Q-dependence of the spin fluctuations in the intermediate valence compound CePd₃

The intermediate valence (IV) metals, which are moderately heavy fermion compounds with effective masses m^* in the range 10–50m_e, continue to contribute to our understanding of correlated electron phenomena [1]. The basic physics is that of the Anderson lattice [2], where localized 4f electrons on a periodic array hybridize with a background band of spd electrons, with onsite Coulomb interactions between 4f electrons giving rise to correlated motion. Despite the fact that the 4f electrons sit on a lattice, the Kondo–Anderson impurity model (K/AIM) does a good job in describing many of the properties of these materials [3, 4]. At low temperatures, however, coherence develops in the 4f lattice giving rise to renormalized band behavior. In YbAl₃, a number of anomalies occur below 50 K that have been attributed to the onset of coherence [5].

Because the Kondo physics is dominated by the spin fluctuations, inelastic neutron scattering (INS) is a key experiment for IV metals. Many such measurements have been done on polycrystals, where the spectra are consistent with the K/AIM [6]. Only a small number of single crystals, such as YbInCu₄ [7], YbAl₃ [8] and CeInSn₂ [9] have been studied by INS. In this paper we present data for a single crystal of the IV compound CePd₃. We show that the neutron spectra at 300 K are independent of momentum transfer Q. This Q-independence is an important necessary condition for the applicability of the K/AIM, which provides excellent fits to the susceptibility χ(T), 4f occupation number n_f, and Q-averaged magnetic neutron spectrum χ''(ΔE) of CePd₃. At low temperatures, however, the INS exhibits a Q-dependence which requires a theory of the coherent state.

Data for our single crystal of CePd₃ are shown in figure 1. The results are very similar to those seen in earlier work on this compound and are very typical of IV metals. The ground state valence z = 4−n_f has the value 3.25 and the magnetic susceptibility χ(T) shows a broad maximum near 125 K [10]. We obtain a value Γ ≈ 30 mJ mol⁻¹ K⁻² (data not shown here) for the linear coefficient of specific heat in close agreement with the value reported by Besnus et al [11]. The Q-averaged magnetic neutron scattering spectrum peaks at E_max = 55 meV [12], implying a Kondo temperature T_K = E_max/k_B of order 600 K. The temperature scale T_coh ≈ 50 K has been identified [13] as that below which coherent renormalized band behavior occurs. The anomalous low temperature upturn in χ(T) has been shown [10] to be intrinsic and to correlate with the onset of a 5d spin susceptibility below 50 K as seen in magnetic form factor measurements [14]; this 5d susceptibility is presumably due to coherent f–d hybridization.

Zoom In Zoom Out Reset image size

Earlier INS experiments on CePd₃ using a time-of-flight (TOF) spectrometer concluded that most of the spectral weight of the low temperature spin fluctuations resides in a broad peak centered near E_max = 53 meV [12]. Measurements on a triple axis spectrometer, however, concluded that the spin fluctuation spectrum could be fit as the sum of two broad peaks centered at ΔE = 0 and 15 meV [15]. We have speculated [16] that the discrepancy might arise from a significant difference between the zone boundary and zone center scattering.

To clarify the Q-dependence of the scattering, we grew an 18 g single crystal of CePd₃ and an 11 g crystal of LaPd₃ by the Czochralski method. The susceptibility, specific heat, and 4f occupation number measurements, as well as the K/AIM calculations using the non-crossing approximation (NCA), were carried out in identical fashion to those reported in [3]. The samples were aligned with a [1,0,0] direction held vertical. INS experiments were performed on the TOF spectrometers MAPS and MERLIN at the spallation source ISIS at the Rutherford Appleton Laboratory. The incident energy was set at 120 meV for both spectrometers. In the MAPS measurements, the angle between the incident neutron beam and the x-axis of the crystal was initially fixed at ϕ = 0 so that only three components of $\vec{Q}$–ΔE space are independent. For example, the Miller index H = a₀Q_x/2π varies with energy transfer ΔE when the Miller indices K and L are fixed. We also measured on MAPS for k_i along [1,1,0] (ϕ = 45°); in this situation the decomposition of the horizontal momentum component Q_H onto both of the Miller indices H and K depends on energy transfer.

We also measured the scattering of LaPd₃ on MAPS to help determine the non-magnetic background scattering, and we measured both samples at T = 7 and 300 K. In the MERLIN measurements (performed at 5 K) the angle ϕ was rotated in discrete steps ('Horace mode') of 2°. Under these circumstances, the scattering intensity over a full 4D volume of $\vec{Q}$–ΔE space is determined by interpolation and the intensity can be plotted for any fixed combinations of momentum and energy transfer; for example, constant-Q spectra can be determined. Measurements on both spectrometers were put on an absolute scale by calibrating the scattering against that of a vanadium standard.

The low temperature spectra of CePd₃ and LaPd₃ at (K, L) = (1/2, 0) as measured on MAPS are compared in figure 2(a). The dashed line shows H(ΔE). The latter compound exhibits the non-magnetic scattering expected for the former: single phonon scattering below a cutoff of 22 meV [17] and multiple scattering extending from the cutoff to about 40 meV. The magnetic scattering S_mag (open circles) in CePd₃ is determined by subtracting the data for LaPd₃, scaled by a factor of 0.7 to represent the difference in the neutron scattering lengths⁸. The diamonds represent S_mag/f²(Q), where the 4f form factor f²(Q) has been divided out to give the scattering in the first Brillouin zone. The momentum-averaged scattering of figure 1(c) was determined by averaging over a number of such spectra for different fixed values of K and L. The low temperature scattering peaks near E_max = 55 meV; in what follows we will refer to this as the 'Kondo scale'.

Zoom In Zoom Out Reset image size

Figure 2(b) exhibits the magnetic scattering at 300 K for a number of fixed values of K and L. The solid line in figure 2(b) represents quasielastic (QE) scattering with a half-width of 23 meV, similar to that seen earlier in polycrystalline samples [12]. The room temperature scattering is clearly independent of momentum transfer. This can also be seen in figure 3(a), which plots the room temperature scattering intensity for energy transfer in the (K, L) plane for energy transfer of order E_max.

Zoom In Zoom Out Reset image size

On the other hand, the low temperature scattering in the energy range of the Kondo scattering E_max shows significant Q-dependence. This can be seen in figures 3(b) and (c), which plot the scattering at 55 meV versus momentum transfer for k_i$\parallel$ [1,0,0] and [1,1,0]; it can also be seen in the MAPS data of figure 2(c), which plots the scattering versus L at K = 1/2 and ΔE = 60 ± 10 meV. While the room temperature data is essentially independent of momentum transfer, the low temperature data oscillates with Q. In figure 3(b), the scattering is most intense near points equivalent to (K, L) = (1/2, 0). As mentioned, H varies with ΔE in figure 2(c) and varies with H² + K² in figure 3(b). Bright spots occur at (1.4, 0.5, 0), the weak spots occur at (0.5, 0.5, 0.5) points, and intermediate intensity spots occur at (1.6, 1, 0). In figure 3(c), the bright spots are observed at (1.4, −0.6, 0). The proximity of these numbers to integers and half integers suggests that the Kondo-scale scattering is strongest at (1/2, 1/2, 0), intermediate at (1/2, 0, 0) and weakest at (1/2, 1/2, 1/2).

To confirm this Q-dependence, we have measured the spectra on MERLIN, which allows us to circumvent the non-constancy of H in figures 2 and 3. We plot the low temperature scattering intensity as measured on MERLIN in the (3/2, K, L) plane as seen near E_max (figure 4(a)) and the scattering near E_max versus L for (H, K) = (3/2, 0) and (3/2, 1/2) (figure 4(b)). Data for a cut along L with H = 1 and K = 1 (not shown here) indicate that the (0, 0, 0) scattering has a similar intensity to that of the (1/2, 0, 0) scattering. In confirmation of the MAPS data, maxima are seen in the MERLIN plots at all Q that are equivalent to (1/2, 1/2, 0) while minima are seen at (1/2, 1/2, 1/2) points; the intensity at 60 meV for (1/2, 0, 0) and (0, 0, 0) scattering is intermediate.

Zoom In Zoom Out Reset image size

Hence we have established that there is significant variation of the Kondo-scale scattering with Q in the low temperature coherent state. On the other hand, the data for the 4f occupation number n_f (T) and the Q-averaged magnetic scattering χ''(ΔE) can be fit very well by the results of a calculation using the NCA to the Anderson impurity model (figure 1). Apart from the low temperature 'tail' (which is a striking coherence effect arising from a correlation between 4f and 5d electrons), the susceptibility is also fit very well by this calculation. The calculation gives 30 mJ mol⁻¹ K⁻² for the specific heat coefficient, which is essentially equal to the measured value 29 mJ mol⁻¹ K⁻². The lack of Q-dependence as well as the QE form of the spectrum that we have observed at room temperature (figure 2(b)) are as expected for the K/AIM at elevated temperature. Given that the calculated results depend mainly on two key parameters, the hybridization V and the f-level energy E_f, and given that the 4f electrons in CePd₃ are not impurities but sit on a lattice, the quality of the fits is perhaps surprising, but similar quality fits have been observed for other IV compounds [3, 5].

We have argued elsewhere [1] that the applicability of the K/AIM to these periodic compounds is a consequence of the fact that the 4f spin fluctuations are highly localized in space. This is supported by the fact that the Q-dependence on the Kondo scale that we do observe at low temperature in CePd₃ is not enormous. To show this, we return to the MAPS data, for which we have subtracted the non-magnetic scattering (which we have not done for the MERLIN data), allowing us to compare the magnitude of the scattering at different Q. In figure 5, we plot the magnetic spectra for k_i$~\parallel$ [1, 0, 0] at fixed values of (K, L) (solid points). The variation of H with energy transfer is shown as the dashed–dotted lines. In figure 6, we plot the spectra for k_i$~\parallel$ [1, 1, 0]. As mentioned above the decomposition of the momentum transfer Q_H onto H and K depends on energy transfer; a typical example is shown in figure 6(c). In figure 5(d), we plot the magnetic scattering at high symmetry points, determined from the data of figures 5(a)–(c) at the energies where H(ΔE) takes on an integral or half-integral value and also from the energies in figure 6 where H and K take on integral or half-integral values. This plot shows that the MAPS data agrees well with the MERLIN data in that the Kondo-scale scattering (i.e. for ΔE~55 meV) is strongest for (1/2, 1/2, 0), intermediate for (1/2, 0, 0) and (0, 0, 0) and weakest for (1/2, 1/2, 1/2). On the other hand, for energy transfer below 40 meV, the situation reverses, with the (1/2, 1/2, 1/2) scattering being strongest.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We note that the data differ by 20–30% from the K/AIM fits to the polycrystalline-averaged data (solid lines in figures 5 and 6). Earlier studies along a single zone direction in YbInCu₄ [7] and CeInSn₂ [9] also concluded that there is only a 20% increase in intensity between zone center and zone boundary scattering. The shape of the spectra was the same for both Q, being essentially of the inelastic Lorentzian form expected based on the K/AIM. In a more recent study of CeInSn₂ using polarized neutrons, Murani et al have shown [19] that the Q-variation of the 4f form factor varies only by 10% from that of a 4f local moment. Such a moderate Q-dependence is very different from the predictions of the Anderson lattice where the Kondo-scale scattering is strongly suppressed at Q = 0 (figure 7(a)). On the other hand, the modest Q-dependence that we observe is not very different from that expected for a localized (Q-independent) spin fluctuation, which helps explain why bulk properties such as χ(T), n_f(T), Γ and χ''(ΔE) which depend primarily on the (nearly localized) spin fluctuation continue to follow K/AIM behavior even at low temperature.

Zoom In Zoom Out Reset image size

The K/AIM cannot, however, capture properties of the coherent IV ground state that depend critically on the 4f periodicity and coherence. In CePd₃ the resistivity behaves as T² below 50 K [13] and the Hall mobility exhibits an anomaly at the same temperature [20], both of which facts indicate that the coherent Fermi liquid state is stable below a coherence temperature of T_coh = 50 K. The low temperature 'tail' in the susceptibility develops on the same scale. As mentioned above, this has been shown to be intrinsic behavior, representing correlation between the 5d and the 4f moments—a truly striking coherence effect. The optical conductivity develops a well-established mid-infrared peak (0.26 eV) at 100 K [21]; this is believed to reflect the onset of the hybridization gap. The full enhancement of the effective mass (m^* =40 m_e), which is determined from the narrow Drude peak in the conductivity and which is indicative of the establishment of the fully renormalized Fermi liquid, appears to occur on the scale T_coh [22]. To understand these properties, the low temperature scale on which they occur, and the Q-dependence in the magnetic scattering requires a lattice-based theory of the coherent state.

A common argument is that oscillations of the neutron spectra with momentum transfer arise from antiferromagnetic (AF) correlations. This statement accords with the fact that the scattering is most intense for zone boundary Q. In the early work on CeInSn₂ [9], this was the explanation given for the oscillation in the scattering intensity. Recently, however, Murani et al [19] have argued that this is not the case, because the oscillation has a similar magnitude in CeInSn₂ where the Kondo temperature is of order T_K~100 K and in CePd₃ where T_K~600 K. The scale of AF correlations in cerium compounds is typically less than 10 K, which is much smaller than the 600 K Kondo scale in CePd₃, making it unlikely that AF correlations play a significant role. Instead, Murani et al [19] propose that Q-dependent hybridization is responsible for the oscillation.

As mentioned in the introduction, the most basic lattice-based theory of IV compounds is the Anderson lattice model (ALM). As shown in figure 7(a) (inset), this predicts an indirect gap Δ_ind, which has the magnitude of the Kondo energy, between zone center and zone boundary due to the hybridization of a narrow f level with a broad conduction band. The INS arises from particle-hole excitations, and is most intense for transitions between regions of high density of states in both the occupied and the unoccupied bands. For the Anderson lattice, this occurs for ΔE~Δ_ind and for zone boundary momentum transfer Q_ZB. Such behavior is seen in Kondo insulators such as YbB₁₂ [23]. The calculated scattering decreases rapidly in intensity and shifts to higher energy transfer as Q decreases; indeed, for Q = 0 it becomes negligible on the scale of the indirect gap and is large on the scale of the direct gap. This is shown in figure 7(a), where we plot the calculations of Aligia and Alascio [24] for the Anderson lattice.

To compare the ALM calculations to our data, we have scaled the energy axis of the calculation so that the zone boundary scattering is on the same energy scale as T_max for CePd₃. The MAPS data for the cut (H(ΔE), 1, 0) (figure 5(c)) traverses the zone center and zone boundary along the [1,0,0] direction. To make a rough comparison to this data, we have drawn closed circles on the open symbol curves of figure 7(a) at the energies where H(ΔE) takes on the reduced value of Q appropriate for each spectrum. The resulting curve (thin line through the symbols) is very different from the data of figure 5(c). In particular, the oscillation with H(ΔE) is much smaller experimentally than predicted by the theory, essentially because the scattering on the Kondo scale does not vanish at zone center points. (The oscillation seen in figure 5(b) for the (H, 1/2, 1/2) cut represents a variation which occurs entirely on the zone face between (1/2, 1/2, 0) and (1/2, 1/2, 0); it occurs because the Kondo-scale scattering at (1/2, 1/2, 1/2) is smaller than at the other high symmetry points.)

Hence there are serious difficulties with the standard Anderson lattice prediction for the dynamic susceptibility. Perhaps the most obvious problem arises from the fact that the calculation of figure 7(a) is carried out for a single parabolic conduction band, crossed by a single mono-energetic f level. This gives rise to the hybridized bands of figure 7(a) inset, with a single hybridization gap. In a real IV material, the energy gaps are governed by the band structure, and correlated band theory is needed to capture the behavior. The hybridized f bands can be much flatter over broad intervals of k-space than in the simple Anderson lattice, and indirect and direct gaps can occur between different high symmetry points. This can be seen in the recent band calculation of Sakai [25]⁹ for CePd₃ which uses dynamic mean field theory (DMFT) to include correlations on top of the standard LDA treatment (figure 8). Unlike the Anderson lattice, there is a direct gap of about 26 meV between occupied and unoccupied states at the Γ point, allowing Q = 0 transitions. There are also indirect gaps of order 30 and 36 meV between Γ and M and between R and X corresponding to a (1/2, 1/2, 0) transition. The R–Γ and Γ–R gaps corresponding to a (1/2, 1/2, 1/2) transition are at 20 meV and 85 meV respectively.

Zoom In Zoom Out Reset image size

Hence, while the band calculation predicts somewhat smaller energies for the excitations than is seen in our data, it predicts the correct order of magnitude for the Kondo energy scale, it shows that the gaps can vary with momentum transfer, it shows that Q = 0 scattering can be significant rather than vanishingly small as predicted by the ALM, and it demonstrates that intense Kondo-scale scattering should be seen at (1/2, 1/2, 0). It also suggests that the (1/2, 1/2, 1/2) scattering will occur at energies that are either substantially larger or smaller than the scattering at (1/2, 1/2, 0). Indeed, it is plausible that the enhanced (1/2, 1/2, 1/2) scattering seen at 30 meV in figure 5(d) represents the lower energy R–Γ scattering.

To improve the comparison to band theory will require two advances. The first is that the scattering should be determined with controlled momentum transfer, as in our MERLIN experiment, but over a broader range of energy transfer than shown here. To determine the magnetic scattering, the non-magnetic scattering should be determined and subtracted from the measured spectra. Second, a calculation of the Q- and E-dependent dynamic susceptibility χ''(Q,E) should be performed, based on the correlated band theory. Such a calculation would include not only the particle-hole excitations between the centroids of the band states, but the full inelasticity of the momentum resolved spectral functions. The point is that the band states ε(k) of the renormalized band theory are only sharp in energy at low temperature and for energies close to the Fermi level. As the temperature increases, or as the distance of the energy from the Fermi level increases, the spectral functions rapidly become quite broad in energy. This broadening results from the strong inelastic scattering that is inherent in correlated electron systems. For our purposes, it has two major consequences. The first is that as the temperature is raised, the crossover to incoherent scattering occurs at a relatively low temperature. An indication of this is that we see Q-independent QE scattering spectra already at 300 K, which is ~T_K/2. The second consequence is that the low temperature spectra at finite ΔE will be considerably broadened in both Q and E compared to the spectra calculated without including the inelasticity. These two features have been seen clearly in calculations for the Anderson lattice [26] and for the Hubbard model [27], and in LDA+DMFT calculations for real materials [28]. We believe that it is the broadening in Q and E that causes the INS spectra that are observed in IV systems to closely resemble those expected for the Anderson impurity model.

As a final point, we note that figure 1 shows that, apart from the low temperature 5d contribution, the Kondo-scale scattering accounts for most of the d.c. susceptibility. Hence our data supports the contention of Murani et al [12] that the primary spectral weight of spin fluctuations is on the scale E_max. Our earlier conjecture [16] that the zone center scattering is shifted to lower energy is not borne out by the current analysis; on the Kondo scale, the zone center data is not very different from the zone boundary data. Our procedure for subtracting the non-magnetic scattering (Figure 2(a)) is not adequate, however, to determine whether magnetic excitations exist at energies below 30 meV where the non-magnetic scattering is important. Hence we cannot rule out the contention of Shapiro et al [15] that significant scattering occurs at lower energy. Indeed, a study using polarized neutrons and a polycrystalline sample of CePd₃ suggested there might be a 15 meV excitation in addition to the one at 55 meV [29]. We believe that the whole issue of the low energy scattering is very much an open question for IV compounds. Most existing studies of single crystals of these compounds [7–9, 19] have explored the scattering on the Kondo scale, where the scattering corresponds to interband scattering between the renormalized bands above and below the Fermi energy. In addition, intraband scattering for particle-hole excitations in occupied bands that cross the Fermi level should be observed. Such excitations give rise to dispersive Fermi liquid scattering with ε~Qv_F where v_F is renormalized by the excitations. While such scattering has been observed in the heavy fermion compound UPt₃ [30], to the best of our knowledge it has not been seen in any IV compound to date. The study of the excitations at energies below the phonon cutoff would be best addressed using polarized neutrons and a single crystal sample.

In conclusion, we have shown that much of the behavior of the IV compound CePd₃ can be understood as that of a Kondo/Anderson impurity. At low temperatures, however, the Kondo scale magnetic scattering exhibits a momentum dependence that can be understood as arising from coherent 4f bands. A full comparison of the Kondo-scale scattering to the predictions of correlated band theory, and a determination of the momentum- and energy-dependence of the scattering below the phonon cutoff (20 meV), remain open questions that will require future measurements to resolve.

Work at the University of California, Irvine (UCI), Temple University, Argonne National Laboratory (ANL), Lawrence Berkeley National Laboratory (LBNL), and Los Alamos National Laboratory (LANL) was supported by the Office of Basic Energy Sciences (BES) of the US Department of Energy (DOE). The work was funded by awards DE-FG02-03ER46036 (UCI), DE-FG02-84ER45872 (Temple), DE-AC02-05CH11231 (LBNL) and DE-AC02-06CH11357 (ANL). Research at Oak Ridge National Laboratory (ORNL) and at the Stanford Synchrotron Radiation Lightsource (SSRL) was sponsored by the Scientific User Facilities Division of BES/DOE.