NMR studies of the incommensurate helical antiferromagnet EuCo2P2 : determination of the antiferromagnetic propagation vector

TL;DR

This study uses NMR techniques to characterize the incommensurate helical antiferromagnetic structure of EuCo2P2, accurately determining its propagation vector and confirming the magnetic ordering pattern.

Contribution

It provides a detailed NMR analysis of EuCo2P2, including the first determination of its AFM propagation vector using multiple NMR isotopes, advancing understanding of its magnetic structure.

Findings

Confirmed incommensurate helical AFM structure in EuCo2P2.

Determined the AFM propagation vector k = (0, 0, 0.73 ± 0.09)2π/c.

Discussed the temperature dependence of the propagation vector.

Abstract

Recently Ding et al. [Phys. Rev. B 95, 184404 (2017)] reported that their nuclear magnetic resonance (NMR) study on EuCo$_2$As$_2$ successfully characterized the antiferromagnetic (AFM) propagation vector of the incommensurate helix AFM state, showing that NMR is a unique tool for determination of the spin structures in incommensurate helical AFMs. Motivated by this work, we have carried out $^{153}$Eu, $^{31}$P and $^{59}$Co NMR measurements on the helical antiferromagnet EuCo$_2$P$_2$ with an AFM ordering temperature $T_{\rm N}$ = 66.5 K. An incommensurate helical AFM structure was clearly confirmed by $^{153}$Eu and $^{31}$P NMR spectra on single crystalline EuCo$_2$P$_2$ in zero magnetic field at 1.6 K and its external magnetic field dependence. Furthermore, based on $^{59}$Co NMR data in both the paramagnetic and the incommensurate AFM states, we have determined the model-independent value of the AFM propagation vector k = (0, 0, 0.73 $\pm$ 0.09)2$\pi$/$c$ where $c$ is the $c$-axis lattice parameter. The temperature dependence of k is also discussed.

Equations

$$\mathcal{H}=-\gamma \hslash \mathbf{I}\cdot {\mathbf{B}}_{\mathrm{int}}+\frac{h{\nu }_{\mathrm{Q}}}{6}[3I_z^2-I(I+1)+\frac{1}{2}\eta (I_{+}^2+I_{-}^2)],$$

$$A=\frac{N_{\mathrm{A}}}{Z}\frac{K(T)}{\chi (T)},$$

$$B_{\mathrm{int}}^{\mathrm{Co}}=2⟨\mu ⟩A^{\mathrm{Co}}\sqrt{2+2\cos \varphi },$$

Full text

I I. Introduction

A great deal of attention in magnetism has been given to systems with geometric frustration Lacroix ; HTD . Such competing magnetic interactions between localized spins often result in noncollinear magnetic structures, such as incommensurate helical magnetic structures. Usually, such noncollinear magnetic structures are determined by using neutron diffraction (ND) measurements.

Very recently nuclear magnetic resonance (NMR) has been shown to be another unique tool to determine spin structures in incommensurate helical antiferromagnets (AFM) Ding2017 . By performing 153Eu, 75As and 59Co NMR measurements on the incommensurate helical AFM EuCo2As2 with an ordering temperature $T_{\rm N}$ = 45 K, the AFM propagation vector of the incommensurate helical AFM state was successfully determined Ding2017 . Such an NMR approach can be used to characterize the magnetic structure in other possible helical magnets such as the isostructural metallic compound EuCo2P2 Reehuis1992 and EuCu1.82Sb2 Anand2015 .

Motivated by the above NMR work, we have carried out NMR measurements to characterize EuCo2P2 with the body-centered tetragonal ThCr2Si2-type structure which is reported to exhibit an incommensurate helical AFM ground state below $T_{\rm N}$ = 66.5 K Marchand1978 ; Morsen1988 ; Reehuis1992 ; Nakama2010 ; Sangeetha2016 . The neutron diffraction measurements Reehuis1992 on EuCo2P2 report that the Eu ordered moment at 15 K is 6.9 $\mu_{\rm B}$/Eu, where $\mu_{\rm B}$ is the Bohr magneton, consistent with Eu2+ ($J=S=$ 7/2) and spectroscopic splitting factor $g$ = 2. The magnetic structure is the same as that in EuCo2As2 where the Eu ordered moments are aligned ferromagnetically in the $ab$ plane with the helix axis along the $c$ axis Reehuis1992 . The AFM propagation vector k = (0, 0, 0.852)2$\pi$/$c$ at 15 K was determined by the ND measurements Reehuis1992 , where $c$ is the $c$-axis lattice parameter. The similar value of k = (0, 0, 0.88)2$\pi$/$c$ at $T$ = 0 K (Ref. Sangeetha2016 ) was also obtained by the analysis of $\chi$ data on a single crystal below $T_{\rm N}$ using molecular field theory which has been recently formulated to apply to planar noncollinear Heisenberg antiferromagnets Johnston2012 ; Gotesch2014 ; Johnston2015 . It is important to independently determine the AFM propagation vector k by using the NMR technique.

Another interesting feature of metallic EuCo2P2 is a change in magnetic properties observed under high pressure Chefki1998 . With the application of pressure, EuCo2P2 exhibits a first order tetragonal to collapsed-tetragonal transition and associated valence transition from Eu2+ to nonmagnetic Eu3+ ($J$ = 0) at 3.1 GPa. Below 3.1 GPa, the AFM ordering originates from Eu 4$f$ local moments where Co moments are considered to be not involved in the magnetic ordering. On the other hand, above 3.1 GPa, the change in the Eu valence from 2+ to 3+ leads to the appearance of itinerant 3$d$ magnetic ordering below 260 K. Thus it is interesting and important to characterize the magnetic and electronic states of each ion in EuCo2P2 from a microscopic point of view.

In this paper, we report NMR results on EuCo2P2, where we succeeded in observing NMR signals from all three 153Eu, 59Co and 31P nuclei, focusing our attention on the spin structure in the incommensurate helical AFM state with the aim of obtaining better understandings of the local magnetic and electronic properties of the three ions in the AFM and paramagnetic states at ambient pressure. From the external field dependence of 153Eu and 31P NMR spectra at 1.6 K, below $T_{\rm N}$ = 66.5 K an incommensurate helical AFM state shown in Fig. 1 was clearly evidenced in EuCo2P2. Furthermore, the AFM propagation vector characterizing the helical AFM state is determined to be k = (0, 0, 0.73 $\pm$ 0.09)2$\pi$/$c$ from the internal magnetic field at the Co site obtained by 59Co NMR under zero magnetic field. The estimated value is slightly smaller than those reported from the neutron diffraction and magnetic susceptibility measurements Reehuis1992 ; Sangeetha2016 . 59Co NMR revealed that no magnetic ordering of the Co $3d$ electron spins occurs in the helix AFM state, evidencing that the magnetism in EuCo2P2 originates from only the Eu spins. The temperature dependence of the Eu ordered moments determined by the internal magnetic induction at the P site $B_{\rm int}^{\rm P}$ can be well reproduced by the Brillouin function with $J$ =$S$ = 7/2, confirming that the magnetic state of the Eu2+ ions is well explained by the local moment picture although the system is metallic. Our NMR study shows again that NMR is a powerful tool for determination of the spin structure in incommensurate helical AFMs.

II II. Experimental

A single crystal ($1\times 1\times 0.3$ mm3) of EuCo2P2 for the NMR measurements was grown using Sn flux Marchand1978 ; Sangeetha2016 . NMR measurements of 153Eu ($I$ = $\frac{5}{2}$, $\frac{\gamma_{\rm N}}{2\pi}$ = 4.632 MHz/T, $Q=$ 2.49 barns), 59Co ($I$ = $\frac{7}{2}$, $\frac{\gamma_{\rm N}}{2\pi}$ = 10.03 MHz/T, $Q=$ 0.4 barns), and 31P ($I$ = $\frac{1}{2}$, $\frac{\gamma_{\rm N}}{2\pi}$ = 17.235 MHz/T) nuclei were conducted using a homemade phase-coherent spin-echo pulse spectrometer. In the AFM state, 153Eu, 31P and 59Co NMR spectra in zero and nonzero magnetic fields $H$ were measured in steps of frequency $f$ by measuring the intensity of the Hahn spin echo. We have used a single crystal for 153Eu, 31P NMR spectrum measurements at the lowest temperature of 1.6 K. Above $T$ = 1.6 K, we performed our measurements using powdered single crystals as intensities of NMR signals with the single crystals were too weak to perform the measurements at higher temperatures. In the paramagnetic (PM) state, 59Co and 31P NMR spectra were obtained by sweeping the magnetic field at $f$ = 51.2 MHz.

III III. Results and discussion

III.1 A. 153Eu NMR spectrum

The bottom panel of Fig. 2 shows the 153Eu NMR spectrum in the AFM state for EuCo2P2 (single crystal) measured in zero magnetic field at a temperature $T$ = 1.6 K. An almost identical 153Eu NMR spectrum was observed on a powder sample EuCo2P2 at 1.6 K (not shown). The observed spectrum is well reproduced by the following nuclear spin Hamiltonian for the case that the Zeeman interaction is much greater than the quadrupole interaction, which produces a spectrum with a central transition line flanked by two satellite peaks on both sides for $I$ = 5/2,

[TABLE]

where $B_{\rm int}$ is the internal magnetic induction at the Eu site, $h$ is Planck’s constant, and $\nu_{\rm Q}$ is the nuclear quadrupole frequency defined by $\nu_{\rm Q}=3e^{2}QV_{ZZ}/2I(2I-1)h$ $(=3e^{2}QV_{ZZ}/20h$ for $I$ = 5/2) where $Q$ is the electric quadrupole moment of the Eu nucleus, $V_{ZZ}$ is the electric field gradient (EFG) at the Eu site, and $\eta$ is the asymmetry parameter of the EFG Slichter_book . Since the Eu site in EuCo2P2 has a tetragonal local symmetry (4$/mmm$), $\eta$ is zero. The red line shown at the bottom panel of Fig. 2 is the calculated spectrum for 153Eu zero-field NMR (ZFNMR) using the parameters $|B_{\rm int}^{\rm Eu}|$ = 25.75(2) T (= 119.3 MHz), $\nu_{\rm Q}$ = 30.2(2) MHz and $\theta=90^{\circ}$. Here $\theta$ represents the angle between $B_{\rm int}^{\rm Eu}$ and the principal axis of the EFG tensor at the Eu sites.

Since the principal axis of the EFG at the Eu site is parallel to the $c$ axis due to the local symmetry Yogi2013 ; Ding2017 , $\theta=90^{\circ}$ indicates that $B_{\rm int}^{\rm Eu}$ is perpendicular to the $c$ axis. This is similar to the case of the Eu nuclei in EuCo2As2 and EuGa4 with the same ThCr2Si2-type crystal structure in which the similar values of $B_{\rm int}^{\rm Eu}$ = 27.5 T and 27.08 T and $\nu_{\rm Q}$ = 30.6 MHz and 30.5 MHz for 153Eu, respectively, have been reported Yogi2013 ; Ding2017 . $B_{\rm int}^{\rm Eu}$ is proportional to A_{\rm hf}$$<$$\mu$$> where $A_{\rm hf}$ is the hyperfine coupling constant and <$$\mu$$> is the ordered Eu magnetic moment. The hyperfine field at the Eu sites mainly originates from core polarization from 4$f$ electrons and is oriented in a direction opposite to that of the Eu moment Freeman1965 . For |$$B_{\rm int}^{\rm Eu}$$| = 25.75(2) T and the reported AFM ordered moment <$$\mu$$> = 6.9(1) $\mu_{\rm B}$/Eu from ND Reehuis1992 , $A_{\rm hf}$ is estimated to be $-$3.73 T/$\mu_{\rm B}$ where the sign is reasonably assumed to be negative due to the core-polarization mechanism. The estimated $A_{\rm hf}$ is very close to $-$3.78 T/$\mu_{\rm B}$ for the case of EuCo2As2 Ding2017 and is not far from the core-polarization hyperfine coupling constant $-$4.5 T/$\mu_{\rm B}$ estimated for Eu2+ ions Freeman1965 . The small difference could be explained by a positive hyperfine coupling contribution due to conduction electrons which cancels part of the negative core polarization field as has been pointed out in the case of EuCo2As2 (Ref. Ding2017 ).

The direction of $B_{\rm int}^{\rm Eu}$ is also directly confirmed by 153Eu NMR spectrum measurements on the single crystal in nonzero $H$. When $H$ is applied along the $c$ axis, almost no change of the 153Eu NMR spectrum is observed (see the top panel in Fig. 2 where the simulated spectum shown by the red line is the same as the case of $H$ = 0). Since the effective field at the Eu site is given by the vector sum of $\bf{B}_{\rm int}^{\rm Eu}$ and $\bf{H}$, i.e., |$$\bf{B}_{\rm eff}$$| = |$$\bf{B}_{\rm int}^{\rm Eu} + \bf{H}$$|, the resonance frequency is expressed for $H$ $\perp$ <$$\mu$$> as $f$ = \frac{\gamma_{\rm N}}{2\pi}$$\sqrt{(B_{\rm int}^{\rm Eu})^{2}+H^{2}}. For our applied field range where $B_{\rm int}^{\rm Eu}$ $>>$ $H$, any shift in the resonance frequency due to $H$ would be small, as observed. Thus, we conclude that $H$ is perpendicular to $B_{\rm int}^{\rm Eu}$ and thus to the ordered Eu moments.

In the case of ${\bf H}$ applied parallel to the $ab$ plane, on the other hand, each line broadens and exhibits a typical two-horn structure expected for an incommensurate planar helical structure as shown in the middle panel of Fig. 2. In fact, the observed spectrum at $H$ = 1 T is well reproduced by a calculated spectrum for an incommensurate helical AFM state shown by the red line. The inset of the middle panel of Fig. 2 shows the external field dependence of the amount of the splitting of the central transition line ($\Delta f$) of the 153Eu ZFNMR spectra. The $\Delta f$ increases with increasing $H$. Since the peak positions of the two-horn shape of the spectrum are given by $B_{\rm eff}$ = $B_{\rm int}$ $\pm$ $H$, the $\Delta f$ is proportional to $H$ according to $\Delta f$ = 2H$$\gamma_{\rm N}/(2$\pi$). As shown by the solid line in the inset, the $H$ dependence of $\Delta f$ is well reproduced by this relation. Thus these NMR results are consistent with an incommensurate helical spin structure with the ordered moments aligned along the $ab$ plane as reported from the ND Reehuis1992 and magnetization Sangeetha2016 measurements. The observed $ab$-plane alignment of the ordered moments is also consistent with the prediction of the moment alignment from magnetic dipole interactions between the Eu spins Johnston2016 .

III.2 B. 31P NMR spectrum

The incommensurate planar helix structure is also clearly evidenced by 31P NMR measurements. The bottom panel of Fig.  3 shows the 31P ZFNMR spectrum at 1.6 K in the AFM state, where the red line is the fit with the parameter $|B_{\rm int}^{\rm P}|$ = 2.69 T. For $I$ =1/2, a single NMR line is expected and observed, because of no quardupole interaction. When ${\bf H}$ is applied along the $c$ axis, almost no change of the spectrum is observed as typically shown in the top panel of Fig. 3 where $H$ = 0.1 T. For comparison, we show the red line calculated for $H$ = 0. This indicates that $\bf H$ is perpendicular to $\bf B_{\rm int}$ at the P site. On the other hand, when $\bf H$ = 0.1 T is applied parallel to the $ab$ plane, similar to the case of the 153Eu NMR spectrum, the line exhibits a characteristic two-horn shape, again expected for the incommensurate planar helix AFM state.

As discussed for EuGa4 (Ref. Yogi2013 ) and EuCo2As2 (Ref. Ding2017 ), the direction of $B_{\rm int}^{\rm P}$ is antiparallel to the Eu ordered moments in the case where the Eu ordered moments are ferromagnetically aligned in the Eu plane, as shown in Fig. 4(a). Therefore, one can expect almost no change of the 31P NMR spectrum when $H$ is perpendicular to the Eu ordered moment, as observed in the 31P NMR spectrum for $H$ $\parallel$ $c$ axis. On the other hand, if one applies $H$ $\parallel$ $ab$ plane, a splitting of the 31P ZFNMR spectrum is expected similar to the case of the 153Eu ZFNMR spectrum. The red line in the middle panel of Fig. 3 is the calculated spectrum of 31P NMR for the planar helix AFM structure under $H$ = 0.1 T, which reproduces the observed spectrum very well.

The $T$ dependence of the 31P ZFNMR spectrum was measured up to 60 K. With increasing $T$, the spectra shift to lower frequency due to reduction of the internal magnetic induction $|B_{\rm int}^{\rm P}|$ which decreases from 2.69 T at 1.6 K to 1.33 T at 60 K. The $T$ dependence of $|B_{\rm int}^{\rm P}|$ is shown in Fig. 4(b), which is the $T$ dependence of the order parameter of the planar helix AFM state, and is well reproduced by a Brillouin function which was calculated based on the Weiss molecular field model with $J$ = $S$ = 7/2, $T_{\rm N}$ = 66.5 K and $B_{\rm int}^{\rm P}$ = 2.69 T at $T$=1.6 K [solid curve in Fig. 4(b)]. This indicates that the magnetic state of the Eu ions is well explained by the local moment picture although the system is metallic as determined from electrical resistivity measurements Sangeetha2016 .

Now we discuss our NMR data for the PM state. Figure 5 shows the temperature dependence of the field-swept 31P NMR spectra for powdered single crystals and $T$ = 150 K to 300 K. With decreasing $T$, the peak position shifts to higher magnetic field and the line becomes broader and asymmetric due to anisotropy in the Knight shifts in the powder sample. We determine the NMR shifts for $H$ parallel to the $c$ axis (${}^{31}K_{c}$) and parallel to the $ab$ plane (${}^{31}K_{ab}$) from fits of the spectra as shown by the red lines. The $T$ dependences of ${}^{31}K_{c}$ and ${}^{31}K_{ab}$ are shown in Fig. 6. The hyperfine coupling constants $A^{\rm P}$ of 31P surrounded by Eu2+ ions can be estimated from the slopes of $K$-$\chi$ plots with the relation

[TABLE]

where the $\chi(T)$ data are from Ref. Sangeetha2016 , and ${N_{\rm A}}$ is Avogadro’s number and $Z=4$ is the number of nearest-neighbor (NN) Eu2+ ions around a P atom. Here we assume the hyperfine field at the P sites mainly originates from the NN Eu spins. As shown in the inset of Fig. 6, both ${}^{31}K_{c}$ and ${}^{31}K_{ab}$ vary linearly with $\chi$. From the respective slopes, the hyperfine coupling constants $A^{\rm P}_{c}$ and $A^{\rm P}_{ab}$ are estimated to be $-$1.43 $\pm$ 0.10 and $-$1.23 $\pm$ 0.09 kOe/$\mu_{\rm B}$/Eu, respectively.

With the value of $|B_{\rm int}^{\rm P}|$ = 2.69 T, the Eu ordered moment is estimated to be <$$\mu$$> = 5.46 $\mu_{\rm B}$ at 1.6 K using the relation $|B_{\rm int}^{\rm P}|$ = 4A^{\rm P}_{ab}$$<$$\mu$$>. This value is smaller than 6.9 $\mu_{\rm B}$/Eu reported from the ND study Reehuis1992 . The difference may suggest that the estimated $A^{\rm P}_{ab}$ in the PM state is slightly greater than that in the AFM state. This would be possible if one takes finite contributions to hyperfine fields from the next-nearest-neighbor (NNN) Eu spins on the next layer into consideration. Since in the AFM state the direction of NNN Eu spins is antiparallel to that of the NN Eu spins, one expects a positive hyperfine field at the P sites which cancels part of the negative hyperfine field produced by the NN Eu spins. Assuming <$$\mu$$> = 6.9 $\mu_{\rm B}$ (Ref. Reehuis1992 ), we thus estimate a $\sim$ 21 % net additional contribution to the hyperfine field from the NNN Eu spins.

III.3 C. 59Co NMR spectrum

Figure 7 shows the temperature dependence of the field-swept 59Co NMR spectra in the PM state of a powder sample where the spectra are seen to broaden with decreasing $T$. Although one expects a central transition line with three satellite lines on both sides for $I$ = 7/2 nuclei, the observed spectra do not show the seven peaks but rather exhibit a single broad line due to inhomogeneous magnetic broadening. Since the powder sample consists of grains with randomly oriented crystal axes, the spectra are powder patterns. From the fitting of the spectra shown by red lines which are calculated from the nuclear spin Hamiltonian with the Zeeman interaction much greater than the quadrupole interaction, we estimate $\nu_{\rm Q}\sim 0.25$ MHz which is nearly independent of $T$. The broadening of the spectra with lowering $T$ originates from magnetic broadening. The $T$ dependence of the NMR shift ${}^{59}K$ determined from the peak position of the spectrum is shown in Fig. 8(a), where we fit the data with the Curie-Weiss law $\frac{C}{T-\theta_{\rm p}}$. The solid line is a fit with $C$ = -533(13) $\%\rm K$ and $\theta_{\rm p}$ = 18(3) K for ${}^{59}K$. The value $\theta_{\rm p}$ = 18(3) K is the same within the error as the powder averaged value obtained for a single crystal of EuCo2P2 from $\chi(T)$ measurements Sangeetha2016 . The positive value of $\theta_{\rm p}$ indicates predominant ferromagnetic (FM) exchange interactions between the Eu spins. This is consistent with the in-plane FM exchange interactions responsible for the planar helix AFM structure. The hyperfine coupling constant $A^{\rm Co}$ for 59Co surrounded by $Z=4$ Eu2+ ions is also estimated from the slope of $K$-$\chi$ plot in Fig. 8(b) with Eq. (2). We thus estimate $A^{\rm Co}$ = ($-$0.98 $\pm$ 0.09) kOe/$\mu_{\rm B}$/Eu. This value is much smaller than a typical value $A$ = $-$105 kOe/$\mu_{\rm B}$ for Co $3d$ electron core polarization Freeman1965 . This indicates that the hyperfine field at the Co site originates from the transferred hyperfine field produced by the Eu spins and that no $3d$ spins on the Co sites contribute to the magnetism of EuCo2P2.

We now consider the influence of the planar helix AFM state on the Co NMR data. We tried to detect the signals in the AFM state using a single crystal but we could not find any. Then we used powder samples for which we succeeded in observing the 59Co ZFNMR spectrum up to 30 K as shown in Fig. 9. From the peak position of the spectrum, the internal magnetic induction at the Co site at 1.6 K is estimated to be $|B_{\rm int}^{\rm Co}|$ = 11.3 kOe which decreases to 10.2 kOe at 30 K. According to the analysis performed in EuCo2As2 Ding2017 , one can estimate the AFM propagation vector in the incommensurate state of EuCo2P2 based on the estimated values of $|B_{\rm int}^{\rm Co}|$ and $A^{\rm Co}$. Here we present a similar discussion as in the previous paper Ding2017 .

In an incommensurate helical AFM state, $|B_{\rm int}|$ at the Co site appears only in the $ab$ plane when the Eu ordered moments lie in the $ab$ plane and is expressed by Ding2017

[TABLE]

where $\phi$ is the turn angle along the $c$ axis between the Eu ordered moments in adjacent Eu planes, which characterizes the helical structure. In the case of $\phi$ = $\pi$ corresponding to a collinear AFM state, $B_{\rm int}^{\rm Co}$ is zero due to a cancellation of the internal magnetic induction from the four nearest-neighbor Eu ordered moments. On the other hand, if $\phi$ deviates from $\pi$ corresponding to a helical state, one can expect a finite $B_{\rm int}^{\rm Co}$ [see Fig. 10(a)]. Thus the observation of the finite $B_{\rm int}^{\rm Co}$ is direct evidence of the planar incommensurate helix AFM state in EuCo2P2. Furthermore, using Eq. (3), we can determine the AFM propagation vector k = (0, 0, $k$)2$\pi$/$c$, where $c$ is the $c$-axis lattice parameter of the body-centered tetragonal Eu sublattice. Since the distance $d$ along the $c$ axis between adjacent layers of FM-aligned Eu moments is $d$ = $c$/2, the turn angle between the ordered moments in adjacent Eu layers is $\phi$ = $kd$, as shown in Fig. 10(b). Using \langle$$\mu$$\rangle = 6.9(1) $\mu_{\rm B}$ Reehuis1992 , $A^{\rm Co}$ = $-$0.98(9) kOe/$\mu_{\rm B}$/Eu and $B_{\rm int}^{\rm Co}$ = 11.3(1) kOe, the turn angle $\phi$ is estimated to be 131$\pm$16*∘* corresponding to a helix wave vector k = (0, 0, 0.73 $\pm$ 0.09)2$\pi$/$c$. This value of k is slightly smaller than than ${\bf k}$ = (0, 0, 0.852)2$\pi$/$c$ obtained from the ND data Reehuis1992 and (0, 0, 0.88)2$\pi$/$c$ estimated from the $\chi$ data Sangeetha2016 on EuCo2P2, and is close to k = (0, 0, 0.73)2$\pi$/$c$ determined by the NMR data in EuCo2As2 Ding2017 . The origin of the small difference in k between the NMR and ND (and $\chi$) data is not clear, but it could be explained, e.g., if one would take other small contributions to the hyperfine field at the Co site from the NNN Eu spins.

The asymmetric shape of the observed 59Co ZFNMR spectrum originates from a distribution of the internal field at the Co sites. The 153Eu ZFNMR lines are sharp as seen in Fig. 2, indicating homogeneous Eu ordered moments. The very sharp 31P ZFNMR line also indicates that the direction of the Eu moments in each ferromagnetic Eu plane is relatively uniform. Therefore, the low-frequency tail of the Co ZFNMR spectrum suggests a distribution of the turn angle $\phi$, i.e., the AFM propagation vector ${\bf k}$. Using the values of the internal field distribution of the Co site from 0.6 T (6 MHz) to 1.2 T (12 MHz), the distribution of the turn angle $\phi$ is estimated to be from 156 to 131*∘*. This corresponds to a change in ${\bf k}$ from (0, 0, 0.86)2$\pi$/$c$ to (0, 0, 0.73)2$\pi$/$c$. It is worth mentioning that the NMR technique determines not only the AFM propagation vector but also its distribution.

Finally we discuss the temperature dependence of the turn angle $\phi$. Assuming the temperature dependence of the Eu ordered moments $\langle\mu\rangle$ is described by the temperature dependence of $B_{\rm int}^{\rm P}$, one can estimate $\phi$ at each temperature based on the temperature dependence of $B_{\rm int}^{\rm Co}$ using Eq. (3). As shown in the inset of Fig. 9, $\phi$ is nearly independent of temperature up to 30 K. According to the ND measurements, on the other hand, $\phi$ changes from 150*∘* at 64 K just below $T_{\rm N}$ to 153*∘* at 15 K Reehuis1992 . Thus the small 2% change in $\phi$ observed in the ND measurements may occur at temperatures higher than 30 K.

IV IV. Summary and concluding remarks

We have carried out 153Eu, 31P and 59Co NMR measurements on the helical antiferromagnet EuCo2P2 with $T_{\rm N}$ = 66.5 K. The external magnetic field dependence of 153Eu and 31P NMR spectra for single crystalline EuCo2P2 clearly evidenced the incommensurate helical AFM structure. The AFM propagation vector characterizing the incommensurate helical AFM state was determined to be k = (0, 0, 0.73 $\pm$ 0.09)2$\pi$/$c$ from the internal magnetic field at the Co site obtained by 59Co NMR under zero magnetic field. The AFM propagation vector is revealed to be nearly independent of temperature up to 30 K, indicating that the small change in the propagation vector observed by ND measurements may occur at temperatures higher than 30 K.

As described in Ding2017 , our NMR approach can be used to study in detail the magnetism originating from the Eu spins in the Co-substituted iron-pnictide high-$T_{\rm c}$ superconductor Eu(Fe1-xCox)2As2. The $x$ = 0 compound, EuFe2As2, exhibits the stripe-type AFM order at 186 K due to the Fe spins. At the same time, the Eu2+ moments order antiferromagnetically below 19 K with the A-type AFM structure where the Eu ordered moments are FM aligned in the $ab$ plane but the moments in adjacent layers along the $c$ axis are antiferrmagnetically aligned Jeevan2008 . With substitution of Co for Fe in Eu(Fe1-xCox)2As2, the magnetic structure of the Eu2+ spins changes from the A-type AFM order in the $x$ = 0 compound, to the A-type canted AFM structure at intermediate Co doping levels around $x\sim$ 0.1, and then to the FM order along the $c$ axis at $x$ $\sim$ 0.18 where superconductivity (SC) appears below $T_{\rm c}$ $\sim$10 K in the range $x$ = 0.1 to 0.18 Jin2016 . Thus it is important to understand the magnetism originating from the Eu and Fe spins and also the SC properties from a microscopic point of view. Our approach based on the NMR technique provides an important avenue to study the origin of the coexistence of SC and magnetism in Eu(Fe1-xCox)2As2 SCs.

V V. Acknowledgments

We thank H. Uehara and F. Kubota for assistance with the experiments The research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358. Part of the work was supported by the Japan Society for the Promotion of Science KAKENHI : J-Physics (Grant Nos. JP15K21732, JP15H05885, and JP16H01078). N. H. also thanks the KAKENHI : J-Physics for financial support to be a visiting scholar at the Ames Laboratory.