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Ab initio studies on magnetism and hybridization of the ternary germanide Ce3Ni2Ge7
Solid State Communications 149 (2009) 2260–2263
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Ab initio studies on magnetism and hybridization of the ternary germanide Ce3 Ni2 Ge7 Y.S. Zhang ∗ , K.L. Yao, Z.L. Liu, X.L. Wang, L.H. Yu Department of Physics, Huazhong University of Science and Technology, WuHan 430074, China
article
info
Article history: Received 21 July 2006 Received in revised form 3 July 2009 Accepted 27 August 2009 by J. A. Brum Available online 31 August 2009 PACS: 71.15.Ap 71.15.Mb 71.20.-b
abstract We perform a theoretical investigation on the magnetism and orbital hybridization in ternary germanide Ce3 Ni2 Ge7 using the full-potential linearized augmented plane wave method (FP_LAPW) based on the density functional theory (DFT). The calculation with local spin density approximation (LSDA) predicts that there are two states for the Ce atoms due to the different environment: one (Ce1 ) is near the nonmagnetic state and the other (Ce2 ) is localized and magnetic. The orbital hybridization plays a key role in determining the state of Ce. On adding on-site Coulomb potential to the localized Ce2 -4f orbit, the magnetic moment obtained from our calculation fits well with the experimental value. © 2009 Elsevier Ltd. All rights reserved.
Keywords: A. Magnetically ordered materials D. Electronic band structure
1. Introduction Ce intermetallic compounds have received a lot of attention from the theoretical and experimental points of view. Many phenomena, such as magnetic order, Kondo effect, intermediate valence and heavy fermion behavior, can occur in these Ce systems [1]. These phenomena are related to the nature of the Ce 4f electron in compounds. But the nature of 4f electrons in these compounds is often controversial. They are localized and subjected to the on-site Coulomb interaction (strongly correlated system) in many cases. They can also hybridize with other orbits and become itinerant when the 4f state is energetically very close to the Fermi level. The competition between these two interactions determines the nature of 4f orbit [2]. Hence the states of Ce in compounds can be magnetic (ferromagnetism, anti-ferromagnetism), intermediate valence and nonmagnetic in different environments. So it may be interesting to explore the magnetism and nature of the 4f electron in the Ce intermetallic system. Ab initio calculation based on the density functional theory (DFT) has become one of the most important tools in condensed physics. It provides a self-consistent framework to calculate the ground-state features of a material. We can gain knowledge about magnetic moments of Ce in compounds from the ab initio
∗
Corresponding author. E-mail address: [email protected] (Y.S. Zhang).
0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.08.038
calculation, which helps us to determine the nature of 4f electrons in Ce compounds. So here we will use this kind of method to calculate the electronic structure of one ternary germanide Ce3 Ni2 Ge7 . As generally known, the exchange and correlation interaction in DFT (Kohn–Sham equation) is often calculated by local spin density approximation (LSDA) or generalized-gradient approximation (GGA). But this is not accurate when there are localized 3d electrons of the transition metal or 4f electrons of the rare-earth metal, and the discrepancy between the experiment and calculation may be obvious. This can be found in the calculation of CeIn3 [1], which has localized 4f electrons, and the magnetic moment of Ce obtained from the calculation based on DFT + GGA is much smaller than that from the experiment. This is due to the underestimation of LSDA (GGA) to the orbital moment: the localization is managed by the on-site Coulomb repulsion U in a strongly localized system, while it is controlled by the Stoner parameter J in DFT with L(S)DA or GGA. The Stoner parameter J is often an order of magnitude smaller than the Coulomb repulsion U. Adding the Coulomb repulsion U on the localized orbit will make the solution correspond to ‘‘split’’ bands, as the occupied states go down by approximately 12 U and unoccupied states go up by 12 U. In the case of the strongly correlated 4f states of a rare-earth metal, such a splitting, as a rule, is very large and the orbital degeneracy will be lifted [3]. So it is always difficult to get quantitatively magnetic results in agreement with the experiment for a Ce-based compound from the DFT + LSDA (GGA) calculation. But in some cases qualitatively describing the state of Ce by theoretical calculation will give us a primary picture about the
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magnetic behavior of Ce. And we can choose different methods to treat different Ce atoms in the compound from this primary understanding. One method to qualitatively describe the state of Ce in the compound has been suggested by V.L. Vildosola et al. They have calculated a series of Ce-based compounds [2] with the full-potential linearized augment plane wave method (FP_LAPW) within the framework of DFT + GGA. They show that one can predict the states of Ce in intermetallic system from spin moment µce of Ce obtained from DFT + GGA calculation: it is magnetic when µce > 0.5 µB , intermediate valence when µce < 0.5µ, and nonmagnetic when µce = 0. This conclusion is verified by the calculation of doped system Ce(Pd1−x Mx ), where M is Rh, Ni and Ag, respectively [4]. But all these situations are simple (contain only one state of Ce atom). It is unknown if this conclusion is also valid to complicated Ce-based compound. Calculation on more complicated Ce compound will help us to solve this. Ce3 Ni2 Ge7 is a ternary germanide compound. Durivault et al. [5] find that there are two different sites occupied by atom Ce in this compound: one is 2d(0, 0, 0.5) and the other is 4i(0, 0.3169, 0) (named as Ce1 and Ce2 in this paper, respectively). These two kinds of Ce have different states: Ce1 is intermediate valence and Ce2 is magnetic. So it is interesting to ask if we can also describe a compound with more than one state of Ce like Ce3 Ni2 Ge7 within the framework of DFT + LSDA. This is one aim of this paper. But just as we have mentioned above and will see below, this DFT + LSDA calculation only can give us a qualitative result. We can get the different states for different Ce atoms, but the magnetic moment of magnetic Ce2 is far from the value observed from experiment because it is localized and acts like a free Ce3+ ion. Quantitative description about it must take into account the on-site Coulomb repulsion U. This will be seen clearly in the following.
Table 1 The magnetic moments of Ce1 and Ce2 obtained from different methods. Spin (Ce1 ) and Orbit (Ce1 ) are the contributions of the spin polarization and orbital polarization to the magnetic moment of Ce1 respectively, while Spin (Ce2 ) and Orbit (Ce2 ) are the contributions of the spin polarization and orbital polarization to the magnetic moment of Ce2 respectively. The unit is µB .
2. Computational details
magnetic moment is almost absent on atom Ce1 . Hence from the spin-polarized calculation without considering the on-site Coulomb repulsion U, we can predict the coexistence of two states for Ce in Ce3 Ni2 Ge7 compound. Here we must point out that, the exchange and correlation potential in the case of Vildosola et al. [2] is the GGA of Perdew and Wang [10], while it is LSDA in our case. This means that their conclusion can also be valid in the LSDA case. Different states of Ce are the result of hybridization between Ce 4f electrons and other orbital electron. The hybridization strength is influenced by the surrounding environment and chemical nature of the near ligand. V.L. Vildosola has discussed the effect of hybridization between Ce and different chemical ligands [2]. But the neighbor chemical ligands of Ce1 and Ce2 are similar in Ce3 Ni2 Ge7 . Hence the different hybridizations mainly come from the surrounding environment. In Ce3 Ni2 Ge7 , all the distance of Ce1 –Ni and Ce1 –Ge are shorter than their homologue concerning the Ce2 atom [5]. So we can imagine that Ce1 hybridize heavier with its neighboring atoms than Ce2 does, which leads to different states for them. This can be seen clearly from the valence charge density near the Fermi level plotted in Fig. 1. The charge density is projected into the (101) plane, which contains those atoms that have the shortest distances with atom Ce1 or Ce2 . The atom that has the shortest distance with Ce1 atoms is Ge1at sit 2b(0.5,0,0), while that has the shortest distance to Ce2 is atom Ge4 at site 4j(0,0.2217,0.5). The plot can give us a direct picture about the hybridization between different atoms. From Fig. 1, we can find that Ce1 hybridize with Ge1 and Ni, while Ni hybridize with Ce1 and Ge4 , Ni and Ge4 hybridize with each other and form a single to hybridize with Ce2 . It can be see clearly that the hybridization between atom Ce1 and its nearest neighbor atom is larger than that between Ce2 and correspondingly nearest neighbor atom. This is in agreement with our assumption. The densities of states (DOS) obtained from the LSDA calculation are plotted in Fig. 2. The valence states of Ce, Ni and Ge near the
Ce3 Ni2 Ge7 compound crystallizes in an orthorhombic cell with the space group Cmmm (a = 4.2327 Å, b = 25.5517 Å, c = 4.2896 Å) [5]. We performed ab initio calculation using the FP_LAPW method implemented into Wien2k code [6]. Exchange– correlation potential is treated with LSDA in all calculations. The calculations have been done with or without on-site Coulomb potential for the 4f orbitals of Ce2 . There are several attempts to improve the L(S)DA in order to take into account strong electron–electron correlations [7], here we will use the method introduced by Czyżyk and Sawatzky [8] which has been implemented into the computer code wien2k (LSDA + U method). Its validity in the Wien2k code computer code has been proven before [9]. We have selected U ≈ 7 ev and J ≈ 0.7 ev. These values have been applied to similar compound CeSb [3]. 300 K-points in the whole Brillouin zone are used for convergence. We have tested with larger number of K-points and there is no obvious influence on the results. The radii of atomic spheres we chosen is (in atomic unit) 2.8, 2.3, 2.0 for Ce, Ni, Ge respectively. The cutoff parameter is taken as Rmt Kmax = 7, where Kmax is the maximal value of the reciprocal lattice vector used in the plane wave expansion. The total energy was converged up to 10−4 Ry/molecular. 3. Results and discussion The spin and/or orbital magnetic moments of Ce1 and Ce2 obtained from calculations are listed in Table 1. The LSDA calculation without on-site Coulomb repulsion yields the spin magnetic moments of Ce1 and Ce2 to be 0.05 µB and 0.67µB , respectively. According to the result of Vildosola, et al. [2], Ce1 should be intermediate valence or close to nonmagnetic while Ce2 should be magnetic. This is in agreement with the magnetization measurement [5]: only Ce2 carries a magnetic moment (1.98 µB ) while
Method
Spin (Ce1 )
Orbit (Ce1 )
Spin (Ce2 )
Orbit (Ce2 )
LSDA LSDA+SO LSDA+SO+U
0.05 0.08 0.01
– – –
0.67 0.61 0.96
–
−0.67 −2.88
Fig. 1. Valence charge density near the Fermi level projected onto the (101) plane of Ce3 Ni2 Ge7 .
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Fig. 3. The partial DOS of Ce1 -4f obtained from LSDA + U calculation. Positive and negative values mean spin-up and spin-down states respectively. X-axis zero level corresponds to the Fermi level.
Fig. 2. The partial DOS of Ce3 Ni2 Ge7 obtained from LSDA spin-polarized calculation. Positive and negative values mean spin-up and spin-down states respectively. X-axis zero level corresponds to the Fermi level.
Fermi level are dominated by 4f, 3d and 4p electrons, respectively. So only the partial DOS of Ce-derived 4f, Ni-derived 3d and Gederived 4p states are plotted. The Ni-derived 3d or Ge-derived 4p states form wide band below Fermi level and hybridize with the Ce-derived 4f states at the Fermi level. The occupied 3d states of the spin-up and spin-down almost have the same weight in the same energy, which means that Ni loses it is permanently magnetic moment and become a nonmagnetic atom. Hence the magnetic moment of Ni is only about 0.02 µB in our calculation. The mainly occupied states near the Fermi level are Ce-derived 4f states. The 4f spin-up and spin-down sates of Ce1 locate at the position about 0.5 ev higher than the Fermi level and cross the Fermi level, while there is a small shift between the spin-up and spin-down 4f states of Ce2 . The mainly occupied states for Ce2 -4f are spin-up states and their center position is closer to the Fermi level than that of Ce1 4f, while correspondingly spin-down state centered above the Ce1 4f states and are almost empty. This is similar with the electronic structure of a free Ce3+ ion: one 4f electron occupies the spinup state and leaves the spin-down state empty. So we can get a conclusion that because of the weaker hybridization between Ce2 and its neighbor ions, Ce2 has preserved more character of a free Ce3+ ion than Ce1 does. Ce1 loses its partial 4f charge and the initially occupied state is pushed up form the Fermi level to the conduction band. The gap between occupied states and unoccupied states becomes smaller and even go to zero. Then partial 4f electron will probably go to fill the initially unoccupied state. So the spindown state of 4f orbit also has a tail in the valence band. From the charge densities and DOS presented above, we have initial understanding about the different states of the Ce ions in
ternary Ce3 Ni2 Ge7 . But there is still some question unresolved. The experimental measurement shows that there is almost no measurable magnetic moment on Ce1 while 1.98 µB on Ce2 in Ce3 Ni2 Ge7 . Obviously, magnetic moment on atom Ce2 obtained from experiment is much larger than the value we have obtained from LSDA calculation (Table 1). Only adding the spin–orbit coupling will not solve this problem (even worse result when taking into account the total magnetic moment, which can be seen from Table 1). This discrepancy comes from treating localized Ce2 with the LSDA. Just like we analyzed above, Ce2 acts like a free ion because of the weaker hybridization with its near neighbor. Its 4f electron behave as localized charge surrounding the free Ce3+ ion other than as Bloch extended electron. So the strongly correlated effect must be taken into account to explain the magnetic behavior of the Ce2 . The contribution of spin polarization and orbital polarization to the magnetic moment with the LSDA + U method mentioned above are also listed in Table 1. Here we add on-site Coulomb repulsion U only on the 4f orbits of Ce2 . The magnetic moment of the spin polarization and orbital polarization of Ce2 are 0.96 µB and −2.88 µB , respectively, then the total moment is −1.96 µB , which is very close to the experimental value (1.98 µB ). The DOS of Ce1 -4f and Ce2 -4f states are plotted in Figs. 3 and 4, respectively (we do not plot other states because the on-site Coulomb U does not affect them much). Ce1 -4f states near the Fermi level now split into several peaks, but the relative position between spin-up and correspondingly spin-down states does not change much. For Ce2 , the spin-up states split into sever peak because of the Coulomb repulsion U: the occupied spin-up states are pushed down below the Fermi level while unoccupied spin-up states are pushed up from the origin position. Unoccupied spindown states also split into several peak above the Fermi level. Because of the band splitting, the orbital degeneracy is lifted, and the Hude rule playing a key role in the free Ce ion is activated. Then a large moment from the orbit polarization is presented, which gives a good fit to the experiment. Here we should mentioned that Samir F. Matar et al. have also calculated the electronic structure of Ce3 Ni2 Ge7 by local spin density functional theory using the augmented spherical wave (ASW) method before [11]. The Spin only moments from their LSDA computations are 0.007 µB and 0.62 µB , respectively. This also confirms that there are two kinds of cerium ions in Ce3 Ni2 Ge7 . To overcome the disagreement between experimental magnetic moment and correspondingly theoretical value of magnetic cerium ion, they have adopted the orbital field (OR) schemes to treat the correlations leading to Hund’s second rule for localized electronic
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we expect our result to give a better and realistic description than that of Samir F. Matar et al. 4. Summary
Fig. 4. The partial DOS of Ce2 -4f obtained from LSDA + U calculation. Positive and negative values mean spin-up and spin-down states respectively. X-axis zero level corresponds to the Fermi level.
states. The orbital moment on magnetic cerium ions is found to be about −2.5 µB , and the corresponding total magnetic moment is about 1.88 µB , which is close to the experimental value. The corresponding DOS distribution obtained from OR schemes is similar with that of our LSDA calculation without on-site correlated potential U: there is a negligible energy shift between the majority and minority DOS in intermediate cerium ion, while a finite shift (exchange splitting) in the magnetic cerium ion. But the energy positions of Ce-derived 4f are also near the Fermi level and not changed dramatically as compared to LSDA case. This may due to only spin exchange interaction that leading to Hund’s second rule is taken in orbital field (OR) schemes. In our LSDA + U approach, the on-site Coulomb potential between localized Co-4f is added. Adding the Coulomb repulsion U on the localized orbit will make occupied states go down and unoccupied states go up. Hence the occupied Ce2 -derived 4f spin-up states are pushed down below the Fermi level while correspondingly unoccupied 4f spin-up states are pushed up from the origin position. Its unoccupied 4f spindown states also split into several peak above the Fermi level. So the states near the Fermi level are decreased largely and now are mainly composed by Ce1 -4f, Ni-3d and Ge-2p orbitals. Since the coupling strength between Ce 4f and conduction electrons (Kondo interaction) is a key factor to determine the features of system containing cerium ions, it is important to determine the relative energy position to Fermi level for cerium ions. Because our calculated results are more in agreement with the experiment ( µce ≈ 1.96 µB ), and the distribution of Ce2 -4f states is similar to the result obtained when treating the 4f as a part of the core [12],
We investigated the electronic structure of Ce-based intermetallic compound Ce3 Ni2 Ge7 based on the DFT calculation in this paper. The calculation with the standard LSDA predicts that there exist two kinds of states of Ce in the compound, which is in agreement with the experimental measurement. The DOSs and charge densities from the LSDA calculation show that the hybridization between Ce1 and its nearest neighbor is stronger than that of its analogue Ce2 with nearest neighbors. So Ce1 is intermediate valence (close to nonmagnetic) while Ce2 preserves the character of a free ion. Adding the on-site Coulomb repulsion U on the Ce2 -4f orbit, the occupied and unoccupied states of 4f orbit shift down and up respectively. Orbital degeneracy is lifted and the orbital polarization and spin polarization combine together to give a good fit to the magnetic moment obtained from the experiment. Acknowledgements The authors would like to acknowledge the support from the National Natural Science Foundation of China (Nos. 10574047 and 10574048) and the Major Project of the National Natural Science Foundation of China (No. 20490210). This work is also supported by the National 973 Project under Grant. No. 2006CB921605 (K.L. Yao) and 2006CB921606 (Z.L. Liu). References [1] M.V. Lalic, J. Mestnik-Filho, A.W. Carbonari, R.N. Saxena, Phys. Rev. B 65 (2001) 054405. [2] V.L. Vildosola, A.M. Llois, Phys. Rev. B 62 (2000) 7027. [3] A.I. Liechtenstein, V.P. Antropov, B.N. Harmon, Phys. Rev. B 49 (1994) 10770. [4] V.L. Vildosola, A.M. Llois, M. Weissmann, J.G. Sereni, J. Magn. Magn. Mater. 236 (2001) 6. [5] L. Durivault, F. Bourée, B. Chevalier, G. André, J. Etourneau, O. Isnard, J. Magn. Magn. Mater. 232 (2001) 139. [6] P. Blaha, K. Scharz, G.K. Madsen, D. Kvasnicka, J. Lutiz, WIEN2k, An augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties, Karlheinz Schwarz, Techn, Universität, Wien, Austria, ISBN: 3-9501031-1-2, 2001. [7] Vladimir I. Anisimov, F. Aryasetiawan, A.I. Lichtenstein, J. Phys. Condens. Matter 9 (1997) 767. [8] M.T. Czyżyk, G.A. Sawatzky, Phys. Rev. B 49 (1994) 14211. [9] P. Novák, F. Bouchér, P. Gressier, P. Blaha, K. Schwarz, Phys. Rev. B 63 (2001) 235114. [10] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [11] Samir F. Matar, Bernard Chevalier, Olivier Isnard, Jean Etourneau, J. Mater. Chem. 13 (2003) 916. [12] E.K.R. Runge, R.C. Albers, Phys. Rev. B 51 (1995) 10375.
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Ab initio studies on magnetism and hybridization of the ternary germanide Ce3 Ni2 Ge7 Y.S. Zhang ∗ , K.L. Yao, Z.L. Liu, X.L. Wang, L.H. Yu Department of Physics, Huazhong University of Science and Technology, WuHan 430074, China
article
info
Article history: Received 21 July 2006 Received in revised form 3 July 2009 Accepted 27 August 2009 by J. A. Brum Available online 31 August 2009 PACS: 71.15.Ap 71.15.Mb 71.20.-b
abstract We perform a theoretical investigation on the magnetism and orbital hybridization in ternary germanide Ce3 Ni2 Ge7 using the full-potential linearized augmented plane wave method (FP_LAPW) based on the density functional theory (DFT). The calculation with local spin density approximation (LSDA) predicts that there are two states for the Ce atoms due to the different environment: one (Ce1 ) is near the nonmagnetic state and the other (Ce2 ) is localized and magnetic. The orbital hybridization plays a key role in determining the state of Ce. On adding on-site Coulomb potential to the localized Ce2 -4f orbit, the magnetic moment obtained from our calculation fits well with the experimental value. © 2009 Elsevier Ltd. All rights reserved.
Keywords: A. Magnetically ordered materials D. Electronic band structure
1. Introduction Ce intermetallic compounds have received a lot of attention from the theoretical and experimental points of view. Many phenomena, such as magnetic order, Kondo effect, intermediate valence and heavy fermion behavior, can occur in these Ce systems [1]. These phenomena are related to the nature of the Ce 4f electron in compounds. But the nature of 4f electrons in these compounds is often controversial. They are localized and subjected to the on-site Coulomb interaction (strongly correlated system) in many cases. They can also hybridize with other orbits and become itinerant when the 4f state is energetically very close to the Fermi level. The competition between these two interactions determines the nature of 4f orbit [2]. Hence the states of Ce in compounds can be magnetic (ferromagnetism, anti-ferromagnetism), intermediate valence and nonmagnetic in different environments. So it may be interesting to explore the magnetism and nature of the 4f electron in the Ce intermetallic system. Ab initio calculation based on the density functional theory (DFT) has become one of the most important tools in condensed physics. It provides a self-consistent framework to calculate the ground-state features of a material. We can gain knowledge about magnetic moments of Ce in compounds from the ab initio
∗
Corresponding author. E-mail address: [email protected] (Y.S. Zhang).
0038-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2009.08.038
calculation, which helps us to determine the nature of 4f electrons in Ce compounds. So here we will use this kind of method to calculate the electronic structure of one ternary germanide Ce3 Ni2 Ge7 . As generally known, the exchange and correlation interaction in DFT (Kohn–Sham equation) is often calculated by local spin density approximation (LSDA) or generalized-gradient approximation (GGA). But this is not accurate when there are localized 3d electrons of the transition metal or 4f electrons of the rare-earth metal, and the discrepancy between the experiment and calculation may be obvious. This can be found in the calculation of CeIn3 [1], which has localized 4f electrons, and the magnetic moment of Ce obtained from the calculation based on DFT + GGA is much smaller than that from the experiment. This is due to the underestimation of LSDA (GGA) to the orbital moment: the localization is managed by the on-site Coulomb repulsion U in a strongly localized system, while it is controlled by the Stoner parameter J in DFT with L(S)DA or GGA. The Stoner parameter J is often an order of magnitude smaller than the Coulomb repulsion U. Adding the Coulomb repulsion U on the localized orbit will make the solution correspond to ‘‘split’’ bands, as the occupied states go down by approximately 12 U and unoccupied states go up by 12 U. In the case of the strongly correlated 4f states of a rare-earth metal, such a splitting, as a rule, is very large and the orbital degeneracy will be lifted [3]. So it is always difficult to get quantitatively magnetic results in agreement with the experiment for a Ce-based compound from the DFT + LSDA (GGA) calculation. But in some cases qualitatively describing the state of Ce by theoretical calculation will give us a primary picture about the
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magnetic behavior of Ce. And we can choose different methods to treat different Ce atoms in the compound from this primary understanding. One method to qualitatively describe the state of Ce in the compound has been suggested by V.L. Vildosola et al. They have calculated a series of Ce-based compounds [2] with the full-potential linearized augment plane wave method (FP_LAPW) within the framework of DFT + GGA. They show that one can predict the states of Ce in intermetallic system from spin moment µce of Ce obtained from DFT + GGA calculation: it is magnetic when µce > 0.5 µB , intermediate valence when µce < 0.5µ, and nonmagnetic when µce = 0. This conclusion is verified by the calculation of doped system Ce(Pd1−x Mx ), where M is Rh, Ni and Ag, respectively [4]. But all these situations are simple (contain only one state of Ce atom). It is unknown if this conclusion is also valid to complicated Ce-based compound. Calculation on more complicated Ce compound will help us to solve this. Ce3 Ni2 Ge7 is a ternary germanide compound. Durivault et al. [5] find that there are two different sites occupied by atom Ce in this compound: one is 2d(0, 0, 0.5) and the other is 4i(0, 0.3169, 0) (named as Ce1 and Ce2 in this paper, respectively). These two kinds of Ce have different states: Ce1 is intermediate valence and Ce2 is magnetic. So it is interesting to ask if we can also describe a compound with more than one state of Ce like Ce3 Ni2 Ge7 within the framework of DFT + LSDA. This is one aim of this paper. But just as we have mentioned above and will see below, this DFT + LSDA calculation only can give us a qualitative result. We can get the different states for different Ce atoms, but the magnetic moment of magnetic Ce2 is far from the value observed from experiment because it is localized and acts like a free Ce3+ ion. Quantitative description about it must take into account the on-site Coulomb repulsion U. This will be seen clearly in the following.
Table 1 The magnetic moments of Ce1 and Ce2 obtained from different methods. Spin (Ce1 ) and Orbit (Ce1 ) are the contributions of the spin polarization and orbital polarization to the magnetic moment of Ce1 respectively, while Spin (Ce2 ) and Orbit (Ce2 ) are the contributions of the spin polarization and orbital polarization to the magnetic moment of Ce2 respectively. The unit is µB .
2. Computational details
magnetic moment is almost absent on atom Ce1 . Hence from the spin-polarized calculation without considering the on-site Coulomb repulsion U, we can predict the coexistence of two states for Ce in Ce3 Ni2 Ge7 compound. Here we must point out that, the exchange and correlation potential in the case of Vildosola et al. [2] is the GGA of Perdew and Wang [10], while it is LSDA in our case. This means that their conclusion can also be valid in the LSDA case. Different states of Ce are the result of hybridization between Ce 4f electrons and other orbital electron. The hybridization strength is influenced by the surrounding environment and chemical nature of the near ligand. V.L. Vildosola has discussed the effect of hybridization between Ce and different chemical ligands [2]. But the neighbor chemical ligands of Ce1 and Ce2 are similar in Ce3 Ni2 Ge7 . Hence the different hybridizations mainly come from the surrounding environment. In Ce3 Ni2 Ge7 , all the distance of Ce1 –Ni and Ce1 –Ge are shorter than their homologue concerning the Ce2 atom [5]. So we can imagine that Ce1 hybridize heavier with its neighboring atoms than Ce2 does, which leads to different states for them. This can be seen clearly from the valence charge density near the Fermi level plotted in Fig. 1. The charge density is projected into the (101) plane, which contains those atoms that have the shortest distances with atom Ce1 or Ce2 . The atom that has the shortest distance with Ce1 atoms is Ge1at sit 2b(0.5,0,0), while that has the shortest distance to Ce2 is atom Ge4 at site 4j(0,0.2217,0.5). The plot can give us a direct picture about the hybridization between different atoms. From Fig. 1, we can find that Ce1 hybridize with Ge1 and Ni, while Ni hybridize with Ce1 and Ge4 , Ni and Ge4 hybridize with each other and form a single to hybridize with Ce2 . It can be see clearly that the hybridization between atom Ce1 and its nearest neighbor atom is larger than that between Ce2 and correspondingly nearest neighbor atom. This is in agreement with our assumption. The densities of states (DOS) obtained from the LSDA calculation are plotted in Fig. 2. The valence states of Ce, Ni and Ge near the
Ce3 Ni2 Ge7 compound crystallizes in an orthorhombic cell with the space group Cmmm (a = 4.2327 Å, b = 25.5517 Å, c = 4.2896 Å) [5]. We performed ab initio calculation using the FP_LAPW method implemented into Wien2k code [6]. Exchange– correlation potential is treated with LSDA in all calculations. The calculations have been done with or without on-site Coulomb potential for the 4f orbitals of Ce2 . There are several attempts to improve the L(S)DA in order to take into account strong electron–electron correlations [7], here we will use the method introduced by Czyżyk and Sawatzky [8] which has been implemented into the computer code wien2k (LSDA + U method). Its validity in the Wien2k code computer code has been proven before [9]. We have selected U ≈ 7 ev and J ≈ 0.7 ev. These values have been applied to similar compound CeSb [3]. 300 K-points in the whole Brillouin zone are used for convergence. We have tested with larger number of K-points and there is no obvious influence on the results. The radii of atomic spheres we chosen is (in atomic unit) 2.8, 2.3, 2.0 for Ce, Ni, Ge respectively. The cutoff parameter is taken as Rmt Kmax = 7, where Kmax is the maximal value of the reciprocal lattice vector used in the plane wave expansion. The total energy was converged up to 10−4 Ry/molecular. 3. Results and discussion The spin and/or orbital magnetic moments of Ce1 and Ce2 obtained from calculations are listed in Table 1. The LSDA calculation without on-site Coulomb repulsion yields the spin magnetic moments of Ce1 and Ce2 to be 0.05 µB and 0.67µB , respectively. According to the result of Vildosola, et al. [2], Ce1 should be intermediate valence or close to nonmagnetic while Ce2 should be magnetic. This is in agreement with the magnetization measurement [5]: only Ce2 carries a magnetic moment (1.98 µB ) while
Method
Spin (Ce1 )
Orbit (Ce1 )
Spin (Ce2 )
Orbit (Ce2 )
LSDA LSDA+SO LSDA+SO+U
0.05 0.08 0.01
– – –
0.67 0.61 0.96
–
−0.67 −2.88
Fig. 1. Valence charge density near the Fermi level projected onto the (101) plane of Ce3 Ni2 Ge7 .
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Fig. 3. The partial DOS of Ce1 -4f obtained from LSDA + U calculation. Positive and negative values mean spin-up and spin-down states respectively. X-axis zero level corresponds to the Fermi level.
Fig. 2. The partial DOS of Ce3 Ni2 Ge7 obtained from LSDA spin-polarized calculation. Positive and negative values mean spin-up and spin-down states respectively. X-axis zero level corresponds to the Fermi level.
Fermi level are dominated by 4f, 3d and 4p electrons, respectively. So only the partial DOS of Ce-derived 4f, Ni-derived 3d and Gederived 4p states are plotted. The Ni-derived 3d or Ge-derived 4p states form wide band below Fermi level and hybridize with the Ce-derived 4f states at the Fermi level. The occupied 3d states of the spin-up and spin-down almost have the same weight in the same energy, which means that Ni loses it is permanently magnetic moment and become a nonmagnetic atom. Hence the magnetic moment of Ni is only about 0.02 µB in our calculation. The mainly occupied states near the Fermi level are Ce-derived 4f states. The 4f spin-up and spin-down sates of Ce1 locate at the position about 0.5 ev higher than the Fermi level and cross the Fermi level, while there is a small shift between the spin-up and spin-down 4f states of Ce2 . The mainly occupied states for Ce2 -4f are spin-up states and their center position is closer to the Fermi level than that of Ce1 4f, while correspondingly spin-down state centered above the Ce1 4f states and are almost empty. This is similar with the electronic structure of a free Ce3+ ion: one 4f electron occupies the spinup state and leaves the spin-down state empty. So we can get a conclusion that because of the weaker hybridization between Ce2 and its neighbor ions, Ce2 has preserved more character of a free Ce3+ ion than Ce1 does. Ce1 loses its partial 4f charge and the initially occupied state is pushed up form the Fermi level to the conduction band. The gap between occupied states and unoccupied states becomes smaller and even go to zero. Then partial 4f electron will probably go to fill the initially unoccupied state. So the spindown state of 4f orbit also has a tail in the valence band. From the charge densities and DOS presented above, we have initial understanding about the different states of the Ce ions in
ternary Ce3 Ni2 Ge7 . But there is still some question unresolved. The experimental measurement shows that there is almost no measurable magnetic moment on Ce1 while 1.98 µB on Ce2 in Ce3 Ni2 Ge7 . Obviously, magnetic moment on atom Ce2 obtained from experiment is much larger than the value we have obtained from LSDA calculation (Table 1). Only adding the spin–orbit coupling will not solve this problem (even worse result when taking into account the total magnetic moment, which can be seen from Table 1). This discrepancy comes from treating localized Ce2 with the LSDA. Just like we analyzed above, Ce2 acts like a free ion because of the weaker hybridization with its near neighbor. Its 4f electron behave as localized charge surrounding the free Ce3+ ion other than as Bloch extended electron. So the strongly correlated effect must be taken into account to explain the magnetic behavior of the Ce2 . The contribution of spin polarization and orbital polarization to the magnetic moment with the LSDA + U method mentioned above are also listed in Table 1. Here we add on-site Coulomb repulsion U only on the 4f orbits of Ce2 . The magnetic moment of the spin polarization and orbital polarization of Ce2 are 0.96 µB and −2.88 µB , respectively, then the total moment is −1.96 µB , which is very close to the experimental value (1.98 µB ). The DOS of Ce1 -4f and Ce2 -4f states are plotted in Figs. 3 and 4, respectively (we do not plot other states because the on-site Coulomb U does not affect them much). Ce1 -4f states near the Fermi level now split into several peaks, but the relative position between spin-up and correspondingly spin-down states does not change much. For Ce2 , the spin-up states split into sever peak because of the Coulomb repulsion U: the occupied spin-up states are pushed down below the Fermi level while unoccupied spin-up states are pushed up from the origin position. Unoccupied spindown states also split into several peak above the Fermi level. Because of the band splitting, the orbital degeneracy is lifted, and the Hude rule playing a key role in the free Ce ion is activated. Then a large moment from the orbit polarization is presented, which gives a good fit to the experiment. Here we should mentioned that Samir F. Matar et al. have also calculated the electronic structure of Ce3 Ni2 Ge7 by local spin density functional theory using the augmented spherical wave (ASW) method before [11]. The Spin only moments from their LSDA computations are 0.007 µB and 0.62 µB , respectively. This also confirms that there are two kinds of cerium ions in Ce3 Ni2 Ge7 . To overcome the disagreement between experimental magnetic moment and correspondingly theoretical value of magnetic cerium ion, they have adopted the orbital field (OR) schemes to treat the correlations leading to Hund’s second rule for localized electronic
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we expect our result to give a better and realistic description than that of Samir F. Matar et al. 4. Summary
Fig. 4. The partial DOS of Ce2 -4f obtained from LSDA + U calculation. Positive and negative values mean spin-up and spin-down states respectively. X-axis zero level corresponds to the Fermi level.
states. The orbital moment on magnetic cerium ions is found to be about −2.5 µB , and the corresponding total magnetic moment is about 1.88 µB , which is close to the experimental value. The corresponding DOS distribution obtained from OR schemes is similar with that of our LSDA calculation without on-site correlated potential U: there is a negligible energy shift between the majority and minority DOS in intermediate cerium ion, while a finite shift (exchange splitting) in the magnetic cerium ion. But the energy positions of Ce-derived 4f are also near the Fermi level and not changed dramatically as compared to LSDA case. This may due to only spin exchange interaction that leading to Hund’s second rule is taken in orbital field (OR) schemes. In our LSDA + U approach, the on-site Coulomb potential between localized Co-4f is added. Adding the Coulomb repulsion U on the localized orbit will make occupied states go down and unoccupied states go up. Hence the occupied Ce2 -derived 4f spin-up states are pushed down below the Fermi level while correspondingly unoccupied 4f spin-up states are pushed up from the origin position. Its unoccupied 4f spindown states also split into several peak above the Fermi level. So the states near the Fermi level are decreased largely and now are mainly composed by Ce1 -4f, Ni-3d and Ge-2p orbitals. Since the coupling strength between Ce 4f and conduction electrons (Kondo interaction) is a key factor to determine the features of system containing cerium ions, it is important to determine the relative energy position to Fermi level for cerium ions. Because our calculated results are more in agreement with the experiment ( µce ≈ 1.96 µB ), and the distribution of Ce2 -4f states is similar to the result obtained when treating the 4f as a part of the core [12],
We investigated the electronic structure of Ce-based intermetallic compound Ce3 Ni2 Ge7 based on the DFT calculation in this paper. The calculation with the standard LSDA predicts that there exist two kinds of states of Ce in the compound, which is in agreement with the experimental measurement. The DOSs and charge densities from the LSDA calculation show that the hybridization between Ce1 and its nearest neighbor is stronger than that of its analogue Ce2 with nearest neighbors. So Ce1 is intermediate valence (close to nonmagnetic) while Ce2 preserves the character of a free ion. Adding the on-site Coulomb repulsion U on the Ce2 -4f orbit, the occupied and unoccupied states of 4f orbit shift down and up respectively. Orbital degeneracy is lifted and the orbital polarization and spin polarization combine together to give a good fit to the magnetic moment obtained from the experiment. Acknowledgements The authors would like to acknowledge the support from the National Natural Science Foundation of China (Nos. 10574047 and 10574048) and the Major Project of the National Natural Science Foundation of China (No. 20490210). This work is also supported by the National 973 Project under Grant. No. 2006CB921605 (K.L. Yao) and 2006CB921606 (Z.L. Liu). References [1] M.V. Lalic, J. Mestnik-Filho, A.W. Carbonari, R.N. Saxena, Phys. Rev. B 65 (2001) 054405. [2] V.L. Vildosola, A.M. Llois, Phys. Rev. B 62 (2000) 7027. [3] A.I. Liechtenstein, V.P. Antropov, B.N. Harmon, Phys. Rev. B 49 (1994) 10770. [4] V.L. Vildosola, A.M. Llois, M. Weissmann, J.G. Sereni, J. Magn. Magn. Mater. 236 (2001) 6. [5] L. Durivault, F. Bourée, B. Chevalier, G. André, J. Etourneau, O. Isnard, J. Magn. Magn. Mater. 232 (2001) 139. [6] P. Blaha, K. Scharz, G.K. Madsen, D. Kvasnicka, J. Lutiz, WIEN2k, An augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties, Karlheinz Schwarz, Techn, Universität, Wien, Austria, ISBN: 3-9501031-1-2, 2001. [7] Vladimir I. Anisimov, F. Aryasetiawan, A.I. Lichtenstein, J. Phys. Condens. Matter 9 (1997) 767. [8] M.T. Czyżyk, G.A. Sawatzky, Phys. Rev. B 49 (1994) 14211. [9] P. Novák, F. Bouchér, P. Gressier, P. Blaha, K. Schwarz, Phys. Rev. B 63 (2001) 235114. [10] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [11] Samir F. Matar, Bernard Chevalier, Olivier Isnard, Jean Etourneau, J. Mater. Chem. 13 (2003) 916. [12] E.K.R. Runge, R.C. Albers, Phys. Rev. B 51 (1995) 10375.