Theory of Magnetic Ordering in the Heavy Rare Earths: Ab Initio Electronic Origin of Pair- and Four-Spin Interactions

We describe a disordered local moment theory for long-period magnetic phases and investigate the temperature and magnetic field dependence of the magnetic states in the heavy rare earth elements (HREs), namely, paramagnetic, conical and helical antiferromagnetic (HAFM), fan, and ferromagnetic (FM) states. We obtain a generic HRE magnetic phase diagram which is consequent on the response of the common HRE valence electronic structure to $f$-electron magnetic moment ordering. The theory directly links the first-order HAFM-FM transition to the loss of Fermi surface nesting, induced by this magnetic ordering, as well as provides a template for analyzing the other phases and exposing where $f$-electron correlation effects are particularly intricate. Gadolinium, for a range of hexagonal, close-packed lattice constants $c$ and $a$, is the prototype, described ab initio, and applications to other HREs are made straightforwardly by scaling the effective pair and quartic local moment interactions that emerge naturally from the theory with de Gennes factors and choosing appropriate lanthanide-contracted $c$ and $a$ values.

Figure 1

The generic magnetic $T\text{−}H$ phase diagram for a heavy lanthanide metal for $\mathbf{H}$ applied along the easy direction constructed from the theory. Continuous (discontinuous) lines correspond to second- (first-) order phase transitions, and a tricritical point is marked ($A$). The calculations were performed for the prototype Gd with lattice constants appropriate to Dy.

Figure 2

The Bloch spectral function in the $LMHK$ plane at the Fermi energy for Gd with Dy’s lattice constants for (a) the PM state and resolved into (b) majority spin and (c) minority spin components when there is an overall net average magnetization of 54% ($m_{\mathrm{FM}}=0.54$) of the $T=0$ K saturation value in the FM state. This is the value in our calculations (Fig. 1) in the FM phase just below the temperature $T_t$ of the HAFM-FM first-order transition. ${\mathbf{q}}_{\text{nest}}$ indicates the nesting wave vector of the FS of the PM state, and the shading represents the broadening from thermally induced local moment disorder.

Figure 3

The lattice Fourier transform of the effective pair interactions ${\mathcal{J}}^{\mathrm{eff}}(\mathbf{q})$ (red line) when $m_{\mathrm{FM}}=0$ and its change when the FS is spin polarized, for finite $m_{\mathrm{FM}}$ (dashed blue line for $m_{\mathrm{FM}}=0.3$ and dot-dashed green line for $m_{\mathrm{FM}}=0.54$). The inset shows the dependence of the pair interactions ${\mathcal{J}}_{n{n}^{\prime }}$ on separation $R_{n{n}^{\prime }}=|{\mathbf{R}}_n-{\mathbf{R}}_{n^{\prime }}|$ for $m_{\mathrm{FM}}=0$.