Exchange and crystal field effects in the ESR spectra of Eu2+ in LaB6 J. G. S. Duque,1 R. R. Urbano,1,2 P. A. Venegas,3 P. G. Pagliuso,1 C. Rettori,1 Z. Fisk,4 and S. B. Oseroff5 1Instituto de Física “Gleb Wataghin,” UNICAMP, 13083-970 Campinas, SP, Brazil 2Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 3Departamento de Física, Universidade Estadual Paulista-Unesp, Caix Postal 473, 17033-360 Bauru, SP, Brazil 4Department of Physics, University of California, Irvine, California 92697-4575, USA 5San Diego State University, San Diego, California 92182, USA �Received 22 May 2007; published 26 September 2007� Electron spin resonance of Eu2+ �4f7, S=7/2� in a La hexaboride �LaB6� single crystal shows a single anisotropic Dysonian resonance. From the observed negative g shift of the resonance, it is inferred that the Eu2+ ions are covalent exchange coupled to the B 2p-like host conduction electrons. From the anisotropy of the spectra �linewidth and field for resonance�, we found that the S ground state of Eu2+ ions experience a cubic crystal field of a negative fourth order crystal field parameter �CFP�, b4=−11.5�2.0� Oe, in agreement with the negative fourth order CFP, A4, found for the non-S ground state R hexaborides. These results support covalency as the dominant contribution to the fourth order CFP for the whole R hexaboride family. DOI: 10.1103/PhysRevB.76.125114 PACS number�s�: 75.50.Gg, 75.50.Tt, 75.60.�d I. INTRODUCTION The cubic hexaboride compounds RB6 �R=rare/alkaline earths� have been the subject of intense experimental and theoretical studies in the last decades. This is due to their variety of interesting physical properties such as magnetic ordering, weak ferromagnetism, metal-insulator transition, magnetic polarons, negative magnetoresistance, quadrupolar ordering, Jahn-Teller effect, superconductivity, heavy fer- mion, fluctuating valence, and Kondo lattice behavior.1–10 Particularly, LaB6 is a stable and hard metal characterized by strong covalent bonds, which is used as a wavelength standard in high resolution x-ray powder diffraction and due to its very low work function, �2.7 eV, as electron emitters.11–13 Besides, for La1−xRxB6 with R=Gd and Ce, spin-glass and Kondo behaviors were, respectively, reported.14,15 Crystal field �CF� effects are known to affect the proper- ties of the hexaborides. For the non-S ground state R �Ce, Nd, and Pr�, low-T anomalies, due to the CF splitting of the ground state multiplet, were observed in various experiments.16–21 There is now a consensus that all non-S ground state R hexaborides present a negative fourth order crystal field parameter �CFP� A4�0.16–21 Also, the experi- mental results in dilute �La,Sm,Ca,Ba,Yb�1−xRxB6 �R =Pr,Er,Dy� are consistent with a negative A4.17,22–25 In this work, we report electron spin resonance �ESR� experiments of Eu2+ �S ground state� ions diluted in the me- tallic hexaboride of LaB6. It is generally observed that for Eu2+ and Gd3+ ions diluted in any cubic metallic/ semimetallic/semiconducting/insulating hosts, the fourth or- der CFP, b4, for both ions, is either positive26–28 or negative,23,29–31 with different absolute values but always with the same sign. It is then expected that the CF acts in the same manner on the S ground state of Eu2+ and Gd3+ �4f7; S=7/2� ions in LaB6. Surprisingly, the analysis of our ESR data allows us to conclude that the b4 parameter of Eu2+ in LaB6 is negative, the same sign as that of A4 for non-S RB6, in contrast to the positive value reported by Luft et al., for Gd3+ in the same compound of LaB6.32 This discrepancy is addressed in this paper. Moreover, it is also a general obser- vation that b4 is positive in metallic hosts and negative in insulators.26,33–37 To the best of our knowledge, our finding of b4�0 for Eu2+ in LaB6 represents a negative value for b4 measured in a metallic host. Although many experimental and theoretical efforts were done trying to correlate the ori- gin of the fourth order CFP b4 �R in an S ground state� with that of the A4 �R in a non-S ground state�, a comprehensive understanding of the dominant contributions to the b4 param- eter has not been achieved yet.33–36 Thus, the R hexaborides form an interesting family of insulating, semiconducting, semimetallic, and metallic systems where the CF effects can be studied. Besides, from our ESR results, we conclude that the Eu2+ and Gd3+ ions, via an exchange interaction, probe different types of conduction electrons at the Fermi level of LaB6. II. EXPERIMENT Single crystals of LaB6 were grown as described in Ref. 3. The cubic structure �space group 221, Pm3m, CsCl type� and phase purity were checked by x-ray powder diffraction and the crystal orientation determined by Laue x-ray diffrac- tion. The ESR spectra were taken in �1�1�0.5 mm3 single crystals in a Bruker X-band �9.48 GHz� and Q-band �34.4 GHz� spectrometers, using appropriated resonators coupled to a T controller of a helium gas flux system for 4.2�T�300 K. The Eu concentration was obtained by fit- ting the susceptibility data to a Curie-Weiss law assuming �ef f =7.94 �B for the Eu2+ ions. Magnetization M�T ,H� mea- surements for 2�T�300 K were taken in a Quantum De- sign Magnetic Properties Measurement System supercon- ducting quantum interference device dc magnetometer. III. EXPERIMENTAL RESULTS Figures 1�a� and 1�b� show, respectively, the low-T ESR spectra for 2200 ppm of Eu in LaB6 at X and Q bands for the PHYSICAL REVIEW B 76, 125114 �2007� 1098-0121/2007/76�12�/125114�6� ©2007 The American Physical Society125114-1 http://dx.doi.org/10.1103/PhysRevB.76.125114 magnetic field H �30° from the �001� direction when H is rotated in the �110� plane. In all our experiments, the ESR spectra showed a single resonance of Dysonian �metallic, A /B�2.24�5�� shape.38 Within the accuracy of the measure- ments and for this orientation, approximately the same line- widths ��Hpp�72�5� Oe� are obtained by both frequencies. This is consistent with a negligible residual inhomogeneous broadening of the resonance. The measured g value indicates that there is a negative g shift ��g=−0.009�2�� relative to that in insulators �g=1.993�, in contrast to the positive g shift found for Gd3+ in LaB6.32 Figures 2�a� and 2�b� display the T dependence of the linewidth �peak to peak of the pure absorption�, �Hpp, and the g value of the resonance of Fig. 1�b�, respectively. The thermal broadening of the linewidth can be fitted to a linear expression, �Hpp=a+bT, with a=63�5� Oe and b =1.70�2� Oe/K. Similar results were obtained for the X band. Within the studied T interval and accuracy of the mea- surements, the g value was found to be T independent, indi- cating the absence of dynamic and/or interaction effects be- tween the localized Eu2+ magnetic moments.39 Figures 3�a� and 3�b� show the Q-band angular depen- dence of the linewidth, �Hpp, and field for resonance, Hr, at T=6 K when H is rotated in the �110� plane, respectively. A strong anisotropic behavior for �Hpp with a minimum for H at ��30° from the �001� direction is observed, while Hr showed a relatively small but still measurable anisotropy. In Fig. 4, we show the angular dependence of the differ- ence, H=Hr���−H0��=30° �, between the resonance field, Hr���, and that corresponding to the minimum linewidth, H0��=30° �=12, 190�10� Oe �see Fig. 3�b��. Although the error bars are large, it is still possible to see that the shift of the field for resonance, H, changes sign around this angle, being H 0 for ��30° and H�0 for � 30°. The experi- mental and calculated �see below� spectra for �=15° and 55° are shown in the inset of Fig. 4. IV. ANALYSIS AND DISCUSSION A. Exchange field Our results show that the Eu2+ resonance in LaB6 present a negative g shift ��g=−0.009�2�� in contrast to the positive g shift ��g=0.018�3�� found for the Gd3+ ions in the same FIG. 1. �Color online� Low-T ESR spectra of Eu2+ in a La1−xEuxB6 �x=0.0022� single crystal for H �30° from �001� in the �110� plane: �a� X band and �b� Q band. The solid lines are the Dyson line shape analysis. FIG. 2. �Color online� T dependence of the Q-band ESR for H �30° �a� line width, �Hpp, and �b� g value of Eu2+ in La1−xEuxB6 �x=0.0022� single crystal. FIG. 3. �Color online� Angular dependence of the Q-band ESR �a� linewidth, �Hpp���, and �b� Hr��� for La1−xEuxB6 �x=0.0022� single crystal. H is rotated in the �110� plane. FIG. 4. �Color online� Q-band angular dependence of the Eu2+ resonance field shift, H=Hr−H0��=30° �. Inset: experimental and calculated Q-band ESR spectra of Eu2+ in La1−xEuxB6 �x=0.0022� single crystal for H ��=15° and 55° from the �001� direction in the �110� plane. DUQUE et al. PHYSICAL REVIEW B 76, 125114 �2007� 125114-2 host.32 The sign change in the g shift may be understood in terms of a two band model40 involving the exchange interac- tion between the localized Eu2+ �and Gd3+� 4f7 electrons with the conduction: �i� R 5d-like electrons and �ii� B 2p-like electrons. The exchange interaction with the 5d-like elec- trons is assumed to be of atomic type, Jat e �q� 0, and that with the B 2p-like electrons is of covalent origin, Jcv h �q��0. Thus, the g shift can be written as �g = �gd + �gp = Jat d �0��F d + Jcv p �0��F p , �1� where Jat d �0� and Jcv p �0� are the q=0 component �zero- conduction electron momentum transfer�41 and �F d and �F p the local densities of states �states/eV mol spin� of d and p elec- trons at the Fermi level, respectively. Although both ions are in the same S ground state �4f7; S=7/2�, we argue that due to their different ionic charges, the local Coulomb repulsion on the 5d conduction electrons of LaB6 will be stronger in the case of the Eu2+ ions. Thus, the Jat d �0��F d term may be- come dominant in the case of Gd3+ and negligible in the case of Eu2+, leading to �g �gd=Jat d �0��F d and �g �gp =Jcv p �0��F p for Gd3+ and Eu2+ in LaB6, respectively. The linear thermal broadening of the homogeneous �Hpp �see Fig. 2�a�� indicates that the spins of the Eu2+ ions relax to the lattice via an exchange coupling between the 4f and conduction electrons �Korringa mechanism�. Therefore, the so-called Korringa rate in the unbottleneck limit, �Hpp /�T,39 should be given by b= ��kB /g�B� �� Jcv p �q��F�F p�2;41 the brackets indicate an average over the Fermi surface. Assuming that there is no q dependence of the exchange interaction, i.e., Jcv p �0�= Jcv p �q��F, the Korringa rate becomes b= ��kB /g�B���gp�2. Using the measured g shift for Eu2+, �gp=−0.009�2� �see Fig. 2�b��, and the involved constants, we find b 1.9�4� Oe/K, in excellent agreement with the value measured experimentally. Hence, we conclude that the Eu2+ ions relax to the lattice basically via an ex- change interaction with the B 2p-like electrons. Similar analysis of the data for the case of Gd3+ in LaB6 �Ref. 32� leads us to conclude that the Gd3+ ions relax to the lattice via an exchange interaction with the R 5d-like electrons. In this case, Luft et al.32 have assumed that �F d may be approxi- mated by the total density of state to be �F d 0.50 states/eV mol spin obtained from the electronic spe- cific heat corrected by the electron-phonon mass enhance- ment of LaB6. With that value, these authors have estimated an exchange parameter of Jat d 40�1� meV. However, band structure calculations suggest that the total density of states at the Fermi level is approximately equally distributed be- tween the La 5d- and B 2p-like electrons.42 Then, assuming that �F d =�F p 0.25 states/eV mol spin, we estimate from the experimental g shifts and Korringa rates values of Jat d 80�2� meV and Jcv p −34�3� meV for Gd3+ and Eu2+ in LaB6, respectively. Note that in the analysis of the g shift, we used the density of states for LaB6 obtained by band struc- ture calculation.42 Moreover, we have assumed that the local density of d states at the Eu site is strongly perturbed by the Coulomb repulsion. Thus, in spite that the Eu d levels will have different energies than those of the La, we argue that the negative g shift will not be much affected by it. Although we are confident in the analysis of our data, in view of the assumptions made, the values for the exchange parameters have to be taken with care. B. Crystal field The angular dependence displayed in Fig. 3�a� for �Hpp shows that besides the homogeneous ESR linewidth, �Hpp���30° �, there is a large contribution to �Hpp from an intrinsic anisotropic inhomogeneous broadening, which re- veals the presence of unresolved cubic CF effects. The mini- mum �Hpp at ��30°, when H is rotated in the �110� plane, identifies the angle where the fine structure of the Eu2+ �4f7; S=7/2� ESR spectra collapses ��=29.7° �.43 It is now well established that diluted localized magnetic moments in me- tallic hosts relax to the lattice via an exchange interaction, Hint=JfsS f ·sce, between the localized spin S f and the host conduction electron spin sce �Korringa mechanism�.39 This mechanism leads to the well known phenomenon of ex- change narrowing of the fine structure and, as a function of � and T, generates a variety of ESR spectra.44,45 We have used this exchange narrowing theory to compute the expected ESR spectra at different angles and temperatures.46,47 Such calculation takes into account the following spin Hamil- tonian: H = g�BH · S f + 1 60 b4�O4 0 + 5O4 4� + JfsS f · sce, �2� where the first term is the Zeeman interaction, the second the fourth order cubic CF potential, and the third the exchange interaction between the localized magnetic moment and the conduction electrons. The sixth order term in the CF poten- tial was not included because the sixth order CFP b6 is al- ways smaller than one-tenth of b4 and, usually, the accuracy of the experiments does not allow one to measure reliable values of b6. To obtain the ESR absorption, the transverse dynamic sus- ceptibility of the local magnetic moment coupled to the con- duction electrons has to be calculated. The transverse dy- namic susceptibility of the local magnetic moments can be calculated following Ref. 45. The model includes the inter- action between local magnetic moments, conduction elec- trons, and CF. Within that model, the susceptibility is ob- tained using the projector formalism in the Liouville space. Our experiments are performed at concentrations where the conduction electron static susceptibility is much smaller than that of the local moments. In that limit the susceptibility for a system in the unbottleneck regime39,45,47 is given by +��� 1 − �0� M,M� PM��−1�M,M�� , �3� where �M,M� −1 is the transition matrix and the quantum num- bers M and M� describe the various Zeeman states �M ,M� =−S ,−S+1,… ,S−1� associated with the S=7/2 Eu2+ spin. The transition probabilities associated with the M↔M +1 transition can be written as EXCHANGE AND CRYSTAL FIELD EFFECTS IN THE ESR… PHYSICAL REVIEW B 76, 125114 �2007� 125114-3 PM = CM exp�M��0/kT�� M� CM� exp�M��0/kT� , �4� where CM =S�S+1�−M�M +1� and k is the Boltzmann con- stant. The elements of the transition matrix, for kT large compared to h�0, are expressed by �M,M� = ���0 g�B − H − HM� M,M� − ia M,M� − i 1 2 bTCM��2 M,M� − M,M�+1 − M,M�−1� , �5� where �0 is the microwave frequency, H the external mag- netic field, a=63�5� Oe the residual linewidth of the various fine structure lines, b=1.70�2� Oe/K the Korringa rate, �B the Bohr magneton, and HM the resonance field of the Eu2+ M↔M +1 transition in a cubic lattice, which is given by HM = H0 + 1 60 b4p M��O4 0 + 5O4 4��M� , �6� where p is the angular dependence in a cubic environment47 and the brackets are the matrix elements of the fourth order CF operator. As we can see in Eq. �5�, the transition matrix �M,M� is tridiagonal. The elements of the main diagonal contain the linewidth and resonance field of each resonance line. The upper and lower diagonal terms represent the fluctuation rates of the local moment between two consecutive reso- nance frequencies. Within the main diagonal, the linewidth that corresponds to the imaginary terms includes the residual linewidth of the various fine structure, a=63�5� Oe, and the Korringa rate b=1.70�2� Oe/K. The real part, related to the magnetic field, contains the magnetic field ��0 /g�B associ- ated with the microwave frequency �0, the external magnetic field H, and the fine structure resonance fields HM. The transition probabilities, with the appropriate Boltz- mann population factors for each of the seven lines, are in- cluded in the calculation.43 We have considered an additional degree of freedom in the calculation that allows for a Gauss- ian distribution of the b4 parameter � b4� ,�b4 �. A distribution of b4 was considered previously by Hardiman et al.,48 when studying Pt:Gd. The Gaussian distribution was symmetri- cally limited around b4� to span only on b4 values with the same sign of b4�. The solid lines shown in Figs. 3�a�, 3�b�, and 4 are the linewidth �Hpp�� ,T=6 K�, the field for reso- nance Hr�� ,T=6 K�, and the shift of the field for resonance H=Hr�� ,T=6 K�−H0��=30° ,T=6 K�, respectively. They were obtained after a Dyson analysis38 of the computed ESR spectra for a=63�5� Oe, b=1.70�2� Oe/K, and A /B =2.24�5�. The best set of values for b4� and 2�b4 obtained from �Hpp�� ,T=6 K� data is b4�=−12�1� Oe and 2�b4 =20�4� Oe, and for H=Hr�� ,T=6 K�−H0��=30° ,T =6 K� data is b4�=−11�1� Oe and 2�b4 =20�4� Oe. Com- bining these results, we obtain b4�=−11.5�2.0� Oe and 2�b4 =20�4� Oe as the most probable set of values for the fourth order CFP and its standard deviation. The inset of Fig. 4 shows the observed experimental ESR spectra at T=6 K for �=15° and 55° and the solid lines correspond to our computed ESR spectra for the same angles and the ESR parameters found for La1−xEuxB6. The negative value found for b4 in LaB6 is consistent with the negative value reported for this parameter in various R1−xEuxB6 and R1−xGdxB6 �R=Ca,Sm�.23,29–31 Therefore, there is a disagreement with the positive value for b4 re- ported by Luft et al.,32 in La1−xGdxB6. Using our method of calculation, including the exchange narrowing mechanism, we have reanalyzed their data for the angular dependence of the resonance �Hpp�� ,T� and shift, H�� ,T�, and we ob- tained b4�=−6�2� Oe and 2�b4 =6�2� Oe for La1−xGdxB6. Thus, their analysis in terms of the first moment of the reso- nance, which does not take into account the exchange nar- rowing mechanism, led them to a misleading conclusion. The fourth order CFP distribution found for R1−xEuxB6 �2�b4 =20�4� Oe� is larger than that found for R1−xGdxB6 �2�b4 =6�2��. That is presumably a consequence of the difference in ionic charge and size between Eu2+, Gd3+, and La3+, which may cause larger local lattice distortions at the Eu2+ site. In our case, the fitting shown in Fig. 4 does not come from the first moment, which is only valid in the extreme narrow re- gime. However, it is obtained using the field for resonance obtained by a Dyson line shape analysis of the calculated spectra, similar to the one used to fit the experimental spec- tra. It should be mentioned that Barnes44 has developed a more complete and involved theory than Plefka.45 To prove the validity of our analysis, using Plefka’s approach, we compared the EPR spectra of Pt:Gd calculated with the Bar- nes theory,48 with the one generated by Plefka, and the agree- ment between the spectra obtained by both theories is good. An eventual difference in the obtained parameters using both models does not compromise the main conclusion of our analysis. Thus, we conclude that the fit of the data of Luft et al.32 using the first moment does not give the correct sign for b4. Most of the reported ESR data for Eu2+ and Gd3+ �S state� show that for any type of local cubic coordination �tetrahedral, octahedral, or simple cubic�, b4 is positive in metallic hosts and negative in insulators and semiconductors.23,26,27,29–31,33–37,47,49 Nonetheless, there are a few low carrier �semimetal/semiconducting� compounds where, still at the lowest T, no CF effects were detected. This may indicate that the maximum crystal field splitting ��40 b4� of the ESR fine structure becomes smaller than the ob- served residual linewidth.51–54 Possibly, this may be due to a subtle cancellation between the various contributions to b4.34,56–58 Hence, to the best of our knowledge, this paper reports a negative value of b4 for both Eu2+ and Gd3+ �S ground state� ions in a good metallic host. Another interesting systematic observed in a few com- pounds of different local cubic coordinations is that the fourth order CFPs, b4 �S ground state� and A4 �non-S ground state�, carry the same sign. That is the case for the R hexaborides,14,16–21,23,31 pnictides,22,50,51 and fluorides,34 where both A4 and b4 are negative, positive, and negative, respectively. However, in simple cubic metals such as Pd, Pt, and Au,34 intermetallic compounds such as �Y,Ce�Pd3 and LaAl2,34,59–61 semiconductors such as CeFe4P12 and DUQUE et al. PHYSICAL REVIEW B 76, 125114 �2007� 125114-4 PbTe,37,49,55 and insulators such as MgO �Ref. 34�, that trend is not satisfied. Therefore, these results indicate that, differ- ently from b4, A4 is more dependent on the type of local cubic coordination than on the metallicity of the material. There has been already a number of efforts to explain the origin of the fourth order CFPs, b4 and A4, and the correla- tion between them. Coles and Orbach56 and Williams and Hirst,57 long ago, have suggested that in metallic hosts, the presence of crystal field splitted 5d conduction electron vir- tual bond state �VBS� may contribute to the screening of the ligand crystal field potential and may account for the sign of fourth order CFP, A4. Chow has introduced the exchange interaction between the 5f and the crystal field splitted 5d VBS to account for the magnitude of the A4 parameter.58 In addition to these Coulombic contributions, Barnes et al.,34 in order to find a correlation between b4 and A4, have consid- ered the covalent contribution to the fourth order CFPs due to the 4f7 valency fluctuation.34 Thus, the balance between all those contributions, as others, should finally determine the sign and magnitude of these parameters. The negative g shift and negative value of b4 found for Eu2+ in LaB6 may indicate the importance of covalency in determining the fourth order CFPs in this metallic material.11 We like to em- phasize that the main purpose of this paper is to report the negative b4 in a metal doped with an S-state impurity. Also, we have addressed for the possible presence of covalent con- tributions to the exchange interaction and the fourth order crystal field parameter in LaB6 doped with Eu and Gd. How- ever, it is out of the scope of this work to elucidate the role of the different covalent contributions to those parameters. V. CONCLUSIONS In summary, using our data and those from others,32 we have shown that in the same metallic host �LaB6�, the Eu2+ and Gd3+ magnetic ion impurities with the same S ground state and electronic configuration �4f7; S=7/2�, but with dif- ferent ionic charges, selectively probe, via an exchange in- teraction, different types of conduction electrons at the host Fermi level. We have argued that this unique behavior may be a consequence of the Coulomb repulsion potential be- tween the ion charges and the host conduction electrons. Most importantly, we have shown that the cubic fourth order CFP b4 for both Eu2+ and Gd3+ S ground state ions is nega- tive in the metallic LaB6 compound. To the best of our knowledge, this paper reports a negative value for the b4 parameter in a metallic host. Following Barnes et al., the negative value of b4 found in the metallic LaB6 compound suggests that in all the R hexaboride family, covalent contri- butions may play an important role in their fourth order CFPs.11,34 ACKNOWLEDGMENTS This work was supported by FAPESP and CNPq, Brazil. 1 R. G. Goodrich, N. Harrison, and Z. Fisk, Phys. Rev. Lett. 97, 146404 �2006�. 2 J. Etourneau and P. Hagenmuller, Philos. Mag. B 52, 589 �1985�. 3 D. P. Young, D. Hall, M. E. Torelli, Z. Fisk, J. L. Sarrao, J. D. Thompson, H. R. Ott, S. B. Oseroff, R. G. Goodrich, and R. Zysler, Nature �London� 397, 412 �1999�. 4 M. E. Zhitomirsky, T. M. Rice, and V. I. Anisimov, Nature �Lon- don� 402, 251 �1999�. 5 L. Degiorgi, E. Felder, H. R. Ott, J. L. Sarrao, and Z. Fisk, Phys. Rev. Lett. 79, 5134 �1997�. 6 S. Massidda, A. Continenza, T. M. de Pascale, and R. Monnier, Z. Phys. B: Condens. Matter 102, 83 �1997�. 7 S. Sullow, I. Prasad, M. C. Aronson, J. L. Sarrao, Z. Fisk, D. Hristova, A. H. Lacerda, M. F. 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