Pressure tuning of competing magnetic interactions in intermetallic CeFe${}_2$

We use high-pressure magnetic x-ray diffraction and numerical simulation to determine the low-temperature magnetic phase diagram of stoichiometric CeFe${}_2$. Near 1.5 GPa we find a transition from ferromagnetism to antiferromagnetism, accompanied by a rhombohedral distortion of the cubic Laves crystal lattice. By comparing pressure and chemical substitution we find that the phase transition is controlled by a shift of magnetic frustration from the Ce-Ce to the Fe-Fe sublattice. Notably the dominant Ce-Fe magnetic interaction, which sets the temperature scale for the onset of long-range order, remains satisfied throughout the phase diagram but does not determine the magnetic ground state. Our results illustrate the complexity of a system with multiple competing magnetic energy scales and lead to a general model for magnetism in cubic Laves phase intermetallic compounds.

Figure 1

(a) Perspective of the cubic Laves lattice of CeFe${}_2$, emphasizing the layered structure and elongated along the $\langle 111\rangle$ axis for clarity. The structure consists of stacked sheets of 3$e$-Fe sites in a kagome lattice, alternating with corrugated sheets of Ce sites and 1$b$-Fe sites in a triangular lattice. Not all atoms are shown for the sake of clarity. Solid lines indicate the nearest-neighbor pairing and dashed lines indicate next-nearest-neighbor pairing. (b)–(d) Clusters of twelve nearest neighbors are shown surrounding each type of atomic site.

Figure 2

(a) Compilation of published data on the magnetic phase behavior of stoichiometric and chemically doped CeFe${}_2$. Magnetic transition temperatures are plotted as a function of lattice constant $a$. The paramagnetic (PM), ferromagnetic (FM), and antiferromagnetic (AF) phases are indicated. The AF phase can be reached with both decreasing and increasing lattice constant. (b) A generic phase diagram for CeFe${}_2$ is created by collapsing the phase diagrams shown in (a) using a tuning parameter $\eta$, which is a linear combination of pressure $P$ (in GPa) and doping $x$ (in $\%$) according to $\eta$ = $P+Ax$. $A$ is a positive, dopant-specific scaling factor with $A$ = 0.20, 0.65, 0.55, and 0.55 GPa/$\%$ for Co, Al, Ru, and Ir, respectively.

Figure 3

Evolution of the CeFe${}_2$ crystal lattice with pressure at $T$ = 3.5 K. (a) Longitudinal ($\theta -2\theta$) scans of the (2, 2, 0) and (3, 1, 1) Bragg peaks at three pressures: in the low-pressure cubic phase, in the regime of phase coexistence, and in the high-pressure rhombohedral phase. The peak splitting at high pressure is evidence of the rhombohedral distortion. The splitting of the (2, 2, 0) peak at $P$ = 1.8 GPa indicates a regime of phase coexistence; for the (3, 1, 1) reflection at 1.8 GPa the peak from the cubic phase is indistinguishable from the rhombohedral (3, 1, $-1$) peak within our measurement resolution (we follow the convention that the rhombohedral distortion compresses the $\langle 111\rangle$ axis). (b) Dependence of the lattice constant $a$ and the cell-axis angle $\alpha$ on pressure. The shaded area marks the phase coexistence regime. Fits to $a$($P$) (dashed lines) are based on the one parameter Birch equation.

Figure 4

Direct observation of antiferromagnetism at $P$ = 3.3 GPa and $T$ = 3.5 K. (a) Longitudinal line scans of magnetic peaks (red) normalized to the lattice peaks (blue) respectively. (b) Quantity $|\Sigma f_m{e}^{i\mathbf{qr}}\vec{\mu }·{\widehat{S}}_2{|}^2$ calculated using Eq. (1) for six measured diffraction peaks (black solid circles). Also shown are the sensitivity limits of null measurements at three positions (black error bars with downward arrows). Calculated values (purple line) are given for the model described in the text.

Figure 5

Spin structure of antiferromagnetic CeFe${}_2$. (a) Schematic showing the bipartite sublattice points ($1,2,3,4$) and ($1^{\prime },2^{\prime },3^{\prime },4^{\prime }$) for a face-centered antiferromagnet.[31, 32] Each sublattice point is associated with a complete Laves basis; the Ce sites for points $1^{\prime }\text{--}{4}^{\prime }$ are omitted for clarity. The magnetic structure on site 1 is explicitly drawn. The spin orientations on site $1^{\prime }$ are the inverse of those on site 1, and likewise for the other pairs. (b) The kagome sheets consisting of 3$e$-Fe spins in a given sublattice are ferromagnetically aligned. (c) The 1$b$-Fe spins have an effective antiferromagnetic interaction and form a plane of frustrated triangular plaquettes. The degeneracy associated with the choice of antiferromagnetic pairs leads to a magnetic domain structure; only a single domain is drawn here for clarity.

Figure 6

Monte Carlo simulation of the generic magnetic phase diagram. Although the energy scale is set by the strongest magnetic interaction, $J_1$, the phase diagram is controlled by competition between the antiferromagnetic Ce-Ce interaction (exchange strength $J_3$), and the ferromagnetic Fe-Fe nearest neighbor interaction ($J_2$). The insets show the spin structure of CeFe${}_2$ in both the ferromagnetic (FM) and antiferromagnetic (AF) phases for the 1$b$-Fe site and its twelve nearest neighbors. The FM spin structure is actually ferrimagnetic, as discussed in Ref. 7. In the ferromagnetic phase, the Ce-Ce bonds are frustrated, while the bonds between 1$b$-Fe and 3$e$-Fe become frustrated in the antiferromagnetic phase.