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

###### Abstract

Recently Ding et al. [Phys. Rev. B 95, 184404 (2017)] reported that their nuclear magnetic resonance (NMR) study on EuCoAs 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 Eu, P and Co NMR measurements on the helical antiferromagnet EuCoP with an AFM ordering temperature = 66.5 K. An incommensurate helical AFM structure was clearly confirmed by Eu and P NMR spectra on single crystalline EuCoP in zero magnetic field at 1.6 K and its external magnetic field dependence. Furthermore, based on 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 0.09)2/ where is the -axis lattice parameter. The temperature dependence of k is also discussed.

###### pacs:

75.25.-j, 75.50.Ee, 76.60.-k## 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 Eu, As and Co NMR measurements on the incommensurate helical AFM EuCoAs with an ordering temperature = 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 EuCoP Reehuis1992 () and EuCuSb Anand2015 ().

Motivated by the above NMR work, we have carried out NMR measurements to characterize EuCoP with the body-centered tetragonal ThCrSi-type structure which is reported to exhibit an incommensurate helical AFM ground state below = 66.5 K Marchand1978 (); Morsen1988 (); Reehuis1992 (); Nakama2010 (); Sangeetha2016 (). The neutron diffraction measurements Reehuis1992 () on EuCoP report that the Eu ordered moment at 15 K is 6.9 /Eu, where is the Bohr magneton, consistent with Eu ( 7/2) and spectroscopic splitting factor = 2. The magnetic structure is the same as that in EuCoAs where the Eu ordered moments are aligned ferromagnetically in the plane with the helix axis along the axis Reehuis1992 (). The AFM propagation vector k = (0, 0, 0.852)2/ at 15 K was determined by the ND measurements Reehuis1992 (), where is the -axis lattice parameter. The similar value of k = (0, 0, 0.88)2/ at = 0 K (Ref. Sangeetha2016 ()) was also obtained by the analysis of data on a single crystal below 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 EuCoP is a change in magnetic properties observed under high pressure Chefki1998 (). With the application of pressure, EuCoP exhibits a first order tetragonal to collapsed-tetragonal transition and associated valence transition from Eu to nonmagnetic Eu ( = 0) at 3.1 GPa. Below 3.1 GPa, the AFM ordering originates from Eu 4 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 magnetic ordering below 260 K. Thus it is interesting and important to characterize the magnetic and electronic states of each ion in EuCoP from a microscopic point of view.

In this paper, we report NMR results on EuCoP, where we succeeded in observing NMR signals from all three Eu, Co and P 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 Eu and P NMR spectra at 1.6 K, below = 66.5 K an incommensurate helical AFM state shown in Fig. 1 was clearly evidenced in EuCoP. Furthermore, the AFM propagation vector characterizing the helical AFM state is determined to be k = (0, 0, 0.73 0.09)2/ from the internal magnetic field at the Co site obtained by Co NMR under zero magnetic field. The estimated value is slightly smaller than those reported from the neutron diffraction and magnetic susceptibility measurements Reehuis1992 (); Sangeetha2016 (). Co NMR revealed that no magnetic ordering of the Co electron spins occurs in the helix AFM state, evidencing that the magnetism in EuCoP originates from only the Eu spins. The temperature dependence of the Eu ordered moments determined by the internal magnetic induction at the P site can be well reproduced by the Brillouin function with = = 7/2, confirming that the magnetic state of the Eu 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 ( mm) of EuCoP for the NMR measurements was grown using Sn flux Marchand1978 (); Sangeetha2016 (). NMR measurements of Eu ( = , = 4.632 MHz/T, 2.49 barns), Co ( = , = 10.03 MHz/T, 0.4 barns), and P ( = , = 17.235 MHz/T) nuclei were conducted using a homemade phase-coherent spin-echo pulse spectrometer. In the AFM state, Eu, P and Co NMR spectra in zero and nonzero magnetic fields were measured in steps of frequency by measuring the intensity of the Hahn spin echo. We have used a single crystal for Eu, P NMR spectrum measurements at the lowest temperature of 1.6 K. Above = 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, Co and P NMR spectra were obtained by sweeping the magnetic field at = 51.2 MHz.

## Iii III. Results and discussion

### iii.1 A. Eu NMR spectrum

The bottom panel of Fig. 2 shows the Eu NMR spectrum in the AFM state for EuCoP (single crystal) measured in zero magnetic field at a temperature = 1.6 K. An almost identical Eu NMR spectrum was observed on a powder sample EuCoP 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 = 5/2,

(1) |

where is the internal magnetic induction at the Eu site, is Planck’s constant, and is the nuclear quadrupole frequency defined by for = 5/2) where is the electric quadrupole moment of the Eu nucleus, is the electric field gradient (EFG) at the Eu site, and is the asymmetry parameter of the EFG Slichter_book (). Since the Eu site in EuCoP has a tetragonal local symmetry (4), is zero. The red line shown at the bottom panel of Fig. 2 is the calculated spectrum for Eu zero-field NMR (ZFNMR) using the parameters = 25.75(2) T (= 119.3 MHz), = 30.2(2) MHz and . Here represents the angle between 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 axis due to the local symmetry Yogi2013 (); Ding2017 (), indicates that is perpendicular to the axis. This is similar to the case of the Eu nuclei in EuCoAs and EuGa with the same ThCrSi-type crystal structure in which the similar values of = 27.5 T and 27.08 T and = 30.6 MHz and 30.5 MHz for Eu, respectively, have been reported Yogi2013 (); Ding2017 (). is proportional to where is the hyperfine coupling constant and is the ordered Eu magnetic moment. The hyperfine field at the Eu sites mainly originates from core polarization from 4 electrons and is oriented in a direction opposite to that of the Eu moment Freeman1965 (). For = 25.75(2) T and the reported AFM ordered moment = 6.9(1) /Eu from ND Reehuis1992 (), is estimated to be 3.73 T/ where the sign is reasonably assumed to be negative due to the core-polarization mechanism. The estimated is very close to 3.78 T/ for the case of EuCoAs Ding2017 () and is not far from the core-polarization hyperfine coupling constant 4.5 T/ estimated for Eu 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 EuCoAs (Ref. Ding2017 ()).

The direction of is also directly confirmed by Eu NMR spectrum measurements on the single crystal in nonzero . When is applied along the axis, almost no change of the Eu 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 = 0). Since the effective field at the Eu site is given by the vector sum of and , i.e., = + , the resonance frequency is expressed for as = . For our applied field range where , any shift in the resonance frequency due to would be small, as observed. Thus, we conclude that is perpendicular to and thus to the ordered Eu moments.

In the case of applied parallel to the 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 = 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 () of the Eu ZFNMR spectra. The increases with increasing . Since the peak positions of the two-horn shape of the spectrum are given by = , the is proportional to according to = 2/(2). As shown by the solid line in the inset, the dependence of 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 plane as reported from the ND Reehuis1992 () and magnetization Sangeetha2016 () measurements. The observed -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. P NMR spectrum

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

As discussed for EuGa (Ref. Yogi2013 ()) and EuCoAs (Ref. Ding2017 ()), the direction of 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 P NMR spectrum when is perpendicular to the Eu ordered moment, as observed in the P NMR spectrum for axis. On the other hand, if one applies plane, a splitting of the P ZFNMR spectrum is expected similar to the case of the Eu ZFNMR spectrum. The red line in the middle panel of Fig. 3 is the calculated spectrum of P NMR for the planar helix AFM structure under = 0.1 T, which reproduces the observed spectrum very well.

The dependence of the P ZFNMR spectrum was measured up to 60 K. With increasing , the spectra shift to lower frequency due to reduction of the internal magnetic induction which decreases from 2.69 T at 1.6 K to 1.33 T at 60 K. The dependence of is shown in Fig. 4(b), which is the 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 = = 7/2, = 66.5 K and = 2.69 T at =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 P NMR spectra for powdered single crystals and = 150 K to 300 K. With decreasing , 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 parallel to the axis () and parallel to the plane () from fits of the spectra as shown by the red lines. The dependences of and are shown in Fig. 6. The hyperfine coupling constants of P surrounded by Eu ions can be estimated from the slopes of - plots with the relation

(2) |

where the data are from Ref. Sangeetha2016 (), and is Avogadro’s number and is the number of nearest-neighbor (NN) Eu 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 and vary linearly with . From the respective slopes, the hyperfine coupling constants and are estimated to be 1.43 0.10 and 1.23 0.09 kOe//Eu, respectively.

With the value of = 2.69 T, the Eu ordered moment is estimated to be = 5.46 at 1.6 K using the relation = 4. This value is smaller than 6.9 /Eu reported from the ND study Reehuis1992 (). The difference may suggest that the estimated 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 = 6.9 (Ref. Reehuis1992 ()), we thus estimate a 21 % net additional contribution to the hyperfine field from the NNN Eu spins.

### iii.3 C. Co NMR spectrum

Figure 7 shows the temperature dependence of the field-swept Co NMR spectra in the PM state of a powder sample where the spectra are seen to broaden with decreasing . Although one expects a central transition line with three satellite lines on both sides for = 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 MHz which is nearly independent of . The broadening of the spectra with lowering originates from magnetic broadening. The dependence of the NMR shift determined from the peak position of the spectrum is shown in Fig. 8(a), where we fit the data with the Curie-Weiss law . The solid line is a fit with = -533(13) and = 18(3) K for . The value = 18(3) K is the same within the error as the powder averaged value obtained for a single crystal of EuCoP from measurements Sangeetha2016 (). The positive value of 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 for Co surrounded by Eu ions is also estimated from the slope of - plot in Fig. 8(b) with Eq. (2). We thus estimate = (0.98 0.09) kOe//Eu. This value is much smaller than a typical value = 105 kOe/ for Co 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 spins on the Co sites contribute to the magnetism of EuCoP.

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 Co 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 = 11.3 kOe which decreases to 10.2 kOe at 30 K. According to the analysis performed in EuCoAs Ding2017 (), one can estimate the AFM propagation vector in the incommensurate state of EuCoP based on the estimated values of and . Here we present a similar discussion as in the previous paper Ding2017 ().

In an incommensurate helical AFM state, at the Co site appears only in the plane when the Eu ordered moments lie in the plane and is expressed by Ding2017 ()

(3) |

where is the turn angle along the axis between the Eu ordered moments in adjacent Eu planes, which characterizes the helical structure. In the case of = corresponding to a collinear AFM state, is zero due to a cancellation of the internal magnetic induction from the four nearest-neighbor Eu ordered moments. On the other hand, if deviates from corresponding to a helical state, one can expect a finite [see Fig. 10(a)]. Thus the observation of the finite is direct evidence of the planar incommensurate helix AFM state in EuCoP. Furthermore, using Eq. (3), we can determine the AFM propagation vector k = (0, 0, )2/, where is the -axis lattice parameter of the body-centered tetragonal Eu sublattice. Since the distance along the axis between adjacent layers of FM-aligned Eu moments is = /2, the turn angle between the ordered moments in adjacent Eu layers is = , as shown in Fig. 10(b). Using = 6.9(1) Reehuis1992 (), = 0.98(9) kOe//Eu and = 11.3(1) kOe, the turn angle is estimated to be 13116 corresponding to a helix wave vector k = (0, 0, 0.73 0.09)2/. This value of k is slightly smaller than than = (0, 0, 0.852)2/ obtained from the ND data Reehuis1992 () and (0, 0, 0.88)2/ estimated from the data Sangeetha2016 () on EuCoP, and is close to k = (0, 0, 0.73)2/ determined by the NMR data in EuCoAs Ding2017 (). The origin of the small difference in k between the NMR and ND (and ) 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 Co ZFNMR spectrum originates from a distribution of the internal field at the Co sites. The Eu ZFNMR lines are sharp as seen in Fig. 2, indicating homogeneous Eu ordered moments. The very sharp P 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 , i.e., the AFM propagation vector . 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 is estimated to be from 156 to 131. This corresponds to a change in from (0, 0, 0.86)2/ to (0, 0, 0.73)2/. 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 . Assuming the temperature dependence of the Eu ordered moments is described by the temperature dependence of , one can estimate at each temperature based on the temperature dependence of using Eq. (3). As shown in the inset of Fig. 9, is nearly independent of temperature up to 30 K. According to the ND measurements, on the other hand, changes from 150 at 64 K just below to 153 at 15 K Reehuis1992 (). Thus the small 2% change in observed in the ND measurements may occur at temperatures higher than 30 K.

## Iv IV. Summary and concluding remarks

We have carried out Eu, P and Co NMR measurements on the helical antiferromagnet EuCoP with = 66.5 K. The external magnetic field dependence of Eu and P NMR spectra for single crystalline EuCoP 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 0.09)2/ from the internal magnetic field at the Co site obtained by Co 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- superconductor Eu(FeCo)As. The = 0 compound, EuFeAs, exhibits the stripe-type AFM order at 186 K due to the Fe spins. At the same time, the Eu moments order antiferromagnetically below 19 K with the A-type AFM structure where the Eu ordered moments are FM aligned in the plane but the moments in adjacent layers along the axis are antiferrmagnetically aligned Jeevan2008 (). With substitution of Co for Fe in Eu(FeCo)As, the magnetic structure of the Eu spins changes from the A-type AFM order in the = 0 compound, to the A-type canted AFM structure at intermediate Co doping levels around 0.1, and then to the FM order along the axis at 0.18 where superconductivity (SC) appears below 10 K in the range = 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(FeCo)As 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.

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