directly with the $a$ and $c$ lattice parameters of their hexagonal-close-packed crystal lattices. We refine our experimental data and obtain magnetic phase diagrams showing magnetic state and transition temperatures versus $a$ and $c$ for Tb and for the alloy Ho${}_{0.4}$Gd${}_{0.6}$. For both systems we mark out the boundaries in $a$ and $c$ space between incommensurate (helical) antiferromagnetic order and ferromagnetic states and find that these agree very well with the theoretical prediction as well as with each other. These data support the proposition of a universal “crystallomagnetic” phase diagram for the heavy rare earths.

Figure 1

(Color online) The Bloch spectral function for the $\Gamma LM$ plane at the Fermi energy for the theoretical paramagnetic rare-earth prototype for $a$ and $c$ values, where the theory indicates (a) ferromagnetic ordering (Gd) and (b) incommensurate magnetic order (Dy). The quantity ${\mathbf{q}}_0$ indicates the nesting wave vector of the Fermi surface and the shading represents the broadening from the thermally induced local moment disorder.

Figure 2

(Color online) The pressure dependence of the magnetic transition temperatures (a) under uniaxial compression along $c$ axis (squares) and (b) under hydrostatic pressure for Ho${}_{0.4}$Gd${}_{0.6}$ (circles). Open dots correspond to the magnetic ordering temperatures $T_{\text{ord}}$, and solid dots to the transition from the helical antiferromagnetic phase to the ferromagnetic one at $T_1$. Dashed lines are linear fits for $T_{\text{ord}}$ and square-root fits for $T_1$. Abbreviations: PM, paramagnetic; HAFM, helical antiferromagnetic; FM, simple ferromagnetic phases. The pressure scales are adjusted in such a manner that the temperature range where the HAFM phase occurs is the same in both (a) and (b) in the any vertical section with the vertical dotted line as an example. Bold-dashed vertical line marks on both figures indicate the triple points where PM, HAFM, and FM phases meet.

Figure 3

(Color online) The magnetic phase diagrams of Fig. 2 replotted as magnetic transition temperature vs $a$ and $c$ hcp crystalline lattice parameters for Ho${}_{0.4}$Gd${}_{0.6}$ (blue hereafter) and the same for Tb (red hereafter, from Ref. 5), viewed in between their triple lines. Open dots correspond to the magnetic ordering temperatures $T_{\text{ord}}$, and solid dots to the transition from the helical antiferromagnetic phase to the ferromagnetic one at $T_1$. Squares represent the uniaxial pressure data, and circles for the hydrostatic ones. Dashed lines (guides to the eye) indicate boundaries between magnetic phases. Bold colored arrows indicate the triple points for both substances at both hydrostatic and uniaxial pressures. Bold-dashed lines are projections of triple lines for Ho${}_{0.4}$Gd${}_{0.6}$ and Tb onto the $a$-$c$ plane; arrows mark the projection of the view direction onto the same plane. Note: The angle between these triple lines is exaggerated in this view; see Fig. 4. Abbreviations: PM, paramagnetic; HAFM, helical antiferromagnetic; FM, simple ferromagnetic phases.

Figure 4

(Color online) Crystallomagnetic phase diagram for Ho${}_{0.4}$Gd${}_{0.6}$ (blue hereafter) and Tb (red hereafter), a projection of Fig. 3 onto the the $a-c$ plane. Solid circles show Ho${}_{0.4}$Gd${}_{0.6}$ and Tb at ambient pressure and the magnetic ordering temperature. Straight colored arrows indicate the behavior of the lattice parameters under uniaxial and hydrostatic compression calculated employing elastic moduli. Open circles correspond to the magnetic triple points under uniaxial and hydrostatic compression. The bold-dashed lines are projections of the triple lines (the same as in Fig. 3) that separate the ranges of ferromagnetic (FM) and helical antiferromagnetic (HAFM) ordering. Arrows mark the projection of the view direction of Fig. 3. Dash-dot-dot and dash-dot lines present the boundary between these types of the magnetic ordering predicted by ab initio theory calculations.