Orbital dependent magnetic exchange interaction in CeX$_{\text{c}}$ (X$_{\text{c}}$=S, Se, Te)

Dispersion relations of the crystal-field excitations in cubic antiferromagnets CeTe, CeSe, and CeS have been investigated by inelastic neutron scattering using single crystalline samples. The Fourier transform of the magnetic exchange interaction $J(q)$ obtained from the crystal-field dispersion is largely different from that of the mean-field interaction obtained from the N\'eel temperature and the Weiss temperature. From detailed reexamination of the magnetic susceptibility and these $J(q)$ relations, we conclude that the magnetic exchange interaction is dependent on the crystal-field levels. The interaction associated with the $\Gamma_8$ excited state is stronger than that with the $\Gamma_7$ ground state.

the anomalous properties observed under high pressures and high magnetic fields. For this purpose, in the present paper, we analyze the dispersion relations of the CEF excitation observed for CeX c 's by inelastic neutron scattering experiments. The temperature dependences of the magnetic susceptibility are reexamined in detail to explain the observations consistently. It is shown that we need to introduce exchange interactions which is dependent on the CEF states.

Experiment
Single crystalline samples were prepared by the Bridgeman method using vacuum sealed tungsten crucibles [6]. Inelastic neutron scattering experiments were performed using the triple-axis thermal neutron spectrometer TOPAN installed at the beam hole 6G of the research reactor JRR-3, Japan Atomic Energy Agency, Tokai. A monochromatized incident beam was obtained using the 002 Bragg reflection of PG crystals. The energy of the scattered beam was analyzed using a PG-002 analyzer in the constant final energy mode at 14.7 meV. Collimators were Blank-30'-30'-Blank. Figure 1 shows the inelastic neutron scattering spectra of the Γ 7 -Γ 8 CEF excitation of CeTe and the dispersion relations at several temperatures. The raw spectrum in Fig. 1(a) was analyzed with a scattering function

Results and Analysis
represents a Lorentzian spectral function centered at E = ∆, the CEF excitation energy, with a half-width Γ and an intensity I. The resultant parameters obtained from the fit is summarized in Fig. 1(b). The q-dependence of the excitation energy is significant at low temperatures and becomes less dispersive at high temperatures. It is interesting that the energy is strongly dependent on temperature, whereas it is independent of temperature at the X-point (1,0,0) in the reciprocal space. The energy width of the excitation peak is almost resolution limited, indicating that the 4 f -electron is well localized in CeTe and the hybridization is weaker than those of CeSe and CeS.
Let us analyze the dispersion relation by considering that it is caused by the inter-site magnetic exchange interaction J i j . The neutron scattering function S (q, E) is related with the imaginary part of the generalized susceptibility χ(q, E) by S (q, E) ∝ χ ′′ (q, E)/{1 − exp(−E/k B T )}, where χ(q, E) can be assumed in the mean-field random-phase (MF-RPA) approximation by (1) J(q) represents the Fourier transform of the magnetic exchange interaction J i, j and χ 0 (E) the frequency (E = ω) dependent single-site dynamic susceptibility due to the CEF split levels. By treating the Γ 8 energy level ∆ and the exchange constants of J 1 (nearest neighbor) and J 2 (next nearest neighbor) as fitting parameters, we analyzed the dispersion relation of Fig. 1. From the result at 2.5 K, J 1 = 0.26 K and J 2 = −0.31 K were obtained. The temperature dependence of the Γ 8 energy level, which decreases with decreasing temperature, is shown in Fig. 1(c). We do not understand the reason for this unusual behavior of ∆(T ) at the moment. The dispersion relation of J(q) = j J j exp(−iq · r j ) deduced from the above obtained J 1 and J 2 values for CeTe is shown in Fig. 2(a). The J(q) relations for CeSe and CeS, which were also deduced from the same procedure as above by inelastic neutron scattering, are also shown in Fig. 2 Table I by J (CEF) . For all the three compounds J(q) takes the maximum at the L-point, which is consistent with the type-II magnetic order at (1/2,1/2,1/2). However, the calculated T N 's from these J(q L ) values, i.e., the temperature where J(q L )χ 0 (0) = g 2 µ 2 B is satisfied in Eq. (1), are different from the actual values. In addition, the positive J(q Γ )'s in Fig. 2(a), which represent (1, 1, 1) the exchange interaction for a uniform field, is inconsistent with the apparently antiferromagnetic exchange observed in the inverse magnetic susceptibility shown in Fig. 3.
Using the single-cite static susceptibility χ 0 due to the CEF split levels, the static susceptibility 1/χ(q) in the mean-field model is written as At the q vector where J(q) takes the maximum χ(q) diverges at the highest temperature, which determines T N in the mean-field approximation. From the experimental T N and the relation of J(q L ) = −6J 2 for the fcc lattice, we can estimate the mean-field J (MF) 2 value. The mean-field J (MF) 1 is obtained from the shift of 1/χ from 1/χ 0 . Since 1/χ(q) = 1/χ 0 − λ(q), where λ(q) = J(q)/g 2 µ 2 B , the meanfield J(q Γ ) = 12J 1 + 6J 2 is associated with the experimental λ(q=0). The mean-field J(q) curves for the three compounds calculated from these parameters are shown in Fig. 2(b). J(q L ) and J(q Γ ) values in this figure reproduces T N and λ just above T N , respectively.
However, again, Eq. (2), with a single parameter λ assuming a uniform exchange constant, cannot well explain the temperature dependences of 1/χ. As shown in Fig. 3 by the lines of uniform exchange, a simple shift of 1/χ 0 to fit the data at low temperatures fails to explain the data at high temperatures. The disagreement between J (CEF) and J (MF) also remains unresolved. To consider these problems, we take into account orbital dependent exchange interactions [11].
The total magnetic moment µ consists of the moments from the Γ 7 , Γ 8 , and from the Van-Vleck contribution through the off-diagonal matrix element, which are represented by µ 7 , µ 8 , and µ [78] ,   respectively. The molecular field for µ 7 arises from µ 7 itself, µ 8 , and µ [78] , each of which has its own exchange constant, λ 77 , λ 78 , and λ 7[78] , respectively. The numbers without and with the brackets represent the Curie and the Van-Vleck terms, respectively. The molecular fields for µ 8  for the local CEF levels, the total magnetic susceptibility χ is obtained by solving the following equations:   The parameters to fit the experimental 1/χ in the whole temperature range from T N to 300 K are listed in Table I. To avoid complexity, we assumed λ 77 = λ [78]7 , λ 88 = λ [78]8 , and λ 78 = (λ 77 + λ 88 )/2. As shown by the solid lines in Fig. 3, the data are well reproduced by considering the orbital dependent exchange constants. It is noted that the exchange constant for the CEF excitation correspond to λ [78][78] because the CEF excitation is associated with the off-diagonal elements between Γ 7 and Γ 8 . The propagation of the CEF excitation (magnetic exciton) is caused by the exchange interaction between the Van-Vleck magnetic moments. Therefore, J(q) in Fig. 2 Fig. 2(a). On the other hand, J(q) in Fig. 2(b) should be written as J 77 (q) because the exchange interaction at low temperatures just above T N mostly arises between the Γ 7 ground states. By treating the exchange interactions as dependent on the CEF states, the difficulties in understanding the exchange constants and the magnetic susceptibilities were removed.

Discussion
The orbital dependent exchange is considered to play an important role in f -electron systems with crystal-field split levels. For example, in the filled-skutterudite compound SmRu 4 P 12 , only the Γ 7 state has a mixing with the conduction band of the p-electrons, which leads to a characteristic exchange interaction and the appearance of a magnetic-field induced charge-ordered phase [12]. In the present analysis on CeX c , it is remarked that the exchange interaction associated with the Γ 8 excited state is larger than that with the Γ 7 ground state. Although this may be associated with the fact that only the Γ 8 state has a mixing with the p-orbital of X c , the details have not been clarified yet. Another point to be noted is that J 2 is larger than J 1 , which can be concluded from Table. I regardless of the evaluation method. This also shows that the exchange interaction through the p-orbital of X c is more important than the nearest neighbor interaction through the conduction electron states of the Ce-5d orbitals. The different exchange parameters obtained from the spin wave dispersion in CeSe is considered to be due to the change in the exchange interaction by forming the magnetic ordered state.
The main contribution from the J 2 term is consistent with the formation of the type-II magnetic order with q=(1/2,1/2,1/2), where all the moments at the second nearest neighbor sites are antiferromagnetically coupled. However, the moments on the nearest neighbor sites are frustrated in the fcc lattice. This could be one of the reasons for the reduced ordered moment in CeTe, where the J 1 and J 2 values are closer to the J 2 = −J 1 line, the boundary between the type-II antiferromagnetic order and the ferromagnetic order in the mean-field model. This is a subject to be studied in future to clarify the effect of frustration.