Structural and Magnetic Phase Transitions in CeCu${}_{6-x}T_{x}$ ($T$ = Ag, Pd)

Energy-Dispersive x-ray Spectroscopy (EDS) analysis performed on several samples indicate that the samples are homogeneous and the elemental composition is in good agreement with their nominal values. Room-temperature laboratory x-ray diffraction was used for phase identification and as a check of sample purity. Synchrotron x-ray diffraction measurements on the sample compositions noted above were utilized as an additional check of the phase purity of the samples. The only evidence of an impurity phase in the samples studied here is the presence in the synchrotron x-ray diffraction patterns of a single unidentified peak smaller than 0.25% of the most intense structural peak of CeCu₆. This peak was not observed in the neutron diffraction or the laboratory x-ray measurements.

DC magnetic susceptibility measurements of several samples are shown in Fig. 2. A linear dependence of the inverse magnetic susceptibility is observed over a large region of temperature indicating Curie-Weiss behavior. Using the Curie-Weiss relation, $\chi=\chi_{0}+\frac{N_{A}\mu_{eff}^{2}}{3k_{B}(T-\theta_{CW})}$, the effective values of the magnetic moment are estimated from the fits of susceptibility data between 200 K $-$ 300 K. The magnetic moments for different compositions of CeCu_6-xT_x (T = Ag, Pd) are close to the expected value ($\mu_{eff}$ = 2.54 $\mu_{B}$) of the Ce³⁺ moment and are 2.42(1) $\mu_{B}$, 2.43(1) $\mu_{B}$, 2.67(1) $\mu_{B}$ and 2.50(1) $\mu_{B}$ for CeCu₆, CeCu_5.65Pd_0.35, CeCu_5.1Ag_0.9 and CeCu_5.85Pd_0.15 respectively.

The crystal structure of CeCu₆ is known to be orthorhombic with space group $Pnma$ at room-temperatureRuck et al. (1993); Larson and Cromer (1961). The orthorhombic unit cell consists of one general $8d$ site and five $4c$ sites. The Cerium atoms occupy one of the $4c$ positions whereas the Cu atoms are distributed among the general $8d$ and four $4c$ sitesRuck et al. (1993); Larson and Cromer (1961). This crystal structure is used as a model for the analysis of the diffraction patterns of CeCu_6-xAg_x and CeCu_6-xPd_x. All the compositions of CeCu_6-xAg_x and CeCu_6-xPd_x that we have studied are isomorphous to the parent compound CeCu₆ at room temperature. The lattice parameter $b$ slightly decreases with Ag/Pd substitution while the parameters $a$ and $c$ increase along with an overall expansion of the unit cell volume in both systems. The synchrotron x-ray diffraction patterns and the fit obtained from the Rietveld refinements of CeCu₆, CeCu_5.7Ag_0.3 and CeCu_5.7Pd_0.3 are shown in Fig. 1, and the results of the fit are summarized in Tables 1 and 2.

The coherent neutron scattering lengths of the dopants, Ag (= 5.92 fm) and Pd (= 5.91 fm), are close to that of copper (= 7.72 fm). This low contrast coupled with the small amount of dopants present renders the determination of the site occupancy with neutron scattering inconclusive. Therefore, high-resolution synchrotron x-ray diffraction was used for this purpose. Rietveld refinement of the synchrotron x-ray diffraction measurement indicates that the Ag-atoms in CeCu_6-xAg_x are not distributed between different copper sites, but prefer the Cu2 site of the $Pnma$ structure. This is similar to CeCu_6-xAu_x system, where Au-atoms occupy the Cu2 site until the site is fully occupiedRuck et al. (1993); Mock et al. (1994). A different situation occurs in CeCu_6-xPd_x, where the Pd-atoms occupy multiple Cu sites. The analysis of the synchrotron x-ray diffraction pattern shows that the majority of the Pd atoms occupy the Cu2 site and the remaining Pd atoms are distributed on Cu1 and Cu4 sites. The precise value of the Pd-occupancies on all other sites except Cu2 are difficult to determine as the occupancies on these sites are very small and are near the limit of what is possible for this analysis. The Pd-occupancies that give the best fit of the diffraction pattern are shown in Fig. 1(c) and are tabulated in the Table 1(b).

Fig. 3 illustrates the different methods used for the characterization of the structural phase transitions: x-ray diffraction, neutron diffraction, and RUS. To check the consistency in these measurements and the quality of the samples, CeCu_5.95Ag_0.05 was grown in both polycrystalline and single crystal forms and measured by all three techniques. A small part of the Czochralski-grown single crystal was measured with neutron diffraction, and different batches of the polycrystalline samples were measured by x-ray diffraction and RUS. The transition temperature (${T_{s}}$) of CeCu_5.95Ag_0.05 was estimated to be at ${T_{s}}$ = 125(1) K using neutron diffraction. The RUS and x-ray diffraction measurements of the polycrystalline CeCu_5.95Ag_0.05 yield ${T_{s}}$ = 122(1) K and ${T_{s}}$ = 125(5) K respectively. This attests to the consistency of the measurement techniques.

The monoclinic order parameter ($(a_{m}c_{m}\cos(\beta))^{2}$) changes smoothly with temperature near ${T_{s}}$ (Fig. 4). The structural peaks that are indexed as (h k l) in space group $Pnma$ split into two structural peaks which are indexed as ( – k l h) and (k l h) in the monoclinic space group $P2_{1}/c$. The splitting of the structural peaks at ${T_{s}}$ can be explicitly observed in the temperature dependence of the diffraction pattern as shown in Fig. 3(a), where the peaks in $Pnma$ phase split into $P2_{1}/c$-peaks as (1 2 2) $\rightarrow$ (– 2 2 1) + (2 2 1), (2 2 0) $\rightarrow$ ( – 2 0 2) + (2 0 2) and (2 2 1) $\rightarrow$ ( – 2 1 2) + (2 1 2). The splitting of these peaks is also confirmed by single crystal neutron diffraction measurements, a part of which is shown in Fig. 3(b).

RUS measurements are one of the most sensitive ways of characterizing structural phase transitions. The resonances occur as the natural frequency of the sample, which is closely related to its elastic properties, matches the incident ultrasonic wave. At ${T_{s}}$, the change in the elastic properties of the sample indicates the occurrence of the structural phase transition. Here, the temperature dependence of the square of the resonant frequency is used to show the structural phase transition. As shown in Fig. 3(c), the square of the resonant frequency versus temperature has a constant slope above ${T_{s}}$. The slope of the curve changes continuously at ${T_{s}}$, and the shift in the resonances below ${T_{s}}$ are much weaker as compared to those at above ${T_{s}}$.

The work presented here indicates the structural phase transition from orthorhombic to monoclinic phase in CeCu₆ occurs at ${T_{s}}$ $\approx$ 240 K, which is somewhat larger compared to the previous studiesVrtis et al. (1986); Suzuki et al. (1985); Yamada et al. (1987). The values of ${T_{s}}$ in CeCu_6-xAg_x drop linearly with Ag concentration until the structural phase transition disappears above the critical composition, $x_{S}$ $\geq$ 0.1(Fig. 5(b)). For 0.1 $\leq$ $x$ $\leq$ 0.85, no structural phase transition was observed above 4 K. The suppression of the structural phase transition due to doping is analogous to CeCu_6-xAu_x, where ${T_{s}}$ drops in similar fashion and the termination of the structural phase transition occurs at a similar Au-composition, $x_{S}$ $\approx$ 0.14Grube et al. (1999); Robinson et al. (2006). However, in CeCu_6-xPd_x, no change in ${T_{s}}$ is observed with Pd-substitution within the range of our investigation, (0 $\leq$ $x$ $\leq$ 0.4). The changes in the transition temperatures with doping in CeCu_6-xAg_x and CeCu_6-xPd_x are summarized in the phase diagrams presented in Fig. 5.

To understand the structural phase transitions, first principles calculations using the planewave code WIEN2K Blaha et al. (2001) have been performed. The generalized gradient approximation (GGA) of Perdew, Burke and Ernzerhof Perdew et al. (1996) was used, with sphere radii for the undoped compound of 2.21 Bohr for Cu and 2.50 for Ce. For the undoped compound, we used the lattice parameters and angles of the orthorhombic and monoclinic phases reported by Asano et al. Asano et al. (1986), and relaxed the internal coordinates until forces were less than 2 mRyd/Bohr.

For the calculations of Ag and Pd doping, several assumptions were made. As detailed in Table 2, in the high temperature orthorhombic phase, for a doping level $x=$ 0.1, the structures for CeCu_5.9Ag_0.1 and CeCu_5.9Pd_0.1 are very similar, differing in volume by less than 0.2%. Since even at this doping level the observed low-temperature structures are very different (orthorhombic for Ag doping, monoclinic for Pd doping), we isolated the effects of charge doping from the small structural differences by using the same lattice parameters and internal coordinates for Ag doping and Pd doping. In each case one of the 24 Cu atoms in the unit cell was replaced by Cu or Pd. Since there is a substantial site preference for Ag doping in this case we used the Cu2 site for the substitution. For Pd doping, two separate sets of calculations were done, with the Pd atom at the Cu1 and Cu2 sites.

For CeCu₆, the monoclinic structure has an energy 33 meV per unit cell lower than the orthorhombic structure, consistent with experimental observation. Despite this energy difference, the changes in electronic structure are scant. Figure 6 plots the calculated densities-of-states (DOS). In both cases there is a DOS peak slightly above the Fermi level, attributable to the Ce states, along with a large contribution between 2 and 5 eV beneath E_F. The Fermi level DOS, at 31.53/eV - u.c. for the orthorhombic cell and 30.82/eV - u.c. for the monoclinic, changes by only 2%.

The relative energies of the doped compounds are presented in table 3. The non-spin polarized (NSP) orthorhombic state was chosen as the zero of energy. To make the spin-polarized (SP) calculations tractable, a ferromagnetic Ce configuration was used rather than the actual antiferromagnetism. The calculations correctly predict the suppression of the monoclinic state with Ag doping - the monoclinic state lies 95 meV higher in energy than the orthorhombic state, and including spin polarization does not appreciably change this result. The SP orthorhombic state, with Ce moment 0.34 $\mu_{B}$, is the groundstate, 4 meV beneath the NSP orthorhombic state and well below both monoclinic states.

In contrast to the experiment, the calculations predict a suppression of the monoclinic state in CeCu_6-xPd_x system. For Pd doping on the Cu2 site this state is 74 meV above the orthorhombic state, and including spin polarization does not remedy this. A similar result is obtained for doping on the Cu1 site (spin polarization was not checked), with an 82 meV energy difference.

While the reason for this disagreement is not known, it likely arises from the heavy fermion state, which tends to confound mean-field based density functional theory. For example, for the base CeCu₆, our calculated T-linear specific heat coefficient $\gamma$ is 18.2 mJ/mol-K², much smaller than the observed value of between 840 and 1600 mJ/mol-K² Stewart et al. (1984). However, since the calculations correctly predict the ground state of CeCu₆, and that for Ag doping, one still has hope for the ability of first principles calculations to describe this system.

One plausible way forward would be to explicitly include correlations through an LDA+U approach or an LDA+DMFT approach, as considered by Shim $et$ $al$ Shim et al. (2007) for CeIrIn₅. Such approaches would likely tend to yield effective masses in better agreement with experiment, as well as more accurate related quantities such as the relative energies of the orthorhombic and monoclinic states. LDA+DMFT approaches have been applied, with a fair degree of success, to structural properties of Ce itself Held et al. (2001); Zölfl et al. (2001); McMahan et al. (2003); Haule et al. (2005) and we anticipate that similar success can be achieved with LDA+DMFT in studying Ag and Pd doping of CeCu₆.

Using the information obtained from the study of the polycrystalline samples together with the CeCu_6-xAu_x literature, magnetic reflections in the (h 0 l) scattering plane were measured in CeCu_6-xAg_x and CeCu_6-xPd_x (for example see Fig. 7). For simplicity and ease of comparison with the work presented here on CeCu_6-xAg_x as well as the extensive work on CeCu_6-xAu_x, we neglect the small monoclinic distortion (less than 2^∘) in CeCu_6-xPd_x and use orthorhombic notation to discuss the magnetic properties.

The magnetic Bragg reflections occurs at the points in the reciprocal space that satisfies the condition $\bf{Q}$ = $\tau$ $\pm$ $\bf{k_{i}}$ (i = 1, 2), where $\tau$ represents a nuclear reciprocal lattice vector and $\bf{k_{i}}$ are the incommensurate propagation vector, $\bf{k_{1}}$ = (0.62 0 0.3) , or the symmetry equivalent $\bf{k_{2}}$ = (0.62 0 $\mathrm{-}$0.3). The magnetic propagation vector for CeCu_6-xAg_x is only weakly dependent on composition. The change in the magnitude of the propagation vector with doping appears to be more pronounced in CeCu_6-xPd_x as presented in Table 4.

The intensities of a structural or a magnetic peak collected with a triple-axis instrument is the convolution of the Bragg intensity with the instrumental resolution function. The software package ResLibZheludev (2007) was utilized to estimate the instrumental resolution and thereby enable the extraction of resolution corrected intensities using the Cooper-Nathans approximationCooper and Nathans (1967). For CeCu_5.25Ag_0.75 the calculated instrumental resolution is indicated by the horizontal lines in Fig. 7(b) and 7(c). The instrumental resolution is not isotropic in the scattering plane and results in an elliptically shaped Bragg peak (e.g. inset of Fig. 7(a). The magnetic Bragg peaks observed for all compositions are found to be resolution limited, consistent with the presence of the long range magnetic order.

After the correction for instrumental resolution, magnetic peaks were normalized to the intensity of the nearby structural peak (0 0 2). The resulting order parameter data is used to characterize the magnetic phase transition. The Neel temperatures ($T_{N}$) are estimated by fitting the power law equation, $y=y_{0}+A(T-T_{N})^{2\beta}$. At temperatures close to $T_{N}$, some rounding of the order parameter is observed. The two most likely reasons for this are the presence of critical scattering or a small compositional variation leading to a distribution of T_Ns. If we attribute the rounding entirely to compositional variation, this indicates a spread of $\Delta T_{N}\leq$ 20%, which, with reference to the phase diagram (Fig. 5), would imply a compositional variation of $\Delta x\leq$ 0.02. On the other hand, a departure from the behavior of a QCP and the associated expectation of mean field behavior ($\beta$ = 0.5) may be the result of the recovery of classical critical behavior and the presence of critical scattering at temperatures close to T_N. With the currently available data we are unable to distinguish between these two possibilities. Therefore, the value of $\beta$ was restricted to the mean field value of 0.5. Mean field behavior has previously been observed near the QCP in other heavy fermion systemsStockert et al. (1999a); Lawrence and Shapiro (1980); Hamann et al. (2013); von Löhneysen et al. (1998). This is likely due to the change in the effective dimension of the system as the quantum dynamics influences the static critical properties–one of the most prominent distinctions from a counterpart classical phase transition. The fits of the order parameter, for several compositions of CeCu_6-xAg_x and CeCu_6-xPd_x, agree well with the mean-field approximation, as shown in figure 8.

To check the consistency with prior work utilizing magnetic susceptibility and heat capacity to determine $T_{N}$ Sieck et al. (1996); Fraunberger et al. (1989); Küchler et al. (2004), a heat capacity measurement was performed for CeCu_5.5Ag_0.5. This gives $T_{N}$ = 0.51(1), which is identical to the value extracted from fitting the order parameter ($T_{N}$ = 0.50(2)) as well as previously published heat capacity measurements Fraunberger et al. (1989); Küchler et al. (2004). For all other compositions, the estimated values of $T_{N}$ from the fit of the order parameter are in good agreement with the magnetic susceptibility and heat capacity measurements of previous studiesSieck et al. (1996); Fraunberger et al. (1989); Küchler et al. (2004). The estimated values of ${T_{N}}$ in CeCu_6-xAg_x and CeCu_6-xPd_x are incorporated in the phase diagrams presented in Fig. 5.

For the wave vector $\bf{k_{1}}$ = (0.62 0 0.3) and space group $Pnma$, representational analysis indicates the four equivalent Ce positions in the unit cell split into separate orbits as given by 1: (0.2586, 0.25, 0.5636), 2: (0.2414, 0.75, 0.06360), 3: (0.7414, 0.75, 0.4364), 4: (0.7586, 0.25, 0.9364). Two irreducible representations (IR) are found for all four orbits with their basis vectors as listed in Table 5. To understand the magnetic structure in greater detail, we focus on CeCu_5.5Ag_0.5 (T_N = 0.51(1) K) and CeCu_5.6Pd_0.4 (T_N = 1.20(1) K). Following the representational analysis, the first IR, $\Gamma_{1}$, restricts the arrangement of Ce-moments to be parallel to the $b$-axis. Given that the propagation vector indicates a modulation in the $ac$-plane, only a transverse modulation of the magnetic moment is possible under this representation. Structures of this type can discarded as the observed intensities do not match with the calculated intensities. The second IR, $\Gamma_{2}$, restricts moments to the $ac$-plane but limits the possible magnetic structures to those with a modulation of the magnetic moment amplitude or a cycloidal modulation of the moment direction. However, the cycloidal model does not account well for the intensities of the observed magnetic reflections. The remaining possibility is a sinusoidal modulation of the moment amplitude in the crystallographic $ac$-plane. To simplify the problem, the directions of the Ce-moments were constrained to be same for all four Ce-orbits but the phases were allowed to vary. The best fit under this assumption was obtained when the moments point along the $c$-axis with a magnitude of 0.61(1) $\mu_{B}$. This yields the magnetic structure shown in the Figure 9(a), which is also consistent with the previous studies on CeCu_6-xAu_x that report a similar structure based on neutron diffraction measurementsLöhneysen et al. (2006); Stockert et al. (1997a); Okumura et al. (1998); Stockert et al. (1999a); Heuser et al. (1998b). The comparison between the observed and the calculated intensities is shown in the Figure 9(b).

In contrast to CeCu_5.5Ag_0.5, the low temperature crystal structure of CeCu_5.6Pd_0.4 is monoclinic with space group $P2_{1}/c$. The representational analysis of CeCu_5.6Pd_0.4 using the monoclinic space group along with the corresponding propagation vector ${\bf Q_{monoclinic}}$ = (0 0.23 0.58) indicates that each orbit constitutes a single representation with three real basis vectors along $a_{m}$, $b_{m}$ and $c_{m}$ axes of the monoclinic structure. The fit of the magnetic structure assuming a sinusoidal modulation of the moment indicates that the moments are pointed along $b_{m}-$ axis of the monoclinic structure ($c-$axis of the orthorhombic structure ¹¹1Note that the structural parameters undergo a cyclic change at the orthorhombic-monoclinic phase transition), identical to the magnetic structure of CeCu_5.5Ag_0.5 described above. The fit of the observed intensity is shown in the figure 9c.

The ordered moment obtained from a neutron diffraction measurement is proportional to the square root of the intensity of the magnetic Bragg peak, which is generally scaled to the nuclear reflections to provide an absolute value. Assuming that the magnetic structure doesn’t vary with the doping composition, the moments of other compositions were determined from a comparison of the normalized magnetic intensity with respect to CeCu_5.5Ag_0.5 and CeCu_5.6Pd_0.4. For the compositions near the QCP, a full saturation of the ordered moments was not observed and the real value of the ordered moment is somewhat larger than the estimated values. The variation of the magnetic moment with Ag/Pd-composition is shown in the figure 9(c).