Half-metallic ferromagnetic ground state in CePdSb

Journal of Alloys and Compounds 423 (2006) 15–20

Half-metallic ferromagnetic ground state in CePdSb ∗ ´ Andrzej Slebarski Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland Available online 8 February 2006

Abstract The CeMX compounds, where M is a transition metal, and X is an sp element, have a number of different types of ground state including magnetic, metallic, and insulating, and they exhibit anomalous physical properties, such as the Kondo effect, heavy-fermion behavior, valence fluctuations, etc. Recently, a Kondo-lattice state has been proposed for ferromagnetic CePdSb, which seems to be unusual. Many Kondo systems have an antiferromagnetic ground state due to the antiferromagnetic nature of the coupling of Ce 4f and the conduction electrons, however, the coexistence of ferromagnetism and Kondo behavior is an unexpected feature. In addition, our band structure calculations for CePdSb show a state very close to a half-metallic ferromagnets (HMF). On the base of Kondo-like HMF ground state properties of CePdSb we discuss its magnetic properties and electrical resistivity data. © 2005 Elsevier B.V. All rights reserved. PACS: 71.27.+a; 71.28.+d; 75.40.Cx; 73.20.At; 79.60.−i Keywords: Rare earth alloys and compounds; Electronic properties; Kondo effect; Heavy-fermions

1. Introduction Cerium-based Kondo- (Anderson-) lattice systems exhibit unusual physical phenomena such as heavy-Fermi (HF) liquid or non-Fermi-liquid types of behavior in a metallic state or a Kondo-lattice insulating type of state. The low-temperature properties of HF materials have remained attractive area of research for the last 20 years. It is generally agreed that there exists in these materials a narrow band of heavy electrons at low temperatures, giving rise to the large electronic specific-heat coefficient γ, the large Pauli susceptibility, and the T 2 T-dependence of the electrical resistivity [1]. At high-temperature limit this narrow band breaks up into localized states, in consequence the magnetic susceptibility usually follows the Curie–Weiss law and the resistivity decreases with increasing T, reminiscent of the Kondo behavior of dilute magnetic alloys. It is also generally accepted that the narrow band originates from the interaction between the conduction band electrons and the tightly bound f electrons. In typical HF systems, the strength of the exchange interaction between f-electrons and conduction

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electrons can be turned by composition or pressure, resulting in either dominant intrasite Kondo or intersite Ruderman–Kittel– Kasuya–Yoshida (RKKY) interactions. The first qualitative picture of both HF metals and Kondo insulators was based on the idea that the ground state results from a competitive character of the Kondo and RKKY interaction [2]. While the RKKY interaction temperature TRKKY ∝ Jsf2 N(F ), where N(F ) is the density of states (DOS) at the Fermi level, the Kondo temperature is given by TK ∝ exp(−1/|Jsf N(F )|). Since in general the hybridization matrix element (and hence the Kondo coupling) between the conduction and Ce f-electrons decreases with increasing volume of the unit cell, the RKKY interaction dominates over the Kondo effect and magnetic ordering can occur as in the case of CePdSn [3], CeCuSn [4], CeAgSn [5], or in CePdSb [6] and CeNiSb [7]. Doniach [2] examined the one-dimensional Kondo-lattice (Kondo necklace) in the mean-field approximation and obtained the magnetic ground state with a simple phase diagram, which gives the magnetic ordering temperature TRKKY as a function of Jsf N(F ). This phase diagram qualitatively describes the magnetic/nonmagnetic ground state properties for many Ce compounds under condition, that the quantity Jsf N(F ) increases smoothly with external pressure P. CePdSb is the first cerium compound where the dependence of TRKKY on P unequivocally

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´ A. Slebarski / Journal of Alloys and Compounds 423 (2006) 15–20

is predicted by the Doniach’s phase diagram [8]. The compound CePdSb is a ferromagnet whose Curie temperature TC = 17 K, however, the resistivity shows the maximum near ∼ 150 K, which would suggest a formation of a coherent Kondo-lattice [6]. CePdSb may be one of a small group of compounds that exhibit both Kondo behavior and ferromagnetism. The relatively large magnetic moment of ∼ 1 µB /Ce and the low value of the Sommerfeld coefficient, γ = 11 mJ/mol K2 [9], however, rather count against a Kondo-lattice ground state, likewise its ferromagnetic behavior cannot be described within a simple model. The stability of para-, ferro-, and antiferromagnetic states in the Kondo-lattice limit was recently discussed by Doradzi´nski and Spałek (DS) [10]. Their theoretical results well describe the ground state properties of the ternary Ce-based intermetallics, in particular CePdSb. The aim of this work is to investigate the electronic structure of CePdSb, which suggests its half-metallic ferromagnetic (HMF) low-temperature behavior. The first part of this presentation represents a discussion of the DS ground state phase diagram for the periodic Anderson model, which gives a good qualitative account of experimental results on the series of Ce-ternary intermetallic compounds. We also have shown that the magnetic behavior of the Kondo-lattice compound CePdSb can be explained well by the DS diagram. In the second part we present the numerical calculations of the electronic density of states which leads to the HMF properties of CePdSb. 2. Magnetic/nonmagnetic phase diagram The stability of paramagnetic versus magnetic ground state in the Kondo-lattice limit [10] is strongly dependent on (i) the on-site hybridization energy among conduction and f electrons, V, (ii) the bare f-level position in the conduction band, Ef , (iii) the magnitude of intrasite Coulomb interaction between two f electrons with opposite spins, U, and (iv) the number of electrons per atom. The last dependence is discussed for CeRhSb, which is known as a heavy-fermion (Kondo) insulator with a narrow energy gap in the heavy-quasiparticle density of states [6]. The energy gap in this compound is very sensitive to the partial heteroelectronic substitution. Previous alloying studies showed that substituting either Sb ions for Sn in CeNiSn (CeNiSn [12] has a Kondo insulating ground state, very similar to that in CeRhSb) or Sn ions for Sb in CeRhSb leads either to a formation of weak ferromagnetic Kondo-lattice [7] or to the non-Fermi-liquid (NFL) ground states [13], respectively. In the series of compounds: CePdSb (CeNiSb), CeRhSb, CeRhSn, and CeRhAl a number of valence electrons per formula is: 19 for both CePdSb and CeNiSb, 18 for CeRhSb, 17 for CeRhSn and 16 for CeRhAl. A paramagnetic Kondo insulator discussed within the periodic Anderson model provides an insulating state for k independent hybridization and for an even number of strongly correlated electrons per unit cell [10,11]. In the case the momentum-dependent hybridization the gap may vanish, this is the case of CeRhSn. On the contrary, the CeRhAl should than be insulator, basing on the assumption that the 4f electrons of Ce are delocalized and contribute essentially to the band filling, this is however, not the case. From the X-ray photoemission spectroscopy (XPS) spectra we estimated the hybridization energy for CeRhAl [14] which

Table 1 The number n of valence electrons, hybridization energy  obtained from the 3d Ce XPS spectra, the hybridization matrix element V = (/(πN(F )))1/2 , the f-shell occupation number nf obtained from the 3d XPS spectra, the number of the conduction electrons nc per atom, and the calculated DOS at the Fermi level for the series of ternary Ce-intermetallic compounds Compound

n

 (meV)

V (meV)

nf

nc

DOS at F (1/eV)

CeRhAl CeRhSn CeRhSb CePdSb CeNiSb

16 17 18 19 19

70 120 150 60 50

0.075 0.147 0.34 0.05 0.09

0.9 0.93 0.9 1 1

0.8 0.8 0.9 0.6 0.7

4.19 1.78 0.4 7.3 1.92

is almost 1/2 of that energy obtained for CeRhSb [15]. This is one reason, while CeRhAl has a metallic ground state (as well as CeRhSn). The electronic properties of the series of Ce-ternary alloys are tabulated in Table 1. The DS phase diagram on the V–ne plane displayed in Fig. 1 has been shown to give a good qualitative account of experimental results on the series of Ce-ternary intermetallics. In this diagram the regions with different magnetic phases form a rather complicated map, depending on the number of conduction electrons, ne , and the hybridization energy V [10]. For an even number of electrons per atom (ne = 2) an antiferromagnetic Kondo insulator (the case of CeRhSn [16]) evolves into a paramagnetic Kondo insulator (the case of CeRhSb and CeNiSn) with increasing hybridization of bare f and conduction-electron states. CeRhAl lies on the V–ne diagram on the line which separates an antiferromagnetic (AFM1) and strongly ferromagnetic metallic (SFM) phases. The competition between the AFM and SFM phases can lead to the NFL behavior, recently observed in CeRhAl [14]. CePdSb and CeNiSb are located on this diagram in the region with strongly ferromagnetic metallic phase. For the strongly ferromagnetic metallic phase the DS model expects also existence of the systems which may have a small magnetic

Fig. 1. Magnetic mean-field phase diagram (details in Ref. [10]). FM, WFM and SFM are ferromagnetic, the weakly ferromagnetic or strongly ferromagnetic metallic phase, respectively. AFM are different antiferromagnetic metallic phases, and PKI or AKI are paramagnetic or antiferromagnetic Kondo insulators, respectively. The points represent the experimental data for different Ceintermetallics, which are experimentally obtained from the 3d Ce XPS spectra.

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moment in spite of strong molecular field [10], this is the case of CePdSb. It should be also emphasized that the DOS at the Fermi level is not substantially enhanced in SFM phase, except for ne → 1 [10]. For this reason the SFM phase cannot be regarded as a good candidate for the Kondo-lattice ground state. However, the DOS at the Fermi energy was obtained for CePdSb and CeNiSb ∼ 7 and ∼ 2 states/eV, respectively, and 1 < ne < 1.7 for the both compounds, therefore the Kondo-lattice ground state cannot be excluded. A Kondo-lattice behavior for this compound is still discussed. In order to determine the ground state f occupation number, nf , and the hybridization energy V from the 3d XPS spectra we used the Gunnarsson–Sch¨onhammer model [17]. Gunnarsson and Sch¨onhammer (GS) explained in Ref. [17] how to determine the initial f-state properties from the Ce 3d XPS spectra, which are related to the final f-states. In particular, they discussed how experimental spectra can be used to estimate the f-occupancy, nf , and the hybridization energy fs between the f-level and the conduction states. The f2 components in the Ce 3d XPS spectra in Fig. 2 are attributed within the Gunnarsson– Sch¨onhammer model to the f-conduction electron hybridization. The hybridization energy , which describes the hybridization part of the Anderson impurity Hamiltonian [18], is defined as πV 2 N(F ), where N(F ) is the maximum in the DOS and V is the hybridization matrix element. It is possible to estimate  from the ratio rf = I(f2 )/(I(f1 ) + I(f2 )), calculated as a function of  in Ref. [17], when the peaks of the Ce 3d XPS spectra that overlap in Fig. 2 are separated (see also Table 1). Displayed in Fig. 2 are plots of the Ce 3d XPS spectra of CeRhSb and CePdSb. Also shown in the figure is the difference between them, which indicates much stronger hybridization effect between 4f and conduction states and mixed valence character of Ce in CeRhSb.

The Ce 3d XPS spectra, however, allow an estimate of the occupation number, nf , and of the energy  within the accuracy of the order of 15%. The errors are due to the uncertainties in the intensity ratios, which can be attributed to the uncertainty of the spectra decomposition [19], the background subtraction, and the surface-to-bulk ratio [20–22] as well as to the approximations in the GS theory (see, discussion in Ref. [11]). Moreover, according to GS, the deviation from the linearity of the intensity ratio of the final to the ground-state f-occupation number depends on , which is also obtained within the same accuracy. With similar accuracy we compare the experimental data with the DS diagram. The magnetic moment compensation resulting from the Kondo screening of the Ce-intermetallic compounds was also analyzed theoretically in Ref. [10] for different phases in the DS diagram. The magnetic moments of f and conduction electrons (c) are plotted in Fig. 3 for a Kondo-insulator states. The compensation ratio r ≡ |me /mf | grows with the increasing hybridization V and achieves the maximal value rmax for the critical value Vc , where AKI transforms into PKI [10]. Vc and rmax depend on U value, suggesting that the effect of magnetic-moment screening is a consequence of the electron correlations present in the system. This is a case of CeRhSb Kondo insulator. The hybridization energy experimentally obtained for CeRhSb is very large of ∼ 370 meV, suggesting in Fig. 3 the nonmagnetic ground state of CeRhSb for the finite value of Coulomb energy 2.5 < U < ∞. For the remaining Ce-ternary intermetallics the hybridization

Fig. 2. Ce 3d XPS spectra of CeRhSb and CePdSn and the difference of the intensities (CeRhSb − CePdSb). The hybridization effect is much stronger in CeRhSb in respect to CePdSb. Also, the f0 satellite lines are observed only for the 3d XPS spectrum of CeRhSb suggesting its mixed valence of Ce, whereas the f shell configuration of Ce in CePdSb is stable.

Fig. 3. The Kondo-insulator state; the magnetic moments of f and conduction electrons versus hybridization energy V, and the compensation ratio r (details in Ref. [10]). This diagram predicts the nonmagnetic ground state for CeRhSb (square point), whereas for the remaining compounds (CeRhSn, CeRhAl and CePdSb) the total magnetic moment µ = 0 at T = 0.

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than that in the AFM phases near ne = 2 (see, Fig. 1) therefore the Kondo-lattice ground state in CePdSb is questionable inside this theoretical model. Fig. 5 displays the f-magnetic moment compensation versus hybridization V, which is expected for CeRhSb. However, the discussion of the Kondo-lattice state of CePdSb seems to be much more difficult than that, proposed by Doradzi´nski and Spałek because of the observed strong interatomic hybridization between Ce f-electrons and Pd d-states. The f–d interatomic hybridization effect in CePdSb is observed either in our XPS valence band spectra or in the LDA calculations of the densities of states. The hybridization effect can lead to the fluctuation between the more localized and itinerant f-states, which complicates the understanding of the nature of the ground state in CePdSb (the discussion in Section 3). 3. Is CePdSb a half-metallic ferromagnet?

Fig. 4. An antiferromagnetic and the weekly ferromagnetic HF metals; the density of states enhancement at the Fermi level for various doping δ ≡ 2 − ne (details in Ref. [10]). This diagram well describes the ground state of CeRhAl, square point in this figure (see, Table 1).

energy V is twice or more smaller (see Table 1), which predicts in Fig. 3 the magnetic ground states with magnetic moment µ = 0. Doradzi´nski and Spałek showed that the moment compensation appears also for a total density of electrons ne < 2, i.e., for AFM and WFM phases. In the case of stable AFM1 state (see Fig. 4) the DOS is strongly enhanced and grows with increasing hybridization. This spectacular growth reflects the narrowing of the magnetic subbands and the growing contribution of the f-states at the Fermi level. CeRhAl is a good example of the antiferromagnetic HF metallic phases displayed in Fig. 4. The moment compensation in the CePdSb compound (this is the SFM phase in DC diagram) of ∼ 1% is far less pronounced

Fig. 5. Magnetization m in the strongly ferromagnetic metal vs. V; U = 2.5, F = −0.75) (details in Ref. [10]). This diagram describes the ferromagnetic ground state and a small Kondo-like hybridization in CePdSb (square point). CeNiSb, having a very similar ground state properties shows much larger Kondolike screening (spherical point) than CePdSb.

The crystal structure of CePdSb was first thought to be of the hexagonal CaIn2 -type (space group P63 /mmc) with Ce atoms occupying 2b crystallographic sites with Pd and Sb atoms randomly distributed on the 4f sites. However, our XPS data compared with the LMTO band structure calculations give preference for the ordered structure (see Fig. 6), also neutron diffraction measurements [23] suggested that the crystal structure is a modification of the CaIn2 structure hexagonal P63 mc GaGeLi, in which the Pd and Sb atoms form ordered sublattices at coordinates (1/3, 2/3, u) and 2/3, 1/3, v), where u = 0.4684 and v = 5164. Very recently [14] we have presented band structure calculations, which indicated the half-metallic ferromagnetic ground state for CePdSb. The main results from Ref. [14] are presented in Fig. 7. The shape of the DOS generally does not depend on the method used for the calculations, excluding the narrow range of energies in the vicinity of the Fermi energy. The local spin density approximation (LDA) and LDA + SO approximation leads to the HMF properties of CePdSb, while the LDA + U potential predicts much more localized f-electron states which are located in the majority band, and the gap at F for the both spin directions. This result is, however, in contradiction to the resistivity ρ(T ) experimental observation, which has not shown an activated behavior [6]. (i) CePdSb is calculated by full potential linear augmented plane wave (LAPW) with the use of LDA exchange correlation (XC) potential to be half-metallic, with a magnetic moment of 1.0 µB /f.u. roughly consistent with that, reported in Ref. [9] (µs = 0.95 µB ), and the electronic specific-heat coefficient γ of 116 mJ/mol K2 that is larger than the γ = 11 mJ/mol K2 experimentally obtained [9]. (ii) The LDA + SO method gives the almost half-metallic ground state. The minority DOS obtained at the Fermi energy in the LDA + U approximation is of about 6% of the total DOS at F . The calculated magnetic moment per formula unit is 0.95 µB . (iii) The bands obtained with the use of LDA + SO + U correction show the gap for the both spin projections at F . For

´ A. Slebarski / Journal of Alloys and Compounds 423 (2006) 15–20

Fig. 6. The numerical calculations of DOS of CePdSb (P63 /mmc) with different occupations of the crystallographic positions: (a) Ce atoms are in 2b (6m2) sites and Pd and Sb accidentally are in 4f (3m) sites, (b) Ce are in 2b and 4f sites are occupied, respectively, by Pd, Sb, Pd, Sb, ..., (c) Ce are in (2b) atomic positions and 4f sites. The total energy has the lowest value for the situation (b). In (d) DOS is calculated for the frozen 4f-states. The total DOS calculated for the paramagnetic CePdSb (dotted curve), convoluted by Lorentzians of halfwidth 0.4 eV, taking into account proper cross sections for bands with different l symmetry. In the situation (b) the calculations are compared with the XPS valence-band data corrected for background (points) (details in Ref. [26]).

Coulomb interaction U = 0, the majority 4f Ce-states are located in the band ∼ 2.7 eV below F , while the minority 4f-DOS is 0. The difference N↑ (E) − N↓ (E) for the f-states can be related to ferromagnetism due to more localized than itinerant f-states. The magnetic moment localized on the Ce atom is 0.98 µB , whereas the total magnetic moment per formula unit of CePdSb is 1 µB . The calculated γ of about 3.4 mJ/mol K2 results from the strong Coulomb correlation energy U, which separates an occupied and empty f-states in the spin polarized bands. However, this γ value is more reasonable in contradiction to (i) and (ii). In the case (iii) we can interpret the f-electron states as the localized Kondo states (the hybridization between localized felectron and conduction states can lead to the gap formation,

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Fig. 7. Calculated total- and atom-spin projected density of states of CePdSb, with majority plotted upward and minority plotted downward (see Ref. [14]).

while both situations (i) and (ii) correspond to the Andersonlattice model and the formation of narrow f-band states of width kB TK , where TK is a Kondo (hybridization) temperature. However, experimental study for CePdSb[6] do not provide direct evidence of the insulating or semiconducting behavior, i.e., the resistivity ρ(T ) is not described by an activated law ρ = ρ0 exp(coh /kB T ). Therefore it has been controversial whether the Ce-ground state in CePdSb is localized or itinerant. One reason is that the LDA + U approximation gives the unphysical interaction of the f-electron with itself, resulting in an extended f-orbital and increasing the hybridization, which can lead to the gap formation. Another explanation could be the following: there is the interference between the ground state calculated by LDA and LDA + U, respectively which leads to the mixed ground state. It is assumed that the narrow f band at ∼ 2.7 eV in the band , interacting via hybridization with a broad Pd d states, delocalizes, which could lead to an enhancement

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´ A. Slebarski / Journal of Alloys and Compounds 423 (2006) 15–20

of the DOS at F . This process could have a character of fluctuation, which well explains the character of the temperature dependence of the resistivity. It has been, however, controversial whether the ground state of CePdSb is in the HMF or in metallic state. The complete spin polarization of the charge carriers at low temperatures would have important consequences for magnetic and electric transport properties: (i) HMF easily achieves the maximum saturation magnetization because a further increase in the spin splitting in this state does not increase the magnetic moment, it is experimentally observed for CePdSb [14]. (ii) The temperature dependence of the resistivity in the range T < TC should be dominated by two-magnon process which give rise to different power exponents [24] in ρ(T ) than one-magnon scattering process (spin-flip scattering) expected for conventional metallic ferromagnets, for which ρ ∝ I 2 N↑ (F )N↓ (F )(T/TC )2 , where I is the d–s exchange parameter and Nσ (F ) is the spin polarized DOS at the Fermi level [25]. Electrical resistivity of CePdSb is very large [6] and ρ(T ) ∝ T  , where  > 2 in the temperature range T < TC , indicative of the HMF state. However, these experimental results only indirectly motivate the HMF properties of CePdSb. Specifically, spin polarized photoemission studies are necessary. Acknowledgements The author acknowledges the support of the State Committee for Scientific Research (KBN), through the Grant No. 1 P03B 052 28. The author is also very grateful to Jozef Spałek, J´ozef Deniszczyk, and Andrzej Jezierski for critical remarks and comments. References [1] G.R. Stewart, Rev. Mod. Phys. 56 (1984) 755. [2] S. Doniach, Physica B 91 (1977) 231 S. Doniach, In: R.D. Parks (Eds.), Valence Instability and Related Narrow Band Phenomena, Plenum, New York, 1977. [3] S.K. Malik, D.T. Adroja, S.K. Dhar, R. Vijayaraghavan, B.D. Padalia, Phys. Rev. B 40 (1989) 2414. [4] J. Sakurai, K. Kegai, K. Nishimura, Y. Ishikawa, K. Mori, Physica B 186– 188 (1993) 583.

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