Enhanced moments of Eu in single crystals of the
metallic helical antiferromagnet EuCo_2-yAs₂

Figures 1 and 2 display the zero-field-cooled (ZFC) magnetic susceptibility $\chi\equiv M/H$ of Sn-flux-grown crystals and CoAs-flux-grown crystals, respectively, as a function of $T$ with $H=~$0.1 T applied along the $c$ axis ($\chi_{c}$, $H\parallel c$) and in the $ab$ plane ($\chi_{ab}$, $H\parallel ab$). The $T_{\rm N}$ of a collinear AFM is given by the temperature of the maximum slope of $\chi T$ versus $T$ for the easy axis direction Fisher1962 ; here, the corresponding field direction is within the easy $ab$ plane of the helical magnetic structure. The inset of each figure shows $d(\chi_{ab}T)/dT$ versus $T$ in the $T$ range 2 to 100 K, with the peak temperature being $T_{\rm N}$. The $T_{\rm N}$ obtained in this way for each crystal is shown in the insets of Figs. 1 and 2 as well as in Table 1 and in Table 3 below. From Table 1 one sees that the $T_{\rm N}$ values correlate with the crystallographic $c/a$ ratio and with the flux used to grow the crystals, but not with the Co-site occupancy. The $T_{\rm N}$ values from previous reports on EuCo_2-yAs₂ are also listed in Table 1 Tan2016 ; Raffius1993 ; Ballinger2012 .

For all four crystals, from the main panels in Figs. 1 and 2 one sees that $\chi_{ab}>\chi_{c}$ in the paramagnetic regime ($T>T_{\rm{N}}$), indicating the presense of a magnetic anisotropy favoring the $ab$ plane. This is consistent with the data for $T\ll T_{\rm N}$ which indicates that the crystallographic $ab$-plane is an AFM easy plane. For $T<T_{\rm N}$, one sees that $\chi_{c}$ is nearly independent of $T$, consistent with the molecular-field theory prediction for a field perpendicular to the ordering axis or plane of a Heisenberg AFM Johnston2012 ; Johnston2015 . Magnetocrystalline anisotropy determines the ordering axis or plane such as for a Heisenberg AFM with dipolar Johnston2016 , uniaxial single-ion $DS_{z}^{2}$ Johnston2017 , and classical field Johnston2017b anisotropies. The observation that $\chi_{ab}$ for $T\to 0$ is a large fraction of $\chi_{c}(T\to 0)$ indicates that EuCo_2-yAs₂ is either a collinear AFM with multiple domains in the $ab$ plane or a coplanar noncollinear $ab$ plane AFM structure. The previous neutron diffraction study on EuCo₂As₂ indeed showed an incommensurate AFM helical structure in which Eu spins are aligned ferromagnetically within the $ab$ plane, where the helix axis is the $c$-axis with an AFM propagation vector of ${\bf k}=(0,0,0.79)\pi/c$ where $c$ is the tetragonal $c$-axis lattice parameter Tan2016 . An incommensurate helical spin structure with almost the same propagation vector was found in the isostructural compound $\rm EuCo_{2}P_{2}$ Reehuis1992 ; Sangeetha2016 .

The inverse susceptibility $\chi^{-1}(T)$ measured in $H=0.1$ T applied along the $c$ axis ($\chi^{-1}_{c}$) and in the $ab$ plane ($\chi^{-1}_{ab}$) for Sn-flux- and CoAs-flux-grown crystals are shown in Figs. 3 and 4, respectively. As one can see from the figures, the $\chi^{-1}$($T$) plots are slightly curved. One can fit this curvature by including a $T$-independent term $\chi_{0}$ in addition to the Curie-Weiss law, giving a so-called modified Curie-Weiss law

As a baseline, we fitted the $\chi_{\alpha}(T)$ data by Eq. (1a) from 100 to 300 K with $\chi_{0}=0$ for each of five of our crystals for each of the two field directions, and the fitted $C_{\alpha}$ and $\theta_{\rm p\alpha}$ values are shown in Table 2 together with $\mu_{\rm eff}$ calculated from $C$ using Eq. (1e). One sees that the values of $\mu_{\rm eff}$ are 4% to 7% larger than the value for $S=7/2$ with $g=2$ given in the table caption, not including the data for outlier crystal #3. These differences are outside the experimental error of $\sim 1$%. Our enhanced values of $\mu_{\rm eff}$ are in qualitative agreement with the previous value in Table 1 reported in Ref. Ballinger2012 . The positive values of $\theta_{\rm p\alpha}$ indicate a net FM exchange interaction between the Eu⁺² spins-7/2.

The value of $\theta_{\rm p\alpha}$ obtained from a fit of experimental $\chi_{\alpha}(T)$ data in the paramagnetic regime at $T>T_{\rm N}$ by Eq. (1a) can be affected by crystal-shape (demagnetization) effects if $\chi_{\alpha}$ is large such as for compounds containing high concentrations of large-spin species such as Eu⁺² with spin $S=7/2$ in EuCo_2-yAs₂. From the treatment in Ref. Johnston2016 , for $\chi_{0}=0$ these affect the Weiss temperature according to

Since $0\leq N_{\rm d\alpha}\leq 1$, a fitted positive value of $\theta_{\rm p\alpha}$ in Table 2 can thus be decreased by up to 1.9 K due to demagnetization effects, which is a maximum of $\sim 10$% of the $\theta_{\rm p\alpha}$ values.

The data for $C$, $\mu_{\rm eff}$, and $\theta_{\rm p}$ for crystal #3 in Table 2 are outliers. We infer that these erroneous values arise from the contribution of a small amount of a ferromagnetic impurity to the magnetization. In particular, including a $\chi_{0}$ in the fits below yields a positive value that includes the FM impurity contribution and leads to $C$, $\mu_{\rm eff}$, and $\theta_{\rm p}$ values in better alignment with those for the other four crystals. From the value of $\chi_{0}$ obtained for crystal #3 below we estimate the contribution of the FM impurity to the magnetization of the crystal in the measuring field of 0.1 T to be $\sim 5\times 10^{-4}~\mu_{\rm B}$/f.u.

Next, we included $\chi_{0}$ in the fits and the three fitting parameters are listed in Table 3 along with the previous reports for this compound. Most of the $\chi_{0}$ values are strongly negative. The fits are shown as the solid curves in Figs. 3 and 4.

Now we obtain an estimate of $\chi_{0}$ expected for ${\rm EuCo_{2}As_{2}}$. ${\rm EuCo_{2}As_{2}}$ is not an ionic compound, so we do not use the ionic values Selwood1956 for the $\chi^{\rm core}$ contributions. Instead, we use the atomic core contributions tabulated in Table 2.1 of Ref. PMS1977 , which are given per mole of atoms as

Assuming the $g$ factor of the conduction carriers is $g=2$, the Pauli spin susceptibility of the conduction carriers in cgs units is given by

where f.u. means the formula unit of EuCo_2-yAs₂ and the density of states at the Fermi energy ${\cal D}(E_{\rm F})$ is for both spin directions, i.e., taking into account the Zeeman degeneracy of the conduction carriers. Taking ${\cal D}(E_{\rm F})\approx 7$ states/eV f.u. obtained from the $C_{\rm p}(T)$ measurements in Table 5 below, one obtains

Then taking into account the Landau diamagnetism of the conduction carriers assuming a free-carrier gas gives the $T$-independent contribution to $\chi$ according to Eq. (1b) as

This value is much smaller in magnitude than the $\chi_{0}$ values listed for crystals #1, #2, #4, and #5 in Table 3, suggesting that these large negative values may instead be reflections of $T$-dependent Curie constants and Weiss temperatures, a possibility examined next.

In order to investigate the possible $T$ dependences of $C_{\alpha}$ and $\theta_{\rm p\alpha}$, we again set $\chi_{0}=0$. We obtained a spline fit to $\chi_{\alpha}(T)$ from 70 to 300 K, and from that we obtained the temperature derivative $\chi_{\alpha}^{\prime}(T)$. Then one has the two simultaneous equations

from which $C_{\alpha}$ and $\theta_{\rm p\alpha}$ were solved for at each $T$. The results are shown in Fig. 5 for Sn-flux-grown crystals #1 $\rm EuCo_{1.90(1)}As_{2}$ and #2 $\rm EuCo_{1.99(4)}As_{2}$ and for CoAs-flux-grown crystal #4 $\rm EuCo_{1.90(2)}As_{2}$. One sees smooth variations in $C$ and $\theta_{\rm p}$ versus $T$ for each crystal, where $C$ increases and $\theta_{\rm p}$ decreases monotonically with decreasing $T$ for each of the three crystals. This behavior of $C$ might be expected if the Eu spins polarize the conduction electrons, since the polarization might be expected to increase with decreasing $T$.

The possibility of conduction-electron polarization due to progressive filling of the $d$ band of the transition metal can lead to a variation of the asphericity of the valence shells of the Eu atoms through a weaker hybridization between the Eu valence states and Co $d$ states. As a result, the itinerant electrons are strongly coupled to the localized moment, leading to an observed effective moment for an $s$-state Eu spin-7/2 given by Stewart1972

Here we take $g=2$, $\rho_{0}(E_{\rm F})$ is the density of states per atom at the Fermi surface for one spin direction, and $J_{sf}$ is the effective $sf$ exchange interaction due to either direct exchange (positive) or $sf$ mixing (negative). The values of $\rho_{0}(E_{\rm F})J_{sf}$ esimated from the effective moments of EuCo_2-yAs₂ compounds are given in the last column of Table 3. The positive sign of the quantity suggests that the $sf$ interaction mechanism in these compounds could be due to direct exchange. These interactions are expected to be affected by the change in lattice parameters $a$ and $c$, and the overall unit-cell volume $V_{\rm cell}$. Another possible reason for the excess Eu moment is related to the contribution of the non-4$f$ electrons of Eu, which is mainly from on-site 5$d$ electrons. This gives rise to dressing of a bare rare-earth spin with a conduction electron spin cloud which for EuCo_2-yAs₂ would add a portion of conduction-electron spin magnetization to the free electron moment. These effects are associated with the indirect RKKY (Ruderman-Kittel-Kasuya-Yosida) exchange interaction Ruderman1954 ; Kasuya1956 ; Yosida1957 and this may affect the $g$ factor. Electron-spin resonance measurements may be useful to confirm or refute the hypothesis that the Curie constant changes with temperature as suggested in Fig. 5.

The $T$- and $H$-dependent magnetic susceptibility $\chi(T,H)$ was measured for one of the two crystals from each of the Sn-flux and CoAs-flux crystal growths. Figures 6 and 7 show $\chi(T)$ of Sn-flux-grown crystal #2$~\rm EuCo_{1.99(2)}As_{2}$ and CoAs-flux-grown crystal #3 $\rm EuCo_{1.92(4)}As_{2}$, respectively, for various values of $H$ applied in the $ab$ plane ($\chi_{ab}$, $H~||~ab$) and along the $c$ axis ($\chi_{c}$, $H~||~c$) for $2~{\rm K}\leq T\leq 100$ K. As shown in Figs. 6(a) and 7(a), the lowest-$T$ data reveal a metamagnetic (MM) transition for $H~||~ab$ between $H=3$ T and 5 T. In addition, breaks in slope of $\chi(T)$ at each field are observed, signifying the $H$-dependent $T_{\rm N}$ which decreases with increasing $H$ as expected for an AFM. Figures 6(b) and 7(b) show that $T_{\rm N}$ is much less sensitive to $H\parallel c$ than to $H\parallel ab$ as seen in Figs. 6(a) and 7(a).

Figures 8 and 9 show $M(H)$ isotherms at $T=2$ K with $H$ applied in the $ab$ plane ($M_{ab},H\parallel ab$) and along the $c$ axis ($M_{c},H\parallel c$) obtained for the Sn-flux-grown crystals #1 $\rm EuCo_{1.90(1)}As_{2}$ and #2 $\rm EuCo_{1.99(2)}As_{2}$ (Fig. 8), and for the CoAs-flux-grown crystals #3 $\rm EuCo_{1.92(4)}As_{2}$ and #4 $\rm EuCo_{1.90(2)}As_{2}$ (Fig. 9). The $M_{c}(H)$ data are nearly linear in field as predicted at $T\ll T\rm_{N}$ by MFT for a helix with the applied field along the helix axis, reaching saturation at the perpendicular critical field $H_{\rm c\perp}\sim 10$–15 T, depending on the sample and field direction.

The $M_{ab}$($H$) isotherms at $T=$ 2 K in Figs. 8 and 9 show what appears to be a field-induced spin-flop (SF) transition at a field $H\rm_{SF}$, with a small hysteresis [see inset of Fig. 8(a)]. The magnetic moment attains its saturation moment $\mu_{\rm sat}$ at the critical field $H\rm_{c}$ which separates the AFM from the paramagnetic (PM) phases. An additional transition of unknown origin at a field $H_{\rm MM}$ is also seen, with $H_{\rm SF}<H_{\rm MM}<H_{\rm c}$.

The detailed $M$($H$) isotherms at many temperatures from 2 K to 300 K of Sn-flux-grown crystals #1 EuCo_1.90(1)As₂ and #2 EuCo_1.99(2)As₂ are shown in Figs. 10 and 11, respectively, and those of CoAs-flux-grown crystals #3 EuCo_1.92(4)As₂ and #4 EuCo_1.90(2)As₂ are shown in Figs. 12 and 13, respectively, where parts (a) and (b) of each of the four figures are for $H\parallel ab\ (M_{ab})$ and $H\parallel c\ (M_{c})$, respectively. For the Sn-grown crystals, $M_{c}(H)$ data in Figs. 10(b) and 11(b) show a negative curvature between 40 and 60 K, but a proportional behaviour of $M_{c}$($H$) is eventually observed at higher temperature ($T>80$ K). On the other hand, $M_{ab}$($H$) in Figs. 10(a) and 11(a) show clear spin flop and metamagnetic transitions at $H\rm_{SF}$ and $H\rm_{MM}$, respectively, for $T\ll 40~$K. These SF and MM transitions shift to lower field with increasing temperature. As shown in Figs. 12 and 13, the CoAs-flux-grown crystals exhibit similar behaviors.

The transition fields $H\rm_{SF}$, $H\rm_{MM}$ and $H\rm_{c}$ versus temperature are taken to be the fields at which $dM$/$dH$ versus $H$ exhibits a peak or a discontinuity (shown in Fig. 14 for Sn-flux-grown crystals and Fig. 15 for CoAs-flux-grown crystals). The results are listed in Table 4. One sees that $H_{\rm c\parallel}$ is different from $H_{\rm c\perp}$ and the saturation moments of these crystals are larger than the theoretical Eu⁺² value $\mu{\rm{}_{sat}}=gS\mu_{\rm B}$/Eu = 7 $\mu_{\rm B}$/Eu, where $g=2$ and $S=7/2$. As seen later in Sec. VIII, this enhancement is due to $d$-electron spin polarization by the ordered Eu spins.

From the above magnetic susceptibility and magnetization data, it is clear that magnetic anisotropy has an important influence on the results. For example, without anisotropy the spin-flop phase for fields in the $ab$ plane would be the stable phase for all fields less than $H_{\rm c}$. Here the anisotropy must give rise to an easy $ab$ plane (XY anisotropy) because the helix axis is $c$ axis and the moments are ferromagnetically-aligned within a given $ab$ plane.

Here we estimate the strength of the anisotropy in terms of a generic classical anisotropy field. The formulas used here are derived in Ref. Johnston2017b . From the value of the anisotropy field parameter $h_{\rm A1}$ to be defined below, we estimate the influence of the anisotropy on the Néel temperature that would occur in the absence of anisotropy.

The definitions and predictions for this type of anisotropy in the presence of Heisenberg exchange interactions are given in Ref. Johnston2017b for systems comprised of identical crystallographically-equivalent spins as applies to the Eu sublattice in EuCo_2-yAs₂. The XY anisotropy field ${\bf H}_{{\rm A}i}$ seen by given moment $\vec{\mu}_{i}$ making an angle $\phi_{i}$ with the positive $x$ axis ($a$ axis here, where the $z$ axis is the $c$ axis) is given by an amplitude $H_{{\rm A0}i}$ times the projection of the moment onto the $xy$ plane, i.e.,

The amplitude is expressed in terms of a more fundamental anisotropy field $H_{\rm A1}$ as

where the reduced ordered and/or field-induced moment $\bar{\mu}_{i}$ is

where $\mu_{i}(T)$ is the $T$-dependent magnitude of $\vec{\mu}_{i}$. Finally, $H_{\rm A1}$ is expressed in reduced form $h_{\rm A1}$ as

where $T_{{\rm N}J}$ is the value that the Néel temperature would have been due to Heisenberg exchange interactions alone (in the absence of anisotropy). Another parameter of the theory is

where $\theta_{{\rm p}J}$ is the Weiss temperature in the Curie-Weiss law due to exchange interactions alone.

The Néel temperature $T_{\rm N}$ in $H=0$ in the presence of both exchange and anisotropy fields is increased in the presence of the XY anisotropy field, as expected, according to the linear relation

The anisotropic Weiss temperatures in the Curie-Weiss law for the paramagnetic susceptibility with XY anisotropy are

where we used Eq. (14) to obtain the third equality. This allows one to easily determine the parameter $h_{\rm A1}$. Usually the ratio on the left side of Eq. (15d) is small, so one can instead use

which is equivalent to the approximation $T_{\rm N}\approx T_{{\rm N}J}$. Using the $T_{\rm N}$ and $\theta_{{\rm p}ab}-\theta_{{\rm p}c}$ values in Table 3, one obtains

Thus the XY anisotropy increases the Néel temperature, also $\theta_{{\rm p}ab}$ by about 5%, or about 2 K for EuCo_2-yAs₂.

In order to fit the low-field $ab$-plane susceptibility $\chi_{ab}(T\leq T_{\rm N})$ by the unified MFT for Heisenberg AFMs in Refs. Johnston2015 and Johnston2012 , we assume that the Curie constant $C_{\alpha}$ and Weiss temperature $\theta_{\rm p\alpha}$ $(\alpha=ab$ or $c$) in the PM state at $T\geq T_{\rm N}$ are independent of $T$ with the values given in Table 3. We first remove the contributions of the $T$-independent susceptibility $\chi_{0}$ and of anisotropy in the PM state to obtain the $\chi_{J\alpha}(T\geq T_{\rm N})$ that would have arisen from exchange interactions alone.

The $T$-independent susceptibility $\chi_{0\alpha}$ is taken into account at all temperatures according to

where $\chi_{\alpha}(T)$ is the measured susceptibility and the $\chi_{0\alpha}$ values are given in Table 3. We assume that the anisotropy in the PM state arises from sources such as magnetic dipole interactions and/or single-ion quantum uniaxial $DS_{z}^{2}$ anistropy, for which the magnetic susceptibility tensor is traceless in the PM state Johnston2016 ; Johnston2017 . Then one obtains the Heisenberg susceptibility $\chi_{J}$ in the PM state given by

as shown in Fig. 16 for one each of the Sn-flux-grown and CoAs-flux grown crystals. As found above in Sec. IV.3, the anisotropy increases $T_{\rm N}$ by about 5% and this small change will henceforth be ignored.

Within MFT, for $T\leq T_{\rm N}$ the perpendicular susceptibility $\chi_{Jc}$ is predicted to be independent of $T$, in good agreement with the data in Fig. 16. The normalized $\chi_{Jab}(T\leq T_{\rm N})/\chi_{J}(T_{\rm N})$ for a helical Heisenberg AFM is given by Johnston2012 ; Johnston2015

the ordered moment versus $T$ in $H=0$ is denoted by $\mu_{0}$, the reduced ordered moment $\bar{\mu}_{0}=\mu_{0}/\mu_{\rm sat}$ is determined by numerically solving the self-consistency equation

$B^{\prime}_{S}(y_{0})=[dB_{S}(y)/dy]|_{y=y_{0}}$ and our definition of the Brillouin function $B_{S}(y)$ is given in Refs. Johnston2015 and Johnston2012 .

We fitted the in-plane $\chi_{Jab}(T)$ data in Fig. 16 by Eqs. (20) using $S=7/2$ and the indicated $f_{J}$ values. For $kd(T)$ we used the neutron diffraction value $kd(T=47~{\rm K})=0.79\pi$ Tan2016 . In order to fit the lowest-$T$ data, we used $kd(T=0)=0.82\pi$ for the Sn-flux-grown crystal and 0.798$\pi$ for the CoAs-flux-grown crystal, calculated from Eqs. (20), which are comparable to the experimentally observed value with respect to neutron diffraction studies Tan2016 . A rough estimated value of $f_{J}$ is $f_{J}\sim$(20 K)/(42 K)  $\sim 0.5$. We treated $f_{J}$ as an adjustable parameter. The $\chi_{Jab}(T\leq T_{\rm N})$ fits thus obtained are plotted as the solid blue curves in Figs. 16(a) and 16(b). Also shown are the $\chi_{Jab}(T\leq T_{\rm N})$ curves using the approximate measured values of $f_{J}$. The discrepancy between the two fitted curves in each figure is a measure of the deficiency of MFT in predicting $\chi_{Jab}(T)$, as previously pointed out in Ref. Johnston2012 .

The heat capacities $C{\rm{}_{p}}(T)$ for Sn-flux-grown crystal #2 EuCo_1.99(2)As₂, CoAs-flux-grown crystal #3 EuCo_1.92(4)As₂, and the nonmagnetic reference compound ${\rm BaCo_{2}As_{2}}$ Sangeetha2016 measured in the temperature range from 1.8 to 300 K are shown in Fig. 17. The data exhibit a prominent peak at $T_{\rm N}=45.1(2)$ K and $T_{\rm N}=40.02(4)$ K for crystals #2 and #3, respectively. Low-temperature $C_{\rm p}/T$ vs $T^{2}$ plots in the range 1.8 to 5 K for the above two crystals and for Sn-flux-grown crystal #6 EuCo_1.94(2)As₂ are shown in the insets of Fig. 17. The data for all three crystals exhibit negative curvature below $\sim 3$ K and hence cannot be fitted by the conventional expression Kittel2005

where $\gamma$ is the Sommerfeld coefficient associated with the conduction electrons and $\beta$ is the coefficent of the $T^{3}$ lattice and three-dimensional AFM spin-wave contributions. Below we attempt to find $\gamma$ by fitting the high-$T$ data. In Table 5 are shown data obtained for similar isostructural compounds.

Shown in Fig. 17(c) are plots of $C{\rm{}_{p}}(T)/T$ versus $T$ for the three crystals #2, #3, and #6. One sees that each crystal shows approximately linear behavior over the $T$ range from 3 to 6 K, i.e., that $C_{\rm p}$ has an approximately $T^{2}$ contribution over this $T$ range. From preliminary linear spin-wave calculations, this behavior may arise from the temperature-dependent heat capacity of AFM spin waves.