Pressure-induced ferromagnetism in the topological semimetal $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$

The antiferromagnet and semimetal $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$ has recently attracted a lot of attention due to a wealth of topological phases arising from the interplay of topology and magnetism. In particular, the presence of a single pair of Weyl points is predicted for a ferromagnetic configuration of Eu spins along the $c$ axis in $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$. In the search for such phases, we investigate here the effects of hydrostatic pressure in $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$. For that, we present specific heat, transport, and $\mu \mathrm{SR}$ measurements under hydrostatic pressure up to $\sim 2.5\mathrm{GPa}$, combined with ab initio density functional theory (DFT) calculations. Experimentally, we establish that the ground state of $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$ changes from in-plane antiferromagnetic $\left({\mathrm{AFM}}_{ab}\right)$ to ferromagnetic at a critical pressure of $\approx 2$ GPa, which is likely characterized by the moments dominantly lying within the $ab$ plane $\left({\mathrm{FM}}_{ab}\right)$. The ${\mathrm{AFM}}_{ab}\text{−}{\mathrm{FM}}_{ab}$ transition at such a relatively low pressure is supported by our DFT calculations. Furthermore, our theoretical results indicate that $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$ moves closer to the sought-for ${\mathrm{FM}}_c$ state (moments $\parallel c$) with increasing pressure further. We predict that a pressure of $\approx 23$ GPa will stabilize the ${\mathrm{FM}}_c$ state if Eu remains in a $2+$ valence state. Thus, our work establishes hydrostatic pressure as a key tuning parameter that (i) allows for a continuous tuning between magnetic ground states in a single sample of $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$ and (ii) enables the exploration of the interplay between magnetism and topology and thereby motivates a series of future experiments on this magnetic Weyl semimetal.

Figure 1

Selected data sets of specific heat divided by temperature, $C/T$, vs $T$ ($8\text{K}\le T\le 11\text{K}$) of $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$ under pressure $p$ for $0.13\mathrm{GPa}\le p\le 1.30\mathrm{GPa}$ (a), for $1.30\mathrm{GPa}\le p\le 1.94\mathrm{GPa}$ (b), and $1.94\mathrm{GPa}\le p\le 2.43\mathrm{GPa}$ (c). The inset in (b) shows an example data set of $d(C/T)$/$dT$, the minimum of which is used to determine the transition temperature (see arrows in the inset and the main panel).

Figure 2

Selected data sets of resistance, normalized to the zero-pressure room-temperature value, $R/R_{300,p=0}$, vs $T$ ($5\text{K}\le T\le 15\text{K}$) of $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$ under pressure $p$ for $0\mathrm{GPa}\le p\le 1.39\mathrm{GPa}$ (a), for $1.39\mathrm{GPa}\le p\le 1.95\mathrm{GPa}$ (b), and for $1.95\mathrm{GPa}\le p\le 2.42\mathrm{GPa}$ (c). Current was applied within the $ab$ plane. The inset in (a) shows all data over the full temperature range up to 300 K. The arrows in (b) indicate the criterion which is used to determine the transition temperature (see text). The inset in (b) depicts the pressure dependence of $R$, normalized to its zero-pressure value $R_{p=0}$, at $T=300\mathrm{K}$ as well as of $R_{\text{max}}.$ The inset in (c) shows an example data set of $dR$/$dT$ at zero pressure, which is used to determine the transition temperatures (see arrow and text).

Figure 3

Temperature-pressure ($T\text{−}p$) phase diagram of $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$, determined from specific heat, resistance, and $\mu \mathrm{SR}$ measurements. Light purple symbols correspond to the transition from the paramagnetic (PM) to the antiferromagnetic (AFM) state. Blue symbols indicate the position of the PM-ferromagnetic (FM) transition. The color shading in the intermediate pressure range indicates that $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$ shows strong tendencies towards FM order (see text for more details).

Figure 4

Evolution of the internal field of the two stopping sites in $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$, $B_{\text{int,1}}$, and $B_{\text{int,2}}$ with temperature $T$ for pressures $p$ of (a) 0 GPa, (b) 1.72 GPa, and (c) 2.56 GPa. Note that the 0 GPa data were taken inside the hydrostatic pressure cell. Lines represent fits to the empirical formula, $B_{\text{int,i}}=B_{\text{int,i}}^0{[1-{\left(\frac{T}{T^{★}}\right)}^{\alpha }]}^{\beta }$ (see text). The insets in (a) and (b) show zero-field spectra (open gray symbols) taken at $T=3.55\mathrm{K}$ and 2.41 K, respectively, together with the fit for two stopping sites (blue lines).

Figure 5

Temperature dependence of the relaxation rate of the pressure cell ${\lambda }_{\text{PC}}$ for pressures of 0 GPa and 2.56 GPa. Lines are guide to the eyes. The insets show the weak-transverse field spectra at 0 GPa and at 2.56 GPa for temperatures below and above the phase transition.

Figure 6

Anisotropic magnetoresistance of $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$ at $T=2\mathrm{K}$ at different pressures. (a) In-plane $R/R_0$, with $R_0$ being the resistance in zero external field, as a function of in-plane field ${\mu }_{0}H$ at different pressures $p$. Arrow indicates the criterion to determine $H_{ab}^{★}$. Note that the data at 2.16 GPa and 2.30 GPa lie on top of each other on this scale. (b) In-plane $R/R_0$ as a function of out-of-plane field ${\mu }_{0}H$ at different pressures $p$. Arrow indicates the criterion to determine $H_c^{★}$. The features at very low field in this data set are likely a small in-plane component that is present due to slight misalignments of the sample with respect to the field. (c) Ambient-pressure anisotropic magnetization as a function of in-plane as well as out-of-plane magnetic field [33]. Arrows indicate the saturation fields $H_{ab}^{★}$ and $H_c^{★}$. (d) Evolution of $H_{ab}^{★}$ and $H_c^{★}$ with pressure. For $p<p_c$, the ground state is AFM with moments lying in the $ab$ plane. Thus, $H_{ab}^{★}$ and $H_c^{★}$ are defined as saturation fields into fully polarized states with moments in the $ab$ and $c$ direction, respectively. For $p>p_c$, the ground state is FM with moments in the $ab$ plane, thus only $H_c^{★}$ can be defined. The light data points in the immediate vicinity of $p_c$ are discussed in Sec. pp1-s2.

Figure 7

Equation of state (EOS) energy difference ($\mathrm{\Delta }{E}_{\text{FM},ab\text{-AFM},ab}$) vs volume ($V$) between the fully relaxed FM and A-type AFM, where in both cases the moment point is in-plane to the nearest-neighbor Eu. The initial ${\mathrm{AFM}}_{ab}$ equilibrium volume of 128.96 ${\AA }^3$/f.u. as well as the phase transition at $\mathrm{\Delta }{E}_{\text{FM},ab\text{-AFM},ab}=0.0$ are indicated by the vertical and horizontal dashed line, respectively. The inset shows an enlarged view around the ${\mathrm{AFM}}_{ab}\text{−}{\mathrm{FM}}_{ab}$ transition region with the $x$ axis converted to pressure ($p$) using the calculated bulk modulus (see text). The dashed line for $\mathrm{\Delta }{E}_{\text{FM},ab\text{-AFM},ab}=0.0$ gives the critical pressure $p_c\approx 3.0$ GPa.

Figure 8

Energy difference of the ${\mathrm{FM}}_c$ order to the ground states (left axis) and corresponding magnetic field (right axis) vs pressure. The solid lines show the energy difference between the ${\mathrm{FM}}_c$ state and the respective ground states, i.e., ${\mathrm{AFM}}_{ab}$ for $p\lesssim 3$ GPa (red) and ${\mathrm{FM}}_{ab}$ for $p\gtrsim 3$ GPa (green). Dashed lines represent the energy differences between non-ground-state magnetic configurations. The ground state transition to ${\mathrm{FM}}_c$ takes place at approximately $p_c^{\prime }\approx 23$ GPa.

Figure 9

Temperature dependence of the dynamical relaxation rate $\mathrm{\Lambda }$ of $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$ for pressures of 0 GPa and 2.56 GPa for $10\mathrm{K}<T<300$ K and $12\mathrm{K}<T<300$ K, respectively, i.e., in the paramagnetic state at each pressure. Lines are guide to the eyes.

Figure 10

Ambient-pressure zero-field $\mu \mathrm{SR}$ asymmetry of an oriented single crystal of $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$ using the GPS spectrometer. The orientation of the sample with respect to the detectors is indicated in the inset of (a). The asymmetry is evaluated from the difference in counts of the up and down detectors (a) as well as the forward and backward detectors (b).

Figure 11

Temperature dependence of the dynamical relaxation rate $\mathrm{\Lambda }$ of $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$ above the magnetic ordering temperature. Data were taken inside the pressure cell at ambient pressure on a arbitrarily oriented aggregate of single crystals (black symbols) as well as an oriented single crystal at ambient pressure outside of the pressure cell (red symbols).

Figure 12

Derivative of $\log (R/R_0$) with respect to field, ${\mu }_{0}H$, for the field oriented along the in-plane direction (a) and out-of-plane direction (b) for selected pressures. The arrows indicate the position of minima that are chosen as criteria for the determination of $H_{ab}^{★}$ and $H_c^{★}$.

Figure 13

Hall resistance $R_{\text{Hall}}$ of $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$ at different pressures at $T=2\mathrm{K}$ (a),(b) and $T=13\mathrm{K}$ (c),(d) in low fields (a),(c) and high fields (b),(d). Data were taken with magnetic field along the in-plane direction. The insets show the high-field slope of the Hall data as a function of pressure at the respective temperatures.

Figure 14

Lattice parameters $a$ (left axis) and $c$ (right axis) of $\mathrm{Eu}{\mathrm{Cd}}_2{\mathrm{As}}_2$ as a function of temperature $T$ at ambient pressure, determined from x-ray diffraction.

Figure 15

Energy difference of the ground state compared to ${\mathrm{FM}}_c$ order (left axis) and corresponding magnetic field (right axis) vs pressure. The red curve describes the difference of ${\mathrm{AFM}}_{ab}$ to ${\mathrm{FM}}_c$, whereas the green curve describes the difference of ${\mathrm{FM}}_{ab}$ to ${\mathrm{FM}}_c$. Compared to the relaxation types in Fig. 7 and Fig. 8, $p_c$ is slightly increased to 4.2 GPa and $p_c^{\prime }$, describing the transition to ${\mathrm{FM}}_c$ ground state, decreased to 20 GPa.