Anisotropic Susceptibility of La_2-xSr_xCoO_4 related to the Spin States of Cobalt
N. Hollmann, M. W. Haverkort, M. Cwik, M. Benomar, M. Reuther, A. Tanaka, T. Lorenz
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Anisotropi Sus eptibility of La − x Sr x CoO related tothe Spin States of CobaltN Hollmann , M W Haverkort , M Cwik , M Benomar ,M Reuther , A Tanaka , T Lorenz II. Physikalis hes Institut, University of Cologne, Zülpi herstrasse 77, D-50937 Köln,Germany Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima739-8530, JapanE-mail: hollmannph2.uni-koeln.deAbstra t.We present a study of the magneti sus eptibility of La − x Sr x CoO single rystalsin a doping range . ≤ x ≤ . . Our data shows a pronoun ed magneti anisotropy forall ompounds. This anisotropy is in agreement with a low-spin ground state ( S = 0 )of Co for x ≥ . and a high-spin ground state ( S = 3 / ) of Co . We ompareour data with a rystal-(cid:28)eld model al ulation assuming lo al moments and (cid:28)nd agood des ription of the magneti behavior for x ≥ . . This in ludes the pronoun edkinks observed in the inverse magneti sus eptibility, whi h result from the anisotropyand low-energy ex ited states of Co and are not related to magneti ordering ortemperature-dependent spin-state transitions.PACS numbers: 71.20.Be, 71.70.Ch, 71.70.Ej, 75.30.Gwnisotropi Sus eptibility of La − x Sr x CoO4
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Anisotropi Sus eptibility of La − x Sr x CoO related tothe Spin States of CobaltN Hollmann , M W Haverkort , M Cwik , M Benomar ,M Reuther , A Tanaka , T Lorenz II. Physikalis hes Institut, University of Cologne, Zülpi herstrasse 77, D-50937 Köln,Germany Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima739-8530, JapanE-mail: hollmannph2.uni-koeln.deAbstra t.We present a study of the magneti sus eptibility of La − x Sr x CoO single rystalsin a doping range . ≤ x ≤ . . Our data shows a pronoun ed magneti anisotropy forall ompounds. This anisotropy is in agreement with a low-spin ground state ( S = 0 )of Co for x ≥ . and a high-spin ground state ( S = 3 / ) of Co . We ompareour data with a rystal-(cid:28)eld model al ulation assuming lo al moments and (cid:28)nd agood des ription of the magneti behavior for x ≥ . . This in ludes the pronoun edkinks observed in the inverse magneti sus eptibility, whi h result from the anisotropyand low-energy ex ited states of Co and are not related to magneti ordering ortemperature-dependent spin-state transitions.PACS numbers: 71.20.Be, 71.70.Ch, 71.70.Ej, 75.30.Gwnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 2Transition-metal oxides are known for their omplex interplay between di(cid:27)erentdegrees of freedom like spin, harge and orbitals. In some of these systems anotherinteresting property is found: ions like Co in a rystal-(cid:28)eld environment an o ur indi(cid:27)erent spin states. Additionally, transitions between di(cid:27)erent spin states are possiblefor some ompounds. A prominent example showing this phenomenon is LaCoO ,whi h has been examined and dis ussed sin e the middle of the last entury (see e.g.Refs. [1(cid:21)11℄).The existen e of di(cid:27)erent spin states arises from the ompetition between rystal-(cid:28)eld e(cid:27)e ts and on-site Coulomb intera tion. Crystal (cid:28)elds lift the degenera y of the d states. If the rystal (cid:28)eld is strong, this an lead to a violation of Hund's rules. In the ase of ubi symmetry and in a one-ele tron pi ture, the (cid:28)ve-fold degenerate d statesare split into a three-fold degenerate t g and a two-fold degenerate e g level. The splittingbetween t g and e g states is alled Dq . With a strong rystal (cid:28)eld, the ele trons willbe for ed into the low-spin state (LS). Regarding a d system, this state onsists of anantiparallel alignment of spins with t g e g and S = 0 . On the other hand, the Coulombintera tion manifests itself in the e(cid:27)e t of Hund's oupling. A parallel arrangement ofspins minimizes the ele tron-ele tron repulsion be ause the Pauli prin iple for es theele trons to o upy di(cid:27)erent orbitals. In a weak ubi rystal (cid:28)eld, this e(cid:27)e t willdominate and lead to the high-spin state (HS), whi h is the on(cid:28)guration with thehighest total spin possible in a ordan e with the Pauli prin iple. For a d system, this on(cid:28)guration is t g e g with S = 2 . For Co the rossover between these two di(cid:27)erentspin states o urs roughly at an energy di(cid:27)eren e of Dq = 2 . eV.In the ase of LaCoO , a low-spin ground state for Co was found [2℄. Interestingly,the di(cid:27)eren e between the rystal (cid:28)eld energies and the promotional energies is so smallthat an ex ited state with di(cid:27)erent spin state an be rea hed by thermal ex itation. Thisex ited state has been a subje t of debate for a long time. Apart from the HS statedes ribed above, the intermediate-spin state (IS, t g e g S = 1 ) was also dis ussed [4℄ andreported in many experiments, but it was shown that it might have been onfused witha spin-orbit oupled HS state [11℄.For the layered obaltates La − x Sr x CoO , mu h less is known about the spin stateof Co . Based on magneti measurements [12℄ a HS ground state for x ≤ . and aspin-state transition to an IS ground state for x > . was proposed. This on lusionwas based on a Curie-Weiss analysis of the sus eptibility in a temperature range of100K to 300K and NMR measurements [13℄. Unrestri ted Hartree-Fo k al ulations [14℄showed a slightly di(cid:27)erent pi ture. Here, three di(cid:27)erent magneti phases were found. Anantiferromagneti HS phase ( x < . ), a ferromagneti HS phase ( . ≤ x ≤ . ) andan antiferromagneti LS-HS-ordered phase ( x > . ) were proposed. These al ulationswere also based on the results of the Curie-Weiss analysis in Ref. [12℄. Taking intoa ount that La − x Sr x CoO is an anisotropi material with rather strong spin-orbit oupling ompared to the tetragonal rystal (cid:28)eld splitting, the validity of the Curie-Weiss law is questionable. The aim of this paper is to analyze the magneti sus eptibilityfrom a di(cid:27)erent perspe tive, on entrating on the spin state of Co .nisotropi Sus eptibility of La − x Sr x CoO4
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Anisotropi Sus eptibility of La − x Sr x CoO related tothe Spin States of CobaltN Hollmann , M W Haverkort , M Cwik , M Benomar ,M Reuther , A Tanaka , T Lorenz II. Physikalis hes Institut, University of Cologne, Zülpi herstrasse 77, D-50937 Köln,Germany Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima739-8530, JapanE-mail: hollmannph2.uni-koeln.deAbstra t.We present a study of the magneti sus eptibility of La − x Sr x CoO single rystalsin a doping range . ≤ x ≤ . . Our data shows a pronoun ed magneti anisotropy forall ompounds. This anisotropy is in agreement with a low-spin ground state ( S = 0 )of Co for x ≥ . and a high-spin ground state ( S = 3 / ) of Co . We ompareour data with a rystal-(cid:28)eld model al ulation assuming lo al moments and (cid:28)nd agood des ription of the magneti behavior for x ≥ . . This in ludes the pronoun edkinks observed in the inverse magneti sus eptibility, whi h result from the anisotropyand low-energy ex ited states of Co and are not related to magneti ordering ortemperature-dependent spin-state transitions.PACS numbers: 71.20.Be, 71.70.Ch, 71.70.Ej, 75.30.Gwnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 2Transition-metal oxides are known for their omplex interplay between di(cid:27)erentdegrees of freedom like spin, harge and orbitals. In some of these systems anotherinteresting property is found: ions like Co in a rystal-(cid:28)eld environment an o ur indi(cid:27)erent spin states. Additionally, transitions between di(cid:27)erent spin states are possiblefor some ompounds. A prominent example showing this phenomenon is LaCoO ,whi h has been examined and dis ussed sin e the middle of the last entury (see e.g.Refs. [1(cid:21)11℄).The existen e of di(cid:27)erent spin states arises from the ompetition between rystal-(cid:28)eld e(cid:27)e ts and on-site Coulomb intera tion. Crystal (cid:28)elds lift the degenera y of the d states. If the rystal (cid:28)eld is strong, this an lead to a violation of Hund's rules. In the ase of ubi symmetry and in a one-ele tron pi ture, the (cid:28)ve-fold degenerate d statesare split into a three-fold degenerate t g and a two-fold degenerate e g level. The splittingbetween t g and e g states is alled Dq . With a strong rystal (cid:28)eld, the ele trons willbe for ed into the low-spin state (LS). Regarding a d system, this state onsists of anantiparallel alignment of spins with t g e g and S = 0 . On the other hand, the Coulombintera tion manifests itself in the e(cid:27)e t of Hund's oupling. A parallel arrangement ofspins minimizes the ele tron-ele tron repulsion be ause the Pauli prin iple for es theele trons to o upy di(cid:27)erent orbitals. In a weak ubi rystal (cid:28)eld, this e(cid:27)e t willdominate and lead to the high-spin state (HS), whi h is the on(cid:28)guration with thehighest total spin possible in a ordan e with the Pauli prin iple. For a d system, this on(cid:28)guration is t g e g with S = 2 . For Co the rossover between these two di(cid:27)erentspin states o urs roughly at an energy di(cid:27)eren e of Dq = 2 . eV.In the ase of LaCoO , a low-spin ground state for Co was found [2℄. Interestingly,the di(cid:27)eren e between the rystal (cid:28)eld energies and the promotional energies is so smallthat an ex ited state with di(cid:27)erent spin state an be rea hed by thermal ex itation. Thisex ited state has been a subje t of debate for a long time. Apart from the HS statedes ribed above, the intermediate-spin state (IS, t g e g S = 1 ) was also dis ussed [4℄ andreported in many experiments, but it was shown that it might have been onfused witha spin-orbit oupled HS state [11℄.For the layered obaltates La − x Sr x CoO , mu h less is known about the spin stateof Co . Based on magneti measurements [12℄ a HS ground state for x ≤ . and aspin-state transition to an IS ground state for x > . was proposed. This on lusionwas based on a Curie-Weiss analysis of the sus eptibility in a temperature range of100K to 300K and NMR measurements [13℄. Unrestri ted Hartree-Fo k al ulations [14℄showed a slightly di(cid:27)erent pi ture. Here, three di(cid:27)erent magneti phases were found. Anantiferromagneti HS phase ( x < . ), a ferromagneti HS phase ( . ≤ x ≤ . ) andan antiferromagneti LS-HS-ordered phase ( x > . ) were proposed. These al ulationswere also based on the results of the Curie-Weiss analysis in Ref. [12℄. Taking intoa ount that La − x Sr x CoO is an anisotropi material with rather strong spin-orbit oupling ompared to the tetragonal rystal (cid:28)eld splitting, the validity of the Curie-Weiss law is questionable. The aim of this paper is to analyze the magneti sus eptibilityfrom a di(cid:27)erent perspe tive, on entrating on the spin state of Co .nisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 3Figure 1: Magneti sus eptibility of La − x Sr x CoO for two di(cid:27)erent dire tions of themagneti (cid:28)eld. The insert is an expanded view of the low-temperature region to showthe di(cid:27)eren e between FC and ZFC measurements for . ≤ x ≤ . . The urves withthe lower sus eptibility refer to the ZFC measurements.The single rystals used for the magneti measurements have been grown using the(cid:29)oating-zone te hnique in an image furna e. A strontium doping range of . ≤ x ≤ . was overed. Resistivity measurements revealed that La − x Sr x CoO is a strong insulatorfor all Sr doping on entrations. The x = 0 . sample turned out to possess the highestresistivity, whi h is in a ordan e with the harge ordering of Co and Co at ≈ Kthat has already been reported [15℄.The magnetization was measured with a Quantum Design vibrating samplemagnetometer (VSM). The (cid:28)eld was aligned parallel to the CoO planes as well asperpendi ular to these planes (the rystallographi c dire tion). The orresponding omponents χ ab and χ c are plotted in Fig. 1. A short-range antiferromagneti orderhas been found in the ompounds . ≤ x ≤ . by neutron measurements [16℄. Thisfrustrated short-range order is also re(cid:29)e ted by the di(cid:27)eren e between (cid:28)eld ooled (FC)and zero-(cid:28)eld ooled (ZFC) measurements at low temperatures. The sus eptibility issmaller for ZFC than FC below a ertain freezing temperature.Figure 2 shows the inverse magneti sus eptibility for both orientations of (cid:28)eld.The main feature of the sus eptibility in the paramagneti regime is the pronoun ednisotropi Sus eptibility of La − x Sr x CoO4
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Anisotropi Sus eptibility of La − x Sr x CoO related tothe Spin States of CobaltN Hollmann , M W Haverkort , M Cwik , M Benomar ,M Reuther , A Tanaka , T Lorenz II. Physikalis hes Institut, University of Cologne, Zülpi herstrasse 77, D-50937 Köln,Germany Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima739-8530, JapanE-mail: hollmannph2.uni-koeln.deAbstra t.We present a study of the magneti sus eptibility of La − x Sr x CoO single rystalsin a doping range . ≤ x ≤ . . Our data shows a pronoun ed magneti anisotropy forall ompounds. This anisotropy is in agreement with a low-spin ground state ( S = 0 )of Co for x ≥ . and a high-spin ground state ( S = 3 / ) of Co . We ompareour data with a rystal-(cid:28)eld model al ulation assuming lo al moments and (cid:28)nd agood des ription of the magneti behavior for x ≥ . . This in ludes the pronoun edkinks observed in the inverse magneti sus eptibility, whi h result from the anisotropyand low-energy ex ited states of Co and are not related to magneti ordering ortemperature-dependent spin-state transitions.PACS numbers: 71.20.Be, 71.70.Ch, 71.70.Ej, 75.30.Gwnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 2Transition-metal oxides are known for their omplex interplay between di(cid:27)erentdegrees of freedom like spin, harge and orbitals. In some of these systems anotherinteresting property is found: ions like Co in a rystal-(cid:28)eld environment an o ur indi(cid:27)erent spin states. Additionally, transitions between di(cid:27)erent spin states are possiblefor some ompounds. A prominent example showing this phenomenon is LaCoO ,whi h has been examined and dis ussed sin e the middle of the last entury (see e.g.Refs. [1(cid:21)11℄).The existen e of di(cid:27)erent spin states arises from the ompetition between rystal-(cid:28)eld e(cid:27)e ts and on-site Coulomb intera tion. Crystal (cid:28)elds lift the degenera y of the d states. If the rystal (cid:28)eld is strong, this an lead to a violation of Hund's rules. In the ase of ubi symmetry and in a one-ele tron pi ture, the (cid:28)ve-fold degenerate d statesare split into a three-fold degenerate t g and a two-fold degenerate e g level. The splittingbetween t g and e g states is alled Dq . With a strong rystal (cid:28)eld, the ele trons willbe for ed into the low-spin state (LS). Regarding a d system, this state onsists of anantiparallel alignment of spins with t g e g and S = 0 . On the other hand, the Coulombintera tion manifests itself in the e(cid:27)e t of Hund's oupling. A parallel arrangement ofspins minimizes the ele tron-ele tron repulsion be ause the Pauli prin iple for es theele trons to o upy di(cid:27)erent orbitals. In a weak ubi rystal (cid:28)eld, this e(cid:27)e t willdominate and lead to the high-spin state (HS), whi h is the on(cid:28)guration with thehighest total spin possible in a ordan e with the Pauli prin iple. For a d system, this on(cid:28)guration is t g e g with S = 2 . For Co the rossover between these two di(cid:27)erentspin states o urs roughly at an energy di(cid:27)eren e of Dq = 2 . eV.In the ase of LaCoO , a low-spin ground state for Co was found [2℄. Interestingly,the di(cid:27)eren e between the rystal (cid:28)eld energies and the promotional energies is so smallthat an ex ited state with di(cid:27)erent spin state an be rea hed by thermal ex itation. Thisex ited state has been a subje t of debate for a long time. Apart from the HS statedes ribed above, the intermediate-spin state (IS, t g e g S = 1 ) was also dis ussed [4℄ andreported in many experiments, but it was shown that it might have been onfused witha spin-orbit oupled HS state [11℄.For the layered obaltates La − x Sr x CoO , mu h less is known about the spin stateof Co . Based on magneti measurements [12℄ a HS ground state for x ≤ . and aspin-state transition to an IS ground state for x > . was proposed. This on lusionwas based on a Curie-Weiss analysis of the sus eptibility in a temperature range of100K to 300K and NMR measurements [13℄. Unrestri ted Hartree-Fo k al ulations [14℄showed a slightly di(cid:27)erent pi ture. Here, three di(cid:27)erent magneti phases were found. Anantiferromagneti HS phase ( x < . ), a ferromagneti HS phase ( . ≤ x ≤ . ) andan antiferromagneti LS-HS-ordered phase ( x > . ) were proposed. These al ulationswere also based on the results of the Curie-Weiss analysis in Ref. [12℄. Taking intoa ount that La − x Sr x CoO is an anisotropi material with rather strong spin-orbit oupling ompared to the tetragonal rystal (cid:28)eld splitting, the validity of the Curie-Weiss law is questionable. The aim of this paper is to analyze the magneti sus eptibilityfrom a di(cid:27)erent perspe tive, on entrating on the spin state of Co .nisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 3Figure 1: Magneti sus eptibility of La − x Sr x CoO for two di(cid:27)erent dire tions of themagneti (cid:28)eld. The insert is an expanded view of the low-temperature region to showthe di(cid:27)eren e between FC and ZFC measurements for . ≤ x ≤ . . The urves withthe lower sus eptibility refer to the ZFC measurements.The single rystals used for the magneti measurements have been grown using the(cid:29)oating-zone te hnique in an image furna e. A strontium doping range of . ≤ x ≤ . was overed. Resistivity measurements revealed that La − x Sr x CoO is a strong insulatorfor all Sr doping on entrations. The x = 0 . sample turned out to possess the highestresistivity, whi h is in a ordan e with the harge ordering of Co and Co at ≈ Kthat has already been reported [15℄.The magnetization was measured with a Quantum Design vibrating samplemagnetometer (VSM). The (cid:28)eld was aligned parallel to the CoO planes as well asperpendi ular to these planes (the rystallographi c dire tion). The orresponding omponents χ ab and χ c are plotted in Fig. 1. A short-range antiferromagneti orderhas been found in the ompounds . ≤ x ≤ . by neutron measurements [16℄. Thisfrustrated short-range order is also re(cid:29)e ted by the di(cid:27)eren e between (cid:28)eld ooled (FC)and zero-(cid:28)eld ooled (ZFC) measurements at low temperatures. The sus eptibility issmaller for ZFC than FC below a ertain freezing temperature.Figure 2 shows the inverse magneti sus eptibility for both orientations of (cid:28)eld.The main feature of the sus eptibility in the paramagneti regime is the pronoun ednisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 4Figure 2: Inverse sus eptibility of La − x Sr x CoO for two di(cid:27)erent dire tions of themagneti (cid:28)eld. The form of the urves and the magneti anisotropy strongly deviatesfrom Curie-Weiss behavior.magneti anisotropy, both in magnitude and form of the urves. The dire tion of theanisotropy is the same for all rystals, (cid:28)nding χ ab to be bigger than χ c .Here, the magneti anisotropy arises from band stru ture and spin-orbit oupling.For Mott and harge-transfer insulators with well-lo alized moments, band-stru tureand ovalen y e(cid:27)e ts an be approximated by an e(cid:27)e tive rystal (cid:28)eld. The rystal (cid:28)eldre(cid:29)e ts the anisotropy of the latti e and lifts the degenera y of the d states, resultingin a new set of linear ombinations of the unperturbed wave fun tions as a basis. Theexpe tation values of the omponents of the orbital moment result from these new linear ombinations. Thus, the rystal's anisotropy may result in an anisotropy of the orbitalmoment. Spin-orbit oupling ties the spin moment to this anisotropy. The spin-orbit oupling Hamiltonian will be written as ζ P i l i · s i where the sum over i runs over allele trons. The dot produ t between spin and orbital momentum tends to align thesemoments antiparallel for ea h ele tron. For an anisotropi orbital momentum, the spinis aligned in the dire tion of maximum orbital momentum [17℄.The full many-body ground-state for a d or d on(cid:28)guration in a rystal-(cid:28)eld al ulation in luding spin-orbit oupling and a tetragonal distortions is not simple [18℄but well known. In order to get an intuitive pi ture one would like to fall ba k to asingle ele tron des ription. In the limit of full spin polarization this an be done andgives important results. In the following we will (cid:28)rst dis uss the magneti anisotropy ofLa − x Sr x CoO in terms of a one-ele tron pi ture. We will show that ea h spin state hasa di(cid:27)erent magneti anisotropy from whi h, by omparison to the experiment, the spinstates of the Co ion an be on luded. In order to obtain also a quantitative des riptionand to verify our simple argumentation we will present a full many-body rystal-(cid:28)eldnisotropi Sus eptibility of La − x Sr x CoO4
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Anisotropi Sus eptibility of La − x Sr x CoO related tothe Spin States of CobaltN Hollmann , M W Haverkort , M Cwik , M Benomar ,M Reuther , A Tanaka , T Lorenz II. Physikalis hes Institut, University of Cologne, Zülpi herstrasse 77, D-50937 Köln,Germany Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima739-8530, JapanE-mail: hollmannph2.uni-koeln.deAbstra t.We present a study of the magneti sus eptibility of La − x Sr x CoO single rystalsin a doping range . ≤ x ≤ . . Our data shows a pronoun ed magneti anisotropy forall ompounds. This anisotropy is in agreement with a low-spin ground state ( S = 0 )of Co for x ≥ . and a high-spin ground state ( S = 3 / ) of Co . We ompareour data with a rystal-(cid:28)eld model al ulation assuming lo al moments and (cid:28)nd agood des ription of the magneti behavior for x ≥ . . This in ludes the pronoun edkinks observed in the inverse magneti sus eptibility, whi h result from the anisotropyand low-energy ex ited states of Co and are not related to magneti ordering ortemperature-dependent spin-state transitions.PACS numbers: 71.20.Be, 71.70.Ch, 71.70.Ej, 75.30.Gwnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 2Transition-metal oxides are known for their omplex interplay between di(cid:27)erentdegrees of freedom like spin, harge and orbitals. In some of these systems anotherinteresting property is found: ions like Co in a rystal-(cid:28)eld environment an o ur indi(cid:27)erent spin states. Additionally, transitions between di(cid:27)erent spin states are possiblefor some ompounds. A prominent example showing this phenomenon is LaCoO ,whi h has been examined and dis ussed sin e the middle of the last entury (see e.g.Refs. [1(cid:21)11℄).The existen e of di(cid:27)erent spin states arises from the ompetition between rystal-(cid:28)eld e(cid:27)e ts and on-site Coulomb intera tion. Crystal (cid:28)elds lift the degenera y of the d states. If the rystal (cid:28)eld is strong, this an lead to a violation of Hund's rules. In the ase of ubi symmetry and in a one-ele tron pi ture, the (cid:28)ve-fold degenerate d statesare split into a three-fold degenerate t g and a two-fold degenerate e g level. The splittingbetween t g and e g states is alled Dq . With a strong rystal (cid:28)eld, the ele trons willbe for ed into the low-spin state (LS). Regarding a d system, this state onsists of anantiparallel alignment of spins with t g e g and S = 0 . On the other hand, the Coulombintera tion manifests itself in the e(cid:27)e t of Hund's oupling. A parallel arrangement ofspins minimizes the ele tron-ele tron repulsion be ause the Pauli prin iple for es theele trons to o upy di(cid:27)erent orbitals. In a weak ubi rystal (cid:28)eld, this e(cid:27)e t willdominate and lead to the high-spin state (HS), whi h is the on(cid:28)guration with thehighest total spin possible in a ordan e with the Pauli prin iple. For a d system, this on(cid:28)guration is t g e g with S = 2 . For Co the rossover between these two di(cid:27)erentspin states o urs roughly at an energy di(cid:27)eren e of Dq = 2 . eV.In the ase of LaCoO , a low-spin ground state for Co was found [2℄. Interestingly,the di(cid:27)eren e between the rystal (cid:28)eld energies and the promotional energies is so smallthat an ex ited state with di(cid:27)erent spin state an be rea hed by thermal ex itation. Thisex ited state has been a subje t of debate for a long time. Apart from the HS statedes ribed above, the intermediate-spin state (IS, t g e g S = 1 ) was also dis ussed [4℄ andreported in many experiments, but it was shown that it might have been onfused witha spin-orbit oupled HS state [11℄.For the layered obaltates La − x Sr x CoO , mu h less is known about the spin stateof Co . Based on magneti measurements [12℄ a HS ground state for x ≤ . and aspin-state transition to an IS ground state for x > . was proposed. This on lusionwas based on a Curie-Weiss analysis of the sus eptibility in a temperature range of100K to 300K and NMR measurements [13℄. Unrestri ted Hartree-Fo k al ulations [14℄showed a slightly di(cid:27)erent pi ture. Here, three di(cid:27)erent magneti phases were found. Anantiferromagneti HS phase ( x < . ), a ferromagneti HS phase ( . ≤ x ≤ . ) andan antiferromagneti LS-HS-ordered phase ( x > . ) were proposed. These al ulationswere also based on the results of the Curie-Weiss analysis in Ref. [12℄. Taking intoa ount that La − x Sr x CoO is an anisotropi material with rather strong spin-orbit oupling ompared to the tetragonal rystal (cid:28)eld splitting, the validity of the Curie-Weiss law is questionable. The aim of this paper is to analyze the magneti sus eptibilityfrom a di(cid:27)erent perspe tive, on entrating on the spin state of Co .nisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 3Figure 1: Magneti sus eptibility of La − x Sr x CoO for two di(cid:27)erent dire tions of themagneti (cid:28)eld. The insert is an expanded view of the low-temperature region to showthe di(cid:27)eren e between FC and ZFC measurements for . ≤ x ≤ . . The urves withthe lower sus eptibility refer to the ZFC measurements.The single rystals used for the magneti measurements have been grown using the(cid:29)oating-zone te hnique in an image furna e. A strontium doping range of . ≤ x ≤ . was overed. Resistivity measurements revealed that La − x Sr x CoO is a strong insulatorfor all Sr doping on entrations. The x = 0 . sample turned out to possess the highestresistivity, whi h is in a ordan e with the harge ordering of Co and Co at ≈ Kthat has already been reported [15℄.The magnetization was measured with a Quantum Design vibrating samplemagnetometer (VSM). The (cid:28)eld was aligned parallel to the CoO planes as well asperpendi ular to these planes (the rystallographi c dire tion). The orresponding omponents χ ab and χ c are plotted in Fig. 1. A short-range antiferromagneti orderhas been found in the ompounds . ≤ x ≤ . by neutron measurements [16℄. Thisfrustrated short-range order is also re(cid:29)e ted by the di(cid:27)eren e between (cid:28)eld ooled (FC)and zero-(cid:28)eld ooled (ZFC) measurements at low temperatures. The sus eptibility issmaller for ZFC than FC below a ertain freezing temperature.Figure 2 shows the inverse magneti sus eptibility for both orientations of (cid:28)eld.The main feature of the sus eptibility in the paramagneti regime is the pronoun ednisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 4Figure 2: Inverse sus eptibility of La − x Sr x CoO for two di(cid:27)erent dire tions of themagneti (cid:28)eld. The form of the urves and the magneti anisotropy strongly deviatesfrom Curie-Weiss behavior.magneti anisotropy, both in magnitude and form of the urves. The dire tion of theanisotropy is the same for all rystals, (cid:28)nding χ ab to be bigger than χ c .Here, the magneti anisotropy arises from band stru ture and spin-orbit oupling.For Mott and harge-transfer insulators with well-lo alized moments, band-stru tureand ovalen y e(cid:27)e ts an be approximated by an e(cid:27)e tive rystal (cid:28)eld. The rystal (cid:28)eldre(cid:29)e ts the anisotropy of the latti e and lifts the degenera y of the d states, resultingin a new set of linear ombinations of the unperturbed wave fun tions as a basis. Theexpe tation values of the omponents of the orbital moment result from these new linear ombinations. Thus, the rystal's anisotropy may result in an anisotropy of the orbitalmoment. Spin-orbit oupling ties the spin moment to this anisotropy. The spin-orbit oupling Hamiltonian will be written as ζ P i l i · s i where the sum over i runs over allele trons. The dot produ t between spin and orbital momentum tends to align thesemoments antiparallel for ea h ele tron. For an anisotropi orbital momentum, the spinis aligned in the dire tion of maximum orbital momentum [17℄.The full many-body ground-state for a d or d on(cid:28)guration in a rystal-(cid:28)eld al ulation in luding spin-orbit oupling and a tetragonal distortions is not simple [18℄but well known. In order to get an intuitive pi ture one would like to fall ba k to asingle ele tron des ription. In the limit of full spin polarization this an be done andgives important results. In the following we will (cid:28)rst dis uss the magneti anisotropy ofLa − x Sr x CoO in terms of a one-ele tron pi ture. We will show that ea h spin state hasa di(cid:27)erent magneti anisotropy from whi h, by omparison to the experiment, the spinstates of the Co ion an be on luded. In order to obtain also a quantitative des riptionand to verify our simple argumentation we will present a full many-body rystal-(cid:28)eldnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 5Figure 3: Splitting of the d levels in a tetragonal rystal (cid:28)eld arising from an elongatedoxygen o tahedron. The two sket hes on the left show the o upation for the LS andHS state of Co in a one-ele tron pi ture. The right sket h refers to HS Co . Thereal orbitals on the right an be used as a basis. al ulation afterwards.Within a ubi rystal stru ture, the d states split into e g orbitals and t g orbitals.The basis an be hosen as { z − r , x − y } and { xy, xz, yz } , respe tively. A partially(cid:28)lled t g shell an produ e a pseudo orbital moment of ˜ L = 1 . Though the individualreal wave fun tions d xy , d yz and d xz of the basis themselves have a ompletely quen hedorbital moment, their linear ombinations are in general omplex. Writing d xm l , d ym l and d zm l as the orbital wave fun tion with the orbital moment quantized along the axes x , y and z , respe tively, one (cid:28)nds d x ± = 1 √ ± d xy + id xz ) (1) d y ± = 1 √ ± d yz + id xy ) (2) d z ± = 1 √ ± d xz + id yz ) . (3)In the limit of very large rystal (cid:28)eld splittings Dq and with a partially (cid:28)lled t g shell, the moment is isotropi , despite the large orbital moment. In fa t, the t g ele trons are sometimes ompared to p ele trons [19℄.Introdu ing a tetragonal distortion, the orbital moment be omes anisotropi . Thetetragonal rystal (cid:28)eld splits the ubi e g states into non-degenerate a g and b g levels,while the t g states split into a non-degenerate b g state and a two-fold degenerate e g level. As a basis for these states, the real orbital fun tions an also be used. The z axis of the system is taken to be identi al with the c axis of the rystal. In the aseof La − x Sr x CoO , the oxygen o tahedron is elongated in the c dire tion. The order oflevels and o upations for the ground states of the two obalt ions is illustrated in Fig. 3.The Co low-spin state is, ex ept for a small Van Vle k sus eptibility, nonmagneti sin e it does not arry any moment. In the high-spin state, Co has spin and orbitalnisotropi Sus eptibility of La − x Sr x CoO4
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Anisotropi Sus eptibility of La − x Sr x CoO related tothe Spin States of CobaltN Hollmann , M W Haverkort , M Cwik , M Benomar ,M Reuther , A Tanaka , T Lorenz II. Physikalis hes Institut, University of Cologne, Zülpi herstrasse 77, D-50937 Köln,Germany Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima739-8530, JapanE-mail: hollmannph2.uni-koeln.deAbstra t.We present a study of the magneti sus eptibility of La − x Sr x CoO single rystalsin a doping range . ≤ x ≤ . . Our data shows a pronoun ed magneti anisotropy forall ompounds. This anisotropy is in agreement with a low-spin ground state ( S = 0 )of Co for x ≥ . and a high-spin ground state ( S = 3 / ) of Co . We ompareour data with a rystal-(cid:28)eld model al ulation assuming lo al moments and (cid:28)nd agood des ription of the magneti behavior for x ≥ . . This in ludes the pronoun edkinks observed in the inverse magneti sus eptibility, whi h result from the anisotropyand low-energy ex ited states of Co and are not related to magneti ordering ortemperature-dependent spin-state transitions.PACS numbers: 71.20.Be, 71.70.Ch, 71.70.Ej, 75.30.Gwnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 2Transition-metal oxides are known for their omplex interplay between di(cid:27)erentdegrees of freedom like spin, harge and orbitals. In some of these systems anotherinteresting property is found: ions like Co in a rystal-(cid:28)eld environment an o ur indi(cid:27)erent spin states. Additionally, transitions between di(cid:27)erent spin states are possiblefor some ompounds. A prominent example showing this phenomenon is LaCoO ,whi h has been examined and dis ussed sin e the middle of the last entury (see e.g.Refs. [1(cid:21)11℄).The existen e of di(cid:27)erent spin states arises from the ompetition between rystal-(cid:28)eld e(cid:27)e ts and on-site Coulomb intera tion. Crystal (cid:28)elds lift the degenera y of the d states. If the rystal (cid:28)eld is strong, this an lead to a violation of Hund's rules. In the ase of ubi symmetry and in a one-ele tron pi ture, the (cid:28)ve-fold degenerate d statesare split into a three-fold degenerate t g and a two-fold degenerate e g level. The splittingbetween t g and e g states is alled Dq . With a strong rystal (cid:28)eld, the ele trons willbe for ed into the low-spin state (LS). Regarding a d system, this state onsists of anantiparallel alignment of spins with t g e g and S = 0 . On the other hand, the Coulombintera tion manifests itself in the e(cid:27)e t of Hund's oupling. A parallel arrangement ofspins minimizes the ele tron-ele tron repulsion be ause the Pauli prin iple for es theele trons to o upy di(cid:27)erent orbitals. In a weak ubi rystal (cid:28)eld, this e(cid:27)e t willdominate and lead to the high-spin state (HS), whi h is the on(cid:28)guration with thehighest total spin possible in a ordan e with the Pauli prin iple. For a d system, this on(cid:28)guration is t g e g with S = 2 . For Co the rossover between these two di(cid:27)erentspin states o urs roughly at an energy di(cid:27)eren e of Dq = 2 . eV.In the ase of LaCoO , a low-spin ground state for Co was found [2℄. Interestingly,the di(cid:27)eren e between the rystal (cid:28)eld energies and the promotional energies is so smallthat an ex ited state with di(cid:27)erent spin state an be rea hed by thermal ex itation. Thisex ited state has been a subje t of debate for a long time. Apart from the HS statedes ribed above, the intermediate-spin state (IS, t g e g S = 1 ) was also dis ussed [4℄ andreported in many experiments, but it was shown that it might have been onfused witha spin-orbit oupled HS state [11℄.For the layered obaltates La − x Sr x CoO , mu h less is known about the spin stateof Co . Based on magneti measurements [12℄ a HS ground state for x ≤ . and aspin-state transition to an IS ground state for x > . was proposed. This on lusionwas based on a Curie-Weiss analysis of the sus eptibility in a temperature range of100K to 300K and NMR measurements [13℄. Unrestri ted Hartree-Fo k al ulations [14℄showed a slightly di(cid:27)erent pi ture. Here, three di(cid:27)erent magneti phases were found. Anantiferromagneti HS phase ( x < . ), a ferromagneti HS phase ( . ≤ x ≤ . ) andan antiferromagneti LS-HS-ordered phase ( x > . ) were proposed. These al ulationswere also based on the results of the Curie-Weiss analysis in Ref. [12℄. Taking intoa ount that La − x Sr x CoO is an anisotropi material with rather strong spin-orbit oupling ompared to the tetragonal rystal (cid:28)eld splitting, the validity of the Curie-Weiss law is questionable. The aim of this paper is to analyze the magneti sus eptibilityfrom a di(cid:27)erent perspe tive, on entrating on the spin state of Co .nisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 3Figure 1: Magneti sus eptibility of La − x Sr x CoO for two di(cid:27)erent dire tions of themagneti (cid:28)eld. The insert is an expanded view of the low-temperature region to showthe di(cid:27)eren e between FC and ZFC measurements for . ≤ x ≤ . . The urves withthe lower sus eptibility refer to the ZFC measurements.The single rystals used for the magneti measurements have been grown using the(cid:29)oating-zone te hnique in an image furna e. A strontium doping range of . ≤ x ≤ . was overed. Resistivity measurements revealed that La − x Sr x CoO is a strong insulatorfor all Sr doping on entrations. The x = 0 . sample turned out to possess the highestresistivity, whi h is in a ordan e with the harge ordering of Co and Co at ≈ Kthat has already been reported [15℄.The magnetization was measured with a Quantum Design vibrating samplemagnetometer (VSM). The (cid:28)eld was aligned parallel to the CoO planes as well asperpendi ular to these planes (the rystallographi c dire tion). The orresponding omponents χ ab and χ c are plotted in Fig. 1. A short-range antiferromagneti orderhas been found in the ompounds . ≤ x ≤ . by neutron measurements [16℄. Thisfrustrated short-range order is also re(cid:29)e ted by the di(cid:27)eren e between (cid:28)eld ooled (FC)and zero-(cid:28)eld ooled (ZFC) measurements at low temperatures. The sus eptibility issmaller for ZFC than FC below a ertain freezing temperature.Figure 2 shows the inverse magneti sus eptibility for both orientations of (cid:28)eld.The main feature of the sus eptibility in the paramagneti regime is the pronoun ednisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 4Figure 2: Inverse sus eptibility of La − x Sr x CoO for two di(cid:27)erent dire tions of themagneti (cid:28)eld. The form of the urves and the magneti anisotropy strongly deviatesfrom Curie-Weiss behavior.magneti anisotropy, both in magnitude and form of the urves. The dire tion of theanisotropy is the same for all rystals, (cid:28)nding χ ab to be bigger than χ c .Here, the magneti anisotropy arises from band stru ture and spin-orbit oupling.For Mott and harge-transfer insulators with well-lo alized moments, band-stru tureand ovalen y e(cid:27)e ts an be approximated by an e(cid:27)e tive rystal (cid:28)eld. The rystal (cid:28)eldre(cid:29)e ts the anisotropy of the latti e and lifts the degenera y of the d states, resultingin a new set of linear ombinations of the unperturbed wave fun tions as a basis. Theexpe tation values of the omponents of the orbital moment result from these new linear ombinations. Thus, the rystal's anisotropy may result in an anisotropy of the orbitalmoment. Spin-orbit oupling ties the spin moment to this anisotropy. The spin-orbit oupling Hamiltonian will be written as ζ P i l i · s i where the sum over i runs over allele trons. The dot produ t between spin and orbital momentum tends to align thesemoments antiparallel for ea h ele tron. For an anisotropi orbital momentum, the spinis aligned in the dire tion of maximum orbital momentum [17℄.The full many-body ground-state for a d or d on(cid:28)guration in a rystal-(cid:28)eld al ulation in luding spin-orbit oupling and a tetragonal distortions is not simple [18℄but well known. In order to get an intuitive pi ture one would like to fall ba k to asingle ele tron des ription. In the limit of full spin polarization this an be done andgives important results. In the following we will (cid:28)rst dis uss the magneti anisotropy ofLa − x Sr x CoO in terms of a one-ele tron pi ture. We will show that ea h spin state hasa di(cid:27)erent magneti anisotropy from whi h, by omparison to the experiment, the spinstates of the Co ion an be on luded. In order to obtain also a quantitative des riptionand to verify our simple argumentation we will present a full many-body rystal-(cid:28)eldnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 5Figure 3: Splitting of the d levels in a tetragonal rystal (cid:28)eld arising from an elongatedoxygen o tahedron. The two sket hes on the left show the o upation for the LS andHS state of Co in a one-ele tron pi ture. The right sket h refers to HS Co . Thereal orbitals on the right an be used as a basis. al ulation afterwards.Within a ubi rystal stru ture, the d states split into e g orbitals and t g orbitals.The basis an be hosen as { z − r , x − y } and { xy, xz, yz } , respe tively. A partially(cid:28)lled t g shell an produ e a pseudo orbital moment of ˜ L = 1 . Though the individualreal wave fun tions d xy , d yz and d xz of the basis themselves have a ompletely quen hedorbital moment, their linear ombinations are in general omplex. Writing d xm l , d ym l and d zm l as the orbital wave fun tion with the orbital moment quantized along the axes x , y and z , respe tively, one (cid:28)nds d x ± = 1 √ ± d xy + id xz ) (1) d y ± = 1 √ ± d yz + id xy ) (2) d z ± = 1 √ ± d xz + id yz ) . (3)In the limit of very large rystal (cid:28)eld splittings Dq and with a partially (cid:28)lled t g shell, the moment is isotropi , despite the large orbital moment. In fa t, the t g ele trons are sometimes ompared to p ele trons [19℄.Introdu ing a tetragonal distortion, the orbital moment be omes anisotropi . Thetetragonal rystal (cid:28)eld splits the ubi e g states into non-degenerate a g and b g levels,while the t g states split into a non-degenerate b g state and a two-fold degenerate e g level. As a basis for these states, the real orbital fun tions an also be used. The z axis of the system is taken to be identi al with the c axis of the rystal. In the aseof La − x Sr x CoO , the oxygen o tahedron is elongated in the c dire tion. The order oflevels and o upations for the ground states of the two obalt ions is illustrated in Fig. 3.The Co low-spin state is, ex ept for a small Van Vle k sus eptibility, nonmagneti sin e it does not arry any moment. In the high-spin state, Co has spin and orbitalnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 6moment. The orbital degenera y is not ompletely quen hed: the degenerate xz and yz are o upied by three ele trons. These real orbitals an thus be re ombined to form omplex orbitals arrying orbital moment in the z dire tion as des ribed in Eq. (3). Notethat the linear ombinations in equations (1) and (2) annot be formed be ause the xy orbital is not degenerate with the xz and yz orbitals. Therefore the orbital moment islarger in the z dire tion making this the easy axis. This anisotropy should be re(cid:29)e tedin the sus eptibility with χ c >χ ab , whi h obviously ontradi ts the anisotropy found inthe measurements. A Co HS system shows the wrong magneti anisotropy.Next, we dis uss the IS state of Co . Due to the elongation of the oxygeno tahedra, the e g ele tron o upies the z − r orbital. This also e(cid:27)e ts the splitting ofthe t g states. The xz and yz orbitals remain degenerate but are not degenerate with the xy orbital. This arises from the di(cid:27)erent harge distributions in relation to the z − r orbital. We have (cid:28)ve ele trons to (cid:28)ll in the t g states, whi h an also be treated as onehole in the t g states. This hole is attra ted to the e g ele tron in the z − r orbital.Fig. 4 shows the harge distribution of this hole and the z − r orbital. In the leftsket h, both the ele tron in the z − r orbital and a hole in the xy orbital is drawn.The sket h in the enter shows the z − r orbital and the hole in a linear ombinationof the degenerate xz and yz orbitals. Regarding the distan es between the ele tron andthe hole, on an on lude that the state with the hole in the xz and yz orbitals is lowerin energy. As the order of levels is reversed when we are speaking about holes insteadof ele trons, this means that the xy orbital is lowered in energy. Thus, the order ando upation of levels is the one shown in Fig. 4. The magneti anisotropy is the same asin the ase of Co HS whi h we treated in the last paragraph. With three ele trons inthe degenerate level of the xz and yz orbitals, we an also make use of Eq. (3) to showthat the dire tion of anisotropy does not (cid:28)t the measurements for La − x Sr x CoO .Neither the HS nor the IS state of Co show the orre t magneti anisotropy. Butbefore we draw further on lusions, Co should be dis ussed. Although Co has alsobeen found in the LS state in some intramole ular ompounds [21℄, for bulk rystals it an be safely assumed that Co is in the HS state [22℄.Regarding the ground state produ ed by the rystal (cid:28)eld in Fig. 3, where the lowestlevel is (cid:28)lled, no linear ombinations like in Eq. (3) an be formed. The orbital momentwould be ompletely quen hed and the magneti moment would be determined by anisotropi spin. If, however, spin-orbit oupling is of the same magnitude as the splittingof the t g orbitals, it will mix these states. Mixing in a state with orbital moment inthe z dire tion would require pla ing a hole in the d z ± orbital (see Eq. 3). This would ost an energy equal to the splitting within the t g orbitals. It is more favorable to mixin a state with orbital moment in the x (or y ) dire tion by pla ing a hole in the d x ± (orthe d y ± ) orbital (see Eq. 1). This osts only half of the rystal-(cid:28)eld splitting within the t g orbitals. An orbital moment in the x (or y ) dire tion osts less energy than in the z dire tion. So, the xy plane will be the easy plane for Co in this elongated tetragonalstru ture [24℄. The spin moment will also be found in the xy plane. Comparing these(cid:28)ndings to the measurement, we see that Co shows the orre t dire tion of magneti nisotropi Sus eptibility of La − x Sr x CoO4
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Anisotropi Sus eptibility of La − x Sr x CoO related tothe Spin States of CobaltN Hollmann , M W Haverkort , M Cwik , M Benomar ,M Reuther , A Tanaka , T Lorenz II. Physikalis hes Institut, University of Cologne, Zülpi herstrasse 77, D-50937 Köln,Germany Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima739-8530, JapanE-mail: hollmannph2.uni-koeln.deAbstra t.We present a study of the magneti sus eptibility of La − x Sr x CoO single rystalsin a doping range . ≤ x ≤ . . Our data shows a pronoun ed magneti anisotropy forall ompounds. This anisotropy is in agreement with a low-spin ground state ( S = 0 )of Co for x ≥ . and a high-spin ground state ( S = 3 / ) of Co . We ompareour data with a rystal-(cid:28)eld model al ulation assuming lo al moments and (cid:28)nd agood des ription of the magneti behavior for x ≥ . . This in ludes the pronoun edkinks observed in the inverse magneti sus eptibility, whi h result from the anisotropyand low-energy ex ited states of Co and are not related to magneti ordering ortemperature-dependent spin-state transitions.PACS numbers: 71.20.Be, 71.70.Ch, 71.70.Ej, 75.30.Gwnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 2Transition-metal oxides are known for their omplex interplay between di(cid:27)erentdegrees of freedom like spin, harge and orbitals. In some of these systems anotherinteresting property is found: ions like Co in a rystal-(cid:28)eld environment an o ur indi(cid:27)erent spin states. Additionally, transitions between di(cid:27)erent spin states are possiblefor some ompounds. A prominent example showing this phenomenon is LaCoO ,whi h has been examined and dis ussed sin e the middle of the last entury (see e.g.Refs. [1(cid:21)11℄).The existen e of di(cid:27)erent spin states arises from the ompetition between rystal-(cid:28)eld e(cid:27)e ts and on-site Coulomb intera tion. Crystal (cid:28)elds lift the degenera y of the d states. If the rystal (cid:28)eld is strong, this an lead to a violation of Hund's rules. In the ase of ubi symmetry and in a one-ele tron pi ture, the (cid:28)ve-fold degenerate d statesare split into a three-fold degenerate t g and a two-fold degenerate e g level. The splittingbetween t g and e g states is alled Dq . With a strong rystal (cid:28)eld, the ele trons willbe for ed into the low-spin state (LS). Regarding a d system, this state onsists of anantiparallel alignment of spins with t g e g and S = 0 . On the other hand, the Coulombintera tion manifests itself in the e(cid:27)e t of Hund's oupling. A parallel arrangement ofspins minimizes the ele tron-ele tron repulsion be ause the Pauli prin iple for es theele trons to o upy di(cid:27)erent orbitals. In a weak ubi rystal (cid:28)eld, this e(cid:27)e t willdominate and lead to the high-spin state (HS), whi h is the on(cid:28)guration with thehighest total spin possible in a ordan e with the Pauli prin iple. For a d system, this on(cid:28)guration is t g e g with S = 2 . For Co the rossover between these two di(cid:27)erentspin states o urs roughly at an energy di(cid:27)eren e of Dq = 2 . eV.In the ase of LaCoO , a low-spin ground state for Co was found [2℄. Interestingly,the di(cid:27)eren e between the rystal (cid:28)eld energies and the promotional energies is so smallthat an ex ited state with di(cid:27)erent spin state an be rea hed by thermal ex itation. Thisex ited state has been a subje t of debate for a long time. Apart from the HS statedes ribed above, the intermediate-spin state (IS, t g e g S = 1 ) was also dis ussed [4℄ andreported in many experiments, but it was shown that it might have been onfused witha spin-orbit oupled HS state [11℄.For the layered obaltates La − x Sr x CoO , mu h less is known about the spin stateof Co . Based on magneti measurements [12℄ a HS ground state for x ≤ . and aspin-state transition to an IS ground state for x > . was proposed. This on lusionwas based on a Curie-Weiss analysis of the sus eptibility in a temperature range of100K to 300K and NMR measurements [13℄. Unrestri ted Hartree-Fo k al ulations [14℄showed a slightly di(cid:27)erent pi ture. Here, three di(cid:27)erent magneti phases were found. Anantiferromagneti HS phase ( x < . ), a ferromagneti HS phase ( . ≤ x ≤ . ) andan antiferromagneti LS-HS-ordered phase ( x > . ) were proposed. These al ulationswere also based on the results of the Curie-Weiss analysis in Ref. [12℄. Taking intoa ount that La − x Sr x CoO is an anisotropi material with rather strong spin-orbit oupling ompared to the tetragonal rystal (cid:28)eld splitting, the validity of the Curie-Weiss law is questionable. The aim of this paper is to analyze the magneti sus eptibilityfrom a di(cid:27)erent perspe tive, on entrating on the spin state of Co .nisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 3Figure 1: Magneti sus eptibility of La − x Sr x CoO for two di(cid:27)erent dire tions of themagneti (cid:28)eld. The insert is an expanded view of the low-temperature region to showthe di(cid:27)eren e between FC and ZFC measurements for . ≤ x ≤ . . The urves withthe lower sus eptibility refer to the ZFC measurements.The single rystals used for the magneti measurements have been grown using the(cid:29)oating-zone te hnique in an image furna e. A strontium doping range of . ≤ x ≤ . was overed. Resistivity measurements revealed that La − x Sr x CoO is a strong insulatorfor all Sr doping on entrations. The x = 0 . sample turned out to possess the highestresistivity, whi h is in a ordan e with the harge ordering of Co and Co at ≈ Kthat has already been reported [15℄.The magnetization was measured with a Quantum Design vibrating samplemagnetometer (VSM). The (cid:28)eld was aligned parallel to the CoO planes as well asperpendi ular to these planes (the rystallographi c dire tion). The orresponding omponents χ ab and χ c are plotted in Fig. 1. A short-range antiferromagneti orderhas been found in the ompounds . ≤ x ≤ . by neutron measurements [16℄. Thisfrustrated short-range order is also re(cid:29)e ted by the di(cid:27)eren e between (cid:28)eld ooled (FC)and zero-(cid:28)eld ooled (ZFC) measurements at low temperatures. The sus eptibility issmaller for ZFC than FC below a ertain freezing temperature.Figure 2 shows the inverse magneti sus eptibility for both orientations of (cid:28)eld.The main feature of the sus eptibility in the paramagneti regime is the pronoun ednisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 4Figure 2: Inverse sus eptibility of La − x Sr x CoO for two di(cid:27)erent dire tions of themagneti (cid:28)eld. The form of the urves and the magneti anisotropy strongly deviatesfrom Curie-Weiss behavior.magneti anisotropy, both in magnitude and form of the urves. The dire tion of theanisotropy is the same for all rystals, (cid:28)nding χ ab to be bigger than χ c .Here, the magneti anisotropy arises from band stru ture and spin-orbit oupling.For Mott and harge-transfer insulators with well-lo alized moments, band-stru tureand ovalen y e(cid:27)e ts an be approximated by an e(cid:27)e tive rystal (cid:28)eld. The rystal (cid:28)eldre(cid:29)e ts the anisotropy of the latti e and lifts the degenera y of the d states, resultingin a new set of linear ombinations of the unperturbed wave fun tions as a basis. Theexpe tation values of the omponents of the orbital moment result from these new linear ombinations. Thus, the rystal's anisotropy may result in an anisotropy of the orbitalmoment. Spin-orbit oupling ties the spin moment to this anisotropy. The spin-orbit oupling Hamiltonian will be written as ζ P i l i · s i where the sum over i runs over allele trons. The dot produ t between spin and orbital momentum tends to align thesemoments antiparallel for ea h ele tron. For an anisotropi orbital momentum, the spinis aligned in the dire tion of maximum orbital momentum [17℄.The full many-body ground-state for a d or d on(cid:28)guration in a rystal-(cid:28)eld al ulation in luding spin-orbit oupling and a tetragonal distortions is not simple [18℄but well known. In order to get an intuitive pi ture one would like to fall ba k to asingle ele tron des ription. In the limit of full spin polarization this an be done andgives important results. In the following we will (cid:28)rst dis uss the magneti anisotropy ofLa − x Sr x CoO in terms of a one-ele tron pi ture. We will show that ea h spin state hasa di(cid:27)erent magneti anisotropy from whi h, by omparison to the experiment, the spinstates of the Co ion an be on luded. In order to obtain also a quantitative des riptionand to verify our simple argumentation we will present a full many-body rystal-(cid:28)eldnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 5Figure 3: Splitting of the d levels in a tetragonal rystal (cid:28)eld arising from an elongatedoxygen o tahedron. The two sket hes on the left show the o upation for the LS andHS state of Co in a one-ele tron pi ture. The right sket h refers to HS Co . Thereal orbitals on the right an be used as a basis. al ulation afterwards.Within a ubi rystal stru ture, the d states split into e g orbitals and t g orbitals.The basis an be hosen as { z − r , x − y } and { xy, xz, yz } , respe tively. A partially(cid:28)lled t g shell an produ e a pseudo orbital moment of ˜ L = 1 . Though the individualreal wave fun tions d xy , d yz and d xz of the basis themselves have a ompletely quen hedorbital moment, their linear ombinations are in general omplex. Writing d xm l , d ym l and d zm l as the orbital wave fun tion with the orbital moment quantized along the axes x , y and z , respe tively, one (cid:28)nds d x ± = 1 √ ± d xy + id xz ) (1) d y ± = 1 √ ± d yz + id xy ) (2) d z ± = 1 √ ± d xz + id yz ) . (3)In the limit of very large rystal (cid:28)eld splittings Dq and with a partially (cid:28)lled t g shell, the moment is isotropi , despite the large orbital moment. In fa t, the t g ele trons are sometimes ompared to p ele trons [19℄.Introdu ing a tetragonal distortion, the orbital moment be omes anisotropi . Thetetragonal rystal (cid:28)eld splits the ubi e g states into non-degenerate a g and b g levels,while the t g states split into a non-degenerate b g state and a two-fold degenerate e g level. As a basis for these states, the real orbital fun tions an also be used. The z axis of the system is taken to be identi al with the c axis of the rystal. In the aseof La − x Sr x CoO , the oxygen o tahedron is elongated in the c dire tion. The order oflevels and o upations for the ground states of the two obalt ions is illustrated in Fig. 3.The Co low-spin state is, ex ept for a small Van Vle k sus eptibility, nonmagneti sin e it does not arry any moment. In the high-spin state, Co has spin and orbitalnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 6moment. The orbital degenera y is not ompletely quen hed: the degenerate xz and yz are o upied by three ele trons. These real orbitals an thus be re ombined to form omplex orbitals arrying orbital moment in the z dire tion as des ribed in Eq. (3). Notethat the linear ombinations in equations (1) and (2) annot be formed be ause the xy orbital is not degenerate with the xz and yz orbitals. Therefore the orbital moment islarger in the z dire tion making this the easy axis. This anisotropy should be re(cid:29)e tedin the sus eptibility with χ c >χ ab , whi h obviously ontradi ts the anisotropy found inthe measurements. A Co HS system shows the wrong magneti anisotropy.Next, we dis uss the IS state of Co . Due to the elongation of the oxygeno tahedra, the e g ele tron o upies the z − r orbital. This also e(cid:27)e ts the splitting ofthe t g states. The xz and yz orbitals remain degenerate but are not degenerate with the xy orbital. This arises from the di(cid:27)erent harge distributions in relation to the z − r orbital. We have (cid:28)ve ele trons to (cid:28)ll in the t g states, whi h an also be treated as onehole in the t g states. This hole is attra ted to the e g ele tron in the z − r orbital.Fig. 4 shows the harge distribution of this hole and the z − r orbital. In the leftsket h, both the ele tron in the z − r orbital and a hole in the xy orbital is drawn.The sket h in the enter shows the z − r orbital and the hole in a linear ombinationof the degenerate xz and yz orbitals. Regarding the distan es between the ele tron andthe hole, on an on lude that the state with the hole in the xz and yz orbitals is lowerin energy. As the order of levels is reversed when we are speaking about holes insteadof ele trons, this means that the xy orbital is lowered in energy. Thus, the order ando upation of levels is the one shown in Fig. 4. The magneti anisotropy is the same asin the ase of Co HS whi h we treated in the last paragraph. With three ele trons inthe degenerate level of the xz and yz orbitals, we an also make use of Eq. (3) to showthat the dire tion of anisotropy does not (cid:28)t the measurements for La − x Sr x CoO .Neither the HS nor the IS state of Co show the orre t magneti anisotropy. Butbefore we draw further on lusions, Co should be dis ussed. Although Co has alsobeen found in the LS state in some intramole ular ompounds [21℄, for bulk rystals it an be safely assumed that Co is in the HS state [22℄.Regarding the ground state produ ed by the rystal (cid:28)eld in Fig. 3, where the lowestlevel is (cid:28)lled, no linear ombinations like in Eq. (3) an be formed. The orbital momentwould be ompletely quen hed and the magneti moment would be determined by anisotropi spin. If, however, spin-orbit oupling is of the same magnitude as the splittingof the t g orbitals, it will mix these states. Mixing in a state with orbital moment inthe z dire tion would require pla ing a hole in the d z ± orbital (see Eq. 3). This would ost an energy equal to the splitting within the t g orbitals. It is more favorable to mixin a state with orbital moment in the x (or y ) dire tion by pla ing a hole in the d x ± (orthe d y ± ) orbital (see Eq. 1). This osts only half of the rystal-(cid:28)eld splitting within the t g orbitals. An orbital moment in the x (or y ) dire tion osts less energy than in the z dire tion. So, the xy plane will be the easy plane for Co in this elongated tetragonalstru ture [24℄. The spin moment will also be found in the xy plane. Comparing these(cid:28)ndings to the measurement, we see that Co shows the orre t dire tion of magneti nisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 7Figure 4: The pi ture on the left shows the harge distribution for an ele tron in the z − r orbital (bla k) and a hole in the xy orbital (white). The harge distribution inthe enter refers to an ele tron in the z + r orbital and a hole in a linear ombinationof the xz and yz orbitals. The splitting and the o upation of the d levels for Co inthe IS state is shown on the right.anisotropy. Now a on lusion about the spin state of Co an be drawn. For x ≥ . , atleast half of the obalt ions in the rystal are Co . Taking into a ount that Co hasa smaller spin moment than HS Co , the latter would dominate the anisotropy. Theopposite is, however, the ase in the data. Thus, for this doping range, the spin state ofCo is identi(cid:28)ed with the LS state and the magnetization must be assigned to Co ,whi h shows the orre t anisotropy. For x = 0 . , we still have 40% Co and, be auseof the pronoun ed anisotropy, we an follow the same argumentation here. Regarding x = 0 . in Fig. 2, the anisotropy is found to be rather small ompared to the other ompounds.It should be stressed that the dire tion of anisotropy in layered perovskites dependson the dire tion in whi h the oxygen o tahedron is distorted. In some systemslike K CoF the o tahedra are ompressed in the c dire tion, resulting in an easy-axis anisotropy [23℄; Co in almost ubi symmetry also tends to have an easy-axisanisotropy like in CoO. However, in our ase of La − x Sr x CoO , the oxygen o tahedronsurrounding the obalt ion is elongated by the inherent tetragonal stru ture. Thissituation is omparable to the hange of anisotropy in CoO (cid:28)lms on di(cid:27)erent substrates[24℄. For a quantitative analysis of the data and a on(cid:28)rmation of the results foundin the qualitative dis ussion of the anisotropy, a full-multiplet al ulation was arriedout within the rystal (cid:28)eld approximation [25℄. A Hamiltonian for the d shell of theCo ion was set up, in luding rystal (cid:28)eld, spin-orbit oupling and magneti (cid:28)eld.The Slater integrals F , F and F hara terizing the Coulomb intera tion and thespin-orbit oupling onstant were taken from a Hartree-Fo k approximation [20℄. Anyex itations from the oxygen p levels and mixing with higher states were negle ted,nisotropi Sus eptibility of La − x Sr x CoO4
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Anisotropi Sus eptibility of La − x Sr x CoO related tothe Spin States of CobaltN Hollmann , M W Haverkort , M Cwik , M Benomar ,M Reuther , A Tanaka , T Lorenz II. Physikalis hes Institut, University of Cologne, Zülpi herstrasse 77, D-50937 Köln,Germany Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima739-8530, JapanE-mail: hollmannph2.uni-koeln.deAbstra t.We present a study of the magneti sus eptibility of La − x Sr x CoO single rystalsin a doping range . ≤ x ≤ . . Our data shows a pronoun ed magneti anisotropy forall ompounds. This anisotropy is in agreement with a low-spin ground state ( S = 0 )of Co for x ≥ . and a high-spin ground state ( S = 3 / ) of Co . We ompareour data with a rystal-(cid:28)eld model al ulation assuming lo al moments and (cid:28)nd agood des ription of the magneti behavior for x ≥ . . This in ludes the pronoun edkinks observed in the inverse magneti sus eptibility, whi h result from the anisotropyand low-energy ex ited states of Co and are not related to magneti ordering ortemperature-dependent spin-state transitions.PACS numbers: 71.20.Be, 71.70.Ch, 71.70.Ej, 75.30.Gwnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 2Transition-metal oxides are known for their omplex interplay between di(cid:27)erentdegrees of freedom like spin, harge and orbitals. In some of these systems anotherinteresting property is found: ions like Co in a rystal-(cid:28)eld environment an o ur indi(cid:27)erent spin states. Additionally, transitions between di(cid:27)erent spin states are possiblefor some ompounds. A prominent example showing this phenomenon is LaCoO ,whi h has been examined and dis ussed sin e the middle of the last entury (see e.g.Refs. [1(cid:21)11℄).The existen e of di(cid:27)erent spin states arises from the ompetition between rystal-(cid:28)eld e(cid:27)e ts and on-site Coulomb intera tion. Crystal (cid:28)elds lift the degenera y of the d states. If the rystal (cid:28)eld is strong, this an lead to a violation of Hund's rules. In the ase of ubi symmetry and in a one-ele tron pi ture, the (cid:28)ve-fold degenerate d statesare split into a three-fold degenerate t g and a two-fold degenerate e g level. The splittingbetween t g and e g states is alled Dq . With a strong rystal (cid:28)eld, the ele trons willbe for ed into the low-spin state (LS). Regarding a d system, this state onsists of anantiparallel alignment of spins with t g e g and S = 0 . On the other hand, the Coulombintera tion manifests itself in the e(cid:27)e t of Hund's oupling. A parallel arrangement ofspins minimizes the ele tron-ele tron repulsion be ause the Pauli prin iple for es theele trons to o upy di(cid:27)erent orbitals. In a weak ubi rystal (cid:28)eld, this e(cid:27)e t willdominate and lead to the high-spin state (HS), whi h is the on(cid:28)guration with thehighest total spin possible in a ordan e with the Pauli prin iple. For a d system, this on(cid:28)guration is t g e g with S = 2 . For Co the rossover between these two di(cid:27)erentspin states o urs roughly at an energy di(cid:27)eren e of Dq = 2 . eV.In the ase of LaCoO , a low-spin ground state for Co was found [2℄. Interestingly,the di(cid:27)eren e between the rystal (cid:28)eld energies and the promotional energies is so smallthat an ex ited state with di(cid:27)erent spin state an be rea hed by thermal ex itation. Thisex ited state has been a subje t of debate for a long time. Apart from the HS statedes ribed above, the intermediate-spin state (IS, t g e g S = 1 ) was also dis ussed [4℄ andreported in many experiments, but it was shown that it might have been onfused witha spin-orbit oupled HS state [11℄.For the layered obaltates La − x Sr x CoO , mu h less is known about the spin stateof Co . Based on magneti measurements [12℄ a HS ground state for x ≤ . and aspin-state transition to an IS ground state for x > . was proposed. This on lusionwas based on a Curie-Weiss analysis of the sus eptibility in a temperature range of100K to 300K and NMR measurements [13℄. Unrestri ted Hartree-Fo k al ulations [14℄showed a slightly di(cid:27)erent pi ture. Here, three di(cid:27)erent magneti phases were found. Anantiferromagneti HS phase ( x < . ), a ferromagneti HS phase ( . ≤ x ≤ . ) andan antiferromagneti LS-HS-ordered phase ( x > . ) were proposed. These al ulationswere also based on the results of the Curie-Weiss analysis in Ref. [12℄. Taking intoa ount that La − x Sr x CoO is an anisotropi material with rather strong spin-orbit oupling ompared to the tetragonal rystal (cid:28)eld splitting, the validity of the Curie-Weiss law is questionable. The aim of this paper is to analyze the magneti sus eptibilityfrom a di(cid:27)erent perspe tive, on entrating on the spin state of Co .nisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 3Figure 1: Magneti sus eptibility of La − x Sr x CoO for two di(cid:27)erent dire tions of themagneti (cid:28)eld. The insert is an expanded view of the low-temperature region to showthe di(cid:27)eren e between FC and ZFC measurements for . ≤ x ≤ . . The urves withthe lower sus eptibility refer to the ZFC measurements.The single rystals used for the magneti measurements have been grown using the(cid:29)oating-zone te hnique in an image furna e. A strontium doping range of . ≤ x ≤ . was overed. Resistivity measurements revealed that La − x Sr x CoO is a strong insulatorfor all Sr doping on entrations. The x = 0 . sample turned out to possess the highestresistivity, whi h is in a ordan e with the harge ordering of Co and Co at ≈ Kthat has already been reported [15℄.The magnetization was measured with a Quantum Design vibrating samplemagnetometer (VSM). The (cid:28)eld was aligned parallel to the CoO planes as well asperpendi ular to these planes (the rystallographi c dire tion). The orresponding omponents χ ab and χ c are plotted in Fig. 1. A short-range antiferromagneti orderhas been found in the ompounds . ≤ x ≤ . by neutron measurements [16℄. Thisfrustrated short-range order is also re(cid:29)e ted by the di(cid:27)eren e between (cid:28)eld ooled (FC)and zero-(cid:28)eld ooled (ZFC) measurements at low temperatures. The sus eptibility issmaller for ZFC than FC below a ertain freezing temperature.Figure 2 shows the inverse magneti sus eptibility for both orientations of (cid:28)eld.The main feature of the sus eptibility in the paramagneti regime is the pronoun ednisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 4Figure 2: Inverse sus eptibility of La − x Sr x CoO for two di(cid:27)erent dire tions of themagneti (cid:28)eld. The form of the urves and the magneti anisotropy strongly deviatesfrom Curie-Weiss behavior.magneti anisotropy, both in magnitude and form of the urves. The dire tion of theanisotropy is the same for all rystals, (cid:28)nding χ ab to be bigger than χ c .Here, the magneti anisotropy arises from band stru ture and spin-orbit oupling.For Mott and harge-transfer insulators with well-lo alized moments, band-stru tureand ovalen y e(cid:27)e ts an be approximated by an e(cid:27)e tive rystal (cid:28)eld. The rystal (cid:28)eldre(cid:29)e ts the anisotropy of the latti e and lifts the degenera y of the d states, resultingin a new set of linear ombinations of the unperturbed wave fun tions as a basis. Theexpe tation values of the omponents of the orbital moment result from these new linear ombinations. Thus, the rystal's anisotropy may result in an anisotropy of the orbitalmoment. Spin-orbit oupling ties the spin moment to this anisotropy. The spin-orbit oupling Hamiltonian will be written as ζ P i l i · s i where the sum over i runs over allele trons. The dot produ t between spin and orbital momentum tends to align thesemoments antiparallel for ea h ele tron. For an anisotropi orbital momentum, the spinis aligned in the dire tion of maximum orbital momentum [17℄.The full many-body ground-state for a d or d on(cid:28)guration in a rystal-(cid:28)eld al ulation in luding spin-orbit oupling and a tetragonal distortions is not simple [18℄but well known. In order to get an intuitive pi ture one would like to fall ba k to asingle ele tron des ription. In the limit of full spin polarization this an be done andgives important results. In the following we will (cid:28)rst dis uss the magneti anisotropy ofLa − x Sr x CoO in terms of a one-ele tron pi ture. We will show that ea h spin state hasa di(cid:27)erent magneti anisotropy from whi h, by omparison to the experiment, the spinstates of the Co ion an be on luded. In order to obtain also a quantitative des riptionand to verify our simple argumentation we will present a full many-body rystal-(cid:28)eldnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 5Figure 3: Splitting of the d levels in a tetragonal rystal (cid:28)eld arising from an elongatedoxygen o tahedron. The two sket hes on the left show the o upation for the LS andHS state of Co in a one-ele tron pi ture. The right sket h refers to HS Co . Thereal orbitals on the right an be used as a basis. al ulation afterwards.Within a ubi rystal stru ture, the d states split into e g orbitals and t g orbitals.The basis an be hosen as { z − r , x − y } and { xy, xz, yz } , respe tively. A partially(cid:28)lled t g shell an produ e a pseudo orbital moment of ˜ L = 1 . Though the individualreal wave fun tions d xy , d yz and d xz of the basis themselves have a ompletely quen hedorbital moment, their linear ombinations are in general omplex. Writing d xm l , d ym l and d zm l as the orbital wave fun tion with the orbital moment quantized along the axes x , y and z , respe tively, one (cid:28)nds d x ± = 1 √ ± d xy + id xz ) (1) d y ± = 1 √ ± d yz + id xy ) (2) d z ± = 1 √ ± d xz + id yz ) . (3)In the limit of very large rystal (cid:28)eld splittings Dq and with a partially (cid:28)lled t g shell, the moment is isotropi , despite the large orbital moment. In fa t, the t g ele trons are sometimes ompared to p ele trons [19℄.Introdu ing a tetragonal distortion, the orbital moment be omes anisotropi . Thetetragonal rystal (cid:28)eld splits the ubi e g states into non-degenerate a g and b g levels,while the t g states split into a non-degenerate b g state and a two-fold degenerate e g level. As a basis for these states, the real orbital fun tions an also be used. The z axis of the system is taken to be identi al with the c axis of the rystal. In the aseof La − x Sr x CoO , the oxygen o tahedron is elongated in the c dire tion. The order oflevels and o upations for the ground states of the two obalt ions is illustrated in Fig. 3.The Co low-spin state is, ex ept for a small Van Vle k sus eptibility, nonmagneti sin e it does not arry any moment. In the high-spin state, Co has spin and orbitalnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 6moment. The orbital degenera y is not ompletely quen hed: the degenerate xz and yz are o upied by three ele trons. These real orbitals an thus be re ombined to form omplex orbitals arrying orbital moment in the z dire tion as des ribed in Eq. (3). Notethat the linear ombinations in equations (1) and (2) annot be formed be ause the xy orbital is not degenerate with the xz and yz orbitals. Therefore the orbital moment islarger in the z dire tion making this the easy axis. This anisotropy should be re(cid:29)e tedin the sus eptibility with χ c >χ ab , whi h obviously ontradi ts the anisotropy found inthe measurements. A Co HS system shows the wrong magneti anisotropy.Next, we dis uss the IS state of Co . Due to the elongation of the oxygeno tahedra, the e g ele tron o upies the z − r orbital. This also e(cid:27)e ts the splitting ofthe t g states. The xz and yz orbitals remain degenerate but are not degenerate with the xy orbital. This arises from the di(cid:27)erent harge distributions in relation to the z − r orbital. We have (cid:28)ve ele trons to (cid:28)ll in the t g states, whi h an also be treated as onehole in the t g states. This hole is attra ted to the e g ele tron in the z − r orbital.Fig. 4 shows the harge distribution of this hole and the z − r orbital. In the leftsket h, both the ele tron in the z − r orbital and a hole in the xy orbital is drawn.The sket h in the enter shows the z − r orbital and the hole in a linear ombinationof the degenerate xz and yz orbitals. Regarding the distan es between the ele tron andthe hole, on an on lude that the state with the hole in the xz and yz orbitals is lowerin energy. As the order of levels is reversed when we are speaking about holes insteadof ele trons, this means that the xy orbital is lowered in energy. Thus, the order ando upation of levels is the one shown in Fig. 4. The magneti anisotropy is the same asin the ase of Co HS whi h we treated in the last paragraph. With three ele trons inthe degenerate level of the xz and yz orbitals, we an also make use of Eq. (3) to showthat the dire tion of anisotropy does not (cid:28)t the measurements for La − x Sr x CoO .Neither the HS nor the IS state of Co show the orre t magneti anisotropy. Butbefore we draw further on lusions, Co should be dis ussed. Although Co has alsobeen found in the LS state in some intramole ular ompounds [21℄, for bulk rystals it an be safely assumed that Co is in the HS state [22℄.Regarding the ground state produ ed by the rystal (cid:28)eld in Fig. 3, where the lowestlevel is (cid:28)lled, no linear ombinations like in Eq. (3) an be formed. The orbital momentwould be ompletely quen hed and the magneti moment would be determined by anisotropi spin. If, however, spin-orbit oupling is of the same magnitude as the splittingof the t g orbitals, it will mix these states. Mixing in a state with orbital moment inthe z dire tion would require pla ing a hole in the d z ± orbital (see Eq. 3). This would ost an energy equal to the splitting within the t g orbitals. It is more favorable to mixin a state with orbital moment in the x (or y ) dire tion by pla ing a hole in the d x ± (orthe d y ± ) orbital (see Eq. 1). This osts only half of the rystal-(cid:28)eld splitting within the t g orbitals. An orbital moment in the x (or y ) dire tion osts less energy than in the z dire tion. So, the xy plane will be the easy plane for Co in this elongated tetragonalstru ture [24℄. The spin moment will also be found in the xy plane. Comparing these(cid:28)ndings to the measurement, we see that Co shows the orre t dire tion of magneti nisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 7Figure 4: The pi ture on the left shows the harge distribution for an ele tron in the z − r orbital (bla k) and a hole in the xy orbital (white). The harge distribution inthe enter refers to an ele tron in the z + r orbital and a hole in a linear ombinationof the xz and yz orbitals. The splitting and the o upation of the d levels for Co inthe IS state is shown on the right.anisotropy. Now a on lusion about the spin state of Co an be drawn. For x ≥ . , atleast half of the obalt ions in the rystal are Co . Taking into a ount that Co hasa smaller spin moment than HS Co , the latter would dominate the anisotropy. Theopposite is, however, the ase in the data. Thus, for this doping range, the spin state ofCo is identi(cid:28)ed with the LS state and the magnetization must be assigned to Co ,whi h shows the orre t anisotropy. For x = 0 . , we still have 40% Co and, be auseof the pronoun ed anisotropy, we an follow the same argumentation here. Regarding x = 0 . in Fig. 2, the anisotropy is found to be rather small ompared to the other ompounds.It should be stressed that the dire tion of anisotropy in layered perovskites dependson the dire tion in whi h the oxygen o tahedron is distorted. In some systemslike K CoF the o tahedra are ompressed in the c dire tion, resulting in an easy-axis anisotropy [23℄; Co in almost ubi symmetry also tends to have an easy-axisanisotropy like in CoO. However, in our ase of La − x Sr x CoO , the oxygen o tahedronsurrounding the obalt ion is elongated by the inherent tetragonal stru ture. Thissituation is omparable to the hange of anisotropy in CoO (cid:28)lms on di(cid:27)erent substrates[24℄. For a quantitative analysis of the data and a on(cid:28)rmation of the results foundin the qualitative dis ussion of the anisotropy, a full-multiplet al ulation was arriedout within the rystal (cid:28)eld approximation [25℄. A Hamiltonian for the d shell of theCo ion was set up, in luding rystal (cid:28)eld, spin-orbit oupling and magneti (cid:28)eld.The Slater integrals F , F and F hara terizing the Coulomb intera tion and thespin-orbit oupling onstant were taken from a Hartree-Fo k approximation [20℄. Anyex itations from the oxygen p levels and mixing with higher states were negle ted,nisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 8Figure 5: Measured magnetization ( olored lines) ompared with the al ulation forCo (bla k lines), plotted inverse. A mean-(cid:28)eld magneti ex hange was added to the al ulation. Note that the magnetization of the ion was weighted with the ontent ofCo ( − x ).as these e(cid:27)e ts are not signi(cid:28) ant when onsidering thermal energies. The parametersfor the rystal (cid:28)eld were treated semi-empiri ally. Regarding the tetragonal distortionas a perturbation of the ubi symmetry, the e(cid:27)e t of the hybridization of the Co d shell with the surrounding oxygen p shells only in reases the splitting of the ubi t g and e g states. Hybridization an thus be in luded into the rystal (cid:28)eld parametersand need not be treated separately. The Hamiltonian was diagonalized and Eigenstatesand Eigenvalues were used to obtain the temperature dependen e of the magnetization.Magneti ex hange was in luded in the al ulation on a mean-(cid:28)eld level. The results an be seen in Fig. 5, where the inverse magnetization is plotted.The rystal-(cid:28)eld parameters were hosen as follows: the Dq splitting and thesplitting within the e g levels were (cid:28)xed at 1.5eV and 0.2eV, respe tively. These two valuesare of minor importan e for the al ulation, be ause the anisotropy of the magnetizationarises from the t g ele trons, as des ribed in the dis ussion above. The splitting of the t g states showed up to be ru ial for the form and the anisotropy of the al ulated urves. The values of the splitting within the t g states were hosen as 80meV, 60meV,68meV, and 50meV for the ompounds x = t g rystal (cid:28)eld splittings for Co are in good agreement with previous measurementson strained CoO thin (cid:28)lms [24℄. They might appear to be surprisingly small omparedto the values found for the Titanates [26, 27℄ in the order of 200 meV. The di(cid:27)eren enisotropi Sus eptibility of La − x Sr x CoO4
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Anisotropi Sus eptibility of La − x Sr x CoO related tothe Spin States of CobaltN Hollmann , M W Haverkort , M Cwik , M Benomar ,M Reuther , A Tanaka , T Lorenz II. Physikalis hes Institut, University of Cologne, Zülpi herstrasse 77, D-50937 Köln,Germany Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima739-8530, JapanE-mail: hollmannph2.uni-koeln.deAbstra t.We present a study of the magneti sus eptibility of La − x Sr x CoO single rystalsin a doping range . ≤ x ≤ . . Our data shows a pronoun ed magneti anisotropy forall ompounds. This anisotropy is in agreement with a low-spin ground state ( S = 0 )of Co for x ≥ . and a high-spin ground state ( S = 3 / ) of Co . We ompareour data with a rystal-(cid:28)eld model al ulation assuming lo al moments and (cid:28)nd agood des ription of the magneti behavior for x ≥ . . This in ludes the pronoun edkinks observed in the inverse magneti sus eptibility, whi h result from the anisotropyand low-energy ex ited states of Co and are not related to magneti ordering ortemperature-dependent spin-state transitions.PACS numbers: 71.20.Be, 71.70.Ch, 71.70.Ej, 75.30.Gwnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 2Transition-metal oxides are known for their omplex interplay between di(cid:27)erentdegrees of freedom like spin, harge and orbitals. In some of these systems anotherinteresting property is found: ions like Co in a rystal-(cid:28)eld environment an o ur indi(cid:27)erent spin states. Additionally, transitions between di(cid:27)erent spin states are possiblefor some ompounds. A prominent example showing this phenomenon is LaCoO ,whi h has been examined and dis ussed sin e the middle of the last entury (see e.g.Refs. [1(cid:21)11℄).The existen e of di(cid:27)erent spin states arises from the ompetition between rystal-(cid:28)eld e(cid:27)e ts and on-site Coulomb intera tion. Crystal (cid:28)elds lift the degenera y of the d states. If the rystal (cid:28)eld is strong, this an lead to a violation of Hund's rules. In the ase of ubi symmetry and in a one-ele tron pi ture, the (cid:28)ve-fold degenerate d statesare split into a three-fold degenerate t g and a two-fold degenerate e g level. The splittingbetween t g and e g states is alled Dq . With a strong rystal (cid:28)eld, the ele trons willbe for ed into the low-spin state (LS). Regarding a d system, this state onsists of anantiparallel alignment of spins with t g e g and S = 0 . On the other hand, the Coulombintera tion manifests itself in the e(cid:27)e t of Hund's oupling. A parallel arrangement ofspins minimizes the ele tron-ele tron repulsion be ause the Pauli prin iple for es theele trons to o upy di(cid:27)erent orbitals. In a weak ubi rystal (cid:28)eld, this e(cid:27)e t willdominate and lead to the high-spin state (HS), whi h is the on(cid:28)guration with thehighest total spin possible in a ordan e with the Pauli prin iple. For a d system, this on(cid:28)guration is t g e g with S = 2 . For Co the rossover between these two di(cid:27)erentspin states o urs roughly at an energy di(cid:27)eren e of Dq = 2 . eV.In the ase of LaCoO , a low-spin ground state for Co was found [2℄. Interestingly,the di(cid:27)eren e between the rystal (cid:28)eld energies and the promotional energies is so smallthat an ex ited state with di(cid:27)erent spin state an be rea hed by thermal ex itation. Thisex ited state has been a subje t of debate for a long time. Apart from the HS statedes ribed above, the intermediate-spin state (IS, t g e g S = 1 ) was also dis ussed [4℄ andreported in many experiments, but it was shown that it might have been onfused witha spin-orbit oupled HS state [11℄.For the layered obaltates La − x Sr x CoO , mu h less is known about the spin stateof Co . Based on magneti measurements [12℄ a HS ground state for x ≤ . and aspin-state transition to an IS ground state for x > . was proposed. This on lusionwas based on a Curie-Weiss analysis of the sus eptibility in a temperature range of100K to 300K and NMR measurements [13℄. Unrestri ted Hartree-Fo k al ulations [14℄showed a slightly di(cid:27)erent pi ture. Here, three di(cid:27)erent magneti phases were found. Anantiferromagneti HS phase ( x < . ), a ferromagneti HS phase ( . ≤ x ≤ . ) andan antiferromagneti LS-HS-ordered phase ( x > . ) were proposed. These al ulationswere also based on the results of the Curie-Weiss analysis in Ref. [12℄. Taking intoa ount that La − x Sr x CoO is an anisotropi material with rather strong spin-orbit oupling ompared to the tetragonal rystal (cid:28)eld splitting, the validity of the Curie-Weiss law is questionable. The aim of this paper is to analyze the magneti sus eptibilityfrom a di(cid:27)erent perspe tive, on entrating on the spin state of Co .nisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 3Figure 1: Magneti sus eptibility of La − x Sr x CoO for two di(cid:27)erent dire tions of themagneti (cid:28)eld. The insert is an expanded view of the low-temperature region to showthe di(cid:27)eren e between FC and ZFC measurements for . ≤ x ≤ . . The urves withthe lower sus eptibility refer to the ZFC measurements.The single rystals used for the magneti measurements have been grown using the(cid:29)oating-zone te hnique in an image furna e. A strontium doping range of . ≤ x ≤ . was overed. Resistivity measurements revealed that La − x Sr x CoO is a strong insulatorfor all Sr doping on entrations. The x = 0 . sample turned out to possess the highestresistivity, whi h is in a ordan e with the harge ordering of Co and Co at ≈ Kthat has already been reported [15℄.The magnetization was measured with a Quantum Design vibrating samplemagnetometer (VSM). The (cid:28)eld was aligned parallel to the CoO planes as well asperpendi ular to these planes (the rystallographi c dire tion). The orresponding omponents χ ab and χ c are plotted in Fig. 1. A short-range antiferromagneti orderhas been found in the ompounds . ≤ x ≤ . by neutron measurements [16℄. Thisfrustrated short-range order is also re(cid:29)e ted by the di(cid:27)eren e between (cid:28)eld ooled (FC)and zero-(cid:28)eld ooled (ZFC) measurements at low temperatures. The sus eptibility issmaller for ZFC than FC below a ertain freezing temperature.Figure 2 shows the inverse magneti sus eptibility for both orientations of (cid:28)eld.The main feature of the sus eptibility in the paramagneti regime is the pronoun ednisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 4Figure 2: Inverse sus eptibility of La − x Sr x CoO for two di(cid:27)erent dire tions of themagneti (cid:28)eld. The form of the urves and the magneti anisotropy strongly deviatesfrom Curie-Weiss behavior.magneti anisotropy, both in magnitude and form of the urves. The dire tion of theanisotropy is the same for all rystals, (cid:28)nding χ ab to be bigger than χ c .Here, the magneti anisotropy arises from band stru ture and spin-orbit oupling.For Mott and harge-transfer insulators with well-lo alized moments, band-stru tureand ovalen y e(cid:27)e ts an be approximated by an e(cid:27)e tive rystal (cid:28)eld. The rystal (cid:28)eldre(cid:29)e ts the anisotropy of the latti e and lifts the degenera y of the d states, resultingin a new set of linear ombinations of the unperturbed wave fun tions as a basis. Theexpe tation values of the omponents of the orbital moment result from these new linear ombinations. Thus, the rystal's anisotropy may result in an anisotropy of the orbitalmoment. Spin-orbit oupling ties the spin moment to this anisotropy. The spin-orbit oupling Hamiltonian will be written as ζ P i l i · s i where the sum over i runs over allele trons. The dot produ t between spin and orbital momentum tends to align thesemoments antiparallel for ea h ele tron. For an anisotropi orbital momentum, the spinis aligned in the dire tion of maximum orbital momentum [17℄.The full many-body ground-state for a d or d on(cid:28)guration in a rystal-(cid:28)eld al ulation in luding spin-orbit oupling and a tetragonal distortions is not simple [18℄but well known. In order to get an intuitive pi ture one would like to fall ba k to asingle ele tron des ription. In the limit of full spin polarization this an be done andgives important results. In the following we will (cid:28)rst dis uss the magneti anisotropy ofLa − x Sr x CoO in terms of a one-ele tron pi ture. We will show that ea h spin state hasa di(cid:27)erent magneti anisotropy from whi h, by omparison to the experiment, the spinstates of the Co ion an be on luded. In order to obtain also a quantitative des riptionand to verify our simple argumentation we will present a full many-body rystal-(cid:28)eldnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 5Figure 3: Splitting of the d levels in a tetragonal rystal (cid:28)eld arising from an elongatedoxygen o tahedron. The two sket hes on the left show the o upation for the LS andHS state of Co in a one-ele tron pi ture. The right sket h refers to HS Co . Thereal orbitals on the right an be used as a basis. al ulation afterwards.Within a ubi rystal stru ture, the d states split into e g orbitals and t g orbitals.The basis an be hosen as { z − r , x − y } and { xy, xz, yz } , respe tively. A partially(cid:28)lled t g shell an produ e a pseudo orbital moment of ˜ L = 1 . Though the individualreal wave fun tions d xy , d yz and d xz of the basis themselves have a ompletely quen hedorbital moment, their linear ombinations are in general omplex. Writing d xm l , d ym l and d zm l as the orbital wave fun tion with the orbital moment quantized along the axes x , y and z , respe tively, one (cid:28)nds d x ± = 1 √ ± d xy + id xz ) (1) d y ± = 1 √ ± d yz + id xy ) (2) d z ± = 1 √ ± d xz + id yz ) . (3)In the limit of very large rystal (cid:28)eld splittings Dq and with a partially (cid:28)lled t g shell, the moment is isotropi , despite the large orbital moment. In fa t, the t g ele trons are sometimes ompared to p ele trons [19℄.Introdu ing a tetragonal distortion, the orbital moment be omes anisotropi . Thetetragonal rystal (cid:28)eld splits the ubi e g states into non-degenerate a g and b g levels,while the t g states split into a non-degenerate b g state and a two-fold degenerate e g level. As a basis for these states, the real orbital fun tions an also be used. The z axis of the system is taken to be identi al with the c axis of the rystal. In the aseof La − x Sr x CoO , the oxygen o tahedron is elongated in the c dire tion. The order oflevels and o upations for the ground states of the two obalt ions is illustrated in Fig. 3.The Co low-spin state is, ex ept for a small Van Vle k sus eptibility, nonmagneti sin e it does not arry any moment. In the high-spin state, Co has spin and orbitalnisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 6moment. The orbital degenera y is not ompletely quen hed: the degenerate xz and yz are o upied by three ele trons. These real orbitals an thus be re ombined to form omplex orbitals arrying orbital moment in the z dire tion as des ribed in Eq. (3). Notethat the linear ombinations in equations (1) and (2) annot be formed be ause the xy orbital is not degenerate with the xz and yz orbitals. Therefore the orbital moment islarger in the z dire tion making this the easy axis. This anisotropy should be re(cid:29)e tedin the sus eptibility with χ c >χ ab , whi h obviously ontradi ts the anisotropy found inthe measurements. A Co HS system shows the wrong magneti anisotropy.Next, we dis uss the IS state of Co . Due to the elongation of the oxygeno tahedra, the e g ele tron o upies the z − r orbital. This also e(cid:27)e ts the splitting ofthe t g states. The xz and yz orbitals remain degenerate but are not degenerate with the xy orbital. This arises from the di(cid:27)erent harge distributions in relation to the z − r orbital. We have (cid:28)ve ele trons to (cid:28)ll in the t g states, whi h an also be treated as onehole in the t g states. This hole is attra ted to the e g ele tron in the z − r orbital.Fig. 4 shows the harge distribution of this hole and the z − r orbital. In the leftsket h, both the ele tron in the z − r orbital and a hole in the xy orbital is drawn.The sket h in the enter shows the z − r orbital and the hole in a linear ombinationof the degenerate xz and yz orbitals. Regarding the distan es between the ele tron andthe hole, on an on lude that the state with the hole in the xz and yz orbitals is lowerin energy. As the order of levels is reversed when we are speaking about holes insteadof ele trons, this means that the xy orbital is lowered in energy. Thus, the order ando upation of levels is the one shown in Fig. 4. The magneti anisotropy is the same asin the ase of Co HS whi h we treated in the last paragraph. With three ele trons inthe degenerate level of the xz and yz orbitals, we an also make use of Eq. (3) to showthat the dire tion of anisotropy does not (cid:28)t the measurements for La − x Sr x CoO .Neither the HS nor the IS state of Co show the orre t magneti anisotropy. Butbefore we draw further on lusions, Co should be dis ussed. Although Co has alsobeen found in the LS state in some intramole ular ompounds [21℄, for bulk rystals it an be safely assumed that Co is in the HS state [22℄.Regarding the ground state produ ed by the rystal (cid:28)eld in Fig. 3, where the lowestlevel is (cid:28)lled, no linear ombinations like in Eq. (3) an be formed. The orbital momentwould be ompletely quen hed and the magneti moment would be determined by anisotropi spin. If, however, spin-orbit oupling is of the same magnitude as the splittingof the t g orbitals, it will mix these states. Mixing in a state with orbital moment inthe z dire tion would require pla ing a hole in the d z ± orbital (see Eq. 3). This would ost an energy equal to the splitting within the t g orbitals. It is more favorable to mixin a state with orbital moment in the x (or y ) dire tion by pla ing a hole in the d x ± (orthe d y ± ) orbital (see Eq. 1). This osts only half of the rystal-(cid:28)eld splitting within the t g orbitals. An orbital moment in the x (or y ) dire tion osts less energy than in the z dire tion. So, the xy plane will be the easy plane for Co in this elongated tetragonalstru ture [24℄. The spin moment will also be found in the xy plane. Comparing these(cid:28)ndings to the measurement, we see that Co shows the orre t dire tion of magneti nisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 7Figure 4: The pi ture on the left shows the harge distribution for an ele tron in the z − r orbital (bla k) and a hole in the xy orbital (white). The harge distribution inthe enter refers to an ele tron in the z + r orbital and a hole in a linear ombinationof the xz and yz orbitals. The splitting and the o upation of the d levels for Co inthe IS state is shown on the right.anisotropy. Now a on lusion about the spin state of Co an be drawn. For x ≥ . , atleast half of the obalt ions in the rystal are Co . Taking into a ount that Co hasa smaller spin moment than HS Co , the latter would dominate the anisotropy. Theopposite is, however, the ase in the data. Thus, for this doping range, the spin state ofCo is identi(cid:28)ed with the LS state and the magnetization must be assigned to Co ,whi h shows the orre t anisotropy. For x = 0 . , we still have 40% Co and, be auseof the pronoun ed anisotropy, we an follow the same argumentation here. Regarding x = 0 . in Fig. 2, the anisotropy is found to be rather small ompared to the other ompounds.It should be stressed that the dire tion of anisotropy in layered perovskites dependson the dire tion in whi h the oxygen o tahedron is distorted. In some systemslike K CoF the o tahedra are ompressed in the c dire tion, resulting in an easy-axis anisotropy [23℄; Co in almost ubi symmetry also tends to have an easy-axisanisotropy like in CoO. However, in our ase of La − x Sr x CoO , the oxygen o tahedronsurrounding the obalt ion is elongated by the inherent tetragonal stru ture. Thissituation is omparable to the hange of anisotropy in CoO (cid:28)lms on di(cid:27)erent substrates[24℄. For a quantitative analysis of the data and a on(cid:28)rmation of the results foundin the qualitative dis ussion of the anisotropy, a full-multiplet al ulation was arriedout within the rystal (cid:28)eld approximation [25℄. A Hamiltonian for the d shell of theCo ion was set up, in luding rystal (cid:28)eld, spin-orbit oupling and magneti (cid:28)eld.The Slater integrals F , F and F hara terizing the Coulomb intera tion and thespin-orbit oupling onstant were taken from a Hartree-Fo k approximation [20℄. Anyex itations from the oxygen p levels and mixing with higher states were negle ted,nisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 8Figure 5: Measured magnetization ( olored lines) ompared with the al ulation forCo (bla k lines), plotted inverse. A mean-(cid:28)eld magneti ex hange was added to the al ulation. Note that the magnetization of the ion was weighted with the ontent ofCo ( − x ).as these e(cid:27)e ts are not signi(cid:28) ant when onsidering thermal energies. The parametersfor the rystal (cid:28)eld were treated semi-empiri ally. Regarding the tetragonal distortionas a perturbation of the ubi symmetry, the e(cid:27)e t of the hybridization of the Co d shell with the surrounding oxygen p shells only in reases the splitting of the ubi t g and e g states. Hybridization an thus be in luded into the rystal (cid:28)eld parametersand need not be treated separately. The Hamiltonian was diagonalized and Eigenstatesand Eigenvalues were used to obtain the temperature dependen e of the magnetization.Magneti ex hange was in luded in the al ulation on a mean-(cid:28)eld level. The results an be seen in Fig. 5, where the inverse magnetization is plotted.The rystal-(cid:28)eld parameters were hosen as follows: the Dq splitting and thesplitting within the e g levels were (cid:28)xed at 1.5eV and 0.2eV, respe tively. These two valuesare of minor importan e for the al ulation, be ause the anisotropy of the magnetizationarises from the t g ele trons, as des ribed in the dis ussion above. The splitting of the t g states showed up to be ru ial for the form and the anisotropy of the al ulated urves. The values of the splitting within the t g states were hosen as 80meV, 60meV,68meV, and 50meV for the ompounds x = t g rystal (cid:28)eld splittings for Co are in good agreement with previous measurementson strained CoO thin (cid:28)lms [24℄. They might appear to be surprisingly small omparedto the values found for the Titanates [26, 27℄ in the order of 200 meV. The di(cid:27)eren enisotropi Sus eptibility of La − x Sr x CoO4 related to the Spin States of Cobalt 9between Co and Ti an be understood by onsidering the radial extent of the d wavefun tion and the transition metal - oxygen (TM-O) distan e. The Co d wave fun tionis mu h smaller than the Ti d wave fun tion, redu ing the ovalen y and thereby thesize of the rystal-(cid:28)eld splitting. At the same time, the average TM-O distan es ofLa − x Sr x CoO and LaTiO are almost equal ( ≈ ≈ e g orbitals in a HSCo ompound whi h pushes the O atoms further away.It an be seen in Fig. 5 that the al ulation gives a good des ription for x = 0 . and x = 0 . . The magnetization of the half-doped sample x = 0 . is reprodu ed well forhigher temperatures. At temperatures below the freezing temperature, the al ulation annot des ribe the system be ause magneti order is not in luded in the model. The x = 0 . sample is (cid:28)tted well for lower temperatures. Above T ≈ K, the magnetizationin the measurement is enhan ed by another e(cid:27)e t and results in a deviation from the al ulation. This e(cid:27)e t an be seen in all urves in Fig. 2, but at signi(cid:28) antly highertemperatures. This in rease of magnetization ould arise from the thermal populationof the Co HS or IS state. But surely further measurements at higher temperaturesare needed to on(cid:28)rm this assumption. Nevertheless, the full-multiplet al ulation of apure Co system already su eeds in des ribing the main features of the magnetizationfor x ≥ . . This on(cid:28)rms that Co is a LS system in this doping range.In summary, the magneti sus eptibility of La − x Sr x CoO has been analyzed for twodi(cid:27)erent dire tions of the external magneti (cid:28)eld. A de(cid:28)nite anisotropy of the magneti moment is found experimentally with χ ab >χ c . This dire tion of anisotropy does notmat h with the behavior of HS or IS Co , whereas Co in the HS state shows the orre t single-ion anisotropy. Thus, the spin state of Co for x ≥ . must be the LSstate. This on lusion is also on(cid:28)rmed by a full-multiplet al ulation.We a knowledge (cid:28)nan ial support by the Deuts he Fors hungsgemeins haft throughSFB 608.Referen es[1℄ Jonker G H, Van Santen J H 1953 Physi a XIX, 120[2℄ Ra ah P M, Goodenough J B 1967 Phys. Rev. 155, 932[3℄ Abbate M, Fuggle J C, Fujimori A, Tjeng L H, Chen C T, Potze R, Sawatzky G A, Eisaki H,U hida S 1993 Phys. Rev. B 47, 16124[4℄ Korotin M A, Ezhov S Y, Solovyev I V, Anisimov V I, Khomskii D I, Sawatzky G A 1996Phys. Rev. B 54, 5309[5℄ Zobel C, Kriener M, Bruns D, Baier J, Grüninger M, Lorenz T, Reutler P, Rev olevs hi A 2002Phys. Rev. B 66, 020402[6℄ Nogu hi S, Kawamata S, Okuda K, Nojiri H, Motokawa M 2002 Phys. Rev. B 66, 94404[7℄ Maris G, Ren Y, Volot haev V, Zobel C, Lorenz T, Palstra T T M 2003 Phys. Rev. B 67, 224423[8℄ Ropka Z, Radwanski R J 2003 Phys. Rev. B 67, 172401[9℄ Lengsdorf R, Ait-Tahar M, Saxena S S, Ellerby M, Khomskii D I, Mi klitz H, Lorenz T, Abd-Elmeguid M M 2004 Phys. Rev. B 69, 140403[10℄ Baier J, Jodlauk S, Kriener M, Rei hl A, Zobel C, Kierspel H, Freimuth A, Lorenz T 2005Phys. Rev. B 71 014443nisotropi Sus eptibility of La − x Sr x CoO4