${\mathrm{EuCo}}_2{\mathrm{P}}_2$: A model molecular-field helical Heisenberg antiferromagnet

The metallic compound ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ with the body-centered tetragonal ${\mathrm{ThCr}}_2{\mathrm{Si}}_2$ structure containing Eu spins-7/2 was previously shown from single-crystal neutron diffraction measurements to exhibit a helical antiferromagnetic (AFM) structure below $T_{\mathrm{N}}=66.5$ K with the helix axis along the $c$ axis and with the ordered moments aligned within the $ab$ plane. Here we report crystallography, electrical resistivity, heat capacity, magnetization, and magnetic susceptibility measurements on single crystals of this compound. We demonstrate that ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ is a model molecular-field helical Heisenberg antiferromagnet from comparisons of the anisotropic magnetic susceptibility $\chi$, high-field magnetization, and magnetic heat capacity of ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ single crystals at temperature $T\le T_{\mathrm{N}}$ with the predictions of our recent formulation of molecular-field theory. Values of the Heisenberg exchange interactions between the Eu spins are derived from the data. The low-$T$ magnetic heat capacity $\sim T^3$ arising from spin-wave excitations with no anisotropy gap is calculated and found to be comparable to the lattice heat capacity. The density of states at the Fermi energy of ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ and the related compound ${\mathrm{BaCo}}_2{\mathrm{P}}_2$ are found from the heat capacity data to be large, 10 and 16 states/eV per formula unit for ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ and ${\mathrm{BaCo}}_2{\mathrm{P}}_2$, respectively. These values are enhanced by a factor of $\sim 2.5$ above those found from DFT electronic structure calculations for the two compounds. The calculations also find ferromagnetic Eu–Eu exchange interactions within the $ab$ plane and AFM interactions between Eu spins in nearest- and next-nearest planes, in agreement with the MFT analysis of ${\chi }_{ab}(T\le T_{\mathrm{N}})$.

Figure 1

Generic helix AFM structure [8]. Each arrow represents a layer of moments perpendicular to the $z$ axis that are ferromagnetically aligned within the $xy$ plane and with interlayer separation $d$. The wave vector k of the helix is directed along the $z$ axis. The magnetic moment turn angle between adjacent magnetic layers is $kd$. The exchange interactions $J_{z1}$ and $J_{z2}$ within the $J_0\text{−}{J}_{z1}\text{−}{J}_{z2}$ Heisenberg MFT model are indicated.

Figure 2

Powder XRD pattern of ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ at 300 K. The solid line through the experimental points is the two-phase Rietveld refinement fit calculated for the ${\mathrm{ThCr}}_2{\mathrm{Si}}_2$-type crystal structure (space group $I4/mmm)$ including the bct $\beta$-Sn impurity phase structure (space group $I{4}_1/amd)$.

Figure 3

(a) In-plane electrical resistivity $\rho \left(T\right)$ of ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ in the temperature $T$ range 1.8 to 300 K for $H=0$ and 10 T applied parallel to the $c$ axis. The vertical arrow indicates the AFM transition. (b) $\rho \left(T\right)$ of ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ in $H=0$. The curve is the fit of Eqs. (4) to the data above 70 K and is extrapolated to $T=0$. The spin-disorder scattering resistivity ${\rho }_{\mathrm{SD}}\left(T\right)$ plotted as filled diamonds is obtained from Eqs. (5).

Figure 4

(a) Temperature dependence of the heat capacity $C_{\mathrm{p}}$ for ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ and ${\mathrm{BaCo}}_2{\mathrm{P}}_2$ in $H=0$. [Insets (1) and (2)] $C_{\mathrm{p}}/T$ vs $T^2$ for ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ and ${\mathrm{BaCo}}_2{\mathrm{P}}_2$, respectively, where the straight lines are fits of the data between 1.8 and 5 K by Eq. (6). (b) $C_{\mathrm{p}}-\gamma T$ versus $T$ for ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ and ${\mathrm{BaCo}}_2{\mathrm{P}}_2$. The black curve is a fit of the data between 200 and 280 K by the Debye lattice heat capacity model in Eqs. (9).

Figure 5

(a) Magnetic contribution $C{}_{\mathrm{mag}}\left(T\right)$ of the Eu spins in ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ obtained after subtraction of the contribution $C{}_{\mathrm{p}}\left(T\right)$ of ${\mathrm{BaCo}}_2{\mathrm{P}}_2$, between 1.8 and 150 K. The MFT prediction for $C_{\mathrm{mag}}\left(T\right)$ with $S=7/2$ and $T_{\mathrm{N}}=65.7$ K is shown as the black solid line. (b) Magnetic entropy $S_{\mathrm{mag}}\left(T\right)$ calculated from $C_{\mathrm{mag}}\left(T\right)$. The horizontal dashed line is the theoretical high-$T$ limit $S_{\mathrm{mag}}=Rln(2S+1)=17.29$ J/mol K for $S=7/2$.

Figure 6

Heat capacity $C_{\mathrm{p}}$ versus $T$ of ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ at various $H\parallel c,C_{\mathrm{p}}(H,T)$. (Inset) Magnetic $H\text{−}T$ phase diagram for ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ as determined from $C_{\mathrm{p}}(H,T)$ measurements. The solid blue curve is a fit of the four data points by Eq. (10a).

Figure 7

Zero-field-cooled (ZFC) magnetic susceptibility $\chi \equiv M/H$ of a ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ single crystal in the temperature $T$ range between 1.8 and 320 K measured with magnetic fields $\mathit{H}$ applied along the $\mathit{c}$ axis $\left({\chi }_c,H\parallel c\right)$ and in the $ab$ plane $\left({\chi }_{ab},H\parallel ab\right)$ for (a) $H=1$ T and (b) $H=3$ T. The insets in (a) and (b) show ${\chi }_{ab}\left(T\right)$ around $T_{\mathrm{N}}=66.6\left(2\right)$ and 65.3(1) K, respectively.

Figure 8

Zero-field-cooled (ZFC) inverse magnetic susceptibility ${\chi }^{-1}$ for single-crystal ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ in the temperature $T$ range from 1.8 to 320 K measured with magnetic fields (a) $H=1$ T and (b) $H=3$ T applied along the $c$ axis $\left({\chi }_c^{-1},H\parallel c\right)$ and in the $ab$ plane $\left({\chi }_{ab}^{-1},H\parallel ab\right)$. The straight lines are respective fits of the ${\chi }^{-1}\left(T\right)$ data by the Curie-Weiss law.

Figure 9

Magnetization $M$ versus applied magnetic field $H$ isotherms for an ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ crystal at the indicated temperatures $T$ for (a) $H\parallel ab$ and (b) $H\parallel c$. The isotherms in (a) for $T<T_{\mathrm{N}}$ exhibit metamagnetic transitions.

Figure 10

Isothermal magnetization $M$ of ${\mathrm{EuCo}}_2{\mathrm{P}}_2$ versus magnetic field $H$ at $T=2$ K for $H\parallel ab$ and $H\parallel c$. Inset: Expanded view of hysteresis in the $M_{ab}\left(H\right)$ data.

Figure 11

(a) Derivative $dM/dH$ versus $H$ for several temperatures $T$ as indicated. (b) Magnetic peak fields $H^{\mathrm{peak}}$ of the $dM/dH$ data in (a) versus $T$. Four of the plots in (a) have two peak temperatures associated with them at a higher (upwards-pointing filled triangle) and lower (downwards-pointing filled triangle) fields. The other two curves only show a single peak at each temperature (filled circles).

Figure 12

(a) ${\chi }_c^{-1}-{\chi }_{ab}^{-1}$ versus temperature $T$ obtained from the data for $H=1$ T in Fig. 8. (b) ${\chi }_J\left(T\right)$ vs $T$ for $H\parallel ab$ and $H\parallel c$ in $H=1$ T after correction for the anisotropies. The fit of ${\chi }_{Jab}\left(T\right)$ for $T\le T_{\mathrm{N}}$ by the MFT prediction for a helix in Eqs. (32) is shown as the blue curve.

Figure 13

Body-centered Eu sublattice, where $c/a=3$. The Heisenberg exchange interactions $J_{\mathrm{A}},J_{\mathrm{B}}$, and $J_{\mathrm{C}}$ are defined in the figure.

Figure 14

Spin-wave dispersion relation $\omega$ versus $q$ in the $ab$ plane and along the $c$ axis for the helix wave vector given in the figure.

Figure 15

Magnetic unit cells of ferromagnetic (FM) and three antiferromagnetic (AFM) ordered Eu spin lattices for which the total energies were calculated. The solid blue and black spheres correspond within these collinear structures to Eu spins up and spins down, respectively. The solid green circles represent P atoms and the smaller solid brown circles represent Co atoms. The collinear ordering axis of the Eu spins is arbitrary with respect to the present DFT calculations.

Figure 16

Electronic band structure for the calculated ordered spin structures. (Top left) FM. (Top right) AFM1. (Bottom left) AFM2. (Bottom right) AFM3. Spin up (down) states are drawn in blue (red). For the AFM figures, the blue curves are largely obscured by the red ones.

Figure 17

Electronic partial density of states (PDOS), showing $s,p,d$, and $f$ contributions in red, blue, green, and cyan, respectively, for the four spin structures for which LDA calculations were done. Also shown is the total DOS (black) and the total contributions of the Eu atoms (black thick dotted curves). The zero of energy is $E_{\mathrm{F}}$ (horizontal dotted lines).