Magnetic phase diagram of pseudo-ternary solid solution URu1−xPdxGe

Journal of Solid State Chemistry 226 (2015) 50–58

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Magnetic phase diagram of pseudo-ternary solid solution URu1
x Pdx Ge D. Gralak, V.H. Tran n Institute of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box 1410, 50-422 Wrocław, Poland

art ic l e i nf o

a b s t r a c t

Article history: Received 30 September 2014 Received in revised form 16 January 2015 Accepted 24 January 2015 Available online 2 February 2015

A systematic study of the solid solution URu1
x Pdx Ge by means of X-ray diffraction, dc-magnetization, specific heat and electrical resistivity measurements is presented. The investigated alloys ð0:0 r x r 0:9Þ crystallize in the orthorhombic TiNiSi-type structure with space group Pnma. The variation of the unit cell volume with Pd content x follows approximately Vegard's law, through the lattice parameters a, b and c exhibit a little anomaly around x ¼0.35, 0.7 and 0.8, respectively. It is found that the magnetic ground state of URu1
x Pdx Ge strongly depends on Pd content. The compositions with x r 0:32 are nonmagnetic down to 0.4 K, while these in the concentration range 0:35 r x r 0:8 order antiferromagnetically. The Néel temperature TN is found to follow a relation T N p ðx
0:32Þ2=3 for the alloys x o 0:45 and to attain a maximum value of 32 K at x ¼0.8. The composition x ¼ 0.9 alike UPdGe manifests a complex magnetic order, probably undergoes two successive magnetic phase transitions: ferromagneticlike at TC ¼ 33 K and unknown ground state characterized by an irreversible magnetization below TIR. The observed development of magnetism has been discussed in terms of changes in the degree of 5f electron localization as well as due to the competition between the Kondo and Ruderman–Kittel–Kausuya–Yosida interactions. Remarkably, in the compound x ¼ 0.32 located at the nonmagnetic-magnetic border, we found a non-Fermi liquid behaviour reflected by thep power-law temperature dependencies of magnetic ffiffiffi susceptibility χ ðTÞ p T
0:48 , specific heat C p ðTÞ p
T and electrical resistivity ρðTÞ p T 3=2 . We compared the results with theoretical predictions for systems with spin fluctuations nearby antiferromagnetic quantum critical point. & 2015 Elsevier Inc. All rights reserved.

Keywords: Magnetic phase diagram Non-Fermi liquid Quantum criticality Magnetic susceptibility Specific heat Electrical resistivity

1. Introduction The U-based series of equiatomic ternaries with the chemical formula UTM, where T is a d-electron transition metal and M¼ Si or Ge, crystallize mainly in the TiNiSi-type structure (space group Pnma) [1]. The U, T and M atoms occupy distinct 4c-positions and form three different polyhedrals respectively consisting of 20 atoms, i.e., ½U8 T 6 M 6 , and of 10 atoms, i.e ½U6 M 4  and ½U6 T 4 . A number of investigations of UTM have indicated that their magnetic ground states spread the continuum from a nonmagnetic to a magnetic order with increasing d-electrons involved in compound. Such a behaviour is believed to arise due to a weakening of the hybridization between the magnetic moment carrying 5f-electrons and the conduction electrons [1]. This result means that the hybridization degree can be tuned by a suitable substitution on either U-atom or the T- or M- atom site and then one can modify magnetic properties of a given UTM compound. In fact, a big change in magnetic properties has been observed in several series of solid solution: URh1
x Rux Ge [2,3],

n

Corresponding author. E-mail address: [email protected] (V.H. Tran).

http://dx.doi.org/10.1016/j.jssc.2015.01.027 0022-4596/& 2015 Elsevier Inc. All rights reserved.

UCo1
x Fex Ge [4] and UCo1
x Rux Ge [5]. The crossover from the ferromagnetic URhGe ðT C ¼ 9:5 KÞ, UCoGe ðT C ¼ 3 KÞ to nonmagnetic ground state appears upon substitution of xcr  38% Rh by Ru and 22% (30%) Co by Fe (Ru), respectively. Moreover, in these systems the physical properties of compositions around xcr, such as electrical resistivity, specific heat and magnetic susceptibility, exhibit nonFermi liquid (nFl) temperature dependencies. Interestingly, Sakarya et al. [2] have shown that the URh0:62 Ru0:38 Ge composition can be regarded as a f-electron system with a ferromagnetic quantum critical point (QCP) at an ambient pressure. Furthermore, our investigations have evidenced spin-fluctuations as a precursor property before the system brings nearer QCP [6]. Similarly,in systems UCo1
x Fex Ge and UCo1
x Rux Ge a ferromagnetic quantum critical point was also found, i.e., at xcr ¼0.22 and 0.3 by Huang et al. [4] and Valiîdka et al. [5], respectively. Inspired by interesting behaviour of URh1
x Rux Ge, UCo1
x Fex Ge and UCo1
x Rux Ge, we undertook to study the low-temperature properties of isostructural pseudo-ternaries URu1
xPdx Ge. To the best of our knowledge, the system has not been investigated yet. In comparison to the ferromagnetic URhGe and UCoGe, the intermetallic UPdGe compound exhibits richer magnetic phase transitions, owing to as many as two phase transitions; an antiferromagnetic (AF) one at TN ¼50 K and a ferromagnetic (F) one at

D. Gralak, V.H. Tran / Journal of Solid State Chemistry 226 (2015) 50–58

TC ¼ 30 K [7]. Magnetic structures of UPdGe investigated by neutron diffraction method have been reported in several works [8–10]. Complicated spin structure was found between 28 and 50 K [8,9], and a simple ferromagnetic structure below 28 K with magnetic moments pointed along the orthorhombic b-axis was determined by Khatibe et al. [10]. Thus, by investigation of solid solution URu1
xPdx Ge we not only look for quantum criticality and nFl properties, which develop in compositions located at the nonmagnetic–magnetic border but we also expect to observe an interesting magnetic phase diagram owing to the existence of different cooperative magnetic interactions. The aim of this work is, moreover, to examine the magnetic properties of the system associated with a change in the strength of the f–d hybridization Vkf between the magnetic f electrons and the conduction electrons. Because the Pd atom possesses two delectrons more than Ru, the change in Vkf should be very noticeable in URu1
x Pdx Ge compared to those in the URh1
x Rux Ge, UCo1
x Fex Ge and UCo1
x Rux Ge systems, where the difference in the d-electrons of parent compounds is only one. In this paper we present a detailed studied of the URu1
x Pdx Ge alloys by means of the measurements of powder X-ray diffraction, dc-magnetization, specific heat and electrical resistivity. Our results, besides exploring magnetic phase diagram showing a continuous change in the magnetic ground state from nonmagnetic URuGe, through nFl state around x¼ 0.3 and antiferromagtic one in x¼ 0.35–0.8 to a complex magnetic order in x Z0:9, highlight that the nFl state develops nearby antiferromagnetic quantum critical point at xcr  0:32.

2. Experimental details Polycrystalline samples of the URu1
x Pdx Ge solid solution with 0.0 r x r 0.9 were synthesized by arc melting of stoichiometric amounts of the constituents in a water cooled copper crucible under a purified argon atmosphere with Ti as a getter. The purity of elements was U: 99.8%, Ru: 99.99%, Pd: 99.9%, Ge: 99.999%. The samples were melted several times and turned over after each melt to improve the homogeneity. Next, the as-cast samples were wrapped in a Ta foil, sealed in a quartz tubes under vacuum and annealed for nine days at 800 1C. Mass losses during the preparation were less than 0.1% for the samples. The quality of the samples was checked before and after annealing by means of powder X-ray diffraction at room temperature, utilizing an X'Pert PRO diffractometer with monochromatized CuK α radiation. A tiny single crystal was mechanically separated from a crushed annealed button of URuGe. For the crystal X-ray intensity data were collected on a KM-4 Xcalibur single-crystal diffractometer (Oxford Diffraction) equipped with a CCD detector employing graphite monochromated MoKα radiation, and then structural parameters of URuGe have been determined and refined by the full least-squares method using the SHELXL97 program [11]. The lattice and atomic parameters of the remaining 0:1 rx r0:9 compositions were obtained by means of the Rietveld refinement using the FULLPROF software [12]. The starting atomic parameters were taken as those of the URuGe (this work) and UPdGe [10]. A single crystalline sample of x¼ 0.32 was grown from a bulk sample, using the Czochralski method. The chemical composition of the sample was checked by a microanalysis using scanning electron microscope FEI Nova NanoSEM 230 equipped with X-ray energy dispersive (EDX) spectrometer. Good agreement was found between the composition values and those determined by EDX analyses. Polycrystalline samples with 0:0 r x r 0:9 for magnetic and electron transport experiments were cut into parallelepipeds with cross-sections about 1  1 mm2 and a length of 5 mm. Dc–magnetization, M, measurements were made with a Quantum Design MPMS

51

magnetometer in the temperature range 1.8–400 K and in magnetic fields μ0 H up to 5.5 T. Magnetic susceptibility, χ ðTÞ, is calculated as χ ðTÞ  MðTÞ=μ0 H. Specific heat, Cp(T), and electrical resistivity, ρðTÞ, measurements were performed with a Quantum Design PPMS platform in the temperature range 1.8–300 K. The Cp data were collected on small samples ( 5 mg) using the 2-τ thermal relaxation method, whereas the ρðTÞ data by the four-probe ac technique. The gold wires were mounted to samples with a electrically conductive silver paste and the alternating current 5 mA of frequency of 97 Hz was applied. The Cp(T) and ρðTÞ measurements of x¼ 0.32 were performed down to 0.4 K, using a 3He PPMS option.

3. Results and discussion 3.1. Structural data The X-ray characterization revealed that the prepared samples are nearly single phases (more than 97%), and crystallize in the same crystal TiNiSi-type structure as that of the parent URuGe and UPdGe compounds. In Fig. 1 we show the powder X-ray diffraction (XRD) pattern for x¼ 0.7 as a representative of the Pd-rich URu1
x Pdx Ge alloys. An analysis of the pattern shows that in addition to the main phase, there is a small amount of an impurity, which is barely detected at un-indexed reflection at 2Θ ¼ 31.51. Unfortunately, after the annealing the quantity of an impurity puts up, since there appear additionally several unindexed reflections, i. e., at 2Θ ¼35.7 and 38.61. We may add that the overall quality of samples is worsened yet, since the background of the X-ray diffraction pattern in the low-angle range becomes enhanced, most presumably due to the short-range order of binaries precipitated during the annealing. Amongst binary compounds of U–Pd, U–Ru, Pd–Ru, U–Ge and Pd–Ge, the Pd5Ge phase seems to be the impurity in our samples. According to Matkovic and

Fig. 1. X-ray powder diffraction patterns of URu0:3 Pd0:7 Ge measured on the sample (a) before and (b) after the annealing. The small symbols and solid lines denote the experimental data and theoretical profiles, respectively. The vertical lines correspond to the position of the Bragg reflections for URu0:3 Pd0:7 Ge and Pd5Ge, respectively. The difference of the XRD profiles are presented at the bottom of the panels.

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D. Gralak, V.H. Tran / Journal of Solid State Chemistry 226 (2015) 50–58

Schubert [13], Pd5Ge crystallizes in the monoclinic structure (space group C121). The Bragg reflections of this phase are denoted by I in Fig. 1. Because the annealing does not improve the quality of the samples, the present work reports physical properties of unannealed samples only. We believe that the presence of Pd5Ge in the studied samples is not expected to affect significantly the observed data, because the 4d-shell of the Pd atoms is completely filled. As already mentioned, refinements of the structural parameters for the single-crystal URuGe have been done. We obtain lattice parameters a¼0.6684(1), b¼ 0.4344(1) and c¼0.7542(2) nm, which agree well with literature values for polycrystalline samples [1]. The atomic parameters of U, Ru and Ge atoms located at the 4c sites were determined to be xU ¼0.0132, xRu ¼0.2639, xGe ¼0.6827, zU ¼ 0.7007, zRu ¼0.4142 and zGe ¼0.4172. The refinement converged satisfactorily to the agreement index R1 ¼0.0492 and ωR2 ¼ 0:087. The room temperature lattice parameters a, b, c and the unit cell volume V, obtained from the Rietveld refinement are plotted as a function of Pd concentration x in Fig. 2. The structural parameters of UPdGe were taken from references given in [1,10]. We observe that a little anomaly in a(x) parameter occurs around x¼0.35, whereas the b (x)- and c(x)- dependencies alter their slope around x¼ 0.7 and 0.8, respectively. In spite of these anomalies, the unit cell volume of URu1
x Pdx Ge follows approximately Vegard's law and expands by ΔV=V  6:6% upon the full substitution of Ru by Pd. The increase in the unit cell volume certainly causes an enlargement of the interatomic distances. Since, the shortest distances between the magnetic U atoms and between the magnetic U and ligand (T; M) atoms determine the magnetic properties of compounds, possible effects accompanied with the changes in dU–U and dU–TM should be considered. While inspecting dU–U of URu1
x Pdx Ge, we recognize two short uranium–uranium distances. One is along the a-axis, d1U–U , and the other, d2U–U being between neighbouring uranium atoms lying in two successive ac-planes, i.e., along the b-axis. To calculate d1U–U and d2U–U of 0.0 r x r 0:9 samples we took the structural parameters obtained from our X-ray refinements. In the case of UPdGe, we assumed the atomic positions of the U atoms as those in the UPdGe compound [10]. In Fig. 3 we plot the concentration dependence of d1U–U and d2U–U . It evidences that d1U–U steadily increases with x,

Fig. 3. The shortest inter-uranium distances d1U–U and d2U–U in URu1
x Pdx Ge as a function of Pd content. The lines are guides to the eye.

though one detects some change around x¼0.8. On the contrary, d2U–U practically does not change with x up to x¼ 0.4 and just after this concentration d2U–U exhibits a faster increase with x.These changes would decisively affect magnetic interactions between uranium magnetic moments, leading to a change in the magnetic properties of the alloys. It is worthwhile to mention that a border discriminating the strong and the weak overlap of the nearest U neighbour 5f-wave functions in uranium-based compounds is assumed to be the phenomenological Hill limit of dHill ¼0.35 nm [14]. Apparently, the shortest uranium–uranium distance in URu1
x Pdx Ge takes value of the Hill limit at compositions around x¼ 0.4. Therefore, it is expected that the direct overlap of the 5f-wave functions becomes weaker in Pd-richer URu1
x Pdx Ge compositions, and a localized magnetism of magnetic moments would develop upon a substitution of Ru by Pd. Furthermore, a certain consequence of the increase in the dU–TM distances is that the wave functions of the 5felectrons of uranium atoms and of spd-electrons of ligands overlap to lesser extent. It results in a decrease in the f-spd hybridization degree, and affects weaker coupling between the 5f- and spd-electrons, J. 3.2. Magnetic susceptibility and magnetization

Fig. 2. Dependence of the lattice parameters a, b, c and unit cell volume V on the Pd content in URu1
x Pdx Ge. The lines are guides to the eye.

In Fig. 4 we show the temperature dependence of the inverse magnetic susceptibility for selected URu1
x Pdx Ge alloys with x r0:9. It is firstly noted that the susceptibility of the studied URuGe sample is more temperature dependent than the previous reported data [7]. In our opinion, the magnetocrystalline effect and the presence of some ferromagnetic impurities in the earlier sample may be the reason for this discrepancy. To analyse the high-temperature susceptibility data of URu1
x Pdx Ge we used a modified Curie–Weiss (mCW) law: χ ðTÞ ¼ χ 0 þ N A μ2eff =½3kB ðT þ Θp Þ, where χ0 is a temperature independent susceptibility, μeff is an effective moment, Θp is a paramagnetic Curie temperature, NA and kB are the Avogadro number and the Boltzmann constant, respectively. The results of the fitting of the data above 100 K are shown as solid lines in Fig. 4. We obtained χ0 of the order of 10
4 emu/mol, which make up mainly the contribution of the Pauli and Van Vleck paramagnetism. The concentration dependencies of μeff and Θp are shown in Fig. 5. It should be noted that due to a significant influence of magnetocrystalline anisotropy on the data of the polycrystalline materials, absolute values of the obtained parameters must be taken with caution. Nonetheless, one can acquire

D. Gralak, V.H. Tran / Journal of Solid State Chemistry 226 (2015) 50–58

Fig. 4. The inverse susceptibility of URu1
x Pdx Ge as a function of temperature. The solid lines are fits to a modified Curie–Weiss law.

53

between Θp and Kondo temperature TK, T K  ð0:251Þ  j Θp j [15,16], the TK is expected to decrease sharply around x  0:3. With further increasing Pd content, Θp is steadily increased and from x¼0.7 Θp takes positive values, hinting at the ferromagnetic interaction is becoming dominant. The dramatic changes in the Θp ðxÞ values across the studied solid solution suggest the serious variation in magnetic interaction types upon varying Pd concentration. Fig. 6(a) displays the low-temperature magnetic susceptibility χ ðTÞ data for URu1
x Pdx Ge ð0:0 r x r 0:32Þ at a low field of 0.1 T. It can be clearly seen, the value of the susceptibility gradually rises with the increasing Pd content. This behaviour points to an increase in the strength of interaction between the magnetic uranium ions. For alloys with 0:35 r x r 0:8 (see Fig. 6(b)), the normalized susceptibility χ ðTÞ=χ ðT N Þ curves exhibit pronounced peak characteristic of an antiferromagnetic ordering. If one assumes the Néel temperature TN to be the position of the maximum of χ ðTÞ, then TN increases with increasing x in this concentration range, and takes a maximum value of 32 K at x¼0.8. Let us focus on the field dependence of the magnetic susceptibility of x¼ 0.32, since this composition is located on the border line between nonmagnetic and magnetic regimes of the URu1
x Pdx Ge solid solution. Fig. 7 displays magnetic susceptibility data ½χ ðTÞ
χ ð0Þ of URu0:68 Pd0:32 Ge measured in several magnetic fields up to 2 T. The independent susceptibility χ ð0Þ ¼ 0:53  10
3 emu=mol was determined previously from the mCW fitting of high-temperature data. The log–log plot of ½χ ðTÞ
χ ð0Þ vs. T clearly indicates that the low-field susceptibility follows a power law of temperature, χ ðTÞ p T
n , impying a non-Fermi liquid behaviour. We found a relation ½χ ðTÞ
χ ð0Þ p T
0:48 for data collected in the temperature range 1.8–20 K and in a field of 0.02 T. This non-Fermi liquid behaviour slowly vanishes as external magnetic field increases and for fields above 0.5 T the susceptibility apparently recovers the Fermi liquid behaviour. The 2 K magnetization curves for selected URu1
x Pdx Ge compositions with x r 0:8 in fields up to 5.5 T are presented in the left

Fig. 5. Concentration dependencies of μeff and Θp of URu1
x Pdx Ge. The lines are guides to the eye. The data for x ¼1.0 are taken from [7].

some valuable information derived from their concentration dependencies. Following the behaviour of μeff ðxÞ, which takes a rather small value of about 1:37 μB inx¼0 and steadily increases with increasing x, and finally attains a relatively large value of 2:9 μB in x¼1.0 [7].A comparison of the concentration dependence of μeff ðxÞ (Fig. 5(a)) to d1U–U ðxÞ and d2U–U ðxÞ (Fig. 3) reveals a consistent relationship between these quantities, and it strongly suggests that the development of a localized magnetism takes place in compositions for x 4 0:35. Usually, the value of Θp is explicit about the strength of magnetic interactions existing in magnetic materials. In the case of uranium intermetallics, Θp reflects not only the character of interactions of U–U exchange but also coupling between localized uranium moments and the spins of conduction electrons. The latter is expressed with a coupling constant J and is designated by both the on-site single-ion Kondo effect and the coupling of the f electrons mediated via conduction electrons, responsible for the intersite RKKY (Ruderman– Kittel-Kasuya–Yosida) interaction. As delineated in Fig. 5(b), large negative values Θp occur in alloys of small Pd content. The negative sign of Θp indicates that the interactions are dominantly antiferromagnetic. Because, the alloys up to x¼ 0.32 do not order magnetically at all (see below), the strongly negative paramagnetic temperatures is indicative for Kondo type interactions. It is surprising that Θp rapidly changes its value in low Pd concentration alloys, i.e., from 
220 K in x¼ 0 up to 
50 K in x E0.3. According to a proportional relation

Fig. 6. (a) dc-susceptibility of URu1
x Pdx Ge with x¼ 0.1–0.32 at a low magnetic field 0.1 T as a function of temperature. (b) The temperature dependence of the normalized susceptibility for URu1
x Pdx Ge with x ¼0.35–0.8. The position of the susceptibility maximum, denoted by arrows, is assumed to be the antiferromagnetic phase transition temperature TN.

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D. Gralak, V.H. Tran / Journal of Solid State Chemistry 226 (2015) 50–58

panel of Fig. 8. The magnetization of x r 0:4 has positive curvature, which is characteristic of paramagnetic materials with ferromagnetic short-range correlations. Contrary to the behaviour of the low Pd-content alloys, the compositions with x ¼0.5–0.8 manifest negative curvatures, thus their antiferromagnetic ground state can be proved. To get more insight into the nature of the compositions in the nonmagnetic–magnetic regime, we evaluate linear and nonlinear susceptibilities from the expression: MðHÞ ¼ χ 1 H þ

χ3 3!

H3 þ

χ5 5!

H5 ;

ð1Þ

where χ1 is the linear susceptibility and χ3, χ5 are nonlinear ones. The latter quantities stand for a measure of the leading nonlinearity of the magnetization. The obtained fitting parameters χ1, χ3, χ5 from fits to Eq. (1) are plotted vs. concentration in the right panel of Fig. 8. With increasing x the linear susceptibility becomes progressively larger. This result is consistent with the interpretation above that interaction between the magnetic uranium ions are being strengthened. An interesting finding is that the nonlinear susceptibilities emerge and show up their extremum at around x¼ 0.3. The negative value of χ3 was previously found in some non-Fermi liquid alloys, e.g., in Nax CoO2 , which is believed to be close to a ferromagnetic quantum critical point [17]. In the case of UBe13 [18], U0:9 Th0:1 Be13 [19], and UCu5
x Pdx [20], the presence of nonlinear susceptibility was taken as an evidence for an electric quadrupole Kondo effect. However, the power-law dependence of the susceptibility presented above for x¼0.32 does not support a quadrupolar Kondo effect scenario, pffiffiffi for which the susceptibility is expected to behave like χ ðTÞ p T [21].

The temperature dependence of the magnetization found in the x¼0.9 sample (Fig. 9(a)) differs dramatically from those of x r0:8. Instead of the antiferromagnetic peak, we observe a broad maximum in the zero-field cooled (ZFC) MZFC(T) curves. As the applied field strength is increased, the position of this maximum shifts down to lower temperatures. By examining the temperature derivative of the ZFC magnetization, dM ZFC ðTÞ=dT, we found two extreme points – maximum and minimum of dM ZFC ðTÞ=dT. We observe that with increasing applied field strength, the position of the maximum decreases, whereas that of the minimum increases. We may ascribe the minimum of dM ZFC ðTÞ=dT to ferromagnetic phase transition temperature TC and we plot the field dependence of TC in Fig. 9(b). The field dependence of the maximum, TM, of dM ZFC ðTÞ=dT can be understood if assuming a high coercivity of the narrow domain walls. From the isotherm measurements (see below), the coercive field is estimated to be about H c  2 T, and it is expected that Hc decreases with increasing temperature in similar manner as TM(H). We have conducted field-cooled (FC) magnetization measurements. The data no longer exhibit maximum but a continue increase in MFC(T) is observed. Below a characteristic temperature TIR there appears difference between the MZFC and the MFC curves. Although such an irreversible effect is found in ferromagnetic systems with a strong magnetocrystalline anisotropy [22], however, based on the field dependence of TIR (see in Fig. 9(b)) and owing to the fact that the x¼0.9 alloy is situated nearby the antiferromagnetic alloys x r0:8, one cannot exclude the coexistence of ferromagnetic and antiferromagnetic components in x¼0.9. Therefore, in the magnetic diagram shown in Fig. 9(b)) we tentatively denote an antiferromagnetic regime established by TIR. The magnetization of URu0:1 Pd0:9 Ge at several selected temperatures and Arrott plots are shown in Fig. 10(a) and (b), respectively. The behaviour of M(H) and M2 vs H/M at temperatures below 30 K indicates a magnetic order in the alloy. Unfortunately, based on the available data the proper magnetic order of the x¼0.9 composition cannot unambiguously be distinguished, and it could be unravelled by more sophisticated techniques like the neutron scattering.

Fig. 7. Temperature dependence of the magnetic susceptibility of URu0:68 Pd0:32 Ge measured in several magnetic fields up to 5 T. The solid line corresponds to a power law T
n with n¼ 0.48.

Fig. 8. Left panel: magnetization of the URu1
x Pdx Ge alloys with x r 0:8 at 2 K in fields up to 5.5 T. The solid lines are fits to Eq. (1). Right panel: the linear and nonlinear susceptibilities of URu0:1 Pd0:9 Ge at 2 K. The dashed lines are guides to the eye.

Fig. 9. (a) Temperature dependence of the magnetization of URu0:1 Pd0:9 Ge measured in several magnetic fields. (b) A magnetic H–T diagram proposed for URu0:1 Pd0:9 Ge.

D. Gralak, V.H. Tran / Journal of Solid State Chemistry 226 (2015) 50–58

Fig. 10. (a) Magnetization of URu0:1 Pd0:9 Ge measured in fields up to 5 T and at selected temperatures between 2–50 K. (b) Arrott plots of URu0:1 Pd0:9 Ge.

3.3. Specific heat The low-temperature specific heat divided by temperature, C p =T, of URuGe and URu0:9 Pd0:1 Ge is plotted in Fig. 11(a) as a function of temperature square. The specific heat data of these alloys follow a standard relation for metals: C p ðTÞ=T ¼ γ þ β T 2 :

ð2Þ

Here, γ and β are the coefficients of the electronic Cel and lattice Cph specific heat, respectively. The fitting of Eq. (2) to the experimental data yields γ ¼ 26.9 mJ/mol K2 and β ¼0.43 mJ/mol K4 for URuGe and γ ¼62.1 mJ/mol K2 and β ¼0.35 mJ/mol K4 for URu0:9 Pd0:1 Ge. Undoubtedly, the Sommerfeld coefficient of the measured alloys is much enhanced as compared to those of an ordinary metal, e.g., to Cu with γ ¼0.694 mJ/mol K2 [23]. According to the relationship γ ¼ ðπ 2 k2B = 3NÞðEF Þð1 þ λe
e þ λe
ph Þ, one would expect that the large γ-value is due to an enhancement in electron–electron coupling, λe–e , and/or a high density of states (DOS) at the Fermi level, NðEF Þ, since it is unlikely that the electron–phonon effect, λe–ph , is extremely large.  1=3 From the relation ΘD ¼ n12π 4 R=5β , where n is the number of atoms per formula unit and R is the gas constant, we derived the lowtemperature Debye temperature ΘD , which amounts to 239 K and 255 K in URuGe and URu0:9 Pd0:1 Ge, respectively. The Debye temperature, denoted as ΘHT D , may be evaluated from high-temperature specific heat data using the Debye function [24]: Z ΘHT D =T x4 expðxÞ HT C ph;D ðTÞ ¼ 9RnD ðT=ΘD Þ3 dx; ð3Þ ½expðxÞ
12 0 where nD is the number of Debye-type vibrators. However, when inspecting the ðC p =T
γ Þ=T 2 vs T plot of URuGe we recognize a maximum at 17 K. The observation invokes consideration of the contribution of optical phonons. The specific heat due to the presence of nE Einstein-type vibrators with a characteristic Einstein temperature ΘE is usually described by the Einstein function [24]: C ph;E ðTÞ ¼ 3RnE

ðΘE =TÞ2 expðΘE =TÞ ½expðΘE =TÞ
12

ð4Þ

55

Fig. 11. (a) Specific heat derived by temperature C p =T of x ¼0 and 0.1 as a function of temperature square. The solid lines are the fits of the data to Eq. (2). (b) The C p =T of x¼ 0.2, 0.3 and 0.32 vs temperature on a logarithmic scale. The solid lines correspond to theoretical fits illustrating spin-fluctuation effect (Eq. (5)) for x ¼0.2 and non-Fermi liquid behaviour (Eq. (6)) for x ¼0.3 and 0.32. (c) Semi-logarithmic plot of the temperature dependence of C p =T of x ¼0.6, 0.8 and 1. The solid lines are fits of the experimental data to Eq. (7).

The experimental data of URuGe between 2 and 300 K (not shown here) can be fitted well with the sum of C ph;D ðTÞ and C ph;E ðTÞ for ΘHT D ¼284 (5) K, nD ¼2.4, ΘE ¼ 89 (3) K, nE ¼0.6. In Fig. 11(b) we show the specific heat divided by temperature C p =T versus temperature on a logarithmic scale for URu1
x Pdx Ge alloys around x¼ 0.3. Clearly, the function in Eq. (2) sole is no longer applicable to the low-temperature specific heat. Below 10 K, the C p =T data show a tendency to rise, indicative of the influence of spin fluctuations. Assuming the formation of narrow band of strongly correlated electrons at the Fermi level, the specific heat takes a form [25]: C p ðTÞ=T ¼ γ þ βT 2 ½1 þ δsf lnðT=T sf Þ;

ð5Þ

where δsf is a parameter, which is related to the Stoner exchangeenhancement coefficient and Tsf is a characteristic spin-fluctuation temperature. In the fitting, we fixed β value to be the same as for URuGe. The best fit for URu0:8 Pd0:2 Ge shown in Fig. 11(b) gives γ ¼120 (1) mJ/mol K2, δsf ¼ 3:4 (0.8) mJ/mol K4 and Tsf ¼ 24 (0.4) K. We have attempted to fit Eq. (5) to the experimental data of x¼ 0.3 and 0.32. However, the discrepancy between the experimental and theoretical data was fairly large. It turns out that the observed upturn in C p =T with decreasing temperature can be well described by the formula: pffiffiffi ð6Þ C p ðTÞ=T ¼ γ þ βT 2
b T ; where the last term accounts for non-Fermi liquid behaviour, and has been found in many intermetallic alloys with magnetic instabilities, e. g., U2Co2Sn [26], ðU0:8 La0:2 Þ2 Zn17 [27], YbCu3:5 Al1:5 [28], U0:07 Th0:93 (Ru, Pt)2Si2 [29]. Least-squarest fits for T o 5 K yield γ ¼173(2) mJ/ mol K2, b¼47(1) mJ /mol K1.5 for x¼0.3 and γ ¼301(2) mJ/mol K2, b¼65(1) mJ /mol K1.5 for x¼0.32. We note that the ptemperature ffiffiffi dependence of the specific heat in the form C p =T 
T has been predicted by Hertz [30], Millis [31], Moriya and Takimoto [32], and Lozanrich [33], for antiferromagnetic spin fluctuations at d¼3,

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D. Gralak, V.H. Tran / Journal of Solid State Chemistry 226 (2015) 50–58

contrary to ferromagnetic spin fluctuations a dependence C p =T 
lnðT=T 0 Þ was anticipated. It may be added that measurement of the field dependence of the specific heat was conducted for x¼0.32 (not shown here). An application of external fields decreases the value of C p =T at 0.4 K, i.e., from 261 mJ/mol K2 in zero field down to 175 mJ/ mol K2 in 9 T. In magnetic fields above 1 T the Fermi-liquid behaviour of the specific heat, i.e., C p ðTÞ ¼ γ T þ β T 2 , is recovered. The temperature dependence of C p =T of selected magnetically ordered alloys URu1
x Pdx Ge is presented in Fig. 11(c). At first glance, the C p =T curve is similar to that of the nonmagnetic x¼ 0 and 0.1 alloys. However, owing to magnetic excitations contributing to the total specific heat, the data cannot be described by Eq. (2). In addition to the electronic and phonon contributions, we must take into account magnetic contribution due to the presence n of magnons, which are described by C mag ¼ kT expð
Δ=kB TÞ [34], where Δ=kB is an energy gap in the magnon spectrum and k is a constant. For the magnetic alloys, we have fitted the experimental data to the following equation: C p ðTÞ=T ¼ γ þ βT 2 þ kT


1þn

expð
Δ=kB TÞ;

ð7Þ

The results of the fits using Eq. (7) are shown as solid lines in Fig. 11(c). The refinements of the exponent n and the energy gap Δ=kB in the spin wave spectrum for the antiferromagnetic alloys (x¼0.6 and 0.8) are provided: n¼1.5 and Δ=kB ¼ 0, while for x¼0.9 (not shown here) and x¼ 1 we obtained n¼ 3 and Δ=kB ¼ 24:3 (0.3) and 33(3) K, respectively. The remaining fitting parameters for x¼0.6, 0.8, 0.9 and 1 are respectively γ ¼134(3), 75(3), 61(2) and 22(1) mJ/mol K2, β ¼0.31 (0.02), 0.88(0.05), 0.32(0.05) and 0.36(0.02) mJ/mol K4 and k¼7.9 (0.7), 7.1(0.5) mJ/mol K5/2, 6.9 (0.7) and 7(2) mJ/mol K4. It is noted that the values of β were found to be comparable with those of URuGe and URu0:9 Pd0:1 Ge, and may suggest that the low-temperature Debye temperatures have a similar magnitude. 3.4. Electrical resistivity In Fig. 12, we show the electrical resistivity ρðTÞ data for selected URu1
x Pdx Ge alloys. Since the samples have many micro-cracks, the reliability of the absolute values of the resistivity is questionable. Therefore, we present ρðTÞ normalized to its room temperature value ρðTÞ=ρRT as a function of temperature. We observe that the ρðTÞ=ρRT curve of x¼0.1 (Fig. 12(a)) is similar to the earlier reported data for URuGe [35]. It is characterized by a knee below 75 K. Below 10 K, the resistivity displays a ρðTÞ p T 2 dependence of the Fermi liquid. With

increasing Pd content up to 0.32, high-temperature resistivity shows up a maximum around 150 K, suggesting the presence of Kondo effect and the development of a coherent state below the resistivity maximum. Remarkably, the low-temperature resistivity no longer follows the T2-dependence. For x¼0.25 and x¼0.3, ρðTÞ can be described by a power law:

ρðTÞ=ρRT ¼ ρ00 þ AT n ;

ð8Þ

where ρ is the normalized residual resistivity and A is a constant. The exponent n is found to take a value of 1.3 at x¼0.25 and 1.5 at x¼0.3, supporting non-Fermi liquid behaviour of these alloys. The resistivity of x¼ 0.32 obeys the power law AT 3=2 fairly well from 3 to 10 K, but inclusion of Kondo-like term
c ln T is required for fitting of the data in a lower temperature range. We can fit the function 0 0

ρðTÞ=ρRT ¼ ρ00 þ AT 3=2
c ln T:

ð9Þ

to the experimental data in the temperature range 0.4–10 K and the solid line in Fig. 12 (a) is the fitting result. The reason for the presence of ln T term in the resistivity may be the contribution of Kondo effect in the sample. In order to shed more light on the coexistence of the Kondo effect and quantum scattering we have measured the resistivity for three principal axes of a single-crystalline x¼0.32 sample. We found T 3=2 -dependence in the resistivity for the current j applied along the c-axis (see inset of Fig. 12(a)), while the
ln T dependence was observed for j //b-axis (not shown here). Thus, the resistivity data of the polycrystalline sample is mixed with resistivities along different axes. The temperature dependence of resistivity of magnetically ordered alloys is seen in Fig. 12(b). For the antiferromagnets, the resistivity shows an upturn below their TN and a tendency to saturate below 10 K. Thus, it is surprising that the resistivity measurements on these samples do not reveal any signal associated with the loss of spindisorder scattering in the ordered phase. There are at least two possibilities for the absence of any drop in ρðTÞ curves. One is due to the magnon gap formation, which usually gives a rise to lowtemperature upturn. The second reason may be a dominating Kondo scattering of conduction electrons on the 5f-electrons. In a simple manner, we have analyzed low-temperature resistivity data using the following formula:

ρðTÞ=ρRT ¼ ρ00
AT n :

ð10Þ

For alloys in the concentrations range x¼0.5–0.8, a satisfactory description of the low-temperature ρðTÞ is obtained with n¼ 0.75(0.05).

Fig. 12. Temperature dependence of the normalized resistivity of selected URu1
x Pdx Ge alloys. The lines are theoretical fits. The inset in the panel (a) shows the resistivity data of a single-crystalline x¼ 0.32 sample with current applied parallel to the c-axis. The residual resistivity ρð0Þ is 195:95μΩ cm.

D. Gralak, V.H. Tran / Journal of Solid State Chemistry 226 (2015) 50–58

Here, we would like to notice that the fulfilment of Eq. (10) has an analytical meaning only, and further work is necessary to elucidate the behaviour of the antiferromagnetic alloys. The important features of the resistivity data of x¼0.9 are (i) the ρðTÞ dependence is very similar to that of UPdGe [35], which exhibits the behaviour of a magnetic Kondo-lattice, (ii) the rapid drop of the resistivity at approximately 30 K goes together with the ferromagnetic phase transition TC from the magnetization measurement and no anomaly at TIR ¼25 K is detected, (iii) the low-temperature resistivity follows the form:

ρðTÞ=ρRT ¼ ρ00 þbT þ AT 2 expð
Δ=kB TÞ;

ð11Þ

with an anisotropy gap Δ=kB ¼ 11 K in magnon dispersion relation. 3.5. Magnetic phase diagram The results of magnetic, specific heat, and electron transport measurements are summarized in Fig. 13. The magnetic phase diagram of URu1
x Pdx Ge (top panel of Fig. 13) consists of several regions. The first region encompasses compositions x o 0:25, where the ground state is nonmagnetic. The physical properties of these alloys are described by the Fermi liquid theory, as evidenced by C el ðTÞ p γ T and ρðTÞ p T 2 . Taking into account the concentration dependencies of χ 1 ðxÞ (right panel of Fig. 8) we can say that the substitution of Pd for Ru gradually increases magnetic correlations. Such a behaviour can be understood assuming that the d-band in the Ru-rich compositions is empty enough and willing to be progressively filled with electrons from the U atoms. This accompanies more and more localization of the 5f-electrons and increasing strength of the f–f interactions. Concerning the concentration dependence of the Sommerfeld ratio at 2 K (bottom panel of Fig. 13), the increase in γ with increasing Pd content surely suggests a development of heavy-fermion state. In the vicinity x¼ 0.3, we found a non-Fermi liquid behaviour, which is characterized by the power-law temperature dependencies of χ ðTÞ, Cp(T) and ρðTÞ. It should be note that the behaviour of the specific heat and electrical resistivity is in good agreement with the theoretical predictions for antiferromagnetic fluctuations nearby quantum critical points [30–33]. The Sommerfeld ratio at 2 K, shown

Fig. 13. Top panel: magnetic phase diagram of URu1
x Pdx Ge. Middle and bottom panel: concentration dependence of the exponent n in the resistivity ATn term and the Sommerfeld ratio of URu1
x Pdx Ge at 2 K, respectively. The lines jointing experimental points are guides to the eye.

57

in the bottom panel of Fig. 13, exhibits an extremum at xcr ¼ 0.32. We believe that a distinctly enhanced γ ð2 KÞ, identifying a large density of states at the Fermi level in xcr is consistent with nFl property of the alloy. Using the values χ and γ of xcr ¼0.32 at 2 K we estimated the 2 Wilson ratio Rw ¼ π 2 kB χ ð0 KÞ=½μ2B μ2eff γ ð0 KÞ to amount to Rw ð2KÞ ¼ 7:7. This result implies further that the thermodynamic properties are under influence of strong spin fluctuations [36]. Unfortunately, the deviation of χ ðTÞ from the Fermi liquid theory cannot unambiguously be explained with models mentioned above for antiferromagnetic spin fluctuations. For the studied system the magnetic susceptibility reveals a temperature exponent 
1=2, which differs from the theoretical value of 3/2. It should be recalled, however, that the power-law dependence of the susceptibility, χ ðTÞ p T n with n 
0:5 has previously been found in some non-Fermi liquid alloys, showing instability due to nearness to antiferromagnetic Th1
x Ux Pt2 Si2 [37], or ferromagnetic Th1
x Ux Cu2 Si2 [38], ordering. The order in the region 0:35 r x r 0:8 is most probably antiferromagnetic. The Néel temperature of alloys in the range xcr – 0.45 is found to follow a relation T N p ðx
xcr Þ2=3 in accordance with the theoretical predictions for antiferromagnetic quantum critical points [30,31]. The Sommerfeld ratio of the antiferromagnetic alloys decreases gradually with increasing x. This behaviour may be referred to an increasing localization of the 5f-electron states. Finally, the x¼ 0.9 and 1.0 alloys exhibit complex magnetic order related with two successive magnetic phase transitions. It is also suggested that the transformation of magnetic structures from antiferromagnetic ðx r 0:8Þ to complex ones at x¼0.9 and 1 implies a significant modification of magnetic interactions between U moments and neutron diffraction experiment will be desirable to determine magnetic structures of the magnetically ordered alloys. Before discussing the evolution of magnetic ground states in URu1
x Pdx Ge, it is worthwhile to compare the behaviour between two related solid solutions: URu1
x Rhx Ge [2,3] and URu1
x Pdx Ge. In both systems, the long-range order appears just at a critical value of valence electrons/f.u., i.e., approximately 8.65 e/f.u. This observation indicates that the number of sd-electrons of the transition metals plays a role in establishing magnetic order. However, the magnetic ground states of the magnetic compositions in these two systems are different, ferromagnetic in URu1
x Rhx Ge but presumably antiferromagnetic in URu1
x Pdx Ge. To explain this difference one evokes to consider another mechanism, besides the total sd-electrons in system. In our opinion the evolution of magnetic ground states in URu1
x Pdx Ge seems to be in line with the increasing 5f-electron localization associated with the increasing the number of sd-electrons in the systems, as well as with a change in competed interactions between the Kondo effect and RKKY exchange. The presence of the latter interaction, which is an oscillating function of the Fermi wave-vector and distance between magnetic ions, may explain why the ferromagnetic or antiferromagnetic state is stable in a given system. Next, let us consider the interplay between the Kondo effect and RKKY exchange. It is well known that the binding energy of Kondo effect is given by kB T K p expð
1=NðEF ÞJÞ, and the strength of RKKY interaction characterized by kB T RKKY p NðEF ÞJ 2 . The coupling constant J is dependent on the hybridization matrix, Vkf and the position of f state Ef with respect to the Fermi level EF, as J p V 2fk =ðEF
Ef Þ [39]. For relatively small Pd concentrations, let say just below x¼0.32, the levels of f electrons with an itinerant character lie nearly at the Fermi level, thus the alloys could have a large Vkf and small EF
Ef . This situation together with the fact that the density of states aims at a high value makes the Kondo behaviour to be dominated at low temperatures. When the Pd concentration increases, the positions of the 5f-electrons shift to far below EF, owing to their higher degree of localization of 5felectrons. This case identifies with a small Vkf and large EF
Ef , providing the RKKY interaction to be dominated. Following this consideration, the position of the nFl liquid alloy x 0.32 just locates between the itinerant electron and local moments regimes.

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4. Summary The physical properties of pseudo-ternary solid solution formed between the nonmagnetic URuGe and F/AF-magnetic compound UPdGe were studied by means of magnetization, specific heat, and electrical resistivity measurements down to 0.4 K and in magnetic fields up to 5.5 T. The substitution of Pd for Ru increases magnetic correlations in URu1
x Pdx Ge, leading to the appearance of different magnetic ground states. Amongst which there are interesting nFl around x¼0.3 and unforeseeably antiferromagnetic with 0:35 rx r 0:8. The increase of the effective moment and the development of magnetic orders reflect the 5f-electrons to be more localized character upon the substitution of Ru by Pd. This observation coincides with increasing U–U distance and change in competed interactions between RKKY and Kondo effects. The compositions with  0.3 exhibit non-Fermi-liquid properties, as one infers from the low-temperature magnetic susceptibility, specific heat and electrical resistivity data. We think that the nFl behaviour in the x¼ 0.32 alloy located at the nonmagnetic-magnetic line makes this alloy to be a good candidate for further research of quantum criticality. Interestingly, in spite of the ferromagnetic ground state of UPdGe, an eventual quantum critical point in the URu1
x Pdx Ge system is presumably of antiferromagnetic character, and different to that in URh1
x Rux Ge, UCo1
x Fex Ge and UCo1
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