Ising-type Magnetic Anisotropy in CePd_2As_2
M. O. Ajeesh, T. Shang, W. B. Jiang, W. Xie, R. D. dos Reis, M. Smidman, C. Geibel, H. Q. Yuan, M. Nicklas
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Ising-type Magnetic Anisotropy in CePd As M. O. Ajeesh , T. Shang , W. B. Jiang , W. Xie , R. D. dos Reis , M. Smidman ,C. Geibel , H. Q. Yuan , and M. Nicklas Max Planck Institute for Chemical Physics of Solids, N ¨othnitzer Str. 40, 01187 Dresden, Germany Center for Correlated Matter and Department of Physics, Zhejiang University, Hangzhou 310058, China * [email protected] ABSTRACT
We investigated the anisotropic magnetic properties of CePd As by magnetic, thermal and electrical transport studies. X-ray diffraction confirmed the tetragonal ThCr Si -type structure and the high-quality of the single crystals. Magnetisation andmagnetic susceptibility data taken along the different crystallographic directions evidence a huge crystalline electric field (CEF)induced Ising-type magneto-crystalline anisotropy with a large c -axis moment and a small in-plane moment at low temperature.A detailed CEF analysis based on the magnetic susceptibility data indicates an almost pure |± / i CEF ground-state doubletwith the dominantly |± / i and the |± / i doublets at 290 K and 330 K, respectively. At low temperature, we observe auniaxial antiferromagnetic (AFM) transition at T N = . K with the crystallographic c -direction being the magnetic easy-axis.The magnetic entropy gain up to T N reaches almost R ln2 indicating localised f -electron magnetism without significant Kondo-type interactions. Below T N , the application of a magnetic field along the c -axis induces a metamagnetic transition from theAFM to a field-polarised phase at µ H c = . T, exhibiting a text-book example of a spin-flip transition as anticipated for anIsing-type AFM.
Introduction
Materials crystallising in the ThCr Si -type structure comprise of such prominent compounds as the first heavy-fermionsuperconductor CeCu Si and BaFe As , a parent compound to the iron-based high-temperature superconductors , makingthem especially attractive for solid-state research in the past decades. The discovery of heavy-fermion superconductivity inCeCu Si resulted in extensive studies which became crucial for the understanding of unconventional superconductivity. InCe-based heavy-fermion systems the strength of the hybridisation between the Ce-4 f electrons and the conduction electronsis particularly important for the physical behaviour at low temperatures. There, the competition between Kondo effect andRuderman-Kittel-Kasuya-Yoshida (RKKY) interaction along with crystalline electric field (CEF) effects lead to a large varietyof different ground-state properties, which might be tuned using external control parameters such as chemical substitution,magnetic field and hydrostatic pressure .A large number of the Ce-based compounds crystallising in the ThCr Si -type structure order antiferromagnetically atlow temperatures. Their magnetism is commonly determined by a large magneto-crystalline anisotropy. This leads to thepresence of distinct field-induced metamagnetic transitions . Depending on the strength of the magnetic anisotropy, thenature of the metamagnetic transition(s) may differ. Additionally, the spin structure in the antiferromagnetic (AFM) phaseplays an important role in the field-induced metamagnetic transition(s). In this regard, CePd As offers the opportunity tostudy magnetism in a localised moment antiferromagnet with a huge magneto-crystalline anisotropy.Recently, the physical properties of polycrystalline CePd As , which crystallises in the ThCr Si -type structure, werereported . CePd As undergoes an antiferromagnetic (AFM) ordering at T N ≈
15 K and shows evidence of a metamagnetictransition. However, a detailed investigation on single crystalline samples is necessary in order to understand the magneticproperties. In this work, we report on the magnetic anisotropy of single crystalline CePd As . To this end, we carried outmagnetic susceptibility, magnetisation, electrical-transport and specific-heat measurements. Our results reveal an Ising-typemagnetic anisotropy which accounts for a text-book-like spin-flip metamagnetic transition. The CEF level scheme could befully resolved based on our experimental data. Furthermore, our analysis suggests a simple, collinear A-type antiferromagneticspin structure in the AFM state.
50 100 150 2000.00.20.40.6 ( e m u O e - m o l - ) CePd As H = 0.15 T
T (K) - ( e m u - O e m o l ) T (K)
Figure 1.
Temperature dependence of the magnetic susceptibility χ = M / H for magnetic fields applied parallel ( χ k ) andperpendicular ( χ ⊥ ) to the c -axis. Here, M is the magnetisation and H is the magnetic field. The inset displays the inversemagnetic susceptibility as a function of temperature. The solid lines represent fits of a Curie-Weiss law, χ ( T ) = C / ( T − θ W ) ,to χ k ( T ) and χ ⊥ ( T ) in the temperature interval 300 K ≤ T ≤
400 K and 350 K ≤ T ≤
400 K, respectively. C m ag / T ( J m o l - K - ) T (K)
CePd As S m ag ( R l n2 ) H = 0
Figure 2.
Temperature dependence of the magnetic contribution to the specific heat C mag of CePd As , plotted as C mag ( T ) / T (left axis). C mag was estimated by subtracting the specific heat of the non-magnetic reference compoundLaPd As from that of CePd As . The calculated magnetic entropy S mag ( T ) is displayed in the unit of R ln 2 (right axis). Results
Magnetic susceptibility and heat capacity
The temperature dependence of magnetic susceptibility χ of CePd As with magnetic fields applied parallel ( χ k ) and perpen-dicular ( χ ⊥ ) to the crystallographic c -axis are depicted in Fig. 1. χ ( T ) shows a sharp peak at T N = . . Remarkably, χ k is two orders of magnitude larger than χ ⊥ implying the presence of a strong magnetic anisotropy.The inverse magnetic susceptibility, χ − k ( T ) and χ − ⊥ ( T ) are plotted in the inset of Fig. 1. Above room temperature, thesusceptibility data can be fit by a Curie-Weiss law, χ ( T ) = C / ( T − θ W ) , where C and θ W are the Curie constant and the Weisstemperature, respectively. We find θ k W = K and µ eff = . µ B for H k c and θ ⊥ W = − K and µ eff = . µ B for H ⊥ c .The obtained effective moments are slightly enhanced compared with the calculated value of 2 . µ B for a free Ce + ion. Thedeviation of χ k , ⊥ ( T ) from a Curie-Weiss law below room temperature can be attributed to CEF effects.For a free Ce + ion with total angular momentum J = /
2, the ground-state consists of 6 − fold degenerate levels. Inthe presence of a CEF with a tetragonal symmetry, these degenerate levels split into three doublets which are energeticallyseparated from each other. The physical properties of CePd As are greatly influenced by the relative thermal population of
10 20 30 40 500.000.010.020.030.040.05 M ( B / C e ) T (K)
CePd As H c H (T) (b)
T (K) M ( B / C e ) CePd As H || c H (T) (a)
Figure 3.
Temperature dependence of the magnetisation M ( T ) measured under various magnetic fields applied (a) paralleland (b) perpendicular to the crystallographic c -axis.these energy levels. In CePd As , the magnetic contribution to the entropy, estimated from the specific heat data, reaches ∼
85% of R ln 2 at T N (see Fig. 2). This indicates that the ground-state is a doublet well-separated from the excited CEF levelsand that the Kondo effect is rather weak. Further evidence for the localised character of the Ce moments comes from themagnetisation data discussed below. -6 -4 -2 0 2 4 6-2-1012 H c M ( B / C e ) H (T) H (T)
2 15 10 20 12 30 14
CePd As H || c M ( B / C e ) T (K) -6 -3 0 3 6-0.050.000.05 T (K) 2 20
Figure 4.
Isothermal magnetisation M ( H ) measured at different temperatures for magnetic fields applied along the c -axisand perpendicular to the c -axis (inset). Magnetisation
Figure 3 presents the temperature dependence of the magnetisation measured under various magnetic fields applied paralleland perpendicular to the crystallographic c -axis. For H k c , T N shifts to lower temperatures upon increasing the magnetic field,which is expected for an antiferromagnet. As the magnetic field approaches 1 T, the peak in M ( T ) corresponding to the AFMtransition disappears and a broad step-like feature with a saturation of M ( T ) toward low temperatures develops. However,for H ⊥ c the position of peak corresponding to T N is independent of the magnetic field and still clearly visible at 7 T. Thesedifferent behaviours reflect the large magnetic anisotropy present in CePd As . We note that, the sudden decrease in themagnetisation below T N for H k c compared to that of H ⊥ c suggests that the crystallographic c -direction is the magneticeasy-axis.The isothermal magnetisation M ( H ) at 2 K, shown in Fig. 4, displays a sudden jump at µ H c ≈ H k c followed by T (K) / K ( c m ) CePd As H || cT (K)
0 3 0.5 5 1 7 1.5 9 H (T) H = 0
Figure 5.
Temperature dependence of electrical resistivity ρ ( T ) of CePd As measured at various magnetic fields appliedalong the c -axis. Inset: normalised resistivity ρ / ρ K as a function of T .an immediate saturation. The observed saturation moment of 2 . µ B / Ce is in reasonable agreement with the theoretical valueof g J J = . µ B (where g J = /
7) expected for a free Ce + ion. The sudden jump in magnetisation to the saturation valueis a typical signature of a spin-flip metamagnetic transition. In the spin-flip process, the spins in the AFM sublattice, whichare antiparallel to the field direction, are flipped at H c . Hence, the antiferromagnetism changes to a field-polarised phase in asudden, single step. The sharp nature of the jump in magnetisation with a small hysteresis point to a first-order type transition.At higher temperatures, the metamagnetic transition in M ( H ) broadens and saturates at much higher fields. In the case of H ⊥ c (inset of Fig. 4), the magnetisation increases monotonously and reaches at 7 T only 2 .
5% of the saturation value for H k c . Furthermore, magnetisation measurements in pulsed fields up to 60 T show a linear increase without any tendencyto saturation (not shown). This suggests the absence of any metamagnetic transition for H ⊥ c , which stipulates the hugemagnetic anisotropy in CePd As . Electrical transport
The electrical resistivity ρ ( T ) of CePd As upon cooling displays a metallic behaviour with a broad curvature at intermediatetemperatures, before showing a pronounced kink at about 15 K indicating the AFM transition (inset of Fig. 5). The broadcurvature in the resistivity may be due either to interband scattering or to weak additional spin scattering originating fromthermal population of excited CEF levels. At low temperatures, the AFM ordering leads to a sudden decrease in ρ ( T ) due tothe loss of spin-disorder scattering contribution below T N . Figure 5 shows the ρ ( T ) data recorded at different magnetic fieldsapplied along the c -axis. Upon increasing the field up to 1 T, the kink indicating T N shifts to lower temperatures and becomeswashed out, in good agreement with the results from the magnetic susceptibility. Moreover, above 1 T the residual resistivityshows a sudden reduction which coincides with the metamagnetic critical field.The field and angular dependencies of the resistivity, plotted in Fig. 6(a-c), give further insights into the nature of themetamagnetic transition. At low temperatures, upon increasing the magnetic field ρ ( H ) suddenly drops at the onset of themetamagnetic transition at the critical field µ H c ≈ ρ ( H ) broadens. Asmall increase in ρ ( H ) is observed just below H c in the AFM phase for temperatures close to T N . This could be due to anincreased scattering during the spin-flip process associated with the transition from the AFM to field-polarised state . Abovethe AFM transition temperature, ρ ( H ) displays a gradual decrease upon increasing magnetic field, suggesting a crossoverfrom the paramagnetic to the field-polarised phase. The variation of ρ as function of the angle ( θ ) between the magneticfield ( µ H = c -axis at different temperatures is shown in Fig. 6(b). The step-like behaviourat lower temperatures changes to a gradual decrease in resistivity above T N , where the system undergoes a crossover fromparamagnetic to the field-polarised phase. Above 30 K, the resistivity becomes independent of the field orientation. Finally,Fig. 6(c) presents the resistivity as a function of field for different angles θ . The metamagnetic critical field H c increases uponincreasing θ and diverges for θ → ◦ . No drop in ρ ( H ) is observed up to 14 T for field perpendicular to the c -axis. This isconsistent with our magnetisation experiments. / C e ) m B cH q Figure 6. (a) Magnetic field dependence of ρ for field parallel to c -axis. (b) ρ as a function of the angle ( θ ) between themagnetic field H and c -axis at different temperatures. (c) Magnetic field dependence of ρ for different angles θ at T = H c as function of θ at T = H c = H c / cos ( θ ) . The inset illustrates the magnetic structures in the A-type antiferromagnetic and in thefield-polarised phase. Discussion
In CePd As , the temperature dependence of the magnetic susceptibility below room temperature strongly deviates from aCurie-Weiss behaviour. This can be attributed to CEF effects. In order to establish the CEF scheme and learn more about themagnetic anisotropy in CePd As , we performed a detailed CEF analysis based on our magnetic susceptibility data. For a Ceatom in a tetragonal site symmetry, the CEF Hamiltonian can be written as, H CEF = B O + B O + B O , (1)where B nm and O nm are the CEF parameters and the Stevens operators, respectively . The magnetic susceptibility includingthe Van Vleck contribution is calculated as, χ CEF , i = N A ( g J µ B ) Z ∑ m = n | h m | J i | n i | − e − β ( E n − E m ) E n − E m e − β E n + ∑ n | h n | J i | n i | β e − β E n ! , (2)where Z = ∑ n e − β E n , β = / k B T and i = x , y , z . The inverse magnetic susceptibility including the molecular field contribution λ i is calculated as χ − i = χ − , i − λ i . χ − i ( T ) is fitted simultaneously to the experimental data for both field orientations (seeFig. 7). The data in the paramagnetic phase are well reproduced by the CEF model with a doublet ground-state | Γ ( ) i = . |± / i + . |∓ / i and the excited doublet states | Γ ( ) i = . |± / i − . |∓ / i and | Γ i = |± / i at 290 K
100 200 300 4000200400600
CePd As H = 0.15 T - ( e m u - O e m o l ) T (K)
330 K 290 K
Figure 7.
Temperature dependence of inverse magnetic susceptibility of CePd As . The solid lines are based on CEFcalculation. A schematic representation of the CEF levels is shown in the inset.and 330 K, respectively. An illustration of the CEF level scheme is shown in the inset of Fig. 7. The crystal field parametersextracted from the model are B = − .
66 K, B = − .
22 K and B = .
67 K, with a molecular field contribution λ c = − − mol along the c -axis. It is clear from our CEF analysis that the ground-state is an almost pure |± / i CEF doubletwhich is well-separated from the excited doublets. The saturation magnetisation along the c -axis for the obtained CEF ground-state is 2.06 µ B / Ce, which is in good agreement with the experimental saturation magnetisation of 2 . µ B / Ce. Furthermore,the CEF parameter B is directly related to the paramagnetic Curie-Weiss temperatures θ ⊥ W and θ k W , along both principalcrystallographic directions, as θ ⊥ CW − θ k CW = B ( J − )( J + ) . Using the experimental values of θ ⊥ W and θ k W , weobtain B = − . K in good agreement to B = − .
66 K from the CEF-model fit.Deeper insights into the magnetic structure of the ordered phases in CePd As can be obtained from the magnetisationand electrical resistivity data measured at different orientations of the magnetic field. The magnetisation data suggest thatthe crystallographic c -direction is the easy-axis of the magnetisation. Moreover, the small magnetisation in the ab -planecompared with the large magnetisation along the c -axis indicates an AFM structure with the spins pointing along the c -axis.In addition, the spins are locked along the c -axis by the magneto-crystalline anisotropy, as indicated by the absence of ametamagnetic transition for magnetic field up to 60 T applied perpendicular to the c -axis. These observations confirm thatthe moments in CePd As are Ising-type. The Ising-nature of the spins is also supported by the angular dependence of themetamagnetic critical field extracted from the electrical resistivity data shown in Fig. 6(c). The resulting angular dependenceof H c is displayed in Fig. 6(d). H c ( θ ) increases sharply for θ → ◦ and H c is not detected for field oriented perpendicular tothe c -axis.In order to understand the angular dependence of H c ( θ ) , we fit the data by the equation, H c = H c cos ( θ ) (3)where H c is the critical field for field parallel to the c -axis. Equation 3 describes the experimental data very well with µ H c = .
95 T. In other words, the metamagnetic transition occurs only when the component of magnetic field along the c -axis reaches the value of H c . Based on these results, we can conclude the following scenario for the spin structure ofCePd As : in the AFM phase, the Ce moments are aligned along the c -axis and are locked along this axis by the magneto-crystalline anisotropy. When the component of external magnetic field along the c -axis exceeds H c , the anti-parallel spinsundergo a spin-flip transition to the field-polarised ferromagnetically ordered phase.The single, sharp jump in the magnetisation with a weak hysteresis at the first-order metamagnetic transition from theAFM to the ferromagnetically polarised state points at a simple spin structure of the AFM phase. The small value of thecritical field H c , compared to the value of T N , indicates that, in terms of a Heisenberg model with a few different inter-siteexchange interactions, the AFM ones are much weaker than the ferromagnetic (FM) ones. Because of the topology of thetetragonal body centered Ce sublattice, a strong FM interaction between atoms in adjacent layers competing with a weakin-plane AFM interaction would always result in a FM ground-state. In contrast, a strong FM in-plane interaction with a weakAFM inter-plane interaction can easily account for all observations. In addition, we note that isovalent substitution of P for Asresults in a FM ground-state . Thus all these properties provide strong indication that the AFM structure of CePd As is just .0 0.5 1.0 1.5 2.0 2.50510152025 H || c nd order 1 st order crossover CePd As PM AFM T ( K ) H (T)
Field-polarized phase T N M H = 1.68 T C p ( J m o l - K - ) T (K) C p ( J m o l - K - ) T (K) H = 0.84 T 13.4 13.6 13.8 14.00.00.20.40.6 C p ( kJ m o l - K - ) H = 1 T
T (K)
Figure 8.
Magnetic phase diagram of CePd As for H k c summarising the results from magnetisation and electricalresistivity data. The lines are guides to the eyes. The three insets present the temperature dependence of the specific heat atthree representative magnetic fields as indicated by the vertical lines. The red and blue arrows in the insets indicate thedirection of the temperature sweep.a simple AFM stacking of FM layers. Substituting P for As is just turning the inter-plane exchange from weakly AFM to FM.Therefore, we propose a magnetic structure with weakly antiferromagnetically coupled FM layers of Ising-spins in the AFMstate of CePd As , as illustrated in the inset of Fig. 6(d). A mean-field approximation based on a two-sublattice model canappropriately describe such a spin system. According to this model, the spin-flip occurs when the applied magnetic field isable to overcome the inter-layer AFM coupling. Therefore, the metamagnetic critical field can be expressed as H c = λ AFM M ,where λ AFM is the inter-sublattice molecular field constant and M is the magnetisation of the ferromagnetic state . Similarly,the intra-sublattice molecular field constant λ FM can be extracted from the relation T N = C ( λ FM − λ AFM ) , where C is theCurie constant C = N A µ g J J ( J + ) µ B / k B . By using the experimentally obtained values H c , M S and T N , the inter-layerAFM exchange strength ( z AFM J AFM ) and intra-layer FM exchange strength ( z FM J FM ) are calculated as − .
25 K and 9.83 K,respectively. Here, z AFM and z FM are the number of nearest-neighbour spins participating in the respective interactions. Thelarge intra-layer FM exchange strength is consistent with the experimental observations and plays a crucial role in the first-order nature of the metamagnetic transition.The T − H phase diagram of CePd As for H k c , presented in Fig. 8, summarises our results. At low temperatures,application of a magnetic field induces a metamagnetic transition at µ H c = .
95 T resulting in a field-polarised phase.Above T N , CePd As shows a crossover behaviour from the paramagnetic to the field-polarised phase, reflected by the broadfeatures in magnetisation and electrical resistivity. Additional information on the nature of the transitions between the variousphases can be obtained from specific heat data. The temperature dependencies of the specific heat C p of CePd As for threerepresentative magnetic fields are plotted in insets of Fig. 8. A cusp in C p ( T ) indicates the transition form the paramagneticto AFM phase at 0.84 T. The second order nature of this transition is evidenced by the absence of any thermal hysteresis inthe data. In contrast, a strong thermal hysteresis and a spike-like anomaly in C p ( T ) at 1 T confirms the first-order nature ofthe metamagnetic transition from the AFM to the field-polarised phase. Finally, the crossover from the paramagnetic to thefield-polarised phase at higher magnetic fields is reflected by a broad, hump-like feature in C p ( T ) .Because of its strong Ising anisotropy and simple magnetic behaviour, CePd As is a nice example to illustrate a misin-terpretation frequently encountered in the analysis and discussion of magnetic properties of Ce- and Yb-based compounds.The Weiss temperatures determined from Curie-Weiss fits to the high temperature part of the susceptibility are frequentlyargued to reflect the anisotropy, the sign and the magnitude of the exchange interactions. In CePd As this would lead to he conclusion that the exchange in the basal plane is strongly antiferromagnetic while the exchange along the c -direction isweaker and ferromagnetic. Our analysis clearly demonstrates that this conclusion would be completely wrong, because theWeiss temperatures determined from Curie-Weiss fits at high temperatures are dominated by the effect of the CEF. Except forspecial cases, CEF generally result in a seemingly AFM, negative θ W for the direction of the small CEF ground-state momentand an apparently FM, positive θ W for the direction of the large CEF ground-state moment. Summary
We have investigated the magnetic properties and the CEF scheme of CePd As by detailed temperature, magnetic fieldand angular dependent magnetic, thermodynamic and electrical transport studies on single crystalline samples. The detailedCEF analysis based on the magnetic-susceptibility data indicates an almost pure |± / i CEF ground-state doublet with thedominantly |± / i and the |± / i doublets at 290 K and 330 K, respectively. CePd As orders antiferromagnetically in asimple A-type order below T N = . c -axis. An external magnetic field applied along the c -axis induces a metamagnetic spin-flip transition at µ H c = .
95 Tleading to a ferromagnetic spin alignment. No metamagnetic transition is observed for a magnetic field perpendicular to the c -axis, proving the huge Ising-like anisotropy in CePd As . Methods
Single crystals of CePd As were synthesised by a self-flux method. Initially, polycrystalline CePd As was obtained by asolid-state reaction as reported previously . Then the polycrystalline pellet was loaded into an alumina crucible and sealedin an evacuated quartz ampule. The ampule was heated up to 1160 ◦ C and held at this temperature for 24 hours, followedby slow cooling to 900 ◦ C at the rate of 1 . ◦ C/h. Shiny plate-like single crystals of CePd As were obtained. The crystalorientation and chemical homogeneity were checked by x-ray diffraction (XRD) and energy dispersive x-ray analysis (EDX),respectively. XRD measurements were carried out on a PANalytical X’pert MRD diffractometer with Cu K α radiation anda graphite monochromator. Magnetisation measurements were carried out in the temperature range 1 . −
400 K and inmagnetic field up to 7 T using a SQUID-VSM (MPMS3, Quantum Design). High-field magnetisation measurements up to60 T in pulsed magnetic fields were performed at the Dresden High Magnetic Field Laboratory, Germany. The electricaltransport experiments were carried out in the temperature range 2 K −
300 K and magnetic fields up to 14 T using a PhysicalProperty Measurement System (PPMS, Quantum Design). The electrical resistivity was measured using a standard four-terminal method, where electrical contacts to the sample were made using 25 µ m gold wires and silver paint. The temperaturedependence of specific heat was also measured using a PPMS. References Steglich, F. et al.
Superconductivity in the presence of strong Pauli paramagnetism: CeCu Si . Phys. Rev. Lett. ,1892–1896 (1979). Rotter, M., Tegel, M. & Johrendt, D. Superconductivity at 38 K in the iron arsenide ( Ba − x K x ) Fe As . Phys. Rev. Lett. , 107006 (2008). Jaccard, D., Behnia, K. & Sierro, J. Pressure induced heavy fermion superconductivity of CeCu Ge . Phys. Lett. A ,475–480 (1992). Movshovich, R. et al.
Superconductivity in heavy-fermion CeRh Si . Phys. Rev. B , 8241–8244 (1996). Grosche, F. M., Julian, S. R., Mathur, N. D. & Lonzarich, G. G. Magnetic and superconducting phases of CePd Si . Physica B: Condensed Matter , 50–52 (1996). Yuan, H. Q. et al.
Observation of two distinct superconducting phases in CeCu Si . Science , 2104 – 2107 (2003). Lengyel, E., Nicklas, M., Jeevan, H. S., Geibel, C. & Steglich, F. Pressure tuning of the interplay of magnetism andsuperconductivity in CeCu Si . Phys. Rev. Lett. , 057001 (2011). Abe, H., Kitazawa, H., Suzuki, H., Kido, G. & Matsumoto, T. High-field magnetization of CeRh Si and CePd Si . Physica B: Condensed Matter , 141–143 (1998). Thamizhavel, A., Kulkarni, R. & Dhar, S. K. Anisotropic magnetic properties of CeAg Ge single crystals. Phys. Rev. B , 144426 (2007). Knafo, W. et al.
High-field metamagnetism in the antiferromagnet CeRh Si . Phys. Rev. B , 094403 (2010). Krimmel, A. et al.
The evolution from long-range magnetic order to spin-glass behaviour in PrAu (Si − x Ge x ) . J. Phys.:Condens. Matter , 6991–7003 (1999). Ota, Y. et al.
Electrical and magnetic properties of CeAu Si . Journal of the Physical Society of Japan , 034714 (2009). Fritsch, V. et al.
Magnetic phase diagram of CeAu Ge : High magnetic anisotropy due to crystal electric field. Phys. Rev.B , 104446 (2011). Luo, Y. et al.
Magnetism and crystalline electric field effect in ThCr Si -type CeNi As . Phys. Rev. B , 245230 (2012). Maurya, A., Kulkarni, R., Dhar, S. K. & Thamizhvel, A. Anisotropic magnetic properties and crystal electric field studieson CePd Ge single crystal. J. Phys.: Condens. Matter , 435603 (2013). Shang, T. et al.
Tunable magnetic orders in CePd As − x P x . J. Phys.: Condens. Matter , 045601 (2014). Yamada, H. & Takada, S. Magnetoresistance due to electron-spin scattering in antiferromagnetic metals at low tempera-tures.
Progress of Theoretical Physics , 1401–1419 (1973). Hutchings, M. T. Point-charge calculations of energy levels of magnetic ions in crystalline electric fields.
Solid StatePhys. , 227 (1964). Stevens, K. W. H. Matrix elements and operator equivalents connected with the magnetic properties of rare earth ions.
Proceedings of the Physical Society A , 209–215 (1952). Wang, Y.-L. Crystal-field effects of paramagnetic curie temperature.
Physics Letters A , 383 – 384 (1971). Bowden, G. J., Bunbury, D. S. P. & McCausland, M. A. H. Crystal fields and magnetic anisotropy in the molecular fieldapproximation. I. general considerations.
Journal of Physics C: Solid State Physics , 1840 (1971). Buschow, K. H. J. & de Boer, F. R.
Physics of Magnetism and Magnetic Materials , 26–34 (Springer US, 2003).
Acknowledgements