Spin-exchange Hamiltonian and topological degeneracies in elemental gadolinium

We present a comprehensive study of the magnetic exchange Hamiltonian of elemental gadolinium. We use neutron scattering to measure the magnon spectrum over the entire Brillouin zone and fit the excitations to a spin wave model to extract the first 26 nearest-neighbor magnetic exchange interactions with rigorously defined uncertainty. We find these exchange interactions to follow RKKY behavior, oscillating from ferromagnetic to antiferromagnetic as a function of distance. Finally, we discuss the topological features and degeneracies in Gd, and HCP ferromagnets in general. We show theoretically how, with asymmetric exchange, topological properties could be tuned with a magnetic field.

Figure 1

HCP crystal structure of Gd, with layers of triangular lattices.

Figure 2

Measured and fitted spin wave spectra of Gd. The top row (a)–(d) shows the measured spin wave spectra of Gd. The $Q$ width in reciprocal lattice units (r.l.u.) in the direction perpendicular to the $x$ axis is shown in the upper right of each panel. The middle row (e)–(h) shows the LSWT calculated spectrum from the best fit Hamiltonian in Table 1. The bottom row (i)–(l) shows a portion of the data points used in the fit (black circles), and the fitted dispersion from this paper (blue solid line) and Lindgård [17] (orange dashed line).

Figure 3

Constant $Q$ cuts of Gd scattering along the $(1,0,\ell )$ direction [top row, from Fig. 2] and $(h,0,1)$ direction [bottom row, from Fig. 2] compared to the best fit LSWT calculation. The magnon peaks are very clearly visible in the data. A phonon is visible in panel (b) at 6 meV, with much smaller intensity than the magnons. LSWT simulations have a energy peak full width half maximum of 1.5 meV, which was chosen to visually match the experimental width at all energies.

Figure 4

Neighbor bond length and magnitude of the refined exchange for Gd. The five panels indicate different layers along the $c$ axis, where $\delta c$ in each panel indicates the bond length along the $c$ axis. The number in each circle denotes the neighbor number. The color indicates the fitted exchange interaction: Ferromagnetic exchange is indicated by blue, antiferromagnetic by red, with the magnitude indicated by the color bar on the right.

Figure 5

Fit versus number of fitted parameters. Panel (a) shows– ${\chi }_{\text{red}}^2$ as a function of number of parameters. Panel (b) shows the absolute value of the fitted exchange constants. Note that a sudden drop in ${\chi }_{\text{red}}^2$ (seen at $n=12,20,23$, and 26) corresponds to a well-constrained parameter. Because ${\chi }_{\text{red}}^2$ only slowly decreases beyond $n=26$, we cut off the fit at $n=26$.

Figure 6

Correlation matrix for $J_n$ from the Gd spin wave mode fits. Red indicates positive correlation, blue indicates negative correlation. Some parameters, like $J_3$, are highly correlated with many other parameters. Others, like $J_4$, have almost no correlation with other parameters, and thus are uniquely constrained by the data.

Figure 7

Refined values of magnetic exchange constants as a function of bond distance $\left|r\right|$ compared with a fitted RKKY exchange [Eq. (1)], see text. A few key points are labeled by their $J_n$ index.

Figure 8

Linear band crossing at $K=\left(\frac{2}{3}\frac{2}{3}0\right)$. Panels (a) and (b) show the measured neutron scattering for two orthogonal cuts through $K$, highlighting the anisotropic intensity around the dispersion cone. Panels (c) and (d) show constant energy slices above and below the band crossing, showing intensity arcs. Panel (e) shows the intensity binned around the circles in (c) and (d), fitted to a sin function.

Figure 9

LSWT calculated spectrum along high-symmetry directions without (a) and with (b) nonzero DM exchange interaction term on the second neighbor site. When $\mathrm{DM}=0, K$ has a linear band crossing and all $\ell =1/2$ is a nodal plane. When $\mathrm{DM}>0$, the $K\to H$ nodal line and the nodal plane degeneracies are broken, leaving nodal lines only along $A\to L$. Panel (c) shows the Brillouin zone nodal lines in (a), and panel (d) shows the same for (b).

Figure 10

Chiral surface mode in the HCP ferromagnet. The top three panels show the bulk (periodic boundary conditions) and surface ($c$-axis termination) modes in a simplified third-nearest neighbor (3NN) exchange model with a second-neighbor DM exchange of $40\mu \mathrm{eV}$. The three panels show the spins polarized at different angles $\theta$ from the $c$ axis. The different spin polarizations induce different mode splitting because of the DM asymmetry. At a certain polarization angle, the two surface modes meet and cross with opposite slopes, evidencing chiral surface magnon modes. For the simplified 3NN model, the chiral angle is $\theta =0.955$. Panel (d) shows this chiral angle as a function of DM strength for different models: Nearest-neighbor exchange only (green), 3NN model (blue), and the full 26-neighbor model (red). In each case, a minimum DM exchange strength is necessary to produce the chiral mode crossing. Panel (e) shows these chiral angle is governed by the energy gap between the surface and bulk modes at $K$.

Figure 11

Comparison between experimental Gd scattering (top row) and the LSWT calculated scattering from a 25-neighbor model (middle row) and a 26-neighbor model (bottom row). As the red circles indicate, certain features in the data require the 26th neighbor $J$ to reproduce.

Figure 12

Possible Hamiltonian solutions within $\mathrm{\Delta }{\chi }_{\text{red}}^2=1$ of the global optimum fit generated by MCMC (see text). Each panel shows the range of such solutions, which we take as an estimate of uncertainty. The small blue circle represents the best fit values.

Figure 13

Hamiltonian solutions within $\mathrm{\Delta }{\chi }_{\text{red}}^2=1$ plotted against $J_5$, as an example of correlations between fitted parameters. The small blue circle represents the best fit values. A circular distribution indicates no correlation, but an ellipsoidal distribution indicates high correlation. $J_1, J_9$, and $J_{11}$ all are highly correlated with $J_5$, and the correlations are not quite linear.

Figure 14

Comparison between experimental Gd scattering (top row) and LSWT calculated scattering from three simple RKKY models: An isotropic RKKY model (second row), an anisotropic RKKY model (third row) where the Fermi wave vector is allowed to vary as a function of bond angle from $c$, and another anisotropic RKKY model (bottom row) where the Fermi wave vector and energy scale are allowed to vary as a function of bond angle from $c$. The last model comes closest, but is still far from reproducing the details of the exchange model.

Figure 15

Origin of the anisotropic intensity around the $K$-point linear band crossing. Panels (a) and (b) show acoustic and optical modes of a $Q=0$ magnon, where the arrows indicate the spin displacement from equilibrium. Panels (c) and (d) show acoustic and optical modes of a $Q=(\frac{1}{3},\frac{1}{3})$ magnon; (e) and (f) show the same for $Q=(\frac{1}{2},\frac{1}{2})$ magnon. With only Heisenberg magnetic exchange, these two modes are related by a degenerate global rotation of a single sublattice at $Q=(\frac{1}{3},\frac{1}{3})$, highlighted by the dark blue circles. Meanwhile, the acoustic mode becomes lower energy at $Q=(\frac{1}{2},\frac{1}{2})$, signaling a mode crossing between $Q=0$ and $Q=(\frac{1}{2},\frac{1}{2})$, which occurs at $Q=(\frac{1}{3},\frac{1}{3})$. The green lines indicate the $\ell$ in Eq. (D1) which give an equivalent $e^{-i\mathbf{Q}·\ell }$ for a neutron scattered along $\left(hh0\right)$. Because of the staggered spin displacements, the optical modes are always invisible, leading to the anisotropic intensity pattern in panel (e), where the color coding on the double cone corresponds to $S(\mathbf{Q},\omega )$.

Figure 16

Linear band crossing at $K=\left(\frac{1}{3}\frac{1}{3}0\right)$ measured with $E_i=100$ meV neutrons. The resolution and statistics are notably worse than Fig. 8 with $E_i=50$ meV, but the anisotropic intensity is still visible.

Figure 17

Mode splitting at $\ell =3/2$ for Gd. The bottom panel shows a constant $Q$ cut on the $(h,h,3/2)$ line where only one mode has intensity. The top panel shows a constant $Q$ cut near $K$ along $(\frac{1}{3}+h,\frac{1}{3}-h,3/2)$, where both modes have intensity. Using the bottom cut to define a resolution profile, we fit two modes of equal intensity to the top cut, yielding a maximum mode splitting of 0.24(5) meV—well beyond the resolution of this experiment.