NMR Determination of an Incommensurate Helical Antiferromagnetic Structure in EuCo2As2

TL;DR

This study uses NMR techniques to identify and characterize an incommensurate helical antiferromagnetic structure in EuCo2As2, providing a model-independent determination of its magnetic propagation vector.

Contribution

The paper demonstrates that NMR can uniquely determine the incommensurate helical AFM structure and its propagation vector without relying on model-dependent assumptions.

Findings

Identification of incommensurate helical AFM structure in EuCo2As2

Model-independent determination of the AFM propagation vector

NMR as a tool for spin structure analysis in incommensurate AFMs

Abstract

We report $^{153}$Eu, $^{75}$As and $^{59}$Co nuclear magnetic resonance (NMR) results on EuCo$_2$As$_2$ single crystal. Observations of $^{153}$Eu and $^{75}$As NMR spectra in zero magnetic field at 4.3 K below an antiferromagnetic (AFM) ordering temperature $T_{\rm N}$ = 45 K and its external magnetic field dependence clearly evidence an incommensurate helical AFM structure in EuCo$_2$As$_2$. 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 ${\bf k}$ = (0, 0, 0.73 $\pm$ 0.07)2$\pi$/$c$ where $c$ is the $c$ lattice parameter. Thus the incommensurate helical AFM state was characterized by only NMR data with model-independent analyses, showing NMR to be a unique tool for determination of the spin structure in incommensurate helical AFMs.

Equations

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

Full text

I I. Introduction

Understanding the magnetism in $A$Fe2As2 ($A$ = Ca, Ba, Sr, Eu) known as $``$122$"$ compounds with a ThCr2Si2-type structure at room temperature became one of the important issues after the discovery of iron-pnictide superconductors Kamihara2008 ; Johnston2010 ; Canfield2010 ; Stewart2011 . These systems undergo coupled structural and magnetic phase transitions at a system-dependent Néel temperature $T_{\rm N}$, below which long-range stripe-type antiferromagnetic (AFM) order emerges originating from Fe $3d$ electron spins. Superconductivity (SC) in these compounds emerges upon suppression of the stripe-type AFM phase by application of pressure and/or carrier doping. Because of the proximity between the AFM and the SC phases, it is believed that stripe-type AFM spin fluctuations play an important role in driving the SC in the iron-based superconductors, although orbital fluctuations are also pointed out to be important Kim2013 . Recently ferromagnetic (FM) correlations were revealed to also play an important role in the iron-based superconductors Johnston2010 ; Nakai2008 ; PaulPRB ; PaulPRL ; JeanPRB .

EuFe2As2, which exhibits SC under the application of 2–3 GPa of pressure and/or carrier doping Tereshima2009 ; Jin2016 , is a special member in the “122” class of compounds, as Eu2+ has a large magnetic moment ($J=S=$ 7/2, $L$ = 0), where $J$, $S$, and $L$ are the total, spin and orbital angular momenta, respectively. EuFe2As2 exhibits the stripe-type AFM order at 186 K due to the Fe spins, while the Eu2+ moments order antiferromagnetically below 19 K with an 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 atoms for the Fe atoms in Eu(Fe1-xCox)2As2, the ground-state magnetic structure of the Eu2+ spins is found to develop from the A-type AFM order in the parent compound, via the A-type canted AFM structure with some net FM moment component along the crystallographic $c$ direction at intermediate Co doping levels around $x\sim$ 0.1, and then to the pure FM order along the $c$ axis at $x$ $\sim$ 0.18 (Ref. Jin2016, ). With further substitution up to $x$ = 1, EuCo2As2 is reported to again exhibit A-type AFM order of the Eu ordered moments below $T_{\rm N}$ $\sim$ 40 K Raffius1993 ; Ballinger2012 , similar to the parent compound. On the other hand, recent neutron diffraction (ND) measurement reported a planar helical AFM structure below 47 K where the Eu ordered moments are aligned in the $ab$ plane with the helical axis along the $c$ axis Tan2016 . Therefore, it is important to elucidate the magnetic state of Eu in EuCo2As2 by using different experimental techniques.

Nuclear magnetic resonance (NMR) is a powerful technique to investigate magnetic properties of materials from a microscopic point of view. In particular, one can obtain direct and local information of magnetic state at nuclear sites. Although Eu, Co and As are NMR active nuclei in EuCo2As2, there have been no NMR studies of this compound up to now to our knowledge.

In this paper, we have carried out NMR measurements to investigate the magnetic and electronic states of each ion in EuCo2As2, where we succeeded in observing NMR signals from all three 151Eu, 59Co and 75As nuclei. From the external field dependence of 153Eu and 75As NMR spectra at 4.3 K, below $T_{\rm N}$ = 45 K an incommensurate helical AFM state shown in Fig. 1(a) was clearly evidenced in EuCo2As2. Furthermore, the AFM propagation vector characterizing the helical AFM state is determined to be k = (0, 0, 0.73 $\pm$ 0.07)2$\pi$/$c$ from the internal magnetic induction at the Co site obtained by 59Co NMR under zero magnetic field. 59Co NMR revealed that no magnetic ordering of the Co $3d$ electron spins occurs in the helical AFM state, evidencing that the magnetism in EuCo2As2 originates from only the Eu spins. These results are consistent with the recent neutron diffraction measurements Tan2016 .

II II. Experiment

A single crystal ($9\times 8\times 1$ mm3) of EuCo2As2 for the NMR measurements was grown using Sn flux Sangeetha2017 . 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 75As ($I$ = $\frac{3}{2}$, $\frac{\gamma_{\rm N}}{2\pi}$ = 7.2919 MHz/T, $Q=$ 0.29 barns) nuclei were conducted using a homemade phase-coherent spin-echo pulse spectrometer. In the AFM state, 153Eu, 75As 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. In the paramagnetic (PM) state, 59Co NMR spectra were obtained by sweeping the magnetic field at $f$ = 51.1 MHz. The 75As nuclear spin-lattice relaxation rate 1/$T_{\rm 1}$ was measured with a saturation recovery method T1 .

III III. Results and discussion

At the bottom panel of Fig. 1(b) the 153Eu NMR spectrum in the AFM state for EuCo2As2 is shown, measured in zero magnetic field at a temperature $T$ = 4.3 K. The observed spectrum is well reproduced by the following nuclear spin Hamiltonian which produces a spectrum with a central transition line flanked by two satellite peaks on both sides for $I$ = 5/2, ${\cal H}={\cal-}\gamma\hbar{\bf I}\cdot{\bf B_{\rm int}}+\frac{h\nu_{\rm Q}}{6}[3I_{z}^{2}-I(I+1)+\frac{1}{2}\eta(I_{+}^{2}+I_{-}^{2})]$, where $B_{\rm int}$ is the internal magnetic induction at the Eu site, $h$ is Planck’s constant, and $\nu_{\rm Q}$ is 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 EuCo2As2 has a tetragonal local symmetry (4$/mmm$), $\eta$ is zero. The blue lines shown at the bottom panel of Fig. 1(b) are the calculated positions for 153Eu zero-field NMR (ZFNMR) lines using the parameters $|B_{\rm int}^{\rm Eu}|$ = 27.5(1) T, $\nu_{\rm Q}$ = 30.6(1) MHz and $\theta=90^{\circ}$. Here $\theta$ represents the angle between $B_{\rm int}^{\rm Eu}$ and the principle axis of the EFG tensor at the Eu sites. As shown by the red curve in the figure, the observed 153Eu ZFNMR spectrum was well reproduced with a broadening of $\sim$ 1.1 T of the calculated lines originated from a distribution of $B_{\rm int}$ probably due to Eu ordered moment distributions.

Since $B_{\rm int}^{\rm Eu}$ is perpendicular to the $c$ axis as will be shown below, the principle axis of the EFG is found to be the $c$ axis, which is similar to the case of the Eu nucleus in EuGa4 with the same ThCr2Si2-type crystal structure in which the similar values of $B_{\rm int}^{\rm Eu}$ = 27.08 T and $\nu_{\rm Q}$ = 30.5 MHz for 153Eu have been reported Yogi2013 . $B_{\rm int}^{\rm Eu}$ 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}$$| = 27.5(1) T and the reported AFM ordered moment <$$\mu$$> = 7.26(8) $\mu_{\rm B}$/Eu from ND Tan2016 , $A_{\rm hf}$ is estimated to be $-$3.78 T/$\mu_{\rm B}$ where the sign is reasonably assumed to be negative due to the core-polarization mechanism. The estimated $A_{\rm hf}$ is 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 cancel part of the negative core polarization field.

In order to determine the direction of $B_{\rm int}^{\rm Eu}$ with respect to the crystal axes, we measured 153Eu NMR in 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. 1(b) where the simulated spectra shown by blue and red lines are the same as the case of $H$ = 0]. This indicates that $H$ is perpendicular to the ordered Eu moments and thus to $B_{\rm int}^{\rm Eu}$. 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.

In the case of ${\bf H}$ applied perpendicular to the $c$ axis, on the other hand, each line broadens as shown in the middle panel of Fig. 1(b). The broadening of each line cannot be explained by the A-type AFM state. In this case, one expects a splitting of each line into two lines corresponding to two Eu planes where the Eu ordered moments are parallel or antiparallel to ${\bf H}$. In order to explain the observed spectrum, we consider a planar helical structure which produces a two dimensional powder pattern. The blue solid line is a calculated spectrum for an incommensurate helical AFM state. With the inhomogeneous magnetic broadening due to the same distribution of $B_{\rm int}^{\rm Eu}$ as in the $H$ = 0 T spectrum, the observed spectrum at $H$ = 1 T is reasonably reproduced as shown by the red solid curve. Thus these NMR results reveal an incommensurate helical spin structure with the ordered moments aligned in the $ab$ plane, consistent with recent ND measurements Tan2016 . 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 . A similar incommensurate helical spin structure has been reported in EuCo2P2 Reehuis1992 ; Sangeetha2016 .

The incommensurate planar helical structure is also revealed by 75As NMR measurements. The bottom panel in Fig.  2 shows the 75As ZFNMR spectrum at 4.3 K in the AFM state, where the blue lines are the expected positions for the three lines (for $I$ = 3/2) calculated with the parameter $|B_{\rm int}^{\rm As}|$ = 5.86 T, $\nu_{\rm Q}$ = 17.0 MHz and $\theta=90^{\circ}$. As in the case of the 153Eu ZFNMR spectrum, the observed 75As ZFNMR spectrum is well reproduced with an inhomogeneous magnetic broadening of 4 kOe, as shown by the red curve. The distribution of $B_{\rm int}^{\rm As}$ originates from the distributions of the Eu ordered moments and its directions. 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. 2 where $H$ = 0.5 T. This indicates that $\bf H$ is perpendicular to $\bf B_{\rm int}$ at the As site. On the other hand, when $\bf H$ is applied parallel to the $ab$ plane, similar to the case of 153Eu ZFNMR spectrum, each line broadens and exhibits a characteristic shape, again expected for the incommensurate planer helical AFM state.

According to Yogi et al., the direction of $B_{\rm int}^{\rm As}$ is parallel to the Eu ordered moments in the case where the Eu ordered moments are ferromagnetically aligned in the Eu plane Yogi2013 . Therefore, one can expect almost no change of the 75As ZFNMR spectrum when $H$ is perpendicular to the Eu ordered moment, as observed in the 75As ZFNMR spectrum for $H$ $\parallel$ $c$ axis. On the other hand, if one applies $H$ $\parallel$ $ab$ plane, a splitting of the 75As ZFNMR spectrum is expected similar to the case of the 153Eu ZFNMR spectrum. The blue lines in the two middle panels of Fig. 2 are calculated spectra of 75As NMR for the planar helical AFM structure under $H$ = 0.5 T and 1 T. With the same inhomogeneous magnetic broadening ($\sim$ 4 kOe) due to a distribution of $B_{\rm int}^{\rm As}$, both spectra are well reproduced as shown by the red curves.

The $T$ dependence of the 75As ZFNMR spectrum is shown in Fig. 3(a). With increasing $T$, the spectra shift to lower frequency due to reduction of the internal magnetic induction $|B_{\rm int}^{\rm As}|$ which decreases from 5.86 T at 4.3 K to 3 T at 40 K. No obvious change in $\nu_{\rm Q}$ = 17.0 MHz is observed. The $T$ dependence of $|B_{\rm int}^{\rm As}|$ shown in Fig. 3(b), which is the $T$ dependence of the order parameter of the planar helical AFM state, is well reproduced by a Brillouin function which was calculated based on the Weiss molecular field model with $J$ = $S$ = 7/2, $T_{\rm N}$ = 45 K and $H_{\rm int}^{\rm As}$ = 5.86 T. 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 Sangeetha2017 . The metallic ground state was confirmed by the $T$ dependence of $1/T_{1}$ measured at the central line of the 75As-ZFNMR spectrum. As shown in Fig. 3(c), $1/T_{1}$ is proportional to $T$, thus obeying a Korringa law $1/T_{1}T$ = 190 (sK)-1. This confirms a metallic state from a microscopic point of view.

Now we discuss our 59Co NMR data for both the PM and AFM ordered states. Figure 4(a) shows the field-swept 59Co NMR spectra in the PM state at $T$ = 60 K for $H\parallel c$ and $H\parallel ab$. For $I$ = 7/2 nuclei, one expects a central transition line with three satellite lines on both sides. The observed spectra, however, do not show the seven distinct lines but rather exhibit a single broad line due to inhomogeneous magnetic broadening. The $T$ dependence of the NMR shift for $H$ $\parallel$ $c$ ($K_{c}$) and $H$ $\parallel$ $ab$ ($K_{ab}$) is shown in Fig. 4(b), where we fit the data with the Curie-Weiss law $\frac{C}{T-\theta_{\rm p}}$. The solid curves are fits with $C$ = $-$553 ($-$440) $\%\rm K$ and $\theta_{\rm p}$ = 24.0 (24.0) K for $K_{c}$ ($K_{ab}$). The values $\theta_{\rm p}$ = 24.0 K for both field directions indicate predominant FM exchange interactions between Eu spins. This is consistent with the in-plane FM exchange interactions responsible for the planar helical AFM structure.

The hyperfine coupling constants $A_{ab}$ and $A_{c}$ for 59Co surrounded by four Eu2+ ions can be estimated from the slopes of $K$-$\chi$ plots with the relation $A$ = $\frac{N_{\rm A}}{Z}\frac{K(T)}{\chi(T)}$, where ${N_{\rm A}}$ is Avogadro’s number and $Z=4$ is the number of nearest-neighbor Eu2+ ions around a Co atom. As shown in the inset of Fig. 4(b), both $K_{ab}$ and $K_{c}$ vary linearly with the corresponding $\chi$. From the respective slopes, we estimate $A_{ab}$ = ($-$0.875 $\pm$ 0.09) kOe/$\mu_{\rm B}$/Eu and $A_{c}$ = ($-$1.09 $\pm$ 0.17) kOe/$\mu_{\rm B}$/Eu, respectively. These values are 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 Eu2+ spins and that no $3d$ spins on the Co sites contribute to the magnetism of EuCo2As2.

We now consider the influence of the planar helical AFM state on the Co NMR data. We have succeeded in observing the 59Co ZFNMR spectrum at 4.3 K as shown in Fig. 4(d), where the internal magnetic induction at the Co site is estimated to be $|B_{\rm int}^{\rm Co}|$ = 10.3 kOe. Based on the analysis for $B_{\rm int}$ by Yogi ${\it et~{}al.}$ for EuGa4 (Ref. Yogi2013, ), we extended their calculation of $B_{\rm int}$ to an incommensurate helical AFM state and found that $|B_{\rm int}|$ at the Co site appears in only the $ab$ plane when the Eu ordered moments lie in the $ab$ plane and is expressed by

[TABLE]

where $\phi$ is the turn angle along the 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 an A-type 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 $B_{\rm int}^{\rm Co}$. Thus the observation of the finite $B_{\rm int}^{\rm Co}$ is direct evidence of the planar incommensurate helical AFM state in EuCo2As2. Furthermore, using Eq. (1), 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. 4(c). Using \langle$$\mu$$\rangle = 7.26(8) $\mu_{\rm B}$ Tan2016 , $A_{ab}$ = $-$0.875 kOe/$\mu_{\rm B}$/Eu and $B_{\rm int}^{\rm Co}$ = 10.3 kOe, the turn angle $\phi$ is estimated to be 132*∘* corresponding to a helical wave vector k = (0, 0, 0.73 $\pm$ 0.07)2$\pi$/$c$. This value of k is in very good agreement with ${\bf k}$ = (0, 0, 0.79)2$\pi$/$c$ obtained from ND data Tan2016 .

IV IV. Summary

In summary, we have shown that by analyzing the NMR spectrum in zero field and its external-field dependence, one can determine directly an incommensurate helical AFM structure in EuCo2As2. The AFM propagation vector characterizing the incommensurate helical AFM state was determined model-independently to be k = (0, 0, 0.73 $\pm$ 0.07)2$\pi$/$c$ from the internal magnetic field at the Co site obtained by 59Co NMR under zero magnetic field. Thus NMR can be a unique tool for a model-independent determination of the spin structure in incommensurate helical antiferromagnets. This should prove valuable for the future investigation of local spin configurations in other europium compounds such as in EuCo2P2 which is also reported to exhibit an incommensurate helical AFM structure below 66 K Reehuis1992 ; Sangeetha2016 . Our NMR approach can also be used to study in detail the magnetism originating from the Eu spins in Eu(Fe1-xCox)2As2 SCs where the magnetic structure of the Eu spins changes from the A-type AFM state to a canted AFM state, and then to the ferromagnetic state with increasing Co substitution Jin2016 . Such a detailed study would provide clues about the origin of the coexistence of SC and magnetism in the Eu(Fe1-xCox)2As2 system.

V V. Acknowledgments

The authors thank Mamoru Yogi at University of the Ryukyus for helpful discussions. 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. N. H. thanks the Japan Society for the Promotion of Science KAKENHI : J-physics (Grant Nos. JP5K21732, JP15H05885, and JP16H01078) for financial support to be a visiting scholar at the Ames Laboratory.