Cerium intermetallics with TiNiSi-type structure

3.2.1 Ferromagnetic CeTX compounds

We start with a summary of the group of materials which shows conventional long-range magnetic ordering. Amongst the large number of CeTX compounds discussed in this review article only three compounds exhibit ferromagnetic ordering. These are CePtAl, CeAgAl and CeAgGa with Curie temperatures of ~6, ~3 and ~5 K, respectively. In the case of CePtAl a complex magnetic system was investigated by neutron powder diffraction and two coexisting, partly ferromagnetic and incommensurate magnetic structures and three magnetic phase transitions at T₁ = 5.9, T₂ = 4.3 and T₃ = 2.2 K have been found [121]. While the transition at T₁ is of the second order, T₂ and T₃ are both first order phase transitions. For T₂ < T < T₁ the two magnetic propagation vectors were determined to be k₁ = 0 and k₃ = [0, 0.463(8), 0], for T < T₂ the vectors k₁ = 0 and k₂ = [0, 1/2, 0] were found. The complexity of the magnetic structure was ascribed to exchange interactions between two chains of Ce atoms with d(Ce–Ce) = 356 pm along the b axis and d(Ce–Ce) = 377 pm along the a axis. The effective magnetic moment was determined to be μ_eff = 2.58 μ_B per Ce atom and the Weiss constant of θ_p = 6.5 K. Simulating the magnetic data revealed a Kramers doublet ground state [122]. In a subsequent paper single crystals of CePtAl grown by the Czochralsky method were investigated by high-field magnetization experiments up to 300 kOe (1 kOe = 7.96 × 10⁴ A m^–1). Pronounced magnetic anisotropy was detected, which was attributed to crystalline–electric field interactions. The a axis corresponds to the easy axis while the hard axis changes from c (T > 39 K) to b (T < 39 K) [33]. Ueda et al. studied the de Haas-van Alphen effect in a high-quality single crystal of CePtAl by detection of the field modulation in fields up to 170 kOe and temperatures down to 30 mK. Their results suggest that the Fermi surfaces in CePtAl consist of multiply connected Fermi surfaces [34]. Finally the pressure dependence of the magnetic susceptibility was studied to determine the critical pressure p_c at which the magnetic ordering temperature becomes zero. In electrical resistivity measurements under pressure they observed a shift of the Kondo features which collapse > 6 GPa. The first ordering temperature follows this trend. It shifts to higher temperatures and disappears above 6 GPa [123–125].

As already discussed in the crystal chemistry part no crystallographic ordering is present in the silver compounds CeAgAl [54] and CeAgGa [126] for Ag and the main group element, thus influencing the physical properties. Both compounds show Curie–Weiss behavior and negative paramagnetic Curie temperatures of –18 [54] and –43 K [70], respectively. The large negative values are not due to antiferromagnetic interactions in the paramagnetic range, but a result of a crystalline electric field (CEF). The presence of a CEF effect has been proven for CeAgGa by inelastic neutron scattering [126]. However, the negative Weiss constants have also been attributed to Kondo interactions [54, 70] (vide infra). Referring to dc-susceptibility measurements magnetic ordering at T = 2.9 for CeAgAl [54], 5.5 [126], and 5.1 K [70] for CeAgGa have been reported. In accordance with the magnetization isotherms the magnetic ordering has ferromagnetic character. However, systematic ac-susceptibility measurements of CeAgGa showed maxima around 5 K with amplitudes and positions depending on the frequency of the applied magnetic field. The dependence of the maximum follows the Vogel–Fulcher law which is characteristic for a spin glass behavior. By measuring the isothermal remanent magnetization M_IRM as a function of time a canonical spin glass could be identified [70]. Further hints to a spin glass-like characteristic were found in the zero-field cooling (ZFC) and field cooling (FC) measurements as well as in the heat capacity. The ZFC/FC curves bifurcate already above T_f and the anomaly in specific heat capacity is a broad peak instead of a λ-anomaly. Furthermore, the shape of the anomaly is rounded and shifted to higher temperatures by applying magnetic fields [70]. Referring to the literature of spin glasses [127, 128] some important spin glass characteristics are fulfilled, however, a broad peak in the specific heat pattern is expected to be approximately 20% above T_f, which is not the case for CeAgGa.

An almost identical behavior can be observed for CeAgAl [54]. Though no ac-susceptiblity investigations were performed and some of the described features have been partially accounted for a Kondo behavior, a somehow spin glass-like behavior can also be predicted for CeAgAl due to the mixing of Ag and Al atoms on the 8h site. CeAgAl [54] and CeAgGa [70] have both been characterized as Kondo lattices. As reasons for this behavior the large negative value of the Weiss constants, the development of the magnetic entropy and the characteristic shape of the anomaly in the specific heat have been stated. In our opinion it should be considered that in cerium based Kondo lattices that exhibit a magnetic ordering, in general the specific heat is dominated by the magnetic ordering in contrast to the Kondo interactions. This, for instance, is the case for CePdSn (vide infra). Consequently, the development of the specific heat with increasing magnetic contribution is most likely caused by the spin glass character which was already considered for CeAgAl [54]. The large negative values of the Weiss constants can also be caused by CEF which has been nicely proven by inelastic neutron scattering [126]. Referring to the original publication by Kondo, a Kondo lattice is present if the resistivity shows a minimum and a logarithmic increase with decreasing temperature due to an additional scattering mechanism [129]. In cerium based Kondo lattices this minimum is often followed by a rapid decrease due to the magnetic ordering at lower temperatures. Such a drop due to the magnetic ordering definitely occurs for both compounds, but only a very weak minimum can be observed for CeAgAl. As a consequence we would be more cautious to define these compounds as Kondo lattices.

3.2.2 Antiferromagnetic CeTX compounds

CeMgSn was reported to possess an antiferromagnetic ground state with an ordering temperature T_N = 12 K [26]. Investigations by high resolution neutron powder diffraction confirmed the AFM ground state and a Néel temperature of T_N = 13 K. No evidence for the Kondo effect was found in the temperature dependent resistivity measurements. The analysis of the difference spectrum between T = 2 K and 20 K revealed multiple magnetic peaks not corresponding to the nuclear Bragg peak positions, confirming the antiferromagnetic coupling. The magnetic propagation vector at 2 K was determined and refined to τ = [0, 0.1886(4), 0.3384(8)]. The maximum magnetic moment of the modulated structure was refined to 2.24(2) μ_B per Ce atom consistent with a strong localization of the 4f¹ electron [50]. Almost the same different magnetic propagation vector of [0, 0.187, 0.340] for CeMgSn was observed at T = 40 K, an indication of an incommensurate antiferromagnetic structure [130].

The orthorhombic high-temperature β-form of CePdZn was found to order antiferromagnetically at T_N = 3.2(1) K. The effective magnetic moment was determined to μ_eff = 2.42(1) μ_B per Ce atom which is close to the theoretical value for the free Ce³⁺ ion. However, a positive Weiss constant was found suggesting ferromagnetic interactions in the paramagnetic region [52].

For the heavier homologue CePtZn [20] no magnetic ordering was observed in susceptibility or heat capacity measurements down to 2 K. The inverse magnetic susceptibility can be described by the Curie–Weiss law resulting in an effective magnetic moment of μ_eff = 2.47(1) μ_B per Ce atom. This clearly indicates trivalent cerium. The Weiss constant was found to be negative with θ_p = –18.7 K. Subsequent work on this system by Dhar and co-workers [21] revealed antiferromagnetic ordering at T_N = 1.7 K and Kondo behavior visible in the low temperature electrical resistivity with the Kondo temperature (T_K ~ 2 K) comparable to the ordering temperature. Field dependent resistivity measurements showed a suppression of the AFM transition around 50–60 kOe. The same suppression has been found in pressure dependent resistivity measurements. The authors suggest the appearance of a quantum critical point (QCP) with T_N → 0 K, the experimental conditions (H = 120 kOe, p = 2.66 GPa) however were not sufficient to experimentally show the superconducting state. However, the QCP is assumed to be at p_c ~ 1.2 GPa and H_c ~ 60 kOe as seen from the non-Fermi liquid behavior [21]. Doping of CePtZn by small amounts of Au leads to an increase of the Néel temperature (T_N = 2.1 K) while in the case of Ni doping a drop to T_N = 1.1 K could be observed [117].

CeAuZn exhibits trivalent cerium with an effective magnetic moment of μ_eff = 2.51(1) μ_B per Ce atom. No Kondo behavior was observed in the electrical resistivity measurements and no magnetic ordering down to 2.1 K was observed by Hermes et al. [65]. Dhar et al. did also not observe a magnetic transition in susceptibility measurements, however heat capacity measurements revealed antiferromagnetic ordering at T_N = 1.7 K [117]. A fit of the region below 1.2 K yielded the electronic specific heat coefficient γ = 0.022 J mol^–1 K^–2. This coefficient is significantly smaller than γ of CePtZn (0.6 J mol^–1 K^–2), which has been explained by the relatively weak electronic correlations in CeAuZn and the changes in the electronic band structure near the Fermi level.

The magnetic properties of CeAuAl were first reported by Hulliger in 1993 [61]. He observed antiferromagnetic ordering with T_N = 3 K, a Weiss constant of θ_p = –13 K and an effective magnetic moment of μ_eff = 2.46 μ_B per Ce atom. A strong deviation from the Curie-Weiss law along with a magnetic transition near 34 kOe in the magnetization isotherms were noted as remarks. Kondo lattice behavior in this compound was reported later by Menon and Malik [62]. They observed AFM ordering at T_N = 3.8 K and the typical Kondo features in the temperature dependent electrical resistivity with a minimum at T_K ~ 20 K [62]. Heat capacity measurements confirmed the AFM ordering temperature. By subtraction of the heat capacity of LaAuAl from the one of CeAuAl the 4f contribution could be approximated. In the temperature range from 8 to 15 K, well above T_N, a large electronic contribution to the heat capacity of γ = 100 mJ mol^–1 K^–2 can be observed. Since crystal field splitting was absent, as known from inelastic neutron studies, the origin of this feature has to be attributed to the Kondo effect [131]. Inelastic neutron scattering studies [132] proved the existence of two well defined crystal field excitations at 24.2 and 45.8 meV. Since the line width of the inelastic peaks is significantly larger than the instrumental resolution, the presence of strong spin-lattice couplings has been deduced, presumably due to the interactions between the 4f electron and the conduction electrons. The magnetization experiments revealed similar results with an ordering temperature of T_N = 3.8 K, a Weiss constant of θ_p = –35 K and an effective magnetic moment μ_eff = 2.5 μ_B per Ce atom. Magnetization experiments at 1.7 K indicated a metamagnetic transition which becomes less pronounced at higher temperatures and disappears at 4.5 K. However, no clear sign for metamagnetism was observed in the 1.7 K magnetoresistance measurements. Between 4.2 and 14 K single-ion Kondo behavior is observed. At low temperatures the cerium ions can be treated as J = 1/2 impurities due to the first excited CF level at 24.2 meV. From the magnetoresistance data the Kondo temperature was determined to T_K = 3.6 K. Finally the thermoelectric power has been determined, which is small and positive above 150 K but changes its sign with decreasing temperature showing a pronounced minimum at 50 K and finally moving towards 0 μV K^–1 for T → 0 K [132].

CePtGa was reported to order antiferromagnetically with T_N = 3.5 K [131]. The influence of static pressure on the electrical resistivity was investigated later and a temperature dependence typical for a concentrated Kondo compound with a Kondo minimum at T_K = 24 K was found [134]. In the electrical resistivity data a decrease of T_N was observed with increasing pressure, however above 0.8 GPa the ordering phenomena disappeared. Magnetoresistance measurements revealed a negative sign which is consistent with an incoherent Kondo system. The sign stays negative with increasing pressure, but an extrapolation of the data suggests a crossover at 3 GPa [134, 135]. Susceptibility investigations on polycrystalline powdered samples showed essentially trivalent cerium (μ_eff = 2.36 μ_B per Ce atom) with a low saturation magnetization of μ_sat = 0.12 μ_B at 5 K and 30 kOe. No magnetic ordering was found by the authors, but a large negative paramagnetic Curie temperature of θ_p = –68 K was observed. Crystal field studies via inelastic neutron scattering experiments at 11 K and 60 meV incident energy displayed two well-defined crystal field transitions near 18 and 36 meV (208 and 417 K). The ground state was found to be a well-separated Kramers doublet [136].

Heat capacity measurements on a single crystal specimen grown by the Czochralski method showed a λ-type anomaly at 3.4 K, consistent with the magnetic ordering temperature from susceptibility measurements. The results of the pressure dependent resistivity measurements conducted at the single crystal specimen were in good agreement with the ones from polycrystalline samples, also showing a decreasing Néel temperature [137]. The magnetic properties and the directional electrical resistivity of CePtGa have finally also been determined on single crystals. By their measurements, Shirakawa and co-workers were the first to explain the magnetic anisotropy caused by the CEF [113]. They investigated the magnetic susceptibility along the three principle axes as a function of temperature and found the antiferromagnetic ordering at T_N = 3.5 K. However, different critical fields for the spin reorientations were found. Along the c axis H_Cr = 35 kOe was observed, while along the a axis a much higher field was needed. Directional resistivity measurements revealed a resistivity along the b axis which is about three times larger compared to the ones along the a and c axes. Furthermore the CEF was calculated from the fits of the χ^–1 data giving energy separations of 294 and 559 K which are quite similar to the data obtained from inelastic neutron scattering experiments.

Measurements of the thermoelectric power of CePtGa showed only a small Seebeck coefficient due to the metallic nature of the material, but an inverse tilda-shaped trend was observed with an inversion of the sign between 50 and 100 K and a minimum at 26.7 K, consistent with the Kondo temperature found from resistivity measurements [138].

A slightly elevated Néel temperature of T_N = 4.2 K has been found for gallium deficient single crystals of CePtGa_1–x [139] along with a strong magnetic response of the susceptibility when measured perpendicular to the b axis. Only a small response was seen when measured parallel to b. Therefore the Ce moments are expected to be in the ac plane. The Ga deficiency finally was estimated to be ~0.6 when assuming an effective magnetic moment equal to the theoretical moment of trivalent cerium. From heat capacity measurements γ = 76 mJ mol^–1 K^–2 was extracted. Pressure dependent resistivity measurements revealed a positive increase of the Néel temperature with increasing pressure, contradictory to the negative shift observed before in polycrystalline CePtGa and stoichiometric single crystalline CePtGa [139]. The authors already noted a structural problem for the gallium deficient sample. Straightforward indexing of the powder pattern was not possible.

Stoichiometric polycrystalline CePtGa was synthesized by Kotsanidis et al. [140]. They measured magnetic susceptibility data above 5 K yielding an experimental magnetic moment of 2.36 μ_B per Ce atom.

CeRhGe was reported to be an antiferromagnetic Kondo compound with an ordering temperature of T_N = 9.3 K [30]. Susceptibility measurements showed an anomaly while magnetization experiments below 11 K with fields up to 50 kOe clearly indicated antiferromagnetism. The reduced effective paramagnetic moment μ_eff = 2.30 μ_B and the large Weiss constant (θ_p = –56 K) were extracted by applying the Curie–Weiss law to the high temperature data (>130 K). The low effective magnetic moment was attributed to crystal field splitting effects while the large negative Weiss constant is consistent with Kondo behavior. Additional heat capacity and electrical resistivity measurements confirmed the ordering phenomena with consistent temperatures of T = 9.3 and T = 9.2 K, respectively. The λ-shape indicates a second-order magnetic phase transition, while the low temperature specific heat (<3.5 K) shows an enhanced electronic contribution γ = 75 mJ mol^–1 K^–2 clearly suggesting Kondo behavior.

Heat capacity measurements on single crystals confirmed the λ-anomaly, the electronic contribution was determined to be γ = 47 mJ mol^–1 K^–2 [141]. The resistivity measurements, however, did not show a clear Kondo minimum but rather a double-peak structure with the maximum at 150 K attributed to the crystal electric field, and the shoulder between 15 and 30 K attributed to the onset of coherency of the Kondo lattice [30]. The transition temperature and effective magnetic moment were confirmed later along with the field for the spin reorientation at H_Cr = 25 kOe [142]. In neutron diffraction experiments at T = 1.7 K one additional magnetic reflection was observed which was indexed as (100). The only possible explanation of the presence of this magnetic reflection is that the magnetic moments are aligned along the b axis. The additional magnetic reflections caused by this model are rather weak and therefore hard to observe [143]. Pressure dependent resistivity studies revealed an increasing peak at T_N in the resistivity with increasing pressure, but above the critical pressure p_c = 1.9 GPa this feature disappears and the Néel temperature becomes zero [144]. A later work, in which single crystals were used, the resistivity peak was associated with the formation of a spin density wave. The magnetic structure, when determined from single crystals, furthermore was found to be incommensurate with k = [–0.0606, –0.1168, 0.9062]. Directional magnetization experiments suggest that the crystallographic a axis is the easy-axis which also shows two spin reorientations at 80 and 130 kOe. The other directions show a linear field dependency. The effective magnetic moment along a is close to the calculated one for trivalent cerium (μ_eff = 2.49 μ_B), however it strongly deviates for H || b and c. The authors suggest strong CEF, roughly estimated to be 1000 K [145]. With the help of theory and the operator method, Li and co-workers calculated the energy levels caused by the CEF for CeRhGe by fitting the experimental data from Ueda et al. [145]. They found that the sixfold degenerated 4f levels of a Ce³⁺ ion are split into three Kramers’ doublets and the total excited energy was calculated to be 970 K (the splitting of these state is usually given on the K scale) which is in good agreement of the values reported earlier [146].

CePdGe was reported to be an antiferromagnetic material with T_N = 3.4 K, determined by susceptibility and heat capacity measurements [30]. The effective moment was determined to be μ_eff = 2.55 μ_B and the Weiss constant of θ_p = –37 K was extracted by applying the Curie–Weiss law to the data >60 K. Resistivity measurements also show an anomaly at T = 3.4 K, consistent with the magnetic data. Neutron diffraction data collected at 2 and 5 K showed two reflections of low intensity indexed by a vector k = (0.42, 0, 0). This vector corresponds to a linear spin wave where the moments are aligned along the b axis [147]. Similar magnetic results were presented by Kotsanidis et al. [148]. At this point it is worthwhile to note that the magnetic structure studies were based on the wrong subcell structure. As emphasized in the crystal chemical part, tripling of the subcell structure is observed for the ordered nuclear structure of CePdGe and thus, the translation in the magnetic structure cannot be shorter than in the nuclear one.

CePtGe orders antiferromagnetically at T_N = 3.4 K with trivalent cerium (μ_eff = 2.54 μ_B) derived from Curie-Weiss fits above 90 K. The large negative Weiss constant θ_p = –82 K was ascribed to the presence of the Kondo effect, however the electrical resistivity shows no Kondo minimum but a double peak structure with a peak at T = 3.4 K, consistent with the susceptibility data. As for CePdGe the magnetic ordering was observed without doubt in the heat capacity measurements showing a λ-shaped anomaly [30]. Further studies regarding the pressure dependence of CePtGe revealed a lowering of the shoulder with increasing pressure, dp(T)/T is zero at p_C = 17.4 kbar. The magnetic transition temperature changes only little from T_N = 3.36 K at ambient pressure to T_N = 3.77 K at p = 13.1 kbar [149]. Investigations regarding the ratio of T_K to T_N showed that CePdGe and CePtGe fit the trend observed for other Kondo compounds like, e.g. CePd₂Ge₂, CePd₂Si₂ or CePd₂Sn₂ [150]. The Seebeck coefficient shows an inversion of the sign with a positive peak at ~100 K and a minimum at 10 K followed by a rise to 0 μV K^–1 for T → 0 K [149], similar to [133].

CePdSn exhibits an antiferromagnetic phase transition at T_N = 7.5 K, much higher than expected, especially when compared to the isostructural Gd compound (T_N = 14.5 K) suggesting strong hybridization between the Ce 4f and the conduction electrons. From the susceptibility data above 50 K the effective magnetic moment was determined to be close to that of free Ce³⁺ ions while the Weiss constant θ_p = –68 K is significantly larger than the Néel temperature. Magnetization experiments revealed a linear field dependency with no hint for saturation up to 8 kOe. Heat capacity measurements confirmed the magnetic data and a λ-shape anomaly was found with a maximum ~7 K. The entropy of this transition was estimated from C/T vs. T plots to be ~4.3 J mol^–1 K^–1. Finally electrical resistivity measurements were conducted proving the Kondo effect in CePdSn due to a minimum in the resistivity data at T_K = 20 K [151, 84]. The magnetic ordering has been confirmed by Kyogaku et al. by ¹¹⁹Sn NMR experiments [152]. The magnetic entropy of CePdSn was obtained by subtracting the value for LaPdSn from the total entropy of CePdSn. Above the ordering temperature R ln 2 behavior is found confirming the postulated doublet ground state. The electronic specific heat coefficient was determined to γ = 60 mJ mol^–1 K^–2, again proving the presence of a dense Kondo effect. Furthermore neutron diffraction experiments were conducted suggesting that the magnetic structure is a simple spiral one with a propagation vector of k = [0, 0.473, 0] [153]. High pressure experiments revealed that above 6 GPa the sharp ordering feature visible in the electrical resistivity disappears [154]. Here we should keep in mind that the structural phase transition to HP-CePdSn proceeds at 10.5 GPa [38].

CePtSn orders antiferromagnetically with two transitions at 7.8 and 5.5 K, respectively. The effective magnetic moment μ_eff = 2.85 μ_B is higher than expected, the Weiss constant θ_p = –9.2 K is in good agreement with the ordering temperature. The magnetization can be described as a linear function of the magnetic field strength at 4.2 K [155]. Neutron diffraction experiments conducted at 1.7 K showed additional reflections in agreement with the antiferromagnetism [156]. Analysis of the data suggests that the moment is parallel to the b axis, as (0 k 0) reflections were found to be systematically absent. Inelastic neutron scattering showed two well defined crystal field excitations at 24.0 and 34.9 meV (278 and 404 K). Resistivity measurements indicated a Kondo minimum at T_K = 20 K followed by a sharp drop at the ordering temperature. The magnetic scattering part of the resistivity was determined by subtracting the resistivity of LaPtSn from the one of CePtSn and shows two different temperature regimes in the ln(T) plots. This was attributed to the Kondo effect arising from the CEF splitting of the ground state [156]. The propagation vector was later determined to be k = [0, 0.428, 0] at 1.7 K with a linear transverse spin wave character and the moments aligned along the c axis which is in contrast to the observations reported before [152]. Single crystal neutron studies of CePtSn showed a magnetic phase transition at T = 3.8 K with an incommensurate wave vector k = [0, 0.466, 0], above the magnetic order at k = [0, 0.418, 0] [157]. ¹¹⁹Sn NMR investigations found the NMR signal to suddenly disappear below T_N. A correlation between the Knight shift of the ¹¹⁹Sn and ¹⁹⁵Pt signals revealed a strong temperature dependence of both signals. However, the shift dependency at higher temperatures is linear. The hyperfine coupling constants were estimated to 9.8 kOe μ_B^–1 for ¹¹⁹Sn and 10.5 kOe μ_B^–1 for ¹⁹⁵Pt. These values tell, compared to 26 kOe μ_B^–1 for CeNiSn, that the s-f hybridization is weaker compared to CeNiSn. The 4f electrons are not so significantly affected by the hybridization, exhibiting more localized character [152].

Sakurai et al. were the first to report that CePdGa shows an antiferromagnetic transition around 2.2 K [133]. The trivalent state of cerium was determined by inelastic neutron scattering experiments. Two well defined transitions can be found at 18.9 and 33.8 meV indicating a well-localized f electron in CePdGa [158]. Susceptibility measurements showed Curie-Weiss behavior in the temperature range of 180–300 K. Below 180 K a strong deviation can be found which was attributed to crystal electric field splitting. Furthermore the magnetic contribution to the electrical resistivity was determined by subtracting the resistivity of LaPdGa from the data obtained from CePdGa. A ln T dependence was found for the high as well as for the low temperature regime. This observation was explained by a Kondo effect in the presence of crystal electric field [159, 160]. Measurements of the Seebeck coefficient revealed a broad maximum at 170 K followed by a sign change at 77 K and a minimum at 23 K [158]. Low temperature investigations down to 70 mK of the specific heat of CePdGa confirmed the antiferromagnetic transition at a slightly lower Neél temperature T_N = 1.8 K. The specific heat data <200 mK was used to determine the electronic contribution of the specific heat to be γ = 372 mJ mol^–1 K^–2. The large value was attributed to heavy fermion behavior. The heat capacity data also shows a small hump at ~1 K. This feature was attributed to the thermal population of spin waves in the antiferromagnetic state. The low temperature specific heat data was interpreted to be either a re-ordering of the localized 4f electrons or a competition of the formation of a heavy fermion state and the antiferromagnetic ordering. Thermal conductivity measurements showed a well-defined temperature dependence even at low temperatures. While at lowest temperatures κ is dominated by electronic transport the magnetic contribution becomes more and more dominant above T > 1 K exhibiting a nearly logarithmic behavior [161]. This logarithmic trend is seen for temperatures that are low when compared to the overall crystal field splitting parameter Δ [162].

3.2.3 CeTX compounds with intermediate valence

Now we want to focus on the second group of compounds – intermediate valence, valence-fluctuating compounds, Kondo compounds and heavy fermion materials. Initially for CeIrAl the cerium atom was reported to be close to tetravalent [55]. Later investigations however revealed mixed-valent behavior with a very high Kondo temperature of T_K ~1300 K obtained by comparing the susceptibility with other results from the literature. At the Kondo temperature hybridization effects between the 4f electrons and the conduction electrons start to take place leading to the formation of a singlet ground state at low temperatures and therefore a quenching of the magnetic moments of the 4f electrons. Consequently no magnetic ordering down to 2 K was detected. The measured resistivity shows, in contrast to CeRhSb, no rise at low temperatures. Instead the typical behavior of a metal is found without a pronounced Kondo minimum between 2 and 300 K [57].

CeRhGa was first reported by Hulliger to be non-magnetic with a cerium valence state being close to tetravalent [68]. More detailed studies showed that the magnetic susceptibility above 75 K is nearly temperature independent. This characterizes CeRhGa as an intermediate valent system with a high Kondo temperature. From the susceptibility measurements a Kondo temperature was estimated to be T_K = 420 K [67]. Goraus et al. [66] investigated CeRhGa by various methods and determined the Weiss constant θ_p ~ 8 K and the effective magnetic moment μ_eff = 1.5 μ_B, which is significantly lower than that of trivalent cerium. With the help of the Sales and Wohlleben model [163] the intrinsic part of the susceptibility and the interconfiguration fluctuation temperature T_sf ~ 500 K were determined. The low temperature part of the heat capacity measurements in zero-field could be fitted and the electronic contribution to the heat capacity of γ = 38.3 mJ mol^–1 K^–2 was obtained. The small value of γ is typical for non-magnetic mixed valent compounds. For the full temperature range C(T) could be fitted by the Debye expression. From the reasonable value of the Debye temperature θ_D ~ 239 K and the number of atoms in the formula unit, the authors concluded that the CEF contribution to the specific heat is small. Resistivity measurements did not indicate a typical Kondo behavior. The low temperature part gave rise to the assumption of Fermi liquid behavior. XPS measurements revealed the presence of 3d⁹_5/2 4f¹ and 3d⁹_3/2 4f¹ states for the Ce³⁺ ions, and additionally 3d⁹ 4f² and 3d⁹ 4f⁰ states were visible. This underlines the interatomic hybridization between the 4f states and the conduction band. The occupation of the 4f state was finally calculated to be n_f ~ 0.9 [66].

Both modifications of CeNiGa [63] have been characterized as intermediate valence systems. While LT-CeNiGa crystallizes in the hexagonal ZrNiAl-type structure, HT-CeNiGa was found to be orthorhombic with TiNiSi-type structure. Magnetic measurements of HT-CeNiGa above 100 K showed Curie–Weiss behavior. An effective magnetic moment of μ_eff = 2.70 μ_B and a large negative Weiss constant of θ_p = –162 K were found. Between 60 and 20 K a slow increase is visible while <20 K a sharp increase with decreasing temperature was observed. The low-temperature feature was attributed to impurities of Ce³⁺ which were determined and subtracted from the low-temperature data. The corrected data can be interpreted as local spin fluctuations as a result of either Kondo interactions or valence fluctuations. The Kondo temperature was furthermore estimated to be T_K = 95(5) K. In comparison, the significantly lower Kondo temperature of HT-CeNiGa compared to CeNiAl (T_K ~ 1000 K) can be interpreted as a less pronounced mixing of the Ce 4f and the conduction band states. The electrical resistivity data confirms the intermediate valence behavior because the low temperature data between 4.2 and 150 K shows a negative curvature while the data above 150 K is temperature independent [63].

CeNiGe was found to exhibit a paramagnetic ground state since no magnetic ordering phenomenon was observed down to 4.2 K. Susceptibility measurements revealed significant contributions of Ce³⁺ impurities masking the actual ground state. High field measurements at 350 kOe were used to estimate the temperature dependence of the intrinsic susceptibility. A rather flat maximum was found indicating an intermediate valence state for cerium. Heat capacity measurements showed an electronic contribution of γ = 32 mJ mol^–1 K^–2 which gives direct evidence to the presence of 4f states at the Fermi level [164]. Seebeck investigations showed that the region below room temperature may be one side of a single and positive peak with a large width. For T → 0 the Seebeck coefficient also approaches zero, as seen for other intermediate valence compounds. The maximum of S could be correlated with the Kondo temperature T_K, while for T < T_K a non-magnetic Kondo state is realized [165].

Susceptibility measurements of CeIrGe revealed non Curie–Weiss behavior and a strong variation of the inverse susceptibility, which was fitted and μ_eff = 0.27 μ_B and a Weiss constant of θ_p = –10 K extracted. The small effective moment can be interpreted as delocalized 4f electrons and therefore a nearly tetravalent cerium ion. This is furthermore in good agreement with the determined lattice parameters of CeIrGe. The resistivity shows a quadratic temperature dependence below 50 K and a negative curvature at high temperatures, similar to other reported compounds [30, 166].

Sasakawa et al. were the first to report the physical properties of CeIrSb in 2005 [167]. They investigated the material with the help of susceptibility, heat capacity, resistivity and Seebeck measurements. However, their samples were contaminated with traces of CeSb which was visible in some of their measurements. From the magnetic measurements no magnetic ordering was visible and a strongly curved shape of the inverse susceptibility suggesting valence fluctuations. The large Weiss constant of θ_P = –1300 K determined above 250 K is in line with the valence fluctuation. Resistivity measurements showed a curved gradual decrease of ρ(T) typical for valence fluctuation compounds along with an anomaly at 20 K which was attributed to the impurity phase CeSb. The increase of the resistivity below 8 K was, in contrast to CeRhSb, not attributed to a gap formation but to carrier localization effects. The reason for this assumption is that the Seebeck coefficient S would show a large peak. Instead S(T) is positive down to 4 K and has a value of 40 μV K^–1 for a wide temperature range (100–300 K).

Heat capacity measurements below 6 K show an electronic contribution with γ = 10 mJ mol^–1 K^–2, suggesting strongly localized 4f states [167]. Later Kawasaki and co-workers investigated CeIrSb with the help of Sb NMR and NQR experiments and found the formation of a pseudogap at the Fermi level. At 4.2 K two transitions are observed for ¹²¹Sb (I = 5/2) and three for ¹²³Sb (I = 7/2), respectively. The temperature dependence of 1/T₁T for CeIrSb has a maximum at 300 K and decreases significantly below this temperature. Below 4 K a short upturn is observed. The decrease of 1/T₁T was attributed to the energy gap at low temperatures. The upturn in 1/T₁T was assigned to the relaxation process of spin fluctuations of the paramagnetic Ce spins. The spin flipping – and hence the upturn – was suppressed by applying a small external magnetic field of 1 kOe. Due to the similarities in the temperature dependences of CeIrSb and CeRhSb the V-shaped gap model was used to determine the bandwidth D = 1800 K and the energy gap Δ = 350 K. Both, D and Δ, are significantly larger (up to one order of magnitude) compared to the values determined for CeRhSb and CeNiSn. The much larger value of D was assigned to the weaker temperature dependence of the magnetic susceptibility and a stronger crystal field hybridization as well as a much larger magnitude of the energy gap. Finally ¹²¹Sb NMR investigations in high magnetic field and at T = 10 K were conducted. Six peaks are found in the field-swept measurement of the polycrystalline sample, which all show neither peak broadening nor splitting down to 5 K indicating a non-magnetic ground state for CeIrSb. The relaxation rates of the central resonance lines are furthermore field-independent up to 100 kOe indicating that the gap structure is not affected by the applied magnetic field [168, 169].

In addition to the structural investigations on equiatomic CeCuZn, the solid solution Ce(Cu_1–xZn_x)₂ with x = 0–1 was investigated with respect to the physical properties. However, for the equiatomic CeCuZn no detailed study was conducted [170].

Similar to CeCuZn the solid solution Ce(Cu_xGa_1–x)₂ was investigated regarding its physical properties. Besides binary CeCu₂, samples with x = 0.95, 0.9–0.5 were prepared and measured. The shallow minimum in the electrical resistivity found for CeCu₂ increases with increasing Ga content x = 0.95–0.6. For x = 0.5 the magnetic contribution was plotted versus the electrical resistivity in a semi logarithmic way. Two different temperature ranges are clearly visible showing –ln T behavior, reflecting Kondo-type interactions in the presence of strong crystal field splitting effects according to Cornut and Coqblin [159].

For CeCuGa a quartet ground state has been found while CeCu₂ exhibits a doublet ground state [171]. No magnetic ordering has been found for any member of the Ce(Cu_xGa_1–x)₂ series with x = 0.95–0.5 caused by the evolution of the Kondo effect. The absence of magnetic ordering was also confirmed by heat capacity measurements. For CeCuGa a large electronic contribution to the heat capacity of γ = 110 mJ mol^–1 K^–2 has been extracted [172]. The magnetic moment of CeCuGa was determined to be μ_eff = 2.14 μ_B and therefore below the theoretical value of μ_eff = 2.54 μ_B for a trivalent cerium ion. However no indication for dynamical mixed valence or valence fluctutations was found. Also no superconductivity was found down to 0.5 K. The ratio of C/T increases remarkably below 10 K with a peak typical for Kondo compounds. This peak indicates a contribution of the localized magnetic moments to the specific heat [173, 174].

Nanocrystalline powders of CeCuGa with an average particle size of 9 nm were prepared by mixalloy processing. From their magnetic susceptibility above 18.5 K an effective magnetic moment of μ_eff = 2.45 μ_B per Ce ion and a Weiss constant of θ_p = –63 K were extracted. From the square of the magnetization versus temperature a Curie temperature of 18.5 K was determined indicating weak antiferromagnetic ordering (considering the negative Weiss constant) below 18.5 K. From magnetization experiments with two external fields the authors concluded that the Kondo effect is not completely suppressed at H = 500 Oe by the disorder within the nanocrystalline material since the M–T curves show two extremes at 3 and 15 K. However, already at H = 5 kOe, this change disappears [175]. Resistivity investigations at very low temperatures (0.5–150 K) showed a minimum at 25 K followed by a slight increase with decreasing temperature. At 3 K the resistivity passes a maximum which is expected if the coherent scattering of the conduction electrons by the magnetic moments leads to a decrease in resistance. The final decrease suggests that the ground state is associated with a Fermi liquid state. The specific heat measurements were carried out down to 80 mK with a hump at 1 K. Below 1 K it decreases before it increases again below 250 mK due to the significant contribution of the nuclear part of the specific heat. The electronic coefficient of the specific heat is extremely high (γ = 1200 mJ mol^–1 K^–2), in line with CeCuGa being a heavy fermion material. γ is reduced when the specific heat is determined in the presence of a magnetic field. The authors stress that this behavior is not caused by a spin glass state as mentioned in previous literature [176], because the peak of the specific heat should be shifting to higher temperature in the presence of applied magnetic fields [177].

Hulliger was the first to report that for CeIrGa the cerium valence is between three and four at room temperature [68]. CeIrGa has been studied furthermore by powder X-ray diffraction and the volume of the unit cell was found to be almost the same as the one reported for CeIrAl. Therefore a mixed-valence state of the cerium ions was assumed. Susceptibility measurements revealed very low values for the magnetic susceptibility (χ ~ 10^–3 emu mol^–1) at low temperatures [71]. A clear increase of χ was found above T > 120 K, suggesting a maximum in the susceptibility well above room temperature. A very similar behavior was found for CeIrAl indicating that CeIrGa also exhibits Kondo interactions. With the help of Rajan’s curve [178] the Kondo temperature was estimated to be T_K ~ 1000 K [71].

By a number of complementary measurements [179] CeRuSi has been characterized as a heavy fermion compound. Susceptibility measurements revealed an effective magnetic moment of μ_eff = 2.43 μ_B and a Weiss constant of θ_P = –91 K, with a Curie–Weiss-type behavior down to 20 K. At low temperatures the susceptibility flattens off reaching a broad maximum at T = 8 K (1 kOe). At 45 kOe the maximum becomes better defined. Resistivity measurements revealed a maximum at slightly higher temperatures compared to the maximum observed at T = 50 K in Seebeck measurements. Furthermore a minimum at 5 K was observed. The reported value of S is about 20% higher compared to literature data [180]. Specific heat measurements have shown a broad maximum centered around 5.5 K. The Sommerfeld coefficient was calculated to be γ ~ 0.18 mJ mol^–1 K² confirming the heavy Fermion character of the compound.

Before we summarize the physical properties of CeNiSn, we need to point out that this stannide has attracted tremendous attention, mostly for its physical properties. More than 230 entries are listed in the Scifinder data base [181]. In the following we focus on the most important contributions keeping the publication dates in mind. The papers dealing with solid solutions are addressed separately in section 3.5.

The first reports on CeNiSn by Scolozdra et al. and Takabatake et al. characterized the compound as an intermediate valence material [182, 183]. The susceptibility deviates from the classical Curie–Weiss behavior below 120 K with a reported effective magnetic moment of μ_eff = 2.86 μ_B [182]. The Weiss constant was determined to be θ_P = –187 K, and the magnetization at 1.3 K and 100 kOe was estimated to be only μ = 0.115 μ_B [183]. Resistivity measurements below 30 K showed an increase by a factor of four with saturation near 0.7 K. A lnρ vs. 1/T plot between 3 and 65 K gave an activation energy of 6 K, corresponding to a small energy band gap of 0.5 meV near the Fermi level. The specific heat measurements did not show any pronounced peak down to 0.45 K, and a C/T vs. T plot revealed a broad peak near 1.3 K, consistent with a gap of 0.5 meV. The electronic contribution was determined to γ = 120 mJ mol^–1 K^–2, suggesting the compound to be a heavy fermion material [183].

Substitution of Ce by La in the solid solution Ce_xLa_1–xNiSn suppresses the gap formation. While for CeNiSn ϵ_g ~ 3 K was determined from resistivity measurements, Ce_0.9La_0.1NiSn has an energy gap of ϵ_g = 1.6 K. With γ = 200 mJ mol^–1 K^–2, the heavy fermion character was confirmed [184]. Following these initial findings, a large number of papers have been published concerning the physical properties of CeNiSn. In this work, often solid solutions have been used to investigate and understand the unique properties of CeNiSn, the properties of the respective members of the solid solutions and the depression of the gap via chemical substitution (vide infra). However, also physical pressure can be used to suppress the gap formation, since the application of pressure increases the hybridization between the 4f orbitals and the conduction electrons. The electrical resistivity under hydrostatic pressure up to 24 kbar was measured on a polycrystalline sample of CeNiSn. The strong increase below 30 K at ambient pressure is suppressed and a maximum at around 100 K arises. No indication for magnetic ordering under pressure has been observed. In order to determine the variation of the energy gap with pressure, lnρ vs. 1/T was plotted, giving a decrease of –0.02 meV kbar^–1, but the linear dependence does not hold below 7 K. The energy gap therefore seems to vanish around 30 kbar [185]. The hybridization of the 4f orbitals and the conduction electrons gives rise to a peak near the Fermi energy E_F and can therefore be investigated by Seebeck measurements, which are sensitive to the energy dependence of the density of states. The investigations of S vs. T on a single crystal of CeNiSn have shown a large anisotropy along the three different crystallographic axes. While S shows very sharp peaks along a and c at 3 K, a broad peak at 100 K along b was observed [186].

¹¹⁹Sn NMR investigations in the temperature range of 0.38–250 K excluded unambiguously the possibility of a magnetic phase transition down to 0.38 K. Analysis of the NMR data indicated furthermore, that the energy gap of CeNiSn is formed at the Fermi level, and that the picture of a Fermi liquid is not applicable. From the hyperfine coupling constant (26 kOe per μ_B), Fermi contact interactions of the 5s electrons with the polarized Ce4f electrons were derived [187–189]. Inelastic neutron studies at 15 and 50 K showed two distinct excitations at 3.5 and 6 meV, whereas the 6 meV excited state was also confirmed by transport property measurements. Interestingly, the spectra recorded at 4.5 and 8.8 K are flat in the range of 2–9 meV indicating the absence of magnetic scattering. This feature was attributed to the partial delocalization of the 4f electrons below 10 K [190]. No well-defined crystal field excitations were observed in other measurements as well [191].

By contrast the inelastic response of CeNiSn was described as rather structureless showing quasi-elastic scattering with large linewidths [192]. Neutron scattering measurements showed that the crossover at T < 10 K from a metallic heavy fermion state to a semiconducting state coincides with the formation of a spin gap in the magnetic excitation spectrum [193]. Thermal expansion measurements revealed an intriguing feature below 2 K. The thermal expansion coefficient α goes through a maximum at T = 1.3 K and changes its sign below 1 K. The amplitude of the negative α(T) anomaly is unusually large but down to 0.5 K no minimum was found. Furthermore a strong phonon line at w ~ 343 cm^–1 was found in energy dependent reflection measurements. The phonon line gets sharper with decreasing temperature, but no splitting was observed below the proposed temperature for magnetic correlations [194]. Pressure dependent thermal expansion coefficient studies on a single crystal were conducted up to 8 kbar in the temperature range 4.2–300 K. The thermal expansion coefficients are highly anisotropic, reflecting the orthorhombic crystal structure, with a sharp peak at 6.5 K for α_b and a shoulder at 8 K for α_a. The anomalies were interpreted to be a result of the opening of the anisotropic energy gap. However, at 8 kbar both anomalies disappear [195].

Theoretical investigations, carried out by self-consistent LAPW calculations, found CeNiSn to be a semimetal with a hole Fermi surface [196, 197]. The temperature dependence of the elastic constants of CeNiSn was studied without and with an applied magnetic field. Without field, the elastic longitudinal constants C₁₁ and C₂₂ which correspond to the crystallographic a and b axes show a pronounced softening around 200 K before becoming stiff at about 40 K again. In contrast, C₃₃ exhibits a shallow minimum at 180 K, showing the anomalous behavior of this compound, and the transverse C₄₄, C₅₅ and C₆₆ modes show no elastic softening. In ultrasonic experiments, no sign of magnetic ordering was detected down to 0.4 K. The anomalous behavior of the longitudinal modes was related to the quasiparticle band at the Fermi level. The monotonic increase of the transverse modes confirms the delocalization of the 4f electrons. Magnetic field effects on the longitudinal C₁₁ mode showed a remarkable effect with respect to the a axis while little dependencies were observed along b and c. By contrast, C₂₂ and C₃₃ show the same anisotropic behavior upon applied field [85].

The mechanism of the gap formation was also studied by Hall and magnetoresistance measurements on single crystals. For the Hall coefficient, all the directions show an increase below 100 K, followed by a sharp drop to negative values. Since the peak temperature of R_H in heavy fermion systems is regarded as the onset of coherence, it was followed that CeNiSn gradually enters a coherent scattering regime below 9 K. However, the strong decrease below 5 K can be ascribed to a diminishing carrier density. Assuming only one type of charge carriers, the concentration was estimated to be 4.6 × 10^–3 charge carriers per formula unit for H || c at 1.3 K. For the magnetoresistance ρ_a, a maximum near 11.4 K was observed, which gets smeared out in a field of 140 kOe for H || a. The upturn below 10 K gets strongly suppressed. A large effect can be found in ρ_b for H || a indicating a metallic behavior, which is in contrast to the semiconductor-like behavior for H || c [198–200].

Photoemission studies on CeNiSn showed two prominent structures in the Ce 4f derived spectra (on-resonance minus off-resonance) around 2.5 eV and near E_F, respectively. The fairly high intensity of the feature near the Fermi level is consistent with the otherwise observed mixed-valent behavior. However, the valence of cerium is very close to +3 with an occupation of the trivalent state >0.95. The valence band photoemission spectra show the Ni 3d emissions at 1.5 eV along with a shoulder near E_F. The shoulder is caused by the hybridization of empty Sn 5p and filled Ni 3d states. This model has been underlined by Ni 3p → 3d on-resonance, off-resonance and Ni 2p_3/2 core level spectra. They exhibit two hole satellites, which occur when the Ni 3d band is not completely filled. Since the Ni 3d band does not substantially cross E_F, the hybridization with the Sn 5p band, which leads to depleting of the 3d states can be derived [201]. Additional investigations of the Ce 4d core level showed a main feature between 105 and 115 eV, originating from the final Ce 4d⁹4f¹ states, along with a hump at 120 eV which can be assigned to the Ce 4d⁹4f⁰ states, underlining the mixed valent character. Ce 3d core level spectra showed two strong peaks at the binding energies of about 884 and 903 eV, which can be assigned to the 3d_5/2-3d_3/2 spin-orbit components of the 3d⁹4f¹ final states [202]. With the help of ¹¹⁹Sn Mössbauer and μSR spectroscopy techniques the so-called transferred hyperfine field was determined. The muon senses the magnitude and the distribution of the magnetic field on an interstitial site. The ¹¹⁹Sn Mössbauer spectra of CeNiSn show no magnetic interaction down to 1.5 K. The spectral shape arises solely from the quadrupole interaction due to the non-cubic local symmetry. For the zero-, longitudinal- and transverse-field μSR measurements of a single crystal of CeNiSn, no influence of the magnetic moments of Ce could be detected above 2 K. At T < 1 K, a continuous change in the muon relaxation is observed, which is typical for a paramagnet approaching the magnetic order. However down to 33 mK no magnetic phase transition was observed [203, 204]. In order to determine the low temperature nuclear orientation, samples with the radioisotope ¹⁴¹Ce were produced by the interaction of thermal neutrons with the sample. The samples were cooled to ~10 mK and the γ-anisotropy was measured. In contrast to CeNi₂Al₅, no anisotropy appeared on CeNiSn, even at applied external fields [205]. Since a collapse of the energy gap was found upon doping CeNiSn, high quality crystals were grown by either the floating-zone or Czochralski methods and with different impurities. They were analyzed by metallography, electron microprobe analysis and physical measurements. The compositions of the crystals were found to be within 1% of the target composition. However, Ce₂O₃, CeNi₂Sn₂ and Ce₂Ni₃Sn₂ were detected as by-products in different ratios. The highest purity of the crystals was achieved by subsequent purification via solid-state electrotransport. In this crystal all by-products were determined to be below 0.1%. The semiconductor-like behavior found below 10 K is readily suppressed in dirty or substituted crystals, the purified sample shows metallic behavior in the resistivity along the a axis. This result along with the large γ = 42 mJ mol^–1 K^–2 supports the presence of a residual density of states at the Fermi level. Therefore the ground state of CeNiSn is not a Kondo semiconducting state, but more a semimetallic state with a pseudogap which closes along the a axis [80, 206]. Evidence for a quasiparticle gap in CeNiSn was generated by tunneling spectroscopy, the most direct probe to investigate the gap. The measurements were conducted using the break-junction technique with a four probe method. The energy gaps determined are nearly of the same magnitude as T_K [207]. Finally we want to mention the comparison of the thermoelectric properties of CeNiSn, CePtSn, CeRhAs, and CeRhSb [208]. All compounds are isotypic and show a δ-shaped DOS in the vicinity of the Fermi level, making them good candidates for LT thermoelectric materials. Due to the high anisotropy of these materials, the thermoelectric properties were determined along the crystallographic b axis. As expected, the resistivity changes from a metallic behavior in CePtSn towards a semimetallic one in CeNiSn and CeRhSb, and to a semiconducting one in CeRhAs. The opposite order is seen in the thermal conductivity κ at 290 K. While for CeRhAs an upturn in the thermal conductivity is seen below 165 K, a much weaker feature is visible for CeRhSb and CeNiSn below 20 and 15 K, respectively. Finally the Seebeck coefficient shows the same trend as the resistivity, being in line with the order of increasing Kondo temperatures T_K. As described before, the Seebeck coefficient drops to 0 for T → 0 K in CeNiSn. The figure of merit zT finally represents again the strong structural anisotropy of these compounds due to large anisotropies in the trend of zT.

In heavy fermion materials the competition between RKKY and Kondo interactions causes several distinct ground states. One of these states is the so-called low carrier Kondo system, also denoted as a Kondo semiconductor [209]. Since an increase in the electrical resistivity at low temperatures was found for CeNiSn [182], CeRhAs [107], and CeRhSb [210], these three compounds were called Kondo semiconductors. The corresponding band gap energies vary from 14 K for CeNiSn to 28 K [211] for CeRhSb [211] and to 282 K for CeRhAs [35]. In the next paragraphs a large number of different gap sizes are mentioned for all three compounds depending on the measurement technique and the sample performance. Thus we would like to emphasize a publication that deals with single crystals of CeNiSn and CeRhSb with different amounts of impurities and the influence of these impurities on different physical properties [206]. The influence of the impurities and related crystallographic defects need to be considered when discussing such complex scattering effects. For instance, Ideka and Miyaka have included impurity scattering within their theory, which will be discussed later on [212].

Similar to CeNiSn, also the pnictides CeRhAs (48 entries in the SciFinder data base) [181] and CeRhSb (115 SciFinder entries) have been studied in great detail. Again, only a representative number of references can be considered herein. CeRhAs is an exceptionally good example for the importance of coherence of crystal structure and related properties. As described in the crystal structure part the lattice parameters show three anomalies (T₁ = 370, T₂ = 235 and T₃ = 165 K), which mainly concern the a lattice parameter. Along the a axis zig-zag chains of cerium are formed, and due to the structural changes the hybridization and consequently the physical properties are influenced significantly [35, 36]. Before the crystal structure had been completely understood, a gap formation has been attributed to the Kondo temperature. The gap size was determined by several methods. Evaluation of photoemission spectroscopy measurements lead to gaps of 180–200 [213] or 90–100 meV [214] below ~210 K, while transport measurements revealed a much smaller gap of 12(3) meV [215, 216]. In none of these publications the gap formation was related to structural changes. Sasakawa et al. could nicely show that the periodic lattice distortions caused by structural phase transitions influence the density of states. These are in line with the anomalies observed in the magnetic susceptibility, heat capacity, electrical resistivity, and the thermopower [35, 217]. According to this publication CeRhAs can be defined as a valence-fluctuating compound due to a broad maximum around 500 K in χ(T) and no significant anisotropic behavior can be observed. The electronic contribution to the heat capacity (C/T = γ + βT² + αT³) could be determined to γ = 3.0 mJ K^–2, which is one order of magnitude smaller than those reported for CeRhSb and CeNiSn. Resistivity measurements of CeRhAs have been performed with a single crystal for all three crystallographic axes. For all directions along the three principal axes, ρ(T) exhibits step-like anomalies and a significant increase can be observed below T₃. Especially, in parallel alignment to the c axis the three anomalies of the structural phase transitions can be observed. One peculiarity that should be mentioned is the anisotropy around the phase transitions. For instance, ρ_a jumps but ρ_c drops at T₃. Though a significant anisotropic character can be observed, all directions exhibit an increase of the electrical resistivity with lower temperatures, clearly proving a band gap in the density of states [35, 217]. Nevertheless, no ideal semiconductive behavior can be observed, most likely due to the structural distortion.

Nevertheless, the gap itself in the so-called “Kondo semiconductors” is not finally resolved and still discussed in the literature. Three different models have been discussed so far, but none of these could be established as the ideal one [36]. In contrast to the isostructural compounds CeRhSb and CeNiSn the gap formation of CeRhAs is related to lattice modulations. Already a substitution of 2.5% of As by Sb is enough to suppress all three structural phase transitions, and consequently no band gap features are observable. Hall coefficient measurements exhibited two different energy scales for the gap structure. Below T₁ a large one of 2000 K and below T₃ a small one with 300 K is reported [217]. This is accordance with optical conductivity measurements [218, 219]. By reference to detailed ⁷⁵As NQR/NMR data a band gap of 272 K with a band width of ≈4000 K was evaluated which is one order of magnitude larger than those in CeRhSb and CeNiSn [220]. As already reported for the antimony substitution in CeRhSb_1–xAs_x, the observed structural phase transitions are intolerant to very small amounts of other elements. A similar behavior is also observed for the substitution of cerium by lanthanum according to Ce_1–xLa_xRhAs. While all anomalies ascribed to the phase transition T₁ remain up to x = 0.1, no features can be observed around the temperatures of T₂ and T₃. Resistivity as well as Seebeck measurements proved that no semiconductivity is observable with x = 0.02 below T₃ [221]. The significant influence of lanthanum substitution on the Ce 4f states between x = 0.003 and 0.02 could also be shown by resonance-photoemission spectroscopy [222]. The very important influence of changes in lattice parameters on the hybridization has nicely been revealed by systematic investigations of the electrical resistivity under pressure. At 250 K the orthorhombic b and c parameters increase slightly, while the a parameter decreases with pressures up to 2.5 GPa [36]. In earlier publications [217] it was argued that the decrease of a enhances the c-f hybridization in analogy to CeNiSn, leading to the gap formation. In general it is expected that the c-f hybridization becomes stronger by applying pressure leading to an enlarged band gap. However, up to 1 GPa the activation energy increases slightly along the c axis, while that along the a axis decreases. At around 2 GPa both gap energies vanish. Consequently, discrepancies between experimental results and the simple scenario based on c-f hybridization can be ascertained [36, 223–227].

In the first publication [210], CeRhSb was classified as a valence-fluctuating compound with a rapid rise in the electrical resistivity below 21 K, which was attributed to a band gap opening in the electronic density of states with a band gap energy of about 4 K. In magnetic susceptibility studies a broad maximum at approximately 113 K, characteristic of a valence-fluctuating compound, could be observed and a fitting with the interconfiguration fluctuation (ICF) model proposed by Sales and Wohlleben led to a convincing agreement [163].

By reference to a heat capacity measurement, CeRhBi can be classified as a heavy fermion material due to the value of γ ≈ 500 mJ K^–2 mol for the electronic contribution to the heat capacity [107]. Electrical resistivity and ²⁰⁹Bi NQR measurements proved the absence of any magnetic ordering down to 150 mK or of a band gap opening like observed for CeRhAs and CeRhSb [18, 228]. Though no magnetic ordering can be observed, cerium can undoubtedly be defined as Ce³⁺ according to Curie–Weiss fitting [107, 229]. The electrical resistivity of CeRhBi is almost constant down to 10 K, and on further cooling a significant drop is observed [229]. One can identify two maxima, the first one around 70 K which is very broad and flat, while the second one at 6 K is more pronounced. The latter one is taken as an onset of coherent Kondo scattering with a Kondo temperature of T_K = 9.2 K, estimated from a C_m/T vs. T plot [18]. Due to the linear temperature dependency of ρ(T) below 3.5 K, CeRhBi is described as a non-Fermi liquid, which is often found in systems in the vicinity of a QCP. Different reasons for the vicinity of a QCP are listed: a T^–1/4 dependency of χ below 7 K, a change of sign in S(T) below 13 K and an increase of C/T below 8 K in proportion to –log T [229].