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Magnetic transport properties of heavy fermionic EuNi2P2
Solid State Communications 300 (2019) 113665
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Communication
Magnetic transport properties of heavy fermionic EuNi2P2 ∗
T
Hirofumi Wada , Kosuke Tanabe, Ibuki Yamamoto, Akihiro Mitsuda Department of Physics, Kyushu University, Motooka 744, Nishi-ku, Fukuoka, 819-0395, Japan
A R T I C LE I N FO
A B S T R A C T
Communicated by H. Akai
Electrical resistivity, magnetoresistance (MR) and Hall resistivity have been measured for Eu-based heavy fermion compound EuNi2P2 (exact composition is Eu1.2Ni2P2.1). It is found that the 4f electron contribution to the resistivity shows −log T above 200 K. The MR is positive at the lowest temperature and shows a negative minimum at around 60 K. The Hall resistivity shows a broad positive peak at around 120 K. These results are very similar to those of typical heavy fermions, CeCu6 and CeAl3. We discuss the results in terms of the Kondo model for heavy fermions. Measurements under high pressures suggest that the scaling of the Hall effect is valid below 1.5 GPa in EuNi2P2.
Keywords: A. EuNi2P2 D. Heavy fermions D. Kondo effect E. Hall effect E. Magnetoresistance
1. Introduction Valence instabilities of the rare-earth compounds have been extensively studied in the past decades. Among the rare-earth metals, Eu can take two kinds of valence states, the divalent state (4f7, J = 7/2) and the trivalent state (4f6, J = 0). When the energy difference between two valence states is sufficiently small, Eu atoms take the intermediate valence state, in which the 4f state fluctuates between two valence states. It is known that the Eu mean valence strongly depends on temperature in the valence fluctuation regime. In most cases, the Eu atoms fall in the nearly trivalent state, when the temperature approaches zero [1]. EuNi2P2 with the ThCr2Si2-type tetragonal structure is an exceptional case, because its Eu valence is about 2.6 at low temperatures [2]. Mössbauer effect and NMR measurements have revealed no magnetic ordering in this compound down to the lowest temperature [3,4]. In 1995, Fisher et al. reported that the electronic specific heat coefficient of EuNi2P2 is about 100 mJ/K2 mol [5]. Then, EuNi2P2 is recognized as the first heavy fermion system of Eu compounds. In recent years, various studies on the heavy fermionic behavior of EuNi2P2 have been reported. The optical conductivity and the angle-resolved photoemission spectroscopy (ARPES) studies have revealed strong hybridization between 4f and conduction electrons in EuNi2P2 [6–8]. Hiranaka et al. observed a steep decrease of the thermal expansion of EuNi2P2 below about 50 K [9]. From the similarity of thermal expansion curve to that of CeRu2Si2, they proposed that the Kondo effect is responsible for the heavy fermionic behavior of EuNi2P2. A recent theoretical study also pointed out the possibility of the Kondo effect in the 4f7 system [10]. To clarify the emergence of the Kondo effect in EuNi2P2, further
∗
experimental evidences are necessary. In this paper, we report the electrical resistivity, magnetoresistance and the Hall effect of EuNi2P2. The results are compared with those of typical Ce-based heavy fermions. 2. Experimental procedure Single crystals of EuNi2P2 and SrNi2P2 were synthesized by a Sn-flux method. The purities of raw materials are 99% for Sr, 99.9% for Eu and Ni, and 99.999% for P and Sn. The mixtures of Eu or Sr, Ni, P, and Sn with a molar ratio of 1:2:2:20 were loaded into an alumina crucible and the crucible was sealed in a quartz tube under vacuum. The ampoule was heated to 1100 °C and cooled down to 600 °C at a rate of 10 °C/h. At 600 °C, the molten flux was removed in a centrifuge. We obtained large single crystals with typical dimensions of 3 × 3 × 0.3 mm3. Powder Xray diffraction measurements have revealed that the samples have the ThCr2Si2-type structure. The lattice parameters are a = 3.942 Å and c = 9.454 Å for EuNi2P2, and a = 3.953 Å and c = 10.469 Å for SrNi2P2. These parameters are in agreement with the previous data [11,12]. Laue diffraction patterns of EuNi2P2 confirmed that the c-axis is perpendicular to the plate surface. The chemical compositions of the compounds were determined from energy dispersive X-ray spectrometer. The obtained results are Eu: Ni: P = 1.16 : 2.00: 2.09 and Sr: Ni: P = 1.02 : 2.00: 1.94. The composition of the Sr compound is very close to SrNi2P2, while that of the Eu compound slightly deviates from the stoichiometry to the Eu-rich side. Electrical resistivity and Hall effect were measured by a four probe method with an ac resistance bridge. The electrical resistivity of the compounds was measured in the c-plane (the electric current I
Corresponding author. E-mail address: [email protected] (H. Wada).
https://doi.org/10.1016/j.ssc.2019.113665 Received 27 March 2019; Received in revised form 18 June 2019; Accepted 25 June 2019 Available online 03 July 2019 0038-1098/ © 2019 Elsevier Ltd. All rights reserved.
Solid State Communications 300 (2019) 113665
H. Wada, et al.
Fig. 1. Temperature dependence of the electrical resistivity ρ of EuNi2P2 and SrNi2P2 in zero magnetic field. The inset shows the 4f electron contribution to the resistivity ρ4f of EuNi2P2 as a function of temperature on the log T scale.
perpendicular to the c-axis). The magnetoresistance (MR) was measured in a magnetic field B up to 12 T. We measured both the longitudinal MR (B//I) and the transverse MR (B ⊥ I) for EuNi2P2. No significant differences between two MRs were observed. In the Hall effect measurements, the magnetic field is applied parallel to the c-axis. The measurement sequence of the temperature dependence of the Hall effect was described in Ref. [13]. The Hall effect of EuNi2P2 was measured under high pressures up to 1.5 GPa, by using a clamp cell with double cylinders made from NiCrAl (inner) and CuBe (outer). Daphne 7373 oil was employed as a pressuretransmitting medium. The relationship between the actual pressure and the pressure load was obtained beforehand from the measurements of the superconducting transition of Sn.
Fig. 2. (a) Field dependence of the longitudinal MR ratio of EuNi2P2 at various temperatures. (b) Temperature dependence of the transverse MR ratio of EuNi2P2 at B = 12 T.
EuNi2P2 fluctuates between Eu2+ and Eu3+, the divalent compound SrNi2P2 is not a precise reference. Therefore, we should be careful to discuss the deviation of ρ4f at low temperatures.
3. Results and discussion 3.2. Magnetoresistance 3.1. Electrical resistivity The longitudinal MR of EuNi2P2 at various temperatures is depicted in Fig. 2 (a) as a function of magnetic field. The MR ratio is defined as Δρ/ρ(0) = { ρ(B) − ρ(0)}/ρ(0). The field dependence of the MR ratio is positive and large at low temperatures, whereas it is negative and small at high temperatures. Similar results were obtained for the transverse MR. To study the temperature dependence of the MR, we have measured the transverse MR in a magnetic field of 12 T from 4.2 to 230 K. The results are displayed in Fig. 2 (b) as a function of temperature. The Δρ/ρ(0) is about +4.5% at 4.2 K. With increasing temperature, the MR ratio decreases rapidly and it changes the sign to negative at 26 K. As the temperature is further increased, the MR ratio shows a broad minimum at around 60 K, followed by a gradual increase. The positive maximum at the lowest temperature and negative minimum in the Δρ/ ρ(0) – T curve are the characteristics of Ce-based heavy fermions, such as CeAl3 and CeCu6 [15–17]. In the case of Ce compounds, the negative MR at high temperatures is explained by the single-impurity Kondo model. The magnetic field suppresses the Kondo effect, which decreases the resistivity. Kawakami and Okiji accounted for the positive MR at low temperatures based on a periodic Anderson model [18]. In the coherent Kondo state, an energy gap is formed in the electronic structure due to the conduction-4f (c-f) electron hybridization. The Fermi level is situated just below the energy gap in zero field, which makes the compounds metallic. When a magnetic field is applied, the Fermi level for either the spin-up or spin-down band shifts to the energy gap, which increases the resistivity. The c-f hybridization and the formation
Fig. 1 shows the temperature dependence of the electrical resistivity ρ of EuNi2P2 and SrNi2P2 in zero magnetic field. The ρ – T curve of EuNi2P2 is very similar to those reported previously. The resistivity follows the Fermi liquid form, ρ = ρ0 + AT2, at low temperatures with large ρ0 (8.76 μΩ cm) and A (2.55 × 10−2 μΩ cm/K2). The residual resistivity ρ0 of our sample is comparable to or smaller than the previous results [6,14]. The residual resistivity ratio, RRR = ρ(300 K)/ρ(4.2 K), is 9.43. At high temperatures, the resistivity tends to saturate. To estimate the nonmagnetic part of the resistivity, we measured the ρ – T curve of SrNi2P2. It is known that the magnetic susceptibility of EuNi2P2 follows the Curie-Weiss law above 200 K with the effective magnetic moment μeff of 7.4 μB/Eu [9]. This μeff is comparable to the theoretical value of Eu2+, 7.94 μB/Eu. Therefore, the divalent compound, SrNi2P2 is a good candidate for a nonmagnetic reference of EuNi2P2, at least, at high temperatures. As shown in Fig. 1, SrNi2P2 shows normal metallic behavior. The residual resistivity and the RRR are 1.17 μΩ cm and 39.6, respectively. The 4f electron contribution to the resistivity ρ4f of EuNi2P2 is obtained by subtracting the resistivity of SrNi2P2 from that of EuNi2P2, which is illustrated in the inset of Fig. 1 as a function of temperature on the log T scale. The temperature dependence of ρ4f resembles that of typical heavy fermion compounds based on Ce. The ρ4f shows −log T above 200 K, suggesting the emergence of the Kondo effect. As the temperature is lowered, ρ4f deviates from −log T and it shows a maximum at around 100 K. Since the Eu valence state of 2
Solid State Communications 300 (2019) 113665
H. Wada, et al.
a positive peak is observed in the ρH – T curve, and, (iii) the Hall resistivity is weakly dependent on temperature at low temperatures [19]. These characteristics have been discussed in terms of the skew scattering due to the Kondo impurities [19]. At temperatures higher than the Kondo temperature TK, the incoherent skew scattering due to the Kondo impurities is dominant and it develops with decreasing temperature. Near T = TK, the c-f hybridization begins, which induces the coherent Kondo state. Then, the skew scattering diminishes and ρH is reduced, as the temperature is lowered. At T ∼0, the coherent Kondo state is realized, where the ordinary Hall effect of the coherent regime is dominant. Therefore, the Hall resistivity becomes constant, as the temperature approaches zero. The large anomalous Hall effect and the maximum in the ρH – T curve observed in EuNi2P2 can be understood in the same scenario. We found that ρH of the compound has a negative minimum at low temperatures. Similar results were reported for CeCu6 [17]. The minimum is related with the nonlinear field dependence of ρH at 4.2 K, shown in Fig. 3 (a). These results are indicative of the competition of positive and negative contributions to the Hall resistivity at low temperatures. The positive contribution is notable at low magnetic fields, while the negative one is dominant at high fields. The latter originates in the ordinary Hall effect at the coherent Kondo state, as described above. We believe that the former arises from the skew scattering due to the magnetic impurities. As described before, the exact composition of the EuNi2P2 sample deviates from the stoichiometry to the Eu-rich side. The large residual resistivity of EuNi2P2 may be associated with the deviation from the stoichiometry. These results suggest the presence of substantial chemical disorder or lattice defects in the sample. We found that the magnetization curve of EuNi2P2 at 4.2 K is expressed as the sum of the linear term and the Brillouin function like term. The latter is attributable to the magnetic impurities. The amount of the impurities is small but not negligible. The magnetization due to the impurities nearly saturates above 5 T. In the ρH – B curve at 4.2 K, the Hall resistivity increases with increasing field up to 5 T. These results strongly support that the positive contribution to ρH originates from the skew scattering due to the magnetic impurities. Above 5 T, ρH decreases with magnetic field. This is because the positive contribution becomes independent of magnetic field at high fields. On the other hand, the negative contribution is proportional to magnetic field in the whole magnetic field range.
Fig. 3. (a) Field dependence of the Hall resistivity ρH of EuNi2P2 at various temperatures. (b) Temperature dependence of ρH of EuNi2P2 at B = 3 T or 12 T and that of SrNi2P2 at B = 12 T. A dashed line is drawn to guide the eye.
of an energy gap in EuNi2P2 were directly observed by ARPES. Danzenbächer et al. observed an energy gap due to the hybridization between the Ni 3d and Eu 4f states of EuNi2P2 at around 20 K [7]. Anzai et al. reported that the spectral weight of the Ni 3d states is rapidly enhanced with decreasing temperature, suggesting the significant c-f hybridization at low temperatures [8]. Therefore, we believe the above explanations proposed by Kawakami and Okiji are applicable to the positive MR of EuNi2P2.
3.4. Hall effect at high pressures We measured the temperature dependence of the Hall resistivity of EuNi2P2 under high pressures up to 1.5 GPa. The results are depicted in Fig. 4 (a). In the whole pressure range studied, the ρH – T curve shows a negative minimum at low temperatures, a steep increase in the intermediate temperature range, and saturation tendency at high temperatures. With increasing pressure, the curve shifts toward high temperatures. The negative peak of the ρH – T curve at ambient pressure is sharper and deeper than that at high pressures. We do not believe that this is intrinsic. The ρH – T curve at ambient pressure was first measured. Then, the measurements under high pressures were carried out for the same sample in the clamp cell after rewiring. Presumably, the different settings gave rise to slight differences in the absolute values of the ρH – T curves. It is known that some physical properties of heavy fermion compounds are determined by a single energy scale [20,21]. This is based on the assumption that the electronic free energy is expressed as a universal function of the reduced temperature T/T0, where T0 is the characteristic temperature. Good scaling of pressure dependent data of specific heat, magnetic susceptibility, or resistivity has been reported in some heavy fermions. We test single energy scaling for the Hall resistivity data of EuNi2P2. Hiranaka et al. estimated TK ≅ 80 K for EuNi2P2 from the thermal expansion curve [14]. This Kondo temperature is adopted as T0 at ambient pressure. Equality of T0 and TK is recognized for Kondo impurity systems but it is controversial for heavy
3.3. Hall effect at ambient pressure Fig. 3 (a) shows the magnetic field dependence of the Hall resistivity ρH of EuNi2P2 at various temperatures. The Hall resistivity shows a nonlinear field dependence at 4.2 K. It first increases and then decreases above 6 T with increasing field. Above 25 K, ρH varies nearly linearly with magnetic field. The ρH – B curve has a negative slope below 50 K, while it has a positive slope above 75 K. We examined the temperature dependence of the Hall resistivity of EuNi2P2 at B = 3 T or 12 T between 4.2 and 200 K, which is illustrated in Fig. 3 (b). As the temperature is lowered from 200 K, the ρH at a constant field gradually increases and have a broad maximum at around 120 K, followed by a decrease. With further cooling, the Hall resistivity changes the sign from positive to negative, and it shows a minimum at around 30 K. We also measured the ρH – B curves of SrNi2P2 at selected temperatures. The ρH of SrNi2P2 at B = 12 T is shown in Fig. 3 (b) as a function of temperature. In nonmagnetic SrNi2P2, the Hall effect is expressed by the ordinary Hall effect, while both the ordinary and anomalous Hall effects contributes to ρH in EuNi2P2 at high temperatures. These results revealed that the anomalous Hall effect is dominant in EuNi2P2 above 100 K. It is known that the Hall effect of heavy fermions is characterized by the following features: (i) the anomalous Hall effect is very large, (ii) 3
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related to the T2 coefficient of resistivity A, as, A ∝ 1/TK2 [22]. In order to investigate the relation between T0 and A, we measured the electrical resistivity at low temperatures under high pressures. The measurements have been done for a different single crystal, which is taken from the same batch as the sample for Hall effect measurements. Fig. 5 shows plots of A vs. 1/T02 of EuNi2P2. In this figure, A data under high pressures reported by Hiranaka et al. were also plotted [14]. It is known that A of EuNi2P2 is sensitive to pressure below 1 GPa [11,14]. Moreover, A values are strongly dependent on the sample [11,14]. Nevertheless, we obtain a nearly linear relation between A vs. 1/T02. Therefore, T0 can be regarded as the Kondo temperature. Finally, we discuss the Grüneisen parameter. The Grüneisen parameter ΩX is defined as the logarithmic derivative of the temperature TX with respect to the volume V, ΩX = −∂logTX/∂logV. It has been reported that the Grüneisen parameter for TK, ΩK = −∂logTK/∂logV, of heavy fermion compounds is very large. For example, ΩK = 115 for CeCu6 and ΩK = 160 for CeAl3 [21]. From the T0 values of EuNi2P2 under high pressures, we have ∂logT0/∂p = 0.61 GPa−1. Combining the bulk modulus B = 147.9 GPa reported by Medvedev et al. [11], the Grüneisen parameter for T0, Ω0 = −∂logT0/∂logV is calculated to be 90. This value is comparable to ΩK of heavy fermion compounds. These results also represent the similarity between T0 and TK. The above discussion suggests that the scaling of the Hall effect is valid at T/ T0 ≤ 1 below 1.5 GPa in EuNi2P2. Naturally, the Hall effect is not a thermodynamic quantity. In this respect, further experiments, such as the pressure dependences of magnetic susceptibility and specific heat are desired to clarify the validity of a single energy scaling in the present compound. 4. Conclusions Fig. 4. (a) Temperature dependence of the Hall resistivity of EuNi2P2 at B = 12 T under high pressures up to 1.5 GPa. (b) Hall resistivity of EuNi2P2 under high pressures plotted as a function of the reduced temperature, T/T0, where T0 is the characteristic temperature depending on pressure.
Electrical resistivity, magnetoresistance and Hall resistivity have been measured for EuNi2P2. We found the magnetic transport properties of EuNi2P2 are very similar to those of typical heavy fermions, CeCu6 and CeAl3. These results suggest that the Kondo model is applicable to heavy fermionic behavior of EuNi2P2. The origin of the Kondo effect in the 4f 7system has been discussed. Nakamura et al. pointed out that the Kondo effect can emerge in the 4f 7system in the j – j coupling scheme [23]. It is known that the j – j coupling is valid, when the spin-orbit interaction λ is larger than the Coulomb interaction among the f orbitals U. In the actual solids, the L – S coupling is realized, because λ < U. Hotta showed that the Kondo effect can occur, when λ/U is of the order of 0.1, by applying a numerical renormalization group method to the seven-orbital Anderson model [10]. This result should be confirmed by different theoretical models. For a better understanding of the Kondo effect or heavy fermion behavior of Eu compounds, further theoretical and experimental studies are highly required. CRediT authorship contribution statement Hirofumi Wada: Supervision, Project administration, Writing original draft. Kosuke Tanabe: Data curation, Investigation. Ibuki Yamamoto: Data curation, Investigation. Akihiro Mitsuda: Methodology.
Fig. 5. Plots of the T2 coefficient of resistivity A vs. 1/T02 of EuNi2P2. T0 was determined from Fig. 4 (b). The A data reported in ref. 14 are also plotted. A dashed line is drawn to guide the eye.
Acknowledgments We are grateful to Yusuke Goki for his technical help in the early stage of the experiments. This work was partly supported by JSPS KAKENHI Grant Number 16K04932.
fermions [20,21]. By choosing appropriate T0 values at high pressures, we found that most of Hall resistivity data at T/T0 ≤ 1 lie on the same curve, which is displayed in Fig. 4 (b). Apparent deviations of the data at ambient pressure at low temperatures are due to differences in the absolute values caused by the different setting. The T0 values used for scaling are 115, 135 and, 155 K at p = 0.5, 1.0 and 1.5 GPa, respectively. For the heavy fermion compounds, the Kondo temperature is
Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ssc.2019.113665. 4
Solid State Communications 300 (2019) 113665
H. Wada, et al.
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5
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Communication
Magnetic transport properties of heavy fermionic EuNi2P2 ∗
T
Hirofumi Wada , Kosuke Tanabe, Ibuki Yamamoto, Akihiro Mitsuda Department of Physics, Kyushu University, Motooka 744, Nishi-ku, Fukuoka, 819-0395, Japan
A R T I C LE I N FO
A B S T R A C T
Communicated by H. Akai
Electrical resistivity, magnetoresistance (MR) and Hall resistivity have been measured for Eu-based heavy fermion compound EuNi2P2 (exact composition is Eu1.2Ni2P2.1). It is found that the 4f electron contribution to the resistivity shows −log T above 200 K. The MR is positive at the lowest temperature and shows a negative minimum at around 60 K. The Hall resistivity shows a broad positive peak at around 120 K. These results are very similar to those of typical heavy fermions, CeCu6 and CeAl3. We discuss the results in terms of the Kondo model for heavy fermions. Measurements under high pressures suggest that the scaling of the Hall effect is valid below 1.5 GPa in EuNi2P2.
Keywords: A. EuNi2P2 D. Heavy fermions D. Kondo effect E. Hall effect E. Magnetoresistance
1. Introduction Valence instabilities of the rare-earth compounds have been extensively studied in the past decades. Among the rare-earth metals, Eu can take two kinds of valence states, the divalent state (4f7, J = 7/2) and the trivalent state (4f6, J = 0). When the energy difference between two valence states is sufficiently small, Eu atoms take the intermediate valence state, in which the 4f state fluctuates between two valence states. It is known that the Eu mean valence strongly depends on temperature in the valence fluctuation regime. In most cases, the Eu atoms fall in the nearly trivalent state, when the temperature approaches zero [1]. EuNi2P2 with the ThCr2Si2-type tetragonal structure is an exceptional case, because its Eu valence is about 2.6 at low temperatures [2]. Mössbauer effect and NMR measurements have revealed no magnetic ordering in this compound down to the lowest temperature [3,4]. In 1995, Fisher et al. reported that the electronic specific heat coefficient of EuNi2P2 is about 100 mJ/K2 mol [5]. Then, EuNi2P2 is recognized as the first heavy fermion system of Eu compounds. In recent years, various studies on the heavy fermionic behavior of EuNi2P2 have been reported. The optical conductivity and the angle-resolved photoemission spectroscopy (ARPES) studies have revealed strong hybridization between 4f and conduction electrons in EuNi2P2 [6–8]. Hiranaka et al. observed a steep decrease of the thermal expansion of EuNi2P2 below about 50 K [9]. From the similarity of thermal expansion curve to that of CeRu2Si2, they proposed that the Kondo effect is responsible for the heavy fermionic behavior of EuNi2P2. A recent theoretical study also pointed out the possibility of the Kondo effect in the 4f7 system [10]. To clarify the emergence of the Kondo effect in EuNi2P2, further
∗
experimental evidences are necessary. In this paper, we report the electrical resistivity, magnetoresistance and the Hall effect of EuNi2P2. The results are compared with those of typical Ce-based heavy fermions. 2. Experimental procedure Single crystals of EuNi2P2 and SrNi2P2 were synthesized by a Sn-flux method. The purities of raw materials are 99% for Sr, 99.9% for Eu and Ni, and 99.999% for P and Sn. The mixtures of Eu or Sr, Ni, P, and Sn with a molar ratio of 1:2:2:20 were loaded into an alumina crucible and the crucible was sealed in a quartz tube under vacuum. The ampoule was heated to 1100 °C and cooled down to 600 °C at a rate of 10 °C/h. At 600 °C, the molten flux was removed in a centrifuge. We obtained large single crystals with typical dimensions of 3 × 3 × 0.3 mm3. Powder Xray diffraction measurements have revealed that the samples have the ThCr2Si2-type structure. The lattice parameters are a = 3.942 Å and c = 9.454 Å for EuNi2P2, and a = 3.953 Å and c = 10.469 Å for SrNi2P2. These parameters are in agreement with the previous data [11,12]. Laue diffraction patterns of EuNi2P2 confirmed that the c-axis is perpendicular to the plate surface. The chemical compositions of the compounds were determined from energy dispersive X-ray spectrometer. The obtained results are Eu: Ni: P = 1.16 : 2.00: 2.09 and Sr: Ni: P = 1.02 : 2.00: 1.94. The composition of the Sr compound is very close to SrNi2P2, while that of the Eu compound slightly deviates from the stoichiometry to the Eu-rich side. Electrical resistivity and Hall effect were measured by a four probe method with an ac resistance bridge. The electrical resistivity of the compounds was measured in the c-plane (the electric current I
Corresponding author. E-mail address: [email protected] (H. Wada).
https://doi.org/10.1016/j.ssc.2019.113665 Received 27 March 2019; Received in revised form 18 June 2019; Accepted 25 June 2019 Available online 03 July 2019 0038-1098/ © 2019 Elsevier Ltd. All rights reserved.
Solid State Communications 300 (2019) 113665
H. Wada, et al.
Fig. 1. Temperature dependence of the electrical resistivity ρ of EuNi2P2 and SrNi2P2 in zero magnetic field. The inset shows the 4f electron contribution to the resistivity ρ4f of EuNi2P2 as a function of temperature on the log T scale.
perpendicular to the c-axis). The magnetoresistance (MR) was measured in a magnetic field B up to 12 T. We measured both the longitudinal MR (B//I) and the transverse MR (B ⊥ I) for EuNi2P2. No significant differences between two MRs were observed. In the Hall effect measurements, the magnetic field is applied parallel to the c-axis. The measurement sequence of the temperature dependence of the Hall effect was described in Ref. [13]. The Hall effect of EuNi2P2 was measured under high pressures up to 1.5 GPa, by using a clamp cell with double cylinders made from NiCrAl (inner) and CuBe (outer). Daphne 7373 oil was employed as a pressuretransmitting medium. The relationship between the actual pressure and the pressure load was obtained beforehand from the measurements of the superconducting transition of Sn.
Fig. 2. (a) Field dependence of the longitudinal MR ratio of EuNi2P2 at various temperatures. (b) Temperature dependence of the transverse MR ratio of EuNi2P2 at B = 12 T.
EuNi2P2 fluctuates between Eu2+ and Eu3+, the divalent compound SrNi2P2 is not a precise reference. Therefore, we should be careful to discuss the deviation of ρ4f at low temperatures.
3. Results and discussion 3.2. Magnetoresistance 3.1. Electrical resistivity The longitudinal MR of EuNi2P2 at various temperatures is depicted in Fig. 2 (a) as a function of magnetic field. The MR ratio is defined as Δρ/ρ(0) = { ρ(B) − ρ(0)}/ρ(0). The field dependence of the MR ratio is positive and large at low temperatures, whereas it is negative and small at high temperatures. Similar results were obtained for the transverse MR. To study the temperature dependence of the MR, we have measured the transverse MR in a magnetic field of 12 T from 4.2 to 230 K. The results are displayed in Fig. 2 (b) as a function of temperature. The Δρ/ρ(0) is about +4.5% at 4.2 K. With increasing temperature, the MR ratio decreases rapidly and it changes the sign to negative at 26 K. As the temperature is further increased, the MR ratio shows a broad minimum at around 60 K, followed by a gradual increase. The positive maximum at the lowest temperature and negative minimum in the Δρ/ ρ(0) – T curve are the characteristics of Ce-based heavy fermions, such as CeAl3 and CeCu6 [15–17]. In the case of Ce compounds, the negative MR at high temperatures is explained by the single-impurity Kondo model. The magnetic field suppresses the Kondo effect, which decreases the resistivity. Kawakami and Okiji accounted for the positive MR at low temperatures based on a periodic Anderson model [18]. In the coherent Kondo state, an energy gap is formed in the electronic structure due to the conduction-4f (c-f) electron hybridization. The Fermi level is situated just below the energy gap in zero field, which makes the compounds metallic. When a magnetic field is applied, the Fermi level for either the spin-up or spin-down band shifts to the energy gap, which increases the resistivity. The c-f hybridization and the formation
Fig. 1 shows the temperature dependence of the electrical resistivity ρ of EuNi2P2 and SrNi2P2 in zero magnetic field. The ρ – T curve of EuNi2P2 is very similar to those reported previously. The resistivity follows the Fermi liquid form, ρ = ρ0 + AT2, at low temperatures with large ρ0 (8.76 μΩ cm) and A (2.55 × 10−2 μΩ cm/K2). The residual resistivity ρ0 of our sample is comparable to or smaller than the previous results [6,14]. The residual resistivity ratio, RRR = ρ(300 K)/ρ(4.2 K), is 9.43. At high temperatures, the resistivity tends to saturate. To estimate the nonmagnetic part of the resistivity, we measured the ρ – T curve of SrNi2P2. It is known that the magnetic susceptibility of EuNi2P2 follows the Curie-Weiss law above 200 K with the effective magnetic moment μeff of 7.4 μB/Eu [9]. This μeff is comparable to the theoretical value of Eu2+, 7.94 μB/Eu. Therefore, the divalent compound, SrNi2P2 is a good candidate for a nonmagnetic reference of EuNi2P2, at least, at high temperatures. As shown in Fig. 1, SrNi2P2 shows normal metallic behavior. The residual resistivity and the RRR are 1.17 μΩ cm and 39.6, respectively. The 4f electron contribution to the resistivity ρ4f of EuNi2P2 is obtained by subtracting the resistivity of SrNi2P2 from that of EuNi2P2, which is illustrated in the inset of Fig. 1 as a function of temperature on the log T scale. The temperature dependence of ρ4f resembles that of typical heavy fermion compounds based on Ce. The ρ4f shows −log T above 200 K, suggesting the emergence of the Kondo effect. As the temperature is lowered, ρ4f deviates from −log T and it shows a maximum at around 100 K. Since the Eu valence state of 2
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a positive peak is observed in the ρH – T curve, and, (iii) the Hall resistivity is weakly dependent on temperature at low temperatures [19]. These characteristics have been discussed in terms of the skew scattering due to the Kondo impurities [19]. At temperatures higher than the Kondo temperature TK, the incoherent skew scattering due to the Kondo impurities is dominant and it develops with decreasing temperature. Near T = TK, the c-f hybridization begins, which induces the coherent Kondo state. Then, the skew scattering diminishes and ρH is reduced, as the temperature is lowered. At T ∼0, the coherent Kondo state is realized, where the ordinary Hall effect of the coherent regime is dominant. Therefore, the Hall resistivity becomes constant, as the temperature approaches zero. The large anomalous Hall effect and the maximum in the ρH – T curve observed in EuNi2P2 can be understood in the same scenario. We found that ρH of the compound has a negative minimum at low temperatures. Similar results were reported for CeCu6 [17]. The minimum is related with the nonlinear field dependence of ρH at 4.2 K, shown in Fig. 3 (a). These results are indicative of the competition of positive and negative contributions to the Hall resistivity at low temperatures. The positive contribution is notable at low magnetic fields, while the negative one is dominant at high fields. The latter originates in the ordinary Hall effect at the coherent Kondo state, as described above. We believe that the former arises from the skew scattering due to the magnetic impurities. As described before, the exact composition of the EuNi2P2 sample deviates from the stoichiometry to the Eu-rich side. The large residual resistivity of EuNi2P2 may be associated with the deviation from the stoichiometry. These results suggest the presence of substantial chemical disorder or lattice defects in the sample. We found that the magnetization curve of EuNi2P2 at 4.2 K is expressed as the sum of the linear term and the Brillouin function like term. The latter is attributable to the magnetic impurities. The amount of the impurities is small but not negligible. The magnetization due to the impurities nearly saturates above 5 T. In the ρH – B curve at 4.2 K, the Hall resistivity increases with increasing field up to 5 T. These results strongly support that the positive contribution to ρH originates from the skew scattering due to the magnetic impurities. Above 5 T, ρH decreases with magnetic field. This is because the positive contribution becomes independent of magnetic field at high fields. On the other hand, the negative contribution is proportional to magnetic field in the whole magnetic field range.
Fig. 3. (a) Field dependence of the Hall resistivity ρH of EuNi2P2 at various temperatures. (b) Temperature dependence of ρH of EuNi2P2 at B = 3 T or 12 T and that of SrNi2P2 at B = 12 T. A dashed line is drawn to guide the eye.
of an energy gap in EuNi2P2 were directly observed by ARPES. Danzenbächer et al. observed an energy gap due to the hybridization between the Ni 3d and Eu 4f states of EuNi2P2 at around 20 K [7]. Anzai et al. reported that the spectral weight of the Ni 3d states is rapidly enhanced with decreasing temperature, suggesting the significant c-f hybridization at low temperatures [8]. Therefore, we believe the above explanations proposed by Kawakami and Okiji are applicable to the positive MR of EuNi2P2.
3.4. Hall effect at high pressures We measured the temperature dependence of the Hall resistivity of EuNi2P2 under high pressures up to 1.5 GPa. The results are depicted in Fig. 4 (a). In the whole pressure range studied, the ρH – T curve shows a negative minimum at low temperatures, a steep increase in the intermediate temperature range, and saturation tendency at high temperatures. With increasing pressure, the curve shifts toward high temperatures. The negative peak of the ρH – T curve at ambient pressure is sharper and deeper than that at high pressures. We do not believe that this is intrinsic. The ρH – T curve at ambient pressure was first measured. Then, the measurements under high pressures were carried out for the same sample in the clamp cell after rewiring. Presumably, the different settings gave rise to slight differences in the absolute values of the ρH – T curves. It is known that some physical properties of heavy fermion compounds are determined by a single energy scale [20,21]. This is based on the assumption that the electronic free energy is expressed as a universal function of the reduced temperature T/T0, where T0 is the characteristic temperature. Good scaling of pressure dependent data of specific heat, magnetic susceptibility, or resistivity has been reported in some heavy fermions. We test single energy scaling for the Hall resistivity data of EuNi2P2. Hiranaka et al. estimated TK ≅ 80 K for EuNi2P2 from the thermal expansion curve [14]. This Kondo temperature is adopted as T0 at ambient pressure. Equality of T0 and TK is recognized for Kondo impurity systems but it is controversial for heavy
3.3. Hall effect at ambient pressure Fig. 3 (a) shows the magnetic field dependence of the Hall resistivity ρH of EuNi2P2 at various temperatures. The Hall resistivity shows a nonlinear field dependence at 4.2 K. It first increases and then decreases above 6 T with increasing field. Above 25 K, ρH varies nearly linearly with magnetic field. The ρH – B curve has a negative slope below 50 K, while it has a positive slope above 75 K. We examined the temperature dependence of the Hall resistivity of EuNi2P2 at B = 3 T or 12 T between 4.2 and 200 K, which is illustrated in Fig. 3 (b). As the temperature is lowered from 200 K, the ρH at a constant field gradually increases and have a broad maximum at around 120 K, followed by a decrease. With further cooling, the Hall resistivity changes the sign from positive to negative, and it shows a minimum at around 30 K. We also measured the ρH – B curves of SrNi2P2 at selected temperatures. The ρH of SrNi2P2 at B = 12 T is shown in Fig. 3 (b) as a function of temperature. In nonmagnetic SrNi2P2, the Hall effect is expressed by the ordinary Hall effect, while both the ordinary and anomalous Hall effects contributes to ρH in EuNi2P2 at high temperatures. These results revealed that the anomalous Hall effect is dominant in EuNi2P2 above 100 K. It is known that the Hall effect of heavy fermions is characterized by the following features: (i) the anomalous Hall effect is very large, (ii) 3
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related to the T2 coefficient of resistivity A, as, A ∝ 1/TK2 [22]. In order to investigate the relation between T0 and A, we measured the electrical resistivity at low temperatures under high pressures. The measurements have been done for a different single crystal, which is taken from the same batch as the sample for Hall effect measurements. Fig. 5 shows plots of A vs. 1/T02 of EuNi2P2. In this figure, A data under high pressures reported by Hiranaka et al. were also plotted [14]. It is known that A of EuNi2P2 is sensitive to pressure below 1 GPa [11,14]. Moreover, A values are strongly dependent on the sample [11,14]. Nevertheless, we obtain a nearly linear relation between A vs. 1/T02. Therefore, T0 can be regarded as the Kondo temperature. Finally, we discuss the Grüneisen parameter. The Grüneisen parameter ΩX is defined as the logarithmic derivative of the temperature TX with respect to the volume V, ΩX = −∂logTX/∂logV. It has been reported that the Grüneisen parameter for TK, ΩK = −∂logTK/∂logV, of heavy fermion compounds is very large. For example, ΩK = 115 for CeCu6 and ΩK = 160 for CeAl3 [21]. From the T0 values of EuNi2P2 under high pressures, we have ∂logT0/∂p = 0.61 GPa−1. Combining the bulk modulus B = 147.9 GPa reported by Medvedev et al. [11], the Grüneisen parameter for T0, Ω0 = −∂logT0/∂logV is calculated to be 90. This value is comparable to ΩK of heavy fermion compounds. These results also represent the similarity between T0 and TK. The above discussion suggests that the scaling of the Hall effect is valid at T/ T0 ≤ 1 below 1.5 GPa in EuNi2P2. Naturally, the Hall effect is not a thermodynamic quantity. In this respect, further experiments, such as the pressure dependences of magnetic susceptibility and specific heat are desired to clarify the validity of a single energy scaling in the present compound. 4. Conclusions Fig. 4. (a) Temperature dependence of the Hall resistivity of EuNi2P2 at B = 12 T under high pressures up to 1.5 GPa. (b) Hall resistivity of EuNi2P2 under high pressures plotted as a function of the reduced temperature, T/T0, where T0 is the characteristic temperature depending on pressure.
Electrical resistivity, magnetoresistance and Hall resistivity have been measured for EuNi2P2. We found the magnetic transport properties of EuNi2P2 are very similar to those of typical heavy fermions, CeCu6 and CeAl3. These results suggest that the Kondo model is applicable to heavy fermionic behavior of EuNi2P2. The origin of the Kondo effect in the 4f 7system has been discussed. Nakamura et al. pointed out that the Kondo effect can emerge in the 4f 7system in the j – j coupling scheme [23]. It is known that the j – j coupling is valid, when the spin-orbit interaction λ is larger than the Coulomb interaction among the f orbitals U. In the actual solids, the L – S coupling is realized, because λ < U. Hotta showed that the Kondo effect can occur, when λ/U is of the order of 0.1, by applying a numerical renormalization group method to the seven-orbital Anderson model [10]. This result should be confirmed by different theoretical models. For a better understanding of the Kondo effect or heavy fermion behavior of Eu compounds, further theoretical and experimental studies are highly required. CRediT authorship contribution statement Hirofumi Wada: Supervision, Project administration, Writing original draft. Kosuke Tanabe: Data curation, Investigation. Ibuki Yamamoto: Data curation, Investigation. Akihiro Mitsuda: Methodology.
Fig. 5. Plots of the T2 coefficient of resistivity A vs. 1/T02 of EuNi2P2. T0 was determined from Fig. 4 (b). The A data reported in ref. 14 are also plotted. A dashed line is drawn to guide the eye.
Acknowledgments We are grateful to Yusuke Goki for his technical help in the early stage of the experiments. This work was partly supported by JSPS KAKENHI Grant Number 16K04932.
fermions [20,21]. By choosing appropriate T0 values at high pressures, we found that most of Hall resistivity data at T/T0 ≤ 1 lie on the same curve, which is displayed in Fig. 4 (b). Apparent deviations of the data at ambient pressure at low temperatures are due to differences in the absolute values caused by the different setting. The T0 values used for scaling are 115, 135 and, 155 K at p = 0.5, 1.0 and 1.5 GPa, respectively. For the heavy fermion compounds, the Kondo temperature is
Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ssc.2019.113665. 4
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