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Electronic structure and magnetism of CeCoGe3
Solid State Communications 141 (2007) 316–319 www.elsevier.com/locate/ssc
Electronic structure and magnetism of CeCoGe3 T. Jeong ∗ DPMC, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland Received 10 July 2006; accepted 15 November 2006 by E.G. Wang Available online 1 December 2006
Abstract The electronic band structure of CeCoGe3 has been calculated using the self-consistent full potential nonorthogonal local orbital minimum basis scheme based on density functional theory. We investigated the electronic structure with the spin–orbit interaction and on-site Coulomb potential for the Ce-derived 4f orbitals to obtain the correct ground state of CeCoGe3 . The exchange interaction between local f electrons and conduction electrons play an important role in their heavy fermion characters. The fully relativistic band structure scheme shows that spin–orbit coupling splits the 4f states into two manifolds, the 4f7/2 and the 4f5/2 multiplets. c 2006 Elsevier Ltd. All rights reserved.
PACS: 71.10.Hf; 71.18.+y; 72.20.Eh; 75.30.Mb Keywords: A. CeCoGe3 ; D. Electronic structure; D. Magnetism
1. Introduction In general, the Ce-based compounds have attracted considerable attention recently because they exhibit a large variety of physical properties such as heavy fermion, Kondo insulating, anisotropic transport and magnetic ordering behaviors. The various physical properties depend on the hybridization between f electrons and conduction electrons, which is characterized by a Kondo temperature TK . With a small hybridization the system exhibits a local magnetic moment and orders magnetically, which can be described by the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. By increasing the hybridization, the Kondo effect increases and the ordered magnetic moment decreases gradually. With further increase of the hybridization, the system will go into a heavy fermion or intermediate valence regime [1]. CeCoGe3 is a heavy fermion system with specific heat coefficient γ = 111 mJ/K2 mol [2]. Two magnetic transitions were found at ∼21 and ∼18 K in the absence of a magnetic state. The upper ordering temperature is believed to be a transition from the paramagnetic state to a c-axis ferrimagnetic state, which in turn transforms into a colinear ∗ Corresponding address: Department of Physics, Konkuk University, 143701 Seoul, Republic of Korea. E-mail address: [email protected].
c 2006 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2006.11.018
antiferromagnetic state (AF) at ∼18 K. The antiferromagnetic phase undergoes an irreversible phase transition to the ferri state under an applied magnetic field. Consequently, magnetic glasslike behaviors such as thermal and magnetic hysterisis, frozen moment, and magnetic relaxation were observed [2]. A nonhysteretic metamagnetic transition to a c-axis ferromagnet occurs at high magnetic fields [2]. The critical spin-flip field increases in the ferri state, and decreases in the AF state as temperature decreases. A close competition of ferromagnetic and antiferromagnetic interactions between Ce moments is believed to be responsible for these two magnetic orderings. Eom et al. studied the magnetic phase diagram of CeCoGe3−x Six using electrical resistivity, specific heat, and dc magnetic susceptibility measurements [3]. Long range antiferromagnetism is found to disappear at x = 1.0. Electrical resistivity of the alloys with x = 1.2, 1.25, and 1.5 demonstrates NFL type linear temperature dependence at low temperatures. CeCoGe1.8 Si1.2 displays logarithmic scaling of C V /T between 2 and 0.2 K. Krishnamurthy et al. studied nonFermi liquid (NFL) spin dynamics using muon spin relaxation in CeCoGe3−x Six [4]. When site disorder is strong, both Griffiths phase, i.e., magnetically inhomogeneous state (at x = 1.2), and a quantum phase transition (at x = 1.5) crucially control the low temperature spin dynamics in a non-Fermi liquid.
T. Jeong / Solid State Communications 141 (2007) 316–319
317
In order to understand the electronic and magnetic properties of CeCoGe3 , we need electronic band structure studies based on density functional theory. In this work, the precise selfconsistent full potential local orbital minimum basis band structure scheme (FPLO) is employed to investigate the electronic and magnetic properties of CeCoGe3 with LDA, LDA + U and fully relativistic schemes. We consider the effects of magnetism on the band structure and compare with experiments. 2. Crystal structure CeCoGe3 crystallizes in the primitive tetragonal bodycentered BaNiSn3 structure. In CeCoGe3 , the Ce atoms lie on the four corners in the body center of the tetragonal crystal structure. The four atoms on the edges in the upper half of the unit cell are connected to a single Co atom at a distance ˚ which is connected to the body-centered Ce atom of 3.414 A in the lower half of the unit cell. The nearest neighbor Ce–Ce ˚ and the next nearest neighbor interatomic distance is 4.32 A ˚ away. It belongs to the I 4mm space group Ce atom is 5.78 A with Ce occupying the 2a site, Co the 2a, and Ge the 2a, 4b ˚ and sites. We used experimental lattice constants, a = 4.32 A ˚ in the calculation described below. c = 9.83 A, 3. Method of calculations We have applied the full-potential nonorthogonal localorbital minimum-basis (FPLO) scheme within the local density approximation (LDA). [5] In these scalar relativistic calculations we used the exchange and correlation potential of Perdew and Wang [6]. Ce 6s, 6p, 5d, Co 4s, 4p, 3d and Ge 4s, 4p, 3d states were included as valence states. All lower states were treated as core states. We included the relatively extended semicore 4s, 4p, 4d, 4f, 5s, 5p states of Ce and 3s, 3p states of Co and 3s, 3p states of Ge as band states because of the considerable overlap of these states on nearest neighbors. This overlap would be otherwise neglected in our FPLO scheme. Ce 6p states were added to increase the quality of the basis set. The spatial extension of the basis orbitals, controlled by a confining potential (r/r0 )4 , was optimized to minimize the total energy. The self-consistent potentials were carried out on a k mesh of 12k points in each direction of the Brillouin zone, which corresponds to 294k points in the irreducible zone. A careful sampling of the Brillouin zone is necessitated by the fine structures in the density of states near the Fermi level E F . 4. Results and discussion We first show the full band structures of CeCoGe3 within LDA scheme in Fig. 1. The Ge 4s bands lie between −12 eV and −7 eV. Between −5 eV and the Fermi level there are mixed Ge 4p and Co 3d states. The very flat bands very above the Fermi level are mainly of Ce-centered 4f character (with bandwidth about 0.1 eV). A prominent feature of the band structure near E F , besides the 4f bands, is the Co 3d character which hybridizes with the Ce 4f bands. We also
Fig. 1. The full LDA nonmagnetic band structure of CeCoGe3 along the symmetry lines.
study the on-site atomic-like correlation effects beyond LDA by using a LDA + U approach in a rotationally invariant, full potential implementation [7]. Minimizing the LDA + U total energy functional with spin–orbit coupling (SOC) treated selfconsistently [8] generates not only the ground state energy and spin densities, but also effective one-electron states and energies that provide the orbital contribution to the moment and Fermi surfaces. The basic difference of LDA + U calculations from LDA is its explicit dependence on the on-site spin and orbitally resolved occupation matrices. The Coulomb potential U and the exchange coupling J for the Ce 4f orbitals have been chosen to be 7.0 and 0.68 eV, respectively. The resulting band structure calculated within the LDA + U scheme is shown in middle panel of Fig. 2. We observe that the crystal field splittings of the Ce 4f bands within LDA are quite small and in fact difficult to identify due to hybridization with itinerant bands. From LDA + U , the 4f bands are still very flat but are split into three manifolds by some combination of the crystal field and the anisotropy of the U interaction. We also calculated the fully relativistic band structure to see spin–orbit coupling effects, which is shown in the lower panel of Fig. 2. As expected spin–orbit coupling splits the 4f states into two manifolds, located +0.05 and −0.4 eV of the Fermi level, the 4f7/2 and the 4f5/2 multiplet, respectively. The atom and symmetry projected densities of states (DOS) shown in Fig. 3 clarify the characters of the bands. Because only the DOS distribution near the Fermi level determines the magnetic properties, we concentrate our attention upon the DOS in the vicinity of the Fermi level. At this range, the valence states for Ce or Co atoms are dominated by 4f and 3d electrons. From the Fig. 3, it can be found that within the LDA calculation, the DOS at the Fermi level have a high peak near the Fermi level and they mainly consist of Ce-derived 4f states. When the spin–orbital coupling along the axis (0 0 1) is taken into account, the 4f orbitals are slightly modified. The spin 4f states become wider and the energy shift between centers becomes larger. This is due to the partial splitting between the degenerate 4f states. Band calculations within the LDA framework cannot yield the correct magnetic moment for many Ce compounds because of the strong correlation interaction between f orbitals. The spin–orbit interaction in these systems is sometimes large
318
T. Jeong / Solid State Communications 141 (2007) 316–319
Fig. 3. Projected density of states of CeCoGe3 . The Ce 4f character dominates the states near the Fermi level.
uniform magnetic susceptibility using the method of Janak [11]. The uniform magnetic susceptibility of a metal can be written as χ=
Fig. 2. Top panel: The LDA band structure of CeCoGe3 along symmetry lines. The very flat bands near the Fermi level are the Ce 4f bands. Middle panel: The band structure within the LDA + U scheme showing that the 4f bands are split into three manifolds. Bottom panel: The fully relativistic band structure of CeCoGe3 showing that spin–orbit coupling splits the 4f states into two manifolds.
and the orbital contribution to the magnetic moment cannot be neglected. The calculated magnetic moment of CeCoGe3 within the LDA scheme is 0.22µ B /Ce. This is mainly from the Ce-derived 4f orbitals. When the on-site correlation potential is added to the Ce 4f electron, the degeneracy between the different f orbits would be lifted and Hund’s rules dominate the locally occupied 4f electrons, which yields the total magnetic moment 0.18µ B /Ce. With a fully relativistic scheme we calculated the magnetic moment with a value 0.17µ B /Ce. Density functional calculations are very reliable in calculating the instability to ferromagnetism. The presence of an electronic instability is signaled by a divergence of the corresponding susceptibility. In the following we study the
χ0 , 1 − N (E F )I
(1)
where the numerator stands for the Pauli susceptibility of a gas of non-interacting electrons proportional to the density of states at the Fermi level N (E F ), and the denominator represents the enhancement due to electron–electron interactions. Within the Kohn–Sham formalism of density functional theory the Stoner parameter I is related to the second derivative of the exchange-correlation functional with respect to the magnetization density. We have evaluated, within the density functional theory formalism, the Stoner enhancement of the susceptibility χ = 1−I χN0(E F ) ≡ Sχ0 , where χ0 = 2µ2B N (E F ) is the non-interacting susceptibility and S gives the electron–electron enhancement in terms of the Stoner constant I . We have calculated I using both the Janak–Vosko–Perdew theory [11] and fixed spin moment calculations [12]. The calculated density of states and normalized Stoner parameter are N (E F ) = 5.08 states/eV and I = 0.24 eV. This gives I N (E F ) = 1.21, larger than unity, corresponding to a ferromagnetic instability. Heavy fermion compounds are characterized by larger electronic specific heat coefficients γ . CeCoGe3 is a moderate heavy fermion compound with γ = 111 mJ/K2 mol [2]. The large specific heat coefficient of CeCoGe3 compounds could not be got from our band calculation. This can be seen from the calculated electronic structure. It can be found that the total number of DOS at the Fermi level is about 5.08 states/eV, which corresponds to γb = 11.9 mJ/K2 mol, underestimating the experimental value by a factor of 9.3. The discrepancy between the band calculation and experiment for the specific heat coefficient is attributed to the formation of quasiparticles. There is an exchange interaction J between the local f and the conduction electrons in CeCoGe3 . The ground state of Ce compounds is determined by the competition between the Kondo and indirect RKKY interaction. With a large J , the Kondo coupling becomes strong and the system locates at the borderline of magnetic–nonmagnetic transition. The exchange interaction between the local f electron and the conduction
T. Jeong / Solid State Communications 141 (2007) 316–319
electrons will result in the formation of quasiparticles. This has a larger mass compared with bare electrons and the enhancement of mass increases with the increase of the exchange. Because of the volume contraction, the exchange interaction between the f and the conduction electrons is large in CeCoGe3 . This will result in the f electrons behaving like itinerant electrons and the narrow f bands locating at the Fermi level. On the other hand, when the exchanging interaction between f and conduction electrons is smaller, the occupied 4f orbitals are located near the Fermi level while the unoccupied 4f orbitals are at the conduction bands. The quasiparticle mass is appropriate to the number of DOS at the Fermi level. So the quasiparticle mass is greatly enhanced in CeCoGe3 . Indeed, it has been shown that when the Ce 4f electrons in CeCoGe3 are treated as localized electrons, the quasiparticle mass was enhanced over the band calculation by a factor 9.3. At this point we need to discuss a remaining question: which band structure (LDA, LDA + U or LDA + SOC) is more realistic? In LDA + SOC (the fully relativistic scheme), SOC is applied at the one-electron level, that is, each 4f orbital acquires its own j = l ± s( 72 or 72 ) label and character. In LDA + UP, one can follow Hund’s rules P to some extent: maximize S = i si , and maximize L = i li , and study various values of J (or even of L). In some respects, such for the ground state energy to determine the equation of state, the LDA+U approach seems most appropriate for rare earth ions. There are several instances of experience with similar 4f configurations which imply that the LDA + U correction is essential; simple LDA often leaves occupied 4f states at too high an energy and unoccupied ones at too low an energy, so there is excessive hybridization at the Fermi level. One example is that of Gd metal [9], another is provided by the Eu monopnictides [10]. On the other hand, 4f single particle-like excitations are likely to be characterized by the 27 , 52 labels. The excitations f0 → f1 or f14 → f13 are likely to show SOC aspects directly. Therefore further experimental investigations, such as the angle-resolved photoemission spectroscopy technique (a powerful technique to directly probe the electronic structure of solids) should be quite interesting. 5. Summary In this article we showed the results of three different electronic band structure calculations. It shows that the
319
Coulomb potential effect on Ce 4f orbitals and the spin–orbit interaction are key factors to understanding the electronic and magnetic properties of CeCoGe3 . When the Coulomb potential is added to the Ce 4f orbitals, the degeneracy between the different f orbits would be lifted and they are split into lower Hubbard bands at the Fermi level and unoccupied upper Hubbard bands in the conduction band. The exchange interaction between local f electrons and conduction electrons plays an important role in their heavy fermion characters. And the fully relativistic band structure scheme shows that spin–orbit coupling splits the 4f states into two manifolds, the 4f7/2 and the 4f5/2 multiplet. The f-electrons can be delocalized through hybridization with conduction electrons. If there exists no hybridization between f and conduction electrons, the felectron number becomes just an integer and the f-electrons are localized. The almost localized f-electrons and delocalized conduction electrons hybridize to deviate the f-electron number from an integer value, and keep the metallic state against the strong electron correlation between f-electrons. Therefore the hybridization between f and conduction d electrons plays an important role in CeCoGe3 . References [1] G.R. Stewart, Rev. Mod. Phys. 73 (2001) 797. [2] V.K. Pecharsky, O.B. Hyun, K.A. Gschneidner, Phys. Rev. B 47 (1993) 11839. [3] D.H. Eom, et al., J. Phys. Soc. Japan 67 (1998) 2495. [4] V.V. Krishnamurthy, K. Nagamine, I. Watanabe, K. Nyshiyama, S. Ohira, M. Ishikawa, D.H. Eom, T. Ishikawa, T.M. Briere, Phys. Rev. Lett. 88 (2002) 046402. [5] K. Koepernik, H. Eschrig, Phys. Rev. B 59 (1999) 1743; H. Eschrig, Optimized LCAO Method and the Electronic Structure of Extended Systems, Springer, Berlin, 1989. [6] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [7] A.I. Liechtenstein, V.I. Anisimov, J. Zaanen, Phys. Rev. B 52 (1995) R5468; A.B. Shick, A.I. Liechtenstein, W.E. Pickett, Phys. Rev. B 60 (1999) 10728. [8] A.B. Shick, D.L. Novikov, A.J. Freeman, Phys. Rev. B 57 (1997) R14259. [9] A.B. Schik, W.E. Pickett, C.S. Fadley, J. Appl. Phys. 87 (2000) 5878. [10] J. Kunes, W. Ku, W.E. Pickett, J. Phys. Soc. Japan 74 (2005) 1408. [11] J.F. Janak, Phys. Rev. B 16 (1977) 255; S.H. Vosko, J.P. Perdew, Can. J. Phys. 53 (1975) 1385. [12] K. Schwarz, P. Mohn, J. Phys. F 14 (1984) L129.
Electronic structure and magnetism of CeCoGe3 T. Jeong ∗ DPMC, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland Received 10 July 2006; accepted 15 November 2006 by E.G. Wang Available online 1 December 2006
Abstract The electronic band structure of CeCoGe3 has been calculated using the self-consistent full potential nonorthogonal local orbital minimum basis scheme based on density functional theory. We investigated the electronic structure with the spin–orbit interaction and on-site Coulomb potential for the Ce-derived 4f orbitals to obtain the correct ground state of CeCoGe3 . The exchange interaction between local f electrons and conduction electrons play an important role in their heavy fermion characters. The fully relativistic band structure scheme shows that spin–orbit coupling splits the 4f states into two manifolds, the 4f7/2 and the 4f5/2 multiplets. c 2006 Elsevier Ltd. All rights reserved.
PACS: 71.10.Hf; 71.18.+y; 72.20.Eh; 75.30.Mb Keywords: A. CeCoGe3 ; D. Electronic structure; D. Magnetism
1. Introduction In general, the Ce-based compounds have attracted considerable attention recently because they exhibit a large variety of physical properties such as heavy fermion, Kondo insulating, anisotropic transport and magnetic ordering behaviors. The various physical properties depend on the hybridization between f electrons and conduction electrons, which is characterized by a Kondo temperature TK . With a small hybridization the system exhibits a local magnetic moment and orders magnetically, which can be described by the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. By increasing the hybridization, the Kondo effect increases and the ordered magnetic moment decreases gradually. With further increase of the hybridization, the system will go into a heavy fermion or intermediate valence regime [1]. CeCoGe3 is a heavy fermion system with specific heat coefficient γ = 111 mJ/K2 mol [2]. Two magnetic transitions were found at ∼21 and ∼18 K in the absence of a magnetic state. The upper ordering temperature is believed to be a transition from the paramagnetic state to a c-axis ferrimagnetic state, which in turn transforms into a colinear ∗ Corresponding address: Department of Physics, Konkuk University, 143701 Seoul, Republic of Korea. E-mail address: [email protected].
c 2006 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2006.11.018
antiferromagnetic state (AF) at ∼18 K. The antiferromagnetic phase undergoes an irreversible phase transition to the ferri state under an applied magnetic field. Consequently, magnetic glasslike behaviors such as thermal and magnetic hysterisis, frozen moment, and magnetic relaxation were observed [2]. A nonhysteretic metamagnetic transition to a c-axis ferromagnet occurs at high magnetic fields [2]. The critical spin-flip field increases in the ferri state, and decreases in the AF state as temperature decreases. A close competition of ferromagnetic and antiferromagnetic interactions between Ce moments is believed to be responsible for these two magnetic orderings. Eom et al. studied the magnetic phase diagram of CeCoGe3−x Six using electrical resistivity, specific heat, and dc magnetic susceptibility measurements [3]. Long range antiferromagnetism is found to disappear at x = 1.0. Electrical resistivity of the alloys with x = 1.2, 1.25, and 1.5 demonstrates NFL type linear temperature dependence at low temperatures. CeCoGe1.8 Si1.2 displays logarithmic scaling of C V /T between 2 and 0.2 K. Krishnamurthy et al. studied nonFermi liquid (NFL) spin dynamics using muon spin relaxation in CeCoGe3−x Six [4]. When site disorder is strong, both Griffiths phase, i.e., magnetically inhomogeneous state (at x = 1.2), and a quantum phase transition (at x = 1.5) crucially control the low temperature spin dynamics in a non-Fermi liquid.
T. Jeong / Solid State Communications 141 (2007) 316–319
317
In order to understand the electronic and magnetic properties of CeCoGe3 , we need electronic band structure studies based on density functional theory. In this work, the precise selfconsistent full potential local orbital minimum basis band structure scheme (FPLO) is employed to investigate the electronic and magnetic properties of CeCoGe3 with LDA, LDA + U and fully relativistic schemes. We consider the effects of magnetism on the band structure and compare with experiments. 2. Crystal structure CeCoGe3 crystallizes in the primitive tetragonal bodycentered BaNiSn3 structure. In CeCoGe3 , the Ce atoms lie on the four corners in the body center of the tetragonal crystal structure. The four atoms on the edges in the upper half of the unit cell are connected to a single Co atom at a distance ˚ which is connected to the body-centered Ce atom of 3.414 A in the lower half of the unit cell. The nearest neighbor Ce–Ce ˚ and the next nearest neighbor interatomic distance is 4.32 A ˚ away. It belongs to the I 4mm space group Ce atom is 5.78 A with Ce occupying the 2a site, Co the 2a, and Ge the 2a, 4b ˚ and sites. We used experimental lattice constants, a = 4.32 A ˚ in the calculation described below. c = 9.83 A, 3. Method of calculations We have applied the full-potential nonorthogonal localorbital minimum-basis (FPLO) scheme within the local density approximation (LDA). [5] In these scalar relativistic calculations we used the exchange and correlation potential of Perdew and Wang [6]. Ce 6s, 6p, 5d, Co 4s, 4p, 3d and Ge 4s, 4p, 3d states were included as valence states. All lower states were treated as core states. We included the relatively extended semicore 4s, 4p, 4d, 4f, 5s, 5p states of Ce and 3s, 3p states of Co and 3s, 3p states of Ge as band states because of the considerable overlap of these states on nearest neighbors. This overlap would be otherwise neglected in our FPLO scheme. Ce 6p states were added to increase the quality of the basis set. The spatial extension of the basis orbitals, controlled by a confining potential (r/r0 )4 , was optimized to minimize the total energy. The self-consistent potentials were carried out on a k mesh of 12k points in each direction of the Brillouin zone, which corresponds to 294k points in the irreducible zone. A careful sampling of the Brillouin zone is necessitated by the fine structures in the density of states near the Fermi level E F . 4. Results and discussion We first show the full band structures of CeCoGe3 within LDA scheme in Fig. 1. The Ge 4s bands lie between −12 eV and −7 eV. Between −5 eV and the Fermi level there are mixed Ge 4p and Co 3d states. The very flat bands very above the Fermi level are mainly of Ce-centered 4f character (with bandwidth about 0.1 eV). A prominent feature of the band structure near E F , besides the 4f bands, is the Co 3d character which hybridizes with the Ce 4f bands. We also
Fig. 1. The full LDA nonmagnetic band structure of CeCoGe3 along the symmetry lines.
study the on-site atomic-like correlation effects beyond LDA by using a LDA + U approach in a rotationally invariant, full potential implementation [7]. Minimizing the LDA + U total energy functional with spin–orbit coupling (SOC) treated selfconsistently [8] generates not only the ground state energy and spin densities, but also effective one-electron states and energies that provide the orbital contribution to the moment and Fermi surfaces. The basic difference of LDA + U calculations from LDA is its explicit dependence on the on-site spin and orbitally resolved occupation matrices. The Coulomb potential U and the exchange coupling J for the Ce 4f orbitals have been chosen to be 7.0 and 0.68 eV, respectively. The resulting band structure calculated within the LDA + U scheme is shown in middle panel of Fig. 2. We observe that the crystal field splittings of the Ce 4f bands within LDA are quite small and in fact difficult to identify due to hybridization with itinerant bands. From LDA + U , the 4f bands are still very flat but are split into three manifolds by some combination of the crystal field and the anisotropy of the U interaction. We also calculated the fully relativistic band structure to see spin–orbit coupling effects, which is shown in the lower panel of Fig. 2. As expected spin–orbit coupling splits the 4f states into two manifolds, located +0.05 and −0.4 eV of the Fermi level, the 4f7/2 and the 4f5/2 multiplet, respectively. The atom and symmetry projected densities of states (DOS) shown in Fig. 3 clarify the characters of the bands. Because only the DOS distribution near the Fermi level determines the magnetic properties, we concentrate our attention upon the DOS in the vicinity of the Fermi level. At this range, the valence states for Ce or Co atoms are dominated by 4f and 3d electrons. From the Fig. 3, it can be found that within the LDA calculation, the DOS at the Fermi level have a high peak near the Fermi level and they mainly consist of Ce-derived 4f states. When the spin–orbital coupling along the axis (0 0 1) is taken into account, the 4f orbitals are slightly modified. The spin 4f states become wider and the energy shift between centers becomes larger. This is due to the partial splitting between the degenerate 4f states. Band calculations within the LDA framework cannot yield the correct magnetic moment for many Ce compounds because of the strong correlation interaction between f orbitals. The spin–orbit interaction in these systems is sometimes large
318
T. Jeong / Solid State Communications 141 (2007) 316–319
Fig. 3. Projected density of states of CeCoGe3 . The Ce 4f character dominates the states near the Fermi level.
uniform magnetic susceptibility using the method of Janak [11]. The uniform magnetic susceptibility of a metal can be written as χ=
Fig. 2. Top panel: The LDA band structure of CeCoGe3 along symmetry lines. The very flat bands near the Fermi level are the Ce 4f bands. Middle panel: The band structure within the LDA + U scheme showing that the 4f bands are split into three manifolds. Bottom panel: The fully relativistic band structure of CeCoGe3 showing that spin–orbit coupling splits the 4f states into two manifolds.
and the orbital contribution to the magnetic moment cannot be neglected. The calculated magnetic moment of CeCoGe3 within the LDA scheme is 0.22µ B /Ce. This is mainly from the Ce-derived 4f orbitals. When the on-site correlation potential is added to the Ce 4f electron, the degeneracy between the different f orbits would be lifted and Hund’s rules dominate the locally occupied 4f electrons, which yields the total magnetic moment 0.18µ B /Ce. With a fully relativistic scheme we calculated the magnetic moment with a value 0.17µ B /Ce. Density functional calculations are very reliable in calculating the instability to ferromagnetism. The presence of an electronic instability is signaled by a divergence of the corresponding susceptibility. In the following we study the
χ0 , 1 − N (E F )I
(1)
where the numerator stands for the Pauli susceptibility of a gas of non-interacting electrons proportional to the density of states at the Fermi level N (E F ), and the denominator represents the enhancement due to electron–electron interactions. Within the Kohn–Sham formalism of density functional theory the Stoner parameter I is related to the second derivative of the exchange-correlation functional with respect to the magnetization density. We have evaluated, within the density functional theory formalism, the Stoner enhancement of the susceptibility χ = 1−I χN0(E F ) ≡ Sχ0 , where χ0 = 2µ2B N (E F ) is the non-interacting susceptibility and S gives the electron–electron enhancement in terms of the Stoner constant I . We have calculated I using both the Janak–Vosko–Perdew theory [11] and fixed spin moment calculations [12]. The calculated density of states and normalized Stoner parameter are N (E F ) = 5.08 states/eV and I = 0.24 eV. This gives I N (E F ) = 1.21, larger than unity, corresponding to a ferromagnetic instability. Heavy fermion compounds are characterized by larger electronic specific heat coefficients γ . CeCoGe3 is a moderate heavy fermion compound with γ = 111 mJ/K2 mol [2]. The large specific heat coefficient of CeCoGe3 compounds could not be got from our band calculation. This can be seen from the calculated electronic structure. It can be found that the total number of DOS at the Fermi level is about 5.08 states/eV, which corresponds to γb = 11.9 mJ/K2 mol, underestimating the experimental value by a factor of 9.3. The discrepancy between the band calculation and experiment for the specific heat coefficient is attributed to the formation of quasiparticles. There is an exchange interaction J between the local f and the conduction electrons in CeCoGe3 . The ground state of Ce compounds is determined by the competition between the Kondo and indirect RKKY interaction. With a large J , the Kondo coupling becomes strong and the system locates at the borderline of magnetic–nonmagnetic transition. The exchange interaction between the local f electron and the conduction
T. Jeong / Solid State Communications 141 (2007) 316–319
electrons will result in the formation of quasiparticles. This has a larger mass compared with bare electrons and the enhancement of mass increases with the increase of the exchange. Because of the volume contraction, the exchange interaction between the f and the conduction electrons is large in CeCoGe3 . This will result in the f electrons behaving like itinerant electrons and the narrow f bands locating at the Fermi level. On the other hand, when the exchanging interaction between f and conduction electrons is smaller, the occupied 4f orbitals are located near the Fermi level while the unoccupied 4f orbitals are at the conduction bands. The quasiparticle mass is appropriate to the number of DOS at the Fermi level. So the quasiparticle mass is greatly enhanced in CeCoGe3 . Indeed, it has been shown that when the Ce 4f electrons in CeCoGe3 are treated as localized electrons, the quasiparticle mass was enhanced over the band calculation by a factor 9.3. At this point we need to discuss a remaining question: which band structure (LDA, LDA + U or LDA + SOC) is more realistic? In LDA + SOC (the fully relativistic scheme), SOC is applied at the one-electron level, that is, each 4f orbital acquires its own j = l ± s( 72 or 72 ) label and character. In LDA + UP, one can follow Hund’s rules P to some extent: maximize S = i si , and maximize L = i li , and study various values of J (or even of L). In some respects, such for the ground state energy to determine the equation of state, the LDA+U approach seems most appropriate for rare earth ions. There are several instances of experience with similar 4f configurations which imply that the LDA + U correction is essential; simple LDA often leaves occupied 4f states at too high an energy and unoccupied ones at too low an energy, so there is excessive hybridization at the Fermi level. One example is that of Gd metal [9], another is provided by the Eu monopnictides [10]. On the other hand, 4f single particle-like excitations are likely to be characterized by the 27 , 52 labels. The excitations f0 → f1 or f14 → f13 are likely to show SOC aspects directly. Therefore further experimental investigations, such as the angle-resolved photoemission spectroscopy technique (a powerful technique to directly probe the electronic structure of solids) should be quite interesting. 5. Summary In this article we showed the results of three different electronic band structure calculations. It shows that the
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Coulomb potential effect on Ce 4f orbitals and the spin–orbit interaction are key factors to understanding the electronic and magnetic properties of CeCoGe3 . When the Coulomb potential is added to the Ce 4f orbitals, the degeneracy between the different f orbits would be lifted and they are split into lower Hubbard bands at the Fermi level and unoccupied upper Hubbard bands in the conduction band. The exchange interaction between local f electrons and conduction electrons plays an important role in their heavy fermion characters. And the fully relativistic band structure scheme shows that spin–orbit coupling splits the 4f states into two manifolds, the 4f7/2 and the 4f5/2 multiplet. The f-electrons can be delocalized through hybridization with conduction electrons. If there exists no hybridization between f and conduction electrons, the felectron number becomes just an integer and the f-electrons are localized. The almost localized f-electrons and delocalized conduction electrons hybridize to deviate the f-electron number from an integer value, and keep the metallic state against the strong electron correlation between f-electrons. Therefore the hybridization between f and conduction d electrons plays an important role in CeCoGe3 . References [1] G.R. Stewart, Rev. Mod. Phys. 73 (2001) 797. [2] V.K. Pecharsky, O.B. Hyun, K.A. Gschneidner, Phys. Rev. B 47 (1993) 11839. [3] D.H. Eom, et al., J. Phys. Soc. Japan 67 (1998) 2495. [4] V.V. Krishnamurthy, K. Nagamine, I. Watanabe, K. Nyshiyama, S. Ohira, M. Ishikawa, D.H. Eom, T. Ishikawa, T.M. Briere, Phys. Rev. Lett. 88 (2002) 046402. [5] K. Koepernik, H. Eschrig, Phys. Rev. B 59 (1999) 1743; H. Eschrig, Optimized LCAO Method and the Electronic Structure of Extended Systems, Springer, Berlin, 1989. [6] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [7] A.I. Liechtenstein, V.I. Anisimov, J. Zaanen, Phys. Rev. B 52 (1995) R5468; A.B. Shick, A.I. Liechtenstein, W.E. Pickett, Phys. Rev. B 60 (1999) 10728. [8] A.B. Shick, D.L. Novikov, A.J. Freeman, Phys. Rev. B 57 (1997) R14259. [9] A.B. Schik, W.E. Pickett, C.S. Fadley, J. Appl. Phys. 87 (2000) 5878. [10] J. Kunes, W. Ku, W.E. Pickett, J. Phys. Soc. Japan 74 (2005) 1408. [11] J.F. Janak, Phys. Rev. B 16 (1977) 255; S.H. Vosko, J.P. Perdew, Can. J. Phys. 53 (1975) 1385. [12] K. Schwarz, P. Mohn, J. Phys. F 14 (1984) L129.